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In forward modeling,
the wavefield recorded at each geophone
along an irregular surface
is the wavefield propagated up to the depth level
where the geophones are located.
It is necessary to stop the wavefield propagation
after recording because the reflection coefficient at surface
is almost 1 Ji and Claerbout (1992).
To do so, I formulate the forward modeling operator
by propagating the wavefield upward
with a filter between extrapolation steps
to stop the wavefield propagation where it is recorded.
Then all wavefields from all depth levels
where the geophones are located
are summed together to produce the wavefield
along the irregular surface.
In order to explain the algorithm clearly and schematically,
I use a simple topography model
that has only eight geophone groups on an irregular surface,
as illustrated in Figure 1.
syngeometry
Figure 1 Synthetic surface recording geometry.
solid squares represent geophone location on an
undulating surface.

 
Figure 2 schematically describes
the forward modeling algorithm
for the simple model.
W_{i} represents upward extrapolation at the ith 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 IF_{3} and IF_{2}F_{3} in Figure 2
stop the wavefield at the locations
where it is recorded below
or at the corresponding depth level
and pass the wavefield at the locations
where it is not yet recorded.
Each small rectangle in Figure 2 represents
an abstract vector that contains wavefields at the corresponding
space location.
The wavefield along the irregular surface is
obtained by summing the wavefields that are grabbed
at the various depth levels.
tpfrdschm
Figure 2 Forward modeling scheme: the schematic diagram for
forward depth extrapolation when the surface is not flat.
W_{i} represents the upward extrapolation operator
at the ith depth level.
F_{1}, F_{2}, and F_{3} are spatial filters shown in the text,
and I is the identity matrix.
The forward modeling scheme
shown in Figure 2 can be algebraically generalized,
if we divide the topography into z levels, as follows:
 
(1) 
 
(2) 
 
(3) 
where
In equation (1), d_{0} and d_{z} are wavefields
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
arrived at the surface because the operator K_{i1}
remove the wavefield if it has arrived at any previous depth level.
All wavefields that arrive
at the surface are saved by the operator F_{i1} for the final output.
For the simple geometry shown in Figure 1,
F_{1}, F_{2}, and F_{3} are just diagonal matrices
whose elements are 1 where the geophones are located
and elsewhere.
Thus their diagonal elements are as follows:
The operator W_{i} in equation (2) can be any extrapolation scheme
including the Kirchhoff, phaseshift, splitstep, or finitedifference method.
If we use the phaseshift extrapolation algorithm for W_{i},
we need an additional inverse Fourier transform in every extrapolation step
because the operator G is in the space domain.
However, all other algorithms, such as
the Kirchhoff, splitstep, and finitedifference methods,
don't need any additional computation except the operation by G,
which is the multiplication of the extrapolated wavefield
by the zero/one filter.
Next: Datuming operator
Up: POSTSTACK DATUMING
Previous: POSTSTACK DATUMING
Stanford Exploration Project
11/17/1997