To test the prestack datuming scheme,
I used a simple flat reflector
with a sinusoidal topography (see Figure 15)
and a medium of constant velocity.
For the forward modeling,
the phase shift extrapolation algorithm was used.
The wavefield on the datumed level is shown
in Figure 16,
and the wavefield on the irregular surface is shown
in Figure 17.
Figures 16 and 17 show
four shot gathers; the location of each shot appears
in Figure 15 as a black dot.
Because the forward extrapolation used the phase shift algorithm,
each shot gather has artifacts caused by the Fourier wraparound.
The prestack datuming was performed using
the phase shift extrapolation as the depth extrapolation operator W.
The four shot gathers after datuming at the
same locations as those in Figure 17
are shown in Figure 18.
We can see that the datuming operator
recovers the hyperbolic event correctly
from the hyperbola distorted by the irregular topography.
The same prestack datuming algorithm was applied
with the finite-difference extrapolation schemes
for the whole prestack data shown in Figure 17;
the datumed results appear in Figure 19.
The results are also satisfactory.
presynmdl
Figure 15 Synthetic model with a flat reflector image (lower)
under an undulating surface (upper). Four black dots
along the surface represent shot locations shown
in Figure 16
csg-at-datum
Figure 16 Four shot gathers at the datumed level
using the phase-shift extrapolation scheme.
csg-at-surface
Figure 17 Four shot gathers at the irregular surface
using the phase-shift extrapolation scheme.
csg-dtmd-gaz
Figure 18 Four shot gathers at the datumed level
after prestack datuming using the phase-shift
extrapolation scheme.
csg-dtmd-wxz45
Figure 19 Four shot gathers at the datumed level
after prestack datuming
using the finite-difference extrapolation scheme.
