Full Waveform Inversion and the Truncated Newton Method

Full waveform inversion (FWI) is a powerful method for reconstructing subsurface parameters from local measurements of the seismic wavefield. This method consists in minimizing the distance between predicted and recorded data. The predicted data are computed as the solution of a wave-propagation problem. Conventional numerical methods for the solution of FWI problems are gradient-based methods, such as the preconditioned steepest descent, the nonlinear conjugate gradient, or more recently the $l$-BFGS quasi-Newton algorithm. In this study, we investigate the desirability of applying a truncated Newton method to FWI. The inverse Hessian operator plays a crucial role in the parameter reconstruction, as it should help to mitigate finite-frequency effects and to better remove artifacts arising from multiscattered waves. For multiparameter reconstruction, the inverse Hessian operator also offers the possibility of better removing trade-offs due to coupling effects between parameter classes. The truncated Newton method allows us to better account for this operator. This method is based on the computation of the Newton descent direction by solving the corresponding linear system through the conjugate gradient method. The large-scale nature of FWI problems requires us, however, to carefully implement this method to avoid prohibitive computational costs. First, this requires working in a matrix-free formalism and the capability of efficiently computing Hessian-vector products. For this purpose, we propose general second-order adjoint state formulas. Second, special attention must be paid to defining the stopping criterion for the inner linear iterations associated with the computation of the Newton descent direction. We propose several possibilities and establish a theoretical link between the Steihaug--Toint method, based on trust regions, and the Eisenstat and Walker stopping criterion, designed for a method globalized by linesearch. We investigate the application of the truncated Newton method to two case studies. The first is a standard case study in seismic imaging based on the Marmousi model. The second is inspired by a near-surface imaging problem for the reconstruction of high-velocity structures. In the latter case, we demonstrate that the presence of large amplitude multiscattered waves prevents standard methods from converging, while the truncated Newton method provides more reliable results.