Local seismic attributes

Definition of local correlation

In a linear algebra notation, the squared correlation coefficient $\gamma$ from equation 8 can be represented as a product of two least-squares inverses

$$\gamma^2 = \gamma_1\,\gamma_2\;,$$ (9)

$$\gamma_1 = \left(\mathbf{a}^T\,\mathbf{a}\right)^{-1}\,\left(\mathbf{a}^T\,\mathbf{b}\right)\;,$$ (10)

$$\gamma_2 = \left(\mathbf{b}^T\,\mathbf{b}\right)^{-1}\,\left(\mathbf{b}^T\,\mathbf{a}\right)\;,$$ (11)

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. Let $\mathbf{A}$ be a diagonal operator composed from the elements of $\mathbf{a}$ and $\mathbf{B}$ be a diagonal operator composed from the elements of $\mathbf{b}$. Localizing equations 10-11 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 , as

$$\mathbf{c}_1 = \left[\lambda^2\,\mathbf{I} + \mathbf{S}\,\left(\mathbf{A}^T\,\... ...bda^2\,\mathbf{I}\right)\right]^{-1}\, \mathbf{S}\,\mathbf{A}^T\,\mathbf{b}\;,$$ (12)

$$\mathbf{c}_2 = \left[\lambda^2\,\mathbf{I} + \mathbf{S}\,\left(\mathbf{B}^T\,\... ...bda^2\,\mathbf{I}\right)\right]^{-1}\, \mathbf{S}\,\mathbf{B}^T\,\mathbf{a}\;.$$ (13)

To define a local similarity measure, I apply the component-wise product of vectors $\mathbf{c}_1$ and $\mathbf{c}_2$. It is interesting to note that, if one applies an iterative conjugate-gradient inversion for computing the inverse operators in equations 12 and 13, the output of the first iteration will be the smoothed product of the two signals $\mathbf{c}_1 = \mathbf{c}_2 = \mathbf{S}\,\mathbf{A}^T\,\mathbf{b}$, which is equivalent, with an appropriate choice of $\mathbf{S}$, to the algorithm of fast local cross-correlation proposed by Hale (2006).

The local similarity attribute is useful for solving the problem of multicomponent image registration. After an initial registration using interpreter's ``nails'' (DeAngelo et al., 2004) or velocities from seismic processing, a useful registration indicator is obtained by squeezing and stretching the warped shear-wave image while measuring its local similarity to the compressional image. Such a technique was named residual $\gamma$ scan and proposed by Fomel et al. (2005). Figure  shows a residual scan for registration of multicomponent images from Figure . Identifying and picking points of high local similarity enables multicomponent registration with high-resolution accuracy. The registration result is visualized in Figure , which shows interleaved traces from PP and SS images before and after registration. The alignment of main seismic events is an indication of successful registration.

Local seismic attributes