Source wavefield downward extrapolation

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:  

$$p^+(z_{i+j}) = w^{+}(z_{i+j},z_i) \, \, p^+(z_i),$$ (2)

initialized by the wavefield at the surface, as follows:

 

where p⁺(z_i) is the source wavefield at depth z_i, p⁺(z₀) is the source wavefield at zero depth, w⁺(z_i+j,z_i) is the downward continuation operator (Green function) that downward propagates the source wavefield from depth z_i to z_i+j, and f is the source wavefield at the surface.

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 (${\bf P}^+$) can be computed as a function of the source signature (${ \bf f}$), as follows:  

$${\bf P}^+ =\left( {\bf I} - {\bf W}^{+} \right)^{-1}{\bf f}.$$ (4)

Defining the the multi-frequency, multi-shot, downward propagation operator as ${\bf B^{+}}= \left( {\bf I} - {\bf W}^{+} \right)^{-1},$ equation (4) can be written as  

$${\bf P}^+ ={\bf B^{+}}{\bf f}.$$ (5)