Deep generative models in inversion : the impact of the generator's nonlinearity and development of a new approach based on a variational autoencoder

When solving inverse problems in geophysical imaging, deep generative models (DGMs) may be used to enforce the solution to display highly structured spatial patterns which are supported by independent information (e.g. the geological setting) of the subsurface. In such case, inversion may be formulated in a latent space where a lowdimensional parameterization of the patterns is defined and where Markov chain Monte Carlo or gradient-based methods may be applied. However, the generative mapping between the latent and the original (pixel) representations is usually highly nonlinear which may cause some difficulties for inversion, especially for gradientbased methods. In this contribution we review the conceptual framework of inversion with DGMs and propose that this nonlinearity is caused mainly by changes in topology and curvature induced by the generative function. As a result, we identify a conflict between two goals: the accuracy of the generated patterns and the feasibility of gradient-based inversion. In addition, we show how some of the training parameters of a variational autoencoder, which is a particular instance of a DGM, may be chosen so that a tradeoff between these two goals is achieved and acceptable inversion results are obtained with a stochastic gradient-descent scheme. A series of test cases using synthetic models with channel patterns of different complexity and cross-borehole traveltime tomographic data involving both a linear and a nonlinear forward operator show that the proposed method provides useful results and performs better compared to previous approaches using DGMs with gradient-based inversion