Reflected P₁S₁ and critically refracted P₁P₂S₁ arrivals near a branch point - higher order approximations in terms of special functions

P. F. Daley

The high frequency solution to the problem of a P1S1 reflected wave at a plane interface between two isotropic homogeneous halfspaces, designated 1 and 2, in the vicinity of the branch point p = p2 = 2-1, is developed. In reflection seismology, sub-critical reflections are often all that is required or wanted for data processing. However, there is a possibility, when a high velocity contrast exists between two layers, resulting in the branch points due to the P and possibly even S - wave velocities in the second layer, that both may influence the recorded seismic traces at relatively small offsets. Apart from a distorted (compared to the zero order geometrical optics approximation to the arrival) wavelet, the presence of critically refracted (head) waves may possibly be seen on the traces. If the time duration of the wavelet is pulse tpulse, the reflected and critically refracted waves are collectively designated as the interference wave in the corresponding offset range. Only the occurrence of a critically refracted P - wave will be considered here. The possibility, given the proper velocity distribution, of a critically refracted wave due to the branch point at p = p4 = 2-1 may be derived in a similar manner.