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Application to the dip estimation problem

The goal of dip estimation is to find a local stepout, $\sigma$, that destroys the local plane wave such that,
0 \approx \frac{\partial u}{\partial h} + \sigma \frac{\partial u}{\partial \tau},\end{displaymath} (9)
where u is the wavefield at time $\tau$ and offset h. For all gathers, we evaluate the slope $\sigma$ with a method based on high-order plane-wave destructor filters Fomel (2002). With the Z transform notation, Fomel (2002) shows that there is a 2-D filter

Cn(Zt,Zh)=Bn(Zt-1)-ZhBn(Zt), (10)

B_n(Z_t)=\sum_{k=-n/2}^{n/2}a_k(\sigma^{n-1})Z_t^{k}, \end{displaymath} (11)
that annihilates the local plane wave. The number of coefficients for the filter Bn is n. The filter coefficients $a_k(\sigma^{n-1})$ are functions of $\sigma^{n-1}$ as detailed in equations (9) and (10) of Fomel (2002). For instance, if n=3, we have
a_{-1}(\sigma^2)&=&\frac{(1-\sigma)(2-\sigma)}{12},\nonumber \\...
 ...,\\  a_{1} (\sigma^2)&=&\frac{(1+\sigma)(2+\sigma)}{12}.\nonumber \end{eqnarray}
In equation (11), $\sigma$ is unknown and is estimated with a non-linear solver.