Probabilistic interpretation of «Bath s Law»

Transcription

1 AALS OF GEOPHYSICS, VOL. 45,. 3/4, June/August 2002 Probabilisti interpretation of «Bath s Law» Anna aria Lombardi Istituto azionale di Geofisia e Vulanologia, Roma, Italy Abstrat Assuming that, in a atalog, all the earthquakes with magnitude larger than or equal to a utoff magnitude follow the Gutenberg-Rihter Law, the ompatibility of this hypothesis with «Bath s Law» is examined. Considering the mainshok 0 and the largest aftershok 1 of a sequene respetively as the first and the seond largest order statisti of a sample of independent and identially distributed exponential random variables, the distribution of 0, 1 and of their differene D 1 is evaluated. In partiular, it is analyzed as the distribution of D 1 hanges when only the sequenes with the magnitude of the mainshok above a seond threshold * are onsidered. It results that the distributions of 0, 1 depend on the differene * and on the number of events in the sequene. oreover, the expeted value of D 1 inreases with inreasing of * for every value of. Then it is shown that «Bath s Law» ould be asribed to seletion of data aused by the two thresholds and * and that it has a qualitative agreement with the model proposed. Key words magnitude distribution luster size b-value order statistis 1. Introdution In 1958, Rihter, in a note to his book, wrote: «Dr. Bath has lately noted that in many instanes the magnitude of the largest aftershok is about 1.2 less than that of the main shok» (Rihter, 1958, pag. 69). Some years later, Bath (1965) onfirmed this issue, asserting that in a atalog for shallow shoks and for magnitudes based on the surfae wave sale the sample mean of the differene D 1 between the magnitude of the mainshok 0 and ailing address: Dr. Anna aria Lombardi, Istituto azionale di Geofisia e Vulanologia, Via di Vigna urata 605, Roma, Italy; lombardi@ingv.it the respetive largest aftershok 1 followed the rule D (1.1) independently of the sequenes examined. He also provided some examples for whih the formula (1.1) did not ompletely fit the data and tried to make hanges to it. In spite of this, from then onwards, eq. (1.1) has beome, under the name of «Bath s Law», one of the most important and mentioned statistial laws onerning the distribution of the earthquakes in a seismi sequene. Several papers have been published onerning the law and many related aspets (i.e. the distribution of the mainshok and the one of the larger aftershok; the distribution of the events in a sequene; the relation of 0, 1 with b or with the number of events in the sequene,...) and in partiular, diverse attempts have been made to provide a statistial explanation of the eq. (1.1). The most ommonly aepted form for the magnitude distribution in a atalog is expressed 455

2 Anna aria Lombardi by the well-known formula (Gutenberg-Rihter law) log 10 [ ] a b (1.2) where () is the number of events with magnitude larger than of equal to (Gutenberg- Rihter, 1954). This equation is equivalent to the statement that the magnitudes of events in a atalog are independent and identially distributed random variables: their distribution is exponential with parameter b ln(10) (Ranalli, 1969). Considering that the estimated value of the parameter b in eq. (1.2) is lose to unity, the value of is about 2.3. If is the utoff magnitude (the lowest magnitude above whih the data set is omplete), the density funtion of the above-mentioned magnitudes is (Ranalli, 1969) ( m f( m) e ) m. (1.3) Utsu (1961, 1969) noted that if the magnitude of the mainshok was inluded in the sample of random variables with density funtion (1.3), the Gutenberg-Rihter law and «Bath s Law» were in strong ontradition. In fat, it is well-known by the theory of order statistis that if 1,... are independent and identially distributed random variables with density funtion (1.3), then the differene between the largest order statisti () and the seond largest order statisti (-1) is an exponential random variable with the same parameter as in (1.3), irrespetive of the sample size ; furthermore, under the same assumptions, this differene is positively orrelated with () (Feller, 1966; Utsu, 1969; Vere-Jones, 1969). Therefore Utsu pointed out that, if the mainshok was inluded in the random sample, the observed sample mean of D 1 (equal to 1.2) was onsiderably larger than the expeted value 1/ (0.5). oreover, the positive orrelation between () and () (-1), predited by the Probability Theory, was in disagreement with the independene, impliit in «Bath s Law», or with the marked negative orrelation, observed by himself on the Japanese aftershok sequenes (Utsu, 1961, 1969), between D 1 and 0. He inferred from these results that the mainshoks had a different distribution from the aftershoks: it was not aeptable to onsider the mainshok as the largest order statisti of an exponential random sample. The statistial interpretation of Utsu was onfuted by Vere-Jones (1969): he asribed the disrepanies observed by Utsu to the bias in seleting data and not to intrinsi properties of the aftershoks. In fat, only the sequenes with the magnitude of the mainshok above a seond threshold * (larger than ) were inluded in the data set. This seletion aused, in his opinion, a different distribution of order statistis, an expeted value larger than 1/ for D 1 and a negative orrelation between 0. Therefore, he onluded that «Bath s Law» and the results of Utsu were ompatible with the hypothesis that the mainshok is the largest member of a sample of independent exponential random variables. He orroborated this thesis in a seond work (Vere-Jones, 1975). It does not seem that the following papers on distribution of D 1 (Papazahos, 1974; Puraru, 1974; Tsapanos, 1990) have aepted the interpretation given by Vere-Jones, preferring to it the one given by Utsu. oreover, in other papers, «Bath s Law» is mentioned as a proof that the mainshok does not ome from the same population as the aftershoks (Evison, 1999; Evison and Rhoades, 2001; Lavenda and Cipollone, 2000). On the ontrary, in my opinion, Vere- Jones provides a satisfying explanation of the problem. The purpose of this paper is to go on with the interpretation of Vere-Jones making a thorough mathematial analysis of «Bath s Law» and of related matters, based only upon basi elements of the Probability Theory, and to verify if the hypothesis that all the events ome from the same population and «Bath s Law» are ompatible. 2. Data and statistial analysis Considering the magnitudes of the mainshok and of its aftershoks as a sample of independent and exponential distributed random variables, it is possible to evaluate the density funtions of 0, 1 onditioned by the event 456

3 Probabilisti interpretation of «Bath s Law» Fig. 1. Gutenberg-Rihter law for lustered events (33 360) of Southern California atalog, with magnitude 2.0 whih ourred in the interval time { 0 * } and the relative expeted values. In Appendix A, the mathematial elaboration is explained in detail: it is shown as the onditional ditributions of 0, 1 depend on b, * and on the number of aftershoks in the sequene (see eqs. (A.5), (A.7), (A.8)). To verify the reliability of the hypothesis of full ompatibility between the Gutenberg-Rihter law and «Bath s law», the atalog ompiled by the Southern California Earthquake Data Center, inluding events with magnitude equal to or larger than 2.0 ourred in the time period , was analyzed; the total number of events is As fig. 1 shows, the Gutenberg-Rihter law fits very well the magnitudes of events and the atalog an be onsidered omplete. To deluster the above-mentioned atalog the Reasenberg algorithm was used (Reasenberg, 1985). The lusters with aftershoks identified number 1763 and the lustered events are To estimate the b-value the maximum likelihood method was used (Utsu, 1966). The value obtained was bˆ The dependene of probability distributions of 0, 1 on implies that, to make a statistial analysis, the data must be divided into groups aording to the size of lusters. Table I shows the results of the analysis of atalog: the data have been divided into groups and only the results relative to all lusters and to groups with at least 30 sequenes have been reported. The lusters with * 4.0 and 2.0 have a very variable size and it is impossible to make a statistial analysis with data relative to a single value of ; then only the results relative to all lusters have been reported. Comparing the sample mean of 0 ( 0, D 1 ), relative to lusters with number of events (foreshoks have been left out) and with thresholds and *, with the orresponding 457

4 Anna aria Lombardi Table I. Results of the statistial analysis of the Southern California atalog divided aording to number of events in every luster and to the differene of the two thresholds. is the number of events in the luster (foreshoks have been left out); l is the number of lusters with number of events (only values with l 30 are shown); D 1 is the sample mean of D 1 ; IE [ D 1 ] is the theoretial expeted value of D 1 ; 0is the sample mean of 0 ; IE [ 0 ] is the theoretial expeted value of 0 ; EstCorr[ 0, D 1 ] is the estimated sample orrelation oeffiient; Corr[ 0, D 1 ] is the theoretial orrelation oeffiient. In olumns relative to and D 0 1, after the symbol ±, there is the standard deviation of the relative variable. l D 1 IE [D 1 ] 0 IE [ 0 ] EstCorr [ 0, D 1 ] Corr [ 0, D 1 ] all ± ± ± ± * ± ± ± ± (1763 lusters) ± ± ± ± ± ± all ± ± * ± ± ± ± (653 lusters) ± ± ± ± * all ± ± (142 lusters) averages expeted by the model (IE[ 0 ], IE[D 1 ]), a substantial agreement an be noted: for the same value of the sample mean for D 1 inreases with inreasing * and the overall mean value 1.2 an be justified only for a differene between the two thresholds equal to 2. The table also lists the estimated values of the orrelation oeffiient between 0 (EstCorr[ 0, D 1 ]) and the orresponding values predited by the model ( Corr [( D, 1 0 )/ { 0 }] ). For the sake of brevity not all alulations to evaluate it by the model are reported. However in fig. 2 its theoretial values are plotted versus for four values of *,100 and for b1. As fig. 2 shows, 0 are always positively orrelated and the orrelation o- effiient is inside the range [ ]. Also by the analysis of the atalog it results that D 1 and 0 are positively orrelated random variables and that the estimated values of the orrelation oeffiient agree with the theoretial ones. As table I shows, the overall estimated orrelation oeffiient between 0 for * 4.0 and 2.0 is very near to 0. As I show in following setion, this results is not in ontradition with the positive values of the theoretial orrelation oeffiient plotted in fig Disussion Unfortunaly, it is very diffiult to verify, with a rigorous statistial test, if the theoretial 458

5 Probabilisti interpretation of «Bath s Law» Fig. 2. Correlation oeffiient between 0, relative to the onditional distributions of Setion 2, versus the sample size for *, + 1, + 2, + 3. distribution evaluated in Appendix fit the data used by Utsu and other authors for their statistial analyses, most of all beause the sample size of every sequene is not known. However some results, presented in the works on «Bath s Law», seem to agree with the model. For example, an explanation of Puraru s results (1974), obtained in his analysis of Japanese and Greek data, an be provided. In his paper, he observed that in both data sets, 0 followed the exponential distribution with the same parameter as the distribution of general earthquakes. Vere-Jones (1969) showed that this result was inompatible with the Probability Theory (the maximum of a sample of independent and identially distributed random variables does not follow the same distribution as the elements of the sample) and that the distribution of 0 was only asymptotially (i.e. when * + ) an exponential one. Figure A.1a-e shows that for every value of, 0 is not an exponential random variable and that the mainshoks have a different distribution from that of any random variable in the sample: in fat it is not a generi variable of the sample, but it is the largest one. However, the distribution of 0 onverges rather fast to an exponential when * + and then the model is ompletely in agreement with their onlusions, onsidering that in the atalogs analyzed by the two authors * it is larger than or equal to 2; to be preise: in Vere-Jones paper, for the Japanese atalog, it is 0 6 and 1 4; in Puraru s paper, for the Japanese atalog, it is 0 6 and 1 3.2; in Puraru s paper for the Greek atalog it is and It has been observed (Solov ev and Solov eva, 1962; Vere-Jones, 1969; Lavenda and Cipollone, 2000) that there is a linear relation between the sample mean of the magnitude of the mainshok and the natural logarithm of the number of events (or only of the aftershoks) in the sequenes of a atalog. oreover, Vere-Jones 459

6 Anna aria Lombardi Fig. 3. Conditional expeted value of 0 by the event { 0 * } versus the natural logarithm of the sample size for *, + 1, + 2, + 3. (1969) demonstrated that there is a linear relation between the expeted value of () and ln() when * is equal to. As fig. 3 shows, eq. (A.6) is onsistent with this result for * equal to, but when * is larger than, the relation is only asymptotially (i.e. when + ) linear. With regard to the distribution of 1, Puraru (1974) noted that it is loser to a normal distribution that an exponential one. Considering the high differene between the two ut-off values of 0 and 1 in two data sets examined by him, this issue is predited by the Gutenberg- Rihter law (see fig. A.2a-d). oreover, there is an impressive similarity between the smoothed histograms for the distribution of 1, shown in figs. 14, 15 and 16 of the Puraru s work, and the theoretial densities plotted in fig. A.2a-d of the present paper, for * 2 and * 3. oreover, omparing fig. A.1a-e to fig. A.2a-d, it is evident that the distribution of 0 is not equal to that of 1. Therefore, the observed differenes between the two above-mentioned distributions do not neessarily have to be asribed to «different onditions in whih the mainshoks and the largest aftershoks our» (Puraru, 1974): 0 and 1 are not variables seleted at random, but they are the first and the seond largest observations of a random sample respetively and, in agreement with the theory of order statistis, they annot have the same distribution. Then, even onsidering the Gutenberg-Rihter law true for all the events, the Probability Theory predits that the mainshok and the largest aftershok ome from different populations. As regards the distribution of D 1, Puraru (1974) showed in his paper that it was not an exponential one: in fat the histograms of the differenes between the mainshoks and the relative largest aftershoks for the atalogs tested by him did not agree with the exponential density. oreover, the evaluated values of the oeffiient of variation (0.52 for Japan and 0.5 for Greee) were disordant with the value (equal to 1) of the oeffiient of variation for an exponential variable. Furthermore, in his onlusions, he pointed out that the observations did not onfirm, 460

7 Probabilisti interpretation of «Bath s Law» generally, «Bath s Law» and that, when the same ut-off was hosen for 0 and 1, the distribution of D 1 seemed to be an exponential one with a sample mean lose to 0.5 (see Puraru, 1974: tables 7 and 8) and an estimated oeffiient of variation lose to 0.8. In the end, he observed that the sample mean of D 1 inreased with the inrease in the differene between the ut-off value of 0 and 1 by a linear dependene. Similar results had been obtained by Vere-Jones (1969) in his analysis of the atalog of Japanese aftershok sequenes ompiled by Utsu in In fat, he had shown that when the differene between the two thresholds dereased from 2 to 0, the sample mean of D 1 dereased from 1.39 to 0.59 and the oeffiient of variation inreased from 0.54 to 0.9. As fig. A.3a-e, figs. 4 and 5 show, all these results ompletely agree with the model. In fat, D 1 is an exponential variable only when * is equal to 0 or when * is positive and +. In fig. 4, the oeffiient of variation is plotted versus for b equal to 1 and * equal to 2.0 (value of * for the Japanese atalog analyzed by Utsu, 1961 and Vere-Jones, 1969), 2.1 and 2.8 (values of * for the Japanese atalog and the Greek atalog, respetively, analyzed by Puraru, 1974): these urves ould be onsistent with the values, lose to 0.5, of the analyses of these authors. Fig. 4. Coeffiient of variation of D 1 relative to the onditional density funtion of eq. (A.8) versus the sample size for * 2.0 (value of the differene between the two thresholds of the Japanese atalog analyzed by Utsu (1961, 1969)) and by Vere-Jones (1969)), for * 2.1 (value of the differene between the two thresholds of the Greek atalog analyzed by Puraru (1974)) and for * 2.8 (value of the differene between the two thresholds of the Japanese atalog analyzed by Puraru, 1974). 461

8 Anna aria Lombardi Fig. 5. Conditional expeted value of D 1 by the event { 0 * } versus the differene between the two thresholds * for 2, 10, 100, The plot in fig. 5 shows, for b equal to 1 and four values of, the relation between the onditional average IE [ D / 1 { 0 }] and the differene of the two thresholds * and : it is onsistent with the linear relation observed by Puraru (1974: fig. 11), onsidering that in both atalogs analyzed by him * is larger than 2.0. In a more reent paper, Drakatos and Latoussakis (2001), in their desription of spatial and temporal harateristis of sequenes in Greee, dealt with the distribution of D 1. They onluded that data were fitted by a normal distribution with an average of 0.9. Considering that the two thresholds hosen were 3.2 and * 5.0, this result is onsistent with the model: the similarity between fig. 5 of the Drakatos- Latoussakis paper and fig. A.3a-d of the present work is evident. The only point that seems to be in disagreement with the results and onlusions of Utsu and Puraru, is the orrelation oeffiient between 0. In fat fig. 2 shows that it is always positive for every value of and *. Also the analysis of the Southern California atalog, in the previous setion, shows a positive orrelation between the two variables, but the overall estimated orrelation oeffiient for 2.0 and * 4.0 is To weigh the ompatibility of this value with the model, a simulation was done of 1000 groups of 142 lusters of independent exponential variables with the same size as lusters identified in the California atalog and the same b-value. The values obtained for the orrelation oeffiient are inside the range [0.010, 0.522]; moreover 15 of them are less than 0.08 and 190 are less than 0.2. Then, onsidering all lusters, apart from 462

9 Probabilisti interpretation of «Bath s Law» their size, the value of the orrelation oeffiient between 0 an be rather low. It is not easy to justify this result with the model proposed: it beomes neessary to onsider a distribution funtion for the luster size and to study the distribution of 0 for a generi luster (aside from ). However it is not impossible for 0 to have a orrelation oeffiient very near to 0 or even negative, as for the data analyzed by Utsu and Puraru. oreover, as the same authors noted, for some lusters of their atalogs, the values of D 1 were not known and then they were exluded in the statistial analysis: this seletion ould have aused a bias in results. In my opinion, there are too many fators that influene the results: the delustering algorithm used, the preision in estimate of magnitudes and then of b-value, the number of events not reorded (it is known that soon after a mainshok with high magnitude, the aftershoks are not reorded). The question needs further enquiries, but, in my opinion, only by analysis of data, without hypotheses about the magnitude distribution, it is evident that the value 1.2 of the «Bath Law» is justified only if the atalog is seleted with thresholds that have a differene equal to or larger than 2.0. oreover, it does not seem to me that the previous studies exhaustively justify the inompatibility between the Gutenberg-Rihter law and the «Bath Law». 4. Conlusions Considering the Gutenberg-Rihter law true for all the earthquakes of a atalog and utilizing only the results of the Probability Theory about the order statistis, the onditioned distributions of 0, 1 by the event { 0 * } are evaluated. As shown in the previous setions, most of the results obtained in the past on «Bath s Law» are onsistent with the probabilisti analysis expounded in the seond setion. In partiular, assuming that the mainshok is the largest event of a sample of independent and identially distributed exponential random variables, it results that: 1) 0 onverges in distribution to an exponential variable when * + : for low values of the onvergene is very fast and this ould justify the frequent onlusion that the mainshok magnitude is an exponential variable. 2) 0 and 1 are random variables with different distributions, not beause the mainshoks have a different nature from aftershoks, but beause the order statistis are not independent and identially distributed random variables as the non-ordered random variables of the sample. 3) The hoie of the two thresholds and * is ruial for the distributions of 0, 1 and D 1. oreover, these distributions depend on and, of ourse, on b. In partiular, when * is larger than, D 1 is not an exponential variable, its expeted value is higher than 1/ and it ould be onsistent with «Bath s Law». 4) There is a substantial agreement between the theoretial model presented and the results obtained in the past on the distribution of 0, 1 and those obtained by the analysis of the Southern California atalog. In partiular, the hypothesis of Vere-Jones, i.e. that «Bath s Law» an be asribed to the seletion of data aused by the hoie of the threshold of the mainshoks *, seems to be onfirmed. 5) Even if the model predits a positive orrelation between 0 for every value of * and, it has been shown by simulations that for all lusters, apart from, the value of the orrelation oeffiient an beome near to 0 or negative. This result ould justify the negative or law orrelation observed by Utsu and Puraru, onsidering also that for atalogs analyzed by them, the values of D 1 were not all known and that a seletion of data was made. The purpose of the present study was to show that «Bath s Law» is justified only if atalogs are suitably seleted and that it does not seem to be in disagreement with the very simple hypothesis that all the events follow the Gutenberg- Rihter law. Of ourse, further analyses must be made on this matter. First of all, the model has to be tested on a suitable data set: in fat the dependene on of the distribution of D 1 in the model requires that this be tested in a atalog with many aftershok sequenes and that the size of every sequene is known. As observed in 463

10 Anna aria Lombardi Setion 2, when * 0, the size of sequenes is highly variable and then it is impossible to make a statistial analysis for a value of. oreover, the influenes of delustering algorithms and of rules used to identify the sequenes ould be studied. In fat, the definitions themselves of «mainshok-aftershok sequenes» hosen by some authors for their analyses (see, i.e. Utsu, 1970; Evison, 1981; Evison and Rhoades, 1993) ould ause arbitrary seletions of data and then they ould further modify the distribution of D 1. It ould also be interesting to study the influene D 1. of the magnitude sale used. In onlusion, the same probabilisti elaboration ould be made utilizing different distributions from the exponential one for the magnitudes of a atalog (i.e. the Kagan distribution; Kagan, 1997). Aknowledgements The author is grateful to Dr. R. Console (IGV) and to reviewers of this paper for their useful omments. Appendix A. athematial bakground. Let 1,..., be independent and identially distributed random variables with density funtion (1.3) and let (1)... () be the orresponding order statistis; then (Feller, 1966; Casella-Berger, 1990) (1),..., () are not independent and the density funtion of (i), 1 i, is f! i1 i m m e ( ) () e ( ) e ( ) 1 ; i ( i 1)!( 1)! m m (A.1) the joint density funtion of (i) and ( j), 1 i < j, is f () m n i ( j )(, ), i j i ( m) ( n) ( m) ( m) ( n) n Ci, je e ( e ) ( e e ) ( e ) j (A.2) where 2! Ci, j. ( i 1)!( j1i)!( j)! Let s onsider the random variables D 1 () (-1), D 2 (-1) (-2),..., D -1 (2) (1), D (1) ; then D 1,...,D are independent exponential random variables with parameter, 2,...,, respetively. Therefore the distribution of D 1 is independent of the sample size and its expeted value is 1 IE [ D 1 ]. (A.3) 464

11 Probabilisti interpretation of «Bath s Law» Furthermore D 1 and () are positively orrelated and the orrelation oeffiient is Var ( D 1 ) ( ) Var (A.4) (Vere-Jones, 1969). From now onwards we shall denote the largest order statisti () and the seond largest order statisti (1) with 0 and 1, respetively. Let now * be a onstant larger than or equal to. Let s ompute the density funtion of 0, 1 onditioned by event { 0 * }. A.1. The density funtion of 0 onditioned by { 0 * } Let s denote the density funtion of 0 onditioned by the event { 0 * } with f. Then { 0 /{ 0 }} f f m { 0 / { 0 }} + f ( ) ( ) ( m) ( x) dx { 0 if m< m m e ( ) 1 e ( e ( 1 1 ) ) 1 if m. (A.5) The relative onditional expeted value is [ / 0 { + IE 0 }] mf ( m) dm 0 /{ 0 } + ( ( ) ) 1 1e 1 { } k 1 ( ( ) ) 1 1e k k. (A.6) The onditional distribution of 0 depends on as well as on the sample size. Furthermore it depends on differene between the two thresholds * and, but this dependene vanishes when + (see fig. A.1a-d). In fig. A.1e the onditional expeted value of 0 (eq. (A.9)) is plotted versus the sample size for four values of * and for b 1: it is evident that the value of IE [ / 0 { 0 }] inreases with the inreasing of and of *. 465

12 Anna aria Lombardi a b d e Fig. A.1a-e. a) Conditional density funtion of 0 by the event { 0 * } (see eq. (A.5)) for *, + 1, + 2, + 3, and for 2. b) The same as (a) but for 10. ) The same as (a) but for 100. d) The same as (a) but for e) Conditional expeted value of 0 by the event { 0 * } (see eq. (A.6)) versus the sample size for four values of differene between * and ( * 0,1,2,3). 466

13 Probabilisti interpretation of «Bath s Law» a b d Fig. A.2a-d. a) Conditional density funtion of 1 by the event { 0 * } (see eq. (A.7)) for *, + 1, + 2, + 3, and for 2. b) The same as (a) but for 10. ) The same as (a) but for 100. d) The same as (a) but for A.2. The density funtion of 1 onditioned by { 0 * }. Let s denote the density funtion of 1 onditioned by the event { 0 * } with Then 0 if m + f m n dn ( ) (, ) 1, + if < m f m { f n dn { 1 / { 0 }} + f m n dn m (, ) 1, + if m> f ( n) dn f. { 1 / { 0 }} 467

14 Anna aria Lombardi { 0 if m 2 m m ( ) ( ) ( ) 1 e e ( e ) 1 if < m ( ) 1 1e 2 2 m m e ( ) ( 1) ( 1 e ) m> ( e ( 1 1 ) if ) (A.7) As eq. (A.7) and figs. A.2a-d show, also the distribution of 1 depends on b, and * but the dependene on * is less marked than that of 0. A.3. The density funtion of D 1 onditioned by { 0 * }. Let s denote the density funtion of D 1 onditioned by the event { 0 * } with Then f. { D1 / { 0 }} IP ( D x/ 1 { 0 }) IP ( D x, 1 0 ) IP ( 0 ) IP ( x,, > )+ IP x,, ( 0 ) IP ( 0 ) IP ( >, + x)+ IP max(, x), + x IP IP ( >, + x )+ IP (, + x ) if x > { IP ( >, + x )+ IP ( x, + x ) if x 468

15 Probabilisti interpretation of «Bath s Law» { + s+ x s+ x s ( 1) ( 1),, ds dt f ( s, t)+ ds dt f ( s, t) if x > + s+ x s+ x s ( 1) x ( 1, ), ds dt f ( s, t)+ ds dt f ( s, t) if x Computing the integrals and deriving the onditional distribution funtion, it results f x / { } { D1 0 } e e ( ) e e ( ) 1 1 e e 1 x x x 1 e ( 1 1 ) { if x * 1 1 e x e ( ( ) ) if x > * (A.8) The relative onditional expeted value is [ D1/ { 0 + }] xf IE 0 D1 /{ 0 } x dx 1 + e 1 1e ( ( ) ) { } 1 k 1 ( ( ) k 1 e ). k (A.9) * The onditional distribution of D 1 depends on, and *. As fig. A.3a-e show, if is equal to, there is no onditioning is an exponential variable. Furthemore, for +, D 1 onverges in distribution to an exponential random variable, for all values of *, but the lower is the value of * the faster is the onvergene. Consequently, it results (see fig. A.3e). 1 + IE [ D1/ { 0 }] (A.10) 469

16 Anna aria Lombardi a b d e Fig. A.3a-e. a) Conditional density funtion of D 1 by the event { 0 * } (see eq. (A.8)) for *, + 1, + 2, + 3, and for 2. b) The same as (a) but for 10. ) The same as (a) but for 100. d) The same as (a) but for e) Conditional expeted value of D 1 by the event { 0 * } (see eq. (A.9)) versus the sample size for four values of differene between * and ( * 0,1,2,3) 470

17 Probabilisti interpretation of «Bath s Law» Table A.I. Values of the expeted value of D 1 onditioned by the event { 0 * } for some values of b,, and *. * * +1 * +2 * +3 b all Given that the estimated values of b are very lose to 1, the asymptoti value of the onditional average of D 1 is about Table A.I shows the dependene of IE [ D / 1 { 0 }] on the b-value for some values of and of * and for b in the range [0.5, 1.5]. The kind of this dependene is highly variable and it does not have a monotone trend (for some values of * and the onditional average of D 1 dereases with the inreasing of b, while, in other ases, it has a minimum for a value of b inside the range examined). However a variation smaller than 0.15 of the onditional expeted value of D 1 orresponds to a variation of 0.1 for b. REFERECES BATH,. (1965): Lateral inhomogeneities of the upper mantle, Tetonophysis, 2, CASELLA, G. and R.L. BERGER (1990): Statistial Inferene (Wadsworth), pp DRAKATOS, G. and J. LATOUSSAKIS (2001): A atalog of aftershok sequenes in Greee ( ): their spatial and temporal harateristis, J. Seismol., 5, EVISO, F. (1981): ultiple earthquakes events at moderateto-large magnitudes in Japan, J. Phys. Earth, 29, EVISO, F. (1999): On the existene of earthquake preursors, Ann. Geofis., 42 (5), EVISO, F. and D. RHOADES (1993): The preursory earthquake swarm in ew Zealand: hypothesis tests,. Z. J. Geol. Geophys., 36, EVISO, F. and D. RHOADES (2001): odel of long-term seismogenesis, Ann. Geofis., 44, (1), FELLER, W. (1956): An Introdution to Probability Theory and its Appliations (Wiley and Sons, ew York), vol. 2, pp GUTEBERG, B. and C.F. RICHTER (1954): Seismiity of the Earth and Assoiated Phenomena, Prineton, pp KAGA, Y.Y. (1997): Seismi moment-frequeny relationship for shallow earthquakes; Regional omparisons, J. Geophys. Res., 102, LAVEDA, B.H. and E. CIPOLLOE (2000): Extreme value statistis and thermodynamis of earthquakes: aftershok sequenes, Ann. Geofis., 43 (5), PAPAZACHOS, B.C.(1974): On ertain aftershok and foreshok parameters in the area of Greee, Ann. Geofis., 27, PURCARU, G. (1974): On the statistial interpretation of the Bath s law and some relations in aftershok statistis, Geol. Inst. Tehn. Ed. Study Geophys. Prospet., Buharest, 10, RAALLI, G. (1969): A statistial study of aftershok 471

18 Anna aria Lombardi sequenes, Ann. Geofis., 22, RICHTER, C.F. (1958): Elementary Seismology (Freeman, San Franiso, Calif.), pp REASEBERG, P. (1985): Seond-order moment of Central California seismiity, , J. Geophys. Res., 90, SOLOV EV, S.L. and O.. SOLOV EVA (1962): Exponential inrease of the total number of an earthquake s aftershoks and the derease of their mean value with inreasing depth, Izv. Akad. auk. SSSR, Ser. Geofiz. 12, , English translation, TSAPAOS,. T. (1990): Spatial distribution of the differene between the magnitudes of the main shok and the largest aftershok in the Cirum-Paifi Belt, Bull. Seismol. So. Am., 80, UTSU, T. (1961): A statistial study on the ourrene of aftershoks, Geophys. ag., 30, UTSU, T. (1966): A statistial signifiane test of the differene in b-value between two earthquake groups, J. Phys. Earth, 14, UTSU, T. (1969): Aftershok and earthquakes statistis (I), J. Fa. Si., Hokkaido Univ., 3, UTSU, T. (1970): Aftershok and earthquakes statistis (II), J. Fa. Si., Hokkaido Univ., 3, VERE-JOES, D. (1969): A note on the statistial interpretation of Bath s Law, Bull. Seismol. So. Am., 59, VERE-JOES, D. (1975): Stohasti models for earthquake sequenes, Geophys. J. R. Astron. So., 42, (reeived Deember 13, 2001; aepted arh 11, 2002) 472