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# Slope estimation

Let us denote by the operator of convolving the data with the 2-D filter C(Zt,Zx) of equation (12) assuming the local slope . In order to determine the slope, we can define the least-squares goal
 (13)
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 (9) and (10) 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 (13). The linearization implies solving the linear system
 (14)
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 (13) 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.

In the case of time- and space-varying slope ,system (14) may lead to undesirably rough slope estimates. Moreover, the solution will be undefined in the regions of unknown or constant data. Both these problems are solved by adding a regularization (styling) goal to system (14). The additional goal takes the form
 (15)
where is an appropriate roughening operator, and is a scaling coefficient. For simplicity, I chose to be the gradient operator. More efficient and sophisticated helical preconditioning techniques are available Fomel et al. (1997); Fomel (2000a).

In theory, estimating two different slopes and from the available data is only marginally more complicated. The convolution operator becomes a cascade of and , and the linearization yields
 (16)
The regularization condition should now be applied to both and :
 (17) (18)
The solution will obviously depend on the initial values of and , which should not be equal to each other. System (16) is generally under-determined, because it contains twice as many estimated parameters as equations, but an appropriate choice of the starting solution and the additional regularization conditions allow us to arrive at a practical solution.

The application examples of the next section demonstrate that when the system of equations (14-15) or (16-18) 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 (15) and (17-18) assure a smooth extrapolation of the slope to the regions of unknown or constant data.

Next: Application examples Up: Fomel: Plane-wave destructors Previous: High-order plane-wave destructors
Stanford Exploration Project
9/5/2000