WKBJ approximation

Zhang (1993) proposes a different surface boundary condition, together with a correction for the source and receiver wavefields extrapolation, aiming to improve the dynamic information of the one-way wave equation. Zhang et al. (2001) apply this surface boundary condition to obtain a true-amplitude shot profile migration result in the WKBJ sense Shan and Biondi (2003). Zhang's surface boundary condition is stated as follows:

$$\begin{eqnarray} p_D({\bf x},z=0;\omega)&=&\frac{1}{2}\Lambda^{-1}W(\omega)\delta( {\bf x}- {\bf x}_s), \end{eqnarray}$$ (2)

where

$$\begin{eqnarray} \Lambda&=&ik_z=i\sqrt{ \frac{\omega^2}{v^2}-{\bf k}^2 }, \end{eqnarray}$$ (3)

$p_D({\bf x},z=0;\omega)$ is the new source wavefield that satisfies the corrected one-way wave equation Zhang et al. (2001), $\Lambda$ is the square-root operator, v is the medium velocity, k_z is the vertical wavenumber, and ${\bf k}=(k_x,k_y)$ is the horizontal wavenumber vector.

This boundary condition is not only an impulse at the shot position (Figure 1a) but also includes a contribution at different times and surface positions depending on the surface velocity (Figure 1b). This appears to resolve the contradiction discussed by Nichols (1994), by creating a V-shaped curve as the surface boundary condition. It mimics a wavefield with a high angle of propagation at the surface, resulting in more homogeneous wavefronts.

The implementation of Zhang's surface boundary condition has the disadvantage that the square root operator ($\Lambda$) is undefined for high values of the spatial wavenumber. That is why there is the need of establishing a cut off for the spatial wavenumber limiting the accuracy of the steep angles.

 

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Figure 1 Surface boundary condition (a) traditional, (b) Zhang's, and (c) Green function.