Seismicity study in Pournari reservoir area (W. Greece) 1981–2010

The spatial-temporal evolution of seismicity is examined, during the initial impoundment of Pournari reservoir located on Arachthos River (Western Greece), as well as for the next 30 years. The results show that, despite the relatively moderate-to-high seismicity from west to east, there is no remarkable earthquake in the vicinity before the first reservoir impoundment. Immediately after the impoundment (January 1981), and during the first 4 months, a considerable number of low-magnitude seismic events were recorded in the broader area of the dam. Moreover, two independent major events occurred on March 10, 1981 (M _L ?=?5.6) and April 10, 1981 (M _L ?=?4.7) with focal depths 13 and 10 km, respectively. The detailed analysis of the two corresponding aftershock sequences shows that they present different behaviors (e.g., larger b-value and lower magnitude of the main aftershock) than that of other aftershock sequences in Greece. This seismicity is probably due to triggering, via the water loading mechanism and the undrained response due to a flysch appearance on the reservoir basement. The activation of the thrust fault may be attributed to the bulging of evaporites that characterize the disordered structure of W. Greece, via possible water intake. The detailed processing of the recorded seismicity during the period 1982–2010, in comparison with the variations of Pournari Dam water level, shows an increase of shallow seismicity (h?≤?5 km) in the vicinity of the reservoir up to a 10-km distance—in contrast to the initial period, characterized by a number of deeper events due to the background response change from undrained to drained status.     

Probabilistic aftershock hazard analysis,two case studies in West and Northwest Iran

Aftershock hazard maps contain the essential information for search and rescue process, and re-occupation after a main-shock. Accordingly, the main purposes of this article are to study the aftershock decay parameters and to estimate the expected high-frequency ground motions (i.e., Peak Ground Acceleration (PGA)) for recent large earthquakes in the Iranian plateau. For this aim, the Ahar-Varzaghan doublet earthquake (August 11, 2012; M _N =6.5, M _N =6.3), and the Ilam (Murmuri) earthquake (August 18, 2014 ; M _N =6.2) have been selected. The earthquake catalogue has been collected based on the Gardner and Knopoff (Bull Seismol Soc Am 64(5), 1363-1367, 1974) temporal and spatial windowing technique. The magnitude of completeness and the seismicity parameters (a,??b) and the modified Omori law parameters (P,??K,??C) have been determined for these two earthquakes in the 14, 30, and 60 days after the mainshocks. Also, the temporal changes of parameters (a,??b,??P,??K,??C) have been studied. The aftershock hazard maps for the probability of exceedance (33%) have been computed in the time periods of 14, 30, and 60 days after the Ahar-Varzaghan and Ilam (Murmuri) earthquakes. For calculating the expected PGA of aftershocks, the regional and global ground motion prediction equations have been utilized. Amplification factor based on the site classes has also been implied in the calculation of PGA. These aftershock hazard maps show an agreement between the PGAs of large aftershocks and the forecasted PGAs. Also, the significant role of b parameter in the Ilam (Murmuri) probabilistic aftershock hazard maps has been investigated.     

The partitioning of radiated energy and the largest aftershock of seismic sequences occurred in the northeastern Italy and western Slovenia

We analyzed the most relevant seismic sequences that occurred from 1977 to 2007 in the Friuli-Venezia Giulia region (northeastern Italy) and western Slovenia. The eight aftershock sequences were triggered by low- to moderate-magnitude earthquakes with mainshock duration magnitude ranging from 3.7 to 5.6. The b-value of the Gutenberg–Richter law varies from 0.8 to 1.1. The modified Omori’s modeling of the sequences evidences values of the p exponent ranging from 0.8 to 1.0. Using the Reasenberg and Jones (Science 243:1173–1176, 1989; Science 265:1251–1252, 1994) approach, we computed the probabilistic estimate of the aftershock rates and the largest aftershock in given time intervals. The difference in magnitude between the mainshock and the largest aftershock is calculated according to the modified Båth law and using an approach that considers the partitioning of the radiated seismic energy between mainshock and aftershocks. The partitioning of the radiated seismic energy appears to play a significant role in the evolution of the sequences. We define the parameter R _ES as the ratio between the radiated seismic energy of the mainshock and the summation of the seismic energy radiated by the aftershocks. The difference in magnitude between the mainshock and the largest aftershock, calculated with the parameter R _ES, agrees well with the observed difference. In most sequences, the parameter R _ES decreases very quickly until the occurrence of the largest aftershock and then becomes constant. By analyzing the values of R _ES during the early hours following the mainshock, we found that the R _ES values after 24 h are well related to the final ones, calculated on the whole sequence, and to the differences in magnitude between the mainshock and the largest aftershock.     

Detailed Study of Time Variations in the Gutenberg–Richter <Emphasis Type="Italic">b</Emphasis>-value Based on Highly Accurate Seismic Observations at the Garm Prognostic Site,Tajikistan

The time variations in the Gutenberg–Richter b-value are minutely studied based on the data of highly accurate seismological observations at the Garm prognostic site, Tajikistan, where a stationary network of seismic stations of the Complex Seismological Expedition (CSE) of Schmidt Institute of Physics of the Earth (IPE) of the USSR (Russian) Academy of Sciences was in operation from 1955 to 1992. A total of 93035 local earthquakes ranging from 0.0 to 6.3 in the Ml magnitudes are considered. The spatiotemporal fluctuations in the minimal magnitude of completeness of the earthquakes, Mc, are analyzed. The study considers a 25-year interval of the observations at the center of the observation system within which Mc = 0.9. It is shown that in most cases, the b-value and log₁₀E^2/3 experience characteristic time variations before the earthquakes with magnitudes higher than the minimal magnitude of the predicted earthquake (MPE). The 6-year anomaly in the parameters’ b-value, log₁₀E^2/3, and log₁₀N associated with the single strongest earthquake with M = 6.3 that occurred in the observation region on October 26, 1984 is revealed. The inversely proportional relationship is established between the time variations in the b-value and the time variations in the velocities of seismic waves Vp and Vp/Vs. It is shown that the exponent p in the power function which links the time variations of the b-value and log₁₀E^2/3 is higher in the zones of crustal compression than in the zones of extension. It is simultaneously confirmed that the average b-value in the zones of compression is lower than in the zones of extension. It is established that in the case of earthquakes with M ≥ 2.6, the time series of seismic activity log₁₀N_i and the time series of the b-value are highly cross correlated with a coefficient of r ≈ 0.75, whereas in the case of earthquakes with M ≥ 0.9, the coefficient of cross correlation between these time series is close to zero (r ≈ 0.06). The law of variations in the slope of the lines approximating the relationship between the log₁₀N_i time series in the different magnitude ranges (M ≥ Mc_i) and b-value time series is obtained. It is hypothesized that the seismic activity of the earthquakes with high magnitudes can be estimated provided that the parameters of the time series of the b-value and time series of the number of earthquakes logN(ΔM_i) in the range of low magnitudes are known. It is concluded that using the parameter log₁₀N for prognostic estimates of the strong earthquakes only makes sense for earthquakes having moderate and large magnitudes. It is inferred that the time variations in the b-value are predominantly contributed by the time variations of the earthquakes with relatively large magnitudes.     

On the possible correlation between the Gutenberg-Richter parameters of the frequency-magnitude relationship

The study of the Gutenberg-Richter (GR) parameters a and b has been very important to describe and characterize the seismicity over the different seismic provinces around the world. As far as we know, the possible correlation between the GR parameters a and b has not received enough attention. Bayrak et al. reported the a and b values for 27 active seismic regions around the boundaries of the main tectonic plates of the world. From these data, we found that there exists a positive correlation between the a and b parameters (R =?0.85, R² =?0.72). On the other hand, we made around 150 computer runs of a spring-block model proposed by Olami et al. (Phys Rev Lett 68(8):1244–1247, 1992). This model roughly emulates the interaction between two fault planes and it reaches a self-organized critical state. With these simulations, we also found that the a and b parameters are positively correlated. Motivated by these results, we propose an analytical demonstration that indeed a and b are positively correlated. In addition, we discuss on other possible applications of the spring-block model to actual seismicity and to frictional experiments made with sandpapers.     

Quantitative Analysis of Seismicity Before Large Taiwanese Earthquakes Using the G-R Law

Seismicity has been identified as an example of a natural, nonlinear system for which the distribution of frequency and event size follow a power law called the “Gutenberg–Richter (G-R) law.” The parameters of the G-R law, namely b- and a-values, have been widely used in many studies about seismic hazards, earthquake forecasting models, and other related topics. However, the plausibility of the power law model and applicability of parameters were mainly verified by statistical error σ of the b-value, the effectiveness of which is still doubtful. In this research, we used a newly defined p value developed by Clausetet al. (Power-Law Distributions in Empirical Data, SIAM Rev. 51, 661–703, 2009) instead of the statistical error σ of the b-value and verified its effectiveness as a plausibility index of the power-law model. Furthermore, we also verified the effectiveness of K–S statistics as a goodness-of-fit test in estimating the crucial parameter \(M_{\text{c}}\) of the power-law model.     

Precursor times of abnormal <Emphasis Type="Italic">b</Emphasis>-values prior to mainshocks

Seismic observations exhibit the presence of abnormal b-values prior to numerous earthquakes. The time interval from the appearance of abnormal b-values to the occurrence of mainshock is called the precursor time. There are two kinds of precursor times in use: the first one denoted by T is the time interval from the moment when the b-value starts to increase from the normal one to the abnormal one to the occurrence time of the forthcoming mainshock, and the second one denoted by T _p is the time interval from the moment when the abnormal b-value reaches the peak one to the occurrence time of the forthcoming mainshock. Let T* be the waiting time from the moment when the abnormal b-value returned to the normal one to the occurrence time of the forthcoming mainshock. The precursor time, T (usually in days), has been found to be related to the magnitude, M, of the mainshock expected in a linear form as log(T)?=?q?+?rM where q and r are the coefficient and slope, respectively. In this study, the values of T, T _p , and T* of 45 earthquakes with 3?≤?M?≤?9 occurred in various tectonic regions are compiled from or measured from the temporal variations in b-values given in numerous source materials. The relationships of T and T _p , respectively, versus M are inferred from compiled data. The difference between the values of T and T _p decreases with increasing M. In addition, the plots of T*/T versus M, T* versus T, and T* versus T-T* will be made and related equations between two quantities will be inferred from given data.     

Gutenberg-Richter <Emphasis Type="Italic">b</Emphasis>-value maximum likelihood estimation and sample size

The Aki-Utsu maximum likelihood method is widely used for estimation of the Gutenberg-Richter b-value, but not all authors are conscious of the method’s limitations and implicit requirements. The Aki/Utsu method requires a representative estimate of the population mean magnitude; a requirement seldom satisfied in b-value studies, particularly in those that use data from small geographic and/or time windows, such as b-mapping and b-vs-time studies. Monte Carlo simulation methods are used to determine how large a sample is necessary to achieve representativity, particularly for rounded magnitudes. The size of a representative sample weakly depends on the actual b-value. It is shown that, for commonly used precisions, small samples give meaningless estimations of b. Our results give estimates on the probabilities of getting correct estimates of b for a given desired precision for samples of different sizes. We submit that all published studies reporting b-value estimations should include information about the size of the samples used.     

The Tolud Burst of Seismicity and the Earthquake of November 30, 2012 (<Emphasis Type="Italic">M</Emphasis><Subscript>C</Subscript> = 5.4, <Emphasis Type="Italic">M</Emphasis><Subscript>W</Subscript> = 4.8) that Accompanied the Start of the 2012‒2013 Tolbachik Eruption

This paper reports a study of the Tolud earthquake sequence; the sequence was a burst of shallow seismicity between November 28 and December 7, 2012; it accompanied the initial phase in the Tolbachik Fissure Eruption of 2012?2013. The largest earthquake (the Tolud earthquake of November 30, 2012, to be referred to as the Tolud Earthquake in what follows, with K_S = 11.3, M_L = 4.9, M_C = 5.4, and M_W = 4.8) is one of the five larger seismic events that have been recorded at depths shallower than 10 km beneath the entire Klyuchevskoi Volcanic Cluster in 1961?2015. It was found that the Tolud earthquake sequence was the foreshock–aftershock process of the Tolud Earthquake. This is one of the larger seismicity episodes ever to have occurred in the volcanic areas of Kamchatka. Data of the Kamchatka seismic stations were used to compute some parameters for the Tolud Earthquake and its largest (M_L = 4.3) aftershock; the parameters include the source parameters and mechanisms, and the moment magnitudes, since no information on these is available at the world seismological data centers. The focal mechanisms for the Tolud Earthquake and for its aftershock are consistent with seismic ruptures at a tension fault in the rift zone. Instrumental data were used to estimate the intensity of shaking due to the Tolud Earthquake. We discuss the sequence of events that was a signature of the time-dependent seismic and volcanic activity that took place in the Tolbachik zone in late November 2012 and terminated in the Tolud burst of seismicity. Based on the current ideas of the tectonics and magma sources for the Tolbachik volcanic zone, we discuss possible causes of these earthquakes.     

Temporal and spatial distributions of precursory seismicity rate changes in the Thailand-Laos-Myanmar border region: implication for upcoming hazardous earthquakes

To study the prospective areas of upcoming strong-to-major earthquakes, i.e., M _w  ≥ 6.0, a catalog of seismicity in the vicinity of the Thailand-Laos-Myanmar border region was generated and then investigated statistically. Based on the successful investigations of previous works, the seismicity rate change (Z value) technique was applied in this study. According to the completeness earthquake dataset, eight available case studies of strong-to-major earthquakes were investigated retrospectively. After iterative tests of the characteristic parameters concerning the number of earthquakes (N) and time window (T _w ), the values of 50 and 1.2 years, respectively, were found to reveal an anomalous high Z-value peak (seismic quiescence) prior to the occurrence of six out of the eight major earthquake events studied. In addition, the location of the Z-value anomalies conformed fairly well to the epicenters of those earthquakes. Based on the investigation of correlation coefficient and the stochastic test of the Z values, the parameters used here (N = 50 events and T _w  = 1.2 years) were suitable to determine the precursory Z value and not random phenomena. The Z values of this study and the frequency-magnitude distribution b values of a previous work both highlighted the same prospective areas that might generate an upcoming major earthquake: (i) some areas in the northern part of Laos and (ii) the eastern part of Myanmar.     

The spatiotemporal analysis of the minimum magnitude of completeness <Emphasis Type="Italic">Mc</Emphasis> and the Gutenberg–Richter law <Emphasis Type="Italic">b</Emphasis>-value parameter using the earthquake catalog of Greece

Spatiotemporal mapping the minimum magnitude of completeness Mc and b-value of the Gutenberg–Richter law is conducted for the earthquake catalog data of Greece. The data were recorded by the seismic network of the Institute of Geodynamics of the National Observatory of Athens (GINOA) in 1970–2010 and by the Hellenic Unified Seismic Network (HUSN) in 2011–2014. It is shown that with the beginning of the measurements at HUSN, the number of the recorded events more than quintupled. The magnitude of completeness Mc of the earthquake catalog for 1970–2010 varies within 2.7 to 3.5, whereas starting from April 2011 it decreases to 1.5–1.8 in the central part of the region and fluctuates around the average of 2.0 in the study region overall. The magnitude of completeness Mc and b-value for the catalogs of the earthquakes recorded by the old (GINOA) and new (HUSN) seismic networks are compared. It is hypothesized that the magnitude of completeness Mc may affect the b-value estimates. The spatial distribution of the b-value determined from the HUSN catalog data generally agrees with the main geotectonic features of the studied territory. It is shown that the b-value is below 1 in the zones of compression and is larger than or equal to 1 in the zones dominated by extension. The established depth dependence of the b-value is pretty much consistent with the hypothesis of a brittle–ductile transition zone existing in the Earth’s crust. It is assumed that the source depth of a strong earthquake can probably be estimated from the depth distribution of the b-value, which can be used for seismic hazard assessment.     

A first-order seismotectonic regionalization of Mexico for seismic hazard and risk estimation

The purpose of this work is to define a seismic regionalization of Mexico for seismic hazard and risk analyses. This seismic regionalization is based on seismic, geologic, and tectonic characteristics. To this end, a seismic catalog was compiled using the more reliable sources available. The catalog was made homogeneous in magnitude in order to avoid the differences in the way this parameter is reported by various agencies. Instead of using a linear regression to converts from m _b and M _d to M _s or M _w , using only events for which estimates of both magnitudes are available (i.e., paired data), we used the frequency-magnitude relations relying on the a and b values of the Gutenberg-Richter relation. The seismic regions are divided into three main categories: seismicity associated with the subduction process along the Pacific coast of Mexico, in-slab events within the down-going COC and RIV plates, and crustal seismicity associated to various geologic and tectonic regions. In total, 18 seismic regions were identified and delimited. For each, the a and b values of the Gutenberg-Richter relation were determined using a maximum likelihood estimation. The a and b parameters were repeatedly estimated as a function of time for each region, in order to confirm their reliability and stability. The recurrence times predicted by the resulting Gutenberg-Richter relations obtained are compared with the observed recurrence times of the larger events in each region of both historical and instrumental earthquakes.     

An experimental study on variation of <Emphasis Type="Italic">b</Emphasis> value during deformation of rock samples with different structural models

A preliminary study of b value of rocks with two kinds of structural models has been made on the base of a new acoustic emission recording system. It shows that b value of the sample decreases obviously when the sample with compressive en echelon faults changes into a tensile one after interchange occurs between stress axis σ ₁ and σ ₂. A similar experiment is observed when the sample with tensile en echelon faults changes into that with a bend fault after two segments of the en echelon fault linking up. These facts indicate that the variation of b value may contain the information of the regional dominant structural model. Therefore, b-value analyses could be a new method for studying regional dominant structural models.     

Bayesian estimation of the Modified Omori Law parameters for the Iranian Plateau

The forecasting of large aftershocks is a preliminary and critical step in seismic hazard analysis and seismic risk management. From a statistical point of view, it relies entirely on the estimation of the properties of aftershock sequences using a set of laws with well-defined parameters. Since the frequentist and Bayesian approaches are common tools to assess these parameter values, we compare the two approaches for the Modified Omori Law and a selection of mainshock–aftershock sequences in the Iranian Plateau. There is a general agreement between the two methods, but the Bayesian appears to be more efficient as the number of recorded aftershocks decreases. Taking into account temporal variations of the b-value, the slope of the frequency-size distribution, the probability for the occurrence of strong aftershock, or larger main shock has been calculated in a finite time window using the parameters of the Modified Omori Law observed in the Iranian Plateau.     

Spatiotemporal distributions of earthquakes in the northeastern segment of the Amur plate in two phases of variations in the modulus of the Earth’s rotation rate

The modeling results are presented on the annual dynamics of seismicity in the northeastern segment of the Amur plate, which are obtained from statistical studies of the number of earthquakes with magnitudes 2 ≤ М ≤ 6 in different phases of variations in the Earth’s rotation rate. We have calculated a degree of relationship between the observed seismicity variations and phases of decrease and increase in the Earth’s rotation rate for the magnitude ranges between 2 ≤ М < 4 and 4 ≤ М < 5 using rank correlation methods. It has been established that epicenters of earthquakes with magnitudes 5 ≤ М ≤ 6 are spatially grouped into a sequence of homogeneous equally spaced, 3.5°–4°, on average, east-westerly oriented clusters.     

Application of the dynamic calibration method to international monitoring system stations in Central Asia using natural seismicity data

The dynamic calibration method (DCM), using natural seismicity data and initially elaborated in [Kedrov, 2001; Kedrov et al., 2001; Kedrov and Kedrov, 2003], is applied to International Monitoring System (IMS) stations in Central Asia. The algorithm of the method is refined and a program is designed for calibrating diagnostic parameters (discriminants) that characterize a seismic source on the source-station traces. The DCM calibration of stations in relation to the region under study is performed by the choice of attenuation coefficients that adapt the diagnostic parameters to the conditions in a reference region. In this method, the stable Eurasia region is used as the latter. The calibration used numerical data samples taken from the archive of the International Data Centre (IDC) for the IMS stations MKAR, BVAR, EIL, ASF, and CMAR. In this paper, we used discriminants in the spectral and time domains that have the form

$D_i = X_i - a_m m_b - b_\Delta \log \Delta $

and are independent of the magnitude m _b and the epicentral distance Δ; these discriminants were elaborated in [Kedrov et al., 1990; Kedrov and Lyuke, 1999] on the basis of a method used for identification of events at regional distances in Eurasia. Prerequisites of the DCM are the assumptions that the coefficient a _m is regionindependent and the coefficient b _Δ depends only on the geotectonic characteristics of the medium and does not depend on the source type. Thus, b _Δ can be evaluated only from a sample of earthquakes in the region studied; it is used for adapting the discriminants D(X _i ) in the region studied to the reference region. The algorithm is constructed in such a way that corrected values of D(X i) are calculated from the found values of the calibration coefficients b _Δ, after which natural events in the region under study are selected by filtering. Empirical estimates of the filtering efficiency as a function of a station vary in a range of 95–100%. The DCM was independently tested using records obtained at the IRIS (Incorporated Research Institutions for Seismology) stations BRVK and MAKZ from explosions detonated in India on May 11, 1998, and Pakistan on May 28, 1998; these stations are similar in location and recording instrumentation characteristics to the IMS stations BVAR and MKAR. This test resulted in correct recognition of the source type and thereby directly confirmed the validity of the proposed calibration method of stations with the use of natural seismicity data. It is shown that the calibration coefficients b _Δ for traces similar in the conditions of signal propagation (e.g., the traces from Iran to the stations EIL and ASF) are comparable for nearly all diagnostic parameters. We arrive at the conclusion that the method of dynamic calibration of stations using natural seismicity data in a region where no explosions were detonated can be significant for a rapid and inexpensive calibration of IMS stations. The DCM can also be used for recognition of industrial chemical explosions that are sometimes erroneously classified in regional catalogs as earthquakes.

     

Determination of source parameters of strong earthquakes of Kamchatka,the Kurile Islands,and Japan from aftershock sequences

Aftershock sequences of some strong earthquakes of Kamchatka, the Kurile Islands, and Japan are examined. Such source parameters as the length L, along-dip width W, motion on fault D, and stress drop Δσ are determined from the aftershock sequences considered. The values of these parameters were obtained by the formal estimation of linear source parameters (lower bound estimates) and visually (upper bound estimates). The correlation dependences of the obtained parameters on the surface wave (M _S ) and seismic moment (M _W ) magnitudes are calculated.     

Scaling Analysis of Time Distribution between Successive Earthquakes in Aftershock Sequences

The earthquake inter-event time distribution is studied, using catalogs for different recent aftershock sequences. For aftershock sequences following the Modified Omori’s Formula (MOF) it seems clear that the inter-event distribution is a power law. The parameters of this law are defined and they prove to be higher than the calculated value (2–1/p). Based on the analysis of the catalogs, it is determined that the probability densities of the inter-event time distribution collapse into a single master curve when the data is rescaled with instantaneous intensity, R(t; M _th ), defined by MOF. The curve is approximated by a gamma distribution. The collapse of the data provides a clear view of aftershock-occurrence self-similarity.     

Study of seismicity in the NW Himalaya and adjoining regions using IMS network

The Reviewed Event Bulletin (REB) of the International Data Center (IDC) has been used in order to investigate the seismicity of the Northwest Himalaya and its neighboring region for the time period June 1999 to March 2015 within the geographical coordinates 25–40° N latitude and 65–85° E longitude. We have used a very precisely located earthquake dataset recorded by the International Monitoring System (IMS) Network containing 7,583 events with body wave magnitudes from 2.5 to 6.3. The study area has been subdivided into six regions based on the Flinn-Engdahl (F-E) seismic and geographical regionalization scheme, which was used as the region classifications of the International Data Center catalog. The examined region includes NW India, Pakistan, Nepal, Xizang, Kashmir, and Hindukush. For each region, Magnitudes of completeness (Mc) and Gutenberg-Richter (GR) recurrence parameters (a and b values) have been estimated. The Gutenberg-Richter analysis is preceded by an overview of the seismotectonics of the study area. The obtained Mc values vary from 3.5 to 3.9. The lower value of Mc was found mainly in Xizang region whereas the higher Mc threshold is evident in Pakistan region. However, the b values vary from 1.19 to 1.48. The lowest b value is recorded in Xizang region, which is mostly related to the Main Karakoram Thrust (MKT) fault, whereas the highest b values are recorded in NW India and Kashmir regions, which are mostly related to the Main Frontal Thrust (MFT) fault. The REB for the selected period has been compared to the most renowned bulletin of global seismicity, namely that issued by the National Earthquake Information Center (NEIC) of the United States Geological Survey (USGS). A study of 4,821 events recorded by USGS in the study region indicates that about 36 % of seismic events were missed and the catalog is considered as complete for events with magnitudes ≥4.0. However, both a and b values are obviously higher than those of IMS catalog. The a and b parameters in the Gutenberg-Richter magnitude–frequency relationship have been utilized to forecast the probability of future earthquakes of different magnitudes and returned periods (recurrence intervals).     

Sampling uncertainties and source <Emphasis Type="Italic">b</Emphasis> likelihood for the Gutenberg-Richter <Emphasis Type="Italic">b</Emphasis> value from the Aki-Utsu method

The Aki-Utsu method of Gutenberg-Richter (G-R) b value estimation is often misapplied so that estimations not using the G-R histogram are often meaningless because they are not based on adequate samples. We propose a method to estimate the likelihood Pr(b?b _m , N, M ₁, M ₂) that an observed b _m estimate, based on a sample of N magnitudes within an [M ₁????≤?ΔM/2,?M ₂?+?ΔM/2) range, where ΔM?=?0.1 is the usual rounding applied to magnitudes, is due to a “true” source b value, b, and use these likelihoods to estimate source b ranges corresponding to various confidence levels. As an example of application of the method, we estimate the b values before and after the occurrence of a 7.4-magnitude earthquake in the Mexican subduction zone, and find a difference of 0.82 between them with 100% confidence that the b values are different.