SEARCH · USGS Science

Results for “Fractals”

Search indexed USGS publications on groundwater, aquifers, geologic maps, mineral resources and earthquakes. Explore source records by subject and place.

SourceResearch connectionFrom yearThrough year

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 19 recordsLinked to original sources

USGS1999

Characterization of branch complexity by fractal analyses and detect plant functional adaptations

The comparison between complexity in the sense of space occupancy (box-counting fractal dimension Dc and information dimension DI ) and heterogeneity in the sense of space distribution (average evenness index and evenness variation coefficient JCV) were investigated in mathematical fractal objects and natural branch ¯ J structures. In general, increased fractal dimension was paired with low heterogeneity. Comparisons between branch architecture in Anthyllis cytisoides under different slope exposure and grazing impact revealed that branches were more complex and more homogeneously distributed for plants on northern exposures than southern, while grazing had no impact during a wet year. Developmental instability was also investigated by the statistical noise of the allometric relation between internode length and node order. In conclusion, our study demonstrated that fractal dimension of branch structure can be used to analyze the structural organization of plants, especially if we consider not only fractal dimension but also shoot distribution within the canopy (lacunarity). These indexes together with developmental instability analyses are good indicators of growth responses to the environment.

International Journal of Plant Sciences↗

USGS1995

Fractal structures and fractal functions as disease indicators

Developmental instability is an early indicator of stress, and has been used to monitor the impacts of human disturbance on natural ecosystems. Here we investigate the use of different measures of developmental instability on two species, green peppers (Capsicum annuum), a plant, and Spanish ibex (Capra pyrenaica), an animal. For green peppers we compared the variance in allometric relationship between control plants, and a treatment group infected with the tomato spotted wilt virus. The results show that infected plants have a greater variance about the allometric regression line than the control plants. We also observed a reduction in complexity of branch structure in green pepper with a viral infection. Box-counting fractal dimension of branch architecture declined under stress infection. We also tested the reduction in complexity of behavioral patterns under stress situations in Spanish ibex (Capra pyrenaica). Fractal dimension of head-lift frequency distribution measures predator detection efficiency. This dimension decreased under stressful conditions, such as advanced pregnancy and parasitic infection. Feeding distribution activities reflect food searching efficiency. Power spectral analysis proves to be the most powerful tool for character- izing fractal behavior, revealing a reduction in complexity of time distribution activity under parasitic infection.

Oikos↗

USGS1991

Fractal patterns of fractures in granites

Fractal measurements using the Cantor's dust method in a linear one-dimensional analysis mode were made on the fracture patterns revealed on two-dimensional, planar surfaces in four granites. This method allows one to conclude that: (1)|The fracture systems seen on two-dimensional surfaces in granites are consistent with the part of fractal theory that predicts a repetition of patterns on different scales of observation, self similarity. Fractal analysis gives essentially the same values of D on the scale of kilometres, metres and centimetres (five orders of magnitude) using mapped, surface fracture patterns in a Sierra Nevada granite batholith (Mt. Abbot quadrangle, Calif.). (2)|Fractures show the same fractal values at different depths in a given batholith. Mapped fractures (main stage ore veins) at three mining levels (over a 700 m depth interval) of the Boulder batholith, Butte, Mont. show the same fractal values although the fracture disposition appears to be different at different levels. (3)|Different sets of fracture planes in a granite batholith, Central France, and in experimental deformation can have different fractal values. In these examples shear and tension modes have the same fractal values while compressional fractures follow a different fractal mode of failure. The composite fracture patterns are also fractal but with a different, median, fractal value compared to the individual values for the fracture plane sets. These observations indicate that the fractal method can possibly be used to distinguish fractures of different origins in a complex system. It is concluded that granites fracture in a fractal manner which can be followed at many scales. It appears that fracture planes of different origins can be characterized using linear fractal analysis.

Earth and Planetary Science Letters↗

USGS1995

The generalized 20/80 law using probabilistic fractals applied to petroleum field size

Fractal properties of the Pareto probability distribution are used to generalize "the 20/80 law." The 20/80 law is a heuristic law that has evolved over the years into the following rule of thumb for many populations: 20 percent of the population accounts for 80 percent of the total value. The general p100/q100 law in probabilistic form is defined with q as a function of p, where p is the population proportion and q is the proportion of total value. Using the Pareto distribution, the p100/q100 law in fractal form is derived with the parameter q being a fractal, where q unexpectedly possesses the scale invariance property. The 20/80 law is a special case of the p100/q100 law in fractal form. The p100/q100 law in fractal form is applied to petroleum fieldsize data to obtain p and q such that p100% of the oil fields greater than any specified scale or size in a geologic play account for q100% of the total oil of the fields. The theoretical percentages of total resources of oil using the fractal q are extremely close to the empirical percentages from the data using the statistic q. Also, the empirical scale invariance property of the statistic q for the petroleum fieldsize data is in excellent agreement with the theoretical scale invariance property of the fractal q. ?? 1995 Oxford University Press.

Nonrenewable Resources↗

USGS2008

Are fractal dimensions of the spatial distribution of mineral deposits meaningful?

It has been proposed that the spatial distribution of mineral deposits is bifractal. An implication of this property is that the number of deposits in a permissive area is a function of the shape of the area. This is because the fractal density functions of deposits are dependent on the distance from known deposits. A long thin permissive area with most of the deposits in one end, such as the Alaskan porphyry permissive area, has a major portion of the area far from known deposits and consequently a low density of deposits associated with most of the permissive area. On the other hand, a more equi-dimensioned permissive area, such as the Arizona porphyry permissive area, has a more uniform density of deposits. Another implication of the fractal distribution is that the Poisson assumption typically used for estimating deposit numbers is invalid. Based on datasets of mineral deposits classified by type as inputs, the distributions of many different deposit types are found to have characteristically two fractal dimensions over separate non-overlapping spatial scales in the range of 5-1000 km. In particular, one typically observes a local dimension at spatial scales less than 30-60 km, and a regional dimension at larger spatial scales. The deposit type, geologic setting, and sample size influence the fractal dimensions. The consequence of the geologic setting can be diminished by using deposits classified by type. The crossover point between the two fractal domains is proportional to the median size of the deposit type. A plot of the crossover points for porphyry copper deposits from different geologic domains against median deposit sizes defines linear relationships and identifies regions that are significantly underexplored. Plots of the fractal dimension can also be used to define density functions from which the number of undiscovered deposits can be estimated. This density function is only dependent on the distribution of deposits and is independent of the definition of the permissive area. Density functions for porphyry copper deposits appear to be significantly different for regions in the Andes, Mexico, United States, and western Canada. Consequently, depending on which regional density function is used, quite different estimates of numbers of undiscovered deposits can be obtained. These fractal properties suggest that geologic studies based on mapping at scales of 1:24,000 to 1:100,000 may not recognize processes that are important in the formation of mineral deposits at scales larger than the crossover points at 30-60 km. ?? 2008 International Association for Mathematical Geology.

Natural Resources Research↗

USGS1989

The frictional properties of a simulated gouge having a fractal particle distribution

The frictional properties of a layer of simulated Westerly granite fault gouge sandwiched between sliding blocks of Westerly granite have been measured in a high-speed servo-controlled double-direct shear apparatus. Most gouge layers were prepared to have a self-similar particle distribution with a fractal dimension of 2.6. The upper fractal limit was varied between 45 and 710 μm. Some gouges were prepared with all particles in the range between 360 and 710 μm. In each experiment the sliding velocity was cyclically alternated between 1 and 10 μms −1 and the coefficient of friction μm and its transient parameters a, b and D c were measured as functions of displacement. In addition to the particle size distribution, the following experimental variables were also investigated: the layer thickness (1 and 3 mm), the roughness of the sliding surfaces (Nos 60 and 600 grit) and the normal stress (10 and 25 MPa). Some of the sample assemblies were epoxy impregnated following a run so the gouge structure could be microscopically examined in thin section. We observed that gouges which were initially non-fractal evolved to a fractal distribution with dimension 2.6. Gouges which had an initial fractal distribution remained fractal. When the sliding blocks had smooth surfaces, the coefficient of friction was relatively low and was independent of the particle distribution. In these cases, strong velocity weakening was observed throughout the experiment and the transient parameters a, b and D c , remained almost constant. When the sliding blocks had rough surfaces, the coefficient of friction was larger and more dependent on the particle distribution. Velocity strengthening was observed initially but evolved to velocity weakening with increased sliding displacement. All three transient parameters changed with increasing displacement. The a and b values were about three times as large for rough surfaces as for smooth. The characteristic displacement D c was not sensitive to surface roughness but was the only transient parameter which was sensitive to the normal stress. For the case of rough surfaces, the coefficient of friction of the 1 mm thick gouge was significantly larger than that for the 3 mm thick layers. Many of these observations can be explained by a micromechanical model in which the stress in the gouge layer is heterogeneous. The applied normal and shear stresses are supported by ‘grain bridges’ which span the layer and which are continually forming and failing. In this model, the frictional properties of the gouge are largely determined by the dominant failure mode of the bridging structures.

Journal of Structural Geology↗

USGS2003

Quantitative characterization of the regressive ecological succession by fractal analysis of plant spatial patterns

We studied the effect of grazing on the degree of regression of successional vegetation dynamic in a semi-arid Mediterranean matorral. We quantified the spatial distribution patterns of the vegetation by fractal analyses, using the fractal information dimension and spatial autocorrelation measured by detrended fluctuation analyses (DFA). It is the first time that fractal analysis of plant spatial patterns has been used to characterize the regressive ecological succession. Plant spatial patterns were compared over a long-term grazing gradient (low, medium and heavy grazing pressure) and on ungrazed sites for two different plant communities: A middle dense matorral of Chamaerops and Periploca at Sabinar-Romeral and a middle dense matorral of Chamaerops, Rhamnus and Ulex at Requena-Montano. The two communities differed also in the microclimatic characteristics (sea oriented at the Sabinar-Romeral site and inland oriented at the Requena-Montano site). The information fractal dimension increased as we moved from a middle dense matorral to discontinuous and scattered matorral and, finally to the late regressive succession, at Stipa steppe stage. At this stage a drastic change in the fractal dimension revealed a change in the vegetation structure, accurately indicating end successional vegetation stages. Long-term correlation analysis (DFA) revealed that an increase in grazing pressure leads to unpredictability (randomness) in species distributions, a reduction in diversity, and an increase in cover of the regressive successional species, e.g. Stipa tenacissima L. These comparisons provide a quantitative characterization of the successional dynamic of plant spatial patterns in response to grazing perturbation gradient. ?? 2002 Elsevier Science B.V. All rights reserved.

Cabo de Gata Natural Park↗

USGS2010

Fat fractal scaling of drainage networks from a random spatial network model

An alternative quantification of the scaling properties of river channel networks is explored using a spatial network model. Whereas scaling descriptions of drainage networks previously have been presented using a fractal analysis primarily of the channel lengths, we illustrate the scaling of the surface area of the channels defining the network pattern with an exponent which is independent of the fractal dimension but not of the fractal nature of the network. The methodology presented is a fat fractal analysis in which the drainage basin minus the channel area is considered the fat fractal. Random channel networks within a fixed basin area are generated on grids of different scales. The sample channel networks generated by the model have a common outlet of fixed width and a rule of upstream channel narrowing specified by a diameter branching exponent using hydraulic and geomorphologic principles. Scaling exponents are computed for each sample network on a given grid size and are regressed against network magnitude. Results indicate that the size of the exponents are related to magnitude of the networks and generally decrease as network magnitude increases. Cases showing differences in scaling exponents with like magnitudes suggest a direction of future work regarding other topologic basin characteristics as potential explanatory variables.

Water Resources Research↗

USGS2007

Fractal topography and subsurface water flows from fluvial bedforms to the continental shield

Surface-subsurface flow interactions are critical to a wide range of geochemical and ecological processes and to the fate of contaminants in freshwater environments. Fractal scaling relationships have been found in distributions of both land surface topography and solute efflux from watersheds, but the linkage between those observations has not been realized. We show that the fractal nature of the land surface in fluvial and glacial systems produces fractal distributions of recharge, discharge, and associated subsurface flow patterns. Interfacial flux tends to be dominated by small-scale features while the flux through deeper subsurface flow paths tends to be controlled by larger-scale features. This scaling behavior holds at all scales, from small fluvial bedforms (tens of centimeters) to the continental landscape (hundreds of kilometers). The fractal nature of surface-subsurface water fluxes yields a single scale-independent distribution of subsurface water residence times for both near-surface fluvial systems and deeper hydrogeological flows. Copyright 2007 by the American Geophysical Union.

Geophysical Research Letters↗

USGS2009

Mapping Curie temperature depth in the western United States with a fractal model for crustal magnetization

We have revisited the problem of mapping depth to the Curie temperature isotherm from magnetic anomalies in an attempt to provide a measure of crustal temperatures in the western United States. Such methods are based on the estimation of the depth to the bottom of magnetic sources, which is assumed to correspond to the temperature at which rocks lose their spontaneous magnetization. In this study, we test and apply a method based on the spectral analysis of magnetic anomalies. Early spectral analysis methods assumed that crustal magnetization is a completely uncorrelated function of position. Our method incorporates a more realistic representation where magnetization has a fractal distribution defined by three independent parameters: the depths to the top and bottom of magnetic sources and a fractal parameter related to the geology. The predictions of this model are compatible with radial power spectra obtained from aeromagnetic data in the western United States. Model parameters are mapped by estimating their value within a sliding window swept over the study area. The method works well on synthetic data sets when one of the three parameters is specified in advance. The application of this method to western United States magnetic compilations, assuming a constant fractal parameter, allowed us to detect robust long-wavelength variations in the depth to the bottom of magnetic sources. Depending on the geologic and geophysical context, these features may result from variations in depth to the Curie temperature isotherm, depth to the mantle, depth to the base of volcanic rocks, or geologic settings that affect the value of the fractal parameter. Depth to the bottom of magnetic sources shows several features correlated with prominent heat flow anomalies. It also shows some features absent in the map of heat flow. Independent geophysical and geologic data sets are examined to determine their origin, thereby providing new insights on the thermal and geologic crustal structure of the western United States.

Journal of Geophysical Research B: Solid Earth↗

USGS1996

Fractal structure of sequential behaviour patterns: an indicator of stress

The detection of stress arising from parasitic infection by Sarcoptes scabieis and from pregnancy is explored, using a fractal analysis of head lifting behaviour and feeding–non-feeding activity sequences in female Spanish ibex, Capra pyrenaica , under natural conditions. Because organisms under stress increase their metabolic rate and, in consequence, energy consumption, it follows that stress will, generally, lead to a reduction in complexity (fractal dimension) of exploratory behaviour. In the present study the fractal dimension of the three measures of complexity used declined with stress, both from pregnancy and from parasitic infection. This observation provides a new and effective way to assess the general state of animals’ health in the field, without the need for capture and handling.

Animal Behaviour↗

USGS1990

Fractal characterization of fracture surfaces in concrete

Fractal geometry is used to characterize the roughness of cracked concrete surfaces through a specially built profilometer, and the fractal dimension is subsequently correlated to the fracture toughness and direction of crack propagation. Preliminary results indicate that the fracture surface is indeed fractal over two orders of magnitudes with a dimension of approximately 1.20.

Engineering Fracture Mechanics↗

USGS1996

The cyclic and fractal seismic series preceding an mb 4.8 earthquake on 1980 February 14 near the Virgin Islands

Seismic activity in the 10 months preceding the 1980 February 14, m b 4.8 earthquake in the Virgin Islands, reported on by Frankel in 1982, consisted of four principal cycles. Each cycle began with a relatively large event or series of closely spaced events, and the duration of the cycles progressively shortened by a factor of about 3/4. Had this regular shortening of the cycles been recognized prior to the earthquake, the time of the next episode of seismicity (the main shock) might have been closely estimated 41 days in advance. That this event could be much larger than the previous events is indicated from time-to-failure analysis of the accelerating rise in released seismic energy, using a non-linear time- and slip-predictable foreshock model. Examination of the timing of all events in the sequence shows an even higher degree of order. Rates of seismicity, measured by consecutive interevent times, when plotted on an iteration diagram of a rate versus the succeeding rate, form a triangular circulating trajectory. The trajectory becomes an ascending helix if extended in a third dimension, time. This construction reveals additional and precise relations among the time intervals between times of relatively high or relatively low rates of seismic activity, including period halving and doubling. The set of 666 time intervals between all possible pairs of the 37 recorded events appears to be a fractal; the set of time points that define the intervals has a finite, non-integer correlation dimension of 0.70. In contrast, the average correlation dimension of 50 random sequences of 37 events is significantly higher, close to 1.0. In a similar analysis, the set of distances between pairs of epicentres has a fractal correlation dimension of 1.52. Well-defined cycles, numerous precise ratios among time intervals, and a non-random temporal fractal dimension suggest that the seismic series is not a random process, but rather the product of a deterministic dynamic system.

Geophysical Journal International↗

USGS2000

MORPH-II, a software package for the analysis of scanning-electron-micrograph images for the assessment of the fractal dimension of exposed stone surfaces

Turcotte, 1997, and Barton and La Pointe, 1995, have identified many potential uses for the fractal dimension in physicochemical models of surface properties. The image-analysis program described in this report is an extension of the program set MORPH-I (Mossotti and others, 1998), which provided the fractal analysis of electron-microscope images of pore profiles (Mossotti and Eldeeb, 1992). MORPH-II, an integration of the modified kernel of the program MORPH-I with image calibration and editing facilities, was designed to measure the fractal dimension of the exposed surfaces of stone specimens as imaged in cross section in an electron microscope.

Open-File Report↗

USGS2007

Active tectonics on Deception Island (West-Antarctica): A new approach by using the fractal anisotropy of lineaments, fault slip measurements and the caldera collapse shape

The tectonic field on Deception Island (South Shetlands, West Antarctica) is determined from structural and fractal analyses. Three different analyses are applied to the study of the strain and stress fields in the area: (1) field measurements of faults (strain analysis), (2) fractal geometry of the spatial distribution of lineaments and (3) the caldera shape (stress analyses). In this work, the identified strain field is extensional with the maximum horizontal shortening trending NE-SW and NW-SE. The fractal technique applied to the spatial distribution of lineaments indicates a stress field with SHMAX oriented NE-SW. The elliptical caldera of Deception Island, determined from field mapping, satellite imagery, vents and fissure eruptions, has an elongate shape and a stress field with SHMAX trending NE-SW.

Open-File Report↗

USGS2000

Stress resistance strategy in an arid land shrub: interactions between developmental instability and fractal dimention

This paper investigates allocation of energy to mechanisms that generate and preserve architectural forms (i.e. developmental stability, complexity of branching patterns) and productivity (growth and reproduction) in response to environmental disturbances (i.e. grazing and resource availability). The statistical error in translational symmetry was used to detect random intra-individual variability during development. This can be thought of as a measure of developmental instability caused by stress. Additionally, we use changes in fractal complexity and shoot distribution of branch structures as an alternate indicator of stress. These methods were applied to Anthyllis cytisoides L., a semi-arid environment shrub, to ascertain the effect of grazing and slope exposure on developmental traits in a 2×2 factorial design. The results show that A. cytisoides maintains developmental stability at the expense of productivity. Anthyllis cytisoides was developmentally more stable when grazed and when on south-facing, as opposed to north-facing slopes. On the contrary, shoot length, leaf area, fractal dimension and reproductive-to-vegetative allocation ratio were larger in north- than in south-facing slopes. As a consequence, under extreme xeric conditions, shrub mortality increased in north-facing slopes, especially when not grazed. The removal of transpiring area and the reduction of plant competition favoured developmental stability and survival in grazed plants. Differences between grazed and ungrazed plants were most evident in more mesic (north-facing) areas.

Journal of Fish Diseases↗

USGS1995

Fractals in petroleum geology and earth processes

In this unique volume, renowned experts discuss the applications of fractals in petroleum research-offering an excellent introduction to the subject. Contributions cover a broad spectrum of applications from petroleum exploration to production. Papers also illustrate how fractal geometry can quantify the spatial heterogeneity of different aspects of geology and how this information can be used to improve exploration and production results.

Book↗

USGS2003

The impact of grazing on plant fractal architecture and fitness of a mediterranean shrub (Anthyllis cytisoides L.)

1. We examined natural grazing by livestock (sheep and goats) on Albaida Anthyllis cytisoides L. with the aim of determining whether variation in the allometric relationships between plant parts provides a sensitive indicator of the impact of grazing. 2. The intra-individual variation in translatory symmetry with scale and increased complexity of fractal structures reflect environmental disturbance under heavy grazing pressure and lack of grazing. 3. Fitness consequences of grazing were also investigated. Grazing promotes growth and adult survival, and a drop in seed production as a consequence of consumption. In spite of that, total inclusive fitness (population rate of change) tends to increase with grazing. 4. Moderate grazing, while promoting growth, also enhances stability of vegetative structures. The favourable effect of moderate levels of herbivory on A. cytisoides is reflected in the homeostatic maintenance of its translatory symmetry and in the increased complexity of its fractal structures.

Almería↗