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Discussion

The data example as well as the previous case studies by Van der Baan et al. (2010b) underline how analysis of the local phase can be used as a complementary attribute to spectral decomposition to highlight variations in wavelet character. There are two complimentary applications, namely analysis of the propagating and locally observed wavelets. The targeted wavelet type is determined by the chosen regularization length: the propagating wavelet is estimated by using long temporal regularization lengths, and the locally observed one from shorter lengths. The underlying assumption is that, for long regularization lengths, variations in the local geology are averaged out, revealing only the propagating wavelet. In this paper, we used relatively short regularization lengths as the aim is to highlight changes in the local reflection character.

Well-log analyses have demonstrated that the Earth's reflectivity series is non-Gaussian (Walden and Hosken, 1986) and, to first order, white (Walden and Hosken, 1985). In addition, impedances tend to increase with depth, hence positive reflection coefficients are slightly more likely than negative one, producing an asymmetric reflectivity distribution. Statistically, the skewness-based criterion assumes that the Earth's reflectivity series are white and asymmetric. This is in contrast to kurtosis used previously by Van der Baan and Fomel (2009), which assumes a non-Gaussian reflectivity series. Both the non-Gaussianity and asymmetry assumptions seem warranted but may fail if the local reflectivity series becomes respectively purely Gaussian or symmetric.

The local skewness attribute has the advantage over kurtosis because of its higher dynamic range, which facilitates picking. Variance is the second statistical order, skewness is the third one, and kurtosis is related to the fourth order. Estimation variances increases with the order of a moment (Mendel, 1991). In other words, less samples are needed to estimate skewness with the same accuracy as kurtosis. We hypothesize that this contributes to the higher dynamic range of the skewness criterion.


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Next: Conclusions Up: Fomel & van der Previous: Application example

2014-02-15