** Next:** Examples of data regularization
** Up:** Regularizing local plane waves
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Let us denote by the operator of convolving the
data with the 2-D filter *C*(*Z*_{t},*Z*_{x}) 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