Approximate Treatment of Elastic Body Waves in Media with Spherical Symmetry

Summary An Earth‐flattening approximation for body waves is given which transforms a spherically symmetrical Earth into a vertically inhomogeneous half‐space. This approximation includes a depth transformation and a transformation of the velocities of compressional and shear waves. It is an optimum approximation in the sense that the displacement amplitudes of corresponding seismic rays, i.e. of rays leaving a point source in the Earth and the image source in the half‐space under the same radiation angle, show maximum agreement in their geometrical optics approximation. The epicentral distances of these rays are identical, and likewise the travel times provided that source and receiver are at the same depth. The amplitude difference is 10 (20,30) per cent at the epicentral distance 60° (82°, 96°). The main field of application of this Earth‐flattening approximation is the computation of theoretical body wave seismograms with methods originally devised for half‐spaces with plane layers. Theoretical P wave seismograms are given for the upper mantle models of Jeffreys, Gutenberg, Lehmann, Johnson and Mayer‐Rosa. The method of calculation is an extension of the ray‐theoretical method which was developed in a previous paper. Even without wave front approximations, it permits rather fast computations. Finally, the Earth‐flattening approximation is applied to vertical reflections from a spherically symmetrical medium, and a method is briefly described which accounts for the Earth's curvature in computations of crustal and mantle transfer functions, respectively.