SEISMIC BODY WAVES IN A 3-D, SLIGHTLY ASPHERICAL EARTH .1. TESTING THE BORN APPROXIMATION

The first stage of a method for calculating long-period body wave seismograms in a laterally heterogeneous Earth is presented. First, the elastodynamic equation of motion in spherical coordinates, which is valid for weak lateral heterogeneity, is established using a generalized spherical harmonic expansion of the displacement and traction fields and the elastic parameters. It is shown that the equations for the displacement-traction components can be simplified by using high-wavenumber asymptotic approximations to the first-order coefficients, which are appropriate for 5s - 20s body waves in the mid- to upper mantle. The Born approximation is then used to develop a computationally efficient method for calculating the first-order scattered field in terms of the high-frequency asymptotic solution for a laterally homogeneous Earth. This efficiency is increased further by decomposing the wavefield into its P, SV, and SH components. Finally, the numerical integration of the displacement-traction components is performed for a few selected frequencies and a restricted band of angular degrees, in order to test the validity of the Born approximation. Comparison of these results with those calculated using the Born approximation indicates that the latter is well suited to models of weak lateral heterogeneity in the Earth, for which the elastic parameters can be expressed as a sum of low-order spherical harmonics. It is demonstrated that, for this particular class of models, the Born approximation yields a good representation of the scattered field for frequencies within the long-period body wave band of 0.05-0.2 Hz.