The one-way approximation to the full wave equation has been widely used for imaging the earth's interior Claerbout (1985). This approximation ignores back-scattering in the wavefield but usually works well with surface seismic data. In the shot-profile migration scheme, two one-way wave equations need to be solved. One downward extrapolates the source wavefield and the other downward extrapolates the receiver wavefield. By using an imaging condition, the subsurface image is formed Claerbout (1971).
Usually, the initial surface boundary condition for downward extrapolating the source wavefield is chosen to be an impulse convolved with a wavelet at the shot position. However, downward extrapolation with the one-way wave equation requires boundary conditions on the surface, z=0, at all locations and all times. Since horizontal (or high-propagation-angle) waves, reflected waves, and overturned waves can all contribute to the surface wavefield, it appears that the full solution must already be known in order to supply these initial data Nichols (1994).
Zhang (1993) proposes two correction terms to the traditional one-way wave equations. One is a correction for the source and receiver wavefields extrapolation, while the other is a new boundary condition for the source wavefield. This new boundary condition is not only an impulse at the shot position but also includes contributions at different times and surface positions, depending on the surface velocity.
In this short note, we introduce an alternative surface boundary condition for the source wavefield downward extrapolation i.e, the Green function corresponding to the Helmholtz equation for a constant velocity medium at z=0. It includes contributions at different times and surface positions depending on the surface velocity, similar to Zhang's boundary condition, but differs in the amplitude.
We compare the wavefront amplitudes calculated using the new surface boundary condition, the traditional surface boundary condition, and Zhang's surface boundary condition. The comparisons are made using three synthetic velocity models: constant, vertically varying, and Marmousi. The results show that the new surface boundary condition produces more balanced wavefronts in the constant-velocity model and in the vertically varying velocity model (VOZ) and behaves similarly to Zhang's in the Marmousi velocity model.