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Slope estimation

Let us denote by $\bold{C}(\bold{\sigma})$ the operator of convolving the data with the 2-D filter C(Zt,Zx) of equation ([*]) assuming the local slope $\bold{\sigma}$. In order to determine the slope, we can define the least-squares goal  
 \bold{C}(\bold{\sigma}) \, \bold{d} \approx 0\;,\end{displaymath} (110)
where $\bold{d}$ is the known data and the approximate equality implies that the solution is found by minimizing the power of the left-hand side. Equations ([*]) and ([*]) show that the slope $\bold{\sigma}$ enters in the filter coefficients in an essentially non-linear way. However, one can still apply the linear iterative optimization methods by an analytical linearization of equation ([*]). The linearization (also known as the Newton iteration) implies solving the linear system  
 \bold{C}'(\bold{\sigma}_0) \, \Delta \bold{\sigma} \, \bold{d} + \bold{C}(\bold{\sigma}_0) \, \bold{d} \approx 0\end{displaymath} (111)
for the slope increment $\Delta \bold{\sigma}$. Here $\bold{\sigma}_0$is the initial slope estimate, and $\bold{C}'(\bold{\sigma})$ is a convolution with the filter, obtained by differentiating the filter coefficients of $\bold{C}(\bold{\sigma})$ with respect to $\bold{\sigma}$. After system ([*]) is solved, the initial slope $\bold{\sigma}_0$ is updated by adding $\Delta \bold{\sigma}$ to it, and one can solve the linear problem again. Depending on the starting solution, the method may require several non-linear iterations to achieve an acceptable convergence. The described linearization approach is similar in idea to tomographic velocity estimation Nolet (1987).

In the case of a time- and space-varying slope $\bold{\sigma}$,system ([*]) may lead to undesirably rough slope estimates. Moreover, the solution will be undefined in regions of unknown or constant data. Both these problems are solved by adding a regularization (styling) goal to system ([*]). The additional goal takes the form analogous to ([*]):  
 \epsilon \bold{D} \, \Delta \bold{\sigma} \approx 0\;,\end{displaymath} (112)
where $\bold{D}$ is an appropriate roughening operator and $\epsilon$is a scaling coefficient. For simplicity, I chose $\bold{D}$ to be the gradient operator. An alternative choice would be to treat local dips as smooth data and to apply to them the tension-spline preconditioning technique from the previous section.

In theory, estimating two different slopes $\bold{\sigma}_1$ and $\bold{\sigma}_2$ from the available data is only marginally more complicated than estimating a single slope. The convolution operator becomes a cascade of $\bold{C}(\bold{\sigma}_1)$ and $\bold{C}(\bold{\sigma}_2)$, and the linearization yields  
 \bold{C}'(\bold{\sigma}_1) \, \bold{C}(\bold{\sigma}_2) \, ...
 ...igma}_1) \,
 \bold{C}(\bold{\sigma}_2) \, \bold{d} \approx 0\;.\end{displaymath} (113)
The regularization condition should now be applied to both $\Delta
\bold{\sigma}_1$ and $\Delta \bold{\sigma}_2$:
 \epsilon \bold{D} \, \Delta \bold{\sigma}_1 & \approx & 0\;; \\  \epsilon \bold{D} \, \Delta \bold{\sigma}_2 & \approx & 0\;.\end{eqnarray} (114)
The solution will obviously depend on the initial values of $\bold{\sigma}_1$ and $\bold{\sigma}_2$, which should not be equal to each other. System ([*]) is generally underdetermined, because it contains twice as many estimated parameters as equations, but an appropriate choice of the starting solution and the additional regularization ([*]-[*]) allow us to arrive at a practical solution.

The application examples of the next subsection demonstrate that when the system of equations ([*]-[*]) or ([*]-[*]) are optimized in the least-squares sense in a cycle of several linearization iterations, it leads to smooth and reliable slope estimates. The regularization conditions ([*]) and ([*]-[*]) assure a smooth extrapolation of the slope to the regions of unknown or constant data.

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Next: Examples of data regularization Up: Regularizing local plane waves Previous: High-order plane-wave destructors
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