The Green's tensor is the displacement vector impulse response at an ``observation'' point due to an impulsive force density located at , and satisfies the auxiliary wave equation:
where is the identity matrix. I will assume the following WKBJ form for the Green's tensor:
where are the polarization vectors evaluated at either endpoint along the ray joining to , and where all have the functional dependency . I refer the reader to Aki and Richards (1980) for a constant velocity overview of the Green's tensor, and to Morse and Feshbach (1953) for a review of the WKBJ method of solution to partial differential equations. Note that the form (14) of implicitly requires an isotropic elastic background propagation model since the two shear components have the same arrival time and amplitude B, although all of the theory previous to this point has been valid for general anisotropy. The traveltimes and satisfy the P and S eikonal equations respectively,
and the WKBJ amplitudes A and B satisfy the P and S transport equations respectively,
The P and S-wave propagation velocities are denoted as and respectively. Incorporating general anisotropy in the background model would require modification of the Green's tensor (14) to have two coupled shear traveltimes and amplitudes B1, B2, given by S1-S2 coupled eikonal and transport equations (Cervený et al., 1977). Since the WKBJ Green's tensor satisfies the zeroth order asymptotic ray eikonal and transport equations, the quantities A, B, , and can be evaluated by numerical ray techniques, providing the migration background model is ray valid (smooth velocity gradients on the order of a seismic wavelength within a sequence of possibly discontinuous parameterized layers).