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# Method

The flattening method described in Lomask et al. (2005) creates a time-shift (or depth-shift) field such that its gradient approximates the dip . The dip is a function of because for any given horizon, the appropriate dips to be summed are the dips along the horizon itself. Using the gradient operator () and the estimated dip (), our regression is
 (1)

To add regularization in the time direction, we apply a 3D gradient operator with a residual weight that controls the amount of vertical regularization defined as
 (2)
where is a large block diagonal matrix consisting of two identity matrices and a diagonal matrix =.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 as
 (3)
The residual is defined as
 (4)

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


iterate {

 (5) (6) (7)

}  ,

where the subscript k denotes the iteration number.

We wish to add a model mask to prevent changes to specific areas of an initial field. This initial 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
 (8) (9) (10) (11) (12)
Our resulting equations are now


iterate {

 (13) (14) (15)

}  .

Typically, we solve equation (6) in the Fourier domain, however in equation (14), 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.