Dispersion Equations of Surface-Waves in a Spherical Layered Earth Through Exact Earth Flattening Transformation and Propagator Matrix

Dispersion equations of Rayleigh and Love waves in a spherical layered Earth for continental and oceanic structures are derived through an ‘exact Earth flattening transformation’ (EEFT). In a spherical shell (layer), EEFT considers P- and S- velocities are proportional to radial distance (r) and Lamé constants are proportional to \({r}^{-2}\) and obtains analytic solutions in terms of exponential functions of wave equation in spherical coordinates \((r,\vartheta ,\varphi )\) with origin at the centre of the Earth. Using EEFT, previous works through generalised reflection-transmission method generated dispersion equations in complex terms which create difficulties in computation; while here, with solutions from EEFT, we obtain a propagator matrix which is used to derive the dispersion equations in real terms increasing computational efficiency. The derived dispersion equations make the computation of phase and group velocities of surface-waves for a spherical layered Earth nearly as simple as that for a flat (plane) layered Earth through propagator matrix. The computational algorithm uses reduced delta (compound) matrix to evade loss of precision and layer reduction procedure to avoid overflow. To simplify computation further, we propose an ‘approximate Earth flattening transformation’ (AEFT) to compute approximate surface wave velocities in a spherical Earth. Surface-wave velocities computed through EEFT for a few Earth models are compared with the corresponding velocities through AEFT and we note that the error of approximate velocities by AEFT is within 0.1% up to significant periods of surface waves.