Self-organized criticality in two-variable models

Hergarten, Stefan; Neugebauer, Horst J.

doi:10.1103/physreve.61.2382

Cited by 34 publications

(34 citation statements)

References 10 publications

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“…This means that every site will finally become unstable, even if its slope gradient is zero. But still more severe, this model was shown to exhibit SOC only in the trivial limiting case where time-dependent weakening becomes negligible (Hergarten and Neugebauer, 2000), at least under driving by tilting the slope. The case of long-term driving by fluvial incision was recently investigated (Hergarten, 2002a).…”

Section: The Role Of Time-dependent Weakeningmentioning

confidence: 99%

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Landslides, sandpiles, and self-organized criticality

Hergarten

2003

Nat. Hazards Earth Syst. Sci.

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Abstract. Power-law distributions of landslides and rockfalls observed under various conditions suggest a relationship of mass movements to self-organized criticality (SOC). The exponents of the distributions show a considerable variability, but neither a unique correlation to the geological or climatic situation nor to the triggering mechanism has been found. Comparing the observed size distributions with models of SOC may help to understand the origin of the variation in the exponent and finally help to distinguish the governing components in long-term landslide dynamics. However, the three most widespread SOC models either overestimate the number of large events drastically or cannot be consistently related to the physics of mass movements. Introducing the process of time-dependent weakening on a long time scale brings the results closer to the observed statistics, so that time-dependent weakening may play a major part in the long-term dynamics of mass movements.

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Section: The Role Of Time-dependent Weakeningmentioning

confidence: 99%

“…The product approach was recently investigated, too (Hergarten and Neugebauer, 2000;Hergarten, 2002a). It was found to differ strongly from the linear combination (Eq.…”

Section: The Role Of Time-dependent Weakeningmentioning

confidence: 99%

Landslides, sandpiles, and self-organized criticality

Hergarten

2003

Nat. Hazards Earth Syst. Sci.

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“…These authors (27) later used a cellular-automata model with time-dependent weakening, similar to the sand pile model, and found a power-law distribution with a L ϳ 2.0. Although it is certainly possible to develop models that reproduce the observed power-law dependence of actual data, there is a real question whether these models are realistic in terms of the governing physics.…”

Section: Landslidesmentioning

confidence: 99%

Self-organization, the cascade model, and natural hazards

Turcotte

Malamud

Guzzetti

et al. 2002

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

117

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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...

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“…Indeed, SOC, epitomised by the Bak-Tang-Wiesenfeld CA (Bak et al, 1987) and generalized by the Olami-Feder-Christensen CA (Olami et al, 1992), is characterized by the emergence of a power-law FSD due to a simple bottom-up triggering process analogue to an avalanche. Although SOC CAs and other CA variants are broadly used for landslide modelling (e.g., D'Ambrosio et al, 2006), the sandpile model (Bak et al, 1987), with slope α = 1.0-1.2 does not explain the steeper slope observed for landslides (Hergarten and Neugebauer, 2000;Turcotte et al, 2002). 20 Pelletier et al, (1997) retrieved α = 2.6±0.1 from a percolation model controlled by a threshold shear stress dependent on a slope based on physical parameters, but only considered the area of the landslide initiation phase, not of the landslide itself.…”

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confidence: 99%

“…Interestingly, Hergarten and Neugebauer (2000) obtained α = 2.1 by applying a two-variable product in a SOC system with one relaxation variable and one time-dependent weakening variable, which was related to the factor of safety as an ad-hoc assumption in Hergarten (2013). 2009) only obtained reasonable α estimates by allowing arbitrary 25 parameter variations to match the data (see discussion in Hergarten, 2013).…”

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confidence: 99%

Origin of the power-law exponent in the landslide frequency-size distribution

Jafarimanesh

Mignan

Danciu

2018

Preprint

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Landslide statistics is characterized by a power-law frequency-size distribution (FSD) with power exponent α centered on 2.2-2.4, independently of the landslide trigger. So far, the origin of the α-value, critical to probabilistic hazard assessment, remains hypothetical. We present a generic landslide cellular automaton (LSgCA) based on the rules of Self 10Organized Criticality and on the Factor of Safety (FS) concept. We show that it reproduces the power-law FSD for realistic parameter ranges (i.e. cohesion, soil unit weight, soil thickness, angle of friction, slope angle, pore water pressure) with LSgCA simulations yielding α = 2.17±0.49, which is in agreement with α = 2.21±0.53 obtained from an updated metaanalysis of the landslide literature. The parameter α remains stable despite changes in the landslide triggering process, with the trigger only influencing the spatial extent of the landslide initiation phase defined from an FS contour. Furthermore, 15 different FS formulations do not significantly alter the results. We find that α is constrained during the initiation phase of the landslide by the fractal properties of the topography, as we observed a positive correlation between fractal dimension and α while α did not change during the propagation phase of the LSgCA. Our results thus suggest that α can be estimated directly from the FS map for probabilistic landslide hazard assessment. However full modeling (including the propagation phase) would be required to combine the spatial distributions of landslide and exposure in probabilistic risk analysis. 20

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Self-organized criticality in two-variable models

Cited by 34 publications

References 10 publications

Landslides, sandpiles, and self-organized criticality

Landslides, sandpiles, and self-organized criticality

Self-organization, the cascade model, and natural hazards

Origin of the power-law exponent in the landslide frequency-size distribution

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