Acoustic Wave Equations for a Virtual Source Shift

Abstract

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. 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.