[1] The ensemble Kalman filter (EnKF) is a powerful tool for assimilating data in earth system models. The approach allows real time Bayesian updating of system states and parameters as new data become available. This paper focuses on EnKF data assimilation in models of groundwater flow through complex geologic media. It has become common to treat the hydraulic conductivity of such media as correlated random fields conditioned on measured conductivity (medium property) and/or hydraulic head (system state) values. This renders the conductivity nonstationary and the corresponding conditional flow equations stochastic. Solving these equations and coupling them with EnKF generally entails computationally intensive Monte Carlo (MC) simulation. We propose to circumvent the need for MC through a direct solution of approximate nonlocal (integrodifferential) equations that govern the space-time evolution of conditional ensemble means (statistical expectations) and covariances of hydraulic heads and fluxes. We illustrate and explore our approach on synthetic two-dimensional examples in which a well pumps water from a randomly heterogeneous aquifer subject to prescribed head and flux boundary conditions. Embedding the solution in EnKF provides sequential updates of conductivity and head estimates throughout the space-time domain of interest. We demonstrate the computational feasibility and accuracy of our methodology, showing that hydraulic conductivity estimates are more sensitive to early than to later head values and improve with increasing assimilation frequency at early time.

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Many earth and environmental (as well as other) variables, Y, and their spatial or temporal increments, ΔY, exhibit non-Gaussian statistical scaling. Previously we were able to capture some key aspects of such scaling by treating Y or ΔY as standard sub-Gaussian random functions. We were however unable to reconcile two seemingly contradictory observations, namely that whereas sample frequency distributions of Y (or its logarithm) exhibit relatively mild non-Gaussian peaks and tails, those of ΔY display peaks that grow sharper and tails that become heavier with decreasing separation distance or lag. Recently we overcame this difficulty by developing a new generalized sub-Gaussian model which captures both behaviors in a unified and consistent manner, exploring it on synthetically generated random functions in one dimension (Riva et al., 2015). Here we extend our generalized sub-Gaussian model to multiple dimensions, present an algorithm to generate corresponding random realizations of statistically isotropic or anisotropic sub-Gaussian functions and illustrate it in two dimensions. We demonstrate the accuracy of our algorithm by comparing ensemble statistics of Y and ΔY (such as, mean, variance, variogram and probability density function) with those of Monte Carlo generated realizations. We end by exploring the feasibility of estimating all relevant parameters of our model by analyzing jointly spatial moments of Y and ΔY obtained from a single realization of Y

[1] We analyze theoretically the ability of model quality (sometimes termed information or discrimination) criteria such as the negative log likelihood NLL, Bayesian criteria BIC and KIC and information theoretic criteria AIC, AICc, and HIC to estimate (1) the parameter vector h of the variogram of hydraulic log conductivity (Y ¼ ln K), and (2) statistical parameters 2 hE and 2 YE proportional to head and log conductivity measurement error variances, respectively, in the context of geostatistical groundwater flow inversion. Our analysis extends the work of Hernandez et al. (2003Hernandez et al. ( , 2006 and Riva et al. (2009), who developed nonlinear stochastic inverse algorithms that allow conditioning estimates of steady state and transient hydraulic heads, fluxes and their associated uncertainty on information about conductivity and head data collected in a randomly heterogeneous confined aquifer. Their algorithms are based on recursive numerical approximations of exact nonlocal conditional equations describing the mean and (co)variance of groundwater flow. Log conductivity is parameterized geostatistically based on measured values at discrete locations and unknown values at discrete ''pilot points.'' Optionally, the maximum likelihood function on which the inverse estimation of Y at pilot points is based may include a regularization term reflecting prior information about Y. The relative weight ¼ YE , as well as h are evaluated separately from other model parameters to avoid bias and instability. This evaluation is done on the basis of criteria such as NLL, KIC, BIC, HIC, AIC, and AICc. We demonstrate theoretically that, whereas all these six criteria make it possible to estimate 2 hE , KIC alone allows one to estimate validly h and 2 YE (and thus ). We illustrate this discriminatory power of KIC numerically by using a differential evolution genetic search algorithm to minimize it in the context of a two-dimensional steady state groundwater flow problem. We find that whereas

We describe a field application of a new data assimilation method recently proposed by Panzeri et al. (2013, 2014). The method couples a modified Ensemble Kalman Filter (EnKF) algorithm with stochastic moment equations (MEs) governing space-time variations of (theoretical ensemble) mean and covariance values of groundwater flow state variables (hydraulic heads and fluxes). Whereas traditional EnKF entails Monte Carlo (MC) simulations and suffers from inbreeding, our approach obviates both. Synthetic case studies have shown the ME-based approach to be computationally efficient and accurate when compared to MC-based results. Here we use our ME-based method to assimilate drawdown data recorded during cross-hole pumping tests in the heterogeneous alluvial Lauswiesen aquifer near Tubingen, Germany. Our results include an estimate of log transmissivity distribution throughout the aquifer and corresponding measures of estimation error. We validate our calibrated model by using it to predict drawdowns recorded during another pumping test at the site and compare its performance with that of standard MC-based EnKF