IMPROVING THE OPERATOR BY USE OF A FILTER

In the first method, discussed in the previous section, the wavefield is continuously propagated to the top of the surface after it is recorded at the lower altitude. This is not physically possible because after recording at the surface the wavefield goes through air and the air velocity is negligible compared to the earth velocity. In order to make the modeling realistic, we introduce a filter after each extrapolation to stop the propagation of the wavefield after recording. If the wavefield only has only vertical component, the filter introduced has no special effect. However, there is a significant difference between examples with and without the filter when the wavefield has large stepout. The improved scheme by use of the filter is explained schematically in Figure .

 

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Figure 7 Modeling scheme 2 : The schematic diagram for the wavefield extrapolation when the surfaces are irregular with spatial filters that stop the wavefield after recording. W_j represents the upward propagating operator, I is the identity matrix, and A,B and C are matrices in equation (2). [ Compare to Figure . ]

The scheme shown in Figure  can be algebraically formulated as

$$\left[ \begin{array} {ccc} A&B&C\end{array}\right] \left[ \b... ...right] = \left[ \begin{array} {c} d_{surface}\end{array}\right]$$ (4)

Now, the datuming operator can be found by taking the conjugate transpose operator of the modeling operator in equation (4). Therefore, the improved datuming of the data gathered on an irregular surface is done by

$$\left[ \begin{array} {ccc} U_1^T&U_2^T&U_3^T\end{array}\righ... ...}\right] = \left[ \begin{array} {c} D_{datum}\end{array}\right]$$ (5)

The schematic diagram of this datuming operator is given in Figure .

 

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Figure 8 Datuming scheme 2 : Schematic diagram for datuming when the surfaces are irregular as the conjugate operator to the modeling scheme 2. W_j^T represents the downward propagating operator. [ Compare to Figure . ]