A Bayesian approach for inverse modeling, data assimilation, and conditional simulation of spatial random fields

Rubin, Yoram; Chen, Xingyuan; Murakami, Haruko; Hahn, M. S.

doi:10.1029/2009wr008799

Cited by 82 publications

(99 citation statements)

References 70 publications

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“…The posterior field uncertainty encompassed a large range of different geostatistical models, which calls into question the common practice in hydrogeology of fixing the variogram model before inversion. For the considered case study, the serial version of our method appears to be more computationally efficient than both the SGS algorithm of Hansen et al [2012Hansen et al [ , 2013aHansen et al [ , 2013b and the MAD method of Rubin et al [2010]. The advantages of the proposed approach are even more apparent when executed on a distributed computing network.…”

Section: Discussionmentioning

confidence: 80%

“…The MAD method (see Rubin et al [2010], for an extensive description) is especially designed for inference of (multi-)Gaussian parameter fields. It differs from classical Bayesian inference methods in the treatment of the likelihood function, L hjd ð Þ p djh ð Þ.…”

Section: Comparison Against the Methods Of Anchored Distributionsmentioning

confidence: 99%

“…In the absence of prior information about the geostatistical parameters (with exception of the ranges of the search space), we assumed either uniform or Jeffreys [1946] (that is, log-uniform) truncated individual priors that span a wide range of values. We selected a bounded uniform prior for m and a bounded Jeffreys prior for v. This is a common choice in the inference of multi-Gaussian fields [e.g., Box and Tiao, 1973;Rubin et al, 2010]. We chose bounded uniform priors for I M , A, and R I , and a bounded Jeffreys prior for m. Also, a Jeffreys prior is selected for r e .…”

Section: Inference Of An Heterogeneous Random Field With Short Integrmentioning

confidence: 99%

“…Furthermore, variogram inference with SGS simply involves sampling of the prior variogram distribution, which may not be efficient if the prior variogram uncertainty range is large and/or the information content of the calibration data is high. Lastly, the method of anchored distributions (MAD) introduced by Zhang and Rubin [2008] and Rubin et al [2010] simultaneously derives variogram parameters and hydraulic conductivity/transmissivity values at selected locations (the so called ''anchors'') within the hydrogeologic domain of interest. In the examples presented by Rubin et al [2010] and Murakami et al [2010], the estimated geostatistical properties are restricted to the mean, sill and range of a 2-D exponential isotropic model.…”

Section: Introductionmentioning

confidence: 99%

“…Lastly, the method of anchored distributions (MAD) introduced by Zhang and Rubin [2008] and Rubin et al [2010] simultaneously derives variogram parameters and hydraulic conductivity/transmissivity values at selected locations (the so called ''anchors'') within the hydrogeologic domain of interest. In the examples presented by Rubin et al [2010] and Murakami et al [2010], the estimated geostatistical properties are restricted to the mean, sill and range of a 2-D exponential isotropic model. Perhaps more importantly, the MAD framework relies on plain Monte Carlo (MC) simulation and is thus computationally very demanding, particularly in high-dimensional parameter spaces [e.g., Murakami et al, 2010].…”

Section: Introductionmentioning

confidence: 99%

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Probabilistic inference of multi‐Gaussian fields from indirect hydrological data using circulant embedding and dimensionality reduction

Laloy

Linde

Jacques

et al. 2015

Water Resources Research

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We present a Bayesian inversion method for the joint inference of high-dimensional multiGaussian hydraulic conductivity fields and associated geostatistical parameters from indirect hydrological data. We combine Gaussian process generation via circulant embedding to decouple the variogram from grid cell specific values, with dimensionality reduction by interpolation to enable Markov chain Monte Carlo (MCMC) simulation. Using the Matern variogram model, this formulation allows inferring the conductivity values simultaneously with the field smoothness (also called Matern shape parameter) and other geostatistical parameters such as the mean, sill, integral scales and anisotropy direction(s) and ratio(s). The proposed dimensionality reduction method systematically honors the underlying variogram and is demonstrated to achieve better performance than the Karhunen-Loè ve expansion. We illustrate our inversion approach using synthetic (error corrupted) data from a tracer experiment in a fairly heterogeneous 10,000-dimensional 2-D conductivity field. A 40-times reduction of the size of the parameter space did not prevent the posterior simulations to appropriately fit the measurement data and the posterior parameter distributions to include the true geostatistical parameter values. Overall, the posterior field realizations covered a wide range of geostatistical models, questioning the common practice of assuming a fixed variogram prior to inference of the hydraulic conductivity values. Our method is shown to be more efficient than sequential Gibbs sampling (SGS) for the considered case study, particularly when implemented on a distributed computing cluster. It is also found to outperform the method of anchored distributions (MAD) for the same computational budget.

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Section: Discussionmentioning

confidence: 80%

Section: Comparison Against the Methods Of Anchored Distributionsmentioning

confidence: 99%

Section: Inference Of An Heterogeneous Random Field With Short Integrmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Probabilistic inference of multi‐Gaussian fields from indirect hydrological data using circulant embedding and dimensionality reduction

Laloy

Linde

Jacques

et al. 2015

Water Resources Research

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Application of ensemble-based data assimilation techniques for aquifer characterization using tracer data at Hanford 300 area

et al. 2013

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[1] Subsurface aquifer characterization often involves high parameter dimensionality and requires tremendous computational resources if employing a full Bayesian approach. Ensemble-based data assimilation techniques, including filtering and smoothing, are computationally efficient alternatives. Despite the increasing use of ensemble-based methods in assimilating flow and transport related data for subsurface aquifer characterization, most applications have been limited to synthetic studies or twodimensional problems. In this study, we applied ensemble-based techniques adapted for parameter estimation, including the p-space ensemble Kalman filter and ensemble smoother, for assimilating field tracer experimental data obtained from the Integrated Field Research Challenge (IFRC) site at the Hanford 300 Area. The forward problem was simulated using the massively parallel three-dimensional flow and transport code PFLOTRAN to effectively deal with the highly transient flow boundary conditions at the site and to meet the computational demands of ensemble-based methods. This study demonstrates the effectiveness of ensemble-based methods for characterizing a heterogeneous aquifer by assimilating experimental tracer data, with refined prior information obtained from assimilating other types of data available at the site. It is demonstrated that high-performance computing enables the use of increasingly mechanistic nonlinear forward simulations for a complex system within the data assimilation framework with reasonable turnaround time.Citation: Chen, X., G. E. Hammond, C. J. Murray, M. L. Rockhold, V. R. Vermeul, and J. M. Zachara (2013), Application of ensemble-based data assimilation techniques for aquifer characterization using tracer data at Hanford 300 area, Water Resour. Res., 49,[7064][7065][7066][7067][7068][7069][7070][7071][7072][7073][7074][7075][7076]

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Joint inversion in coupled quasi-static poroelasticity

Hesse

Stadler

2014

J. Geophys. Res. Solid Earth

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[1] Geodetic surveys now provide detailed time series maps of anthropogenic land subsidence and uplift due to injection and withdrawal of pore fluids from the subsurface. A coupled poroelastic model allows the integration of geodetic and hydraulic data in a joint inversion and has therefore the potential to improve the characterization of the subsurface and our ability to monitor pore pressure evolution. We formulate a Bayesian inverse problem to infer the lateral permeability variation in an aquifer from geodetic and hydraulic data and from prior information. We compute the maximum a posteriori (MAP) estimate of the posterior permeability distribution and a Gaussian approximation of the posterior. Computing the MAP estimate requires the solution of a large-scale minimization problem subject to the poroelastic equations, for which we propose an efficient Newton-conjugate gradient optimization algorithm. The covariance matrix of the Gaussian approximation of the posterior is given by the inverse Hessian of the log posterior, which we construct by exploiting low-rank properties of the data misfit Hessian. First and second derivatives are computed using adjoints of the time-dependent poroelastic equations, allowing us to fully exploit transient data. Using three increasingly complex model problems, we find the following general properties of poroelastic inversions: Augmenting standard hydraulic well data by surface deformation data improves the aquifer characterization. Surface deformation contributes the most in shallow aquifers but provides useful information even for the characterization of aquifers down to 1 km. In general, it is more difficult to infer high-permeability regions, and their characterization requires frequent measurement to resolve the associated short-response timescales. In horizontal aquifers, the vertical component of the surface deformation provides a smoothed image of the pressure distribution in the aquifer. Provided that the mechanical properties are known, coupled poroelastic inversion is therefore a promising approach to detect flow barriers and to monitor pore pressure evolution.

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A Bayesian approach for inverse modeling, data assimilation, and conditional simulation of spatial random fields

Cited by 82 publications

References 70 publications

Probabilistic inference of multi‐Gaussian fields from indirect hydrological data using circulant embedding and dimensionality reduction

Probabilistic inference of multi‐Gaussian fields from indirect hydrological data using circulant embedding and dimensionality reduction

Application of ensemble-based data assimilation techniques for aquifer characterization using tracer data at Hanford 300 area

Joint inversion in coupled quasi-static poroelasticity

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