Acoustic wavefield evolution as function of source location perturbation

Conclusions

The transformation in the wavefield shape as a function of source location is directly related to the lateral velocity variation. Such transformation is described by partial differential equations that have forms similar to the conventional acoustic wave equation in which their solutions provide the coefficients needed for a Taylor's series type expansion. The source function, for the perturbation equation, depends on the background wavefield of the original source as well as lateral derivatives of the velocity of the medium. 2As a result, while the second order expansion, which requires solving two PDEs, provide the best approximation of the perturbation in generally smooth velocity models, as expected, and similar to the Born approximation, the accuracy of the approximation here reduces with the size of the source perturbation. However, unlike the Born approximation, the accuracy here depends only on the amount of lateral velocity variation, not on the velocity perturbation acting as a secondary source.

As a result, lateral discontinuities in the velocity model can impose problems in the evaluation. As a result, while the second order expansion, which requires solving two PDEs, provide the best approximation of the perturbation in generally smooth velocity models, the first order version provided the best result for the unsmoothed Marmousi.Another version of the perturbation equations is independent of the velocity derivatives and dependent on higher order derivatives of the wavefield. These new equations yield good results for the unsmoothed Marmousi model in all cases.

Acoustic wavefield evolution as function of source location perturbation