Slope estimation

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

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

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

$$\epsilon \bold{D} \, \Delta \bold{\sigma} \approx 0\;,$$ (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  

$$\bold{C}'(\bold{\sigma}_1) \, \bold{C}(\bold{\sigma}_2) \, ... ...igma}_1) \, \bold{C}(\bold{\sigma}_2) \, \bold{d} \approx 0\;.$$ (113)

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

$$\begin{eqnarray} \epsilon \bold{D} \, \Delta \bold{\sigma}_1 & \approx & 0\;; \\ \epsilon \bold{D} \, \Delta \bold{\sigma}_2 & \approx & 0\;.\end{eqnarray}$$ (114) (115)

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.