Receiver wavefield extrapolation

The second step of shot-profile modeling consists of upward continuation of the receiver wavefield to the surface, which is done by using the following equation:

$$\begin{eqnarray} p^-(z_{i}) &=& w^{-}(z_{i},z_{i+j}) \, \, p^-(z_{i+j})+ p^+(z_{i})\, \,r(z_i)+ p_r^-(z_{i}) \end{eqnarray}$$ (6)

where  

$$p_r^-(z_{i})=\left\{ \begin{array} {c} \sum_{m=1}^{j-1}w^... ...d \quad \quad \quad \mbox{otherwise}, \end{array} \right.$$ (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 (${\bf P}^-$) can be computed as follows:  

$${\bf P}^-=[{\bf I}-{\bf W}^{-}]^{-1}{\bf P}^+\, \Sigma^t_{\omega s} \, {\bf r}.$$ (9)

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

$${\bf P}^-={\bf B}^-{\bf P}^+\, \Sigma^t_{\omega s} \, {\bf r}.$$ (10)