![next](icons/next.png) |
![up](icons/up.png) |
![previous](icons/previous.png) |
![](icons/left.png) | Nonlinear structure-enhancing filtering using plane-wave prediction | ![](icons/right.png) |
![[pdf]](icons/pdf.png) |
Next: Appendex B: Lower-upper-middle filter
Up: Liu etc.: Structurally nonlinear
Previous: Acknowledgments
Fomel (2007a) defined local similarity as follows. The global
correlation coefficient between two different signals
and
is the functional
![\begin{displaymath}
\gamma = \frac {\langle a(t),b(t)\rangle}{\sqrt{\langle a(t),a(t)\rangle \langle b(t),b(t)\rangle}}\;,
\end{displaymath}](img33.png) |
(5) |
where
denotes the dot product between two signals
![\begin{displaymath}
\langle x(t),y(t)\rangle = \int x(t)y(t)dt\;.
\end{displaymath}](img35.png) |
(6) |
In a linear algebra notation, the squared correlation coefficient
from equation A-1 can be represented as a product of two
least-squares inverses
![\begin{displaymath}
\gamma^2 = \gamma_1 \gamma_2\;,
\end{displaymath}](img37.png) |
(7) |
![\begin{displaymath}
\gamma_1 = (\mathbf{a}^T \mathbf{a})^{-1}(\mathbf{a}^T \mathbf{b})\;,
\end{displaymath}](img38.png) |
(8) |
![\begin{displaymath}
\gamma_2 = (\mathbf{b}^T \mathbf{b})^{-1}(\mathbf{b}^T \mathbf{a})\;,
\end{displaymath}](img39.png) |
(9) |
where
is a vector notation for
,
is a
vector notation for
, and
denotes the
dot product operation defined in equation A-2. Let
be a diagonal operator composed of the elements of
and
be a diagonal operator composed of the
elements of
. Localizing equations A-4
and A-5 amounts to adding regularization to
inversion. Scalars
and
turn into vectors
and
defined, using shaping
regularization (Fomel, 2007b)
![\begin{displaymath}
\mathbf{c}_1 = [\lambda^2 \mathbf{I} + \mathbf{S}(\mathbf...
...\lambda^2 \mathbf{I})]^{-1}\mathbf{S}\mathbf{A}^T\mathbf{b}\;,
\end{displaymath}](img49.png) |
(10) |
![\begin{displaymath}
\mathbf{c}_2 = [\lambda^2 \mathbf{I} + \mathbf{S}(\mathbf...
...\lambda^2 \mathbf{I})]^{-1}\mathbf{S}\mathbf{B}^T\mathbf{a}\;,
\end{displaymath}](img50.png) |
(11) |
where
scaling controls the relative scaling of operators
and
. Finally, the componentwise product of
vectors
and
defines the local similarity
measure.
For using time-dependent smooth weights in the stacking process, the
local similarity amplitude can be chosen as a weight for stacking
seismic data. We thus stack only those parts of the predicted data whose
similarity to the reference one is comparatively large (Liu et al., 2009a).
![next](icons/next.png) |
![up](icons/up.png) |
![previous](icons/previous.png) |
![](icons/left.png) | Nonlinear structure-enhancing filtering using plane-wave prediction | ![](icons/right.png) |
![[pdf]](icons/pdf.png) |
Next: Appendex B: Lower-upper-middle filter
Up: Liu etc.: Structurally nonlinear
Previous: Acknowledgments
2013-03-02