Borehole seismic modeling

In many borehole seismic experiments, the velocity of the tube wave is higher than that of the surrounding rock shear wave. This fast tube wave creates a strong conical shear wave in the surrounding rock, similar to the Mach wave in supersonic aviation and the Cherenkov radiation in electrodynamics. Many geophysicists have tried to utilize the conical signal in VSP (vertical seismic profiling) and cross borehole data interpretation, using quasi static approximations to model the borehole effect. Two popular quasi static approximations are: the effective source array method for source borehole modeling and the squeeze strain method for receiver borehole modeling. These quasi static approximations are sensible as they qualitatively conform to Hueygen's principle and the typical wavelength of a VSP or a cross borehole seismic experiment is much larger than the borehole radius. However, they have not been quantitatively benchmarked against other non approximation method such as the frequency wave number method. The frequency wave number method is a rigorous, non approximation method for modeling straight boreholes without lengthwise variation. The boreholes may consist of many coaxial, homogeneous and axially symmetric shells. In this thesis, the results of the quasi static approximations are compared to the results obtained from the frequency wave number method. The comparison demonstrates that both the effective source array method and squeeze strain method gives the correct arrival time. The effective source array method gives incorrect amplitude and waveform for direct arrivals and tube waves due to its arbitrary assumption of the elementary source radiation pattern. The squeeze strain method gives fairly accurate amplitude and waveform for P and S direct arrivals but it fails to match the tube wave results obtained from the frequency wave number method. The omission of tube wave dispersion and amplitude loss by the quasi static approximation methods also contributes to the inconsistency between the results from the quasi static approximation methods and the results from the frequency wave number method.