  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 flavors 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 (LSM). 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 datasets and a field dataset.
