An extended Gauss-Newton method for full waveform inversion

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.

💡 Research Summary

The paper introduces an Extended Gauss‑Newton (EGN) algorithm designed to overcome the longstanding challenges of Full Waveform Inversion (FWI), namely the severe non‑linearity of the inverse problem, the high dimensionality of the model space, and the tendency of Newton‑type methods to become trapped in local minima when low‑wavenumber information or low‑frequency data are missing. The authors begin by reformulating the conventional Gauss‑Newton (GN) system, which normally solves a vector‑valued linear system JᵀJ δm = –Jᵀδd, into an equivalent matrix equation that exploits the Hadamard (element‑wise) product. By expressing the GN Hessian as (SᵀS) ∘ (UᵀU), where S and U are respectively the receiver‑side Green’s function matrix and the source‑side wavefield matrix, they reveal a separable structure that isolates source and receiver contributions.

The key innovation is to relax the search direction from a diagonal matrix (or vector) to a full matrix Δm. This relaxation introduces an extra degree of freedom along the subsurface‑offset axis, effectively sampling the model in a two‑dimensional offset space rather than only along depth. Consequently, the update Δm can be interpreted as a “blur‑deblur” operator that simultaneously regularizes the data residual matrix Δd in both source and receiver dimensions. The resulting linear system can be solved analytically by inverting only two small matrices: one of size Ns × Ns (number of sources) and another of size Nr × Nr (number of receivers). These inverses act as de‑blurring filters along the source and receiver axes, respectively. Unlike traditional extended‑source FWI, which applies a one‑dimensional de‑blurring filter only on the receiver side, EGN employs a two‑dimensional kernel, thereby achieving a more balanced and robust correction of the residuals.

By integrating this relaxed search direction with the separable Hessian, EGN bridges three major FWI frameworks: reduced (standard) FWI, extended‑source FWI, and extended‑model FWI. It inherits the computational efficiency of reduced FWI because only two modest‑size matrix inversions are required per iteration, yet it gains the robustness of extended formulations through the additional offset degree of freedom and the explicit de‑blurring of residuals.

The authors validate the method with three numerical experiments, including a 2‑D Marmousi‑like model and a 3‑D synthetic geological scenario. Comparisons against Preconditioned Steepest Descent, truncated Newton with Conjugate Gradient, and the classic extended‑source approach demonstrate that EGN converges in fewer iterations, yields higher‑fidelity velocity reconstructions, and remains stable even when low‑frequency content is deliberately removed from the data. Complexity analysis shows that the overall cost scales as O(Ns³ + Nr³) for the two matrix inversions, which is negligible compared with the O(Ns · Nr) cost of forward and adjoint wavefield simulations that dominate any FWI workflow. Memory requirements are modest because Δm is stored as a matrix rather than a large vector, and the algorithm is amenable to GPU acceleration for the small‑matrix solves.

In conclusion, the Extended Gauss‑Newton method offers a practical, theoretically sound, and computationally tractable solution to the ill‑posedness and local‑minimum issues that have limited the broader adoption of FWI. By exploiting the separable structure of the GN Hessian and by extending the search direction into the subsurface‑offset domain, EGN achieves a compelling balance between robustness and efficiency, paving the way for more reliable seismic imaging in realistic, low‑frequency‑poor acquisition scenarios. Future work will explore adaptive regularization of the offset space, multi‑frequency continuation strategies, and real‑data applications in exploration seismology.