An inverse-cascade model for self-organized critical behavior

Proc. Natl. Acad. Sci. U.S.A.

Malamud

Guzzetti

et al. 2002

117

We consider the frequency-size statistics of two natural hazards, forest fires and landslides. Both appear to satisfy power-law (fractal) distributions to a good approximation under a wide variety of conditions. Two simple cellular-automata models have been proposed as analogs for this observed behavior, the forest fire model for forest fires and the sand pile model for landslides. The behavior of these models can be understood in terms of a self-similar inverse cascade. For the forest fire model the cascade consists of the coalescence of clusters of trees; for the sand pile model the cascade consists of the coalescence of metastable regions.A number of complex natural phenomena exhibit power-law frequency-area statistics in a very robust manner. The classic example is earthquakes. For over 50 years, it has been accepted that earthquakes universally obey Gutenberg-Richter scaling; the number of earthquakes in a region with magnitudes greater than m, N CE , is related to m by (1)where b E and C are constants; b E is known as the b value and has a near-universal value b E ϭ 0.90 Ϯ 0.15. The constant C is a measure of the intensity of the regional seismicity. When Eq. 1 is expressed in terms of the earthquake rupture area A E instead of earthquake magnitude, this relation becomes a power law (2)Earthquakes satisfy fractal (power-law) scaling in a very robust manner, despite their complexity. Earthquakes occur in hot regions and cold regions, with compressional, shear, and tensional stresses, yet they universally satisfy the power-law scaling given in Eq. 2. In this article we will focus our attention on two other natural phenomena, forest fires and landslides. We will show that both also exhibit power-law frequency-area statistics in a robust manner.A fundamental question is: Why do earthquakes, forest fires, and landslides exhibit a power-law frequency-area dependence? In each case a simple cellular automata model has been proposed to explain the observed behavior. For earthquakes it is the slider-block model. The standard slider-block model (3, 4) considers an array of slider blocks each with a mass m. The blocks are attached to a constant velocity driver plate with puller springs, spring constant k p , and to each other with connector springs, spring constant k c . The blocks interact nonlinearly with the surface over which they are dragged through friction.If the sliding (dynamic) friction coefficient f d is less than the static friction coefficient f s , stick-slip behavior is observed. The area of a slip event is the number of blocks involved in the event. Under a wide variety (but not all) conditions, the frequency-area statistics of the slip events are well approximated by the powerlaw relationwhere N E is the number of events with area A E , the number of blocks slipping in an event. The exponent a E is generally found to be a E ϳ 1.0. For a noncumulative power-law distribution (Eq. 3) with a ϭ 1, the corresponding cumulative distribution obtained by summing or integrating will be logarithmic. For a noncum...

Section: Cascade Modelmentioning

confidence: 99%

Self-organization, the cascade model, and natural hazards

Proc. Natl. Acad. Sci. U.S.A.

Malamud

Guzzetti

et al. 2002

117

“…An explanation for this robust behavior is given in terms of an inverse cascade of metastable clusters (Gabrielov, et al, 1999;Turcotte, et al, 1999;Yakovlev, et al, 2005). A metastable cluster is the region over which an avalanche spreads once triggered.…”

Section: Introductionmentioning

confidence: 99%

Recurrence and interoccurrence behavior of self-organized complex phenomena

Abaimov

Nonlin. Processes Geophys.

Shcherbakov

et al. 2007

Abstract. The sandpile, forest-fire and slider-block models are said to exhibit self-organized criticality. Associated natural phenomena include landslides, wildfires, and earthquakes. In all cases the frequency-size distributions are well approximated by power laws (fractals). Another important aspect of both the models and natural phenomena is the statistics of interval times. These statistics are particularly important for earthquakes. For earthquakes it is important to make a distinction between interoccurrence and recurrence times. Interoccurrence times are the interval times between earthquakes on all faults in a region whereas recurrence times are interval times between earthquakes on a single fault or fault segment. In many, but not all cases, interoccurrence time statistics are exponential (Poissonian) and the events occur randomly. However, the distribution of recurrence times are often Weibull to a good approximation. In this paper we study the interval statistics of slip events using a slider-block model. The behavior of this model is sensitive to the stiffness α of the system, α=k C /k L where k C is the spring constant of the connector springs and k L is the spring constant of the loader plate springs. For a soft system (small α) there are no system-wide events and interoccurrence time statistics of the larger events are Poissonian. For a stiff system (large α), system-wide events dominate the energy dissipation and the statistics of the recurrence times between these system-wide events satisfy the Weibull distribution to a good approximation. We argue that this applicability of the Weibull distribution is due to the power-law (scale invariant) behavior of the hazard function, i.e. the probability that the next event will occur at a time t 0 after the last event has a power-law dependence on t 0 . The Weibull distribution is the only distribution that has a scale invariant hazard function. We further show that the onset of system-wide events is a well defined critical point. We find that the number of system-wide events N SWE Correspondence to: S. G. Abaimov (abaimov@geology.ucdavis.edu) satisfies the scaling relation N SWE ∝ (α−α C ) δ where α C is the critical value of the stiffness. The system-wide events represent a new phase for the slider-block system.

“…This distribution of cluster sizes is the same as that found for the forest-fire model . This power-law distribution of cluster sizes can be explained by the inverse cascade model introduced by Turcotte et al (1999), Turcotte (1999), and Gabrielov et al (1999), as well as Newman and Knopoff (1982, and Knopoff and Newman (1983). In this cascade model, smaller tree clusters coalesce to form larger clusters.…”

Section: The Modelmentioning

confidence: 98%

A simple model for the earthquake cycle combining self-organized complexity with critical point behavior

Newman

Nonlin. Processes Geophys.

2002

Abstract.We have studied a hybrid model combining the forest-fire model with the site-percolation model in order to better understand the earthquake cycle. We consider a square array of sites. At each time step, a "tree" is dropped on a randomly chosen site and is planted if the site is unoccupied. When a cluster of "trees" spans the site (a percolating cluster), all the trees in the cluster are removed ("burned") in a "fire." The removal of the cluster is analogous to a characteristic earthquake and planting "trees" is analogous to increasing the regional stress. The clusters are analogous to the metastable regions of a fault over which an earthquake rupture can propagate once triggered. We find that the frequency-area statistics of the metastable regions are powerlaw with a negative exponent of two (as in the forest-fire model). This is analogous to the Gutenberg-Richter distribution of seismicity. This "self-organized critical behavior" can be explained in terms of an inverse cascade of clusters. Small clusters of "trees" coalesce to form larger clusters. Individual trees move from small to larger clusters until they are destroyed. This inverse cascade of clusters is self-similar and the power-law distribution of cluster sizes has been shown to have an exponent of two. We have quantified the forecasting of the spanning fires using error diagrams. The assumption that "fires" (earthquakes) are quasi-periodic has moderate predictability. The density of trees gives an improved degree of predictability, while the size of the largest cluster of trees provides a substantial improvement in forecasting a "fire."