Acoustic wavefield evolution as function of source location perturbation

Discussion

The transformation of the wavefield's shape as a function of source location provides valuable information for many applications, and in particular interpolation, velocity estimation, and imaging. All three applications rely in one way or another on the relation between the wavefields and the source location. To predict the content of a missing shot (or receiver) gather, we usually rely on reflection slope information of nearby common shot gathers [Fomel (2002)] to extend the information to the missing locations. For homogeneous and vertically inhomogeneous media, the process of interpolation is trivial as the common shot gathers are the same regardless of surface position. The complication occurs when the velocity varies laterally and differences, as we have seen above, can be large. Using these perturbation partial differential equations we can estimate the changes needed to fill in the gaps. This can be done as part of a finite-difference modeling or a reverse time migration process. It also can be done using a point source to generate the wavefield in a forward finite-difference approach or using a boundary condition, as typically the case for reverse time extrapolation of the receiver wavefields recored at the surface for the purpose of imaging. The equations shown above have no source restrictions and their development are not based on a particular source.

A major drawback of using conventional methods to solve the wave equation is that typically the velocity information and complexity have no baring on the efficiency of obtaining such solutions. In the development here, the perturbed wavefields are only excited by lateral velocity variation and in the absence of such variations we do not need to evaluate the perturbations. This allows us to implement selective computations that depends on the wavefield complexity and isolate areas of contribution based on the velocity field.

Nevertheless, the accuracy of the first and second order expansion approximations introduced here depends on the size of the source (or velocity) shift. Unlike, the traveltime version [Alkhalifah and Fomel (2009)] the wavefield is highly 2oscillatory (sinusoidal components)and thus their Taylor's series approximation accuracy is dependent on the wavelength of the perturbed wavefield within the context of the lateral velocity complexity. 2The accuracy here is synonymous to what we encounter using the Born approximation. However, unlike Born approximation, the source functions in equations (4) and (7) depend on the lateral velocity variation, not the source perturbation. Specifically, if the lateral velocity change induces perturbations in the wavefield that exceeds a half wavelength, we will encounter aliasing in the construction. This issue effects, more frequently, large dips and large perturbations with respect to the wavelength. However, unlike conventional source or velocity perturbation developments, this wavefield shape perturbation approach is far more stable and explicitly depends on the complexity of the lateral velocity variation as for the case of lateral homogeneity the approach is exact independent of amount of perturbation.

timeSSSide6F48
Figure 12. A snap shot of the difference between the 5 km source wavefield and the 5.05 km source wavefield (left), and the 5.075 km source wavefield (middle), and the 5.1 km source wavefield (right) after using the modified second-order expansion for 50, 75 and 100 meters perturbations, respectively. The velocity model is the original smoothed Marmousi model.

Figure 12 shows a 1.2 second snap shot of the differences between the wavefields obtained directly from a source and that obtained from nearby sources and perturbed a distance of 50 meters (left), 75 meters (middle), and 100 meters (right) for the smoothed Marmousi model using the second-order expansion. As expected the difference (error) is larger for the bigger perturbation. Also, we can observe that the dipping parts of the wavefield have larger errors as the effective change is bigger. Of course, we have to remind our selves that we are dealing with the Marmousi model, which is highly complex, and we can expect better results for smoother models. Also, we can observe that the difference is mainly manifested in the amplitude, where the kinematics (phase) show little difference.

Acoustic wavefield evolution as function of source location perturbation