syngeometry
Figure 1 Synthetic surface recording geometry. Solid squares represent geophone location on an undulating surface. |
Figure schematically describes the forward modeling algorithm for the simple model. W_{i} represents upward extrapolation at the i-th depth level and F_{1}, F_{2}, and F_{3} are spatial filters for grabbing the wavefield where the geophones are located at the corresponding depth levels. The operators I-F_{3} and I-F_{2}-F_{3} in Figure stop propagating the wavefield at the locations where it is recorded below or at the corresponding depth level and keep propagating the wavefield at the locations where it is not yet recorded. Each small rectangle in Figure represents an abstract vector that contains wave fields at the corresponding space location. The wavefield along the irregular surface is obtained by summing the wave fields that are grabbed at the various depth levels.
The forward modeling scheme shown in Figure can be generalized algebraically, if we divide the topography into z levels, as follows:
(61) |
(62) |
(63) |
(64) |
In equation (), d_{0} and d_{z} are wave fields on the irregular surface and the datum level, respectively. The extrapolation operator E is followed by the spatial filter G at every depth level. We can see that the upward extrapolation operator W_{i} is applied to the wavefield that does not arrive at the surface because the operator K_{i-1} removes the wavefield if it has arrived at any previous depth level. All wave fields that arrive at the surface are saved by the operator F_{i-1} for the final output. If a datum is located in the subsurface instead of at the surface, the operators K_{i} should not be used so that the wavefield keep propagating through. The operator K_{i} described in this appendix is a crude type of transmission coefficient. In order to implement the operator K_{i} more correctly, its diagonal elements should be the transmission coefficient according to the property of the interface and the incidence angle of wavefield instead of 1 or . However such a rigorous implementation of the transmission coefficient is not practical since it requires exhaustive ray tracing to find the incidence angle. It is generally accepted that the transmission coefficient is 1 on the interface between two solids and on the interface between the solid and the air.
For the simple geometry shown in Figure , F_{1}, F_{2}, and F_{3} are simply diagonal matrices whose elements are 1 where the geophones are located and elsewhere. Thus, their diagonal elements are as follows: