stochastic uncertainty analysis for unconfined flow systems 1 a new stochastic approach proposed by zhang and lu 2004 called the karhunen loeve decomposition based moment equation klme has been extended to solving nonlinear unconfined flow problems in randomly heterogeneous aquifers this approach is on the basis of an innovative combination of karhunen loeve decomposition polynomial expansion and perturbation methods the random log transformed hydraulic conductivity field lnks is first expanded into a series in terms of orthogonal gaussian standard random variables with their coefficients obtained as the eigenvalues and eigenfunctions of the covariance function of lnks next head h is decomposed as a perturbation expansion series h m where h m represents the mth order head term with respect to the standard deviation of lnks then h m is further expanded into a polynomial series of m products of orthogonal gaussian standard random variables whose coefficients are deterministic and solved sequentially from low to high expansion orders using modflow 2000 finally the statistics of head and flux are computed using simple algebraic operations on a series of numerical test results in 2 d and 3 d unconfined flow systems indicated that the klme approach is effective in estimating the mean and co variance of both heads and fluxes and requires much less computational effort as compared to the traditional monte carlo simulation technique

stochastic modeling of macrodispersion in heterogeneous porous media a method for the prediction of dispersion processes of inert solutes occurring in heterogeneous porous media is presented its main features are as follows 1 the velocity field is assumed to be a space random function its moments are expressed through the physical parameters of the log transmissivity and head random fields by linearizing the flow equation the random velocity being a linear function of the head and log transmissivity which are assumed to be jointly multivariate normal mvn is thus also mvn its pdf is completely defined by its first two moments 2 the statistics of the dispersion process are obtained by the particle tracking method through monte carlo simulations which are based on gaussian conditioning the method is applied to the borden natural gradient tracer test freyberg 1986 showing favorable results

comparison of gaussian conditional mean and kriging estimation in the geostatistical solution of the inverse problem two separate applications of the geostatistical solution to the inverse problem in groundwater modeling are presented both applications estimate the transmissivity field for a two dimensional model of a confined aquifer under steady flow conditions the estimates are based on point observations of transmissivity and hydraulic head and also on a model of the aquifer which includes prescribed head boundaries leakage and steady state pumping the model used to describe the spatial variability of the log transmissivity describes large scale fluctuations through a linear mean or drift intermediate and small scale fluctuations through a two parameter covariance function the first application presented estimates the log transmissivities using gaussian conditional mean estimation the second application uses an extended form of cokriging the two methods are compared and their relative merits discussed the extended cokriging application is applied to the jordan aquifer of iowa a comparison is also made between the conditional mean application and an analytical approach

estimating the variance and integral scale of the transmissivity field using head residual increments a modification of previously published solutions regarding the spatial variation of hydraulic heads is discussed whereby the semivariogram of increments of head residuals termed head residual increments hris are related to the variance and integral scale of the transmissivity field a first order solution is developed for the case of a transmissivity field which is isotropic and whose second order behavior can be characterized by an exponential covariance structure the estimates of the variance y2 and the integral scale of the log transmissivity field are then obtained via fitting a theoretical semivariogram for the hri to its sample semivariogram this approach is applied to head data sampled from a series of two dimensional simulated aquifers with isotropic exponential covariance structures and varying degrees of heterogeneity y2 0 25 0 5 1 0 2 0 and 5 0 the results show that this method provided reliable estimates for both and y2 in aquifers with the value of y2 up to 2 0 but the errors in those estimates were higher for y2 equal to 5 0 it is also demonstrated through numerical experiments and theoretical arguments that the head residual increments will provide a sample semivariogram with a lower variance than will the use of the head residuals without calculation of increments

block incomplete factorization preconditioners for a symmetric block tridiagonal m matrix we propose new block incomplete factorization preconditioners for a symmetric block tridiagonal m matrix which can be computed in parallel and then theoretical properties for these block preconditioners are studied spectral properties of the transformed coefficient matrices with the block incomplete factorization preconditioners are also examined to see the convergence rate of the preconditioned cg pcg method lastly numerical results of the pcg using the block incomplete factorization preconditioners are compared with those of the pcg using a standard incomplete factorization preconditioner to see how effective the block incomplete factorization preconditioners are

quasi linear geostatistical theory for inversing a quasi linear theory is presented for the geostatistical solution to the inverse problem the archetypal problem is to estimate the log transmissivity function from observations of head and log transmissivity at selected locations the unknown is parameterized as a realization of a random field and the estimation problem is solved in two phases structural analysis where the random field is characterized followed by estimation of the log transmissivity conditional on all observations the proposed method generalizes the linear approach of kitanidis and vomvoris 1983 the generalized method is superior to the linear method in cases of large contrast in formation properties but informative measurements i e there are enough observations that the variance of estimation error of the log transmissivity is small the methodology deals rigorously with unknown drift coefficients and yields estimates of covariance parameters that are unbiased and grid independent the applicability of the methodology is demonstrated through an example that includes structural analysis determination of best estimates and conditional simulations

a proposed stepwise regression method for model structure identification this paper proposes a new methodology for constructing groundwater models the proposed methodology which determines simultaneously both model structure and model parameters is based on the following ideas 1 when solving the inverse problem different model structures always produce different model parameters 2 since the number of possible model structures of an aquifer is infinite the number of possible representative parameters is also infinite 3 to obtain a set of appropriate representative model parameters we must have an appropriate model structure and 4 an appropriate model structure should be determined not only by observation data and prior information but also by the accuracy requirements of model applications in this proposed methodology we start with a homogeneous model structure and step by step gradually increase the complexity of the model structure at each level of complexity we calculate not only the fitting residual of parameter identification but also the error of model structure to determine if a more complex model structure is needed the model structure error of using one model structure to replace another model structure is defined by a maximum minimum max min problem that is based on the distance between the two models and is measured in parameter observation and prediction or decision spaces this proposed methodology is used to solve a hypothetical remediation design problem in which the true hydraulic conductivity is a random field with a certain trend we have found that for the example problem virtually identical pumping policy is obtained when a five zone model with an optimized zonation pattern is used to represent the nonstationary random field we have also found that observation errors have minimum impact on management solution in comparison with structure errors to calculate the model structure error for this example the inverse solution is coupled with a management problem we have also developed an effective iteration method to handle nonlinear water quality constraints

boundedness of calderon zygmund operators on non homogeneous metric measure spaces equivalent characterizations national natural science foundation of china 10871025 changjiang scholars and innovative research team in university of china fundamental research funds for the central universities of china 2011121040

stochastic capture zone delineation within the generalized likelihood uncertainty estimation methodology conditioning on head observations a stochastic methodology to evaluate the predictive uncertainty in well capture zones in heterogeneous aquifers with uncertain parameters is presented the approach is based on the generalized likelihood uncertainty estimation methodology the hydraulic conductivity is modeled as a random space function allowing for the uncertainty that stems from the imperfect knowledge of the parameters of the correlation structure parameters are sampled from prior distributions and are used for the generation of a large number of hydraulic conductivity fields which are subsequently used to solve the groundwater flow equation a likelihood is calculated for every simulation based on some goodness of fit measure between simulated heads and available observations using inverse particle tracking a capture zone is determined which is assigned the likelihood calculated for that particular simulation statistical analysis of the ensemble of all simulations enables the predictive uncertainty of the well capture zones to be defined results are presented for a hypothetical test case and different likelihood definitions used in the conditioning process the results show that the delineated capture zones are most sensitive to the mean hydraulic conductivity and the variance whereas the integral scale of the variogram is the parameter with the smallest influence for all likelihood measures the prior uncertainty is reduced considerably by introducing the observation heads but the reduction is most effective for the very selective likelihood definition the method presented can be used in real applications to quantify the uncertainty in the location and extent of well capture zones when little or no information is available about the hydraulic properties through the conditioning on head observations

stochastic analysis of boundaries effects on head spatial variability in heterogeneous aquifers 2 impervious boundary the spatial variability of the head h in a two dimensional flow in the vicinity of an impervious boundary is investigated this study is a continuation of a previous one rubin and dagan 1988 which dealt with the influence of a given head boundary by representing h and the log transmissivity y as space random function and by solving the flow equation first order closed form analytical expressions for the head variogram and the head log transmissivity y covariance are developed these expressions offer the means to evaluate the influence of the boundary on the joint y h random field the assumption of unbounded domain is examined and it is found to be a good approximation at distances larger than three log transmissivity integral scales from the boundary by using conditional probabilities a simple method to estimate the statistical moments close to the boundary is presented it requires the knowledge of the moments of y and h in the unbounded domain only the method is compared with the exact analytical solution showing favorable results

stochastic analysis of head spatial variability in bounded rectangular heterogeneous aquifers groundwater flow through bounded rectangular aquifers is analyzed in a stochastic framework new analytical expressions of the head covariance and variance and the log transmissivity head cross covariance are derived for the simplified first order problem effects of different boundary conditions the aquifer size and the log transmissivity correlation scale upon head moments are studied the results are compared to analytical half plane solutions demonstrating significant differences in several cases finally the problem is solved numerically using the monte carlo simulation method the realizations of the log transmissivity field are generated in two different ways through comparisons with the analytical results the accuracy of the numerical methods is shown to be excellent

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

stochastic analysis of three dimensional flow in a bounded domain a commonly accepted first order approximation of the equation for steady state flow in a fully saturated spatially random medium has the form of poisson s equation this form allows for the advantageous use of green s functions to solve for the random output hydraulic heads in terms of a convolution over the random input the logarithm of hydraulic conductivity a solution for steady state three dimensional flow in an aquifer bounded above and below is presented consideration of these boundaries is made possible by use of green s functions to solve poisson s equation within the bounded domain the medium hydraulic conductivity is assumed to be a second order stationary random process as represented by a simple three dimensional covariance function upper and lower boundaries are taken to be no flow boundaries the mean flow vector lies entirely in the horizontal dimensions the resulting hydraulic head covariance function exhibits nonstationary effects resulting from the imposition of boundary conditions comparisons are made with existing infinite domain solutions

a stochastic approach to the problem of upscaling of conductivity in disordered media theory and unconditional numerical simulations a general problem in groundwater modeling is how to assign conductivity values to blocks of a numerical model in this study a method is proposed to upscale the conductivity measurements observed at a given scale to block conductivity values for arbitrary block size assuming a multilognormal distribution for the point conductivities the block conductivity appears as a random function whose distribution conditional to point measurements can be derived this conditional distribution allows filling the flow simulator blocks with equiprobable realizations of block conductivities locally conditioned to actual data these realizations could also be conditioned to conductivity values measured at any scale if the size of the measurement support is known the analytical results are compared to results obtained through a monte carlo analysis and a good agreement was found even for variances of ln t larger than 1

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

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

statistical inference of spatial random functions a methodology to infer the spatial distribution of the statistical moments of a stationary normal random spatial function based on a finite set of measured values is described the uncertainty of estimation of the parameters characterizing the probability density function pdf is incorporated in the process of inferring and conditioning relationships generalizing the traditional approach of stochastic interpolation and simulation have been derived they yield i an estimate of the conditional expectation ii an estimate of the conditional multivariate normal mvn covariance and variance and iii a new quantity the variance of the estimated variance the method is illustrated for a two dimensional synthetic example with both estimated mean and variance regarded as random a set of 60 points was generated within a square of sides equal to unity from a mvn population with zero mean unity variance and exponential covariance with a linear integral scale of 0 3 contour maps of the inferred expected values and variances are drawn and compared with maps of kriged values and maps of kriged variances contour maps of variances of estimation of the variance are given to demonstrate the uncertainty of variance estimates the unconditional and conditional statistical moments of the space average block value were also evaluated the reduction of the uncertainty of the inferred random variable as the number of measured points increases is demonstrated with the same synthetic example

optimal data acquisition strategy for the development of a transport model for groundwater remediation the reliability of groundwater quality management algorithms is limited in large part by the uncertainty present in the model parameters because the field parameter measurement costs and the remediation costs must be supplied by the same financial source the classical optimization procedure does not minimize the real total remediation investment this research presents an algorithm able to find the total minimum for the sum of both the measurement and the pumping costs a chance constrained technique is used to cast the optimization problem in stochastic form relating the concentration covariance matrix to the log transmissivity covariance matrix by means of the transport equations and a first order approximation for the uncertainty the simulation model solves the steady state flow equations on a finite element triangular mesh and the transport equations using the backward method of characteristics the resulting nonlinearly constrained optimization problem is solved using the quasi linearity algorithm this algorithm is designed to find a good initial point for the local minimum search when the feasible domain is not convex

cokriged estimation of aquifer transmissivity as an indirect solution of the inverse problem a practical approach a simple approach to indirectly address the inverse problem in steady state for aquifer transmissivity has been studied the data on transmissivity and water level have simultaneously been used to produce the spatial distribution of transrnissivity through cokriging a simple relation between transmissivity and head was used to calculate coherent head covariance and head transmissivity cross covariance based on a chosen transmissivity covariance two theoretical aquifer models were studied one with simple and almost parallel flow and the other with a complex flow and source terms the results were compared with the theoretical values the comparison shows that a better estimation both in terms of the estimated values of transmissivities and in terms of the standard deviation of the estimation errors is obtained when water levels combined with transmissivity are used than in case of the estimation with transmissivity alone provided the groundwater flow direction in a close neighborhood of the estimation point is carefully calculated

simultaneous inversion of hydrogeologic and thermal data 1 theory and application using hydraulic head data one goal in solving the inverse problem in groundwater hydrology is to obtain parameter estimates for use in a simulation model a philosophy about the inverse is adopted here that takes the form of constructing a variety of models under different norms large differences in parameter estimates obtained using different norms suggests the possibility of an incorrect interpretational model an optimization procedure is used that is based on the constrained simplex method this algorithm easily handles constraints does not require derivatives of the functional with respect to the parameters and can be employed to minimize complicated functionals the algorithm produces a cluster of search points in parameter space about the global minimum of the objective function each of these points generates data that matches observed values within a prescribed tolerance the centroid of the search points is taken as the best estimate of the parameters the covariances of the estimated parameters and estimated data can also be computed this inverse technique is applied to a sparse inaccurate hydraulic head data set observed at downie slide british columbia canada for this cross sectional problem with noisy data and a strong drift component interpolation of hydraulic head values using kriging techniques was not possible the domain is represented as a homogeneous isotropic medium four parameters are estimated the free surface position of the water table and three boundary conditions further simulations using a synthetic data set with assumed properties similar to that of downie slide show that with noise free data and an adequate spacing between measurement points it is possible to interpolate an unbiased estimate of hydraulic head data at all nodes in the equivalent discretized domain when the inverse technique is applied the hydraulic conductivity structure is correctly identified the implications for downie slide are that in order to resolve more than a simple hydraulic conductivity structure accurate field measurements of hydraulic head a better spacing between measurements and more detailed knowledge of the boundary conditions are required

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

stochastic analysis of spatial variability in two dimensional steady groundwater flow assuming stationary and nonstationary heads two dimensional steady groundwater flow in a confined aquifer with spatially variable transmissivity t is analyzed stochastically using spectral analysis and the theory of intrinsic random functions conditions that ensure a stationary statistically homogeneous head process are derived and using two convenient forms for the covariance function of the ln t process the head covariance function is studied in addition the head variogram is obtained for a particular nonstationary case and the asymptotic head variogram is derived under very general conditions results are compared to those obtained by gelhar 1976 for one and two dimensional phreatic flow and bakr et al 1978 for one and three dimensional confined flow multidimensional flow analysis results in a significantly reduced head variance the head correlation remains high over much greater distances than the ln t correlation the variogram obtained when stationary heads are assumed is identical to that obtained for nonstationary heads for dimensionless lag distances up to 2 times the correlation scale of the log transmissivity the variogram for nonstationary heads continues to grow logarithmically as lag distance increases independent of the form of the input covariance in the nonstationary case the conditions for stationarity are contrasted with the corresponding results obtained for the one and three dimensional cases of gutjahr and gelhar 1981 the head variance calculated from the stationary theory is found to agree with that of previous monte carlo simulations

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

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

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

review of parameter identification procedures in groundwater hydrology the inverse problem the purpose of this survey is to review parameter identification procedures in groundwater hydrology and to examine computational techniques which have been developed to solve the inverse problem parameter identification methods are classified under the error criterion used in the formulation of the inverse problem the problem of ill posedness in connection with the inverse problem is addressed typical inverse solution techniques are highlighted the review also includes the evaluation of methods used for computing the sensitivity matrix statistics which can be used to estimate the parameter uncertainty are outlined attempts have been made to compare and contrast representative inverse procedures and direction for future research is suggested

a reassessment of the groundwater inverse problem this paper presents a functional formulation of the groundwaterflow inverse problem that is sufficiently general to accommodate most commonly used inverse algorithms unknown hydrogeological properties are assumed to be spatial functions that can be represented in terms of a possibly infinite basis function expansion with random coefficients the unknown parameter function is related to the measurements used for estimation by a forward operator which describes the measurement process in the particular case considered here the parameter of interest is the large scale log hydraulic conductivity the measurements are point values of log conductivity and piezometric head and the forward operator is derived from an upscaled groundwaterflow equation the inverse algorithm seeks the most probable or maximum a posteriori estimate of the unknown parameter function when the measurement errors and parameter function are gaussian and independent the maximum a posteriori estimate may be obtained by minimizing a least squares performance index which can be partitioned into goodness of fit and prior terms when the parameter is a stationary random function the prior portion of the performance index is equivalent to a regularization term which imposes a smoothness constraint on the estimate this constraint tends to make the problem well posed by limiting the range of admissible solutions the gaussian maximum a posteriori problem may be solved with variational methods using functional generalizations of gauss newton or gradient based search techniques several popular groundwater inverse algorithms are either special cases of or variants on the functional maximum a posteriori algorithm these algorithms differ primarily with respect to the way they describe spatial variability and the type of search technique they use linear versus nonlinear the accuracy of estimates produced by both linear and nonlinear inverse algorithms may be measured in terms of a bayesian extension of the cramer rao lower bound on the estimation error covariance this bound suggests how parameter identifiability can be improved by modifying the problem structure and adding new measurements

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

a continuous time inverse operator for groundwater and contaminant transport modeling model identifiability general identifiability conditions are derived for the model coefficients of the spatially discretized continuous time diffusion equation these conditions in turn determine some sufficient and necessary conditions for the existence and uniqueness of the continuous time inverse operator described in a previous article the identifiability conditions are in terms of the initial conditions and the flux vector containing boundary condition information and the eigenspace of the spatially discretized system the derivation of these conditions gives some insight into the location of the solution of the inverse problem in terms of the direct operator under investigation in the space of the laplace transform and limited insight into the stability of the solution near that location

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

depth averaging effects on hydraulic head for media with stochastic hydraulic conductivity hydraulic conductivity of a porous medium frequently is considered to be a single realization of a three dimensional spatial stochastic process the most common observations of flow in porous media are hydraulic head measurements obtained from wells which are screened over extensive sections of the medium these measurements represent approximately a one dimensional spatial average of the actual three dimensional head distribution the actual head distribution being a stochastic process resulting from flow through a random hydraulic conductivity field this paper examines via ensemble averages the effect of such spatial averages of groundwater flow on the spatial autocovariance function for a simple yet viable stochastic model of a bounded medium the model is taken to be three dimensional flow in a medium that is bounded above and below and in which the hydraulic conductivity is a second order stationary stochastic process spatial averaging of the hydraulic heads is considered to take place over the entire thickness of the medium ensemble variances and autocorrelations for depth averaged heads are computed for the resulting two dimensional flow system and compared with those from a fully three dimensional flow system

stochastic modeling of groundwater flow by unconditional and conditional probabilities 1 conditional simulation and the direct problem mathematical modeling of groundwater flow in heterogeneous porous formations of large extent is investigated the formation properties hydraulic conductivity transmissivity as well as flow variables head specific discharge solute concentration are regarded as random variables subjected to uncertainty the main aim of the study is to analyze the influence of conditional probability of input variables upon the statistical structure of the dependent variables the unconditional probability density functions are supposed to be stationary multivariate normal while conditioning accounts for the measured values at a few points of the formation two problems of groundwater flow are discussed in part 1 conditional simulation and the direct problem for steady flow analytical results are obtained by using perturbation approximations average uniform head gradient flows as well as recharge and flow to wells are discussed in unconditional modeling the input variable the conductivity or transmissivity log is regarded as stationary and is represented by its constant mean and variogram the ensemble of formations on which statistical calculations are carried out represents all aquifers with same probability density distributions in conditional modeling the measured values at a few points are kept fixed and uncertainty prevails only at other points consequently statistical computations are performed for the subensemble of aquifers which preserve the measured values and as a result both input and output variables are nonstationary the main effect of conditioning is to reduce the variance i e the uncertainty of these variables this effect is particularly important in the case of flow toward wells application of the modeling method to field problems is outlined