An Earth-flattening Transformation for Waves from a Point Source* by David P. Hill

An earth-flattening transformation is developed for wave-propagation problems that can be formulated in terms of uncoupled scalar Helmholtz equations. Through the transformation, wave problems in isotropic, spherically symmetric media with a specified radial heterogeneity can be expressed in terms of a flat geometry with a suitably vertical heterogeneity. The transformation is exact for homogeneous (no source) problems and is useful for normal mode studies. When a point source of waves is present, the earth-flattening transformation together with the Watson transform converts the reflected wave field from a sum over discrete, spherical eigenfunctions to an integral over continuous wave numbers in a flat geometry. The far-field form of this integral shares many properties with the Weyl integral and is useful for body-wave studies in a spherical earth. INTRODUCTION Earth-flattening transformations provide a means for including the Earth's curvature in plane layer formulations of wave-propagation problems. Through such transformations, the existing store of analytical and numerical methods for treating flat problems become available for treating problems with spherical geometry. With emphasis on the resolution of progressively more subtle details of the Earth's structure using both phase and amplitude information on seismograms, it becomes important to include the effects of curvature for even relatively short propagation paths and wave lengths. Earth-flattening approximations based on a linear modification of the index of refraction have been used for some time in making corrections for the Earth's curvature in radio-wave propagation problems (Schelling et al., 1933; Budden, 1960). Alterman et al. (1961) introduced an analogous approximation into the seismological literature for short-period Love waves. Kovach and Anderson (1962) and Anderson and Toks6z (1963) developed a more general earth-flattening transformation for Love waves that involves transforming radially symmetric, isotropic shells into flat, vertically heterogeneous, anisotropic layers. More recently, Sat6 (1968) and Biswas and Knopoff (I 970) introduced transformations that exactly convert toroidal, or SH-wave motion in a spherically symmetric, radially heterogeneous earth to S H motion in plane, vertically heterogeneous layers. The exact earth-flattening transformations developed by these authors for homogeneous (no sources) equations of S H motion are most useful for normal mode or surface-wave problems. Analogous exact transformations for homogeneous spheroidal, or P S V wave motion have not been reported; one might expect that such a transformation cannot be made exactly because the velocity gradients that serve to map spherical to plane geometries would introduce spurious P S V coupling. Muller (1971) has developed an approximate earth-flattening transformation for body waves using arguments based on geometric ray theory. In this paper, we develop an earth-flattening transformation suitable for body waves from a wave-theoretical approach by incorporating an exact flattening transformation into the inhomogeneous *Publication authorized by the Director, U.S. Geological Survey. 1195 1196 BULLETIN OF THE SEISMOLOGICAL SOCIETY OF AMERICA equations of motion for waves generated by a steady point source in a radially heterogeneous earth. The transformation is developed for wave problems that can be formulated in terms of uncoupled scalar Helmholtz equations in spherically symmetric, radially heterogeneous media. Because acoustic, electromagnetic, and elastic wave propagation problems can be expressed in terms of uncoupled Helmholtz equations under a wide range of conditions, this earth-flattening operator applies to a variety of geophysical applications. Analogous earth-flattening transformations are readily developed for any wave-propagation problem in which the equations of motion are separable in a spherical coordinate system and in which the radial eigenfunctions are associated with a Sturm-Liouville operator. Our approach will be first to summarize briefly the potential representations known to reduce the acoustic and elastic equations of motion in radially heterogeneous media to canonical wave equations (or Helmholtz equations in the steady state). Because a primary motivation for this paper is to obtain an earth-flattening transformation for seismic body waves, we give particular attention to the elastic displacement potentials recently introduced by Richards (1971) that result in high-frequency decoupling of P S V motion in both radially and vertically heterogeneous media. Next, we develop an exact earth-flattening transformation for the homogeneous Helmholtz equation for waves in a spherical medium in which the wave velocity and density may vary smoothly with radius (this transformation is parallel to those obtained by Sat6 (1968) and Biswas and Knopoff (1970) for the SH-wave equation). In addition, we will indicate how the continuity conditions across an abrupt change in velocity and density behave under the transformation using S H motion as an example. Finally, taking acoustic motion as a simple example and introducing the exact earthflattening transformation together with the Watson transform, we convert the response of a radially heterogeneous body to a steady point source from an infinite sum over discrete order numbers to an integral over continuous wave numbers in a vertically heterogeneous half-space. Here we will show that for conditions generally satisfied by body waves in the crust and upper mantle, this integral reduces to the form of a Weyl (or Sommerfield) integral in which the effects of curvature are included in a generalized reflection coefficient. EQUATIONS OF MOTION AND POTENTIAL REPRESENTATIONS We will restrict our considerations to plane layered and spherically layered heterogeneous, isotropic media. Specifically, we will consider acoustic and elastic isotropic media in which the scalar material parameters may vary as a function of either depth or radius beneath the reference surface z = 0 or r = a (across which the material parameters change discontinuously). Furthermore, we will consider a single, isotropic steady point source of angular frequency, co, so that the resulting field depends on only two coordinates (z and p in a cylindrical system and r and 0 in a spherical system, see Figures 1 and 2). In inviscid fluid media, the acoustic approximation to the equations of motion can be written as V 2 p + k 2 p p tVP'Vp = 0 (1) where P is the deviation from the static ambient pressure, p is the density and k = o9/c, c being the acoustic velocity. Following Brekhovskikh (1960), we introduce the pressure potential = P p ~ (2) AN EARTH-FLATTENING TRANSFORMATION FOR WAVES FROM A POINT SOURCE l 197