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Recent developments in the computation of theoretical seismograms have shown the advantages of numerical integrations in the complex ray parameter plane. We here extend the approach, to calculate the amplitude of body waves interacting with a discontinuous velocity increase. The method incorporates a complex velocity profile to account for attenuation and the uniformly asymptotic method of Langer to account for frequency-dependence of reflection-transmission coefficients near grazing incidence. In the integrand of reflection-transmission coefficients, the Debye ray expansion is not made where it is poorly convergent near critical incidence. The method thereby correctly evaluates all the waves in the high velocity medium that are repeatedly refracted back up to, and reflected from, the low velocity medium. Calculations are equally efficient both for Earth models specified on discrete radii or as analytic functions of radius. The method is used to calculate theoretical seismograms for PKP waves in the PEM Earth model in the distance range 110°–152° and in Earth models 1066B and C2 in the distance range 112°–132°. Calculations of the amplitude near point D of the travel time curve for PKP (which is the critical distance of K waves incident from the fluid core upon the solid inner core) indicate that at finite frequencies the amplitude maximum associated with critical incidence is smaller than that that would be inferred from frequency independent reflection-transmission coefficients. At lower frequencies the maximum is displaced to a longer distance than D. Comparison of observed and calculated seismograms indicates that the PEM model having a P velocity jump of 0.83 km/s at the inner core can fit the observed amplitudes. Consequently the data do not require an anomalous P velocity gradient at the top of the inner core. A low frequency precursor to the DF branch of PKP at distances shorter than the B caustic at 146° agrees with Buchbinder's (1974) explanation of diffraction from the B caustic.
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