Robust elastic full-waveform inversion using an alternating direction method of multipliers with reconstructed wavefields

TL;DR

This paper presents a robust elastic full-waveform inversion (EFWI) framework based on the alternating direction method of multipliers (ADMM) with reconstructed wavefields. It introduces a three-block ADMM scheme comprising wavefield reconstruction, model estimation, and dual ascent, and exploits a block-diagonal Gauss-Newton Hessian to suppress inter-parameter cross-talk while accommodating physical constraints through convex sets. A damping strategy for Lagrange multipliers and source-sketching are proposed to improve convergence efficiency and stability, enabling large-scale and noisy-data applications. Numerical experiments on a double circular toy model and a SEG/EAGE overthrust model demonstrate superior convergence, resilience to rough initial models, and enhanced resolution compared with classical FWI and WRI, with successful handling of multi-physical constraints and free-surface effects.

Abstract

Elastic full-waveform inversion (EFWI) is a process used to estimate subsurface properties by fitting seismic data while satisfying wave propagation physics. The problem is formulated as a least-squares data fitting minimization problem with two sets of constraints: Partial-differential equation (PDE) constraints governing elastic wave propagation and physical model constraints implementing prior information. The alternating direction method of multipliers is used to solve the problem, resulting in an iterative algorithm with well-conditioned subproblems. Although wavefield reconstruction is the most challenging part of the iteration, sparse linear algebra techniques can be used for moderate-sized problems and frequency domain formulations. The Hessian matrix is blocky with diagonal blocks, making model updates fast. Gradient ascent is used to update Lagrange multipliers by summing PDE violations. Various numerical examples are used to investigate algorithmic components, including model parameterizations, physical model constraints, the role of the Hessian matrix in suppressing interparameter cross-talk, computational efficiency with the source sketching method, and the effect of noise and near-surface effects.

Figures (25)

• Figure 1: The sets $C_1$ and $C_2$ and the related parameters. The filled area shows the desired set.
• Figure 2: Hessian structure for two-parameters problem. The main-diagonal blocks and off-diagonal blocks of $\beta \bold{L}^{T}\bold{L}$ respectively include $\frac{\partial^2 \mathcal{L}}{\partial \bold{m}_{i} \partial \bold{m}_{i}}$ and $\frac{\partial^2 \mathcal{L}}{\partial \bold{m}_{i} \partial \bold{m}_{j}}$, in which $i,j$ denote the index of each parameter class.
• Figure 3: Double circular model. (a-b) True ($\bold{V}_\text{P}$, $\bold{V}_\text{S}$) and ($\boldsymbol{\lambda}, \boldsymbol{\mu}$) models, in which parameters with superscripts (b) indicate the background values. The estimated models obtained with parameterization ($\boldsymbol{\lambda}, \boldsymbol{\mu}$) (c-d) , ($\bold{V}_\text{P}$, $\bold{V}_\text{S}$) (e-f) and ($\bold{V}_\text{P}^2, \bold{V}_\text{S}^2$) (g-h) after 70 iterations.
• Figure 4: Double circular model. The evolution of the computed model errors during iteration for different model parameterizations: (a) ($\boldsymbol{\lambda}, \bold{V}_\text{P}, \bold{V}_\text{P}^{2}$), and (b) ($\boldsymbol{\mu}, \bold{V}_\text{S}, \bold{V}_\text{S}^{2}$).
• Figure 5: Sensitivity analysis for $(\boldsymbol{\lambda}, \boldsymbol{\mu})$ and $(\bold{V}_\text{P}^2, \bold{V}_\text{S}^2)$ parameterizations. Relative error in the gradient vector versus the error in each parameter for (a) $(\boldsymbol{\lambda}, \boldsymbol{\mu})$ and (b) $(\bold{V}_\text{P}^2, \bold{V}_\text{S}^2)$. The condition number of the Hessian matrix versus relative error of the input parameters for (c) $(\boldsymbol{\lambda}, \boldsymbol{\mu})$ and (d) $(\bold{V}_\text{P}^2, \bold{V}_\text{S}^2)$.