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.

翻译：本文针对基于Eikonal方程的反问题提出了一种直接探测方法。对于含点源的Eikonal方程，粘性解代表高频极限下波场从源点到观测点的最小走时。相应的反问题是从分布式点源对应的测量面初至走时数据确定非均匀波速分布，这被称为透射走时层析成像。在低频条件下，重建结果近似频率相关的波速分布。我们分析了Eikonal反问题，证明其具有高度不适定性。随后我们发展了一种直接探测方法，该方法融合了Eikonal方程的解分析与速度模型的多个特征。当波速分布相对于均匀介质存在小扰动时，我们采用滤波反投影法重建非均匀介质。对于高对比度介质，我们假设存在背景介质，并发展基于伴随的反投影方法以识别介质相对于假设背景的变化。