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

analysis of hydraulic tomography using temporal moments of drawdown recovery data received 2 june 2005 revised 30 september 2005 accepted 21 october 2005 published 1 february 2006 1 transient hydraulic tomography tht is a potentially cost effective and highresolution technique for mapping spatial distributions of the hydraulic conductivity and specific storage in aquifers interpretation of abundant well hydrographs of a tht survey however is a computational challenge we take on this challenge by developing an estimation approach that utilizes the zeroth and first temporal moments of well hydrographs instead of drawdown itself the governing equations for the temporal moments are poisson s equations these equations demand less computational resources as opposed to the parabolic equation that governs drawdown evolution likewise the adjoint equations for evaluating sensitivities of the moments for parameter estimation also take the same forms therefore a temporal moment approach is expected to expedite the interpretation of tht surveys on the basis of this premise we extend our sequential successive linear estimator to use the zeroth moment and characteristic time of the drawdown recovery data generated by tht surveys we subsequently investigate computational efficiency and accuracy of the moment approach results of the investigation show that the temporal moment approach yields results similar to those from the approach that uses transient heads but at significantly less computational costs limitations using temporal moments are discussed subsequently

principal component geostatistical approach for large dimensional inverse problems the quasi linear geostatistical approach is for weakly nonlinear underdetermined inverse problems such as hydraulic tomography and electrical resistivity tomography it provides best estimates as well as measures for uncertainty quantification however for its textbook implementation the approach involves iterations to reach an optimum and requires the determination of the jacobian matrix i e the derivative of the observation function with respect to the unknown although there are elegant methods for the determination of the jacobian the cost is high when the number of unknowns m and the number of observations n is high it is also wasteful to compute the jacobian for points away from the optimum irrespective of the issue of computing derivatives the computational cost of implementing the method is generally of the order of m2n though there are methods to reduce the computational cost in this work we present an implementation that utilizes a matrix free in terms of the jacobian matrix gauss newton method and improves the scalability of the geostatistical inverse problem for each iteration it is required to perform k runs of the forward problem where k is not just much smaller than m but can be smaller that n the computational and storage cost of implementation of the inverse procedure scales roughly linearly with m instead of m2 as in the textbook approach for problems of very large m this implementation constitutes a dramatic reduction in computational cost compared to the textbook approach results illustrate the validity of the approach and provide insight in the conditions under which this method perform best

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

temporal moments revisited why there is no better way for physically based model reduction in time 1 many hydro geo logical problems are highly complex in space and time coupled with scale issues variability and uncertainty especially time dependent models often consume enormous computational resources but model reduction techniques can alleviate this problem temporal moments tm offer an approach to reduce the time demands of transient hydro geo logical simulations tm reduce transient governing equations to steady state and directly simulate the temporal characteristics of the system if the equations are linear and coefficients are time independent this is achieved by an integral transform projecting the dynamic system response onto monomials in time in comparison to classical approaches of model reduction that involve orthogonal base functions however the monomials for tm are nonorthogonal which might impair the quality and efficiency of model reduction thus we raise the question of whether there are more suitable temporal base functions than the monomials that lead to tm in this work we will derive theoretically that there is only a limited class of temporal base functions that can reduce hydro geo logical models by comparing those to tm we conclude that in terms of gained efficiency versus maintained accuracy tm are the best possible choice while our theoretical results hold for all systems of linear partial or ordinary differential equations pdes odes with any order of space and time derivatives we illustrate our study with an example of pumping tests in a confined aquifer for that case we demonstrate that two four tm are sufficient to represent more than 80 90 of the dynamic behavior and that the information content strictly increases with increasing tm order

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

a strategy for improved computational efficiency of the method of anchored distributions 1 this paper proposes a strategy for improving the computational efficiency of model inversion using the method of anchored distributions mad by bundling similar model parametrizations in the likelihood function inferring the likelihood function typically requires a large number of forward model fm simulations for each possible model parametrization as a result the process is quite expensive to ease this prohibitive cost we present an approximation for the likelihood function called bundling that relaxes the requirement for high quantities of fm simulations this approximation redefines the conditional statement of the likelihood function as the probability of a set of similar model parametrizations bundle replicating field measurements which we show is neither a model reduction nor a sampling approach to improving the computational efficiency of model inversion to evaluate the effectiveness of these modifications we compare the quality of predictions and computational cost of bundling relative to a baseline mad inversion of 3 d flow and transport model parameters additionally to aid understanding of the implementation we provide a tutorial for bundling in the form of a sample data set and script for the r statistical computing language for our synthetic experiment bundling achieved a 35 reduction in overall computational cost and had a limited negative impact on predicted probability distributions of the model parameters strategies for minimizing error in the bundling approximation for enforcing similarity among the sets of model parametrizations and for identifying convergence of the likelihood function are also presented

lie symmetries and group classification of a class of time fractional evolution systems in this paper the lie group method is developed to study a class of time fractional evolution systems as an application the complete group classification of a system of two component fractional evolution equations with an arbitrary function is performed the admitted symmetries of all inequivalent systems belonging to the class under study are determined what is more all reduced fractional systems of ordinary differential equations and even explicit exact solutions are presented

assessment of parametric uncertainty for groundwater reactive transport modeling the validity of using gaussian assumptions for model residuals in uncertainty quantification of a groundwater reactive transport model was evaluated in this study least squares regression methods explicitly assume gaussian residuals and the assumption leads to gaussian likelihood functions model parameters and model predictions while the bayesian methods do not explicitly require the gaussian assumption gaussian residuals are widely used this paper shows that the residuals of the reactive transport model are non gaussian heteroscedastic and correlated in time characterizing them requires using a generalized likelihood function such as the formal generalized likelihood function developed by schoups and vrugt 2010 for the surface complexation model considered in this study for simulating uranium reactive transport in groundwater parametric uncertainty is quantified using the least squares regression methods and bayesian methods with both gaussian and formal generalized likelihood functions while the least squares methods and bayesian methods with gaussian likelihood function produce similar gaussian parameter distributions the parameter distributions of bayesian uncertainty quantification using the formal generalized likelihood function are non gaussian in addition predictive performance of formal generalized likelihood function is superior to that of least squares regression and bayesian methods with gaussian likelihood function the bayesian uncertainty quantification is conducted using the differential evolution adaptive metropolis dream zs algorithm as a markov chain monte carlo mcmc method it is a robust tool for quantifying uncertainty in groundwater reactive transport models for the surface complexation model the regression based local sensitivity analysis and morris and dream zs based global sensitivity analysis yield almost identical ranking of parameter importance the uncertainty analysis may help select appropriate likelihood functions improve model calibration and reduce predictive uncertainty in other groundwater reactive transport and environmental modeling

effects of error covariance structure on estimation of model averaging weights and predictive performance 1 when conducting model averaging for assessing groundwater conceptual model uncertainty the averaging weights are often evaluated using model selection criteria such as aic aicc bic and kic akaike information criterion corrected akaike information criterion bayesian information criterion and kashyap information criterion respectively however this method often leads to an unrealistic situation in which the best model receives overwhelmingly large averaging weight close to 100 which cannot be justified by available data and knowledge it was found in this study that this problem was caused by using the covariance matrix c of measurement errors for estimating the negative log likelihood function common to all the model selection criteria this problem can be resolved by using the covariance matrix cek of total errors including model errors and measurement errors to account for the correlation between the total errors an iterative two stage method was developed in the context of maximum likelihood inverse modeling to iteratively infer the unknown cek from the residuals during model calibration the inferred cek was then used in the evaluation of model selection criteria and model averaging weights while this method was limited to serial data using time series techniques in this study it can be extended to spatial data using geostatistical techniques the method was first evaluated in a synthetic study and then applied to an experimental study in which alternative surface complexation models were developed to simulate column experiments of uranium reactive transport it was found that the total errors of the alternative models were temporally correlated due to the model errors the iterative two stage method using cek resolved the problem that the best model receives 100 model averaging weight and the resulting model averaging weights were supported by the calibration results and physical understanding of the alternative models using cek obtained from the iterative two stage method also improved predictive performance of the individual models and model averaging in both synthetic and experimental studies

assessing the impacts of parameter uncertainty for computationally expensive groundwater models 1 computational requirements often limit the assessment of uncertainty in complex environmental models monte carlo based techniques such as generalized likelihood uncertainty estimation glue and bayesian uncertainty assessment are limited in applicability because of their heavy computational demands this paper presents a computationally efficient method that combines automatic calibration using function approximation based global optimization with importance weighting for assessing parametric uncertainty in model outputs for computationally expensive models we call this method automatic calibration and uncertainty assessment using response surfaces acuars the bias introduced during optimization because of oversampling of high goodness of fit regions of the parameter space is corrected using a multidimensional cell declustering technique the proposed method is applied to two hypothetical examples in enhanced bioremediation of chlorinated ethenes in groundwater that differ in computational requirements and the number of parameters that were calibrated the results from the simpler example in which uncertainty of three model parameters in model outputs was considered showed that for a given acceptability threshold and a given number of permissible model simulations acuars finds many more representative parameter sets than glue glue produced only half the number of acceptable parameter sets for uncertainty assessment even after 10 times as many simulations as acuars was performed for glue further the median output of acuars predicted the true value of the outputs more closely than glue for a more complicated numerical example where noisy data were used for calibration of seven model parameters glue required 2400 simulations to find 31 acceptable parameter sets while acuars found 46 acceptable parameter sets in only 300 simulations for a moderately stringent threshold this enabled a more accurate estimate of uncertainty by providing closer estimates of the true outputs

minimum relative entropy forward probabilistic modeling the pioneering work of jaynes in bayesian maximum entropy methods has been successfully explored in many disciplines the principle of maximum entropy pme is a powerful and versatile tool of inferring a probability distribution from constraints that do not completely characterize that distribution minimum relative entropy mre is a method which has all the important attributes of the maximum entropy approach with the advantage that prior information may be easily included in this paper we use mre to determine the prior probability density function pdf of a set of model parameters based on limited information the resulting pdf is used in monte carlo simulations to provide expected values in field variables such as concentration and confidence limits we compare the probabilistic results from a traditional advection dispersion ade model based on volumetric averaging concepts with that of a model based on the assumption that the hydraulic conductivity is a stationary stochastic process the results suggest that although naff s 1990 model satisfies the observed data to a better degree than ade model the upper and lower confidence bands about the mean value are larger than the ade results this result we attribute to the fact that naff s 1990 model simply contains more parameters each of which is unknown and has to be estimated there is no statistical difference between the expected values of second spatial moments for the two models the examples presented in this paper illustrate problems associated with assigning gaussian pdfs as priors in a probabilistic model first such an assumption for the input parameters actually injects more information into the problem than may actually exist whether consciously or unconsciously this fact is borne out by comparison with minimum relative entropy theory second the output parameters as suggested from the monte carlo analysis cannot be assumed to be gaussian distributed even when the prior pdf is gaussian in form in a practical setting the significance of this result and the approximation of gaussian form would depend on the toxicity and environmental standards that apply to the problem

a glimm effros dichotomy for borel equivalence relations a basic dichotomy concerning the structure of the orbit space of a transformation group has been discovered by glimm g12 in the locally compact group action case and extended by effros e 1 e2 in the polish group action case when additionally the induced equivalence relation is f it is the purpose of this paper to extend the glimm effros dichotomy to the very general context of an arbitrary borel equivalence relation not even necessarily induced by a group action despite the totally classical descriptive set theoretic nature of our result our proof requires the employment of methods of effective descriptive set theory and thus ultimately makes crucial use of computability or recursion theory on the integers

effective and efficient global optimization for conceptual rainfall runoff models the successful application of a conceptual rainfall runoff crr model depends on how well it is calibrated despite the popularity of crr models reports in the literature indicate that it is typically difficult if not impossible to obtain unique optimal values for their parameters using automatic calibration methods unless the best set of parameters associated with a given calibration data set can be found it is difficult to determine how sensitive the parameter estimates and hence the model forecasts are to factors such as input and output data error model error quantity and quality of data objective function used and so on results are presented that establish clearly the nature of the multiple optima problem for the research crr model sixpar these results suggest that the crr model optimization problem is more difficult than had been previously thought and that currently used local search procedures have a very low probability of successfully finding the optimal parameter sets next the performance of three existing global search procedures are evaluated on the model sixpar finally a powerful new global optimization procedure is presented entitled the shuffled complex evolution sce ua method which was able to consistently locate the global optimum of the sixpar model and appears to be capable of efficiently and effectively solving the crr model optimization problem

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

recent developments in nonparametric density estimation advances in computation and the fast and cheap computational facilities now available to statisticians have had a significant impact upon statistical research and especially the development of nonparametric data analysis procedures in particular theoretical and applied research on nonparametric density estimation has had a noticeable influence on related topics such as nonparametric regression nonparametric discrimination and nonparametric pattern recognition this article reviews recent developments in nonparametric density estimation and includes topics that have been omitted from review articles and books on the subject the early density estimation methods such as the histogram kernel estimators and orthogonal series estimators are still very popular and recent research on them is described different types of restricted maximum likelihood density estimators including order restricted estimators maximum penalized likelihood estimators and sieve estimators are discussed where restrictions are imposed upon the class of densities or on the form of the likelihood function nonparametric density estimators that are data adaptive and lead to locally smoothed estimators are also discussed these include variable partition histograms estimators based on statistically equivalent blocks nearest neighbor estimators variable kernel estimators and adaptive kernel estimators for the multivariate case extensions of methods of univariate density estimation are usually straightforward but can be computationally expensive a method of multivariate density estimation that did not spring from a univariate generalization is described namely projection pursuit density estimation in which both dimensionality reduction and density estimation can be pursued at the same time finally some areas of related research are mentioned such as nonparametric estimation of functionals of a density robust parametric estimation semiparametric models and density estimation for censored and incomplete data directional and spherical data and density estimation for dependent sequences of observations

essential countability of treeable equivalence relations we establish a dichotomy theorem characterizing the circumstances under which a treeable borel equivalence relation e is essentially countable under additional topological assumptions on the treeing we in fact show that e is essentially countable if and only if there is no continuous embedding of e1 into e our techniques also yield the first classical proof of the analogous result for hypersmooth equivalence relations and allow us to show that up to continuous kakutani embeddability there is a minimum borel function which is not essentially countable to one

an adaptive sparse grid high order stochastic collocation method for bayesian inference in groundwater reactive transport modeling 1 bayesian analysis has become vital to uncertainty quantification in groundwater modeling but its application has been hindered by the computational cost associated with numerous model executions required by exploring the posterior probability density function ppdf of model parameters this is particularly the case when the ppdf is estimated using markov chain monte carlo mcmc sampling in this study a new approach is developed to improve the computational efficiency of bayesian inference by constructing a surrogate of the ppdf using an adaptive sparse grid high order stochastic collocation asg hsc method unlike previous works using first order hierarchical basis this paper utilizes a compactly supported higher order hierarchical basis to construct the surrogate system resulting in a significant reduction in the number of required model executions in addition using the hierarchical surplus as an error indicator allows locally adaptive refinement of sparse grids in the parameter space which further improves computational efficiency to efficiently build the surrogate system for the ppdf with multiple significant modes optimization techniques are used to identify the modes for which high probability regions are defined and components of the asg hsc approximation are constructed after the surrogate is determined the ppdf can be evaluated by sampling the surrogate system directly without model execution resulting in improved efficiency of the surrogate based mcmc compared with conventional mcmc the developed method is evaluated using two synthetic groundwater reactive transport models the first example involves coupled linear reactions and demonstrates the accuracy of our high order hierarchical basis approach in approximating high dimensional posteriori distribution the second example is highly nonlinear because of the reactions of uranium surface complexation and demonstrates how the iterative asg hsc method is able to capture multimodal and non gaussian features of ppdf caused by model nonlinearity both experiments show that asg hsc is an effective and efficient tool for bayesian inference

bayesian geostatistical design task driven optimal site investigation when the geostatistical model is uncertain 1 geostatistical optimal design optimizes subsurface exploration for maximum information toward task specific prediction goals until recently most geostatistical design studies have assumed that the geostatistical description i e the mean trends covariance models and their parameters is given a priori this contradicts as emphasized by rubin and dagan 1987a the fact that only few or even no data at all offer support for such assumptions prior to the bulk of exploration effort we believe that geostatistical design should 1 avoid unjustified a priori assumptions on the geostatistical description 2 instead reduce geostatistical model uncertainty as secondary design objective 3 rate this secondary objective optimal for the overall prediction goal and 4 be robust even under inaccurate geostatistical assumptions bayesian geostatistical design follows these guidelines by considering uncertain covariance model parameters we transfer this concept from kriging like applications to geostatistical inverse problems we also deem it inappropriate to consider parametric uncertainty only within a single covariance model the matern family of covariance functions has an additional shape parameter controlling model shape by a parameter converts covariance model selection to parameter identification and resembles bayesian model averaging over a continuous spectrum of covariance models this is appealing since it generalizes bayesian model averaging from a finite number to an infinite number of models we illustrate how our approach fulfills the above four guidelines in a series of synthetic test cases the underlying scenarios are to minimize the prediction variance of 1 contaminant concentration or 2 arrival time at an ecologically sensitive location by optimal placement of hydraulic head and log conductivity measurements results highlight how both the impact of geostatistical model uncertainty and the sampling network design vary according to the choice of objective function

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

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

water quality modeling a review of the analysis of uncertainty this paper reviews the role of uncertainty in the identification of mathematical models of water quality and in the application of these models to problems of prediction more specifically four problem areas are examined in detail uncertainty about model structure uncertainty in the estimated model parameter values the propagation of prediction errors and the design of experiments in order to reduce the critical uncertainties associated with a model the review is rather lengthy and it has therefore been prepared in effect as two papers there is a shorter largely nontechnical version which gives a quick impression of the current and future issues in the analysis of uncertainty in water quality modeling enclosed by this shorter discussion is the main body of the review dealing in turn with 1 identifiability and experimental design 2 the generation of preliminary model hypotheses under conditions of sparse grossly uncertain field data 3 the selection and evaluation of model structure 4 parameter estimation model calibration 5 checks and balances on the identified model i e model verification and model discrimination and 6 prediction error propagation much time is spent in discussing the algorithms of system identification in particular the methods of recursive estimation and in relating these algorithms and the subject of identification to the problems of prediction uncertainty and first order error analysis there are two obvious omissions from the review it is not concerned primarily with either the development and solution of stochastic differential equations or the issue of decision making under uncertainty although clearly some reference must be made to these topics in brief the review concludes not surprisingly that much work has been done on the analysis of uncertainty in the development of mathematical models of water quality and much remains to be done a lack of model identifiability has been an outstanding difficulty in the interpretation and explanation of past observed system behavior and there is ample evidence to show that the larger more comprehensive models are easily capable of generating highly uncertain predictions of future behavior for the future of the subject it is speculated that there is the possibility of progress in the development of novel algorithms for model structure identification a need for new questions to be posed in the problem of prediction and a distinct challenge to the conventional views of this review in the new forms of knowledge representation and manipulation now emerging from the field of artificial intelligence

analysis of regression confidence intervals and bayesian credible intervals for uncertainty quantification 1 confidence intervals based on classical regression theories augmented to include prior information and credible intervals based on bayesian theories are conceptually different ways to quantify parametric and predictive uncertainties because both confidence and credible intervals are used in environmental modeling we seek to understand their differences and similarities this is of interest in part because calculating confidence intervals typically requires tens to thousands of model runs while bayesian credible intervals typically require tens of thousands to millions of model runs given multi gaussian distributed observation errors our theoretical analysis shows that for linear or linearized nonlinear models confidence and credible intervals are always numerically identical when consistent prior information is used for nonlinear models nonlinear confidence and credible intervals can be numerically identical if parameter confidence regions defined using the approximate likelihood method and parameter credible regions estimated using markov chain monte carlo realizations are numerically identical and predictions are a smooth monotonic function of the parameters both occur if intrinsic model nonlinearity is small while the conditions of gaussian errors and small intrinsic model nonlinearity are violated by many environmental models heuristic tests using analytical and numerical models suggest that linear and nonlinear confidence intervals can be useful approximations of uncertainty even under significantly nonideal conditions in the context of epistemic model error for a complex synthetic nonlinear groundwater problem the linear and nonlinear confidence and credible intervals for individual models performed similarly enough to indicate that the computationally frugal confidence intervals can be useful in many circumstances experiences with these groundwater models are expected to be broadly applicable to many environmental models we suggest that for environmental problems with lengthy execution times that make credible intervals inconvenient or prohibitive confidence intervals can provide important insight during model development when frequent calculation of uncertainty intervals is important to understanding the consequences of various model construction alternatives and data collection strategies strategic use of both confidence and credible intervals can be critical

pumping tests in heterogeneous aquifers an analytical study of what can be obtained from their interpretation using jacob s method interpretation of pumping tests to estimate hydraulic parameter values is typically based on the assumption of aquifer homogeneity the applicability of the traditional methods of interpretation in real aquifers can be questioned since the evaluation of the drawdown curves observed at different locations in a single test may not result in one consistent set of hydraulic parameters thus most hydrogeologists tend to look at estimated transmissivities t as some average property of the medium while estimated storativities s are disregarded in some cases particularly when they are obtained from data measured at the pumping well an analytical study of drawdown under radially convergent flow toward a single point in heterogeneous aquifers shows that large time drawdown values form a straight line on a drawdown versus log time plot jacob s method consists of obtaining estimates for t and s from the slope and intercept of this line we find that even in a heterogeneous field these estimates provide valuable information about the aquifer estimated t values for different observation points tend to converge to a single value which corresponds to the effective t derived under parallel flow conditions estimated storativities however display higher variability but the geometric mean of the sest values can be used as an unbiased estimator of the actual s thus it appears that although jacob s method was originally derived for homogeneous media it can provide valuable information in real aquifers

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

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

inference of reactive transport model parameters using a bayesian multivariate approach parameter estimation of subsurface transport models from multispecies data requires the definition of an objective function that includes different types of measurements common approaches are weighted least squares wls where weights are specified a priori for each measurement and weighted least squares with weight estimation wls we where weights are estimated from the data together with the parameters in this study we formulate the parameter estimation task as a multivariate bayesian inference problem the wls and wls we methods are special cases in this framework corresponding to specific prior assumptions about the residual covariance matrix the bayesian perspective allows for generalizations to cases where residual correlation is important and for efficient inference by analytically integrating out the variances weights and selected covariances from the joint posterior specifically the wls and wls we methods are compared to a multivariate mv approach that accounts for specific residual correlations without the need for explicit estimation of the error parameters when applied to inference of reactive transport model parameters from column scale data on dissolved species concentrations the following results were obtained 1 accounting for residual correlation between species provides more accurate parameter estimation for high residual correlation levels whereas its influence for predictive uncertainty is negligible 2 integrating out the co variances leads to an efficient estimation of the full joint posterior with a reduced computational effort compared to the wls we method and 3 in the presence of model structural errors none of the methods is able to identify the correct parameter values

maximum likelihood bayesian averaging of spatial variability models in unsaturated fractured tuff 1 hydrologic analyses typically rely on a single conceptual mathematical model yet hydrologic environments are open and complex rendering them prone to multiple interpretations and mathematical descriptions adopting only one of these may lead to statistical bias and underestimation of uncertainty bayesian model averaging bma hoeting et al 1999 provides an optimal way to combine the predictions of several competing models and to assess their joint predictive uncertainty however it tends to be computationally demanding and relies heavily on prior information about model parameters neuman 2002 2003 proposed a maximum likelihood version mlbma of bma to render it computationally feasible and to allow dealing with cases where reliable prior information is lacking we apply mlbma to seven alternative variogram models of log air permeability data from single hole pneumatic injection tests in six boreholes at the apache leap research site alrs in central arizona unbiased ml estimates of variogram and drift parameters are obtained using adjoint state maximum likelihood cross validation samper and neuman 1989a in conjunction with universal kriging and generalized least squares standard information criteria provide an ambiguous ranking of the models which does not justify selecting one of them and discarding all others as is commonly done in practice instead we eliminate some of the models based on their negligibly small posterior probabilities and use the rest to project the measured log permeabilities by kriging onto a rock volume containing the six boreholes we then average these four projections and associated kriging variances using the posterior probability of each model as weight finally we cross validate the results by eliminating from consideration all data from one borehole at a time repeating the above process and comparing the predictive capability of mlbma with that of each individual model we find that mlbma is superior to any individual geostatistical model of log permeability among those we consider at the alrs

a comparison of three methods for selecting values of input variables in the analysis of output from a computer code two types of sampling plans are examined as alternatives to simple random sampling in monte carlo studies these plans are shown to be improvements over simple random sampling with respect to variance for a class of estimators which includes the sample mean and the empirical distribution function

an analysis of the pilot point methodology for automated calibration of an ensemble of conditionally simulated transmissivity fields an analysis of the pilot point method for automated calibration of an ensemble of conditionally simulated transmissivity fields was conducted on the basis of the simplifying assumption that the flow model is a linear function of log transmissivity the analysis shows that the pilot point and conditional simulation method of model calibration and uncertainty analysis can produce accurate uncertainty measures if it can be assumed that errors of unknown origin in the differences between observed and model computed water pressures are small when this assumption is not met the method could yield significant errors from overparameterization and the neglect of potential sources of model inaccuracy the conditional simulation part of the method is also shown to be a variant of the percentile bootstrap method so that when applied to a nonlinear model the method is subject to bootstrap errors these sources of error must be considered when using the method

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

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

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

the classification of hypersmooth borel equivalence relations this paper is a contribution to the study of borel equivalence relations in standard borel spaces i e polish spaces equipped with their borel structure a class of such equivalence relations which has received particular attention is the class of hyperfinite borel equivalence relations these can be defined as the increasing unions of sequences of borel equivalence relations all of whose equivalence classes are finite or as it turns out equivalently those induced by the orbits of a single borel auto morphism hyperfinite equivalence relations have been classified in djk under two notions of equivalence borel bi reducibility and borel isomorphism

review papers recent developments in nonparametric density estimation abstract advances in computation and the fast and cheap computational facilities now available to statisticians have had a significant impact upon statistical research and especially the development of nonparametric data analysis procedures in particular theoretical and applied research on nonparametric density estimation has had a noticeable influence on related topics such as nonparametric regression nonparametric discrimination and nonparametric pattern recognition this article reviews recent developments in nonparametric density estimation and includes topics that have been omitted from review articles and books on the subject the early density estimation methods such as the histogram kernel estimators and orthogonal series estimators are still very popular and recent research on them is described different types of restricted maximum likelihood density estimators including order restricted estimators maximum penalized likelihood estimators and sieve estimators are discussed where re

approximate bayesian inference with the weighted likelihood bootstrap we introduce the weighted likelihood bootstrap wlb as a way to simulate approximately from a posterior distribution this method is often easy to implement requiring only an algorithm for calculating the maximum likelihood estimator such as iteratively reweighted least squares in the generic weighting scheme the wlb is first order correct under quite general conditions inaccuracies can be removed by using the wlb as a source of samples in the sampling importance resampling sir algorithm which also allows incorporation of particular prior information the sir adjusted wlb can be a competitive alternative to other integration methods in certain models asymptotic expansions elucidate the second order properties of the wlb which is a generalization of rubin s bayesian bootstrap the calculation of approximate bayes factors for model comparison is also considered we note that given a sample simulated from the posterior distribution the required marginal likelihood may be simulation consistently estimated by the harmonic mean of the associated likelihood values a modification of this estimator that avoids instability is also noted these methods provide simple ways of calculating approximate bayes factors and posterior model probabilities for a very wide class of models

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

evaluating marginal likelihood with thermodynamic integration method and comparison with several other numerical methods evaluating marginal likelihood is the most critical and computationally expensive task when conducting bayesian model averaging to quantify parametric and model uncertainties the evaluation is commonly done by using laplace approximations to evaluate semianalytical expressions of the marginal likelihood or by using monte carlo mc methods to evaluate arithmetic or harmonic mean of a joint likelihood function this study introduces a new mc method i e thermodynamic integration which has not been attempted in environmental modeling instead of using samples only from prior parameter space as in arithmetic mean evaluation or posterior parameter space as in harmonic mean evaluation the thermodynamic integration method uses samples generated gradually from the prior to posterior parameter space this is done through a path sampling that conducts markov chain monte carlo simulation with different power coefficient values applied to the joint likelihood function the thermodynamic integration method is evaluated using three analytical functions by comparing the method with two variants of the laplace approximation method and three mc methods including the nested sampling method that is recently introduced into environmental modeling the thermodynamic integration method outperforms the other methods in terms of their accuracy convergence and consistency the thermodynamic integration method is also applied to a synthetic case of groundwater modeling with four alternative models the application shows that model probabilities obtained using the thermodynamic integration method improves predictive performance of bayesian model averaging the thermodynamic integration method is mathematically rigorous and its mc implementation is computationally general for a wide range of environmental problems

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

aquifer parameter identification with optimum dimension in parameterization this paper presents a systematic procedure whereby the functional coefficients imbedded in a two dimensional partial differential equation which governs unsteady groundwater flow are optimally identified the coefficients to be identified are transmissivities which vary spatially finite elements are used to represent the unknown transmissivity function parametrically in terms of nodal values over a suitable discretization of a flow region a modified gauss newton algorithm is used for parameter optimization covariance analysis is used to estimate the reliability of the estimated parameters as the dimension of the unknown parameter increases the modeling error represented by a least squares criterion will generally decrease but errors in data would be propagated to a greater degree into the estimated parameters thus reducing the reliability of estimation the reliability of the estimated parameters is characterized by a norm of the covariance matrix this information is used for the determination of the optimum dimension in parameterization

comment on lie symmetries and group classification of a class of time fractional evolution systems j math phys 56 123504 2015 in this article the prolongation formulae proposed in huang and shen j math phys 56 123504 2015 for the lie symmetry analysis of time fractional systems of partial differential equations are proved incomplete and the required correct prolongation operators are suggested with the help of some examples the efficiency of the operators introduced in this study is illustrated

bayesian inversion of mualem van genuchten parameters in a multilayer soil profile a data driven assumption free likelihood function this paper introduces a hierarchical simulation and modeling framework that allows for inference and validation of the likelihood function in bayesian inversion of vadose zone hydraulic properties the likelihood function or its analogs objective functions and likelihood measures are commonly assumed to be multivariate gaussian in form however this assumption is not possible to verify without a hierarchical simulation and modeling framework in this paper we present the necessary statistical mechanisms for utilizing the hierarchical framework we apply the hierarchical framework to the inversion of the vadose zone hydraulic properties within a multilayer soil profile conditioned on moisture content observations collected in the uppermost four layers the key result of our work is that the goodness of fit validated likelihood function form provides empirical justification for the assumption of multivariate gaussian likelihood functions in past and future inversions at similar sites as an alternative the likelihood function need not be assumed to follow a parametric statistical distribution and can be computed directly using nonparametric methods the nonparametric methods are considerably more computationally demanding and to demonstrate this approach we present a smaller dimension synthetic case study of evaporation from a soil column the main drawback of our work is the increased computational expense of the inversion

a comparison between gauss newton and markov chain monte carlo based methods for inverting spectral induced polarization data for cole cole parameters we have developed a bayesian model to invert spectral induced polarization sip data for cole cole parameters using markov chain monte carlo mcmc sampling methods we comparedtheperformanceofthemcmc basedstochasticmethod with an iterative gauss newton based deterministic method for cole cole parameter estimation through inversion of syntheticandlaboratorysipdata thegauss newton basedmethod canprovideanoptimalsolutionforgivenobjectivefunctionsunder constraints but the obtained optimal solution generally dependsonthechoiceofinitialvaluesandtheestimateduncertainty information often is inaccurate or insufficient in contrast the mcmc based inversion method provides extensive global informationonunknownparameters suchasthemarginalprobability distribution functions from which we can obtain better estimates and tighter uncertainty bounds of the parameters than with the deterministic method in addition the results obtained with the mcmc method are independent of the choice of initial values becausethemcmc basedmethoddoesnotexplicitlyofferasingleoptimalsolutionforgivenobjectivefunctions thedeterministic and stochastic methods can complement each other forexample thestochasticmethodcanbeusedfirsttoobtainthe mediansofunknownparametersbystartingfromanarbitraryset of initial values the deterministic method then can be initiated using the medians as starting values to obtain the optimal estimatesofthecole coleparameters

stochastic spectral methods for efficient bayesian solution of inverse problems we present a reformulation of the bayesian approach to inverse problems that seeks to accelerate bayesian inference by using polynomial chaos pc expansions to represent random variables evaluation of integrals over the unknown parameter space is recast more efficiently as monte carlo sampling of the random variables underlying the pc expansion we evaluate the utility of this technique on a transient diffusion problem arising in contaminant source inversion the accuracy of posterior estimates is examined with respect to the order of the pc representation the choice of pc basis and the decomposition of the support of the prior the computational cost of the new scheme shows significant gains over direct sampling

the selection of prior distributions by formal rules abstract subjectivism has become the dominant philosophical foundation for bayesian inference yet in practice most bayesian analyses are performed with so called noninformative priors that is priors constructed by some formal rule we review the plethora of techniques for constructing such priors and discuss some of the practical and philosophical issues that arise when they are used we give special emphasis to jeffreys s rules and discuss the evolution of his viewpoint about the interpretation of priors away from unique representation of ignorance toward the notion that they should be chosen by convention we conclude that the problems raised by the research on priors chosen by formal rules are serious and may not be dismissed lightly when sample sizes are small relative to the number of parameters being estimated it is dangerous to put faith in any default solution but when asymptotics take over jeffreys s rules and their variants remain reasonable choices we also provide an annotated b

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

simulation of dispersion in heterogeneous porous formations statistics first order theories convergence of computations this paper discusses the results of numerical analysis of dispersion of passive solutes in two dimensional heterogeneous porous formations statistics of flow and transport variables the accuracy and the role of approximations implicit in existing first order theories and the convergence of computational results are investigated the results suggest that quite different rates of convergence with monte carlo runs hold for different spatial moments and that over 1000 realizations are required to stabilize second moments even for relatively mild heterogeneity sigma y 2 1 6 this has implications for the extent of the spatial domain for single realization numerical studies of the same type a comparison of the variance of plumes with the results of linear theories 0 05 sigma y 2 1 6 shows an unexpectedly broad validity field for the theoretical solution obtained from a suitable linearization of flow and transport reformulation of the same problem linearizing in tum the flow or the transport equations shows opposite deviations from the linear theory the interesting consequence is that the errors induced by linearizations in the flow or the transport equations have different signs and their effects on the moments of dispersing plumes are compensating thereby yielding consistent formulations unexpected features of the statistics of probability distributions of longitudinal and transverse velocities and travel times are also computed and discussed

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

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

a bayesian approach to transformations to normality summary the analysis of transformation of observations in the linear model with normal errors proposed by box cox 1964 is considered a different choice of noninformative unnormed prior is advocated which is not outcome dependent this new selection of prior leads to a formal identity between likelihood and bayesian inference both for the estimation of the best transformation to normality and for the presence of homoscedasticity and additivity under this transformation extension to a related problem is mentioned

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

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

reservoir parameter identification using minimum relative entropy based bayesian inversion of seismic ava and marine csem data a stochastic joint inversion approach for estimating reservoir fluid saturations and porosity is proposed the approach couples seismic amplitude variation with angle ava and marine controlled source electromagnetic csem forward models into a bayesian framework which allows for integration of complementary information to obtain minimally subjective prior probabilities required for the bayesian approach the principle of minimum relative entropy mre is employed instead of single value estimates provided by deterministic methods the approach gives a probability distribution for any unknown parameter of interest such as reservoir fluid saturations or porosity at various locations the distribution means modes and confidence intervals can be calculated providing a more complete understanding of the uncertainty in the parameter estimates the approach is demonstrated using synthetic and field data sets results show that joint inversion using seismic and em data gives better estimates of reservoir parameters than estimates from either geophysical data set used in isolation moreover a more informative prior leads to much narrower predictive intervals of the target parameters with mean values of the posterior distributions closer to logged values

the structure of hyperfinite borel equivalence relations we study the structure of the equivalence relations induced by the orbits of a single borel automorphism on a standard borel space we show that any two such equivalence relations which are not smooth i e do not admit borel selectors are borel embeddable into each other this utilizes among other things work of effros and weiss using this and also results of dye varadarajan and recent work of nadkarni we show that the cardinality of the set of ergodic invariant measures is a complete invariant for borel isomorphism of aperiodic nonsmooth such equivalence relations in particular since the only possible such cardinalities are the finite ones countable infinity and the cardinality of the continuum there are exactly countably infinitely many isomorphism types canonical examples of each type are also discussed

dichotomy theorems for families of non cofinal essential complexity we prove that for every borel equivalence relation e either e is borel reducible to mathbb e 0 or the family of borel equivalence relations incompatible with e has cofinal essential complexity it follows that if f is a borel equivalence relation and cal f is a family of borel equivalence relations of non cofinal essential complexity which together satisfy the dichotomy that for every borel equivalence relation e either e in cal f or f is borel reducible to e then cal f consists solely of smooth equivalence relations thus the dichotomy is equivalent to a known theorem

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

a formal likelihood function for parameter and predictive inference of hydrologic models with correlated heteroscedastic and non gaussian errors estimation of parameter and predictive uncertainty of hydrologic models has traditionally relied on several simplifying assumptions residual errors are often assumed to be independent and to be adequately described by a gaussian probability distribution with a mean of zero and a constant variance here we investigate to what extent estimates of parameter and predictive uncertainty are affected when these assumptions are relaxed a formal generalized likelihood function is presented which extends the applicability of previously used likelihood functions to situations where residual errors are correlated heteroscedastic and non gaussian with varying degrees of kurtosis and skewness the approach focuses on a correct statistical description of the data and the total model residuals without separating out various error sources application to bayesian uncertainty analysis of a conceptual rainfall runoff model simultaneously identifies the hydrologic model parameters and the appropriate statistical distribution of the residual errors when applied to daily rainfall runoff data from a humid basin we find that 1 residual errors are much better described by a heteroscedastic first order auto correlated error model with a laplacian distribution function characterized by heavier tails than a gaussian distribution and 2 compared to a standard least squares approach proper representation of the statistical distribution of residual errors yields tighter predictive uncertainty bands and different parameter uncertainty estimates that are less sensitive to the particular time period used for inference application to daily rainfall runoff data from a semiarid basin with more significant residual errors and systematic underprediction of peak flows shows that 1 multiplicative bias factors can be used to compensate for some of the largest errors and 2 a skewed error distribution yields improved estimates of predictive uncertainty in this semiarid basin with near zero flows we conclude that the presented methodology provides improved estimates of parameter and total prediction uncertainty and should be useful for handling complex residual errors in other hydrologic regression models as well

erratum and addendum lie symmetries and group classification of a class of time fractional evolution systems j math phys 56 123504 2015

a method for enforcing parameter nonnegativity in bayesian inverse problems with an application to contaminant source identification received 28 may 2002 revised 18 september 2002 accepted 24 september 2002 published 14 february 2003 1 when an inverse problem is solved to estimate an unknown function such as the hydraulic conductivity in an aquifer or the contamination history at a site one constraint is that the unknown function is known to be everywhere nonnegative in this work we develop a statistically rigorous method for enforcing function nonnegativity in bayesian inverse problems the proposed method behaves similarly to a gaussian process with a linear variogram i e unrestricted brownian motion for parameter values significantly greater than zero the method uses the method of images to define a prior probability density function based on reflected brownian motion that implicitly enforces nonnegativity this work focuses on problems where the unknown is a function of a single variable e g time a markov chain monte carlo mcmc method specifically a highly efficient gibbs sampler is implemented to generate conditional realizations of the unknown function the new method is applied to the estimation of the trichloroethylene tce and perchloroethylene pce contamination history in an aquifer at dover air force base delaware based on concentration profiles obtained from an underlying aquitard index terms 1831 hydrology groundwater quality 1869 hydrology stochastic processes 3260 mathematical geophysics inverse theory keywords stochastic inverse modeling contaminant source identification inference under constraints markov chain monte carlo mcmc gibbs sampling bayesian inference