b value (marzziotti sandri 2003)

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 onGR Law) (Gutenberg and Richter, 1954) is cer-tainly one of the most remarkable and ubiqui-tous features of worldwide seismicity. In themost 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 islinked to its apparent ubiquity. It emerges in avariety of tectonic settings and depth ranges, inseismic catalogs ranging from a few months tocenturies, and in natural as well as induced seis-micity. This ubiquity makes the GR law veryuseful in seismic hazard studies, and very at-tractive from a theoretical point of view. In pastyears, many efforts were devoted to under-standing the physical meaning of such a powerlaw, but the conclusions are diverging. Someauthors (e.g., Bak and Tang, 1989; Ito and Mat-suzaki, 1990), by assuming a presumed univer-sality of the parameter b and the implicit scaleindependence in eq. (1.1), propose a model ofSelf-Organized Criticality (SOC) to explainearthquake occurrence.

A review and new insights on the estimation of the b-value

and its uncertainty

Warner Marzocchi and Laura SandriIstituto 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-RichterLaw. 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 itsuncertainty, 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 theestimation 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 nowmostly ignored, provide estimations without significant biases; iv) the effect on the bias due to the measurementerrors 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, 40128Bologna, Italy; e-mail: [email protected]

1272

Warner Marzocchi and Laura Sandri

In this frame, the fractal geometry distribu-tion and the earthquake dynamics are the spatialand temporal signatures of the same phenome-non. On the other hand, several lines of empiri-cal evidence dispute the universality of the GRLaw. In fact, in spite of a «first order» validityof 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), thestress level imposed on rocks (Scholz, 1968),and pore pressure variations. Another importantdiscrepancy of the hypothesis of the universali-ty of the GR Law is that the latter seems to holdonly 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) foundthat a gamma distribution better matches the re-quirement of a maximum seismic energy re-leased by an earthquake (cf. Wyss, 1973). In allthe cases reported above the b-value should notbe a constant in the earthquake catalog.

The empirical validation of the «universal-ity» hypothesis, as well as the identificationof spatial and temporal changes, passesthrough the estimation of the b-value and itsuncertainty. 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 acontinuous random variable, are still the mostused. Implicitly, this means that, at least forinstrumental measurements, the effect of thebinning is considered negligible, and that allthe formulas mentioned above give almostcomparable 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 reliabilityand the possible bias of the formulas as a

function of the number of data and of themeasurement errors in the magnitude.

Here, we do not consider the estimationsprovided through the least squares technique. Infact, 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 techniquedoes not have any statistical foundation (e.g.,Page, 1968; Bender, 1983).

2. Estimation of b and bv t through theMaximum Likelihood technique

In the first papers that described the MaximumLikelihood (ML) estimation (Aki, 1965; Utsu,1965), the magnitude M was considered a contin-uous Random Variable (RV). If eq. (1.1) holds, theprobability density function (pdf) of M is

lnf M b 1010 10

10bM bM

bM

min max=

-- -

-

^ ^h h (2.1)

where Mmin and Mmax are, respectively, the min-imum and the maximum magnitude allowed. IfMmax >> 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 fora range of magnitudes Mmax – Mmin ≥ 3. This as-sumption might be questionable in studyingvariations of the b-value in smaller ranges (e.g.,Knopoff, 2000).

The ML estimation of eq. (2.2) consists ofchoosing the b-value which maximizes the like-lihood function (Fisher, 1950), that is

lnb

M10

1

thresh

)=-n

tt^ `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 minimummagnitude for the completeness of the seismiccatalog. Hereinafter, the symbol ^ distinguishesthe 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 estimationwhich we shall introduce later. The uncertaintyis estimated by (Aki, 1965)

N

bb =v

))tt

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

. bN N

M2 30

1( )

b

SBi

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 reliableestimation also in the presence of possible (timeand/or spatial) variations of the b-value (Shiand Bolt, 1982).

As a matter of fact, the magnitude is not acontinuous variable and it is not devoid of meas-urement errors. In practical cases, the uncertain-ties on the measured magnitudes lead to the useof «binned» magnitudes, i.e. the magnitudes aregrouped by using a selected interval ∆M. For in-stance, for instrumental measurements, themagnitude interval used for the grouping is ∆M= 0.1; for magnitude estimation of historicalevents, 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 estimatethe b-value and its uncertainty (e.g., Jin andAki, 1989; Henderson et al., 1994; Öncel etal., 1996; Mori and Abercrombie, 1997; Utsu,1999; Hiramatsu et al., 2000; Gerstenbergeret al., 2001; Klein et al., 2001; Vinciguerra etal., 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 themagnitudes and the measurement errors onthe estimations of the b-value and its uncer-tainty. We remark that a wrong choice of thecompleteness threshold magnitude Mthresh canalso cause a significant bias, as shown byWiemer and Wyss (2000). Here, we do notdeal with this issue, but it should be kept inmind 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 inthe estimation of the b-value and its uncertain-ty made through eqs. (2.3) and (2.4). Then, wecheck through numerical simulations the relia-bility of some formulas reported in scientificliterature which either assume the magnitudesas 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 twofactors: i) the average µ of a continuous RVwith a power law distribution is different fromthe average of the same «binned» RV; ii)Mthresh ≠ Mmin. Hereinafter we will call thesebiases θ1, and θ2, respectively.

The influence of θ1 was estimated by Ben-der (1983). She found that the sample averagent computed from binned data is systematicallyhigher than the true value µ. This is due to thefact that the real (continuous) magnitudes in theinterval Mi − ∆Μ/2 ≤ M < Mi + ∆Μ/2 are notsymmetrically distributed around the centralvalue 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-valuefor historical catalogs.

1274


The bias θ 2 was considered by the first pa-per which addressed the use of the binnedmagnitudes (Utsu, 1966), suggesting a slightmodification of eq. (2.3). Since the lowestbinned magnitude, i.e. the threshold magni-tude, contains all the magnitudes in the rangeMthresh − ∆Μ/2 ≤ M < Mthresh + ∆Μ/2, then Mmin == Mthresh − ∆Μ/2 < Mthresh (e.g., Bender, 1983).Then, eq. (2.3) becomes

.ln

bM M10 2

1

thresh

=- -n ∆

tt^ `h j: D

(3.1)

Remarkably, this «corrected» formula was notlargely 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 useof 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). Yetsurprisingly the correction reported in eq. (3.1) israrely explicitly mentioned in the scientific liter-ature (e.g., Guo and Ogata, 1997). As a matter offact, we suspect that, in some cases, the correc-tion reported in eq. (3.1) is used in the analysiswithout explicitly mention of it in the manuscript.For instance, Gerstenberger et al. (2001) quotedAki’s (1965) formulas (eq. (2.3)) in the manu-script, but the code ZMAP written by one of theauthors (see above) contains the «corrected» for-mula given by eq. (3.1).

In any case, we argue that this correctioncould have not been largely employed also be-cause the influence of the bias θ 2 is erroneouslyconsidered 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 thedistribution

M= +

ln

ln

b M dM

b

10 10

101

min

b M M

M

min

min

$= =n3

- -#^ ]

^

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 completelyneglected, concerns the effect of the bias θ 2 onthe estimation of the uncertainty given by eq.(2.4) and the «corrected» form which reads

.N

bb=vtt

t (3.4)

The positive bias θ2 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

b1b b

2= +v i v)t

ttt td n (3.5)

therefore, > .bbv v)t tt t On the other hand, the biasθ 2 leads to an increase of the dispersion of theRV b)t around its expected value. By neglectingthe bias θ1, the true variance of b)t is

lnb var.

E .=ln

E b b

M M101

10

b

thresh thresh

22

2

4 2

= - =

- -

-

v

n n

n n

n

) )

)

)t

t

t

t t

t

J

L

KKK

_

^ ` `

` ^

N

P

OOO

i

h j j

j h

R

T

SSSS

9V

X

WWWW

C

(3.6)

where var (nt ) is the variance of the RV nt . Inthe same way, we have

.

lnE b b E

M M M M

101

2 2

10

b

thresh thresh

22

2

4 2

$

$

= - =

- + - +

-

v

n n

n n

n

∆ ∆

lnb var.

t

t

t

t t

t

J

L

KK_

^

` `

` ^

N

P

OOO

ih

j j

j h

R

T

SSSSS

9

V

X

WWWW

C

(3.7)

1275


The validity of eqs. (3.6) and (3.7) deserves fur-ther explanations. In particular, these equationsassume that E (b)t ) = b) and E(bt ) = b, respec-tively. If we take the expected value of Taylor’sexpansion around the true value µ of eqs. (2.3)and (3.1), we see that these assumptions holdonly for small deviations of nt , i.e. for largedatasets. Numerical investigations have shownthat the biases are negligible for datasets with50 or more earthquakes.

By comparing eqs. (3.6) and (3.7) we obtain

b1b b

2

2

= +v i v)

tt 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 theRV b bv)

)t t^ h increases more than the increase inthe estimation of the uncertainty bv )t t . In otherwords, eq. (2.4) provides an underestimation ofthe 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 thegrouping of the magnitudes. Remarkably, be-sides very few exceptions (e.g., Frohlich andDavis, 1983), these formulas were almost ig-nored in subsequent applications. We arguethat the reasons are mainly of a technical na-ture. Bender’s (1983) formula, for example,can be solved only numerically. Moreover, inher analysis she gave more emphasis to thebias θ1 introduced by the use of the continu-ous approximation (eq. (2.3)), concluding thatthe latter provides almost unbiased estima-tions of the b-value if the magnitude intervalfor the grouping is ∆M = 0.1.

A definite improvement to the estimationof the b-value was provided by Tinti and Mu-largia (1987). Their formula reads

1lnb p

10=

ln M∆TMt

^_

hi (3.9)

where

pMM

1thresh

= +-n∆

t

J

L

KK

N

P

OO (3.10)

and the associated asymptotic error is

Npln M

p

10

1b =

-v

∆TMt t

^ h(3.11)

where N is the number of earthquakes. In thiscase, 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 formulasdescribed above, we simulate 1000 seismic cata-logs, for different catalog sizes. The magnitudes Mare obtained by binning, with ∆M = 0.1 (as for theinstrumental magnitudes), a continuous RV dis-tributed with a pdf given by eq. (2.2); in otherwords, 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 andbTMt calculated in 1000 synthetic catalogs as a func-tion of the number of data, for the case b = 1 andb = 2. To each median is attached the 95% confi-dence interval, given by the interval between the2.5 and 97.5 percentile. From fig. 1a,b, we can seethat the estimation bTM

t (Tinti and Mulargia, 1987)is bias free, also for a small dataset. As regards thecontinuous formulas, with and without correction(respectively eqs. (3.1) and (2.3)), we can see thatthe bias θ2 reported in fig. 1a,b is comparable to thetheoretical expectation given by eq. (3.3). The cor-rected estimation bt is undoubtedly much closer tothe real b-value. The slight underestimation of bt

(much less than 1% of the real b-value) is due to thebias 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 theestimations of the uncertainty, it is necessaryto compare each estimation with the true dis-

1276


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 theF test values relative to the same uncertainties calculated for bt instead of b)t (i.e. eq. (3.4)). Asterisks are the Ftest values relative to the uncertainty calculated through eq. (3.11). The dotted line represents the critical valueto 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 solidthin line represents the true b-value.

a b

1277


persion of estimation of the b-value around itscentral value. In particular, we compare thedispersion of the b-value estimation aroundits average with the average of the estimateduncertainties, through the Fisher test (e.g.,Kalb- fleisch, 1979)

= Average of thesquare of the uncertaintyVariance of theestimation of the valueF b -

(3.12)

The null hypothesis is that the two variances arethe same.

The results are reported in fig. 2a,b. The mostinteresting 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 thenon-corrected estimation b)t (eq. (2.3)) leads to anunderestimation of its real dispersion. This fact isvery important because it might suggest varia-tions in the b-value whereas they do not exist.Note that the amplitude of this bias is correctlyestimated by eq. (3.8). On the contrary, all theother estimations of the uncertainty do not havesignificant biases. It is important to note that theerror estimated through eq. (2.5) (Shi and Bolt,1982) is unbiased if the b-value is estimatedthrough 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 devoidof measurement errors distributed with a pdfgiven by eq. (2.1), and ε simulates the meas-urement errors distributed as a Gaussiannoise. The choice of the normal distributionhas 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 ofM+

can be obtained as follows (Ventsel, 1983;Tinti and Mulargia, 1985; Rhoades, 1996)

d dG M f M f MM

M M M

1 2

min

max

= f f3-

-+

+# #c ^ ^m h h

R

T

SSS

V

X

WWW

(4.2)

where G( M+

) is the cumulative distribution ofM+

, f1(M) is given by eq. (2.1), and f2 (ε) is N(0, 2vf ) distributed. By differentiating withrespect to the variable M

+, we obtain the pdf

.-

exp$ $

+$

ln

ln ln

ln

ln

g Mb

b M b

M M b

M M b

erf

erf

21

10 10

10

21

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^

^ ^

^

^

mh

h h

h

h

R

T

SSS

R

T

SSS

;

V

X

WWW

V

X

WWW

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 berf

b

b

g

10

2

101

10

10

21

10 2

21

min

bM

2

2

min$

- +-

- +

=

v

v

v

-

f

f

f

+

+

+

^

^

^

^

c

h

h

h

h

m

R

T

SSS

;

V

X

WWW

E

Z

[

\

]]

]]

)

_

`

a

bb

bb

3

(4.4)

The ML estimation of eq. (4.4) gives +

0=

+

ln ln

lnln

ln

expln

bM

bN

M M b

M M b

erf

110 10

102 10

12

10

2

10

min

min

min

i

i

i

N

22

2

2

2

1

$

$

- + +

-

-- +

-- +

n

v rv

v

v

v

v

=

f

f

f

f

f

f

+

+

/

!

^ ^

^`^

^

^

h h

hjh

h

h

R

T

SSSS

R

T

SSSS

V

X

WWWW

V

X

WWWW

Z

[

\

]]]]

]]]]

Z

[

\

]]

]]

_

`

a

bb

bb

_

`

a

bbbb

bbbb

(4.5)

1278


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

b

Fig. 4a,b. As for fig. 2a,b, but relative to the case with added error to the magnitudes. The standard deviationsof the measurement errors are σε = 0.1, 0.3, 0.5 (from left to right).

1280


where +n/

is the sample average of M+

. The firstthree 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 thebracket in eq. (4.5) to the b-value estimationfor 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 allthe estimations for b = 2 and σε = 0.5. This canbe due to the improper binning of ∆M = 0.1 inthe 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 previouscase, by plotting the results of the F test (fig.4a,b). The results are almost the same as thoseobtained for the case without measurement er-rors (fig. 2a,b). Note that also in this case theuse of eq. (2.4) leads to a significant underesti-mation of the uncertainty.

5. Final remarks

The purpose of this paper was to review themain concepts of the estimation of the b-valueand its uncertainty, and to provide new insightson the reliability of different formulas reportedin scientific literature. The detection and quan-tification of possible significant biases in suchestimations is a crucial issue in many scientificapplications, such as hazard studies and any at-tempt to test the constancy or the universalityof the b-value. Here we studied analyticallyand numerically the possible biases introducedby the use of the binned magnitudes and meas-urement errors on different formulas.

We found that, in general, the influence ofthe measurement errors appears negligible com-

pared to the effects of the binned magnitudes ifimproperly dealt with. In particular, we foundthat the most commonly used formulas (Aki,1965; see eqs. (2.3) and (2.4)), which assumethe magnitude as a continuous random vari-able, produce a strong bias in the estimation ofthe b-value, and a significant underestimationof its uncertainty. This means that any spatialor temporal variation of the b-value obtainedby eqs. (2.3) and (2.4) has to be regarded witha strong skepticism. On the contrary, we showhow the same continuous formula with a smallcorrection 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. We also verified that other very sim-ple formulas provided in the past (e.g., Tintiand Mulargia, 1987) do not have significant bi-ases and they work very well also with a smalldataset contaminated by realistic measurementerrors.

Acknowledgements

The authors thank Stefano Tinti for thehelpful comments.

REFERENCES

AKI, K. (1965): Maximum likelihood estimate of b in theformula log (N) = a − bM and its confidence limits,Bull. Earthq. Res. Inst. Tokyo Univ., 43, 237-239.

BAK, P. and C. TANG (1989): Earthquakes as a self-organ-ized criticality phenomenon, J. Geophys. Res., 94,15635-15637.

BENDER, B. (1983): Maximum likelihood estimation of bvalues for magnitude grouped data, Bull. Seismol. Soc.Am., 73, 831-851.

FISHER, R.A. (1950): Contribution to Mathematical Statis-tics (Wiley, New York).

FROHLICH, C. and S. DAVIS (1993): Teleseismic b-values: or,much ado about 1.0, J. Geophys. Res., 98, 631-644.

GERSTENBERGER, M., S. WIEMER and D. GIARDINI (2001): Asystematic test of the hypothesis that the b-value varieswith depth in California, Geophys. Res. Lett., 28, 57-60.

GUO, Z. and Y. OGATA (1997): Statistical relations betweenthe parameters of aftershocks in time, space, and mag-nitude, J. Geophys. Res., 102, 2857-2873.

GUTENBERG, B. and C. RICHTER (1954): Seismicity of theEarth and Associated Phenomena (Princeton Universi-ty Press, Princeton, New Jersey), 2nd edition, pp. 310.

HEALY, J.H., W.W. RUBEY, D.T. GRIGGS and C.B. RALEIGH

(1968): The Denver earthquakes, Science, 161, 1301-1310.

1281


HENDERSON, J., I.G. MAIN, R.G. PEARCE and M. TAKEYA

(1994): Seismicity in the North-Eastern Brazil: Fractalclustering and the evolution of the b-value, Geophys. J.Int., 116, 217-226.

HIRAMATSU, Y., N. HAYASHI and M. FURUMOTO (2000): Tem-poral changes in coda Q −1 and b-value due to the staticstress change associated with the 1995 Hyogo-KenNanbu earthquake, J. Geophys. Res., 105, 6141-6151.

ITO, K. and M. MATSUZAKI (1990): Earthquakes as self-or-ganized critical phenomena, J. Geophys. Res., 95,6853-6860.

JIN, A. and K. AKI (1989): Spatial and temporal correlationbetween coda Q −1 and seismicity and its physicalmechanisms, J. Geophys. Res., 94, 14041-14059.

KAGAN, Y. (1993): Statistics of characteristic earthquakes,Bull. Seismol. Soc. Am., 83, 7-24.

KAGAN,Y. (1994): Observational evidence for earthquakes asa nonlinear dynamic process, Physica D, 77, 160-192.

KAGAN, Y. (1997): Seismic moment-frequency relation forshallow earthquakes: regional comparison, J. Geophys.Res., 102, 2835-2852.

KALBFLEISCH, J.G. (1979): Probability and Statistical Infer-ence (Springer, Berlin), voll. 2, pp. 658.

KARNIK, V. and K. KLIMA (1993): Magnitude-frequencydistribution in the European-Mediterranean earthquakeregions, Tectonophys, 220, 309-323.

KLEIN, F.K., A.D. FRENKEL, C.S. MUELLER, B.L. WESSON

and P.G. OKUBO (2001): Seismic hazard in Hawaii:high rate of large earthquakes and probabilistic groundmotion maps, Bull. Seismol. Soc. Am., 91, 479-498.

KNOPOFF, L. (2000): The magnitude distribution of declus-tered earthquakes in Southern California, Proc. Nat.Acad. Sci. U.S.A., 97, 11880-11884.

MAIN, I. (2000): Apparent breaks in scaling in the earth-quake cumulative frequency-magnitude distribution:fact or artifact?, Bull. Seismol. Soc. Am., 90, 86-97.

MIYAMURA, S. (1962): The magnitude-frequency relation ofearthquakes and its bearing on its geotectonics, Proc.Japan Acad., 38, 27.

MOGI, K. (1962): Study of the elastic shocks caused by thefracture of heterogeneous materials and its relation toearthquake phenomena, Bull. Earthq. Res. Inst., 40,125-173.

MORI, J. and R.E. ABERCROMBIE (1997): Depth dependenceof earthquake frequency-magnitude distributions inCalifornia: implications for the rupture initiation, J.Geophys. Res., 102, 15081-15090.

OKAL, E.A. and S.H. KIRBY (1995): Frequency-momentdistribution of deep earthquake; implications for theseismogenic zone at the bottom of Slabs, Phys. EarthPlanet. Inter., 92, 169-187.

ÖNCEL, A.O., I.G. MAIN, Ö. ALPTEKIN and P. COWIE (1996):Spatial variations of the fractal properties of seismicityin the Anatolian fault zones, Tectonophys, 257, 189-202.

PACHECO, J.F. and L.R. SYKES (1992): Seismic moment cat-alog of large shallow earthquakes, 1900 to 1989, Bull.Seismol. Soc. Am., 82, 1306-1349.

PACHECO, J.F., C.H. SCHOLTZ and L.R. SYKES (1992):Changes in frequency-size relationship from small tolarge earthquakes, Nature, 355, 71-73.

PAGE, R. (1968): Aftershocks and microaftershocks of theGreat Alaska earthquake of 1964, Bull. Seismol. Soc.Am., 58, 1131-1168.

REASENBERG, P. (1994): Computer programs ASPAR,GSAS and ENAS and APROB for the statisticalmodeling of aftershock sequences and estimation ofaftershock hazard, U.S. Geol. Surv. Open File Rep.,94-122.

RHOADES, D.A. (1996): Estimation of the Gutenberg-Richter relation allowing for individual earthquakemagnitude uncertainties, Tectonophysics, 258, 71-83.

SCHOLTZ, C.H. (1968): The frequency-magnitude rela-tion of microfracturing in rock and its relation toearthquakes, Bull. Seismol. Soc. Am., 58, 399-416.

SCHOLTZ, C.H. (1997): Size distributions for large andsmall earthquakes, Bull. Seismol. Soc. Am., 87,1074-1077.

SHI, Y. and B.A. BOLT (1982): The standard error of themagnitude-frequency b-value, Bull. Seismol. Soc. Am.,72, 1677-1687.

TINTI, S. and F. MULARGIA (1985): Effects of magnitude un-certainties on estimating the parameters in the Guten-berg-Richter frequency-magnitude law, Bull. Seismol.Soc. Am., 75, 1681-1697.

TINTI, S. and F. MULARGIA (1987): Confidence intervals ofb-values for grouped magnitudes, Bull. Seismol. Soc.Am., 77, 2125-2134.

TRIEP, E.G. and L.R. SYKES (1997): Frequency of occur-rence of moderate to great earthquakes in intracontine-tal regions: implications for changes in stress, earth-quake prediction, and hazard assessment, J. Geophys.Res., 102, 9923-9948.

UTSU, T. (1965): A method for determining the value of b ina formula log n = a – bM showing the magnitude-fre-quency relation for earthquakes, Geophys. Bull., Hok-kaido Univ., Hokkaido, Japan, 13, 99-103 (in Japanese).

UTSU, T. (1966): A statistical signficance test of the differ-ence in b-value between two earthquake groups, J.Phys. Earth, 14, 34-40.

UTSU, T. (1999): Representation and analysis of the earth-quake size distribution: a historical review and somenew approaches, Pure Appl. Geoph., 155, 509-535.

VENTSEL, E. (1983): Teoria delle Probabilità (Mir, Moscow),pp. 550.

VINCIGUERRA, S., S. GRESTA, M.S. BARBANO and G. DI STE-FANO (2001): The two behaviour of Mt. Etna volcanobefore and after a large intrusive episode: evidencesform b-value and fractal dimension of seismicity, Geo-phys. Res. Lett., 28, 2257-2260.

WIEMER, S. and J.P. BENOIT (1996): Mapping the b-valueanomaly at 100 km depth in the Alaska and New Zea-land subduction zones, Geophys. Res. Lett., 23, 1557-1560.

WIEMER, S. and K. KATSUMATA (1999): Spatial variability ofseismicity parameters in aftershock zones, J. Geophys.Res., 104, 13135-13151.

WIEMER, S. and S.R. MCNUTT (1997): Variations in fre-quency-magnitude distribution with depth in two vol-canic areas: Mount St. Helens, Washington, and Mt.Spurr, Alaska, Geophys. Res. Lett., 24, 189-192.

WIEMER, S. and M. WYSS (1997): Mapping the frequency-magnitude distribution in asperities: an improved tech-nique to calculate recurrence times, J. Geophys. Res.,102, 15115-15128.

WIEMER, S. and M. WYSS (2000): Minimum magnitude ofcompleteness in earthquake catalogs: examples from

1282


Alaska, the Western United States and Japan, Bull.Seismol. Soc. Am., 90, 859-869.

WIEMER, S., S.R. MCNUTT and M. WYSS (1998): Temporaland three-dimensional spatial analysis of the frequen-cy-magnitude distribution near Long Valley Caldera,California, Geophys. J. Int., 134, 409-421.

WYSS, M. (1973): Towards a physical understanding of the

earthquake frequency distribution, Geophys. J. R. As-tron. Soc., 31, 341-359.

(received March 25, 2003;accepted August 1, 2003)

b value (marzziotti sandri 2003)

Documents