Seismic ray theory is analogous to optical ray theory and has been applied for over 100 years to interpret seismic data. It continues to be used extensively today, owing to its simplicity and applicability to a wide range of problems. These applications include most earthquake location algorithms, body-wave focal mechanism determinations, and inversions for velocity structure in the crust and mantle. Ray theory is intuitively easy to understand, simple to program, and very efficient. Compared to more complete solutions, it is relatively straightforward to generalize to three-dimensional velocity models. However, ray theory also has several important limitations. It is a high-frequency approximation, which may fail at long periods or within steep velocity gradients, and it does not easily predict any “non-geometrical” effects, such as head waves or diffracted waves. The ray geometries must be completely specified, making it difficult to study the effects of reverberation and resonance due to multiple reflections within a layer.

In this chapter, we will be concerned only with the timing of seismic arrivals, deferring the consideration of amplitudes and other details to later. This narrow focus is nonetheless very useful for many problems; a significant fraction of current research in seismology uses only travel time information. The theoretical basis for much of ray theory is derived from the eikonal equation (see Appendix C); however, because these results are not required for most applications we do not describe them here.