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

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