A 2-D example

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.

 

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

 

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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.

 

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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.