Mainshock Magnitude Research Articles

A probabilistic framework for quantification of aftershock ground‐motion hazard in California: Methodology and parametric study

AbstractThis paper presents a proposed method of aftershock probabilistic seismic hazard analysis (APSHA) similar to conventional ‘mainshock’ PSHA in that it estimates the likelihoods of ground motion intensity (in terms of peak ground accelerations, spectral accelerations or other ground motion intensity measures) due to aftershocks following a mainshock occurrence. This proposed methodology differs from the conventional mainshock PSHA in that mainshock occurrence rates remain constant for a conventional (homogeneous Poisson) earthquake occurrence model, whereas aftershock occurrence rates decrease with increased elapsed time from the initial occurrence of the mainshock. In addition, the aftershock ground motion hazard at a site depends on the magnitude and location of the causative mainshock, and the location of aftershocks is limited to an aftershock zone, which is also dependent on the location and magnitude of the initial mainshock. APSHA is useful for post‐earthquake safety evaluation where there is a need to quantify the rates of occurrence of ground motions caused by aftershocks following the initial rupture. This knowledge will permit, for example, more informed decisions to be made for building tagging and entry of damaged buildings for rescue, repair or normal occupancy. Copyright © 2008 John Wiley & Sons, Ltd.

Improved tests reveal that the accelerating moment release hypothesis is statistically insignificant

We test the hypothesis that accelerating moment release (AMR) is a precursor to large earthquakes, using data from California, Nevada, and Sumatra. Spurious cases of AMR can arise from data fitting because the time period, area, and sometimes magnitude range analyzed before each main shock are often optimized to produce the strongest AMR signal. Optimizing the search criteria can identify apparent AMR even if no robust signal exists. For both 1950–2006 California‐Nevada M ≥ 6.5 earthquakes and the 2004 M9.3 Sumatra earthquake, we can find two contradictory patterns in the pre–main shock earthquakes by data fitting: AMR and decelerating moment release. We compare the apparent AMR found in the real data to the apparent AMR found in four types of synthetic catalogs with no inherent AMR. When spatiotemporal clustering is included in the simulations, similar AMR signals are found by data fitting in both the real and synthetic data sets even though the synthetic data sets contain no real AMR. These tests demonstrate that apparent AMR may arise from a combination of data fitting and normal foreshock and aftershock activity. In principle, data‐fitting artifacts could be avoided if the free parameters were determined from scaling relationships between the duration and spatial extent of the AMR pattern and the magnitude of the earthquake that follows it. However, we demonstrate that previously proposed scaling relationships are unstable, statistical artifacts caused by the use of a minimum magnitude for the earthquake catalog that scales with the main shock magnitude. Some recent AMR studies have used spatial regions based on hypothetical stress loading patterns, rather than circles, to select the data. We show that previous tests were biased and that unbiased tests do not find this change to the method to be an improvement. The use of declustered catalogs has also been proposed to eliminate the effect of clustering but we demonstrate that this does not increase the statistical significance of AMR. Given the ease with which data fitting can find desired patterns in seismicity, future studies of AMR‐like observations must include complete tests against synthetic catalogs that include spatiotemporal clustering.

Seismicity patterns in Vrancea and predictive features

Notable clustering effects in seismic activity are observed at the sharp bend of the South-Eastern Carpathians both at subcrustal and crustal domains. The seismic energy release is strongly increasing with depth, where apparently the principal source controlling the geodynamics of the entire system is located. New developments of instrumental monitoring and relatively extended historical data spanning several seismic cycles in Vrancea allow refined analyses of earthquake patterns, outlining high-resolution images of specific alignments of seismicity in the Vrancea region. Refined joint locations are carried out for the most recent earthquakes, produced after 1995. In parallel, a new approach has been implemented to produce numerical simulations of the Vrancea seismic cycle and to reproduce its main characteristics: return period, main shock magnitude, depth location, non-linear frequency-magnitude distribution, aftershock activity and their variability from one cycle to other.

Stress relations in three recent seismic series in the Murcia region, southeastern Spain

In this work we study the relation between the seismicity belonging to three recent seismic series in the Murcia region, southeastern Spain, with the coseismic stress changes due to their largest shocks. The first series occurred in 1999 near the town of Mula, with a mainshock magnitude of Mw = 5.1; the second series occurred in 2002, in the region of Bullas 35 km SW from the 1999 series, with events of magnitudes Mw ≤ 5.0. The last one occurred three years later (2005), 5 km SE from the 2002 series with magnitudes of events Mw ≤ 4.8. For the analysis, we relocated the seismicity of each series, next we computed the coseismic stress change produced by its main events and a foreshock and finally tested the possible stress relations inside and among the series based on the Coulomb Failure Stress changes. Our results show that the Mula 1999 main earthquake, associated in different publications to the Crevillente fault, was positively influenced for triggering by the foreshock. We also find that the 1999 series aftershocks were jointly influenced for triggering by both the foreshock and the mainshock events. From a comparative analysis of P wave arrivals, the series seems to be associated to different neighbouring seismogenic zones belonging to the Crevillente system. For the Bullas 2002 and the Bullas–La Paca series, we found that it is unlikely that any of them have been triggered by the previous Mula 1999 event, but it appears that the main event in 2002 had an important influence on the triggering of its aftershock activity and the promotion of the mainshock and aftershock seismicity in the zone of the 2005 series. Summing the effects of the last two main events, it may have been produced different zones of high stress increase on the aftershock area of the 2002 and 2005 series. Finally, the seismicity of the two latest series could be associated to different neighbouring seismogenic zones.

New Approach to the Characterization of M max and of the Tail of the Distribution of Earthquake Magnitudes

We develop a new method for the statistical estimation of the tail of the distribution of earthquake sizes recorded in the Harvard catalog of seismic moments converted to m W -magnitudes (1977–2004 and 1977–2006). For this, we suggest a new parametric model for the distribution of main-shock magnitudes, which is composed of two branches, the pure Gutenberg-Richter distribution up to an upper magnitude threshold m 1, followed by another branch with a maximum upper magnitude bound M max, which we refer to as the two-branch model. We find that the number of main events in the catalog (N = 3975 for 1977–2004 and N = 4193 for 1977–2006) is insufficient for a direct estimation of the parameters of this model, due to the inherent instability of the estimation problem. This problem is likely to be the same for any other two-branch model. This inherent limitation can be explained by the fact that only a small fraction of the empirical data populates the second branch. We then show that using the set of maximum magnitudes (the set of T-maxima) in windows of duration T days provides a significant improvement, in particular (i) by minimizing the negative impact of time-clustering of foreshock/main shock/aftershock sequences in the estimation of the tail of magnitude distribution, and (ii) by providing via a simulation method reliable estimates of the biases in the Moment estimation procedure (which turns out to be more efficient than the Maximum Likelihood estimation). We propose a method for the determination of the optimal choice of the T value minimizing the mean-squares-error of the estimation of the form parameter of the GEV distribution approximating the sample distribution of T-maxima, which yields T optimal = 500 days. We have estimated the following quantiles of the distribution of T-maxima for the whole period 1977–2006: Q 16%(M max) = 9.3, Q 50%(M max) = 9.7 and Q 84%(M max) = 10.3. Finally, we suggest two more stable statistical characteristics of the tail of the distribution of earthquake magnitudes: The quantile Q T (q) of a high probability level q for the T-maxima, and the probability of exceedance of a high threshold magnitude ρ T (m*) = P{m k ≥ m*}. We obtained the following sample estimates for the global Harvard catalog $$ \hat{Q}_T (q=0.98)=8.6 {\pm}0.2 $$ and $$ \hat{\rho}_T (8)=0.13-0.20. $$ The comparison between our estimates for the two periods 1977–2004 and 1977–2006, where the latter period included the great Sumatra earthquake 24.12.2004, m W = 9.0 confirms the instability of the estimation of the parameter M max and the stability of Q T (q) and ρ T (m*) = P{m k ≥ m*}.

Dynamical scaling and generalized Omori law

The power law decay of the aftershocks rate is observed only after a characteristic time scale c. The dependence of c on the mainshock magnitude MM and on the lower cut‐off magnitude MI is well established. By considering ten sequences recorded in the California Catalog we show that the aftershock number distribution becomes independent of both MM and MI if time is rescaled by an appropriate time scale fixed by the difference MM − MI. This result is interpreted within a more general dynamical scaling hypothesis recently formulated, relating time differences to magnitude differences. The above hypothesis gives predictions in good agreement with the recent findings by Peng et al. (2007).

Statistical Features of ETAS Model Parameters in Early Stage and the Seismological Meanings

AbstractThe average statistical features of ETAS (Epidemic Type Aftershock Sequence) model parameters during the early stage (15 days) after the moderate or large earthquakes in Chinese mainland have been studied and the decay and generation of aftershocks under the different conditions have been discussed. Macroscopically, the numerical distribution of primary parameters of the ETAS model, b, p and α , are dispersed and the differences of the average value of parameters under different conditions are not very remarkable. In detail, b‐value increases with the magnitude of the mainshock, but the differences of b‐value are not obvious for different regions, for different mainshock fracture modes, as well as for different sequence types. p‐ and α ‐value show some regional features, average p‐values in southwest and northwest region of China are a little larger than that in Xinjiang and in North China, this means that the decay of aftershock activity is a little slower in southwest and in northwest region than in Xinjiang and in North China. Average α‐value is smaller in northwest region, and is larger in North China, which means that the earthquake in North China has a high ability to generate high‐order aftershocks, despite the aftershock decay in North China is something quick. And for northwest region of China, despite the aftershock decay is slow, due to the simplex framework of aftershock sequences, the ability of high‐order aftershock generation is something weak. It is not evident for a relation between p‐value and the fracture mode of the mainshock, in other words, the mainshock fracture mode is not the key factor controlling the aftershock decay rate. α‐value is related to the mainshock fracture mode to some extent. Averagely, α‐value is small for strike‐slip, middle for oblique slip, and large for dip slip. This expresses that the strike slip has the higher ability to generate aftershocks and it is weaker for dip slip and middle for oblique slip. p‐ and α‐value decreases with the increment of the mainshock magnitude, which means that the larger the mainshock, the slower the sequence decay and the higher the ability to generate offspring. There are some differences in p‐ and α‐values of different sequence types. p‐value is smaller for mainshock‐aftershock type (MAT) and is larger for isolated earthquake type (IET), expressing that aftershock decay rate is slow for MAT, is quick for IET and is in the middle for multiple mainshock type (MMT). α‐values of IET and MAT are about the same, both two are larger than that of MMT, this means that MMT has a higher ability to generate high‐order aftershocks than IET and MAT.

Application of the time-predictable model in Peninsular India for future seismic hazard assessment

It has been the belief among Earth scientists that the Peninsular Shield is aseismic, as the region attained stability long ago. However, the earthquake at Koyna (10 December 1967), Bhadrachalam (13 April 1969), Broach (23 March 1970), Hyderabad (30 June 1983), Khillari (30 September 1993), Jabalpur (22 May 1997), Gujarat (26 January 2001), and additional ones of smaller magnitudes, altered this concept. This area has experienced many widely distributed shallow earthquakes, some of them having large magnitudes. It is now widely accepted that seismic activity still continues with moderate events. Therefore, a need has arisen to take into consideration recent seismological data to assess the future seismic status of Peninsular India. Earthquake generation model has been studied to develop the statistical relations with surface wave magnitude (MS ≥ 4.5). Five seismogenic sources showing clustering of earthquakes and including at least three main shocks of magnitude 4.5 ≤ MS ≤ 6.5 giving two repeat times, have been identified. It is mainly based on the so-called “regional time-predictable model”. For the considered region it is observed that the time interval between two consecutive main shocks depends on the preceding main shock magnitude (Mp) not on the following main shocks magnitude Mf suggesting the validity of time predictable model in the region.

Quantifying the Media Bias in Intensity Surveys: Lessons from the 2001 Bhuj, India, Earthquake

Many seismologists have looked at the 26 January 2001 Bhuj earthquake as a key modern calibration event that could be used to improve estimates of magnitudes of large historic mainshocks in stable continental regions. Since no instrumental data are available for important historic events such as the 1819 Allah Bund, India, and the 1811–1812 New Madrid, central U.S. mainshocks, calibration hinges on comparisons of the macroseismic effects of these earthquakes with those of comparable modern earthquakes for which a reliable, instrumentally determined moment magnitude is available. However, although such a comparison is conceptually straightforward, in practice it is complicated by potentially significant inconsistencies in methods used to quantify macroseismic effects in different regions and/ or times. For the Bhuj earthquake, extensive intensity data sets have been compiled and published from both media accounts and detailed direct surveys. Comparing the two provides a quantification of the previously suspected media bias, whereby earthquake effects can be exaggerated in media accounts. This bias is a strong function of intensity level, with substantial bias at the highest shaking levels and significantly less bias at low intensities. Because only sparse documentary data are in general available for older historic earthquakes, the results of this study suggest that their inferred intensity distributions might be similarly biased. We further use the survey- based intensity values to develop a new relationship between intensities and ground motions.

Paleoseismological evidence for non-characteristic behavior of surface rupture associated with the 2004 Mid-Niigata Prefecture earthquake, central Japan

The 2004 Mid-Niigata Prefecture earthquake sequence (mainshock magnitude, M JMA 6.8), which occurred in an active fold-and-thrust belt in northern central Japan, generated a small thrust surface rupture (< 20 cm of vertical displacement) along a previously unmapped northern extension of the active Muikamachi–Bonchi–Seien fault zone, on the eastern margin of the epicentral region. To better understand past seismic behavior of the rupture, we conducted a paleoseismic trenching study across the 10-cm-high west-side-up surface rupture at the foot of a pre-existing 1.8-m-high east-facing scarp, which probably resulted from past earthquake(s). A well-defined west-dipping thrust fault zone accompanied by drag folding and displacing the upper Pliocene to lower Pleistocene strata and the unconformably overlying upper Pleistocene (?) to Holocene strata was exposed. The principal fault zone is connected directly to the 2004 surface rupture. From the deformational characteristics of the strata and radiocarbon dating, we inferred that two large paleoseismic events occurred during the past 9000 years prior to the 2004 event. These two pre-2004 events have a nearly identical fault slip (at minimum, 1.5 m), which is ≥ 15 times that of the 2004 event (∼ 10 cm). These paleoseismic data, coupled with the geological and geomorphological features, suggest that the 2004 event represented non-characteristic behavior of the fault, which can potentially generate a more destructive earthquake accompanied by meter-scale surface displacement. This study provides insight into the interpretation of past faulting events and increases our understanding of rupture behavior.

Does aftershock duration scale with mainshock size?

It has been claimed that aftershock duration scales with the mainshock recurrence interval [Dieterich, 1994]. This implies a scaling between aftershock duration and mainshock magnitude. To see if such a scaling exist, we compare properties of aftershock sequences associated with mainshocks whose size span several magnitude units. Earthquake properties were examined along two northern California fault segments, one is the Calaveras fault, another is a northern segment of the San Andreas fault. The first study area contains the rupture area of the M6.2, 1984 Morgan Hill, and the latter is adjacent the southern end of the M7.1, 1989 Loma Prieta rupture. Both the Morgan Hill and the Loma Prieta aftershock activities lasted 1–3 × 108 seconds. On the other hand, aftershock activities of small‐to‐moderate mainshocks lasted about 1–2 × 106 seconds. Nevertheless, there is no evidence for scaling between mainshock size and aftershock duration for moderate‐to‐small mainshocks. It thus seems that large earthquakes, because they break the entire seismogenic depth and interact with deeper parts of the crust, trigger a relaxation mechanism that is inactive during aftershock activity due to moderate‐to‐small quakes, and whose characteristic relaxation time is much longer than that of the brittle crust. Current seismicity models do not provide explanation for these observations.

Relocation of Early and Late Aftershocks of the 2001 Bhuj Earthquake Using Joint Hypocentral Determination (JHD) Technique: Implication toward the Continued Aftershock Activity for more than Four Years

We employed layered model joint hypocentral determination (JHD) with station corrections to improve location identification for the 26 January, 2001 Mw 7.7 Bhuj early and late aftershock sequence. We relocated 999 early aftershocks using the data from a close combined network (National Geophysical Research Institute, India and Center for Earthquake Research Institute, USA) of 8–18 digital seismographs during 12–28 February, 2001. Additionally, 350 late aftershocks were also relocated using the data from 4–10 digital seismographs/accelerographs during August 2002 to December 2004. These precisely relocated aftershocks (error in the epicentral location<30 meter, error in the focal depth estimation < 50 meter) delineate an east-west trending blind thrust (North Wagad Fault, NWF) dipping (~ 45°) southward, about 25 km north of Kachchh main land fault (KMF), as the causative fault for the 2001 Bhuj earthquake. The aftershock zone is confined to a 60-km long and 40-km wide region lying between the KMF to the south and NWF to the north, extending from 2 to 45 km depth. Estimated focal depths suggest that the aftershock zone became deeper with the passage of time. The P- and S-wave station corrections determined from the JHD technique indicate that the larger values (both +ve and -ve) characterize the central aftershock zone, which is surrounded by the zones of smaller values. The station corrections vary from −0.9 to +1.1 sec for the P waves and from −0.7 to +1.4 sec for the S waves. The b-value and p-value of the whole aftershock (2001–2004) sequences of Mw ≥ 3 are estimated to be 0.77 ± 0.02 and 0.99 ± 0.02, respectively. The p-value indicates a smaller value than the global median of 1.1, suggesting a relatively slow decay of aftershocks, whereas, the relatively lower b-value (less than the average b-value of 1.0 for stable continental region earthquakes of India) suggests a relatively higher probability for larger earthquakes in Kachchh in comparison to other stable continental regions of the Indian Peninsula. Further, based on the b-value, mainshock magnitude and maximum aftershock magnitude, the Bhuj aftershock sequence is categorized as the Mogi's type II sequence, indicating the region to be of intermediate level of stresses and heterogeneous rocks. It is inferred that the decrease in p-value and increase in aftershock zone, both spatially as well as depth over the passage of time, suggests that the decay of aftershocks perhaps could be controlled by visco-elastic creep in the lower crust.

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.

Correlation between the parameters of the aftershock rate equation: Implications for the forecasting of future sequences

We analyzed the correlations among the parameters of the Reasenberg and Jones [Reasenberg, P.A., Jones, L.M., 1989. Earthquake hazard after a mainshock in California, Science 243, 1173–1176] formula describing the aftershock rate after a mainshock as a function of time and magnitude, on the basis of parameter estimates made in previous works for New Zealand, Italy and California. For all of three datasets we found that the magnitude-independent productivity a is significantly correlated with the b-value of the Gutenberg–Richter law and, in some cases, with parameters p and c of the modified Omori's law. We also found significant correlations between p and c but, different from some previous works, not between p and b. We verified that assuming a coefficient for mainshock magnitude α ≈ 2/3 b (instead of b) removes the correlation between a and b and improves the ability to forecast the behavior of Italian sequences occurred from 1997 to 2003 on the basis of average parameters estimated from sequences occurred from 1981 to 1996. This assumption well agrees with direct α estimates made in the framework of an epidemic type model (ETAS) from the data of some large Italian sequences. Our results suggest a modification of the original Reasenberg and Jones (1989) formulation leading to predict lower rates (and probabilities) for stronger mainshocks and conversely higher rates for weaker ones. We also inferred that the correlation of a with p and c might be the consequence of the trade-off between the two parameters of the modified Omori's law. In this case the correlation can be partially removed by renormalizing the time-dependent part of the rate equation. Finally, the absence of correlation between p and b, observed for all the examined datasets, indicates that such correlation, previously inferred from theoretical considerations and empirical results in some regions, does not represent a common property of aftershock sequences in different part of the world.

Perspectives for earthquake prediction in the Mediterranean and contribution of geological observations

Abstract Accelerating seismic strain caused by the generation of intermediate-magnitude preshocks in a broad (critical) region, accompanied by decelerating seismic strain caused by the generation of smaller preshocks in the seismogenic region are systematically observed before strong mainshocks. On the basis of this seismicity pattern a model has been developed that seems promising for intermediate-term earthquake prediction, called the ‘Decelerating in-Accelerating out Seismic Strain Model’. Recent seismological data for the Mediterranean region are used here for backward and forward testing of this model. The selection of the broader Mediterranean region as a test area was motivated not only by the interest of time-dependent seismic hazard assessment in a high-seismicity and highly populated region but also by the fact that the Mediterranean is a natural geophysical and geological laboratory where both complex multi-plate and continuum tectonics are found in a more or less convergent zone. Within this complex geotectonic setting several geological phenomena such as subduction, collision, orogen collapse and back-arc extension take place, leading to the generation of a broad spectrum of mainshocks, reaching M W = 8.0 or greater for subduction-related thrust events and a variety of corresponding seismicity levels and neotectonic activity ranging from very low (e.g. large parts of Iberian peninsula) to very high (broader Aegean area). The backward procedure shows that all six strong (M ≥ 6.8) mainshocks that have occurred in the Mediterranean since 1980 had been preceded by preshock sequences that followed this seismicity pattern and satisfy all model constraints. Application of the model for future mainshocks has led to the identification of nine regions (in the Pyrenees, Calabria, NE Adriatic, Albania, Northern Greece, SE Aegean, NW Anatolia, western Anatolia, NE Anatolia) where current intermediate-magnitude seismicity satisfies the constraints of the model and corresponds to strong (M ≥ 6.2) mainshocks. The magnitudes, epicentres and origin times of these probably ensuing mainshocks, as well as their corresponding uncertainties, are estimated, so that it is possible to evaluate the model potential during the next decade (2006–2015). Furthermore, it is shown that geological observations of surface fault traces can contribute to the accurate location of the foci of future strong mainshocks in the Mediterranean and to an estimation of their sizes. For this purpose, globally valid relations between fault parameters based on geological observations (surface fault length, L S , and fault slip, u S ) and measures of mainshock size (mainshock magnitude, subsurface fault length, L , and fault slip, u ) are proposed.

Global Observational Properties of the Critical Earthquake Model

The preshock (critical) regions of 20 mainshocks with magnitudes between 6.4 and 8.3, which occurred recently (since 1980) in a variety of seismotectonic regimes (Greece, Anatolia, Himalayas, Japan, California), were identified and investigated. All these strong earthquakes were preceded by accelerating time-to-mainshock seismic crustal deformation (Benioff strain). The time variation of the cumulative Benioff strain follows a power law with a power value ( m = 0.3) in very good agreement with theoretical considerations. We observed that the dimension of the critical region increased with increasing mainshock magnitude and with decreasing long-term seismicity rate of the region. An increase of the duration of this critical (preshock) phenomenon with decreasing long-term seismicity rate was also observed. This spatial and temporal scaling expresses characteristics of the critical earthquake model, which are of importance for earthquake prediction research. We also showed that the critical region of an oncoming mainshock coincides with the preparing region of this shock, where other precursory phenomena can be observed.

An application of the time- and magnitude-predictable model to long-term earthquake prediction in eastern Anatolia

In order to estimate the recurrence intervals for large earthquakes occurring in eastern Anatolia, this region enclosed within the coordinates of 36∘–42∘N, 35∘–45∘E has been separated into nine seismogenic sources on the basis of certain seismological and geomorphological criteria, and a regional time- and magnitude-predictable model has been applied for these sources. This model implies that the magnitude of the preceding main shock which is the largest earthquake during a seismic excitation in a seismogenic source governs the time of occurrence and the magnitude of the expected main shock in this source. The data belonging to both the instrumental period (MS≥ 5.5) until 2003 and the historical period (I0≥ 9.0 corresponding to MS≥ 7.0) before 1900 have been used in the analysis. The interevent time between successive main shocks with magnitude equal to or larger than a certain minimum magnitude threshold were considered in each of the nine source regions within the study area. These interevent times as well as the magnitudes of the main shocks have been used to determine the following relations: $$\log T_t= 0.11M_{\min} - 0.12M_p - 0.11\log M_0 + 2.51$$ $$M_f= 0.89M_{\min} - 0.24M_p + 0.31\log M_0 - 4.99$$ fwawhere Tt is the interevent time measured in years, Mmin is the surface wave magnitude of the smallest main shock considered, Mp is the magnitude of the preceding main shock, Mf is magnitude of the following main shock, and M0 is the released seismic moment per year in each source. Multiple correlation coefficient and standard deviation have been computed as 0.50 and 0.28, respectively for the first relation. The corresponding values for the second relation are 0.64 and 0.32, respectively. It was found that the magnitude of the following main shock Mf does not depend on the preceding interevent time Tt. This case is an interesting property for earthquake prediction since it provides the ability to predict the time of occurrence of the next strong earthquake. On the other hand, a strong negative dependence of Mf on Mp was found. This result indicates that a large main shock is followed by a smaller magnitude one and vice versa. On the basis of the first one of the relations above and taking into account the occurrence time and magnitude of the last main shock, the probabilities of occurrence P(Δ t) of main shocks in each seismogenic source of the east Anatolia during the next 10, 20, 30, 40 and 50 years for earthquakes with magnitudes equal 6.0 and 7.0 were determined. The second of these relations has been used to estimate the magnitude of the expected main shock. According to the time- and magnitude-predictable model, it is expected that a strong and a large earthquake can occur in seismogenic Source 2 (Erzincan) with the highest probabilities of P10 = 66% (Mf = 6.9 and Tt = 12 years) and P10 = 44% (Mf = 7.3 and Tt = 24 years) during the future decade, respectively.

Distribution of the largest aftershocks in branching models of triggered seismicity: Theory of the universal Båth law

Using the epidemic-type aftershock sequence (ETAS) branching model of triggered seismicity, we apply the formalism of generating probability functions to calculate exactly the average difference between the magnitude of a mainshock and the magnitude of its largest aftershock over all generations. This average magnitude difference is found empirically to be independent of the mainshock magnitude and equal to 1.2, a universal behavior known as Båth's law. Our theory shows that Båth's law holds only sufficiently close to the critical regime of the ETAS branching process. Allowing for error bars +/- 0.1 for Båth's constant value around 1.2, our exact analytical treatment of Båth's law provides new constraints on the productivity exponent alpha and the branching ratio n: 0.9 approximately < alpha < or =1. We propose a method for measuring alpha based on the predicted renormalization of the Gutenberg-Richter distribution of the magnitudes of the largest aftershock. We also introduce the "second Båth law for foreshocks:" the probability that a main earthquake turns out to be the foreshock does not depend on its magnitude rho.

Magnitude‐dependent Omori law: Theory and empirical study

We propose a new physically based “multifractal stress activation” model of earthquake interaction and triggering based on two simple ingredients: (1) a seismic rupture results from activated processes giving an exponential dependence on the local stress and (2) the stress relaxation has a long memory. The combination of these two effects predicts in a rather general way that seismic decay rates after main shocks follow the Omori law ∼1/tpwith exponentsplinearly increasing with the magnitudeMLof the main shock. We carefully test the prediction on the magnitude dependence ofpby a detailed analysis of earthquake sequences in the southern California earthquake catalog. We find power law relaxations of seismic sequences triggered by main shocks with exponentspincreasing with the main shock magnitude by approximately 0.1–0.15 for each magnitude unit increase, fromp(ML= 3) ≈ 0.6 top(ML= 7) ≈ 1.1, in good agreement with the prediction of the multifractal model. The results are robust with respect to different time intervals, magnitude ranges, and declustering methods. When applied to synthetic catalogs generated by the epidemic‐type aftershock sequence (ETAS) model constituting a strong null hypothesis with built‐in magnitude‐independentpvalues, our procedure recovers the correct magnitude‐independentpvalues. Our analysis thus suggests that a new important fact of seismicity has been unearthed. We discuss alternative interpretations of the data and describe other predictions of the model.

Analysis of the Spatial Distribution Between Successive Earthquakes

Spatial distances between subsequent earthquakes in southern California exhibit scale-free statistics, with a critical exponent delta approximately 0.6, as well as finite size scaling. The statistics are independent of the threshold magnitude as long as the catalog is complete, but depend strongly on the temporal ordering of events, rather than the geometry of the spatial epicenter distribution. Nevertheless, the spatial distance and waiting time between subsequent earthquakes are uncorrelated with each other. These observations contradict the theory of aftershock zone scaling with main shock magnitude.