JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 106, NO. B9, PAGES 19,391-19,410,SEPTEMBER 10, 2001
Normal fault interaction caused by coseismic and postseismic
stresschanges
Concetta Nostro, Antonio Piersanti, and Massimo Cocco
IstitutoNazionaledi Geofisicae Vulcanologia,Rome, Italy
Abstract. We studycoseismicandpostseismic
stressfieldscausedby a normalfaulting
earthquakein a self-gravitating,stratified,viscoelasticsphericalEarth over distancesfrom a
few to hundredsof kilometers.We investigatethe contributionof postseismicrelaxationon
the inducedCoulombstressfor extensionaltectonicsettingsaccountingfor the effectsof the
Earthstratification.We usea numericalcodebasedon the sphericalself-gravitatingEarth
model developedby Piersanti et al. [ 1995, 1997]. We studyhow postseismicrelaxationcan
modify the stateof stressat thebaseof the seismogenic
layer wherelargeearthquakes
are
believedto nucleate.We compareour resultswith thoseobtainedby meansof a threedimensionaldislocationmodel in an elastichalf-space,which doesnot accountfor the timedependentpostseismic
stresstransfer.The viscoelasticrelaxationprocessmodifiesthe
coseismicstresschangesduringtime periodsfrom severaldecadesto centuries.The
postseismic
stressis generallygreaterthanthe coseismicstresschange.Postseismic
relaxation increases the Coulomb stress near the causative faults and tends to reduce the
stressshadowareas.The temporalevolutionof Coulombstressrevealsthat in additionto the
viscosityvalue,the thicknessof the elasticlayer controlsthe time at which the relaxation
processis completed.A largerthicknessof the elasticlayer yields a fasterrelaxationin the
first few decadesafterthe seismiceventbut smallerpostseismic
stressamplitudesat longer
timescales.One of the most significantresultsof this studyis the extremesensitivityof the
timescalesof theviscoelasticrelaxationto smallchangesin the thicknessanddepthof the
shallowestviscoelasticlayer aswell as in variationof the viscosity.Sucha resultsuggests
that the interpretationof the time evolutionof the postseismicsignalsonly in termsof
viscosityvaluescouldlead to misleadingconclusions.
1. Introduction
The understandingof the processesrelated to repeated
earthquake occurrence along segmented seismogenic
structuresis a major goal of geophysicalinvestigations.Fault
interactionand triggeringcausedby earthquakeruptureshave
been studied in recent years by analyzing static as well as
dynamic stresschanges[Stein et al., 1992; King et al., 1994;
Gomberget al., 1997, 1998; Belardinelli et al., 1999; see also
Harris, 1998; King and Cocco, 2001, and referencestherein].
The basic assumptionthat motivates these studies is that
earthquakesperturbthe stressstateon adjacentfaults and can
promote, as well as delay, subsequentearthquakeruptures
[see Stein, 1999, and referencestherein]. Great earthquakes
can increase as well as decrease the stressover wide areas, in
the latter case creating a stress shadows which delays
subsequent moderate magnitude seismicity [Harris and
Simpson,1993, 1996, 1998]. It is well acceptedthat the stress
redistributionprocessoccursat different spatialand temporal
scales and several process6s are responsible for fault
interaction [Harris, 1998; King and Cocco, 2001]. Static
stress changes can provide information on earthquake
interactionson adjacent faults from several tens of seconds,
when the dynamic stress field has reached the static
Copyright
2001bytheAmerican
Geophysical
Union.
Papernumber2001JB000426.
0148-0227/01/2001 JB000426509.00
configuration[e.g., Harris and Day, 1993; Belardinelli et al.,
1999], to years, when postseismic viscoelastic diffusion
processesbecomerelevant [e.g., Ben-Zion et al., 1993; Pollitz
and Sacks,1997]. However,othertime-dependent
postseismic
effects,suchas aseismicslip or fluid migration,can modify
the induced
static
stress field
also at shorter
timescales
[Hudnut et al., 1989; Ghosh et al., 1992; Noir et al., 1997;
Amelungand King, 1997;Parsonset al., 1999].
Fault interaction through stress transfer is currently
modeled by considering the Coulomb stress changes on
assignedfault planes,as well as on optimally orientedplanes,
causedby slip on a fault [King and Cocco,2001]. These static
stresschangesare often calculatedfrom an elastic dislocation
model in an isotropic and homogeneous medium. The
redistributionof coseismicstressafter a strongearthquakehas
attct shOCKS as
as
been proposed to cxptatn 011-lctUlt
variations of seismicityrates [Rybicki, 1973; Das and Scholz,
1981; Dieterich, 1972, 1994; Stein et al., 1992; Reasenberg
and Simpson,1992; Simpsonand Reasenberg,1994; Toda et
al., 1998]. Severalrecentstudieshave also tried to statistically
correlate the Coulomb stress changes with the aftershock
distribution
and
mechanisms
and
with
the
variation
of
seismicity[Stein, 1999; King and Cocco,2001, and references
therein]. Toda et al. [1998] studied the stress redistribution
causedby the 1995 Kobe (Japan)earthquake.They found an
increase of seismicity rate in areas of calculated stress
increaseand a decreaseof seismicityin regionsof calculated
stressdecrease.Such correlationbetween seismicityrate and
Coulomb stress changes is found by considering the
19,391
19,392
NOSTRO ET AL.' NORMAL
FAULT AND POSTSEISMIC
backgroundseismicity in the 8 years before the Kobe main
shock and the aftershocksin the following 1.5 years. These
results raise the question of what is the contribution of
viscoelasticrelaxation to the induced stressfield during this
time interval.
Most
of the stress interaction
studies
have
focused
on
coseismic stress changes immediately following a seismic
event. Several authorshave applied Coulomb stressanalysis
to historicalearthquakesin Turkey and in the Aegean [Barka,
1996; Stein et al., 1997; Nalbant et al., 1998; Hubert-Ferrari
et al., 2000] as well as in California [Jaume and Sykes,1996;
Deng and Sykes, 1997a, 1997b]. The elasticmodelsallow to
estimate the coseismicstressperturbations,but they do not
include the effects of long-term loading and relaxation of the
lithosphereand asthenosphere
[Harris, 1998]. Only in the last
decade, several authorsbegan to model the mechanicsof the
postseismicstresstransferconsideringviscoelasticprocesses
and the Earth'stime-dependentmaterialproperties.
The first investigations that included the viscoelastic
relaxationprocessin stressinteractionanalyseswere focused
on coupled subduction zones. These studies attempted to
model the completeearthquakecycle [Thatcher and Rundle,
1984; Dmowska et al., 1988; Taylor et al., 1996, 1998], as
well as to explain the delay in time betweenthe initial event
and a triggeredearthquake[Rydelekand Sacks, 1990,'Pollitz
and Sacks, 1995, 1997]. Viscoelastic relaxation has also been
consideredto model the induced stress changes for crustal
earthquakes[Roth, 1988; Ghoshet al., 1992; Ben-Zion et al.,
1993; Freed and Lin, 1998; Kenner and Segall, 1999]. The
main motivation
of these studies is that the relaxation
of the
coseismic stress in the lower part of the crust or the
lithospherecan transferstressto the seismogenicstructuresin
the upper crust. Following this idea, Deng et al. [1999]
investigatedthe stressloadingfrom viscousflow in the lower
crust and the triggering of the aftershocks of the 1994
Northridge thrust earthquake.They found that the decay of
aftershockssequenceexhibits a temporal dependencesimilar
to the evolution of the viscoelasticloading govemed by the
relaxation of the lower crust. Freed and Lin [1998] studied
the time-dependent changes in Coulomb stress following
thrust earthquakesdue to the relaxation of a viscous lower
crust or upper mantle as well as postseismiccreep. Kenner
and Segall [1999] investigated how lower crustal structure
can affect the magnitude, duration, and spatial extent of the
stress shadowing effect for strike-slip faults in northern
Califomia.
Most fault interaction studies considered strike-slip or
thrust earthquakes,but only a few were limited to coseismic
regime investigations concerning normal faulting events.
Namely, Nostro et al. [1997] have studied coseismicstress
changesin an extensionaltectonicsetting.They investigated
the interactions among fault segments belonging to the
southemApenninesseismogenicbelt (Italy), computingthe
stressredistributioncausedby large earthquakes(M'•_6)which
occurredin the last three centuries(see Plate l a). Computing
Coulomb stresschangesfor historicalearthquakes,distributed
during a time interval of centuries,requires us to properly
accountfor the contributionof postseismicrelaxation. In this
studywe investigatethe effect of viscoelasticstressrelaxation
in an extensionaltectonic settingover timescalesof the order
STRESS CHANGES
in assessingthe spatiotemporalevolutionof the postseismic
Coulomb
stress field.
2. Modeling Approach
We use the spherical self-gravitating Earth model
developedby Piersanti et al. [1995, 1997] and Boschi et al.
[2000], which consider a layered medium with viscoelastic
rheology. All the layers have their own physical and
rheologicalparameters.In general,we considerfour layers.
The top is a purely elastic crust and overlies two Maxwell
viscoelasticlayers,while the bottomis perfectlyinviscid.This
model is based on the equationsgoverning the quasi-static
deformationsof an incompressible,self-gravitatingMaxwell
body with sphericalsymmetry,which is initially in a stateof
hydrostatic equilibrium and takes advantage of the
correspondence
principleof linearviscoelasticity[e.g., Fung,
1965]. The basic equationis the momentumconservationfor
a Maxwell body:
-p0V•JtV(ll.p0goer)
+V.T: p0r,
(1)
where the subscriptzero and 1 refer to equilibrium and
incremental quantities, respectively, and the tilde denote
Laplace-transformedvariables. In (1), p0 and g0 are the
referencedensityand gravityfield, •J•is the perturbationto
the geopotential induced by deformation, n is the
displacement
field, er is the unit vector in the outer radial
direction,
andT denotes
incremental
stress
tensor.
With P0'1•
we indicatethe body force equivalentto a point dislocation
[Smylieand Mansinha,1971]. Owing to incompressibility,
the
displacementfield, n, is subjectto the constraint
v.n=0,
(2)
whereasthe incrementalpart of the gravitationalpotential
obeysthe Laplaceequation
V2•j,=0
(3)
both inside and outside the Earth. The incremental stress
tensor is related to the tensor of infinitesimal deformations E
by the constitutive
equationof an incompressible
viscoelastic
bodywith linearrheology
T = 2;tl•+ ,Di
I,
(4)
where ;t is the Laplace-transformedshear modulus for a
Maxwellbody,,•l is theincremental
pressure
fieldandI is
the identitymatrix.
The methodof solutionof theseequationsis basedon a
spectralapproach,
whichrequiresdecomposition
of the field
variables(displacementsand stresses)on the basis of the
sphericalharmonicfunctions.As shownby Piersantiet al.
[1995], this procedure allows us to obtain two linear
nonhomogenous
systems
of first-order
differentialequations
d
_ y•n= S/y•n
+ t•n
dr
d
m
m
- Z•n'--r/Zl q-gl ,
dr
(5a)
(5b)
of 102yearsandspatialscalesof •4x102km (the same where the poloidal equations(5a) are completelydecoupled
timescalesand spatial scalesused by Nostro et al. [1997]).
from the toroidal ones (5b). The poloidal six-component
We will point out the dominantrole of the Earth stratification vector y'•
and the toroidaltwo-componentvector z•' contain
!
NOSTROET AL.' NORMALFAULTAND POSTSEISMIC
STRESSCHANGES
19,393
ß
,,
':..:.t•_r•.',
' -, ß
• .•.:
%
30'••
.
. . .ß
. • . ßß...
•
',
•..
'
-""
•
'
.-
'
.
!
.
30'
ß
.
30'
ao'
•ao
-1.0
ao'
•4ø
-0.5
ao'
•6: "'i•:"'""
....... •'' .... ' ao'
....... •70
' .... •:
0.0
0.5
1.0
CFFchanges
(bars)
Plate 1. Coseismic
Coulombstresschanges
causedby a sequence
of subsequent
normalfaultingearthquakes
mapped
atdepths
of (a)7 kmand(b) 17kmprojected
ona normal
faultstriking
310øalongtheApennines
and
dipping
60øtoward
theNE.Thedepths
of themapsin Platesl a andlb nearlycorrespond
tothemiddleandto
thebaseof theseismogenic
layer,respectively.
Theregionalstress
hasnotbeenincluded
in thesecalculations.
Thesource
parameters
of thesehistorical
earthquakes
arelistedbyNostroet al. [ 1997,1998].It isemphasized
thatnormalfaultingearthquakes
increase
thestaticstress
bothin thestrikedirection
andat thebaseof the
seismogenic
layer(whereearthquakes
arebelievedto nucleate).
19,394
NOSTRO ET AL.: NORMAL
FAULT AND POSTSEISMIC
STRESS CHANGES
information regarding postseismic displacements u,
incrementalstressesT, and gravity potential 0. The elements
of the matricesS•and T•, which dependon the mechanicaland
rheologicalparametersdescribingthe Earth model, are listed
by Piersanti et al. [1995]. The sourcelocation and geometry
displacementat depth dependson the distancebetweenthe
sourceand the samplingdepth.This is in agreementwith the
finding of Sun and Okubo [1993], who studiedthe potential
and gravity changescausedby dislocationsin spherically
symmetric,elastic,Earth model. A satisfactoryconvergence
areaccounted
for on the sourcetermstt and gt.
Thesesetsof equationsare solvedfor eachharmonicorder
of the solution for a 5 km distance between the source and the
I and degree rn by means of propagatormatrix techniques,
imposing appropriate boundary conditions at the internal
interfaces(continuity of all of the field variables) and at the
free surface (vanishing vertical traction). Because of the
analyticalstructureof the propagatormatricesinvolvedin the
solutionof (Sa) and (Sb) [e.g., Spada, 1992], the postseismic
displacementfield u can be inverted from the Laplace to the
time domainby meansof simplemethods(seePiersanti et al.
[1995] andBoschiet al. [2000] for details).
Assuming a Heavyside time history for the dislocationof
arbitrary harmonic degree I and order m, the poloidal and
toroidal time-dependent generic observable (e.g.,
displacementand stresscomponents,gravitationalfield) takes
the form
M
xv(t)=
.gp
+
)fp
e-'
t=l
sampling depth in the lithosphere is now ensured.
Nevertheless,the resultsat depthstill show someoscillations
due to an incomplete convergenceaffecting the spherical
harmonicdecomposition.
We will show resultsfor the spaceand time evolutionof
the stressfield generatedby a normal fault. The fault plane
has been modeled by a line source represented by a
distributionof point sources.Previous studies[Piersanti et
al., 1997; Nostro et al., 1999] have demonstratedthat this
approximation works excellently for epicentral distances
exceedinga few kilometers.
We compute the stress tensor by a numerical
differentiationof the coseismicand postseismicdisplacements
and by applying the Hooke's law for an incompressible,
isotropic linear elastic body. The Coulomb stress, Ac•f is
thereforecomputedas follows:
(6)
Acy•= A z' + /•' Acy•
(7)
N
xr (t)=
AT
+
AT • T
e '"*•- 1 '
[King et al., 1994; Harris, 1998; King and Cocco, 2001],
where A'•is the fault shear stresschangein the direction of
slip on the assignedfault plane, A(Jn is the outward normal
stresschangeon the fault (positive for extension),and [t' is
the apparentfrictioncoefficientthat is definedby
t=I
As seen in (6), both poloidal (P) and toroidal (T)
postseismicobservablefields for a viscoelasticEarth may be
decomposedin the sum of two contributions.The first one,
described
by xe© andXr©, is relatedto the coseismic
instantaneous(t=0) response of the Earth to the seismic
dislocation. Its value is completely unaffected by mantle
viscoelastic rheology and only depends upon the elastic
parametersof the model. The second contribution, which
accountsfor delayedviscoelasticresponse,is characterizedby
(8)
where B (Skempton coefficient) takes into account the
modificationsof the effectivenormalstresscausedby pore
fluid pressure
[seeHarris, 1998].The problemof findingthe
theviscous
amplitudes
(xe(øandxr% andby therelaxation best expressionfor the effectivefriction coefficient[Harris,
times
(re© andVr%,which
provide
therelaxation
spectrum
of 1998; Cocco and Rice, 1999; Beeler et al., 2000] is not
the model under study. The magnitudeof poloidal and relevant for the discussionpresentedin this paper. We
toroidal relaxation times and their number (M and N, thereforeuse (7) as donecurrentlyin literaturewithoutany
respectively)are completelyindependentof the natureof the further comment.
The adoptionof a sphericalgeometryapproach
mightseem
sourceandare only affectedby the mechanical
profileof the
Earth.For a detailedanalysisof the relaxationspectraand not justified by the range of distances involved in our
modal repartition of postseismicdeformation,we refer to analysis;however,previousworkshaveshownthat long-term
Spada [1992] and Piersanti et al. [1995]. The total deformationcan be affected by Earth sphericity also at
displacement
field u(r,t) at a givenpointr of the shallowest distancesof few hundredsof kilometers [Antonioli et al.,
layer of the Earth may be retrievedby summationover the 1998; Nostro et al., 1999]. In particular,our model exhibits
sphericalharmonicfunctionsaccordingto the procedure some peculiar advantages:For instance,it allows us to take
describedby Piersanti et al. [1995, 1997]. The solutionfor into account in a self-consistentway the effects of selfsource and/or observer located within the viscoelastic lower
gravitation.Many seismological
applicationsdo not dedicate
layers requires a further developmentof the modeling muchattentionto theseeffects.This is probablydue to the
approach,
whichis not discussed
in thispaper[seeBoschiet fact that they considerthe short-termdynamicand/or static
al., 2000].
part of the deformation,for whichthe gravitationaleffectsare
regimethe
Thenumerical
codedeveloped
by Piersantiet al. [1995]to indeedsmall.On the contrary,in the postseismic
compute
theradialcoefficients
of thespherical
decompositiongravitationaleffects play a major role in determiningthe
field [Piersantiet al., 1997;Nostroet al., 1999].
hasbeenimprovedby Nostroet al. [ 1999]in orderto increase deformation
the maximumdegreeof summationup to /=9000, while in Alternative modeling approachessimulatethe gravitational
previous analyses it was limited to /=1000. Nostro et al.
loading by applying some external forces at the internal
[1999] have found that the rate of convergenceof the boundaries,but this proceduredoes not always guarantee
harmonicdecomposition
of the displacement
at the surface goodresults,especiallyfor compressiblematerials[e.g., C.
dependson the sourcedepth.In thiswork we find thattherate Giunchi,personalcommunication,
2000; Giunchiand Spada,
of convergenceof the harmonic decompositionof the 2000].
19,395
NOSTRO
ETAL.:NORMALFAULTANDPOSTSEISMIC
•STRESS
CHANGES
3. Postseismic
Stress Evolution
fault (rake 270ø) dipping60ø NE and striking310ø (parallel to
the Apennines).It canbe consideredas a typical examplefor
a normal faulting earthquakein the southernApennines[see
for a Normal
Fault
Nostroet al. [ 1997]computedthe coseismicstresschanges
causedby several historical earthquakesin the southern
Apennines
(Italy) usinga three-dimensional
(3-D) dislocation
modelin an elastichalf-space.
Theydiscussed
the sequence
of
11 earthquakes
thatrupturednormalfaultsorientedalongthe
Apenninesin the last three centuries(1688-1990) and
concluded that each event increased the static stress on the
adjacentfaultspromotingsubsequent
failures.Plate l a shows
the Coulombstresschangesat the middleof the seismogenic
layer (7 km depth) causedby a sequenceof historical
earthquakes
(more completethanthat usedby Nostro et al.
[1997]) and without considering the regional stress.
Moreover, in Plate lb we show the same calculation at a
depthof 17 km, whichnearlycorresponds
to the baseof the
seismogenic
layer.Theseresultspointoutthatnormalfaulting
earthquakes
increasethe staticstressboth in the directionof
the fault strike(that is, alongthe Apennines)and,as expected,
at the baseof the seismogeniclayer.
Nostro et al., 1997, and references therein]. As a reference
case for the elastic solution, Plate 2 shows the coseismic
Coulombstresschangescausedby a normal fault 35 km long
and 15 km wide, dipping60ø northeastward,
mappedat 7 km
(midfaultdepth,Plate 2a) and 17 km depth(at the baseof the
seismogeniclayer, Plate 2b). For thesecalculationswe used
the elastic dislocationequationsproposedby Okada [1992]
and the numericalcodedevelopedby Nostro et al. [ 1997]. We
will use this configuration to compute coseismic and
postseismicstresschangesresulting from the viscoelastic
code previouslydiscussed(equation(6)). The static stress
changesshown in Plate 2 representa referencecase for the
elastic solution. We therefore study the space and time
evolutionof the stressfield for a normal fault in a spherical,
self-gravitating, viscoelastic, stratified Earth. In the
viscoelasticmodel we assumea line source35 km long and
locatedat a depthof 12 km. We computethe Coulombstress
changesfor secondarynormal faults parallel to the master
fault; we map the stress changes at a depth of 17 km,
thereforebelow the downdipextensionof the extendedfault.
The seismicmoment is that of the 1980 Irpinia earthquake
Although these results suggestthat interaction exists
betweennormalfaultingevents,the time separationbetween
subsequent
earthquakes
(severaldecades)
requiresto properly
accountfor the contributionof postseismic
relaxation.In this (M0=2.5
10•gNm).Thelinesource
condition
(l-D) represents
studywe investigate
the effectof viscoelastic
stressrelaxation an extremely efficient numerical approximation for what
causedby normalfaults over timescalesof severalcenturies concern the CPU time; moreover, previous analyses have
and spatialscalesof hundredsof kilometers,analyzingthe shown that it gives very good results for static and
effectsof the Earth stratificationon Coulombstresschanges.
postseismicdisplacements
also near the causativefault (i.e.,
In our calculationswe choosethe sourceparametersfor within two fault lengths[see Piersanti et al., 1997; Nostro et
computingcoseismicand postseismic
stresschangesto be al., 1999]). In the present applicationsthe line sourcehas
thoseproposedfor the 1980 Irpinia earthquake:A normal been discretizedby 31 point sourcesalignedalong the strike
200 -
b)
2oo!
100
100
ß
e-
%.
0-
-100
-100
! ,,,
-200
-100
•)
100
20•)
-200
distance
(kin)
-0.10
-100
0
100
200
d,stance
(kin)
-0.0õ
0.00
0.0õ
0.10
CFF changes(bars)
Plate2. Coseismic
Coulomb
stress
changes
(ACFF)at depths
of (a) 7 km and(b) 17km(asin Plate1) caused
bya normal
fault(striking
310øN anddipping
60øNE,rakeof 270ø)computed
bymeans
of a purely
elastic
uniform
flatmodel[Okada,1992;Nostroet al., 1997]andprojected
onfaultshavingthesamemechanism
of
the master fault.
19,396
NOSTRO ET AL.: NORMAL FAULT AND POSTSEISMIC STRESS CHANGES
Table1. Incompressible
Four-Layer
ModelParameters
forModl
Layer
Thickness,
Density,
km
kgm'3
Rigidity,
10•øPa
Viscosity,
1021Pas
Model B
Elasticlithosphere
Viscoelastic
asthenosphere
80
200
3115
3400
5.60
6.75
0.01
Viscoelasticmantle
Fluid core
2620
3471
4695
10931
18.60
0.00
1.00
0.00
Elasticlithosphere
Viscoelastic
asthenosphere
80
200
2520
2520
3.00
3.00
0.01
Viscoelasticmantle
Fluid core
2620
3471
2520
10931
3.00
0.00
1.00
0.00
Model A
Model 0
Elastic
lithosphere
Viscoelastic
asthenosphere
Viscoelasticmantle
Fluid core
100
150
3193
3385
5.86
6.60
0.01
2650
3471
4662
10931
18.30
0.00
1.00
0.00
of the fault (this correspondsto a linear fault densityof 0.9
sources/km).
Sincethe time-dependentpostseismicstressfield depends
on the assumedEarth model, we will computecoseismicand
postseismic stress changes using several different Earth
stratifications
both
for
the
elastic
and
the
viscoelastic
The second class of Earth models is characterizedby an
elastic crust overlying two Maxwell viscoelasticlayers (see
Table 2 and Plate 3b). In these models the thicknessof the
elastic layer is 35 km, thinner than in Modl, since only the
crust is assumedto be elastic. In the following, we refer to
this classof modelsasMod2X (X=P, E in Table 2).
The
parameters.We will discussthe main featuresof the adopted
Earth models in section 4.
last class of models
considered
here consists
of an
elasticuppercrust,whosethicknessrangesbetween18 and 28
km, and a viscoelastic lower crust and mantle (see Table 3
4. Adopted Earth Models
To compute postseismic stress perturbations, we use
severalmodelsconsistingof four layers with different elastic
and/or viscoelasticparameters.Extremely deep layersplay a
minor role at such short distances [Piersanti et al., 1995,
1997; Pollitz, 1992; Pollitz et al., 1998], therefore in our
analysisthe inner layer (an inviscid core) is always the same
for all the simulations.The viscoelasticlower mantle always
extendsdown to a depthof 2900 km (see Tables 1, 2, and 3).
To compare the coseismic and postseismicstresschanges
resulting from the viscoelasticmodel with those from the
homogeneouselasticmodel (Plate 2), we use a classof Earth
models characterizedby an elastic lithosphere overlying a
Maxwell viscoelasticasthenosphere(see Table 1 and Plate
3a). The thicknessof the elastic lithosphererangesbetween
80 and 100 km. In the rest of the paper we refer to this class
of modelsasModlX (X=A, B, O in Table 1).
and Plate 3c). Among all the models adoptedin this study,
this class, Mod3, is characterized by the shallowest
viscoelasticlayer.
The parametersadoptedin the differentclassof modelsare
described in Tables 1, 2, and 3. The viscosity for the
shallower
viscoelastic
layeriskeptfixedto 1019
Pas,andthat
of thedeeper
oneis keptfixedat 1021
Pas [e.g.,Piersanti,
1999],exceptwhereotherwiseindicated.
5. Modeling Results
In this sectionwe presentthe resultsof a detailedanalysisof
the spatial patterns of coseismic and postseismic stress
perturbationsdue to a normal fault, using the different Earth
modelsdescribedin section4. We separatelyshowthe coseismic
andthe additionalpostseismic
stresschangescomputedfrom (6)
as well as the total stress perturbation (coseismic and
postseismic)by using the seminumericalapproachproposedby
Table 2. Incompressible
PREM-AveragedFour-LayerModelParameters
for Mod2
Layer
Thickness,
Density,
km
kgm'3
Rigidity,
101øPa
Viscosity,
1021
Pas
Model P
Elastic crust
35
2855
625
3592
8.30
Viscoelastic lower mantle
2240
4953
21.10
1.00
Fluid core
3471
10931
0.00
0.00
Viscoelasticuppermantle
4.46
0.01
Model E
Elastic crust
35
2855
Viscoelastic
uppermantle
45
3379
4.46
6.75
Viscoelastic lower mantle
2820
4562
17.40
1.00
Fluid core
3471
10931
0.00
0.00
0.01
NOSTRO ET AL.' NORMAL FAULT AND POSTSEISMIC STRESS CHANGES
19,397
Table 3. IncompressiblePREM-Averaged Four-Layer Model Parametersfor Mod3
Layer
Thickness,
CrustalThickness,
Density,
km
km
kgm'3
Rigidity,
10!øPa
Viscosity,
102•
Pas
Model L
Elastic crust
18
Viscoelastic
lower crust
Viscoelastic
mantle
Fluid core
29
5.90
2871
4526
17.00
1.00
3471
10931
0.00
0.00
2520
3.00
18
Viscoelastic
lower crust
Viscoelastic
mantle
Fluid core
17
3199
5.90
2865
4526
17.00
1.00
3471
10931
0.00
0.00
24
Viscoelastic
lower crust
Viscoelastic
mantle
Fluid core
0.01
C
35
Model
Elastic crust
3.00
3199
Model
Elastic crust
2520
11
0.01
G
35
2520
3.00
11
3199
5.90
2865
4526
17.00
1.00
3471
10931
0.00
0.00
2855
4.50
•o
3381
6.80
0.01
2861
4526
17.00
1.00
3471
10931
0.00
0.00
2520
3.00
0.01
Model M
Elastic crust
28
Viscoelastic
lower crust
Viscoelastic
mantle
Fluid core
39
11
Model I
Elastic crust
18
Viscoelastic
lower crust
Viscoelastic
mantle
Fluid core
33
15
3199
5.90
2867
4526
17.00
0.01
1.00
3471
10931
0.00
0.00
Model F
Elastic crust
24
Viscoelastic
lower crust
Viscoelastic
mantle
Fluid core
2520
3.00
56
80
3309
6.40
2820
4562
17.00
1.00
3471
10931
0.00
0.00
2520
3.00
0.01
Model N a
Elastic crust
24
Viscoelastic lower crust
15
3199
5.90
2861
4526
17.00
1.00
3471
10931
0.00
0.00
2855
4.50
,o
3380
6.80
0.01
2857
4536
17.20
1.00
3471
10931
0.00
0.00
Viscoelastic
mantle
Fluid core
39
0.01 a
Model Q
Elastic crust
28
Viscoelastic
lower crust
Viscoelastic
mantle
Fluid core
43
15
Model H
Elastic crust
20
Viscoelastic
lower crust
Viscoelastic
mantle
Fluid core
2520
3.00
15
35
3199
5.90
2865
4526
17.00
1.00
3471
10931
0.00
0.00
0.01
with*.... other
Piersanti et al. [1995]. We first discussthe spatialpatternsof
coseismicand postseismic
stresschangesusingthe samespatial
scaleadoptedfor the elasticcalculationsshownin Plate 2. In this
section,we will showthe spatialpatternsof postseismicstress
changesdue to the viscoelasticrelaxationof the ductile layers,
after 100 years, not consideringother potential causesof stress
loading,suchas aseismicafterslipand tectonicloading.A time
interval of 100 years can be consideredas representativeof the
subsequent
occurrenceof normal faulting earthquakesalong the
Apennines[see Nostro et al., 1997]. In section 6 we discussin
greaterdetail the temporalevolution of the postseismicstress
field.
5.1. Simulationsfor an Elastic Lithosphere
Here we consider the coseismic and postseismicstress
fields for a normal fault in a spherical, self-gravitating,
viscoelastic, stratified Earth model, whose lithosphere
consists of a 80-km-thick elastic plate underlain by a
•9
viscoelasticasthenosphere
with
viscosity
of 10 Pa s and a
ß
ß
ß
.21
viscoelasticmantle whosewscos•ty•s 10 Pa s (see Table 1).
Among the different models considered,Plate 4 shows the
calculations resulting from two Earth models: ModlB has
four layerswith differentviscosities,densities,and rigidities,
whereasModlA has uniform elastic parametersthrough all
19,398
NOSTRO ET AL.' NORMAL FAULT AND POSTSEISMIC STRESS CHANGES
a)
Lithosphere
Asthenosphere
Mantle
Crust
'
Not to scale
Modl
b)
Crust
Crust
'
/(nvis½i
Core
Not to scale
To scale
Mod2
c)
Crust
Lower
crust
Mantle
Core
Not to scale
To scale
Mod3
Plate3. Earthmodels
usedinthisstudy.
All themodels
consist
of fourlayers
withaninviscid
core.(a)Models
characterized
by an elasticlithosphere
andviscoelastic
asthenosphere
andmantle(Modl).(b) Models
characterized
byanelastic
crustanda viscoelastic
mantle
(Mod2).(c) Models
characterized
by anelastic
upper
crust
andviscoelastic
lower
crust
andmantle.
Thedifferent
viscosities,
densities,
andrigidities
foreach
modelarelistedin Tables1,2, and3, respectively.
In all themodels
considered
in thisstudy
thedensity
and
therigidity
areobtained
through
avolume
average
ofPREMcorresponding
values
[Dziewonski
andAnderson,
1981] (exceptModlA).
NOSTRO ET AL.' NORMAL
FAULT AND POSTSEISMIC
STRESS CHANGES
o
'8
o
o
-8
!
-o
o
o
o
o
t
o
i
o
o
o
d:•
6
(tU>l):a:•u04.s•p
19,399
19,400
NOSTRO
ET AL.: NORMAL
FAULT
AND POSTSEISMIC
STRESS CHANGES
the layers(Table 1). The relevantthicknessof the elasticlayer
(the whole lithosphere)adoptedin theseEarth modelsallows
km-thick lithosphere),the coseismicstresschangesare quite
similar, while the postseismicstresschangesare sensibly
a comparisonwith the elasticcalculationsshownin Plate 2.
different. Namely, in the case of a thick lithospherethe
As expected, the pattern of coseismic Coulomb stress postseismicadditionalstressshowsan elliptic shape,while
changesresultingfrom the two methods,and shownin Plates for a thin lithospherethe postseismicstressexhibitsevident
2 and 4, is very similar. We recall that in Plates 2 and 4 we off-fault lobes typical of the coseismicpattern. This means
plot the Coulomb stress changes at the base of the that the depth of the first viscoelastic interface mostly
seismogeniclayer, that is, below the seismogenic
faults.The contributesto modificationof the postseismicevolutionof the
additionalpostseismicCoulomb stresschanges,computed stresstransfer at the base of the seismogeniclayer [Cohen,
after 1O0years, are quite different from the coseismicones. 1980, 1984; Matsu 'ura et al., 1981].
Plate 4 showsthat the additionalpostseismicstresschanges
causedby a normal faulting earthquakeincreasein the fault 5.3. Simulations for a Viscoelastic Lower Crust
strikedirection,whereasthe coseismicstresschangepattern
In this sectionwe investigatethe effects on the induced
showsonly positive off-fault lobes. The off-fault stresslobes Coulomb stressfield of including a viscoelasticlower crust.
to the northeastof the fault are larger than the southwestern This implies that in these casesthe viscoelasticlayer is
onesbecausethe fault dipstowardthe northeast,althoughthe shallower than in the simulationspreviously shown. As a
symmetryof the spatialpatternof the additionalpostseismic consequence,we expectto have even more evident effects on
Coulomb stressdoesnot changeconsiderablywith the dip thepostseismic
evolutionof the stressperturbation.
angle of causative normal fault. The largest additional
We considerdifferentmodelshavingan elasticuppercrust
postseismicCoulomb stresschangesare located below the whosethicknessrangesbetween18 and 28 km overlyinga
causativefault and are elongatedin the strike direction.As a Maxwell viscoelasticlower crust, whose thicknessranges
consequence,the total postseismic stress changes are between 11 and 17 km (see Table 3). The viscosity of the
increased in the direction of the fault strike.
lowercrustis l019Pa s andthatof the viscoelastic
mantleis
102t Pa s. We list the main featuresof the differentEarth
considered,
100yearsof viscoelastic
relaxationis not enough modelsadoptedin this case in Table 3, and the resulting
to recoverthe stressshadowingeffect(exceptalongthe strike coseismicand postseismic(after 100 years) Coulomb stress
direction). The comparisonbetween the stress patterns changesare shown in Plate 6 for the first four models. As in
resultingfrom the two different Earth models (ModlA and the previouscalculations,the stresschangesare mappedat a
These examples show that for the Earth models here
ModlB in Plate 4) shows that both the coseismicand the
postseismicstresschangesare quite similar. This is an
expectedresultconfirmingthatvaryingthe elasticparameters
only changesthe magnitudeof viscoelasticrelaxation(see
Plate 4) that reloadsthe elastic layer [Cohen, 1980, 1984;
Rundle, 1982].
These results also confirm that under appropriate
conditions,our viscoelasticmodel yields coseismicstress
changessimilar to thoseof the purely elasticmodels[see
Nostro et al., 1999] and that the depth of the viscoelastic
layersbelowthe elasticone,ratherthanthe depthvariationof
the elasticparameters,
is more importantin controllingthe
viscoelastic relaxation process [e.g. Cohen, 1980; 1984;
Nostro et al., 1999, and referencestherein]. We will further
investigatethis in section5.2.
5.2. Simulations
for an Elastic Crust
We have computedcoseismicand postseismicCoulomb
stresschangesfor different Earth modelsconsistingof an
elasticcrust(whosethicknessis 35 km) overlyinga Maxwell
viscoelasticmantle whoselayers have different thicknesses.
Table2 liststhe parameters
for two Earthmodelsconsisting
respectivelyof a viscoelasticupper mantle (625-km-thick,
Mod2P)anda thinlow viscositylayer(45-km-thick,Mod2E).
depthof 17 km.
The coseismic stress changes for a normal faulting
earthquakeexhibit the four characteristiclobes of Coulomb
stress increase and two major lobes of Coulomb stress
decreaseyielding very similar patterns for all the models
consideredin this study.We first comparetwo modelswhich
have the same elastic upper crust (18 km thick) but different
thicknessfor the viscoelasticlower crust(Mod3C and Mod3L
in Table 3). The resultingpatternof the additionalpostseismic
stressafter 100 years is quite similar to that of the coseismic
stress,but the magnitudeand spatialextentof the postseismic
off-fault lobes (both for stressincreaseand shadowing)are
greater than those of the coseismic changes. This spatial
similarity decreases or disappears for the other models
(Mod3G and Mod3M) shownin Plate 6. Mod3G and Mod3M
are characterizedby the same thicknessof the viscoelastic
lower crust (11 km as in Mod3L) but different thicknessof
the elasticupper crust, in suchway that Mod3C and Mod3G
have the same total crustal thickness but in different
elastic
and viscoelastic portions. These results have important
implicationsfor those applicationsconcerningnormal fault
interaction
where
most
of the Coulomb
stress reduction
(shadowing effect) is located off-fault and oriented
perpendicularlyto the fault direction(see Plate 6 and Nostro
For both the models the viscoelastic lower mantle extends to
et al. [1997]). This emphasizesthat the stress shadowing
2900 km (seeTable2). We showtheresultingCoulombstress effectsare time-dependentfunctionsof the layer's thickness
andthe rheologyof the lithosphere.
patterns calculated at a depth of 17 km in Plate 5. We
consider these models in order to discuss the effect of the
The resultsshown in Plate 6 suggestthat the thicknessof
thicknessof the shallowestviscoelasticlayer within the the elasticcrusthas a greatinfluenceon the postseismicstress
mantle.
distribution.Looking at the additional postseismicCoulomb
Plate 5 showsthat with the crustthicknessfixed to 35 km, functionafter 100 years,we can concludethat the viscoelastic
increasing
the thickness
of the firstviscoelastic
layerfrom45 relaxation tends to reduce the off-fault coseismic shadow
to 625 km, has a moderateimpact on the Coulomb stress zones,althoughfor models G and M appeartwo small stress
changespatternafter 100 years.With respectto Modl (80- shadowareasparallelto the fault. We attributethis sensitivity
NOSTRO
ET AL.' NORMAL
FAULT
AND POSTSEISMIC
STRESS CHANGES
I.U
n_
-8
'
i
i
i
19,401
19,402
NOSTRO ET AL.' NORMAL FAULT AND POSTSEISMIC STRESS CHANGES
AdditionalpostseismicCFF
after 100 years
Coseismic CFF
PostseismicCFF after 100 years
200
100
Mod3L
200
/
I00
'•
f
";b
/_.
-o.•o
1'-ø'øød io.,o'5
1
003,'• 1'00•1'00
--"
-0.03'
'
-O.03
)
/,
!
•
i
- , -
! ....
ß
-0.10
•. •
/ o.00
\
J, ,.•'!•ø'.., .• ;.....•... ,\..
i
•,.. Mod3C
200
100
Mod3G
-200*
20O
100
Mod3M
-100
d•stance(kin)
distance(kin)
-0.10
-0.05
0.00
0.05
O. tO
CFFchanges(•rs)
Plate 6. Same calculationsof Plate 4 but for an Earth model consistingof an elastic upper crest overlying a
Maxwell viscoelasticlower crust (whose thicknessrangesbetween 11 and 17 km) with a viscosityof about
10•9Pas (seePlate3candTable3). Mod3LandMod3Chavethesamethickness
of thefirstelastic
layer(the
uppercrest).Mod3L,Mod3G,andMod3M havethe samethickness
of theviscoelastic
lowercrest.The whole
thicknessof the crestis the same(35 km) for Mod3C andMod3G
NOSTRO ET AL.: NORMAL FAULT AND POSTSEISMIC STRESS CHANGES
to the fact that by changingthe thicknessof the elastic layer,
we modify the distance between the source and the first
elastic-viscoelastic
interface.This yields evident variationsto
the spatialpatternof thepostseismic
stress.
We
first
consider
the models
19,403
used in the calculations
shownin Plates4 and 5, whoseparametersare listed in Tables
1 and 2. The resultsof thesecalculationsare shownin Figure
1. We remark that ModlA and ModlB (Table 1 and Plate 4)
are characterizedby a completely elastic lithosphere,while
Mod2E and Mod2P are characterizedonly by an elastic crust.
6. Temporal StressEvolution:
Thus all thesemodelshave two viscoelasticlayerswithin the
Effects of Rheological Stratification
mantle, but they differ in the depth of the elastic-viscoelastic
In section 5 we discussedthe spatial dependenceof the interface. Figure 1 shows that the magnitude of the
viscoelasticrelaxation of the Coulomb stresschangescaused postseismicstressdependson the thicknessof the elasticlayer
by a normalfault for differentEarth models.The postseismic and, consequently, on the depth of the first viscoelastic
evolutionhasbeenpresentedonly at a fixed time (100 years) interface:The viscoelasticrelaxationyields larger postseismic
after the earthquakein orderto facilitatethe comparisonwith Coulomb stressfor shallowerviscoelasticlayers. In Figure 1
the coseismic Coulomb stress changes. In this section we the smallest viscoelasticstressperturbation is reached for a
100-km-thickelasticlithosphere(Mod 1O).
focus our discussion on the temporal evolution of the
viscoelastic relaxation.
The Earth models consistingof a fully elastic lithosphere
The previous results have shown that in general, we yield quite similar temporal evolution of the postseismic
observethat the relaxation of the viscoelasticlayers reloads stresschanges(ModlA and ModlB). A similar conclusionis
the area around and below the coseismic fault for all the
also valid for the models characterizedby a fully elastic crust
considered Earth models. However, the time-dependent and a viscoelasticmantle (Mod2E and Mod2P). This is also
postseismic relaxation modifies the distribution of the evident looking at the behaviorin the first 300 years, shown
Coulomb stresschanges(of fault-end and off-fault lobes) in in the Figure 1 inset. Moreover, the thicknessof the first
differentways dependingon the adoptedEarth model.
viscoelasticlayer controlsthe rate of relaxation:The thicker
We now compute the temporal evolution of Coulomb and shallowerviscoelasticlayer adoptedin Mod2P yields the
stresschangesin a fixed point, locatedat (55, -10) km in the fastestrelaxationwithin the uppercrust(rememberthat for all
maps shown before (which always has positive values of the models consideredin Figure 1 the viscosity value of the
Coulombstresschanges,seethe starin Plates4, 5, and 6), and first viscoelastic
layeris fixedto 1019
Pa s). This is in
we analyze its behavior during 1000 years after the agreement with the findings of Soldati et al. [1998]
earthquake. Our goal is to understand the effects of the concerningthepostseismic
gravitationalperturbations.
thicknessof the shallowerlayersand the distancebetweenthe
More interesting behaviors are obtained when a
sourcedepth and the first elastic-viscoelasticinterfaceas well viscoelasticlower crustis considered(see the models listed in
asthe rheology(viscosityandrigidity) of the adoptedmodels.
Table 3 and shown in Plate 6). Figure 2a shows the
0.30
0.25
0.20
0.15
0
100
200
300
400
500
600
700
800
900
Years after the earthquake
Figure 1. Temporalevolutionof Coulombstresschangesfor 1000 yearsafter the earthquakein the point
x=+55 km,y=-10 km at 17 km depth(seestarin the mapsshownin Plates4 and5) for five differenttypesof
Earthmodels(seeTables1 and2). ModlO is characterized
by the sameelasticandviscoelastic
parameters
of
Mod 1B, but the thicknessof'the first two layersis different:A lithosphereof 100 km and an asthenosphere
of
150 km.
1000
19,404
NOSTRO
ET AL.' NORMAL
FAULT
AND POSTSEISMIC
STRESS CHANGES
0.8
0.6
0.2
0.0
0
tO0
200
300
400
500
600
700
800
900
1000
0.8 I
0.6
0.2
0.0'
0
100
200
300
400
500
600
700
800
900
1000
Yearsafter the earthquake
Figure 2. Temporalevolutionof Coulombstress
changes
for modelsconsisting
of anelasticuppercrustwith
thickness
fixedto (a) 18 km and(b) 24 km overlyinga Maxwellviscoelastic
lowercrust(seeTable3). The
thicknessof the viscoelastic
lowercrustrangesfrom 11 to 56 km andis listedcloseto eachcurve.The rateof
changeof Coulombstress
for eachEarthmodelin thefirst300yearsaftertheearthquake
is shownin theinset.
The viscoelastic mantle extends to 2900 km for all these models.
postseismic evolution of Coulomb stress for Earth models
after the earthquakeshownin Figure2a are differentfrom
(Mod3C,Mod3L andMod3I in Table3) consisting
of an thoseshownin Figure 2b: Increasingthe thicknessof the
elasticcrust(18-km-thick)overlyinga Maxwell viscoelastic elastic upper crust yields a faster relaxation of the lower
lower crustwith differentthicknesses.
Figure2b showsthe viscoelastic
crust.For a givenelasticlayer, in the first five
samefor a 24-km-thickelasticcrust(Mod3G,Mod3N, and decades
afterthe earthquake
thepostseismic
stressrelaxation
Mod3F;seeTable3 for Earthmodelparameters).
Figure2 occursat higher rates for increasingthicknessof the lower
clearly emphasizesthat the relaxationprocessesof the viscoelastic
crust(seeinsetsin Figures2a and2b). After the
viscoelastic
layers,whichcausethe reloadingof the elastic first 100 years the temporalderivativesof the evolution
crust,involvetimescales
thatdepend
notonlyontheviscosity curvestend to coincide.However, while for a thicker elastic
butalsoonthethickness
of thedifferent
layers.It is evident
in crust the relaxation process is almost completed (the
Figure2 that whenthe elasticcrustthicknessis constant, derivatives
vanishin Figure2b), for a thinnerelasticlayerthe
increasing the thickness of the lower crust increases the temporaldistributionof Coulombstresshasnot yet reached
magnitude of postseismicCoulomb stressfor both the two its postseismic
fluid limit (seeFigure2a). The comparison
setsof models.
Therelaxation
ratesduringthefirst300years betweenthetimeevolutionof Coulombstresschanges
for the
NOSTRO ET AL.' NORMAL FAULT AND POSTSEISMIC STRESS CHANGES
19,405
0.8
0.6
0.4
0.2
0.0
0
100
0
100
200
300
400
500
600
700
800
900
1000
0.8•
0.6
0.2
0.0
200
300
400
õ00
600
700
800
900
1000
Yeorsofter the eorthquoke
Figure 3. Temporalevolutionof Coulomb stresschangesfor different Earth modelshaving a fixed thickness
of the lower crust,equalto (a) 11 km and (b) 15 km, and an uppercrustthicknessrangingbetween18 and 28
km (seeTable3). The temporalevolutioncurvesfor the first 300 yearsafterthe earthquake
is shownin Figure
3a inset.The temporalderivativeof the evolutioncurvesfor the first 300 yearsafterthe earthquakeis shownin
Figure3b inset.The viscoelasticmantleextendsto 2900 km for all the modelsshown.
two setsof Earth models(Figures2a and 2b) pointsout that
the distancebetweenthe sourcedepth and the first elastic-
Figure 3). For longer time intervals, the models with the
largestelastic thicknessreach the postseismicfluid limit,
viscoelastic
meaningthat for theseEarthmodelsthe reloadingprocessis
completely finished after nearly 60 years. Therefore
Coulomb
interface
controls
the relaxation
rates of the
stress.
In Figure 3 we compare the temporal patterns of
postseismic
Coulombstresscomputedfor a fixed thicknessof
the viscoelastic lower crust but different upper crust
thicknesses
(rangingbetween18 and28 km, seeTable 3). The
thicknessof the elastic crust (and therefore the distance
betweenthe sourcedepthandthe first interface)considerably
affectsthe shapeof the reloadingcurve. The thinner elastic
crustis characterized
by a much slowerviscoelasticrelaxation
(see Figures 2 and 3). This is particularly evident in the
behaviorduring the first 50 years (see the annexedboxes in
increasingthe thicknessof the elastic crust also affects the
time at whichthe postseismic
fluid limit is reached.
Figure 4 shows the comparisonbetween Earth models
havingthe sametotal crustalthickness(35 km) anda different
partitioningbetweenelasticuppercrustandviscoelasticlower
crust(Mod3C, Mod3G, and Mod3H in Table 3). Figure 4
summarizes the results discussed above and shows that for a
given crustalthicknessthe absolutevalue of postseismic
Coulomb stress is not largely modified by the crust
partitioningbetweenelasticandviscoelastic
layers.However,
19,406
NOSTRO
ET AL.: NORMAL
FAULT
AND POSTSEISMIC
STRESS CHANGES
0.8
0.6
0.2
0.0
0
100
200
300
400
õ00
600
700
800
900
1000
Yearsafter the earthquake
Figure 4. Temporal patternsof Coulombstresschangesfor Mod3 models(see Table 3 and Plate 3c) having
the total thicknessof the two crustallayersfixed to 35 km (C, H, andG).
the relaxationrate largely dependson the thicknessof the two
shallowestlayers.
An important,thoughexpectedeffect, concernsthe choice
of the viscosityof the first viscoelasticlayer: The functional
dependenceon the viscosityof postseismicdisplacementand
gravitationalfields hasbeenlargelydiscussed
in the scientific
literature [see, e.g., Cohen, 1980, 1984; Matsu'ura et al.,
characterizedby a 24-km-thick elasticupper crustand a 15km-thick viscoelasticlower crustbut using different viscosity
values(Mod3N in Table 3). As expected,the lower viscosity
yieldsa fasterrelaxationof the viscoelasticcrust.As distinct
from the behavior of a uniform plane model [e.g., Rundle,
1981, 1982] the relaxationratesdo not scalelinearly with the
viscosityvalue, indicatingthat not only the lower crust but
1981; Melosh, 1983; Soldati et al., 1998; Piersanti, 1999].
alsothe uppermantleplay a role in thepostseismic
relaxation
Figure 5 showsthe postseismicevolutionof Coulombstress process[Piersanti, 1999]. This is also in agreementwith the
for a point located at 17 km of depth for Earth models findingsof Soldati et al. [1998] concerningthe postseismic
0.81
0.6
0.2
0.0
0
100
200
300
400
õ00
600
700
800
900
1000
Yearsafter the earthquake
Figure 5. Temporal evolution of Coulomb stresschangesfor Mod3N calculatedby using three different
viscosity
valuesfor thelowercrust:10•a 10•9 and1020Pa s, respectively
(seeTable3). Thetemporal
derivative of the evolution curvesfor eachEarth model in the first 300 years after the earthquakeis shownin
the inset.
NOSTRO
ET AL.: NORMAL
FAULT
AND POSTSEISMIC
gravitationalperturbations,and with Nostro et al. [1999], who
studiedthe deformationrate for the 1960 Chileanearthquake.
Recent papers have emphasized the importance of
including the effects of long-term loading and relaxation of
the lithosphere and asthenosphere in fault interaction
investigations[seeHarris, 1998; King and Cocco, 2001, and
referencestherein]. The resultspresentedin this study further
supportthesefindings.The applicationsto the study of fault
interaction for historical earthquakes[see,e.g., Jaurne and
Sykes,1996; Nostro et at., 1997; Stein et at., 1997; Deng and
Sykes, 1997a, 1997b; Natbant et at., 1998] also require to
properlyincludethe contributionof viscoelasticrelaxationto
the induced stress field. The present study is focused on
postseismic
relaxationcausedby normalfaultingearthquakes.
The main feature emerging from the parametric tests
above
is that the viscoelastic
relaxation
increases
the Coulomb stress near the causative faults (at distances
within few fault lengths) and tends to reduce the stress
shadowareas.This latter effect hasalready been suggestedby
Kennerand Segatl[1999] for verticalstrike-slipfaults.
Nostro et al. concluded that out of 1 1 earthquakes, 10
occurredin areasof Coulomb stressincrease.In this studywe
show that for all the adoptedEarth models the postseismic
stress relaxation
does not reduce the induced Coulomb
/
-100
0
viscoelasticpostseismic
relaxationfurthersupportsthe results
of Nostro et al. [1997] becauseit increasesthe values of the
inducedCoulomb stressfor normal faulting at the baseof the
seismogenic layer enlarging the enhanced stress area.
Moreover, postseismicrelaxation increases the Coulomb
stressin the off-fault lobes, promoting interaction between
paralleland/orantitheticfaults.
Anotherfeatureconcernsthe optimaflyorientedplanesfor
Coulomb failure. In fact, by assumingan extensionaltectonic
stressfield perpendicularto the Apennines (see Plate 7),
Nostro et al. [1997] have shown that the strike of optimal
oriented planes for normal faulting is parallel to the
Apennines (which correspondsto the orientation of the
100L
200
distcmce
(km)
-0.10
-0.05
0.00
stress
in the strike direction, which identifies the zones of enhanced
Coulomb stress for elastic interaction. In other words,
0
-200
19,407
The resultspresentedin sections5 and 6 have important
implicationsfor the interpretationof elasticfault interaction
between the historical earthquakesin the Apennines(Italy)
shown in Plate 1 and discussedby Nostro et al. [1997].
7. Discussion
discussed
STRESS CHANGES
0.05
0.10
CFF changes(b(]rs)
Plate ?, Cos½ismicand postseismicCoulomb stresschanges(after 100 years) causedby a normal fault (with
the samemechanismsand geometrydescribedin Plate 2). The arrows show the optimaflyorientedplanesfor
failure calculatedassumingan extensionaltectonicstressof' 20 bars orientedN216ø as drawn in the map (scc
Nostro et al. [ 1997] for details).White arrowsindicatethe strike directionspredictedby the coseismicstress
changesonly, while black arrowsthosepredictedby the total postseismicstress.The red and blue fault plane
solutionsshow the expected focal mechanismsassociatedto coseismicand postseismicstresschanges,
respectively.
19,408
NOSTRO ET AL.: NORMAL FAULT AND POSTSEISMIC STRESSCHANGES
causative faults). Moreover, when the coseismic stress
changesare comparableto the regional stressamplitudes,
there is a rotation of the strike direction of the optimally
orientedplanescloseto the rupturingfaults [seealsoKing et
al., 1994; King and Cocco,2001]. It might be interestingto
verify if the redistributionof postseismicstresscausedby
viscoelasticrelaxation of the lithospherecan change (with
time) the geometry of the optimally oriented planes for
Coulombfailure. In orderto testthis effect we have computed
the optimally orientedplanesfor failure (shownin Plate 7)
resultingfrom the total Coulombstressfield after 100 years
given by the coseismicand the postseismicstresschanges.
The comparisonbetween the predicted optimal Coulomb
planes and those calculatedonly from the coseismicstress
perturbations(white arrowsin Plate 7) providesa measureof
the possible changesof geometry caused by viscoelastic
relaxation: The changesin dip are of the order of +10 ø,while
factor 100. A similar range of variability for the amplitudeof
postseismicstress can also be obtained varying only the
thickness of the shallow layers (see Figures 2 and 3).
However, it should be noted that the assumed value of
viscosity of the shallowestviscoelasticlayer is in any case
more importantin controllingthe relaxationrates(seeFigures
3b and 5) than the crustal layer thickness.This allows us to
conclude that, while this trade-off can limit the constraint of
the amplitudeof postseismicstressfew centuriesafter a large
magnitude earthquake,the choice of the viscosity value is
crucial to determine the relaxation rate during this time
interval.
Some of the results presented in this work for normal
faulting earthquakescould be also inferred from previous
analysesconcerningthe effects of viscoelasticrelaxation on
the surface postseismic displacement and strain field for
different tectonic settings (from those of early 1980s [e.g.,
those for the rake are +20 ø. The viscoelastic
relaxation
Cohen, 1980, 1984; Matsu'ura et al., 1981; Rundle, 1982;
enlargesthe areaswhere the inducedstressyields a rotationof Melosh, 1983] to those presented in the last years [e.g.,
the optimally orientedplanes,sinceit increasesthe amplitude Pollitz, 1992; Piersanti et al, 1995; Deng et al., 1998].
of the total induced stresschangesrelatively to the regional Nevertheless, it must be remembered that we analyze the
tectonic stressover a wide area. Plate 7 showsa comparison effects of viscoelastic relaxation on the Coulomb stress field
of the fault plane solutionspredictedby the coseismicstress on specifiedas well as optimally orientedplanes for failure,
changes (red mechanisms) and the total postseismicstress while previousinvestigationsfocusedmainly on postseismic
changes after 100 years (blue mechanisms)for particular displacementsor strain field. Moreover, we investigate the
locations.This comparisonsuggeststhat the generalpatternof spatiotemporalpattern of the postseismicstressfield at depth
strike orientations does not differ very much from that (i.e., at the base of the seismogeniclayer) and not at the
calculated from only the coseismic stress changes. This surface.
further confirms that viscoelastic
relaxation
increases the
The simulations presented here strengthen the results
Coulomb stressfor normal faulting earthquakeswith a spatial obtained by Nostro et al. [1997] concerningthe interaction
patternnot too different from thoseresultingfrom coseismic between historical events in the Apennines. Viscoelastic
stressperturbations.Finally, we must note that all the effects relaxation processesin the lithospheretend to increase the
discussed above would be increased by the presence of Coulomb stressalong the Apennines,where normal faulting
shallow
ductilelayerswithviscosity
smaller1019
Pa [e.g., earthquakesoccur.Moreover, they also increasethe off-fault
lobe of positiveCoulomb stresschangesallowing interactions
Deng et al., 1998;Piersanti, 1999].
among the faults belonging to the southern Apennines
seismogenicbelt. It is also evidentthat viscoelasticrelaxation
8. Conclusions
contributesin recoveringthe stressshadowingeffects caused
by
coseismicruptures.However, the trade-off betweenmodel
The additionalpostseismicstresschangesfor a fully elastic
lithosphereare sensiblydifferent from thoseresultingfrom a parametersdiscussedabove limits the applicability of such
viscoelastic upper crust or mantle (compare Plate 4 with modeling to compute long-term stressevolution at regional
Plates 5 and 6). This is also true for the long-term temporal distances.Even if postseismicrelaxation can substantially
relaxation at a fixed point (see Figures 1 and 2). The results inc/'ease the Coulomb stress perturbation, its temporal
presentedin Plates 5 and 6 point out that for a normal fault evolution dependsboth on the assumedviscosity and on the
the viscoelasticrelaxation processstronglyreducesthe stress thicknessof the shallowerlayers.
shadow
zone in the strike direction
and it modifies
the off-
fault lobes of Coulomb stress increase. Analysis of the
temporal evolution of total postseismic stress, shown in
Figures 1, 2, and 3, points out that the thicknessof the elastic
layer controls the time ar which the relaxation process is
completed(i.e., the relaxationprocessreachesthe fluid limit).
Increasing the thickness of the elastic layer causesa faster
relaxation in the first decadesbut smaller postseismicstress
amplitudes(see Figures 1 and 3) at longer timescales.Larger
thickness of the shallowest viscoelastic layer increasesthe
amplitudeof the postseismicstress.
The most important result emerging from our simulations
is the trade-off between viscosity values and thicknessof the
shallowerlayers.In fact, as expected,decreasingthe viscosity
of the shallowest viscoelastic layer increasesthe relaxation
rates leading to larger postseismicstressperturbation (see
Figure 5): The amplitude of the postseismicCoulomb stress
after 100 yearschangesby ---41%for a viscosityvariationof a
Acknowledgments. We thank Maurizio Bonafede, Giorgio
Spada,Ross Stein, Fred Pollitz, and TortstenDahm for the useful
discussions
and for the commentsto the motivationsand applications
of this study. We thank Enzo Boschifor encouragingthis research.
This research has been partially supported by the European
Community, contract ENV4-CT97-0528;
Faust: Faults as
SeismologicalTool.
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