Stable Distribution of Fractional Fluctuations of Well Log Data in Japan

For the statistical characterization of the random heterogeneity of the solid earth medium, we have measured the distribution of fractional fluctuations of well log data after removing the slowly varying background variation. The well log data used are P- and S-wave velocities and mass density acquired in Japan: three data sets for the thick sedimentary layers of the Kanto Plain and two data sets for the Kujyu volcanic area of Kyushu. The estimated kurtosis is much greater than 3 for most cases, which is difficult to explain with a normal distribution. This suggests that the random fluctuations should be looked at from a broader perspective. We note the fact that the generalized central limit theorem leads to stable distributions, which are broad enough to include normal and Cauchy distributions as special cases. When the stable, normal, and Cauchy distributions are applied to the histograms of the observed fractional fluctuations, the stable distribution is found to be the most appropriate among the three distributions in most cases, where the stability parameter \(\alpha\) ranges from 1.5 to 2. These case studies demonstrate that the stable distribution faithfully describes the thick tails of the observed fractional fluctuations as the result of a geological sedimentation process without removing them as outliers.