Seismograms from Epstein Transition Zones

Abstract

Spectral amplitudes and theoretical seismograms in the vicinity of caustic and critical points are evaluated numerically for acoustic waves propagating in Epstein media. The amplitude decay into the region beyond the caustic is compared with the asymptotic results and good agreement is found at high frequency. At low frequencies the partial reflection enhances the amplitudes at sub-critical angles and the amplitude increases with decreasing thickness of the transition zone. For a thin transition and low frequencies a head wave exists whose amplitude is inversely proportional to frequency. The study of amplitudes near caustics and critical point is of significant theoretical and experimental interest to seismologists. Improved coverage by stations of the USCGS worldwide network has provided better data and their interpretation has lead to more accurate estimates of elastic wave velocities within the Earth (Johnson 1967). In regions of high velocity gradient in the upper mantle caustics are formed and triplications occur on the travel-time curves. In this paper the amplitude properties near caustics and critical point are studied for the Epstein velocity transitions (Epstein 1930). Epstein profiles represent a group of models with transition zones for which solutions of the wave equation exist in terms of familiar tabulated functions. Wave propagation has also been studied analytically for linear transition layers (Nakamura 1964; Gupta 1966; Hirasawa & Berry 1971; Ward 1973) and exponential and parabolic layers (Merzer 1971 ; Rydbeck 1943). Linear layers have discontinuities in velocity gradient which may not be realistic. Merzer’s study is concerned only with head waves and their amplitude dependence on frequency, layer thickness and shape of the transition. Using Epstein models the caustics, head waves and shadow can be studied. Epstein (1930), Rawer (1939) and Phinney (1970) studied the modulus of the reflection coefficient in order to estimate the partial reflection. The modulus of the reflection coefficient itself, however, cannot give a full description of the energy propagation when the waves interact with the strong velocity gradient. That can only be studied when the spectral amplitudes and theoretical seismograms are evaluated. In this paper the exact spectral amplitudes are computed by contour integration using the method of Phinney & Cathles (1969). For evaluation of the theoretical seismograms the technique of Chapman & Phinney (1972) is employed. The amplitude curves are evaluated in the neighbourhood of both ends of the triplication. At high frequencies the results agree with the asymptotic results. At low