Upper-truncated power laws and self-similar criticality in geophysical processes

Cumulative number-size distributions associated with many natural phenomena follow a power law. A data set that follows a power law may be fractal since both power laws and fractals are scale invariant. For many data sets of natural processes, the cumulative number-size distribution exhibits a “fall-off” from a power law as the measured object size increases. Previous attempts to analyze such distributions have often either ignored the fall-off region or described this region with a different function. We show that when a data set is abruptly truncated at large object size, fall-off from a power law is expected for the cumulative distribution. We derive functions to describe this fall-off for both linearly and logarithmically binned data. These functions lead to a generalized function, the upper-truncated power law, that is independent of binning method. Fitting the upper-truncated power law to a cumulative number-size distribution determines the parameters of the power law, thus providing the scaling exponent of the data. Using the upper-truncated power law, we analyze distributions associated with the following natural processes: forest fire areas in the Australian Capital Territory, fault offsets in the Vernejoul coal field, hydrocarbon volumes in the Frio Strand Plain exploration play, fault lengths on the plains of Venus, earthquake magnitudes associated with subduction of the Nazca plate, and hotspot seamount volumes in the Easter Island/Salas y Gomez seamount chain (ESC). In all cases, the upper-truncated power law provides a better description of the data than does a single power law. Applying the upper-truncated power law to earthquake cumulative frequency-magnitude distributions provides new insight into the reported change in b-value preceding large earthquakes. To understand hotspot seamount volume distributions, we develop a model where uniform energy input produces events initiated on a self-similar distribution of critical cells. We call this model Self-Similar Criticality (SSC). By allowing the spatial distribution of magma migration to be self-similar, the SSC model recreates the observed ESC seamount volume distribution. The SSC model may provide a connection between fractal geometry and observed power law distributions for other natural systems such as forest fires and landslides.