Next: Linear forward operator
Up: Forward operator
Previous: Source wavefield downward extrapolation
The second step of shotprofile modeling consists of upward continuation of the receiver wavefield
to the surface, which is done by using the following equation:
 
(6) 
where
 
(7) 
with the final condition
where p^{}(z_{i}) is the receiver wavefield at depth z_{i}, p^{+}(z_{i}) is the source wavefield at depth z_{i}, w^{}(z_{i},z_{i+j}) is the upward continuation operator (Green function) from depth z_{i+j} to z_{i}, r(z_{i}) is the reflectivity at depth z_{i}, and p_{r}^{}(z_{i}) is the contribution by the source wavefield and the reflectivity when jumps bigger than a depth step are used.
The Green functions can be computed recursively at each depth step (from z_{i+1} to z_{i}) (Figure 3), or precomputed and stored to allow jumps (from z_{i+j} to z_{i}) (Figure 4). In addition, the third term in equation (6) must also be stored.
Equation (6) can
be written also in matrix form (see APPENDIX B).
figure3
Figure 3 Upward continuation of the receiver wavefield by recursive computation of the Green functions at each depth step.

 
figure4
Figure 4 Upward continuation of the receiver wavefield with the precomputed Green functions.

 
Including all the frequencies and all the shot positions in the data,
it follows from equation (19) that the receiver
wavefield () can be computed as follows:
 
(9) 
Defining the the multifrequency, multishot, upward propagation operator as equation (4) can be written as
 
(10) 
Next: Linear forward operator
Up: Forward operator
Previous: Source wavefield downward extrapolation
Stanford Exploration Project
5/23/2004