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:

(13) |

where is the identity matrix. I will assume the following WKBJ form for the Green's tensor:

(14) |

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,

(15) |

and the WKBJ amplitudes *A* and *B* satisfy the *P* and *S* transport
equations respectively,

(16) |

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 *B _{1}*,

11/17/1997