** Next:** Application examples
** Up:** Fomel: Plane-wave destructors
** Previous:** High-order plane-wave destructors

Let us denote by the operator of convolving the
data with the 2-D filter *C*(*Z*_{t},*Z*_{x}) 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