An extended Gauss-Newton method for full waveform inversion

TL;DR

The paper tackles the challenge of nonlinear, ill-posed full waveform inversion by extending the Gauss-Newton direction beyond a diagonal constraint, yielding an Extended Gauss-Newton (EGN) formulation. By recasting the GN system as a matrix equation and relaxing diagonality, it derives a separable, explicit solution that deblurs data residuals along source and receiver axes via $\Delta d^e = H_r^{-1}\Delta d H_s^{-1}$ and $\Delta m = S^T \Delta d^e U$, with Hessians $H_r$ and $H_s$. The work further extends to extended-source FWI through a penalty formulation with $Q(\bold{m})^{-1}$ weighting, providing an explicit extended direction $\Delta m = S^T \Delta d^e U_{\beta}$ and connecting reduced, model-extended, and source-extended formulations. Numerical experiments on Camembert, Marmousi, and Overthrust demonstrate that EGN improves robustness and convergence speed, and randomized EGN via sketching significantly reduces computational cost while preserving inversion quality. Overall, EGN offers a computationally efficient, robust framework that unifies and enhances extended and reduced FWI approaches for challenging seismic inversion problems.

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.

Figures (11)

• Figure 1: Simplified subsurface model with a single source and receiver. The black wave represents the incident wave propagating from the source to the $i$-th reflecting point in the subsurface. At the receiver location, two reflected waves are observed. The blue wave represents the reflection from the same $i$-th point, characterized by the diagonal reflection coefficient $\Delta \bold{m}_{ii}$. The red wave represents a reflection from the $j$-th point, characterized by the off-diagonal reflection coefficient $\Delta \bold{m}_{ij}$.
• Figure 2: Subsurface model and pattern in matrix $\Delta\bold{m}$. (a) Visualization of a 5 by 5 subsurface model. The black cell (3, 3) serves as the reference cell. Different colors are assigned to cells neighboring (3, 3) along different directions and distances, highlighting their relationships in the model. (b) The corresponding pattern in matrix $\Delta\bold{m}$, constructed to represent the relationships and positions of cells in the subsurface model. Matrix $\Delta\bold{m}$ has a size of 25 by 25, with each column containing a reshaped column vector from the subsurface model, aligning the respective reference cell along the main diagonal of $\Delta\bold{m}$. The pattern in $\Delta\bold{m}$ reveals the alignment of neighboring cells based on their distances and directions from the reference cell (3, 3).
• Figure 3: Camembert model example. The (a) true velocity model and (b) initial model. (c-h) Final velocity models inferred after 50 iterations of (c) PSD, (d) GN, (e) EGN solving the reduced FWI with $\bold{h}=\bold{0}$, (f) EGN solving the penalty FWI with $\bold{h}=\bold{0}$, (g) EGN solving the reduced FWI with $|\bold{h}|\leq \frac{1}{4}\lambda$, (h) EGN solving the penalty FWI with $|\bold{h}|\leq \frac{1}{4}\lambda$.
• Figure 4: Trajectory of the misfit function versus iteration for the Camembert model obtained by different methods (Figures \ref{['Camembert']}c-f). The curves represent the misfit function evolution for the following methods: PSD (red line), GN (black line), EGN solving the reduced FWI with $\bold{h}= \bold{0}$ (green line), EGN solving the penalty FWI with $\bold{h}= \bold{0}$ (blue line), EGN solving the reduced FWI with $|\bold{h}|\leq \frac{1}{4}\lambda$ (magenta line), and EGN solving the penalty FWI with $|\bold{h}|\leq \frac{1}{4}\lambda$ (cyan line).
• Figure 5: Comparison of model perturbation accumulated in the first 5 iterations for different FWI methods. The figure displays subplots (a) to (f), each corresponding to a different method used in constructing the Camembert model. The rows represent the model perturbation obtained after a specific number of iterations (from the top row, representing the first iteration, to the bottom row, representing the fifth iteration). The methods include (a) PSD, (b) GN, (c) EGN applied to the reduced FWI with $\bold{h}= \bold{0}$, (d) EGN applied to the penalty FWI with $\bold{h}= \bold{0}$, (e) EGN applied to the reduced FWI with $|\bold{h}|\leq \frac{1}{4}\lambda$, and (f) EGN applied to the penalty FWI with $|\bold{h}|\leq \frac{1}{4}\lambda$.