In order to fully appreciate this book, it is necessary to recall some of the recent history of body wave seismology. Until the late 1960s, most of our knowledge of subsurface structure came from travel time studies. Pekeris [1948] and Haskell [1953] had shown how to model seismic data, but existing computers limited the use of their methods to the computation of dispersion curves for simple earth models. Then Helmberger [1968] used the Cagniard-de Hoop method [de Hoop, 1960] to model refraction arrivals and thereby demonstrated the practicality of seismic modeling in the time domain. The Cagniard-de Hoop method is a generalized ray method, which means (in practical terms) that it is good for synthesizing first motions but not so good for the later parts of the seismogram. Accordingly, Fuchs and Muller [1971] returned to the methods of Pekeris and Haskell and showed that with large modern computers, the whole seismogram could be synthesized. However, problems remained, because Haskell matrices are numerically unstable when used to synthesize SV body waves. Methods of overcoming this instability were found (and are still being found), but they are all, with the possible exception of the methods of Schmidt and Tango [1986] and of Chin et al. [1984], complicated, difficult to program, and lacking in physical insight.