We present Lift and Relax for Waveform Inversion (LRWI), an approach that mitigates the cycle-skipping issue in full waveform inversion via a combination of two convexification techniques. The first technique (Lift) extends the set of variables in the optimization problem to products of those variables, arranged as a moment matrix. This algebraic idea is a celebrated way to replace a hard polynomial optimization problem by a semidefinite programming approximation. Concretely, both the model and the wavefield are lifted from vectors to rank-2 matrices. The second technique (Relax) invites to consider the wave equation, not as a hard constraint, but as a soft constraint to be satisfied only approximately – a technique known as wavefield reconstruction inversion (WRI). WRI weakens wave-equation constraints by introducing wave-equation misfits as a weighted penalty term in the objective function. The relaxed penalty formulation enables balancing the data and wave-equation misfits by tuning a penalty parameter. Together, “Lift” and “Relax” help reformulate FWI as a set of constraints on a rank-2 moment matrix in a higher dimensional space. Such a lifting strategy permits a good data and wave-equation fit throughout the inversion process, while leaving the numerical rank of the rank-2 moment matrix to be minimized down to one. Numerical examples indicate that the proposed rank-2 lifting approach with the wave-equation relaxation is capable of reconstructing geologically plausible subsurface models from poor initial models without stagnation at suboptimal models encountered by conventional FWI.
Presentation Date: Monday, October 12, 2020
Session Start Time: 1:50 PM
Presentation Time: 3:55 PM
Location: Poster Station 3
Presentation Type: Poster
