Aftershock Rate Decay Research Articles

Properties of the Zagreb 22 March 2020 earthquake sequence

Although strong and damaging earthquakes have hit Zagreb in the past, the 22 March 2020 earthquake (Mw 5.4) is the first one that was recorded by a modern digital local seismic network, and which could be analysed not only by macroseismic methods, but also by microseismic ones. Herewith we used the 3003 carefully analysed and located events from the first year of the aftershock sequence to learn more about the aftershock rate decay, their magnitude distribution, focal mechanisms and hypocentral locations. The aftershock activity rate was found to closely follow the modified Omori law, and fault-plane solutions for 10 events indicated prevailing pure-reverse faulting. Our analyses suggest that the reverse North Medvednica boundary fault (NMBF) was the causative fault, as it fits with the focal mechanisms and with the geometry of aftershock locations. The epicentral area was of a triangular shape with the mainshock in one vertex, and the opposite side of the triangle lying parallel to the surface trace of the NMBF. The hypocentres of aftershocks were predominantly located in the hanging wall of the fault. No surface break was observed, so the rupture is assumed to be buried. These facts were interpreted as a combination of the effect of conservation of mass (seismic flow) requiring some fault-parallel stress redistribution and transfer of material, and the fault loading and activation in the compressive environment controlled by the stress partition at the brittle-ductile transition zone within the crust. The later process involves compression within the hanging wall during the interseismic stage when the fault segment in the brittle crust is locked, followed by sudden dilatation during the rupture phase.

Residual Strain Mechanism of Aftershocks and Exponents of the Modified Omori's Law

AbstractThe rate of aftershocks caused by residual stresses left after a major seismic event or material failure is considered. It is assumed that the local (or “micro” with respect to the scale of the major event) failures registered as aftershocks are produced by static fatigue (delayed fracture). The key point is that the post failure unloading leaves residual strain; the latter induces stresses proportional to the effective (average) moduli. The moduli decay with accumulated microfailures (microcracks or defects or sliding zones), which explains the aftershock rate decay observed by the (modified) Omori's law. It is found that the value of Omori's exponent, p, reveals one of three types of effective moduli decay with the number of aftershocks: p = 1 (the original Omori's law) corresponds to the exponential reduction of the effective moduli, typical for isotropic distribution of accumulated microcracks or sliding zones; p < 1 (the modified Omori's law) corresponds to the power law reduction of the effective moduli, typical for parallel sliding zones; p > 1 (the modified Omori's law) corresponds to the emergence of critical number of defects. This can be indicative of approaching critical event, that is, major seismic event or failure.

The Omori Law: The 150-Year Birthday Jubilee of Fusakichi Omori

The outstanding Japanese seismologist Fusakichi Omori was born 150 years ago, on October 30, 1868. He discovered his first law in earthquake physics that now bears his name when he was 26. Essentially, Omori’s law tells us that the decay of aftershock rate follows a hyperbolic law. This paper provides a brief account of how that discovery was made. Attention is focused on two approaches in an attempt at generalizing Omori’s law. One of these has been known for many years and to many researchers. It is based on the concept that the aftershock rate depends on time as a power law. The other approach was discovered recently by using a formal analogy between the decay of aftershock activity in the lithosphere and the recombination of charged particles in the ionosphere. A differential equation for aftershocks has been proposed. The general solution to the equation retains the hyperbolic structure of Omori’s law, but also enables a flexible simulation of rock instability at the earthquake source that is “cooling down” after the main shock. The general solution also incorporates the non-monotonic function that describes the time-dependent decay of aftershock rate, which occurs when an earthquake source is acted on by endogenous and exogenous triggers. The aftershock equation is used to formulate the inverse problem in earthquake source physics.

Insights on earthquake triggering processes from early aftershocks of repeating microearthquakes

AbstractCharacterizing the evolution of seismicity rate of early aftershocks can yield important information about earthquake nucleation and triggering. However, this task is challenging because early aftershock seismic signals are obscured by those of the mainshock. Previous studies of early aftershocks employed high‐pass filtering and template matching but had limited performance and completeness at very short times. Here we take advantage of repeating events previously identified on the San Andreas Fault at Parkfield and apply empirical Green's function deconvolution techniques. Both Landweber and sparse deconvolution methods reveal the occurrence of aftershocks as early as few tenths of a second after the mainshock. These events occur close to their mainshock, within one to two rupture lengths away. The aftershock rate derived from this enhanced catalog is consistent with Omori's law, with no flattening of the aftershock rate down to the shortest resolvable timescale ∼0.3 s. The early aftershock rate decay determined here matches seamlessly the decay at later times derived from the original earthquake catalog, yielding a continuous aftershock decay over timescales spanning nearly 8 orders of magnitude. Aftershocks of repeating microearthquakes may hence be governed by the same mechanisms from the earliest time resolved here, up to the end of the aftershock sequence. Our results suggest that these early aftershocks are triggered by relatively large stress perturbations, possibly induced by aseismic afterslip with very short characteristic time. Consistent with previous observations on bimaterial faults, the relative location of early aftershocks shows asymmetry along strike, persistent over long periods.

Effects of a revised rate- and state-dependent friction law on aftershock triggering model

Abstract Seismicity may be considered as a sequence of earthquake nucleation events controlled by regional loading history. On the basis of this concept, Dieterich (1994) modeled aftershock seismicity following an imposed stress step, by using a laboratory-derived rate- and state-dependent friction law (RSF). Although this model predicts the canonical 1/ t decay of aftershock rate, two huge gaps from observations are known; the model, with frictional parameters assumed to be laboratory-observed values, predicts too low aftershock productivity and also too long a delay before the decay onset. These gaps are by orders of magnitude. We suspected that the problem might be the incorrectness of traditional RSFs, none of which was free from contradictions with laboratory data. Hence we modeled aftershock triggering with a revised RSF ( Nagata et al., 2012 ), which seems to have resolved the previously known flaws in reproducing laboratory data. The original analytic approach of Dieterich (1994) was found invalid for the revised RSF, so we did an equivalent analysis by numerically tracking individual nucleation sources. The revised RSF produced generally similar aftershock seismicity, with the gaps mentioned above narrowed by a factor, though these are far too small improvements of the huge gaps, that is, the revised RSF did not fully resolve the problem. On the other hand, our simulations found a counterintuitive response of a fault obeying the revised RSF; if imposed during a certain stage in the seismic cycle, a positive stress step can cause oscillatory slow slip events before eventual seismic instability, instead of a usual response of further monotonic acceleration to seismic instability. This delays the timing of the next earthquake. Due to this behavior, the exponent of the aftershock decay can be greater than unity. Also, the decay can once overshoot below the background seismicity before eventually returning to the background level.

Random stress and Omori's law

We consider two statistical regularities that were used to explain Omori′s law of the aftershock rate decay: the Lévy and Inverse Gaussian (IGD) distributions. These distributions are thought to describe stress behaviour influenced by various random factors: post-earthquake stress time history is described by a Brownian motion. Both distributions decay to zero for time intervals close to zero. But this feature contradicts the high immediate aftershock level according to Omori′s law. We propose that these statistical distributions are influenced by the power-law stress distribution near the earthquake focal zone and we derive new distributions as a mixture of power-law stress with the exponent ψ and Lévy as well as IGD distributions. Such new distributions describe the resulting inter-earthquake time intervals and closely resemble Omori′s law. The new Lévy distribution has a pure power law form with the exponent −(1 +ψ/2) and the mixed IGD has two exponents: the same as Lévy for small time intervals and −(1 +ψ) for longer times. For even longer time intervals this power-law behaviour should be replaced by a uniform seismicity rate corresponding to the long-term tectonic deformation. We compute these background rates using our former analysis of earthquake size distribution and its connection to plate tectonics. We analyse several earthquake catalogues to confirm and illustrate our theoretical results. Finally, we discuss how the parameters of random stress dynamics can be determined through a more detailed statistical analysis of earthquake occurrence or by new laboratory experiments.

Statistical properties of aftershock rate decay: Implications for the assessment of continuing activity

Aftershock rates seem to follow a power law decay, but the assessment of the aftershock frequency immediately after an earthquake, as well as during the evolution of a seismic excitation remains a demand for the imminent seismic hazard. The purpose of this work is to study the temporal distribution of triggered earthquakes in short time scales following a strong event, and thus a multiple seismic sequence was chosen for this purpose. Statistical models are applied to the 1981 Corinth Gulf sequence, comprising three strong (M = 6.7, M = 6.5, and M = 6.3) events between 24 February and 4 March. The non-homogeneous Poisson process outperforms the simple Poisson process in order to model the aftershock sequence, whereas the Weibull process is more appropriate to capture the features of the short-term behavior, but not the most proper for describing the seismicity in long term. The aftershock data defines a smooth curve of the declining rate and a long-tail theoretical model is more appropriate to fit the data than a rapidly declining exponential function, as supported by the quantitative results derived from the survival function. An autoregressive model is also applied to the seismic sequence, shedding more light on the stationarity of the time series.

Common dependence on stress for the two fundamental laws of statistical seismology

Two of the long-standing relationships of statistical seismology are power laws: the Gutenberg-Richter relation describing the earthquake frequency-magnitude distribution, and the Omori-Utsu law characterizing the temporal decay of aftershock rate following a main shock. Recently, the effect of stress on the slope (the b value) of the earthquake frequency-magnitude distribution was determined by investigations of the faulting-style dependence of the b value. In a similar manner, we study here aftershock sequences according to the faulting style of their main shocks. We show that the time delay before the onset of the power-law aftershock decay rate (the c value) is on average shorter for thrust main shocks than for normal fault earthquakes, taking intermediate values for strike-slip events. These similar dependences on the faulting style indicate that both of the fundamental power laws are governed by the state of stress. Focal mechanisms are known for only 2 per cent of aftershocks. Therefore, c and b values are independent estimates and can be used as new tools to infer the stress field, which remains difficult to measure directly.

An empirical comparison among aftershock decay models

We compare the ability of three aftershock decay models proposed in the literature to reproduce the behavior of 24 real aftershock sequences of Southern California and Italy. In particular, we consider the Modified Omori Model (MOM), the Modified Stretched Exponential model (MSE) and the Band Limited Power Law (LPL). We show that, if the background rate is modeled properly, the MSE or the LPL reproduce the aftershock rate decay generally better than the MOM and are preferable, on the basis of the Akaike and Bayesian information criteria, for about one half of the sequences. In particular the LPL, which is usually preferable with respect to the MSE and fits well the data of most sequences, might represent a valid alternative to the MOM in real-time forecasts of aftershock probabilities. We also show that the LPL generally fits the data better than a purely empirical formula equivalent to the aftershock rate equation predicted by the rate- and state-dependent friction model. This indicates that the emergence of a negative exponential decay at long times is a general property of many aftershock sequences but also that the process of aftershock generation is not fully described by current physical models.

A comparative analysis of different models of aftershock rate decay by maximum likelihood estimation of simulated sequences

Most empirical studies on the decay of the rate of aftershock with time after a main shock assume the simple power law described by the modified Omori model (MOM). A couple of alternative models, also including an exponential decay at long times, have been proposed in the last decades: the modified stretched exponential (MSE) model and the band‐limited power law (LPL). The first aim of this work is to discuss the functional properties of such models and the relations existing on their parameters. In particular, we derive, in term of common transcendental functions, the analytical integrals of the LPL and MSE rate functions, which are useful to simplify and speed up computations. We also define, as a function of the parameters of the LPL, two characteristic times tb and ta, which correspond approximately to the delay time c of the MOM and the exponential decay relaxation time t0 of the MSE, respectively. Hence, the MOM, the MSE, and the LPL models can be compared among each other in terms of the same set of four general parameters: (1) the power law exponent, (2) the initial delay time, (3) the exponential relaxation time (∞ for the MOM), and (4) a normalization parameter, which can be related in some cases to the total number of potential aftershocks. A second aim of this paper is to test the ability of maximum likelihood methods to detecting the most appropriate decay model among alternatives. By the analysis of sequences simulated according to a MSE or a LPL we show that if the assumed exponential decay relaxation time is shorter than the time window over which the sequence is observed, the Akaike and Bayesian information criteria select correctly the true model (MSE or LPL). Conversely, when the relaxation time is definitely longer than the observing window, the information criteria usually prefer the MOM, but the maximum likelihood of the true model is higher, and if the data set of shocks is sufficiently large, the order of magnitude of the simulated characteristic time of the exponential decay can be estimated quite consistently. Hence, when analyzing real sequences, the possible emergence of the exponential decay might be hidden by the short duration of the time window if the standard information criteria are considered. Moreover, when the relaxation time is short, the estimated power law exponent p for the MOM results definitely higher than that assumed in the simulation. This indicates that high values of p (>1.5–2.0), sometimes observed in real sequences, might be due to the (not modeled) early startup of the negative exponential decay. Our analysis cannot decide which model is the most appropriate in describing the temporal behavior of aftershock rate after a main shock but suggests that the assumption of a model also including the exponential decay is generally preferable as it allows capture of all of the features of sequence decay.

Comparing different models of aftershock rate decay: The role of catalog incompleteness in the first times after main shock

We evaluated the efficiency of various models in describing the time decay of aftershock rate of 47 simple sequences occurred in California (37) from 1933 to 2004 and in Italy (10) from 1976 to 2004. We compared the models by the corrected Akaike Information Criterion (AIC c) and the Bayesian Information Criterion (BIC), both based on the log-likelihood function and also including a penalty term that takes into account the number of independent observations and of free parameters of each model. These criteria follow two different approaches (probabilistic and Bayesian respectively) well covering the wide spectra of current views on model comparison. To evaluate the role of catalog incompleteness in the first times after the main shock, we compared the performance of different models by varying the starting time T s and the minimum magnitude threshold M min for each sequence. We found that Omori-type models including parameter c are preferable to those not including it, only for short T s and low M min while the latters generally perform better than the formers for T s longer than a few hours and M min larger than the main shock magnitude M m minus 3 units. For T s > 1 day or M min > M m − 2.5, only about 15% of the sequences still give a preference to models including c. This clearly indicates that a value of parameter c different from zero does not represent a general property of aftershock sequences in California and Italy but it is very likely induced in most cases by catalog incompleteness in the first times after the main shock. We also considered other models of aftershock decay proposed in the literature: the Stretched Exponential Law in two forms (including and not including a time shift) and the band Limited Power Law (LPL). We found that such models perform worse than the Modified Omori Model (MOM) and other Omori-type models for the large majority of sequences, although for LPL, the relatively short duration of the analyzed sequences (one year) might also contribute to its poor performance. Our analysis demonstrates that the MOM with c kept fixed to 0 represent the better choice for the modeling (and the forecasting) of simple sequence behavior in California and Italy.

Tests of models of aftershock rate decay

Six rate decay functions have been fit to 14 aftershock sequences, mostly from southern California. The best fit was usually by the modified Omori function, but five of the sequences were best fit by new variants of the stretched exponential model, and three more were best fit by the modified Omori with a steady background rate added. Background rate terms in the modified Omori model never improved the fit to sequences best fit by exponential models. The variety of decay laws observed can be explained if the underlying aftershock generation process follows stretched exponential relations with spatial inhomogeneities in q value. Tests of a large number of synthetic sequences show that if different parts of a fault zone produced aftershocks decaying with stretched exponentials having a wide variety of q values, the combined sequence will be best fit by a variant of the modified Omori relation. The sequences best fit by modified exponential may be associated with more homogeneous material properties or simpler geometry.

The stretched exponential function as an alternative model for aftershock decay rate

The stretched exponential (Williams‐Watts) relaxation function is N*(t)=N*(0)exp[−(t/t0)q], 0<q≤1. As applied to aftershocks, N* (0) is the number of aftershocks that have not yet occurred at time t, starting with a finite number of potential events, N* (0). This function describes slow relaxation in a number of distinct physical systems. Whether it is the appropriate function for aftershocks, as an alternative to the modified Omori power law function, depends on the long‐time behavior of the mean interevent time. The behavior of this function has been tested on synthetic sequences in comparison with the Omori relation to determine the feasibility of determining which is the better model of aftershock rate decay. The differences are practically indistinguishable for very slowly decaying sequences but are measurable for normally and rapidly decaying sequences if excellent data are available. Maximum likelihood estimates of the parameters in the stretched exponential function were determined for 29 sequences and the fit of the model to the data was compared with the modified Omori fit by the Akaike Information Criterion. If the time of the rnainshock is taken as the start time for the sequence, the Omori fit is almost always better. If a later start time is used, 15 min to 2.5 hours for the cases tested, the stretched exponential fit is better in about half the cases. The selected start time is the time at which the rate of occurrence begins to decrease, after an initial apparent increase probably due to missed events during the very active interval. As expected, the relaxation time t0 is inversely related to the Omori p value and may prove to be more easily interpretable in terms of the physics of the aftershock‐generating process.