THE ``WAVEKILL'' FILTER

Jon Claerbout suggested that I should use a filter that had the property that it would destroy plane waves of a given dip. Given such an operator I could perform a joint estimation of the local dip and the missing data values. He suggested a form for such a filter, which he calls the ``plane wave destruction'' filter or the ``wavekill'' operator.

I can reject a dip, p, with the operator,

$${\bf A} \equiv (\partial_x-p\partial_t).$$

An approximation to this operator can be implemented on a $2 \times 2$ differencing star. The best estimate of the dip of an array of data can be found by minimizing

$$\vert A \cdot \phi \vert^2$$

If ${\bf x}$ is the array $\partial \phi / \partial x$ and ${\bf t}$ is the array $\partial \phi / \partial t$ then the minimum value of $\vert {\bf x} - p {\bf t}\vert^2$ is at $p = {\bf x} \cdot {\bf t} / {\bf t}\cdot{\bf t}.$

To use this operator for interpolation when the dip is known I minimize the residual,

$$\vert A\cdot u + A\cdot k \vert^2.$$

This is a linear least squares problem of the type described earlier. If the dip is not known I have to solve a non-linear problem for both the dip and the missing data values.