Publications

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We investigate Bayesian techniques that can be used to reconstruct field variables from partial observations. In particular, we target fields that exhibit spatial structures with a large spectrum of lengthscales. Contemporary methods typically describe the field on a grid and estimate structures which can be resolved by it. In contrast, we address the reconstruction of grid-resolved structures as well as estimation of statistical summaries of subgrid structures, which are smaller than the grid resolution. We perform this in two different ways (a) via a physical (phenomenological), parameterized subgrid model that summarizes the impact of the unresolved scales at the coarse level and (b) via multiscale finite elements, where specially designed prolongation and restriction operators establish the interscale link between the same problem defined on a coarse and fine mesh. The estimation problem is posed as a Bayesian inverse problem. Dimensionality reduction is performed by projecting the field to be inferred on a suitable orthogonal basis set, viz. the Karhunen-Loeve expansion of a multiGaussian. We first demonstrate our techniques on the reconstruction of a binary medium consisting of a matrix with embedded inclusions, which are too small to be grid-resolved. The reconstruction is performed using an adaptive Markov chain Monte Carlo method. We find that the posterior distributions of the inferred parameters are approximately Gaussian. We exploit this finding to reconstruct a permeability field with long, but narrow embedded fractures (which are too fine to be grid-resolved) using scalable ensemble Kalman filters; this also allows us to address larger grids. Ensemble Kalman filtering is then used to estimate the values of hydraulic conductivity and specific yield in a model of the High Plains Aquifer in Kansas. Strong conditioning of the spatial structure of the parameters and the non-linear aspects of the water table aquifer create difficulty for the ensemble Kalman filter. We conclude with a demonstration of the use of multiscale stochastic finite elements to reconstruct permeability fields. This method, though computationally intensive, is general and can be used for multiscale inference in cases where a subgrid model cannot be constructed.

Previously developed techniques that comprise statistical parametric mapping, with applications focused on human brain imaging, are examined and tested here for new applications in anomaly detection within remotely-sensed imagery. Two approaches to analysis are developed: online, regression-based anomaly detection and conditional differences. These approaches are applied to two example spatial-temporal data sets: data simulated with a Gaussian field deformation approach and weekly NDVI images derived from global satellite coverage. Results indicate that anomalies can be identified in spatial temporal data with the regression-based approach. Additionally, la Nina and el Nino climatic conditions are used as different stimuli applied to the earth and this comparison shows that el Nino conditions lead to significant decreases in NDVI in both the Amazon Basin and in Southern India.

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Truncated Gaussian fields provide a flexible model for defining binary media with dispersed (as opposed to layered) inclusions. General properties of excursion sets on these truncated fields are coupled with a distance-based upscaling algorithm and approximations of point process theory to develop an estimation approach for effective conductivity in two-dimensions. Estimation of effective conductivity is derived directly from knowledge of the kernel size used to create the multiGaussian field, defined as the full-width at half maximum (FWHM), the truncation threshold and conductance values of the two modes. Therefore, instantiation of the multiGaussian field is not necessary for estimation of the effective conductance. The critical component of the effective medium approximation developed here is the mean distance between high conductivity inclusions. This mean distance is characterized as a function of the FWHM, the truncation threshold and the ratio of the two modal conductivities. Sensitivity of the resulting effective conductivity to this mean distance is examined for two levels of contrast in the modal conductances and different FWHM sizes. Results demonstrate that the FWHM is a robust measure of mean travel distance in the background medium. The resulting effective conductivities are accurate when compared to numerical results and results obtained from effective media theory, distance-based upscaling and numerical simulation. © 2011 Elsevier Ltd.

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The risk assessment approach has been applied to support numerous radioactive waste management activities over the last 30 years. A risk assessment methodology provides a solid and readily adaptable framework for evaluating the risks of CO2 sequestration in geologic formations to prioritize research, data collection, and monitoring schemes. This paper reviews the tasks of a risk assessment, and provides a few examples related to each task. This paper then describes an application of sensitivity analysis to identify important parameters to reduce the uncertainty in the performance of a geologic repository for radioactive waste repository, which because of importance of the geologic barrier, is similar to CO2 sequestration. The paper ends with a simple stochastic analysis of idealized CO2 sequestration site with a leaking abandoned well and a set of monitoring wells in an aquifer above the CO2 sequestration unit in order to evaluate the efficacy of monitoring wells to detect adverse leakage.

Injection of CO2 into formations containing brine is proposed as a long-term sequestration solution. A significant obstacle to sequestration performance is the presence of existing wells providing a transport pathway out of the sequestration formation. To understand how heterogeneity impacts the leakage rate, we employ two dimensional models of the CO2 injection process into a sandstone aquifer with shale inclusions to examine the parameters controlling release through an existing well. This scenario is modeled as a constant-rate injection of super-critical CO2 into the existing formation where buoyancy effects, relative permeabilities, and capillary pressures are employed. Three geologic controls are considered: stratigraphic dip angle, shale inclusion size and shale fraction. In this study, we examine the impact of heterogeneity on the amount and timing of CO2 released through a leaky well. Sensitivity analysis is performed to classify how various geologic controls influence CO2 loss. A 'Design of Experiments' approach is used to identify the most important parameters and combinations of parameters to control CO2 migration while making efficient use of a limited number of computations. Results are used to construct a low-dimensional description of the transport scenario. The goal of this exploration is to develop a small set of parametric descriptors that can be generalized to similar scenarios. Results of this work will allow for estimation of the amount of CO2 that will be lost for a given scenario prior to commencing injection. Additionally, two-dimensional and three-dimensional simulations are compared to quantify the influence that surrounding geologic media has on the CO2 leakage rate.

We present results from a recently developed multiscale inversion technique for binary media, with emphasis on the effect of subgrid model errors on the inversion. Binary media are a useful fine-scale representation of heterogeneous porous media. Averaged properties of the binary field representations can be used to characterize flow through the porous medium at the macroscale. Both direct measurements of the averaged properties and upscaling are complicated and may not provide accurate results. However, it may be possible to infer upscaled properties of the binary medium from indirect measurements at the coarse scale. Multiscale inversion, performed with a subgrid model to connect disparate scales together, can also yield information on the fine-scale properties. We model the binary medium using truncated Gaussian fields, and develop a subgrid model for the upscaled permeability based on excursion sets of those fields. The subgrid model requires an estimate of the proportion of inclusions at the block scale as well as some geometrical parameters of the inclusions as inputs, and predicts the effective permeability. The inclusion proportion is assumed to be spatially varying, modeled using Gaussian processes and represented using a truncated Karhunen-Louve (KL) expansion. This expansion is used, along with the subgrid model, to pose as a Bayesian inverse problem for the KL weights and the geometrical parameters of the inclusions. The model error is represented in two different ways: (1) as a homoscedastic error and (2) as a heteroscedastic error, dependent on inclusion proportionality and geometry. The error models impact the form of the likelihood function in the expression for the posterior density of the objects of inference. The problem is solved using an adaptive Markov Chain Monte Carlo method, and joint posterior distributions are developed for the KL weights and inclusion geometry. Effective permeabilities and tracer breakthrough times at a few 'sensor' locations (obtained by simulating a pump test) form the observables used in the inversion. The inferred quantities can be used to generate an ensemble of permeability fields, both upscaled and fine-scale, which are consistent with the observations. We compare the inferences developed using the two error models, in terms of the KL weights and fine-scale realizations that could be supported by the coarse-scale inferences. Permeability differences are observed mainly in regions where the inclusions proportion is near the percolation threshold, and the subgrid model incurs its largest approximation. These differences also reflected in the tracer breakthrough times and the geometry of flow streamlines, as obtained from a permeameter simulation. The uncertainty due to subgrid model error is also compared to the uncertainty in the inversion due to incomplete data.

We consider the problem of placing a limited number of sensors in a municipal water distribution network to minimize the impact over a given suite of contamination incidents. In its simplest form, the sensor placement problem is a p-median problem that has structure extremely amenable to exact and heuristic solution methods. We describe the solution of real-world instances using integer programming or local search or a Lagrangian method. The Lagrangian method is necessary for solution of large problems on small PCs. We summarize a number of other heuristic methods for effectively addressing issues such as sensor failures, tuning sensors based on local water quality variability, and problem size/approximation quality tradeoffs. These algorithms are incorporated into the TEVA-SPOT toolkit, a software suite that the US Environmental Protection Agency has used and is using to design contamination warning systems for US municipal water systems.

Heterogeneity plays an important role in groundwater flow and contaminant transport in natural systems. Since it is impossible to directly measure spatial variability of hydraulic conductivity, predictions of solute transport based on mathematical models are always uncertain. While in most cases groundwater flow and tracer transport problems are investigated in two-dimensional (2D) systems, it is important to study more realistic and well-controlled 3D systems to fully evaluate inverse parameter estimation techniques and evaluate uncertainty in the resulting estimates. We used tracer concentration breakthrough curves (BTCs) obtained from a magnetic resonance imaging (MRI) technique in a small flow cell (14 x 8 x 8 cm) that was packed with a known pattern of five different sands (i.e., zones) having cm-scale variability. In contrast to typical inversion systems with head, conductivity and concentration measurements at limited points, the MRI data included BTCs measured at a voxel scale ({approx}0.2 cm in each dimension) over 13 x 8 x 8 cm with a well controlled boundary condition, but did not have direct measurements of head and conductivity. Hydraulic conductivity and porosity were conceptualized as spatial random fields and estimated using pilot points along layers of the 3D medium. The steady state water flow and solute transport were solved using MODFLOW and MODPATH. The inversion problem was solved with a nonlinear parameter estimation package - PEST. Two approaches to parameterization of the spatial fields are evaluated: (1) The detailed zone information was used as prior information to constrain the spatial impact of the pilot points and reduce the number of parameters; and (2) highly parameterized inversion at cm scale (e.g., 1664 parameters) using singular value decomposition (SVD) methodology to significantly reduce the run-time demands. Both results will be compared to measured BTCs. With MRI, it is easy to change the averaging scale of the observed concentration from point to cross-section. This comparison allows us to evaluate which method best matches experimental results at different scales. To evaluate the uncertainty in parameter estimation, the null space Monte Carlo method will be used to reduce computational burden of the development of calibration-constrained Monte Carlo based parameter fields. This study will illustrate how accurately a well-calibrated model can predict contaminant transport.

Multi-scale binary permeability field estimation from static and dynamic data is completed using Markov Chain Monte Carlo (MCMC) sampling. The binary permeability field is defined as high permeability inclusions within a lower permeability matrix. Static data are obtained as measurements of permeability with support consistent to the coarse scale discretization. Dynamic data are advective travel times along streamlines calculated through a fine-scale field and averaged for each observation point at the coarse scale. Parameters estimated at the coarse scale (30 x 20 grid) are the spatially varying proportion of the high permeability phase and the inclusion length and aspect ratio of the high permeability inclusions. From the non-parametric, posterior distributions estimated for these parameters, a recently developed sub-grid algorithm is employed to create an ensemble of realizations representing the fine-scale (3000 x 2000), binary permeability field. Each fine-scale ensemble member is instantiated by convolution of an uncorrelated multiGaussian random field with a Gaussian kernel defined by the estimated inclusion length and aspect ratio. Since the multiGaussian random field is itself a realization of a stochastic process, the procedure for generating fine-scale binary permeability field realizations is also stochastic. Two different methods are hypothesized to perform posterior predictive tests. Different mechanisms for combining multi Gaussian random fields with kernels defined from the MCMC sampling are examined. Posterior predictive accuracy of the estimated parameters is assessed against a simulated ground truth for predictions at both the coarse scale (effective permeabilities) and at the fine scale (advective travel time distributions). The two techniques for conducting posterior predictive tests are compared by their ability to recover the static and dynamic data. The skill of the inference and the method for generating fine-scale binary permeability fields are evaluated through flow calculations on the resulting fields using fine-scale realizations and comparing them against results obtained with the ground truth fine-scale and coarse-scale permeability fields.