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 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).