[Paper Review] A note on extended full waveform inversion

Tristan van Leeuwen|arXiv (Cornell University)|Mar 31, 2019

Seismic Imaging and Inversion Techniques16 references4 citations

TL;DR

This paper presents an equivalent reformulation of extended full waveform inversion (FWI) as a conventional FWI with a medium-dependent residual weighting matrix, $Σ(\mathbf{m}) + \Sigma_m$, derived from the physics and measurement uncertainties. The key contribution is showing that this reweighting reduces non-linearity and enables efficient implementation using standard FWI workflows with only an additional weight matrix per iteration.

ABSTRACT

Full waveform inversion (FWI) aims at estimating subsurface medium properties from measured seismic data. It is usually cast as a non-linear least-squares problem that incorporates uncertainties in the measurements. In exploration seismology, extended formulations of FWI that allow for uncertaties in the physics have been proposed. Even when the physics is modelled accurately, these extensions have been shown to be beneficial because they reduce the non-lineary of the resulting data-fitting problem. In this note, I derive an alternative (but equivalent) formulation of extended full waveform inversion. This re-formulation takes the form of a conventional FWI formulation that includes a medium-dependent weight on the residuals. I discuss the implications of this re-formulation and illustrate its properties with a simple numerical example.

Motivation & Objective

• To provide an alternative, equivalent formulation of extended full waveform inversion (FWI) that retains its benefits while simplifying implementation.
• To demonstrate that the extended FWI formulation can be recast as a conventional FWI with a parameter-dependent residual weighting matrix.
• To show that this reweighting reduces the non-linearity of the data-fitting problem, improving convergence properties.
• To enable practical implementation by leveraging existing FWI software stacks with minimal modifications.
• To connect the extended FWI framework with Bayesian uncertainty quantification through the marginal posterior covariance.

Proposed method

• Derives an equivalent reduced formulation of extended FWI by eliminating the wavefield variable $\mathbf{u}$, leading to an objective function with a medium-dependent weight matrix $\Sigma(\mathbf{m}) + \Sigma_m$.
• Expresses the residual weighting matrix as $\Sigma(\mathbf{m}) = P(A^*(\mathbf{m})\Sigma_p^{-1}A(\mathbf{m}))^{-1}P^*$, representing the correlation of receiver-side Green's functions.
• Derives a gradient expression similar to conventional FWI, requiring only one additional forward solve per iteration.
• Proposes practical approximations for $\Sigma(\mathbf{m})$, including randomized matrix probing, data-based covariance estimation, and analytic forms for simple models.
• Uses a toy 1D acoustic problem with varying velocity to numerically illustrate the reduced non-linearity in the new objective function.
• Demonstrates equivalence between the extended FWI and the residual-weighted FWI formulation in both time and frequency domains.

Experimental results

Research questions

• RQ1 Can extended FWI be reformulated as a conventional FWI with a medium-dependent residual weighting matrix?
• RQ2 How does the inclusion of $\Sigma(\mathbf{m})$ affect the non-linearity of the FWI objective function?
• RQ3 What is the computational cost and practical feasibility of implementing the new residual-weighting scheme in standard FWI workflows?
• RQ4 Can the residual-weighting matrix $\Sigma(\mathbf{m})$ be efficiently approximated without full matrix inversion?
• RQ5 What is the connection between the new formulation and Bayesian uncertainty quantification in FWI?

Key findings

• The extended FWI formulation is mathematically equivalent to a conventional FWI with a parameter-dependent residual weighting matrix $\Sigma(\mathbf{m}) + \Sigma_m$, which accounts for both process and measurement uncertainties.
• The new formulation reduces the non-linearity of the data-fitting problem, as demonstrated by a numerical toy example showing smoother objective function contours.
• The gradient of the new objective can be computed with only one additional forward simulation, making it computationally efficient for implementation.
• The residual weighting matrix $\Sigma(\mathbf{m})$ can be interpreted as the covariance of the marginal posterior distribution in a Bayesian FWI framework.
• Approximations such as data-based covariance estimation and randomized probing can be used to compute $\Sigma(\mathbf{m})$ efficiently in practice.
• The reformulation reveals a direct connection between extended FWI and the equation error method, especially when the sampling operator $P$ is invertible.