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Appendix A: Similarity-mean filter

Fomel (2007a) defined local similarity as follows. The global correlation coefficient between two different signals $a(t)$ and $b(t)$ 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} (5)

where $\langle x(t),y(t)\rangle$ denotes the dot product between two signals
\begin{displaymath}
\langle x(t),y(t)\rangle = \int x(t)y(t)dt\;.
\end{displaymath} (6)

In a linear algebra notation, the squared correlation coefficient $\gamma$ 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} (7)


\begin{displaymath}
\gamma_1 = (\mathbf{a}^T \mathbf{a})^{-1}(\mathbf{a}^T \mathbf{b})\;,
\end{displaymath} (8)


\begin{displaymath}
\gamma_2 = (\mathbf{b}^T \mathbf{b})^{-1}(\mathbf{b}^T \mathbf{a})\;,
\end{displaymath} (9)

where $\mathbf{a}$ is a vector notation for $a(t)$, $\mathbf{b}$ is a vector notation for $b(t)$, and $\mathbf{x}^T \mathbf{y}$ denotes the dot product operation defined in equation A-2. Let $\mathbf{A}$ be a diagonal operator composed of the elements of $\mathbf{a}$ and $\mathbf{B}$ be a diagonal operator composed of the elements of $\mathbf{b}$. Localizing equations A-4 and A-5 amounts to adding regularization to inversion. Scalars $\gamma_1$ and $\gamma_2$ turn into vectors $\mathbf{c}_1$ and $\mathbf{c}_2$ 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} (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} (11)

where $\lambda$ scaling controls the relative scaling of operators $\mathbf{A}$ and $\mathbf{B}$. Finally, the componentwise product of vectors $\mathbf{c}_1$ and $\mathbf{c}_2$ 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 up previous [pdf]

Next: Appendex B: Lower-upper-middle filter Up: Liu etc.: Structurally nonlinear Previous: Acknowledgments

2013-07-26