Omori Law Research Articles

Signatures of criticality in turning avalanches of schooling fish

Moving animal groups transmit information through propagating waves or behavioral cascades, exhibiting characteristics akin to systems near a critical point from statistical physics. Using data from freely swimming schooling fish in an experimental tank, we investigate spontaneous behavioral cascades involving turning avalanches, where large directional shifts propagate across the group. We analyze several avalanche metrics and provide a detailed picture of the dynamics associated with turning avalanches, employing tools from avalanche behavior in condensed-matter physics and seismology. Our results identify power-law distributions and robust scale-free behavior through data collapses and scaling relationships, confirming a necessary condition for criticality in fish schools. We explore the biological function of turning avalanches and link them to collective decision-making processes in selecting a new movement direction for the school. We report relevant boundary effects arising from interactions with the tank walls and influential roles of boundary individuals. Finally, spatial and temporal correlations in avalanches are explored using the concept of aftershocks from seismology, revealing clustering of avalanche events below a designated timescale and an Omori law with a faster decay rate than observed in earthquakes. Published by the American Physical Society 2024

Exploring the Dynamic Behavior of Crude Oil Prices in Times of Crisis: Quantifying the Aftershock Sequence of the COVID-19 Pandemic

Crude oil prices crashed and dropped into negative territory at the onset of the COVID-19 pandemic. This extreme event triggered a series of great-magnitude aftershocks. We seek to investigate the cascading dynamics and the characteristics of the series immediately following the oil market crash. Utilizing a robust method named the Omori law, we quantify the correlations of these events. This research presents empirical regularity concerning the number of times that the absolute value of the percentage change in the oil index exceeds a given threshold value. During the COVID-19 crisis, the West Texas Intermediate (WTI) oil prices exhibit greater volatility compared to the Brent oil prices, with higher relaxation values at all threshold levels. This indicates that larger aftershocks decay more rapidly, and the period of turbulence for the WTI is shorter than that of Brent and the stock market indices. We also demonstrate that the power law’s exponent value increases with the threshold value’s magnitude. By proposing this alternative method of modeling extreme events, we add to the current body of literature, and the findings demonstrate its practical use for decision-making authorities—particularly financial traders who model high-volatility products like derivatives.

Some characteristics of foreshocks and aftershocks of the 2022 ML6.8 Chihshang, Taiwan, earthquake sequence

Foreshocks and aftershocks occurred before and after the ML6.8 (Mw7.0) earthquake in eastern Taiwan on 18 September 2022. We explore the epicentral distribution and temporal variations for the mainshock, foreshocks, and aftershocks. Most of the events were located in the area around the Longitudinal Valley. Most foreshocks occurred around the mainshock, while the aftershocks happened outwards from the foreshock area. The temporal variations in seismic-wave energy show that the largest foreshock and the mainshock were responsible for releasing most of the energy during the earthquake sequence. In addition, the b values of the Gutenberg-Richter frequency-magnitude law were 0.62 for foreshocks, 0.87 for aftershocks, and 0.71 for the whole seismic activity by using the least squares method and 0.52 for foreshocks, 0.84 for aftershocks, and 0.65 for the whole seismic activity by using the maximum likelihood method. The b values increase from foreshocks to aftershocks, suggesting the possibility that the fluid pressure of faults during foreshocks is higher than that of the faults during aftershocks due to the outward migration of water. The p-value of the Omori-Utsu law for the aftershock sequence was estimated to be 0.92 for all aftershocks in the study, 1.39 for the aftershocks occurred in the first 6 days, and 1.30 for the aftershocks occurred in the first 12 days. The foreshock sequence could not be described by the inverse Omori law.

Aftershock probabilistic seismic hazard analysis based on enhanced Bayesian network considering frequency information

AbstractBayesian network (BN) is an important tool in probabilistic seismic risk analysis (PSRA) due to its holistic nature and powerful probabilistic inference capabilities. However, while the information that can be stored by BN includes variable values, probability distributions, and relationships between variables, it does not involve event quantities. The results obtained from PSRA and probabilistic seismic hazard analysis (PSHA) are commonly frequency‐related (the number of occurrences per unit time), fundamentally describing event numbers. BN assumes a fixed event number relationship between nodes, but the aftershock counts vary for different mainshock magnitudes. This limitation further restricts its effectiveness in lifespan PSRA for structures. Therefore, this study aims to propose an enhanced BN (EBN) for modeling aftershock PSHA (APSHA) considering frequency information. This paper first introduces the modeling method using BN, including PSHA and a simplified model of APSHA using Båth's law. Next, an EBN that can effectively address frequency issues between the mainshock and aftershocks is presented for modeling APSHA considering the modified Omori's law, while the mathematical principles of both forward and backward inferences for the EBN are elaborated in detail. Then, the article carefully analyzes the utilization of the EBN for APSHA modeling under different scenarios where the information regarding the mainshock is known to varying extents, followed by the extension of the EBN in a time domain to obtain time‐dependent aftershock hazard. Finally, an example of APSHA in a specific location in China is illustrated.

Transient postseismic slip and aftershock triggering: A case study of the 2008 MW7.9 Wenchuan Earthquake, China

In this study, we investigate how a stress variation generated by a fault that experiences transient postseismic slip (TPS) affects the rate of aftershocks. First, we show that the postseismic slip from Rubin-Ampuero model is a TPS that can occur on the main fault with a velocity-weakening frictional motion, that the resultant slip function is similar to the generalized Jeffreys-Lomnitz creep law, and that the TPS can be explained by a continuous creep process undergoing reloading. Second, we obtain an approximate solution based on the Helmstetter-Shaw seismicity model relating the rate of aftershocks to such TPS. For the Wenchuan sequence, we perform a numerical fitting of the cumulative number of aftershocks using the Modified Omori Law (MOL), the Dieterich model, and the specific TPS model. The fitting curves indicate that the data can be better explained by the TPS model with a B/A ratio of approximately 1.12, where A and B are the parameters in the rate- and state-dependent friction law respectively. Moreover, the p and c that appear in the MOL can be interpreted by the B/A and the critical slip distance, respectively. Because the B/A ratio in the current model is always larger than 1, the model could become a possible candidate to explain aftershock rate commonly decay as a power law with a p-value larger than 1. Finally, the influence of the background seismicity rate r on parameters is studied; the results show that except for the apparent aftershock duration, other parameters are insensitive to r.

Increased earthquake rate prior to mainshocks

According to the Omori-Utsu law, the rate of aftershocks after a mainshock decays as a power law with an exponent close to 1. This well-established law was intensively used in the past to study and model the statistical properties of earthquakes. Moreover, according to the so-called inverse Omori law, the rate of earthquakes should also increase prior to a mainshock—this law has received much less attention due to its large uncertainty. Here, we mainly study the inverse Omori law based on a highly detailed Southern California earthquake catalog, which is complete for magnitudes mc≥1 or even lower. First, we develop a technique to identify mainshocks, foreshocks, and aftershocks. We then find, based on a statistical procedure we developed, that the rate of earthquakes is higher a few days prior to a mainshock. We find that this increase is much smaller for (a) a catalog with a magnitude threshold of mc=2.5 and (b) for the Epidemic-Type Aftershocks Sequence (ETAS) model catalogs, even when used with a small magnitude threshold (i.e., mc=1). We also analyze the rate of aftershocks after mainshocks and find that the Omori-Utsu law does not hold for many individual mainshocks and that it may be valid only statistically when considering many mainshocks together. Yet, the analysis of the ETAS model based on the Omori-Utsu law exhibits similar behavior as that of the real catalogs, indicating the validity of this law.

Deep Postseismic Creep Following Large Earthquakes Revealed by Repeating Aftershocks in the Southeastern Tibetan Plateau

Abstract Large earthquake occurrence and the subsequent postseismic period are the most dramatic part of a seismic cycle that usually lasts months to years. However, the fault dynamics that account for the postseismic events are yet to be fully understood. Here, we use the repeating aftershock sequences (RASs) to investigate postseismic slips following the Mw 6.6 Lushan, Mw 6.5 Jiuzhaigou, Mw 6.1 Jinggu, and Mw 6.2 Ludian earthquakes in the southeastern Tibetan Plateau and find 135 RASs following the mainshocks. The RAS seismicity suggests that seismogenic faults began to creep in depth within a few hours after the Lushan, Jiuzhaigou, and Jinggu mainshocks. The deep creeps mainly follow a velocity-strengthening friction mode and decay with an Omori law p-value of ∼1. The results suggest that the combination of fault healing and geometry together controls deep fault behaviors. We develop two conceptual models to explain our observations. Our results provide new insights into spatiotemporal fault evolution after large earthquakes.

On the predictability of localization instabilities of quasibrittle materials from accelerating rates of acoustic emission

Forecasting quasibrittle failure in geomaterials is of paramount importance in the physics of fractures but rarely succeeds. To clarify the physics relating to the failure predictability in a quasibrittle regime, a combined AE (acoustic emission)-DIC (digital image correlation) technique is used in uniaxial compression experiments of flawed granite to trace its damage conditions in real time. The mechanistic correlations of the AE rate-process with the internal FPZ (fracture process zone) development are available. It is shown that the complete record of the AE activities does not meet a deterministic scaling law (e.g., typical time‐reversed Omori law), indicating a challenge in predicting ultimate failure via AE. Subsets of a power law acceleration registered much earlier than ultimate failure imply that the procedural sample-sized localization instability is predictable but still depends on the FPZ dynamic evolutions. More importantly, we find that the coexistence of FPZs and the intermittency of crack growth are the two main reasons for the difficulty of forecasting failure in flawed granite. This study sheds light on upcoming investigations in which the roles of FPZs, especially their spatiotemporal patterns, should be highlighted.

Testing the forecasting skills of aftershock models using a Bayesian framework

The Epidemic Type Aftershock Sequence (ETAS) model and the modified Omori law (MOL) are two aftershock rate models that are used for operational earthquake/aftershock forecasting. Previous studies have investigated the relative performance of the two models for specific case studies. However, a rigorous comparative evaluation of the forecasting performance of the basic aftershock rate models for several different earthquake sequences has not been done before. In this study, forecasts of five prominent aftershock sequences from multiple catalogs are computed using the Bayesian predictive distribution, which fully accounts for the uncertainties in the model parameters. This is done by the Markov Chain Monte Carlo (MCMC) sampling of the model parameters and forward simulation of the ETAS or MOL models to compute the aftershock forecasts. The forecasting results are evaluated using five different statistical tests, including two comparison tests. The forecasting skill tests indicate that the ETAS model tends to perform consistently well on the first three tests. The MOL fails the same tests for certain forecasting time intervals. However, in the comparison tests, it is not definite whether the ETAS model is the better performing model. This work demonstrates the use of forecast testing for different catalogs, which is also applicable to catalogs with a higher magnitude of completeness.

Modeling Exact Frequency-Energy Distribution for Quakes by a Probabilistic Cellular Automaton.

We develop the notion of Random Domino Automaton, a simple probabilistic cellular automaton model for earthquake statistics, in order to provide a mechanistic basis for the interrelation of Gutenberg-Richter law and Omori law with the waiting time distribution for earthquakes. In this work, we provide a general algebraic solution to the inverse problem for the model and apply the proposed procedure to seismic data recorded in the Legnica-Głogów Copper District in Poland, which demonstrate the adequacy of the method. The solution of the inverse problem enables adjustment of the model to localization-dependent seismic properties manifested by deviations from Gutenberg-Richter law.

Stress‐Based and Convolutional Forecasting of Injection‐Induced Seismicity: Application to the Otaniemi Geothermal Reservoir Stimulation

AbstractInduced seismicity observed during Enhanced Geothermal Stimulation at Otaniemi, Finland is modeled using both statistical and physical approaches. The physical model produces simulations closest to the observations when assuming rate‐and‐state friction for shear failure with diffusivity matching the pressure build‐up at the well‐head at onset of injections. Rate‐and‐state friction implies a time‐dependent earthquake nucleation process which is found to be essential in reproducing the spatial pattern of seismicity. This implies that permeability inferred from the expansion of the seismicity triggering front (Shapiro et al., 1997, https://doi.org/10.1111/j.1365-246x.1997.tb01215.x) can be biased. We suggest a heuristic method to account for this bias that is independent of the earthquake magnitude detection threshold. Our modeling suggests that the Omori law decay during injection shut‐ins results mainly from stress relaxation by pore pressure diffusion. During successive stimulations, seismicity should only be induced where the previous maximum of Coulomb stress changes is exceeded. This effect, commonly referred to as the Kaiser effect, is not clearly visible in the data from Otaniemi. The different injection locations at the various stimulation stages may have resulted in sufficiently different effective stress distributions that the effect was muted. We describe a statistical model whereby seismicity rate is estimated from convolution of the injection history with a kernel which approximates earthquake triggering by fluid diffusion. The statistical method has superior computational efficiency to the physical model and fits the observations as well as the physical model. This approach is applicable provided the Kaiser effect is not strong, as was the case in Otaniemi.

Earthquake magnitude distribution and aftershocks: A statistical geometry explanation.

The emergence of a power-law distribution for the energy released during an earthquake is investigated in several models. Generic features are identified which are based on the self-affine behavior of the stress field prior to an event. This field behaves at large scale as a random trajectory in one dimension of space and a random surface in two dimensions. Using concepts of statistical mechanics and results on the properties of these random objects, several predictions are obtained and verified, in particular the value of the power-law exponent of the earthquake energy distribution (the Gutenberg-Richter law) as well as a mechanism for the existence of aftershocks after a large earthquake (the Omori law).

Earthquake Detection Using a Nodal Array on the San Jacinto Fault in California: Evidence for High Foreshock Rates Preceding Many Events

AbstractWe use a dense seismic array of 1,108 vertical‐component geophones within a 600‐m footprint to detect thousands of small earthquakes near an active strand of the San Jacinto fault zone in southern California during a 26‐day period. We first correct site effects using multichannel cross‐correlations of the P‐waves of 256 cataloged earthquakes, and then perform beamforming analysis on the continuous waveforms in a slowness range from −0.4 to 0.4 s/km in both the east and north directions. At each time step, we identify the beam slowness with maximum amplitude and apply a picking algorithm to identify 13,408 events. These detections include over 55.6% of the events in the Quake Template Matching (QTM) catalog for all of southern California during the same time period and 70% of those within 100 km of the array. In addition, we detect over 10,000 new events, not in the QTM catalog. Many of these events can also be seen in records from nearby borehole seismic stations. Measured slownesses for the catalog and newly‐discovered events group into clusters that can be associated with QTM earthquake locations, but with slowness values considerably distorted from predictions based on a 1‐D velocity model, presumably owing to strong velocity heterogeneity near the San Jacinto Fault. Amplitudes of the detected events obey a Gutenberg‐Richter distribution with a b‐value close to one. Foreshocks are common among these detected events, increasing in rate before mainshocks following an inverse Omori's law.

Hazard Model: Epidemic-Type Aftershock Sequence (ETAS) Model for Hungary

In this article we present a space–time epidemic-type aftershock sequence (ETAS) model for the area of Hungary, motivated by the goal of its application in insurance risk models. High-quality recent instrumental data from the period 1996–2021 are used for model parameterization, including data from the recent nearby Zagreb and Petrinja event sequences. In the earthquake-triggering equations of our ETAS model, we replace the commonly used modified Omori law with the more recently proposed stretched exponential time response form, and a Gaussian space response function is applied with a variance add-on for epicenter error. After this model was tested against the observations, an appropriate overall fit for magnitudes M≥3.0 was found, which is sufficient for insurance applications, although the tests also show deviations at the M=2.5 threshold. Since the data used for parameterization are dominated by Croatian earthquake sequences, we also downscale the model to regional zones via parameter adjustments. In the downscaling older historical data are incorporated for a better representation of the key events within Hungary itself. Comparison of long-term large event numbers in simulated catalogues versus historical data shows that the model fit by zone is improved by the downscaling.

An Analytic Expression for the Volcanic Seismic Swarms Occurrence Rate. A Case Study of Some Volcanoes in the World

AbstractSeismic swarms are defined as a group of earthquakes occurring very close in time and space but without any distinctively large event triggering their occurrence. Up to now no simple law has been found to describe the swarm occurrence rate. Here we find an expression able to fit the average occurrence rate on some volcanic areas. This expression exhibits some differences in respect to the classical Omori law. Namely the c parameter of the Omori law is equal to zero and the power law decay of the average occurrence rate of the earthquakes is followed by an exponential decaying regime. Both the results can be interpreted in term of fluid injection and/or movements. Indeed this is a more impulsive phenomenon compared to the occurrence of a large earthquake, with a duration compatible with a c = 0. The exponential decay following the power law one could be explained by a viscoelastic relaxation of the stress induced by the injection and/or movements of fluids in the earth crust.

History dependence of avalanche dynamics of ferroelectric phase transition in BaTiO3 under external bias field

A ferroelectric phase transition under an external bias electric field after field cooling and zero field cooling in barium titanate single crystals shows scale invariant nucleation and growth of complex domain structures. The avalanche energy exponents vary with the external bias, depending on the cooling history of the sample under applied field or zero field cooling. After field cooling, resulting in a single domain sample, the energy exponent is near the integrated mean field value of 5/3, namely, 1.68 ± 0.022. The sample after field cooling shows the same exponent with an external bias electric field range of up to 4.5 kV/cm. The exponent of a multi-domain sample, after zero field cooling, decreases from ε = 1.85 to the fixed point of 1.66 under high fields. The different behavior is attributed to the greater complexity of domain patterns in the multi-domain sample, which reduces to a single domain state under strong applied fields. Compared with the single-domain state, the multi-domain state has more domain boundaries and the intersections formed by these domain boundaries. These domain boundaries and their intersections will hinder the movement of the phase boundary and act as a pinning effect on the front of the phase interface. The effect is to generate more small energy signals, making the critical exponent high. At the same time, the aftershock time distribution (Omori law) remains the same for all switching conditions with an Omori exponent near −1 and switching time correlations of −1 ± 0.05 for short times (<1 s) and −2 ± 0.10 for long times (>1 s).

Precursory predictors of the onset of stick-slip frictional instability

Defining precursory phenomena that forewarn the onset of stick slip events remains an elusive but central goal in earthquake and other failure forecasting. We present a method to define the timing of instability by separately decoupling velocity and inertial effects in a spring-slider system. System state is defined in phase-space by velocity and inertia that in turn control critical stiffness and define the onset of instability while sliding velocity V = VLk/(k + dF/du) remains finite. A general energy analysis defines velocity and inertial effects (coefficients) that act against each other to promote/suppress instability by respectively increasing/decreasing critical stiffness. Repetitive stick-slip experiments define features of stick-slip cycles and identify a precursory trend in accelerating slip that precedes the onset of unstable sliding. We represent this precursory acceleration (dV/dt) immediately preceding instability by a general power-law relation dV/dt = AVα that reduces to V = B(ci + ti-t)-β. This represents the “true” timing of the onset of stick slip, ti relative to that “projected” from the linearization of the precursory data, tf, with tf=ci + ti. ci represents the small offset between the “predicted” and “true” timing of the onset of stick slip resulting from velocity and inertial effects and may be considered as the error in prediction. This error is one-order of magnitude smaller (∼3 ms) than the lead-time that forewarns of the event (∼40 ms) in our experiments and thus is useful as a predictor. Three independent methods using power-law, linear and criticality relationships confirm the fidelity of the timing of the slip instability transition evident in the stick slip data. This theoretical treatment suggests that the underlying physical meaning of the parameter c in the modified Omori law R∼(c + tM-t)-q is the time of the main shock in advance of the timing of the singularity predicted for an ideal response.

Foreshocks of the 2010 Mw 6.7 Yushu, China Earthquake Occurred Near an Extensional Step‐Over

AbstractWe conduct a detailed study of the foreshock sequence preceding the 2010 Mw 6.7 Yushu, Qinghai earthquake in the Tibetan plateau by examining continuous waveforms recorded at a seismic station near the mainshock rupture zone. By using a deep learning phase picker—EQTransformer and a matched‐filter technique, we identify 120 foreshocks with magnitude ranging from −0.7 to 1.6, starting with a Mw 4.6 foreshock approximately 2 hr before the Mw 6.7 Yushu mainshock. Our analyses show that the foreshock sequence follows a typical Omori's law decay with a p‐value of 0.73 and the Gutenberg‐Richer frequency‐magnitude b‐value of 0.66. We do not find any evidence of accelerating events leading up to the Yushu mainshock. Hence, they could be considered as aftershocks of the Mw 4.6 earthquake. We further invert for the focal mechanisms and rupture directions for both the largest foreshock and the mainshock. The Mw 4.6 foreshock likely occurred on a NE‐SW trending fault conjugating to the NW‐SE trending fault of the mainshock. Coulomb stress analysis suggests the Mw 4.6 foreshock induces negative stress on the mainshock source area. These observations do not support either the pre‐slip or the cascade triggering model for foreshock generation. The occurrence of the foreshock, mainshock and large aftershocks appear to be modulated by the Earth's tidal forces, likely reflecting the role of high pore‐fluid pressures. Our observations, together with other recent studies, suggest that extensional step‐overs and conjugate faults along major strike‐slip faults play an important role in generating short‐term foreshock sequences.

“Aftershock Faults” and What They Could Mean for Seismic Hazard Assessment

Abstract We study stress-loading mechanisms for the California faults used in rupture forecasts. Stress accumulation drives earthquakes, and that accumulation mechanism governs recurrence. Most moment release in California occurs because of relative motion between the Pacific plate and the Sierra Nevada block; we calculate relative motion directions at fault centers and compare with fault displacement directions. Dot products between these vectors reveal that some displacement directions are poorly aligned with plate motions. We displace a 3D finite-element model according to relative motions and resolve stress tensors onto defined fault surfaces, which reveal that poorly aligned faults receive no tectonic loading. Because these faults are known to be active, we search for other loading mechanisms. We find that nearly all faults with no tectonic loading show increase in stress caused by slip on the San Andreas fault, according to an elastic dislocation model. Globally, faults that receive a sudden stress change respond with triggered earthquakes that obey an Omori law rate decay with time. We therefore term this class of faults as “aftershock faults.” These faults release ∼4% of the moment release in California, have ∼0.1%–5% probability of M 6.7 earthquakes in 30 yr, and have a 0.001%–1% 30 yr M 7.7 probability range.

Mw ≥ 5 aftershocks of the 2008 Sichuan earthquake: Analysis of temporal variation of Omori Law p-value

Strong to great earthquakes considerably weaken the structural strength of the subsurface strata and civil constructions. The occurrence of Mw ≥ 5 aftershocks further contributes to the loss of human lives and damage to property. Therefore, the estimate of the duration of such aftershocks occurring near a metropolitan area is of major concern. In a previous study (Gupta et al., 2008), aftershocks for the destructive Sichuan earthquake, China (12 May 2008 Mw 7.9) for 23 days were used to estimatep-value and it was proposed that the M ≥ 5 aftershocks may continue for 7 months. In the present study, we analyze aftershock activity of Mw≥ 5 for 2-year duration within an area of 300 km radius from the epicenter, using modified Omori Law, which is well-known for analyzing aftershock rate and duration. We compare our results with the duration of aftershocks estimated using the exponential scaling law for the M ≥ 5 aftershock duration for major to great Himalayan earthquakes. Our analysis suggests that a highp-value (>1) indicates a slow decay of aftershocks as observed. As thep-value changes with time, the estimates of the duration of the occurrence of aftershocks of M ≥ 5 may change. Using a dynamic approach, the data of the first 100 days has been used in this study and the duration of occurrence of Mw ≥ 5 earthquakes is estimated to be ∼300 days, which is found to be true. Our analysis suggests that the temporal variation of thep-value from the onset of the mainshock and the slope of its variation are the key elements in comprehending the duration of the occurrence of the aftershocks of a certain magnitude. We infer that a highp-value (p>1) after the main earthquake, indicates a longer duration of larger aftershocks, which has been found in the case of the Sichuan earthquake. Thep-value decreases with time, however, an intermittent increase in thep-value could be an indicator of an increased chance for the occurrence of a higher magnitude aftershock. The proposed dynamic approach to estimating temporalp-values may help in forecasting the occurrence of stronger aftershocks more effectively. We believe that this is the first time, where the temporal variation ofp-values is estimated and related to the occurrence of aftershocks.