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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
| ![\begin{displaymath}
\boldsymbol{\nabla} {\boldsymbol \tau}(x,y,t)\quad = \quad {\bf p}(x,y,{\boldsymbol \tau}).\end{displaymath}](img6.gif) |
(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
| ![\begin{displaymath}
{\bf W}_\epsilon\boldsymbol{\nabla}=
\left[
\begin{array}
{c...
... \ \frac{ \partial }{ \partial t} \end{array} \right] \quad ,\end{displaymath}](img8.gif) |
(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
| ![\begin{displaymath}
\boldsymbol{\nabla}_\epsilon \quad = \quad \left[ \begin{arr...
...ilon \frac{ \partial }{ \partial t} \end{array} \right] \quad .\end{displaymath}](img13.gif) |
(3) |
The residual is defined as
| ![\begin{displaymath}
\bf r \quad = \quad { \boldsymbol{\nabla}_\epsilon \boldsymb...
...}{{\bf p}_x} \ {{\bf p}_y} \ {\bf 0} \end{array} \right] }.\end{displaymath}](img14.gif) |
(4) |
We solve this using a Gauss-Newton approach by iterating over equations (5)-(7), i.e.,
iterate {
| ![\begin{eqnarray}
\bf r \quad &=& \quad [\boldsymbol{\nabla}_\epsilon \boldsymbol...
...bol{\boldsymbol{\tau}}_{k} + \Delta \boldsymbol{\boldsymbol{\tau}}\end{eqnarray}](img15.gif) |
(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
| ![\begin{eqnarray}
\bold 0 &\approx& \boldsymbol{\nabla}_\epsilon \boldsymbol{\ta...
...mbol{\nabla}_\epsilon\bold K\boldsymbol{\tau} + \bold r_0 - \bf p.\end{eqnarray}](img18.gif) |
(8) |
| (9) |
| (10) |
| (11) |
| (12) |
Our resulting equations are now
iterate {
| ![\begin{eqnarray}
\bf r \quad &=& \quad \boldsymbol{\nabla}_\epsilon {\bf K}\bold...
...bol{\boldsymbol{\tau}}_{k} + \Delta \boldsymbol{\boldsymbol{\tau}}\end{eqnarray}](img19.gif) |
(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.