3D Frequency-Domain Seismic Inversion with Controlled Sloppiness

Seismic waveform inversion aims at obtaining detailed estimates of subsurface medium parameters, such as the spatial distribution of soundspeed, from multi-experiment seismic data. A formulation of this inverse problem in the frequency-domain leads to an optimization problem constrained by a Helmholtz equation with many right-hand-sides. Application of this technique to industry-scale problem faces several challenges: Firstly, we need to solve the Helmholtz equation for high wavenumbers over large computational domains. Secondly, the data consists of many independent experiments, leading to a large number of PDE-solves. This results in high computational complexity both in terms of memory and CPU time as well as i/o costs. Finally, the inverse problem is highly non-linear and a lot of art goes into preprocessing and regularization. Ideally, an inversion needs to be run several times with different initial guesses and/or tuning parameters. In this paper, we discuss the requirements of the various components (PDE-solver, optimization method, ...) when applied to large-scale 3D seismic waveform inversion and combine several existing approaches into a flexible inversion scheme for seismic waveform inversion. The scheme is based on the idea that in the early stages of the inversion we do not need all the data or very accurate PDE-solves. We base our method on an existing preconditioned Krylov solver (CARP-CG) and use ideas from stochastic optimization to formulate a gradient-based (Quasi-Newton) optimization algorithm that works with small subsets of the right-hand-sides and uses inexact PDE solves for the gradient calculations. We proposed novel heuristics to adaptively control both the accuracy and the number of right-hand-sides. We illustrate the algorithms on synthetic benchmark models for which significant computational gains can be made without being sensitive to noise and without loosing accuracy of the inverted model.