Multiplier waveform inversion: A reduced-space full-waveform inversion by the method of multipliers

Abstract

Full-waveform inversion (FWI) is a nonlinear optimization problem that addresses the estimation of subsurface model parameters by matching the predicted to the observed seismograms. We formulate FWI as a constrained optimization problem in which the regularization term is minimized subject to the nonlinear data matching constraints. Unlike standard FWI, which solves this regularized problem with a penalty method, we use an augmented Lagrangian formulation in the framework of the method of multipliers (MM). This leads to a two-step recursive algorithm, which is called multiplier waveform inversion (MWI). First, the primal step solves an unconstrained minimization problem, which is similar to the standard FWI but with a modified data term. These modified data are obtained by adding the Lagrange multipliers to the data residuals of the classical FWI. Then, the dual step updates the Lagrange multipliers with basic gradient ascent steps. In this framework, they reduce to the running sum of the data residuals of the previous iterations. The performance of the overall algorithm is improved by considering that the MM does not require the exact solution of the primal step at each iteration. In fact, convergence is attained when the primal subproblem is solved with one (without an inner loop) iteration of a gradient-based method at each iteration of the MWI. The new algorithm can be easily implemented in existing FWI codes without computational overhead by adding the Lagrange multipliers to the data residuals at each iteration. We find with numerical examples that MWI has a faster convergence speed and much improved stability compared with FWI and can converge to an accurate solution of the inverse problem in the absence of low-frequency data, even with a constant step size.