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Catastrophic failures have momentous impact in many scientific and technological fields but remain challenging to understand and predict. One key difficulty lies in the burstiness of rupture phenomena, which typically involve a series of progressively shorter quiescent phases punctuated by sudden bursts, rather than a smooth continuous progression. This seemingly erratic pattern challenges the conventional power law assumption of continuous scale invariance. Here, we propose a generalized material failure law based on the log-periodic power law, which better captures the discrete scale invariance inherent in intermittent rupture dynamics. Our method's superiority is demonstrated through testing on 109 historical geohazard events, including landslides, rockbursts, glacier breakoffs, and volcanic eruptions. The results indicate that our method is general and robust, offering significant potential to forecast catastrophic failures., Comment: Main Text (14 pages, 3 figures, and 2 tables) and Supplementary Materials (54 pages, Materials and Methods, 18 figures, and 16 tables)

A systematic quantitative investigation into whether the mechanisms of large earthquakes are unique could significantly deepen our understanding of fault rupture and seismicity patterns. This research holds the potential to advance our ability to predict large earthquakes and enhance the effectiveness of disaster prevention and mitigation strategies. In 2009, one of us introduced the dragon-king theory, offering a quantitative framework for identifying and testing extreme outliers-referred to as dragon-king events-that are endogenously generated. This theory provides valuable tools for explaining, predicting, and managing the risks associated with these rare but highly impactful events. The present paper discusses the feasibility of applying this theory to seismology, proposing that dragon-king earthquake events can be identified as outliers to the Gutenberg-Richter law. It also examines several seismological mechanisms that may contribute to the occurrence of these extraordinary events. Although applying the dragon-king theory to seismology presents practical challenges, it offers the potential to significantly enrich statistical seismology. By reexamining the classification of earthquake rupture types through a statistical testing lens and integrating these insights with underlying physical mechanisms, this approach can greatly enhance the analytical tools and depth of research in the field of statistical seismology., Comment: 21 pages, 2 figures

The Epidemic-Type Aftershock Sequences (ETAS) model and its variants effectively capture the space-time clustering of seismicity, setting the standard for earthquake forecasting. Accurate unbiased ETAS calibration is thus crucial. But we identify three sources of bias, (i) boundary effects, (ii) finite-size effects, and (iii) censorship, which are often overlooked or misinterpreted, causing errors in seismic analysis and predictions. By employing an ETAS model variant with variable spatial background rates, we propose a method to correct for these biases, focusing on the branching ratio n, a key indicator of earthquake triggering potential. Our approach quantifies the variation in the apparent branching ratio (napp) with increased cut-off magnitude (Mco) above the optimal cut-off (Mcobest). The napp(Mco) function yields insights superior to traditional point estimates. We validate our method using synthetic earthquake catalogs, accurately recovering the true branching ratio (ntrue) after correcting biases with napp(Mco). Additionally, our method introduces a refined estimation of the minimum triggering magnitude (m0), a crucial parameter in the ETAS model. Applying our framework to the earthquake catalogs of California, New Zealand, and the China Seismic Experimental Site (CSES) in Sichuan and Yunnan provinces, we find that seismicity hovers away from the critical point, nc = 1, remaining distinctly subcritical, however with values tending to be larger than recent reports that do not consider the above biases. It is interesting that, m0 is found around 4 for California, 3 for New Zealand and 2 for CSES, suggesting that many small triggered earthquakes may not be fertile. Understanding seismicity's critical state significantly enhances our comprehension of seismic patterns, aftershock predictability, and informs earthquake risk mitigation and management strategies., Comment: 36 pages, 7 figures, 5 tables

Landslides exhibit intermittent gravity-driven downslope movements developing over days to years before a possible major collapse, commonly boosted by external events like precipitations and earthquakes. The reasons behind these episodic movements and how they relate to the final instability remain poorly understood. Here, we develop a novel 'endo-exo' theory to quantitatively diagnose landslide dynamics, capturing the interplay between exogenous stressors such as rainfall and endogenous damage/healing processes. We predict four distinct types of episodic landslide dynamics (endogenous/exogenous-subcritical/critical), characterized by power law relaxations with different exponents, all related to a single parameter $\vartheta$. These predictions are tested on the dataset of the Preonzo landslide, which exhibited multi-year episodic movements prior to a catastrophic collapse. All its episodic activities can be accounted for within this classification with $\vartheta \approx 0.45\pm0.1$, providing strong support for our parsimonious theory. We find that the final collapse of this landslide is clearly preceded over 1-2 months by an increased frequency of medium/large velocities, signaling the transition into a catastrophic regime with amplifying positive feedbacks., Comment: 5 figures

We develop a statistical thermodynamic approach for understanding earthquake nucleation on homogeneous faults, explaining the occurrence of both self-arresting and run-away unstable ruptures (subshear and supershear events) previously observed in numerical simulations. Our theory identifies the conditions under which self-arresting earthquakes occur, based on critical sliding distance and dynamical stress drop. We also derive the Gutenberg-Richter distribution for these earthquakes, linking the fractal nature of faulting with the nucleation physics through the critical size's dependence on dynamical stress drop. Furthermore, we connect our findings to the dragon-king theory, which suggests that the largest earthquakes differ significantly in physical and statistical properties from smaller ones, offering new insights for earthquake prediction and risk assessment., Comment: 5 pages, 1 figure

The new broadband seismic network in China will increase the number of stations from approximately 950 to 2000. The higher-resolution monitoring of the frequent smaller earthquakes expected inside Mainland China can be quantified via the completeness magnitude (Mc) metric. Using the Bayesian Magnitude of Completeness (BMC) method, we generate the spatial distribution of Mc predicted for the new network, based on the prior model calibrated on the current earthquake catalog (2012 to 2021) and network configuration. If 99% of Mainland China is at present covered down to Mc = 2.7, this threshold will soon fall to Mc = 2.0. This means approximately 3 times more earthquakes available per year. Based on the observation that seismic precursors are most likely to be observed at least at 3 units below the mainshock magnitude, the new seismic network shall achieve the goal of almost total coverage for optimal seismic-based earthquake prediction research., Comment: 20 pages, 5 figures, 1 table

The China Earthquake Administration (CEA) has launched an ambitious nationwide earthquake early warning (EEW) system project currently under development, which will include approximately 15,000 seismic stations and be the largest EEW system in the world. The new EEW system is planned to go online at the end of 2023. In approximately 50%, 30% and 20% of Chinese mainland, the inter-station distance will soon be smaller than 50 km, 25 km and 15 km, respectively. The expected effectiveness of this EEW system can be quantified via the metric determined from the radius of the blind zone, which refers to the area near the epicenter where there is insufficient time to issue a warning before the arrival of strong S- and surface waves. This study uses a theoretical network-based method together with Monte Carlo simulation to obtain the spatial distribution of the blind zone radii and their associated uncertainties for the new seismic network based on its configuration. We find that the densified new seismic network is expected to have excellent EEW performance as the area covered by small blind zones with radius less than 30 km increases dramatically from approximately 2% to 22%, that is by 2.4 million km2 inside Chinese mainland. We also find that every 1,000,000 RMB (about 146,000 USD) invested to densify the planned network will lead to an areal increase of 3,000 km2 of small blind zones. Continuing to increase the density of stations in some key regions with blind zone radii ranging from 15 to 40 km is still necessary to control the unexpected expansion of blind zones due to possible (and common) stations failure. Our work provides insights into the expected performance of the upcoming EEW network in Chinese mainland, and our proposed evaluation approach is broadly applicable for predicting the performance of EEW systems during their planning, design, and implementation stages., Comment: 41 pages, 14 figures, 1 table

Fundamentally related to the UV divergence problem in Physics, conventional wisdom in seismology is that the smallest earthquakes, which are numerous and often go undetected, dominate the triggering of major earthquakes, making the prediction of the latter difficult if not inherently impossible. By developing a rigorous validation procedure, we show that, in fact, large earthquakes (above magnitude 6.3 in California) are preferentially triggered by large events. Because of the magnitude correlations intrinsic in the validated model, we further rationalize the existence of earthquake doublets. These findings have far-reaching implications for short-term and medium-term seismic risk assessment, as well as for the development of a deeper theory without UV cut-off that is locally self-similar.

We study two-dimensional tensorial elastic wave transport in densely fractured media and document transitions from propagation to diffusion and to localization/delocalization. For large fracture stiffness, waves are propagative at the scale of the system. For small stiffness, multiple scattering prevails, such that waves are diffusive in disconnected fracture networks, and localized in connected ones with a strong multifractality of the intensity field. A reentrant delocalization is found in well-connected networks due to energy leakage via evanescent waves and cascades of mode conversion., Comment: 5 pages, 5 figures

Seismicity and faulting within the Earth crust are characterized by many scaling laws that are usually interpreted as qualifying the existence of underlying physical mechanisms associated with some kind of criticality in the sense of phase transitions. Using an augmented Epidemic-Type Aftershock Sequence (ETAS) model that accounts for the spatial variability of the background rates $\mu(x,y)$, we present a direct quantitative test of criticality. We calibrate the model to the ANSS catalog of the entire globe, the region around California, and the Geonet catalog for the region around New Zealand using an extended Expectation-Maximization (EM) algorithm including the determination of $\mu(x,y)$. We demonstrate that the criticality reported in previous studies is spurious and can be attributed to a systematic upward bias in the calibration of the branching ratio of the ETAS model, when not accounting correctly for spatial variability. We validate the version of the ETAS model which possesses a space varying background rate $\mu(x,y)$ by performing pseudo prospective forecasting tests. The non-criticality of seismicity has major implications for the prediction of large events.

We analyse the compiled set of precursory data that were reported to be available in real time before the Ms 7.5 Haicheng earthquake in Feb. 1975 and the Ms 7.6-7.8 Tangshan earthquake in July 1976. We propose a robust and simple coarse-graining method consisting in aggregating and counting how all the anomalies together (geodesy, levelling, geomagnetism, soil resistivity, Earth currents, gravity, Earth stress, well water radon, well water level) develop as a function of time. We demonstrate a strong evidence for the existence of an acceleration of the number of anomalies leading up to the major Haicheng and Tangshan earthquakes. In particular for the Tangshan earthquake, the frequency of occurrence of anomalies is found to be well described by the log-periodic power law singularity (LPPLS) model, previously proposed for the prediction of engineering failures and later adapted to the prediction of financial crashes. Based on a mock real-time prediction experiment, and simulation study, we show the potential for an early warning system with lead-time of a few days, based on this methodology of monitoring accelerated rates of anomalies., Comment: 18 pages, 13 figures, 3 tables

We introduce the first fully self-consistent model combining the seismic micro-ruptures occurring within a generalized Burridge-Knopoff spring-block model with the nucleation and propagation of electric charge pulses within a coupled mechano-electrokinetic system. This model provides a general theoretical framework for modeling and analyzing geoelectric precursors to earthquakes. In particular, it can reproduce the unipolar pulses that have often been reported before large seismic events, as well as various observed anomalies in the statistical moments of the ambient electric fields and the power-law exponent transition of the power spectra of electric fields.

We examine the precursory behavior of geoelectric signals before large earthquakes by means of an algorithm including an alarm-based model and binary classification. This algorithm, introduced originally by Chen and Chen [Nat. Hazards., 84, 2016], is improved by removing a time parameter for coarse-graining of earthquake occurrences, as well as by extending the single station method into a joint stations method. We also determine the optimal frequency bands of earthquake-related geoelectric signals with the highest signal-to-noise ratio. Using significance tests, we also provide evidence of an underlying seismoelectric relationship. It is appropriate for machine learning to extract this underlying relationship, which could be used to quantify probabilistic forecasts of impending earthquakes, and to get closer to operational earthquake prediction.

Forecasting the full distribution of the number of earthquakes is revealed to be inherently superior to forecasting their mean. Forecasting the full distribution of earthquake numbers is also shown to yield robust projections in the presence of "surprise" large earthquakes, which in the past have strongly deteriorated the scores of existing models. We show this with pseudo-prospective experiments on synthetic as well as real data from the Advanced National Seismic System (ANSS) database for California, with earthquakes with magnitude larger than 2.95 that occurred between the period 1971-2016. Our results call in question the testing methodology of the Collaboratory for the study of earthquake predictability (CSEP), which amounts to assuming a Poisson distribution of earthquake numbers, which is known to be a poor representation of the heavy-tailed distribution of earthquake numbers. Using a spatially varying ETAS model, we demonstrate a remarkable stability of the forecasting performance, when using the full distribution of earthquake numbers for the forecasts, even in the presence of large earthquakes such as Mw 7.1 Hector Mine, Mw 7.2 El Mayor-Cucapah, Mw 6.6 Sam Simeon earthquakes, or in the presence of intense swarm activity in Northwest Nevada in 2014. While our results have been derived for ETAS type models, we propose that all earthquake forecasting models of any type should embrace the full distribution of earthquake numbers, such that their true forecasting potential is revealed.

Currently, one of the best performing and most popular earthquake forecasting models rely on the working hypothesis that: "locations of past background earthquakes reveal the probable location of future seismicity". As an alternative, we present a class of smoothed seismicity models (SSMs) based on the principles of the Epidemic Type Aftershock Sequence (ETAS) model, which forecast the location, time and magnitude of all future earthquakes using the estimates of the background seismicity rate and the rates of future aftershocks of all generations. Using the Californian earthquake catalog, we formulate six controlled pseudo-prospective experiments with different combination of three target magnitude thresholds: 2.95, 3.95 or 4.95 and two forecasting time horizons: 1 or 5 year. In these experiments, we compare the performance of:(1) ETAS model with spatially homogenous parameters or GETAS (2) ETAS model with spatially variable parameters or SVETAS (3) three declustering based SSMs (4) a simple SSM based on undeclustered data and (5) a model based on strain rate data, in forecasting the location and magnitude of all (undeclustered) target earthquakes during many testing periods. In all conducted experiments, the SVETAS model comes out with consistent superiority compared to all the competing models. Consistently better performance of SVETAS model with respect to declustering based SSMs highlights the importance of forecasting the future aftershocks of all generations for developing better earthquake forecasting models. Among the two ETAS models themselves, accounting for the optimal spatial variation of the parameters leads to strong and statistically significant improvements in forecasting performance.

The driving concept behind one of the most successful statistical forecasting models, the ETAS model, has been that the seismicity is driven by spontaneously occurring background earthquakes that cascade into multitudes of triggered earthquakes. In nearly all generalizations of the ETAS model, the magnitudes of the background and the triggered earthquakes are assumed to follow Gutenberg-Richter law with the same exponent (\b{eta}-value). Furthermore, the magnitudes of the triggered earthquakes are always assumed to be independent of the magnitude of the triggering earthquake. Using an EM algorithm applied to the Californian earthquake catalogue, we show that the distribution of earthquake magnitudes exhibits three distinct \b{eta}-values: \b{eta}_b for background events; \b{eta}_a-{\delta} and \b{eta}_a+{\delta}, respectively, for triggered events below and above the magnitude of the triggering earthquake; the two last values express a correlation between the magnitudes of triggered events with that of the triggering earthquake, a feature so far absent in all proposed operational generalizations of the ETAS model. The ETAS model incorporating this kinked magnitude distribution provides by far the best description of seismic catalogs and could thus have the best forecasting potential. We speculate that the kinked magnitude distribution may result from the system tending to restore the symmetry of the regional displacement gradient tensor that has been broken by the initiating event. The general emerging concept could be that while the background events occur primarily to accommodate the symmetric stress tensor at the boundaries of the system, the triggered earthquakes are quasi-Goldstone fluctuations of a self-organized critical deformation state., Comment: Accepted for publication in JGR: Solid Earth

The starting point of the present review is to acknowledge that there are innumerable reports of non-seismic types of earthquake precursory phenomena that are intermittent and seem not to occur systematically, while associated reports are not widely accepted by the geoscience community at large because no one could explain their origins. We review a unifying theory for a solid-state mechanism, based on decades of research bridging semi-conductor physics, chemistry and rock physics. A synthesis has emerged that all pre-earthquake phenomena could trace back to one fundamental physical process: the activation of electronic charges (electrons and positive holes) in rocks subjected to ever-increasing tectonic stresses prior to any major seismic activity, via the rupture of peroxy bonds. In the second part of the review, we critically examine satellite and ground station data, recorded before past large earthquakes, as they have been claimed to provide evidence that precursory signals tend to become measurable days, sometimes weeks before the disasters. We review some of the various phenomena that can be directly predicted by the peroxy defect theory , namely, radon gas emanations, corona discharges, thermal infrared emissions, air ionization, ion and electron content in the ionosphere, and electro-magnetic anomalies. Our analysis demonstrates the need for further systematic investigations, in particular with strong continuous statistical testing of the relevance and confidence of the precursors. Only then, the scientific community will be able to assess and improve the performance of earthquake forecasts.

The ETAS model is widely employed to model the spatio-temporal distribution of earthquakes, generally using spatially invariant parameters. We propose an efficient method for the estimation of spatially varying parameters, using the Expectation-Maximization (EM) algorithm and spatial Voronoi tessellation ensembles. We use the Bayesian Information Criterion (BIC) to rank inverted models given their likelihood and complexity and select the best models to finally compute an ensemble model at any location. Using a synthetic catalog, we also check that the proposed method correctly inverts the known parameters. We apply the proposed method to earthquakes included in the ANSS catalog that occurred within the time period 1981-2015 in a spatial polygon around California. The results indicate a significant spatial variation of the ETAS parameters. We find that the efficiency of earthquakes to trigger future ones (quantified by the branching ratio) positively correlates with surface heat flow. In contrast, the rate of earthquakes triggered by far-field tectonic loading or background seismicity rate shows no such correlation, suggesting the relevance of triggering possibly through fluid-induced activation. Furthermore, the branching ratio and background seismicity rate are found to be uncorrelated with hypocentral depths, indicating that the seismic coupling remains invariant of hypocentral depths in the study region. Additionally, triggering seems to be mostly dominated by small earthquakes. Consequently, the static stress change studies should not only focus on the Coulomb stress changes caused by specific moderate to large earthquakes but also account for the secondary static stress changes caused by smaller earthquakes.

A likely source of earthquake clustering is static stress transfer between individual events. Previous attempts to quantify the role of static stress for earthquake triggering generally considered only the stress changes caused by large events, and often discarded data uncertainties. We conducted a robust two-fold empirical test of the static stress change hypothesis by accounting for all events of magnitude M>=2.5 and their location and focal mechanism uncertainties provided by catalogs for Southern California between 1981 and 2010, first after resolving the focal plane ambiguity and second after randomly choosing one of the two nodal planes. For both cases, we find compelling evidence supporting the static triggering with stronger evidence after resolving the focal plane ambiguity above significantly small (about 10 Pa) but consistently observed stress thresholds. The evidence for the static triggering hypothesis is robust with respect to the choice of the friction coefficient, Skempton's coefficient and magnitude threshold. Weak correlations between the Coulomb Index (fraction of earthquakes that received positive Coulomb stress change) and the coefficient of friction indicate that the role of normal stress in triggering is rather limited. Last but not the least, we determined that the characteristic time for the loss of the stress change memory of a single event is nearly independent of the amplitude of the Coulomb stress change and varies between ~95 and ~180 days implying that forecasts based on static stress changes will have poor predictive skills beyond times that are larger than a few hundred days on average.

We introduce the Anisotropic Clustering of Location Uncertainty Distributions (ACLUD) method to reconstruct active fault networks on the basis of both earthquake locations and their estimated individual uncertainties. After a massive search through the large solution space of possible reconstructed fault networks, we apply six different validation procedures in order to select the corresponding best fault network. Two of the validation steps (cross-validation and Bayesian Information Criterion (BIC) process the fit residuals, while the four others look for solutions that provide the best agreement with independently observed focal mechanisms. Tests on synthetic catalogs allow us to qualify the performance of the fitting method and of the various validation procedures. The ACLUD method is able to provide solutions that are close to the expected ones, especially for the BIC and focal mechanismbased techniques. The clustering method complemented by the validation step based on focal mechanisms provides good solutions even in the presence of a significant spatial background seismicity rate. Our new fault reconstruction method is then applied to the Landers area in Southern California and compared with previous clustering methods. The results stress the importance of taking into account undersampled sub-fault structures as well as of the spatially inhomogeneous location uncertainties., Comment: 36 pages, 17 figures

Mechanical and sliding instabilities are the two processes which may lead to breaking off events of large ice masses. Mechanical instabilities mainly affect unbalanced cold hanging glaciers. For the latter case, a prediction could be achieved based on data of surface velocities and seismic activity. The case of sliding instabilities is more problematic. This phenomenon occurs on temperate glacier tongues. Such instabilities are strongly affected by the subglacial hydrology: melt water may cause (i) a lubrication of the bed and (ii) a decrease of the effective pressure and consequently a decrease of basal friction. Available data from Allalingletscher (Valais) indicate that the glacier tongue experienced an active phase during 2-3 weeks with enhanced basal motion in late summer in most years. In order to scrutinize in more detail the processes governing the sliding instabilities, a numerical model developed to investigate gravitational instabilities in heterogeneous media was applied to Allalingletscher. This model enables to account for various geometric configurations, interaction between sliding and tension cracking and water flow at the bedrock. We could show that both a critical geometrical configuration of the glacier tongue and the existence of a distributed drainage network were the main causes of this catastrophic break-off. Moreover, the analysis of the modeling results diagnose the phenomenon of recoupling of the glacier to its bed as a potential new precursory sign announcing the final break-off. This model casts a gleam of hope for a better understanding of the ultimate rupture of such glacier sliding instabilities., Comment: 34 pages, 22 figures

We report new tests on the Taiwan earthquake catalog of the prediction by the Multifractal Stress Activation (MSA) model that the p-value of the Omori law for the rate of aftershocks following a mainshock is an increasing function of its magnitude Mm. This effort is motivated by the quest to validate this crucial prediction of the MSA model and to investigate its possible dependence on local tectonic conditions. With careful attention to the long-term as well as short-term time-dependent magnitude completeness of the Taiwan catalog, and with the use of three different declustering techniques, we confirm the universality of the prediction p(Mm) = (0.09 \pm 0.03) \times Mm + (0.47 \pm 0.10), valid for the SCEC Southern California catalog, the Harvard-CMT worldwide catalog, the JMA Japan catalog and the Taiwan catalog. The observed deviations of the two coefficients of the p(Mm) linear dependence on Mm from catalog to catalog are not significant enough to correlate meaningfully with any tectonic features., Comment: This paper has been withdrawn after a request from the Editor of the published version. You can now find it in BSSA (Bulletin of Seismological Society of America)

The Altels hanging glacier broke off on September 11, 1895. The ice volume of this catastrophic rupture was estimated at $\rm 4.10^6$ cubic meters and is the largest ever observed ice fall event in the Alps. The causes of this collapse are however not entirely clear. Based on previous studies, we reanalyzed this break-off event, with the help of a new numerical model, initially developed by Faillettaz and others (2010) for gravity-driven instabilities. The simulations indicate that a break-off event is only possible when the basal friction at the bedrock is reduced in a restricted area, possibly induced by the storage of infiltrated water within the glacier. Moreover, our simulations reveal a two-step behavior: (i) A first quiescent phase, without visible changes, with a duration depending on the rate of basal changes; (ii) An active phase with a rapid increase of basal motion over a few days. The general lesson obtained from the comparison between the simulations and the available evidence is that visible signs of the destabilization process of a hanging glacier, resulting from a progressive warming of the ice/bed interface towards a temperate regime, will appear just a few days prior to the collapse., Comment: 27 pages, 18 figures

A hanging glacier at the east face of Weisshorn (Switzerland) broke off in 2005. We were able to monitor and measure surface motion and icequake activity for 25 days up to three days prior to the break-off. The analysis of seismic waves generated by the glacier during the rupture maturation process revealed four types of precursory signals of the imminent catastrophic rupture: (i) an increase in seismic activity within the glacier, (ii) a decrease in the waiting time between two successive icequakes, (iii) a change in the size-frequency distribution of icequake energy, and (iv) a modification in the structure of the waiting time distributions between two successive icequakes. Morevover, it was possible to demonstrate the existence of a correlation between the seismic activity and the log-periodic oscillations of the surface velocities superimposed on the global acceleration of the glacier during the rupture maturation. Analysis of the seismic activity led us to the identification of two regimes: a stable phase with diffuse damage, and an unstable and dangerous phase characterized by a hierarchical cascade of rupture instabilities where large icequakes are triggered., Comment: 16 pages, 7 figures

This chapter first presents a rather personal view of some different aspects of predictability, going in crescendo from simple linear systems to high-dimensional nonlinear systems with stochastic forcing, which exhibit emergent properties such as phase transitions and regime shifts. Then, a detailed correspondence between the phenomenology of earthquakes, financial crashes and epileptic seizures is offered. The presented statistical evidence provides the substance of a general phase diagram for understanding the many facets of the spatio-temporal organization of these systems. A key insight is to organize the evidence and mechanisms in terms of two summarizing measures: (i) amplitude of disorder or heterogeneity in the system and (ii) level of coupling or interaction strength among the system's components. On the basis of the recently identified remarkable correspondence between earthquakes and seizures, we present detailed information on a class of stochastic point processes that has been found to be particularly powerful in describing earthquake phenomenology and which, we think, has a promising future in epileptology. The so-called self-exciting Hawkes point processes capture parsimoniously the idea that events can trigger other events, and their cascades of interactions and mutual influence are essential to understand the behavior of these systems., Comment: 44 pages with 16 figures and 167 references, chapter in "Epilepsy: The Intersection of Neurosciences, Mathematics, and Engineering",Taylor & Francis Group, Ivan Osorio, Mark G. Frei, Hitten Zaveri, Susan Arthurs, eds (2010)

We present a new method of data clustering applied to earthquake catalogs, with the goal of reconstructing the seismically active part of fault networks. We first use an original method to separate clustered events from uncorrelated seismicity using the distribution of volumes of tetrahedra defined by closest neighbor events in the original and randomized seismic catalogs. The spatial disorder of the complex geometry of fault networks is then taken into account by defining faults as probabilistic anisotropic kernels, whose structures are motivated by properties of discontinuous tectonic deformation and previous empirical observations of the geometry of faults and of earthquake clusters at many spatial and temporal scales. Combining this a priori knowledge with information theoretical arguments, we propose the Gaussian mixture approach implemented in an Expectation-Maximization (EM) procedure. A cross-validation scheme is then used and allows the determination of the number of kernels that should be used to provide an optimal data clustering of the catalog. This three-steps approach is applied to a high quality relocated catalog of the seismicity following the 1986 Mount Lewis ($M_l=5.7$) event in California and reveals that events cluster along planar patches of about 2 km$^2$, i.e. comparable to the size of the main event. The finite thickness of those clusters (about 290 m) suggests that events do not occur on well-defined euclidean fault core surfaces, but rather that the damage zone surrounding faults may be seismically active at depth. Finally, we propose a connection between our methodology and multi-scale spatial analysis, based on the derivation of spatial fractal dimension of about 1.8 for the set of hypocenters in the Mnt Lewis area, consistent with recent observations on relocated catalogs.

We model the progressive maturation of a heterogeneous mass towards a gravity-driven instability, characterized by the competition between frictional sliding and tension cracking, using array of slider blocks on an inclined basal surface, which interact via elastic-brittle springs. A realistic state- and rate-dependent friction law describes the block-surface interaction. The inner material damage occurs via stress corrosion. Three regimes, controlling the mass instability and its precursory behavior, are classified as a function of the ratio $T_c/T_f$ of two characteristic time scales associated with internal damage/creep and with frictional sliding. For $T_c/T_f \gg 1$, the whole mass undergoes a series of internal stick and slip events, associated with an initial slow average downward motion of the whole mass, and progressively accelerates until a global coherent runaway is observed. For $T_c/T_f \ll 1$, creep/damage occurs sufficiently fast compared with nucleation of sliding, causing bonds to break, and the bottom part of the mass undergoes a fragmentation process with the creation of a heterogeneous population of sliding blocks. For the intermediate regime $T_c/T_f \sim 1$, a macroscopic crack nucleates and propagates along the location of the largest curvature associated with the change of slope from the stable frictional state in the upper part to the unstable frictional sliding state in the lower part. The other important parameter is the Young modulus $Y$ which controls the correlation length of displacements in the system., Comment: 40 pages, 13 figures

The concept of universality proposes that dynamical systems with the same power law behaviors are equivalent at large scales. We test this hypothesis on the Earth's crust and the epileptic brain, and discover that power laws also govern the distributions of seizure energies and recurrence times. This robust correspondence is extended over seven statistics, including the direct and inverse Omori laws. We also verify in an animal seizure model the earthquake-driven hypothesis that power law statistics co-exist with characteristic scales, as coupling between constitutive elements increases towards the synchronization regime. These observations point to the universality of the dynamics of coupled threshold oscillators for systems even as diverse as Earth and brain and suggest a general strategy for forecasting seizures, one of neurosciences' grails., Comment: 21 pages including 6 figures

Validation is often defined as the process of determining the degree to which a model is an accurate representation of the real world from the perspective of its intended uses. Validation is crucial as industries and governments depend increasingly on predictions by computer models to justify their decisions. In this article, we survey the model validation literature and propose to formulate validation as an iterative construction process that mimics the process occurring implicitly in the minds of scientists. We thus offer a formal representation of the progressive build-up of trust in the model, and thereby replace incapacitating claims on the impossibility of validating a given model by an adaptive process of constructive approximation. This approach is better adapted to the fuzzy, coarse-grained nature of validation. Our procedure factors in the degree of redundancy versus novelty of the experiments used for validation as well as the degree to which the model predicts the observations. We illustrate the new methodology first with the maturation of Quantum Mechanics as the arguably best established physics theory and then with several concrete examples drawn from some of our primary scientific interests: a cellular automaton model for earthquakes, an anomalous diffusion model for solar radiation transport in the cloudy atmosphere, and a computational fluid dynamics code for the Richtmyer-Meshkov instability. This article is an augmented version of Sornette et al. [2007] that appeared in Proceedings of the National Academy of Sciences in 2007 (doi: 10.1073/pnas.0611677104), with an electronic supplement at URL http://www.pnas.org/cgi/content/full/0611677104/DC1. Sornette et al. [2007] is also available in preprint form at physics/0511219., Comment: 50 pages, 5 figures, be published in the Lecture Notes in Computational Science and Engineering, Vol. TBD), Proceedings of in Computational Methods in Transport, Granlibakken 2006, F. Graziani and D. Swesty (Eds.), Springer-Verlag, New York (NY), 2007

We propose a new pattern recognition method that is able to reconstruct the 3D structure of the active part of a fault network using the spatial location of earthquakes. The method is a generalization of the so-called dynamic clustering method, that originally partitions a set of datapoints into clusters, using a global minimization criterion over the spatial inertia of those clusters. The new method improves on it by taking into account the full spatial inertia tensor of each cluster, in order to partition the dataset into fault-like, anisotropic clusters. Given a catalog of seismic events, the output is the optimal set of plane segments that fits the spatial structure of the data. Each plane segment is fully characterized by its location, size and orientation. The main tunable parameter is the accuracy of the earthquake localizations, which fixes the resolution, i.e. the residual variance of the fit. The resolution determines the number of fault segments needed to describe the earthquake catalog, the better the resolution, the finer the structure of the reconstructed fault segments. The algorithm reconstructs successfully the fault segments of synthetic earthquake catalogs. Applied to the real catalog constituted of a subset of the aftershocks sequence of the 28th June 1992 Landers earthquake in Southern California, the reconstructed plane segments fully agree with faults already known on geological maps, or with blind faults that appear quite obvious on longer-term catalogs. Future improvements of the method are discussed, as well as its potential use in the multi-scale study of the inner structure of fault zones.

Mega et al.(cond-mat/0212529) proposed to use the ``diffusion entropy'' (DE) method to demonstrate that the distribution of time intervals between a large earthquake (the mainshock of a given seismic sequence) and the next one does not obey Poisson statistics. We have performed synthetic tests which show that the DE is unable to detect correlations between clusters, thus negating the claimed possibility of detecting an intercluster correlation. We also show that the LR model, proposed by Mega et al. to reproduce inter-cluster correlation, is insufficient to account for the correlation observed in the data., Comment: Comment on Mega et al., Phys. Rev. Lett. 90. 188501 (2003) (cond-mat/0212529)

The empirical Bath's law states that the average difference in magnitude between a mainshock and its largest aftershock is 1.2, regardless of the mainshock magnitude. Following Vere-Jones [1969] and Console et al. [2003], we show that the origin of Bath's law is to be found in the selection procedure used to define mainshocks and aftershocks rather than in any difference in the mechanisms controlling the magnitude of the mainshock and of the aftershocks. We use the ETAS model of seismicity, which provides a more realistic model of aftershocks, based on (i) a universal Gutenberg-Richter (GR) law for all earthquakes, and on (ii) the increase of the number of aftershocks with the mainshock magnitude. Using numerical simulations of the ETAS model, we show that this model is in good agreement with Bath's law in a certain range of the model parameters., Comment: major revisions, in press in Geophys. Res. Lett

We analyze 21 aftershock sequences of California to test for evidence of space-time diffusion. Aftershock diffusion may result from stress diffusion and is also predicted by any mechanism of stress weakening. Here, we test an alternative mechanism to explain aftershock diffusion, based on multiple cascades of triggering. In order to characterize aftershock diffusion, we develop two methods, one based on a suitable time and space windowing, the other using a wavelet transform adapted to the removal of background seismicity. Both methods confirm that diffusion of seismic activity is very weak, much weaker than reported in previous studies. A possible mechanism explaining the weakness of observed diffusion is the effect of geometry, including the localization of aftershocks on a fractal fault network and the impact of extended rupture lengths which control the typical distances of interaction between earthquakes., Comment: latex file of 34 pages, 15 postscript figures, minor revision. In press in J. Geophys. Res

Using the simple ETAS branching model of seismicity, which assumes that each earthquake can trigger other earthquakes, we quantify the role played by the cascade of triggered seismicity in controlling the rate of aftershock decay as well as the overall level of seismicity in the presence of a constant external seismicity source. We show that, in this model, the fraction of earthquakes in the population that are aftershocks is equal to the fraction of aftershocks that are indirectly triggered and is given by the average number of triggered events per earthquake. Previous observations that a significant fraction of earthquakes are triggered earthquakes therefore imply that most aftershocks are indirectly triggered by the mainshock., Comment: Latex document of 17 pages + 2 postscript figures

The observation of foreshocks preceding large earthquakes and the suggestion that foreshocks have specific properties that may be used to distinguish them from other earthquakes have raised the hope that large earthquakes may be predictable. Among proposed anomalous properties are the larger proportion than normal of large versus small foreshocks, the power law acceleration of seismicity rate as a function of time to the mainshock and the spatial migration of foreshocks toward the mainshock, when averaging over many sequences. Using Southern California seismicity, we show that these properties and others arise naturally from the simple model that any earthquake may trigger other earthquakes, without arbitrary distinction between foreshocks, aftershocks and mainshocks. We find that foreshocks precursory properties are independent of the mainshock size. This implies that earthquakes (large or small) are predictable to the same degree as seismicity rate is predictable from past seismicity by taking into account cascades of triggering. The cascades of triggering give rise naturally to long-range and long-time interactions, which can explain the observations of correlations in seismicity over surprisingly large length scales., Comment: 20 pages, 6 figures, minor revision, in press in J. Geophys. Res