Time-domain elastic Gauss–Newton full-waveform inversion: a matrix-free approach

Abstract

SUMMARY We present a time-domain matrix-free elastic Gauss–Newton full-waveform inversion (FWI) algorithm. Our algorithm consists of a Gauss–Newton update with a search direction calculated via elastic least-squares reverse time migration (LSRTM). The conjugate gradient least-squares (CGLS) method solves the LSRTM problem with forward and adjoint operators derived via the elastic Born approximation. The Hessian of the Gauss–Newton method is never explicitly formed or saved in memory. In other words, the CGLS algorithm solves for the Gauss–Newton direction via the application of implicit-form forward and adjoint operators which are equivalent to elastic Born modelling and elastic reverse time migration, respectively. We provide numerical examples to test the proposed algorithm where we invert for P- and S-wave velocities simultaneously. The proposed algorithm performs positively on mid-size problems where we report solutions of slight improvement than those computed using the conventional non-linear conjugate gradient method. In spite of the aforementioned limited gain, the theory developed in this paper contributes to a better understanding of time-domain elastic Gauss–Newton FWI.