Nonlinear Electrical Resistivity Tomography with a Fourier Neural Operator Surrogate Model

Electrical resistivity tomography (ERT) is commonly applied for shallow subsurface imaging. Inversion techniques generate images of the subsurface resistivity structure to interpret the data, with applications including the imaging of permafrost soils. While linearized inversion is a common method, nonlinear treatment provides advantages in terms of parametrization and model selection. However, it often incurs prohibitive computational costs.

 

Markov Chain Monte Carlo (MCMC) methods offer nonlinear uncertainty quantification for ERT, where the computational cost is dominated by the forward model evaluations. Surrogate models advance the physics forward model with a considerable speedup; therefore, they have the potential to enable MCMC applications for inverse problems that were not previously possible.

 

We introduce a surrogate forward model for 2D ERT based on a Fourier Neural Operator (FNO). This model leverages the FNO's capability to learn and generalize mappings between infinite-dimensional function spaces, making it particularly suitable for solving PDE-driven problems like ERT. Based on the inputs of electrode geometry and subsurface resistivity distribution, FNO predicts potentials from which apparent resistivities are computed. This process reduces evaluation times of a subsurface resistivity distribution from seconds to milliseconds with prediction errors below 5%.

 

This efficiency gain enables applying the FNO in MCMC sampling. We show several examples of MCMC sampling results with simulated data for pole-dipole arrays and realistic subsurface models. The subsurface parametrization of resistivity considers irregular grids based on Gaussian random fields or Voronoi cells. The results demonstrate that nonlinear inversion and uncertainty quantification are computationally feasible for typical field survey scales.