Nonlinear structure-enhancing filtering using plane-wave prediction

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

$$\gamma = \frac {\langle a(t),b(t)\rangle}{\sqrt{\langle a(t),a(t)\rangle \langle b(t),b(t)\rangle}}\;,$$ (5)

where $\langle x(t),y(t)\rangle$ denotes the dot product between two signals

$$\langle x(t),y(t)\rangle = \int x(t)y(t)dt\;.$$ (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

$$\gamma^2 = \gamma_1 \gamma_2\;,$$ (7)

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

$$\gamma_2 = (\mathbf{b}^T \mathbf{b})^{-1}(\mathbf{b}^T \mathbf{a})\;,$$ (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)

$$\mathbf{c}_1 = [\lambda^2 \mathbf{I} + \mathbf{S}(\mathbf... ...\lambda^2 \mathbf{I})]^{-1}\mathbf{S}\mathbf{A}^T\mathbf{b}\;,$$ (10)

$$\mathbf{c}_2 = [\lambda^2 \mathbf{I} + \mathbf{S}(\mathbf... ...\lambda^2 \mathbf{I})]^{-1}\mathbf{S}\mathbf{B}^T\mathbf{a}\;,$$ (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).

Nonlinear structure-enhancing filtering using plane-wave prediction