Synthetic examples

I test this datuming scheme with phase-shift, split-step, and finite-difference depth extrapolation algorithms. For testing, I use a simple model, Figure 4, that has a syncline reflector under an undulating surface. The irregular surface is modeled to have not only gradual topographic changes but also a discontinuity.

 

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

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.

 

datumgsw
Figure 5 (a) Wave field recorded on the irregular surface using phase-shift extrapolation scheme. (b) Datumed wavefield using phase-shift algorithm. (c) Datumed wavefield using split-step algorithm. (d) Datumed wavefield using 45-degree finite-difference algorithm.

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.+0.2x+0.2z (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.

 

vxz-split-modl
Figure 6 (a) velocity model ( v(x,z)=1500.+0.2x+0.2z ). (b) Wave field recorded on the irregular surface using split-step extrapolation scheme.

 

vxz-split-wxz-datum
Figure 7 (a) Datumed wavefield using split-step algorithm. (b) migrated image using split-step algorithm. (c) Datumed wavefield using finite-difference algorithm. (d) migrated image using finite-difference algorithm.