Time domain full waveform inversion with decomposed Gauss-Newton Hessian

Gaoshan Guo , Stephane Operto

Full waveform inversion (FWI), a nonlinear data fitting approach for parameter estimation, is generally implemented with local optimization methods. The convergence of the iterations can be slow when the steepest descent direction provided by the negative gradient of the data misfit function is not preconditioned by the inverse full Newton Hessian or its linear approximation, namely the Gauss-Newton (GN) Hessian. Here, we present a novel generalized (or weighted) form of time-domain FWI allowing for a robust and computationally-efficient implementation of Gauss-Newton FWI. This weighted form of FWI is inspired by the extended-source formulation of FWI. In this framework, the GN Hessian of FWI can be decomposed into a source-side diagonal pseudo-Hessian in the model domain and a receiver-side Hessian in the data domain. The pseudo-Hessian corresponds to the auto-correlation of the so-called virtual sources, which is classically used to precondition the descent direction of FWI by scaling the amplitudes of the gradient. The receiver-side data-domain Hessian represents the Hessian of an underdetermined source problem. This source problem aims at estimating the volume scattering sources that would be generated by the interaction of the true incident wavefields with the unknown perturbation model at the current FWI iteration. The so-called adjoint fields of FWI are the adjoint approximations of these scattering sources. The pseudo-Hessian does not generate computational overhead, while the inverse data-domain Hessian is approximated by a multi-dimensional non-stationary matching filter for each source. Finally, the computational cost of one decomposed GN descent direction is two times that of the steepest descent. Numerical tests validate that the decomposed GN Hessian provides a reliable approximation of the GN Hessian. Alternatively, the weighted gradient by the inverse data-domain Hessian can be used in quasi-Newton methods with the diagonal pseudo-Hessian as preconditioner. Synthetic tests show that the decomposed GN Hessian not only accelerates the convergence of classical FWI but can also reconstruct more accurate velocity models when the initial model is not accurate.