In this chapter, we investigate approximate solutions for many canonical signals, e.g. direct rays, partial and total reflections, transmissions, head waves, interface waves, tunnelling waves, caustics, Fresnel and deep shadows. Unfortunately, often these canonical signals do not occur in isolation and for realistic, complicated models it is necessary to use numerical methods as described in the last chapter (Section 8.5.2) for the complete response. With modern computers, these numerical computations are entirely practical and are preferred for forward modelling. Nevertheless, it is worth investigating approximations for canonical signals as it aids an understanding of wave propagation, and for interpreting the results of modelling and real data. While it is rare that the approximate results described here are used for complete, forward modelling, they are frequently employed to check more complete methods, and as an aid to interpretation or in an inverse problem, when complete modelling is too expensive.

The nomenclature for signals, particularly non-geometrical, is not standardized. In this chapter, we denote rays in multi-layered media with a sequence of letters with a subscript for each ray segment, e.g. P1S2 is a ray with a P ray segment in layer 1 and an S segment in layer 2 (therefore the source must be in layer 1, receiver in layer 2 and it must have a converted transmission at the interface, z = z2).