Upper-truncated Power Laws in Natural Systems

Abstract — When a cumulative number-size distribution of data follows a power law, the data set is often considered fractal since both power laws and fractals are scale invariant. Cumulative number-size distributions for data sets of many natural phenomena exhibit a “fall-off ” from a power law as the measured object size increases. We demonstrate that this fall-off is expected when a cumulative data set is truncated at large object size. We provide a generalized equation, herein called the General Fitting Function (GFF), that describes an upper-truncated cumulative number-size distribution based on a power law. Fitting the GFF to a cumulative number-size distribution yields the coefficient and exponent of the underlying power law and a parameter that characterizes the upper truncation. Possible causes of upper truncation include data sampling limitations (spatial or temporal) and changes in the physics controlling the object sizes. We use the GFF method to analyze four natural systems that have been studied by other approaches: forest fire area in the Australian Capital Territory; fault offsets in the Vernejoul coal field; hydrocarbon volumes in the Frio Strand Plain exploration play; and fault lengths on Venus. We demonstrate that a traditional approach of fitting a power law directly to the cumulative number-size distribution estimates too negative an exponent for the power law and overestimates the fractal dimension of the data set. The four systems we consider are well fit by the GFF method, suggesting they have properties characterized by upper-truncated power laws.