Calculation of the amplitudes of elastic waves in anisotropic media in Cartesian or ray-centred coordinates
Abstrakt
We derive various expressions for the amplitude of the ray-theory approximation of elastic waves in heterogeneous anisotropic media, and show their mutual relations. The amplitude of a wavefield with general initial conditions can be expressed in terms of two paraxial vectors of geometrical spreading in Cartesian coordinates, and in terms of the 2x2 matrix of geometrical spreading in ray-centred coordinates.
The amplitude of the Green tensor can be 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 2x2 paraxial matrices corresponding to two ray parameters in ray-centred coordinates, (c) in terms of the 3x3 upper right submatrix of the 6x6 propagator matrix of geodesic deviation in Cartesian coordinates, (d) in terms of the 2x2 upper right submatrix of the 4x4 propagator matrix of geodesic deviation in ray-centred coordinates, (e) in terms of the 3x3 matrix of the mixed second-order spatial derivatives of the characteristic function with respect to the source and receiver Cartesian coordinates, and (f) in terms of the 2x2 matrix of the mixed second-order spatial derivatives of the characteristic function with respect to the 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.