Acoustic wavefield evolution as function of source location perturbation

The Marmousi model

The geometry of the Marmousi is based, somewhat, on a profile through the Cuanza basin Versteeg (1993). The target zone is a reservoir located at a depth of about 2500 m. The model contains many reflectors, steep dips, and strong velocity variations in both the lateral and the vertical directions (with a minimum velocity of 1500 m/s and a maximum of 5500 m/s). However, the Marmousi model includes complex discontinuities that pose problems to the perturbation formulation. As a result, we smooth the velocity model to obtain the model in Figure 7 (right). 2The point source considered here is a Ricker wavelet with a 15 Hz peak frequency.

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Figure 7. The Marmousi velocity model (left) and a smoothed version of it (right).

Again using the fourth-order finite difference approximation in space and second order in time we solve the wave equation for a source located at the surface at lateral position 5 km, A snap shot of the resulting wavefield at time 1.2 s is shown in Figure 8 (left). Solving the wave equation for a source located 25 meters away results in the snap shot of the wavefield at 1.2 s shown in Figure 8 (middle). Superimposing the sources for the two fields and subtracting them yields the difference shown in Figure 8 (right). All three snap shots are plotted at the same scale 2(and this scale is maintained for all Figures in this section) and thus the difference, which is totally due to lateral inhomogeneity, is relatively large. It is especially large for the parts of the wavefront that were exposed to large lateral variations in the smoothed Marmousi model. This difference represents the wavefield we anticipate from the solution of new perturbation equation.

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Figure 8. A snap shot of the wavefield obtained from solving the conventional wave equation in the smoothed Marmousi model for a source located at surface at 5 km (left) and at 5.025 km (middle). The difference between the two wavefields when we shift one of them to make the sources coincide is shown on the right. All three plots are displayed using the same range and this range is maintained in all Figures corresponding to Marmousi example.

Using the new perturbation partial differential equations, I predict this difference from the original wavefield with source at location 5 km. We then add this difference to that original wavefield using equation (8), which provides an approximate to the wavefield for a source located at 5.025 km. Figure 9 shows a 1.2 s snap shot of the wavefield computed directly from a source at 5.025 km (left) and that obtained from the first-order perturbation expansion (middle), as well as, the difference between the two wavefields (right). Clearly, the difference is now less than that in Figure 8 in which perturbation was not used.

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Figure 9. A snap shot of the wavefield obtained from solving the conventional wave equation in the smoothed Marmousi model for a source located at surface at 5.025 km (left), and the snap shot by perturbing the 5 km source wavefield to approximate the 5.025 km one using the first-order equation (middle). The difference between the two wavefields is shown on the right.

In fact, if we display the differences side by side along with that associated with the second-order perturbation approximation, Figure 10 demonstrates that the difference decreases considerably for the higher-order perturbation approximation, shown on the right.

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Figure 10. A snap shot of the difference between the 5 km source wavefield and the 5.025 km source wavefield (left), the difference after using the first order perturbation (middle), and after using the second-order perturbation (right).

One of the main objectives of solving the wave equation is to simulate the behavior of the wavefield at the surface (the measurement plane) as a function of time. Figure 11 shows the difference between the 5.025 km source and the 5 km source common-shot gathers after superimposing the sources (left) and compares it with difference between the 5.025 km source and first-order (middle) and the second order (right) perturbed versions. Clearly, the source gather extracted from using the perturbation equations better resemble the directly evaluated one than the source gather that does not include the perturbation. Specifically, most of the primary reflections in the section are seemingly well modeled by the perturbation approximation, as evidence by the small difference between the directly extracted gather and the perturbed one.

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Figure 11. The difference between a shot gather for a source located at surface at 5.025 km and one located at 5 km after superimposing the sources (left), the difference between a shot gather for a source located at surface at 5.025 km and one extracted from the expansion of the 5 km source location using the first-order perturbation approximation (middle), and using the second-order perturbation approximation (right) for the smoothed Marmousi model.

Acoustic wavefield evolution as function of source location perturbation