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 *B*_{o2} and *B*_{o3} are the equivalent *S _{1}* and

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 *A*_{s} 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.

11/17/1997