Full-waveform inversion (FWI) is a nonlinear optimisation procedure, seeking to match synthetically-generated seismograms with those observed in field data by iteratively updating a model of the subsurface seismic parameters, typically compressional wave (P-wave) velocity.
Advances in high-performance computing have made FWI of 3-dimensional models feasible, but the low sensitivity of the objective function to deeper, low-wavenumber components of velocity makes these difficult to recover using FWI relative to more traditional, less automated, techniques.
While the use of inadequate physics during the synthetic modelling stage is a contributing factor, I propose that this weakness is substantially one of ill-conditioning, and that efforts to remedy it should focus on the development of both more efficient seismic modelling techniques, and more sophisticated preconditioners for the optimisation iterations. I demonstrate that the problem of poor low-wavenumber velocity recovery can be reproduced in an analogous one-dimensional inversion problem, and that in this case it can be remedied by making full use of the available curvature information, in the form of the Hessian matrix.
In two or three dimensions, this curvature information is prohibitively expensive to obtain and store as part of an inversion procedure. I obtain the complete Hessian matrices for a realistically-sized, two-dimensional, towed-streamer inversion problem at several stages during the inversion and link properties of these matrices to the behaviour of the inversion. Based on these observations, I propose a method for approximating the action of the Hessian and suggest it as a path forward for more sophisticated preconditioning of the inversion process.