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The 2-D velocity model in Figure is composed of
three regions with variable velocity separated by
a dipping linear interface and a curved interface. Above
the first interface, the velocity has a horizontal and a vertical gradient.
Between the first dipping interface and the curved interface the
horizontal gradient is reversed. Under the curved interface there
is only a vertical gradient.
There is a velocity discontinuity at the location of the
dipping event and another velocity jump under the curved boundary.
The structural events are a dipping interface at the first
velocity boundary
and nine diffractors positioned at equal depth intervals.
Both modeling algorithms are implemented using absorbing
boundaries (Cerjan et al., 1985).
The size of the absorbing region is a 20 gridpoint strip
on each side of the grid.
**mv
**

Figure 4 The 2-D reflection model and the velocity model. Velocity in the
upper layer *v*(*x*,*z*)=1500 + 0.8*x* + 0.7*z* (m/s),
in the intermediate layer *v*(*x*,*z*)=3000-0.8*x*+0.7*z* (m/s) and
in the lower layer *v*(*x*,*z*)=4000+0.7*z* (m/s).

While in the migration case the PSPI appears to perform better than
Split-step in the presence of lateral velocity discontinuities,
in the modeling case I found the Split-Step algorithm to
achieve better results.
Figure presents the results of modeling
with the two algorithms using the velocity model shown
in Figure . The PSPI modeling in this figure actually
was implemented without the additional phase shift subtraction
and addition introduced by Gazdag in the original PSPI
algorithm. When using PSPI modeling, the amplitudes of the
diffractions appear weaker than in the Split-Step algorithm.

**pssp2D
**

Figure 5 2-D comparison of the two modeling algorithms:

**a.** PSPI modeling. The individual diffractors
appear less defined compared to the Split-Step image.

**b.** Split-Step Fourier modeling.

Figure shows the results of running
PSPI modeling with the additional phase shift trick.
The difference
between the two figures is that in Figure a the
extra phase-shift term from equation (11)
is applied **after** the upward propagation step
which corresponds to the conjugate transpose of the original
algorithm. In Figure b the extra phase-shift term is
applied **before** the upward propagation step, which is equivalent
to applying the Gazdag trick after downward propagation in the
migration algorithm, in the
same manner as it is done in the Split-Step migration.
As seen here the two figures look almost identical.
However it appears that while the individual diffractions are
better delineated than in the simple PSPI algorithm
(without the phase-shift trick), the dipping reflector is not.
Overall the faster Split-Step appears to give better results for this
particular velocity model.

**pspidv2D
**

Figure 6 2-D comparison of the two modeling algorithms:

**a.** PSPI modeling with extra phase-shift **after**
upward propagation.

**b.** PSPI modeling with extra phase-shift **before** upward propagation.

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Stanford Exploration Project

11/18/1997