Scaling Methods in Soil Physics

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.

© 2003 by CRC Press LLC

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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

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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

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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

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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

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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

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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

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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

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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<=r i])

x=log(ri) -1 0

-0.5

0

0.5

Series1 Linear fit

-1 -2

y = 0.88x - 1.22 R2 = 0.997

-3

Log cumulative mass of solids of size < rs(i)

a) Density scaling 2.5

2.0

1.5

1.0 -4

-3

-2

-1

0

Log size rs(i) of solids

b) Aggregate size distribution

c) Particle size distributions

FIGURE 1.1 Scaling measures and distributions: a) density data (clay soil [From Chepil, W.S., Soil Sci., 70, 351, 1950.]) fitted by a powerlaw model and associated log/log data fitted by a linear model; b) aggregate size distribution (From Perrier, E.M.A. and Bird, N.R.A., Soil Tillage Res., 64, 91, 2002. With permission); c) particle size distributions from several soils. (From Bird, N.R.A., Perrier, E., and Rieu, M., Eur. J. Soil Sci., 55, 55, 2000. With permission.)

[ ]

N ≥ r ∝ r −D

(1.2)

In a way similar to that applied to measures, when a cumulative distribution of natural objects (e.g., soil particles, pores or aggregates) is fitted by a powerlaw model (Figure 1.1b,c), it is commonly called fractal in the soil science literature despite some formal objections.9,10 For this reason, many soils have been called fractal, due to successful linear regression on log–log plots of many types of data (Figure 1.1). This view is reinforced when the D parameter proves to be a new indicator of structure that discriminates between different types of soil structures or different structural states for the same type of soil.11–13 Sometimes the log–log plot deviates strongly from linearity and making impossible a fractal description. Nevertheless, the measure generally does vary with scale, which means that the concept of a measure viewed as a scaling function remains valid, and deviations from a powerlaw scaling function suggest a more complex, multiscale organization in terms of soil structure. Even when good linear fits are obtained for log–log plots a degree of caution is needed in their interpretation. The estimation of a D value from such plots is sensitive to data precision and to the type of powerlaw model tested, as illustrated in Figure 1.2. Because this analysis is carried out on © 2003 by CRC Press LLC

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Scaling Methods in Soil Physics

Series1

EQ.MassFractal

Series2

0.3

Linear (Series1)

a)

LOG(ρr/ρL)

0.25

Linear (Series2)

0.2 Series1 : 0.15

y = -0.05x + 0.0006

0.1

R2 = 0.9785

0.05 Series2 : 0 -3

-2

-1

0

y = -0.2582x - 0.3588 R2 = 0.9613

LOG(r/L)

EQ.PSF

b)

LOG((ρr-ρPSF)/(ρL-ρPSF))

1.5 1.2 0.9

y = -0.6177x + 0.037

0.6

R2 = 0.9676

0.3 0 -3

-2

-1

0

LOG(r/L)

FIGURE 1.2 (a) Density scaling data (Sharpsburg soil [From Wittmus, H.D. and Mazurak, A.P., Soil Sci. Soc. Am. J., 22, 1, 1958.) fitted by Equation 1.17, solid line, or by Equation 1.1, dashed line; (b) two domains in the log–log plot associated with Equation 1.1, that is, two estimated fractal dimensions, similar to Rieu, M. and Sposito, G., Soil Sci. Soc. Am. J., 55, 1231, 1991b.

real data from soils whose type of fractal organization and dimension is obviously unknown a priori, the validity of the result is always questionable and research is ongoing to address this issue. B. DIFFERENT MEANINGS

FOR

“THE” SOIL FRACTAL DIMENSION AND SEARCH FOR LINKS

A major problem occurs when using fractal dimensions to quantify and classify soils because fractal dimensions may be derived from independent fractal interpretations of different observed scaling laws. If one chooses arbitrarily one type of fractal interpretation, and if the associated fractal dimension appears as a good parameter to discriminate between different soils from a practical point of view and in a particular operational study, this is useful; the meaning of the fractal dimension D may not matter in this specific, limited context. However the different fractal interpretations of scaling measures or distributions can lead to confusion if we mix different soil classifications based on indicators having the same name D but of different origin and meaning. A key question is: are there some links or no links between the fractal dimensions used in soil science?14–20 The more we find such links, the closer we are to the goal of a unified theory. Our aim is to clarify this point to at least avoid misuses, and also to investigate how far a single scaling process may apply to different properties of the same soil structure, in order to establish whether links between different types of structural scaling properties exist. If the possibility of links is corroborated by a theoretical model, the aim is to look for validation through appropriate © 2003 by CRC Press LLC

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The PSF Model of Soil Structure: A Multiscale Approach

5

experiments, in order to infer missing structural characteristics from measured ones, or hydraulic properties from structural properties. A way to proceed is to move from a mere conceptual fractal model of scaling measures or size distributions to a geometrical model representing simultaneously the different components of soil structure. Rieu and Sposito led pioneering work in that direction21 by considering a mass fractal model including theoretical links between different powerlaw scaling properties. For instance, the fractal dimension D derived from the powerlaw density scaling law was theoretically the same as the D associated with the powerlaw pore-size distribution and consequently the same as the D that appeared in their expression for the water retention curve. C. THE PSF APPROACH:

A

GEOMETRICAL, MULTISCALE MODEL

OF

SOIL STRUCTURE

With the PSF approach, we follow that of Rieu and Sposito and move from a purely conceptual fractal model of a particular scaling property (for instance, a Von Koch curve to model a fractal interface surface between solids and pores in, for example, References 22 and 23) to an explicit geometrical model of an entire soil structure (Figure 1.3). Whereas the traditional solid mass fractal represents a powerlaw pore-size distribution embedded in a conceptual solid space, which vanishes when the model is developed towards arbitrary small scales (and conversely for a pore mass fractal model), the PSF is a more general approach that represents pore and solid distributions, and that includes previous cases as limiting cases. Because it is a geometrical approach, from any assumption on the spatial arrangement of one phase one can derive consequences for the other phase, the boundary between the two phases and other related properties. Beyond the search for geometrical coherence, the PSF approach is defined first as a multiscale approach, to be open to future developments where self-similarity may occur only over a limited range of scales.

Geometrical objects used to model a given powerlaw scaling measure or distribution The PSF approach: looking for an actual geometrical object to model the soil structure

A Sierpinski carpet illustrating a fractal pore size distribution

A Sierpinski carpet illustrating a fractal particle size distribution

A Von Koch curve illustrating a fractal void solid interface

A Sierpinski carpet illustrating a fractal solid density

FIGURE 1.3 From independent fractal models and interpretations of powerlaw curve-fitting of scaling measures or distributions to a fractal model of a very porous object.

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Scaling Methods in Soil Physics

III. THE PSF MODEL A. DEFINITION The PSF model can potentially describe any multiscale porous medium. It is based on the iterative partitioning of a bound region in a space of Euclidean dimension d. We start with a representative region R of linear extent L, which is divided into three sets P, S and F (Figure 1.4a). The sets P and S represent pore and solid phases, respectively, which are considered to be well identified at any given level of resolution — for example, as white and black parts in an initially nonbinarized image of a porous medium. F represents the undefined “gray” complement where an increased resolution is needed to identify again P and S components. At each resolution level, i.e., at each iteration i in the modeling process, new P, S and F components appear within the F set, and the proportion occupied by these components is defined in a general way by probabilities pi, si, and fi respectively. At level i = 1, the initial region R can be depicted as a square (d = 2) or a cube (d = 3) composed of n subregions of linear size r1 = αL, so n = α −d

(1.3)

where α is called the similarity ratio, and np1 subregions belong to P, ns1 subregions belong to S, and nf1 subregions belong to F. At level i = 2, the set F is partitioned in a geometrically similar way yielding new pore and solid structures and a new set F composed of nf1nf1 regions of size r2 = α2L. At each level i, ri = α i L

(1.4)

r1 = Lα

L

a)

P

Fz

S

S

P

P

P

S

Fz

Level i=1 (p1=4/9, s1=3/9, f1=2/9)

Level i=2 (p2=2/9, s2=4/9, f2=3/9)

s=0

p=0

b)

Solid (mass) Fractal (p=1/9,s=0,f=8/9)

Pore Solid Fractal (p=1/9,s=1/9,f=7/9)

Pore (mass) Fractal (p=0,s=1/9,f=8/9)

FIGURE 1.4 (a) Definition of the PSF approach: multiscale formalism. P: pore set, probability pi; S: solid set, probability si; F: fractal set, probability fi; (b) the PSF model generalizes mass fractal models.

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The PSF Model of Soil Structure: A Multiscale Approach

7

This process is repeated T times. The resulting multiscale structure is composed of the pores and particles (sets P and S) generated at each iteration over the range of scales (ri)1≤i≤T and a set F composed of i =T

nT

∏f

i

i =1

subregions of size rT = αTL. In the simplest case, where the iterative process is strictly self-similar, we restrict ourselves to constant p, s and f probabilities at each level i, p + s + f =1

(1.5)

and to simplify further the definition and the subsequent calculations, we may consider constant p, s and f proportions instead of probabilities at each level. Thus the PSF multiscale definition includes that of a fractal set F defined by successive iterations, creating at each level i (nf)i subregions of linear size ri = αiL. We note that F is in fact a so-called prefractal set when the total number T of iterations is finite. The fractal dimension of F is given by D=

log( nf ) log(1 / α )

(1.6)

It is clear that, in the PSF multiscale approach, a pure fractal structural model is only the particular case where strict self-similarity occurs across a broad range of scale. Our following mathematical results have been obtained only in this case and should be considered as a useful, idealized reference case for future extensions. B. GENERATING PSF STRUCTURE MODELS 1. Different Geometrical Patterns Examples of different PSF structures are provided in References 3 and 4. The simplest originates from Neimark,24 who described a multiscale percolation system (Figure 1.5a) with black and white sites randomly distributed on a regular grid as in standard percolation theory, then smaller black and white cells appearing with increasing resolution. Figure 1.5b exhibits a variant where the cells exhibit irregular polygonal shapes (generated by a Voronoi tesselation around randomly distributed seed points, with an extension to three-dimensional polyhedral shapes in Figure 1.5k). In Figure 1.5c the pores are located at each level around the solids in a way that has been shown3 to be equivalent to the version where pores and solids have a symmetrical geometry; the equivalence relates to the global, fractal or nonfractal, measures and distributions and not to the topology or connectivity, which is obviously quite different. 2. With Only Pores or Only Solids The PSF model includes two special cases: when s = 0, the PSF reduces to a solid mass fractal and when p = 0, the PSF reduces to a pore mass fractal (Figure 1.4b). The PSF approach is thus a generalization of previous studies made about mass fractal models, with solid mass fractals used mainly in soil science, and pore mass fractals used mainly in geology. The two cases can occur whatever the geometrical pattern (Figures 1.5d,e,f). If T goes to infinity these two special cases yield a total void (the porosity equals 1) and a total solid structure (the porosity equals 0) respectively and hence cease to function as practical models of porous media. Thus the fractal domain must be © 2003 by CRC Press LLC

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Scaling Methods in Soil Physics

a)

b)

c)

d)

e)

f)

g)

h)

i)

j)

k)

FIGURE 1.5 The PSF model of soil structure can model different geometrical patterns: (a) is built upon a regular grid whereas (b) uses an irregular polygonal grid (Figures 1.5a,b [From Perrier, E., Bird, N., and Rieu, M., Geoderma, 88, 137, 1999.]) and in (c) the pores are located “around” the solids. (d) and (e) exhibit, respectively, pure solid and pore mass fractals to be compared with (a). (f) shows a solid mass fractal to be compared to the pore/solid composition in (c). (g) and (h) are, respectively, associated with (f) and (a) where the fractal (gray) set has been replaced by a solid phase at the last level representing the lower cutoff of scale. Similarly, (i) is associated with (c), but the fractal set at the last level has been replaced by a mixture of pores and solids. (j) and (k) are three-dimensional versions of (a) and (b), again with a mixture of pores and solids at the last level.

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The PSF Model of Soil Structure: A Multiscale Approach

9

bounded and, in the first case, the resulting prefractal set F at level T is de facto associated with the solid phase (e.g., the gray phase in Figure 1.5f becomes black in Figure 1.5g), and with the pore phase in the second case. 3. With Lower Bound or No Lower Bound In the PSF approach, there is no necessity for a lower cutoff of scale in order to model realistic porosities. When infinite iterations are carried out, the PSF remains a valid model for a porous medium, the porosity of which has been shown to depend only on the p and s parameters as follows: φ PSF =

p p+s

(1.7)

A PSF model can easily match any experimental porosity value while representing a broad distribution of pores and solids of different sizes — the properties of which will be given later on in the chapter. Nevertheless, we will generally introduce a lower bound in the PSF model because we consider that there should be a lower cutoff of scale in real porous media; thus we have a finite number of iterations T and a resulting prefractal set F. In order to complete the description of a porous medium we must identify this set with the solid phase (Figure 1.5h), the pore phase or a combination of both phases. The treatment of the set F is important in determining the scaling behavior of density and porosity. Indeed, with the inclusion of a lower bound to the PSF it is possible to cover a range of monotonic scaling behavior from increasing density and decreasing porosity through constant density and porosity to decreasing density and increasing porosity with increasing sample size.6 C. FRAGMENTATION

OF A

PSF STRUCTURE

In the first studies about the PSF model of soil structure, we considered only the distributions of pores and solid and related measures that could be made on a soil sample, but we did not consider the distribution of aggregates that may result from incomplete fragmentation of the sample. In fact, a fragmentation process is partly independent of the underlying structure. For example, a fractal number size distribution of fragments may result from a very simple, multiscale fragmentation process applied to a homogeneous bulk material.25 We showed later5 that the superimposition of a self-similar fragmentation process of dimension Dfrag on a PSF structure of fractal dimension D leads to a generalization of previous results obtained in soil science, with the novel and realistic feature that both aggregates and particles are released during the fragmentation process. As illustrated in Figure 1.6, one keeps (1 – xfrag) unfragmented aggregates within the F subparts of the PSF model, and one defines a fragmentation fractal dimension as:

D frag =

(

D • Log nfx frag Log( nf )

)

(1.8)

IV. INFERRING DETERMINISTIC LINKS BETWEEN SEVERAL SCALEDEPENDENT SOIL PHYSICAL PROPERTIES A. LINKS

BETWEEN

DIFFERENT SCALING STRUCTURAL PROPERTIES

In this section we provide a list of mathematical results obtained for the self-similar PSF model in previous studies, using unified notations.

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Scaling Methods in Soil Physics

FIGURE 1.6 The PSF model generalizes fractal models for soil fragmentation, for example, where p = 7/16, s = 3/16, f = 6/16, xfrag = 4/6. (From Perrier, E.M.A. and Bird, N.R.A., Soil Tillage Res., 64, 91, 2002.)

1. Pore and Particle Size Distributions It has been shown4 that the number Ns[r = ri] of solid particles of size ri in a PSF structure is:

[

]

s N S r = ri = f

[

 ri   L  

−D

(1.9)

]

and that the cumulative number N S r ≥ ri of solid particles of size greater than ri is:   r  −D    i  − 1  L  N S r ≥ ri = ns  nf − 1       

[

]

(1.10)

For ri << L we obtain, in a simplified and continuous version, the following approximation: r NS ≥ r ∝   L

[ ]

−D

(1.11)

In a totally symmetrical way, the number Np of pores of size ri is given by: pr  N P r = ri =  i  f  L

[

and, if ri << L

© 2003 by CRC Press LLC

]

−D

(1.12)

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The PSF Model of Soil Structure: A Multiscale Approach

11

−D

r NP ≥ r ∝    L

[ ]

(1.13)

Equations 1.11 and 1.13 commonly define a fractal number-size distribution of solids and pores, as previously reported (see, for example, Equation 1.2). 2. Aggregate or Fragment Size Distributions When a constant probability xfrag of fragmentation is considered to apply at each scale on a PSF structure (Figure 1.6), it has been shown5 that the number Nfrag[r = ri] of fragments of size ri is:

N frag

 s  r  r = ri =  1 − x frag +   i  f  L  

[

]

− D frag

(1.14)

where the fragmentation dimension Dfrag → D when xfrag → 1. In the PSF approach, the fragments are a mix of porous aggregates and particles, as in real fragmentation experiments. If xfrag = 1, the fragmentation is complete; all the fragments represent primary particles and Equation 1.14 reduces to Equation 1.9. If s = 0 there are only porous aggregates made of monosized particles and Equation 1.14 reduces to the formula established for the number NA[r = ri] of aggregates from solid mass fractal theory:

] (

[

N A r = ri = 1 − x frag

 ri   L  

)

− D frag

(1.15)

For the cumulative number of aggregates or fragments, when ri << L, we obtain the approximation,

N frag

r ≥ r ∝   L

[ ]

− D frag

(1.16)

which commonly defines a fractal number size distribution of fragments, aggregates or particles. 3. Mass, Density/Porosity Scaling The mass M (of the solid phase) in a PSF structure scales as a function of the sample size L as follows:6 D

M L = L ρ PSF d

 L + (ρ r − ρ PSF )  r d r

where  s  ρ PSF =  ρ  p + s s © 2003 by CRC Press LLC

(1.17)

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12

Scaling Methods in Soil Physics

is the density of the PSF model when the model is developed ad infinitum, ρs is the density of the solid phase, where ρr is the density of the F parts of size r. Let us note that, when s = 0, that is, in the special case of a solid mass fractal model, ρPSF = 0, and only in that case, Equation 1.17 reverts to the classical powerlaw  L ML = Mr   r

D

given by Equation 1.1. From Equation 1.17, the bulk density is given by:  L ρ L = ρ PSF + (ρ r − ρ PSF )  r

D −d

(1.18)

and, similarly, the porosity of the PSF follows as:

φ L = φ PSF

 L + ( φ r − φ PSF )  r

D −d

(1.19)

Except in the special cases already quoted, the PSF is not a mass fractal, but in every case its mass, porosity and density are scaling functions, expressed in terms of the sample size, or observation scale, L, and resolution level, or yardstick scale, r. The above expressions are generalizations of those developed for mass fractal models, incorporating the extra fitting parameters ρPSF and φPSF. 4. Solid-Pore Interface Area Scaling It has been shown3 that the area S of the pore solid interface in a PSF structure of size L scales as a function of the resolution r as a logarithmic function when D = d – 1, and when D ≠ d – 1, and ps ≠ d – 0 as: D

SL =

 L 2 dn − d ps ( Ld −1 −   r d −1 ) −d p + s (1 − n f ) r

(1.20)

and that Equation 1.20 becomes a simple powerlaw scaling function D

 L SL ∝   Sr r if and only if D > d − 1 and the number T of iterations is high enough, for the first term in the right-hand side then becomes negligible. A similar result holds for the two special mass fractal cases (when p = 0 or s = 0 ). 5. Overview Discussion A self-similar PSF structure can model at the same time: © 2003 by CRC Press LLC

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13

• A (fractal) powerlaw pore size distribution and a (fractal) powerlaw particle size distribution, the exponents of which involve the same D • A fractal pore-solid interface of dimension D when D > d – 1 • A (fractal) powerlaw fragment or aggregate size distribution involving a dimension Dfrag where D frag < D , and Dfrag → D when the fragmentation process is complete • A fractal mass for the solid or the pore space in two special cases and, in the general case, mass, density or porosity scaling laws involving exponent D (and at least one additional parameter) The equations derived for the PSF are identical to or generalize previous equations derived for classical fractal models of soil physical properties. Thus, in many cases, they can be used in the same context and will provide the same fits and estimated fractal dimensions as in previous studies (e.g., Figure 1.1) except when a generalized equation introduces one more parameter to fit (Figure 1.2). The PSF approach allows us to establish, then to check, a list of coherent, theoretical links, on a geometrical basis, between separate studies and estimated fractal dimensions. If a comprehensive set of data were available on the same soil, the theoretical links inferred by the PSF equations could be validated, and the fractal behavior of a certain class of soils could be better established. Moreover, a general scaling trend exhibited simultaneously by a large set of complementary data would appear more reliable than a collection of independent, log–log plots, each of them leading to the estimation of a fractal dimension without “cross validation.” B. LINKS

BETWEEN

STRUCTURAL

AND

HYDRAULIC SCALING PROPERTIES

In the same vein, the PSF approach allows us to check or to establish theoretical links between structural and hydraulic properties in soils. The main link relies on the physically based, though very simplified, capillary model that has been used for a long time to associate the water retention curve or the hydraulic conductivity curve with the pore-size distribution. This association integrates more complex, but deterministic, links between structure and dynamics at a microscopic scale, and should at least account for the connectivity of the pore network,26,27 especially with regard to the crucial effect of water paths on the hydraulic conductivity, and also on the hysteresis of the water retention curve. Nevertheless, in a first modeling approach, working as many other authors, one could introduce mere weighting coefficients accounting for tortuosity or connectivity in the calculation of a hydraulic conductivity curve from a pore-size distribution (for example, References 21 and 28). One can look again, mainly for trends in a one-to-one association between a water retention curve and a pore-size distribution. Using the latter point of view, we obtained an analytical expression for the water retention curve in a PSF model, the first interest of which lies in the comparison with previous theoretical expressions obtained using the same assumptions. Two expressions were available to model the water retention curve in a fractal soil (Equations 1.21 and 1.22). By establishing Equation 1.21, θ θ max

h  =  min   h 

d −D

(1.21)

where θ is the volumetric water content and h is the capillary pressure. Tyler and Wheatcraft29 gave a fractal interpretation of the widely used Brooks and Corey30 empirical expression, whereas Rieu and Sposito21 obtained a different expression in the same context:

θ + 1 − θ max

© 2003 by CRC Press LLC

h  =  min   h 

d −D

(1.22)

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Scaling Methods in Soil Physics

Then Perrier31 showed that only Equation 1.22 applied on a mass fractal model and that the general equation for the water retention curve associated with any fractal pore-size distribution has the following general expression: h  θ + A − θ max = A min   h 

d −D

where hmin ≤ h ≤ hmax

(1.23)

where A is the upper limit of the fractal porosity32 and Equation 1.23 has two special cases (Equation 1.21 when A = 1 and Equation 1.22 when A = θmax). By finally establishing the following expression for the water retention curve in the selfsimilar PSF model:   h  d −D  θ = θ max − φ PSF  1 −  min   where hmin ≤ h ≤ hmax *   h    

(1.24)

Bird et al.4 gave a geometrical interpretation of Equation 1.23 that has several implications as regards the links between hydraulic and structural data. Let us consider the exponent D that can be estimated from a fit of Equation 1.24 to water retention data. This D value could give the mass or the density or the porosity scaling exponent involved in Equations 1.17, 1.18, and 1.19, using a PSF theory generalizing that of Rieu and Sposito.21 The value of D could also give the exponent of the particle-size distribution, using a PSF theory that shows that the particle- and the pore-size distributions scale in an identical way. Conversely, as illustrated in Figure 1.7, the value of D estimated from simple measures of particle sizes and masses, obtained from mechanical sieving, could give the exponent in Equation 100

(c)

(a)

80

Water suction /kPa

Log cumulative mass of solids of size
2.5

2.0

1.5

60 (b)

40

20

1.0 -4

-3

-2

-1

0

Log size rs(i ) of solids Estimated D from the particle distribution: best fit 2.73, confidence interval at 90% [2.55,2.91]

a)

0

0

0.1

0.2

0.3

Water content

0.4

0.5

/m3m-3

Estimated water retention curve using D: a) D=2.73, b) D=2.55 c) D=2.91

b)

FIGURE 1.7 Structural data scaling vs. hydraulic data scaling, example, where the value of D calculated from the particle size distribution (a) of a silty clay loam (Ariana soil [From Rieu, M. and Sposito, G., Soil Sci. Soc. Am. J., 55, 1231, 1991b.]) is used to predict the water retention curve (b). (From Bird, N.R.A., Perrier, E., and Rieu, M., Eur. J. Soil Sci., 55, 55, 2000. With permission.) * Let us note that the condition hmin ≤ h ≤ hmax, associated with a limited range of scale over which a fractal structure exists, applies to Equations 1.21 and 1.22. In Equation 1.21 hmax = ∞. © 2003 by CRC Press LLC

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1.24 of the variation of the capillary pressure as a function of the water content. Of course we cannot judge the success of the approach on one example, and the log–log fit of the five structural data is less than perfect. But this example, published by Bird et al.,4 illustrates the strong principle of the PSF approach, where deterministic links between structural and hydraulic properties could be inferred by a simplified geometrical and physical modeling approach. V. CONCLUSION AND PERSPECTIVES: TOWARD EXTENDED PSF VIRTUAL STRUCTURES AND PORE NETWORK MODELING Further work has to be done to validate and to calibrate the PSF model on real data. This will require, at least, the acquisition of pore-, particle- and aggregate-size distributions on the same soil sample to test the validity of the PSF model and to estimate its parameters p,s, (thus f = 1-p-s), n and xfrag. The more constraints that will be available, the better, and porosity or density data sets over a large range of scale would be very useful. Then, if successful, PSF virtual soil structures matching real soil samples’ properties can be built to conduct numerical experiments. The formalism developed in the PSF approach to define in a very general way a multiscale porous medium provides scope for further theoretical studies. On the one hand, different geometrical patterns (Figure 1.5) can be created, first to check how far the mathematical results obtained on regular space a) air entering water-saturated soil

3000

b) water entering air-saturated soil

1.0

pressure h (mbars)

drainage

K/K sat

c)

2000

d)

simulated data 0.5

experimental data 1000

imbibition water content

Θ

0 0.1

0.2

0.3

0.4

0.5

(cm3/cm3)

0.0 0.0

0.2

0.4

0.6

0.8

1.0

θ/θ sat

FIGURE 1.8 Pore network simulation in a two-dimensional space. (a) and (b) simulation of drainage and imbibition in a mass fractal structure calibrated to represent a given pore size distribution (Ariana soil [From Rieu, M. and Sposito, G., Soil Sci. Soc. Am. J., 55, 1231, 1991b.]); (c) hysteresis of the simulated water retention curves; (d) the simulated hydraulic conductivity curve is unreliable because of the difficulty in modeling the connectivity of the pore network in a two-dimensional space. (From Rieu, M. and Perrier, E., C.R. Acad. d’Agricul. France, 80(6), 21, 1994.)

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pressure (arbitrary units)

a)

b) 0.00001 0.000008 0.000006 0.000004 0.000002 0

0

200000

400000

600000

800000

water content (voxels) c)

FIGURE 1.9 Pore network simulation in a three-dimensional space (courtesy of Jean-François Delerue). (a) an arbitrary image of a PSF structure exhibiting polyhedral pores over three scale levels; (b) extraction of a pore network: the balls represent the pore location and sizes; (c) simulation of the water retention curve in imbibition.

partitions hold when variations are introduced concerning pore and solid shapes, as well as sizes within each discrete level to match better real variability. Such work has already been done31 on two-dimensional computer constructions such as those shown in Figures 1.5g and 1.5i, where analytical expressions Equations 1.22 and 1.24 for the water retention curve proved to give quite successful fits and led to very good estimations of the underlying fractal dimension D of the simulated fractal model structures. This could be extended to other patterns or to three-dimensional fractal models (Figures 1.5j and 1.5k). On the other hand, using simulations on virtual soil structures, one can account for the effect of the connectivity of the pore network on the hysteresis of the water retention curve (Figure 1.8c) and on the value of the hydraulic conductivity (Figure 1.8d). Such results33 were obtained directly by applying classical methods in the field of pore network modeling31 The novelty involves extracting a pore network (Figure 1.9) from any type of two- or threedimensional porous structure by means of new image analysis tools that have been initially tested on real volumetric images of soils.34 These new algorithms will first allow us to simulate the hydraulic properties of a large set of PSF models, including the hydraulic conductivity, which depends not only on the pore size distribution but also on the geometrical pattern of the structure and on the associated topology of the pore network. The mathematical scaling expressions summarized in this chapter and obtained in the self-similar, fractal case will provide useful limiting cases to check the results of the simulations. VI. ACKNOWLEDGMENTS This work is dedicated to the memory of the late Michel Rieu, who initiated and greatly encouraged it. © 2003 by CRC Press LLC

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REFERENCES 1. Baveye, P., Parlange, J.Y., and Stewart, B.A., Eds., Fractals in Soil Science, Advances in Soil Science, CRC Press, Boca Raton, FL, 1998. 2. Pachepsky, Y.A., Crawford, J.W., and Rawls, J.W., Eds., Fractals in Soil Science, Developments in Soil Science 27, reprinted from Geoderma, 88. 3–4, Elsevier, 2000. 3. Perrier, E., Bird, N. and Rieu, M., Generalizing the fractal model of soil structure: the PSF approach. Geoderma, 88, 137, 1999. 4. Bird, N.R.A., Perrier, E., and Rieu, M., The water retention curve for a model of soil structure with Pore and Solid Fractal distributions. Eur. J. Soil Sci., 55, 55, 2000. 5. Perrier, E.M.A. and Bird, N.R.A., Modeling soil fragmentation: the PSF approach. Soil Tillage Res., 64, 91, 2002. 6. Bird, N.R.A. and Perrier, E.M.A., The PSF model and soil density scaling. Eur. J. Soil Sci., to appear in 2003. 7. Baveye, P., Boast, C.W., Ogawa,S., Parlange, J.Y., and Steenhuis, T., Influence of image resolution and thresholding on the apparent mass fractal characteristics of preferential flow patterns in field soils. Water Resour. Res., 34, 2763, 1998. 8. Mandelbrot, B.B., The fractal geometry of nature. Ed. Tech. Doc. Lavoisier, Paris, 1983. 9. Crawford, J.W., Sleeman, B.D., and Young, I.M., On the relation between number-size distribution and the fractal dimension of aggregates. J. Soil Sci., 44, 555, 1993. 10. Baveye, P. and Boast, C.W., Concepts of “fractals” in soil science: demixing apples and oranges. Soil Sci. Soc. Am. J., 62, 1469, 1998. 11. Bartoli, F., Bird, N., Gomendy, V., and Vivier, H., The relationship between silty soil structures and their mercury porosimetry curve counterparts: fractals and percolation. Eur. J. Soil Sci., 50, 9, 1999. 12. Young, I.M. and Crawford, J.W., The analysis of fracture profiles of soil using fractal geometry, Aust. J. Soil. Res. 30, 291, 1992. 13. Perfect, E. and Blevins, R.L., Fractal characterization of soil aggregation and fragmentation as influenced by tillage treatment. Soil Sci. Soc. Am. J., 61, 896, 1997. 14. Rieu, M. and Sposito, G., Relation pression capillaire-teneur en eau dans les milieux poreux fragmentés et identification du caractère fractal de la structure des sols (in French). Comptes Rendus Acad. Sci., Paris, Série II, 312, 1483, 1991. 15. Borkovec, M., Wu, Q., Dedovics, G., Laggner, P., and Sticher, H., Surface area and size distributions of soil particles. Colloids Surf. A, 73, 65, 1993. 16. Crawford, J.W., Matsui, N., and Young, I.M., The relation between the moisture release curve and the structure of soil. Eur. J. Soil Sci., 46, 369, 1995. 18. Perfect, E., On the relationship between mass and fragmentation fractal dimensions. In: Novak, M.M. and Dewey, T.G., Eds., Fractal Frontiers, World Scientific, Singapore, 349, 1997. 19. Pachepsky, Ya.A., Gimenez, D., Logsdon, S., Allmaras, R., and Kozak, E., On interpretation and misinterpretation of fractal models: a reply to “comment on number-size distributions, soil structure and fractals,” Soil Sci. Soc. Am. J., 61, 1800, 1997. 20. Gimenez, D., Allmaras, R.R., Huggins, D.R., and Nater, E.A., Mass, surface and fragmentation fractal dimensions of soil fragments produced by tillage. Geoderma, 86, 261, 1998. 21. Rieu, M. and Sposito, G., Fractal fragmentation, soil porosity, and soil water properties: I Theory. II Applications. Soil Sci. Soc. Am. J., 55, 1231, 1991. 22. de Gennes, P.G., Partial filling of a fractal structure by a wetting fluid. In: Physics of Disordered Materials, Adler et al., Eds., Plenum Press, New York, 227, 1985. 23. Avnir, D., Farin, D., and Pfeifer, P., Surface geometric irregularity of particulate materials: the fractal approach. J. Colloid Interface Sci., 103, 1985. 24. Neimark, A.V., Multiscale percolation systems. Soviet Phys.-JETP, 69, 786, 1989. 25. Turcotte, D.L., Fractals and fragmentation. J. Geophys. Res., 91, 1921, 1986. 26. Perrier, E., Mullon, C., Rieu, M., and de Marsily, G., Computer construction of fractal soil structures. Simulation of their hydraulic and shrinkage properties. Water Resour. Res., 31, 12, 2927–294, 1995. 27. Bird, N.R.A. and Dexter, A.R., Simulation of soil water retention using random fractal networks. Eur. J. Soil Sci., 48, 633, 1997.

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Scaling Methods in Soil Physics 28. Fuentes, C., Vauclin, M., Parlange, J-Y., and Haverkamp, R., Soil-water conductivity of a fractal soil. In: Baveye, P., Parlange, J.Y., and Stewart, B.A., Eds., Fractals in Soil Science, Advances in Soil Science, CRC Press, Boca Raton, FL, 1998. 29. Tyler, S.W. and Wheatcraft, S.W., Fractal processes in soil water retention. Water Resour. Res., 26, 1045, 1990. 30. Brooks, R.H. and Corey, A.T., Hydraulic Properties of Porous Media. Hydrology Paper 3, Colorado State University, Fort Collins, CO, 1964. 31. Perrier, E., Structure géométrique et fonctionnement hydrique des sols. Ph.D. Université Pierre et Marie Curie, Paris VI, 1994 (Editions ORSTOM, Paris, 1995). 32. Perrier, E., Rieu, M., Sposito, G., and de Marsily, G., Models of the water retention curve for soils with a fractal pore-size distribution. Water Resour. Res., 32(10), 3025, 1996. 33. Rieu, M. and Perrier, E., Modelisation fractale de la structure des sols. C.R. Acad.Agric. France, 80(6), 21, 1994. 34. Delerue, J.F. and Perrier, E., DXSoil, a library for image analysis in soil science. Comput. Geosci. 28, 1041, 2002. 35. Chepil, W.S., Methods of estimating apparent density of discrete soil grains and aggregates. Soil Sci., 70, 351, 1950. 36. Wittmus, H.D. and Mazurak, A.P., Physical and chemical properties of soil aggregates in a Bunizem soil, Soil Sci. Soc. Am. J., 22, 1, 1958.

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Scaling and Multiscaling of Soil Pore Systems Determined by Image Analysis A.M. Tarquis, D. Giménez, A. Saa, M.C. Díaz, and J.M. Gascó

CONTENTS I. Introduction.............................................................................................................................19 II. Background.............................................................................................................................20 A. Box-Counting Method .................................................................................................20 B. Dilation Method...........................................................................................................20 C. Random Walk ..............................................................................................................21 D. Generalized Dimensions, Dq .......................................................................................22 E. The f(α)-Singularity Spectrum ....................................................................................24 F. The Configuration Entropy..........................................................................................25 III. Practical Considerations .........................................................................................................26 IV. Scaling of Pores in Soils and Rocks......................................................................................28 V. Future Research Lines............................................................................................................29 VI. Acknowledgments ..................................................................................................................31 References ........................................................................................................................................31

I. INTRODUCTION One of the most direct methods of characterizing soil structure is the analysis of the spatial arrangement of pore and solid spaces on images of sections of resin-impregnated soil. Recent technological advances in digital imagery and computers have greatly facilitated the application of image analysis techniques in soil science.1,2 Thick sections (soil-blocks) are analyzed by reflected light, and thin sections are analyzed by transmitted light to obtain images from which pores (filled with a resin) and solid spaces can be separated using image analyses techniques.3,4 Direct measurements on images together with application of set theory are used to quantify connectivity, size, and shape of pores.5 Statistical analyses of the geometry of soil structure can then provide indices of a more general applicability to the characterization of soil structure than any physical property because they can lead to a mechanistic understanding of relationships between geometry of soil structure and soil processes. However, several authors have pointed out that results are a function of image resolution and of the threshold value used to separate pore from solid space.6–8 Scaling of pore systems could potentially be characterized with fractal and multifractal techniques. The fractal approach assumes a hierarchical distribution of mass in space such that at any resolution the fractal structure is seen as the union of subsets similar to the whole. These subsets are either identical (deterministic fractal) or statistically identical (stochastic fractal). In this instance 19 © 2003 by CRC Press LLC

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a single fractal dimension serves to characterize the mass distribution.9 Fractal techniques have been successfully applied to the characterization of pore systems in soil.10–14 In an object exhibiting multifractal scaling, mass is distributed in a hierarchical pattern as in the fractal case. The whole is formed by the union of similar subsets, but the subsets are related to the whole through different scaling factors. As a result, multifractal distributions cannot be characterized with a single fractal dimension.15 The multifractal character of a system is intuitively related to the complexity (deterministic or stochastic) of the processes generating it. Multifractal scaling has been found in rock pore systems16 and suggested as possible in soils based on the distribution of pore sizes17,18 or on their spatial pattern.19–21 The distinction between fractal and multifractal scaling in soils is needed for modeling links among fractal dimensions and soil processes and properties.22,23 Even though comprehensive reviews have been written on fractal models in soil science,24–28 there is significantly less information on multifractal techniques as they apply to soils. Thus, the objectives of this work are to review multifractal techniques as they relate to image analysis of the spatial distribution of soil pores, and to discuss potential applications.

II. BACKGROUND The main purpose of this section is to introduce some of the fractal and multifractal methods used in the context of image analysis of soil structure. A complete account of fractal and multifractal theories can be found, among others, in Feder’s29 and Baveye and Boast’s books.26

A. BOX-COUNTING METHOD This methodology is classical in this field and has generated a large volume of work. If a fractal line in a two-dimensional space is covered by boxes of side length δ, the number of such boxes, n(δ), needed to cover the line when δ → 0 is:30 n(δ ) = cδ − D L

(2.1)

The length of the line studied (e.g., the pore–solid interface), L(δ), can be defined at different scales and is equal to δn(δ). At small δ values the method provides a good approximation to the length of the line because the resolution of the image is approached. At larger sizes the difference between δn(δ) and the “true” length increases. Thus, DL is estimated using small δ values.13 The box-counting method is also used to obtain a fractal dimension of pore space by counting boxes that are occupied for at least one pixel belonging to the class “pore.”13

B. DILATION METHOD The study by Dathe et al.8 is the only published report of this method in soil science. The dilation method follows essentially the same procedure as the box-counting method, but instead of using boxes it uses other structuring elements to cover the object under study, e.g., circles.8 The image is formed by pixels, which are either square or rectangular in shape. If circles are used, the measure of scale is their diameter (as is the side length of the box in the box-counting technique). If we want to have the same dilation in any direction, the orthogonal and diagonal increments should be biased by √2, which corresponds to the hypotenuse of a square of unit side length.31 The length of the studied object is counted by numbers of circles, and then the slope of the regression line between the log of the object length and the log of the object diameter is defined by the relation: L(δ ) = cδ 1 −D L © 2003 by CRC Press LLC

(2.2)

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Dathe et al.8 applied the box-counting and dilation methods to the same images and found nonsignificant differences in the values of the fractal dimensions obtained with both methods. They pointed out, however, that fractal dimensions estimated with both methods are different: the box counting dimension is the Kolmogorov dimension while the dimension obtained with the dilation method is the embedding dimension (Mikowski-Bouligand). For further details see Takayasu’s work.32

C. RANDOM WALK Fractal methods can also be used to describe the dynamic properties of fractal networks.12,33 Characterization of fractals involving space and time is achieved through the use of fractons31 or the spectral dimension.34 For example, Crawford et al.33 related measurements of the spectral dimension d to diffusion through soil, associating d with the resistance degree to which the network delays the diffusing particle in a given direction. The determination of d is based on random walks, where in each walk the number of steps taken (ns) and the number of different pore pixels visited (Sn) are computed. At the beginning of the random walk a pore pixel is randomly chosen, then a random step is taken to another pore pixel from the eight pixels surrounding the present one (see Figure 2.1A). If the new pore pixel has not been visited by the random walk, the Sn and ns are increased by one, otherwise only ns is increased. The random walk stops when a certain number of null steps (the step goes into a site used previously during the walk) are achieved or the random walk arrives at an edge of the image (for further details see Crawford et al.33). A graphical representation of these random walks is shown in Figure 2.2. The number of walks and the maximum number of null steps for each walk can vary.31 Also a four-connected random walk (Figure 2.1B) can be used instead of eight-connected one.12 For each random walk, d is calculated based on the relation:

ns = cSn

d 2

(2.3)

where c is a constant. The mean value of the d calculated for each walk is the spectral dimension. A)

B)

x

x

pore pixel

possible step

soil pixel

pixel nonconsidered

FIGURE 2.1 Possible steps taken in a) eight-connected random walk, and b) four-connected random walk. The present position of the pore pixel is marked by an ×, and the arrows indicate the possible next pore pixel. © 2003 by CRC Press LLC

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FIGURE 2.2 A simplified example of one random walk through the pore space. (From Anderson, A.N. et al., Soil Sci. Soc. Am. J., 60, 962, 1996. With permission.)

D. GENERALIZED DIMENSIONS, DQ In the box-counting technique, a box is counted regardless of the proportion of its area covered with pixels of the pore class. Thus, boxes entirely occupied by pore class pixels have the same weight as boxes containing one pore pixel. On the other hand, generalized dimensions are calculated using the box-counting technique by accounting for the mass contained in each box. An image is divided into n boxes of size δ (n(δ)), and for each box the fraction of pore space in that box, Pi, is calculated:

Pi =

mi = M

mi

(2.4)

n (δ )

∑m

i

i =1

where mi is the number of pore class pixels and M is the total number of pore class pixels in an image. In this case, the pore space area is the measure whereas the support is the pore space itself. The next step is to define the generating function (χ(q, δ)) as: © 2003 by CRC Press LLC

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n(δ )

χ(q, δ ) =

∑ [ χ (q, δ)]

q ∈R

i

(2.5)

i =1

where    m q χ i ( q, δ ) = Pi =  n δ i ()  mi   i=1

∑

q

      

(2.6)

and n(δ) is the number of boxes of size δ in the whole image, χi is a weighted measure that represents the percentage of pore space in the ith box, and q is the weight or moment of the measure. A log–log plot of a self-similar measure, χ (q, δ), vs. δ at various values for q gives: χ( q, δ ) ~ δ

()

−τ q

(2.7)

where τ(q) is the qth mass exponent,29 sometimes called the Rényi exponent.35 We can express τ(q) as: log( χ( q, δ )) log(δ )

τ( q) = − limδ→0

(2.8)

Then the generalized dimension, Dq, can be introduced by the following scaling relationship:35

D q = limδ→0

[

log χ( q, δ )

]

(q - 1) logδ

(2.9)

and, therefore, τ(q) = (q – 1)Dq

(2.10)

For the case that q = 1, Equation 2.9 cannot be applied, using instead:

()

nδ

∑ χ (1, δ)log[χ (1, δ)] i

D1 = limδ→0

i =1

i

logδ

(2.11)

The generalized dimensions, Dq, for q = 0, q = 1 and q = 2 are known as the capacity, the information, and the correlation dimensions, respectively.36 The capacity dimension is the boxcounting or fractal dimension. Two images with the same capacity dimension can have different distribution of mass (i.e., different spectra). The information dimension is related to the entropy of the system, whereas the correlation dimension computes the correlation of measures contained in intervals of various sizes. A graph representing the values of Dq vs. q presents a characteristic shape of an inverted S (see Figure 2.3B) in the case of a multifractal measure. Muller et al.37 defined the © 2003 by CRC Press LLC

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B)

A)

2.33

2.00

2.23 Dq

f(α)

1.50 1.00

2.03

0.50 0.00 1.60

2.13

1.93

1.80

2.00

2.20

2.40

1.83 -12

2.60

-9

-6

α

-3

0

3

6

9

12

q

FIGURE 2.3 Spectra obtained with a multifractal analysis of the gray-level measure from a soil thin section image: A) spectrum of the fractal dimensions f(α) and B) spectrum of generalized dimensions Dq.

width of the multifractal spectra as w = D0 – D1 and suggested that w could be an important predictive parameter. A greater w means a wider spectrum and a more heterogeneous distribution on the pore space. When f(α) and τ (q) are smooth functions, we can express38 α ( q) = −

(

d τ( q) dq

(2.12)

)

f α ( q) = qα ( q) + τ( q)

(2.13)

where Equation 2.13 is obtained by Legendre’s transformation. A multifractal measure will show an f(α) curve with a parabolic shape (see Figure 2.3A).

E. THE f(α)-SINGULARITY SPECTRUM There are several ways to calculate the f(α)-singularity spectrum besides the general methodology of Evertsz and Mandelbrot39 and Feder,29 and the specific techniques of Meneveau and Sreenivasan.40 Once the generating function (χ(q, δ)) has been defined and taking µi as: χ i ( q, δ) χ( q, δ )

µ i (q, δ ) =

(2.14)

the coarse Hölder exponent, α(q), and the Hausdorff measure, f(q) are calculated for a series of diminishing box sizes δ and over a range of values of q as:

()

nδ

α ( q) = limδ→0

∑ µ (q, δ) log[µ (1, δ)] i

i =1

i

log δ

(2.15)

()

nδ

f ( q) = limδ→0

© 2003 by CRC Press LLC

∑ µ (q, δ)log[µ (q, δ)] i

i =1

i

logδ

(2.16)

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A relation between f and α is thus established, with q as a parameter. Notice that in the case that q = 1, then f(1) = α(1) = D1. For further details on Equations 2.15 and 2.16 see Chhabra and Jensen.41 The f(α) vs. α function contains three characteristic points: (αmin), (αmax) and fmax(α). Some authors used the amplitude, a, of the spectrum, i.e., a = (αmax – αmin), to analyze the complexity of a distribution.42,43 The amplitude is used in the same way as the width, w, but is a more sensitive parameter because its value is determined by the whole spectrum. Because a larger number of dimensions are needed to define a measure, larger amplitudes imply higher levels of complexity. Notice that the amplitude is determined by the positive (αmin) and negative (αmax) parts of a spectrum. From a probabilistic point of view, the negative part of the spectrum (q < 0) characterizes the rare events, or the values of the measure with lower frequencies (smaller values). As a result, the negative part of a spectrum contains always larger calculation errors (see Equation 2.6), and is typically not used for analyses other than for estimations of the amplitude. The symmetry of the spectrum provides information on a system. For example, a symmetric spectrum indicates that higher and lower frequencies of events are contributing the same information. An asymmetric spectrum shifted to the right/left area indicates that lower/higher frequencies scale over a larger range than the opposite part of the spectrum.43 The analysis of the symmetry of a spectrum requires a large number of points to assure that errors are minimized. To avoid or minimize calculation errors, some authors select a minimum threshold value of frequency to analyze the negative portion of the spectrum.44 Muller and McCauley16 obtained the f(α)-singularity spectra of simulated Cantor sets using two different scaling ratios, or Pi (weights). The symmetry and width of the simulated spectra increased as the difference in the Pi values increased. Thus, larger amplitude simulates a more diverse and unbalanced hierarchy of pore distribution in the soil.

F. THE CONFIGURATION ENTROPY There are situations in which multifractal methods are not enough to characterize a system entirely. For instance, Beghdadi et al.45 used two artificial images with subtle differences to demonstrate that the configuration entropy could separate them, while their multifractal spectra were identical. Another advantage of this parameter is that there is no need to select a range of scales for the calculations.45,46 Most of the developments presented here are based in the work of Andraud et al.,46,47 who estimated configuration entropies from planar thin sections of gold and sandstone. The basis of this method is to study the effect of scale in some geometrical quantities, for example, porosity. Estimation of porosity from a binary image implies counting the pixels representing pores and expressing this count as a percentage of the total number of pixels in the image. If an image is divided in an arbitrary number of smaller areas (e.g., boxes) and porosity is estimated in each subarea, a distribution of the measure “porosity” is obtained for the image. The basic idea of local porosity distributions (Local is related to each box into which the image is divided.) is to turn a global measure into a distribution of local measures.48 Even though we are focusing on porosity, any geometrical quantity is susceptible to the local concept. Imagine an image formed by pixels arranged in a square lattice of side L. A distribution of local porosity is obtained if this lattice is subdivided in n(δ) boxes of size δ from δ = 1 to δ = L/4 and in every box the number of pixels belonging to the pore class, Nj, is recorded. (Obviously, δ cannot approach the size of the lattice because the distribution of local porosity would lose meaning.) The probability associated with a case of j pore pixels in a box of size δ (pj(δ)) is defined as:

p j (δ ) =

© 2003 by CRC Press LLC

N j (δ ) n(δ )

(2.17)

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where Nj(δ) is the number of boxes with j pore pixels and n(δ) is the number of boxes of size δ. In this probability function the value j = 0 has a meaning and should be considered. The configuration entropy, H,46 δx∂

H ( δ) = −

∑ p (δ) log(p (δ)) j

(2.18)

j

j=0

measures the uncertainty associated with the porosity that can be attained by a set of boxes of size δ. Andraud et al.47 provided a rigorous connection between the configuration entropy H(δ) and the local porosity concept. This last one was first defined by Boger et al.48 Because the underlying probability changes with the number of pixels inside the box (δ × δ), H(δ) needs to be normalized for comparing entropy values corresponding to different sizes δ.48 This is done through47 H * ( δ) =

H ( δ) H max ( δ )

(2.19)

where H max ( δ ) = log(δ 2 + 1) H*(δ) is called the normalized configuration entropy of the two-dimensional morphology of the image. Andraud et al.47 calculated H* for two generated images of simple models (Bernoulli site percolation model and Poisson grain model) and two microstructures from experimental images (gold film deposit on a glass substrate and a natural sandstone). Except for the Bernoulli site percolation model (corresponding to a random microstructure), all H* curves have a local maximum at a characteristic length L* (Figure 2.4). The authors concluded that L*, and the curvature of H* around this value, characterize the microstructure. A more heterogeneous structure shows a wider extremum than those in which the pore and solid regions are more compact.47

III. PRACTICAL CONSIDERATIONS The application of a multifractal analysis requires first defining a measure. Several measures can be defined depending on the objective of the study. For instance, the interest could be in the spatial distribution of the gray-levels where any pixel can have an integer value ranging from 0 to 255.49,50 Also possible is to define a measure in a binary image in which a pixel can only have a value of 0 or 1. In the latter case the calculation is simpler than with the spectrum of gray-levels. Barnsley et al.51 describe a measure for a binary case, which is similar to other probability measures (e.g., Mandelbrot30; Hentschel and Procaccia52) but computationally simpler to implement. When computing boxes of size δ, the possible values of mi are from 0 to δ × δ. So let Nj be the number of boxes containing j pixels of pore space. Equations 2.5 and 2.6 will then be: q

n(δ )

χ(q, δ ) =

∑ (P ) = ∑ i

i =1

© 2003 by CRC Press LLC

n(δ )

i =1

q

 mi   M =  

δx∂

∑ j=1

 j N j   M

q

(2.20)

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FIGURE 2.4 Configuration entropy (H*(L)) as a function of the side length of the measurement cell calculated (L) for random microstructures: a) site percolation image (circles); b) random image with a constant bulk porosity (triangles); c) Poisson grain model (diamonds); and d) experimental image of observed gold film morphology (squares). (From Andraud, C., et al., Physica A, 235, 307, 1997. With permission.)

Using the distribution function Nj, calculations become simpler and computational errors are smaller. It is obvious that one should start with j = 1, because j = 0 means that there is no pore mass to account for. After defining a measure, attention should be paid to the determination of the scales over which a generation function vs. the length is linear when a log–log scale is used, as well as to the range of q’s over which the multifractal scaling is valid. For instance, Muller and McCauley16 found that, from the maximum possible range (1 to 512 pixels), the generation function was linear only between scales of 16 and 128 pixels (four points). This limited range of scales compromises the estimation of parameters through fitting. In Muller and McCauley16 q spanned from –15 to 15 at steps of 0.1. In a later work, Muller et al.37 used the generalized dimensions Dq to study the effect of image manipulation such as digitization and thresholding. They reduced the range of q from 0 to 5 (covered in increments of 0.2) because they considered that within this range errors are relatively small. Saucier and Muller53 studied the choice of the scaling range by selecting the range based on the best Chi-square statistics. Recently, Saucier and Muller54 proposed a systematic method to choose the scaling range for multifractal analysis and illustrated it with pore space of sedimentary chalks. They described χ(q,δ), τ(q) and Dq and then defined a “global” generating function that was obtained from several samples of equal size, from which the χ(q,δ) was previously calculated. The next step is to obtain the global τ(q) among the intervals that satisfies the well-known multifractality conditions for τ′(q) and τ′′(q).55 The intervals that satisfy these conditions are selected and then the reduced Chi-square of the regression is calculated individually; the one that presents the minimum value in the statistic test applied is selected. They stressed that further work should be done on this methodology. © 2003 by CRC Press LLC

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Other issues that require further investigation are whether a relatively small image (e.g., 512 × 512-pixel size) is representative of a pore system, and if instead of several replicates in a same image a better resolution can improve the results. Another possible problem is that thresholding could vary among replicates, creating more variation.

IV. SCALING OF PORES IN SOILS AND ROCKS Among the most commonly estimated fractal dimensions from images of soil pore systems are the mass fractal dimension and the fractal dimension of the pore–solid interface (surface fractal dimension).8,12,13,56–63 In more rare cases, the capacity and spectral dimensions have also been estimated.12,64 Generalized fractal dimensions have been used to characterize images of preferential flow patterns in field soils. Baveye et al.7 used the information dimension (DI) and the correlation dimension (DC) to sow the influence of image manipulation on the values of the generalized and of the mass and surface dimensions. This work was the first to apply the concept of generalized dimensions to images of soil profiles, and could be used as a model for studies with images of soil sections. In the first application of multifractal concepts to images of natural porous media, Muller and McCauley16 and Muller et al.37 demonstrated that the structure of pore space in sedimentary rocks can be successfully characterized by multifractal analysis. In Muller and McCauley,16 values of f(α) and their error bars were calculated based on ten replicate images from the same thin section. Differences in texture are reflected in the amplitude of the multifractal spectra (f(α)) calculated for each rock (Figure 2.5). Muller et al.65 found that the information dimension (DI) was linearly correlated (positive slope) to air permeability. The authors hypothesized that higher D1 values

FIGURE 2.5 The f(α) spectra for three different sedimentary rocks reflect different texture of the rocks. (From Muller, J. and McCauley, J.L., Transport Porous Media, 8, 133, 1992. With permission.)

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A)

B)

29

C)

FIGURE 2.6 Gray images (520 × 480-pixel resolution) of soil sections of a Normania clay loam soil sampled before harvest from a long-term tillage system from plots that had disc in the previous fall followed by either: a) chisel (NC2), or b) no till (NC3) in the spring, and c) a sample from an artificially packed column (AU).

FIGURE 2.7 Binary version of images in Figure 2.6.

(homogeneous distribution of pores in space) could mean higher connectivity. This hypothesis warrants more investigation. The concept of configuration entropy has not been applied to soils. To illustrate the method we selected three images of soil samples representing contrasting soil void patterns (Figures 2.6 and 2.7). Images are from a tilled Normania clay loam (fine loamy, mixed mesic, Aquic Haplustoll) soil sampled before harvest from a long-term tillage system experiment (natural/consolidated structure, NC). In addition, a sample from an artificially packed column (artificial/unconsolidated sample, AU) was used in the analysis. More details on the soil and experimental conditions can be found in Giménez et al.13 Gray images were converted to binary images (Figure 2.8) by visually comparing the derived binary image with the original gray image. The normalized configuration entropies (H*(δ)) were calculated at different δ on the binary images following the methodology in Andraud et al.47 Results of H*(δ) for the three samples show a common pattern: there is a range of δ values where H*(δ) reaches a maximum value after which it decreases and tends to stabilize around 0.2 (Figure 2.8). The maximum values of the normalized configuration entropy H*(L*) and of the characteristic length or the entropy length (L*) are different for each image. Curves in Figure 2.8 are similar to the one obtained from an image created by simulating a Poisson grain distribution model (Figure 2.4). In the example of the Poisson grain distribution model, the characteristic length is close to the circle diameter used in the Poisson simulation.

V. FUTURE RESEARCH LINES What is more interesting: to characterize the fractality of given physical objects or to identify multifractal measures defined on these objects? This question posed by Baveye and Boast26 has no easy answer. There are situations in which multifractal measures are found in supports that are not fractal. For instance, Folorunso et al.66 found that penetration resistance measured along a transect was a multifractal measure in a support that was a Euclidean line. Similarly, the distribution of © 2003 by CRC Press LLC

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1 0.9 0.8

H*(φ)

0.7 0.6

NC2

0.5

NC3

0.4

AU

0.3 0.2 0.1 0 0

30

60

90

120

150

φ(pixels) FIGURE 2.8 Configuration entropy (H*(δ)) as a function of the side length of the measurement cell (δ) for the binary images in Figure 2.7.

gray-levels in images shows multifractal scaling, but the maximum value of the spectrum is 2, indicating that the support of this distribution is a plane.49,50 There are also situations where the support of a multifractal measure is fractal, for example, a cluster generated by a DLA model is also the basis for constructing multifractal measures, as growth probability by particle collision, with fractal support.29 In image analysis of soil pore systems, the effort should be concentrated in applying multifractal statistics to search for measurements that can quantify soil properties and soil processes. The relationship between multifractal parameters and the configuration entropy could lead to the definition of a scale characteristic of the processes of soil genesis and of alterations introduced by soil management.19–21,50 The use of images of soil sections for fractal analysis and eventually for prediction of soil processes is limited by the two-dimensional nature of the images. Three-dimensional representation of soil structure from two-dimensional space requires assuming isotropy of the original soil sample.64 On the other hand, direct measurement of three-dimensional pore systems using x-ray computed tomography is becoming an important technique for analysis of soil structure.67 The relationship of multifractal parameters obtained from thin sections with processes determined by the three-dimensional configuration of soil structure remains to be tested. The local porosity concept has already been applied in three-dimensional images,68 opening the possibility of similar research in soils. An alternative avenue to the study of pore systems with multifractal techniques is to focus on the distribution of pore sizes obtained from images of soil sections.18 In a way, this approach reduces two-dimensional information to one dimension. The advantage of this approach might be in improving models of soil processes influenced by size distribution rather than by the spatial distribution of pores, e.g., hydraulic properties at the dry range of water contents. In general, however, image analyses of a pore system should take full advantage of the spatial distribution of pores in the two-dimensional space.

VI. ACKNOWLEDGMENTS In memory of the late Professor Richard Protz, for the enthusiasm that he always put in this work and his friendship. We are also very much obliged to the referees for their time and dedication to this manuscript. © 2003 by CRC Press LLC

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REFERENCES 1. Protz, R. and VandenBygaart, A.J. Towards systematic image analysis in the study of soil micromorphology. Sciences of Soils, 3, 1998. (available online at http://link.springer.de/link/service/journals/). 2. VandenBygaart, A.J. and Protz, R. The representative elementary area (REA) in studies of quantitative soil micromorphology. Geoderma, 89, 333, 1999. 3. Moran, C.J., McBratney, A.B., and Koppi, A.J. A rapid method for analysis of soil macropore structure. I. Specimen preparation and digital binary production. Soil Sci. Soc. Am. J., 53, 921, 1989. 4. Vogel, H.J. and Kretzschmar, A. Topological characterization of pore space in soil-sample preparation and digital image-processing. Geoderma, 73, 23, 1996. 5. Horgan, G.W. Mathematical morphology for analysing soil structure from images. Eur. J. Soil. Sci., 49, 161, 1998. 6. Ogawa, S., Baveye, P., Boast, C.W., Parlange, J.Y., and Steenhuis, T. Surface fractal characteristics of preferential flow patterns in field soils: evaluation and effect of image processing. Geoderma, 88, 109, 1999. 7. Baveye, P,, Boast, C.W., Ogawa, S., Parlange, J.Y., and Steenhuis, T. Influence of image resolution and thresholding on the apparent mass fractal characteristics of preferential flow patterns in field soils. Water Resour. Res., 34, 2783, 1998. 8. Dathe, A., Eins, S., Niemeyer, J., and Gerold, G. The surface fractal dimension of the soil-pore interface as measured by image analysis. Geoderma, 103, 203, 2001. 9. Perrier, E., Bird, N., and Rieu, M. Generalizing the fractal model of soil structure: the pore–solid fractal approach. In Fractals in Soil Science, Pachepsky, Crawford, and Rawls, Eds., Elsevier Science, Amsterdam, 2000, 47. 10. Hatano, R., Kawamura, N., Ikeda, J., and Sakuma, T. Evaluation of the effect of morphological features of flow paths on solute transport by using fractal dimensions of methylene blue staining patterns. Geoderma, 53, 31, 1992. 11. Brakensiek, D.L., Rawls, W.J., Logsdon, S.D., and Edwards, W.M. Fractal description of macroporosity. Soil Sci. Soc. Am. J., 56, 1721–1723, 1992. 12. Anderson, A.N., McBratney, A.B., and FitzPatrick, E.A. Soil mass, surface, and spectral fractal dimensions estimated from thin section photographs. Soil Sci. Soc. Am. J., 60, 962, 1996. 13. Giménez, D., Allmaras, R.R., Nater, E.A., and Huggins, D.R. Fractal dimensions for volume and surface of interaggregate pores — scale effects. Geoderma, 77, 19, 1997. 14. Bartoli, F., Dutartre, P., Gomendy, V., Niquet, S., and Vivier, H. Fractal and soil structures. In Fractals in Soil Science, Baveye, Parlange and Stewart, Eds., CRC Press, Boca Raton, FL, 1998, 203. 15. Gouyet, J.G. Physics and Fractal Structures. Masson, Paris, 1996. 16. Muller, J. and McCauley, J.L. Implication of fractal geometry for fluid flow properties of sedimentary rocks. Transport Porous Med., 8, 133, 1992. 17. Taguas, F.J., Martín, M.A., Caniego, F.J., and Tarquis, A.M. A mathematical model in particle-size distribution and multifractal analysis of soil porosity, presented in International Meeting on Fractal Geometry, Chaos and Ergodic Theory (ERGOFRACT ’95), Las Palmas de Gran Canaria, May, Spain, 1995. 18. Caniego, F.J., Martín, M.A., and San José, F. Singularity features of pore-size soil distribution: singularity strength analysis and entropy spectrum. Fractals, 9, 305, 2001. 19. Tarquis, A.M., Saa, A., Díaz, M.C., Gascó, J.M., Protz, R., Giménez, D., Duke, C., and Vandenbygaart, A.J. Fractal values of pore distributions in soils using four sizes of pixels, presented at 11th International Soil Conservation Organization Conference, Buenos Aires, Argentina, October 22–25,2000. 20. Protz, R., Giménez, D., Tarquis, A.M., VandenBygaart, A.J., Duke, C., Saa, A., Díaz, M.C., and Gascó, J.M. Assessing the influence of pixel size and subsampling interval on fractal values of soil voids, presented at Soil Research into the Next Millennium, Canadian Society of Soil Science. Guelph, ON, Canada, 2–4 August, 2001. 21. Tarquis, A.M., Giménez, D., Saa, A., and Díaz, M.C. Analysis of soil pore images: thresholding and configuration entropy, in Proc. 12th Int. Soil Conserv. Organ. Conf. Vol. IV, W. Lianxiang, W. Deyi, T. Xiaoning, and N. Jing, Eds., Tsinghua University Press, Beiging, China, 2002, 291. 22. Saucier, A. Effective permeability of multifractal porous media. Physica A, 183, 381, 1992.

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Scaling Methods in Soil Physics 23. Lovejoy, S., Schertzer, D., and Silas, P. Diffusion in one dimensional multifractal porous media. Water Resour. Res., 34, 3283, 1998. 24. Giménez, D., Perfect, E., Rawls, W.J., and Pachepsky, Y. Fractal models for predicting soil hydraulic properties: a review. Eng. Geol., 48, 161, 1997. 25. Anderson, A.N., McBratney, A.B., and Crawford, J.W. Applications of fractals to soil studies. Adv. Agron., 63, 1, 1998. 26. Baveye, P. and Boast, C.W. Fractal geometry, fragmentation processes and the physics of scaleinvariance: an introduction. In Fractals in Soil Science, Baveye, Parlange, and Stewart, Eds., CRC Press, Boca Raton, FL, 1998, 1. 27. Crawford, J.W., Baveye, P., Grindrod, P., and Rappoldt, C. Application of fractals to soil properties, landscape patterns, and solute transport in porous media, in Assessment of Non-Point Source Pollution in the Vadose Zone. Geophysical Monograph 108, Corwin, Loague, and Ellsworth, Eds., American Geophysical Union, Washington, D.C., 1999, 151. 28. Pachepsky, Y.A., Giménez, D., Crawford, J.W., and Rawls, W.J. Conventional and fractal geometry in soil science. In Fractals in Soil Science, Pachepsky, Crawford, and Rawls, Eds., Elsevier Science, Amsterdam, 2000, 7. 29. Feder, J. Fractals, Plenum Press, New York, 1989, 66. 30. Mandelbrot, B.B. The Fractal Geometry of Nature. W.H. Freeman, San Francisco, CA, 1982. 31. Kaye, B.G. A Random Walk through Fractal Dimensions. VCH Verlagsgesellschaft, Weinheim, Germany, 1989, 297. 32. Takayasu, H. Fractals in the Physical Sciences. Manchester University Press, Manchester, 1990. 33. Crawford, J.W., Ritz, K., and Young, I.M. Quantification of fungal morphology, gaseous transport and microbial dynamics in soil: an integrated framework utilising fractal geometry. Geoderma, 56, 1578, 1993. 34. Orbach, R. Dynamics of fractal networks. Science (Washington, D.C.), 231, 814, 1986. 35. Holley, R. and Waymire, E.C. Multifractal dimensions and scaling exponents for strongly bounded random cascades. Ann. Appl. Probab., 2, 819, 1992. 36. Posadas, A.N.D., Giménez, D., Bittelli, M., Vaz, C.M.P., and Flury, M. Multifractal characterization of soil particle-size distributions. Soil Sci. Soc. Am. J., 65, 1361, 2001. 37. Muller, J., Huseby, O.K., and Saucier, A. Influence of multifractal scaling of pore geometry on permeabilities of sedimentary rocks. Chaos, Solitons Fractals, 5, 1485, 1995. 38. Falconer, K. Fractal Geometry, John Wiley & Sons, Chichester, England, 1990. 39. Evertsz, C.J.G. and Mandelbrot, B.B. Multifractal measures. In: Chaos and Fractals: New Frontiers of Science, Peitgen, H.-D., Jurgens, H. and Saupe, D., Eds., Springer, New York, 1992, 921. 40. Meneveau, C. and Sreenivasan, K.R. Measurement of f(α) from scaling of histograms, and applications to dynamical systems and fully developed turbulence. Phys. Lett., A137, 103, 1989. 41. Chhabra, A. and Jensen, R.V. Direct determination of the f(α) singularity spectrum. Phys. Rev. Lett., 62, 1327, 1989. 42. Hou, J.G., Yan, W., Rui, X., Xiaoguang, Z., Haiqian, W., and Wu, Z.Q. Multifractal analysis of the spatial distribution of the film surfaces with different roughening mechanisms. Phys. Rev. E., 58, 2213, 1998. 43. Tarquis, A.M., Losada J.C., Sarmiento, R., Benito, R., and Borondo, F. Multifractal analysis of the tori destruction in a molecular Hamiltonian system. Phys. Rev. E, 65, 016213, 2002. 44. Solé, R.V. and Manrubia, S.C. Are rainforests self-organized in a critical state? J. Theor. Biol., 173, 31, 1995. 45. Beghdadi, A., Andraud, C., Lafait, J., Peiro, J., and Perreau, M. Entropic and multifractal analysis of disordered morphologies. Fractals, 1, 671, 1993. 46. Andraud, C., Beghdadi, A., and Lafait, J. Entropic analysis of random morphologies. Physica A, 207, 208, 1994. 47. Andraud, C., Beghdadi, A., Haslund, E., Hilfer, R., Lafait, J., and Virgin, B. Local entropy characterization of correlated random microstructures. Physica A, 235, 307, 1997. 48. Boger, F., Feder, J., Jossang, R., and Hilfer, R. Microstructural sensitivity of local porosity distributions. Physica A, 187, 55, 1992. 49. Parrinello, T. and Vaugham, R.A. Multifractal analysis and feature extraction in satellite imagery, Int. J. Remote Sens., 23, 1799, 2002.

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50. Tarquis, A.M., Saa, A., Giménez, D., Protz, R., Díaz, M.C., Hontoria, Ch., and Gascó, J.M. The nature of the gray tones distribution in soil image. In: Emergent Nature. Patterns, Growth and Scaling in the Sciences, M.M. Novak, Ed., World Scientific, Singapore, 2002. 51. Barnsley, M.F., Devaney, R.L., Mandelbrot, B.B., Peitgen, H.O., Saupe, D., and Voss, R.F. The Science of Fractal Images. Peitgen, H.O. and Saupe, D., Eds., Springer-Verlag, New York, 1988. 52. Hentschel, H.G.R. and Procaccia, I. The infinite number of generalized dimensions of fractals and strange attractors. Physica D, 8, 435, 1983. 53. Saucier, A. and Muller, J. Remarks on some properties of multifractals. Physica A, 199, 350, 1993. 54. Saucier, A. and Muller, J. Textural analysis of disordered materials with multifractals. Physica A, 267, 221, 1999. 55. Meneveau, C. and Sreenivasan, K.R. The multifractal nature of turbulent energy dissipation. J. Fluid. Mech., 224, 429, 1991. 56. Peyton, R.L., Gantzer, C.J., Anderson, S.H., Haeffner, B.A., and Pfeifer, P. Fractal dimension to describe soil macropore structure using x-ray computed tomography. Water Resour. Res., 30, 691, 1994. 57. Giménez, D., Allmaras, R.R., Huggins, D.R., and Nater, E.A. Mass, surface and fragmentation fractal dimensions of soil fragments produced by tillage. Geoderma, 86, 261, 1998. 58. Pachepsky, Y.A., Yakovchenko, V., Rabenhorst, M.C., Pooley, C., and Sikora, L.J. Fractal parameters of pore surfaces as derived from micromorphological data: effect of long term management practices. Geoderma, 74, 305, 1996. 59. Oleschko, K., Fuentes, C., Brambila, F., and Alvarez, R. Linear fractal analysis of three Mexican soils in different management systems. Soil Technol., 10, 185, 1997. 60. Oleschko, K. Delesse principle and statistical fractal sets: 1. Dimensional equivalents. Soil Till. Res., 49, 255,1998a. 61. Oleschko, K., Brambila, F., Aceff, F., and Mora, L.P. From fractal analysis along a line to fractals on the plane. Soil Till. Res., 45, 389, 1998. 62. Bartoli, F., Bird, N.R., Gomendy, V., Vivier, H., and Niquet, S. The relation between silty soil structures and their mercury porosimetry curve counterparts: fractals and percolation. Eur. J. Soil Sci., 50, 9, 1999. 63. Gomendy, V., Bartoli, F., Burtin, G., Doirisse, M., Philippy, R., Niquet, S., and Vivier, H. Silty topsoil structure and its dynamics: the fractal approach. In Fractals in Soil Science, Pachepsky, Crawford, and Rawls, Eds., Elsevier, Amsterdam, 2000, 75. 64. Crawford, J.W., Sleeman, B.D., and Young, I.M. On the relation between number-size distributions and the fractal dimension of aggregates. J. Soil Sci., 44, 555, 1993. 65. Muller, J. Characterization of pore space in chalk by multifractal analysis. J. Hydrol., 187, 215, 1996. 66. Folorunso, O.A., Puente, C.E., Rolston, D.E., and Pinzon, J.E. Statistical and fractal evaluation of the spatial characteristics of soil surface strength. Soil Sci. Soc. Am. J., 58, 284, 1994. 67. Young I.M., Crawford J.W., and Rappoldt, C. New methods and models for characterising structural heterogeneity of soil. Soil Till. Res., 61, 33, 2001. 68. Virgin, B., Haslund, E., and Hilfer, R. Rescaling relations between two- and three-dimensional local porosity distributions for natural and artificial porous media. Physica A, 232, 1, 1996.

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Scaling and Estimating the Soil Water Characteristic Using a One-Parameter Model R.D. Williams and L.R. Ahuja

CONTENTS I. Introduction.............................................................................................................................35 II. Scaling Approach....................................................................................................................36 A. Determination of Generalized p and q Values ............................................................37 B. Relationship to Similar-Media Scaling .......................................................................38 III. Soil Databases ........................................................................................................................38 A. Generated Textural Class Data ....................................................................................38 B. Real Soil Databases .....................................................................................................39 C. Data Analysis ...............................................................................................................40 IV. Applications............................................................................................................................40 A. Scaling Soil Water Retention with Textural Group p and q Values ...........................40 B. Generalized p and q Values and Scaling of Textural Class Mean ψ(θ).....................41 C. Scaling Individual Soils with Generalized p and q Values ........................................43 D. Scaling Combined Soils with Generalized p and q Values ........................................43 1. The Small Soil Database .......................................................................................43 2. The Large Database...............................................................................................44 V. Concluding Remarks ..............................................................................................................45 References ........................................................................................................................................47

I. INTRODUCTION Scaling provides a means to relate soil properties of different soil types or spatial locations by use of simple conversion factors called the scaling factors. It is a useful technique for approximately describing field spatial variability of soil hydraulic properties —matric potential and unsaturated hydraulic conductivity as a function of soil water content1–3 — as well as characteristics derived from these, such as infiltration and drainage.2,4 The measured curves for these properties, from many experimental sites and different depths, are combined into a representative mean curve by choosing a single scaling factor for each site and depth. Scaling can also be used to estimate soil hydraulic properties at different locations in a watershed from measurement of these properties at a representative location and limited data at other locations.5,6 There are basically two ways to derive the scaling factors: dimensional analysis and an empirical method. Dimensional analysis is based on the existence of physical similarity in the system, while the empirical methods, termed functional normalization, are based on regression analysis. Miller and Miller7 were the first to present physically based scaling factors for soil hydraulic properties

35 © 2003 by CRC Press LLC

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that were based on the assumption of a geometric similarity existing among different soils or porous media (called similar media). More recent attempts at scaling have extended the similar-media approach to field soils that are generally nonsimilar by invoking an additional assumption concerning effects of soil porosity and using a regression approach. This technique has been very useful for describing spatial variability, watershed modeling, and estimation of hydraulic properties. Gregson et al.8 presented a one-parameter model for the soil water characteristic (water content as a function of matric potential) for potentials below the air-entry value. This was achieved by showing that the two parameters of the soil water characteristic, obtained by fitting the commonly used Brooks and Corey9 log–log relationship, were strongly correlated and that a linear relationship between these parameters seemed to be independent of soil type. The results were based on a database of 41 Australian and British soils. Gregson et al.8 anticipated the possibility of scaling the soil water characteristic based on their model, but their main purpose was to use the one-parameter model for estimating the soil water characteristic from only one point measured on the curve (e.g., –10 or –33 kPa water content). Earlier we demonstrated that the soil water characteristic and the hydraulic conductivity for individual soils could be scaled using the one-parameter model10 and that the one-parameter model could effectively estimate the soil water retention curve from one known value (at –33 kPa).10 Here we present new results using the one-parameter model to scale the soil water characteristic over several soil types with a wide range of textures.

II. SCALING APPROACH Gregson et al.8 represented the relationship between the volumetric soil water content, θ, and the matric potential, ψ, for ψ≤ –5 kPa at any spatial location i as: ln[–ψi(θ)] = ai + bi ln(θ)

(3.1)

This is the inverse of the well known Brooks and Corey9 relationship with the residual water content, θr, equal to zero. This relationship was fitted to experimental data for eight soil textural groups of Australia, 19 textural classes of England and Wales, and 14 classes of Scotland. Equation 3.1 provided a good fit for all the data with the correlation coefficients exceeding 0.97 in every case. Gregson et al.8 then found a very close linear relationship between parameters ai and bi of Equation 3.1 obtained from the fitted equations for each textural class: ai = p + q bi

(3.2)

The coefficient of correlation exceeded 0.99 in each case. They also found that the ai vs. bi relationship derived from all the soils coalesced together with a small scatter, indicating a possibility of obtaining values for the constants of p and q that would be applicable for all soils. Substituting Equation 3.2 in Equation 3.1 yields: ln[–ψi(θ)] = p + bi [ln(θ) + q]

(3.3)

Here p and q are constants independent of the spatial location, and bi is the only parameter that is dependent on the location. Thus, bi is a scaling factor for each location i. Rearranging Equation 3.3 we obtain: ln (θ) = [ln[–ψi(θ)] – p]/bi – q = scaled ln[––ψi(θ)] © 2003 by CRC Press LLC

(3.4)

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Thus, at a fixed θ value, the right-hand term of Equation 3.4 should be the same for all spatial locations. This term is here called the scaled ln(–ψi). If the correlation between ai and bi in Equation 3.2 is perfect, plots of ln(θ) and the scaled ln(–ψi) for all different locations should coalesce into a single 1:1 relationship. Equation 3.3 can also be used to estimate the entire ψ(θ) relationship, below the air-entry (ψb) value, at a given location from just one measured value of the ψ(θ) curve.8 The known ψ,Θ pair value is used to determine the one unknown parameter bi in Equation 3.3. This method would enable the estimation of ψ (θ) for modeling purposes from soil survey data that generally contains either a –33 or –10 kPa water-content value and soil bulk density. If the constants p and q in Equation 3.2 were found to be approximately the same in different soils, the one-parameter model would be a generalized method of scaling and estimation across soil types. This would be a distinct advantage over the similar-media scaling approach, which is generally restricted in its application to within soil type. A. DETERMINATION

OF

GENERALIZED p

AND

q VALUES

Earlier, for the purpose of estimating ψ(θ) from one known value, we found it better to have several p and q values based on texture groups11 However, we also found that we could use Gregson et al.8 p and q values based on their larger database even though the results were somewhat less precise. To use the one-parameter model for scaling across soil textural classes, p and q values based on a number of soils or textural classes would be of more value. Given the general form of Equation 3.1 modified to include the residual water content (θr),10 we have: ln[–ψ(θ)] = a + b ln(θ – θr)

(3.5)

rearranging: b ln(θ – θr) = a – ln(–ψ) ln(θ – θr) = – a/b + 1/b ln(–ψ) (θ – θr) = e – a/b * (–ψ)1/b

(3.6)

ln(–ψb) = a + b ln(θs – θr)

(3.7)

Equation 3.5 also implies:

where ψb is the bubbling pressure (kPa) and θs is the total porosity (cm3/cm3). Rearranging this equation, as was done in Equation 3.5, we can express Equation 3.7 as: (θs – θr) = e – a/b * (–ψb)1/b

(3.8)

Combining Equations 3.6 and 3.8 we have (θ – θr)/(θs – θr) = (ψb/ψ)–1/b

(3.9)

Equation 3.9 is equivalent to the following form of the Brooks and Corey equation for ψ < ψb: (θ – θr)/(θs – θr) = (ψb/ψ)λ where λ is the pore size distribution. © 2003 by CRC Press LLC

(3.10)

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Given the relationship between Equations 3.9 and 3.10 then b = –1/λ

(3.11)

When this relationship is known, the intercept, a, can be estimated with Equation 3.7 by substituting –1/λ for the slope, b, and rearranging: a = ln(–ψb) +(1/λ * ln(θs – θr))

(3.12)

Using Equations 3.11 and 3.12 to determine the slope and intercept of log–log expression of the ψ(θ) characteristic is dependent upon having information on the average ψb, λ, θs, and θr for a soil or group of soils. Fortunately, Rawls et al.12 have provided the mean values of these properties for 11 textural classes. For each textural class, the intercept a was obtained using Equation 3.12 and known mean values of ψb, λ, θs, and θr for the class. Equation 3.2 was regressed on a vs. (–1/λ) set of values for 11 textural classes to obtain the generalized p and q values for all classes.

B. RELATIONSHIP

TO

SIMILAR-MEDIA SCALING

In the similar-media scaling approach, the value of ψ at location i for any fixed value of θ is related to ψj at location j at the same θ as: ψi(θ) = (αj/αi) ψj(θ)

(3.13)

where αi and αj are the respective scaling factors for locations i and j, which are independent of θ. For any θ, the difference in ln(–ψi) and ln(–ψj) is then given as: ln(–ψi) – ln(–ψj) = ln(αj/αi)

(3.14)

which indicates that this difference remains the same for all θ values. In light of this, if the θ(ψ), for ψ less than the air-entry value, at each location were to be represented by Equation 3.1, ln–ln straight lines for all different locations would be parallel. On the other hand, the use of the one-parameter approach in scaling for any fixed θ, Equation 3.3 leads to: ln(–ψi) – ln(–ψj) = (bi – bj)(ln θ + q)

(3.15)

where the slopes bi and bj are independent of θ. In order to interpret Equation 3.15 in terms of the similar-media scaling approach, we equate the right-hand terms of Equations 3.14 and 3.15 to obtain: ln(αj/αi) = (bi – bj)(ln θ + q)

(3.16)

This result shows that the use of the Gregson et al.8 approach allows ln(αj/αi) to vary linearly with lnθ. This approach is thus more flexible and could improve scaling for nonsimilar soils, compared with the use of similar-media scaling approach for such soils.

III. SOIL DATABASES A. GENERATED TEXTURAL CLASS DATA Using Equation 3.10 and the mean hydrologic parameters in Table 3.1, soil water contents were calculated for each of the 11 textural classes at selected matric potentials (–5, –10, –20, –50, –100, © 2003 by CRC Press LLC

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TABLE 3.1 Selected Mean Hydrologic Soil Parameters Classified by Soil Texturea

Texture Sand Loamy sand Sandy loam Loam Silt loam Sandy clay loam Clay loam Silty clay loam Sandy clay Silty clay Clay a

Bubbling Pressure (ψb) kPa 0.726 0.869 1.466 1.115 2.076 2.808 2.589 3.256 2.917 3.419 3.730

Residual Saturation (θr) cm3/cm3 0.020 0.035 0.041 0.270 0.015 0.068 0.075 0.040 0.109 0.056 0.090

Effective Porosity (θs) cm3/cm3 0.417 0.402 0.412 0.436 0.486 0.330 0.389 0.431 0.321 0.423 0.385

Pore Size Distribution (8λ) 0.591 0.474 0.322 0.220 0.211 0.250 0.194 0.151 0.168 0.127 0.131

Water Retained at –33 kPa cm3/cm3 0.0634 0.1064 0.1917 0.2335 0.2856 0.2458 0.3119 0.3434 0.3222 0.3728 0.3790

Based on Rawls, W.J., Brakensiek, D.L., and Saxton, K.E., Trans. ASAE, 25, 1316, 1982.

–500, –1000, and –1500 kPa). The resulting data were combined in the test of scaling on the inverse process of estimating a ψ(θ) curve from one known value. The data were scaled using Equation 3.4 modified to include θr, as: ln(θ–θr) = [ln[–ψ (θ)]–p]/b–q

(3.17)

where the textural class b = –1/λ (Equation 3.11) and the generalized p and q values calculated as described in the theory section above. To estimate water content at different matric potentials using the one-parameter model (Equation 3.1), the average soil water content at –33 kPa for each texture class was used as the one ψ(θ) value required by the model. Using this ψ(θ) value, b was estimated for each textural class using the generalized p and q derived earlier. Soil water content was calculated for each matric potential and the data combined for plotting regardless of textural class.

B. REAL SOIL DATABASES Two soil data sets were used. The first data set was a small database consisting of five soils used by Ahuja and Williams10 and Williams and Ahuja.11 The soils in this database provided a wide range of textures including sand, sandy loam, sandy clay loam, loam, silt loam, silty clay loam, clay loam, and clay. Each soil was represented by 48 to 155 samples and when combined for analysis the data set contained 481 samples. The second data set was a large database consisting of 35 soils selected from the database of Rawls et al.12 Each soil was represented by a minimum of 6 samples and when the soils were combined for analysis the database contained 556 samples that ranged widely in texture. Descriptions of the soils and complete data sources are available from Williams and Ahuja11 and Rawls et al.12 Because in both data sets the θ,ψ data pairs for individual locations corresponded to values of ψ that differed from soil to soil, and sometimes within a soil, a log–log interpolation was used to calculate θ at specific matric potentials (–5, –10, –20, –100, –500, –1000, and –1500 kPa). As with the textural class data the water contents were scaled using Equation 3.17. Again, to test the inverse estimation of ψ(θ), the volumetric water content at –33 kPa, which was available in both data sets, was used as the known ψ(θ) value required by the model. Using this ψ(θ) value, b was estimated using the generalized p and q values derived earlier. Soil water content was calculated for each matric potential and the data were combined for plotting regardless of soil or matric potential. © 2003 by CRC Press LLC

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C. DATA ANALYSIS To determine how well the one-parameter model scaled and estimated ψ(θ), ln(θ – θr) was plotted against scaled (ψ), and the estimated water contents were plotted against the observed water contents, for all matric potentials, so that the scatter around and the relationship of the data to the 1:1 line could be observed. In addition, a mean error and root-mean-square error were determined for estimated water content for each data set, for all matric potentials. The mean error was calculated by summing the differences between the estimated and observed soil water content; dividing by the number of observations calculated the mean error. The root-mean-square error was calculated by taking the square root of the sum of squares of the differences between the estimated and observed soil water contents, divided by the number of observations minus one.

IV. APPLICATIONS A. SCALING SOIL WATER RETENTION

WITH

TEXTURAL GROUP p

AND

q VALUES

Examples of scaling the soil water characteristic of selected individual soils from the small database with the one-parameter model are presented in Figure 3.1. These soils range in texture from sandy to clay loam and encompass a range of scaling efficiencies from best (Renfrow) to worst (Pima). The scaling is not perfect, but it compresses the data variability considerably, and the scaled data fall along the expected 1:1 line. The p and q values used in the analysis were based on texture groups (Table 3.2) rather than p and q values developed for the individual soils.10 If the p and q values developed for the individual soils had been used, the scatter around the 1:1 line would have been slightly less than that shown in Figure 3.1. However, when the relative scaling efficiency (percent reduction in mean sum of squares deviations by scaling based on linear regressions) was determined for the group vs. individual p and q values, the scaling efficiency was only slightly less for the group values.10 Scaling efficiency for the soils in Figure 3.2 were 81, 74, 61, and 42% for Renfrow, Teller, Lakeland, and Pima, respectively. Similarmedia scaling efficiencies for these soils were 45, 74, 7, and 36% for Renfrow, Teller, Lakeland, and Pima, respectively. When the estimated water contents were compared the one-parameter 0

A

-2

-4

-6

B

-1

LN(θ-θr)

LN(θ-θr)

0

-6

-4

-2

-2 -3 -4

0

-4

-3

Scaled ψ 0

C LN(θ-θr)

LN(θ-θr)

0

-1

-2

-3

-3

-2

-1

Scaled ψ

-2

-1

0

Scaled ψ

0

D

-1 -2 -3 -4

-4

-3

-2

-1

0

Scaled ψ

FIGURE 3.1 Results of scaling the soil water characteristic for Lakeland (A), Pima (B), Renfrow (C) and Teller (D) soils using the one-parameter model and p and q values based on textural groups.

© 2003 by CRC Press LLC

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41

TABLE 3.2 Average p and q Values Based on Soil Texture Groupsa p ln(kPa)

Group

Soils

Textural range

1 2 3 —

Oxisols, Kirkland, Renfrow, Pima Norfolk, Teller, Bernow (45–90 cm) Lakeland, Bernow (0–45 cm) Australian and British soils

Loam-silty clay loam-clay loam Sandy loam-sandy clay loam Sand A mixture of textures

1.415 0.343 0.541 –0.982

q ln(cm3/cm3) 0.839 1.072 1.469 0.585

a Based on Ahuja, L.R. and Williams, R.D., Soil Sci. Soc. Am. J., 55, 308, 1991.

-1 -2

Intercept, a

-3 -4 -5 -6

a = -0.5236 + 0.6691b r2= 0.86

-7 -8 -8

-7

-6

-5

-4

-3

-2

-1

Slope, b FIGURE 3.2 Relationship of the intercept, a, and slope, b, (solid line) calculated with Equations 3.11 and 3.12 and the mean hydrologic parameters for each texture class. Using the p and q values of Gregson, D., Hector, D.J., and McGowan, M. (J. Soil Sci., 38, 483, 1987), the slope (dash line) was calculated for each textural class based on Equation 3.1.

model results showed less scatter in the relationship to the 1:1 line, concomitant with smaller calculated error terms than the similar-media scaling method.11

B. GENERALIZED p AND q VALUES AND SCALING OF TEXTURAL CLASS MEAN ψ(θ) There was a strong linear relationship between the textural class mean intercept (a) and slope (b) that could be described by Equation 3.2 (Figure 3.2). Based on the hydrologic properties for the 11 textural classes, p equaled –0.5236 ln kPa and q equaled 0.6691 cm3/cm3 with an r2 of 0.86. Using Gregson et al.8 p and q values, –0.982 ln kPa and 0.5852 ln cm3/cm3, respectively, and the calculated slope for each texture class based on Equation 3.11, a line was superimposed on the results for the textural classes (Figure 3.2, dashed line). Both lines are similar, showing the universal nature of the relationship, a vs. b. This supports our earlier contention that a common set of p and q values may be used for scaling across soil types and textural classes. Using the Brooks and Corey equation (Equation 3.10) and the average hydrologic parameters of the texture classes in Table 3.1, we calculated the soil water characteristic curve for each texture class (Figure 3.3). These soil water characteristic curves were scaled using Equation 3.17; the slope, b, was estimated using the average λ (Equation 3.11); the p and q values were based on the slope–intercept relationship determined in Figure 3.2. The results of scaling the ψ(θ) across texture classes are presented in Figure 3.4. The scaling in technique with the one-parameter model coalesced the data into a tight group surrounding the 1:1 line. Inversely, we also estimated the volumetric water content at selected © 2003 by CRC Press LLC

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0

LN(θ)

-1

-2

-3

-4

0

1

2

3

4

5

6

7

8

LN(ψ) FIGURE 3.3 Soil water characteristic curves calculated for each texture class using Equation 3.10 and average hydrologic properties.

0

LN(θ-θr)

-1 -2 -3 -4 -5 -6 -6

-5

-4

-3

-2

-1

0

Scaled ψ

Estimated water content (cm3/cm3)

FIGURE 3.4 Results of scaling the soil water characteristic for the 11 texture classes.

0.5 0.4 0.3 0.2 0.1 0.0 0.0

0.1

0.2

0.3

Observed water content

0.4 0.5 3 (cm /cm3)

FIGURE 3.5 Estimated vs. observed volumetric water content for the 11 texture classes for all matric potentials.

© 2003 by CRC Press LLC

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43

matric potentials using the one-parameter model. In this case, we used Equation 3.3 modified to include θr, using the water content at –33 kPa as the known ψ(θ) value required by the model, and the generalized p and q values in Figure 3.2. The model estimated the soil water content quite well and, with a few exceptions, the data coalesced tightly around the 1:1 line (Figure 3.5). The mean error was –0.005 cm3/cm3, while the root mean square error was 0.032 cm3/cm3, regardless of matric potential. C. SCALING INDIVIDUAL SOILS

WITH

GENERALIZED p

q VALUES

AND

As a comparison with group p and q values (Table 3.2) previously discussed (Figure 3.1), we scaled the soil water characteristic for these same soils using the one-parameter model and generalized p and q values (Figure 3.6). A comparison of Figures 3.1 and 3.6 shows very little, if any, difference in the results. Inversely, when the estimated vs. the observed volumetric water content was plotted for each soil for all matric potentials, the plots (results not shown) for the two sets of p and q values and the errors were similar. Using the generalized p and q values, the mean errors in estimating θs for Lakeland, Pima, Renfrow, and Teller soils were 0.026, 0.012, –0.047, and 0.004 cm3/cm3, respectively, while the root-mean-square errors were 0.032, 0.052, 0.066, and 0.025 cm3/cm3, respectively. For Lakeland and Pima soils these errors are very similar to those obtained with the group p and q values (Table 3.2), while the mean error and the root-mean-square error were larger for Renfrow (compared to –0.006 cm3/cm3) and Teller (compared to 0.012 cm3/cm3), respectively. However, the overall results for these soils using the generalized p and q are quite similar to those reported earlier for individual and group values.10,11 D. SCALING COMBINED SOILS

WITH

GENERALIZED p

AND

q VALUES

1. The Small Soil Database Distribution of textures represented in the small soil database from Williams and Ahuja6,11 is presented in Figure 3.7, and the soil water characteristic curves represented in the database are presented in Figure 3.8. There is considerable variability in the unscaled data. The data were scaled 0

A

B

-1 -2

LN(θ-θr)

LN(θ-θr)

0

-4

-6

-6

-4

-2

0

-2 -3 -4

-4

-3

0

-1

-2

-3

-3

-2

-1

Scaled ψ

-1

0

D

C LN(θ-θr)

LN(θ-θr)

0

-2

Scaled ψ

Scaled ψ

0

-1 -2 -3 -4

-4

-3

-2

-1

0

Scaled ψ

FIGURE 3.6 Results of scaling the soil water characteristic for Lakeland (A), Pima (B), Renfrow (C) and Teller (D) soils using the one-parameter model and generalized p and q values. © 2003 by CRC Press LLC

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100

0 10

90

20

80

30

70

40

60

Clay

Silt

50

50

60

40

70

30

80

20

90

10

100

0 100

90

80

70

60

50

40

30

20

10

0

Sand FIGURE 3.7 Soil texture distribution of the 5 soils and 481 samples in the small database. 0

LN(θ)

-1

-2

-3

-4 -5

0

5

10

LN(ψ) FIGURE 3.8 Variability in the soil water retention curves (unscaled) in the small database. 0

LN(θ-θr)

-1 -2 -3 -4 -5 -6 -6

-5

-4

-3

-2

-1

0

Scaled ψ FIGURE 3.9 Results of scaling the soil water characteristic for the small database with Equation 3.4 using b estimated from ψ(θ) at –33 kPa and generalized p and q values. © 2003 by CRC Press LLC

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

45

0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

Observed water content (cm3/cm3) FIGURE 3.10 Estimated vs. observed water content for the small database, for all matric potentials, using Equation 3.17 with the generalized p and q values and b estimated from ψ(θ) at –33 kPa.

using Equation 3.17 with b estimated from ψ(θ) at –33 kPa as the known value and the generalized p and q values given in Figure 3.2. There is more scatter in the results (probably due to errors in estimating b), but the data coalesce around the 1:1 line (Figure 3.9). If the inversely estimated vs. observed water content is plotted, regardless of matric potential, most of the data coalesces near the 1:1 line (Figure 3.10). The calculated mean error in this case is 0.026 cm3/cm3, while the root mean square error is 0.063 cm3/cm3. When we scaled these data and estimated the water content using Gregson et al.,8 p and q values and the results were similar to those resulting from using the generalized p and q values (data not shown) and the estimation errors were essentially the same. With the range in textures and variability in soil data, we would not expect as tight a relationship to the 1:1 line as presented for the texture classes. 2. The Large Database Distribution of soil textures represented in the large database12 is presented in Figure 3.11. The soil water characteristic was scaled as described above for the small database. Again the scaled data coalesces around the 1:1 line compressing the variability in the data set (Figure 3.12). The soil water characteristic was also scaled using Gregson et al.8 p and q values. The results (Figure 3.13) are essentially the same as those using the generalized p and q values (Figure 3.12). The estimated vs. observed water contents using Equation 3.3 and the generalized p and q values are shown in Figure 3.14; error terms, regardless of matric potential, were 0.012 cm3/cm3 for the mean error and 0.064 cm3/cm3 for the root-mean-square error. Similar error terms resulted if Gregson et al.8 values were used in the calculations. An interesting thing to note from the results in Figures 3.10 and 3.12 is that the generalized p and q values did a better job in scaling the variability across 35 soil types (Figure 3.12) than in scaling the variability within five soil types (Figure 3.9).

V. CONCLUDING REMARKS Earlier we demonstrated that scaling with the one-parameter model was more efficient than a similar-media technique within soil types, where we used p and q values developed for individual soils or broad-based texture groups. Here we extend the earlier work by showing that generalized, universal, p and q values based on the mean hydrologic parameters of the 11 textural classes can be used to scale the soil water characteristic. Using these p and q values, the scaling results for the small database were similar to those for the individual soil using specific p and q values based on

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0

100

10

90

20

80

30

70

40

60

Clay

50

50

Silt 60

40

70

30

80

20

90

10

100

0 100

90

80

70

60

50

40

30

20

10

0

Sand

FIGURE 3.11 Soil texture distribution of the 35 soils and 556 samples in the large database.

LN (θ-θr)

0

-5

-10

-15 -15

-10

-5

0

SCALED (ψ) FIGURE 3.12 Results of scaling the soil water characteristic for the large database using the generalized p and q values. 0

LN (θ-θr)

-5

-10

-15 -15

-10

-5

0

SCALED (ψ) FIGURE 3.13 Results of scaling the soil water characteristic for the large database using p and q values from Gregson, D., Hector, D.J., and McGowan, M. (J. Soil Sci., 38, 483, 1987). © 2003 by CRC Press LLC

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Estimated water content (cm3/cm3)

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

47

0.8

0.6

0.4

0.2

0.0 0.0

0.2

0.4

0.6

Observed water content

0.8

(cm3/cm3)

FIGURE 3.14 Estimated vs. observed water content for the large database, for all matric potentials, using Equation 3.17 with the generalized p and q values and b estimated from ψ(θ) at –33 kPa.

texture groups. The scaling results reported here are similar to, or are better than, the results reported earlier for similar-media scaling. Although there are some errors in the inversely estimated vs. observed water content for the two databases, these errors are usually smaller than those when multiple regression equations or similar-media scaling is used to calculate soil water content (results not shown). It is also interesting that the generalized p and q values developed here are similar to those developed by Gregson et al.8 for Australian and British soils, and the results from the two sets of p and q values are similar. This provides further indication that this technique is quite general and applies to different soil types and texture groups. Overall, the one-parameter model is a useful technique for scaling soil hydraulic properties, including the soil water characteristic (presented here), the unsaturated hydraulic conductivity,10 and estimating the soil water content.11,13

REFERENCES 1. Warrick, A.W., Mullen, G.J., and Nielsen, D.R., Scaling field-measured soil hydraulic properties using a similar-media concept, Water Resour. Res., 13, 355, 1977. 2. Simmons, C.S., Nielsen, D.R., and Biggar, J.W., Scaling of field-measured soil water properties, Hilgardia, 47, 77, 1979. 3. Russo, D. and Bresler, E., Scaling soil hydraulic properties of a heterogeneous field, Soil Sci. Soc. Am. J., 44, 681, 1980. 4. Sharma, M.L., Gander, G.A., and Hunt, C.G., Spatial variability of infiltration in a watershed, J. Hydrol., 45, 101, 1980. 5. Ahuja, L.R., Naney, J.W., and Williams, R.D., Estimating soil water characteristics from simpler soil properties or limited data, Soil Sci. Soc. Am. J., 49, 1100, 1985. 6. Williams, R.D. and Ahuja, L.R., Estimating soil water characteristics using physical properties and limited data, in Indirect Methods for Estimating the Hydraulic Properties of Unsaturated Soils, van Genuchten, M.Th., Leij, F.J., and Lind, L.J., Eds., Univ. California Press, Riverside, 1991, 405. 7. Miller, E.E. and Miller, R.D., Physical theory for capillary flow phenomena, J. Appl. Phys., 27, 324, 1956. 8. Gregson, D., Hector, D.J., and McGowan, M., A one-parameter model for the soil water characteristic, J. Soil Sci., 38, 483, 1987. 9. Brooks, R.H. and Corey, A.T., Hydraulic properties of porous media, Hydrol. Pap. No.3, Colorado State Univ., Ft. Collins, 1964. 10. Ahuja, L.R. and Williams, R.D., Scaling water characteristic and hydraulic conductivity based on Gregson-Hector-McGowan approach, Soil Sci. Soc. Am. J., 55, 308, 1991.

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Scaling Methods in Soil Physics 11. Williams, R.D. and Ahuja, L.R., Evaluation of similar-media scaling and a one-parameter model for estimating the soil water characteristic, J. Soil Sci., 43, 237, 1992. 12. Rawls, W.J., Brakensiek, D.L., and Saxton, K.E., Estimation of soil water properties, Trans. ASAE, 25, 1316, 1982. 13. Williams, R.D. and Ahuja, L.R., Using a one-parameter model to estimate the soil water characteristic, in Advances in Hydro-Science and -Engineering, Volume I, Part A, Wang, Sam S.Y., Ed., Center for Computational Hydroscience and Engineering, School of Engineering, University of Mississippi, University, 1993, 485.

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4

Diffusion Scaling in Low Connectivity Porous Media R. P. Ewing and R. Horton

CONTENTS I. II. III. IV.

Introduction.............................................................................................................................49 Connectivity: The Missing Link ............................................................................................50 Scaling Methodology .............................................................................................................51 Diffusive Transport Features ..................................................................................................52 A. Diffusivity ....................................................................................................................52 B. Porosity and Pore Accessibility...................................................................................53 C. Tortuosity at the Pore Scale ........................................................................................56 V. Relevance and Conclusion .....................................................................................................59 IV. Acknowledgments ..................................................................................................................60 References ........................................................................................................................................60

I. INTRODUCTION “Scaling,” in the context of soil physics, generally refers to the scale dependence of some transport phenomena, such as dispersivity increasing with distance. Because scaling has been observed in many hydrogeological processes, its origin is of some interest. What are the underlying mechanisms that give rise to scaling? Dispersion is caused by a combination of fluid and medium effects: the fluid diffuses, and heterogeneity in the medium gives rise to streamlines that vary in passage time. However, these combined processes do not necessarily result in dispersivity increasing with distance. A common explanation, presented in an REV-type formulation by Bear1 and popularized in fractal form by Wheatcraft and Tyler, 26 is that scaling is caused by some form of spatial structure whose variance increases with size. For example, if the semivariogram of clay content does not have a sill but rather rises continually with lag, the soil would be described as having scale-dependent spatial structure. This structure is then invoked as a potential cause of scaling in the transport processes of interest. However, an increasing scale of structure is apparently not necessary for scaling in transport. For example, Molz et al.16 pointed out that scale-dependent dispersion coefficients could simply result from flow within parallel layers that have little diffusive interaction. Berkowitz et al.3 demonstrated scale-dependent dispersion in a laboratory setting, where the medium had a simple structure with a single characteristic length. If increasing scale of structure is not needed in order to have scale-dependent dispersion, is structure even necessary? In this study, we address that question by demonstrating a form of diffusion-based scaling that is not dependent on structure in the porous medium. Additionally, we show that two different forms of scaling can stem from the same underlying cause. The scaling we address here results from low connectivity at the pore scale. 49 © 2003 by CRC Press LLC

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II. CONNECTIVITY: THE MISSING LINK Papert18 noted that the field of mathematics has three main branches — number theory, analysis, and topology — and that students from elementary school onward are generally taught only the first two. In soils and hydrology, as in mathematics, concepts of topology have historically received short shrift. Topology is essentially the mathematics of connections independent of the specifics of size. For example, a bracelet and a coffee mug are topologically identical, because each has a single loop; likewise a river network and a tree have no loops. Various topological metrics have been derived, such as topological genus23 and coordination number. However, the question of greatest interest for transport processes is whether two given points (locations in a maze, on a map, in a porous medium) are connected: is it possible to go from point A to point B? The evidence in hydrogeology is that facies continuity and interconnection may drive dispersion.7 Likewise, wormholes are commonly considered to be sources of bypass flow, but in fact they may often be poorly interconnected.17,19 Transport requires connectivity. When a medium is topologically random — that is, the probability p of a given connection being present is independent of position — then the study of how the connections affect the resultant macroscopic properties belongs to a mathematical specialty called percolation theory. Percolation theory has as a fundamental theorem that, somewhere between disconnected and densely connected, there exists a connection probability at which a network is just barely connected by a single “infinite cluster.” This probability, called the critical probability (pc) or the percolation threshold, is clearly a critical point for transport: below the threshold, transport is limited to local shuffling within isolated clusters, while above it, arbitrarily long-distance movement can take place (Figure 4.1). Percolation theory largely focuses on properties of infinite random media at and near the percolation threshold. Two concepts important to this study have developed in percolation theory.24 First is the idea that near and at the percolation threshold, some properties obey scaling laws, e.g., the conductivity K of the largest connected cluster (the infinite cluster) near the percolation threshold shows K ∼ (p – pc)µ. Second is the concept of universality: the actual value of the scaling exponent (µ in the example above) is dependent only on the dimension, not on the specifics of the lattice. In other words, we can study square (two-dimensional) lattices because they are relatively easy to program, and apply our results to triangular or honeycomb lattices; likewise, results from cubic lattices can be applied to three-dimensional irregular lattices.

A

B

FIGURE 4.1 A square lattice with site occupancy of (A) 50% and (B) 66%. Bonds are shown as “active” (thick gray) if they have an occupied site at each end. At an intermediate occupancy of approximately 59%, the edges would be only just connected by a sequence of active bonds. In (B), the infinite cluster is shown with thick black bonds, but sites that form dangling clusters on the infinite lattice are shown in white.

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Because soils are not infinite, random, or near the percolation threshold,2 one might reasonably wonder how percolation theory could possibly apply to them. With respect to infinity, the number of pores in, for example, a cubic meter of soil may be thought sufficiently large to approximate an infinite medium. But more strictly, percolation theory often deals with finite size scaling: how does a result derived at one scale apply to a different scale? Finite size scaling is clearly not limited to infinite media. Additionally, some properties of interest to soil scientists and hydrologists are controlled at the “edge” of a medium, such as the edge of a pebble or the inlet face of a core sample in the laboratory, and percolation theory has ways of approaching this also, as will be shown below. As for the issues of randomness and proximity to the percolation threshold, soils are generally not random media because they are often structured (like a river basin) in such a way as to minimize energy, or correlated by localized forces. But correlated pore spaces generally have a relatively small correlation length, and percolation concepts still apply at scales above that length if the medium is macroscopically at the percolation threshold.20 Approximately random pore arrangements can be found inside sand and gravel particles and rock matrices, where slow diffusive processes dominate the transport. It is also in these locations that pore spaces are most likely to be at, or close to, the percolation threshold. This research is therefore most clearly applicable to diffusive processes in those rock matrices whose pore space is close to the percolation threshold, such as igneous rocks with intragranular cooling crack networks,4,8 rock aggregates subjected to high temperatures and pressures,13 or welded tuff and well-cemented limestones.10 The clearest applicability to soils might therefore be diffusion inside the low-porosity primary particles, sand and silt grains.27 We note, however, that connectivity in the real world may be a function of size. Pores that are accessible to a contaminant molecule may not be accessible to bacteria, for example, so only some subset of the total porosity is bacteria accessible. The bacteria-accessible pore space may be at the percolation threshold by virtue of the proportion of pore throats that are sufficiently large for bacterial transport, and so percolation-based behaviors may be observed in biodegradation even if all pores are accessible to the contaminant. A soil may therefore be at the percolation threshold with respect to a specific process, even if its pore space seems at first to be too well connected (or too finite, or too nonrandom); it is therefore useful for soil physicists to have some acquaintance with percolation processes. Low pore connectivity in rock matrices can lead to non-Fickian diffusion,6,14,21 which may manifest itself macroscopically as scaling. Our ultimate objective here is to examine the effects of pore connectivity on diffusion at multiple spatial scales. However, because pore connectivity affects accessible porosity and tortuosity, each an important component of diffusivity, we will examine all three parameters: diffusivity, accessible porosity, and tortuosity.

III. SCALING METHODOLOGY Simulations were conducted using a cubic lattice, which has a site coordination number (mean number of intersections, or pore bodies, to which each body is connected) z = 6. Rocks with wellinterconnected pore spaces, such as noncemented sandstones, have coordination in the range of z = 4 to 8.15,28 At this value connectivity is not limiting, so (for example) accessible porosity equals total porosity. Lower coordinations are achieved by randomly pruning (turning off) bonds between neighboring sites. This strategy is called “bond percolation,” and the cubic lattice thus pruned is at the percolation threshold when bonds are turned on with a probability p = 0.2488,24 or equivalently when the lattice has a mean coordination of 1.4928. Once the lattice has been pruned, it is analyzed to determine which sites are on the infinite cluster (accessible from the inlet and outlet ends), which are accessible from the inlet end only, and which are inaccessible. If the lattice is below the percolation threshold it is discarded and the process is started over. Diffusion is then modeled using random walk methods.11,22 The walkers are initially placed at random locations on the inlet face, and at each time step each walker randomly attempts to move © 2003 by CRC Press LLC

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in one of six directions. Movement is permitted only along active bonds, and the walker is permitted to exit the lattice only through the outlet face. After a walker exits through the outlet face, the number of time steps for its passage is recorded and a new walker is introduced at the inlet face. Faces orthogonal to the inlet are periodic boundaries. The diffusivity of the system can be calculated using a steady-state equation,5,11 C − C out Q , = D in 12 2 L A

(4.1)

where Q is the number of walkers, A is the area of the inlet face, t is the time of transit for a particle and the angle brackets denote an arithmetic mean, D is the diffusion coefficient, L is the Euclidean distance from the inlet to the outlet face, and Cout and Cin refer to the walker concentrations at the outlet and inlet faces, respectively. When p = 1.0, the theoretical value of the diffusion coefficient is D = 1/6.11 In the program, Cout is maintained at 0, and Cin can be easily calculated for the p = 1.0 case and used for all subsequent cases.6 For each case, 100 realizations were run, with lattices smaller than 2563 using 1000 random walkers, and lattices 2563 or larger using 100. More details of the methods used and results of model validation tests can be found in Ewing and Horton.6

IV. DIFFUSIVE TRANSPORT FEATURES A. DIFFUSIVITY Diffusivity at low pore connectivity (p < 1/3, or equivalently z < 2) shows two distinct kinds of size scaling (Figure 4.2). Most obviously, diffusivity at the percolation threshold is described by a power law that scales with lattice size: specifically, D ∼ L–1.68. Secondly, diffusivity above the percolation threshold also shows some scaling in what appears to be a finite-size effect, that is, an effect that diminishes as size increases.

Diffusivity,L2T-1

10-1

p = 1.0

10-2

p = 0.333

10-3

p = 0.27

10-4

p = 0.255

10-5 p = 0.2488 10

100

Lattice Size FIGURE 4.2 Diffusivity as a function of connectivity and lattice size. Notice that at connectivities above the percolation threshold, finite size effects disappear at larger sizes, but at the percolation threshold scaling is seen at all sizes. © 2003 by CRC Press LLC

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Scaling of diffusivity at the percolation threshold is well documented in percolation theory.24 Ignoring porosity, tortuosity, concentration gradients and other “details,” diffusion can be described as D = L2/.

(4.2)

At criticality, the mean time required for diffusing a distance L is known to scale by an exponent k, whose value varies depending on whether diffusion is taking place through the entire medium (k = 0.32 in three-dimensions), or through the infinite cluster only (k = 0.27 in three-dimensions).24 Putting this exponent, k, into Equation 4.2, we see that D ∼ L2–k, in agreement with the result mentioned above. Restricting diffusion to the infinite cluster would result in an exponent of –1.83 rather than the –1.68 obtained here.6,24 What is the cause of the finite size effect? Does it show up in real systems? How can it be predicted? The answers to these questions are somewhat intricate, and we approach them using the common formalism Deff = Daq φ / τ,

(4.3)

where Deff is the effective diffusion coefficient, Daq is aqueous diffusivity, φ is porosity, and τ is tortuosity (following the convention that τ > 1). We shall take the known value of diffusivity on a completely connected cubic lattice to represent aqueous diffusivity, leaving us two unknowns (φ and τ) with which to explain the simulation results Deff.

B. POROSITY

AND

PORE ACCESSIBILITY

Porosity can be subdivided into accessible and inaccessible porosity. Most measurement methods measure accessible porosity; inaccessible porosity occurs in disconnected pores and pore clusters, like the insides of soap bubbles that touch each other but through which air cannot pass from one bubble to another. Accessible porosity can be usefully further subdivided into edge-only accessible, backbone, and dangling end porosity (Figure 4.3); all these forms are illustrated in Figure 4.1B.

Total Porosity

Accessible Porosity

Edge-only Accessible Porosity

Inaccessible Porosity

Infinite Cluster

Backbone FIGURE 4.3 Conceptual subdivision of total porosity. © 2003 by CRC Press LLC

Dangling Ends

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1.0

Accessible Porosity

pc 0.8

0.6

0.4

0.2

0.0 0.0

0.2

0.4

p

0.6

0.8

1.0

FIGURE 4.4 Accessible porosity as a function of bond connection probability p in a 2563 lattice.

Edge-only accessible pores do not form part of the infinite cluster but can be accessed from one or more edges. The infinite cluster can be subdivided into the backbone, which would conduct fluid if a pressure difference were applied across the medium, and dangling ends that connect to the backbone at only a single point and so would conduct no fluid. At high connectivity most or all pores are part of the infinite cluster, while at the percolation threshold the inaccessible fraction is quite large. Most of the decrease in accessible porosity occurs immediately above the percolation threshold (Figure 4.4), as is also the case with diffusivity (Figure 4.2). The implication is that, at high connection probabilities (say, p > 1/3), most sites are redundantly connected. This further implies that, at high connection probabilities, changing one bond from active to inactive is unlikely to have an effect beyond that bond, but at low connection probabilities, inactivating one bond may disconnect a large group of singly connected sites. In real-world applications and noninfinite media, the spatial distribution of the different fractions may be of more interest than the actual value. The infinite cluster and the edge-only accessible pores have characteristic spatial distributions (Figure 4.5): edge-only accessible pores are (of course) concentrated near the edge, while the infinite cluster is smaller near the edge because some potential pathways are intercepted by the edge. We clarify here that what is shown is single-edge-only accessibility, that is, accessibility from only one of the six sides of the cubic lattice. The spatial distribution of the sum of infinite cluster and edge-only accessible porosities has a characteristic two-slope distribution when plotted on log–log axes (Figure 4.6A). A similar profile was recently observed (Figure 4.6B) in high-resolution measurements by Hu9 at LBNL. Hu vacuumsaturated samples of welded tuff from Yucca Mountain, Nevada, with a perrhenate tracer (a Technetium analog), and used laser ablation — inductively coupled plasma — mass spectrometry (LA-ICP-MS) to measure perrhenate and intrinsic Al concentrations at depth increments as small as 0.85 µm at two nearby locations, labeled Hole 1 and Hole 2. Previous experiments10 had suggested that welded tuff has a connectivity just above the percolation threshold, consistent with the similarity between the simulated p = 0.255 curve (Figure 4.6A) and the experimental data (Figure 4.6B). However, the numbers raise a new difficulty. The porosity of the tuff is approximately 0.09 and the characteristic pore radius is on the order of 1 µm, so a cubic “pore unit” (volume occupied by a single pore of characteristic size plus rock to give the correct porosity) would be approximately 2.8 µm on a side, and the depth of the edge effect would be approximately 42 µm (15 pores, Figure 4.6A). This is about right for Hole 1, but for Hole 2 it is low by a factor of approximately 24 (Figure 4.6B). An explanation is offered by Knackstedt et al.,12 who show that pore-size correlation © 2003 by CRC Press LLC

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1.0

Inaccessible

Porosity Fraction

0.8

0.6

p = 0.27

EdgeAccessible

0.4

0.2 p = 0.2488

Infinite

0.0 0

50

100

150

200

250

Distance from Edge, Pores FIGURE 4.5 Spatial distribution of infinite cluster, edge-only accessible, and inaccessible porosity in a 2563 lattice at two connection probabilities. For each probability (0.27 or 0.2488), the fraction above the lines is inaccessible, the fraction below is the infinite cluster, and the portion between the lines near the left edge of the figure is the edge-only accessible fraction.

is likely to exist down to the pore scale. Such correlation, with a correlation length of 24 pore units, provides an explanation for Hu’s9 results, and suggests that future work should examine effects of pore-scale correlation on accessibility statistics. If we calculate the depth at which the edge-only accessible porosity becomes negligible (conveniently defined here as the depth, in pores, at which the edge-only-accessible porosity is no more than 1% of the mean accessible porosity), we see that this depth, Dedge, scales with proximity to the percolation threshold as Dedge ∼ (p – pc)–0.83 (Figure 4.7). Percolation theory has a related concept: the characteristic cluster radius or “radius of gyration.” We can imagine that the edgeonly accessible pores were originally isolated clusters within a larger lattice, but they were intersected when the edge of the current lattice was imposed. Knowing their characteristic size allows us to calculate the depth of the edge effect. It is known that just above the percolation threshold the radius of gyration scales as ξ ~ (p – pc)–ν for ν = 0.88 in three dimensions.24 The exponents differ slightly, perhaps because our definition of Dedge was chosen more for convenience than rigor, but the two approaches give similar results (Figure 4.7). Total porosity also scales (Figure 4.8). Percolation theory predicts that at the percolation threshold the size of the infinite cluster scales as M ~ LD, where M is the mass of the cluster (e.g., the number of sites on it), L is the size of the lattice, and D is an exponent whose value for threedimensional systems is 2.53.24 If effective porosity, εeff, is the number of sites in the infinite cluster pores divided by the number in the whole medium, then percolation theory predicts εeff = LD/L3 ~ L3-D. The exponent obtained from the simulations shown here is 0.49, close to the theoretical value of 0.47. If, however, effective porosity also includes the edge-only accessible pores at the inlet and outlet ends, the exponent is decreased to 0.41 (Figure 4.8), at least for the relatively small sample sizes tested here. The above discussions of porosity, and specifically of the depth of the edge effect, have thus far avoided the question of how these parameters influence diffusivity. Before proceeding to tortuosity, we need to address the question of which components of porosity should be used in Equation 4.3. Clearly the infinite cluster is the portion that actually carries the diffusing solute, but the diffusion is surely reduced by the infinite cluster porosity pinched off by the edge-only accessible porosity right at the edge (Figure 4.5): resistance is often controlled by the most restrictive portion © 2003 by CRC Press LLC

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1

p = 0.33

Edge-Accessible Porosity Fraction

p = 0.27 p = 0.255

0.1

p = 0.2488

A 0.01 1

10

100

Distance from Edge, Pores Intensity Ratio (dimensionless)

10-1

10-2

Hole 2

10-3

Hole 1

Cut Face

B 10-4 100

101

102

103

104

Depth from Rock Surface (µm) FIGURE 4.6 Single-edge-accessible porosity (A) simulated and (B) observed.

of a pathway. We suggest that any formulation that uses porosity to predict diffusivity should consider the mean porosity of the infinite cluste, and the infinite cluster’s minimum cross-sectional porosity. In this chapter we use a composite porosity, calculated as the harmonic mean of the mean porosity of the infinite cluster, weighted by (lattice size – 2), and the minimum cross-sectional porosity of the infinite cluster, weighted by 2 (Figure 4.9). Like the infinite cluster porosity, this composite porosity as εw ~ L–0.49.

C. TORTUOSITY

AT THE

PORE SCALE

Tortuosity is generally defined as the ratio of two lengths: microscopic length (mean distance actually traveled by a diffusing molecule) divided by macroscopic length (straight-line distance from inlet face to outlet face). In practice, though, it is usually treated as a residual term, calculated from aqueous diffusivity, effective diffusivity and porosity (Equation 4.3). Calculating the composite porosity mentioned immediately above from our simulation results yields the tortuosities shown in Figure 4.10. This calculated tortuosity scales at criticality as τ ~ L1.18. Above the percolation © 2003 by CRC Press LLC

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100

10

DEdge ~ (p - pc)-0.83

1 0.0001

0.001

0.01

0.1

1

p - pc FIGURE 4.7 Depth of the edge effect as a function of bond connection probability. The dotted line has a slope of –0.88, the theoretical value. 1

Porosity Fraction

Inaccessible

Accessible Infinite Cluster 0.1

Edge-only Accessible 0.01 10

100

Lattice Size, Pores FIGURE 4.8 Distribution of porosity subclasses at the percolation threshold as a function of lattice size. Note that accessible = infinite cluster + edge-only accessible. The scaling exponent for accessible porosity differs slightly from that of the infinite cluster.

threshold, tortuosity becomes constant at some finite size that is larger for lower probabilities, consistent with tortuosity having an edge effect in addition to scaling at criticality. Replacing the composite porosity in Equation 4.3 with the mean or minimum porosity also results in calculated tortuosities with edge effects (not shown). Why is there an edge effect in tortuosity, if tortuosity is simply a ratio of distances? A diffusing molecule has no “knowledge” of which pore-scale pathways are dead-end, and which are through pathways. (As an aside, a nonreactive tracer molecule moving primarily under advective forces would have some knowledge of which pathways to enter because there would © 2003 by CRC Press LLC

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1 p = 1.0

Weighted Porosity

p = 0.33 p = 0.27 p = 0.255

0.1 p = 0.2488

10

100

Lattice Size FIGURE 4.9 Porosity used in diffusion, calculated as a harmonic mean of average and minimum porosities of the infinite cluster.

104

Calculated Tortuosity

p = 0.2488 p = 0.255

103

p = 0.27 102

p = 0.33

101 p = 1.0

100 10

100

Lattice Size FIGURE 4.10 Tortuosity calculated via Equation 4.3 from assumed Daq, simulated Deff, and harmonic mean weighted porosity.

be no flow into dead-end pathways; for this reason diffusive tortuosity is never less than, and generally greater than, advective tortuosity.) This means that, when pores along the inlet face belong to the infinite cluster with probability (for example) 0.1 and to edge-only accessible porosity with probability 0.3, a molecule is initially three times as likely to head down a deadend as to move into a “useful” path. Furthermore, the longer the dead-end path, the more time will be wasted traversing it. This implies that the ratio of infinite cluster and edge-only accessible porosities at the inlet face, and the lesser of the mean dead-end path length (which can be calculated from the radius of gyration) and the medium thickness, should enter into a calculation of tortuosity. This suggests the empirical equation © 2003 by CRC Press LLC

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  Edge − only at inlet   τ = A1 + ξ *    Total accessible at inlet   

(4.4)

where ξ* = Min(2 ξ, L–1) and A = 1 for p = 1.0; A = 5 for p < 0.5. From Equations 4.3 and 4.4 and the composite porosity, we obtain a reasonable fit between simulated and calculated diffusivities (Figure 4.11). Note that, except for the empirical factor A (which only serves to separate redundantly connected from sparsely connected media), the calculated porosity, tortuosity, and diffusivity are based entirely on pore connectivity-derived figures. The empirical tortuosity scales as τ ~ L1.11 at criticality, so it is slightly off: by difference; tortuosity should scale as τ ~ φ/D ~ L–0.49/L–1.68 ~ L1.19. Nonetheless, it provides reasonable estimates of diffusivity, and illustrates the combined roles of connectivity and sample size in tortuosity. When L–1 < 2 ξ, changing the lattice size has a direct influence on tortuosity by varying the length of the edge-only accessible pores in which molecules wander for long times without reaching the outlet. Recall that, when the porous medium is correlated at the pore scale, ξ may be large enough (Figure 4.6) that edge effects may lead to samples of different thickness (in the mm to cm range) having different measured diffusivity values, even somewhat above the percolation threshold.

V. RELEVANCE AND CONCLUSION The existence of an inherent edge effect in rock matrices with respect to diffusivity, tortuosity, and especially porosity has interesting implications for contaminant hydrology. For example, if the accessible porosity decreases with distance from a borehole, short-term diffusion tests may yield higher estimates than longer-duration tests. Likewise, laboratory diffusivity tests may yield different results for different sample sizes. However, the greatest effect is likely to occur in granular media because such media have the greatest surface-to-volume ratio and so a significant proportion of the volume is likely to be less than Dedge from the surface. For example, a 2-mm diameter sand grain whose pore space is near criticality and that has an edge effect 0.5 mm deep will have the majority of its accessible porosity near the surface. Diffusive exchange of contaminants between the bulk solution and these near-surface pores may be fairly rapid. However, a small portion of the accessible porosity will be deep inside the particle, on the infinite cluster. Contaminant in this small fraction of the accessible pore space will equilibrate only very slowly with the bulk solution (recall that, p = 1.0

Diffusivity, L2T-1

10-1 p = 0.333 10-2 p = 0.27 10-3

10-4

p = 0.255

10-5 p = 0.2488 10

100

Lattice Size FIGURE 4.11 Diffusivity simulated (circles) and calculated via Equation 4.4 (triangles). © 2003 by CRC Press LLC

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near criticality, tortuosity increases with distance) so breakthrough experiments will show longer tailing than expected from a simple spherical diffusion model. It is even possible that this behavior will be interpreted as slow sorption25 or perhaps surface sorption coupled with intraparticle diffusion, even though no explicit sorption mechanism is involved. The ability to detect and predict scaling behavior can help avoid potentially large errors. In this chapter, we have shown through Monte Carlo simulation that two kinds of scaling can occur in low-connectivity media. One kind, obeying simple scaling laws, shows up at the percolation threshold; the other is an edge effect that is apparent mainly in small samples. This edge effect can be measured in laboratory samples and, like the scaling laws, is caused by low connectivity. A reasonable estimate of diffusivity can be made purely based upon pore connectivity and accessibility considerations, showing the pivotal role of pore accessibility in determining macroscopic behavior in low-connectivity media.

IV. ACKNOWLEDGMENTS Paper No. J-19250 of the Iowa Agriculture and Home Economics Experiment Station, Ames, Iowa, Project No. 3287, and supported by Hatch Act and State of Iowa. The authors gratefully acknowledge support from the Iowa State University Agronomy Department Endowment Fund.

REFERENCES 1. Bear, J., Dynamics of Fluids in Porous Media, Dover Publications, Inc., New York, 1972. 2. Berkowitz, B. and Ewing, R.P., Percolation theory and network modeling applications in soil physics, Surv. Geophysics, 19, 23, 1998. 3. Berkowitz, B., Scher, H., and Silliman, S.E., Anomalous transport in laboratory-scale, heterogeneous porous media, Water Resour. Res., 36, 149, 2000. 4. Chelidze, T.L., Percolation and fractures, Phys. Earth Plant. Interiors, 28, 93, 1982. 5. Crank, J., The Mathematics of Diffusion (2nd ed.), Clarendon Press, Oxford, 1975. 6. Ewing, R.P. and Horton, R., Diffusion in sparsely connected porespaces: temporal and spatial scaling, Water Resour. Res., 2003. 7. Fogg, G.E., Carle, S.F., and Green, C., Connected-network paradigm for the alluvial aquifer system, in Zhang, D. and Winter, C.I. (Eds.), Theory, Modeling and Field Investigation in Hydrogeology: a Special Volume in Honor of Schlomo P. Neuman’s 60th Birthday, Boulder, CO, Geol. Soc. Am. Spec. Paper 348, p. 25–42, 2000. 8. Gueguen, Y., David, C., and Gavrilenko, P., Percolation networks and fluid transport in the crust, Geophys. Res. Lett., 18, 931, 1991. 9. Hu, Q., Personal communication, 2001. 10. Hu, Q. and Ewing, R.P., Pore connectivity effects on solute transport in rocks, Proc. 2002 Int. Groundwater Symp., Berkeley, CA, March, CD-ROM, IAHR, Madrid, Spain, 2003. 11. Kärger, J. and Ruthven, D.M., Diffusion in Zeolites and Other Microporous Solids, John Wiley & Sons, New York, 1992. 12. Knackstedt, M.A., Sheppard, A.P., and Sahimi, M., Pore network modeling of two-phase flow in porous rock: the effect of correlated heterogeneity, Adv. Water Resour., 24, 257, 2001. 13. Knackstedt, M.A. and Cox, S.F., Percolation and the pore geometry of crustal rocks, Phys. Rev. E, 51, R5181, 1995. 14. Knackstedt, M.A., Ninham, B.W., and Monduzzi, M., Diffusion in model disordered media, Phys. Rev. Lett., 75, 653, 1995. 15. Lin, C. and Cohen, M.H., Quantitative methods for microgeometric modeling, J. Appl. Phys., 53, 4152, 1982. 16. Molz, F.J., Güven, O., and Melville, J. G., An examination of scale-dependent dispersion coefficients, Ground Water, 21, 715, 1983.

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17. Munyankusi, E., Gupta, S.C., Moncrief, J. F., and Berry, E.C., Earthworm macropores and preferential transport in a long-term manure applied typic hapludalf, J. Env. Qual., 23, 773, 1994. 18. Papert, S., Mindstorms: Children, Computers, and Powerful Ideas, Basic Books, New York, 1980. 19. Perret, J., Prasher, S.O., Kantzas, A., and Langford, C., Three-dimensional quantification of macropore networks in undisturbed soil cores, Soil Sci. Soc. Am. J., 63, 1530, 1999. 20. Renault, P., The effect of spatially correlated blocking-up of some bonds of a network on the percolation threshold, Trans. Porous Med., 6, 451, 1991. 21. Sahimi, M., Flow and Transport in Porous Media and Fractured Rock, VCH, Weinheim, 1995. 22. Scheidegger, A.E., Statistical hydrodynamics in porous media, J. Geophys. Res., 66, 3273, 1954. 23. Scott, G.J.T., Webster, R., and Nortcliff, S., The topology of pore structure in cracking clay soil. II. Connectivity density and its estimation, J. Soil Sci., 39, 315, 1988. 24. Stauffer, D. and Aharony, A., Introduction to Percolation Theory, Taylor and Francis, London, 1992. 25. Steinberg, S.M., Pignatello, J.J., and Sawhney, B.L, Persistence of 1,2-dibromoethane in soils: entrapment in intraparticle micropores, Environ. Sci. Technol., 21, 1201, 1987. 26. Wheatcraft, S.W. and Tyler, S.W., An explanation of scale-dependent dispersivity in heterogeneous aquifers using concepts of fractal geometry, Water Resour. Res., 24, 566, 1988. 27. Wood, W.W., Kraemer, T.F., and Hearns, P.P., Jr., Intragranular diffusion: an important mechanism influencing solute transport in clastic aquifers?, Science, 247, 1569, 1990. 28. Yanuka, M., Dullien, F.A.L., and Elrick, D.E., Serial sectioning digitization of porous media for twoand three-dimensional analysis and reconstruction, J. Micros., 135, 159, 1984.

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5

Solute Transport in Porous Media: Scale Effects L. Zhou and H.M. Selim

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

Introduction.............................................................................................................................63 Definition of Scale Effects .....................................................................................................64 Mean Travel Distance and Distance from Source .................................................................65 Confusion................................................................................................................................65 Inadequacies in Studies on Scale Effects ..............................................................................67 Clarification of Scale and Scale Effects ................................................................................68 Rational and Case Studies......................................................................................................69 Generalized Equations and Simulations ................................................................................69 A. CDE with a Linearly Time-Dependent Dispersivity ..................................................69 B. CDE with a Linearly Distance-Dependent Dispersivity.............................................70 C. CDE with a Nonlinearly Time-Dependent Dispersivity .............................................71 D. CDE with a Nonlinearly Distance-Dependent Dispersivity .......................................72 E. Comparison of Models and Simulations.....................................................................72 1. Effects of Nonlinearity of Dispersivity Model .....................................................74 IX. Distance-Dependent Dispersivity: Derivation and Simulation..............................................76 A. Travel Time Variance vs. Distance..............................................................................79 B. Apparent Dispersivity vs. Distance.............................................................................79 X. Summary.................................................................................................................................81 XI. Appendix A. Derivation of Finite Difference Equations for CDE with a Linearly Time-Dependent Dispersivity.................................................................................................82 XII. Appendix B. Derivations of Finite Difference Equations for CDE with a Linearly Distance-Dependent Dispersivity ...........................................................................................83 XIII. Appendix C. Derivations of Finite Difference Equations for CDE with a Nonlinearly Time-Dependent Dispersivity.................................................................................................84 XIV. Appendix D. Derivations of Finite Difference Equations for CDE with a Nonlinearly Distance-Dependent Dispersivity ...........................................................................................85 References ........................................................................................................................................86

I. INTRODUCTION The advection-dispersion equation (ADE) is often used to describe solute transport in geologic systems under saturated and unsaturated conditions. Dispersivity, one of the parameters of the ADE, is a measure of dispersive properties of a geologic system. Traditionally, it has been considered as a characteristic single-valued parameter for an entire medium.1 However, during the last three decades a number of studies have shown that a constant dispersivity is not always adequate, and a dispersivity that is dependent on the mean travel distance or scale of the geologic system is often needed.2–6 63 © 2003 by CRC Press LLC

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The dependence of dispersivity on the mean travel distance or scale of the geologic system is referred to as “scale effects.” Pickens and Grisak4 provided a detailed review of the scale effects in field dispersion investigations. They summarized results from several computer simulations, and laboratory and field transport studies. They found that dispersivities obtained from computer modeling studies of contamination zones ranged from 12 to 61 m and tended to increase with the scale of the contamination zone (Table 1 in Pickens and Grisak4). In contrast, dispersivities obtained from the analysis of laboratory breakthrough-curve (BTC) data on repacked materials were of the order of 0.01 to 1.0 cm, and those obtained from analysis of various types of field tracer tests ranged between 0.012 and 15.2 m (Table 2 in Pickens and Grisak4). Gelhar et al.6 provided a critical review of some 104 dispersivity values determined from 59 different sites. The longitudinal dispersivities ranged from 10–2 to 104 m for scales ranging from 10–1 to 105 m. Although fairly scattered, the data indicated a trend of increase in the longitudinal dispersivity with observation scale. Case studies conducted by Peaudecerf and Sauty7 showed that the dispersivity changes with distance. From a field transport experiment, Sudicky and Cherry3 found that dispersivity values for chloride based on analytical solutions increased with mean travel distance in the groundwater flow domain. Fried2 reported longitudinal dispersivities from several sites. He reported several values ranging from 0.1 to 0.6 m for the local (aquifer stratum) scale, 5 to 11 m for the global (aquifer thickness) scale, and 12.2 m for the regional (several kilometers) scale (cited in Pickens and Grisak4). Later, Fried8 defined several scales in terms of the “mean travel distance” of a tracer or contaminant. These scales are: • • • •

Local scale, ranging between 2 and 4 m Global scale 1, ranging between 4 and 20 m Global scale 2, ranging between 20 and 100 m Regional scale, larger than 100 m (usually several kilometers)

Interestingly, some field as well as laboratory studies questioned whether the “scale effect” exists. For example, Taylor and Howard9 concluded based on their study in a sandy aquifer that a distance-dependent dispersivity was not observed. Based on column experiments, Khan and Jury5 found that the dispersivity increased with increasing column length for undisturbed soil columns but was length independent for repacked columns. Based on these studies one may conclude that scale effects may exist for some systems but not for others.

II. DEFINITION OF SCALE EFFECTS Based on the above discussion, it is conceivable that the so-called scale effect carries two meanings: one refers to the dispersivity (α) as a function of mean travel distance ( x ) of a tracer; the other is a function of distance (x) from the source of a tracer solute. When plotted against a test scale with scale either mean travel distance or distance from a source, α increases with x or x or both. Specifically, the relation between α and mean travel distance x is obtained by fitting solute concentration profiles at different times with appropriate analytical or numerical solution of the ADE. Conversely, the relation between α and distance x is obtained from column studies by fitting breakthrough curves sampled at different distances or depths x. Therefore, it appears that the concept of scale effect is not well defined. If a clear definition of the concept of scale effect is not realized, one expects to encounter misunderstanding and misinterpretation of results. For example, one recognizes that in a geologic system, if α increases with x then there exists a scale effect for such a system. Likewise, if α increases with distance x from source, we also recognize that there exists a scale effect. On the other hand, if we only know that a scale effect exists for a geological formation, we cannot a priori distinguish whether α increases with mean travel distance x or with distance from source x. This situation is awkward, and the ambiguous meaning of the scale effects makes it difficult to implement. © 2003 by CRC Press LLC

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Suppose that we have sufficient data such that we can examine the relationship between α and x as well as that between α and x for the same geologic system. If it happens that α increases with x but not with distance x or α increases with distance x but not with mean travel distance x , can one conclude that the system has a scale effect? Can one ignore that the system has a scale effect because α does not increase with distance from source x? We are not able to answer these questions because there is no unique concept of the scale effects.

III. MEAN TRAVEL DISTANCE AND DISTANCE FROM SOURCE The above-mentioned confusion stems from the ambiguous meaning of the term “scale.” When one refers to the term scale, one generally means the space scale such as the length of a soil column, the dimension (length and width) of an aquifer in a field transport experiment, or the area of the aquifer covered by a monitoring instrument. It should be pointed out that scale is a physically measurable quantitative property. In other words, a scale is a characteristic index associated with transport processes. For laboratory experiments, a scale could be the length of the soil column used. For field transport experiments, the potential scale could thus be infinite. Analogous to laboratory experiments, under certain circumstances, the distance between an observation well and an injection well could also be taken as a scale. In general, a space coordinate could be treated as a scale as long as the origin is set at the inlet of the soil column or the injection well. Obviously, scale is associated with distance from a source or a space coordinate and has nothing to do with time. In other words, scale should depend only on distance but not on time. Therefore, it is not appropriate to define scale in terms of x . Mean travel distance x is actually not a distance from source or space coordinate in the physical sense. On the contrary, x is a function of time or an expression for time. If x is taken as a scale, we encounter incorrectly the situation that scale varies with time. Like a space coordinate, scale is an independent variable and should not depend on any other variables. In laboratory experiments, this concept is straightforward and easily understood because a soil column length is the obvious parameter that could be associated with scale. However, when one deals with field experiments, we have difficulties in using this concept. In fact, several researchers use the mean travel distance x as a scale indiscriminately. Mean travel distance is theoretically where the solute front is at a certain time t. If one considers x as a scale, it turns out that the scale for a field experiment does not exist prior to the application of a solute pulse. Besides, because of the linear increase of x with time, scale is also a linear function of time in this case. These two deductions definitely do not sound correct. As mentioned above, scale is a characteristic parameter for a field site and is determined for a given system, for example, an experimental setup, regardless of whether a transport experiment is actually conducted or when a transport experiment is initiated. If we have observation wells at different distances from an injection well, potentially we can say that this experiment is monitored at different scales. If the scale is defined based on x , one is unable to predetermine the scale without prior knowledge of the schedule of the experimental sampling scheme. What x tells us is how far a solute front advances after release. So the validity of determining a scale based solely on x is questionable.

IV. CONFUSION The term scale is frequently misrepresented because it often refers to the mean travel distance as well as distance from source. It appears that both x and x are unified under the umbrella of scales. However, this unification is somewhat misleading and causes confusion. One source of confusion is that x and x are used interchangeably. In other words, x can be replaced by x and vice versa. All too often in the literature, this type of interchange has been done. For example, Equation 16 in Wheatcraft and Tyler10 is a relationship between α and x . However, when this relationship was cited by Su,11 it was converted to a relationship between α and x. Similar interchange has been © 2003 by CRC Press LLC

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conducted by Yates.12,13 We recognize that a relationship between dispersivity α and the space coordinate x may exist. However, we want to emphasize here that one cannot derive a relationship between α and x on the basis of a relationship between α and x . Another example of confusion is the reconstruction of the variance of travel distance σ 2x based on dispersivity α at a distance x rather than x . Pickens and Grisak1 reconstructed the variance mean travel distance relationship based on dispersivity values measured at different distances by Peaudecerf and Sauty.7 Sudicky and Cherry3 plotted reconstructed variances based on dispersivity from BTCs at different distances and those estimated from snapshots at different times in one graph to discuss the scale effects. The reconstruction of variance can be described as follows. Based on the assumption of homogeneous media, the variance of travel distance increases linearly with time such that σ 2x = 2 Dt

(5.1)

where D is the dispersion coefficient. Since D can be expressed as D = αv ,

(5.2)

where v is the pore water velocity. We thus have σ 2x = 2αx

(5.3)

The relationship given by Equation 5.3 is often used to estimate α given both x and σ 2x . However, we found that this relationship is also employed to reconstruct σ 2x .1,3 A dispersivity α measured at distance xis substituted into Equation 5.3. Therefore, the actual equation implemented reads, σ 2x = 2αx

(5.4)

Thus an interchange from x to x is implicitly carried out. It should be pointed out that σ 2x is related only to x and not x. In other words, for a given time t or mean travel distance x , one can compute σ 2x given α is known. However, no variance of travel distance σ 2x exists for a given distance x. The reason is because one needs a set or a collection of points to estimate a variance. On the contrary, a variance of travel time ( σ 2t ) exists with respect to a distance x. For example, one can estimate σ 2t from a breakthrough curve as follows.

σ 2t

∫ =

∞

0

(t − t ) c(t)dt 2

∫

∞

0

c (t ) dt

where c(t) is the solute concentration change over time at a fixed distance x (BTC), and t is the mean travel time and can be estimated as follows ∞

t=

∫ tc(t)dt ∫ c(t)dt 0

∞

0

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Therefore, σ 2x is associated with time (t) or mean travel distance ( x ) whereas the variance of travel time, σ 2t , is associated with distance (x). A linear relationship between σ 2x and t or x implies that a constant α for all times up to the maximum time in consideration could be estimated based on experimentally measured solute concentration profiles. Such an estimate of α applies to all distances from the source. Conversely, a dispersivity α estimated at a certain distance based on BTCs means a constant dispersivity is needed to described these BTCs at that specific distance for all times in consideration. If one should express the estimated dispersivity from a BTC at a certain distance in the variance-time (mean travel distance) format, it should be a straight line with a slope of 2α. Using Equation 5.3 to reconstruct σ 2x simply means that the obtained dispersivity α only applies to that specific x or time t. Therefore, to reconstruct variance based on α and distance xat which the dispersivity is estimated violates the assumption based on which the dispersivity is obtained. Clearly the examples of confusion described above stem from the ambiguous definition of scale.

V. INADEQUACIES IN STUDIES ON SCALE EFFECTS A literature search reveals that dispersivity values are often compared without discrimination. Specifically, dispersivities measured at different distances from a given source are often compared with dispersivities measured at different x or time t.4,6,14,15 In addition, variation in dispersivities is almost always attributed to the scale under which it is estimated. The heterogeneity or types of formation of the geological systems is often ignored. The other problem is that the integrity of transport processes is often ignored when a regression model is applied to dispersivities estimated based on different transport processes and from different media. Dispersivity has been estimated in both laboratory and field studies because of its importance to the governing convection-dispersion equation. A comparison is often made among measured dispersivity values for different scales to support the finding that dispersivity is scale dependent. No matter whether the dispersivities were estimated for a distance from a source or for certain time in terms of mean travel distance, they were compared in terms of a quantitative index: scale.6,15 As discussed above, the definition of the term scale is not clear. Therefore, whether this comparison is reasonable remains open. Although distance from a source x and mean travel distance x share a common dimension of length, these two terms are quite different according to their physical meanings. Conceptually, dispersivities obtained under different mean travel distance and different distance are not comparable in terms of length scale, which is the quantitative value of either x or x . The reason is that the underlying assumptions for the estimation of dispersivity with respect to distance and those with respect to mean travel distance are actually exclusive to each other as discussed above. When comparisons among dispersivities are made, values corresponding to different scales are often from different media, i.e., porous media and fractures, instead of the same medium. This is understandable because of difficulties in obtaining dispersivity information from the same medium with scales varying from tens of centimeters to hundreds of meters. Beyond the comparison of dispersivity is the regression of dispersivity vs. scale. Based on dispersivity data available in the literature, several studies have been conducted to develop a functional relationship between dispersivity values and associated scale based on regression analysis.14–16 Furthermore, a universal dispersivity value for a specific scale is determined based on the function obtained. Mathematically, this kind of regression analysis is feasible. However, rigorous theoretical development is needed in support of the regression analysis. Currently, several models are available to describe the relationship between α and x . For example, Zhou and Selim17 developed a model to describe timedependent dispersivity. In addition, others are stochastic models that describe dispersivity mean travel distance based on heterogeneity of the hydraulic conductivity in porous media.18,19 These models are commonly used to support regression of measured α vs. different scales representing several distances and mean travel distances over different sites.10,15 © 2003 by CRC Press LLC

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To justify the regression of data from different sites, a separate theory must be developed; in fact, fractal and stochastic models describe an instantaneous relationship between dispersivity and time or mean travel distance. Dispersivity takes a distinct value for a specific time. The measured α available in the literature, however, is an apparent or average α over the period up to the time at which the dispersivity is estimated. Thus, the estimated dispersivity under such conditions is not actually applicable to the models discussed above.

VI. CLARIFICATION OF SCALE AND SCALE EFFECTS As pointed out above, discrepancies regarding use of the terms scale and scale effects are mainly caused by the ambiguous definition of the term scale. In fact, we are dealing with four types of relations between dispersivity and time or distance rather than one universal dispersivity scale relationship: 1. The first is a time-averaged dispersivity vs. time (or mean travel distance). Here apparent dispersivity is estimated at different discrete times or mean travel distances. When plotted, dispersivities may increase with time as reported by several studies.20,21 Though dispersivity is estimated according to a snapshot at a certain time t, such α represents an average value for the time period up to the time t. 2. The second is a time-dependent dispersivity as described by fractal and stochastic models.14,15,17 This relationship reveals that α is a continuous function of time. For each time t, one has an instantaneous value of α for that specific time. 3. The third is α as a continuous function with distance from source. This type of relationship is not actually observed experimentally or developed theoretically. Rather it is derived by replacing the mean travel distance x with x distance from source in the second case.11–13 Removal of the bar from mean travel distance ( x ) in Equation 5.5 changes the relationship to a dispersivity–distance relationship. One may argue that the first type of relationship is a description of dispersivity vs. scale and this derived relationship should hold in terms of scale (distance from source is also a scale). As discussed above, the distance from the source and mean travel distance are not interchangeable. If dispersivity can vary with distance from the source according to such a trend, theoretical support is needed to support such an expression. 4. The fourth type of dispersivity vs. scale is a distance-averaged α vs. the length scale in consideration.5,23–25 In this case, dispersivity for different column lengths or distances between observation wells and an injection well is obtained through fitting of observed BTCs. In some cases, dispersivity values were found to increase with the length scale in consideration, i.e., the column length or the distance between the observation and injection wells. What such a relationship means is that a constant dispersivity for the whole column is needed to predict the breakthrough process. However, for each column length, a different (constant) dispersivity must be used. Thus, one cannot conclude from this observation that dispersivity is distance dependent. Actually, in the fourth case, for each individual length, the basic assumption is that the dispersivity is constant along the whole column up to the length considered. The dispersivity value could thus be considered as an average or apparent value. Rather than one so-called generic dispersivity–scale relationship, we emphasize here four distinct dispersivity–time or dispersivity–distance relationships. Among these four relationships, only the fourth could be possibly characterized by the so-called dispersvity–scale relationship if one defines scale as length of column or distance from source.

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VII. RATIONAL AND CASE STUDIES As pointed out earlier, distance-dependent dispersivity has been used rather than time-dependent dispersivity in terms of dispersivity mean travel distance relationship. In the previous section, we emphasized the differences between mean travel distance x and distance from source x. Such differences have been often ignored. Given an expression for dispersivity mean travel distance, for example, α = 0.1x , we investigated the effect on BTCs and solute concentration profiles if the bar from x is removed and thus yields a dispersivity–distance relationship, i.e., α = 0.1x. We used the finite difference approach to solve the convection-dispersion equation (CDE) with a time-dependent (in terms of x ) or distance-dependent dispersivity. We focused on the differences in BTCs and concentration profiles resulting from different dispersivity models. We also propose a procedure for obtaining a distance-dependent dispersivity model. To compare the differences in transport processes in media with time-dependent dispersivity and distance-dependent dispersivity, we have to solve the CDE with time-dependent dispersivity or distance-dependent dispersivity. Analytical solution is available for several dispersivity-distance models. Yates12 suggested an analytical solution for one-dimensional transport in heterogeneous porous media with a linear distance-dependent dispersion function. Su11 developed a similarity solution for media with a linearly distance-dependent dispersivity. Yates13 gave an analytical solution for one-dimensional transport in porous media with an exponential dispersion function. Logan26 extended the work of Yates12,13 and developed an analytical solution for transport in porous media with an exponential dispersion function and decay. Transport in porous media with time-dependent dispersion function has been studied27 as well. Generally, dispersivity is expressed directly as a function of time t instead of mean travel distance x . As far as we know, analytical solutions for transport in porous media with timedependent dispersion function in terms of x are not available. Although analytical solutions for some cases are available, they were not used because they are difficult to use. Besides, numerical evaluation is often necessary to compute the analytical solution. In this investigation, we solved the CDE with time-dependent dispersivity or distance-dependent dispersivity using numerical methods. As an illustration, we chose linear or power law form as a model for time-dependent dispersivity. The advantage of using a power law model lies in that it will recover a linear model or reduce to a constant (homogeneous case) if we set the exponent term to proper values.

VIII. GENERALIZED EQUATIONS AND SIMULATIONS A. CDE

WITH A

LINEARLY TIME-DEPENDENT DISPERSIVITY

For a linearly time-dependent dispersivity in terms of mean travel distance, x , a representative model can be expressed as1 α( x ) = a1 x

(5.5)

where a1 is a dimensionless constant. Physically, a1 is the slope of dispersivity mean travel distance line. If we ignore molecular diffusion, the dispersion coefficient can be written as D(t ) = α ( x ) v = a1 xv = a1 v 2 t

(5.6)

where v is mean pore water velocity. Accordingly, the governing equation in a heterogeneous system with a time-dependent dispersion coefficient is given by ∂c ∂  ∂c  ∂c =  D(t )  − v ∂t ∂x  ∂x  ∂x © 2003 by CRC Press LLC

(5.7)

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Clearly, the dispersion coefficient as determined by Equation 5.6 is constant for the entire domain at any fixed time. Therefore, the governing equation can be rewritten as ∂c ∂2 c ∂c = a1 v 2 t 2 − v ∂t ∂x ∂x

(5.8)

In addition, the appropriate initial and boundary conditions for a finitely long soil column can be expressed as c( x, t ) = 0,

t=0

(5.9)

vc ( x , t ) = vc0 − a1 v 2 t

∂d , ∂x

x = 0, 0 < t ≤ T

(5.10)

vc ( x , t ) = −a1 v 2 t

∂c , ∂x

x = 0, t > 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)

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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

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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

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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

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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

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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

Solute Concentration (mg/L)

10

2b 8

Time = 10 hr

6

Distance-dependent

4

Time-dependent

2

0 0

20

40

60

80

100

Solute Concentration (mg/L)

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.

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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

Relative Concentration (C/Co)

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 ) =

© 2003 by CRC Press LLC

V 2 dσ 2t 2 dz

(5.29)

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77

Relative Concentration (C/Co)

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

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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

© 2003 by CRC Press LLC

d

(5.35)

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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

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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

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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

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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=

© 2003 by CRC Press LLC

1 a v 2 ( j + 1)β2 − vβ 2 1

(5.A6)

(5.A7)

(5.A8)

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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′ =

© 2003 by CRC Press LLC

1 a viβ − vβ(1 − a2 ) 2 2

(5.B4)

(5.B5)

(5.B6)

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Scaling Methods in Soil Physics

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)

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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 ′′′ =

© 2003 by CRC Press LLC

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)

(

)

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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.

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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.

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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.

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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 <10 cm to >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

© 2003 by CRC Press LLC

1.46

(6.2)

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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

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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

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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

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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

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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.

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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-

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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

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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

[

© 2003 by CRC Press LLC

]

]

m 2

]

n −m

(7.6)   

2

(7.7)

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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

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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

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  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

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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.

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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

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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

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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

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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

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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

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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

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Capillary Pressure Head (-cm)

Capillary Pressure Head (-cm)

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

Infiltration 0.0 200.0 0.0

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

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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.

© 2003 by CRC Press LLC

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Capillary Pressure Head (-cm)

Capillary Pressure Head (-cm)

Capillary Pressure Head (-cm)

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

Infiltration 0.0 200.0 0.0

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

Infiltration 0.0 200.0 0.0

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 7.9 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

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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

Infiltration, Ps=30(-cm) |R|2=0.1

0.10

c) 0.08

Infiltration, Ps=150(-cm) |R|2=1.0

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

Evaporation, Ps=240(-cm) |R|2=0.1

0.00 0.0

0.08

Evaporation, Ps=480(-cm) |R|2=1.0

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

0.10

g)

0.06

0.00 0.0

Infiltration, Ps=150(-cm) |R|2=0.1

0.00 0.0

100.0

200.0

0.00 0.0

Evaporation, Ps=480(-cm) |R|2=0.1

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)

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

0.15

0.18

b)

Infiltration, Ps=30(-cm) |R|2=0.1

0.18

c)

0.15

Infiltration, Ps=150(-cm) |R|2=1.0

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

0.03

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)

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

0.15

0.00 0.0

100.0

200.0

0.18

f)

Evaporation, Ps=240(-cm) |R|2=0.1

0.00 0.0

g)

0.15

Evaporation, Ps=480(-cm) |R|2=1.0

0.15

0.12

0.12

0.12

0.09

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

Infiltration, Ps=150(-cm) |R|2=0.1

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)

Evaporation, Ps=480(-cm) |R|2=0.1

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

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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

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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.

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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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60

a)

50

40

30 20 10 60

b)

50

40

30 20

1 Jul 2000

1 Jan 2000

1 Jul 1999

1 Jan 1999

1 Jul 1998

1 Jan 1998

10

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

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27 Sep 1995

Random pattern 13 Apr 1996

28 Mar 1996

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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

OF

SOIL MOISTURE

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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1

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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

Saturation

Spatial mean moisture

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

60 50 40 30 20 10 0

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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

OF

SOIL MOISTURE

IN

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

OF

VARIABILITY

IN

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

OF

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.

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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.

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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.

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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

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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

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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

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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

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k

k

t

)

2

(

α 2 + x j3 − x k3

)

2

 α3  

P

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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

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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.

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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 ) =

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1 a

ψ a ,τ (

x −τ ) a

(10.2)

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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

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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.

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• 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.

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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.

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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.

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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 (<60 m), the local wavelet spectrum of log(α) shows three prominent features at locations from 1 to 130 m, around 240, and around 450 m, respectively. An oscillation exists at locations from 1 to 130 m and then disappears at locations beyond 130 m. At locations around 240 and 450 m, another two features appear. Comparison of these features with soil texture indicated that all these features are associated with fine soil texture. Soil with a finer soil texture has large aggregates and cracks, resulting in strong small-scale spatial variation in the values of α. As a result, strong amplitude change would be expected for those locations, resulting in high wavelet variance as shown in Figure 10.5a. Strong variations at locations from 0 to 240 m with periods between 180 and 300 m exist in the local wavelet spectrum of log(α) (Figure 10.5a). This is because, from location 0 to location 300

a

Period

240 180 120 60

b Scale-averaged power spectrum

Variance

2

1

0 0

120

240

360

480

600

Location FIGURE 10.5 Normalized local wavelet spectrum (a) and scale-averaged wavelet power spectrum (b) of measured log(α). The normalization was performed by dividing the local wavelet spectrum by the sample variance. © 2003 by CRC Press LLC

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300

a

Period

240 180 120 60

b Scale-averaged power spectrum

Variance

2

1

0 0

120

240

360

480

600

Location FIGURE 10.6 Normalized local wavelet spectrum (a) and scale-averaged wavelet power spectrum (b) of measured log(Ks). The normalization was performed by dividing the local wavelet spectrum by the sample variance.

240 m, the soil texture changes from fine to coarse, then to fine, and again to coarse. This alternation of coarse and fine soil texture results in large amplitudes, thus, large variance in log(α). From location 240 m to location 600 m, the wavelet spectrum of log(α) did not show pronounced variation because soil texture is generally coarse and did not change drastically. There is a strong variation from location 0 to location 120 m and from location 360 m to location 480 m with periods varying from 60 to 80 m; causes of the strong variation are not known (Figure 10.5a). The local wavelet spectrum of log(Ks) is very similar to that of log(α) (Figure 10.6a) at periods less than 180 m. However, there was a strong variation in wavelet coefficients from location 0 to location 600 m at periods between 160 and 220 m. This cyclic behavior has a decreasing evolving period indicating cycles speeding up with increasing distance from the origin. At locations around 360 m, soil was highly eroded with lower soil organic matter content and higher compaction, which may result in relatively low Ks. However, α may be less sensitive to the change of organic matter content. Therefore, there was a quasi-cyclic behavior in log(Ks), but not in log(α). The wavelet transform and Fourier transform can also provide equivalent information in terms of the global power spectrum. The global wavelet power spectrum can be obtained by integrating the local wavelet spectrum across the transect for each scale (frequency) (Figure 10.7). There are three peaks in the local wavelet spectrum of log(α) at scales of 24, 96, and 190 m, respectively, with the maximum variance at a period of 190 m (Figure 10.7b). These distinct periods or scales suggested that there might be different processes embedded in the spatial variation in log(α). Like that of the local wavelet spectrum, the global wavelet power spectrum shows similar features with that of log(α) (Figure 10.7a). The only difference in the global wavelet power spectra of log(α) and log(Ks) is that the third peak is much more pronounced. However, no conclusion can be derived from the global wavelet power spectrum regarding whether a peak is an indication of a global event or a local event because a peak may be a result of local changes or global cycles. © 2003 by CRC Press LLC

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Wavelet Spectrum

5 log(Ks) 4 3 2 1

Wavelet Spectrum

0 5 4

log(α)

3 2 1

Wavelet Spectrum

0 5 4

log(Ks) vs. log(α)

3 2 1 0 0

60

120

180

240

Scale(m) FIGURE 10.7 Normalized global wavelet power spectrum as a function of scale: (a) log(α); (b) log(Ks); and (c) cross-wavelet power spectrum between log(α) and log (Ks). The normalization was performed by dividing the local wavelet spectrum by the sample variance.

On the other hand, the local wavelet spectrum has greater detail and global features and local features can be identified quite easily. The advantage of the global wavelet power spectrum is that a statistical test can be carried out on cycles. Therefore, the best strategy to isolate global and local features is to examine the local wavelet spectrum first. The local wavelet power spectrum shows the location of change. If the variations in the local wavelet power spectra exist across the whole spatial domain at a certain period or scale, then this cycle is a global cycle. If strong variations in local wavelet power spectrum show up only at a few locations, these strong variations are local. If a global cycle is identified, then the global wavelet power spectrum should be constructed and statistical significance of the cycles be examined; however, the reverse is not true. For example, a significant cycle (at a 95% confidence level) with a scale of 24 m exists in the global wavelet spectrum for log(α). However, the local wavelet spectrum of log(α) did not show any consistent cycles, ruling out existence of cycles in log(α) along the transect. On the other hand, the local wavelet power spectrum of log(Ks) shows a consistent cycle at a period of 180 m (Figure 10.6). Therefore, we can make a hypothesis that a cycle exists. Statistical tests (not shown here) on the global power spectrum reject the hypothesis at a 95% confidence level, again ruling out the existence of a cycle with a period of 180 m. © 2003 by CRC Press LLC

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300

a Period

240 180 120 60

b Scale-averaged cross spectrum

Variance

2

1

0 0

120

240

360

480

600

Location (m) FIGURE 10.8 Normalized cross-wavelet spectrum between measured log(α) and log(Ks) along a transect: (a) local cross-wavelet spectrum; (b) scale-averaged cross-wavelet spectrum. The normalization was performed by dividing the local wavelet cross spectrum by the sample covariance between log(α) and log (Ks).

To examine the spatial information of scales, the local wavelet power spectrum can be integrated across all scales or a certain range of scales and plotted as a function of distance. As a demonstration, only the averaged power spectrum across all scales is calculated. The scale-averaged variance shows a strong variation at distance from 0 to 120 m and bulge at a distance of 420 m (Figure 10.6b). This uneven distribution of variance across the transect is evidence of nonstationarity in log(α) along the transect, which is consistent with the visual observation of the measured data (Figure 10.4). The cross-wavelet spectrum of log(Ks) and log(α) is shown in Figure 10.8. There were strong covariances at locations from 0 to 360 m with periods between 180 and 240 m and at three locations (0 to 120, 240 to 260, and 420 to 480 m) with a period of 30 m. This may be related to the fact that changes in log(Ks) and log(α) are positively correlated to soil sand content. Therefore, there is a strong cross-covariance at locations from 0 to 120 m at scales of 30, 60, and 200 m. However, strong covariance only exists at small scales (30 m) at locations from 400 to 600 m (Figure 10.8b). In summary, soil hydraulic properties and their covariance are not only scale dependent, but also location dependent. Examination of raw data can only give the location dependence, but nothing about scales. Fourier spectrum gives scale dependence, but nothing about location dependence. Wavelet analysis is a promising tool for revealing scale and location dependence of soil hydraulic properties and their covariate relationships. VII. CONCLUSIONS In this chapter, the continuous wavelet transform is introduced and wavelet analysis demonstrated using analytical signals and real soil hydraulic properties. The local wavelet power spectrum provided detailed information regarding global features as well as localized features. Integration of the local wavelet spectrum across all locations provides the global wavelet power spectrum, © 2003 by CRC Press LLC

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allowing statistical testing of global features or cycles. On the other hand, integration of the local wavelet power spectrum over a range of scales of interest allows examination of spatial content of scale information. Soil hydraulic conductivity and inverse microscopic capillary length scale along a transect in hummocky landscapes were determined using tension infiltrometers. These hydraulic properties exhibit multiscale variations and localized features due to the change of soil texture. Local variances were nonuniform, indicating strong nonstationarity in variance of soil hydraulic properties along the transect. No significant cyclic behaviors were observed from these soil hydraulic properties. The localized features and nonstationarity may have significant impacts on modeling soil water flow and chemical transport using stochastic approaches based on the stationarity assumption of soil hydraulic properties.

REFERENCES 1. Nielsen, D.R., Bigger, J.W., and Erh, K.T. Spatial variability of field measured soil-water properties, Hilgardia, 42, 215, 1973. 2. Webster, R. Is soil variation random? Geoderma, 97, 149, 2000. 3. Cressie, N. Statistics for Spatial Data, John Wiley & Sons, New York, 1991. 4. Deutsch, C.V. and Journel, A.G., GSLIB: Geostatistical Software Library and User’s Guide, 2nd ed., Oxford University Press, New York, 1998. 5. Shumway, -R.H. et al. Time and frequency domain analysis of field observations, Soil Sci., 147, 286, 1989. 6. King, G.I., Acton, D.F., and Acnaud, R.J. St. Soil landscape analysis in relation to soil distribution and mapping at a sight within the Weyburn Association, Can. J. Soil Sci., 63, 657, 1983. 7. Webster, R. Spectral analysis of gilgai soil, Austr. J. Soil Res., 15,191, 1977. 8. Nielsen, D.R., Tillotson, P.M., and Vieira, S.R. Analyzing field-measured soil-water properties, Agric. Water Manage., 6, 93, 1983. 9. Kachanoski, R. G., Rolston, D.E., and de Jong, E. Spatial and spectral relationships of soil properties and microtopography: I. Density and thickness of A horizon, Soil Sci. Soc. Am.J., 49, 804, 1985. 10. Kachanoski, R.G. and de Jong, E. Scale dependence and temporal persistence of spatial patterns of soil water storage, Water Resour. Res., 24, 85, 1988. 11. Folorunso, A.O. and Rolston, D.E. Spatial and spectral relationships between field-measured denitrification gas fluxes and soil properties, Soil Sci. Soc. Am. J., 49,1087, 1985. 12. Kumar, P. and Foufoula-Georgiou, E. Wavelet analysis for geophysical applications, Rev. Geophys., 35, 485, 1997. 13. Morlet, J. et al. Wave propagation and sampling theory, 1. Complex signal and scattering in multilayered media, Geophysics, 47, 203, 1982. 14. Farge, M. Wavelet transform and their applications to turbulence, Annu. Rev. Fluid Mech., 24, 395, 1992. 15. Gao, W. and Li, B.L. Wavelet analysis of coherent structure at the atmosphere–forest interface, J. Appl. Meteor., 32, 171, 1993. 16. Liu, P.C. Wavelet spectrum analysis and ocean wind waves, in Wavelets in Geophysics, FoufoulaGeorgiou, E. and Kumar. P., Eds., Academic Press, 1994, 151. 17. Katual, G.G. and Vidakovic, B. The partitioning of attached and detached eddy motion in the atmospheric surface layer using Lorentz wavelet filtering, Boundary Layer Meterol., 77, 153, 1996. 18. Kumar, P. and Foufoula-Georgiou, E. A multicomponent decomposition of spatial rainfall fields: 1. Segregation of large- and small-scale features using wavelet transforms, Water Resour. Res., 29, 2515, 1993. 19. Lindsay, R.W., Perceval, D.B., and Rothrock, D.A. The discrete wavelet transform and the scale analysis of the surface properties of sea ice, IEEE Trans. Geosci. Remote Sens., 34, 771, 1996. 20. Labat, D., Ababou, R., and Mangin, A. Rainfall-runoff relations for karstic springs. Part II: Continuous wavelet and discrete orthogonal multiresolution analyses, J. Hydro., 238, 149, 2000.

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21. Labat, D., Ababou, R., and Mangin, A. Introduction of wavelet analyses to rainfall/runoffs relationship for a karstic basin: The case of Licq-Atherey karstic system (France), Groundwater, 39, 605, 2001. 22. Lark, R.M. and Webster, R. Analysis and elucidation of soil variation using wavelets, Eur.J. Soil Sci., 50, 185, 1999. 23. Lark, R.M. and Webster, R. Changes in variance and correlation of soil properties with scale and location: analysis using an adapted maximal overlap discrete wavelet transform, Eur.J. Soil Sci., 52,547, 2001. 24. Lau, K.-M. and Weng, H.-Y. Climate signal detection using wavelet transform: how to make a time series sing, Bull. Am. Meteor. Soc., 76, 2391, 1995. 25. Torrence, T. and Compo, G.P. A practical guide to wavelet analysis, Bull. Am. Meteor. Soc., 79, 61, 1998. 26. Topp, G.C., Davis, J. L., and Annan, A.P. Electromagnetic determination of soil water content: measurements in coaxial transmission lines, Water Resour. Res., 16, 574, 1980. 27. Gardner, W.R. Some steady-state solutions of the unsaturated moisture flow equation with application to evaporation from a water table, Soil Sci., 85, 228, 1958. 28. White, I. and Sully, M.J. Macroscopic and microscopic capillary length and time scales from field infiltration, Water Resour. Res., 23,1514, 1987.

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11

Multiple Scale Conditional Simulation P. Kumar

CONTENTS I. Introduction...........................................................................................................................179 II. Review of Multiple Scale Estimation ..................................................................................180 III. Multiple Scale Conditional Simulation................................................................................185 A. Model Formulation ....................................................................................................185 B. Application to Fractional Brownian Random Fields ................................................187 IV. Conclusions...........................................................................................................................189 References ......................................................................................................................................190

I. INTRODUCTION Measurements of a process are often made at multiple scales. For example, in situ measurements of near-surface soil moisture is made in conjunction with remote sensing measurements1,2 to characterize its variability. For groundwater flow studies measurements of hydraulic conductivity are often available at two or more scales depending on the sampling technique or aquifer test.3 The problem of characterizing a process through measurements made at multiple scales is becoming increasingly important as we continue to study processes at ever increasing scales. Due to cost constraints, observations made at multiple scales have the property that several regions of fine scale measurements with limited coverage are embedded within coarse scale measurements of larger coverage (see Figure 11.1). Further analysis needs to be performed to integrate the information across scales. Combining the information at multiple scales to obtain estimates that are consistent across scales has been addressed by various authors in References 4 through 7. Their methodology is based on a multiple scale Kalman filtering algorithm. The key to their algorithm is the development of a state-space model evolving over the scales, i.e., the scale parameter is treated akin to time parameter of the usual state-space models, such that description at a particular scale captures the features of the process up to that scale that are relevant for the prediction of finer scale features. Kalman filtering technique is used to obtain optimal estimates of the states described by the multiple scale model using observations at a hierarchy of scales.6,7 Estimation techniques are based on minimizing a cost function that typically involves the variance and therefore underestimates the variability; the resulting estimated field is smoother than the true field. Therefore, conditional simulation8 rather than estimation is used to characterize the variability of the underlying field. The random fields obtained by this technique preserve the observed values where they are known and at the same time simulate the intrinsic fluctuations at other locations that are consistent with the mean and the covariance of the process. Given the wide utility of conditional simulation technique for the usual one- and two-dimensional processes, an algorithm that utilizes measurements at multiple scales was introduced by Kumar2 and is 179 © 2003 by CRC Press LLC

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Coarse Scale Grid

Fine Scale Grid

FIGURE 11.1 Schematic showing an idealized embedding of fine scale measurement grid of limited coverage within a coarse scale grid of larger coverage for a two-level scheme.

reviewed in this chapter. The method relies on the multiple scale state-space model for estimation and simulation. This chapter is organized as follows. Optimal estimation using the multiple scale Kalman filtering is briefly reviewed in Section II. The methodology of multiple scale conditional simulation is then developed in Section III and illustrated using fractional Brownian fields. Concluding remarks are given in Section IV.

II. REVIEW OF MULTIPLE SCALE ESTIMATION In this section the algorithm of multiple scale Kalman filtering developed by authors of References 4 through 7, as well as the author of Reference 9 is briefly reviewed with the aim of establishing the necessary equations needed for the conditional simulation algorithm. The reader is encouraged to consult the original references for details. Consider the problem of disaggregating a two-dimensional vector valued random field from coarse to fine resolution (large to small scale). At the coarsest resolution the field will be represented by a single vector (see Figure 11.2). At the next resolution there will be q = 4 vectors and, in general, at the mth resolution we obtain qm vectors. The values of the random field can be described on the index set (m,i,j) where m represents the resolution and (i,j) the location index. The scale to scale decomposition can be schematically depicted as a tree structure (quadtree for two-dimensional processes) as shown in Figure 11.2. To describe the model let us use an abstract index λ to specify the nodes on the tree and let γλ specify the parent node of λ (see Figure 11.2). Then the multiple scale stochastic process can be represented as X (λ) = A(λ) X ( γλ) + B(λ) W (λ) ( n ×1)

( n ×n )

( n ×1)

( n ×n )

( n ×1)

(11.1)

Here X(λ)∈Rn is the state vector and A(λ) and B(λ) are matrices of appropriate dimensions. The root node X(0), which represents the properties of the mean state over the entire domain, and the driving noise W(λ) are normally distributed with known covariance matrices, i.e., X (0) ~ N (0, PX (0)) PX (0) = E[ X (0) X T (0) ] ( n ×n )

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(11.2)

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X0

X (1,1) 1

181

Resolution 0

γλ

X (1,2) 1 Resolution 1

X (2,1) 1

X (2,2) 1

λ X2 (1,1) X2 (1,2) X2 (1,3) X2 (1,4)

X2 (2,1) X2 (2,2) X2 (2,3) X2 (2,4)

Resolution 2

X2 (3,1) X2 (3,2) X2 (3,3) X2 (3,4)

X2 (4,1) X2 (4,2) X2 (4,3) X2 (4,4)

FIGURE 11.2 The structure of a multiple scale random field is shown. The values at various grid locations (i,j) are given as xm(i,j) where m is the resolution index. At the coarsest resolution (m = 0), the field is represented by a single state vector; generally, at the mth resolution, there are 4m state vectors. (right) Abstract representation of the multiple scale decomposition. The abstract index λ refers to a node in the tree and γλ refers to the parent node.

W (λ) ~ N (0, I )

(11.3)

where I is the identity matrix and W(λ) and X(0) are independent. Interpreting the states at a given level of the tree as a representation of the process at one scale, we see that Equation 11.1 describes the evolution of the process from coarse to fine scale. The term A(λ)X(γλ) represents the interpolation or prediction down to the next finer level and B(λ)W(λ) represents new information added as the process evolves from one scale to the next.10 It is further imposed that, conditioned on any node on the tree, each of the subtrees connected to this node are conditionally independent, i.e., the model is Markovian when going from the coarse to fine scale. For this to hold it is only required that W(λ) be independent from scale to scale. It is easy to verify that the covariance PX (λ) ≡ E[X (λ) X T (λ) ] ( n ×n )

of the state at node λ evolves as © 2003 by CRC Press LLC

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PX (λ) = A(λ) PX ( γλ) A T (λ) + B(λ) B T (λ)

(11.4)

Following Verghese and Kailath,11 the corresponding fine to coarse scale evolution model can be written as6,7 X ( γλ) = F (λ) X (λ) + W (λ) ( n ×1)

( n ×n )

( n ×1)

(11.5)

( n ×1)

where −1

F (λ) = PX ( γλ) A T (λ) PX (λ)

(11.6)

and W(λ) is uncorrelated along all upward paths of the tree and has a variance given by

[

]

T −1   E W (λ)W (λ)  ≡ Q(λ) = PX ( γλ) I − A T (λ) PX (λ) A(λ) PX ( γλ) .  

(11.7)

Multiple scale stochastic models can be used to make estimates of processes from noisy measurements obtained at different resolutions. If we are given noisy measurements Y(λ) of the state X(λ), we can develop an estimation problem using the combination of Equation 11.1 and the following equation: Y (λ) = C(λ) X (λ) + V (λ) ( N ×1)

( N ×n ) (n ×1)

(11.8)

( N ×1)

where V(λ) characterizes the measurement variance. It is independent of X(λ) and W(λ) for all λ, and V(λ) ~ N(0,R(λ)). The matrix C(λ) can specify, in a very general way, measurements taken at different spatial locations and at different scales. R(λ)specifies the covariance of the measurement errors V(λ). Here R(λ) is diagonal but not a multiple of the identity, i.e., σ 21  0 R(λ) ≡ E[V (λ)V T (λ) ] =  M (N ×N )  0

0 σ 22 M 0

L L O L

0   0  M   σ 2N 

(11.9)

indicating each measurement component can have a different variance. As explained next, Equations ˆ λ) . Notice 11.1 and 11.8 can be solved jointly to obtain estimates of X(λ), hereafter denoted as X( that this is a very attractive technique because it enables us to combine estimation and filtering while exploiting hierarchical measurements at different scales. The basic scheme for the optimal estimation of X(λ) based on the measurements Y(λ) proceeds in two steps: an upward sweep and a downward sweep. The upward sweep step carries information from the fine scale to the coarse scale which, during the downward sweep, is passed onto other nodes that are not under the same subtree. For example, in Figure 11.2, for X1(2,1) to contribute to X1(2,2) (or its descendents) one needs first to carry the information up to X0 and then down the tree. 1. Upward or fine-to-coarse-scale sweep. The first step of the estimation applies Kalman filter prediction going from fine to coarse scale using Equation 11.5 (see Figure 11.3).

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a) Upward Pass

b) Downward Pass

FIGURE 11.3 Schematic of two-pass estimation in the multiple scale framework for a binary tree. First the upward pass propogates information up the tree. At each tree node the estimate incorporates all measurements on that node and its descendants. Then during the downward pass the information is propagated down the tree. The estimates at each tree node are now based on information on all nodes on the tree.

Therefore, for each coarse scale node there will be q predicted values, one from each of the q child nodes. These q predictions are then merged to obtain a single predicted value. This merge step does not have any counterparts in the usual time domain Kalman filtering schemes. Using this merged prediction, the update step that incorporates the observations (Equation 11.8) is the usual Kalman filter step involving the Kalman gain matrix. 2. Downward or coarse-to-fine-scale sweep. After the above procedure is carried out from the finest to the coarsest scale, the downward sweep generalizes the Rauch-Tung-Stribel algorithm12 and produces smoothed best estimates and error variances on every node based on all the data. The details of these computations are given below. We assume that each node has q children. We denote by αiλ the ith child node of λ for i = 1, … ,q. Also define Xˆ (λ | α iλ) : predicted value of X(λ) using the estimate of child node αi (i = 1, … ,q) of λ during the upward sweep ˆ X(λ | λ−) : predicted value of X(λ) after merging the predictions of the q child nodes of λ during the upward sweep ˆ ˆ λ | λ−) and the measurement Y(λ) during the X(λ | λ+) : updated value of x(λ) using X( upward sweep ˆ λ) : estimated value of X(λ) after smoothing during the downward sweep X( ˆ λ | λ−) , P( ˆ λ | λ+) , P( ˆ λ) analogously. Define the error variances Pˆ (λ | α iλ) , P( The estimation proceeds in the following steps: • Initialization: assign the following prior values at λ corresponding to the finest scale node. ˆ λ | λ−) = 0 X( Pˆ (λ | λ −) = PX (λ) where PX(λ)is the prior error variance at the node λ. • Upward sweep ˆ λ | λ−) from the child • Measurement update: the predicted (and merged) values, X( nodes are combined with the measurements to get estimates of the state vectors and the error variances using:

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[

Xˆ (λ | λ +) = Xˆ (λ | λ −) + K (λ) Y (λ) − C(λ) Xˆ (λ | λ −)

[

]

]

(11.11)

Pˆ (λ | λ +) = I − K (λ)C(λ) Pˆ (λ | λ −) where the Kalman gain matrix K(λ) is given by

[

K (λ) = Pˆ (λ | λ −)C T (λ) C(λ) Pˆ (λ | λ −)C T (λ) + R(λ)

]

−1

(11.12)

• One step prediction: moving up to the parent node we apply the Kalman filter prediction to get predictions from each child node using: Xˆ (λ | α iλ) = F (α iλ) Xˆ (α iλ | α iλ +)

(11.13)

Pˆ (λ | α iλ) = F (α iλ) Pˆ (α iλ | α iλ +) F T (α iλ) + Qˆ (α iλ)

(11.14)

Qˆ (α iλ) = A −1 (α iλ) B(α iλ)Q(α iλ) B T (α iλ) A − T (α iλ)

(11.15)

• Merge step: Note that each node gets q predictions from each of the q child nodes. These are then merged to obtain a single prediction using: q

∑ Pˆ

Xˆ (λ | λ −) = Pˆ (λ | λ −)

−1

(λ | α iλ) Xˆ (λ | α iλ)

(11.16)

i =1

 −1 Pˆ (λ | λ −) = (1 − q) PX (λ) + 

q

∑ i =1

 Pˆ (λ | α iλ)  

−1

−1

(11.17)

• The upward sweep terminates when the recursion of update, predict and merge reaches the root node and we obtain the estimate Xˆ (0) = Xˆ (0 | 0 +) . • Downward sweep: the information is propagated downward through a smoothing process after the upward sweep is complete. The estimators are:

[

Xˆ (λ) = Xˆ (λ | λ +) + J (λ) Xˆ ( γλ) − Xˆ ( γλ | λ +)

[

]

]

(11.18)

Pˆ (λ) = Pˆ (λ | λ +) + J (λ) Pˆ ( γλ) − Pˆ ( γλ | λ +) J T (λ)

(11.19)

J (λ) = Pˆ (λ | λ +) F T (λ) Pˆ −1 ( γλ | λ +)

(11.20)

where

Equations 11.18 and 11.19 indicate that the estimate at a particular node in the downward sweep is equal to the sum of its estimate in the upward sweep and the difference in the © 2003 by CRC Press LLC

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estimates of the parent node in the downward and upward sweep (value in the brackets) weighted by a coefficient. Notice that: • Calculations on every node are performed once on each of the upward and downward sweeps. • Missing observations at any node are handled easily by setting the updated estimates to be the same as the predicted value. • The estimation errors are computed at each node during the upward and the downward pass. • Nonhomogeneities are handled easily by specifying parameters that vary from one node to another. We will take advantage of this in developing the conditional simulation model. • This method considers random field frozen in time and no temporal evolution is involved. The model considers only variability across scales.

III. MULTIPLE SCALE CONDITIONAL SIMULATION A. MODEL FORMULATION The multiple scale random field X(λ) admits several realizations. The true field X0(λ) is one of such realizations that is commensurate with the modeling objective, such as preservation of mean and neighborhood dependence (the covariance in a two-dimensional random field). When measurements Y(λi) are available for selected λi, (i = 1, …, M), the objective of the estimation algorithm is to ˆ λ) for all λ using the observations, thus incorporating their measureobtain an estimated field X( ment uncertainty and the neighborhood dependence. This neighborhood dependence can take into account the spatial proximity when considering a fixed scale, or the proximity in scale when considering a fixed location, or both. Our objective in multiple scale simulation is to generate equiprobable fields {Xs(λ)}s that exhibit similar dependence in space and scale. The real and the simulated realizations, X0(λ) and Xs(λ), respectively, of X(λ) differ from each other at given locations but preserve the neighborhood dependence. From the infinitely many realizations of {Xs(λ)}s, the objective of conditional simulation is to choose that field Xc(λ) whose values at the measurement locations are as close to the estimated values as possible (exact when there are no measurement errors). The conditionally simulated and the real fields have the same cluster of large and small values. Let us first consider the standard approach used in geostatistics13 for the generation of a conditionally simulated two-dimensional random field. The true field X0(λ), where λ now denotes locations in a two-dimensional space, can be broken into the optimally estimated Xˆ 0 (λ) (using kriging14,15) and an orthogonal error term [ X0 (λ) − Xˆ 0 (λ) ] , i.e., X0 (λ) = Xˆ 0 (λ) + [ X0 (λ) − Xˆ 0 (λ) ]

(11.21)

Because Xˆ 0 (λ) is estimated and therefore known for all λ, it suffices to substitute [ X0 (λ) − Xˆ 0 (λ) ] by a field that is isomorphic to it; that is, it has the same covariance. This is easily accomplished by kriging a simulated field Xs(λ)with values retained only at spatial locations where data are available and then taking the difference between the kriged field and the simulated field, i.e., by obtaining [ X s (λ) − Xˆ s (λ) ] . The error term [ X s (λ) − Xˆ s (λ) ] is isomorphic to [ X0 (λ) − Xˆ 0 (λ) ] and independent of Xˆ 0 (λ) . The unconditionally simulated field Xs(λ) can be generated through any number of techniques such as the turning bands method.23 We can, therefore, generate our conditionally simulated field as © 2003 by CRC Press LLC

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X c (λ) = Xˆ 0 (λ) + [ X s (λ) − Xˆ s (λ) ] .

(11.22)

This field is now known for all locations λ. Every new instance of the simulation field Xs(λ) gives a new Xc(λ) for the same estimated field Xˆ 0 (λ) , thereby giving several equiprobable fields. One can think of the above procedure more abstractly. Whereas the estimated field Xˆ 0 (λ) is the projection of the process X(λ) onto the space spanned by information available from the measurements, the error term [ X s (λ) − Xˆ s (λ) ] provides a modeled projection of X(λ) onto the corresponding null space. The conditional simulation therefore adds to the range space the unknown null space information by generating it through a model. Hence, the key for conditional simulation is to find a model for the error process. The modeling of the error process is done implicitly in the procedure described above using kriging for two-dimensional random fields. For our case of multiple scale processes characterized through a tree structure, we can adopt the same idea provided we can generate the error term [ X s (λ) − Xˆ s (λ) ] through a model that preserves the same neighborhood dependence as [ X0 (λ) − Xˆ 0 (λ) ] . ˆ λ | λ+) denotes the optimal Recall that in the multiple scale Kalman filtering algorithm, X( estimate at node λ conditioned on measurements on node λ and its descendents during the upward ˆ λ | λ+) denotes the corresponding error variance. Also X( ˆ λ) and P( ˆ λ) denote the pass, and P( optimal smoothed estimates during the downward pass and the corresponding error variances, respectively. Define the estimation error as

[

]

X˜ (λ) = X (λ) − Xˆ (λ) .

(11.23)

Pˆ (λ) = E {X˜ (λ) X˜ T (λ)}

(11.24)

Therefore by definition

ˆ λ) obeys Equation 11.19. where P( ˜ λ) that reproduces Equation 11.19 can indeed be developed and cast as a A model for X( multiple scale model of the form in Equation 11.1.16 Without going through the technicalities of the derivation, consider the model X˜ s (λ) = J (λ) X˜ s ( γλ) + G(λ)W (λ)

(11.25)

where X˜ s (λ) denotes the simulated error process and W(λ) is a white noise process independent ˆ λ) satisfies Equation 11.19 provided of X˜ s (0) . Using Equation 11.4, it is easy to verify that P(

[

]

G(λ)G T (λ) = Pˆ (λ | λ +) I − F T (λ) Pˆ −1 ( γλ | λ +) F (λ) Pˆ (λ | λ +) .

(11.26)

The above can be alternatively written as sum of two positive definite matrices17 G(λ)G T (λ) = [1 − J (λ) ]P(λ | λ +)[1 − J (λ) ] T + J (λ)Qˆ (λ) J T (λ).

(11.27)

ˆ λ) is a positive definite matrix (see Equation 11.15). G(λ) can then be obtained using where Q( the Cholesky decomposition of the RHS. Therefore, Equation 11.25 provides the necessary model for the error process. The conditionally simulated field is simply © 2003 by CRC Press LLC

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X c (λ) = Xˆ (λ) + X˜ s (λ)

(11.28)

X c (λ) = Xˆ (λ) + [J (λ) X˜ s ( γλ) + G(λ)W (λ) ].

(11.29)

or

Equations 11.28 and 11.29 will henceforth be referred to as the multiple scale conditional simulation model. By recognizing that when no measurements are available Pˆ (λ | λ +) = PX (λ) and Pˆ ( γλ | λ +) = PX ( γλ) , it is easy to verify that J(λ) = A(λ) and G(λ) = B(λ) and the model of the error X˜ s (λ) is exactly the same as that of X(λ). This is intuitive because, when there is lack of measurements, the unconditional model simulation provides the best characterization of the process. We should expect that the error term X˜ s (λ) will be large in magnitude for nodes with no measurements because the estimation errors for these nodes will be large. Consequently, the error field X˜ s (λ) will be nonhomogeneous even if X(λ) is homogeneous but sparsely sampled. The multiple scale model (Equation 11.25) enables us to incorporate this easily by specifying a different J(λ) for each λ that can be computed. Notice that the above algorithm is general and applicable to a wide range of multiple scale models as determined by the model parameters A(λ) and B(λ).

B. APPLICATION

TO

FRACTIONAL BROWNIAN RANDOM FIELDS

In order to assess the applicability of the estimation and simulation techniques discussed above we apply them to a simulated fractional Brownian field. Fractional Brownian models have found applications in a variety of physical processes. For example, log conductivity fields for groundwater flow3 and near-surface soil-moisture distribution18 are often described by such models. Due to their generality, the results presented in this section are applicable to all such scenarios where fractional Brownian models are useful. Equation 11.1 provides a very general modeling framework. It is possible to choose A(λ) and B(λ) such that Equation 11.1 leads to a fractional Brownian model. Consider the case where the state is a scalar and the model depends only on the resolution index m(λ) of node λ (m(λ) = 0 for the root node and increases sequentially as we go down the tree) and not on locations i and j. Then a canonical fractional Brownian model can be written as X (λ) = X ( γλ) + B0 Θ m (λ )W (λ)

(11.30)

where B0 is a constant that characterizes the total variance of the process and Θ depends on the fractal dimension or spectral decay rate.19,20 For a two-dimensional process with spectrum 1/fµ + 1, Θ is given by Θ=4

−

µ 2

(11.31)

For example, µ = 2 corresponds to Brownian field with Hurst exponent H = 1/2. The validity of this result rests on wavelet theory where it is shown that wavelet coefficients of a fractional Brownian process are approximately uncorrelated with the variance decaying as a power law as a function of scale.21 We first discuss some parameter estimation issues for the model (Equation 11.30). We then demonstrate the estimation and conditional simulation results using sparsely sampled synthetic fractional Brownian fields.

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Parameter estimation for the model (Equation 11.30) can be obtained using a maximum likelihood or a method of moments technique. The maximum likelihood technique for the multiscale model was developed by authors of Reference 22. It maximizes the likelihood function obtained from a set of noisy measurements assuming that the data correspond to the particular multiscale model. For the fractal model (Equation 11.30), where we need to estimate the two parameters B0 and µ, it is found that the method gives a set of combinations that have approximately the same large likelihood.20 Also, the estimation becomes computationally intensive with increasing number of scales. For this work we use a simpler method of moments technique. Using Equation 11.4 for the model (Equation 11.30), we can write 2 m (λ )  2 ) 2  Θ (1 − Θ PX (λ) = PX (0) + B0   2 1−Θ  

(11.32)

If we know the variance σ2(m(λ)) of the process at scale m(λ), and ignoring PX(0), which is the prior uncertainty in the estimation of the global mean state or the root node, we see that 2 m (λ )  2 ) 2  Θ (1 − Θ σ 2 ( m(λ)) = B0  . 2 1−Θ  

(11.33)

The parameter µ, which completely determines Θ, can be estimated through the decay rate of the spectrum or through the slope of the log–log plot of variance at different scales vs. scale. We use the latter, i.e., if log σ 2 ( m(λ)) = 2α log m(λ) + constant

(11.34)

then µ = 2α + 1. For a two-dimensional process α is related to the fractal dimension D as α = 3 – D. Once the estimated values µ, and consequently Θ is determined, Bˆ0 can be obtained using Equation 11.33. For the measurement model (Equation 11.8), we use C(λ) = 1 for all λ. The parameter PX(0) can be specified as an arbitrary large value or evaluated as the uncertainty in the estimation of the global mean state. The usual estimator for the mean has an uncertainty 2 2 2 2 given as (σ X + σ e ) / M where σ X and σ e are the variances of the process and the measurement errors, respectively. We use this value to specify the parameter PX(0). An isotropic fractional Brownian field on a 64 × 64 grid with fractal dimension D = 2.7 and 2 variance σ X = 3.3 was generated using a spectral technique.24 The estimated parameters for the ˜ = 0.33 and Bˆ = 5.2 . The synthetic field is shown in Figure 11.4a. generated field are µˆ = 1.6 , Θ 0 Figure 11.4b shows a randomly subsampled field where only 10% of the original points of data shown in Figure 11.4a are retained. Estimated field obtained through the multiple scale estimation algorithm using the subsampled data is shown in Figure 11.4c. The estimated field remarkably captures the large scale variability of the original field. The corresponding estimation errors are shown in Figure 11.4d. The large scale average field obtained by averaging the original data is shown in Figure 11.5a. Estimated field at the finest scale by combining this large scale average with the sampled field (Figure 11.4b) is shown in Figure 11.5b. The estimated field shows a little improvement over that obtained using just the fine scale subsampled data (compare with Figure 11.4c). Notice from Figure 11.4d that the estimation error is nonhomogeneous with large values at locations with no measurements. The parameter J(λ) used for conditional simulation, which is a function of these errors, is now different for each node. This provides an example where the model is useful for fields whose parameters vary at different locations at a particular scale.

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FIGURE 11.4 (See color insert following page 144.) Estimation from subsampled fractional Brownian field using the multiple scale algorithm: (a) original field on a 64 × 64 grid (σ = 1.8); (b) subsampled fractional Brownian field (σ = 1.9); (c) estimated field using the subsampled field (σ = 1.58) and (d) estimation errors (σ = 0.19).

A conditionally simulated field using just the subsampled field at the fine scale (Figure 11.4b) is shown in Figure 11.5c. Notice the large variability in regions with missing observations captured by this field. Figure 11.5d shows a conditionally simulated field using data at the two levels. The improved representation of the variability in the conditionally simulated fields is evident. The above examples serve to illustrate the following points: 1. The multiple scale model provides an elegant estimation technique when sparsely sampled data are available at a single scale or in conjunction with data at multiple scales. The estimated field, however, is smoother than the original field. 2. The conditionally simulated field provides a better representation of the variability. Significant improvement is achieved when measurements at multiple scales are incorporated.

IV. CONCLUSIONS Estimation of a process and characterization of its variability using measurements at multiple scales is a problem of considerable interest in several areas of geophysics. Elegant estimation techniques, based on multiresolution trees, to combine measurements across scales were developed by the authors of References 4 through 7. Due to the nature of the optimization, the estimation is a smoothing process that may not provide a good representation of the variability, particularly in regions where there are no observations. A complementary conditional simulation technique is developed that allows us to construct synthetic fields representative of the intrinsic variability of the process. This uses a multiple scale model for the simulation of the estimation error field whose

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FIGURE 11.5 (See color insert following page 144.) Illustration of multiple scale conditional simulation using subsampled fractional Brownian field: (a) large scale average field; (b) estimated field using the subsampled (see Figure 11.4b) and large scale average field (σ = 1.67); (c) conditionally simulated field using only the subsampled field (σ = 1.82) and (d) conditionally simulated field using the subsampled and large scale average field (σ = 1.74).

parameters can be explicitly computed. The conditional model reduces to the unconditional simulation model in the absence of measurements, as should be expected. A challenging problem is that of formulating a particular process model into a multiple scale framework and determining the model parameters A(λ) and B(λ). However, once these are established, the formulations presented in this chapter are directly applicable. A simple example using synthetic fractional Brownian motion fields, which has applications in a wide variety of fields, demonstrates the models utility for scalar valued fields. However, the formulations are general and adaptable to covarying vector valued multiple scale random fields.

REFERENCES 1. Jackson, T.J. and Le Vine, D.E., Mapping surface soil moisture using an aircraft-based passive microwave instrument: algorithm and example, J. Hydrol., 184 (1/2), 1996. 2. Kumar, P., A multiple scale state-space model for characterizing subgrid scale variability of nearsurface soil moisture, IEEE Trans. Geosci. Remote Sensing, 37(1), 182–197, 1999. 3. Brewer, K.E. and Wheatcraft, S.W., Including multi-scale information in the characterization of hydraulic conductivity distribution, in Foufoula-Georgiou, E. and P. Kumar (Eds), Wavelets in Geophysics, 213–248, Academic Press, 1994. 4. Basseville, M., Benveniste, A., and Willsky, A.S., Multiscale autoregressive processes, Part 1, IEEE Trans. Signal Process., 40, 1915–1934, 1992a. 5. Benveniste, A., Nikoukhah, R. and Willsky, A.S., Multiscale system theory, IEEE Trans. Circuits Systems — 1: Fundamental Theory Appl., 41(1), 2–15, 1994.

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6. Chou, K.C., Willsky, A.S., Benveniste, A., Multiscale recursive estimation, data fusion, and regularization, IEEE Trans. Auto. Control, 39(3), 464 – 478, 1994a. 7. Chou, K.C., Willsky, A.S., and Nikoukhah, R., Mutiscale systems, Kalman filters, and Riccati equations, IEEE Trans. Auto. Control, 39(3), 479–492, 1994b. 8. Journel, A.G., Geostatistics for conditional simulation of ore bodies, Econ. Geol., 69, 673–687, 1974. 9. Basseville, M., Benveniste, A., and Willsky, A.S., Multiscale autoregressive processes, Parts 1 and 2, IEEE Trans. Signal Process., 40, 1935–1954, 1992b. 10. Chou, K.C., A stochastic modeling approach to multiscale signal processing, Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, May, 1991. 11. Verghese, G. and Kailath, T., A further note on backward markovian models, IEEE Trans. Info. Theory, 25(1), 121–124, 1979. 12. Rauch, H.E., Tung, F., and Striebel, C.T., Maximum likelihood estimates of linear dynamic systems, AIAA J., 3(8), 1445–1450, 1965. 13. Deutsch, C.V., and Journel, A.G., GSLIB: Geostatistical Software Library and User’s Guide, Oxford University Press, pp. 340, 1992. 14. Journel, A.G., Fundamentals of Geostatistics in Five Lessons, American Geophysical Union, Washington, D.C., pp.40, 1989. 15. Journel, A.G. and Huijbregts, Ch. J., Mining Geostatistics, Academic Press, Boston, MA, pp. 600, 1978. 16. Luettgen, M.R. and Willsky, A.S., Multiscale smoothing error models, IEEE Trans. Auto. Control, 40(1), 173–175, 1995b. 17. Luettgen, M.R., Personal communication, 1996. 18. Rodriguez-Iturbe, I., Vogel,G.K., Rigon, R., Entekhabi, D., Castelli, F., and Rinaldo, A., On the spatial organization of soil moisture fields, Geophys. Res. Lett., 22(20), 2757–2760, 1995. 19. Luettgen, M.R., Image processing with multiscale stochastic models, Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, May, 1993. 20. Fieguth, P.W., Application of multiscale estimation to large scale multidimensional imaging and remote sensing problems, Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, May, 1995. 21. Wornell, G., A Karhunen-Loeve like expansion for 1/f processes, IEEE Trans. Inf. Theory, 36, 859–861, 1990. 22. Luettgen, M.R. and Willsky, A.S., Likelihood calculation for a class of multiscale stochastic models, with application to texture discrimination, IEEE Trans. Image Process., 4(2), 1995a. 23. Tompson, A.F.B., Abanou, R., and Gelhar, L., Application and use of the three-dimensional tuning bands random field generator: single realization problems, MIT Dept. of Civ. Eng., Report Number 313, 1987. 24. Voss, R.F., Fractals in nature: from characterization to simulation, in Science of Fractal Images, H.O. Peitgen and D. Saupe, Eds., Springer-Verlag, New York, 1988.

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12

Effects of Topography, Soil Properties and Mean Soil Moisture on the Spatial Distribution of Soil Moisture: A Stochastic Analysis D.-H. Chang and S. Islam

CONTENTS I. II. III. IV.

Introduction...........................................................................................................................193 Methodology.........................................................................................................................195 Stochastic Analysis of Steady-State Unsaturated Flow.......................................................196 Preliminary Analysis ............................................................................................................198 A. Evaluation of the Soil Moisture Variance and Its Covariance with Topography and Soil Properties.....................................................................................................198 B. One-Dimensional Analysis ........................................................................................202 V. Applications and Discussion Based on Preliminary Analysis.............................................205 A. General Discussion of Relevant Parameters .............................................................205 B. Relationship between Soil Moisture Distribution and Topography .........................207 C. Relationship between Soil Moisture Distribution and Soil Properties.....................209 D. The Impact of Mean Soil Moisture on the Soil Moisture Variation ........................211 E. Comparison with Previous Studies ...........................................................................217 VI. Conclusions...........................................................................................................................218 VII. Appendix A: Derivation of the Large-Scale and Perturbation Models...............................219 VIII. Appendix B: Solving the Perturbation Equations Using Spectral Techniques ...................222 References ......................................................................................................................................224

I. INTRODUCTION For each kilogram of water on Earth, only one milligram is stored as soil moisture. Yet this miniscule amount of water exerts significant control over various hydrological, ecological and meteorological processes ranging from boundary layer dynamics to the global water cycle. There is a growing consensus that a unified approach is necessary to monitor, characterize, and model distribution of soil moisture over a range of scales. Analysis of several field-measured soil moisture data1–13 has been carried out to characterize and model the spatial variation of soil moisture. Such analyses provide fundamental knowledge about the dependencies between soil moisture distribution and environmental factors. Several recent

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studies suggest that topography, soil physical properties, vegetation, and climate are the key environmental factors that control soil moisture variations over large scales.11,14 A considerable amount of apparent contradiction, however, appears in the literature about the influence of these factors. With respect to topography, several studies6,11 noted the influence of topography on soil moisture variability. Charpentier and Groffman,8 Niemann and Edgell9 and Ladson and Moore7 found no obvious relationship between topography and soil moisture. Hawley et al.,6 however, reported that the variation of soil types has minimum impact on the soil moisture variation. Rodriguez-Iturbe et al.,15 on the other hand, suggested that the spatial organization of soil moisture is a consequence of that of soil properties. Chang and Islam16 illustrated the capability of inferring the spatial map of soil properties through remotely sensed brightness temperature maps, in which soil moisture is strongly related with brightness temperature.17 Several investigations have suggested that the variance of soil moisture increases with increasing mean soil moisture.1–5,10,11 Charpentier and Groffman,8 however, found no systematic relationship between the variance of soil moisture and mean soil moisture. Famiglietti et al.12 reported that variance of soil moisture decreases with increasing mean soil moisture. In order to perform a reliable statistical analysis that can decipher dependencies between soil moisture and other environmental factors, one needs a large number of samples under different environmental conditions. Due to the time and economic constraints related to field measurement and remote sensing data, inferred relationships between soil moisture and other environmental factors are often site specific. It is possible that the reported controversies or apparent inconsistencies in the interpretation of soil moisture data come from certain site-specific parameters, for example, the differences in mean soil-textural type and the correlation scale of topography, among others. To overcome the problem of limited information about the spatial soil variability and topography, a stochastic framework has been suggested. This approach assumes that spatial variability of an attribute (e.g., soil physical properties or topography) is a realization of a random field. The large-scale model structure is then derived by averaging local governing flow equation over the ensemble of realization of the underlying soil properties or topography random field. Mantoglou and Gelhar18 presented a stochastic methodology to derive a large-scale model of transient unsaturated flow in spatially variable soil formations. Yeh and Eltahir14 formulated the stochastic problem of water flow in unsaturated zone such that topography becomes the forcing term for the movement and distribution of soil moisture. In reality, soil physical properties and topography will control spatial variations of soil moisture over large areas. It is conceivable under certain situations that topographic control will dictate the distribution of soil moisture while in some other cases soil physical properties will be the key factor that controls variations of soil moisture. A primary objective of this study is to develop a general stochastic methodology that explicitly considers the effects of spatial variability of soil physical properties and topography. A product of this chapter will be a theoretical framework, with tractable analytical solutions, that relates the spatial distribution of soil moisture to statistical properties of soil physical properties and topography. Clearly, the problem is quite complex and several assumptions will be needed to arrive at analytically tractable results. Nevertheless, a general stochastic framework would allow us to better understand the spatial organization of soil moisture with explicit considerations for heterogeneity in soil physical properties and topography. In particular, such a framework would help us explore, for example, under what conditions (1) the relationship between soil moisture and topography will be enhanced or reduced, (2) the relationship between soil moisture and soil properties will be enhanced or reduced, and (3) the variation of soil moisture will decrease (or increase) with increasing mean soil moisture. We will examine the effects of topography, soil properties and mean soil moisture on the soil moisture distribution at the root zone of shallow unsaturated soil. The interaction between root zone and groundwater table as well as the effects from other environmental factors, such as vegetation, will not be addressed in this study. This chapter is organized as follows. In the second section, a general stochastic framework for characterizing the soil moisture distribution is outlined. © 2003 by CRC Press LLC

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In the third, the general problem considered in this study is formulated and the perturbation method is applied to solve the problem. In the fourth section, a preliminary analysis based on onedimensional analytical solution of the problem is provided. The discussion and conclusions are included in the last two sections.

II. METHODOLOGY The general methodology to characterize soil moisture distribution is summarized schematically in Figure 12.1. The problem is formulated as a soil-water continuum equation including source and sink terms. The input is statistical information about topography, soil type, and climatic variables (precipitation and evaporation). The water source is from effective precipitation, while evaporation — formulated as functions of precipitation and mean soil moisture — serves as the sink of mean soil moisture. The perturbation method is applied to solve the soil moisture dynamics. The resulting model formulates the spatial distribution of soil moisture as a function of topography, soil properties, and mean soil moisture. We will compare and contrast the modeled results with previous studies based on analysis and interpretation of field measurements. In particular, we will focus on the effects of topography, soil properties and mean soil moisture on the spatial distribution of soil moisture. We are not trying to render a comprehensive study that includes all related effects from these three environmental factors. Rather, this study focuses on the following aspects. For soil properties, we are interested in the soil parameters needed to characterize the soil-water characteristic curve. Other properties such as soil color and the existence of macroporosity and organic matters are not discussed in this study. For topography, we focus on the effects of relative elevation. Topographic indices, such as aspects, specific contributing areas Stochastic Analysis 1. Formulate the soil-water continuum equation relating moisture flow with topography, soil properties, precipitation, and evaporation. 2. Decompose the continuum equation into large-scale (spatial mean) and small-scale (perturbation) equations. 3. A spectral theory is applied to convert the small-scale equation into statistical expressions that relate the distribution of soil wetness with that of topography, soil properties, and mean soil moisture.

Observed Data Analysis Observed relationship between the distribution of soil moisture and that of topography, soil properties, and information about mean soil moisture.

Validation Compare the proposed statistical expressions with observed data analysis.

Resulting Model Characterization of the spatial distribution of soil wetness as a function of topography, soil properties, and mean soil moisture.

FIGURE 12.1 Schematics of the proposed stochastic framework.

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and curvature, are beyond the scope of this study. Furthermore, we are interested in the following spatial scales: (1) horizontal scale smaller than 10 km so that the heterogeneous precipitation is assumed to have minor impacts at such scales, and (2) vertical spatial scale (soil depth) of upper 1 m so that we assume that the vertical moisture below 1 m is much smaller than that in upper depth. In this study the soil moisture content is represented by normalized soil moisture (η), defined as η = (θ – θr)/(θs – θr) where s and r represent the saturated and residual values of volumetric soil moisture (θ), respectively.

III. STOCHASTIC ANALYSIS OF STEADY-STATE UNSATURATED FLOW In this section, we develop a stochastic framework for characterizing the spatial distribution of soil moisture. The proposed approach assumes that the spatial variables (soil properties, unsaturated soil hydraulic properties and elevation) are realizations of a two-dimensional, crosscorrelated, second-order stationary random field. Similar assumption has also been made by other stochastic analyses.14,18 We begin with a Richardson-type equation derived by Yeh and Eltahir.14 They have combined the continuum equation with Darcy’s law to develop the equation of moisture flux in shallow soil in a two-dimensional domain:

D

∂θ ∂  ∂( −ψ + z )  =D K −s+R =0 ∂t ∂x i  ∂x i 

(12.1)

where the direction xi (i = 1,2) represents the horizontal coordinates, D is the depth of root zone with the assumption that the soil below such depth is completely impermeable, θ is volumetric soil moisture, K is unsaturated hydraulic conductivity, ψ is hydraulic suction of water, z is elevation (positive upward), s is evaporation flux, parameterized as s = βK by assuming that evaporation is proportional to unsaturated hydraulic conductivity where β is equivalent to the vertical hydraulic gradient near land surface, and R is effective rainfall defined as the portion of rainfall infiltrating into soil. The typical ranges of above relevant parameters will be discussed in Section V. In the following, a steady-state case is considered. This assumption is reasonable when we consider a long time scale. Similar to Yeh and Eltahir14 we also assume that the parameters D, β, and R are homogeneous in space. A key difference between Yeh and Eltahir14 and this study is the inclusion of effects of soil heterogeneity. Three unsaturated soil hydraulic parameters (K, θ, ψ) of Equation 12.1 are correlated with each other through the so-called soil hydraulic functions (i.e., K–θ–ψ relationship). A significant knowledge base exists in the soil physics and groundwater literature that focuses on the development and refinement of methods to describe such K–θ–ψ relationship. Most of them, however, are expressed in a highly nonlinear fashion and consequently analytical result is extremely difficult to obtain when such relationships are used with other modeling approaches. As an alternative, a linear approximation of ψ–θ relationship,18 θ = θ s − Cψ

(12.2)

and a quasilinear approximation of ψ–K relationship,19 ψ = B ln( K / K s )

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(12.3)

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have been applied,14,18,20 where C is the specific capacity of soil moisture, B is the thickness of capillary fringe and Ks is the saturated hydraulic conductivity. Such approximations of soil hydraulic functions provide a dramatic simplification for various modeling applications. Although the gains in simplification are achieved at the cost of accuracy, the reality is that even the most complicated soil hydraulic functions cannot precisely address the behavior of soil-moisture flow because so much is unknown and uncertain in the unsaturated porous media. By converting the volumetric soil moisture into normalized soil moisture, Equation 12.3 becomes: η = 1 − Csψ

(12.4)

where Cs = C/(θs – θr) represents the specific capacity of normalized soil moisture. By combining Equations 12.3 and 12.4 to eliminate ψ, one can derive the K–η relationship as below: η = 1 + A ln( K / K s )

(12.5)

where A, a dimensionless soil parameter, is defined as the product of B and Cs. Equation 12.5 shows a power-law relationship between normalized soil moisture and hydraulic conductivity. Similar power-law functions have also been reported by Gardner et al.21 and Campbell.22 Note that Equations 12.1 through 12.3 involve three soil parameters (A, B, and Ks), while the parameter Cs can be termed as Cs = A/B. Usually, these parameters are considered independent of unsaturated soil hydraulic properties (ψ, K and θ) and assumed to vary only as a function of soil type. Mantoglou and Gelhar18 argue that such constant assumption is valid for an intermediate range of ψ values provided local hysteresis is relatively small. In this study, the linear-type expressions of K–ψ relationship (Equation 12.3) and K–η relationship (Equation 12.5) will be applied to Equation 12.1 in order to represent the soil-water flow equation with a single dependent variable η. Following Mantoglou and Gelhar,18 a stochastic analysis is applied to the steady-state unsaturated flow equation (Equation 12.1), where the soil hydraulic functions are described by Equations 12.3 through 12.5. The proposed stochastic analysis involves a two-step approach: (1) derivation of the large-scale and perturbation model (see Appendix A), and (2) solution of the perturbation equations using spectral techniques (see Appendix B). Consequently, we derive the following statistical relationship that characterizes the spatial distribution of normalized soil moisture as a function of topography, soil properties and mean soil moisture: σ 2η = VK sK s σ 2K s + VBB σ 2B + Vzz σ 2z + V AAσ 2A

(

+2 V AB σ Aσ B + V AKs σ Aσ K s + V Az σ Aσ z + VBKs σ B σ K s + VBz σ Aσ z + VK sz σ K s σ z

© 2003 by CRC Press LLC

)

(12.6a)

σ ηA = X AAσ 2A + X BAσ Aσ B + X K s Aσ Aσ K s + X zAσ Aσ z

(12.6b)

σ ηB = X AB σ Aσ B + X BB σ 2B + X K sB σ K s σ B + X zB σ z σ B

(12.6c)

σ ηK s = X AKs σ Aσ K s + X BKs σ B σ K s + X K sK s σ 2K s + X zKs σ z σ K s

(12.6d)

σ ηz = X Az σ Aσ z + X Bz σ B σ z + X K sz σ K s σ z + X zz σ 2z

(12.6e)

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To simplify the notation, here we use the variables u and v to represent Ks, A, B or z. In Equation 12.6, σ 2u and σu represent the variance and standard deviation of variable u, respectively, σuv represents the covariance between variable u and v, and the scalars V uv and X uv are defined as:

Vuv = Vvu =

∫∫

e

(

(

)WW

)W

Suv k x , k y

Suv k x , k y

i k x + ky τ

u

σ uσ v

v

−∞→∞

X uv =

∫∫ e

(

i k x + ky τ

(

σ uσ v

u

−∞→∞

x

) dk dk x

) dk dk

y τ =0

(12.7) y τ =0

where x, y are spatial coordinates, kx, ky are wave numbers in x and y directions, i = −1 , Suv is the spectral density function between u and v, and WU is a function of large-scale soil and topographic properties ( u ), wave numbers, effective rainfall (R), root zone soil depth (D) and vertical hydraulic gradient (β): WK s = W0 =

    A (W − 1), WB =  1 −B η W0 , Wz = − BA W0 , WA = − 1 −A η  , Ks 0

(k

(k 2 x

2 x

+ k 2y

+ k 2y

)

)

β = , β1 = BD + β1

(12.8)

R0 Kse

−

1− η A

, R0 = R / D B

IV. PRELIMINARY ANALYSIS A general stochastic theory for characterizing the spatial distribution of soil moisture was provided in the previous section. The resulting model is a function of the variance of soil properties and the variance of topography as well as the covariance between topography and soil properties. Evaluation of the resulting model (Equation 12.6) requires knowledge of the covariance and cross-covariance functions of soil properties (A, B, Ks) and topography (z). In this section, we provide some analytical results of the proposed stochastic model (Equation 12.6) based on the frequently used exponential covariance functions23 in one-dimensional horizontal field. A comparison between our analytical results and those of Yeh and Eltahir14 will also be discussed.

A. EVALUATION OF THE SOIL MOISTURE VARIANCE TOPOGRAPHY AND SOIL PROPERTIES

AND ITS

COVARIANCE

WITH

Previous studies have used exponential covariance functions for the topographic14 and soil distribution.24 Here, we consider the one-dimensional unsaturated flow with explicit considerations for heterogeneous topography and heterogeneous soil properties. The covariance-spectrum pair for the exponential covariance function is expressed as: Ru (τ) σ 2u Su ( k) σ 2u

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 τ  = exp −   λu  =

(

λu

π 1 + λ2u k 2

)

(12.9)

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where the variable u represents Ks, A, B or z; τ is the lag, λ is the correlation scale and k is the wave number in the horizontal flow direction. The assignments of cross-covariance functions of soil properties and topography need attention. Since the work of Jenny,25 it has been a well-known fact in soil science that topography is an important factor in influencing soil distribution. There are virtually no systematic field studies that quantify cross covariance properties of topography and soil properties. To develop some insights regarding such relationships between topography and soil properties, here we examine the relationship among A, B, Ks and z for the following three extreme cases: (1) A, B, Ks and z are uncorrelated, (2) the A, B and Ks are perfectly correlated but the soil properties are uncorrelated with z and (3) A, B, Ks and z are perfectly correlated. Above, three extreme cases are selected in order to evaluate the relationship between σ 2η and σηu (u = A, B, Ks or z). Let ξ2, ζ2 and ω2 be the ratios of the variance of B, Ks and z to the variance of A, i.e., ξ2 = σ 2K s σ 2A ς 2 = σ 2B σ 2A

(12.10)

ω 2 = σ 2z σ 2A The cross-spectral density functions of (B, Ks and z) are then related to that of A by: Case 1: A, B, Ks, z uncorrelated SKsKs = ξ2 SAA SBB = ξ2 SAA SAKs = 0 SAB = 0 SBKs = 0 SzA = 0 SzKs = 0 SzB = 0 Case 2: A, B, Ks perfectly correlated but uncorrelated with z SKsKs = ξ2 SAA SBB = ξ2 SAA

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(12.11)

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SAKs = ξ SAA SAB = –ξ SAA SBKs = –ξζ SAA

(12.12)

SzA = 0 SzKs = 0 SzB = 0 Case 3: A, B, Ks and z perfectly correlated SKsKs = ξ2 SAA SBB = ζ2 SAA Szz = ω2 SAA SAKs = ξ SAA SAB = –ζ SAA

(12.13)

SBKs = –ξζ SAA SzA = s ω SAA SzKs = s ωξ SAA SzB = –s ωζ SAA where s is a switch with a value of either +1 or –1. The switch s = +1 represents that the study area contains more coarse-texture soil in higher elevation than that in lower elevation; s = –1 represents the opposite case. Such a switch is necessary to explicitly acknowledge the nature of correlation between soil physical properties and topography. For instance, if higher elevation has coarser-texture soil and lower elevation has finer-texture soil then the correlation between saturated hydraulic conductivity and topography is positive. On the other hand, if higher elevation has finertexture soil and lower elevation has coarser-texture soil then the correlation between saturated hydraulic conductivity and topography will be negative. It is commonly accepted in soil science that finer-texture soil often appears in low-lying places and better-drained soils are located in higher lying areas.25 Therefore, we will use s = +1 for subsequent calculation. Also note that finer-texture soils generally have smaller A and K s values but larger B values; coarser-texture soils have larger ( A , K s ) values but smaller ( B ).26 Consequently, the negative signs appear in SBKs and SAB. Applying Equations 12.11, 12.12 and 12.13 to Equation 12.6 yields the following results: © 2003 by CRC Press LLC

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Case 1: A, B, Ks, z uncorrelated

(

)

σ 2η = ξ2V˜K sK s + ς 2V˜BB + V˜AA σ 2A + Vzz σ 2z

(12.14a)

σ ηA = X˜ Aσ 2A

(12.14b)

σ ηB = ς 2 X˜ B σ 2A

(12.14c)

σ ηK s = ξ2 X˜ K s σ 2A

(12.14d)

σ ηz = X zz σ 2z

(12.14e)

Case 2: A, B, Ks perfectly correlated but uncorrelated with z

(

)

σ 2η = ξ2V˜K sK s + ζ2V˜BB + V˜AA − 2ξζV˜K sB + 2ξV˜K s A − 2ζV˜BA σ 2A + Vzz σ 2z

(

)

σ ηA = X˜ A + ξX˜ K s − ζX˜ B σ 2A

(12.15a)

(12.15b)

(

)

(12.15c)

σ ηK s = ξ X˜ A + ξX˜ K s − ζX˜ B σ 2A

(

)

(12.15d)

σ ηz = X zz σ 2z

(12.15e)

σ ηB = −ζ X˜ A + ξX˜ K s − ζX˜ B σ 2A

Case 3: A, B, Ks and z perfectly correlated

(

σ 2η = ξ 2 V˜Ks Ks + ζ 2 V˜BB + ω 2 V˜zz + VAA − 2ξζV˜Ks B + 2ξV˜Ks A

)

(12.16a)

σ ηA = X˜ A + ξX˜ Ks − ζX˜ B + ωX˜ z σ 2A

(12.16b)

−2ζV˜BA σ BA + 2ωξV˜Ks z − 2ωζV˜Bz + 2ωV˜zA σ 2A

(

)

(

)

(12.16c)

(

)

(12.16d)

(

)

(12.16e)

σ ηB = −ζ X˜ A + ξX˜ Ks − ζX˜ B + ωX˜ z σ 2A σ ηKs = ξ X˜ A + ξX˜ Ks − ζX˜ B + ωX˜ z σ 2A σ ηz = ω X˜ A + ξX˜ Ks − ζX˜ B + ωX˜ z σ 2A © 2003 by CRC Press LLC

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In Equations 12.14 through 12.16, the scalars V˜uv , X˜ u , Vzz and Xzz are defined as:

∫∫ e

V˜uv = V˜vu =

) W W SAA (k x , k y )dk dk u v x y 2

(

i kx + ky τ

σA

−∞→∞

X˜ u

∫∫

e

) W SAA (k x , k y ) dk dk u x y 2

(

i kx + ky τ

σA

−∞→∞

Vzz =

∫∫ e ∫∫

e

(

τ=0

) dk dk

(12.17)

(

) W2 z

(

) W Szz (k x , k y ) dk dk x y z 2

i kx + ky τ

Szz k x , k y

−∞→∞

X zz =

τ=0

σ 2z

x

y τ=0

i kx + ky τ

σz

−∞→∞

τ=0

where the variables u and v represent Ks, A, B or z. Several important features regarding the relationship between soil properties and topography may be identified from Equations 12.14 through 12.16. First, if A, B, Ks and z are uncorrelated (Case 1), variability in soil moisture is composed of individual variability of soil properties and topography (i.e., Equation 12.14a). In other words, variability of soil moisture can be viewed as a sum of four random variables (A, B, Ks and z). In such cases, covariance between soil moisture and the attributes (i.e., A, B, Ks or z) will be a function of those attributes only. For instance, covariance between soil moisture and topography (Equation 12.14e) is only a function of variability in topography. If soil properties are correlated among themselves but uncorrelated with topography (Case 2), variability in soil moisture is composed of individual variability of soil properties and topography as well as the covariance among soil properties (i.e., Equation 12.15a). Similarly, covariance between soil moisture and individual soil property (A, B or Ks) will be the results in Case 1, plus the terms with respect to the correlation among soil properties. However, for the covariance between soil moisture and topography, Case 1 and Case 2 yield the same result (i.e., Equations 12.14e and 12.15e). In other words, if topography is uncorrelated with soil properties, the correlation among soil properties has no influence on the relationship between topography and soil moisture. In the third case (A, B, Ks and z are perfectly correlated), variability in soil moisture is the result in Case 2, plus the terms with respect to the covariance between soil properties and topography. In such cases, the covariance between soil moisture and other attributes (A, B, Ks or z) shows the following relationship: σ ηA σ ησ A

=−

σ ηB σ ησ B

=

σ ηKs σ η σ Ks

=

σ ηz σ ησ z

(12.18)

In other words, in Case 3 the correlation coefficients between soil moisture and any attributes (A, B, Ks or z) are the same.

B. ONE-DIMENSIONAL ANALYSIS By substituting the covariance functions of soil parameter (SAA) and topography (Szz) in Equation 12.17 with an exponential type covariance model (Equation 12.9) and after integration, we obtain the analytical results of the scalars V˜uv , X˜ u , Vszz and Xszz as: © 2003 by CRC Press LLC

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 1 − η V˜AA =  A 

203

2

1/ 2  2  3/ 2  1 − η   g λ , A − 3gλ , A + 2  ˜ VBB =  B   2 g −1 2  λ,A  

(

)

2 g 3/ 2 − 3gλ1/,2A + 2   A 2 V˜Ks Ks =   1 − 1/ 2 + λ,A 2   K s   gλ , A + 1 2 gλ , A − 1  

(

)

2  3/ 2  1/ 2  A   gλ , A − 3gλ , A + 2  ˜ Vzz =   2   B   2 g −1 λ,A  

(

)

 1 − η 1  V˜Ks A = V˜AKs =  1 − 1/ 2   gλ , A + 1  Ks  

(1 − η)2  1  V˜BA = V˜AB = 1 AB  gλ1/,2A + 1  1 − η 1  V˜zA = V˜Az =  B   gλ1/,2A + 1 V˜Ks B = V˜BKs

 3/ 2  1/ 2 A (1 − η) gλ , A − 3gλ , A + 2 1   − 1/ 2 = 2 BK s  2 g − 1 gλ , A + 1   λ,A

(

)

 3/ 2  1/ 2 A 2  gλ , A − 3gλ , A + 2 1  V˜Ks z = V˜zKs = − 1/ 2 2 BK s  2 g − 1 gλ , A + 1  λ,A  

(

)

 3/ 2  1/ 2 A (1 − η)  gλ , A − 3gλ , A + 2  V˜Bz = V˜zB = − B2  2 g −1 2  λ,A  

(

 1 − η X˜ A = −  A   1 − η 1  X˜ B =  B   gλ1/,2A + 1

© 2003 by CRC Press LLC

)

(12.19)

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 A  1  X˜ Ks = −  1 − 1/ 2   K s   gλ , A + 1   A  1  X˜ z = −   1/ 2   B   gλ , A + 1  2  3/ 2  1/ 2  A  g − 3gλ ,z + 2  Vzz =    λ ,z 2   B   2 g −1 λ ,z  

(

)

 A  1  X zz = −   1/ 2   B   gλ ,z + 1 where gλ,A and gλ,A are dimensionless parameters defined as: gλ ,i = λ2i β1 , i = A or z

(12.20)

A computer code (not shown) in Mathematica was written and executed for the required calculation. In the following section, we will apply and discuss the proposed stochastic model based on the preliminary analysis described in this section. The results will be used to address the three main issues defined in the introduction of this chapter. Beside the three extreme cases discussed in Section IV.A, there are two particular cases of interest: (1) homogeneous soil properties and (2) homogeneous topography. The results of these two cases can be easily obtained by setting σ 2A = 0 , A = A , B = B and K s = K s for Case 1 and σ 2z = 0 for Case 2. The assumption of homogeneous soil properties has been made by Yeh and Eltahir14 in developing their stochastic model. Their two main results for characterizing the variation of soil moisture and the covariance between soil moisture and topography, respectively, are as follows:  g 3/ 2 − 3g1/ 2 + 2  λ ,z σ 2 σ = C  λ ,z  2 g −1 2  z λ ,z  

(12.21a)

 1  2 σ θz = −C 1/ 2  σz  gλ , z + 1 

(12.21b)

2 o

2

(

)

where C is the specific capacity of soil moisture, θ is the volumetric soil moisture and gλ,z is the same as Equation 12.20. Here we provide a brief comparison between our proposed stochastic model in the case of homogeneous soil properties with the stochastic model developed by Yeh and Eltahir.14 In the case of homogeneous soil properties, the three extreme cases are simplified into one single result: 2  3/ 2 g − 3gλ1/,2z + 2  2 A σ 2η =    λ ,z σ  B  2 g − 1 2  z λ , z  

(

© 2003 by CRC Press LLC

)

(12.22a)

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A  1  2 σ ηz = −   1/ 2 σ  B   g + 1 z  λ ,z 

205

(12.22b)

We note here that the results in Equation 12.22 are exactly the same compared to those of Yeh and Eltahir14 (Equation 12.21) given the following relationship: σ θ2

σ 2η =

(θ s − θ r )

σ ηz =

2

σ θz θs − θr

(12.23)

A = Cs B Cs =

C θs − θr

V. APPLICATIONS AND DISCUSSION BASED ON PRELIMINARY ANALYSIS A. GENERAL DISCUSSION

OF

RELEVANT PARAMETERS

A general stochastic theory for characterizing soil moisture distribution as a function of soil physical properties and topography is developed in this study. The relevant parameters associated with the steady-state horizontal distribution of soil moisture can be categorized into four groups: (1) the correlation scales of soil properties (λA) and relative elevation (λZ), (2) large-scale soil properties ( A , B and K s ), (3) large-scale water source and mean soil moisture (R0, η ), and (4) variability of relative elevation ( σ 2z ) and soil properties ( σ 2A , σ 2B and σ 2K ). In this subsection we discuss s the physical meaning and the typical range of these parameters. First, the correlation scales, λA and λZ, are related to the geologic process and undulation of soil and topographic fields. Previous study14 has shown that smooth undulating fields of relative elevation (large λZ) tend to result in smaller variability of soil moisture. Similarly, large correlation scale of soil properties will have smaller variability in moisture distribution. It is possible that in a study area with large λZ the soil moisture distribution will be controlled by λA, provided λAλA << λZλz. To our knowledge, the comparative effects of λA and λZ on soil moisture distribution have not drawn much attention. The relative scale of λA and λZ, however, is likely to have significant impact on resulting soil moisture distribution. A typical value of λA is about 1 m18 and the typical range of λZ is 102 to104 m.14 In the following three subsections, different values of both correlation scales will be used to evaluate the comparative effects of λA and λZ on spatial distribution of soil moisture. Second, parametric groups include the large-scale soil properties, namely A , B and K s . In general, finer-texture soils have smaller A and K s values but larger B values; and coarser-texture soils have larger A and K s values but smaller B .26 Philip27 suggested that if the thickness of capillary fringe ( B ) were large (i.e., finer-texture soil) the gravity flow introduced by elevation gradient would be less obvious. Thus, in an area mainly composed of finer-texture soils, the heterogeneous soil properties may have more control than topography on the distribution of soil moisture. Therefore, large-scale distribution of soil properties must be carefully considered in examining the soil moisture distribution. © 2003 by CRC Press LLC

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TABLE 12.1 Soil Properties of Sandy Loam, Silt Loam, Clay Loam and Mixed Soil

A (dimensionless) B (m) Ks (m/h)

Sandy loam

Silt loam

0.179 0.058 0.0442

0.072 0.290 0.0045

Clay loam 0.061 0.416 0.0026

Mixed soil 0.104 0.255 0.0171

We will evaluate the effects of large-scale soil properties by comparing the distribution of soil moisture based on four soil types (1) sandy loam, (2) silty loam, (3) clay loam and (4) mixed soil. The mixed soil is assumed to contain even portions of sandy loam, silty loam and clay loam. The soil properties of these four soils are summarized in Table 12.1, in which the properties ( A and B ) were estimated26 based on the soil database compiled by Carsel et al.28 The soil properties of mixed soil are assumed to be the average values of the soil properties of sandy loam, silty loam and clay loam. The third parametric group includes the mean soil moisture ( η ) and effective rainfall per depth (R0). In Equation 12.20, we have derived the expression of vertical hydraulic gradient (β) as functions of η and R0 and mean soil properties ( A , B , K s ). This expression contains the information of large-scale water balance. A large vertical gradient creates a large vertical moisture sink and η approaches small limiting values. In such cases, variability of soil moisture is primarily controlled by variability in soil properties. In this study, typical ranges for evaporation (0.5 ~ 1 m yr–1) and root zone depth (D = 1 m) were used following Yeh and Eltahir.14 From these values we can estimate the range of effective rainfall per depth, where effective rainfall is taken to be 50% of evaporation and R0 is a ratio of effective rainfall and root-zone depth. Parameters in the last group ( σ 2A , σ 2B , σ 2K s and σ 2z ) represent variations of heterogeneous soil properties and topography. Here we express the variability in terms of coefficient of variation (CV): σ 2A = A 2 CVA2 σ 2B = B 2 CVB2 σ 2Ks = K s 2 CVK2s

(12.24)

σ 2z = z 2 CVz2 Note that CV is a dimensionless parameter. To make a fair comparison between the contributions from topography and soil properties, in the following experiment we will assume that the coefficients of variation are equivalent for topography and soil properties, i.e., CVA = CVB = CVK s = CVz = cv. The mean soil properties ( A , B and K s ) are as in Table 12.1 for four different soil types and the mean relative elevation is assumed to be z = 30 m. In such cases, Equation 12.10 is approximated as:

(

ξ 2 = Ks A

(

)

2

(

)

2

ς2 = B A ω2 = z A © 2003 by CRC Press LLC

)

2

(12.25)

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207

FIGURE 12.2 The normalized covariance between topography and soil moisture (σηz/cv2) for Case 1 and Case 2 as a function of mean soil moisture (E[η]) and two different correlation scales of topography [λz = 102 m (solid line) and λz = 104 m (dashed line)]. Figure 12.2a refers to sandy loam. Figure 12.2b refers to silt loam. Figure 12.2c refers to clay loam. Figure 12.2d refers to mixed soil.

B. RELATIONSHIP

BETWEEN

SOIL MOISTURE DISTRIBUTION

AND

TOPOGRAPHY

Here we evaluate the covariance between soil moisture and topography, σηz, based on the preliminary analysis (Equations 12.14e, 12.15e and 12.16e) and the typical values of relevant parameters discussed above. We first look at the situation when soil properties are entirely uncorrelated with topography (Case 1 and Case 2). Note that the expression of σηz in Case 1 is the same as that in Case 2 (see Subsection V.A for discussion). The σηz in Case 1 and Case 2 is shown in Figure 12.2 for sandy loam, silty loam, clay loam and mixed soil. For each soil type, the normalized covariance between topography and soil moisture, σηz/cv2, is plotted as a function of mean soil moisture ( η ) for two different correlation scales (λz = 102 and 104 m). Several general features are identified in Figure 12.2. First, soil moisture is negatively correlated with relative elevation for different soil types. The negative covariance between soil moisture and topography has also been identified in a number of previous studies.3,6,10,29,30 Second, |σηz| is smaller for finer-texture soil and larger for coarser-texture soil, where “| |” denotes the absolute value. A weaker covariance between soil moisture and topography for finer-texture soil results from smaller pore and larger surface area, which yields more resistance on the topographically induced moisture flow. On the other hand, coarser-texture soil with larger pore size and smaller surface area appears to enhance the topographically induced moisture flow. Third, |σηz| is smaller if the topographic field is gently rolling (i.e., large λz) and larger if the topographic field is highly undulated (i.e., small λz). Usually, soil moisture tends to accumulate in the valley area and dissipate at the ridge area. Larger λz tends to smooth the appearance of valley and ridge and consequently reduce the soil moisture variability. Finally, |σηz| decreases for dry soil condition and increases for wet soil condition. This feature stems from direct impact of conductivity on the topographically induced moisture flow. In dry soil, low unsaturated hydraulic conductivity prohibits the topographically induced moisture flow. On the other hand, high conductivity (wet soil) reinforces such redistribution. © 2003 by CRC Press LLC

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FIGURE 12.3 Similar to Figure 12.2, but for Case 3. Note that solid line refers to λA = λζ = 102 m and dashed line refers to λA = λz = 104 m.

The σηz in Case 3 (topography and soil properties perfectly correlated) are shown in Figure 12.3 by assigning equivalent correlation scales for topography and soil properties (λA = λz = 102 m and λA = λZ = 104 m). It can be seen in Figure 12.3 that the appearance of negative covariance and the effect of mean soil properties (i.e., smaller |σηz| for finer-texture soil) as well as the effect of correlation scale of topography (i.e., smaller |σηz| for larger λz) are similar to those in Figure 12.2. However, the impacts from η (i.e., increasing |σηz| for wetter soil moisture) are altered by the soil-heterogeneity effects on σηz. To inspect the effects of soil properties on σηz, we decompose Equation 12.16e into two terms: σ ηz = fc + fv

(

)

fc = ω X˜ A + ξX˜ Ks − ζX˜ B σ 2A

(12.26)

fv = ω 2 X˜ z σ 2A where fc represents the impact from correlation between topography and soil properties, and fv represents the impact from variability of topography. Figure 12.4 plots the fc/cv2 and fv/cv2 for four different soil types as in Figure 12.3. A comparison between Figures 12.3 and 12.4 shows that in coarser-texture soil (e.g., sandy loam) fv has more dominant control on σηz. In other words, for coarser-texture soils σηz in Case 3 is similar to that in Cases 1 and 2. Also for areas with fine-texture soil (i.e., clay loam) or with a large λz, fc will have more dominant control on σηz. In such cases, |σηz| increases in dry soil condition and decreases in wet soil condition. In a mixed soil and small λz, fv has more dominant control on σηz in wet soil condition and fc has more dominant control for dry soil condition. In such cases, the negative covariance between soil moisture and topography is initially reduced and then enhanced as the soil changes from dry to wet. © 2003 by CRC Press LLC

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209

FIGURE 12.4 The fc/cv2 and fv/cv2 for four different soil types as in Figure 12.3, where fc represents the impact from correlation between topography and soil properties, and fv represents the impact from variability of topography. The relationship between Figures 12.3 and 12.4 is “Figure 12.3x = Figure 12.4x[1] + Figure 12.4[2],” where the figure index “x” represents “a” for sandy loam, “b” for silt loam, “c” for clay loam and “d” for mixed soil.

C. RELATIONSHIP

BETWEEN

SOIL MOISTURE DISTRIBUTION

AND

SOIL PROPERTIES

The covariances between soil moisture and soil properties (A, B and Ks) are represented respectively by (σηA, σηB and σ ηKs ). For illustration purposes, we only discuss the results for the covariance between soil moisture and saturated hydraulic conductivity σ ηKs . The results of Cases 1 and 2 are shown in Figures 12.5 and 12.6, respectively, for four soil types described before. For each soil © 2003 by CRC Press LLC

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FIGURE 12.5 The normalized covariance between saturated conductivity and soil moisture (σηKs/cv2) for Case 1 as a function of mean soil moisture (η) and two different correlation scales of soil property A [λA = 10–1 m (solid line) and λA 102 m (dashed line)]. Figure 12.5a refers to sandy loam. Figure 12.5b refers to silt loam. Figure 12.5c refers to clay loam. Figure 5d refers to mixed soil.

FIGURE 12.6 Similar to Figure 12.5, but for Case 2. © 2003 by CRC Press LLC

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211

type, the normalized covariance between soil moisture and saturated hydraulic conductivity ( σ ηKs ) is plotted as a function of mean soil moisture ( η ). In Figures 12.5 and 12.6, two different correlation scales of soil properties (λA = 0.1 and 10 m) are investigated. Because the evaporation rate is higher in soils with higher conductivity and slower in smaller conductivity, soil moisture and saturated hydraulic conductivity show negative covariance. A general trend in Cases 1 and 2 is that the negative covariance between saturated hydraulic conductivity and soil moisture is enhanced as the soil becomes dry. In other words, the heterogeneous soil properties have more pronounced influence on the distribution of soil moisture for dry soil condition. An enhancement of soil-heterogeneity effect on soil moisture distribution under wet soil condition has also been observed by Hawley et al.6 We note here that | σ ηKs | in Case 1 comes from the variance of Ks only, whereas | σ ηKs | in Case 2 is the net result from the variability of three soil properties (A, B and Ks). Consequently, a comparison between Figures 12.5 and 12.6 shows that | σ ηKs | in Case 2 is about one order of magnitude larger than that in Case 1. We also note in Case 2 that the correlation scale of soil properties (λA) has a relatively minor impact on σ ηKs . As discussed in Section IV.A, the following relationship holds if soil properties and topography are perfectly correlated (Case 3): σ ηKs σ η σ Ks

=

σ ηz σ ησ z

(12.27)

Therefore, the result of σ ηKs in Case 3 as shown in Figure 12.7 is similar to that of σηz in Case 3.

D. THE IMPACT

OF

MEAN SOIL MOISTURE

ON THE

SOIL MOISTURE VARIATION

In the previous two subsections, we have shown that the mean soil moisture has different influences on the topographically induced or soil-heterogeneity induced moisture distribution. Specifically, in the case when soil properties and topography are entirely uncorrelated (Cases 1 and 2), |σηz| decreases for dry soil condition and increases in wet soil condition. On the other hand, | σ ηKs | increases for dry soil condition and decreases in wet soil condition. If soil properties and topography

FIGURE 12.7 Similar to Figure 12.5. but for Case 3.

© 2003 by CRC Press LLC

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are perfectly correlated (Case 3), |σηz| and | σ ηKs | show similar response with mean soil moisture. Because |σηz| and | σ ηKs | in Case 3 is the net result of topographically induced and soil-heterogeneity induced moisture distribution. The response of |σηz| and | σ ηKs | with mean soil moisture will depend on whether topography or soil properties have more dominant control on soil moisture distribution. In coarser-texture soil with small λz, topography tends to have more dominant control on soil moisture distribution. In such situations, |σηz| and | σ ηKs | in Case 3 decreases for dry soil condition and increases in wet soil condition, which is similar to the response of |σηz| with mean soil moisture in Cases 1 and 2. In fine-texture soil or with large λz, soil properties tend to have more dominant control on soil moisture distribution. In such situations, |σηz| and | σ ηKs | in Case 3 increase for dry soil condition and decrease in wet soil condition, which is similar to the response of | σ ηKs | with mean soil moisture in Cases 1 and 2. In a mixed soil, topography has more dominant control in wet soil condition and soil properties have more dominant control on soil moisture distribution for dry soil condition. Therefore, for a mixed soil in Case 3, |σηz| and | σ ηKs | will decrease initially and then increase as the soil changes from dry to wet. Similar to |σηz| and | σ ηKs | in Case 3, the variance of soil moisture ( σ 2η ) is the net result of topographically induced and the soil-heterogeneity induced moisture distribution. An important implication from the above discussion is that the impact of mean soil moisture on σ 2η will also depend on whether topography or soil properties have more dominant control. Based on above discussion, we summarize that (1) topography will have dominant control on soil moisture distribution when the area is dominated by coarse-texture soil or by mixed soil with small λz; (2) soil properties will have dominant control on soil moisture distribution for fine-texture soil or by mixed soil with large λz; and (3) both topography and soil properties will have comparable influence for medium-texture soil with moderate value of λz. Figures 12.8 through 12.10 show the variances of soil moisture as a function of different soil types for three different cases. For each soil type the normalized soil moisture variability ( σ 2η / cv 2 ) is plotted as a function of mean soil moisture ( η ). In Cases 1 and 2, two different correlation scales of topography (λz = 102 and 104 m) as well as two different correlation scales of soil property A (λA = 10–1 and 10 m) are evaluated. In Case 3, equivalent correlation scales are assigned for topography and soil property A (λA = λz = 102 m and λA = λz = 104 m). A common and important feature is identified in Figures 12.8 through 12.10. If topography has more dominant control on soil moisture distribution, the soil moisture variability increases as the soil becomes wet. On the other hand, if soil properties have more dominant control on soil moisture distribution, the soil moisture variability decreases as the soil becomes wet. In cases when topography and soil properties have similar control on soil moisture distribution, the soil moisture variability initially decreases and then increases as the soil changes from dry to wet. For instance, if the area is dominated by fine-texture soil such as clay loam (Figures 12.8(c), 12.9(c), and 12.10(c)), soil properties tend to have more dominant control on soil moisture distribution and the soil moisture variability decreases as the soil becomes wet. We also note that the correlation between soil properties will reinforce the impact of soil properties on the soil moisture distribution. Take the sandy loam with λz = 104 m for example (Figure 12.8.2(a) for Case 1 and Figure 12.9.2(a) for Case 2). If soil properties are entirely uncorrelated (Case 1), topography has more dominant control on soil moisture distribution and thus soil moisture variability increases as the soil becomes wet. However, if soil properties are entirely uncorrelated (Case 1), the impact of soil properties on the soil moisture distribution becomes more significant and thus the soil moisture variability initially decreases and then increases as the soil changes from dry to wet. Table 12.2 summarizes the relative importance of topography and soil properties as well as the role of mean soil moisture on the variability of soil moisture based on different environmental conditions, such as in Figures 12.2 through 12.10.

© 2003 by CRC Press LLC

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213

FIGURE 12.8.1 The normalized variance of soil moisture ( σ 2n cv 2 ) for Case 1 as a function of mean soil moisture ( η ) and different correlation scales of Ks and z. This figure is for small correlation scale of topography (λz = 102 m). The solid line refers to λA = 10–1 m and the dash line refers to λA = 102 m. (a) refers to sandy loam; (b) refers to silt loam; (c) refers to clay loam; and (d) refers to mixed soil.

FIGURE 12.8.2 Similar to Figure 12.8.1, but for large correlation scale of topography (λz = 104 m).

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FIGURE 12.9.1 Similar to Figure 12.8.1, but for Case 2.

FIGURE 12.9.2 Similar to Figure 12.8.2, but for Case 2. © 2003 by CRC Press LLC

Scaling Methods in Soil Physics

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FIGURE 12.10.1 Similar to Figure 12.8.1, but for Case 3. Note that λA = λz = 102 m.

FIGURE 12.10.2 Similar to Figure 12.8.2, but for Case 3. Note that λA = λz = 104 m. © 2003 by CRC Press LLC

215

Sandy Loam

Undulated topographic field and highly heterogeneous soil (λz = 102 m λA = 10–1 m) Undulated topographic field and less heterogeneous soil (λz = 102 m λA = 10 m) Gently rolling topographic field and highly heterogeneous soil (λz = 104 m λA = 10–1 m) Gently rolling topographic field and less heterogeneous soil (λz = 104 m λA = 10 m)

Dominance Relationship Dominance Relationship Dominance Relationship Dominance Relationship

Cases 1 and 2 Topo. + Topo. + Topo. →+ Soil. → Topo. →+

Silt Loam

Clay Loam

Mixed Soil

Soil →Topo. –→+ Soil →Topo. –→+ Soil Soil -

Soil Soil Soil Soil -

Topo. + Topo. + Soil Soil -

Referred Figures

Figs. 12.2, 12.5, 12.6 Figs.12.8.1, 12.9.1 Figs. 12.2, 12.5, 12.6 Figs.12.8.1, 12.9.1 Figs. 12.2, 12.5, 12.6 Figs.12.8.2, 12.9.2 Figs. 12.2, 12.5, 12.6 Figs.12.8.2, 12.9.2

© 2003 by CRC Press LLC

Scaling Methods in Soil Physics

Case 3 Dominance Topo. Soil →Topo. Soil Topo. Figs. 12.3, 12.7 Undulated topographic field and less heterogeneous soil Relationship + –→+ + Fig. 12.10.1 λA = λz = 102 m Dominance Topo. Soil Soil Soil Figs 12.3, 12.7 Gently rolling topographic field and less heterogeneous soil Relationship + Fig. 12.10.2 λA = λz = 104 m Notes: The notation “topo.” represents that topography has more dominant control on soil moisture distribution; “soil” represents that soil properties have more dominant control on soil moisture distribution; and “soil→topo” represents that the soil moisture distribution is initially dominated by soil properties and then dominated by topography as the soil changes from dry to wet. The notation “+” represents that soil moisture variability increases as the soil becomes wet; “-” represents that soil moisture variability decreases as the soil becomes wet; and “- → +” represents that the soil moisture variability initially decreases and then increases as the soil changes from dry to wet.

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TABLE 12.2 Summary of the Features Identified from Figures 12.2 through 12.10

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E. COMPARISON

WITH

217

PREVIOUS STUDIES

To evaluate the robustness of the proposed stochastic theory in modeling the soil moisture distribution, it is essential to compare the results to relevant field observations. Unfortunately, few and often incomplete observational data exist. It is therefore extremely difficult to provide a quantitative comparison. Here we discuss a number of field observations that are in qualitative agreement with the results of this study. We will focus on the effects of mean soil properties and the correlation scale of topography on the relative importance of topographically induced moisture flow and soil-heterogeneity induced moisture flow as well as the impact of mean soil moisture on the soil moisture distribution. Qiu et al.31 discussed the soil moisture variation in relation to topography and land-use in a hillslope catchment in the Loess Plateau (China). The altitude ranges from 1000 to 1350 m in a 3.5 km2 area. A highly undulated topographic field is expected with such elevation difference. In such cases, topography is likely to control the soil moisture distribution. However, they have reported the land use and soils have more pronounced control on soil moisture distribution than topography. We note here that the catchment is mostly composed of fine-texture soil (fine silt and silt). Consequently, results of Qiu et al.31 are consistent with the suggestions in Subsection V.C in that finer-texture soil tends to reduce the influence of topographically induced moisture distribution. Western et al.13 investigated the relationship between soil moisture distribution and several topographic indices in Tarrawarra catchment, which is dominated by finer-texture soil. They found weak correlation in most topographic indices (i.e., aspect, specific contribution area), with the exception of tangent curvature. According to the observation by Famiglietti et al.,11 aspect and specific contribution area have much higher correlation with relative elevation than that with curvature. In other words, the results of Western et al.13 suggest that the finer-texture soil might result in weak correlation between soil moisture and elevation. These findings are also consistent with our results. Crave and Gascuel-Odoux32 analyzed the influence of elevation difference on soil moisture distribution within a subcatchment of the Coët-Dan catchment (Brittany, France). The topographic field is observed as highly undulated (small λz). The average saturated hydraulic conductivity in their study area is 3.01∞10–5 m/s in the hillslop domain and 2.15 ∞10–6 m/s in the valley. Data suggest that the correlation between topography and soil properties exists in this area and the soil type ranges from coarse to medium-coarse. One of their results shows (in their Figure 4c and 4d) that the relationship between topography and soil moisture is enhanced in wet condition and reduced in dry condition. Such a relationship is similar to the solid line in Figure 12.3a. (Soil properties and topography are perfectly correlated and the study area is dominated by coarse-texture soil with small λz.) Famiglietti et al.12 investigated the soil moisture variability within remote sensing footprints obtained in Southern Great Plains 1997 hydrology experiment (SGP97). They have found, in sites LW13 and LW21, no correlation between surface soil moisture and downslope decrease in elevation. In addition, the variances of soil moisture in both sites are found to decrease as the soil becomes wet. We note here that the soil types in both sites are dominated by medium-fine soil (i.e., LW13 is dominated by loam and LW21 by silt loam). In addition, the topographic fields are gently rolling in LW13 and flat in LW21 (i.e., large λz). This result agrees with our finding in that a smooth topography field and the domination of fine-texture soil will lead to weak relationship between elevation and soil moisture. In such cases, the variance of soil moisture will decrease as the soil becomes wet. The field observations discussed above show qualitative agreement with the findings from proposed stochastic theory with respect to the effects of mean soil properties and the correlation scale of topography on the relative importance of topographically induced moisture flow and soilheterogeneity induced moisture flow. We note that inference from the proposed stochastic theory will not work as well if precipitation is highly heterogeneous in spatial and temporal scale. For instance, Famiglietti et al.11 investigated the soil moisture variability along a hillslope transact at

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Rattlesnake Hill (Texas). The topographic field is gently rolling (i.e., large λz) and the soil field is evenly composed of clay, silt and sand. A strong correlation between topography and soil properties appears in the study area. Therefore the soil and topographic conditions in this area are similar to those in the solid line of Figure 12.3d. The solid line of Figure 12.3d shows that the negative correlation between soil moisture and topography is minimized as the soil becomes wet. This result partially agrees with the observation of Famiglietti et al.11 They found that the negative correlation is minimized as the soil changes from dry to medium wet. However, as soil changes from medium wet to extreme wet, they found that the correlation between topography and soil moisture alters from negative to positive correlation. Probably, such positive correlation results from highly spatialtemporal variation in precipitation during their experiment.

VI. CONCLUSIONS A stochastic framework is proposed in this study for characterizing the steady-state soil moisture distribution in a heterogeneous-soil and -topography field under the influence of precipitation and evaporation. The problem of water flow in an unsaturated zone is expressed in a partial differential equation that depends on three stochastic variables: the heterogeneity of soil properties, the variability of topography and the change of mean soil moisture. This is perhaps the first attempt to include the effects of topography and soil properties on soil moisture distribution. A perturbation method and spectral technique are applied to solve the soil moisture dynamics. The resulting model provides closed form analytical solutions for (a) the variance of soil moisture distribution ( σ 2η ), and (b) the covariance between soil moisture distribution and soil properties (σηA, σηB, σηKs,), and (c) the covariance between soil moisture distribution and topography (σηz) as a function of soil heterogeneity, topography and mean soil moisture. The proposed approach assumes that the spatial variables (soil properties, unsaturated soil hydraulic properties and elevation) are realizations of a two-dimensional, cross-correlated, secondorder stationary random field. Evaluation of the influences of these spatial variables on the soil moisture distribution requires knowledge of the cross-covariance functions of these variables. Very few field observations can quantify cross-covariance functions of topography and soil properties. To develop an insight regarding the interdependencies of these variables and their influence on variability of soil moisture, we focus on three limiting cases in this study: (1) soil properties (A, B, Ks) and topography (z) are uncorrelated; (2) the soil properties are correlated among themselves but uncorrelated with topography; and (3) soil properties and topography are perfectly correlated. Several important features regarding the relationship between soil properties and topography may be identified from these three limiting cases. First, if soil properties and topography are uncorrelated, variability of soil moisture can be viewed as a sum of the individual variability of soil properties and topography. In such cases, covariance between soil moisture and the attributes (i.e., A, B, Ks or z) will be a function of those attributes only. If soil properties are correlated among themselves but uncorrelated with topography, variability in soil moisture is composed of individual variability of soil properties and topography as well as the covariance among soil properties. In the third case, when A, B, Ks and z are perfectly correlated, cross correlation between soil moisture and soil physical properties or soil moisture and topography is equivalent. The proposed stochastic model is then used to evaluate the influence of topography, soil physical properties, and mean soil moisture on the variation of soil moisture distribution. To do this, we have investigated the role of correlation scales for different attributes, large-scale properties, and variance of soil properties and topography on the variability of soil moisture distribution. Our results suggest that, first, topography appears to have dominant control on soil moisture distribution when the area is dominated by coarse-texture soil or by mixed soil with small correlation scale for topography (i.e., small λz). In such cases the soil moisture variability increases as the soil becomes wet. Second, soil properties are likely to have dominant control on soil moisture distribution for fine-texture soil or for mixed soil with large λz. In such cases, the soil moisture variability decreases © 2003 by CRC Press LLC

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219

as the soil becomes wet. Finally, topography and soil properties appear to have similar control for medium-texture soil with moderate value of λz. In such cases, the soil moisture variability initially decreases and then increases as the soil changes from dry to wet. To evaluate the robustness of the stochastic modeling theory, it is necessary to compare the results to relevant field observation. Preliminary comparisons with several field observations suggest that our modeling results are consistent with observed data. We must emphasize, however, that few and often incomplete observational datasets exist to evaluate the adequacy of the proposed methodology fully. Thus, the conclusions should be viewed as tentative. Our proposed analytical approach, however, can provide a systematic framework to evaluate the importance of different interacting variables, for example, coupling between topography and saturated hydraulic conductivity, on the variation of soil moisture distribution. To keep the analytical approach tractable, we have examined three limiting cases in detail. Clearly, observed data will fall somewhere in between these limiting cases; the extension of these analytical results to actual field situations should be done with caution. Also, we have considered the steady-state distribution of soil moisture under given climatic conditions. Future work will address the corresponding transient problem, and the issues of spatially variable climatic forcing and vegetation on the soil moisture distribution.

VII. APPENDIX A: DERIVATION OF THE LARGE-SCALE AND PERTURBATION MODELS The general form of the unsaturated flow model (Equation 12.1) at steady state can be divided into the large-scale and small-scale components. The large-scale model represents the “mean” water balance at large scales. On the other hand, the small-scale perturbation model relates the distribution of soil moisture with heterogeneous topography and soil properties. The first step in the derivation of the large-scale and perturbation models is to decompose the variables K, ψ, and z into their spatial mean ( K , ψ , z ) and perturbation terms (K′, ψ′, z′), so that Equation 12.1 becomes: ∂( − ψ ′ + z ′) ∂( − ψ + z ) ∂( − ψ ′ + z ′)  ∂  ∂( − ψ + z ) +K + K′ + K′ K  ∂xi  ∂xi ∂xi ∂xi ∂xi 

(12.A1)

β R K + K′ + = 0 − D D

(

)

The large-scale model is obtained by averaging the local Equation 12.A1 over the entire domain; it becomes: ∂ ∂xi

 ∂  ∂( − ψ + z )  β R K  − K + = −E D ∂xi  ∂xi   D

 ∂( − ψ ′ + z ′)    K′ ∂xi   

(12.A2)

If the spatial-dependnt variables are assumed as the realization of second-order stationary random field and the expected value of higher-order perturbation terms in the right-hand side of Equation 12.A2 is relatively small to other large-scale terms, Equation 12.A2 can be further simplified following Yeh and Eltahir:14 β= R/ K

(12.A3)

Equation 12.A3 implies that the mean soil moisture, which is related to mean unsaturated hydraulic conductivity K , is the net result of effective rainfall (R) and vertical hydraulic gradient (β). Such © 2003 by CRC Press LLC

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a relationship is reasonable based on the theory of mass conservation, which indicates that the residual moisture is primarily controlled by the input (i.e., effective rainfall) and output (i.e., evaporation) terms. The perturbation model is obtained by subtracting the large-scale Equation 12.A2 from 12.A1: ∂ ∂xi

 ∂( − ψ ′ + z ′)  β K  − K′ = 0 ∂xi   D

(12.A4)

Equation 12.A4 is derived under the assumption that the higher-order perturbation terms can be approximated by their expected values. Similar assumption is also made by other stochastic analyses.14,18 The derivation of the perturbation equation is not complete without considering the effects of heterogeneous soil properties, which are introduced into Equation 12.A4 through the K–ψ relationship (Equation 12.3) and K–η relationship (Equation 12.5). By decomposing the variables B, ψ, K and Ks in Equation 12.3 into their spatial mean and perturbation term, it becomes:     K′ K′ K  ψ + ψ ′ = − B ln1 +  − ln1 + s  + ln1 +  K Ks  K s      

(12.A5)

    K′ K′ K  − B′ ln1 +  − ln1 + s  + ln1 +  K Ks  K s       Assuming that K ′ / K << 1 , by using Maclaurin series (ln(1 + ξ) = ξ – ξ2/2 + ξ3/3 – + … ≈ ξ for |ξ| << 1), Equation 12.A5 is approximated as:  K′ K′  K′ K′  K   K  ψ + ψ ′ = − B  − s + ln   − B′  − s + ln    K s    K s    K K s  K K s

(12.A6)

By taking the expected value of Equation 12.A6, we obtain the following large-scale ψ – K relationship:   K ′ K s′   K ψ = − B ln  − E  B′ −   Ks    K K s  

(12.A7)

Using the same process to derive Equation 12.A4, the perturbation expression of hydraulic suction is derived as:  K ′ K s′  K ψ ′ = − B ′ −  − B′ ln  K K   Ks  s

(12.A8)

By substituting the perturbation expression of K–ψ relationship (Equation 12.A8) into Equation 12.A4, we have the following differential equation: ∂ ∂xi © 2003 by CRC Press LLC

 ∂   K ′ K ′    β K − s  + B′ ln  + z ′   − K ′ = 0 B  K ∂xi   K K s   Ks    D 

(12.A9)

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Similar to the derivation of Equation 12.A4, we assume that all spatially related variables follow the second-order stationary process, and thus Equation 12.A9 becomes: −

2 β 1 ∂ K s′ 1 ∂ 2 K ′ 1  K  ∂ 2 B′ 1 ∂ 2 z ′ + + ln  + − K′ = 0 2 K s ∂xi K s ∂xi2 B  K s  ∂xi2 B ∂xi2 KBD

(12.A10)

Equation 12.A10 is the perturbation equation of unsaturated hydraulic conductivity, which can be rewritten as the perturbation equation of soil moisture through the K–η relationship (Equation 12.5). By decomposing the variables A, K and Ks in Equation 12.5 into their spatial mean and perturbation term, and using similar process for deriving the ψ – K and ψ′ – K′ relationship as in Equations 12.A7 and 12.A8, respectively, the following η – K and η′ – K′ relationships are derived as:   K ′ K s′   K η = 1 + A ln  + E  A′ −   Ks    K K s  

(12.A11)

 K ′ K s′  K η′ = A  −  + A ln  K K   Ks  s

(12.A12)

Equation 12.A11 can be further simplified by assuming that the expected value of higher-order perturbation is much smaller than other large-scale terms. In such cases, the mean relative hydraulic conductivity can be estimated as:  K  1− η ln  = A  Ks 

(12.A13)

The final form of the perturbation model is represented by two simultaneous Equations 12.A14a and 12.A14b, through substituting the large-scale Equations 12.A3 and 12.A13 into Equation 12.A10 and through substituting the large-scale Equation 12.A13 into Equation 12.A12, respectively: −

2 1 ∂ K s′ 1 ∂ 2 K ′ 1  1 − η  ∂ 2 B′ 1 ∂ 2 z ′ β1 + − + − K′ = 0 K s ∂xi2 K ∂xi2 B  A  ∂xi2 B ∂xi2 K

 K ′ K s′   1 − η − −A η′ = A   A   K Ks 

(12.A14a)

(12.A14b)

where β1 =

β = BD

R0 Ks e

−

1− η A

(12.A15) B

The parameter R0 is defined as the portion of effective rainfall infiltrating to a unit depth of soil (R0 = R/D). Based on previously derived Equation 12.A3, the effective rainfall (R) is equivalent to large-scale evaporation (β K ). © 2003 by CRC Press LLC

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VIII. APPENDIX B: SOLVING THE PERTURBATION EQUATIONS USING SPECTRAL TECHNIQUES The spectral techniques are applied to solve the above simultaneous perturbation equations. Following Gelhar,23 the two-dimensional stationary random fields K′, η′, A′, B′, Ks′, and z′ can be expressed in their wave number domain as below: ∞

K′ =

∫e

(

) dZ k , k K( x y)

(

) dZ k , k y) η( x

(

) dZ k , k A( x y)

i kx x + ky y

−∞ ∞

∫e

η′ =

i kx x + ky y

−∞ ∞

∫e

A′ =

i kx x + ky y

−∞ ∞

(12.B1)

∫e

B′ =

(

) dZ k , k B( x y)

(

) dZ

i kx x + ky y

−∞ ∞

K s′ =

∫e

i kx x + ky y

Ks

−∞ ∞

z′ =

∫e

(

i kx x + ky y

−∞

(k , k ) x

y

) dZ k , k z( x y)

where x, y are spatial coordinates, kx, ky are wave numbers in x and y directions, i = −1 and dZ is the Fourier-Stieltjes spectral amplitudes. By substituting the spectral relationships (Equation 12.B1) into Equation 12.A14a, we can derive the Fourier-Stieltjes spectral amplitude of unsaturated hydraulic conductivity as: dZ K =

K K  1 − η K W0 dZ Ks + W0 dZ B − W0 dZ z   Ks B A B

(12.B2)

where

W0 =

(k

(k 2 x

2 x

+ k y2

+k

2 y

)

)+β

(12.B3)

1

Similarly, by substituting the wave number relationship into Equation 12.A14b, we can derive the Fourier-Stieltjes spectral amplitude of soil moisture as: 1   1 − η 1 dZ η = A  dZ K − dZ Ks  − dZ A K K    A  s © 2003 by CRC Press LLC

(12.B4)

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By substituting Equation 12.B2 into Equation 12.B4, we can relate dZη with dZA, dZB, dZKs, and dZz as the following: dZ η = WKs dZ Ks + WB dZ B + Wz dZ z + WA dZ A

(12.B5)

where WKs =

A (W − 1), WB =  1 −B η  W0 , Wz = − BA W0 , WA = − 1 −A η  Ks 0

(12.B6)

Equation 12.B5 relates the normalized soil moisture (η), soil properties (A, B, Ks) and topography (z) in Fourier-Stieltjes Domain. An inverse transform is applied to convert Equation 12.B5 from wave-number domain into the real domain. Following Gelhar,23 the variation of soil moisture, σ 2η , can be derived through the inverse Fourier-Stieltjes transformation of the spectral density function of soil moisture (Sηη) as: σ 2η =

∫∫ e

(

) S k , k dk dk ηη ( x y) x y

i kx + ky τ

−∞→∞

(12.B7a)

τ=0

where Sηηdkxdky = E[dZη ⋅ dZη*] and the prescript “*” indicates the conjugative form. Similarly, the covariance between soil moisture and variable u (where u = parameters A, B, Ks or z), σηu, can be derived through the following: σ ηu =

∫∫ e

−∞→∞

(

) S k , k dk dk ηu ( x y) x y

i kx + ky τ

(12.B7b)

τ=0

where Sηudkxdky = E[dZη ⋅ dZu*]. Using Equation 12.B5, the spectral density function of soil moisture is derived as: Sηη = WKs 2 SKs Ks + WB 2 SBB + Wz 2 Szz + WA 2 SAA

(

+ 2 WKs WB SKs B + WKs Wz SKs z + WKs WA SKs A + WB WA SBA + Wz WA SzA

)

(12.B8a)

Similarly, the cross-spectral density function of soil moisture and the variable u (where u = parameters A, B, Ks or z) is derived as: SηA = WA SAA + WB SBA + WKs SKs A + Wz SzA

(12.B8b)

SηB = WA SBA + WB SBB + WKs SKs B + Wz SBz

(12.B8c)

SηKs = WA SKs A + WB SKs B + WKs SKs Ks + Wz SKs z

(12.B8d)

Sηz = WA SzA + WB SBz + WKs SKs z + Wz Szz

(12.B8e)

By substituting Equation 12.B8a into Equation 12.B7a and substituting Equations 12.B8b through B8e into Equation 12.B7b, Equation 12.6 is thus derived. © 2003 by CRC Press LLC

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REFERENCES 1. Hills, T.C. and Reynolds, S.G., Illustrations of soil moisture variability in selected areas and plots of different size, J. Hydrol., 8, 27, 1969. 2. Reynolds, S.G., The gravimetric method of soil moisture determination. I: A study of equipment, and methodological problems, J. Hydrogol., 11, 258, 1970. 3. Henninger, D.L., Peterson, G.W., and Engman, E.T., Surface soil moisture within a watershed: Variations, factors influencing, and relationships to surface runoff, Soil Sci. Soc. Am. J., 40, 773, 1976. 4. Bell, K.R., Blanchard, B.J., Schmugge, T.J., and Witczak, M.W., Analysis of surface moisture variations within large field sites, Water Resour. Res., 16, 796, 1980. 5. Hawley, M.E., McCuen, R.H., and Jackson, T.J., Volume-accuracy relationships in soil moisture sampling, J. Irrig. Drain. Div., Proc. Am. Soc. Civ. Eng., 108(IR1) 1–1, 1982. 6. Hawley, M.E., Jackson, T.J., and McCuen, R.H., Surface soil moisture variation on small agricultural watersheds, J. Hydrol., 62, 179, 1983. 7. 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. 8. Charpentier, M.A. and Groffman, P.M., Soil moisture variability within remote sensing pixels, J. Geophys. Res., 97, 18987, 1992. 9. Niemann, K.O. and Edgell, M.C.R., Preliminary analysis of spatial and temporal distribution of soil moisture on a deforested slope, Phys. Geogr., 14, 449, 1993. 10. Robinson, M. and Dean, T.J., Measurement of near surface soil water content using a capacitance probe, Hydrol. Process., 7, 77, 1993. 11. 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. 12. Famiglietti, J.S., Devereaux, J.A., Laymon, C.A., Tsegaye, T., Houser, P.R., Jackson, T.J., Graham, S.T., Rodell, M., and van Oevelen, P.J., Ground-based investigation of soil moisture variability within remote sensing footprints during the Southern Great Plains (SGP97) hydrology experiment, Water Resour. Res., 35, 1839, 1999. 13. Western, A.W., Grayson, R.B., Bloschl, G., Willgoose, G.R., and McMahon, T.A., Observed spatial organization of soil moisture and its relation to terrain indices, Water Resour. Res., 35, 797, 1999. 14. Yeh, P.J.-F. and Eltahir, E.A.B., Stochastic analysis of the relationship between topography and the spatial distribution of soil moisture, Water Resour. Res., 34, 1251, 1998. 15. Rodriguez-Iturbe, I., Vogel, G.K., Rigon, R., Entekhabi, D., Castelli, F., and Rinaldo, A., On the spatial organization of soil moisture fields, Geophys. Res. Lett., 22, 1995. 16. Chang, D.-H. and Islam, S., Estimation of soil physical properties using remote sensing and artificial neural network, Remote Sensing Environ., 74, 534, 2000. 17. Jackson, T.J., Le Vine, D.M., Swift, C.T., Schmugge, T.J., and Schiebe, F.R., Large scale mapping of soil moisture using the ESTAR passive microwave radiometer in Washita ‘92, Remote Sensing Environ., 53, 27, 1995. 18. Mantoglou, A. and Gelhar, L.W., Stochastic modeling of large-scale transient unsaturated flow system, Water Resour. Res., 23, 37, 1987. 19. Gardner, W.R., Some steady state solutions of the unsaturated moisture flow equation with application to evaporation from a water table, Soil Sci., 85, 228, 1958. 20. Yeh, T.C., Gelhar, L.W., and Gutjahr, A.L., Stochastic analysis of unsaturated flow in heterogeneous soils, 1, Statistically isotropic media, Water Resour. Res., 21, 447, 1985. 21. Gardner, W.R., Hiller, D., and Benyamini, Y., Post-irrigation movement of soil water, 1. Redistribution, Water Resour. Res., 6, 851, 1970. 22. Campbell, G.S, A simple method for determining unsaturated conductivity from moisture retention data, Soil. Sci., 117, 311, 1974. 23. Gelhar, L.W., Stochastic Subsurface Hydrology. NJ: Prentice-Hall, Englewood Cliffs, 1993. 24. Przewlocki, J., Two-dimensional random field of mechanical soil properties, J. Geotech. Geoenviron., 126, 373, 2000. 25. Jenny, H., Factors of Soil Formation. NY: McGraw Hill, 1941. 26. Chang, D.-H., Analysis and modeling of space-time organization of remotely sensed soil moisture, in Dept. of Civil and Environmental Engineering. Ph.D. thesis: University of Cincinnati, 2001.

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27. Philip, J.R., The theory of infiltration, Adv. Hydrosci., 5, 215, 1969. 28. Carsel, R.F. and Parrish, R.S., Developing joint probability distributions of soil water retention characteristic, Water Resour. Res., 24, 755, 1988. 29. Krumback, A.W.J., Effects of microrelief on distribution of soil moisture and bulk density, J. Geophys. Res., 64, 1587, 1959. 30. Nyberg, L., Spatial variability of soil water content in the covered catchment of Gardsjon, Sweden, Hydrol. Process., 10, 89, 1996. 31. Qiu, Y., Fu, B., Wang, J., and Chen, L., Soil moisture variation in relation to topography and land use in a hillslope catchment of the Loess Plateau, China, J. Hydrol., 240, 243, 2001. 32. Crave, A. and Gascuel-Odoux, C., The influence of topography on time and space distribution of soil surface water content, Hydrol. Process., 11, 203, 1997.

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13

A Mix of Scales: Topography, Point Samples and Yield Maps D.J. Timlin, Y.A. Pachepsky, and C.L. Walthall

CONTENTS I. Introduction...........................................................................................................................227 II. Analysis of Data from Various Sources...............................................................................228 A. Site Description .........................................................................................................228 B. Spatial Autoregression ...............................................................................................229 1. Regression Methodology .....................................................................................231 2. Prediction of WHC at Different Scales...............................................................231 C. Yield Data ..................................................................................................................232 III. Interpolation and Mapping with Auxiliary Variables ..........................................................232 A. Topography and Water Flow in the Landscape ........................................................232 B. Terrain Variables and Soil Properties........................................................................233 C. Spatial Autoregression ...............................................................................................233 D. Interpolation and Mapping of WHC at Different Scales..........................................236 E. Measured Crop Yields and Interpolated WHC .........................................................237 IV. Concluding Remarks ............................................................................................................239 V. Acknowledgments ................................................................................................................240 References ......................................................................................................................................240

I. INTRODUCTION Agricultural land management has requirements for many kinds of data, including information on soil hydraulic properties, soil texture, terrain attributes, and crop cover. The sources of these data are varied and range from manual collection of sparse point samples to acquisition of highly dense remotely sensed soil and crop canopy reflectance, and yield monitor data. Samples collected to characterize soil properties, however, are often collected manually and are likely to be discontinuous and represent localized sites and small scales. Elevation and topographic parameters may be useful to help interpolate discontinuous values of soil properties because they provide a more or less continuous distribution of measurements over large areas and usually represent a range in scales.1,2 Terrain indices have also been shown to be related to moisture distributions in soils.3,4 Soil texture or water holding capacity data collected as point samples may provide an adequate statistical representation of a local variable but there may not be enough data to accurately represent the overall pattern of variability in order to map the data. Terrain attributes such as soil surface curvature, slope and elevation are related to soil texture, soil water content and crop yields.5–9 Slope and soil surface curvatures have been shown to be good predictors of distributions of soil texture in the landscape.7,10 Elevation can be easily measured and generally has a reasonably continuous variance over space in agricultural fields. By relating soil water holding capacity to topographic variables it may be possible to interpolate sparse samples of soil water holding capacity.10 Thus 227 © 2003 by CRC Press LLC

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one could obtain a more accurate distribution of soil water holding capacity than from interpolation of the measured variables alone. Regression analysis is commonly used to develop predictive equations for interpolation or to explain variability in terms of known variables. Regression is commonly used to develop pedotransfer functions to relate easily measured soil attributes to soil hydraulic properties.11,12 The measured variables are often not collected with spatial information and are assumed to be independent. When ordinary least squares regression (OLS) is used on data that are spatially autocorrelated, the standard errors can be underestimated, tests of significance overinflated,13 and prediction errors large.14 Spatial autoregression can be a useful tool for quantifying large scale measurements, terrain variables, and point measurements where spatial autocorrelation exists.13,15 Knowledge of spatial relationships can be used to "fill in" or interpolate sparse measurements of soil properties. Redundant information in the variables and local variances can be exploited to make better estimates. Previous research at the site studied in this chapter7 reported that the strongest relationship between soil water retention and topographic variables was observed at capillary pressures of 10 and 33 kPa, i.e., in the range where the soil reaches its “field capacity.” Because the water content at 15 MPa (wilting point) did not vary substantially, the study proposed that the water holding capacity depended on landscape position and these available water holding capacities predicted from terrain variables could be used as interpretive attributes for yield maps in precision agriculture. The objective of this study was to characterize soil water holding capacity (WHC) at the field scale using topographic variables as predictors and applying a spatial autoregressive response model to account for spatial dependence of the water holding capacities on terrain attributes. Several scales of interpolation of the elevation data were used to investigate the effect of interpolated DEM (digital elevation model) grid size on the relationship between calculated terrain variables and WHC. The spatially estimated water holding capacities were compared to measured yields to determine their usefulness as interpretive attributes for precision agriculture.

II. ANALYSIS OF DATA FROM VARIOUS SOURCES A. SITE DESCRIPTION The experimental site was a 6-ha corn field located on the USDA, Henry A. Wallace Beltsville Agricultural Research Center in Beltsville, Maryland. The data set described here was collected during the growing seasons of 1997 and 1998. The investigation carried out on this field is part of a larger study entitled Optimizing Production Inputs for Economic and Environmental Enhancement (OPE3). The OPE3 project is addressing: 1) watershed scale fluxes of chemical inputs and biological agents from conventional, precision, and sustainable farming practices, 2) environmental impact of chemical inputs and biological agents on a wooded riparian wetland, 3) development of remotely sensed data products and analytical techniques for measuring and managing the spatial variability of crops and soils, and 4) long-term economic and environmental impacts and tradeoffs of precision and sustainable agricultural production practices. The study site has a gentle slope running from the northwest part of the site to southeast with an elevation difference of approximately 4 m (Figure 13.1). Sandy loam soils predominate with silt and clay contents increasing down slope. The corn was planted north to south in 0.74-m wide rows. At the time of the data collection there were no known major pest infestations. The yields were recorded using an AgLeader 2000 yield monitor (Ag Leader Technology, Ames, Iowa). A topographic survey of the site was obtained from a total of 555 photogrammetric mass points from airborne stereophotography (Air Survey Corp., Virginia). The average mass of sampled points was about one measurement per 140 m2. Elevation values were obtained from a digital elevation model constructed by interpolation of the photogrammetric points’ data to nodes of various grid sizes that ranged from 5 × 5 m to 55 × 55 m. The interpolation method of minimum curvature in © 2003 by CRC Press LLC

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Transects

B

D

229

Locations of grid samples

Field boundary

C

A FIGURE 13.1 Elevation map of study site showing locations of transects, grid sampling locations, and field boundary.

Surfer (Golden Software, Golden, Colorado) was used to interpolate the grids. The method of minimum curvature was reported to give good results based on comparisons among five commonly used interpolation methods.7 Values of maximum slope, profile curvature and tangential curvature were used as topographic attributes for the study area. Profile curvature is defined as curvature of the surface cross section made in the direction of maximum slope. This is the uphill rate of change in slope. Negative (positive) values of profile curvature indicate convex (concave) flow paths where a surface flow accelerates (slows down). Tangential curvature is defined as curvature of the vertical surface cross section made perpendicular to the direction of maximum slope. Negative (positive) values of tangential curvature represent areas of divergent (convergent) flow.16 Equations for calculation of the terrain parameters are given in Pachepsky et al.7 Soil was sampled along four transects positioned in different landscape elements (i.e., along slopes and at foot slopes) and at nodes of a 30- × –30-m grid as shown in Figure 13.1 . The sampling locations were 2 m apart within each transect. All samples were taken in duplicate from points 30 cm apart. There were 54 duplicated transect samples and 39 duplicated grid samples. Sampling depth was 4 to 10 cm. The same topographic variables were assigned to the duplicated samples. A transect sampling scheme was used to provide detailed information on changes in soil properties along a gradient of slope, which would not have been possible with more widely spaced samples. However, the transects have been augmented by grid samples that provide a (pseudo) nested sampling structure.

B. SPATIAL AUTOREGRESSION The regressions were carried out using a spatial autoregressive response model. A more complete description of autoregressive models applied to spatial data can be found in References 13 and 17 through 19. In ordinary least squares regression, the dependent variable is a function of the independent variable as Y = Xβ + ε where Y and X are n × 1 vectors, β is a 1 × k +1 vector of © 2003 by CRC Press LLC

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coefficients and ε is error with mean of 0 and σ2 = constant. There are n observations(dependent, Y) and k predictors (independent, X). If Y or X exhibit spatial autocorrelation, then they must be corrected for values of X or Y at nearby locations. Because the data are collected on a twodimensional grid, a value of X or Y at a lag of i – 1 cannot be used to correct for autocorrelation as it can for a one-dimensional series of data. Instead, a connectivity matrix D is used.18,20 The autoregressive model becomes: Y - αDY = β0 + X 1 β1 + X 2 β2 + X 3 β3 + ε

(13.1)

The value α is the autoregressive parameter and lies between 0 and 1. It is assumed here that α is a constant for the field. This may or may not be appropriate but is a good first approximation. The connectivity matrix, D, is an n × n weighting matrix with 0 on the diagonal so that only neighboring values and not the value itself are used for predictions. The independent variables are the three terrain attributes (slope, profile curvature and tangential curvature). The error, ε, is distributed as ε ~ N(0,σ2I). We used an n × n connectivity matrix (D) described in Pace and Barry.19 The properties of this matrix are 1) the diagonals are zero, and 2) the rows sum to one. The strength of the connection is related to distance between points and does not require uniform separation distances and can thus accommodate nonuniform grid spacings. Figure 13.2 shows the relationship between a 3 × 3 grid (A) with equally spaced cells and the spatial weights (B) calculated by the nearest neighbor method. For example, the first row of the adjusted Y (αDY), Y1a, would be calculated as α0.5Y2 + α0.5Y4. This is a two-dimensional analog of a time series model where the current Y value is adjusted for past values. The spatial connectivity matrix adjusts the Y value for its neighbors. The autoregressive parameter, α, is solved using maximum likelihood computations.19 The profile likelihood function is defined as:  n L( β, α , σ 2 ) = C + ln | I - αD | -  ln(SSE)  2

(13.2)

where C is a constant and ln|I – αD| is the log determinant. The SSE is the sum of squared errors: SSE = (Y - αDY - X βα ) T (Y - αDY - X βα )

(13.3)

where T denotes transpose of the matrix and β_α is an n × 1 vector of coefficients β0 – β3 in Equation 13.1). An efficient and rapid method to evaluate the maximum of the profile likelihood function and hence solve for α and β was used here.19 The log determinant, ln|I – αD|, is calculated for a number (usually 100) of values of α and then a lookup table is used to find the minimum of L in Equation 13.2. Matlab (The Mathworks, Natick, Massachusetts) programs from the SpatialToolBox, v1.1* were used to perform the calculations. The SpatialToolBox function FPAR1 was used to compute the spatial autoregression. After α was determined, the error sums of squares for α were then calculated. Spatially adjusted values of Y (WHC) were calculated by subtracting αDY from Y. These adjusted Y values (Ya) were next used in an OLS regression with the topographic variables (X1 – X3). The sums of squares for α were calculated by subtracting the total sums of squares for the OLS [Ya – mean (Ya)] from the total sums of squares for the AR model [Y – mean(Y)]. Next, the sums of squares for the OLS regression were obtained by subtracting the error sums of squares for the OLS (Ya – Yˆ a ) from the total sums of squares for OLS [Ya – mean(Ya)]. * Spatial Statistics Toolbox 1.1. R. K. Pace and R. Barry, available at: http:\\spatial-statistics.com

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A)

B)

Cell

1

2 1/2

1 0 2

1/3

0

1/2

0

4

1/2

5 0

1

2

3

4

5

6

7

8

9

4 1/2

1/3 1/2

3 0

3 0

231

5 0 1/2

0

6

7

8

0

0

0

0

0

0

0

0

0

0

0

1/2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 1/2

9

0

1/2

1/2

0 1/2

6 0

0

7 0

0

0

1

0

0

0

0

0

8 0

0

0

0

1

0

0

0

0

9 0

0

0

0

0

1

0

0

0

FIGURE 13.2 Representation of the spatial weighting matrix (D) for a 3 × 3 grid. The weights sum to 1 in the rows. The Spatial Statistics Toolbox, v1.1 was used to obtain the weights.

1. Regression Methodology As described above, terrain variables (slope and curvature) were calculated from the measured elevation data that were interpolated to different sizes of regular grids (5 × 5 m to 55 × 55 m). The calculated terrain variables were interpolated to the measured locations and the regressions were carried out using the WHC data in their measured locations, i.e., no gridding was performed for the WHC data. The same number of independent and dependent variables were used for each scale at which the regressions were carried out. 2. Prediction of WHC at Different Scales In order to calculate WHC, predicted WHC, ( Yˆ ), as a function of terrain variables at different scales, Equation 13.1 can be rewritten as: β + X 1 β1 + X 2 β2 + X 3 β3 Yˆ = 0 1- αD

(13.4)

The value of α is determined from the spatial autoregression of the measured water holding capacities vs. the terrain variables calculated as a function of a specific grid size. The predictions are an interpolative process to fill in water holding capacities on regular grids at locations where measurements are missing. In actuality, Equation 13.4 is a smoothing interpolator for WHC because the predicted values are divided by the means of neighboring values. The grid sizes for interpolation range from 5 × 5 (4725 values) to 55 × 55 (20) values. © 2003 by CRC Press LLC

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C. YIELD DATA For comparison with the interpolated WHCs, the 6- × 2-m yield data were gridded to 20- × 20-, 25- × 25-, 30- × 30-, and 35- ×y 35-m grids by block kriging using Surfer. The yield data were then smoothed by using an average of the neighboring blocks (one grid cell on each side). The linear default semivariogram was used. The linear semivariogram provides a smoothing of the data similar to a moving average. Because the data were nearly continuous, the effect of the kriging was to smooth the data rather than interpolate and fill in missing locations. For this reason, an isotrophic, linear semivariogram was thought to be adequate. The WHC predicted from terrain attributes were interpolated to the yield measurement locations using Surfer, which uses a bilinear interpolation method.

III. INTERPOLATION AND MAPPING WITH AUXILIARY VARIABLES A. TOPOGRAPHY

AND

WATER FLOW

IN THE

LANDSCAPE

The field slopes gently from the north to the southeast (Figure 13.1). This generates flow of water and materials toward the riparian area on the eastern border of the field. Zones of high positive tangential curvature [convergent flow (red)] are aligned with the flow vectors (Figure 13.3) so that zones are more continuous in the east to west direction than in the north to south direction. These data suggest that landscape properties would have some relation to the correlation scales of crop yield and soil hydraulic properties. Semivariance for crop yields has a longer range in the east to west direction than in the north to south direction and suggests that the soil conditions that 150550

Convergent Flow

150500

0.01

150450

150400

0

150350

-0.05

150300

-0.4

Divergent Flow

150250

413200

413250

413300

413350

413400

FIGURE 13.3 (See color insert following page 144.) Tangential curvature and water flow vectors calculated from a 5- × 5-m DEM grid.

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Semivariance (Kg/ha)2

2000000

1500000

1000000

500000

0 0

50

100

150

200

Distance (m) East/West

North/South

North/South Measured

East/West Measured

FIGURE 13.4 Semivariogram of the 1998 yield data for N–S and E–W directions

affect crop growth also have this correlation structure (Figure 13.4). Fine soil material with higher WHC and nutrients would tend to accumulate in the areas that are concave and converge water flow and be removed from areas that are convex and shed water.

B. TERRAIN VARIABLES

AND

SOIL PROPERTIES

Results from the same field7 showed that topographic variables complemented each other in distinguishing zones of different texture within the landscape. Sands and silts were separated reasonably well by slope, and tangential curvature discriminated transects by clay. Tangential curvature and slope were significantly related to water contents at 10 and 33 kPa.7 Figure 13.5 shows the relationship between WHC (the difference between the 1500 and 10 kPa water contents) and soil terrain variables: slope, tangential curvature, and profile curvature. There was only a weak relationship between profile curvature and WHC. The relationships between WHC and slope, and WHC and tangential curvature are stronger. In this site, the clay and silt contents of the soil increase from the north section of the field to the south as elevation decreases. Slow erosion of soil over a long period of time has resulted in movement of fines downslope and an increase in WHC. The relationship with slope is stronger than with tangential curvature. Tangential curvature rather than slope, however, was useful for distinguishing transects A and C, which differed with respect to soil texture. The contour map of WHC in Figure 13.6 illustrates the relationship with slope where WHC is increasing downslope with increasing silt content. Areas of equal WHC are distributed perpendicular to the slope.

C. SPATIAL AUTOREGRESSION Spatial autoregressions were carried out with terrain variables calculated from elevation grids for a range in scales. Because the relationship with profile curvature was weak, only slope and tangential curvature were used as regressors. The relationships shown in Figure 13.5 appear nonlinear. A linear relationship was used here as a first approximation because there was little prior information on how the linearity would be affected at different scales. The regression results in Table 13.1 show that a large part of the variation had a spatial component. The value of α, the spatial autoregression parameter, varied from 0.49 to 0.75 and was largest for terrain variables calculated on the finest © 2003 by CRC Press LLC

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0.20 Transect A Transect B Transect C Transect D Grid

0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.0

0.9

1.8

2.7

3.6 -0.4

Slope x 100 (m m-1)

-0.2

0.0

0.2

0.4 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4

Profile curvature x1000 (m-1)

Tangential curvature x 1000 (m-1)

FIGURE 13.5 Relationships between WHC and the terrain attributes, slope, profile curvature, and tangential curvature.

0.06

0.08

0.10

0.12

0.14

0.16

0.18

WHC (cm3/cm3) FIGURE 13.6 Elevation map of study site with contours of measured WHC. The field boundary is shown.

grids. The amount of error explained by α was about five times that explained by the regression. The proportion of the sums of squared differences explained by α was fairly constant among all the scales, about 0.041. The error explained by regression varied from 0.02 to 0.012 cm3 cm–3. The total sums of squares, which is the variation about the mean of WHC, was constant for all scales because the same values of WHC were used in all the regressions; only the terrain variables differed among scales. The error sums of squares were also fairly constant, which is a reflection of the large component of spatial variability and also shows that, as the error explained by α decreases, the © 2003 by CRC Press LLC

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TABLE 13.1 Regression Parameters for Slope and Tangential Curvature from Spatial Autoregression and Ordinary Least Squares Regression Ordinary least squares parameters

Spatial autoregression parameters Scale m×m

α

Intercept

P

Slope

P

× × × × × × × × × × ×

0.75 0.76 0.72 0.64 0.56 0.61 0.50 0.63 0.61 0.59 0.58

0.037 0.034 0.041 0.052 0.061 0.055 0.066 0.052 0.057 0.059 0.064

0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

–0.325 –0.294 –0.400 –0.505 –0.478 –0.512 –0.340 –0.439 –0.631 –0.667 –0.942

0.005 0.004 0.000 0.000 0.002 0.003 0.056 0.045 0.004 0.002 0.000

05 10 15 20 25 30 35 40 45 50 55

05 10 15 20 25 30 35 40 45 50 55

Tangential curvature 2.027 1.030 –1.140 10.635 21.238 16.236 39.052 14.196 10.072 13.280 0.629

p

Intercept

Slope

0.129 0.499 0.651 0.057 0.002 0.050 0.000 0.127 0.458 0.298 0.971

0.141 0.136 0.141 0.143 0.137 0.141 0.132 0.136 0.144 0.143 0.150

–1.137 –0.948 –1.276 –1.399 –1.108 –1.320 –0.647 –0.957 –1.505 –1.560 –2.149

Tangential curvature 2.330 2.123 –3.196 21.230 44.495 36.532 77.906 48.011 31.250 36.319 8.394

Note: The slopes and tangential curvatures were calculated from elevations interpolated to the scales given below. The units for WHC are cm3 cm–3, for the slope parameter are (cm3 cm–3) per (m m–1)and for the tangential curvature parameter are (cm3 cm–3) per m–1.

error explained by regression increases. The root mean squared error of the pure OLS model was somewhat more than the root mean square error of the spatial autoregressive model. A large component of the error is explained by the spatial location and shows the importance of using local information to reduce error in regression relationships. A previous study13 investigated relationships between yield and NDVI (normalized difference vegetation index) and reported that, for a 9- × 9-m grid, elevation and NDVI explained only 7% of the variance in yield, while the autocorrelation parameter, α, explained 86% of the variance. The spatial autocorrelation coefficient still explained a large portion of the error in spite of including elevation, a locational parameter in the regression. The parameters for the regression models are given in Table 13.1. The parameter for slope was significant at all scales except the 35- × 35-m scale. Slope is a large scale variable and changes smoothly over distance in this field. The parameter for tangential curvature was not significant at the larger or smaller scales. The tangential curvature, however, is calculated as second differences of elevation and can be roughly interpreted as a change in slope. At the largest scales, the small scale changes in slope are largely smoothed and the effect of tangential curvature (changes in slope) is lost. At small scales, there is so much noise in the data that a meaningful relationship between tangential curvature and WHC cannot be determined. Note that, among the different scales, the largest weight was assigned to tangential curvature at the 35- × 35-m scale where the r2 was highest. The parameter for slope for this scale, however, was not significant at p = 0.05. Also, the parameter for slope is smallest where the parameter for tangential curvature is highest. Some of this effect can be attributed to colinearity between the two variables, which is not unexpected because they are both calculated from elevation. The best fit to the data in terms of r2, error, and significance of parameters was provided by the terrain variables calculated at the 25- × 25-m scale. The terrain variables calculated at the 35- × 35m scale had the lowest error but not all regression parameters were significant. Within the range of the 25- × 25-m and 35- × 35-m scales there is not a clear advantage in terms of r2 and error to calculating terrain variables from any one scale of elevation data. This may be due to the selection © 2003 by CRC Press LLC

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TABLE 13.2 Output Statistics for the Spatial Autoregression of Water Holding Capacity vs. Terrain Parameters Where Terrain Parameters Are Calculated from Different Scales of Interpolated Elevation Data RMSE (cm3 cm–3)

Scale of landscape parameters m×m

α

α

Regression

Error

Total

α

Regression

Overall

Spatial

OLS

× × × × × × × × × × ×

0.75 0.76 0.71 0.64 0.55 0.61 0.49 0.63 0.61 0.58 0.58

0.046 0.046 0.045 0.043 0.039 0.042 0.036 0.042 0.042 0.040 0.040

0.002 0.002 0.003 0.005 0.009 0.006 0.012 0.005 0.006 0.007 0.007

0.016 0.017 0.017 0.017 0.017 0.017 0.016 0.017 0.017 0.017 0.017

0.065 0.065 0.065 0.065 0.065 0.065 0.065 0.065 0.065 0.065 0.065

71.1 71.5 69.6 66.2 60.7 64.5 56.4 65.6 64.5 62.7 62.7

3.3 2.3 4.2 8.2 13.6 9.7 18.0 7.9 9.4 11.0 11.3

75.4 75.0 74.5 74.6 74.5 74.4 74.5 73.8 74.1 73.8 74.1

0.013 0.013 0.013 0.013 0.013 0.013 0.013 0.013 0.013 0.013 0.013

0.022 0.023 0.020 0.018 0.016 0.017 0.015 0.017 0.017 0.016 0.016

05 10 15 20 25 30 35 40 45 50 55

05 10 15 20 25 30 35 40 45 50 55

Sums of squares (cm3 cm–3)2

r2

of the same distance for the grid dimensions on the east to west and north to south sides. Because of the anisotropy in the slope and curvature data, an anisotropic grid may have been a better choice. The regression parameters from the pure OLS model were higher than from the spatial model (Table 13.1). This is because values of WHC predicted by the spatial autoregression have been adjusted by removing the effects of neighboring values. This reduces the magnitude of WHC in the regression relationship. Only the effect of calculating slope and tangential curvature from different scales of elevation data has been investigated. The scale of the measured WHC is the measured scale and has not been changed. Thus the change in α over the different scales reflects the effects of calculating terrain variables from grids interpolated to varying levels of detail from a given set of measured elevations. The lowest value of α was associated with the scale with the highest regression r2 (Table 13.2). This suggests that more spatial error can be explained by stronger relationships with the regression variables. Note that the RMSE for the OLS regression varied over the scales and was generally lower when terrain variables were calculated from the more coarse scales of elevation data (Table 13.2). The RMSE for the spatial autoregression was almost constant (the differences were only seen in the fourth significant digit after the zero). This indicates that the total information content in terrain parameters and WHC was roughly the same among all the scales when spatial relationships were taken into account.

D. INTERPOLATION

AND

MAPPING

OF

WHC

AT

DIFFERENT SCALES

The value of α and the regression coefficients determined from the terrain variables calculated from the elevation grids were used to generate contour plots of WHC at 20- × 20-m, 25- × 25-m and 30- × 30-m resolutions (Figure 13.7). Smoothness and loss of detail increase as the size of the grid on which terrain variables were calculated increases. The coarsest scale of 30 × 30 m in Figure 13.7 shows larger, connected areas of uniform water availability. WHC in all the maps shows a general pattern of bands of increasing WHC from the upper section of the field to the lower section in the direction of decreasing slope. This is in accord with the measured WHC (Figure 13.6). Overall, as the scale of the elevation data from which the terrain variables are calculated becomes more coarse, the level of spatial autocorrelation for WHC decreases and the amount of © 2003 by CRC Press LLC

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20 x 20

25 x 25

237

30 x 30

150550 150500 150450 150400 150350 150300 150250

413150

413250

413350

413150

413250

413350 413150

413250

413350

0.14 0.12 0.1 0.08 0.06 0.04 0.02

FIGURE 13.7 Contour maps of predicted and interpolated WHC at three scales.

error explained by the regression of WHC on slope and tangential curvature increases. This may be related, in part, to the aggregation of the elevations as terrain parameters are calculated from elevations averaged over larger distances, an effect termed the modifiable unit area problem.13 The significance of the dependence of WHC on slope and tangential curvature also changes with scale. The dependence on tangential curvature is only significant at the midrange scales similar to that of the scale of the elevation data. Interpolation of tangential curvature to scales much finer than the scale at which elevations were measured probably increases the error so very little real information is related to WHC. At the coarsest scales, the effect of smoothing elevation results in a loss of detail in the tangential curvature, reducing its effect on WHC. The dependence of WHC on slope is not greatly different among scales although its effect is least significant at the midscales, 35 × 35 m and 40 × 40 m. At these scales, the effect of tangential curvature is highly significant and the tangential curvature coefficients are largest. This suggests that tangential curvature is accounting for some of the effects of slope, especially at the 35- × 35m scale. These results reflect the scale of the two predictors, slope and tangential curvature. At this site, slope is a large scale variable and varies slowly over large distances. Because tangential curvature is calculated as a derivative, it is sensitive to small changes in elevation and would have a smaller scale than elevation.

E. MEASURED CROP YIELDS

AND INTERPOLATED

WHC

The interpolated water holding capacities were compared to corn grain yields obtained from a yield monitor for the 1997 and 1998 growing seasons. Figure 13.8 shows the relationships between predicted WHC and yield. Second-order polynomial regressions were carried out for the WHC and yield relationships (Figure 13.8). These relationships predict a plateau at higher water holding capacities where increasing WHC no longer results in increasing yields. This relationship is supported by measured water holding capacities and yields at this site.21 Root mean square errors and r2 values for these OLS polynomial regressions (Table 13.3) are presented for approximate comparison purposes only because they are likely to be inflated due to spatial correlation. The errors and r2 values are largely similar among all the scales. Generally, all © 2003 by CRC Press LLC

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1997

1998

8000

20 X 20

6000 4000 2000 8000

Grain yield (kg ha-1)

6000

25 X 25

4000 2000 8000

30 X 30

6000 4000 2000 8000 6000

35 X 35

4000 2000 0 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.06 0.08 0.10 0.12 0.14 0.16

Soil water holding capacity (cm3 cm-3) FIGURE 13.8 Relationships between predicted WHC and crop yields from 1997 and 1998 for several scales.

TABLE 13.3 Root Mean Square Errors (RMSE (cm3cm–3)2) and r2 Values from the Second-Order Polynomial Regression between Interpolated Yield and Predicted Water Holding Capacity at Four Scales 1997

1998

Scale (m × m)

RMSE

r

× × × ×

568.6 543.9 474.2 472.8

43.9 40.3 39.5 24.4

20 25 30 35

© 2003 by CRC Press LLC

20 25 30 35

2

RMSE

r2

1037.3 905.0 900.2 898.9

54.6 55.1 46.6 37.5

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the scales to 30 × 30 m seem to give similar results; only the 35- × 35-m scale seems very different from the others in terms of r2 and RMSE. The minimum values and range of WHC vary by scale (Figure 13.8). The minimum values of WHC predicted at the 20- × 20- and 25- × 25-m grids were lower than in the input data. The 20- × 20- and 20- 25-m scales had the widest range of predicted WHC values and the measured WHC ranged from 0.068 to 0.19 cm3 cm–3. In spite of the extrapolation, however, the relationship of these lowest values with grain yield is consistent with the remainder of the yields at higher WHC. Others22 have reported correlations on the order of 0.13 and 0.20 for corn and soybean yields, respectively, with soil water storage. The strength of the correlation varied by elevation. The larger range of WHC values in the finer scales probably contributes to the higher r2 values. Also, the r2 value is proportional to the ratio between the amount of error explained by the model and the total error. As in the previous data, there is no clear advantage to choice of scale. When comparing the use of a single scale of yield data to scaling the yield data to the WHC scale, the latter method appears to result in lower RMSE over all the scales. A water budget model has been used21 to estimate soil water availability from yield map data collected after seasons with below average rainfall. The correspondence between grain yield and WHC found here is in agreement with the results of that study. This also suggests that the optimization method used21 could be improved by including terrain variables and using some method of stochastic simulation to distribute the optimized water holding capacities spatially in the landscape.

IV. CONCLUDING REMARKS Water holding capacities in the upper 10 cm of soil were sampled on transects and a grid on a 6-ha field. The objective of the study was to interpolate these sparsely sampled data to a more dense grid using terrain variables calculated from more readily available elevation data. Terrain variables (slope, tangential curvature and plan curvature) were calculated from elevations interpolated to scales from 5 × 5 m to 55 × 55 m. Spatial autoregression was used to predict WHC at the measured locations using calculated terrain parameters as predictors. Slope and tangential curvature were found to be significant predictors of surface WHC. The spatial autoregression parameter, α, explained 60 to 70% of the variance in the relationship between terrain variables and WHC. The 25- × 25-m scale of terrain variables gave the best fit. At finer scales, there was too much noise and at more coarse scales too much smoothing of the terrain attributes. The water holding capacities were predicted on regular grids of 20- × 20-, 25- × 25-, 30- × 30- and 35- × 35-m dimensions and compared with crop yields measured with a yield monitor and interpolated to the same grids using block kriging. There was a good correspondence between predicted WHC and measured corn grain yields. Prediction of WHC to fill in “holes” using spatial autoregression is similar to forecasting time series analysis. Error is minimized by using neighboring values to better estimate WHC at a specific location.15 In this sense, local information is taken into account. Estimated points that are further apart would have a higher prediction error than points closer together. Other geostatistical methods, such as co-kriging, have been used to relate two variables spatially, but this requires that the semivariograms for the variables be modeled as one semivariogram.14 One advantage of spatial autoregression is that it is primarily a statistical tool as opposed to geostatistics where the principal goal is to produce a generalized map surface (Griffith and Layne,15 page 469). Both methods, however, seek to quantify spatial autocorrelation and so have many similarities15 describing linkages between autoregressive and semivariogram models (Griffith and Layne,15 page 469). As shown in this chapter, spatial autoregression is a useful tool to develop prediction equations for spatial data. Water holding capacities used in spatial models are often estimated from soil maps and soil texture.23 Although textural components, as well as soil map units, are closely related to topography, the estimated water holding capacities may not be realistically spatially distributed. This spatial © 2003 by CRC Press LLC

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distribution can be better accounted for by distributing the water holding capacities as a function of topography. The WHC predicted from terrain attributes using spatial autoregression can be used to generate a map of WHC. Such a map will be useful to develop management zones for this field or to use as input in crop models.

V. ACKNOWLEDGMENTS The authors wish to express their gratitude to Dr. Kelley Pace and Dr. Jhonathan Ephrath, who provided very helpful suggestions for early drafts of the paper, and to the anonymous reviewers for their highly insightful comments and suggestions. These greatly improved this chapter.

REFERENCES 1. Yeh, P.J.F. and Eltahir, E.A.B., Stochastic analysis of the relationship between topography and the spatial distribution of soil moisture, Water Resour. Res., 34, 1251, 1998. 2. Western, A.W., Grayson, R.B., Bloschl, G., Willgoose, G., and McMahon, T.A., Observed spatial organization of soil moisture and its relation to terrain indices, Water Resour. Res., 35, 797, 1999. 3. Tomer, M.D. and Anderson, J.L., Variation of soil water storage across a sand plain hillslope, Soil Sci. Soc. Am. J., 59, 1091, 1995. 4. Hanna, A.Y., Harlan, P.W., and Lewis, D.T., Soil available water as influenced by landscape position and aspect, Agron. J., 74, 999, 1982. 5. Moulin, A.P., Anderson, D.W., and Mellinger, M., Spatial variability of wheat yield, soil properties and erosion in hummocky terrain, Can. J. Soil Sci., 74, 219, 1994. 6. Sudduth, K.A., Drummond, S.T., Birrell, S.J., Kitchen, N.R., and Stafford, J.V., Spatial modeling of crop yield using soil and topographic data, in Precision Agriculture '97. Volume I. Spatial Variability in Soil and Crop. Papers presented at the First European Conference on Precision Agriculture, Warwick University, UK, 7-10 September, 1997, 439–447. 7. Pachepsky, Ya., Timlin, D.J., and Rawls, W.J., Soil water retention as related to topographic variables, Soil Sci. Soc. of Am. J., 65, 1787, 2001. 8. Timlin, D.J., Pachepsky, Ya., Snyder, V., and Bryant, R.B., Spatial and temporal variability of corn yield on a hillslope, Soil Sci. Soc. Am. J., 62, 764, 1998. 9. Sinai, G., Zaslavsky, D., and Golany, P., The effect of soil surface curvature on moisture and yield — Beer Sheba observation, Soil Sci., 132, 367, 1981. 10. Moore, I., Gessler, P.E., Nielsen, G.A., and Peterson, G.A., Soil attribute prediction using terrain analysis, Soil Soc. Am. J., 57, 443, 1993. 11. Bouma, J., Using soil survey data for quantitative land evaluation, Adv. Soil Sci., 9, 177, 1989. 12. Pachepsky, Ya., Rawls, W.J., and Timlin, D.J., The current status of pedotransfer functions: their accuracy, reliability, and utility in field- and regional-scale modeling, in Assessment of Non-point Source Pollution in the Vadose Zone, Corwin, D.L., League, K., and Ellsworth, T.R., Eds., American Geophysical Union, Washington, D.C., Geophys. Monograph, 108: 223-234, 1999. 13. Long, D.S., Spatial autoregression modeling of site-specific wheat, Geoderma, 85, 181, 1998. 14. Goovaerts, P., Using elevation to aid the geostatistical mapping of rainfall erosivity, Catena, 34, 227, 1999. 15. Griffith, D.A. and Layne, L., A Casebook for Spatial Statistical Data Analysis. Spatial Information Systems. Oxford University Press, 1999, 506 pps. 16. Mitásova, H. and Hofierka, J., Interpolation by regularized spline with tension: II. Application to terrain modeling and surface geometry analysis, Math. Geol., 25, 657, 1993. 17. Griffith, D.A., Estimating spatial autoregressive model parameters with commercial statistical software, Geogr. Anal., 20, 176, 1988. 18. Griffith, D.A., A numerical simplification of estimating parameters of spatial autoregressive models, in Spatial Statistics: Past, Present, and Future, Griffith, D.A., Ed., Institute of Mathematical Geography. Ann Arbor, MI, 1990, 185–196.

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19. Pace, K.R. and Barry, R., Quick computation of regressions with a spatially autoregressive dependent variable, Geogr. Anal., 29, 232 ,1997. 20. Pace, K.R. and Gilley, O.W., Using the spatial configuration of the data to improve estimation, J. Real Estate Finance Econ., 14, 333, 1997. 21. Timlin, D.J., Pachepsky, Ya., Walthall, C.L., and Loechel, S.E., The use of a water budget model and yield maps to characterize water availability in a landscape, Soil Till. Res., 58, 219, 2001. 22. Logsdon, S., Pruger, J., Meek,, D., Colvin, T., Jaynes, D., and Milner, M., Crop yield variability as influenced by water in rain-fed agriculture, in Proceedings of the Fourth International Conference on Precision Agriculture, St. Paul, Minnesota, USA, 19-22 July 1998. Part A and Part B. Robert, P.C., Rust, R.H., and Larson, W.E., Eds., 1999, 453-465. 23. Turner, D.P., Dodson, R., and Marks, D., Comparison of alternative spatial resolutions in the application of spatially distributed biogeochemical model over complex terrain, Ecol. Model, 90, 53, 1996.

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14

Evaluating Soil Data from Several Sources Using a Landscape Model C.L.S. Morgan, J.M. Norman, C.C. Molling, K. McSweeney, and B. Lowery

CONTENTS I. Introduction...........................................................................................................................243 A. The Precision Agricultural-Landscape Modeling System ........................................244 B. Modern Technological and Analytical Tools ............................................................246 II. Case Study Site ....................................................................................................................248 A. Soil Mapping Strategy 1............................................................................................249 B. Soil Mapping Strategy 2............................................................................................250 C. Soil Mapping Strategy 3............................................................................................250 D. Soil Mapping Strategy 4............................................................................................252 E. Map Comparisons......................................................................................................253 III. Scale Effects: Results and Discussion .................................................................................253 V. Concluding Remarks ............................................................................................................257 References ......................................................................................................................................258

I. INTRODUCTION Over the past 50 years, much advancement has been made in analytical and technical tools for exploring the soil–plant–atmosphere continuum, but the transfer of these innovations to good land stewardship has been minimal. These modern analytical and technical tools include spatial statistics, remote sensing, process-level modeling, user-friendly computer software interfaces, and improved soil mapping techniques. Without a vehicle to deliver modern advances in research and technology to farm managers, continued innovation in crop production and environmental protection will have limited impact. Of the various tools listed above, integrative process-level modeling offers the possibility of providing a framework for incorporating all the other tools into a package for use by farm managers to improve land resource stewardship while maintaining productivity and reducing environmental degradation. The use of process-based models and related innovations requires four categories of input information: weather, landscape, plant, and management. Of these four categories, weather, plant, and management information will continuously change, anywhere from hours (weather) to days (management, such as planting, tillage, harvest) to seasons (plant hybrid). The weather, plant and management information needs to be collected, and fortunately, that is not excessively costly or time consuming in most situations. Landscape information, on the other hand, can be very costly and time consuming to collect. Some landscape information, such as soil nutrient concentration, 243 © 2003 by CRC Press LLC

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changes continuously and should be collected regularly. Other landscape data, such as topography and parameters that describe soil hydraulic properties, are temporally stable from year to year (except in cases of severe erosion or compaction) and can be collected once or at least infrequently. Unfortunately, little is known about what soil information is needed for process-based landscape modeling at the agricultural field scale. Uncertainties remain regarding what soil properties are required; how precisely these properties need to be measured; and on what spatial scale they need to be collected.1 For example, sources for soil information vary from having a relatively high vertical resolution and low spatial resolution (such as USDA soil surveys) to a high spatial resolution and low vertical resolution (such as remote sensing or an electromagnetic landscape survey sensor). Ideally these sources of soil information at different scales should be combined to make the best possible three-dimensional soil landscape map. The objective of this chapter is to evaluate several sources of landscape information by comparing maps of predicted corn yield to measured corn yield using a precision agricultural-landscape modeling system (PALMS).

A. THE PRECISION AGRICULTURAL-LANDSCAPE MODELING SYSTEM A brief description of PALMS is required because we will use it to evaluate various levels of soil information. PALMS is an example of a quasi-three-dimensional modeling system that was designed to transfer new technology to farm managers. PALMS was developed as a primary component of a precision agriculture decision support system under NASA’s Regional Earth Science Applications Center (RESAC) program. The user-interface for PALMS is still under development; therefore, PALMS is not yet available for distribution. The objective guiding PALMS development was to simulate key hydrologic and biophysical processes at a level of physical realism and spatio-temporal detail (spatially, the subfield scale, 5- to 20-m resolution) sufficient to evaluate the physical and economic consequences of specific cropping, tillage, and fertilizer management strategies. PALMS is a combination of two models: 1) a two-dimensional, diffusive wave, runoff model2 with ponding, and 2) a one-dimensional, point-column, land-process or biophysical model called the integrated biosphere simulator (IBIS),3,4 which has been extensively tested4,5 (Figure 14.1). PALMS is structured on a three-dimensional grid, with the horizontal dimensions (easting and northing) consisting of a constant-sized grid, and a third dimension consisting of vertical soil layers. Using a digital elevation map, a grid with a grid-cell size of 5 to 20 m is set up on the field of interest and the one-dimensional IBIS model is run at each grid point. PALMS incorporates varying soil texture at the subfield scale and assigns variable hydrological properties to each grid location according to values typical of its textural class. Hourly weather data provide the atmospheric forcing that drives the PALMS physical system. When precipitation occurs in excess of infiltration and detention storage, the diffusive-wave model is activated and rainfall is simultaneously routed over the landscape and infiltrated into the soil. PALMS includes ponding and reinfiltration on a field, something missing from most existing runoff models. PALMS simulates runoff patterns, as affected by anisotropic surface roughness (caused by row tillage), till-angle interactions with topography, and the change of random roughness with accumulated precipitation.6 The influence of surface sealing, which has been demonstrated by various authors,7 is also included in PALMS. The IBIS component of PALMS predicts a relatively complete hierarchy of ecosystem phenomena, including a) land surface physics (energy, water, and momentum exchange within the soil–vegetation–atmosphere system); b) canopy gas exchange (photosynthesis, respiration, and stomatal behavior); c) vegetation phenology (seasonal cycles of leaf development, reproductive development, and leaf senescence); d) whole-plant physiology (allocation of carbon and nitrogen, plant growth, tissue turnover, and age-dependent changes); and e) carbon and nitrogen cycling (flow of carbon and nitrogen between the atmosphere, vegetation, litter, and soils including mineralization and decomposition). Agricultural crops are simulated based on the approaches of CERES-Maize8 and EPIC9 so that both yield and harvest index are predicted.

© 2003 by CRC Press LLC

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Data Inputs Observed weather Soil landscape map Topography

245

Management

Surface effects • roughness • row anisotropy • crusting

Tillage type and timing Crop cultivar Planting date

Land Surface Module - IBIS

Diffusive Wave Module Water runoff Water ponding

Vegetation physiology Canopy physics • energy balance • photosynthesis • primary production • water balance • aerodynamics Vegetation phenology • maturity stages • C, N allocation • grain drying

Soil physics • energy balance • water balance • C, N cycling

Outputs • Soil moisture • Soil temperature • Surface crusting • Infiltration

• Biomass-yield • Transpiration • Grain moisture • N-Stress

• Runoff • Ponding • Evaporation • Drainage

FIGURE 14.1 The precision agricultural-landscape modeling system (PALMS) represented in a flowchart. IBIS is the integrated biosphere simulator, a module in PALMS.

PALMS uses various inputs and predicts, among other quantities, a yield map for a particular field for a particular season. The plant growth and development model in PALMS, like the models it is based on, is complex and generally will not fit yield data from a given field accurately unless the model is calibrated. Calibration means parameters in the model are varied until the various quantities predicted by the model more closely match field measurements. This is not an acceptable procedure for farm application because research data needed for calibration will not be available; therefore, PALMS contains generic corn and soybean crop models4 that may contain biases for a given field but should produce appropriate spatial patterns. This bias can be reduced by considering a measured yield map, measured by a combine, as an input and using the difference between PALMS’ predicted yield map and the measured yield map to refine the PALMS predictions. The challenge here is to know which quantities in the model to vary to reduce the difference between predicted and measured yield maps. Additional data, such as remote sensing images, may also help to refine this procedure. For PALMS to simulate plant, soil, and microclimate processes that occur in commercial farm fields, it must have an “adequate” three-dimensional soil map on a 5- to 20-m grid with detailed topography. PALMS requires information describing the horizontal and vertical distribution of the following soil characteristics: porosity, organic matter, texture, hydraulic, and thermal characteristics. Sources for this information vary from readily available published material (such as USGS topographic quads and USDA soil surveys), to most intensive and time consuming in situ field measurements (such as hydraulic conductivity from a tension infiltrometer). Although PALMS clearly needs an adequate description of soil properties as input, a major research subject is to determine the meaning of “adequate” in this context. Sadler et al.1 investigated the effect of improving the precision of soil data requirements for the CERES-Maize model with the goal of determining the required spatial precision of soil information for predicting spatial variation of yield. Sadler et al.1 found no significant improvement in the model results when precision of soil data was increased; however, the model used was not © 2003 by CRC Press LLC

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specifically designed for landscape-scale interactions of crop and soil processes. The intent of designing PALMS was to incorporate biophysical interactions at a landscape scale, creating a connection between processes at a location with processes of surrounding locations. For example, a field may receive uniform rainfall, but one area in the field with a low infiltration rate may cause some of the water to run down slope to an area that can infiltrate the excess water. The results detailed in this chapter represent an attempt to evaluate various sources of soil data, collected at different spatial scales, using PALMS. From these soil data, soil hydrological,10 thermal,10 and surface-sealing11 properties are estimated along with rooting depth. Soil survey information provided by the USDA soil survey is generally the first resource used when beginning to map soils across a landscape. However, soil survey information alone usually is not adequate for precision soil landscape mapping because of its coarse scale. In addition, the soil boundaries are discrete and soil pedological properties (horizonation, color, and structure) vary in their relationship to soil functional properties (hydraulic and thermal conductivity, bulk density, and soil moisture characteristic curve). A common scale for U.S. national soil survey maps is 1:20,000 with the smallest scale at 1:12,000 and a practical minimum delineation of 0.3 to 0.4 ha.12 The spatial scale associated with variations in soil water storage is less than 0.3 ha in many instances. The range of spatial correlation of soil hydraulic properties has been measured to be less than 30 m.13,14 In addition, the variations of soil pedological properties may have different scales from functional soil properties related to management.15 Sadler et al.16 found that soil map units are not homogenous with respect to water relations in water-stressed corn. Soil survey information may be most useful for supplying general information about the vertical and horizontal soil heterogeneity to assist with design of more intensive soil mapping activities. A few in situ soil measurements may characterize a uniform field at low cost, but in situ measurements may be a prohibitively expensive way to characterize a heterogeneous field. Optimal spatial resolution of soil sampling represents a trade-off between the cost of sampling and the information content required for reliable decision-making. Field-to-field differences in spatial variability suggest that a single optimal grid spacing is not likely to be developed. Wollenhaupt et al.17 investigated 32- to 96-m grid spacings and concluded grid spacings less than 96 m were required for precision agriculture. In contrast, Franzen and Peck18 suggested a grid spacing of less than 30 m was required for accurate fertilizer application. In one field, Cahn et al.19 reported that by reducing the sampling interval from 50 to 1 m, the variance of soil water content, soil organic carbon, nitrate, phosphorous and potassium were reduced by 74, 95, 25, 64, and 58%, respectively. Although this finding is logical, the additional cost of 1-m soil sampling is prohibitive.

B. MODERN TECHNOLOGICAL

AND

ANALYTICAL TOOLS

Clearly, soil-sampling requirements for characterizing soil across the field will vary depending on the field variation. A high spatial resolution of soil sampling is not generally feasible. Therefore, it is essential that we capitalize on methods that can efficiently and noninvasively characterize soils. Geostatistics are an essential methodology that can capture inherent heterogeneity of natural systems, particularly soil. Geostatistics and more traditional statistical methods can link information collected through sparse soil core sampling with data collected using landscape survey sensors, remote sensing, and other landscape mapping techniques, such as penetrometers. Application of geostatistical tools to soil science has increased greatly over recent years. Kriging and its variations have been the most popular geostatistical tool for interpolation in soil science. Other interpolation methods include nearest neighbor, Laplacian smoothing splines, and inverse distance. Kriging, however, has the advantage of using spatial structure of the data to aid in predictions. Whelan et al. summarized from other research that when more than 100 sample points are measured, which is the case when using remote sensing or survey sensors, kriging and Laplacian smoothing splines have the lowest prediction sums of squares.20 In soil science, kriging is generally the preferred method over Laplacian smoothing splines for two reasons: 1) kriging has a method © 2003 by CRC Press LLC

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for error estimation,20,21 and 2) ordinary point kriging has many hybrids and forms, such as local kriging,22 cokriging,23,24 universal kriging,25 regression kriging26 and block kriging,27 for improving estimations under different conditions. Instrumentation advances such as yield monitors, electrical conductivity terrain meters, and penetrometers allow us to collect information with a high spatial resolution (meters) over large areas; however, the information is only indirectly related to the soil properties we require for precision modeling. Geostatistics can then be used to couple this lower level of information with high-level soils information collected by taking soil cores at relatively few locations. Many large-scale producers have been incorporating yield monitors into their management strategies for the past 6 years. The popularity of yield monitors among producers will continue to increase as the price of instrumentation decreases. Some fields, especially those involved in research, have 6 years or more of yield monitor data collected. Yield monitor data are relatively easy to collect at high spatial resolution, once the yield monitor is calibrated. Because of these characteristics, yield monitor data have much potential for precision agriculture. Yield monitor data can be used for mapping soils in two ways. One way is to correlate yield data directly with a measured soil property from soil cores. Some researchers have found relationships between yield and rooting depth,28 depth to a Bt horizon,29,30 and depth to a calcic horizon;30 however, these correlations depended on the weather conditions and required soil coring to establish the relationship. A second method for using yield monitor data, which does not require the collection of soil cores, combines the yield data with a crop model and weather data to create a map of the amount of water the soil had to supply in order to get the measured yield. This second method, referred to as an inverse method, provides an integrative measure of the water-supplying capacity of the soil, which can vary across the field for many reasons. A simple inverse yield model (SIYM) was created by Morgan et al.31–33 to map plant-available water from yield maps by using inverse water-budget modeling. The input requirements for SIYM are solar radiation, temperature, precipitation, and daytime vapor pressure deficit. This plantavailable water map can then be used to calculate the depth to glacial till horizon by using the textural class and volumetric water content measurements reported by Rawls et al.34 A similar inverse method was proposed by Timlin et al.35,36 Although the estimates of plant-available water from yield-monitor inverse methods provide crude information in the vertical dimension compared to core sampling methods, the horizontal resolution is better by several orders of magnitude. Reducing the need for soil coring make these inverse methods relatively inexpensive and convenient to use. Landscape survey sensors are also valuable tools for creating soil maps. A survey sensor sensitive to a soil characteristic, such as electrical conductivity, can be calibrated with 10 to 20 soil cores, fixing the cost of sampling while not compromising on spatial resolution. The spatial variability of some soil properties over a field determines if a survey sensor improves the overall knowledge of the field. Two instruments widely used for measuring bulk soil electrical conductivity (EC) in situ are the EM38 terrain meter (Geonics Limited, Mississauga, Ontario), the Veris System (Veris Technologies, Salina, Kansas). Both instruments are ideally constructed for the purpose of rapidly collecting georeferenced measurements of soil properties across the landscape. The EC meters respond to the average bulk soil electrical conductivity between the surface and a maximum depth of 1 m (Veris) or 1.5 m (EM38). The EC of a soil is dependent on the salt, water content, and the clay mineralogy present in the soil. Bulk soil electrical conductivity measurements were first applied for mapping soil salinity in arid regions where accumulation of salts in soil hinders crop performance.37,38 The need to map soil features over large areas has increased the popularity and diversified the applications of EC measurements. The direct response of EC to soil clay and water content has naturally led to uses of the Veris and EM38 for mapping these properties, for example, clay content39 and depth to a clay pan.40 If a field has low salinity and uniform clay minerology, EC measurements © 2003 by CRC Press LLC

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should be dependent on clay amount and water content, which are likely correlated in a field with uniform management. Kachanoski et al.41 demonstrated a curvilinear relationship (r2 = 0.96) and Sheets and Hendrickx42 demonstrated a linear relationship (r2 range 0.19 to 0.64) between EC measurements and soil water content. Sheets and Hendrickx42 also determined five neutron probe measurements to be adequate to calibrate the EC readings to soil volumetric water content. EC measurements are also correlated to soil variables not directly measured by EC. These soil variables include organic matter content (r2 = 0.59),43 the soil pesticide partitioning coefficient (r2 = 0.34),43 exchangeable cations (r2 = 0.64),44 and depth of the A horizon.45 Because EC does not directly measure soil physical and chemical properties, it is often the case that correlations between these soil properties and EC are not consistent in every field. A penetrometer can be used to collect soil profile information at discrete locations on a landscape. While the information is not continuous across the landscape like EC-meter data, the penetrometer data contain continuous vertical information about the soil profile. The cone index is defined as force per unit area and the measurement is based on the force of soil against the tip of the penetrometer. The cone index has been related to soil properties such as bulk density,46–48 particle size distribution,48 compaction,49 and cementation.50 More recently, research has addressed using the penetrometer to map soil horizons. Rooney and Lowery51 demonstrated that a cone penetrometer was able to show changes in the soil properties with depth. These authors suggested the penetrometer may be used for mapping soil horizon thickness and boundary delineations. Grunwald et al.52,53 also concluded cone index profiles could be used for landscape mapping of contrasting soil materials, such as differentiating glacial till from loess parent materials. Most penetrometer surveys have been conducted using a penetrometer that measures tip force only. Grunwald et al.52 suggested the use of a tip-and-sleeve penetrometer to decrease the associated uncertainties with using penetrometers to map soils. A tip-and-sleeve penetrometer measures tip force and sleeve resistance; the sleeve resistance is the resistance of the soil on the side (or sleeve) of the penetrometer. The additional sleeve resistance measurement can facilitate differentiation between a compacted fine-textured soil and a coarse-textured soil.54 Soil horizons with relatively more sand will create less friction on the sleeve portion while an increase in fine particles will increase the friction on the sleeve.54 The tip-and-sleeve penetrometer was able to map a compacted layer in a field using the additional sleeve information.54 Further penetrometer developments and research offer exciting improvements in the ability to map soils across the landscape rapidly and accurately. Grunwald et al.48 demonstrated the likelihood of developing pedotransfer functions to map soil texture and bulk density using the profile cone penetrometer. Soil imaging equipment has been added to the penetrometer with the purpose of quantifying macropore, color, and structural information.55 Further research to standardize these methods will ensure the use of penetrometers as part of the foundation of methods for soil landscape mapping.

II. CASE STUDY SITE A 1.8-ha field (43°18′’01.45″N, 89°23′05.57″W) located on the University of Wisconsin’s Arlington Research Station in Columbia County was used in this study. The field landscape included a closed depression and less than 1% slope with an overall elevation change of 1.4 m (Figure 14.2). The two soil-mapping units in the field included the Plano silt loam (fine-silty, mixed, mesic Typic Argiudoll 0 to 2% slope) and the Ringwood silt loam (fine-loamy, mixed, mesic Typic Argiudoll 1 to 6% slope). The Plano silt loam was the dominant soil mapping unit found in the field; it was described by the USDA soil survey as a moderately well drained soil with high available-water holding capacity and high fertility. The Plano silt loam was located throughout the majority of the field except the northeast corner. The Ringwood silt loam was located at the most eroded area of the field, the northeast corner. The Ringwood soil was described as well drained with medium available-water capacity and high fertility. The field was managed in a cropping rotation of corn © 2003 by CRC Press LLC

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N

269.2

269.4

269.6

269.8

270

270.2

meters

FIGURE 14.2 The surface map of the field elevation — measurements have better than 0.1 m precision.

(Zea mays L.) and soybeans (Glycine max (L.) Merr.). For years during which corn was planted (1999 and 2000), the row spacing was 0.76 m and the corn was harvested for grain with a combine equipped with a yield monitor. Four different soil mapping strategies were created to evaluate which measurable attributes were most important for improving the utility of landscape models. Each of the four strategies was created to illustrate an increasing knowledge of the field and an increase in effort to obtain the knowledge. The first strategy assumed a basic knowledge of the field obtained without making any direct or indirect measurements of the soil or landscape. The USDA soil survey and USGS topographic map were the two sources of information used to describe the landscape for the model for Strategy 1. The second strategy added to the basic information acquired from the USDA soil survey by using yield maps from the producer and surveying the field for accurate elevation data (< 0.1 m vertical precision) with a survey grade GPS (Trimble 4600LS Antenna, Trimble, California). A third strategy used the yield monitor data and GPS elevation, and included surveys of electrical conductivity and a penetrometer. Additionally, the third strategy used ten soil cores to correlate the instrument surveys to soil properties such as a horizon location, depth, and texture based on field observations. The fourth strategy used all the previous information plus laboratory measurements of texture and bulk density from the soil cores and field measurements of saturated and unsaturated hydraulic conductivity.

A. SOIL MAPPING STRATEGY 1 The first soil mapping strategy (Strategy 1) used the USDA soil survey map and the USGS topography map. The soil survey map for this particular field was hand-digitized by approximating the boundary between the two soil map units found in the field. Because of the geometry of the soil map units in the field, hand digitizing was accomplished with minimal error. The USGS topography map for the field showed no change in elevation; therefore, the landscape map for Strategy 1 was characterized as having no slope. The descriptions of the soil horizons were taken directly from the soil survey description of the soil series and map unit (Figure 14.3A). The soil series and map unit descriptions provided textural classification; clay and sand percentages were selected from the USDA soil texture triangle according to the soil texture classification given in © 2003 by CRC Press LLC

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TABLE 14.1 Soil Horizon Descriptions Translated from the USDA Soil Survey of Columbia County, Wisconsin, for Input into the Precision Agricultural-Landscape Modeling System Texture

Silt loam Silt loam Silt loam (heavy) Silty clay loam Loam Sandy loam

Silt loam Silty clay loam Sandy clay loam Loam Sandy Loam

Lower depth cm

% Clay

Plano silt loam 0 to 2% slope 46 18 56 18 79 25 127 30 140 20 >165 10 Ringwood silt loam 1 to 6% slope 25 18 50 30 60 30 73 20 >165 10

% Sand

10 10 10 10 45 60

10 10 55 45 60

the description (Table 14.1). Soil physical properties required for running PALMS were obtained by matching the textural class of each horizon to physical properties published in Rawls et al.34 The physical properties used are listed in Table 14.2.

B. SOIL MAPPING STRATEGY 2 The second strategy (Strategy 2) modified the lower depth of the loam horizon using the yield maps and more precise elevation information from a survey grade GPS (Figure 14.3B). The lower depth of the loam horizon was modified on the following basis: The sandy loam (glacial till/outwash) layer below the loam horizon was root limiting; therefore, the zone of soil from the soil surface to the bottom of the loam horizon provided all the soil-stored plant-available water. A plant-available water map for field was estimated from the 1999 yield map using inverse water-budget modeling based on measurements of solar radiation, temperature, precipitation, and vapor pressure deficit available online through the Wisconsin Agricultural Weather Observing Network.31–33 The plantavailable-water map created for the field had the same spatial resolution as the yield monitor data (~5 m) and was converted into a map of the lower depth of the loam horizon. The results from using the water budget model produced a map of the lower depth of the loam horizon with a mean of 0.9 m and range of 0.6 to 1.9 m deep. The elevation data were collected with the survey grade GPS on 5-m transects while collecting continuous elevation measurements every 1 s, a speed of approximately 4 m s–1. The yield map (and subsequently the lower boundary of the loam horizon) data were interpolated into a regular 5-m grid using block kriging. The elevation data were interpolated into a regular 5-m grid using ordinary kriging. Like Strategy 1, the soil physical properties from Rawls et al.34 were used (Table 14.2).

C. SOIL MAPPING STRATEGY 3 The third strategy did not follow the concept of the two previous strategies, in which two discrete soil map units were located in the field. Strategy 3 was different because it assumed a continuous change of soil horizonation across the field (Figure 14.3C). The soil survey was still used for a © 2003 by CRC Press LLC

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Plano

N

251

Ringwood A

A Bt 1

Bt 2Bt 1

Bt 2

2Bt 2

Bt 3 2Bt

2BC

2BC

B)

Plano A Bt 1 Bt 2 Bt 3 2Bt 2BC

C)

Ringwood A Bt 2Bt 1 2Bt 2 2BC

A E Bt

2C

FIGURE 14.3 Three soil-mapping strategies illustrated in three-dimensions A) Strategy 1 (Plano and Ringwood soils), B) Strategy 2 (Plano and Ringwood soils), C) Strategies 3 and 4 (four soil horizons). The soil maps are rotated clockwise from north to show the east and south edges of the field; these edges best illustrate the soil changes.

general description of how soil varied across the field, but landscape survey sensor measurements, such as bulk soil electrical conductivity and penetrometer resistance, were used to determine where the soil varied across the field. The EM38 terrain meter was pulled across the field on 5-m transects while making continuous georeferenced measurements. Ten soil core subsamples were collected to a maximum depth of 1.7 m at the same time the EM38 survey was conducted. The soil cores were collected with a 2-cm diameter push probe. For these ten samples, the lower depth of the A horizon, the upper boundary of the till/outwash horizon and a manual determination of texture for each horizon were recorded. In addition, the penetrometer was pushed through the ground to a depth of 2 m at 36 locations in the field that were selected after the EM38 survey was evaluated. After evaluating the core subsamples, the landscape sensor information, and soil survey descriptions, four major soil horizons were chosen to represent the soil horizonation and physical characteristics of the field. Taxonomically, more than four major soil horizons existed in the field, just as the soil survey indicated; however, we chose to combine soil horizons that were similar in textural class or could not be easily differentiated with soil coring techniques. For example, when soil © 2003 by CRC Press LLC

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TABLE 14.2 Measured Soil Physical Properties of the Four Mapped Horizons in Strategy 4 Compared to the Look-Up Table Values Used in the Strategies 1 through 3 Soil horizon A

Porosity fraction Percent clay Percent sand Campbell’s B exponent Air entry potential (m) Saturated hydraulic conductivity (m s–1)

0.50 22 5 4.7 0.21 1.9 × 10–6

Porosity fraction Percent clay Percent sand Campbell’s B exponent Air entry potential (m) Saturated hydraulic conductivity (m s–1)

0.54 23 5 4.9 0.42 1.1 ×x 10–6

E

Bt

2C

Look-up table 0.50 10 4 4.7 0.21 1.9 × 10–6

0.50 26 8 4.7 0.21 4.2 × 10–7

0.44 10 58 3.1 0.15 7.19 × 10–6

Measured values 0.41 19 3 13.9 0.25 2.5 × 10–7

0.51 27 8 5.2 0.43 8.6 × 10- 7

0.50 10 56 1.6 0.44 7.19 × 10–6

coring, the loam horizon above the till/outwash horizon was difficult to discern because it was so thin and almost undetectable in some locations; therefore, this horizon was not included in the third mapping strategy. The four horizons that were included are the following: 1) an A horizon, 2) a Bt silt loam horizon, 3) a 2C till/outwash horizon, and 4) a compacted E silt loam horizon that had a very high bulk density (difficult to auger through). This fourth E horizon was only found in the basin area of the field, but it was considered to be a root-limiting horizon and was found directly under the A horizon; therefore, it was important to include. For mapping purposes, only the lower depths of the A, E, and Bt and the boundary of the E horizon needed to be quantified. The lower depth of the A horizon across the field was quantified using the EM38 measurements. A linear relationship existed between the A horizon depths from the core samples and the electrical conductivity with an r2 of 0.69 (Y = –3.58 * × + 95.978, pvalue <0.05). One outlier in this regression that occurred in the basin where the E horizon existed was removed. The A-horizon depth measurements were far enough apart that they were not spatially correlated; therefore, the simple linear regression was used to map the A horizon. The E horizon was mapped using penetrometer and core samples. The EM38 also responded to the depth of the E horizon, but the penetrometer was used to map its depth and thickness. The lower depth of the Bt horizon marked the transition from the silt loam loess parent material to the glacial till and outwash parent material. The lower depth of the Bt horizon was measured at a total of 36 locations using penetrometer measurements. The lower depth of the Bt horizon was kriged using two relationships: 1) the linear correlation between the depth of the Bt horizon and elevation, and 2) the spatial structure of the elevation data.

D. SOIL MAPPING STRATEGY 4 The fourth strategy used the same soil landscape map created in Strategy 3, but instead of using the look-up table of properties,34 actual measurements of bulk density, hydraulic conductivity, and the soil moisture release curve were used (Figure 14.3C). Laboratory measurements of sand © 2003 by CRC Press LLC

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and clay fraction were also used to replace the estimated particle size from the soil survey book and the USDA textural triangle. Soil cores were collected at six locations in the field, two replications per location, at 15 cm depth intervals from the soil surface to the till/outwash horizon. Each core sample was 4.75 cm in diameter and 3.6 cm long and was categorized to represent one of the four horizons (i.e., A, E, Bt or 2C). The collective bulk densities of each horizon category were averaged to represent the bulk density of each horizon (A, E, Bt, or 2C). The same core samples that were used to measure bulk density were also used to measure the soil moisture release curve. The volumetric water content of each core sample was measured at about 0, 20, 40, 80, 120, 230, 300, and 330 cm of water tension. Soil moisture retention parameters, air entry potential in J kg–1, ψe, and the Campbell b parameter, were calculated using the Campbell equation, θ ψ m = ψ e   θs 

−b

(14.1)

where ψm is matric potential in J kg–1, θ is volumetric water content in m m–3, and θs is volumetric water content at saturation in m m–3.10 Particle size fractions of sand and clay were also measured from these same soil cores using the hydrometer method.56 The sand and clay fractions measured were also categorized by the horizon they represented and averaged (Table 14.2). The hydraulic conductivities of the A, E, and Bt horizons were measured with a tension infiltrometer.57 The base of the infiltrometer was 20 cm in diameter, and infiltration rates were measured at –15, –10, and –5 cm of tension. The saturated hydraulic conductivity was calculated using the Wooding and Gardner methods outlined in the tension infiltrometer instruction manual (Table 14.2).57

E. MAP COMPARISONS The resulting three-dimensional soil landscape maps created by each of the four strategies were used to initialize the soil requirements for PALMS. Each soil landscape map was run for the 1999 and 2000 growing seasons. The model performed its landscape simulation using a 5-m grid. The resulting yield maps from each of the PALMS simulations were compared to the yield monitor data by visually comparing the yield distributions and spatial patterns in each map and using the root mean squared error (RMSE) calculated with the equation

RMSE =

1 n ∑ ( z ( x i ) −z * ( x i ))2 n i=1

(14.2)

where z(xi) is the yield measured by the yield monitor, and z*(xi) is the predicted yield from PALMS.

III. SCALE EFFECTS: RESULTS AND DISCUSSION The 1999 growing season had average rainfall quantities and intensities for the region including soil moisture near field capacity at planting in the spring, and adequate rainfall distribution throughout the season to prevent severe drought stress. Regardless of the favorable rainfall distribution in 1999, water availability was a yield-limiting factor for some areas in the field. The 1999 measured yield map shows a wide range of yield across the field including lower yields in the northeast corner and in the closed basin (Figure 14.4A). The low yields around the north, south, and east edges of the field were from combine turn-around (north and south side) and the combine not harvesting a full width (east side). © 2003 by CRC Press LLC

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A) Mean = 8.0(0.11) Mg ha-1

N

B) Mean = 9.9(0.03) Mg ha-1 RMSE = 2.0 Mg ha-1

C) Mean = 9.4(0.03) Mg ha-1 RMSE = 1.3 Mg ha-1

D) Mean = 9.6(0.04) Mg ha-1 RMSE = 1.8 Mg ha-1

E) Mean = 9.5(0.06) Mg ha-1 RMSE = 1.6 Mg ha-1

5.7

6.3

6.9

7.5

8.2

8.8

9.4

Mg ha -1 FIGURE 14.4 (See color insert following page 144.) Surface maps including the mean (coefficient of variation) of the 1999 corn yields for A) the measured yield and the yield simulated by the PALMS for the soil landscape maps of B) Strategy 1, C) Strategy 2, D) Strategy 3, and E) strategy 4. RMSE = root mean squared error calculated between simulated and measured yield.

Because water availability did limit yield in some locations of the field, the spatial distribution and depth of the root-limiting 2C and E horizons became influential in determining yields. Yield results from Strategy 1 were uniform for each of the two soil types (Figure 14.4B). In the northeast corner, where the Ringwood soil was located, yield was slightly lower, which was expected because the silt loam layer was shallower there. The yield in Strategy 2 varied with the depth of the 2C horizon because the depth of the 2C horizon affected the available water stored by the soil across the field (Figure 14.4C). Nonetheless, the coefficients of variation for yield of Strategy 1 and Strategy 2 were the same. Strategies 3 and 4 had a different overall spatial pattern of yield than the first two strategies (Figures 14.4D and 14.4E). Many of the differences in the spatial patterns were credited to the differences in the depth and distributions of the root limiting horizons (E and 2C). On average, the 2C horizon was mapped shallower in Strategy 2 (average of 0.9 m) than in Strategies 3 and 4 (average 1.69 m); as a result, Strategy 2 had a lower average yield (Figure 14.4). In Strategies 3 and 4, the area of lower yield in the northeast corner was larger than that of Strategy 2; the yield also had more of a gradient as the elevation decreases into the basin. This decrease in yield with decrease in elevation more closely matched the pattern of the measured yield map. The presence of the E horizon in Strategies 3 and 4 decreased the yield in the basin for 1999, because it limited © 2003 by CRC Press LLC

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water availability to the crop. This yield decrease was present in the yield monitor data (Figure 14.4A) and in the mapping strategies that included the E horizon (Figures 14.4D and 14.4E). The differences in yield between Strategies 3 and 4 were a result of different values for saturated hydraulic conductivity (Ksat). In Strategy 4, the Ksat was smaller in the A horizon and larger in B horizon than in the look-up table Ksat values used for Strategy 3 (Table 14.2). The smaller Ksat value in the A horizon reduced infiltration and the larger Ksat value in the B horizon increased drainage of water. Overall, less water was available for crop growth and yield was reduced by 2% overall. According to the RMSE calculations, Strategy 2 was closest to actual yield measurements at a RMSE of 1.29 Mg ha–1. The 1999 yield map was used to create the Strategy 2 soil map through use of SIYM; however, the lower RMSE was more likely attributed to Strategy 2 having the lowest average yield because the 2C horizon was mapped so shallow. The use of the 1999 yield map to get the depth of the 2C horizon should be noted when evaluating the RMSE for Strategy 2 in 1999. A yield map from a different year would have been preferred, but was not available. The 1999 yield map was not used to create any other soil landscape maps. Because the patterns and spatial variability of the simulated yield are most useful to evaluating the effects of the scales of soil information, the CV is a more descriptive index than the RMSE. Strategy 2 had a much lower variability of yield (CV = 0.03) than the measured yield (0.11). Strategies 3 and 4 had the highest coefficients of variation, 0.04 and 0.06, respectively, and their yield patterns more closely corresponded to the yield monitor yield patterns. The 1999 model results showed the significance of soil type and composition compared to the significance of topography in a season with little runoff of water. The addition of topography from Strategy 1 to Strategy 2 did not produce a more variable yield (CV of 0.03 for both years). However, when the soils were more accurately described, such as with Strategies 3 and 4, the yield reduction in the higher elevations became more prominent and the reduction of yield due to the root limiting E horizon became evident. All of the modeled yield results for 1999 were higher than the yield monitor measurements by 14 to 19%. Many crop growth models reported in the literature are much closer to predicting yield than 14 to 19% because the models have been calibrated for individual fields. Because PALMS was created for use by consultants and the agricultural sector, the PALMS crop growth and yield model have not been calibrated for any particular field and the crop growth model is quite generic. Nonetheless, rigorous testing has been implemented to ensure the quality of the PALMS soil water and runoff routines. We believe that predicting yield is only one of many purposes for which PALMS will be implemented. PALMS will be used for other management decisions such as timing for planting, compaction, and harvest date based on grain dry-down. Currently, the important aspect of the PALMS yield results is the spatial variation and distribution of yield over the landscape of the field. If yield results from Strategy 4 were multiplied by a constant so that the yield from Strategy 4 and the measured yield had the same average, the RMSE between the two yield maps would be 0.73 Mg ha–1, which is an acceptable result for a yield model. Hence the fact that the yield portion of PALMS is not calibrated does not hinder the usefulness in determining the important soil properties — their spatial distribution and effect on yield across the landscape. Rainfall and soil moisture conditions in the spring of 2000 were very different from 1999. The crop emergence in the spring of 2000 was followed by weeks of intense rainfall that resulted in water runoff in sloped areas and ponding of water in the closed basin of the field. This ponding of water killed the crop in a portion of the closed basin (Figure 14.5A). Though rainfall was above average in 2000, the northeast corner of the field did exhibit some water stress towards the end of the growing season. The 2000 yield results for Strategy 1 were uniform across the field within soil mapping units and had the same CV as 1999 (Figure 14.5B). Strategy 1 did not have any water runoff or runon because the landscape was flat; however, the Ringwood soil still demonstrated lower yield. Though rainfall was frequent and plentiful, the Ringwood soil could not supply enough water to the crop © 2003 by CRC Press LLC

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A) Mean = 6.6(0.44) Mg ha-1

N

B) Mean = 9.0(0.03) Mg ha-1 RMSE = 3.7 Mg ha-1

C) Mean = 7.8(0.09) Mg ha-1 RMSE = 3.5 Mg ha-1

D) Mean = 7.1(0.38) Mg ha-1 RMSE = 2.6 Mg ha-1

E) Mean = 7.5(0.19) Mg ha-1 RMSE = 3.2 Mg ha-1

0

1.3

2.5

3.8

5.0

Mg ha

6.3

7.5

8.9

10.4

-1

FIGURE 14.5 (See color insert following page 144.) Surface maps including the mean (coefficient of variation) of the 2000 corn yields for A) the measured yield and the yield simulated by the PALMS for the soil landscape maps of B) Strategy 1, C) Strategy 2, D) Strategy 3, and E) Strategy 4. RMSE = root mean squared error calculated between simulated and measured yield.

for optimum crop yield. Even in a wet year, favorable soil hydraulic properties (such as water storage) were still required to attain maximum crop yield. The inclusion of an elevation change and a shallower 2C horizon in the soil landscape map (Strategy 2) affected the amount of water available for crop growth, particularly in the northeast corner, because of increased runoff and lower water holding capacity (Figure 14.5C). The addition of topography and a depth to 2C in Strategy 2 also increased the CV for yield from 0.03 to 0.09. Even though slope was added to the landscape map in Strategy 2, ponding of water in the basin did not occur long enough to drown the crop. The yield patterns and CVs of Strategies 3 and 4 were more like the measured yield patterns. Strategies 3 and 4 included the compact E horizon in the basin, which caused the crop to drown from ponding water in the basin (Figures 14.5D and 14.5E). The ponded area in Strategy 3 was very similar to the area detectable by yield monitor measurements, while the ponded area in Strategy 4 was much smaller. The CV of Strategy 3 (0.38) had the closest match to the measured yield, and the CV of Strategy 4 (0.19) was the second highest of the four strategies. Strategy 3 overpredicted the measured yield by 6%; Strategy 4 overpredicted yield by 12%. Using measured soil physical properties (Strategy 4) vs. the look-up table (Strategy 3) considerably affected the yield result in 2000. The crop die-out area from ponded water in Strategy 3 was © 2003 by CRC Press LLC

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larger than in Strategy 4. Strategy 4 also had a larger decrease in yield on the hilltop compared to the yield monitor measurements and Strategy 3. The yield was lower on the hilltop in Strategy 4 for the same reason the ponding size decreased; the hydraulic conductivity of the B horizon was higher in Strategy 4. The higher hydraulic conductivity of the B horizon in Strategy 4 drained water faster than Strategy 3, leaving less water for plant growth on the hilltop. Likewise, ponded water in Stategy 4 drained faster than in Strategy 3. Since Strategy 4 had a B-horizon Ksat twice as large as Strategy 3’s B-horizon Ksat, the ponded water drained twice as fast where the E horizon was not mapped in Strategy 4. This faster draining allowed the water ponded over the E horizon in Strategy 4 to flow to the areas that were draining faster. Draining of ponded water in Strategy 4 was fast enough not to kill corn plants in some areas, while in Strategy 3; ponded water could not drain fast enough for corn plant survival in most of the basin area. The faster draining of ponded water in Strategy 4 may have been due to mapping the E horizon improperly, or errors in hydraulic conductivity and lab retention curve measurements. When mapping the E horizon with the penetrometer, discerning whether the increased cone index was a result of the E horizon or the B horizon in the basin was sometimes difficult. When in doubt, the estimate was conservative regarding the extent of the E horizon and may have been too conservative. In addition, the hydraulic property measurements could have been biased. The retention data were measured on 4.75-cm cores, which were relatively small compared to the area that they represented. Pachepsky et al.58 have shown a substantial difference between soil water retention measurements made in the laboratory and those in the field. The difference could be caused by a number of factors including spatial variability, overburden pressure, hysteresis, possible nonequilibrium and bulkdensity scaling.58 The measurements of hydraulic conductivity could also contain biases because of the scale at which they were measured (20 cm diameter disk infiltrometer). In addition, uncertainty arises with the estimate of the hydraulic conductivity; namely, the value of unsaturated hydraulic conductivity depends on the model used to represent the measurement geometry and convert raw data to conductivity estimates.57,59

V. CONCLUDING REMARKS Easily obtainable soil and topography information such as the USDA soil survey and USGS topography was not specific enough for detailed landscape modeling for PALMS. For this particular field, the two most important data requirements were depth of the root-limiting horizons (the E and 2C) and topography. The root-limiting horizon information was most important in 1999 when plant-available water was the major factor limiting crop performance. In 2000, the topography and the E horizon were equally important field-specific data requirements. The topography was necessary to route the excess water and the E horizon was necessary because it limited water infiltration. The technology to map the elevation accurately is available and straightforward to obtain. Methods to measure the root-/hydraulically limiting horizons, such as electrical conductivity and penetrometers, are still evolving and not routine to implement. In this specific exercise, the mapping strategy with the most accurately measured soil properties did not have the lowest RMSE in either year. In 1999, Strategy 4 had the spatial pattern of yield most similar to measured yield, but in 2000 it did not. The hydraulic conductivity of the B horizon in the look-up table was lower than the measured hydraulic conductivity — the main factor for differences between Strategies 3 and 4. Predicting whether or not the same results would occur in other fields is difficult. The differences in Strategies 3 and 4 for the 2 years are an excellent example of how difficult it is to get an accurate representation of hydraulic properties of the whole field. Even if many measurements of hydraulic conductivity could be made, the method for representing an accurate statistical distribution of hydraulic conductivity across the field must be addressed. Most importantly, using methods available today, making hydraulic property measurements (hydraulic conductivity, air entry potential, and Campbell b) across a field at several depths is not practical (time- and expense-wise). Clearly, using a look-up table, such as Rawls et al.34 was just © 2003 by CRC Press LLC

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as effective as field measurements and definitely more practical for representing the soil physical properties (in Table 14.2) needed to run PALMS. Soil survey maps have very detailed vertical profile information; however, more detailed horizontal and less detailed vertical resolution of soil information is more useful. The USDA soil survey information alone was not adequate for precision modeling data needs. Augmenting soil survey information with methods like inverse modeling and landscape survey sensors improves the horizontal resolution required for creating soil landscape maps vital to models like PALMS. Not only do landscape survey sensors and inverse modeling techniques make soil mapping possible for models like PALMS, but they also make models like PALMS potentially feasible for farm management. Emerging technologies, such as tractor-mounted soil sensors, optical penetrometers, improved geostatistical methods, and improved remote sensing methods will be useful.

REFERENCES 1. Sadler, E.J. et al., Site-specificity of CERES-Maize model parameters: a case study in the South Eastern U.S. Coastal Plain, in Proc. Int. Conf. Precis. Agric., Stafford, J. V., Ed., Sheffield Academic Press, UK, 1999, 551. 2. Julien, P.Y., Saghafian, B., and Ogden, F.L., Raster-based hydrologic modeling of spatially-varied surface runoff, Water Resour. Bull., 31, 523, 1995. 3. Foley, J. A. et al., An integrated biosphere model of land surface processes, terrestrial carbon balance, and vegetation dynamics, Global Biogeochem. Cy., 10, 603, 1996. 4. Kucharik, C.J. et al., Testing the performance of a dynamic global ecosystem model: water balance, carbon balance and vegetation structure, Global Biogeochem. Cy., 14, 795, 2000. 5. Delire, C. and Foley, J.A., Evaluating the performance of a land surface/ecosystem model with biophysical measurements from contrasting environments, J. Geophys. Res-Atmos., 104-D14, 895, 1999. 6. Zobeck, T.M. and Onstad, C.A., Tillage and rainfall effects on random roughness: a review, Soil Till. Res., 9, 1, 1987. 7. Norton, L.D., Shainberg, I., and King, K.W., Utilization of gypsiferous amendments to reduce surface sealing in some humid soils of eastern USA, in Soil Surface Sealing and Crusting, Poesen, J.W.A. and Nearing, A., Eds., Catena Suppl. 24, Catena Verlag, Cremlingen-Destedt, Germany, 1993, 77. 8. Jones, C.A. and Kiniry, J. R., Eds., CERES-Maize: A Simulation Model of Maize Growth and Development, Texas A&M University Press, College Station, 1986. 9. Sharpley, A.N. and Williams, J. R., Eds., EPIC — Erosion/productivity impact calculator: 1. Model documentation, U.S. Department of Agriculture Technical Bulletin No. 1768, 1990, 235. 10. Campbell, G.S., Soil Physics with Basic, Elsevier Sci. Publ., New York, 1985. 11. Smith, R.E., OPUS: An Integrated Simulation Model for Transport of Nonpoint-Source Pollutants at the Field Scale, Vol I, U.S. Department of Agriculture, Agricultural Research Service, ARS-98, 1992. 12. Mausbach, M.J., Lytle, D.J., and Spivey, L.D., Application of soil survey information to soil specific farming, in Soil Specific Crop Management, Robert, P.C. and Pierce, F.J., Eds., ASA, CSSA, SSSA, Madison, WI, 1993, 57. 13. Burden, D.S. and Selim, H.M., Correlation of spatially variable soil water retention for a surface soil, Soil Sci., 148, 436, 1989. 14. Shouse, P.J. et al., Spatial variability of soil water retention functions in a silt loam soil, Soil Sci. 159, 1, 1995. 15. Verhagen, J. and Bouma, J., Modeling soil variability, in The State of Site-Specific Management for Agricultural Systems, Pierce, F.J. and Sadler, E.J., Eds., ASA, CSSA, SSSA, Madison, 1997, 55. 16. Sadler, E.J. et al., Spatial scale requirements for precision farming: a case study in the southeastern USA, Agron. J., 90, 191, 2000. 17. Wollenhaupt, N.C., Wolkowski, R.P., and Clayton, M.K., Mapping soil test phosphorus and potassium for variable-rate fertilizer application, J. Prod. Agric., 7, 441, 1994. 18. Franzen, D.W. and Peck, T.R., Field soil sampling density for variable rate fertilization, J. Prod. Agric., 8, 568, 1995.

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19. Cahn, M.D., Hummel, J.W., and Brouer, B.H., Spatial analysis of soil fertility for site-specific crop management, Soil Sci. Soc. Am. J., 58, 1240, 1994. 20. Whelan, B.M., McBratney, A.B., and Rossel, V.A., Spatial prediction for precision agriculture, Precis. Agric., ASA, CSSA, SSSA, Madison, WI, 1996, 331. 21. Isaaks, E.H. and Srivastava, R.M., An Introduction to Applied Geostatistics, Oxford University, Oxford, 1990. 22. Haas, T.C., Lognormal and moving window methods of estimating deposition,J. Am. Stat. Assoc., 85, 950, 1990. 23. Vauclin, M. et al., The use of cokriging with limited field soil observations, Soil Sci. Soc. Am. J., 47, 175, 1983. 24. Oliver, M.A., Geostatistics and its applications to soil science, Soil Use Manage., 3, 8, 1987. 25. Webster, R. and Burgess, T.M., Optimal interpolation and isarithmic mapping: III. Changing drift and universal kriging, J. Soil Sci., 31, 505, 1980. 26. Odeh, I.O.A., McBratney, A.B., and Chittleborough, D.J., Further results on prediction of soil properties from terrain attributes: heterotropic cokriging and regression-kriging, Geoderma, 67, 215, 1995. 27. Burgess, T.M. and Webster, R., Optimal interpolation and isarithmic mapping of soil properties: II. Block kriging, J. Soil Sci., 31, 333, 1980. 28. Timlin, D.J. et al., Spatial and temporal variability of corn grain yield on a hillslope, Soil Sci. Soc. Am. J., 62, 764, 1998. 29. Karlen, D.L., Sadler, E.J., and Busscher, W.J., Crop yield variation associated with Costal Plain soil map units, Soil Sci. Soc. Am. J., 54, 859, 1990. 30. Tolk, J.A., Howell, T.A., and Evett, S.R., Evapotranspiration and yield of corn growth in three high plains soils, Agron. J., 90, 447, 1998. 31. Morgan, C.L.S., Quantifying plant-available water across landscapes using an inverse yield model and electromagnetic induction, M.S. thesis, Univ. of Wisconsin, Madison, 2000. 32. Morgan, C.L.S. et al., Two approaches to mapping plant-available water: EM-38 measurements and inverse yield modeling. Proc. Int. Conf. Precis. Agric., 5th, ASA, CSSA, SSSA, Madison, 2000. 33. Morgan, C.L.S., Norman, J.M., and Lowery, B., Estimating plant-available water across a field with an inverse yield model, Soil Sci. Soc. Am. J., 67, 2003. 34. Rawls, W.J., Ahuja, L.R., and Brakensiek, D.L., Estimating soil hydraulic properties from soil data, in Indirect Methods for Estimating Hydraulic Properties of Unsaturated Soils, Van Genuchten, M.T. et al., Eds., U.C. Riverside Press, Riverside, 1992. 35. Timlin, D.J. et al., Water budget approach to quantifying corn grain yields under variable rooting depths, Soil Sci. Soc. Am. J., 65, 1219, 2001. 36. Timlin, D.J. et al., The use of a water budget model and yield maps to characterize water availability in a landscape, Soil Till. Res., 58, 219, 2001. 37. De Jong, E., Measurement of apparent electrical conductivity of soils by electromagnetic inductive methods, Soil Sci. Soc. Am. J., 43, 810, 1979. 38. Rhoades, J.D. and Corwin, D.L., Determining soil electrical conductivity-depth relations using an inductive electromagnetic soil conductivity meter, Soil Sci. Soc. Am. J., 45, 255, 1981. 39. Williams, B.G. and Hoey, D., The use of electrical induction to detect the spatial variability of the salt and clay contents of soils, Aust. J. Soil Res., 25, 21, 1987. 40. Doolittle, J.A. et al., Estimating depths to claypans using electromagnetic induction methods, J. Soil Water Conserv., 49, 572, 1994. 41. Kachanoski, R.G., Gregorich, E.G., and Van-Wesenbeeck, I.J., Estimating spatial variations of soil water content using noncontacting electromagnetic inductive methods, Can. J. Soil Sci., 68, 715, 1988. 42. Sheets, K.R. and Hendrickx, J.M.H., Noninvasive soil water content measurement using electromagnetic induction, Water Resour. Res, 31, 2401, 1995. 43. Jaynes, D.B. et al., Estimating herbicide partition coefficients from electromagnetic induction measurements, J. Environ. Qual., 24, 36, 1995. 44. McBride, R.A., Gordon, A.M., and Shrive S.C., Estimating forest soil quality from terrain measurements of apparent electrical conductivity, Soil Sci. Soc. Am. J., 54, 290, 1990. 45. Khakural, B.R., Robert, P.C., and Huggins, D.R., Use of non-contacting electromagnetic inductive method for estimating soil moisture across a landscape, Comm. Soil Sci. Plant Anal., 29, 2055, 1998.

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46. Ayers, P.D. and Perumpral, J.V., Moisture and density effect on cone index, Trans. ASAE, 24, 1169, 1982. 47. Henderson, C., Levett, A., and Lisle, D., The effects of soil water content and bulk density on the compactability and soil penetration resistance of some Western Australian sandy soils, Aust. J. Soil Res., 26, 391, 1988. 48. Grunwald, S. et al., Development of pedotransfer functions for a profile cone penetrometer, Geoderma, 100, 25, 2001. 49. Lowery, B. and Schuler, R.T., Duration and effects of compaction on soil and plant growth in Wisconsin, Soil Till. Res., 29, 205, 1994. 50. Puppala, A.J., Acar, Y.B., and Tumay, M.T., Cone penetration in a very weakly cemented sand, J. Geotech. Engr., 121, 589, 1995. 51. Rooney, D.J. and Lowery, B., A profile cone penetrometer for mapping soil horizons, Soil Sci. Soc. Am. J., 64, 2136, 2000. 52. Grunwald S. et al., Profile cone penetrometer data used to distinguish between soil materials, Soil Till. Res., 62, 27, 2001. 53. Grunwald S. et al., Soil layer models created with profile cone penetrometer data, Geoderma, 103, 181, 2001. 54. Rooney, D.J., Norman, J.M., and Stelford, M., Mapping soils with a multiple-sensor penetrometer, in P. WI Aglime Pest Manage. Conf., Madison, 2001, 213. 55. Rooney, D.J., Norman J.M., and Grunwald, S., Soil imaging penetrometer: a tool for obtaining realtime in situ soil images, Paper 013107, Proc. ASAE, Sacramento, 2001. 56. Methods of Soil Analysis: Physical and Mineralogical Methods, 2nd ed., Klute, A., Ed., ASA, SSSA, Madison, 1986, chap. 15. 57. Soil Moisture Systems, instruction manual for the tension infiltrometer, Soil Moisture Systems, Tucson, p. 12. 58. Pachepsky, Y., Rawls, W.J., and Gimenez, D., Comparison of soil water retention at field and laboratory scales, Soil. Sci. Soc. Am. J., 65, 460, 2001. 59. Klute, A., Ed., Methods of Soil Analysis: Physical and Mineralogical Methods, 2nd ed., ASA, SSSA, Madison, WI, 1986, chap. 30, 31.

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Application of a Neural Network-Based Spatial Disaggregation Scheme for Addressing Scaling of Soil Moisture T.D. Tsegaye, W.L. Crosson, C.A. Laymon, M.P. Schamschula, and A.B. Johnson

CONTENTS I. II. III. IV. V.

Introduction...........................................................................................................................261 Statement of Problem ...........................................................................................................262 Scientific Objective and Approach.......................................................................................262 Sampling Site Description....................................................................................................263 Sampling Location and Analysis .........................................................................................263 A. Analysis of Soil Physical Properties .........................................................................264 VI. Models and Data ..................................................................................................................264 A. Models........................................................................................................................264 1. SHEELS...............................................................................................................264 2. Forward Radiative Transfer Model .....................................................................265 3. Disaggregation Neural Network (DisaggNet).....................................................265 B. Model Domain and Data ...........................................................................................265 VII. DisaggNet Training ..............................................................................................................267 VIII. Validation of DisaggNet Soil Moisture Estimation .............................................................269 IX. Conclusions...........................................................................................................................273 X. Acknowledgments ................................................................................................................276 References ......................................................................................................................................276

I. INTRODUCTION Soil moisture plays an important role in the near-surface meteorology, locally as well as globally, by regulating the surface–atmosphere energy exchange. Variation of soil moisture directly affects plant growth and crop yields. Most researchers in soil science, hydrology, meteorology, and remote sensing seek better quantification of soil moisture variation, in time and in space, in order to improve model performance. Thus, improving the techniques to estimate soil moisture on a local and regional scale is the main focus of recent research efforts in hydrology1 and microwave remote sensing of soil moisture.2–5 261 © 2003 by CRC Press LLC

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Soil properties differ widely in type and distribution.6 This variation, in turn, causes large differences in the way dissimilar soil types store and transmit water, and dictates the need for hydrologic investigations on projects such as those required to relate remotely sensed data to surface and subsurface hydrology. Hydrological prediction at the micro- and mesoscales is intimately dependent on the ability to characterize the spatial variability of soil hydraulic properties. Several soil and plant properties are needed for modeling surface and subsurface phenomena.7–9 The lack of these data at the required spatial resolution is the greatest impediment to the successful application of local and regional scale hydrology models. Soil scientists and hydrologists are well aware of the nature of spatial variability in hydraulic properties. Saturated hydraulic conductivity of a single soil sample may vary considerably.6 There is even greater variability of soil hydraulic properties among samples of different soil textures. Soil hydraulic properties are generally observed in the field over a wide range of scales from a few centimeters to kilometers, depending on the intended use. Characterizing and understanding these properties is important for improved performance of local and regional scale hydrology models and accurate estimation of the distribution of soil moisture in space and time.7

II. STATEMENT OF PROBLEM Estimation of surface soil moisture using microwave remote sensors holds great promise for many applications, including numerical weather prediction and agriculture. However, there exists a scale disparity between the resolutions of future satellite-borne microwave remote sensors (30 to 60 km) and the much finer scales at which soil moisture estimates are desired (~1 km). Hydrology models may be useful for bridging this gap, as the factors controlling soil moisture variability (precipitation, soil and vegetation properties, terrain slope) are known with reasonable accuracy at fine spatial scales and can be used in models to estimate the spatial distribution of soil moisture at high resolutions. Therefore, to facilitate the assimilation of remote sensing data, it is important to explore ways to disaggregate low-resolution passive microwave remote sensing data to the higher resolution of a hydrologic model.

III. SCIENTIFIC OBJECTIVE AND APPROACH In this section we describe tests of the performance of a neural network-based model, called DisaggNet, developed by the authors to disaggregate low-resolution satellite microwave remote sensing data for the purpose of estimating soil moisture at finer scales used in hydrologic models. We also quantify estimation errors as a function of input data resolution. Ideally, the purpose of a disaggregation scheme is to produce the “correct” high-resolution (subpixel) pattern of soil moisture from lower-resolution remotely sensed observations. For several reasons, it is difficult to develop and validate such a scheme. First, the correct subpixel soil moisture pattern, or ground truth, within a satellite footprint is rarely if ever known within acceptable error bounds. Thus, adequate data for developing statistical models or more complex models such as neural networks, both of which rely on some type of data fitting, do not exist, and may never exist, for areas larger than field scale. When satellite data become available operationally on a global scale, it may be possible to develop a disaggregation scheme using a combination of remotely sensed data and output from coupled hydrology–radiative transfer models (RTM). Currently, however, high-resolution data from aircraft platforms are available for only limited areas and times during intensive field experiments. While these could theoretically be used to develop a disaggregation scheme, the results would likely not be transferable to other geographical areas or even to different hydrometeorological conditions in the same region. Furthermore, the amount of data needed to train a neural network adequately exceeds the amount obtained in a typical field campaign. © 2003 by CRC Press LLC

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Because of this paucity of remotely sensed observations, we believe that the most viable approach is to train a neural network using solely model output, and then test its performance using actual remotely-sensed data. In this scenario, model-simulated data serves as a proxy for satellite-borne microwave remote sensor data. This approach necessitates the following assumptions: 1. The surface hydrology–radiative transfer model accurately simulates the spatial patterns of soil moisture and brightness temperature within an actual or hypothetical low-resolution satellite footprint, although the model estimates averaged over the footprint may be biased with respect to the satellite estimates. 2. Low-resolution brightness temperature observations are unbiased and have a known noise variance with respect to the ground truth at the scale of the observations. 3. The functional relationship between brightness temperature and soil moisture “learned” by the neural network is consistent with the relationship simulated by the RTM. This can be easily verified from the outputs of the neural network and RTM. Following these assumptions, we designed the neural network to simulate the model (highresolution) soil moisture pattern within each satellite footprint while preserving the mean remotely sensed brightness temperature (TB) or microwave emissivity (ε), the main neural network input, which may differ significantly from the model mean over the footprint. To the extent that the emissivity–soil moisture relationship is linear, the neural network will also preserve the footprintmean soil moisture. The neural network, once trained, will be applicable to a range of hydrometeorological conditions within a geographic domain, but would need to be retrained in order to be transferred to another domain. Here we present a description of the disaggregation methodology and results related to training and testing of the scheme using solely model data. In future research we will apply the scheme to aircraft remote sensing data as a more relevant and revealing application of the method.

IV. SAMPLING SITE DESCRIPTION The research was conducted in LWRB in Southwest Oklahoma during the Southern Great Plains 1997 (SGP97) Hydrology Field Experiment. Further detailed description of the experimental plan can be found elsewhere (http://hydrolab.arsusda.gov/sgp97/). The watershed covers 603 km2 and is a tributary of the Washita River. The climate of this location is classified as moist and subhumid, and the average annual rainfall is approximately 747 mm. In this watershed the summer is typically long, hot, and relatively dry. The average daily high temperature for July is 94°F (35°C), and the average accumulative rainfall for July is 56 mm. The topography of the land is gently to moderately rolling and the average slope within the entire sampling location ranges from 1 to 5%. Additional information about the LWRB can be found in the works of Allen and Naney.10

V. SAMPLING LOCATION AND ANALYSIS To characterize and assess the spatial distribution of soil hydraulic properties in the LWRB, soil core samples were collected from four sites: Apache, Berg, DOE-EF26, and NOAA. Multiple undisturbed soil cores, 3.0-cm length by 7.6-cm diameter, were taken from 5, 15, 30, 50, and 70 cm soil depths using a Uhland core sampler12 from these sites to characterize the spatial distribution of soil hydraulic properties including hydraulic conductivity (Ksat), bulk density, and soil water retention. Using a water content reflectometer, soil moisture measurements were also made at 30min intervals at ten depths, from 3 to 70 cm to continuously examine the soil moisture distribution for the various soils in the LWRB. © 2003 by CRC Press LLC

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A. ANALYSIS

OF

SOIL PHYSICAL PROPERTIES

Saturated hydraulic conductivity (Ksat) was determined by ponding 3.6 cm of water on top of each core under a constant head. A 6- to 8-h equilibration period was allowed, after which preliminary outflow measurements were taken to determine if the rate of outflow was consistent. Outflow was determined for a 15-min period, after which Ksat was calculated using Darcy’s equation.11 Following Ksat measurements, soil water retention was determined for each core at matric potential values of 0, –33.3, –100, –500, –1000 and –1500 kPa. A pressure plate apparatus was used to determine soil water retention for all matric potentials. Soil water retention was expressed in terms of volumetric water content using the bulk density of individual cores for the conversion. Bulk density was calculated for each soil core.12

VI. MODELS AND DATA A. MODELS 1. SHEELS We have used the land surface flux–hydrology model SHEELS (simulator for hydrology and energy exchange at the land surface), the physics of which are based on the biosphere–atmosphere transfer scheme (BATS) of Dickinson et al.13 Figure 15.1 depicts the main physical processes simulated in SHEELS. Variables such as surface energy fluxes and temperatures are modeled similarly to an earlier version of the model.14 Subsurface processes in SHEELS differ significantly from BATS.15 In SHEELS, the number and depth of soil layers is user defined, permitting higher vertical resolution near the surface where temperature and moisture gradients are large. The soil water dynamics algorithms in SHEELS include Darcy flow to model vertical subsurface fluxes and a kinematic

Wind

Radiative fluxes Precipitation

Shortwave Longwave

Interception by canopy Bare soil energy fluxes Sensible Latent Surface runoff Infiltration

Subsurface lateral flow

Diffusion/drainage

Canopy energy fluxes Sensible Latent Top of canopy

Throughfall Ground heat flux

Heat exchange

Upper zone Soil layers Root zone Bottom zone

FIGURE 15.1 (See color insert following page 144.) Schematic representation of the basic energy flux and hydrologic processes in the simulator for hydrology and energy exchange at the land surface (SHEELS). © 2003 by CRC Press LLC

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wave approach to simulate overland flow, thus providing a mechanism for estimating the threedimensional soil water fluxes. 2. Forward Radiative Transfer Model The forward radiative transfer model coupled with SHEELS is based on the coherent wave model of Njoku and Kong16 and is used to estimate L-band (1.413 GHz) microwave brightness temperature. The effects of surface roughness and vegetation are corrected for using methods described in Choudhury et al.17 and Jackson and Schmugge.18 SHEELS supplies the required RTM inputs of soil moisture and temperature profiles and surface temperature. The remaining input variables (surface roughness, vegetation water content and soil density profiles) are based on measurements. The RTM has been tested and validated with field data.19 3. Disaggregation Neural Network (DisaggNet) We have approached the problem of disaggregation using a linear artificial neural network (ANN). The ANN chosen is the simplest one imaginable, consisting of a single neuron. All of the inputs are weighted and then summed; thus the input to output mapping function is linear. Inputs and outputs of DisaggNet are described in Section III.

B. MODEL DOMAIN

AND

DATA

We applied the disaggregation scheme using data collected across the LWRB in south central Oklahoma (Figure 15.2) during the Southern Great Plains 1997 Hydrology Experiment (SGP ’97) conducted during June and July 1997.20 Aircraft remote sensing brightness temperature data were collected nearly daily by the electronically steered thinned array radiometer (ESTAR) for a region encompassing the LWRB. ESTAR observations are used here only to initialize the surface soil moisture state in SHEELS. We restricted our simulations to the approximate 600-km2 area of the LWRB because it contains the highest concentration of meteorological and soil moisture measurements in the SGP ’97 experimental domain. We have applied the disaggregation scheme for the period from June 18 (day 169) through July 20 (day 201).

FIGURE 15.2 (See color insert following page 144.) Digital elevation model and stream network for the Little Washita River Basin, Oklahoma. © 2003 by CRC Press LLC

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A model grid of 800 m, coincident with the ESTAR grid, was used in SHEELS simulations. Land surface properties were specified on that grid in SHEELS by the following data sets: • Elevation, slope: U.S. Dept. of Agriculture/Agricultural Research Service (USDA/ARS) 30 m DEM • Hydrography: U.S. Geological Survey digital line graphs (DLGs) • Vegetation parameters: SGP ’97 30-m land cover • Soil properties: CONUS 1-km multilayer soil hydrologic characteristics • Meteorological and soil moisture and temperature data: Oklahoma Mesonet, USDA/ARS Micronet, SGP ’97 soil profile stations • Precipitation: USDA/ARS Micronet rain gauge data The raw data having native resolutions finer than 800 m were aggregated to the model grid using the mean value, or in the case of categorical data such as soil and land cover classes, the mode. The CONUS soil properties were resampled from 1 km to the 800-m model grid; surface soil texture classes are shown in Figure 15.3. Meteorological data, with the exception of rainfall, were averaged across all sites and applied uniformly across the LWRB. Distributed rainfall estimates for the model grid were obtained from the Micronet point measurements and converted to 800-m gridded data by constructing Thiessen polygons around each gauge location. Selected physical properties of the LWRW soils, as sampled during the SGP’97 field hydrology experiment, are presented in Table 15.1. Except for the Apache and Berg sites, bulk

Sand Sandy Clay Loam Silt Loam

Sandy Loam

FIGURE 15.3 (See color insert following page 144.) Surface soil texture classes for the Little Washita River Basin.

TABLE 15.1 Bulk Density (Pb) (g/cm3) and Saturated Hydraulic Conductivity (Ksat) (cm/hr) for 5-, 15-, 30-, 50- and 70-cm Soil Depths for the LWRB Site 1-Apache

Site 2- Berg

Site 3- DOE-EF26

Site 4- NOAA

Depth (cm)

Pb

Ksat

Pb

Ksat

Pb

Ksat

Pb

Ksat

5 15 30 50 70

1.52 1.60 1.50 1.53 1.61

61.26 12.37 1.92 0.89 0.03

1.60 1.59 1.48 1.28 1.49

0.22 0.93 0.01 64.80 5.28

1.52 1.51 1.59 1.71 1.73

8.54 1.41 0.38 0.01 0.01

1.56 1.52 1.56 1.64 1.67

0.48 0.18 1.06 0.09 0.06

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D5

0.40

VWC (cm3/cm3)

D15 0.30

D30 D30

0.20

D70 0.10 0.00 0

400

800

1200

1600

Soil Water Matric Potential (kPa) FIGURE 15.4 (See color insert following page 144.) Volumetric water content (cm3/cm3) for 5-, 15-, 30-, 50, and 70-cm soil depths for NOAA site in the LWRB.

density in general increased with depth. Saturated hydraulic conductivity (Ksat) in general decreased with depth for all locations except the Berg site. The upper 5 and 15 cm soil depths indicated higher spatial variability of Ksat. Differences due to soil type and pore size distribution caused the volumetric water content to vary under different soil depths (Figure 15.4) and vegetation cover (data not shown). The temporal variations of soil moisture decreased with depth as the soil dries down (Figure 15.5). A gradual drying is observed at the deeper layers through the experimental period. Figure 15.6 shows the temporal behavior of basin-mean, near-surface (0 to 5 cm) fractional water content (FWC) estimated at hourly time steps by SHEELS. This quantity is the proportion of saturation and is defined as volumetric water content (VWC) divided by soil porosity. From the beginning of the period until day 191, there was a general drying trend, interrupted by four minor rain events. On days 191 to 192, a substantial basin-wide rain event occurred, with a basin mean rainfall of 48 mm. This resulted in the wettest observed conditions with much of the watershed, especially the western end, briefly reaching saturation. 45 40 35

VWC (%)

30 25 20 5 cm 15

30 cm

10

50 cm 70 cm

5 0 167

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1997 Day of Year FIGURE 15.5 (See color insert following page 144.) Temporal variability of volumetric water content (%) for 5-, 30-, 50-, and 70-cm soil depths for NOAA site in the LWRB. © 2003 by CRC Press LLC

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0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 169

173

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1997 Day of Year FIGURE 15.6 Basin-mean fractional soil water content estimated by SHEELS for the 0- to 5-cm layer, for each hour during the study period.

VII. DISAGGNET TRAINING As discussed previously, our approach was to train DisaggNet, using soil water content and emissivity output from the coupled SHEELS/RTM model. Thus, the DisaggNet learns “mapping” from low (sensor) resolution ε to high (model) resolution soil water content that is conservative in ε at the footprint scale and seeks to replicate the model patterns of soil water content at each time step. Use of ε instead of TB as input eliminates the diurnal cycle caused by surface temperature variations. The accuracy of this relationship depends on how well the SHEELS/RTM characterizes these subpixel scale patterns, i.e., the validity of our first assumption. Once DisaggNet is trained, this mapping can be applied to actual remotely sensed observations. Because the mapping preserves the pixel-scale means, any large-scale errors in the model estimates will be “corrected” via application of DisaggNet, based on our second assumption that the remotely sensed measurements are unbiased with respect to ground truth. Model outputs used to train and validate DisaggNet were generated by running SHEELS/RTM at an hourly time step over the LWRB for the 33-day study period beginning at 0:00 UTC on day 169. Initial soil moisture conditions were specified using the ESTAR estimates from day 169. The model produces, among other variables, fractional soil water content, TB and skin temperature at each model time step on the 800-m model grid. L-band emissivity was calculated by dividing TB by skin temperature and then aggregated to various resolutions by averaging over 2 × 2, 4 × 4, 8 × 8, 16 × 16 and 32 × 32 800-m grid cells. An independent normal random deviate with zero mean and a standard deviation of 0.02 was added to each aggregated emissivity value to represent actual remotely sensed microwave observations more realistically. The emissivity standard error of 0.02 corresponds to a standard error in TB of 6 K for a skin temperature of 300 K and was chosen to approximate errors expected from future satellite-borne microwave remote sensors. An emissivity error of 0.02 corresponds to an error of approximately 2% in volumetric water content. DisaggNet was trained to predict high-resolution SHEELS upper zone (0 to 5 cm) FWC using approximately one-half of the study period (350 consecutive hours from days 179 through 193) over all pixels simultaneously. Training was performed separately for each emissivity aggregation (2 × 2 pixels, 4 × 4, etc.) using the following inputs: © 2003 by CRC Press LLC

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ESTAR high-resolution (800 m) emissivity Average spatially and add noise Low-resolution emissivity

Sand and clay contents Vegetation water content

Upstream contributing area Antecedent precipitation

DisaggNet

Model-resolution water content

Compare with SHEELS 800 m water content

FIGURE 15.7 (See color insert following page 144.) Schematic illustrating the procedure used to apply DisaggNet.

• Remotely sensed (low-resolution) emissivity with noise • Antecedent precipitation for the following time periods (in hours) prior to current time: 0 to 1, 1 to 3, 3 to 6, 6 to 12, 12 to 24, 24 to 48, 48 to 96 and 96 to 192 • Clay content • Sand content • Vegetation water content • Upstream contributing area (surface area draining into a grid cell) Once trained, DisaggNet generalizes for times outside the training period according to the procedure illustrated in Figure 15.7. Outputs from DisaggNet are estimates of FWC on the model grid.

VIII. VALIDATION OF DISAGGNET SOIL MOISTURE ESTIMATION At each model time step, the trained DisaggNet generates estimates of fractional soil water content at each grid cell using the inputs listed above. Two points in time were selected to demonstrate the performance of DisaggNet. These times fall within the period used to train DisaggNet, so this is not an independent test; however, they were selected because they are close to the driest and wettest times in the study period. Output from DisaggNet for two input resolutions (2 × 2, or 1.6 km, and 16 × 16, or 12.8 km) is compared with SHEELS FWC estimates for these two times in Figures 15.8 and 15.9. As shown in Figure 15.8 for dry soil conditions at 1400 UTC on day 184, the largescale SHEELS soil moisture pattern is captured by DisaggNet using 1.6 or 12.8 km input. This is not unexpected because the input emissivities are derived from the SHEELS soil water content via the RTM. If the emissivity–soil moisture relationship learned by DisaggNet is consistent with the relationship in the RTM, agreement should be excellent, at least when the emissivity input is not © 2003 by CRC Press LLC

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DisaggNet Soil Moisture 1.6 km

Day 184 - Dry

12.8 km

SHEELS 0-5 cm Soil Moisture

FIGURE 15.8 (See color insert following page 144.) DisaggNet soil moisture estimates for the dry soil case (1997, day 184) for 1.6 km (top left) and 12.8 km (top right) aggregated emissivity input, compared to SHEELS 0- to 5-cm soil moisture (bottom).

highly aggregated. The sources of differences between FWC estimated by DisaggNet and SHEELS are (1) aggregation of emissivity, (2) random noise added to the emissivity, and (3) inherent error associated with the neural network. In both DisaggNet output maps, the random noise is evident at the respective scale, superimposed on the overall soil moisture pattern. The wet case corresponds to 1400 UTC on day 192 and is shown in Figure 15.9. The overall pattern is again well simulated, particularly for the high-resolution (1.6 km) case. For the lowresolution (12.8 km) case, the saturated soil in the western part of the domain is slightly underestimated. However, in the eastern half of the basin, the small-scale areas of relatively dry and wet soils are captured nicely in both cases. For example, the area of lower soil moisture associated with sandy soil in the north central part of the basin is clearly indicated by DisaggNet. A quantitative evaluation of the agreement between soil moisture estimated by DisaggNet and by SHEELS is shown in Figures 15.10a and 15.10b in the form of root-mean-square differences (RMSD) across the LWRB at each model time step (hour). Outside of the very wet periods, RMS differences for the 1.6 and 12.8 km resolutions tend to be between 0.03 and 0.07 (3 to 7% FWC, or 1.5 to 3.5% VWC). However, during and immediately following rain periods, errors are quite large — typically greater than 0.07 (3.5% VWC) and occasionally above 0.08 (4% VWC) for the 1.6-km case and above 0.12 (6.0% VWC) for the 12.8-km case. RMS differences are slightly higher for the 12.8-km case than for the 1.6-km case, with mean values of 0.0569 and 0.0527, respectively. Table 15.2 lists the overall RMS errors for all resolutions, with and without random noise in the input emissivities. The spatial distribution of DisaggNet-SHEELS root-mean-square differences in FWC, averaged over the 3-day study period, is shown in Figure 15.11 for the low-resolution case. RMS differences are slightly higher in the western part of the basin, where rainfall was greater, but are less than 0.1 © 2003 by CRC Press LLC

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DisaggNet Soil Moisture 12.8 km

1.6 km

Day 192 - Wet

SHEELS 0-5 cm Soil Moisture

FIGURE 15.9 (See color insert following page 144.) Same as Figure 15.8 except for the wet soil case (1997, day 192).

TABLE 15.2 RMS Differences between Fractional Water Content Estimated by SHEELS and by DisaggNet, with and without Random Noise in the Emissivity Inputs, for Each Level of Emissivity Input Aggregation Aggregation level

RMS without emissivity noise

RMS with emissivity noise

1 × 1 (0.8 km) 2 × 2 (1.6 km) 4 × 4 (3.2 km) 8 × 8 (6.4 km) 16 × 16 (12.8 km) 32 × 32 (25.6 km)

0.0266 0.0396 0.0437 0.0473 0.0493 0.0542

0.0475 0.0527 0.0545 0.0560 0.0569 0.0598

for almost the entire basin. The two points shown as having very high RMSD are, in fact, classified as water bodies in SHEELS, where the soil is treated as always saturated. This condition is not well simulated because the radiative transfer model is not configured to estimate emissivity from water surfaces properly; therefore the emissivities supplied to DisaggNet are inconsistent with the permanently saturated conditions at these grid points in SHEELS. The spatial distribution of RMSD is very similar for the high-resolution case. The agreement between DisaggNet- and SHEELS-estimated FWC is shown in Figures 15.12a and 15.12b for the high- and low-resolution cases, for each hour during the 3-day study period, produced using noisy emissivity inputs. There is slightly more scatter in the low-resolution case, consistent with the slightly higher RMS value. Versions of the scatter plots smoothed by binning © 2003 by CRC Press LLC

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1997 Day of Year FIGURE 15.10A Root-mean-square difference time series between fractional water content estimated by SHEELS and by DisaggNet using emissivity input aggregated to 1.6-km grid cells. The time period used for DisaggNet training is shaded.

Fractional water content RMSE

0.15

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FIGURE 15.11 (See color insert following page 144.) Root-mean-square differences, averaged over the 3day study period, between fractional water content estimated by SHEELS and by DisaggNet using inputs aggregated to 12.8-km grid cells.

FWC values are shown in Figures 15.13a and 15.13b to show better any systematic biases between DisaggNet and SHEELS estimates. In both cases, for low to moderate soil moisture conditions, DisaggNet estimates are unbiased with respect to SHEELS. At higher moisture levels (above 0.5 (0.6) for the low (high) resolution case), DisaggNet estimates become negatively biased. This is particularly pronounced for SHEELS FWC values around 0.8 and above 0.9. This result is not yet understood, but we speculate that it is due to nonlinear rainfall–FWC relationships associated with heavy rainfall that are not handled properly in DisaggNet due to its linear weighting functions. A second possible explanation is the spatial extent of saturated areas. As the input resolution becomes coarser, input emissivities are averaged, making extremely high DisaggNet estimates of FWC less likely.

IX. CONCLUSIONS A neural-network based scheme called DisaggNet has been developed for disaggregating lowresolution satellite microwave remote sensing data to higher resolutions compatible with hydrologic data requirements. DisaggNet has been trained using output from a coupled hydrologic–radiative transfer model using input data from the SGP ’97 field experiment. Results shown here focus on the driest and wettest days for the study period. In this procedure, microwave emissivity was simulated by the coupled model and used as input to train the disaggregation scheme. Emissivity data were degraded to various resolutions by simple averaging from the model resolution of 800 m, and random Gaussian noise was added. Results are shown here for the cases using 1.6-km data (2 × 2 pixel averaging) and 12.8-km data (16 × 16 averaging). As expected, RMS differences between DisaggNet and model-simulated soil water content increased with aggregation level. RMS differences are quite low during dry periods, but somewhat larger under very wet conditions. We believe that this is due to an underestimation of high soil moisture values following heavy rainfall, due to spatial averaging of inputs or the linear nature of the rainfall–soil moisture relationship inherent in DisaggNet. Future refinements to DisaggNet will focus on modifying these linear mapping functions. © 2003 by CRC Press LLC

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High resolution (1.6 km)

DisaggNet fractional water content

1.0

0.8

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0.0 0.0

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SHEELS fractional water content FIGURE 15.12A Scatter diagram between fractional water content estimated by SHEELS and by DisaggNet using emissivity input aggregated to 1.6-km grid cells.

Low resolution (12.8 km) DisaggNet fractional water content

1.0

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SHEELS fractional water content FIGURE 15.12B Same as Figure 15.12a except for 12.8 km emissivity input. © 2003 by CRC Press LLC

1.0

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SHEELS fractional water content FIGURE 15.13A Smoothed scatter diagram between fractional water content estimated by SHEELS and by DisaggNet using emissivity input aggregated to 1.6-km grid cells.

Low resolution (12.8 km) DisaggNet fractional water content

1.0

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SHEELS fractional water content FIGURE 15.13B Same as Figure 15.13a except for 12.8 km emissivity input. © 2003 by CRC Press LLC

1.0

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X. ACKNOWLEDGMENTS We acknowledge Frank Archer, Ahmed Fahsi, Andrew Manu, Jimmy Moore, Narayan Rajbhandari, Garland Robertson, Vishwas Soman, and Wubishet Tadesse for their assistance during the field experiment. We thank Ashutosh Limaye for providing the gridded rainfall data product. Data were obtained from the Atmospheric Radiation Measurement (ARM) Program sponsored by the U.S. Department of Energy, Office of Energy Research, Office of Health and Environmental Research, Environmental Sciences Division. LWRB Micronet meteorological data were obtained from the USDA Agricultural Research Service. Department of Plant and Soil Sciences, Alabama A&M University, Normal, AL 35762. Contributed by the Agricultural Experiment Station, Alabama A&M University, manuscript No. 483. This work was supported by NASA Grant NCCW0084 with Alabama A&M University’s Center for Hydrology, Soil Climatology and Remote Sensing (HSCaRS).

REFERENCES 1. Engman, E.T., Angus, G., and Kustas, W.P. Relationship between the hydrologic balance of a small watershed and remotely sensed soil moisture. Remote sensing and large-scale global processes, In: Proc. IAHS 3rd Int. Assembly, Baltimore, MD, May, 1989. IAHS Publ. No. 186, IAHS, Wallingford, 75–84, 1989. 2. Jackson, T.J., Schmugge, T.J., and Wang, J.R. Passive microwave sensing of soil moisture under vegetation canopies, Water Resour. Res., 18, 1137–1142, 1982. 3. O’Neill, P.E., Chauhan, N.S., and Jackson, T.J. Use of active and passive microwave remote sensing for soil moisture estimation through corn, Int. J. Remote Sensing, 17, 1851–1865, 1996. 4. Schmugge, T.J., Jackson, T.J., and McKim, H.L. Survey of methods for soil moisture determination, Water Resour. Res., 16, 961–970. 1980. 5. Schmugge, T.J., O’Neill, P.E., and Wang, J. R. Passive microwave soil moisture research, IEEE Trans. Geosci. Rem. Sens, 14, 12–22, 1986. 6. Tsegaye, T. and Hill, R.L. Intensive tillage effects on spatial variability of soil physical properties, Soil Sci. J. 163:143–154,1998. 7. Tsegaye, T.D., Coleman, T.L., Crosson, W.L., Manu, A., and Boggs, J. L. Surface and subsurface characterization of soil physical properties for hydrology models, in Agron, Abstract. 184, 1998. 8. Tsegaye, T., Coleman, T.L., Senwo, Z.N., Tadesse, W., Rajbhandari, N.B, and Surrency, J. A. Southern Great Plains ‘97 Hydrology Experiment: the spatial and temporal distribution of soil moisture within a quarter section pasture field, Preprint NASA-URC Technical Conference Huntsville, AL,76–82,1998. 9. Tsegaye, T.D., Coleman, T.L., Manu, A., Senwo, Z., Fahsi, A., Belisle, W., Tedesse, W., Robertson, G., Boggs, J., Archer, F., Surrency, J., Birgan, L., Laymon, C., Crosson, W., and Miller, J. The spatial and temporal variability of soil moisture with and without cover crop and its impact on remote sensing, 13th Conference on Hydrology, American Meteorological Society, 349–351, 1997. 10. Allen, P.B. and Naney, J.W. Hydrology of the Little Washita river watershed, Oklahoma: data and analyses, Rep. ARS-90, U.S. Dept. of Agriculture., Agric. Res. Serv., Washington, D.C., 1991. 11. Darcy, H. Les Fontaines Publiques de la Ville de Dijon, Dalmont, Paris. 1856. 12. Blake, G.R. and Hartge, K.H. Bulk density: In Methods of Soil Analysis, Part 1, 2nd ed.,A. Klute (Ed.), Agronomy monograph no. 9, ASA, Madison, WI, 363–375, 1986. 13. Dickinson, R.E., Henderson-Sellers, A., Kennedy, P.J., and Giorgi, F. Biosphere atmosphere transfer scheme (BATS) version 1e as coupled to the NCAR community climate model, NCAR/TN-387+STR, 72, 1993. 14. Smith, E.A., Crosson, W.L., Cooper, H.J., and Weng, H.-Y. Estimation of surface heat and moisture fluxes over a prairie grassland. Part III: Design of a hybrid physical/remote sensing biosphere model, J. Geophys. Res., 98, 4951–4978,1993 15. Crosson, W.L., Laymon, C.A., Inguva, R., and Schamschula, M. Assimilating remote sensing data in a surface flux-soil moisture model, Hydrological Processes, 16, 1645, 2002.

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16. Njoku, E.G. and Kong, N.-A. Theory for passive microwave remote sensing of near-surface soil moisture, J. Geophys. Res., 82, 3108–3118, 1977. 17. Choudhury, B.J., Schmugge, T.J., Chang, A., and Newton, R.W. Effect of surface roughness on the microwave emission from soils, J. Geophys. Res., 84, 5699–5706, 1979. 18. Jackson, T.J. and Schmugge, T.J. Vegetation effects on the microwave emission of soils, Rem. Sens. Environ., 36, 203–212, 1991. 19. Crosson, W.L., Laymon, C.A., Inguva, R., and Bowman, C. Comparison of two microwave radiobrightness models and validation with field measurements, IEEE Trans. Geosci. and Rem. Sens., 40, 143–152, 2002. 20. Jackson, T.J., LeVine, D.M., Hsu, A.Y., Oldak, A., Starks, P.J., Swift, C.T., Isham, J.D., and Haken, M. Soil moisture mapping at regional scales using microwave radiometry: the Southern Great Plains Hydrology Experiment, IEEE Trans. Geosci. Rem. Sens., 37, 2136–2151, 1999.

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Scaling Soil Mechanical Properties to Predict Plant Responses D.K. Cassel and E.C. Edwards

CONTENTS I. Introduction...........................................................................................................................279 II. Scaling Considerations .........................................................................................................280 III. Mechanical Impedance Investigations .................................................................................281 IV. Plot Scale Investigations of MI–Plant Response .................................................................284 V. Regional Scale Variability in MI..........................................................................................286 VI. Field- and Landscape-Scale MI–Plant Response ................................................................287 VII. Transference of MI–Plant Response Relationships from One Scale to Another................291 References ......................................................................................................................................293

I. INTRODUCTION Tillage, both primary and secondary operations, has been standard practice in crop production systems for hundreds of years. Tillage is performed for a number of reasons, but one of the most important is to modify soil properties so that they are favorable for seed germination and root growth. In the past few decades, there has been a tendency to minimize the number of tillage operations and to leave plant residues on the soil surface. This procedure has been successful in many locales, but not all. Regardless of the number of tillage operations or the frequency of tillage, all tillage practices alter some soil physical properties. One soil property that often has been ignored is the mechanical impedance (MI) of the soil, that is, the resistance the soil matrix offers to shoot and root growth. In the late 1960s and early 1970s, it was observed that growth and yields of row crops such as corn (Zea mays L.), cotton (Gossypium hirsutum L.), and soybean (Glycine max (L.) Merr.), were very nonuniform in many tilled fields in large regions throughout the Atlantic Coastal Plain states, especially in North Carolina, South Carolina, and Georgia. In the case of soybean, large differences in plant height were observed in many fields and led to the descriptive term, “up and down soybeans.” A number of scientists in the southeastern U.S. conducted research to evaluate the cause of this effect and to develop remedies. The MI of the sandy-surfaced Ultisols on the Atlantic Coastal Plain was found to be a very important factor in crop production and tillage practices were often altered to remedy the problem.1–4 This chapter describes MI research (tillage research) conducted at various scales in North Carolina during a period of three decades. In 1974, a team of North Carolina State University researchers, composed of a plant breeder, an entomologist, a plant pathologist, a crop physiologist, a soil physicist, and two soil fertility experts, began to study the uneven soybean growth phenomenon at two locations. A summary of

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this research published by Kamprath et al.5 stated that although rainfall in the Atlantic Coastal Plain generally is adequate, insufficient uptake of soil water by soybean plants was the main cause of uneven growth in sandy soils. The main reason for the water deficiency was the presence of a tillage-induced pan, hereafter referred to as “tillage pan.” The tillage pan occurred 20 to 25 cm below the soil surface, immediately below the depth of moldboard plowing that had been a common tillage practice on these soils for years. The MI of this tillage pan often was so great that roots could not penetrate it to access available water stored in the subsoil.1,2 During the past 28 years, research addressing the tillage pan and its effect on row crop production in North Carolina has been conducted at a number of scales. The plot scale research beginning in 1974 was initiated to establish the cause for the observed uneven soybean plant growth and yield, and to evaluate several different tillage practices that were hypothesized to alleviate the MI problem. Today one question asked is “How does a farm operator optimally manage an entire field that has a yield-limiting tillage pan of variable MI in some parts of it, but not in other parts?” This ”precision agriculture” problem is representative of a larger group of soil-related problems for which scaling up is usually attempted using insufficient information collected at a smaller scale. In most cases, resources are not available to collect the requisite number of samples required for scaling up. The objectives of this chapter are (1) to review results of studies relating plant response to MI conducted at various scales on Atlantic Coastal Plain soils, (2) to examine the variability of MI and plant response patterns, and (3) to discuss transference of information collected at one scale to a different scale.

II. SCALING CONSIDERATIONS Ecological processes, of which agricultural production is one, can be studied in a broad range of areas or domains (e.g., microplots, plots, fields, landscapes). To evaluate soil and crop phenomena (in our case, MI and crop responses) one first decides over what region or domain the process is of concern. Once the domain is established, one decides which variables or processes to measure within the domain of concern. Typically, investigators use their best educated guess to determine which variables affect the process.6 A sampling regime is then developed to collect a sufficient number of measurements of the selected variables and processes. The sampling intensity and support size for individual measurements of each variable must also be selected. Here we consider the effect of MI of tillage pans on plant response. This problem can be studied at many scales, ranging from the effect of a tillage pan on the growth of a single plant to the effect of tillage pans of variable MI on crop production over an entire field or farm. Regardless of the scale one chooses to evaluate this problem, one must select (1) one or more factors related to the plant and (2) one or more soil properties related to MI. The support size for each soil and crop parameter, for example, will be different when evaluating an entire field of soybeans as opposed to evaluating a 0.5 m2 microplot having only a few soybean plants. It is critical to choose the appropriate scale of observation to address the specific question asked. Because the scale at which one works is determined by the specific question asked, it is a difficult and sometimes futile exercise to attempt to scale this information up or down. In essence, when scaling up, for example, the problem becomes “How can one minimize the errors caused by using measurements collected at one scale to answer a different question that requires measurements at a different scale?” The scale was likely correct to answer the original research question, but when another question is posed, the original scale may not be the best scale. A number of tools for analyzing scale are available and were recently summarized by Withers and Meentemeyer.7 Among this list of tools are multiple regression, remote sensing, geographic information systems, fractal geometry, variography, and simulation methods. In the discussion to follow, multiple regression and variography coupled with kriging are the primary tools used. © 2003 by CRC Press LLC

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TABLE 16.1 Questions Related to the Scale of Observation to Assess Mechanical Impedance (MI) How How How How How How How

does does does does does does does

a farmer maximize net income? MI affect plant growth? MI affect plant root intensity? MI vary with depth? MI vary with tillage? MI vary with landscape? MI interact with other soil and plant properties?

III. MECHANICAL IMPEDANCE INVESTIGATIONS Mechanical impedance of soils is affected by a number of soil properties and processes. A number of studies have shown that MI is directly related to bulk density and inversely related to soil water content.8,9 Any process that increases soil bulk density or reduces soil water content affects MI. Prior to initiating the first study on MI in 1974, our research team addressed a number of questions posed by the team. The hypothesis that MI was a major factor causing uneven plant growth was developed. Questions addressed with respect to the proper scales of research for this and future studies are listed in Table 16.1. Information on the effects of bulk density and soil water content on root elongation of single plants has been summarized.10–12 Many of the studies cited in the above publications were conducted at the single plant scale. The current chapter focuses on scales larger than a single plant. Mechanical impedance was measured with a constant rate, hydraulically driven cone penetrometer in all studies discussed below. Although the penetration of a pointed metal rod into the soil does not simulate exactly the ability of roots to grow into the soil, penetrometers have been used for many years and continue to be used to study soil strength and plant performance. The complexity of MI–plant response relationships and scale choices is illustrated in Figure 16.1 and assists in the selection of the appropriate domain, sampling intensity, and support size to address the question of concern. At the ”whole field” scale (Figure 16.1A), maximization of net profit in growing a crop is of paramount importance. A given field on the Upper or Middle Atlantic Coastal Plain may vary in elevation, drainage class, slope and a number of other factors, and likely will be composed of one or more mapping units (Figure 16.1B), typically mapped and published at the 1:24,000 scale. It is also common for small but important differences in landscape position (Figure 16.1C) to occur within mapping units even though the differences in elevation in the field or mapping unit are small. The areas represented by each landscape position are so small that they are usually not delineated. Often several landscape positions are included in a given mapping unit. We acknowledge for many soils and processes that the landscape scale is considered larger than field scale, but for these soils, this is not usually the case. A root-limiting tillage pan might exist throughout the entire field, be absent throughout the entire field, or exist at only scattered locations throughout the field. If a tillage pan is present in one or more parts of the field, its thickness and MI often vary spatially. If this is the case, it might be economically beneficial to manage the field in a manner to selectively perform some management practices to mitigate the tillage pan in some portions of the field but not in the whole field. From the practical standpoint, it is fortunate that soil map units are based on factors (e.g., depth of solum, soil texture, and degree of wetness) that also influence plant performance.13,14 These factors, in addition to others, interact to influence the development and MI of tillage pans. In light of the above discussion, the effect of MI on plant response might be evaluated at the map unit scale (Figure 16.1B) or at the landscape scale (Figure 16.1C) that, as stated above, might include several landscape positions in the Coastal Plain. Differences in plant height, leaf color, © 2003 by CRC Press LLC

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A. Field

B. Map Unit

C. Landscape

Ap E B

D. Plot or Microplot FIGURE 16.1 Schematic diagram illustrating several possible scales for evaluating mechanical impedance (MI)–plant response relationships.

plant population, and leaf canopy density are often observed in response to minor changes in elevation or landscape position. Are these plant response differences at the landscape scale due to changes in variations in soil water retention, water table height, MI, or another variable, or a combination of several variables? Variation in MI with depth below a given point or locus on the soil surface occurs in response to differences in soil texture, soil structure, and associated differences in chemical and biological properties in various soil horizons (Figure 16.1D). Finally, MI at a given depth varies at short range (decimeter scale) due to bulk density variations caused by the processes of wheel and machinery compaction, differences in spatial geometry of tillage practices, soil texture, and local differences in soil water content. For example, “down-the-row” and “across-the-row” MI at tillage pan depth was determined at 24-cm intervals 3 months after performing a subsoil tillage operation in a Typic Paleudults at Snow Hill, North Carolina. The semivariance of MI in the “in-the-row” direction was about 1 MPa2, also equal to the sill, and exhibited no spatial structure (Figure 16.2A). The sill of the semivariogram in the across-the-row direction was much greater; the semivariance with strong spatial structure exhibited a cyclical pattern with a 96-cm period (Figure 16.2B). The period is equal to the row spacing between adjacent subsoiler shanks on the toolbar used to break the tillage pan. This decimeter scale directional variability in MI likely will have impacts on plant response on soils prone to tillage pan development. When assessing MI in tilled fields, © 2003 by CRC Press LLC

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3 PARALLEL TO ROW

A

γ(MPa2)

2

1

0

0

2.4

4.8

7.2

9.6

Distance (m)

T

3

TO SUBSOILED DIRECTION

2

γ(MPa2)

B

1

0

0

2.4

4.8

7.2

9.6

Distance (m) FIGURE 16.2 Semivariogram for mechanical impedance (MI) at tillage pan depth for an in-row-subsoiled Typic Paleudults in the (A) ”down-the-row” and the (B) “across-the-row” directions.

one must be cognizant that such short-scale variation can exist and include this knowledge in designing MI measurement schemes for larger scale investigations. If one wishes to construct a semivariogram that will later be used to krige some response variable likely to exhibit short-scale (or sublag) variability or periodicity, one should take a few samples at a separation distance (or lag) shorter than the standard chosen. For example, if one chooses a standard separation distance of 10 m for sampling, then every so often along each transect samples should be collected every 0.5 or 1.0 m between two 10-m spaced samples. This modification to the sampling scheme permits one to model short-scale (<10 m) variability in the semivariogram. Recognition that MI varies vertically with depth and laterally at a given soil depth is essential and must be considered in selecting the appropriate support size for MI measurements. Table 16.2 lists a number of hierarchical possibilities for schemes to measure MI. At the lowest hierarchical level is the measurement at a given point. In the strictest sense, it is not possible to measure MI with a cone penetrometer at a point because penetration of the penetrometer tip into some volume or segment of soil is required. This possibility is listed as Scheme 2 in Table 16.2. An extension © 2003 by CRC Press LLC

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TABLE 16.2 Hierarchical Possibilities for Schemes to Measure Mechanical Impedance (MI) 1. MI value at a given point in the soil 2. Segment of MI profile at a given location (point) in the field 3. Mean of MI profile at a given point in the field 4. Horizontal transect composed of MI values: (1) point values, (2) segment values, or (3) profile means MI 5. Two-dimensional array of MI values: (1), (2), or (3) (plot level) 6. One or more transects or a two-dimensional array of MI: (1), (2), or (3) (field level) 7. An irregular array of MI measurements composed of (1), (2), (3), (4), (5), or (6) (field, township or county level)

of this concept is to take the mean (or, alternatively, the maximum or even the minimum) value of the MI profile measured within a specified depth increment through a point as indicated in Scheme 3. The support size and geometry become more complex as we descend in the table. In the following discussion the MI–plant response studies were conducted using Schemes 1, 5, 6, and 7 in Table 16.2.

IV. PLOT SCALE INVESTIGATIONS OF MI–PLANT RESPONSE The plot scale study by Kamprath et al.5 was designed to evaluate soybean plant response and yield for three tillage practices. Three replicated tillage practices were conventional (double disking), chiselplow to the 27-cm depth with shanks spaced 30 cm apart, and subsoil to the 40-cm depth followed by a disk-bedder to form a 10-cm-thick mound of soil over the subsoil slit. Within each plot, MI was measured with a tractor-mounted, recording cone penetrometer, hydraulically driven at a constant rate into the profile. Mechanical impedance was continuously recorded to the 42-cm depth at 28 points (Figure 16.3), i.e., at seven equally spaced points about 15 cm apart, on four transects perpendicular to the row. The distance between adjacent transects was 1 to 2 m. The MI data, plotted as a function of depth and position perpendicular to the row, for four replicated sets of transect measurements for three tillage practices are shown in Figure 16.4.15 Each transect was constructed using Scheme 2 in Table 16.2, based on mean MI in 1-cm-long depth segments for each of the seven penetrometer probings. The measurements were taken in December, PLOT SAMPLING SCHEME

POSITIONS 1

2

3

4

(M)

5

6

7 (MT)

SET 1

SET 2

SET 3

SET 4

ROW 3

ROW 4

FIGURE 16.3 Penetrometer sampling locations within each soybean plot. Twenty-eight probings were made in each plot. M and MT indicate “nontrafficked” and “trafficked” interrows, respectively.

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CONVENTIONAL

SUBSOIL

CHISEL PLOW

(Plot 12)

(Plot 11)

(Plot 10)

0 .25

Soil Depth (m)

0 .25

0 .25

0 .25

R

MT

Kg/cm2

M

M

R

MT

M

0 -14

42 -56

14 -28

> 56

R

MT

28 -42

FIGURE 16.4 Mechanical impedance of Norfolk sandy loam on December 12, 1975, as a function of soil depth and position normal to the row for four replications of conventional, subsoil, and chiselplow tillage. M, MT, and R indicate nontrafficked interrow, trafficked interrow, and row, respectively. (Adapted from Cassel, D.K. and Nelson, L.A., Soil Sci. Soc. Am. J., 43, 450, 1979. With permission.)

1975, 6 months after the tillage treatments were imposed and after the soybean crop was harvested. Comparison of MI at a specific depth and specific position for the four replicated transects for a given tillage treatment reveals high variability. Large differences in MI within the distance of one or two decimeters occur at certain depths, particularly at tillage pan depth. The degree of variability, however, is dependent on the particular tillage practice. For example, MI at the 15- to 25-cm depth (in the tillage pan) for conventional tillage exhibits higher variability than for chiselplow tillage. However, even with the high degree of MI variability for the conventional tillage, the overall pattern of MI is similar for the four replicated transects within a given plot, that is, a tillage pan exists even though its strength may vary somewhat at a given depth. Compared to conventional tillage, the degree of MI variability with depth and position was reduced by chiselplow tillage. Subsoil tillage created a zone of low MI in the tillage pan but did not reduce the overall variability in MI at tillage pan depth. Selected crop response data were collected in the above experiment. Mean dry weight of secondary soybean roots at full bloom in 10- × 10- × 75-cm deep soil samples taken at all seven transect positions was 86 µg cm–3 for conventional tillage compared to 114 and 132 µg cm–3 for chiselplow and subsoil tillage, respectively.5 Of the secondary roots, 52% occurred in the upper 10 cm of soil for conventional tillage compared to 35% for the two deep tillage practices. The tillage pan severely restricted root penetration for conventional tillage where only 4% of the secondary root mass was found below the 30-cm soil depth compared to 20% for the deep tillage practices. Both subsoil and chiselplow tillage © 2003 by CRC Press LLC

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increased soybean grain yield. Mean yields for the 1974 to 1975 period were 3.70, 5.66, and 4.93 Mg ha–1 for conventional, chiselplow, and subsoil tillage, respectively.

V. REGIONAL SCALE VARIABILITY IN MI

Mechanical Impedance (kg/cm2)

Observations of soil at various locations on the Coastal Plain led us to believe that soil texture was an important factor in tillage pan formation. Tillage pans appeared to be more prevalent when the sand fraction was about 70% of the oven-dry soil mass. If the sand fraction was well graded, the smaller sand particles fit into large pores created by the larger particles. After the smaller sand particles fit into the large pores, the silt and clay fractions could, in turn, fit into the remaining smaller pores. The result is that the particular sand, silt, and clay fractions coupled with low organic matter content and very weak soil structure made these soils very susceptible to tillage pan formation as well as to severe compaction from wheeling and farm machinery. However, we also noticed that even if soils had nearly the same percentages of sand, silt and clay, some of them packed to higher densities than others. Upon noticing that MI of tillage pans varied with location on the Coastal Plain, we hypothesized that particle shape and particle surface roughness of the sand fraction, in addition to particle size distribution, affect the bulk density to which a given soil will pack. The bulk density, in turn, would partially control the resulting MI and root penetration. To test this hypothesis, the dense soil angle of repose (DSAR) was measured on soil samples from the Ap, E, and B horizons for a number of soils throughout the Coastal Plain in North Carolina.16,17 Soil material was air-dried and placed into small pans and vibrated to attain a range of bulk density values. The angle of the pan with respect to the horizontal was slowly increased until the dry soil material sheared. The angle at which shearing occurred is defined as the DSAR for that particular soil at that particular bulk density. Soils from different geographical regions in the Coastal Plain when packed to the same bulk density had different DSAR values (Figure 16.5). Moreover, MI plotted on the log scale was linearly related to DSAR. In addition, inspection of the sand grains under a microscope showed that soils having rougher sand grains had higher DSAR values and did not compact to bulk densities as high as those soils with smoother sand grains. These relationships, we believed, would help us predict which soils on the Coastal Plain would be most likely to form root-limiting tillage pans. Furthermore, we might be able to identify those geographical locations where the soils would be expected to develop the densest tillage pans. 10 Db = 1.60 g/cm3 4 2 1

0.4 0.2 50

60

70

80

DSAR (Degrees) FIGURE 16.5 Mechanical impedance as a function of dense soil angle of repose (DSAR) for 14 Coastal Plain soil materials at a bulk density of 1.60 Mg m–3 (From Cruse, R.M. et al., Soil Sci. Soc. Am. J., 45, 1210, 1981. With permission.) © 2003 by CRC Press LLC

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Data from this DSAR regional scale investigation were analyzed using multivariate statistics to develop mathematical relationships between MI and commonly measured soil factors. These relationships, if they existed, might prove useful in predicting those regions in the Coastal Plain where soils would be highly susceptible to tillage pan formation. Regions might be as large as townships or counties. One disadvantage of this approach is that DSAR data are not routinely measured in soil laboratories, but are not difficult to measure. The work by Cruse et al.17 clearly showed that bulk density, soil water content and DSAR affected MI. However, other factors might have an important effect on MI and might be useful in answering questions concerning the locations of soils that likely would benefit from deep tillage. Realizing that any soil property that affects soil structure affects MI, Stitt et al.18 conducted a regional scale study to evaluate the relationship of MI to a broad range of soil chemical and physical properties and to try to develop the mathematical relationship between MI and soil variables further. Six undisturbed cylindrical soil cores were collected from tillage pan depth at each of 50 tilled fields throughout a large area in the Coastal Plain region of North Carolina (Figure 16.6A). In the laboratory, the soil cores were equilibrated at soil water pressures of –10 and –100 kPa, and a constant rate penetrometer was used to measure MI at 2-mm increments. A partial list of measured soil properties for each core included soil water content, coefficient of particle uniformity, depth to tillage pan, depth to B horizon, percentages of sand, silt and clay, saturated hydraulic conductivity, DSAR, pH, CEC, total surface area, external surface area, percent organic matter, and buffer capacity. The resulting data led to development of the multiple regression models in Table 16.3. Of the soil properties investigated, the best three variable models for MI, Model #3, included soil water content, bulk density, and DSAR. Because MI tends to be distributed lognormally,19 the model using log10 MI, Model #4, had a greater coefficient of determination than the R2 for Model #3. Figure 16.6 shows regional kriged maps of MI and selected soil properties. At the regional scale no single soil property was correlated highly enough with MI to be useful in predicting those areas where MI of tillage pans was greatest. No plant response information was obtained for the above regional scale study. Too many factors other than MI (e.g., rainfall, insects, weeds, hail damage) vary at the regional scale and influence yields. Thus, it is extremely difficult experimentally to obtain consistent MI–plant response relationships at the regional level. Clearly, the larger the area or scale, the more data are required and the more difficult it becomes to develop reliable relationships among various processes.

VI. FIELD- AND LANDSCAPE-SCALE MI–PLANT RESPONSE Realizing that prediction of MI at the regional scale was difficult and that it was not feasible to relate MI to regional plant responses, attention was focused at the field and landscape scales. The complex question of concern is “In Coastal Plain regions where we expect to find fields with tillage pans that limit crop production, is the limitation equal throughout the field, or does it vary with location in the field or on the landscape?” This question is currently being asked in the development of precision farming management systems. To address this question a field scale–landscape scale study evaluating corn response to MI as a function of landscape position was conducted in 1984 and 1985 on a Norfolk-Wagram (Typic Kandiudults-Arenic Kandiudults) complex in a 2.6-ha field at Oak City, North Carolina.20,21 The slope within the field was less that 2% and change in elevation was less than 2 m. The field had been disked for many years and no deep tillage had ever been done. Total thickness of the A plus E horizon ranged from 0.1 to 1.1 m. Bulk density at tillage pan depth was 1.57 Mg m–3 at the foot slope compared to 1.64 Mg at the linear slope. Prior to tillage a significant difference in MI measurements of the E horizon existed as a function of landscape position. Mean MI ranged from 3.26 MPa at the linear and interfluve positions to 2.15 MPa at the foot slope. Replicated disk only, chiselplow, and subsoil plus bedding tillage practices were stripped across the field. The strips crossed several landscape positions that served as subplots in the data analysis. © 2003 by CRC Press LLC

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Sampling locations N

# # # #

# # ## ##

###

## # ## # # ### # # #

# ##

#

A.

#

##

# ## # # # ### ##

#

## #

# ## #

## #

## # #

###

# # ##

Dense soil angle of repose 49 - 55 55 - 60 60 - 65 65 - 70 60 - 75

B.

Bulk density (gm/cm3) 1.63 - 1.66 1.66 - 1.68 1.68 - 1.70 1.70 - 1.72 1.72 - 1.74

C.

Mechanical impedance at 100kpa (kg/cm2)

D.

41 - 45 45 - 48 48 - 52 52 - 56 56 - 58

FIGURE 16.6 (A) Fifty sampling locations on the Coastal Plain; (B) dense soil angle of repose; (C) bulk density; and (D) mechanical impedance.

Each tillage strip was subdivided into adjoining plots 2.2-m long × 5.5-m wide (six-row plots). Mean corn grain and stover yields for the 2-year period for the disk and subsoil plus bed treatments are shown in Figure 16.7. The landscape effect on yield is indirectly related to MI of the tillage pan. Moreover, subsoiling increased corn grain yield about 1.5 Mg ha–1 at the interfluve, shallow (A plus E horizon thickness <0.75-cm thick) linear slope, and deep (A plus E thickness >75 cm) linear slope positions, but only about 0.8 Mg ha–1 at the foot slope. Mechanical impedance was least at the foot slope and presumably had less effect on root growth and water uptake. Subsoiling increased stover production (Figure 16.7B) approximately 0.7 to 0.8 Mg ha–1 for all landscape positions except the foot slope where it increased 1.4 Mg ha–1. The higher stover yield for subsoil tillage is related to more rapid growth of the young corn plant early in the growing season. © 2003 by CRC Press LLC

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TABLE 16.3 Regression Models to Describe MI (kg cm–2) for 50 Tillage Pans in the Upper and Middle Coastal Plains in North Carolina Model No

Model

R2

1 2 3 4

MI = 240–1160 w MI = –30 –3370 w + 6.81 DSAR MI = 1225 –3800 w + 692 Db + 7.77 DSAR Log10MI = –1.55 –12 w + 1.76 Db + 0.025 DSAR

0.15*** 0.47*** 0.67*** 0.74***

Notes: MI = mechanical impedance, w = mass wetness (kg kg–1), DSAR = dense soil angle of repose, Db = bulk density (Mg m–3). (Adapted from Stitt, R.E. et al., Soil Sci. Soc. Amer. J., 46, 100, 1982. With permission.)

8

A.

Disk

Corn grain yield (Mg/ha)

Subsoil and bed 6

4

2

0

6

B.

Disk

Stover yield (Mg/ha)

Subsoil and bed 4

2

0

Interfluve

Shallow linear slope

Deep linear slope

Foot slope

FIGURE 16.7 Mean corn grain yield and mean stover production in 1984 and 1985 as influenced by landscape and disk and subsoil plus bed tillage.

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UNIT I

VERTICAL HEIGHT (m)

1.0

UNIT I I

UNIT I I I

.8

A.

.6 A .4 E

.2

B .0 0

40

80

120

160

200

DISTANCE FROM SUMMIT (m)

MODEL

R2

MI = 11.0 -0.146 D

0.74

TILLAGE CD

7

B.

MI (MPa)

6 5 4 3

0.27

0.35

0.43

0.51

DEPTH TO B HORIZON (m) FIGURE 16.8 (A) Relative change in elevation and A and E horizon thickness across a 198-m long field and (B) mechanical impedance of the tillage pan as affected by depth to B horizon (From Anderson, S.H. and Cassel, D.K., Soil Sci. Soc. Am. J., 48, 1411, 1984. With permission.)

Another field scale–landscape scale study involving MI and corn response on a Coastal Plain field was conducted in 1981 and 1982 on Norfolk (Typic Kandiudults) soils at Snow Hill, North Carolina.22,23 Texture of the soil surface was loamy sand to sandy loam, and depth to the B horizon (depth of the A plus E horizon), ranged from 27 to 51 cm (Figure 16.8A). A tillage pan was present in the entire field although its strength varied with landscape position as shown in Figure 16.8B. Likewise, bulk density of the E horizon varied with position across the field; bulk density was inversely related to depth to B horizon. Based on the physical property data, the field was arbitrarily divided into three units of similar soil properties. Four tillage practices (disk, chiselplow, subsoil and bed, and bed only), replicated three times, were imposed as strips extending across the 198-m-long field. At harvest, each transect was subdivided into 13 adjoining 15- × 7.6-m plots. Grain and stover were hand harvested from an area 15 m × 2.5 m in each plot. This procedure produced 39 plot yield estimates for each tillage practice (3 strips × 13 plots). © 2003 by CRC Press LLC

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DISK

KRIGED YIELD -1 Mg ha 9 - 10

291

MEASURED YIELD 7.44 Mg ha

-1

CHISEL

8-9 7-8

8.50 Mg ha

6-7

-1

5-6 <5 SUBSOIL

7.94 Mg ha

-1

BED

7.67 Mg ha

-1

FIGURE 16.9 Corn yield distribution in a 69- × 198-m field as affected by tillage practice on Norfolk soil with variable depth to B horizon and variable mechanical impedance of the tillage pan (From Cassel, D.K., Upchurch, D.R., and Anderson, S.H., Soil Sci. Soc. Am. J., 52, 222, 1988. With permission.)

Mean corn grain yield in 1981 was about 8 Mg ha–1 and did not produce the expected variation by treatment, however, as shown in Figure 16.9, the yield distribution throughout the field was different for each tillage practice. Directional semivariograms based on relative yields were computed for each tillage practice, and the yield distribution throughout the entire field for each practice was estimated using block kriging.23 Computation of the semivariograms in the down-the-row direction for each tillage practice was straightforward; each tillage treatment had a different variance structure in this direction. Computation of the across-the-row variogram for each tillage practice was problematic in that only a small number of yield data was available to calculate the semivariogram in this direction. This dilemma was solved by developing an across-the-field directional semivariogram based on pooling relative corn yield values calculated for each tillage practice. First, actual grain yield values for each tillage practice were converted to relative yields. The relative yield values were used to develop semivariograms for each tillage practice and used to krige the relative yield distribution for each practice. Finally, the kriged relative yield values for each practice were transformed back to actual yields for each practice. The resulting actual corn grain yield distribution for each tillage practice for the 1981 crop is shown in Figure 16.9. It is obvious that the detailed analysis of the field in the long dimension direction was not sufficient to predict the resulting plant response variability. Variability in MI and other soil properties in the short dimension direction caused large yield differences for each tillage practice that were not initially hypothesized.

VII. TRANSFERENCE OF MI–PLANT RESPONSE RELATIONSHIPS FROM ONE SCALE TO ANOTHER In this chapter we have reviewed research conducted at several scales on sandy soils of the Coastal Plain in North Carolina. Mechanical impedance was evaluated in various tillage studies and, when © 2003 by CRC Press LLC

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possible, the effects of MI on plant responses were assessed. Studies were conducted at the decimeter, plot, field–landscape, and regional scales. Plant response data for corn and soybean crops were collected for all except the regional scale study. The question asked and the hypothesis tested were different at the different scales. Consequently, the sampling regimes, including support sizes for MI and plant responses, were different at the various scales. The question still remains, “How do we bridge these scales?” The discussion that follows addresses this question and integrates the three decades of research. For example, at the decimeter scale, soybean roots were quantified in soil blocks with dimensions of 10 × 10 × 10 or 15 cm, and in the same study, grain yield was measured in 1.8- × 12-m areas. The accompanying MI data were collected at the decimeter scale and the plot scale to ascertain the short-range and plot level variability in MI for a given tillage treatment. At the field and landscape scale, corn grain yields were measured on long transects segmented to form contiguous 2.2- × 5.5m or 2.5- × 15-m blocks. Mechanical impedance was measured within those blocks, giving particular attention that all MI measurements were taken at the same position with respect to the crop row and wheel compaction, i.e., using information obtained from the decimeter scale study. These choices of support size for the sampling regimes appear to be arbitrary, but they were based on our best knowledge of the processes under study. Wiens6 acknowledged that personal bias in scaling exists because scales and the sampling patterns within them tend to be arbitrary and reflect the hierarchies of spatial scales based on one’s perception of the variation in the parameters and processes of concern. To address the transference of the MI–plant response relationship from one scale to another, let us consider an arbitrarily selected farm on the Coastal Plain in North Carolina. Students and farm managers often ask if they can expect to obtain an increase in crop yield using deep tillage, i.e., by reducing MI at tillage pan depth for entire fields on their individual farms. This is a fieldlandscape scale question. This question can be answered by downscaling information collected at different scales. In answering, the following factors are considered: (1) location (region) on the Coastal Plain of the farm, (2) soil texture at tillage pan depth, and (3) depth to the B horizon. Information derived from the previously discussed regional scale study by Stitt et al.18 indicated the regions on the Coastal Plain where the strongest tillage pans are expected to occur (Figure 16.6). Information derived from the field–landscape scale studies at Oak City and Snow Hill, and the regional scale study by Cruse et al.16,17 suggest that, if particle size distribution at tillage pan depth is loamy sand or sandy loam, it likely will have a high DSAR and a high MI. Therefore, if a soil occurs in a tillage pan-prone area, and the texture at the pan depth is loamy sand or coarse sandy loam, and if the depth to B horizon is ≤ 0.4 m, there is a high probability that tillage pans will form on these soils. The relative magnitude of MI in a tillage pan within a given field can be estimated using information derived from the two field–landscape scale studies. Soil strength of the pan likely will increase as depth to the B horizon decreases but, unfortunately, properties of Coastal Plains soils change over such short distances that soil mapping units delineated in conventional soil surveys are neither small enough nor sufficiently pure to use them to predict MI of pans in localized areas of a given field. However, soil maps play a complementary role in the process of estimating where tillage pans will occur. If a tillage pan does exist in a soil, qualitative estimates of the rooting pattern of row crops growing near wheel track-compacted areas or subsoiled zones of the soil profile can be predicted based on the decimeter scale soybean root data published by Kamprath et al.5 and experience gained in making hundreds of field observations of row crop root development in soils with tillage pans over the past quarter-century. A different approach to transfer information on corn and soybean yield response to deep tillage systems was reported by Denton et al.14 Using research results from 29 deep tillage experiments conducted on 18 on-farm locations in the Piedmont and Costal Plain of North Carolina from 1977 through 1981. The authors created a technical soil classification adapted from the fertility capability © 2003 by CRC Press LLC

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classification (FCC) initially developed by Sanchez et al.24 A total of 32 soil series were present. A technical classification based on soil properties thought to affect yield responses was selected. Properties reported to affect the yield response were texture and thickness of the Ap and E horizons, and texture of Bt horizons. The largest yield increases for corn (up to 36%) and soybean (up to 25%), compared to conventional tillage, occurred on Coastal Plain soils with sandy Ap and E horizons underlain by loamy Bt horizons. This is a different approach but reaches the same conclusions discussed in this chapter. The sandy soil texture of the E horizon has poor soil structure and is easily compacted, increasing MI and reducing the capability of root to penetrate to extract water from the soil beneath the tillage pan. In conclusion, studies relating plant responses to MI conducted at different scales were used to develop qualitative and quantitative relationships that, when taken collectively, form the basis to scale the MI-plant responses to the farm, field, and landscape scales. The cited information developed at the various scales, when considered together, should prove useful to address questions in specific cases about MI-plant response relationships in soils with tillage pans on Coastal Plain soils.

REFERENCES 1. Campbell, R.B., Reicosky, D.C., and Doty, C.W., Physical properties and tillage of Paleudults in the southeastern Coastal Plains, J. Soil Water Conserv., 29, 220, 1974. 2. Doty, C.W., Campbell, R.B., and Reicosky, D.C., Crop response to chiseling and irrigation in soils with a compact A2 horizon, Trans. ASAE, 18, 668, 1975. 3. Campbell, R.B. and Phene, C.J., Tillage, matric potential, oxygen and millet yield relations in a layered soil, Trans. ASAE, 20, 251, 1977. 4. Camp, C.R., Christenbury, G.D., and Doty, C.W., Tillage effects on crop yield in Coastal Plain soils, Trans. ASAE, 27, 1729,1984. 5. Kamprath, E.J. et al., Tillage effects on biomass production and moisture utilization by soybeans on Coastal Plain soils, Agron. J., 71, 1001, 1979. 6. Wiens, J.A., Spatial scaling in ecology, Functional Ecol., 3, 385, 1989. 7. Withers, M.A. and Meentemeyer, M.V., Concepts of scale in landscape ecology, in Landscape Ecological Analysis: Issues and Applications, Klopatek, J. M. and Gardner, R.H., Eds., Springer, N.Y., 1999, 205. 8. Taylor, H.M. and Bruce, R.R., Effects of soil strength on root growth and crop yield in the Southern United States, Trans. 9th Int. Congr. Soil Sci., 1, 803, 1968. 9. Taylor, H.M., and Gardner, H.R., Penetration of cotton seedling taproots as influenced by bulk density, moisture content and strength of soil, Soil Sci., 96, 153, 1963. 10. Glinski, J. and Lipiec, J., Soil Physical Conditions and Plant Roots, CRC Press, Inc., Boca Raton, FL., 1990. 11. Bowen, H.D., Alleviating mechanical impedance, in Modifying the Root Environment to Reduce Crop Stress, Arkin, G.F. and Taylor, H.M., Eds., Am. Soc. Agric. Eng., St. Joseph, MI, 1981, 21. 12. Taylor, H.M., Managing root systems for efficient water use: an overview, in Limitations to Efficient Water Use in Crop Production, Taylor, H.M., Jordan, W.R., and Sinclair, T.R., Eds., Am. Soc. Agron., Madison, WI, 1983, 87. 13. Buol, S.W. and Denton, H.P., The role of soil classification in technology transfer, in Soil Taxonomy – Achievements and Challenges, Grossman, R.B., Rust, R.H., and Eswaran, H., Eds, Soil Sci. Soc. Am. Special Publication Number14, Soil Sci Soc. Am., Madison, WI, 1984, 29. 14. Denton, H.P., Naderman, G.C., Buol, S.W., and Nelson, L.A.,. Use of a technical soil classification system in evaluation of corn and soybean response to deep tillage, Soil Sci. Soc. Am. J., 50:1309–1314, 1986. 15. Cassel, D.K. and Nelson, L.A., Spatial and temporal variability of soil physical properties of Norfolk loamy sand as affected by tillage, Soil Tillage Res., 5, 5, 1985. 16. Cruse, R.M., Cassel, D.K., and Averette, F.G., Effect of particle surface roughness on densification of coarse-textured soil, Soil Sci. Soc. Am. J., 52, 222, 1980. © 2003 by CRC Press LLC

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17. Cruse, R.M. et al., The effect of particle surface roughness on mechanical impedance of coarsetextured soil materials, Soil Sci. Soc. Am. J., 45, 1210, 1980. 18. Stitt, R.E. et al., Mechanical impedance of tillage pans in Atlantic Coastal Plains soils and relationships with soil physical, chemical, and mineralogical properties, Soil Sci. Soc. Am. J., 46, 100, 1982. 19. Cassel, D.K. and Nelson, L.A., Variability of mechanical impedance in a tilled one-hectare field of Norfolk sandy loam, Soil Sci. Soc. Am. J., 43, 450, 1979. 20. Simmons, F.W. and Cassel, D.K., Cone index and soil physical property relationships on a sloping Paleudult complex, Soil Sci., 147, 40, 1989. 21. Simmons, F.W., Cassel, D.K., and Daniels, R.B., Landscape and soil property effects on corn grain yield response to tillage, Soil Sci. Soc. Am. J., 3, 534, 1989. 22. Anderson, S.H. and Cassel, D.K., Effect of soil variability on response to tillage of an Atlantic coastal plain ultisol, Soil Sci. Soc. Am. J., 48, 1411, 1984. 23. Cassel, D.K., Upchurch, D.R., and Anderson, S.H., Using regionalized variables to estimate field variability of corn yield for four tillage regimes, Soil Sci. Soc. Am. J., 52, 222, 1988. 24. Sanchez, P.A., Couto, W., and Buol, S.W., The fertility capability soil classification system; interpretation, application, and modification, Geoderma, 27, 83, 1982.

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17

Estimating Nitrate-N Losses at Different Spatial Scales in Agricultural Watersheds D.J. Mulla, P.H. Gowda, A.S. Birr, and B.J. Dalzell

CONTENTS I. II. III. IV.

Introduction...........................................................................................................................295 Scales and Scale Issues ........................................................................................................296 Processes, Pathways, and Data Needs: Nitrogen Example .................................................296 Nitrogen Losses at Various Scales .......................................................................................297 A. Nitrogen Losses at the Plot Scale .............................................................................298 B. Nitrogen Losses at the Field Scale ...........................................................................299 C. Nitrogen Fluxes at the Minor Watershed Scale ........................................................301 D. Nitrogen Fluxes at the Major Watershed Scale ........................................................302 V. Upscaling Nitrogen Losses ..................................................................................................304 VI. Conclusions...........................................................................................................................305 References ......................................................................................................................................306

I. INTRODUCTION Nonpoint source pollution from agriculture is a widespread problem in Europe and North America. Concerns typically include the nutrients nitrogen and phosphorus, as well as herbicides and pathogens. Researchers working on transport and fate of agricultural nutrients have typically focused on the spatial scale of the small research plot, hill slope, or field, at the temporal scale of several months or years. The aim of most researchers is to describe the processes and pathways that control the concentration and load of nutrients leaving the plot, hill slope, or field in response to various factors that might include climate, soil, or landscape properties, and management alternatives. In contrast, agency leaders and policy makers are typically concerned about decade- or centurylong regional trends in water quality at the scale of watersheds, basins, counties, provinces, states, countries, and continents. They are typically interested in monitoring regional trends in water quality, identifying the level of impairment, identifying the sources of pollution, and developing goals and strategies for restoring good water quality. A few researchers have started to focus their research at the spatial and temporal scale of greatest concern to agency leaders and policy makers. They have encountered significant challenges arising from an inability to model and collect data at the fine resolution needed for evaluation of regional water quality trends. This chapter provides an overview of some of these challenges, and gives examples being used to address nitrogen transport and fate at various scales in Minnesota.

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II. SCALES AND SCALE ISSUES The scales at which transport and transformation of nutrients occur are diverse. Spatial scales of typical interest include the pore, pedon, hill slope, landscape, minor watershed, major watershed, or basin. Temporal scales include the picosecond, microsecond, second, minute, hour, day, month, year, decade, century, millenium, or eon. Typically, we are interested in spatial and temporal scales that are congruent. Studies of nitrogen transformation kinetics at the pore scale occur in milliseconds, rather than eons. Studies of nitrogen transport at the watershed scale occur in days or months. Changes in global soil organic nitrogen and carbon sequestration occur in decades or centuries. It is rare to study incongruent spatial and temporal scales — for instance, processes at the pore scale over a time scale of centuries or studies of nitrogen loads to the Gulf of Mexico at the scale of milliseconds. Scale issues have been discussed in several previous papers.1–6 Some of the more important issues involve spatial and temporal variability, complexity, nonuniqueness, nonlinearity, patchiness, and change in process. All of these factors contribute to uncertainty in estimates of transport. Various approaches for upscaling, downscaling, combinations of up- and downscaling, and strategic cycling across scales have been developed.7 The appropriateness of statistical methods changes with scale. At the pore scale, statistical mechanics might be appropriate, but this is not true at the watershed scale. At the plot and field scale, controlled experiments involving variation of nitrogen rates and measurement of water quality impacts are typically evaluated using analysis of variance. The latter experiments would not be possible at the pore or basin scales. At the watershed scale, experimental replication becomes difficult. Paired watershed studies may be used to study water quality impacts of varying nitrogen application rates at this scale. Either geostatistical or regression methods are appropriate at the hill slope scale and at coarser resolutions. Temporal statistical methods traditionally involve time series analysis. Examples include smoothing techniques, forecasting techniques, and autoregressive techniques. Combined spatial and temporal statistical tools are rare. Examples include state–space analysis and wavelet analysis.8,9

III. PROCESSES, PATHWAYS, AND DATA NEEDS: NITROGEN EXAMPLE Nitrogen is an essential plant nutrient. In agricultural regions, addition of nitrogen to growing crops often leads to significant increases in yield. Addition of excess nitrogen can lead to serious water quality degradation. Determining what constitutes an excess application rate is complicated because of the numerous processes and pathways that affect the fate and transport of nitrogen. These include plant uptake, nitrification, denitrification, mineralization, immobilization, runoff, erosion, and leaching or drainage. All of these are subject to spatial and to temporal variability. The soil factors that influence leaching or drainage losses of nitrate may vary by several orders of magnitude within the same field.10,11 Data needs for evaluating nitrogen fate and transport must typically address multiple pathways. As the scale of study becomes coarser, it becomes more difficult to obtain accurate information concerning nitrogen fate and transport pathways and processes. For example, what is the spatial and temporal variability in denitrification at the scale of a major watershed? Does it matter if we use a spatially or temporally average denitrification value across the entire watershed? What is the spatial variation in rates of nitrogen applied from fertilizer and manure across the watershed? Does it matter if we use the average rate for modeling watershed scale losses of nitrate? In addition to the processes and pathways, it is essential to have accurate information concerning the inputs of nitrogen from fertilizer, manure, atmospheric deposition, and fixation. For plots, hill slopes, and fields, these inputs can be reasonably controlled through management. At the scale of watersheds and large regions, we must increasingly rely on statistical survey information for the sales of fertilizer, number and species of farm animals, average rates of manure production and © 2003 by CRC Press LLC

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manure nutrient content, and types of confinement, storage, or land application methods. Other useful information includes land use, dates of crop planting and harvest, and residue cover. At coarse scales, information concerning spatial and temporal variations in precipitation becomes important. During a particular storm, one area of the watershed may experience much more intense precipitation than the rest of the watershed. For this reason, having an accurate network of precipitation gauges is important. New processes occur at the scale of watersheds that are not important at the scale of plots or hill slopes, including ground water base flow, transformations in ditches, streams, lakes, and wetlands, and uptake by grass and trees. Multiple authors have found that the spatial variability in soil properties within soil mapping units has little influence on predictions of nitrate losses at the scale of watersheds and aquifers.6,12–14 This suggests that commonly available soil databases are appropriate for understanding the impact of soil variability on regional variations in nitrate leaching potential.

IV. NITROGEN LOSSES AT VARIOUS SCALES In the four subsections that follow, various approaches are described for estimating nitrate losses from agricultural landscapes in Minnesota. The scales discussed range from a few square meters to almost a million hectares. The examples include university plot-scale research, farm-scale research, minor watershed-scale modeling, and major watershed-scale nitrogen balances. The general location of each study is shown in Figure 17.1. For each study, distinctly different approaches are used, including the spatial resolution of input data, the method of estimating spatial variability, the method for flow routing, the types of processes accounted for, and the extent of mechanistic modeling vs. mass balancing.

FIGURE 17.1 Location of study sites at various scales.

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AT THE

PLOT SCALE

At the plot scale (m2 resolution), experiments can be conducted to determine the leaching or drainage losses of nitrogen under conditions that are rigorously controlled and measured. As an example, consider experimental measurements of nitrate in tile drainage water conducted at the Waseca Agricultural Experiment Station in Minnesota over a period of 13 years.15 Three plots of 13.5 × 15.0 m in area were treated with 202 kg/ha nitrogen as a broadcast application in spring. Plots were planted to continuous corn (Zea mays L.) and subjected to conventional moldboard plowing. Tile drain spacing and depth were 27 and 1.2 m, respectively. These plots were managed using University of Minnesota best management practices, so nitrate losses may be considerably less than those observed on more poorly managed farm fields receiving similar rates of nitrogen fertilizer or manure. A process-based simulation model, the agricultural drainage and pesticide transport (ADAPT) model,16, was used to evaluate the relative impacts of rate of applied nitrogen fertilizer, and depth or spacing of tile drains, on nitrate losses in drainage water. The ADAPT model operates at a daily time step, and accounts for hydrology (precipitation, soil frost, snowmelt, evapotranspiration, infiltration, runoff, drainage, seepage), nutrient cycling (fertilizer dissolution, nitrification, denitrification, mineralization, immobilization, leaching, uptake), erosion, and crop growth. The ADAPT model climatic inputs include rainfall, temperature, wind speed, relative humidity, and solar radiation. Soil inputs include horizon thickness, porosity, moisture characteristic curve, organic matter content, saturated hydraulic conductivity, and texture. Management parameters needed by the model include rate, timing, and method of fertilizer application, dates and types of tillage, depth and spacing of tile drains, and dates of planting and harvesting. Other important input parameters include slope steepness, runoff curve number, maximum crop rooting depth, and depth to impermeable layer. Half of the experimental years were used to calibrate the model, the other half to validate the model.17 Four statistical procedures were used to evaluate the model performance during calibration and validation; only two will be reported here. These include 1) the observed and predicted means and standard deviations, and 2) the slope and intercept of a least squares regression between the predicted and observed values. After validation, the model was used to evaluate nitrogen application rate (100, 125, 150, 175, 200, and 225 kg/ha), tile spacing (15, 27, 40, 80, 100, and 200 m), and tile depth (0.9, 1.2, and 1.5 m) using a continuous corn rotation with a 50-year climatic record for the site. Results for nitrate losses in drainage were ranked in decreasing order, and summarized using exceedance probabilities. There was excellent agreement between observed and predicted flow or nitrate losses in drainage during calibration and validation of the model.17 During the calibration period, observed and measured monthly tile flows averaged 4.56 and 4.57 cm/month, respectively, with an R2 value of 0.9 (Figure 17.2). Observed monthly nitrate losses during the calibration period were 6.87 kg/ha/month, while predicted losses were 6.69 kg/ha/month with an R2 value of 0.71 between observed and predicted nitrate losses. For a tile drain spacing of 27 m and a tile drain depth of 1.2 m, regression analysis was used to show the average relationship between growing season precipitation (using data for the period from 1915 to 1996) and nitrate losses in drainage for different simulated rates of applied nitrogen fertilizer. In general, nitrate losses increased linearly with precipitation. The rate of increase depended on the rate applied, with faster increases (greater slopes for the regression lines) with higher rates of application. Once nitrogen application rates were increased above the recommended guidelines (125 kg/ha), the excess nitrogen was very susceptible to being lost in drainage water. At a nitrogen application rate of 150 kg/ha, the annual nitrate losses averaged about 10 kg/ha. For an application rate of 175 kg/ha, nitrate losses in drainage averaged about 23 kg/ha/yr. In addition to evaluating average behavior, we also evaluated the probability of a particular nitrate loss using exceedance probabilities. For any applied rate of fertilizer nitrogen, an exceedance

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Observed Subsurface Drainage (cm/month) FIGURE 17.2 Comparison between measured and predicted tile drain flows (cm/month) for the plot scale study at Waseca.

probability curve was developed in response to variations in precipitation over the period from 1915 to 1996. An exceedance probability of 0.5 corresponds to the median behavior; the nitrate losses at this probability level occur half the time across a wide range of climatic conditions. An exceedance probability of 0.2 means nitrate losses at this level occur 20% of the time. An exceedance probability of 0.8 corresponds to nitrate losses that occur 80% of the time. As an example of this approach, with an applied fertilizer rate of 175 kg/ha, the exceedance probabilities for a 10, 20, or 30 kg/ha/year nitrate loss were 0.87, 0.63, and 0.34, respectively, meaning that a 10 kg/ha/year loss occurs 87% of the time, a 20 kg/ha/year loss occurs 63% of the time, and a 30 kg/ha/year loss occurs 34% of the time. Clearly, if policy demanded that nitrate losses of 20 kg/ha/year occur in less than 50% of the years, an application rate of 175 kg/ha would not be allowed.

B. NITROGEN LOSSES

AT THE

FIELD SCALE

At the field scale (32 ha), we measured nitrate losses through tile drainage for 1 year in a field located in the Beauford watershed in southern Minnesota.18 The field was divided into a steeper southern portion and a flatter northern portion. The southern portion has less subsurface tile drainage than the northern portion. Runoff and tile drainage for each portion were measured separately. Nitrogen fertilizer was applied uniformly to the field in fall at typical rates. The losses of nitrate were simulated with the ADAPT model using a corn–soybean rotation and a soybean–corn rotation with a 50-year climatic record, and the results from the two sequences of simulation averaged to avoid bias from having corn or soybeans in a particular year. Input data included spatial variability in soil organic carbon (from soil sampling and kriging), topography (from elevation surveys), and soil physical properties (saturated hydraulic conductivity, moisture retention curves, porosity, texture) from soil databases. The site has landscapes ranging from 0 to 6% slope steepness, and soil organic carbon contents ranging from 2 to 5%. The field has seven mapped soil series, mostly finer textured loams, silt loams, silty clay loams, and clay loams, with © 2003 by CRC Press LLC

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Julian Day FIGURE 17.3 Comparison between measured and predicted nitrate-N losses (kg/day) in tile drains from the 14 ha northern portion of the field scale study in the Beauford watershed.

hydrologic classes B, C, B/D, and C/D, where B indicates good internal drainage, C indicates somewhat poor drainage, and D indicates poor internal drainage. The model internally calculates nitrogen transformation processes using algorithms from the GLEAMS model19 that are based on precipitation and air temperature data and internally generated soil moisture content. The field was subdivided into 128 50 × 50-m cells, and each cell was assigned input parameters based on available information. Routing of flows was necessary so that the contribution from cells was properly accounted for at the location where water quality data were collected and to account for effects of upslope runoff on infiltration in downslope cells. Surface runoff was routed from cell to cell based on surface topography. Subsurface drainage was routed from cell to cell based on available maps of subsurface tile drainage. Roughly 60% of the field is tile drained. Calibrated model predictions of flow and nitrate loss agreed closely with measured data (Figure 17.3). Following calibration, simulations with a 50-year record of precipitation and temperature data were run using uniform application rates of 120, 140, and 160 kg/ha nitrogen fertilizer applied in the fall. The northern and southern portions of the field were kept separate, due to a topographic crest dividing the two portions. Exceedance probability curves were developed for each applied rate of fertilizer on each portion of the field in response to long-term variations in precipitation. The median long-term rates of nitrate loss through tile drainage increased with application rate, with losses of 26, 28, and 31 kg/ha/year nitrate in the southern portion of the field at rates of 120, 140, and 160 kg/ha, respectively. Median losses of 33, 36, and 39 kg/ha/year were predicted on the northern portion of the field at applied rates of 120, 140, and 160 kg/ha, respectively. As we shift from simulating nitrate losses at the plot scale to losses at the field scale, there is a similarity in methods used for the simulation, but a difference in the handling of input parameters. At the plot scale we do not use information concerning spatial variability of soil properties or management inputs, and routing of flows across the topography is not necessary. At the field scale, information about spatial variability and routing of flows becomes important. © 2003 by CRC Press LLC

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At the minor watershed scale, we used the ADAPT model to simulate flow and nitrate losses20 at the mouth of the Bevens Creek watershed (340 km2). Model inputs were obtained from various sources. The Minnesota State Climatology Office provided daily climatic information from ten recording stations in and around the watershed. Daily results from these stations were averaged and used as input to the model. Information concerning average fertilizer application rates and dates of planting and harvesting of crops was obtained from the Minnesota Agricultural Statistics Office. Water quality and flow monitoring data for 3 years (1994 to 1996) were available at the mouth of Bevens Creek from the Twin Cities Metropolitan Council for Environmental Services. Monthly loadings of nitrate were estimated using monitoring data and a regression model based on daily flow. Soil physical properties were obtained from the Map Unit User File (MUUF) soils database associated with the State Soil Geographic (STATSGO) database maintained by the USDA-NRCS. Roughly 58% of the soils in Bevens Creek watershed are classified by the Soil Survey as having poor internal drainage. These soils are assumed to be artificially drained with subsurface tile drainage systems to reduce excess water and improve soil productivity. Land use information (corn, soybean, grass, forest, urban areas, roads, open water) was obtained using LANDSAT remote sensing images and ground truth sampling at 85 locations from July, 1995. Of this watershed, 84% is in cultivated cropland, with lesser portions in grass and pasture, and forest. Crop residue cover was estimated using LANDSAT TM band 5 reflectance values from May, 1997, ground truth data from 85 locations, and a logistic regression equation.21 Slope steepness was estimated using a 30m resolution digital elevation model for the watershed. Hydrologically unique spatial data units were identified by overlaying GIS layers of slope, soil properties, land use, and residue cover. Each resulting polygon is unique in terms of model inputs and spatial location, and is referred to as a hydrologic response unit (HRU). Running a separate model simulation for each HRU would be prohibitively time consuming, because there are over 25,000 HRUs for Bevens Creek. Spatial data units (HRUs) were arranged into functional modeling units called transformed hydrologic response units (THRUs). The THRUs are distinct in terms of model input parameters, but include HRUs at different locations with identical input parameters. The ADAPT model is run for each THRU, a total of only 63 simulations. Output from each THRU is routed to the mouth of Bevens Creek within 1 day, and the routing algorithm accounts for denitrification losses of nitrate beyond the edge-of-field. The ADAPT model for Bevens Creek predicted nitrate loads accurately over the 3-year period from 1994 to 1997 (Figure 17.4). The regression line between predicted and observed nitrate (Figure 17.4) had a coefficient of determination (R2) of 0.67, with a slope of 0.90 and an intercept of 1.98 tons/month. Under baseline conditions (130 kg/ha of applied N fertilizer applied half in fall and half in spring), the simulated loading to Bevens Creek was 641 tons/year (an average loss of 19 kg/ha over all types of land uses, including corn, soybeans, grass and alfalfa, and forest). Alternative scenarios with a 20% increase or decrease in applied nitrogen fertilizer rate were also simulated. The model showed that a 20% reduction in N fertilizer gave an 8% reduction in nitrate loads in Bevens Creek. A 30% increase in N fertilizer gave a 13% increase in nitrate loads for the watershed. Modeling nitrate losses at the minor watershed scale, as implemented here, involves the concept of THRUs, as opposed to the use of simple landscape based HRUs, which are used at the field scale. Also, there are differences in the spatial resolution of model input data. Spatial resolution of model input data is 30 m for digital elevation models at the minor watershed scale, and 1 m for field topography. Spatial resolution of soil mapping units is at a scale of 1:5,000 at the field-scale and 1:125,000 for the minor watershed scale. Spatial resolution for crop planting and harvesting dates is at the scale of counties (~1,800 km2) for the minor watershed modeling, in comparison to the scale of a single field. In addition to these differences, the combination of modeling and GIS database analysis becomes much more important at the minor watershed scale than the modeling © 2003 by CRC Press LLC

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Observed Nitrate Loss (ton/month) FIGURE 17.4 Comparison between measured and predicted nitrate-N losses (tons/month) in the Bevens Creek minor watershed.

approach at the field scale. Finally, routing algorithms at the minor watershed scale account for instream denitrification losses, while routing algorithms at the field scale do not.

D. NITROGEN FLUXES

AT THE

MAJOR WATERSHED SCALE

Nitrogen fluxes have been monitored at the scale of major watersheds in the Minnesota River Basin for roughly 20 years (1974 to 1994). The Minnesota River Basin has 12 major watersheds, of which three watersheds generate two-thirds of the nitrate loads22 carried to the mouth of the Minnesota River (31,670 tons nitrate/year out of a total of 50,270 tons/year). These three watersheds, which cover only 20% of the area in the Minnesota River Basin (8,176 km2), are the Blue Earth, Le Sueur, and Watonwan watersheds (Figure 17.1). According to Soil Survey information, about 55% of the soils in these watersheds are poorly drained, and have artificial tile drainage. Row crop agriculture accounts for 92% of the land uses, and cropland receives high rates of manure and nitrogen fertilizer, has high organic matter contents (> 4%), and is in a humid climatic region with greater than 750 mm/year of precipitation. All of these conditions contribute to excess nitrogen leached from cropland, which is then transported through tile drains to surface waters in large quantities. We calculated the excess nitrogen applied to row crops in these three watersheds using a mass balance approach, and compared the excess nitrogen with the monitored nitrate loads carried by surface waters.23 We assume that most of the excess nitrogen (the exception being N loss by denitrification) is transported to surface waters after leaching to subsurface tile drains and discharge in surface ditches. Calculations of excess nitrogen were accomplished using a combination of nutrient recommendations from the University of Minnesota, agricultural statistical data on fertilizer applications and fertilized cropland area at the scale of counties, land use layers for crop types, statistical surveys of farmers applying both manure and fertilizer to cropland, animal inventory data for each feedlot in the watersheds, a state feedlot-permitting database, © 2003 by CRC Press LLC

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research data on manure N content by animal species and weight, research data on manure N losses for various types of animal confinement, storage, and application methods, and GIS analysis of these datalayers. The first and most complicated step in this process was to estimate the rates of N applied to cropland from manure. To do this we estimated the manure produced by animals in each feedlot from the feedlot inventory based on species type, numbers, and weight distributions. This manure production was converted to nitrogen production using average manure N contents. Losses of manure N were estimated based on a weighted average of the losses occurring in various types of animal confinement and storage types, with the information about confinement and storage types obtained from the state-permitting database for feedlots. The permit database also contained information about the area of land available for spreading of manure. Linear regression was used for each animal species to relate animal units to the acres of available land for manure spreading. Good linear relationships were found for the predominant animal species in this area, namely; hogs, beef, and dairy cattle, with coefficients of determination ranging from 0.53 to 0.63. For each feedlot, the regression line for the dominant species was used to estimate the area of land available for manure spreading. Manure N losses during application were estimated based on a weighted average of application methods from the permit database. The remaining manure N was multiplied by 0.55 to give first year available N applied to the land. This gives the amount of manure N applied to the land that was potentially available for leaching or crop uptake. The land available for manure spreading in any single year was estimated as a fraction of the total land available for manure spreading in multiple years based on farmer surveys. These surveys provided the crop mixes used by animal operations, and the percent of each crop receiving manure in any single year. Half of the cropland in this region is in corn; the other half is in soybeans. University of Minnesota fertilizer guidelines recommend 134 kg/ha nitrogen for corn, and no nitrogen for soybeans. Farmers typically apply manure to 25% of their corn land at annual rates of 65 kg/ha, and to 10% of their soybeans at annual rates of 55 kg/ha in any single year. The 25% of corn land receiving manure also receives commercial N fertilizer at annual rates of 161 kg/ha. The rate of manure N applied in a single year was estimated based on the amount of manure N applied to the land divided by the cropland area available for manure spreading in a single year. Aside from manured cropland, there is also unmanured cropland to consider. Potentially manured corn land that does not receive manure in any single year (75% of potentially manured corn land) is fertilized with commercial N fertilizer at rates of 185 kg/ha. In addition, about 43% of the total cropland in the watersheds is corn land that never receives manure, and this land also receives N fertilizer at rates of 185 kg/ha. The remaining cropland (47% of the total cropland area) is soybeans, which do not receive nitrogen fertilizer applications. Excess N applications for the three major watersheds were obtained by subtracting the recommended amounts of N from the fertilizer or manure N applied to cropland (whether manured or unmanured). This approach gives 28,701 tons/year of excess nitrogen applied within the Blue Earth, Le Sueur, and Watonwan major watersheds, which, for such a simple approach, is surprisingly close to the measured nitrate loads (31,670 tons/year). The N excess of 28,701 tons/year translates to about 59 kg/ha/year of excess N applied on corn acres, which means farmers are applying 43% more nitrogen than the University of Minnesota-recommended rate of 134 kg/ha/year for corn. There are major differences in the approaches for estimating N losses at the major watershed scale vs. the minor watershed scale. At the major watershed scale there is an increased reliance on GIS and statistical mass balancing techniques, and a reduced reliance on process based simulation. Process-based modeling becomes unwieldy at the scale of major watersheds, and the uncertainties and coarse spatial resolution in input parameters do not justify a modeling approach. A major limitation of the N excess method used at the major watershed scale is that it is not possible to estimate N leaching losses directly. Also, processes such as denitrification, volatilization, mineralization, and immoblilization of N are not accounted for with the excess N estimation approach. © 2003 by CRC Press LLC

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V. UPSCALING NITROGEN LOSSES The nitrogen losses estimated at the plot, field, and minor watershed scales can be used to independently estimate nitrogen losses at the major watershed scale through upscaling. The simplest approach to upscaling is to simply multiply the N loss rates at each scale (for a scenario involving from 160 to 170 kg N/ha applied to corn) by the total area of the three watersheds in corn (408,810 ha). Another 408,810 ha is in unfertilized soybeans, which receives no fertilizer. Leaching losses from soybean land also need to be accounted for. At the plot scale, the long-term simulated N loss from a 175 kg/ha spring fertilizer application to corn was 23 kg/ha/year. At the field scale, the long-term N simulated loss from a 160 kg/ha fall fertilizer application to corn was 35 kg/ha/year, while the long-term loss during the soybean portion of the rotation was 23 kg/ha/year. At the minor watershed scale, the N losses from 168 kg/ha applied half in fall and half in spring to corn were 21.5 kg/ha/year. Multiplying the watershed area by these losses gives total N losses of 18,805 tons N/year based on the plot scale modeling, 23,710 tons N/year based on field scale modeling, and 17,579 tons N/year based on minor watershed scale modeling. As mentioned previously, long-term water quality monitoring shows that the Blue Earth, Le Sueur, and Watonwan major watersheds carry 31,741 tons of nitrate/year. We would not, however, expect tile drainage nitrate loads from these three watersheds to equal 31,741 tons/year exactly because sources such as atmospheric deposition and point source discharges should also contribute to the total nitrate load monitored. It is estimated that atmospheric deposition and point source discharges account for about 25% of the N loads monitored in the Blue Earth, Le Sueur, and Watonwan watersheds.24 Subtracting these contributions leaves 23,806 tons/year for losses of nitrate from agricultural landscapes, about the same amount estimated using the upscaled field scale modeling method (23,710 tons/year). The agreement between these two is satisfactory, in view of the vastly differing approaches used. The upscaled nitrate losses from modeling at the plot or minor watershed scales are significantly lower than the monitored nitrate loads at the scale of major watersheds (Figure 17.5). Plot scale simulations are based on spring-applied nitrogen, which is less susceptible to leaching than nitrogen applied in the fall at the field scale study site. Switching from spring to fall nitrogen applications typically increases tile drain losses by about 30%.25 A 30% increase in N losses for the upscaled plot scale data gives 24,447 tons/year, very similar to the estimated 23,806 tons/year from monitored nonpoint source pollution in the Blue Earth, Le Sueur, and Watonwan watersheds. The level of agreement between these two results (after corrections for N timing) is satisfactory, given the vastly different scales of observation. The upscaled results from Bevens Creek minor watershed are about 25% lower than the monitored loads for the Blue Earth, Le Sueur, and Watonwan watersheds (after adjusting for point 35000 30000 25000

Nitrate Load (tons/yr)

20000 15000 10000 5000 0 Plot

Field

Minor

Major

Monitored

FIGURE 17.5 Comparison between nitrate-N loads (tons/year) at different scales after up-scaling.

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sources). Intuitively, we would expect that the accuracy of up-scaling, relative to the water quality monitoring data, would improve as the similarity in scales increases. Although Bevens Creek is located 50 km from the Blue Earth, Le Sueur, and Watonwan watersheds, the soils, topography, and crops grown in the two areas are quite similar. Because the accuracy did not improve, there must be some differences between the two scales that help explain the results. One important difference between Bevens Creek watershed and the Blue Earth, Le Sueur, and Watonwan watersheds is that far fewer livestock are in Bevens Creek watershed. Carver county (Bevens Creek) has only about 30,000 hogs (0.31 hogs/ha), whereas Blue Earth and Martin counties alone (Blue Earth watershed) have 450,000 hogs (2.3 hogs/ha). Our upscaled simulations based on Bevens Creek attempted to account for these differences by increasing the rate of applied N from 130 to 168 kg/ha. Apparently, this approach was not sufficient; there must be effects associated with the increased animal agriculture other than rate of applied N. One possibility is an increase in the amount of N entering streams directly from feedlots as a result of runoff and spills of liquid manure. Our modeling approach does not account for these processes. A second difference between Bevens Creek and the Blue Earth, Le Sueur, and Watonwan watersheds is precipitation. Annual precipitation for the 3 years of simulation in Bevens Creek averaged 79 cm. In contrast, annual precipitation for the same 3 years from the Blue Earth, Le Sueur, and Watonwan watersheds averaged 83 cm. The increased precipitation would lead to increased losses of nitrate in tile drainage.

VI. CONCLUSIONS Process-based simulation models are increasingly used across a wide range of spatial scales to investigate nitrate losses to the environment. As spatial scale becomes coarser, model input data become less precise in terms of spatial resolution. At coarser spatial scales the importance of routing algorithms increases relative to finer spatial scales. Input data aggregation and classification become more important at coarser spatial scales than at finer spatial scales. Mass balancing approaches seem more appropriate than mechanistic modeling at the coarsest spatial scale. The performance of spatial up-scaling techniques does not seem to depend as much on the magnitude of upscaling as on the relative similarity between the smaller unit that is upscaled and the larger unit. In the examples presented in this chapter, the fraction of land that is tile drained, the rates and timing of fertilizer application, and the climates must be similar. In this chapter, best results for up-scaling to the major watershed scale (10,000 km2) were obtained from long-term field scale (32 ha) simulations, rather than from the short-term minor watershed scale (800 km2) or long-term plot scale (m2) simulations. This is because the field scale conditions (soils, landscapes, climate, N management) were more similar to the major watershed conditions than conditions at the plot or minor watershed scale. At the major watershed scale, nitrate simulation modeling was not attempted due to issues of complexity and spatial variability of model input parameters. Instead, nitrate losses were estimated using a calculation of excess applied N involving statistical databases and a GIS approach. Nitrate leaching potentials with the latter approach (28,701 tons/year) compared satisfactorily with the upscaled results from field scale simulations (23,710 tons/year), and with long-term water quality monitoring data (31,741 tons/year). The computational effort to simulate nitrate losses does not change significantly in the transition from field scale to minor watershed scale; however, the computational effort to simulate nitrate losses at either scale is greater than the effort to simulate losses from small research plots. At the field scale, we ran 128 separate simulations to account for spatial variability across the field. In contrast, we ran 63 simulations to account for spatial variability across the minor watershed scale. In both cases, model predictions were reasonably close to measured nitrate losses. This shows that watershed scale simulations of nitrate loss can be surprisingly accurate even in the absence of detailed information about spatial variability within each field in the watershed. © 2003 by CRC Press LLC

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Further research is warranted to identify the major sources of uncertainty in simulations of nitrate loss at various scales. Key questions for this research could include: How do results change with the extent of spatial discretization at a particular scale? How does the change in uncertainty of model input parameters with scale affect uncertainty in model results? How does nitrate leaching in natural spatial units (pedons, catenas, minor watersheds) differ from leaching in artificial spatial units such as fields, farms, and counties? As these questions indicate, significant challenges remain in simulating nitrate losses at various scales.

REFERENCES 1. Addiscott T.M., Smith, J., and Bradbury, N., Critical evaluation of models and their parameters, J. Environ. Qual., 24, 803–807,1995. 2. Jury, W.A. and Gruber, J., A stochastic analysis of the influence of soil and climatic variability on the estimate of pesticide groundwater contamination potential, Water Resour. Res., 25, 2465–2474, 1989. 3. Kirby, M.J., Imeson, A.C., Bergkamp, G., and Cammeraat, L.H., Scaling up processes and models from the field plot to the watershed and regional areas, J. Soil Water Conserv., 51, 391–396, 1996. 4. Loague, K., Bernknopf, R.L., Green, R.E., and Giambelluca, T.W., Uncertainty of groundwater vulnerability assessments for agricultural regions in Hawaii: review, J. Environ. Qual., 15, 15–32, 1996. 5. Mulla, D.J. and Addiscott, T.M., Validation approaches for field-, basin-, and regional-scale water quality models, in: Geophysical Mono. 108, Assessment of Non-Point Source Pollution in the Vadose Zone, Corwin, D.L., Loague, K., and Ellsworth, T.R. (Eds.), Am. Geophys. Union, 63–78, 1999. 6. Wagenet, R.J. and Hutson, J.L., Scale dependency of solute transport modeling/GIS applications, J. Environ. Qual., 25, 499–510, 1996. 7. Root T.L. and Schneider, S.H., Ecology and climate: research strategies and implications, Science, 269, 334–341, 1995. 8. Shumway, R.H., Biggar, J.W., Morkoc, F., Bazza, M., and Nielsen, D.R., Time and frequency domain analysis of field observations, Soil Sci., 147, 286–298, 1989. 9. Lark, R.M. and Webster, R., Analysis and elucidation of soil variation using wavelets, Eur. J. Soil Sci., 50, 185–206, 1999. 10. Biggar, J.W. and Nielsen, D.R., Spatial variability of the leaching characteristics of a field soil, Water Resour. Res., 12, 78–84, 1976. 11. Wagenet, R.J. and Rao, B.K., Description of nitrogen movement in the presence of spatially variable soil hydraulic properties, Agric. Water Manage., 6, 227–242, 1983. 12. Refsgaard, J.C., Thorsen, M., Jensen, J.B., Kleeschulte, S., and Hansen, S., Large-scale modeling of groundwater contamination from nitrate leaching, J. Hydrol., 221, 117–140, 1999. 13. Wylie, B.K., Shaffer, M.J., Brodahl, M.K., Dubois, D., and Wagner, D.G., Predicting spatial distributions of nitrate leaching in northeastern Colorado, J. Soil Water Conserv., 49: 288–293, 1994. 14. Gorres, J. and Gold, A.J., Incorporating spatial variability into GIS to estimate nitrate leaching at the aquifer scale, J. Environ. Qual., 25, 491–498, 1996. 15. Randall, G.W. and Iragavarapu, T.K., Impact of long-term tillage systems for continuous corn on nitrate-leaching to tile drainage, J. Environ. Qual., 24, 360–366, 1995. 16. Chung, S.O., Ward, A.D., and Shalk, C.W., Evaluation of the hydrologic component of the ADAPT water table management model, Trans. Am. Soc. Ag. Eng., 35, 571–579, 1992. 17. Davis, D.M., Gowda, P.H., Mulla, D.J., and Randall, G.W., Modeling nitrate nitrogen leaching in response to nitrogen fertilizer rate and tile drain depth or spacing for southern Minnesota, J. Environ. Qual., 29, 1568–1581, 2000. 18. Mulla, D.J., Gowda, P.H., Koskinen, W.C., Khakural, B.R., Johnson, G., and Robert, P.C., Modeling the effect of precision agriculture: pesticide losses to surface waters, in: Terrestrial Field Dissipation Studies: Purpose, Design, and Interpretation, Arthur, E.L., Barefoot, A.C. and Clay, V.E. (Eds.), Am. Chem. Soc., 2002 (In press). 19. Leonard, R.A., Knisel, W.G., and Still, D.A., GLEAMS: groundwater loading effects of agricultural management systems, Trans. ASAE, 30, 1403–1418, 1987.

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20. Dalzell, B.J., Gowda, P.H., and Mulla, D.J., Predicting nonpoint source pollution for an agricultural watershed in southern Minnesota, Paper No. 99–2215, Am. Soc. Ag. Eng., 1999. 21. Gowda, P.H., Dalzell, B.J., Mulla, D.J., and Kollman, F., Mapping tillage practices with Landsat Thematic Mapper based logistic regression models, J. Soil Water Conserv., 56,14–19, 2001. 22. Mulla, D.J. and Mallawatantri, A.P., Minnesota river basin water quality overview, Minnesota Extension Serv., FO-7079-E, Univ. Minnesota, 1997. 23. Mulla, D.J., Birr, A.S., Randall, G.W., Moncrief, J. F., Schmitt, M., Sekely, A., and Kerre, E., Impacts of animal agriculture on water quality, in: Technical Work Papers on Animal Agriculture, Johnson, G. (Ed.), Minnesota Planning Office, 2001. 24. Goolsby, D.A. and Battaglin, W.A., Nitrogen in the Mississippi Basin — estimating sources and predicting flux to the Gulf of Mexico, U.S.G.S. Fact Sheet, 135, 1–6, 2000. 25. Randall, G.W. and Mulla, D.J., Nitrate nitrogen in surface waters as influenced by climatic conditions and agricultural practices, J. Environ. Qual., 30, 337–344, 2001.

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Incorporation of Remote Sensing Data in an Upscaled Soil Water Model M.S. Seyfried

CONTENTS I. Introduction...........................................................................................................................309 II. General Considerations ........................................................................................................310 A. Scale and Scaling.......................................................................................................310 B. Some Direct Implications of Upscaling Soil Water Models ....................................311 C. Scale and the Nature of Spatial Variability...............................................................312 D. Model Extent and Sources of Deterministic Variability...........................................312 E. Spatial Aggregation ...................................................................................................313 F. Role of Remote Sensing............................................................................................314 G. Model-Scale-Remote Sensing Linkage .....................................................................315 III. Applications..........................................................................................................................317 A. Model Description in the Context of Scale and Spatial Variability .........................317 1. Model Extent and Sources of Spatial Variability................................................317 2. Modeling Approach .............................................................................................319 B. Remote Sensing Applications....................................................................................325 1. Vegetation Cover Type ........................................................................................325 2. Leaf Area Index ...................................................................................................329 3. Model Remote Sensing Data Compatibility .......................................................335 IV. Concluding Remarks ............................................................................................................340 A. Future Work ...............................................................................................................340 B. Final Thoughts ...........................................................................................................341 V. Acknowledgments ................................................................................................................341 References ......................................................................................................................................341

I. INTRODUCTION Soil water content (θ, m3/m3) affects a number of important soil hydrologic processes. Groundwater recharge and overland flow, which may transport pollutants and sediments into ground and surface waters; mineralization of soil organic matter, which releases carbon into the atmosphere and plant nutrients into the soil solution; and transpiration, which directly affects plant growth and the atmospheric circulation of carbon and water vapor, are directly affected by θ. These processes are critical to a number of practical applications including water supply and flood forecasting, regional and global climate simulation, natural resource management and agronomic crop management.

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There is a substantial mismatch between the scale at which these processes have generally been studied, mostly in laboratory columns and small experimental plots ranging in size from 1 to perhaps 100 m, and the application scale, which ranges from 1000 m up to continental or even global scales. Thus, small-scale simulation models and measurement technologies are commonly the scientific basis for applied simulation models that require soil water content information for areas much larger than the scales at which they were developed.1 In this chapter, models developed on small plots will be referred to as point-scale models because they are one dimensional and were developed and tested assuming horizontal homogeneity. Increasing modeling scale would be straightforward if important processes did not change with scale, if spatial variations of model parameters and inputs were easily described or insignificant, and if model parameters scaled linearly. In fact, processes do change with scale, spatial variations may dictate model results, and model parameters are often nonlinear with respect to scale. Bridging the research to application scale gap has become an important research issue in soil science2 and related fields of hydrology3 and ecology.4 Relatively recent technology developments related to computational power, software development (e.g., geographical information systems) and remote sensing have resulted in tools that make implementation of point-scale models at large scales possible. A commensurate development in our understanding of scale and spatial variability on the landscape is required if those tools are to be used effectively. In this chapter, ongoing research intended to partially bridge that gap for soil water modeling is described. The emphasis is on soil water within the plant root zone because that is relevant to most applications. In many cases, data are collected only for part of the root zone for practical reasons and this will be noted. The chapter is organized in two major sections. In the first section, we provide as background a general conceptual framework for model development in a spatially variable, scale-dependent environment and describe how remote sensing data can be incorporated into that framework. In the second section, we provide field data that demonstrate how scale and spatial variability affect model design and describe how remote sensing data can be integrated into a point-scale model. A distinction between this work and many others published is that we provide field data, which are commonly lacking,5 to demonstrate the applicability of these approaches.

II. GENERAL CONSIDERATIONS A. SCALE

AND

SCALING

We use the term “scale” roughly as proposed by Bloshl and Sivapalan,6 in which scale refers to a characteristic length or time of a process or model. In this chapter we focus on spatial as opposed to temporal scales. In the modeling context we consider two aspects of spatial scale: the overall spatial extent of the model, and the degree of spatial resolution or aggregation represented as model elements within the model. “Scaling,” in this context, means changing scale, either in extent or degree of aggregation. Temporal scaling, which is also critical and in many ways inseparable from spatial considerations, is addressed in this chapter only in the context of spatial scaling. Downscaling refers to inferring spatial patterns or statistical distributions on the landscape from relatively large-scale data. Examples of such data for soil water modeling include stream flow at a weir or the value of a relatively large remote sensing pixel. A wide range of θ patterns or distributions may be consistent with a given level of stream flow or large-scale pixel value. The challenge is to use additional information or relationships with more easily obtained data (e.g., with topography7) to estimate the internal distribution or variability of θ. Upscaling a model in this context involves increasing the spatial extent of the model or increasing the degree of aggregation of the model elements. In this chapter we are concerned with upscaling soil water models that were generally developed at the point scale to watersheds of about 109 m2 in extent. The intent is to be able to map root-zone θ variations that affect plant production and stream flow in diverse watersheds. Because the degree of spatial aggregation represented by © 2003 by CRC Press LLC

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model elements is invariably greater than the point or small plot scale used for model development, effects of spatial aggregation must also be considered. Upscaling and downscaling are conceptually complementary approaches that ultimately should produce the same result. That is, the proper spatial integration of many small pixels should equal the value of a single large pixel, or the summed outflow from many model elements within a watershed should be equal to the outflow at the watershed outlet. In practice, such comparisons often are not straightforward. For example, information from remote sensing may change with pixel size in a nonlinear manner that is dependent on land surface properties; it is not necessarily clear how they should be spatially integrated. In the case of stream flow, the difficulty of summing up smaller portions of the watershed has been well documented (e.g., Woolhiser8). Clearly, there are advantages and disadvantages to both approaches.7,9 We are pursuing an upscaling approach for two reasons. First, the critical relationships between easily measured landscape features and θ required for downscaling are not apparent for many regions. Second, we are interested in estimating the effects of management at specific sites both locally, at the site, and for larger areas. In choosing this approach it is important to recognize the limitations of the approach and devise a modeling approach that minimizes the effects of those limitations.

B. SOME DIRECT IMPLICATIONS

OF

UPSCALING SOIL WATER MODELS

Upscaling point-scale models to larger areas inevitably leads to a loss of simulation accuracy and precision relative to what can be achieved at the point scale. This is not often acknowledged and is difficult to quantify, but nonetheless has important implications for upscaling soil water models. The success of model upscaling is partly dependent on minimizing these effects. Upscaled model accuracy is partly reduced due a reduction in model input parameter estimation accuracy. No matter how well the model represents the processes of interest, if the required parameters are of reduced accuracy, the resultant simulation must also have reduced accuracy. Most models require input values for a variety of parameters that can be directly and accurately measured at the small plot scale. However, it is impractical to measure those variables, which may vary considerably in space, when the spatial extent is increased. These values must therefore be estimated from some alternative data source (e.g., using pedotransfer functions) that inevitably results in a less accurate parameter value. This problem is exacerbated for parameters that scale nonlinearly in space and have no well-defined procedure for spatial averaging.1 Model accuracy is also reduced because the accuracy of input driving variables decreases as the area they represent increases with scale. Rainfall, for example, can be directly measured with high accuracy at a site with a precipitation gauge, but that accuracy decays as the area represented by that gauge increases (e.g., Houser et al.10). Changing scale also affects the accuracy with which processes affecting θ can be simulated. For example, small-scale processes related to the complex interactions between individual shrubs, or specific paths of infiltration related to rooting habit, are practically impossible to simulate explicitly at larger scales, partly due to computing power required and partly due to a lack of sufficient information describing the shape and position of these features. It is usually assumed (or hoped) that these small-scale processes do not substantially impact the larger scale patterns of interest, but this may not be the case. On the other hand, processes that may be insignificant at the small plot scale may gain importance at larger scales. Surface runoff may be small enough to be neglected at the small plot scale without causing significant errors in simulation of θ. However, this neglect could result in substantial errors at the larger watershed scale following major runoff events that cause substantial horizontal redistribution of water. In the most fundamental sense, the loss of precision with upscaling is due to the increase of spatial variability with scale. That is, θ, which is a point measurement, generally varies more as the size of spatial aggregates increases. This takes place regardless of model structure or accuracy. Thus, even if the model accurately calculates the average θ for a spatial aggregate, the actual θ at © 2003 by CRC Press LLC

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a given point within that aggregate will deviate from that value to some degree. In fact, the utility of the modeled results depends, to some degree, on the amount of that deviation. Considering model precision more specifically, there is corresponding loss of measurement precision that accompanies the loss of accuracy with scale. Similarly, the various simplifications and assumptions built into models that tend to increase with scale may also be considered to increase the model uncertainty. The total model variance is the sum of the variance due to input uncertainty and that due to model uncertainty.11 Both terms are difficult to quantify,1 but can generally be expected to increase with scale. (In some cases, nonlinear parameters may propagate errors such that the output variance decreases with scale.12)

C. SCALE

AND THE

NATURE

OF

SPATIAL VARIABILITY

Upscaled model precision and accuracy depends, to some extent, on how effectively spatial variability is represented at the model scale. Spatial variability may be represented as deterministic or as stochastic.13–15 With deterministic variability, soil properties or inputs vary spatially in a known way.14 Spatial variation may be “known,” or deterministic, from: (1) theoretically derived relationships; (2) empirically observed and described relationships, and (3) mappable (e.g., soil survey) trends. Deterministic descriptions of variability result in specific parameter values for each field location. These parameter values may be used in a model to simulate spatial patterns of θ in the landscape. Stochastic variability is random. It may be further subdivided into spatially dependent and spatially independent variability. Spatial dependence is generally quantified by a variogram that shows how variability changes (generally increases or remains constant) as the space between measurements in a stationary environment increases. Spatially independent data are not affected by position on the landscape. This variability includes, or is indistinguishable from, measurement variability (assuming no bias), small-scale variability and other “unexplained” variability, and may be described by statistical parameters such as the mean and variance or a probability distribution function. In many models stochastic variability is ignored. The total variability in the field is a composite of both kinds of variability. Where the deterministic component can be described as a function of position or is constant, the relationship between these parameters is summarized with the following equation: Z(x) = m(x) + ε′(x) + ε″(x)

(18.1)

where x is the position in 1, 2 or 3 dimensions, m(x) is the deterministic function, ε′(x) is the stochastic, spatially dependent component, and ε″(x) is the spatially independent, Gaussian noise term having zero mean and variance.16 Other deterministic variability, such as that described by maps, may not be described with Equation 18.1 but may be thought of as being partitioned as in Equation 18.1.

D. MODEL EXTENT

AND

SOURCES

OF

DETERMINISTIC VARIABILITY

The treatment of variability as deterministic or stochastic is an important but usually implicit part of model design. Soil water exhibits deterministic spatial variability at scales ranging from 10–9 m, where variations are due to molecular interactions at the solid–water interface17 to the continental scale,18 where climatic patterns are determined by the position of mountain ranges or proximity to the ocean. Point-scale soil water models have been developed at the macroscopic or Darcian scale, which is delineated at the lower, small-scale end by the representative elementary volume (REV). The REV is the scale at which the number of pore combinations is sufficient for variability to be considered stochastic and a single, composite value can be used.19 Thus, all field-scale soil water models consider soil water variations due to individual particle interactions to be stochastic. In so

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doing the ability to model spatial variations within the REV is lost. For purposes of measuring θ, a scale of several cm is generally considered to be sufficiently large for the REV. For descriptive purposes, deterministic variability may be related to “sources” that are associated with increases in spatial variability as scale increases. For example, in sagebrush-(Artemsia spp.) dominated semiarid regions, it has been observed that the structure of soil located underneath shrubs, which is typically covered with moss, protected from direct raindrop impact and experiences relatively few freeze–thaw cycles, is quite different from that in the adjacent exposed interspaces, which have little vegetative protection.20,21 The result is a greater infiltration capacity22,23 under shrubs. Higher water contents and deeper wetting fronts under shrubs are observed after runoff events.24 These variations occur on the order of 1 m. As the model extent expands from a few cm it becomes important to consider variability due to shrub–interspace relationships because they have a large impact on θ. This requires explicit consideration of each. However, increasing the model extent further, to 20 m, for example, results in the inclusion of a large number of individual shrub–interspace combinations (a 20-m pixel would have about 400 shrub–interspace combinations). It becomes reasonable to consider the variability to be stochastic and represented by statistical parameters such as the mean and standard deviation or probability distribution function. With this approach, model complexity is greatly reduced but small-scale variations are lost and the cause of those variations ignored. Thus we may refer to the variability introduced by shrub cover as a shrub effect without explicitly considering its cause. Seyfried and Wilcox25 used the term deterministic length scale (DLS) to define the scales at which variability due to a given source may be described as deterministic and the scale or sample size is insufficient for stochastic descriptions. Increasing the model extent will likely introduce additional sources of variability. A 20 by 20 m area may have a uniform soil type, land use and topography, but as the spatial extent is increased, those conditions will eventually change and the amount of variability will increase. This added source may be treated deterministically so that, for a given model, some sources of variability will be modeled as stochastic, others as deterministic. For many hydrologic applications it has been established that the spatial pattern of θ is critical.26 For these applications, the sources of variability that cause the spatial patterns must be treated deterministically. Bloschl27 has suggested that, because the sources of variability in large-extent models may be numerous, the model should focus on capturing the most dominant sources.

E. SPATIAL AGGREGATION Conceptually, the modeled area may be composed of an arbitrary number of spatial aggregates in a regular (e.g., grid) or irregular (e.g., soil map) pattern. The size and shape of the spatial aggregates relative to the DLS determine if those sources of variability will be effectively represented as deterministic or stochastic. Minimum aggregate sizes within the DLS can delineate deterministic sources while aggregate sizes greater than the DLS can characterize stochastic variability. Intermediate aggregate sizes are not effective for either approach. That is, they are too large to delineate and simulate the causes of the soil water variations and too small to characterize that variability effectively in a statistical sense. In terms of shrub–interspace interactions, 10-cm aggregates can be characterized as having shrub or interspace properties and 20-m aggregates have a composite value. However, an aggregate size of 1 m, when distributed across the landscape, may have any composition, from 100% shrub to 100% interspace. In this case, the best estimate of soil properties is some kind of average value and the model effectively simulates at 20-m resolution even though 1 m aggregates are used. The process of aggregating is straightforward for parameters that scale linearly. In this context, that means that the composite value is the arithmetic average of many smaller aggregates. That is the case for θ. However, it may not be true for parameters used to calculate it. A well-known example of this is hydraulic conductivity (K), which is a highly nonlinear function of θ. For K, © 2003 by CRC Press LLC

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the composite, of effective aggregate value is not the arithmetic average and, in fact, is not currently well defined. (This will be discussed in greater length in the next section). In practice, determination of the minimum aggregate size is commonly governed by the computational demands (limits the number of delineations) or by the scale of available for input data (e.g., a 30-digital elevation model (DEM)). These practical considerations do not alter the effects of scale and the nature of variability. If the minimum model aggregate is 30 m because that is the maximum resolution of topographic information, then shrub–interspace interactions must be considered in a stochastic sense. If the minimum model aggregate is 1 km, then many topographic effects must be considered as stochastic because, in most landscapes, 1 km includes a wide variety of slopes and aspects.

F. ROLE

OF

REMOTE SENSING

Upscaled models intended to portray spatial variations require spatially extensive data as data input and for model testing. Lack of availability of such data is one of the major limitations to the extension of soil water models to large areas. Considering that DEMs and soil maps are essentially remote sensing products (largely derived from aerial photography), remote sensing is the only source of such data. Hence, progress in this model upscaling virtually requires the incorporation of remote sensing data. Recent advances in computer technology, geographical information system software and global positioning system technologies, along with improved satellite data quality, have lead to the expectation that remote sensing data will be used to parameterize and drive large-scale models, or perhaps even replace models with direct measurements. In fact, considerable progress has been made in these directions for data relevant for soil water models, including the measurement of surface soil water content28,29 and evaporation.30 Even so, it is safe to say that, in general, the promise of remote sensing has so far been largely unfulfilled.31 There are some very serious limitations to the use of remote sensing. These limitations vary to some degree with sensor and application but, in general, are somewhat generic for natural resource modeling and involve temporal and spatial scale problems. The temporal problems stem from the fact that remote sensing data are always a retrospective snapshot. They can only provide information concerning a past instant in time. The time lag between image acquisition and availability varies considerably with sensor and platform, but for measurements using visible wavelengths and therefore limited by cloud cover, the lag can be multiple months even in a semiarid climate. In any event, these data cannot be used to drive models in real time or for future projections. That each image represents only an instant in time may be critical for applications simulating fluxes, which can vary dramatically in a short time because only a tiny fragment of the day is sampled. There are several aspects to the spatial scale problem. Ideally, because hydrologic systems of interest and the models used to describe them generally exhibit significant nonlinearities with scale, input data and model parameters would be defined at a consistent scale that coincides with the scale of interest.11 Unfortunately, such approaches are not practical at this time. From the upscaling perspective, we are left with the problem that remote sensing pixels are generally larger than the traditional measurements needed for models. Interpretation of remote sensing data in this context involves two interrelated problems: the conversion of the raw data collected remotely (e.g., backscatter) to model-required variables (e.g., θ) with associated accuracy and precision, and the establishment of a relationship between remotely acquired spatial data and standard point scale measurements. The conversion of raw remote sensing data usually involves using simplified representations of the land surface or empirical relationships. This process is subject to errors from a variety of sources. In most cases, some degree of standard, on the ground measurement (determination of “ground truth”) is required to determine the validity of simplifying assumptions or empirical relationships in the context of the specific land surface properties of interest. In many cases the

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measurement error associated with remote sensing data is dependent on those properties and not known in advance. The process of establishing ground truth is greatly complicated by spatial variability and scale interactions. If the ground truth data are highly variable within a pixel, evaluation of that pixel in terms of ground measurement is problematic. This is especially true if the variables scale nonlinearly. In addition, variability beyond the pixel must be considered because of registration errors inherent in all remote sensing data. That is, remote sensing data are registered to specific coordinates on the ground, but the accuracy of that registration is usually on the order of one to a few pixels. That means that the data collected remotely from a given location may actually represent land surface conditions two or three pixels away from the true position. Finally, remotely sensed pixel size must be considered in the context of the DLS of critical variability in the field. Pixel size affects the treatment of spatial variability in much the same way that the size of model aggregates does. If the pixels are larger than the DLS, then a deterministic representation of θ due to that variability will be difficult to achieve. For remote sensing of θ, for example, the ideal measurement for our applications would be of the entire root zone on a daily or more frequent basis. Although much good research has been done, at present, this is not possible. The most promising approaches use active or passive microwave sensors.32,33 Current active microwave sensor–platform combinations provide data after a considerable time lag and only for the upper 2 cm of soil. Although these data are valuable from a model testing and input perspective, they do not provide subsurface information. The backscatter signal from active microwave remote sensing, which must be interpreted to obtain θ, is subject to variations due to topography, surface roughness and vegetation, which must be taken into account.34 In addition, the typical pixel size of 30 m presents a challenge for ground truth determination. Upcoming passive microwave sensors will have pixel sizes much larger (multiple km) than the DLS of many critical sources of variability. As with data from any measurement, data from remote sensing need to be evaluated in the context of the intended purpose. Merely plugging remote sensing data into models may be worse than not using them at all because they may add unknown amounts of information and unknown amounts of error out of context with the rest of the model. In order for remote sensing data to be useful for modeling purposes, they should be spatially and functionally compatible with the model. By spatial compatibility, we mean that the spatial resolution of remote sensing-derived land surface characteristics causing or resulting from critical variability in θ should be at least as great as that of the model (i.e., the pixel size should be smaller than the model element). Functional compatibility is intended to convey the idea that the accuracy and precision of remote sensing-derived data are sufficient for the model to use them effectively to describe θ distribution on the landscape.

G. MODEL-SCALE-REMOTE SENSING LINKAGE Figure 18.1 was included in this chapter in an effort to provide a framework for model description and remote sensing application in the context of scale and spatial variability. It is a slight modification of a figure presented by Hoosbeek and Bryant,35 who had the similar purpose of describing soil genesis models. The depictions of model extent, degree of aggregation and model approach are intended to facilitate transfer of models among users and adaptation of model components, as well as providing a context for evaluating remote sensing inputs. Although somewhat subjective, it is an improvement over commonly used descriptions in current usage, such as “fully distributed,” “process oriented” or “large scale” because it highlights three aspects of spatial models that, although critical in terms of model application and data applicability, are often not acknowledged explicitly. In this three-dimensional representation, the vertical axis represents the spatial extent. This axis was originally described as organizational hierarchy by Hoosbeek and Bryant,35 implying that natural scale breaks, such as the ped, soil series, catena or watershed, are somewhat akin to the

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FIGURE 18.1 A three-dimensional diagram describing models in the context of scale, spatial variability and modeling approach. The circle indicates where the modeling described in this chapter falls.

ecological concept of ecosystem, within which spatial variability is relatively uniform. Although this is a useful concept, we have simply used the model extent because it is less ambiguous (i.e., soil series, watersheds, etc. exist at a wide range of scales). This axis may be the most important because, to a large degree, it determines which processes and sources of variability must be simulated and tends to drive choices of model approach and degree of spatial aggregation. One of the horizontal axes portrays the complexity of the modeling approach with the understanding that, for any given spatial extent, a variety of modeling approaches may be applied. Following Hoosbeek and Bryant35 we describe these as ranging from mechanistic to functional. Mechanistic soil water models explicitly simulate the processes affecting θ while functional models use empirical relationships to represent those processes. At the Darcian scale, soil water models at the mechanistic extreme are represented by three-dimensional solutions to the Richards equation with provisions for hysteresis and macropore or bypass flow and including heat transport. Simple “black box” soil water models that treat soil as an empirical storage term represent the functional model extreme. In general, more mechanistic approaches are more computationally intensive because they require numerical solutions to partial differential equations. The axis is represented as a continuum because functional models vary considerably in complexity and mechanistic models often employ empirical relationships, especially when applied to larger scales. Mechanistic models are more dominant in a research mode because the processes they explicitly simulate are often the objects of interest. Functional models tend to be more management oriented because they are easier to apply computationally and in terms of data requirements. The degree of spatial aggregation is represented on the other horizontal axis. The intent is to provide a description of the spatial complexity simulated. A crude but informative measure of this is simply the number of delineations per km2 within the modeled extent. This will tend to decrease as the model extent increases, so mesoscale models might use model elements of 104 km2 or larger, which correspond to aggregate densities of 0.0001 delineations/km2 or smaller, while other models may use the 30 m DEM size common in the U.S., resulting in density of about 103 delineations/km2. There is a considerable range in the degree of model aggregation used by different models for a given model extent. This depends on several factors, including the intent of the model, data availability and the nature of spatial variability within the modeled extent. The model level on all three axes has important implications for the use of remote sensing data. For relatively large spatial extents, some kinds of data may be impractical. There are over 1000 Landsat pixels in 1 km2. As the spatial extent increases, the amount of information acquired © 2003 by CRC Press LLC

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can become difficult to manage. In addition, as increasing extent involves increasing sources of spatial variability to be modeled, it also includes increasing sources of variability to confound remote sensing interpretation. In some cases, the model extent may exceed the size of a remote sensing scene. This adds complications associated with stitching together scenes acquired with different sun angles and cover properties. The level of spatial aggregation needs to be considered in the context of the pixel size. If the remote sensing pixels and model element aggregates are the same size, no additional considerations are required. If the pixels are smaller than the model aggregates, then an averaging procedure needs to be determined. If the pixels are larger than the model elements, some means of disaggregating the pixel values needs to be devised. In both cases this may be problematic if there is large spatial variability or nonlinear scaling of the measured values. In terms of model approach, in general, mechanistic models tend to be more demanding in terms of data requirements (remote and otherwise) than functional models, in the kind of data that can be used and the frequency they are needed. Functional models, which are empirical to start with, are more adapted to using empirical relationships between remote sensing data and the land surface properties. The critical consideration is whether the accuracy and precision of the data derived from remote sensing are sufficient for the needs of the model.

III. APPLICATIONS A. MODEL DESCRIPTION

IN THE

CONTEXT

OF

SCALE

AND

SPATIAL VARIABILITY

In the remainder of this chapter we describe work conducted at the Reynolds Creek Experimental Watershed (RCEW) directed toward the development of a soil water model upscaled to a watershed scale. This model is intended to provide information useful for estimating plant production and for evaluating the effects of management, especially prescribed fire, on soil water balance in predominantly semiarid terrain. In these regions, variations in the availability of soil water for plant growth are the primary determinant of year-to-year variations in plant growth. The work is ongoing, and we will not describe a finished product. Instead we describe how remote sensing, despite its limitations, may be integrated in an upscaled soil water model. The remote sensing data we intend to use are “standard” in that they are widely available and not under experimental development (mostly Landsat thematic mapper data with a pixel size of 30 m). The incorporation of these data into the modeling approach is done in the context of the modeled spatial extent, modeling approach, and degree of aggregation relative to the scale and nature of critical variability as described in the previous section. Although the examples are from a relatively small area (the RCEW), it is expected that some generalizations may be drawn that will be beneficial for others working at different sites. 1. Model Extent and Sources of Spatial Variability There are several advantages to the use of the RCEW in this context. First, there is a long history of research at the site and many critical processes have been well described (see Seyfried and Wilcox25 for examples). Second, there is a strong infrastructure for research, including long-term monitoring of a variety of environmental inputs.36 Third, the size of the watershed is approximately the scale at which many management decisions must be made. Finally, conditions within the watershed roughly bracket those found throughout a large part of the western U.S. including significant portions of the states of Washington, Oregon, Nevada, Wyoming, Montana, Utah, Idaho and parts of California and Colorado. Although the model testing and development are taking place in a data-rich environment, we intend the model to be applicable to a large geographic area. The RCEW, located in southwest Idaho, is 238 km2 in extent (Figure 18.2). Excepting the alluvial valley near the northern end of the watershed, the topography is rugged, and the elevation ranges from about 1100 m at the outlet to over 2100 m at peaks at the southern end (Figure 18.2). © 2003 by CRC Press LLC

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FIGURE 18.2 (See color insert following page 144.) The location, elevation and topography (shaded relief), of the RCEW. Elevation data is in 30-m pixels. Reynolds Creek flows roughly up the center of the RCEW from south to north.

The precipitation regime is predominantly semiarid. However, mean annual precipitation varies linearly with elevation, increasing about 85.5 mm per 100 m of elevation from about 250 mm at the lowest elevations to over 1000 mm at the highest elevations, where most of the precipitation is snow.37 The differences in precipitation between the high and low elevations occur mostly in the winter months; the entire watershed is quite dry in the summer months.37 The geology of the RCEW is dominated by basalt with large outcroppings of granite along the west-central portion of the watershed and alluvium in the valley. The vegetation is dominated by different subspecies of sagebrush that occupy different elevation bands. Plant coverage of most of the watershed is about 50% in late summer. Conifers and aspen (Populus tremuloides) are found in the highest precipitation portions of the watershed, and riparian plant communities along Reynolds Creek and other sources of perennial water. Much of the watershed has steep, shallow, rocky soils, but there are areas of deep, loamy soils that are rock free. Surface textures are generally loam and silt loam on soils derived from basalt and sandy loam on soils derived from granite. Subsoil textures vary considerably. Saline soils of high pH can be found in the valley and low pH soils under a forest floor in the higher elevations. The soil temperature regime is mesic in most of the watershed and cryic at upper elevations (> 1750 m). The soil moisture regime at the lowest elevations is borderline to aridic, but most of the RCEW soils have a xeric moisture regime. Udic moisture regimes are found at the upper elevations, particularly in snow accumulation areas and in the southwest corner of the watershed. Seyfried et al.38 provide a more detailed description of the distribution of soils, geology and vegetation on the RCEW. This high spatial variability environment is typical of the region. Previous studies at the RCEW have demonstrated that different sources of variability may be expected to dominate hydrologic processes at different scales.25 With regard to θ, Pierson et al.39 showed that position relative to © 2003 by CRC Press LLC

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shrubs or interspaces (the space between shrubs) is a source of small-scale spatial variability. Seyfried40 showed that different soils are a source of somewhat larger-scale spatial variability. At a similar scale, Flerchinger et al.41 showed that different vegetation types can also be a source of deterministic variability. At somewhat larger scales, Rawls et al.42 showed that climate is an important source of spatial variability. Other potential sources of variability include topography and geology. 2. Modeling Approach Among point-scale soil water models two basic approaches have been identified:43 those based on transport parameters (mechanistic) and those based on capacity parameters (functional). Capacity parameters are generally static, such as texture, porosity or permanent wilting point. Transport parameters are used to calculate fluxes as a function of time and include hydraulic conductivity (K), which may be expressed as a function of soil water potential (h) or θ. Transport models are mechanistic because they calculate soil water movement as the result of a soil water potential gradient. In unsaturated soils, the basic relationship is the Richards equation. The advantages of the mechanistic approach are that: 1. It allows for the study of processes involved in soil water movement because those processes are explicitly simulated. 2. It can, in principle, be transferred to any site because the basic processes are universal. 3. It can, in principle, be used to simulate related processes such as solute transport and plant water uptake that are fundamentally linked to soil water movement and also driven by potential gradients. In the case of plant water uptake, the mechanistic approach links water movement from the soil through plants to the atmosphere via a water potential continuum sometimes referred to as the soil–plant–atmosphere continuum. In contrast, models based on capacity parameters approximate processes using parameters that are empirically or conceptually related to them. For example, in vertically uniform soils with no evaporation permitted, it has been demonstrated that a saturated field soil with a deep water table will drain at an ever decreasing rate such that the log of the amount of soil water stored is proportional to the log of the drainage time as K decreases nonlinearly with each increment of water loss.44 A capacitance approach is to approximate this process (e.g., Seyfried and Rao45) as an exponential decrease in water content with time to a fixed, empirically determined value, commonly called field capacity (θfc). The drainage rate is defined by an empirical time constant and neither h nor K is considered. Similarly, from a mechanistic approach, h at the soil–root interface decreases as the soil loses water to the roots. This generates steeper h gradients that allow the plants to meet atmospheric demand as K decreases with h. As h approaches the limit of what the plant can produce, fluxes begin decline and various plant responses are triggered by the lack of water. This is approximated empirically in functional models as proceeding to an empirically determined θ, the permanent wilting point θpwp, after which transpiration is zero. Evaporation and transpiration are calculated from a potential evapotranspiration (e.g., Priestly and Taylor46) as affected by leaf area index (LAI) and θ, and the soil–water–plant continuum is not explicitly considered. The capacitance parameter approach may be characterized as a series of bracketing parameters (saturated θ (θs), θfc, θpwp) that control infiltration depth, deep percolation and evapotranspiration. The approach has been criticized because it creates “constants,” such as θfc, which are not truly constant but vary with conditions such as texture, vertical morphology, antecedent water content and especially time of consideration44 and because it ignores critical processes. For example, because fluxes are not considered, Horton overland flow cannot be simulated. Also, because all flow is implicitly assumed to be gravity driven, upward fluxes, such as occur from a shallow water table, cannot be accommodated without additional empirical modification. Prior to modern

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computers, functional approaches were used widely because they were computationall