The first step of shot-profile modeling with the one-way wave equation consists of downward continuation of the source wavefield (for each shot, at each frequency), which is done by using the following equation:
(2) |
p^{+}(z_{0}) = f, |
(3) |
The Green functions can be computed recursively at each depth step (from z_{i} to z_{i+1}) (Figure 1), or precomputed and stored to allow jumps (from z_{i} to z_{i+j}) (Figure 2). The equations (2) and (3) can be written also in matrix form (see APPENDIX A).
figure1
Figure 1 Downward continuation of the source wavefield by recursive computation of the Green functions at each depth step. |
figure2
Figure 2 Downward continuation of the source wavefield with the precomputed Green functions. |
Including all the frequencies and all the shot positions in the data, it follows from equation (15) that the source wavefield () can be computed as a function of the source signature (), as follows:
(4) |
(5) |