Reflectivity modelling by finite difference
P. F. Daley, Gary F. Margrave
The reflectivity method has been used for several decades to create exact synthetic traces for wave propagation in plane parallel layered media (e.g. Müller, 1985). For the problem of coupled P-SV wave propagation in a radially symmetric isotropic homogenous plane parallel layered medium, the radial coordinate is temporarily removed by Hankel transforms, a Fourier time transform applied, and the resultant depth problem is handled by propagator matrix theory or variations thereof.
We take an alternate approach suggested by Mikhailenko (Mikhailenko, 1985), that uses finite Hankel transforms to temporarily remove the radial coordinate and finite difference methods to deal with the resultant problem in depth and time. A finite difference problem in one spatial dimension and time avoids many of the numerical difficulties inherent in problems with higher order spatial dimensions. If the additional assumption that the time dependence of the source wavelet is band limited in the frequency domain is made, the problem is fairly well defined, apart from initial and finite boundary conditions. The approach indicated here was referred to as the pseudo-spectral method in past decades. In recent CREWES volumes there are reports by P.F. Daley dealing with certain aspects of this theory. CREWES is releasing Matlab software this year that implements this theory.