Reflection of an incident spherical P wave on a free surface (near-vertical incidence)

Abstract

A high-frequency, first order saddle point approximation is derived for the PS and PP reflected disturbances due to the incidence of a P wave from a buried point source on a free interface in an isotropic homogeneous half-space. The first order approximation, i.e., a solution in which a term in ( i ω)−1 is present (for an harmonic source), was required to explain the differences in synthetic seismograms when the exact Alekseev-Mikhailenko method was compared with any high-frequency approximation in which only the first term (the zero order term) in the asymptotic series expansion of the solution was used. These differences occurred mainly near-vertical incidence and near-grazing incidence, the first of which will be treated in this paper. The modifications produced in the amplitude and phase of the geometrical PP and PS reflected arrivals due to the addition of the second leading term in the asymptotic expansion are discussed, and comparisons of exact synthetic seismograms using the Alekseev-Mikhailenko method and the higher order approximate solution are presented.