Teleseismic b values

1. JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 98, NO. B1, PAGES 631-644, JANUARY 10, 1993
Teleseismic b Values; Or, Much Ado About 1.0
CUFF F•OHUCH
Institutefor Geophysics,UniversityofTexasatAustin
Scozr D. DAVIS1
CooperativeInstitutefor ResearchintheEnvironmentalSciencesandNationalGeophysicalData Center,Boulder,Colorado
In thispaperweinvestigatethevalueof bintheGutenberg-Richterrelationforfourteleseismiccatalogsof
earthquakes:Abe'shistoricalcatalog,theHarvardCentroidMomentTensor(CMT) catalog,thecatalogof the
InternationalSeismologicalCentre(ISC),andtheBlacknestcatalog.Anunfortunateresultisthatbdiffersby30%
ormorewhendeterminedindifferentmagnituderanges,indifferentcatalogs,orusingdifferentmethods.Forglobal
catalogsseparatedintoshallow,intermediate,anddeepearthquakegroups,allvaluesdeterminedforbliebetween
0.72and1.34.Wecanidentifynosystematicglobalvariationofbwithdepth.Forteleseismiccatalogsitisdifficult
tobelievemeasuredgeographicvariationsinbbecausesystematicerrorscauseproblemsofearthquakedetection,
earthquakelocation,aftershockidentification,andmagnitudedetermination.However,somevariationsinbareso
persistentandlargethattheymustbereal.FordeepearthquakesinTonga-Fiji,forexample,variousmeasurements
of blie between1.06and1.57,comparabletob forshallowearthquakes,whereasmeasurementsof b fordeep
earthquakesintherestoftheworldaremuchlower,between0.53and0.96.ForshallowearthquakesintheHarvard
CMTcatalog,earthquakeswiththrustandstrikeslipfocalmechanismshavesignificantlylowerbvalues(0.86and
0.77)thanearthquakeswithnormalfaultingmechanisms(1.06).WhenweseparatetheISCcatalogintoprimary
events(mainshocksandearthquakeswithnoaftershocksorforeshocks)andsecondaryevents(aftershocksand
foreshocks),weobservethatbfor secondaryeventsisnearlyalwayssignificantlyhigherthanbformainshocks.
However,weshowthatthedifferencehasnophysicalsignificance,asit arisessimplyfromtheactof choosing
mainshocksasthelargestearthquakein aforeshock-mainshock-aftershocksequence.Whenwecorrectforthis
systematiceffectbycomparingtherealcatalogstoidenticalcatalogswithrandomlyreassignedmagnitudes,wefind
thatb for secondaryeventsin therealcatalogis actuallylowerthanexpected.Thusamongaftershockslarge
earthquakesarerelativelymorecommonthanexpected,perhapsbecausethemainshockraptureloadsasperitiesin
adjacentregions.
INTRODUCTION
A trivia]observationaboutthenatureof earthquakesis that
not all earthquakesare the same size. Useful parameters
which quantify earthquakesize includemoment[Hanks and
Kanamori, 1979], potency[Heaton and Heaton, 1989], and
radiated energy. However, for historical reasonsthe most
commonlyusedmeasureof earthquakesize is magnitude,as
determinedfrom a variety of differentmagnitudescales. In
practice the statisticaldistributionof sizes for a group of
earthquakescan be quite complicated. For purposesof
comparison,the so-calledGutenberg-Richter[1954] model
hasprovideda quitesuccessfulstartingmodel. Over a limited
range of earthquakemagnitudes,for most magnitudescales
the distributionoften appearsto satisfy
1og10N=a-bM (1)
where N is the numberof earthquakesin the grouphaving
magnitudeslargerthanM, anda andb areconstants.Clearly,
b is a statisticmeasuringtheproportionsof largeandsmall
earthquakesin the group;if b is large,small earthquakesare
relatively common, whereas when b is small,. small
earthquakesarerelativelyrare.
There is a surprisinglylargeliteratureconcerningthese"b
'NowatInstituteforGeophysics,UniversityofTexasatAustin
Copyfight1993bytheAmericanGeophysicalUnion.
Papernumber9211101891.
0148-0227/93/9ZIB-01891505.00
631
values"(seeBath [1981] for a review). A few papersattempt
to relate the value of b to the physical processeswhich.
generategroupsof earthquakes.One approachis to assume
thatthenumberof earthquakesof a givensizeis proportional
to the areaor volumeavailablefor faultingandthusthat theb
value is closelyrelated to the dimensionalityof this volume
[Kanarnori and Anderson, 1975; Aki, 1981]. Another
approachis to attemptto derive a relationbetweenthe stress
or stressdrop in the sourcearea and the b value [Scholz,
1968; Wyss,1973;Hanks, 1979]; generallythe resultis that
higherstressescorrespondto lower b values.
If such physical factors do affect b, then we ought to
observe variations in b that depend on tectonic and
theological conditions near the earthquake source. The
primary objectiveof this paperwill be to determineb values
for selected earthquakegroups in several different global
earthquake catalogs. In particular, we shall addressthe
following questions:
1. How similar or different are b values measured in
different global catalogs? How large must observed
differencesbe to be significant?
2. How-different are b valuesfor earthquakesin different
geographicand tectonic groups, and are these differences
significant?
3. Does bsec, the b value for catalogs of earthquake
foreshocksor aftershocks,differ from bMain,theb value for
events which are mainshocks or which possess no
aftershocks?
The next section describes several different global
earthquakecatalogs,the magnitudescalesusedfor evaluating

2. 632 FRoauca^NDD^v•s' Muc• ADoABotrrT•ss•c b
earthquakesize,andvariousproceduresfor determiningb. To
separateearthquakesinto geographicgroups,we usethesame
subgroupsas in a companionpaperon p values[Davis and
Frohlich, 1991a]. To separateearthquakesin terms of
tectonicgroupsor sourcetype we utilize methodsdescribed
by Frohlich and Apperson[1992].
To investigatedifferencesbetween fore- and aftershocks
and other earthquakes,we utilize methods incorporating
single-linkclusteranalysis(SLC), describedin somedetail
previously[Frohlich and Davis, 1990;Davis and Frohlich,
199lb]. We defineaftdrshocksasin ourpreviouswork,
namely,if a specifiedstatisticaltestsuggeststhata sequence
of earthquakesisrelated,thenwe definethelargestearthquake
as the "primary event" or mainshock, and all other
earthquakesare "secondaryevents", with aftershocksand
foreshocksbeing those which occur after, or before, the
primary event.
Somedifficultiesarisebecause,asweshallshow,thevery
actof identifyingthe primaryeventasthe largestearthquake
in each sequencesystematically affects the values of b
determinedfor primaryandsecondaryevents. Thusmeasured
valuesof bsecfor aftershocksor secondaryeventsmustnearly
always be higher than bMain for mainshocks, even in the
absenceof a physicalcause. To overcomethisproblemand
determinewhetherobserveddifferencesbetweenbMainandbsec
are significant,we developa methodwhich comparesbMain
andbSecfromreal earthquakecatalogswith valuesdetermined
from a correspondinghypotheticalcatalog with randomly
reassignedmagnitudes.
DATA AND METHODS
GlobalSeismicityCatalogsandMagnitudeScales
Because b value is a statistical measure of the relative
occurrenceratesof big and small earthquakes,we prefer
magnitudescalesthatmeasureearthquakesizefaithfully,and
we require catalogsthat are as completeand uniform as
possible.Forthepresentstudy,wehaveutilizeda varietyof
magnitudedatafromfourdifferentglobalseismicitycatalogs.
Abe's catalog. The first of theseis thecatalogof large
earthquakespreparedby Abe [1981, 1984]. He carefully
reevaluatedamplitudeand period observationsfrom several
sourcesincludingworksheetsin the archivesat the California
Institute of Technology. For 1614 large earthquakes
occurring between 1897 and 1980, he redetermined the
magnitudeMs from surfacewaveshavingperiodsof about
20s,andGutenberg's[1945]magnitudemB frombodywaves
having periodsof 6-8 s (Figure 1). The distributionof
magnitudesin Abe'scatalogsuggeststhatMs andmB saturate
at about8.5 and8.2, respectively.As magnitudedecreases,
therateof activityincreasesdownto7.0forbothMs andmB.
Abe's [1981, p. 72] objectivein this work wasto "remove
major inhomogeneitiesin the existing catalogs." For
earthquakeswithmagnitudesof 7'.0andabove,it is likely
thatAbe'scatalogis moreuniformandcompletethanother
availablecatalogs,at leastfor earthquakesoccurringbetween
1904 and 1976.
Harvard catalog. The secondcatalogusedis the Harvard
CentroidMoment Tensor (CMT) catalog. This consistsof
recordsof 8719 earthquakesoccurringbetweenJanuary1977
and June1989. For this periodthe Harvardgrouputilized
3.0
2.5
1.0
0.5
0.0
I I I I I
6.0 7.0 •"' 8.0
rn B
Fig.1.DistributionofmagnitudesmsinAbe'scatalogoflargeearthquakes
occurringbetween1897and1980. Thesolidlineisthecumulativenumber
ofearthquakeshavingmsequaltoorexceedingthegivenvalue,thedashed
lineisthenumberinbinsofwidth0.1.Theverticallinesbetweenmsof7.0
and7.5delineatetherangeusedtodetermineba•,eandbc•,winTable1.
long-period (50-200 s) body and/or surface waves to
determinescalarmomentsandCMT for mostearthquakes
havingmagnitudesof about5.5 andgreater(Figure2). For
thisstudywe convertedthe scalarmomentsMo (in dyne
centimeters)to momentmagnitudesMw andmwusingthe
relationshipssuggestedby Hanksand Kanamori[1979]and
Kanamori [1983]:
2
Mw=T 1øgloMø- 10.7, (2)
mw=(1OgloMo- 10.1)./2.4. (3)
In practice,the momentmagnitudescalesdo not saturate
becausetheydependon suchlongperiodobservations.For
magnitudesbetween7.0 and7.5 wherenoneof thescalesMw,
mw,Ms andmB arenearsaturation,Kanamori [1983] notes
thatMw andmw correspondto Ms andmB, respectively.
WhileHarvard'smethodsfordeterminingCMTshavechanged
somewhat over time [Dziewonski et al., 1981; Dziewonski
and Woodhouse,1983], generallymomentmagnitudesare
becomingacceptedover othermagnitudesbecausemoment
magnitudesare determinedfrom digital data recordedat a
relativelyfew carefullyoperatedseismographstations.
ISC catalog. The third catalog used is that of the
International Seismological Centre (ISC). While this
containsrecordsof severalhundredthousandearthquakes,
betweenJanuary1964andFebruary1986only about47,553
possessmagnitudesmb of 4.8 or larger(Figure3). Because
m/, utilizesbodywaveswith periodsof approximately1.0
second,it saturatesat a magnitudeof about 6.8. In the
National Oceanographicand AtmosphericAdministration
HypocenterData File, which includesmany of the same
earthquakesas the ISC catalog,analysisby Habermann
[1982, 1983] andHabermannet al. [1986]foundthatin most
geographicregions reporting of earthquakeactivity is
homogeneousforearthquakeswithmbof approximately5.0.

3. FROHLICHANDDAVIS' MUCH ADO ABOUTTELESEISMICb VALUES 633
4
I I
Harvard CMT
I I I I I
Fig. 2. Distributionof magnitudesMwin the HarvardCentroidMoment
Tensorcatalogforearthquakesoccurringbetween1977and1990.Solidand
dashedlines are the cumulativenumberand numberin bins of width 0.1,
respectively.TheverticallinesbetweenMwof 5.65and6.55delineatethe
rangeusedtodeterminebHrvin Table1.
4 ISC
I I I I I I I I
5.0 6.0 7.0 õ.0
baselinedifferencesbetweenindividualstations.Like mb, the
redeterminedmaximum likelihood magnitudesrnb (rnlike)
saturateat about 6.8. As magnitudedecreases,the rate of
activityincreasesdownto about4.2 (Figure4). However,for
the Blacknest catalog the semilog plot of number versus
magnitudeis not so straightas with the Harvard or Abe
catalogs. This is probably becausein certain geographic
regions the global network often missessome earthquakes
havingmagnitudesbetweenabout4.3 and5.0. Nevertheless,
for the period 1964 to 1983, we presumethat magnitudesin
the Blacknest catalog are more uniform temporally and
geographicallythanin theISC catalog. For earthquakeswith
magnitudesof about 5.0 and below rnb (rnlike) averages
approximately0.2 smallerthanmb.
"Altered" magnitudes: Combining the Abe and Harvard
catalogs. Nearly all earthquakecatalogsexhibit temporal
changesin the rate of activity within particularmagnitude
ranges. While some investigatorsattribute this to intrinsic
changesin the rate of occurrenceof earthquakes[Abe and
Kanamori, 1979], another plausible explanation is that
changesin global network coverageand seismographstation
practicesystematicallyaffectmagnitudes.Perez and Scholz
[1984] show that these apparentrate changesdisappearif
Abe'sMs is adjustedby subtracting0.5 from all magnitudes
for earthquakes prior to 1908, and subtracting 0.2 for
earthquakesbetween1908 and 1948.
Using an approachsimilar to that of Perez and Scholz
[1984], we canadjustAbe'srn/• so that for earthquakeswith
magnitudebetween 6.95 and 7.55, the occurrencerate is
nearly constantand in agreementwith the Harvard CMT
catalog(Figure5).Wedefinethe"adjusted"magnitudernB(a•lj)
for anearthquakein yearYby
mB(a•ij) =mB- 0.2forYbetween1904and1939;
= mB- 0.3 for Y between1940 and 1949;
= mB - 0.2 for Y between1950 and 1958;
= mw- 0.1 for Y between1959 and 1976;
= mwfor Y between 1977 and 1990.
Here mw is calculated as above from the CMT scalar
momentMo. SimilarapparentratechangesalsoafflicttheISC
catalogand(to a lesserextent)theHarvardCMT catalog.For
earthquakeswithmagnitudesrnb (mlike) of 4.5 andgreater,
occurrenceratesin theBlacknestcatalogarenearlyconstant
over time.
IIl b
Fig.3. Distributionof magnitudesmbexceeding4.8 in theInternational
SeismimologicalCentrecatalogfor earthquakesoccurringbetween1964
and 1986. Solid and dashed lines are the cumulative number and number in
binsofwidth0.1,respectively.Theverticallinesbetweenmbof5.0and6.0
delineatetherangeusedtodetermineblsc in Table1.
Blacknestcatalog. The fourthcatalogusedis theBlacknest
catalog[Lilwall andNeary, 1986],whichattemptsto remove
stationreportingbiasesin mb asdeterminedby theISC. As
discussedby Habermann[1991] andnumerousothers,thereis
considerableevidencethat both regionaland time-dependent
nonuniformitiesin rnb doexist. Lilwall andNeary [1986] use
a maximum likelihood method similar to that developedby
Ringdal [1986] that purportsto eliminate bias occurring
becauseof data truncation at low readings and becauseof
Methodsfor Measuringb Values
Determining the relative proportionsof large and small
earthquakeswithin a grouprequiresknowledgeof rates of
seismic activity for earthquakesof different sizes. One
approachis to measurethe slope(b value) of the distribution
of log-number versus magnitude, taking care to select a
relatively small magnituderange. Over this rangethe slope
mustbe reasonablystraight,andwe mustselecta rangewhere
themagnitudescalemeasuresearthquakesizefaithfully. Utsu
[1966] and Aki [1965] proposedthe maximum likelihood
methodto measureb for a catalogwheremagnitudesexceed
some minimum value Minin, andPage [1968] presented
modificationsto apply when the catalog is complete only
with a rangeof magnitu.desbetweenMininandM,nax.Bender
[1983] generalizedthe method to apply when the data are
occurrenceratesmeasuredin discretemagnitudeintervals.
Bender's method. Bender [1983] explains her method

4. 634 FROHLICHANDDAVIS: MUCH ADo A•otrr TELESFaSMICb VALUES
clearly and in some detail, therefore here we will only
summarizeher result. In particular,supposethat for a group
of earthquakeswe wishtodetermineb for magnitudesreported
in n discretebinsof width8m betweena minimummagnitude
of Mmin anda maximummagnitudeMm•x. Thenthemaximum
likelihood estimate of b satisfies
q q• (M>-Minin
• -n • = (4)
1-q 1-q &n '
whereq = exp(- 23026 bSm)and(M) is themeanvalueof all
magnitudeswithin the stated ranges. Generally it is not
possibleto solve this equationexplicitly for q, insteadwe
determineit numericallyß
Forexample,supposewe wishto findb for all Aberna with
magnitudesbetween7.0 and 7.5ßthenM min= 6.95, M max =
7.55, n = 6 and8m = 0.1. For this groupof earthquakes
(Figure1) we findthat(M) = 7.18, andthusb = 1.15.
Compositemethod.'A completelydifferentapproachis to
determineb directlyby comparingthe rate of occurrenceof
large and small earthquakes.As we would like "large" and
"small" to encompass the greatest possible range of
magnitudes,we will utilize different magnitudescalesfor
determiningthe relative rates. Since individual magnitude
scalessaturatefor earthquakesoutsidesome optimum size
range, Frohlich [1989] suggestedmeasuringan effective
valueforbeffusingfrequencyofoccurrencedeterminedfortwo
differentmagnitudescales,eachwithin their optimumrange.
In particular,shpposethatrlarge is the frequencyfor
earthquakeshavingmagnitudesbetweenMlargeandMlarge +
•M andrsmatt isthefrequencyforearthquakeswithmagnitudes
betweenMsmalI andMsmatt+ •M. If eachgroupis distributed
asr = A 10't'M thenit is reasonableto define
rlarge
b,y=M -M ' (5)large small
Indeed,if eachgroupis distributedwith the samevalue of b,
thenbeff = b exactly,independentof thebinwidthbM. Of
course,becausedifferentmagnitudescalesmeasuredifferent
properties,oftenbeffdiffersfromtheb valuesdetermined
from either scale.
In this paper we employ a slight generalizationof this
schemeby utilizing three different magnitudescales. To
determinea compositeb valuehereaftercalledbcom•,,we
utilizethemB(adj)scaledescribedabove,themw moment
magnitudesdeterminedfrom scalarmomentsreportedHarvard,
andthemb (mlike) magnitudesfrom the Blacknestcatalog.
We determinerAbe(adj)forearthquakeshavingmagnitudes
between7.0 and 7.5, rHrv for magnitudesbetween5.90 and
6.50, andrBtl•for magnitudesbetween4.85 and5.45. Then
bMis0.6,andbco,,,•,istheslopeoftheleastsquaresfit tothe
magnitudelog-ratepairs (4.85, log10ratk),(5.90, loglorHrv),
and(6.95,10910rAbe(aclj)) (Figure6).
GeographicandTectonicGroupingof Earthquakes
Subsequently we determine b values for a variety of
earthquakesubgroups.-Thus we separateearthquakesinto
groupsdeterminedby (1) geographicregion,(2) focal depth,
(3) nearest tectonic feature, (4) focal mechanism, and (5)
4
o2
I I I I I I I I I
3.0 4.0 5.0 6.0 7.0
Illmlike
Fig.4. Distributionofmagnitudesmr,(m/ike)in theBlacknestcatalogfor
earthquakesoccurringbetween1964and1983.Solidanddashedlinesare
thecumulativenumberandnumberinbinsof width0.1,respectively.The
verticallinesbetweenmr,(m/ike)of4.86and6.15delineatetherangeused
todeterminebst• in Table1.
primary (mainshocks)or secondaryevents(aftershocksand
foreshocks).
For the geographic regionalization, we form regions
identicalto thosedefinedbyDavisandFrohlich[1991a]. For
focal depth, we separately consider three depth ranges:
shallow (depth h lessthan 70 km), intermediate(h between
70 and 300 km), and deep (h greater than 300 kin). To
determinethe distancebetween earthquakesand the nearest
majortectonicfeature,we useddigitizedmapsof trenchesand
ridge-transforms supplied by the Paleo-Oceanographic
MappingProject(POMP) groupat the Universityof Texasat
Austin [Royeret al., 1992].
1000
800
600
400
200
ß
!
!
!
-.1
m W
19100 19120 19•40 19•60 19180
Year
Fig.5. Cumulativenumberofearthquakeswithmagnitudesbetween6.95
and7 55forthe"adjusted"magnitudescalerns.a•-(solidline) andwithoutß •, •J ,
adjustment(dashedline).ThemagnitudemB(adj}ISdeterminedfrommsby
subtracting0.2priorto 1940andbetween1950and1958,subtracting0.3
between1940and1949,andsubtracting0.1between1959and1976. After
1977,mB(adj)equalsmw,themomentmagnitudedeterminedfromthe
Controid Moment Tensor scalar moment.

5. FRO•ILICI-IANDDAVIS: MUCHADo Anotrr TELESEISMICb VALUES 635
Focal mechanismsare available generally only for the
earthquakesin the HarvardCMT catalog. We classifythese
usingthe trianglediagrams(Figure7) proposedby Frohlich
and Apperson[1992] and describedin detail by Frohlich
[1992]. In this scheme,the three vertices of the triangle
represent "pure" thrust, strike slip, or normal faulting
earthquakeswith vertical T, B, andP axes,respectively. If
ST, •B, and5/, arethe dip anglesof theT, B, andP axes
reportedin theHarvardCMT catalog,then
2 • +sin6?=1. (6)sin•r+sin2 2
Thus for any earthquake, we can define its fractional
proportionsof thrust, strike slip, or normal faulting to be
sin2ST,sin2bB,andsin25/,. In thispaper,weshallclassify
aCMT mechanismas"thrust"if ST isgreaterthan50ø,"strike
slip"if bBis greaterthan60ø,and"normal"if 5/, is greater
than 60ø. All other mechanismswe classifyas "other." In
addition,we includeas"other" all CMT mechanismshaving a
compensatedlinearvectordipole(CLVD) fractionfcivaof 0.2
andgreater.The CLVD fractionfctvdis definedastheabsolute
value of the ratio of the largest and smallest principal
moments[e.g., Frohlich and Apperson,1992].
To separate earthquakes into "primary events"
(mainshocks)and"secondaryevents"(fore- and aftershocks),
we use singlelink clustering(SLC) algorithmsdescribedby
Frohlich and Davis [1990] andDavis and Frohlich [1991a,
1991b]. To identify aftershockswe employ a conversion
factorC of 1.0km/dayanda "cutoffdistance"D of 40 "space-
time km" (ST lcm,seeFrohlich and Davis [1990]) for the ISC
catalog,50 ST km for theBlacknestcatalog,and 100 ST km
for theHarvardCMT catalog.
Measuring b for Mainshocksand SecondaryEvents; a
Tutorial Example. We cannot compare b values for
mainshocksand secondaryeventsin the usualway, because
separating an earthquake catalog into mainshocks and
Three Global Caf•alogs
I I
30
o• / ' ,0
Z 0• / Blackne '
• oo
o
-lo ', , ,"3 4 5 6 ? 8
m•lik, M•, mp(•dj)
Fig.6.O•fhitionofbc•. • thcksoldanddashedlhesshow•e activity
rate•r 0.1ma•itude unitforea•quakes • •e Blacknest,Hazard, and
Abe(adjustS)catalogs.•e solidciml•sam• acdvityrateswi•h •
rangeof ma•itudes •dimted by •e bars;for Blacknest,4.85-5.45;for
Hazard,5.•-6.50;forA• (adjustS),6.95-?.55.Wedefinebc• asthe
slopeof•e dashedstraightl•e whichistheleastsquaresfitto•ese acfivi•
•tes.
secondaryeventscreatesa systematicbiascausingbsec to be
greaterthanamain.To illustratethis(Figure8), we generatea
hypotheticalcatalogof 10,000 earthquakeswhich satisfythe
Gutenberg-Richterlaw exactly,with b = 1.0 andMinin= 4.75.
Supposefurtherthat in this catalogeachpair of earthquakes
comprisesa primary-secondaryevent sequence,so that there
are 5000 primary and5000 secondaryevents. If we identify
thelargereventof eachpair astheprimaryevent,thenamain
is 0.91 and bSec is 2.12. These b values differ from 1.0
because our selection method for identifying secondary
events systematically discriminates against selecting the
largeeventsin the catalog. In contrast,for thepopulationof
primary events, it discriminatesin favor of selectinglarge
events. Clearly, this systematic bias also acts when we
distinguish mainshocksand secondaryearthquakesin real
catalogs as well, although it has apparently not been
recognizedexplicitly in previousstudies.
To take accountof this systematicbias, we comparethe
observedvaluesof amainandbsec in eachcatalogwith values
determined for an "identical catalog with randomized
magnitudes"(ICRM). The ICRM has the samenumber of
earthquakes,the samedistributionof magnitudes,the same
number of primary events, and for each primary the same
number of secondaryevents as the correspondingprimary
event in the real catalog. However, we have randomly
reorderedthe magnitudesin the entire ICRM to insure that
possible physical differences generating mainshocks and
secondaryeventscannotaffect measurementsof amainand
bSec.
Notethatby themselvesmeasurementsof amainandbsec in
the ICRM have absolutelyno physicalsignificance;they are
NORMAL THRUST
OFig.7. Trianglediagramfordisplayingearthquakefocalmechanisms.The
threeverticescorrespondtopurestrikeslip(top),thrust(right),andnormal
fault(left)mechanisms.Alsoshownarethemechanismswhichplotatthe
midpointsofthethreesidesandinthecenter.ForanearthquakewithT,B,
andP axeshavingdipanglesrelativetohorizontalof br, 15a,andfit',the
proportionofthrust,strikeslip,andnormalfaultingaresin2fyr,sin2fia,and
sin2fi•.ThecurvedlinesonthefigurecorrespondtomechanismshavingP
andB axeswithdipsof60ø,andtoaT axiswithadipof50ø.In thispaper
wedefineearthquakesasnormal,strikeslip,orthrustif thedipoftheP orB
axesexceeds60øorif thedipoftheT axisexceeds50ø.Oddearthquakesare
all other events.

6. 636 FROHLICHANDDAVIS: MUCH ADO Aaotrr TELESEtSM•Cb VALUES
useful only as standardsof comparisonfor evaluating the
difference(bsec-bMain)in the real catalog. Thusphysically
significantdifferencesbetweenmainshocksand aftershocks
showup asa differencebetweenthequantity(bsec- bMai,,)in
therealcatalogand(bsec -bMain) in theICRM.
WhendeterminingbMainandbsec for theICRM by Bender's
method,a minor difficulty occursif we apply a fixed upper
magnitudecutoffasdescribedabove(e.g.,seeTable1). That
is, a fixed upper magnitude cutoff may exclude some
mainshockswithout excluding their associatedsecondary
events, thus preventingthe ICRM from having the same
distributionof magnitudesas the original data catalog. To
avoidthisproblem,whendeterminingbMainandbsec in each
geographicregionwe separatelydeterminedthe maximum
magnitudesfor primary andsecondaryevents,and we used
thesemaximaastheuppercutoffvaluesfor Bender'smethod.
To include as many secondaryeventsas possible,for the
analysisof bMainandbsecin theISC catalogwe useda smaller
lower magnitudecutoff (mr,= 4.8) thanin Table 1. For the
BlacknestandHarvardCMT analyseswe usedthesamelower
cutoffs as in Table 1.
For the ISC catalog,a secondminor difficulty may occur
becausesaturationof themr,magnitudescalecausesthelarger
earthquakesto be assignedunrealisticallylow magnitudes.
As thiswouldsystematicallyaffectprimaryeventsandbMain,
asin thestudyof Davis and Frohlich [1991a] we utilized
surfacewaveor momentmagnitudesfor all earthquakeswith
mr, exceeding6.5. In the analysesof the Blacknestand
HarvardCMT catalog,we utilizedmr,(mlike) andMw for all
earthquakes,regardlessof theirmagnitude.
RESULTS
Global Measurements
Evenfor globalcatalogs,thevalueof b measureddepends
stronglyon the particularcatalogchosenand the rangeof
magnitudesusedin thedetermination(Table1). Forexample,
for earthquakesof all depthsconsideredtogether,blsc is 1.25
for the body wave magnitudemr,, whereasfor the moment
magnitude Mw, bHrv is 0.90, or 28% lower. While blsc
exceeds bHrv in all other depth ranges as well, the
relationship between the other scales differs somewhat
amongthe differentdepthgroupings.For example,babe is
nearlyidenticalto blsC for intermediateanddeepearthquakes,
but it is 0.15 smaller for shallow events. However, for the
Harvardcatalog,bHrvmeasuredfromMw ishigherfor shallow
earthquakesthanfor deepandintermediateevents,whilebHrv
measuredfrommwis actuallylowerfor theshallowevents.
These observed differences in measured b are not due
primarilyto statisticalvariations.For bothBender's[1983]
andPage's [1968] method,the standarddeviationfor b
determined from the magnitudes of a catalog with-N
earthquakesis approximatelyproportionaltob/N1/2. Thus
for the entries in Table 1 the statistical uncertainties are less
than0.01 evenfor babe, andlessstill for b determinedfrom
the other catalogs.
A moreimportantsourceof variationin b arisesbecause.in
most catalogsthe relationshipbetweenthe log of number
versusmagnitudeis not rea.llylinear, i.e., the Gutenberg-
Richter"law" is not strictlytrue. This is especiallyevident
in the ISC catalog (Figure 3) and the Blacknest catalog
(Figure4). Forexample,if wemeasureb forearthquakesof all
Synthetic Catalog
:3.5
3.0
• 2.5
2.0
ol.5
_o
0.5
0.0
• • for entirecatalogb - 1.0
- • 1 aftershock/mainshock
_ ••ainshockwithlargermb
bM•u=0'91k,,• •
- bsee=2'12• •'•A
I I I I I I
5.0 6.0 ?.0
rn b
Fig.8. Defining"mainshocks"asthelargestearthquakein amainshock-
aftershockssequencecausesdifferentbvaluesformainshocksandaftershocks.
Formagnitudeincrementsof 0.1, graphshowsthenumberofmainshocks
(bottomdashedline),aftershocks(solidline),andall events(topdashed
line).Thissyntheticearthquakecataloghas10,000eventswithmagnitude
exceeding4.75andanoverallb valueof 1.0. Eachmainshock-aftershock
sequenceconsistsofonemainshockandonesecondaryevent(aftershock),
with themainshockbeingthelargerof thetwo stochasticallygenerated
events.Fortheportionofthecatalogwithmagnitudesexceeding5.25(tothe
rightof shortverticalline),bseeis 2.12andbMainis0.91.
depthshaving magnitudesbetween5.25 and 6.35, we find
blscis 1.44 (comparedto 1.25 in Table 1) andbBt•:is 1.32
(comparedto 1.13 in Table 1).
Regional and TectonicVariations
Measurementsof b in different regional and focal depth
groupings(Table 2) rangebetween0.64 and 1.76, which are
the minimumandmaximumof bBll:in the Blacknestcatalog.
The compositevaluebcomppossessesthesmallestrange
(0.73 to 1.16). However, there is surprisingly little
correlation between the values of b determined by the
differentmethods. For example,the correlationcoefficients
relatingcolumnsof Table2 areblsC andbBt•:= 0.53;blsC and
bHrv= 0.66;b•sc andbcomp= 0.36;bt•t•:andbHrv= 0.36;
andbComp= 0.60;andbHrvandbcomp=0.29.InTable2there
is no apparentsystematicdifference between b valuesfor
shallow earthquakesand those for intermediate and deep
earthquakes.
Individual geographic regions generally possess
earthquakeswith a variety of tectonic features and focal
mechanisms. While there seems to be no clear difference in b
whenearthquakesare groupedby proximity to nearestmajor
tectonic feature (Table 3), bHrv for thrust and strike slip
earthquakes(0.86 and0.77) is significantlylessthanbHrvfor
normaland"other"earthquakes(1.06 and 1.05).
The Tonga-Fiji region is unusualin that it possessesa
significantlylargenumberof shallow,intermediate,anddeep
earthquakes.In all depthranges,all four of the b valuesthat
we determined(Table 3) are larger in the Tonga-Fiji region
thanelsewherein the world. As notedpreviouslyby Giardini
[1988] and Frohlich [1989], this is especiallytrue for the

7. FRoroaCHANDDAVIS: Muc• ADo ABOUTT•es•s•c b VALUES 637
deep earthquakes. In Tonga-Fiji, b for deep earthquakes
rangesbetween1.06forbCom•,and1.57forb;sc andbBlk,
valueswhich are comparableto the b determinedfor shallow
earthquakeselsewhere.For therestof theworld,b for deep
earthquakesis extremelylow, rangingfrom 0.53 for bHrvto
0.96 for b;sc.
Mainshocksand SecondaryEvents
In every geographic region considered (Table 4 and
Figure9), bMainfor primary eventswas lessthan bSec for
secondaryevents(aftershocksandforeshocks).This wastrue
not only for the observationswhere the magnitudes for
individualeventswere as assignedin catalogs,but alsoin the
ICRM catalogwhere thesemagnitudeswere randomizedand
reassigned (Table 4 and Figure 10). These results are
unsurprisingconsideringthat our selection of mainshocks
createsa systematicbias which increasesb for secondary
events(Figure 8).
However, for the ISC data what is surprisingis that the
difference(bsec -bMain ) is nearly always larger in the
randomized (ICRM) catalog than in the observations
(Table4, Obs-ICRM column, and Figure 11). Thus in a
relativesensewe canconcludethatbsec for secondaryevents
is actually"smaller than" expected,i.e., it is smallerthan the
valuedeterminedif magnitudesareassignedrandomly.
We obtain the sameresultsfrom analysisof mainshocks
and secondaryevents within geographicsubregionsof the
BlacknestCMT catalog(Figure 12 andTable 5) andwithin
geographicsubregionsof the Harvard CMT catalog(Figure
12).In particular,bMainwasgenerallylessthanbsec;however
the differencewas lessthan expectedas determinedfrom an
ICRM.
ISC Observations
2.4
2.2
2.0
1.8
1.6
1.4
1.2
1.0
0.8
O_oO
I I I I I I I I I
0.8 1.2 1.6 2.0 2.4
bMain
Fig.9. MeasurementofbMa•nforprincipalevents(mainshocks)andbse½for
secondaryevents(aftershocksandforeshocks)for earthquakesin the
InternationalSeismologicalCentrecatalogfor 34 geographicregions
investigatedby DavisandFrohlich[1991a](seeTable4). Straightline
indicatesbMa/n= bsec.Notethatgenerallyb$ec> bMain.

8. 638 FROHLICHANDDAVIS: MUCH Ar>o ABOUTT•aSM•C b VALUF_,•
TABLE 2. Comparisonof b ValuesDeterminedby SeveralMethodsin VariousGeographicRegions
Geographic
Regions blsC bB1k hi_itv bComp
ShallowEarthquakes(h < 70 km)
1 1.09 1.06 0.78 1.12
2 1.34 0.78 1.07 0.95
3 1.25 1.1 1 0.88 0.7 8
4 1.45 1.15 -- 0.87
5 and 6 1.19 1.03 0.91 0.81
7 1.33 1.35 0.89 1.14
8 0.86 1.28 -- 0.76
9-11 1.31 1.46 1.09 1.12
12 1.32 1.76 0.87 •
13 and 14 1.16 1.21 0.65 1.02
15 and 16 1.08 1.06 0.68 0.92
17 1.17 1.19 0.85 0.97
18 1.53 1.17 1.26 1.1 1
19-21 1.15 1.04 0.89 0.95
22 1.34 1.08 1.11 0.83
23 1.27 1.04 0.66 0.88
24 1.10 0.98 0.88 0.99
25 and26 1.10 1.14 0.79 1.16
27 1.01 0.88 0.89 0.89
28 1.37 1.04 1.01 1.02
29 1.47 1.25 0.73 •
30 1.46 1.46 1.14 1.06
31 1.35 1.13 • •
32 1.06 1.02 -- 1.08
33 1.48 1.60 1.21 1.20
34 1.71 1.24 • 0.98
Average 1.27 1.17 0.92 0.98
IntermediateDepthEarthquakes(70 km _•h < 300 kin)
1 1.57 1.11 • •
3 1.50 1.27 • --
5 and 6 1.26 1.03 0.67 0.92
7 0.94 0.87 -- •
9-11 1.29 1.38 0.96 1.12
13 and 14 1.21 0.99 1.09 0.79
15 1.08 1.01 • 1.13
17 1.20 1.01 • 0.90
18 1.50 1.23 • 0.97
19-21 1.29 1.05 -- 0.88
22 1.12 0.65 • 0.68
23 1.62 1.24 • 1.08
24 1.51 1.33 • 1.12
25 and 26 1.24 1.07 0.98 1.01
34 1.41 0.91 • 0.73
Average 1.32 1.08 0.93 0.94
DeepEarthquakes(300 km _<h)
9-11 1.48 1.41 1.09 --
12 1.63 1.68 1.29 0.99
18 0.92 0.64 • 0.77
25 and 26 1.00 • • •
Average 1.26 1.24 1.19 0.88
Themagnitudescalescatalogsetc.for theb valuesblsC bB1k blJavandbCorn--areasin Table 1
withbHrvutiliz•gtheMwscale,andbCornpdeterminedusingthecompositemetliOddescribedmthe
text. Geographicregionsare as givenby DavisandFrohlich [1991a]andTable 4, exceptseveral
regionsare combinedas indicated.The tabledoesnot includeb valuesfor regionswhere there are
fewerthan 100 eventsin the ISC catalog,75 eventsin the Blacknestcatalog,or 40 eventsin the
HarvardCMT catalog.Averagesareunweightedmeanof numbersin eachcolumn.

9. FROHLICHANDDAMS: MUC• ADOABotrrTELES•SMICb VALUE$ 639
TABLE 3: Comparisonof b Valuesfor EarthquakesGroupedbyNearestTectonicFeature,
Focal MechanismType,andby Proximityto Tonga-Fiji
bComp
Nearest Tectonic Feature a
RidgeTransform-FractureZone 1.56 1.17 O.91
Subduction Zone 1.55 1.14 O.89
1.22
0.91
FocalMechanismTypeb(h_<50km)
Thrust -- -- 0.86
StrikeSlip • • 0.77
Normal -- • 1.06
Other • • 1.05
Proximity to Tonga-Fiji (Flinn-EngdahlRegions12-13)
Shallow(h < 70 km)
Tonga-Fiji 1.31 1.47 1.05 1.11
Rest of World 1.22 1.11 0.89 0.95
Intermediate(70 < h < 300 km)
Tonga-Fiji 1.29 1.37 0.96 1.08
Rest of World 1.33 1.08 0.79 0.93
Deep(h < 300 km)
Tonga-Fiji 1.57 1.57 1.22 1.06
Rest of World 0.96 0.58 0.53 0.69
Themagnitudescales,catalogs,etc.,forthebvaluesbisC, bBlk,bHrv,andbCom areasinTable1p ,
withbHrvutilizingtheMwscale,andbCompdeterminedusingthecompositemethoddescribedinthe
text.
aHypocentersareearthquakeswithfocaldepthh < 50 kmoccurringwithin200 km of a spreading
ridge,transform,fracturezoneor subductionzoneasindicatedbythePOMPGroup.
bseeFigure7fordefinitionsofthrust,strikeslip,etc.
ISC ICRM Synthetics
2.4
2.2
2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.8 1.2 1.6 2.0 2.4
bMain
Fig.10.MeasurementofbMainformainshocksandbsecforsecondaryevents
asin Figure9 exceptthatwithineachgeographicregion,eventmagnitudes
are reassignedrandomly(i.e., theyconstitutean IdenticalCatalogwith
RandomizedMagnitudes,or ICRM). StraightlineindicatesbMain=bsec.
Notethatbsec> bMalneventhoughtherandomizationremovesanypossible
effectcausedbyphysicaldifferencesinmainshocksandsecondaryevents.
DISCUSSIONAND CONCLUSIONS
How Well Do We Know b?
Recently,severalinvestigatorshave generatedsynthetic
earthquake catalogs having b values of 1.0 using
generalizationsof Burridge and Knopoff[1967] block and
springmodels[Bak and Tang, 1989;Carlson, 1991a; 1991b;
Shawet al., 1992]. The presentpapersuggeststhat while
observedvaluesof b equal1.0 in thesensethatb is neveror
almostneverlessthan0.5 andneveror almostnevergreater
than 2.0, its precise value is uncertain or indeterminate.
Exceptin thisabovelimitedsense,it is simplynottruethatb
is 1.0.
This result shouldnot be surprising,as there is no one
"best"or "correct"magnitudescale. The Gutenberg-Richter
"law" is essentiallyempirical,earthquakesaresimplyoneof
a hostof phenomenafor whichlogarithmicplotsof number
versussize are approximatelystraight(e.g., Frohlich and
Buskirk,1982]. Furthermore,studiesseekingto find a linear
relationshipbetween various magnitudescalesseldomfind
thatthecoefficientof b is 1.0 (e.g.,seeHeatonet al., [1986],
and Bath's [1981] comprehensive table of these
relationships).
An importantconclusionof thispaperis thatoneshouldbe
very wary aboutattachingunduesignificanceto differencesin
measurements of b values, unless these differences are
extremelylarge,suchas25% or evengreater. Herewe have
usedseveraldifferentmethodstosystematicallydeterminethe
parameterb of theGutenberg-Richterlaw for severalcatalogs

10. 640 FROHLICHANDDAVIS: MUCH ADO A•otrr TELESmSMtCb VALUES
TABLE 4. Comparisonof b Valuesfor MainshockandSecondaryEventsin the
GeographicRegionsInvestigatedby DavisandFrohlich[1991a]
Geographic Flinn-Engdahl Observations ICRMa
Regions Regions bMainObs bSeeobsbMainR bSecRObs-ICRMb
1 Alaska 1 1.06 1.31 1.02 1.42 -0.15
2 West CoastNorth America 2,3,4 1.16 1.76 1.16 1.77 -0.01
3 Mexico-Central America 5,6 1.25 1.55 1.21 1.86 -0.35
4 Caribbean 7 1.26 2.05 1.27 2.00 0.06
5 SouthAmerica 1 8,9 1.06 1.20 1.04 1.44 -0.26
6 SouthAmerica 2 8,9 1.23 1.34 1.18 1.66 -0.37
7 South Sandwich Islands 10 1.03 1.32 1.01 1.43 -0.13
8 New Zealand 11 1.08 1.16 1.07 1.20 -0.05
9 Tonga1 12 1.18 1.48 1.16 1.66 -0.20
10 Tonga2 12 1.11 1.34 1.07 1.52 -0.22
11 Tonga3 12 1.18 1.36 1.12 1.63 -0.33
12 Fiji 13 1.25 1.68 1.21 2.12 -0.48
13 Vanuatu(NewHebrides)1 14 1.01 1.33 0.95 1.54 -0.27
14 Vanuatu(NewHebrides)2 14 0.98 1.27 0.94 1.39 -0.16
15 Solomon Islands 1 15 0.99 1.26 0.98 1.29 -0.04
16 Solomon Islands 2 15 0.94 1.07 0.85 1.28 -0.30
17 New Guinea 16 0.98 1.13 0.96 1.21 -0.10
18 Carolines- Marianas 17,18 1.22 1.82 1.25 1.65 0.20
19 Kuriles 1 19 0.99 1.20 0.92 1.34 -0.21
20 Kuriles 2 19 1.16 1.39 1.11 1.48 -0.14
21 Kuriles 3 19 1.16 1.49 1.12 1.66 -0.21
22 Japan- Taiwan 20,21 1.13 1.58 1.10 1.84 -0.29
23 Philippines 22 1.07 1.38 1.05 1.44 -0.08
24 Molucca-Celebes 23 1.01 1.29 0.99 1.35 -0.08
25 Java 1 24 0.90 0.87 0.88 1.05 -0.20
26 Java 2 24 1.16 1.10 1.08 1.62 -0.60
27 Asia 25,26,27,28 1.11 1.18 1.02 1.69 -0.60
28 Asia Minor 29,30 1.26 1.87 1.21 2.17 -0.35
29 Africa 31,37 1.38 1.58 1.32 1.84 -0.32
30 Atlantic-Indian oceans 32,33,34,35 1.31 2.05 1.31 2.13 -0.08
31 PolarRegions 36,40,42,49,50 1.08 1.57 1.06 1.85 -0.30
32 PacificOcean 38,39,41 0.95 1.31 0.99 1.20 0.15
33 SouthPacificOcean 43,44,45,46 1.22 1.82 1.25 1.61 0.24
34 Pakistan-HinduKush 47,48 1.44 1.76 1.34 2.53 -0.87
DataareISCdatawithmagnitudemb,exceptmagnitudesexceeding6.5,areadjustedasinDavisandFrohlich[1991a].Allb
valuesaredeterminedusingBender'smethod,witha lowermagnitudecutoffof 4.8 andanuppermagnitudecutoffdeterminedfromthe
data as described in the text.
aIdenticalcatalogswithrandomizedmagnitudes,asdescribedin text.
bObs-ICRM=(bSeeOb•-bMainOb•)-(bSecR-bMainR)
of teleseismicallyrecordedearthquakes. For the different
global catalogs,we find valuesof b rangingfrom 0.79 to
1.25. Becausethe log numberversusmagnituderelation is
not linear(Figures1 to 4), evenfor individualcatalogstheb
valuevariesby 15% or moredependingon theexactrangeof
magnitudesusedfor its determination.
There are alsoserioussystematicproblemsthat afflict the
measurement of b values and that seem to be overlooked in
many of the papers interpretingobservationsof b. The
determination of b rests on the accurate determination of
values for magnitudes. Even thoughmagnitudescalesare
logarithmic, magnitudesdeterminedby different stations
typicallydiffer by asmuchas0.5. Habermann [1991] and
othershave shownthat reportedmagnitudesoften appearto
vary over time, as does the level of completenessof
earthquake catalogs, apparently for systematic reasons.
There is excellentevidencethatnumerousearthquakeswith mr,
between5.0 and 5.5 are missedregularly even in the ISC
catalog[e.g.,WalkerandMcCreary, 1985].
Furthermore,individual magnitude scales typically are
meaningful only over a limited range of earthquakesize,
which is problematicsince one desiresas large a range as
possible to obtain good information about b. Finally,
methods to calculate statisticaluncertaintiesin b [Aki, 1965;
Page, 1968; Bender, 1983] invariably assumethat the log
number versus magnitude curve is straight over some
magnituderange, and thus they seriouslyunderestimatethe
trueuncertaintyin thedeterminationof b. Log numberversus
magnitudecurves(Figures1-4) generallyare not thatstraight
overany magnituderangeunlesstheyhavebeen"adjusted"to
makethemstraight,asin thestudyof Pachecoet al. [1992].
Becauseof theseproblems,thereare legitimatereasonsto
doubtmany publishedinterpretationsof b values,especially
when these interpretations rest on the measurementof
differencesin b of 0.1 or less[e.g.,Andersonet al., 1980].
This is especiallytrue sincevariousparametersquantifying
the size of earthquakes'measuredifferent propertiesof the
earthquake source. Thus we should not expect close
agreement between b values determined from different
magnitudescales. We shouldalsonot expecttoo muchfrom
equationsattemptingto describethe relationshipbetween
magnitudescalesandothermeasuresof earthquakesize,such
asmoment,potency,or radiatedenergy.
Regional,Tectonic,andDepthDependenceof b
In spiteof ourpessimismaboutobtaininghighlyprecise

11. FROHLICHANDDAVIS' MUCHADOABOUTTF.LES•SMICb VALUES 641
ISC Catalog
1.6 -
•08 -
I
• 0.6-e
?,0.4
I I I I I I I I I
0.0 0.4: 0.8 1•2 1.6
[bSec- b•lein]observeUons
Fig.11. Thedifferencebetweenbvaluesfor secondaryevents(bsec)and
primaryevents(bM•n)forthe InternationalSeismologicalCentre(I$C)
catalog(horizontalaxis)andfor theIdenticalCataloõwithRandornized
Magnitudes(ICRM) (verticalaxis).Notethatthedifferenceis generally
smaller for I$C observations than for the ICtLM. The data are from Table 4.
Blacknest and Harvard Catalogs
1.6
1.4
•12• ß
•08• ß
I
o0.6
?, 0.4
0.2
0.0
_ o
_ o6 o•
.,
o
0.0 0.4 0.8 1.2 1.6
[bSec-b•ain]ob$ervation$
Fig. 12. Thedifferencebetweenb valuesfor secondaryevents(b$zc)and
primaryevents(bM•n)fortwoearthquakecatalogs(horizontalaxis)andfor
theIdenticalCatalogwithRandomizodMagnitudes(verticalaxis).Circles
areBlacknestdatafor28geographicregionswhicharenearlyidenticaltothe
geographicregionsevaluatexiinTable4 andFigures9to 11. Trianglesare
HarvardControidMomentTensordataforsixgeographicregions.Forthe
Blacknestcatalog,the b valuesare determinedfrom m/, (m/ike)using
Bender'smethodwith a minimummagnitudeof 4.85. For theHarvard
catalogthebvaluesaredeterminedfromMwusingBender'smethodwitha
minimummagnitudeof 5.65.
and accuratemeasurementsof b for real earthquakecatalogs,
there exist some differences in the size distributions of
earthquakegroupswhichwebelievearereal. Oneof themost
strikingdifferencesis for deepearthquakesin Tongarelative
to deep earthquakeselsewherein the world. For deep
earthquakesin Tonga-Fiji(Table3) we find thatb is 1.0 or
greater,whilefor deepearthquakeselsewhere,b issmallerby
30% or more. Clearly, small earthquakesare especially
commonin Tonga-Fijirelativeto thenumberof earthquakes
withmttorMwexceeding7.0. In contrast,elsewherethereis
evidence that large deep earthquakesare more common
relative to the number of small earthquakesobserved. The
Tonga-Fijiregionis generallyconsideredto be thearchetypal
deepearthquakezonebecausetwo thirdsof the world'sdeep
earthquakesoccurthere;however,thesedatasuggestthatit is
in fact anomalousin comparisonto otherregionswheredeep
earthquakesoccur.
Anotherpossiblysignificantobservationis that bHrv for
shallowthrusting(0.86) and strike slip earthquakes(0.77)
seemsto be somewhatlessthan bHrv for earthquakeswith
normalfaulting(1.06) focal mechanisms(Table3). These
differencesareno largerthansomeof theregionaldifferences
noted in Table 2; however, we tend to find them more
believablebecausetheyaredeterminedfrom a largernumber
of eventsand becausethe CMT catalogpossessesa nearly
linear log numberversusmagnituderelationover a larger
rangethanmostothercatalogs(Figure2). The higherb for
thenormalfaultingearthquakesmaycomeaboutif theyoccur
underconditionsof lower stressor stressdrop than thrustor
strikeslip earthquakes[Scholz,1968;Wyss,1973].
Mainshocksand SecondaryEvents
An important result in this paper is that earthquake
aftershocksand foreshocksnearly always have higher b
valuesthanthosereportedin catalogsof mainshocks(Figure
9) but that this is primarily a systematiceffect due to
selectingmainshocksas the largest events in sequences
(Figures8 and10). This apparentlyhasnot beenrecognized
previously,althoughKnopoffet al. [1982] did suggestthat
undiscoveredsystematicerrors might be responsiblefor
many reported differences in b before and after large
earthquakes.The particularsystematicproblemdiscoveredin
this paper shouldoccur wheneveraftershocksor dependent
events are removed from earthquake sequencesprior to
statisticalanalysis. However, it shouldnot affect analyses
like thoseof Smith [1986], where earthquakesare separated
only into temporalor spatialgroups.
Ourresultis thatbSec is actuallylower thanexpectedif the
only influenceon b comesfrom defining the mainshockas
the largestearthquakein eachsequence.This resultis quite
robust, as it seems to hold for earthquakesin the ISC,
Blacknest,and Harvard CMT catalogs(Figures11 and 12).
However,thisresultis quitesubtle;to determineit, we hadto
comparethe observedmagnitudesof primaryand secondary
events with a hypothetical catalog having an identical
distributionof magnitudes,but assignedrandomlyto primary
and secondaryevents. Since the occurrenceof a primary
eventundoubtedlyaltersthe stressfield in the neighborhood
of the sourceregion, it is unsurprisingthat it affects the
relative distributionsof small and large eventsas well. Our
resultimpliesthat amongaftershocks,b valuesare lower than
expected, and thus large aftershocksare relatively more

12. 642 FgotmIct• A•D DAYre: Moctt ADo A•otrr T•-Lmsmssacb VALUmS
TABLE 5. Comparisonof b Valuesfor MainshockandSecondaryEvents
for GeographicRegionsin theBlacknestCatalog
•ographic Observations
egions bMainObs bSeeobsbMainR bSeeR Obs-ICRMb
1 0.93 1.34 0.96 1.28 0.09
2 0.76 0.92 0.61 1.25 -0.48
3 1.08 1.41 1.02 1.81 -0.46
4 1.02 1.65 0.99 1.95 -0.33
5 and 6 0.96 1.23 0.96 1.25 -0.02
7 1.09 1.73 1.06 1.85 -0.15
8 1.16 1.49 1.12 1.78 -0.33
9 1.35 1.92 1.29 2.44 -0.58
10 and 11 1.19 1.68 1.16 1.83 -0.18
12 1.36 2.32 1.32 2.9 8 -0.70
13 and 14 1.05 1.51 1.06 1.49 0.03
15 and 16 0.90 1.33 0.88 1.37 -0.06
17 1.01 1.51 0.97 1.70 -0.23
18 1.04 1.66 1.01 1.97 -0.34
19 0.92 1.26 0.94 1.20 0.08
20 and 21 0.94 1.28 0.91 1.36 -0.11
22 0.91 1.50 0.87 1.85 -0.39
23 0.98 1.45 0.98 1.48 -0.03
24 1.08 1.45 1.06 1.52 -0.09
25 and 26 1.07 1.25 1.05 1.33 -0.10
27 0.89 0.96 0.79 1.42 -0.56
28 0.96 1.54 0.94 1.65 -0.13
29 1.17 1.54 1.08 1.97 -0.52
30 1.42 2.10 1.40 2.45 -0.37
31 0.88 1.34 0.88 1.36 -0.02
32 0.90 1.23 0.95 1.07 0.21
33 1.46 2.40 1.44 2.71 -0.33
34 0.97 1.17 0.91 1.65 -0.54
Magnitudesarem4b'(mlike),andvaluesaredeterminedusingBender'smethod,withalowermagnitudecutoffof 75 andanuppermagnitudecutoffdeterminedfromthedataasdescribedin the
text. Regionnumberscorrespondto geographicregionsin Table4.
aIdenticalcatalogswithrandomizedmagnitudes,asdescribedintext.
bObs-ICRM=(bSecobs-bMainObs)-(bSeer-bMainR)
common than expected. If we interpret this in terms of
analysessuchas thoseof Scholz [1968], Wyss [1973], and
Hanks [1979], this suggestsstressesor stressdropsbecome
higher near the source region following a mainshock,
perhaps because the mainshock rupture process loads
asperitieswithin the sourceregion and in adjacentregions.
At presentwe offerno furtherexplanationfor thisresult.
This problem of when bias may be present in the
determinationof b is fairly subtle,asb is subjectto numerous
systematic and statistical errors which afflict earthquake
detection,earthquakelocation,aftershockidentification,and
magnitude determination. Moreover, we show in the
appendixthat the situationis not straightforwardeven if we
constructa syntheticcatalogwith mainshocksand synthetic
aftershock sequences, where the mainshocks follow a
Gutenberg-RichterlawwithbvaluebMain,andeachindividual
mainshockgeneratesaftershocksequenceswithbvaluebAft.
In this case, when we separate the entire catalog into
mainshocks and aftershocks, we find that the observed b
valuebsecfor aftershocksin the catalog as a whole is not
usuallyequaltobAft. Rather,bsec forthecatalogmayequal
bMain,bAft, or anyvaluein between.Mostobservational
studiesfind thataftershocksmakeup a substantial,buthighly
variable fraction of the eventsin earthquakecatalogs[e.g.,
Reasenberg, 1985; Eneva and Pavlis, 1988; Davis and
Frohlich, 1991a]. Thusif bMainandbsecdiffer, this suggests
that someof the regionalandtemporalvariationsin b values
observedin previousstudiesmay be due to differencesin the
proportionof aftershocks.
APPENDIX:bMainANDbs•½FORPRIMARY
AND SECONDARY EvF2qrs
Supposethata catalogof earthquakesconsistsof two types
of events: primaryevents,whichoccurindependentlyof one
another;and secondaryevents,whosenumber,size, and time
of occurrencemay dependon the occurrenceof the primary
events. In the equations that follow, we shall use the
subscript"Main" to label primaryevents,andthe subscripts
"Sec" and "Aft" to label secondaryevents. We shallusethe
subscript"Aft" only to refer to propertiesof secondaryevents
of individual earthquakes,whereasthe subscript"Sec" will
refer to properties of all secondaryevents in the catalog,
consideredtogether. Usually we studycatalogsconsistingof
mainshock-aftershocksequenceswherethe mainshockis the
largest event and also the primary event. However, more
generally, secondaryeventsmay include foreshocksas well
as aftershocks.
Supposethatprimary eventsfollow the Gutenberg-Richter
relation up to some maximummagnitudeMm,•x. Then the
expectednumbernMain(M)dM of mainshocks having
magnitudesbetweenM andM + dM is
-b M
0a Ma/nnMain(M)dM=1 10 dM. (A1)

13. FROHLICHANnDAVIS' MUCH Ano ABotrr TELESEISMICb VALUES 643
SupposealsothattheexpectednumbernAft(MAft,M)dMAftof
secondaryeventshavingmagnitudesbetweenMaftandMAft+
dMAftforeachprimaryeventofmagnitudeM scalesas
nAft(MAft,M)dMAft=f(M)g(M-MAft)dMAft. (A2)
For the purposesof this paper we shall assumethat the
functionsf andg taketheparticularlysimpleformsf(M)= c, a
constant and that
This impliesthat eachindividualaftershocksequencefollows
aGutenberg-RichterlawwithabvalueequaltobAft. Thenin
the entire earthquake catalog the expected number of
secondary events nobs(M $ec)dMsec observed to have
magnitudesbetweenMsecandMsec+ dMSecwill be
S•½
(A4)
If bMaindiffersfrombnft , thisisa
c 10 -b MMa• $•c
- l• ß
nøbs(Msec)riMSee(baft-bMain)lnlO
c'
As. the ratio of the two terms in the brackets is
10•'bAftMain)('Mmax-MSec)weseethatwhenbAft>>bMainthe
secondterm will be negligiblewith respectto the first, and
thebvaluebsec for the entiregroupof secondaryeventswill
be approximatelybAft. Similarly,whenbMain>>bAft the
distribution of primary events dominates bsec, which
approachesbMain. Moregenerally,notethatif wedefinebsec
= - d(loglonobs)/dMSec,then:
(bAft-bMain)
bsec=b•ft+[]. (A6)
An important conclusion is that for a catalog of
earthquakesdivided into primary and secondaryevents,the
observedb valuebsec for secondaryeventsin the catalogis
notsimplyequalto thebAft, theb valueforsequencesof
secondaryevents for individual earthquakeswithin the
catalog.
Acknowledgments.WethankRoyLilwall,TedHabermann,andvarious
colleaguesattheInternationalSeismologicalCentreandHarvardUniver-
sityforprovidinguswiththeearthquakecatalogsanalyzedin thispaper.
WearegratefulforfinancialsupportfromtheNationalScienceFoundation
undergrantsEAR-89-16665andEAR-91-105069,andfrom the U.S.
GeologicalSurvey. Finally, we are thankfulfor conversationswith
numerousindividualswhoprovokedustowritethispaper,eventhoughin
thepastoneofus(C.F.)statedthathewouldnever,everwriteapaperabout
bvalues.Thispaperis contribution928of theInstitutefor Geophysics,
UniversityofTexas.
REFERENCES
Abe,K.,Magnitudesoflargeshallowearthquakesfrom1904to1980,Phys.
Earth Planet.Inter., 27, 72-92, 1981.
Abe,K.,Complementsto"Magnitudesoflargeshallowearthquakesfrom
1904to 1980",Phys.EarthPlanet.Inter.,34, 17-23,1984.
Abe,K.,andH.Kanamori,Temporalvariationsoftheactivityofintermedi-
ateanddeepfocusearthquakes,J.Geophys.Res.,84,3589-3595,1979.
Aki, K., Maximumlikelihoodestimateof bin theformulalogN '- a - bM
anditsconfidencelimits,Bull.EarthquakeRes.Inst.Univ.Tokyo,43,
237-239, 1965.
Aki,K.,Aprobabilisticsynthesisofprecursoryphenomena,inEarthquake
Prediction:AnInternationalReview,MauriceEwingSer.,vol.4, edited
byD.W. SimpsonandP.G.Richards,pp.566-574,AGU,Washington,
D.C., 1981.
Anderson,R. N., A. Hasegawa,N. Umino,andA. Takagi,Phasechanges
andthefrequency-magnitudedistributionintheupperplaneofthedeep
seismiczonebeneathTohoku,Japan,J. Geophys.Res.,85, 1389-1398,
1980.
Bak,P.,andC.Tang,Earthquakesasaself-organizedcriticalphenomenon,
J. Geophys.Res.,94, 15,635-15,637,1989.
Bath,M., Earthquakemagnitude-Recentresearchandcurrenttrends,Earth
Sci.Rev., 17, 315-398, 1981.
Bender,B., Maximumlikelihoodestimationof b valuesfor magnitude
groupeddata,Bull.Seismol.Soc.Am.,73,831-851,1983.
Burridge,R., andL. Knopoff,Modelandtheoreticalseismology,Bull.
Seismol.Soc.Am., 57, 341-371, 1967.
Carlson,J. M., Time intervalsbetweencharacteristicearthquakesand
correlationswith smallerevents:An analysisbasedon a mechanical
modelof afault,J. Geophys.Res.,96,4255-4267,199la.
Carlson,J.M., A two-dimensionalmodelofafault,Phys.Rev.A,44, 6226,
1991b.
Davis,S. D., andC. Frohlich,Single-linkclusteranalysisof earthquake
aftershocks:Decaylawsandregionalvariations,J. Geophys.Res.,96,
6335-6350, 199la.
Davis,S. D., and C. Frohlich,Single-linkclusteranalysis,synthetic
earthquakecatalogs,andaftershockidentification,Geophys.J. Int.,
104,289-306, 1991b.
Dziewonski,A.M., T. A. Chou,andJ. H. Woodhouse,Determinationof
earthquakesourceparametersfromwaveformdataforstudiesofglobal
andregionalseismicity,J. Geophys.Res.,86, 2825-2852,1981.
Dziewonski,A.M., andJ. H. Woodhouse,An experimentin systematic
studyof globalseismicity:Centroid-momenttensorsolutionsfor 201
moderateandlargeearthquakesof 1981,J. Geophys.Res.,88, 3247-
3271, 1983.
Eneva,M., andG. L. Pavlis,Applicationof pair analysisstatisticsto
aftershocksofthe1984MorganHill,California,earthquake,J.Geophys.
Res.,93, 9113-9125, 1988.
Frohlich,C., Thenatureof deep-focusearthquakes,Annu.Rev.Earth
Planet. Sci.,17, 227-254, 1989.
Frohlich,C.,Trianglediagrams:Ternarygraphstodisplaysimilarityand
diversityofearthquakefocalmechanisms,Phys.EarthPlanet.Inter.,in
press,1992.
Frohlich,C.,andK. D.Apperson,Earthquakefocalmechanisms,moment
tensors,andtheconsistencyof seismicactivitynearplateboundaries,
Tectonics,11,279-296, 1992.
Frohlich,C., andR. E. Buskirk,Survivorshipcurvesandgrowthratesfor
apopulationof mustachehairs,TexasJ. Sci.,34, 207-214,1982.
Frohlich,C., andS.D, Davis,Single-linkclusteranalysisasamethodto
evaluatespatialandtemporalpropertiesofearthquakecatalogs,Geophys.
J. Int., 100, 19-32, 1990.
Giardini,D., Frequencydistributionandquantificationof deepearth-
quakes,J. Geophys.Res.,93,2095-2105,1988.
Gutenberg,B., Amplitudesof P, PP andS andmagnitudesof shallow
earthquakes,Bull.Seismol.Soc.Am.,35, 57-69,1945.
Gutenberg,B., andC. F. Richter,SeismicityoftheEarthandAssociated
Phenomena,PrincetonUniversityPress,Princeton,N.J., 1954.
Habermann,R. E.,Consistencyofteleseismicreportingsince1963,Bull.
Seismol.Soc.Am., 72, 93-111, 1982.
Habermann,R. E., Teleseismicdetectionin theAleutianIslandarc,J.
Geophys.Res.,88, 5056-5064,1983.
Habermann,R. E., Seismicityratevariationsandsystematicchangesin
magnitudeinteleseismiccatalogs,Tectonophysics,193,277-289,1991.
Habermann,R. E., W. R. McCann,andB. Perin,Spatialseismicity
variationsalongconvergentplateboundaries,Geophys.J. R. Astron.
Soc.,85, 43-68, 1986.
Hanks,T. C.,b valuesandtonseismicsourcemodels:Implicationsfor
tectonicstressvariationsalongactivecrustalfaultzonesandtheestima-
tionof high-frequencystronggroundmotion,J. Geophys.Res.,84,
2235-2242, 1979.

14. 644 FROitL•CltASr•DAVIS: Muc• Ar•o ArtouT TF.L•S•SM•C b VALUES
Hanks,T. C., andH. Kanamori,A momentmagnitudescale,J. Geophys.
Res.,84, 2348-2350, 1979.
Heaton,T. H.,andR.E.Heaton,Staticdeformationsfrompointforcesand
forcecoupleslocatedinweldedelasticPoissonianhalf-spaces:Implica-
tionsfor seismicmomenttensors,Bull. Seisrnol.Soc.Am.,79, 813-841,
1989.
Heaton,T. H.,F.Tajima,andA. W.Mori,Estimatinggroundmotionsusing
recordedaccelerograms,Surv.Geophys.,8, 25-83,1986.
Kanamori,H., Magnitudescaleand quantificationof earthquakes,
Tectonophysics,93, 185-199,1983.
Kanamori,H., andD. L. Anderson,Theoreticalbasisof someempirical
relationsinseismology,BUll.Seismol.Soc.Amer.,65,1073-1095,1975.
Knopoff,L., Y. Y. Kagan,andR. Knopoff,bvaluesforforeshocksand
aftershocksinrealandsimulatedearthquakesequences,Bull.Seisrnol.
Soc.Am., 72, 1663-1676, 1982.
Lilwall,R.C.,andJ.M. Neary,Redeterminationofearthquakebody-wave
magnitudes(mb)usingISCbulletindata,Rep.O 21/85,38pp.,At.
WeaponsRes.Estab.,Aldermaston,England,1986.
Pacheco,J.F., C. H. Scholz,andL. R. Sykes,Changesin frequency-size
relationshipfromsmalltolargeearthquakes,Nature,355,71-73,1992.
Page,R.,AftershocksandmicroaftershocksofthegreatAlaskaearthquake
of 1964,Bull. Seismol.Soc.Am., 58, 1131-1168, 1968.
Perez,O.J.,andC. H. Scholz,Heterogeneitiesoftheinstrumentalseismic-
ity catalog(1904-1980)for strongshallowearthquakes,Bull.Seismol.
Soc.Am., 74,669-686, 1984.
Reasenberg,P., Second-ordermomentof centralCaliforniaseismicity,
1969-1982,J. Geophys.Res.,90,5479-5495,1985.
Ringdal,F.,Studyofmagnitudes,seismicity,andearthquakedetectability
usingaglobalnetwork,Bu//.Seismol.Soc.Am.,76, 1641-1659,1986.
Royer,J.Y., R.D. Muller,L. M. Gahagan,L. A. Lawver,D. Numberg,and
J.G.Sclater,A GlobalIsochronChart,UniversityofTexasInstitutefor
GeophysicsTechnicalReportNo. 117,pp.38, 1992.
Scholz,C.H.,Thefrequency-magnituderelationofmicrofracturinginrock
anditsrelationto earthquakes,Bull.Seismol.Soc.Am.,58, 399-415,
1968.
Shaw,B. E.,J.M. Carlson,andJ.S.Langer,Pauemsofactivitypreceding
largeearthquakes,J. Geophys.Res.,97,479-488,1992.
Smith,W.D.,Evidenceforprecursorychangesinthefrequency-magnitude
b-value,Geophys.J. R. Astron.Soc.,86, 815-838,1986.
Utsu,T., A statisticalsignificancetestofthedifferenceinb-valuebetween
twoearthquakegroups,J.Phys.Earth,14,37-40,1966.
Walker,D. A.,andC.S.McCreary,Significantunreportedearthquakesin
"aseismic"regionsofthewesternPacific,Geophys.Res.Lett.,12,433-
436, 1985.
Wyss,M., Towardsaphysicalunderstandingoftheearthquakefrequency
distribution,Geophys.J. R.Astron.Soc.,31, 341-359,1973.
S.D. Davis and C. Frohlieh, Institute for Geophysics,
Universityof Texasat Austin,8701NorthMopaeBoulevard,
Austin, TX 78759-8345.
(ReceivedJanuary20, 1992;
revisedJuly27, 1992;
acceptedAugust10,1992.)