We present a method of constructing low-dimensional nonlinear models describing the main dynamical features of a discrete 2D cellular fault zone, with many degrees of freedom, embedded in a 3D elastic solid. A given fault system is characterized by a set of parameters that describe the dynamics, rheology, property disorder, and fault geometry. Depending on the location in the system parameter space we show that the coarse dynamics of the fault can be confined to an attractor whose dimension is significantly smaller than the space in which the dynamics takes place. Our strategy of system reduction is to search for a few coherent structures that dominate the dynamics and to capture the interaction between these coherent structures. The identification of the basic interacting structures is obtained by applying the Proper Orthogonal Decomposition (POD) to the surface deformations fields that accompany strike-slip faulting accumulated over equal time intervals. We use a feed-forward artificial neural network (ANN) architecture for the identification of the system dynamics projected onto the subspace (model space) spanned by the most energetic coherent structures. The ANN is trained using a standard back-propagation algorithm to predict (map) the values of the observed model state at a future time given the observed model state at the present time. This ANN provides an approximate, large scale, dynamical model for the fault.
A common use of Markov Chains is the simulation of the seismic cycle in a fault, i.e. as a renewal model for the repetition of its characteristic earthquakes. This representation is consistent with Reid's elastic rebound theory. Here it is proved that in {\it any} one-way Markov cycle, the aperiodicity of the corresponding distribution of cycle lengths is always lower than one. This fact concurs with observations of large earthquakes in faults all over the world.
We discuss various methods used to investigate the geometric complexity of earthquakes and earthquake faults, based both on a point-source representation and the study of interrelations between earthquake focal mechanisms. We briefly review the seismic moment tensor formalism and discuss in some detail the representation of double-couple (DC) earthquake sources by normalized quaternions. Non-DC earthquake sources like the CLVD focal mechanism are also considered. We obtain the characterization of the earthquake complex source caused by summation of disoriented DC sources. We show that commonly…
Numerical models are starting to be used for determining the future behaviour of seismic faults and fault networks. Their final goal would be to forecast future large earthquakes. In order to use them for this task, it is necessary to synchronize each model with the current status of the actual fault or fault network it simulates (just as, for example, meteorologists synchronize their models with the atmosphere by incorporating current atmospheric data in them). However, lithospheric dynamics is largely unobservable: important parameters cannot (or can rarely) be measured in Nature. Earthquake…
A prominent feature of earthquakes is their empirical laws including memory (clustering) in time and space. Several earthquake forecasting models, like the EpidemicType Aftershock Sequence (ETAS) model, were developed based on earthquake empirical laws. Yet, a recent study showed that the ETAS model fails in reproducing significant long-term memory characteristics found in real earthquake catalogs. Here we modify and generalize the ETAS model to include short- and long-term triggering mechanisms, to account for the short- and long-time memory (exponents) recently discovered in the data. Our ge…
The burgeoning need to sequester anthropogenic CO$_2$ for climate mitigation and the need for energy sustenance leading upto enhanced geothermal energy production has made it incredibly critical to study potential earthquakes due to fluid activity in the subsurface. These earthquakes result from reactivation of faults in the subsurface due to pore pressure perturbations. In this work, we provide a framework to model fault slip due to pore pressure change leading upto quantifying the earthquake magnitude.
In meteorology, engineering and computer sciences, data assimilation is routinely employed as the optimal way to combine noisy observations with prior model information for obtaining better estimates of a state, and thus better forecasts, than can be achieved by ignoring data uncertainties. Earthquake forecasting, too, suffers from measurement errors and partial model information and may thus gain significantly from data assimilation. We present perhaps the first fully implementable data assimilation method for earthquake forecasts generated by a point-process model of seismicity. We test the …
We present an axiomatic approach to earthquake forecasting in terms of multi-component random fields on a lattice. This approach provides a method for constructing point estimates and confidence intervals for conditional probabilities of strong earthquakes under conditions on the levels of precursors. Also, it provides an approach for setting multilevel alarm system and hypothesis testing for binary alarms. We use a method of comparison for different earthquake forecasts in terms of the increase of Shannon information. 'Forecasting' and 'prediction' of earthquakes are equivalent in this approa…
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.…
We illustrate how a simple statistical model can describe the quasiperiodic occurrence of large earthquakes. The model idealizes the loading of elastic energy in a seismic fault by the stochastic filling of a box. The emptying of the box after it is full is analogous to the generation of a large earthquake in which the fault relaxes after having been loaded to its failure threshold. The duration of the filling process is analogous to the seismic cycle, the time interval between two successive large earthquakes in a particular fault. The simplicity of the model enables us to derive the statisti…
Earthquake ruptures dynamically activate coseismic off-fault damage around fault cores. Systematic field observation efforts have shown the distribution of off-fault damage around main faults, while numerical modeling using elastic-plastic off-fault material models has demonstrated the evolution of coseismic off-fault damage during earthquake ruptures. Laboratory scale micro-earthquake experiments have pointed out the enhanced high-frequency radiation due to the coseismic off-fault damage. However, the detailed off-fault fracturing mechanisms, subsequent radiation and its contribution to the o…
Quantifying the Role of 3D Fault Geometry Complexities on Slow and Fast Earthquakes
Laboratoire de Géologie, Ecole Normale Superieure, CNRS-UMR 8538, PSL Research University（巴黎高等师范学院地质实验室）；Geo-Energy Lab - Gaznat Chair on Geo-Energy, Swiss Federal Institute of Technology in Lausanne, EPFL-ENAC-IIC-GEL（洛桑联邦理工学院能源地质实验室）；Division of Geological and Planetary Sciences, California Institute of Technology（加州理工学院地质与行星科学系）
Active faults release elastic strain energy via a whole continuum of modes of slip, ranging from devastating earthquakes to Slow Slip Events and persistent creep. Understanding the mechanisms controlling the occurrence of rapid, dynamic slip radiating seismic waves (i.e. earthquakes) or slow, silent slip (i.e. SSEs) is a fundamental point in the estimation of seismic hazard along subduction zones. On top of showing slower rupture propagation velocity than earthquakes, SSEs exhibit different scaling relationships, which could reflect either different physical mechanisms or an intriguing lack of…
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, t…
Fault-zone fluids control effective normal stress and fault strength. While most earthquake models assume a fixed pore fluid pressure distribution, geologists have documented fault valving behavior, that is, cyclic changes in pressure and unsteady fluid migration along faults. Here we quantify fault valving through 2-D antiplane shear simulations of earthquake sequences on a strike-slip fault with rate-and-state friction, upward Darcy flow along a permeable fault zone, and permeability evolution. Fluid overpressure develops during the interseismic period, when healing/sealing reduces fault per…
Recent studies in the literature have introduced a new approach to earthquake forecasting based on representing the space-time patterns of localized seismicity by a time-dependent system state vector in a real-valued Hilbert space and deducing information about future space-time fluctuations from the phase angle of the state vector. While the success rate of this Pattern Informatics (PI) method has been encouraging, the method is still in its infancy. Procedural analysis, statistical testing, parameter sensitivity investigation and optimization all still need to be performed. In this paper, we…
The dynamic energy balance is essential for earthquake studies. The energy balance approach is one of the most famous developments in fracture mechanics. To interpret seismological data, crack models and sliding on a frictional surface (fault) models are widely used. The macroscopically observable energy budget and the microscopic processes can be related through the fracture energy $G_c$. The fault surface morphology is the direct result of the microscopic processes near the crack tip or on the frictional interface. Here we show that the dynamic energy balance in earthquakes can be expressed …
Anticipating the maximum magnitude of injection-triggered earthquakes is highly valuable for the safe and efficient exploitation of geoenergies. The recent work by Li et al.(2021) reached the conclusion that unexpectedly large injection-triggered earthquakes are primarily caused by large pre-existing critical shear stresses on seismogenic faults. Also of great interest is the proposal of the ratio of fault slip to dilation as an index to anticipate the fault rupture energetics. However, their fluid injection experiments were conducted under fully drained conditions, where the fluid pressure di…
Earthquakes induced during hydraulic fracturing operations have occurred in a number of locales. However, in-situ studies aimed to discern the triggering mechanism remains exclusively statistical in their nature. Here, we calculate the fault slip-tendencies of eleven hydraulic fracturing induced earthquakes in a historically aseismic area using a recently constructed quantitative model for in-situ stresses. It is shown that the ambient pore pressures of the nearby Duvernay unconventional reservoirs can provide enough Pf triggering fault movement. The local fluid pressures acting on the fault c…
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 f…
When a dynamic earthquake rupture propagates on a fault in the Earth's crust, the medium around the fault is dynamically damaged due to stress concentrations around the rupture tip. Recent field observations, laboratory experiments and canonical numerical models show the coseismic off-fault damage is essential to describe the coseismic off-fault deformation, rupture dynamics, radiation and overall energy budget. However, the numerical modeling of "localized" off-fault fractures remains a challenge mainly because of computational limitations and model formulation shortcomings. We thus developed…
Machine learning regression can predict macroscopic fault properties such as shear stress, friction, and time to failure using continuous records of fault zone acoustic emissions. Here we show that a similar approach is successful using event catalogs derived from the continuous data. Our methods are applicable to catalogs of arbitrary scale and magnitude of completeness. We investigate how machine learning regression from an event catalog of laboratory earthquakes performs as a function of the catalog magnitude of completeness. We find that strong model performance requires a sufficiently low…
Past severe earthquakes, such as Bam earthquake of 2003 and Tabas earthquake of 1978, have demonstrated that many cities in Iran are prone to be struck by near-fault earthquakes. Such earthquakes are impulsive in nature, and therefore, they are more destructive than the ordinary ground shaking. In the fourth edition of Iranian seismic code (Standard No. 2800), some changes, including a modification factor for the elastic acceleration response spectrum (EARS) have been recently recommended to reflect the effects of such probable near-fault earthquakes for the designing procedure. In this study,…
Following N.Kozyrev's idea about the influence of the gravitational fields of the Sun and the Moon on the Earth's crust, we consider a low-frequency resonance of the Earth's crust blocks is happening before the occurrence of the earthquake. The resonance affects several geophysical parameters and is associated with many short-term precursory phenomena; the most notable are the release of underground gases (radon), the air temperature rises, change of thermal radiation in the troposphere, and an increase in the electron density in the ionosphere. During the final stage of the earthquake genesis…
One of the main interests in seismology is the formulation of models able to describe the clustering in time occurrence of earthquakes. Analysis of the Southern California Catalog shows magnitude clustering in correspondence to temporal clustering. Here we propose a dynamical scaling hypothesis in which time is rescaled in terms of magnitude. This hypothesis is introduced in the context of a generalized trigger model and gives account for clustering in time and magnitude for earthquake occurrence. The model is able to generate a synthetic catalog reproducing magnitude and inter-even time distr…
Recent studies have shown that real-valued principal component analysis can be applied to earthquake fault systems for forecasting and prediction. In addition, theoretical analysis indicates that earthquake stresses may obey a wave-like equation, having solutions with inverse frequencies for a given fault similar to those that characterize the time intervals between the largest events on the fault. It is therefore desirable to apply complex principal component analysis to develop earthquake forecast algorithms. In this paper we modify the Pattern Informatics method of earthquake forecasting to…
A major goal in earthquake physics is to derive a constitutive framework for fault slip that captures the dependence of shear strength on fault rheology, sliding velocity, and pore-fluid pressure. In this study, we present H-MEC (Hydro-Mechanical Earthquake Cycles), a newly-developed two-phase flow numerical code - which couples solid rock deformation and pervasive fluid flow - to simulate how crustal stress and fluid pressure evolve during the earthquake cycle on a fluid-bearing fault structure. This unified, continuum-based model, incorporates a staggered finite difference-marker-in-cell (SF…
The subsequent series of responses to big events may exhibit a synchronicity of event number, frequency and energy release in different fault zones. This synchronicity is a reliable source for probing non-intuitive geological structures, assessing regional seismicity hazard map and even predicting the next big events. The synchronicity of main faults in the eastern margin of the Qinghai-Tibetan Plateau is still unknown to us. We propose to examine the correlation of earthquake occurrence among different fault zones to indicate this synchronicity, and to obtain a preliminary understanding of ge…
Large continental earthquakes activate multiple faults in a complex fault system, dynamically inducing co-seismic damage around them. The 2016 Mw 7.8 Kaikoura earthquake in the northern South Island of New Zealand has been reported as one of the most complex continental earthquakes ever documented1, which resulted in a distinctive on and off-fault deformation pattern. Previous geophysical studies confirm that the rupture globally propagated northward from epicenter. However, the exact rupture- propagation path is still not well understood because of the geometrical complexity, partly at sea, a…
Pattern informatics (PI) technique can be used to detect precursory seismic activation or quiescence and make earthquake forecast. Here we apply the PI method for optimal forecasting of large earthquakes in Japan, using the data catalogue maintained by the Japan Meteorological Agency. The PI method is tested to forecast large (magnitude m >= 5) earthquakes for the time period 1995-2004 in the Kobe region. Visual inspection and statistical testing show that the optimized PI method has forecasting skill, relative to the seismic intensity data often used as a standard null hypothesis. Moreover, w…
One of the most suitable methods for modeling fully dynamic earthquake cycle simulations is the spectral boundary integral element method (sBIEM), which takes advantage of the fast Fourier transform (FFT) to make a complex numerical dynamic rupture tractable. However, this method has the serious drawback of requiring a flat fault geometry due to the FFT approach. Here we present an analytical formulation that extends the sBIEM to a mildly non-planar fault. We start from a regularized boundary element method and apply a small-slope approximation of the fault geometry. Making this assumption, it…
We systematically study effects of external perturbations on models describing earthquake fault dynamics. The latter are based on the framework of the Burridge-Knopoff spring-block system, including the cases of a simple mono-block fault, as well as the paradigmatic complex faults made up of two identical or distinct blocks. The blocks exhibit relaxation oscillations, which are representative for the stick-slip behavior typical for earthquake dynamics. Our analysis is carried out by determining the phase response curves of first and second order. For a mono-block fault, we consider the impact …
On 15 May 2020 an MWW 6.5 earthquake occurred beneath the Monte Cristo Range in the Mina Deflection region of western Nevada. Rapid deployment of eight temporary seismic stations enables detailed analysis of its productive and slowly decaying aftershock sequence (p=0.8) which included ~18,000 autodetected events in 3.5 months. Double-difference, waveform-based relative relocation of 16,714 earthquakes reveals a complex network of faults, many of which cross the inferred 35-km long east-northeast-striking, left-lateral mainshock rupture. Seismicity aligns with left-lateral, right-lateral, and n…
Forecasting fault failure is a fundamental but elusive goal in earthquake science. Here we show that by listening to the acoustic signal emitted by a laboratory fault, machine learning can predict the time remaining before it fails with great accuracy. These predictions are based solely on the instantaneous physical characteristics of the acoustical signal, and do not make use of its history. Surprisingly, machine learning identifies a signal emitted from the fault zone previously thought to be low-amplitude noise that enables failure forecasting throughout the laboratory quake cycle. We hypot…
Aftershocks of the 2012 Off-Coast of Sumatra Earthquake Sequence exhibit a complex and diffuse spatial distribution. The first-order complexity in aftershock distribution is clear and well beyond the influence of typical earthquake location uncertainty. The sequence included rupture of multiple fault segments, spatially separated. We use surface-wave based relative centroid locations to examine whether, at the small scale, the distribution of the aftershocks was influenced by location errors. Surface-wave based relative location has delineated precise oceanic transform fault earthquake locatio…
Natural earthquake fault systems are highly non-homogeneous. The inhomogeneities occur be- cause the earth is made of a variety of materials which hold and dissipate stress differently. In this work, we study scaling in earthquake fault models which are variations of the Olami-Feder- Christensen (OFC) and Rundle-Jackson-Brown (RJB) models. We use the scaling to explore the effect of spatial inhomogeneities due to damage and inhomogeneous stress dissipation in the earthquake-fault-like systems when the stress transfer range is long, but not necessarily longer than the length scale associated wi…
In this study, we propose an analysis of the earthquake clusters that occurred in North-Eastern Italy and western Slovenia from 1977 to today. Given a mainshock generating alarm in the population, we are interested in forecasting if a similar magnitude earthquake will follow. We classify the earthquake clusters associated with mainshocks of magnitude Mm into two classes: if the strongest aftershock has a magnitude >=Mm-1 (swarms or large aftershock seismic sequences) as type A, otherwise (smaller aftershocks seismic sequences) as type B. A large aftershock following a main shock can cause sign…
The inhomogeneity of the materials with different physical properties in the Earth is responsible for a wide variety of spatial and temporal behaviors. In this work, we study an earthquake fault model based on Olami-Feder-Christensen (OFC) and Rundle-Jackson-Brown (RJB) cellular automata models with particular aspects of spatial heterogeneities and long-range stress interactions. In our model some localized stress accumulators are added into the system by converting a percentage of randomly selected sites into stronger sites which are called asperity cells. These asperity cells support much hi…
How to quantify earthquake predictability? Advances in earthquake forecasting and predictability limits
The Institute of Statistical Mathematics（统计数学研究所）；the Graduate University for Adavanced Studies (SOKENDai)（advanced studies 研究生院（SOKENDai））；Institute of Risk Analysis, Prediction and Management, Southern University of Science and Technology（南方科技大学风险分析、预测与管理研究院）
We propose two methods to calibrate the parameters of the epidemic-type aftershock sequence (ETAS) model based on expectation maximization (EM) while accounting for temporal variation of catalog completeness. The first method allows for model calibration on long-term earthquake catalogs with temporal variation of the completeness magnitude, $m_c$. This calibration technique is beneficial for long-term probabilistic seismic hazard assessment (PSHA), which is often based on a mixture of instrumental and historical catalogs. The second method generalizes the concept of $m_c$, considering rate- an…
The North Anatolian Fault (NAF) has produced numerous major earthquakes. After decades of quiescence, the $\rm{M_w}$ 6.8 Elazığ earthquake (January 24, 2020) has recently reminded us that the East Anatolian Fault (EAF) is also capable of producing significant earthquakes. To better estimate the seismic hazard associated with these two faults, we jointly invert Interferometric Synthetic Aperture Radar (InSAR) and GPS data to image the spatial distribution of interseismic coupling along the eastern part of both the North and East Anatolian Faults. We perform the inversion in a Bayesian framework…
Imaging the anelastic deformation within the crust and lithosphere using surface geophysical data remains a significant challenge in part due to the wide range of physical processes operating at different depths and to various levels of localization that they embody. Models of Earth's elastic properties from seismological imaging combined with geodetic modeling may form the basis of comprehensive rheological models of Earth's interior. However, representing the structural complexity of faults and shear zones in numerical models of deformation still constitutes a major difficulty. Here, we pres…
Based on waveform data from a profile of aftershocks following the north-south trace of the June 28, 1992 Landers rupture across the Mojave desert, we construct a new velocity model for the Mojave region which features a thin, slow crust. Using this model, we obtain source parameters, including depth and duration, for each of the aftershocks in the profile, and in addition, any significant (M>3.7) Joshua Tree--Landers aftershock between April, 1992 and October, 1994 for which coherent TERRAscope data were available. In all, we determine source parameters and stress-drops for 45 significant (M_…
During an earthquake, part of the released elastic strain energy is dissipated within the slip zone by frictional and fracturing processes, the rest being radiated away via elastic waves. Frictional heating thus plays a crucial role in the energy budget of earthquakes, but, to date, it cannot be resolved by seismological data. Here we investigate the dynamics of laboratory earthquakes by measuring frictional heat dissipated during the propagation of shear instabilities at typical seismogenic depth stress conditions. We perform, for the first time, the full energy budget of earthquake rupture a…
A major part of the seismicity striking the Mediterranean area and other regions worldwide is hosted in carbonate rocks. Recent examples are the destructive earthquakes of L'Aquila Mw 6.1 2009 and Norcia Mw 6.5 2016 in Central Italy. Surprisingly, within this region, fast (\approx 3km/s) and destructive seismic ruptures coexist with slow (maximum 10 m/s) and non-destructive rupture phenomena. Despite of its relevance for seismic hazard studies, the transitions from fault creep to slow and fast seismic rupture propagation are still poorly constrained by seismological and laboratory observations…
