A Direct Probing Method of an Inverse Problem for the Eikonal Equation

.In this paper, we propose a direct probing method to solve the inverse problem of the Eikonal equation. This problem involves the determination of the inhomogeneous wave-speed distribution from first-arrival time data at measurement surfaces corresponding to distributed point sources. The viscosity solution of the point-source Eikonal equation represents the least traveltime of wave fields from the source to the point at the high-frequency limit. We show that this inverse problem is highly ill-posed. To address this issue, we develop a direct probing method that incorporates solution analysis of the Eikonal equation and several aspects of the velocity models. Specifically, we use the filtered back-projection method to reconstruct the inhomogeneous wave-speed distribution when it has a small variation from the homogeneous medium. For the scenarios involving high-contrast media, we assume a background medium and develop an adjoint-based back-projection method to identify the variations of the medium from the assumed background.