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

Slope estimation

Let us denote by the operator of convolving the data with the 2-D filter C(Zt,Zx) of equation ( ) assuming the local slope . In order to determine the slope, we can define the least-squares goal (110)
where 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 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 (111)
for the slope increment . Here is the initial slope estimate, and is a convolution with the filter, obtained by differentiating the filter coefficients of with respect to . After system ( ) is solved, the initial slope is updated by adding 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 ,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 ( ): (112)
where is an appropriate roughening operator and is a scaling coefficient. For simplicity, I chose 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 and from the available data is only marginally more complicated than estimating a single slope. The convolution operator becomes a cascade of and , and the linearization yields (113)
The regularization condition should now be applied to both and : (114) (115)
The solution will obviously depend on the initial values of and , 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.     Next: Examples of data regularization Up: Regularizing local plane waves Previous: High-order plane-wave destructors
Stanford Exploration Project
12/28/2000