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Investigation of the national weather service soil moisture accounting models for flood prediction in the northeast floods of january 1996Extensive flooding occurred throughout the northeastern United States during January of 1996. The flood event cost the lives of 33 people and over a billion dollars in flood damage. Following the `Blizzard of `96 ", a warm front moved into the MidAtlantic region bringing extensive rainfall and causing significant melting and flooding to occur. Flood forecasting is a vital part of the National Weather Service (NWS) hydrologic responsibilities. Currently, the NWS River Forecast Centers use either the Antecedent Precipitation Index (API) or the Sacramento Soil Moisture Accounting Model (SACSMA). This study evaluates the API and SAC SMA models for their effectiveness in flood forecasting during this rain on snow event. The SAC SMA, in conjunction with the SNOW17 model, is calibrated for five basins in the Mid Atlantic region using the Shuffled Complex Evolution (SCEUA) automatic algorithm developed at the University of Arizona. NashSutcliffe forecasting efficiencies (Ef) for the calibration period range from 0.79 to 0.87, with verification values from 0.42 to 0.95. Flood simulations were performed on the five basins using the API and calibrated SACSMA model. The SACSMA model does a better job of estimating observed flood discharge on three of the five study basins, while two of the basins experience flood simulation problems with both models. Study results indicate the SACSMA has the potential for better flood forecasting during complex rainonsnow events such as during the January 1996 floods in the Northeast.

Investigations into the availability of additional water supplies and water storage areas for the Santa Cruz active management area, ArizonaThe desert climate of Southern Arizona coupled with the overdraft of its groundwater resources, led to the passing of the 1980, Groundwater Management Act. The Act mandates the creation of management plans in designated areas of heavy overdraft. Of the four initial Active Management Areas (AMAs, three had management plans that were designed to secure sustainable yield of the aquifer by 2025. In 1994, the Arizona legislature created a fifth AMA by designating the southern part of the Tucson AMA as the Santa Cruz AMA (SCAMA). The purpose for this subdivision was to facilitate the bi national negotiations for coordinated water resource management in this internationally shared basin. Additionally, the SCAMA is to coordinate the management of surface water and groundwater rights for public health, safety and welfare. A.R.S. § 45411.04. The legislature also assigned the SCAMA the management goals of maintaining safe yield conditions and preventing long term declines in local water table levels. A.R.S. § 45 562(C) (ADWR, 1999). This study is a result of a grant award from the 1999 Augmentation and Conservation Assistance Program in an attempt to investigate the availability of additional water supplies and water storage areas within the SCAMA.

Investigations of streamaquifer interactions using a coupled surfacewater and groundwater flow modelA finite element numerical model is developed for the modeling of coupled surfacewater flow and groundwater flow. The mathematical treatment of subsurface flows follows the confined aquifer theory or the classical Dupuit approximation for unconfined aquifers whereas surfacewater flows are treated with the kinematic wave approximation for open channel flow. A detailed discussion of the standard approaches to represent the coupling term is provided. In this work, a mathematical expression similar to Ohm's law is used to simulate the interacting term between the two major hydrological components. Contrary to the standard approach, the coupling term is incorporated through a boundary flux integral that arises naturally in the weak form of the governing equations rather than through a source term. It is found that in some cases, a branch cut needs to be introduced along the internal boundary representing the stream in order to define a simply connected domain, which is an essential requirement in the derivation of the weak form of the groundwater flow equation. The fast time scale characteristic of surfacewater flows and the slow time scale characteristic of groundwater flows are clearly established, leading to the definition of three dimensionless parameters, namely, a Peclet number that inherits the disparity between both time scales, a flow number that relates the pumping rate and the streamflow, and a Biot number that relates the conductance at the riveraquifer interface to the aquifer conductance. The model, implemented in the Bill Williams River Basin, reproduces the observed streamflow patterns and the groundwater flow patterns. Fairly good results are obtained using multiple time steps in the simulation process.

An Iterative Geostatistical Inverse Method For SteadyFlow In The Vadose ZoneAn iterative stochastic inverse technique utilizing both primary and secondary information is developed to estimate conditional means of unsaturated hydraulic conductivity parameters (saturated hydraulic conductivity and pore size distribution parameters) in the vadose zone. Measurements of saturated hydraulic conductivity and pore size distribution parameter are considered as the primary information, while measurements of steady state flow processes (soil water pressure head and degree of saturation) are regarded as the secondary information. This inverse approach is similar to the classical geostatistical approach, which utilizing a linear estimator that depends on the cross covariance and covariance functions of unsaturated hydraulic conductivity parameters and flow processes. The linear estimator is, however, improved successively by solving the governing flow equation and by updating the residual covariance and cross covariance functions, in an iterative manner. Using an approximate perturbation solution for steady, variably saturated flow under general boundary conditions, the covariances of secondary information and the cross covariance between the primary and secondary information are derived. The approximate solution is formulated based on a first order Taylor series expansion of a discretized finite element equation. The sensitivity matrices in the solution are evaluated by an adjoint state sensitivity approach for flow in heterogeneous media under variably saturated conditions. As a result, the nonlinear relationships between unsaturated hydraulic conductivity parameters and flow processes are incorporated in the estimation. Through some numerical examples, the iterative inverse model demonstrates its ability to improve the estimates of the spatial distribution of saturated hydraulic conductivity and pore size distribution parameters compared to the classical geostatistical inverse approach. In addition, the inconsistency problem existing in classical geostatistical inverse approach is alleviated. The estimated fields of unsaturated hydraulic conductivity parameters and flow fields not only retain their observed values at sample locations, but satisfy the governing flow equation as well.

AN ITERATIVE STOCHASTIC INVERSE METHOD: CONDITIONAL EFFECTIVE TRANSMISSIVITY AND HYDRAULIC HEAD FIELDSAn iterative stochastic approach is developed to estimate transmissivity and head distributions in heterogeneous aquifers. This approach is similar to the classical cokriging technique and it uses a linear estimator that depends on the covariances of transmissivity and hydraulic head and their cross covariance. The linear estimator is, however, improved successively by solving the governing flow equation and by updating the covariances and cross covariance function of transmissivity and hydraulic head fields in an iterative manner. As a result, the nonlinear relationship between transmissivity and head is incorporated in the estimation and the estimated fields are approximate conditional means. The ability of the iterative approach is tested with some deterministic and stochastic inverse problems. The results show that the estimated transmissivity and hydraulic head fields have smaller mean square errors than those obtained by classical cokriging even in the aquifer with variance of transmissivity up to 3.

Literature Pertaining to Water Quality and Quantity in Unsaturated Porous MediaIntroduction: The movement of moisture and the simultaneous transfer of water and solutes in unsaturated porous media are problems of practical interest in ground water hydrology and soil physics. A large fraction of the water falling as rain on the land surfaces of the earth moves through unsaturated zone of soil during the subsequent processes of infiltration, drainage, evaporation, and absorption of soil water by plant roots. A soil profile is characteristically nonuniform in its properties, nonisothermal, and may be nonrigid. Microorganisms and the roots of higher plants are a part of the system. This region is characterized by cylic fluctuation of water content as water is removed from the soil profile by evaportranspiration and replenished by recharge, irrigation, or rainfall. In unsaturated porous media the problem of movement and retention of water may be approached from (1) the molecular, (2) the microscopic, or (3) the macroscopic standpoint. In the molecular viewpoint theories of the mechanisms of flow and retention in terms of the behavior of water molecules are devised. At microscopic level a theory of flow treating the fluid in pores as a continuum and applying the principles of continuum mechanics to understand the detailed behavior of fluid within the pores is developed. The complicated pore geometry and consequent impossibility of specifying the boundary conditions on flow, preclude any practical progress by this appraoch. Since the behavior of individual molecules and the distributions of fluid velocity and pressure cannot be observed in porous media, a macroscopic theory of flow is needed. In the macroscopic approach, all variables are treated continuous functions of time and space. Velocity, pressure, and other variables are assumed as point functions. Thus, any theory of water transport to be useful must be developed to the point of describing the transfer of water on the macroscopic level. The coefficients of transport such as permeability and diffusivity can be defined microscopically. In many investigations which involve the transport of pesticides and fertilizes along with water , the simultaneous movement of water and solutes is of primary concern. These pollutants when mixed with water move in the unsaturated soil and finally join the region of saturated soil or water table, resulting in the contamination of fresh water existing below the water table. The scope of this report is to review the available literature, that may be categorized into two parts; one, the movement of water in unsaturated soil, and the other, the simultaneous movement of water and solutes in unsaturated soil. The papers, reviewed in this report, pertain to the theoretical study, laboratory study and field study on the two problems. At the end, an appendix appears which lists the references, categorizing the kind of study by various investigators.

A lower San Pedro river basin groundwater flow modelWater issues in the Lower San Pedro River basin in southeastern Arizona are becoming increasingly contentious as urban development, agriculture, and mining needs compete with the needs of the riparian habitat. To better understand the water demands in this basin, a new groundwater flow model has been created. First, the conceptual model was produced using various Geographic Information System (GIS) applications. A new method allocating digital precipitation data to the smaller drainages within the watershed was used to estimate mountain front recharge. Well data was gathered from both the United States Geological Survey (USGS) and Arizona Department of Water Resources (ADWR). Depth to bedrock was interpolated from an earlier gravity survey of the area. The current extent of riparian vegetation was determined by recent United States Forest Service aerial photography. GIS shapefiles were created depicting the data necessary for MODFLOW. Second, the numerical MODFLOW model was formed using GMS (Groundwater Modeling System), a graphical user interface for MODFLOW. GMS was used to create the grid, allocate the information from the shapefiles into MODFLOW input files, create the MODFLOW numerical model, and calibrate the model. The model results project potential impacts to the overall sustainability of groundwater within the basin. In the future, the model will be used as an administrative tool to assess alternative land management scenarios and their abilities to sustain or improve the riparian habitat along the San Pedro River.

Management Model for Electrical Power Production from a HotWater Geothermal ReservoirA management model is developed that determines the optimum economic recoverability of a particular hot water geothermal reservoir undergoing exploitation for electric power generation. The management model integrates a physical model of the reservoir that predicts the areas of pressure decline due to withdrawals, and pressure rise due to reinjection of spent fluid, with a model of a two stage steam turbine power plant that determines the quantity of electricity generated for a rate of hot water extraction. Capital costs, variable costs and annual fixed costs are obtained for the reservoir development, extraction and reinjection, the transmission system, and the power plant. Revenues are determined for electrical power production. Application of the management model to a simplified, yet realistic example reservoir demonstrates that the methodology developed in this report can be used for analyzing the management of an integrated geothermal reservoirpower plant system.

A MATHEMATICAL MODEL OF PRIMARY PRODUCTIVITY AND LIMNOLOGICAL PATTERNS IN LAKE MEADThe temporal and spatial changes in chemical and biological properties of Lake Mead have been investigated, thereby indicating the sources of water pollution and the time of highest pollution potential. Planktonic organisms have been shown to indicate the presence of water problems. Macro and micronutrient analyses have shown that primary productivity is not inhibited by limiting concentrations. A mathematical model has been developed, tested with one set of independent data, and shown worthy of management utility. Although the model works very well for the Lake Mead area, the physical reality of the Multiple Linear Regression equation should be tested on independent data.

MATHEMATICAL SYSTEM THEORY AND THE ECOSYSTEM CONCEPT, AN APPROACH TO MODELLING WATERSHED BEHAVIORThis study explores the possible role of mathematical system theory in integrating existing ecological knowledge within the existing concepts of the structure of the biosphere. The objective of this integration is a theory of ecosystems which must include interactions. The basic unit of the biosphere is the biogeocoenose; similar to the ecosystem, but homogeneous with respect to topographic, microclimatic, vegetation, animal, pedalogical, hydrological and geochemical conditions. The role of the biogeocoenose in a theory of ecosystems based on system theory is discussed. The biogeocoenose may serve as the building block for modeling watersheds as ecosystems. The fundamentals of system theory are reviewed. As an example, an analysis and synthesis of the arid zone water balance follows. The water balance is resolved into twenty components which represent the water balance of (1) the canopy, (2) the mulch, (3) the soil surface, (4) the soil, and (5) the plant, including interactions. The twenty components were modeled as separate systems which were later coupled into one overall, complex, well defined ecosystem water balance system. The example illustrates the role of system theory in integrating ecological knowledge. Further discussion indicates the need for explicitly including plant behavior in the water balance model.

Model Choice in Multiobjective Decision Making in Water and Mineral Resource SystemsThe problem of model choice in multiobjective decision making, that is, the selection of the appropriate multiobjective solution technique to solve an arbitrary multiobjective decision problem, is considered. Classifications of the available techniques are discussed, leading to the development of a set of 27 model choice criteria and an algorithm for model choice. This algorithm divides the criteria into four groups, oily one of which must be reevaluated for each decision problem encountered. Through the evaluation of the available multiobjective techniques with respect to each of the model choice criteria, the model choice problem is modeled as a multiobjective decision problem. Compromise programming is then used to select the appropriate technique for implementation. Two case studies are presented to demonstrate the use of this algorithm. The first is a river basin planning problem where a predefined set of alternatives is to be ranked with respect to a set of criteria, some of which cannot be quantified. The second is a coal blending problem modeled as a mathematical programming problem with two linear objective functions and a set of linear constraints. An appropriate multiobjective solution technique is selected for each of these case studies. In addition, an approach for the solution of dynamic multiobjective problems, one area where solution techniques are not available, is presented. This approach, known as dynamic compromise programming, essentially transforms a multiobjective dynamic programming problem into a classical dynamic programming problem of higher dimension. A dynamic programming problem, modeled in terms of three objectives, is used to demonstrate an application of this technique.

MODEL UNCERTAINTY IN THE DESIGN OF A FLOOD PROTECTION LEVEEThe model choice problem in Hydrology is illustrated by means of the optimum levee design for flat rivers along a confluence reach. Special attention is given to the selection of a probability distribution for the joint flood stages. The optimality criterion used is the minimization of construction plus expected flood damage costs. The main assumption in the mathematical model is that the levee profile is uniquely determined as a function of the levee heights at the extremes of the reach; thus the problem is reduced to the determination of the optimum pair of extreme levee heights. The selection of a probability distribution of flood stages, from a set of distributions estimated from the partial duration series, is performed using either one of two selection procedures: likelihood of the Chi square statistic and sample likelihoods. A composite distribution, taking into account the model uncertainty, is also derived. The methodology presented is applied to the remodeling of the levee on the west bank of the Zagyva River, in Hungary. A sensitivity analysis is performed, using the best ranking distributions according to the two model choice procedures. The composite distribution appears to offer a reasonable choice.

Modeling of GroundWater Flow and Surface/GroundWater Interaction for the San Pedro River Basin Part I Mexican Border to Fairbank, ArizonaMany hydrologic basins in the southwest have seen their perennial streamflows turn to ephemeral, their riparian communities disappear or be jeopardized, and their aquifers suffer from severe overdrafts. Under management of ground water exploitation and of conjunctive use of surface and ground waters are the main reasons for these events.

MODRSP: a program to calculate drawdown, velocity, storage and capture response functions for multiaquifer systemsMODRSP is program used for calculating drawdown, velocity, storage losses and capture response functions for multi  aquifer ground water flow systems. Capture is defined as the sum of the increase in aquifer recharge and decrease in aquifer discharge as a result of an applied stress from pumping [Bredehoeft et al., 19821. The capture phenomena treated by MODRSP are stream aquifer leakance, reduction of evapotranspiration losses, leakance from adjacent aquifers, flows to and from prescribed head boundaries and increases or decreases in natural recharge or discharge from head dependent boundaries. The response functions are independent of the magnitude of the stresses and are dependent on the type of partial differential equation, the boundary and initial conditions and the parameters thereof, and the spatial and temporal location of stresses. The aquifers modeled may have irregular shaped areal boundaries and non homogeneous transmissive and storage qualities. For regional aquifers, the stresses are generally pumpages from wells. The utility of response functions arises from their capacity to be embedded in management models. The management models consist of a mathematical expression of a criterion to measure preference, and sets of constraints which act to limit the preferred actions. The response functions are incorporated into constraints that couple the hydrologic system with the management system (Maddock, 1972). MODRSP is a modification of MODFLOW (McDonald and Harbaugh, 1984,1988). MODRSP uses many of the data input structures of MODFLOW, but there are major differences between the two programs. The differences are discussed in Chapters 4 and 5. An abbreviated theoretical development is presented in Chapter 2, a more complete theoretical development may be found in Maddock and Lacher (1991). The finite difference technique discussion presented in Chapter 3 is a synopsis of that covered more completely in McDonald and Harbaugh (1988). Subprogram organization is presented in Chapter 4 with the data requirements explained in Chapter 5. Chapter 6 contains three example applications of MODRSP.

MR2K: A program to calculate drawdown, velocity, storage and capture response functionsA program, MR2K, used for calculating drawdown, velocity, storage loss, and capture response functions for multi aquifer groundwater flow systems was developed. Capture is defined as the sum of the increase in aquifer recharge and decrease in aquifer discharge as a result of an applied stress from groundwater pumping. The capture phenomena treated are streamaquifer leakance, reduction of evapotranspiration losses, reduction of drain flows, flows to and from prescribed head boundaries, and increases or decreases in natural recharge or discharge from headdependent boundaries. The response functions are independent of the magnitude of the pumping stresses, and are dependent on the type of partial differential equation, boundary and initial conditions and the parameters thereof, and the spatial and temporal locations of stresses. The aquifers modeled may have irregular shaped boundaries and nonhomogeneous transmissive and storage qualities. The stresses are groundwater withdrawals from wells. The utility of response functions arises from their capacity to be embedded in management models such as decision support systems. The response functions are incorporated into the objective function or constraints that couple the hydrologic system with the management system. Three response function examples are presented for a hypothetic basin.

MULTIPARAMETER SENSITIVITY ANALYSIS AND OPTIMIZATION OF THE ALPINE HYDROCHEMICAL MODELThe University of Arizona's Alpine Hydrochemical Model (AHM) is an integrated set of algorithms for water and chemical balances that describes hydrologic and chemical processes in a headwater catchment. We developed AHM for use both as a research tool and as a predictive model for estimating effects of natural and anthropogenic changes in climate or in atmospheric pollutant loading on alpine watersheds. We initially applied AHM to Emerald Lake watershed in the southern Sierra Nevada, and estimated model parameters by trial and error using a single water year of data and process level studies. Using the same parameters, AHM successfully reproduced stream chemistry and discharge for a second water year. We have extended that empirical analysis by doing a systematic analysis of parameter sensitivity and an automatic optimization of model parameters. In the sensitivity analysis, a large number of Monte Carlo simulations done on the multi dimensional function field were used to identify the sensitive parameters and to set an appropriate range for each parameter. These results were then used to reduce the computational load in the automatic optimization, which is based on the downhill simplex method in multiple dimensions; we estimate the global optimum parameter set according to the fluctuation of the sum of squared errors between observed and modeled stream discharge and chemistry. Sensitive physical and chemical parameters were identified, including those describing evapotranspiration, hydraulic conductivity and soil depth or porosity; and those describing mineral weathering, ion release from the snow  pack, ion exchange, soil CO2 and nitrogen reactions. The automatic optimization method succeeded in estimating a global optimum parameter set from a single water year of data that improved the fitting compared to the set from trial and error manipulation.

A multistep automatic calibration scheme (MACS) for river forecasting models utilizing the national weather service river forecast system (NWSRFS)Traditional model calibration by National Weather Service (NWS) River Forecast Center (RFC) hydrologists involves a laborious and time consuming manual estimation of numerous parameters. The National Weather Service River Forecasting System (NWSRFS), a software system used by the RFCs for hydrologic forecasting, includes an automatic optimization program (OPT3) to aid in model calibration. The OPT3 program is not used operationally by the majority of RFC hydrologists who perform calibration studies. Lack of success with the traditional single  step, singlecriterion automatic calibration approach has left hydrologists more comfortable employing a manual stepbystep process to estimate parameters. This study develops a Multistep Automatic Calibration Scheme (MACS), utilizing OPT3, for the river forecasting models used by the RFCs: the Sacramento Soil Moisture Accounting (SACSMA). and SNOW17 models. Sixteen parameters are calibrated in three steps, replicating the progression of manual calibration steps used by NWS hydrologists. MACS is developed by minimizing different objective functions for different parameters in a step wise manner. Model runs are compared using the MACS optimized parameters and the manually estimated parameters for six basins in the North Central River Forecast Center (NCRFC) forecast area. Results demonstrate that the parameters obtained via the MACS procedure generally yield better model performance than those obtained by manual calibration. The MACS methodology is a timesaving approach that can provide prompt model forecasts for NWS watersheds.

A multiobjective global optimization algorithm with application to calibration of hydrologic modelsThis report presents a new multiple objective optimization algorithm that is capable of solving for the entire Pareto set in one single optimization run. The multiobjective complex evolution (MOCOMUA) procedure is based on the following three concepts: (1) population, (2) rankbased selection, and (3) competitive evolution. In the MOCOMUA algorithm, a population of candidate solutions is evolved in the feasible space to search for the Pareto set. Ranking of the population is accomplished through Pareto ranking, where all points are successively placed on different Pareto fronts. Competitive evolution consists of selecting subsets of points (including all worst points in the population) based on their ranks and moving the worst points toward the Pareto set using the newly developed multiobjective simplex (MOSIM) procedure. Test analysis on the MOCOMUA algorithm is accomplished on mathematical problems of increasing complexity and based on a bicriterion measure of performance. The two performance criteria are: (1) efficiency, as measured by the ability of the algorithm to converge quickly, and (2) effectiveness, as measured by the ability of the algorithm to locate the Pareto set. Comparison of the MOCOMUA algorithm against three multiobjective genetic algorithms (MOGAs) favors the former. In a realistic application, the MOCOMUA algorithm is used to calibrate the Soil Moisture Accounting model of the National Weather Service River Forecasting Systems (NWSRFSSMA). Multiobjective calibration of this model is accomplished using two bicriterion objective functions, namely the Daily Root Mean SquareHeteroscedastic Maximum Likelihood Estimator (DRMSHMLE) and rising limb /falling limb (RISE/FALL) objective functions. These two multiobjective calibrations provide some interesting insights into the influence of different objectives in the location of final parameter values, as well as limitations in the structure of the NWSRFSSMA model.

Nonlocal and localized finite element solution of conditional mean flow in randomly heterogeneous mediaThis report considers the effect of measuring randomly varying local hydraulic conductivities K(x) on one's ability to predict deterministically, without upscaling, steady state flow in bounded domains driven by random source and boundary terms. Our aim is to allow optimum unbiased prediction of hydraulic heads h(x) and Darcy fluxes q(x) by means of their ensemble moments, , and c, conditioned on measurements of K(x). It has been shown earlier that these predictors satisfy a deterministic flow equation which contains an integrodifferential "residual flux" term. This term renders c nonlocal and nonDarcian so that the concept of effective hydraulic conductivity looses meaning in all but a few special cases. Instead, the residual flux contains kernels which constitute nonlocal parameters that are conditional on hydraulic conductivity data and therefore nonunique. The kernels include symmetric and nonsymmetric second rank tensors as well as vectors. We derive exact integrodifferential equations for second conditional moments of head and flux which constitute measures of predictive uncertainty. We then develop recursive closure approximations for the moment equations through expansion in powers of a small parameter ay which represents the standard estimation error of In K(x). Finally, we solve these nonlocal equations to first order in a by finite elements on a rectangular grid in two dimensions. We also solve the original stochastic flow equations by conditional Monte Carlo simulation using finite elements on the same grid. Upon comparing our nonlocal finite element and conditional Monte Carlo results we find that the former are highly accurate, under either mean uniform or convergent flows, for both mildly and strongly heterogeneous media with a as large as 4  5 and spatial correlation scales as large as the length of the domain. Since conditional mean quantities are smooth relative to their random counterparts our method allows, in principle, resolving them on relatively coarse grids without upscaling. We also examine the quc on under what conditions can the residual flux be localized so as to render it approximately Darcian. One way to achieve such localization is to treat ' "draulic conductivity as if it was locally homogeneous and mean flow as if it was locally uniform. This renders the flux predictor Darcian according to c _  Kc(x) \7c where Kc(x) is a conditional hydraulic conductivity tensor which depends on measurements of K(x) and is therefore a nonunique function of space. This function can be estimated by means of either stochastically derived analytical formulae or standard inverse methods (in which case localization coincides with common groundwater modeling practice). We use the first approach and solve the corresponding localized conditional mean equation by finite elements on the same grid as before. Here the conditional hydraulic conductivity is given by the geometric mean KG(x). Upon comparing our localized finite element solution with a nonlocal finite element solution and conditional Monte Carlo results, we find that the first is generally less accurate than the second. The accuracy of the localized solution deteriorates rel tive to that of the nonlocal solution as one approaches points of conditioning and singularity, or as the variance and correla': ^n scale of the log hydraulic conductivity increase. Contrary to the nonlocal solution, locàlzation does not yield information about predictive uncertainty.