a full bayesian approach to the groundwater inverse problem for steady state flow a full bayesian approach to the estimation of transmissivity from hydraulic head and transmissivity measurements is developed for two dimensional steady state groundwater flow the approach combines both bayesian and maximum entropy viewpoints of probability in the first phase log transmissivity measurements are incorporated into bayes theorem and the prior probability density function is updated yielding posterior estimates of the mean value of the log transmissivity field and covariance the two central moments are generated assuming that the prior mean variance and integral scales are hyperparameters that is they are treated as random variables in themselves which is contrary to classical statistical approaches the probability density functions pdfs of these hyperparameters are in turn determined from maximum entropy considerations in other words pdfs are chosen for each of the hyperparameters that are maximally uncommitted with respect to unknown information this methodology is quite general and provides an alternative to kriging for spatial interpolation the final step consists of updating the conditioned natural logarithm transmissivity ln t field with hydraulic head measurements utilizing a linearized aquifer equation it is assumed that the statistical properties of the noise in the hydraulic head measurements are also uncertain at each step uncertainties in all pertinent hyperparameters are removed by marginalization finally what is produced is a ln t field conditioned on measurements of both hydraulic heads and log transmissivity and covariances of the ln t field in addition we can also produce resolution matrices confidence credibility limits and the like for the ln t field it is shown that the application of the methodology yields good estimates of transmissivities even when hydraulic head measurements are noisy and little or no information is specified on mean values of ln t variance of ln t and integral scales

application of the multiscale finite element method to flow in heterogeneous porous media 1 the multiscale finite element method is applied in this paper to simulate groundwater flow in heterogeneous porous media the method can efficiently capture the large scale behavior of the solution without resolving all the small scale features by constructing the multiscale finite element base functions that are adaptive to the local property of the differential operator both multiscale finite element method and conventional finite element method are applied to five 2 d and two 3 d groundwater flow problems including a 2 d steady flow problem with continuous coefficients a 2 d steady flow with gradual change in coefficients a 2 d transient flow with gradual change in coefficients a 2 d steady flow problem with an abrupt change in coefficients a 2 d transient flow problem with an abrupt change in coefficients and a 3 d steady and a 3 d transient flow problem with gradual change in coefficients in the horizontal direction and with abrupt change in the vertical direction respectively the applications demonstrate the main advantages of the multiscale finite element method i e significantly reducing computational efforts and improving the accuracy of the solutions

the matching of a prescribed linear input output behavior in a nonlinear system in this paper we deal with the problem of finding a feedback under which the input output behavior of a nonlinear system becomes exactly the same as that of a specified linear model the solvability of this problem is shown to depend on a formal infinite zero structure associated with the system

transient flow in bounded randomly heterogeneous domains 1 exact conditional moment equations and recursive approximations in a recent paper we developed exact nonlocal conditional moment equations for transient flow in bounded domains driven by random forcing terms sources initial head and boundary conditions whereas our conditional mean equations took into account the randomness of forcing terms our conditional second moment equations did not we do so in this brief addendum

on the relative error probabilities of linear multiuser detectors the relative error probability performance of three linear multiuser detectors the minimum mean square error mmse detector the decorrelator and the conventional matched filter mf detector is investigated under nonorthogonal signaling and additive white gaussian noise conditions it is shown that contrary to the general belief the mmse detector does not uniformly outperform the other two detectors in fact even for the two user case one can find counterexamples where the matched filter is significantly better

on the probability of error in linear multiuser detection the problem of determining the bit error probability in linear multiuser detection analytically is considered a number of upper and lower bounds are developed some of which are very tight and issues of asymptotic behavior optimality and gaussian approximation are explored several special cases including those of equi energy and equi correlated signals are treated in some detail

nonlinear decoupling via feedback a differential geometric approach the paper deals with the nonlinear decoupling and noninteracting control problems a complete solution to those problems is made possible via a suitable nonlinear generalization of several powerful geometric concepts already introduced in studying linear multivariable control systems the paper also includes algorithms concerned with the actual construction of the appropriate control laws

comparative analysis of formulations for conservative transport in porous media through sensitivity based parameter calibration 1 we apply a general strategy based on global sensitivity analysis gsa and model discrimination criteria to a calibrate the parameters embedded in competing models employed to interpret laboratory scale tracer experiments b rank these models and c estimate the relative degree of likelihood of each model through a posterior probability weight we consider a conservative transport experiment in a uniform porous medium we apply gsa to three transport models based on the classical advection dispersion equation ade a dual porosity dp formulation with mass transfer between mobile and immobile regions and the continuous time random walk ctrw approach gsa is performed through polynomial chaos expansion of the governing equations treating key model parameters as independent random variables we show how this approach allows identification of a the relative importance of model dependent parameters and b the space time locations where the models are most sensitive to these parameters gsa is then employed to assist parameter estimates within a maximum likelihood framework finally formal model identification criteria are employed to a rank the alternative models and b associate each model with a posterior probability weight for the specific case study the gsa based calibration of each model returns an acceptable approximation remarkably accurate in the case of the ctrw model of all available concentration data with calibration being performed using minimum sets of observations corresponding to the most sensitive space time locations

effect of local dispersion on solute transport in randomly heterogeneous media recently an exact eulerian lagrangian theory of advective transport in space time random velocity fields was developed by one of us we present a formal extension of this theory that accounts for anisotropic local dispersion the resultant conditional mean transport equation is generally nonlocal in space time to assess the impact of local dispersion on the prediction of transport under uncertainty we adopt a first order pseudo fickian approximation for this equation we then solve it numerically by galerkin finite elements for two dimensional transport from an instantaneous square source in a uniform unconditional mean flow field subject to isotropic local dispersion we use a higher order approximation to compute explicitly the standard deviation and coefficient of variation of the predicted concentrations our theory shows in an exact manner and our numerical results demonstrate under the above closure approximations that the effect of local dispersion on first and second concentration moments varies monotonically with the magnitude of the local dispersion coefficient when this coefficient is small relative to macrodispersion its effect on the prediction of nonreactive transport under uncertainty can for all practical purposes be disregarded this is contrary to some recent assertions in the literature that local dispersion must always be taken into account no matter how small

relationship between juxtaposed overlapping and fractal representations of multimodal spatial variability 1 it has been shown by neuman 1990 di federico and neuman 1997 and di federico et al 1999 that the power variogram of a statistically isotropic or anisotropic fractal field can be constructed as a weighted integral from zero to infinity of exponential or gaussian variograms of overlapping homogeneous random fields modes having mutually uncorrelated increments and variance proportional to a power 2h of the integral spatial correlation scale where h is the hurst coefficient low and high frequency cutoffs are related to length scales of the sampling window domain and data support sample volume respectively intermediate cutoffs account for lacunarity due to gaps in the multiscale hierarchy created by a hiatus of modes associated with discrete ranges of scales alternative mathematical representations of multimodal spatial variability were formulated by various authors in which space is filled by a discrete number of juxtaposed mutually exclusive materials or categories each having its own architecture and attributes the spatial distribution of categories is characterized by indicator random variables and their attributes by random fields this paper focuses on expressions developed by lu and zhang 2002 in which the indicator variables and their attributes are mutually uncorrelated while each is autocorrelated and cross correlated within and between categories upon rewriting their expressions for statistically homogeneous and anisotropic media it is demonstrated mathematically that in the limit as the categories stretch to occupy each point in space overlap in a way that preserves their local architecture their attributes become mutually uncorrelated categories are said to overlap if they are found in fixed proportions within a representative sampling volume centered about any mathematical point throughout space the idea is analogous to the well known dual continuum concept in which two categories most commonly fractures and porous blocks are considered to overlap in reality the categories do not overlap but are considered to preserve their relative architectural arrangement throughout space the variogram of an attribute sampled jointly over overlapping statistically homogeneous and anisotropic categories is the sum of the variograms of statistically homogeneous and anisotropic components of this attribute sampled over individual categories weighted by their volumetric proportions it is further demonstrated that when the overlapping categories represent an infinite or truncated hierarchy of modes whose attributes have exponential or gaussian variograms such that the product of category probability density function pdf and attribute variance is proportional to a power 2h 1 of the attribute integral scale they represent a fractal field characterized by a power or truncated power variogram as in the work of di federico and neuman any category pdf that satisfies this condition and any attribute pdf having a variance that satisfies the same condition are admissible neither of them need to be gaussian to form fractional brownian motion or levy like to approximate fractional levy motion for the attribute to form a random fractal intermediate cutoffs render the latter model discrete as is that of lu and zhang hence the two multimodal models are mutually consistent and complementary

estimation of aquifer parameters under transient and steady state conditions 2 uniqueness stability and solution algorithms paper 2 of this three part series starts with a discussion of the question under what conditions is the aquifer inverse problem well posed after defining the terms uniqueness identifiability and stability theoretical considerations and synthetic examples are used to demonstrate that ill posedness can be mitigated by including prior information about the parameters in the estimation criterion to be minimized at the same time the inclusion of such information is shown to be insufficient to guarantee uniqueness and stability in all cases several test problems in the recent literature which have resulted in pessimistic conclusions about the solvability of the aquifer inverse problem are shown to be ill posed a question is thus raised about the validity of these conclusions in the general case various conjugate gradient algorithms coupled with the adjoint state finite element method for computing the gradient of the estimation criterion and with newton s method for determining the optimum step size downgradient are compared a marked improvement in the rate at which these algorithms converge is shown to be achieved by switching from one method to another when the former slows down or fails to converge

on model selection criteria in multimodel analysis 1 hydrologic systems are open and complex rendering them prone to multiple conceptualizations and mathematical descriptions there has been a growing tendency to postulate several alternative hydrologic models for a site and use model selection criteria to 1 rank these models 2 eliminate some of them and or 3 weigh and average predictions and statistics generated by multiple models this has led to some debate among hydrogeologists about the merits and demerits of common model selection also known as model discrimination or information criteria such as aic aicc bic and kic and some lack of clarity about the proper interpretation and mathematical representation of each criterion we examine the model selection literature to find that 1 all published rigorous derivations of aic and aicc require that the true model having generated the observational data be in the set of candidate models 2 though bic and kic were originally derived by assuming that such a model is in the set bic has been rederived by cavanaugh and neath 1999 without the need for such an assumption and 3 kic reduces to bic as the number of observations becomes large relative to the number of adjustable model parameters implying that it likewise does not require the existence of a true model in the set of alternatives we explain why kic is the only criterion accounting validly for the likelihood of prior parameter estimates elucidate the unique role that the fisher information matrix plays in kic and demonstrate through an example that it imbues kic with desirable model selection properties not shared by aic aicc or bic our example appears to provide the first comprehensive test of how aic aicc bic and kic weigh and rank alternative models in light of the models predictive performance under cross validation with real hydrologic data

turbulence control simulation using the variational multiscale method the capabilities of the variational multiscale vms method are explored in the context of turbulence control by applying vms to the simulation of a simple opposition control strategy for turbulent channel flow with the results compared to prior direct numerical simulations and large eddy simulations based on the dynamic subgrid scale model in all cases the vms method is found to be more efficient and more accurate than the dynamic model and the simplicity accuracy and generality of vms makes it particularly attractive for turbulence control investigations

estimation of spatial covariance structures by adjoint state maximum likelihood cross validation 1 theory this series of three papers describes a cross validation method to estimate the spatial covariance structure of intrinsic or nonintrinsic random functions from point or spatially averaged data that may be corrupted by noise any number of relevant parameters including nugget effect can be estimated the theory described in this paper is based on a maximum likelihood approach which treats the cross validation errors as gaussian various a posteriori statistical tests are used to verify this hypothesis and to show that in many cases correlation between these errors is weak the log likelihood criterion is optimized through a combination of conjugate gradient algorithms an adjoint state theory is used to efficiently calculate the gradient of the estimation criterion optimize the step size downgradient and compute a lower bound for the covariance matrix of the estimation errors issues related to the identifiability stability and uniqueness of the resulting adjoint state maximum likelihood cross validation asmlcv method are discussed this paper also describes the manner in which asmlcv allows one to use model structure identification criteria to select the best covariance model among a given set of alternatives practical aspects of asmlcv and its application to synthetic data are presented in paper 2 samper and neuman this issue a applications to real hydrogeological data transmissivities and groundwater levels have been presented elsewhere while hydrochemical and isotopic data are analyzed by asmlcv in paper 3 samper and neuman this issue b

the variational multiscale method a paradigm for computational mechanics abstract we present a general treatment of the variational multiscale method in the context of an abstract dirichlet problem we show how the exact theory represents a paradigm for subgrid scale models and a posteriori error estimation we examine hierarchical p methods and bubbles in order to understand and ultimately approximate the fine scale green s function which appears in the theory we review relationships between residual free bubbles element green s functions and stabilized methods these suggest the applicability of the methodology to physically interesting problems in fluid mechanics acoustics and electromagnetics

convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients we propose a multiscale finite element method for solving second order elliptic equations with rapidly oscillating coefficients the main purpose is to design a numerical method which is capable of correctly capturing the large scale components of the solution on a coarse grid without accurately resolving all the small scale features in the solution this is accomplished by incorporating the local microstructures of the differential operator into the finite element base functions as a consequence the base functions are adapted to the local properties of the differential operator in this paper we provide a detailed convergence analysis of our method under the assumption that the oscillating coefficient is of two scales and is periodic in the fast scale while such a simplifying assumption is not required by our method it allows us to use homogenization theory to obtain a useful asymptotic solution structure the issue of boundary conditions for the base functions will be discussed our numerical experiments demonstrate convincingly that our multiscale method indeed converges to the correct solution independently of the small scale in the homogenization limit application of our method to problems with continuous scales is also considered

porous media with linearly variable hydraulic properties the similar media concept is reformulated and extended to allow for analysis of unsaturated or saturated flow in a system of parallel nonhomogeneous one two or three dimensional soil profiles the resulting invariance of richards equation for a set of soil profiles is valid for certain initial and boundary conditions only simple techniques are outlined and demonstrated by examples for direct identification of scaling factors based on measured soil hydraulic data and indirect identification based on measured dynamic characteristics of a soil water system suggested applications of the discussed concept are 1 the classification of soils according to the shape of their reference hydraulic properties and 2 the description of time and space variability of soil water properties for numerical modeling purposes

parameter estimation and predictive uncertainty in stochastic inverse modeling of groundwater flow comparing null space monte carlo and multiple starting point methods 1 given a highly parameterized groundwater model in which the conceptual model of the heterogeneity is stochastic a set of inverse calibrations from multiple starting points msps provide an ensemble of calibrated parameters and follow on transport predictions however the multiple calibrations are computationally expensive a recently developed null space monte carlo nsmc method combines the calibration solution space parameters with the ensemble of null space parameters creating sets of calibration constrained parameters for input to follow on transport predictions the consistency between probabilistic ensembles of parameter estimates and predictions created using the msp calibration and the nsmc approaches is examined using a highly parameterized 1300 parameters model of the culebra dolomite previously developed for the waste isolation pilot plant project in new mexico as a test case a total of 100 estimated fields are retained from the msp approach and the ensemble of results defining the model fit to the data and prediction of an advective travel time are compared with the same results obtained using nsmc we demonstrate that the nsmc fields based on a single calibrated model can be significantly constrained by the calibrated solution space and the resulting distribution of advective travel times is biased toward the travel time from the single calibrated field to overcome this newly proposed strategies to employ a multiple calibration constrained nsmc m nsmc approach are evaluated comparison of the m nsmc and msp methods demonstrates that m nsmc can provide a computationally efficient and practical solution for predictive uncertainty analysis in highly nonlinear and complex subsurface flow and transport models

uncertainty quantification in scale dependent models of flow in porous media equations governing flow and transport in randomly heterogeneous porous media are stochastic and scale dependent in the moment equations me method exact deterministic equations for the leading moments of state variables are obtained at the same support scale as the governing equations computable approximations of the mes can be derived via perturbation expansion in orders of the standard deviation of the random model parameters as such their convergence is guaranteed only for standard deviation smaller than one here we consider steady state saturated flow in a porous medium with random second order stationary conductivity field we show it is possible to identify a support scale where the typically employed approximate formulations of me yield accurate statistical moments of a target state variable therefore at support scale and larger me presents an attractive alternative to slowly convergent monte carlo mc methods whenever lead order statistical moments of a target state variable are needed we also demonstrate that a surrogate model for statistical moments could be constructed from mc simulations at larger support scales and be used to accurately estimate moments at smaller scales where mc simulations are expensive and the me method is not applicable

on minimum ber linear multiuser detection for ds cdma channels the minimum bit error rate mber linear multiuser detection of direct sequence code division multiple access ds cdma communications is considered it is shown that the problem of mber linear multiuser detection is equivalent to a constrained optimization problem where the constrained space is the surface and interior of a hyperellipsoid it is further shown that any solution of the constrained optimization problem must be a solution of a set of nonlinear equations based on these the mber linear multiuser detector of the two user case is obtained for general cases we propose a convex subspace of the constrained space and show that the ber cost function is convex on the subspace and therefore has a unique solution some sufficient conditions that guarantee the convex subspace always contains the global minimizer are provided which show that if the signal to noise ratio snr is sufficiently large the convex subspace always contains the global minimizer a newton method is developed for the set of nonlinear equations and computer simulation gives some illustrative examples

variational multiscale stabilized fem formulations for transport equations stochastic advection diffusion and incompressible stochastic navier stokes equations an extension of the deterministic variational multiscale vms approach with algebraic subgrid scale sgs modeling is considered for developing stabilized finite element formulations for the stochastic advection and the incompressible stochastic navier stokes equations the stabilized formulations are numerically implemented using the spectral stochastic formulation of the finite element method ssfem generalized polynomial chaos and karhunen loeve expansion techniques are used for representation of uncertain quantities the proposed stabilized method is then applied to various standard advection diffusion and fluid flow examples with uncertainty in essential boundary conditions comparisons are drawn between the numerical solutions and monte carlo analytical solutions wherever possible

eulerian lagrangian theory of transport in space time nonstationary velocity fields exact nonlocal formalism by conditional moments and weak approximation a unified eulerian lagrangian theory is presented for the transport of a conservative solute in a random velocity field that satisfies locally v x t f x t where f x t is a random function including sources and or the time derivative of head solute concentration satisfies locally the eulerian equation c x t t j x t g x t where j x t is advective solute flux and g x t is a random source independent of f x t we consider the prediction of c x t and j x t by means of their unbiased ensemble moments c x t and j x t conditioned as implied by the subscript on local hydraulic measurements through the use of the latter in obtaining a relatively smooth unbiased estimate x t of v x t these predictors satisfy c x t v t j x t g x t where j x t x t c x t q x t and q x t is a dispersive flux we show that q is given exactly by three space time convolution integrals of conditional lagrangian kernels with q with c and with c for a broad class of v x t fields including fractals this implies that q x t is generally nonlocal and non fickian rendering c x t non gaussian the direct contribution of random variations in f to q depends on c rather than on c we elucidate the nature of the above kernels discuss conditions under which the convolution of and c becomes pseudo fickian with a lagrangian dispersion tensor similar to that derived in 1921 by taylor recall a 1952 result by batchelor which yields an exact expression for c at early time use the latter to conclude that linearizations which predict that c bifurcates at early time when the probability density function of v is unimodal cannot be correct propose instead a weak approximation which leads to a nonlinear integro differential equation for c due to an instantaneous point source and which improves with the quantity and quality of hydraulic data demonstrate that the weak approximation is analogous to the direct interaction closure of turbulence theory offer non fickian and pseudo fickian weak approximations for the second conditional moment of the concentration prediction error demonstrate that it yields the so called two particle covariance as a special case conclude that the conditional variance of c does not become infinite merely as a consequence of disregarding local dispersion and discuss how to estimate explicitly the cumulative release of a contaminant across a compliance surface together with the associated estimation error

a finiteness criterion for nonlinear input output differential systems we solve the following problem which has been dealt with in several publications when does a system have the same input output behaviour as a system depending linearly on the state variables precise definitions of this question are supplied in the text a particular case of the problem is the equivalence of a system with a bilinear one at the end we illustrate our methods and the scope of our results by a brief study of several examples

blueprint for perturbative solution of flow and transport in strongly heterogeneous composite media using fractal and variational multiscale decomposition 1 this paper offers a blueprint for extending perturbative ensemble moment and potentially other e g gaussian or perturbative karhunen loeve solutions of stochastic flow and transport problems in composite randomly heterogeneous porous media to cases in which medium heterogeneity is arbitrarily large the proposed approach is based on a novel idea of combined fractal and variational multiscale decomposition it considers log hydraulic conductivity to be a composite random field with piecewise statistically homogeneous increments characterized by a truncated power variogram i e a truncated fractal this consideration is supported either exactly or approximately by a growing amount of observational data it allows decomposing the field into two or more mutually uncorrelated components having relatively small variances associated with a hierarchy of spatial correlation scales combining this fractal decomposition with a variational multiscale decomposition formalism motivated by the multiscale nature of turbulence allows decomposing flow and transport problems into two or more coupled sets of variational problems each of which is amenable to perturbative solution regardless of how heterogeneous the original field was the problems are defined on a hierarchy of computational grids having discretization intervals proportional to the correlation scales the proposed approach does not require introducing effective parameters on any scale instead it allows resolving flow and transport ensemble moments fully on multiple scales of spatial resolution the approach nevertheless allows subgrid closure another idea inspired by the literature on turbulence through subgrid refinement or the use of effective subgrid parameters it additionally offers the critically important capability of conditioning flow and transport analyses on multiscale measurements details are presented in the context of steady state single phase flow in a composite medium

stochastic modeling of groundwater flow by unconditional and conditional probabilities the inverse problem the inverse problem is defined here as follows determine the transmissivity at varius points given the shape and boundary of the aquifer and recharge intensity and given a set of measured log transmissivity y and head h values at a few points the log transmissivity distribution is regarded as a realization of a random function of normal and stationary unconditional probability density function pdf the solution of the inverse problem is the conditional normal pdf of y conditioned on measured h and y which is expressed in terms of the unconditional joint pdf of y and h the problem is reduced to determining the unconditional head log transmissivity covariance and head variogram for a selected y covariance which depends on a few unknown parameters this is achieved by solving a first order approximation of the flow equations the method is illustrated for an exponential y covariance and the effect of head and transmissivity measurements upon the reduction of uncertainty of y is investigated systematically it is shown that measurement of h has a lesser impact than those of y but a judicious combination may lead to significant reduction of the predicted variance of y possible applications to real aquifers are outlined

convergence of a nonconforming multiscale finite element method the multiscale finite element method msfem t y hou x h wu and z cai math comp 1998 to appear t y hou and x h wu j comput phys 134 1997 pp 169 189 has been introduced to capture the large scale solutions of elliptic equations with highly oscillatory coefficients this is accomplished by constructing the multiscale base functions from the local solutions of the elliptic operator our previous study reveals that the leading order error in this approach is caused by the resonant sampling which leads to large error when the mesh size is close to the small scale of the continuous problem similar difficulty also arises in numerical upscaling methods an oversampling technique has been introduced to alleviate this difficulty t y hou and x h wu j comput phys 134 1997 pp 169 189 a consequence of the oversampling method is that the resulting finite element method is no longer conforming here we give a detailed analysis of the nonconforming error our analysis also reveals a new cell resonance error which is caused by the mismatch between the mesh size and the wavelength of the small scale we show that the cell resonance error is of lower order our numerical experiments demonstrate that the cell resonance error is generically small and is difficult to observe in practice

a fractal based stochastic interpolation scheme in subsurface hydrology real porosity and hydraulic conductivity data do not vary smoothly over space so an interpolation scheme that preserves irregularity is desirable such a scheme based on the properties of fractional brownian motion fbm and fractional gaussian noise fgn is presented following the methodology of hewett 1986 the authors test for the presence of fgn in a set of 459 hydraulic conductivity k measurements the use of rescaled range analysis strongly indicated the presence of fgn when applied to the natural logs of the k data and the resulting hurst coefficient h was determined to be 0 82 this h value was then used along with the methodology for successive random additions to generate a fbm k interpolation realization in the vertical cross section between two wells the results appeared realistic and the overall methodology presented herein may serve as an improved basis for a conditional simulation approach to the study of various transport processes in porous media copyright c 1993 american geophysical union

nonlocal and localized analyses of conditional mean steady state flow in bounded randomly nonuniform domains 2 computational examples in paper 1 guadagnini and neuman this issue of this two part series we presented exact nonlocal equations for first and second conditional ensemble moments of hydraulic head and flux under steady state groundwater flow in bounded randomly nonuniform domains in the presence of uncertain source and boundary terms we derived recursive approximations for these equations based on expansion in powers of a small parameter y which represents the standard estimation error of log hydraulic conductivity developed a two dimensional finite element scheme for their solution under the action of deterministic forcing terms to first order in y2 showed how to localize the conditional mean flow equations so as to yield familiar looking darcian differential equations in which however all quantities are conditional on data and therefore nonunique and proposed that traditional deterministic flow models be reinterpreted in light of these localized equations we stressed that all quantities which enter into our nonlocal and localized conditional moment equations are defined on a consistent support scale thereby obviating the need for upscaling from support to grid scale or the a priori definition of a computational grid that such a grid can therefore be defined a posteriori based on the degree of smoothness one expects the conditional moments to exhibit and that conditioning points are often spaced widely enough to render these moments relatively smooth and hence resolvable on a coarser grid than is necessary for monte carlo simulations this notwithstanding we use in this paper a high resolution grid to explore superimposed mean uniform and convergent flows in unconditional and conditional log hydraulic conductivity domains and to compare the relative accuracies of corresponding nonlocal localized and monte carlo finite element solutions whereas our nonlocal solution is nominally restricted to mildly nonuniform media with y2 1 we find that it actually yields remarkably accurate results for strongly nonuniform media with y2 at least as large as 4 localized solutions deteriorate relative to nonlocal results as one approaches some points of conditioning and singularity or as y2 increases though they require much less computer time than do nonlocal solutions localized solutions have the added disadvantage that they yield no information about predictive uncertainty

a comparison of seven geostatistically based inverse approaches to estimate transmissivities for modeling advective transport by groundwater flow this paper describes the first major attempt to compare seven different inverse approaches for identifying aquifer transmissivity the ultimate objective was to determine which of several geostatistical inverse techniques is better suited for making probabilistic forecasts of the potential transport of solutes in an aquifer where spatial variability and uncertainty in hydrogeologic properties are significant seven geostatistical methods fast fourier transform ff fractal simulation fs linearized cokriging lc linearized semianalytical ls maximum likelihood ml pilot point pp and sequential self calibration ss were compared on four synthetic data sets each data set had specific features meeting or not classical assumptions about stationarity amenability to a geostatistical description etc the comparison of the outcome of the methods is based on the prediction of travel times and travel paths taken by conservative solutes migrating in the aquifer for a distance of 5 km four of the methods ls ml pp and ss were identified as being approximately equivalent for the specific problems considered the magnitude of the variance of the transmissivity fields which went as high as 10 times the generally accepted range for linearized approaches was not a problem for the linearized methods when applied to stationary fields that is their inverse solutions and travel time predictions were as accurate as those of the nonlinear methods nonstationarity of the true transmissivity field or the presence of anomalies such as high permeability fracture zones was however more of a problem for the linearized methods the importance of the proper selection of the semivariogram of the log10 t field or the ability of the method to optimize this variogram iteratively was found to have a significant impact on the accuracy and precision of the travel time predictions use of additional transient information from pumping tests did not result in major changes in the outcome while the methods differ in their underlying theory and the codes developed to implement the theories were limited to varying degrees the most important factor for achieving a successful solution was the time and experience devoted by the user of the method

two dimensional characterization of hydraulic heterogeneity by multiple pumping tests 1 the conventional analysis of pumping tests by type curve methods is based on the assumption of a homogeneous aquifer applying these techniques to pumping test data from real heterogeneous aquifers leads to estimates of the hydraulic parameters that depend on the choice of the pumping and observation well positions in this paper we test whether these values may be viewed as pseudo local values of transmissivity and storativity which can be interpolated by kriging we compare such estimates to those obtained by geostatistical inverse modeling where heterogeneity is assumed in all stages of estimation we use drawdown data from multiple pumping tests conducted at the test site in krauthausen germany the geometric mean values of transmissivity and storativity determined by type curve analysis are very close to those obtained by geostatistical inversion but the conventional approach failed to resolve the spatial variability of transmissivity in contrast the estimate from geostatistical inversion reveals more structure this indicates that the estimates of the type curve approaches can not be treated as pseudo local values concerning storativity both analysis methods show strong fluctuations because the variability of all terms making up the storativity is small we believe that the estimated variability of storativity is biased we examine the influence of measurement error on estimating structural parameters of covariance functions in the inversion we obtain larger correlation lengths and smaller prior variances if we trust the measured data less

transport in multiscale log conductivity fields with truncated power variograms di federico and neuman this issue investigated mean uniform steady state groundwater flow in an unbounded domain with log hydraulic conductivity that forms a truncated multiscale hierarchy of statistically homogeneous and isotropic gaussian fields each associated with an exponential autocovariance here we present leading order expressions for the displacement covariance and dispersion coefficient of an ensemble of solute particles advected through such a flow field in two or three dimensions both quantities are functions of the mean travel distance s the hurst coefficient h and the low and high frequency cutoff integral scales l and u the latter two are related to the length scales of the sampling window region under investigation and sample volume data support respectively if one considers transport to be affected by a finite domain much larger than the mean travel distance so that s l then an early preasymptotic regime develops during which longitudinal and transverse dispersivities grow linearly with s if one considers transport to be affected by a domain which increases in proportion to s then l and s are of similar order and a preasymptotic regime never develops instead transport occurs under a regime that is perpetually close to asymptotic under the control of an evolving scale l s we show that if additionally u l then the corresponding longitudinal dispersivity grows in proportion to l1 2h or equivalently s1 2h both these preasymptotic and asymptotic theoretical growth rates are shown to be consistent with the observed variation of apparent longitudinal fickian dispersivities with scale we conclude by investigating the effect of variable separations between cutoff scales on dispersion

bayesian calibration of a large scale geothermal reservoir model by a new adaptive delayed acceptance metropolis hastings algorithm 1 the aim of this research is to estimate the parameters of a large scale numerical model of a geothermal reservoir using markov chain monte carlo mcmc sampling within the framework of bayesian inference all feasible parameters that are consistent with the measured data are summarized by the posterior distribution and hence parameter estimation and uncertainty quantification are both given by calculating expected values of statistics of interest over the posterior distribution it appears to be computationally infeasible to use the standard metropolis hastings algorithm mh to sample the high dimensional computationally expensive posterior distribution to improve the sampling efficiency a new adaptive delayed acceptance mh algorithm adamh is implemented to adaptively build a stochastic model of the error introduced by the use of a reduced order model this use of adaptivity differs from existing adaptive mcmc algorithms that tune proposal distributions of the metropolis hastings algorithm mh though adamh also implements that technique for the 3 d geothermal reservoir model we present here adamh shows a great improvement in the computational efficiency of the mcmc sampling and promising results for parameter estimation and uncertainty quantification are obtained this algorithm could offer significant improvement in computational efficiency when implementing sample based inference in other large scale inverse problems

an application of the geostatistical approach to the inverse problem in two dimensional groundwater modeling the geostatistical approach to the estimation of transmissivity from head and transmissivity measurements is developed for two dimensional steady flow the field of the logarithm of transmissivity log transmissivity is represented as a zero order intrinsic random field its spatial structure is described in this application through a two term covariance function that is linear in the parameters 1 and 2 linearization of the discretized flow equations allows the construction of the joint covariance matrix of the head and log transmissivity measurements as a linear function of 1 and 2 in this particular application the coefficient matrices are calculated numerically in a noniterative fashion maximum likelihood estimation is employed to estimate 1 and 2 as well as additional parameters from measurements linear estimation theory cokriging then yields point or block averaged estimates of transmissivity the approach is first applied to a test case with favorable results it is shown that the application of the methodology gives good estimates of transmissivities it is also shown that when the transmissivities are used in a numerical model they reproduce the head measurements quite well results from the application of the methodology to the jordan aquifer in iowa are also presented

estimation of aquifer parameters under transient and steady state conditions 1 maximum likelihood method incorporating prior information in this series of three papers a method is presented to estimate the parameters of groundwater flow models under steady and nonsteady state conditions the parameters include values and directions of principal hydraulic conductivities or transmissivities in anisotropic media specific storage or storativity interior and boundary recharge or leakage rates coefficients of head dependent interior and boundary sources and boundary heads in transient situations the initial head distribution can also be estimated if the system is originally at a steady state paper 1 of the series discusses some of the advantage in treating the inverse problem statistically and in regularizing its solution by means of penalty criteria based on prior estimates of the parameters the inverse problem is posed in the framework of maximum likelihood theory cast in a manner that accounts for prior information about the parameters since not all the factors which contribute to the prior errors can be quantified statistically at the outset the covariance matrices of these errors are expressed in terms of several parameters which if unknown can be estimated jointly with the hydraulic parameters by a stagewise optimization process when transient head data are separated by a fixed time interval the temporal structure of these data is approximated by a lag one autoregressive model with a correlation coefficient that can be treated as another unknown parameter estimation errors are analyzed by examining the lower bound of their covariance matrix in the eigenspace paper 1 concludes by suggesting that certain model identification criteria developed in the time series literature all of which are based on the maximum likelihood concept might be useful for selecting the best groundwater model or the best method of parameterizing a particular model among a number of given alternatives

a higher order approximation to effective conductivity in media of anisotropic random structure properties of the effective conductivity tensor keff are studied by deriving the second order terms in its expansion in the variance 2 of normally distributed log conductivity it is shown that for media of anisotropic structure the components of the effective conductivity tensor are expressed by a functional of the log conductivity covariance that is it depends on the shape of the correlation function and not only on anisotropy ratios variance 2 and space dimensions however the trace of keff is independent of the log conductivity autocovariance and for a given mean conductivity depends only on 2 the dependence of the effective conductivity on the correlation structure is illustrated for gaussian and exponential autocovariances of log conductivity and for two and three dimensional flows

nonlocal and localized analyses of conditional mean steady state flow in bounded randomly nonuniform domains 1 theory and computational approach we consider the effect of measuring randomly varying hydraulic conductivitiesk x on one s ability to predict numerically without resorting to either monte carlo simulation or 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 fluxes q x by means of their ensemble moments h x c and q x c respectively conditioned on measurements of k x these predictors have been shown by neuman and orr 1993a to satisfy exactly an integrodifferential conditional mean flow equation in which q x c is nonlocal and non darcian here we develop complementary integrodifferential equations for second conditional moments of head and flux which serve as measures of predictive uncertainty obtain recursive closure approximations for both the first and second conditional moment equations through expansion in powers of a small parameter y which represents the standard estimation error of ln k x and show how to solve these equations to first order in y2 by finite elements on a rectangular grid in two dimensions in the special case where one treats k x as if it was locally homogeneous and mean flow as if it was locally uniform one obtains a localized darcian approximation q x c kc x h x c in which kc x is a space dependent conditional hydraulic conductivity tensor this leads to the traditional deterministic darcian steady state flow equation which however acquires a nontraditional meaning in that its parameters and state variables are data dependent and therefore inherently nonunique it further explains why parameter estimates obtained by traditional inverse methods tend to vary as one modifies the database localized equations yield no information about predictive uncertainty our stochastic derivation of these otherwise standard deterministic flow equations makes clear that uncertainty measures associated with estimates of head and flux obtained by traditional inverse methods are generally smaller often considerably so than measures of corresponding predictive uncertainty which can be assessed only by means of stochastic models such as ours we present a detailed comparison between finite element solutions of nonlocal and localized moment equations and monte carlo simulations under superimposed mean uniform and convergent flow regimes in two dimensions paper 1 presents the theory and computational approach and paper 2 guadagnini and neuman this issue describes unconditional and conditional computational results

on the use of block effective macrodispersion for numerical simulations of transport in heterogeneous formations 1 the fundamental question we consider in this paper is how to allow flexibility in numerical grid design without discounting the dispersive action of the unmodeled variability in doing that we wish to preserve the interplay between all relevant length scales those relevant to the spatial variability as well as those created by design in this study we extend and test the concept of block scale macrodispersion introduced by rubin et al 1999 for modeling unresolved hydraulic property variations at scales smaller than the numerical grid blocks we present closed form analytical results for the block scale macrodispersion and test them numerically closed form analytical results are presented for the large time aymptotic limits and it is shown that these limits are attained very fast the conditions of applicability are investigated and we show that ergodicity with regard to block scale heterogeneity is attained surprisingly fast

universal scaling of hydraulic conductivities and dispersivities in geologic media an interpretation is offered for the observation that dispersivities increase with scale apparent longitudinal dispersivity data from a variety of hydrogeologic settings are assumed to represent a continuous hierarchy of log hydraulic conductivity fields with mutually uncorrelated increments each field having its own exponential autocovariance associated integral scale and variance that increases as a power of scale such a hierarchy is shown theoretically to form a self similar random field with homogeneous increments regardless of whether or not the underlying assumption is valid one can justify interpreting the apparent dispersivities in a manner consistent with a recent quasi linear theory of non fickian and fickian dispersion in homogeneous media which supports the notion of a self similar hierarchy a posteriori the hierarchy is revealed to possess a semivariogram s cs where c is a constant and a fractal dimension d e 0 75 where e is the topological dimension of interest this can be viewed as a universal scaling rule about which large deviations occur due to local influences including the existence of discrete natural scales at which log hydraulic conductivity is statistically homogeneous as such homogeneity is at best a local phenomenon occurring intermittently over narrow bands of the scale spectrum one must question the utility of associating medium properties with representative elementary volumes and relying on fickian models of dispersion over more than relatively narrow scale intervals porous and fractured media appear to follow the same idealized scaling rule for both flow and transport raising a question about the validity of many distinctions commonly drawn between such media finally the data suggest that conditioning transport models through calibration against hydraulic measurements has the effect of filtering out large scale modes from the hierarchy

discrimination of fractional brownian movement and fractional gaussian noise structures in permeability and related property distributions with range analyses recent studies have shown that a useful scheme to characterize subsurface heterogeneity is to use fractional brownian motion fbm and fractional gaussian noise fgn commonly used techniques to detect these fractal distributions include rescaled range spectral and semivariogram analyses although rescaled range analysis is considered to be superior to spectral or semivariogram techniques it cannot be used to clearly discriminate fgn and fbm from a given data set in some cases on the basis of the fact that both adjusted range and rescaled range analyses for a true fgn data set should yield the same slope value in the relevant log log plots a simple scheme is developed to discriminate fgn and fbm using a combination of the two analyses the usefulness of the proposed scheme is evaluated with one dimensional computational experiments and by an application to a data set of observed air permeability logs lack of clear discrimination may have contributed to inconsistent past results and it is recommended that old data sets be reanalyzed

a geostatistical approach to the inverse problem in groundwater modeling steady state and one dimensional simulations the problem of estimating hydrogeologic parameters in particular permeability from input output measurements is reexamined in a geostatistical framework the field of the unknown parameters is represented as a random field and the estimation procedure consists of two main steps first the structure of the parameter field is identified i e mathematical representations of the variogram and the trend are selected and their parameters are established by using all available information including measurements of hydraulic head and permeability second linear estimation theory is applied to provide minimum variance and unbiased point estimates of hydrogeologic parameters kriging structure identification is achieved iteratively in three substeps structure selection maximum likelihood estimation and model validation and diagnostic checking the methodology was extensively tested through simulations on a simple one dimensional case the results are remarkably stable and well behaved the estimated field is smooth while small scale variability is statistically described as the quality of measurements improves the procedure reproduces more features of the original field the results are also shown to be rather insensitive to deviations from assumptions about the geostatistical structure of the field

scaling of random fields by means of truncated power variograms and associated spectra an interpretation is offered for the observation that the log hydraulic conductivity of geologic media often appears to be statistically homogeneous but with variance and integral scale which grow with domain size we first demonstrate that the power semi variogram and associated spectra of random fields having homogeneous isotropic increments can be constructed as weighted integrals from zero to infinity an infinite hierarchy of exponential or gaussian variograms and spectra of mutually uncorrelated homogeneous isotropic fields modes we then analyze the effect of filtering out truncating high and low frequency modes from this infinite hierarchy in the real and spectral domains a low frequency cutoff renders the truncated hierarchy homogeneous with an autocovariance function that varies monotonically with separation distance in a manner not too dissimilar than that of its constituent modes the integral scales of the lowest and highest frequency modes cutoffs are related respectively to the length scales of the sampling window domain and data support sample volume taking each relationship to be one of proportionality renders our expressions for the integral scale and variance of a truncated field dependent on window and support scales in a manner consistent with observations the traditional approach of truncating power spectral densities yields autocovariance functions that oscillate about zero with finite in one and two dimensions or vanishing in one dimension integral scales our hierarchical theory allows bridging across scales at a specific locale by calibrating a truncated variogram model to data observed on a given support in one domain and predicting the autocovariance structure of the corresponding multiscale field in domains that are either smaller or larger one may also venture we suspect with less predictive power to bridge across both domain scales and locales by adopting generalized variogram parameters derived on the basis of juxtaposed hydraulic and tracer data from many sites

bayesian inverse problem and optimization with iterative spatial resampling 1 measurements are often unable to uniquely characterize the subsurface at a desired modeling resolution in particular inverse problems involving the characterization of hydraulic properties are typically ill posed since they generally present more unknowns than data in a bayesian context solutions to such problems consist of a posterior ensemble of models that fit the data up to a certain precision specified by a likelihood function and that are a subset of a prior distribution two possible approaches for this problem are markov chain monte carlo mcmc techniques and optimization calibration methods both frameworks rely on a perturbation mechanism to steer the search for solutions when the model parameters are spatially dependent variable fields obtained using geostatistical realizations such as hydraulic conductivity or porosity it is not trivial to incur perturbations that respect the prior spatial model to overcome this problem we propose a general transition kernel iterative spatial resampling isr that preserves any spatial model produced by conditional simulation we also present a stochastic stopping criterion for the optimizations inspired from importance sampling in the studied cases this yields posterior distributions reasonably close to the ones obtained by a rejection sampler but with a greatly reduced number of forward model runs the technique is general in the sense that it can be used with any conditional geostatistical simulation method whether it generates continuous or discrete variables therefore it allows sampling of different priors and conditioning to a variety of data types several examples are provided based on either multi gaussian or multiple point statistics

transient flow in bounded randomly heterogeneous domains 2 localization of conditional mean equations and temporal nonlocality effects in randomly heterogeneous porous media one cannot predict flow behavior with certainty one can however render optimum unbiased predictions of such behavior by means of conditional ensemble mean hydraulic heads and fluxes we have shown in paper 1 tartakovsky and neuman this issue that under transient flow these optimum predictors are governed by nonlocal equations in particular the conditional mean flux is generally nonlocal in space time and therefore non darcian as such it cannot be associated with an effective hydraulic conductivity except in special cases here we explore analytically situations under which localization is possible so that darcy s law applies in real laplace and or infinite fourier transformed spaces approximately or exactly with or without conditioning we show that the corresponding conditional effective hydraulic conductivity tensor is generally nonsymmetric an alternative to darcy s law in each case valid under mean no flow conditions along neumann boundaries is a quasi darcian form that includes only a symmetric tensor which however does not constitute a bona fide effective hydraulic conductivity both lack of symmetry and differences between darcian and quasi darcian forms disappear to first but not necessarily higher order of approximation in the conditional variance of natural log hydraulic conductivity we adopt such an approximation to investigate analytically the effect of temporal nonlocality on one and three dimensional mean flows in infinite statistically homogeneous media our results show that temporal nonlocality may manifest itself under either monotonic or oscillatory time variations in the mean hydraulic gradient the effect of temporal nonlocality increases with the variance of log hydraulic conductivity and is more pronounced in one dimension than in three

effects of kriging and inverse modeling on conditional simulation of the avra valley aquifer in southern arizona the avra valley aquifer in southern arizona is modeled stochastically at three levels of uncertainty the highest level of uncertainty occurs when log transmissivity estimates are based on measured values of this parameter but without regard to the geographic location of each measurement point the resulting steady state hydraulic heads in the aquifer computed by unconditional simulation with the aid of a multivariate normal random number generator coupled with a finite element model have a relatively large variance this variance can be reduced by conditioning the log transmissivity estimates on the spatial arrangement of the data by means of kriging when conditional simulation of the aquifer is performed at this intermediate level of uncertainty with the aid of the same technique as in the unconditional case the variance of the predicted head values is reduced by a factor of 3 2 the lowest level of uncertainty is achieved when the log transmissivity estimates are further conditioned on data relating to the flow regime such as flow rates and water levels in observation wells by means of a statistical inverse procedure conditional simulation at this level of uncertainty is a novel concept it results in a hydraulic head prediction variance that is 14 3 times lower than the corresponding variance based on kriged log transmissivities the net effect of conditioning by means of kriging and inverse modeling is to reduce the prediction variance by a factor of 46 0 a similar study performed by binsarti 1980 on the cortaro aquifer in southern arizona showed insignificant variance reduction due to kriging but a factor of four reduction after inverse modeling these results indicate that the conditioning effect of the inverse method may be much greater than that of kriging the fact that conditioning can reduce the hydraulic head prediction variance by as much as a factor of 46 implies that one should be cautious when dealing with the results of stochastic aquifer models in which conditioning effects are disregarded

a bayesian approach for inverse modeling data assimilation and conditional simulation of spatial random fields 1 this paper addresses the inverse problem in spatially variable fields such as hydraulic conductivity in groundwater aquifers or rainfall intensity in hydrology common to all these problems is the existence of a complex pattern of spatial variability of the target variables and observations the multiple sources of data available for characterizing the fields the complex relations between the observed and target variables and the multiple scales and frequencies of the observations the method of anchored distributions mad that we propose here is a general bayesian method of inverse modeling of spatial random fields that addresses this complexity the central elements of mad are a modular classification of all relevant data and a new concept called anchors data types are classified by the way they relate to the target variable as either local or nonlocal and as either direct or indirect anchors are devices for localization of data they are used to convert nonlocal indirect data into local distributions of the target variables the target of the inversion is the derivation of the joint distribution of the anchors and structural parameters conditional to all measurements regardless of scale or frequency of measurement the structural parameters describe large scale trends of the target variable fields whereas the anchors capture local inhomogeneities following inversion the joint distribution of anchors and structural parameters is used for generating random fields of the target variable s that are conditioned on the nonlocal indirect data through their anchor representation we demonstrate mad through a detailed case study that assimilates point measurements of the conductivity with head measurements from natural gradient flow the resulting statistical distributions of the parameters are non gaussian similarly the moments of the estimates of the hydraulic head are non gaussian we provide an extended discussion of mad vis a vis other inversion methods including maximum likelihood and maximum a posteriori with an emphasis on the differences between mad and the pilot points method

inverse stochastic moment analysis of steady state flow in randomly heterogeneous media 1 nonlocal stochastic moment equations have been used successfully to analyze steady state and transient flow in randomly heterogeneous media conditional on measured values of medium properties we present a nonlinear geostatistical inverse algorithm for steady state flow that makes it possible to further condition such analyses on measured values of state variables our approach accounts for all scales of spatial variability resolvable by the computational grid it is based on recursive finite element approximations of exact nonlocal first and second conditional moment equations hydraulic conductivity is parameterized geostatistically on the basis of measured values at discrete locations and unknown values at discrete pilot points prior estimates of pilot point values are obtained optionally by generalized kriging posterior estimates at pilot points and optionally at measurement points are obtained by calibrating mean flow against measured values of head the parameters are projected onto a computational grid via kriging maximum likelihood calibration allows estimating not only hydraulic but also optionally unknown variogram parameters with or without prior information about the former the approach yields covariance matrices for parameter estimation as well as head and flux prediction errors the latter being obtained a posteriori we implement our inverse approach on highly and mildly heterogeneous media under superimposed mean uniform and convergent flows in a bounded two dimensional domain our examples illustrate how conductivity and head data act separately and jointly to reduce parameter estimation errors and to model predictive uncertainty we also evaluate the functional form of the log conductivity variogram and its parameters using likelihood based model discrimination criteria

numerical solutions of moment equations for flow in heterogeneous composite aquifers 1 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 tartakovksy 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 within material 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

predictive uncertainty analysis of a saltwater intrusion model using null space monte carlo 1 because of the extensive computational burden and perhaps a lack of awareness of existing methods rigorous uncertainty analyses are rarely conducted for variable density flow and transport models for this reason a recently developed null space monte carlo nsmc method for quantifying prediction uncertainty was tested for a synthetic saltwater intrusion model patterned after the henry problem saltwater intrusion caused by a reduction in fresh groundwater discharge was simulated for 1000 randomly generated hydraulic conductivity distributions representing a mildly heterogeneous aquifer from these 1000 simulations the hydraulic conductivity distribution giving rise to the most extreme case of saltwater intrusion was selected and was assumed to represent the true system head and salinity values from this true model were then extracted and used as observations for subsequent model calibration random noise was added to the observations to approximate realistic field conditions the nsmc method was used to calculate 1000 calibration constrained parameter fields if the dimensionality of the solution space was set appropriately the estimated uncertainty range from the nsmc analysis encompassed the truth several variants of the method were implemented to investigate their effect on the efficiency of the nsmc method reducing the dimensionality of the null space for the processing of the random parameter sets did not result in any significant gains in efficiency and compromised the ability of the nsmc method to encompass the true prediction value the addition of intrapilot point heterogeneity to the nsmc process was also tested according to a variogram comparison this provided the same scale of heterogeneity that was used to generate the truth however incorporation of intrapilot point variability did not make a noticeable difference to the uncertainty of the prediction with this higher level of heterogeneity however the computational burden of generating calibration constrained parameter fields approximately doubled predictive uncertainty variance computed through the nsmc method was compared with that computed through linear analysis the results were in good agreement with the nsmc method estimate showing a slightly smaller range of prediction uncertainty than was calculated by the linear method

multimodel bayesian analysis of groundwater data worth we explore the way in which uncertain descriptions of aquifer heterogeneity and groundwater flow impact one s ability to assess the worth of collecting additional data we do so on the basis of maximum likelihood bayesian model averaging mlbma by accounting jointly for uncertainties in geostatistical and flow model structures and parameter hydraulic conductivity as well as system state hydraulic head estimates given uncertain measurements of one or both variables previous description of our approach was limited to geostatistical models based solely on hydraulic conductivity data here we implement the approach on a synthetic example of steady state flow in a two dimensional random log hydraulic conductivity field with and without recharge by embedding an inverse stochastic moment solution of groundwater flow in mlbma a moment equations based geostatistical inversion method is utilized to circumvent the need for computationally expensive numerical monte carlo simulations the approach is compatible with either deterministic or stochastic flow models and consistent with modern statistical methods of parameter estimation admitting but not requiring prior information about the parameters it allows but does not require approximating lead predictive statistical moments of system states by linearization while updating model posterior probabilities and parameter estimates on the basis of potential new data both before and after such data are actually collected

conditional stochastic averaging of steady state unsaturated flow by means of kirchhoff transformation we consider the effect of measuring randomly varying soil hydraulic properties on one s ability to predict steady state unsaturated flow subject to random sources and or initial and boundary conditions our aim is to allow optimum unbiased prediction of system states pressure head water content and fluxes deterministically without upscaling and without linearizing the constitutive characteristics of the soil it has been shown by neuman et al 1999 that such prediction is possible by means of first ensemble moments of system states and fluxes conditioned on measured values of soil properties when the latter scale in a linearly separable fashion as proposed by vogel et al 1991 the uncertainty associated with such predictions can be quantified by means of the corresponding conditional second moments the derivation of moment equations for soils whose properties do not scale in the above manner requires linearizing the corresponding constitutive relations which may lead to major inaccuracies when these relations are highly nonlinear as is often the case in nature when the scaling parameter of pressure head is a random variable independent of location the steady state unsaturated flow equation can be linearized by means of the kirchhoff transformation for gravity free flow linearization is also possible in the presence of gravity when hydraulic conductivity varies exponentially with pressure head for the latter case we develop exact conditional first and second moment equations which are nonlocal and therefore non darcian we solve these equations analytically by perturbation for unconditional vertical infiltration and compare our solution with the results of numerical monte carlo simulations our analytical solution demonstrates in a rigorous manner that the concept of effective hydraulic conductivity does not apply to ensemble averaged unsaturated flow except when gravity is the sole driving force

strategic placement of localization devices such as pilot points and anchors in inverse modeling schemes 1 localization schemes using devices such as pilot points or anchors localization devices are used in inversion schemes as a tool for gleaning information from measurements that can be used to constrain inversion schemes with the added benefit in the form of a potential for reduction in the number of parameters employed in the inversion scheme this paper proposes and demonstrates a method for strategic placement of the localization devices the method combines stochastic singular value decomposition with parameter rejection methods to produce a map of intensity scores which allows the inverse modeler to place the localization devices at locations defined by their probability for producing informative probabilistic constraints on the statistical distributions of the target variables we also show that the method can be used for selecting measurement locations when designing data acquisition campaigns the method is demonstrated for the case of steady state flow in a heterogeneous conductivity field using mad method of anchored distributions in summary the proposed localization scheme intends to increase the information gleaned from measurements while reducing the associated computational costs associated with stochastic inversion

analysis of a two scale locally conservative subgrid upscaling for elliptic problems we present a two scale theoretical framework for approximating the solution of a second order elliptic problem the elliptic coefficient is assumed to vary on a scale that can be resolved on a fine numerical grid but limits on computational power require that computations be performed on a coarse grid we consider the elliptic problem in mixed variational form over w times bf v subset l 2 times h rm div we base our scale expansion on local mass conservation over the coarse grid it is used to define a direct sum decomposition of w times bf v into coarse and subgrid subspaces w c times bf v c and delta w times delta bf v such that 1 nabla cdot bf v c w c and nabla cdot delta bf v delta w and 2 the space delta bf v is locally supported over the coarse mesh we then explicitly decompose the variational problem into coarse and subgrid scale problems the subgrid problem gives a well defined operator taking w c times bf v c to delta w times delta bf v which is localized in space and it is used to upscale that is to remove the subgrid from the coarse scale problem using standard mixed finite element spaces two scale mixed spaces are defined a mixed approximation is defined which can be viewed as a type of variational multiscale method or a residual free bubble technique a numerical green s function approach is used to make the approximation to the subgrid operator efficient to compute a mixed method pi operator is defined for the two scale approximation spaces and used to show optimal order error estimates

a finite element variational multiscale method for the navier stokes equations this paper presents a variational multiscale method vms for the incompressible navier stokes equations which is defined by a large scale space lh for the velocity deformation tensor and a turbulent viscosity nu t the connection of this method to the standard formulation of a vms is explained the conditions on lh under which the vms can be implemented easily and efficiently into an existing finite element code for solving the navier stokes equations are studied numerical tests with the smagorinsky large eddy simulation model for nu t are presented

anisotropy lacunarity and upscaled conductivity and its autocovariance in multiscale random fields with truncated power variograms it has been shown by di federico and neuman 1997 1998a b that observed multiscale behaviors of subsurface fluid flow and transport variables can be explained within the context of a unified stochastic framework which views hydraulic conductivity as a random fractal characterized by a power variogram such a random field is statistically nonhomogeneous but possesses homogeneous spatial increments di federico and neuman 1997 have demonstrated that the power variogram and associated spectra of a statistically isotropic fractal field can be constructed as a weighted integral from zero to infinity an infinite hierarchy of exponential or gaussian variograms and spectra of mutually uncorrelated fields modes that are homogeneous and isotropic we show in this paper that the same holds true when the field and its constituent modes are statistically anisotropic provided the ratios between principal integral spatial correlation scales are the same for all modes we then analyze the effect of filtering out truncating modes of low high and intermediate spatial frequency from this infinite hierarchy in the real and spectral domains a low frequency cutoff renders the truncated hierarchy homogeneous the integral scales of the lowest and highest frequency cutoff modes are related to length scales of the sampling window domain and data support sample volume respectively taking the former to be proportional to the latter renders expressions for the integral scale and variance of the truncated field dependent on window and support scale in a manner previously shown to be consistent with observations in the isotropic case it also allows in principle bridging across scales at a specific locale as well as among locales by adopting either site specific or generalized variogram parameters the introduction of intermediate cutoffs allows us to account in a straightforward manner for lacunarity due to gaps in the multiscale hierarchy created by the absence of modes associated with discrete ranges of scales for example where textural and structural features are associated with distinct ranges of scale such as fractures having discrete ranges of trace length and density which dissect the rock into matrix blocks having corresponding ranges of sizes we explore mathematically and graphically the effects that anisotropy and lacunarity have on the integral scale variance covariance and spectra of a truncated fractal field we then develop an expression for the equivalent hydraulic conductivity of a box shaped porous block embedded within a multiscale log hydraulic conductivity field under mean uniform flow the block is larger than the support scale of the field but is smaller than a surrounding sampling window consequently its equivalent hydraulic conductivity is a random variable whose variance and spatial autocorrelation function conditioned on a known mean value of support scale conductivity across the window are given explicitly by our multiscale theory

efficient posterior exploration of a high dimensional groundwater model from two stage markov chain monte carlo simulation and polynomial chaos expansion 1 this study reports on two strategies for accelerating posterior inference of a highly parameterized and cpu demanding groundwater flow model our method builds on previous stochastic collocation approaches e g marzouk and xiu 2009 and marzouk and najm 2009 and uses generalized polynomial chaos gpc theory and dimensionality reduction to emulate the output of a large scale groundwater flow model the resulting surrogate model is cpu efficient and serves to explore the posterior distribution at a much lower computational cost using two stage mcmc simulation the case study reported in this paper demonstrates a two to five times speed up in sampling efficiency

nonlocal and localized analyses of conditional mean transient flow in bounded randomly heterogeneous porous media 1 we consider the numerical prediction of transient flow in bounded randomly heterogeneous porous media driven by random sources initial heads and boundary conditions without resorting to monte carlo simulation after applying the laplace transform to the governing stochastic flow equations we derive exact nonlocal integro differential equations for the mean and variance covariance of transformed head and flux conditioned on measured values of log conductivity y ln k approximating these conditional moment equations recursively to second order in the standard deviation y of y we solve them by finite elements for superimposed mean uniform and convergent flows in a two dimensional domain an alternative conditional mean solution is obtained through localization of the exact moment expressions the nonlocal and localized solutions are obtained using a highly efficient parallel algorithm and inverted numerically back into the time domain a comparison with monte carlo simulations demonstrates that the moment solutions are remarkably accurate for strongly heterogeneous media with y2 as large as 4 the nonlocal solution is only slightly more accurate than the much simpler localized solution but the latter does not yield information about predictive uncertainty the accuracy of each solution improves markedly with conditioning a preliminary comparison of computational efficiency suggests that both the nonlocal and localized solutions for mean head and its variance require significantly less computer time than is required for monte carlo statistics to stabilize when the same direct matrix solver is used for all three we do not presently know how using iterative solvers would have affected this conclusion this is true whether the laplace inversion and monte carlo simulations are conducted sequentially or in parallel on multiple processors and regardless of problem size the underlying exact and recursive moment equations as well as the proposed computational algorithm are valid in both two and three dimensions only the numerical implementation of our algorithm is two dimensional

exact sensitivity matrix and influence of the number of pilot points in the geostatistical inversion of moment equations of groundwater flow received 3 august 2009 revised 12 july 2010 accepted 21 july 2010 published 4 november 2010 1 we present novel equations for the exact sensitivity matrix of the ensemble mean hydraulic head under steady state groundwater flow conditions these equations are embedded in a geostatistical inverse procedure to condition approximations of stochastic moment equations of flow on measured hydraulic conductivities and heads our formulation allows considerable improvement of the methodology proposed by hernandez et al 2003 2006 and renders the inversion of moment equations feasible for a large number of unknown hydraulic parameters the spatial distribution of the natural logarithm y of conductivity is parameterized within the pilot points framework whereas prior values of y at pilot points are obtained by a variant of kriging posterior estimates at pilot points are obtained through a maximum likelihood fit of computed to measured heads the maximum likelihood function also includes a regularization term by means of a synthetic example and upon adopting formal model information criteria we explore the influence of 1 the number of pilot points and 2 the order of approximation of the governing mean flow equation on our ability to properly estimate the log conductivity and head fields and identify the relative weight of the regularization term and the parameters of the underlying y variogram we find that none of the adopted information criteria can identify the optimum number of pilot points and the plausibility weight and variogram parameters values can be determined by the kashyap s bayesian measure

on the synthesis of linear input output responses for nonlinear systems necessary and sufficient conditions are given for the existence of a feedback control law under which the input dependent part of the response of a given nonlinear system becomes linear in the input and independent of the initial state

an efficient high order perturbation approach for flow in random porous media via karhunen loeve and polynomial expansions in this study we attempt to obtain higher order solutions of the means and co variances of hydraulic head for saturated flow in randomly heterogeneous porous media on the basis of the combination of karhunen loeve decomposition polynomial expansion and perturbation methods we first decompose the log hydraulic conductivity y ln ks as an infinite series on the basis of a set of orthogonal gaussian standard random variables i the coefficients of the series are related to eigenvalues and eigenfunctions of the covariance function of log hydraulic conductivity we then write head as an infinite series whose terms h n represent head of nth order in terms of y the standard deviation of y and derive a set of recursive equations for h n we then decompose h n with polynomial expansions in terms of the products of n gaussian random variables the coefficients in these series are determined by substituting decompositions of y and h n into those recursive equations we solve the mean head up to fourth order in y and the head variances up to third order in y2 we conduct monte carlo mc simulation and compare mc results against approximations of different orders from the moment equation approach based on karhunen loeve decomposition klme we also explore the validity of the klme approach for different degrees of medium variability and various correlation scales it is evident that the klme approach with higher order corrections is superior to the conventional first order approximations and is computationally more efficient than the monte carlo simulation

role of model selection criteria in geostatistical inverse estimation of statistical data and model parameters 1 we analyze theoretically the ability of model quality sometimes termed information or discrimination criteria such as the negative log likelihood nll bayesian criteria bic and kic and information theoretic criteria aic aicc and hic to estimate 1 the parameter vector of the variogram of hydraulic log conductivity y ln k and 2 statistical parameters and proportional to head and log conductivity measurement error variances respectively in the context of geostatistical groundwater flow inversion our analysis extends the work of hernandez et al 2003 2006 and riva et al 2009 who developed nonlinear stochastic inverse algorithms that allow conditioning estimates of steady state and transient hydraulic heads fluxes and their associated uncertainty on information about conductivity and head data collected in a randomly heterogeneous confined aquifer their algorithms are based on recursive numerical approximations of exact nonlocal conditional equations describing the mean and co variance of groundwater flow log conductivity is parameterized geostatistically based on measured values at discrete locations and unknown values at discrete pilot points optionally the maximum likelihood function on which the inverse estimation of y at pilot points is based may include a regularization term reflecting prior information about y the relative weight assigned to this term and its components and as well as are evaluated separately from other model parameters to avoid bias and instability this evaluation is done on the basis of criteria such as nll kic bic hic aic and aicc we demonstrate theoretically that whereas all these six criteria make it possible to estimate kic alone allows one to estimate validly and and thus we illustrate this discriminatory power of kic numerically by using a differential evolution genetic search algorithm to minimize it in the context of a two dimensional steady state groundwater flow problem we find that whereas and the integral scale of y can be estimated on the basis of a zero order mean flow equation the sill of the y variogram is estimated more accurately by a second order approximation of flow this notwithstanding kic prefers the simpler zero order moment over the more complex second order version

prediction of steady state flow in nonuniform geologic media by conditional moments exact nonlocal formalism effective conductivities and weak approximation we consider the effect of measuring randomly varying local hydraulic conductivities k x on one s ability to predict steady state flow within a bounded domain driven by random source and boundary functions more precisely we consider the prediction of local hydraulic head h x and darcy flux q x by means of their unbiased ensemble moments h x and q x conditioned on measurements of k x these predictors satisfy a deterministic flow equation in which q x x h x r x where x is a relatively smooth unbiased estimate of k x and r x is a residual flux we derive a compact integral expression for r x which is rigorously valid for a broad class of k x fields including fractals it demonstrates that q x is nonlocal and non darcian so that an effective hydraulic conductivity does not generally exist we show analytically that under uniform mean flow the effective conductivity may be a scalar a symmetric or a nonsymmetric tensor or a set of directional scalars which do not form a tensor we demonstrate numerically that in two dimensional mean radial flow it may increase from the harmonic mean of k x near interior and boundary sources to the geometric mean far from such sources for cases where r x can neither be expressed nor approximated by a local expression we propose a weak integral approximation closure which appears to work well in media with pronounced heterogeneity and improves as the quantity and quality of k x measurements increase the nonlocal deterministic flow equation can be solved numerically by standard methods our theory shows clearly how the scale of grid discretization should relate to the scale quantity and quality of available data after providing explicit approximations for the second moments of head and flux prediction errors we conclude by discussing practical methods to compute x from noisy measurements of k x and to calculate required second moments of the associated estimation errors when k x is lognormal

erratum transient flow in bounded randomly heterogeneous domains 1 exact conditional moment equations and recursive approximations water resources research 1998 34 1 1 12

application of multi scale finite element methods to the solution of the fokker planck equation abstract this paper presents an application of multi scale finite element methods to the solution of the multi dimensional fokker planck equation the fokker planck or forward kolmogorov equation is a degenerate convective diffusion equation arising in markov process theory it governs the evolution of the transition probability density function of the response of a broad class of dynamical systems driven by gaussian noise and completely describes the response process analytical solutions for the fokker planck equation have been developed for only a limited number of low dimensional systems leading to a large body of approximation theory one such approach successfully applied to the solution of these problems in the past is the finite element method though for systems of dimension three or less in this paper a multi scale finite element method is applied to the fokker planck equation in an effort to develop a formulation that can yield higher accuracy on cruder spatial discretizations thus reducing the computational overhead associated with large scale problems that arise in higher dimensions

multi scale finite volume method for elliptic problems in subsurface flow simulation in this paper we present a multi scale finite volume msfv method to solve elliptic problems with many spatial scales arising from flow in porous media the method efficiently captures the effects of small scales on a coarse grid is conservative and treats tensor permeabilities correctly the underlying idea is to construct transmissibilities that capture the local properties of the differential operator this leads to a multi point discretization scheme for the finite volume solution algorithm transmissibilities for the msfv have to be constructed only once as a preprocessing step and can be computed locally therefore this step is perfectly suited for massively parallel computers furthermore a conservative fine scale velocity field can be constructed from the coarse scale pressure solution two sets of locally computed basis functions are employed the first set of basis functions captures the small scale heterogeneity of the underlying permeability field and it is computed in order to construct the effective coarse scale transmissibilities a second set of basis functions is required to construct a conservative fine scale velocity field the accuracy and efficiency of our method is demonstrated by various numerical experiments

calibration constrained monte carlo analysis of highly parameterized models using subspace techniques we describe a subspace monte carlo ssmc technique that reduces the burden of calibration constrained monte carlo when undertaken with highly parameterized models when monte carlo methods are used to evaluate the uncertainty in model outputs ensuring that parameter realizations reproduce the calibration data requires many model runs to condition each realization in the new ssmc approach the model is first calibrated using a subspace regularization method ideally the hybrid tikhonov tsvd superparameter approach described by tonkin and doherty 2005 sensitivities calculated with the calibrated model are used to define the calibration null space which is spanned by parameter combinations that have no effect on simulated equivalents to available observations next a stochastic parameter generator is used to produce parameter realizations and for each a difference is formed between the stochastic parameters and the calibrated parameters this difference is projected onto the calibration null space and added to the calibrated parameters if the model is no longer calibrated parameter combinations that span the calibration solution space are reestimated while retaining the null space projected parameter differences as additive values the recalibration can often be undertaken using existing sensitivities so that conditioning requires only a small number of model runs using synthetic and real world model applications we demonstrate that the ssmc approach is general it is not limited to any particular model or any particular parameterization scheme and that it can rapidly produce a large number of conditioned parameter sets

conditional moment analysis of steady state unsaturated flow in bounded randomly heterogeneous soils 1 we consider steady state unsaturated flow in bounded randomly heterogeneous soils under the influence of random boundary and source terms our aim is to predict pressure heads and fluxes without resorting to monte carlo simulation upscaling or linearization of the constitutive relationship between unsaturated hydraulic conductivity and pressure head we represent this relationship through gardner s exponential model while treating its exponent as a random constant and saturated hydraulic conductivity ks as a spatially correlated random field we linearize the steady state unsaturated flow equations by means of the kirchhoff transformation and integrate them in probability space to obtain exact integro differential equations for the conditional mean and variance covariance of transformed pressure head and flux after approximating these equations recursively to second order in the standard deviation y of y ln ks we solve them by finite elements for superimposed mean uniform and divergent flows in the vertical plane with and without conditioning on measured y values comparison with monte carlo solutions demonstrates that whereas our nonlocal solution is nominally restricted to mildly nonuniform media with y2 1 it yields remarkably accurate results for strongly nonuniform media with y2 at least as large as 2 this accords well with a previous theoretical analysis which shows that the solution may remain asymptotic for values of y2 as large as 2

application of inverse system for linearization and decoupling abstract we show that for a special class of nonlinear systems the linearization and or decoupling of nonlinear dynamics by immersion under feedback is in fact an application of the right inverse system

a stochastic analysis of steady state two phase flow in heterogeneous media 1 we present a novel approach to modeling stochastic multiphase flow problems for example nonaqueous phase liquid flow in a heterogeneous subsurface medium with random soil properties in particular with randomly heterogeneous intrinsic permeability and soil pore size distribution a stochastic numerical model for steady state water oil flow in a random soil property field is developed using the karhunen loeve moment equation klme approach and is numerically implemented an exponential model is adopted to define the constitutive relationship between phase relative permeability and capillary pressure the log transformed intrinsic permeability y x and soil pore size distribution b x are assumed to be gaussian random functions with a separable exponential covariance function the perturbation part of these two log transformed soil properties is then decomposed into an infinite series based on a set of orthogonal normal random variables xn the phase pressure capillary pressure and phase mobility are decomposed by polynomial expansions and the perturbation method combining these expansions of y x b x and dependent pressures the steady state water oil flow equations and corresponding boundary conditions are reformulated as a series of differential equations up to second order these differential equations are solved numerically and the solutions are directly used to construct moments of phase pressure and capillary pressure we demonstrate the validity of the proposed klme model by favorably comparing firstand second order approximations to monte carlo simulations the significant computational efficiency of the klme approach over monte carlo simulation is also illustrated

stochastic identification of transmissivity and effective recharge in steady groundwater flow 2 case study the study is a continuation and extension of a previous work dagan 1985a whose aim was to identify the values of the log transmissivity y for steady flow the common basic assumptions are that y is a normal and stationary random space function the aquifer is unbounded and a first order approximation of the flow equation is adopted the expected value of the water head h as well as the y unconditional autocovariance are supposed to have analytical expressions which depend on a parameters vector the proposed solution of the inverse problem consists of identifying with the aid of the model and of the measurements of y and h and subsequently computing the statistical moments of y conditioned on the same data the additional features of the present study are 1 incorporation of a constant but random effective recharge and its identification and 2 accounting for the fact that estimation is associated with some uncertainty whereas before was assumed to be identified with certainty analytical expressions are derived for the y and h covariances for an exponential autocovariance of y paper 2 rubin and dagan this issue of the study illustrates the applications of the method to a real life case

a general fractal model of flow and solute transport in randomly heterogeneous porous media 1 one of the challenges of hydrology is modeling flow and solute transport through media that are not uniformly porous this paper proposes a general fractal model of flow and solute transport in randomly heterogeneous porous media we describe the random field using fractional levy motion flm which more effectively represents a field with a high degree of variability understanding flow and solute transport in the flm field expands the applicability of stochastic theory following the work of v di federico and s p neuman 1997 and v di federico et al 1999 we derive truncated isotropic and anisotropic power variograms for the flm fields the results are analogous to but more general than those of fractional brownian motion fbm v di federico and s p neuman 1997 and v di federico et al 1999 for a random field when the levy index in flm is replaced by a constant of 2 fbm is recovered this study employs first order perturbation analyses of flow and solute transport in two dimensional flm fields for the stationary fbm and flm fields macrodispersion coefficients are compared the plume is found to grow slowest in the flm field but will show highest concentration the proposed fractal model is general in the sense that when the levy index of flm equals 2 the results of flow and solute transport for the commonly used fbm model are recovered when the upper cutoff of the truncated power variogram is close to the lower cutoff the results for the stationary exponential or gaussian model can be well approximated since the proposed general fractal model has broader applications than the stationary and fbm models it simulates flow and solute transport in more different scenarios to yield more accurate modeling results

estimation of spatial covariance structures by adjoint state maximum likelihood cross validation 2 synthetic experiments paper 2 of this three part series uses synthetic data to investigate the properties of the adjoint state maximum likelihood cross validation asmlcv method presented in paper 1 samper and neuman this issue a more than 40 synthetic experiments are performed to compare various conjugate gradient algorithms investigate the manner in which computer time varies with asmlcv parameters study the effect of sample size and choice of kriging points on asmlcv estimates evaluate the ability of various model structure identification criteria to help select the most appropriate semivariogram model among given alternatives study the conditions required for parameter identifiability uniqueness and stability quantify the statistics of cross validation errors test hypotheses concerning the distribution and autocorrelation of these errors and illustrate the computation of approximate quality indicators for asmlcv parameter estimates

highly parameterized inverse estimation of hydraulic conductivity and porosity in a three dimensional heterogeneous transport experiment 1 assessing the impact of parameter estimation accuracy in models of heterogeneous three dimensional 3 d groundwater systems is critical for predictions of solute transport a unique experimental data set provides concentration breakthrough curves btcs measured at a 0 253 cm3 scale over the 13 8 8 cm3 domain 53 000 measurement locations advective transport is used to match the first temporal moments of btcs or mean arrival times m1 averaged at 0 253 and 1 0 cm3 scales through simultaneous inversion of highly parameterized heterogeneous hydraulic conductivity k and porosity fields pilot points parameterize the fields within eight layers of the 3 d medium and estimations are completed with six different models of the k relationship parameter estimation through advective transport shows accurate estimation of the observed m1 values results across the six different k relationships have statistically similar fits to the observed m1 values and similar spatial estimates of m1 along the main flow direction the resulting fields provide the basis for forward transport modeling of the advection dispersion equation ade using the estimated k and fields demonstrates that advective transport coupled with inversion using dense spatial field parameterization provides an efficient surrogate for the ade these results indicate that there is not a single set of model parameters or a single k relationship that leads to a best representation of the actual experimental sand packing pattern i e nonuniqueness additionally knowledge of the individual sand k and values along with their arrangement in the 3 d experiment does not reproduce the observed transport results at small scales small scale variation in the packing and mixing of the sands causes large deviations from the expected transport results as highlighted in forward ade simulations highly parameterized inverse estimation is able to identify those regions where variations in mixing and packing alter the expected property values and significantly improve results relative to the naive application of the experimentally derived property values impacts of the observation scale the scale over which results are averaged and the number of observations and parameters on the final estimations are also examined results indicate existence of a representative element volume rev at 0 25 cm3 the existence of subgrid scale heterogeneity that impacts transport and the accuracy of highly parameterized models with even relatively small amounts of observations finally this work suggests that local heterogeneity features below the rev scale are difficult to incorporate into parameterized models highlighting the importance of addressing prediction uncertainty for small scale variability i e uncaptured variability in modeling practice

stochastic uncertainty analysis for solute transport in randomly heterogeneous media using a karhunen loeve based moment equation approach 1 a new approach has been developed for solving solute transport problems in randomly heterogeneous media using the karhunen loeve based moment equation klme technique proposed by zhang and lu 2004 the klme approach combines the karhunen loeve decomposition of the underlying random conductivity field and the perturbative and polynomial expansions of dependent variables including the hydraulic head flow velocity dispersion coefficient and solute concentration the equations obtained in this approach are sequential and their structure is formulated in the same form as the original governing equations such that any existing simulator such as modular three dimensional multispecies transport model for simulation of advection dispersion and chemical reactions of contaminants in groundwater systems mt3dms can be directly applied as the solver through a series of two dimensional examples the validity of the klme approach is evaluated against the classical monte carlo simulations results indicate that under the flow and transport conditions examined in this work the klme approach provides an accurate representation of the mean concentration for the concentration variance the accuracy of the klme approach is good when the conductivity variance is 0 5 as the conductivity variance increases up to 1 0 the mismatch on the concentration variance becomes large although the mean concentration can still be accurately reproduced by the klme approach our results also indicate that when the conductivity variance is relatively large neglecting the effects of the cross terms between velocity fluctuations and local dispersivities as done in some previous studies can produce noticeable errors and a rigorous treatment of the dispersion terms becomes more appropriate