Given the elastodynamic integral solution (12) and the WKBJ Green's
tensor (14), a closed form solution for the incident wavefield
can be derived. I assume the source can be represented as
a body point force, and thus evaluated by the volume integral portion only
of (12):
![]() |
(17) |
A spatially compact impulsive body force may take the general form:
![]() |
(18) |
where are all evaluated at the source location
.The amplitude
is the scale and radiation pattern of the
P source displacement at
, and may vary as a function of the take-off
angle which can be obtained from
,
and in static strength as a function of shot location
.The terms Bo2 and Bo3 are the equivalent S1 and S2 factors
of Ao. The term
is the P waveform, and may vary with
source location
. The terms w2(t') and w3(t') are the S1
and S2 waveforms respectively, and may differ from w1 in
both phase and frequency content as a function of
.
The body force (18) can be substituted into the integral solution
(17) for the incident wavefield. The dot product is
evaluated from (18) and (14), for
, as
![]() |
||
(19) |
where the remaining polarization vectors are all evaluated at the
observation point
. Substitution of
(19) into (17) and performing the t' integration results in:
![]() |
(20) |
The final volume integration over V' yields a compact form for the incident wavefield solution:
![]() |
(21) |
where the notation As means and
means
,i.e., the value at
due to a source at
.
I remind you again that the polarization vectors
in (21) are
to be explicitly evaluated at each subsurface location
.To evaluate (21),
the WKBJ amplitudes and traveltimes
,
and the polarization vectors
, need to be
raytraced from each source position
to each subsurface position
,
by numerically solving systems (15)-(16) with a rayracing or
finite difference algorithm.