Related work of other people

Few researchers have applied Fourier-integral operator theory or similar branches of mathematics to seismic problems. De Hoop and Hörmann (2001, 2003) and Stolk and de Hoop (2002) have done some work applying microlocal analysis, a field of mathematics which includes Fourier-integral operators and pseudo-differential operators, to seismic problems. The elastic-wave equation they use is equivalent to the equations of motion (5) but with body forces. They use pseudo-differential operators and Fourier-integral operators to do forward modelling with the Kirchhoff and Born approximations separately and together, and also do inverse scattering. Their mathematical development should smooth my way somewhat, but there is some room for further original developments beyond their work, perhaps again in analytic development to a point, followed by numerical completion, without such approximations. However their work may mean that on its own my application of FIO methods to equations (5) might not be quite enough for a Ph.D. but when combined with the method of characteristics, principal-symbol analysis, and comparison to numerical results and real data for one test case it should be enough. If not I could also add a consideration of nonlinearity and possibly anelasticity.