Model inference and uncertainty quantification in complex resistivity imaging

Abstract

This thesis contributes to the solution of the complex resistivity imaging (CRI) problem by introducing an inversion workflow that emphasizes accurate model inference and uncertainty quantification in a probabilistic framework. In applied geophysics, CRI is a frequency-domain (FD) technique for the analysis of tomographic induced polarization (IP) measurements, inferring the complex electrical conductivity distribution in the subsurface from measurements of the complex electrical impedance. Applications can be found across earth sciences. As earth scientists gain a better understanding of how the macroscopic geoelectric properties of earth materials relate to the structures and mechanisms at the pore scale, CRI must meet the increasing demand for accuracy and uncertainty quantification, to ensure that advances made in laboratory studies translate into the field setting.<br /> Despite an increase in the practical utilization of FD measurements during recent years, time-domain (TD) measurements remain the predominant methodology used in field surveys. The first contribution of this thesis is an approach for the conversion of IP measurements from the TD into the FD, which is based on the Debye decomposition of the transients into relaxation-time distributions and the subsequent calculation of the equivalent spectra. The conversion scheme is tested in a synthetic study, confirming its accuracy and the validity of propagated measurement uncertainties. The field application is demonstrated on a data set from Kamchatka (Russia), followed by the inversion of the obtained complex electrical impedances into subsurface images using the established CRI technique.<br /> The second contribution of this thesis introduces a probabilistic framework for the solution of non-linear geophysical inverse problems in complex variables, specifically focusing on the application to CRI. By formulating the likelihood and prior terms as complex probability distributions and combining them into a posterior distribution using Bayes' theorem, the approach can simultaneously account for individual data errors of the real and imaginary data parts, independently regularize the real and imaginary parts of the complex model, and still take into account cross-sensitivities that originate from a complex forward calculation. The complex conductivity image with the highest probability is determined using a Gauss-Newton scheme. The variances and covariances of the inversion result are approximated locally under the simplifying assumption of a linearized forward calculation and normally distributed model parameters. In a synthetic study, the advantages of the probabilistic framework over the established inversion approach are demonstrated.<br /> In the third contribution, the probabilistic framework is used as the basis for a probabilistic inversion of CR measurements using the Hamiltonian Monte Carlo (HMC) method, aiming at a comprehensive characterization of the posterior distribution and an accurate global uncertainty quantification, both of which have not been fully achieved within the second contribution. Convergence criteria are monitored to assess the quality of the probabilistic inversion result, and the final sample is analyzed. Based on the HMC inversion result, the validity of the locally approximated variances and covariances obtained on the basis of the deterministic inversion result is assessed.<br /> The collective contribution of this thesis is an enhanced workflow for CRI based on TD and FD IP measurements.