MFR

1. JOURNAL OF GEOPHYSICAL RESEARCH,VOL. 92, NO. B2, PAGES 1349-1355,FEBRUARY 10, 1987
Magnitude-FrequencyRelationfor SmallEarthquakes'
A Clueto theOriginoffmaxOf'LargeEarthquakes
KEIITI AKI
Departmentof Geolo•7icalSciences,Universityof SouthernCalifornia,Los An•7eles
In order to test the hyothesisthat the frnaxof large earthquakesis due to the sourceeffectand is
causallyrelated to the corner frequencyof small earthquakeswhen it becomesconstantbelow magnitude
about 3, we constructedthe frequency-magnituderelation for earthquakes with magnitude range from
- 1/3 to 4 recordedat a borehole seismographstation operatedby the University of Southern California
at Baldwin Hills in the middle of the Newport-Inglewood fault. A procedure was developed for an
accurateestimationof momentmagnitudefrom the measurementof codaamplitudeas a functionof
lapse time. The cumulative number of earthquakes per unit time period and area was estimated by
findingthedetectionlimitfor ts_vforeachmagnitude.We founda cleardepatureof theobservedlog
frequency-magnituderelation for M < 3 from the linear relation extrapolatedfrom the data for M > 3.
The observedfrequencyof earthquakeswith M '-' 0.5 is almost 10 timeslessthan that expectedfrom the
extrapolationof empiricalrelation for earthquakeswith magnitudeslarger than 3. We concludethat the
observedfreqt•ency-magnituderelation departs from self-similarity for earthquakes with magnitude
smallerthan about 3. This departure coincideswith the widely observeddeparture of the moment-corner
frequencyrelationfromself-similarity.Theseobservationsareconsistentwith theinterpretationOffmaxof
large earthquakesin terms of the slip-weakeninginstability model in which the critical slip may be a few
tenths of a meter to a few meters and the cohesivezone size may be a few hundred meters to a few
kilometers.
INTRODUCTION
Theeffectiveupperboundfrequencyfmaxoftheacceleration
spectrumgeneratedby an earthquakeis a practicallyimpor-
tantparameterforearthquakeengineering;fmaxissometimes
confusingandcontroversialbecauseit couldbeattributedto
source,path,recordingsiteeffects,or a combinationof all
three.Brune[1970]introduceda cutofffrequencyofabout10
H in theaccelerationspectrumin accordancewiththegeneral
appearanceof observedstrongmotionrecords.Ida [1973]
interpretedthecutofffrequencyasthesourceeffectandesti-
matedthecriticalslipin hisslip-weakeningmodelto beofthe
orderof 10cm.A similarvalueofcriticalweakeningslipwas
adoptedby Day [1982] with somejustificationfor three-
dimensionedsimulationof spontaneousruptureon an earth-
quakefault.On theotherhand,Hanks[1979]originallyat-
tributedfm•xto anelasticattenuationalongthetotaltravel
distance,althoughlaterHanks[1982]attributedit primarily
to thelocalrecordingsiteeffect.A strongeffectofattenuation
onhigh-frequencyspectrawasalsobroughtup by Anderson
andHough [1984].
Papa•teorqiouandAki[1983]foundsharpcutofffrequencies
rangingfrom2.5to 5 Hz in theobservedaccelerationpower
spectraofseveralmajorCaliforniaearthquakesaftercorrect-
ingfor theattenuationpatheffects.In thecaseof theSan
Fernandoearthquakeof 1971,A. S. Papageorgiou(personal
communication,1986)found that fm•x determinedfrom re-
cordsat soilstationsis nearlythe sameas that determined
from recordsat hard rock stations.FollowingIda [1973] and
Aki [1979],Papa•ieor•IiouandAki [1983]interpretedtheob-
servedfm•xintermsofaslip-weakeningmodel.Theyfoundfor
majorCaliforniaearthquakesthatthecriticalsliprangesfrom
a fewtenthsof a meterto a fewmetersand the sizeof cohesive
Copyright1987bytheAmericanGeophysicalUnion.
Papernumber6B6038.
0148-0227/87/006B-6038505.00
zone from a few hundred meters to a few kilometers. We note
thattheupperfractallimitoftheSanAndreasfaultmeasured
byOkuboandAki[1987]isofthesameorderofmagnitudeas
the sizeof cohesivezone.Sincethe upperfractallimit as de-
finedby the methodusedby Okuboand Aki is clearlya
measureof width of the zonecontainingsubfaults,thefmaxof
largeearthquakesmaybedeterminedbythewidthofa fault
zone.Theslipneara cracktipassociatedwitha majorearth-
quakemaybesmearedoutovera distancecomparabletothe
faultzonewidth,andhigh-frequencywavesmaybesmoothed
out. The distanceover which the smearingtakes place may
dependon theamountof slipin an earthquake.We then
expecta slightdecreaseinfm•xwithincreasingmagnitudeas
observedby Papa•Ieor•IiouandAki [1983]andIrikuraand
Yokoi [ 1984].
Anothersubtlebutextremelyinterestingobservationisthat
thefma•for largeearthquakesis roughlythesameasthe
cornerfrequencyofsmallearthquakeswhenthelatterbecome
nearlyconstantindependentof seismicmomentbelowa cer-
tainmagnitude.Thetendencyforthecornerfrequencybecom-
ingconstantbelowseismicmomentof about102• dyncm
(magnitudeabout3)hasbeenobservedwidely(Chouetetal.
[1978],RautianandKhalturin[1978],Fletcher[1980],Bakun
et al. [1976],SpottiswoodeandMcGarr[1975],Tuckerand
Brune[1973],andArchuletaetal.[1982]amongothers).The
cornerfrequencyof smallearthquakeswiththefaultlength
lessthan the fault widthmay be primarilycontrolledby the
fault zone width.
J. Boatwright(personalcommunication,1986)suggested
thatif theaboveideaonfmaxiscorrectthereshouldbea kink
in themagnitude-frequencyrelationat magnitudearound3,
reflectinga departurefromself-similaritydueto theeffectof
fault zonewidth. Dietrich's[1979] estimationof a minimum
earthquakefora givencriticalweakeningslipisalsorelevant
tothisproblemandisdiscussedlaterin comparisonwithour
result.Thepurposeofthepresentpaperisto examineif there
1349

2. 1350 AKI' ORIGINOFfmax
5O
4O
z 20
• I0
<• 5
4
3
0 2
uJ I
0.5
I I
E 3 4 5 I0 20 30 40,50 I00 200 300
LAPSE TIME (MEASURED FROM ORIGIN TIME) IN SECONDS
Fig. 1. Observedrelation betweencoda amplitudeand lapsetime
(measuredfrom the earthquake origin time) for selectedindividual
events and a family of curves corresponding to equation (4) for
variousM•,.
is such a kink in the magnitude-frequencyrelation using the
seismogramsobtained at a borehole station in the Newport-
Inglewood fault zone operatedby the University of Southern
California (USC). As shownin Figure 5, we indeedfound the
kink.
CONSTRUCTION OF FREQUENCY-MAGNITUDE RELATION
In order to examine the departure from self-similarityfor
the frequency-magnituderelation, it is essentialto developan
accurateprocedureto estimatethe seismicmomentfrom ob-
served seismograms.We cannot use the magnitude reported
routinely for small earthquakesbecausethey are basedon an
arbitrary extrapolation of some empirical relation toward
smallmagnitudes.
It is also important to know the sampledarea or volume
and time period for each magnitude so that the frequencyfor
all magnitudesrefers to the common spatial and temporal
extent. In order to meet the above requirements,we applied
the coda method [Aki, 1969; Aki and Chouet,1975] to single-
station data in the following manner.
First, we developeda set of calibration curveswhich relate
the coda amplitude A at a lapse time t (measuredfrom the
origintime)withthemomen•tmagnitude.Second,weselected
a time period over which a continuous visible recording is
available. For all seismic events visible on the record, the
momentmagnitudeandts_•,weredetermined.Third,exam-
ining seismicsignalson the record, we determined the maxi-
mumvalueof ts_• for eachmomentmagnitudefor which
signalswere unambiguouslydetectedon the record.Fourth,
we countedthe number of earthquakesfor eachmoment mag-
nitude interval and normalized it to unit area usingthe maxi-
mumts_pinestimatingthetotalsampledarea(orvolume).
Becauseof the useof single-stationdata describedabove,it
is possibleto underestimatethe frequencyof occurrencefor
small earthquakesif we chooseour station in a stableaseismic
block. In order to avoid this possibility(which may introduce
the kink for a different reason),we choosea station within the
Newport-Inglewood fault zone, where the activity of mi-
croearthquakeshasbeenknown [Teng et al., 1973].
Data
We usedseismogramsrecorded at station DHB (34ø01.05'N,
118ø23.1YW), located at Baldwin Hills, Los Angeles,where a
vertical componentseismograph(L-1B-HT) with natural fre-
quency4.5 Hz hasbeenoperatedin a boreholeat depth of 1.7
km for many years[Teng et al., 1980]. SinceApril 1984,the
routine processingand archival of local earthquakedata have
beensystematizedon the computer,and detailedinformation
about the epicenterlocation,arrival time, and magnitudehas
becomereadily availablefor reference.We thereforeexamined
continuousrecordsat DHB for the period from June 1984 to
August 1985.During this period the gain on the visiblecon-
tinuousrecordchangedfrom time to time, but we wereable to
correctfor the changeusingthe magnetictape recordingsfor
whichthe gain hasbeenfixed for the aboveperiod.
MOMENT MAGNITUDE BASED ON CODA AMPLITUDE
Hanks andKanamori [1979] introducedmoment magnitude
M wmotivatedby Kanamori[1977] for greatearthquakesfor
whichMsis an inadequatemeasureof thesize.Their formula
is consistentwith the Gutenberg-Richterenergyformula log
W = 1.5Mw+ 11.8,whereenergyW isassumedto beequalto
seismicmomentModividedby2 x 104,andisgivenby
log M o= 1.5Mw+ 16.1 (1)
This equation is very closeto the following relation between
seismicmomentand localmagnitudeM Lfor southernCalifor-
nia obtained by Thatcher andHanks [1973]:
log M o= 1.5ML + 16.0 (2)
Furthermore, this expressionis remarkably similar to the fol-
lowing relation betweenlocal magnitude and seismicmoment
estimatedfrom the amplitude of coda wavesby Aki [1969] for
the aftershocksof Parkfieldearthquakeof 1966with M• from
3to5:
log M o= 1.5M• + 15.8 (3)
Equation (3) wasderivedon the assumptionthat the predomi-
nant frequenciesof coda waves are lower than the corner
o
u
o
I I I I I
2 3 4 5 6
BULLETIN ML
Fig. 2. Comparison between the moment magnitude determined
from coda amplitude using equation (4) and the local magnitude re-
ported routinely at USC and CIT-USGS. Seetext for the explanation
of discrepancyfor M < 3.

3. AKI' ORIGINOFfmax 1351
frequencyof sourcespectrumand the coda amplitude is di-
rectly proportional to seismicmoment.
The consistencyamong the above three equationssupports
the assumptionthat coda amplitudeis proportional to seismic
moment for the Parkfield aftershocks.Sincethe validity of this
assumptiondependson the instrument response,it is neces-
saryto demonstrateits validity for the presentcase.
We measurethe peak to peak amplitude A of coda wavesas
a function of lapse time t from the origin time (not from the
time of P arrival) for selectedearthquakes of various mag-
nitudes. Plotting A as a function of t as shown in Figure 1, we
found that a remarkably simple straight line relation exists
betweenlog A and log t for all earthquakes.We did not find
any systematicchange of the slope of the straight line as a
function of t.
Assuming that A is proportional to seismic moment and
defining moment magnitude M w as proportional to 2/3 log
M o to be consistentwith equations(1), (2), and (3), we can
write observed relations between A (in millimeters) and t (in
seconds)as
log A = 1.5Mw+ 4.2 log t + 2.5 (4)
The constant term in equation (4) was adjusted in such a
way that M wagreeon the averagewith M Lreportedroutinely
by California Institute of Technology/U.S. Geological Survey
(CIT-USGS) and USC. A comparisonof M wand M L for indi-
vidual eventsis shownin Figure 2. The agreementis adequate
for M > 3, but not for M < 3. This •'.•n be explained as fol-
lows.
The magnituderoutinelydeterminedby USC is basedon an
empiricalformulabetweenM L determinedby Richter'sdefini-
tion and the signaldurationtr_ Pbetweenthe arrival time of P
wave and the tail end [Real and Tenq, 1973] for 3 <
Thus,for 3 < M L,M wfrom codaamplitudeshouldagreewith
M L estimatedfrom tr_Pif our assumptionabout codaampli-
tude is valid. The empirical formula of Real and Teng is given
by
M = 0.1 + 1.59log tF_P-+-0.001A
where M is the equivalentRicnter magnitude, tr_ P is in sec-
onds,and the epicentraldistanceA isin kilometers.
The relation between magnitude and signal duration can
also be inferredfrom our equation(4) for coda amplitude by
settingA at a smallconstantvalue.This will give
M w= const+ 2.8 log tr_ 0 (6)
M
' ' I ' '''1 I
, , I , , , , I• I I I , , , , ,
>_ 5 io 20 30 40
DURATION IN SECONDS
Fig. 3. Schematic explanation for the difference in the slope of
magnitude-log(duration)relationbetweentr_ oand tr_ e.
k
4.0
3.0
2.o
1.0
0.5
0 I 2 3 4 5
Mw FROMCODAAMPLITUDE
Fig. 4. Plotsof ts_•,and themomentmagnitudedeterminedby
the coda method. For earthquakesbelow the detection limit, seismic
signals are strong and identified unambiguously as earthquakes on
the record.
wheretr_ ois the durationbetweenthe origintime and the tail
end. The above coefficient 2.8 is mcuh greater than the one
determinedby Real and Teng, 1.59in equation(5).
The reason for the above discrepancymay be explained by
the differencein definition of signal duration. For example, if
the travel time of P waves is on the average 10 s, we find a
significantly different magnitude-duration relation between
tr_ P and tr_ o as illustrated in Figure 3. The slope will be
gentlerfor tF_Pthan for tr_ o.
From the point of physical theory of coda waves based on
the back-scattering model [Aki, 1969] the coda amplitude
shouldbe a function of time tr_ 0 measuredfrom the origin
time. The use of tr_ P, especiallyfor small duration, will be
inconsistent with the theory. Smaller events in our data are
closereventswith smallerdifferencebetweentr_ o and tr_ e.
Thus the formula (5) developedfor earthquakeswith greater
differencebetween them would not apply. If the formula (5) is
forced to them, the magnitude reduction for shorter duration
is smaller than the one based on the formula (6) and the
discrepancyshown in Figure 2 results.
Sinceour assumptionabout the coda frequencybeing lower
than the corner frequency is more valid for smaller earth-
quakes, we feel justified to extrapolate our equation (4) to
magnitudes smaller than 1, for which direct measurementsof
A versust are not available (Figure 1).
Using the calibration curve, we determined graphically
moment magnitudefor earthquakeslistedin Table 1. Table 1
alsoliststs_, foreachearthquake.
OBSERVEDFREQUENCY-MAGNITUDE RELATION
Figure4 showsts_, andMwdeterminedfromcodeampli-
tude for 225 earthquakes listed in Table 1. In Figure 4 we
draw a line showing detection limit below which signals are
strong and unambiguously identified as earthquakes on the
record. We are confident that earthquakes located below the
line in Figure 4 were not missed.

4. 1352 AK•'OreGONOFfmax
TABLE 1. Time of Earthquake,ts_p, one Moment Magnitude
From Coda Amplitudes,and M L DeterminedRoutinelyat USC and
CIT-USGS
Time, ts_p,
Date LT s M w M•.*
June29,1984 0128 9.5 2-}3
1
June29, 1984 2010 2.2
June30,1984 1153 10.2 2•
June30,1984 0634 20.8 3-•
July3, 1984 1237 1.6 1• 2.0
July 4, 1984 1300 13.5 3
July6, 1984 1200 22.6 3«
July6, !984 1003 11.8 2•
July8, 1984 1141 5.0 2-• 2.9
July8, 1984 1309 6.0 2• 2.8
July8, 1984 1650 3.1 1!3 2.1
July9, 1984 1551 8.8 2-• 3.0
July9, 1984 1909 19.0 3• 3.6
July 9, 1984 1913 19.0 3
July9, 1984 1920 19.0 3«
July 9, 1984 1931 19.0 3
July 9, 1984 1932 19.0 3
1
July 13, 1984 1539 1.0
July17,1984 0250 4.2 1-} 1.9
July 18, 1984 0010 0.2 0
July18,1984 0213 17.2 2!3
July23,1984 1442 0.6 1«
July24,1984 0212 1.5 1-} 2.1
July 25, 1984 1530 9.4 4
July25,1984 1556 12.4 2« 2.6
July25, 1984 1843 18.2 3-}
July 25, 1984 2356 6.5 2 2.5
July 29, 1984 0914 2.2 2
July31, 1984 1932 1.5 lX3
Aug. 2, 1984 0543 0.4 0
Aug.5, 1984 0935 5.8 2• 2.8
Aug.5, 1984 0938 5.8 1• 2.0
Aug. 5, 1984 0938 5.5 1 1.9
Aug. 5, 1984 0938 5.8 3 2.7
Aug.5, 1984 1528 5.5 2• 2.6
Aug. 5, 1984 1529 5.8 2
Aug. 6, 1984 0814 19.2 4 4.4
Aug.11,1984 1158 3.5 1-} 1.7
1
Aug. 12, 1984 1528 1.6
Aug.14,1984 0124 2.3
1
Aug. 19, 1984 1833 0.4
Aug.23,1984 0202 3.1 1• 2.1
Aug.23,1984 1242 3.0 1-} 1.7
1
Aug. 23, 1984 1442 0.2
Aug. 25, 1984 1311 0.1 0
Aug. 25, 1984 1912 17.2 3
Aug.26,1984 1512 3.8 2-• 2.8
Aug.29,1984 0214 3.1
Aug. 31, 1984 0515 2.3 1
Aug.31,1984 0630 2.1 1«
Sept.6, 1984 1004 16.5 3-} 3.0
Sept. 7, 1984 1103 18.0 4 4.1
Sept. 7, 1984 1757 23.2 4 4.0
1
Sept.11, 1984 0428 1.2
Sept.12,1984 0413 5.2 2-•
Sept.12, 1984 1000 2.6 _23
Sept.12,1984 2313 10.5 2-• 2.9
Sept. 13, 1984 0643 1.5 1
Sept.17,1984 0201 3.2 1
2
Sept. 17, 1984 0801 0.2
Sept.19,1984 0849 2.2 13-}
1
Sept.21, 1984 1946 0.1
TABLE 1. (continued)
Date
Time, ts_p,
LT s m w
Sept. 23, 1984 2137 0.8
Sept. 23, 1984 2337 2.1
Sept. 25, 1984 0027 1.8
Oct. 2, 1984 0259 12.7
Oct. 2, 1984 0938 12.2
Oct. 3, 1984 1249 0.4
Oct. 5, 1984 0737 20.2
Oct. 5, 1984 1824 5.5
Oct. 6, 1984 0821 13.4
Oct. 6, 1984 2001 3.5
Oct. 7, 1984 0953 0.5
Oct. 7, 1984 1018 2.4
Oct. 13, 1984 0210 4.0
Oct. 14, 1984 0805 3.1
Oct. 15, 1984 1744 0.6
Oct. 15, 1984 1806 2.8
Oct. 17, 1984 0057 2.5
Oct. 19, 1984 1403 1.3
Oct. 22, 1984 2029 0.5
Oct. 22, 1984 2029 1.2
Oct. 22, 1984 2212 5.0
Oct. 25, 1984 0551 2.4
Oct. 25, 1984 0655 2.2
Oct. 26, 1984 1720 0.5
Oct. 27, 1984 1722 1.4
Oct. 27, 1984 2310 7.2
Oct. 28, 1984 1552 6.8
Oct. 29, 1984 0533 7.0
Oct. 30, 1984 1411 6.1
Oct. 30, 1984 1443 6.1
Nov. 4, 1984 0100 5.3
Nov. 5, 1984 0259 11.0
Nov. 5, 1984 0315 11.0
Jan. 2, 1985 0524 21.0
Jan. 7, 1985 0219 2.0
Jan. 13, 1985 0850 3.0
Jan. 14, 1985 0343 5.0
Jan. 16, 1985 0253 0.5
Jan. 18, 1985 1308 1.7
Jan. 18, 1985 1309 1.7
Jan. 19, 1985 0030 21.0
Jan. 19, 1985 0324 24.0
Jan. 19, 1985 1608 2.2
Jan. 19, 1985 1732 2.0
Jan. 24, 1985 1024 1.2
Jan. 26, 1985 0641 8.0
Jan. 28, 1985 0720 0.8
Feb. 6, 1985 0441 4.5
Feb. 8, 1985 1601 1.7
Feb. 6, 1985 1821 1.5
Feb. 8, 1985 0226 1.8
Feb. 8, 1985 0658 18.0
Feb. 9, 1985 0532 0.8
Feb. 10, 1985 0920 24.0
Feb. 10, 1985 1359 24.0
Feb. 10, 1985 1823 1.2
Feb. 11, 1985 1711 9.5
Feb. 11, 1985 1737 10.0
Feb. 12, 1985 1135 2.0
Feb. 14, 1985 2322 5.0
Feb. 15, 1985 1626 10.8
Feb. 15, 1985 2326 22.0
Feb. 16, 1985 0910 10.0
Feb. 16, 1985 2330 11.5
1
3-}
3-23
3
4
3«
3
3«
o
4-}
2{
3x3
3
3
2
2
3-•
33•
3
3«
1
2
2
4
3
3•
1
3-}
3
4
3
ML•
2.3
2.3
2.9
2.8
2.8
2.6
3.0
2.3
3.4
2.3
3.8
2.6
2.0
3.5
3.6
2.7
2.0
1.6
2.9 (3.3)
2.5
2.1
4.3
3.7 (3.5)
4.2 (3.6)
!.4
2.7
1.7
3.6 (3.3)
4.0
3.2
2.8

5. AI•: OreGONOFfmax 1353
TABLE 1. (continued)
Date
Time, ts-v,
LT s M w
Feb. 17, 1985
Feb. 19, 1985
Feb. 1% 1985
Feb. 19, 1985
Feb. 19, 1985
Feb. 20, 1985
Feb. 21, 1985
Feb. 21, 1985
Feb. 25, 1985
Feb. 25, 1985
Feb. 28, 1985
March 1, 1985
March 2, 1985
March 3, 1985
March 4, 1985
March 4, 1985
March 4, 1985
March 4, 1985
March 5, 1985
March 5, 1985
March 6, 1985
March 7, 1985
March 9, 1985
March 9, 1985
March 9, 1985
March 11, 1985
March 11, 1985
March 13, 1985
March 14, 1985
March 15, 1985
March 17, 1985
March 18, 1985
March 18, 1985
March 18, 1985
March 28, 1985
April 2, 1985
April 3, 1985
April 3, 1985
April 4, 1985
April 6, 1985
April 8, 1985
April 8, 1985
April 8, 1985
April 8, 1985
April 9, 1985
April 12, 1985
April 12, 1985
April 13, 1985
April 20, 1985
April 23, 1985
April 30, 1985
May 1, 1985
May 3, 1985
May 4, 1985
May 4, 1985
May 17, 1985
May 17, 1985
May 10, 1985
May 13, 1985
May 14, 1985
June 11, 1985
June 14, 1985
June 16, 1985
0913 1.0
0509 16.5
1637 16.0
1726 1.8
1731 2.2
1321 9.6
0327 10.0
0633 7.0
0909 1.8
1438 6.0
0809 3.5
1223 2.2
0453 1.0
0126 20.0
0339 6.0
0512 16.0
1150 2.5
2041 1.0
0105 6.0
0515 6.5
0554 1.0
0414 5.0
0711 3.0
0732 3.0
0955 3.1
1304 1.1
1621 3.0
0120 0.8
1624 10.0
0503 4.0
0658 1.2
0743 2.8
0930 2.8
1558 1.2
0623 1.5
2346 0.8
0405 9.0
0410 9.0
1157 0.7
0623 1.5
0109 6.5
0915 6.5
0121 6.5
1320 6.5
1043 12.0
0943 10.0
2004 1.0
1222 1.8
1533 6.0
0144 2.5
1833 0.6
1618 3.0
1508 3.0
0808 3.0
0817 3.0
0534 3.0
1218 9.0
1216 1.0
1219 4.5
1035 20.0
1723 2.5
1404 1.0
0626 2.2
0
3
3«
3
2
1
2
4
3
3•
3
2
3
3
2
2
o
3
2
3•
3
3•
3•-
3
2
2
2
2
3
TABLE 1. (continued)
ML*
Time, ts_•,
Date LT s M w ML*
3.4
3.1
2.5
2.8
2.1
2.6
2.0
2.4
3.7
2.5
2.8
3.2
2.3
3.0 (2.8)
2.7
2.5
2.0
2.8
2.6
2.8
2.7
2.5
1.5
2.8
3.6
2.4
2.5
3.4
2.7
3.0
2.8 (2.6)
2.1
2.4
2.5
2.2
3.0 (2.6)
2.2
3.9
June16,1985 0650 1.5
June 16, 1985 1026 14.5 4 4.0
June21, 1985 0050 14.0 3« 3.2
June27, 1985 1811 1.5 1• 1.7
July3, 1985 0603 2.8 1« 1.5
July 3, 1985 2055 2.0 2 2.1
July3, 1985 2231 6.0 2• 2.5
July8, 1985 0625 5.0 2«
July8, 1985 2323 8.0 2• 2.7
July 9, 1985 2307 1.0 _23
July10,1985 1223 8.5 2• 2.6
July11,1985 0139 3.0
July11,1985 1000 3.5 1«
July 11, 1985 1402 1.5 3
July13,1985 0416 1.0 2« 2.4
1
July15,1985 0609 0.8
July16,1985 1758 18.0 3• 3.8
July 18, 1985 0323 1.0 0
July 18, 1985 1405 22.0 4 4.1
July19,1985 1617 10.0 2• 2.6
July20, 1985 0555 0.6 -«
July21,1985 1407 3.5
July 22, 1985 2150 1.8 1
July 23, 1985 1155 1.0 0
Aug. 1, 1985 1942 6.0 2 2.3
July 23, 1985 1931 1.8 23
Aug.3, 1985 0919 2.0 1«
Aug.3, 1985 1421 2.1 1«
Aug.4, 1985 1201 30.0 5• (5.9)
Aug.5, 1985 1448 1.7 1• 1.9
Aug.12,1985 2101 14.0 3z3 3.4(3.5)
Aug.16,1985 1209 3.2 1• 1.8
*Values in parenthesesare from CIT-USGS.
We,then,countthenumberofearthquakeN fallingin the«
magnitude unit interval below the detection line, as shown in
Table2. Table2 alsoliststhedetectionlimit/-s-t,foreach
mangnitude.
Since the hypocentral distance (in kilometers) is about 8
timests_t,(inseconds)andfocaldepthsofourearthquakesare
in the range from 0 to about 15 km, the sampledvolume of
seismicrerion for a given magnitudemay be proportional to
(is_p)2foris-t,> 2sandto(is-t,)3foris-t,< 2s.
First, we calculatedthe areal densityof earthquakesby di-
vidingthenumberof earthquakesbelowis_t,by(is_t,)2,as
shownin Table 2. Then, we calculatedcumulativedensityE as
shown in Table 2 and in Figure 5 by the curve labeled
"AREA." We then corrected the curve for M < 1 for which
is-t,islessthan2 sandthevolumedensityisproportionalto
(i-s_p)3.Thecorrectedcurveislabeled"volume"inFigure5.
The observedcurvesshown in Figure 5 correspondto the
cumulativenumberof earthquakesin a time periodeffectively
about 1 yearlongan in an areaof about200 km2; the areaof
acirclewithradius8kmcorrespondingtois_t,= 1s.
For comparison,we showan extrapolationfrom the empiri-
cal result for earthquakeswith magnitude greater than 3 oc-
curring in the Los Angelesbasin in the 29 year period from
1934 to 1963 [Allen et al., 1965]. Sincethe result of Allen et al.
wasgivenin termsof the number of earthquakesper year per
1000km2,wedividedtheirresultbya factorof5.Theslopeof

6. 1354 AKI' ORIGINOFfmax
the line, so-called b value, is 1.0 for the Los Angelesbasin in
agreementwith our resultfor M > 3. We seea cleardeparture
of the observedcurve for M < 3 from the line extrapolated
from the data for M > 3. The observedfrequencyof earth-
quakes with M -• -0.5 is almost 10 times lessthan that ex-
pected from the extrapolation of empirical relation for earth-
quakeswith magnitudelarger than 3.
DISCUSSION
Our conclusion is very simple. The observed frequency
magnitude relation departs from self-similarity for earth-
quakes with magnitude smaller than about 3. Although we
have not yet confirmed with earthquakes in the Newport-
Inglewood fault zone, this departure coincideswith the widely
observeddeparture of the moment-corner frequencyrelation
from self-similarity; the corner frequency tends to become
constantfor earthquakeswith magnitude smaller than about
3. As mentioned in the introduction, this constant corner fre-
quency agrees roughly with the extrapolation of the
moment-fmaxrelationfromlargeearthquakes.We alsosuggest-
ed that the absolutevaluesof fmaxdeterminedfrom actual
strong motion acceleration spectra are consistentwith the
slip-weakeningfailure criterion in which the critical slip is a
few tenths of a meter to a few meters and the cohesive zone
size is a few hundred meters to a few kilometers, roughly
comparable to the width of a fault zone.
Relevant to this conclusion is the discussiongiven by
Dietrich [1979] on the existenceof a minimum earthquakefor
a fixed value of critical weakeningslip. He estimates,for ex-
ample, the minimum magnitude to be 1 for the critical wea-
kening slip of 2 mm. Extrapolating his result, the minimum
magnitudewould be 3 for the critical weakeningslip of 2 cm.
Sincetheseestimatesare rough order of magnitude estimates,
his estimatescan probably be made consistentwith ours by
adjusting the parametersof the instability model. Of course,
earthquakessmaller than the minimum earthquake can existif
they are associatedwith a differentinstability mechanismwith
a smallercritical slip.
Some of the observations which supported the constant
corner frequencyfor M < 3 as the sourceeffectmight be un-
founded, but others in increasing numbers have been well
founded. If the suggestedequivalencebetween the constant
TABLE 2. Calculation of Cumulative Number of Earthquakes per
Area of 200 km 2
M s ([•_•)2 N N/([•_•)2 E
_! 1.0 1 6 6.0 22.83
0 1.2 1.4 7 5.0 16.8
ñ 1.5 2.3 9 3.9 11.83
_2 2.0 4 8 2.0 7.933
1 2.5 6.2 13 2.1 5.93
1« 3.5 12.2 21 1.72 3.83
132- 4.6 21.1 21 1.00 2.11
1 6.0 36 18 0.50 1.11
2« 8.0 64 16 0.25 0.61
23• 10.0 100 24 0.24 0.356
3 13.0 169 7 0.041 0.116
3« 16.0 256 11 0.043 0.075
3•3 20.0 400 5 0.013 0.032
4 24.0 546 10 0.018 0.019
4-} 30.0 900 1 0.001 0.001
z
• i
I I I I
TO DATA FOR M>3
(b=l.O)
•-L.A. BASIN(1934-63)
(ALLEN ET AL,1965)
FOR M>3
0 I 2 3 4
Mw FROM CODAAMPLITUDE
Fig. 5. Cumulative number of earthquakes per year and per 200
km2obtainedfromthedatashownin Figure4 lyingbelowthedetec-
tion limit. The curve labeled "Area" is obtained by dividing the
numberof earthquakesby thesquareof the rs_p at detectionlimit.
The curvelabeled"Volume" correctsfor the effectthat the total depth
range of seismogenicone is not sampled for M < 1. The observed
curve for M < 3 showsa clear departure from the extrapolation of
empirical relation for M > 3.
cornerof smallearthquakesand thefmaxof largeearthquakes
is well established, we shall have a direct means to estimate
one of the mostimportant parametersof strongmotionsfrom
the study of weak motions of small earthquakeswhich are
more easilyobservable.
Furthermore, there will be a possibility to determine the
parametersof slip-weakeningand friction laws from seismo-
logicalstudiesof thefm•xof large earthquakes,the constant
corner frequency of small earthquakes, and the kink in
magnitude-frequencyrelation. Those estimated parameters
can be usedin predictinglong-term fault behavior including
the precursory phenomena before major earthquakes ICao
and Aki, 1985].
Acknowledgments.It has been a pleasure to work with excellent
records obtained at a borehole in the middle of the Newport-
Inglewood fault by the able technical group under the leadershipof
Ta-liang Teng and supervisionof Egill Hauksson.I would alsolike to
thank Steve Day for pointing out the relevanceof Dietrich's (1979)
work to our conclusionand Tianqing Cao, SyhhongChand, and Lily
Hsu for their assistancein analysis.This work was supportedby the
National ScienceFoundation under grant ENG-8408227.
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(ReceivedApril 21, 1986;
revisedAugust 26, 1986;
acceptedOctober 13, 1986.)