aa r X i v : . [ phy s i c s . g e o - ph ] J un Bath’s law, correlations and magnitudedistributions
B. F. ApostolInstitute of Earth’s Physics, Magurele-Bucharest MG-6,POBox MG-35, Romaniaemail: [email protected]
Abstract
The empirical Bath’s law is derived from the magnitude-differencestatistical distribution of earthquake pairs. It is shown that earthquakecorrelations can be expressed by means of the magnitude-differencedistribution. We introduce a distinction between dynamical correla-tions, which imply an "earthquake interaction", and purely statisticalcorrelations, generated by other, unknown, causes. The pair distri-bution related to earthquake correlations is presented. The single-event distribution of dynamically correlated earthquakes is derivedfrom the statistical fluctuations of the accumulation time, by meansof the geometric-growth model of energy accumulation in the focal re-gion. The derivation of the Gutenberg-Richter statistical distributionsin energy and magnitude is presented, as resulting from this model.The dynamical correlations may account, at least partially, for theroff-off effect in the Gutenberg-Richter distributions. It is shown thatthe most suitable framework for understanding the origin of the Bath’slaw is the extension of the statistical distributions to earthquake pairs,where the difference in magnitude is allowed to take negative values.The seismic activity which accompanies a main shock, including boththe aftershocks and the foreshocks, can be viewed as fluctuations inmagnitude. The extension of the magnitude difference to negative val-ues leads to a vanishing mean value of the fluctuations and to thestandard deviation as a measure of these fluctuations. It is suggestedthat the standard deviation of the magnitude difference is the average ifference in magnitude between the main shock and its largest after-shock (foreshock), thus providing an insight into the nature and theorigin of the Bath’s law. It is shown that moderate-magnitude dou-blets may be viewed as Bath partners. Deterministic time-magnitudecorrelations of the accompanying seismic activity are also presented. Bath’s law states that the average difference ∆ M between the magnitude ofa main shock and the magnitude of its largest aftershock is independent ofthe magnitude of the main shock (Bath, 1965; see also Richter, 1958). Thereference value of the average magnitude difference is ∆ M = 1 . . Deviationsfrom this value have been reported (see, for instance, Lombardi, 2002; Felzeret al, 2002; Console et al, 2003), some being discussed by Bath, 1965.The Bath’s law is an empirical statistical law. The earliest advance in un-derstanding its origin was made by Vere-Jones, 1969, who viewed the mainshock and its aftershocks as statistical events of the same statistical ensemble,distributed in magnitude. The magnitude-difference distribution introducedby Vere-Jones, 1969, may include correlations, which are viewed sometimesas indicating the opinion that the main shocks are statistically distinct fromthe aftershocks, or the foreshocks (Utsu, 1969, Evison and Rhoades, 2001).The Bath’s law enjoyed many discussions and attempts of elucidation (Pa-pazachos, 1974; Purcaru, 1974; Tsapanos, 1990; Kisslinger and Jones, 1991;Evison, 1999; Lavenda and Cipollone, 2000; Lombardi, 2002; Helmstetterand Sornette, 2003). The prevailing opinion ascribes the variations in ∆ M to the bias in selecting data and the insufficiency of the realizations of thestatistical ensemble. This standpoint was substantiated by means of the bino-mial distribution (Console, 2003; Lombardi, 2002; Helmstetter and Sornette,2003). In order to account for the deviations of ∆ M Helmstetter and Sor-nette, 2003, employed the ETAS (epidemic-type aftershock sequence) modelfor the differences in the selection procedure of the mainshocks and the af-tershocks. According to this model the variations in the number ∆ M arerelated to the realizations of the statistical ensemble and the values of thefitting parameters (see also Lombardi, 2002; Console, 2003).The statistical hypothesis for the distributions of earthquake time series is far-reaching. Usually, a statistical distribution is (quasi-) independent of time (it2s an equilibrium distribution), such that the aftershocks, distributed in the"future", should be identical with the foreshocks, distributed in the "past".However, it seems that there are differences between these two empirical dis-tributions (Utsu, 2002; Shearer, 2012). Moreover, in order the statisticallaws to be operational in practice, we need to have a large statistical ensem-ble of accompanying (associated) earthquakes, identified as aftershocks andforeshocks, which is a difficult issue in empirical studies. In addition, theempirical realizations of the statistical ensembles of earthquakes should berepetitive, a point which raises problems of principle, considering the varia-tions in time and locations. In order to get meaningful results by applyingthe statistical laws to physical ensembles, it is necessary to prepare, in thesame conditions, identical realizations of such ensembles. This is possible inStatistical Physics for various physical systems (like gases, liquids, solids,.etc). However, an ensemble of earthquakes, collected from a given zone ina (long) period of time, is not reproducible, because the conditions of itsrealization cannot be reproduced: the distribution of the focal regions maychange in the given zone, or in the next period of time, including the changein physical (geological) conditions. The application of the statistical methodto earthquakes exhibits limitations.We show in this paper that the appropriate tool of approaching the accom-panying seismic activity (foreshocks and aftershocks) of the main shocks isthe distribution function of the difference in magnitude. The derivation ofthis distribution is made herein by means of the conditional probabilities (theBayes theorem), as well as by using pair distributions. The earthquake corre-lations can be expressed by means of the magnitude-difference distribution.These correlations may be dynamical, or purely statistical. The dynamicalcorrelations arise from an "earthquake interaction", while the purely sta-tistical correlations originate in other, unknown, causes. Pair distributionsrelated to statistical correlations are presented. The singe-event distributionof dynamically correlated earthquakes is derived from the statistical fluctu-ations of the accumulation time, by means of the geometric-growth modelof energy accumulation in the focal region. The dynamical correlations mayaccount, at least partially, for the roll-off effect in the Gutenberg-Richterstatistical distributions. The difference in magnitude is extended to negativevalues, leading to a symmetric distribution for the foreshocks and aftershocks,with a vanishing mean value for the magnitude difference. This suggests toview the accompanying seismic activity as representing fluctuations, and to3ake their standard deviation as a measure for the Bath’s average difference ∆ M between the magnitude of the main shock and its largest aftershock(foreshock). This way, the Bath’s law is deduced. It is shown that moderate-magnitude doublets may be viewed as "Bath partners". Deterministic time-magnitude correlations of the associated seismic activity are also presented. According to the geometric-growth model of energy accumulation in a local-ized focal region (Apostol, 2006a,b), the accumulated energy E is related tothe accumulation time t by t/t = (1 + E/E ) r , (1)where t and E are time and energy thresholds and r is a geometrical pa-rameter which characterizes the focal region. This parameter is related tothe reciprocal of the number of effective dimensions of the focal region and tothe strain accumulation rate (which, in general, is anisotropic). Very likely,the parameter r varies in the range / < r < . For a pointlike focal regionwith a uniform accumulation rate r = 1 / (three dimensions), for a two-dimensional uniform focal region r = 1 / , while for a one-dimensional focalregion r tends to unity. An average parameter r may take any value in thisrange.In equation (1) the threshold parameters should be viewed as very small,such that t/t , E/E ≫ and equation (1) may be written as t/t ≃ ( E/E ) r . (2)A uniform frequency of events ∼ t /t in time t indicates that the parameter t is the reciprocal of the seismicity rate /t . It follows immediately thetime distribution P ( t ) dt = 1( t/t ) dtt (3)and, making use of equation (2), the energy distribution P ( E ) dE = r ( E/E ) r dEE . (4)4t this point we may use an exponential law E/E = e bM , where M isthe earthquake magnitude and b = · ln 10 = 3 . , according to Kanamori,1977, and Hanks and Kanamori, 1979 (see also, Gutenberg and Richter, 1944,1956); we get the (normalized) magnitude distribution P ( M ) dM = βe − βM dM , (5)where β = br . In decimal logarithms, P ( M ) = (1 . r ) · − (1 . r ) M , where . < . r < . (for / < r < ). Usually, the average value . r = 1 ( β = 2 . ) is currently used as a reference value, corresponding to r = 2 / (see, for instance, Stein and Wysesssion, 2003; Udias, 1999; Lay and Wallace,1995; Frohlich and Davis, 1993). Since t ≫ t , E ≫ E , the magnitudedistribution given by equation (5) is not adequate for very small magnitudes( M −→ ).It is worth noting that the magnitude distribution (equation (5)) has theproperty P ( M + M ) ∼ P ( M ) P ( M ) , while the time and energy distribu-tions (equations (3) and (4)) have not this property. This is viewed some-times as indicating that the earthquakes would be correlated in occurrencetime and in energy (Corral, 2006).The magnitude distribution is particularly important because it can be usedto analyze the empirical distribution P ( M ) = ∆ NN ∆ M = t ∆ NT ∆ M , (6)of ∆ N earthquakes with magnitude in the range ( M, M + ∆ M ) out of atotal number N = T /t of earthquakes which occurred in time T . We get ln (∆ N/T ) = ln β ∆ Mt ! − βM . (7)From the magnitude frequency ∆ N/T (equation (6)) we get the mean recur-rence time t r = t β ∆ M e βM (8)for an earthquake with magnitude M ( i.e. in the interval ( M, M + ∆ M ) ).This time should be compared with the accumulation time t a = t e βM for an5arthquake with magnitude M , given by equation (2) and the exponentiallaw E/E = e bM . These times are related by t a = ( β ∆ M ) t r , whence onecan see that t a < t r (for β ∆ M < ), a relationship which shows that theenergy corresponding to a magnitude M may be lost by seismic events lowerin magnitude, as expected. Moreover, by the definition of the seismicityrate, an earthquake with magnitude M is equivalent with a total number t a /t = e βM of earthquakes with zero magnitude (energy E ) (Michael andJones, 1998; Felzer et al, 2002). We may call these earthquakes (defined bythe seismicity rate) "fundamental earthquakes" corresponding to magnitude M . It is worth noting that the magnitude distribution βe − βM implies anerror of the order (cid:16) √ M − M (cid:17) /M = √ − , at least, i.e. , ∆ t r /t r ≃ . .For a maximal entropy with mean recurrence time t r we get easily a Poissondistribution (1 /t r ) e − t/t r for the recurrence time, which has a (large) standarddeviation q ( t − t r ) = t r .Similarly, from equation (5) we get the excedence rate (the so-called re-currence law), which gives the number N ex of earthquakes with magnitudegreater than M . The corresponding probability is readily obtained from (5)as P ex = e − βM , such that the excedence rate reads ln N ex = ln N − βM . (9)The distributions given above may be called Gutenberg-Richter statisticaldistributions (equations (7) and (9) and, implicitly, (5)). They are currentlyused in statistical analysis of the earthquakes. The parameter β is derivedby fitting these distributions to data. For / < r < it varies in the range . < β < . (for decimal logarithms . < . r < . ). For instance, ananalysis of a large set of global earthquakes with . < M < . ( ∆ M = 0 . )indicates β = 1 . (and /t = 10 . per year), corresponding to r = 0 . , avalue which suggests an intermediate two/three-dimensional focal mechanism(Bullen, 1963). For r = 1 / , corresponding to a uniform pointlike focalgeometry, we get β = 1 . . Equations (5), (7) and (9) have been fitted to aset of earthquakes with magnitude M ≥ ( ∆ M = 0 . ), which occurredin Vrancea between − ( years) (Apostol 2006a,b). The meanvalues of the fitting parameters are − ln t = 9 . and β = 1 . ( r = 0 . ).A similar fit has been done for a set of earthquakes with magnitude M ≥ which occurred in Vrancea during − ( years). The fittingparameters for this set are − ln t = 11 . and β = 2 . ( r = 0 . ). The data6or Vrancea have been taken from the Romanian Earthquake Catalog, 2018.The parameter β varies from region to region, depends on the nature of thefocal mechanism (parameter r in β = br ), the size and the time period of thedata set. The range of empirical values . < . r < . coincides with thetheoretical range ( / < r < ).The statistical analysis gives a generic image of a collective, global earthquakefocal region (a distribution of foci). Particularly interesting is the parameter r , which is related to the reciprocal of the (average) number of effectivedimensions of the focal region and the rate of energy accumulation. The value r = 0 . (Vrancea, period − ) indicates a (quasi-) two-dimensionalgeometry of the focal region in Vrancea, while the more recent value r = 0 . for the same region suggests an evolution of this (average) geometry towardsone dimension. At the same time, we note an increase of the seismicityrate /t in the recent period in Vrancea. The increase of the geometricalparameter r determines an increase of the parameter β , which dominates themean recurrence time. For instance, the accumulation time for magnitude M = 7 is increased from t a ≃ . years (period − to at least t a ≃ years. This large variability indicates the great sensitivity of thestatistical analysis to the data set. In particular, for any fixed M we mayview the exponential M e − Mβ as a distribution of the parameter β , whichindicates an error ≃ . in determining this parameter.We note that inherent errors occur in statistical analysis. For instance, an er-ror is associated to the threshold magnitude ( e.g. , M = 3 ), because the largeamount of data with small magnitude may affect the fit. Also, it is difficultto include events with high magnitude in a set with statistical significance,because such events are rare. The size of the statistical set may affect theresults. The fitting values given above for Vrancea have an error of approx-imately . Such difficulties are carefully analyzed on various occasions( e.g. , Felzer et al, 2002; Console et al, 2003; Lombardi, 2002; Helmstetterand Sornette, 2003).The statistical distributions given above may be employed to estimate con-ditional probabilities, and to derive Omori laws for the associated (accom-panying) seismic activity. Also, the conditional probabilities can be used foranalyzing the next-earthquake distributions (inter-event time distributions),(Apostol and Cune, 2020) which may offer information for seismic hazardand risk estimation. We present here another example of using these distri-butions, in analyzing the Bath’s empirical law.7 Bath’s law
In general, two or more earthquakes may appear as being associated in timeand space with, or without, a mutual interaction between their focal re-gions. In both cases they form a foreshock-main shock-aftershock sequencewhich exhibits correlations. The correlations which appear as a consequenceof an interaction imply an energy transfer (exchange) between the focal re-gions ( e.g ., a static stress). These correlations may be called dynamical (or"causal") correlations. Another type of correlations may appear without thisinteraction, from unknown causes. For instance, an earthquake may producechanges in the neighbourhood of its focal region (adjacent regions), and thesechanges may influence the occurrence of another earthquake. Similarly, anassociated seismic activity may be triggered by a "dynamic stress", not astatic one (Felzer and Brodsky, 2006). "Unknown causes" is used here in thesense that the model employed for describing these earthquakes does not ac-count for such causes. The correlations arising from "unknown causes" maybe called purely statistical (or "acausal") correlations. It is worth notingthat all the correlations discussed here have a statistical character.Let us first discuss the dynamical correlations. If an amount δE of en-ergy (positive or negative) is provided to a focal region by neighbouringfocal regions, the accumulation time t in the time-energy accumulation law t/t = ( E/E ) r (equation (2)) changes. For a given energy we may assignthis variation to the parameter r . This may correspond to a change in thefocal region subject to interaction (as, for instance, one produced by a staticstress). We are led to examine the changes in the accumulation time t . Wehave shown above that the ratio t/t is the number n = e βM of "fundamen-tal earthquakes" corresponding to the magnitude M , where t is the cutofftime (time threshold; equations (2) to (5)). We may write, approximately,this (large) number as the sum n = n X i =1 n i , n i = 1 . (10)In this equation we may view n i as statistical variables, with mean value n i = 1 . Therefore, we consider the number of fundamental earthquakes n = n X i =1 n i , (11)8ith the mean value n = n . The variables n i are viewed as independentstatistical variables; their fluctuations are due to the interaction of these fun-damental earthquakes with other fundamental earthquakes, correspondingto different magnitudes (exchange of numbers of fundamental earthquakes).These interactions play the role of an external "thermal bath" for the funda-mental earthquakes corresponding to a given magnitude M . Therefore, thedeviation ∆ n = q ( δn ) (where δn = n − n , δn i = n i − n i ) is the number n c of (dynamically) correlated fundamental earthquakes corresponding to M .We get immediately ( δn ) = n X i,j =1 δn i δn j = n X i =1 ( δn i ) = a n , (12)where a = ( δn i ) (independent of i ). It follows n c = a √ n = ae βM . (13)Obviously, this number is t c /at , where t c is the accumulation time for thecorrelated earthquakes with magnitude M (in order to preserve the corre-spondence of the cutoff time and energy parameters we need to re-define thecutoff time t as at ). Now, we can repeat the derivation from equations (2)to (5) and get the magnitude probability of the correlated earthquakes P c ( M ) = 12 βe − βM . (14)We note that in the energy-accumulation law given by equation (2) the pa-rameter r changes to r/ for dynamically correlated earthquakes ( t c /at =( E/E ) r/ ).The dynamically correlated earthquakes may be present in foreshock-mainshock-aftershock sequences. It is difficult to test empirically the probability P c ( M ) , because we cannot see any means to separate the dynamical corre-lations from the purely statistical correlations. We note that the dynam-ically correlated earthquakes are distinct from the rest of the earthquakes(their distribution ∼ e − βM is different from the distribution ∼ e − βM ). Usu-ally, in empirical studies we do not make this difference (see, for instance,Kisslinger, 1996), but the error may be neglected, because the total num-ber of earthquakes N is much larger than the total number of dynami-cally correlated earthquakes N c , according to equations (5), (6) and (14)9 N c = (4∆ N c / ∆ N ∆ M ) N ); N c is, simply, proportional to the statistical er-ror √ N . Like the total number of earthquakes, the dynamically correlatedearthquakes are concentrated in the region of small magnitudes, where theslope of the function ln P c ( M ) is changed from − β to − β . Such a devi-ation is well known in empirical studies (the roll-off effect; Pelletier, 2000;Bhattacharya et al, 2009), and is attributed usually to an insufficient de-termination of the small-magnitude data. Moreover, small-magnitude cor-related earthquakes in foreshock-main shock-aftershock sequences are asso-ciated with high-magnitude main shocks, which have a large productivityof accompanying events. Therefore, dynamically correlated earthquakes areexpected in earthquake clusters with high-magnitude main shocks.The Bath’s law is expressed in terms of the difference in magnitude betweenthe main shock and its largest aftershock. Let us first consider two earth-quakes with magnitudes M , , with the distribution law ∼ e − βM , M = M , ,and seek the distribution of the difference in magnitude m = M − M . Inthis law the magnitude M is positive, but for the difference M − M weneed to extend this variable to negative values. Since M = M − M + M and M = M − M + M , the law ∼ e − βM suggests a magnitude-differencedistribution ∼ e − β ( M − M ) for M > M and fixed M , and a distribution ∼ e − β ( M − M ) for M > M and fixed M . These are conditional probabilities(related to the Bayes theorem). In both cases, this distribution can be writtenas ∼ e − β | m | , where m = M − M (or m = M − M ), | m | < max ( M , M ) ,irrespective of which M , is fixed. The condition | m | < max ( M , M ) is es-sential for statistical correlations. We pass now to the Bath’s law. Let us as-sume a main shock, with magnitude M s , and the accompanying earthquakes(foreshocks and aftershocks), with magnitude M . We define the foreshocksand aftershocks as those correlated earthquakes with magnitude M smallerthan M s . We refer the magnitudes M of the foreshocks and aftershocks to themagnitude M s of the main shock, by observing the ordering of the partnersin each pair. This can be done by defining m = M s − M > for foreshocksand m = M − M s < for aftershocks. According to the above discussion, thedistribution of the magnitude difference is ∼ e − β | m | , | m | < M s . Therefore,the total distribution is P ( M s , m ) = β e − βM s e − β | m | , | m | < M s , M s > . (15)This is a pair distribution, for two events M s and m .Let us apply first this law to dynamically-correlated earthquakes, by replacing10 in equation (15) by β/ . Since these earthquake clusters are associated withhigh-magnitude main shocks we may omit the condition | m | < M s , and let | m | go to infinity. In this case the statistical correlations are lost; we areleft only with the dynamical correlations. The distribution given by equation(15) becomes a distribution of two independent events, identifed by M s and m ; we may use only the magnitude difference distribution p c ( m ) = 14 βe − β | m | , −∞ < m < + ∞ . (16)This distribution has a vansihing mean value m ( m = 0 ). The next correctionto this mean value, i.e. the smallest deviation of m , is the standard deviation ∆ m = q m = 2 √ β . (17)Therefore, we may conclude that the average difference in magnitude be-tween the main shock and its largest aftershock (or foreshock) is given bythe standard deviation ∆ M = ∆ m = 2 √ /β . This is the Bath’s law. Thenumber √ /β does not depend on the magnitude M s (but it depends on theparameter β , corresponding to various realizations of the statistical ensem-ble). It is worth noting that ∆ m given by equation (17) implies an averaging(of the squared magnitude differences). Making use of the reference value β = 2 . we get ∆ M = 1 . , which is the Bath’s reference value for the mag-nitude difference. In the geometric-growth model the reference value β = 2 . corresponds to the parameter r = 2 / . We can check that higher-order mo-ments m n , n = 2 , , ... are larger than m (for any value of β in the range . < β < . ).The result ∆ M = 2 √ /β could be tested empirically, although, as it is wellknown, there exist difficulties. In empirical studies the magnitude difference ∆ M is variable, depending on the fitting parameter β , which can be obtainedfrom the statistical analysis of the data. The results may tend to the value ∆ M = 1 . by adjusting the cutoff magnitudes (Lombardi, 2002; Console,2003), or by choosing particular values of fitting parameters (Helmstetterand Sornette, 2003); there are cases when the data exhibit values close to ∆ M = 1 . (Felzer et al, 2002). It seems that values closer to ∆ M = 1 . occur more frequently in the small number of sequences which include high-magnitude main shocks. 11f we extend the dynamical correlations to moderate-magnitude main shocks,we need to keep the condition | m | < M s , such that the distribution is P c ( M s , m ) = 14 β e − βM s e − β | m | , | m | < M s , M s > . (18)The standard deviation is now ∆ M = ∆ m = √ /β , which leads to ∆ M ≃ . for the reference value β = 2 . . Such a variablility of ∆ M can of-ten be found in empirical studies. For instance, from the analysis made byLombardi, 2002, of Southern California earthquakes 1990-2001 we may infere β ≃ and an average ∆ M ≃ . (with large errors). From Console et al,2003, New Zealand catalog (1962-1999) and Preliminary Determination ofEpicentres catalog (1973-2001), we may infere β ≃ . − . and an average ∆ M = 0 . − . , respectively, while ∆ M = √ /β gives . − . . Inother cases, like the California-Nevada data analyzed by Felzer et al, 2002,the parameters are β = 2 . and ∆ M ≃ . , in agreement with ∆ M = 2 √ /β .We note that ∆ M = √ /β given here is an over-estimate, because it extends,in fact, the dynamical correlations (equation (14)) to small-magnitude mainshocks.Leaving aside the dynamical correlations we are left with purely statisticalcorrelations for clusters with moderate-magnitude main shocks. In this casewe use the distribution given by equation (15), which leads to ∆ M =∆ m =1 / √ β and ∆ M = 0 . for the reference vale β = 2 . . The Bath partnerfor such a small value of the magnitude difference looks rather as a doublet(Poupinet et al, 1984; Felzer et al, 2004).For statistical correlations we can compute the correlation coefficient (vari-ance). The correlation coefficient R = M s M / ∆ M s ∆ M between the mainshock and an accompanying event M = M s − | m | ( | m | < M s ) computedby using the distribution given in equation (15) is R = 2 / √ . For the cor-relation coefficient beteen two accompanying events M and M we need thethree-events distribution (which includes M , and M s ). In general, the correlations are visible in the pair (two-event, bivariate) distri-butions. Such distributions are obtained as the mixed second-order deriva-12ive of a generating function of two variables. We give here the pair dis-tribution derived from the geometric-growth model of energy accumulation.We show that it coincides, practically, with the pair distribution used above(equation (15)). Moreover, the pair distribution derived here exhibits thedynamical correlations. The single-event probability P ( t ) given by equation(3) is obtained as the derivative P ( t ) = − ∂F/∂t of the frequency func-tion F ( t ) = t /t . Similarly, by using the change of variable t/t = e βM ,we get the Gutenberg-Richter magnitude distribution P ( M ) = βe − βM from P ( M ) = − ∂F/∂M , where F ( M ) = e − βM (equation (5)).Let us assume that two successive earthquakes may occur in time t , one aftertime t = t e βM , another after time t = t e βM from the occurrence of theformer. Using the partition t = t + t we get the distribution P ( t , t ) ∼ ∂ F∂t ∂t = 2 t ( t + t ) , (19)or, properly normalized, P ( M , M ) = 4 β e β ( M + M ) ( e βM + e βM ) . (20)We can see that this distribution is different from P ( M ) P ( M ) = β e − β ( M + M ) ,which indicates that the two events M , are correlated.Let M = M + m and M > M , < m < M ; equation (20) becomes P ( M , M ) = 4 β e − βM e − βm (1 + e − βm ) ; (21)similarly, for M > M , − M < m < we get P ( M , M ) = 4 β e − βM e βm (1 + e βm ) . (22)It follows that we may write P ( M , M ) = 4 β e − βmax ( M ,M ) e − β | m | (1 + e − β | m | ) , (23)which highlights the magnitude-difference distribution, with the constraint | m | < max ( M , M ) . 13et M s and M be the magnitudes of the main shock and an acompanyingearthquake (foreshock or aftershock), respectively. We define the orderedmagnitude difference m = M s − M > for foreshocks and m = M − M s < for aftershocks, | m | < M s . According to equation (23), the distribution ofthe pair consisting of the main shock and an acompanying event is P ( M s , m ) = 4 β e − βM s e − β | m | (1 + e − β | m | ) , | m | < M s . (24)The exponential e − β | m | falls off rapidly to zero for increasing m , so we mayneglect it in the denominator in equation (24). We are left with the pairdistribution given by equation (15) (properly normalized).If we integrate equation (20) with respect to M (and redefine M = M ), weget the so-called marginal distribution P mg ( M ) = βe − βM e − βM ) . (25)This distribution differs appreciably from the Gutenberg-Richter distribution βe − βM for βM ≪ and only slightly (by an almost constant factor ≃ ) formoderate and large magnitudes. The corresponding cummulative distribu-tion for all magnitudes greater than MP exmg ( M ) = e − βM
21 + e − βM (26)can be written as P exmg ( M ) ≃ e − βM − βM ≃ e − βM (27)in the limit M −→ , which indicates that the slope of the excedence rate ln P exmg ( M ) deviates from − β , corresponding to the usual Gutenberg-Richterexponential distribution, to − β (the roll-off effect). This deviation indi-cates the presence of dynamical correlations governed by the distributionlaw P c ( M ) = βe − βM (equation (14)).The pair distribution given above can be written both for the earthquakesgoverned by the Gutenberg-Richter distribution ∼ e − βM and for the sub-set of dynamically-correlated earthquakes governed by the distribution ∼ − βM . The procedure of extracting dynamically-correlated earthquakes canbe iterated, passing from β/ to β/ , etc; however, the number of affectedearthquakes tends rapidly to zero, and the procedure becomes irrelevant.We may assume that energy E is released by two successive earthquakes withenergies E , = E e bM , , such that E = E + E . The time correspondingto the energy E is t = t ( E /E + E /E ) r , or t = t (cid:16) e bM + e bM (cid:17) . Wecannot derive a pair distribution from the second-order derivative of t /t = (cid:16) e bM + e bM (cid:17) − , because the variation of the magnitudes M , implies thevariation of the energies E , , and not of the times t , . This would contradictthe geometric-growth model which assumes that the probabilities are givenby time derivatives. Let us assume that the amount of energy E accumulated in time t is releasedby two successive earthquakes with energies E , , such as E = E + E . Since,according to equation (2), t/t = ( E/E ) r = ( E /E + E /E ) r << ( E /E ) r + ( E /E ) r = t /t + t /t , (28)where t , are the accumulation times for the energies E , , we can see thatthe time corresponding to the pair energy is shorter than the sum of theindependent accumulation times of the members of the pair, as expected forcorrelated earthquakes. This is another type of correlations, different fromdynamical or statistical correlations. They are deterministic correlations,arising from the non-linearity of the accumulation law given by equation (2).The time interval τ between the two successive earthquakes, τ = t [(1 + E /E ) r − , (29)given by t = t + τ , depends on the accumulation time t . If we introducethe magnitudes M , in equation (29), we get τ = t h(cid:16) e − bm (cid:17) r − i , (30)15igure 1: The magnitude M of the accompanying seismic events vs the time τ elapsed from the main event with magnitude M and accumulation time t (equation (34) for M = 5 , b = 3 . , r = 2 / ). The Bath partner M ≃ . corresponds to τ /t ≃ × − . Higher values of the magnitude M occurat much longer times, where the correlations are unlikely.where m = M − M . We can see that this equation relates the time τ tothe magnitude difference. The same equation can be applied to dynamically-correlated earthquakes, by replacing r by r/ . We get τ = t (cid:20)(cid:16) e − bm (cid:17) r/ − (cid:21) , (31)These correlations can be called time-magnitude correlations.We apply this equation to a main shock-aftershock sequence, where M isthe magnitude of the main shock ( m > ); similar results are valid for theforeshock-main shock sequence. For the largest aftershock (foreshock), where m may be replaced by ∆ m = 2 √ /β , we get τ = t (cid:20)(cid:16) e − b ∆ m (cid:17) r/ − (cid:21) ≃≃ rt e − b ∆ m = rt e − √ /r (32)(for b ∆ m ≫ ). This is the occurrence time of the Bath partner, measuredfrom the occurrence of the main shock. The ratio τ /t varies between . × − ( r = 1 / ) and × − ( r = 1 ); for r = 2 / we get τ /t = 5 × − .It is worth noting, according to equation (28), that a partner close to the16ain shock in magnitude ( bm ≪ ) occurs after a lapse of time ∆ t ≃ t (cid:16) r/ − (cid:17) , (33)which is much greater than τ ( ∆ t/t varies between . and . for / The author is indebted to the colleagues in the Institute of Earth’s Physics,Magurele, to members of the Laboratory of Theoretical Physics, Magurele,for many enlightening discussions, and to the anonymous reviewer for useful19omments. This work was partially carried out within the Program Nucleu2016-2019, funded by Romanian Ministry of Research and Innovation, Re-search Grant REFERENCES Apostol, B. F. (2006a). A Model of Seismic Focus and Related StatisticalDistributions of Earthquakes. Phys. Lett. , A357, 462-466Apostol, B. F. (2006b). A Model of Seismic Focus and Related StatisticalDistributions of Earthquakes. Roum. Reps. Phys. , 58, 583-600Apostol, B. F. (2006c). Euler’s transform and a generalized Omori law. Phys.Lett. , A351, 175-176Apostol, B. F. & Cune, L. C. (2020). Short-term seismic activity in Vrancea.Inter-event time distributions. Ann. Geophys. , to appearBath, M. (1965). Lateral inhomogeneities of the upper mantle. Tectono-physics , 2, 483-514Bhattacharya, P., Chakrabarti, C. K., Kamal & Samanta, K. D. (2009).Fractal models of earthquake dynamics. Schuster, H. G. ed. Reviews ofNolinear Dynamics and Complexity pp.107-150. NY: Wiley.Bullen, K. E. (1963). An Introduction to the Theory of Seismology . London:Cambridge University Press.Console, R., Lombardi, A. M., Murru, M. & Rhoades, D. (2003). Bath’slaw and the self-similarity of earthquakes. J. Geophys. Res. , 108, 212810.1029/2001JB001651Corral, A. (2006). Dependence of earthquake recurrence times and indepen-dence of magnitudes on seismicity history. Tectonophysics , 424, 177-193Evison, F. (1999). On the existence of earthquake precursors. Ann. Geofis .,42, 763-770Evison, F. & and Rhoades, D. (2001). Model of long term seismogenesis. Ann. Geophys. , 44, 81-93Felzer, K. R., Becker, T. W., Abercrombie, R. E., Ekstrom, G. & Rice,J. R. (2002). Triggering of the 1999 M w M w J. Geophys. Res. , 107,2190 10.1029/2001JB000911Felzer, K. R., Abercrombie, R. E. & Ekstrom, G. (2004). A common originfor aftershocks, foreshocks and multiplets. Bull. Seism. Soc. Am. , 94, 88-98Felzer, K. R. & Brodsky, E. E. (2006). Decay of aftershock density withdistance indicates triggerring by dynamic stress. Nature , 441,735-738Frohich, C. & Davis, S. D. (1993). Teleseismic b values; or much ado about . . J. Geophys. Res. , 98, 631-644Gutenberg, B. & Richter, C. (1944). Frequency of earthquakes in California, Bull. Seism. Soc. Am. , 34, 185-188Gutenberg, B. & Richter, C. (1956). Magnitude and energy of earthquakes. Annali di Geofisica , 9, 1-15 ((2010) . Ann. Geophys. , 53, 7-12Hanks, T. C. & Kanamori, H. (1979). A moment magnitude scale. J. Geo-phys. Res. , 84, 2348-2350Helmstetter, A. & Sornette, D. (2003). Bath’s law derived from the Gutenberg-Richter law and from aftershock properties. Geophsy. Res. Lett. , 30, 206910.1029/2003GL018186Kanamori, H. (1977). The energy release in earthquakes. J. Geophys. Res. ,82, 2981-2987Kisslinger, C. (1996). Aftershocks and fault-zone properties. Adv. Geophys. ,38 1-36Kisslinger, C. & Jones, L. M. (1991). Properties of aftershock sequences inSouthern California. J. Geophys. Res. , 96, 11947-11958Lavenda, B. H. & Cipollone, E. (2000). Extreme value statistics and thermo-dynamics of earthquakes: aftershock sequences. Ann. Geofis. , 43, 967-982Lay, T. & Wallace, T. C. (1995). Modern Global Seismology . San Diego, CA:Academic.Lombardi, A. M. (2002). Probability interpretation of "Bath’s law. Ann.Geophys. , 45, 455-472Michael, A. J. & Jones, L. M. (1998). Seismicity alert probability at Park-field, California, revisited. Bull. Seism. Soc. Am. , 88, 117-13021gata, Y. & Tsuruoka, H. (2016). Statistical monitoring of aftershock se-quences: a case study of the 2015 M w Earth,Planets and Space , 68:44, 10.1186/s40623-016-0410-8Papazachos, P. C. (1974). On certain aftershock and foreshock parametersin the area of Greece. Ann. Geofis. , 24, 497-515Pelletier, J. D. (2000). Spring-block models of seismicity: review and analysisof a structurally heterogeneous model coupled to the viscous asthenosphere.Rundle, J. B., Turcote, D. L. & Klein, W. eds. Geocomplexity and the Physicsof Earthquakes . vol. 120. NY: Am. Geophys. Union.Poupinet, G., Elsworth, W. L. & Frechet, J. (1984). Monitoring velocity vari-ations in the crust using earthquake doublets: an application to the Calaverasfault, California. J. Geophys. Res ., 89, 5719-5731Purcaru, G. (1974). On the statistical interpretation of the Bath’s law andsome relations in aftershock statistics. Geol. Inst. Tech. Ec. Stud. Geophys.Prospect. (Bucharest), 10, 35-84Richter, C. F. (1958). Elementary Seismology (p.69). San Francisco, CA:Freeman. Romanian Earthquake Catalogue (ROMPLUS Catalog). (2018). NationalInstitute for Earth Physics, Romania.Shearer, P. M. (2012). Self-similar earthquake triggering, Bath’s law, andforeshock/aftershock magnitudes: simulations, theory, and results for SouthermCalifornia. J. Geophys. Res. , 117, B06310, 10.1029/2011JB008957Stein, S. & Wysession, M. (2003). An Introduction to Seismology, Earth-quakes, and Earth Structure . NY: Blackwell.Tsapanos, T. M. (1990). Spatial distribution of the difference between mag-nitudes of the main shock and the largest aftershock in the circum-Pacificbelt. Bull. Seism. Soc. Am. , 80, 1180-1189Udias, A. (1999). Principles of Seismology . NY: Cambridge University Press.Utsu, T. (1969). Aftershocks and earthquake statistics (I,II): Source param-eters which characterize an aftershock sequence and their interrelations. J.Fac. Sci. Hokkaido Univ. , Ser. VII, 2, 129-195, 196-266Utsu, T. (2002). Statistical features of seismicity. International Geophysics ,81, Part A, 719-732 22ere-Jones, D. (1969). A note on the statistical interpretation of Bath’s law.