Lift and Relax for PDE-Constrained Inverse Problems in Seismic Imaging

We present lift and relax for waveform inversion (LRWI), an approach that mitigates the local minima issue in seismic full waveform inversion (FWI) via a combination of two convexification techniques. The first technique (Lift) extends the set of unknown variables to their products, 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 relax the strict wave-equation constraint-a technique known as wavefield reconstruction inversion (WRI), which introduces 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 the inverse problem as a set of constraints on a rank-2 moment matrix. Such a lifting strategy permits good data and wave equation fits throughout the inversion process while leaving the numerical rank of the rank-2 moment matrix to be minimized down to one. Moreover, LRWI does not require adjoint wavefield to compute the gradient, which mitigates computational burdens. Numerical examples indicate that starting with a poor initial model, LRWI can conduct successful inversions with a starting frequency that is higher than that required by FWI and WRI.