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.
Abstract. A new Chile ridge tectonic framework is developed based on satellite altimetry data, shipboard geophysical data and, primarily, 38,500 km of magnetic data gathered on a joint U.S.-Chilean aeromagnetic survey. Eighteen active transforms with fossil fracture zones (FZs), including two complex systems (the Chile FZ and Valdivia FZ systems), have been mapped between the northern end of the Antarctic-Nazca plate boundary (Chile ridge) at 35øS and the Chile margin triple junction at 47øS. Chile ridge spreading rates from 23 Ma to Present have been determined and show slowdowns in spreading rates that correspond to times of ridge-trench collisions. The Valdivia FZ system, previously mapped as two FZs with an uncharted seismically active region between them, is now recognized to be a multiple-offset FZ system composed of six FZs separated by short ridge segments 22 to 27 km long. At chron 5A (-12 Ma), the Chile ridge propagated from the Valdivia FZ system northward into the Nazca plate through crust formed 5 Myr earlier at the Pacific-Nazca ridge. Evidence for this propagation event includes the Friday and Crusoe troughs, located at discontinuities in the magnetic anomaly sequence and interpreted as pseudofaults. This propagation event led to the formation of the Friday microplate, which resulted in the transferal of crust from the Nazca plate to the Antarctic plate, and in a 500-km northward stepwise migration of the Pacific-Antarctic-Nazca triple junction. Rift propagation, microplate formation, microplate extinction, and stepwise triple junction migration are found to occur during large-scale plate motion changes and plate boundary changes in the southeast Pacific.
[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.
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