Calculation of the amplitudes of elastic waves in anisotropic media in Cartesian or ray-centred coordinates

Ludek Klimes

Summary

We derive and compare various expressions for the amplitude of the ray-theory approximation of elastic waves in heterogeneous anisotropic media. The amplitude of a wavefield with general initial conditions is expressed in terms of two paraxial vectors of geometrical spreading in Cartesian coordinates, and in terms of the 2× matrix of geometrical spreading in ray-centred coordinates. The amplitude of the Green tensor is expressed in six different ways: (a) in terms of the paraxial vectors corresponding to two ray parameters in Cartesian coordinates, (b) in terms of the 2×2 paraxial matrices corresponding to two ray parameters in ray-centred coordinates, (c) in terms of the 3×3 upper right submatrix of the 6×6 propagator matrix of geodesic deviation in Cartesian coordinates, (d) in terms of the 2×2 upper right submatrix of the 4×4 propagator matrix of geodesic deviation in ray-centred coordinates, (e) in terms of the 3×3 matrix of the mixed second-order spatial derivatives of the characteristic function with respect to source and receiver Cartesian coordinates, and (f) in terms of the 2×2 matrix of the mixed second-order spatial derivatives of the characteristic function with respect to source and receiver ray-centred coordinates. The step-by-step derivation of various equivalent expressions, both known or novel, elucidates the mutual relations between these expressions.

Keywords

Amplitude, transport equation, elastic Green tensor, geodesic deviation, paraxial ray approximation, second-order derivatives of the characteristic function, anisotropy, heterogeneity.


Stud. geophys. geod., 63 (2019), in press.