The concept of self-organized criticality was introduced to explain the behaviour of the sandpile model. In this model, particles are randomly dropped onto a square grid of boxes. When a box accumulates four particles they are redistributed to the four adjacent boxes or lost off the edge of the grid. Redistributions can lead to further instabilities with the possibility of more particles being lost from the grid, contributing to the size of each ‘avalanche’. These model ‘avalanches’ satisfied a power-law frequency‐area distribution with a slope near unity. Other cellular-automata models, including the slider-block and forest-fire models, are also said to exhibit self-organized critical behaviour. It has been argued that earthquakes, landslides, forest fires, and species extinctions are examples of self-organized criticality in nature. In addition, wars and stock market crashes have been associated with this behaviour. The forest-fire model is particularly interesting in terms of its relation to the critical-point behaviour of the sitepercolation model. In the basic forest-fire model, trees are randomly planted on a grid of points. Periodically in time, sparks are randomly dropped on the grid. If a spark drops on a tree, that tree and adjacent trees burn in a model fire. The fires are the ‘avalanches’ and they are found to satisfy power-law frequency‐area distributions with slopes near unity. This forest-fire model is closely related to the site-percolation model, that exhibits critical behaviour. In the forest-fire model there is an inverse cascade of trees from small clusters to large clusters, trees are lost primarily from model fires that destroy the largest clusters. This quasi steady-state cascade gives a power-law frequency‐area distribution for both clusters of trees and smaller fires. The site-percolation model is equivalent to the forest-fire model without fires. In this case there is a transient cascade of trees from small to large clusters and a power-law distribution is found only at a critical density of trees.

/pdf/self-organized-criticality-4v87dkoa15.pdf

Self-organized criticality

Many aspects of earthquake mechanics remain an enigma as we enter the closing years of the twentieth century. One potential bright spot is the realization that simple calculations of stress changes may explain some earthquake interactions, just as previous and on going studies of stress changes have begun to explain human-induced seismicity. This paper, which introduces the special section “Stress Triggers, Stress Shadows, and Implications for Seismic Hazard,” reviews many published works and presents a compilation of quantitative earthquake interaction studies from a stress change perspective. This synthesis supplies some clues about certain aspects of earthquake mechanics. It also demonstrates that much work remains before we can understand the complete story of how earthquakes work.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/98JB01576

Introduction to Special Section: Stress Triggers, Stress Shadows, and Implications for Seismic Hazard

Expression of active tectonics in erosional landscapes

[1] Crustal deformation patterns are affected by multiscale granulation and healing processes associated with phase transitions between continuum and discrete states of rocks. The ongoing continuum-discrete transitions are accompanied by progressive evolution of disordered fault networks to dominant localized fault zones, development of bimaterial interfaces, and increasing dynamic weakening of fault surfaces. Results on individual fault zones point to three general dynamic regimes. The first is associated with broad range of heterogeneities, little dynamic weakening, power law frequency-size statistics, temporal clustering of intermediate and large events, and accelerated seismic release before large earthquakes. The second is associated with relatively uniform localized structures, significant dynamic weakening, characteristic earthquake statistics, and quasi-periodic temporal occurrence of large events without precursory accelerated release. For a range of conditions, the fault zone response can switch back and forth between the foregoing two dynamic regimes. Higher temperature, fluid content, and thickness of sedimentary cover reduce the seismic coupling in a region and change the properties of local earthquake sequences. Brittle regions with high seismic coupling have few foreshocks and long-duration aftershock sequences with high event productivity, whereas more viscous regions with low seismic coupling have increased foreshocks activity and low-productivity aftershock sequences or swarms. The results provide criteria for organizing data in classes associated with different evolutionary stages and different regional conditions. An ability to recognize the dynamic regime of a given fault zone or a region can increase the information content of the data and lead to improved strategies for seismic hazard assessment.

Collective behavior of earthquakes and faults: Continuum-discrete transitions, progressive evolutionary changes, and different dynamic regimes

[1] Earthquakes and the faults upon which they occur interact over a wide range of spatial and temporal scales. In addition, many aspects of regional seismicity appear to be stochastic both in space and time. However, within this complexity, there is considerable self-organization. We argue that the occurrence of earthquakes is a problem that can be attacked using the fundamentals of statistical physics. Concepts of statistical physics associated with phase changes and critical points have been successfully applied to a variety of cellular automata models. Examples include sandpile models, forest fire models, and, particularly, slider block models. These models exhibit avalanche behavior very similar to observed seismicity. A fundamental question is whether variations in seismicity can be used to successfully forecast the occurrence of earthquakes. Several attempts have been made to utilize precursory seismic activation and quiescence to make earthquake forecasts, some of which show promise.

https://www.eoas.ubc.ca/~mjelline/453website/eosc453/E_prints/2003RG000135.pdf

Statistical physics approach to understanding the multiscale dynamics of earthquake fault systems

We test the concept that seismicity prior to a large earthquake can be understood in terms of the statistical physics of a critical phase transition. In this model, the cumulative seismic strain release increases as a power law time to failure before the final event. Furthermore, the region of correlated seismicity predicted by this model is much greater than would be predicted from simple elastodynamic interactions. We present a systematic procedure to test for the accelerating seismicity predicted by the critical point model and to identify the region approaching criticality, based on a comparison between the observed cumulative energy (Benioff strain) release and the power law behavior predicted by theory. This method is used to find the critical region before all earthquakes along the San Andreas system since 1950 with M≥6.5. The statistical significance of our results is assessed by performing the same procedure on a large number of randomly generated synthetic catalogs. The null hypothesis, that the observed acceleration in all these earthquakes could result from spurious patterns generated by our procedure in purely random catalogs, is rejected with 99.5% confidence. An empirical relation between the logarithm of the critical region radius (R) and the magnitude of the final event (M) is found, such that log R∝0.5M, suggesting that the largest probable event in a given region scales with the size of the regional fault network.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/98JB00792

An observational test of the critical earthquake concept

Paleoseismological data suggest the occurrence of four bursts of seismic moment release in the Los Angeles region during the past 12,000 yr. The historic period appears to be part of an ongoing lull that has persisted for about the past 1000 yr. These periods of rapid seismic displacement in the Los Angeles region have occurred during the lulls between similar bursts of activity observed on the eastern California shear zone in the Mojave Desert, which is now seismically active. A kinematic model in which the faults of the greater San Andreas system suppress activity on faults in the eastern California shear zone, and vice versa, can explain the apparent switching of activity between the two fault networks. Combined with the observation that short-term geodetic and longer-term geologic rates co-vary on major southern California fault systems, this suggests that either (1) a temporal cluster of seismic displacements on upper-crustal faults increases ductile deformation on their downward extensions, or (2) rapid ductile slip in the lower crust beneath faults loads the upper crust, driving a seismic cluster. We suggest that alternating periods of rapid seismic displacement may be the expected mode of seismicity when two fault systems accommodate the same plate-boundary motion, and slip on one system suppresses slip on the other.

/pdf/long-range-and-long-term-fault-interactions-in-southern-jmcw50hqz3.pdf

Long-range and long-term fault interactions in Southern California

Long-Range and Long-Term Fault Interactions in Southern California

The stress field that existed before a large earth- quake can be calculated based on the known source parame- ters of the event. This stress field can be used to define a region that shows greater seismic moment rate changes prior to the event than arbitrarily shaped regions, allowing us to link two previously unrelated subjects: Coulomb stress interactions and accelerating seismicity before large earthquakes. As an example, we have examined all M≥6.5 earthquakes in California since 1950. While we illustrate the model using seismicity in California, the technique i s general and can be applied to any tectonically active re- gion. We show that where sufficient knowledge of the re- gional tectonics exists, this method can be used to aug- ment current techniques for seismic hazard estimation.

Accelerating seismicity and stress accumulation before large earthquakes

Oblique motion along tectonic boundaries is commonly partitioned into slip on faults with different senses of motion. The origin of slip partitioning is important to structural geology, tectonophysics, and earthquake mechanics. Partitioning can be explained by the upward elastoplastic propagation of oblique slip from a fault or shear zone at depth. The strain field ahead of the propagating fault separates into zones of predominantly normal, reverse, and strike-slip faulting. The model successfully predicts the distribution of fault types along parts of the San Andreas and Haiyuan faults.

http://science.sciencemag.org/content/sci/300/5622/1121.full.pdf

D. D. Bowman

Papers

An observational test of the critical earthquake concept

Long-range and long-term fault interactions in Southern California

Long-Range and Long-Term Fault Interactions in Southern California

Accelerating seismicity and stress accumulation before large earthquakes

Slip partitioning by elastoplastic propagation of oblique slip at depth.