Probabilistic inversion of geoelectric and induced polarization measurements on reduced model spaces using Hamiltonian Monte Carlo

The probabilistic formulation of geoelectric and induced polarization inverse problems using Bayes’ theorem inherently accounts for data errors and uncertainties in the prior assumptions, both of which are propagated naturally into the solution. Due to the non-linearity of the physics underlying the geoelectric forward calculation, the inverse problem must be solved numerically. Markov chain Monte Carlo (MCMC) methods provide the capability to create a sample of the corresponding posterior distribution, based on which statistical estimators of interest can be approximated. In a typical geoelectric imaging application, the subsurface is discretized as a 2-D mesh with the model parameters representing the averaged values of the imaged electrical conductivity within the individual cells. The resulting model space is often of high dimensionality and usually insufficiently resolved by the measurements, posing a challenge to the efficient application of MCMC methods. In our work, we use the Hamiltonian Monte Carlo (HMC) method to sample from the posterior distribution and operate on a reduced model space to enhance the efficiency of the inversion. The basis of the reduced model space is constructed via a principal component analysis of the model prior term. We consider different resolution measures to ensure that the information lost by operating in the reduced model space is negligible. In addition to the inversion of electrical resistivity tomography measurements in real variables, we also demonstrate the model space reduction and subsequent application of HMC for the solution of the complex resistivity tomography inverse problem in complex variables, imaging the distribution of the complex electrical conductivity in the subsurface. This study contributes to the needed increase of uncertainty quantification in the inversion of geoelectric and induced polarization measurements, aiming to provide a reliable basis for the processing and interpretation of geophysical imaging results.