[Paper Review] Seismic Imaging and Optimal Transport

Björn Engquist, Yunan Yang|arXiv (Cornell University)|Aug 14, 2018

Seismic Imaging and Inversion Techniques3 references4 citations

TL;DR

This paper proposes using the quadratic Wasserstein metric ($W_2$) as a misfit function in Full Waveform Inversion (FWI) to overcome the local minima problem inherent in traditional $L^2$-norm methods. By leveraging optimal transport theory and solving the Monge-Ampère equation numerically, $W_2$ provides better convexity and robustness to noise, enabling accurate inversion even from highly smoothed initial models, as demonstrated in 2D synthetic Marmousi and layered models with trace-by-trace $W_2$ computation outperforming $L^2$ in convergence and artifact suppression.

ABSTRACT

Seismology has been an active science for a long time. It changed character about 50 years ago when the earth's vibrations could be measured on the surface more accurately and more frequently in space and time. The full wave field could be determined, and partial differential equations (PDE) started to be used in the inverse process of finding properties of the interior of the earth. We will briefly review earlier techniques but mainly focus on Full Waveform Inversion (FWI) for the acoustic formulation. FWI is a PDE constrained optimization in which the variable velocity in a forward wave equation is adjusted such that the solution matches measured data on the surface. The minimization of the mismatch is usually coupled with the adjoint state method, which also includes the solution to an adjoint wave equation. The least-squares norm is the conventional objective function measuring the difference between simulated and measured data, but it often results in the minimization trapped in local minima. One way to mitigate this is by selecting another misfit function with better convexity properties. Here we propose using the quadratic Wasserstein metric as a new misfit function in FWI. The optimal map defining the quadratic Wasserstein metric can be computed by solving a Monge-Ampere equation. Theorems pointing to the advantages of using optimal transport over the least-squares norm will be discussed, and a number of large-scale computational examples will be presented.

Motivation & Objective

• To address the persistent challenge of local minima in Full Waveform Inversion (FWI) using conventional $L^2$ misfit functions.
• To improve the convexity and robustness of FWI by replacing the $L^2$ norm with the quadratic Wasserstein metric ($W_2$).
• To develop and validate numerical methods for computing $W_2$ via the Monge-Ampère equation in seismic inversion contexts.
• To demonstrate the effectiveness of $W_2$ in recovering accurate subsurface velocity models from noisy and highly smoothed initial data.
• To compare trace-by-trace $W_2$ with global $W_2$ and $L^2$ in terms of convergence speed, artifact suppression, and noise insensitivity.

Proposed method

• Proposes the quadratic Wasserstein metric ($W_2$) as a new misfit function in FWI, replacing the standard $L^2$ norm.
• Uses the Monge-Ampère equation to compute the optimal transport map that defines $W_2$, enabling gradient computation via the adjoint-state method.
• Applies a finite difference solver to numerically approximate solutions to the Monge-Ampère equation in each FWI iteration.
• Introduces a trace-by-trace $W_2$ approach that computes $W_2$ independently for each receiver trace using exact 1D optimal transport solutions.
• Employs L-BFGS as the optimization algorithm for model updates, with adjoint sources derived from the $W_2$ gradient.
• Applies data filtering to ensure regularity for numerical stability in solving the Monge-Ampère equation, particularly in higher dimensions.

Experimental results

Research questions

• RQ1 Can the $W_2$ metric significantly reduce the likelihood of converging to local minima in FWI compared to the $L^2$ norm?
• RQ2 How does the computational cost and accuracy of global $W_2$ compare to the trace-by-trace $W_2$ approximation in 2D seismic models?
• RQ3 To what extent does $W_2$ improve robustness to noise in seismic data, especially when signal-to-noise ratio is low?
• RQ4 How does the $W_2$-based FWI perform when initialized from a highly smoothed velocity model far from the true solution?
• RQ5 What is the impact of data regularization on the numerical solution of the Monge-Ampère equation in practical FWI applications?

Key findings

• The $W_2$-based FWI successfully avoids local minima, producing artifact-free inversion results even from a highly smoothed initial model, whereas $L^2$ produces spurious high-frequency artifacts.
• Trace-by-trace $W_2$ achieves a relative misfit reduction to 0.1 in just 20 iterations, while $L^2$ converges slowly to a local minimum with persistent artifacts.
• The $W_2$ metric demonstrates insensitivity to noise: with a signal-to-noise ratio of -3.47 dB, the inversion still recovers most features of the true Marmousi model.
• Global $W_2$ computation is limited by the need for regular data and numerical stability of the Monge-Ampère solver, making the trace-by-trace approach more practical for real-world applications.
• The adjoint source for $W_2$ is derived from the solution of the Monge-Ampère equation and involves a non-local, non-linear transformation of the data difference.
• Numerical results show that $W_2$ provides better convexity properties than $L^2$, enabling faster and more reliable convergence in full-waveform inversion.