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USGS1995

Earthquakes in the Los Angeles metropolitan region: A possible fractal distribution of rupture size

Although there is debate on the maximum size of earthquake that is possible on any of several known fault systems in the greater Los Angeles metropolitan region, it is reasonable to assume that the distribution of earthquakes will follow a fractal distribution of rupture areas. For this assumption and an overall slip-rate for the region of approximately 1 centimeter per year, roughly one magnitude 7.4 to 7.5 event is expected to occur every 245 to 325 years. A model in which the earthquake distribution is fractal predicts that, additionally, there should be approximately six events in the range of magnitude 6.6 in this same span of time, a higher rate than has occurred in the historic record.

California↗

USGS1998

MORPH-I (Ver 1.0) a software package for the analysis of scanning electron micrograph (binary formatted) images for the assessment of the fractal dimension of enclosed pore surfaces

MORPH-I is a set of C-language computer programs for the IBM PC and compatible minicomputers. The programs in MORPH-I are used for the fractal analysis of scanning electron microscope and electron microprobe images of pore profiles exposed in cross-section. The program isolates and traces the cross-sectional profiles of exposed pores and computes the Richardson fractal dimension for each pore. Other programs in the set provide for image calibration, display, and statistical analysis of the computed dimensions for highly complex porous materials. Requirements: IBM PC or compatible; minimum 640 K RAM; mathcoprocessor; SVGA graphics board providing mode 103 display.

Open-File Report↗

USGS2001

River flow mass exponents with fractal channel networks and rainfall

An important problem in hydrologic science is understanding how river flow is influenced by rainfall properties and drainage basin characteristics. In this paper we consider one approach, the use of mass exponents, in examining the relation of river flow to rainfall and the channel network, which provides the primary conduit for transport of water to the outlet in a large basin. Mass exponents, which characterize the power-law behavior of moments as a function of scale, are ideally suited for defining scaling behavior of processes that exhibit a high degree of variability or intermittency. The main result in this paper is an expression relating the mass exponent of flow resulting from an instantaneous burst of rainfall to the mass exponents of spatial rainfall and that of the network width function. Spatial rainfall is modeled as a random multiplicative cascade and the channel network as a recursive replacement tree; these fractal models reproduce certain types of self-similar behavior seen in actual rainfall and networks. It is shown that under these modeling assumptions the scaling behavior of flow mirrors that of rainfall if rainfall is highly variable in space, and on the other hand flow mirrors the structure of the network if rainfall is not so highly variable. ?? 2001 Elsevier Science Ltd. All rights reserved.

Advances in Water Resources↗

USGS2012

High-frequency spectral falloff of earthquakes, fractal dimension of complex rupture, b value, and the scaling of strength on faults

The high-frequency falloff ω −γ of earthquake displacement spectra and the b value of aftershock sequences are attributed to the character of spatially varying strength along fault zones. I assume that the high frequency energy of a main shock is produced by a self-similar distribution of subevents, where the number of subevents with radii greater than R is proportional to R −D , D being the fractal dimension. In this model, an earthquake is composed of a hierarchical set of smaller earthquakes. The static stress drop is parameterized to be proportional to R η , and strength is assumed to be proportional to static stress drop. I find that a distribution of subevents with D = 2 and stress drop independent of seismic moment (η = 0) produces a main shock with an ω −2 falloff, if the subevent areas fill the rupture area of the main shock. By equating subevents to “islands” of high stress of a random, self-similar stress field on a fault, I relate D to the scaling of strength on a fault, such that D = 2 − η. Thus D = 2 corresponds to constant stress drop scaling (η = 0) and scale-invariant fault strength. A self-similar model of aftershock rupture zones on a fault is used to determine the relationship between the b value, the size distribution of aftershock rupture zones, and the scaling of strength on a fault. The b value for aftershock sequences on a fault is found to equal (3 − 1.5η)/(3 + η). Therefore this model indicates that the typically observed spectral falloffs of ω −2 and b values of 1 can be entirely caused by scale-invariant strength (η = 0) along fault zones.

Journal of Geophysical Research↗

USGS2012

Fractal properties of tremor and gas piston events observed at Kilauea Volcano, Hawaii

Studies the fractal properties of shallow volcanic tremor and gas piston events associated with magma degassing at Kilauea Volcano, Hawaii, using data from two dense short-baseline arrays of seismographs deployed near the active crater of Puu Oo on the east rift of the volcano. The existence of a categorically stable attractor characterizing both types of activities strongly suggests that the excitation mechanism of tremor is similar to that of gas piston events, which in turn are correlated with visual observations at the volcanic vent. Low values derived for the dimension of the attractor in phase space points to significant self-organization in the process of generation of tremor and offers general constraints on the dimensionality of attractors derived from models of acoustic emission associated with magma flow, vesiculation, and degassing. -from Authors

Hawaii↗

USGS1991

Fractal analysis of multiscale spatial autocorrelation among point data

The analysis of spatial autocorrelation among point-data quadrats is a well-developed technique that has made limited but intriguing use of the multiscale aspects of pattern. In this paper are presented theoretical and algorithmic approaches to the analysis of aggregations of quadrats at or above a given density, in which these sets are treated as multifractal regions whose fractal dimension, D, may vary with phenomenon intensity, scale, and location. The technique is illustrated with Matui's quadrat house-count data, which yield measurements consistent with a nonautocorrelated simulated Poisson process but not with an orthogonal unit-step random walk. The paper concludes with a discussion of the implications of such analysis for multiscale geographic analysis systems. -Author

Environment and Planning A↗

USGS1995

Fluctuating asymmetry and fractal dimension of the sagittal suture as indicators of inbreeding depression in dama and dorcas gazelles

The effects of inbreeding on the developmental instability of skulls of dorcas ( Gazella dorcas ) and dama ( G . dama ) gazelles were investigated. In total, 132 dorcas gazelle skulls and 74 dama gazelle skulls from the Estación Experimental de Zonas Aridas in Almera, Spain, were measured. The fluctuating asymmetry of 9 meristic characters, consisting of the numbers of foramina on the two sides of the skull and mandible, was calculated. Although only the foramen infraorbitalis showed a significant increase in asymmetry with inbreeding in dorcas gazelles, the sum of the foramina in 5 of the skull regions clearly indicates an increase in asymmetry with inbreeding in both dorcas and dama gazelles. The fractal dimension of the sagittal suture was calculated by means of the coastline method. A greater effect of inbreeding on the sagittal suture in dama than in dorcas gazelle was observed, in concordance with the more evident deleterious effects of inbreeding depression in dama than in dorcas gazelles.

Canadian Journal of Zoology↗

USGS2011

Fractal analysis of the Navassa Island seascape

This release provides the numerical results of the fractal analyses discussed in Zawada and others (2010) for the Navassa Island reefscape. The project represents the continuation of a U.S. Geological Survey (USGS) research effort begun in 2006 (Zawada and others, 2006) to understand the patterns and scalability of roughness and topographic complexity from individual corals to complete reefscapes.

Data Series↗

USGS1998

Instability of development and fractal architecture in dryland plants as an index of grazing pressure

Developmental instability has been used to monitor the well-being of natural populations exposed to physical, chemical and biological stressors. Here, we use developmental instability to assess the impact of grazing on Chrysothamnus greenii and Seriphidium novum shrubs, and Oryzopsis hymenoides grass, common in the arid intermountain west of the U.S.A. Statistical noise in allometric relations was used as an indicator of developmental instability arising from grazing-induced stress. Unpalatable species that are not grazed ( Chrysothamnus greenii ) or species that are dormant during the winter–spring grazing period ( Oryzopsis hymenoides ) show lower allometric variability under high grazing pressure. Palatable species ( Seriphidium novum ) exhibit high developmental instability under low and high grazing pressure. Grazing pressure imposed by presumably co-adapted wild herbivores enhances developmental stability in species habituated to moderate grazing, like Oryzopsis hymenoides , but stresses plants such as Chrysothamnus greenii that prefer disturbed environments. These grazing effects are probably due to the impact grazing has on competitive relationships and not to the direct action of the herbivore on the plants.

Journal of Arid Environments↗

USGS1993