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## The incident wavefield

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.

Next: The reflected wavefield Up: THEORY Previous: The WKBJ Green's tensor
Stanford Exploration Project
11/17/1997