Bayesian inversion with total variation prior for discrete geologic structure identification

Abstract

.[1] The characterization of geologic heterogeneity that affects flow and transport phenomena in the subsurface is essential for cost-effective and reliable decision-making in applications such as groundwater supply and contaminant cleanup. In the last decades, geostatistical inversion approaches have been widely used to tackle subsurface characterization problems and quantify their corresponding uncertainty. Some wellestablished geostatistical methods use models that assume gradually varying parameters. However, in many cases, the subsurface can often be better represented as consisting of a few relatively uniform geologic facies or zones with abrupt changes at their boundaries. We advance a Bayesian inversion approach with the gradient represented not through a Gaussian but a Laplace prior, also known as total variation prior, for the case that there are reasons to believe that discrete geologic structures with relatively homogeneous properties predominate in the subsurface but their number, locations, and shapes are unknown a priori. Structural parameters (or hyperparameters of the inversion scheme) are determined in a Bayesian framework by maximizing the marginal distribution of these parameters using an expectation-maximization approach; this allows proper weighting of prior versus data information and produces results with realistic uncertainty quantification. We present here three applications of the method: a time-varying extraction rate estimation at a well, a linear cross-well seismic tomography, and a nonlinear hydraulic tomography. These results are compared with those achieved in the classical geostatistical method and it is shown that the Bayesian inversion approach with total variation prior can be a useful tool to identify discrete geologic structures.Citation: Lee, J., and P. K. Kitanidis (2013), Bayesian inversion with total variation prior for discrete geologic structure identification, Water Resour. Res., 49,[7658][7659][7660][7661][7662][7663][7664][7665][7666][7667][7668][7669]

A Sparse Bayesian Imaging Technique for Efficient Recovery of Reservoir Channels With Time-Lapse Seismic Measurements

“…This probabilistic approach is derived from a machine learning algorithm for linear inverse problems and avoids the need for regularization. A similar scheme was adopted in [22] where it was combined with an expectation-maximization (EM) algorithm for maximizing the marginal distribution of the estimated parameters.…”

Reconstruction of channelized geological facies based on RIPless compressed sensing

“…On this, we applied the classical basis pursuit (BP) algorithm (Candès, 2008;Candès et al, 2006a,b;Elad, 2010) on the whole image using the DCT (as explored in , the Haar basis with 5 levels of iterations (Vetterli and Kovacevic, 1995), and the bi-orthogonal Wavelet CDF9/7 (Cohen et al, 1992) with 5 level of iterations (used for image compression in JPEG2000, Wallace, 1991;Vetterli and Kovacevic, 1995). Furthermore, we explore two other regularized solutions for ill-posed image reconstruction: the total variations (TV) minimization (subject to linear measurements) (Needell and Ward, 2013;Lee and Kitanidis, 2013), and the linear least square (LLS) (Elad, 2010, Chapter 1.1) (also explored in .…”

“…In particular, several contributions have been developed in the problem of subsurface flow model calibrations based on non-linear flow measurements (Jafarpour et al, 2010;Jafarpour, 2011;Khaninezhad et al, 2012;Elsheikh et al, 2013;Khaninezhad and Jafarpour, 2014;Lee and Kitanidis, 2013). In these contributions, CS results are used to propose new sparsity promoting algorithms that recover subsurface models based on the formulation of complexity regularization problem.…”

Linear functional minimization for inverse modeling

“…Our implementation uses measurements of a model parameter and a single state, but can be extended to include multiple states. Lee and Kitanidis [2013] have employed the TV regularization as a prior for Bayesian inverse modeling and structure identification in the context of hydrogeological applications. Their work employs a TV prior to sample the Bayesian posterior, and does not use transient measurements of system states.…”