Stacking seismic data using local correlation |

The global uncentered correlation coefficient between two discrete
signals and can be defined as the functional

where 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 and :

where and are vector notions for and . Let and 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 and turn into vectors and , defined as

where scaling controls relative scaling of operators and and where is a shaping operator such as Gaussian smoothing with an adjustable radius. The component-wise product of vectors and 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 |

2013-03-02