Scaling Methods in Soil Physics

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scaling methods IN

soil physics

© 2003 by CRC Press LLC

scaling methods IN

soil physics Edited by

Yakov Pachepsky David E. Radcliffe H. Magdi Selim

CRC PR E S S Boca Raton London New York Washington, D.C.


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Library of Congress Cataloging-in-Publication Data Scaling methods in soil physics / edited by Yakov Pachepsky, David Radcliffe, H. Magdi Selim. p. cm. Includes bibliographical references and index. ISBN 0-8493-1374-0 (alk. paper) 1. Soil physics. 2. Scaling laws (Statistical physics) I. Pachepsky, Y. II. Radcliffe, David Elliot, 1948- III. Selim, Hussein Magd Eldin, 1944S592.3 .S32 2003 631.4′3--dc21

2002191161

This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. All rights reserved. Authorization to photocopy items for internal or personal use, or the personal or internal use of specific clients, may be granted by CRC Press LLC, provided that $1.50 per page photocopied is paid directly to Copyright clearance Center, 222 Rosewood Drive, Danvers, MA 01923 USA. The fee code for users of the Transactional Reporting Service is ISBN 0-8493-1374-0/03/$0.00+$1.50. The fee is subject to change without notice. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe.

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Preface Soil physical properties are needed to understand and manage natural systems spanning an extremely wide range of scales: from microbial habitats to plant root zone environment to field crop productivity to watershed processes to regional weather modeling and global circulation models. Capabilities of soil measurements at those scales are vastly different. This creates a fundamental problem for soil physicists and for multiple users of soil physics data. Many soil data are obtained from small soil samples and cores, monoliths, or small field plots, yet the goal is to reconstruct soil physical properties across fields, watersheds, and landforms, or to predict physical properties of pore surfaces and structure of pore space. The representation of processes and properties at a scale different from the one at which observations and property measurements are made is a pervasive problem in soil physics, as well as in soil science in general. This scale-transfer problem must be solved, in particular, in order: To integrate chemical, biological and physical processes affecting soil quality and environmental health To describe effectively the coupled fluxes of heat, moisture, gases and solutes across land surfaces To establish appropriate soil parameters for describing the long-term fate of pollutants To interpret various remote sensing data To delineate management zones in agricultural fields To estimate water yield and geochemical fluxes in ungauged watersheds To understand sources and importance of diversity and patchiness in terrestrial ecosystems To provide parameters for estimating biogeochemical trends related to climate change The multiscale characterization of processes and parameters of soil physics needs to be addressed as a research issue of scale dependencies in soil physical properties and as a practical/operational issue of data assimilation or data fusion in environmental monitoring and prediction. Scale is a complex concept having multiple connotations reflected in the majority of chapters in this book. A notion of support is important to characterize and relate different scales in soil physics. Support is the length, area, or volume for which a single value of soil property is defined and no variations in this and other properties are taken into account. Size of an individual soil sample and size of the discrete spatial element in a soil model are typical examples of supports. The term “resolution” is often used for supports defined in terms of length, and the term “representative elementary volume” is applied for supports defined as volumes. The terms “pixel size” and “grid size” are also used to define support. An area or a volume that is sampled with given support determines the extent of measurements. Yet another notion, spacing, i.e., distance between sampling locations, is of importance in characterization of the scale of research or an application. Any research of soil physical properties is made with specific support, extent and spacing. If those properties are to be used with different support, extent or spacing, scaling becomes necessary. Scaling is used as a noun to denote a relationship between soil physics data at different scales or as a verb to denote an action of relating such data on different scales. Upscaling (downscaling) usually refers to soil physical properties at a support that is larger (smaller) than the one at which data are available. Two general approaches to scaling are represented in this book. One approach assumes that a physical model can be invoked or developed to perform scaling. The most prominent examples of


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this approach are fractal models and soil-landscape models. Another approach relies on establishing empirical scaling relationships from a large database. Both approaches have obvious advantages and limitations. The accuracy of a scaling can be broadly defined as a correspondence between measured and estimated data for the data set from which a scaling has been developed. The reliability of scaling can be assessed in terms of the correspondence between measured and estimated data for the data sets other than the one used to develop a scaling. Models in physics-based scaling cannot capture all factors of inherent variability in soils, and therefore scaling is not as accurate in simulating data as an empirical model might be. These models, however, have a potential to be more reliable, whereas the reliability of empirical scaling is essentially unknown. In many cases, empirical scaling in soil physical properties has eventually led to the development of physical models to explain this scaling. This book is organized across the hierarchy of spatial scales in soils. The first three chapters deal with scaling in properties of soil pore space spanning pore radii range from 10–6 to 10–2 m. Fractal models of soil physical properties have become popular sources of scaling relationships for those support sizes. Fractal geometry was developed to describe the hierarchy of ever-finer detail in the real world. Natural objects often have similar features at different scales. Measures of these features, e.g., total number, total length, total mass, average roughness, total surface area etc. are dependent on the scale on which the features are observed. Fractal geometry assumes that this dependence is the same over a range of scales, i.e., it is scale invariant within this range. This dependence is used for scaling. To apply fractal geometry, one must have in mind a physical or mathematical model that explains the process involved in formation of fractal features in the objects under study. In Chapter 1, Perrier and Bird present a pore solid fractal (PSF) model that can be used as a reference model to describe the number-size distributions of soil particles, pores, aggregates and the scaling of measures such as solid–pore interface areas, solid and pore volumes, density and porosity, in soils or in any porous medium exhibiting hierarchical heterogeneities over a broad range of scales. This model provides an explicit geometrical description of scaling in soil structure and leads to deterministic links between the scaling laws of different structural properties and soil hydraulic properties. In Chapter 2, Tarquis, Giménez, Saa, Díaz and Gascó, present an overview of scaling of soil porosity data using multifractal models and configuration entropy. The importance of such scaling methods increases as more two- and three-dimensional data on soil pore space become available; reconstruction of pore connectivity will become feasible, thus opening an avenue to explain and predict preferential flow patterns. In Chapter 3, Williams and Ahuja show that the assumption of similarity is not crucial for development of an empirical scaling law for soil pore space properties. They propose a oneparameter model of the soil water retention curve that is based on a strong, linear relationship observed between the intercept and slope of a log–log plot of matric potential and soil water content below the air-entry value. Furthermore, for widely different soils this relationship is found to coalesce into one common relationship. The following three chapters explore scaling in solute diffusion and dispersion in soils using the travel distance as a measure of scale. Chapter 4, by Ewing and Horton, explores scaling laws that emerge from diffusion in porous media with sparsely connected pore spaces, of which soils are an example. Monte Carlo simulations using pore network models, in conjunction with percolation theory, show that, at the percolation threshold, accessible porosity, tortuosity and diffusivity are described by equations that scale with time, distance or proximity to the percolation threshold. Slightly above the percolation threshold, a different kind of scaling appears. From porosity and diffusivity a residual tortuosity can be calculated, which also shows both kinds of scaling. In Chapter 5, Zhou and Selim examine the notion of scale in soil solute dispersion studies. They present four distinct types of dispersivity–time or dispersivity–distance relationships that are appropriate to describe the relationship between dispersivity and time or distance. These types of scaling were analyzed using simulations and analysis of literature data. In Chapter 6, Perfect demonstrates the © 2003 by CRC Press LLC

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applicability of the power law scaling of dispersivity in geological materials to soils. This scaling law allows him to scale up the dispersivity predicted from water retention properties. As the scales become coarser and the soil profile is included in the extent of study, two different approaches to the scaling problem can be found in the literature and in this book. One is to assume that the same parameters of soil physical properties can be used at the laboratory sample scale and at the pedon/plot scale. Then an effective averaging procedure can be found to upscale soil properties to the pedon scale. Such an approach is taken in Chapter 7, written by Zhu and Mohanty. They compare commonly used averaging schemes for the hydraulic parameters and compare their capability to generate effective parameters for the ensemble behavior of heterogeneous soils. It appears that the efficiency of the upscaling procedure depends on the degree of correlation between different hydraulic parameters and boundary conditions. Another approach to the transition to the plot/pedon scale is to change the parameter used to characterize the same soil property. A routine example is using a soil water retention curve for the sample scale and field water capacity for the pedon/plot scale to characterize a soil’s ability to retain water. Soils are inherently variable. A model of spatial variability of soil properties has to be known if upscaling is performed by aggregating the additive soil properties. Chapter 8, written by Western, Grayson, Blöschl, and Wilson, provides an introduction to the topic. The authors present a variety of statistical approaches for representing variability and for the spatial scaling of soil moisture, for spatially distributed deterministic modeling of soil moisture patterns at the small catchment scale, and for using remote sensing and topography to interpret variability in soil moisture at larger scales. The next three chapters outlay specific techniques to model and characterize the variability for scaling purposes. Ellsworth, Reed, and Hudson in Chapter 9 examine the performance of six interpolation methods applied to soil and groundwater solute concentrations. Spacing appears to be an important scale parameter. A nonlinear geostatistical method, referred to as quantile kriging, was found to be optimal for the sparse, clustered sample designs, whereas ordinary kriging and a deterministic calibrated variant of inverse distance interpolation performed the best with dense, regularly spaced sample data. Chapter 10, written by Si, shows opportunities in analyzing spacescale dependencies in soil properties with wavelet analysis that can handle the spatial nonstationarity common in field soils. The localized features and nonstationarity may have significant impacts on modeling soil water flow and chemical transport. The wavelet analysis of the soil hydraulic conductivity and the inverse microscopic capillary length transects exhibits the multiscale variations and localized features seen at different scales. In Chapter 11, Kumar shows that a model of spatial variability can be established that spans several scales. Such a model can be used to relate measurements of soil properties made at multiple scales with different measurement techniques. Typically, several regions of fine-scale measurements of limited coverage are embedded within coarse-scale measurements of larger coverage. Consequently, in regions at the fine scale that are devoid of measurements, inferences about the statistical variability can be made only through conditional simulation. This chapter describes a conditional simulation technique that utilizes measurements at multiple scales and its application to remote sensing data of soil moisture. Soil properties are known to be related to landscape position; scaling at field, landscape or regional scales can take advantage of soil-landscape relationships. Chang and Islam in Chapter 12 present a stochastic framework for characterizing the steady-state soil moisture distribution in a heterogeneous-soil and -topography field under the influence of precipitation and evaporation. Upscaling is accomplished by applying a perturbation method and spectral techniques to a stochastic partial differential equation that depends on three main factors: the heterogeneity of soil properties, the variability of topography and the change of mean soil moisture. Results suggest that topography (soil properties) controls soil moisture distributions when the area is dominated by coarse-texture (fine-texture) soil or by soils with small (large) correlation lengths of topography. Timlin, Pachepsky, and Walthall in Chapter 13 use spatial autoregression and terrain variables to estimate water holding capacity across a field over a range of spacings. Slope and tangential © 2003 by CRC Press LLC

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curvature were found to be significant predictors of surface water holding capacity and spacing had an optimum value. A good correspondence was found between predicted water holding capacity and measured corn grain yields across the field. High-density data other than topography can be used to define soil properties at field or landscape scales. Chapter 14, written by Morgan, Norman, Molling, McSweeney, and Lowery, presents an overview of measurement techniques available for that purpose. The authors have built a hierarchical example of using data sets of different availability in a model to predict crop yield. They found that at the field scale the USDA soil survey information alone will not be adequate for data needs. Augmenting soil survey information with methods such as inverse modeling to infer soil properties from spatially dense data and landscape survey sensors improves the horizontal resolution required for input in biophysical crop productivity models. In Chapter 15, Tsegaye, Crosson, Laymon, Schamschula, and Johnson show that temporal highdensity data on rainfall can be used along with basic soil and vegetation properties to downscale remote sensing measurements of soil moisture made at coarse scales. An artificial neural network trained with three sources of input, i.e., high-density rainfall data, coarse scale spatial data on soil moisture, and fine-scale soil and vegetation data, generates values of soil moisture contents at a fine scale. Performance of the neural network becomes worse as the difference increases between coarse-scale and fine-scale supports. It still can be sufficient for applications in which temporal aggregation can be made to match the coarse spatial scale of remote sensing data. Upscaling and downscaling need to be applied in projects at the field scale where both regional and sample-scale observations appear to be useful to provide input for specific predictions. This book contains several case studies of this type. Chapter 16, written by Cassel and Edwards, explores accumulating and using information about plant response to soil mechanical impedance at sample, plot, field and regional scales. These authors emphasize that research and management questions as well as the relevant soil physical properties are different at different scales. They demonstrate how a management problem at the field scale can be addressed by using regional data to find a probable solution, using field scale data to define soil parameters controlling the usefulness of the proposed solution, and upscaling plot and small-sample scale data to tailor the management practice to a particular combination of soil physical properties. In Chapter 17, Mulla, Gowda, Birr, and Dalzell describe applications of process-based models to simulate nitrate losses from agricultural fields across a wide range of spatial scales. The authors observe that, as spatial scale becomes coarser, upscaling and aggregation lead to progressively larger uncertainty of model input data. Using simple mass balance equations appears to be more appropriate at the coarsest spatial scale than mechanistic modeling. The performance of spatial upscaling techniques does not seem to depend as much on the magnitude of upscaling as on the relative similarity between the smaller units being upscaled and the larger unit. Chapter 18, written by Seyfried, examines techniques to combine remote sensing data on vegetation with hydrologic modeling. The techniques involve upscaling point-scale soil water models, the incorporation of scale and spatial variability effects on model parameters and the measurements used as input and for model testing, delineating vegetation types, and inferring leaf area index from the vegetation index. Soil mapping units, used in the model to delineate the critical deterministic variability of soil water content, aggregate LANDSAT remote sensing pixels sufficiently that vegetation cover type and vegetation index are effectively described within mapping units while delineating differences among them. Finally, Chapter 19 by Lin and Rathbun shows that the scaling concept can be used to integrate knowledge and data on soil hydrologic properties and regimes in a self-consistent system of concepts and techniques. The quest of soil physicists to bridge scales is by no means unique. Many scientific disciplines strive to relate observation and models from different scales. One of the closest to soil physics disciplines is represented by Chapter 20, written by Faybishenko, Bodvarsson, Hinds and Witherspoon. The chapter presents a panoramic view of scaling problems in large and complex subsurface volumes of unsaturated fractured rock. Using several examples from experimental investigations © 2003 by CRC Press LLC

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in fractured basalt and tuff, the authors show how the concept of the hierarchy of scales becomes instrumental in measuring and modeling flow transport processes. For a given scale, boundary conditions can be defined from studies at a coarser scale whereas determining model parameters requires information from a finer scale. This chapter illustrates the wide opportunities for interdisciplinary cross-pollination in approaching the scale conundrums. This book does not contain all available ideas, conceptual approaches, techniques or methodologies for scaling of soil physical properties. The list of suggested reading at the end of this preface, as well as references in individual chapters, will help the interested reader. Scaling of soil physical properties is a burgeoning field, responding to the increasing need in environmental modeling and prediction and to the progress in remote sensing technologies to estimate environmental parameters at large scales, in spatially intensive methods to measure indirect indicators of soil physical properties, in in situ measurement techniques to obtain small-scale soil data, and in integration of georeferenced data collected at various scales. The contributions in this volume by some of the pioneers in the field represent a broad spectrum of techniques developed and tested to facilitate the use of soil physics data in a wide variety of soil–land–earth-related applications. Y.A. Pachepsky D.E. Radcliffe H.M. Selim

SUGGESTED READING Bierkens, F.P., P.A. Finke and Peter de Wiligen. 2000. Upscaling and Downscaling Methods for Environmental Research. Developments in Plant and Soil Sciences, vol. 88. Kluwer Academic Publishers, Dordrecht/Boston, London. Hillel, D. and D.E. Elrick (Eds.). 1990. Scaling in Soil Physics: Principles and Applications. SSSA Special Publication 25. Soil Science Society of America. Madison, WI. Pachepsky, Ya., J.W. Crawford and W.J. Rawls. 2002. Scaling effects, in: Rattan, L., Ed. Encyclopedia of Soil Science. Marcel Dekker, New York, 1175–1179. Sposito, G. (Ed.). 1998. Scale Dependence and Scale Invariance in Hydrology. Cambridge University Press, Cambridge. Stewart, J.B., E.T. Engman, R.A. Feddes and Y. Kerr (Eds.). 1996. Scaling Up in Hydrology Using Remote Sensing. John Wiley & Sons, New York, 1996.


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The Editors Yakov Pachepsky is a soil scientist with USDA Agricultural Research Service in Beltsville, Maryland. He received his M.S. in mechanics, Ph.D. in physics and mathematics and Ph.D. in soil science from Moscow State University, Russia. Dr. Pachepsky has published more than 180 papers and book chapters, written four books and edited two books. His research interests focus on relationships among structure, composition, hydrologic processes, and contaminant transport in soils at a variety of scales. Dr. Pachepsky serves as an associate editor of Soil Science Society of America Journal and Vadose Zone Journal. He is a member of editorial or advisory boards of Geoderma, Catena, Ecological Modeling, Land Degradation and Development and International Agrophysics journals. David Radcliffe has been with the University of Georgia since 1983, where he is a professor in the Crop and Soil Sciences Department. He teaches an undergraduate and an advanced graduate course in soil physics, and team teaches a course in site assessment. He received a B.S. in Naval Science from the U.S. Naval Academy and an M.S. and Ph.D in soil physics from the University of Kentucky. His research is focused on phosphorus, bacterial and sediment losses to surface water from agricultural sources. Dr. Radcliffe has published more than 60 journal articles and book chapters, and has been an associate editor for the Soil Science Society of America Journal for 5 years and a technical editor for 3 years. He is a Fellow of the Soil Science Society of America, and is chair-elect of the Soil Physics Division of the Soil Science Society of America. H. Magdi Selim is professor of soil physics at Louisiana State University, Baton Rouge, Louisiana. Dr. Selim received his M.S. and Ph.D. degrees in soil physics from Iowa State University, Ames, Iowa, in 1969 and 1971, respectively, and his B.S. in soil science from Alexandria University in 1964. Dr. Selim has published more than 100 papers and book chapters, is a co-author of one book and co-editor of three books. His research interests focus on modeling the mobility of dissolved chemicals and their reactivity in soils and groundwater, and also include saturated and unsaturated water flow in multilayered soils. Dr. Selim is the recipient of several awards including the Phi Kappa Phi Award, the First Mississippi Research Award for Outstanding Research, Gamma Sigma Delta Outstanding Research Award, the Doyle Chambers Achievement Award and the Sedberry Teaching Award. Professor Selim has organized and co-organized several international conferences, workshops and symposia. He has served as associate editor of Water Resources Research and the Soil Science Society of America Journal. Dr. Selim served as a member of the executive board of the International Society of Trace Element Biogeochemistry and as chair of the Soil Physics Division of the Soil Science Society of America. He is a fellow of the American Society of Agronomy and the Soil Science Society of America.


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Contributors Lajpat R. Ahuja USDA-ARS Ft. Collins, Colorado N.R.A. Bird Silsoe Research Institute, Soil Science Group Silsoe, Bedford, England A.S. Birr Department of Soil, Water, and Climate University of Minnesota St. Paul, Minnesota Günter Blöschl Institut für Hydraulik, Gewässerkunde und Wasserwirtschaft Technische Universität Wien, Austria Gudmundur S. Bodvarsson Lawrence Berkeley National Laboratory Berkeley, California D. Keith Cassel Department of Soil Science North Carolina State University Raleigh, North Carolina Dyi-Huey Chang Cincinnati Earth System Science Program Department of Civil and Environmental Engineering University of Cincinnati Cincinnati, Ohio William L. Crosson National Space Science and Technology Center Huntsville, Alabama B.J. Dalzell Department of Soil, Water, and Climate University of Minnesota St. Paul, Minnesota


M.C. Díaz Dpto. de Edafología, E.T.S. de Ingenieros Agrónomos Ciudad Universitaria Madrid, Spain Ellis C. Edwards Department of Soil Science North Carolina State University Raleigh, North Carolina T.R. Ellsworth Department of Natural Resources and Environmental Sciences University of Illinois at Urbana-Champaign Urbana, Illinois Robert P. Ewing Department of Agronomy Iowa State University Ames, Iowa Boris Faybishenko Lawrence Berkeley National Laboratory Berkeley, California J.M. Gascó Dpto. de Edafología, E.T.S. de Ingenieros Agrónomos Ciudad Universitaria Madrid, Spain Daniel Giménez Department of Environmental Sciences Rutgers, The State University of New Jersey New Brunswick, New Jersey P.H. Gowda Department of Soil, Water, and Climate University of Minnesota St. Paul, Minnesota

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Rodger B. Grayson Cooperative Research Centre for Catchment Hydrology and Centre for Environmental Applied Hydrology Department of Civil and Environmental Engineering The University of Melbourne Melbourne, Australia Jennifer Hinds Geological Engineering University of Idaho Moscow, Idaho Robert Horton Department of Agronomy Iowa State University Ames, Iowa Robert J.M. Hudson Department of Natural Resources and Environmental Sciences University of Illinois at Urbana-Champaign Urbana, Illinois Shafiqul Islam Cincinnati Earth System Science Program Department of Civil and Environmental Engineering University of Cincinnati Cincinnati, Ohio Alton B. Johnson Mississippi Delta Center Alcorn State University Alcorn, Mississippi Praveen Kumar Environmental Hydrology and Hydraulic Engineering Department of Civil and Environmental Engineering University of Illinois Urbana, Illinois Charles A. Laymon National Space Science and Technology Center Huntsville, Alabama


Hangsheng Lin Department of Crop and Soil Sciences The Pennsylvania State University University Park, Pennsylvania Birl Lowery Department of Soil Science University of Wisconsin Madison, Wisconsin Kevin McSweeney Department of Soil Science University of Wisconsin Madison, Wisconsin Binayak P. Mohanty Department of Biological and Agricultural Engineering Texas A&M University College Station, Texas Cristine C. Molling Space Science and Engineering Center University of Wisconsin Madison, Wisconsin Christine L.S. Morgan Department of Soil Science University of Wisconsin Madison, Wisconsin David J. Mulla Department of Soil, Water, and Climate University of Minnesota St. Paul, Minnesota John M. Norman Department of Soil Science University of Wisconsin Madison, Wisconsin Yakov A. Pachepsky USDA/ARS USDA-ARS Animal Waste Pathogen Laboratory Beltsville, Maryland

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Edmund Perfect Department of Geological Sciences University of Tennessee Knoxville, Tennessee Edith M.A. Perrier UR Geodes, IRD Bondy Cedex, France Stephen Rathbun Department of Statistics The Pennsylvania State University University Park, Pennsylvania Patrick M. Reed Department of Civil and Environmental Engineering University of Illinois at Urbana-Champaign Urbana, Illinois Antonio Saa Dpto. de Edafología, E.T.S. de Ingenieros Agrónomos Ciudad Universitaria Madrid, Spain Marius P. Schamschula Center for Applied Optical Sciences, Center for Hydrology, Soil Climatology and Remote Sensing Alabama A&M University Normal, Alabama H. Magdi Selim Department of Agronomy Louisiana State University Baton Rouge, Louisiana Mark S. Seyfried Northwest Watershed Research Center USDA-Agricultural Research Service Boise, Idaho Bing Cheng Si Department of Soil Science University of Saskatchewan Saskatoon, Saskatchewan, Canada


Ana M. Tarquis Dpto. de Matemática Aplicada a la Ingeniería Agronómica, E.T.S. de Ingenieros Agrónomos Ciudad Universitaria Madrid, Spain Dennis J. Timlin USDA/ARS Alternate Crops and Systems Laboratory Beltsville, Maryland Teferi D. Tsegaye Center for Hydrology, Soil Climatology and Remote Sensing Alabama A&M University Normal, Alabama Charles L. Walthall USDA/ARS Hydrology and Remote Sensing Laboratory Beltsville, Maryland Andrew W. Western Cooperative Research Centre for Catchment Hydrology and Centre for Environmental Applied Hydrology Department of Civil and Environmental Engineering The University of Melbourne Melbourne, Australia Robert D. Williams USDA-ARS, Langston University Langston, Oklahoma David J. Wilson Cooperative Research Centre for Catchment Hydrology and Centre for Environmental Applied Hydrology Department of Civil and Environmental Engineering The University of Melbourne Melbourne, Australia Paul A. Witherspoon Lawrence Berkeley National Laboratory Berkeley, California

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Liuzong Zhou Department of Agronomy Louisiana State University Baton Rouge, Louisiana


Jianting Zhu Department of Biological and Agricultural Engineering Texas A&M University College Station, Texas

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Contents Chapter 1

The PSF Model of Soil Structure: A Multiscale Approach .......................................1

E.M.A. Perrier and N.R.A. Bird Chapter 2

Scaling and Multiscaling of Soil Pore Systems Determined by Image Analysis.....................................................................................................19

A.M. Tarquis, D. Giménez, A. Saa, M.C. Díaz, and J.M. Gascó Chapter 3

Scaling and Estimating the Soil Water Characteristic Using a One-Parameter Model ............................................................................................35

R.D. Williams and L.R. Ahuja Chapter 4

Diffusion Scaling in Low Connectivity Porous Media ............................................49

R. P. Ewing and R. Horton Chapter 5

Solute Transport in Porous Media: Scale Effects.....................................................63

L. Zhou and H.M. Selim Chapter 6

A Pedotransfer Function for Predicting Solute Dispersivity: Model Testing and Upscaling ...................................................................................89

E. Perfect Chapter 7

Upscaling of Hydraulic Properties of Heterogeneous Soils.....................................97

J. Zhu and B.P. Mohanty Chapter 8

Spatial Variability of Soil Moisture and Its Implications for Scaling ...................119

A.W. Western, R.B. Grayson, G. Blöschl, and D.J. Wilson Chapter 9

An Evaluation of Interpolation Methods for Local Estimation of Solute Concentration ..........................................................................................143

T.R. Ellsworth, P.M. Reed, and R.J.M. Hudson Chapter 10 Scale- and Location-Dependent Soil Hydraulic Properties in a Hummocky Landscape: A Wavelet Approach.................................................163 B.C. Si


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Chapter 11 Multiple Scale Conditional Simulation ..................................................................179 P. Kumar Chapter 12 Effects of Topography, Soil Properties and Mean Soil Moisture on the Spatial Distribution of Soil Moisture: A Stochastic Analysis.....................193 D.-H. Chang and S. Islam Chapter 13 A Mix of Scales: Topography, Point Samples and Yield Maps.............................227 D.J. Timlin, Y.A. Pachepsky, and C.L. Walthall Chapter 14 Evaluating Soil Data from Several Sources Using a Landscape Model................243 C.L.S. Morgan, J.M. Norman, C.C. Molling, K. McSweeney, and B. Lowery Chapter 15 Application of a Neural Network-Based Spatial Disaggregation Scheme for Addressing Scaling of Soil Moisture...................................................261 T.D. Tsegaye, W.L. Crosson, C.A. Laymon, M.P. Schamschula, and A.B. Johnson Chapter 16 Scaling Soil Mechanical Properties to Predict Plant Responses............................279 D.K. Cassel and E.C. Edwards Chapter 17 Estimating Nitrate-N Losses at Different Spatial Scales in Agricultural Watersheds ......................................................................................295 D.J. Mulla, P.H. Gowda, A.S. Birr, and B.J. Dalzell Chapter 18 Incorporation of Remote Sensing Data in an Upscaled Soil Water Model ....................................................................................................309 M.S. Seyfried Chapter 19 Hierarchical Frameworks for Multiscale Bridging in Hydropedology...................347 H. Lin and S. Rathbun Chapter 20 Scaling and Hierarchy of Models for Flow Processes in Unsaturated Fractured Rock ...............................................................................373 B. Faybishenko, G.S. Bodvarsson, J. Hinds, and P.A. Witherspoon


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1

The PSF Model of Soil Structure: A Multiscale Approach E.M.A. Perrier and N.R.A. Bird

CONTENTS I. Introduction...............................................................................................................................1 II. From Fractal Scaling to a Multiscale Model of Soil Structure ...............................................2 A. Fractals: A Theory of Measure and Powerlaw Scaling Laws ......................................2 B. Different Meanings for “the” Soil Fractal Dimension and Search for Links ..............4 C. The PSF Approach: A Geometrical, Multiscale Model of Soil Structure....................5 III. The PSF Model ........................................................................................................................5 A. Definition .......................................................................................................................5 B. Generating PSF Structure Models.................................................................................7 1. Different Geometrical Patterns................................................................................7 2. With Only Pores or Only Solids .............................................................................7 3. With Lower Bound or No Lower Bound ................................................................9 C. Fragmentation of a PSF Structure.................................................................................9 IV. Inferring Deterministic Links between Several Scale-Dependent Soil Physical Properties..........................................................................................................10 A. Links between Different Scaling Structural Properties ..............................................10 1. Pore and Particle Size Distributions......................................................................10 2. Aggregate or Fragment Size Distributions............................................................11 3. Mass, Density/Porosity Scaling.............................................................................11 4. Solid-Pore Interface Area Scaling .........................................................................12 5. Overview Discussion .............................................................................................13 B. Links between Structural and Hydraulic Scaling Properties......................................13 V. Conclusion and Perspectives: Toward Extended PSF Virtual Structures and Pore Network Modeling ..................................................................................................15 VI. Acknowledgments ..................................................................................................................16 References ........................................................................................................................................16

I. INTRODUCTION Many papers and books have been written about fractals in general and fractals in soil science in particular1,2 in the past decade. Central to this theme are the notions of a multiscale structure and a scaling symmetry imposed on this structure. The PSF (pore solid fractal) model is a development of this theme, representing a generalization of the fractal models currently used to model soil structure. While accommodating these models as special, albeit degenerate, cases, it overcomes

1 © 2003 by CRC Press LLC


2

Scaling Methods in Soil Physics

some of their shortcomings, providing a geometrical, unified framework which exhibits broad poresize and particle-size distributions, and permitting structure to be modeled over a much wider range of scales and indeed to arbitrarily small scales. In the second section of this chapter we shall review some of the issues that arise in the fractal modeling of soil structure. In the third section we define the PSF model as a simplified but concrete representation of multiscale organizations of pores and solids occurring in soils, which reduces to a fractal model when strict self-similarity occurs at every scale. In the fourth section, we give a comprehensive list of the properties of the PSF model arising from previous studies3–6 and infer possible deterministic dependencies between different real soil scaling properties. In the concluding section of the chapter, we give an overview of possible extensions of the PSF approach to more complex types of structures, which cannot be analyzed mathematically but by simulation and pore network modeling. II. FROM FRACTAL SCALING TO A MULTISCALE MODEL OF SOIL STRUCTURE A. FRACTALS:

A

THEORY

OF

MEASURE

AND

POWERLAW SCALING LAWS

Fractal geometry has brought new concepts to the search for a better quantification of scaledependent soil characteristics. Scaling effects have been observed for a long time in soil physics, for example, soil bulk density varying with the sample size (Figure 1.1a), specific surface areas varying as a function of observation scale, or an increasing number of small voids revealed with increased resolution. There may be different ways to cope with the technical difficulties that such effects produce on measurements.7 Fractal theory suggests that these scaling phenomena may be more the rule than the exception and can be explained by an underlying multiscale structure. Similar but theoretical measurements made on a large set of very simple mathematical objects — generated by iterative copies of simple patterns at successive scales — give the same type of results as those obtained on many natural objects. A measure appears to be no longer a single number, nor a mean value within a confidence interval, but a function of scale. In the simplest fractal case, associated with self-similarity at every scale, this function is a powerlaw, and the powerlaw exponent — i.e., the slope of the associated straight line in a log–log plot — depends only on the so-called fractal dimension D of the object (Figure 1.1a). Many formulae of the same type have been derived for different types of measures M (lengths, surfaces, volumes, densities, etc.). A very simple example is the measure of the mass M of a sample of size L which, for a solid mass fractal structure, varies as a power law of L: M L = M 0 LD

(1.1)

This does not mean that we forget the actual and classical value of the measure at a given scale (e.g., M0 for L = 1) to compare different objects measured at the same resolution, but the fractal dimension D appears to be a key parameter, a second fundamental descriptor of the measure. Conversely, because many measures obtained on natural objects appear also as straight lines in a log(measure) vs. log(scale), such as solid–void interface areas or masses of soil samples, this suggests that many soils can be considered as fractal objects, even if we are ignorant of the genesis of the natural object, and even if the scaling behavior can be observed only over a narrow range of scales. In addition, the same type of fractal conceptual model may be extended from a theory of measures to the simple characterization of number-size distributions for a collection of objects. Because the cumulative number-size distribution of the holes in a lacunar fractal model such as a Sierpinski carpet or a Menger sponge varies as a powerlaw, by extension8 termed as fractal a collection of objects with a cumulative number size distribution N[≥ r] varying as a powerlaw of the size, r © 2003 by CRC Press LLC


The PSF Model of Soil Structure: A Multiscale Approach

3

1.9 0.25 0.2

1.8

0.15 0.1

density

1.75

0.05

Log(density)

0.3

1.85

Data Linear fit

0

1.7

-1

-0.5

0

0.5

1

Log(size)

1.65 1.6

Data Powerlaw fit

1.55 1.5 0

5

sample size

y=log(M[r T

(5.11)

∂c ∂x

x=L

=0

(5.12)

where L is the length of the soil column; T is the input pulse duration; c0 is the solute concentration in the input pulse. The governing Equation 5.8 subject to initial and boundary conditions (Equations 5.9 through 5.12) can be solved using finite difference methods. The detailed finite difference scheme is shown in the Section XI (Appendix A). The resulting tri-diagonal linear equation system was solved using the Thomas algorithm.28

B. CDE

WITH A

LINEARLY DISTANCE-DEPENDENT DISPERSIVITY

By the removal of the bar from the mean travel distance ( x ) in the dispersivity mean travel distance relationship, we change the mean travel distance to a distance from source (x) and obtain a distancedependent dispersivity as given by α( x ) = a2 x

(5.13)

where now a2 is a constant as a1 is in Equation 5.5. If we also ignore molecular diffusion, the dispersion coefficient becomes a function of distance from source and is thus given by D( x ) = α ( x ) v = a2 xv

(5.14)

Therefore, the transport for a tracer solute or nonreactive chemical in a one-dimensional heterogeneous soil system with distance-dependent dispersion coefficient, under steady state water flow, is governed by the following equation: ∂c ∂c  ∂  ∂c =  D( x )  − v ∂t ∂x  ∂x  ∂x © 2003 by CRC Press LLC

(5.15)


Solute Transport in Porous Media: Scale Effects

71

Substituting Equation 5.14 into the above governing equation and expanding gives ∂c ∂2 c ∂c = a2 vx 2 − (1 − a2 ) v ∂t ∂x ∂x

(5.16)

The corresponding initial and boundary conditions for a finite soil column can be expressed as c( x, t ) = 0, c ( x , t ) = c0 ,

t=0

(5.17)

x = 0, 0 < t ≤ T

(5.18)

x = 0, t > T

(5.19)

=0

(5.20)

c( x, t ) = 0, ∂c ∂x

x=L

where L, T, and c0 are the same as in Equations 5.10 through 5.12. The third-type boundary condition is applied to the upper boundary. However, because the dispersion coefficient vanishes at x = 0, the third-type boundary condition formally reduces to the first-type boundary condition. The governing Equation 5.16 subject to initial and boundary conditions 5.17 through 5.20 were solved numerically. (See Section XII, Appendix B for the detailed finite difference scheme.)

C. CDE

WITH A

NONLINEARLY TIME-DEPENDENT DISPERSIVITY

Zhou and Selim17 developed a fractal model to describe a time-dependent dispersivity in terms of mean travel distance x . The fractal model reads α( x ) = a3 x

D fr −1

(5.21)

2−D

where now a3 is a constant with dimension L fr , and Dfr is the fractal dimension of the tortuous stream tubes in the media. Dfr varies from 1 to 2. If Dfr = 1, we recover the time-invariant constant dispersivity. Similarly, if Dfr = 2, Equation 5.21 reduces to Equation 5.5. Again, we assume molecular diffusion can be ignored and dispersion coefficient for a nonlinear dispersion function is given by D(t ) = α ( x ) v = a3 x

D fr −1

v = a3 v

D fr D fr −1

t

(5.22)

Substituting Equation 5.22 into Equation 5.7 and rearranging yields the following governing equation: 2 ∂c ∂c D D −1 ∂ c = −v + a3 v fr t fr ∂t ∂x ∂x 2

(5.23)

Equation 5.8 is recovered if we let Dfr = 2 in the above equation. The upper boundary conditions are © 2003 by CRC Press LLC


72


vc ( x , t ) = vc0 − a3 v

D fr D fr −1

vc ( x , t ) = −a3 v

D fr D fr −1

t

t

∂c , ∂x

x = 0, 0 < t ≤ T

(5.24)

∂c , ∂x

x = 0, t > T

(5.25)

The remaining initial and lower boundary conditions are the same as those for linear dispersivity model (Equations 5.9 and 5.12). The above system was also solved using finite difference method. (See Section XIII, Appendix C.)

D. CDE

WITH A

NONLINEARLY DISTANCE-DEPENDENT DISPERSIVITY

If we remove the bar from x in Equation 5.21, we obtain the following nonlinearly distancedependent dispersivity α( x ) = a4 x 2−D fr

where now a4 is a constant with dimension L D is given by

D fr −1

(5.26)

. Under this condition, the dispersion coefficient

D( x ) = α ( x ) v = a4 x

D fr −1

(5.27)

v

Accordingly, the governing equation now reads

[

]

2 ∂c ∂c D −1 ∂ c D −1 = a4 vx fr − 1 − a4 ( D fr − 1) x fr v 2 ∂t ∂x ∂x

(5.28)

Equation 5.28, subject to initial and boundary conditions 5.17 through 2.20, was solved with finite difference method. (See Section XIV, Appendix D.)

E. COMPARISON

OF

MODELS

AND

SIMULATIONS

For the CDE with a time-dependent dispersivity, the magnitude of the dispersivity α increases with mean travel distance or time. Under this situation, the dispersivity value remains constant over the entire spatial domain. In other words, the entire medium is treated as a homogeneous system with a fixed constant dispersivity value for each specific time. On the contrary, if one removes the bar from the mean travel distance in the dispersivity–mean travel distance relationship, the dispersive property of the medium is completely altered. As a result, dispersivity becomes a function of distance from source instead of mean travel distance or time. Under such conditions, the dispersivity is held constant over the time of consideration for any location but increases with distance from the source where the solute is released. Therefore, the resulting parameter fields and thus the governing equations are quite different and depend on whether or not the bar in mean travel distance is removed. An extra term occurs for the distance-dependent dispersivity models in the governing equation to account for the dependency of dispersivity on distance. Differences in the governing equations inevitably induce the differences in the numerical scheme (finite difference equations; see Sections XI through XIV for details). The dependence of dispersivity on time or distance is carried over to the finite difference approximations. Corresponding to time dependence, index j, which indicates time domain discretization, occurs in the finite © 2003 by CRC Press LLC



73

TABLE 5.1 Parameters Used for Simulations of Time-Dependent and Distance-Dependent Dispersivities (α) Parameter

Time-Dependent Dispersivity

Distance-Dependent Dispersivity

Moisture content (cm3/cm3) Column length (cm) (short/long) Water flux rate (cm/h) Initial concentration (mg/L) Concentration in input pulse (mg/L) Pulse duration (hour) (L= 50 cm/100 cm) Dispersivity α (cm)

0.40 50.0/100.0 5.0 0.0 10.0 2.0/16.0 0.5 x

0.40 50.0/100.0 5.0 0.0 10.0 2.0/16.0 0.5x

difference approximation for the governing equation with dispersivity as a function of mean travel distance. Accordingly, index i appears for a distance-dependent dispersivity. Comparison between the finite difference equations only is not enough to confirm the difference between the two processes described by these two different governing equations. One needs to show differences in BTCs as well as solute distribution along spatial coordinate or flow direction to achieve a generalized conclusion. For comparison, both governing equations subject to the same initial and boundary conditions were considered here. Because of the complexity of the difference equations, it is difficult to assess convergence conditions. The time and space increments were based on the governing equation with a constant dispersivity, which in our case equals the coefficient a in the dispersivity function. The assessment of the convergence of numerical approximation was achieved through mass balance calculations as well as the magnitude and oscillation of resulting numerical solutions for solute concentration. The parameters used for our simulations are given in Table 5.1. Similar parameter values were selected for both cases. The only difference lies in that the variable mean travel distance for the timedependent dispersivity is replaced with distance from source to generate the distance-dependent dispersivity. Two different column lengths were considered. One is 50 cm in length, the other 100 cm. A longer pulse length is used for the 100-cm column to obtain comparable BTCs. Comparison of BTCs as well as distribution profiles for a solute tracer was made for all different scenarios. Simulated BTCs with linearly time-dependent and distance-dependent α are shown in Figure 5.1 for 50- and 100-cm soil columns. The BTCs from time-dependent or distance-dependent α appear somewhat similar. Nevertheless, several distinct features are apparent. Based on our simulations, the column length showed modest influence in the relative relationship between BTCs of the media with a time-dependent dispersivity and those with a distance-dependent dispersivity. For both long and short columns, distance-dependent dispersivity resulted in earlier arrival of the BTC than the time-dependent counterpart. Both BTCs exhibited similar leading edge, however. BTCs for the distance-dependent cases showed higher peak concentrations than those for the time-dependent cases. In general, timedependent α resulted in enhanced tailing compared with distance-dependent counterparts. Snapshots for solute distributions at different times for the 100-cm column with linear timedependent or distance-dependent α are shown in Figure 5.2. Because the dispersion coefficient vanishes at the inlet (x = 0) for the distance-dependent α, the concentration at the inlet is always at c0 for all times during pulse application (16 h). At early times, solute distribution profiles appear similar for both types of α models. However, the two snapshots separated gradually over time. Generally, solute concentration profiles exhibited a rapid decrease or sharp fronts when distance-dependent α was used. In other words, distance-dependent α resulted in steeper solute concentration profile than time-dependent α. At later times, solute fronts advanced further in the porous media when distance-dependent α was used. Our results clearly demonstrate the differ© 2003 by CRC Press LLC


74


Relative concentration (C/Co)

0.5

1a

Distance-dependent

0.4 Time-dependent

0.3

L= 50 cm

0.2

0.1

0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Pore Volume

Relative Concentration (C/Co)

1.0

1b

Distance-dependent

0.8 Time-dependent

0.6 L=100 cm

0.4

0.2

0.0 0

1

2

3

4

5

6

Pore Volume FIGURE 5.1 Comparison of simulated BTCs based on time-dependent (Equation 5.8) and distance-dependent dispersivity (Equation 5.16) for 50-cm (top) and 100-cm (bottom) columns.

ences between transport processes in a medium with time-dependent and that with distancedependent α. 1. Effects of Nonlinearity of Dispersivity Model The nonlinear dispersivity models (Equations 5.21 through 5.28) provide an opportunity to investigate the effects of fractal dimension on solute BTCs. For computational convenience, only 10-cm long soil columns were considered in this section (Table 5.2). For time-dependent dispersivity, the BTCs for different fractal dimension Dfr are compared in Figure 5.3. From this figure, we can see that the fractal dimension has significant influences on the overall shape of © 2003 by CRC Press LLC



75

Solute Concentration (mg/L)

10

2a Time = 2 hr

8 Distance-dependent

6

Time-dependent

4

2

0 0

20

40

60

80

100


10

2b 8

Time = 10 hr

6

Distance-dependent

4

Time-dependent

2

0 0

20

40

60

80

100


10

2c Time = 20 hr

8

6

4 Distance-dependent

2 Time-dependent

0 0

20

40

60

80

100

Distance (cm) FIGURE 5.2 Comparison of simulated solute concentration profiles of a pulse tracer based on time-dependent (Equation 5.8) and distance-dependent (Equation 5.16) dispersivity after 2, 10 and 20 h of transport. Column length L = 100 cm and pulse duration = 16 h.



76


TABLE 5.2 Parameters Used in Simulation to Examine Effect of Exponent in the Dispersivity Expressions of Equations 5.21 and 5.26 Time-Dependent α

Parameter Moisture content (cm3/cm3) Column length (cm) Water flux rate (cm/hr) Initial concentration (mg/L) Concentration in input pulse (mg/L) Pulse duration (hour) Dispersivity α (cm)

0.5 x

Dfr −1

,

Distance-Dependent α

0.40 10.0 5.0 0.0 10.0 2.0 Dfr = 1.25, 1.50, 1.75, 2.0

0.5 x

Dfr −1

,

0.40 10.0 5.0 0.0 10.0 2.0 Dfr = 1.25, 1.50, 1.75, 2.0


1.0

D fr

0.8

1.25 1.5

0.6

1.75 2

0.4

α = 0.5 x

0.2

Dfr −1

0.0 0

1

2

3

4

5

6

Pore Volume FIGURE 5.3 Comparison of simulated BTCs based on time-dependent dispersivity (Equation 5.23) with different fractal dimensions Dfr.

BTCs. As Dfr increases, the initial arrival time becomes shorter with lower peak concentrations. Moreover, the BTCs exhibited enhanced tailing or increased spreading as Dfr value equals to 2.0. Comparison of BTCs for distance-dependent α having different exponent values is shown in Figure 5.4. For distance-dependent dispersivity, higher values of Dfr resulted in earlier arrival of BTC, a lower peak concentration and an enhanced tailing in BTCs. However, our simulations clearly show that the differences among BTCs for different Dfr shown in Figure 5.4 are relatively small compared with the time-dependent counterpart shown in Figure 5.3.

IX. DISTANCE-DEPENDENT DISPERSIVITY: DERIVATION AND SIMULATION Based on stochastic analysis, Simmons29 defined a generalized macro-dispersivity λ( z ) as λ( z ) =


V 2 dσ 2t 2 dz

(5.29)



77


1.0

D fr

0.8

1.25 1.5

0.6

1.75 2

0.4

α = 0.5 x

D fr −1

0.2

0.0 0

1

2

3

4

5

6

Pore Volume FIGURE 5.4 Comparison of simulated BTCs based on distance-dependent dispersivity (Equation 5.28) with different exponents Dfr.

where z is the distance or depth from the solute source, V is the harmonic mean value of v(z), which is the velocity of the solute particles, and σ 2t is the variance of the random travel time from z = 0 to z. To use Equation 5.29 to obtain the macrodispersivity, one needs to assess the velocity fluctuation. Equation 5.29 has not been used because the velocity fluctuation may not be independently measurable, according to Jury and Roth.30 However, we followed the idea embedded in Equation 5.29 to investigate solute transport in heterogeneous soil. If one can obtain a functional relationship between σ 2t and distance z, one may be able to develop an expression for a distance-dependent dispersivity by taking derivative of σ 2t with respect to z. An average velocity V is assumed. The variance of travel time σ 2t for different distances could thus be estimated from BCTs based on moment analysis. An alternative way to obtain σ 2t is to back-calculate it from the optimized apparent dispersivity. The latter method is preferred because of the availability of data sets in literature. According to Jury and Roth,30 the travel time variance σ 2t for the CDE is given by σ 2t ( z ) =

2 Dz V3

(5.30)

where D is the dispersion coefficient. If we assume D = Vλ , where λ is the apparent dispersivity, we obtain the following equation to back-calculate σ 2t : σ 2t ( z ) =

2 λz V2

(5.31)

According to Equation 5.31, σ 2t can be computed for different distances if apparent dispersivity λ is known. The obtained σ 2t values at different distances or depths are then plotted and a functional expression for the relationship between σ 2t and z can be obtained through regression analysis. As long as a functional relationship between σ 2t and z is available, we are able to obtain a functional relationship of dispersivity to distance using Equation 5.29. It should be pointed out that the obtained dispersivity–distance relationship is a distance-specific dispersivity function. In other words, dispersivity varies from point to point and the medium is heterogeneous. If σ 2t is a © 2003 by CRC Press LLC


78


linear function of distance, then we obtain a constant value of dispersivity. Otherwise, a distancedependent dispersivity is obtained. In addition, we can observe how dispersivity behaves based upon the fitted trend curves. According to transfer function theory,30 the variance of travel time may increase linearly with distance or grow proportionally to the square of distance for different transfer function models. Without loss of generality, we may assume that the relationship of travel time variance to distance could be described by the following power law form as: σ 2t ( z ) = gz h

(5.32)

where h is a dimensionless constant and g is a constant with dimension of T2L–h. Following Equation 5.29, we can develop a distance-dependent dispersivity function as λ( z ) =

V 2 ghz h −1 2

(5.33)

The travel time variance-distance relationship based on experimental data could be fitted using Equation 5.32 to obtain g and h. Substituting estimated g and h to Equation 5.33 yields a functional distance-dependent dispersivity. Since the average pore water velocity V may not always be available, we propose an alternative method to obtain the distance-dependent dispersivity based on apparent dispersivities at different depths or distances. If we assume the average value of λ for depth from 0 to z described by the distance-dependent dispersivity function is the same as the estimated apparent λ at depth z, we can obtain a distance-dependent dispersivity based on simple regression. Because the CDE with a constant λ describes a Fickian process, whereas that with a distance-dependent dispersivity describes a non-Fickian process, the average λ from a nonFickian process is not necessarily comparable with that from a Fickian process. Although our assumptions may not be strictly valid, this approach provides a quick way to obtain a distancedependent dispersivity. Inspection of apparent dispersivity data listed in Pachepsky et al.31 reveals that a power law function is perhaps suitable for most data sets. It should be pointed out that the maximum length of soil columns used for laboratory studies is limited to tens of meters. Therefore, for distances longer than tens of meters, say, hundreds of meters, the power law function may not be appropriate. For distances in the range of hundreds of meters, an asymptotic function is perhaps more appropriate. Assume a distance-dependent dispersivity is given by λ( z ) = cz d

(5.34)

where c and d are constants and d is dimensionless whereas c has a dimension of L1-d. Although Equation 5.34 takes on the same form as the fractal model (Equation 5.21), the parameter d is not necessarily associated with the fractal dimension of the media and the range of d cannot be determined in advance. Denoting the average dispersivity up to distance z by λ AV ( z ) , we can develop an expression for λ AV ( z ) as follows: z

z

0

0

∫ λ(z ′)dz ′ = ∫ cz ′ dz ′ =  c z (z) = z  d + 1 ∫ dz ′ d

λ AV

z

0


d

(5.35)



79

TABLE 5.3 Optimized Parameters (with One Standard Error) for the Relationship of Travel Time Variance σt2 vs. Distance (Equation 5.29) g 0.0471 0.0385 0.0195 0.0279 0.0323 0.0141 0.00153

± ± ± ± ± ± ±

R2

h 0.0189 0.0125 0.0152 0.0466 0.0621 0.0225 0.00147

1.3369 1.3471 1.5830 1.8802 1.8285 2.0679 2.3130

± ± ± ± ± ± ±

0.1047 0.0844 0.2018 0.4306 0.4945 0.4097 0.1384

0.9976 0.9980 0.9938 0.9775 0.9692 0.9823 0.9869

Data source (ref.)

Remarks Sandbox, Test1, probes 20, 15, 11, 7, Sandbox, Test2, probes 20, 15, 11, 7, Sandbox, Test3, probes 20, 15, 11, 7, Sandbox, Test1, probes 21, 16, 12, 8, Sandbox, Test2, probes 21, 16, 12, 8, Sandbox, Test3, probes 21, 16, 12, 8, Homogeneous column, simultaneous fitting of V and D

3 3 3 4 4 4

24 24 24 24 24 24 22

where z′ is a dummy integral variable. Equation 5.35 could be used to fit the estimated apparent dispersivity-distance relationship to obtain parameters c and d. We used PROC NLIN in SAS32 (Equations 5.32 and 5.34) to fit travel time variance–distance and apparent dispersivity–distance relationships to the power law models. We also discussed the range of exponents h and d.

A. TRAVEL TIME VARIANCE

VS.

DISTANCE

Results of regression analysis for travel time variance σ 2t vs. distance (Equation 5.32) are given in Table 5.3. These results are based on published data from Burns24 and Zhang et al.22 Based on our regression analysis, the exponent h of Equation 5.32) ranged from 1.3 to 2.3 for the different cases considered. Two data sets (probes 20, 15, 11, 7, and 3 in test 1 and test 2) from Burns24 gave an exponent h significantly higher than 1.0 (at the 0.05 level); others were not significantly different from 1.0. Nevertheless, the estimated h values for those cases were much larger than 1. Data from a homogeneous column from Zhang et al.22 gave an h value significantly greater than 2.0. Based on these experimental data sets, the exponent h cannot be associated with the fractal dimension of stream tubes. In fact, as depicted in Table 3.3, for half the cases considered, the nonlinear relationship between σ 2t and distance is not well supported statistically. On the other hand, it appears that the power law function fails to give a good description of the relationship between σ 2t and distance. An example of the relationship between σ 2t and distance based on the data from Zhang et al.22 is shown in Figure 5.5.

B. APPARENT DISPERSIVITY

VS.

DISTANCE

The results of regression analysis on apparent dispersivity (Equation 5.35) from published data are given in Table 5.4. The estimated exponent parameter d of Equation 5.34 varied from -0.098 to 1.0058. Most of the cases considered give an exponent d not significantly different from 0.0 (at 0.05 level) because 0.0 was within the 95% confidence interval. The actual relationship between apparent dispersivity and distance is at best complex (Figure 5.6). In fact, it is difficult to describe all data sets with a single function, e.g., a power law function. Mishra and Parker33 proposed an asymptotic scale-dependent dispersion model to describe the relationship between dispersivity and distance. They fitted the numerical solution of CDE with an asymptotic distance-dependent dispersion model to the data set from Butters.34 However, it is difficult to judge whether an asymptotic model is superior to a power law model. Similar analysis to that discussed previously was carried out by Pachepsky et al.31 For some cases, the slopes of the log–log plot as obtained by Pachepsky et al.31 were larger than the © 2003 by CRC Press LLC


80


21000

Travel Time Variance (min2)

Measured data Fitted curve

14000

7000

0 0

300

600

900

1200

Distance (cm) FIGURE 5.5 Estimated travel time variance at different distances (based on data in Zhang, R., Huang, R., and Xiang, J., Adv. Water Resour., 17, 317, 1994.).

TABLE 5.4 Optimized Parameters (with One Standard Error) for the Relationship of Apparent Dispersivity λ vs. Distance (Equation 5.32) c/(d + 1)

d

R2

Remarks

Data source (ref.)

0.0987 ± 0.2327 1.3018 ± 2.5944 0.1868 ± 0.0412 0.0102 ± 0.000559 0.0868 ± (0.1103) 0.0791 ± 0.00587 0.0864 ± 0.00896 0.0287 ± 0.0118 0.3042 ± 0.4340 0.1329 ± 0.2498 0.2782 ± 0.3372 0.1984 ± 0.3532 1.7846 ± 1.6843

1.0039 ± 0.3473 0.6817 ± 0.2977 0.3853 ± 0.2082 0.3123 ± 0.0320 1.0058 ± 0.3812 0.2788 ± 0.1047 0.3247 ± 0.1531 –0.1967 ± 0.1444 0.4893 ± 0.2468 0.5264 ± 0.3243 0.2974 ± 0.2141 0.3073 ± 0.3141 –0.0981 ± 0.1799

0.8591 0.8429 0.8911 0.9999 0.9905 0.9861 0.9686 0.9430 0.9410 0.9069 0.9285 0.8598 0.9190

Heterogeneous column, using V at 100 cm Heterogeneous column, using V at 1200 cm Field data 8-m long column Field data Heterogeneous sandbox, Run 12 Heterogeneous sandbox, Run 13 Sandy aquifer Uniform column, Tritium data Uniform column, Bromide data Layered column, Tritium data Layered column, Bromide data Layered column, Chloride data

22 22 34 35 7 36 36 9 37 37 37 37 37

exponent d given in Table 5.4. Their results were obtained through linear regression on logtransformed data, whereas we conducted nonlinear regression analysis on the original data. The discrepancies may stem from different approaches employed. On the other hand, results from our analysis revealed that the rate of increase in dispersivity for a homogeneously packed column is higher than that for heterogeneous column22 (exponent of 1.3 for homogeneous column from Table 5.3 vs. 1.0 and 0.68 for heterogeneous columns from Table 5.4). Based on this, it appears that reasons for distance-dependent dispersivity are complex and may not be solely associated with heterogeneity of the medium. © 2003 by CRC Press LLC



81

Apparent Dispersivity (m)

2.5

2.0

1.5

1.0

Zhang et al. [22] Butters [34] Peaudecerf and Sauty [7]

0.5

Talyor and Howard [9] Pang and Hunt [35]

0.0 0

5

10

15

20

25

30

35

40

Distance (m) FIGURE 5.6 Observed apparent dispersivity vs. distance in soils and aquifers. Data are multiplied size by a factor of 100. (Data from Taylor, S.R. and Howard, K.W.F., J. Hydrol., 90, 11, 1987 and Pang, L. and Hunt, B., J. Cont. Hydrol., 53, 21, 2001.)

X. SUMMARY In this contribution, the definition of the term scale was closely examined and representations in the literature in terms of mean travel distance and distance are discussed. We discussed the implications of the use of mean travel distance vs. distance in solute transport. The fundamental difference between these two terms lies in that mean travel distance is a dependent variable depending on time, whereas distance is an independent variable. Ambiguity in the definition of scale caused several confusions in studies of transport processes in porous media. A source of confusion is the equivalence and interchange of mean travel distance and distance from the source, which implies an interchange of time and distance. After careful inspection of generic scale-dependent dispersivity in the literature, we emphasized four distinct types of dispersivity–time or dispersivity–distance relationships that are appropriate to describe the relationship between dispersivity and time or distance. Specifically, they are: instantaneous time-dependent dispersivity, time-averaged dispersivity, distance-specific dispersivity, and distance-averaged dispersivity. Time-dependent or time-averaged dispersivity is supported by stochastic theory. However, the distance-specific dispersivity is not supported theoretically. Although the value of distance-averaged dispersivity may vary with distance, no consistent relationship between these two could be established based on experimental data available in the literature. Transport processes in porous media with time-dependent and distance-dependent dispersivities were simulated using finite difference methods. The solute distribution profiles and BTCs were compared. Simulation results show that transport in a system with a time-dependent dispersivity consistently exhibited more spreading of the BTCs than that in a system with distance-dependent dispersivity, e.g., α (t ) = 0.1x vs. α ( x ) = 0.1x . Our simulations clearly illustrate that x and x are not interchangeable. Because the distance-specific dispersivity is not supported by currently available theory, we proposed a justification for the use of a distance-specific dispersivity based on experimental data from published data in the literature. This attempt was not conclusive. Based on experimental © 2003 by CRC Press LLC


82


measurements, the relationship between apparent dispersivity and distance appears to be rather complex, with inconsistent trends for different media. Moreover, rigorous theoretical justification for the use of distance-specific dispersivity is needed.

XI. APPENDIX A. DERIVATION OF FINITE DIFFERENCE EQUATIONS FOR CDE WITH A LINEARLY TIME-DEPENDENT DISPERSIVITY The governing equation for a linear dispersivity model reads (Equation 5.8), ∂c ∂c ∂2 c = −vx + a1 v 2 t 2 ∂t ∂x ∂x

(5.A1)

Denoting time and space increments by ∆t and ∆x, we can establish the finite difference scheme for point (i∆x,j∆t), where i and j are integers and denote space and time steps, respectively. Finite difference approximations of each partial derivative in Equation 5.A1 are as follows:

t

where

c ij

j+1 j ∂c c i − c i = ∂t ∆t

(5.A2)

j+1 j+1 ∂c c i+1 − c i = ∂x ∆x

(5.A3)

c j+1 − 2 c ij+1 + c ij−+11 c j − 2 c ij + c ij−1  ∂2 c 1  + j∆t i+1 = ( j + 1) ∆t i+1  2 2 2  ∆x ∆x 2 ∂x 

(5.A4)

stands for solute concentration at node (i∆x,j∆t). For convenience, let β=

∆t ∆x

Substituting Equations 5.A2 through 5.A4 to 5.A1 and rearranging gives Ac ij−+11 + Bc ij+1 + Cc ij++11 = E

(5.A5)

where A=

1 a v 2 ( j + 1)β2 2 1

B = −1 + vβ − a1 v 2 ( j + 1)β2 C=


1 a v 2 ( j + 1)β2 − vβ 2 1

(5.A6)

(5.A7)

(5.A8)



E = −c ij −

83

1 a v 2 jβ2 ( c ij+1 − 2 c ij + c ij−1 ) 2 1

(5.A9)

After discretization, the upper boundary conditions (Equations 5.10 and 5.11) read

[1 + a v ( j + 1)β]c 2

1

j+1 1

[1 + a v ( j + 1)β]c 2

1

− a1 v ( j + 1)βc2j+1 = c0 , 0 ≤ ( j + 1) ∆t ≤ T j+1 1

− a1 v ( j + 1)βc2j+1 = 0, ( j + 1) ∆t > T

(5.A10)

(5.A11)

where c0 is the solute concentration in input pulse, and T is the pulse duration.

XII. APPENDIX B. DERIVATIONS OF FINITE DIFFERENCE EQUATIONS FOR CDE WITH A LINEARLY DISTANCE-DEPENDENT DISPERSIVITY In this case, the governing equation becomes (Equation 5.16) ∂2 c ∂c ∂c = a2 vx 2 − (1 − a2 ) v ∂t ∂x ∂x

(5.B1)

The finite difference approximation of the above equation for point (i∆x , j∆t ) can be developed using the same notation as above. The approximation of the first partial derivatives of c with respect to time t and distance x are the same as Equations 5.A2 and 5.A3, respectively. The second derivative of concentration with respect to x is given by j+1 j+1 j+1 j j j ∂2 c 1  c i+1 − 2 c i + c i−1 c i+1 − 2 c i + c i−1  = +   ∂x 2 2  ∆x 2 ∆x 2 

(5.B2)

Substituting Equation 5.B2 together with Equations 5.A2 and 5.A3 into Equation 5.B1, replacing x with (i∆x), and rearranging gives an equation similar to Equation 5.A5: A′ c ij−+11 + B ′c ij+1 + C ′c ij++11 = E ′

(5.B3)

where

A′ =

1 a viβ 2 2

B ′ = −1 + vβ(1 − a2 ) − a2 viβ C′ =


1 a viβ − vβ(1 − a2 ) 2 2

(5.B4)

(5.B5)

(5.B6)


84


E ′ = −c ij −

(

1 a viβ c ij+1 − 2 c ij + c ij−1 2 2

)

(5.B7)

where β is defined as the ratio of time increment to space increment as above.

XIII. APPENDIX C. DERIVATIONS OF FINITE DIFFERENCE EQUATIONS FOR CDE WITH A NONLINEARLY TIME-DEPENDENT DISPERSIVITY For a power law dispersivity-time model, the governing equation is as follows (Equation 5.23): 2 ∂c ∂c D D −1 ∂ c + a3 v fr t fr = −v ∂x ∂t ∂x 2

(5.C1)

Approximations of first partial derivatives are given in Equations 5.A2 and 5.A3. The second derivative with respect to x is given by

t

D fr −1

j+1 j+1 j j j + c ij−+11 D fr −1 D fr −1 c i +1 − 2 c i + c i −1  ∂2 c 1  D fr −1 c i +1 − 2 c i = + 1 ∆ ∆ + j t j t   ( ) ( ) ∂x 2 2  ∆x 2 ∆x 2 

(5.C2)

For convenience, we let D

∆t ∆t fr and γ = . β= ∆x ∆x 2 Notice that γ = β2 for Dfr = 2. Substituting Equation 5.C2 together with Equations 5.A2 and 5.A3 to 5.C1 and rearranging yields A′′ c ij−+11 + B ′′c ij+1 + C ′′c ij++11 = E ′′

(5.C3)

where A′′ =

D fr −1 1 D a v fr ( j + 1) γ 2 3

B ′′ = −1 + vβ − a3 v C ′′ =

E ′′ = −c ij −

D fr

( j + 1)

D fr −1

γ

(5.C5)

D fr −1 1 D a v fr ( j + 1) γ − vβ 2 3

(

1 D D −1 a3 v fr j fr γ c ij+1 − 2 c ij + c ij−1 2

Obviously, Equation 5.C3 reduces to Equation 5.A5 for Dfr = 2. © 2003 by CRC Press LLC

(5.C4)

(5.C6)

)

(5.C7)



85

For this case, the upper boundary conditions after discretization are given by D fr −1   D −1 ∆t D −1 1 + a3 v fr ( j + 1) fr  c1j+1 x ∆  

− a3 v

D fr −1

( j + 1)

D fr −1

D fr −1

∆t ∆x

(5.C8)

c2j+1 = c0 , 0 ≤ ( j + 1) ∆t ≤ T

D fr −1   D −1 ∆t D −1 1 + a3 v fr ( j + 1) fr  c j+1 ∆x  1 

− a3 v

D fr −1

( j + 1)

D fr −1

D fr −1

∆t ∆x

(5.C9)

( j + 1)∆t > T

c2j+1 = 0,

where c0 is the solute concentration in input pulse, and T is the pulse duration.

XIV. APPENDIX D. DERIVATIONS OF FINITE DIFFERENCE EQUATIONS FOR CDE WITH A NONLINEARLY DISTANCEDEPENDENT DISPERSIVITY For a parabolic dispersivity-distance model, the governing equation is (Equation 5.28)

[

(

]

)

2 ∂c ∂c D −1 ∂ c D −1 = a4 vx fr − 1 − a4 D fr − 1 x fr v 2 ∂t ∂x ∂x

(5.D1)

For convenience, we let β=

∆t ∆t and ξ = 3 −D . ∆x ∆x fr

Notice that ξ = β for Dfr = 2. Replacing x with (i∆x) and substituting approximation for partial derivatives, e.g., Equations 5.A2, 5.A3 and 5.B2, to the above equation and rearranging produces A′′′ c ij−+11 + B ′′′c ij+1 + C ′′′c ij++11 = E ′′′

(5.D2)

where A′′′ =

1 D −1 a4 vi fr ξ 2

B ′′′ = −1 + vβ − a4 vξ[ i

C ′′′ =


D fr −1

(5.D3)

+ ( D fr − 1)i

D fr −2

]

(5.D4)

1 D −1 D −2 a vξi fr + a4 vξ D fr − 1 i fr − vβ 2 4

(5.D5)

(

)


86


E ′′′ = −c ij −

(

1 D −1 a vξi fr c ij+1 − 2 c ij + c ij−1 2 4

)

(5.D6)

Apparently, Equation 5.D2 reduces to Equation 5.B3 for Dfr = 2.

REFERENCES 1. Pickens, J.F. and G.E. Grisak, Modeling of scale-dependent dispersion in hydrogeologic systems, Water Resour. Res., 17, 1701, 1981. 2. Fried, J.J., Miscible pollution of ground water: a study of methodology, in Proc. of the International Symposium on Modelling Techniques in Water Resources Systems, Vol. 2, A.K. Biswas, Ed., Environment Canada, Ottawa, 1972, 362. 3. Sudicky, S.A. and J.A. Cherry, Field observations of tracer dispersion under natural flow conditions in an unconfined sandy aquifer, Water Pollut. Res. Can., 14, 1, 1979. 4. Pickens, J.F. and G.E. Grisak, Scale-dependent dispersion in a stratified granular aquifer, Water Resour. Res., 17, 1191, 1981. 5. Khan, A. Ul-Hassan and W.A. Jury, A laboratory study of the dispersion scale effect in column outflow experiments, J. Cont. Hydrol., 5, 119, 1990. 6. Gelhar, L.W., C. Welty, and K.R. Rehfeldt, A critical review of data on field-scale dispersion in aquifers, Water Resour. Res., 28, 1955, 1992. 7. Peaudecerf, P. and J.P. Sauty, Application of a mathematical model to the characterization of dispersion effects of groundwater quality, Prog. Water Technol., 10, 443, 1978. 8. Fried, J.J., Groundwater Pollution: Theory, Methodology, Modeling and Practical Rules, Elsevier Scientific Publishing Company, Amsterdam--Oxford--New York, 1975. 9. Taylor, S.R. and K.W.F. Howard, A field study of scale-dependent dispersion in a sandy aquifer, J. Hydrol., 90, 11, 1987. 10. Wheatcraft, S.W. and S.W. Tyler, An explanation of scale-dependent dispersivity in heterogeneous aquifers using concepts of fractal geometry, Water Resour. Res., 24, 566, 1988. 11. Su, N.-H., Development of the Fokker-Planck equation and its solutions for modeling transport of conservative and reactive solutes in physically heterogeneous media, Water Resour. Res., 31, 3025, 1995. 12. Yates, S.R., An analytical solution for one-dimension transport in heterogeneous porous media, Water Resour. Res., 26, 2331, 1990. 13. Yates, S.R., An analytical solution for one-dimension transport in porous media with an exponential dispersion function, Water Resour. Res., 28, 2149, 1992. 14. Arya, A., T.A. Hewett, R.G. Larson, and L.W. Lake, Dispersion and reservoir heterogeneity, SPE Reservoir Eng., Feb. 1988, 139. 15. Neuman, S.P., Universal scaling of hydraulic conductivities and dispersivities in geologic media, Water Resour. Res., 26, 1749, 1990. 16. Xu, M. and Y. Eckstein, Use of weighted least-squares method in evaluation of the relationship between dispersivity and field scale, Gound Water, 33, 905, 1995. 17. Zhou, L.-Z. and H.M. Selim, A conceptual fractal model for describing time-dependent dispersivity, Soil Sci., 167, 173, 2002. 18. Neuman, S.P. and Y.-K. Zhang, A quasi-linear theory of non-Fickian and Fickian subsurface dispersion. 1. Theoretical analysis with application to isotropic media, Water Resour. Res., 26, 887, 1990. 19. Zhang, Y.-K. and S.P. Neuman, A quasi-linear theory of non-Fickian and Fickian subsurface dispersion, 2, Application to anisotropic media and the Borden site, Water Resour. Res., 26, 903, 1990. 20. Freyberg, D.L., A natural gradient experiment on solute transport in a sand aquifer. 2. Spatial moments and the advection and dispersion of nonreactive tracers, Water Resour. Res., 22, 2031, 1986. 21. Sudicky, S.A., J.A. Cherry, and E.O. Frind, Migration of contaminants in groundwater at a landfill: a case study. 4. A natural-gradient dispersion test, J. Hydrol., 63, 81, 1983. 22. Zhang, R., K. Huang, and J. Xiang, Solute movement through homogeneous and heterogeneous soil columns, Adv. Water Resour., 17, 317, 1994.




87

23. Butters, G.L. and W.A. Jury, Field scale transport of bromide in an unsaturated soil. 2. Dispersion modeling, Water Resour. Res., 25, 1583, 1989. 24. Burns, E., Results of 2-dimensional sandbox experiments: Longitudinal dispersivity determination and seawater intrusion of coastal aquifers, Master's thesis, Univ. of Nevada, Reno, 1996. 25. Pang, L. and M. Close, Field-scale physical non-equilibrium transport in an alluvial gravel aquifer, J. Cont. Hydrol., 38, 447, 1999. 26. Logan, J.D., Solute transport in porous media with scale-dependent dispersion and periodic boundary conditions, J. Hydrol., 184, 261, 1996. 27. Zou, S., J. Xia, and A.D. Koussis, Analytical solutions to non-Fickian subsurface dispersion in uniform groundwater flow, J. Hydrol., 179, 237, 1996. 28. Press, W.H., S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes in FORTRAN: the Art of Scientific Computing, 2nd Ed., Cambridge Univ. Press, New York, 1992. 29. Simmons, C.S., Scale dependent effective dispersion coefficients for one dimensional solute transport, in Proc. 6th Annu. AGU Front Range Branch Hydrol. Days Pub., Fort Collins, Co., 1986. 30. Jury, W.A. and K. Roth, Transfer Functions and Solute Movement through Soil: Theory and Applications, Birkhauser Verlag, Basel, 1990. 31. Pachepsky, Y., D. Benson, and W. Rawls, Simulating scale-dependent solute transport in soils with the fractional advective-dispersive equation, Soil Sci. Soc. Am. J., 64, 1234, 2000. 32. SAS Institute Inc., The SAS System for Windows, Release 8.2, SAS Institute Inc., Cary, NC, 2001. 33. Mishra, S. and J.C. Parker, Analysis of solute transport with a hyperbolic scale-dependent dispersion model, Hydrol. Proc., 4, 45, 1990. 34. Butters, G.L., Field scale transport of bromide in unsaturated soil, Ph.D. dissertation, Univ. of California, Riverside, 243 pp., 1987. 35. Pang, L. and B. Hunt, Solutions and verification of a scale-dependent dispersion model, J. Cont. Hydrol., 53, 21, 2001. 36. Silliman, S.E. and E.S. Simpson, Laboratory evidence of the scale effect in dispersion of solutes in porous media, Water Resour. Res., 23, 1667, 1987. 37. Porro, I., P.J. Wierenga, and R.G. Hills, Solute transport through large uniform and layered soil columns, Water Resour. Res., 29, 1321, 1993.



6

A Pedotransfer Function for Predicting Solute Dispersivity: Model Testing and Upscaling E. Perfect

CONTENTS I. Introduction.............................................................................................................................89 II. Pedotransfer Functions for Dispersivity ................................................................................90 III. Scale Dependency of Dispersivity .........................................................................................90 IV. Upscaling Procedures .............................................................................................................91 V. Data Sets, Methods, and Models ...........................................................................................91 VI. Comparison of Predicted and Observed Dispersivities .........................................................92 VII. Example Application and Discussion ....................................................................................93 VIII. Conclusions.............................................................................................................................94 References ........................................................................................................................................95

I. INTRODUCTION Nonreactive solutes moving through saturated porous media are dispersed by the multiplicity of pore-water velocities resulting from different pore shapes, sizes and connections.1 At high Peclet numbers, the extent of mixing that occurs in response to a given flow rate can be characterized by2 λ≡

D v

(6.1)

where λ is the longitudinal dispersivity, and D and v are the dispersion coefficient and mean pore water velocity, respectively, from the one-dimensional advection-dispersion equation (ADE). Dispersivity is a required input parameter for contaminant transport models based on the ADE.3 Estimates of λ can be obtained from laboratory or field tests. In the laboratory, dispersivity is usually determined by conducting miscible displacement experiments on soil columns.4 In the field, it can be measured by mapping a plume of known contamination or through the use of tracers injected into an aquifer via a well.2 All three approaches rely on solving the ADE with D and v as unknowns. This dependency on inverse procedures imposes a severe limitation on our predictive capability. As a result, interest is growing in the use of statistically based approaches to estimate λ from independent measurements of soil/aquifer properties.



90


II. PEDOTRANSFER FUNCTIONS FOR DISPERSIVITY Pedotransfer functions are regression equations that are used to predict difficult-to-obtain parameters from more easily measured soil properties.5 They have been widely used to predict input parameters for soil hydrological models from basic soil physical properties such as particle-size distribution and bulk density (e.g., Rawls et al.6). Recently, Xu and Eckstein,7 Gonçalves et al.,8 and Perfect et al.9 have applied this approach to estimate the input parameters for solute transport models. Xu and Eckstein,7 working with mixtures of glass beads, showed that λ was positively correlated (r = 0.84**) with the uniformity coefficient describing the mass-size distribution of glass beads. A limitation of this study was the choice of characteristics describing the solid phase as independent variables when solute dispersion is more directly related to pore space geometry.1 Furthermore, packed beds of glass beads do not simulate the heterogeneity of natural porous media. Using multiple regression and neural network analyses, Gonçalves et al.8 investigated relationships between solute transport parameters for the two-region (nonequilibrium) form of the ADE and a suite of predictor variables measured on 24 undisturbed soil columns. A pedotransfer function was developed for predicting the log of the dispersion coefficient based on soil organic matter content, saturated hydraulic conductivity and parameters describing the water retention curve; the coefficient of determination (R2) for this function was 0.90. However, this function is not very useful because it does not account for variations in v. Most of the variability in D observed by Gonçalves et al.8 can be accounted for by differences in v (see their Table 2). In fact, reanalysis of the data in Tables 1 and 3 of Gonçalves et al.,8 taking into account variations in v through the use of Equation 6.1, resulted in no significant correlations (at p < 0.05) between λ and any of their potential predictor variables. This result may be related to the relatively small number of columns employed in this study. Perfect et al.,9 also working with undisturbed samples, investigated relations between λ and soil hydraulic properties based on 69 soil columns. The soil water content at saturation, saturated hydraulic conductivity and Campbell10 water retention parameters, ψa and b, were chosen as the independent variables. This decision was based on: 1. The transport and retention of water in soil is directly related to void characteristics. 2. Soil hydraulic properties are more frequently measured than void characteristics. 3. Databases of soil hydraulic properties are widely available. A pedotransfer function was derived by stepwise linear multiple regression analysis. This function explained ~50% of the variability in λ and indicated that λ increased as ψa and b increased.9

III. SCALE DEPENDENCY OF DISPERSIVITY Predictions of λ made using the pedotransfer function of Perfect et al.9 are only valid for 6-cm long soil columns. To be useful this function should also be able to predict the dispersivities of differently sized samples. However, it is well known that inverse estimates of λ tend to increase as the volume of soil or aquifer material sampled in a solute transport experiment increases (e.g., Pickens and Grisak11, Khan and Jury12). Neuman13 statistically analyzed 131 longitudinal dispersivities from laboratory and field tracer studies conducted in porous and fractured media at scales ranging from 100 km. Regression analysis showed that although the data were widely scattered, a log–log relationship between dispersivity and apparent length scale, la, was able to account for 74% of the total variation in λ. This relationship can be expressed as: λ = 0.0175 l a


1.46

(6.2)


A Pedotransfer Function for Predicting Solute Dispersivity: Model Testing and Upscaling

91

Equation 6.2 holds for la < 3500 m. Similar relationships between λ and scale have been reported by Gelhar et al.14 and Xu and Eckstein.15 The reasons for this observed scale dependency are unclear, although several studies (e.g., Dagan,16 Zhan and Wheatcraft17) have related increases in λ to increasing spatial variability of soil hydraulic properties with increasing scale. Because the magnitude of λ depends upon the measurement scale, methods are needed for upscaling the pedotransfer function predictions of Perfect et al.9

IV. UPSCALING PROCEDURES Upscaling is a mathematical procedure whereby effective field-scale transport parameters are derived from data collected at smaller spatial scales.18 Several different approaches have been applied to the problem of upscaling λ; these include simple spatial averaging,19 real space renormalization,20,21 the fractional advection dispersion equation or FADE,22,23 and fractal geometry, which is currently the most widely accepted method. Fractals are based upon the idea that a spatial pattern observed at one scale is repeated at other scales.24,25 Using this approach, several authors have derived scale-dependent expressions for λ (e.g., Neuman,13,26 Zhan and Wheatcraft,17 Kemblowski and Wen,27 Massan et al.28). In the dispersion model of Neuman13,26 the spatial distribution of log saturated hydraulic conductivity is assumed to be selfsimilar with homogeneous increments characterized by the fractal dimension D = E + 1 – H, where E is the Euclidean dimension of interest and H is the Hurst coefficient. Under these conditions the exponent in Equation 6.2 is equivalent to 1 + 2H, and thus can be used to calculate the fractal dimension of the underlying conductivity field.

V. DATA SETS, METHODS, AND MODELS The major objectives of this study were: (1) to verify estimates of λ predicted using the pedotransfer function of Perfect et al.9 against independent data, and (2) to develop a method for upscaling the predictions. To facilitate these goals, data from Schwartz29 and Vervoort et al.,30 and the fractal upscaling model of Neuman were used.13,26 Perfect et al.9 used stepwise multiple linear regression analysis to establish the following pedotransfer function relating dispersivity to the air-entry value (ψa) and exponent (b) in the Campbell10 water retention model: λ = –2.91 + 0.23ψa + 1.27b

(6.3)

where λ is in cm, ψa is in kPa and b is dimensionless. They worked with 6-cm long by 5.35-cm diameter undisturbed soil columns from six soil types ranging in texture from loamy sand to silty clay. Equation 6.3 is based on data for 69 soil columns; the root mean square error and R2 values were 4.11 cm and 0.47, respectively. The standard errors associated with the parameter estimates were 1.40 for the intercept, 0.04 for ψa, and 0.21 for b. Schwartz29 measured λ under saturated conditions and water retention curves on seven undisturbed soil columns from a fine-textured Ultisol. Water retention data were obtained close to saturation for each column. In addition, pressure plate measurements were performed on undisturbed subsamples extracted from the columns after the miscible displacement experiments were concluded. Readers are referred to the original reference for details on the experimental techniques employed. The Campbell10 model was fitted to the pooled water retention data of Schwartz29 using segmented nonlinear regression analysis.31 The resulting parameter estimates are given in Table 6.1 along with the measured values of λ and column dimensions. Vervoort et al.30 measured λ under saturated conditions on seven undisturbed columns from the Ap/Bt1, Bt2, Bt3, and BC1 horizons of Esto and Faceville sandy loam soils. Water retention curves were measured on replicated undisturbed cores from the same horizons. The experimental © 2003 by CRC Press LLC


92


TABLE 6.1 Column Lengths, Solute Dispersivities, and Corresponding Campbell10 Water Retention Parameters Used for Model Verification and Upscaling Column ID

Length (cm)

λ (cm)

ψa (kPa)

b

Column #1a Column #2a Column #3a Column #5a Column #6a Column #8a Column #14a Esto Ap/Bt1b Esto Bt2b Esto Bt3b Esto BC1b Faceville Ap/Bt1b Faceville Bt2b Faceville BC1b

15 15 20 15 20 15 15 30 30 30 30 30 30 30

13.7 19.6 73.0 40.9 61.5 177.2 77.7 5.8 14.8 45.5 80.6 3.8 16.5 47.9

3.3 × 10–2 8.0 × 10–3 4.0 × 10–3 3.4 × 10–2 5.3 × 10–2 1.3 × 10–1 1.0 × 10–2 5.8 × 100 4.7 × 100 5.7 × 100 1.3 × 101 6.4 × 100 1.9 × 101 1.2 × 101

11.7 16.5 19.2 15.7 18.8 20.4 18.9 2.0 4.6 5.6 5.9 1.0 4.0 5.3

a b

Schwartz, R.C., Ph.D. dissertation, Texas A&M University, College Station, 1998. Vervoort, R.W. et al.,Water Resour. Res., 35, 913, 1999.

techniques employed are described in the original reference. Vervoort et al.30 fitted their water retention data to the van Genuchten32 equation; estimates of the model parameters (α, n, θs, and θr) were then averaged. Assuming θr = 0, I calculated equivalent average parameters for the Campbell10 equation using the following relations: ψa = 1/α and b = 1/(n – 1).6 The resulting values of λ, ψa, and b are listed in Table 6.1 along with the column dimensions. The estimates of ψa and b given in Table 6.1 were used in Equation 6.3 to predict a λ value for each of the 14 soil columns. The columns used by Schwartz29 and Vervoort et al.30 were longer than those used by Perfect et al.9 to develop Equation 6.3. Because λ is known to depend upon measurement scale, the pedotransfer function predictions were upscaled using the fractal expression proposed by Neuman.13,26 After some manipulation, Equation 6.2 can be rewritten as: l  λ u = λ p u  l  p

1.46

(6.4)

where λu is the upscaled dispersivity, λp is the dispersivity predicted using Equation 6.3, lu is the length of the columns in the transport experiments performed by Schwartz29 and Vervoort et al.,30 and lp is the length of the columns used by Perfect9 to develop Equation 6.3. The raw and upscaled predictions were then compared with the measured values of λ using linear regression analysis. Equations 6.3 and 6.4 were also used to predict λ as a function of soil textural class and column length based on the averaged Campbell10 model parameters reported by Cosby et al.33

VI. COMPARISON OF PREDICTED AND OBSERVED DISPERSIVITIES Measured values of λ for the 14 soil columns ranged from 3.8 to 177.2 cm (Table 6.1), while the raw predictions of λ based on Equation 6.3 ranged from –0.1 to 23.1 cm (Figure 6.1). Negative values of λ are not physically possible; the one negative value encountered here can be explained © 2003 by CRC Press LLC



93

190 raw predictions

Predicted λ (cm)

upscaled predictions

140

90

1:1 line

points excluded from fits

y = 1.01x + 24.85 R 2 = 0.62

40 y = 0.17x + 4.36 2 R = 0.36

-10 -10

40

90

140

190

Observed λ (cm) FIGURE 6.1 Raw and upscaled predicted dispersivities as related to the measured values from Schwartz, R.C., Ph.D. dissertation, Texas A&M University, College Station, 1998, and Vervoort, R.W. et al.,Water Resour. Res., 35, 913, 1999.

by the intercept in Equation 6.3, which results in negative predictions of λ when ψa and b are both small. Because of the relatively narrow range in predicted values, there was only a weak positive relationship between the observed dispersivities and raw predictions, even when the data point for column 8 of Schwartz,29 which had an exceptionally large dispersivity, was excluded from the analysis (Figure 6.1). The resulting regression equation deviated significantly from a 1:1 line at p < 0.05, indicating that the raw predicted values of λ consistently underestimated the measured dispersivities. The upscaled predictions of λ ranged from –1.3 to 124.9 cm, and were positively correlated with the observed dispersivities (Figure 6.1). Regression analysis explained 62% of the variation in observed and upscaled λ values when the data point for column 8 of Schwartz29 was ignored. The root mean square error for this analysis was 23.3 cm. The regression equation was close to a 1:1 line. However, an F-test for the joint hypothesis that the intercept equaled zero and the slope equaled one31 indicated the regression equation was significantly different from a 1:1 relation at p < 0.05. The upscaling procedure overestimated the observed dispersivities (Figure 6.1). Despite this tendency, Equations 6.3 and 6.4 still represent an improvement over existing approaches for predicting λ. For example, longitudinal dispersivities are sometimes estimated by multiplying the flow length of a transport experiment by 0.1 (e.g., Fetter2). Applying this approach to the data in Table 6.1 yielded estimates of λ of between 1.5 and 3.0 cm for the columns used by Schwartz29 and Vervoort et al.30 Clearly, these values are much less accurate than the upscaled predictions in Figure 6.1. This is because Equations 6.3 and 6.4 take into account differences in soil hydraulic properties and flow length, whereas the 0.1λ rule of thumb is based exclusively on the scale effect.

VII. EXAMPLE APPLICATION AND DISCUSSION In an example application, Equations 6.3 and 6.4 were used to predict λ from the soil water retention database of Cosby et al.33 These authors presented mean values of ψa and b for 11 soil textural classes ranging from sand to clay. Their data are for 4.6-cm long soil columns. The predicted dispersivities are shown in Figure 6.2; given Figure 6.1 the actual values of λ for these soils can be expected to be slightly lower than the predicted values. There is a clear trend toward increasing © 2003 by CRC Press LLC


94


300

250

200

Dispersivity 150 (cm) 100

50 50 40 Column 30 20 length (cm) 10 Sand

Loamy sand

Sandy loam

Loam

Silt loam

Sandy clay loam

Clay loam

Silty clay loam

Sandy clay

Silty clay

Clay

0

Textural Class

FIGURE 8.2 Dispersivities predicted from the data of Cosby, B.J. et al., Water Resour. Res., 20, 682, 1984, as a function of soil textural class and column length.

dispersivity with increasing clay content, and this trend becomes more pronounced as the column length increases. The increase in λ with increasing clay content is probably a soil structural effect. Equation 6.3 was developed using undisturbed columns; in this condition fine-textured soils tend to have a much wider range of pore sizes than coarse-textured ones. It should be noted that Equation 6.3 may not hold for disturbed or remolded samples in which differences in pore-size distribution due to texture have been removed. The use of pedotransfer functions has been widely investigated in the soils literature, yet relatively little work has been done to assess their usefulness for predicting input parameters for solute transport models such as the ADE. This is somewhat surprising because the potential advantages of such an approach are clear. Soil water retention databases are widely available, while miscible displacement experiments, conducted on a case-by-case basis, are tedious and time consuming. The present study demonstrates that water retention curves determined in the laboratory, coupled with a simple fractal upscaling technique, can be used to predict dispersivities over a range of scales up to ~1 m. It is assumed that parameters describing the water retention curve are scale invariant. Further research along these lines may ultimately reduce our reliance on solute transport experiments and inverse procedures.

VIII. CONCLUSIONS The pedotransfer function developed by Perfect et al.9 was verified for predicting longitudinal dispersivity from the Campbell10 water retention parameters. The function was tested against an independent data set derived from Schwartz29 and Vervoort et al.30 The raw predictions (applicable to 6-cm long columns) were upscaled to the dimensions of the columns used by Schwartz29 and Vervoort et al.30 with the fractal model of Neuman.13,26 For dispersivities up to ~1 m, the upscaled predictions were positively related to the measured values (R2 = 0.62**). However, there was a © 2003 by CRC Press LLC



95

significant trend towards overestimation. More accurate pedotransfer functions could be developed by collecting additional paired miscible displacement and water retention data sets over a wider range of soil types. In an example application the pedotransfer function and fractal upscaling equation was used to predict λ as a function of column length for the 11 textural classes in the soil water retention database of Cosby et al.33 The predicted dispersivities increased, moving from coarse- to finetextured soils, and with increasing column length. In the absence of more detailed information for a particular soil, such an approach might be used to estimate the dispersivity parameter required for modeling solute transport using the ADE.

REFERENCES 1. Perfect, E. and Sukop, M.C., Models relating solute dispersion to pore space geometry in saturated media: a review. In: Physical and Chemical Processes of Water and Solute Transport/Retention in Soils, Special Publ. 56, H.M. Selim and D.L. Sparks (Eds.), Soil Sci. Soc. Am., Madison, WI, 77, 2001. 2. Fetter, C.W., Contaminant Hydrogeology. 2nd Ed., Prentice Hall Inc., Upper Saddle River, NJ, 1999. 3. Zheng, C. and Bennett, G.D., Applied Contaminant Transport Modeling. Van Nostrand Reinhold, New York, 1995. 4. van Genuchten, M.T. and Wierenga, P.J., Solute dispersion coefficients and retardation factors. In: Methods of Soil Analysis. Part 1. Physical and Mineralogical Methods. 2nd ed., Agron. Monogr. 9, A. Klute (Ed.), ASA/SSSA, Madison, WI, 1986, 1025–1054. 5. Bouma, J., Using soil survey data for quantitative land evaluation. Adv. Soil Sci., 9, 177, 1989. 6. Rawls, W.J., Gish, T.J., and Brakensiek, D.L., Estimating soil water retention from soil physical properties and characteristics. Adv. Agron., 16, 213, 1991. 7. Xu, M. and Eckstein, Y., Statistical analysis of the relationships between dispersivity and other physical properties of porous media. Hydrogeology J., 5, 4, 1997. 8. Gonçalves, M.C., Leij, F.J., and Schaap, M.G., Pedotransfer functions for solute transport parameters of Portuguese soils. Eur. J. Soil Sci., 52, 563, 2001. 9. Perfect, E., Sukop, M.C., and Haszler, G.R., Prediction of dispersivity for undisturbed soil columns from water retention parameters. Soil Sci. Soc. Am. J., 66, 696, 2002. 10. Campbell, G.S., A simple method for determining unsaturated conductivity from moisture retention data. Soil Sci., 117, 311, 1974. 11. Pickens, J.F. and Grisak, G.E., Scale-dependent dispersion in a stratified granular aquifer. Water Resour. Res., 17, 1191, 1981. 12. Khan, A.U.-H. and Jury, W.-A., A laboratory study of the dispersion scale effect in column outflow experiments. J. Contaminant Hydrol., 5, 119, 1990. 13. Neuman, S.P., Universal scaling of hydraulic conductivities and dispersivities in geologic media. Water Resour. Res., 26, 1749, 1990. 14. Gelhar, L.W., Welty, C., and Rehfeldt, K.R., A critical review of data on field-scale dispersion in aquifers. Water Resour. Res., 28, 1955, 1992. 15. Xu, M. and Eckstein, Y., Use of weighted least-squares method in evaluation of the relationship between dispersivity and field scale. Ground Water, 33, 905, 1995. 16. Dagan, G., The significance of heterogeneity of evolving scales to transport in porous formations. Water Resour. Res., 30, 3327, 1994. 17. Zhan, H. and Wheatcraft, S.W, Macrodispersivity tensor for nonreactive solute transport in isotropic and anisotropic fractal porous media: analytical solutions. Water Resour. Res., 32, 3461, 1996. 18. Sposito, G. (Ed.), Scale Dependence and Scale Invariance in Hydrology. Cambridge University Press, Cambridge, UK, 1998. 19. Mishra, S. and Parker, J.C., Analysis of solute transport as a hyperbolic scale-dependent model. Hydrol. Process., 4, 45, 1990. 20. Morris, M.I. and Ball, R.C., Renormalization of miscible flow functions. J. Phys. A 23, 4199, 1990. 21. King, P.R, Muggeridge, A.H., and Price, W.G., Renormalization calculations of immiscible flow. Transp. Por. Media, 12, 237, 1993.



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Scaling Methods in Soil Physics 22. Benson, D.A., Wheatcraft, S.W., and Meerschaert, M.M., Application of a fractional advectiondispersion equation. Water Resour. Res., 36, 1403, 2000. 23. Pachepsky, Y., Timlin, C., and Benson, D.A., Transport of water and solutes in soil as in fractal porous media. In: Physical and Chemical Processes of Water and Solute Transport/Retention in Soils, Special Publ. No. 56, Selim, H.M. and Sparks, D.L. (Eds.), Soil Sci. Soc. Am., Madison, WI, 2001, 51–75. 24. Korvin, G., Fractal Models in the Earth Sciences. Elsevier, Amsterdam, The Netherlands, 1992. 25. Turcotte, D.L., Fractals and Chaos in Geology and Geophysics. 2nd ed., Cambridge University Press, Cambridge, UK, 1997. 26. Neuman, S.P., On advective transport in fractal permeability and velocity fields. Water Resour. Res., 31, 1455,1995. 27. Kemblowski, M.W. and Wen, J.-C., Contaminant spreading in stratified soils with fractal permeability distribution. Water Resour. Res., 29, 419, 1993. 28. Hassan, A.E., Cushman, J.H., and Delleur, J.W., Monte Carlo studies of flow and transport in fractal conductivity fields: comparison with stochastic perturbation theory. Water Resour. Res., 33, 2519, 1997. 29. Schwartz, R.C., Reactive transport of tracers in a fine textured Ultisol. Ph.D. dissertation, Texas A&M University, College Station, 1998. 30. Vervoort, R.W., Radcliffe, D.E., and West, L.T., Soil structure development and preferential solute flow. Water Resour. Res., 35, 913, 1999. 31. SAS Institute Inc., SAS/STAT User’s Guide, Version 6, 4th ed., Vols. 1 and 2, Cary, NC, 1990. 32. van Genuchten, M.T., A closed form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J., 44, 892, 1980. 33. Cosby, B.J., Hornberger, G.M., Clapp, R.B., and Ginn, T.R., A statistical exploration of the relationships of soil moisture characteristics to the physical properties of soils. Water Resour. Res., 20, 682, 1984.



7

Upscaling of Hydraulic Properties of Heterogeneous Soils J. Zhu and B.P. Mohanty

CONTENTS I. Introduction.............................................................................................................................97 II. Hydraulic Property Models ....................................................................................................98 A. Gardner-Russo Model..................................................................................................98 B. Brooks-Corey Model ...................................................................................................99 C. Van Genuchten Model .................................................................................................99 III. Steady State Flow at Local Scale ........................................................................................100 IV. Spatial Variability of Hydraulic Parameters and Its Influence on Flux Rate......................102 V. Parameter Averaging Schemes .............................................................................................106 VI. Validity of Stream-Tube Flow Assumption .........................................................................111 VII. Summary...............................................................................................................................114 References ......................................................................................................................................115

I. INTRODUCTION Simulations of variably-saturated flow and solute transport in soil typically use closed-form functional relationships to represent water-retention characteristics and unsaturated hydraulic conductivities. The Gardner1 and Russo2 exponential model, Brooks and Corey piecewise continuous model3 and van Genuchten model4 represent some of the most widely used and practical hydraulic property models. These parameter models are valid at point or local scale based on a point-scale hydrologic process. When these models are used in larger (plot, field, watershed or regional) scale processes, major questions arise about how to average hydrologic processes over a heterogeneous soil volume5–10 and what averages of hydraulic property shape parameters to use for these models.11,12 Smith and Diekkruger13 studied one-dimensional vertical flows through spatially heterogeneous areas and treated the soil heterogeneity using the distributions of parameters describing the soil characteristic relationships. Their results demonstrated that hydraulic characteristics measured from a heterogeneous sample could not be used to describe unsteady flow through that sample. They treated the random variation in soil characteristic parameters as independent of each other. Green et al.12 investigated methods for determining the upscaled water retention characteristics of stratified soil formations using the van Genuchten model for soil hydraulic properties. They compared between linear volume average (LVA) and direct parameter average for an upscaled water retention curve of periodically layered soils. Chen et al.14,15 developed the spatially horizontally averaged Richards equation model for the mean water saturation in each horizontal soil layer and the cross-



98


covariance of the saturated hydraulic conductivity and the water saturation in each horizontal soil layer in a heterogeneous field. Their approach is restricted to the uncertainty from spatial variability in the saturated hydraulic conductivity. Govindaraju et al.10 studied field-scale infiltration over heterogeneous soils. They considered the spatial variability of saturated hydraulic conductivity, which is represented by a homogeneous correlated lognormal random field. Kim et al.16 investigated the significance of soil hydraulic heterogeneity on the water budget of the unsaturated zone using the Brooks and Corey model, based on a framework of approximate analytical solutions. In their work, the geometrical scaling theory was assumed appropriate and the air entry value (1/α) was assumed to be deterministic. For a majority of these previous studies, it was typically assumed that the flow is virtually vertical. One-dimensional vertical flow at measurement scale is a practical assumption because the gradients would be very small in the horizontal direction for flows from wetting at the surface. In this chapter, we discuss the fundamental principles of hydraulic property upscaling based on some of our recent work.17–20 We investigate several hydraulic parameter averaging schemes and their appropriateness in describing the ensemble behavior of heterogeneous formations. Our main objective is to determine how well the commonly used averaging schemes perform in simulating the average hydrologic behavior of heterogeneous soils when compared with the effective parameters and suggest some practical guidelines of the conditions when we could use one of the averaging schemes in lieu of the “effective parameters.” The calculated effective parameters are the parameters that will discharge approximately the same ensemble-mean flux as the heterogeneous soil. We consider the influence of parameter correlation on upscaled effective parameters. Three widely used hydraulic conductivity models were employed, i.e., the Gardner exponential model, the Brooks and Corey piecewise-continuous model and the van Genuchten model. The impact of parameter correlation, boundary condition (surface pressure head) and elevation above water table on effective saturated hydraulic conductivity Ks and shape parameter α are examined and discussed here.

II. HYDRAULIC PROPERTY MODELS Soil hydraulic behavior is characterized by the soil water retention curve, which defines the water content (θ) as a function of the capillary pressure head (ψ), and the hydraulic conductivity function, which establishes relationship between the hydraulic conductivity (K) and water content or capillary pressure head. Simulations of unsaturated flow and solute transport typically use closed-form functions to represent water-retention characteristics and unsaturated hydraulic conductivities. Some of the commonly used functional relationships include the Gardner-Russo model,1,2 the BrooksCorey model3 and the van Genuchten model.4 A brief review of these models is given below. Interested readers are refered to Leij et al.21 for more comprehensive review and discussion on various closed-form expressions of hydraulic properties, including the models given below.

A. GARDNER-RUSSO

MODEL

The unsaturated hydraulic conductivity (K)-capillary pressure head (ψ) and the reduced water content (Se)-capillary pressure (ψ) are assumed to be represented by the Gardner model:1,2

[

]

Se ( ψ ) = e −0.5αψ (1 + 0.5αψ ) K = K s e −αψ

2 ( l+2 )

(7.1) (7.2)

where Ks is the saturated hydraulic conductivity, α is known as the pore-size distribution parameter, l is a parameter that accounts for the dependence of the tortuosity and the various correlation © 2003 by CRC Press LLC


Upscaling of Hydraulic Properties of Heterogeneous Soils

99

factors on the water content estimated to be about 0.5 as an average for many soils,22 Se = (θ – θr)/(θs – θr) is the effective, dimensionless reduced water content, θ is the total volumetric water content and θs and θr are the saturated and residual (irreducible) water contents, respectively.

B. BROOKS-COREY MODEL Brooks and Corey3 established the relationship between K and ψ using the following empirical equations from analysis of a large database, Se ( ψ ) = (αψ )

−λ

Se ( ψ ) =1 K ( ψ ) = K s (αψ )

−β

K (ψ ) = Ks

when αψ > 1

(7.3a)

when αψ ≤ 1

(7.3b)

when αψ > 1

(7.4a)

when αψ ≤ 1

(7.4b)

where β = λ(l + 2) + 2 . λ is a parameter used by Brooks and Corey to define the relationship between water content and ψ affecting the slope of the retention function. This model has been successfully used to describe retention data for relatively homogeneous and isotropic samples. The model may not describe the data well near saturation, where a discontinuity occurs at ψ = 1/α.

C. VAN GENUCHTEN MODEL Van Genuchten4 identified an S-shaped function that fits measured water-retention characteristics of many type of soils very well. The function was also combined with Mualem’s hydraulic conductivity function22 to predict unsaturated hydraulic conductivity. Subsequently, the van Genuchten function has become one of the most widely used curves for characterizing soil hydraulic properties. The van Genuchten equation of soil water retention curve can be expressed as follows, Se ( ψ ) =

1

[

1 + (αψ )

(7.5)

]

n m

where ψ, n and m are parameters which determine the shape of the soil water retention curve. Assuming m = 1 –1 /n, van Genuchten4 combined above soil water retention function with the theoretical pore-size distribution model of Mualem22 and obtained the following relationships for the hydraulic conductivity in terms of the reduced water content or the capillary pressure head:

[

(

K ( Se ) = K s Sel 1 − 1 − Se1/ m

[

)

K s 1 − (αψ ) 1 + (αψ ) K (ψ ) =  n ml 1 + (αψ ) mn

[


]

]

m 2

]

n −m

(7.6)   

2

(7.7)


100


TABLE 7.1 Statistics of Ks, α and n for Different Textural Classesa Class Sand Loamy sand Loam Sandy loam Silt loam Sandy clay loam Silty clay loam Clay loam Silt Clay Sand clay Silty clay a

Nα

&n

308 205 249 481 332 181 89 150 6 92 12 29

α) log(α log(cm–1)

Log(n)

NKs

log(Ks) log (cm/day–1)

–1.45(0.25) –1.46(0.47) –1.95(0.73) –1.57(0.45) –2.30(0.57) –1.68(0.71) –2.08(0.59) –1.80(0.69) –2.18(0.30) –1.82(0.68) –1.48(0.57) –1.79(0.64)

0.50(0.18) 0.24(0.16) 0.17(0.13) 0.16(0.11) 0.22(0.14) 0.12(0.12) 0.18(0.13) 0.15(0.12) 0.22(0.13) 0.10(0.07) 0.08(0.06) 0.12(0.10)

253 167 113 314 135 135 40 62 3 60 10 14

2.81(0.59) 2.02(0.64) 1.08(0.92) 1.58(0.66) 1.26(0.74) 1.12(0.85) 1.05(0.76) 0.91(1.09) 1.64(0.27) 1.17(0.92) 1.06(0.89) 0.98(0.57)

Standard deviations are given in parentheses. (Extracted from Schaap, M.G. and Leij, F.J., Soil Sci., 163, 765, 1998.)

There are then four parameters for the Gardner-Russo model to describe the soil hydraulic characteristics of each sample: Ks, α, θs and θr; five parameters for Brooks-Corey model: Ks, α, λ,θs and θr; and five parameters for van Genuchten model: Ks, α, n, θs and θr. In this study, we will consider the spatial variability introduced by the spatial variation of the parameters Ks and α for Gardner-Russo and Brooks-Corey models and the spatial variation of the parameters Ks, α and n for van Genuchten model. Table 7.1 lists the average and the standard deviation values of Ks, α, and n for van Genuchten function for different USDA textural classes.23 There are some conflicting reports about the correlation between the hydraulic parameters of soil in the literature. For example, after analyzing soil samples gathered in the Krummbach and Eisenbach catchments in northern Germany and from a field experiment near Las Cruces, New Mexico, Smith and Diekkruger13 concluded that no significant correlation was observed among any of the characteristic parameters and suggested that most random variation in soil characteristic parameters could be treated as independent. However, in another study Wang and Narasimhan24 indicated that Ks and α were correlated with Ks ∝ α2. We discuss both correlated and independent cases and the significance of their correlation on the ensemble behavior of soil dynamic characteristic of unsaturated flow.

III. STEADY STATE FLOW AT LOCAL SCALE General equation relating capillary pressure head and elevation above the water table for steady state vertical flows can be expressed as25,26 ψ

z=

K ( ψ ) dψ

∫ K (ψ ) + q

(7.8)

0

where ψ is the capillary pressure head, z is the vertical distance (positive upward) with the water table location being at z = 0 and q is the steady state evaporation (positive) or infiltration (negative) rate. Its dimensionless form can be expressed as © 2003 by CRC Press LLC


Upscaling of Hydraulic Properties of Heterogeneous Soils αψ

αz =

101

K r ( x )dx

∫ K ( x) + q′ 0

(7.9)

r

where the dimensionless hydraulic conductivity Kr = K/Ks, the dimensionless pressure head x = αψ, and the dimensionless flux rate q′ = q/Ks. When the pressure head at the surface, ψL, is known, the dimensionless state steady flux q/Ks can be found out from the following equation αψ L

αL =

∫ 0

K r ( x )dx Kr ( x ) + q ′

(7.10)

where L is the elevation of the ground surface above the water table. From Equation 7.10, it can be seen that the dimensionless steady state flux rate q′ itself is not related to the saturated hydraulic conductivity Ks. In other words, the flux rate q is a linear function of Ks. In turn, we can infer from Equation 7.9 that the capillary pressure head, ψ, is not related to the saturated hydraulic conductivity, Ks. When the Gardner-Russo hydraulic conductivity model is used, the capillary pressure profile (ψ) and the dimensionless flux rate (q′ = q/Ks) can be analytically expressed as   e αL − 1 ψ = ln   e − αz + e − αψ L − e − αL − e − α ( z + ψ L )  q′ =

1α

−L

−α ψ − L 1− e ( L ) e αL − 1

(7.11)

(7.12)

For the Brooks-Corey model, analytical solutions are also possible. But the evaporation and infiltration cases need to be analyzed separately. The capillary pressure head (ψ) can be related to the elevation above the water table (z) as the following series relationship for steady state evaporation:27 αz =

1  1 1 1 q′  βq ′ ; (q ′)−1 β • Bu  , 1 −  −  2 • 2 F1 1, 2; 2 + β β  (1 + β)(1 + q ′) β 1 + q′  β 

(7.13)

where Bu is the incomplete Beta function with u = q′αβψβ/(1 + q′αβψβ) and 2F1 is the Gaussian hypergeometric function. The relationship between the dimensionless evaporation rate q′ and the surface pressure head ψL can be established iteratively by the following equation: αL =

 1 1 1 1 q′  βq ′ ; (q ′)−1 β • BuL  , 1 −  −  2 • 2 F1 1, 2; 2 + β β  (1 + β)(1 + q ′) β 1 + q′   β

(7.14)

where

(

u L = q ′α β ψ βL 1 + q ′α β ψ βL

)

while for steady state infiltration, the relationship can be established as following17 with p′ = -q′. © 2003 by CRC Press LLC


102


  1 1 1 βp ′ 1  αz = αψ • 2 F1  , 1;1 + ; p ′α β ψ β  + • F 1, ; 2 + ; p ′ β β   (1 − p ′)(1 + β) 2 1  β β

(7.15)

The relationship between the dimensionless infiltration rate p′ and the surface pressure head ψL can be established iteratively by the following equation:  1  1 1 βp ′ 1  αL = αψ • 2 F1  , 1;1 + ; p ′α β ψ βL  + • 2 F1 1, ; 2 + ; p ′ β β   (1 − p ′)(1 + β) β  β

(7.16)

For van Genuchten model, the integrations in Equations 7.9 and 7.10 were carried out numerically.

IV. SPATIAL VARIABILITY OF HYDRAULIC PARAMETERS AND ITS INFLUENCE ON FLUX RATE Parameters Ks and α could be satisfactorily fit by lognormal distribution.28 Both Ks and α were assumed to obey the log normal distribution. Because the van Genuchten parameter n has to be greater than 1, we assume (n – 1) rather than n to be lognormally distributed to ensure n > 1 in considering the spatial variability of parameter n. The cross-correlated random fields of the parameter Ks, α and n – 1 were generated using the spectral method proposed by Robin et al.29 The random fields were produced with the power spectral density function, which was based on the exponentially decay covariance functions. The coherency spectrum defined as follows is an indicator of parameter correlation: R(f ) =

[φ

φ12 (f )

11 ( f )φ 22 ( f )]

12

(7.17)

where φ11(f), φ22(f) are the power spectra of random fields log(Ks) and log(α) or log(Ks) and log(n – 1) respectively. φ12(f) is the cross spectrum between log(Ks) and log(α) or log(Ks) and log(n – 1). When |R|2 = 1, it indicates perfect linear correlation between the random fields. The random fields are assumed to be isotropic with domain length being equal to 10 correlation length that in turn corresponds to 50 grid length. A random field of 2500 (50 × 50) values has been generated for log(Ks), log(α) or log(n – 1) field. We investigate two main themes of hydraulic parameter spatial variability in calculating dynamic flow characteristics in heterogeneous unsaturated soil: (1) variable saturated hydraulic conductivity Ks and variable van Genuchten parameter α with constant van Genuchten parameter n and (2) variable saturated hydraulic conductivity Ks and variable n with constant α. For each theme, we consider three hydraulic parameter averaging schemes and compare them with the effective parameters calculated according to the ensemble flux behavior, i.e., mean behavior of flow dynamics. The hydraulic parameter averaging schemes considered are: (1) arithmetic means for both spatial variables, (2) arithmetic mean for Ks, and geometric mean for α or n; and (3) arithmetic mean for Ks, and harmonic mean for α or n. From the nature of areally heterogeneous parallel stream-tube type of flow that we consider in this study, the arithmetic average (mean) for the saturated hydraulic conductivity can be considered as an appropriate averaging scheme.12 In the numerical experiments demonstrating the results for dynamic flow characteristics in the following, a water table depth of 180 cm has been used. In Figure 7.1 we plot a few representative images of randomly generated van Genuchten parameter fields (log(Ks), log(α) and log(n – 1)) used in the simulations. The means and the standard deviations used in the random field generator are based on the values for the loam class from Table




103

a) 700 4.5 4 3.5

600

Y (m)

500

3 2.5 2

400

1.5 1 0.5 0

300 200

-0.5

100

log(Ks) 100

200

300

400

500

600

700

X (m)

b)

c)

700

-2.5

600

600

-3

500

500

700

Y (m)

Y (m)

-3.5 -4.5

300

300

-5

200

200

-5.5

|R|2=1.0

X (m)

X (m)

e)

700

500

500

Y (m)

600

400

300

300

200

200

100

100

100 200 300 400 500 600 700 X (m) |R|2=1.0

|R|2=0.1

700

600

400

-6.5 100 200 300 400 500 600 700

100 200 300 400 500 600 700

Y (m)

-6

100

100

d)

-4

400

400

100 200 300 400 500 600 700

X (m)

|R|2=0.1

-7

log(alpha)

-0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1 -1.1 -1.2 -1.3 -1.4

log(n-1)

FIGURE 7.1 Images of randomly generated van Genuchten parameter fields for loam used in the simulations. (a): log(Ks) with Ks in (cm/d); (b): log(α) with α in (1/cm), |R|2 = 1.0; (c): log(α) with α in (1/cm), |R|2 = 0.1; (d): log(n – 1), |R|2 = 1.0; (e): log(n – 1), |R|2 = 0.1.



104


Infiltration

Evaporation

a)

-3

700

700

600

600

500

500

Y (m)

Y (m)

|R|2=1.0

400

b) |R|2=1.0

400

300

300

200

200

100

100

-3.5 -4 -4.5

100 200 300 400 500 600 700

100 200 300 400 500 600 700

X (m)

-5

X (m)

-5.5 700

-6.5

600

600

-7

500

500

-8

log(q)

c)

Y (m)

-7.5

Y (m)

700

-6

400

400

300

300

200

200

100

100

|R|2=0.1 100 200 300 400 500 600 700

X (m)

d) 100 200 300 400 500 600 700

|R|2=0.1

X (m)

FIGURE 7.2 Calculated log(q) fields for steady state infiltration and evaporation when Ks and α are spatially variable fields. (a): infiltration with |R|2 = 1.0; (b): evaporation with |R|2 = 1.0; (c): infiltration with |R|2 = 0.1; (d): evaporation with |R|2 = 0.1.

7.1, assuming exponentially decayed covariance functions. Image (a) represents spatially variable log(Ks) with Ks in (cm/day). Image (b) plots log(α) with α in (1/cm) when it is fully correlated with log(Ks) field |R|2 = 1.0; while image (c) represents log(α) when |R|2 = 0.1. Similarly, image (d) is log(n – 1) when it is fully correlated with log(Ks) (i.e., |R|2 between log(Ks) and log(n – 1) is 1.0) and image (e) represents log(n – 1) when |R|2 = 0.1. As expected, when the two random fields are fully correlated, their images follow very similar patterns (compare (a) and (b) as well as (a) and (d)), while the other images (i.e., images (c) and (e)) for a much lower degree of correlations have little resemblance with log(Ks) image (i.e., image (a)). Figure 7.2 plots the corresponding log(q) fields for both steady state infiltration and evaporation calculated using input parameters as Figure 7.1 when Ks and α are assumed to be spatially variable fields. The images (a) and (c) are the results for infiltration flux rate, while the images (b) and (d) are for evaporation flux rate. The images (a) and (b) represent results when two spatially variable fields (log(Ks) and log(α)) are fully correlated, i.e., |R|2 = 1.0, while the images (c) and (d) are the results for |R|2 = 0.1. It can be observed that the infiltration flux rate is mainly dictated by the saturated hydraulic conductivity field, Ks. The infiltration flux rate is typically larger where Ks is larger (compare Figures 7.2(a) and (c) with Figure 7.1(a)). The evaporation flux rate is mainly dictated by α field. The evaporation flux rate is typically larger where α is smaller (compare Figure 7.2(b) with Figure 7.1(b) and Figure 7.2(d) with Figure 7.1(c)). For both infiltration and evaporation, the flux rate field is less variable when two random parameter fields are more correlated, i.e., the variation range © 2003 by CRC Press LLC



105

Evaporation

Infiltration

a)

b)

-3 -3.5 -4

700

700

600

600

500

500

Y (m)

Y (m)

|R|2=1.0

400

400

300

300

200

200

100

100

-4.5 -5

|R|2=1.0

100 200 300 400 500 600 700

100 200 300 400 500 600 700

X (m)

X (m)

-5.5 700

-6.5

600

600

-7

500

500

-7.5 -8

log(q)

c)

Y (m)

700

Y (m)

-6

400

400

300

300

200

200

100

100

|R|2=0.1

d)

100 200 300 400 500 600 700

100 200 300 400 500 600 700

X (m)

X (m)

|R|2=0.1

FIGURE 7.3 Calculated log(q) fields for both steady state infiltration and evaporation when Ks and n – 1 are spatially variable fields. (a): infiltration with |R|2 = 1.0; (b): evaporation with |R|2 = 1.0; (c): infiltration with |R|2 = 0.1; (d): evaporation with |R|2 = 0.1.

of log(q) is significantly smaller when |R|2 = 1.0 compared with |R|2 = 0.1 (compare Figure 7.2(a) with Figure 7.2(c) and Figure 7.2(b) with Figure 7.2(d)). The reduced variability of the flux rate field due to the correlation of Ks and α fields can be explained by considering the separate effects of Ks and α on the flux and the implication of the correlation between Ks and α. On one hand, a larger Ks would lead to a larger flux rate, while a larger α would result in a smaller rate on the other hand. A higher degree of correlation between Ks and α means that the values Ks and α in each cell would simultaneously be either high or low. The two opposite mechanisms neutralize each other and lead to a reduced flux rate variability across the cells seen in Figures 7.2(a) and (b). Figure 7.3 plots the corresponding log(q) fields for both steady state infiltration and evaporation calculated using input of parameters as Figure 7.1 when Ks and n – 1 are assumed to be spatially variable fields. The images (a) and (c) are the results for infiltration, while the images (b) and (d) are those for evaporation. The images (a) and (b) represent results when two variable fields are fully correlated, i.e., |R|2 = 1.0, while the images (c) and (d) are the results for |R|2 = 0.1. It can be seen that all the flux field images follow the pattern of Ks field with larger Ks resulting in a larger flux rate. The main reason for this phenomenon is that the variance of log(n) is quite small compared with that of Ks and its variability is not large enough to have a significant impact on the flux field. In practice, the parameter n can be determined with greater certainty than the other parameters involved in the van Genuchten model. In the study relating van Genuchten hydraulic property model, Hills et al.30 also demonstrated that the random variability in water retention



106


characteristics could be adequately modeled using a variable van Genuchten parameter α with a constant van Genuchten parameter n, or a variable n with a constant α, with better results when α was variable.

V. PARAMETER AVERAGING SCHEMES After generating the random fields for Ks and α, the ensemble characteristics of hydraulic properties can be calculated as follows: Se ( ψ ) = (1 10000)

10000

∑ S (ψ ) ei

(7.18)

i =1

10000

K ( ψ ) = (1 10000)

∑K

i

(7.19)

i =1

where 〈Se(ψ)〉 and 〈K(ψ)〉 are the ensemble reduced water content and ensemble hydraulic conductivity; Sei and Ki are the reduced water content and hydraulic conductivity based on the parameters Ksi and αi, respectively. Our approach is based on the ergodic hypothesis, which requires the field to be big enough. Figure 7.4 depicts the hydraulic conductivity as function of the capillary pressure head for Gardner-Russo model at three different values of coherency (correlation), including the comparison between their ensemble characteristics and that of a sample having the arithmetic or geometric mean value of the characteristic parameter α. As might be expected, it is not possible to characterize ensemble hydraulic property characteristics by effective parameters. However, it can be said that the geometric mean of α is a better indicator (i.e., closer to ensemble hydraulic property) of an effective parameter. Apparently, the effectiveness of average parameters improves as the degree of correlation between the parameters (|R|2) increases. The higher parameter correlation between Ks and α makes the ensemble hydraulic behavior more sand-like (i.e., the ensemble hydraulic conductivity curve is steeper). Figure 7.5 depicts a conceptual scheme of parameter correlation between Ks and α. Compared to the bottom arrangement, the top arrangement would represent a higher degree of correlation between Ks and α because, for each rectangular pixel, the values Ks and α would be either simultaneously high or low which, in turn, indicates a high degree of correlation. The top configuration, however, would exhibit ensemble of more sand-like textures characterized by a higher value of effective parameter α. The bottom arrangement (Figure 7.5(b)) indicates a less correlated configuration where the values Ks and α would show a lower degree of correlation than Figure 7.5(a). Its ensemble hydraulic behavior would be characterized by a lower value of effective parameter α. In reality, textural composition of a real field is more complicated than those shown in Figure 7.5. Figure 7.6 shows a soil texture map for a site in Las Nutrias, New Mexico. The results indicate that the correlation between Ks and α is an important factor in describing the effective hydraulic behavior of soil. Although the pixel-scale hydraulic property is generally estimated by area-weighted average or dominant soil type approach, the hydraulic parameter correlation also plays an important role in large area soil hydrologic behavior. Figure 7.7 shows the hydraulic conductivity as a function of the capillary pressure head for the Brooks-Corey model at three different values of coherency (correlation), including their ensemble characteristics. The hydraulic conductivity based on average parameters (geometric or arithmetic mean) is unable to catch the smooth-out effect of the ensemble hydraulic conductivity. For the Gardner-Russo model, the averaging scheme based on the geometric mean for α shows more promising results for the hydraulic conductivity than arithmetic average. © 2003 by CRC Press LLC


Hydraulic Conductivity (cm/s)




107

1.0E-04

a)

1.0E-05

2

R = 0.1

1.0E-06 1.0E-07 1.0E-08 1.0E-09 0.0

50.0

100.0

150.0

200.0

250.0

300.0

350.0

1.0E-04

b)

1.0E-05

2

R = 0.5

1.0E-06 1.0E-07 1.0E-08 1.0E-09 0.0

50.0

100.0

150.0

200.0

250.0

300.0

350.0

1.0E-04

c)

1.0E-05

2

R = 1.0

1.0E-06 1.0E-07 1.0E-08 1.0E-09 0.0

50.0

100.0

150.0

200.0

250.0

300.0

350.0

Capillary Pressure Head (-cm) ensemble hydraulic conductivity based on geometric mean alpha

based on arithmetic mean alpha

FIGURE 7.4 Hydraulic conductivity vs. capillary pressure head (static) for Gardner-Russo model.

Below we compare the resulting ensemble characteristics to that for mean values of the parameters and for the static ensemble soil characteristics. Specifically, four types of averaging schemes have been used in calculating dynamic characteristics of flow in unsaturated soil: (1) ensemble behavior, i.e., mean behavior of flow dynamics; (2) flow dynamics based on arithmetic means for the saturated hydraulic conductivity, Ks, and the pore-size distribution parameter, α; (3) flow dynamics based on arithmetic means for the saturated hydraulic conductivity, Ks, and geometric mean for the pore-size distribution parameter, α; and (4) flow dynamics based on the ensemble characteristic of unsaturated hydraulic conductivity. Based on stream-tube type vertical and heterogeneous flow that we consider in this study, the arithmetic average (mean) for the saturated hydraulic conductivity can be considered as an appropriate averaging scheme. © 2003 by CRC Press LLC


108


FIGURE 7.5 Schematic arrangement of soil samples with different degrees of parameter correlation between Ks and α: (a) higher degree; (b) lower degree.

FIGURE 7.6 Soil texture of a field in Las Nutrias, New Mexico. © 2003 by CRC Press LLC






109

1.0E-04

a)

2

R = 0.1

1.0E-05

1.0E-06

1.0E-07

1.0E-08 0.0

50.0

100.0

150.0

200.0

250.0

300.0

350.0

1.0E-04

b)

2

R = 0.5

1.0E-05

1.0E-06

1.0E-07

1.0E-08 0.0

50.0

100.0

150.0

200.0

250.0

300.0

350.0

1.0E-04

c)

2

R = 1.0

1.0E-05

1.0E-06

1.0E-07

1.0E-08 0.0

50.0

100.0

150.0

200.0

250.0

300.0

350.0

Capillary Pressure Head (-cm) ensemble hydraulic conductivity based on geometric mean alpha

based on arithmetic mean alpha

FIGURE 7.7 Hydraulic conductivity vs. capillary pressure head (static) for the Brooks-Corey model.

Figure 7.8 shows capillary pressure head profiles vs. distance above the water table for evaporation and infiltration cases for Gardner-Russo model for various coherencies. For evaporation case, the steady evaporation rate was given as one-fourth of the maximum possible value for each cell; while for infiltration case the steady infiltration rate was given as one-fourth its saturated hydraulic conductivity for each cell. The solid curve is the profile ψ ( z ) for the field. The two dashed curves are obtained from use of a single set of hydraulic curves defined by the mean values of each parameter, with the mean for α taking arithmetic and geometric averages respectively. The fourth curve is for results using the static ensemble hydraulic conductivity. For evaporation case, the capillary pressure head distribution is very sensitive to the value of the parameter α. © 2003 by CRC Press LLC



Capillary Pressure Head (-cm)



110

350.0

80.0

300.0

2

70.0

2

R = 0.1

R = 0.1

60.0

250.0

50.0

200.0

40.0 150.0 30.0 100.0

20.0

50.0

10.0

Evaporation 0.0 0.0

50.0

100.0

150.0

Infiltration 0.0 200.0 0.0

350.0

50.0

100.0

150.0

200.0

80.0 2

2

R = 0.5

70.0

R = 0.5

300.0

60.0

250.0

50.0

200.0

40.0 150.0 30.0 100.0

20.0

50.0

10.0

Evaporation 0.0 0.0

50.0

100.0

150.0


350.0

50.0

100.0

150.0

200.0

80.0 2

2

R = 1.0

70.0

R = 1.0

300.0

60.0

250.0

50.0

200.0

40.0 150.0

30.0

100.0

20.0

50.0

10.0

Evaporation 0.0 0.0

50.0

100.0

150.0

Infiltration

0.0 0.0 200.0

Distance above Water Table (cm)

ensemble pressure head based on geometric mean alpha

50.0

100.0

150.0

200.0

Distance above Water Table (cm) based on arithmetic mean alpha based on ensemble conductivity

FIGURE 7.8 Capillary pressure head vs. distance above water table. Left for evaporation given steady evaporation rate = 0.25qmax. Right for infiltration given steady infiltration rate = 0.25Ks for GardnerRusso model.

Overall, the geometric mean of α is found to be a more effective averaging scheme compared with the arithmetic mean of α. This is generally true in case of a log normal isotropic medium. In practice, the geometric mean can be considered as a good effective value in two dimensions.31 For a log normal distribution, the geometric mean is smaller than the arithmetic mean. Our result indicates, therefore, that an effective value (geometric mean) of α is smaller than the expected value (arithmetic mean). Although for evaporation the ensemble hydraulic conductivity is a good representation for the dynamic characteristics, its effectiveness is somewhat questionable for the infiltration case. It shows that ensemble profiles are represented better by mean parameters than




111

by the static ensemble hydraulic conductivity for infiltration. The heterogeneous convection due to the differences in soil hydraulic properties at large scale behaves similarly to hydrodynamic dispersion at small scale. The effective media generally underestimate this type of smearing effect due to heterogeneous convection. Consistent with the static case, the higher degree of correlation between the parameters results in a more sand-like behavior for the flow dynamics based on the ensemble hydraulic conductivity. In other words, higher degree of correlation between Ks and α results in a smaller capillary pressure and a larger reduced water content at the same elevation. Figure 7.9 shows capillary pressure head profiles vs. distance above the water table for evaporation and infiltration cases for Brooks-Corey model at different coherencies. Similar conclusions to Gardner-Russo model can be said for Brooks-Corey model. It should also be noted that, for the heterogeneous flow scenario considered, the static ensemble hydraulic conductivity is a better approximation of overall dynamic hydraulic conductivity for the evaporation than for the infiltration; the effective characteristics are conditional to flow conditions.

VI. VALIDITY OF STREAM-TUBE FLOW ASSUMPTION In the previous discussion, we made two major assumptions: (1) the hydraulic parameter heterogeneity is horizontal and (2) the flow is vertical. These two assumptions seem to be contradictory, since the assumption of horizontal heterogeneity would result in a different pressure profiles across different cells within a pixel which, in turn, would lead to horizontal flow induced by the pressure differential across different cells. The maximum ratio of horizontal flux rate over vertical flux rate can be defined as follows  q  Max  Ki, j ψ i +1, j − ψ i −1, j Ki, j ψ i, j +1 − ψ i, j −1 Max  h  = , i, j  qi, j 2 ∆x 2 ∆y  q qi, j 

   

(7.20)

where (∆x, ∆y) is the cell size, qi,j is the evaporation or infiltration rate of cell (i,j), and Ki,j is the unsaturated hydraulic conductivity of cell (i,j). Plotted in Figure 7.10 are the maximum ratios of horizontal flux rate over vertical flux rate of the entire field of 10,000 cells for some selected values of surface pressure heads when Ks and α are assumed to be the spatially variable fields. Figures 7.10(a), (c), (e) and (g) represent results when the two random fields are fully correlated (|R|2 = 1.0), while Figures 7.10(b), (d), (f) and (h) demonstrate the results for |R|2 = 0.1. The top four plots are the results for infiltration, while the bottom four figures are those for evaporation. The horizontal flux rate was calculated as the flow rate induced by the pressure differential between two adjacent neighboring cells and the hydraulic conductivity at a local depth. When Ks and α are spatially variable, the sand class always produces largest maximum ratio of horizontal over vertical flows. The maximum ratio typically appears not far from the water table. The location at which the ratio reaches the maximum, zmax, is related to the height of capillary fringe for each individual soil class. From the figures, it is observed that the silt loam class has the largest zmax, which contributes to the smallest mean value of α (or the highest mean capillary fringe) for that soil class. A higher capillary fringe would mean larger hydraulic conductivity at higher location, a condition that would favor larger horizontal flow at higher location. Another distinct feature seen from these figures is that infiltration at small surface pressure head ((a) and (b)) leads to a small horizontal and vertical flow ratio (no larger than 2%). It is due to diminishing pressure differential across different cells as the flow scenario switches from large surface pressure to small surface pressure. The free drainage scenario is an extreme case where pressure head across the formation, including at the surface, is always zero, indicating no horizontal flow.







112

350.0

80.0

300.0

2

70.0

2

R = 0.1

R = 0.1

60.0

250.0

50.0

200.0

40.0 150.0

30.0

100.0

20.0

50.0

10.0

Evaporation 0.0 0.0

50.0

100.0

150.0


350.0

50.0

100.0

150.0

200.0

80.0

R = 0.5

300.0

2

R = 0.5

70.0

2

60.0

250.0

50.0

200.0

40.0 150.0

30.0

100.0

20.0

50.0

10.0

Evaporation 0.0 0.0

50.0

100.0

150.0


350.0

50.0

100.0

150.0

200.0

80.0 2

2

R = 1.0

70.0

R = 1.0

300.0

60.0

250.0

50.0 200.0 40.0 150.0

30.0

100.0

20.0

50.0

10.0

Evaporation 0.0 0.0

50.0

100.0

150.0


Distance above Water Table (cm)

ensemble pressure head based on geometric mean alpha

50.0

100.0

150.0

200.0

Distance above Water Table (cm) based on arithmetic mean alpha based on ensemble conductivity

FIGURE 7.9 Capillary pressure head vs. distance above water table. Left for evaporation given steady evaporation rate = 0.25qmax. Right for infiltration given steady infiltration rate = 0.25Ks for Brooks-Corey model.

Figure 7.11 plots maximum ratio of horizontal flux over vertical flux of the entire field of 10,000 cells for some selected values of surface pressure heads when Ks and n – 1 are spatially variable fields. Figures 7.11(a), (c), (e) and (g) represent results when two random fields are fully correlated (|R|2 = 1.0), while Figures 7.11(b), (d), (f) and (h) show the results for |R|2 = 0.1. The top four plots are results for infiltration, while the bottom four figures are for evaporation. When Ks and n – 1 are variable, the sand class also produces largest maximum ratio except for small surface pressure head condition, which means a situation close to the free drainage. In this case, the silt loam class produces largest maximum ratio. From Table 7.1, we can see that the silt loam




0.10

0.10

a) Maximum Flux Ratio (Horizontal/Vertical)

0.08

Infiltration, Ps=30(-cm) |R|2=1.0

113

0.10

b) 0.08


0.10

c) 0.08


d) 0.08

0.06

0.06

0.06

0.06

0.04

0.04

0.04

0.04

0.02

0.02

0.02

0.02

0.00 0.0

100.0

200.0

0.10

0.00 0.0

100.0

200.0

0.10

e) 0.08

Evaporation, Ps=240(-cm) |R|2=1.0

0.00 0.0

100.0

200.0

0.10

f) 0.08


0.00 0.0

0.08


h) 0.08

0.06

0.06

0.06

0.04

0.04

0.04

0.04

0.02

0.02

0.02

0.02

100.0

200.0

0.00 0.0

100.0

200.0

100.0

200.0

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g)

0.06

0.00 0.0


0.00 0.0

100.0

200.0

0.00 0.0


100.0

200.0

Elevation above Water Table (cm) sand silty cl. loam

loamy sand clay loam

loam silt

sandy loam clay

silt loam sandy clay

sandy cl. loam silty clay

FIGURE 7.10 Maximum ratio of the horizontal flux over the vertical flux (qh/q) for selected values of the surface pressure head when Ks and α are spatially variable fields. 0.18

Maximum Flux Ratio (Horizontal/Vertical)

0.15

0.18

a)


0.15

0.18

b)


0.18

c)

0.15


0.15

0.12

0.12

0.12

0.12

0.09

0.09

0.09

0.09

0.06

0.06

0.06

0.06

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0.03

0.03

0.03

0.00 0.0

100.0

200.0

0.18 0.15

0.00 0.0

100.0

200.0

0.18

e)


0.15

0.00 0.0

100.0

200.0

0.18

f)


0.00 0.0

g)

0.15


0.15

0.12

0.12

0.12

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0.09

0.09

0.09

0.06

0.06

0.06

0.06

0.03

0.03

0.03

0.03

100.0

200.0

0.00 0.0

100.0

200.0


100.0

200.0

0.18

0.12

0.00 0.0

d)

0.00 0.0

100.0

200.0

0.00 0.0

h)


100.0

200.0

Elevation above Water Table (cm) sand silty cl. loam

loamy sand clay loam

loam silt

sandy loam clay

silt loam sandy clay

sandy cl. loam silty clay

FIGURE 7.11 Maximum ratio of the horizontal flux over the vertical flux (qh/q) for selected values of surface pressure head when Ks and n – 1 are spatially variable fields. © 2003 by CRC Press LLC


114


0.020

0.040

0.020

0.015

0.030

0.015

0.010

0.020

0.010

0.005

0.010

0.000 0.0

180.0

360.0

0.005

SANDY CLAY

SILTY CLAY 540.0

0.008

0.000 0.0

180.0

360.0

CLAY 540.0

180.0

360.0

540.0

0.012

0.025 0.020

0.006

0.000 0.0

0.009

0.015 0.004

0.006 0.010

0.002

0.003

0.005

SILT 0.000 0.0

180.0

360.0

SILTY CLAY LOAM

CLAY LOAM 540.0

0.000 0.0

180.0

360.0

540.0

180.0

360.0

540.0

0.030

0.008

0.025

0.000 0.0

0.025

0.020

0.006 0.020

0.015 0.004

0.015

0.010

0.010 0.002

0.005 0.000 0.0

180.0

360.0

540.0

0.005

SILT LOAM

SANDY CLAY LOAM 0.000 0.0

180.0

360.0

540.0

0.016

0.040

0.040

0.012

0.030

0.030

0.008

0.020

0.020

0.004

0.010

LOAM 0.000 0.0

180.0

360.0

SANDY LOAM

0.000 0.0

540.0

360.0

540.0

0.010

SAND

LOAMY SAND 0.000 0.0

180.0

180.0

360.0

540.0

0.000 0.0

180.0

360.0

540.0

Surface Pressure Head (-cm) Eff. alpha (|R|2=1.0) alpha(geo. mean)

Eff. alpha (|R|2=0.1) alpha(harm. mean)

alpha(arith. mean)

FIGURE 7.12 Effective parameter αeff vs. surface pressure for various soil textural classes.

class has the smallest α or largest bubbling pressure head, i.e., largest capillarity. The maximum ratio is expected to be related to variance (variability) of hydraulic parameters, textural classes (mean hydraulic parameter values). From Figures 7.10 and 7.11, it can be seen that the maximum ratio is no more than 17% for all cases, thus supporting that the stream-tube type of flow is a reasonable assumption, which makes our analysis significantly more tractable. The effective parameter αeff as a function of the surface pressure head for various soil textural classes is plotted in Figure 7.12. Notice that αeff decreases as the value of surface pressure head increases (or as flow switches from infiltration to evaporation). High correlation between Ks and α results in consistently large effective parameter αeff. These results are consistent with our previous findings that correlation between Ks and α makes the soil more sand-like, i.e., having large effective parameter αeff.18 Although variability of q is smaller as a result of parameter correlation, it makes © 2003 by CRC Press LLC



1.400

115

1.300

1.300

1.200

1.200

1.100

1.100

1.300 1.200 1.100

SANDY CLAY

SILTY CLAY 1.000 0.0

180.0

360.0

540.0

1.800

1.000 0.0

180.0

360.0

CLAY 540.0

1.000 0.0

1.600

1.700

1.500

1.600

1.600

180.0

360.0

540.0

1.500

1.400

1.400 1.400

1.300 1.300 1.200

1.200

1.200 1.100

SILT 1.000 0.0

180.0

360.0

540.0

1.500

1.000 0.0

1.100

CLAY LOAM 180.0

360.0

540.0

1.000 0.0

1.900

1.600

1.600

1.400

1.300

1.200

SILTY CLAY LOAM 180.0

360.0

540.0

1.400 1.300 1.200 1.100

SILT LOAM

SANDY CLAY LOAM 1.000 0.0

180.0

360.0

540.0

1.700

1.000 0.0

180.0

360.0

SANDY LOAM 540.0

1.000 0.0

2.000

3.500

1.800

3.000

1.400

1.600

2.500

1.300

1.400

2.000

1.600

180.0

360.0

540.0

1.500

1.200 1.100 1.000 0.0

1.200

LOAM 180.0

360.0

1.500

SAND

LOAMY SAND 540.0

1.000 0.0

180.0

360.0

540.0

1.000 0.0

180.0

360.0

540.0

Surface Pressure Head (-cm) Eff. n (|R|2=1.0 ) n (geo. mean)

Eff. n (|R|2=0.1) n (harm. mean)

n (arith. mean)

FIGURE 7.13 Effective parameter neff vs. surface pressure for various soil textural classes.

ensemble behavior more sand-like. A reasonable practical guide for most soil textural classes is that the effective α falls between the arithmetic mean and the geometric mean for the highly correlated case, and between the geometric mean and the harmonic mean for the less correlated case, with the exception of two coarser textural classes (loamy sand and sand). For infiltration, the effective value would be near the top limit of that range. Figure 7.13 plots for various soil textural classes the effective parameter neff as a function of the surface pressure head. The influence of surface pressure head on neff is not as strong as on αeff. This is partly because the variance of log(n) is small. The higher correlation between Ks and n – 1 usually leads to a slightly larger effective parameter neff. The results hence also demonstrate that correlation between Ks and n – 1 makes the soil more sand-like, i.e., giving a larger effective parameter neff. The effect, however, is typically small. For practical applications it will be reasonable © 2003 by CRC Press LLC


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to ignore the effect of spatial variability in the parameter n, given its small impact on the effective values, and the fact that it can be determined with greater certainty than the other parameters in van Genuchten’s model as mentioned earlier.

VII. SUMMARY We summarize the important findings based on our results: 1. We suggest the following guidelines for the practical use of averaging van Genuchten parameters in dealing with large scale steady state infiltration and evaporation: arithmetic means for Ks and n, between arithmetic and geometric means for α when Ks and α are highly correlated, between geometric and harmonic means for α when Ks and α are little correlated. 2. For predominantly vertical evaporation and infiltration, the use of a geometric mean value for α simulates the ensemble behavior better than an arithmetic average of α. The effectiveness of the average parameters depends on the degree of correlation between parameters and flow conditions. With parameters perfectly correlated, average parameters are most effective. The correlation between the hydraulic conductivity Ks and the parameter α results in an ensemble soil behavior more like a sand. 3. For sand dominant fields, it is more difficult to define average parameters in lieu of effective parameters to simulate ensemble soil behavior, since the effective parameters tend to change more rapidly with surface pressure conditions. 4. For van Genuchten model, spatial variability of α has larger impact on ensemble behavior of soils than that of n, partly because n can be determined with greater certainty in practice. Therefore, it is reasonable to treat n as deterministic. 5. For typical applications, the assumption of stream-tube type vertical flow for large fields is reasonable because the horizontal pressure discontinuity would cause little horizontal flow compared with the vertical flux.

REFERENCES 1. Gardner, W.R., Some steady state solutions of unsaturated moisture flow equations with applications to evaporation from a water table, Soil Sci., 85(4), 228, 1958. 2. Russo, D., Determining soil hydraulic properties by parameter estimation: on the selection of a model for the hydraulic properties, Water Resour. Res., 24(3), 453, 1988. 3. Brooks, R.H. and Corey, A.T., Hydraulic properties of porous media, Colorado State Univ., Hydrology Paper No. 3, 27 pp, 1964. 4. van Genuchten, M.Th., A closed-form equation for predicting the hydraulic conductivity of unsaturated soils, Soil Sci. Soc. Am. J., 44, 892, 1980. 5. Yeh, T.-C.J., Gelhar, L.W., and Gutjahr, A.L., Stochastic analysis of unsaturated flow in heterogeneous soils. 1. Statistically isotropic media, Water Resour. Res., 21(4), 447, 1985. 6. Yeh, T.-C.J., Gelhar, L.W., and Gutjahr, A.L., Stochastic analysis of unsaturated flow in heterogeneous soils. 2. Statistically anisotropic media with variable α, Water Resour. Res., 21(4), 457, 1985. 7. Yeh, T.-C.J., Gelhar, L.W., and Gutjahr, A.L., Stochastic analysis of unsaturated flow in heterogeneous soils. 3. Observation and applications, Water Resour. Res., 21(4), 465, 1985. 8. Russo, D., Upscaling of hydraulic conductivity in partially saturated heterogeneous porous formation, Water Resour. Res., 28(2), 397, 1992. 9. Desbarats, A.J., Scaling of constitutive relationships in unsaturated heterogeneous media: A numerical investigation, Water Resour. Res., 34(6), 1427, 1998. 10. Govindaraju, R.S., Morbidelli, R., and Corradini, C., Areal infiltration modeling over soils with spatially correlated hydraulic conductivities, J. Hydrol. Eng., ASCE, 6(2), 150, 2001.




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11. Yeh, T.-C.J., One-dimensional steady state infiltration in heterogeneous soils, Water Resour. Res., 25(10), 2149, 1989. 12. Green, T.R., Contantz, J. E., and Freyberg, D.L., Upscaled soil-water retention using van Genuchten’s function, J. Hydrologic Eng., ASCE, 1(3), 123, 1996. 13. Smith, R.E. and Diekkruger, B., Effective soil water characteristics and ensemble soil water profiles in heterogeneous soils, Water Resour. Res., 32(7), 1993, 1996. 14. Chen, Z., Govindaraju, R.S., and Kavvas, M.L., Spatial averaging of unsaturated flow equations under infiltration conditions over areally heterogeneous fileds. 1. Development of models, Water Resour. Res., 30(2), 523, 1994. 15. Chen, Z., Govindaraju, R.S., and Kavvas, M.L., Spatial averaging of unsaturated flow equations under infiltration conditions over areally heterogeneous fields. 2. Numerical simulations, Water Resour. Res., 30(2), 535, 1994. 16. Kim, C.P., Stricker, J.N.M., and Feddes, R.A., Impact of soil heterogeneity on the water budget of the unsaturated zone, Water Resour. Res., 33(5), 991, 1997. 17. Zhu, J. and Mohanty, B.P., Analytical solutions for steady state vertical infiltration, Water Resour. Res., 38(8) 20, 2002; DOI 10.1029/2001WR000398. 18. Zhu, J. and Mohanty, B.P., Upscaling of soil hydraulic properties for steady state evaporation and infiltration, Water Resour. Res., 38(9), 17, 2002; DOI 10.1029/2001/WR000704. 19. Zhu, J. and Mohanty, B.P., Spatial averaging of van Genuchten hydraulic parameters for steady state flow in heterogeneous soils, revised for Vadose Zone J., 1, 261, 2002. 20. Zhu, J. and Mohanty, B.P., Effective hydraulic parameters for steady state vertical flow in heterogeneous soils, revised for Water Resour. Res., 2002. 21. Leij, F.J., Russell, W.B., and Lesch, S.M., Closed-form expressions for water retention and conductivity data, Ground Water, 35(5), 848, 1997. 22. Mualem, Y., A new model for predicting the hydraulic conductivity of unsaturated porous media, Water Resour. Res., 12(3), 513, 1976. 23. Schaap, M.G. and Leij, F.J., Database-related accuracy and uncertainty of pedotransfer functions, Soil Sci., 163, 765, 1998. 24. Wang, J.S.Y. and Narasimhan, T.N., Distribution and correlations of hydrologic parameters of rocks and soils, in Indirect Methods for Estimating the Hydraulic Properties of Unsaturated Soils, van Genuchten, M.Th., Leij, F.J., and Lund, L.J., Eds., University of California, Riverside, 1992,169. 25. Zaslavsky, D., Theory of unsaturated flow into a non-uniform soil profile, Soil Sci., 97, 400, 1964. 26. Warrick, A.W. and Yeh, T.-C.J., One-dimensional, steady vertical flow in a layered soil profile, Adv. Water Resources, 13(4), 207, 1990. 27. Warrick, A.W., Additional solutions for steady-state evaporation from a shallow water table, Soil Sci., 146(2), 63, 1988. 28. Nielsen, D.R., Biggar, J. W., and Erh, K.T., Spatial variability of field-measured soil-water properties, Hilgardia, 42(7), 215, 1973. 29. Robin, M.J.L., Gutjahr, A.L., Sudicky, E.A., and Silson, J. L., Cross-correlated random field generation with the direct Fourier transform method, Water Resour. Res., 29(7), 2385, 1993. 30. Hills, R.G., Hudson, D.B., and Wierenga, P.J., Spatial variability at the Las Cruces trench site, in Indirect Methods for Estimating the Hydraulic Properties of Unsaturated Soils, van Genuchten, M.Th., Leij, F.J., and Lund L.J., Eds., University of California, Riverside, 1992, 529. 31. Renard, P., Le Loch, G., Ledoux, E., de Marsily, G., and Mackay, R., A fast algorithm for the estimation of the hydraulic conductivity of heterogeneous media, Water Resour. Res., 36(12), 3567, 2000.



8

Spatial Variability of Soil Moisture and Its Implications for Scaling A.W. Western, R.B. Grayson, G. Blöschl, and D.J. Wilson

CONTENTS I. Introduction...........................................................................................................................119 II. Characteristics of Spatial Patterns of Soil Moisture ............................................................121 A. Soil Moisture Processes at Small Catchment Scales................................................122 B. Statistical Representation of Soil Moisture Patterns ................................................125 C. Relationships of Soil Moisture to Other Variables ...................................................127 III. Modeling Soil Moisture .......................................................................................................130 A. Representation of Soil Moisture in Models ..............................................................130 B. Representation of Variability in Models ...................................................................131 C. Modeling Spatial Patterns of Soil Moisture..............................................................132 IV. Moving to Larger Scales ......................................................................................................133 V. Summary and Conclusions...................................................................................................138 VI. Acknowledgments ................................................................................................................138 References ......................................................................................................................................138

I. INTRODUCTION Soil moisture has an important influence on hydrological and ecological processes, although the volume of water stored as soil moisture represents only a small proportion of liquid freshwater on the Earth. Soil moisture is important in processes that partition rainfall into runoff and infiltration. It is also the major source of water associated with the latent heat flux from land to the atmosphere and hence is important in partitioning incoming energy into latent, sensible, and other heat and radiative fluxes at the ground. These fluxes have important impacts on the atmospheric boundary layer and in turn on climate and weather and the prediction of each. Soil moisture modulates plant growth and hence primary production in terrestrial ecosystems and has an important influence on a variety of soil processes including erosion (by controlling runoff), soil chemical processes and solute transport, and ultimately pedogenesis. Western et al.1 summarize soil water hydrologic processes and their influence on surface energy balance processes in more detail. We have been pursuing an active soil moisture research program incorporating field and modeling studies at small catchment scales (1 km2) over the past decade. In this chapter we aim to summarize the outcomes of that program, outline ongoing research at much larger scales and to review the availability of information on soil hydrologic properties. We believe the latter area is a major constraint on our ability to predict the hydrologic and energy balance, especially at larger scales where it is not possible to implement intensive soil property measurement campaigns. Our



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research program has been motivated by fundamental questions about spatial and, to a lesser extent, temporal variability and associated scale issues. We have focused on soil moisture because it is arguably the most important near-surface state controlling variability in the surface and near surface water and energy balances. Here we are using the term state to refer to variables that change in time. These are generally the stores and fluxes within the system of interest. We will use the term parameter to refer to characteristics of the system that are constant in time, or that can be assumed constant over the time scales of interest. A typical example of a parameter is the saturated hydraulic conductivity of the soil, although it may not always be appropriate to consider this constant in, for example, cracking soils.2 Variability is omnipresent in the environment. Here we will concentrate on spatial variability, although many of the concepts discussed also apply to temporal variability. Variability is important for interpreting (and planning) measurements and for understanding processes, especially the relationships between processes at different scales. Due to spatial variability, knowledge of a property at one location does not provide perfect knowledge of the same property at another location, although, depending on the characteristics of the variability and the proximity (in some sense) of the points, the measurement may provide some information. For example the soil moisture at two points separated by one meter is likely to be more similar than the soil moisture at two points separated by one thousand kilometers. If one understands the characteristics of the variation (by understanding the processes that lead to the variation, by understanding the statistical properties of the variation, or both), these characteristics can be utilized in predicting the likely relationships between points. Spatial variability has an important impact on processes. For example, soil moisture availability controls evapotranspiration during drier conditions and therefore evapotranspiration varies in space due in part to soil moisture variations. Saturation excess runoff is another example. If the dependence of a particular process on soil moisture is nonlinear, which is usually the case, then knowledge of the variability and the mean soil moisture state is necessary to predict the processes. In the above examples the nonlinearity of the evapotranspiration process (a continuous process) is less than the saturation excess runoff processes (a threshold process) and hence variability is likely to be less important for evapotranspiration than for saturation excess runoff. Spatial variability usually increases with spatial scale, at least across some scale range. This scale dependence means that the characteristics of moisture dependent processes are also influenced by spatial scale. It is not only the amount of variability (the variance) that is important but also its spatial characteristics. Spatial variation can be random or organized or a combination of the two.3–5 Here we use random to mean variability that is not predictable in detail but that has predictable statistical properties and organized to mean variability that has regularity or order. Spatial organization implies variation that is characterized by consistent spatial patterns.5 The ultimate in disorder is white noise. With increasing organization, processes may exhibit (a) continuity (which is captured statistically by the variogram or the autocorrelation function); (b) connectivity (i.e., connected thin bands such as saturated source areas in drainage lines); or (c) convergence (i.e., a branching structure of drainage lines and hillslopes). Our work has demonstrated that soil moisture exhibits organized features under some but by no means all circumstances and that the degree of organization depends on the catchment’s current and past wetness state6,7 Similarly, the amount of variation is also dependent on the wetness state of the catchment8 Various studies have shown that the proper representation of organization can be critically important for accurate modeling and prediction of hydrologic processes. A corollary of this is that hydrologic processes depend on the amount and the characteristics of the spatial variation. Grayson et al.9 showed that simulated runoff was quite different for antecedent soil moisture patterns with the same variance (probability density function) and continuity (as measured by the variogram) but differing in terms of the presence or absence of connectivity and convergence. Their simulations were dominated by saturation excess runoff. Saturation excess runoff occurs when a drainage constraint (usually at depth) combined with large accumulated rainfall depths results in a water © 2003 by CRC Press LLC


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table that reaches the surface, causing the whole profile to saturate and the production of runoff. Merz and Plate10 also found that organization is important where infiltration excess runoff is the dominant runoff mechanism. Infiltration excess runoff occurs when the rainfall intensity exceeds the capacity of the surface soil to transmit infiltrating water, causing the soil surface to saturate and the production of runoff. Western et al.11 obtained similar results in a study that utilized observed rather than synthetic9 or interpolated10 soil moisture patterns. The characteristics of spatial variability can also be important in influencing lower atmospheric behavior.12 Weaver and Avissar13 showed that circulation in the atmospheric boundary layer is enhanced by a patchy distribution of latent and sensible heat fluxes. They argued that this enhanced circulation influences cloud formation and ultimately precipitation. Mills14 has shown that variations in soil moisture specification in the initial state of a mesoscale numerical weather prediction (NWP) forecast can make sufficient difference to surface temperatures over land that the movement of a cold front through Victoria, Australia, is well forecast with a realistic soil moisture field, but the front is completely absent if a climatological soil moisture field is specified. This sensitivity of near-surface temperatures to soil moisture specification in the NWP forecast has been demonstrated in many other studies. For atmospheric effects, it is the size of patches relative to the atmospheric boundary layer properties that is important.15 Ultimately we want to be able to understand and predict the overall result of a variety of interacting processes on the water and energy balance of the landscape at different scales. We always use a model to assist in this, sometimes just a perceptual model but more often a quantitative mathematical model. Practical applications of modeling include the prediction of runoff yield and floods, contaminant generation and transport, and land surface–atmosphere interactions as required in NWP models and general circulation models. Because variability has an important impact on processes, we need to consider whether it is necessary to incorporate the effects of variability in our models and, if so, the appropriate way of doing this. From a theoretical standpoint spatial variability can be represented in models deterministically or statistically, although most models can be considered to be a combination of the two approaches, with most emphasis placed on one or the other. This will be discussed further below. In either case, an understanding of the characteristics of spatial variability is important for deciding on the details of the model structure and algorithms. In addition, modeling can be used to help understand and interpret field observations of spatial variability, which are always limited to some extent in their spatial and temporal coverage. The remainder of this chapter is organized as follows. First we discuss our results on the spatial characteristics of soil moisture patterns at small catchment scales. Then we consider a variety of statistical approaches for representing that variability and for the spatial scaling of soil moisture. Next we discuss results of spatially distributed deterministic modeling of soil moisture patterns, also at the small catchment scale. As mentioned above, modeling of variability is particularly important from a practical perspective, and is an important link between soil physics and wider environmental science. Then we change scales and discuss some new research that aims to improve our soil moisture modeling capability at larger scales in a way that appropriately translates our understanding of small-scales. As part of this we consider the problem of obtaining reliable soil parameters at large scales, which is currently a major constraint. Western et al.1 provide a broader review of the scaling of soil moisture.

II. CHARACTERISTICS OF SPATIAL PATTERNS OF SOIL MOISTURE We have conducted a series of small catchment experiments motivated by the need to understand the characteristics and hydrologic process implications of the spatial variation of soil moisture.16,17 Central to these experiments has been the measurement and modeling of detailed spatial patterns of soil moisture. The resolution of these patterns has been sufficient to resolve organized variation © 2003 by CRC Press LLC


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at the hillslope scale. To detect organization high resolution is required18,19 because the shape of features such as narrow bands of high soil moisture in drainage lines must be resolved. Williams18 argues that much of the perceived randomness in hydrology is a consequence of inadequate spatial resolution of measurements. Our spatial patterns of soil moisture have been measured using time domain reflectometry probes mounted on an all-terrain vehicle fitted with differential global positioning system (DGPS) instruments and hydraulics for sensor insertion. Soil moisture is measured in the 0 to 30 cm layer in all cases considered here. This depth was chosen because at each site, the root zone was concentrated in this depth. The spatial resolution (separation of points) of our measurements varies among sites, depending on the site characteristics. The following discussions are based on results from the Tarrawarra catchment, Victoria, Australia,16 three sites in the Mahurangi River catchment, New Zealand,17 and the Point Nepean site in Victoria, Australia.20 These sites are described in more detail in the relevant references. All three study areas have a relatively uniform rainfall distribution through the year and a summer peak in potential evapotranspiration. The New Zealand sites are humid with rainfall exceeding potential evapotranspiration in nearly all months, whereas the Australia sites are somewhat drier with rainfall exceeding potential evapotranspiration for half of the year and potential evapotranspiration exceeding rainfall for the other half. Tarrawarra has clayey duplex soils and the Mahurangi River sites have clayey gradational soils, whereas Point Nepean has deep uniform sandy soils. The terrain at all the sites is undulating to hilly.

A. SOIL MOISTURE PROCESSES

AT

SMALL CATCHMENT SCALES

Figure 8.1 shows examples of spatial patterns of soil moisture. Contours of elevation are also shown. The top row of patterns is from Tarrawarra, Australia. From left to right the patterns represent a) wet conditions, b) dry conditions and c) the transition from dry to wet. There is a major contrast in the characteristics of the patterns between wet conditions and dry conditions. Under wet conditions the wettest parts of the catchment are concentrated in the drainage lines, whereas under dry conditions the wettest parts of the catchment are distributed randomly across the measurement area. There is also evidence of slightly wetter south-facing slopes associated with lower radiation input on these slopes.7 In the classification of types of variability described above the wet pattern shows connectivity and convergence, while the dry pattern is random with some continuity. That is, the wet pattern is much more highly organized. This suggests that the spatial characteristics of the soil moisture patterns depend on the soil moisture state. Clearly under wet conditions there is a relationship between the topography and the soil moisture7 due to subsurface lateral flow, which concentrates water in the drainage lines.6,7,19 Hence soil moisture depends on upslope processes controlling the delivery of water and local or downslope processes affecting drainage of water. The presence of connected wet bands in the drainage lines significantly affects the rainfall-runoff response of the catchment.11 During dry conditions, the dependence on topography disappears because the subsurface lateral flow decreases rapidly to zero as the soil dries and the unsaturated hydraulic conductivity decreases. Under these conditions the soil moisture is controlled locally. Grayson et al.6 show that seasonal changes in the balance between precipitation and evapotranspiration lead to persistent periods of low and high soil moisture in many parts of Australia. Where shallow subsurface lateral flow occurs under wet conditions, this seasonal shift between high and low moisture will be associated with a change between local and nonlocal controls on soil moisture. This has important implications for modeling patterns of soil moisture. Figure 8.1c shows a soil moisture pattern during the transition from dry to wet conditions. Here the wettest areas of the catchment are in the areas of greatest local convergence (the heads of the drainage lines). These areas become wet first because lateral flow has only recently become significant and only part of the hillslope is contributing flow to any given point.21,22 Grayson et al.6 argue that where there is a rapid change from precipitation to evaporation dominance and viceversa, there will be a relatively rapid change in the soil moisture state from wet to dry or dry to




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FIGURE 8.1 (See color insert following page 144.) Example of soil moisture patterns. Each rectangle represents a single point measurement of average soil moisture in the top 30 cm of the soil profile. The measurements are taken at the center of the rectangle using time domain reflectometry. Note that the scales vary between patterns and that map scales vary between sites. a) Tarrawarra, September, 25, 1995; b) Tarrawarra, February 14, 1996; c) Tarrawarra, April 13, 1996; d) Point Nepean, July 16, 1998; e) Point Nepean, April 13, 1999; f) Satellite Station, Mahurangi River, March 26, 1998; g) Clayden’s, Mahurangi River, March 30, 1998; h) Carran’s Mahurangi River, April 1, 1998. Contour lines represent elevation.

wet. The speed of this change depends on the magnitude of the imbalance between these two fluxes and the size of the soil water store. At Tarrawarra the changes are generally quite rapid, especially the change from dry to wet where a couple of large precipitation events can rapidly fill the soil water store. At the Mahurangi field sites the soil moisture remains at high levels for more extended periods than at Tarrawarra, due to the more humid climate (Figure 8.2). In some environments and soil conditions, soil moisture behavior will essentially always be in one state or another. Cool high rainfall areas can be always “wet” while arid regions can be always “dry.” In semiarid regions, the dominant lateral flow process can be infiltration excess runoff. This can lead to similar soil moisture patterns to those above, because the infiltration opportunity time is greatest in the depression areas. Such patterns, at least in surface soil moisture, are generally short lived because evaporation removes differences in surface soil moisture, although areas where recharge to groundwater occurs can have distinct differences in total profile soil water storage (e.g., riparian zones in semiarid regions where bands of vegetation indicate significant available soil water storage). The soil profile at Tarrawarra favors lateral flow of water and the climate is such that the soil moisture conditions change from wet to dry seasonally. At Point Nepean the climate is similar but the soil moisture behavior is quite different as a consequence of deep sandy soils at this site. These soils allow rapid vertical drainage to depth and as a result significant shallow lateral subsurface flow does © 2003 by CRC Press LLC


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FIGURE 8.2 Time series of soil moisture in the top 30 cm of the soil profile at a) Tarrawarra, Australia, and b) Satellite Station, Mahurangi River, New Zealand.

not occur and typical volumetric soil moisture levels are much lower than at Tarrawarra. However, the spatial patterns of soil moisture at Point Nepean do show consistent behavior over time. In this case it is a consequence of spatial variation in soil particle size distribution. While the soils are sandy, there are two areas (labeled A and B on Figure 8.1d and 8.1e) that have higher fines content and possibly greater organic matter content. This leads to larger moisture-holding capacity for the surface soils in these areas and thus consistently higher volumetric soil moisture contents. It is not clear whether or to what degree these higher moisture contents translate into higher moisture availability in these two areas. Both the climate and soil profiles in the Mahurangi River should favor lateral flow and soil moisture patterns like those under wet conditions at Tarrawarra. Figure 8.1 f through h shows some evidence of topographic control of soil moisture patterns at Satellite Station but not at Carran’s and Clayden’s. Most of the topographic control at Satellite Station is a consequence of the floodplains being wetter and the behavior on the hillslopes at this site is actually similar to Carran’s and Clayden’s. The wetter flood plains at Satellite Station are likely to be due to shallow groundwater or higher water retention capabilities of the floodplain soils (clay content = 60%) compared to the hillslope soils (clay content = 30%). At Tarrawarra there are increasing soil moisture trends down the hillslopes (as evidenced by correlations between soil moisture and upslope area23) but similar © 2003 by CRC Press LLC



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trends do not occur at the Mahurangi sites.24 Nevertheless perennial saturated areas are maintained within the drainage lines (which are often not resolved by the measurements due to their very narrow nature) and these must be fed from water moving laterally from the hillslopes. The reasons why this lateral movement of water is not evident in the measured hillslope soil moisture patterns for the top 30 cm are unclear. A number of possible explanations include: • Lateral flow is occurring deep within the regolith. • Variability (of soils) on the hillslopes masks any trends. • A soil pipe network is sufficiently extensive to rapidly drain the entire hillslope. From visual inspection, there is no clear layer impeding drainage of the soils in the Mahurangi, whereas there is at Tarrawarra at about 30 to 40 cm. In addition, the drainage lines at Satellite Station and Clayden’s are quite narrow and incised compared with the gentle swales at Tarrawarra. This tends to contain the exfiltration of subsurface water within the drainage lines themselves. These factors may lead to quite deep lateral flow pathways that do not influence the surface moisture here. It is also possible that variable soils may mask any soil moisture trends on the hillslopes in the Mahurangi, although the overall variability of the soil moisture patterns is similar to Tarrawarra, except for Satellite Station. We have also observed soil piping at Satellite Station and Carran’s. These pipes do transport water laterally at relatively shallow depths; however, it is not clear that these pipes are common enough to affect the drainage of water from the hillslopes substantially. We have performed detailed spatial pattern measurements at several locations in Australia and New Zealand. These measurements have demonstrated quite different behavior among sites. The results at Point Nepean and Tarrawarra are as might be expected on the basis of the soil characteristics and our understanding of processes, while those in New Zealand were somewhat contrary to our expectations. Only at Tarrawarra is the topographic control of soil moisture so often assumed in hydrologic models clearly evident and even here it only occurs during part of the year. At the New Zealand sites where similar behavior would be expected, it is much less obvious. It is often assumed that topographically induced lateral flow is the dominant source of spatial soil moisture variability in sloping landscapes25 — clearly a questionable assumption that we will return to later. It is worth noting that others have studied the spatial distribution of soil moisture using various approaches. Where links to terrain have been studied in the field (e.g., References 21 and 26 through 32) they have been of quite variable strength over time and between sites, but it is rare that terrain explains more than half of the spatial variation in soil moisture, which is quite consistent with our results. This suggests that terrain is only one of a number of important influences on the spatial distribution of soil moisture and that in some circumstances terrain is actually unimportant.25

B. STATISTICAL REPRESENTATION

OF

SOIL MOISTURE PATTERNS

Before discussing the experiments and results, it is worth considering for a moment a hierarchy of characteristics that we might want to use to characterize soil moisture variability. In order of increasing detail of knowledge we might characterize spatial soil moisture fields with: • Measures of central tendency (i.e., mean, median, etc.) • Measure of spread (i.e., variance, interquartile range, etc.) • Measures of extreme behavior (i.e., percentage above or below a threshold, a high or low percentile) • The probability density function (pdf) • Spatial relationships (variograms, cross correlations or covariograms with, for example, terrain, connectivity [see below]) • The actual pattern



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The level of knowledge required depends on the situation. If we want to know about the spatial average of linear or weakly nonlinear processes it is probably sufficient to know the average moisture. As the degree of nonlinearity and the amount of spatial dependence of a process increase, the level of detail that we need to effectively characterize the effect of soil moisture on that process increases. One of the important processes that we are often interested in is saturation excess runoff, which is a threshold (i.e., extremely nonlinear) process. This means that we would usually want to know about the shape of the pdf, at least in the upper tail region, and ideally we would also need information on the degree of connectivity of wet areas. For some management situations we need to know where to place on-ground works (e.g., erosion controls). This implies that we need to know about the actual pattern of soil moisture. We have applied a variety of techniques for characterizing spatial patterns of soil moisture. These fall into two categories. The first considers relationships between soil moisture at different locations across a catchment, with the ultimate aim of characterizing the organization present in the patterns. The second examines the relationship between soil moisture and catchment topography explicitly. The degree of organization of spatial soil moisture patterns varies. Here we discuss statistical measures that capture continuity and connectivity, starting with standard spatial statistics and then moving to more sophisticated approaches. Continuity can be characterized using the tools of geostatistics, especially the variogram, which quantifies the spatial correlation structure. Quantitative estimates of this structure are required for a number of purposes, including the interpolation of spatial patterns from point data and estimation of the catchment average soil moisture, and for analyzing the effects of scale changes. We have used variography to quantify the correlation structure of soil moisture at Tarrawarra and showed that the soil moisture fields were stationary and that the correlation length was shorter and the sill (i.e., total variance) much larger during wet conditions than during dry conditions.33 A resampling analysis showed that to quantify the correlation structure reliably, a large number (~300) of measurements in space is required. This is many more points than are often used in published studies. We also analyzed the utility of the geostatistical approach for predicting the effects of scale changes (where the scale is defined quantitatively as extent, spacing and support — see Western and Blöschl34) on variability and correlation length and found that the approach performed adequately despite breaking underlying assumptions of randomness for the organized (connected) patterns. More recent analyses of data from Point Nepean and the Mahurangi River catchments have also shown that soil moisture is approximately stationary at the scales considered.20,35 We have also been able to show that the correlation lengths of the topographic parameters and soil moisture are closely linked at Tarrawarra, Carran’s and Clayden’s. At Satellite Station the soil moisture has shorter correlation lengths than the topography, which suggests that there are smaller scale processes controlling the soil moisture there. At Point Nepean soil moisture correlation lengths are longer than those for topographic parameters, as a consequence of the larger scale variation in soil particle size distribution.20 Variograms only capture continuity in spatial patterns and an assumption of maximum disorder consistent with the correlation structure imposed by the variogram is usually made in analyses utilizing the variogram, for example, when generating stochastic spatial patterns. Visual observations of some of the patterns in Figure 8.1 suggest that this is not always realistic. A different approach is required to capture the connected features in some soil moisture patterns. Some authors have argued that indicator variograms can be used to capture connectivity. Indicator variograms are similar to standard variograms, except that the spatial pattern is reduced to indicator values (ones and zeros) that represent high and low soil moisture. Indicator variograms are calculated and compared for a number of different percentile thresholds and the pattern of indicator variogram ranges is compared to infer whether the spatial pattern behaves in a similar way to that expected for disordered patterns.36 Some authors have argued that a long correlation length at extreme percentiles, compared with the expected range for random patterns (which can be estimated from the range for moderate percentiles), indicates connectivity. However, Western et al.37 clearly showed © 2003 by CRC Press LLC


Connectivity


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Separation (m) FIGURE 8.3 Connectivity functions for the soil moisture patterns shown in Figure 8.1a through c. The connectivity function for a random spatial field is also shown.

that this approach was unable to quantify the connectivity observed at Tarrawarra. In reality the approach actually quantifies continuity at different thresholds and thus does provide useful information on the continuity characteristics of the field, but not its connectivity. We then tried an alternative approach based on connectivity functions, which come from percolation theory.38–41 Connectivity functions were very useful in distinguishing between random and connected patterns.11 Connectivity functions are calculated using patterns of indicator statistics and they quantify the probability that two high soil moisture pixels are connected by an arbitrary continuous path of high soil moisture pixels. Figure 8.3 shows example connectivity functions for the soil moisture patterns shown in Figure 8.1a through c. To express the information in a connectivity function in terms of a single value, Western et al.11 proposed an integral connectivity scale, which is analogous to the correlation scale of the variogram. Simulations of event rainfall-runoff response using a distributed hydrologic model showed that peak and total runoff were related to the integral connectivity scale of the antecedent soil moisture pattern.

C. RELATIONSHIPS

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TO

OTHER VARIABLES

While the above approaches provide useful information on the statistical structure of spatial patterns, they (at least in their simple forms) ignore relationships with other spatial fields. There are a variety of process considerations that suggest soil moisture should be related to topography. These are discussed in detail in Western et al.7 The general hypothesis is that topography modulates processes of lateral flow and evaporation and that this is a dominant source of variation. If this is the case, it should be possible to predict spatial patterns from appropriate terrain parameters, such as upslope area, slope, aspect, curvature or terrain indices including the topographic wetness index42–44 and radiation indices.7,45 The terrain analysis required to obtain such indices is discussed in detail in References 46 and 47. Up to now most terrain analysis has tried to characterize hillslopes. A significant feature of many landscapes is the presence of valley floors with major accumulations of alluvial fill where soil moisture and other hydrologic behavior may be different. A recent advance is the development of a multiple-resolution index of valley bottom flatness (MRVBF) that can be used to divide a landscape into hillslopes and valley bottoms and that provides a measure of the local extent of the valley bottom.48 Of the terrain indices that aim to represent the effects of subsurface lateral flow on hillslopes, the topographic wetness index of Beven and Kirkby42 is the most widely used as a surrogate for the spatial soil moisture pattern and it is based on a simplified model of water movement in the landscape. This index assumes steady state conditions, which implies that the entire upstream hillslope contributes to subsurface flow. It also assumes saturated lateral flow, areally uniform recharge to the saturated zone, an exponential decrease in saturated hydraulic conductivity with © 2003 by CRC Press LLC


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Proportion less than FIGURE 8.4 Cumulative frequency distribution of R2 values between soil moisture and the topographic wetness index (lower line) and a combination of the topographic wetness index and potential solar radiation index.

depth and a hydraulic gradient (water table slope) equal in magnitude and direction to the surface slope. These various assumptions have been relaxed for specific situations.1 A number of empirical studies have examined the correlation between terrain properties and a measure of soil water storage status. Rarely has the percentage of variance explained exceeded 50% and at no site has it consistently exceeded 50%.29,30,32,45,49–52 At Tarrawarra we found that the explained variance ranged between 0 and 50% for individual terrain parameters (Figure 8.4). A combination of wetness index and potential solar radiation index explained up to 61% of the soil moisture variance and it performed best under moderately wet conditions where lateral flow was active but saturated areas were confined to the drainage lines.7 This combination did a good job of capturing the spatial organization present in the soil moisture patterns. The unexplained variance remaining after regressing soil moisture against the two indices appeared random and was mainly at spatial scales smaller than the hillslope scale. Thus the limit on ability of terrain indices to predict soil moisture patterns is linked to the amount of topographic organization present in the pattern, as would be expected. The wetness index is worse at predicting depth to water table than the soil moisture at Tarrawarra.53 Our results from Mahurangi and Point Nepean indicate very low correlations between soil moisture and terrain.24 This is explicable for Point Nepean in terms of the processes and soil characteristics at this site, but it was unexpected at Mahurangi. Some possible explanations for the behavior in the Mahurangi have already been discussed above. The rather weak relationships between soil moisture and terrain have some important implications for hydrologic modeling. One of the most common uses of the wetness index is to represent subelement variability statistically in models such as Topmodel42 and its derivatives. It is assumed that this subelement variability results primarily from topographically driven lateral flow, that the characteristics of the patterns are largely temporally invariant, and that terrain indices are a good predictor of the spatial patterns, or at least their associated probability density functions (pdfs). The empirical literature suggests that the topographic control and spatial predictability assumptions are often poor. Our analysis of the Tarrawarra data also shows that the relationships between the terrain and soil moisture pdfs are poor. An implication of the temporally constant relationship between saturation deficits (closely related to depth integrated soil moisture) that is often assumed is that the variance of soil moisture should be temporally constant. At all our sites except Satellite Station, there are major temporal changes in the soil moisture variance. © 2003 by CRC Press LLC



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Spatial variance

Our analysis of spatial soil moisture measurements from sites around the world shows that there are typically substantial changes in spatial moisture variance with time,8 which suggests that assumptions of temporally invariant spatial patterns are unrealistic. Figure 8.1a through c is a clear example of temporally varying pattern characteristics. These assumptions can be overcome in traditional distributed models (e.g., Western et al.19) Also Beven and Freer54 have recently developed a dynamic Topmodel that relaxes the assumption of temporally invariant patterns while retaining a fairly high level of spatial simplification by assuming the landscape can be divided into hydrological response units that are discontiguous but hydrologically similar in terms of their response. Fundamentally, models such as Topmodel represent subelement variability using a pdf. Given that the pdf of soil moisture is poorly predicted by terrain indices, an alternative approach should be considered. One possibility is to develop a generic pdf for soil moisture. On the basis of the results discussed above, we would expect that the parameters (e.g., the variance) of such a pdf would depend on the soil moisture state, which can be represented by the spatial mean soil moisture. In practice soil moisture is physically bounded by porosity and wilting point. Therefore, theoretical considerations imply that the variance of soil moisture should be equal to the variance of porosity when the spatial mean moisture is equal to the spatial mean porosity. Similar considerations apply at the wilting point. In statistical terms soil moisture can be considered to be a bounded distribution. Typically, bounded distributions exhibit minimums of variance at the boundary and a peak in variance between the boundaries. This is illustrated conceptually for the case of soil moisture in Figure 8.5. Another consequence of bounding is that it tends to induce skewness in the pdf. For example, it would be expected that as the mean moisture approaches wilting point, the driest locations would be affected first, the lower tail would be compressed and the skewness would become more positive. The characteristics of the spatial soil moisture pdf have been studied by a number of authors (e.g., References 30 and 55 through 59). Generally the soil moisture pdf has been found to be reasonably symmetric. Results from hypothesis tests for normality have been equivocal and there is a tendency for larger samples to fail the test due to increased statistical power. Studies with a larger number of sample occasions have tended to find that 50 to 80% of soil moisture patterns can be approximated by a normal distribution.56–58 Systematic increases and decreases in variance with mean soil moisture have been reported by Bell et al.57 and Famiglietti et al.,58 respectively. We have analyzed the spatial soil moisture pdf from 13 study areas around the world with climates ranging from semiarid to humid, soils ranging from sands to clays, vegetation ranging from sparse rangelands to tall wet Eucalyptus forests, and topography from gently undulating to

Wilting point

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FIGURE 8.5 Conceptual relationship between spatial variance and spatial mean soil moisture. © 2003 by CRC Press LLC


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Spatial Variance (%V/V)2

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Spatial Mean Moisture (%V/V) FIGURE 8.6 Relationships between spatial mean soil moisture and spatial variance of soil moisture for 13 study areas around the world.

steep. The soil moisture is measured over different (and sometimes multiple) depths at each site, but is always sufficiently deep to be representative of at least a significant proportion of the root zone. Figure 8.6 shows plots of variance against mean moisture. The curves are smoothed relationships calculated using LOWESS60 and the data points have been omitted for clarity. While there are differences between the curves for a given mean moisture, the data do behave in a similar manner to that illustrated in Figure 8.5. That is, variance increases with average moisture in dry catchments and it decreases in wet catchments. Where the spatial mean moisture has a sufficiently large range over time, the variance peaks at intermediate values. The location and magnitude of the peak in variance change between catchments and further analysis is required to understand why this is so. Where multiple depths have been measured at a particular catchment, depth appears to have only a small effect on the relationship.

III. MODELING SOIL MOISTURE A. REPRESENTATION

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MODELS

So far we have discussed statistical (sometimes called behavioral) approaches to describing and predicting spatial soil moisture patterns. Hydrological models can also be used to predict hydrologic processes and patterns and are an important practical tool applied at scales from point to global. A primary motivation for understanding the variability and scaling of soil moisture is to improve these models. In addition, model development can assist in interpreting observations. Soil moisture is represented in models in a variety of ways. The simplest is a conceptual bucket that represents the depth-integrated soil water store. Such models can be applied at a point, but more often they are used within conceptual catchment models to represent soil moisture storage averaged over the whole catchment. They do not try to represent physical soil moisture that can be measured in the field but rather to represent the temporal variation in the content of the soil water store. Conceptual buckets connected in series are also sometimes used to represent shallow and deep layers within the soil profile, with the aim, for example, of distinguishing between the root zone and the deeper soil zone. Most of the models used to predict floods and water yield around the world are based on these simplified “bucket style” models. In these models, spatial variability is ignored or represented simplistically by functions that relate average soil water storage to runoff. Other models based on Richard’s equation, or simplifications thereof, represent the vertical distribution of soil moisture and the movement of liquid water under the influence of gravitational © 2003 by CRC Press LLC



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and capillary (suction) gradients. Such models are often applied at a point (the scale for which they were originally derived) and they do try to represent the physical processes and physically realistic soil moisture values. They assume that vertical flow is via the soil matrix and that there is no net lateral flow. Sometimes they are implemented as coupled water and energy (heat) models. Flow in macropores and cracks can be important and needs additional model components for its representation. Often such coupled models are used to represent spatial average soil moisture content (and vertical fluxes of water and heat) over larger areas (anything from plot scale to roughly 1000 km2). When applied at the large scales (such as in numerical weather prediction models) spatial variability is largely ignored, and uniform soil parameter values are used over areas from hundreds to tens of thousands of square kilometers). In such cases the soil moisture in the model becomes difficult to interpret physically. This is the case where lateral flow processes occur or where spatial variations in soil or vegetation properties exist (i.e., most practical applications). It is especially the case in applications such as numerical weather prediction, where the soil moisture value is often changed in an effort to obtain the correct latent and sensible heat flux predictions from the model. While at first glance it may seem a little odd to change soil moisture arbitrarily (and thus break the mass conservation law), it can be easily justified by two facts. First, the object of such modeling is to provide the correct flux boundary conditions for the atmospheric model that is used to predict the weather and, second, there is uncertainty in the surface model forcing (especially the rainfall), as well as the surface model physics. As a rule, these problems of interpretation become greater at large scales and in complex terrain. A key challenge for the soil physics and hydrological communities is to translate detailed understanding of variability to these larger scales so that the important effects of variability can be represented without the need for explicit description.

B. REPRESENTATION

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MODELS

The above approaches do not attempt to represent spatial variability in soil moisture or landscape properties, but spatial variability does exist, as observed in the field, and usually strongly affects the model results. So how should spatial variability be represented in a model? It is worthwhile discussing the representation of variability from a more theoretical perspective for a moment. From a modeling perspective, variability can be represented explicitly or implicitly. Implicit representation implies that the effects of variability are modeled but that the variability is not resolved by individual model elements. Variability is represented explicitly in models by discretizing or breaking the system up into elements. Differences in soil moisture and hydrologic fluxes between elements can then represent spatial variation in the hydrologic system. However, because there is some variation even at small scales, some variability always remains unresolved at the model element scale. This “subelement” variability can be treated in a variety of different ways.61 Although a wide range of models exist, from those that have only one or a few large spatial elements to those with a large number of quite small spatial elements, most models use implicit and explicit representations of variability and it is the emphasis on one or the other that changes. The approach of explicitly representing spatial variability is conceptually similar in all models, but there are some differences in practice. Spatial variation can be represented at larger scales by dividing the catchment into large grid cells or subcatchments and implementing any (e.g., conceptual buckets, Richard’s equation) of the above representations of soil moisture independently for each element. Any lateral interaction within the grid cell or subcatchment is ignored in the explicit representation of variability. At smaller scales (e.g., on hillslopes), spatial variability can be represented by using a grid of soil water stores and allowing lateral flow between those stores, which is the approach used in distributed hillslope models such as Thales.19,62 A more detailed vertical representation in the unsaturated zone is possible in such distributed models by employing Richard’s equation (e.g., Zhang et al.63) Two- and three-dimensional implementations of saturated–unsaturated flow models also exist that can be used at the hillslope scale.64,65 Where models attempt to represent



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variation on a hillslope explicitly, they generally do attempt to simulate physically realistic soil moisture values that can be compared (almost) directly with soil moisture measurements (e.g., Western et al.19). Variability in the whole range of parameters of the soil water–vegetation system can be incorporated at the element scale in these models by varying parameters among elements. However setting realistic spatially variable parameter values is a very difficult task and a core problem for turning our understanding of soil moisture variability and scale effects into practically applicable tools. This leaves the subgrid variability, which can be represented in a variety of ways. Two important approaches in hydrologic models are the use of effective parameters and distribution functions. Sometimes subelement variability is simply ignored, as in the case of detailed spatially distributed hillslope models (e.g., Western et al.19) or conceptual hydrologic models (“multibucket models” referred to above) like the Stanford watershed model.66 In the case of distributed hillslope models it is often argued that nearly all the spatial variability is resolved and the remaining subelement variability is unimportant. The validity of this argument is doubtful in some cases; for example, where rainfall rates exceed infiltration rates, subelement scale variability in hydraulic conductivity can be important.67 In the case of conceptual hydrologic models it is argued that the effects of variability are of secondary importance and they can be incorporated in the model parameter values, usually by a process of calibration. In both these cases the model parameters really become effective parameters in that they are assumed to incorporate the effects of subgrid variability. In these cases parameters lose their physical meaning.68,69 It is also possible to develop effective parameter representations of the effects of subelement variability by more formal process analysis (e.g., Haverkamp et al.70). Conceptual bucket storages can be manipulated to represent spatial variability in a variety of ways. Multiple buckets of different sizes can be used in parallel71 or a continuous distribution of bucket sizes can be used.72,73 Another approach is to use a fixed-size bucket but a distribution to represent spatial variation in the bucket content at any time, which is the approach taken in Topmodel.42 These approaches retain a lumped conceptualization of the catchment (or subcatchment) but utilize a statistical representation of variability within that lumped catchment. Comparisons of measured soil moisture and simulated soil moisture can be made for some of these models but they need to be done statistically, for example, by comparing pdfs (e.g., References 74 through 76). In the examples where a continuous function is used to represent subelement variability, the function can be formally interpreted as a pdf that can be integrated with a point process model to derive the resulting behavior at the grid scale. This is the approach taken for calculating saturated area in models like Topmodel and the VIC72 or Xianxiang73 model, which use terrain-based and empirical pdfs, respectively. This is the so-called distribution function approach. Other examples of this approach include the infiltration model of Hawkins and Cundy,77 which utilizes an exponential distribution of saturated hydraulic conductivity and assumes that infiltration occurs at the lesser of the precipitation intensity or the saturated hydraulic conductivity. Scaling approaches exist that aim to represent spatial variability within Richard’s equation models (e.g., Sposito78). These scaling approaches still assume vertical flow, so they are not applicable where a significant source of spatial variability is lateral flow. The developing understanding of the statistical properties of spatial soil moisture described in the earlier sections will provide a basis for improving the representation of soil moisture variability using distribution function approaches within models.

C. MODELING SPATIAL PATTERNS

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SOIL MOISTURE

Soil moisture patterns are often used to set patterns of initial conditions in distributed models for event simulation (e.g., for flood prediction) and often produced as model outputs where distributed models are run for longer periods. These patterns can be at grid scales from meters to hundreds of kilometers, depending on the models employed. Most often initial condition patterns have been obtained by making some assumptions about the characteristics of the patterns9 or by interpolating




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observed point soil moisture patterns.10,79 Most often, simulated patterns from longer runs have not been tested against detailed spatial soil moisture data. An emerging area is the combined use of models and data within formal data assimilation approaches.80 These attempt to combine our process understanding (as expressed in a model) with measurements, both of which are uncertain, to make an optimal estimate of the system state, in this case the pattern of soil moisture. We have used lumped (the VIC model)75 and distributed (Thales)19,62,75 models at Tarrawarra. VIC is a lumped model that represents subelement variability using an empirical pdf of saturation deficit, while Thales represents a catchment as a large number of small elements and ignores the small amount of subelement variability in soil moisture. In each cell, the processes of infiltration, evapotranspiration and lateral flow are represented so parameters representing soil hydraulic behavior and vegetation characteristics are needed. Surface runoff was adequately represented by both models, but spatial patterns could be represented only by Thales. In the following discussion we will concentrate on the simulation of spatial patterns of soil moisture using Thales as an example of how the understanding of soil moisture variability described above can be used to develop a practical tool for runoff prediction, and how the process of modeling can identify shortcomings in our representation of variability. The parameter values in Thales primarily represent soil and soil–plant characteristics. They include the saturated hydraulic conductivity for lateral flow and a second value for deep seepage, porosity, wilting point, field capacity and soil depth (two layers). The two soil layers are an upper laterally transmissive layer and a deeper soil water reservoir that provides additional moisture for transpiration. Lateral flow hydraulic conductivity was set on the basis of field measurements, with some fine tuning by calibration. The porosity, wilting point, field capacity and soil depth were all set by analysis of wet, dry, and transition patterns and storage derived from soil moisture profiles, respectively. The depth of the transmissive layer was set on the basis of mapped A horizon depths. Thales did a good job of capturing the characteristics of the wet and dry patterns with very limited calibration (Figure 8.7). However, this is a relatively simple model where the effects of individual parameters can be understood and we had detailed observations of the soil water behavior, which is the dominant control on the rainfall-runoff process in this catchment. Inclusion of deep seepage and calibration of the deep seepage hydraulic conductivity was necessary to simulate runoff correctly. Further details of the modeling are provided in Western et al.19 Some shortcoming of the Thales model were identified by using a combination of spatial patterns of soil moisture in the top 30 cm and soil moisture profile data at 20 points around the catchment. The simulated transition from typical dry (uniform) to wet (topographically controlled) patterns (i.e., random and organized) was too slow, implying that lateral redistribution of water was underestimated. This was possibly related to extensive cracking of the surface soils, which may have provided rapid preferential flow paths until they closed shortly after the soil moisture increased. The catchment also tended to dry too quickly during the spring period, probably as a consequence of the model using the total soil water store to estimate the soil moisture control on transpiration, which is likely to be more sensitive to moisture in the upper part of the soil profile.

IV. MOVING TO LARGER SCALES While these studies show that modeling soil moisture patterns is practical in small, intensively monitored watersheds, a key challenge is to obtain similar performance out of these models without detailed soil moisture data. This requires reliable soil property information. At small scales this can be achieved with intensive studies, but for routine application this is not practical. An even greater challenge lies in large-scale simulations. In climate and weather modeling systems, land surface models are used to predict surface response to atmospheric forcing at grid scales from one to hundreds of kilometers and for continental and global model domains. There are three key challenges with this type of model.



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FIGURE 8.7 (See color insert following page 144.) Observed and simulated soil moisture patterns for Tarrawarra. The Thales model was used to simulate the soil moisture patterns.

• The first is to develop appropriate process descriptions at the grid scale. This will require understanding the importance of variability at different scales, including subelement variability and developing appropriate representations of the unresolved variability in space where necessary. (Temporal variability is well resolved as a consequence of short time steps imposed by numerical considerations in the atmospheric models.) • The second is to develop appropriate parameter sets describing the hydrologic and thermodynamic properties of the land surface, particularly soils and vegetation. These parameters need to be at the model grid scale, not the point scale typical of many soil property measurements. If point measurements are to be used, this scale transformation represents a major challenge for the soil physics and hydrology communities. • The third is to develop appropriate observational data sets for model testing (and parameter identification) and to reconcile the fundamental differences between the model state variable (soil moisture is really a conceptual construct in models at these scales) and the




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measurements that we make. These developments will require involvement of the soil physics community, as well as hydrologists and atmospheric modelers, if they are to be successful. Remote sensing has revolutionized the level of detail we have available about vegetation over the globe. The differential absorption and reflection of solar radiation at different wavelengths due to photosynthetic processes and the fact that this can be seen from space make the spatial and temporal distribution of vegetation well known in comparison to soils. It is difficult to apply remote sensing to soils because electromagnetic signals do not penetrate the soil at most wavelengths and even the soil surface is often obscured by vegetation. There is also no “signature” process analogous to photosynthesis that contributes distinctive information to the remotely sensed signal. One key simplification, though, is that soils are relatively stable over time, unlike vegetation, which is quite dynamic seasonally and even at shorter time scales. This temporal stability may enable effective soil parameters to be inferred from temporal changes in moisture content81 or moisture availability. Another challenge that arises when moving to larger scales is that new sources of variability become important and variability occurs in different factors at different characteristics scales. Figure 8.8 shows variability in elevation, slope, topographic wetness index, and precipitation for a large area centered at Austria. To examine the spatial scale of a number of variables relevant for the spatial distribution of soil moisture we performed an aggregation analysis. Specifically, we aggregated topographic attributes (topographic elevation, topographic slope, and the topographic wetness index, after Beven and Kirkby42), and mean annual precipitation from the resolution of 250 × 250 m to 256 × 256 km, quadrupling the grid scale in each step. In each aggregation step, a pixel value was taken as the arithmetic average of the 16 pixels at the finer resolution. In the terminology of Blöschl and Sivapalan,82 this aggregation is an increase in support (or spatial averaging) scale. For each aggregation step we now calculated what Blöschl83 termed the “variance between” and the “variance within.” The variance between is simply the spatial variance of the patterns. The variance within for, e.g., the 1-km aggregation step, has been calculated by first estimating the spatial variance

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FIGURE 8.8 The effect of aggregation on variance of a) topographic elevation, b) topographic slope, c) topographic wetness index, and d) mean annual precipitation for an area centered at Austria.



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of the 16 pixels of 250-m size contained in the 1-km pixel. This resulted in a spatial field of variances for all of Austria at a resolution of 1 × 1 km. The spatial average of this field then is termed the variance within. The results are shown in Figure 8.8. As would be expected, the variance between (solid lines) decreases with grid scale. Interestingly, the topographic wetness index, which provides a measure of topographic influences on lateral subsurface flow, exhibits the strongest reduction in variance, which implies the largest loss in information when one aggregates. In contrast, topographic elevation and precipitation show a much more modest decrease in variance because these tend to vary at larger scales. The “variance within” (dashed lines) shows the opposite trend because this is the portion of variability lost in the aggregation process. The trends of variance reduction and increase can be used to estimate a characteristic spatial scale of the variable examined. The intersection scale of the two variances is a robust estimator of the integral scale of the spatial field. The integral scale is a measure of the average spatial correlation. Figure 8.8 suggests that topographic elevation and precipitation are relatively large scale processes because the integral scale is on the order of 130 km in both cases. Topographic slope is a slightly smaller scale (integral scale of about 80 km) and the topographic wetness index is an even smaller scale variable (integral scale of about 2 km), which is consistent with the small scale variability of soil moisture mentioned above. The key point here is that a number of sources of variability with quite different spatial scales influence the likely soil moisture response of the land surface. While the above analysis is in the context of large-scale responses, similar arguments can be made at smaller scales.84 Most soil scientists and hydrologists agree that soil characteristics have an important impact on the hydrologic response of the landscape. Comparing Figure 8.1b and e is a clear if somewhat extreme demonstration of this for the key state variable soil moisture. Yet it is unclear exactly how much of the complex variation we see in the landscape we need to represent and for many parts of the world the soil hydrologic properties are poorly known. It is likely that the most important hydrologic soil parameters in models are properties, such as soil water storage capacity, or parameters such as soil/root depth and soil water release curves, that ultimately determine the soil water storage capacity in the model, as well as the parameters specifying the relationships between evapotranspiration and the soil moisture status. These parameters are extremely sparsely measured and quite variable in space. Developing continuous maps of them is extremely challenging. The standard approach to specifying them in large scale modeling is to utilize parameter sets that have ultimately been derived from soil mapping information supplemented by soil samples. Pedotransfer functions are often used to convert maps of soil landscape (which may contain more than one pedological soil type) or more commonly measured properties (e.g., texture) to soil hydrologic parameters. The resulting parameter maps contain sharp boundaries between uniform patches and, when used in models, they tend to impose an unrealistic zonal structure on the simulated soil moisture.85 One reason for this is that there is often as much within unit variability as between unit variability in the soil properties,86,87 which modelers usually ignore. This variability is likely to smooth out the transition in soil properties and soil moisture between soil units to some degree. The variability also means that there is significant statistical uncertainty in pedotransfer functions and even more statistical uncertainty when one tries to predict model parameters. One of the most important parameters in soil moisture models is the overall water-holding capacity of the soil. We have compared plant available water-holding capacities derived from analysis of soil moisture profile measurement time series and pedotransfer function-based interpretations88 of the soil landscapes mapped over the continent of Australia89 and found predictions biased low and great uncertainty between the observations and predictions.90 The waterholding capacities are estimated by combining estimates of field capacity, wilting point and soil depth (A and B horizons) for each soil landscape. The results indicate that the bias comes from underestimating the soil/regolith depth over which plants access water and that the uncertainty comes from all three soil parameters. © 2003 by CRC Press LLC



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Part of the uncertainty is related to our assumption that our samples could be represented by the dominant soil profile type for each landscape; actually a number of different soil types is in each soil landscape. Also, part undoubtedly comes from the uncertainty in the properties of a given soil profile type, which is related to spatial variability and sampling issues. None of this is meant as criticism of the interpretations of McKenzie et al.88 (all these issues are emphasized in their documentation of their interpretations), which are a substantial step forward. Rather, these results are an indication of the challenge involved in providing this sort of information at continental and global scales, scales at which modeling is undertaken to inform issues of great social and political importance (such as weather prediction and climate change). There are a number of potential paths forward. Relationships exist between soil characteristics and terrain and it may be possible to combine terrain information with information in soils databases.91–93 This requires accurate positioning of soil samples, which can be a significant impediment to utilizing historic data (N. McKenzie, personal communication), but with the advent of accurate GPS equipment there is no excuse for not providing sample coordinates (along with information on mapping datum) to an accuracy better than 20 m on the ground in the future. This sort of accuracy is required to match the resolution of modern digital terrain models and to give a good indication of position on the hillslope in complex terrain. Other sources of data (e.g., airborne radiometrics often flown for mineral exploration93) also exist and are exploited as covariates to assist in mapping soils. An alternative approach is to try to use information on system response to infer the system properties. A possible information source here is to use remote sensing of the surface energy balance and surface soil moisture linked with models. Such information can be used in an assimilation approach to estimate the system state better. For example, surface soil moisture80,94 or surface temperature95 can be used to estimate root zone soil moisture. It is also possible to estimate evaporative fluxes using remote sensing,96–98 from which information about soil moisture availability can be inferred. With sufficiently long time series of remotely sensed information, it should also be possible to apply calibration or inverse modeling techniques to estimate surface properties.99,100 One advantage of using thermal remote sensing information is that it tells us something about the availability of water to plants, which is more important than the absolute soil moisture for many applications. An exciting possibility is to try to use information on multiple system responses to constrain parameters further and to identify model deficiencies. For example, by using spatial and temporal measurements of soil moisture and temporal measurements of runoff, we were able to identify problems with the Thales model; we would not have been able to do this with a single source of data.19 Others are working on bringing together more diverse information from multiple cycles, including the water, energy and carbon cycles101 affecting the land surface. Because these cycles are interlinked through transpiration and photosynthesis processes, information on each can complement the other. Progress is certainly being made in the scaling of soil moisture and its representation at small and at large scales. At small scales modern instruments enabling detailed ground measurements of soil moisture and the possibility of studying soil and vegetation properties intensively have led to improved understanding. At large scales there are many important applications of knowledge on the characteristics of soils and soil moisture, many of which require modeling of one form or another. There are some significant challenges in representing the effects of variability and in obtaining relevant information on the land surface at these scales. However, we believe that a number of exciting possibilities, relating primarily to new data sources, are emerging and will lead to significant advances in our understanding of the response of land surface systems in the future. To fully realize these possibilities, we will need enhanced cooperation and sharing of ideas and perspectives, particularly among soil physicists, hydrologists, and atmospheric modelers. © 2003 by CRC Press LLC


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V. SUMMARY AND CONCLUSIONS Spatial variation in soil moisture is ubiquitous and many moisture-dependent processes are nonlinear, which leads to significant scale effects. The characteristics of spatial patterns of soil moisture change between landscapes and over time as different processes influence the soil moisture pattern. There is a general tendency for variance to be low for dry (close to wilting point) and wet (close to saturation point) conditions and to be higher for intermediate conditions. Climate, particularly the balance between potential evapotranspiration and rainfall at seasonal time scales, largely determines the seasonal average temporal pattern of soil moisture. In arid landscapes moisture is typically low and variance increases with increasing moisture content. In humid landscapes moisture is typically high and variance decreases with increasing moisture content. In landscapes with an intermediate climate, particularly those where rainfall dominates in one season and evapotranspiration dominates in the other, a wider range of spatial mean moisture content is likely to be observed over time, and a peak in variance may occur. Due to temporal changes in soil moisture, the processes that are dominant in determining the spatial pattern of soil moisture can change over time. This is essentially related to dramatic decreases in hydraulic conductivity and increases in suction as the soil dries. A consequence of this is that effective flow length scales become very short during dry conditions and the spatial soil water balance is essentially controlled by differences in the point scale vertical water balance. During wet conditions flow length scales get much longer and in some landscapes lateral flow down hillslopes becomes the dominant control on the spatial pattern of soil moisture. This can lead to topographically controlled connected patterns of soil moisture in some landscapes. The connectivity has an important impact on the surface runoff response of the landscape. Connectivity can be analyzed using connectivity functions described above. Although topographic control of soil moisture patterns does occur in some landscapes for part or all of the time, terrain is a relatively poor predictor of soil moisture patterns and variability. The common assumption that terrain is the dominant control on soil moisture patterns is often incorrect. Spatial patterns of soil properties and vegetation can be a dominant control on the soil moisture pattern in some landscapes. Land surface modeling is an important tool for prediction and understanding of system behavior. Models must deal with scale effects related to the variability of soil moisture and soil properties in an appropriate manner if they are to represent the system behavior effectively. This is particularly true at large scales where there are significant constraints related to algorithms and soil property information. Measurement and analysis of the behavior of soil moisture variability is informing the development of new modeling structures but new measurements at larger scales are required. In particular significant interest in the representation of subgrid variability in large scale models exists. There is a significant opportunity for interdisciplinary efforts including the soil physics and hydrologic/atmospheric modeling communities to address in particular the problem of developing reliable information on soil properties relevant to large scale modeling.

VI. ACKNOWLEDGMENTS Our soil moisture research has been supported financially by the Australian Research Council, the Oesterreichische National Bank, Vienna, and the Australian Department of Industry Science and Tourism. Rodger Young and Sen-Lin Zhou have assisted in various aspects of the work. Tom McMahon has provided enthusiastic support, encouragement and guidance for the projects.

REFERENCES 1. Western, A.W., Grayson, R.B., and Blöschl, G., Scaling of soil moisture: a hydrologic perspective, Annu. Rev. Earth Planetary Sci., 30, 149, 2002.




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2. Bethune, M. and Turral, H., Modeling water movement in cracking soils: a review, in Modeling Water Movement in Cracking Soils, Bethune, M. and Kirby, M., Eds., Dept. Natural Resources and Environment, Victoria, Melbourne, 2001, 29. 3. Gutknecht, D., Grundphänomene hydrologischer prozesse, Zürcher Geographische Schriften, 53, 25, 1993. 4. Blöschl, G. et al., Organization and randomness in catchments and the verification of distributed hydrologic models, Eos, Trans. Am. Geophys. Union, 74, 317, 1993. 5. Grayson, R.B. and Blöschl, G., Spatial Patterns in Hydrological Processes: Observations and Modeling, Cambridge University Press, 2000, 406. 6. Grayson, R.B. et al., Preferred states in spatial soil moisture patterns: local and non-local controls, Water Resour. Res., 33, 2897, 1997. 7. Western, A.W. et al., Observed spatial organization of soil moisture and its relation to terrain indices, Water Resour. Res., 35, 797, 1999. 8. Western, A.W., Grayson, R.B., and Blöschl, G., Spatial scaling of soil moisture: a review and some recent results, in Modsim 2001, Canberra, Australia, 2001. 9. Grayson, R.B., Blöschl, G., and Moore, I.D., Distributed parameter hydrologic modeling using vector elevation data: Thales and TAPES-C, in Computer Models of Watershed Hydrology, Singh, V.P., Ed., Water Resources Pub., Highlands Ranch, CO, 1995, pp. 669. 10. Merz, B. and Plate, E.J., An analysis of the effects of spatial variability of soil and soil moisture on runoff, Water Resour. Res., 33, 2909, 1997. 11. Western, A.W., Blöschl, G., and Grayson, R.B., Toward capturing hydrologically significant connectivity in spatial patterns, Water Resour. Res., 37, 83, 2001. 12. Pielke, Sr., R.A., Influence of the spatial distribution of vegetation and soils on the prediction of cumulus convective rainfall, Rev. Geophys., 39, 151, 2001. 13. Weaver, C.P. and Avissar, R., Atmospheric disturbances caused by human modification of the landscape, Bull. Am. Meteorol. Soc., 82, 269, 2001. 14. Mills, G.A., The Enfield fire — LAPS model results, in Proc., 6th Fire Weather Workshop, Bureau of Meteorology, Australia, Hahndorf, South Australia, 1995. 15. Raupach, M.R. and Finnigan, J. J., Scale issues in boundary-layer meteorology: surface energy balances in heterogeneous terrain, Hydrol. Process., 9, 589, 1995. 16. Western, A.W. and Grayson, R.B., The Tarrawarra data set: soil moisture patterns, soil characteristics and hydrological flux measurements, Water Resour. Res., 34, 2765, 1998. 17. Woods, R.A. et al., Experimental design and initial results from the Mahurangi River variability experiment: MARVEX, in Observations and Modeling of Land Surface Hydrological Processes, Lakshmi, V., Albertson, J. D., and Schaake, J., Eds., American Geophysical Union, 2001. 18. Williams, R.E., Comment on “statistical theory of groundwater flow and transport: pore to laboratory, laboratory to formation and formation to regional scale” by Gedeon Dagan, Water Resour. Res., 24, 1197, 1988. 19. Western, A.W. and Grayson, R.B., Soil moisture and runoff processes at Tarrawarra, in Spatial Patterns in Catchment Hydrology — Observations and Modeling, Grayson, R.B. and Blöschl, G., Eds., Cambridge University Press, 2000, 209. 20. Western, A.W. et al., spatial correlation of soil moisture in small catchments and its relationship to dominant spatial hydrological processes, J. Hydrol., in review. 21. Barling, R.D., Moore, I.D., and Grayson, R.B., A quasi-dynamic wetness index for characterizing the spatial distribution of zones of surface saturation and soil water content, Water Resour. Res., 30, 1029, 1994. 22. Chirico, G.B., Terrain-based distributed modeling for investigating scale issues in hydrology, in Ingegneria Idraulica, Univesità Degli Studi Napoli Federico II, Napoli, 2001, 182. 23. Western, A.W. et al., Field investigations of spatial organization of soil moisture in a small catchment, in Water and the Environment, 23rd Hydrology and Water Resources Symposium, Vol. 2, i.e., Aust. Nat. Conf. Pub. 96–05, 21–24 May, 1996, Hobart, 1996, 547. 24. Wilson, D. et al., Spatial and temporal soil moisture distribution, Eos, Trans. Am. Geophys. Union, 80, F323, 1999. 25. Grayson, R.B. and Western, A.W., Terrain and the distribution of soil moisture, Hydrol. Process., 15, 2689, 2001.



140


26. Anderson, M.G. and Burt, T.P., The role of topography in controlling throughflow generation, Earth Surf. Process., 3, 331, 1978. 27. Bárdossy, A. and Lehmann, W., Spatial distribution of soil moisture in a small catchment. Part 1: Geostatistical analysis, J. Hydrol., 206, 1, 1998. 28. Burt, T.P. and Butcher, D.P., Development of topographic indices for use in semi-distributed hillslope runoff models, Zietschrift für Geomorphologie N.F., Suppl.-Bd. 58, 1, 1986. 29. Famiglietti, J. S., Rudnicki, J. W., and Rodell, M., Variability in surface moisture content along a hillslope transect: Rattlesnake Hill, Texas, J. Hydrol., 210, 259, 1998. 30. Nyberg, L., Spatial variability of soil water content in the covered catchment at Gårdsjön, Sweden, Hydrol. Process., 10, 89, 1996. 31. Walker, J. P., Willgoose, G.R., and Kalma, J. D., The Nerrigundah data set: soil moisture patterns, soil characteristics and hydrological flux measurements, Water Resour. Res., 37, 2653, 2001. 32. Zavlasky, D. and Sinai, G., Surface hydrology: I — explanation of phenomena, J. Hydraulics Div., Proc. Am. Soc. Civil Eng., 107, 1, 1981. 33. Western, A.W., Blöschl, G., and Grayson, R.B., Geostatistical characterization of soil moisture patterns in the Tarrawarra Catchment, J. Hydrol., 205, 20, 1998. 34. Western, A.W. and Blöschl, G., On the spatial scaling of soil moisture, J. Hydrol., 217, 203, 1999. 35. Wilson, D.J. et al., Spatial distribution of soil moisture at 6 cm and 30 cm depth, Mahurangi River Catchment, New Zealand, J. Hydrol., in review, 2002. 36. Deutsch, C.V. and Journel, A.G., GSLIB Geostatistical Software Library and User’s Guide, Oxford University Press, New York, 1992. 37. Western, A.W., Blöschl, G., and Grayson, R.B., How well do indicator variograms capture the spatial connectivity of soil moisture? Hydrol. Process., 12, 1851, 1998. 38. Stauffer, D. and Aharony, A., Introduction to Percolation Theory, Taylor and Francis, London, 1991. 39. Grimmet, G., Percolation, Springer-Verlag, New York, 1989. 40. Allard, D. and Group, H., On the connectivity of two random set models: the truncated Gaussian and the Boolean, in Geostatistics Tróia ‘92, Vol. 1, Soares, A., Ed., Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993, 467. 41. Allard, D., Simulating a geological lithofacies with respect to connectivity information using the truncated Gaussian model, in Geostatistical Simulations. Proceedings of the Geostatistical Simulation Workshop, Fontainbleau, France, 27–28 May 1993, Armstrong, M. and Down, P.A., Eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, Fontainbleau, France, 1994, 197. 42. Beven, K.J. and Kirkby, M.J., A physically based variable contributing area model of basin hydrology, Hydrol. Sci. Bull., 24, 43, 1979. 43. O’Loughlin, E.M., Saturation regions in catchments and their relations to soil and topographic properties, J. Hydrol., 53, 229, 1981. 44. O’Loughlin, E.M., Prediction of surface saturation zones in natural catchments by topographic analysis, Water Resour. Res., 22, 794, 1986. 45. Moore, I.D., Burch, G.J., and Mackenzie, D.H., Topographic effects on the distribution of surface soil water and the location of ephemeral gullies, Trans. Am. Soc. Agric. Eng., 31, 1098, 1988. 46. Wilson, J.P. and Gallant, J.C., Terrain Analysis: Principles and Applications, John Wiley & Sons, New York, 2000. 47. Moore, I.D., Grayson, R.B., and Ladson, A.R., Digital terrain modeling: a review of hydrological, geomorphological, and biological applications, Hydrol. Process., 5, 3, 1991. 48. Gallant, J.C. and Dowling, T.D., A multi-resolution index of valley bottom flatness for mapping depositional areas, Water Resour. Res., in review. 49. Ladson, A.R. and Moore, I.D., Soil water prediction on the Konza Prairie by microwave remote sensing and topographic attributes, J. Hydrol., 138, 385, 1992. 50. Burt, T.P. and Butcher, D.P., Topographic controls of soil moisture distributions, J. Soil Sci., 36, 469, 1985. 51. Moore, R.D. and Thompson, J. C., Are water table variations in a shallow forest soil consistent with the TOPMODEL concept? Water Resour. Res., 32, 663, 1996. 52. Jordan, J. P., Spatial and temporal variability of stormflow generation processes on a Swiss catchment, J. Hydrol., 152, 357, 1994.




141

53. Western, A.W., Grayson, R.B., and Green, T.R., The Tarrawarra Project: high resolution spatial measurement, modeling and analysis of soil moisture and hydrological response, Hydrol. Process., 13, 633, 1999. 54. Beven, K. and Freer, J., A dynamic TOPMODEL, Hydrol. Process., 15, 1993, 2001. 55. Loague, K., Soil water content at R-5. Part 1. Spatial and temporal variability, J. Hydrol., 139, 233, 1992. 56. Charpentier, M.A. and Groffman, P.M., Soil moisture variability within remote sensing pixels, J. Geophys. Res., 97, 18987, 1992. 57. Bell, K.R. et al., Analysis of surface moisture variations within large-field sites, Water Resour. Res., 16, 796, 1980. 58. Famiglietti, J. S. et al., Ground-based investigation of soil moisture variability within remote sensing footprints during the Southern Great Plains 1997 (SGP97) Hydrology Experiment, Water Resour. Res., 35, 1839, 1999. 59. Mohanty, B.P., Skaggs, T.H., and Famiglietti, J.S., Analysis and mapping of field-scale soil moisture variability using high-resolution, ground-based data during the Southern Great Plains 1997 (SGP97) Hydrology Experiment, Water Resour. Res., 36, 1023, 2000. 60. Cleveland, W.S., Robust locally weighted regression and smoothing scatterplots, J. Am. Stat. Assoc., 74, 829, 1979. 61. Grayson, R. and Blöschl, G., Spatial modeling of catchment dynamics, in Spatial Patterns in Catchment Hydrology — Observations and Modeling, Grayson, R.B. and Blöschl, G., Eds., Cambridge University Press, 2000, 51. 62. Chirico, G.B., Grayson, R.B., and Western, A.W., Lateral flow processes in a small experimental catchment: analysis and distributed modeling, Hydrol. Process., in press, 2003. 63. Zhang, L. et al., Estimation of soil moisture and groundwater recharge using the TOPOG_IRM model, Water Resour. Res., 35, 149, 1999. 64. Binley, A., Elgy, J., and Beven, K., A physically based model of heterogeneous hillslopes 1. Runoff production, Water Resour. Res., 25, 1219, 1989. 65. VanderKwaak, J. E. and Loague, K., Hydrologic-response simulations for the R-5 catchment with a comprehensive physics-based model, Water Resour. Res., 37, 999, 2001. 66. Crawford, N.H. and Linsley, R.K., Digital simulation in hydrology, Stanford Watershed Model IV, Dept. of Civil Eng., Stanford Univ., Stanford, 1966. 67. Yu, B., Cakurs, U., and Rose, C.W., An assessment of methods for estimating runoff rates at the plot scale, Trans. Am. Soc. Agric. Eng., 41, 653, 1998. 68. Beven, K., Changing ideas in hydrology — the case of physically based models, J. Hydrol., 105, 157, 1989. 69. Grayson, R.B. and Nathan, R.J., On the role of physically based models in engineering hydrology, in WATERCOMP, IEAust, Melbourne, March 30–April 1, 1993, 45. 70. Haverkamp, R. et al., Scaling of the Richards equation and its application to watershed modeling, in Scale Dependence and Scale Invariance in Hydrology, Sposito, G., Ed., Cambridge University Press, Cambridge, 1998, 190. 71. Boughton, W.C., An Australian water-balance model for semiarid watersheds, J. Soil Water Conserv., 50, 454, 1995. 72. Wood, E.F., Lettenmaier, D.P., and Zartarian, V.G., A land-surface hydrology parameterization with subgrid variability for general circulation models, J. Geophys. Res., 97(D3), 2717, 1992. 73. Zhao, R.-J., The Xinanjiang model applied in China, J. Hydrol., 135, 371, 1992. 74. Western, A.W. et al., Wimmera River: hydrology, data and modeling, in Hydrology and Water Resources Symposium, IEAust, Newcastle, June 30 - July 2 1993, 1993, 59. 75. Western, A.W. et al., The Tarrawarra Project: high resolution spatial measurement and analysis of hydrological response, in MODSIM 97 International Congress on Modeling and Simulation, Vol. 1, McDonald, A.D. and McAleer, M., Eds., The Modeling and Simulation Society of Australia, Inc, Hobart, Tasmania, 1997, 403. 76. Kalma, J. D., Bates, B.C. and Woods, R.A., Predicting catchment-scale soil moisture status with limited field measurements, Hydrol. Process., 9, 445, 1995. 77. Hawkins, R.H. and Cundy, T.W., Steady-state analysis of infiltration and overland flow for spatiallyvaried hillslopes, Hydrol. Sci. Bull., 23, 251, 1987.



142


78. Sposito, G., Scale Dependence and Scale Invariance in Hydrology, Cambridge University Press, Cambridge, 1998, 423. 79. Merz, B., Effects of spatial variability on the rainfall runoff process, J. Hydrol., 213, 304, 1998. 80. Walker, J. P. and Houser, P.R., A methodology for initializing soil moisture in a global climate model: assimilation of near-surface soil moisture observations, J. Geophys. Res. — Atmos., 106, 11761, 2001. 81. Hollenbeck, K.J. et al., Identifying soil hydraulic heterogeneity by detection of relative change in passive microwave remote sensing observations, Water Resour. Res., 32, 139, 1996. 82. Blöschl, G. and Sivapalan, M., Scale issues in hydrological modeling: a review, Hydrol. Process., 9, 251, 1995. 83. Blöschl, G., Scaling issues in snow hydrology, Hydrol. Process., 13, 2149, 1999. 84. Seyfried, M.S. and Wilcox, B.P., Scale and the nature of spatial variability: Field examples having implications for hydrologic modeling, Water Resour. Res., 31, 173, 1995. 85. Houser, P., Goodrich, D., and Syed, K., Runoff, precipitation, and soil moisture at Walnut Gulch, in Spatial Patterns in Catchment Hydrology: Observations and Modeling, Grayson, R. and Blöschl, G., Eds., Cambridge University Press, Cambridge, 2000, 125. 86. Warrick, A.W. et al., Kriging versus alternative interpolators: errors and sensitivity to model inputs, in Field-Scale Water and Solute Flux in Soils, Roth, K., Flühler, H., Jury, W.A. and Parker, J. C., Eds., Birkhäuser Verlag, Basel, 1990, 157. 87. Vertessy, R. et al., Storm runoff generation at La Cuenca, in Spatial Patterns in Catchment Hydrology: Observations and Modeling, Grayson, R. and Blöschl, G., Eds., Cambridge University Press, Cambridge, 2000, 247. 88. McKenzie, N.J. et al., Estimation of soil properties using the Atlas of Australian Soils, CSIRO Land and Water, Canberra, 2000. 89. Bureau of Rural Sciences after Commonwealth Scientific and Industrial Research Organization, Digital Atlas of Australian Soils (ARC/INFO® vector format). [Online] Available HTML: http://www.brs.gov.au/data/datasets, 1991. 90. Ladson, A. et al., Estimating extractable soil moisture content for Australian soils, in Hydrol. Water Resour. Symp. 2002 [CD-ROM], i.e., Aust, Melbourne, Australia, 2002, 9. 91. Gessler, P.E. et al., Soil-landscape modeling in southeastern Australia, in 2nd Int. Conf./Wshop. Integrating Geogr. Inf. Syst. Environ. Modeling, Sept 26–30, 1993, Breckenridge, CO, 1993. 92. Gessler, P.E. et al., Soil-landscape modeling and spatial prediction of soil attributes, Int. J. Geogr. Inf. Syst., 9, 421, 1995. 93. McKenzie, N.J. and Ryan, P.J., Spatial prediction of soil properties using environmental correlation, Geoderma, 89, 67, 1999. 94. Walker, J. P., Willgoose, G.R., and Kalma, J. D., One-dimensional soil moisture profile retrieval by assimilation of near-surface observations: a comparison of retrieval algorithms, Adv. Water Resour., 24, 631, 2001. 95. Lakshmi, V., A simple surface temperature assimilation scheme for use in land surface models, Water Resour. Res., 36, 3687, 2000. 96. Bastiaanssen, W.G.M., SEBAL-based sensible and latent heat fluxes in the irrigated Gediz Basin, Turkey, J. Hydrol., 229, 87, 2000. 97. Bastiaanssen, W.G.M., Molden, D.J., and Makin, I.W., Remote sensing for irrigated agriculture: examples from research and possible applications, Agric. Water Manage., 46, 137, 2000. 98. Roerink, G.J., Su, Z., and Menenti, M., S-SEBI: a simple remote sensing algorithm to estimate the surface energy balance, Phys. Chem. Earth, Part B — Hydrol. Oceans Atmos., 25, 147, 2000. 99. Franks, S.W., Beven, K.J., and Gash, J. H.C., Multi-objective conditioning of a simple SVAT model, Hydrol. Earth Syst. Sci., 3, 477, 1999. 100. McCabe, M.F., Franks, S.W., and Kalma, J. D., On the estimation of land surface evapotranspiration: parameter inference in SVAT modeling using a temporal record of thermal data, in Water 99 Joint Cong., Vol. 1, IEAust, Brisbane, 1999, 16. 101. Raupach, M.R. et al., Terrestrial biosphere models and forest-atmosphere interactions, in Forests and Water, Vertessy, R. and Elsenbeer, H., Eds., IUFRO, 2002, in press.1.Western, A.W., Grayson, R.B., and Blöschl, G., Scaling of soil moisture: a hydrologic perspective, Annu. Rev. Earth Planetary Sci., 30, 149, 2002.



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An Evaluation of Interpolation Methods for Local Estimation of Solute Concentration T.R. Ellsworth, P.M. Reed, and R.J.M. Hudson

CONTENTS I. Introduction...........................................................................................................................143 II. Interpolation Methods ..........................................................................................................145 A. Spatial Data Sets........................................................................................................145 1. Unsaturated Zone, Experimental Data ................................................................145 2. Saturated Zone, Simulated Data..........................................................................145 B. Search Neighborhood ................................................................................................147 C. Deterministic Interpolation Methods.........................................................................148 1. Traditional Inverse Distance (ID2)......................................................................148 2. Nonlinear Least Squares Inverse Distance (NLS) ..............................................148 D. Geostatistical Interpolation Methods.........................................................................149 1. Linear Geostatistics .............................................................................................150 2. Nonlinear Geostatistics........................................................................................151 E. Evaluation Criteria.....................................................................................................153 F. Case Study .................................................................................................................154 G. Cross-Validation Scores.............................................................................................156 H. Validation Scores .......................................................................................................158 I. Estimation Uncertainty ..............................................................................................159 III. Summary...............................................................................................................................160 References ......................................................................................................................................161

I. INTRODUCTION The growing concern about contamination of water resources in the United States has led to increased research, management, and remediation activity aimed at preserving these resources for present and future generations. Many of these efforts require considerable scientific research to quantify and model chemical fate and transport processes, with each type of water resource having different challenges to be solved. In every case, demonstrating that we understand these processes well enough to manage natural systems requires that we simulate observed distributions of contaminants in the field. Thus, the challenge of understanding the processes is intimately tied to the challenge of characterizing the true spatio-temporal distributions of solutes in the field. Arguably, the difficulty of observing these distributions is greatest in soil and groundwater systems, which is one reason that we have chosen to apply geostatistical methods to the study of this subject. The spatio-temporal characterization of chemical con-



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centrations in saturated and unsaturated porous media porous media is an essential component in each of these three endeavors (i.e., research, monitoring, and remediation). This characterization is also perhaps one of the most challenging tasks facing those involved in this work. There are several reasons why this task is so challenging. The first is that the mechanisms governing the fate and transport of chemicals in porous media are coupled nonlinear processes that occur in highly heterogeneous environments. This results in spatial and temporal distributions of contaminant concentrations that are highly irregular, and situations in which concentrations may vary over several orders of magnitude within relatively small scales (i.e., cm/min). Second, it is a relatively difficult and expensive task to sample and measure chemical concentrations in porous media. Relatively few indirect and noninvasive methods exist for measuring solute concentrations in porous media, although there is hope in that such methods are on the increase. Some methods include radiation radiochemical methods (i.e., Geiger counters to study radionuclide transport), bulk solution electrical conductivity measurement methods (frequency and timedomain reflectometry), and microscale methods primarily suited for laboratory research (x-ray computed tomography). However, in general, direct measurements are necessary. This requires the installation and monitoring of multilevel sampling wells, monitoring tile effluent, soil coring, or excavation. When the difficult and costly effort associated with obtaining samples is coupled with the additional expense associated with chemical extractions and subsequent analysis in the laboratory, it is apparent that obtaining measurements of chemical concentrations in porous media is an arduous task. Thus, it is often the case in the associated research, monitoring/assessment, and remediation efforts that we are faced with a situation in which we must characterize the spatial and temporal distribution of contaminants relying on sparse, often irregularly and preferentially located (in space and time) data. Given the highly skewed, erratic nature of solute distributions in porous media and the relatively sparse data available for characterization, it is clear that this task represents perhaps one of the greatest challenges to the scientists and engineers who work in these disciplines. The present chapter addresses this task of plume characterization. Much of this chapter is primarily based on previous work that has been reported elsewhere.1,2 We summarize these results here and also include two new case studies to expand the analyses. The primary focus of this work is on spatial plume interpolation, although the methods could be applied with modification to spatio-temporal characterization. The methods we investigate are those that would be readily available to practitioners involved with assessment and remediation efforts in these disciplines; they include various linear and nonlinear geostatistical methods, as well as inverse distance interpolation and a nonlinear variant of inverse distance interpolation. As an alternative to what we discuss here, we also note recent advances in what has been termed “data assimilation” and also “modern spatiotemporal geostatistics.” These latter approaches couple traditional geostatistical tools with assumptions about physical process models and what is known of the governing physical laws to achieve spatial/temporal estimation (see, for example, McLaughlin3 and Christakos 4). In the present chapter, we focus primarily on local estimation (i.e., estimating the chemical concentration and the uncertainty in this concentration at any specific spatial location), which is a precursor to the task of global estimation.5 We do not address global estimation, which is essentially an upscaling exercise in which, for example, the goal is to Estimate the total mass of chemical within a spatial region Estimate the plume center of mass or Quantify various measures related to the spread of the contaminant about the plume center of mass, such as the total mass of chemical within a spatial region or a plume’s center of mass © 2003 by CRC Press LLC


An Evaluation of Interpolation Methods for Local Estimation of Solute Concentration

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II. INTERPOLATION METHODS A. SPATIAL DATA SETS We have selected several data sets to evaluate the various plume interpolation methods. These data sets differ with respect to the underlying porous media, the spatial extent of the contamination and chemical residence time. Also, and perhaps most importantly, they differ with respect to the spatial density of the sampling network. Of the following, the saturated zone data sets were evaluated in the research reported by Reed et al.2 We summarize these findings here and also augment this earlier work by providing an evaluation of the interpolation methods with respect to the following unsaturated case studies as well. 1. Unsaturated Zone, Experimental Data The experimental data for solute spatial distributions in unsaturated soil are obtained from a field study performed near Champaign, Illinois. The soils at the research site are a silty clay loam that overlies glacial till. The experimental procedures are briefly described here. Two solutes (chloride and bromide) were applied in series under steady, unsaturated water flow to nine plots. Solutes were applied to achieve a pulse of finite duration with a two-dimensional surface solute boundary condition. The solute source area at the surface of three of the nine plots was 0.15 × 0.15 m2, 0.30 × 0.30 m2 for three additional plots, and the final three plots had a source area of 0.60 × 0.60 m2, with plot sizes of 0.45 × 0.45 m2, 0.90 × 0.90 m2, and 1.8 × 1.8 m2, respectively. Each plot was destructively sampled with soil coring (2.5-, 5-, and 10-cm diameter cores for the small, intermediate, and large plots, respectively). Approximately 20% of each plot was excavated and analyzed with this sampling method. Soil solution for analysis was obtained via suction filtration.6 The timing of solute applications and the cumulative net water infiltration at the time of sampling were staggered so that the data provide insight into the temporal evolution of solute plumes in unsaturated soil. This sampling scheme produced approximately 15,000 resident concentration measurements for each solute, with each sample representing a specific location in space and time. These data thus provide a relatively detailed spatial characterization of 18 separate three-dimensional solute plumes. Two subsets of these data will be used in the present effort to evaluate the various interpolation methods, with the subsets varying with respect to spatial density of the sampling networks. Two of these eighteen plumes are used in the present effort to evaluate the various interpolation methods. One of the two plumes selected was sampled shortly after application and thus has spread relatively little; this plume will be noted as small plume, unsaturated soil, SPU, with 1049 observations while the other plume encompasses a greater spatial extent (large plume, unsaturated soil, LPU, with 1852 spatial observations). As an illustration of the data, Figure 9.1 provides a three-dimensional perspective of the sampled spatial distribution of bromide for the LPU. Each dot within the frame represents a sample location, and the size of the dot is a relative measure of the observed concentration at that spatial location. 2. Saturated Zone, Simulated Data The saturated zone test case data were obtained from a tetrachloroethylene (PCE) transport simulation. The hydrogeology for the simulation was generated based upon a field site located at Lawrence Livermore National Laboratory (LLNL) in Livermore, California (for details, see Maxwell et al.7 This site is under federal management (under the U.S. Comprehensive Environmental Response, Compensation, and Liability Act, i.e., CERCLA) with extensive measurements and monitoring data available. These data were used to condition the simulated aquifer, which is a highly heterogeneous alluvial system. The flow and transport domain was discretized into approximately 50 million nodes. Flow was modeled using a parallel, steady-state flow code. The transport simulation, for a continuous PCE source located below the land surface, was achieved




Depth (cm)

146

FIGURE 9.1 The spatial distribution of the measured bromide concentration for case study LPU. Each symbol within the frame indicates a sample location, and the size of the symbol indicates the relative bromide concentration.

using a LaGrangian particle-tracking code and was performed on a massively parallel computing facility at LLNL. One of the objectives of this work is to evaluate the influence of data sparsity on the performance of the interpolation methods, similar to that which a practitioner would encounter in a real monitoring/assessment situation. To achieve this, several test cases were developed from the Maxwell et al.7 simulation. The data sets for each test case differ in two primary aspects. First, the number of samples differs considerably, with a total of 26, 58, or 124 sampling locations. Second, the maturity and extent of the plume at the time of sampling also differs among test cases, with the assumption that an increase in monitoring wells (and thus sample locations) will occur as the plume evolves in space and in time. The first, and smallest test case is obtained by sampling 20 hypothetical monitoring wells relatively early in the contaminant simulation. Several of these monitoring wells are assumed to be multilevel sampling wells, thus providing a total of 26 sample locations within approximately 1 million cubic meters of aquifer. This small plume, saturated case (SPS) represents a worst case scenario for spatial plume interpolation. A medium plume, saturated case (MPS) was obtained by a hypothetical sampling with 29 monitoring wells and provided a total of 58 sample locations within 6 million cubic meters of contaminated aquifer. The final case, denoted large plume, saturated case (LPS), assumes 59 monitoring wells, and provides a total of 124 sample locations for representing 16 million cubic meters of contaminated aquifer. Figure 9.2 illustrates a three-dimensional perspective of the spatial distribution of the LPS plume at the time of the simulated sampling, and Figure 9.3 provides a two-dimensional projection of the plume onto the horizontal plane, showing the extent of the plume (dots) as well as the sample locations (asterisks). © 2003 by CRC Press LLC



147

Z (m)

110

90

70 100

600

X(

700

m) 1100

500

m)

Y(

600. 450.

Y (m)

750.

FIGURE 9.2 The spatial distribution of the simulated PCE concentration for the case study LPS. Each symbol indicates the spatial location of a nonzero PCE concentration.

0.

200.

400.

600.

800.

1000.

1200.

X (m) FIGURE 9.3 Two-dimensional perspective (in the horizontal plane) of the monitoring well locations (shown with asterisk) relative to the spatial plume distribution (shown with dots).

B. SEARCH NEIGHBORHOOD To implement the subsequent estimation approaches, it was necessary to define a search neighborhood. The characteristics of the search neighborhood determine how many neighboring samples to include in the estimation procedure. These characteristics have a significant influence on the results of the estimation. The presence of a plume within the interpolation domain presents a significant challenge in terms of search neighborhoods. We followed the recommendations of Cooper and Istok8 and Goovaerts9 in defining the neighborhood for each of the subsequent estimation methods. This led us to define an ellipsoid search neighborhood, with the major and minor axes defined to be equal to a preliminary estimate of one half the plume lengths in each of the principal directions (for details, see Reed et al.2 Defining our search neighborhood in this fashion implicitly assumes that the orthogonal coordinates of the experimental sample sets are congruent with the principal directions defining the plume orientation. The basis for such a coordinate transformation, if required, was achieved visually by examining a proportional location map of the measured values. In addition, because of the preferential sampling that often occurs in spatial plume sampling, we employed an octant search method, with a minimum of one and a maximum of three observed values within each octant. This provided a minimum of 8 and a maximum of 24 neighboring values for each spatial estimate. If the minimum neighboring data were not present, then we did not provide an estimate at that spatial location. We used the same search neighborhood for each of the following interpolation methods, with the exception of the intrinsic kriging and nonlinear least squares methods. © 2003 by CRC Press LLC


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The only exception was for SPS and MPS, since data sparsity prevented us from employing a local search neighborhood for the intrinsic kriging and nonlinear least squares regression methods (discussed subsequently); thus a global search neighborhood was used in these two cases. A motivating factor that leads to the chosen characteristics of the local search neighborhood is that this neighborhood “scales” with the spatial extent of the plume, and can be readily applied by a practicing engineer to any spatial plume estimation problem. We believe that, given the wide range of spatial plume extents examined in the present study, this type of search neighborhood makes the following evaluation more general than it would be if the search neighborhood were tailored for each specific situation.

C. DETERMINISTIC INTERPOLATION METHODS Variants of the classic inverse distance estimation approach are very popular in practice, due primarily to three factors: 1) the ease of implementation, 2) the exactness of the interpolator, which reproduces the observed value at each sample location, and 3) the bounded nature, i.e., all of the estimated values lie between the minimum and maximum observed values. Numerous studies have compared the spatial estimation performance of inverse distance interpolation methods to various kriging variants.10–14 However, to our knowledge, none of these study comparisons has examined performance of these interpolators for spatial plume estimation. As discussed above, spatial plume estimation presents a significant challenge in that sample locations are within the plume as well as outside the plume boundaries. Furthermore, as has been theoretically shown by Kitanidis,15 and experimentally shown by Ellsworth,6 the concentration data are generally heteroskedastic. Thus the data are, statistically speaking, strongly nonstationary. A drawback of the inverse distance approaches as they are commonly implemented is that they are deterministic methods, in that they do not provide an estimate of the uncertainty associated with the spatial estimation. However, we note that it is possible in principle, through resampling methods such as the bootstrap, to obtain an estimate of the uncertainty in spatial estimation. Such an approach would be similar to that employed by Barry and Sposito,16 who characterized the uncertainty in global plume estimates using a bootstrap method coupled with a variant of inverse distance interpolation. 1. Traditional Inverse Distance (ID2) The first deterministic method we examined is perhaps most often used in practice, due to the simplicity of implementation. In this approach, the value at unsampled locations is estimated as inversely proportional to the square of the distance between neighboring observations; thus the method is termed inverse distance to the power of 2 (ID2). We note that it possibly makes more sense to view the inverse distance exponential weighting coefficient as a parameter. Thus the optimal weighting coefficient for spatial estimation would depend on the specific nature of the spatial plume distribution. This concept underlies our second deterministic method. 2. Nonlinear Least Squares Inverse Distance (NLS) The approach we employ here is a special case of that used by Barry and Sposito,6 in that we arbitrarily set one of their exponential weighting coefficients to zero (i.e., their symbol γ in their Equation 3 is arbitrarily set to zero). The resulting system is given as: w ( x j , x k ) = 1 D( x j , x k )

(

 D( x j , x k ) =  x j1 − x k1 

)

2

(

α 1 + x j2 − x k2

N

c est ( x j ) =

∑ w(x , x ) c(x ) w , ∀j j

k =1


k

k

t

)

2

(

α 2 + x j3 − x k3

)

2

 α3  

P

(9.1)



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In Equation 9.1, c denotes concentration, xk is spatial location of the kth measurement (of the N measurements in the search neighborhood), wt is the sum of the individual datum weights, and xj is the location of the concentration value being estimated. The general form of this expression allows an anisotropic interpolation that implicitly accounts for the differing spatial correlation along the assumed principal directions of the plume. Note that Equation 9.1 has four adjustable parameters (α1, α2, α3, and P). The “distance” D is a dimensionless measure, which implies αi has units of distance squared and P may be any positive real number. For each experimental data set, an optimal set of values for each of these parameters was determined by Levenberg-Marquadt nonlinear least squares (NLS) optimization, based on minimizing the cross validation (discussed below) sum of squares.2

D. GEOSTATISTICAL INTERPOLATION METHODS Geostatistics is a practical tool that has been developed to solve problems related to the evaluation and management of spatially distributed properties. Examples of problems where geostatistical methods have been successfully applied include estimating the concentration of lead in the soil surrounding a battery production facility based on a sparse soil sampling, or estimating the fertilizer nutrient requirements in an agricultural field from composite data sources including remotely sensed images and soil samples, etc.9 From a statistical perspective, there are two primary features of such physically distributed attributes, discussed in some detail by Matheron.17 The first and perhaps foremost characteristic is that each case is unique. In other words, there are no two “fields” that are exactly alike anywhere in nature. A second characteristic of these spatial data is that the properties of interest are physically distributed, and often vary in a complex manner (i.e., they are regionalized variables, and exhibit what often appear to be both random and deterministic features). Despite the fact that each particular field or situation is found nowhere else in nature, we often find, after repeated study, many similarities among the “unique” objects. In addition, it has been repeatedly shown that geostatistics provides a reliable tool for solving many practical problems related to the analysis of such spatially distributed properties. It is this “sanction of practice” that Matheron17 terms “external objectivity” and which has been very appropriately bestowed on the geostatistical framework. In his essay on probability in practice, Matheron17 goes well beyond this external objectivity in the quest of internal objectivity, and provides a clear demarcation between the subjective and objective portions of a geostatistical analysis. He further explores the problems encountered with global and with local geostatistical estimation, and provides insight into the theoretical underpinnings of each. The reader is referred to this excellent reference for further discussion of these issues. In the present case, we employ the practical tool of geostatistics to perform spatial plume interpolation. Also, note that we are dealing with a nonstationary problem in a statistical sense, in that the local mean and local variance (i.e., the average value, and the variance of the values, within a local search neighborhood) change in a somewhat systematic manner with the spatial location within the plume. Thus, we are required to employ methods that accommodate this nonstationary behavior. Several methods can be used to achieve this. First, one can fit a trend model to the plume, such as a numerical or analytical solute transport model or an arbitrary local or global trend function. One can then perform kriging on the residuals, and finally add back the trend value. Also, as discussed by Gotway et al.,13 this can be done in an iterative manner to improve the estimates. However, as also noted by Gotway et al.,13 trend removal methods result in biased estimates of the residuals, and hence the variogram and resulting kriging system are also biased. Chilés and Delfiner18 discuss at considerable length the subjective nature of trend removal, as does Cressie.19 Both authors note that trend removal methods are subjective with respect to the decomposition of the observed variability into random and deterministic features. For this reason, we avoid such methods here.



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In addition to explicit trend removal, other methods for interpolating nonstationary data include intrinsic kriging of order k, ordinary kriging, and nonlinear transforms coupled with ordinary kriging of the transformed data. The objective of the nonlinear data transformations is to generate a mapping of the plume data into a data set that more closely satisfies the stationarity and symmetrical distribution assumptions desired for the kriging system.20 These approaches are briefly discussed here. 1. Linear Geostatistics By the term “linear geostatistics,” we mean that the local estimate is equal to a weighted sum of the neighboring observation points and that the weights assigned to each neighboring point do not depend on the actual data values. a. Ordinary Kriging (OK) Ordinary kriging is an interpolation method that relies on local search neighborhoods and the behavior of the variogram near the origin to perform spatial estimation. The underlying property need not have an average value that is the same everywhere within the global domain, although the average value is assumed to be constant, albeit unknown, within any given local estimation neighborhood. In practice, the global variogram is used to perform this local estimation, which implies that the local variogram is reasonably represented by its global counterpart, an approximation that is reasonable for the short lag spacings within the local neighborhood.9 This approach also only requires the intrinsic hypothesis (i.e., that the variance of first-order differences are stationary, E[(Z(xo) – Z(xo+h))2 = E[(Z(x1) – Z(x1+h))2, rather than second-order stationarity. In the ordinary kriging method, the local, unknown mean is filtered from the kriging estimation using a constraint on the kriging weights. The method is rather robust in that the resulting estimate relies primarily on the behavior of the variogram near the origin. b. Intrinsic Kriging (ItK) Intrinsic kriging is an extension of the ordinary kriging approach. As discussed above, in OK the local unknown mean is filtered from the kriging system, and the spatial variability structure of the property is characterized using two-point first-order differences (i.e., the variogram). Intrinsic kriging generalizes the OK approach by working with allowable linear combinations of order k, also referred to as generalized increments of order k.18 This method assumes that the property of interest can be represented as an intrinsic random function of order k (IRF-k). An IRF-k is a random function whose increments of order k are stationary. These increments of order k implicitly filter out any arbitrary polynomial of order k (i.e., any trend or drift that can be modeled as a polynomial of order k; see Chilés and Delfiner18). Thus this approach overcomes the difficulties and biases associated with estimating the drift and computing a variogram of residuals, and allows spatial estimation without requiring the subjective task of separating variability into random and deterministic features. The simplest example of kriging with IRF-k is ordinary kriging, which is actually an IRF-0, and filters out the unknown polynomial of order 0, i.e., the unknown mean, which is assumed to be a constant. The IRF-k approach can be used with a local or global neighborhood, with the understanding that within the corresponding neighborhood the authorized increments are stationary. The variance of these authorized increments is called a generalized covariance, and when an appropriate model is fitted for the generalized covariance, kriging is performed in a fashion analogous in form to the universal kriging approach. The approach is very attractive in theory, but in practice suffers from several limitations. First, there must be sufficient data available for obtaining the allowable linear combinations for estimating the generalized covariance within the corresponding estimation neighborhood. This required us to use global neighborhoods for the smaller data sets. Second, if the data are not on a regular grid, an implicit fitting of the generalized covariance is required. Such an automatic structure identification method was used in this research, as provided in the ISATIS™ software program © 2003 by CRC Press LLC



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(http://www.geovariances.com). In this approach, several alternative polynomial basis functions are selected, and the one most consistent with the chosen search neighborhood and the data is identified. Then, given this kth order polynomial basis function, a generalized covariance model is implicitly fit to the data based on cross validation/kriging performance. This implicit fitting method, or automatic structure identification approach, is not robust when working with highly skewed or heterogeneous data. For instance, a few isolated extreme concentration values can strongly influence the criteria used to fit the generalized covariance.18 2. Nonlinear Geostatistics The goal here is to perform a nonlinear transform of the data, followed by kriging of these transformed values, and subsequently an inverse transformation on the kriged estimates to obtain estimates of the concentration. Note that the class of linear estimators Z* = Σλi Zi is a subclass of a more general estimator of the form Z* = f(Zi). As such, the best linear unbiased estimator cannot, at least in theory, be better than the best nonlinear unbiased estimator.21 Also, recall that if, after transformation, the data are multivariate Gaussian with known mean, then the simple kriging estimate is equivalent to the conditional expectation of the unknown value given the n observed data, and in this special case, represents the best estimator possible. (For a single realization, such a statement about the nature of the multivariate distribution lacks objective meaning and is an assumption, although we can at least evaluate whether the data are consistent with such an assumption). Consider the influence of a data transformation on the tasks of variogram estimation and ordinary kriging. For illustration, consider the rank transform, which is discussed in greater detail below. For highly skewed, sparse and heterogeneous data, computing a variogram on the raw values is fraught with difficulty. One or two outliers may have an overwhelming influence on the traditional method of moments variogram estimator. In addition, in the presence of extreme values, the resulting sum of the linear weighting of the local neighborhood sample values that occurs in ordinary kriging may be dominated as well by one or two extreme values. In contrast, the process of variogram estimation on the rank transformed values, as well as ordinary kriging of these rank values, is not nearly as sensitive to the raw data extreme values. Despite this apparent advantage, the process of data transformation, kriging with transformed values, and back transforming the estimates often results in biased estimates7,18,22 as a consequence of several factors, some of which are discussed here. First, the data may be more consistent with a nonstationary model, assuming a global univariate cumulative distribution function (cdf) in the data transformation as well as the back transformation at each spatial location where values are estimated results in a bias. A notable exception to this is when the data are multivariate lognormal, in which case the raw data may exhibit a proportional effect that suggests nonstationarity, with the local mean squared proportional to the local variance, and yet the data are stationary. Second, if the mean value is assumed to be unknown, and ordinary kriging is used, the resulting inverse transformed values are often biased. Third, the data may not be well represented by the multivariate distribution that is assumed in the transformation, which also leads to a biased estimator. Finally, the back transformed values often depend on the variogram sill, and the estimation variance of the back transformed value often depends nonlinearly on the kriging variance. Such dependence generally results in exaggerated estimates of uncertainty. As noted by Saito and Goovaerts,21 there are methods for reducing the bias resulting from the nonlinear back transform of kriged values. However, such approaches often rely on distributional assumptions regarding the data or on adjustment factors that are not theoretically supported. Thus, because the primary focus of this work is an evaluation of methods to guide a practitioner, and for the sake of ease with respect to implementation, we do not incorporate these bias reduction methods here. Note that nonlinear geostatistics requires more significant assumptions regarding the distributional properties of the regionalized variable. This leads to a potentially more powerful estimation © 2003 by CRC Press LLC


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method, and also one that is more prone to error. The utility of such methods is perhaps best evaluated by comparing them to linear kriging, such as that done by Kravchenko and Bullock.14 In their work, for data that were reasonably consistent with a stationary model, lognormal kriging outperformed both deterministic interpolation approaches and linear geostatistical estimation. It is with these considerations in mind that we evaluate the following nonlinear estimators. a. Multi-Gaussian Kriging (MGK) Multi-Gaussian kriging is a generalization of lognormal kriging. To use this approach, we first transformed the data using the GSLIB subroutine NSCORE23 to be normally distributed with mean zero and variance of 1 (i.e., N(0,1)). Then we calculated a variogram of the normal score data, which was subsequently used to perform ordinary kriging. The values were then back transformed using the GSLIB routine BACKTR. Confidence intervals and estimation variances for the concentration estimates were also obtained by back transformation by assuming that the normal score estimate is the conditional mean of a Gaussian distribution with variance given by the ordinary kriging estimation variance. To be strictly correct, if the normal score data are multiGaussian, the simple kriging variance is the appropriate conditional variance (for example, see Saito and Goovaerts21). Thus, using the ordinary kriging variance results in a conservative estimate of uncertainty.18 We point out several drawbacks to the MGK method. First, it only accounts for the nonstationary plume distribution by using OK of the transformed values This results in a bias when making the back transform that will vary with spatial location within the plume. Second, it ignores the fact that the transformation is not unique because a large proportion of each data set is essentially zero in value. This results in a discontinuity in the univariate cdf, resulting in a back transform that is not unique. (In addition to simply ignoring this discontinuity, we evaluated the performance of an alternative approach that treats the data as deriving from two separate populations, one population being observed values that were not significantly greater than background and the other being the observed significant nonzero values. This method of identifying two separate populations only slightly improved the performance over simply ignoring the large proportion of zero values). Finally, even with the univariate Gaussian transformation the data do not adhere to a multivariate Gaussian distribution. Thus, when we perform OK, especially for search neighborhoods that include extreme values, the resulting regression equation results in weights assigned to the observed values that are biased, in that they do not provide the conditional mean value. b. Rank Order Kriging (QK) This approach, termed rank order kriging or quantile kriging, is similar in concept to MGK. It was originally proposed by Journel and Deutsch23 for integrating diverse data sources and supports, and was recently applied with reasonable success to estimate soil cadmium concentrations by Juang et al.24 We briefly outline the approach here, and refer the reader to References 18 or 24 for details. In the present case of spatial plume interpolation, with a relatively large proportion of nondetect or background concentration values, the data were first sorted into two classes: zero and nonzero values. Then the nonzero values were sorted again, and a standardized rank was assigned to each (i.e., a value between 0 and 1). A value of zero was assigned to each nondetect value. Using the combined zero and standardized rank values, we computed an experimental variogram and fit a corresponding model. This model was then used to perform OK on the transformed values, followed by a back transform using the standardized rank transform. A 95% confidence interval (CI) was also computed in the transform space, assuming the data are uniformly distributed, using the kriging variance. The CI was also nonlinearly back transformed to provide a corresponding uncertainty estimate for the estimated concentration value. With respect to performing the structural analysis (i.e., selecting the variogram model) for the geostatistical methods (with the exception of the ItK method) for each case study, we used a twostep process. First, the appropriate method of moments experimental variogram was computed and




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several candidate models were fitted via least-squares and visual observation. Then an optimal model was selected from among the candidate models for each test case and kriging method based on an evaluation of the cross-validation scores. This evaluation included plots of estimated vs. true values, measures of robustness, and histograms of standard errors in CV scores. The figures of the fitted models and CV scores are given in Appendices B and C of Reed.1 The ItK approach for irregularly spaced data requires a black box fitting, which relies almost entirely upon the CV scores (see Chilès and Delfiner18 for further discussion).

E. EVALUATION CRITERIA The interpolation methods were evaluated in two ways. The first evaluation approach was cross validation, which is perhaps the most commonly used in practice. In cross validation, one datum at a time is removed, and the remaining data are used to estimate the removed value. For each of the five case studies, the corresponding entire sample data set was used for calibrating each specific, i.e., specific to that data set, interpolation model (except, of course, for ID2, which does not require calibration). This calibrated interpolation model was then used for estimating the cross-validation values. One potential problem with this method of evaluation is that the selection of the calibration model relies heavily on the cross-validation scores. This is true for the geostatistical methods and for the NLS method. Thus, the optimal calibrated model may be somewhat dependent on the design of the sampling network, since the relative location of the samples within the domain will strongly influence the overall cross-validation sum of squares. Therefore, the CV scores may be misleading in terms of the true average interpolation errors, especially for irregular and clustered sampling networks (i.e., SPS, MPS, and LPS case studies). The second evaluation approach we employed does not suffer from the above limitation. In this approach, we used a validation data set (“true” data) to evaluate interpolation model performance. For the SPS, MPS, and LPS case studies, validation data were readily available because we have the true value at each spatial location within the domain from the simulation. Rather than work with the approximately 50 million nodes of the transport domain, a subset was taken in each case on a regular three-dimensional grid. For LPS, the test data were obtained from a regular grid with spacing of 20, 24, and 4.8 m in the longitudinal, transverse, and vertical directions, resulting in approximately 6500 values. For MPS, a regular grid with the same spacing but with a smaller overall dimension was used, resulting in approximately 2500 values. For the smallest test case, SPS, a regular grid with spacing of 10, 12, and 4.8 m was used, which resulted in approximately 2000 samples. For these three case studies, the ability of the various interpolation methods to predict these true values thus provides a very straightforward and robust assessment of model performance. However, for the two unsaturated zone data sets, no such true data were available. Resampling approaches provide the only alternative for evaluating the interpolation methods for these two cases. We employed a jackknife (Deutsch and Journel7) approach to obtain a validation, or true, data set for the SPU and LPU case studies. In this approach, the data are portioned into two subsets; one subset, referred to as the calibration data set, is used for model calibration. The calibration data and model are then used to predict the values of the other subset (referred to as the validation, or true, data set). For cases SPU and LPU, the true data were obtained by performing a stratified random selection of five soil cores from the sampled data for each experiment. In the original sampling, a soil core was taken at each node of a 20-cm square horizontal grid, with approximately 64 cores taken in total. Each core was further divided into approximately 30 samples, one for each 5-cm depth increment. Thus, the validation data for SPU and LPU represent a worst case scenario in terms of spatial interpolation in that these spatial locations are farther removed from neighboring sample data than are any of the other possible interpolated values within the interpolation domain. However, because these validation sets represent relatively few, randomly selected values within the interpolation domain, the associated prediction errors may not be representative of the actual prediction errors incurred in the spatial interpolation of the unsaturated zone plumes. A more robust



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method for evaluating uncertainty would be to obtain a series of random subsets of the original data set as jackknife data and perform the evaluation for each subset. Saito and Goovaerts21 employ such a method in evaluating several geostatistical interpolation methods with respect to the spatial distribution of dioxin. In their analysis, to conserve time and computation burden, they employ the experimental and model variograms obtained from the entire data set, which limits the utility of the analysis. Several measures were used to compare the relative performance of the various interpolation methods with respect to the true data estimation as well as the cross-validation estimation. These measures included estimation error, standardized estimation error, and root-mean-square error (RMSE). In addition, for the geostatistical methods, confidence intervals were calculated from the kriging variances to evaluate the measures of uncertainty that these models provide. The estimation error is simply defined as E ( x ) ≡ c true( x ) − Cest ( x ) , and directly reflects the performance of the estimation. The standardized estimation error is defined as the estimation error divided by the square root of the kriging estimation variance, and is thus relevant only to the geostatistical methods. The final measure of model performance is evaluated using the root-meansquare error. RMSE is the average estimation error of the estimated values, either the true data or the cross-validation data, and is given by Equation 9.2:

)∑

1 RMSE = n

n

[ Cest ( x j ) − c true( x j )]2

(9.2)

j=1

In addition to the measures discussed above, the kriging methods, via the kriging variance, provide an estimate of local uncertainty (“local” in that we refer to a specific spatial location). As discussed in detail by Goovaerts,9 several rather restrictive assumptions are implicit in the use of the kriging variance as an estimate of local uncertainty. In the present application, we evaluate the performance of the kriging variance as a local uncertainty measure by computing a 95% CI for each of the kriging methods.2 An estimate is defined as robust if it lies within this 95% CI. The fraction of the estimated values that were robust was computed for each of the kriging methods.

F. CASE STUDY The several data sets were chosen so as to represent a reasonably wide spectrum of situations. This allows us to examine the relative performance of the various interpolation schemes for such diverse conditions. For instance, the sampling networks for the SPU and LPU data sets are relatively uniform grids within a rather small spatial domain (~5 m3), and are also rather large data sets (n = 1049 and 1852, respectively). In contrast, the SPS, MPS, and LPS sampled data are sparse data sets (n = 26, 58, and 124, respectively) that encompass very large spatial extents (106 m3, 6 × 106 m3, and 16 × 106 m3, respectively). As shown by the coefficients of variation and skewness in Table 9.1, all of the data sets are highly heterogeneous and asymmetric. Note that each of the data sets

TABLE 9.1 Summary Statistics of the Observed Spatial Plumes (mg/m3) SPU Mean Median Coefficient of Variation Coefficient of Skewness Number of Data


28,000.0 440.0 4.0 5.2 1,049

LPU 12,000.0 440.0 2.8 4.0 1,852

SPS

MPS

LPS

355.0 9.0 1.8 1.5 26

164.0 11.0 3.6 5.0 58

29.0 3.6 11.4 7.9 124



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is positively skewed, with a preponderance of background/nondetect values. For example, approximately 75% of the SPU and 41% of the LPS data sets were not significantly greater than the background concentration. We note that nondetects play an important role in the spatial plume interpolation, in that they are necessary to identify the plume boundaries clearly. The nature of the true or validation data sets is another important contrast between the unsaturated and saturated test cases (in addition to the much greater regularity and density of the unsaturated zone sampling networks discussed above). Recall that the validation data sets used for SPU and LPU are relatively small subsets of the observed data. In each of these cases, a subset of the data was used for calibrating the interpolation model (n = 946 for SPU and n = 1659 for LPU) while a randomly selected subset was used as the validation data (n = 103 and 193, respectively). Thus the prediction errors associated with the validation data for these two cases may differ significantly from the true, yet unknown, average prediction errors associated with the plume interpolation. However, this is not the situation for the saturated zone case studies (SPS, MPS, and LPS). In each of these cases, the spatial distribution of the plume is known with great detail, given the output generated by the 50-million node simulation. The regular and relatively dense grids used to generate the true data for each of the saturated case studies thus provide a very reliable assessment of the average prediction errors associated with each interpolation method. For all except the ID2 interpolation schemes, model calibration was required. For the NLS scheme, this was simply identifying an optimal (in a least-squares sense) set of parameter values as indicated in Equation 9.1. The optimized NLS parameter values for each case study are given in Table 9.2. Note that for the SPU and LPU cases, the α parameters are very similar, and reflect the general horizontal block-like structure of the plumes that is a consequence of the two-dimensional surface boundary condition associated with solute application. Also note that for the more densely sampled data sets SPU, LPU, and LPS, the P parameter is relatively large. It is relatively straightforward to show that, as P approaches infinity, Equation 9.1 reverts to a nearest neighbor interpolation scheme for finite values of α. As P decreases, Equation 9.1 reverts to a much smoother spatial filter, which is the result for the SPS and MPS cases. As discussed previously, for each geostatistical method, the calibration entailed estimating and evaluating a variogram model. For a given geostatistical approach and case study, each variogram model was first evaluated based on a combination of least-squares/visual fitting and then upon cross-validation scores. The CV scores included histograms of errors, scatterplots, and measures of robustness. Thus, four variogram models were calibrated, for each method, to each of the five case studies, employing all of the observed data in each case. In addition, to evaluate performance with respect to estimating the true values, four additional variogram models were calibrated to each of the calibration subsets for the SPU and LPU case studies, making a total of 24 variogram models.

TABLE 9.2 Nonlinear Least Squares Fitting Parameters (See Equation 9.1)

SPU LPU SPS MPS LPS

α1

α2

α3

1.75 1.93 36.00 306.00 1.80

1.74 1.82 60.00 304.00 0.10

0.49 0.49 0.50 305.00 0.40

P 22.8 22.9 1.20 0.50 30.00

Note: The parameter α1 is with respect to the principal flow direction, while α2 and α3 are relative to the principal coordinates in the transverse plane. (Furthermore, for the saturated plumes, α3 is with respect to the vertical coordinate.) For SPU and LPU, αi have units of cm2, whereas for SPS, MPS, and LPS, αi have units of m2.



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In general, as the density of the sampling network increased, the ability to characterize the spatial structure in terms of variogram models improved. For example, for all except the ItK generalized covariance models (which were isotropic) anisotropic variogram models were identified for the SPU, LPU, and LPS data sets. In contrast, for the MPS and SPS data sets, the variogram models were isotropic. Furthermore, for the smallest data set (SPS), only nugget variogram models were identified for each geostatistical method. In addition, as sample size decreased, the influence of extreme values on variogram estimation became much more pronounced (i.e., for the SPS, MPS, and LPS data sets). This was particularly true for the OK and ItK approaches. For instance, no spatial structure could be detected for OK or ItK models with respect to the LPS data until the four largest values were removed. (However, these same values did not significantly influence the variogram estimates for the MGK and QK methods.) When fitting a variogram model to the SPU and LPU data sets, the role of sample support should be considered. Each of the observations in these two cases represents the average solute concentration within approximately 350 cm3 of soil (a core sample with both radius and length of 5 cm). Thus, the experimental variograms are “regularized” variograms (see Ellsworth6 for details). Therefore, to obtain an estimate of the theoretical point variogram model actually requires a fitting to the theoretical deconvolution of the regularized variograms. This is especially true if the required analysis involves a change of support, such as with block kriging. However, for the present evaluation, we can avoid this complication because the kriged values are desired at the same sample support as the observed values, and fitting a variogram model directly to the observed regularized variograms provides a relatively accurate representation of the required values for the kriging system. With respect to SPS, MPS, and LPS, we assume the wells are passive and noninvasive thus the sample support corresponds to the scale of the simulation node. In light of the differences highlighted above among the several test cases, the following discussion first examines the cross-validation scores for each test case and each interpolation scheme. We then examine the relative performance of the interpolations on the basis of the validation data sets. Finally, we consider the performance of the geostatistical methods with respect to assessing uncertainty in the spatial estimates.

G. CROSS-VALIDATION SCORES The approach of reestimating observed values in order to evaluate a prediction model’s performance is a well-developed subject in statistics and in geostatistics. Cross validation is one of the most common methods of computing such reestimation scores. Prior to discussing the cross-validation results, it is instructive to consider the following comment taken from page 94 of Journel and Deutsch:23 “The exercise of cross validation is analogous to a dress rehearsal: It is intended to detect what could go wrong but it does not ensure that the show will be successful.” This statement has particular relevance to the present endeavor, as will be illustrated with several case studies that provide examples of when cross validation may be expected to be reasonably successful and when it may not. Table 9.3 gives the RMSE values for each interpolation method and each test case. These data show that ID2 consistently has the largest RMSE for any of the interpolation methods and for each of the case studies. This is not surprising given that each of the other methods involves a model calibration specific to the cross-validation scores associated with each data set. The CV performance is also illustrated in Figure 9.4, which gives the relative ranking of the various interpolation schemes for each case study. This figure indicates the OK and QK are consistently among the best performing interpolation methods for each case study in terms of cross-validation scores. However, these results should be treated with caution. We point out that there are several reasons why the cross-validation results may not be representative of how well the various models perform in terms of spatial plume interpolation. For instance, consider the influence of sample clustering on cross-validation scores. If the observed data are clustered at various locations within the domain of interest, then the average separation distance between adjacent samples will be relatively short © 2003 by CRC Press LLC



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TABLE 9.3 Cross-Validation RMSE Values (mg/m3) for Each Test Case and Interpolation Method

SPU LPU SPS MPS LPS

NLS

ID2

OK

ItK

MGK

QK

32.5 7.6 680.0 110.0 61.0

55.3 19.2 876.0 949.0 596.0

28.9 7.0 840.0 46.0 13.0

53.5 6.5 715.0 94.0 13.0

35.9 10.4 881.0 80.0 14.0

28.2 7.4 895.0 44.0 14.0

Note: The units for SPU and LPU are scaled by 10–3.

2

NLS ID2 OK ItK MGK QK

1.5

1

0.5

SPU

LPU

SPS

MPS

LPS

FIGURE 9.4 Cross-validation scores shown as relative RMSE (relative to the RMSE for the best performing interpolation method for each test case).

compared to the average separation distance associated with spatial plume interpolation within the given domain. This is particularly true when samples are preferentially obtained near a source area, or from either multilevel sampling wells or soil coring. In these situations, when performing cross validation, neighboring samples are often relatively close. Therefore, because the estimation error generally increases as the separation distances between the estimate location and the locations of observed values increases, the RMSE for cross-validation scores would be less than the (unknown) RMSE for spatial plume interpolation. The converse of this may also occur. For instance, consider a sampling network that is a relatively regular grid with uniform spacing in each spatial direction. When performing cross validation, the average separation distance between the location of the estimated value and the nearest neighbors used in the interpolation would be greater (and hence the CV RMSE also greater) than the average separation distance associated with interpolating values on a dense grid within the sampling network. Data sparsity is another factor that may result in reestimation scores not accurately reflecting errors associated with spatial plume interpolation. This is especially so for highly heterogeneous, skewed data such as solute concentrations in porous media. Hence, the observed data may not accurately reflect the spatial structure of the plume. Therefore, reestimation scores for a model calibrated, or fitted, to the observed sparse data set may be greatly different from the true estimation errors for interpolating within the domain. © 2003 by CRC Press LLC


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TABLE 9.4 Interpolation Method Performance as RMSE with Respect to “True” Data Sets

SPU LPU SPS MPS LPS

NLS

ID2

OK

ItK

MGK

QK

60 31 580 342 17

75 32 580 386 123

61 28 559 379 86

80 29 683 339 136

83 35 288 69 20

95 34 322 52 8

Note: For SPS, MPS, and LPS, the observed data sets (n = 26, 58, and 124, respectively) were used to predict true data values (n = 2090, 2496, and 6372, respectively). For SPU and LPU, a calibration data set (n = 946 and 1659, respectively) was used to predict values for a validation data set (n = 103 and 193, respectively). The units for SPU and LPU are scaled by 10–3.

Thus, given the above discussion, the relatively dense and regular sampling networks for SPU and LPU suggest that the cross-validation results for these two cases provide a reasonable assessment of interpolation model performance. However, for the remaining three cases, the CV results are likely not as accurate in evaluating performance. Thus, for relatively dense, regular sampling grids (i.e., SPU and LPU) NLS, OK, and QK appear to perform reasonably well, while the ID2 and MGK methods are less reliable. It appears that as the number of observations increases, so does the performance of the ItK approach. Specifically, note that ItK is optimal with respect to the CV scores for the LPU data set. In addition, it also performs very well with respect to the validation scores for this case. We now examine the performance of the various methods with respect to the validation data sets.

H. VALIDATION SCORES The results for the validation data sets are shown in Table 9.4 and Figure 9.5. Recall that the true data for SPS, MPS, and LPS case studies provide a very robust evaluation of interpolation 2

NLS ID2 OK ItK MGK QK

1.5

1

0.5

SPU

LPU

SPS

MPS

LPS

FIGURE 9.5 Interpolation performance for “true” test data, shown as relative RMSE (relative to the RMSE for the best performing interpolation method for each test case). © 2003 by CRC Press LLC



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model performance. This is so because these data are obtained from the simulation, and provide the concentration values at the nodes of dense, regularly spaced grids. For these three cases in which the sampled data are obtained from sparse, clustered and irregular sampling networks, QK is clearly the optimal interpolation method. For example, QK greatly outperforms all of the other methods for the LPS case; this is true for MPS as well, although MGK provides a close second. Finally, for the SPS case study, QK performs almost equally as well as the optimal MGK method. Examination of the interpolation performance with respect to the validation data sets for SPU and LPU indicates that OK and NLS are optimal for these dense, somewhat regularly spaced sample grids. This is consistent with the CV results above for SPU and LPU as well, with the exception that QK, which performed well in terms of CV scores, performed poorly at predicting the validation values for these two cases. The poor performance of QK in this case may be due to one of the following factors. First, because the validation sets in these two cases are resampled data, being randomly selected from the observation data sets and relatively small subsets of the original data (i.e., n = 103 and 193, respectively), the associated prediction scores may not be indicative of the true prediction errors. A second, and perhaps more plausible, explanation is that the data were obtained via a stratified random sampling, with the strata preferentially located within the central region of the plume. This resulted in the validation data having a considerably higher mean as well as less variability (i.e., compared to Table 9.1, the validation data sets for SPU and LPU had mean values of 37,000 and 20,000 mg m-3, and coefficient of variations of 2.6 and 2, respectively). This disparity between the distributions of the calibration and validation data sets was especially problematic for the MGK and QK methods. These two approaches rely on the assumption that the sample histogram is representative of the true, or jackknife, data.

I.

ESTIMATION UNCERTAINTY

The geostatistical approach is perhaps most useful in that it provides not only an estimate of the unknown value, but often more importantly, an estimate of the uncertainty in the predicted value. Within the geostatistics framework, there are various methods for estimating local uncertainty (see Goovaerts9). The simplest of these is the kriging variance, which is also perhaps the least robust in that it depends only on the variogram model and geometry of the interpolation problem, and not on the actual data values. Herein we evaluate the ability of the kriging variance associated with the four geostatistical methods to reflect the estimation uncertainty accurately with respect to the true data. Reed et al.2 provide a detailed discussion of the estimation errors associated with each interpolation method for cases SPS, MPS, and LPS. This includes detailed three-dimensional spatial maps that illustrate the spatial distribution of errors associated with each of the interpolation methods. We summarize these findings here. Table 9.5 shows the percent of the true data values that lie within the estimated 95% confidence intervals for each of the four methods and for each case study. In examining this table, it is important to make the distinction between the jackknife validation results, which rely on dense, regularly spaced calibration data sets and sparse validation sets (SPU and LPU), and the simulation validation results, which rely on sparse, irregularly spaced calibration sets and very dense, regularly spaced simulated data sets (SPS, MPS, and LPS). For the simulation results, which provide a much more accurate measure of the true prediction errors, it is seen that QK and MGK provide relatively conservative uncertainty estimates, while OK and ItK perform less satisfactorily. The regular sampling grids associated with SPU and LPU resulted in values for the kriging variances that were relatively uniform within the interpolation domain. However, the irregular and clustered sample networks for the SPS, MPS, and LPS cases resulted in a widely contrasting spatial distribution of estimation variances. Given that, especially for the SPS, MPS, and LPS



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TABLE 9.5 Performance of Geostatistical Interpolation Methods to Assess Estimation Uncertainty, Shown as Percent of the “True” Values That Fall within the Estimated 95% Confidence Interval

OK ItK MGK QK

SPU

LPU

97 89 96 87

91 85 95 85

SPS 100 89 100 100

MPS

LPS

66 76 100 100

92 92 95 99

Note: For SPS, MPS, and LPS, the observed data sets (n = 26, 58, and 124, respectively) were used to predict true data values (n = 2090, 2496, and 6372, respectively). For SPU and LPU, calibration data sets (n = 946 and 1659, respectively) were used to predict values for validation data sets (n = 103 and 193, respectively).

cases, the QK method was the optimal interpolation method in terms of spatial estimation, we examined the rank correlation (see page 21 in Goovaerts9) between the QK kriging variances and the true prediction errors for these three cases. The rank correlation coefficients between the true prediction errors and the QK kriging variances were 0.97, 0.97, and 0.98 for SPS, MPS and LPS, respectively. Thus, the QK approach provided a very accurate ranking of the spatial distribution of the estimation uncertainty.

III. SUMMARY Spatial distributions of solute in porous media are often highly heterogeneous and asymmetric, as illustrated by the five case studies in the present work. In addition, sample collection is often costly and time consuming, which often results in data sparsity. The task of estimating the spatial distribution of solute from experimental data is thus very challenging indeed. This chapter has evaluated six interpolation methods for performing spatial interpolation of solute concentrations. The evaluation was based on two highly contrasting situations, as represented in the five case studies. The first situation is a somewhat regularly spaced, relatively dense, spatial sampling grid (case studies SPU and LPU). The second situation, as represented in case studies SPS, MPS, and LPS, is essentially the converse of this, with irregular, sparse spatial sampling grids. The evaluation of cases SPU and LPU relied primarily on CV scores because the true spatial distribution of the solute was unknown. However, the simulated data for the SPS, MPS, and LPS cases provided an ideal situation to examine the true prediction errors associated with spatial plume interpolation, and thus these true errors were used to evaluate interpolation model performance. The results indicate that for relatively uniform, dense sampling grids, OK and NLS methods appear to be optimal. Since in the present effort we do not have an irregular, dense sampling grid to evaluate, it is not clear whether the satisfactory performance of these two methods is due to the uniform spatial coverage or whether it is a consequence of data density. We hypothesize that it is a consequence of the relatively large number of observations, which lessens the influence of extreme values on model calibration and spatial interpolation. For irregular, sparse spatial sampling grids, the nonlinear geostatistical QK method was optimal. For the sparse data sets, the variogram of the rank transformed values more clearly illustrated spatial structure in the data relative to the variogram of the original concentration values. Also, the presence of extreme values apparently explains the smaller bias associated with the nonlinear transformation and subsequent linear weighting of the transformed values (via OK of the rank values) relative to performing OK directly on the original values in the presence of extremes. In © 2003 by CRC Press LLC



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addition, for these sparse sampling grids, QK provided a very accurate, albeit overly conservative, spatial map of the relative ranking of uncertainty in the spatial estimates. This work also illustrates the difficulty in evaluating model performance based solely on the observed values (or, in other words, for all practical situations in which we want to examine model performance). A possible future effort in this respect would be to perform a more thorough resampling of the SPU and LPU data to generate multiple calibration/validation subsets. If the observed data provide a reasonably uniform and dense coverage of the contaminant plume, the average prediction errors over all subsets may then be reasonably indicative of the true interpolation errors. However, the utility of such a method for sparse, irregular data is questionable. Also, note that for this situation, i.e., for sparse, clustered sampling grids, our work shows that cross-validation scores provided a highly erroneous classification of interpolation model performance. In such a situation, the practitioner must draw upon previous experience and research such as this to decide what interpolation scheme to employ.

REFERENCES 1. Reed, P. Striking the balance: long-term groundwater monitoring design for multiple conflicting objectives, Doctoral Dissertation, University of Illinois, Urbana, 2002. 2. Reed, P., Ellsworth, T.R., and Minsker, B., Spatial interpolation methods for nonstationary plume data, submitted to Groundwater, 2002. 3. McLaughlin, D., Recent developments in hydrologic data assimilation, Rev. Geophys., Supplement, Am. Geophysical Union, 977,1995 4. Christakos, G., Modern Spatiotemporal Geostatistics, Oxford University Press, New York, 2000. 5. Bierkens, M.F.P., Finke, P.A., and de Willigen, P. Upscaling and Downscaling Methods of Environmental Research, Kluwer Academic Publishers, Netherlands, 2000, chap.2. 6. Ellsworth, T.R. Influence of transport variability structure on parameter estimation and model discrimination. In Corwin and Loague (Eds.). Application of GIS to the Modeling of Non-Point Source Pollutants in the Vadose Zone. Soil Sci. Soc. Am. Special Publ. 48, 101, 1996. 7. Maxwell, R.M., Carle, S.F., and Tompson, F.B., Contamination, risk, and heterogeneity: on the effectiveness of aquifer remediation, Lawrence Livermore National Laboratory Report, UCRL-JC139664, Livermore, CA., 2000. 8. Cooper, R.M. and Istok, J.D., Geostatistics applied to groundwater contamination I: Methodology, J. Environ. Eng., 114, 2, 270, 1988. 9. Goovaerts, P., Geostatistics for Natural Resource Evaluation, Oxford University Press, New York, 1997 10. Warrick, A.W., Zhang, R., El-Harris, M.K., and Myers, D.E., Direct comparisons between kriging and other interpolators. In Wierenga, P.J. and Bachelet, D., (Eds.), Validation of Flow and Transport Models for the Unsaturated Zone: Conference Proceedings. Report 88-SS-04, New Mexico State University, Las Cruces, NM. 505, 1988. 11. Weber, D. and Englund, E., Evaluation and comparison of spatial interpolators, Math. Geol., 24, 381, 1992. 12. Wollenhaupt, N.C., Wolkowski, R.P., and Clayton, M.K., Mapping soil test phosphorous and potassium for variable-rate fertilizer application, J. Prod. Agric., 7, 441, 1994. 13. Gotway, C.A., Ferguson, R.B., Hergert, G.W., and Peterson, T.A., Comparisons of kriging and inversedistance methods for mapping soil parameters, Soil Sci. Soc. Am. J., 60, 1237, 1996. 14. Kravchenko, A. and Bullock, D.G., Comparison of interpolation methods for mapping soil P and K contents, Agron. J., 91, 393, 1999. 15. Kitanidis, P.K., The concept of the dilution index, Water Resour. Res., 30(7), 2011, 1994. 16. Barry, D.A. and Sposito, G., Three-dimensional statistical moment analysis of the Stanford/Waterloo Borden tracer data, Water Resour. Res., 26, 1735, 1990. 17. Matheron, G., Estimer et Choisir. Cahiers du Centre de Morphologie Mathematique de Fontainebleau, Fasc. 7, Ecole des Mines de Paris, 1978, Translation to English, Estimating and Choosing-An Essay on Probability in Practice, Springer Verlag, Berlin, 1989.



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18. Chilès, J.P. and Delfiner, P., Geostatistics: Modeling Spatial Uncertainty, Wiley Series in Probability and Statistics, John Wiley & Sons, Inc., New York, 1999. 19. Cressie, N., Statistics for Spatial Data, John Wiley & Sons, New York, 1993. 20. Journel, A.G. and Huijbregts, J., Mining Geostatistics, Academic Press, New York, 1978. 21. Saito, H. and Goovaerts, P., Geostatistical interpolation of positively skewed and censored data in a dioxin contaminated site, Environ. Sci. Technol., 35, 4223, 2001. 22. Deutsch, C.V. and Journel, A.G., GSLIB: Geostatistical Software Library and User’s Guide, Oxford University Press, New York, 1998. 23. Journel, A.G. and Deutsch, C.V., Rank order geostatistics: a proposal for a unique coding and common processing of diverse data. In Baafi, E.Y. and Schofield, N.A., (Eds.) Geostatistics Wollongong ’96, Vol. 1, Proceedings of the 5th International Geostatistics Congress, Wollongong, Australia, Kluwer Academy Publ., Dordrecht, The Netherlands, 1996. 24. Juang, K., Lee, D., and Ellsworth, T.R., Using rank order geostatistics for spatial interpolation of highly skewed data in a heavy metal contaminated site, J. Environ. Qual., 30, 894, 2001.



10

Scale- and LocationDependent Soil Hydraulic Properties in a Hummocky Landscape: A Wavelet Approach B.C. Si

CONTENTS I. II. III. IV. V. VI.

Introduction...........................................................................................................................163 Wavelets................................................................................................................................165 Continuous Wavelet Transform ............................................................................................166 Wavelet Power Spectrum .....................................................................................................167 Demonstrations Using Analytical Signals ...........................................................................167 Application of Wavelet Analysis in Characterizing Scale Dependence of Hydraulic Properties ........................................................................................................171 VII. Conclusions...........................................................................................................................175 References ......................................................................................................................................176

I. INTRODUCTION Understanding spatial variability has important applications in agriculture, environmental sciences, hydrology, and earth sciences. Studies on spatial distribution of soil properties have in most cases indicated considerable variation, especially for soil hydraulic properties.1 In general, spatial variability of soil properties represents the interactions among soil physical, chemical, and biological processes that operate on a wide range of spatial and temporal scales. Some of these processes vary frequently in time and/or space and are referred to as high-frequency (small-scale) processes, while other processes vary slowly, and are called low-frequency (large-scale) processes. The scales of these processes extend over spatial scales of a few centimeters to tens of kilometers and over time scales from seconds to decades. Elucidation of the scales of these processes is essential for understanding and predicting soil hydrological, biological, and chemical processes. Different scales of variation in soils have long been recognized and utilized in soil classification. Large-scale variations are represented by orders and meso-scale variations by suborders or groups and small-scale variations by series. However, this classification is qualitative and empirical, and does not allow prediction of one class from another at places near a boundary.2 In order to reveal the true nature of soil and hydrological processes, there has been increasing interest in exploring the nature of spatial variability of soil hydraulic properties and how it changes with scales.



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Spatial variations are generally not purely white noise. Values are much more similar when the spatial locations of the two points are close to each other. Geostatistics offers theory and techniques for describing these scale-dependent soil variations.3,4 Semivariance and autocorrelation functions can be used to describe this increasing similarity of soil properties as distance between two points decreases. Development of these statistical functions has led to tools such as kriging or cokriging for interpolating and predicting values of soil properties at unsampled points.4 The advantage of kriging or cokriging techniques is that one can conduct smoothing and joint estimation with a very sparse collection of observed data points, whereas conventional time-series techniques require that one collect relatively equally spaced data from the random field. The disadvantage is a great reliance placed on the assumed or estimated covariance function, the variogram.5 Scale dependence is not the only characteristic of spatial variation in soil properties. Cyclic behavior has long been recognized and used in soil classification. The soil catena concept, for example, explains the observed cyclic pattern of soil types in hummock landscapes.6 Methods revealing this behavior, particularly frequency-domain analysis, have been used in soil science.7–11 Frequency-domain analysis is realized through the Fourier transform. The sum of squares of Fourier coefficients at different frequencies (power spectra) are a measure of the contribution of processes at a given frequency to the total variability; power spectra break the total variation in soil properties into processes of different frequencies. Because the components of variance at different frequencies are orthogonal to each other, the traditional analysis of variance and statistical tests can be used to analyze the data in the frequency domain. Therefore, quantitative analysis or analysis of variance is possible on cycles at different frequencies by hypothesis test. Webster7 analyzed the apparently periodic pattern of soil in a gilgai terrain in Australia. He revealed a cyclic component of variation in the data in addition to a strong nonperiodic (random) one with a short range. Nielsen et al.8 have identified furrows, tractor compaction, and preplant irrigation as possible causes for cyclic variation in soil and have predicted the behavior to be expected in the power spectrum from such causes. Kachanoski et al.9 analyzed soil spatial variations of A horizon thickness, density, and mass. These authors found thickness and mass had significant spectral peaks indicating cycling. Frequency-domain analysis can also be used to reveal scale-dependent soil properties. Kachanoski and de Jong10 used coherency spectrum to break the autocovariance into different frequencies and analyzed the scale-dependent temporal stability of soil moisture along a transect in a rolling landscape. The common feature of these frequency analyses is the assumption of stationarity. If all the moments of a distribution of a random process are the same everywhere, we have full stationarity. If the first two moments are required to be the same, we have second order stationarity. Many methods require second-order stationarity: (1) soils are considered as stochastic processes with a constant mean and (2) spatial covariance depends on the separation distance between two points, regardless of their spatial locations. The second order stationarity assumption allows these statistical methods to deal with the average behavior of soil processes. In essence, if a data series is chopped up into different segments (sufficiently long), the means, variances, and spatial covariances of these segments tend to be the same. Frequency domain analysis deals with global information or mean states. Often, two completely different spatial series with different local information may result in very similar mean states. Therefore, spatial information is completely lost in frequency domain analysis. More often than not, soil spatial variation is nonstationary, consisting of a variety of frequency regimes that may be localized in space (relative to the entire spatial domain) or may span a large portion of the data record. If a nonstationary data series is chopped up into different segments, different means, variances, and spatial covariances will be associated with each segment. Common nonstationarity includes a natural trend in a spatial series — the values of soil properties tend to increase or decrease with distance in a deterministic manner. Some soil properties have localized features. For example, depressions in semiarid environment may be a discharge area, while the majority of the landscape is a recharge area. Similarly, some properties have transient features, or © 2003 by CRC Press LLC


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features that are short-lived. These natural trends, localized features, or transient features may in fact reflect important parts of soil processes and spatial variability and are worthy of investigation on their own right. Hence, it is important to recognize that it is the totality of both the local and the global information that constitutes a true spatial series. For nonstationary soil variations (nonperiodic), the summation of a finite number of the periodic functions, sine, and cosine as used in the Fourier transform does not accurately represent the soil variation, because an infinite number of periodic features is needed to represent sharp changes. In addition, a major issue in spectral analysis and geostatistical analysis is the loss of localized information because these analyses operate globally on the spatial variation in a statistical way. This means that although we might be able to determine all the frequencies present in a spatial series or the spatial covariance for a given sampling interval, we do not know when or where they are present. A better tool for representing the nonperiodic soil variations and local variations is the windowed Fourier transform (WFT).12 With WFT, the spatial series is chopped up into sections and each section is analyzed for its frequency content separately. The WFT is advantageous because it provides information about signals simultaneously in the time domain and in the frequency domain. However, the window size determines the detail one can look at. Large windows allow one to look at large-scale features, while small windows allow one to look at small-scale features. The WFT does not allow analyzing features according to their scales. A promising method introduced by Morlet et al.13 is called wavelet analysis. An important property of wavelets is called time-frequency localization. It enables one to study features on the spatial series locally with a detail matched to their scale, i.e., broad features on a large scale and fine features on small scales. This property is especially useful for spatial variations that are nonstationary, have short-lived transient components and features at different scales, or have singularities. Due to this property, wavelet analysis has wide applications, from fluid dynamics14–17 to geophysics or hydrology.18–21 However, the application of wavelet analysis in soil science has been limited. Lark and Webster22,23 used wavelet analysis to reveal strongly contrasting local features of the variation. These authors used wavelet correlation to describe scale dependence in the correlation between two variables. However, these authors used a discrete wavelet transform, which does not allow for detailed scale analysis of soil processes. Few studies have been reported on application of wavelet analysis in soil hydraulic properties. The objective of this study is to introduce the continuous wavelet analysis in nonstationary fields to demonstrate the advantage of wavelet analysis through analytical signals. Then, the wavelet transform is applied to the analysis of spatial variations in soil hydraulic properties. II. WAVELETS Wavelets have two properties that overcome the limitation of Fourier transform: (1) compact support of basis functions and (2) basis functions that are obtained through dilations and modulations of a basis function. The basis functions sin(nt) and cos(nt) in the Fourier transforms have infinite support, while the wavelets have a compact support (i.e., they are zero everywhere outside the domain of finite size). This enables the localization in time or space. Like Fourier transforms, wavelets allow localization in frequency. However, the mechanism for frequency localization is different from that of Fourier functions. The basis functions in Fourier transform are constructed by the modulation of a single function (i.e., sin(t) and cos(t)), whereas the wavelet basis functions are dilates and translates of a “mother wavelet.” By dilation and contraction, the size of the support of wavelet functions is proportional to the “size of the feature” it represents.17 Although WFT also allows some localization in frequency and space, wavelet functions are much more desirable. By analogy with the uncertainty principle, high-resolution frequency and spatial localization cannot be achieved simultaneously. High-frequency resolution can be obtained © 2003 by CRC Press LLC


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through long windows in the spatial domain and high spatial resolution through long windows in the frequency domain. WFT uses the same window sizes in the frequency domain and the spatial domain. Because high-frequency resolution can only be obtained through small windows in spatial domain and high spatial resolution through small windows in the frequency domain (uncertainty principle), high resolution in the frequency domain or in the spatial domain cannot be obtained using WFT. Wavelet transforms, on the other hand, have varying windows in frequency domain and spatial domain. One can have some very short basis functions (short windows) in the spatial domain so that the functions can fit sharp changes in space. At the same time, one can have some very long basis functions (long windows) such that high frequency components can be identified. Thus, we have small support for high-frequency features and large support for low-frequency or large wavelength features. This property enables one to zoom into the irregularities of a function and characterize them locally in spatial domain or frequency domain. A fully scalable modulated window is shifted along the spatial series and the spectrum is calculated for every position. Then this process is repeated many times with a slightly shorter (or longer) window for every new cycle. By varying the window size of wavelet functions, fluctuations at different scales can also be obtained. This is the basis for application of wavelets in process scale analysis. Wavelet transform can be classified as continuous or discrete wavelet transforms. In the discrete wavelet transform (DWT), wavelet transform is carried out by skipping along the spatial series such that the wavelet does not overlap with the previous ones. Depending on the algorithm and wavelet selected, DWT can be further divided into orthogonal and nonorthogonal wavelet transform. Orthogonal wavelet transforms re-express a correlated series in terms of some combination of uncorrelated variables. In general, orthogonal wavelet transforms are desirable for use in decomposition and in reconstruction of spatial series with the minimum number of scales. This is important for statistical analysis and efficient computation. However, the orthogonal wavelet transforms may not always yield the most physically meaningful scale analysis, because the scales are analyzed only at integer powers of two, not at fractional powers of two. Wavelet functions used in discrete wavelet transforms include Daubechies, Coiflet, and others. In the continuous wavelet transform (CWT), the wavelet is shifted along the spatial series. Therefore, each wavelet overlaps the ones next to it, which provides a redundant representation of a signal; that is, the CWT of a function at a scale and location can be obtained from the continuous wavelet transform of the same function at other scales and locations. This redundancy implies correlation between coefficients, which is intrinsic to the wavelets and not the analyzed signal. The redundancy of the continuous wavelets yields enhanced information on the spatial-scale localization (sharp changes). However, the information may not offer a perfect reconstruction.24 Thus, orthogonal wavelets are better used for synthesis and data compression, while continuous wavelets are better used for scale analysis. We choose CWT in this study because our focus is on process scale analysis. III. CONTINUOUS WAVELET TRANSFORM The integral wavelet transform is defined by W ( a, τ) =

∫

∞

−∞

y ( x )ψ * a ,τ ( x ) dx

(10.1)

where ψ a ,τ ( x ) =


1 a

ψ a ,τ (

x −τ ) a

(10.2)



167

and the asterisk corresponds to the complex conjugate. The function ψ(x), which can be a real or complex function, is called a wavelet. The parameter, a, can be interpreted as a dilation (a > 1) or contraction (a < 1) factor of the wavelet function ψ(x), corresponding to different scales of observation. The parameter, τ, can be interpreted as a temporal or spatial translation or shift of the function ψ(x), which allows the study of the signal, y(x), locally around the location τ. The wavelet function must fulfill some strict mathematical conditions, called admissibility conditions, implying, for example, a sufficiently rapid decrease of ψ(x) around the origin of time, or equivalently a rapid decrease of Fourier transform ψ(x) around the origin of the frequencies (localized in space and in frequency). In addition, this function must have a zero mean.12,14 Wavelet functions used in continuous wavelet transforms include Morlet, Daubechies, Mexican hat, etc. One popular wavelet is the Morlet wavelet, which is obtained by localizing a complex exponential function with a Gaussian (bell-shaped) envelope. The Morlet wavelet can be expressed as: ψ ( x ) = π −0.25 ⋅ exp(6 ⋅ i ⋅ x − 0.5 ⋅ x 2 )

(10.3)

The Morlet wavelet does not lead to an orthogonal basis because of the long tail of the Gaussian envelope. However, the Morlet wavelet is complex, thus allowing us to detect location-dependent amplitude and phase for different frequencies exhibited in the spatial series.15,24 In addition, Morlet wavelets are naturally robust against shifting a feature in space, making the feature appear in the same way regardless of spatial locations. All orthogonal wavelets present great challenges in ensuring consistency across spatial locations. The Morlet wavelet allows for a good frequency resolution as well as a good spatial/temporal resolution. Therefore, we use the Morlet wavelet in the following analysis. IV. WAVELET POWER SPECTRUM The Fourier power spectrum can be constructed from the sum of squares of Fourier coefficients at a certain frequency. We can also derive the wavelet power spectrum from local wavelet coefficients. Because the wavelet function ψ(t) is in general complex, the wavelet transform W(a,τ) is also complex. The transform can then be divided into the real part and imaginary part, or amplitude, Wn (a,τ), and phase, tan–1[ℑ{Wn(a,τ)}/ℜ{Wn (a,τ)}]. Consequently, one can also define the wavelet power spectrum as Wn (a,τ)2. As indicated by Torrence and Compo,25 for real-valued wavelet functions such as the derivatives of a Gaussian, the imaginary part is zero and the phase is undefined. To facilitate comparison of wavelet spectrum between different variables, all the wavelet spectra presented below are normalized by the sample variance of the data series. Therefore, the normalized wavelet spectra are unitless. V. DEMONSTRATIONS USING ANALYTICAL SIGNALS In order to illustrate the power of wavelet analysis, we use three generic signals commonly encountered in soil sciences. • Abrupt changes in space or time are the result of sudden changes in soil and hydrological processes at a location or time. One example is N2O emission. N2O emission is strongly related to freezing and thawing. Over the winter, subzero temperature and a freezing layer that is not permeable to gases limit N2O emission result in accumulation of N2O below ground. In spring, as weather warms up, the frozen layer disappears, and the N2O accumulated over the winter and spring is released suddenly. Shortly N2O emission becomes low again. This phenomenon is called N2O “burp.” We can represent this “burp” using the following equation. © 2003 by CRC Press LLC


168


y = A ⋅ exp(t − 400)

(10.4)

where t (minute) is time, A is a constant, and y is the N2O emission rate. This signal has a sudden change at 400 minutes, resembling a delta function with no long-term effect (Figure 10.1a). The local wavelet spectrum of this signal shows that the sudden change is well defined by concentration of high wavelet variance within the “cone of influence” centered at the time of occurrence of the sudden change at 400 minutes (Figure 10.1b). As seen in the corresponding Fourier spectrum, a large number of components with a wide range of frequencies are needed to represent the signal (Figure 10.1c). The time of occurrence is completely lost and this sudden change in a small time period cannot be determined to the desired accuracy without using a large number of Fourier components. However, the local wavelet spectrum provides detailed information regarding the location and frequency of the sudden change.

Power spectrum

2

a 1

0

Period

-1 60

b

Location

40

20

0 300

350

400

450

500

Location

0.0035

Power spectrum

0.0030

c

0.0025 0.0020 0.0015 0.0010 0.0005 0.0000 0

100

200

300

400

Period FIGURE 10.1 Sudden change in time: (a) data series; (b) local wavelet spectrum; and (c) Fourier power spectrum.




169

• Sudden change in frequency is a result of one process changed by an external force. In Saskatchewan, Canada, crop row spacing ranges from 12.5 to 25 cm. Suppose half of the field is planted with a row spacing of 12.5 cm and the other half with a row spacing of 25 cm. Water content in the inter-row will be lower and between rows will be higher. As a result, half of the field has cycles with a period of 12.5 cm and the other has 25 cm. This sudden change of frequency at the boundary of two cropping systems cannot be shown in Fourier spectrum. However, as seen from the local wavelet spectrum of this spatial series, the second half of the spatial domain shows a sudden change of frequencies at a distance of 512 cm, with the first frequency existing in the first half of the spatial domain and the second at the second half of the spatial domain (Figure 10.2b). However, the Fourier power spectrum only indicates that there are two processes with periods of 12.5 and 25 cm, respectively. • Signals with superimposition of two frequencies occur commonly under natural conditions. For example, the spacing between furrows is usually 0.9 to 1.5 m for furrow irrigation. In addition, spacing between crop rows varies from 5 to 30 cm. Due to crop 8

a

Value

4 0 -4 -8 60

b

Period

50 40 30 20 10 300

400

500

700

Location

300

Power spectrum

600

c 200

100

0 0

10

20

30

40

50

60

Period FIGURE 10.2 Abrupt change in frequencies: (a) data series; (b) local wavelet spectrum; and (c) Fourier power spectrum.



170


a

10

Data series

Value

5 0 -5 -10 150

b Period (cm)

120 90 60 30 0 240

300

360

420

480

540

120

150

Location (cm)

Power spectrum

1000

c

800 600 400 200 0 0

30

60

90

Period FIGURE 10.3 Superimposing two processes with different frequencies: (a) data series; (b) local wavelet spectrum; and (c) Fourier power spectrum.

root extraction of soil water, soil water content will change between furrows and rows or inter-row. Therefore, soil water content will have two frequencies due to furrow and crop row. The local wavelet spectrum of this spatial series shows the two lines across the whole spatial domain (Figure 10.3b), indicating two overlapping signals with different frequencies. Although Fourier transform tells us that two frequencies are present in the signal (Figure 10.3c), it is unable to distinguish between the two signals: one with two frequencies superimposed over the entire domain (Figure 10.3a) and the other with one frequency present in the first half of its domain and the other frequency present over the second half of its domain (Figure 10.2a). The local wavelet spectra of these signals clearly show the difference.




171

VI. APPLICATION OF WAVELET ANALYSIS IN CHARACTERIZING SCALE DEPENDENCE OF HYDRAULIC PROPERTIES A 600-m long transect was identified at Alvena, 45 km north of Saskatoon, SK, Canada. The soil is classified as an Aridic Ustoll. Soil texture varied from sandy loam to clay loam. A topographic survey was conducted using a laser theoderlite. The landscape is generally hummocky with the maximum relief of about 7m along the transect. Two tension infiltrometers (Soil Measurement Systems, Tucson, AZ) with disk size of 20 cm in diameter were used to measure soil hydraulic properties at 100 locations at 6-m intervals along the transect. Cumulative infiltration rate at transient infiltration and steady state infiltration was recorded at the 3-cm tension. Both the initial water content (before the infiltration run) and steady state soil water content were measured using a Tectronix 1502B time domain reflectometry with a 10-cm long probe was installed horizontally at the 2.5 depth. Topp’s equation26 was used to calculate soil volumetric soil water content. Soil unsaturated hydraulic conductivity K as a function of soil suction h, K(h), is expressed using the Gardner relationship:27 K ( h) = Ks ⋅ exp( −α ⋅ h)

(10.5)

where Ks is the saturated hydraulic conductivity (m s-1), and α is the inverse microscopic capillary length scale (m-1). Saturated hydraulic conductivity and inverse microscopic length scale along the transect were determined using the sorptivity method.28 Soil texture was determined by simple hand touch method. Since Ks and α are generally lognormally distributed, the following analysis is based on logtransformed Ks and α. Figure 10.4 shows the log-transformed measured saturated hydraulic conductivity (Ks) and inverse capillary length scale (α), and elevation along the transect. There was considerable spatial variability in Ks and α along the transect with coefficients of variation of 30 and 40% for Ks and α, respectively. Furthermore, the spatial variation in log(Ks), and log(α) exhibits a natural trend, reflecting a change in soil texture from a clay loam to a loam soil along the transect. In addition, a high correlation coefficient (R = 0.8) exists between log(Ks) and log(α). This is not unexpected because Ks and α are all positively correlated to soil sand content. Finer soil texture is generally associated with smaller Ks and α values. As seen from Figure 10.4, depressions along this transect have finer soil texture, thus smaller values of Ks and α 4 log(Ks) Log(α) Elevation

Value

2

0

-2

0

120

240

360

480

600

Location (m) FIGURE 10.4 Standardized log(α) measurement, standardized log(Ks) measurement, and relative elevation as a function of distance along a transect. The standardization was performed by subtracting mean values and then dividing the difference by their standard deviation.



172


The continuous wavelet transforms were implemented using the built-in fast Fourier transform functions in MathCad (Mathsoft Inc., version 2000, Cambridge, MA). Fourier transform by default assumes that the data series repeats itself, connecting the beginning of the data series with its end. This is referred to as wrap-around. The implementation of wavelet transform through Fourier transform creates the wrap-around effects. Therefore, wavelet coefficients at one end are affected by the data far away at the other end. This defeats the purpose of wavelet analysis for time-frequency locations. To avoid the wrap-around effect, it is generally recommended that the data series be padded with zeros to the next closest binary number. Our number of measurement is 100, so the next closest binary number is 128. To avoid wrap-around effects further in this study, the 100 measurements of log(Ks) and log(α) were padded with zeroes to a total number of data points of 256. Another benefit of zero padding is to speed up the wavelet transforms, because Fourier transform on binary numbers is relatively fast. The wavelet coefficients obtained from CWT were used to calculate wavelet power spectrum. Figure 10.5a shows the local wavelet power spectrum of log(α) as a function of period (or scale) and distance. At small periods or scales (

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