Nowcasting Avalanches as Earthquakes and the Predictability of Strong Avalanches in the Olami-Feder-Christensen Model

Nowcasting earthquakes, suggested recently as a method to estimate the state of a fault and hence the seismic risk, is based on the concept of natural time. Here, we generalize nowcasting to a prediction method the merits of which are evaluated by means of the receiver operating characteristics. This new prediction method is applied to a simple (toy) model for the waiting (natural) time of the stronger earthquakes, real seismicity, and the Olami-Feder-Christensen earthquake model with interesting results revealing acceptable to excellent or even outstanding performance.     

Prediction problem for target events based on the inter-event waiting time

In this paper we address the problem of forecasting the target events of a time series given the distribution ξ of time gaps between target events. Strong earthquakes and stock market crashes are the two types of such events that we are focusing on. In the series of earthquakes, as McCann et al. show [W.R. Mc Cann, S.P. Nishenko, L.R. Sykes, J. Krause, Seismic gaps and plate tectonics: seismic potential for major boundaries, Pure and Applied Geophysics 117 (1979) 1082-1147], there are well-defined gaps (called seismic gaps) between strong earthquakes. On the other hand, usually there are no regular gaps in the series of stock market crashes [M. Raberto, E. Scalas, F. Mainardi, Waiting-times and returns in high-frequency financial data: an empirical study, Physica A 314 (2002) 749-755]. For the case of seismic gaps, we analytically derive an upper bound of prediction efficiency given the coefficient of variation of the distribution ξ. For the case of stock market crashes, we develop an algorithm that predicts the next crash within a certain time interval after the previous one. We show that this algorithm outperforms random prediction. The efficiency of our algorithm sets up a lower bound of efficiency for effective prediction of stock market crashes.     

Fragmentation dynamics within shear bands--a model for aging tectonic faults?

A numerical model for packing of fragmenting blocks in a shear band is introduced, and its dynamics is compared with that of a tectonic fault. The shear band undergoes a slow aging process in which the blocks are being grinded by the shear motion and the compression. The dynamics of the model have the same statistical characteristics as the seismic activity in faults. The characteristic magnitude distribution of earthquakes appears to result from frictional slips at small and medium magnitudes, and from fragmentation of blocks at the largest magnitudes. Aftershocks to large-magnitude earthquakes are local recombinations of the fragments before they reach a new quasi-static equilibrium. The aftershocks satisfy Omori's law. Local precursor activity at a few times the normal background level appears at a short time before a major earthquake. Seismic gaps appear as a natural consequence of the aging process of a fault. Explanation of the heat flux and principal stress direction anomalies at the faults both involve the value of fracture stress of the blocks in the gouge. The final form of a tectonic fault is predicted to involve a gouge dominated by fine-grained and rather rounded blocks so that it cannot withstand large shear stresses. Received 26 July 2000     

Optimization of green-times at an isolated urban crossroads

We propose a model for the intersection of two urban streets. The traffic status of the crossroads is controlled by a set of traffic lights which periodically switch to red and green with a total period of T. Two different types of crossroads are discussed. The first one describes the intersection of two one-way streets, while the second type models the intersection of a two-way street with an one-way street. We assume that the vehicles approach the crossroads with constant rates in time which are taken as the model parameters. We optimize the traffic flow at the crossroads by minimizing the total waiting time of the vehicles per cycle of the traffic light. This leads to the determination of the optimum green-time allocated to each phase. Received 19 October 2000 and Received in final form 25 May 2001     

Power-law time distribution of large earthquakes

We study the statistical properties of time distribution of seismicity in California by means of a new method of analysis, the diffusion entropy. We find that the distribution of time intervals between a large earthquake (the main shock of a given seismic sequence) and the next one does not obey Poisson statistics, as assumed by the current models. We prove that this distribution is an inverse power law with an exponent mu=2.06+/-0.01. We propose the long-range model, reproducing the main properties of the diffusion entropy and describing the seismic triggering mechanisms induced by large earthquakes.     

The Weibull-log Weibull distribution for interoccurrence times of earthquakes

By analyzing the Japan Meteorological Agency (JMA) seismic catalog for different tectonic settings, we have found that the probability distributions of time intervals between successive earthquakes-interoccurrence times-can be described by the superposition of the Weibull distribution and the log-Weibull distribution. In particular, the distribution of large earthquakes obeys the Weibull distribution with the exponent α₁<1, indicating the fact that the sequence of large earthquakes is not a Poisson process. It is found that the ratio of the Weibull distribution to the probability distribution of the interoccurrence time gradually increases with increase in the threshold of magnitude. Our results infer that Weibull statistics and log-Weibull statistics coexist in the interoccurrence time statistics, and that the change of the distribution is considered as the change of the dominant distribution. In this case, the dominant distribution changes from the log-Weibull distribution to the Weibull distribution, allowing us to reinforce the view that the interoccurrence time exhibits the transition from the Weibull regime to the log-Weibull regime.     

论地震发生机制

地震发生的物理机理和过程是还没有认识清楚的问题. 此前人们将浅源地震归因于弹性回跳,根据这一观点和岩石实验结果计算得到的地震能量与实际观测结果有很大矛盾,被称之为“热流佯谬”. 中源和深源地震发生在地幔区域,其成因也没有合理的解释. 考虑到地壳和地幔是离散集合态物质体系及其慢动力学运动行为的基本特点,本文根据物理学原理,特别是近年凝聚态物理发展出来的相关新观念,并依据已有观测事实,从新的视角探究地震发生的物理机制. 1) 关于地壳岩石层中的应力分布:在不考虑构造力时,依据万物皆流的流变学原理,原始地壳岩石在自重压强长时间作用下,纵向和横向应力相同,没有差应力. 大地构造力推动岩块滞滑移动挤压断层泥,施加于其他岩块,逐渐传递和积累. 这种附加的横向构造力与原始岩石中应力叠加,形成地壳岩石层中的实时应力. 由于断层泥属于颗粒物质体系,具有与岩石不同的力学特征,其弹性模量比岩石小得多,且随压强而增大,导致构造作用力随深度非线性增大. 给出了地壳中构造应力分布及其变化规律. 2) 关于地壳岩石层强度:地壳岩石的自重会使岩石发生弹性–塑性转变. 通过对弹性–塑性转变深度的计算,并根据实际情况分析,给出了地壳岩石弹性、部分塑性和完全塑性三个区域的典型深度范围. 在部分塑性区,塑性体比例达到约10%以上时,发生塑性连通,这时岩石剪切强度由塑性特征决定. 塑性滑移的等效摩擦系数比脆性破裂小一个数量级以上,致使塑性滑移时岩石剪切强度比脆性破裂小得多. 同时,随深度增大,有多种因素使得岩石剪切屈服强度减小. 另一方面,地震是大范围岩石破坏,破坏必然沿薄弱路径发生. 因此,浅源地震岩石的实际破坏强度必定比通常观测到的岩石剪切强度值低. 给出了地壳岩石平均强度和实际破坏强度典型值随深度的分布规律. 3) 关于地震发生的条件和机制:地震发生必定产生体积膨胀,只有突破阻挡才可膨胀. 地震发生的条件是:大地构造力超过岩石破坏强度、断层边界摩擦力以及所受阻挡力之和. 因此,浅源地震是岩石突破阻挡发生的塑性滑移. 在此基础上提出了浅源地震发生的四种可能模式. 深源地震是冲破阻挡发生的大范围岩块流. 浅源地震和深源地震都是堵塞–解堵塞转变,是解堵塞后岩石层块滑移或流动造成的能量释放. 4) 关于地震能量和临震前兆信息:地震能量即为堵塞–解堵塞转变过程释放的动能. 以实例估算表明,地震岩石滑移动能与使岩块剪切破坏和克服周围摩擦阻力所需做的功相一致,不会出现热流佯谬. 同时指出,通过观测地震发生前构造力的积累过程、局域地区地质变迁以及岩石状态变化等所产生的效应,均可能获得有价值的地震前兆信息. 关键词： 地震发生机制 热流佯谬 地壳岩石应力和强度 堵塞–解堵塞转变     

Renormalization of branching models of triggered seismicity from total to observable seismicity

Several recent works point out that the crowd of small unobservable earthquakes (with magnitudes below the detection threshold m_d) may play a significant and perhaps dominant role in triggering future seismicity. Using the ETAS branching model of triggered seismicity, we apply the formalism of generating probability functions to investigate how the statistical properties of observable earthquakes differ from the statistics of all events. The ETAS (epidemic-type aftershock sequence) model assumes that each earthquake can trigger other earthquakes (“aftershocks”). An aftershock sequence results in this model from the cascade of aftershocks of each past earthquake. The triggering efficiency of earthquakes is assumed to vanish below a lower magnitude limit m₀, in order to ensure the convergence of the theory and may reflect the physics of state-and-velocity frictional rupture. We show that, to a good approximation, the statistical distribution of seismic rates of events with magnitudes above m_d generated by an ETAS model with branching ratio n is the same as that of events generated by another ETAS model with effective parameter n(m_d). Our present analysis thus confirms, for the full statistical (time-independent or large time-window approximation) properties, the results obtained previously by one of us and Werner, based solely on the average seismic rates (the first-order moment of the statistics). Our analysis also demonstrates that this correspondence is not exact, as there are small corrections which can be systematically calculated, in terms of additional contributions that can be mapped onto a different branching model. We also show that this approximate correspondence of the ETAS model onto itself obtained by changing m₀ into m_d, and n into n(m_d) holds only with respect to its statistical properties and not for all its space-time properties.     

Variation of radon flux along active fault zones in association with earthquake occurrence

Radon flux measurements were carried out at three radon stations along an active fault zone in the Langadas basin, Northern Greece by various techniques for earthquake prediction studies. Specially made devices with alpha track-etch detectors (ATDs) were installed by using LR-115, type II, non-strippable cellulose nitrate films (integrating method of measurements). Continuous monitoring of radon gas exhaling from the ground was also performed by using silicon diode detectors, Barasol and Clipperton type, in association with various probes and sensors including simultaneously registration of the meteorological parameters, such as precipitation height (rainfall events), temperature and barometric pressure. The obtained radon data were studied in parallel with the data of seismic events, such as the magnitude, M_L of earthquakes, the epicentral distance, the hypocentral distance and the energy released during the earthquake event occurred at the fault zone during the period of measurements to find out any association between the rad on flux and the meteorological and seismological parameters. Seismic events with magnitude M_L ≥ 4.0 appeared to be preceded by large precursory signals produced a well-defined “anomaly” (peak) of radon flux prior to the event. In the results, the radon peaks in the obtained spectra appeared to be sharp and narrow. The rise time of a radon peak, that is the time period from the onset of a radon peak until the time of radon flux maximum is about a week, while the after time, that is the time interval between the time of radon flux maximum and the time of a seismic event ranges from about 3 weeks or more.     

Dynamical analogy between epileptic seizures and seismogenic electromagnetic emissions by means of nonextensive statistical mechanics

The field of study of complex systems considers that the dynamics of complex systems are founded on universal principles that may be used to describe a great variety of scientific and technological approaches of different types of natural, artificial, and social systems. Several authors have suggested that earthquake dynamics and neurodynamics can be analyzed within similar mathematical frameworks. Recently, authors have shown that a dynamical analogy supported by scale-free statistics exists between seizures and earthquakes, analyzing populations of different seizures and earthquakes, respectively. The purpose of this paper is to suggest a shift in emphasis from the large to the small scale: our analyses focus on a single epileptic seizure generation and the activation of a single fault (earthquake) and not on the statistics of sequences of different seizures and earthquakes. We apply the concepts of the nonextensive statistical physics to support the suggestion that a dynamical analogy exists between the two different extreme events, seizures and earthquakes. We also investigate the existence of such an analogy by means of scale-free statistics (the Gutenberg–Richter distribution of event sizes and the distribution of the waiting time until the next event). The performed analysis confirms the existence of a dynamic analogy between earthquakes and seizures, which moreover follow the dynamics of magnetic storms and solar flares.     

EARTHQUAKE SCALING PARADOX

Two measures of earthquakes, the seismic moment and the broadband radiated energy, show completely different scaling relations. For shallow earthquakes worldwide from January 1987 to December 1998, the frequency distribution of the seismic moment shows a clear kink between moderate and large earthquakes, as revealed by previous works. But the frequency distribution of the broadband radiated energy shows a single power law, a classical Gutenberg-Richter relation. This inconsistency raises a paradox in the self-organized criticality model of earthquakes.     

The distribution of first-passage times and durations in FOREX and future markets

Possible distributions are discussed for intertrade durations and first-passage processes in financial markets. The view-point of renewal theory is assumed. In order to represent market data with relatively long durations, two types of distributions are used, namely a distribution derived from the Mittag-Leffler survival function and the Weibull distribution. For the Mittag-Leffler type distribution, the average waiting time (residual life time) is strongly dependent on the choice of a cut-off parameter t_max, whereas the results based on the Weibull distribution do not depend on such a cut-off. Therefore, a Weibull distribution is more convenient than a Mittag-Leffler type if one wishes to evaluate relevant statistics such as average waiting time in financial markets with long durations. On the other hand, we find that the Gini index is rather independent of the cut-off parameter. Based on the above considerations, we propose a good candidate for describing the distribution of first-passage time in a market: The Weibull distribution with a power-law tail. This distribution compensates the gap between theoretical and empirical results more efficiently than a simple Weibull distribution. It should be stressed that a Weibull distribution with a power-law tail is more flexible than the Mittag-Leffler distribution, which itself can be approximated by a Weibull distribution and a power-law. Indeed, the key point is that in the former case there is freedom of choice for the exponent of the power-law attached to the Weibull distribution, which can exceed 1 in order to reproduce decays faster than possible with a Mittag-Leffler distribution. We also give a useful formula to determine an optimal crossover point minimizing the difference between the empirical average waiting time and the one predicted from renewal theory. Moreover, we discuss the limitation of our distributions by applying our distribution to the analysis of the BTP future and calculating the average waiting time. We find that our distribution is applicable as long as durations follow a Weibull law for short times and do not have too heavy a tail.     

Scale-invariant structure of energy fluctuations in real earthquakes

Earthquakes are obviously complex phenomena associated with complicated spatiotemporal correlations, and they are generally characterized by two power laws: the Gutenberg-Richter (GR) and the Omori-Utsu laws. However, an important challenge has been to explain two apparently contrasting features: the GR and Omori-Utsu laws are scale-invariant and unaffected by energy or time scales, whereas earthquakes occasionally exhibit a characteristic energy or time scale, such as with asperity events. In this paper, three high-quality datasets on earthquakes were used to calculate the earthquake energy fluctuations at various spatiotemporal scales, and the results reveal the correlations between seismic events regardless of their critical or characteristic features. The probability density functions (PDFs) of the fluctuations exhibit evidence of another scaling that behaves as a q-Gaussian rather than random process. The scaling behaviors are observed for scales spanning three orders of magnitude. Considering the spatial heterogeneities in a real earthquake fault, we propose an inhomogeneous Olami-Feder-Christensen (OFC) model to describe the statistical properties of real earthquakes. The numerical simulations show that the inhomogeneous OFC model shares the same statistical properties with real earthquakes.     

New approach to Gutenberg-Richter scaling

We introduce a new model for an earthquake fault system that is composed of noninteracting simple lattice models with different levels of damage denoted by q. The undamaged lattice models (q=0) have Gutenberg-Richter scaling with a cumulative exponent β=1/2, whereas the damaged models do not have well defined scaling. However, if we consider the "fault system" consisting of all models, damaged and undamaged, we get excellent scaling with the exponent depending on the relative frequency with which faults with a particular amount of damage occur in the fault system. This paradigm combines the idea that Gutenberg-Richter scaling is associated with an underlying critical point with the notion that the structure of a fault system also affects the statistical distribution of earthquakes. In addition, it provides a framework in which the variation, from one tectonic region to another, of the scaling exponent, or b value, can be understood.     

The construction of an Ito model for geoelectrical signals

The Ito stochastic differential equation governs the one-dimensional diffusive Markov process. Geoelectrical signals measured in seismic areas can be considered as the result of competitive and collective interactions among system elements. The Ito equation may constitute a good macroscopic model of such a phenomenon in which microscopic interactions are adequately averaged. The present study shows how to construct an Ito model for a geoelectrical time series measured in a seismic area of southern Italy. Our results reveal that the Ito model describes the whole time series quite well, but it performs better when one considers fragments of the data set with lower variability range (absent or rare large fluctuations). Our findings show that generally detrended geoelectrical time series can be considered as approximations of Markov diffusion processes.     

Analysis of the spatial distribution between successive earthquakes

Spatial distances between subsequent earthquakes in southern California exhibit scale-free statistics, with a critical exponent delta approximately 0.6, as well as finite size scaling. The statistics are independent of the threshold magnitude as long as the catalog is complete, but depend strongly on the temporal ordering of events, rather than the geometry of the spatial epicenter distribution. Nevertheless, the spatial distance and waiting time between subsequent earthquakes are uncorrelated with each other. These observations contradict the theory of aftershock zone scaling with main shock magnitude.     

Solar, geomagnetic and seismic activity

Summary An 11-y modulation of large Italian earthquakes has been successfully identified and found to be positively linked to sunspot activity. The seismic activity appears to be modulated by the 11-y sunspot cycle through the coherent variation of geomagnetic activity. It is proposed that the two phenomena are linked by the influence of a magnetostriction process on stresses in the crust. An implication of this model is that geomagnetic storms may directly trigger large earthquakes.

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Riassunto L'analisi dei valori annuali delle macchie solari, della variazione geomagnetica interdiurna e dei grossi terremoti italiani ha permesso di identificare una significativa diretta modulazione undecennale dell'attività sismica da parte dell'attività solare attraverso la coerente variazione dell'attività geomagnetica con un relativo effetto magnetostrittivo sulle rocce crostali. Il modello proposto si rivela di pratica importanza per la previsione dei terremoti innescati da tempeste magnetiche.

     

Unified scaling law for earthquakes

We show that the distribution of waiting times between earthquakes occurring in California obeys a simple unified scaling law valid from tens of seconds to tens of years. The short time clustering, commonly referred to as aftershocks, is nothing but the short time limit of the general hierarchical properties of earthquakes. There is no unique operational way of distinguishing between main shocks and aftershocks. In the unified law, the Gutenberg-Richter b value, the exponent -1 of the Omori law for aftershocks, and the fractal dimension d(f) of earthquakes appear as critical indices.     

Intermittent criticality in the long-range connective sandpile (LRCS) model

We here propose a long-range connective sandpile model with variable connection probability Pc which has an important impact on the slope of the power-law frequency-size distribution of avalanches. The long-range connection probability Pc is changed according to an explicit function of the size of the latest event, although the evolution rule of Pc may be different in various physical systems. Such version of the sandpile model demonstrates large fluctuations in the dynamical variable 〈Z〉(t) (the spatially averaged amount of grains retained within the grid at each time step), indicating the state of intermittent criticality in the system. Many researches have suggested that the earthquake fault system is an intermittent criticality system, which would imply some level of statistical predictability of great events. Our modified sandpile model thus provides a testing ground for many proposed precursory measures related to great earthquakes.     

Scaling and universality in rock fracture

We present a detailed statistical analysis of acoustic emission time series from laboratory rock fracture obtained from different experiments on different materials including acoustic emission controlled triaxial fracture and punch-through tests. In all considered cases, the waiting time distribution can be described by a unique scaling function indicating its universality. This scaling function is even indistinguishable from that for earthquakes suggesting its general validity for fracture processes independent of time, space, and magnitude scales.