There are many reports in the literature of anomalies in traveltime data when an isotropic homogeneous Earth model is used to interpret field data. In several cases, the introduction of a layered transversely isotropic model has successfully explained these kinematic irregularities. However, it is useful, in fact essential, to confirm the kinematic fit with a dynamic (amplitude) comparison.In this paper the problem of SH waves propagating in a transversely isotropic plane layered medium is discussed through the use of integral transforms and evaluation by steepest descents. This procedure yields not only the asymptotic solution which is also attainable using an asymptotic ray series approach, but also allows for the investigation of the interference of the reflected and head waves in the vicinity of the critical point (point of critical refraction). It is in this region that asymptotic ray theory breaks down or at least introduces significant error in the displacement amplitudes.It can be shown that a simple transformation will reduce this problem to one that may be solved exactly by the Cagnaird de Hoop technique but it is instructive to examine nonspherical wave-fronts in order to obtain an insight into more complicated anisotropic media.

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