Nonlinear structure-enhancing filtering using plane-wave prediction |

Fomel (2007a) defined local similarity as follows. The global
correlation coefficient between two different signals and
is the functional

In a linear algebra notation, the squared correlation coefficient
from equation A-1 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 defined in equation A-2. Let be a diagonal operator composed of the elements of and be a diagonal operator composed of the elements of . Localizing equations A-4 and A-5 amounts to adding regularization to inversion. Scalars and turn into vectors and defined, using shaping regularization (Fomel, 2007b)

where scaling controls the relative scaling of operators and . Finally, the componentwise product of vectors and 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 |

2013-07-26