Local seismic attributes |

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

where is a vector notation for , is a vector notation for , and denotes the dot product operation. Let be a diagonal operator composed from the elements of and be a diagonal operator composed from the elements of . Localizing equations 10-11 amounts to adding regularization to inversion. Scalars and turn into vectors and defined, using shaping regularization , as

To define a local similarity measure, I apply the component-wise product of vectors and . 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 , which is equivalent, with an appropriate choice of , 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 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 |

2013-03-02