Pulse distortion and Hilbert transformation in multiply reflected and refracted body waves

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Abstract
Many seismic body waves are associated with rays which are not minimum travel-time paths. Such arrivals contain pulse deformation due to a phase shift in each frequency component. For sufficiently high frequencies, the phase shift each time a ray touches an internal caustic is π/2 and frequency-independent. The distorting effect of a frequency-independent phase shift is successfully observed in seismograms from events in several regions. The data examined are long-period (T > 9 sec). They include deep earthquakes (depth > 500 km), in which a series of well-separated S phases (S, sS, SS and sSS) are available. These show that the wave form of SS, which has been distorted in propagation through the Earth, can be derived from the wave form of sS, which is not distorted. Shallow events, in which multiple S phases overlap, also exhibit behavior predicted by phase distortion. Rays supercritically reflected or refracted at a discontinuity in the Earth also suffer a constant phase shift, which in general can have any value. An important case is SKKS: its undistorted wave form resembles that of SKS, which has a minimum travel-time path. Without exception, all the distorted wave forms bear little or no resemblance to the original wave form. That is, neither the first arrival of energy nor the subsequent relative position of peaks and troughs on a distorted wave form appear at the ray theoretical times. Thus, T-Δ curves constructed by choosing arrival times to correspond to the first arrival of energy may be biased. Similarly, doubt is cast on differential travel times chosen from first motions, or from averaging several points on what appear to be corresponding peaks and troughs of two wave forms. Some of the rays most important to seismology, in which the distortion phenomenon occurs, include P and S (where d2T/dΔ2 > 0), PKPAB, PP, SS, and SKKS. Removal of phase distortion in the data is computationally straightforward. By exploiting the resulting wave forms to full advantage in correctly picking arrival times, we may hope to improve velocity models of the Earth. It is shown that matched filtering to obtain differential travel times is appropriate for certain pairs of body waves if they are phase-corrected.

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