Acoustic wavefield evolution as function of source location perturbation

A lens

Since the differential equation depends on velocity changes in the direction of the perturbation, we test the methodology on a model that contains a lens anomaly in an otherwise homogeneous medium with a velocity of 2 km/s (Figure 3). The lens apex is located at 600 meters laterally and 500 meters in depth with a velocity perturbation of +250 m/s (or 12.5%). The lens has a diameter of 300 meters. Using this model we will test the accuracy of the first- and second-order perturbation equations.

model Figure 3. A velocity model containing a lens in an otherwise homogeneous background with a velocity of 2 km/s.

For a source located at the surface 0.3 km from the origin, I apply a second-order in time and fourth-order in space finite difference approximation to the wave equation as well as the perturbation equations to simulate a source 50 meters away from the original source position along the surface. A separate direct finite difference calculation using the wave equation is done for a source at 0.35 km location for comparison. Figure 4 shows a snap shot at time 0.5 seconds of the wavefield generated for the source at 0.35 km (left), as well as the snap shot at the same time for the perturbed wavefields using the first-order approximation (middle) and the second order approximation (right). All three wavefields look similar.

However, if we subtract the actual wavefield for the 0.3 km source from that of the 0.35 km one after we superpose the sources we obtain the difference between the wavefields. This difference occurs only if there is lateral velocity variation. Since there are no lateral variations in the velocity field in Figure 3 prior to wavefront from the source crossing the lens we expect that the difference snapshot plots are zero. However, at time 0.3 s the difference, where the wavefront starts to interact with the lens as shown in Figure 5, appears as expected largest for the unperturbed case (left), while the differences for the perturbed case are much smaller, especially in the case of the second-order expansion. 2All three plots in Figure 5 are displayed using the same range, for easy comparison, and this range is maintained for all Figures in this section.

timeSide20
Figure 4. A snap shot at time 0.5 seconds of the wavefield obtained from solving the conventional wave equation using the velocity model in Figure 3 for a source located at surface at 0.35 km (left), a snap shot of the wavefield by perturbing the 0.3 km source wavefield to approximate the 0.35 km one using the first order approximation (middle), and using the second order approximation (right).

timeSide212
Figure 5. A snap shot at time 0.3 seconds of the difference between the 0.35 km source wavefield and the 0.3 km source wavefield after superimposing the sources (left), the difference after using the first order perturbation on the 0.3 km source wavefield (middle), and after using the second-order perturbation on it (right). All three plots are displayed using the same range, for comparison purposes, and this range is maintained in all Figures corresponding to this lens example.

Figure 6 shows a snap shot at 0.5 s (at the same time as wavefields shown in Figure 4) of the difference. Again, the second-order approximation shows less difference, and thus, a better match than the first order approximation and definitely the unperturbed wavefield. In fact, in the unperturbed wavefield a clear polarity reversal at the anomaly apex is evident.

timeSide220
Figure 6. A snap shot at time 0.5 seconds of the difference between the 0.35 km source wavefield and the 0.3 km source wavefield after superposing the sources (left), the difference after using the first order perturbation on the 0.3 km source wavefield (middle), and after using the second-order perturbation on it (right).

Clearly, the perturbation formulas help reduce the difference between the actual wavefield and the perturbed one. An even closer look suggests that most of the difference is amplitude related.

Acoustic wavefield evolution as function of source location perturbation