Newton optimization and the Hessian matrix in scale-separation waveform inversion problems with reflected and transmitted waves

Abstract

SUMMARY Full waveform inversion (FWI) is often utilized for determining the medium parameters of the Earth by minimizing the misfit between observed and modelled waves. The conventional formulation of FWI is well suited to utilize transmitted waves to determine long-to-intermediate wavelength properties of the medium. Hovewer, the penetration depth of transmitted waves may be limited for a given acquisition. As a remedy, reflection waveform inversion (RWI) exploits the kinematic information in reflected waves to retrieve these long-wavelength properties, particularly below the maximum illumination depth of transmitted waves. Previous works have demonstrated the benefits of utilizing both transmitted and reflected waves, thereby considering a reconciliation of conventional FWI and RWI. One such formulation is known as joint full waveform inversion (JFWI), where reflected and transmitted waves are separately compared in the misfit functional. A disadvantage of gradient-based optimization schemes in RWI and JFWI is the presence of strong crosstalk due to mapping of reflected data residuals along the entire reflection wavepaths. Because RWI and JFWI require frequent least-squares migration, we are incentivized to determine well-focused search directions that allow large movements between the migration steps. Thus, we propose an improvement to RWI and JFWI by utilizing Newton methods to directly account for crosstalk between parameters. The crosstalk effects are described by the second-order sensitivities of the misfit functional, i.e. the Hessian matrix. We adopt the adjoint-state method to compute the first- and second order sensitivities of the misfit functional for any of these waveform inversion procedures in a unified framework and for arbitrary misfit functionals. Furthermore, we conceptually study the Hessian matrix in least-squares based full-, reflection- and joint full waveform inversion from a data-domain point of view. Synthetic examples and a three-dimensional field data example demonstrate that the truncated Gauss-Newton method is effective in attenuating artefacts due to crosstalk between spatially separated variables, thus substantially improving the spatial focusing of the search directions.