Point source

The traditional boundary condition for the source wavefield extrapolation with the one-way wave equation Claerbout (1985) can be written as follows:  

$$D({\bf x},z=0;\omega)=W(\omega)\delta( {\bf x}- {\bf x}_s),$$ (1)

where $D({\bf x},z=0;\omega)$ is the source wavefield at the surface, $W(\omega)$ is the wavelet, and $\delta( {\bf x}- {\bf x}_s)$ is an impulse at the shot position ${\bf x}_s$. Although kinematically correct, Claerbout's scheme fails to provide the correct wavefield amplitudes.