Applications of plane-wave destruction filters |

Let us denote by
the operator of convolving the
data with the 2-D filter of equation (12),
assuming the local slope
is known. In order to determine
the slope, we can define the least-squares goal

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 (14) 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 slope in equation (14) does not have to be
constant. We can consider it as varying in both time and space
coordinates. This eliminates the need for local windows but may lead
to undesirably rough (oscillatory) local slope estimates. Moreover,
the solution will be undefined in regions of unknown or constant data,
because for these regions the local slope is not constrained. Both
these problems are solved by adding a regularization (styling) goal to
system (14). The additional goal takes the form

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

The solution will obviously depend on the initial values of and , which should not be equal to each other. System (16) is generally underdetermined, because it contains twice as many estimated parameters as equations: The number of equations corresponds to the grid size of the data , while characterizing variable slopes and on the same grid involves two gridded functions. However, an appropriate choice of the starting solution and the additional regularization (17-18) 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.

Applications of plane-wave destruction filters |

2014-03-29