Omori-Utsu Law Research Articles

Onset of Aftershocks: Constraints on the Rate-and-State Model

Abstract Aftershock rates typically decay with time t after the mainshock according to the Omori–Utsu law, R(t)=K(c+t)−p, with parameters K, c, and p. The rate-and-state (RS) model, which is currently the most popular physics-based seismicity model, also predicts an Omori–Utsu decay with p = 1 and a c-value that depends on the size of the coseismic stress change. Because the mainshock-induced stresses strongly vary in space, the c-value should vary accordingly. Short-time aftershock incompleteness (STAI) in earthquake catalogs has prevented a detailed test of this prediction so far, but the newly developed a-positive method for reconstructing the true earthquake rate now allows its testing. Using previously published slip models, we calculate the coseismic stress changes for the six largest mainshocks in Southern California in recent decades and estimate the maximum shear as a scalar proxy of the coseismic stress tensor. Aftershock rates reconstructed for events in different stress ranges show that the rates follow a power law with p = 1 independent of stress with no clear sign of a c-value. The onset of the power-law decay is abrupt and more delayed in areas with smaller stress changes. The observations do not necessarily contradict the RS model, as STAI limits the resolution for early aftershocks, and the RS model can reproduce the observations for specific Aσ values. However, the observations lead to strong constraints, namely Aσ<10 kPa and a power-law decay of the background rate with distance to the fault, with exponent 2.7.

A fractional Hawkes process model for earthquake aftershock sequences

Abstract A new type of Hawkes process, known as the fractional Hawkes Process (FHP), has been recently introduced. This process uses a Mittag-Leffler density as the kernel function which is asymptotically a power law and so similar to the Omori–Utsu law, suggesting the FHP may be an appropriate earthquake model. However, it is currently an unmarked point process meaning it is independent of an earthquake’s magnitude. We extend the existing FHP, by incorporating Utsu’s aftershock productivity law and a time-scaling parameter from the fractional Zener Model to a marked version so that it may better model earthquake aftershock sequences. We call this model the ‘Seismic Fractional Hawkes Process’ (SFHP). We then estimate parameters via maximum likelihood and provide evidence for these estimates being consistent and asymptotically normal via a simulation study. The SFHP is then compared to the epidemic type aftershock sequence and FHP models on four aftershock sequences from Southern California and New Zealand. While it is inconclusive if the seismic fractional Hawkes process performs better in a retrospective predictive performance experiment, it does perform favourably against both models in terms of information criteria and residual diagnostics especially when the aftershock clustering is stronger.

Reduced seismic activity after mega earthquakes

Mainshocks are often followed by increased earthquake activity (aftershocks). According to the Omori-Utsu law, the rate of aftershocks decays as a power law over time. While aftershocks typically occur in the vicinity of the mainshock, previous studies have suggested that mainshocks can also trigger earthquakes in remote locations, beyond the range of aftershocks. Here we analyze the rate of earthquakes that occurred after mega-earthquakes (with a magnitude of 7.5 or higher) and show that there is a significantly higher occurrence of mega-earthquakes that are followed by reduced activity beyond a certain distance from the epicenter compared to the expected frequency; the results are based on statistical tests we developed. However, the remote earthquake rate after the strongest earthquakes (magnitude ≥8) can also be significantly higher than the expected rate. Comparing our findings to the global Epidemic-Type Aftershock Sequence model, we find that the model does not capture the above findings, hinting at a potential missing mechanism. We suggest that the reduced earthquake rate is due to the release of global energy/tension after substantial mainshock events. This conjecture holds the potential to enhance our comprehension of the intricacies governing post-seismic activity.

Stress spatial distributions, the Gutenberg–Richter and Omori–Utsu laws

We investigate several earthquake models in one and two dimensions of space and analyze in these models the stress spatial distribution. We show that the statistical properties of stress distribution are responsible for the distribution of earthquake magnitudes, as described by the Gutenberg–Richter (GR) law. A series of predictions is made based on the analogies between the stress profile and one-dimensional random curves or two-dimensional random surfaces. These predictions include the b-value, which determines the ratio of small to large seismic events and, in two-dimensional models, we predict the existence of aftershocks and their temporal distribution, known as the Omori–Utsu law. Both the GR and Omori–Utsu law are properties which have been extensively validated by earthquake observations in nature.

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.

Analysis of the b, p values, and the fractal dimension of aftershocks sequences following two major earthquakes in central Himalaya

On April 25 of 2015, earthquake of 7.6 ML struck the central Himalayan region having epicenter at Barpak village in the Gorkha district of Nepal. The event was followed by 7.0 ML earthquake known as Dolakha earthquake, epicenter on the border of Dolakha and Sindhupalchowk. In this study, the b-value, aftershock decay rate p-value, and correlation fractal dimension were estimated for the aftershock sequences. The data used were obtained from the National Earthquake Monitoring and Research Centre (NEMRC), Nepal. The b–values 0.89 ± 0.05 and 0.90 ± 0.07 were computed for aftershocks after Gorkha and Dolakha event, respectively. A significant increase in b-values were reported following the Gorkha earthquake and Dolakha earthquake. The slip procured in the primary fault is respectively 60 % and 56 % after the events. The aftershocks sequences after both events were modelled by the Omori-Utsu law with p = 1.2 ± 0.11 and p = 0.76 ± 0.04. The observed b and p values after the earthquake sequences may correlate well with the large slip experienced by the seismogenic fault. This study sheds light on the mechanism behind the preparation of the significant earthquakes in the tectonic regions of the Himalaya.

How Injection History Can Affect Hydraulic Fracturing–Induced Seismicity: Insights from Downhole Monitoring at Preston New Road, United Kingdom

ABSTRACT In August 2019, a multistage hydraulic fracturing (HF) operation was carried out at Preston New Road, United Kingdom. HF caused abundant seismic activity that culminated with an ML 2.9 event. The seismic activity was recorded by a downhole array of 12 sensors located in a nearby monitoring well. About 55,556 events were detected and located in real time during the operation by a service company. In this study, we first improve the number of detections by applying template matching and later calculate the moment magnitude of the associated earthquakes. Then we show that by separately analyzing the periods during and immediately after injection, distinct patterns can be identified. We observe an increase in the delay and decrease in amplitude of peak seismicity during subsequent phases of injection. After injection, the seismicity decay can be described by the Omori–Utsu law. The decay rate tends to slow with each successive injection, in particular during the later injection stages. In addition, the frequency–magnitude distribution evolves from a tapered distribution (lack of large events) to a bilinear distribution (excess of large events). This evolution is gradual, with the corner magnitude increasing with each injection. We interpret these patterns as the result of the combined effect of two factors: (1) the stimulated volume becoming increasingly aseismic and (2) the gradual increase in its size, which increases the probability of triggered events on preexisting faults. More generally, these patterns suggest that seismic activity during injection is strongly influenced by the injection history and is modulated by local conditions such as stress state, fault structure, and permeability.

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.

Fast radio bursts trigger aftershocks resembling earthquakes, but not solar flares

ABSTRACT The production mechanism of repeating fast radio bursts (FRBs) is still a mystery, and correlations between burst occurrence times and energies may provide important clues to elucidate it. While time correlation studies of FRBs have been mainly performed using wait time distributions, here we report the results of a correlation function analysis of repeating FRBs in the 2D space of time and energy. We analyse nearly 7,000 bursts reported in the literature for the three most active sources of FRB 20121102A, 20201124A, and 20220912A, and find the following characteristics that are universal in the three sources. A clear power-law signal of the correlation function is seen, extending to the typical burst duration (∼ 10 msec) towards shorter time intervals (Δt). The correlation function indicates that every single burst has about a 10–60 per cent chance of producing an aftershock at a rate decaying by a power law as ∝ (Δt)−p with p = 1.5–2.5, like the Omori–Utsu law of earthquakes. The correlated aftershock rate is stable regardless of source activity changes, and there is no correlation between emitted energy and Δt. We demonstrate that all these properties are quantitatively common to earthquakes, but different from solar flares in many aspects, by applying the same analysis method for the data on these phenomena. These results suggest that repeater FRBs are a phenomenon in which energy stored in rigid neutron star crusts is released by seismic activity. This may provide a new opportunity for future studies to explore the physical properties of the neutron star crust.

A neural encoder for earthquake rate forecasting

Forecasting the timing of earthquakes is a long-standing challenge. Moreover, it is still debated how to formulate this problem in a useful manner, or to compare the predictive power of different models. Here, we develop a versatile neural encoder of earthquake catalogs, and apply it to the fundamental problem of earthquake rate prediction, in the spatio-temporal point process framework. The epidemic type aftershock sequence model (ETAS) effectively learns a small number of parameters to constrain the assumed functional forms for the space and time correlations of earthquake sequences (e.g., Omori-Utsu law). Here we introduce learned spatial and temporal embeddings for point process earthquake forecasting models that capture complex correlation structures. We demonstrate the generality of this neural representation as compared with ETAS model using train-test data splits and how it enables the incorporation additional geophysical information. In rate prediction tasks, the generalized model shows >4% improvement in information gain per earthquake and the simultaneous learning of anisotropic spatial structures analogous to fault traces. The trained network can be also used to perform short-term prediction tasks, showing similar improvement while providing a 1000-fold reduction in run-time.

Aftershocks and Fluctuating Diffusivity.

The Omori-Utsu law shows the temporal power-law-like decrease of the frequency of earthquake aftershocks and, interestingly, is found in a variety of complex systems/phenomena exhibiting catastrophes. Now, it may be interpreted as a characteristic response of such systems to large events. Here, hierarchical dynamics with the fast and slow degrees of freedom is studied on the basis of the Fokker-Planck theory for the load-state distribution to formulate the law as a relaxation process, in which diffusion coefficient in the space of the load state is treated as a fluctuating slow variable. The evolution equation reduced from the full Fokker-Planck equation and its Green's function are analyzed for the subdynamics governing the load state as the fast degree of freedom. It is shown that the subsystem has the temporal translational invariance in the logarithmic time, not in the conventional time, and consequently the aging phenomenon appears.

Hidden jerk in universal creep and aftershocks.

Most materials exhibit creep memory under the action of a constant load. The memory behavior is governed by Andrade's creep law, which also has an inherent connection with the Omori-Utsu law of earthquake aftershocks. Both empirical laws lack a deterministic interpretation. Coincidentally, the Andrade law is similar to the time-varying part of the creep compliance of the fractional dashpot in anomalous viscoelastic modeling. Consequently, fractional derivatives are invoked, but since they lack a physical interpretation, the physical parameters of the two laws extracted from curve fit lack confidence. In this Letter, we establish an analogous linear physical mechanism that underlies both laws and relates its parameters with the material's macroscopic properties. Surprisingly, the explanation does not require the property of viscosity. Instead, it necessitates the existence of a rheological property that relates strain with the first order time derivative of stress, which involves jerk. Further, we justify the constant quality factor model of acoustic attenuation in complex media. The obtained results are validated in light of the established observations.

High probability of successive occurrence of Nankai megathrust earthquakes

Great earthquakes along the Nankai megathrust in south-western Japan feature in the top priority list of Japan’s disaster management agenda. In May 2019, an alert system was incepted to issue public warnings when the probability of an earthquake occurrence along the Nankai megathrust became higher than usual. One of the cases that trigger the issuance of public warnings is when a great earthquake occurred and another one of the same scale is anticipated within a short period of time. Although such “twin ruptures” have occurred multiple times along the Nankai megathrust, the quantification of the probability of such twin ruptures has never been attempted. Based on global statistics and local earthquake occurrence history, we estimated the probability of a successive occurrence of two M8 or larger earthquakes within 3 years globally and along the Nankai megathrust to be 5.3–18% and 4.3–96%, respectively. The timing of the second earthquake followed the Omori–Utsu law in global statistics, which allowed the estimation of the probability for the successive occurrence of Nankai megathrust earthquakes in arbitrary time frames. The predicted probability for the one-week timeframe was 100–3600-fold higher than that of the norm, endorsing the necessity for the warning scheme.

Investigation of the Factors Controlling the Duration and Productivity of Aftershocks Following Strong Earthquakes in Greece

Strong crustal earthquakes in Greece are typically followed by aftershocks, the properties of which are important factors in seismic hazard assessment. In order to examine the properties of earthquake sequences, we prepared an earthquake catalog comprising aftershock sequences with mainshocks of Mw ≥ 5.5 from 1995 to 2021. Regional aftershock parameters were estimated to highlight variations in aftershock decay and productivity among regions with similar seismotectonic characteristics. A statistically based method of estimating aftershock duration and a metric of relative aftershock productivity to examine the variations among the different cases were employed. From the detailed analysis of the selected seismic sequences, we attempt to unravel the physical mechanisms behind deviations in aftershock duration and productivity and resolve the relative contribution of background seismicity, the Omori–Utsu law parameters and the mainshock faulting properties. From our analysis, the duration of aftershock sequences depends upon the rupture process of the mainshock, independently of its magnitude. The same applies to aftershock productivity, however, other tectonic setting (e.g., seismic coupling) or source-related (e.g., focal depth, stress drop) parameters also contribute. The estimated regional parameters of the aftershock rate models could be utilized as initial ones to forecast the aftershock occurrence rates at the early stage following a mainshock.

The Analysis of the Aftershock Sequences of the Recent Mainshocks in Alaska

The forecasting of the evolution of natural hazards is an important and critical problem in natural sciences and engineering. Earthquake forecasting is one such example and is a difficult task due to the complexity of the occurrence of earthquakes. Since earthquake forecasting is typically based on the seismic history of a given region, the analysis of the past seismicity plays a critical role in modern statistical seismology. In this respect, the recent three significant mainshocks that occurred in Alaska (the 2002, Mw 7.9 Denali; the 2018, Mw 7.9 Kodiak; and the 2018, Mw 7.1 Anchorage earthquakes) presented an opportunity to analyze these sequences in detail. This included the modelling of the frequency-magnitude statistics of the corresponding aftershock sequences. In addition, the aftershock occurrence rates were modelled using the Omori–Utsu (OU) law and the Epidemic Type Aftershock Sequence (ETAS) model. For each sequence, the calculation of the probability to have the largest expected aftershock during a given forecasting time interval was performed using both the extreme value theory and the Bayesian predictive framework. For the Bayesian approach, the Markov Chain Monte Carlo (MCMC) sampling of the posterior distribution was performed to generate the chains of the model parameters. These MCMC chains were used to simulate the models forward in time to compute the predictive distributions. The calculation of the probabilities to have the largest expected aftershock to be above a certain magnitude after a mainshock using the Bayesian predictive framework fully takes into account the uncertainties of the model parameters. Moreover, in order to investigate the credibility of the obtained forecasts, several statistical tests were conducted to compare the performance of the earthquake rate models based on the OU formula and the ETAS model. The results indicate that the Bayesian approach combined with the ETAS model produced more robust results than the standard approach based on the extreme value distribution and the OU law.

Aftershock Rate Changes at Different Ocean Tide Heights

The differential probability gain approach is used to estimate quantitatively the change in aftershock rate at various levels of ocean tides relative to the average rate model. An aftershock sequences are analyzed from two regions with high ocean tides, Kamchatka and New Zealand. The Omori-Utsu law is used to model the decay over time, hypothesizing an invariable spatial distribution. Ocean tide heights are considered rather than phases. A total of 16 sequences of M ≥6 aftershocks off Kamchatka and 15 sequences of M ≥6 aftershocks off New Zealand are examined. The heights of the ocean tides at various locations were modeled using FES 2004. Vertical stress changes due to ocean tides are here about 10–20 kPa, that is, at least several times greater than the effect due to Earth tides. An increase in aftershock rate is observed by more than two times at high water after main M ≥6 shocks in Kamchatka, with slightly less pronounced effect for the earthquakes of M = 7.8, December 15, 1971 and M = 7.8, December 5, 1997. For those two earthquakes, the maximum of the differential probability gain function is also observed at low water. For New Zealand, we also observed an increase in aftershock rate at high water after thrust type main shocks with M ≥6. After normal-faulting main shocks there was the tendency of the rate increasing at low water. For the aftershocks of the strike-slip main shocks we observed a less evident impact of the ocean tides on their rate. This suggests two main mechanisms of the impact of ocean tides on seismicity rate, an increase in pore pressure at high water, or a decrease in normal stress at low water, both resulting in a decrease of the effective friction in the fault zone.

Aftershocks are fluid-driven and decay rates controlled by permeability dynamics

One aspect of earthquake physics not adequately addressed is why some earthquakes generate thousands of aftershocks while other earthquakes generate few, if any, aftershocks. It also remains unknown why aftershock rates decay as ~1/time. Here, I show that these two are linked, with a dearth of aftershocks reflecting the absence of high-pressure fluid sources at depth, while rich and long-lasting aftershock sequences reflect tapping high-pressure fluid reservoirs that drive aftershock sequences. Using a physical model that captures the dominant aspects of permeability dynamics in the crust, I show that the model generates superior fits to observations than widely used empirical fits such as the Omori-Utsu Law, and find a functional relationship between aftershock decay rates and the tectonic ability to heal the co- and post-seismically generated fracture networks. These results have far-reaching implications, and can help interpret other observations such as seismic velocity recovery, attenuation, and migration.

A Typical Foreshock and Aftershock Anomaly: Observations, Interpretation, and Applications

Previous publications by this author (Rodkin, 2008a, 2008b, 2012; Rodkin and Tikhonov, 2016; Rodkin and Rundkvist, 2017) proposed and actually used a method for constructing and analyzing the generalized vicinity of large earthquakes (GVLE). The GVLE method provides an enormous increase in the amount of data that is used and, hence, a much more detailed description of the typical features in the foreshock–aftershock process. The GVLE clearly detects the power-law foreshock seismicity increase and the aftershock process obeying the Omori–Utsu law. In addition, many parameters (the slope of the recurrence plot and the average depths of focus, apparent stresses, the duration of the seismic process, and several other quantities) show, when treated in the GVLE framework, similar anomalies whose amplitudes are increasing toward the time of the generalized large earthquake minus the logarithm of the time remaining until the occurrence. Similar anomalies (more frequently with different parameters) are observed both in the foreshock area and in the aftershock area. Because the anomalies are so similar we are entitled to treat them as a seismicity pattern. Precursory anomalies are of special interest. We discuss the question of which of these are primary and which are secondary. The anomaly is interpreted in terms of the multiplicative cascade model. An analogy in the character of this anomaly with Zhurkov’s concept of kinetic failure exists. We discuss the approaches to the use of the anomaly in earthquake prediction.

Forecasting Aftershock Activity: 5. Estimating the Duration of a Hazardous Period

Continuing the series of publications on aftershock hazard assessment, we consider the problem of estimating the time interval after a strong earthquake that is prone to aftershocks which may pose an independent hazard. The distribution model of this quantity, which depends on three parameters of the Omori–Utsu law, is constructed. With the appropriate averaged parameter estimates, the model fairly closely fits the real (empirical) distributions of this quantity on the global and regional scale. A key parameter in the model is the expected number of aftershocks of a given magnitude. This number broadly varies from earthquake-to-earthquake, which determines the wide confidence variant of the estimates based on the averaged parameters. Therefore, for forecasting the duration of the hazardous aftershock-prone period, we propose to use two variants of the estimates. The first variant is based only on the averaged parameter estimates for the region under study and on the value of the magnitude of the earthquake. This variant is applicable immediately after a strong earthquake. The second variant employs information about the aftershocks that occurred during the first few hours after an earthquake, which improves the forecast considerably.

Seismiclike organization of avalanches in a driven long-range elastic string as a paradigm of brittle cracks.

Crack growth in heterogeneous materials sometimes exhibits crackling dynamics, made of successive impulselike events with specific scale-invariant time and size organization reminiscent of earthquakes. Here, we examine this dynamics in a model which identifies the crack front with a long-range elastic line driven in a random potential. We demonstrate that, under some circumstances, fracture grows intermittently, via scale-free impulse organized into aftershock sequences obeying the fundamental laws of statistical seismology. We examine the effects of the driving rate and system overall stiffness (unloading factor) onto the scaling exponents and cutoffs associated with the time and size organization. We unravel the specific conditions required to observe a seismiclike organization in the crack propagation problem. Beyond failure problems, implications of these results to other crackling systems are finally discussed.