Geophys. J. Int. (2) 142, 193 198 Is the seisic oent frequency relation universal? C. Godano1 and F. Pingue2 1 Dipartiento di Scienze Abientali, Seconda Università di Napoli, via V ivaldi 43, 811 Caserta, Italy. E-ail: godano@osve.unina.it 2 Osservatorio Vesuviano, via A. Manzoni 249, 8123 Napoli, Italy. E-ail: pingue@osve.unina.it Accepted 2 February 28. Received 2 February 28; in original for 1999 August 16 SUMMARY We evaluate the seisic oent frequency relation for the Harvard catalogue in the period 1977 1994. This catalogue is coposed of about 12 earthquakes. After selection of events in ters of depth and energy, we retain about 8 data points. We estiate two paraeters of the seisic oent distribution: the power exponent b and the cut-off value M. The ethod used is a least-squares linear fit on a log log scale perfored over a range selected on the basis of the standard deviation fro the histogra. The analysis is carried out for different subdivisions of the Earth in square grids of different sizes. Neither paraeter exhibits a dependence on cell size, suggesting the universality of their values and the interpretation of the existence of a cut-off as a finite size effect linked to a finite catalogue length. The variations of the paraeters are investigated as a function of tie (duration of the catalogue) and versus the nuber of events used for building up the distribution. Again, b and M do not depend on tie, but M depends on the nuber of events, reaching a stable value for N#1. The only significant change in the paraeters is observed for different values of M in the catalogue, revealing the existence of universality classes. Key words: scaling laws, universality classes. INTRODUCTION Frohlich & Davis 1993; Okal & Roanowicz 1994; Sornette et al. 1996), which is a ore physical quantity than the The scaling properties of earthquakes are generally described agnitude. The cuulative distribution of the seisic oent in ters of a power law distribution usually referred as exhibits a power law behaviour followed by a sharp roll-off the Gutenberg Richter distribution fro the naes of the for values greater than M. Soe authors (see Anderson & ax researchers who first observed this power law distribution of Luco 1983) have proposed cutting the distribution after M. ax earthquake agnitudes. They found that the frequency of Against this hard truncation, Kagan (1991) proposed that the occurrence of earthquakes with agnitude less than M follows seisic oent distribution is well represented by a gaa a power law distribution. The gaa distribution has a probability density log N=a bm, f (M )=C 1MbM 1 b exp( M /M ), (1) c where C is a noralizing coefficient, M is the iniu c threshold in the data and M is the seisic oent over which the exponential decay becoes doinant with respect to the power law distribution for M <M. In this sense M represent a soft liit of the scaling region. Using a axiu likelihood ethod Kagan (1997, 1999) found that this distribution is universal; in fact, b assues a value of.63 everywhere with the exception of the id-oceanic ridge zones, where it is equal to.93. This result is consistent with that of Okal & Roanowicz (1994). The results of Kagan (1997, 1999) present soe probles in the estiation of M due to the non- quadratic shape of the axiu likelihood for soe regions. Moreover, soe questions ay arise fro the regionalization procedure. The first concerns the different results obtained by where a represents the overall seisicity and b is typically close to 1. Fluctuations (up to 3 per cent) in the value of b around its typical value are widely observed depending on the catalogue, the estiating ethod and the agnitude range over which it is estiated (Frohlich & Davis 1993). Regional variation of the b-value have been observed by any authors (see e.g. Hattori 1974; Kronrod 1984), who ade use of different regionalizations (see also Utsu 1971). Such features could be an indication that scaling properties of earthquakes are not universal, but soe authors have suggested that this difference in b-values could be due to systeatic errors in agnitude deterination (Utsu 1971; Kagan 1991). For this reason, other authors have preferred to study the distribution of the seisic oent (Kagan 1991; Pacheco et al. 1992; 2 RAS 193 Downloaded fro https://acadeic.oup.co/gji/article-abstract/142/1/193/593566 on 31 Deceber 217
194 C. Godano and F. Pingue other authors (see e.g. Cornell 1994 and Johnston 1994), 1 who have obtained a non-universality of the size distribution of earthquakes using different regionalizations. The second concerns the variability of the nuber of seisic events in 1 different regions; the estiates are thus differently weighted for different areas. In this paper we try to obviate both of these probles using a copletely different approach that estiates the two paraeters independently of the distribution and does not ake 1 use of a regionalization based on tectonic arguents, but uses earthquakes as a pure statistical object. 1 b=.73 DATA 1 We analyse the global catalogue of the oent tensor inversion 1 1 24 1 1 25 1 1 26 1 1 27 1 1 28 1 1 29 copiled by the Harvard group (Dziewonski et al. 1994) Seisic oent (dyne c) in the period fro 1977 January 1 to 1994 Deceber 31. The catalogue contains about 12 events whose CMT solutions Figure 1. Cuulative frequency of the seisic oent for the whole were obtained using the sae technique. Thus we expect that catalogue. The dashed line represents the linear fit on the log log scale this is a catalogue that ensures good stability in the estiates in the range 5 124[5.2 127 (see text for details). of seisic oent. In order to obtain a better hoogeneity of the catalogue, we liited our analysis to shallow earthquakes, the first 1 points (at least one order of agnitude) of the for which we can hypothesize the sae fracture echanis; distribution. In this range we assue that the scaling properties we thus cut off the events with focal depths greater than 7 k, of the distribution have a power law feature. The sae linear where we assue that there is a transition between brittle and fit is then perfored for a nuber of points increased by one, ductile regies. A second cut-off is with respect to the seisic step by step up to the end of the distribution. For each step oent we excluded events with M <2. 124 dyne c, the standard deviation of the fit fro the real data, which corresponds to M =5.5. Here M is estiated using W W the relation (Okal & Roanowicz 1994): s i =S (n(th) n(ex) )2/(np 1) j j j M = 2 W 3 log M 1.73, (2) is calculated, where n(th) is the nuber of events predicted by j the fit for the jth class of the distribution, n(ex) is the nuber where M is expressed in dyne c. This liit is lower than the j of earthquakes experientally observed in the sae class liit of copleteness indicated by Kagan (1997), but we and np is the nuber of points used for the fit. In ore detail, decided to take this into account in the phase of the estiation we first calculate the slope of the distribution using 1 points of the power law exponent. Here our liit represents that and we then increase the nuber of points used for the fit to where the frequency oent curve exhibits a change in the n=1+i, where i=1, 2,, N, where N is the nuber of sign of the slope, indicating a departure fro self-siilarity points of the histogra inus 1. Using such an approach we (absence of scaling). obtain any slopes and any standard deviations b and s, i i respectively. In the power law range, s reains approxiately i constant up to a certain value of M, corresponding to M ESTIMATION OF b AND M (hereafter, when we refer to the estiated value of M it will As stated in the Introduction, we want to estiate both of the be denoted M ), where it suddenly increases due to the change paraeters, b and M, of the seisic oent distribution of slope (Fig. 2). This eans that, in the scaling region, the (relation 1). Often the axiu likelihood does not allow the differences between the single s (hereafter Ds ), fluctuate i i estiation of M ; we thus decided to use the ore standard randoly around zero. Thus, for a ore quantitative choice technique of least-squares linear fit on a log log scale. We use we decided to evaluate the average of the Ds (D ) and their i A the weighted least-squares ethod, which gives us a correct standard deviation (s ). Note that s should not be confused with the s ; the first one is the standard deviation of i D D estiation of the slope, but a biased estiation of the standard error, which is not used in this paper. As can be seen in Fig. 1, the distribution of the Ds, while the second represents a i the greatest proble with using a linear fit is the individuation easure of how good the fit is. When we have the condition of the scaling region (the range of M for which we can Ds >D ±2s, we have found M. Note that in using such i A D consider valid the power law behaviour) due to the sharp an approach, our M represents a lower liit of the true M change of slope occurring at about M =5 127 and to the estiated using the axiu likelihood ethod (Kagan usual probles of copleteness of the catalogue, which affect 1997). The values we obtain for b and M are, respectively, the first part of the distribution. The lower liit was chosen.73 and 5.17 127 dyne c (M =7.8). These values are in W on the basis of the copleteness analysis of Kagan (1997), very good agreeent with the estiate of Okal & Roanowicz who put this liit at M =124.7 dyne c. The choice of the (1994), but they are not coparable with the results of Kagan upper liit was ore coplex and is based on a trial-anderror technique used to indicate where the linear fit becae and to the regionalization used by Kagan (here we estiated (1997, 1999) due to differences in the estiating ethod for M worse that the previous one. We first perfor a linear fit for the values for the whole catalogue) for b. Cuulative frequency 2 RAS, GJI 142, 193 198 Downloaded fro https://acadeic.oup.co/gji/article-abstract/142/1/193/593566 on 31 Deceber 217
Seisic oent frequency relation 195 Standard deviation.3.2.1. 5.17 1 27 1 1 26 1 1 27 1 1 28 Seisic oent (dyne c) Figure 2. Standard deviation of the linear fit fro the data versus M. In order to avoid any dependence of our estiates on the chosen regionalization, we decided to adopt a criterion based on the equal size of the analysed areas. First we divided the Earth s surface into equal-sized square cells with the side length varying between 5 and 25 k (steps of 25 k), because the estiates could depend on the choice of cell size. We retained for our analysis only the cells with nuber of events greater than 1 (generally n is in the range 25[6) because we observed that the distribution shape becoes stable for nuber of events greater than or equal to 1. In each cell we sapled randoly 1 independent groups with 1 events in each group. For each group we estiated b and M. The sapling allowed us to analyse the estiated paraeters with statistical ethods, which ade the evaluation of errors for our estiates easy. Fig. 3 shows an exaple (a single saple in a single cell) of the cuulative distribution of M for all the cell sizes; all the curves exhibit very siilar shapes. The x2 test for the distribution of the estiated values for each cell size (the distribution was evaluated for all the saple and for all the cells of a given size) revealed a noral distribution for b and a log-noral one for M. This allowed us to plot in Figs 4 and 5 the average values (error bars are the standard deviation of each distribution) of the two paraeters as a function of cell size. Within errors there are no differences in the two paraeters of the distribution for all cell sizes. This confirs the results of Kagan (1997, 1999) concerning the universality of the b-value, and reveals the existence of the universality of the logarith of M. This latter result iplies that the quantity that is universal is the cut-off agnitude. On the right-hand scale of Fig. 5 we give M estiated using W relation (2). Moreover, the technique of estiation used here showed the robustness of the universality of the paraeters and their independence of the regionalization. In other words, our fundaental result is that the two paraeters b and M are valid worldwide and are independent of the tectonic regie of the seisogenic area. A bias in estiating the paraeters of the distribution could be introduced by the teporal extension of the catalogue. Thus, we decided to study the teporal dependence of b and M. The procedure was as follows. First we cut the catalogue at different ties and then (as previously done) we randoly b.8.7.6.5 5 1 15 Cell size (k) 2 25 Figure 4. Average values of the exponent values versus the cell size. The error bars represent the standard deviation of the b distribution. 27.5 Cuulative frequency 1 1 1 1 1 23 5k 75k 1k 125k 15k 175k 2k 225k 25k 1 1 24 1 1 25 1 1 26 1 1 27 Seisic oent (dyne c) Figure 3. Exaples of cuulative frequency for different saples for different cell sizes. Log of seisic oent cut-off 27. 26.5 26. 25.5 5 1 15 2 Cell size (k) 25 Figure 5. Average values of the seisic oent cut-off versus the cell size. The error bars represent the standard deviation of the M distribution. 7.5 7. 6.5 M W 2 RAS, GJI 142, 193 198 Downloaded fro https://acadeic.oup.co/gji/article-abstract/142/1/193/593566 on 31 Deceber 217
196 C. Godano and F. Pingue selected 1 independent groups of 1 events. For each group, b and M were estiated. The values, averaged over the groups, are given in Figs 6 and 7, which show a very clear independence of the two paraeters on the duration of the catalogue. In fact, although the first three points in Fig. 7 see to exhibit an increase of M with tie, but this is sharply asked by the standard deviation. This result indicates, as expected, that the scaling laws are stationary. A stronger dependence could arise fro the nuber of points in the catalogue, siply because there should exist a iniu nuber of points that ake the estiation stable. The procedure was as usual: we selected 1 independent groups of a variable nubers of events and then b and M were estiated for each group. The values, averaged over the groups, are given in Figs 8 and 9. In this case we see that b can be considered independent of the nuber of events, while M varies with the nuber of events in the group, reaching a stable (asyptotic) value at x in the range 1[15. We interpret this result as being due to statistical reasons; in fact, there is no correlation between the variation of M and that.5.6 b.7.8.9 1 2 3 4 Nuber of events Figure 8. Average values of the exponent values versus the nuber of events in the catalogue. The error bars represent the standard deviation of the b distribution..4 29. b.6.8 Log of seisic oent cut-off 28.5 28. 27.5 27. 1. 5 1 Tie (years) Figure 6. Average values of the exponent values versus tie. The error bars represent the standard deviation of the b distribution. Log of seisic oent cut-off 27.2 26.8 26.4 26. 25.6 5 1 Tie (years) Figure 7. Average values of the seisic oent cut-off versus tie. The error bars represent the standard deviation of the M distribution. 15 15 2 2 26.5 1 2 3 Nuber of events 4 Figure 9. Average values of the seisic oent cut-off versus the nuber of events in the catalogue. The error bars represent the standard deviation of the distribution. of b. This eans that the variations of M are linked to statistical instabilities due to the low nuber of events; in other words, this paraeter is statistically less stable (Kagan 1997) than b. The ost intriguing result that we obtained is the variation of the paraeters as a function of M in the catalogue. Note that M is a copletely different paraeter fro M : the first is the axiu seisic oent in a given catalogue (in our procedure a subcatalogue), while the second is a paraeter of the studied distribution. The procedure was as follows. First we selected 1 subcatalogues with different M (in the range 5 126 6 127 dyne c) and then for each subcatalogue we estiated the paraeters for 1 groups, randoly selected, each of 1 events; our final estiation is represented by the average of the 1 values of b and log(m ) for each subcatalogue. Fig. 1 shows the clear correlation between M and b (r=.97), showing a power law functional shape typical of phenoena for which there exist universality 2 RAS, GJI 142, 193 198 Downloaded fro https://acadeic.oup.co/gji/article-abstract/142/1/193/593566 on 31 Deceber 217
b 2 1.5 1 1 25 1 1 26 1 1 27 Seisic oent cut-off 1 1 28 Figure 1. Average value of the exponent value versus average value of the seisic oent cut-off. The error bars represent the standard deviation of the b distribution. DISCUSSION AND CONCLUSIONS Seisic oent frequency relation 197 classes. Note that here universal does not ean valid world- wide, but rather has the ore physical sense of scale-invariant. Thus a universality class defines the scaling for a given class of events. This is the sense of the variability of b with M. because we have not waited a sufficiently long period of tie. In other words, we would like to advance the hypothesis that the finite size effect has no physical eaning, but is linked to the length of the catalogues. The investigation of such a hypothesis could be a topic for future studies. On the other hand, the teporal stationarity of b and M has a physical interpretation: there is no reason for invoking a change of the dynaics generating a given scaling law. The independence of b on the nuber of events in the catalogue indicates the great statistical stability of this paraeter, while the variability of M indicates its statistical instability. The observation of the existence of universality classes has soe iplications that need to be discussed in detail. In fact, the variations of b with M (the slope of the power law distribution decreases as M increases) iply that as M increases, the nuber of large events is lower than that expected if the distribution were universal (not dependent on M ). The physical reason could be a less efficient earthquake-generating echanis at larger scales. In other words, the propagation of fractures could becoe less efficient as the scale of the earthquakes increase. This supposition is supported by the existence of two different scaling laws (one at low energies and the other at larger energies) between earthquake energy and fault length (Kanaori & Anderson 1975). Moreover, this could be an indication that earthquakes do not have any asyptotic distribution law; in fact, Vespignani & Pietronero (1991) showed that the universality of diffusion liited aggregation (DLA) and that of the dielectric breakdown odel (DBM) are strictly linked to the existence of asyptotic dynaics doinating the entire process. The coparison of DLA and DBM with the earthquake dynaics is siply due to the assuption that the expansion of the rock fractures is adiffusive process as, for exaple, in self-organized criticality and spring chain odels (Bak & Tang 1989; Burridge & Knopoff 1967), which could easily be copared to DLA and DBM. We have estiated the two paraeters of the distribution of the seisic oent (b and M in relation 1) using the Harvard catalogue. b and M were estiated independently. b was evaluated using a standard linear fit perfored fro the copleteness liit up to M, which was selected by eans of a trial-and-error technique based on the standard deviation of the fit fro the distribution. This was first studied by selecting earthquakes occurring in windows of various sizes, and for all cell sizes we estiated the values of b and M. We found that worldwide both paraeters assue the sae ACKNOWLEDGMENTS value, confiring and extending previous results (Okal & We would like to acknowledge Y. Y. Kagan, who kindly Roanowicz 1994; Kagan 1997, 1999). Moreover, the values reviewed an earlier version of our anuscript. of b and M are independent of the regionalization and thus of the tectonic regie [a siilar result was first obtained by Kagan (1997, 1999), but with a different technique of regionalization, which ade the estiates unequally weighted]. REFERENCES The two paraeters are also independent of the teporal duration and the nuber of events in the catalogue (the Anderson, J.G. & Luco, J.E., 1983. Consequences of slip rate constraints variation of M has no physical eaning, but is due only to on earthquake occurrence relation, Bull. seis. Soc. A., 73, statistical instabilities). 471 496. The regionalization independence is an indication that the Bak, P. & Tang, C., 1989. Earthquakes as a self-organised critical phenoenon, J. geophys. Res., 94, 15 635 15 637. change of slope observed in the distributions of the seisic Burridge, R. & Knopoff, L., 1967. Model and theoretical seisicity, oent should be interpreted not as being due to physical Bull. seis. Soc. A., 57, 341 371. factors, but siply as a finite size effect: generally this change Cornell, C.A., 1994. Statistical analysis of axiu agnitudes, in in slope has been interpreted as being linked to the saturation T he Earthquakes of Stable Continental Regions, pp. 5-1 5-27, ed. of the width of the faults (for a ore detailed discussion see Schneider, J., Electr. Power Res. Inst., Palo Alto, CA. Roanowicz & Rundle 1993 and Okal & Roanowicz 1994); Dziewonski, A.M., Ekstro, G. & Salganik, M.P., 1994. Centroidhowever, here we advance another hypothesis, interpreting this oent tensor solutions for January-March, Phys. Earth planet. feature of the distribution as being due to a teporal finite Inter., 86, 253 261. size effect. We suggest that our distribution has a lower nuber Frohlich, C. & Davis, S.D., 1993. Telesisic b values; or uch ado of large earthquakes than expected for a self-siilar process about 1., J. geophys. Res., 98, 631 644. 2 RAS, GJI 142, 193 198 Downloaded fro https://acadeic.oup.co/gji/article-abstract/142/1/193/593566 on 31 Deceber 217
198 C. Godano and F. Pingue Hattori, S., 1974. Regional distribution of b value in the world, Bull. Okal, E. & Roanowicz, B.A., 1994. On the variation of b values with Int. Inst. seis. Earthq. Eng., 12, 39 57. earthquake size, Phys Earth planet. Inter., 87, 55 76. Johnston, A.C., 1994. Seisotectonic interpretations and con- Pacheco, J.F., Scholz, C.H. & Sykes, L.R., 1992. Changes in frequencysize clusions fro stable continental region seisicity database, in The relationship fro sall to large earthquakes, Nature, 355, 71 73. Earthquakes of Stable Continental Regions, pp. 4-1 4-13, ed. Roanowicz, B. & Rundle, J.B., 1993. On scaling relations for large Schneider, J., Electr. Power Res. Inst., Palo Alto, CA. earthquakes, Bull. seis. Soc. A., 83, 1294 1297. Kagan, Y.Y., 1991. Seisic oent distribution, Geophys. J. Int., Sornette, D., Knopoff, L., Kagan, Y.Y. & Vanneste, C., 1996. Rankordering 16, 123 134. statistics of extree events: application to the distribution Kagan, Y.Y., 1997. Seisic oent-frequency relation for shallow of large earthquakes, J. geophys. Res., 11, 13 883 13 893. earthquakes: regional coparison, J. geophys. Res., 12, 2835 2852. Utsu, T., 1971. Aftershocks and earthquakes statistics (III), Analysis Kagan, Y.Y., 1999. Universality of the oent-frequency relation, of the distribution of earthquakes in agnitude, tie and space PAGEOPH, 155, 537 573. with special considerations to clustering characteristics of earth- Kanaori, H. & Anderson, D.L., 1975. Theoretical basis of soe quakes occurrence, J. Fac. Sci. Hokkaido Univ. Japan, Ser. VII, 3, epirical relations in seisology, Bull.seis.Soc.A.,65, 173 196. 379, 441. Kronrod, T.L., 1984. Seisicity paraeters for the ain high-seisicity Vespignani, A. & Pietronero, L., 1991. Fixed scale transforation regions of the world, Vychislitel naya Seisologiya, 17, 36 58 applied to diffusion liited aggregation and dielectric breakdown (Coput. Seis., 17, 35 54, English translation). odel in three diensions, Physica A, 173, 1. 2 RAS, GJI 142, 193 198 Downloaded fro https://acadeic.oup.co/gji/article-abstract/142/1/193/593566 on 31 Deceber 217