Enabling numerically exact local solver for waveform inversion—a low-rank approach

As in many fields, in seismic imaging, the data in the field is collected over a relatively large medium even though only a part of that medium is truly of interest. This results in significant waste in computation as a typical inversion algorithm requires many solutions of the wave equation throughout the entire domain, even if only a small part of the domain is being updated. One way to mitigate this is to use a numerically exact local wave equation solver to perform waveform inversions in an area of interest, where the idea is to compute accurate solutions of the wave equation within a subdomain of interest. Although such solvers exist, many require the computation of Green’s function matrices in the background domain. For large-scale seismic data acquisition, the computation of the Green’s function matrices is prohibitively expensive since it involves solving thousands of partial differential equations in the background model. To mitigate this, in this work, we propose to exploit the low-rank structure of the full subsurface Green’s function. Using carefully selected 2D stylized models, we first show that the full subsurface Green’s function tensor organized as a matrix exhibits the low-rank structure in a transform domain. We then propose a randomized singular value decomposition–based framework to compute the low-rank approximation of the Green’s functions, where the cost of wave equation solves depends on the rank of the underlying Green’s function matrix instead of the number of grid points at the surface of the background model and on the boundary of the local domain. Next, we validate the proposed framework by performing time-lapse waveform inversion using the 2D Marmousi model. Finally, we demonstrate a rank-minimization–based framework to compute the low-rank factorized form of the Green’s function matrices in large-scale 3D seismic data acquisition.