Rapid calculation of seismic amplitudes: SHORT NOTE

The ability to calculate traveltimes and amplitutdes of seismic waves is useful for many reflection seismology applications such as migration and tomography. Traditionally, ray tracing (cerveny et al., 1977; Julian, 1977), paraxial methods (Claerbout, 1971), or full-wave methods (Alterman and Karal, 1968) are used for such calculations. These methods have in common considerable computational expense. Recently, Vidale (1988, 1990a) presented two-dimensional and three-dimensional methods to efficiently compute traveltimes of the first arrivals to every point in a regularly spaced grid of points, given an arbitrary velocity field sampled at these points. The computational cost of finding each traveltime is roughly one square root operation. This note describes how the amplitudes of the first arrivals can be calculated in two dimensions without explicit knowledge of the raypaths. The traveltimes for a set of four closely spaced sources are sufficient to determine the geometric amplitudes for each point in the grid in the case of a smoothly varying velocity structure. The computation cost for each amplitude is six square roots and an arctangent. This low cost allows the computation of many more amplitudes than is possible with ray tracing or waveform methods. Alternatively, this method of amplitude estimation can be done using other less efficient but more accurate traveltime calculations, such as ray tracing or ray bending (Prothero et al., 1988; Thurber, 1983) and needs four to sixteen traveltimes per amplitude.