Accounting for Model Errors Using Deep Generative Neural Networks and Markov Chain Monte Carlo Inversion

Abstract:

Most geophysical forward problems are non-linear and prone to errors as a result of discretization or simplification of physical processes. The resulting model errors are difficult to assess and they are often disregarded or accounted for in a simplified manner during inversion by relying on linear least-squares theory. One alternative to account for the model error that arises from using a low-fidelity forward solver is to learn a compact parameterization of the discrepancy between high-fidelity and low-fidelity solvers and to add its parameters to those being inferred. Our approach combines Markov chain Monte Carlo (MCMC) inference with a pre-trained convolutional neural network of the spatial generative adversarial network (SGAN) type. At each MCMC step, the low-fidelity forward solver response is corrected by a model-error realization generated by the pre-trained SGAN. The non-linear transformation between the low-dimensional latent space and the model-error space, through a series of convolution operations, offers a substantial reduction of the number of model error parameters to infer. Moreover, the latent space has localization properties, such that each parameter controls specific regions in the model space, making the inference more efficient. The network was successfully trained on images of anisotropic multi-Gaussian ground-penetrating radar slowness models and associated non-Gaussian model errors describing the discrepancy between two cross-borehole GPR forward solvers: curved-ray (high fidelity) and straight-ray (low fidelity). Our methodology is compared against approaches in which the model error is ignored, as well as simplified by adding a bias correction and an additional covariance distribution to the likelihood function. We conclude that our method is of particular value in moderate to highly nonlinear situations. In the near future, we will use finite-difference time-domain wave field simulations to represent the high fidelity forward model.