Upper-truncated power laws in natural systems

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 ar e scale invariant. Cumulative number-size distributions for data sets of ma ny 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 genera lized equation, herein called the General Fitting Function (GFF), that desc ribes an upper-truncated cumulative number-size distribution based on a pow er 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 inc lude data sampling limitations (spatial or temporal) and changes in the phy sics controlling the object sizes. We use the GFF method to analyze four na tural systems that have been studied by other approaches: forest fire area in the Australian Capital Territory; fault offsets in the Vernejoul coal fi eld; hydrocarbon volumes in the Frio Strand Plain exploration play; and fau lt lengths on Venus. We demonstrate that a traditional approach of fitting a power law directly to the cumulative number-size distribution estimates t oo negative an exponent for the power law and overestimates the fractal dim ension of the data set. The four systems we consider are well fit by the GF F method, suggesting they have properties characterized by upper-truncated power laws.