Frequency-size Distribution Research Articles

The mechanism of melt extraction from lower continental crust of orogens

ABSTRACTMelt extraction is a process with a length scale that spans many orders of magnitude. Studies of residual migmatites and granulites suggest that melt has migrated from grain boundaries to networks of leucosome-filled structures to steeply inclined cylindrical or tabular granites inferred to have infilled ascent conduits. For example, in anatectic rocks from southern Brittany, France, during decompression-induced biotite-breakdown melting, melt is inferred to have been expressed from foliation-parallel structures analogous to compaction bands to dilation and shear bands, based on location of residual leucosome, and from this network of structures to ascent conduits, preserved as dykes of granite. The leucosome-filled deformation band network is elongated parallel to a sub-horizontal lineation, suggesting that mesoscale melt flow was focused primarily in the plane of the foliation along the lineation to developing dilatant transverse structures. The leucosome network connects with petrographic continuity to granite in dykes; however, the orientation of dykes discordant to fabric anisotropy suggests that their formation was controlled by stress, which indicates that the process is a fracture phenomenon. Blunt fracture tips and zigzag propagation paths indicate that the dykes represent ductile opening-mode fractures; these are postulated to have formed by coalescence of melt pockets. The structures record a transition from accumulation to draining; quantitative volume fluxes are calculated and presented for the generalised extraction process. The anatectic system may have converged to a critical state at some combination of melt fraction and melt distribution that enabled formation of ductile opening-mode fractures, but fractal distribution of inferred mesoscale melt-filled structures has not been demonstrated; this may reflect the inherent anisotropy and/or residual nature of the drained source. Melt extraction has been modelled as a self-organised critical phenomenon, but the mechanism of extraction is not described and the relationship between these models and the spatial and temporal granularity of lower continental crust is not addressed. Self-organised critical phenomena are driven systems involving ‘avalanches’ with a fractal frequency-size distribution; thus, the distribution of melt batch sizes might be expected to be fractal, but this has not yet been demonstrated in nature.

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.

A sandpile experiment and its implications for self-organized criticality and characteristic earthquake

We have performed an experiment in which a conical sandpile was built by slowly dropping sand onto a circular disk through a funnel with a small outlet. Avalanches (sand dropping off the disk) occurred, the size and the number of which were observed. It was seen that the behavior of avalanches (frequency-size distribution) was determined solely by the ratio of grain size to disk size, which is consistent with earlier studies. We categorize the behavior into three types: (1) the self-organized criticality (SOC) type (obeying Gutenberg-Richter’s law), (2) the characteristic earthquake (CE) type where only large avalanches are almost periodically generated, and (3) the transition type. The transition from SOC to CE type drastically occurs when the ratio of grain diameter to disk radius is reduced to about 0.02. The underlying mechanism to cause the transition is considered. Results of simulation by cellular automaton models, an experimental result showing that a conical pile has a stress dip near its center, and a two-dimensional simulation building up a conical pile, all suggest that the transition occurs due to a change in stress profile inside and near the surface of the pile. Although we are unfortunately not able to understand the detailed mechanism at the present stage, it seems very important to further investigate the underlying physics of the transition because it presumably provides us a clue to understand the mechanism of the periodicity of large characteristic earthquakes and may open a way for earthquake prediction.

Power-law distribution of city fires

This paper presents results of fire distributions in China and Switzerland. The frequency-size (loss) distribution and the rank-size distribution (or Zipf plot) of city fires are plotted to check their fit with a power-law relationship. The results indicate that fire data in both countries have good power-law distributions and the power-law relation is invariant with scale and time, i.e. fires in different places or in different periods will have the same distribution characteristics. Steady power-law distribution corresponds with self-organized criticality (SOC) and is expected to provide practical implications for fire protection.

Transition regimes for growing crack populations.

Numerous observational papers on crack populations in the material and geological sciences suggest that cracks evolve in such a way as to organize in specific patterns. However, very little is known about how and why the self-organization comes about. We use a model of tensile-like cracks with friction in order to study the time and space evolution of normal faults. The premise of this spring-block analog is that one could model crustal deformation for long time scales assuming a brittle layer coupled to a ductile substrate. The long time-scale physics incorporated into the model are slip-weakening friction, strain-hardening rheology for coupling the two layers, and randomly distributed yield strength of the brittle layer. We investigate how the evolution of populations of cracks depends on these three effects, using linear stability analysis to calculate the stable regimes for the friction as well as numerical simulations to model the nonlinear interactions of the cracks. We find that we can scale the problem to reduce the relevant parameters to a single one, the slip weakening. We show that the distribution of lengths of active cracks makes a transition from an exponential at very low strains, where crack nucleation prevails, to a power law at low to intermediate strains, where crack growth prevails, to an exponential distribution of the largest cracks at higher strains, where coalescence dominates. There is evidence of these different length distributions in continental and oceanic normal faults. For continental deformation the strain is low, and the faults have power-law frequency-size distributions. For mid-ocean ridge flanks the strain is greater, up to an order of magnitude higher than the continental strain, and faults have exponential-like frequency-size distributions. No theory has been offered to explain this difference in the distributions of continental and mid-ocean faults. In this paper we argue that they are indicative of different stages of evolution. The former faults are at an early stage of relatively small deformation, while the latter are at a later stage of the evolution. For high strain the faults reach a saturation regime with system size cracks evenly spaced in proportion to the brittle layer thickness. We asymptotically approximate the time space evolution of faults as a long time-scale phenomenon, thereby avoiding modeling the short time-scale earthquakes. We show that this assumption is valid, which implies that the faults that creep and faults with earthquakes display the same time and space evolutions.

Self-organized complexity in the physical, biological, and social sciences.

The National Academy of Sciences convened an Arthur M. Sackler Colloquium on “Self-organized complexity in the physical, biological, and social sciences” at the NAS Beckman Center, Irvine, CA, on March 23–24, 2001. The organizers were D.L.T. (Cornell), J.B.R. (Colorado), and Hans Frauenfelder (Los Alamos National Laboratory, Los Alamos, NM). The organizers had no difficulty in finding many examples of complexity in subjects ranging from fluid turbulence to social networks. However, an acceptable definition for self-organizing complexity is much more elusive. Symptoms of systems that exhibit self-organizing complexity include fractal statistics and chaotic behavior. Some examples of such systems are completely deterministic (i.e., fluid turbulence), whereas others have a large stochastic component (i.e., exchange rates). The governing equations (if they exist) are generally nonlinear and may also have a stochastic driver. Many of the concepts that have evolved in statistical physics are applicable (i.e., renormalization group theory and self-organized criticality). As a brief introduction, we consider a few of the symptoms that are associated with self-organizing complexity. The classic example of self-organizing complexity is the frequency-size distribution of earthquakes. Earthquakes are certainly complex yet they universally satisfy the relation (to a good approximation) ![Formula][1] where N is the number of earthquakes in a specified time interval and region with their rupture area greater than A . This is the well known Guttenberg–Richter relation (1). The scaling exponent D is the fractal dimension introduced by B. Mandelbrot (2). In his book, Mandelbrot (3) pointed out the wide range of validity of the fractal scaling relation ![Formula][2] where Ni is the number of objects of size ri . Power-law scaling is second only to the Gaussian distribution in terms of applicability. Its relatively recent acceptance, in terms of the fractal paradigm, can be attributed to the fact that it cannot be used as … [1]: /embed/tex-math-1.gif [2]: /embed/tex-math-2.gif

Frequency-size distribution of global seismicity seen frombroad-band radiated energy

The frequency–energy distribution of global seismicity is studied using broad-band radiated energy of shallow earthquakes from January 1987 to December 1994 estimated by NEIC. Rank-ordering statistics are applied to enhance the resolution in retrieving the power-law distribution with undersampled data, namely a few tens of events. Seen in the perspective of broad-band radiated energy with higher resolution, a single (Gutenberg–Richter-type) power-law distribution can fit the data. For earthquakes with energy larger than 1014 J, the number N of events with energy E depends on E via N∝E−B, with the scaling constant B = 0.64 ± 0.04, corresponding to b = 0.95 ± 0.06. This relation is different from that of scalar seismic moment, which shows a transition of power-law distributions between small and large earthquakes. To demonstrate such a difference we use the same set of earthquakes with both broad-band energy estimation and CMT estimation. It is found that for the same data set, the energy distribution and the moment distribution show different patterns. The moment distribution has a clear kink between small and large earthquakes, while the energy distribution shows a single power law with no convincing kink between small and large earthquakes. To investigate the effect of different focal mechanisms and different seismic regions, events with strike-slip mechanisms and events within the Japan–Kuril region are considered. For these subsets of events, a similar pattern exists, in which the moment distribution shows a kink between small and large earthquakes, while the energy distribution shows a single power law.

How many plumes are there?

The objective of this paper is to quantitatively assess the role of mantle plumes in transporting heat to the base of the lithosphere. We first review the mechanisms responsible for mantle heat flow. We take the total global surface heat flow to be 4.43×10 13 W. Of this, we attribute 0.68×10 13 W (15%) to radiogenic heat production in the continental crust and 3.75×10 13 W (85%) to heat loss from the mantle. Of the heat loss from the mantle, 2.17×10 13 W (58%) is attributed to the subduction of the oceanic lithosphere and the remainder, 1.58×10 13 W (42%), heats the base of the oceanic and continental lithosphere. Prior buoyancy studies of plumes give a plume heat flux of 0.24×10 13 W, which is only 15% of the total heat flux associated with basal heating of the lithosphere. Thus, the amount that remains unaccounted for is a basal heat flux of 1.34×10 13 W. The missing heat flux can be attributed either to plumes that do not have a significant surface expression, or to secondary mantle convection beneath the plates. We show that the cumulative frequency-size distribution of the large and intermediate size plume fluxes can be reasonably well approximated by a power-law distribution. We then extrapolate this distribution to smaller plumes in order to estimate a total plume heat flux. This requires about 5200 plumes, with the smallest plume fluxes about 10 9 W. This compares with 12×10 9 W for the smallest plume fluxes previously reported in the literature. We suggest that the large number of seamounts represents surface evidence for small plumes, and conclude that it is reasonable to attribute the entire basal heat flux to plumes.

Statistical self‐similarity of magmatism and volcanism

Magmatism and volcanism exhibit spatial and temporal clustering on a wide range of scales. Using the spatial pair‐correlation function, the number of pairs of magmatic or volcanic events whose separation is between r − ½Δr and r + ½Δr per unit area, we quantify the spatial clustering of magmatism and volcanism in several data sets. Statistically self‐similar clustering characterized by power law spatial pair‐correlation functions is observed. The temporal pair‐correlation function is used to identify self‐similar temporal clustering of magmatism and volcanism in the Radiometric Age Data Bank of 11,986 dated intrusive and extrusive rocks in the North American Cordillera. The clustering of magmatism and volcanism in space and time in this data set is found to be statistically self‐similar and identical to those of distributed seismicity. The frequency‐size distributions of eruption volume and areal extent of basaltic flows are also found to be self‐similar with power laws analogous to the Gutenburg‐Richter distribution for earthquakes. In an attempt to understand the origin of statistical self‐similarity in magmatism and volcanism we present one end‐member model in which the ascent of magma through a disordered crust of variable macroscopic permeability is modeled with a cellular‐automaton model to create a distribution of magma supply regions which erupt with equal probability per unit time. The model exhibits statistical self‐similarity similar to that observed in the real data sets.

An inverse-cascade model for self-organized critical behavior

We introduce an inverse-cascade model to explain self-organized critical behavior. This model is motivated by the forest-re model. In the forest-re model trees are randomly planted on a grid, sparks are also dropped on the grid resulting in res in which trees are lost. In the inverse-cascade model single trees are introduced and these combine to form larger and larger clusters. This is the inverse cascade and gives a power-law (fractal) frequency-size distribution of clusters. Model res eliminate trees from all cluster sizes but signicant numbers of trees are lost only from the largest clusters and this loss terminates the power-law scaling. Finally, our model illustrates important dierences between critical and self-organized critical behavior. c 1999 Elsevier Science B.V. All rights reserved.

Frequency-Size Distribution of Model Seismicity in a Statistical Mechanical Model of Earthquakes

The frequency-size distribution of model earthquakes in a one-dimensional Burridge-Knopoff spring-block model [Phys. Rev. Lett. 22 (1989) 2632] is studied. Analytical approach is taken to avoid the limitations on the size of models and computation time in numerical calculations. By using the Hamiltonian of the spring-block system a concept of `seismic phonon' is proposed which is similar to the concept of phonon in solid state physics. The result of the occurrence frequency of model earthquakes shows a distribution similar to the modified gamma distribution proposed by Main and Burton using information theory.

Scale-invariance of soil moisture variability and its implications for the frequency-size distribution of landslides

Power spectral analyses of soil moisture variability are carried out from scales of 100 m to 10 km on the microwave remotely-sensed data from the Washita experimental watershed during 1992. The power spectrum S(k) has an approximate power-law dependence on wave number k with the exponent −1.8. This behavior is consistent with the behavior of a stochastic differential equation for soil moisture at a point, and it has important consequences for the frequency-size distribution of landslides. We present the cumulative frequency-size distributions of landslides induced by precipitation in Japan and Bolivia as well as landslides triggered by the 1994 Northridge, California earthquake. Large landslides in these regions, despite being triggered by different mechanisms, have a cumulative frequency-size distribution with a power-law dependence on area with an exponent ranging from −1.5 to −2. We use a soil moisture field with the above statistics in conjunction with a slope stability analysis to model the frequency-size distribution of landslides. In our model, landslides occur when a threshold shear stress dependent on cohesion, pore pressure, internal friction and slope angle is exceeded. This implies a threshold dependence on soil moisture and slope angle since cohesion, pore pressure and internal friction are primarily dependent on soil moisture. The cumulative frequency-size distribution of domains of shear stress greater than a threshold value with soil moisture modeled as above and topography modeled as a Brownian walk is a power-law function of area with an exponent of −1.8 for large landslide areas. This distribution is similar to that observed for landslides. The effect of strong ground motion from earthquakes lowers the shear stress necessary for failure, but does not change the frequency-size distribution of failed areas. This is consistent with observations. This work suggests that remote sensing of soil moisture can be of great importance in monitoring landslide hazards and proposes a specific quantitative model for landslide hazard assessment.

A self-consistent description of ruptures in an elastic medium: an application to earthquakes

We present a general self-consistent procedure using lattice Green's functions to find the stress redistribution due to multiple ruptures in a discrete elastic medium. We use this method to study a dipole version of the quasi-static crack-propagation model for earthquakes proposed by Chen et al. (1991) Unlike the earlier treatment we do not assume that ruptures occur independently. We obtain the stresses self-consistently and this changes the scaling behaviours of the frequency-size (seismic moment, energy) distribution. We make a careful comparison of our results with recent analysis of seismological data.

Earthquake mechanism and predictability shown by a laboratory fault

Slip events generated in a laboratory fault model consisting of a circulinear chain of eight spring-connected blocks of approximately equal weight elastically driven to slide on a frictional surface are studied. It is found that most of the input strain energy is released by a relatively few large events, which are approximately time predictable. A large event tends to roughen stress distribution along the fault, whereas the subsequent smaller events tend to smooth the stress distribution and prepare a condition of simultaneous criticality for the occurrence of the next large event. The frequency-size distribution resembles the Gutenberg-Richter relation for earthquakes, except for a falloff for the largest events due to the finite energy-storage capacity of the fault system. Slip distributions, in different events are commonly dissimilar. Stress drop, slip velocity, and rupture velocity all tend to increase with event size. Rupture-initiation locations are usually not close to the maximum-slip locations.

A Review of Methods for Estimating Future Hydrocarbon Supply

Techniques to estimate the size of the undiscovered hydrocarbon resource base and produce discovery rate forecasts are reviewed and assessed. Accurate assessments of the resource base are needed by exploration firms and the government in comparing the potential of competing areas of exploration and by policy makers in developing appropriate policies to address possible hydrocarbon shortages. The techniques reviewed include judgemental prediction, extrapolative methods, discovery process models, econometric models, and probabilistic models. A critical distinction exists between econometric and geologically-based approaches. The geological approaches aim at explicitly determining the size of the undiscovered resource base and its frequency-size distribution, producing discovery rate forecasts as a by-product of the adopted methodology. The econometric approaches aim explicitly at forecasting future discovery rates, producing estimates of the resource base as byproduct.

Seismic moment catalog of large shallow earthquakes, 1900 to 1989

AbstractWe compile a worldwide catalog of shallow (depth < 70 km) and large (Ms ≥ 7) earthquakes recorded between 1900 and 1989. The catalog is shown to be complete and uniform at the 20-sec surface-wave magnitude Ms ≥ 7.0. We base our catalog on those of Abe (1981, 1984) and Abe and Noguchi (1983a, b) for events with Ms ≥ 7.0. Those catalogs, however, are not homogeneous in seismicity rates for the entire 90-year period. We assume that global rates of seismicity are constant on a time scale of decades and most inhomogeneities arise from changes in instrumentation and/or reporting. We correct the magnitudes to produce a homogeneous catalog. The catalog is accompanied by a reference list for all the events with seismic moment determined at periods longer than 20 sec. Using these seismic moments for great and giant earthquakes and a moment-magnitude relationship for smaller events, we produce a seismic moment catalog for large earthquakes from 1900 to 1989. The catalog is used to study the distribution of moment released worldwide. Although we assumed a constant rate of seismicity on a global basis, the rate of moment release has not been constant for the 90-year period because the latter is dominated by the few largest earthquakes. We find that the seismic moment released at subduction zones during this century constitutes 90% of all the moment released by large, shallow earthquakes on a global basis. The seismic moment released in the largest event that occurred during this century, the 1960 southern Chile earthquake, represents about 30 to 45% of the total moment released from 1900 through 1989. A frequency-size distribution of earthquakes with seismic moment yields an average slope (b value) that changes from 1.04 for magnitudes between 7.0 and 7.5 to b = 1.51 for magnitudes between 7.6 and 8.0. This change in the b value is attributed to different scaling relationships between bounded (large) and unbounded (small) earthquakes. Thus, the earthquake process does have a characteristic length scale that is set by the downdip width over which rupture in earthquakes can occur. That width is typically greater for thrust events at subduction zones than for earthquakes along transform faults and other tectonic environments.

Fractals in geology and geophysics

The definition of a fractal distribution is that the number of objectsN with a characteristic size greater thanr scales with the relationN∼r −D. The frequency-size distributions for islands, earthquakes, fragments, ore deposits, and oil fields often satisfy this relation. Fractals were originally introduced by Mandelbrot to relate the length of a coastline to the length of the measuring stick. This application illustrates a fundamental aspect of fractal distributions, scale invariance. The requirement of an object to define a scale in photographs of many geological features is one indication of the wide applicability of scale invariance to geological problems, scale invariance can lead to fractal clustering. Geophysical spectra can also be related to fractals; these are self-affine fractals rather than self-similar fractals. Examples include the earth's topography and geoid.

The balloon observations of small cosmic X-ray bursts

In the record of the balloon observation which was performed on 27 September, 1970, a transient burst of X-rays was found. This event is concluded to be a cosmic gamma-ray burst of a smaller size or of a larger distance compared to the Vela bursts observed over the X-ray energy range. The energy spectrum is consistent with that of some of the Vela bursts. The time profile of the event is qualitatively similar to the 27 April, 1972 event studied by Apollo 16. The detection of small bursts over the X-ray energy range by the balloon observation during a period of the order of 10 to 100 h is not surprising considering a probable frequency-size distribution of the burst.

Radio echoes and cosmic ray showers

It is suggested that the origin of some of the transient ionic clouds, generally assumed to be responsible for the low level sporadic radio reflexions, may be due to large cosmic ray showers. It is shown that cascade cosmic ray showers of sufficient energy to produce some of these radio reflexions certainly exist, but there is insufficient published evidence to decide whether any of the echoes already observed are actually due to such showers. More conclusive evidence could be obtained from the frequency-size distribution of the radio echoes observed from a horizontal or vertically directed beam.