Seismic modeling in the domain of intercept time and ray parameter

In recent years, seismic modeling has been based primarily on two methods: generalized ray theory and the reflectivity method. Variations within these methods are characterized by the domains of computation and the approximations applied. One common feature of both approaches is the straightforward translation of a mathematical solution of the wave equation into a computational algorithm, which develops the seismic response for a given earth structure in the observation plane of seismic experiments, source-receiver offset and travel time (X-T). We present an alternative approach in which we develop acoustic and seismic models in the domain of vertical delay, or intercept, time, and horizontal ray parameter (τ-p). This is accomplished by a modified version of the reflectivity method followed by a Fourier transform over frequency. In these models, the τ-p seismic response for compressional and shear waves is readily interpreted for all primary and multiple arrivals. The τ-p response can also be used to compute broad-band synthetic X-T seismograms by an appropriate integration through the τ-p plane. A fast method of computing seismograms is suggested, based on a weighted integration designed to automatically evaluate the points of stationary phase.