Nonlinear structure-enhancing filtering using plane-wave prediction

Next: Appendex B: Lower-upper-middle filter Up: Liu etc.: Structurally nonlinear Previous: Acknowledgments

# Appendix A: Similarity-mean filter

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

 (5)

where denotes the dot product between two signals
 (6)

In a linear algebra notation, the squared correlation coefficient from equation A-1 can be represented as a product of two least-squares inverses

 (7)

 (8)

 (9)

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)
 (10)

 (11)

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

Next: Appendex B: Lower-upper-middle filter Up: Liu etc.: Structurally nonlinear Previous: Acknowledgments

2013-07-26