Datuming operator

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

 

$$\left[ \begin{array} {c} d_z \end{array} \right] = \left[ \b... ...array} \right] \left[ \begin{array} {c} d_0 \end{array} \right]$$ (65)

$$E_i^T = \left[ \begin{array} {cc} I&0\\ 0&W_i^T\\ \end{array} \right]$$ (66)

$$G_i^T = \left[ \begin{array} {cc} I&0\\ F_i&K_i\\ \end{array} \right]$$ (67)

In equation (), 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  shows this datuming operator, while the topography is given by Figure .

 

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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. By having both the forward operator and its adjoint we are prepared for inversion and velocity estimation.

 

synmdl Figure 4 Synthetic model with a syncline reflector image (lower) under an undulating surface (upper).

 

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Figure 5 Synthetic wave field on the irregular surface in v(z) (a) Wave field recorded on the irregular surface using phase-shift extrapolation scheme. (b) Datumed wave field using phase-shift algorithm. (c) Datumed wave field using split-step algorithm. (d) Datumed wave field using 45-degree finite-difference algorithm.