Power Law Size Distributions in Geoscience Revisited

Abstract

The size or energy of diverse structures or phenomena in geoscience appears to follow power law distributions. A rigorous statistical analysis of such observations is tricky, though. Observables can span several orders of magnitude, but the range for which the power law may be valid is typically truncated, usually because the smallest events are too tiny to be detected and the largest ones are limited by the system size. We revisit several examples of proposed power law distributions dealing with potentially damaging natural phenomena. Adequate fits of the distributions of sizes are especially important in these cases, given that they may be used to assess long‐term hazard. After reviewing the theoretical background for power law distributions, we improve an objective statistical fitting method and apply it to diverse data sets. The method is described in full detail, and it is easy to implement. Our analysis elucidates the range of validity of the power law fit and the corresponding exponent and whether a power law tail is improved by a truncated lognormal. We confirm that impact fireballs and Californian earthquakes show untruncated power law behavior, whereas global earthquakes follow a double power law. Rain precipitation over space and time and tropical cyclones show a truncated power law regime. Karst sinkholes and wildfires, in contrast, are better described by truncated lognormals, although wildfires also may show power law regimes. Our conclusions only apply to the analyzed data sets but show the potential of applying this robust statistical technique in the future.

Some Experiments in Extreme‐Value Statistical Modeling of Magnetic Superstorm Intensities

“…For a lognormal model fitted to the data for a superstorm threshold of −Dst c = 283 nT, Figure 4f, the p value is 0.97 (a high probability). But this result, like those we have previously reported for similar analyses (e.g., Love et al, 2015), is based on data that were used to estimate the model parameters-it is circular logic (we now recognize) to calculate a p value using the same data used to estimate a model's parameters (e.g., Chave, 2017, Chapter 8.2.4;Clauset et al, 2009, Section 3.4;Corral & González, 2019, Section 3.2; Steinskog et al, 2007).…”

“…Consider, next, a rescaling of the power law and Fréchet distributions ( > 0) by a positive factor r, This means that the power law distribution and the extreme-value end of the Fréchet distribution are "regularly varying" (e.g., Beirlant et al, 2004, Chapter 2.9.2;Bingham et al, 1987;Zwart, 2009, Chapter 2.1), "scale-invariant," or "self-similar" (e.g., Corral & González, 2019;Ghil et al, 2011;Sornette, 2006)-that is, a scaling of the independent variable x gives a corresponding power scaling of the distribution. In contrast, the lognormal distribution does not have this self-similar property,…”

Scale-free networks well done

“…Specifically, it is quite unclear what it really means for a degree sequence in a given real-world network to be power-law or "close" to a power law. This lack of rigor has led and still leads to confused controversy and never-ending heated debates [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. This controversy has culminated in the recent work [20] that concluded that "scale-free networks are rare."…”

No Significant Effect of Coulomb Stress on the Gutenberg-Richter Law after the Landers Earthquake

“…where the distribution of the test statistic and, from it, the p−value of the fit, p f it , is calculated using 10 4 Monte Carlo simulations [61,62]. Although some fitting procedures look for the value of m min that optimizes the fit for a given data set [61][62][63], we have opted for a fixed m min in order to compare the different subsets on the same footing. So, in all cases the exponential fit for m ≥ 3 cannot be rejected (p−value of the test larger than 0.05).…”