Reflected PP arrival in anelastic media
P. F. Daley, Edward S. Krebes
A homogeneous wave incident on an interface between two anelastic halfspaces in welded contact is considered. In the anelastic sense, a homogeneous wave is defined by the condition that the P ropagation and attenuation vectors are collinear. It has been indicated in a number of papers over the past several decades that the proper definition of the real and imaginary parts of the vertical components of the slowness vector in the reflection coefficients are not obvious for some distributions of the quality factor, Q. This can result in anomalous behaviours of both or either of the amplitude and phase of the PP reflection coefficient when displayed versus the incident propagation angle or equivalently the real part of the horizontal component of the incident slowness vector.
In an earlier work (Krebes and Daley, 2007) the question of anomalies in the amplitude and phase of the PP plane wave reflection coefficient for these distributions of the quality factor Q in adjacent anelastic halfspaces was discussed in considerable detail. In what follows, the above paper (Paper 1) will be referred to often to minimize repetition of previous discussions.
The problem of the PP reflection coefficient is addressed again here. This is done within the context of two selected approximate methods, of varying complexity, which produce acceptable behaviour for the anomalous quantities, from a numerical viewpoint. What causes this behaviour in the PP reflection coefficient may be attributed, at least in part, to improper signs being assigned to the real and imaginary parts of the radical defining the transmitted wave P -vertical slowness vector component. However, this may be looked upon as a symptom rather than the actual cause of the problem.
Consideration of the PP plane wave reflection coefficient is the first matter dealt with and the discussion is then extended to the high frequency geometrical optics solution of a Sommerfeld type integral, using zero order saddle point methods for determining the particle displacement vector of the reflected PP disturbance due to a P -wave point source incident at an interface separating two anelastic media.
One approximation of the saddle point method was presented in detail in Paper 1 and another approximate approach was suggested and is expanded on here. The accuracy of approximations to the saddle point method are established through comparison with an "exact" (numerical integration) solution.