Stephen M. Burroughs

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.

Cumulative versus transient shoreline change: Dependencies on temporal and spatial scale

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.

Dune Retreat and Shoreline Change on the Outer Banks of North Carolina

Abstract Barrier islands are popular recreational areas of economic importance and are constantly undergoing change. Costly efforts are made to maintain beaches and stabilize dunes within this dynamic environment. Light detection and ranging data collected in September 1997 and 1998 along a 175-km stretch of the Atlantic coast of the Outer Banks, North Carolina, provide the basis for quantitative determination of the changes in beach morphology. The 1998 survey was conducted just after the passage of Hurricane Bonnie. During the 1-year study interval, beach widths throughout the study region tended to decrease. Maximum dune retreat was determined for each 1-m bin of 1997 beach width. For comparable beach widths, maximum dune retreat increased from south to north. The maximum dune retreat was greatest for supratidal beaches with widths of ∼20 m. For wider supratidal beaches, from 20 to 60 m, the associated maximum dune retreat gradually decreased. There was no further decrease in maximum dune retreat for beaches wider than ∼60 m. Relatively little change in beach width, dune height, and dune base position occurred along the less developed beaches of the Core Banks. The greatest morphological changes occurred on Ocracoke Island and Hatteras Island. Of the geomorphic parameters examined, preexisting beach width and the dune base elevation were the best indicators of vulnerability to dune retreat.

Predicting Water Table Response to Rainfall Events, Central Florida

A rise in water table in response to a rainfall event is a complex function of permeability, specific yield, antecedent soil-water conditions, water table level, evapotranspiration, vegetation, lateral groundwater flow, and rainfall volume and intensity. Predictions of water table response, however, commonly assume a linear relationship between response and rainfall based on cumulative analysis of water level and rainfall logs. By identifying individual rainfall events and responses, we examine how the response/rainfall ratio varies as a function of antecedent water table level (stage) and rainfall event size. For wells in wetlands and uplands in central Florida, incorporating stage and event size improves forecasting of water table rise by more than 30%, based on 10 years of data. At the 11 sites studied, the water table is generally least responsive to rainfall at smallest and largest rainfall event sizes and at lower stages. At most sites the minimum amount of rainfall required to induce a rise in water table is fairly uniform when the water table is within 50 to 100 cm of land surface. Below this depth, the minimum typically gradually increases with depth. These observations can be qualitatively explained by unsaturated zone flow processes. Overall, response/rainfall ratios are higher in wetlands and lower in uplands, presumably reflecting lower specific yields and greater lateral influx in wetland sites. Pronounced depth variations in rainfall/response ratios appear to correlate with soil layer boundaries, where corroborating data are available.

Statistical Self-Similarity of Hotspot Seamount Volumes Modeled as Self-Similar Criticality

The processes responsible for hotspot seamount formation are complex, yet the cumulative frequency-volume distribution of hotspot seamounts in the Easter Island/Salas y Gomez Chain (ESC) is found to be well-described by an upper-truncated power law. We develop a model for hotspot seamount formation 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 have broad applicability to other natural systems.

Time-Frequency Methods for Characterizing Cuspate Landforms in Lidar Data

Abstract Time-frequency techniques to characterize cuspate patterns in light detection and ranging (lidar) data are introduced using examples from the Atlantic coast of Florida, United States. These techniques permit the efficient study of beach face landforms over many kilometers of coastline at multiple spatial scales. From a lidar image, a beach-parallel spatial series is generated. Here, this series is the shore-normal position of a specific elevation (contour line). Well-established time-frequency analysis techniques, wavelet transforms, and S-Transforms, are then applied to the spatial series. These methods yield results entirely compatible with the traditional method of estimating the spacing of cuspate features. In addition, confidence intervals are readily established for the spatial extent and wavelengths of cuspate landforms simultaneously at multiple scales. Examples show this method is useful for capturing transitions in cuspate shapes. With the advent of land-based time-lapse lidar, such techniques should be particularly useful for characterizing the evolution of cuspate landforms and testing models for beach face dynamics.

Self-Similar Criticality

Cumulative frequency-size distributions associated with many natural phenomena follow a power law. Self-organized criticality (SOC) models have been used to model characteristics associated with these natural systems. As originally proposed, SOC models generate event frequency-size distributions that follow a power law with a single scaling exponent. Natural systems often exhibit power law frequency-size distributions with a range of scaling exponents. We modify the forest fire SOC model to produce a range of scaling exponents. In our model, uniform energy (material) input produces events initiated on a self-similar distribution of critical grid cells. An event occurs when material is added to a critical cell, causing that material and all material in occupied non-diagonal adjacent cells to leave the grid. The scaling exponent of the resulting cumulative frequency-size distribution depends on the fractal dimension of the critical cells. Since events occur on a self-similar distribution of critical cells, we call this model Self-Similar Criticality (SSC). The SSC model may provide a link between fractal geometry in nature and observed power law frequency-size distributions for many natural systems.