Analysis of Environmental Radionuclides, Volume 11 (Radioactivity in the Environment)

ANALYSIS OF ENVIRONMENTAL RADIONUCLIDES RADIOACTIVITY IN THE ENVIRONMENT A companion series to the Journal of Environ...

Author: P. Povinec

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ANALYSIS OF ENVIRONMENTAL RADIONUCLIDES

RADIOACTIVITY IN THE ENVIRONMENT A companion series to the Journal of Environmental Radioactivity Series Editor M.S. Baxter Ampfield House Clachan Seil Argyll, Scotland, UK Volume 1: Plutonium in the Environment (A. Kudo, Editor) Volume 2: Interactions of Microorganisms with Radionuclides (F.R. Livens and M. Keith-Roach, Editors) Volume 3: Radioactive Fallout after Nuclear Explosions and Accidents (Yu.A. Izrael, Author) Volume 4: Modelling Radioactivity in the Environment (E.M. Scott, Editor) Volume 5: Sedimentary Processes: Quantification Using Radionuclides (J. Carroll and I. Lerche, Authors) Volume 6: Marine Radioactivity (H.D. Livingston, Editor) Volume 7: The Natural Radiation Environment VII (J.P. McLaughlin, S.E. Simopoulos and F. Steinhäusler, Editors) Volume 8: Radionuclides in the Environment (P.P. Povinec and J.A. Sanchez-Cabeza, Editors) Volume 9: Deep Geological Disposal of Radioactive Waste (R. Alexander and L. McKinley, Editors) Volume 10: Radioactivity in the Terrestrial Environment (G. Shaw, Editor) Volume 11: Analysis of Environmental Radionuclides (P.P. Povinec, Editor)

ANALYSIS OF ENVIRONMENTAL RADIONUCLIDES

Editor

Pavel P. Povinec Faculty of Mathematics, Physics and Informatics Comenius University Bratislava, Slovakia

AMSTERDAM – BOSTON – HEIDELBERG – LONDON – NEW YORK – PARIS SAN DIEGO – SAN FRANCISCO – SINGAPORE – SYDNEY – TOKYO

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK First edition 2008 Copyright © 2008 Elsevier B.V. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: [email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-08-044988-3 ISSN: 1569-4860

For information on all Elsevier publications visit our website at books.elsevier.com

Printed and bound in Hungary 08 09 10 11 12

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Contents

Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.

1

Statistical sampling design for radionuclides by E. Marian Scott and Philip M. Dixon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

2.

Sampling techniques by Fedor Macášek . . . . . . . . . . . . . . . . . . . . .

17

3.

Detection and quantification capabilities in nuclear analytical measurements by L.A. Currie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

Radiometric determination of anthropogenic radionuclides in seawater by M. Aoyama and K. Hirose . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

137

Monte Carlo simulation of background characteristics of gamma-ray spectrometers—a comparison with experiment by Pavel P. Povinec, Pavol Vojtyla and Jean-François Comanducci . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163

Underground laboratories for low-level radioactivity measurements by Siegfried Niese . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

209

Accelerator mass spectrometry of long-lived light radionuclides by A.J. Timothy Jull, George S. Burr, J. Warren Beck, Gregory W.L. Hodgins, Dana L. Biddulph, Lanny R. McHargue and Todd E. Lange . . . . . . . . . . . . . . .

241

Accelerator mass spectrometry of long-lived heavy radionuclides by L.K. Fifield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

263

Analysis of radionuclides using ICP-MS by Per Roos . . . . . . . . . . . . . .

295

4.

5.

6.

7.

8.

9.

10. Resonance ionization mass spectrometry for trace analysis of long-lived radionuclides by N. Erdmann, G. Passler, N. Trautmann and K. Wendt . . . . .

331

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Contents

11. Environmental radioactive particles: A new challenge for modern analytical instrumental techniques in support of radioecology by Maria Betti, Mats Eriksson, Jussi Jernström and Gabriele Tamborini . . . . . . . . . . . . . . . . . . .

355

12. Activation analysis for the determination of long-lived radionuclides by Xiaolin Hou . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

371

13. In situ and airborne gamma-ray spectrometry by Andrew N. Tyler . . . . . .

407

14. Underwater gamma-ray spectrometry by Pavel P. Povinec, Iolanda Osvath and Jean-François Comanducci . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

449

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

481

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

525

1

Foreword The Radioactivity in the Environment series is an ambitious project covering recent progress in this rapidly developing field, which has included aspects such as the behavior of radionuclides in the environment, the use of natural and anthropogenic radionuclides as tracers of environmental processes, marine radioactivity studies, radiation protection, radioecology, etc. to mention at least a few. State of the art radioanalytical environmental technologies have always been a limiting factor for environmental radioactivity studies, either because the available sensitivity was not high enough to get meaningful results or the required sample size was too big to carry out such investigations, very often with limiting financial resources. There has in recent years been great progress in the development of analytical tools related to sampling strategies, development of rapid and efficient radiochemical separation methods, radiometric counting systems utilizing high sensitivity Ge detectors often working underground, and mass spectrometry technologies based on ICPMS (inductively coupled plasma mass spectrometry) and AMS (accelerator mass spectrometry) for sensitive analysis of natural and anthropogenic radionuclides in the environment. For example, in the marine environment, where research work has been heavily dependent on the new technologies, we have seen a replacement of time-consuming and expensive large volume water sampling (500 L) from several km water depths by Rosette multisampling systems enabling high resolution water sampling within one or two casts with 12 L bottles only. The sampling strategies are often developed and controlled using satellite information for the optimization of the sampling programs. Further, the philosophy of sampling and laboratory measurements has changed, where appropriate, to in situ analysis of radionuclides in the air, on land, in water and in the sediment, thus developing isoline maps of radionuclide distributions in the investigated environment. In the field of analytical technologies we have moved from simple radiochemical methods and gas counters to robotic radiochemical technologies and sophisticated detectors working on line with powerful computers, often situated underground or having anticosmic and/or anti-Compton shielding to protect them against the cosmic radiation, and thus considerably decreasing their background and increasing their sensitivity for analysis of radionuclides in the environment at very low levels. The philosophy of analysis of long-lived radionuclides has also changed considerably from the old concept of counting of decays (and thus waiting for them) to the direct counting of atoms (as if they were stable elements) using highly sensitive mass spectrometry techniques such as AMS, ICPMS, TIMS (thermal ionization mass spectrometry), RIMS (resonance ionization mass spectrometry) and SIMS (secondary ionization mass spectrometry). There have also been considerable changes in the philosophy and organization of research as institutional and national investigations have been replaced by global international projects

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P.P. Povinec

such as WOCE (world ocean circulation experiment), CLIVAR (climate variability and predictability study), PAGES (past global changes), WOMARS (worldwide marine radioactivity studies), GEOTRACES (global marine biochemistry of trace elements and isotopes), SHOTS (southern hemisphere ocean tracer studies), to mention at least a few. Although the topic of the analysis of environmental radionuclides has already been covered in several reviews, there has not been available a book covering critical progress in recent years. The present collection of review papers covers a wide range of topics starting with the development of statistically based sampling strategies to study radionuclides in the environment (Chapter 1 by Scott and Dixon), followed by description of sampling techniques and pre-concentration of samples (Chapter 2 by Macášek). Statistical evaluation of data has been a crucial point in correct interpretation of measurements, especially when dealing with counting rates very close to the detector background (Chapter 3 by Currie). Recent progress in environmental studies is documented by the analysis of 137 Cs, 90 Sr and Pu isotopes in the seawater column (Chapter 4 by Aoyama and Hirose). Monte Carlo simulations of detector background characteristics have been an important pre-requisite when designing low-level counting systems (Chapter 5 by Povinec et al.), also important when working in laboratories situated hundreds of meters underground, where radioactive purity of construction materials and radon concentration in the air become dominant factors controlling the detector background (Chapter 6 by Niese). AMS has been a revolutionary breakthrough in analytical methodologies for long-lived environmental radionuclides, as described by Jull et al. in Chapter 7 for light elements, and Fifield in Chapter 8 for heavy elements. However, the most widely used mass spectrometry technique for analysis of long-lived environmental radionuclides has been ICPMS, as documented by Ross in Chapter 9. Another new trend in analytical techniques has been an introduction of resonance ionization mass spectrometry for radionuclide analysis (Chapter 10 by Erdemann et al.), and a change from bulk sample analysis to particle sensitive analysis, as described by Betti et al. in Chapter 11 using SIMS, scanning electron microscopy (SEM), and synchrotron based techniques like µ-XRF and 3D-µ tomography. Neutron activation analysis (NAA) has been contributing in specific applications with long-lived radionuclides, and usually this is the only alternative technique for certification of reference materials (Chapter 12 by Hou). In situ techniques represent a new approach to analysis of environmental radionuclides and these have been recently widely applied for surface monitoring of radionuclides using either mobile gamma-ray spectrometers, helicopters and airplanes (Chapter 13 by Tyler) or measurements carried out under the water, e.g., for radionuclide mapping of seabed sediments and/or stationary monitoring of radionuclides in the aquatic environment as described in Chapter 14 by Povinec et al. The Editor would like to thank all authors for their fruitful collaboration during preparation of this compilation and Prof. Baxter, the Radioactivity in the Environment Series Editor, for his patience when working on this book. In publishing this book we hope to further stimulate work in the exciting field of environmental radioactivity and the use of radionuclides as tools for investigations of environmental processes. Pavel P. Povinec Editor Comenius University Bratislava, Slovakia

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Statistical sampling design for radionuclides E. Marian Scotta,∗ , Philip M. Dixonb a Department of Statistics, University of Glasgow, Glasgow G12 8QW, UK b Department of Statistics, Iowa State University, Ames, IA 50011-1210, USA

1. Introduction This chapter presents some of the key ideas for designing statistically based sampling strategies to study radionuclides in the environment. Environmental samples will naturally vary in their specific activity, no matter how precise the radionuclide measurement. This variability is caused by natural variations in the processes that control radionuclide transport and uptake in the environment. A statistically based sampling design quantifies this variability and allows information from specific samples to be generalized to a larger population. The statistical sampling principles discussed here are detailed in many textbooks and papers about environmental sampling, such as the general sampling textbooks by Cochran (1977) and Thompson (2000), the environmental statistics textbook by Gilbert (1987), and government agency guidance documents (US EPA, 2002). A recent ICRU report (2006) presents a thorough presentation of sampling issues for environmental radionuclides. This chapter draws heavily on these reference works. It provides only a taster to the issues; the reader is strongly encouraged to read more in-depth descriptions. Environmental sampling should not be considered as a ‘recipe’ based activity. The best (most efficient, valid and reliable) sampling schemes use environmental knowledge to guide the sampling. Changes in objectives (apparently small) may also lead to quite significant changes to the sampling scheme. 1.1. General sampling concepts and principles Statistical sampling is a process that allows inferences about properties of a large collection of things (the population) to be made from observations (the sample) made on a relatively small number of individuals (sampling units) belonging to the population. The population is the set of all items that could be sampled, such as all deer in a forest, all people living in the UK, etc. A sampling unit is a unique member of the population that can be selected as an individual ∗ Corresponding author. E-mail address: [email protected]

RADIOACTIVITY IN THE ENVIRONMENT VOLUME 11 ISSN 1569-4860/DOI: 10.1016/S1569-4860(07)11001-9

© 2008 Elsevier B.V. All rights reserved.

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E.M. Scott and P.M. Dixon

sample for collection and measurement. The sample is then the set of sampling units that are measured. Sampling units might be individual deer, individual people, trees, garden plots, or soil cores of a given dimension. An essential concept is that a statistically based sample of a sufficient number of individual sampling units is necessary to make inferences about the population. Statistical sampling also allows a quantification of the precision with which inferences or conclusions can be drawn about the population. The focus of this chapter is on statistical sampling design, namely how to select specific sampling units from a population or sampling locations within a larger area, and how to determine the number of individual units to collect. Sampling has many purposes, including estimation of the distribution (and mean) concentration of a radionuclide (Bq l−1 ) in a river, or in fruit in a region (Bq kg−1 ), or a map of radionuclide deposition (Bq m−2 ). Different purposes require different sampling strategies and different sampling efforts in order to be effective and efficient, so it is important that the purpose(s) of the sampling program be clearly specified. The environmental context also plays an important part in determining the choice of sampling method. Statistical sampling requires information about the nature of the population and characteristics to be described. 1.2. Methods of sampling A statistical sampling design is based on probability sampling, in which every sampling unit has a known and non-zero probability of being selected. The actual sample (set of sampling units to be measured) is chosen by randomization, using published tables of random numbers or computer algorithms. Selecting a probability sample is easy when the population can be enumerated. As a simple example, imagine sampling 10 adults from a specified geographic area for whole body monitoring. We could use an electoral register or census information to enumerate all individuals. Suppose that the population comprised 972 such individuals, then we could generate 10 random numbers lying between 1 and 972, such as 253, 871, 15, 911, 520, 555, 106, 83, 614, 932 to identify the 10 individuals. If the same number was generated more than once, then we would simply continue the process till we had 10 unique random numbers and these would then identify the individuals to be called for monitoring. The actual numbers (253, 871, etc.) are read from random number tables or may be generated by statistical software. There are many sampling designs. We describe simple random sampling, stratified random sampling and systematic sampling because these are the three most common in environmental studies. For each design, we discuss how to select a sample and how to estimate the population mean and its sampling error. A brief review of the advantages and disadvantages of the different methods is also included. More detail can be found in ICRU (2006). 1.2.1. Simple random sampling In a simple random sample, every sampling unit in the population has an equal probability of being included in the sample and all pairs of sampling units have the same probability of being included in the sample. One way to select a simple random sample is to enumerate all sampling units in the population, then use random numbers to select the desired number of sampling units. Simple random sampling is easy to describe but may be difficult to achieve

Statistical sampling design for radionuclides

5

in practice. Some common problems include lack of response from some individuals, inaccessibility of some plots of ground, and long travel times between sampling locations when sampling large areas. Example: Estimation of the average baseline 14 C level in the food-chain An estimate of the dose to the general public due to 14 C in the food-chain is an important radiological quantity for regulatory impact assessment since many nuclear power stations discharge 14 CO2 which is rapidly taken up. At a given station, the task would be to select representative environmental samples that enter the food-chain, e.g., root or cereal crops or fruits. For this particular problem, definition of the population should include identification of the species and information on where and when it grew and its spatial context. For the choice of species, it should be widely available and a suitable material for 14 C assay; a material such as soft fruit, mushrooms or grain would be ideal. The analysis requirements would then define how much material needed to be collected for each sampling unit. In terms of the temporal extent, it would be logical for the samples to be selected from a single growing season and in a specific year such as 2004. This results in a clear definition of the population, namely all selected crop in the vicinity of the site growing in a specific year and of a sampling unit, namely a bulked sample of berries, vegetables or wheat harvested from a specific location. The next step requires identification of the locations at which the samples will be collected and determination of how many sampling units will be required to satisfy the objectives of the study. A map of the vicinity in terms of all locations where the crops grow, would allow the numbering of all the locations from 1 to N , and random numbers would then be used to identify which actual locations would be sampled. The reader might care to consider whether there are more efficient but equally valid sampling approaches to this problem. Analysis of the results from a simple random sample Suppose that the 14 C activity density is measured in each of the n sampling units in the sample. ¯ given by Equation (1) The value from sampling unit i is denoted as yi . The sample average, y, is a good (unbiased) estimate of the population mean 14 C activity density and the sample variance, s 2 , given by Equation (2) is a good estimate of the population variance: yi , y¯ = (1) n and ¯ 2 (yi − y) . s2 = (2) n−1 The sample average is a random quantity; it will be a different number if different sampling units were chosen for the sample, because of the natural variation in 14 C activity densities among sampling units. The uncertainty in the sample average is quantified by the sampling variance, given by Equation (3) or its square root, the estimated standard error, e.s.e. The sampling fraction, f , is the fraction of the population included in the sample, n/N, which is usually very small. 2 1−f . Var(y) ¯ =s (3) n

6


In the 14 C example, there is no environmental evidence to believe that the population of crop is non-homogeneous. This means that we have no reason to expect any sub-groups with distinctly different 14 C levels and so simple random sampling is a reasonable sampling strategy. However, there may be spatial information which could prove important, such as distance and direction from the stack and which might lead the scientist to believe that the population is heterogeneous and so a more directed sampling scheme might use contextual information such as wind rose data to determine a different sampling scheme. Next, we consider an example with a similar objective, but where the environmental context would suggest that the population could be non-homogeneous, hence a different sampling scheme could be better (Dixon et al., 2005). 1.2.2. Stratified sampling Stratified sampling designs provide two important advantages over simple random sampling designs, namely, efficiency and improved estimates for meaningful subdivisions of the population. We must assume that the population can be divided into strata, each of which is more homogeneous than the entire population. In other words, the individual strata have characteristics that allow them to be distinguished from the other strata, and such characteristics are known to affect the measured attribute of interest, namely the radioactivity. Usually, the proportion of sample observations taken in each stratum is similar to the stratum proportion of the population, but this is not a requirement. Stratified sampling is more complex and requires more prior knowledge than simple random sampling, and estimates of the population quantities can be biased if the stratum proportions are incorrectly specified. Example: 60 Co activity in an estuary Mapping radioactive contamination in specific locations or areas is a common objective in radioecological investigations (ICRU, 2001). Suppose one wished to map 60 Co in the sediments of an estuary. The population could be conceived to be all possible sediment cores (depth 30 cm, diameter 10 cm) (N in total) within the estuary; a sampling unit would be a single core. A simple random sample, i.e. a random selection of n of the N possible core locations, is a valid sampling design, but it may not be the best. Stratified sampling, using additional information about the estuary and the environmental behavior of 60 Co, can provide a more precise estimate of the mean activity density in the estuary. The distribution of 60 Co in the estuary may not be homogeneous because sediment type and particle size distribution are associated with 60 Co activity. If a map of the sediment type within the estuary is available, it will indicate areas of mud, sand, etc., each of which would be expected to have a different 60 Co activity density. A stratified random sample estimates the mean 60 Co activity in each sediment type. Estimates from each stratum and the area of each sediment type in the estuary are combined to estimate the overall mean activity for the estuary and the sampling error. In stratified sampling, the population is divided into two or more strata that individually are more homogeneous than the entire population, and a sampling method is used to estimate the properties of each stratum. Usually, the proportion of sample observations in each stratum is similar to the stratum proportion in the population. In the example above, we might consider the estuary as being composed of areas of mud, sand and rock. These would then define the


7

strata. This is an example where environmental knowledge and the problem context lead to a better sampling scheme. In stratified sampling, the population of N units is first divided into sub-populations of N1 , N2 , . . . , NL units representing the sampling units in each of the different strata. These sub-populations are non-overlapping and together comprise the whole population. They need not have the same number of units, but, to obtain the full benefit of stratification, the subpopulation sizes or areas must be known. In stratified sampling, a sample of units is drawn from each of the strata. Often, simple random sampling is used in each stratum. The number of sampling units allocated to a stratum is often proportional to the population size or area of that stratum. When each stratum has the same within-stratum variance, proportional allocation leads to the most precise estimate of the population mean (Cochran, 1977). For the sediment example, the strata might be defined as distinct sediment types. Knowledge of the fractional areas of each sediment type within the estuary would be needed to ensure appropriate sampling fractions within each stratum. Simple random samples of size n1 , n2 , . . . , nl would be taken from each strata. Thus if the estuary was 60% sand, 30% silt and 10% mud, then 60% of the sampling units would be selected in the sandy areas, 30% in the silty areas and 10% in the muddy areas. To estimate the average and variance of each stratum, one would use Equations (1) and (2). The population mean activity, Ac , and its sampling error, Var(Ac ), are weighted averages of the average, y¯1 , and variance, sl2 , for each stratum, l. The weights, Wl , are the fractions of each stratum in the population, i.e. Wl = Nl /N . (Nl y¯l ) Ac = l (4) , N and Var(Ac ) =

l

Wl2

sl2 (1 − fl ) . nl

(5)

The equation for the sampling error (5) assumes that Wl , the stratum weight, is known. This would be the case when the strata are areas on a map. When proportional allocation for the sampling fraction is used (i.e., nl /n = Nl /N ), then in Equation (5), Nl is replaced by nl and N is replaced by n. It is not necessary for the sediment map to be accurate, but inaccuracy in the definition of the strata increases the sampling error and decreases the benefit of stratification. Stratified random sampling will, with appropriate use, provide more precise (i.e. less uncertain) estimates than simple random sampling, but more information is required before the specific strategy can be carried out. Simple random and stratified random sampling may be impractical, say in the sediment example, if finding precise sampling locations is difficult. A more practical method of sampling might involve covering the area in a systematic manner, say in parallel-line transects and this final sampling method, systematic sampling, which is often easier to execute than simple or stratified random sampling, and which in some cases is more representative than a random sample is described below. One disadvantage of systematic sampling is that the analysis of the results is often more complex. Again this example also illustrates how the theoretical description of the sampling scheme needs to be modified by the practical reality of sampling in the environment.

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1.2.3. Systematic sampling Systematic sampling is probably the most commonly used method for field sampling. It is generally unbiased as long as the starting point is randomly selected and the systematic rules are followed with care. Transects and two dimensional grids are specific types of systematic samples. Consider sampling sediment in an estuary. One possible systematic sample of 15 locations might be obtained by randomly choosing a transect across the estuary and taking 5 core samples along this transect, and then placing a further two transects equally spaced along the estuary. Systematic sampling is often more practical than random sampling because the procedures are relatively easy to implement in practice, but this approach may miss important features if the quantity being sampled varies with regular periodicity. Systematic sampling differs from the methods of random sampling in terms of practical implementation and in terms of coverage. Again, assume there are N (= nk) units in the population. Then to sample n units, a unit is selected for sampling at random. Then, subsequent samples are taken at every k units. Systematic sampling has a number of advantages over simple random sampling, not least of which is convenience of collection. A systematic sample is thus spread evenly over the population. In a spatial context such as the sediment sampling problem, this would involve laying out a regular grid of points, which are fixed distances apart in both directions within a plane surface. Data from systematic designs are more difficult to analyze, especially in the most common case of a single systematic sample (Gilbert, 1987; Thompson, 2000). Consider first the simpler case of multiple systematic samples. For example, 60 Co activity in estuary sediment could be sampled using transects across the estuary from one shoreline to the other. Samples are collected every 5 m along the transect. The locations of the transects are randomly chosen. Each transect is a single systematic sample. Each sample is identified by the transect number and the location along the transect. Suppose there are i = 1, . . . , t systematic samples (i.e. samtransects in the estuary example) and the yij is the j th observation on the ith systematic i yij /ni . ple for j = 1, . . . , ni . The average of the samples from the ith transect is y¯i = nj =1 The population mean is estimated by t ni t i=1 j =1 yij i=1 ni y¯i = . y¯sy = t (6) t i=1 ni i=1 ni The estimator of the population mean from a systematic sample is exactly the same as the estimator for a simple random sample but it is more difficult to estimate the variance. When there are multiple systematic samples, each with n observations, the variance of the mean can be estimated by Var(y¯sy ) =

t 1 − t/T (y¯i − y¯sy )2 , t (t − 1)

(7)

i=1

where T is the number of transects in the population (Gilbert, 1987; Thompson, 2000). The term in the numerator, 1 − t/T , is a finite population correction factor that can be ignored if the number of transects in the systematic sample, t, is small relative to the number in the population. The variance estimator given by Equation (7) cannot be used in the common case of a single systematic sample, i.e. when t = 1. Many different estimators have been proposed


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(summarized in Cochran, 1977; Gilbert, 1987). If the population can be assumed to be in random order, then the variance can be estimated as if the systematic sample were a simple random sample, i.e. using Equation (3). That equation is not appropriate when the population has any form of non-random structure. More details of these and other problems are given in Cochran (1977), Gilbert (1987) and Thompson (2000), and in ICRU (2006). Other sampling schemes exist, but they are often intended for rather specialized situations. These include cluster sampling, double sampling and adaptive sampling. These are beyond the scope of this chapter but details can be found in ICRU (2006). Although estimation of the average is probably the most common objective for a sampling campaign, there are other quantities that are of interest in the population, and the basic sampling designs are equally applicable. Perhaps one of the most common sampling purposes is to map the spatial extent of a pollutant, or to estimate the spatial pattern. This is described in more detail in the next section since most radionuclide problems have a spatial context and there is growing use of geographic information systems (GIS) within the radioecology community.

2. Sampling to estimate spatial pattern 2.1. Introduction In many sampling problems, especially in the environmental context, we must consider the spatial nature of the samples collected. It is only common sense that samples, e.g., plants, animals, or soil that are close together are more similar to each other than other samples that are farther apart. Euclidean distance between them can be measured in one dimension if samples are taken along a single transect, or in two dimensions if samples are taken over an area. Spatial sampling methods use the possible relationship between nearby sampling units as additional information to better estimate the quantities of interest. One other important consideration in spatial sampling concerns the ‘spatial size’ of the sampling unit, e.g., a soil sample is of small ‘spatial size’ in the context of mapping a valley. In remote sensing applications, the spatial size’ of the sampling unit may be several hundred square meters and small scale features and variation within a sampling unit would not be observable. In environmental radioactivity, as in general spatial problems, there are two quite different general cases: C ASE 1. We assume that in principle it is possible to measure the radionuclide at any location defined by coordinates (x, y) over the domain or area of interest. This case would generally be appropriate for radionuclides in soil, water and air, such as mapping the Chernobyl fallout, where x and y would typically be latitude and longitude or some other positioning metric. C ASE 2. We assume that in principle it is not possible to measure the radionuclide at all locations defined by coordinates (x, y) over the domain or area of interest, but that it can be measured only at specific locations. For example, consider 137 Cs concentrations in trees. It can only be measured at locations of trees.

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In this chapter, we focus on Case 1, and refer the reader to ICRU (2006) and the more specialized textbooks such as Webster and Oliver (2001) or Cressie (2000) for further discussion of Case 2. Methods of statistical sampling and analysis that take a spatial perspective, such as kriging (Cressie, 2000), form part of the broad topic known as geostatistics, which has a long history (from mining engineering) and is becoming increasingly popular (Wackernagel, 2003; Webster and Oliver, 2001) in radioecology. 2.1.1. Spatial scale Consider the Europe-wide mapping of Chernobyl fallout, where maps for different countries were produced based on very different sampling techniques with very different spatial extent. In some countries, there was a detailed and intensive sampling scheme based on some of the schemes described in Section 1, thus quite detailed inferences could be drawn. In other countries, helicopters and fixed wing aircraft were used to map the fallout in a more continuous and systematic manner. This case is closest to that described in Case 1 above. These two scenarios also emphasis the importance of the spatial scale. If we look at this example in more detail, taking as an example the use of soil samples, consider a survey designed to explore the levels of spatial variation in Chernobyl fallout radionuclides in a small area. A sampling unit is defined to be a 38-mm diameter soil core. Consider nine such cores collected at random within a 1-m2 area. This experiment provides information about small-scale spatial variation within that 1-m2 area. The results could also be combined to provide an areal average and to complete the map of presumably quite a small area, there would need to be many 1-m2 areas surveyed in such a way. A second approach might use a sampling unit of larger dimension (e.g., a 400 m2 area) to sample from a much bigger area, e.g., of 10,000 m2 . This second approach could provide detail about variation over a larger spatial scale and would more easily lend itself to providing a country map. A third study might be to map the entire continent of Europe where the sampling units are ‘counties’ or ‘regions’ within countries. If the environment is heterogeneous at the moderate spatial scale, perhaps because 137 Cs in the soil cores is influenced by soil type and vegetation cover that vary considerably within the sampling area, then these three studies provide quite different descriptions of the spatial pattern and variability. The small scale study might find no spatial pattern, i.e. no correlation between nearby samples, within the 1-m2 area. The intermediate scale study might find a strong spatial pattern, because nearby samples are from the same soil type and vegetation cover. The large scale study will identify a different spatial pattern that depends on the large scale geographic variation between countries. If there were only a few samples per country, the large scale study could not detect the moderate scale pattern due to vegetation and soil characteristics within a country. The use of aerial radiometrics to map large scale radioactivity and its relationship to smaller spatial scale techniques is dealt with in Sanderson and Scott (2001). The extent of the area to be sampled defines the population for which inferences will be made. This will be defined by the context and purpose of the study. The physical area of the study could range from locating hot spots in a former radium factory site to mapping Chernobyl fallout throughout Europe. Practicality and feasibility also affect the optimal sampling design. Although sampling grids and sampling effort may be prescribed on statistical grounds according to some criterion of optimality, the sampling design must also be practical.


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2.1.2. Sampling objectives Specific sampling objectives that arise in the spatial context are usually similar to those described earlier but with added consideration of area, for example estimation of the average activity over a specified area, or estimation of the inventory within an area (or volume). Other objectives could include mapping the radionuclide distribution over an area, estimating the percentage of an area that exceeds a given level and mapping these locations, estimating the scale of spatial variation, and detecting hot spots. These are only possible with spatially referenced data. Finally, on the basis of observations made one might also wish to predict activity at unsampled (target) locations using a spatial interpolation scheme based on the observed measurements. There is a great deal of specialized terminology and notation used in spatial sampling, and these are best studied in some specialized texts, including Cressie (2000), Webster and Oliver (2001), Wackernagel (2003), and Burrough and McDonnell (1998). 2.2. Classical spatial sampling methods The classical sampling designs and analyses described in Section 1 can be used in a spatial context. The sampling procedure remains probabilistic (random) with the assumption that we have identified the full set of ‘individuals’ or locations within the target population, and sampling involves a selection of individual sites to be examined. In random sampling, a random sample of locations at which the attribute is to be measured is chosen from the target population of locations. If there is knowledge of different strata over the sampling domain (such as soil type), the use of a stratified sample would be recommended and a random sample of locations would be selected within each strata. The number of samples to be collected in each stratum would be defined based on the relative areas of each stratum. The data set is then given by the spatial coordinates of each measurement location and the measured value of the attribute at that location. However, systematic sampling is more commonly used in the spatial setting due to its practicality. Usually, for systematic sampling the region is considered as being overlaid by a grid (rectangular or otherwise), and sampling locations are at gridline intersections at fixed distance apart in each of the two directions. The starting location is expected to be randomly selected. Both the extent of the grid and the spacing between locations are important. The sampling grid should span the area of interest (the population), so that any part of the population could become a sampling location. A systematic grid may also include additional points at short distances from some of the grid points. These points can provide additional information about small-scale spatial correlations. Some commonly used sampling schemes are based on quadrats and transects. A quadrat is a well-defined area within which one or more samples are taken; it is usually square or rectangular in shape, with fixed dimensions. The position and orientation of the quadrat will be chosen as part of the sampling scheme. A line transect is a straight line along which samples are taken, the starting point and orientation of which will be chosen as part of the sampling scheme. When it is necessary to sample a large area with closely spaced sampling locations, a systematic grid can require a very large number of sampling locations. Transect sampling can provide almost as much information with many fewer sampling locations. A transect sample is a random or, more commonly, a systematic sample along a line. Two types of transect

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samples are common. In one, transects are short relative to the size of the study area. These transects are randomly located within the study area. Both the starting point and directional orientation of the transect should be randomly chosen. Ideally, data are collected from more than one transect; each transect has a new starting point and direction. In the second form of transect sampling, transects extend completely across one dimension of the study area, and all transects have the same orientation. Often the spacing of sampling locations along the transect is much shorter than the distance between transects. This form of transect sampling results in a rectangular grid sample. The measurements generated from such sampling can be used to estimate population averages, proportions and other percentiles of the distribution. 2.3. Geostatistical based sampling methods Some specialized techniques have been developed for spatial data. In statistical terminology, these are model-based techniques, since underlying the analysis there is a well-defined statistical model. The most commonly used approach is that of kriging, which is an optimal interpolation method, often used to provide estimates of an assumed continuous attribute, based on a measurements made at a sample of locations (such as a map of Chernobyl fallout). Estimates of the map surface are based on a weighted average of the attribute of interest at neighboring sites. The weights are based on the spatial correlation, which must be estimated. The spatial correlation is often presented as a function of the distance separating the sites and presented graphically in a variogram or semi-variogram. The kriging algorithm can provide a spatially continuous map and can also be used to provide predictions of the attribute at unsampled locations. An important part of the calculation includes production of a map of the uncertainty which can then be used to better design subsequent sampling schemes. Geostatistics is a large and active research topic in its own right and the interested reader is referred to the specialized literature referenced at the start of the section for further detail. 2.4. Conclusions Spatially referenced data are becoming an increasingly common part of many radioecological problems. Many choices of sampling design are possible. The most common classical choices are simple random sampling, systematic sampling using transects or grids, or stratified sampling. Other sampling methods, based on geostatistical ideas, can be used both to estimate population characteristics and to predict values at unobserved locations using interpolation schemes such as kriging. A dense grid of predicted values can be used to draw maps. Visualization of the sampling locations and also the measurements are useful exploratory tools, and with increasing availability of GIS software, these methods will gain increasing use in radioecology. 3. Other sampling methods and some common problems 3.1. Other sampling methods Other, more specialized and hence less common sampling methods include two-stage sampling, which involves definition of primary units, some fraction of which are selected ran-


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domly. Then, the selected primary units are sub-divided and a fraction of the sub-units are selected randomly. Cluster sampling is most frequently applied in situations where members of the population are found in clusters or colonies. Then, clusters of individuals are selected randomly and all individuals within each cluster are selected and measured. Double sampling can be useful when one characteristic may be difficult or expensive to measure but another related characteristic is simple or easy to measure. This might involve making a relatively large number of analyses using the more efficient technique, and selecting a few specimens from this sample on which to make the more expensive analysis. These and other sampling schemes can be found in Thompson (2000) and Thompson and Seber (1996). 3.2. Number of replicate samples One of the most commonly asked questions is “how many individual samples are required?” To answer this question, we must ask a further series of questions. The questions given here focus on the mean, but the approach can be applied for any population parameter. (a) First we must determine if it is the population mean or the difference in two population means which is of interest. If it is the difference in two means, then (b) we must ask how much of a difference would be of real world importance and hence is important to be able to detect. (c) Next, how variable is the quantity in the population, i.e. what is the variance in the population? (d) Finally, how sure do we want/need to be in the answer, i.e. what is the desired standard error (for an estimate) or statistical power (for detecting a difference)? If a quantity is very variable in the population, then we are likely to detect only very large effects. A precise estimate of the population mean will require a large sample size. Alternatively, if we want to study a small effect, then we will need to increase the sample size perhaps to an unmanageable level. In both cases, there is a trade-off between sample size, precision and size of effect. The exact relationships depend on the sampling design and the quantities of interest. Formulae for sample size determination can be found in many statistical textbooks; the computations are implemented in general statistical software and specialized programs (e.g., Visual Sample Plan; Pulsipher et al., 2004). 3.3. Practical sampling issues One common practical issue that might arise concerns the problem of not being able to follow exactly the pre-determined statistical sampling design. This happens for various good reasons, such as problems with sample selection (e.g., material not present at the selected site), the presence of obstacles or conditions preventing a sample being taken at a given location, and analytical problems (e.g., insufficient amount of material for analysis). Absence of suitable material is a common source of missing values in environmental sampling. For example, if lichen are being sampled, what should be done if the designated sampling location has no lichen? One common solution is to select a nearby location that does contain appropriate material. Another is to ignore the location and reduce the sample size. A less common solution is to select another location using the original sample design. This

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approach differs from the first in that the alternate location may be far from the original location. There are no definitive guidelines about a best strategy to adopt. Good sense and knowledge of the environmental context will provide guidance.

4. Conclusions Sampling for radionuclides in the environment is similar to sampling for other attributes of environmental media. Statistical sampling is pertinent and necessary in radioecology because of the natural stochastic variation that occurs, and the fact that this variation is usually much larger than variations associated with measurement uncertainties. The environmental context of the problem affects the nature of the sampling to be carried out. Individual cases vary in objective and environmental context. Design of sampling schemes requires problem-specific environmental knowledge, statistical knowledge about the choice of the sampling design and practical knowledge concerning implementation. Frequently, practical issues can limit sampling designs. Good discussions of the statistical aspects of sampling include the general sampling textbooks by Cochran (1977) and Thompson (2000), the environmental statistics textbook by Gilbert (1987), and many scientific papers including ICRU (2006).

Acknowledgements This work draws heavily on the preparation work for the ICRU report (2006). The drafting committee included Ward Whicker, Kurt Bunzl, and Gabi Voigt. The many helpful discussions with them are gratefully acknowledged.

References Burrough, P.A., McDonnell, R.A. (1998). Principles of Geographical Information Systems. Oxford Univ. Press, Oxford, UK. Cochran, W.G. (1977). Sampling Techniques, third ed. John Wiley and Sons, New York. Cressie, N. (2000). Statistics for Spatial Data, second ed. John Wiley and Sons, New York. Dixon, P.M., Ellison, A.M., Gotelli, N.J. (2005). Improving the precision of estimates of the frequency of rare events. Ecology 86, 1114–1123. Gilbert, R.O. (1987). Statistical Methods for Environmental Pollution Monitoring. Van Nostrand Reinhold Company, New York. International Commission on Radiation Units and Measurements, ICRU (2001). Quantities, Units and Terms in Radioecology. ICRU Report 65, ICRU 1 (2). Nuclear Technology Publishing, Ashford Kent, UK, pp. 1–48. International Commission on Radiation Units and Measurements, ICRU (2006). Sampling for Radionuclides in the Environment. ICRU Report 75. Oxford Univ. Press, Oxford, UK. Pulsipher, B.A., Gilbert, R.O., Wilson, J.E. (2004). Visual Sample Plan (VSP): A tool for balancing sampling requirements against decision error risk. In: Pahl-Wostl, C., Schmidt, S., Rizzoli, A.E., Jakeman, A.J. (Eds.), Complexity and Integrated Resources Management, Transactions of the 2nd Biennial Meeting of the International Environmental Modelling and Software Society. iEMSs, Manno, Switzerland. Sanderson, D.C.W., Scott, E.M. (2001). Special issue: Environmental radiometrics. J. Environ. Radioact. 53 (3), 269–363.


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Thompson, S.K. (2000). Sampling. John Wiley and Sons, New York. Thompson, S.K., Seber, G.A.F. (1996). Adaptive Sampling. John Wiley and Sons, New York. United States Environmental Protection Agency, US EPA (2002). Guidance on Choosing a Sampling Design for Environmental Data Collection for Use in Developing a Quality Assurance Project Plan. EPA QA/G-5S, EPA/240/R02/005. EPA, New York. Wackernagel, H. (2003). Multivariate Geostatistics: An Introduction with Applications, third ed. Springer-Verlag, New York. Webster, R., Oliver, M.A. (2001). Geostatistics for Environmental Scientists. John Wiley and Sons, New York.

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Sampling techniques Fedor Macášek∗ Department of Nuclear Chemistry, Faculty of Natural Sciences, Comenius University, Mlynská dolina CH-1, SK-84125 Bratislava, Slovakia

1. Introduction There are several ways to analyze radionuclides in the environment. Direct radioactivity measurements for environmental surveys have been carried out by placing a detector near the media being surveyed and inferring radionuclide levels directly from the detector response. Scanning is a measurement technique performed by moving a portable radiation detector at a constant speed above a surface to assess areas of elevated activity. However, there are certain radionuclides that will be impossible to analyze using this simple approach because of their decay properties. Examples of such radionuclides include pure beta emitters such as 3 H, 14 C, 90 Sr and 63 Ni and low-energy photon emitters such as 55 Fe and 125 I. Analysis of alpha emitters is also restricted to surface contamination of relatively smooth, impermeable surfaces. Although direct measurements are practical for space–time monitoring and obtaining averaged radionuclide levels they cannot provide information on structural peculiarities of radioactive contamination. Therefore, a detailed environmental survey usually starts by sampling and preparation for laboratory analyses and is characterized by inspection of the technically and financially feasible fraction of the entire population of objects under investigation. The purpose of sampling is to achieve a mass reduction by collection of a certain subpopulation of the objects of analysis while preserving complete similitude of the crucial parameters of the sample and object. The ideal sample has the same composition as the sampled object; in the case of radiometric and radiochemical analysis of environmental samples, this means the same massic activity (e.g., Bq kg−1 ) or activity concentration (e.g., Bq m−3 ) of radionuclides, and the same relative abundance of their physical and chemical forms. The persistent difference between the true (or total population) value and the sampled one (subpopulation) is an exhibition of bias—a systematic distortion of the measurement in one direction. Sampling for radionuclide analysis is thus defined as the process of collecting a portion of an environmental medium which is representative of the locally remaining medium. Represen∗ Present address: BIONT, Karloveská 63, SK-84229 Bratislava, Slovakia. E-mail address: [email protected]



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tativeness is characterized as a measure of the degree to which data accurately and precisely represent a characteristic of a population, parameter variations at a sampling point, a process condition, or an environmental condition (EPA, 2002a, 2002b, 2002c, 2002d; ANSI, 1994). Still, the obstacles in practical interpretation of such a definition are obvious when you imagine any sampling of your actual close-by environment. Therefore the definition of sampling as the process of gaining information about a population from a portion of that population called a sample (Hassig et al., 2004) is more universal and acceptable. The sampling is the foundation block on which any analytical result is built. Without reliable out-of-lab and in-lab sampling plans no quality assurance of results can be achieved. Most traditional strategies of environmental sampling were implanted from the field of geochemical assay, which should provide precise information on the ore and other raw material resources within the area designed for exploitation (Journel and Huijbregts, 1978; Isaaks and Srivastava, 1989). Even more, the same random sampling tests are incorrectly applied in the same way as in the chemical and pharmaceutical industry where their goals are to provide evidence of declared identity and homogeneity of the whole batch sampled (Duncan, 1986; Schilling, 1982). For such purposes the reliable statistical standards are well elaborated (ISO, 1991, 1995, 2000, 2001; ASTM, 2002). Specific sampling analysis plans are necessary for survey of the environment where the field is random by its nature (Gilbert, 1987; Cressie, 1993; Myers, 1997; Byrnes, 2000; USACE, 2001; EPA, 2001). The MARSSIM (the EPA’s MultiAgency Radiation Surveys and Site Investigations Manual) guide focuses specifically on residual radioactive contamination in surface soils and on building surfaces (EPA, 2002a, 2002b, 2002c, 2002d). The natural and industrialized environment is neither homogeneous nor unchanging. Even the most illusive homogeneous system like the atmosphere is in fact a colloidal/particulate system—typical indoor air contains about 107 –108 particles in the 0.5 to 100 µm size range per cubic meter. Air sampling needs the size fractionation and characterization of particles, the radioaerosols specifically: at least, distinguishing the radioaerosol size below 1 µm and in the interval 1–5 µm is necessary for radiation protection assessment purposes (ICRP, 1966; Koprda, 1986; ISO, 1975; IAEA, 1996). The measurement of pollutants, the radioactive ones included, may be oriented less to their inventory and more towards the estimation of their role in litho- and biosphere migration which cannot be expressed unequivocally either in space or time. Monitoring of contamination of soil by radiocesium serves well as an illustration. The global contamination with 137 Cs characterizes the radiation field above the ground and it can be easily determined by in situ gamma spectrometry and aerospace scanning. However, when there is the concern for radiocesium bioavailability or soil remediation (Cremers et al., 1988; Navratil et al., 1997; Wauters et al., 1996) the assay of the soil should include determination of the mobile forms of cesium (Macášek and Shaban, 1998; Bartoš and Macášek, 1999, 2002). The same applies to other radionuclides and metal ions (Ure et al., 1992; Smith-Briggs, 1992; Hlavay et al., 2004) that can exist in several different soil or sediment phases, e.g., in solution (ionic or colloidal), organic or inorganic exchange complexes, insoluble mineral/organic phases, precipitated major metal oxides or in resistant secondary minerals. Very often, environmental authorities and decision-makers long for reproducible and reliable (= accurate) data on pollution, i.e. smooth data with a narrow statistical deviation. From this request the traditional features of the sampling and preparatory treat-

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Table 1 Requested parameters for food and environmental radioactivity monitoring (IAEA, 1988) Monitoring

Tolerable bias

Assessment time

Screening Very fast Fast

10× 2–3× 20–50%

5–15 min 1–6 h 6–24 h

ment of environmental samples for trace and radiochemical analysis follow (Green, 1979; Alberts and Horwitz, 1988), such as the concern for representativeness of the results. However, the principal question in environmental assay, and that for radioactive pollution in particular, is whether all the requirements for a radiation survey should be satisfied with representative samples and the indication of a low uncertainty of the analytical techniques or by a scattered plenitude of data on naturally occurring samples? It is evident, for example, that the subsamples containing hot particles of burnt nuclear fuel after the Chernobyl accident (Tcherkezian et al., 1994; Salbu et al., 1994) should be statistically found irrelevant by going far beyond the confidence limits of a normal distribution of activity of samples collected from the contaminated areas. The total variance of sample assay σt2 is σt2 = σd2 + σs2 + σl2 ,

(1)

where σd2 , σs2 and σl2 are the total variances of the sampling design, field sampling and laboratory technique, respectively (Smith et al., 1988; Kirchhoff et al., 1993). The time factor is significant in general sampling design. For various decision-making tasks the required accuracy/precision of radionuclide monitoring is compromised by the sampling, processing and measurement times. For example, during a nuclear incident, some compromise must be reached between the ideal sample coverage and the need to gather larger numbers of samples as rapidly as possible, as in Table 1 (IAEA, 1988). 1.1. Sampling homogenization versus speciation Ordinary application of mean values of massic activities and standard deviations characterizing the laboratory technique is imperfect because variation in the object of sampling is usually much larger than the variation of laboratory techniques. It could resemble a situation when a tailor would use his customer’s family’s average size and the tape measure distortion as a legitimate bias instead of the individual and changing differences between folk—most probably his garments will fit nobody. The effort for representativeness incorporates: (1) Taking of “sufficiently large” samples (Mika, 1928; Baule and Benedetti-Pichler, 1928; Gy, 1992), which is a must when the specific activity is below the detection limit of radiometry and pre-concentration (matrix removal) is necessary (2) homogenization and/or other processing of samples (Remedy and Woodruff, 1974), and (quite often)

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(3) spiking or isotopic labeling of the sample (isotope dilution) if the reprocessing of the sample leads to losses of analyte (Tölgyessy et al., 1972; Macášek, 2000). As shown below, the homogenization in principle leads to a loss of information contained in the sample. Hence, it is substantiated only when the sample is defined as homogeneous and its specific activity, as an average value, is deemed important, the assay of the layers of ground investigated for penetration of radionuclides to depth being an example. An averaged radionuclide content in the sample results from a population of n different sorts of entities. Each ith sort of entity represents a homogeneous component, such as a chemical compound, species or particulate form of radionuclide, of identical composition. When i indexes these constitutive species, the average concentration can be expressed as a function of the massic activities of radionuclide species ai and the fractional abundances of the species xi , a¯ =

n

xi ai .

(2)

i=1

The massic activity of any j th subsample is given by a subset of entities, a¯ j =

ν

yij ai ,

(3)

i

where “i”s are ν random values from the interval 1, n and yij is the abundance of the ith entity in the j th subsample. The subsample is representative in respect of massic activity, when a¯ ≈ a¯ j which is merely a statistical task except for a simple form of radionuclide in a homogeneous matrix (i = 1), a true solution of radionuclide present in a single particulate and chemical form. Also the evaluation of various species in heterogeneous objects usually proceeds through replacement of the full set of parameters by assuming that yij = xi .

(4)

This traditional access is still most widespread and representativeness is ensured by a large sample and random mass (volume) reduction to ensure its validity. Also, a composite sample is formed by collecting several samples and combining them (or selected portions of them) into a new sample which is then thoroughly mixed. The approach is strongly favored, e.g., by the tasks of geological resource assessment, technological material balances, artificial preparation of modified materials and standards for validation of analytical methods. The approach always looks attractive because of the ability of current analytical techniques to supply reproducible data of low uncertainty for a set of replicate samples—the relation to true value of radionuclide concentration (the massic activity) remains the ultimate test for quality of measurements. In fact, the objects of environmental origin are far from being either homogeneous or uniform and it is therefore difficult to present precise data. More sophisticated procedures may include: (i) an estimation of specific forms of radionuclide in various fractions of the representative sample (i > 1, j = 1), i.e. a physico-chemical speciation that is the “species distribution or abundance studies which provide a description of the numerical distribution (or

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abundance) of different species containing the same central element, in a given sample” (Pickering, 1995; Macášek, 1994, 1996). (ii) a treatment of a more complete subset of analytical data on the partial concentrations and matrix composition (m > 1, j > 1). Such a procedure is applied when variability of the samples appears to be too high, e.g., in analysis of acid extracts of sediments, or for stratified (segregated) random sampling (ACS, 1980). Reduced uncertainty of environmental radionuclide data is artificially achieved in the course of thoroughly applied but entropy generating homogenization procedures. The grinding, milling, mincing, blending, chopping, mixing, melting, mineralization, burning and dissolution procedures are substantiated when the analyte concentration is below the detection limit of an analytical technique and a pre-concentration is necessary. However, in the course of homogenization, considerable information content is lost according to the Shannon information entropy (Eckschlager and Danzer, 1994), pi ln pi H =− (5) i

(information can be considered as negative entropy). H becomes minimal (zero) for an ideally homogenized sample when radionuclide should be in a single form (xi = 1) and the probability of its presence is either pi = 1 or pi = 0 (in the latter case, conditionally, pi ln pi = 0). A sample preserving, e.g., two particulate forms of radionuclide at equal abundance (pi = 1/2) has the information content for 0.693 entropic units higher. 1.2. Ontology of environmental analysis The question is “why?” and “when?” the sample should be “representative” and “precise”. This attribution is transformed to the problem of “what?” should be determined in the sense of what information (not identical to the “activity” or “massic activity”!) should be extracted from the analysis of environmental objects and how should it be properly presented. The final question is a real ontological problem of the purpose of analysis (Macášek, 2000). Therefore, analytical scenarios must be designed from the very beginning by co-operation of environmentalists and (radio)analytical chemists or other nuclear specialists. In other words, the way to determine radionuclides in environmental objects strongly depends upon the mode of application of the data received; such an assessment may be shown to vary from a random sampling and sample homogenization to a stratified sampling and sophisticated speciation of radionuclides in the environment. The aim of the analysis is conditioned by the purpose for which the data are being collected. Obviously in many cases, instead of an averaged value of analyte determination in a homogenized sample and an uncertainty derived from standard deviations of the analytical procedures, more complex issues should be derived. It should be stressed that quality assurance, good laboratory practices, reference methods and reference materials should be applied throughout the analytical procedures (Povinec, 2004). We shall characterize (Table 2) the goals of data collection and related sampling and sample treatment by the preservation of information content, i.e. the consequent change of entropy in

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Table 2 Links between the objectives of analysis, respective sampling and pre-analytical treatment strategy Sampling and treatment Homogenization Composite sampling

Random replicate sample analysis

Stratified random sampling and analysis Speciation

Strongly entropic mode

Medium entropic mode

Low entropic mode

Objectives (examples) (1) Radioactivity inventory, regional abundance and availability of radionuclide, differences between areas and contour maps of GIS (2) Regional environmental impact evaluation (3) Regional spatial dynamics of radionuclide in pseudocontinuous environment (4) To determine source and undeclared releases of radioactivity

(1) Natural distribution and environmental heterogeneity indication by a non-parametric evaluation

(1) Physico-chemical disclosure of species and their relations

(2) Test statistics for hypothesis tests related to particular parameter (3) Assessment of a stochastic action of radionuclide

(2) Specific action of radionuclide species

(4) Average mobility of radionuclide in multicomponent media

(3) Distribution between environmental compartments and food chains (4) Specific mobility and transfer of species (5) Future behavior forecast

the course of reprocessing—see Section 1.2. Obviously, the associated costs increase to the right and down the list. Direct physical and chemical speciation is desirable, e.g., in ecotoxicological evaluation (function) of analyte (Remedy and Woodruff, 1974): 1. functionally – biological availability – toxicity – mobility and transfer 2. operationally – leachability – exchangeability 3. particulate, morphological and chemical state. The native state of species can be seriously disturbed not only by a destructive pre-analytical treatment but also in the course of exposure of samples to air oxygen and microbial flora, light, heat and contact with sampling devices and vessels. Then the task of analytical speciation is either to distinguish “stable” species or to get a fingerprint of original native abundance (Macášek, 1994, 1996). The time factor is also important; due to their heterogeneity, photochemical and metabolic processes the natural objects occur much more frequently in a steady-state than in thermodynamic equilibrium.

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2. Optimization of sampling 2.1. Data quality objectives Data quality objectives (DQOs) are “qualitative and quantitative statements derived from the process that clarify study technical and quality objectives, define the appropriate type of data, and specify tolerable levels of potential decision errors that will be used as the basis for establishing the quality and quantity of data needed to support decisions” (EPA, 2000, 2002a, 2002b, 2002c, 2002d). Data quality objectives derive from a systematic scientific planning that defines the type, quality and quantity of data needed to satisfy a specified use (EPA, 2000). The key elements of the process include: • • • • • • •

concisely defining the problem identifying the decision to be made identifying the inputs to that decision defining the boundaries of the study developing the decision rule specifying tolerate limits on potential decision errors, and selecting the most resource efficient data collection design.

Data validation is often defined by the following data descriptors: • • • • • •

reports to the decision maker documentation data sources analytical method and detection limit data review, and data quality indicators.

The principal data quality indicators according to EPA (1998) are: • • • • •

precision bias representativeness comparability, and completeness.

Uncomplicated spikes, repeated measurements and blanks are used to assess bias, precision and contamination, respectively. Other data quality indicators affecting the radiation survey process include the selection and classification of survey units, uncertainty rates, the variability in the radionuclide concentration measured within the survey unit, and the lower bound of the gray region. Of the six principal data quality indicators, precision and bias are quantitative measures, representativeness and comparability are qualitative, completeness is a combination of both qualitative and quantitative measures, and accuracy is a combination of precision and bias. The selection and classification of survey units is qualitative, while decision error rates, variability, and the lower bound of the gray region are quantitative measures. Data qualifiers (codes applied by

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F. Macášek

the data validator) help quickly and critically to judge the collected data and give the manager a sign on how to use them (EPA, 2002c). Environmental data verification and the validation processes test for non-conformance in project planning, field activities and sample management. The process for determining the utility of the obtained data is based on scientific and statistical evaluation of whether they are of the right type, quality and quantity to support their intended use (EPA, 2002d). 2.2. Sampling plan The sampling plan can be divided into design and field sampling components. The design part of sampling solves the questions • • • • • •

what is the objective of sampling what type of samples to pick in respect of radionuclides of interest minimal amount of sample necessary for its laboratory and statistical assay how many samples to collect where to sample, and when and at what intervals to sample.

Special care should be addressed on choice of blank samples. The field sampling phase is concerned with • • • • • • • • • • • •

site preparation who takes the samples how to describe them how to identify and label samples how to collect them how to avoid cross-contamination how to stabilize samples how to pack samples how to transport samples how to store, and how to advance samples to the laboratory, and last but not least what are the sampling and treatment costs.

The sampling plan is usually a compromise between various demands of data users, sampling enterprise and analytical laboratories on accessibility and availability of samples and cost-effectiveness of sampling and measurement procedures. The last strongly differs for low and high activity samples, type (alpha, beta and gamma) of radionuclides, type of matrix and required precision of individual data. Basically, two alternate procedures for sampling plans and test hypothesis are developed: (1) Classical procedures of random sampling when nothing is considered about the population under investigation (i.e., absence of prior data or “zero hypothesis”), each individual sample in a population has the same chance of being selected as any other, and just the information received experimentally is taken into account (“frequentist” approach). (2) Bayesian methods when the prior data (i.e., information collated prior to the main study) are used in the establishment of a priory probability laws for sample parameters, i.e. new

Sampling techniques

25

Table 3 Number of samples for a frequentist approach Number

n

Substantiation

Fixed

1 2 3 5 10 21 or 28

Minimally imaginable Enabling a rough assessment of average value and standard deviation Median can be assessed with its approximate deviation A non-parametric estimate of deviation becomes possible Adapted for populations 10 N 100 Evaluation of median and distribution in percentiles (quantiles)

Population N related N/20 . . . N/10 √ N

To be used when N < 10 A “reasonable” fraction (5–10%) of a medium size of population, 30 N 100 Deviations diminish with size of analyzed samples, suitable for larger populations (N > 100) 20 + (N − 20)/10 Good evaluation of median and distribution in percentiles (quantiles) in populations (N > 100)

samples are planned according to available information or a working hypothesis on the former ones. Bayesian methods might be further classified according to their approach to prior information. The uninformative approach implies there is no information concerning the assayed prior sample parameters. Empirical methods extract the prior distributions from the available statistical information on the studied samples. Subjective Bayesian methods use the prior information based on expert personal beliefs (Carlin and Louis, 1996). Application of the Bayesian approach enables minimization of the number of samples necessary for confirmation of a working hypothesis. The determination of the number of samples that should be gathered is one of the most important practical tasks of a sampling plan (Aitken, 1999). The number of samples may issue from the allowed decision uncertainty and estimated variability of the samples. The variance of parameters x in a population of n samples is usually derived from the standard deviation based on a Gaussian (“bell-shaped”) distribution, n (xi − x) ¯ 2 2 σ = i=1 (6) , n−1 i.e. n > 1 is necessary. However, the Student coefficient as large as 12.7 for a 95% confidence interval should be considered for the standard deviation obtained from two samples! Thus, two-item replicates are the customer’s favorite as a minimal number to estimate variability. Five sample replicates may be sufficient for a non-parametric evaluation—see below. A replicate set of 28 samples is considered suitable for non-Gaussian estimates of the 10-, 25-, 50-, 75- and 90-percentile points (the P -percentile point is that value at or below which P percent of total population lies)—see Table 3. When the estimate is not available from a pilot study of the same population, a conservatively large preliminary value of the population variance should be estimated by another study conducted with a similar population, or based on a variance model.

26

F. Macášek

Table 4 Number of samples to be taken for one-sample t-test (Tukey, 1977) Significance

Test power

10%

Relative size of gray region 20%

30%

40%

50%

Risk level 5%

95% 90% 80%

1084 858 620

272 216 156

122 97 71

69 55 40

45 36 27

Risk level 10%

95% 90% 80%

858 658 452

215 166 114

96 74 51

55 42 29

36 28 19

To determine the minimum replicate sample size needed to estimate a population mean (for example, a mean contaminant concentration), in the total absence of prior information, the rough standard deviation expectation can be calculated by dividing the expected range of the population (between expected maximal and minimal values of x) by six, i.e. xmax − xmin . (7) 6 Using either Equation (6) or (7), the minimum sample size needed to achieve a specified precision for estimates of the population means and proportions is further evaluated. Table 4 gives the number of samples necessary for a hypothesis test for the number of samples to be taken with acceptable risk level of a wrong decision. It illustrates the reliability of a test when the relative width of the gray region towards the standard deviation is within 10 to 50% (broader data are available for various sampling designs; EPA, 2002a, 2002b, 2002c, 2002d). As seen, the number of samples strongly increases when the gray zone of an ambiguous decision is formulated too narrowly. For example, the limit for contamination of construction walls with 239 Pu is 3 kBq m−2 . In preliminary investigation of a building, the activity of samples did not exceed a rough value of 2.6 kBq m−2 . The sampling should ensure that neither false contamination under a true value of 3.03 kBq m−2 nor a false clean facility, actually contaminated above 2.8 kBq m−2 , will be announced with an acceptable level of risk 5%. From the last figures, the gray area width is 3.03 − 2.85 = 0.18 kBq m−2 . According to Equation (7), the standard deviation may be expected to be (2.6 − 0)/6 = 0.43 kBq m−2 . The relative width of the gray area is 100 × 0.18/0.43 = 45%. If the test reliability (test power) should be 90%, we found from Table 4 that the necessary number of random samples should be between 36 and 55 (say 45) to confirm the hypothesis of the mean contamination level of facility walls. The same value will harmonize with a higher, 95% reliability of test, but also a higher acceptable 10% level of risk. Two statistical tests are usually used to evaluate data from final status surveys. For radionuclides also present in background, the non-parametric Wilcoxon rank sum (WRS) test is used. When contaminants are not present in background, the sign test is used (EPA, 2002a). A desired sample size in the Bayesian approach is derived from the hypothesis of the occurrence of non-conforming units of population (e.g., the fraction of those contaminated by σ =

Sampling techniques

27

Table 5 Required sample size n to guarantee with 95% confidence that at least a fraction θ = q/n of population units is clean if all samples are clean (q = n) Population, N

n θ = 50%

θ = 70%

θ = 90%

10 20 30 40 50 90 100 1000 5000 10000

3 4 4 4 4 5 5 5 5 5

5 6 7 7 8 8 8 9 9 9

8 12 15 18 19 23 23 28 29 29

radioactivity) which should be accepted or rejected on an empirical base. Let the zero hypothesis says that in the whole population of samples there are Q clean units and the rest R = N − Q is contaminated. The question is what is the probability to find q clean samples in the set of n samples (and r = contaminated ones), while the sample set is representative, i.e. the fraction θ is the same as in the whole population, θ = Q/N = q/n? Such conditional probability P can be obtained as Q R P (q|N, Q, n) =

q

r

N .

(8)

n

Some data are illustrated for the case when it is necessary to confirm various fractions θ of clean population units by obtaining n totally conforming (clean) samples in Table 5. It follows that for confirmation of, e.g., area cleanness, it is advantageous to withdraw random samples from a large set of population units, and with almost the same effort applied for their analysis, the statement on contamination may cover a much larger area. For large populations (N > 100), the calculation by hypergeometric distribution (Equation (8)) can be easily approximated by a binomial distribution, n q P (q|θ, n) = θ (1 − θ )n−q , (9) q which for the case of all negative samples (n = q) gives P = θ n and n can be calculated for probability P and fraction θ simply as n = log P / log θ . For example, to guarantee with 95% confidence (P = 0.95) the occurrence of 0.2% or less of contaminated units (θ = 0.998), the number of all negative samples from a large population should be at least n = log(0.95)/ log(0.998) = 2.23 × 10−2 /8.69 × 10−4 ≈ 27. 2.2.1. Simple random sampling A simple random sampling design is the most common way of obtaining data for statistical analysis methods, which also assume that the data were obtained in this way. For a simple

28

F. Macášek

random sampling plan it is typical that the population (field) is considered systematic (not random) and its premises are irrelevant; it is the mode of sampling that is kept random. It is clear that simple random sampling is appropriate when the population is indeed relatively uniform or homogeneous. Visually, the classical simple random sampling is the “basket of apples” method. If one wants to measure the parameters of apples (activity in particular) without using the whole population of apples on a tree, just a basket of apples is taken for analysis. The randomness lies in arranging the choice of the apples and random numbers are used to determine which sample of the whole set is to be taken. Then the averaged parameter of the subpopulation in the basket and its variance is supposed to perfectly imitate the “true” parameters of the whole population—see discussion for Equation (4). Random sampling would meet numerous logistic difficulties in reaching the sites picked on a map by random numbers. Sometimes further on-site randomization (classically, the blind throwing of a rock hammer) is necessary in this case. From the information point of view, any spatial interrelation existing in the original population of samples (“apples on a tree”) is destroyed by randomization and information is discarded. 2.2.2. Systematic-random sampling A special random sampling technique to obtain regional coverage is systematic sampling. The sampling locations vary in a systematic manner and the distance between the samples is uniform. However, the first location in the tissue is random; a rectangular Cartesian grid is laid over the investigated region, the origin and axis orientation of co-ordinates being obtained from random numbers. Both systematic-random and pure random sampling are used to converge to the expected parameter. However, estimates with systematic-random sampling converge to the expected value sooner. Systematic-random sampling makes more precise estimates in less time; therefore, systematic-random sampling is more efficient than pure random sampling. 2.2.3. Stratified-random sampling Stratified random sampling consists in a preliminary subdivision of the samples into more homogeneous groups, subgroups or strata. There can be no sample that does not belong to any of the strata and no sample that belongs to more than one stratum. The subgroups are expected to exhibit smaller variance than the whole population and the weighted combination of strata means gives the value for the total population—see Equation (2). Preserving the rhetoric of apples, the strata can be composed of the fruits separately collected from the top and bottom of a tree or from its north and south sides. In the case of biota they may be lichens or mushrooms chosen from the flora, etc. Most verbatim, various soil layers can be sampled as strata. In most situations the vegetative cover is not considered part of the surface soil sample and is removed in the field. A specific stratum for sampling is a cell of a Cartesian (also an irregular) grid, from which a specified number of random samples are collected. GPS (Global Positioning System) technology facilitates irregular grid sampling. In the working environment, strata are more imaginary, e.g., exposure zones based on similarity of jobs, environment, etc. Temporal strata permit dif-

Sampling techniques

29

ferent samples to be selected for specified time periods and support accurate monitoring of trends. The stratification of objects happens to be affected by the strong personal opinions of planners led by their intuition. Such discretionary samples may be helpful but certainly cannot be used validly within the random sampling plans. 2.2.4. Representative sampling As discussed in Section 1.1, the representativeness of a particular sample cannot be quantified because the “truth” is not known. It can be verified through application of statistical approaches based on the actual measurements. When the average concentration of analyte is of main interest, Mika’s formula (Mika, 1928) can be applied for estimation of the minimal mass m representing an inhomogeneous object, m>K

d3 , δ2

(10)

where K (g mm−3 ) depends on relative size and relative weights of species, d is the diameter (mm) of the largest particles and δ is the tolerated relative uncertainty. This equation is simplified in the “Gy’s safety rule” (Pitard, 1993) which calls for the minimum mass of solid samples m 125d 3 .

(11)

For example, for the sieved soil samples with particle diameter d 1 mm the minimum representative size of sample is calculated as 125 g, and when 0.25 g subsamples are assayed the sample should be powdered below 0.13 mm. 2.2.5. Geostatistical (random field) sampling When there is a spatial or temporal dependence, samples close together in space and time scale will tend to have more similar values than samples far apart. This is often the case in an environmental setting. Geostatistical sampling resides in such a sampling procedure that is guided by the assumed properties of some region via random space–time coordinates and prior estimates of the covariance functions (statistical measures of the correlation between two variables) to account for the pattern of spatial continuity (Borgman and Quimby, 1988; Armstrong, 1998). Initial estimation of the covariance structure of the field and its stationary or non-stationary character is of principal importance. Therefore, the design phase of a geostatistical sampling plan when a model for probability law is to be developed is rather laborious but the field phase and gross sampling are less costly than for random sampling. A primary advantage of geostatistical sampling method is that sample sites may be relatively freely selected by personal judgment to best cover the investigated area. On the other hand, the sample sites should allow estimation of the covariance structure, the correlation of data in space and time. Site preparation involves obtaining consent for performing the survey, establishing the property boundaries, evaluating the physical characteristics of the site, accessing surfaces and land areas of interest, and establishing a reference coordinate system. A typical reference system spacing for open land areas is 10 meters and can be established by a commercially available GPS, while a differential GPS provides precision on the order of a few centimeters.

30

F. Macášek

2.2.6. Ranked set sampling In this two-phased approach, r subsets of m subsamples each are selected and ranked according to some feature that is a good indicator of the parameter of interest using professional judgment or a rough estimate (Bohn and Wolfe, 1994; Patil et al., 1994). From the first subset only the first ranked unit (rank m = 1) is chosen and measured. Another set is chosen, and the (m + 1)th ranked unit is chosen and measured, etc. The advantage is that only r samples are sufficient to estimate an overall mean and variance, instead of the full set r × m. For example, suppose that nine samples would be randomly selected and grouped into three groups of three each. The three samples in each group would be ranked by inspection (assumed to be correlated with the parameter of interest). The sample with rank 1 in group 1, the sample in group 2 with rank 2, and the sample in group 3 with rank 3 would be composited and analyzed. The initial group of nine samples reduces to only one composite sample of size three. In terms of the precision of the estimated mean, such an approach should perform better than a simple random sample of size three (though worse than a simple random sample of size nine). 2.2.7. Adaptive cluster sampling Choosing an adaptive cluster sampling design has two key elements: (1) choosing an initial sample of units and (2) choosing a rule or condition for determining adjacent units to be added to the sample (Thompson, 1990, 2002; Seber and Thompson, 1994). Initial samples are selected randomly and evaluated; it is most useful when the field radioactivity measurement can be used for this step. Then additional samples are taken at locations surrounding those sites where the measurements exceed some threshold value. Several rounds of such sampling may be required. Adaptive cluster sampling is similar in some ways to the kind of “oversampling” done in many geostatistical studies. Therefore, selection probabilities are used to calculate unbiased estimates to compensate for oversampling in some areas via either declustering techniques, polygons of influence, or kriging. Kriging is the extrapolation method to estimate a field at an unobserved location as an optimized linear combination of the data at the observed locations (Stein, 1999). Kriging also allows an estimate of the standard error of the mean once the pattern of spatial covariance has been modeled. 2.2.8. Hot spot identification The Bayesian method can be also demonstrated in the identification of hot spots, relatively small areas of elevated activity (EPA, 1989). The sampling plan for identification of a hot spot of the radius R is made with a distance D between adjacent quadratic grid points. It is supposed that at least one grid point in any square of area D 2 will fall inside the hot spot. The prior probability P (H |E) of hitting a hot spot at its existence E by such sampling (at large distance from sampling points as compared with the hot spot area) is P (H |E) = 1, when D 2R. The most frequent sampling situation is that the grid distance is large compared with the hot spot radius. If R < D/2 then the probability to hit the spot is P (H |E) = (πR 2 )/D 2 .

(12)

Sampling techniques

31

√ For a less frequent case of medium distances R 2 D 2R the probability is √ R 2 [π − 2 arccos(D/2R) + (D/4) 4R 2 − D 2 ] P (H |E) = (13) D2 where the angle θ = D/(2R) is expressed in radians. Geometrical similitude exists for the same θ . When the probability based on previous experience that a hot spot exists is P (E), then the posterior probability that it does not exist can be found by the Bayes formula. The probability of existence of the hot spot E when there was no hit by the sampling (H¯ ) is P (E|H¯ ) =

P (E)P (H¯ |E) . ¯ (H¯ |E) ¯ P (E)P (H¯ |E) + P (E)P

(14)

¯ = 1, and Because the probability that the hot spot is not hit when it does not exist is P (H¯ |E) ¯ the sum of the prior probabilities to find or not find the spot is P (E) + P (E) = 1, and also P (H |E) + P (H¯ |E) = 1, there is P (E|H¯ ) = P (E)

1 − P (H |E) . 1 − P (E)P (H |E)

(15)

For example, let the sampling plan to find hot spots of radius 2 m have a grid distance of 10 m. The prior probability P (H |E) = 0.126 according to Equation (12). Previous experience from the equal sub-area sampling indicated contamination in 8% of the collected samples, i.e. the probability of hot spot existence is P (E) = 0.08. When at last sampling there was no hit of a hot spot, still there is the probability of its hidden existence, though only slightly lower, 1 − 0.126 = 0.071, 1 − 0.08 × 0.126 and the absence of hot spot probability is 1 − 0.071 = 0.929. P (E|H¯ ) = 0.08

2.2.9. Hot particle sampling Hot particles originated mostly from world-wide fallout after the nuclear weapon tests carried out in the atmosphere (Mamuro et al., 1968), and also from nuclear reactor accidents, principally that at Chernobyl (Sandalls et al., 1993). Their size varies between 0.1 up to the “giant” particles of 30–1000 µm with the occurrence of 1–100 particles per 1000 m3 . Hot particles have much smaller dimensions than a hot spot, but their estimation is important to radiation exposure of a population after a fallout event (ICRU, 2000). A statistical evaluation for detecting hot particles in environmental samples by sample splitting was performed by Bunzl (1997). The presence of hot particles in the environment could be detected with fairly high probability in replicate or collocate samples. The wider the frequency distribution of the activities of the hot particles, the smaller the number of parallel sample measurements is necessary to detect their presence. 2.2.10. Visual Sample Plan VSP 3.0 The Pacific Northwest National Laboratory offers to upload the Visual Sample Plan software (VSP 3.0) (Hassig et al., 2004) elaborated for the U.S. Department of Energy and the U.S. Environmental Protection Agency. VSP is designed for selecting the right number and location

32

F. Macášek

of environmental samples so that the results of statistical tests performed on the data collected via the sampling plan have the required confidence for decision making. VSP allows selection of a design from the following list (all but the judgment sampling are probability-based designs): • simple random sampling • systematic grid sampling on a regular pattern (e.g., on a square grid, on a triangular grid, along a line) • stratified sampling • adaptive cluster sampling • sequential sampling, requiring the user to take a few samples (randomly placed) and enter the results into the program before determining whether further sampling is necessary to meet the sampling objectives • collaborative sampling design, also called “double sampling”, uses two measurement techniques to obtain an estimate of the mean—one technique is the regular analysis method (usually more expensive), the other is inexpensive but less accurate. It is actually not a type of sampling design but rather a method for selecting measurement method • ranked set sampling • sampling along a swath or transect—continuous sampling to find circular or elliptical targets is done along straight lines (swaths) using geophysical sensors capable of continuous detection • sampling along a boundary in segments, which combines the samples for a segment, and analyzes each segment to see if contamination has spread beyond the boundary • judgment sampling—the sampling locations are based on the judgment of the user such as looking in the most likely spot for evidence of contamination or taking samples at predefined locations. 2.3. Replicates, composite and collocated samples An averaged value of the analyte estimation in the homogenized sample, and the uncertainty derived from the standard deviations of the analytical procedures are the regular presentation format for analytical results. However, the distribution of the analyte in fractions of the original sample and its non-parametric statistical (Horn, 1983; Efron, 1981) and physico-chemical speciation (Macášek, 1994, 2000; Pickering, 1995) should be referred to as reflecting the true similarity between the subsamples and virgin matrix. Field replicates are samples obtained from one location, homogenized, divided into separate containers and treated as separate samples throughout the remaining sample handling and analytical processes. They are used to assess variance associated with sample heterogeneity, sample methodology and analytical procedures. Conversely, composite samples consist of several samples, which are physically combined and mixed, in an effort to form a single homogeneous sample, which is then analyzed. They are considered as the most cost effective when analysis costs are large relative to sampling costs. However, their information value is dubious. Collocated samples are two or more specimens collected at the same spot (typically, at a distance of about 10–100 cm away from the selected sample location) and at the same time and these can be considered as identical. Collocated samples should be handled in an

Sampling techniques

33

identical way. Analytical data from collocated samples can be used to assess site variation in the sampling area. From the total variance (Equation (1)) the measure of identity of collocated samples can be obtained as σs2 = σt2 − σd2 − σl2 .

(16)

The real situation is well documented by the fact that has resulted from a growing number of intercomparison analysis—the reliability of the overall means decreases though the statistical uncertainty reaches unrealistically small values, without relation to the real variability of the individual samples (Seber and Thompson, 1994; Thompson, 2002). This indicates that a similarity of averaged values of total and sub-populations is not sufficient, and a similarity of population fields should be considered instead, the latter not being achievable using homogenized gross samples. Anyone’s effort to reduce the uncertainty of radionuclide data does a disservice to the purpose of analytical quality assurance. As usual, the larger the set of samples the better it reflects the abundance of species in the total population of entities. However, the uncertainty expressed by the expected standard deviation (s) of the mean (x) or median (μ) gives just a vague idea of the natural sample variance and creates an illusion of high certainty in the assessment. The treatment of unevenly distributed data by a normal (and even more so for a lognormal) distribution approach means an underestimation of each species abundance. Hence, not only sample homogenization but also normal distribution statistics exhibit the tendency to smooth out the picture of the radionuclide distribution in the investigated area. Still, an environmental decision-maker should be provided not only with averaged values but also with their realistic statistics of environmental abundance. The results of replicate sampling analysis when expressed through percentiles and a Tukey (1977) box plot better reflect the distribution of an analyte species (or the matrix heterogeneity). In routine analysis, usually the criteria derived from a normal distribution are applied to indicate the outlying results for 2s (95% confidence) or 3s (99% confidence) from a mean value. In the Hoaglin statistics which considers the normal distribution of non-rejected results (Patil et al., 1994; Meloun and Militký, 1994) the lower and upper fences, BL and BU , for outliers are established from the 0.25 and 0.75 percentiles as follows, BL∗ = x¯0.25 − K(x¯0.75 − x¯0.25 ),

(17)

BU∗

(18)

= x¯0.25 + K(x¯0.25 − x¯0.75 ),

where K is found for 8 n 100 without any posterior data, just from their number n as 3.6 . K∼ = 2.25 − n

(19)

For small subsets, the limits are non-parametrically assessed as a distance from the median on the value sμ =

x(n−k+1) − x(k) 2u

(20)

34

F. Macášek

which is the non-parametric mathematical expectation of the median dispersion (Meloun and Militký, 1994). For the 95% confidence interval u = 1.96, and k is found from

n+1 n k = int (21) − |u| . 2 4 The dispersion of the values indicates the real situation in the samples, but the fetishism for a mean value mostly prevails and the result would probably be issued and accepted by the client with a standard deviation, but may be corrected by Student coefficients, as indicating a minor “error”. A small set of the smallest subsamples, e.g., 5 randomly collected entities, can be easily treated by four different non-parametric methods: (i) Finding a median and the fences according to Equations (17) and (18); for the set of n = 5 data, k = 1 and n − k + 1 = 5, i.e. the dispersion of the median at 95% confidence is easily obtained from the set. (ii) Marritz–Jarret evaluation (Meloun and Militký, 1994) of the small set of data median dispersion is performed as n 2 2 sμ = (22) wi x − wi xi , i

i

where the weights wi of measurements xi are found from the Bessel functions J , J i−0.5 n wi = n j −0.5 . j =1 J n

(23)

For the set of n = 5 sorted values i

1

2

3

4

5

J wi

0.24 0.05

1.32 0.26

1.88 0.37

1.32 0.26

0.24 0.05

(iii) The median is assessed by pivot range according to the Horn statistics (Patil et al., 1994; Meloun and Militký, 1994) for small subsets of data; indexes of sorted pivot data are derived for 1 n+1 h = int (24) 2 2 or n+1 1 +1 h = int (25) 2 2 —depending on which one is an integer—and iL = h,

(26)

iU = n + 1 − h.

(27)

Sampling techniques

35

For n = 5 the pivot indexes are simply iL = 2 and iU = 4, and from the corresponding xi values the median and deviations can be obtained. (iv) The median and its dispersion obtained by the “bootstrap” method (Bohn and Wolfe, 1994; Thompson, 1990) may provide a large set of subsets created by multiple (200– 1000 times) random repeating of n original data to calculate more reliably the median and the dispersion. Most surprisingly, it may be just the small random set of the “most non-representative subsamples” which gives the smallest bias of the median from the true value of the total population. Also its ordinary dispersion assessment by Student statistics or non-parametric methods looks much more realistic in reflecting the true variance of samples composition (Macášek, 2000).

3. Field sampling 3.1. Sampling protocol and standard operation procedures The sampling operation should be based on Standard Operating Procedures (SOPs) or protocols especially developed for the specific problem. Design of protocols is very important for the quality assessment and comparativeness. Sample takers should be trained to understand and fulfill the protocols carefully. All deviations from the survey design as documented in the standard operation procedures should be recorded as part of the field sample documentation. 3.2. Field sample preparation and preservation Proper sample preparation and preservation are essential parts of any radionuclide-sampling program. Storage at reduced temperatures (i.e., cooling or freezing) to reduce biological activity may be necessary for some samples. Addition of chemical preservatives for specific radionuclides or media may also be required (Kratochvil et al., 1984; DOE, 1997). Water samples may need filtering and acidification. Samples taken for tritium analysis should be sealed in air-tight glass or HPPE containers to avoid isotope exchange with atmospheric water vapor. Convenient and economical containers for soil samples are the polyethylene bottles with screw caps and wide mouths. Glass containers are fragile and tend to break during transportation. Soil and sediment sample protocols for organic-bound radionuclide analysis require cooling of soil samples at least to 4 ◦ C within the day of collection and during shipping and storage though it is not a practice normally followed for radiochemical analysis. When storage to −20 ◦ C is demanded, resistant plastic sample bottles should be used and be no more than 80% full to allow for expansion when freezing. Specific demands for sample conservation arose for long-term environmental specimen banking, such as “shock freezing” by cooling to below −150 ◦ C by liquid nitrogen and parallel high-pressure mineralization of a portion of the samples (Rossbach et al., 1992).

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F. Macášek

3.3. Packaging and transporting, traceability Chain-of-custody procedures are necessary for quality assurance of the sampling process. The sample collector is responsible for the care and custody of the samples until they are properly transferred or dispatched. This means that samples are in his/her possession, under constant observation, or secured. All samples should be accompanied by a chain-of-custody record, which documents sample custody transfer from the sampler, often through another person, to the laboratory. Quality assurance of all sampling procedures should be recorded in sufficient detail to enable the finding of the records on space, time, mode of sampling, storage, treatment and transfer of samples. Appropriate photographs and drawings should be provided in the quality assurance report. Field personnel are responsible for maintaining field logbooks with adequate information to relate the sample identifier (sample number) to its location and for recording other information necessary to adequately interpret the results of sample analytical data. Identification and designation of the sample are critical to being able to relate the analytical result to a site location. However, samples are typically collected by one group, and analyzed by a second group. Shipping containers should be sealed (metal containers can present a problem) and include a tamper-indicating seal that will indicate if the container seal has been disturbed. Even if the identity of samples is well declared and communications between the field personnel and laboratory is ensured, an accredited analytical laboratory should not certify the link of the analytical result to a sampled field object and can only account for the uncertainties that occur after sample receipt. For the field aspect is always the responsibility of the sampling body (a sample taker). Global quality assurance means that the field sampling phase was performed or supervised and recorded by the same staff or under the same project manager. 3.4. Sample spiking and isotope labeling Environmental samples are spiked with a known concentration of a target analyte(s) to verify percent recoveries. This procedure can be done during the sampling, but laboratory procedures are more practical and reliable. Spiking is used primarily to check sample matrix interference but can also be used to monitor radionuclide behavior in laboratory performance. Spiking in analysis of radionuclides is applied either as labeling with non-active carrier (reverse isotope dilution analysis) or adding a well measured (usually gamma-emitting) radioisotope (Tölgyessy et al., 1972; Macášek, 2000; Navrátil et al., 1992). It enables a simple assessment of many components of the combined uncertainty of the final result, especially those caused by a varying yield of analyte recovery R. This can be evaluated from the added isotope label amount S0 (in mass or activity units) and isolated amount S as R = S/S0 . A well-known formula is applied to calculate the unknown radionuclide activity A0 in the original sample from the amount A measured in reprocessed sample, A0 = A

S0 . S

(28)

Most customary gamma indicators are 85 Sr for 90 Sr, 243 Am for 241 Am, 232 U for 235,238 U, and 236 Pu for 238,239,240 Pu.

Sampling techniques

37

A further advantage of labeling (spiking) with a gamma radionuclide is to assay the realistic variability of the contaminant distribution in environmental objects (Macášek, 2000). To determine the actual level of natural variance of contamination by radioactive and trace substances is a very difficult task because large samples are to be processed and the measurements are performed very often in the vicinity of detection limits. The chance that the variance of label distribution and isotope exchange rate will lead to indistinguishable massic activities of different species is sufficiently low even for two species. Now, the advantage of radioisotope labeling has been made plain; it is the technique which may enhance laboratory replicate sampling and non-parametric assessment, which normally sharply increases the cost of analysis because of the great number of analyzed subsamples. No doubt this procedure deserves more attention in the future development of radiochemical analysis and radioanalytical methods, especially when combined with rapid monitoring techniques and new statistical methods.

4. Sampling technologies Improper sampling devices and inexperienced staff may cause serious bias in sampling operations. The important considerations for field work should be fulfilling a sampling protocol to avoid disturbance of the samples, especially when volatile, labile or metabolized species are collected. Luckily, in the case of environmental radioactivity, there is no danger of serious contamination of samples by the device used, except if it is carelessly transferred from a zone of heavy contamination to a clean area. Cross-contamination is more probable and undetectable in systematic sampling. However, it can be minimized by executing the sampling plan from the clean or less contaminated sectors. For determining bias resulting from cross-contamination, the field blanks that are certified clean samples of soil or sand are used. Background samples of soils are collected from a reference area offsite of a contaminated section. The “reference background water” from deep wells and glacial aquifers is the basis of any tritium sampling program. For an increased tritium assay, a method blank sample of deionized water is simply used in the case of water samples. Probably, the most comprehensive list of recommended sampling procedures for natural materials is contained in the US Department of Energy manual HASL-300 (DOE, 1997). 4.1. Air and aerosols Except for integral in-line/flow-through detection, there are the following methods for obtaining samples or measurements of airborne radioactivity concentrations: • • • • • •

filtration container filling impaction/impingement adsorption on solids absorption in liquids condensation/dehumidification.

38

F. Macášek

These techniques can be designed as integral and size-selective in nature (Hinds, 1982). In all cases, to calculate the concentrations of radionuclides in air, it is necessary to accurately determine the total volume of the air sampled. The criteria for filter selection are good collection efficiency, high particle loading capacity, low-flow resistance, low cost, high mechanical strength, low-background activity, compressibility, low-ash content, solubility in organic solvents, non-hygroscopicity, temperature stability, and availability in a variety of sizes. Dense cellulose and cellulose-asbestos filters, glass fiber filters, or membrane filters are used. The latter have the advantage of dissolving in organic solvents and then analyzed in a counter, e.g., by liquid scintillation, or they can be burnt. An air mover, such as a vacuum pump, should be used to draw air through the removable filter medium. To sample large volumes of air and obtain total particle collection in a filter battery it is necessary to use an appropriate filter material and an air mover. A suitable air mover should reach a flow rate of 0.5–2 m3 min−1 at pressure drops across the filter ranging from ∼5 to ∼20 kPa, but for small filters (up to 5 cm in diameters) a flow of 5–20 dm3 min−1 is sufficient. To generate complete particle size spectra for the chemical species of interest (radon progeny, Aitken particles, etc.), multichannel (parallel) or multistage (series) screens or disk filters batteries are used. The sizes of particles passing through a stack of filters are calculated from the hydrodynamic parameters of sampling (Cheng et al., 1980; Maher and Laird, 1985). When evacuated containers are used for air sampling, they are opened at the sample location to draw the air into the container. The sample is sealed in the container and removed for analysis or its activity is measured directly in the vessel. To ensure the sample is representative, the flow rate in the sample device inlet must be the same as the flow rate in the system, such as the duct or stack. When the sample line velocity is equal to the system velocity at the sampling point, it is called isokinetic sampling. In other cases discrimination can occur for smaller or larger particles. This occurs because the inertia of the more massive particles prevents them from following an airstream that makes an abrupt directional change. If the velocity of the sample airstream is bigger than the velocity of the system airstream, then the larger particles cannot make the abrupt change and are discriminated against in the sample, i.e., the smaller particles are collected more efficiently, and vice versa. Voluntary discrimination of particles is used in impingers or impactors. Particles are collected on a selected surface as the airstream is sharply deflected. The surface on which the particles are collected must be able to trap the particles and retain them after impaction. This is achieved by coating the collection surface with a thin layer of grease or adhesive or wetting by water or a higher alcohol. High volume cascade inertial impactors operating with filter paper treated with light mineral oil are used to measure the particle size distribution between 0.2 and 20 µm for both indoor and outdoor aerosols (Fuchs, 1964; Theodore and Buonicore, 1976). A dichotomous sampler is capable of separating the “inhalable” particles LC | L = Lo ) α.

(3.4)

Further discussion of the sociotechnical issues and the role of such a non-zero null is given in (Currie, 1988) in connection with discrimination limits. Also relevant is Section 8.2 of this chapter, which addresses bias and information loss in the Reporting of Low-Level Data.

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3.3. Simplified relations3 Under the very special circumstances where the distribution of Lˆ can be taken as Normal, with constant standard deviation σo (homoscedastic) and the default values for parameters α, β, kQ , the foregoing defining relations yield the following expressions: LC = z1−α σo → 1.645σo , LC = t1−α,ν so → 2.132so

(3.5) [4 df],

(3.6)

LD = LC + z1−β σD → 2LC = 3.29σo , LD = δα,β,ν σo ≈ 2t1−α,ν σo → 4.26σo LQ = kQ σQ → 10σo ,

(3.7) [4 df],

(3.8) (3.9)

ˆ under the null where σo , σD , and σQ represent the standard deviation of the estimator L: hypothesis, at the Detection Limit, and at the Quantification Limit, respectively; so2 represents an estimate of σo2 , based on ν degrees of freedom (df); z1−α (or zP ) and t1−α,ν (or tP ,ν ) represent the appropriate 1-sided critical values (or percentage points) of the standard normal variate and Student’s-t, respectively; and δ represents the non-centrality parameter of the noncentral-t distribution. The symbol ν indicates the number of degrees of freedom. For the illustrative case (4 degrees of freedom), the actual value for δ[0.05, 0.05, 4] appearing above (Equation (3.8)) is 4.07.4 The above relations represent the simplest possible case, based on restrictive assumptions; they should in no way be taken, however, as the defining relations for detection and quantification capabilities. Some interesting complications arise when the simplifying assumptions do not apply. These will be discussed below. It should be noted, in the second expressions for LC and LD given above, that although so may be used for a rigorous detection test,5 σo is required to calculate the detection limit. If so is used for this purpose, the calculated detection limit must be viewed as an estimate with an uncertainty derived from that of σ/s. (See Equation (4.8).) Finally, for chemical measurements a fundamental contributing factor to σo , and hence to the detection and quantification performance characteristics, is the variability of the blank. This is introduced formally below, together with the issue of heteroscedasticity. 3 Once the defining relations are transformed into algebraic expressions for particular distributions, it becomes necessary to introduce critical values of those distributions—in particular: zP , tP ,ν , δα,β,ν , and χP2 ,ν . Meaning and

relevance will be indicated as the symbols are introduced; and a table of critical values used in this chapter is given in Appendix A (Table A1). 4 When ν is large, 2t is an excellent approximation for δ. For ν 25, with α = β = 0.05, the approximation is good to 1% or better. For fewer degrees of freedom, a very simple correction factor for 2t is 4ν/(4ν + 1). This takes into account the bias in s, and gives values that are within 1% of δ for ν 5. For the above example where ν = 4, δ would be approximated as 2(2.132)(16/17) which equals 4.013. 5 The use of ts for multiple detection decisions is valid only if a new estimate s is obtained for each decision. In o o the case of a large or unlimited number of null hypothesis tests, it is advantageous to substitute the tolerance interval factor K for Student’s-t. See Currie (1997) for an extended discussion of this issue.

Detection and quantification capabilities in nuclear analytical measurements

59

4. The signal (S-) domain 4.1. Basic equations In many cases the smallest signal SD that can be reliably distinguished from the blank, given the critical level SC , is desired—as in the operation of radiation monitors. Assuming normality ˆ simple expressions can be given for the two quantities involved in and knowledge of σ [S], Signal Detection. Equation (3.5) takes the following form for the Critical Value, SC = z1−α σo → 1.645σo ,

(4.1)

where the expression to the right of the arrow results for α = 0.05. In the Signal Domain, L is equated to the net signal S, which in turn equals E(y) − B. In this expression, y represents the observed (gross) signal or response, and B represents the expectation of the background or blank. The variance of the estimated net signal is then given by ˆ = V [y] + V [B] ˆ → V [B] + V [B] ˆ = Vo . V [S]

(4.2)

The quantity to the right of the arrow is the variance of the estimated net signal Sˆ when its true value (expectation) S is zero—i.e., when E(y) = B. If the variance of Bˆ is negliof the Blank. If B is estimated in a “paired” gible, then σo = σB , the standard deviation √ √ ˆ = VB , then σo = σB 2. Note that σo = σB , and σo = σB 2, are experiment—i.e., V [B] √ ˆ limiting cases. More generally, σo = σB η, where η = 1 + (V [B]/V [B]). Thus, η reflects different numbers of replicates, or, for particle or ion counting, different counting times for ˆ or the “sample” vs blank measurements. Taking n to represent the number of replicates for B, ratio of counting times, we find that η equals (n + 1)/n. The Minimum Detectable Signal SD derives similarly from Equation (3.7), that is, σo2 ,

SD = SC + z1−β σD ,

(4.3)

where σD2 represents the variance of Sˆ when S = SD . For the special case where the variance is constant between S = 0 and S = SD , and α = β = 0.05, the Minimum Detectable Signal √ SD becomes 2SC = 3.29σo = 3.29σB η, or 4.65σB for paired observations. The treatment using an estimated variance, so2 and Student’s-t follows that given above in Section 3.3 (Equations (3.6), (3.8) with LC,D set equal to SC,D ). 4.2. Heteroscedasticity and counting experiments The above result that equates SD to 2SC is not correct if the variance of Sˆ depends on the magnitude of the signal. A case in point is the counting of particles in radiation detectors, or the counting of ions in accelerators or mass spectrometers, where the number of counts accumulated follows the Poisson distribution, for which the variance equals the expected number of counts. Taking B to be the expectation for the number of background counts, σo2 = ηB; and for the normal approximation to the Poisson distribution with α = β = 0.05 and kQ = 10, resulting expressions for SC , SD , and SQ (units of counts) are given by (Currie, 1968) SC = zP (ηB) = 1.645 (ηB), (4.4)

60

L.A. Currie

SD = zP2 + 2zP (ηB) = 2.71 + 3.29 (ηB), 2 2) S Q = kQ /2 1 + 1 + (4ηB/kQ = 50 1 + 1 + (ηB/25) .

(4.5) (4.6)

Asymptotic expressions for SQ , for negligible √ B and for B 1, are simply 100 counts and 10σo , respectively. For SD , we get 3.29 (ηB) for the large B asymptotic expression.6 As shown in Currie (1968), these large B asymptotes for SD and SQ lie within 10% of the complete expressions (Equations (4.5), (4.6)) for (ηB) > 67 and 2500 counts, respectively. Equating the background variance with the expected number of counts B, often leads to an underestimation—i.e., it ignores non-Poisson variance components. When B is large, over 104 counts for example, additional components tend to dominate; and these can and should be represented in VB and Vo . A common practice is to add an extra variance component for background variability, VxB = (ϕxB B)2 , where ϕxB represents the relative standard deviation of the non-Poisson B-variation. VB then becomes B + VxB , and the variance of the null signal becomes 2 B . Vo = ηVB = η B + (ϕxB B)2 = ηB 1 + ϕxB (4.7) In more complicated cases where net signals are estimated in the presence of chromatographic or spectroscopic baselines, or where they must be deconvolved from overlapping peaks, the limiting standard deviations (σo , σD , and σQ ) must be estimated by the same procedures used to calculate the standard deviation of the estimated (net) signal of interest. Examples can be found in Currie (1988) (see also Sections 5 and 6). Other cases, to be considered in later sections, include the treatment of the discrete, non-normal Poisson distribution function for small numbers of counts, and the fitting of empirical variance functions. Regarding the latter, an added quadratic term, (ϕxS S)2 , representing asymptotic constant relative variance, is often appropriate. (See Section 7.1.2.) Accuracy of the (Poisson–normal) expressions for SC , SD Expressions for SC and SD above treat the discrete Poisson distribution with parameter μ (mean and variance) as though it were continuous normal. The question posed here is: How accurate is the Poisson–normal approximation for small values of μB , where μB is equal to B, the expectation of the background counts? For the well-known background case, where η = 1 in Equations (4.4) and (4.5), an exact comparison can be made. To illustrate, taking B as 10.3 counts, the Poisson critical value for the observed number of counts, nC , is found to be 16, necessarily an integer. The corresponding value for the false positive probability α is 0.034, consistent with the inequality requirement of Equation (3.1). SC for the exact Poisson distribution is thus 16 − 10.3 = 5.7 counts. The normal approximation for SC is given by √ Equation (4.4) (with η = 1)—i.e., 1.645 10.3 which equals 5.3 counts. The detection limit, calculated from the exact Poisson distribution, derives from the value of (SD + B) for which the lower tail area of the Poisson distribution equals β (0.05) for n nC or 16 counts. The result is 24.3 counts, so SD = 24.3 −√10.3 or 14.0 counts. The normal approximation, as given by Equation (4.5), is 2.71 + 3.29 10.3 = 13.3 counts. In each case (SC and SD ) we find that the result given by the Poisson–normal approximation lies within 6 Note that S , which equals z σ √η, is the same quantity that appears in the 1-sided Prediction Limit for the C P B + zP σB √η). (For σ 2 estimated as s 2 , the corresponding expression is tP ,ν sB √η.) background, (B


61

one count of the exact Poisson result. Similar results obtain for other small values for B. For B = 5.0 counts, for example, exact (Poisson) values of SC and SD equal 4.0 and 10.7 counts; the normal approximation gives 3.7 and 10.1 counts, respectively. For B = 20.0 counts, exact (Poisson) values of SC and SD equal 8.0 and 18.4 counts; the normal approximation gives 7.4 and 17.4 counts, respectively The case of extreme low-level counting, involving (1) the difference between a Poisson variable and a known or presumed mean background, and (2) the difference between two Poisson variables with unknown means, is treated in more detail in Section 7.4. 4.3. Uncertainty of estimated values for SD , SQ , and α For normally distributed variables, the expressions in Equations (3.5)–(3.9) give expected (true) values for LC [α], LD [α, β], and LQ [kQ ]—immediately useful for judging the absolute and comparative capabilities of alternative measurement processes as a function of the background or blank, the type-I, II risks, and the quantifiability parameter. Evaluation of these performance characteristics using experimental data from a particular measurement process, however, transforms these quantities into data-based estimates, with associated uncertainties. Uncertainties will addressed here for two classes of measurements: those for which σˆ is based on replication variance (s 2 ), and those for which it derives from “counting statistics” and the Poisson assumption. Applications requiring a combination can be met through variance (“error”) propagation, taking into account effective degrees of freedom (ISO, 1995). First, we consider the case of replication, where variance is estimated as s 2 (with ν degrees for freedom), SC is given correctly as t1−α,ν so , given and the selected values for ν and α—i.e., an uncertainty in the estimated value is not an issue, provided that each SC decision is based on an independent variance estimate. (The critical value SC in this case is a random variable, but the overall probability that (Sˆ > SC ) remains α, through the application of the critical value of the t-distribution.) SD and SQ , on the other hand, can only be estimated, barring knowledge of σo . Considering the homoscedastic case (constant-σ ), the relative uncertainties of the estimates—SˆD = δα,β,ν so and SˆQ = kQ so —are given by those of estimated standard deviations (so , sQ ). For normal data, bounds for σ 2 , given s 2 and ν, come directly from the chi-square distribution. If the observations are distributed normally, s 2 /σ 2 is distributed as χ 2 /ν. A 95% interval estimate for this ratio is therefore given by (χ 2 /ν)0.025 < s 2 /σ 2 < (χ 2 /ν)0.975 .

(4.8)

A useful approximation, excellent for large ν, for rapidly estimating the standard uncertainty √ of s/σ is 1/ (2ν). Thus, about 50 degrees of freedom are required before the relative standard uncertainty in σˆ is decreased to about 10%. Using Equation (4.8) and taking roots, we get a quite consistent 95% (expanded) uncertainty (ν = 50) of 0.804 < s/σ < 1.20. For more modest, perhaps more typical, degrees of freedom, e.g., ν = 10, this can be a major source √ of SD , SQ uncertainty. (The approximate relative standard uncertainty of σˆ for ν = 10 is 1/ 20, or 22%.) 4.3.1. Confidence intervals (normal, Poisson) for B, given Bobs For the Poisson case, experiment-based estimates for the expressions for SC , SD , and SQ require, at a minimum, estimated values of B. (If non-Poisson variance must also be considered, then ϕxB and the variance function may require additional replication-based variance

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estimates.) In the context of uncertainties in SC , SD , and SQ , this section (4.3.1) has two purposes: (1) to determine the uncertainty interval for B, given the observed number of counts Bobs (integer), as this will directly impact the uncertainty of estimated values for α, SD and SQ ; (2) to evaluate the accuracy of the Poisson–normal approximation for that interval, in comparison to the exact Poisson limits. We first consider the latter approximation. Poisson–normal lower (BL ) and upper (BU ) bounds for the 90% confidence interval for B (background expectation and Poisson mean, variance) are given as the solutions to the 2-sided equality BL + zP BL = Bobs = BU − zP BU . (4.9) The dual (Poisson–normal) solutions to Equation (4.9) are given in columns 4 and 5 of the table below, taking zP = 1.645 and the series of Bobs integer counts in column 1. The exact (Poisson) 90% confidence limits are given in columns 2 and 3. Although the normal confidence limits in the table were calculated from√ the exact solution to Equation (4.9), they can 2 /2 (Cox and Lewis, 1966). The be very closely approximated by Bobs ± z0.95 Bobs + z0.95 2 7 Poisson limits were derived from the χ distribution. Although Bobs values are necessarily integers for the discrete Poisson distribution, the confidence limits for the Poisson parameter are real numbers.

Bobs

BL (Poisson)

BU (Poisson)

BL (normal)

BU (normal)

4 counts 9 counts 25 counts 100 counts

1.37 4.70 17.38 84.01

9.16 15.70 34.97 118.28

1.80 5.24 18.02 84.85

8.91 15.47 34.69 117.86

The first conclusion from this small study is that the Poisson–normal approximation for the 90% confidence interval is remarkably good, differing from the exact Poisson interval by no more than 10%, for as few as 4 counts (Bobs ). The second conclusion is that estimates of SC , SD based on the observed count, Bobs , may be seriously misleading, especially for small numbers of counts observed—e.g., xC | x = 0) α,

(5.4)

Pr(xˆ xC | x = xD ) = β,

(5.5)

xQ =

(x) kQ σ Q .

(5.6)

For Case I, we treat xˆ as approximately normal, using the Taylor expansion to generate its variance by “error-propagation”.10 The Taylor expansion for x is given as

x(y, ˆ B, A) = x + xy δy + xB δB + xA δA + xBA δB δA + · · · ,

( )

(5.7)

(

)

and second derivatives, and all derivatives, including where the primes represent first the zeroth, are evaluated at the expected values E(y), B, and A. Higher order derivatives (with respect to A, only) exist but have no consequence in calculating the variance of x, ˆ because y, B, and A are taken as normal, and moments of higher order than three are neglected in this approximation. (The third moment for normal variables is zero.) The second moment (variance) of xˆ follows by calculating E(x − x) ˆ 2 using the above expansion in first and second ˆ Aˆ covariance is derivatives. The result, which takes into account possible B, Vx = (1/A2 ) (Vy [x] + VBˆ )J + x 2 VAˆ + 2xVBA , (5.8) 2 ). (NB: ϕ denotes the relative standard deviation of A.) ˆ It should where J is equal to (1 + ϕA A be noted that (1) (Vy [0] + VBˆ ) is, by definition Vo , the variance of the estimated net signal under the null hypothesis, and (2) for small ϕA , such that J ≈ 1, Vxˆ ≈ Vy−yˆ /A2 —a result that will be of interest when we consider Case II. Equation (5.8) provides the basis for deriving expressions for xC,D,Q , under the normal approximation. The basic relations are √ √ xC = z1−α σo(x) = z1−α σo J /A = SC J /A, (5.9) (x)

xD = xC + z1−β σD ,

(5.10)

10 The parenthetical exponent notation (x) in Equation (5.6) is used to indicate an x-domain (concentration-domain)

standard deviation. Also, to reduce notational clutter, the circumflex is omitted from subscripts in the more extended expressions.

Detection and quantification capabilities in nuclear analytical measurements (x)

xQ = k Q σ Q .

69

(5.11) (x)

In Equation (5.9) σo (standard deviation of xˆ when x = 0) is expressed in terms of σo (standard deviation ofSˆ when S = 0) using Equation (5.8) with x set to equal to zero. (Note that σo in this case is (Vy [0] + VBˆ ).) The Detection Decision is made by comparing xˆ with xC , according the defining relation (5.4). That is, √ ˆ Aˆ = S/ ˆ Aˆ is compared with xC = (SC /A) J . xˆ = (y − B)/ (5.12) No longer is the decision made in the signal domain—i.e., we do not compare Sˆ with SC to make the decision; and the x-critical value is, in fact, increased in comparison with the de facto value (SC /A) when A has negligible error. The increase derives from the variance of the denominator estimator Aˆ which affects the dispersion of xˆ even when the true value of x is zero. In the limit of negligible Aˆ variance, the quantity J goes to unity, giving the result xC = SC /A. A conceptually very important distinction between this treatment of the concentration detection decision and the following one (case II) is that J is absent from the latter, where in fact, the decision remains in the signal domain, regardless of the magnitude of ϕA . The absence of J —i.e., J → 1—would make the numerator in Equation (5.8) identical with the variance of (y − y). ˆ The Detection Limit requires the solution of Equation (5.10), which in turn requires knowledge of Vy [x] if the response is heteroscedastic. Although an analytic solution may be possible in certain cases of heteroscedasticity, an iterative solution is generally required. Here, we treat only the homoscedastic case where Vo = Vy + VBˆ is constant. However, even in this case, Vxˆ is not constant, because of the x-dependence shown on the right side of Equation (5.8). Taking this into account, the solution of Equation (5.10) leads to the following expression for the minimum detectable concentration11 : √ √ xD = (2z1−α σo J /A)(K/I ) = (2SC J /A)(K/I ), (5.13) √ √ where K = 1 + z1−α VBA /(Aσo J ) = 1 + r[B, A](σB /σo )(z1−α ϕA )/ J , I = 1 − (z1−α ϕA )2 ,

constraint: I > 0.

When B and A are estimated independently, VBA equals zero, so K = 1; and in the limit of negligible A uncertainty, J , K, and I all equal unity; in that case xD is identical to the expression given in Section 5.1. At the other extreme, xD will be unattainable if the relative standard deviation of Aˆ becomes too large. In particular, the factor I in the denominator goes to zero when ϕA approaches 1/z1−α , or 0.61 using the default value for z1−α . When Vy is estimated as s 2 , SC is replaced with t1−α,ν so in Equation (5.9), and 2SC is replaced with δ1−α,ν σo in Equation (5.13). Also, in this equation, the z’s in K and I are replaced with δ1−α,ν /2. In this case, the ϕA limit for finite xD is 2/δ1−α,ν or 0.49 for the default α (0.05) and 4 degrees of freedom. 11 When V is constant, the change in V with concentration can be both negative and positive. At low concentrations, y xˆ negative correlation between Bˆ and Aˆ tends to decrease it; at higher concentrations the term x 2 VAˆ tends to dominate, causing Vxˆ to increase. Except for bad calibration designs, these effects tend to be small. Variation of Vy [x], however, has the potential to cause significant differences among Vy [0], Vy [xD ], and Vy [xQ ]. (See Section 7.1.2 for discussion of variance function models and non-conventional weighting schemes for Vy [x] estimation.)

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The Quantification Limit is given as the solution of Equation (5.11), using Equation (5.8) with the substitution of xQ for x. For the homoscedastic case the resulting expression is √ 1/2 + gQ , xQ = (kQ σo J )/(A IQ ) 1 + gQ (VBA /Vo ) 2 2 2 2 where gQ = kQ VBA (A IQ ), IQ = 1 − kQ ϕA , and ϕA = σA /A. (5.14) When independent estimates are made for A and B, VBA = 0, and xQ takes the simpler form √ xQ = kQ σo J /(A IQ ). (5.15) As with xD , xQ may be unattainable if the relative standard deviation of Aˆ becomes too large. In particular, the factor in the denominator IQ goes to zero when ϕA approaches 1/kQ , or 0.10 when using the default value for kQ . In the limit of negligible A uncertainty, xQ reverts to the form given in Equation (5.3). For the heteroscedastic (Poisson) case, where Vy = S + B counts, a closed solution obtains when VBA = 0—i.e., in the case of independent estimates for σB and σA . Then, 1/2 . xQ = k 2 /(2AIQ ) 1 + 1 + (4Vo IQ )/(k 2 J ) (5.16) For k = 10, the asymptotic solutions for xQ are then 100/(AIQ ) and (10σo )/(A (J IQ )). The latter is approached more rapidly for large background counts—i.e., when Vo > 100 counts. 5.2.2. Case II: Quasi-concentration domain: calibration-based limits Undoubtedly the most popular, “calibration-based” approach to detection limits is that inspired by the established method for calculating the confidence interval for x through inversion of the linear calibration curve. By extending the equations given for computing lower and upper confidence bounds for x (Natrella, 1963), given an observed response y, Hubaux and Vos (1970) developed expressions for the critical y-value (yC ) and an x-“detection limit” for a particular realization of the linear calibration curve. The beauty of the method is that it is based on strictly linear relationships, so that normality is fully preserved. This comes about because the operations are actually performed in the signal domain, hence the “quasiconcentration” domain designation. The major drawback with the method is that it produces “detection limits” that are random variables, as acknowledged in the original publication of the method (Hubaux and Vos, 1970; p. 850), different results being obtained for each realization of the calibration curve, even though the underlying measurement process is fixed. For that reason, in the following text we label such limits “maximum upper limits”, denoted by symbol xu . The defining relations, which are linked in this case to the estimated (fitted) calibration curve and its limit xu , are Pr(y > yC | x = 0) α,

(5.17)

Pr(y yC | x = xu ) = β.

(5.18)

The conditions are thus equivalent to points on the fitted calibration curve corresponding to x = 0 and x = xu .


71

To derive the critical (signal) value SC and maximum (concentration) upper limit xu , we ˆ need the variance of (y − y) ˆ as a function of concentration. Noting that (y − y) ˆ = (y − Bˆ − Ax), and applying variance propagation to the relation that, unlike case I, is linear in the random variables, we obtain V [y − y] ˆ = Vy [x] + VBˆ + x 2 VAˆ + 2xVBA .

(5.19)

If y is normally distributed, and if Bˆ and Aˆ are derived from linear operations, then (y − y) ˆ is also normal, and SC and xu can be calculated using the central and non-central-t distributions, ˆ follows from Equation (5.17) with x = 0, respectively. SC or (yC = SC + B) SC = t1−α,ν so , or z1−α σo if σo is known, (5.20) √ where σo = Vo , and Vo = Vy [0] + VBˆ = VB + VBˆ = VB η. The development thus far is similar to that in Sections 3 and 4, as it must be, because ˆ does not appear of the correspondence between Equations (3.1) and (5.17). Since A (or A) in Equation (5.20), its value does not influence the detection decision, made in the signal domain; so, in contrast to Case I, the α of Case II has no A-induced uncertainty. The maximum (concentration) upper limit (“detection limit”) xu , for the known-σ situation, derives from the signal domain variance of (y − y) ˆ given by Equation (5.19) with x = xu . This limit may then be derived from the solution of Equation (5.21): ˆ u ) − (z1−α σo + B) ˆ (yu − yC ) = (Bˆ + Ax 1/2 . = z1−β Vy [xu ] + VBˆ + xu2 VAˆ + 2xu VBA

(5.21)

The solution to Equation (5.21) for the homoscedastic case (σy = const.), with α = β, is straightforward. The result is ˆ ˆ xu = (2z1−α σo /A)(K/I ) = (2SC /A)(K/I ), ˆ , where K = 1 + r[B, A](σBˆ /σo ) z1−α (σAˆ /A) ˆ 2, I = 1 − z1−α (σAˆ /A) constraint: xu > 0.

(5.22)

The constraint is imposed to suppress rare, but physically meaningless negative values for xu ˆ Unlike Case I, the above solution that could arise from the effects of the random variable A. does not depend on approximate normality of x, ˆ nor does it depend on an unknown sensitivity parameter, since the estimated value Aˆ is used in the solution. If σo is not assumed known, the detection decision is made using (t1−α so ) for SC , and the non-central-t distribution is employed to compute xu . (In that case substitute t1−α for z1−α for approximate correction of K and I .) Notes 1. The distinction between case I and case II is important. The former gives a result xD that represents the fixed detection limit for a fully specified measurement process, albeit with unknown A. The latter gives a variable upper limit xu that is directly calculable, given the ˆ but which is applicable only to the specific calibration experiment that proobserved A, duced it. This means for the measurement process as a whole that there is a distribution of ˆ corresponding to the distribution of A’s. ˆ When A is used in Equation (5.22), limits xu [A]

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the resulting xD can be shown to be approximately equal to the median value of the distriˆ bution of the maximum upper limits. (The mean does not exist for 1/A.) 2. When B and A are estimated from the same calibration data set, the estimates will be negatively correlated with r[B, A] = −x/x ¯ q , xq being the quadratic mean (or root mean square). The ratio K/I may then range from slightly less than one to very much greater, depending on the calibration design and the magnitude of σy . The effect of the factor I in particular, can cause xD (Case I) or xu (Case II) to differ substantially from 2t1−α,ν σo /A. The extreme occurs when the relative standard deviation of Aˆ approaches 1/t1−α,ν (or 1/kQ ); then xD (or xQ ) is unbounded. When B and A are estimated independently, then r[B, A] = 0, and K = 1. If the relative standard deviation of Aˆ is negligible compared to 1/t1−α,ν , then K and I both approach unity, and xD reduces to the form given in Equation (5.2). 3. A note of caution: If the parameters used in Equations (5.20) and (5.22) derive from a calibration operation that fails to encompass the entire measurement process, the resulting values for SC and xu are likely to be much too small. Such would be the case, for example, if the response variance and that of the estimated intercept, based on instrument calibration data alone, were taken as representative of the entire measurement process, which may have major response and blank variations associated with reagents and the sample preparation environment. This makes a strong argument for estimating B from an independent background/blank study, rather than relying heavily on the B-magnitude and uncertainty from the intercept of a linear calibration experiment. 4. An alternative approach for estimating xD , developed by Liteanu and Riˇca (1980), is based on empirical frequencies for the type-II error as a function x. Using a regression– interpolation technique these authors obtain a direct interval estimate for xD corresponding to β = 0.05, given xC . This “frequentometric” technique for estimating detection limits is sample intensive, but it has the advantage that, since it operates directly on the empirical xˆ distribution, it can be free from distributional or calibration shape assumptions, apart from monotonicity. It foreshadows the theoretical approach to the x-distribution ˆ that comprises case III. 5.2.3. Case III: Exact treatment for xC , xD A special transformation procedure developed for the ratio of random variables (Eisenhart and Zelen, 1958) was adapted to treat the exact, non-normal distribution of x, ˆ in an attempt to overcome certain limitations of the foregoing methods. Details of the method and the relatively complicated results for the general case involving covariance are given in Currie (1997). Here, we present results for xC and xD for the simpler case involving independent estimates ˆ can be taken of B and A (i.e., VBA = 0). In the dominant background situation, where V [S] as approximately constant between zero and xD , the results for xC and xD are √ xC = (z1−α σo )/(A I ), where I = 1 − (z1−α ϕA )2 , (5.23) xD = 2xC .

(5.24)

The remarkable, exact relationship of Equation (5.24) holds despite large deviations from normality, as related to ϕA , and also in the presence of covariance (VBA = 0). As a consistency test, a modest Monte Carlo experiment was performed twice to examine the empirical distribution of xˆ for both the null state (x = 0) and at the detection limit (x =


73

Fig. 3. Concentration domain detection limits (case III). Null and alternative hypothesis empirical density histograms (top) and distributions (bottom). When x = 0 the distribution is symmetric but kurtotic; increasing concentration leads to increasing asymmetry as shown by the distribution on the right. The expected medians equal the true concentrations (x = 0 and x = xD = 0.52), and xC = 0.26 marks the upper 0.95 tail of the null distribution and the lower 0.05 tail of the alternate (detection limit) distribution.

ˆ A, ˆ based on two sets of 1000 xD ). The empirical distribution of xˆ was obtained as the ratio S/ ˆ ˆ variables, simulating the normal random samples of the net signal (S) and sensitivity (A) “direct reading” (or self-calibrating) type of experiment mentioned at the beginning of this ˆ was purposely pushed section. The relative standard deviation of the non-linear variable (A) close to its limit, with σAˆ /A = 0.33; resulting in strikingly non-normal distributions for x. ˆ The results proved to be consistent with Equation (5.24), despite an (empirical) null distribution that was kurtotic, though symmetric, and a distribution for x = xD that had sub-

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stantial positive skew (Figure 3) (Currie, 1997; Figure 3). Such distributional deviations from normality are expected to give smaller values for xD than the preceding methods. Comparisons with the exact expression for xD (Equations (5.23), (5.24)), including significant nonlinearity from 1/Aˆ (with σAˆ /A = 0.33), confirmed this. Both methods I and II overpredicted xD (xu , expected median): case I (Equation (5.13)), overpredicted xD by 26%, whereas for case II (Equation (5.22)) the overprediction (xu ) was about 20%. With more precise calibration (σAˆ /A = 0.10), the overpredictions were trivial: 1.9% for case I, 1.4% for case II. 6. Multiple detection decisions; multicomponent and multivariate detection limits Emphasis in this section is placed on concepts underlying multiple detection decisions and the corresponding detection limits, such as arise in the interpretation of multicomponent chemical chromatograms and nuclear and optical spectra. The simplest case, which might be described as multi-univariate detection, relates to the detection of a series of non-overlapping gamma ray peaks, for example, or a series of independent monitoring tests for a specific radionuclide or chemical pollutant over time. When the null hypothesis dominates, such multiple detection tests can lead to an overall false positive probability far in excess of the default 5% (Section 6.2). When spectral peaks overlap, a matrix solution is required, where the critical levels and detection limits for each component depend on the levels of other components—this is the case where pure component detection limits can be seriously degraded as a result of multicomponent interference. Finally, the problem of multivariate detection is addressed where a measured component is characterized by a multiple variable (multivariate) pattern or fingerprint (Section 6.3). Only the basic concepts, and approaches for normal variates with known or assumed covariance matrices will be presented in any detail. 6.1. Multicomponent detection When a sensing instrument responds simultaneously to several analytes, one is faced with the problem of multicomponent detection and analysis. This is a very important situation in chemical analysis, having many facets and a large literature, including such topics as “errorsin-variables-regression” and “multivariate calibration”; but only a brief descriptive outline will be offered here. For the simplest case, where blanks and sensitivities are known and signals additive, S can be written asthe summation of responses of the individual chemical Sij = Aij xj , where the summation index j is the chemical components—i.e., Si = component index, and i, a time index (chromatography, decay curves), or an energy or mass index (optical, mass spectrometry). In order to obtain a solution, S must be a vector with at least as many elements Si as there are unknown chemical components. Two approaches are common: (1) When the “peak-baseline” situation obtains, as in certain spectroscopies and chromatography, for each non-overlapping peak, the sum Ax can be partitioned into a one component “peak” and a smooth (constant, linear) baseline composed of all other (interfering) components. This is analogous to the simple “signal–background” problem, such that for each isolated peak, it can be treated as a pseudo one component problem. (2) In the absence of such a partition, the full matrix equation, S = Ax, must be treated, with xkC and xkD computed for component k, given the complete sensitivity matrix A and concentrations of all


75

other (interfering) components. These quantities can be calculated by iteratively computing, from the Weighted Least Squares covariance matrix, the variance of component k as a function of its concentration, keeping all interfering components constant, and using the defining equations (3.1) and (3.2), or their normal forms, Equations (3.5) and (3.7). Further discussion can be found in Currie (1988) and references therein. An additional topic of some importance for multicomponent analysis is the development of optimal designs and measurement strategies for minimizing multicomponent detection limits. A key element for many of these approaches is the selection of measurement variable values (selected sampling times, optical wavelengths, ion masses, etc.) that produce a sensitivity matrix A satisfying certain optimality criteria. Pioneering work in this field was done by Kaiser (1966, 1972); a review of later advances is given by Massart et al. (1988). 6.2. Multiple and collective detection decisions When several independent null or alternative hypothesis tests are made, error probabilities necessarily magnify. This is a common occurrence in spectrometry or chromatography at extremely low levels where null hypothesis dominance is the rule. A striking illustration is a quality assurance test spectrum for gamma ray spectrometry, distributed by the International Atomic Energy Agency, where an international comparison resulted in up to 23 false positives and 22 false negatives from a single (simulated) gamma ray spectrum (Parr et al., 1979; Currie, 1985a). In what follows we present a brief introduction into two ways for controlling these hypothesis testing errors when treating the limited, collective decision problem, such as occurs in the analysis of a single nuclear, optical or mass spectrum, a single liquid or gas chromatogram, or a limited space–time field study. Also included is a treatment of a logical (and essential) extension to the case of unlimited detection decisions. In order to convey basic concepts and dilemmas in this brief treatment of the multiple decision problem, we restrict discussion in this section to known (or assumed) distribution functions, including as a consequence, known σ . Realistic examples, that come close to meeting this restriction, are drawn from counting experiments where the Poisson distribution is nominally satisfied. There is a growing literature on the topics of multiple and multivariate detection decisions and limits, and the closely related topic of simultaneous inference, both in the statistical and chemometrics literature (Miller, 1981; Gibbons, 1994; Davis, 1994). 6.2.1. Setting the stage As a focus for developing the topic, we present in Figure 4 a portion of a measured spectrum of the NIST Mixed Emission Source Gamma Ray Standard Reference Material, SRM 42156. The spectrum extract shows two of the energy-scale calibration peaks at 1332 keV (60 Co) and 1836 keV (88 Y, 41.7% of the 60 Co peak height), as well as a small impurity peak at 1460 keV (40 K, 1.6% of the 60 Co peak height). Were it not for the impurity, the region between the two major peaks should have nothing other that the Compton baseline and detector background, the sum of which amounts to about 265 counts per channel in the region of the 40 K peak, or 1.3% of the 60 Co peak height. (See also Appendix A.4.) There are several relevant lessons to be drawn from this snapshot of a gamma ray spectrum. First, the detection limit (counts per peak) is determined by the counting statistics of the baseline and the peak estimation algorithm, when no impurity peak is present. The baseline is the

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Fig. 4. Portion of a spectrum of the NIST Mixed Emission Standard gamma ray spectrum, SRM 4215f, showing energy calibration peaks at 1332 keV (60 Co) and 1836 keV (88 Y), together with an impurity peak at 1460 keV (40 K). The counting data should be approximately Poisson, so the square root transformed data shown here should have variance ≈0.25.

summation of the Compton counts and the detector background counts, possibly perturbed by the presence of the counting sample. The baseline counts derive from single and multiple scattering of “sample” (measurand) and interference gamma rays in the sample matrix and detector system. Added to the 265 baseline counts per channel (at 1460 keV) is the 40 K impurity peak, which probably has dual sources: background potassium in the detector system, and a chemical impurity in the master solution for the standard, possibly derived from leaching of potassium from the glass container. (40 K, which has a half-life of 1.2 × 109 years, is found in all naturally occurring potassium.) The magnitude and variability of each of these three types of “blank” must be considered when deriving critical levels and detection limits. Estimation algorithm dependence is simply demonstrated by considering alternatives for the response:


77

peak height, peak area (given fraction or full integration), or fitted peak parameter(s). To further illustrate, we take one of the simplest “square wave” (trapezoidal) estimators, which uses as a weighting function the symmetric 12 channel filter (−1 −1 −1 1 1 1 1 1 1 −1 −1 −1). This leads to an estimate of the net peak area as the summation of the central 6 channels corrected with baseline counts from the 6 channels from the left and right regions adjacent to the peak—here totaling 1589 counts. (The filter width obviously must adapt to the peak width, which is often a slow and smoothly changing function of gamma ray energy.) In the absence of the 40 K peak and using this filter, the standard deviation of the estimated net signal (null hypothesis) would be approximately (2 · 1589)1/2 = 56.37 counts (σo ). Assuming approximate normality for the Poisson distribution, and taking α = β = 0.05, gives a critical level SC of 1.645σo or 92.7 net peak counts; SD would be twice that, or 185.4 counts, which is just 0.34% of the 60 Co peak area estimated by the same “3/6/3” filter algorithm. Since Sˆ for the 40 K peak (839 counts) exceeds S , it is considered “detected” above the (Poisson) baseline. C (One small assumption has been made in the foregoing calculation; namely, that the integrated number of baseline counts (1589) is a good approximation for its expected value which, in turn, equals the Poisson variance; see Section 4.3.) 6.2.2. Multiple, independent detection decisions In many cases, in both the research and the regulatory environments, it is necessary to make a number of detection decisions simultaneously. An example comes from the Technical Specifications required by the US Nuclear Regulatory Commission for environmental radioactivity in the vicinity of operating nuclear power reactors. These specifications mandate the capability to detect prescribed levels of, for example, up to 9 fission and activation products in the same gamma ray spectrum.12 Let us suppose that the 9 detection decisions are based on an “empty” spectral region, such as depicted in the baseline portion of Figure 4. For k = 9 independent Ho tests with the error of the first kind set equal to α , the binomial distribution gives the probability of r false positives as (α )r (1 − α )k−r . The collective risk α of one or more false positives is thus α = 1 − Pr(r = 0) = 1 − (1 − α )k , or approximately kα , if α is sufficiently small. Thus, for k = 9, α must be adjusted to 1 − (1 − α)1/k = 1 − (0.95)1/9 = 0.00568 to attain a collective risk α of 0.05. For normal data, that results in an expansion of zC from 1.645 to 2.531. (For the 40 K example in the preceding paragraph, the critical level would be increased from 92.7 counts per peak to 2.531 · 56.37 = 142.7 counts, and the detection limit would increase from 185.4 to (2.531 + 1.645) · 56.37 = 4.176 · 56.37 = 235.4 counts per integrated peak.) If α is not adjusted, the collective false positive risk α in an empty spectrum then becomes 1 − (0.95)9 or 0.37; and the expected number of false positives for a nine-fold independent test scenario is then 9 · (0.05) or 0.45. A 90% two-sided interval for the number of false positives would range from zero to two counts, as given by the binomial distribution with N = 9 and p = 0.05. A small extension to this example would be to inquire not just whether a specific set of (9) peaks is present, but rather whether any gamma ray peaks, besides 40 K, are present in the apparently empty region between the 60 Co and 88 Y gamma ray peaks. This can be addressed by determining the maximum number of independent null tests that can be made using the 12 An early draft for detection capabilities for environmental sample analysis for radionuclides in water lists requisite

“lower limits of detection” ranging between 1 and 30 pCi/L for Mn-54, Fe-59, Co-58,60, Zn-65, Zr-Nb-95, I-131, Cs-134, Cs-137, and Ba-La-140 (U.S. Nuclear Regulatory Commission, 1982).

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selected algorithm. Taking a search region of 216 channels (1.40–1.60 MeV) in the baseline portion of Figure 4, and using the 3/6/3 algorithm, we can perform a total of 17 independent tests of the null hypothesis. To control the false positive error rate to 0.05 would then require α = 0.0030, or zC = 2.745. That means an increase in detection limit by a factor of (2.745 + 1.645)/3.29 = 1.33 over the single peak test value. Life, in fact, may be a bit more complicated: the number of potential gamma rays in the search region from nuclides with half-lives of an hour or more is approximately 54; so totally independent detection testing is impossible. What this means, in such cases of intensive peak searching, is that the multiple independent test technique must be replaced by one that takes into account the full covariance matrix. This issue will be touched upon at the end of this section. 6.2.3. Multiple false negatives (quasi-Bayesian approach) Thus far, we have used the foreknowledge concerning the number of null states likely present to adjust α. Although an analogous treatment of multiple false negatives was not found in the chemical literature, it is clear that this situation also must be controlled if we are to achieve an overall error rate of, say, 0.05. If “subjective” probabilistic information were available on relative HA abundance, we could give a Bayesian formulation for the detection limit. The prefix “quasi-” is used here to illustrate the extension of the treatment of Ho state multiplicity to include that of HA state multiplicity where the relative abundance of each is known (or assumed) in advance. A case in point is the IAEA quality assurance gamma ray spectrum mentioned earlier. Here, it was known (by those who created the data) that the 2048 channel spectrum was empty (baseline only) except for 22 peaks. Given such knowledge, and defining the search window as 12 channels, a zeroth order method of analysis would comprise (2048 − 12)/12 ≈ 170 tests, of which k = 148 would be applied to the null state and m = 22 to the alternate state. (See the following sub-section for commentary on so large a number of tests of the null hypothesis.) To preserve an overall (expected) error rate of 0.05 for signals lying outside the deadzone between zero and SD , we would need to set α to 0.00034 and β to 0.0023. The corresponding values of z1−α and z1−β are 3.39 and 2.83, resulting in nearly a doubling of the detection limit (6.22σo vs 3.29σo )—the price of adequately limiting the probability of both false positives and negatives.13 The coefficient 2.83 is derived for the worst case, where the HA state is m-fold degenerate, all 22 peaks having the same amplitude. In effect we are developing an operating characteristic for the doubly degenerate case (Ho : k-fold; HA : m-fold). The inequality (β < 0.05) applies to all other (HA ) cases: this is clear when one or more of the 22 peaks actually present exceeds the detection limit SD ; if one or more is less than SD , then m has been overestimated, and the probability of one or more false negatives for peaks of magnitude SD must again be BG

>Ambient

>Ambient (rsd = 0.12)

LD 40 cm3 (SG) 40 cm3 (NaI) 5 cm3 (NaI)

0.79 0.40 0.30

0.95 0.67 0.71

1.23 1.03 1.06

LQ 40 cm3 (SG) 40 cm3 (NaI) 5 cm3 (NaI)

2.83 1.68 1.68

3.32 2.48 2.79

4.17 3.56 3.83

Excursions (Bq m−3 ): Prague (June 1986): 1.4; Freiburg (April 1995): 3.2.

(NaI anticoincidence) GM tube with its higher background (0.07 cpm) but improved counting efficiency (90%). The complete discussion, including basic equations and data is found in Currie and Klouda (2001). In particular, Figure 3 in that publication highlights the diminishing returns that set in for background reduction with increased counting time and increased precision requirements—in connection with the conventional “Figure of Merit”, S 2 /B. Of special interest is the comparison of an ideal “zero background” counter, which shows little advantage over the background of the 5 mL counter with increasing counts (counting time). The transition is seen also in Table 3, where the detection limit advantage of the smaller (5 mL) counter nearly disappears at the quantification limit. To complete the ultimate detection discussion for 85 Kr, mention should be made of the need to measure 85 Kr contamination at extreme low levels in connection with the BOREXINO solar neutrino experiment (Simgen et al., 2004). The contamination issue relates to the potential problem of gaseous radionuclide impurities in N2 used for sparging the organic liquid scintillator used to detect 7 Be solar neutrinos (expected rate: 34 ν-induced events per day). The 85 Kr impurity limit for the experiment is set at 0.13 µBq/m3 of nitrogen. Such a low detection limit demands both massive sampling (here, 750 m3 ) and ultra-low level counting. The latter is being achieved in an exceptional underground laboratory with 1 cm3 proportional counters having a detection limit of just 100 µBq. This raises another interesting detection capability comparison, for the NIST 5 cm3 counter (Table 3) has an estimated detection limit of about 0.30 Bq/m3 air, which for 2 L air is equivalent to 600 µBq. This gives a direct measure of the improvement that may be expected by adopting the smallest practical counter and moving to an underground laboratory. The comparative detection limits above, however, leave a question that is most pertinent to the central topic of this chapter: “What is meant by the “Detection Limit?”20 The 600 µBq value is based on the IUPAC definition, with α and β equal to 0.05, the Poisson–normal distribution, η = 2, and T = 1000 min, with a background of 0.02 cpm and 65% counting efficiency. If, for example, the 100 µBq were based on the minimum significant signal (critical 20 That question may be answered in work referenced by Simgen and coworkers, but such references were not

searched.


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level), that would yield a factor of two between the two “detection limits”. Changing from √ paired measurements to η = 1 could account for another factor of 2, and extension of counting time beyond 1000 min could easily yield another factor of two. 7.3. Further exploration of laboratory (blank) and environmental (baseline) distributions Empirical “B” distributions, as shown in Figure 7, may be used directly to estimate critical levels and detection limits, especially when there are many observations, or degrees of freedom—for S (Figure 7b), n = 179; for 85 Kr (Figure 7d) n = 109. When such frequency histograms display positive skewness, however, caution is advised, for it may be an indication of non-stationarity and contamination. Time series analysis and physicochemical analysis are powerful tools in such cases. We illustrate below some insights derived from the S and 85 Kr distributions. A third example, explored in Section 7.5 (low-level counter backgrounds), demonstrates that, even with symmetric frequency histograms, exploration of “hidden” dimensions—beyond the blank frequency distribution, per se—can provide very important insight concerning non-random phenomena. 7.3.1. Distribution of the sulfur laboratory blank In a remarkable series of experiments on the laboratory blank associated with trace analysis of sulfur by thermal ionization mass spectrometry, driven by the impacts of trace amounts of sulfur on fuel emissions (coal) and materials properties (steel), Kelly and coworkers have systematically collected blank data over nearly a 20 year period (Kelly et al., 1994). A histogram showing the data distribution for the first decade of this record (n = 179) is given in Figure 7b. The distribution is clearly skewed. Although there is no rigorous theoretical basis for the form of the distribution, the distribution of sulfur blanks is well fitted with a 2-parameter lognormal function: x¯ = 0.27 µg S, s = 0.20 (p = 0.39).21 A normal distribution yields a very poor fit (p < 0.000001). At this point, it is important to inject a note of caution, regarding assumptions. Use of the fitted distribution for significance testing, or more generally for generating uncertainty intervals for low-level measurements, depends on assumptions that: (1) the presumed form of the distribution function is correct22 and (2) the blank is stationary (fixed mean, dispersion) and random (independent, or “white” noise). If the blank has multiple components (generally the case for environmental blanks), stationarity implies that the “mix” of components is fixed (not generally the case for environmental blanks). Even in application of a multi-step sample preparation process, the structure of the blank distribution may change as a result of multiple injections and multiple losses of blank components along the way. Testing for stationarity and randomness requires additional information, such as a meaningful time/space series, or exploration of some other type of informing variable, or “hidden dimension”. More generally, such external information relates to potential sources and the 21 Unless otherwise indicated, all “p-values” for lack of fit refer to the χ 2 test. Unless the histogram fit is very poor, p(χ 2 ) should interpreted cautiously, because the p-value will change with the number of classes and location of class boundaries, sometimes substantially. When there are too few data classes to perform the χ 2 test, the Kolmogorov–

Smirnov test, indicated p(K–S), is applied. 22 Note that a “good fit”, per se, does not constitute proof, except perhaps in the case that all other models can be

ruled out. If the randomness assumption is valid, however, distribution-free techniques may be applied, such as the use of non-parametric tolerance limits (see also the following footnote).

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chemistry and physics of the blank, i.e., the science underlying the introduction of contaminant. For sulfur there are several interesting sources, transport routes, oxidation states, and phase changes to be considered. To quote Kelly, regarding the shape (positive skew) of the empirical frequency distribution of the sulfur blank: “. . . this distribution is probably a combination of a normal distribution plus random events that add sulfur to our system. These events could be changes in SO2 gas concentration in the air in the laboratory, changes in sulfate particulates in the laboratory, and other unknown events. I would expect all blank distributions to be positively skewed to some degree. We have noticed that on days when it rains, that the blanks appear to be lower, and we are tracking this. The large Dickerson Power Station and the new county incinerator are located about 20 miles to our west. Negative skewness in a blank distribution is an impossibility in my mind” (W.R. Kelly, personal communication, 14 September 2000). The essential point is that for rigorous application, the empirical blank frequency distribution must be at least tested for randomness. Far better, however, for reduction and control, is the injection of scientific understanding of the sources and pathways for the introduction of the blank. 7.3.2. Distribution of the 85 Kr environmental background The distribution of a second series of blanks, also noteworthy for its length, is given in Figures 7d and 14a. As with the sulfur blanks, the data distribution is skewed to the right, perhaps lognormal, and it again represents an accumulation of data over a period of about a decade, by Wilhelmova et al. (1994) (see also Csongor et al., 1988). In contrast to the sulfur data, however, Figure 14a represents a field blank, in this case atmospheric (85 Kr) background radioactivity in central Europe (Prague). The data, which were collected during the period 1984–1992, contain hints of some of the most important events in modern European history. The histogram of the complete dataset (n = 109) can be fit to a lognormal distribution (p = 0.12), whereas a normal fit is unacceptable (p ≈ 0.0002). Omitting the “unusual” point at 2.23 Bq m−3 brings about an improved lognormal fit (p = 0.59) which is the one shown in Figure 14a. The fitted parameters (x, ¯ s) are (0.930, 0.185) Bq m−3 , equivalent to a relative standard deviation (rsd) of 19.9%. Although the frequency distribution is still noticeably skew, a normal distribution cannot be rejected.23 Looking beyond summary statistics and empirical frequency distributions can be remarkably revealing. Through the added dimension of time (Figure 14b) the 85 Kr data project a multicomponent background structure that is neither stationary, nor simply random. Visually, even, there appear to be sharp, positive excursions, and a complex secular trend, including quasi-seasonal variations and an apparent level shift of nearly 30% during 1989. In Figure 14b it is seen that the unusual datum at 2.23 Bq m−3 resulted from sampling in spring 1986, apparently shortly after the fateful event of 26 April 1986 at Chernobyl! Trend modeling of the 85 Kr background series was done by Wilhelmova et al. (1994) with linear and quadratic baselines, but the resulting residuals were far from random. In principle, much more might be learned via scientific and “political” modeling, linked, for example, 23 A “good fit” of an environmental blank frequency distribution to any model distribution function must be treated

with some circumspection. Not only does it not constitute proof that the blank data have been drawn from such a distribution, it does not even imply that the data are random or independent. Since environmental blanks generally reflect complex, non-stationary processes that are multicomponent in nature, it is risky to consider fitted model distributions more than smooth descriptions for the particular blank data set. Even the concept of outliers, apart from outright blunders, deserves cautious interpretation.


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Fig. 14. Environmental (field) background data for 85 Kr (n = 109) in the central European atmosphere covering the period 1984–1992. (a) (upper plot) shows the radioactivity concentration frequency distribution and the best fit lognormal distribution (excluding the highest point). Estimated parameters are (x, s) = (0.930, 0.185) Bq m−3 , equivalent to a relative standard deviation (rsd) of 19.9% (p = 0.59). (b) (lower plot) shows the 9 year time series of measured concentrations, revealing a non-stationary mean and several positive excursions. Prior to 1989, data were derived from daily samples taken at monthly intervals; later data correspond to integrated monthly samples. (Figure (b) is adapted from Figure 1 in Wilhelmova et al. (1994), with permission.)

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to tracer applications and to anthropogenic releases from spent nuclear fuel, combined with transport and mixing in the troposphere. An interesting alternative, as a first step in understanding such complicated data series, is the application of the techniques of Exploratory Data Analysis as pioneered by the late J.W. Tukey (1977, 1984). We have taken the latter course. The result of iteratively applying Tukey’s “5RSSH” resistant, non-linear smoother to the 85 Kr baseline data of Figure 14b is shown as the smooth in Figure 15a. The 5RSSH algorithm is resistant to outliers and generally more flexible than simple moving averages; it is based on 5 point running medians, supplemented by “splitting” and “Hanning” running weighted averaging (Velleman and Hoaglin, 1981). Because of the magnitudes of the excursions above the baseline, the smoother was applied twice, with those values replaced by corresponding smoothed values produced by the first pass of 5RSSH on the raw data. The resulting temporal pattern is striking. The principle features displayed by the smooth in Figure 15a include: (1) a secular trend that rises from a mean level of 0.81 Bq m−3 prior to 1989, to an average level of ca. 1.04 Bq m−3 after 1989; (2) a quasi-annual cycle with maxima tending to occur in mid-year; and (3) a dramatic, nearly 30% level shift (increase) during a single year, 1989.24 One wonders whether there might be an indirect historical link with the tumultuous sociopolitical events in Eastern Europe during that same year, when the “Iron Curtain” was rent. About the time that the smooth began to rise above values of the preceding 5 years, the collapse of Communism in Eastern Europe began—with the dismantling of the barbed wire fence separating Hungary and Austria on 2 May 1989, and culminating with the destruction of the Berlin Wall on 9 November 1989. Complementing the 5RSSH description of the systematic background variation is the Tukey rough or residual component (raw data minus the smooth, excluding artifact zeroes); this provides an approximate description of the random component of the background variations. In contrast to the skew frequency distribution of Figure 14a, the distribution of the rough (Figure 15b) is consistent with a normal distribution (x, s, p = 0.00, 0.084, 0.50, respectively). Normal tolerance limits (P , γ : 0.90, 0.90 bounding lines in the figure) show that gross concentrations that exceed the local value of the smooth by ca. 0.16 Bq m−3 would be judged significant. The rough is not homoscedastic, however, as seen by the pre- and post-1989 tolerance limits in Figure 15b. The relative standard deviations (rsd) before and after 1989, however, are approximately the same, at ≈8.7%. This value serves as the fundamental constraint to the capability to detect excursions above the smoothed environmental baseline. (See Section 7.2 and Currie and Klouda (2001) for a detailed theoretical treatment of the impacts of Poisson, instrumental, and environmental “noise” on detection and quantification limits for atmospheric 85 Kr.) Although the rsd of the rough is about twice the rsd from counting statistics, as shown in Figure 14b, removal of the structure underlying the histogram of Figure 14a has resulted in an rsd reduction (from 19.9%) by more than a factor of two. Conclusion. An interesting and central feature of laboratory and environmental blank distributions is their tendency to fall into two classes: those that tend to be controlled (endogenous) and more or less normally distributed, and those that are uncontrolled (exogenous) that tend to 24 A significant level shift (nearly 20%) remains even if the large excursions from the baseline are included in the

pre-1989 mean.


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Fig. 15. Decomposition of the 85 Kr background time series into “signal” and “noise” components. Exploratory data analysis was utilized to extract a complex signal with minimal assumptions from the raw time series data of Figure 14b, using the Tukey resistant non-linear smoothing procedure 5RSSH (Tukey, 1977, 1984; Velleman and Hoaglin, 1981). (a) (upper plot) shows the estimated signal (the smooth). Important features are the significant shift in mean concentration during the momentous year 1989, and the quasi-periodic structure that tends to favor higher atmospheric concentrations in mid-year. (b) (lower plot) shows the residuals (the rough) remaining after the smooth is extracted from the raw baseline data of Figure 14b.

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exhibit positive skewness. The first class is found in generally controlled physical and biological systems, such as (1) self-regulating vegetative and human organisms whose health depends on limited concentration ranges of “essential” elements or compounds, and (2) environmental (geochemical) compartments and laboratory reagents having stabilized blank components that tend to be well mixed and uniformly distributed. The second class is the “wild class” where toxic or non-essential contaminants are found in widely varying concentrations, depending of the whims of contaminant (source) strengths and environmental transport processes. A striking contrast in this regard is found in the distributions and multivariate patterns of toxic and essential elements found in different environmental/global regions in the Daily Diet Study of the International Atomic Energy Agency (Currie, 1992). 7.4. Extreme low-level counting: the Poisson perspective This final part of the “B” section of the chapter treats the situation where few counts are observed, and the Poisson distribution is far from normal. Detection decisions and detection limits are considered in Sections 7.4.1 and 7.4.2 for the two asymptotic cases where tB

ty = tS+B (well-known blank, η = 1) and tB = ty (paired counting, η = 2), respectively. While not presented explicitly here, the intermediate cases are readily treated by the theory in Section 7.4.2. The last segment, Section 7.5, considers the nature of the distribution of (background) counts, which is not necessarily Poissonian. 7.4.1. Poisson detection decisions and limits for the well-known background asymptote (η = 1) (Currie, 1972) When the background is “well-known,” with expectation μB , the exact Poisson formalism for SC and SD (Section 7.4.1.1) is quite straightforward, since Sˆ is represented by the difference between a single Poisson variable and a constant (μB ). Because of the discrete nature of the Poisson distribution, however, the defining equation for the detection decision must be cast as an inequality (Equation (3.1)). If one wishes to realize a specific target value for α (as 0.05), a little-used Bernoulli weighting procedure can be employed to overcome the inequality (Section 7.4.1.2). Finally, to complement the algebraic solutions presented, a graphical representation of η = 1, extreme Poisson critical values and detection limits is given in Section 7.4.1.3. 7.4.1.1. Exact Poisson solution Notation. For the treatment of the extreme Poisson distribution the following notation is adopted.25 • Expectation of background counts: μB (real). • Expectation of “sample” (gross counts) at the detection limit: μD (real)—alternatively: yD . • P (n | μB ) = (μB )n /n! · exp(−μB ): Poisson density function, Prob of n counts (integer), given μB . • P (n | μD ) = (μD )n /n! · exp(−μD ): Poisson density function, Prob of n counts (integer), given μD . 25 Note that the Poisson parameter (μ) is continuous (real), whereas the observable counts (n) are necessarily discrete

(integers).


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Table 4 Exact Poisson confidence limitsa Observed counts (n)

Lower limit (μ− )

Upper limit (μ+ )

0 1 2 3 4 5 6 7 8 9

0 0.0513 0.355 0.818 1.37 1.97 2.61 3.28 3.98 4.70b

3.00 4.74 6.30 7.75 9.15 10.51 11.84 13.15 14.44 15.70b

a P = 0.05, 0.95; μ = χ 2 /2 (ν = 2n, ν = 2n + 2). ± − + P b Poisson–normal approximation (Section 4.3.1): 5.24 (μ ), 15.47 (μ ). The mid-point of the confidence interval − + (solution of Equation (4.9)) equals (n + z2 /2), or (n + 1.35) for the 90% CI.

• P (n | μ) = (μ)n /n! · exp(−μ): general expression for the Poisson density function. k Pr(n k | μ) = (7.3) P (n | μ): • 0

Poisson (cumulative) distribution, for integers zero to k. Given the expectation for the background counts (μB ), we can apply the exact cumulative Poisson distribution to calculate the critical number of counts nC , considering the error of the first kind (α), and then the detection limit for the expectation of the gross sample counts μD , considering the error of the second kind (β). These values follow from the defining equations (3.1) and (3.2), adapted to this special case. For the net signal, it follows that SC = nC − μB , and SD = μD − μB . Critical value: Pr(n > nC | μB ) 0.05

(7.4)

(an inequality because of the discrete Poisson distribution). Detection limit: Pr(n nC | μD ) = 0.05.

(7.5)

The solutions to Equation (7.4) and (7.5), for any particular choice of μB , present no problem, due to the wide availability of mathematical–statistical software. For a convenient overview, however, we present brief tabular and graphical renditions for the problem. The solutions, in fact, can be easily extracted from a table of the confidence limits for the Poisson parameter vs the integer values of n. These confidence limits can be determined readily from the chi-squared table. (See Section 4.3 for a more complete exposition, including a detailed comparison with the results of the Poisson–normal approximation, which gives remarkably good interval estimates for μ, even for very few counts observed.) Table 4 gives exact Poisson lower (0.05) and upper (0.95) confidence limits for μ for integer values of n, ranging from zero to 9. Table 5 gives the derived values for the critical number of counts (nC ) and the gross count detection limits (μD ).

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Table 5 Poisson critical values and detection limits (η = 1; tB tS+B ) α (minimum)a

μB (range)

nC = SC + μB (α 0.05)

μD = SD + μB (β = 0.05)

– 0.0013 0.0060 0.0089 0.013 0.016 0.018 0.020 0.021 0.022

0–0.051 0.052–0.35 0.36–0.81 0.82–1.36 1.37–1.96 1.97–2.60 2.61–3.28 3.29–3.97 3.98–4.69 4.70–5.42

0 1 2 3 4 5 6 7 8 9

3.00 4.74 6.30 7.75 9.15 10.51 11.84 13.15 14.44 15.70

a For each μ

B range, α varies monotonically from αmin to 0.05.

To illustrate, take the expectation of the background count to be 2.30. Referring to Table 5, we find that nC = 5 counts, and μD = 10.51 counts (expectation). Thus, SC = 5 − 2.30 = 2.70 counts; and SD = 10.51 − 2.30 = 8.21 counts. The net count detection limit, expressed in units of background equivalent activity (BEA) is 8.21/2.30 = 3.57. The first column of Table 5 shows us that the actual value of α falls in the range of 0.016 to 0.050. The expression for the cumulative Poisson distribution (Equation (7.3)), with μB = 2.30 and k = nC = 5, gives the result α = 0.030. For k = 4, α = 0.084. If only an estimate for μB is available—e.g., from nB counts observed, we can compute an exact Poisson confidence interval (CI) for μB , and from that, an interval for nC and μD . If the μB CI falls entirely within one of the μB bands in column 2 of Table 5, then nC and μD have unique values, but the actual value for α is uncertain. Such an event is unlikely for small numbers of counts observed, however. For example, if nB = 4 counts, then the 90% CI for μB equals 1.37 to 9.15 counts. The “Bayesian” estimate of (nB + 1) is no less uncertain, but it does give a somewhat better point estimate for μB as the approximate mid-point of the (90%) confidence interval. (Compare: Est(μB ) = 4 + 1 = 5 counts, with mid-CI = (1.37 + 9.15)/2 = 5.26 counts.) The exact Poisson–normal solution for the mid-point of the μB CI (solution to Equation (4.9)) gives the mid-point as (nB + z2 /2) = (4 + 1.35) = 5.35 counts for the 90% CI. 7.4.1.2. Overcoming the inequality Equation (7.4) can be transformed into an equality, such that α = 0.05, by applying Bernoulli weights for the α’s from the cumulative Poisson distribution for k = nC and k = nC − 1. Taking the previous example where μB = 2.30 counts (nC = 5), we can force a value α = 0.05, by selecting a value of p such that p · α(k = 4) + (1 − p) · α(k = 5) = 0.05. Using the α’s (k = 4, k = 5) found above, the expression becomes α = 0.05 = p · (0.084) + (1 − p) · (0.030),

giving p = 0.37.

In practice we would use a series of Bernoulli random numbers (0’s and 1’s), with p = 0.37, to select 4 counts as a modified critical value 37% of the time, and 5 counts 63% of the time.


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Fig. 16. Poisson critical values (nC ) and detection limits (ρD ) vs background expectation (μB ). The y-axis corresponds to the background equivalent activity (BEA or ρ)—i.e., the ratio of the net signal to the background, S/μB . The sawtooth envelope represents the detection limit SD /μB , with nC indicated by the integers just above the envelope. The dashed line corresponds to the Poisson–normal approximation for ρD .

7.4.1.3. Graphical representation; approach to normality A global perspective is presented in Figure 16, which shows, as a function of μB : (1) the critical numbers of counts nC , (2) the detection limit, expressed here as the ratio SD /μB —i.e., ρD , in units of background equivalent activity (BEA), and (3) the approach to normality. The latter is indicated by the dashed line that is, in effect, the Poisson–normal extension of the discontinuous “sawtooth” envelope of the exact Poisson detection limits (ρD ). The results for previous example (μB = 2.3 counts), based on the limited tabular (Table 5) equivalent of Figure 16, are evident also from the plot: the ordinate for the ρD curve corresponding to 2.3 counts on the abscissa is seen visually to be approximately 3.6 (SD /μB ), giving a net signal detection limit of about 8.3 counts. The corresponding value for nC , indicated just above the ρD envelope is, again, 5 counts. The Poisson–normal approximation is already reasonably good for μB = 2.3 counts; application of Equation (4.5) gives SD = 7.7 counts, or ρD = 3.4. The asymptotic function for ρD (dashed curve), corresponding to Equa√ tion (4.5), equals (2.71/μB + 3.29/ μB ). 7.4.2. Extreme low-level (Poisson) detection decisions and limits for paired counting (η = 2) The solution for the other asymptotic case, where detection decisions and limits must be evaluated for pairs of Poisson variables, was published more than 65 years ago, by Przyborowski and Wilenski (1939). The stimulus for the work of these authors was the practical problem of

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detecting small numbers of dodder seeds in clover. This was of some practical import, because the contaminating seeds belong to the class of twining herbs that are parasitic to plants. In the context of this chapter, the detection of the rarely occurring dodder seeds is the analog of the detection of rarely occurring nuclear particles or decays. Unlike the previous section, where the challenge was to detect a significant signal above a well-known background, we now address the problem of detecting a significant (1-sided) difference between two extreme low-level Poisson variables. The relative simplicity of the previous section is gone, since the distribution of the difference between two Poisson variables is no longer Poissonian. In fact, the solution space is now 2-dimensional, with a critical region replacing the critical level of the single Poisson variable. Przyborowski and Wilenski formulate the problem by first expressing the joint probability law for observations x, y as y p(x, y | μx , μy ) = μxx μy /x!y! · exp −(μx + μy ) ,

(7.6)

where, in the context of low-level counting, x represents counts observed from the background variable (B) having expectation (mean) μx , and y represents counts observed from the gross count (signal + background) variable having expectation μy . The density function follows from the fact that the distribution of the sum of Poisson variable is itself Poissonian.26 Equation (7.6) can be transformed into a more interesting form (7.7) using the following substitutions: ρ = μy /(μx + μy ), μ = μ x + μy , n p(x, y | ρ, μ) = (μ /n!) · exp(−μ) n!/ y!(n − y)! ρ y (1 − ρ)n−y .

n = x + y,

(7.7)

Critical region. The sample space is necessarily a 2-dimensional (integer) grid, with the possible sample points (E) defined by the discrete observable values of x (background counts) and y (gross sample counts). For a given n, the partition into y and x = n − y is governed only by the second factor in Equation (7.7), which is a term in the binomial expansion of [(1 − ρ) + ρ]n . For the null hypothesis, μy = μx , so ρ = 1/2; thus, for each n, the critical value for y is given by Pr(y > yC | n, ρ = 0.5) 0.05, independent of μ. The yC are simply the 1-sided critical values for proportions, which may be determined from the binomial (cumulative) distribution (n, ρ). To give a specific illustration, consider an observation pair for which x + y = n = 12 counts. Then the integer yC derives from the 95+ percentile of the binomial distribution (12, 0.5), which equals 9 counts. (The probability that y > 9 counts, given n, ρ, is 0.019.) The full set of gross count critical values (yC ) for background counts from x = 0 to x = 30 for α 0.05 is given below.27 26 For more convenient application to the low-level counting problem, slight changes have been made from the

original formulation (in notation, only). 27 Extension to larger values of x can be accomplished using the Poisson–normal approximation, including the cor-

rection of 1/2 for “discontinuity” (Przyborowski and Wilenski, 1939; p. 323), also called the correction for “continuity” (Cox and Lewis, 1966, pp. 21 and 225, respectively). This correction has not been applied to the Poisson–normal approximations appearing elsewhere in this chapter.


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(x, yC ) (0, 4)

(1, 6) (11, 21) (21, 34)

(2, 8) (12, 22) (22, 35)

(3, 9) (13, 23) (23, 36)

(4, 11) (14, 25) (24, 37)

(5, 12) (15, 26) (25, 39)

(6, 14) (16, 27) (26, 40)

(7, 15) (17, 29) (27, 41)

(8, 17) (18, 30) (28, 42)

(9, 18) (19, 31) (29, 44)

(10, 19) (20, 32) (30, 45)

Detection limit. Evaluation of the detection limit for the exact paired count Poisson problem is not so simple, in that the full probability equation (7.7) must be considered. Przyborowski and Wilenski calculate the power function (1 − β, given α) from the expression ∞ P {E ∈ w | ρ, μ} = n!/ y!(n − y)! ρ y (1 − ρ)n−y , (7.8) (μn /n!) · exp(−μ) n=0

w

where (E ∈ w) refers to all observable pairs of observations (x, y) that lie within the critical region w—i.e. beyond the critical contour. Numerical data given for power (1 − β) = 0.95, for α 0.05, were combined with critical values yC to construct plots showing the critical values (yC , discrete; dashed curve) and detection limits (yD , continuous; solid curve) as a function of μx (which is μB ). This information is given in the upper plot in Figure 17a. Note that although the detection test is based strictly on the observed count pair (x, y), the detection limit necessarily depends on μB . The lower plot, Figure 17b, gives similar information for the well-known blank case (η = 1), based on the derivations of Section 7.4.1. To give a clearer representation of the two-dimensional distribution of x, y for the null case, with a given mean μ, a Monte Carlo simulation is shown in Figure 18. The three-dimensional frequency histogram was created for the null case, with a common mean (μx , μy ) equal to (5.00, 5.00), and 1000 random samples from the pair of Poisson distributions. The lower part of the figure shows a series of (x, y) pairs extending beyond the critical region; these constituted 2.3% of the random pairs. This is consistent with 0.05 as the upper limit for the actual α of the null hypothesis test, and the value of 0.024 given for the true significance level when μ = 5 (Table IIa in Przyborowski and Wilenski, 1939). α-Control. Like the well-known blank problem, the paired Poisson variable problem is characterized by discrete observables (counts), resulting in discontinuities in α. An alternative to the “Bernoulli trick” to force the equality, α = 0.05, is the possibility of exercising modest control over the wide, small count swings at the critical value (nC ) by the selection of that member of the bracketing pair having α closer to 0.05. Such a practice moderates the swings to extremely small α, without the need to generate and apply series of Bernoulli random numbers—but at the cost of modest dispersion (+, −) about the target value of 0.05. To illustrate, the bracketing (1 − α)’s for n = x + y from 10–15 counts, for (nC , nC − 1) are: n = 10 (0.989, 0.949), n = 11 (0.967, 0.887), n = 12 (0.981, 0.927), n = 13 (0.954, 0.866), n = 14 (0.971, 0.910), n = 15 (0.982, 0.941). Using the conventional inequality (α 0.05) gives (α±sd) ¯ = 0.026±0.013; for the alternate, “closest approach” rule, (α±sd) ¯ = 0.049±0.017. Evidently the quite simple, alternate rule is a good choice, at least for this part of the critical region. (See also footnote 9.)

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Fig. 17. Extreme low-level counting: critical regions (dashed curves) and detection limits (solid curves) for (1) paired counting (η = 2)—upper plot (a), and (2) well-known background (η = 1)—lower plot (b). In (a), both axes have integer values only for the dashed (critical value) curve, where x = observed background counts, and y = observed (gross) sample counts. The solid curve represents the detection limit as a function of the background expectation; both axes are then continuous. In (b), the x-axis represents the expectation of the blank counts, and is continuous. y (gross counts) has integer values only, for the dashed (critical value) curve, but it is continuous for the solid curve which represents the detection limit (expectation) as a function of the background expectation.

7.4.3. Some closing observations, and reference to other work Some summary observations can be drawn from the two sets of yC critical boundaries and two sets of yD detection limit curves in Figure 17. For the paired case (η = 2, Figure 17a), the minimum value for yC (4 counts) occurs when x = 0; thus, the smallest integer pair that would indicate “detection” would be (x, y) = (0, 5). Similarly, for x = 4 counts, the smallest significant value for y would be 12 counts. Looking at the detection limit curve, we see that for a truly zero background, the minimum detectable gross count (yD ) is approximately 9.0 counts, which here also equals SD . For μB = 4.0 counts, the minimum detectable yD ≈ 20.0 counts (SD ≈ 16.0 counts). To detect a net signal (S) equal to 4 times the background, the intersection of line of slope 5 (μy /μx ) with the (gross signal) detection limit curve is needed. That occurs at μB = 4.0 counts. Considering the same background values (μB = 0, μB = 4.0 counts) for the well-known background case (η = 1, Figure 17b), the minimum detectable gross counts are yD = 3.00


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Fig. 18. Empirical (Monte Carlo) test of the critical region for extreme low-level counting (paired Poisson variables, η = 2).

counts and yD = 14.44 counts, respectively. The corresponding values for the minimum detectable net signal are SD = 3.00 counts and SD = 10.44 counts, respectively. So, for this small count range, excellent knowledge of the background expectation carries a benefit (reduction in SD ) of a factor of two to three. (The Poisson–normal approximation, Equation (4.5), applied to μB = 4.0, gives SD = 9.29 counts (η = 1) and SD = 12.02 counts (η = 2).) Although the critical value boundary curve allows rigorous testing for significant signal, background differences in the two asymptotic (η = 1, 2) extreme Poisson cases, the corresponding detection limit curves are essential for planning for successful extreme low-level studies. A statement from Przyborowski and Wilenski (1939) captures the thought: “It might with reason be regarded as undesirable to plan an experiment in which the chance was less than 0.5 of detecting from two random samples that differences . . . of practical importance existed”. The extreme Poisson, well-known blank case has achieved practical importance in very low-level monitoring and spectroscopic applications, such as in the large scale gamma ray monitoring activities of the Comprehensive Nuclear-Test-Ban Treaty Organization (CTBTO) (De Geer, 2004). Three factors are at play in the low-level monitoring activities in question:

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(1) the backgrounds (and gamma ray baselines) are generally empty and smooth, such that large regions can be utilized for background estimation, yielding a background that is rather well known; (2) significant peak areas may fall in the very small count region, such that the rigorous Poisson distribution should be applied; (3) continual monitoring with null hypothesis dominance means that large numbers of false positives will occur, unless the effect of multiple detection decisions is taken into account. For these special circumstances, De Geer proposes application of the rigorous Poisson distribution with the well-known blank, with the α reduced from 0.05 to reflect the large number of null decisions. Table 2 of De Geer (2004) is basically an extension of our Table 5 (Section 7.4.1) for type-1 risk levels (α) ranging from 5% to 0.0001% and LC (Poisson critical counts) ranging from 1 to 50. Multiple null decision α reduction is governed by the binomial distribution (Section 6.2.2); for small α, the reduction required is approximately α/n, where n is the expected number of null hypothesis tests. There has been a resurgence of interest in recent years in the low-level counting communities in the problem addressed by Przyborowski and Wilenski so long ago, in part because of major advances in, and needs for, low-level radiation measurement, and in part because of then un-dreamed of advances in computing power. A review of such work is impracticable in this chapter, but a recent inter-agency resource document may be consulted for an excellent critical review with pertinent references, exhibits, and examples (MARLAP, 2004). For special topics related to extreme low-level gamma-ray spectrometry, see also the Proceedings of the Conference on Low-Level Radionuclide Measurement Techniques (ICRM, 2004), CELLAR (2004), Laubenstein et al., (2004); Povinec et al. (2004), and Méray (1994). 7.5. On the validity of the Poisson hypothesis for background counts It seems to be conventional wisdom, at least for low to moderate counting rates, that counting background can be described as a Poisson process. Such an assumption, of course, affects uncertainty estimates, as well as computed critical levels and detection limits, where the variance of the background is dominant. For those background components from long-lived radioactivity, the assumption for the decay process is doubtless valid, but the assumption for the counting process deserves rigorous testing for possible environmental and/or measurement process artifacts. The following summary is drawn from a comprehensive study of time series of some 1.4 × 106 individual low-level GM background events (coincidence and anticoincidence) spanning a period of nearly one month, where the amplitude and time of occurrence of each pulse was recorded to the nearest 100 µs. Additionally, the system provided pulse-pair resolution to the nearest 1 µs, and full waveform analysis (Currie et al., 1998). 7.5.1. Background as a Poisson process; expected characteristics Nuclear decay can be described as a Bernoulli process, which, in the limit of a large pool of decaying nuclei each with a small but fixed decay probability, can be approximated as a Poisson process. Measurements of long-lived radionuclides are therefore expected to follow: (1) the Poisson distribution of counts, which is asymptotically normal, but discrete and positively skewed for small numbers of counts, and for which the expectation of the Index of Dispersion (variance/mean) is unity (Cox and Lewis, 1966); (2) the Uniform distribution of arrival times; and (3) the Exponential distribution of inter-arrival times, also positively skewed. Thus, we


109

have the situation where skewed data distributions are grounded in the very nature of the physical process; and tests of the fundamental assumptions for actual measurement processes may be made by performing significance tests on the corresponding empirical data distributions. For low-level measurements of radioactivity, where background is dominant, one finds that the assumption of an underlying Poisson process is commonly extended to the background radiation. The validity of such an assumption is crucial for making low-level detection decisions and in estimating detection limits. It is central also, in the “propagation of uncertainty” for low-level radioactivity results. Problems are not expected for background components derived from long-lived radioactive contaminants. If, however, there are other significant background components, in particular cosmic radiation and background artifacts characteristic of the real, laboratory measurement process, the assumption should be approached cautiously. A comprehensive investigation of the validity of the Poisson assumption for low-level, anticoincidence background radiation was carried out with the NIST individual pulse analysis system, with 40 cm3 cylindrical gas counters operated in the Geiger mode (Currie et al., 1998). The individual pulse analysis capability of the NIST system is unique, in that it captures the shape and time of arrival of individual low-level coincidence (muon) and anticoincidence pulses with the capability of time stamping each counting event and accumulating up to 105 events in a single file. Without such a capability it would not be possible to construct the arrival and inter-arrival time distributions. Essential characteristics of the system are: (1) ca. 80% of the gross background was eliminated by surrounding the counting tube with 25 cm of pre-nuclear era steel (World War I naval gun barrel); (2) the bulk (98%) of the residual background (ca. 20 cpm), due to penetrating cosmic ray muons, was “canceled” by anticoincidence shielding. This is achieved by placing the sample counting tube within an outer cylindrical tube (guard counter), such that an external cosmic ray (muon) which penetrates both tubes results in a coincidence pulse, whereas the lower energy net background (BG) radiation in the inner sample tube does not. The fundamental assumptions to be tested are whether the muon (coincidence) background and the net (anticoincidence) background can be treated as independent Poisson processes. 7.5.2. Empirical distributions and tests of the Poisson (background) hypothesis Distributional highlights from the NIST study are given in Figures 19–21. The basic information is generated from the individual pulse arrival and inter-arrival times, as shown in Figure 19 which represents a 150 s glimpse of the pulse data stream (arrival times) for coincident (C) and anticoincident (A) events from a 980 min background counting experiment. (The event labeled “G” represents a rare giant pulse, atypical of the ordinary GM process.) Considering first the Poisson distribution of counts and the corresponding count rates, we examined results for C (muon) and A (net background) events spanning a period about sixteen thousand times as long as that shown in Figure 19, nearly a month (7 May–5 June 1997). At the beginning of this period the GM counting tube was filled with purified counting gas; following that, background (C and A) counts were aggregated for 21 individual counting periods, 16 of which has relatively uniform counting times of about 1600 min. The Poisson hypothesis was tested with χ 2 , considering the ratio of the observed variance to the theoretical variance based on the number of counts accumulated in each counting period. (This variance ratio can be viewed as a weighted estimate for the index of dispersion Iw .) The A-events passed the test (Iw = 1.36, p = 0.16) but the C-events failed (Iw = 4.92, p < 0.000001).

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Fig. 19. A 150 s snapshot of the individual pulse data stream from the low-level GM counting tube, showing times of arrival (TOA, x-axis) and pulse amplitudes (z-axis), extracted from a 980 min background measurement during May 1997. Two anticoincidence pulses (A) occurred during this interval, the remainder being coincidence pulses (C). DT represents the interval between successive anticoincidence background events, while dt represents the interval between anticoincidence events and preceding (coincidence) events. (DT and dt distributional analysis for the entire, 588 A-count sequence is shown in Figure 21.)

Insight is gained by looking beyond the summary statistics. The presumably random, symmetric distribution of the 1600 min C-background counts, shown in Figure 7a, is repeated in Figure 20a, together with further exploration in two additional dimensions: time (Figure 20b) and barometric pressure (Figures 20c, 20d). The time series shows immediately that the extra C-background variance (above Poisson) is not due simply to an added component of normal, random error. Rather, a distinctive quasi-periodic pattern is evident, well in excess of the Poisson–normal standard uncertainty bars. In fact, the source of the non-Poisson, non-random C-background variance represents an interesting blend of meteorology and physics, with time as surrogate: increased barometric pressure leads to increased attenuation of mesons in the atmosphere. Removal of the barometric pressure effect reduces Iw to 2.19. Arrival time distributions for the C- and A-pulses (not shown) from the 980 min experiment referenced in Figure 19 showed no significant difference from a Poisson process (uniform distribution), but inter-arrival times were a different matter. The inter-arrival times (DT) for the low-level anticoincidence background did show a good fit (p = 0.40) to the exponential distribution as shown in Figure 21a, but the A vs C inter-arrival times (dt) did not. As seen in Figure 21b, there was an excessive number of very short intervals falling within the first (1 s) histogram bin. Upon expansion of the time scale, it was found that the excess was concentrated almost entirely between dt = 150 µs and dt = 350 µs (43 counts). The physics of this phenomenon is well known. It relates to a counting process artifact whereby a secondary event (“afterpulse”), which occurs within a few hundred microseconds of the primary event, is occasionally generated by a complex photo-electric process within


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Fig. 20. Upper left (a): frequency distribution of gross background pulses in the NIST low-level gas counting system. (Each of the 16 background measurements corresponds to a counting period of 1600 min.) Although the histogram appears symmetric that does guarantee that the background events are normally distributed. The time series of background events (b, upper right) reveals a non-random, quasi-periodic structure significantly in excess of the bounds of the Poisson–normal error bars. (c, lower left) and (d, lower right) demonstrate a clear dependence on barometric pressure. (N and p curves represent the count rate and barometric pressure time series in (c), and the y- and x-axes in (d).)

the counting tube. For the NIST system the afterpulse probability is ca. 0.1%. Because of their absolute abundance and time constant, the spurious pulses are scarcely detectable in the DT distribution. They have a good chance of escaping cancellation during the C-pulse time gate, however, and appearing in the dt distribution where intervals may be as short as the intrinsic deadtime of the GM tube (≈140 µs). Since the primary C-pulses are more abundant than the valid A-pulses by a factor of 70, the contamination probability is raised to ca. 7% for the anticoincidence background. A further, interlaboratory complication is that the afterpulse contamination of the A-background depends on the electronic deadtime of the low-level counting system. Conclusion. Fundamental (theoretical) skewed data distributions are firmly established for radioactive decay, in the form of the Poisson distribution of counts and the exponential distribution of inter-arrival (decay) times. Extension of this model to low-level background radiation is widely assumed, but it does not necessarily follow, especially for the major gas counting background component (cosmic ray muons) and for the extremely important low-

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Fig. 21. Empirical (exponential) distribution of inter-arrival times: (DT, left) between anticoincidence background events, and (dt, right) between anticoincidence and preceding coincidence events. The histogram on the left (a) summarizes DT inter-arrival times between zero and 1000 s; mean = 100.7 s, equivalent to a mean background rate of 0.60 cpm (total for the two GM counters that were operational during the 980 min period). The fit is good (p = 0.40). The histogram on the right (b) summarizes dt inter-arrival times between zero and 30 s; mean = 2.72 s, equivalent to a mean rate of 22.0 cpm. The fit is not good (p = 0.0006), showing excessive counts in the first 1 s class (histogram bin).

level anticoincidence background component. Because of the enormous excess of C-events over A-events in low-level anticoincidence counting, it is essential to provide extremely effective muon (anticoincidence) shielding and afterpulse control via special timing and pulse shape circuitry. Note that the afterpulse artifact was manifest as a departure from the theoretical skewed (exponential) data distribution. The non-random character of the muon background, together with its effect on afterpulse amplification, makes it clear that strict validity of the Poisson assumption for counting background requires systems where “meson leakage” (Theodórsson, 1992) can be minimized by extremely efficient anticoincidence shielding, or better, by largely eliminating the cosmic ray meson background by going deep underground (CELLAR, 2004).

8. Two low-level data quality issues 8.1. Detection limits: intralaboratory vs interlaboratory perspectives The issue of single laboratory vs multi-laboratory Detection and Quantification Limits has, on occasion, become contentious, resulting in what might be considered “two cultures”. Some of the statistical and conceptual issues, and controversy involved may grasped from an article by


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Fig. 22. Sampled (S) and target (T ) populations.

Gibbons (1995), which treats the detection of environmental pollutants. Of particular interest are insightful and provocative contributions from the Discussants. The purpose of this brief section is to give some perspective on the two views—both of which are correct, and incorrect; and to suggest how they might be reconciled. To begin, it is useful to consider the error structure of the Compound Measurement Process which, in its simplest manifestation, has been presented by Natrella (1963). In this view, the measurement process is treated in terms of two populations which represent, respectively, the population (of potential measurements) actually sampled (S-population), and that which would be sampled in the ideal, bias-free case (T -population). The corresponding S and T populations are shown schematically in Figure 22, for a two-step measurement process. When only the S-population is randomly sampled (left side of the figure), the error e1 from the first step is systematic while e2 is random. In this case, the estimated uncertainty is likely to be wrong, because (a) the apparent imprecision (σS ) is too small, and (b) an unmeasured bias (e1 ) has been introduced. Realization of the T -population (right side of the figure) requires that all steps of the MP be random—i.e., e1 and e2 in the figure behave as random, independent errors; T thus represents a Compound Probability Distribution. If the contributing errors combine linearly and are themselves normal, then the T -distribution also is normal. The concept of the S and T populations is absolutely central to all hierarchical measurement processes (Compound MPs), whether intralaboratory or interlaboratory. Strict attention to the concept is essential if one is to obtain consistent uncertainty estimates for Compound MPs involving different samples, different instruments, different operators, or even different methods. In the context of (material) sampling, an excellent exposition of the nature and importance of the hierarchical structure has been presented by Horwitz (1990). From the perspective of a single laboratory, the S-population represents that for which the type-“A” uncertainty component can be estimated via replication. The link to the T -

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population, and the total uncertainty, is the uncertainty resulting from the unmeasured (nonsampled) distribution (upper left, error e1 ). Difficulties arise when that uncertainty component is ignored, and the single laboratory’s result is accompanied only by the replication-based type-“A” uncertainty. In particular, uncertainty (confidence) intervals would be unrealistically small, and derived values for detection and quantification limits would be overly optimistic. The approach taken by the National Metrological Institutes (NMIs) leads to the resolution of the problem. That is, the mandated policy for all NMIs is that all uncertainty components must be taken into account, for all measured results. The process begins with a complete uncertainty (error) budget for the measurement process, and all non-sampled component(s), deemed type-“B” uncertainties, must be estimated and combined with the “A” components to provide a total combined standard uncertainty (uC ). The expanded uncertainty (U = kuC ) may be given, using a “k”, commonly 2, to generate an approximate 95% confidence interval. Note that neither absolute nor relative uncertainties are necessarily the same for different instrumentation or laboratories. Each measured result, when accompanied by a valid combined standard uncertainty, is expected to be consistent with the “truth” or assigned value for the measurand. A complete presentation of the principles and methods for deriving combined standard uncertainties may be found in the “Guide to the Expression of Uncertainty in Measurement” (ISO-GUM) (ISO, 1995). From the interlaboratory perspective, the first population in Figure 22 (e1 ) would represent the distribution of errors among laboratories; the second [S] would reflect intralaboratory variation (“repeatability”); and the third [T ], overall variation (“reproducibility”) (ISO, 1993). If the sample of collaborating laboratories can be taken as unbiased, representative, and homogeneous, then the interlaboratory “process” can be treated as a compound MP. In this fortunate (doubtless asymptotic) situation, results from individual laboratories are considered random, independent variates from the compound MP population. For parameter estimation (means, variances) in the interlaboratory environment it may be appropriate to use weights—for example, when member laboratories employ different numbers of replicates (Mandel and Paule, 1970). As in the case of the individual laboratory following the ISO Guide (ISO-GUM), unbiased T -distribution results would be expected from the multilaboratory consortium that satisfies the foregoing conditions. In these very special circumstances, the different perspectives of the two cultures should be reconciled—each providing meaningful measurement uncertainties and unbiased estimates of (detection, quantification) measurement capabilities. For detection decisions and limits, a small problem exists when there are residual systematic error components—intra- or inter-laboratory. If a measurement error is fixed—e.g., through the repeated use of a faulty instrument or calibration source, or background- or blank- or σ -estimate, then the ideal random error formulation of hypothesis testing cannot apply. For a single measurement, the question is moot; but for a suite of data from the same (biased) laboratory, or sharing the same numerical estimate of the blank, or imprecision, or calibration factor, then the independence (randomness) assumption is not satisfied. A conservative treatment of the problem, resulting in inequalities for α and β, is given in Currie (1988). In Figure 23, we attempt to reconcile the intra- and inter-laboratory environments, by presenting an explicit partitioning of error components that can be grasped from both perspec-


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Fig. 23. Partitioning of method, interlaboratory, and intralaboratory error. (Adapted from Currie (1978).)

tives. Here, we see that specification of the Performance Characteristics of a compound or hierarchical MP depends upon one’s viewing point or position in the hierarchy. That is, at least for the “tree” structure, all segments below the viewing (or “null”) node consist of multiple branches or replicates—essential for direct assessment of random error. Only a single path lies above the null node; this path necessarily fixes the bias of the MP. By moving up in the hierarchy, one has an opportunity to convert bias into imprecision—put another way, what is viewed as a fixed (albeit unknown, perhaps “type-B”) error at one level of a compound MP, becomes random at a higher level. This is very important, for random error may be estimated statistically through replication, but bias may not; yet inaccuracy (total error) necessarily comprises both components. Collaborative or interlaboratory tests, which under the best of circumstances may be found at the uppermost node of the Compound MP, provide one of the best means for accuracy assessment. In a sense, such intercomparisons epitomize W.J. Youden’s recommendation that we vary all possible factors (that might influence analytical results), so that the observed dispersion can give us a direct experimental (statistical) measure of inaccuracy (Youden, 1969). The basic concept, as indicated in Figure 23, is that fixed intralaboratory biases are converted into random errors from the interlaboratory perspective. If the overall interlaboratory mean is free from bias, then the observed interlaboratory dispersion is the measure of both imprecision and inaccuracy. An apt metaphor that has been applied to the two perspectives is that the intra-laboratory approach (following ISO-GUM) uses a “bottom up” evaluation of total uncertainty, whereas the inter-laboratory approach uses “top down” total uncertainty evaluation. Guidelines to the derivation of detection (quantification) limits that would apply to the

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individual laboratory situation have been prepared by IUPAC (1995, 1998) and ISO (1997, 2000), while ASTM (1997, 2000) documents treat the interlaboratory situation. Total reconciliation between the two cultures is guaranteed, if the two approaches are routinely combined, somewhat in the spirit of the NMI “key comparisons” policy. That is, (1) individual laboratories would be advised always to consider the complete error budget for their specific MPs, taking into account both type-A (statistical) and type-B (experiential, theoretical) errors in assessing detection, quantification capabilities, and (2) at the same time, the consortium of such laboratories should adopt the Youden philosophy, to utilize interlaboratory data, to provide a cross-check on the intralaboratory type-B uncertainty bounds. If the two approaches are self-consistent, meaningful estimates of uncertainty and measurement limits can be expected; if not, research into the source(s) of the discrepancy is called for.28 8.2. Reporting of low-level data 8.2.1. Statement of the problem; values and non-values Quantifying measurement uncertainty for low-level results—i.e., those that are close to detection limits—deserves very careful attention: (a) because of the impact of the blank and its variability, and (b) because of the tendency of some to report such data simply as “nondetects” (Lambert et al., 1991) or “zeroes” or “less than” (upper limits). The recommendations of IUPAC (1995, 1998) in such cases are unambiguous: experimental results should not be censored, and they should always include quantitative estimates of uncertainty, following the guidelines of ISO-GUM (ISO, 1995). When a result is indistinguishable from the blank, ˆ with the critical level (LC ), then it is important also based on comparison of the result (L) to indicate that fact, perhaps with an asterisk or “ND” for not detected. But “ND” should never be used alone. Otherwise there will be information loss, and possibly bias if “ND” is interpreted as “zero”. Data from an early IAEA intercomparison exercise on radioactivity in seawater illustrates the point (Fukai et al., 1973). Results obtained from fifteen laboratories for the fission products Zr-Nb-95 in low-level sample SW-1-1 consisted of eight “values” and seven “non-values” as follows. (The data are expressed as reported, in pCi/L; multiplication by 0.037 yields Bq/L.) Values: 2.2 ± 0.3 9.5 9.2 ± 8.6 77 ± 11 0.38 ± 0.23 84 ± 7 14.1 0.44 ± 0.06. Non-values: xC . For smaller m it leads always to positive bias, becoming practically significant for absent analytes. The alternative rule, where x ≈ xC /4 (solid line) is attractive in that its average bias is small and its bias extrema, more acceptable at b− = −0.20 and b+ = +0.54. It is perhaps close to optimal in the Bayesian sense, if the m values for non-detects are believed to be uniformly distributed between 0 and xD . Impact of hardware/software thresholds. Major negative bias may result from data-destroying thresholds or discriminators that are set well in excess of the intrinsic critical level. Such thresholds are not uncommon in instrumental measurement systems, particularly chromatographic systems designed for monitoring organic pollutants, where the threshold discriminator rejects signals below a set point, and returns non-values, or zeroes.29 Although such an oper29 Because of the widespread use of “black box” chromatographic monitoring systems for the detection of organic

pollutants in air and water, it seems likely that the severe negative bias problem may be responsible for the spate of


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ation can be viewed as applying xC σo , such that α 0.05, the substitution of zero for an observed value xˆ and its uncertainty makes for severe negative reporting bias. Referring to the discussion of the impact of thresholds in Section 4.3.4, with a slight change in notation (xT in place of ST ), it was seen that setting xT far above the 0.05 critical level xC caused a significant degradation in detection capability. Failure to report numerical values (“non-detected” observations, x) ˆ compounds the problem, by creating negative bias. We consider here the magnitude of that bias, based on Equations (8.1) and (8.2), and the chromatographic data mentioned in Section 4.3.4 (Kurtz, 1985). The measurement problem in that case was the detection of residues of the synthetic pesticide, fenvalerate, in chickens and eggs. Electron capture gas chromatographic data were produced from a system that included a “black box” hardware/software peak integrator, that had a built-in threshold that returned nonzero areas only for peaks exceeding 1.00 cm2 —equivalent to a fenvalerate mass (xT ) equal to 42 pg. (The measurement precision at this artificial, de facto critical level, was significantly better than that of the conventional quantification limit!) By comparison, using σo and the variance function, with α = β = 0.05, the intrinsic critical value and detection limit were 2.0 and 4.2 pg, respectively. With zero-substitution for non-reported signals, the xT -induced negative bias function equals "xT b(0, m) = −

xϕ(x − m) dx.

−∞

Thus, when the actual amount m is well below the threshold, e.g., −3σ , the negative bias is equal to the mean—i.e., b(0, m) = −m (0 m xT ). For m = xT , and xT σ , the bias approaches −m/2, and it vanishes when m approaches zero and when m xT . For the specific fenvalerate measurement process, taking into account the calibration and variance functions, the maximum bias (−37 pg) occurs for m ≈ 37 pg, and it becomes negligible for m > 49 pg. Thus, the combination of an artificially high threshold and the suppression of data below that threshold has resulted in a maximum negative bias nearly nine times the detection limit!

Appendix A: Worked examples, and critical valves of test statistic A.1: Examples of applications The following subsections give an overview of selected applications, illustrating some critical issues arising in the estimation of detection and quantification limits, together with references to source literature. For the two examples, based on actual data for “simple” low-level counting and gamma-ray peak detection, respectively, expanded treatments showing numerical details (“worked examples”) are given in Sections A.3 and A.4. zeroes in the organic emissions database discussed at the beginning of this section. Also, it may partly underlie the significant current controversy concerning the meaning and use “MDL” (method detection limit) in connection with water pollution monitoring (Waterscience, 2005).

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Table A1 Selected critical values of test statistics used in this chapter Degrees of freedom (ν)

t (0.95, ν)a

1 2 5 10 20 50 100 ∞

6.314 2.920 2.015 1.812 1.725 1.676 1.661 1.645

δ(0.95, 0.95, ν)

χ 2 (0.05, ν)

χ 2 (0.95, ν)

5.516 3.870 3.543 3.408 3.335

0.0039 0.1026 1.15 3.94 10.85 34.76 77.84

3.84 5.99 11.07 18.31 31.41 67.60 124.50

a z(0.95) = t (0.95, ∞).

Special note: Because of the tutorial, numerical intent of the examples in Sections A.3 and A.4, all calculated results are given to at least one decimal digit. The objective is to preserve computational (numerical) accuracy even though the convention concerning significant figures may be violated. By contrast, observed counts will always be integers because of the discrete nature of the Poisson process. In practice, of course, final results should always be reported to the correct number of significant figures. Selected critical values of test statistics used in this chapter are given in Table A1. A.2: Overview Low-level beta counting (39 Ar) Issues (Currie et al., 1998): Normal background (small RSD) Tests of Poisson hypothesis, randomness Heteroscedasticity, non-central-t Combined uncertainty (concentration domain) Two cases are presented using anticoincidence data (Set I) and coincidence data (Set II) from Currie et al. (1998). For Set I data the background variance is found to be consistent with that of a Poisson process and the observed counts are sufficiently large that the Poisson– normal approximation can be used for detection decisions and for the calculation of detection and quantification limits. The Set II data illustrate the contrasting situation where there is an extra variance component, requiring the use of the replication variance estimate s 2 and information drawn from the t (central and non-central) and χ 2 distributions. This is a common occurrence for non-anticoincidence, background limited, low-level counting where the primary background component derives from an external process such as cosmic ray interactions. For the Set II data, the background variance was primarily due to cosmic ray variations, which in the short term (intervals of a few days) are not even random, reflecting short term trends in barometric pressure. Peak detection (γ -ray spectrometry) Issues (Currie, 1997): Algorithm dependence Background, baseline, blank


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Null hypothesis dominance Multiple detection decisions Low-level data reporting Several of the above issues are illustrated in the numerical example given in Section A.4. Multiple detection decisions, mentioned but not covered in the example, require an adjustment of z or t to take into account the number of null hypothesis tests, because the overall probability for one or more false positives when k-tests are made is 1 − (1 − α)k ≈ kα

(α 1).

(A.1)

Thus, to maintain an overall false positive probability of 0.05, one must use 1-sided z(α ) or t (ν, α ), where α ≈ 0.05/k. For k = 5 independent tests, for example, z(0.01) = 2.326, and t (9, 0.01) = 2.821. In the latter case, it is also necessary to obtain an independent estimate of σo for each application of the t-test.1 Extensions The applications discussed above highlight some of the more common issues involving detection, quantification, and data reporting for low-level, nuclear-related measurements as found in the “simple” scalar, sample-background situation (see Section A.3) and simple peak detection, including multiple detection decisions and the treatment of background vs baseline vs blank (see Section A.4). Although the influence of interfering components is illustrated by the effect of an elevated baseline on isolated peaks, this latter example does not address more complicated cases such as overlapping peaks, where the impact of interference is amplified by a “variance inflation factor” due a poorly conditioned inverse matrix. Some discussion of this topic may be found in IUPAC (1995) and Currie (1997). Other noteworthy situations which will not be illustrated here include: (1) calibration-based detection limits, which capture the covariance between the estimated blank (as intercept) and sensitivity (as calibration factor or slope) as well as the variance function (dependence of σ 2 on concentration); and (2) more complicated functional and blank models, which arise in environmental isotope ratio measurements such as accelerator and stable isotope mass spectrometry, where non-linear blank correction functions are the rule, and multivariate and nonnormal blanks are often encountered. Some discussion of these advanced topics is given in Currie (1997, 2001). Numerical examples for calibration-based detection limits, based on real experimental data, may be found in ISO (1997) and Currie (1985b). A.3: Example 1: low-level beta counting (39 Ar) The first example illustrates the treatment of simple counting data for making detection decisions and the estimation of detection and quantification limits in the signal (counts) and radioactivity concentration (Bq/mol) domains. Two sets of actual background counting data, drawn from Currie et al. (1998), are used to illustrate the Poisson–normal case, where the standard deviation is (1) well represented by the square root of the mean number of counts (Set 1 data) and (2) based on replication, due to the presence of non-Poisson variance components (Set 2 data). Set 1 data are in fact the observed number of anticoincidence counts in ten, 1000 min counting periods; Set 2 are the corresponding cosmic ray induced coincidence counts.

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A.3.1: The data (counts; t = 1000 min) Set 1: 265 305 277 277 263 312 310 318 286 270 Mean (B) = 288.3, √ Poisson-σB = 288.3 = 17.0, Replication-sB = 21.0. Using the χ 2 test, we find (s/σ )2 to be non-significant (p = 0.13), so the Poisson value for σB will be used. Set 2: 18730 19310 19100 19250 19350 19210 19490 19190 19640 18780 Mean (B) = 19205.0, √ Poisson-σB = 19205.0 = 138.6, Replication-sB = 283.2. Using the χ 2 test, we find (s/σ )2 to be very significant (p < 0.0001), so the replication value sB must be used as an estimate for σB . A.3.2: Equations for Poisson–normal variables (counts) ˆ Sˆ = y − B, ˆ = V (y) + V (B) ˆ = S + B + B/n = S + Bη, V (S) √ σo = (Bη) . . . OR . . . so = sB η [TEST: χ 2 ],

(A.2)

LC = zC σo

(A.5)

. . . OR . . .

LC = tC so ,

LD = LC + zσD = LC + z(z + σo ), . . . OR . . . LD ≈ δσo ≈ 2LC , where δ ≈ 2t 4ν/(4ν + 1) ≈ 2t, ν = n − 1, # $ LQ = kQ σQ = kQ (kQ /2) 1 + SQRT 1 + (2σo /kQ )2 , $ # LQ = 10σQ = 10 5 1 + SQRT 1 + (σo /5)2 .

(A.3) (A.4) (A.6a)

LD = z + 2zσo 2

(A.6b) (A.7a) (A.7b)

Note that the facts that σD > σo (Equation (A.6)) and σQ > σo (Equation (A.7)) derive from the variance function for Poisson counting data (Equation (A.3)). Also, when this is applied to actual counting data, a subtle but necessary approximation is made by calculating the Poisson variances from observed counts which serve as estimates for the (unobservable) population means, S and B. Parameter values for the current example are: n = 10,

ν = n − 1 = 9,

α = β = 0.05,

η = (1 + 1/n) = 11/10 = 1.10,

kQ = 10,

z = 1.645, t = 1.833, δ = 3.57, √ √ σo = σB η = σB 1.10,

√ √ so = sB η = sB 1.10.


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Note that LD and LQ require information on the dependence of the variance σ 2 on signal (or concentration) level (variance function). For the Set-1 data, this is given directly from Poisson “counting statistics” where the variance is equal to the (true) mean number of counts. For the Set-2 data, where we must use s 2 to estimate σ 2 , one really needs full replication data as a function of signal (or concentration) level. For the purpose of this example, however, we shall assume that the variance is approximately independent of concentration over the range of interest—i.e., σ ≈ σo for L LQ . In practice, such an assumption should always be tested. A.3.3: Signal domain (all units are counts) Set-1 data √ σB = 288.3 = 17.0, sB = 21.0 [χ 2 : NS], √ √ therefore: σo = σB η = 17.0 (11/10) = 17.8. SC = 1.645σo = 29.3, SD = 2.71 + 3.29σo = 61.3, SQ = 10σQ = 10(5) 1 + SQRT 1 + (σo /5)2 = 10(23.49) = 234.9. In the case of SQ , we see that the variance function for this particular Poisson process yields a ratio (σQ /σo ) of 1.32. New observation: y = 814; then Sˆ = 814 − 288.3 = 525.7. Sˆ exceeds SC , so “detected”. Report: Sˆ = 525.7, u = (814 + [288.3/10])1/2 = 29.0 [relative u = 5.5%]. Set-2 data

√ 19205.0 = 138.6, sB = 283.2 χ 2 : significant (p < 0.00001) , √ √ therefore: so = sB η = 283.2 (11/10) = 297.0 (σ assumed constant for S SQ ; homoscedastic). σB =

SC = 1.833so = 544.4, SD = δσo = 3.57σo , SˆD = 3.57so = 1060.3, SQ = 10σQ ≈ 10σo , SˆQ = 10so = 2970.0. The relative (standard) uncertainty for SˆD and SˆQ derive from that for so /σo . Rigorous bounds 2 for the latter can be obtained √ from the χ distribution, but a good approximation for ν > 5 is given by u(so /σo ) ≈ 1/ (2ν); for 9 degrees of freedom, this gives u = 23.6%. The minimum possible value for σo , of course, is the Poisson value (Currie, 1994). New observation: y = 19004; then Sˆ = 19004 − 19205.0 = −201.0. Sˆ does not exceed SC , so “not detected”. Report: Sˆ = −201.0, u = so = 297.0.

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A.3.4: Concentration domain (Bq/mol)2 ˆ ˆ A, ˆ xˆ = (y − B)/A = S/ Aˆ = t Vˆ Eˆ = (60000 s)(0.000252 mol)(0.939) = 14.19 mol/Bq, where t is the counting time; V , the moles of Ar in the counting tube; and E, the counting efficiency for 39 Ar. # 2 2 $ uc (A)/A = SQRT u(V )/V + u(E)/E $ # = SQRT [0.52%]2 + [1.28%]2 = 1.4%. Set-1 data xˆD = 61.3/14.19 = 4.32 (uc = 1.4%), xˆQ = 234.9/14.19 = 16.55 (uc = 1.4%). New observation: xˆ = 525.7/14.19 = 37.0, # $ uc = SQRT [5.5%]2 + [1.4%]2 = 5.7% or 2.1 Bq/mol. Set-2 data xˆD = 1060.3/14.19 = 74.7, xˆQ = 2970.0/14.19 = 209.3. For the set-2 data, the relative uncertainties of xD , xQ√are dominated those SD , SQ , shown √ in Section A.3.3 to be approximately 1/ (2ν) = 1/ 18, or 23.6%. Combining this with the 1.4% relative uncertainty for A, as above, has little effect; the combined relative uncertainty for the concentration detection and quantification limits is still about 23.6%. See IAEA (2004) for a discussion of additional error components that must be considered for low-level β counting. New observation3 : x = −201.0/14.19 = −14.2, uc = 297.0/14.19 = 20.9. A.4: Example 2: gamma-ray peak detection The second example treats detection and quantification capabilities for isolated (“baseline resolved”) gamma ray peaks. The data were derived from a one hour measurement of the gamma ray spectrum of NIST Mixed Emission Source Gamma Ray Standard Reference Material (SRM) 4215F. The full spectrum for this SRM contains several major pure radionuclide peaks used for calibration of the energy scale. For the purposes of this example, however, we consider only the high energy region dominated by the Compton baseline plus 60 Co peaks at 1173 and 1332 keV and the 88 Y peak at 1836 keV. Within this region, shown in Figure A1,


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60 88 Fig. A1. SRM 4215F: γ -ray spectrum segment showing √ energy calibration ( Co [1173, 1332 keV], Y [1836 keV]) and impurity (40 K [1460 keV]) peaks. (Y -axis equals counts.)

there is a small impurity peak at 1460 keV due to 40 K, the subject of this example. The ordinate in Figure A1 uses the Poisson variance stabilizing transformation, counts1/2 , corresponding to a constant y-standard deviation of 0.50, which is 5% of the smallest y-axis interval (tic mark) shown. (In the figure, the y-axis ranges from 0 to 200, corresponding to a count range of 0 to 40,000 counts; and the x-axis ranges from channel 1100 to 2100, with the last datum in channel 2048.) Peak channel gross counts are 26184 (1173 keV), 20918 (1332 keV), 581 (1460 keV), and 8881 (1836 keV). The baseline in the region of the 40 K peak is ca. 265 counts/channel. The data shown in Figure A1 will be used to estimate detection and quantification limits for 40 K under the three “B” limiting conditions (background, baseline, blank), and to illustrate the reporting of low-level peak area data. To focus on these issues we limit the discussion here to the signal domain (peak area detection, quantification, and estimation). See IAEA (2004) for a discussion of additional error components that must be considered in radionuclide or element uncertainty estimation for gamma-ray spectrometry and neutron activation analysis. A.4.1: Background limiting This case is exactly analogous to the low-level counting example above, except that signal and background are here based on the summation of counts over a k-channel spectrum window. Given the approximate peak width of 3.5 channels (FWHM) in this region of the spectrum, we take a 6-channel window centered at the 40 K peak location for background and net peak area estimation. For a 60 min counting period, this gives a long-term average B = 3.6 counts for the 6-channel window. For the purposes of this example, we take this to be the “well-known” background case, where B has negligible uncertainty. Unlike the first example, the expected value of the background is so small that the normal approximation to the Poisson distribution is inadequate. Also, because of the discrete nature of the Poisson distribution, we must use the inequality relationship indicated in Equation (7.4), such that α 0.05. For a mean of 3.6 counts, this gives a critical (integer) value for the

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gross counts of 7, for which α = 0.031. (For 6 counts, α = 0.073.) Using 7 counts for the gross counts critical value, Equation (7.5) gives 13.15 as the gross counts detection limit, for which β = 0.05. Subtracting the background of 3.6 gives the corresponding net signal (peak area) values for SC (3.4 counts) and SD (9.55 counts). Had the √ normal approximation to the Poisson distribution been used, we would have obtained 1.645 3.6 = 3.12 counts for SC , √ and 2.71 + 3.29 3.6 = 8.95 counts for SD . The net count quantification limit SQ is given by 50[1 + (1 + B/25)1/2 ] = 103.5 counts. In this case B is sufficiently small that it has little influence on the value of SQ . (See also Table 5, Section 7.4.) A.4.2: Baseline limiting Detection and Quantification capabilities under background limiting conditions tend to be overly optimistic, reflecting interference-free conditions. When the Compton baseline is nonnegligible, we get the situation depicted in Figure A1. Here, the limits are governed by both the baseline noise and the peak estimation algorithm and its estimated parameters. To illustrate, we take the simplest of estimation algorithms, the 3/6/3 square wave filter, for the estimation of the net peak area. This carries the assumption that the peak is located entirely within the central 6 channels, and that two 3-channel tails can serve as a reliable correction for a linear baseline. The channel count data for the 40 K peak in Figure A1 are: Baseline counts, Sum,

B = 1589.

Peak counts, Sum,

Bi = 266 270 266 [left tail], 279 258 250 [right tail];

Pi = 328 473 581 501 301 244;

P = 2428. Sˆ = P − B = 839;

Net peak area (counts), u(S) = (2428 + 1589)

1/2

= 63.4 (Poisson).

A.4.2.1: Detection, quantification limits (Poisson variance) SC , SD and SQ can be calculated using the equations given in Section A.3, using η = 2 for the paired signal-background case, since the baseline is estimated with the same number of channels as the gross peak. Thus, in units of counts, √ √ σB = 1589 = 39.9, σo = σB 2 = 56.4, SC = zσo = 1.645 · 56.4 = 92.8, SD = z2 + 2SC = 2.71 + 2(92.8) = 188.3, 1/2 SQ = 50 1 + 1 + (σo /5)2 = 616.2 net counts [ca. 10σo ]. Since Sˆ (839) exceeds SC (92.8), we conclude “detected”. A significance test can be applied also to Sˆ in comparison with SQ . Since the lower (0.05) tail of the 90% confidence interval, which equals 839 − 1.645 · 63.4 = 734.7, exceeds SQ , we conclude that the “true” (expected) value of S exceeds the quantification limit. Algorithm dependence. The above calculations apply to the simplest of all peak estimation algorithms, one that works reliably with isolated peaks supported by linear baselines. The same data, however, could be evaluated with far more sophisticated algorithms, based, for


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example, on a non-linear fit to a 3 or 4 parameter peak function (Gaussian–exponential, or Gaussian–Lorentzian). In such cases, the detection and quantification limits would be different, as they depend upon the entire measurement process, including data reduction algorithms. These performance characteristics would be further altered, of course, if the level or nature of the interference were greater or more complex, as with the case of overlapping peaks. Influence of the background. If the background spectrum is linear over the peak + baseline spectral region used for the above estimate, its influence is automatically taken into account by the baseline correction algorithm. If the background, itself, contains a significant 40 K peak, however, the magnitude and uncertainty of that peak must be considered in the estimation of S and its uncertainty, and in the calculation of SC,D,Q . It is not a problem for the present example, however, because of the relative magnitude of the Compton interference. That is, even if the entire 3.6 count background in the six channel peak window were 40 K, it would be equivalent to less than 0.5% of the baseline correction and less than 10% of the uncertainty for that correction. A.4.2.2: Detection, quantification limits (replication variance) Alternatively, the detection decisions and detection and quantification limits may be based on the replication variance s 2 . Comparison of the Poisson-σ to the replication-s is accomplished by taking the sum of the Poisson weighted squared residuals from fitting the baseline model. The sum should be distributed as chi-square if there is no extra (non-Poisson) variance component. The result of such a fit, using the 3 left and 3 right tail Bi ’s gives a non-significant slope (p = 0.45) and si = 10.28 counts per channel with 4 degrees of freedom. (Again, extra, non-significant digits are shown in the interest of numerical accuracy.) The same conclusion follows from calculating the difference between the means of the 3 left and 3 right Bi ’s compared to the Poisson-σ for that difference, which is 5.0 ± 23.0 counts. We can gain a degree of freedom, therefore, by computing s from the mean (per channel) background; this gives si = 9.97 counts per channel with 5 degrees of freedom. The estimated replication standard deviation √ 6 = 24.4 counts. Comparing this for the 6-channel background sum is therefore s = 9.97 √ with the Poisson-σ ( 1589) from Section A.4.2.1, we get (s/σ ) = (24.4/39.9) = 0.61, for which p(χ 2 ) = 0.87, consistent with Poisson variance only. That conclusion for this particular set of experimental observations could be the end of this example, but we continue because of the important tutorial value of illustrating the treatment of replication variance. When we choose to use the replication variance,4 which would be necessary if p(χ 2 ) < 0.05, we get the following results (units of counts): √ so = sB 2 = 34.4, ν = 5 degrees of freedom, α = β = 0.05, sB = 24.4, SC = t1−α,ν so = t0.95,5 · 34.4 = 2.015 · 34.4 = 69.3, SD = δσo = 3.87σo δ ≈ 2t 4ν/(4ν + 1) = 2t (20/21) = 1.905t = 3.84 , σo ≈ so ;

bounds are given by Chi square.

For 5 degrees of freedom, (s/σ )min = 0.48, (s/σ )max = 1.49 [90% interval]. Thus, SˆD = 3.87 · 34.4 = 132.0 net counts; 90% interval: 88.6 to 275.1 net counts. SˆQ = 344.0, SQ = 10σQ ≈ 10σo ,

90% interval: 230.9 to 716.7 counts.

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The approximation assumes σ (S) to be constant. In practice the variance function should be determined. ˆ Conclusions. Since S(839) > SC (69.3), 40 K has been detected. Since the lower (0.05) limit for S (769.7 counts) exceeds the upper limit for SQ (716.7 counts), we conclude that S exceeds the quantification limit. A.4.3: Blank limiting5 This represents the most difficult case, where the very component of interest is present as a blank component of the reference spectrum. That, in fact, was the case for SRM 4215F, where the unintended 40 K peak was present as a blank contaminant. Calculation of SC , SD , and SQ then requires an estimate of σB for 40 K which must be obtained by replication of the blank. The distribution of the blank, and the magnitude of σB depend on the sources of the blank. For the particular SRM 4215F, it is believed that 40 K arose from background potassium in the detector system and a chemical impurity in the master solution for the standard, possibly derived from leaching of potassium from the glass container (Currie, 1997). In more complicated cases of sample preparation, trace levels of potassium can be derived from the sampling and sample handling environment. Since 40 K is a natural radionuclide having a half-life of ca. 1.2 × 109 years, it is therefore present in all potassium, and in favorable cases it can be used for the non-destructive analysis of potassium. In fact, such an application is relevant to research on atmospheric aerosol in remote regions, where it is of great practical importance to determine the minimum quantifiable amount of potassium by a non-destructive technique, such as the direct measurement of 40 K. Although σB can be determined only through blank replication of the actual measurement process, some observations can be offered in relation to the above mentioned sources of the blank. The detection system component, which likely represents a fixed contamination level of this long-lived radionuclide, is expected to exhibit a Poisson distribution, with a gross (peak + baseline) blank variance equal to the expected number of counts. If the blank derives from processing reagents and if its relative standard deviation is rather small (e.g., 20%), and the distribution, asymmetric and non-normal, since negative blank contributions cannot occur. To offer a brief numerical treatment, let us suppose that 10 replicates of the potassium blank gave an average net peak area of 620 √ counts with an estimated standard deviation sB of 75 counts. Since the ratio s/σ = 75/ 620 = 3.01 exceeds the 0.95 critical value ((s/σ )C = √ 1.88 = 1.37 for 9 df) from the chi-square distribution, we conclude that non-Poisson error components (chemical blank variations) are present; so we shall use the sB for estimation of detection and quantification limits. For this particular example, √ we take measurements of gross signal and blank to be paired.6 Thus, η = 2 and so = sB 2 = 106.1 counts. Values for SC , SD , and SQ (units of counts) are calculated as follows. t-Statistics For ν = 9 degrees of freedom, t (ν, α) = t (9, 0.05) = 1.833 non-central-t,

δ(ν, α, β) = δ(9, 0.05, 0.05) = 3.575,


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δ-approximation: 2t 4ν/(4ν + 1) = 1.946t = 3.567, SC = tso = 1.833(106.1) = 194.5 counts. Note that SC calculated in this way is a “single use” statistic. Repeated detection decisions require independent estimates of σo , as well as an adjustment of t as discussed in Section A.2. SD = δσo

provided that σ is constant between S = 0 and S = SD .

If σ (S) is not constant, the variance function must be determined. For the present illustration, we shall treat σ as constant (homoscedastic). Not knowing σo , we use so as an estimate, and set bounds from bounds for (s/σ ) from the χ 2 distribution. Thus, SˆD = δso = 3.575(106.1) = 379.3 counts, (s/σ )0.05 = 0.607,

(s/σ )0.95 = 1.37 (from χ 2 with ν = 9),

276.9 < SD < 624.9

(90% confidence interval),

SQ = kQ σQ = 10σo

provided that σ is constant between S = 0 and S = SQ .

(See the above note, concerning the variance function.) Thus, SˆQ = 10so = 1061.0 counts, 774.4 < SQ < 1747.9 (90% confidence interval). If σ is an increasing function of signal level, the above estimates will be biased low, but they can be viewed as lower bounds. The ultimate lower bounds, for counting data,√are given by the Poisson-based values for SD and SQ . For the present example, taking σB = 620 (negligible baseline and/or background case) these are √ SD (Poisson) = 2.71 + 3.29 2B = 118.6 counts, 1/2 = 405.7 counts. SQ (Poisson) = 50 1 + 1 + (35.2/5)2 A.4.4: Summary The comparative results for the 40 K detection and quantification capabilities as limited by the three B’s (background, baseline, blank), expressed in terms of the 1460 keV peak area counts are as follows: Background limiting: Baseline limiting: Blank limiting:

SD = 9.55, SD = 188.3, SˆD = 379.3,

SQ = 103.5. SQ = 616.2. SˆQ = 1061.0.

For the blank limiting case, estimated values for SD and SQ are given, because (non-Poisson) blank variability made it necessary to use a statistical estimate (so ) for σo . As shown in Section A.4.3, SD and SQ are consequently uncertain by about a factor of 1.5. Although this example was purposely limited to the signal (peak area) domain, it is interesting to consider the results in terms of radioactivity (Bq) for 40 K, and mass (mg) for naturally

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radioactive potassium. Conversion factors (sensitivities), taking into account the 1460 keV gamma counting efficiency, 40 K gamma branching ratio, and K activity concentration (specific activity) are 42.2 peak area counts/Bq, and 1.26 counts/mg, respectively. Detection limits for 40 K for the three cases are thus 0.23 Bq (background limiting), 4.46 Bq (baseline limiting), and 8.99 Bq (blank limiting). The corresponding detection limit for potassium in the background limiting case is 7.58 mg. Uncertainties in the estimated sensitivities would introduce equivalent relative uncertainties in the estimated detection and quantification limits.5 Because of the applicability of 40 K counting to the direct, non-destructive assay of potassium, it is interesting also to compare the last result with what could be obtained by low-level β counting. For the measurement process discussed in Section A.3 the corresponding sensitivity is 1.62 counts/µg; since SD in that case was 61.3 counts, the mass detection limit for potassium would be (61.3/1.62) or 37.8 µg. The gain of more than a factor of one thousand in sensitivity came about because of major increases in branching ratio, counting efficiency, and counting time for the low-level β counting technique. Because β-particles are far more readily absorbed than gamma rays, this latter technique is most attractive for inherently thin samples (such as air particulate matter) or potassium that has been concentrated chemically, as opposed to the non-destructive analysis of large samples by gamma spectrometry. For selected applications, therefore, low-level β counting can be useful for the assay of natural levels of 40 K, as it is for 14 C and 3 H.

Notes 1. The use of tso for multiple detection decisions is valid only if a new estimate so is obtained for each decision. In the case of a large or unlimited number of null hypothesis tests, it is advantageous to substitute the tolerance interval factor K for Student’s-t. See Currie (1997) for an extended discussion of this issue. 2. The transformation from the signal domain (observed response, in the present case, counts) to the concentration domain is fraught with subtle “error propagation” difficulties, especially for calibration factors having large relative uncertainties. Following the discussion of Section 5 we take the simplest approach in this example, estimating the uncertainty in detection and quantification limits from the uncertainty in the calibration factor A. For small relative uncertainty in A, the result differs little from more sophisticated approaches employing “calibration-based” limits (Currie, 1997). When the relative uncertainty of A is large (e.g., >20%) there is a further complication due to increasing non-normality of the xˆ distribution. For the special case where the background and standard, and therefore x, are measured for each sample, the simplified expressions in Section A.3 can be applied directly to the observed series, with xi = Si /Ai = ((y − B)/A)i , provided that the relative standard deviation of the Ai is not so large that the normality assumption for the xi is seriously violated. 3. Set 2 data illustrate a special xˆ error propagation problem that arises for low-level data, when the net signal is at or near zero. The simplified form of the Taylor expansion does not work, because of (implicit) division by zero. The simplified form gives the relative variance of the estimated concentration xˆ as the sum of the relative variances of the net


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ˆ and the calibration factor (denominator, A): ˆ signal (numerator, S) (σx /x)2 = (σA /A)2 + (σS /S)2 . The S = 0 problem can be avoided by using the proper expression for the uncertainty of a ratio of random variables (Section 5.2, for independent S, A) 2 2 Vx = (1/A2 ) VS (1 + ϕA , ) + S 2 ϕA where ϕA is the relative standard deviation of A. 2 1, the result If S = 0, the last term in the brackets vanishes. If, in addition ϕA becomes σx ≈ σS /A, as with the final result for the Set-2 data. 4. The replication variance s 2 may always be selected as the variance estimator, but it is quite imprecise when ν is small. σ 2 (Poisson) is more precise and sets the lower bound for the true variance, but it underestimates the true variance if extra, non-Poisson components are present. A conservative but potentially biased approach, generally applied in low-level counting and accelerator mass spectrometry, is to use the larger of s 2 or σ 2 (Poisson) (Currie, 1994). 5. In keeping with the peak area (signal domain) treatment for the background and baseline parts of this example, we continue with the reference to the blank expressed as counts, and the symbols SC,D,Q . If the replications of the blank were expressed directly in x-units (radioactivity, Bq for 40 K; or mass, kg for K) the appropriate symbol set would be xC,D,Q . For a comprehensive treatment of all significant uncertainty components in the measurement of 40 K, see the worked example on Radionuclides in Marine Sediment Samples in the contribution of Dovlete and Povinec (2004) to IAEA (2004). 6. Paired measurements are always advisable, to offset possible errors from a drifting or changing blank. For a drifting blank, so may be calculated directly from paired blank observations; alternatively, sB may be calculated from “local” Bi replicates, or from the dispersion about a fitted trend line. Furthermore, if a non-normal blank distribution is likely, paired observations will force a symmetric distribution for Sˆnull (S = 0), but in such a case it is advantageous also to estimate the net signal from k-replicate pairs. That brings about approximate normality as a result of the central limit theorem. In the present example, the estimated relative standard deviation of the blank (75/620) is 0.12, so the assumption of normality may be considered acceptable.

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Radiometric determination of anthropogenic radionuclides in seawater M. Aoyama∗ , K. Hirose Geochemical Research Department, Meteorological Research Institute (MRI), Tsukuba, 305-0052 Japan

Anthropogenic radionuclides in seawater have been concerned with their ecological effects and have been oceanographically used as a tracer. Current concentrations of anthropogenic radionuclides in the oceanic waters are generally extremely low. Determination of anthropogenic radionuclides in seawater has been traditionally performed with radiometric methods such as γ -spectrometry, β-counting and α-spectrometry. The radiometric method is still a useful tool to determine concentrations of anthropogenic radionuclides, although recently mass spectrometric methods have been developed extensively. Here we describe the radiometric methods to determine typical anthropogenic radionuclides such as 137 Cs, 90 Sr and plutonium in seawater, which includes recent development of the radiometric methods such as extremely low background γ -spectrometry at the Ogoya Underground Laboratory (OUL).

1. Introduction Huge amounts of anthropogenic radionuclides have been introduced into marine environments as global fallout from large-scale atmospheric nuclear-weapon testing, discharge from nuclear facilities and ocean dumping of nuclear wastes (UNSCEAR, 2000). The radiological and ecological effects of anthropogenic radionuclides are still of world concern. To assess the marine environmental effects of anthropogenic radionuclides, it is significant to clarify their behavior and fate in the marine environments. Therefore, concentrations of anthropogenic radionuclides in seawater are an important tool to evaluate the ecological effect of anthropogenic radionuclides. 137 Cs is one of the most important anthropogenic radionuclides in the field of environmental radioactivity because of its long physical half-life of 30.0 years. It is a major fission product (fission yield: 6–7%) from both plutonium and uranium (UNSCEAR, 2000). 137 Cs in the ocean has been mainly derived from global fallout (Reiter, 1978; Bowen et al., 1980; UNSCEAR, 2000; Livingston and Povinec, 2002), together with close-in fallout from the ∗ Corresponding author. E-mail address: [email protected]



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Pacific Proving Ground nuclear explosions (Bowen et al., 1980; Livingston and Povinec, 2002), discharge of radioactive wastes from nuclear facilities and others (Sugiura et al., 1975; Pentreath, 1988; Hirose et al., 1999). 137 Cs in seawater has been determined since 1957 to elucidate the radioecological effects of anthropogenic radioactivity in the marine environment (Miyake and Sugiura, 1955; Rocco and Broecker, 1963; Shirasawa and Schuert, 1968; Saruhashi et al., 1975; Bowen et al., 1980; Folsom, 1980; Nagaya and Nakamura, 1987a, 1987b; Miyake et al., 1988; Hirose et al., 1992; Aoyama and Hirose, 1995; Hirose et al., 1999; Aoyama et al., 2000, 2001; Aoyama and Hirose, 2003; Hirose and Aoyama, 2003a, 2003b; Ito et al., 2003; Povinec et al., 2003, 2004; Hirose et al., 2005). Additionally, 137 Cs in seawater is a powerful chemical tracer of water mass motion on the time scale of several decades (Bowen et al., 1980; Folsom, 1980; Miyake et al., 1988; Miyao et al., 2000; Tsumune et al., 2001, 2003a, 2003b) because most of the 137 Cs in water columns is present in dissolved form. Another advantage of the use of 137 Cs as an oceanographic tracer is the quantity and accessibility of marine radioactivity during the past four decades in contrast with other chemical tracers such as CFCs (Warner et al., 1996). Another important fission product is 90 Sr, which is a β-emitter with a half-life of 28.5 years. It has been believed that the oceanic behavior of 90 Sr is very similar to that of 137 Cs because both 137 Cs and 90 Sr, a typical non-particle-reactive element, exist as ionic forms in seawater. In contrast to 137 Cs, measurements of 90 Sr in seawater are generally inconvenient because of complicated radioanalytical processes. However, the behavior of 90 Sr in the ocean differs from that of 137 Cs; for example, a significant amount of 90 Sr has been introduced to the ocean via river discharge in contrast to 137 Cs (Livingston, 1988), which is tightly retained in soil mineral surfaces. In fact, the 90 Sr/137 Cs ratios in oceanic waters have varied spatially and temporally. 90 Sr may be considered to be a tracer of the effect of river discharge to the ocean. These findings suggest that there is a need independently to determine 90 Sr in seawater. Plutonium in the ocean has been introduced by global fallout from large-scale atmospheric nuclear weapons testing programs, from which the major oceanic input occurred in the early 1960s (Harley, 1980; Perkins and Thomas, 1980; Hirose et al., 2001a). However, 239,240 Pu inventories in water columns in the North Pacific are significantly greater than estimates based on global fallout deposition (Bowen et al., 1980). This additional source of plutonium in the North Pacific has been attributed to inputs from close-in fallout from the U.S. nuclear explosions conducted at the Pacific Proving Grounds in the Marshall Islands in the 1950s (Bowen et al., 1980; Livingston et al., 2001). In order to have better understanding of the oceanic behavior of 239,240 Pu, it is important to elucidate the effect of close-in fallout of 239,240 Pu in the western North Pacific. Plutonium in the ocean is transported by physical and biogeochemical processes. The residence time of plutonium in surface waters of the open ocean ranges from 6 to 21 years and is generally shorter than that of the corresponding surface 137 Cs (Hirose et al., 1992, 2001b, 2003). In contrast to 137 Cs, plutonium is a typical particle-reactive radionuclide; plutonium in particulate matter of surface waters comprises from 1 to 10% of the total (Hirose et al., 2001b; Livingston et al., 1987), whereas particulate 137 Cs is less than 0.1% of the total (Aoyama and Hirose, 1995). Plutonium vertically moves with sinking biogenic particles (Fowler et al., 1983; Livingston and Anderson, 1983) and regenerates into soluble forms in deep waters as a result of microbial decomposition of particles. These biogeochemical processes typically produce vertical concentration profiles of plutonium which show a surface minimum, a mid-depth

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maximum and, thereafter, a decrease with increasing water depth (Hirose, 1997; Tsumune et al., 2003a, 2003b). Although plutonium moves vertically by biogeochemical processes, most of the plutonium introduced into the Pacific Ocean still exists in the water column; plutonium inventories in sediments of the open ocean are generally less than 10% of the total inventories except in sea areas near Bikini and Enewetak (Livingston et al., 2001). However, plutonium behavior in oceanic water is further affected by physical processes such as advection and upwelling (Hirose et al., 2002). For example, the observation that the plutonium maximum layer in the mid-latitude region of the North Pacific has deepened with time (Livingston et al., 2001) is explained by a simple one-dimensional biogeochemical model (Hirose, 1997), but does not provide a mechanism for reducing the overall water-column inventory. Consequently, advection must play a significant role in determining the temporal trends in regional oceanic inventories of plutonium. The North Pacific deep waters (more than 2000 m depth) contain a significant amount of plutonium (Bowen et al., 1980), however, the input processes are still unknown. In this paper, we describe traditional radiometric methods of 137 Cs, 90 Sr and plutonium assay in seawater and exhibit typical recent results on the oceanic behaviors of 137 Cs, 90 Sr and plutonium. 2. Analytical method for 137 Cs analysis in seawater 2.1. Background 137 Cs

decays to 137 Ba by emitting β-rays (188 keV) and γ -rays (661.7 keV). Cs, that exists in ionic form in natural water, is one of the alkali metals and chemically shows less affinity with other chemicals. The concentration of stable Cs in the ocean is only 3 nM. Known adsorbents to collect Cs in seawater are limited, e.g., ammonium phosphomolybdate (AMP) and hexacyanoferrate compounds (Folsom and Sreekumaran, 1966; La Rosa et al., 2001). The AMP has been an effective ion exchanger of alkali metals (Van R. Smit et al., 1959). It has been known that AMP forms insoluble compound with Cs. AMP, therefore, has been used to separate other ions and concentrate Cs in environmental samples. In the late 1950s, determination of 137 Cs in seawater was carried out with β-counting because of underdevelopment of γ -spectrometry. Two to several ten mg of Cs carrier is usually added when the radiocesium is determined by β-counting because of the formation of a precipitate of Cs2 PtCl6 and calculation of chemical yields of cesium throughout the procedure (Yamagata and Yamagata, 1958; Rocco and Broecker, 1963). After the development of γ -spectrometry using Ge detectors, the AMP procedure with γ -spectrometry became a convenient concentration procedure for the determination in environmental samples. In Japan, the AMP method is recommended for radiocesium measurements in seawaters, in which Cs carrier is stated to be unnecessary (Science and Technology Agency, 1982). It, however, must be noted that large volumes of seawater samples (more than 100 liters) were required to determine 137 Cs because of the relatively low efficiency of Ge-detectors (around 10%). In the previous literature, the weight yield of AMP has not been used because the chemical yield of Cs could be obtained and the loss of a small amount of AMP during the treatment did not cause serious problems. Actually, the use of AMP reagent produced in the 1960s and

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Table 1 Performance of well-type Ge detectors operated in a ground-level lab (MRI) and in a underground lab (Ogoya) Institute

Type

Active volume (cm3 )

Absolute efficiencya (%)

Backgroundb (cpm/1 keV)

MRI

ORTEC 6 7 8 9 Canberra EYRISYS

280 80 280 600 199 315

20.5 10.8 16.5 23.7 14.5 20

0.092 0.033 0.109 0.074 0.0003 0.0016

Ogoya

a The absolute efficiencies of HPGE are calculated at the 660 keV photo-peak of 137 Cs. b The background values were calculated as a sum from 660 to 664 keV corresponding to the 662 keV photo-peak of 137 Cs.

in the mid 1980s gave a range from 70% to 90% as the weight yields of AMP without Cs carrier in laboratory experiments in 1996. These weight yields are in good agreement with the records of weight yields of AMP in our laboratory during the 1970s and 1980s. However, the weight yield of AMP without Cs carrier had been decreasing from the end of the 1980s and it sometimes became very low, less than 10%, in the mid 1990s. To improve 137 Cs determination in seawater, Aoyama et al. (2000) re-examined the AMP procedure. Their experiments revealed that stable Cs carrier of the same equivalent amount as AMP is required to form the insoluble Cs-AMP compound in an acidic solution (pH = 1.2 to 2.2). The improved method has achieved high chemical yields of more than 95% for sample volumes of less than 100 liters. Another improvement is to reduce the amount of AMP from several tens of grams to 4 g to adsorb 137 Cs from the seawater samples. As a result, the sample volume has been reduced from around 100 liters to less than 20 liters to be able to use high-efficiency well-type Ge-detectors (Table 1). This improvement for 137 Cs is favorable to its use as a tracer in the oceanographic field. However, there has been a serious problem regarding 137 Cs measurement; i.e., large-volume sampling of more than 100 liters has been required to determine 137 Cs concentrations in deep waters because of the very low concentrations of 137 Cs (less than 0.1 Bq m−3 ). A major problem not improving sensitivity is that high-efficiency well-type Ge-detectors result in higher backgrounds during γ -spectrometry in ground-level laboratories. Especially, it is difficult to determine accurate 137 Cs concentrations in deep waters (>1000 m) because of the difficulty of acquiring large, non-contaminated samples. However, recently, Komura (Komura, 2004; Komura and Hamajima, 2004) has established an underground facility (Ogoya Underground Laboratory: OUL) to achieve extremely low background γ -spectrometry using Ge detectors with high efficiency and low background materials. The OUL has been constructed in the tunnel of the former Ogoya copper mine (235 m height below sea level, Ishikawa prefecture) in 1995 by the Low Level Radioactivity Laboratory, Kanazawa University. The depth of the OUL is 270 m water equivalent and the contributions of muons and neutrons are more than two orders of magnitude lower than those at ground level. In order to achieve extremely low background γ -spectrometry, high efficiency well type Ge detectors specially designed for low level counting were shielded with extremely


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low background lead prepared from the very old roof tiles of Kanazawa Castle. As a result, the background of γ -rays corresponding to the energy range of 137 Cs is two orders of magnitude lower than that in ground-level facilities as shown in Table 1. The detection limit of 137 Cs at the OUL is 0.18 mBq for a counting time of 10,000 min (Hirose et al., 2005). There is a residual problem of underground γ -spectrometry for 137 Cs measurements. AMP adsorbs trace amounts of potassium when Cs is extracted from seawater because K is a major component in seawater and radioactive potassium (40 K) contains 0.0118% of total K in natural materials. Trace amounts of 40 K cause elevation of the background corresponding to the energy range of 137 Cs due to Compton scattering of 40 K. If 40 K can be removed in AMP/Cs compound samples, the full performance of underground γ -spectrometry for 137 Cs measurements is established. To remove 40 K from the AMP/Cs compound, a precipitation method including the insoluble platinate salt of Cs was applied for purification of Cs. This method performed to remove trace amounts of 40 K from the AMP/Cs compound has a chemical yield of around 90% for 137 Cs (Hirose et al., 2006b). 2.2. Method 2.2.1. Sampling and materials Seawater samples are collected using a CTD-rosette sampler, which collects seawater of 12 liters volume at each of 24–36 different depth layers. All reagents used for 137 Cs, 90 Sr and Pu assay are special (G.R.) grade for analytical use. All experiments and sample treatments are carried out at ambient temperatures. It is very important to know the background γ activities of reagents. The 137 Cs activity in CsCl was less than 0.03 mBq g−1 using extremely low background γ -spectrometry. The 137 Cs activity in AMP was less than 0.008 mBq g−1 . There is no serious contamination of 137 Cs from other reagents. 2.2.2. γ -Spectrometry 137 Cs measurements were carried out by γ -spectrometry using well-type Ge detectors coupled with multi-channel pulse height analyzers. The performance of the well-type Ge detectors is summarized in Table 1. Detector energy calibration was done using IPL mixed γ -ray sources, while the geometry calibration was done using the “reference” material with the same tube and very similar density. 2.2.3. Recommended procedure We propose an improved AMP procedure with the ground-level γ -spectrometer as follows: (1) Measure the seawater volume (5–100 L) and put into a tank of appropriate size. (2) pH should be adjusted to 1.6–2.0 by adding concentrated HNO3 (addition of 40 mL conc. HNO3 for 20 L seawater sample makes pH of sample seawater about 1.6). (3) Add 0.26 g of CsCl to form an insoluble compound and stir at a rate of 25 L per minute for several minutes. (4) Weigh 4 g of AMP and pour it into a tank to disperse the AMP with seawater. (5) 1 h stirring at a rate of 25 L air per minute. (6) Settle until the supernate becomes clear. The settling time is usually 6 h to overnight, but no longer than 24 h.

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(7) Take an aliquot of 50 mL supernate to calculate the amount of the residual cesium in the supernate. (8) Loosen the AMP/Cs compound from the bottom of the tank and transfer into a 1–2 L beaker; if necessary, do additional step of decantation. (9) Collect the AMP/Cs compound onto 5B filter by filtration and wash the compound with 1 M HNO3 . (10) Dry up the AMP/Cs compound for several days at room temperature. (11) Weigh the AMP/Cs compound and determine weight yield. (12) Transfer the AMP/Cs compound into a Teflon tube of 4 mL volume and subject to γ -ray spectrometry. 2.2.4. Underground γ -spectrometry (1) The same procedure from step (1) to step (12). (2) Dissolve AMP/Cs compound by adding alkali solution. (3) pH should be adjusted to ca. 8.1 by adding 2 M HCl and adjust the volume of solution to ca. 70–100 mL. (4) Perform precipitation of Cs2 PtCl6 by adding chloroplatinic acid (1 g/5 mL D.W.) at pH = 8.1 and keep in refrigerator for a half-day. (5) Collect the Cs2 PtCl6 precipitate onto a filter and wash the compound with solution (pH = 8.1). (10) Dry the Cs2 PtCl6 precipitate for several days at room temperature. (11) Weigh the Cs2 PtCl6 precipitate and determine the weight yield. (12) Transfer the Cs2 PtCl6 precipitate into a Teflon tube of 4 mL volume and subject to underground γ -spectrometry. 2.3. Application to ocean samples 2.3.1. Database Marine radioactivity databases have been constructed in several institutes (e.g., IAEA, Marine Environmental Laboratory, Monaco (GLOMARD database, IAEA, 2000; Povinec et al., 2004, 2005), Meteorological Research Institute, Japan (HAM database, Aoyama and Hirose, 2004)). Aoyama and Hirose (2004) have been preparing a comprehensive database (HAM) including 137 Cs, 90 Sr and 239,240 Pu concentrations in seawater of the Pacific and marginal seas. The present total numbers of 137 Cs, 90 Sr and 239,240 Pu concentrations in the HAM database are 7737, 3972 and 2666 records, respectively. The numbers of surface 137 Cs, 90 Sr and 239,240 Pu concentrations in the HAM database are 4312, 1897 and 629 records, respectively. For data analysis, we carried out no special quality controls except removal of 239,240 Pu data in highly contaminated areas (the Pacific Proving). 2.3.2. Surface 137 Cs in the ocean The level of the 137 Cs concentrations in surface waters of the world ocean has been summarized in the IAEA-MEL research project (Livingston and Povinec, 2002; Povinec et al., 2005). The higher 137 Cs concentrations in the surface waters in 2000, ranging from 10 to 100 Bq m−3 , occurred in the Baltic Sea, Irish Sea, Black Sea and Arctic ocean, which reflects the effect of the Chernobyl fallout as does the discharge of radioactive wastes from the


143

Fig. 1. Schematic map of Pacific boxes.

Sellafield nuclear reprocessing plants. The level of surface 137 Cs in 2000 ranged from 1 to 10 Bq m−3 for most of the ocean areas in the world such as the Pacific and Indian Oceans, except the South Atlantic and Antarctic Oceans, whose 137 Cs concentrations was less than 1 Bq m−3 . The temporal change of chemical tracers can provide significant information on the physical and biogeochemical processes in the ocean. However, we have no complete data set of anthropogenic radionuclides temporally and spatially in the Pacific because only a few current data are present and the geographic distribution of sampling stations is heterogeneous. In order to elucidate the present levels of anthropogenic radionuclides in the Pacific, it is important to examine the temporal change of concentrations of anthropogenic radionuclides in surface waters. The temporal changes of 137 Cs and 239,240 Pu concentrations in surface waters differ for the various sea areas of the Pacific (Aoyama and Hirose, 1995; Hirose et al., 2001a). Therefore, the Pacific Ocean basin is divided into twelve boxes as shown in Figure 1, taking into account the latitudinal and longitudinal distributions of anthropogenic radionuclides (Hirose and Aoyama, 2003a, 2003b), the oceanographic knowledge and the latitudinal distribution of global fallout (UNSCEAR, 2000; Hirose et al., 1987, 2001b). Box 1 (north of 40◦ N) is the Subarctic Pacific Ocean, in which greater depositions of 137 Cs, 90 Sr and 239,240 Pu occurred in the 1960s (UNSCEAR, 2000; Hirose et al., 1987, 2001b). Boxes 2 and 3 (25◦ N– 40◦ N) are upstream and downstream of the Kuroshio extension, respectively, which correspond to the mixing area between the water masses of the Kuroshio and Oyashio currents. The greatest deposition of 137 Cs, 90 Sr and 239,240 Pu throughout the Pacific occurred in Box 2 in the 1960s. Boxes 4 and 5 (5◦ N–25◦ N) are downstream and upstream of the North Equatorial current, which is a typical oligotrophic ocean corresponding to the subtropical gyre. Box 5 includes the California current. These boxes include the highly radioactivitycontaminated islands (Bikini, Enewetak, Johnston islands and others) of the Pacific Proving

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Fig. 2. Calculated concentrations of 137 Cs in surface waters of the Pacific boxes. Small numbers show the observed values, which were obtained in the year indicated in parentheses.

ground nuclear weapons test sites, especially typical for plutonium (Livingston et al., 2001; Buesseler, 1997). The subtropical region of the western North Pacific was affected by close-in fallout compared with that in the eastern North Pacific. Boxes 6 and 7 (5◦ S–5◦ N) are downstream and upstream of the South Equatorial current, respectively. The boundary between Boxes 4, 5 and 6, 7 is approximately the Equatorial Counter current. Box 7 includes the equatorial upwelling region. Boxes 8 and 9 (5◦ S–25◦ S) are downstream and upstream of the weak South Equatorial current, respectively. Box 9 includes the French nuclear weapons test site (Mururoa). Box 10 (25◦ S–40◦ S) corresponds to the Tasman Sea. Box 11 (25◦ S–40◦ S) is the mid-latitude region of the South Pacific. Box 12 (40◦ S–60◦ S), including the Indian Ocean, is the Southern Ocean corresponding to the Antarctic circumpolar current. Temporal changes of 137 Cs concentrations in the surface waters of each box were examined using the HAM database. Surface 137 Cs decreased exponentially throughout the period from 1971 to 1998. Similar exponential decreases of surface 137 Cs occurred in other boxes of the North Pacific (Hirose and Aoyama, 2003a, 2003b). Although the observed data are scant in the South Pacific, it is expected that surface 137 Cs decreased exponentially after 1971 because the major input due to global fallout occurred in the mid 1960s and the major French nuclear weapons tests were conducted until 1971 (UNSCEAR, 2000). The concentrations of surface 137 Cs in 2000 can be extrapolated from the regression curves of the time-series data in each box. However, the confidence levels of the regression curves vary greatly between the boxes because the numbers of data differ significantly among the boxes. In the North Pacific, we obtain the estimated concentrations of anthropogenic radionuclides with higher quality. The calculated 137 Cs concentrations in surface waters of each box in 2000, together with the most recent observed data, are shown in Figure 2. The calculated 137 Cs concentrations in surface waters of each box in 2000 ranged from 1.2 to 2.8 Bq m−3 . The 137 Cs concentrations in the North Pacific surface waters are still higher than those in the South Pacific, although about four decades have passed since the major injection of 137 Cs in the Northern Hemisphere.


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Fig. 3. Section of 137 Cs concentration (Bq m−3 ). Dots show positions of samples. Measurements of about sixty samples in the southern part in this section have not been completed yet.

Higher surface 137 Cs concentration occurs in the eastern Equatorial Pacific (Box 7). The lower values appear in the Tasman Sea (Box 10). It, however, is noteworthy that the surface 137 Cs concentration is homogeneous in the Pacific compared with those in the 1960s and 1970s. 2.3.3. Cross-section of 137 Cs along 165◦ E in the Pacific The cross-section of 137 Cs concentrations obtained from the Ryofu-Maru 2002 cruises along 165 deg. E is presented in Figure 3. A remarkable core with high 137 Cs concentrations in the 137 Cs cross-section which exceeded 2 Bq m−3 was found (Aoyama et al., 2004). This 137 Cs core occurred around 20◦ N, corresponding to the southern part of the subtropical gyre in the Northern Pacific. The depth of the 137 Cs core was between 200 and 500 m. The 137 Cs concentrations in the surface waters were 1.5 Bq m−3 or less. We also found a subsurface maximum at 100–150 m south of 15◦ N. The 137 Cs concentration in deep waters below 1500 m ranged around 0.01–0.1 Bq m−3 in the northern part of this section. In terms of density, the core of 137 Cs (200–500 m depth) corresponds to a sigma of 25–26.2 or 3. We can consider the gyre-scale subduction as a potential cause of the observed 137 Cs core. The subduction is a principal upper ocean ventilation process. The water masses related to subduction are characterized through the exchange of heat, moisture, and dissolved gases and transient tracers such as 137 Cs in our study. After the water masses were formed and transferred beneath the mixed layer, they are shielded by the upper water masses and subsequently their properties modified by mixing in the ocean interior. At present, two opportunities for the subduction in the North Pacific can be considered. One is a lighter variety of central mode water with a density of sigma around 26.0, whose outcrop occurs in the central North Pacific (35◦ –40◦ ) in winter. Another is a denser variety of central mode water with density of 26.4, whose outcrop occurs in the central northern North Pacific (>40◦ N). Overlaying the 137 Cs contour onto a potential vorticity section reveals that the 137 Cs core corresponds to a southern part of lower potential vorticity at density level of 26.0. The lower potential vorticity reflects the property of the origin of the water mass corresponding to a lighter variety of the central mode water. For the density corresponding to the denser variety of the central mode water, there was no marked signal in the 137 Cs concentrations. Comparing with previous data, the

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137 Cs concentrations in this core seem to be still increasing. Another typical feature is that the

subsurface 137 Cs maximum occurred in the 100–150 m depth layer around 20◦ N in the western North Pacific, which phenomenon was observed in previous papers (Bowen et al., 1980; Hirose and Aoyama, 2003a). The subsurface 137 Cs maximum corresponds to the North Pacific subtropical mode water (NPSTMW) with the 25.4 isopycnal, whose winter outcrop occurs around 35◦ N. The NPSTMW is transported south westward accompanied by subductions. Southward transport of the NPSTMW might control a longer half-residence time of the 137 Cs surface water concentrations and an increasing tendency of the 137 Cs water column inventory in the equatorial region (Aoyama and Hirose, 2003; Hirose and Aoyama, 2003a). The 137 Cs concentration in surface waters of the NW Pacific has decreased steadily to present levels between 2 and 4 Bq m−3 since the early 1960s. In a previous study (Saruhashi et al., 1975) a typical latitudinal distribution of 137 Cs was observed, which was high in midlatitudes (10–20 Bq m−3 ) and low in the Equatorial region (around 5 Bq m−3 ). The differences of the 137 Cs concentrations in the surface waters between the mid-latitudes and the Equatorial region have decreased over the past two decades. In the Equatorial region, the 137 Cs concentration in the surface waters showed no temporal change apart from radioactive decay (Aoyama and Hirose, 1995; Aoyama et al., 2001). Therefore, the current geographical distribution of surface 137 Cs in the NW Pacific reflects that of global fallout input re-distributed by surface/subsurface circulation in the North Pacific which includes subduction and equatorial upwelling processes. 3. Analytical method for 90 Sr 3.1. Background 90 Sr

decays to stable 90 Zr via 90 Y (half-life: 2.67 day; maximum energy of β-particles: 2.28 MeV). Strontium comprises about 0.025% of the Earth’s crust and the concentration of stable Sr in the ocean is 8.7 × 10−5 M. It is widely distributed with calcium. The chemistry of strontium is quite similar to that of calcium, of which the concentration in the ocean is 10−2 M. The biological behavior of strontium in the ocean is also very close to that of calcium, and thus the behavior of Sr in the ocean can be considered to be different from that of Cs. The only radiometric method for 90 Sr is β-counting. Therefore, radiochemical separation is required for determination of 90 Sr. An essential step in 90 Sr analytical methodologies is the separation and purification of the strontium, both to remove radionuclides which may interfere with subsequent β-counting and to free it from the large quantities of inactive substances typically present, i.e., calcium in seawater. In the 1950s, the oxalate technique was used to separate Sr and Ca (Miyake and Sugiura, 1955). They applied the fuming HNO3 method for purification of Sr. On the other hand, carbonate techniques (Sugihara et al., 1959) were used for 90 Sr determination for the Atlantic samples in the 1950s. Shirasawa and Schuert (1968) used similar procedures to those developed by Rocco and Broecker (1963) in which the oxalate technique was adapted. During the GEOSECS period, the oxalate technique was applied for 90 Sr determination in the world’s oceans (Bowen et al., 1980). Classical methods for the separation of strontium from calcium rely upon the greater solubility of calcium nitrate in


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fuming nitric acid. Those procedures require numerous steps including repeated precipitation in strong nitric acid. Therefore, various alternative methods for separation have been proposed; precipitation methods of strontium sulfate and strontium rhodizonate (Weiss and Shipman, 1957), sorption of strontium on an ion-exchange resin from a solution of chelating agent such as CyDTA and EDTA (Noshkin and Mott, 1967). These methods, however, had not improved into shorter analytical procedures because the precipitation and extraction methods yield strontium fractions containing significant amounts of calcium. In the late 1970s, Kimura et al. (1979) proposed the use of macrocyclic polyethers for the separation of strontium and calcium. In the 1990s, extraction chromatography, using a solution of 4,4 (5 )-bis(tertbutylcyclohexano)-18-crown-6 in 1-octanol sorbed on an inert substrate, has been developed for the separation of strontium and calcium (Horwitz et al., 1990). Recently membrane filter coating crown ether was developed for separation of strontium from others (Lee et al., 2000; Miró et al., 2002). These modern techniques have contributed to down-sizing to small volumes of samples. On the other hand, large volumes of seawater have been used for determination of 90 Sr in seawater due to the low concentrations of 90 Sr. Therefore, the current practical method for separation and purification of 90 Sr in seawater still contains precipitation at the first step. There are some problems in the current practical methods using the carbonate technique; one is a lower recovery of Sr with a range from 30 to 60% and another is a long radiochemical separation. Since the radioactivity of 90 Sr in seawater is even lower in the surface water at present, improvement of the Sr recovery is one of the key issues for determination of 90 Sr activity in seawater. Another key point is that the Ca/Sr ratio in the carbonate precipitates is remarkably reduced from that in seawater. To improve 90 Sr determination in seawater, we re-examined the Sr separation technique for seawater samples. 3.2. Method 3.2.1. Sampling and materials Since 90 Sr concentrations in seawater are low, large volumes (50 to 100 L) of seawater are still required for assay of 90 Sr. Surface water samples of 100 L were collected with a pumping system on board. Deep water samples were collected with 100 L GoFlo type samplers. All of the water samples were filtered through a fine membrane filter (Millipore HA, 0.45 µm pore size) immediately after sampling. 90 Sr was assayed as 90 Y using β-counting following the radiochemical separation described in detail as follows: 3.2.2. Pre-concentration of 90 Sr The co-precipitation method has generally been used for extracting Sr from large volumes of seawater. Both the oxalate and carbonate techniques have been applied for pre-concentration of Sr from large volume water samples. The pre-concentration of 90 Sr with carbonate is performed by adding 500 g NH4 Cl and 500 g Na2 CO3 to 100 L seawater. To improve the recovery of 90 Sr using the carbonate technique, it is essential to remove Mg as hydroxide at pH = 12 from the sample seawater before performing the carbonate precipitation. 3.2.3. Radiochemical separation In the first step of the radiochemical separation, Sr is separated from calcium as an oxalate precipitate at pH = 4. After dissolution of the oxalate precipitate, Ra and Ba are removed

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with Ba chromate. After Ra and Ba are removed as precipitates, Sr is recovered as a carbonate precipitate. Further purification of Sr is carried out using fuming nitric acid to remove Ca. After 90 Sr–90 Y equilibrium has been attained, co-precipitation of 90 Y with ferric hydroxide is carried out and the solid is then mounted on a disk for counting. 3.2.4. β-Counting β-Counting of 90 Y is carried out by gas proportional counting with external solid samples. A typical efficiency of 90 Y counting by gas proportional counting with a thin window is ca. 40%. The detection limit of 90 Y is several mBq when the counting time is 360 min. 3.3.

90 Sr

concentration in ocean surface water

The distribution of 90 Sr concentrations in surface waters of the world ocean has been summarized in the IAEA-MEL research project (Livingston and Povinec, 2002; Povinec et al., 2005). The higher 90 Sr concentrations in the surface waters, ranging from 10 to 100 Bq m−3 , occurred in the Irish Sea, Black Sea and Baltic Sea, which reflects the effect of the Chernobyl fallout and the discharge of radioactive wastes from the Sellafield nuclear reprocessing plants. The level of surface 90 Sr ranged from 1 to 10 Bq m−3 for most of the ocean areas in the Northern Hemisphere, such as the North Pacific and North Atlantic Oceans, while for the oceans in the Southern Hemisphere such as the South Atlantic and Antarctic Oceans, 90 Sr concentrations were less than 1 Bq m−3 . The 90 Sr concentrations measured in surface waters in 1997 were in the range of 1.5– 2.2 Bq m−3 (Povinec et al., 2003). The highest 90 Sr surface concentration was observed in the mid-latitude region of the NW Pacific at St. 3. The geographical distribution of 90 Sr in the NW Pacific surface waters in 1997 is homogeneous, as for 137 Cs. However, still lower surface 90 Sr was observed in the Equatorial region near 10◦ N. The difference between maximum and minimum surface 90 Sr concentrations in the NW Pacific is smaller than for 137 Cs. 3.4.

90 Sr

concentrations in the water column in the Pacific Ocean

90 Sr profiles as well as 137 Cs in the water columns of the central NW Pacific Ocean at IAEA’97

(Figure 4), as well as GEOSECS and Hakuho-Maru stations are very similar to those of 137 Cs (Povinec et al., 2003). In the Kuroshio Extension and subtropical regions, compared with 90 Sr, the 137 Cs concentrations are higher by a factor of about 1.6, which represents a typical global fallout ratio for these radionuclides, whereas significantly low 137 Cs/90 Sr activity ratios (around 1) occurred in the Equatorial Pacific near 10◦ N, as more 90 Sr attached to CaCO3 particles from the atolls was deposited in close-in fallout than for 137 Cs. Low 137 Cs/90 Sr ratios were also observed in soil and sediment samples collected in Bikini, Enewetak and Rongelap atolls (Robison and Noshkin, 1999). 90 Sr in the surface water has been transported to deeper water layers than 137 Cs due to the co-precipitation with Ca in carbonate materials, which are later dissolved in the deep ocean. In the western North Pacific, there was also a remarkable decrease in surface and sub-surface concentrations between the GEOSECS and IAEA samplings-from about 4 to 1.5 Bq m−3 for 90 Sr and from about 6.5 to 2 Bq m−3 for 137 Cs, respectively. As we have seen in Figures 4 and 6, the distribution patterns of 137 Cs and 90 Sr in the water column were quite different from 239,240 Pu, because these elements


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Fig. 4. 90 Sr profiles in the Pacific Ocean (using data of Povinec et al., 2003).

are soluble and conservative with much lower distribution coefficients (IAEA, 1985) than Pu. Therefore, most of the 137 Cs and 90 Sr deposited from global fallout into the ocean still remains in the water column with the highest concentrations in surface and sub-surface waters.

4. Analytical method for plutonium in seawater 4.1. Background Plutonium in the environments consists of a number of different isotopes: 238 Pu, 239 Pu, 240 Pu, and 241 Pu. 239 Pu and 240 Pu are the most abundant plutonium isotopes, and have radioactive half-lives of 24,110 and 6563 years, respectively. 238 Pu and 241 Pu, with radioactive half-lives of 87.7 and 14.1 years, respectively, are also important isotopes of plutonium despite having lower atom concentrations. All of the plutonium isotopes except 241 Pu are α-emitters; the αparticle energies of 238 Pu, 239 Pu and 240 Pu are 5.49, 5.10 and 5.15 MeV, respectively, while the maximum β-particle energy of 241 Pu is 5.2 keV. Therefore, α-spectrometry is effective for determining plutonium concentrations in seawater. However, the sum of 239 Pu and 240 Pu activities has generally been measured as the plutonium concentration because the separation between 239 Pu and 240 Pu is difficult using α-spectrometry due to the similar energy range of their α particles. On the other hand, measurements of 241 Pu were carried out by counting in situ growth 241 Am (half-life: 433 years). The activity of 241 Pu was calculated from the following equation # $−1 A(241 Pu) = A(241 Am){1 − T1/2,2 /T1/2,1 } exp(−λ1 t) − exp(−λ2 t) , (1) where A(241 Pu) and A(241 Am) are the activities of 241 Pu at separation time and ingrown 241 Am measured at re-count time, respectively, T 241 Pu and 1/2,1 and T1/2,2 the half-lives of 241 Am, respectively, and λ and λ the decay constants of 241 Pu and 241 Am, respectively. 1 2

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Recently, liquid scintillation spectrometers equipped with pulse height analysis and pulse shape analysis techniques have been developed. The low-level liquid scintillation counting system allows us to measure the spectra of both β- and α-emitters simultaneously, i.e., 241 Pu and 239,240 Pu, even at the low-level activity of environmental samples (Lee and Lee, 1999). However, the sensitivity of liquid scintillation counting is not enough to measure Pu isotopes in seawater due to the higher background of liquid scintillation counting (about 1 cpm). The chemistry of plutonium in natural waters is generally complicated because plutonium shows four valences (III, VI, V and VI) and forms complexes with inorganic and organic ligands (Coleman, 1965). Choppin and Morgenstern (2001) summarized the chemical behavior of plutonium in the aqueous environment. The most important chemical property is its oxidation state because many chemical processes, such as solubility, hydrolysis and others, are largely affected by valence. The oxidation state of plutonium affects its biogeochemical processes in the marine environment and analytical processes such as concentration and separation. The oxidation of plutonium depends on pH in solution; lower oxidation states are stabilized in acidic solution, whereas higher oxidation states are more stable forms in alkaline solution. The reactivity of Pu(IV) and Pu(III) is generally greater than that of Pu(V) and 2+ Pu(VI) because Pu(V) and Pu(VI) exist as PuO+ 2 and PuO2 , respectively. However, Pu(IV) in natural aquatic conditions is difficult to form complexes with organic ligands such as humic acids because the free Pu(IV) ion concentration is extremely low due to strong hydrolysis. Speciation studies (Nelson and Lovett, 1978; Nelson et al., 1984; Hirose and Sugimura, 1985; Fukai et al., 1987) revealed that the dominant species of Pu dissolved in seawater are oxidized forms (V and/or VI). Orlandini et al. (1986) and Choppin (1991) showed that Pu(V) dominates the oxidized fraction and demonstrated the instability of Pu(VI) in natural waters. In most natural systems the interesting oxidation states are Pu(IV) and Pu(V), while Pu(III) and Pu(VI) are generally insignificant. Taking into account hydrolysis and complexation with carbonate species, a dominant species of Pu(V) under the conditions of artificial seawater is PuO+ 2 whereas Pu(IV) exists as hydroxo complexes (Pu(OH)4 ) (Choppin and Morgenstern, 2001). On the other hand, the chemical form of Pu in marine particulate matter is an organic complex of Pu(IV) (Hirose and Aoyama, 2002). The 240 Pu/239 Pu atom ratio and 238 Pu/239,240 Pu and 241 Pu/239,240 Pu activity ratios in fallout, which depend on the specific nuclear weapons design and test yield, are variable (Koide et al., 1985). The global fallout average 240 Pu/239 Pu atom ratio is 0.18, based on aerosols, soil samples and ice core data (HASL, 1973; Krey et al., 1976; Muramatsu et al., 2001; Warneke et al., 2002). However, different nuclear test series can be characterized by either higher or lower ratios; i.e., fallout from Nagasaki, the Nevada test site and the Semipalatinsk test site is characterized by generally lower 240 Pu/239 Pu ratios, 0.042, 0.035 as an average value, and 0.036, respectively (Komura et al., 1984; Hicks and Barr, 1984; Buesseler and Scholkovitz, 1987; Yamamoto et al., 1996), whereas elevated 240 Pu/239 Pu ratios (0.21–0.36) have been measured in soil samples from the Bikini atoll (Hisamatsu and Sakanoue, 1978; Muramatsu et al., 2001). Plutonium isotope signatures, therefore, are a useful tool to identify the sources of plutonium and to have a better understanding of their behavior in the marine environments. Originally the plutonium isotope signatures in environmental sample were determined by thermal ionization mass spectrometry (Krey et al., 1976; Buesseler and Halverson, 1987). Recent development of Inductively Coupled Plasma Mass Spectrometry (ICP-MS) has provided more data on the 240 Pu/239 Pu atom ratio in marine sam-


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Fig. 5. Calculated concentrations of 239,240 Pu in surface waters of the Pacific box. Small numbers show observed values, which were obtained in the year indicated in parentheses.

ples (Kim et al., 2002; Kim et al., 2003, 2004). By analyzing the 240 Pu/239 Pu atom ratios in seawater samples, Buesseler (1997) revealed that there is a significant contribution of close-in fallout from the Pacific Proving Ground nuclear testing to the North Pacific deep water. The Pu isotope ratios such as the 240 Pu/239 Pu atom ratio and 238 Pu/239,240 Pu activity ratio were used to estimate the contribution of close-in fallout Pu to total Pu water column inventory in the western North Pacific (Hirose et al., 2006a). 4.2. Method 4.2.1. Sampling Pu concentrations in seawater are extremely low (see Figure 5). Large volumes (50 to 500 L) of seawater are still required for assay of plutonium as for americium too. Surface water samples of 100 to 500 L were collected with a pumping system on board. Deep water samples were collected with 100 L GoFlo type samplers. All water samples were filtered through a fine membrane filter (Millipore HA, 0.45 µm pore size) immediately after sampling. Both dissolved and particulate plutonium were assayed using α-spectrometry following radiochemical separation using anion exchange resin, described in detail as follows: 4.2.2. Pre-concentration of Pu Co-precipitation methods have been generally used for extracting trace amounts of Pu from large volumes of seawater, which process is usually performed on-board. In the early days, co-precipitation of Pu with ferric hydroxide or bismuth phosphate was carried out (Bowen et al., 1971; Imai and Sakanoue, 1973). However, low and erratic recoveries of trace Pu in the environmental samples led Hodge et al. (1974) to propose a simple concentration method including the partial precipitation of (Mg, Ca) hydroxides/carbonates by the addition of small amounts of sodium hydroxide. Typically 34 mL of 50% NaOH solution was added to a 200 L

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seawater sample after addition of tracer into a rapidly stirred solution. The resulting milky mixture of (Mg, Ca) hydroxides/carbonates was stirred for 1 to 2 h and the fine precipitate allowed to settle overnight. After that, the clear water was removed from the container and the precipitate separated from the residual solution by centrifuging at 2500 rpm. A further pre-concentration method of Pu in seawater, co-precipitation with MnO2 has recently been proposed (La Rosa et al., 2001; Sidhu, 2003). After the seawater sample is acidified, 242 Pu is added as a yield tracer and KMnO4 is added to change the chemical forms of Pu to a dissolved form (Pu(VI)). After adding NaOH and MnCl2 solution to the solution, Pu co-precipitates with the bulky manganese dioxide at pH = 9. Finally the settled MnO2 precipitate is collected. Another pre-concentration method for Pu has been developed (Chen et al., 2001). A seawater sample of 200 L is acidified to pH = 2 with 12 M HCl (200 mL). After addition of ferric chloride (2 g), a known amount of tracer (242 Pu) and K2 S2 O5 (100 g), the solution is stirred for 1 h. In this stage, all of the Pu species in solution are reduced to Pu(III). Co-precipitation of Pu with ferric hydroxide occurs at pH = 10 by adding dilute NaOH solution (0.5–1 M). The ferric oxide precipitate formed contains small amounts of Ca(OH)2 and Mg(OH)2 . 4.2.3. Radiochemical separation Relatively large amounts of precipitates were obtained at the pre-concentration stage. Although a process to reduce precipitate is usually required, a simple separation method was proposed (Hodge et al., 1974; Hirose and Sugimura, 1985). Precipitates ((Mg, Ca) hydroxides/carbonates) were dissolved with 12 M HCl and added to bring the acid strength to 9 M (three times the dissolved materials). One drop of 30% H2 O2 was added for each 10 mL solution, and the solution was heated just below boiling for 1 h. After the solution had cooled, Pu was isolated by anion exchange techniques (Dowex 1-X2 resin, 100 mesh; a large column (15 mm of diameter and 250 mm long) was used). The sample solution was passed through the column and was washed with 50 mL 9 M HCl. At this stage, Pu, Fe and U were retained on the resin, whereas Am and Th eluted. Fe and U fractions were sequentially eluted with 8 M HNO3 . After elution of U, the column was washed with 5 mL 1.2 M HCl. Finally the Pu fraction was eluted with 1.2 M HCl (100 mL) containing 2 mL of 30% H2 O2 . The solution was dried on a hot plate. The chemical yield was around 70%. In order to remove the large amounts of MnO2 or Ca(OH)2 and Mg(OH)2 , additional coprecipitation techniques were performed (La Rosa et al., 2001). The MnO2 precipitates were separated and dissolved in 2 M HCl with excess NH2 OH HCl to reduce it to Mn(II). To scavenge Pu with ferric hydroxide, 50 mg of Fe(III) was added to the solution and oxidized from Pu(III) to Pu(IV) with NaNO2 . The Fe(OH)3 was formed by the addition of ammonium hydroxide to pH = 8–9. After briefly boiling, the pH in the solution was adjusted to 6–7, at which residual amounts of MnO2 changed to Mn(II). The Fe(OH)3 precipitate was separated from the supernatant solution by centrifugation. The precipitate was dissolved in 14 M HNO3 . Finally the acid concentration of the solution was adjusted to 8 M HNO3 . The solution was passed through the anion exchange column (Dowex 1 × 8, NO3 form; 10 mm diameter and 150 mm long), so that Pu and Th were adsorbed onto the resin, whereas Am, U and Fe remained in the effluent. The column was washed with 8 M HNO3 and the Th fraction was eluted with 10 M HCl. The Pu fraction was eluted with 9 M HCl solution containing 0.1 M NH4 I. The chemical recoveries were 40 to 60%.


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The Fe(OH)3 precipitates including Ca(OH)2 and Mg(OH)2 were dissolved with 12 M HCl and the acid concentration of the solution was about 0.1 M HCl (Chen et al., 2001). The solution was adjusted to pH = 9–10 with 6% ammonium hydroxide. The precipitates were settled and the supernatant solution was discarded. The precipitates were collected onto GF/A filter paper by filtration. The Ca2+ and Mg2+ were removed by repeating this precipitation process. The precipitates were dissolved in 12 M HCl and the acid concentration was adjusted to about 6 M. U, Po and Fe in the solution were removed by extraction with 80 mL 10% TIOA/xylene. After separation of the organic phase, 10 mg of Fe(III) was added to the aqueous solution and formed precipitates with ammonium solution at pH = 10. The precipitates were dissolved in 2–3 mL 12 M HCl. 50 mg NaNO2 was added to the solution. The HNO3 concentration in the solution was adjusted to 8 M with 14 M HNO3 . The solution was passed through an anion exchange column (10 mm diameter and 130 mm long, 100–200 mesh AG 1 × 4, NO3 form). The column was washed with 20 mL 8 M HNO3 and 50 mL 12 M HCl. The Pu fraction was eluted with 80 mL 2 M HCl containing 2 g NH2 OH HCl and 0.1 g NaCl. After the addition of 4 mL 14 M HNO3 , the eluate was evaporated to dryness. The chemical yield was 70 to 85%. Recently new types of extraction chromatographic resins, instead of anion exchange resins, have been proposed for radiochemical separation and purification of the actinides (Horwitz et al., 1990). Kim et al. (2002) have developed analytical techniques for Pu determination in seawater using ICP-MS, incorporating a sequential injection system including a TEVA-Spec (Eichrom Industries) column (6.6 mm diameter and 25 mm long) for radiochemical separation of Pu. TRU Resin™ (comprised of a 0.75 M solution of the bifunctional organophosphorus extractant octylphenyl-N ,N -di-isobutyl carbamoylphosphine oxide in tri-n-butyl phosphate immobilized on an inert porous polymeric resin) is used for separation of the actinides (Horwitz et al., 1990; Grate and Egorov, 1998; La Rosa et al., 2001; Sidhu, 2003). 4.2.4. Radiochemical separation of plutonium in particulate matter Filtered samples (100–1000 L) were digested with 14 M HNO3 and 12 M HCl in an appropriate beaker on a hot plate. 242 Pu was added as a tracer of the chemical yield. The solution was evaporated to dryness on a hot plate. The residue was dissolved with 8 M HNO3 and NaNO2 added on a hot plate. After filtration, the solution was passed through an anion exchange column (8 mm diameter and 150 mm long: Dowex 1 × 8, NO3 form). The column was successively washed with 20 mL 8 M HNO3 and 50 mL 9 M HCl. The Pu fraction was eluted with 100 mL of 1.2 M HCl including 1 mL of 30% H2 O2 . The chemical yield was more than 90%. 4.2.5. Electrodeposition Pu samples for α counting were usually electroplated onto stainless steel disks. The diameter of the stainless steel disk depends on the active surface area of the detectors. The electrodeposition was performed using an electrolysis apparatus with an electrodeposition cell consisting of a Teflon cylinder, a cathode of platinum and the anode of the stainless steel disk. The electrodeposition of Pu has been carried out at two conditions, i.e., in aqueous sulfuric acid medium (Talvitie, 1972) and in ethanol medium (Hirose and Sugimura, 1985). To allow homogeneous plating onto the disk, the disk surface has to be carefully cleaned by appropriate washing processes.

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The purified Pu fraction was dissolved in 7 mL of 0.05 M H2 SO4 and transferred into an electrodeposition cell using 3 mL of 0.05 M H2 SO4 solution three times. Two drops of methyl blue and 25% ammonium solution were added to the solution. The pH of the solution was adjusted to 2.5. The Pu was electroplated onto a stainless steel disk at 0.7 A/m2 for 4–5 h. The purified Pu fraction was dissolved in 1 mL 2 M HCl and transferred into an electrodeposition cell using 20 mL ethanol. The Pu was electroplated onto a stainless steel disk (30 mm diameter) at 15 V and 250 mA for 2 h. 4.2.6. α-Spectrometry The α-spectrometer consists of several vacuum chambers including a solid-state detector, a pulse height analyzer and a computer system. The detector, which is of the silicon surface barrier type (PIPS, energy resolution: 100 mBq m−3 ) occurred in the Irish Sea, in which the effect of the Sellafield discharge (Pentreath, 1988) has still continued. The relatively high 239,240 Pu concentrations in the surface waters (10–50 mBq m−3 ) have been observed in the northern North Atlantic, Barents Sea, North Sea, Mediterranean Sea and English Channel. The 239,240 Pu concentrations in the surface waters of the Pacific, Atlantic and other oceans ranged from 1 to 10 mBq m−3 except in the Antarctic Ocean, which is less than 0.5 mBq m−3 . The behavior of surface 239,240 Pu in the Pacific has been reexamined (Hirose et al., 1992; Hirose and Aoyama, 2003a, 2003b). The 239,240 Pu concentration in surface water has decreased exponentially since 1970 although the decrease rates depend on the sea area. The decrease rates for surface 239,240 Pu concentration are generally faster than those of 137 Cs, which can be explained by biogeochemical processes. The 239,240 Pu concentrations in surface waters of the Pacific in 2000 (Figure 5) are estimated to be in the range from 0.3 to 2.7 mBq m−3 , which spatial variation is larger than that of 137 Cs. Higher surface 239,240 Pu concentrations appear in the Subarctic Pacific (Box 1; see Figure 1) and the eastern equatorial Pacific (Box 7). Higher surface 239,240 Pu concentrations in the Subarctic Pacific can be explained by rapid recycling of scavenged 239,240 Pu due to large vertical mixing in the winter (Martin, 1985; Hirose et al., 1999), whereas surface 239,240 Pu in the eastern equatorial Pacific may be maintained at a higher level by equatorial upwelling (Quay et al., 1983). The lower values appear in the mid-latitude region of the eastern North Pacific (Box 3), which is consistent with the higher decrease rate of 137 Cs in the Box 3. Another point is that estimated surface 239,240 Pu concentrations in Boxes 2 and 4 of the western North Pacific are higher than those in Boxes 3 and 5 of the eastern North Pacific. Furthermore, there is only a small difference of surface 239,240 Pu concentrations between the Northern Hemisphere and Southern Hemisphere mid-latitude regions, although its causes are still


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unknown. As a result, in addition to physical processes, biogeochemical processes are significant in controlling the current geographic distribution of 239,240 Pu concentrations in surface waters of the Pacific (Bowen et al., 1980; Livingston and Anderson, 1983; Hirose et al., 1992; Hirose, 1997; Hirose et al., 2001a), taking into account the relatively short residence time of surface 239,240 Pu (Hirose and Aoyama, 2003a, 2003b). 4.3.2. Vertical profiles of 239,240 Pu in the ocean Figure 6 shows vertical profiles of 239,240 Pu observed in the mid-latitude and subtropical regions of the western North Pacific in 1997 (Povinec et al., 2003). All vertical profiles of 239,240 Pu in the ocean are characterized by a typical distribution pattern with surface minimum, subsurface maximum and gradual decrease with increasing depth (Bowen et al., 1980; Nagaya and Nakamura, 1984, 1987a, 1987b; Livingston et al., 2001; Hirose et al., 2001a). Maximum 239,240 Pu concentrations in the water column appeared at around 700 m depth (Povinec et al., 2003), which is consistent with those observed in the mid-latitudes of the North Pacific in the 1980s (Hirose et al., 2001a) and the early 1990s (Livingston et al., 2001). The shape of the 239,240 Pu peak depends on the sea area, that is, a rather broad peak of 239,240 Pu maximum has been observed in the Sea of Japan (Yamada et al., 1996; Hirose et al., 1999, 2002; Ito et al., 2003), and a smaller 239,240 Pu maximum compared with that in the North Pacific occurred in the South Pacific (Chiappini et al., 1999). It is reported that the subsurface maximum of 239,240 Pu in mid-latitudes of the North Pacific has moved into deeper water during the past two decades (Livingston et al., 2001). This pattern of vertical 239,240 Pu profiles and their temporal change has been demonstrated by particle-scavenging processes, the removal of 239,240 Pu by sinking particles following regeneration of dissolved 239,240 Pu due to the biological degradation of particles (Livingston and Anderson, 1983; Hirose, 1997). The water column inventory of 239,240 Pu in the western North Pacific was 100–130 Bq m−2 (Hirose et al., 2001a; Povinec et al., 2003), which is the same order of magnitude as previous values observed in the western North Pacific (Bowen et al., 1980; Livingston et al., 2001). However, although the sediment inventories of 239,240 Pu were not taken into account, this water column 239,240 Pu inventory is about three times larger than that estimated from global fallout (Hirose et al., 1987, 2001b; UNSCEAR, 2000). This finding suggests that most of the 239,240 Pu inventory in the water column in the western North Pacific has originated from sources other than global fallout, such as close-in fallout, which is consistent with the fact that close-in fallout from testing at the Pacific Proving Grounds is the most important source of plutonium in the North Pacific according to a plutonium isotope study (Buesseler, 1997). It is probable that the present water column inventories of 239,240 Pu are controlled by horizontal advection and the bottom topography of the North Pacific instead of deposition due to global fallout and close-in fallout, because the major deposition of 239,240 Pu originating from atmospheric nuclear weapon testing occurred in the 1950s and early 1960s. There is a possibility that the apparent inventories of plutonium depend on depth. Therefore, 2300 m depth inventories of 239,240 Pu have been investigated to elucidate the characteristics of the spatial distribution of the 239,240 Pu water column inventories (Livingston et al., 2001; Ikeuchi et al., 1999). We calculated the 2300 m 239,240 Pu inventories at two stations (27◦ 48 N 130◦ 44 E: 66 Bq m−2 , 27◦ 45 N 130◦ 45 E: 73 Bq m−2 ) of the Shikoku Basin (Ikeuchi et al., 1999). The water column inventory of 239,240 Pu in the ocean is calculated from the 239,240 Pu

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Fig. 6. Depth distribution of 239,240 Pu concentrations (dots) and 238 Pu/239,240 Pu activity ratios (squares) in seawater of the NW Pacific Ocean (IAEA’97 expedition data).


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vertical profile (Livingston et al., 2001). The higher 2300 m 239,240 Pu inventory was observed in the Sikoku Basin (Ikeuchi et al., 1999). The 2300 m 239,240 Pu inventories seem to be greater than those in mid-latitudes of the central North Pacific (35–53 Bq m−2 ) and the same order of magnitude as in the sea area near Bikini and Enewetak Atolls observed in 1997 (61 and 74 Bq m−2 ) (Livingston et al., 2001).

5. Sequential treatment of seawater samples It is cost-effective to use the same seawater sample for 137 Cs, Pu and 90 Sr determinations. In the first step, 137 Cs extraction by the AMP method described in Section 2.2 is performed. Then, Pu is extracted by the method described in Section 4.2. The last step is to perform carbonate precipitation to separate Sr from the seawater samples by the method described in Section 3.2.

6. Conclusion The analysis of anthropogenic radionuclides in seawater is still a challenging issue in the fields of oceanography and marine ecology because anthropogenic radionuclides are powerful tracers of ocean circulation and biogeochemical processes and are directly related to the effects on marine ecosystems. The analytical methods for anthropogenic radionuclides have been developed over the past four decades. In these studies, several kinds of analytical techniques have been proposed for each radionuclide; the major methods involve radiometric techniques such as γ -spectrometry, β-counting and α-spectrometry. Recently mass spectrometry such as ICP-MS and Accelerator Mass Spectrometry (AMS) and thermal ionization mass spectrometry (TIMS) have been developed for analysis of long-lived radionuclides such as plutonium (Buesseler and Halverson, 1987; Povinec et al., 2001). Pu analysis using sector field-ICPMS has succeeded in reducing sampling volumes to around 10 L of seawater (Kim et al., 2002). However, the radiometric methods are still more effective for 137 Cs and 90 Sr analyses in seawater. The analyses of anthropogenic radionuclides in seawater require large volumes of sample seawater and relatively long radiochemical separation and purification processes. Longer radiochemical processes cause lower and unstable chemical yields. The recent improvement of the 137 Cs analysis leads to higher and stable chemical yields (>90%) for a simple 137 Cs separation and concentration process. Therefore, the recommended procedure can be described for determination of 137 Cs in seawater. On the other hand, it is difficult to describe the recommended procedures for Pu and Sr analyses because of their lower and unstable chemical yields, although Pu analysis will be improved by using ICP-MS and small volumes of samples in future. Further studies to develop more convenient methodologies for Pu and Sr are needed. The comprehensive studies of radioactivity in the world-wide marine environment during the past decade have been achieved to have better understanding of the present levels of anthropogenic radionuclides in the world oceans and to establish a database of marine environmental radioactivity. These results reveal that anthropogenic radionuclides are a powerful tool to understand the decadal change of the oceans; that is, time-series and cross-section data

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have been used to verify the general ocean circulation model and biogeochemical modeling (Nakano and Povinec, 2003a, 2003b; Tsumune et al., 2003a, 2003b). Therefore, marine radioactivity studies should provide important information for understanding climate change and its resulting effects on the marine environment.

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Monte Carlo simulation of background characteristics of gamma-ray spectrometers— a comparison with experiment Pavel P. Povineca,b,∗ , Pavol Vojtylac , Jean-François Comanduccia a International Atomic Energy Agency, Marine Environment Laboratory, Monaco b Comenius University, Faculty of Mathematics, Physics and Informatics, Bratislava, Slovakia c European Organization for Nuclear Research (CERN), Geneva, Switzerland

1. Introduction Although the most effective way of increasing the factor of merit of a counting system is to increase counting efficiency and the amount of sample in a detection system, usually the only possible way is to decrease the detector background. Therefore in the past development of low level counting techniques investigations of detector background were the most frequent studies (Heusser, 1994, 1995; Povinec, 1994, 2004; Zvara et al., 1994). The background components of a typical low-level GE detector, not situated deep underground, are cosmic radiation (cosmic muons, neutrons and activation products of construction materials), radionuclides present in construction material (including the detector itself), radon and its progenies. For a present-day, carefully designed low-level HPGe spectrometer, the dominating background component is cosmic radiation, mainly cosmic muons (Heusser, 1995; Theodorsson, 1996; Vojtyla and Povinec, 2000; Laubenstein et al., 2004; Povinec, 2004). High energy cosmic particles are initiating a large number of complicated physical processes leading to background induction. In order to quantify the background, it is necessary to deal with rather complicated detector–shield assembles and physics. Obviously analytical solutions are not possible and the only way to solve this problem is Monte Carlo simulation of interaction processes. The development of a simulation code for background induction by cosmic rays is useful for optimization of a counting system in respect to its background characteristics. The prediction of detector background characteristics can be made before the system is built, that ∗ Corresponding author. Present address: Comenius University, Faculty of Mathematics, Physics and Informatics,

SK-84248 Bratislava, Slovakia. E-mail address: [email protected] RADIOACTIVITY IN THE ENVIRONMENT VOLUME 11 ISSN 1569-4860/DOI: 10.1016/S1569-4860(07)11005-6


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can optimize the system characteristics, and it can also save money during the construction. Systematic investigations of the influence of various parameters on the detector background can be carried out as well (Vojtyla and Povinec, 2006). Monte Carlo simulations of background have been facilitated by the development of physics software and availability of powerful computers. As the processes leading to background induction are very weak processes that occur with low efficiencies, large numbers of events have to be processed to obtain results with acceptable statistical uncertainties. In a single Ge gamma-ray spectrometer there is no protection against cosmic muons, therefore a spectrometer with anticosmic shielding will greatly reduce background levels. The anticosmic shield can be made of gas or plastic scintillation detectors, which surround the lead shield housing the HPGe detector (Heusser, 1994; Vojtyla et al., 1994; Heusser, 1995; Semkow et al., 2002; Povinec et al., 2004; Schroettner et al., 2004). Another possibility is to use an anti-Compton spectrometer, which is a powerful tool for reducing the detector’s background as it combines both anticosmic and anti-Compton background suppression (Debertin and Helmer, 1998; Povinec et al., 2004). The most sophisticated spectrometric system is, however, the multi-dimensional anti-Compton gamma-ray spectrometer (Cooper and Perkins, 1971; Povinec, 1982) in which signals from the analyzing detectors (nowadays HPGe) create three-dimensional spectra (volumetric peaks) which can contain both coincidence and anticoincidence events.

2. Sources of detector background For better understanding of cosmic ray induced background and its effect on counting sensitivity of the detection system we need to review sources of the detector background. The main sources of background in low-level counting systems are: (i) (ii) (iii) (iv) (v)

cosmic radiation environmental radioactivity outside of the counting system radioactivity of construction materials of the shielding radioactivity of construction materials of the detector radon and its progenies.

2.1. Cosmic radiation Cosmic ray induced background is the most important component of the detector background, especially in surface and medium depth underground laboratories. Primary cosmic ray particles are consisting mainly of protons (90%), alpha particles (9%) and heavier nuclei that bombard the upper atmosphere with a flux of about 1000 particles per m2 per second. During their interactions with atmospheric nuclei (O, N, Ar, Ne, Kr, Xe) they produce secondary particles (neutrons, protons, pions, electrons, positrons, photons, muons and neutrinos), as well as cosmogenic radionuclides (e.g., 3 H, 7 Be, 14 C, etc.). The relative intensities of charged particles at sea level are as follows:

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1 for muons 0.34 for neutrons 0.24 for electrons 0.009 for protons and 0.0007 for pions. The muon flux at sea level is about 1.9 cm−2 s−1 (Gaiser, 1990). The fluxes of cosmic ray particles depend on the geomagnetic latitude and on variations of solar activity during the eleven year solar cycle. From the variety of cosmic ray particles observed at sea level only muons, neutrons, electrons and photons are important for background induction. The soft component of cosmic rays (electrons and photons) can be easily shielded by heavy materials such as lead, concrete, steel, copper, etc. Protons from the nucleonic component of cosmic radiation have low intensity compared to neutrons and, moreover, they are converted to neutrons in nuclear reactions with shielding materials of the building and a metallic shield. Therefore only muons and neutrons (the hard component of cosmic rays) are important for background induction, as especially in the case of muons they can easily penetrate the shielding materials surrounding the detector. Buildings and other shielding structures absorb secondary neutrons (200 g cm−2 ) better than muons (2000 g cm−2 at sea level). Therefore, the relative composition of hard cosmic radiation varies according to the distance below the surface. At low depths (∼10 m w.e. (water equivalent is a unit frequently used in underground physics to normalize the depth for different rock overburden)), muons are by far the prevailing component of cosmic radiation. The muon energy spectrum is shifting to higher energies with increasing depth as the softer spectrum is gradually filtered out. Secondary neutrons (actually we should call them tertiary, to distinguish them from the neutrons produced in the upper atmosphere) are produced in the shielding (building materials, metallic shielding) by muon capture and fast muon nuclear interactions. At moderate depths (∼30 m w.e.), the negative muon capture prevails as the muon energy spectrum is rich in low energy muons that are likely to be stopped. On the other hand, fast muon nuclear interactions dominate deep underground (>100 m w.e.). However, not only particle flux magnitudes, but also mechanisms and efficiencies of background induction processes are important. Muons deposit tens of MeV energy in a HPGe detector by direct ionization. This is outside of interest for most of low-level counting applications. The main processes by which cosmic muons induce detector background are electromagnetic processes which are taking place in the metallic shield. Muons generate delta electrons that radiate bremsstrahlung initiating electromagnetic particle showers. Photons from showers are then forming a background radiation as they can penetrate through thicker layers of material. For muons of higher energies there is also a contribution from direct electron– positron pair formation and muon bremsstrahlung. For muons of lowest energies, muon decay positrons and electrons with energies up to 52 MeV, and negative muon nuclear capture, contribute as well. Processes including a production of nuclei are even more complicated to study. During muon capture, the atomic number Z of a nucleus decreases by one, and the nucleus radiates gamma-rays and fast neutrons when re-arranging its structure. These processes lead to activation of shielding and detector materials. Heusser (1995) estimated the flux of neutrons produced by muon capture to be 1.1×10−4 cm−2 s−1 for the muon flux of 8×10−3 cm−2 s−1 .

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A part of high energy muons (>100 Ge) also interacts with nuclei via virtual photons, causing their spallation accompanied by emission of numerous hadrons. These reactions are another source of tertiary neutrons which are important in deep underground laboratories. A muon-induced background gamma-spectrum of a Ge detector is characterized by a dominant 511 keV annihilation peak and a continuum peaking at about 200 keV. The integral background count rates are within 0.6–1.6 s−1 at sea level, depending on the size of the detector, the detector and shield materials and the shield lining. Annihilation peak count rates are usually ∼0.01 s−1 . The comparison of background spectra measured with various shielding material shows that the continuum decreases with the increase in the atomic number Z of the shielding material and its maximum shifts to higher energies. The reason is in a stronger self-absorption of induced radiation in higher Z materials. Thin layers (mm) of lining have a significant influence on the shape and background counting rate (Vojtyla et al., 1994). An efficient way to eliminate the muon induced background is the use of anticosmic (guard) detectors surrounding the main analyzing detector (Heusser, 1994, 1995; Vojtyla et al., 1994; Heusser, 1995; Semkow et al., 2002; Povinec et al., 2004; Schroettner et al., 2004). These detectors register cosmic muons before they generate secondary particles in the shield, and anticoincidence electronics rejects such background counts. Gas proportional chambers placed inside the shield have a small efficiency for gamma-rays, and they can introduce additional radioactive contamination into the shield (Heusser, 1994, 1995; Vojtyla et al., 1994). NaI(Tl) detectors used as anticosmic shielding have much better efficiency for gamma-rays, however, due to contamination of crystals and glass of photomultipliers by 40 K, and a lower efficiency for cascade gamma-quanta, it is not advisable to use them inside a shield, unless they are part of an anti-Compton spectrometer (Povinec, 1982; Povinec et al., 2004). At present mostly plastic scintillators or large proportional chambers placed outside of the shield are used. The advantage of proportional chambers is their small count rate (∼200 s−1 ), and hence negligible corrections for dead time. Plastic scintillation detectors, although having higher counting rates, they can detect neutrons. The basic mechanisms of background induction by neutrons in Ge detectors are their capture and nuclear excitation by inelastic scattering. A neutron induced gamma-ray spectrum of a Ge detector contains a number of lines that can be attributed to nuclei in a detector working medium (Ge isotopes), and to nuclei of materials in the vicinity of the detector. Their intensities depend on the size of the detector, the cryostat and shielding materials, and most importantly on the shielding depth below the surface. Heusser (1993) compiled a large number of gamma-lines observed so far in background gamma-ray spectra. A characteristic feature of neutron induced background spectra are saw-tooth shaped peaks with energies of 595.8 and 691.0 keV corresponding to inelastic scattering of fast neutrons on 74 Ge and 72 Ge. The Ge nuclei de-excite almost immediately compared to the time of charge collection and the de-excitation energy is summed with the continuously distributed recoil energy of the nucleus. Therefore, these peaks have sharp edges from a lower energy side and tails towards higher energies. The summation does not take place during de-excitation of long-lived isomer states of Ge isotopes that de-excite after collection of the charge from the primary process. The most intensive lines of this kind are those of 75m Ge and 71m Ge (T1/2 = 47.7 s and 21.9 ms, respectively) with energies of 139.7 and 198.3 keV and count rates of up to 2 × 10−3 s−1 . Further, lines coming from the shielding materials can be identi-


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fied in neutron induced background spectra, e.g., 206 Pb, 207 Pb, 63 Cu and 65 Cu (Heusser, 1993; Vojtyla et al., 1994). Neutrons also induce a background continuum and the annihilation peak, probably via hard gamma-rays emitted from highly excited nuclei. Hard gamma-rays loose their energy by Compton scattering, and initiate electromagnetic showers during their passage through shield materials. The shaped of the continuum is ruled by Z of the material and lining used, similarly as in the case of muon induced background. The neutron induced background can be minimized with antineutron shields consisting of layers made of low Z materials for their slowing down (e.g., polyethylene, paraffin) with admixture of boron or lithium for their capturing. However, their usefulness depends on shielding layers situated above the laboratory. If neutrons are mostly tertiary, i.e. produced in the shield, the neutron absorbing layer should be the innermost layer. This leads, however, to large shield dimensions and hence higher shield costs, as the neutron absorbing layer should be tens of cm thick to be efficient enough. For shields situated deeper underground (>10 m w.e.) where tertiary neutrons produced by cosmic muons prevail, a better solution would be to use an anticosmic shielding place outside of the shield. Although the anticosmic shielding is very effective way to decrease the detector background, it cannot be 100% effective in eliminating the effects of cosmic rays. However, a substantial overburden by placing the detection system deep underground can have profound effects on the background reduction. Down to a few tens of m w.e. cosmic rays still prevails, and an anticosmic shielding would be still useful. In the transition region to a few hundreds of m w.e., cosmic rays are still of importance, however, radioactivity free materials for the construction of detectors and shields become more and more important. Below about a thousand m w.e. the cosmic ray background is already negligible. The hadronic component of cosmic rays is efficiently removed by the first 10 m w.e. (Heusser, 1995), and comic muon induced particle fluxes (tertiary neutrons, photons) are ruled by the penetrating muon component. There are two other background components that were not covered earlier as they are negligible for surface laboratories, but important for specific applications carried out deep underground: neutrons can also be created in uranium fission and (α, n) reactions in surrounding materials. The estimated flux of about 4.5 × 10−5 cm−2 s−1 corresponding to the average U and Th concentrations in the Earth’s crust (Heusser, 1992) is negligible compared to that generated by cosmic rays in shallow depths, but it is important in experiments carried deep underground. These neutrons can also penetrate thick shields and induce background either directly or via a long-term activation of the construction material. The simplest way of eliminating this background source is to shield the U and Th containing walls with a neutron shielding material of sufficient thickness (at least 20 cm). The secondary cosmic ray particles can induce background not only in one event by interactions in the detector and surrounding materials that are shorter in time than the resolution time of electronics, but also by production of radioactive nuclei in materials intended to be used underground but processed and stored at sites with higher cosmic ray fluxes. The cosmic activation component is caused by short-lived radionuclides (e.g., 56–58 Co, 60 Co) produced primarily in other target elements such as copper or iron. The activated contamination can be sometimes more important than the residual primordial contamination. The most important mechanisms of the activation are secondary neutrons (at shallow depths or higher altitudes), and by negative muon capture and fast muon interactions (at underground laboratories). Many

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short-lived radionuclides can be produced during the time of an intercontinental airplane flight due to high fluxes of secondary cosmic ray particles (mostly neutrons) at altitudes of about 10 km. At shallow and deep shielding depths, after filtering out secondary neutrons, fast muon interactions and negative muon capture activate materials either directly by disintegration and changing nuclei, or through tertiary neutrons. Compared to the direct background induction, the fast muon interactions at shallow shielding depths are more important than the muon capture. 65 Zn, 57 Co, 58 Co and 54 Mn are produced at saturated production rates of tens to hundreds of µBq kg−1 by the activation of germanium at sea level. 54 Mn is also produced at a high saturated production rate of 4.5 mBq kg−1 in iron used for the construction of shieldings (Brodzinski et al., 1990). The magnitude of background due to cosmic ray activation is, e.g., in the case of saturated 65 Zn activity of 0.44 mBq kg−1 in germanium at sea level in a 1 kg Ge detector, which gives an integral background rate above 50 keV of only 11 counts per day (Brodzinski et al., 1988). 2.2. Environmental radioactivity outside of the counting system The sources of background radiation are decay product of nuclei in U and Th decay series and 40 K. Generated gamma-radiation is critical and therefore a massive metallic shielding is required to protect detectors. A typical sea level gamma-ray flux above 50 keV at 1 m above ground is 10 photons cm−2 s−1 , representing the most intensive photons flux to which an unshielded detector is exposed. The cosmic photon flux is less than 1% of the environmental flux. A contribution from all other primordial radionuclides to the detector background is negligible. The average concentrations of primordial radionuclides in the continental upper crust are 850 Bq kg−1 of 40 K, 100 Bq kg−1 of 87 Rb, 44 Bq kg−1 of 232 Th and 36 Bq kg−1 of 238 U. Except for 232 Th, the concentrations are about twice as small in soils, but large variations can occur. The highest concentrations are found in granites and pegmatites. Secular equilibrium in decay series is rarely achieved in surface and near surface geological environments because the radionuclides in the chain migrate and take part in various chemical and physical processes. A special case is radon which breaks each series and can, due to its inert nature, migrate far from the rock or soil surface. Therefore radon will be treated separately in this chapter later. The impact of cosmogenic radionuclides produced in the atmosphere on the background of Ge detectors is usually negligible. Large amounts of radionuclides were released to the environment during the Chernobyl accident. Relatively long-lived 137 Cs is still present as a surface contamination of construction materials, therefore it should be checked and removed if present, especially in materials used for the detector construction. 2.3. Radioactivity of construction materials of the shielding A detector should be placed in a massive shield, best made of high-Z material, to minimize the environmental background. An anticoincidence detector with high efficiency for gammaradiation (e.g., NaI(Tl), BGO—bismuth germanate) can be used to further decrease the detector background, and usually also scattered photons, when working as an anti-Compton


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gamma-ray spectrometer (Debertin and Helmer, 1998). A comparison of background gammaray spectra with and without massive shield can be found, e.g., in Sýkora et al. (1992). A background spectrum is usually dominated by 40 K (1461 keV), 214 Bi (609.3 and 1120.3 keV) and 208 Tl (583.1 and 2614.7 keV). The integral background of an unshielded Ge gamma detector of usual size is ∼100 s−1 . In the energy region above 200 keV and under the Compton edge of the 208 Tl peak, the background is reduced by two orders of magnitude by placing the detector inside the shield. Except for the 511 keV annihilation peak, attenuation factors for lines of the radionuclides in question are in agreement with the assumed absorption properties of the shielding layers for gamma-radiation of corresponding energy. On the other hand, the continuum in the spectrum is attenuated almost independently of the energy. It could be expected that the photons of lower energies would be absorbed more efficiently than those with higher energies. The reason that this does not occur is that the lower energy photons are continuously replenished by Compton-scattered photons of higher energies when passing through the thick layer of material. In this way, the continuous spectrum reaches an equilibrium shape with a general attenuation coefficient. Smith and Levin (1990) using Monte Carlo simulation found a mass attenuation coefficient of 0.045–0.050 cm2 g−1 for both high- and low-Z shielding materials. It is not worth increasing the thickness of the shielding over about 150 g cm−2 because other background components begin to dominate the total background. Muon-induced background together with a usually smaller neutron component rule the background beyond this thickness. Moreover, increasing the thickness of a high-Z material like lead enhances the production of tertiary neutrons by negative muon capture (mostly at low altitudes below sea level) and fast muon interactions (deeper underground). Paradoxically, the background increases slightly due to interactions of tertiary neutrons when increasing the shield thickness over 150 g cm−2 . Lead has proven to be the best material for bulk shields as it is cheap, has good mechanical properties, a high atomic number, low neutron capture cross-section, small neutron gamma yields and a low interaction probability for cosmic rays including the forming of radionuclides by activation. However, its intrinsic radioactivity can often be disturbing. Many authors investigated the origin of lead contamination in the past (Kolb, 1988; Eisenbud, 1973; Heusser, 1993). When using gamma spectrometry to check lead contamination, 210 Pb and its progenies 210 Bi and 210 Po are hidden contaminants as they cannot be clearly identified by gamma lines. Due to the long half-life of 210 Pb (22 years), the abundance of these radionuclides in lead can be much higher than expected from secular equilibrium. 210 Pb itself does not contribute to the background of Ge gamma-spectrometers as its very soft beta (Emax of 16.5 and 63 keV) and gamma (46.5 keV) radiations cannot escape self-absorption. The background is induced by the progeny 210 Bi emitting hard beta radiation with end-point energy of 1161 keV. Beta-radiation generates bremsstrahlung in the high-Z material, and also lead Xrays with energies of 72.8, 75.0, 84.9 and 87.4 keV. The continuum peaks at around 170 keV. As the end-point energy of 210 Bi is not sufficient for efficient pair production, the annihilation peak cannot be observed in the spectrum. Concentrations of 210 Pb reported in the literature range from 0.001 to 2.5 kBq kg−1 (Povinec et al., 2004). Ordinary contemporary lead has a 210 Pb massic activity of ∼100 Bq kg−1 . Lead with 210 Pb massic activity of about 1 Bq kg−1 is commercially available, lead with a lower activity is very expensive. Once 210 Pb is present in lead it cannot be removed by any chemical procedure and the only way to eliminate it is to wait several half-lives for its de-

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cay. Old lead is an alternative to modern radioactive pure lead, however, its supply from water pipes, sunken shiploads or the ballast of sailing ships are limited. Moreover, old lead may not always be radioactive pure. If uranium was not removed, 210 Pb will have been supplied from the uranium decay series from the beginning of lead production. Another possibility is to use old iron plates or rods for the detector shielding. A recent iron may be dangerous because of 137 Cs contamination from Chernobyl fallout, or due to a contamination during its production. Therefore iron should be always tested before purchasing. Electrolytic copper is a better material, however, it is much more expensive than iron or ordinary lead. Also in situ production of cosmogenic radionuclides in copper may be disadvantageous when operating at shallow depths. 2.4. Radioactivity of construction materials of the detector The most sophisticated techniques for reducing the background will be unsuccessful if the detector is made of contaminated materials. Selecting radio-pure materials for the detector is important as these materials are close to the detector sensitive volume and radiation from them is detected with high efficiency. Most materials are contaminated with the primordial radionuclides, 40 K, 238 U and 232 Th. Complex production processes and contacts with various reagents increase the probability of contamination with primordial radionuclides. Usually, each production step must be checked to be sure that the produced material will be free of contaminants to the desired level. Physical and chemical procedures to which to processed material is subjected, result in isolated impurities (e.g., 210 Pb, 226 Ra, 228 Tl) as elements of the decay series have different properties and the chain breaks down. In fabricating metals, the redox potential is generally an important factor influencing the degree of radio-purity that can be achieved. That is probably the reason that copper can be produced very cleanly and is widely used in low-level counting applications. Its redox potential of 0.337 V is high compared to those of K, U and Th and their progenies. Cooper is routinely purified after melting by electrolytic dissolution and subsequent redeposition in solution even in large scale production. The contribution of radioactivity of detector materials to the background depends significantly on the detector type. It is usually lowest for semiconductor detectors as in this case a detection medium has to be very pure merely to enable the proper function of a semiconductor diode (a charge collection). The material is cleaned from primordial radionuclides and also from 137 Cs as a by-product through zone refining and crystal pulling. The background originates in parts placed around the crystal, and closer a part is the cleaner it should be. Materials used for crystal holders, end-caps and signal pins have to be selected most carefully. To obtain the best results the crystal holders should be made of electrolytic copper ( Tcut has the form (CERN, 1992) √ 1 dE Tmax Tcut z2 Z β2 Tcut C δ = D 2 ln − 1+ , − − ρ dx 1 2 Tmax 2 Z Aβ where D = 4πNa re2 mc2 = 0.307 MeV cm2 g−1 . A, Z and ρ are the mass number, atomic number and density of the material, respectively; z is the particle charge, and I = 16Z 0.9 eV is the effective ionization potential. The factor δ is a correction for the density effect, and C is the shell correction term. For Tmax < Tcut the formula has a different form (CERN, 1992). It has been known for many years that most of the muon induced background in Ge gammaspectrometers is caused by bremsstrahlung radiation of delta electrons. As the energy region of interest is above a few tens of keV, the electron kinetic energy cut above which delta electrons should be explicitly generated should also be a few tens of keV, at least in the inner layers of the shield and in the detector itself. Therefore Tmax will always be greater than Tcut for cosmic muons. The differential cross-section of delta electron production in interactions of muons (spin 1/2) can be written as (CERN, 1992) T2 dσ 2 1 1 2 T + = 2πre m 1−β , dT β T2 Tmax 2E 2 where T is the electron kinetic energy, E is the muon energy, and the other symbols have the usual meaning. Far below the kinematic limit, Tmax , the delta electron spectrum is inversely proportional to the square of its kinetic energy, and the number of electrons produced above a given threshold is inversely proportional to the threshold energy. There are well established facts on delta electron production. The lower the delta electron cut, Tcut , is, the more realistic the simulation of the ionization process is achieved. However, the process requires more computing time as more particles have to be processed. 4.1.2. Direct electron–positron pair production When a muon of energy E moves in the field of a nucleus of charge Z, it can create an electron–positron pair through a fourth order QED process with a differential cross-section (CERN, 1985), 2 2 1−ν m ∂ 2σ φμ , = α 4 (Zλ- 2 ) φe + ∂v∂ρ 3π ν M where E+ − E− E+ + E− and ρ = . ν= E E


177

α is the fine structure constant, λ- is the Compton wavelength, E + and E − are the total energies of the positron and the electron, respectively; the terms φe and φμ can be found in the CERN (1985) report. There are kinematic ranges for the fraction of muon energy transferred to pair ν: 4m 3 √ M 1/3 e Z = νmin ν νmax = 1 − E 4 E and 0 = ρmin

% % % % ρ(ν) ρmax (ν) = 1 −

6m E 2 (ν − 1)

1−

4m . νE

There is an energy threshold of 2.04 MeV (4mc2 ) for this process. The dominant contribution comes from the low-ν region. In GEANT, this process is simulated for the muon energy range 1–104 GeV using parametrized expressions for cross-sections with an inaccuracy of no more than 10% (CERN, 1990). It is possible to set an energy cut for the resulting pair higher than the kinematic limit. In this case, the energy expended by the muon for the production of a pair is treated as a quasi-continuous loss, and the muon is absorbed at the interaction point. There is no reason to use higher cuts in simulations of background induction as the direct pair production is an important process, and an acceleration of computation is not significant. 4.1.3. Muon bremsstrahlung The muon bremsstrahlung is not often discussed, although muons are massive particles, and the process takes a place in the Coulomb field of the nucleus. We shall not present here any formulae for calculating a differential cross-section of this process as they are too complicated (CERN, 1985). Muon bremsstrahlung events are quite rare but very powerful so that a muon can lose a large fraction of its energy in one interaction. Therefore, it contributes significantly to mean energy losses of high energy muons (about 3 MeV/(g cm−2 )) for 1000 GeV muons in iron (CERN, 1985). The hard photon produced generates an electromagnetic shower in the material. Due to its high energy, a shower can contain up to 105 particles with energies above 50 keV. GEANT simulates this process in the muon energy range 1–104 GeV with accuracy of parametrized cross-sections not worse than 12% (CERN, 1990). As for the direct pair production, it is possible to set an energy cut below which the muon bremsstrahlung is treated as a quasi-continuous energy loss. However, this process is rare and the computation acceleration is negligible. 4.1.4. Multiple scattering To complete the list of electromagnetic processes with muons, multiple scattering of muons on atoms in the material should be covered as well. This process is characterized by numerous elastic collisions with the Coulomb field of nuclei, and to a lesser degree with the electron field, which leads to a statistical declination of a muon from its original direction. It is important only for muons below about 1 GeV/c, and it gains importance as the muon approaches zero kinetic energy. However, the expressions are too complex to be presented in this review. The first three electromagnetic processes with muons are compared in Figure 3, where macroscopic cross-sections in lead for muons with momenta up to 100 GeV/c are shown. The

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Fig. 3. Macroscopic cross-sections of electromagnetic processes with muons in lead as used by the GEANT code.

cross-section of delta electron production is given for electrons with kinetic energies above 10 MeV. At this energy the ionization and radiation energy losses of electrons are equal. The cut for muon bremsstrahlung of 50 keV, and electron–positron pair production just above the kinematic limit are considered. The dominance of production of delta electrons for muons momenta below 100 GeV/c is dominant. 4.1.5. Muon decay Muon decay is a well-known weak process that occurs mostly when a muon has been stopped within the set-up. The cause is a relatively long mean life-time of 2.197 µs, which is prolonged due to the relativistic time dilatation for energetic cosmic muons. A positive (negative) muon decays to a positron (electron) and neutrinos escaping from the set-up according to the decay schemes μ+ → e+ + νe + ν μ

and

μ− → e− + ν e + νμ ,

respectively. A decay accompanied by a photon emission occurs in 1.4 ± 0.4% cases. There is a kinematic limit for the positron (electron) momentum in the muon rest system given as 1 m2 p0 = Mc 1 − 2 = 52.8 MeV/c 2 M if we consider mass-less neutrinos. The standard V –A weak interaction model gives for the positron (electron) momentum, pe , spectrum, neglecting terms of m/E (Morita, 1973) M 5 gμ2 c4 2pe 2 p(x) = , (1 − x) + ρ (4x − 3) x 2 with x = 7 3 9 Mc 16π h¯ which is in good agreement with experiment for ρ = 3/4.


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Most of the positrons (electrons) have high energies and are likely to generate electromagnetic showers if a muon decays in a high-Z material like lead. GEANT treats decay of particles in a general way so that the four-momenta of the decay products are generated with isotropic angular distribution in the center-of-mass system, and then transformed into the laboratory system. The consequence of such a general approach is the simplification of the momentum spectra of the decay products. The decay accompanied by gamma-ray emission is not simulated. However, it was found in a detailed simulation that the muon decay is only a minor process in background induction by cosmic muons. 4.1.6. Negative muon capture A concurrent process to negative muon decay in a material is muon capture. A negative muon is attracted by the nucleus, and if it does not succeed in decaying, it finally falls to the 1s level of a muonic atom. Due to its 207 times bigger mass compared to electron, a large part of the muon probability distribution function in the 1s state rests within the nucleus, especially in heavy nuclei. The relatively long life-time of muons allows the basic weak process μ− + p → n + νμ to take place. Simple kinematic calculations show that a free neutron could gain only 537 MeV of kinetic energy while the rest escapes with the muon neutrino. However, in a nucleus, a higher energy is transferred to the nucleus (typically 10–20 MeV) because of the non-zero proton momentum and nuclear effects. The binding energy of negative muons in heavy nuclei can be quite high, e.g., 10.66 MeV in lead. On the bases of experimental data (Suzuki et al., 1987) it can be calculated that muon capture without muon decay occurs in natural Pb, Cd, Cu, Fe and Al in 97%, 96%, 93%, 90% and 61%, respectively. As more than 99% of muons in a counting system equipped with a lead shield stop in lead due to its high mass fraction in the whole system (calculated in a simulation), practically all the stopped negative muons are captured without electron emission. The muon capture is therefore the most intensive source of tertiary neutrons in laboratories placed in moderate shielding depths. After the negative muon capture the excited nucleus de-excites by the emission of several neutrons. Emission of charged particles in heavy elements is suppressed due to the Coulomb barrier at these energies. In lead, 1.64 ± 0.16 neutrons are emitted in average (Morita, 1973). The probabilities of multiplicities of the emitted neutrons are 0.6% for no neutron, 59.1% for one neutron, 23.6% for two, 5.1% for three and 12.4% for four neutrons. The neutron energy spectrum for lead was measured by Schröder et al. (1974). It contains an evaporation part expressed as dN(E) ∝ E 5/11 e−E/θ dE in the energy region of 1 to 4 MeV. The effective nuclear temperature θ is 1.22 MeV. The part of the neutron spectrum above 4.5 MeV decreases exponentially (dN/dE ∝ exp(−E/Ed )), and is attributed to direct pre-compound emission. The decrement energy Ed is 8 ± 1 MeV, and the fraction of the high-energy distribution integrated from 4.5 to 20 MeV accounts for 9.7–1.0% of the total spectrum. Later Kozlowski et al. (1985) extended the measurements of the neutron spectrum up to 50 MeV and found a decrement energy Ed in the energy region

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10–50 MeV of 8.6 ± 0.5 MeV, which is in accordance with that measured by Schröder et al. (1974). The de-excitation of a nucleus may be completed by emission of several gamma-rays, however, there are no reliable data on gamma-ray energies and intensities for the materials in question. Also X-rays with correspondingly higher energies can be emitted from muon-atomic levels during the descent of the muon to the 1s level. Although such X-rays are used for material analysis, they have never been identified in background spectra, and their contribution is considered to be negligible. 4.1.7. Fast muon nuclear interactions These interactions, also called photonuclear muon interactions, occur at high muon energies, and they are mediated via a virtual photon, interacting with nucleons. The total cross-section per nucleon can be expressed for muons above 30 GeV as (CERN, 1990) σ = 0.3(E/30)0.25 . It is considered to be constant (0.3 µb/nucleon) for muons below 60 GeV. The macroscopic cross-section for muons below 30 GeV in lead is only 2.0 × 10−6 cm−1 and increases only slowly for higher muon energies. As in muon bremsstrahlung, the fast muon interaction is characterized by rare but very hard events. Therefore, despite its small cross-section, it contributes to mean energy losses of high-energy muons. However, the contribution is still only about 5% (

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