Using deep generative neural networks to account for model errors in Markov chain Monte Carlo inversion

Abstract

Summary Most geophysical inverse problems are nonlinear and rely upon numerical forward solvers involving discretization and simplified representations of the underlying physics. As a result, forward modeling errors are inevitable. In practice, such model errors tend to be either completely ignored, which leads to biased and over-confident inversion results, or only partly taken into account using restrictive Gaussian assumptions. Here, we rely on deep generative neural networks to learn problem-specific low-dimensional probabilistic representations of the discrepancy between high-fidelity and low-fidelity forward solvers. These representations are then used to probabilistically invert for the model error jointly with the target geophysical property field, using the computationally-cheap, low-fidelity forward solver. To this end, we combine a Markov-chain-Monte-Carlo (MCMC) inversion algorithm with a trained convolutional neural network of the spatial generative adversarial network (SGAN) type, whereby at each MCMC step, the simulated low-fidelity forward response is corrected using a proposed model-error realization. Considering the crosshole ground-penetrating radar traveltime tomography inverse problem, we train SGAN networks on traveltime discrepancy images between: (1) curved-ray (high fidelity) and straight-ray (low fidelity) forward solvers; and (2) finite-difference-time-domain (high fidelity) and straight-ray (low fidelity) forward solvers. We demonstrate that the SGAN is able to learn the spatial statistics of the model error and that suitable representations of both the subsurface model and model error can be recovered by MCMC. In comparison with inversion results obtained when model errors are either ignored or approximated by a Gaussian distribution, we find that our method has lower posterior parameter bias and better explains the observed traveltime data. Our method is most advantageous when high-fidelity forward solvers involve heavy computational costs and the Gaussian assumption of model errors is inappropriate. Unstable MCMC convergence due to nonlinearities introduced by our method remain a challenge to be addressed in future work.

Accounting for Modeling Errors in Linear Inversion of Crosshole Ground‐Penetrating Radar Amplitude Data: Detecting Sand in Clayey Till

“…We use N = 800 realizations, since we experienced that above this number the obtained inversion tomograms did not change significantly. This is consistent with Hansen et al (2014) who used N = 600 realizations as a basis for the model error characterization and Levy et al (2021) who chose to use N = 800 samples.…”

“…(2014) who used N = 600 realizations as a basis for the model error characterization and Levy et al. (2021) who chose to use N = 800 samples.…”

“…Choosing such an approximate model for the forward problem will give rise to modeling errors, which will, if their magnitude is significant compared to the measurement errors, introduce systematic errors in the solution space. In recent years, the importance of accounting for modeling errors when using approximate forward solvers has been acknowledged (Hansen & Cordua, 2017; Hansen et al., 2014; Köpke et al., 2018; Levy et al., 2021; Linde et al., 2017, etc. ).…”

“…Finally, Levy et al. (2021) used a neural network approach to handle the modeling error in a Monte Carlo Markov Chain (MCMC) based inversion of GPR traveltime. Regardless of the fidelity of the best forward solver, model discrepancy, that is, the difference between reality and the model output, may persist (Brynjarsdóttir & Ohagan, 2014), and therefore we can never expect to account for all errors with these approaches.…”

“…Köpke et al (2018) accounted for the modeling error term during the inversion through occasional runs of the more-accurate forward solver, whereas Köpke et al (2019) used principal component analysis on a number of modeling error realizations and constructed a general description prior to the inversion. Finally, Levy et al (2021) used a neural network approach to handle the modeling error in a Monte Carlo Markov Chain (MCMC) based inversion of GPR traveltime. Regardless of the fidelity of the best forward solver, model discrepancy, that is, the difference between reality and the model output, may persist (Brynjarsdóttir & Ohagan, 2014), and therefore we can never expect to account for all errors with these approaches.…”