The Earth Flattening Transformation in Body Wave Theory

Abstract

Summary A simple class of power law transformations between spherical and plane models is investigated to determine the effect of the Earth flattening approximation in body wave studies. The transformations lead to the correct kinematic properties and should be chosen to reduce the inaccuracies in the dynamic properties. It is concluded that except in a fluid, the Love wave transformation of Biswas & Knopoff is the most satisfactory for SH, SV and P ray studies, Biswas's transformation for Rayleigh waves is not suitable for body wave studies, and the transformation used by Helmberger is optimum in a fluid. htrodnction The computation of theoretical seismograms in an inhomogeneous, spherically symmetric, Earth model has received considerable attention recently. Gilbert & Helmberger (1972) have applied the Cagniard-de Hoop generalized ray method to a layered sphere and the method has been used with considerable success by Helmberger & Wiggins (1971) in upper mantle studies. Gilbert & Helmberger (1972) describe their method as a direct approximation to the spherical wave functions. These approximations are invalid in layers where the ray is travelling nearly tangentially, e.g. near the turning point of the ray. If an Earth flattening approximation is applied first, the Cagniard-de Hoop method can be applied exactly to the approximate model of homogeneous plane layers. This method has been used by Miiller (1970) and is in fact the method used by Helmberger & Wiggins (1971) (Wiggins, private communication). Helmberger (1973) has used an identical transformation for velocity and density, and compared the exact solution in a homogeneous model with the Cagniardde Hoop solution. In this paper we shall discuss the nature of the approximation made in the Earth flattening approximation. In a recent paper, Hill (1972) has considered the approximations to the spherical wave functions in more detail. He obtains phase corrections to the plane wave approximations and frequency dependent corrections to the plane wave reflection coefficients. The combined effect of these approximations on the dynamic properties of the body wave and the choice of density transformation is not investigated. Chapman & Phinney (1972) have described an alternative spectral method for computing body wave seismograms. Although their method includes a solution of the fourth order spherical wave equations, it is necessary to match this to an approximate WKBJ solution. The results in this paper can also be used to indicate the approximation made here.