Constrained electrical resistivity tomography Bayesian inversion using inverse Matérn covariance matrix

Abstract

Bayesian inversion using maximum a posteriori (MAP) estimator is a quantitative approach that has been successfully applied to the electrical resistivity tomography inverse problem. In most approaches, model covariance parameters are generally chosen stationary and isotropic, which assumes statistical homogeneity of studied field.However, the statistical properties of resistivity within the Earth are, in reality, location depend due to spatially varying processes that control the bulk resistivity of rocks, such as water content, porosity, clay content, etc. In order to take into account the spatial variability of the resistivity field, we propose to use the non-stationary Matérn covariance family that is defined through linear stochastic partial differential equations. Two types of prior information are considered, structure orientation and spatially increasing range with increasing depth. The latter is applied successfully on the first synthetic model which aims at retrieving the depth of bedrock and the shape of conductive lens. In the second synthetic example, a conductive dyke model embedded into four layers is used to study the performance of structure orientation. Finally, the proposed approach is used to invert real data measured over an extensively characterized sandy-to-silty aquifer. Structure orientation of this aquifer was firstly determined by applying structure tensor calculated using gradients of gpr image. The introduction of this information gives a resistivity model that is more compatible with the aquifer structure.

Assessing potential impacts of shale gas development on shallow aquifers through upward fluid migration: A multi-disciplinary approach applied to the Utica Shale in eastern Canada

“…First, a classical 1-D layered inversion using seven layers based on data from resistivity logs (see section 5.2) was used, but it did not provide results that seemed to fit with the current knowledge of the study area, particularly in the vicinity of the RJC fault. Second, HTEM data were inverted using laterally constrains (Auken et al, 2005) and Bayesian inversion using non-stationary Matérn matrix (Bouchedda et al, 2017). This allowed for a more laterally continuous and more accurate spatial distribution of the electrical resistivity of the ground.…”

Estimation of a non‐stationary prior covariance from seismic data

“…Most often the statistical properties are assumed to be the same everywhere in the model space, which leads to an implicit assumption of stationarity of the prior model (Bouchedda et al . ). In this paper, we use a linearized Bayesian inversion scheme as described in, for example, Buland and Omre () for prediction.…”

“…This spatial dependency is similarly expected for the statistical properties of the physical parameters for rocks (acoustic impedance, density, porosity, resistivity etc. ), which can generally be considered to be spatially non‐stationary (Bouchedda, Bernard and Gloaguen ; Sabeti et al . ).…”

Hydrogeophysical model calibration and uncertainty analysis via full integration of PEST/PEST++ and COMSOL

“…Previous authors have shown limitations of the capability of geophysical inversion to accurately resolve subsurface features (Day-Lewis et al, 2005;Singha and Moysey, 2006), typically providing a blurry, smoother, and sometimes distorted representation of reality, as well as significant uncertainties in the values of the parameter measured. Although alternative regularization approaches may improve the inverted model (Hermans et al, 2012;Bouchedda et al, 2017;Thibaut et al, 2021), recent studies (e.g. Revil et al, 2017;Brunetti and Linde, 2018;González-Quirós and Comte, 2020) have also shown the importance of conceptual and structural errors in the hydrogeophysical workflow, such as the assumption of homogeneity in heterogeneous systems when parameterizing the petrophysical model or the incorrect selection of the petrophysical model itself.…”