wavefield reconstruction

Gazdag and Sguazzero (1984) use the following method for merging the extrapolated wavefield. Assume that the two extrapolated wavefields with the different reference velocities v_ref¹and v_ref² are as follows:  

$$P_{1}\left( x_{i},y_{j},z+\triangle z;\omega \right)=A_{1} e^{i\theta_{1}},$$ (18)

 

$$P_{2}\left( x_{i},y_{j},z+\triangle z;\omega \right)=A_{2} e^{i\theta_{2}},$$ (19)

where A₁ and A₂, and $\theta_{1}$ and $\theta_{2}$ represent amplitude and phase, respectively. The merged wavefield at a spatial point is  

$$P\left( x_{i}, y_{j}, z+ \triangle z; \omega \right)= Ae^{i \theta}.$$ (20)

The amplitude and phase of the merged wavefield is cauculated with  

$$A=\frac{ A_{1}\left( v_{ref}^{2}-v\left( x_{i},y_{j},z+\tri... ...angle z\right)-v_{ref}^{1}\right) }{v_{ref}^{2}-v_{ref}^{1}},$$ (21)

and  

$$\theta= \frac{ \theta_{1}\left( v_{ref}^{2}-v\left( x_{i},... ...ngle z\right)-v_{ref}^{1}\right) } {v_{ref}^{2}-v_{ref}^{1}}.$$ (22)

where $v\left( x_{i},y_{j},z+\triangle z\right)$ is the velocity at the spatial point $\left( x_{i},y_{j},z+\triangle z\right)$.

Kessinger (1992) shows that the wavefield at a spatial point can be directly replaced by the extrapolated wavefield with the reference velocity corresponding to the spatial point. The following formula is used for merging the extrapolated wavefields with SSF operator

$$\begin{eqnarray} P\left( x_{i},y_{j},z+\triangle z;\omega \right) &=& \delta \l... ...angle z}P\left( x_{i},y_{j},z+\triangle z;\omega \right) \right],\end{eqnarray}$$ (23)

where $v_{ref}\left( x_{i},y_{j},z+\triangle z\right)$ stands for the reference velocity at a spatial point $\left( x_{i},y_{j}\right)$, v_ref^l is the member of the set of chosen reference velocities with the index l. The delta function $\delta\left(\bullet \right) =1$ if $v_{ref}\left( x_{i},y_{j},z+\triangle z\right)=v_{ref}^{l}$; otherwise, $\delta\left(\bullet \right) =0$. With this method, the extrapolated wavefield can be directly inserted into the relevant position without storing it. However, the amplitude and phase of the merged wavefield is not so accurate in the case of severe lateral velocity variations. We use quadratic interpolation to reconstruct the extrapolated wavefield with the following equation:

where $P^{l-1}\left( x_{i},y_{j},z+\triangle z;\omega \right)$, $P^{l}\left(x_{i},y_{j},z+\triangle z;\omega \right)$ and $P^{l+1}\left(x_{i},y_{j},z+\triangle z;\omega \right)$ are the extrapolated wavefield at point $\left( x_{i},y_{j}\right)$ with the three adjacent reference velocities v_ref^l-1, v_ref^l and v_ref^l+1, respectively.