A direct probing method of an inverse problem for the Eikonal equation

TL;DR

This paper introduces a direct probing method for the inverse Eikonal problem, enabling the reconstruction of wave-speed distributions from first-arrival time data, with approaches tailored for both small variations and high-contrast media.

Contribution

It develops a novel direct probing approach for the inverse Eikonal problem, combining filtered back projection and adjoint-based methods for different media contrasts.

Findings

Effective reconstruction for small variations using filtered back projection.

Successful identification of high-contrast media variations with adjoint-based methods.

Analysis showing the inverse problem's high ill-posedness.

Abstract

In this paper, we propose a direct probing method for the inverse problem based on the Eikonal equation. For the Eikonal equation with a point source, the viscosity solution represents the least travel time of wave fields from the source to the point at the high-frequency limit. The corresponding inverse problem is to determine the inhomogeneous wave-speed distribution from the first-arrival time data at the measurement surfaces corresponding to distributed point sources, which is called transmission travel-time tomography. At the low-frequency regime, the reconstruction approximates the frequency-depend wave-speed distribution. We analyze the Eikonal inverse problem and show that it is highly ill-posed. Then we developed a direct probing method that incorporates the solution analysis of the Eikonal equation and several aspects of the velocity models. When the wave-speed distribution has a small variation from the homogeneous medium, we reconstruct the inhomogeneous media using the filtered back projection method. For the high-contrast media, we assume a background medium and develop the adjoint-based back projection method to identify the variations of the medium from the assumed background.