Ð 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-o '' from a power law as the measured object size increases. We demonstrate that this fall-o 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 coecient 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 ®re area in the Australian Capital Territory; fault osets in the Vernejoul coal ®eld; hydrocarbon volumes in the Frio Strand Plain exploration play; and fault lengths on Venus. We demonstrate that a traditional approach of ®tting 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 ®t by the GFF method, suggesting they have properties characterized by upper-truncated power laws.

[1] Using shoreline change measurements of two oceanside reaches of the North Carolina Outer Banks, USA, we explore an existing premise that shoreline change on a sandy coast is a self-affine signal, wherein patterns of change are scale invariant. Wavelet analysis confirms that the mean variance (spectral power) of shoreline change can be approximated by a power law at alongshore scales from tens of meters up to ∼4-8 km. However, the possibility of a power law relationship does not necessarily reveal a unifying, scale-free, dominant process, and deviations from power law scaling at scales of kilometers to tens of kilometers may suggest further insights into shoreline change processes. Specifically, the maximum of the variance in shoreline change and the scale at which that maximum occurs both increase when shoreline change is measured over longer time scales. This suggests a temporal control on the magnitude of change possible at a given spatial scale and, by extension, that aggregation of shoreline change over time is an important component of large-scale shifts in shoreline position. We also find a consistent difference in variance magnitude between the two survey reaches at large spatial scales, which may be related to differences in oceanographic forcing conditions or may involve hydrodynamic interactions with nearshore geologic bathymetric structures. Overall, the findings suggest that shoreline change at small spatial scales (less than kilometers) does not represent a peak in the shoreline change signal and that change at larger spatial scales dominates the signal, emphasizing the need for studies that target long-term, large-scale shoreline change.

A power-law scaling relationship describes tsunami runup heights at ten locations in Japan. Knowledge of the scaling law for tsunamis can be the basis for probabilistic forecasting of the size and number of future events and for estimating probabilities of extremely large events. Using tsunami runup data archived by the U.S. National Geophysical Data Center, we study ten locations where the tsunami record spans at least one order of magnitude in runup height and the temporal record extends back several decades. A power law or upper-truncated power law describes the cumulative frequency-size distribution of tsunami runup heights at all ten locations. Where the record is sufficient to examine shorter time intervals within the record, the scaling relationship for the shorter time intervals is consistent with the scaling relationship for the entire record. The scaling relationship is used to determine recurrence intervals for tsunami runup heights at each location. In addition to the tsunami record used to determine the scaling relationship, at some of the locations a record of large events (>5 m) extends back several centuries. We find that the recurrence intervals of these large events are consistent with the frequency predicted from the more recent record. For tsunami prone locations where a scaling relationship is determined, the predicted recurrence intervals may be useful for planning by coastal engineers and emergency management agencies.

The horizontal, shore-perpendicular change in shoreline position along the Outer Banks of North Carolina is found to be a self-affine signal. We measure shoreline change by determining the horizontal change in position of the 0.8-m contour sampled from shoreperpendicular profiles spaced at 20-m intervals along the coast. The profiles are obtained from two light detection and ranging surveys performed in September 1997 and September 1998. For six selected sections of coast, wavelet analysis of the shoreline change signal indicates the signal is self-affine with a scaling exponent that varies from 1.2 to 2.1. This self-affine behavior indicates that the shoreline change signal is nonstationary with long-range persistence. A stochastic diffusion model of sediment transport replicates the observed self-affine behavior observed south of Cape Hatteras (scaling exponent between 1.2 and 1.6) whereas a random walk model replicates the signal observed north of Cape Hatteras (scaling exponent Ϸ2.0). Because of the finite nature of the data set, there are limits in space and time to the power law behavior of the system. Characteristics of such systems can be described by upper-truncated power laws, which yield the upper limits of power law behavior. Applying an upper-truncated power law to the data for one section of coast, we find an upper limit of 7 km for the maximum continuous alongshore distance eroding or accreting. For the same section of coast, we find upper limits of 25 m for the maximum shore-perpendicular erosion and 11 m for the maximum shore-perpendicular accretion during the study period.