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Given the filter D2 (k), defined in formula (10), we can
construct an accurate approximation to the second derivative operator
-k2 by considering a filter ratio (another Padé-type
approximation) of the form
| ![\begin{displaymath}
-k^2 \approx \frac{D_2(k)}{1 + \beta D_2 (k)}\;,\end{displaymath}](img55.gif) |
(25) |
where
is an adjustable constant Claerbout (1985).
The actual Padé coefficient is
. As pointed out by
Francis Muir, the value of
gives an exact fit at the Nyquist frequency
. Fitting the
derivative operator in the L1 norm yields the value of
. All these approximations are shown in Figure
10.
sixth
Figure 10 The second-derivative operator
in the wavenumber domain and its approximations.
|
| ![sixth](../Gif/sixth.gif) |
B
Next: Constructing an ``isotropic'' Laplacian
Up: Fomel & Claerbout: Implicit
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Stanford Exploration Project
9/12/2000