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

multiresponse multilayer vadose zone model calibration using markov chain monte carlo simulation and field water retention data 1 in the past two decades significant progress has been made toward the application of inverse modeling to estimate the water retention and hydraulic conductivity functions of the vadose zone at different spatial scales many of these contributions have focused on estimating only a few soil hydraulic parameters without recourse to appropriately capturing and addressing spatial variability the assumption of a homogeneous medium significantly simplifies the complexity of the resulting inverse problem allowing the use of classical parameter estimation algorithms here we present an inverse modeling study with a high degree of vertical complexity that involves calibration of a 25 parameter richards based hydrus 1d model using in situ measurements of volumetric water content and pressure head from multiple depths in a heterogeneous vadose zone in new zealand we first determine the trade off in the fitting of both data types using the amalgam multiple objective evolutionary search algorithm then we adopt a bayesian framework and derive posterior probability density functions of parameter and model predictive uncertainty using the recently developed differential evolution adaptive metropolis dreamzs adaptive markov chain monte carlo scheme we use four different formulations of the likelihood function each differing in their underlying assumption about the statistical properties of the error residual and data used for calibration we show that amalgam and dreamzs can solve for the 25 hydraulic parameters describing the water retention and hydraulic conductivity functions of the multilayer heterogeneous vadose zone our study clearly highlights that multiple data types are simultaneously required in the likelihood function to result in an accurate soil hydraulic characterization of the vadose zone of interest remaining error residuals are most likely caused by model deficiencies that are not encapsulated by the multilayer model and can not be accessed by the statistics and likelihood function used the utilization of an explicit autoregressive error model of the remaining error residuals does not work well for the water content data with hydrus 1d prediction uncertainty bounds that become unrealistically large

a bayesian mixture modeling approach for flow conditioned multiple point statistical facies simulation from uncertain training images 1 multiple point statistics mps provides a systematic approach for pattern based simulation of complex discrete geologic objects from a conceptual training image ti as prior model the ti contains the general shape geometry and connectivity structures of complex patterns and encodes the related higher order spatial statistics of the expected features conditioning mps simulated facies on flow data poses a challenging nonlinear inverse problem for estimating discrete parameter fields additionally the pattern imitating nature of mps simulation implies that the simulated facies inherit the spatial structure of the features in the ti since tis are constructed from uncertain geologic information and imperfect assumptions the resulting simulated facies may fail to predict the correct flow and transport behavior in the subsurface environment it is therefore prudent to account for the full range of structural variability in describing the geologic facies distribution by considering multiple tis here we present a bayesian mixture model for adaptive and efficient sampling of conditional facies from multiple uncertain tis we partition the posterior distribution of facies into individual conditional densities of the tis and estimate the corresponding mixture weights from the likelihood function for each ti to implement the conditional sampling we apply a recently developed ensemble kalman filter enkf based probability conditioning method whereby enkf is used to invert the flow data and obtain a facies probability map soft data to guide conditional facies simulation from each ti we demonstrate the suitability of the proposed bayesian mixture modeling approach using several numerical experiments in fluvial formations with uncertain orientation and structural connectivity

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

jacquet modules of strongly positive representations of the metaplectic group strongly positive discrete series represent a particularly important class of irreducible square integrable representations of p adic groups indeed these representations are used as basic building blocks in known constructions of general discrete series in this paper we explicitly describe jacquet modules of strongly positive discrete series the obtained description of jacquet modules which relies on the classification of strongly positive discrete series given in 10 is valid in both classical and metaplectic case we expect that our results besides being interesting by themselves should be relevant to some potential applications in the theory of automorphic forms where both representations of metaplectic groups and structure of jacquet modules play an important part

improved parameter inference in catchment models 1 evaluating parameter uncertainty a bayesian methodology is developed to evaluate parameter uncertainty in catchment models fitted to a hydrologic response such as runoff the goal being to improve the chance of successful regionalization the catchment model is posed as a nonlinear regression model with stochastic errors possibly being both autocorrelated and heteroscedastic the end result of this methodology which may use box cox power transformations and arma error models is the posterior distribution which summarizes what is known about the catchment model parameters this can be simplified to a multivariate normal provided a linearization in parameter space is acceptable means of checking and improving this assumption are discussed the posterior standard deviations give a direct measure of parameter uncertainty and study of the posterior correlation matrix can indicate what kinds of data are required to improve the precision of poorly determined parameters finally a case study involving a nine parameter catchment model fitted to monthly runoff and soil moisture data is presented it is shown that use of ordinary least squares when its underlying error assumptions are violated gives an erroneous description of parameter uncertainty

analysis of accuracy of approximate simultaneous nonlinear confidence intervals on hydraulic heads in analytical and numerical test cases inaccuracies in parameter values parameterization stresses and boundary conditions of analytical solutions and numerical models of groundwater flow produce errors in simulated hydraulic heads these errors can be quantified in terms of approximate simultaneous nonlinear confidence intervals presented in the literature approximate confidence intervals can be applied in both error and sensitivity analysis and can be used prior to calibration or when calibration was accomplished by trial and error in this paper the method is expanded for use in numerical problems and the accuracy of the approximate intervals is evaluated using monte carlo runs four test cases are used to compare results in two numerical test cases the approximate simultaneous confidence intervals are generally accurate and are calculated efficiently enough to indicate that the method is useful for real world problems in one of the two analytical test cases considered the approximate intervals are too wide to be of practical interest improving the method used to approximate the uncertainty of the model inputs could make the confidence intervals useful in more problems

role of the calibration process in reducing model predictive error 1 an equation is derived through which the variance of predictive error of a calibrated model can be calculated this equation has two terms the first term represents the contribution to predictive error variance that results from an inability of the calibration process to capture all of the parameterization detail necessary for the making of an accurate prediction if a model is uncalibrated with parameter values being supplied solely through outside information this is the only term required the second term represents the contribution to predictive error variance arising from measurement noise in an overdetermined system such as that which may be obtained through parameter lumping e g through the introduction of a spatial zonation scheme this is the only term required it is shown however that parameter lumping is a form of implicit regularization and that ignoring the implied first term of the predictive error variance equation can potentially lead to underestimation of predictive error variance a model s role as a predictor of environmental behavior can be enhanced if it is calibrated in such a way as to reduce the variance of those predictions which it is required to make it is shown that in some circumstances this can be accomplished through overfitting against historical field data it can also be accomplished by giving greater weight to those measurements which carry the greatest information content with respect to a required prediction this suggests that a departure may be necessary from the custom of using a single calibrated model for the making of many different predictions instead model calibration may need to be repeated many times so that in each case the calibration process is optimized for the making of a specific model prediction

a direct method of parameter estimation for steady state flow in heterogeneous aquifers with unknown boundary conditions 1 we propose a novel direct method for estimating steady state hydrogeological model parameters and model state variables in an aquifer where boundary conditions are unknown the method is adapted from a recently developed potential theory technique for solving general inverse reconstruction problems unlike many inverse techniques used for groundwater model calibration the new method is not based on fitting and optimizing an objective function which usually requires forward simulation and iterative parameter updates instead it directly incorporates noisy observed data hydraulic heads and flow rates at the measurement points in a single step without solving a boundary value problem the new method is computationally efficient and is robust to the presence of observation errors it has been tested on two dimensional groundwater flow problems with regular and irregular geometries different heterogeneity patterns variances of heterogeneity and error magnitudes in all cases parameters hydraulic conductivities converge to the correct or expected values and are thus unique based on which heads and flow fields are constructed directly via a set of analytical expressions accurate boundary conditions are then inferred from these fields the accuracy of the direct method also improves with increasing amount of observed data lower measurement errors and grid refinement under natural flow i e no pumping the direct method yields an equivalent conductivity of the aquifer suggesting that the method can be used as an inexpensive characterization tool with which both aquifer parameters and aquifer boundary conditions can be inferred

inverse groundwater modeling with emphasis on model parameterization 1 this study develops an inverse method aiming to circumvent the subjective decision regarding model parameterization and complexity in inverse groundwater modeling the number of parameters is included as a decision variable along with parameter values a parameterization based on b spline surfaces bss is selected to approximate transmissivity and genetic algorithms were selected to perform error minimization a transform based on linear least squares lls is developed so that different parameterizations may be combined by standard genetic algorithm operators first three applications with isotropic anisotropic and zoned aquifer parameters are examined in a single objective optimization problem and the estimated transmissivity is found to be near the true one interestingly in the anisotropic case the algorithm converged to a solution with an anisotropic distribution of control points next a single objective optimization with regularization penalizing complex models is considered and last the problem is expressed in a multiobjective optimization framework moo where the goals are simultaneous minimization of calibration error and model complexity the result of moo is a pareto set of potential solutions where the user can examine the tradeoffs between calibration error and model complexity and select the most suitable model by comparing calibration with prediction errors it appears that the most promising models are the ones near a region where the rate of decrease of calibration error as model complexity increases drops bend of error curve this is a useful result of practical interest in real inverse modeling applications

strongly positive representations of metaplectic groups abstract in this paper we obtain a classification of irreducible strongly positive square integrable genuine representations of metaplectic groups over p adic fields using a purely algebraic approach our results parallel those of m glin and tadic for classical groups but their work relies on certain conjectures on the other hand our results are complete and there are no additional conditions or hypotheses the important point to note here is that our results and techniques can be used in the case of classical p adic groups in a completely analogous manner

strongly positive representations of gspin_ 2n 1 and the jacquet module method we explicitly construct the structure of jacquet modules of parabolically induced representations of gspin 2n 1 over a p adic field f of any characteristic using this construction of the jacquet module we obtain a classification of strongly positive representations of gspin 2n 1 over f and describe the general discrete series representations of gspin 2n 1 over f assuming the half integer conjecture one application of this paper is the proof of the equality of l functions from the langlands shahidi method and artin l functions through the local langlands correspondence kim in langlands shahidi l functions for gspin groups and the generic arthur packet conjecture preprint

core scale solute transport model selection using monte carlo analysis 1 model applicability to core scale solute transport is evaluated using breakthrough data from column experiments conducted with conservative tracers tritium 3h and sodium 22 22na and the retarding solute uranium 232 232u the three models considered are single porosity double porosity with single rate mobile immobile mass exchange and the multirate model which is a deterministic model that admits the statistics of a random mobile immobile mass exchange rate coefficient the experiments were conducted on intact culebra dolomite core samples previously data were analyzed using single porosity and double porosity models although the culebra dolomite is known to possess multiple types and scales of porosity and to exhibit multirate mobile immobile domain mass transfer characteristics at field scale the data are reanalyzed here and null space monte carlo analysis is used to facilitate objective model selection prediction or residual bias is adopted as a measure of the model structural error the analysis clearly shows single porosity and double porosity models are structurally deficient yielding late time residual bias that grows with time on the other hand the multirate model yields unbiased predictions consistent with the late time 5 2 slope diagnostic of multirate mass transfer the analysis indicates the multirate model is better suited to describing core scale solute breakthrough in the culebra dolomite than the other two models

a full bayesian approach to parameter inference from tracer travel time moments and investigation of scale effects at the cape cod experimental site a method for inverting the travel time moments of solutes in heterogeneous aquifers is presented and is based on peak concentration arrival times as measured at various samplers in an aquifer the approach combines a lagrangian rubin and dagan 1992 solute transport framework with full bayesian hydrogeological parameter inference in the full bayesian approach the noise values in the observed data are treated as hyperparameters and their effects are removed by marginalization the prior probability density functions pdfs for the model parameters horizontal integral scale velocity and log k variance and noise values are represented by prior pdfs developed from minimum relative entropy considerations analysis of the cape cod massachusetts field experiment is presented inverse results for the hydraulic parameters indicate an expected value for the velocity variance of log hydraulic conductivity and horizontal integral scale of 0 42 m d 0 26 and 3 0 m respectively while these results are consistent with various direct field determinations the importance of the findings is in the reduction of confidence range about the various expected values on selected control planes we compare observed travel time frequency histograms with the theoretical pdf conditioned on the observed travel time moments we observe a positive skew in the travel time pdf which tends to decrease as the travel time distance grows we also test the hypothesis that there is no scale dependence of the integral scale with the scale of the experiment at cape cod we adopt two strategies the first strategy is to use subsets of the full data set and then to see if the resulting parameter fits are different as we use different data from control planes at expanding distances from the source the second approach is from the viewpoint of entropy concentration no increase in integral scale with distance is inferred from either approach over the range of the cape cod tracer experiment

a hybrid regularized inversion methodology for highly parameterized environmental models 1 a hybrid approach to the regularized inversion of highly parameterized environmental models is described the method is based on constructing a highly parameterized base model calculating base parameter sensitivities and decomposing the base parameter normal matrix into eigenvectors representing principal orthogonal directions in parameter space the decomposition is used to construct super parameters super parameters are factors by which principal eigenvectors of the base parameter normal matrix are multiplied in order to minimize a composite least squares objective function these eigenvectors define orthogonal axes of a parameter subspace for which information is available from the calibration data the coordinates of the solution are sought within this subspace super parameters are estimated using a regularized nonlinear gauss marquardt levenberg scheme though super parameters are estimated tikhonov regularization constraints are imposed on base parameters tikhonov regularization mitigates over fitting and promotes the estimation of reasonable base parameters use of a large number of base parameters enables the inversion process to be receptive to the information content of the calibration data including aspects pertaining to small scale parameter variations because the number of super parameters sustainable by the calibration data may be far less than the number of base parameters used to define the original problem the computational burden for solution of the inverse problem is reduced the hybrid methodology is described and applied to a simple synthetic groundwater flow model it is then applied to a real world groundwater flow and contaminant transport model the approach and programs described are applicable to a range of modeling disciplines copyright 2005 by the american geophysical union

efficient regularization and uncertainty analysis using a global optimization methodology 1 the concept of the pareto front has received considerable attention in the model calibration literature particularly in conjunction with global search optimizers that have been developed for use in contexts where objective function surfaces are pitted with local optima or characterized by multiple broad regions of attraction in parameter space in this paper use of the pareto concept in such calibration contexts is extended to include regularization and model predictive uncertainty analysis both of these processes can be formulated as constrained optimization problems in which a trade off is analyzed between a set of constraints on model parameters on the one hand and maximization minimization of one or a number of model outputs of interest on the other hand in both cases the optimal trade off point though being calculable on a theoretical basis for synthetic cases must be chosen subjectively when working with real world models two cases are presented to illustrate the methodology one a synthetic groundwater model and the other a real world surface water model

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

optimization and uncertainty assessment of strongly nonlinear groundwater models with high parameter dimensionality 1 highly parameterized and cpu intensive groundwater models are increasingly being used to understand and predict flow and transport through aquifers despite their frequent use these models pose significant challenges for parameter estimation and predictive uncertainty analysis algorithms particularly global methods which usually require very large numbers of forward runs here we present a general methodology for parameter estimation and uncertainty analysis that can be utilized in these situations our proposed method includes extraction of a surrogate model that mimics key characteristics of a full process model followed by testing and implementation of a pragmatic uncertainty analysis technique called null space monte carlo nsmc that merges the strengths of gradient based search and parameter dimensionality reduction as part of the surrogate model analysis the results of nsmc are compared with a formal bayesian approach using the differential evolution adaptive metropolis dream algorithm such a comparison has never been accomplished before especially in the context of high parameter dimensionality despite the highly nonlinear nature of the inverse problem the existence of multiple local minima and the relatively large parameter dimensionality both methods performed well and results compare favorably with each other experiences gained from the surrogate model analysis are then transferred to calibrate the full highly parameterized and cpu intensive groundwater model and to explore predictive uncertainty of predictions made by that model the methodology presented here is generally applicable to any highly parameterized and cpu intensive environmental model where efficient methods such as nsmc provide the only practical means for conducting predictive uncertainty analysis

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

on an algebraic approach to the zelevinsky classification for classical p adic groups abstract in this paper we classify irreducible admissible representations of p adic classical groups in a purely algebraic way we present a new and simple algorithm for constructing the classifying data which is also useful in other contexts where the computation with jacquet modules or constant terms of eisenstein series appears we construct and study non standard intertwining operators they are defined in a purely algebraic way although their rationality follows from g muic a geometric construction of intertwining operators for reductive p adic groups manuscripta math 125 2008 241 272

on regular square integrable representations of p adic groups we study the problem of construction of noncuspidal irreducible square integrable representations of classical p adic groups and questions related to it starting from cuspidal reducibilities in the generalized rank one case we give a construction of regular irreducible square integrable representations of p adic gsp n sp n and so 2 n 1 under an assumption expected to hold in general we show that all such representations should come from this construction further we obtain a number of general results about places where square integrable representations can appear in parabolically induced representation among others we show that in irreducible cuspidal representations of general linear groups only the self contragredient ones play a role in the construction of square integrable representations of the above groups at the end we use our regular square integrable representations to study some basic properties of whittaker models we also show that there exists nondegenerate standard modules of classical groups which contain degenerate irreducible subrepresentations

a markov chain monte carlo scheme for parameter estimation and inference in conceptual rainfall runoff modeling a fully bayesian approach to parameter estimation and inference in conceptual rainfall runoff models crrms is presented computations are performed using a markov chain monte carlo mcmc method based on the metropolis hastings algorithm single site and block updating schemes are used for model parameters subject to nonnegativity restrictions as well as interval equality and order constraints diagnostics for the convergence of the markov chain and crrm assessment are also considered the mcmc approach produces samples from the joint posterior distribution of the model parameters this provides more information than single point estimates and avoids the need to use a normal approximation to the posterior as the basis for inference the methodology is demonstrated using an eight parameter conceptual rainfall runoff model and two case studies from southeastern australia the first case study considers a watershed with high runoff yield over a 12 year period the second case study considers a watershed with low yield over a 17 year period the results indicate that 1 bayesian methods provide an objective framework for model criticism and choice 2 the proposed strategies for handling constraints on model parameters are effective 3 the model parameters are sensitive to likelihood function selection 4 the conventional approach of using a power transformation and an autoregressive process to stabilize error variance and model dependence in the residuals may have limited success and 5 some care is required in the implementation of the mcmc approach and reliable results will be difficult to obtain when crrm complexity exceeds the limitations of the rainfall runoff data at hand a key finding is that the mcmc scheme presented herein provides a powerful means of identifying specific inadequacies in the structure of crrms

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

impact of numerical artifact of the forward model in the inverse solution of density dependent flow problem we investigate the effect of computational error on the inversion of a density dependent flow and transport model using seawat and ucode 2005 in an inverse identification of hydraulic conductivity and dispersivity using head and concentration data from a 2 d laboratory experiment we investigated inversions using three different solution schemes including variation of number of particles and time step length in terms of the three aspects the shape and smoothness of the objective function surface the consequent impacts to the optimization and the resulting pareto analyses this study demonstrates that the inversion is very sensitive to the choice of the forward model solution scheme in particular standard finite difference methods provide the smoothest objective function surface however this is obtained at the cost of numerical artifacts that can lead to erroneous warping of the objective function surface total variation diminishing tvd schemes limit these impacts at the cost of more computation time while the hybrid method of characteristics hmoc approach with increased particle numbers and or reduced time step gives both smoothed and accurate objective function surface use of the most accurate methods tvd and hmoc did lead to successful inversion of the two parameters however with distinct results for pareto analyses these results illuminate the sensitivity of the inversion to a number of aspects of the forward solution of the density driven flow problem and reveal that parameter values may result that are erroneous but that counteract numerical errors in the solution

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

strongly positive subquotients in a class of induced representations of classical p adic groups we determine under which conditions the induced representation of the form 1 2 where 1 2 are irreducible essentially square integrable representations of a general linear group and is a discrete series representation of classical p adic group contains an irreducible strongly positive subquotient

the zelevinsky classification of unramified representations of the metaplectic group abstract in this paper a zelevinsky type classification of genuine unramified irreducible representations of the metaplectic group over a p adic field with p 2 is obtained the classification consists of three steps firstly it is proved that every genuine irreducible unramified representation is a fully parabolically induced representation from unramified characters of general linear groups and a genuine irreducible negative unramified representation of a smaller metaplectic group genuine irreducible negative unramified representations are described in terms of parabolic induction from unramified characters of general linear groups and a genuine irreducible strongly negative unramified representation of a smaller metaplectic group finally genuine irreducible strongly negative unramified representations are classified in terms of jordan blocks the main technical tool is the theory of jacquet modules

an iterative stochastic inverse method conditional effective transmissivity and hydraulic head fields an iterative stochastic approach is developed to estimate transmissivity and head distributions in heterogeneous aquifers this approach is similar to the classical cokriging technique it uses a linear estimator that depends on the covariances of transmissivity and hydraulic head and their cross covariance the linear estimator is however improved successively by solving the governing flow equation and by updating the covariances and cross covariance function of transmissivity and hydraulic head fields in an iterative manner as a result the nonlinear relationship between transmissivity and head is incorporated in the estimation and the estimated fields are approximate conditional means the ability of the iterative approach is tested with some deterministic and stochastic inverse problems the results show that the estimated transmissivity and hydraulic head fields have smaller mean square errors than those obtained by classical cokriging even in the aquifer with variance of transmissivity up to 3

steady flows of inviscid fluids under localized perturbations on essaie d obtenir tous les ecoulements stationnaires dans un canal avec un petit obstacle pour des nombres de froude proches de la valeur critique fo

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

a generalized grid connectivity based parameterization for subsurface flow model calibration 1 we develop a novel method of parameterization for spatial hydraulic property characterization to mitigate the challenges associated with the nonlinear inverse problem of subsurface flow model calibration the parameterization is performed by the projection of the estimable hydraulic property field onto an orthonormal basis derived from the grid connectivity structure the basis functions represent the modal shapes or harmonics of the grid are defined by a modal frequency and converge to special cases of the discrete fourier series under certain grid geometries and boundary assumptions therefore hydraulic property updates are performed in the spectral domain and merge with fourier analysis in ideal cases dependence on the grid alone implies that the basis may characterize any grid geometry including corner point and unstructured is model independent and is constructed off line and only once prior to flow data assimilation we apply the parameterization in an adaptive multiscale model calibration workflow for three subsurface flow models several different grid geometries are considered in each case the prior hydraulic property model is updated using a parameterized multiplier field that is superimposed onto the grid and assigned an initial value of unity at each cell the special case corresponding to a constant multiplier is always applied through the constant basis function higher modes are adaptively employed during minimization of data misfit to resolve multiscale heterogeneity in the geomodel the parameterization demonstrates selective updating of heterogeneity at locations and spatial scales sensitive to the available data otherwise leaving the prior model unchanged as desired

two dimensional travelling waves over a moving bottom abstract two dimensional travelling waves on an ideal fluid with gravity and surface tension over a periodically moving bottom with a small amplitude are studied the bottom and the wave travel with a same speed the exact euler equations are formulated as a spatial dynamic system by using the stream function a manifold reduction technique is applied to reduce the system into one of ordinary differential equations with finite dimensions a homoclinic solution to the normal form of this reduced system persists when higher order terms are added which gives a generalized solitary wave the homoclinic solution connecting a periodic solution

the minimum structure solution to the inverse problem the inverse problem of estimating the conductivity function from head observations is generally ill posed many conductivity functions are consistent with the data it is widely accepted now that a well defined estimate can be obtained only if additional information about the function structure is introduced into the problem formulation this work presents a method to obtain a stable and reasonable estimate that utilizes only the data and the flow or transport model with the minimum possible suppositions about the unknown function or its structure the motivation is to develop a solution that has only characteristics that are traced directly to the data and the flow or transport model without taking advantage of spatial continuity or other prior information the solution is obtained by minimizing the upper bound to the error or in a stochastic conceptual framework as the most likely solution given the data this solution although generally not the most accurate since it neglects to utilize structural information that may be available is of fundamental importance and may be useful as a benchmark for example by comparing this solution with other solutions one can become aware of how prior information or the model of spatial structure affects the solution to the inverse problem

on the non unitary unramified dual for classical p adic groups in this paper we give a zelevinsky type classification of unramified irreducible representations of split classical groups

construction of discrete series for classical p adic groups in this paper the authors complete the classification of irreducible square integrable representations of classical p adic groups assuming a natural assumption which is expected to hold in general this classification implies a parameterization of irreducible tempered representations of these groups and it implies a classification of the non unitary duals of these groups modulo cuspidal data

exact theory of secondary supercritical solutions for steady surface waves over a bump abstract two dimensional steady surface waves on an ideal fluid over a small and symmetric bump are studied when the froude number f of the uniform flow far upstream is greater than unity recent numerical results for a symmetric bump indicate that corresponding to f 1 there may exist two solutions for the problem one approaches the uniform flow and the other a solitary wave as the bump size goes to zero in this paper we use a two parameter asymptotic expansion to derive an explicit approximate expression for the later solution and show that the expansion is indeed an asymptotic approximation to the solution of the exact nonlinear equations

evaluation of prediction intervals for expressing uncertainties in groundwater flow model predictions we tested the accuracy of 95 individual prediction intervals for hydraulic heads streamflow gains and effective transmissivities computed by groundwater models of two danish aquifers to compute the intervals we assumed that each predicted value can be written as the sum of a computed dependent variable and a random error testing was accomplished by using a cross validation method and by using new field measurements of hydraulic heads and transmissivities that were not used to develop or calibrate the models the tested null hypotheses are that the coverage probability of the prediction intervals is not significantly smaller than the assumed probability 95 and that each tail probability is not significantly different from the assumed probability 2 5 in all cases tested these hypotheses were accepted at the 5 level of significance we therefore conclude that for the groundwater models of two real aquifers the individual prediction intervals appear to be accurate

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

direct secant updates of matrix factorizations this paper presents a new context for using the sparse broyden update method to solve systems of nonlinear equations the setting for this work is that a newton like algorithm is assumed to be available which incorporates a workable strategy for improving poor initial guesses and providing a satisfactory jacobian matrix approximation whenever required the total cost of obtaining each jacobian matrix or the cost of factoring it to solve for the newton step is assumed to be sufficiently high to make it attractive to keep the same jacobian approximation for several steps this paper suggests the extremely convenient and apparently effective technique of applying the sparse broyden update directly to the matrix factors in the iterations between reevaluations in the hope that fewer fresh factorizations will be required the strategy is shown to be locally and q superlinearly convergent and some encouraging numerical results are presented

parabolic induction and jacquet functors for metaplectic groups abstract in this paper we give a new formulation of a geometric lemma for the metaplectic groups over p adic field f analogous to the existing one for classical groups this enables us to give a zelevinsky type classification of irreducible admissible genuine representations of metaplectic groups as an application of our jacquet module technique we explicitly calculate jacquet modules of even and odd weil representations

simultaneous confidence and prediction intervals for nonlinear regression models with application to a groundwater flow model methods are presented for computing three types of simultaneous confidence and prediction intervals exact likelihood ratio and linearized on output from nonlinear regression models with normally distributed residuals the confidence intervals can be placed on individual regression parameters or on the true regression function at any number of points in the domain of the independent variables and the prediction intervals can be placed on any number of future observations the confidence intervals are analogous to simultaneous scheffe intervals for linear models and the prediction intervals are analogous to the prediction intervals of hahn 1972 all three types of intervals can be computed efficiently by using the same straightforward lagrangian optimization scheme the prediction intervals can be treated in the same computational framework as the confidence intervals by including the random errors as pseudoparameters in the lagrangian scheme the methods are applied to a hypothetical groundwater model for flow to a well penetrating a leaky aquifer three different data sets are used to demonstrate the effect of sampling strategies on the intervals for all three data sets the linearized confidence intervals are inferior to the exact and likelihood ratio intervals with the latter two being very similar however all three types of prediction intervals yielded similar results the third data set time drawdown data at only a single observation well points out many of the problems that can arise from extreme nonlinear behavior of the regression model

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

posterior population expansion for solving inverse problems solving inverse problems in a complex geologically realistic and discrete model space and from a sparse set of observations is a very challenging task extensive exploration by markov chain monte carlo mcmc methods often results in considerable computational efforts most optimization methods on the other hand are limited to linear continuous model spaces and the minimization of an objective function what often proves to be insufficient to overcome these problems we propose a new ensemble based exploration scheme for geostatistical prior models generated by a multiple point statistics mps tool the principle of our method is to expand an existing set of models by using posterior facies information for conditioning new mps realizations the algorithm is independent of the physical parametrization it is tested on a simple synthetic inverse problem when compared to two existing mcmc methods iterative spatial resampling isr and interrupted markov chain monte carlo imcmc the required number of forward model runs was divided by a factor of 8 12

a bayesian approach to stochastic capture zone delineation incorporating tracer arrival times conductivity measurements and hydraulic head observations this paper presents a methodology to invert tracer arrival times and to incorporate travel time data in the delineation of well capture zones within a bayesian framework the observed arrival times are used to obtain probability based weights for each realization of the hydraulic conductivity field realizations that closely reproduce the observed arrival times are more likely to represent the true or real conductivity field than realizations yielding worse predictions and are consequently characterized by a higher conditional probability in the prediction of the capture zones the realizations are weighted by their respective probability we combine the arrival time data with conductivity measurements and head observations to delineate stochastic capture zones the conductivity measurements update the prior distributions specified for the unknown structural parameters of the conductivity field and are used as conditioning data in the generation of conditional conductivity fields the parameter values used to generate the conductivity realizations are sampled by monte carlo from the updated parameter distributions the head and travel time observations are subsequently used to assign probability based weights to the conductivity realizations by applying bayes theorem the hyperparameters of the error model are assumed unknown and their effect is accounted for by marginalization a synthetic flow setup consisting of a single extraction well in a fully confined aquifer under a regional gradient is used to illustrate the method we evaluate the relative worth of the different data types by introducing them sequentially in the inverse methodology results indicate that the different data types are complementary and that the travel time data are essential to improve the predictions of the capture zones this is confirmed by the results for the case where uncertainty in the homogeneous porosity is accounted for

occam s inversion to generate smooth two dimensional models from magnetotelluric data magnetotelluric mt data are inverted for smooth 2 d models using an extension of the existing 1 d algorithm occam s inversion since an mt data set consists of a finite number of imprecise data an infinity of solutions to the inverse problem exists fitting field or synthetic electromagnetic data as closely as possible results in theoretical models with a maximum amount of roughness or structure however by relaxing the misfit criterion only a small amount models which are maximally smooth may be generated smooth models are less likely to result in overinterpretation of the data and reflect the true resolving power of the mt method the models are composed of a large number of rectangular prisms each having a constant conductivity apriori information in the form of boundary locations only or both boundary locations and conductivity may be included providing a powerful tool for improving the resolving power of the data joint inversion of te and tm synthetic data generated from known models al

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

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