Robust full waveform inversion with deep Hessian deblurring

SUMMARY Full waveform inversion (FWI) is a technique widely used in geophysics to obtain high-resolution subsurface velocity models from waveform seismic data. Due to its large computation cost, most flavours of FWI rely only on the computation of the gradient of the loss function to estimate the update direction, therefore ignoring the contribution of the Hessian. Depending on the level of computational resources one can afford, an approximate of the inverse of the Hessian can be calculated and used to speed up the convergence of FWI towards the global (or a plausible local) minimum. In this work, we propose to use an approximate Hessian computed from a linearization of the wave equation as commonly done in least-squares migration. More precisely, we rely on the link between a migrated image and a doubly migrated image (i.e. an image obtained by demigration–migration of the migrated image) to estimate the inverse of the Hessian. However, instead of using non-stationary compact filters to link the two images and approximate the Hessian, we propose to use a deep neural network to directly learn the mapping between the FWI gradient (output) and its Hessian (blurred) counterpart (input). By doing so, the network learns to act as an approximate inverse Hessian: as such, when the trained network is applied to the FWI gradient, an enhanced update direction is obtained, which is shown to be beneficial for the convergence of FWI. The weights of the trained (deblurring) network are then transferred to the next FWI iteration to expedite convergence. We demonstrate the effectiveness of the proposed approach on two synthetic data sets and a field data set.