Slope estimation

Let us denote by $\bold{C}(\bold{s})$ the operator of convolving the data with the 2-D filter C(Z_t,Z_x) of equation (12) assuming the local slope $\bold{s}$. In order to determine the slope, we can define the least-squares goal  

$$\bold{C}(\bold{s}) \, \bold{d} \approx 0\;,$$ (13)

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 (9) and (10) show that the slope $\bold{s}$ 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  

$$\bold{C}'(\bold{s}_0) \, \Delta \bold{s} \, \bold{d} + \bold{C}(\bold{s}_0) \, \bold{d} \approx 0$$ (14)

for the slope increment $\Delta \bold{s}$. Here $\bold{s}_0$ is the initial slope estimate, and $\bold{C}'(\bold{s})$ is a convolution with the filter, obtained by differentiating the filter coefficients of $\bold{C}(\bold{s})$ with respect to $\bold{s}$. After system (13) is solved, the initial slope $\bold{s}_0$ is updated by adding $\Delta \bold{s}$ 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 $\bold{s}$,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  

$$\epsilon \bold{D} \, \Delta \bold{s} \approx 0\;,$$ (15)

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. More efficient and sophisticated helical preconditioning techniques are available Fomel et al. (1997); Fomel (2000a).

In theory, estimating two different slopes $\bold{s}_1$ and $\bold{s}_2$ from the available data is only marginally more complicated. The convolution operator becomes a cascade of $\bold{C}(\bold{s}_1)$ and $\bold{C}(\bold{s}_2)$, and the linearization yields  

$$\bold{C}'(\bold{s}_1) \, \bold{C}(\bold{s}_2) \, \Delta \... ...}(\bold{s}_1) \, \bold{C}(\bold{s}_2) \, \bold{d} \approx 0\;.$$ (16)

The regularization condition should now be applied to both $\Delta \bold{s}_1$and $\Delta \bold{s}_2$:

$$\begin{eqnarray} \epsilon \bold{D} \, \Delta \bold{s}_1 & \approx & 0\;; \ \epsilon \bold{D} \, \Delta \bold{s}_2 & \approx & 0\;.\end{eqnarray}$$ (17) (18)

The solution will obviously depend on the initial values of $\bold{s}_1$ and $\bold{s}_2$, 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.