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The flattening method described in Lomask et al. (2005) creates a time-shift (or depth-shift) field $\boldsymbol{\tau}(x,y,t)$ such that its gradient approximates the dip $\bf{p}\rm (x,y,\boldsymbol{\tau})$. The dip is a function of $\boldsymbol{\tau}$ because for any given horizon, the appropriate dips to be summed are the dips along the horizon itself. Using the gradient operator ($\boldsymbol\nabla=[ \frac{\partial }{\partial x} \; \; \frac{\partial }{\partial y}]^T$) and the estimated dip (${\bf p} = [{{\bf p}_x} \; \; {{\bf p}_y}]^T$), our regression is
\boldsymbol{\nabla} {\boldsymbol \tau}(x,y,t)\quad = \quad {\bf p}(x,y,{\boldsymbol \tau}).\end{displaymath} (1)

To add regularization in the time direction, we apply a 3D gradient operator with a residual weight ${\bf W}_\epsilon$ that controls the amount of vertical regularization defined as
{\bf W}_\epsilon\boldsymbol{\nabla}=
 ... \  \frac{ \partial }{ \partial t} \end{array} \right] \quad ,\end{displaymath} (2)
where ${\bf W}_\epsilon$ is a large block diagonal matrix consisting of two identity matrices $\bold I$ and a diagonal matrix $\boldsymbol{\epsilon}$=$\epsilon \bold I$.For simplicity, we implicitly chain this weight operator to the gradient operator to create a new operator now defined as a 3D gradient with an weighting parameter $\epsilon$ as  
\boldsymbol{\nabla}_\epsilon \quad = \quad \left[ \begin{arr...
 ...ilon \frac{ \partial }{ \partial t} \end{array} \right] \quad .\end{displaymath} (3)
The residual is defined as  
\bf r \quad = \quad { \boldsymbol{\nabla}_\epsilon \boldsymb...
 ...}{{\bf p}_x} \  {{\bf p}_y} \  {\bf 0} \end{array} \right] }.\end{displaymath} (4)

We solve this using a Gauss-Newton approach by iterating over equations (5)-(7), i.e.,

		 iterate {    
\bf r \quad &=& \quad [\boldsymbol{\nabla}_\epsilon \boldsymbol...
 ...bol{\boldsymbol{\tau}}_{k} + \Delta \boldsymbol{\boldsymbol{\tau}}\end{eqnarray} (5)

		  }  ,
where the subscript k denotes the iteration number.

We wish to add a model mask $\bf K $ to prevent changes to specific areas of an initial $\boldsymbol{\tau}_0$ field. This initial $\boldsymbol{\tau}_0$ field can be picks from any source. In general, they may come from a manually picked horizon or group of horizons. These initial constraints do not have to be a continuous surfaces but instead could be isolated picks, such as well-to-seismic ties. To apply the mask we follow the same the development as the operator approach to missing data in Claerbout (1999) as
 \bold 0 &\approx& \boldsymbol{\nabla}_\epsilon \boldsymbol{\ta...
 ...mbol{\nabla}_\epsilon\bold K\boldsymbol{\tau} + \bold r_0 - \bf p.\end{eqnarray} (8)
Our resulting equations are now

		 iterate {    
\bf r \quad &=& \quad \boldsymbol{\nabla}_\epsilon {\bf K}\bold...
 ...bol{\boldsymbol{\tau}}_{k} + \Delta \boldsymbol{\boldsymbol{\tau}}\end{eqnarray} (13)

		  }  .
Typically, we solve equation (6) in the Fourier domain, however in equation (14), $\bf K $ is non-stationary making its application in the Fourier domain difficult if not impossible. Therefore, for now we solve it in the time domain using conjugate gradients.