SourcePertPDE
Acoustic wave equations for a virtual source shift
The wavefield is typically simulated for seismic exploration applications through solving the wave equation for a specific seismic source location. The direct relation between the shape of the wavefield and the source location can provide insights useful for velocity estimation and interpolation. As a result, I derive partial differential equations that relate changes in the wavefield shape to perturbations in the source location, especially along the Earth's surface. These partial differential equations have the same structure as the wave equation with a source function that depends on the background (original source) wavefield. The similarity in form implies that we can use familiar numerical methods to solve the perturbation equations, including finite difference and downward continuation. In fact, we can use the same Green's function to solve the wave equation and its source perturbations by simply incorporating source functions derived from the background field. The solutions of the perturbation equations represent the coefficients of a Taylor's series type expansion for the wavefield. As a result, we can speed up the wavefield calculation as we can approximate the wavefield shape in the vicinity of the original source. The new formula introduces changes to the background wavefield only in the presence of lateral velocity variation or in general terms velocity variations in the perturbation direction. The accuracy of the representation, as demonstrated on the Marmousi model, is kinematically high, with some amplitude short comings due to its approximation nature and its dependence on derivatives of the velocity field. Another form of the perturbation partial differential wave equation is independent of direct velocity derivatives, and thus, provide possibilities for wavefield continuation in complex media. The caveat here is that the medium complexity information is embedded in the wavefield and thus the wavefield shape evolution as a function of shift in the velocity or source can be extracted from the background wavefield and produce wavefield shapes for nearby sources.
A 0.5 s snap shot of the Wavefield difference for wavefields through the Marmousi Model. The wavefields were generated from sources that are 25 meters apart, then we superimpose the source and take the difference between the wavefields. Thus, the difference is solely attributable to the lateral velocity changes in the Marmousi model. The difference plot on the left is without perturbation to correct for the shift, the one in the middle is with first order perturbation and the one on the left is with second order perturbation, where the difference is obviously less.
Here are the synthetic seismograms for the three situations. Again, the second order perturbation correction shows the least coherent difference.
Department of Physical Science and Engineering 4700 King Abdullah University of Science and Technology
Thuwal 23955-6900 Saudi Arabia