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Power-law size distributions in geoscience revisited

Álvaro Corral, Álvaro González

arXiv:1810.07868v2physics.geo-phnlin.AOphysics.ao-phphysics.data-anstat.AP

TL;DR

Power-law fits for geoscience hazards are difficult because valid ranges are truncated by detection limits and system size, while prior fitting practices can be statistically unreliable. The paper improves an objective fitting protocol, tests power-law and truncated log-normal alternatives across diverse data sets, and finds distinct distributional regimes whose conclusions remain limited to the analyzed data.

  • Problem

    Power-law behavior is difficult to assess rigorously because apparent log-log linearity and least-squares fits can misidentify distributions, especially when valid ranges are truncated.

    Source: §1 · p. 3, §2.1 · p. 7, §3.1 · p. 9

  • Method

    The paper extends an objective maximum-likelihood and Kolmogorov-Smirnov fitting procedure to identify cutoffs, compare power-law regimes with truncated log-normal tails, and quantify cutoff uncertainty.

    Source: §3.2 · p. 10, §3.3 · p. 12, §5 · pp. 19–20

  • Results

    The analysis finds untruncated power laws for impact fireballs and Southern-California earthquakes, a double power law for global earthquakes, truncated power-law regimes for rainfall and tropical cyclones, and truncated log-normals for sinkholes and wildfires.

    Source: §5 · p. 20

  • Takeaways & Limitations

    The fitting protocol clarifies the supported range and model type for potentially hazardous natural-phenomenon size distributions in the analyzed data sets.

    Source: §5 · pp. 19–20

    The conclusions apply only to the analyzed data sets, and some results may change with larger databases or improved completeness.

    Source: §5 · p. 20

Abstract

from arXiv · show

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 log-normal. 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 log-normals, 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.

1 Introduction

Power-law claims are widespread across geoscience and other fields, but apparent log-log linearity and least-squares fits can yield biased inferences. This paper therefore applies a rigorous, objective protocol to representative potentially hazardous geoscience data sets, while limiting its scope to size and severity distributions.

1 Introduction

  • Power-law-like behavior has been reported across social, technological, biological, and geoscientific systems, with exponents varying substantially by context.Geoscience examples include geological structures and catastrophic phenomena, while Zipf-like systems often have exponents near 2.Source: §1 · pp. 2–3
  • Statistical evidence for power laws has often relied on apparent log-log linearity or least-squares regression, which can bias probability-distribution inference.More recent studies have sought improved statistical methodology.Source: §1 · p. 3
  • The paper revises representative geoscience examples with an improved rigorous statistical protocol because heterogeneous prior approaches prevent fair comparisons.The authors describe their method as reasonable and fully objective, while noting that no definitive fitting recipe exists.Source: §1 · p. 3
  • The analysis targets sizes or energies of potentially damaging structures and natural events because fitted distributions contribute to long-term hazard assessment.Examples include sinkholes, rain clusters, earthquakes, wildfires, tropical cyclones, rainfall, and impact fireballs.Source: §1 · p. 3
  • The paper does not address mechanistic or generative explanations for power-law-like distributions, although it summarizes proposed alternatives.Its statistical analysis focuses on fitting and comparison rather than resolving the origin of these distributions.Source: §1 · pp. 3–4

2 Power-law distributions and power-law-like distributions

This section distinguishes probability-density and complementary-cumulative descriptions of power-law-like data, then develops untruncated, truncated, and double power-law models. It emphasizes that unknown lower and upper deviations, finite-size effects, incompleteness, and variable definitions determine how power-law ranges and exponents should be interpreted.

2 Power-law distributions and power-law-like distributions

  • A power-law density is defined above a lower cutoff, while its complementary cumulative function uses exponent β−1 rather than the density exponent β.The complementary cumulative function gives the probability that a value exceeds x.Source: §2.1 · pp. 4–5, §2 · p. 4

2.1 Untruncated power-law distribution

  • For an untruncated power law, maximum likelihood estimates the exponent from the sample and sets the lower cutoff to the smallest included observation.Least-squares fitting of log-transformed empirical densities or cumulative functions is inappropriate for probability-distribution inference.Source: §2.1 · pp. 5–6
  • Power-law-like distributions only resemble power laws over a possibly finite range, so the unknown lower cutoff is the central fitting problem.Small-value incompleteness and large-value finite-size effects can both create deviations from ideal power-law behavior.Source: §2.1 · pp. 6–7

2.2 Truncated power-law distribution

  • Truncated power laws model a central interval [a, b], requiring both cutoffs to be selected because deviations may occur at small and large x.The upper cutoff changes the complementary cumulative function from a simple log-log linear form, and direct endpoint estimates are not useful when the fitting subset is unknown.Source: §2.2 · pp. 7–8

2.3 Double power-law distribution

  • Double power laws represent two regimes separated by a crossover θ, with continuity enforced between their exponents.A sudden slope change at θ may poorly describe real-world distributions with many observations.Source: §2.3 · p. 8

2.4 Change of variables

  • Changing the size variable transforms the exponent, so geological objects have larger exponents when measured by linear size than by volume.The simple transformation does not apply when volumes are inferred from random cross sections.Source: §2.4 · p. 8

3 Fitting of power-law-like distributions

The section contrasts the Clauset et al. procedure with an objective alternative for fitting truncated power laws, estimating uncertainties, and comparing power-law tails with truncated log-normals.

3.1 The Clauset et al.’s procedure

  • The Clauset et al. procedure selects the lower cutoff minimizing the Kolmogorov-Smirnov distance after maximum-likelihood fitting, then evaluates the fit with bootstrap and synthetic power-law samples.The paper criticizes this method as ad hoc, inapplicable to truncated power laws, and unreliable on some simulated power-law data.Source: §3.1 · p. 9

3.2 Alternative power-law fitting procedure

  • The alternative procedure searches cutoff pairs, accepts fits with p-values above pmin = 0.20, and selects the accepted range with the largest ratio b/a.It fits β by maximum likelihood, evaluates Kolmogorov-Smirnov distances, and uses simulation to obtain p-values.Source: §3.2 · pp. 9–10
  • Bootstrap resampling supplies uncertainty distributions for both cutoffs and the exponent β, addressing the original procedure’s inability to estimate cutoff uncertainty.Because cutoff distributions are asymmetric, cutoff uncertainty is reported using the standard deviation of their logarithms.Source: §3.2 · p. 10
  • Exponent stability across increasing a and decreasing b helps distinguish a real power law from a spurious one within the accepted fitting range.A trend in β with the cutoffs can indicate an alternative distribution, especially when sample sizes are small.Source: §3.2 · p. 10

3.3 Comparison of untruncated power law versus truncated log-normal

  • The power-law-versus-truncated-log-normal comparison uses the logarithmic coefficient of variation, rejecting the power law below the fifth percentile and retaining it between the fifth and ninety-fifth percentiles.The test compares relative likelihoods rather than determining whether either candidate is an adequate fit; a log-normal may still be preferred when it fits a wider range.Source: §3.3 · p. 12

4 Data and results

The paper reanalyzes natural-hazard size distributions with objective fitting, comparing untruncated and truncated power laws with truncated log-normals across earthquakes, sinkholes, wildfires, tropical cyclones, rain, and impact fireballs. Results range from untruncated and double power laws to truncated regimes and preferred log-normal descriptions, with conclusions restricted to the analyzed data sets.

4 Data and results

  • The analysis reports results in tables of fits, bootstrap uncertainties, preferred distributions, and illustrative empirical-versus-fit figures that do not determine the fits.Figures 1 and 2 provide visual intuition only; the fitting procedure uses the statistical protocol and tabulated results.Source: §4 · pp. 12–13

4.1 Earthquakes

  • Global earthquakes are best described by a double power law, whereas Southern California earthquakes follow an untruncated power law with β = 1.655 across almost 6 orders of magnitude.The global crossover occurs near magnitude 7.6; the tail exponent exceeds 2, avoiding divergence of the mean energy.Source: §4.1 · pp. 13–14

4.2 Karst sinkholes and closed topographic depressions

  • Florida and Kentucky sinkholes show short untruncated power-law ranges with β close to 2.5, but truncated log-normals are preferred overall because they fit much larger ranges.Dead-Sea sinkholes likewise favor a globally preferred log-normal despite a weak final-order-of-magnitude power-law fit.Source: §4.2 · pp. 14–15

4.3 Wildfires

  • Wildfires lack nearly universal power-law exponents: Angola admits both power-law and log-normal fits, while Canada supports either a small-size truncated power law or a large-size log-normal.The Angola log-normal is preferred because it covers a larger range; the Canada fits have different ranges and no clear transition.Source: §4.3 · pp. 15–16

4.4 Tropical cyclones

  • Tropical-cyclone PDI distributions show marginal untruncated tails but a truncated power law with β = 1.1–1.2 over about 2 orders of magnitude.The untruncated tails span less than one order of magnitude and show increasing apparent exponents with the lower cutoff, indicating they are not genuine power laws.Source: §4.4 · pp. 16–17

4.5 Rain

  • Ocean clusters follow a truncated power-law distribution with β = 1.7 over two orders of magnitude, comprising more than one million data points, whereas the other two files show no such behavior.Source: §4.5 · p. 18

4.6 Impact fireballs

  • Impact fireballs follow an untruncated power law with β ≃2 over about 3 orders of magnitude, preferred over a log-normal across that range.The truncated power law gives practically the same result, while the exponent remains highly uncertain.Source: §4.6 · p. 19

5 Conclusions and Discussion

The paper applies a unified, objective fitting framework to diverse geoscience size distributions, revealing substantially different regimes and model preferences. Its conclusions are dataset-specific, while physical explanations such as self-organized criticality remain plausible but difficult to establish.

Method

  • The improved method constrains fitting ranges, compares power-law and log-normal tails, studies exponent stability, and bootstraps cutoff uncertainty.It extends maximum-likelihood and goodness-of-fit procedures to both untruncated and truncated power laws.Source: §5 · pp. 19–20

Earthquakes

  • Global earthquakes follow two power-law regimes, whereas Southern-California seismicity follows one untruncated power law over almost six orders of magnitude.The global regimes have β = 1.655 below about magnitude 7.6 and β ≃2 above it; the Californian range begins near magnitude 3.4.Source: §5 · p. 20

Tropical cyclones

  • Tropical-cyclone energy exhibits truncated power-law behavior in central ranges, constrained by small-storm incompleteness and finite basin boundaries.Small databases prevent certainty that the regime would persist with much larger records.Source: §5 · p. 20

Rainfall

  • Rainfall clusters generally resist accepted fits, but oceanic spatio-temporal precipitation shows a truncated power law spanning two decades across more than one million events.The very large rainfall databases make goodness-of-fit tests highly difficult to satisfy.Source: §5 · p. 20

Sinkholes

  • Sinkholes are better described by truncated log-normal distributions because their fits cover a much larger range than narrow power-law tails.The minimum-area thresholds are probably controlled by database resolution.Source: §5 · pp. 20–21

Wildfires

  • Wildfires are generally well fitted by truncated log-normals, although the Canadian record remains compatible with a power-law regime.For Canada, the two fits overlap substantially, partly because few wildfires were recorded.Source: §5 · p. 21

Physical interpretation

  • Self-organized criticality is a plausible mechanism for earthquakes and wildfires, but the complexity gap between models and real phenomena makes attribution difficult.Log-normal behavior may instead arise naturally from multiplicative random growth, while several other mechanisms can also generate power-law distributions.Source: §5 · pp. 21–22

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