Recent advances in sensor technologies, field methodologies, numerical modeling, and inversion approaches have contributed to unprecedented imaging of hydrogeological properties and detailed predictions at multiple temporal and spatial scales. Nevertheless, imaging results and predictions will always remain imprecise, which calls for appropriate uncertainty quantification (UQ). In this paper, we outline selected methodological developments together with pioneering UQ applications in hydrogeology and hydrogeophysics. The applied mathematics and statistics literature is not easy to penetrate and this review aims at helping hydrogeologists and hydrogeophysicists to identify suitable approaches for UQ that can be applied and further developed to their specific needs. To bypass the tremendous computational costs associated with forward UQ based on full-physics simulations, we discuss proxy-modeling strategies and multi-resolution (Multi-level Monte Carlo) methods. We consider Bayesian inversion for non-linear and non-Gaussian state-space problems and discuss how Sequential Monte Carlo may become a practical alternative. We also describe

[1] Quantifying the spatial configuration of hydraulic conductivity (K) in heterogeneous geological environments is essential for accurate predictions of contaminant transport, but is difficult because of the inherent limitations in resolution and coverage associated with traditional hydrological measurements. To address this issue, we consider crosshole and surface-based electrical resistivity geophysical measurements, collected in time during a saline tracer experiment. We use a Bayesian Markov-chain-Monte-Carlo (McMC) methodology to jointly invert the dynamic resistivity data, together with borehole tracer concentration data, to generate multiple posterior realizations of K that are consistent with all available information. We do this within a coupled inversion framework, whereby the geophysical and hydrological forward models are linked through an uncertain relationship between electrical resistivity and concentration. To minimize computational expense, a facies-based subsurface parameterization is developed. The Bayesian-McMC methodology allows us to explore the potential benefits of including the geophysical data into the inverse problem by examining their effect on our ability to identify fast flowpaths in the subsurface, and their impact on hydrological prediction uncertainty. Using a complex, geostatistically generated, two-dimensional numerical example representative of a fluvial environment, we demonstrate that flow model calibration is improved and prediction error is decreased when the electrical resistivity data are included. The worth of the geophysical data is found to be greatest for long spatial correlation lengths of subsurface heterogeneity with respect to wellbore separation, where flow and transport are largely controlled by highly connected flowpaths.Citation: Irving, J., and K. Singha (2010), Stochastic inversion of tracer test and electrical geophysical data to estimate hydraulic conductivities, Water Resour. Res., 46, W11514,

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Wavelet dispersion caused by frequency-dependent attenuation is a common occurrence in groundpenetrating radar (GPR) data, and is displayed in the radar image as a characteristic "blurriness" that increases with depth. Correcting for wavelet dispersion is an important step that should be performed before GPR data are used for either qualitative interpretation or the quantitative determination of subsurface electrical properties. Over the bandwidth of a GPR wavelet, the attenuation of electromagnetic waves in many geological materials is approximately linear with frequency. As a result, the change in shape of a radar pulse as it propagates through these materials can be well described using one parameter, Q * , related to the slope of the linear region. Assuming that all subsurface materials can be characterized by some Q * value, the problem of estimating and correcting for wavelet dispersion becomes one of determining Q * in the subsurface and deconvolving its effects using an inverse-Q filter. We present a method for the estimation of subsurface Q * from reflection GPR data based on a technique developed for seismic attenuation tomography. Essentially, Q * is computed from the downshift in the dominant frequency of the GPR signal with time. Once Q * has been obtained, we propose a damped-least-squares inverse-Q filtering scheme based on a causal, linear model for constant-Q wave propagation as a means of removing wavelet dispersion. Tests on synthetic and field data indicate that these steps can be very effective at enhancing the resolution of the GPR image.