Annals n.6/2003 ok 23/04 1271 ANNALS OF GEOPHYSICS, VOL. 46, N. 6, December 2003 Key words b-value – seismology – statistical meth- ods – synthetic-earthquake catalogs 1. Introduction The Gutenberg-Richter Law (from now on GR Law) (Gutenberg and Richter, 1954) is cer- tainly one of the most remarkable and ubiqui- tous features of worldwide seismicity. In the most common form it reads N M a bMLog = -^ h8 B (1.1) where N is the number of events with magni- tude M, and a and b are two constant coeffi- cients. The scientific importance of the GR Law is linked to its apparent ubiquity. It emerges in a variety of tectonic settings and depth ranges, in seismic catalogs ranging from a few months to centuries, and in natural as well as induced seis- micity. This ubiquity makes the GR law very useful in seismic hazard studies, and very at- tractive from a theoretical point of view. In past years, many efforts were devoted to under- standing the physical meaning of such a power law, but the conclusions are diverging. Some authors (e.g., Bak and Tang, 1989; Ito and Mat- suzaki, 1990), by assuming a presumed univer- sality of the parameter b and the implicit scale independence in eq. (1.1), propose a model of Self-Organized Criticality (SOC) to explain earthquake occurrence. A review and new insights on the estimation of the b-value and its uncertainty Warner Marzocchi and Laura Sandri Istituto Nazionale di Geofisica e Vulcanologia, Bologna, Italy Abstract The estimation of the b-value of the Gutenberg-Richter Law and its uncertainty is crucial in seismic hazard stud- ies, as well as in verifying theoretical assertions, such as, for example, the universality of the Gutenberg-Richter Law. In spite of the importance of this issue, many scientific papers still adopt formulas that lead to different es- timations. The aim of this paper is to review the main concepts relative to the estimation of the b-value and its uncertainty, and to provide some new analytical and numerical insights on the biases introduced by the un- avoidable use of binned magnitudes, and by the measurement errors on the magnitude. It is remarked that, al- though corrections for binned magnitudes were suggested in the past, they are still very often neglected in the estimation of the b-value, implicitly by assuming that the magnitude is a continuous random variable. In partic- ular, we show that: i) the assumption of continuous magnitude can lead to strong bias in the b-value estimation, and to a significant underestimation of its uncertainty, also for binning of ∆M = 0.1; ii) a simple correction ap- plied to the continuous formula causes a drastic reduction of both biases; iii) very simple formulas, until now mostly ignored, provide estimations without significant biases; iv) the effect on the bias due to the measurement errors is negligible compared to the use of binned magnitudes. Mailing address: Dr. Warner Marzocchi, Istituto Na- zionale di Geofisica e Vulcanologia, Via D. Creti 12, 40128 Bologna, Italy; e-mail: marzocchi@bo.ingv.it 1272 Warner Marzocchi and Laura Sandri In this frame, the fractal geometry distribu- tion and the earthquake dynamics are the spatial and temporal signatures of the same phenome- non. On the other hand, several lines of empiri- cal evidence dispute the universality of the GR Law. In fact, in spite of a «first order» validity of the GR Law with a constant b-value ≈ 1, sig- nificant spatial and temporal variations in the b- value seem to take place (Miyamura, 1962; Healy et al., 1968; Pacheco and Sykes, 1992; Kagan, 1997; Wiemer and McNutt, 1997; Wiemer and Wyss, 1997; Wiemer et al., 1998; Wiemer and Katsumata, 1999). These varia- tions are usually attributed to many processes, such as fault heterogeneity (Mogi, 1962), the stress level imposed on rocks (Scholz, 1968), and pore pressure variations. Another important discrepancy of the hypothesis of the universali- ty of the GR Law is that the latter seems to hold only for a finite range of magnitudes (e.g., Pacheco and Sykes, 1992; Pacheco et al., 1992; Scholz, 1997; Triep and Sykes, 1997; Knopoff, 2000). For example, Kagan (1993, 1994) found that a gamma distribution better matches the re- quirement of a maximum seismic energy re- leased by an earthquake (cf. Wyss, 1973). In all the cases reported above the b-value should not be a constant in the earthquake catalog. The empirical validation of the «universal- ity» hypothesis, as well as the identification of spatial and temporal changes, passes through the estimation of the b-value and its uncertainty. Different formulas were pro- posed in the past (e.g., Aki, 1965; Utsu, 1965, 1966; Shi and Bolt, 1982; Bender, 1983; Tin- ti and Mulargia, 1987) which take into ac- count the unavoidable binning of the magni- tudes in different ways. As a matter of fact, Aki’s formulas, which consider magnitude a continuous random variable, are still the most used. Implicitly, this means that, at least for instrumental measurements, the effect of the binning is considered negligible, and that all the formulas mentioned above give almost comparable estimations. In this paper, we re- view the most diffuse formulas so far pro- posed and investigate analytically and numer- ically on the main properties of such estima- tions. In particular, we estimate the reliability and the possible bias of the formulas as a function of the number of data and of the measurement errors in the magnitude. Here, we do not consider the estimations provided through the least squares technique. In fact, even though this approach was still em- ployed in many recent scientific papers (Pache- co and Sykes, 1992; Pacheco et al., 1992; Karnik and Klima, 1993; Okal and Kirby, 1995; Scholz, 1997; Triep and Sykes, 1997; Main, 2000), the use of the least squares technique does not have any statistical foundation (e.g., Page, 1968; Bender, 1983). 2. Estimation of b and bv t through the Maximum Likelihood technique In the first papers that described the Maximum Likelihood (ML) estimation (Aki, 1965; Utsu, 1965), the magnitude M was considered a contin- uous Random Variable (RV). If eq. (1.1) holds, the probability density function (pdf) of M is lnf M b 10 10 10 10 bM bM bM min max = - - - - ^ ^h h (2.1) where Mmin and Mmax are, respectively, the min- imum and the maximum magnitude allowed. If Mmax >> Mmin, eq. (2.1) becomes M 10 b M M min- - .lnf b 10=^ ^ ]h h g (2.2) Note that the passage from eq. (2.1) to eq. (2.2) requires that, in practice, the GR Law holds for a range of magnitudes Mmax – Mmin ≥ 3. This as- sumption might be questionable in studying variations of the b-value in smaller ranges (e.g., Knopoff, 2000). The ML estimation of eq. (2.2) consists of choosing the b-value which maximizes the like- lihood function (Fisher, 1950), that is ln b M10 1 thresh ) = -n t t^ `h j (2.3) where nt is the sampling average of the magni- tudes, and Mthresh is the threshold magnitude 1273 A review and new insights on the estimation of the b-value and its uncertainty which usually corresponds to the minimum magnitude for the completeness of the seismic catalog. Hereinafter, the symbol ^ distinguishes the estimation by the true value of the parame- ter. We have also attached an asterisk to b* in or- der to distinguish it from another estimation which we shall introduce later. The uncertainty is estimated by (Aki, 1965) N b b =v ) )t t t (2.4) where N is the number of earthquakes. A major contribution was provided later on by Shi and Bolt (1982) who provided a new for- mula to estimate the error of the b-value . b N N M 2 30 1 ( ) b SB i i N 2 2 1) = - - v n = ) ! t t t t ^ ` h j (2.5) where N is the number of earthquakes. Com- pared to eq. (2.4), eq. (2.5) provides a reliable estimation also in the presence of possible (time and/or spatial) variations of the b-value (Shi and Bolt, 1982). As a matter of fact, the magnitude is not a continuous variable and it is not devoid of meas- urement errors. In practical cases, the uncertain- ties on the measured magnitudes lead to the use of «binned» magnitudes, i.e. the magnitudes are grouped by using a selected interval ∆M. For in- stance, for instrumental measurements, the magnitude interval used for the grouping is ∆M = 0.1; for magnitude estimation of historical events, the grouping can even be ∆M ≥ 0.5. In spite of such unavoidable binning, the «continuous» Aki’s (1965) formulas (eqs. (2.3) and (2.4)) are probably the most em- ployed in practical applications to estimate the b-value and its uncertainty (e.g., Jin and Aki, 1989; Henderson et al., 1994; Öncel et al., 1996; Mori and Abercrombie, 1997; Utsu, 1999; Hiramatsu et al., 2000; Gerstenberger et al., 2001; Klein et al., 2001; Vinciguerra et al., 2001). In such cases, an implicit assump- tion is that the binned magnitudes can be con- sidered a continuous RV. While this approxi- mation can be strictly justifed only for ∆M → → 0, in practice it is usually assumed also for ∆M = 0.1 (instrumental magnitudes). In the following sections, we consider sep- arately the effects due to the binning of the magnitudes and the measurement errors on the estimations of the b-value and its uncer- tainty. We remark that a wrong choice of the completeness threshold magnitude Mthresh can also cause a significant bias, as shown by Wiemer and Wyss (2000). Here, we do not deal with this issue, but it should be kept in mind in any practical analysis. 3. Influence of the binned magnitudes In this section, we first study the biases in- troduced by the binning of the magnitudes in the estimation of the b-value and its uncertain- ty made through eqs. (2.3) and (2.4). Then, we check through numerical simulations the relia- bility of some formulas reported in scientific literature which either assume the magnitudes as a continuous RV or take properly into ac- count the binning of the magnitudes. 3.1. Effects of the binned magnitudes on Aki’s (continuous) formulas The use of eq. (2.3) produces a biased es- timation of the real b-value, because of two factors: i) the average µ of a continuous RV with a power law distribution is different from the average of the same «binned» RV; ii) Mthresh ≠ Mmin. Hereinafter we will call these biases θ1, and θ2, respectively. The influence of θ1 was estimated by Ben- der (1983). She found that the sample average nt computed from binned data is systematically higher than the true value µ. This is due to the fact that the real (continuous) magnitudes in the interval Mi − ∆Μ/2 ≤ M < Mi + ∆Μ/2 are not symmetrically distributed around the central value Mi. Moreover, she showed that the bias θ1 is negligible for ∆M = 0.1, while it is very im- portant for larger ∆M (for example, ∆M = 0.6), that might be necessary to evaluate the b-value for historical catalogs. 1274 Warner Marzocchi and Laura Sandri The bias θ 2 was considered by the first pa- per which addressed the use of the binned magnitudes (Utsu, 1966), suggesting a slight modification of eq. (2.3). Since the lowest binned magnitude, i.e. the threshold magni- tude, contains all the magnitudes in the range Mthresh − ∆Μ/2 ≤ M < Mthresh + ∆Μ/2, then Mmin = = Mthresh − ∆Μ/2 < Mthresh (e.g., Bender, 1983). Then, eq. (2.3) becomes . ln b M M10 2 1 thresh = - -n ∆ t t^ `h j: D (3.1) Remarkably, this «corrected» formula was not largely employed. Utsu himself, in a recent paper (Utsu, 1999), reported eq. (2.3) to estimate the b- value. A major exception is the works making use of ZMAP (e.g., Wiemer and Benoit, 1996; Wie- mer and McNutt, 1997; Wiemer and Wyss, 1997, 2000) and ASPAR (Reasenberg, 1994) codes, which estimate the b-value through eq. (3.1). Yet surprisingly the correction reported in eq. (3.1) is rarely explicitly mentioned in the scientific liter- ature (e.g., Guo and Ogata, 1997). As a matter of fact, we suspect that, in some cases, the correc- tion reported in eq. (3.1) is used in the analysis without explicitly mention of it in the manuscript. For instance, Gerstenberger et al. (2001) quoted Aki’s (1965) formulas (eq. (2.3)) in the manu- script, but the code ZMAP written by one of the authors (see above) contains the «corrected» for- mula given by eq. (3.1). In any case, we argue that this correction could have not been largely employed also be- cause the influence of the bias θ 2 is erroneously considered negligible. Indeed, the difference be- tween b)t and bt , respectively the «non-corrected» and «corrected» estimation, is certainly consider- able because in a power law distribution the aver- age µ is very close to the minimum value of the distribution M= + ln ln b M dM b 10 10 10 1 min b M M M min min $= =n 3 - -#^ ] ^ h g h (3.2) In particular, 2 . ln b b M M M 10 2 thresh thresh 2 = - = = - - + i n n ∆ ) M∆ t t t t^ ` `h j j: D (3.3) For ∆M = 0.1 (as for instrumental magnitudes), and .M 0 38thresh.-nt (obtained by eq. (3.2) with b = 1) we obtain θ2 ≈ 0.13. Another crucial aspect, until now completely neglected, concerns the effect of the bias θ 2 on the estimation of the uncertainty given by eq. (2.4) and the «corrected» form which reads . N b b =vt t t (3.4) The positive bias θ 2 has two competitive effects. It leads to an increase in the estimated uncertain- ty; in fact, comparing eqs. (3.4) and (2.4) we ob- tain b 1b b 2 = +v i v)t t tt td n (3.5) therefore, > .bbv v)t tt t On the other hand, the bias θ 2 leads to an increase of the dispersion of the RV b)t around its expected value. By neglecting the bias θ 1, the true variance of b)t is lnb var. E .= ln E b b M M10 1 10 b thresh thresh 2 2 2 4 2 = - = - - - v n n n n n ) ) ) ) t t t t t t J L K KK _ ^ ` ` ` ^ N P O OO i h j j j h R T S S S S 9 V X W W W W C (3.6) where var ( nt ) is the variance of the RV nt . In the same way, we have . ln E b b E M M M M 10 1 2 2 10 b thresh thresh 2 2 2 4 2 $ $ = - = - + - + - v n n n n n ∆ ∆ lnb var. t t t t t t J L K K_ ^ ` ` ` ^ N P O O O i h j j j h R T S S S S S 9 V X W W W W C (3.7) 1275 A review and new insights on the estimation of the b-value and its uncertainty The validity of eqs. (3.6) and (3.7) deserves fur- ther explanations. In particular, these equations assume that E ( b)t ) = b) and E( bt) = b, respec- tively. If we take the expected value of Taylor’s expansion around the true value µ of eqs. (2.3) and (3.1), we see that these assumptions hold only for small deviations of nt , i.e. for large datasets. Numerical investigations have shown that the biases are negligible for datasets with 50 or more earthquakes. By comparing eqs. (3.6) and (3.7) we obtain b 1b b 2 2 = +v i v) t t td n (3.8) therefore, > .b bv v)t t From eqs. (3.5) and (3.8), we can conclude that the true dispersion of the RV b bv) )t t^ h increases more than the increase in the estimation of the uncertainty bv )t t . In other words, eq. (2.4) provides an underestimation of the true dispersion. 3.2. Binned formulas After the correction suggested by Utsu (1966), Bender (1983), Tinti and Mulargia (1987) provided formulas to estimate the b- value, by properly taking into account the grouping of the magnitudes. Remarkably, be- sides very few exceptions (e.g., Frohlich and Davis, 1983), these formulas were almost ig- nored in subsequent applications. We argue that the reasons are mainly of a technical na- ture. Bender’s (1983) formula, for example, can be solved only numerically. Moreover, in her analysis she gave more emphasis to the bias θ1 introduced by the use of the continu- ous approximation (eq. (2.3)), concluding that the latter provides almost unbiased estima- tions of the b-value if the magnitude interval for the grouping is ∆M = 0.1. A definite improvement to the estimation of the b-value was provided by Tinti and Mu- largia (1987). Their formula reads 1 lnb p 10 = ln M∆ TM t ^ _ h i (3.9) where p M M 1 thresh = + -n ∆ t J L K K N P O O (3.10) and the associated asymptotic error is Npln M p 10 1 b = - v ∆ TM t t ^ h (3.11) where N is the number of earthquakes. In this case, we think the very scarce use of these for- mulas was probably due to some kind of crypti- cism of the paper. 3.3. Numerical check In order to check the reliability of the formulas described above, we simulate 1000 seismic cata- logs, for different catalog sizes. The magnitudes M are obtained by binning, with ∆M = 0.1 (as for the instrumental magnitudes), a continuous RV dis- tributed with a pdf given by eq. (2.2); in other words, Mi is the magnitude attached to all the syn- thetic seismic events with real continuous magni- tude in the range Mi – 0.05 ≤ M < Mi + 0.05. In fig. 1a,b we report the medians of b)t , bt and bTMt calculated in 1000 synthetic catalogs as a func- tion of the number of data, for the case b = 1 and b = 2. To each median is attached the 95% confi- dence interval, given by the interval between the 2.5 and 97.5 percentile. From fig. 1a,b, we can see that the estimation bTMt (Tinti and Mulargia, 1987) is bias free, also for a small dataset. As regards the continuous formulas, with and without correction (respectively eqs. (3.1) and (2.3)), we can see that the bias θ2 reported in fig. 1a,b is comparable to the theoretical expectation given by eq. (3.3). The cor- rected estimation bt is undoubtedly much closer to the real b-value. The slight underestimation of bt (much less than 1% of the real b-value) is due to the bias θ1 previously discussed (Bender, 1983). Therefore, at least for ∆M = 0.1, θ1 can be neglect- ed (e.g., Bender, 1983), but θ2 is certainly relevant. In order to evaluate the reliability of the estimations of the uncertainty, it is necessary to compare each estimation with the true dis- 1276 Warner Marzocchi and Laura Sandri Fig. 2a,b. F test values (see eq. (3.12)) for the case of fig. 1a,b (see text). Open and solid squares are, respec- tively, the F test values relative to the uncertainties given by eqs. (2.4) and (2.5). Open and solid circles are the F test values relative to the same uncertainties calculated for bt instead of b )t (i.e. eq. (3.4)). Asterisks are the F test values relative to the uncertainty calculated through eq. (3.11). The dotted line represents the critical value to reject the null hypothesis at a significance level of 0.05. a b Fig. 1a,b. Medians and 95% confidence bands of b)t (dashed lines), bt (dotted lines) and bTMt (thicker solid line) from 1000 synthetic catalogs, as a function of the catalog size, for the cases b = 1 (a) and b = 2 (b). The solid thin line represents the true b-value. a b 1277 A review and new insights on the estimation of the b-value and its uncertainty persion of estimation of the b-value around its central value. In particular, we compare the dispersion of the b-value estimation around its average with the average of the estimated uncertainties, through the Fisher test (e.g., Kalb- fleisch, 1979) = Average of the square of the uncertainty Variance of the estimation of the valueF b - (3.12) The null hypothesis is that the two variances are the same. The results are reported in fig. 2a,b. The most interesting result is relative to the systematic re- jection of the F test for the non-corrected case, given by eq. (2.4). This means that the use of the non-corrected estimation b)t (eq. (2.3)) leads to an underestimation of its real dispersion. This fact is very important because it might suggest varia- tions in the b-value whereas they do not exist. Note that the amplitude of this bias is correctly estimated by eq. (3.8). On the contrary, all the other estimations of the uncertainty do not have significant biases. It is important to note that the error estimated through eq. (2.5) (Shi and Bolt, 1982) is unbiased if the b-value is estimated through the corrected (eq. (2.3)) and non-correct- ed (eq. (3.1)) formulas. 4. Effects of the measurement errors In order to take into account the measure- ment errors, we consider the «real» magni- tude M + as a sum of two independent RVs M M= + f + (4.1) where M is the earthquake magnitude devoid of measurement errors distributed with a pdf given by eq. (2.1), and ε simulates the meas- urement errors distributed as a Gaussian noise. The choice of the normal distribution has been discussed in detail by Tinti and Mu- largia (1985), and Rhoades (1996). In the case the two RVs, M and ε, are inde- pendent, the statistical cumulative distribution of M + can be obtained as follows (Ventsel, 1983; Tinti and Mulargia, 1985; Rhoades, 1996) d dG M f M f M M M M M 1 2 min max = f f 3- - + + # #c ^ ^m h h R T S S S V X W W W (4.2) where G( M + ) is the cumulative distribution of M + , f1(M) is given by eq. (2.1), and f2 (ε) is N (0, 2vf ) distributed. By differentiating with respect to the variable M + , we obtain the pdf .- exp$ $ +$ ln ln ln ln ln g M b b M b M M b M M b erf erf 2 1 10 10 10 2 1 10 2 10 2 10 2 10 max min bM bM 2 2 2 min max $= - - + - + - + v v v v v - - f f f f f + + + + c ^ ^ ^ ^ ^ m h h h h h R T S S S R T S S S ; V X W W W V X W W W E Z [ \ ]] ]] ) _ ` a bb bb 3 (4.3) In the case Mmax >> Mmin, eq. (4.3) becomes .$ exp $$ ln ln ln ln M M b M M b erf b b g 10 2 10 1 10 10 2 1 10 2 2 1 min bM 2 2 min $ - + - - + = v v v - f f f + + + ^ ^ ^ ^ c h h h h m R T S S S ; V X W W W E Z [ \ ]] ]] ) _ ` a bb bb 3 (4.4) The ML estimation of eq. (4.4) gives + 0= + ln ln ln ln ln exp ln b M b N M M b M M b erf 1 10 10 10 2 10 1 2 10 2 10 min min min i i i N 2 2 2 2 2 1 $ $ - + + - - - + - - + n v r v v v v v = f f f f f f + + / ! ^ ^ ^` ^ ^ ^ h h hj h h h R T S S SS R T S S SS V X W W WW V X W W WW Z [ \ ] ] ]] ] ] ]] Z [ \ ]] ]] _ ` a bb bb _ ` a b b b b b b b b (4.5) 1278 Warner Marzocchi and Laura Sandri Fig. 3a,b. As for fig. 1a,b, but relative to the case with added measurement error to the magnitudes. The stan- dard deviations of the measurement errors are σε = 0.1, 0.3, 0.5 (from left to right). a b 1279 A review and new insights on the estimation of the b-value and its uncertainty a b Fig. 4a,b. As for fig. 2a,b, but relative to the case with added error to the magnitudes. The standard deviations of the measurement errors are σε = 0.1, 0.3, 0.5 (from left to right). 1280 Warner Marzocchi and Laura Sandri where +n / is the sample average of M + . The first three addenda represent the estimation of the b- value without measurement errors (cf. eq. (2.3)), while the term inside the braces is the addition- al part which takes into account the measure- ment errors. 4.1. Numerical check The contribution of the term inside the bracket in eq. (4.5) to the b-value estimation for different σε can be studied by adding meas- urement errors to the synthetic catalogs gener- ated as described before. In fig. 3a,b we report the results for differ- ent σε. Compared to the case without measure- ment errors (fig. 1a,b), the only observable dif- ference is a global negative bias present in all the estimations for b = 2 and σε = 0.5. This can be due to the improper binning of ∆M = 0.1 in the presence of larger measurement errors (σε = 0.5). As regards the estimation of the uncertain- ties bvt t and bv )t t , we operate as in the previous case, by plotting the results of the F test (fig. 4a,b). The results are almost the same as those obtained for the case without measurement er- rors (fig. 2a,b). Note that also in this case the use of eq. (2.4) leads to a significant underesti- mation of the uncertainty. 5. Final remarks The purpose of this paper was to review the main concepts of the estimation of the b-value and its uncertainty, and to provide new insights on the reliability of different formulas reported in scientific literature. The detection and quan- tification of possible significant biases in such estimations is a crucial issue in many scientific applications, such as hazard studies and any at- tempt to test the constancy or the universality of the b-value. Here we studied analytically and numerically the possible biases introduced by the use of the binned magnitudes and meas- urement errors on different formulas. We found that, in general, the influence of the measurement errors appears negligible com- pared to the effects of the binned magnitudes if improperly dealt with. In particular, we found that the most commonly used formulas (Aki, 1965; see eqs. (2.3) and (2.4)), which assume the magnitude as a continuous random vari- able, produce a strong bias in the estimation of the b-value, and a significant underestimation of its uncertainty. This means that any spatial or temporal variation of the b-value obtained by eqs. (2.3) and (2.4) has to be regarded with a strong skepticism. On the contrary, we show how the same continuous formula with a small correction to take into account the binned mag- nitudes (eqs. (3.1) and (3.4)) drastically re- duces the biases of the b-value and of its un- certainty. 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