Datuming operator

The datuming operator can now be easily found by transposing and taking the complex conjugate of each matrix in the forward operator shown in equation (7). Thus we obtain the prestack datuming operator for the data gathered on an irregular surface as follows:

$$\left[ \begin{array} {c} d_z \end{array} \right] = \left[ \b... ...array} \right] \left[ \begin{array} {c} d_0 \end{array} \right]$$ (11)

$$E_{s,i}^T = \left[ \begin{array} {ccc} I&0&0 \\ 0&I&0\\ 0&0&W_{s,i}^T\\ \end{array} \right]$$ (12)

$$G_{s,i}^T = \left[ \begin{array} {ccc} I&0&0\\ 0&0&I\\ H_{... ...{g,i-1}K_{s,i}&K_{g,i-1}(F_{s,i}+K_{s,i})\\ \end{array} \right]$$ (13)

$$G_{g,i}^T = \left[ \begin{array} {ccc} I&0&0\\ 0&0&I\\ H_{s... ...& H_{s,i}K_{g,i}&K_{s,i}(F_{g,i}+K_{g,i})\\ \end{array} \right]$$ (14)

Figure 14 is a schematic diagram of the prestack datuming operator when the topography is given by Figure 12. Even though it looks very complicated, the basic algorithm is that which introduces wavefield at each surface location at the corresponding extrapolation level in both the shot and geophone spaces before the shot and the geophone extrapolation and excludes wavefields in the extrapolation in either shot or geophone space if the wavefield was introduced only in the other space.

 

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Figure 14 Prestack datuming scheme: The schematic diagram for prestack datuming when the surface is given by Figure 12. W_s,i and W_g,i represent downward extrapolation for the common-geophone-gathers (CGG) and the common-shot-gathers (CSG), respectively. at the i-th depth level. K_g/s,i, H_g/s,i and F_g/s,i are spatial filters. The solid and empty rectangles in the filter represent numerically 1 and , respectively.