Datuming operator

Now the datuming operator can be easily found by transposing and taking the complex conjugate of each matrix in the forward modeling operator shown in equation (1). The datuming operator for a poststack data set gathered on an irregular surface is thus

$$\left[ \begin{array} {c} d_z \end{array} \right] = \left[ \b... ...array} \right] \left[ \begin{array} {c} d_0 \end{array} \right]$$ (4)

$$E_i^T = \left[ \begin{array} {cc} I&0\\ 0&W_i^T\\ \end{array} \right]$$ (5)

$$G_i^T = \left[ \begin{array} {cc} I&0\\ F_i&K_i\\ \end{array} \right]$$ (6)

In equation (4), we can see that the downward extrapolation E^T is preceded by the filter G^T at every depth level. We then apply the downward extrapolation operator W_i^T to the wavefield that was introduced by the operator F_i up to a given depth level. The portion of the wavefield that is not introduced until a given depth level is removed by the operator K_i. Figure 3 shows this datuming operator when the topography is given by Figure 1.

 

tpdtmschm
Figure 3 Datuming scheme: schematic diagram for the datuming operator as the conjugate to the forward extrapolation scheme, when the surfaces are irregular. W₁^T represents the downward propagation operator at each depth level.

Coincidentally, this datuming scheme is the same as Reshef's Reshef (1991). He used the algorithm for depth migration from irregular surfaces with depth extrapolation. We can deduce that Reshef's algorithm assumes the same forward modeling algorithm as this one.