Three fundamental laws of the physics of earthquakes, bearing the names of their discoverers Omori, Gutenberg, Richter and Bath, are widely used in original, review, monographic, and encyclopedic literature. In this paper, we have tried to look at laws from an unusual point of view. The idea behind our approach is as follows. Each of the three laws was established in the course of work on solving a specific research problem, and the result was expressed in a specific mathematical form. We asked ourselves whether it is possible to modify the statement of the problem, and/or modify the form of expression of the law in order to see new facets of a well-known scientific statement? The result was successful in relation to Omori's law, according to which the frequency of aftershocks hyperbolically decreases over time. The modified formulation of the problem was to find a differential equation describing the time evolution of aftershocks. We found a simple differential model, within the framework of which the concept of deactivation of the earthquake source is introduced. The question of dependence of the deactivation factor on the magnitude of main shock is analyzed theoretically and experimentally. A monotonic decrease in the deactivation factor with an increase in the magnitude of the main shock has been reliably established. Other possible applications of the proposed methodological technique are considered. In particular, it is indicated how to overcome the contradiction between the Omori law and the law of energy conservation. Keywords: Omori law, Gutenberg-Richter law, Bath law, main shock, aftershock, deactivation factor, Omori epoch, differential equation of evolution.
The earthquake source after the main shock can theoretically be represented as a black box without an entrance. At the output, there is a signal in the form of aftershocks, the frequency of which decreases on average with time according to the Omori law. The task of the researcher is to evaluate the structure, state and mechanism of functioning of a dynamic system simulating the earthquake source based on the output signal. In this paper, we outline an approach to a partial solution of this general problem. Omori's law is presented as a differential equation of aftershock evolution. An inverse…
Strong deep earthquakes occurred in the Sea of Okhotsk, May 24, 2013 at 05:44 UTC. The magnitude of the earthquake is M = 8.3, the depth of the hypocenter is 629 km. The earthquake was accompanied by the deep-focus aftershocks. The paper is devoted to the study of the space-time properties of aftershocks. An estimate is made of the deactivation coefficient of the earthquake source, which "cools down" after the main shock. The modulation of aftershocks by toroidal oscillations of the Earth was discovered. The questions about the round-the-world seismic echo and the ribbed structure of the spati…
Recently, simple differential models of aftershocks have attracted increased attention of researchers at the Institute of Physics of the Earth, Russian Academy of Sciences for their remarkable properties. The shortened Bernoulli equation allows us to introduce the concept of the source deactivation factor and phenomenologically describe the nonstationarity of the geological environment in the earthquake source, "cooling down" after the main shock. The Verhulst logistic equation makes it possible to take into account the transition of the aftershock flow to the background seismicity mode. The K…
The paper is based on the report read by the author on October 24, 2018 at the meeting of the Scientific Council of the Institute of Earth Physics of the Russian Academy of Sciences. The report was dedicated to the 150th anniversary of the outstanding Japanese seismologist Fusakichi Omori. As is known, Omori established the first empirical law of the earthquakes physics, bearing his name. The Omori law states that the frequency of aftershocks on average decreases hyperbolically over the time. Three versions of Omori law are described briefly. The recent version allows to poses the inverse prob…
One hundred years ago, Fusakichi Omori died. Our paper is dedicated to his memory. Omori made an outstanding contribution to the physics of earthquakes. In 1894 he formulated the law of aftershock evolution. Omori's Law states that after the main shock of an earthquake, the frequency of aftershocks decreases hyperbolically with time. In this paper, we briefly describe one of the directions of modern aftershock research. Keywords: earthquake, mainshock, aftershock, evolution equation, deactivation coefficient, inverse problem, trigger, proper time.
The presented paper is devoted to the search for mathematical basis for describing the aftershock evolution of strong earthquakes. We consider the experimental facts and heuristic arguments that allow to make a choice and to focus on the nonlinear diffusion equation as the master equation. Analysis of the master equation indicates that, apparently, the selected mathematical basis makes it possible to simulate two important properties of the aftershock evolution known from the experiment. We are talking about the Omori law and the slow propagation of aftershocks from the epicenter of the main s…
Recently, the physics of aftershocks has been enriched by three new problems. We will conditionally call them dynamic, inverse, and morphological problems. Dynamic problem is to search for the cumulative effect of a round-the-world seismic echo appearing after the main shock of earthquake. According to the theory, a converging surface seismic wave excited by the main shock returns to the epicenter about 3 hours after the main shock and stimulates the excitation of a strong aftershock. The second problem is an adequate description of the averaged evolution of the aftershock flow. We introduced …
Omori Law describes the evolution of the aftershocks of a strong earthquake. Established at the end of the century before last, it is characterized by the beauty of its form, quite definite clarity, as a result of which it still attracts considerable attention of the geophysical community. In recent years, we have accumulated considerable experience in studying the Omori law by theoretical and experimental methods. This paper summarizes the results of our study briefly. The main attention is focused on the phenomenological theory of aftershocks, the foundations of which were laid by Fusakichi …
The idea of the global daily magnitude (GDM, unit designation is Mg) of earthquakes is introduced. The formula is given for calculating Mg from earthquake strength data, characterized by the classic magnitude M on the Richter scale. An example of the calculation of Mg is given. It is noted that the Mg index may be useful in studying a number of seismological problems.
Today, observations of earthquake precursors remain widely debated. While precursory slow slip is an important feature of earthquake nucleation, foreshock sequences are not always observed and their temporal evolution remains unconstrained. Here, we report on stick-slip experiments (laboratory earthquakes) conducted under seismogenic stresses in dry and fluid pressure conditions. We show that the precursory moment release scales with mainshock magnitude irrespective of the slip behavior (seismic or aseismic), the presence of fluid and the fault's slip history. Importantly, this observation is …
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…
In this paper, we used the Global Catalog of the National Earthquake Information Center US Geological Survey (NEIC USGS) for analysis of the magnitude-frequency distribution of earthquakes. We selected the unimodal part of the distribution and proposed an empirical formula that approximates this part in a wide range of magnitudes. The formula may be useful in studying the problems seismology, in which the expected effect is manifested in relatively weak earthquakes. The paper will be sent to the journal Geodynamics and Tectonophysics (ISSN 2078-502X). Keywords: planetary seismicity, Gutenberg-…
We estimate the relative importance of small and large earthquakes for static stress changes and for earthquake triggering, assuming that earthquakes are triggered by static stress changes and that earthquakes are located on a fractal network of dimension D. This model predicts that both the number of events triggered by an earthquake of magnitude m and the stress change induced by this earthquake at the location of other earthquakes increase with m as \~10^(Dm/2). The stronger the spatial clustering, the larger the influence of small earthquakes on stress changes at the location of a future e…
We consider three questions related to the 2011 Tohoku mega-earthquake: (1) Why was the event size so grossly under-estimated? (2) How should we evaluate the chances of giant earthquakes in subduction zones? and (3) What is the repeat time for magnitude 9 earthquakes off the Tohoku coast? The "maximum earthquake size" is often guessed from the available history of earthquakes, a method known for its significant downward bias. There are two quantitative methods for estimating the maximum magnitude in any region: a statistical analysis of the available earthquake record, and the moment conservat…
The $b$-value in the Gutenberg-Richter (GR) law contains information that is essential for evaluating earthquake hazard and predicting the occurrence of large earthquakes. Estimates of $b$ are often based on seismic events whose magnitude exceed a certain threshold, the so called magnitude of completeness. Such estimates are sensitive to the choice of threshold and often ignore a substantial portion of available data. We present a general model for the distribution of observable earthquake magnitudes and an estimation procedure that takes all measurements into account. The model is obtained by…
With the present study we introduce a fast and robust method to calculate the source displacement spectra of small earthquakes on a local to regional scale. The work is based on the publicly available Qopen method of full envelope inversion which is further tuned for the given purpose. Important source parameters -- seismic moment, moment magnitude, corner frequency and high-frequency fall-off -- are determined from the source spectra by fitting a simple earthquake source model. The method is demonstrated by means of a data set comprising the 2018 West Bohemia earthquake swarm. We report momen…
The physics of earthquake triggering together with simple assumptions of self-similarity impose the existence of a minimum magnitude m0 below which earthquakes do not trigger other earthquakes. Noting that the magnitude md of completeness of seismic catalogs has no reason to be the same as the magnitude m0 of the smallest triggering earthquake, we use quantitative fits and maximum likelihood inversions of observed aftershock sequences as well as Baath's law, compare with ETAS model predictions and thereby constrain the value of m0. We show that the branching ratio $n$ (average number of trigge…
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…
Calendar Time Local Earthquake Forecasts from Earthquake Nowcasts: A Do-It-Yourself (DIY) Ensemble Method
University of California, Davis（加州大学戴维斯分校）；Santa Fe Institute（圣塔菲研究所）；Jet Propulsion Laboratory（喷气推进实验室）；Purdue University（普渡大学）；University of California, Irvine（加州大学尔湾分校）；University of Virginia（弗吉尼亚大学）；Shizuoka University（静冈大学）
By using the M>=5 global earthquake data for Jan. 1950 to Dec. 2015, we performed statistical analyses for the parameters magnitude, time, and depth on a yearly scale. The magnitude spectrum, which is the earthquake number accumulated at different magnitudes, had an exponential distribution. For the first time, we report a very significant characteristic of the sinusoidal periodic variation in the spectral index. The cycle of the sine function fitting was 30.98 years. The concept of annual equivalent total magnitude (AETM) of total released energy for each year was introduced and the trend var…
In this short note, I comment on the research of Pisarenko et al. (2014) regarding the extreme value theory and statistics in case of earthquake magnitudes. The link between the generalized extreme value distribution (GEVD) as an asymptotic model for the block maxima of a random variable and the generalized Pareto distribution (GPD) as a model for the peak over thresholds (POT) of the same random variable is presented more clearly. Pisarenko et al. (2014) have inappropriately neglected that the approximations by GEVD and GPD work only asymptotically in most cases. This applies particularly for…
The Tohoku EQ (11th of March, 2011, M = 9.0) in Japan falsified the proposed EQ magnitude range (M = 7.0 - 8.5) of the same seismogenic regional area that had been determined by the compiled hazard maps, study of historical data, or other probabilistic methods while a larger magnitude (M > 9.0) had been proposed for all subduction zones. The observed discrepancy between the proposed EQ magnitude range and the actual one of the Tohoku EQ is studied in this work in terms of the cumulative seismic energy release of the study area and by the use of the Lithospheric Seismic Energy Flow Model (LSEFM…
Spatial and temporal data for earthquakes with magnitude M greater than or equal to 6.5 can provide crucial information about the seismic history and potential for large earthquakes in a region. We analyzed approximately 313,500 events that occurred in the Sunda-arc region during the last 56 years, from 1964 to 2020, reported by the International Seismological Center. We report a persistent increase in the annual number of the events with mb greater than or equal to 6.5. We tested this increase against the null hypothesis and discarded the possibility of the increase being due to random groupi…
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…
We solve the traditional problems of earthquake location and magnitude estimation through a supervised learning approach, where we train a Graph Neural Network to predict estimates directly from input pick data, and each input allows a distinct seismic network with variable number of stations and positions. We train the model using synthetic simulations from assumed travel-time and amplitude-distance attenuation models. The architecture uses one graph to represent the station set, and another to represent the model space. The input includes theoretical predictions of data, given model paramete…
The time dependence of the parameter of the Gutenberg-Richter (GR) magnitude distribution is computed for foreshock sequences of earthquakes, correlated with the main shock, by using the geometric-growth model of earthquake focus, the magnitude distribution of correlated earthquakes and the time-magnitude correlations, derived recently. It is shown that this parameter decreases in time in the foreshock sequence, from the background values down to the main shock. If correlations are present, this time dependence and the time-magnitude correlations provide a tool of monitoring the foreshock seis…
Reliable earthquake detection and seismic phase classification is often challenging especially in the circumstances of low magnitude events or poor signal-to-noise ratio. With improved seismometers and better global coverage, a sharp increase in the volume of recorded seismic data is witnessed. This makes the handling of the seismic data rather daunting based on traditional approaches and therefore fuels the need for a more robust and reliable method. In this study, we investigate two deep learningbased models, termed 1D ResidualNeuralNetwork (ResNet) and multi-branch ResNet, for tackling the …
Trapped particles bursts have long been observed to be frequently occurred several hours before earthquakes, especially for strong earthquakes, from several space experiments during past decades. However, the validity of earthquake origin of particles bursts events is still unsolved. In this paper, we firstly reported the frequency distribution and time evolution of particles bursts within different time windows centered around earthquakes for various magnitudes. The results showed nearly the same systematic dependence of particle bursts frequency on earthquake magnitude and characteristic tim…
Automatic detection of low-magnitude earthquakes has become an increasingly important research topic in recent years due to a sharp increase in induced seismicity around the globe. The detection of low-magnitude seismic events is essential for microseismic monitoring of hydraulic fracturing, carbon capture and storage, and geothermal operations for hazard detection and mitigation. Moreover, the detection of micro-earthquakes is crucial to understand the underlying mechanisms of larger earthquakes. Various algorithms, including deep learning methods, have been proposed over the years to detect …
The detection and rapid characterisation of earthquake parameters such as magnitude are of prime importance in seismology, particularly in applications such as Earthquake Early Warning (EEW). Traditionally, algorithms such as STA/LTA are used for event detection, while frequency or amplitude domain parameters calculated from 1-3 seconds of first P-arrival data are sometimes used to provide a first estimate of (body wave) magnitude. Owing to extensive involvement of human experts in parameter determination, these approaches are often found to be insufficient. Moreover, these methods are sensiti…
Coulomb-stress theory has been used for years in seismology to understand how earthquakes trigger each other. Whenever an earthquake occurs, the stress field changes, and places with positive increases are brought closer to failure. Earthquake models that relate earthquake rates and Coulomb stress after a main event, such as the rate-and-state model, assume that the magnitude distribution of earthquakes is not affected by the change in the Coulomb stress. By using different slip models, we calculate the change in Coulomb stress in the fault plane for every aftershock after the Landers event (C…
Geodetic earthquake early warning (EEW) algorithms complement point-source seismic systems by estimating fault-finiteness and unsaturated moment magnitude for the largest, most damaging earthquakes. Because such earthquakes are rare, it has been difficult to demonstrate that geodetic warnings improve ground motion estimation significantly. Here, we quantify and compare timeliness and accuracy of magnitude and ground motion estimates in simulated real time from seismic and geodetic observations for a suite of globally-distributed, large earthquakes. Magnitude solutions saturate for the seismic …
A crucial point in the debate on feasibility of earthquake prediction is the dependence of an earthquake magnitude from past seismicity. Indeed, whilst clustering in time and space is widely accepted, much more questionable is the existence of magnitude correlations. The standard approach generally assumes that magnitudes are independent and therefore in principle unpredictable. Here we show the existence of clustering in magnitude: earthquakes occur with higher probability close in time, space and magnitude to previous events. More precisely, the next earthquake tends to have a magnitude simi…
The upper bound earthquake magnitude (maximum possible magnitude) of a truncated Gutenberg-Richter relation is the right truncation point (right end-point) of a truncated exponential distribution and is important in the probabilistic seismic hazard analysis. It is frequently estimated by the Bayesian inference. This is a non-regular case and suffers some shortcomings in contrast to the Bayesian inference for the regular case of likelihood function. Here previous non-Bayesian inference methods are outlined and discussed as alternatives including, the formulation of the corresponding confidence …
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）
Predicting earthquakes is of the utmost importance, especially to those countries of high risk, and although much effort has been made, it has yet to be realised. Nevertheless, there is a paucity of statistical approaches in seismic studies to the extent that an old theory is believed without verification. Seismic records of time and magnitude in Japan were analysed by exploratory data analysis (EDA). EDA is a parametric statistical approach based on the characteristics of data and is suitable for data-driven investigations. The distribution style of each dataset was determined, and the import…
