C Larrabee Winter
 Professor
 Professor, Atmospheric Sciences
 Professor, Applied Mathematics  GIDP
 Member of the Graduate Faculty
 John W. Harshbarger Building, Rm. 318B
 Tucson, AZ 85721
 winter@arizona.edu
Biography
2009Present: University of Arizona, Tucson, AZ
Professor and Head, Department of Hydrology and Water Resources
Professor, Atmospheric Sciences
Member, Steering Committee Program in Applied Mathematics
Adjunct Professor, Department of Mathematics, University of Wyoming
Interim CEO, National Ecological Observatory Network, Boulder, CO
2003 – 2009: NCAR, Boulder, CO
Deputy Director
Senior Scientist, Institute for Mathematics Applied to Geoscience, NCAR
1990  2003: Los Alamos National Laboratory, Los Alamos, NM
Science Advisor, Office of the Governor, State Capitol, Santa Fe, NM
Leader, Computer Research and Applications Group (C3), Computing and
Computational Sciences Division
Leader, Geoanalysis Group (EES5), Earth and Environmental Sciences Division
Leader, Applied Mathematics and Statistics Team (T7), Theoretical Division
1985  1990: SAIC Advanced Computing Division, Tucson, AZ
Chief Scientist
1983  1985: Idaho State University, Pocatello, ID
Assistant Professor of Mathematics and Computer Science
1982  1983 University of Arizona, Tucson, AZ
PostDoctoral Associate, Department of Hydrology and Water Resources
Degrees
 Ph.D. Applied Mathematics
 University of Arizona, Tucson, Arizona, USA
 Asymptotic properties of diffusion in random velocity fields
Work Experience
 National Center for Atmospheric Research (2003  2009)
Interests
Research
Risk assessment for environmental systems, groundwater, general hydrology, applied math and stochastic hydrology, flow through networks of pores
Teaching
Risk assessment for environmental systems, groundwater, general hydrology, applied math and stochastic hydrology, flow through networks of pores
Courses
202223 Courses

Water Science+Environmnt
HWRS 201 (Fall 2022)
202122 Courses

Statistical Hydrology
CE 549 (Spring 2022) 
Statistical Hydrology
HWRS 449 (Spring 2022) 
Statistical Hydrology
HWRS 549 (Spring 2022) 
Risk Asmnt for Enviroment Sys
HWRS 443A (Fall 2021) 
Risk Asmnt for Environment Sys
HWRS 543A (Fall 2021)
202021 Courses

Statistical Hydrology
CE 449 (Spring 2021) 
Statistical Hydrology
CE 549 (Spring 2021) 
Statistical Hydrology
HWRS 449 (Spring 2021) 
Statistical Hydrology
HWRS 549 (Spring 2021) 
Water Science+Environmnt
HWRS 201 (Fall 2020)
201920 Courses

Statistical Hydrology
CE 449 (Spring 2020) 
Statistical Hydrology
HWRS 449 (Spring 2020) 
Statistical Hydrology
HWRS 549 (Spring 2020) 
Risk Asmnt for Enviroment Sys
HWRS 443A (Fall 2019) 
Risk Asmnt for Environment Sys
HWRS 543A (Fall 2019)
201819 Courses

Dissertation
MATH 920 (Spring 2019) 
Statistical Hydrology
CE 549 (Spring 2019) 
Statistical Hydrology
HWRS 449 (Spring 2019) 
Statistical Hydrology
HWRS 549 (Spring 2019) 
Dissertation
MATH 920 (Fall 2018) 
Thesis
HWRS 910 (Fall 2018) 
Water Science+Environmnt
HWRS 201 (Fall 2018)
201718 Courses

Thesis
HWRS 910 (Summer I 2018) 
Dissertation
HWRS 920 (Spring 2018) 
Dissertation
MATH 920 (Spring 2018) 
Statistical Hydrology
CE 449 (Spring 2018) 
Statistical Hydrology
CE 549 (Spring 2018) 
Statistical Hydrology
HWRS 449 (Spring 2018) 
Statistical Hydrology
HWRS 549 (Spring 2018) 
Thesis
HWRS 910 (Spring 2018) 
Dissertation
HWRS 920 (Fall 2017) 
Dissertation
MATH 920 (Fall 2017) 
Risk Asmnt for Enviroment Sys
HWRS 443A (Fall 2017) 
Risk Asmnt for Environment Sys
HWRS 543A (Fall 2017)
201617 Courses

Dissertation
HWRS 920 (Spring 2017) 
Dissertation
MATH 920 (Spring 2017) 
Thesis
HWRS 910 (Spring 2017) 
Dissertation
HWRS 920 (Fall 2016) 
Dissertation
MATH 920 (Fall 2016) 
Fund Of Subsurface Hydr
HWRS 518 (Fall 2016) 
Risk Asmnt for Enviroment Sys
HWRS 443A (Fall 2016) 
Risk Asmnt for Environment Sys
HWRS 543A (Fall 2016) 
Thesis
HWRS 910 (Fall 2016)
201516 Courses

Dissertation
HWRS 920 (Spring 2016) 
Dissertation
MATH 920 (Spring 2016) 
Thesis
HWRS 910 (Spring 2016)
Scholarly Contributions
Chapters
 Winter, C. L. (2019). Stochastic Hydrogeology: Chuck Newman had a good idea about where to start. In Sojourns in Probability Theory and Statistical Physics – II Brownian Web and Percolation,. Springer.
 Winter, C. L., & Tartakovsky, D. M. (2017). Probabilistic Risk Assessment. In Handbook of Groundwater Engineering, ed. by J. Cushman. CRC Press.
Journals/Publications
 Adeyemi, B., Ghanbarian, B., Winter, C. L., & King, P. R. (2022). Determining effective permeability at reservoir scale: Application of critical path analysis. Advances in Water Resources, 159.
 Chang, L., Yuan, R., Gupta, H. V., Winter, C. L., & Niu, G. (2020). Why Is the Terrestrial Water Storage in Dryland Regions Declining? A Perspective Based on Gravity Recovery and Climate Experiment Satellite Observations and Noah Land Surface Model With Multiparameterization Schemes Model Simulations. Water Resources Research, 56(11).
 Clark, C. L., Winter, C. L., & Corley, T. (2020). Effects of percolation on the effective conductivity of irregular composite porous media. Advances in Water Resources, 137.
 Li, L., Nearing, M. A., Nichols, M. H., Polyakov, V. O., Larrabee Winter, C., & Cavanaugh, M. L. (2020). Temporal and spatial evolution of soil surface roughness on stony plots. Soil and Tillage Research, 200.
 Guadagnini, A. A., Neuman, S. P., Nan, T., Riva, M., & Winter, C. L. (2015). Scalable statistics of correlated random variables and extremes applied to deep borehole porosities. Hydrol. Earth Syst. Sci., 19, 729745,.
 Hyman, J. D., & Winter, C. L. (2014). Stochastic generation of explicit pore structures by thresholding Gaussian random fields. Journal of Computational Physics, 277, 1631.
 Hyman, J. D., Guadagnini, A., & Winter, C. L. (2015). Statistical scaling of geometric characteristics in stochastically generated pore microstructures. Journal Computational Geosciences, Volume 19(Issue 4), 845854.
 Siena, M., Hyman, J. D., Riva, M., Guadagnini, A., & Winter, C. L. (2015). Direct numerical simulation of fully saturated flow in natural porous media at the pore scale: a comparison of three computational systems. Computational Geosciences.
 Zhu, J., Winter, C. L., & Wang, Z. (2015). Nonlinear effects of locally heterogeneous hydraulic conductivity fields on regional stream–aquifer exchanges. Journal Hydrology and Earth System Sciences, Volume 19(11), 45314545.
 Guadagnini, A., Neuman, S. P., Nan, T., Riv, M., & Winter, C. L. (2014). Extreme value statistics of scalable data exemplified by neutron porosities in deep boreholes. Hydrology and Earth System Sciences Discussions, 11, 1163711686.
 Siena, M., Riva, M., Hyman, J. D., Winter, C. L., & Guadagnini, A. (2014). Relationship between pore size and velocity probability distributions in stochastically generated porous media. hysical Review E Volume 89, 89.
 Hyman, J. D., & Winter, C. L. (2013). Hyperbolic regions in flows through threedimensional pore structures. Physical Review E  Statistical, Nonlinear, and Soft Matter Physics, 88(6).More infoAbstract: Finite time Lyapunov exponents are used to determine expanding, contracting, and hyperbolic regions in computational simulations of laminar steadystate fluid flows within realistic three dimensional pore structures embedded within an impermeable matrix. These regions correspond approximately to pores where flow converges (contraction) or diverges (expansion), and to throats between pores where the flow mixes (hyperbolic). The regions are sparse and disjoint from one another, occupying only a small percentage of the pore space. Nonetheless, nearly every percolating fluid particle trajectory passes through several hyperbolic regions indicating that the effects of inpore mixing are distributed throughout an entire pore structure. Furthermore, the observed range of fluid dynamics evidences two scales of heterogeneity within each of these flow fields. There is a larger scale that affects dispersion of fluid particle trajectories across the connected network of pores and a relatively small scale of nonuniform distributions of velocities within an individual pore. © 2013 American Physical Society.
 Winter, C. L., Smolarkiewicz, P. K., & Hyman, J. D. (2013). Pedotransfer functions for permeability: A computational study at pore scales. Water Resources Research, 49(4), 20802092. doi:10.1002/wrcr.20170More info[1] Three phenomenological power law models for the permeability of porous media are derived from computational experiments with flow through explicit pore spaces. The pore spaces are represented by threedimensional pore networks in 63 virtual porous media along with 15 physical pore networks. The power laws relate permeability to (i) porosity, (ii) squared mean hydraulic radius of pores, and (iii) their product. Their performance is compared to estimates derived via the Kozeny equation, which also uses the product of porosity with squared mean hydraulic pore radius to estimate permeability. The power laws provide tighter estimates than the Kozeny equation even after adjusting for the extra parameter they each require. The best fit is with the power law based on the Kozeny predictor, that is, the product of porosity with the square of mean hydraulic pore radius.
 Winter, C., Hyman, J. D., Smolarkiewicz, P. K., & Winter, C. L. (2012). Heterogeneities of flow in stochastically generated porous media. Physical review. E, Statistical, nonlinear, and soft matter physics, 86(5 Pt 2).More infoHeterogeneous flows are observed to result from variations in the geometry and topology of pore structures within stochastically generated three dimensional porous media. A stochastic procedure generates media comprising complex networks of connected pores. Inside each pore space, the NavierStokes equations are numerically integrated until steady state velocity and pressure fields are attained. The intricate pore structures exert spatially variable resistance on the fluid, and resulting velocity fields have a wide range of magnitudes and directions. Spatially nonuniform fluid fluxes are observed, resulting in principal pathways of flow through the media. In some realizations, up to 25% of the flux occurs in 5% of the pore space depending on porosity. The degree of heterogeneity in the flow is quantified over a range of porosities by tracking particle trajectories and calculating their attributes including tortuosity, length, and first passage time. A representative elementary volume is first computed so the dependence of particle based attributes on the size of the domain through which they are followed is minimal. High correlations between the dimensionless quantities of porosity and tortuosity are calculated and a logarithmic relationship is proposed. As the porosity of a medium increases the flow field becomes more uniform.
 Winter, C. L. (2010). Normalized Mahalanobis distance for comparing processbased stochastic models. Stochastic Environmental Research and Risk Assessment, 24(6), 917923.More infoAbstract: We investigate a method based on normalized Mahalanobis distance, D, for comparing the performance of alternate stochastic models of a given environmental system. The approach is appropriate in cases where data are too limited to calculate either likelihood ratios or Bayes factors. Computational experiments based on simulated data are used to evaluate D's ability to identify a true model and to single out good models. Data are simulated for two populations with different signalnoise ratios (S/N) The expected value of D is decomposed to evaluate the effects of normalization, model bias, and model correlation structure on D's discriminatory power. Normalization compensates for the advantage one model may have over another due to technical features of its hypothesized correlation structure. The relative effects of bias and correlation structure vary with S/N, model bias being most important when S/N is relatively high and correlation structure increasing in importance as S/N decreases. © 2010 SpringerVerlag.
 Winter, C. L., & Nychka, D. (2010). Forecasting skill of model averages. Stochastic Environmental Research and Risk Assessment, 24(5), 633638.More infoAbstract: Given a collection of sciencebased computational models that all estimate states of the same environmental system, we compare the forecast skill of the average of the collection to the skills of the individual members. We illustrate our results through an analysis of regional climate model data and give general criteria for the average to perform more or less skillfully than the most skillful individual model, the "best" model. The average will only be more skillful than the best model if the individual models in the collection produce very different forecasts; if the individual forecasts generally agree, the average will not be as skillful as the best model. © 2009 SpringerVerlag.
 Winter, C. L., & Smolarkiewicz, P. K. (2010). Pores resolving simulation of Darcy flows. Journal of Computational Physics, 229(9), 31213133. doi:10.1016/j.jcp.2009.12.031More infoA theoretical formulation and corresponding numerical solutions are presented for microscopic fluid flows in porous media with the domain sufficiently large to reproduce integral Darcy scale effects. Pore space geometry and topology influence flow through media, but the difficulty of observing the configurations of real pore spaces limits understanding of their effects. A rigorous direct numerical simulation (DNS) of percolating flows is a formidable task due to intricacies of internal boundaries of the pore space. Representing the grain size distribution by means of repelling body forces in the equations governing fluid motion greatly simplifies computational efforts. An accurate representation of porescale geometry requires that within the solid the repelling forces attenuate flow to stagnation in a short time compared to the characteristic time scale of the porescale flow. In the computational model this is achieved by adopting an implicit immersedboundary method with the attenuation time scale smaller than the time step of an explicit fluid model. A series of numerical simulations of the flow through randomly generated media of different porosities show that computational experiments can be equivalent to physical experiments with the added advantage of nearly complete observability. Besides obtaining macroscopic measures of permeability and tortuosity, numerical experiments can shed light on the effect of the pore space structure on bulk properties of Darcy scale flows.
 Damron, M., & Winter, C. L. (2009). A nonMarkovian model of rill erosion. Networks and Heterogeneous Media, 4(4), 731753.More infoAbstract: We introduce a new model for rill erosion. We start with a network similar to that in the Discrete Web [7, 11] and instantiate a dynamics which makes the process highly nonMarkovian. The behavior of nodes in the streams is similar to the behavior of Polya urns with timedependent input. In this paper we use a combination of rigorous arguments and simulation results to show that the model exhibits many properties of rill erosion; in particular, nodes which are deeper in the network tend to switch less quickly. © American Institute of Mathematical Sciences.
 Sviercoski, R. F., Warrick, A. W., & Winter, C. L. (2009). Twoscale analytical homogenization of Richards' equation for flows through block inclusions. Water Resources Research, 45(5).More infoAbstract: In this paper we propose an analytical form for the upscaled coefficient applicable to the nonlinear Richards' equation. Block inclusions are symmetrically centered in a grid cell, and flow is dominated by capillary forces on the small scale. The nonlinear boundary value problems can be defined in a threedimensional bounded domain, with a periodic and rapidly oscillating saturated hydraulic conductivity coefficient. The new result is derived by applying a corrector to an analytical approximation of the wellknown cell problem obtained by the twoscale asymptotic expansion to the original heterogeneous nonlinear problem. The previously known analytical results for the upscaled coefficient, including the geometric average for the checkerboard geometry, can be regarded as particular cases of this new form. We perform numerical simulations to obtain the error between the finescale solution and the upscaled solution, respectively, as well as convergence properties of the approximation, corroborating results in the literature. There is no limitation regarding either the ratio of permeability between the matrix and inclusion or its shape for both computing the upscaled coefficient and obtaining the upscaled equation. This is illustrated by the comparison with known results and by numerically demonstrating convergence properties with a medium having square and circular inclusions in a main matrix with heterogeneity ratio of 1:100 and 1:10. Even though the derivation and applications are presented for single phase and steady state, the results are valid for multiphase and transient flow conditions. We also show that the approximation is independent of the relative saturated hydraulic relationship. Copyright 2009 by the American Geophysical Union.
 Sviercoski, R. F., Winter, C. L., & Warrick, A. W. (2008). Analytical approximation for the generalized laplace equation with step function coefficient. SIAM Journal on Applied Mathematics, 68(5), 12681281.More infoAbstract: Many problems in science and engineering require the solution of thsteadystate diffusion equation with a highly oscillatory coefficient. In this paper, we propose an analytical approximation ũ(x) ε L P(ω), 1 ≥ p ≥ ∞, for the generalized Laplace equation ▼ · (Κ (x) ▼u(x)) = 0 in ω s{cyrillic} Rn, with prescribed boundary conditions and the coefficient function Κ(x) ε LP(ω) defined as a step function, not necessarily periodic. The proposed solution can be regarded as an approximation to theweak solution belonging to W1,p (ω), the Sobolev space. When the coefficient function describes inclusions in a main matrix, then Κ(x) is a periodic function, and such formulation leads to an approximation, in Lp(ω), to the solution of the periodic cellproblem, ▼ · (Κ(ε1x) = ▼w(ε1 x)) = ▼ · (Κ(ε1 x)1). The solution to the cellproblem is the key information needed to obtain the upscaled coefficient and therefore the zerothorder approximation for a generalized elliptic equation with highly oscillating coefficient in ω s{cyrillic} Rn. Our numerical computation of the error between the proposed analytical approximation for the cellproblem, w̃(ε 1x) in Lp(Ω), and the solution w(ε 1x) in W1,2 Ω), demonstrates to converge in the L2 norm, when the scale parameter ε approacheszero. The proposed approximation leads to the lower bound of the generalized VoigtReiss inequality, which is a more accurate twosided estimate than the classical VoigtReiss inequality. As an application, we compute our approximate value for the homogenized coefficient when theheterogeneous coefficients are inclusions such as squares, circles, and lozenges, and we demonstrate that the results underestimate the effective coefficient with an error of 10% on average, when compared with publishednumerical results. ©2008 Society for Industrial and Applied Mathematics.
 Tartakovsky, D. M., & Winter, C. L. (2008). Uncertain future of hydrogeology. Journal of Hydrologic Engineering, 13(1), 3739.More infoAbstract: Many of hydrogeology's most fundamental questions remain unresolved today, a hundred years after the basic governing equations for groundwater flow and transport were formulated. This paper provides a brief overview of the field and outlines the future directions, with a special emphasis on uncertainty quantification. © 2008 ASCE.
 Winter, C. L., & Tartakovsky, D. M. (2008). A reduced complexity model for probabilistic risk assessment of groundwater contamination. Water Resources Research, 44(6).More infoAbstract: We present a model of reduced complexity for assessing the risk of groundwater pollution from a point source. The progress of contamination is represented as a sequence of transitions among coarsely resolved states corresponding to simple statements like "a spill has occurred." Transitions between states are modeled as a Markov jump process, and a general expression for the probability of aquifer contamination is obtained from two basic assumptions: that the sequence of transitions leading to contamination is Markovian and that the time when a given transition occurs is independent of its end state. Additionally, we derive an asymptotic value for the probability of contamination that is equivalent to the socalled rare event approximation. First we develop the model for sites in statistically homogeneous natural porous media, and then we extend it to highly heterogeneous media composed of multiple materials. Finally, we apply the model to a simple example to illustrate the method and its potential. Copyright 2008 by the American Geophysical Union.
 Winter, C. L., Guadagnini, A., Nychka, D., & Tartakovsky, D. M. (2006). Multivariate sensitivity analysis of saturated flow through simulated highly heterogeneous groundwater aquifers. Journal of Computational Physics, 217(1), 166175.More infoAbstract: A multivariate Analysis of Variance (ANOVA) is used to measure the relative sensitivity of groundwater flow to two factors that indicate different dimensions of aquifer heterogeneity. An aquifer is modeled as the union of disjoint volumes, or blocks, composed of different materials with different hydraulic conductivities. The factors are correlation between the hydraulic conductivities of the different materials and the contrast between mean conductivities in the different materials. The precise values of aquifer properties are usually uncertain because they are only sparsely sampled, yet are highly heterogeneous. Hence, the spatial distribution of blocks and the distribution of materials in blocks are uncertain and are modeled as stochastic processes. The ANOVA is performed on a large sample of Monte Carlo simulations of a simple model flow system composed of two materials distributed within three disjoint blocks. Our key finding is that simulated flow is much more sensitive to the contrast between mean conductivities of the blocks than it is to the intensity of correlation, although both factors are statistically significant. The methodology of the experiment  ANOVA performed on Monte Carlo simulations of a multimaterial flow system  constitutes the basis of additional studies of more complicated interactions between factors that define flow and transport in aquifers with uncertain properties. © 2006.
 Guadagnini, A., & Winter, C. L. (2004). Introduction: Stochastic Models of Flow and Transport in Multiplescale Heterogeneous Porous Media. Journal of Hydrology, 294(13), 13.
 Winter, C. L., Springer, E. P., Costigan, K., Fasel, P., Mniszewski, S., & Zyvoloski, G. (2004). Virtual watersheds: Simulating the water balance of the Rio Grande basin. Computing in Science and Engineering, 6(3), 1826.More infoAbstract: The Los Alamos Distributed Hydrologic System (LADHS), a virtual watershed model which contains the essential physics of all elements of a regional hydrospheres and allows feedback between them, is discussed. LADHS emphasizes natural processes, but its components can be extended to include such anthropogenic effects as municipal, industrial, and agricultural demands. The system is embedded in the Parallel Application Work Space (PAWS), a software infrastructure for connecting separate parallel applications with in multicomponent model. LADHS is composed of four interacting components such as a regional atmospheric model, a landsurface hydrology model, a subsurface hydrology model, and a riverrouting model.
 Guadagnini, A., Guadagnini, L., Tartakovsky, D. M., & Winter, C. L. (2003). Random domain decomposition for flow in heterogeneous stratified aquifers. Stochastic Environmental Research and Risk Assessment, 17(6), 394407.More infoAbstract: We study twodimensional flow in a layered heterogeneous medium composed of two materials whose hydraulic properties and spatial distribution are known statistically but are otherwise uncertain. Our analysis relies on the composite media theory, which employs random domain decomposition in the context of groundwater flow moment equations to explicitly account for the separate effects of material and geometric uncertainty on ensemble moments of head and flux. Flow parallel and perpendicular to the layering in a twomaterial composite layered medium is considered. The hydraulic conductivity of each material is lognormally distributed with a much higher mean in one material than in the other. The hydraulic conductivities of points within different materials are uncorrelated. The location of the internal boundary between the two contrasting materials is random and normally distributed with given mean and variance. We solve the equations for (ensemble) moments of hydraulic head and flux and analyze the impact of unknown geometry of materials on statistical moments of head and flux. We compare the composite media approach to approximations that replace statistically inhomogeneous conductivity fields with pseudohomogeneous random fields.
 Winter, C. L., Tartakovsky, D. M., & Guadagnini, A. (2003). Moment differential equations for flow in highly heterogeneous porous media. Surveys in Geophysics, 24(1), 81106.More infoAbstract: Quantitative descriptions of flow and transport in subsurface environments are often hampered by uncertainty in the input parameters. Treating such parameters as random fields represents a useful tool for dealing with uncertainty. We review the state of the art of stochastic description of hydrogeology with an emphasis on statistically inhomogeneous (nonstationary) models. Our focus is on composite media models that allow one to estimate uncertainties both in geometrical structure of geological media consisting of various materials and in physical properties of these materials.
 Guadagnini, A., Guadagnini, L., Tartakovsky, D. M., & Winter, C. L. (2002). Moment analysis of flow in heterogeneous layered aquifers. Acta Universitatis Carolinae, Geologica, 46(23), 120123.More infoAbstract: We study twodimensional flow in a layered heterogeneous medium composed of two materials whose hydraulic properties and spatial distribution are known statistically but are otherwise uncertain. Our analysis relies on the composite media theory, which explicitly accounts for the separate effects of material and geometric uncertainty on statistical moments of head and flux. Flow parallel and perpendicular to the layering in a composite layered medium is considered. The hydraulic conductivity of each material is lognormally distributed with a much higher mean in one material than in the other. The hydraulic conductivities of points within different materials are uncorrelated. The location of the internal boundary between the two contrasting materials is normally distributed random with given mean and variance. We solve the equations for (ensemble) moments of hydraulic head and flux and analyze the impact of unknown geometry of materials on distribution of head variance.
 Guadagnini, A., Guadagnini, L., Tartakovsky, D. M., & Winter, C. L. (2002). Solution of moment equations of groundwater flow in random composite layered aquifers. IAHSAISH Publication, 108114.More infoAbstract: We study twodimensional flow in a layered heterogeneous medium composed of two materials whose hydraulic properties and spatial distribution are known statistically but are otherwise uncertain. Our analysis relies on the composite media theory, which explicitly accounts for the separate effects of material and geometric uncertainty on statistical moments of head and flux. Flow parallel and perpendicular to the layering in a composite layered medium is considered. The hydraulic conductivity of each material is lognormally distributed with a much higher mean in one material than in the other. The hydraulic conductivities of points within different materials are uncorrelated. The location of the internal boundary between the two contrasting materials is usually distributed randomly with given mean and variance. We solve the equations for (ensemble) moments of hydraulic head and flux and analyse the impact of unknown geometry of materials on the distribution of head variance.
 Tartakovsky, D. M., & Winter, C. L. (2002). Radial flow in heterogeneous aquifers with uncertain hydraulic parameters. Acta Universitatis Carolinae, Geologica, 46(23), 155157.More infoAbstract: We consider flow to a pumping well operating in a heterogeneous aquifer. In recent years this problem has attracted considerable attention due to its importance in many practical applications. A standard approach to dealing with aquifer's heterogeneity and ubiquitous lack of data is to treat hydraulic parameters, such as hydraulic conductivity and transmissivity, as random fields. In this stochastic framework, significant progress has been made in predicting flow and transport toward wells, and in deriving expressions for the corresponding equivalent (upscaled) parameters. The major theoretical difficulty in analyzing radial flows in heterogeneous media is that the equipotential lines (lines of constant hydraulic head) are no longer circular. Yet, most existing studies use this simplifying assumption. To demonstrate this point, we start by reviewing the existing literature. We then present a stochastic variational approach that avoids this oversimplification. This approach is further used to derive bounds for equivalent (upscaled) transmissivity.
 Winter, C. L., & Tartakovsky, D. M. (2002). Groundwater flow in heterogeneous composite aquifers. Water Resources Research, 38(8), 2312311.More infoAbstract: We introduce a stochastic model of flow through highly heterogeneous, composite porous media that greatly improves estimates of pressure head statistics. Composite porous media consist of disjoint blocks of permeable materials, each block comprising a single material type. Within a composite medium, hydraulic conductivity can be represented through a pair of random processes: (1) a boundary process that determines block arrangement and extent and (2) a stationary process that defines conductivity within a given block. We obtain secondorder statistics for hydraulic conductivity in the composite model and then contrast them with statistics obtained from a standard univariate model that ignores the boundary process and treats a composite medium as if it were statistically homogeneous. Next, we develop perturbation expansions for the first two moments of head and contrast them with expansions based on the homogeneous approximation. In most cases the bivariate model leads to much sharper perturbation approximations than does the usual model of flow through an undifferentiated material when both are applied to highly heterogeneous media. We make this statement precise. We illustrate the composite model with examples of onedimensional flows which are interesting in their own right and which allow us to compare the accuracy of perturbation approximations of head statistics to exact analytical solutions. We also show the boundary process of our bivariate model is equivalent to the indicator functions often used to represent composite media in Monte Carlo simulations.
 Winter, C. L., Tartakovsky, D. M., & Guadagnini, A. (2002). Numerical solutions of moment equations for flow in heterogeneous composite aquifers. Water Resources Research, 38(5), 131138.More infoAbstract: We analyze flow in heterogeneous media composed of multiple materials whose hydraulic properties and geometries are uncertain. Our analysis relies On the composite media theory of Winter and Tartakovsky [2000, 2002], which allows one to derive and solve moment equations even when the medium is highly heterogeneous. We use numerical solutions of Darcy flows through a representative composite medium to investigate the robustness of perturbation approximations in porous medium with total log conductivity variances as high as 20. We also investigate the relative importance of the two sources of uncertainty in composite media, material properties, and geometry. In our examples the uncertain geometry by itself captures the main features of the mean head estimated by the full composite model even when the withinmaterial conductivities are deterministic. However, neglecting randomness within materials leads to head variance estimates that are qualitatively and quantitatively wrong. We Compare the composite media approach to approximations that replace statistically inhomogeneous conductivity fields with pseudohomogeneous random fields with deterministic trends. We demonstrate that models with a deterministic trend can be expected to give a poor estimate of the statistics of head.
 Tartakovsky, D. M., & Winter, C. L. (2001). Dynamics of free surfaces in random porous media. SIAM Journal on Applied Mathematics, 61(6), 18571876.More infoAbstract: We consider free surface flow in random porous media by treating hydraulic conductivity of a medium as a random field with known statistics. We start by recasting the boundaryvalue problem in the form of an integral equation where the parameters and domain of integration are random. Our analysis of this equation consists of expanding the random integrals in Taylor's series about the mean position of the free boundary and taking the ensemble mean. To quantify the uncertainty associated with such predictions, we also develop a set of integrodifferential equations satisfied by the corresponding second ensemble moments. The resulting moment equations require closure approximations to be workable. We derive such closures by means of perturbation expansions in powers of the variance of the logarithm of hydraulic conductivity. Though this formally limits our solutions to mildly heterogeneous porous media, our analytical solutions for onedimensional flows demonstrate that such perturbation expansions may remain robust for relatively large values of the variance of the logarithm of hydraulic conductivity.
 Winter, C. L., & Tartakovsky, D. M. (2001). Theoretical foundation for conductivity scaling. Geophysical Research Letters, 28(23), 43674369.More infoAbstract: Scaling of conductivity with the support volume of experiments has been the subject of many recent experimental and theoretical studies. However, to date there have been few attempts to relate such scaling, or the lack thereof, to microscopic properties of porous media through theory. We demonstrate that when a pore network can be represented as a collection of hierarchical trees, scalability of the pore geometry leads to scalability of conductivity. We also derive geometrical and topological conditions under which the scaling exponent takes on specific values 1/2 and 3/4. The former is consistent with universal scaling observed by Neuman [1994], while the latter agrees with the allometric scaling laws derived by West et al. [1997].
 Mitkov, I., Tartakovsky, D. M., & Winter, C. L. (2000). Reply to "comment on 'dynamics of wetting fronts in porous media'". Physical Review E  Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, 61(2), 21522153.More infoAbstract: In our work [Phys. Rev, E 58, R5245 (1998)] we introduced a dynamic phenomenological approach to model propagation of localized wetting fronts in porous media. Gray and Miller in their Comment [Phys. Rev. E 61, 2150 (2000)] criticize our approach on several issues. The main criticism addresses the problem of mass conservation in our model. In this Reply we argue that their criticism is incorrect.
 Winter, C. L., & Tartakovsky, D. M. (2000). Mean flow in composite porous media. Geophysical Research Letters, 27(12), 17591762.More infoAbstract: We develop probabilities and statistics for the parameters of Darcy flows through saturated porous media composed of units of different materials. Our probability model has two levels. On the local level, a porous medium is composed of disjoint, statistically homogeneous volumes (or blocks) each of which consists of a single type of material. On a larger scale, a porous medium is an arrangement of blocks whose extent and location are uncertain. Using this twoscaled model, we derive general formulae for the probability distribution of hydraulic conductivity and its mean; then we develop general perturbation expansions for mean head. We express distributions and parameters in terms of mixtures of locally homogeneous block densities weighted by largescale block membership probabilities.
 Zhang, D., & Winter, C. L. (1999). Momentequation approach to single phase fluid flow in heterogeneous reservoirs. SPE Journal, 4(2), 118127.More infoAbstract: In this paper, we study single phase, steadystate flow in bounded, heterogeneous reservoirs. We derive general equations governing the statistical moments of flow quantities by perturbation expansions. These moments may be used to construct confidence intervals for the flow quantities. Due to their mathematical complexity, we solve the moment differential equations (MDEs) by the numerical technique of finite differences. The numerical MDE approach renders the flexibility in handling complex flow configurations, different boundary conditions, various covariance functions of the independent variables, and moderately irregular geometry, all of which are important factors to consider for realworld applications. The other method with these flexibilities is Monte Carlo simulation (MCS) which has been widely used in the industry. These two approaches are complementary, and each has its own advantages and disadvantages. The numerical MDE approach is compared with published results of MCS and analytical MDE approaches and is demonstrated with two examples involving in injection/production wells.
 Hagelberg, C. R., Cooper, D. I., Winter, C. L., & Eichinger, W. E. (1998). Scale properties of microscale convection in the marine surface layer. Journal of Geophysical Research D: Atmospheres, 103(D14), 1689716907.More infoAbstract: We analyze the scale distribution of coherent water vapor structures in the marine atmospheric boundary layer as measured by a shipboard Raman lidar during the Combined Sensor Program (March 1996) using a twodimensional continuous wavelet transform. Coherent structures in the lidar measured water vapor concentration field correspond to locations where covariance with the wavelet is a local extremum. Scales of the significant structures are identified using a filtered wavelet variance (detection density) derived from 24 "images" in a horizontal plane. A dominant radius of 14 m is identified using complimentary approaches to the analysis. Copyright 1998 by the American Geophysical Union.
 Zhang, D., & Winter, C. L. (1998). Nonstationary stochastic analysis of steady state flow through variably saturated, heterogeneous media. Water Resources Research, 34(5), 10911100.More infoAbstract: In this study we develop a firstorder, nonstationary stochastic model for steady state, unsaturated flow in randomly heterogeneous media. The model is applicable to the entire domain of a bounded vadose zone, unlike most of the existing stochastic models. Because of its nonstationarity, we solve it by the numerical technique of finite differences, which renders the flexibility in handling different boundary conditions, input covariance structures, and soil constitutive relationships. We illustrate the model results in one and two dimensions for soils described by the BrooksCorey constitutive model. It is found that the flow quantities such as suction head, effective water content, unsaturated hydraulic conductivity, and velocity are nonstationary near the water table and approach stationarity as the vertical distance from the water table increases. The stationary limits and the critical vertical distance at which stationarity is attained depend on soil types and recharge rates. The smaller the recharge rate is, the larger the critical distance; and the coarser the soil texture is, the smaller the distance. One important implication of this is that the existing simpler, gravitydominated flow models may provide good approximations for flow in vadose zones of large thickness and/or coarsetextured soils although they may not be valid for vadose zones of finetextured soils with a shallow water table. It is also found that the vertical extent of a domain where nonstationarity is important may be estimated by solving the onedimensional Richards equation for mean head with average soil properties and appropriate boundary conditions. On the basis of the mean head, one may then determine whether the full, nonstationary model must be solved or whether a simpler, gravitydominated model will suffice. The flow quantities are also nonstationary in the horizontal direction near the lateral boundaries, as found for flow in saturated zones.
 Zhang, D., Wallstrom, T. C., & Winter, C. L. (1998). Stochastic analysis of steadystate unsaturated flow in heterogeneous media: Comparison of the BrooksCorey and GardnerRusso models. Water Resources Research, 34(6), 14371449.More infoAbstract: Existing stochastic models of unsaturated flow and transport are usually developed using the simple GardnerRusso constitutive relationship though it is generally accepted that the more complex van Genuchten and BrooksCorey relationships may perform better in describing experimental data. In this paper, we develop firstorder stochastic models for gravitydominated flow in secondorder stationary media with both the BrooksCorey and the GardnerRusso constitutive relationships. These models also account for the spatial variability in effective water content, while the spatial variability is generally neglected in most existing stochastic models. Analytical solutions are obtained for the case of onedimensional gravitydominated flow. On the basis of the solutions, we illustrate the differences between results from these two constitutive models through some onedimensional examples. It is found that the impacts of the constitutive models on the statistical moments of suction head, effective water content, unsaturated hydraulic conductivity, and velocity depend on the saturation ranges. For example, the mean head and the mean effective water content for the BrooksCorey model differ in a great manner with their counterparts for the GardnerRusso model near the dry and wet limits while the differences are small at the intermediate range of saturation. This finding is confirmed with some twodimensional examples. It is also found that the BrookCorey model has certain advantages over the GardnerRusso model in analyzing unsaturated flow in heterogeneous media. For example, the stochastic model developed based on the BrooksCorey function requires the coefficient of variation of head and soil parameter 'α(Bc)' to be small (
 Carter, K., & Winter, C. L. (1995). Fractal nature and scaling of normal faults in the Española Basin, Rio Grande rift, New Mexico: implications for fault growth and brittle strain. Journal of Structural Geology, 17(6), 863873.More infoAbstract: Quaternary faults in the western Española Basin of the Rio Grande rift show a powerlaw size (displacement) distribution suggesting that faulting in this region is scale invariant, and that faults are self similar. The power law, or fractal, distribution is characterized by fractal dimension of 0.66 to 0.79 and represents a young, immature, active fault population in a continental extensional regime. Based on this distribution, it is estimated that unobserved faults with very small displacements account for up to 6% of the total strain. Since 1.2 Ma. total extension in this part of the basin has been at least 5%. A direct correlation exists between maximum displacement and length of faults in this area suggesting that they obey a scaling relationship in which the ratio of log dmax/log L is 5 × 103. This ratio is nearly constant for faults whose lengths span threeorders of magnitude, indicating that there is no difference in the scaling relationship of displacement and length between faults of all sizes. Considering previous models, these fault characteristics suggest that, in the western Española Basin: (1) host rock shear strengths are low; (2) remote shear stresses were probably high: and (3) most faults do not extend throughout the brittle crust. Finally, displacement profiles on five of the largest faults arc asymmetric and show a rapid decrease in displacement from the point of maximum displacement toward the fault tip. The fractal nature, scaling relationship and distribution of displacement on faults are used to suggest that faults grew by nearly proportional increases in displacement and length, perhaps by mechanisms dominated by propagating shear fractures rather than by linkage of preexisting joints or faults. © 1995 Elsevier Science Ltd.
 Levitt, T. S., Winter, C. L., Turner, C. J., Chestek, R. A., Ettinger, G. J., & Sayre, S. M. (1995). Bayesian inferencebased fusion of radar imagery, military forces and tactical terrain models in the image exploitation system/balanced technology initiative. International Journal of Human  Computer Studies, 42(6), 667686.More infoAbstract: The Imagery Exploitation System/Balanced Technology Initiative (IES/BTI) inputs synthetic aperture radar (SAR) imagery and outputs probabilistically ranked interpretations of the presence and location of military force membership, organization, and expected ground formations. There are also probabilistic models of underlying terrain types from a tactical perspective that provide evidence supporting or denying the presence of forces at a location. The system compares sets of detected military vehicles extracted from imagery against the models of military units and their formations to create evidence of force type and location. Based on this evidence, the system dynamically forms hypotheses of the presence, location and formations of military forces on the ground, which it represents in a dynamically modified Bayesian network. The IES/BTI functional design is based on a decision theoretic model in which processing choices are determined as a utility function of the current state of interpretation of imagery and a toplevel goal to exploit imagery as accurately and rapidly as possible, given the available data, current state of the interpretation of force hypotheses and the system processing suite. In order to obtain sufficient throughput in processing multimegabyte SAR imagery, and also to take advantage of natural parallelism in 2Dspatial reasoning, the system is hosted on a heterogeneous network of multiple parallel computers including a SIMD Connection Machine 2 and a MIMD Encore Multimax. Independent testing by the US Army using imagery of Iraqi forces taken during Desert Storm, indicated an average 260% improvement in the performance of expert SAR imagery analysts using IES/BTI as a front end to their image exploitation. © 1995 Academic Press. All rights reserved.
 Cooke, B. J., Lackner, K. S., Palounek, A. P., Sharp, D. H., Winter, L., & Ziock, H. (1992). Efficient data transmission from silicon wafer strip detectors. IEEE Transactions on Nuclear Science, 39(5 pt 1), 12541258.More infoAbstract: An architecture for onwafer processing is proposed for central siliconstrip tracker systems as they are currently designed for high energy physics experiments at the Superconducting Super Collider (SSC), and for heavy ion experiments at the Relativistic Heavy Ion Collider (RHIC). The authors discuss a concrete example which is based on the preliminary design for a central silicon tracker. The complete digital information generated on a silicon wafer for use in the silicon tracker system can be compressed by as much as a factor of 40. A set of data which completely describes the state of the wafer for low occupancy events and which contains important statistical information for more complex events can be transmitted immediately. This information could be used in early trigger decisions. Additional data packages which complete the description of the state of the wafer vary in size and are sent through a second channel.
 Winter, C. L. (1989). Bugs: A realtime adaptive network that responds to motion. IJCNN Int Jt Conf Neural Network, 622.More infoAbstract: Summary form only given, as follows. Bugs is an artificial neural system which detects motion in an external environment and can be trained by a teacher to react according to a desired strategy. System elements are a hardwired motion detection network and an adaptive controller network. The motion detection net is based on a model of motionsensitive ganglia found in rabbit eyes, while the controller learns strategies from a teacher in the form of finitestate transitions. Bugs's subnetworks are integrated by continuoustime dynamics, and elements are synchronized through simple timescaling parameters. The author reports two experiments in which Bugs learned quite different responses to moving objects.
 Guarino, D. R., Kruger, R. P., Sayre, S., Sos, T., Turner, C. J., & Winter, C. L. (1988). DARPA sensor national testbed: Hardware and software architecture. Array, 295301.More infoAbstract: A heterogeneous network of parallel computers developed for complex distributedprocessing applications is described. Network computers include a Connection Machine, a Butterfly multiprocessor, a WARP systolic array, and a Symbolics and several SUN workstations. An ethernet and a highbandwidth APTEC bus support data transfers. Distributed applications are built from individual processes executing on computers in the network. A powerful asynchronous communication facility is built upon the multiple computer operating systems to provide uniform message passing, global memory variables, and remote process execution services to processes. An executive controller and the LISP+ functional language provide a method of integrating distributed processes into an application with transparent control of network resources and communications. Additional applications can be rapidly built from existing processing to support experiments in distributed and parallel applications.
 Winter, C. L. (1988). Adaptive network that flees pursuit. Neural Networks, 1(1 SUPPL), 367.More infoAbstract: We describe a hierarchical artificial neural system (ANS) that learns to flee predators in a world without obstacles. The ANS is composed of a motion detection network (MD) an adaptive controller and body. Currently we simulate the activity of the body. The network operates in continuous time and is controlled solely through the activities of nodes. The controller obtains sensory information from MD in the form of reduced resolution maps of motion in the prey's visual field and determines a desirable next position by evaluating both motion information and the prey's current position as obtained from the body. The next position is passed as a goal, or instruction, to the body which moves accordingly.
 Neuman, S. P., Winter, C. L., Winter, C. L., Newman, C. M., & Neuman, S. P. (1987). Stochastic theory of field‐scale fickian dispersion in anisotropic porous media. Water Resources Research, 23(3), 453466. doi:10.1029/wr023i003p00453More infoA threedimensional theory is described for fieldscale Fickian dispersion in anisotropic porous media due to the spatial variability of hydraulic conductivities. The study relies partly on earlier work by the authors the attributes of which are briefly reviewed. It leads to results which differ in important ways from earlier theoretical conclusions about dispersion in anisotropic media. We express the dispersion tensor D as the sum of a local component d and a fieldscale component Δ. The local component is assumed to be independent of velocity (which is most appropriate if it represents molecular diffusion) and its principal terms are taken to act parallel and normal to the mean velocity vector μ. The fieldscale component is written as αμ, where α is a dispersivity tensor and μ= μ. We show that at large Peclet numbers P, the dispersivity tensor reduces to a single principal component parallel to the mean velocity, regardless of how μ is oriented. This result, valid for arbitrary covariance functions of loghydraulic conductivity, differs from that of L. W. Gelhar and C. L. Axness (1983), according to whom the asymptotic dispersivity tensor may possess more than one nonzero eigen value. They calculate the direction of the largest principal dispersivity to be offset from the mean velocity toward the direction of least spatial correlation (or away from the stratification in typical layered media). We show that this principal dispersivity is offset in the opposite direction at small and intermediate Peclet numbers but rotates toward the mean velocity as P increases. The largest eigen value is constant and dominated by fieldscale velocity fluctuations at large P values. The other two eigen values diminish asymptotically in proportion to P−1 and are controlled by d as well as by fieldscale differential convection. The range of small Peclet numbers has not been previously investigated under anisotropic conditions yet is of much importance for transport in lowpermeability rocks or soils. We show that at low P values all three principal dispersivities are proportional to P and thus Δ is proportional to μ2 (a phenomenon reminiscent of Taylor diffusion). When the mean velocity is inclined to the axes of anisotropy, the eigen values of Δ are neither parallel nor normal to μ. However, since D is dominated by d at small Peclet numbers, the principal dispersion coefficients are asymptotically (as P→0) parallel and normal to the mean velocity just like when P is large; their maximum deviation from these directions occurs at intermediate P values.
 Reynolds, T. D., Shepard, R. B., Laundre, J. W., & Winter, C. L. (1987). Calibrating resistancetype soil moisture units in a highclaycontent soil. Soil Science, 144(4), 237241.More infoAbstract: A calibration equation describing the relationship between electrical resistance and gravimetric soil moisture was generated for resistancetype soilmoisturesensing units to be used in a highclaycontent soil. from Authors
 Ryan, T. W., & Winter, C. L. (1987). VARIATIONS ON ADAPTIVE RESONANCE.. Array, ii/767775.More infoAbstract: In the process of implementing adaptive resonance circuits (ARCs) for a particular application, the authors have previously considered several circuit modifications and alternative processing conditions. They report here on some of these variations. First they examine an adaptive thresholding technique that prevents inadvertent encoding of recognition nodes which can occur when novel patterns are presented. Next, they consider the behavior of an ARC when patterns are iteratively presented for relatively short periods of time. Finally, they discuss the case of continuous ARC operation in which 'neural' activity is not reinitialized with each pattern presentation. The adaptive thresholding technique provides a novel clustering algorithm applicable to both binary and multilevel data.
 Ryan, T. W., Winter, C. L., & Turner, C. J. (1987). Dynamic control of an artificial neural system: The property inheritance network. Applied Optics, 26(23), 49614971.
 Winter, C. L., Ryan, T. W., & Turner, C. J. (1987). TIN: A TRAINABLE INFERENCE NETWORK.. Array, ii/777785.More infoAbstract: The Trainable Inference Network (TIN) is an adaptive network that can learn the functionality of any finitestate machine. TIN is composed of two modified adaptive resonance circuits (ARCs) that learn transition and output tables and an auxiliary assembly of control nodes that facilitates state transitions. The first ARC learns to recognize currentstate, input, nextstate patterns appearing on separate slabs; the other learns currentstate, input, output patterns. Features of TIN's macrocircuit and dynamics are described, focusing on the first, statetransition ARC. TIN is then taught the states, input, and transitions of two simple finitestate machines. The results are summarized, and future research directions are indicated.
 Winter, C. L., & Cook, W. L. (1986). INTERVAL ESTIMATES FOR YIELD MODELING.. IEEE Journal of SolidState Circuits, SC21(4), 590591.More infoAbstract: In a recent paper C. H. Stapper (1983) notes that semiconductor yield cannot be accurately modeled by a single critical area or a single defect density. Instead he considers an expected number of faults to which each type of defect contributes and then suggests a model for the average number of faults. Stapper's approach illuminates several important features of yield modeling, but it also makes clear the inappropriateness of using a single value to estimate yield. The authors follow statistical convention by calling such estimates point estimates. Defect density is a random phenomenon; thus a function of defect density, namely yield, will also be a random variable. An analysis is proposed of the yield model that allows calculation of interval bounds for yield, based on flexible defect models. An examination is also made of the interval estimates for yield from an individual wafer, and the confidence intervals for average yield for a given type of wafer.
 Winter, C. L., & Girse, R. D. (1985). EVOLUTION AND MODIFICATION OF PROBABILITY IN KNOWLEDGE BASES.. Array, 2731.More infoAbstract: The assignment of certainties to rules, their modification and their interaction in chains of inference, are important issues in knowledge engineering. The authors argue that probability theory, because its results can be compared to the external world, is an appropriate model of certainty. The rich convergence results available in probability also allow rigorous updating of uncertainties. A simple modification of the Law of Large Numbers which provides updating of initial estimates of certainties is given. The independence requirements of probability also constrain the design of rule bases in ways which prove useful.
Presentations
 Winter, C. L. (2017, Summer). Uncertainty Quantification for Groundwater Hydrology. QUIET 2017 Workshop. Trieste, Italy: NSF/EU.
Reviews
 Winter, C. L. (2018. Multiple reviews during 20172018 for WRR, AWR, SIAM J Multiscale Systems, Phys Rev E.