Stacking seismic data using local correlation

Review of local correlation

The global uncentered correlation coefficient between two discrete signals $\mathbf{a}_i$ and $\mathbf{b}_i$ can be defined as the functional

$$\gamma = \frac{\displaystyle\sum_{i=1}^{N}\mathbf{a}_i \mat... ...m_{i=1}^{N}\mathbf{a}^2_i \sum_{i=1}^{N}\mathbf{b}^2_i}}\;,$$ (1)

where N is the length of a signal. The global correlation in equation 1 supplies only one number for the whole signal. For measuring the similarity between two signals locally, one can define the sliding-window correlation coefficient

$$\gamma_w(t) = \frac{\displaystyle\sum_{i=t-w/2}^{t+w/2} \ma... ...^2 \displaystyle\sum_{i=t-w/2}^{t+w/2} \mathbf{b}_i^2 }}\;,$$ (2)

where $w$ is window length.

Fomel (2007a) proposes the local correlation attribute that identifies local changes in signal similarity in a more elegant way. In a linear algebra notation, the correlation coefficient in equation 1 can be represented as a product of two least-squares inverses $\gamma_1$ and $\gamma_2$:

$$\gamma^2 = \gamma_1 \gamma_2\;,$$ (3)

$$\gamma_1 = \textrm{arg} \min_{\gamma_1}\parallel \mathbf{b}... ... = (\mathbf{a}^T\mathbf{a})^{-1}(\mathbf{a}^T\mathbf{b})\;,$$ (4)

$$\gamma_2 = \textrm{arg} \min_{\gamma_2}\parallel \mathbf{... ...^2 = (\mathbf{b}^T\mathbf{b})^{-1}(\mathbf{b}^T\mathbf{a})\;,$$ (5)

where $\mathbf{a}$ and $\mathbf{b}$ are vector notions for $\mathbf{a}_i$ and $\mathbf{b}_i$. Let $\mathbf{A}$ and $\mathbf{B}$ be two diagonal operators composed of the elements of a and b. Localizing equations 4 and 5 amounts to adding regularization to inversion. Using shaping regularization (Fomel, 2007b), scalars $\gamma_1$ and $\gamma_2$ turn into vectors $\mathbf{c}_1$ and $\mathbf{c}_2$, defined as

$$\mathbf{c}_1 = [\lambda^2 \mathbf{I} + \mathbf{S}(\mathbf{A... ...mbda^2 \mathbf{I})]^{-1}\mathbf{S}\mathbf{A}^T\mathbf{b}\;,$$ (6)

$$\mathbf{c}_2 = [\lambda^2 \mathbf{I} + \mathbf{S}(\mathbf{B... ...mbda^2 \mathbf{I})]^{-1}\mathbf{S}\mathbf{B}^T\mathbf{a}\;,$$ (7)

where $\lambda$ scaling controls relative scaling of operators $\mathbf{A}$ and $\mathbf{B}$ and where $\mathbf{S}$ is a shaping operator such as Gaussian smoothing with an adjustable radius. The component-wise product of vectors $\mathbf{c}_1$ and $\mathbf{c}_2$ defines the local correlation measure. Local correlation is a measure of the similarity between two signals. An iterative, conjugate-gradient inversion for computing the inverse operators can be applied in equations 6 and 7. Interestingly, the output of the first iteration is equivalent to the algorithm of fast local cross-correlation proposed by Hale (2006).

Stacking seismic data using local correlation