An extended Gauss-Newton method for full waveform inversion

TL;DR

This paper introduces an extended Gauss-Newton method for full waveform inversion that improves robustness and computational efficiency by reformulating the system and relaxing certain constraints, effectively addressing local minima issues.

Contribution

The paper proposes a novel extended Gauss-Newton method that reformulates the FWI system into a matrix form and relaxes diagonality constraints, enhancing robustness and efficiency.

Findings

Numerical demonstrations show improved robustness of the EGN method.

The EGN method simplifies Hessian inversion using small matrix inversions.

The approach effectively combines benefits of model and source extension.

Abstract

Full waveform inversion (FWI) is a large-scale nonlinear ill-posed problem for which computationally expensive Newton-type methods can become trapped in undesirable local minima, particularly when the initial model lacks a low-wavenumber component and the recorded data lacks low-frequency content. A modification to the Gauss-Newton (GN) method is proposed to address these issues. The standard GN system for multisource multireceiver FWI is reformulated into an equivalent matrix equation form, with the solution becoming a diagonal matrix rather than a vector as in the standard system. The search direction is transformed from a vector to a matrix by relaxing the diagonality constraint, effectively adding a degree of freedom to the subsurface offset axis. The relaxed system can be explicitly solved with only the inversion of two small matrices that deblur the data residual matrix along the source and receiver dimensions, which simplifies the inversion of the Hessian matrix. When used to solve the extended source FWI objective function, the Extended GN (EGN) method integrates the benefits of both model and source extension. The EGN method effectively combines the computational effectiveness of the reduced FWI method with the robustness characteristics of extended formulations and offers a promising solution for addressing the challenges of FWI. It bridges the gap between these extended formulations and the reduced FWI method, enhancing inversion robustness while maintaining computational efficiency. The robustness and stability of the EGN algorithm for waveform inversion are demonstrated numerically.