JOURNAL
OF GEOPHYSICAL
RESEARCH,
VOL. 98, NO. B7, PAGES 12,141-12,151, JULY 10, 1993
On the Frequency-Length
Distributionof the SanAndreasFault System
P. DAVY
GeoSciences
Rennes,CentreNationalde la Recherche
$cientifique,
Campusde Beaulieu,Rennes,France
The frequency-length
distributionof the San Andreasfault systemwas analyzedand comparedwith
theoreticaldistributions.
Both.density
andcumulativedistributions
werecalculated,anderrorswereestimated.
Neitherexponential
functions
norpowerlawsareconsistent
withthecalculated
distributions
overtherangeof
studiedlengths.Thebestfit on bothdensityandcumulative
distributions
wasachievedwith a gammafunction
which mixesa powerlaw and an exponentialfunction.At smalllengths,the gammafunctionbehavesas a
power law with an exponentof-1.3_-_4-0.3.
At large lengths(above 10 km), the distributionis a mixed
exponential-power
law functionwith a characteristic
lengthscaleof about23+6 km. The gammadistributionis
proposedto resultfrom a length-dependent
segmentation
of a fractalfault pattern.This studyshowsthe
importanceof comparingboth cumulativeand densitydistributions.
It alsoshowsthat the studiedrangeof
lengths(1-100 km) is not appropriatefor measuringpowerlaw exponents.
INTRODUCTION
The frequency-lengthdistributionis a basic characteristicof
fault growth processes.Faults occur in natureon a very wide
range of scales.Recent work [e.g., Turcotte, 1986; Scholzand
Cowie, 1990; Main et al., 1990; Hirata, 1989] has looked for
evaluatingdensitydistributionis presented
in the nextparagraph.
The confidence
on the measured
distributions
andon parameters
derivedfrom proposeddistributions
are alsodiscussed.
DENSITY AND CUMULATIVE DISTRIBUTIONS
FROM THE SAN ANDREAS FAULT SYSTEM
powerlawsn(1).-I-a asproofof a fractaldistribution
of faultsin
natural systems.Such laws refer to physicalprocesses
of rock
fragmentation,where all scalesare involvedduring fault growth
with repetitivesubdivisionof breakagepatterns[Turcotte, 1986;
King, 1983]. The laws have been found to hold in experiments
modellinglithospheredeformation[Davy et al., 1990; Sornetteet
al., 1993]. In the geological(and even astronomical)literature,
other fitting law have been used, for example, the Poisson
distribution
(n(1).-exp(-l/lo))
or lognormal
relationships,
which
mainly differ from the power law by the introductionof a
characteristic
lengthscalelo [e.g., Korvin, 1989]. In this paper,
the significanceof the fault frequency-lengthdistributionis
discussed
at lithospherescale.This studyis basedon the analysis
of the San Andreasfault system(California),a very well-studied
exampleof lithosphericdeformation,coveredby rathercomplete
data.
Amongstthe problemsthat the analysesof the fault length
distributionaddress,is the existenceof characteristiclength
scales.For instance,the finite thicknessof the brittle crustmay
control the fault growth process.This was demonstrated
from
earthquakedistributionsby Pachecoet al. [1992], who argued
that crustal thicknessdefines a transition in the frequencymagnitudedistribution of earthquakes.The related transition
magnitude(around 7.5-8) correspondsto a point where the
slippedfault areainvolvesthe whole seismogenic
layer.A recent
review on the scalingof seismicmomentas a function of the
lengthof rupture[Rornanowicz,1992] emphasizes
modelswhere
slip is controlledby the verticallengthof the fault. Becausethe
vertical length of transcurrentfaults is no greater than the
Data
For this study,I used the most completemap of the San
Andreas fault pattern at continental scale, which is the
compilation(CDMG) madeby Jennings[1988], drawnto a scale
of 1:750,000. This tectonic map of California presents
interpretations
of geologicor geomorphologic
featuresin termsof
recently or historically active faults, Quaternary faults, and
preQuaternaryfaults.The maphasbeenperiodicallyrevised(this
is a fourthedition)[Wallace, 1990]. Only faultslocatedbetween
the 39th parallel and the Mexican border have been included.A
more completedescriptionof the fault systemand the derived
statisticalanalysisof the differentunits will be presentedby
0.Bouret al. (manuscript
in preparation,1993).The totalnumber
of digitisedfaultsis 5772 representing
a cumulativelengthof
39,050km (Figure1). The largestfault segmentis 160km long
and the smallest 0.25 km.
It is important,thoughdifficult,to estimatethe accuracyof the
data.The samplingis definitelyincompletefor faultssmallerthan
1 km. However,becausethe dataarebasedon a compilation,
it is
alsonot possibleto judge the accuracyof faultslargerthan 1 km,
without a comparisonwith a model of distribution.Another
difficultycomesfrom the definitionof a fault which requiresa
connectivitycriterionbetweensegments.In other words, a fault
can appearas severalseparatesegmentswhich may have,in fact,
no individual identity. Although an effort is made to check
maximum connectivities, the errors can introduce a bias on the
frequency-length
distributionof faults.
thickness
of thebrittlecrust(lbc),thesemodels
predicta major Frequency-Length
Distribution
change
of therupture
process
around
lbc.Thepresent
analysis
can
give someinsightson the long-termevolutionof earthquakes,
that
is, on tectonicfault patterns.
Both densityand cumulativedistributionsof fault lengthsare
calculatedin this paper in order to go thoroughlyinto the
As in similar studiesof fault systems[e.g., Villerninand
Sunwoo, 1987; Hirata, 1989; Gudrnundsson,1987], the first
parameterto be calculatedis the cumulativedistributionC(L), i.e.,
thenumberof faultswhoselengthis greaterthanL:
comparisonwith theoreticaldistributions.Also, a new methodfor
C(L) =
(l)dl,
(1)
Copyfight
1993bytheAmerican
Geophysical
Union.
Papernumber93JB00372.
0148-0227/93/93JB-00372505.00
wheren(l) is the densitydistribution.Becauseof n(l) variations,
the integralis generallysensitiveto its lower boundL. Thus an
12,141
12,142
DAVY:FREQUENCY-LENGTH
DISTRIBUTION
OFSANANDREAS
FAULTSYSTEM
Sail
x,
I''''
0
I
200
'
•
•
•
I
400 km
Fig. 1. Map of theSanAndreasfaultsystem(right)with anoverviewof thedigitizedfault(left) with a scalein kilometers.
Only faultslargerthan10 km
arefigured.
integralover lengthsmallerthanL would be not pertinentsinceit
If dl is smallenough,n(l) is equalto N/dl. The errormadeby
depends
on theresolutionscaleof the sampling.The advantage
of eliminatingthe two other termsof (3) dependson the assumed
using the cumulativefunction is to smooththe large variations distribution
function.For a powerlaw n(l)=Al-a, therelativeerror
which may be observed for n(/). Although the cumulative is adl/21; for an exponentialfunction n(l)=A*exp(-l/lo), it is
distribution is convenient, one cannot avoid the calculations of
dl/21o.For a given length,a functionf(l,N)=N/dl is calculatedfor
density distributions.At first, the comparisonbetween both differentvaluesof N. In practice,the faults are first sortedby
distributionscan be very useful for estimatingfitting parameters.
A secondreasonis that the fluctuationsof the densitydistribution
will be helpful for estimatingthe amountof confidenceon the
parametersof theoreticaldistributions.
The most intuitive method for calculating the density
distributionn(/) is to dividethe lengthrangeby a constantinterval
dl and to countthe numberof faults N in eachinterval (assuming
that n(/)= N/dl). The choice of the interval dl is critical becauseit
definesthe smoothingfactor of the obtainedvalues.Ideally, dl
shouldbe determinedfrom the uncertaintyon data and alsofrom
the expectedtheoreticaldistribution.Here is presenteda method
yieldingn(l) andan estimationof the associated
errors.
By definition,thenumberof faultsbetweenI andl+dl is
fl+dl
N=Jln(l
')dl'.
(2)
A second-orderTaylor series expansionof (2) around l'=l
gives
n(l)
=dl
N •'dn
I dlø(d12)
(3)
increasing
lengths.
ThefirstfaultlengthlargerthanI is termed
ll;
thesecond,/2;
andsoon.f(l,N) is calculated
by selecting
thefault
lengthlN andis equalto N/(lN-l).Figure2 showsvariations
of
f(l,N) for differentvaluesof dl=lN-l.Eachsymbolon thecurves
representsa singlefault. The curvespresentthreemain domains:
(1) For the first 10 faults, there are huge variationswhich are
characteristic
of a small numberof data; (2) for larger N values
and for dl/l lessthan 8-10%, f(l,N) fluctuatesaroundan average;
and (3) for larger dl/l, the functionsystematically
decreases
as
predictedby (3).
n(l) waschosenas the averageof f (/,N) in the seconddomain
(N>10 and dl/l<10%). The fluctuationAn(l) aroundthe average
wascalculatedas an indicatorof the uncertaintyand presentedin
Figure3 for differentlengthsI. The relativefluctuationsAn(l)/n(l)
are around5-10% without any strongsystematicvariationwith
length I. This result will be used further for adjusting a
minimization procedure applied to different theoretical
distribution functions.
To accountfor the large variationsof the length scales(from
0.2 to 100 km), n(l) and C(L) are calculated for both a linear
variation of lengths (constant differences between two
consecutivelengths)and a logarithmicvariation(constantratio
DAVY:FREQUENCY-LENGTH
DISTRIBUTION
OFSANANDREAS
FAULTSYSTEM
a
2500
12,143
62=
1l•ld(1)-A*f(l,a,lø
2
¾
2000
xa53
(4)
',
whereNl is thenumber
of sampled
lengths,
d(1)is themeasured
N/dl
distribution,Ad(/) is its error, and A*f(l,a, lo....) is a fitting
functionand its adjustableparameters.
The measured
distribution
can be either the density distributionn(1) or the cumulative
1500
distribution
C(1).Theresolution
is obtained
byminiraising
62
1000
0
5
10
15
20
with respectto A, a, lo, etc. NoticethatA canbe obtaineddirectly
by
din (%)
E f(l,a,lo,...)
*d(l)
A= I
(5)
E f(l,a,
lo,...)2
'
b 75
70
65
Then, the fitting procedurefurther needstwo assessments
(1)
on the samplingof the lengthscaleand (2) on the distributionof
60
N/dl55
errorsAt/(/).
50
45
40
.
0
10
20
30
dl/l (%)
C 8
7
6
5
N/dl
4
3
2
1
0
,
0
10
20
30
dl/l (%)
Fig.2. Variations
of N/dl (seetext)asfunctionof dl/l for differentlengths
l, (a) 1 km, (b) 15 km, and(c) 45 km. N is thenumberof faultsbetweenl
andl+dl. Eachsquareonthecurverepresents
a faultbelonging
to theSan
The lengthsampling.Ideally, a good samplingof the length
scaleshouldbe adaptedto the modelof distributionthatis looked
for. For instance,linear or exponentialregressionsrequire an
arithmeticprogression
in/, while powerlaws requirea geometric
progression
in I (i.e., an arithmeticprogression
in log(1)).In this
study,the densitydistributionn(/) and the cumulativedistribution
C(1) were calculated for both an arithmetic and a geometric
progressionin I. The minimization was then applied on both
length sets together.Becauseno confidencecan be given for
faults smallerthan 1 km, the smallestlength for the minimization
was setat 2 km. The problemof the scaleresolutionis, however,
addressed further by comparison with some theoretical
distributions.The largestlength for the minimizationwas set at
65 km becausethenumberof largerfaults(25) becomessmall.
The distributionof errors. On the basis of the observed
fluctuations,the error distributionon the densitydistributionn(1)
was set to 10% of the measuredvalue whateverthe lengthI. This
modelgives similarweighting,with respectto the minimization,
for largeand smallfaults.Notice that this impliesa similarmodel
of error distributions for the cumulative distribution C(/):
AC(/)=0.1*C(1).It is alsopossibleto calculatea minimumvalue
30
Andreassystem.
betweentwo consecutivelengths).The lengthrangeis 0.25 km
(for the smallestfault) to 65 km (over which thereexist only 25
•
20
faults).
The resultsarepresented
in Figure4 (densitydistribution
n(/),
top;C(/) cumulativedistribution,
bottom)with a comparison
with
exponentialfits (left), power laws (fight), and lognormalfit
10
(dashedlines).
0
Fitting Procedure
0
20
40
60
The next stepof the dataprocessing
is the testingof different
l (km)
distributionssuch as exponential functions, power laws or
lognormal.A classicalminimizationprocedurewasusedto find Fig. 3. Variationsof the relativefluctuations
An(l)/n(l)of the density
parameters
of thesedistributions.
This fitting procedureconsists distribution
n(l) asa functionof I. Thewayof calculating
thefluctuation
inminimising
thequantity
62given
by
An is described in the text.
12,144
DAVY: FREQUENCY-LENGTH
DISTRIBUTION
OFSANANDREAS
FAULTSYSTEM
n(1)]
density
distribution n(1)
3
2
lo=14km
ld
0
10
-1
10
.........,.........•.....'"', .........,.........,.........,'" .......
0
C(1)
20
40
cumulative
60 1
10
10
C(1)
distribution
mm
3
mm
mmm
mmm
103
:
2
lo=16km
2
1
1
,
.
0
0
0
40
80
,
III
I I
10•0 ..... ""•
10 1 ' ' '
.......
l
1
Fig. 4. Fault lengthdistributionof the San Andreasfault: to the top, the densitydistributionand, to the bottom,the cumulativedistribution.The
horizontalaxis (length)is eitherin a normalprogression
(left) or in a logarithmicprogression
(right).Also drawnare exponential
regression
(left
diagrams),
powerlaw regression
(rightdiagrams),
andlognormal
fit (dashed
lines).
fluctuationsobservedon the densitydistribution.Notice that the
foro2 associated
with
thefluctuations
A/n(/)
calculated
earlier
cumulativedistributiongives a slight deviationof parametersa
and lo. A test for decidingof the compatibilitybetweenboth
regressions
is to reinject parametersobtainedfrom the density
andpresentedin Figure3. This valueis
øm
='•/'/ An(l)
)'
distribution into the cumulative distribution and vice versa. Then,
(6) thecalculation
of c•2 givesvalues
of about
3, which
remains
andis about2.0.Thatmeans
thatall distributions
withsimilar02
values are consistent with the data.
reasonable.
The tectonic (or mechanical)significanceof the lognormal
distributionis, however,questionable.
The lognormaldistribution
predicts the density to decrease for lengths smaller than
An example: The fit with a lognormal distribution.To
illustratethe fitting procedure,the minimizationis first appliedto lo*exp(-a
2) (aboutl km). This is contradictory
with the
the lognormalfunctionwhich is often usedfor fitting geological geologicalevidence,which showsa hugenumberof small faults.
data.The lognormalfunctionis expressedas
In fact, the superpositionof both data and regressionlaw shows
thatthesmallvalueof lo ensures
theflattening
of the'curves
at
smalllengthsand thusquantifiesthe lack of observationof small
faultsas alreadypointedout by Einsteinand Baecher[1983]. In
(7)
2a2
the following section,anothergeneticlaw is proposedwhich, in
my opinion, gives a better physicalinsight of the fault growth
This functiondependson three parameters:a dimensionless mechanismandalsoimprovesthe fit for intermediatescales.
exponenta, a characteristiclengthlo, and a normalizingconstant
ON POWER LAWS AND EXPONENTIAL FUNCTIONS
A. The lognormaldistributionensuresan efficientfit for data(see
dashed line in Figure 4). For the density distribution,the
WhyAre PowerLawsandExponentialFunctionsNot Suitable?
n(/)
=4exp(1øg•ø2
).
minimization
wasobtained
with02 of 1.90(a=l.16,1o=3
km).
Forthecumulative
distribution,
thefit wasobtained
forc•2 of
1.32 (a=0.86, /o=7.5km).
These results agree with the
Power law and exponential functions represent tested
distributions with the smallest degree of freedom. If the
DAVY:FREQUENCY-LENGTH
DISTRIBUTION
OFSANANDREAS
FAULTSYSTEM
12,145
normalizing constant A is not taken into account, these and Allmendinger[1991]. More precisely,the probabilityfor
distributions
have only one degreeof freedom,eithera length observinglarge faults at the Earth's surfaceis 1 and becomes
scalelo (exponential)
or an exponenta (powerlaw). The density proportionalto the fault lengthI if the verticalsizeof the fault is
distributiontendsto fit exponentialfunctionsfor large lengths smaller
thanlbc.It is thenimpossible
to describe
thefaultlength
(>20 km in Figure 4). For small scales,however,the numberof distributionby a simplepower law, for instance,in the entire
faults is significantlylarger than predictedby the exponential rangeof lengths.The simplestmodel for powerlaw distribution
function.A regressionappliedon either the densityor the (hereaftercalledtwo-/three-dimensional
powerlaw) mustpresent
cumulativedistributionwith an exponentialfunctiongave lo of two regimesseparated
by a characteristic
lengthscalelo suchas
10kmor14kmrespectively,
withaminimum
ofo2 at4.0or2.6.
A power law can also be visually found over one order of
magnitude
between5 and60 km. The exponentof the powerlaw
(8)
n(l)=A(lllo)-a+l,
(9)
for l>lo and
is2.1(02=4.0)
forthedensity
distribution
and2.8(o2=5.15)
for
the cumulativedistribution.
The obtainedvaluesare significantly
differentsincean exponentof 2.8, reinjectedinto the density
n(l)=A (lllo)-a,
for l<lo.
distribution,
leadsto a o2 factorof 8.0.Boththese
distributions This distributionwas testedfor the San Andreasfault system.
are,by far, lessefficientthanthelognormalfit (seeabove).
For the densitydistribution,the bestfit is shownin Figure6 (left)
The possibility of using both density and cumulative withparameters
a=2.7 and/o=17km. The'fit is obtained
with
distributionis powerful for testingthe validity of thesetwo 02=2.3,
showing
a significant
improvement
withrespect
to a
distributions.
Indeed,the ratio betweenthe densitydistribution simple
power
law(02=4).Notice
thatthefit isnotsignificantly
and the cumulativedistributionn(l)/C(l) shouldbe independent improvedif one considerstwo independentexponents-b (l<lo)
on lengthfor exponentialfunctionsandshouldvaryasthe inverse and-a
(l>lo).Inthatcase,
02is2.0andcalculated
parameters
are
ofthelength
(/-1)forpower
laws.
Thelog-log
plot(Figure
5,
right)showsthatneitheran exponential
functionnor a powerlaw
canfit thisfrequency-length
distributionof faults.This is truefor
any lengthfrom at least 1 km to 100 km. This test contradictsthe
b=l.6, a=3.0 and /o=19 km. The two-/three-dimensional
power
law is, however, not very efficient for fitting the cumulative
distribution(Figure 6, right); the fit is obtained with a bad
minimization
factor
02(3.4)andanupsetting
deviation
ofa (3.4)
validity of power laws found, by many authors,from simple and lo (34 km). As a confirmation,the reinjection of these
analysesof the cumulativedistribution.Noticethat the density- parameters
intothedensity
distribution
gives
a verybado2 of6.
cumulativeratio seemsto vary as the inverseof the squareof the This result confirms the analysismade on the ratio between
length/-0.5
which
gives
adistribution
ofthetype
exp(-a(l/lo)
0'5)
density and cumulativedistributionswhich showedno evidence
whose fit gives lo values of a few kilometers.Becausethis
distributionhasno full meaningfor me, it will not be discussed
any further.
ofl-1variations
even
atlarge
lengths.
More on Power Laws
Preliminaries
The rangeof lengthsusedfor this study(1-100 km) brackets
A MODEL FOR LARGE LENGTH DISTRIBUTION
Them is no doubt that fault systemshave an internal self-
thethickness
of thebrittlecrustlbc(about10km).Because
faults similarityat differentscaleswhich shouldresultin power law
One can masonablyassumethat the power law
cannothaveverticallengthlargerthanlbc, theremustbe a distributions.
shouldhold up to the largestscaleof the system1oand that the
transition
aroundlbc as observed
for the frequency-magnitude
distributionof earthquakes
[Rundle,1989; Pachecoet al., 1992;
Romanovicz,
1992]. For instance,one necessary
consequence
is
finite thickness
of the brittlecrustcouldbe a goodcandidatefor
1o.Theminimumnumberof degrees
of freedomof thefaultlength
thatfaultslargerthanlbcarealways
visible
ontheEarth's
surface, distributionis then 3 (a normalizingconstantA, an exponenta,
while for smallerfaults,the samplingof the Earth'ssurfaceonly anda lengthscalelo). If a completelydifferentdistributionexist
partiallyrepresents
the fault population,as suggested
by Marrett for largerlengths,the distributionwill havemorethan3 degrees
n(O
n(O
ß
10
1-i
1(•'
0 20 40 60!
1
1(• 10
10
Fig.5. Variation
of theration(1)/C(1)
asa function
ofthefaultlengthl, wheren(1)isthedensity
distribution
andC(1)isthecumulative
distribution
The
curve
1-1(corresponding
toanypower
laws)
and
l0(exponential
functions)
arealso
drawn.
12,146
DAVY: FREQUENCY-LENGTH
DISTRIBUTION
OFSANANDREASFAULTSYSTEM
n(1)
c(/):
2
lO
2
1
lO
1
ø
2D/3D power law
-1
10
0
10
........
•
1
10
........
'
o
1
10
10
Fig. 6. A comparison
betweenthe measured
faultlengthdistribution
(greycurve)andits fit by a two-/three-dimensional
powerlaw (seetext and
equations
(8) and(9)). (Right)The densitydistribution;
(left) thecumulative
distribution.
of freedom.In my opinion,it is possibleto describethe datawith
distributions
with only 3 degreesof freedom,i.e., wherelargeand
smallfaultsarestatistically
linked.
The questionof the fault lengthdistributionat large length
addresses
theproblem
of •aultgrowth
in a brittle
layerof finite
thickness.
In sandexperiments
modellinglithosphere
deformation
[Davy et aL, 1990; Sornetteet aL, 1993], the fault length
distribution
is a powerlaw for lengthslargerthanthethickness
of
the upper brittle layer. However in experiments,faults are
continuousstraightlines while largenaturalfaultswhenmapped
in detail often appearvery segmented
into strandson all scales
[Wallace,1973;Scholz,1990].Therehavebeensomeattemptsto
considerthe existenceof differentscalesas a proofof the fractal
distributionof faults [Scholz, 1990], but it is not clearly
demonstrated
by statisticson strandlengths.There exist other
experiments
whereclay, sand,or limestonelayersare deformed
on top of a basementfault [Tchalenko,1970; Bartlett et al., 1981;
Richard et al., 1989; Baslie, 1990]. The surfacetracesof faults
give a differentand compleximageof the deep fault pattern.In
these experiments,the length of the "Riedel shears"is clearly
theprobabilityp(l,A)dl of observinga segmentof lengthbetween
I and l+dl is
pN(l,A
)=NA+1exp(NA+1l).
(10)
The completestatistics
wouldbe described
by considering
the
probability
forhaving
N breakpoints
PNandbysumming
oneach
possible
valuesof N the product
PN(I,A)*P
N. It is possible
to
simplifythis summationsincethe probabilityfor havingN break
points in a segmentA is maximum for (N+l)*Io--A. Then a
reasonablesimplificationis to consideronly the numberof break
points N such as (N+l)IA=111o. This assumptionhas been
successfullytested by numerical calculations. With this
simplification,the probability of having a segmentof length
between I and l+dl,
p(l,A)=exp(-I/lo) dl/lo,
(11)
becomes
independent
on A. The numberof segments
of lengthI is
obtainedby summingoverall suprafaults
A largerthanI
relatedto the thicknessof the brittle layer. It suggests
that the
distributionof fault strandswithinonelargecrustalfaultcouldbe
length dependent.This assumptionis corroboratedby the
(12)
n(l) dl= F(A) p(l, A) dA dl,
measurements
on stranddistributionof the single San Andreas
fault [Wallace,1973] that haveshownthatthe lengthandnumber
where F(A)dA, the density of suprafaults,is a power law
of fault strandsfollow an exponentialdistribution.Because
A(A/lo)-a. Thusn(/) becomes
strandsmay be only a complexsurfaceexpressionof a rather
continuousprocessat depth,this assumption
is not incompatible
with the fractaldistributionof largecrustalfaultsas suggested
by
(13)
lithosphere
experiments[Davyet al., 1990].
n(l)dl
=•l (lIlo)
-a+l
exp(-//1o)
dl.
This function is an extrapolationof the gammafunction
[Papoulis,1984].The previousanalysisis valid if the numberof
An interestingmodel could be to considerindependently breakpointsN is large,i.e., if the suprafaultlengthA is much
(statisticallyspeaking)the faults (hereaftercalled suprafaultsto largerthanlo. If A is smallerthanlo, the probabilityof the fault
avoidproblemsof terminology)and the strandswhichcompose remainingunbroken(i.e., exp(-A/lo)) becomeslarger than the
the suprafault.Suprafaultsare consideredto follow powerlaw probabilityof havingone or morebreakpoints.The contribution
distributions.
Here,I assumethat strandshavea length-dependent of lengthslargerthanI on n(/) is alsonegligibleandn(1)dlis thus
distribution.An useful assumptionis to considerPoissonian equal
toA (I/lo)
-a+l,where
theexponent-a+l
alsoincludes
the
statisticsfor segmentlengths,postulatingthat the breaks(points probabilityof observinga fault at the Earth surface(seeabove).
separatingsegments)are randomlydistributedwith an average The gamma function is, then, appropriatefor describingthe
densityof l/lo. For a suprafaultof lengthA with N fault breaks, distributionof small and large faults(with a correctingfactor
Gamma Function
DAVY: FREQUENCY-LENGTH
DISTRIBUTIONOFSANANDREASFAULTSYSTEM
l/(a-1) for largefaults)and is worthtestingwith respectto the
faultlengthdistribution.
At first,a simplegammadistribution,
n(/)=Al-a+1exp(-l/lo),
12,147
proposedabove: lognormal,exponentialfunction, power law,
two-/three-dimensional
power law and the gamma function
respectively.
Theevolution
oftheerror
o2 with
Iresisobviously
(14) dependent
on the assumedtheoreticaldistribution(Figure 8a),
betweenerrorestimationand
was tested with respectto the density distributionand the showingthe stronginterdependence
For exponential
functionor simplepower
cumulativedistributionof the San Andreas fault system modelsof distribution.
law,thelarger
isIres,thesmaller
iso2.Forother
distributions,
there
exits
a minimum
of o2 around
Ires=2
kmwitha huge
increase
of o2 forsmaller
Ires. Therobustness
of thefit
(Figure7).
Fit Efficiencyof the GammaFunction
The fit on the densitydistributiongavea=2.3,/o=22 km, and
parametersa and lo are also presented(Figures 8b, 8c) with
to Ires. For the gammadistribution,
a andlo are not
o2=1.80;
andforthedensity
distribution,
a=2.3,/o=24
km,and respect
varying very much for resolutionscale larger than 3 km. In
02=0.65.
Thecomparison
withother
tested
distribution
(Table
1) contrast,theotherdistributionsshownotsucha goodstabilisation
showsthat the gammafunctionis muchmoreefficientfor fitting
thedata.
Inparticular,
theregression
factor
o2 obtained
withthe
cumulativedistributionis 2 timessmallerthanfor the lognorma!.
distribution,4 timessmallerthanfor the exponentialdistribution,
and 6 times smaller than for the two-/three-dimensional
power
law. Moreover, the parametersderivedfrom the densityand the
cumulativedistributions
arecompletelycompatible.
It is possibleto build other models by superimposinga
poissoniansegmentation
on a fractalfault pattern.Thesemodels
!0sethe analyticalsimplicityof the gammafunctionandgenerally
give similarresultswith exponentsa varyingfrom 2.3 to 2.7.
of theirparameters.
Amountof Confidenceon Parameters.
Here is presenteda way for estimatingthe amount of
confidence
on regression
parameters
from the testeddistributions.
The analysisfocusedmainly on the densitydistributionwhich
presentlarger variationsthan the cumulativedistribution.Only
two functionsare discussed:the gammafunctionand the two/three-dimensional
powerlaw (equations
(8) and(9)). The amount
of confidenceon the distributionparametersa and 1o can be
approached
bya mapo2(a,lo)
(Figure
9).Theminimization
was
FURTHER REMARKS ON THE FAULT LENGTH DISTRIBUTION
calculatedon the densitydistributionwith a resolutionscaleof
2 km. For the gammafunction(Figure9, left), a and lo are not
completelyindependentparametersWith respectto the
Resolution Scale
The derivation of any statistical parameters from fault
distributionsrequiresan estimationof the completenessof the
sampling.For the fault length distribution,the only parameter
which can be approachedis the resolutionscale of the fault
sampling,i.e., the smallestlength where the samplingis rather
complete. I first assumedthat this scale was about 2 km but
without objectivearguments.From the digitized map, it clearly
appearsthat faults smallerthan 1 km are not completelysampled.
It is possibleto use an a posterioriargumentwith theoretical
distributions
by varyingthe rangeof lengthusedfor fitting data.
Here, the fit was calculatedfor lengthsvarying from a variable
resolution
lengthIres to the maximumlengthavailable,i.e.,
minimization;there existsa positivecorrelationbetweenthese
two parameters.For this model, the uncertaintiesare about0.3 for
a and5 kmforlo. Similaruncertainty
canbederivedforexponent
a with the two-/three-dimensional
powerlaw (Figure9, fight);
However,in thatcase,the minimization
is not verysensitive
to
the crossoverlength lo and values between7 and 30 km are
possible.
WhyMustGreatCareBe TakenWithCumulative
Regression?
It is postulated
thatthe densitydistribution
variesas a power
law for lengths
smallerthana characteristic
length'
scalelo.
Generally, it is easiest to calculate only the cumulative
65 km (see discussionon data). Results are presentedfor the distribution.However, the cumulativedistributionC(1) should
densitydistributionfits only but are similar for the cumulative neverbehaveas a powerlaw sinceit integrates
faultslargerthan
distribution.
This
fit
was calculated
with
the distributions
n(l)
lo; but, as C(1) is mainlydependenton the lower boundof the
C(1)
ld
3
2
2
1
1
lOo
-'1Gamma
........
•
10 0
10
10
0
lO
lO
Fig.7. A comparison
between
themeasured
faultlength
distribution
(greycurve)
anditsfit bya gamma
function
(equation
(14)in thetext).(Right)the
densitydistribution;(left) the cumulativedistribution.
12,148
DAVY:FREQUENCY-LENGTH
DISTRIBUTION
OFSANANDREAS
FAULTSYSTEM
TABLE 1. Comparison
betweenthefive testeddistributions
Lognormal
a
lo, km
o2
1.16-
Exponential
0.86
3.0-
Two-/Three-Dimensional
Power
Law
Power
law
2.1 - 2.8
7.5
10-
14
Gamma
2.7-
3.4
2.3-
2.3
17-
35
22-
24
1.9- 1.3
4.0- 2.6
4.0- 5.1
2.3- 3.4
1.8- 0.6
3
5
7.5
6
1.5
(reinjection)
Regression
parameters
calculated
for eachmodelof distribution.
Theleft numberresultsfromfit on thede•nsity
distribution;
therightnumberresults
fromfit onthecumulative
distribution.
a isa dimensionless
exponent
andlo,a characteristic
length
scale.
t•z isthevariance
of theminimization.
The
lastrow is the varianceof the minimizationwhenparameters
obtainedfrom the cumulativefit are reinjectedinto the densitydistribution.The reverse
operation(densityparameters
intocumulativefit) givesimilarresults.
a
5
- - a - Exponential
4.5
4
c• -
•2 3.5
3
Power law
Gamma
2.5
2D/3D
2
powerlaw
1.5
0
5
10
15
-'
Resolutionscale (krn)
b
Lognormal
3.5
3
. . rn
- - rn - Power law
2.5
a
2
1.5
1
ß
Gamma
ß
2D/3D
powerlaw
-'
Lognormal
0.5
0
5
10
15
Resolutionscale (krn)
c
25
Exponential
20
15
to
10
2D/3D
5
power law
0
1
0
5
10
15
-
Lognormal
Resolutionscale (krn)
Fig.8.Effects
ofvarying
theresolution
scale
Iresonthefit.Thefitisapplied
from
Irestothemaximum
length
available.
(a)Thevariance
02,(b)the
exponent
a, and(c) thecharacteristic
length
lo arecalculated
at different
Ires forthedensity
distribution.
Thedifferent
curves
represent
fitsbyan
exponential
function
(opentriangle),
a simplepowerlaw (opensquare),
a gamma
model(solidcircles),
a two-/three-dimensional
powerlaw (solid
triangle)or a lognormal
distribution
(solidsquare).
integrall, C(O is expectedto varyasa powerlaw for sufficiently values,predictingfewer large faults. This differencecan be
small lengths.An attemptis made, here, to estimatethe limit expressed
asa ratio[5,where
underwhich C(/) can be considered
as a powerlaw. This limit
mustfix therangeof lengthswherea powerlaw canbe accurately
[5=
,
(15)
measuredwith C(/).
The densitydistributionis assumed
to be n(l)=A/lo*(l/lo)-a for
J
lengthssmallerthanlo anda function
f(/) for largerlengths.In the
limit of smalllengths,a is predicted
to be about1.3-1.7.For any with0<[l.<l.An analytical
valueof [5canbederivedwiththetwogivenlength,f(/) is smallerthan the corresponding
powerlaw /three-dimensional
power
lawwhere
f(/) varies
asl-a-l' In that
•(l)dl
DAVY:
FREQUENCY-LENGTH
DISTRIBUTION
OFSANANDREAS
FAULT
SYSTEM
12,149
go
25
20
20
15
15
10
10
1.0
1.5
2.0
2.5
5
2.0
a
2.5
Gamma distribution
3.0
a
2D/3D power law
Fig.9.Mapoftheminimization
factor
c•2versus
a andlo(see
text)
for(left)
thegamma
function
and
(fight)
thetwo-/three-dimensional
power
law.
corresponding
to t•=l.5*(1-a), i.e., with a relativedifferenceof
50% between the local slope and the expected exponent.
Assuming•=a/(1-a) and a=l.3, one obtainsa lengthlc equalto
case,[5is equalto (a-1)/a.
For lengthssmallerthanlo, C(1)is
/o'0.06, i.e., 1 km for/o=20 km. Even smaller values of lc are
C(l)=.Jlo•,lo
j d/+ (1)dl=-•--•_l(k•oj
- 1+[5).(16)
l
C(1) is, thus,the sum of a power law and constantterm. The
exponentis generally calculatedby measuringthe local slope
d(log(C)/d(log(/)).Accordingto (16), thissloper• is
1
(x=(l-a)*
[ I ,•a-1
'
(17)
As expected,
r• tendsto 1-a whenI decreases.
To fix thelimit
of validity of the relation {x=l-a, I calculatethe length lc
n(1)
obtainedfor exponents
a closeto 1. An illustrationof thispointis
givenin Figure10 wheregammafunctions(solidline) andtheir
asymptoticpowerlaws (dashedlines) are shownfor lengthsare
varying from 0.01 km to 100km. The exponent(1.3) of the
gammafunctionis the one obtainedfrom the minimisation.For
the densitydistribution,the gammafunctiondepartsfrom the
power law for lengthslarger than 10 km; for the cumulative
distribution,for lengthslargerthan 1 km.
This simplecalculationshowsthat gmat care mustbe taken
when analysingcumulativedistributions.
It also showswhy the
cumulativedistributionneveris a powerlaw in the studiedlength
range(2-100 km).
C(1)]
4
4
10
10
3
10
2
10
2
10
1
o
lO
lO
1=10
I
! ! I Illl]
-2
10
10
km
-i ....... • 0 ........ 1 ........ I
10
10
I
lO
0]
-2
lO
I
I I Illll
I
]o
!
-1
! I IIIII
I
]o
0
! ! Illll
I
1
]o
!
! ' IIIII
I
1
Fig.10.Comparison
between
thegamma
function
(solidlines)andtheasymptotic
powerlawat smalllengths
(dashed
lines)(left)for thedensity
distribution
and(righ0for thecumulative
distribution.
12,150
DAVY: FREQUENCY-LENGTH
DISTRIBUTION
OFSANANDREASFAULTSYSTEM
CharacteristicLengthScales
The length scales obtained with mixed distributions(two/three-dimensional
powerlaw and gammafunction)are about1520 km, in accordance with the thickness of the brittle crust. In
Acknowledgements.
I thank L. Knopoff who providedvery
thoroughand constructive
reviewswhich considerably
improved
the final draft. I also thank O. Bour, who digitized the fault
pattern, J. de Bremond d'Ars, D. Gibtrt, M. Benes and
P.R.Cobbold,
for usefuldiscussion.
This work wassupported
by
California,the thicknessof the upperseismogenic
crusthasbeen
theFrenchCNRS (programDBT).
estimatedfrom the depth distributionof earthquakes.It varies
from 15 to 25 km in California for earthquakesrecordedduring
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