A modified, sparsity promoting, Gauss-Newton algorithm for seismic waveform inversion

Images obtained from seismic data are used by the oil and gas industry for geophysical exploration. Cutting-edge methods for transforming the data into interpretable images are moving away from linear approximations and high-frequency asymptotics towards Full Waveform Inversion (FWI), a nonlinear data-fitting procedure based on full data modeling using the wave-equation. The size of the problem, the nonlinearity of the forward model, and ill-posedness of the formulation all contribute to a pressing need for fast algorithms and novel regularization techniques to speed up and improve inversion results. In this paper, we design a modified Gauss-Newton algorithm to solve the PDE- constrained optimization problem using ideas from stochastic optimization and compressive sensing. More specifically, we replace the Gauss-Newton subproblems by randomly subsampled, -$\ell_1$ regularized subproblems. This allows us us significantly reduce the computational cost of calculating the updates and exploit the compressibility of wavefields in Curvelets. We explain the relationships and connections between the new method and stochastic optimization and compressive sensing (CS), and demonstrate the efficacy of the new method on a large-scale synthetic seismic example.