Full Waveform Inversion and the Truncated Newton Method

Abstract

Abstract. Full Waveform Inversion (FWI) is a powerful method for reconstructing subsurface parameters from local measurements of the seismic wavefield. This method consists in minimizing a distance between predicted and recorded data. The predicted data is computed as the solution of a wave propagation problem. Conventional numerical methods for the resolution of FWI problems are gradient-based methods, such as the preconditioned steepest-descent, or more recently the l-BFGS quasi-Newton algorithm. In this study, we investigate the interest of applying a truncated Newton method to FWI. The inverse Hessian operator plays a crucial role in the parameter reconstruction. The truncated Newton method allows one 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 an iterative procedure such as the conjugate gradient method. The large-scale nature of FWI problems requires however to carefully implement this method to avoid prohibitive computational costs. First, this requires to work in a matrix-free formalism, and the capability of computing efficiently Hessian-vector products. To this purpose, we propose general second-order adjoint state formulas. Second, special attention must be payed to define 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 stopping criterion, designed for method globalized by linesearch. We investigate the application of the truncated Newton method to two test cases: the first is a standard test case in seismic imaging based on the Marmousi II model. The second one is inspired by a nearsurface imaging problem for the reconstruction of high velocity structures. In the latter case, we demonstrate that the presence of large amplitude multi-scattered waves prevents standard methods from converging while the truncated Newton method provides more reliable results.

A multigrid solver to the Helmholtz equation with a point source based on travel time and amplitude

“…The Helmholtz equation with a point source is common in geophysical applications, for example, seismic modeling and full-waveform inversion (FWI). [1][2][3][4][5][6][7][8] The Helmholtz equation is accompanied with boundary conditions (BCs), which can be Neumann or Dirichlet for example. In many cases, the equation is involved with absorbing BCs that mimic the propagation of a wave in an open domain.…”

“…The resulting approximation includes only the first-arrival information of the wave propagation, and since (7) does not depend on , this approximation aims to be valid for multiple frequencies. Our approach is different in the way that the amplitude is defined, according to (4) instead of (7). This way, the amplitude is specific for a given frequency and contains all the information of the wave propagation, for example, it includes reflections and interferences.…”

“…However, the methods mentioned above are not suitable for solving (6) for our purpose. To get an accurate solution for the amplitude in (7), the numerical approximation for (⃗ x) has to be very accurate. 33 Because the analytical (⃗ x) is nonsmooth at the point source, the numerical solution of Equation (6) is polluted with errors when it is computed using the aforementioned standard finite difference methods.…”

“…Approximating the second derivative of a nonsmooth function may yield high numerical errors. For this reason, the third-order Lax-Friedrichs scheme, which adds smoothness to 1 , is used in the works of Luo and Qian 32 and Luo et al 33 However, both Equations (7) and (5) show that the problems can be formulated without Δ 1 , and there is no real need to approximate it numerically.…”

Multimodal 3D Shape Reconstruction under Calibration Uncertainty Using Parametric Level Set Methods

“…Introduction. The reconstruction of 3D objects is an important task in many disciplines like computer graphics [1,4,6,52,54], computer vision [27,16,26], geophysics [32], computational biology [41], civil engineering, medical applications, architecture, and archaeology, among others. Typically, we have incomplete data measurements of a physical object, and wish to reconstruct the full and continuous object that corresponds to these measurements.…”