Theoretical vertical seismic profiling seismograms

Offset vertical seismic profiling (VSP) theoretical seismograms which include multiples and mode conversions can be computed using a modified 'reflectivity' method. In this method, the transformed displacement potentials are first calculated by multiplying the source spectrum by the composite reflectivity function. Integration over wavenumber, followed by inverse Fourier transformation over the frequency range of the signal, yields the synthetic trace. The composite reflectivity function for a buried receiver is derived from Kennett's matrices (Kennett, 1974, 1979) which are synthesized to form phase-related reflection and transmission coefficients from a layer stack.Both conventional fixed source-moving receiver and fixed receiver-walkaway source (multioffset) VSP geometries can be handled easily. The method can also readily accommodate deviated-hole VSP. The method is general in that no ray needs to be specified. Because the order of the multiples can be controlled, wraparound problems with the discrete Fourier transform can be avoided. The normal-incidence VSP seismograms can be rapidly generated as a special case. Several examples illustrate the method. Some classes of laterally varying structures can be approximately handled by restricting the range of ray-angle integration and by using the principle of superposition.