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

The forward modeling experiment was done
using the algorithm explained in Figure 2
for a constant velocity; Figure 5(a) shows the result.
Datuming was then performed using
the algorithm shown in Figure 3
with the phase-shift extrapolation
as the depth extrapolation operator *W*.
The result appears in Figure 5(b);
the exact bow-tie shaped wavefield is
the characteristic of the syncline reflector on a flat datum.
I then applied the same algorithm
with the other depth extrapolation schemes.
Figures 5(c) and (d) show the datumed
results for the split-step and
the 45-degree finite-difference methods, respectively.
When the velocity is constant, the split-step algorithm
is identical to the phase shift algorithm.
Therefore we can see that the datumed wavefields
in Figures 5(b) and (c) are the same.
The result of the finite-difference method,
Figure 5(d), also shows a correctly located
bow-tie shaped wavefield
except very weak artifacts
in the region under the undulating surface.
These artifacts can be explained
as the energy from the evanescent region
which has not removed.

Figure 5

To illustrate the effectiveness of the datuming algorithm,
a velocity function which varies in depth and lateral extent
is tested for the same reflector and topographic model
shown in Figure 4.
The velocity model used in this experiment
has linear increasing both in depth and laterally:
*v*(*x*,*z*)=1500.+0.2*x*+0.2*z* (Figure 6(a)).
The zero-offset data are modeled
using split-step extrapolation;
Figure 6(b) shows the result.
The datuming algorithm is then applied to the data,
Figure 6(b), using the split-step extrapolation;
the result is shown in Figure 7(a).
Figure 7(b) shows the migrated image.
By comparing Figure 4 and Figure 7(b)
we can see the effect of the datuming algorithm.
The datuming algorithm using the finite-difference extrapolation
are also tested for the same data Figure 6(b);
the datumed wavefield is shown in Figure 7(c)
and the migrated image is shown Figure 7(d).
In Figure 7(d)
we can see the reflector is imaged correctly
execpt the steep dip portion, which is
limited by the 45-degree wave equation.

Figure 6

Figure 7

11/17/1997