Efficient uncertainty analysis of the 3D electrical tomography inverse problem

Abstract

We have evaluated the uncertainty analysis of the 3D electrical tomography inverse problem using model reduction via singular-value decomposition and performed sampling of the nonlinear equivalence region via an explorative member of the particle swarm optimization (PSO) family. The procedure begins with the local inversion of the observed data to find a good resistivity model located in the nonlinear equivalence region. Then, the dimensionality is reduced via the spectral decomposition of the 3D geophysical model. Finally, the exploration of the uncertainty space is performed via an exploratory version of PSO (RR-PSO). This sampling methodology does not prejudge where the initial model comes from as long as this model has a geologic meaning. The 3D subsurface conductivity distribution is arranged as a 2D matrix by ordering the conductivity values contained in a given earth section as a column array and stacking parallel sections as columns of the matrix. There are three basic modes of ordering: mode 1 and mode 2, by using vertical sections in two perpendicular directions, and mode 3, by using horizontal sections. The spectral decomposition is then performed using these three 2D modes. Using this approach, it is possible to sample the uncertainty space of the 3D electrical resistivity inverse problem very efficiently. This methodology is intrinsically parallelizable and could be run for different initial models simultaneously. We found the application to a synthetic data set that is well-known in the literature related to this subject, obtaining a set of surviving geophysical models located in the nonlinear equivalence region that can be used to approximate numerically the posterior distribution of the geophysical model parameters (frequentist approach). Based on these models, it is possible to perform the probabilistic segmentation of the inverse solution found, meanwhile answering geophysical questions with its corresponding uncertainty assessment. This methodology has a general character could be applied to any other 3D nonlinear inverse problems by implementing their corresponding forward model.