On approximate solutions for reflection of waves in a stratified medium

Summary The properties of the asymptotic and iterative solutions of the wave equation are investigated for a medium containing discontinuities. The asymptotic method involves matching different solutions on either side of a discontinuity in order to satisfy the boundary conditions. The zeroth iteration, on the other hand, is a single approximation that can be used at all depths. The full effects of discontinuities are introduced only at higher iterations via the singularities and discontinuities of a coupling parameter, which may be described as a frequency‐independent differential reflection coefficient. This iterative solution, proposed by Chapman, has the attraction that time domain results are relatively simple to obtain and interpret when it is sufficient to include only the first iteration. If a turning point and a wavespeed gradient discontinuity coincide, the coupling parameter is infinite at that point. The coupling parameter is also infinite at a first‐order discontinuity. Provided the turning point does not lie at a first‐order discontinuity, the iterative series for such cases takes on a limiting form in which it is easy to include many terms in the frequency domain. Ths amounts to an expansion of the reflection coefficient in logarithms of the material properties. However, simple time domain expressions cannot be obtained for terms beyond the first in this expansion. The iterative solution is not well suited to the case where the turning point lies at a first‐order discontinuity. Both the first‐order asymptotic solution and the first iteration predict the same, step‐like time domain signal from a second‐order discontinuity which is well above any turning point. As the turning point nears the second‐order discontinuity, the major contribution to the reflection comes from the wavespeed gradient discontinuity and can be approximated by a zeroth‐order asymptotic solution. Numerical results are obtained for a particular, geophysically relevant model meant to represent a Moho with finite thickness. The differences between results from the two methods can be explained in some detail, often by giving a physical interpretation to terms appearing in the equations.