Christopher C. Barton

Fractals in the Earth Sciences

Fractal Distributions in Geology, Scale Invariance, and Deterministic Chaos (D.L. Turcotte, J. Huang). Some Long Run Properties of Geophysical Records (B.B. Mandelbrot, J.R. Wallis). Some Remarks on the Numerical Estimation of Fractal Dimension (S.A. Pruess). Measuring the Dimension of Selfaffine Fractals: The Example of Rough Surfaces (S.R. Brown). A Review of the Fractal Character of Natural Fault Surfaces with Implications for Friction and the Evolution of Fault Zones (W.L. Power, T.E. Tullis). Fractals and Ocean Floor Topography: A Review and a Model (A. Malinverno). Fractal Transitions on Geological Surfaces (C.H. Scholz). Fractal Scaling of Fracture Networks (C.C. Barton). Fractal Fragmentation in Crustal Shear Zones (C.G. Sammis, S.J. Steacy). Fractal Distribution of Fault Length and Offsets: Implications on Brittle Deformation Evaluation: The Lorraine Coal Basin (T. Villemin et al.). Fractal Dynamics of Earthquakes (P. Bak, K. Chen). Mineral Crystallinity in Igneous Rocks: Fractal Method (A.D. Fowler). Fractal Structure of Electrum Dendrites in Bonanza Epithermal AuAg Deposits (J.A. Saunders, P.A. Schoenly). Appendix. Index.

Fractal Characterization of Fracture Surfaces in Concrete

Abstract 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.

Morphology and distribution of seamounts surrounding Easter Island

We investigate the morphology and distribution of a seamount population on a section of seafloor influenced by both superfast seafloor spreading and hotspot volcanism. The population under investigation is part of a broad chain of seamounts extending eastward from the East Pacific Rise, near Easter Island. In order to define the morphological variability of the seamounts, basal shape, cross-sectional area, volume, flatness, and flank slope are plotted against height for 383 seamounts with heights greater than 200 m, based on bathymetry data collected by GLORI-B and SeaBeam 2000, during three cruises onboard the R/V Melville in the spring of 1993. Nearly complete swath mapping coverage of the seamounts is available for the analysis of size and shape distribution. We quantitatively describe the seamount population of this active region, in which seamounts cover ∼27% of the seafloor, and account for ∼4.2% of the total crustal volume. Over 50% of the total volume (61,000 km3) of seamounts used in this study is made up by the 14 largest seamounts, and the remaining volume is made up by the 369 smaller seamounts (>200 m in height). Our analysis indicates there are at least two seamount populations in the Easter Island-Salas y Gomez Island (25°–29°S, 113°–104°W) study area. One population of seamounts is composed of short seamounts ( 1200 m), shield-like, pointy cones (flatness ∼1200 m) originate exclusively from a hotspot source, but only a portion of the smaller volcanoes (<∼1200 m) are formed from a hotspot source. The remainder would be presumably formed by a normal mantle or mixed source.

Model estimates hurricane wind speed probabilities

In the United States, intense hurricanes (category 3, 4, and 5 on the Saffir/Simpson scale) with winds greater than 50 m s −1 have caused more damage than any other natural disaster [Pielke and Pielke, 1997]. Accurate estimates of wind speed exceedance probabilities (WSEP) due to intense hurricanes are therefore of great interest to (re)insurers, emergency planners, government officials, and populations in vulnerable coastal areas. The historical record of U.S. hurricane landfall is relatively complete only from about 1900, and most model estimates of WSEP are derived from this record. During the 1899–1998 period, only two category-5 and 16 category-4 hurricanes made landfall in the United States. The historical record therefore provides only a limited sample of the most intense hurricanes.

Where in the world are my field plots? Using GPS effectively in environmental field studies

Global positioning system (GPS) technology is rapidly replacing tape, compass, and traditional surveying instruments as the preferred tool for estimating the positions of environmental research sites. One important problem, however, is that it can be difficult to estimate the uncertainty of GPS-derived positions. Sources of error include various satellite- and site-related factors, such as forest canopy and topographic obstructions. In a case study from the Hubbard Brook Experimental Forest in New Hampshire, hand-held, mapping-grade GPS receivers generally estimated positions with 1–5 m precision in open, unobstructed settings, and 20–30 m precision under forest canopy. Surveying-grade receivers achieved precisions of 10 cm or less, even in challenging terrain. Users can maximize the quality of their GPS measurements by “mission planning” to take advantage of high-quality satellite conditions. Repeated measurements and simultaneous data collection at multiple points can be used to assess accuracy and prec...

The Fractal Size and Spatial Distribution of Hydrocarbon Accumulations

Objects in nature are often very irregular, so that, within the constraints of Euclidean geometry, one is forced to approximate their shape grossly. A rock fragment, for example, is treated as being spherical, and a coastline straight or smoothly curved. Upon examination, however, these objects are found to be jagged, and their jaggedness does not diminish when viewed at ever finer scales. In his monumental work, Mandelbrot (1982) developed and popularized fractal geometry, a geometry that applies to many irregular natural objects. The analytical techniques for treating fractal geometry and their inter-relationships have undergone rapid development and broad application (Feder, 1988). Fractal geometry has been applied to a wide variety of geological and geophysical objects and phenomena: for surveys see Scholz and Mandelbrot (1989), Turcotte (1989), Turcotte (1992)), and Barton and La Pointe (1995). During this same time, there have also been revolutionary developments in understanding the types of physical processes that produce patterns described by fractal geometry. Fractal patterns have been found to arise from a wide variety of nonlinear dynamical systems, particularly those that exhibit certain types of chaotic behavior (e.g., Schuster, 1988). Examples are dissipative systems with many degrees of freedom that show self-organized criticality, such as avalanches on the surface of a sand pile and earthquakes (Bak et al., 1988; Bak and Chen, 1991).

Nonlinear forecasting analysis of inflation-deflation patterns of an active caldera (Campi Flegrei, Italy)

The ground level in Pozzuoli, Italy, at the center of the Campi Flegrei caldera, was monitored by tide gauges between 1970 and 1976 and then continuously since 1982. Tide gauges offer a long record of a variable that is believed to be related to the activity of an underlying shallow magma chamber. Previous work suggests that the dynamics of the Campi Flegrei system, as reconstructed from the tide gauge record, is chaotic and low dimensional. According to this suggestion, in spite of the complexity of the system, at a time scale of days the ground motion is driven by a deterministic mechanism with few degrees of freedom; however, the interactions of the system may never be describable in full detail. Our new analysis of the tide gauge record from January 1987 to June 1989, using Nonlinear Forecasting, confirms low-dimensional chaos in the ground elevation record at Campi Flegrei and suggests that Nonlinear Forecasting could be a useful tool in volcanic surveillance.

Characterizing fractured rock for fluid-flow, geomechanical, and paleostress modeling; methods and preliminary results from Yucca Mountain, Nevada

Fractures have been characterized for fluid-flow, geomechanical, and paleostress modeling at three localities in the vicinity of drill hole USW G-4 at Yucca Mountain in southwestern Nevada. A method for fracture characterization is introduced that integrates mapping fracture-trace networks and quantifying eight fracture parameters: trace length, orientation, connectivity, aperture, roughness, shear offset, trace-length density, and mineralization. A complex network of fractures was exposed on three 214- to 260-m 2 pavements cleared of debris in the upper lithophysal unit of the Tiva Canyon Member of the Miocene Paint-brush Tuff. The pavements are two-dimensional sections through the three-dimensional network of strata-bound fractures. All fractures with trace lengths greater than 0.2 m were mapped and studied.

Chaos in geomagnetic reversal records: A comparison between Earth's magnetic field data and model disk dynamo data

The Earths geomagnetic field reverses its polarity at irregular time intervals. However, it is not clear whether a reversal is a deterministic (low-dimensional) or a random (high-dimensional) process; the duration-frequency distribution of the polarity time intervals resembles those generated by random processes, but many models suggest that a geomagnetic field reversal can be the outcome of a deterministic dynamics, that of the convection in the Earths outer core. The latter, in turn, is only a part of an extremely complex system, made up of both terrestrial and extraterrestrial subsystems nonlinearly interacting with each other over a wide range of time scales. We studied the geomagnetic field reversal patterns by means of several techniques of nonlinear dynamics and compared the results obtained on actual geomagnetic reversal data with synthetic reversal sequences generated by the Rikitake and Chillingworth-Holmes models of the Earths magnetic field. We analyzed both the geomagnetic and the synthetic reversal scales by nonlinear forecasting and found that we cannot predict the geomagnetic reversal sequence with nonlinear forecasting. Predictability of the synthetic data varies widely depending on the model parameters. Phase portraits of data obtained from the magnetic field models show fractal structures similar to those associated with the Lorenz attractor. We measured the correlation dimension DC of the synthetic and geomagnetic data by means of the Grassberger-Procaccia method and found that DC always has a value of about one for the synthetic data. The correlation integrals for the geomagnetic reversal sequence behave very differently from those of randomized reversal sequences and suggest that the Earths geomagnetic field reversal dynamics is not random. However, the limited size of the magnetic reversal data set (282 points) and the poor convergence of the correlation integrals make a quantitative assessment of low-dimensional chaos impossible. Our analysis sets a lower limit to the correlation dimension of the geomagnetic reversal dynamics: DC > 3.

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.