Calculating Ray Paths for First-Arrival Travel Times Using a Topography-Dependent Eikonal Equation Solver

Abstract

Abstract Irregular surfaces cause a number of problems for seismic processing and interpretation. The problems lie in the proper treatment of topography in the first‐arrival travel‐time calculation and ray‐path tracing, both of which are subject to preconditions in ray‐based seismogram synthesis, seismic tomography, and seismic migration calculations. Two treatment schemes for irregular surfaces have been used previously: (1) model expansion with the irregular surface treated as an inner discontinuity, and (2) flattening of the irregular surface using a transformation between curvilinear and Cartesian coordinates, while maintaining it as a free surface. In the first approach, first‐arrival travel times can be calculated using an eikonal equation solver, and rays are traced backward from the receiver to the source along the direction of the gradient of the travel‐time field. In the second scheme, a topography‐dependent eikonal equation is used to calculate irregular‐surface first‐arrival travel times. We present a ray‐path tracing scheme for irregular surfaces, which applies a travel‐time field calculated using a topography‐dependent eikonal equation. The scheme is realized using travel‐time gradients in curvilinear coordinates. The validity of the scheme for tracing ray paths in the presence of irregular topography is illustrated by five models containing differing degrees of topographical complexity. Comparison of ray paths and first‐arrival travel times between the two irregular‐surface treatment schemes suggests the topography‐flattened scheme avoids the difficulties of both discretizing the irregular surface and the velocity selection of the infill medium in the model expansion scheme. For the treatment of the inner discontinuity by model expansion, there is a possibility that ray paths will be traced outside the real physical model. Using the topography‐dependent eikonal equation solver with our ray‐path tracing scheme should provide an efficient means of dealing with irregular surfaces, which could be applied in the fields of seismic tomography, seismic migration, and tomographic static correction.