Travel-time computations for true-amplitude migration of constant-offset seismic data

Accurate travel times are essential for 3-D imaging. This is particularly true when the obliquity factor in the Kirchhoff integral is designed to preserve the amplitude of the wavelet. The required attributes involve partial derivatives of the travel-time function, and thus are extremely sensitive to travel-time errors. This paper describes an approach to computing travel times and their derivatives with high accuracy. It is based a on a ray tracing method in which ray paths and travel times are computed using exact analytic formulas. The velocity function is defined on a 3-D Cartesian grid and treated as a system of discrete cubic cells, each with a constant velocity gradient. Rays enter and exit at cell boundaries. Within such a cell the ray path is a circle segment. Rays are traced through a sequence of cells, and consequently, the entire ray path becomes a contiguous set of circle segments or straight lines. It is demonstrated that travel times along a circle segment can be computed from an exact analytic formula. As a result, travel time errors are limited to a fraction of a millisecond. This order of precision facilitates the estimation of geometric spreading by monitoring the behavior of a beam composed of three rays with take off angles differing only by a fraction of a degree. The effectiveness of this approach to computing travel times, raypaths, and geometric spreading is demonstrated on velocity models in which these attributes can also be computed analytically and can thus be used to estimate the accuracy of the proposed method.