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rL~gL

0AV IS

STP 1283

Geostatistics for Environmental and Geotechnical Applications Shahrokh Rouhani, R. Mohan Srivastava, Alexander J. Desbarats, Marc V. Cromer, and A. Ivan Johnson, editors

ASTM Publication Code Number (PCN): 04-012830-38

ASTM 100 Barr Harbor Drive West Conshohocken, PA 19428-2959 Printed in the U.S.A.

Library of Congress Cataloging-in-Publication Data Geostatistics for environmental and geotechnical applications/ Shahrokh Rouhani ... let al.l. p. cm. - (STP: 1283) Papers presented at the symposium held in Phoenix, Arizona on 26-27 Jan. 1995, sponsored by ASTM Committee on 018 on Soil and Rock. Includes bibliographical references and index. ISBN 0-8031-2414-7 1. Environmental geology-Statistical methods-Congresses. 2. Environmental geotechnology-Statistical methods-Congresses. I. Rouhani, Shahrokh. II. ASTM Committee 0-18 on Soil and Rock. III. Series: ASTM special technical publication: 1283. QE38.G47 1996 96-42381 628.5'01 '5195-dc20 CIP

Copyright © 1996 AMERICAN SOCIETY FOR TESTING AND MATERIALS, West Conshohocken, PA. All rights reserved. This material may not be reproduced or copied, in whole or in part, in any printed, mechanical, electronic, film, or other distribution and storage media, without the written consent of the publisher.

Photocopy Rights Authorization to photocopy items for internal or personal use, or the internal or personal use of specific clients, is granted by the AMERICAN SOCIETY FOR TESTING AND MATERIALS for users registered with the Copyright Clearance Center (CCC) Transactional Reporting Service, provided that the base fee of $2.50 per copy, plus $0.50 per page is paid directly to CCC, 222 Rosewood Dr., Danvers, MA 01923; Phone: (508) 750-8400; Fax: (508) 750-4744. For those organizations that have been granted a photocopy license by CCC, a separate system of payment has been arranged. The fee code for users of the Transactional Reporting Service is 0-8031-2414-7/96 $2.50 + .50

Peer Review Policy Each paper published in this volume was evaluated by three peer reviewers. The authors addressed all of the reviewers' comments to the satisfaction of both the technical editor(s) and the ASTM Committee on Publications. To make technical information available as quickly as possible, the peer-reviewed papers in this publication were printed "camera-ready" as submitted by the authors. The quality of the papers in this publication reflects not only the obvious efforts of the authors and the technical editor(s), but also the work of these peer reviewers. The ASTM Committee on Publications acknowledges with appreciation their dedication and contribution to time and effort on behalf of ASTM.

Printed in Ann Arbor, MI October t 996

Foreword This publication, Geostatistics for Environmental and Geotechnical Applications, contains papers presented at the symposium of the same name held in Phoenix, Arizona on 26-27 Jan. 1995. The symposium was sponsored by ASTM Committee on DIS on Soil and Rock. The symposium co-chairmen were: R. Mohan Srivastava, FSS International; Dr. Shahrokh Rouhani, Georgia Institute of Technology; Marc V. Cromer, Sandia National Laboratories; and A. Ivan Johnson, A. Ivan Johnson, Inc.

Contents OVERVIEW PAPERS Geostatistics for Environmental and Geotechnical Applications: A Technology Transferred-MARc V. CROMER

3

Describing Spatial Variability Using Geostatistical Analysis-R. MOHAN SRI VASTA VA

13

Geostatistical Estimation: Kriging-sHAHROKH ROUHANI

20

Modeling Spatial Variability Using Geostatistical Simulation-ALEXANDER J. DESBARATS

32

ENVIRONMENTAL ApPLICATIONS Geostatistical Site Characterization of Hydraulic Head and Uranium Concentration in Groundwater-BRUcE E. BUXTON, DARLENE E. WELLS, AND ALAN D. PATE

51

Integrating Geophysical Data for Mapping the Contamination of Industrial Sites by Polycyclic Aromatic Hydrocarbons: A Geostatistical Approach-PIERRE COLIN, ROLAND FROIDEVAUX, MICHEL GARCIA, AND SERGE NICOLETIS

69

Effective Use of Field Screening Techniques in Environmental Investigations: A Multivariate Geostatistical Approach-MIcHAEL R. WILD AND SHAHROKH ROUHANI

88

A BayesianiGeostatistical Approach to the Design of Adaptive Sampling ProgramsROBERT L. JOHNSON

102

Importance of Stationarity of Geostatistical Assessment of Environmental Contamination-KADRI DAGDELEN AND A. KEITH TURNER

117

Evaluation of a Soil Contaminated Site and Clean-Up Criteria: A Geostatistical Approach-DANIELA LEONE AND NEIL SCHOFIELD

133

Stochastic Simulation of Space-Time Series: Application to a River Water Quality Modelling-AMILcAR o. SOARES, PEDRO J. PATINHA, AND MARIA J. PEREIRA

146

Solid Waste Disposal Site Characterization Using Non-Intrusive Electromagnetic Survey Techniques and Geostatistics-GARY N. KUHN, WAYNE E. WOLDT, DAVID D. JONES, AND DENNIS D. SCHULTE

162

GEOTECHNICAL AND EARTH SCIENCES ApPLICATIONS

Enhanced Subsurface Characterization for Prediction of Contaminant Transport Using Co-Kriging---CRAIG H. BENSON AND SALWA M. RASHAD

181

Geostatistical Characterization of Unsaturated Hydraulic Conductivity Using Field Infiltrometer Data-sTANLEY M. MILLER AND ANJA J. KANNENGIESER

200

Geostatistical Simulation of Rock Quality Designation (RQD) to Support Facilities Design at Yucca Mountain, Nevada-MARc V. CROMER, CHRISTOPHER A. 218

RAUTMAN, AND WILLIAM P. ZELINSKI

Revisiting the Characterization of Seismic Hazard Using Geostatistics: A Perspective after the 1994 Northridge, California Earthquake-JAMES R. CARR Spatial Patterns Analysis of Field Measured Soil

236

Nitrate-FARIDA S. GODERY A, M. F.

DAHAB, W. E. WOLDT, AND I. BOGARD!

248

Geostatistical Joint Modeling and Probabilistic Stability Analysis for ExcavationsDAE S. YOUNG

Indexes

262 277

Overview Papers

Marc V. Cromer l

Geostatistics for Environmental and Geotechnical Applications: A Technology Transferred

REFERENCE: Cromer, M. V., "Geostatistics for Environmental and Geotechnical Applications: A Technology Transferred," Geostatistics for Environmental and Geotechnical Applications. ASTM STP 1283, R. M. Srivastava, S. Rouhani, M. V. Cromer, A. J. Desbarats, A. I. Johnson, Eds., American Society for Testing and Materials, 1996. ABSTRACT: Although successfully applied during the past few decades for predIcting the spatial occurrences of properties that are cloaked from direct observation, geostatistical methods remain somewhat of a mystery to practitioners in the environmental and geotechnical fields. The techniques are powerful analytical tools that integrate numerical and statistical methods with scientific intuition and professional judgment to resolve conflicts between conceptual interpretation and direct measurement. This paper examines the practicality of these techniques within the entitles field of study and concludes by introducing a practical case study in which the geostatistical approach is thoroughly executed. KEYWORDS: Geostatistics, environmental investigations, decision analysis tool

INTRODUCTION Although, geostatistics is emerging on environmental and geotechnical fronts as an invaluable tool for characterizing spatial or temporal phenomena, it is still not generally considered "standard practice" in these fields. The technology is borrowed from the mining and petroleum exploration industries, starting with the pioneering work of Danie Krige in the 1950's, and the mathematical formalization by Georges Matheron in the early 1960's. In these industries, it has found acceptance through successful application to cases where decisions concerning high capital costs and operating practices are based on interpretations derived from sparse spatial data. The application of geostatistical methods has since extended to many fields relating to the earth sciences. As many geotechnical and, certainly, environmental studies are faced with identical "high-stakes" decisions, geostatistics appears to be a natural transfer of technology. This paper outlines the unique characteristics of this sophisticated technology and discusses its applicability to geotechnical and environmental studies.

1 Principal Investigator, Sandia National Laboratories/Spectra Research Institute, MS 1324. P.O. Box 5800, Albuquerque, NM 87185-1342

3

4

GEOSTATISTICAL APPLICATIONS

IT'S GEOSTATISTICS The field of statistics is generally devoted to the analysis and interpretation of uncertainty caused by limited sampling of a property under study. Geostatistical approaches deviate from more "classical" methods in statistical data analyses in that they are not wholly tied to a population distribution model that assumes samples to be normally distributed and uncorrelated. Most earth science data sets, in fact, do not satisfy these assumptions as they often tend to have highly skewed distributions and spatially correlated samples. Whereas classical statistical approaches are concerned with only examining the statistical distribution of sample data, geostatistics incorporates the interpretations of both the statistical distribution of data and the spatial relationships (correlation) between the sample data. Because of these differences, environmental and geotechnical problems are more effectively addressed using geostatistical methods when interpretation derived from the spatial distribution of data have impact on decision making risk. Geostatistical methods provide the tools to capture, through rigorous examination, the descriptive information on a phenomenon from sparse, often biased, and often expensive sample data. The continued examination and quantitative rigor of the procedure provide a vehicle for integrating qualitative and quantitative understanding by allowing the data to "speak for themselves." In effect, the process produces the most plausible interpretation by continued examination of the data in response to conflicting interpretations.

A GOAL-ORIENTED, PROJECT COORDINATION TOOL The application of geostatistics to large geotechnical or environmental problems has also proven to be a powerful integration tool, allowing coordination of activities from the acquisition offield data to design analysis (Ryti, 1993; Rautman and Cromer, 1994; Wild and Rouhani, 1995). Geostatistical methods encourage a clear statement of objectives to be set prior to any study. With these study objectives defined, the flow of information, the appropriate use of interpretations and assumptions, and the customer/supplier feedback channels are defined. This type of coordination provides a desirable level of tractability that is often not realized. With environmental restoration projects, the information collected during the remedial investigation is the sole basis for evaluating the applicability of various remedial strategies, yet this information is often incomplete. Incomplete information translates to uncertainty in bounding the problem and increases the risk of regulatory failure. While this type of uncertainty can often be reduced with additional sampling, these benefits must be balanced with increasing costs of characterization. The probabilistic roots deeply entrenched into geostatistical theory offer a means to quantify this uncertainty, while leveraging existing data in support of sampling optimization and riskbased decision analyses. For example, a geostatistically-based, costlrisklbenefit approach to sample optimization has been shown to provide a framework for examining the many tradeoffs encountered when juggling the risks associated with remedial investigation, remedial

CROMER ON A TECHNOLOGY TRANSFERRED

5

design, and regulatory compliance (Rautman et. aI., 1994). An approach such as this explicitly recognizes the value of information provided by the remedial investigation, in that additional measurements are only valuable to the extent that the information they provide reduces total cost. GEOSTATISTICAL PREDICTION

The ultimate goal of geostatistical examination and interpretation, in the context of risk assessment, is to provide a prediction of the probable or possible spatial distribution of the property under study. This prediction most commonly takes the form of a map or series of maps showing the magnitude and/or distribution of the property within the study. There are two basic forms of geostatistical prediction, estimation and simulation. In estimation, a single, statistically "best" estimate of the spatial occurrence of the property is produced based on the sample data and on the model determined to most accurately represent the spatial correlation of the sample data. This single estimate (map) is produced by the geostatistical technique commonly referred to as kriging. With simulation, many equally-likely, high-resolution images of the property distribution can be produced using the same model of spatial correlation as developed for kriging. The images have a realistic texture that mimics an exhaustive characterization, while maintaining the overall statistical character of the sample data. Differences between the many alternative images (models) provides a measure of joint spatial uncertainty that allows one to resolve risk-based questions ... an option not available with estimation. Like estimation, simulation can be accomplished using a variety of techniques and the development of alternative simulation methods is currently an area of active research. NOT A BLACK BOX

Despite successful application during the past few decades, geostatistical methods remain somewhat of a mystery to practitioners in the geotechnical and environmental fields. The theoretical complexity and effort required to produce the intermediate analysis tools needed to complete a geostatistical study has often deterred the novice from this approach. Unfortunately, to many earth scientists, geostatistics is considered to be a "black box." Although this is far from the truth, such perceptions are often the Achilles' heel of many mathematical/numeric analytical procedures that harness data to yield their true worth because they require a commitment in time and training from the practitioner to develop some baseline proficiency. Geostatistics is not a solution, only a tool. It cannot produce good results from bad data, but it will allow one to maximize that information. Geostatistics cannot replace common sense, good judgment, or professional insight, in fact it demands these skills to be brought to bare. The procedures often take one down a blind alley, only to cause a redirection to be made because of an earlier miss-interpretation. While these exercises are nothing more than cycling through the scientific method, they are often more than the novice is willing to commit to. The time and frustration associated with continually rubbing one's nose in the

6


details of data must also take into account the risks to the decision maker. Given the tremendous level of financial resources being committed to field investigation, data collection, and information management to provide decision making power, it appears that such exercises are warranted. CASE STUDY INTRODUCTION This introductory paper only attempts to provide a gross overview of geostatistical concepts with some hints to practical application for these tools within the entitled fields of scientific study. Although geostatistics has been practiced for several decades, it has also evolved both practically and theoretically with the advent offaster, more powerful computers. During this time a number of practical methods and various algorithms have been developed and tested, many of which still have merit and are practiced, but many have been left behind in favor of promising research developments. Some of the concepts that I have touched upon will come to better light in the context of the practical examination addressed in the following suite of three overview papers provided by Srivastava (1996), Rouhani (1996), and Desbarats (1996). In this case study, a hypothetical database has been developed that represents sampling of two contaminants of concern: lead and arsenic. Both contaminants have been exhaustively characterized as a baseline for comparison as shown in Figures 1 and 2. The example scenario proposes a remedial action threshold (performance measure) of 500 ppm for lead and 30 ppm for arsenic for the particular remediation unit or "VSR" (as discussed by Desbarats, 1996). Examination of the exhaustive sample histogram and univariate statistics in Figures 1 and 2 indicate about one fifth of the area is contaminated with lead, and one quarter is contaminated with arsenic. The two exhaustive databases have been sampled in two phases, the first of which was on a pseudo-regular grid (square symbols in Figure 3) at roughly a separation distance of 50 m. In this first phase, only lead was analyzed. In the second sampling phase, each first-phase sample location determined to have a lead concentration exceeding the threshold was targeted with eight additional samples (circle symbols of Figure 3) to delineate the direction of propagation of the contaminant. To mimic a problem often encountered in an actual field investigation, during the second phase of sampling arsenic contamination was detected and subsequently included in the characterization process. Arsenic concentrations are posted in Figure 4 with accompanying sample statistics. The second phase samples, therefore, all have recorded values for both arsenic and lead. Correlation between lead and arsenic is explored by examining the co-located exhaustive data which are plotted in Figure 5. This comparison indicates moderately good correlation between the two constituents with a correlation coefficient of 0.66, as compared to the slightly higher correlation coefficient of 0.70 derived from the co-located sample data plotted in Figure 6. There are a total of 77 samples from the first phase of sampling and 13 5 from the second phase. The second sampling phase, though, has been biased because of its focus on "hot-


7

FIGURE 1: EXHAUSTIVE PB DATA

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spot" delineation. This poses some difficult questions/problems from the perspective of spatial data analysis: What data are truly representative of the entire site and should be used for variography or for developing distributional models? What data are redundant or create bias? Have we characterized arsenic contamination adequately? These questions are frequently encountered, especially in the initial phases of a project that has not exercised careful pre-planning. The co-located undersampling of arsenic presents an interesting twist to a hypothetical, yet realistic, problem from we can explore the paths traveled by the geostatistician.

REFERENCES Desbarats, A.J., "Modeling Spatial Variability Using Geostatistical Simulation," Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R. Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Eds., American Society for Testing and Materials, Philadelphia, 1996. Rautman, C.A., M.A. McGraw, J.D. Istok, 1M. Sigda, and P.G. Kaplan, "Probabilistic Comparison of Alternative Characterization Technologies at the Fernald Uranium-InSoils Integrated Demonstration Project", Vol. 3, Technolo~y and Pro~rams for Radioactive Waste Mana~ement and Environmental Restoration, proceedings of the Symposium on Waste Management, Tucson, AZ, 1994. Rautman, C.A. and M.V. Cromer, 1994, "Three-Dimensional Rock Characteristics Models Study Plan: Yucca Mountain Site Characterization Plan SP 8.3.1.4.3.2", U.S. Department of Energy, Office of Civilian Radioactive Waste Management, Washington, DC. Rouhani, S., "Geostatistical Estimation: Kriging," Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R. Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Eds., American Society for Testing and Materials, Philadelphia, 1996. Ryti, R., "Superfund Soil Cleanup: Developing the Piazza Road Remedial Design," .Im!1:nill. Air and Waste Mana~ement, Vol. 43, February 1993. Srivastava, R.M., "Describing Spatial Variability Using Geostatistical Analysis," Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R. Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Eds., American Society for Testing and Materials, Philadelphia, 1996. Wild, M. and S. Rouhani, "Taking a Statistical Approach: Geostatistics Brings Logic to Environmental Sampling and Analysis," Pollution En~ineerin~, February 1995.

R. Mohan Srivastava! DESCRIBING SPATIAL VARIABILITY USING GEOSTATISTICAL ANALYSIS

REFERENCE: Srivastava, R. M., "Describing Spatial Variability Using Geostatistical Analysis," Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R. M. Srivastava, S. Rouhani, M. V. Cromer, A. J. Desbarats, A. I. Johnson, Eds., American Society for Testing and Materials, 1996. ABSTRACT: The description, analysis and interpretation of spatial variability is one of the cornerstones of a geostatistical study. When analyzed and interpreted propp-r'j', the pattern of spatial variability can be used to plan further sampling programs, to improve estimates and to build geologically realistic models of rock, soil and fluid properties. This paper discusses the tools that geostatisticians use to study spatial variability. It focuses on two of the most common measures of spatial variability, the variogram and the correlogram, describes their appropriate uses, their strengths and their weaknesses. The interpretation and modelling of experimental measures of spatial variability are discussed and demonstrated with examples based on a hypothetical data set consisting of lead and arsenic measurements collected from a contaminated soil site.

KEYWORDS: Spatial variation, variogram, correlogram.

INTRODUCTION

Unlike most classical statistical studies, in which samples are commonly assumed to be statistically independent, environmental and geotechnical studies involve data that are not statistically independent. Whether we are studying contaminant concentrations in soil, rock and fluid properties in an aquifer, or the physical and mechanical properties of soil, data values from locations that are close together tend to be more similar than data values from locations that are far apart. To most geologists, the fact that closely lManager, FSS Canada Consultants, 800 Millbank, Vancouver, 13

Be, Canada V5V 3K8

14


spaced samples tend to be similar is hardly surprising since samples from closely spaced locations have been influenced by similar physical and chemical processes. This overview paper addresses the description and analysis of spatial dependence in geostatistical studies, the interpretation of the results and the development of a mathematical model that can be used in spatial estimation and simulation. More specific guidance on the details of analysis, interpretation and modelling of spatial variation can be found in the ASTM draft standard guide entitled Standard Guide for Analysis of Spatial Variation in Geostatistical Site Investigations. DESCRIBING AND ANALYZING SPATIAL VARIATION

Using the sample data set presented earlier in this volume in the paper by Cromer, Figure 1 shows an example of a "variogam", the tool that is most commonly used in geostatistical studies to describe spatial variation. A variogram is a plot of the average squared differences between data values as a function of separation distance. If the phenomenon being studied was very continuous over short distances, then the differences between closely spaced data values would be small, and would increase gradually as we compared pairs of data further and further apart. On the other hand, if the phenomenon was completely erratic, then pairs of closely spaced data values might be as wildly different as pairs of widely spaced data values. By plotting the average squared differences between data values (the squaring just makes everything positive so that large negative differences do not cancel out large positive ones) against the separation distance, we can study the general pattern of spatial variability in a spatial phenomenon. Figure 2 shows an example of another tool that can be used to describe spatial variation, the "correlogram" or "correlation function". On this type of plot, we again group all of the available data into different classes according to their separation distance, but rather than plotting the average squared difference between the paired data values, we plot their correlation coefficient. If the phenomenon under study was very continuous over short distances, then closely spaced data values would correlate very well, and would gradually decrease as we compared pairs of data further and further apart. On the other hand, if the phenomenon was completely erratic, then pairs of closely spaced data values might be as uncorrelated as pairs of widely spaced data values. A plot of the correlation coefficient between pairs of data values as a function of the separation distance provides a description of the general pattern of spatial continuity.

SRIVASTAVA ON SPATIAL VARIABILITY

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Figure 1. An example of a variogram using the sample lead data set described by Cromer (1996).

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Figure 2. An example of a correlogram using the sample lead data set described by Cromer (1996).

As can be seen by the examples in Figures 1 and 2, the variogram and the correlogram are, in an approximate sense, mirror images. As the variogram gradually rises and reaches a plateau, the correlogram gradually drops and also reaches a plateau. They are not exactly mirror images of one another, however, and a geostatistical study of spatial continuity often involves both types of plots. There are other tools that geostatistician use to describe spatial continuity, but they all fall into two broad categories: measures of dissimilarity and measures of similarity. The measures of similarity record how dif. ferent the data values are as a function of separation distance and tend to rise like the variogram. The measures of dissimilarity record how similar the data values are as a function of separation distance and tend to fall like the correlogram. INTERPRETING SPATIAL VARIATION

Variograms are often summarized by the three characteristics shown in Figure 3:

Sill: The plateau that the variogram reaches; for the traditional definition of the variogram - the average squared difference between paired data values - the sill is approximately equal to twice the variance of the data. 3 3The "semivariogram", which is simply the variogram divided by two, has a sill that is approximately equal to the variance of the data.

16


Range : The distance at which the variogram reaches the sill; this is often thought of as the "range of influence" or the "range of correlation" of data values. Up to the range, a sample will have some correlation with the unsampled values nearby. Beyond the range, a sample is no longer correlated with other values. Nugget Effect: The vertical height of the discontinuity at the origin. For a separation distance of zero (i.e. samples that are at exactly the same location), the average squared differences are zero. In practice, however, the variogram does not converge to zero as the separation distance gets smaller. The nugget effect is a combination of:

• short-scale variations that occur at a scale smaller than the closest sample spacing • sampling error due to the way that samples are collected, prepared and analyzed

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Figure 3. Terminology commonly used to describe the main features of a variogram. Of the three characteristics commonly used to summarize the variogram, it is the range and the nugget effect that are most directly linked to our intuitive sense of whether the phenomenon under study is "continuous" or "erratic". Phenomena whose variograms have a long range of correlation and a low nugget effect are those that we think of as "well behaved" or "spatially continuous"; attributes such as hydrostatic head, thickness of a soil layer and topographic elevation typically have long ranges and low nugget effects. Phenomena whose variograms have a short range of correlation and a high nugget

17


effect are those that we think of as "spatially erratic" or "discontinuous" j contaminant concentrations and permeability typically have short ranges and high nugget effects. Figure 4 compares the lead and arsenic variograms for the data set presented earlier in this volume by Cromer. For these two attributes, the higher nugget effect and shorter range on the arsenic variogram could be used as quantitative support for the view that the lead concentrations are somewhat more continuous than the arsenic concentrations. (b) Arsenic

(a) Lead

1600 1400 E 1200 ~ 1000

60000 50000 E 40000 ~

.2 30000

·2 CG

~ 20000

>

10000

o

o

20

40

60

80

soo

600 400 200 0

100 120

0


20

40

60

80

100 120


Figure 4. Lead and arsenic variograms for the sample data described by Cromer (1996).

(b) Northeast - Southwest

(a) Northwest - Southeast 60000

60000

50000

50000

~

~ 40000

40000

g, 30000 .c:

g, 30000 .c:

~ 20000

~ 20000

10000

10000

o

o

o 20

40

60

80

100 120


o

20

40

60

80

100 120


Figure 5. Directional variograms for the sample lead data described by Cromer (1996). In many earth science data sets, the pattern of spatial variation is directionally dependent . In terms of the variogram, the range of correlation often depends on direction.

18


Using the example presented earlier in this volume by Cromer, the lead values appear to be more continuous in the NW-SE direction than in the NE-SW direction. Geostatistical studies typically involve the calculation of separate variograms and correlograms for different directions. Figure 5 shows directional variograms for the sample lead data presented by Cromer. The range of correlation shown by the NW-SE variogram (Figure 5a) is roughly 80 meters , but only 35 meters on the NE-SW variogram (Figure 5b). This longer range on the NW-SE variogram provides quantitative support for the observation that the lead values are, indeed, more continuous in this direction and more erratic in the perpendicular direction. MODELLING SPATIAL VARIATION

Once the pattern of spatial variation has been described using directional variograrns or correlograms, this information can be used to geostatistical estimation or simulation procedures. Unfortunately, variograms and correlograms based on sample data cannot provide information on the degree of spatial continuity for every possible distance and in every possible direction. The directional variograms shown in Figure 5, for example, provided information on the spatial continuity every 10 m in two specific directions. The estimation and simulation algorithms used by geostatisticians require information on the degree of spatial continuity for every possible distance and direction. To create a model of spatial variation that can be used for estimation and simulation, it is necessary to fit a mathematical curve to the sample variograms.

(a) Northwest - Southeast 60000

60000

50000

50000

~ 40000

~ ·c

(b) Northeast - Southwest

~ 40000

0-

~

30000

30000

~ 20000

·c ~ 20000

10000

10000

o

o

o 20

40

60

80

100 120


o

20

40

60

80

100 120


Figure 6. Variogram models for the directional sample variograms shown in Figure 5.


19

The traditional practice of variogram modelling makes use of a handful of mathematical functions whose shapes approximate the general character of most sample variograms. The basic functions - the "spherical", "exponential" and "gaussian" variogram models - can be combined to capture the important details of almost any sample variogram. Figure 6 shows variogram models for the directional variograms of lead (Figure 5). Both of these use a combination of two spherical variogram models, one to capture short range behavior and the other to capture longer range behavior, along with a small nugget effect to model the essential details of the sample variograms. In kriging algorithms such as those described later in this volume by Rouhani, it is these mathematical models of the spatial variation that are used to calculate the variogram value between any pair of samples, and between any sample and the location being estimated.

REFERENCES ASTM, Standard Guide for Analysis of Spatial Variation in Geostatistical Site Investigations, 1996, Draft standard from D18.01.07 Section on Geostatistics. Cromer, M.V., 1996, "Geostatistics for Environmental and Geotechnical Applications: A Technology Transfer," Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R. Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Ed., American Society for Testing and Materials, West Conshohocken, PA. Deutsch, C.V. and Journel, A.G., 1992, GSLIB: Geostatistical Software Library and User's Guide, Oxford University Press, New York, 340 p. Isaaks, E.H. and Srivastava, R.M., 1989, An Introduction to Applied Geostatistics, Oxford University Press, New York, 561 p. Journel, A.G. and Huijbregts, C., 1978, Mining Geostatistics, Academic Press, London, 600p. Rouhani, S., 1996, "Geostatistical Estimation: Kriging," Geostatistics for Environmental and Geotechnical Applications, ASTM STP ma, R. Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Ed., American Society for Testing and Materials, West Conshohocken, PA. Srivastava, R.M. and Parker, H.M., 1988, "Robust measures of spatial continuity," Geostatistics, M. Armstrong (ed.), Reidel, Dordrecht, p. 295-308.

Shahrokh Rouhani 1 GEOSTATISTICAL ESTIMATION: KRIGING

REFERENCE: Rouhani, S., "Geostatistical Estimation: Kriging," Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R. Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Alexander 1. Desbarats, Eds., American Society for Testing and Materials, 1996. ABSTRACT: Geostatistics offers a variety of spatial estimation procedures which are known as

kriging. These techniques are commonly used for interpolation of point values at unsampled locations and estimation of average block values. Kriging techniques provide a measure of accuracy in the form of an estimation variance. These estimates are dependent on the model of spatial variability and the relative geometry of measured and estimated locations. Ordinary kriging is a linear minimum-variance interpolator that assumes a constant, but unknown global mean. Other forms of linear kriging includes simple and universal kriging, as well as co-kriging. If measured data display non-Gaussian tendencies, more accurate interpolation may be obtained through non-linear kriging techniques, such as lognormal and indicator kriging. KEYWORDS: Geostatistics, kriging, spatial variability, mapping, environmental investigations. Many environmental and geotechnical investigations are driven by biased or preferential sampling plans. Such plans usually generate correlated, and often clustered, data. Geostatistical procedures recognize these difficulties and provide tools for various forms of spatial estimations. These techniques are COllectively known as kriging in honor of D. G. Krige, a South African mining engineer who pioneered the use of weighted moving averages in the assessment of ore bodies. Common applications of kriging in environmental and geotechnical engineering include: delineation of contaminated media, estimation of average concentrations over exposure domains, as well as mapping of soil parameters and piezometric surfaces (Joumel and Huijbregts, 1978; Delhomme, 1978; ASCE, 1990). The present STP offers a number of papers that cover various forms of geostatistical estimations, such Benson and Rashad (1996), Buxton (1996), Goderya et at. (1996), and Wild and Rouhani (1996). Comparison of kriging to other commonly used interpolation techniques, such as distanceweighting functions, reveals a number of advantages (Rouhani, 1986). Kriging directly incorporates the model of the spatial variability of data. This allows kriging to produce sitespecific and variable-specific interpolation schemes. Estimation criteria of kriging are based on IAssociate Professor, School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0355.

20

ROUHANI ON KRIGING

21

well-defmed statistical conditions, and thus, are superior to subjective interpolation techniques. Furthennore, the automatic declustering of data by kriging makes it a suitable technique to process typical environmental and geotechnical measurements. Kriging also yields a measure for the accuracy of its interpolated values in the fonn of estimation variances. These variances have been used in the design of sampling plans because of two factors: (1) each estimate comes with an estimation variance, and (2) the estimation variance does not depend on the individual observations (Loaiciga et al., 1992). Therefore, the impact of a new sampling location can be evaluated before any new measurements are actually conducted (Rouhani, 1985). Rouhani and Hall (1988), however, noted that in most field cases the use of estimation variance, alone, is not sufficient to expand a sampling plan. Such plans usually require consideration of many factors in addition to the estimation variance. To use the estimation variance as a basis for sampling design, additional assumptions must be made about the probability density function of the estimation error. A common practice is to assume that, at any location in the sampling area, the errors are normally distributed with a mean of zero and a standard deviation equal to the square root of the estimation variance, referred to as kriging standard deviation. The nonnal distribution of the errors has been supported by practical evidence (Journal and Huijbregts, 1978, p. 50 and 60).

Ordinary Kriging Among geostatistical estimation methods, ordinary kriging is the most widely used in practice. This procedure produces minimum-variance estimates by taking into account: (1) the distance vector between the estimated point and the data points; (2) the distance vector between data points themselves; and (3) the statistical structure of the variable. This structure is represented by either the variogram, the covariance or the correlogram function. Ordinary kriging is also capable of processing data averaged over different volumes and sizes. Ordinary kriging is a "linear" estimator. This means that its estimate, Z', is computed as a weighted sum of the nearby measured values, denoted as z!, ~, ... , and Zy,. The fonn of the estimation is n

2:A;Z;

1

;=}

where Ai'S are the estimation weights. Z· can either represent a point or a block-averaged value, as shown in Fig. 1. Point kriging provides the interpolated value at an unsampled location. Block kriging yields an areal or a volumetric average over a given domain. • The kriging weights, Ai' are chosen so as to satisfy two suitable statistical conditions. These conditions are: ! (1) Non-bias condition: This condition requires that the estimator Z· to be free of any ! systematic error, which translates into

I !

i f

~

22


•

Zl

•

z· • Z4

.~

Z:l

• (a)

(b)

Fig. 1. Example of Spatial Estimation: (a) Point Kriging; (b) Block Kriging.

ROUHANI ON KRIGING

8

~

E

Q, Q,

.5 5os Cl 'g .., ....l

'0

..,;.

rIl

."

'./:3 L1 N o.i)

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24


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=1

2

i==/

(2)

Minimum-variance condition: This requires that the estimator Z' have minimum variance of estimation. The estimation variance of Z', d-, is defmed as where Yio is the variogram between the i-th measured point and the estimated location and n

0.2

=

n

n

iLAiY io - 2:LAiAjYij + Y oo i:}

3

i:/ j : }

Yij is the variogram between the i-th and j-th measured points. The kriging weights are computed by minimizing the estimation variance (Eq. 3) subject to the non-bias condition (Eq. 2). The computed weights are then used to calculate the interpolated value (Eq. 1). As Delhomme (1978) notes: "the kriging weights are tailored to the variability of the phenomenon. With regular variables, kriging gives higher weights to the closest data points, precisely since continuity means that two points close to each other have similar values. When the phenomenon is irregular, this does not hold true and the weights given to the closest data points are dampened." Such flexibility does not exist in methods, such as distance weighting, where the weights are pre-defmed as functions of the distance between the estimated point and the data point.

Case Study: Kriging of Lead Data

As noted in Cromer (1996), a soil lead field is simulated as a case study as shown in Fig. 2. The measured values are collected from this simulated field. Similar to most environmental investigations, the sampling activities are conducted in two phases. During the first phase a pseudo-regular grid of 50x50 m is used for soil sampling. In the second phase, locations with elevated lead concentrations are targeted for additional irregular sampling, as indicated in Fig. 3. The analysis of the spatial variability of the simulated field is presented in the previous paper (Srivastava, 1996). Using this information, ordinary kriging is conducted. Fig. 4 displays the kriging results of point estimations. The comparison of the original simulated field (Fig. 2) and the kriged map (Fig. 4) shows that the kriged map captures the main spatial features of lead contamination. This comparison, however, indicates a degree of smoothing in the kriged map which is a consequence of the interpolation process. In cases where the preservation of the spatial variability of the measured field is critical to the study objectives, then the use of kriging for estimation alone is inappropriate and simulation methods are recommended (Desbarats, 1996). Each kriged map is accompanied by its accuracy map. Fig. 5 displays the kriging

25

ROUHANI ON KRIGING

• • • • • • • • • • • •• • • •• • • • • • • • • • • • • • ffII#fI' • fI' • • • • •••~ C • • • • •• •• • •

e

0. 0.

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.",

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Fig. 4. Soil Lead Concentration Map by Ordinary Kriging in ppm (Blank spaces are not estimated)

:Il

oC I

~

Z

o

120

240

oZ A :Il

Fig. 5. Kriging Standard Deviation of Soil Lead Concentration in ppm

Gi Z

Gl

I\)

-..J

28


standard deviation map of soil lead data. zones of high versus poor data coverage.

This latter map can be used to distinguish between

Block Kriging In many instances, available measurements represent point or quasi-point values, but the study requires the computation of the areal or volumetric value over a lager domain. For instance, in environmental risk assessments, the desired concentration term should represent the Depending on the computed average average contamination over an exposure domain. concentration or its upper confidence limit, a block is declared impacted or not-impacted. This shows that the decision is based on the estimated block value, and not its true value. So there is a chance of making error in two forms: (1) Wrong Rejection: Certain blocks will be considered impacted, while their true average concentration is below the target level, and (2) Wrong Acceptance: Certain blocks will be considered not-impacted when their true average concentrations are above the target level. As shown in Iournel and Huijbregts (1978, p. 459), the kriging block estimator, Z', is the linear estimator that minimizes the sum of the above two errors. Therefore, the block kriging procedure is preferred to any other linear estimator for such selection problems.

Alternative Fonns of Kriging As noted before, ordinary kriging is a linear minimum-variance estimator. There are other folms of linear kriging. For example, if the global mean of the variable is known, the nonbias condition (Eq. 2) is not required. This leads to simple kriging. If, on the other hand, the global mean is not constant and can be expressed as a polynomial function of spatial coordinates, then universal kriging may be used. In many instances, added information is available whenever more than one variable is sampled, provided that some relationship exists between these variables. Co-kriging uses a linear estimation procedure to estimate Z' as n

"'

;~J

j~J

Z' = LA,;Z; + LWjY j

4

were Zj is the i-th measured value of the "primary" variable with a kriging weight of A;, and Yj is the j-th "auxiliary" measured value with a kriging weight of {OJ. Co-kriging is specially advantageous in cases where the primary measurements are limited and expensive, while

ROUHANI ON KRIGING

29

~auxiliary measurements are available at low cost. Ahmed and de Marsily (1987) enhanced their 1limited transmissivity data based on pumping tests with the more abundant specific capacity data. This resulted in an improved transmissivity map. The present STP provides examples of co~'kriging, such as Benson and Rashad (1996) and Wild and Rouhani (1996).

i

(Non-linea, Kriging

~'

i

The above linear kriging techniques do not require any implicit assumptions about the distribution of the interpolated variable. If the investigated variable is multivariate . :normal (Gaussian), then linear estimates have the minimum variance. In many cases where \:the histogram of the measured values displays a skewed tendency a simple transformation may ~,produce normally distributed values. After such a transformation, linear kriging may be used. If Ithe desired transformation is logarithmic, then the estimation process is referred to as lognormal ""kriging. Although lognormal kriging can be applied to many field cases, its estimation process ':requires back-transformation of the estimated values. These back transformation are complicated ,~and must be performed with caution (e.g. Buxton, 1996). ~t Sometimes, the observed data clearly exhibit non-Gaussian characteristics, whose log~transforms are also non-Gaussian. Examples of such data sets include cases of measurements i'with multi-modal histograms, highly skewed histograms, or data sets with large number of \~'below-detection measurements. These cases have motivated the development of a set of r~techniques to deal with non-Gaussian random functions. One of these methods is indicator ~'kriging. In this procedure, the original values are transformed into indicator values, such that ~l:they are zero if the datum value are less than a pre-defmed cutoff level or unity if greater. The ~.',I,'.,stimated value by indicator kriging represents the probability of not-exceedence at a location. "I::!,his technique provides a simple, yet powerful procedure, for generating probability maps "I~OUhani and Dillon, 1990). ~:"underlying

, I

:~~

:~i

~Recommended Sources

'l

i

~or more information on ~iging, readers are referred to Journel and Huijbregts (1978), Marslly (1986), Isaaks and Snvastava (1989), and ASCE (1990). ASTM Standard D 5549, titled: "Standard Guide for Content of Geostatistical Site Investigations," provides information on the various elements of a kriging report. ASTM DI8.0l.07 on Geostatistics has so drafted a guide titled: "Standard Guide for Selection of Kriging Methods in Geostatistical . ite Investigations." This guide provides recommendations for selecting appropriate kriging methods based on study objectives and common situations encountered in geostatistical site ;investigations .

~de

30


References

(1)

(2)

(3)

(4)

(5)

(6) (7)

(8)

(9) (10) (11)

(12)

ASCE Task Committee on Geostatistical Techniques in Geohydrology, "Review of Geostatistics in Geohydrology, 1. Basic Concepts, 2. Applications," ASCE Journal of Hydraulic Engineering, 116(5), 612-658, 1990. Ahmed, S., and G. de Marsily, "Comparison of geostatistical methods for estimating transmissivity using data on transmissivity and specific capacity," Water Resources Research, 23(9), 1717-1737, 1987. Benson, C.H., and S.M. Rashad, "Using Co-kriging to Enhance Subsurface Characterization for Prediction of Contaminant Transport," Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Eds., American Society for testing and Materials, Philadelphia, 1996. Buxton, B.E., "Two Geostatistical Studies of Environmental Site Assessments," Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Eds., American Society for testing and Materials, Philadelphia, 1996. Cromer, M., "Geostatistics for Environmental and Geotechnical Applications," Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Eds., American Society for testing and Materials, Philadelphia, 1996. Delhomme, J.P., "Kriging in the hydro sciences , " Advances in Water Resources, 1(5), 251-266, 1978. Desbarats, A., "Modeling of Spatial Variability Using Geostatistical Simulation," Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Eds., American Society for testing and Materials, Philadelphia, 1996. Goderya, F.S., M.F. Dahab, and W.E. Woldt, "Geostatistical Mapping and Analysis of Spatial Patterns for Farm Fields Measured Residual Soils Nitrates," Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Eds., American Society for testing and Materials, Philadelphia, 1996. Isaaks, E. H. and R M. Srivastava, An Introduction to Applied Geostatistics, Oxford University Press, New York, 561 p., 1989. Joumel, A. G. and C. Huijbregts, Mining Geostatistics, Academic Press, London, 600 p.,1978. Loaiciga, H.A., RJ. Charbeneau, L.G. Everett, G.E. Fogg, B.F. Hobbs, and S. Rouhani, "Review of Ground-Water Quality Monitoring Network Design," ASCE Journal of Hydraulic Engineering, 118(1), 11-37, 1992. Marsily, G. de, Quantitative Hydrogeology, Academic Press, Orlando, 1986.

ROUHANI ON KRIGING

(13) {14) (15)

. (16)

(17)

(18)

31

Rouhani, S., "Variance Reduction Analysis", Water Resources Research, Vol. 21, No.6, pp. 837-846, June, 1985. Rouhani, S., "Comparative study of ground water mapping techniques," Journal of ~ Ground Water, 24(2), 207-216, 1986. Rouhani, S., and M. E. Dillon, "Geostatistical Risk Mapping for Regional Water Resources Studies," Use of Computers in Water Management, Vol. 1, pp. 216-228, V/O "Syuzvodproekt", Moscow, USSR, 1989 . Rouhani, S., and Hall, T.J., "Geostatistical Schemes for Groundwater Sampling," Journal of Hydrology, Vol. 103, 85-102, 1988. Srivastava, R. M., "Describing Spatial Variability Using Geostatistical Analysis," Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R. Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Eds., American Society for testing and Materials, Philadelphia, 1996. Wild, M.R., and S. Rouhani, "Effective Use of Field Screening Techniques in Environmental Investigations: A Multivariate Geostatistical Approach," Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R. Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Eds., American Society for testing and Materials, Philadelphia, 1996.

'7

Alexander J. Desbarats 1

MODELING SPATIAL VARIABILITY USING GEOSTATISTICAL SIMULATION

REFERENCE: Desbarats, J. A., "Modeling Spatial Variability Using Geostatistical Simulation," Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R. M. Srivastava, S. Rouhani, M. V. Cromer, A. I. Johnson, 1. A. Desbarats, Eds., American Society for Testing and Materials, 1996. ABSTRACT: This paper, the last in a four part introduction to geostatistics, describes the application of simulation to site investigation problems. Geostatistical simulation is a method for generating digital representations or "maps" of a variable that are consistent with its values at sampled locations and with its in situ spatial variability, as characterized by histogram and variogram models. Continuing the synthetic case study of the three previous papers, the reader is lead through the steps of a gebstatistical simulation. The si;nulated fields are then compared with the exhaustive data sets describing the synthetic site. Finally, it is shown how simulated fields can be used to answer questions concerning alternative site remediation strategies. KEYWORDS : Geostatistics, kriging, simulation, variogram

INTRODUCTION

In a geostatistical site investigation, after we have performed an exploratory analysis of our data and we have modeled its spatial variation structure, the next step is usually to produce a digital image or "map" of the variables of interest from a set of measurements at scattered sample locations. We are then faced with a choice between two possible approaches, estimation and simulation. This choice is largely dictated by study objectives. Detailed guidance for selecting between these two approaches and among the various types of simulation is provided in the draft ASTM Guide for the Selection of Simulation Approaches in Geostatistical Site Investigations. Producing a map from scattered measurements is a classical spatial estimation problem that can be addressed using a non-geostatistical interpolation method such IGeological Survey of Canada, 601 Booth st., Ottawa, ON KIA DES, Canada

32

DESBARATS ON SPATIAL VARIABILITY

33

as inverse-distance weighting or, preferably, using one of the least-squares weighting methods collectively known as kriging discussed in Ro~hani (this volume). Regardless of the interpolation method that is selected, the result is a representation of our variable in which its spatial variability has been smoothed compared to in situ reality. Along with this map of estimated values, we can also produce a map of estimation (or error) variances associated with the estimates at each unsampled location. This map provides a qualitative or, at best, semi-quantitative measure of the degree of uncertainty in our estimates and the corresponding level of smoothing we can expect. Unfortunately, maps of estimated values, even when accompanied by maps of estimation variances, are often an inadequate basis for decision-making in environmental or geotechnical site investigations. This is because they fail to convey a realistic picture of the uncertainty and the true spatial variability of the parameters that affect the planning of remediation strategies or the design of engineered structures. The alternative to estimation is simulation. Geostatistical simulation (Srivastava, 1994) is a Monte-Carlo procedure for generating outcomes of digital maps based on the statistical models chosen to represent the probability distribution function and the spatial variation structure of a regionalized variable. The simulated outcomes can be further constrained to honor observed data values at sampled locations on the map. Therefore, not only does geostatistical simulation allow us to produce a map of our variable that more faithfully reproduces its true spatial variability, but we can generate many equally probable alternative maps, each one consistent with our field observations. A set of such alternative maps allows a more realistic assessment of the uncertainty associated with sampling in heterogeneous geological media. This paper presents an introduction to the geostatistical tool of simulation. Its goals are to provide a basic understanding of the method and to illustrate how it can be used in site investigation problems. To do this, we will continue the synthetic soil contamination case study started in the three previous papers. We will proceed step by step through the simulation study, pausing here and there to compare our results with the underlying reality and the results of the kriging study (Rouhani, this volume). Finally, we will use our simulated fields to answer some questions that can arise in actual soil remediation studies. STUDY OBJECTIVES

The objective of our simulation study is to generate digital images or maps of lead (Pb ) and arsenic (As) concentrations in soil. We will then use these maps to determine the proportion of the site area in which Pb or As concentrations exceed the remediation thresholds of 150 ppm and 30 ppm, respectively. The maps are to reproduce the histograms and variograms of Pb and As in addition to observed measurements at sampled locations. Although the full potential of the simulation method is truly achieved only in sensitivity or risk analysis studies involving multiple outcomes of the simulated maps, we will focus on the generation of a single outcome. In many respects, even a sin~l~Il!~p of simulated concentrations is more useful than a map of kriged values. This is because a realistic portrayal of in situ spatial variability is often a sobering warning to planners whereas maps of kriged values are easily

'I

I

,I

"I "!

!

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I"~

iII'!.!; :

i

ii,

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34


misinterpreted as showing much smoother spatial variations. For our study, we have chosen the concentrations of Pb and As as the two regionalized variables to work with. This may seem like an obvious choice however we could have taken another approach based on an indicator or binary transformation of our original variables. The new indicator variables corresponding to each contaminant would take a value of 1 if the concentration exceeds the remediation threshold and a value of 0 otherwise. Proceeding in a somewhat different manner than shown here, we could then generate maps of simulated indicator variables for the two contaminants. From such maps, the proportion of the site requiring remediation is readily determined. The drawback with an indicator approach is that we have sacrificed detailed knowledge of contaminant concentrations in exchange for simplicity and conciseness. Should the remediation thresholds change, new indicator variables would have to be defined and the study repeated. Here, we will stick with the more involved but also more flexible approach of simulating contaminant concentrations. An application of indicator simulation is described in Cromer et al. (this volume). HISTOGRAM MODELS

The first step in our simulation study is to decide what probability distribution functions or, more prosaically, what histogram models are to be honored by our simulated concentrations. We would like these histograms to be representative of the entire site. Often, the raw histograms of sample data are the most appropriate choice. However, here this isn't the case: The sampling of our contaminated site was carried out in two stages. In the first stage, we obtained 77 measurements of Pb distributed on a fairly regular grid. In the second stage, we focused our sampling on areas identified in the first stage as having high Pb concentrations. Furthermore, by then we had become aware that arsenic contamination was present and we analyzed an additional 135 samples for both Pb and As. Thus, our Pb data consist of 77 values that are probably representative of the entire site area and another 135 values drawn from the most contaminated region. As for arsenic, our 135 samples were obtained exclusively from the most contaminated region and are probably not representative of the entire site. The raw histograms of Pb and As shown in Cromer (this volume) reflect the preferential or biased sampling procedure and do not provide adequate models for our simulation. The answer to this problem is to weight our sample data in such a way as to decrease the influence of clustered measurements while increasing that of more isolated values. In geostatistics, this exercise is known as "declustering" and can be accomplished several ways (Isaaks and Srivastava, 1989; Deutsch and Journel, 1992). Here we used a cell declustering scheme to find sample weights. This involved moving a 10 x 10 unit cell over N non-overlapping positions covering the study area. At a each cell position, the number n of samples within the cell was counted and each sample was then assigned a relative weight of liNn. This procedure may be expected to work well for Pb but for As there is no escaping the fact that our samples are restricted to a few small, highly contaminated patches and are hardly representative of the site as a whole. Obtaining a reasonably representative histogram is crucial for a simulation

35


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100.

1.0.

Figure 1: Declustered histograms of a) Pb and b) As . study therefore desperate measures are called for. Although no geostatistical method can truly compensate for lack of data, the following "fix" was attempted here: Using our knowledge of the correlation between Pb and As provided by the 135 samples of the second sampling campaign (Cromer, this volume), we filled in the missing As values at the 77 locations of the first campaign. For each of the 77 Pb values, we looked up the closest Pb value from the second campaign and read off the corresponding As value. Thus, all 212 sample locations have both Pb and As measurements and the same declustering weights can be used for both variables. The resulting histograms of weighted Pb and As samples are shown in Figures 1 a) and b), respectively. They should be compared with the un-declustered histograms shown in Cromer (this volume). We now have histograms that, we think, provide reasonable models of the exhaustive distributions of Pb and As that we are trying to replicate in our simulated fields. A peek at the true exhaustive distributions (Cromer, this volume) shows that our declustered Pb histogram does a fairly good job of reproducing the main statistical parameters whereas our As histogram does a rather mediocre job despite our best efforts. Further discussion of the declustering issue can be found in Rossi and Dresel (this volume). NORMAL-SCORE TRANSFORMATION OF VARIABLES

The next step of our study involves transforming our Pb and As sample values into standard Normal deviates (Deutsch and Journel, 1992). This "normal-score" transformation is required because the simulation algorithm we will be using is based on the multivariate Normal (or Gaussian) distribution model and assumes that all sam-

200.

36


pIe data are drawn from such a distribution. In simple terms, this transformation is performed by replacing the value correponding to a given quantile of the original distribution with the value from a standard Normal distibution associated with the same quantile. For example, a Pb value of 261 ppm correponding to a quantile of 0.50 (i.e. the median) in the sample histogram is transformed into a value of 0 corresponding to the median of a standard Normal distribution. In mathematical terms, we seek the transformations ZI and Z2 of Pb and As such that :

G(Zd

(1 )

where G( ) is the cumulative distribution function (cdf) of a standard Normal distribution and FI ( ) and F 2 ( ) are the sample cdfs for lead and arsenic, respectively. Implementation of this transformation is fairly straightforward except when identical sample values are encountered. In such cases, ties are broken by adding a small random perturbation to each sample value and ranking them accordingly (Deutsch and Journel, 1992). Here, this "despiking" procedure was required to deal with a large number of below-detection As values. In general however, it is good practice to avoid extensive recourse to this procedure. If, for example, large numbers of samples have values below detection limits, it is better to subdivide the data set into two populations, above and below detection, and analyze each group seperately, or adopt an indicator approach (Zuber and Kulkarni, this volume). PRINCIPAL COMPONENT TRANSFORMATION

Before we can proceed to the analysis of spatial variation, one last step is required. Our simulation algorithm can only be used to generate fields of one variable at a time. However, we wish to simulate two variables ZI and Z2, reproducing not only their respective spatial variation structures but also the relationship between them shown in Figure 2. We must therefore "decouple" the variables ZI and Z2 so that we can simulate them independently. To do this, we use the following principal component transformation which yields the independent variables Yi and 12 from the correlated variables ZI and Z2 :

(2)

12 = where p is the correlation coefficient between ZI and Z2 which is found to be 0.839.

DESBARATS ON SPATIAL VARIABILITY Pb

II'S As

37

: Declustered Normal Scores Nurrber 01 daIa 135 Nurrbertrlmmed n

3.00

X Varillb..: mean 0.887 otd. daY. 0.783

0 0

.;i-';.

2.00

0

1.00

~ 0.00

Y Varillb..: mean 0.80S Old. dey. 0.806

00

00

0

,

0 0

..

0

0

correlation 0.839 rank correlation 0.869

0

0

0

-1.00 0

0

-2.00

.3.00-h-T""I"''TTTT"'TTTT"......,..,...'''..,..,.....,..,..,..,....,..,..,..,...,.... -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 Z1

Figure 2: Scatter plot of Zl and Z2 for 135 sample values. The correlation coefficient is 0.839. VARIOGRAM MODELS

In this section, we examine and model the spatial variation structure of the two independent variables Y1 == Zl and l'2. The jargon and the steps involved in an analysis of spatial variation are described in more detail by Srivastava (this volume) so that only a summary of results is given here. Directional variograms, or more specfically correlograms, were calculated for Y1 using all 212 data values, and for 1'2 using the 135 values of the second sampling campaign. For each variable, eight directional correlograms were calculated at azimuth intervals of 22.5° using overlapping angular tolerances of 22.5°. Lag intervals and distance tolerances were 10 grid units and 5 grid units, respectively for Yi and 5 grid units and 2.5 grid units, respectively for 1'2. The purpose of these directional correlograms is to reveal general features of spatial variation such as directional anisotropies and nested structures. Results for Yi. and Y2 are shown in Figures 3 a) and b), respectively. These figures provide a planimetric representation of spatial correlation structure, displaying correlogram values as a surface, function of location in the plane of East~West (x) and North-South (y) lag components. For Yi., we observe, in addition to a significant nugget effect, what we interpret as two nested structures with different principal directions of spatial continuity. The first, shorter scale, structure has a direction of maximum continuity approximately North North-West, a maximum range of about 20 grid units and an anisotropy ratio of about 1.4 : 1. The second, larger scale structure has a direction of maximum continuity approximately West North-West, an indeterminate maximum range and a minimum range of at least 40 grid units.

38


--'§.s

40.0

20.0

"d .~

-

20.0

10.0

...:I

0.0

0.0

-20.0

-10.0

~

~

::s0 rn I

:E

Z

-40.0 -40.0 -20.0

0.0

20.0

-20.0 40.0 -20.0 -10.0

East-West Lag (grid units)

0.0

10.0

20.0


Figure 3: Directional correlograms (planimetric view) for a)

Yi and b) Y2.

Detailed experimental correlograms were calculated in the directions of maximum and minimum continuity of the larger scale structure. They are shown in Figures 4 a) and b), respectively. These correlograms were used in the fitting of a model to the spatial variation structure of Yi. The fitted model is the sum of three components: 1. A nugget effect accounting for 35% of the spatial variance.

2. A short-scale structure accounting for 50% of the spatial variance. It is represented by an exponential model with maximum continuity in the North NorthWest direction. It has a range parameter of 6 grid units and an anisotropy ratio of 1.43 : 1. 3. A large-scale structure accounting for 15% of the spatial variance. This structure is also represented by an exponential model with, however, maximum continuity in the West North- West direction. The model range parameter is 300 grid units with an anisotropy ratio of 10 : 1. Such a large maximum range ensures that the "sill" value of the structure is not reached in the direction of maximum continuity within the limits of the site. What we have done here is model a "zonal anisotropy" (Journel and Huijbregts, 1978) as a geometric anisotropy with an arbitrarily large range in the direction of greatest continuity. This model is also ·shown in Figures 4 a) and b) for comparison with the experimental results. For Y2 , there are less data and we are careful not to over-interpret the directional correlograms. Indeed, the apparent periodicity in the NNE direction is probably an artifact of the sampling pattern . With somewhat more confidence, we note a

39


0.800 Co".log,.m

,*,d:Z1

1I1/:Z1

dlrecHon 1

0.600 Comlog,.m

0.700

0.500

0.600

0.

direction 2

0.300

0.500

~

/wd:Zf

II/I/:Z1

~

f

0.«10

>

0.300

0.200 0.100

0.200

0.000

0.100

.Q.loo .Q.2OO

0.000 0.0

10.0

20.0

30.0

40.0

SO.O

60.0

Dstanoe

0.0

10.0

20.0

30.0

40.0

SO.O

I

60.0

Dstanoe

Figure 4: Experimental correlograms for }'J in a) direction WNW; b) direction NNE. The fitted model is shown in dashed lines. strong nugget effect and a structure with maximum continuity in the West NorthWest direction, a maximum range of about 30 grid units and an anisotropy ratio of about 3 : 1. Detailed directional correlograms were calculated for the directions of maximum and minimum continuity and are shown in Figures 5 a) and b), respectively. The model fitted to these correlograms is the sum of two components: 1. A nugget effect accounting for 55% of the spatial variance. 2. A structure accounting for 45% of the spatial variance. This structure is represented by an exponential model with greatest continuity in the West North-West direction . The model has a range parameter of 8 grid units and an anisotropy ratio of 3 : 1. This mQdel is shown in Figures 5 a) and b) for comparison with the experimental results. SEQUENTIAL GAUSSIAN SIMULATION

We are now ready to simulate fields of the two independent standard Normal variables, Y1 and Y2 • The simulations of Y1 and Y2 are to be conditioned on 212 and 135 sample values, respectively. Both fields are simulated on the same 110 x 70 grid as the exhaustive data sets for Pb and As . To perform our simulations, we are going to use the Sequential Gaussian method. This method is based on two important theoretical properties of the multivariate Normal (or Gaussian) distribution: First, the conditional distribution of an unknown

40


0.600 Corre/ogram

IaIl:Y2

heIId:Y2

dlreclJon 1

0.600 Cotrelogram

0.500

direction 2

0.400

\

0.300

~

_:Y2

0.500

0.400

! 8'

IIIII:Y2

0.300

" "- "

0.200 0.100

! 8'

~

'"

0.200 0.100

---

0.000

"-

·0.100

-0.100

·0.200

-0.200 0.0

5.0

10.0

15.0

20.0

25.0

30.0

0.0

5.0

Distance

Distanct

Figure 5: Experimental correlograms for Y2 in a) direction WNW; b) direction NNE. The fitted model is shown in dashed lines. variable at a particular location, given a set of known values at nearby locations, is Normal. Second, the mean and variance of this conditional distribution are given by the simple kriging (SK) estimate of the unknown value and its associated error variance. Simple kriging is a variant of ordinary kriging (OK) described in Rouhani (this volume). Then, it follows that since the conditional distribution is Normal, it is completely determined by the mean and variance provided by simple kriging. The Sequential Gaussian Simulation algorithm is described in detail elsewhere (Deutsch and Journel, 1992; Srivastava, 1994) ; however, because of its simplicity, it is briefly outlined here: 1. Start with a set of conditioning data values at scattered locations over the field to be simulated. 2. Select at random a point on the grid discretizing the field where there is not yet any simulated or conditioning data value. 3. Using both conditioning data and values already simulated from the surrounding area, calculate the Simple Kriging estimate and corresponding error variance. These are the mean and variance of the conditional distribution of the unknown value at the point given the set of known values from the surrounding area. 4. Select at random a value from this conditional distribution. 5. Add this value to the set of already simulated values.


41

6. Return to step 2 and repeat these steps recursively until all points of the discretized field have been assigned simulated values. Thus, in many ways, the Sequential Gaussian simulation method is similar to the point kriging process described by Rouhani (this volume). The difference is that we are drawing our simulated value at random from a distribution having the kriged estimate as its mean, rather than using the kriged estimate itself as a "simulated" value. Intuitively, we see how this process leads to fields having greater spatial variability than fields of kriged values. BACK-TRANSFORMATIONS

We now have simulated fields of the two independent standard Normal variables Y1 and )12. In order to obtain the corresponding fields of Pb and As , we must reverse the earlier transformations. First, we reverse equations (2) to get the correlated standard Normal variables Zl and Z2 from Yi and Y2. Then we reverse equations (1) to get the variables Pb and As from the standard Normal deviates Zl and Z2. Finally, we are left with simulated fields of Pb and As on a dense 110 x 70 grid discretizing the site. Although here we are focusing on single realizations of each of these fields, multiple realizations can be generated by repeating the simulation step using different seed values for the random number generator. COMPARISON OF TRUE AND SIMULATED FIELDS

In addition to honoring values of Pb and As at sampled locations, the simulated fields should reproduce the histogram and correlogram models that we used to characterize contaminant spatial variability. These fields should also reproduce the correlation between Pb and As . Because this is a synthetic case study, we have exhaustive knowledge of Pb and As contamination levels over the entire site, something we would never have in practice. Therefore, we can conduct a postmortem of our study, comparing our simulated fields with the exhaustive fields described in Cromer (this volume). In order to compare the distributions of true and simulated values, we will use what is known as a Q-Q plot. This involves plotting the quantile of one data set against the same quantile of the other data set. Thus, we would plot the median (0.5 quantile) of our simulated values against the median of our true values. Therefore, if the histograms of the two data sets are similar, all points should plot close to the 45° line. Figures 6 a) and b) show Q-Q plots between exhaustive and simulated values of Pb and As , respectively. These results show that while we did a rather good job of reproducing the exhaustive histogram of Pb , we can claim no great success for As . Although we did our best to correct for the effects of a grossly unrepresentative sampling of As , in the end, this was not good enough. This failure serves as a reminder that geostatistics alone cannot compensate for a biased site sampling campaign. Next, we check how well our simulation reproduced the correlation between Pb and As concentrations. Figure 7 shows a scatter plot of simulated Pb versus As values. This figure is to be compared with the scatter plot of true values given by

42

GEOSTATISTICAL APPLICATIONS 200. Q-Q Plot for A. Slmul.tlon

1000. Q-Q Plot for Pb Slmul.tlon

160.

800.

]

600.

."

i

~ r/)

I

120.

1~

80.

j

~

400.

r/)

40.

200.

o.

200.

400.

600.

True valu••

800.

1000.

o.

40.

80.

120.

160.

200.

True vw..

Figure 6: Q-Q quantile plots of exhaustive and simulated data: a) Pb ; b) As Cromer (this volume). The comparison shows that we did quite a respectable job of reproducing the relationship between Pb and As in our simulated fields. Directional correlograms calculated on our two simulated fields are shown in Figures 8 a) and b). The main features of these correlograms compare favorably with those observed in the correlograms presented by Srivastava (this volume). Given the limited number of data and their spatial clustering, the models we fitted to the experimental correlograms were quite successful in representing the true spatial variation structures of Pb and As . No comparison of true and simulated fields would be complete without looking at images or maps of the simulated fields. Although qualitative, the visual comparison of simulated and true fields is in fact the most stringent measure of the success of our simulation. We must check how well we have captured the character of contaminant spatial variability at the site, its "noisyness", the grain of any spatial patterns, and any trends. We should also check to see what our simulated values are doing in areas far from conditioning data values. The spatial variability in such areas should be consistent in character with that observed in more densely sampled areas.'" Grey-scale digital images of simulated Pb and As fields are shown in Figures 9 and 10, respectively. Comparison with the corresponding true images in Cromer (this volume) shows that we have reason to be satisfied with our simulation. Discrepancies between simulated and true fields exist; however, these are manifestations of the uncertainty associated with our knowledge of site contamination as provided by the rather limited sampling data. It should be emphasized that we are looking at but one pair of images of contamination from amongst the many equally possible alternatives that would be consistent with sampling information. We can also compare Figure 9


43

vs As : Simulated values Number of data 7700 X Variable: mean 296.479 std. dey. 225.313

160.

Y Variable: mean 15.619 std. dey. 30.076 correlation 0.722 rank correlation 0.881

o.

200.

600.

400.

800.

1000.

Pb

Figure 7: Scatter plot of simulated Ph and As values

40.0 - -- - - "u-.::----,'-J,..,----.;:,-----, 20.0 t-=l

~

0.0

as~

-20.0

~ Z

20.0

-20.0 0

-40. 0 ~----r--"'r----Y'-----,""::"-fL-.----Y----, -40. 0 ~~--f.:""""';~,uLr--.-'-~..c:; -40.0 -20.0 0.0 20.0 40.0 -40.0 -20.0 0.0 20.0 40.0



Figure 8: Directional correlograms (planimetric view ) for simulated a) Ph and h) As .

44


0.0 .-.

.....s

§

~ 0 0 00

20.0

'1:l .~

.-.

S p.

'-'

til)

.S

40.0

p.

'-'

~ 0 0

~

Z

S!

~

60.0 0.0

20.0

40.0

60.0

80.0

100.0

Easting (grid units)

0

0

Figure 9: Grey-scale digital image of the simulated Pb field. with the kriged field shown in Rouhani (this volume) . We see that kriging smoothes spatial variations in a non-uniform manner: less in regions with abundant sample control, more in unsampled regions . This may lead the unsuspecting to conclude that large portions of the site are quite homogeneous! Simulation, on the other hand, preserves in-situ spatial variability regardless of the proximity of sampling points. APPLICATION

Now that we have simulated fields of Pb and As that we confidently assume are representative of the true yet unknown contamination at the site, we can use these fields to answer some simple questions . Perhaps the most basic question that we may ask is what fraction of the site requires remediation given the contamination thresholds of 150 ppm and 30 ppm for Pb and As , respectively? However, before we can attempt to answer that question, we must decide on a "volume of selective remediation" or VSR. Note that the concept of volume of selective remediation is identical to that of selective mining unit (smu) described in the mining geostatistics literature (chapter 6 of Journel and Huijbregts, 1978; chapter 19 of Isaaks and Srivastava, 1989). The volume, or in the present case, area of selective remediation is the smallest portion of soil that can be either left in place or removed for treatment, based upon its average contaminant concentration. The VSR may depend on several factors including the size of equipment being used in the remediation and the sampling information ultimately available for the selection process. It is an important· design parameter because the variance of spatially averaged concentrations decreases as the


45

0

0

0.0

to

,-..

~ .....

§

20.0 0

'0 .~

0

"

'E Q)

3.0

:

(/)

2.0

:

450.0

500.0

......... : .......... .:. .... . .

1.0 0.0 0.0

:

......... '[ " ......... -;- .......... ! .... . .. . . . ~

50.0

100.0

150.0

200.0

250.0

300.0

350.0

400.0

separation distance (Days) Fig. 3-- Tempora l Semivariogram from joi nt spatial- t emporal a n alysis of hydraulic he ad l evels . No t e tha t 1 foot; 30 cm.

BUXTON ET AL. ON SITE CHARACTERIZATION

57

TABLE 1--Fitted semivariogram models of spatial and temporal correlation. (Note that 1 foot = 30 cm.) Data

Semivariogram Model

Joint Spatial-Temporal Hydraulic Head Pressure in ft.

K = 3 Nested Structures, Nugget Variance o ft.2 1. Geometric Anisotro~ic Spherical, variance = 1.5 ft. , Spatial Range 1200 ft., Temporal Range = 30 days 2. Geometric Anisotro~ic Spherical, variance = 5.5 ft. , Spatial Range 7000 ft., Temporal Range = 700 days 3. Zonal Gaussian in Spatial NS Direction, Variance = 22 ft.2, Spatial NS Range = 6060 ft.

Steady-State Hydraulic Head Pressure in ft. (June, 1993 Data)

K = 2 Nested Structures, Nugget Variance ft.2 1. Isotropic Linear, Slope = 0.00045 ft. 2/ft. 2. Zonal Gaussian in Horizontal NS Direction, Variance = 13 ft.2, NS Range = 8660 ft.

1990 Uranium Levels in flg/L

K = 1 Structure, Nugget Variance = 0.3 [In(flg/ L )]2 1. Geometric Anisotropic Spherical, Variance = 2.7 [In(flg/L)]2, Horizontal Range = 3000 ft., Vertical Range = 120 ft.

0

Steady-State Analysis One primary reason for performing the joint spatial-temporal analysis in the previous section was to select, for the steady-state analysis, a single month which was representative of average hydraulic head levels during the 1990-1993 time period. Examining the results in Figure 4, it appears that three months can be considered representative: January, 1990; November, 1991; and June, 1993. Hydraulic heads in each of these three months appear to be approximately equal to the average head levels across the entire 1990-1993 time period. However, several new wells were installed in the area in 1993, particularly in the southeastern part of the modeling grid. Therefore, a significantly greater number of head measurements were available for June, 1993 in comparison with January, 1990 and November, 1991. As a result, June, 1993 was selected as the month to represent steady-state conditions. Hydraulic head measurements for June, 1993 were available for the steady-state kriging analysis in 202 wells at various depths. The horizontal spatial semivariograms for these data are shown in Figure 5. As in the joint spatial-temporal analysis (Figure 2), horizontal semivariograms in Figure 5 were calculated in four primary directions. The fitted model is also shown in Figure 5, where the bold line denotes the model in the north direction; and the dashed line denotes the model in the east direction. A kriging analysis was performed with the June, 1993 data and the semivariogram model shown in Figure 5, as well as Table 1. This analysis estimated steady-state head levels across the groundwater modeling grid at regular 5 ft (1.5 m) vertical intervals from 390 to 540 ft (117 to 162 m) above sea level. The horizontal variability in steady-state head levels at the 490 ft (147 m) elevation is shown in

!

il

58

i

.! ...... ~Q)

:r:

"0

....Q)a1 E :;:; CJ)

W


527 526 525 524 523 522 521 520 519 518 517 516 515 514 513 512 511 510 509 508 507 506

0

10

20

30

40

50

Time (months)

Grid Location

>!!= 50

Fig. 13 - -Mean estimate of 1990 uranium concentrations

(~g / L).

68


120 110 100

90

~

80

8

70

·c

"0

60

l:

50

:0

,g -e 0 z

40 30 20 10

o

10

20

30

40

50

60

70

80

90

100

110

120

East (grid blocks)

Krg. Prec.

•••

< 150 < 240

•••

< 180

••• >

= 240

•••

< 210

Fig . 14--Additive uncertainty factor (width of 95% confidence interv al) in ~g / L for mean estimates of 1 9 90 uranium concentrations .

Pierre Colin/ Roland Froidevaux/ Michel Garcia3 and Serge Nicoletis4

INTEGRATING GEOPHYSICAL DATA FOR MAPPING THE CONTAMINATION OF INDUSTRIAL SITES BY POLYCYCLIC AROMATIC HYDROCARBONS: A GEOSTA TISTICAL APPROACH

REFERENCE: Carr, J. R., "Revisiting the Characterization of Seismic Hazard Using Geostatistics: A Perspective After the 1994 Northridge, California Earthquake," Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R. Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, and Alexander J. Desbarats, Eds., American Society for Testing and Materials, 1996. ABSTRACT: A case study is presented of building a map showing the probability that the concentration in polycyclic aromatic hydrocarbon (P AH) exceeds a critical threshold. This assessment is based on existing PAH sample data (direct information) and on an electrical resistivity survey (indirect information). Simulated annealing is used to build a model of the range of possible values for P AH concentrations and of the bivariate relationship between P AH concentrations and electrical resistivity. The geostatistical technique of simple indicator kriging is then used, together with the probabilistic model, to infer, at each node of a grid, the range of possible values which the P AH concentration can take. The risk map is then extracted for this characterization of the local uncertainty. The difference between this risk map and a traditional iso-concentration map is then discussed in terms of decision-making. KEYWORDS: polycyclic aromatic hydrocarbons contamination, geostatistics, integration of geophysical data, uncertainty characterization, probability maps, simulated annealing, bivariate distributions, indicator kriging.

IHead, Environmental and Industrial Risks, Geostock, Rueil-Malmaison, France 2Manager, FSS Consultants SA, Geneva, Switzerland 3Manager, FSS International r&d, Chaville, France 4Head, Geophysical Services, Geostock, Rueil-Malmaison, France

69

70


Steelwork and coal processing sites are prone to contamination by polycyclic aromatic hydroccu"bons (P}Jl), SOnte of which are known to be carcinogenic. Consequently, local and state regulatory agencies require that all contaminated sites be characterized and that remediation solutions be proposed. The traditional approach for delineating the horizontal and vertical extent of the contamination is to use wells and boreholes to construct vertical profiles of the contamination at several locations. This approach, however, is both time consuming and expensive. Recent work has shown that, in some situations, electrical conductivity and resistivity surveys could be used as a pathfinder for delineating the contaminated areas. These geophysical surveys, which are both expedient and cost effective, could be used to reduce the number of wells and boreholes to be drilled. Geophysical data, however, do not provide direct information on soil chemistry. They are indicative of the ground nature, which in turn may reflect human activities (backfill material, tar tanks) and potential sources of ground pollution. These geophysical data have to be treated as indirect and imprecise infonnation. The mapping of the contamination therefore requires that imprecise geophysical data be correctly integrated with precise chemical analyses from wells and boreholes. Geostatistics offers an ideal framework for addressing such problems. Different types of information can be integrated in a manner which takes into consideration not only the statistical correlation between the different types of infonnation, but also the spatial continuity characteristics of both. Using this approach, it is possible to provide maps showing the probability that the P AH exceeds some critical level. A case study from an industrial site in Lorraine (northern France) is used to compare the geostatistical approach to the traditional approach of directly contouring the data from wells and boreholes. AVAILABLE DATA The available information consisted of chemical measurements of P AH concentrations (in ppm) from 51 boreholes. Figure 1 shows the location of these boreholes. As can be seen, the coverage is not even, and the lower right quadrant of the map is undersampled. In terms of distribution of the concentration values, it is seen that the highest concentrations are located toward the middle, where the cokeworks were located. The geophysical information included both conductivity measurements (electromagnetic survey) and resistivity measurements (dipole-dipole electrical measurements). The electromagnetic survey made it possible to investigate the overall site area and to identify anomalous zones where more accurate resistivity measurements were carried out. These resistivity measures are average values over large volumes of soil and are directly related to the soil nature (recent alluvium, slag deposits and other backfill material). They depend also on soil heterogeneities and the spatial arrangement of these heterogeneities. Although the physical phenomena that govern P AH transport, and the reactions between P AH and soils of different nature are not yet well understood (they are the subject of ongoing research projects), the presence of P AH in significant amounts has been found to be be

COLIN ET AL. ON GEOPHYSICAL DATA

71

associated with low resistivity values (i.e it increases, locally, the soil conductivity). This relationship, however, remains site specific and cannot be considered as a general law. The available resistivity measurements (in ohm-meter) come from 14 electrical lines, tightly criss-crossing the contaminated area. Gaps between electrical lines were filled, first, by sequential simulation to produce the full resistivity map shown on Figure 2.

500 rn

o

0

0 00 0 0 0

••

PAH concentration :

0 0

0

• " 0 0" " 0 0 " 0 • " 0 o a- "· 0 0 0 " • • " • 0

·.

•

...

0 0

0

> 200 ppm 40 - 200 ppm < 40 ppm

0 0

" 0

0 0

0 0

600rn

Figure 1 : Sample Location Map

500

15 ohm· m 25

~a~

• •• •• II

0 0

600

Figure 2 : Electrical Resistivity Map

40 95 250 375 750 1500 3000

If

72


OBJECTIVE OF THE STUDY The problem at hand is to delineate (on a 10 by 10 meter grid) areas where the risk that the P AH contamination is in excess of a critical threshold is deemed large enough to warrant either remediation or further testing. The critical threshold used for this study is 200 ppm PAH, and three classes of risk were considered: Low Risk :

The probability that the concentration in P AH exceeds 200 ppm is less than 20 percent.

Medium Risk: The probability of exceedance is in the 20 to 50 percent range. High Risk:

The probability of exceedance is over 50 percent.

From a methodological point of view, assessing the risk of exceedance implies that, at each node of the grid, the range of possible values for the P AH concentration, along with their associated probability, be available. The challenge therefore is to take advantage of both the direct measurements of P AH concentration and the indirect information provided by electrical resistivity to infer, at each grid node, the range of possible values that P AH concentration could take. These ranges of possible values, called also local conditional distribution functions, can be viewed as measures of the local uncertainty in the P AH concentration. Once these local uncertainties are established, the risk maps can be produced.

EXPLORATORY DATA ANALYSIS Given the objectives of the study, we need to understand the following critical features from the available data: 1- The range of possible values for the PAH concentrations which may be encountered away from sampled locations; 2- The relationship which exists between P AH concentrations and electrical resistivity. In other words, knowing the resistivity value at a particular location, what can we say about the possible range of P AH concentration values at the same location? 3- The spatial correlation structure of P AH concentrations?

Univariate distribution ofPAH concentrations Any geostatistical estimation or simulation algorithm requires a model describing the probability distribution function of the variable under consideration, i.e. an enumerated list of possible values with their associated probabilities. Traditionally, this probability distribution function is based on the experimental histogram built on the available data. Figure 3 shows the experimental histogram, and the corresponding summary statistics, of the available PAH


73

concentration data. We can see that: -

The number of data available to construct the histogram is fairly limited, resulting in a lack of sufficient resolution : the class frequencies tend to jump up and down erratically and there are gaps between data values.

-

The histogram is extremely skewed to the left, the bulk of the values being below 300 ppm, with some erratic high values extending all the way up to 6500 ppm (Figure 3a). Not surprisingly, the coefficient of variation is very high (3 .03).

-

The mean and variance of the data are severely affected by this high variability and they cannot be established at any acceptable level of reliability: removing the two largest values reduces the average by a factor 3 and the variance by a factor 65!

-

Ifwe use a logarithmic scale to visualize the same histogram (Figure 3b), we see clearly the existence of three populations: a first one below 50 ppm and accounting for 61 percent of the total population, a second one in the range 50 to 500 ppm and including 34 percent of the population, and a third small (5 %) population characterized by extreme P AH values ranging from 600 to 6500 ppm.

f

Summary statistics Number of data : Mean : Standard deviation : Coer variation : Minimum : 1st quartile : Median: 3rd quartile : Maximum :

0.5

0.25

0.0

-Jlllll:ilib:tlhl...frlliL..Illl.....IDlla,.__ 250

51 382 ppm 1159 ppm 3.03 2 ppm 9 ppm 33 ppm 206 ppm 6500 ppm

PAH (ppm)

500

Figure 3a : Histogram and summary statistics for P AH concentrations (in ppm)

74


f

0. 1

0.0

..,....""'-'''''-+ PAH wpm] 50

500

Figure 3b : Histogram ofPAH concentrations (logarithmic scale)

The first population can be interpreted as representing the background concentration level on the site. The second population seems clearly related to contamination itself, with the bulk of it above the critical threshold of 200 ppm. The third population is a little more dubious to interpret, primarily because it is represented only by three samples. Although it is obviously associated with the contamination, it is not entirely clear whether it represents a different factor of contamination or if it is merely the tail end of the second population. From this analysis it is clear that the experimental histogram cannot be used as is as a model of the distribution function of P AH concentrations over the site area. This probability distribution function should, instead, be modelled and have the following features :

It should not be based on parameters like the mean and variance, which are highly affected by extreme values and are, as a result, not known with any degree of reliability; It should provide probabilities for the entire range of possible values, from the absolute minimum to the absolute maximum and fill the gaps between existing data values;

It should reproduce the existence of the three populations and their respective frequencies.

Bivariate Distribution The cross-plot shown on Figure 4 describes the relationship which exists between PAH concentrations and electrical resistivity.


75

PAH [ppm[ 1000

100

•

•

•• • ••• • • •• •••

• • • • •• • • • • •

10

10

100

Resist . ohm. m

1000

Covariance : -0.799 Correlation (pearson) : -0.360 Correlation (Spearman) : -0.266

Figure 4 : Cross plot and bivariate statistics ofPAH vs Electrical Resistivity

The most important feature ofthis plot is the existence oftwo distinct clouds of points, which is a direct consequence of the multi-modality of the PAH distribution, and of the bimodality of the electrical resistivity: An upper cloud where P AH values are in excess of 3 5 ppm and the electrical resistivity ranges from 15 to 150 ohmomo Within this cloud, the correlation between the two attributes is positive.

A lower cloud with PAH values below 35 ppm and with electrical resistivities in the 30 to 1600 ohmom range. The correlation, again, appears to be positive, but less significantly so . From this cross-plot it seems that high concentrations ofPAH (over 35 ppm) are associated to rather low resitivity values. One possible explanation of this feature, which stilJ remains to be confirmed, is that P AH, which are viscous fluids, tend to flow down through backfilJ materials until they reach the top of the natural soil. At this level they filJ up the soil pore volume, thus creating a flow barrier to water.

76


Traditionally, bivariate distributions are parametrized by the mean and variance of their n;arginal distributions together with the correlation coefficient. Such an approach is inapplicable in our case, since it will fail completely to reflect the most important feature of the cross-plot which is the existence of the two populations. The solution adopted for this study consists of using directly a bivariate histogram to describe the bivariate distribution model. Because of the sparsity of data, the experimental cross plot is not sufficient to inform all the possible bivariate probabilities: it is spiky and full of gaps. The required bivariate histogram, therefore, will be obtained by an appropriate smoothing of this experimental cross plot, making sure that the two clouds of points are correctly reproduced. Spatial Continuity Analysis The variogram analysis performed on the natural logarithm ofPAH concentrations (Figure 5) shows that the phenomenon is reasonably well structured, with a maximum correlation distance (range) of approximately 70 meters. There was no evidence of anisotropy and the shape of the variogram was exponential.

y(h) 6

... .. ,.. .. .. i · .. · .. ... .... ..................... .......... .. ~

..

~

~

'

.. ~' 3

0r-----------r----------.-----------r~ hIm) so

100

ISO

Figure 5 : Experimental variogram for Ln(P AH) concentrations


77

BUILDING THE PROBABILISTIC MODEL Based on the results of the exploratory analysis, the probabilistic model to be used in estimation and uncertainty assessment will consist of the following: -

A smooth univariate histogram approximating the marginal distribution of the P AH concentration, and

-

A smooth bivariate histogram describing the bivariate distribution of P AH concentration and electrical resistivity.

Several approaches have been proposed to produce smooth histograms and cross-plots: quadratic programming (Xu 1994), fitting of kernel functions (Silverman 1986; Tran 1994) and simulated annealing (Deutsch 1994). The technique selected for this study is simulated annealing, because it was perceived to be the most flexible to accommodate all the requirements of the probabilistic model. Simulated annealing is a constrained optimization technique which is increasingly used in earth sciences to produce models which reflect complex multivariate statistics. A detailed discussion of the technique can be found in (Press et al. 1992; Deutsch and Journel 1992; Deutsch and Cockerham 1994). In this study the modelling of the bivariate probabilistic model was done in two steps: first the marginal distribution of the P AH concentration was modelled, and then the cross-plot between PAH and electrical resisitivity (there was no need to model the marginal distribution of resistivity, since it was directly available from the resistivity map).

,., i. II:

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The modelling of the marginal distribution ofPAH via simulated annealing was implemented as follows: 1- An initial histogram is created by subdividing the range of possible values into 100 classes, and assigning initial frequency values to each of these classes by performing a moving average of the original experimental histogram and then rescaling these frequencies so that they sum up to one. 2- An energy function (Deutsch 1994) is defined to measure how close to desired features of the final histogram the current histogram is. In the present case the energy function takes into consideration the reproduction of the mean, variance, selected quantiles and a smoothing index devised to eliminate spurious spikes in the histogram. 3- The original probability values are then perturbated by choosing at random a pair of classes, adding an incremental value b.p to the first class and substracting it from the second, hence ensuring that the sum of the frequencies is still one. 4- The perturbation is accepted if it improves the histogram, i.e. if the energy function decreases. Ifnot, the perturbation may still be accepted with a small probability. This will ensure that the process will not converge in some local minimum. 5- This perturbation procedure is repeated until the resulting histogram

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deemed

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78


satisfactory (the energy function has reached a minimum value) or until no further progress is possible. The modelling of the cross plot followed a similar general approach: 1- An initial bivariate histogram is created by subdividing the range of possible values along both axes into 100 classes, and assigning initial bivariate frequency values to each of these cells by performing a moving average of the original cross-plot followed by a rescaling of the frequencies to ensure that they sum up to one. 2- An energy function is defined to measure the goodness of fit of the current bivariate histogram to desired features of the final one. In the present case the energy function takes into consideration the reproduction ofthe marginal distributions defined previously, the correct reproductions of some critical bivariate quantiles and, again, a smoothing index devised to eliminate spurious spikes in the cross plot. 3- The original bivariate frequencies are perturbated by randomly selecting a pair of cells, and adding an incremental probability t,p to the first class and subtracting it from the second, leaving therefore the sum of frequencies unchanged. 4- As before the perturbation is accepted if it decreases the energy function and accepted with a certain probability if not. 5- The perturbation mechanism is iterated until the energy function has converged to some minimum value. A detailed discussion on how to use simulating annealing for modelling histograms and cross plots can be found in Deutsch (1994). The result of this modelling is shown on Figure 6 (experimental histogram of P AH concentrations and smooth model) and Figure 7 (experimental cross plot of P AH versus resistivity and corresponding smooth bivariate histogram). As can be seen, all the important features appear to be well reproduced: the multi-modality of P AH, the bi-modality of the resistivity and the existence of the two clouds on the cross plot.

ESTIMA TING THE LOCAL DISTRIBUTION FUNCTIONS Having developed the bivariate probabilistic model for P AH and electrical resistivity, we will now use it to infer the local conditional cumulative distribution function (ccdt) of the P AH concentration. This inference (see Appendix I) involves two steps: 1- The local a priori distribution function (cdt) of the P AH, given the local resistivity value, is extracted from the bivariate histogram. This local cdf characterizes the uncertainty of the P AH value based on the overall relationship existing between P AH and resistivity, but


before using the local P AH data values themselves.

f

0.1

...,...~~--+

0.0

50

PAH [ppm]

500

f

...,...--.... PAH[ppm]

0.0

50

500

Experimental Figure 6 : P AH Concentration histogram and smooth histogram model (logarithmic scale)

79

80


2- The local a priori cdf is then conditioned to the nearby P AH data values via simple indicator kriging (Journel \989) . This ccdf now describes the uncertainty on the P AH concentration once the local conditioning information has been accounted for. Note that simple indicator kriging calls for a model of the spatial continuity of the residual indicators (see Appendix I). This model is shown on Figure 8.

y

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01) = 0.01 + 0.25Gaus 70 (h)

.. ." ........................... " ......... " ....... " .. •

r-----.---~r---~----~-----r-----r~h[ml 70

140

210

Figure 8 : Variogram model for the indicator residuals

In this approach, the two types of information (direct measurements ofP AH concentrations and electrical resistivities) are mixed in a smooth, transitional fashion: when there is abundant nearby sample data, the simple indicator kriging system will put a lot of emphasis on this conditioning information and down play the influence of the indirect information, whereas when there is little or no conditioning data, the range of possible outcome for P AH will be primarily controlled by the local resistivity value.

PROBABILITY MAPS Having inferred the local distribution functions of P AH concentrations, we can now build the probability map (Figure 9c) showing the risk that this concentration exceeds the critical threshold of200 ppm .

COLIN ET AL. ON GEOPHYSICAL DATA PAH (ppm )

• 1000

•

100

..... •

• •• •••

• • • • • •• • • ••

10

10

100

ResilL (ohm-m 1

1000

PAH (pp m(

Freguency 0 0.01 0.02

1000

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II

100

••

10

0.03 0.04 0.05 0.06 0.07 0.08 0.09

=="-~ Resistivity [o1un . m] 10

100

1000

Figure 7 : P AH vs Electrical Resistivity Experimental cross-plot and bivariate histogram model

81

82


500 m

a) Iso-concentration map

II

PAH > 200 ppm

[j

PAH Zk

(5)

where Zk is the cutoff level. By using kriging, the interpolated indicator variable at any point Xo can be estimated by n

I: (xo)

=

L

(6)

AjOik(x)

j=l

where

1*k = the estimate of the conditional probability at Xj

'\0 = the kriging weight for the indicator value at point Xj

(Rouhani and Dillon 1989).

The conditional probability in this case is defined as:

(7) By varying zk' the cumulative probability can then be constructed.

CASE STUDY The case site has been in operation for over 50 fifty years and is currently a commercial site. The area under investigation has a total of nine underground storage tanks (USTs) situated around the site. Some of the tanks have been in operation for up to 30 years and were suspected to have

WILD AND ROUHANI ON FIELD SCREENING

93

leaked for an unknown number of years. The site is approximately 90 to 95 percent covered with concrete and is relatively level. The investigated data and some site characteristics have been altered to maintain confidentiality. The horizontal and vertical contamination at the site were assessed by three boring programs. A total of 82 borings were advanced. All USTs were eventually excavated and removed after the environmental investigation. A confirmatory campaign was performed after the tanks were removed and produced 12 additional sample locations. Figure 1 shows the locations of the 94 borings and the previous locations of the tanks. A Foxboro® OV A3 was used to screen for VOCs in each boring for all four investigations. The OV A was used to screen over 300 samples from these borings.

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Figure 1-- Site Map. In addition to the OV A screening, laboratory analysis was performed on 35 of the 300 samples. Typically, laboratory testing is performed on high OVA readings if one wants to identify the highest concentrations for risk assessment. Alternatively, when OVA readings are neither high nor low, laboratory results can be used to determine potential contamination of a sample. In this case study, the boring campaigns collected VOC information over the entire site. Laboratory samples were severely under collected for the size of the site. From an agency standpoint, this site characterization would be incomplete and unacceptable. Figure 2 demonstrates the collected laboratory samples in the surficial soils.

3A Foxboro® OVA uses the principle of hydrogen flame ionization for detection and measurement of total organic vapors. The OVA meter has a scale from 0 to 10 which can be set to read at 1 X, 10 X or 100 X or 0-10 ppm, 10-100 ppm and 100-1000 ppm, respectively. The OVA is factory-calibrated to methane.

94


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Figure 2-- Laboratory analyzed sample locations; ethylbenzene concentrations are shown. This paper salvages this extensive investigation by providing means to extract the maximum level of information from the existing data.

ANALYSIS OF DATA SET

Cross-correlations were computed between the laboratory analyzed compounds in order to determine applicable indicator compounds, if any. The available data were also analyzed for correlation between the laboratory-analyzed data and the OV A samples. Next, the characteristic distribution of the data was determined to select the most appropriate geostatistical technique for the spatial or structural analysis. Only benzene, ethylbenzene, toluene and xylene (BTEX) were investigated. All other compounds detected in the samples, together, made up less than eight percent of the total volatile compounds detected (EPA Method 8240 or Priority Pollutant compounds were tested) . These compounds, which were mostly methylene chloride measurements, were consistently measured at low levels. Both the OVA and VOC measurements were grouped into 3-foot intervals. Because the three boring campaigns did not always collect data at consistent depths, the intervals had varying amounts of data and spatial distribution. However, these measurements were distributed over various depths. Figure 2 shows only 13 samples in the surficial layer which was the most impacted layer in terms of horizontal extent. Cross-Correlation Analysis A cross-correlation analysis was performed between each variable to determine applicability of

95


indicator compounds. For this purpose, the entire data set of 35 samples, which spanned various depths, was used. Ethylbenzene had the highest average correlation value with the other BTEX compounds, R2 = 0.92 (R2 is the correlation coefficient). In addition, ethylbenzene correlated well with total BTEX (R2 = 0.94). As mentioned previously, the soil-gas probes measure total VOCs. Therefore, ethylbenzene was used as an indicator of the other three parameters and for total VOCs . The correlation analysis perfonned for each BTEX compound versus the corresponding OVA reading produced poor correlations (Figure 3). The highest correlation coefficient was for the

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1200

8' WOO E; 800 ' ~

o

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600 400 200

o~

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10,000

20,000

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Ethylbenzene (Ppb)

Figure 3-- OVA to ethylbenzene correlation analysis. complete OVA data set versus ethylbenzene yielding an R2 = 0.37 . This low correlation coefficient indicates that direct correlation between laboratory and OVA measurements could not be justified. As discussed previously, similar results were also found by many investigators. Structural Analysis of OVA Measurements A structural analysis was perfonned on the surficial soil OVA measurements to determine their spatial correlation. Due to the qualitative nature of the OV A, the structural analysis exhibited a high degree of variability. It was therefore concluded that, given the non-gaussian shape of the histogram of OVA measurements, an indicator transfonnation was preferable. Two approaches were considered for this analysis. The first approach, suggested by Isaaks and Srivastava (1989), uses the median value of the data as the cutoff. Given the qualitative nature of the OVA data, the median cutoff value may have no real significance. Therefore, a second approach was developed. This approach identifies an OVA cutoff that would provide a high degree of confidence that the soil is less contaminated than an established regulatory threshold for petroleum hydrocarbons. This threshold was based on a review of a number of government guidelines on petroleum-contaminated soils. The threshold or target value could then be used to develop the conditional probability Prob[Ethylbenzene ~ Target

/ OVA]

(8)

96


where Target =the cleanup or suggested maximum-allowable hydrocarbon contamination level. Calculation of the conditional probability for varying target levels (Figure 4) showed that there was a greater than 95 percent chance that the ethylbenzene level in the soil was equal to or less than a 100% 95%

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OVA Readings (ppm) Figure 4-- Conditional probability based on OV A readings and regulatory cleanup standards. 20 parts per billion (ppb) target level, given an OVA reading of 20 parts per million (ppm) or less. Therefore, a target level of 20 ppb for ethylbenzene was selected as a conservative cleanup standard. The 20 ppm cutoff value for the OVA readings was similar to the median values for the surficial soils. Using the surficial soils data, the indicator variogram of OVA at 20 ppm (Y) is shown as Figure 5. 0.4 r - - - - - - - - - - - - - - - - - - - - - ,

•

0.3

•

0.3

!

0.2

o

~ 0.2 0.1 0.1

Distance (meters)

Figure 5-- Indicator variogram of OVA measurements at 20 ppm threshold


97

As shown by this figure, the variogram demonstrated a well-defined spatial structure.

Structural Analysis of Ethylbenzene Recalling Figure 2, only 13 surficial ethylbenzene measurements were available for mapping soil contamination. As detennined in exploratory data analysis, the ethylbenzene measurements exhibited a tendency toward a log normal distribution. To account for this tendency, the natural log of the ethylbenzene measurements were taken. Furthermore, in order to avoid the possibility of numerical errors in the cokriging process, the log-transformed values were then normalized (mean =0, standard deviation =1). This made the latter data set numerically consistent with the indicator OVA values, thus minimizing the chance of numerical errors in cokriging. Unlike the OVA measurements, the variogram for the standardized, log-transformed ethylbenzene measurements (Z) demonstrated a relatively poor spatial structure (Figure 6). This short range prohibited accurate mapping with the current ethylbenzene data set. 1.4,-------------------,

•

1.2

~ 1.0 ~

0.8 0.6

10

20 30 Distance (meters)

40

50

Figure 6-- Direct variogram of normalized ethylbenzene measurements . All the above direct variographies were performed using the U.S. Environmental Protection Agency (EPA) public domain program, GEO-EAS (Englund and Sparks 1988). Cross-Variography of OVA and Ethylbenzene Measurements Cross-variography between the above two variables was conducted based on the linear model of co-regionalization (Rouhani and WackernageI1990). These computations were conducted using EPA's program, GEOPACK (Yates and Yates 1989). In this approach, the relationship between the direct and cross variograms is defined as

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DAGDELEN AND TURNER ON STATIONARITY

129

Indicator Kriging Using Da·ta from Entire Site

To further explore applicability of estimators based on second-order stationary models for the data, indicator kriging was used to analyze the entire data set. The TCE data values were transformed into 0 and 1 values, depending on their values relative to the 3.0 ppm (the median of sample values)TCE threshold. Directional variograms were produced. Indicator kriging was then applied to produce a map of probabilities of any location exceeding the 3.0 ppm threshold (Figure 10). Figure 11 shows the cross-validation plot of these probability estimates against the actual occurrence of sample values greater than 3.0 ppm ( using 50% or greater probabilities). There are 19 locations (26%) with false positives and 17 locations (24%) with false negatives (Table 1). The bias toward over-estimation has been further reduced, but additional reduction in the numbers of false positive and false negative locations appeared desirable. Indicator Kriging Using Data by Sub-regions

The indicator kriging process was then independently applied to the individual sub-regional data sets. Figure 12 shows the estimated probability of exceeding 3 ppm in each block. Figure 13 shows the cross-validation plot for these resulting probability estimates against the actual occurrence of sample values greater than 3.0 ppm. A further reduction in the degree of overestimation of the area of contamination exceeding the 3.0 ppm threshold is evident. There are 13 false positives, (18%)and 12 false negatives (17%) (see also Table 1).

ttlll 11,11

CONCLUSIONS

Mis-interpretation of the extent and degree of contamination at a site is likely to occur when traditional kriging is applied to a sample data set that does not consider the geological complexity. This result is likely because: •

Kriging should not be applied to data sets having a coefficient of variation greater than 1.0, since a few high concentration samples in such skewed data sets make the model assumptions inappropriate for the data at hand, resulting in biased estimation of local averages.

•

One of the important assumptions of geostatistics is second-order stationarity. In order to be able to apply kriging, a given data set must combine samples so that they can be conceptualized as an outcome of a second-order stationary of random function. This means that the data being processed by geostatistical kriging should come from a single consistent population. Only data from similar contaminant sources and geologic environments are likely to satisfy the stationarity assumption of the model.

For the example site discussed in this paper, the coefficient of variation for the bedrock TCE data is approximately 3.96, much greater than the limit of 1.0 defined above. At this site, because the TCE contamination appears to come from multiple point sources and to be

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DAGDELEN AND TURNER ON STATIONARITY

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132


,~CClt rollEd cy dLO!ererlt ground-water flow regimes, the assumption of stationarity was not satisfied. Hence, application of the ordinary kriging technique to the entire site without subdivisions gave biased and erroneously high estimates of local TCE values, "smearing" high TCE values into locations where they do not actually occur. Such a "smearing" effect presents a false impression of widespread TCE contamination throughout the site, and suggests the presence of a large contaminant plume. Cross validation analysis provides a means for assessing the degree of bias and therefore the appropriateness of the kriged estimates at existing sample locations.

Indicator kriging was used to analyze bedrock TCE contamination data with a threshold limit of 3.0 ppm. (median of the data values). This method indicates that many areas within th~ site have low probabilities of being contaminated with TCE above this very low threshold level. Application of indicator kriging at higher threshold levels will define even more restricted areas as having significant probabilities for higher levels of TCE contamination. Analysis of the entire data set by indicator kriging procedures still resulted in slightly biased estimation; better results were obtained when indicator kriging was applied to subregional data sets. These results are summarized in Table 1. Indicator kriging is thus proposed as an appropriate method for developing realistic estimates of contamination levels at many geologically complex sites. It provides a mechanism for substantially meeting the underlying assumptions of stationarity in the model. Coupled with a complete conceptualization of the geological and hydrological framework for the site, optimal estimates may be achieved by applying indicator kriging methods to subregional data sets that reflect geologic controls. This approach will identify location of the misclassification bias both with respect to overestimation and underestimation and provide more accurate assessment of contamination limits. REFERENCES

Journel, A.G., and Huijbregts, C.J., 1975, Mining Geostatistics, Academic Press, New York, NY. Isaaks, E., and Srivastava, R., 1989, An Introduction to Applied Geostatistics, Oxford University Press, New York, NY. Journel, A.G., 1983,"Non Parametric Estimation of Spatial Distribution". Mathematical Geology: Vol. 15, No.3; 1983, pp. 445-468. Isaaks, E., 1984,"Risk Qualified Mappings for Hazardous Waste Sites: A case study in Distribution-free Geostatistics". Master's thesis, Stanford University, CA. Deutsch, C.V., and Journel, A.G., 1992, GSLIB: Geostatistical Software Library and User's Guide. Oxford University Press, New York.

Daniela Leonte 1 , and Neil Schofield 2

EVALUATrON OF A sorL CONTAMrNATED A GEOSTATrSTrCAL APPROACH

srTE

AND

CLEAN-UP

CRrTERrA:

REFERENCE: Leonte, D. and Schofield, N., "Evaluation of a Soil Contaminated Site and Cleanup Criteria: A Geostatistical Approach," Geostatistics foI' EnviI'onmental and Geotechnical Applications, ASTM STP 128:5, R. Mohan SI'ivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Alexander J. Desbarats, Eds., American Society for Testing and Materials, 1996.

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Figure 3--Qrnni-directional horizontal sample yariograrns for l ead and for several indicator thresholds. Estimates of the local conditional probability for the lead concentration to exceed given thresholds were made using indicato r point kriging on a 10 meter square grid. Contours of the probability of exceedance for the 300 ppm lead thre shold for layer 1 and layer 2 are shown in Figure 4. The previously identified hotspots of high lead contamination are also shown with grey shading. The maps indicate

142


: . E. .~£f-= areas wi th a probability of at least 70 percent that the lead concentration in samples exceeds 300 ppm. The hotspots identified for clean-up lie close to the contaminated centres of two of these areas. However, a large area of high lead contamination in the south-eastern part of the site (20 m x 120 m northing and 270 m x 320 m easting) has been ignored completely, most likely because remediation of the previously identified hotspots would ensure compliance under the 75th percentile of a global lognormal criteria for lead at the site.

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Figure 4 Cont ours of the conditional probability for lead concentration to exceed the recommended level of 300 ppm. Hotspots are shown by grey shading.

LEONE AND SCHOFIELD ON CONTAMINATED SITE

143

Figure 5 presents contour maps 0;: t he: '=s ·,:il r,at. e~1 pr::>(.clinl i ty for lead concentration to exceed 500 ppm in soils for layers 1 and 2. On these maps, the areas with very low probability of contamination are clearly shown. The areas with potentially high contamination are also clear with the southern area (20 m x 120 m northing and 270 m x 320 m easting) again standing out as unrecognised by the previous investigation.


220

~, 0.1-,-1

1 7 -- - , 1

I 01 ~

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120

t

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Z

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Layer 2,

Pr (Pb > 500 ppm)

120

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320

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t

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320

Figure 5 Contours of the conditional probability for lead concentration to exceed the recommended level of 500 ppm. Hotspots are shown by grey shading .

144


CONCLtl'SrONS

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The recommendation of the ANZECC/NHMRC document for the use of both criteria-based and site-specific standards to assess soil contamination and clean-up is supported by the authors of this paper. The use of a universal or blanket standard for assessment of all sites appears inappropriate. This conclusion is supported by the outcome of applying the 75th percentile of a lognormal model criteria to the site in question in this paper. The cleaning of small areas of extreme contamination may often reduce the global level of contamination below some acceptance threshold. However, large areas carrying a significant risk of contamination above the acceptance threshold may go unrecognised and uncleaned. The application of geostatistical methods to analyse and model the lead contamination at this site appears appropriate. The dumping of lead contaminated material at the site does not seem to have been highly organised introducing considerable uncertainty as to the exact location of the contamination. Subsequent migration of the lead in soil due to natural processes has likely modified the spatial distribution of lead introducing greater complexity and uncertainty into its spatial geometry. Indicator kriging has enabled a mapping of the lead contamination at a local scale which permits an assessment of the risk associated with certain levels of contamination. When compared to previous attempts to identify areas of significant contamination (hotspots), the IK mapping indicates much larger areas associated with those hotspots where the risk of contamination is high. In addition, a large area of significant contamination which has previously gone unrecognised due to a naive decision rule, has been identified through geostatistical analysis. Although other techniques would have enabled estimation of the global lead contamination at the site by accounting for its specific directions of spatial continuity, the IK tool uniquely introduces the risk factor through the quantification of the uncertainty associated with the estimation process. Therefore making decisions on the extent and nature of remedial action to be implemented becomes a more informed process in which clean-up cost and potential liability associated with it can be evaluated.

REFERENCES

Lll

[~l

M.G. Knight. "Scale of the hazardous waste problem in Australia and disposal practice," Symposium on Environmental Geotechnics and Pyoblematic Soils and Rocks, Bangkok: Asian Institute of Technology, South-east Asian Geomechanics Society, 1985. Australian and New Zealand Environment and Conservation Council, and National Health and Medical Research Council (ANZECC/NHMRC). Australian and New Zealand Guidelines for the Assessment and Management of Contaminated Sites, January 1992.

LEONE AND SCHOFIELD ON CONTAMINATED SITE

U.l

[~]

[~]

[.§.]

[1]

[.a]

[.2.]

145

J. Daffern, C.M. Gerard and R.McFarland. "Regulatory and nonregulatory control of contaminated sites," Geotechnical Management of Waste and Contamination, Fell, Phillips and Gerrard (editors), Balkema, Rotterdam, 1993. E.H. Isaaks. Risk Oualified Mappings for Hazardous Wasce Sites; A Case Study in Distribution Free Geostatiscics. Master's thesis, Stanford University, 1984. E.H. Isaaks and R.M. Srivastava. An Introduction to Applied Geostatistics. Oxford University Press, 1989. A.G. Journel. "Non-parametric geostatistics for risk and additional sampling assessment," Principles of Environmental Sampling, L. Keith (ed.), American Chemical Society, 1988. N.A. Schofield. "Using the entropy statistic to infer population parameters from spatially clustered sampling," Proceedings of the 4th International Geostatistical Congress, Troia 92, pages 109-120, Kluwer, Holland, 1992. R.M. Srivastava and H. Parker. "Robust measures of spatial continuity," M. Armstrong (ed.), Third International Geostatistics Congress, D. Reidel, Dordrecht, Holland, 1988. M. Swane, I.C. Dunbavan and Riddell P. "Remediation of contaminated sites in Australia," Fell, Phillips and Gerrard, (editors), Geotechnical Management of Waste and Contamination, Balkerna, Rotterdam, 1993, pp 127-141.

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Amilcar O. Soares', Pedro 1. Patinha2, Maria J. Pereira

2

STOCHASTIC SIMULATION OF SPACE-TIME SERIES: APPLICATION TO A RIVER WATER QUALITY MODELLING

REFERENCE: Soares, A. 0., Patinha, P. J., Pereira, M. 1., "Stochastic Simulation of Space-Time Series: Application to a River Water Quality Modelling," Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R. Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Alexander J. Desbarats, Eds., American Society for Testing and Materials, 1996.

ABSTRACT: This study aims to develop a methodology to simulate the joint behaviour, in space and time, of some water quality indicators of a river, resulting from a mine effluent discharge, in order to enable the prediction of extreme scenarios for the entire system. Considering one pollutant characteristic measured in N monitoring stations along the time T, a random function X(e, t), e=l, ... ,N, t=1, .. . ,T, can be defined. The proposed methodology, a data driven approach, intends to simulate the realisation of a variable located in station e in time t, based on values located before e and t, and using a sequential algorithm. To simulate one value from the cumulative distribution F{X(e,t) I X(e-J,t), ... ,xU, t),x(e,t-I), ... ,x(e, I)}, the basic idea of the proposed methodology is to replace the e.t conditioning values by a linear combination of those:

* [x(e, t)1

=

e-I

(-I

u=1

~=I

L: au X( u, t} + L: brt X( e, ~) which allows the values to be drawn sequentially I-'

from bidistributions. The final simulated time series of pH and dissolved oxygen reproduce the basic statistics and the experimental time and spatial covariances calculated from historical data recorded over 15 months at a selected number of monitoring stations on a river with an effluent discharge of a copper mine located in the south of Portugal.

KEYWORDS: stochastic simulation, space-time series, water quality

'Professor, CVRM - Centro de Valoriza

~

.

400

.!l

! 300 ~

.,== N' = .... Q

Q

e:. .,(j

Q

i!i

~200

0

z 100

100

'"'" rJJ n> t'l ....

S'

= ..., .,~

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200

300

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200

300

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400

KUHN ET AL. ON SOLID WASTE DISPOSAL

175

Intemreted Subsurface Environment Based on the information obtained from this study, it appears that leachate may have migrated from the landfill and impacted the groundwater table. The plume appears to be migrating horizontally in the northeast direction with a vertical component. Figure 5 consists of a plan view and cross section illustrating an interpreted leachate plume located relative to the identified waste.

PROPERTY LINE

Figure S. Schematic of Interpreted Subsurface Environment

Computer Software Support All data description and estimation efforts requiring computer software support was completed utilizing an IBM 80386 processor. Software support utilized in this study consisted of Geo-EAS, GSLIB and TecPlot Version 6.0 (TecPlot). Probability and histogram plots describing the 1679 observed conductivity values were prepared utilizing Geo-EAS. Geo-EAS generated on-screen plots within minutes allowing efforts to be focused on the descriptive analyses. The expected values. are presented on cross section contour maps included as Figures 3 and 4. The three dimensional expected value data set was imported into TecPlot. TecPlot utilizes linear interpolation to construct each contour line. Tecplot generated cross section maps based on a three dimensional data set. By fixing one dimension, a cross section at a desired location was generated within minutes.

176


CONCLUSIONS Geostatistical analysis demonstrated that the data is spatial correlated which allowed for an interpreted subsurface model to be developed based on kriged estimated values. As an alternative to traditional intrusive characterization techniques, surface based electromagnetic surveying techniques proved to be a key non-intrusive, cost-effective element in the refinement of the second phase of the hydrogeologic investigation. Review of kriging error maps can further refine this second phase by focusing on the areas with the largest error. This study demonstrated that this methodology, as a preliminary field screening tool, can provide sufficient information to optimize the placement and minimize the number of permanent groundwater monitoring wells.

REFERENCES Barlow, P.M., Ryan, B.J., 1985,"An Electromagnetic Method of Delineating GroundWater Contamination, Wood River Junction, Rhode Island," Selected Papers in Hydrologic Sciences, U.S. Geological Survey Water-Supply Paper 2270, pp. 35-49. Cooper, R.M., Istok, J.D., 1988, "Geostatistics Applied to Groundwater Contamination. I: Methodology, "Journal of Environmental Engineering, Vol. 114, No.2., pp. 270-285. Cressie, N.A., 1989, "Geostatistics," American Statistician, Vol. 43, pp. 197-202. Deutsch, C.V., Journel, A.G., 1992, "GSLIB - Geostatistical Software Library and User's Guide," Oxford University Press. Environmental Protection Agency, 1994, "EPA Criteria for Municipal Solid Waste Landfills," The Bureau of National Affairs, Inc., 40 CFR Part 258. Englund, E. and Sparks, A., 1988, "Geo-EAS 1,2,1 User's Guide", EPA Report #60018911008, EPA-EMSL, Las Vegas, Nevada. French, R.B., Williams, T.R., Foster, A.R., 1988, "Geophysical Surveys at a Superfund Site, Western Processing, Washington," Symposium on the Application of Geophysics to Engineering and Environmental Problems, Golden, Colorado, pp. 747-753. Greenhouse, J.P., Slaine, DD, 1986, "Geophysical Modeling and Mapping of Contaminated Groundwater Around Three Waste Disposal Sites in Southern Ontario," Canadian Geotechnical Journal, Vol. 23, pp. 372-384. Hagemeister, M.E., 1993, "Systems Approach to Landfill Hazard Assessment with Geophysics (SALHAG)," Unpublished Masters Thesis, University of Nebraska - Lincoln. 1994, "Handbook of Solid Waste Management," McGraw-Hill Publishing

It

Ii11

KUHN ET AL. ON SOLID WASTE DISPOSAL

177

Isaaks E.H., Srivastava R.M., 1989, "An Introduction to Applied Geostatistics," Oxford University Press, New York. Joumel, A., Huijbregts, C., 1978, "Mining Geostatistics," Academica Press, New York. McNeill J.D., October 1980, "Electromagnetic Terrain Conductivity Measurement at Low Induction Numbers," Geonics Limited Technical Note TN-6. NDEQ, February 1990, "Ground Water Quality Investigation of Five Solid Waste Disposal Sites in Nebraska", Nebraska Department of Environmental Quality. SCS Engineers, December 1991, "Volume 1 - Recommendations to State and Local Governments," Nebraska Solid Waste Mana~ement Plan, Nebraska Department of Environemtnal Quality. Woldt, W.E., 1990, "Ground Water Contamination Control: Detection and Remedial Planning," Ph.D. Dissertation, University of Nebraska - Lincoln.

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Geotechnical and Earth Sciences Applications

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Craig H. Benson1 and Salwa M. Rashad 2 ENHANCED SUBSURFACE CHARACTERIZATION FOR PREDICTION OF CONTAMINANT TRANSPORT USING CO-KRIGING REFERENCE: Benson, C. H. and Rashad, S. M., "Enhanced Subsurface Characterization for Prediction of Contaminant Transport Using Co-Kriging," Geostatisticsfor Environmental and Geotechnical Applications, ASTM STP 1283, R. Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Alexander J. Desbarats, Eds., American Society for Testing and Materials, 1996. ABSTRACT: Groundwater flow and advective transport were simulated in a heterogeneous synthetic aquifer. These simulations were conducted when the aquifer was fully defined and when it was characterized using a limited amount of hard and soft data (hydraulic conductivity data and soil classifications). Co-kriging was used to combine the data types when estimating the hydraulic conductivity field throughout the aquifer. Results of the flow and transport simulations showed that soil classifications were useful in characterizing the hydraulic conductivity field and reducing errors in statistics describing the plume. KEYWORDS: kriging, co-kriging, ground water, contaminant transport, hydraulic conductivity, soil classifications INTRODUCTION

Simulating flow and contaminant transport is often an essential feature of remediation projects dealing with contaminated groundwater. In recent years, numerous sophisticated groundwater models have been developed to conduct such simulations. The complexity of these models allows one to realistically simulate the fate of contaminants provided properties of the aquifer affecting transport are adequately characterized. Unfortunately, what level of characterization is "adequate" is unknown, especially at sites where the subsurface is heterogeneous. Thus, when limited data are available to describe subsurface conditions, predictions of contaminant transport can be uncertain even when sophisticated models are used. Although many factors affect the fate of groundwater contaminants, the spatial distribution of hydraulic conductivity is the primary factor affecting which pathways are active in transport (Webb and Anderson 1996). To better define these pathways, additional data must be collected and analyzed. The most useful data are hydraulic conductivity measurements. However, "hard" data such as hydraulic conductivity measurements are expensive to obtain, especially if the data are to be collected from a site that is contaminated. It is advantageous, therefore, to investigate the effectiveness of using less expensive "soft" data, such as soil classifications, to reduce uncertainty. Soft data can be readily collected using less expensive exploration techniques such a ground penetrating radar, terrain resistivity surveys, or cone penetrometer soundings. The objective of the project described in this paper was to evaluate how characterizing the subsurface affects predictions of contaminant transport. Simulations of 1Assoc. Prof., Dept. of Civil & Environ. Eng., Univ. of Wisconsin, Madison, WI, 53706, USA. 2Asst. Scientist, Dept. of Civil & Environ. Eng., Univ. of Wisconsin, Madison, WI, 53706, USA.

181

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182


groundwater flow and advective transport were conducted in a heterogeneous "synthetic aquifer." The aquifer was characterized using various amounts of hard data (hydraulic conductivities) and soft data (soil classifications). Co-kriging was used to combine the two data types when estimating the hydraulic conductivity field. Similar uses of cokriging have been described by Seo et al. (1990 a,b) and Istok et al. (1993).

SYNTHETIC AQUIFER Characteristics

A "synthetic aquifer" was used in this study because it can be fully-defined; that is, the hydraulic properties throughout the aquifer are defined with certainty. In this particular application, fully-defined means that hydraulic conductivities and soil classifications can be assigned to every cell in the finite-difference grid used in simulating flow and transport in the aquifer. Thus, flow and transport simulations conducted with the "fully-defined" aquifer are representative of its "true" behavior. Comparisons can then be made between results obtained using the fully-defined case and cases where the aquifer has been characterized with a limited amount of sub-surface data. This comparison provides a direct means to evaluate the inherent inaccuracies associated with estimating subsurface conditions from a limited amount of information. A schematic illustration of the aquifer is shown in Fig. 1. It is extremely heterogeneous, as might be encountered in a supra-glacial depositional environment such as those occurring in the upper midwestern United States (Mickelson 1986, Simpkins et al. 1987). Details of the method used to design the aquifer are in Cooper and Benson (1993). Although an attempt was made to create a realistic aquifer, the synthetic aquifer was created without any site-specific data and thus may not be "geologically correct." The reader should keep this limitation in mind when considering the results and conclusions described later.

6 Clay 5 Clayey Sill • Silty Sand 3 Ane Sand

2 Coarse-Medium Sand 1 Clean Gravel

Upstream Boundary

Hydraulic Gradient. 0.01

Downstream

Boundary • FIG. 1 - Synthetic aquifer.

The aquifer is discretized into 12,500 cells that comprise a finite-difference grid used in simulating flow and transport. The aquifer is segregated into 25 layers. Each layer contains 20 rows and 25 columns of finite-difference cells. Each cell is 100 cm

BENSON AND RASHAD ON CO-KRIGING

183

long per side. Groundwater flow was induced by applying an average hydraulic gradient of 0.01. Constant head boundary conditions were applied at the upstream and downstream boundaries of the aquifer. No flow boundaries were applied along the remaining surfaces of the aquifer. An important feature of the aquifer is that soil types are layered to create continuous and non-continuous soil lenses. Lenses with high hydraulic conductivity, such as clean gravel and coarse to medium sand, simulate preferential flow paths that might not be detected during a subsurface investigation. Low hydraulic conductivity soils such as clayey silt and clay are layered to create pinches and stagnation points that may cause the flow of groundwater to slow or even stop. These intricacies of the aquifer also might not be detected during a subsurface investigation. Hydraulic Conductivity of Geologic Units

A soil classification was assigned to each geologic unit (i.e., the geologic facies) in the fully-defined synthetic aquifer. The soil classifications used to describe geology of the aquifer are: (1) clean gravel, (2) coarse to medium sand, (3) fine sand, (4) silty sand, (5) clayey silt, and (6) clay. These soil classifications are represented numerically using the integers 1-6. The writers note that the integer ordering of these classifications is arbitrary. Consequently, results somewhat different than those described herein may have been obtained had a different categorical scheme been used. Each cell in a given geologic unit was assigned a single realization from the distribution of hydraulic conductivity corresponding to the unit. Single realizations were generated using Monte Carlo simulation via inversion. In addition, no spatial correlation was assumed to exists within a geologic unit. Thus, the correlation structure inherent in the aquifer is due primarily to the relative location and size of the geologic units. The triangular distribution (Fig. 2) was used to describe spatial variability in hydraulic conductivity for a given soil type. The distribution is defined using an upper bound (Kmax), a lower bound (Kmin), and the peak of the density function (Kp). To select K max , Kmin, and Kp for each soil classification, a chart was developed that summarizes hydraulic conductivities assigned to various soil types in thirteen publications (Fig. 3). The hydraulic conductivities recommended by others were synthesized into a single "composite chart" having the six different soil types that comprise the synthetic aquifer, each with a corresponding range of hydraulic conductivities (Table 1).

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FIG. 2 - Distribution of hydraulic conductivity in a geologic unit.

184


Oaa (1985) Bowl .. (1984) Domenico & Schwartz (1990) HoH.&

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FIG. 3 - Range in hydraulic conductivities for different soil types. TABLE 1 - Parameters describing hydraulic conductivity distributions. Soil Type Clean Gravel

Kmin (em/sec) 5 x 10- 1

Kp (em/sec) 5 x 100

I x 10-3 I x 10-4

5 x 10-2 5 x 10- 3

5 x 10- 2

Clayey Silt

5 x 10-5 I x 10-7

5 x 10-4 I x 10-6

5 x 10- 3 5 x 10- 5

Clay

I x 10- 10

I x 10- 8

1 x 10- 6

Coarse - Med. Sand Fine Sand Silty Sand

Kmax (em/sec) 5 x 102 I x 100

Spatial Correlation Structure

The spatial correlation structure inherent in the soil type and hydraulic conductivity fields was characterized by computing directional experimental variograms in three dimensions. A model was then fit to the experimental variograms. A similar approach was also used to characterize the spatial cross-correlation structure between hydraulic conductivity and soil type. Experimental variograms were computed using the program GAM3 from the GSLIB geostatisticallibrary (Deutsch and Joumel 1992). The experimental variograms were computed by: N(h)

1 ~ Y*In K (h) = 2 N(h) £.J [In K(xi + h) - In K(xi) ] 2 1=1

(1)


185 (2)

In Eqs. 1-2, Y*lnK(h) is the estimated variogram for InKs separated by the vector h, y* s(h) is the estimated variogram for soil classifications (S), N(h) is the number of data pairs separated approximately by the same vector h, and Xi is a generic location in the aquifer. The cross-variogram between InK and S is computed as;

1 Y~nK,S (h) = 2 N(h)

N(h)

L [S(Xj + h) - S(Xj)] [lnK(xj+h)-lnK(xD]

(3)

1=1

The principal axes for soil classification were identified by computing a series of experimental variograms each having a different orientation relative to the traditional Cartesian axes. The analysis showed that mild anisotropy exists in the X- Y plane, with the principal axis oriented 45° counterclockwise from the X-axis. For the vertical direction, the principal axis coincided with the vertical (Z) axis (Benson and Rashad 1994). The principal axes for the hydraulic conductivity field were assumed to correspond to the principal axes for soil type because the hydraulic conductivity field was generated directly from the soil type field. A similar assumption was made regarding the cross-variogram (InK-soil type). Experimental directional variograms for soil type and hydraulic conductivity corresponding to the principal axes are shown in Figs. 4 a & b. The experimental crossvariogram (InK vs. S) is shown in Fig. 4c. For each set of variograms, the range is largest in the Y' direction and smallest in the Z' direction, which is consistent with the size and shape of the geologic units shown in Fig. 1. In contrast, the sill is essentially the same for the Y' and Z' directions, but is smaller in the X' direction. A spherical model with no nugget was found to best represent the experimental variograms. The spherical variogram is described by (Issaks and Srivastava 1989); y(h) = C[ I.S(h/a) -D.S(h/a)3] y(h)

=C

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187

TABLE 2 - Sill (C) and range (a) for variograms

X'

Parameter InK - Sill InK - Range (em) Soil Class. - Sill Soil Class. - Range (em) Cross-Sill Cross - Range (em)

25 925 1.5 925 -5.8 925

Direction Y'

32 1840 2.25 1840 -7.7 1840

Z' 27 500 1.75 500

!;

-6.5

500

A three-level nested structure was used to combine the geometric and zonal anisotropies into a single variogram model. The model has the form: (5)

=

The model YI,j is a spherical function (Eq. 4) having C 1; it provides the basis for the nested structure. The subscript j denotes the variable being described (S, InK, or S-lnK for the cross-variogram). The separation distance hi is an "equivalent" distance:

h 1-

J

hX')2 h y,)2 h ,)2 ( - + ( - + (zax' ay' az'

(6)

The weight WI,S corresponds to the smallest sill for soil classification (X' axis: sill = WI,S = 1.5 for S). The models Y2 and Y3 are also spherical models and they are used to ensure that sills corresponding to the Z' and Y' axes are preserved. That is: W2,S = 1.75 - WI,S = 0.25

(7a)

w3,S = 2.25 - (WI,S + w2,S) = 0.50

(7b)

and

The equivalent distances h2 and h3 are:

" ":1 (8a)

and

:1

II

:1 r' (8b)

A summary of the weights used for the soil classifications, hydraulic conductivity, and cross-variogram models is contained in Table 3. It is important to note that data for hydraulic conductivity and soil classification were available for each cell in the aquifer when computing the cross-variogram. In more realistic cases, both types of data will probably not be available at each location. Problems associated with this disparity can be resolved by using the pseudo-cross variogram (Meyers 1991).

188


TABLE 3 - Summary of variogram weights.

Variable Soil classification, S Hydraulic Conductivity, InK Cross-Variogram InK vs. S

wI 1.5 25.0

w2 0.25 4.0

w3 0.5 3.0

-5.75

-0.75

-1.2

FLOW AND ADVECTIVE TRANSPORT MODELS MODFLOW

The three-dimensional finite difference program MODFLOW was used to simulate steady-state saturated flow in the aquifer. MODFLOW uses a block-centered finite difference scheme to solve the groundwater flow equation. A detailed description of MODFLOW can be found in MacDonald and Harbaugh (1988). MODFLOW was modified for use in this study by adding subprograms and changing the existing data collection and storage procedures. A detailed description of these modifications can be found in Benson and Rashad (1994). For each field of hydraulic conductivity that was simulated, MODFLOW was used to compute the total heads at each node and the total flow rate (Q) emanating from cells at the downstream end. The hydraulic head field is used by the advective transport model for simulating contaminant transport. PATH3D

The program PATH3D (Zheng 1988) was used to simulate advective contaminant transport. PATH3D is a general particle-tracking program for calculating groundwater paths and travel times in steady-state or transient, two- or three-dimensional flow fields. A detailed description of PATH3D can be found in Zheng (1988). Changes in PATH3D were required before it could be used in this study. These changes included modifying the algorithm for time step adjustment and modifying the post-processor to describe characteristics of the plume. Details of these changes can be found in Benson and Rashad (1994). Transport was initiated by releasing contaminant fluid particles along a vertical profile located at X =0, Y = 1000 cm, and Z =0 to 2500 cm. Eight particles were placed in each cell along this profile (200 total particles in aquifer). RESULTS AND ANALYSIS Ground water flow and contaminant transport simulations were conducted for three different conditions: (1) fully-defined aquifer, (2) partially defined aquifer using only hydraulic conductivity data, and (3) partially-defined aquifer using hydraulic conductivity and soil classification data. In the partially-defined cases, a co-kriging program (based on the program COKB3D, Deutsch and Journel 1992) was used to


189

estimate hydraulic conductivity for each finite-difference cell in the aquifer using a liner co-regionalization model for the variograms (see previous sections). In this application, co-kriging was only used to estimate the primary variable, hydraulic conductivity. In addition, point kriging was used instead of block kriging because it was more easily implemented and the cells used to discretize the aquifer were small. Nevertheless, a small error was introduced by point kriging. The variogram models previously discussed were used to describe the spatial correlation structure. Input for the co-kriging program included subsurface information consisting of profiles of hydraulic conductivity or soil classifications. A description of the co-kriging implementation can be found in Benson and Rashad (1994). Estimating the Hydraulic Conductivity Field

Initial Comparison: Kriging vs. Co-Kriging An initial comparison was made between the estimated hydraulic conductivity field and the fully-defined hydraulic conductivity field along a transect (single row of cells; X=50 cm, Z=50 cm, Y=O to 2500 cm) through the aquifer. In one case, the hydraulic conductivity field was estimated with kriging using only hydraulic conductivity data; co-kriging using hydraulic conductivity and soil classification data was used for the other case. u

The hydraulic conductivity fields estimated using kriging and co-kriging are shown in Fig. 5. Hydraulic conductivities measured along three vertical profiles were used as input when only kriging was conducted. The resulting estimated InK field is a smooth, nearly linear interpolation between the profiles at which measurements were made. The estimated InK field is very different from the "true" hydraulic conductivity field obtained from the fully-defined synthetic aquifer. That is , the irregular spatial variations in InK are not preserved.

~------,

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500

1000

1500

2000

2500

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FIG. 5 - Estimated hydraulic conductivities along a transect. Co-kriging was conducted using three profiles of hydraulic conductivity (primary variable) and 11 profiles of soil classifications (secondary variable). Figure 5 shows that

190


addition of the secondary variable greatly improves the estimated hydraulic conductivity field. The estimated InKs along the transect more closely resemble the "true" InKs. However: even with co-kriging, the estimated field is smoother then the "true" field. Selecting Profiles -- Various exploration schemes (i.e., collections of hydraulic conductivity and soil classification profiles) were selected to evaluate the relative effectiveness of different types of subsurface data in estimating the accuracy of the hydraulic conductivity field. The initial exploration scheme consisted of five profiles of hydraulic conductivity and five profiles of soil classifications. Subsequent schemes incorporating more data were constructed by adding more profiles of either hydraulic conductivities or soil classifications. A consistent method was needed to select locations for additional profiles. The writers chose to select subsequent profiles at locations where the co-kriging variance is largest. These locations have the greatest uncertainty in the estimated hydraulic conductivity. The writers note, however, that locations where the co-kriging variance is largest are not necessarily the critical locations where uncertainty in hydraulic conductivity has the greatest impact on contaminant transport. However, these critical locations cannot be identified a priori, because under normal circumstances the detailed characteristics of the aquifer are unknown. The aforementioned methodology was used to select 14 exploration schemes. The first seven schemes consist of five profiles of hydraulic conductivity (NK=5) and a varying number of soil classification profiles (NS=O, 5, 9, 15,22,32, 125). The second set of seven schemes was similar, except ten hydraulic conductivity profiles (NK=lO) were used. The layout of each exploration scheme is contained in Benson and Rashad (1994). Precision of the Hydraulic Conductivity Field -- In this section, two statistics are used to characterize the precision of the estimated field. These statistics are: the maximum co-kriging variance and the mean co-kriging variance. The mean co-kriging variance (cr~) is used to quantify the global estimation error. The mean co-kriging variance is computed as: H

.... 2

_~ ~ 2 N c 1= .1..1 cr cki'

uci ,

(9)

where cr~k,i is the co-kriging variance at the ith cell in the finite-difference grid and Nc is the total number of grid points (Nc = 12,500). In each case, cr c2k,I· is the variance of the primary variable (hydraulic conductivity) being estimated, as described in Issaks in Srivastava (1989, p. 404). The maximum co-kriging variance is shown in Fig. 6. The maximum co-kriging variance is essentially the same for the schemes using five and ten profiles of hydraulic conductivity; the co-kriging variance is only slightly larger for the scheme using five profiles, More importantly, however, addition of soil classification profiles results in a significant reduction in the maximum co-kriging variance. The mean co-kriging variance is shown in Fig. 7 as a function of the number of soil classification profiles. In this case, the error is significantly larger when five hydraulic conductivity profiles are used instead of ten profiles. A significant difference is


C>

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.~

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No. of Soil Classification Profiles

FIG. 6 - Maximum co-kriging variance for InK.

,

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expected, because the mean co-kriging variance represents a global measure of uncertainty whereas the maximum co-kriging variance is a local measure of uncertainty. Adding more hydraulic conductivity profiles will have a significant effect on the maximum co-kriging variance only if the profiles are located near or directly at the location where the maximum co-kriging variance exists because the co-kriging variance is a point measure. In contrast, because the mean co-kriging variance is a global measure of uncertainty, it will be reduced by adding more profiles, regardless of their location. Figure 7 also shows that exploration schemes employing more soil classification profiles with fewer hydraulic conductivity profiles can be as effective in reducing uncertainty as schemes that simply use more hydraulic conductivity profiles. For example, the scheme consisting of five hydraulic conductivity profiles and five soil classification profiles has similar mean co-kriging variance as the scheme using ten hydraulic conductivity profiles and no soil classification profiles. Furthermore, the scheme using more soil classification profiles and fewer hydraulic conductivity profiles is likely to be less expensive. Thus, a similar reduction in uncertainty can be obtained at less cost. Total Flow

One means to evaluate how well the aquifer is characterized is to compare the total flow rate across the compliance surface for the fully-defined condition with the total flow rate when the aquifer is characterized using a limited amount of subsurface data. For the synthetic aquifer, the compliance surface was defined as the downstream boundary (Fig. 1). If the flow rates are not nearly equal, then the aquifer is not adequately characterized. If the flow rate is too high, low conductivity regions blocking flow have been missed. In contrast, a flow rate that is too low is indicative of missing preferential pathways (Fogg 1986). Figure 8 shows total flow rate when the aquifer is characterized with 5 or 10 profiles of hydraulic conductivity and a variable number of soil classification profiles. When no profiles of soil classifications are used (kriging only), the flow rate is one-third to one-half the total flow rate. Apparently, the sampling program has inadequately defined the preferential pathways controlling true total flow. However, when more soil classification profiles are added, the flow rate begins to rise and then becomes equal (i.e., > 10 profiles) to the flow rate for the fully-defined condition. Two other characteristics of Fig. 8 are notable. First, similar flow rates were obtained when five or ten profiles of hydraulic conductivity (but no soil classifications) were used to characterize the aquifer. Apparently, neither set of measurements is of sufficient extent to capture the key features controlling flow. Second, the aquifer was better characterized (in terms of total flow rate) using five hydraulic conductivity profiles and 15 soil classification profiles then 10 hydraulic conductivity profiles and 15 soil classification profiles. This indicates that focusing on collecting a greater quantity of index measurements (i.e., soil classifications) may be more useful in characterization than collecting a fewer number of more precise measurements (i.e., hydraulic conductivities). In this case, hydraulic conductivity inferred from a soil classification had a precision of two to three orders of magnitude, whereas the hydraulic conductivity "measurements" were exact. Thus, in this case, simply defining the existence of critical flow paths apparently is more important than precisely defining their hydraulic conductivity.


193

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FIG. 8 - Total Flow Rate Through the Synthetic Aquifer Trajectory of the Plume - Centroid

Trajectory of the plume can be characterized by the coordinates (X, Y, Z) of its centroid. Trajectories for several different explorations schemes are shown in Fig. 9. In each case, the trajectory is recorded for only four years. For longer times, a portion of the plume has passed the downstream edge of the aquifer. Consequently, the statistics used to describe the plume (centroid and variance) are ambiguous. At early times, the trajectory of the centroid does not depend greatly on the exploration scheme. However, as the plume evolves, different trajectories of the centroid are obtained. In particular, the plume moves more slowly in the down-gradient (Xdirection) when the aquifer is characterized with a limited amount of subsurface data. Apparently, the preferential pathways controlling down-gradient movement were inadequately characterized. Addition of soil classification data did not consistently improve the trajectory in the X-direction. Adding five soil classification profiles improved the trajectory significantly, but the worst trajectory was obtained when 22 soil classification profiles were used. Adding even more soil classification profiles (NS = 32 or 125) improved the trajectory only slightly. This is particularly discouraging because 125 soil classification profiles corresponds to sampling 25% of the entire aquifer. The cause of this discrepancy is inadequate representation of subsurface anomalies that affect movement of the plume. At approximately 0.2 years, the centroid of the plume moves dramatically as the particles flow around a low conductivity region. The Y-coordinate increases and the Z-coordinate decreases (i.e., the plume moves upward and towards the rear face of the aquifer). None of the exploration schemes provided enough information to adequately characterize this movement. However, adding soil classification profiles did improve the prediction. When only hydraulic conductivity profiles were used (kriging only), the plume moved downward and to the front which is the exact opposite behavior occurring in the fully-defined case. Adding soil classification profiles did prevent the centroid from moving in the opposite direction

194

GEOSTATISTICAL APPLICATIONS 900 - NK =10, Ns=5

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FIG, 9 - Centroid of plume: (a) X-coordinate, (b) Y-coordinate, and (c) Z-coordinate


195

and, in the case where 125 profiles were used, did result in a subsurface where the plume moved upward and to the rear of the aquifer. Unfortunately, the degree of plume movement existing when 125 soil classification profiles were used is still too small to simulate the fully-defined condition. For brevity, graphs of trajectory of the centroid are not shown for the exploration schemes where 10 hydraulic conductivity profiles were used. These graphs can be found in Benson and Rashad (1994). Smaller errors in the predicted trajectory occurred when ten hydraulic conductivity profiles were used in the exploration scheme. In this case, addition of soil classification profiles also had a smaller impact on the predicted down-gradient movement of the plume. However, adding soil classification profiles did improve the Y and Z-coordinates of the centroid. When only ten hydraulic conductivity profiles were used (kriging only), the plume moved in the opposite direction as was observed in Fig. 9 (NK=5, NS=O). But, when soil classification profiles were added, the plume moved in the correct direction. Nevertheless, the movement occurred more slowly than in the fully-defined case, which caused the down-gradient movement of the plume in the estimated aquifers to lag behind the down-gradient movement of the plume in the fullydefined aquifer. Spreading - Variance of the Plume

Spreading of the plume is characterized by the variance (or second central moment); a larger variance corresponds to a greater amount of spreading. Evolution of the variance of the plume is shown for various exploration schemes in Fig. 10. Some general features of Fig. 10 are noteworthy of mention. First, the variance is larger in the X and Z-directions. The Z-variance is large because the particles are uniformly distributed along a vertical profile (i.e., Z-direction) at the onset of the flow and transport simulation. A large X-variance occurs because down-gradient movement of the plume occurs in the X-direction and thus the X-variance corresponds to longitudinal spreading of the plume. Accordingly, the Y-variance is much smaller because it corresponds to lateral spreading orthogonal to the average hydraulic gradient, which is generally smaller than spreading in the longitudinal direction. The X-variance increases with time. At short times, the variance is small because little spreading of the plume has occurred. However, as the plume moves down-gradient, the variance increases as more spreading occurs. Furthermore, the ability to capture the true amount of spreading depends on the amount of subsurface information used in characterization (Fig. 10). When less information is used (e.g., kriging only, NK=5, NS=O), the variance is smallest and when more information is used (i.e., by adding soil classification profiles) the variance increases. This is expected, because a smoother subsurface containing fewer heterogeneities exists when less data are used in characterization. However, adding more soil classification profiles does not consistently improve the X-variance. In fact, the X-variance for NS=5 is closer to the X-variance in the fully-defined case than the schemes having NS=22, 32, and 125. Similar behavior was noted for the Y-variance. However, adding more soil classification profiles had a more consistent effect on the Z-variance. Adding more soil classification profiles consistently resulted in a Z-variance that was closer to the Zvariance in the fully-defined case.

196


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Time (years)

FIG. 10 - Variance of plume: (a) X-variance. (b) Y-variance. and (c) Z-variance


197

When ten hydraulic conductivity profiles were used for characterization, the Xvariance in the estimated aquifers was similar to the X-variance in the fully-defined case regardless of the number of soil classification profiles used in characterization (Benson and Rashad 1994). Apparently, the ten hydraulic conductivity profiles used for characterization resulted in a sufficiently heterogeneous subsurface such that spreading in the down-gradient direction was preserved. In contrast, spreading in the Y and Z-directions for the fully-defined case was distinctly different from spreading that occurred in these directions when ten hydraulic conductivity profiles were used for characterization (Benson and Rashad 1994). Adding five soil classification profiles resulted in a V-variance that was closer to the V-variance in the fully-defined case. However, as even more soil classification profiles were added (NS=22, 32, 125), the V-variance became much different than was observed in the fullydefined case. Apparently, the heterogeneities causing spreading in the V-direction were inadequately represented when the subsurface was characterized with additional soil classification profiles. Kriging with only ten soil classification profiles resulted in a Z-variance that differed greatly from the Z-variance in the fully-defined case. For larger times, the Zvariance was much smaller than the Z-variance for the fully-defined condition. However, when soil classifications were added, the Z-variance more closely resembled the Zvariance for the fully-defined case. Thus, using soil classifications apparently resulted in heterogeneities that were similar to those controlling spreading in the Z-direction in the fully-defined aquifer. SUMMARY AND CONCLUSIONS The objective of this study was to illustrate how predictions of contaminant transport differ as the quantity and type of information used to characterize the subsurface changes. Groundwater flow and advective contaminant transport were simulated through a heterogeneous synthetic aquifer that was fully defined. The aquifer was highly heterogeneous, as might be encountered in supraglacial sediments, such as those found in the upper mid-western United States. Additional flow and transport simulations were conducted using versions of the aquifer that were characterized using a limited amount of subsurface data. Comparisons were then made between the true movement of the plume (in the fully defined aquifer) and movement of the plume in versions of the aquifer that were characterized with limited subsurface data. Results of the flow and transport simulations show that soil classifications can be used to augment or replace more costly hydraulic conductivity measurements while maintaining similar accuracy in terms of total flow through the aquifer. However, the geologic details that govern transport through the synthetic aquifer apparently were never sufficiently characterized. Bulk movement of the plume (i.e., the centroid) and spreading (i.e., variance) of the plume were never simulated accurately, regardless of the amount of subsurface data (hard or soft) that were used for characterization.

198


ACKNOWLEDGMENT

,,

The study described in this paper was sponsored by the U.S. Dept. of Energy (DOE), Environmental Restoration and Waste Management Young Faculty Award Program. This program is administered by Oak Ridge Associate Universities (ORAU). Neither DOE or ORAU have reviewed this paper, and no endorsement should be implied. REFERENCES

Benson, C. and S. Rashad (1994), "Using Co-Kriging to Enhance Hydrogeologic Characterization," Environmental Geotechnics Report No. 94-1, Dept. of Civil and Environmental Engineering, University of Wisconsin, Madison, WI. Bowles, J. (1984), Physical and Geotechnical Properties of Soils, 2nd Edition, McGrawHill, New York. Cooper, S. and C. Benson (1993), "An Evaluation of How Subsurface Characterization U sing Soil Classifications Affects Predictions of Contaminant Transport," Environmental Geotechnics Report No. 93-1, Dept. of Civil and Environmental Engineering, University of Wisconsin, Madison, WI. Das, B. (1985), Principles of Geotechnical Engineering, PWS-Kent Publishing, Boston. Deutsch, C. and A. Joumel (1992), GSLIB Geostatistical Software Library and User's Guide Book, Oxford University Press, New York. Domenico, P. and F. Schwartz (1990), Physical and Chemical Hydrogeology, John Wiley, New York. Fogg, G. (1986), "Groundwater Flow and Sand Body Interconnectedness in a Thick, Multiple-Aquifer system," Water Resources Research, 22(5), 679-694. Holtz, R. and W. Kovacs (1981), An Introduction to Geotechnical Engineering, PrenticeHall, Englewood Cliffs, N1. Hough, B. (1969), Basic Soils Engineering, 2nd Edition, Ronald Press Co., New York. Isaaks, E. and R. Srivastava (1989), Applied Geostatistics, Oxford Univ. Press, New York. Istok, J., Smyth, J., and Flint, A. (1993), "Multivariate Geostatistical Analysis of GroundWater Contamination: A Case History," Ground Water, 31(3), 63-73. Lee, I., White, W., and Ingles, O. (1983), Geotechnical Engineering, Pitman Co., Boston. McCarthy, D. (1982), Essentials of Soil Mechanics and Foundations, Basic Geotechnics, Reston Publishing, Reston, VA. MacDonald, M. and A. Harbaugh (1988), "A Modular Three-Dimensional Finite Difference Ground-Water Flow Model," Techniques of Water-Resources Investigations of the United States Geological Survey, USGS, Reston, VA. Means, R. and 1. Parcher (1963), Physical Properties of Soils, Merrill Books, Columbus.


199

Meyers, D. (1991), "Pseudo-Cross Variograms, Positive Definiteness and Co-Kriging," Mathematical Geology, 23, 805-816. Mickelson, D. (1986), "Glacial and Related Deposits of Langlade County, Wisconsin," Information Circular 52, Wisc. Geologic and Natural History Survey, Madison, WI. Mitchell, J. (1976), Fundamentals of Soil Behavior, John Wiley and Sons, New York. Scott, C. (1980), An Introduction to Soil Mechanics and Foundations, 3rd Edition, Applied Science Publishers, London. Simpkins, W., McCartney, M., and D. Mickelson (1987), "Pleistocene Geology of Forest County, Wisconsin," Information Circular 61, Wisconsin Geologic and Natural History Survey, Madison, WI. Smith, G. (1978), Elements of Soil Mechanics for Civil and Mining Engineers, 4th Ed., Granada Publishing, London. Seo, D-J, Krajewski, W., and Bowles, D. (1990a), "Stochastic Interpolation of Rainfall Data from Rain Gages and Radar Using Co-Kriging," Water Resources Research, 26(3),469-477. Seo, D-J, Krajewski, W., Azimi-Zonooz, A., and Bowles, D. (1990b), "Stochastic Interpolation of Rainfall Data from Rain Gages and Radar Using Co-Kriging. Results," Water Resources Research, 26(5), 915-924. Sowers, G. and G. Sowers (1970), Introductory Soil Mechanics and Foundations, 3rd Ed., McMillan Co., New York. Webb, E. and Anderson, M. (1996), "Simulation of Preferential Flow in ThreeDimensional, Heterogeneous Conductivity Fields with Realistic Internal Architecture," Water Resources Research, 31(3), 63-73. Whitlow, R. (1983), Basic Soil Mechanics, Construction Press, New York. Zheng, C. (1988), "PATH3D, A Groundwater Path and Travel Time Simulator, User's Manual," S. S. Papadopulos and Associates, Inc., Rockville, MD.

Stanley M. Miller 1 and Anja J. Kannengieser 2

GEOSTATISTICAL CHARACTERIZATION OP HYDRAULIC CONDUCTIVITY USING PIELD

UNSATURATED INPILTROMETER

DATA

REFERENCE: Miller, S. M., and Kannengteser, A. J., "Geostatistical Characterization of Unsaturated Hydraulic Conductivity Using Field Infiltrometer Data," Geostatlstlos for Environmental and Geoteohnioal Applioations, ASTM STP 1283, R. Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, Alexander J. Desbarats, Eds., American Society for Testing and Materials, 1996.

ABSTRACT: Estimation of water infiltration and retention in surficial soils is a critical aspect of many geotechnical and environmental site evaluations. The recent development of field-usable tension infiltrometers now allows insitu measurements of unsaturated hydraulic conductivity (K u ), thus avoiding some uncertainties associated with remolded soil samples tested in the laboratory. Several different geostatistical "mapping" methods can be used to spatially characterize Ku , including ordinary and indicator kriging, as well as spatial simulations that provide realizations (stochastic images) of Ku that exhibit more natural variability than do the smoothed spatial estimations of kriging. Multivariate procedures, such as cokriging and Markov-Bayes simulation, can incorporate information from a secondary attribute (e.g., particle size information) to enhance the spatial characterization of an undersampled Ku field. These geostatistical procedures are demonstrated and compared for a case study at a 700 sq. meter site comprised of coarse soil material. Results indicate that percent-by-weight fractions can be used effectively to enhance insitu spatial characterization of Ku' KEY WORDS: unsaturated hydraulic conductivity, particle size, characterization, geostatistics, kriging, spatial simulation.

site

An important physical property to be measured when investigating water infiltration through surficial soils is the insitu unsaturated hydraulic conductivity (K u ). The recent development of field-usable tension infiltrometers now provides the capability to measure insitu Ku , thus avoiding some of the uncertainties associated with remolded soil specimens tested in the laboratory (e.g., loss of insitu soil structure). Even though field measurements of unsaturated hydraulic conductivity exhibit spatial variability, enough spatial dependence typically 10ept. of Geology and Geol. Engrg., Univ. of Idaho, Moscow, 10 20ept . of Mathematics and Statistics, Univ. of Idaho, Moscow, 10

200

83844 83844

MILLER AND KANNENGIESER ON CONDUCTIVITY

201

is observed to warrant a geostatistical investigation to characterize Ku across the study site. Because of a fairly rapid sampling time, as many as 20 to 30 Ku field measurements can be obtained in two days, a much faster and more time-efficient procedure than that associated with laboratory testing of remolded specimens. This provides an adequate number of data for many geostatistical assessments, and the data base can be supplemented by additional data on other physical properties related to Ku (particle-size distribution attributes or insitu density) . Several different geostatistical "mapping" methods can be used to spatially characterize Ku. Univariate procedures include: 1) ordinary kriging, which provides a smoothed map of Ku estimates at unsampled locations across the site, 2) indicator kriging, which provides local estimates of conditional probability distributions of Ku at specified grid locations across the site, and 3) gaussian-based simulations, which provide spatial realizations (stochastic images) of Ku that exhibit more natural variability than do the smoothed spatial estimations of kriging. Multivariate procedures, such as cokriging and Markov-Bayes simulation can incorporate spatial information from a secondary attribute (e.g., the median particle size) to enhance the spatial characterization of an undersampled Ku field. To demonstrate and evaluate these various spatial characterization methods, a case study at a 700 sq. meter site was conducted. The coarse-grained soil material, as represented by 20 sampling locations across the site, had a median particle size of approximately 7 mm and averaged less than 6% by weight fines (i.e., finer than a No. 200 sieve). Insitu Ku values at a 5-cm tension were obtained at each of the 20 sites to provide a minimally sized sample for geostatistical studies. Background information, analytical procedures, and results of the study are presented below.

TENSION

INPILTROMETER

For nearly a hundred years, soil scientists have been describing the flow of water through unsaturated materials. Estimating the amount and rate of such water flow requires knowledge of the Ku/moisture content relationship or the Ku/soil tension (water potential) relationship. The most commonly used method to define these relationships relies on laboratory measurements obtained by pressure desorption of a saturated core of soil material, which leads to the construction of a moisture characteristic curve of moisture content vs. soil tension (Klute 1986). However, there are three problems associated with such testing: 1) the time required to setup samples and then test over a wide range of soil tensions; 2) the cost of field sampling, remolding specimens, and monitoring the laboratory tests, which may take several weeks; and 3) potentially unrealistic results due to the remolding of specimens, which destroys any insitu soil structure or packing arrangements that may have strong influence on flow characteristics. This latter concern is especially applicable to coarse-grained soil materials, such as those containing a significant amount of gravel or coarse sand. Because of these concerns, there has been considerable interest in recent years among soil scientists to develop methods for field measurements of unsaturated flow properties (Ankeny et al. 1988; Perroux and

I: I

,i

t

202


White 1988; Clothier and Smettem 1990; Ankeny et al. 1991, Reynolds and Elrick 1991}. Field capable devices for such work are characterized as "tension infiltrometers" or "disk permeameters." They allow direct measurement of insitu infiltration (flow rate) as a function of tension, which leads to estimation of the insitu Ku value. The tension infiltrometer used in this study was manufactured by Soil Measurement Systems of Tucson, Arizona. It has a 20-cm diameter infiltration head that supplies water to the soil under tension from a Marriotte tube arrangement with a 5-cm diameter water tower and a 3.8-cm diameter bubbling tower (Fig. I). Three air-entry tubes in the stopper on top of the bubbling tower are used to set the operating tension. All major parts are constructed of polycarbonate plastic, with a very fine nylon mesh fabric covering the infiltration head. Pressure transducers installed at the top and bottom of the water tower are used to measure accurately the infiltration rate. Output from the transducers is fed electronically to a field datalogger for real-time data acquisition and storage. Procedures for field setup and use of the instrument are given in the SMS User Manual (1992). Using the measured flow rates, 0 (cm 3 /hr), from the field tests, values of Ku can be obtained using formulae given by Ankeny et al. (1988), Ankeny et al. (1991), and Reynolds and Elrick (1991). The first step is to calculate the pore-size distribution parameter, a, for a pair of tension settings:

where: h1 = first soil tension value, h2 = second soil tension value (higher than h1), 01 = volumetric infiltration rate for the first tension setting, 02 = volumetric infiltration rate for the second tension setting. Next, a parameter known as Ks (akin to saturated hydraulic conductivity) is calculated as follows: (2 )

r

where: r = effective radius of wetted area beneath the infiltration disk (cm), h1 = selected soil tension value in the testing range, 01 volumetric infiltration rate corresponding to h1' Then, an exponential relationship is used to calculate the desired Ku(h} given the results from Eqns. (I) and (2):

II

I

(3 )

For our field study, measured infiltration rates were recorded at soil tensions (suctions) of -3, -6, and -15 cm of water. The pair of tensions at -3 and -15 cm of tension was used in Eqns. (I) and (2) to obtain estimates of a and Ks , respectively. Values of Ku then could be calculated at any soil tension h desired.

i:


203

em

Transducer Pinch Clomp

Water reservoir Water level

Three-hole stopper Bubble tower

8 1 em

56 em

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Air tube

Infiltration disc Shut-off valve

1--20 em---l

3.8 eM

I

I

I

I

I

==0:::::'=:':::::::::::::::':==---- ---- ------- -FIG. 1--Schematic diagram of tension infiltrometer.

S BLBCTBD

GBOSTATISTICAL

MAPPING

MBTHODS

Geostatistical spatial characterization of a specified attribute generally involves the generation of maps by "filling in" values of the attribute at numerous unsampled locations. Such filling-in processes that honor the available data can be achieved by one of two methods, interpolation or simulation. Spatial interpolation methods tend to smooth the spatial pattern of the attribute (causing the set of estimates to have a smaller variance than t h e actual data set) , but generally provide good local esti mations. Spatial simulations, on the other hand, provide more realistic fluct u ations, with the set of simulated values having a variance that approximates that of the actual data set. The theoretical basis and important practical considerations of ordinary kriging, the common geostatistical estimation method, have been described in published literature over recent years (for example , see David 1977; Journel and Huijbregts 1978; Clark 1979; Isaaks and srivastava 1989). In essence, the procedure involves calculating a weighted average of neighborhood data, where the weights represent least-squares regression coefficients obtained by incorporating spatial covariances between the data locations and the estimation location (CSi's) and those between the pairs of the data values (Cij ' S). Ordinary kriging provides unbiased and minimum-error estimates, and it can be used to estimate values at point locations or to estimate the average value of blocks (areas or vol umes) . The estimated value of the block (or point , if point kriging is used) is obtained by a weighted average of n data in the immediate neighborhood (x's at locations u i):

204


VB

n

=

L ai x(Ui)

where the ai'S are the kriging weights.

(4)

i=l In practice, the number of neighborhood data used in kriging estimation is limited so that only those data locations within a range of influence (or so) of the block or point location are used. Range of influence is defined at that distance beyond which data values are not dependent (i.e., covariance is zero). The block covariance CBB is a constant value for all blocks of identical dimensions; it is estimated by averaging the calculated covariance values between location pairs in the block defined by 4, 9, 16, or 25 locations. For point kriging, CBB

= s2 , the sample variance. The CBi values are obtained by averaging the covariances between 4, 9, 16, or 25 locations in the block with each i-th data location in the neighborhood. Any given Cij value is the covariance calculated for the lag and direction defined by the i-th and j-th data locations in the neighborhood. In all cases, the desired covariance value is obtained from the modeled variogram or complementary covariance at the specified lag distance and direction of the pair of locations being considered. Ordinary kriging is a useful spatial interpolation and mapping tool, because it honors the data locations, provides unbiased estimates at unsampled locations, and provides for minimum estimation variance. It also produces a measure of the goodness of estimates via the calculated kriging variance or kriging standard deviation. Because kriging is an interpolator, it produces a smoothed representation of the spatial attribute being mapped. Consequently, the variance of kriged estimates often is considerably less than the sample variance, and a kriged map will appear smoother than a map of the raw data. Kriging also accounts for redundancy in sample locations through the incorporation of the Cij information. Thus, kriging weights assigned to data locations clustered in the neighborhood will be less than those assigned to solitary data locations. In fact, data ·overloading" to one side of the estimation point or block may result in the calculation of small negative kriging weights. The ordinary kriging system of equations to be solved for the kriging weights is given by (a Lagrange term, A, is used to preserve unbiased conditions and to optimize estimates by minimizing the estimation error): n

L aj Cij + A. = CBi

(Sa)

j=l

(5b)

This system of equations is solved to obtain the ai weights and A. In addition, the estimation variance, or kriging variance, can be obtained at each estimation location.


205

When describing the spatial dependence of an attribute of interest (i.e., the covariance values needed for kriging), either the semivariogram function or the spatial covariance function can be used (for example, see Isaaks and Srivastava 1989). Sometimes, difficulties are encountered when estimating these functions using skewed data sets that contain outliers. The influence of such outliers can be mitigated in many cases by using monotonous data transforms or by using an indicator-transform framework that leads to computing indicator variograms for use in indicator kriging (Journel 1983). The goal when kriging indicator-transformed data is not to estimate the unsampled value at location Uo, X(Uo), nor its indicator transform ;(110; Xk), which equals 1 if X(Uo).s;Xkand equals 0 ifX(Uo»Xk' Instead, indicator kriging yields a least-squares estimate of the local, conditional cumulative distribution function (cdf) for each cutoffXk, this estimate being valued between 0 and 1 and obtained by ordinary kriging of indicator values. Thus, at each ofkcutoffs, an estimated (designated by *) conditional cdf value forXkis obtained, which is equivalent to the indicator kriged value (a weighted average of neighboring 0' sand l' s) at location 110 with cutoff

Xk: F*[Xkl (n)] = P*[{X(uo) .s;Xk} I (n)]

= E*[ {I(uo; Xk)}

I (n)]

= [;(uo; Xk)] *

(6 )

where (n) represents the local conditioning information in the neighborhood surrounding the unsampled location Do. Once local conditional cdf's are estimated and then post-smoothed (if needed), maps of probability information or percentiles can be constructed to characterize the site. When bivariate data sets are available, cokriging can be used to provide estimates of the main attribute of interest that incorporate additional information from a secondary attribute. This requires computing semivariograms or spatial covariances for each individual attribute, as well as the cross-semivariogram between the two attributes (for example, see Isaaks and Srivastava 1989). Linear coregionalization of the semivariogram models allows for the cokriging covariance matrix to be positive definite and thus avoid theoretical and computational problems, such as estimating negative kriging variances (Issaks 1984). When adequate data are available and sufficient intervariable relationships observed, cokriging may provide a more comprehensive estimation than univariate kriging. Several types of spatial simulations also are available for mapping a spatial attribute. As discussed earlier, simulations provide natural-looking fluctuations in spatial patterns, while still honoring known data locations and preserving the desired variance and spatial covariance. Thus, simulations do not provide the smoothed appearance on maps typical to kriging estimations. For the case study that follows, we wanted to compare a straightforward simulation procedure to a more complicated approach. Therefore, we investigated sequential Gaussian simulation and MarkovBayes simulation, respectively. In addition, we used simulated annealing to generate numerous "data" values to supplement the available 20 "hard" data, and thus, provide a reference or training image to be used as a standard basis for comparisons. Discussions of these simulation methods and related software are given by Deutsch and Journel (1992).

206 CASE


STUDY

The site selected for the case study was a portion of a heap leaching pad at a base-metal mine in the Western U.S. Material at the site consisted of blasted ore, with particle sizes ranging from several microns up to several tens of millimeters. Although not a typical soil in agricultural terms (i.e., possessing necessary organic materials to support plant life), this material would be classified by engineers as a coarse gravel with some sand and fines. This type of coarse material would provide a rigorous test for the SMS tension infiltrometer, which was designed to be used primarily for agricultural-type soils. Field and Laboratory Work Due to time and budgetary limitations, only 20 locations were sampled over the study site, which was approximately 30 m (E-W) by 20 m (N-S). Prior to selecting the sampling locations in the field, various sampling layouts were studied by investigating their lag (separation distance between any two locations) distributions. The goal was to have a sampling plan that would provide adequate numbers of data pairs at short and intermediate lags to facilitate the computation and modeling of semivariograms. At the same time, fairly uniform coverage across the site was desired to establish a solid basis for kriging and for simulation. Numerous sampling layouts were evaluated by a trial-and-error method before the final layout was selected. Even this plan was not final, because some changes would be needed in the field, such as when a specified location lay directly over a large cobble. At each of the 20 sampling locations, an infiltrometer test pad was leveled by hand, large rocks were removed (those greater than about 8 cm across), and a 3-mm layer of fine sand was laid down to provide proper contact between the infiltrometer head and the ground surface. The 20-cm diameter head of the SMS tension infiltrometer device then was placed on the prepared pad and the infiltration test conducted. Water flow quantities were measured for three different tensions (suctions): -3 cm, -6 cm, and -15 cm of water head. Pressure transducers and electronic data-acquisition hardware were used to record the flow data on a storage module for later use. Once the infiltration test was completed at a given location, the wetted soil material directly beneath the infiltration disc was sampled. Several kilograms of the material were placed in sealed sample bags for subsequent analysis at the University of Idaho. Insitu measurements of density were not attempted at this particular site, due to the amount of gravel and larger-sized rocks. However, such measurements with a neutron moisture/density gage are recommended for similar studies of unsaturated hydraulic conductivity. At the University of Idaho Soils Laboratory, the 20 specimens were air-dried and rolled to break-up aggregated fines prior to sieve analyses conducted according to procedure ASTM-D422, excluding the hydrometer analyses. A stack of 13 sieves was used to sieve the granular materials, and particle-size distribution curves then were plotted to display the sieve results. All specimens showed fairly well-graded particle size distributions over size ranges from less than 0.075 mm (fines) to 75 mm (coarse gravel). None of the specimens had more than 8% by weight passing the No. 200 sieve (0.075 mm). Consequently,


207

hydrometer analyses were not deemed necessary. Based on the Unified soil Classification system, the materials were identified as sandy gravel with nonplastic fines. Data Analysis Given the measured volumetric flow rates at -3 and -IS cm tensions, values of a and Ks were calculated according to Eqns. (1) and (2). Values of Ku at several selected tensions, h, then were computed and compared. Desiring to stay within the field measurement range and yet wanting to approximate field behavior at near-saturation conditions (such as after a heavy precipitation event or during spring snow-melt), we eventually selected a soil tension of -S cm for all subsequent calculations of the unsaturated hydraulic conductivity. Resulting values of Ku(-S), expressed in cm/day, for the 20 sampling locations are shown in the postplot of Fig. 2. Sample statistics for Ku(-S) are summarized below (units are cm/day): mean S.d.n-l

s.d. n

183 36.9 3S.9

minimum median maximum

124 189 238

various particle-size attributes were studied to evaluate their influence on Ku(-S), including the 010, 025, and 050 sizes, as well as the percent-by-weight finer than 2.0 mm (No. 10 sieve) and 4.7S mm (No. 4 sieve). Scatterplots of Ku(-S) vs. each of these attributes were generated and fitted with linear regression models. The three characteristics showing the strongest linear relationship were the percent-by-weight finer than 2.0 mm and finer than 4.7S mm, and the 025 size. Linear correlation coefficients were in the O.SS - 0.60 range, a positive value for the first two attributes and a negative value for the third one, respectively. Subsequent computations of experimental semivariograms for these three characteristics indicated that only the percent-by-weight finer than 2.0 mm (PF2) showed any significant univariate spatial dependence and cross spatial dependence with Ku(-S). Thus, this parameter from the particle-size distributions was selected as a secondary attribute to help estimate and map the primary attribute, Ku(-S). Sample statistics for PF2 are summarized below (units are %): mean s .d.n-l

s.d. n

24.S 2.S0 2.44


17.S 2S.1 27.9

Computing usable semivariograms with small data sets can be a challenging task. With only 20 data locations for this study, it was difficult to select computational lag bins that would provide adequate numbers of data pairs for the irregularly spaced data set. Therefore, we decided to use a ·sliding lag window" approach for computing the experimental semivariograms. A sliding window S-m wide was used for both the Ku(-S) and the PF2 data sets. Thus, the plotted points shown on the semivariogram graphs in Fig. 3 represent overlapping lag bins of O-S m, 1-6 m, 2-7 m, and so on. Because of the limited number of data, only isotropic (omnidirectional) semivariograms were computed. The two

208


3396. + 124.

3394.

+ 211. + 138.

3392.

+ 198.

3390.

+ 194.

+

3388. + 215.

3386.

+ 132.

238.

+ 183.

+ 205.+ 148.

+ 210.

+ 209.+ 127. + 174.+ 162.

3384.

+ 161 .

3382

+

+ 231.

+ 233.

161.

3378.L-~--~--~--~--~--~~--~--~--~--L-~--~--~~

664. 666.

668.

670.

672.

674.

676. 678. 680. 682. 664. 686. 668.

690.

692.

694.

FIG. 2--Postplot of e s timated insitu values of Ku(-S) in cm/day; northing and easting coordinates are in meter s .

1800

"i 1600

++

_7

+

~1400

+

"0

1; 1200

"E 5

u

-1000

~

:.'"

.2

.~

"

(/)

+

!i8

~

~4

800 600 400 200 0

0

P2 ·e

.

(/) 1 0 0

2

4

6

8

10

12

14

Lag Oi.lance (m)

(a)

16

18

20

22

0

2

4

6

10 12

8

14

16

18

20

22

Lag Distance (m)

(b)

FIG. 3--Estimated semivariograms and fitted spherical model s ; (a) Ku(-S) model: y(b)=671+620Sph I2 (b); (b) PF2 model: Y(h)=O.4+5.55Sph I2 (h).

experimental semivariograms were fitted with sph erical variogram models, as de scribed and annotated in Fig . 3. Sills on the models were set equal to the sample variance in both cases . The Ku(-S) data set wa s fitted with a fir s t-order trend s urface


209

model to determine if there was any significant trend in mean across the site. Calculated F-statistics for this regression fit were not large enough to induce a rejection of the null hypothesis that a significant trend was not present. Thus, one of the primary considerations (i.e., that the mean does not depend on spatial location) of the covariance stationarity model for spatial random functions could be readily accepted for subsequent spatial estimations and simulations. Site Characterization and Mapping Using Geostatistics The isotropic semivariogram model shown in Fig. 3a provided the spatial covariance model to conduct ordinary point kriging on regular grids to generate estimates of Ku(-5) across the study site. GeoEAS (Englund and Sparks 1991) computer software was used. A regular 25 x 35 grid at 1-m spacings was used, because at field sampling locations an area approximately 0.7 to 1 m in diameter was wetted during each infiltrometer test. The shaded contour map of Fig. 4 clearly shows the smoothing characteristics of kriging. Summary statistics for these estimations are presented in Table 1. Comparisons of the sample variances again reflect the significant amount of spatial smoothing inherent to kriging estimations. Given similar estimates for storativity and soil-layer thickness, water-balance computations that incorporate annual precipitation and evapotranspiration values can be conducted to predict total annual recharge at each grid location at the site (Miller et al. 1990). Point cokriging of Ku(-5) also was conducted on the 1-m grid, incorporating the spatial information and codependence of PF2 (percentby-weight finer than 2.0 mm). The two semivariogram models of Fig. 3, as well as a cross-semivariogram between the two attributes (Fig. 5), were used in the GSLIB cokriging software (Deutsch and Journel 1992). Cokriged estimates were plotted and contoured to produce the shaded contour map given in Fig. 6. Summary statistics of the estimates are reported in Table 1. Cokriging did yield estimates with greater variance than ordinary point kriging, but not with as great of relief (i.e., difference between maximum and minimum). This estimation method would be especially applicable in situations where numerous sampling sites with particle-size analyses could be used to supplement a few actual Ku sampling sites.

TABLE 1--Summarized statistics of kriging results for Ku (-5), em/day.

Data n mean s.d. var. min. max.

20 183.0 35.9 1291.0 124.0 238.0

Ordinary Point Kriging

875 183.2 11.8 140.1 127.2 237.8

Cokriging

875 182.3 13.1 170.8 139.6 215.7

210


207 .00 199.00 191.00 183.00

§.

'"c

175 .00

0

167 .00

€ z

159 .00 151.00 143 .00 135.00

Eas1ing 1m)

FIG . 4 - -Shaded contour map of Kul-5) point k ri ging e s tima t es Ic m/ d ay) on a 1-m regular grid .

80 E ~

'"

70

+

.~ 50

..

+ + ++

60

+

.~ 40 C/)

30

~ 20

u

10 0 0

2

4

6

8

10

12

14

16

18 20

22

Lag Distance 1m)

FIG . 5--Es tima ted cross s emivariogram betwee n Kul-5) and PF2 with fit ted spher ical mode l: Y(h) = 8.0 + 42.5 Sph12(h) .

Indicator kriging yield s a different kind of mapping informatio n that often i s u s eful to characteri z e a s patial attribute . As di s cu ss ed previously, local conditional cdf 's are e s timated acro ss the s ite by indicator krig ing at s everal different data thre s hold value s. For th is s tudy, five thre s hold s were a ss igned for the Kul-5) data : 140, 161.5, 190, 211 . 5, and 235 cm / day. Computer s oftware in GSLIB IDeut s ch and Journel 199 2 ) was u s ed to c o nduct the indicator kriging, s moo th the


211

207 .00 I 99 .00 191.00 183 .00 175 .0 0 167 .00 159.0 0 151 .00 143 .00 135 .00

Eosling 1m)

FIG. 6--Shaded contour map of Ku(-5) cokriging estimates on a 1-m regular grid, using PF2 as the secondary attribute.

estimated cdf's, and produce E-type estimates (expectation, or mean values) for mapping purposes (Fig. 7a). The probability of exceeding 200 cm/day also was calculated at each estimation location, and a s haded contour map of this exceedance probability wa s produced (Fig. 7b). This exceedance cutoff value was selected arbitrarily, but serves to illustrate the types of probability maps that can be generated to help characteri ze Ku at the site and provide input for cost-benefit studies to assist in treatment or remediation designs. Another advantage of the indicator kriging framework is that "soft" in format ion (inequal i ty relation s, professional judgments, etc.) can be coded probabilistically and used to supplement available "hard" indicator data (Journel 1986). As shown in the Ku(-5) postplot of Fig. 2, spatial variability in the unsaturated hydraulic conductivity is typical. For example, at a northing coordinate of 3386 m, a value of 205 cm/day is adjacent to a value of 148 cm/day, and a 209-cm/day value is adjacent to one of 127 cm/day. This spatial variability is to be expected , especially for coarser grained soil materials. Therefore, smoothed kriging maps of the type presented thus far may not always be the most appropriate way to characterize Ku. Spatial simulations that honor available data and also pre serve the sample variance provide quite a different prediction of spatial pattern s. To compare the performances of two type s of spatial simulators for small data sets of Ku , we first used simulated annealing (Deutsch and Journel 1992) on a 1-m grid to generate a pseudo ground-truth image of Ku(-5) that could serve as a reference base-map (Fig. 8). Sample statistics for the 875 simulated values are summarized below (in cm/day) :

212

GEOSTAT ISTICAL APPLICATIONS mean s.d. var.

184 35.8 1280


115 189 248

25.00 15.00 05.00 195.00 185.00 175 .00 165 .0 0 155 .0 0 145 .00 135 .00

Easling (meIer)

(a)

0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00

Easting (meIer)

(b)

FIG . 7--Shaded contour maps of Ku {-5) indicator kriging results on a 1-m regular grid; (a) E-type map s howing cdf expectation values, cm/day; (b) probability of exceeding 200 cm/day.

213

MI LLER AND KANNENGIESER ON CONDUCTIVITY

250.00 230.00 210.00 190 .00 170.00 150.00

130.00 110.00

Eosling (meier)

FIG. 8--Shaded contour map of Ku (-5) "reference data" based on simulated annealing.

The isotropic semi variogram for the simulated values was similar to that shown in Fig. 3a. Sequential Gaussian and Markov-Bayes simulations of Ku(-5) were conducted on a I-m grid u sing software from GSLIB (Deutsch and Journel 1992), based on the 20 known data values and on the semivariogram model of Fig. 3a. The Markov-Bayes procedure also uses secondary information (PF2 in this case), but when both the primary and seconda ry attributes have data values at the same locations, the primary information is given precedence over the secondary (Zhu 1991; Miller and Luark 1993). Thus, results of the two different simulation methods for four trials (simulation passes, or iterations) were quite similar, s howing average meansquare-errors on the order of 2,300 cm/day squared. These errors were calculated as squared differences between the 875 simulated values and the 875 values of the ground-truth image. It was not surprising to observe that the largest of these errors occurred in the most s parsely sampled areas of the study site , where uncertainties are greatest for the simulated annealing approach and the other simulation methods. Advantages of a bivariate simulation method, such as the MarkovBayes (MB) procedure, become apparent when the primary attribute i s undersampled in regard to the secondary attribute. To illustrate this, we selected several subsets (reduced data sets) containing 10 of the original 20 Ku sampling sites. The goal became one of simulating 875 Ku (-5) values, given 20 PF2 sites and 10 Ku(-5) sites. The s e results then could be compared to those ba sed on sequential Gaussian (SG) simulation u sing only the 10 Ku(-5) data. Basic statistica l information for the three different subsets , A, B, and C, is presented in Table 2.

214


TABLE 2--Summarized statistics for the three subsets of Ku (-5) and for the corresponding simulation results (units are cm/day).

Original Data n

mean s.d. var. min. max.

20 183.0 35.9 1291.0 124.0 238.0

Subset A

Subset B

Subset C

10

10 194.4 30.3 918.6 138.4 232.8

176.5 40.1 1610.0 123.7 232.8

177.4 35.6 1264.0 123.7 237.8

10

SO simulation results (ayera~d from four trials);

n mean s.d. mean sq. err. using ref. image

875 176.4 34.5 2391.

875 194.2 2176.

875 173.0 40.3 2763.

875 182.1 33.4 2344.

875 178.8 37.2 2476.

31.1

MB simulation results (ayera~ed from four trials); n

mean s.d. mean sq. err. using ref. image

875 175.6 36.2 2588.

Note that Set B has smaller variance than the original data set, and that Set C has a larger variance than the original data set. In terms of overall mean squared error, SG simulation outperformed MB simulation for Subsets A and B where the sample variance was relatively small. However, when the subset had a larger variance, MB simulation was the better procedure according to this criterion. The beneficial influence of secondary data as used in the MB simulation method is shown clearly by the sample means and standard deviations of simulated sets based on the three different subsets. Note especially the more consistent results for Subsets Band C shown by the MB procedure compared to more inconsistent results of the SG procedure. Examples of MB simulation results for these two subsets are presented as shaded contour maps in Fig. 9.

CONCLUSIONS A variety of geostatistical tools are available for mapping and characterizing unsaturated hydraulic conductivity. Using recently developed tension infiltrometers for field use, measurements of volumetric infiltration rates provide a basis for estimating Ku values that reflect insitu conditions of soil density, packing, and structure. Although a vast improvement over laboratory testing of disturbed specimens, such insitu testing still requires enough time and effort that numerous measurements (greater than 30) at a study site likely will

MI LLER AND KANNENGIESER ON CONDUCTIVITY

215

250 .00

230.00 210.00 190 .00 170 .00

150.00

130 .00 110 .00

Easling (melor)

(a)

250 .00

230 .00 210 .00 190.00 170 .00

150.00 130 .00

110.00

Easling (meier) (b)

FIG. 9--Examples of Markov-Bayes s imulations of Ku (-5), cm / day; (a) bas ed on a subset (Set B) of 10 Ku data with variance lower than that of original data; (b) based on a subset (Set C) of 10 Ku data with variance higher than that of original data.

not be affordable except for large-budget investigations . However, s econdary information more economical to obtain, especially particlesi z e characteristics s uch a s the percent-by-weight finer than 2.0 mm ,

216


can be used in bivariate types of kriging and simulation to fill-in Ku values at unsampled locations and provide enhanced spatial mappings. The case study presented here dealt only with surface measuremen and two-dimensional maps. However, trenching with benched sidewalls could be used to provide insitu Ku assessments at various elevations add a third dimension of elevation into the characterization scheme. The kriging and simulation methods described herein are readily adapt to three-dimensional situations. If point estimates are desired for generating contour maps of estimated Ku , then ordinary point kriging (or indicator kriging for local cdf's) would be preferred. When local cdf's are estimated by indicator kriging, a variety of probabilistic type maps can be gene rat to characterize spatial patterns of Ku across the study site. When secondary data are available, and a recognizable relationsh is present between secondary and primary data, Markov-Bayes simulation often will provide better results than those produced by univariate simulations, such as the sequential Gaussian method. The former meth particularly has advantages when primary sample data are sparse and perhaps not representative of the entire population, and when a larger sample is available of the secondary attribute.

ACKNOWLEDGEMENTS Portions of this research work were supported by the Idaho Cente for Hazardous Waste Remediation Research under Grant No. 676-X405. authors also express appreciation to John Hammel, John Cooper, and Mit Linne of the University of Idaho for their technical advice and assistance in the operation of the tension infiltrometer, analysis of its measurements, and in the laboratory testing program. The Universi of Idaho does not endorse the use of any specific commercial material product mentioned in this paper.

REFERENCES Ankeny, M.D., M. Ahmed, T.C. Kaspar, and R. Horton, 1991, 'Simple Fie Method for Determining Unsaturated Hydraulic Conductivity," £oil Sci. Soc. of America Jour., Vol. 55, No.2, p. 467-470. Ankeny, M.D., T.C. Kaspar, and R. Horton, 1988, "Design for Automated Tension Infiltrometer," Soil Sci. Soc. of America Jour., Vol. 52, No. p. 893-896. Clark, I., 1979, 129 p.

Practical Geostatistics, Applied Sci. Publ., London,

Clothier, B.E., and K.R.J. Smettem, 1990, 'Combining Laboratory and Field Measurements to Define the Hydraulic Properties of Soil," ~ Sci. Soc. of America Jour., Vol. 54, No.2, p. 299-304. David, M., 1977, Geostatistical Ore Reserve Estimation, Elsevier, Amsterdam, 364 p.


217

Deutsch, C.V., and A.G. Journel, 1992, GSLIB: Geostatistical Software Library and User's Guide, Oxford Univ. Press, New York, 340 p. Englund, E., and A. Sparks, 1991, Geostatistical Environmental Assessment Software User's Guide (GeoEAS 1.2.1), USEPA Env. Monitoring Systems Lab., Las Vegas, NV. Isaaks, E.H., 1984, "Risk Qualified Mappings for Hazardous Waste Sites: A Case Study in Distribution-Free Geostatistics,· M.S. Thesis, Stanford Univ., Stanford, CA, 111 p. Isaaks, E.H., and R.M. Srivastava, 1989, An Introduction to Applied Geostatistics, Oxford Univ. Press, New York, 561 p. Journel, A.G., 1983, "Nonparametric Estimation of Spatial Distributions," Math. Geology, Vol. 15, No.3, p. 445-468. Journel, A.G., 1986, "Constrained Interpolation and Qualitative Information -- the Soft Kriging Approach,· Math. Geology, Vol. 18, No. 3, p. 269-286. Journel, A.G., and C.J. Huijbregts, 1978, Academic Press, New York, 600 p.

Mining Geostatistics,

Klute, A., 1986, "Methods of Soil Analysis, Part 1," Amer. Soc. of Agronomy. Monograph 9. Miller, S.M., J.E. Hammel, and L.F. Hall, 1990, HCharacterization of Soil Cover and Estimation of Water Infiltration at CFA Landfill II, Idaho National Engineering Laboratory; Res. Report C85-110544, Idaho Water Resources Research Inst., Univ. of Idaho, Moscow, ID, 216 p. Miller, S.M., and R.D. Luark, 1993, HSpatial Simulation of Rock Strength Properties Using a Markov-Bayes Method," Int. Jour. Rock Mech. Min. Sci. & Geomech. Abstr., Vol. 30, No.7, p. 1631-1637. Perroux, K.M., and I. White, 1988, HDesigns for Disk Permeameters," Soil Sci. Soc. of America Jour., Vol. 52, No.5, p. 1205-1215. Reynolds, W.D., and D.E. Elrick, 1991, HDetermination of Hydraulic Conductivity Using a Tension Infiltrometer,· Soil Sci. Soc. of America Jour., Vol. 55, No.3, p. 633-639. Soil Measurement Systems, 1992, "Tension Infiltrometer User Manual,· Soil Measurement Systems, Tucson, AZ. Zhu, H. 1991, HModeling Mixture of Spatial Distributions with Integration of Soft Data," Ph.D. dissertation, Dept. of Applied Earth Sci., Stanford Univ., Stanford, CA.

l

2

Marc V. Cromer , Christopher A. Rautman , and William P. Zelinski

3

Geostatistical Simulation of Rock Quality Designation (RQD) to Support Facilities Design at Yucca Mountain, Nevada

REFERENCE: Cromer, M. V., Rautman, C. A., and Zelinski, W. P., "Geostatistical Simulation of Rock Quality Designation (RQD) to Support Facilities Design at Yucca Mountain, Nevada," Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R. M. Srivastava, S. Rouhani, M. V. Cromer, A. 1. Johnson, A. 1. Desbarats, Eds., American Society for Testing and Materials, 1996. ABSTRACT: The conceptual design of the proposed Yucca Mountain nuclear waste repository facility includes shafts and ramps as access to the repository horizon, located 200 to 400 m below ground surface. Geostatistical simulation techniques are being employed to produce numerical models of selected material properties (rock characteristics) in their proper spatial positions. These numerical models will be used to evaluate behavior of various engineered features, the effects of construction and operating practices, and the waste-isolation performance of the overall repository system. The work presented here represents the first attempt to evaluate the spatial character of the rock strength index known as rock quality designation (RQD). Although it is likely that RQD reflects an intrinsic component of the rock matrix, this component becomes difficult to resolve given the frequency and orientation of data made available from vertical core records. The constraints of the two-dimensional study along the axis of an exploratory drift allow bounds to be placed upon the resulting interpretations, while the use of an indicator transformation allows focus to be placed on specific details that may be of interest to design engineers. The analytical process and subsequent development of material property models is anticipated to become one of the principal means of summarizing, integrating, and reconciling the diverse suite of earth-science data acquired through site characterization and of recasting the data in formats specifically designed for use in further modeling of various physical processes. KEYWORDS: indicator simulation, rock quality designation, variogram, core data 1 Principal Investigator, Sandia National Laboratories/Spectra Research Institute, MS 1324, P.O. Box 5800, Albuquerque, NM 87185-1342 2 Principal Investigator and Senior Member Technical Staff, Sandia National Laboratories, MS 1324, P.O. Box 5800, Albuquerque, NM 87185-1342 3 Principal Investigator, Sandia National Laboratories/Spectra Research Institute, MS 1324, P.O. Box 5800, Albuquerque, NM 87185-1342

218

CROMER ET AL. ON ROCK QUALITY DESIGNATION

219

INTRODUCTION Yucca Mountain, Nevada is currently being studied by the U. S. Department of Energy as a potential site for the location of a high-level nuclear waste repository. Geologic, hydrologic, and geotechnical information about the site will be required for both engineering design studies and activities directed toward assessing the waste-isolation performance of the overall repository system. The focus of the overall Yucca Mountain Site Characterization Project is the acquisition of basic geologic and other information through a multidisciplinary effort being conducted on behalf of the U. S. Department of Energy by several federal agencies and other organizations The location of the proposed underground facilities and the proposed subsurface access drift are shown on Figure 1. Also shown are the locations for the bore holes used in this two-dimensional study. The Yucca Mountain site consists of a gently-eastward dipping sequence of volcanic tuffs (principally welded ash flows with intercalated nonwelded and reworked units). Various types of alteration phenomena, including devitrification, zeolitization, and the formation of clays, appear as superimposed upon the primary lithologies. The units are variably fractured and faulted. This faulting has complicated characterization efforts by offsetting the various units, locally juxtaposing markedly different lithologies. Most design interest is focused on the Topopah Spring Member and immediately adjacent units. By comparison, the waste-isolation performance of the repository system must be evaluated within a larger geographic region termed the "controlled area" (Figure 1). The region evaluated by this study is contained entirely within the controlled area. In general, this study is further restricted to the location of the subsurface access drift known as the North Ramp, in keeping with a general engineering orientation. This twodimensional study represents the first attempt to identify local uncertainty in the rock structural index known as Rock Quality Designation (RQD).

CONCEPTUAL MODEL The U.S. Geological Survey provided the original geological cross-section model along the North Ramp (USGS, 1993). That model was subsequently modified by others and new cross-sections have also been prepared manually. For this study, the cross-section shown in Figure 2 was recreated interactively using the Lynx GMS Geosciences Modeling System, to insure that all of the new bore hole data and corroborative surface control (Scott and Bonk, 1984) was honored. The cross-section shown in Figure 2 is consistent with the conventional assumption that all faults in the repository area are generally down-thrown on the west side. This interpretation requires a variable, but relatively steep, dip to the beds that can locally exceed 6 degrees (10% grade). This cross-section also suggests the possible existence of one or more faults with the east side down thrown. The eight bore holes noted in Figure 2 are of variable lengths and are shown in their proper orientation with respect to the

220


Nellis Air Force Range

NRG-7A

t

N

(Not to Scale) Conceptual Perimiter Drift Boundary,

- - --- --

~- --) -- - -- -

ELM (Public Lands)

NRG-l"

North Ramp

Nevada Test Site

---.0 20.0 15.0

10.0

5.0 .0

Figure 9 Three alternative 2-D images (realizations) ofRQD along the axis of the North Ramp. The angular trace in the middle of each image represents the vertical orientation of the ramp within the cross-section. Each image can be considered equally probable given the state of existing knowledge, because each is conditioned to the same sample data and honor the same spatial statistics. The differences between the images provides a measure of joint spatial uncertainty.


231

value above, or below, a threshold. A map showing the value at which an individual pixel reaches a specified cumulative probability, for example, would provide valuable information for quantifying risk. Figure 11 shows the probability of exceeding an RQD value of 25. Although this map looks very similar to the expected value map of Figure 10, it is revealing very different information. The gray-scale in Figure 11 ranges between zero (0% probability) and one (100% probability), unlike the expected value map.

CONCLUSIONS Unfortunately, the 2-D simulated images along the North Ramp cross-section do not explicitly focus information on the expected variability to be encountered along the drift itself. To evaluate anticipated conditions specifically along the drift, the designed inclination of the drift has been projected from the tunnel entrance and is shown as the trace super-imposed on the images in Figure 9. The expected (mean) value ofRQD along the tunnel projection has been extracted on a pixel-by-pixel basis for comparison against each of the three simulations presented in Figure 9. The graphs shown in Figure 12 allow us to compare the variability in simulated RQD along the three tunnel projections (taken from Figure 9) as a function of distance from the right (east) edge of the cross-section. As a point of reference, the east edge of the crosssection also corresponds to the location of boring NRG-l. The most immediate observation in Figure 12 is the widespread, erratic fluctuations in simulated values about their expected (mean) value. This was to be expected following our variography exercises and discovery of the limited horizontal correlation range (approx. 800.0 ft (243.8 m)) for lower RQD values and negligible spatial correlation in the higher RQD values. What is not so apparent is the performance of the simulation in areas that are conditioned by the available boring logs. At distances ofless than 3200 ft (975.4m) from NRG-l the simulations, in general, tend to deviate less from the expected value. Boring log data in this region are available to constrain uncertainty and, therefore, reduce the spread oflikely outcomes for a local prediction

FINAL THOUGHTS Basic exploratory data analysis identified a great deal oflocal variability in RQD. Although very low RQD (i.e. less than 25) can be anticipated periodically along the entire length of the North Ramp, it would not be prudent to extrapolate this interpretation to the entire mountain. Three factors were found to influence the interpretation ofRQD: 1) stratigraphic setting, 2) proximity to major fault/fracture zones, and 3) very local footby-foot factors (likely due to individual high-angle fractures sub-parallel to the drill core). The high degree of variability over very short distances may require design planning to accommodate the worst rock conditions along the entire length of excavation

232


Figure 10. Mean (expected) value map developed from 100 individual simulations of RQD.

1 .0 .90 .80 .70 .60 .50

.40 .30 .20 .10 .0

Figure 11 Probability map reflecting the likelihood of exceeding an RQD value of 25 . Note the scale reflects a probability range from 0% to 100%.

CROMER ET AL. ON ROCK QUALITY DESIGNATION RN - 112063

RN-30157

10

_________ _

Horlzontlol dl,lance frvm "RO·t

RN-22475

10

"

~~~a~~~~§ .",IM_ """" NRO·'

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Figure 12 Simulated RQD values along the proposed North Ramp taken from the three fields shown in Figure 9. For comparison, also shown (in bold) are their expected values derived from the 100 simulations.

233

234


Investigative work on rock properties in the exploratory studies facility is underway to supplement drill hole data with an adequate number and distribution of data pairs collected in a fashion that will support geostatistical analyses. In the meantime, simulation analyses has provided a preliminary assessment of the conditions that could be encountered during the excavation of the North Ramp. Indicator simulation along the axis of this drift identifies the need for additional information if this study, or similar studies, are to forecast engineering requirements for facilities design, especially with respect to spatial continuity of higher RQD values.

i

I

This study has demonstrated how the measurement and analysis of data may lead to interpretations that are not obvious or apparent using other means of research. Although, many statistical tools are useful in developing insights into a wide variety of natural phenomena; many others can be used to develop quantitative answers to a specific questions. Unfortunately, most classical statistical methods make no use of the spatial information in earth science data sets. However, like classical statistical tests, geostatistical techniques are based on the premise that information about a phenomenon can be deduced from an examination of a small sample collected from a vastly larger set of potential observations on the phenomena. Geostatistics offers a way of describing the spatial continuity that is an essential feature of many natural phenomena and provides adaptations of classical regression techniques to take advantage of this continuity. The quantitative methodology found in applications of geostatistical modeling techniques can reveal the insufficiency of data, the tenuousness of assumptions, or the paucity of information contained in most geologic studies.

II REFERENCES Cecil III, O.S., 1970, "Correlations of Rock Bolt--Shotcrete Support and Rock Quality Parameters in Scandinavian TWmels," Ph.D. Thesis, University of Illinois, Urbana. Cromer, M. V. and R. M. Srivastava, 1992, "Indicator Variography for Spatial Characterization of Aquifer Heterogeneities," in Water Resources Plannjn~ and Mana~ement. Proceedin"s of the Water Resources sessions at Water Forum '92. Au~ust 2-5.1992, American Society of Civil Engineers, Baltimore, MD, pp 420-425.

i' I;

I

Deere, D.U., and D.W. Deere, 1989, "Rock Quality Designation (RQD) after twenty years: US Army Corps of Engineers," Contract Report GI-89-1. Deutsch C.V. and A.G. Journel, 1992, "GSLIB Geostatistical Software Library and User's Guide," Oxford Univ. Press, New York, New York. Isaaks, E. H., and R. M. Srivastava, 1989, "An Introduction to Applied Geostatistics," New York: Oxford University Press.


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Lin, M., M. P. Hardy, and S. 1. Bauer, 1993, "Fracture Analysis and Rock Quality Designation Estimation for the Yucca Mountain Site Characterization Project: Sandia Report SAND92-0449," Sandia National Laboratories, Albuquerque, NM. Olea, R.A. ,1991, "Geostatistical Glossary and Multilingual Dictionary," International Association of Mathematical Geology Studies in Mathematical Geology No.3, Oxford University Press. Scott, R.B. and J. Bonk, 1984, "Preliminary Geologic Map of Yucca Mountain, Nye County, Nevada, with Geologic Sections," U.S. Geol. Survey Open-File Report 84-494. US Geological Survey, 1993 "Methodology and source data used to construct the demonstration lithostratigraphic model: second progress report". Wingle, W. L., E. P. Poeter, and S. A. McKenna, 1994 "UNCERT User's Guide: A Geostatistical Uncertainty Analysis Package Applied to Ground Water Flow and Contaminant Transport Modeling," draft report to the United States Bureau of Reclamation, Colorado School of Mines.

James R. Carr! REVISITING THE CHARACTERIZATION OF SEISMIC HAZARD USING GEOSTATISTICS: A PERSPECTIVE AFTER THE 1994 NORTHRIDGE, CALIFORNIA EARTHQUAKE

REFERENCE: Carr, 1. R., "Revisiting the Characterization of Seismic Hazard Using Geostatistics: A Perspective After the 1994 Northridge, California Earthquake," Geostatistics for Environmental and Geotechnical Applications, ASTM STP 1283, R. Mohan Srivastava, Shahrokh Rouhani, Marc V. Cromer, A. Ivan Johnson, and Alexander J. Desbarats, Eds., American Society for Testing and Materials, 1996.

ABSTRACT: An indicator kriging model of seismic hazard for southern California, based on the time period 1930 - 1971, is developed. This hazard assessment is evaluated in light of the occurrence of more recent, moderate earthquakes: the 1987 Whittier Narrows, the 1990 Upland, and the 1994 Northridge earthquakes. The hazard map shows relatively poor spatial correlation between regions of high hazard and known, active faults. A hypothesis is developed, however, suggesting that high seismic hazard in southern California is a function of spatial proximity to all active faults, not to anyone active fault. KEYWORDS: seismic hazard, modified Mercalli intensity, southern California, kriging, semivariogram, indicator functions

Geostatistical analysis of earthquake ground motion was first attempted by Glass (1978). Therein, modified Mercalli intensity data for the 1872 Pacific Northwest earthquake were analyzed using semivariogram analysis, then regularized (gridded) using kriging and contoured. Glass (1978) demonstrates the usefulness of geostatistics vis-a-vis semivariogram analysis and kriging for analyzing earthquake ground motion. Based on the success of Glass (1978), an experiment was attempted to characterize seismic hazard for southern California (Carr 1983; Carr and Glass 1984). Kriging was used to form digital rasters of modified Mercalli intensity data for all earthquakes in the time period, 1930 - 1971, that occurred within a 125 km radius of San Fernando, California (an arbitrary choice). These digital rasters !Professor, Department of Geological Sciences/l72, University of Nevada, Reno, NV 89557 236

CARR ON NORTHRIDGE EARTHQUAKE

237

were geographically registered and, as such, served as input to a Gumbel (1958) extreme events model for computing seismic hazard. Procedures for developing this model consisted of the following steps: 1) kriging was used to form a digital raster for each earthquake in the aforementioned time frame; all of these rasters were geographically registered; 2) for each year, 1930 - 1971, if more than one earthquake occurred, then the maximum kriged intensity for each cell of the raster was found and a summary raster formed reflecting maximum intensity for the year; this process resulted in 42 digital rasters, each a record of maximum intensity values for an entire year; 3) Gumbel (1958) statistics of extreme values was used to compute the probability that an intensity VI was exceeded for a raster cell over the 1930 - 1971 time period; an intensity VI was an arbitrary choice, but this is the intensity value at which exterior damage to buildings begins. These exceedance probabilities constitute the seismic hazard (Fig. 1).

119

118

Fig . 1. Seismic hazard model developed using Gumbel (1958); from Carr (1983) and also published in Carr and Glass (1984). Contoured values are probabilities (%) of exceeding an intensity VI over a 50 year period.

A Gumbel (1958) model requires that certain decisions be made when computing the probability of exceeding a particular level of ground motion. For example, a minimum, or threshold, ground motion value must be chosen for calculations. In Carr and Glass (1984), for instance, a minimum intensity value of III was chosen, yet in many years, the minimum value was actually O. The choice of an intensity III was entirely arbitrary . As an alternative to a Gumbel (1958) model, Carr and Bailey (1985) developed an indicator kriging (c.f., Iournel 1983) seismic hazard model. This model does not use Gumbel's statistics of extremes method for computing exceedance probabilities. Instead, modified Mercalli intensity data are rust converted to indicator values as is described later. Once converted to indicator values, kriging is applied to the indicator data to form digital rasters. As in the Carr and Glass (1984) model, these rasters

238


were geographically registered during the kriging process. Because the rasters are registered, the final step in the indicator kriging model is simply a summing of all rasters to form one, combined raster. A contour map of the combined raster shows the frequency of exceeding a threshold VI over a particular time period. This frequency constitutes the seismic hazard for a particular geographic region. Carr and Bailey (1985) applied the indicator kriging model to the New Madrid, Missouri seismic zone in the time period, 1811 - 1980. Because the indicator kriging model is considerably easier to apply in comparison to one using the Gumbel (1958) method, the seismic hazard in southern California in the time frame 1930 - 1971 is revisited herein using indicator kriging. One objective of this study is to compare the seismic hazard map from indicator kriging to that obtained using a Gumbel calculation. Another aspect of this analysis is to compare the occurrence of recent southern California earthquakes, in particular the 1987 Whittier Narrows, the 1990 Upland, and the 1994 Northridge earthquakes, to the seismic activity that preceded them (1930 - 1971).

A BRIEF REVIEW OF GEOST ATISTICS In general, geostatistical methods are useful for characterizing the spatial variation of regionalized phenomena. Other than earthquake ground motion, geotechnical applications include soil density and strength, ground water level, and ground water salinity; of course, there are many more examples. The term, geostatistics, is often considered synonymous with the spatial estimation technique known as kriging (Matheron 1963). This estimator is a relatively simple, weighted average of the form: N

Z*(xo }

= Laiz(xi } i=l

wherein Z(x;) are data values at N nearest data locations to the estimation location, Xc; Z'(Xc) is the estimated value at the estimation location, Xc, and the values, a, are weights applied to the N data values to obtain the estimate. A restriction is placed on the weights, a, in ordinary kriging such that their sum is 1; this assures unbiased estimation. That kriging is a relatively simple estimator is seen in its equation form, a simple weighted average. Obtaining the weights, a, for this equation is more complicated. The weights are obtained by solving: [COVu]{a} = {COVo;}. Notice that these matrices are functions of spatial covariance (COV). Covariance in this case is the autocovariance of the spatial data, Z, between two locations in space. Knowledge of spatial covariance is obtainable from what is known as the semivariogram (often referred to simply as the variogram; see Matheron 1963; or 10urnel and Huijbregts 1978). The semivariogram is estimated from the spatial data


239

as follows: N

Y (h)

=

-.l L 2N i

[Z(x)

- Z(xi+h) J 2

=1

which is a function of average squared difference in Z as a function of spatial separation distance (lag), h. Once the semivariogram is calculated, it must be modeled for use in kriging. Only a few functions , those that are negative semi-definite, qualify as valid models (see Journel and Huijbregts 1978). The most useful semivariogram model is known as the spherical model and is graphed (Fig. 2) . To model a calculated semivariogram (Fig. 2), values for the nugget, sill, and range (Fig. 2) are interpreted, allowing the spherical model equation to fit the calculated semivariogram as closely as possible (Fig . 2) . Then, spatial covariance is obtainable from the semivariogram model as follows: COV(h)

= s i l l - y (h)

In kriging, once a semivariogram model is selected and parameters defined (nugget, sill, and range), covariance entries in the foregoing matrix system are computed using the semivariogram model. How these calculations are performed is described in Carr (1995) using hand calculation examples. Once the covariance matrix entries are obtained, the matrix system is solved for the weights, a, using an equation solver, such as Gauss elimination or LV decomposition. Software for semivariogram calculation and kriging is given in Deutsch and Journel (1992), including diskettes containing FORTRAN source code. Software is also given in Carr (1995) along with graphics routines for displaying results. 300 225 SiII - 165 r(h)

?"'..,...------

Spherical model

7S R:mge - 2.3 3

4

h nag distance. km)

Fig. 2. A calculated semivariogram modeled using a spherical model; note nugget (CO), sill, and range (from Carr 1995).

A BRIEF NOTE REGARDING THE DATA Herein is presented a seismic hazard model of southern California that is based on modified Mercalli intensity data. Such data are subjectively assigned and are integer values in the range 0 to XII (12). A value, 0, represents no ground motion; a

240


value, XII (12), represents total damage, landsliding, fissuring, liquefaction, and so on. A value, VI (6), is that value at which exterior structural damage is noticed, such as cracked chimneys. Interior damage is noted with a value, V (5). Subsequent to an earthquake, the United States Geological Survey distributes questionnaires to citizens living within the region experiencing the earthquake. They are asked to describe what they experienced during the earthquake. Examples include: 1) Did you observe damage and, if so, what was the damage?; 2) Did you feel the earthquake and, if so, where were you when you felt it? Intensity values are then assigned [subjectively] to each questionnaire. That modified Mercalli intensity data are subjective is obvious. What is not obvious is that geostatistics (kriging) is validly applied to grid (estimate) such data. Clearly, Glass (1978) showed this empirically. Joumel (1986) discusses the application of geostatistics to "soft," or subjective, data in considerable detail.

INDICATOR KRIGING SEISMIC HAZARD MODEL Indicator kriging is a form of kriging that does not entail a change in the equation for the kriging estimator, but does entail a change in the data to which kriging is applied. With indicator kriging, a transform is applied to the data, in this case modified Mercalli intensity values. This transform is a simple one: i(x) = 0, if Z(x) < c; i(x) = 1 otherwise; this simple transform yields the indicator function, i. Notice that the indicator function is a binary one, taking on only two possible values, o and 1. Because of this, the indicator function is said to be a nonparametric function, because the notion of a probability distribution for such a function is not pertinent. The nonparametric nature of the indicator function has certain advantages in geostatistics (Joumel 1983), chiefly the minimization of the influence of extreme data values on the calculation of the semivariogram and in kriging. The value, c, used to define the indicator function is called a threshold value. In this study of seismic hazard, c is that critical ground motion value chosen to define the hazard. In this study, c is chosen to be an intensity value of VI (6) because this intensity value is that at which exterior structural damage is first noticed. When performing indicator kriging, the indicator function, i, is used rather than the raw data, Z. Other than this substitution, the kriging estimator is applied using the same equation as shown before. Weights, a, are calculated using the matrix system shown previously; covariance entries in this matrix system are obtained using the semivariogram for the function, i. When performing kriging on i, estimates are obtained that range between 0 and 1, inclusive. As the function, i, is defined for seismic hazard analysis, the estimate of i is interpreted as the probability at the estimation location that ground motion exceeds the threshold value, c, used to define the indicator function. An indicator kriging model for assessing seismic hazard is a simple one. Modified Mercalli intensity data for each earthquake in a particular time period are


241

transformed to indicator values as follows: if intensity is VI or greater, the intensity value is converted to 1, otherwise the intensity value is converted to zero. Kriging is used to form a regular grid (a digital raster) of the indicator values. For this study, 50 x 50 rasters were designed, registered to geographic coordinates as shown in various figures herein (for example, Fig. 3). Once rasters are formed for each earthquake in the time period, the digital rasters are simply added together to form a final, composite-sum map. Higher hazard is realized in this map by noticing regions that are associated with a higher sum.

APPLICATION TO SOUTHERN CALIFORNIA SEISMICITY, 1930-1971 Indicator kriging has been used to characterize southern California earthquake hazard previously (van der Meer and Carr 1992). The present study uses all 46 earthquakes that occurred between 1930 and 1971 that were associated with intensity values of VI or greater (see Carr 1983 for a list of these earthquakes). Van der Meer and Carr (1992) used only the 11 largest magnitude earthquakes of this 46. Hence, one objective of this current study is to revisit the earlier indicator kriging model and to update it using more information. Another objective, one not considered by van der Meer and Carr (1992), is to compare recent, large earthquakes with the seismic patterns analyzed in the indicator kriging model that is based on the time period, 1930-1971. An indicator kriging seismic hazard model based on the 46 earthquakes is shown (Fig. 3). It shares some similarities to that obtained previously (Fig. 1). In particular, a region of high hazard is found in each map near Oxnard/Santa Barbara. However, the indicator kriging hazard map finds particularly high hazard north to northeast of Long Beach. Both maps (Figs. 1 and 3) are also associated with relatively low hazard near Mojave, California. Van der Meer and Carr (1992) focused analytical attention on whether high hazard correlated spatially with known, active faults. That study found that higher hazard could not be directly related to anyone active fault in southern California. This study verifies this conclusion. Higher hazard does not directly correlate spatially with known active faults (Fig. 3). Because southern California is associated with so many active faults, it is perhaps not surprising that higher hazard sometimes occurs spatially where it is not expected. A hypothesis (Fig. 4) is forwarded as a possible explanation. This figure shows three hypothetical earthquakes. A circle encloses each epicenter and ground motion intensity of at least MMI VI (6) is assumed to have occurred everywhere within each circle. The dark, gray patterned area is that affected by all three earthquakes and therefore has a higher hazard because three episodes of damaging

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119

Fig. 3. Indicator kriging hazard map with major active faults superimposed. Regions associated with at least 6 episodes of intensity VI or higher ground motion in the time period, 19301971 , are highlighted in gray. The faults are coded as follows : A) White Wolf Fault; B) Garlock Fault; C) Big Pine; D) Santa Ynez; E) Oak Ridge; F) San Andreas ; G) San Gabriel; H) NewportInglewood ; I) San Jacinto .

ground motion were experienced . But, a higher hazard would not necessarily be expected within this gray-patterned region because it is not near anyone fault . Its proximity to three active faults, however, makes it vulnerable to damage during earthquakes occurring on all three faults . This hypothetical model is thought to explain the regions of higher hazard in Figure 3. With respect to Long Beach, it has experienced damaging ground motion from earthquakes occurring on the Newport-Inglewood fault (the 1933 Long Beach Earthquake), faults in the San Fernando Valley (e.g., the 2 February 1971 earthquake), the White Wolf fault (the 1952 Kern County earthquake) , and also earthquakes occurring on the San Gabriel, San Andreas, San Jacinto, Oak Ridge, and Santa Ynez faults. With respect to Oxnard, it has been affected by earthquakes on the Newport-Inglewood fault (1933 Long Beach earthquake), the Oak Ridge fault (1941 Santa Barbara and 1957 Ventura earthquakes), the White Wolf fault (1952 Kern County earthquake), and to a lesser extent by earthquakes in the San Fernando valley. As a test of the hypothesis (Fig. 4) , the active faults shown in Figure 3 are idealized as shown in Figure 5. A digital raster is developed for each of these faults as follows : 1) an attenuation function was designed from a general formula given in Cornell (1968): intensity = 5.4 + M - 31nR, where M is Richter magnitude and R is the distance from the fault; 2) a typical Richter magnitude was chosen for each of the nine (9) faults (Table 1); 3) a 34 x 34 digital raster (an arbitrary choice of size) was


243

E- EPICENTER - -' FAULT

119

Fig. 4. Three hypothetical earthquakes occurring on the faults shown. Notice that the gray-shaded region is affected by all three earthquakes.

118

Fig. 5. Idealized active fault locations. Codes for faults are the same as described in the caption to Fig. 3.

developed, geographically registered to the kriged seismic hazard rasters (note that this grid size is smaller than that used for indicator kriging; both grid sizes, however, are of arbitrary size and merely facilitate the construction of contour maps). An intensity value was estimated for each cell of the raster using the foregoing attenuation formula (not by indicator kriging in this case); 4) if the estimated intensity was VI or greater, the raster cell was assigned a value of 1; otherwise the cell was assigned the value O. Once a digital raster was developed by this procedure for each of the nine (9) active faults (Fig. 6), a composite raster was formed as a sum of all nine rasters. Frequency of intensity VI or greater ground motion was then contoured (Fig. 7). Gray shading highlights the geographic regions associated with the highest frequency of damaging ground motion. A comparison of Figure 7 to Figure 3 shows that regions of higher hazard found in the hypothetical map (Fig. 7) do not exactly match those in the indicator kriging hazard map (Fig. 3). But, the region of higher hazard near Long Beach (Fig. 3) is near one of the higher hazard regions of Figure 7; and, the higher hazard found near Oxnard (Fig. 3) is near another region of higher hazard found in Figure 7. Both Figures 4 and 8 identify lower hazard near Mojave. Similarities between these two maps are interesting and lend credibility to the foregoing hypothesis (Figure 4).

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TABLE l--Richter magnitudes used for nine active faults. Fault White Wolf Garlock Big Pine Santa Ynez Oak Ridge San Andreas San Gabriel Newport-Inglewood San Jacinto

Magnitude 7.0 7.0 6.5 6.5 6.5 8.3 6.5 6.5 6.5

RECENT SOUTHERN CALIFORNIA EARTHQUAKES Epicenters of three, recent southern California earthquakes are plotted (Fig. 8): 1) the 1987 Whittier Narrows earthquake, magnitude 5.5 to 6.0; 2) the 1990 Upland earthquake, magnitude 5.0 to 5.4; and the 1994 Northridge earthquake, magnitude 6.6 (estimated). It is interesting that these three earthquakes occur closely to the San Gabriel Fault. With respect to the indicator kriging result, none of these earthquakes occurs within a region identified as having a high seismic hazard. Of course, this is the point made with the foregoing hypothesis (Figures 5 and 8) that higher hazard cannot be spatially correlated with anyone active fault in southern California. The 1987 Whittier Narrows and the 1994 Northridge earthquakes, for example, caused damaging levels of ground motion within the region of higher hazard found north of Long Beach; these earthquakes increased the hazard within this region. Furthermore, the 1994 Northridge earthquake caused damaging levels of ground motion in the Oxnard area, another region identified as having higher hazard. Only the 1990 Upland earthquake occurred in a lower hazard area and did not have a large enough magnitude to influence any of the higher hazard regions. Figure 8 also shows these three epicenters plotted on the hypothetical hazard map (Figure 7). The epicenters for the 1990 Upland and 1994 Northridge earthquakes occur just outside regions of highest hazard, whereas the epicenter for the 1987 Whittier Narrows earthquake occurs within the region of high hazard north of Long Beach.

CONCLUSION An indicator kriging seismic hazard model is much more easily developed in comparison to one based on Gumbel's statistics of extreme values (Gumbel 1958). With the indicator kriging model, modified Mercalli intensity data are first transformed to indicator values, transformed to 1 if the intensity is VI (6) or greater, transformed to 0 otherwise. Kriging is used to estimate the Oil indicator data at nodes of a regular grid, hence forming a raster. Once rasters are formed for all


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Multiaxial Fatigue and Deformation: Testing and Prediction (ASTM Special Technical Publication, 1387)

Fatigue and Fracture of Medical Metallic Materials and Devices (ASTM special technical publication, 1481)

Concrete Pipe for the New Millennium (ASTM Special Technical Publication, 1368)

Evaluation and Remediation of Low Permeability and Dual Porosity Environments (ASTM Special Technical Publication, 1415)

Performance and Durability of the Window-Wall Interface (ASTM special technical publication, 1484)

Contaminated Sediments: Evaluation and Remediation Techniques (ASTM special technical publication, 1482)

Acoustic Emission: Standards and Technology Update (ASTM Special Technical Publication, 1353)

Joining And Repair Of Composite Structures (ASTM Special Technical Publication, 1455)

Constructing and Controlling Compaction of Earth Fills (ASTM Special Technical Publication, 1384)

Fretting Fatigue: Current Technology and Practices (ASTM Special Technical Publication, 1367)

Durability 2000: Accelerated and Outdoor Weathering Testing (ASTM Special Technical Publication, 1385)

Isocyanates: Sampling, Analysis, and Health Effects (ASTM Special Technical Publication, 1408)

Technology of Floor Maintenance and Current Trends (ASTM special technical publication, 1448)

Fatigue Testing and Analysis Under Variable Amplitude Loading Conditions (ASTM special technical publication, 1439)

Testing and Performance of Geosynthetics in Subsurface Drainage (ASTM Special Technical Publication, 1390)

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