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.

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