Solutions in exponential functions for Rayleigh wave equations of motion with gravity in a radially heterogeneous spherical Earth

Abstract

SUMMARY With the availability of broad-band waveform data, the inversion of long-period surface wave velocities play an important role in evaluating the crust and mantle. For forward modelling, the flattening transformation allows us to compute theoretical velocities in a spherical Earth through effective algorithms of a flat-layered Earth. Using this transformation, the phase and group velocities of Rayleigh waves of period up to 300 s are computed for a non-gravitating Earth and compared with the corresponding results for a gravitating Earth. The differences in first few modes are seen to be small but not negligible. In order to obtain such transformation for a gravitating layered spherical Earth, we require analytic solutions of the corresponding equations in terms of exponential functions. To achieve this goal, we consider the equations of motion of spheroidal oscillations in an isotropic elastic radially heterogeneous self-gravitating sphere. The density variation in the crust and mantle is small compared to variation from mantle to core; so if we consider Rayleigh waves confined to crust and mantle, we may neglect the perturbations in gravitational potential due to insignificant variation in density distribution during oscillations. Thus the equations of motion consist of two second-order differential equations with gravity terms included in it. In each shell we assume Lame’s parameters λ and μ ∝ r p and density ρ ∝ r p−2 , gravity g ∝ r , where r is the radial distance and p is an arbitrary constant. With such heterogeneity in a shell of a layered Earth, we get solutions for displacement in terms of exponential functions. The numerical results of Rayleigh wave velocities through flattening transformation based on the present solutions are expected to be close to those of a gravitating spherical Earth.