Encyclopedia of Earthquake Engineering

3953 Pages·2015·211.626 MB·English

by  Michael Beer, Ioannis A. Kougioumtzoglou, Edoardo Patelli, Siu-Kui Au (eds.)

N
Noise-Based Seismic Imaging and seismic noise sequences from regional networks
Monitoring of Volcanoes inordertoextractandtoinvertRayleighwavesto
produce high-resolution seismic images of the
FlorentBrenguier shallow crustal layers under these networks
InstitutdesSciencesdelaTerre,Universityof (Shapiro et al. 2005). The first part of this entry
Grenoble,Grenoble,France willshow how noise-based seismicimagingcan
beemployedtoimagevolcanointeriors.
Theearlydetectionofvolcanicunrestmainly
Synonyms relies on the monitoring of volcanic seismicity
andgrounddeformation.Thosemethodsprovide
Eruption forecasting; PitondelaFournaiseVol- insightsintothedynamicsofmagmapressuriza-
cano; Seismic imaging; Seismic noise; Volcano tionandtransport.However,despiteconsiderable
monitoring effort, the precise forecasting of eruptions and
their intensity has proven to be difficult. There-
fore, there is a constant need for novel observa-
Introduction tional methods to obtain information about
ongoingvolcanicprocesses.Pressurizedvolcanic
Volcanic eruptions represent a great hazard to fluids(magma,water)orgasesinducedeformation
population leaving nearby volcanoes. In order andthusperturbationsoftheelasticpropertiesof
to better understand volcanic activity and thus volcanicedifices.Thesesmallperturbationscanbe
to improve eruption forecasting, it is of primary detected as changes of seismic wave properties
importancetobetterimageandmonitorthestruc- using repetitive seismic sources (Ratdomopurbo
tureofvolcanicedifices. and Poupinet 1995; Greˆt et al. 2005; Wegler
In the field of ultrasonics, it has been shown et al. 2006). However, none of these approaches
both theoretically and experimentally that were apt to provide a continuous monitoring of
a random wavefield has correlations which, on volcano elastic properties. In a pioneering work,
average,taketheformoftheGreen’sfunctionof Sens-SchoenfelderandWegler(2006)proposedto
themedia(WeaverandLobkis2001).Inseismol- use the repetitive waveforms of seismic noise
ogy,recentstudiesalsoshowedthattheGreen’s cross-correlationstotrackforsubsurfacevolcanic
function between pairs of seismographs can be edifice velocity changes. In this manner, the
extracted from cross-correlations of coda waves continuous recording of ambient seismic noise
and ambient noise (Campillo 2006). Moreover, allowscontinuousmonitoringofvolcanointeriors.
some authors used correlations of long ambient The second part of this paper will show how
#Springer-VerlagBerlinHeidelberg2015
M.Beeretal.(eds.),EncyclopediaofEarthquakeEngineering,
DOI10.1007/978-3-642-35344-4
1562 Noise-BasedSeismicImagingandMonitoringofVolcanoes
Noise-Based Seismic Imaging and Monitoring of triangles, GPS as crosses, and observatory short-period
Volcanoes, Fig. 1 Location of (a) La Re´union island, seismic stations as circles. The inset map shows lava
(b)PitondelaFournaiseVolcano,and(c)theUnderVolc flows from 2000 to 2010 (Courtesy of OVPF,
and observatory seismic and GPS networks. UnderVolc T.Staudacher,Z.Servadio)
broadband seismic stations are shown as inverted
noise-basedseismicmonitoringcanbeemployed PitondelaFournaiseVolcanoismonitoredby
tomonitorvolcanointeriors. OVPF/IPGPvolcanoobservatory.In2009,inthe
framework of UnderVolc project (Brenguier
et al. 2012), 15 new broadband seismic stations
Noise-BasedSeismicImagingofPiton complemented the observatory seismic network
delaFournaiseVolcano thatwas composed ofabout 20seismic stations.
The intense eruptive activity together with
PitondelaFournaiseVolcano(PdF)isahotspot, a weak tectonic activity makes Piton de la
shield volcano located on La Re´union island in Fournaise Volcano well suited for studies
the Indian Ocean (Fig. 1). It erupted more than focused on the processes of magma pressuriza-
30 times between 2000 and 2010. These erup- tion and injection and for the development of
tions lasted from a few hours to a few months innovativeimagingandmonitoringmethods.
andwereassociatedwiththeemissionofmainly By using continuous records of ambient seis-
basalticlavawithvolumerangingfromlessthan mic noise from the PdF Volcano observatory
one to tens of million cubic meters (Peltier short-period seismic network, Brenguier
et al. 2009). The time period that is considered et al. (2007) were able to measure Rayleigh
here (1999–2011) started and ended with two wave velocity dispersion curves obtained from
major eruptions, namely, the March 1998 erup- the cross-correlations of 18 months of ambient
tion (60 million of cubic meter of lava emitted) seismicnoise.Thetomographyofthesemeasure-
and the April 2007 eruption associated to the mentsaswellasthedepthinversionledtoahigh-
300mhighcollapseofthemainDolomieucrater resolutionimageoftheshallowstructureofPdF
(130 million of cubic meter of lava emitted) Volcano(Fig.2).Theresultsshowahighseismic
(Staudacheretal.2009). velocity body located in the central part of the
Noise-BasedSeismicImagingandMonitoringofVolcanoes 1563
timedelaysasaconsequenceofaseismicveloc-
ity change in the propagating medium. Measur-
ing travel time perturbations in the coda thus
allows detecting very small velocity changes
that would not bedetectableby a classical mea-
sureoffirstarrivaltimedelays.Thedrawbackof
thatapproachisthatitisdifficulttoestimatethe
travel paths of the scattered waves constituting
thecoda.However,recentpromisingresultssug-
gest it may be possible to produce refined 3D
maps of small changes in a near future (Larose
etal.2010).
In a previous work Brenguier et al. (2012)
measured seismic velocity variations within the
Piton de la Fournaise edifice from June 2010 to
April 2011 following Brenguier et al. (2008).
Continuous seismic velocity changes as well as
the daily seismicity rate are shown on Figs. 3
and 4 for two different time periods at PdF vol-
Noise-Based Seismic Imaging and Monitoring of cano. Figure 3 shows small (0.1 %) seismic
Volcanoes,Fig.2 3Dtomographicimageoftheshallow velocity drops preceding eruptions at PdF vol-
centralpartofPitondelaFournaiseVolcanousingcross-
cano. On Fig. 4, the authors observe a velocity
correlations of ambient seismic noise (Brenguier
etal.2007) dropstartingabout1monthbeforethe2010Octo-
ber 14th eruption. The velocity decrease corre-
lates in time with an increase of seismicity and
volcano. This body is being interpreted as has been interpreted as the deformation of the N
a preferential zone of magma injection within edifice induced by magma pressure buildup and
the edifice. The high velocities are associated injection.Thesevelocitydropsareinterpretedas
with intrusive magma slowly cooling within the beingassociatedwiththedeformationoftheedi-
surroundingeffusivelowvelocity material.This fice induced by magma pressure buildup. Inter-
information is crucial in order to define a more estingly, seismic velocity increases during the
precise structural model for the volcano edifice eruption of October 2010 and after the short
andthustobettermodelandthenunderstandthe eruption of December 2010 indicating
responseofthevolcanotomagmaticactivity. a possible mechanism of stress relaxation
induced bythe emptyingofthe magmaticreser-
voir. Furthermore, to improve the accuracy and
Noise-BasedSeismicMonitoringof thus time resolution of velocity measurements,
PitondelaFournaiseVolcano Baig et al. (2009) developed a cross-correlation
filtering method based on time-frequency trans-
The approach consists in measuring very small formsandphasecoherencefiltering.
waveformtimedelaysinthecodaofnoisecross- Noise-based seismic velocity monitoring is
correlations. This method is described as the thus a unique method that allows to precisely
so-called Moving Window Cross Spectrum monitor volcanic activity. It must however
(Ratdomopurbo and Poupinet 1995; Clarke be mentioned that other phenomena perturb
et al. 2011) or Coda Wave Interferometry seismic velocities of the rock mass such as the
(Snieder et al. 2002) techniques. Coda waves presence of water in the medium, temperature
(late part of seismograms) are scattered waves and barometric atmospheric pressure changes,
that travel long distances and thus accumulate andsolidandoceanictides.Itisthusimportant
1564 Noise-BasedSeismicImagingandMonitoringofVolcanoes
Noise-Based Seismic Imaging and Monitoring of 0.1–0.9 Hz. For details, see Brenguier et al. (2008) (b)
Volcanoes, Fig. 3 (a) Relative velocity changes com- Inter-eruptive seismicity (pre-eruptive swarms are
paredtoextensometer(FORX) locatedonPdFvolcano. excluded)
Thetraveltimeshiftsaremeasuredinthefrequencyrange
Noise-BasedSeismicImagingandMonitoringofVolcanoes,Fig.4 Relativeseismicvelocitychangeswitherror
barsanddailyseismicityratebetweenJune2010andApril2011.Eruptionperiodsareshownasgrayrectangles
Noise-BasedSeismicImagingandMonitoringofVolcanoes 1565
to correctlymodeltheseeffectsinordertocor- ThismethodisnowusedroutinelyatthePiton
rect them from the observed seismic velocity delaFournaiseObservatoryinordertoimprove
changes in order to extract the volcanic-related eruption forecasting together with earthquake
signal. and deformation observations. Lecocq
In April 2007, a major eruption occurred at et al. (2013) developed an integrated package
PitondelaFournaiseVolcanoejectingmorethan that includes all processing steps required to
250 million cubic meters of lava and located obtain continuous seismic velocity changes
a few kilometers east of the main active central from continuous seismic records. This package
partofthevolcano.Clarkeetal.(2013)observed isavailableforfreeatwww.msnoise.org.
anunusualhighseismicvelocitydropassociated
with this volcanic episode that could not be
explained by the pre-eruptive edifice inflation Summary
due to magma pressure buildup as described by
(Brenguieretal.2008).Also,(Clarkeetal.2013) Ambient seismic noise continuously travels
showed surface displacements images obtained along the surface and the interiors of the Earth.
from InSAR data inversion by J-L. Froger Theseseismicwavesthuscarrycrucialinforma-
(OPGC, France). These results show that tion about the medium they propagate through.
betweenMarchandMay2007,awidespreadvol- When crossing volcanoes, these waves are
canic edifice flank movement occurred with affected by the mechanical heterogeneities of
a maximum of 1.4 m of eastward displacement. thevolcanicedificeandarealsoperturbedduring
However, the timing of this movement and, in unrest periods by temporal changes of volcano
particular,itsrelationwiththeoccurrenceofthe interiors.Noise-basedseismicimagingandmon-
April 2007 eruption could not be identified by itoringallowsimagingthestructureandtemporal
a lack of temporal resolution of the InSAR evolution of volcanic edifices. In particular, it
images. Clarke et al. (2013) proved using high proved success in detecting very small changes
temporalresolutionseismicvelocitychangemea- ofseismicvelocitiesofvolcanicedifices(0.1%) N
surements that the strong dropof seismic veloc- that have been shown, on Piton de la Fournaise
ities started at the time of a small eruption Volcano(LaRe´unionisland),tobeprecursorsof
preceding the main April eruption by a few volcaniceruptions.
days.Inferringtheassociationbetweenthishigh
seismicvelocity dropandthewidespreadvolca-
nic flank movement, the authors concluded that
Cross-References
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▶SeismicNoise
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▶SeismicTomographyofVolcanoes
et al. (2013), it is likely that the small eruption
▶TrackingChangesinVolcanicSystemswith
preceding the main one triggered a large
SeismicInterferometry
elastoplastic deformation of the edifice flank
▶VolcanicEruptions,Real-TimeForecastingof
that released stresses and favored the horizontal
migration of magma to a long distance through
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Staudacher T (2013) Timing of a large volcanic
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that do not fit the DC model. Earthquakes that
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Ratdomopurbo A, Poupinet G (1995) Monitoring
mon in volcanic and geothermal areas. More-
a temporal change of seismic velocity in a volcano:
over, minor departures from the DC model are
application to the 1992 eruption of Mt. Merapi
(Indonesia).GeophysResLett22(7):775–778 detectedincreasinglyfrequentlyinstudiesusing
Sens-Schoenfelder C, Wegler U (2006) Passive image high-quality data. These observations probably
interferometry and seasonal variations of seismic
reflectdeparturesfromidealizedmodels,caused
velocities at Merapi Volcano. Indones Geophys Res
by effects such as rock anisotropy or fault
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industrialactivitiessuchasoilandgasproduction
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and storage, hydrofracturing, geothermal energy
interferometry for estimating nonlinear behavior in
seismicvelocity.Science295(5563):2253 exploitation, CO sequestration, and waste
2
Non-Double-CoupleEarthquakes 1567
disposal can induce earthquakes, and that these RepresentationofEarthquake
eventsoftenhavenon-DCmechanisms.Induced Mechanisms
earthquakes pose legal hazards because of their
destructivepotential,buttheycanalsobebenefi- TheEquivalentForceSystem
cial, because they can be used to monitoring Physically, an earthquake involves a nonlinear
physical processes that accompany industrial failureprocessoccurringwithinalimitedregion.
activity. The equivalent force system, acting in an intact
Non-DC earthquakes are therefore important (unfaulted)“model”medium,wouldproducethe
for improving our understanding of how faults same displacement field outside the source
and volcanoes work, perhaps leading some day region.Anyphysicalsourcehasauniqueequiv-
toanabilitytopredictearthquakesanderuptions, alentforcesysteminagivenmodelmedium,but
for avoiding nuisance seismicity in connection theconversestatementisnottrue.Differentphys-
with industrial activity, and for monitoring such ical processes can have identical force systems
activityindetail. and therefore identical static and dynamic dis-
Non-DC earthquake mechanisms, by defini- placementfields.
tion,requirefortheirdescriptionamoregeneral Theequivalentforcesystemisallthatcanbe
mathematical formalism than the DC model. deduced from observations of displacement
The most widely used such formalism is the fields, so it constitutes a phenomenological
expansion of the elastodynamic field in terms description of the source. There is a one-to-one
of the spatial moments of the equivalent-force correspondence between equivalent force sys-
system (Gilbert 1970). Usually, attention is temsandelastodynamicfieldsoutsidethesource
restricted to second moments, and thus to region,sowecan,inprincipleatleast,determine
second-rank moment tensors, and moreover force systems from observations. On the other
thesetensorsareusuallyassumedtobesymmet- hand,thecorrespondencebetweenforcesystems
ric, so that they have six independent compo- andphysicalsourceprocessesisone-to-many,so
nents(ADChasfourindependentcomponents.). equivalent force systems (and therefore seismic N
Thisrestrictionisnotalwaysjustified,however. andgeodeticobservations)cannotuniquelydiag-
Sourcesinvolvingnetforcesortorquesarethe- nosephysicalsourceprocesses.
oreticallypossible,andphenomenasuchasland- Failurecanberegardedasasuddenlocalized
slides and volcanic eruptions provide clear change in the constitutive relation (stress–strain
examplesofthem. law)intheEarth(Backus1977).Beforeanearth-
Because it has two extra adjustable parame- quake, the stress field satisfies the equations of
ters, a moment tensor can describe volume equilibrium.Atthetimeoffailure,arapidchange
changesandgeneralkindsofsheardeformation. intheconstitutiverelationcausesthestressfield
This increased generality allows the moment to change. The resulting disequilibrium causes
tensor to encompassphysicalprocesses such as dynamic motions that radiate elastic waves. We
geometricallycomplexfaulting,tensilefaulting, disregard the effect of gravity in the following
faultinginanisotropicmedia,faultinginhetero- discussion.Intheabsenceofexternalforces,the
geneous media (e.g., near an interface), and equationofmotionis
polymorphicphasetransformations.
The moment tensor also has an important ru€i ¼sij,j; (1)
computational property: Mathematical expres-
sionsforstaticanddynamicdisplacementfields where r(x) is density, u(x, t) is the particle
are linear in the moment-tensor components. displacementvector,s(x,t)isthephysicalstress
This property facilitates the evaluation of these tensor, x is position, and t is time. Dots indicate
fields and enormously simplifies the inverse differentiation with respect to t, ordinary
problem of deducing source mechanisms from subscriptsindicateCartesiancomponentsofvec-
observations. tors or tensors, the subscript, j indicates
1568 Non-Double-CoupleEarthquakes
differentiation with respect to the jth Cartesian momentum;ifthesourceregionisatrestbefore
spatialcoordinatex,andduplicatedindicesindi- andaftertheearthquake,thetotalimpulseofthe
j
cate summation. The true stress is unknown, equivalent force (its time integral) must vanish.
however, so in theoretical calculations we use No such requirement holds for the torque. The
thestresss(x,t),givenbytheconstitutivelawof total torque exerted by gravitational forces need
the model medium(usuallyHooke’s law). If we notvanish even after theearthquake.Horizontal
replace s by s in the equations of motion, displacementofthecenterofmassofthesource
ij ij
though, we must also introduce a correction regionleadstoagravitationaltorque,whichmust
term,f(x,t): be balanced by stresses on the boundary of the
sourceandcausestheradiationofelasticwaves.
ru€i ¼sij,jþfi; (2) Because gravity acts vertically, there can be no
nettorqueaboutaverticalaxis.
(cid:1) (cid:3)
fid¼ef sij(cid:2)sij ,j: (3) TheMomentTensor
Wecannotusetheequivalentforcesystemf(x,t)
This term has the form of a body-force density and the elastodynamic Eq. 2 to determine the
and is the equivalent force system of the earth- displacement field for a hypothetical source.
quake.Itdiffersfromzeroonlywithinthesource The equivalent force system itself depends on
region. the displacement field that is being sought. Two
different approaches are commonly used: (i) In
NetForcesandTorques thekinematicapproachweassumesomemathe-
Nearlyallanalysesofearthquakesourcemecha- matically tractable displacement field in the
nisms explicitly exclude net forces and torques. source region (e.g., suddenly imposed slip, con-
TheequivalentforcesgivenbyEq.3,whicharise stant over a rectangular fault plane), derive the
from the imbalance between true physical equivalent force system from Eq. 3, and solve
stresses and those in the model, are consistent Eq.2fortheresultingdisplacementfieldoutside
with these restrictions. The stress glut s (cid:2) s the source region; and (ii) in the inverse
ij ij
is symmetric, so f exerts no net torque at any approach, we use Eq. 2 to determine the force
point. Furthermore, s (cid:2) s vanishes outside system f(x, t) from the observed displacement
ij ij
the source region, so Gauss’s theorem implies field and compare the result with force systems
thatthetotalforcevanishesateachinstant. predicted theoretically for hypothesized source
More complete analysis, however, including processes. The most useful way to parameterize
the effects of gravitation and mass advection, the force system in this approach is to use its
shows that Eq. 3 is based on overly restrictive spatialmoments.
assumptions and that net force and torque com-
ponents are possible for realistic sources within TheMoment-TensorExpansionfortheResponse
the Earth (Takei and Kumazawa 1994). These Given the equivalent force system f(x, t), com-
forces and torques transfer linear and angular putingtheresponseoftheEarthisalinearprob-
momentum between the source region and the lem, and its solution can be expressed as an
rest of the Earth, with both types of momentum integral over the source volume V (Aki and
conserved for the entire Earth. An easily under- Richards 2002, Eq. 3.1, omitting displacement
stood example is the collapse of a cavity, in andtractiondiscontinuitiesforsimplicity):
which rocks fall from the ceiling to the floor.
ððð
Whiletherocksarefalling,theEarthoutsidethe
uðx,tÞ¼ G ðx,x,tÞ(cid:3)f ðx,tÞd3x; (4)
cavityexperiencesanetupwardforce,relativeto i ij j
V
thestatebeforeandaftertheevent.
The net force component in any source is where G (x, x, t) is the Green’s function, which
ij
constrained by the principle of conservation of gives the ith component of displacement at
Non-Double-CoupleEarthquakes 1569
positionxandtimetcausedbyanimpulsiveforce Themagnitudesofthesix(ornine)elementary
in the j direction applied at position x and time force systems (the moment-tensor components)
0,andthesymbol*indicatestemporalconvolu- transform according to standard tensor laws
tion. If we expand the Green’s function in a under rotations of the coordinate system, so
Taylorseriesinthesourcepositionx, there exist many different combinations of ele-
mentaryforcesthatareequivalent.Inparticular,
Gijðx,x,tÞ¼Gijðx,0,tÞþGij,kðx,0,tÞxkþ... ; forasymmetric(six-element)momenttensorone
canalwayschooseacoordinatesysteminwhich
(5)
theforcesystemconsistsofthreeorthogonallin-
Eq.4fortheresponsebecomes eardipoles,sothatthemomenttensorisdiagonal.
In other words, a general point source can be
described by three values (the principal
uiðx,tÞ¼Gijðx,0,tÞ(cid:3)FjðtÞþGij,kðx,0,tÞ
(cid:3)M ðtÞþ... ; (6) moments) that describe its physics and three
jk
valuesthatspecifyitsorientation.
Themomenttensorhasthreeimportantprop-
where
ertiesthatmakeitusefulforrepresentingseismic
ððð
sources. (1) It makes the “forward problem” of
FjðtÞd¼ef fjðx,tÞd3x (7) computing theoretical seismic-wave excitation
V
linear. A general source is represented as a
weighted sum of elementary force systems, so
isthetotalforceexertedbythesourceand
anyseismicwaveisjustthesameweightedsum
ððð
of the waves excited by the elementary sources.
MjkðtÞd¼ef xkfjðx,tÞd3x (8) The linearity of the forward problem in turn
V
makes muchmore tractable the inverse problem
is the moment tensor. If the equivalent force is of determining source mechanisms from obser-
derivable from a stress glut via Eq. 3, then the vations.(2)Itsimplifiesthecomputationofwave N
moment tensor is the negative of the volume excitation. By transforming the moment tensor
integralofthestressglut: intoanappropriatelyorientedcoordinatesystem,
theanglesdefiningtheobservationdirectioncan
ððð
(cid:1) (cid:3) bemadetotakeonspecialvaluessuchas0andp/
M ðtÞ¼(cid:2) s (cid:2)s d3x (9)
jk ij ij 2.Thusradiationbytheelementarysourcesmust
V
be computed not for a general direction, but for
The moment tensor is a second-rank tensor, onlyafewdirectionsforwhichthecomputation
which describesasuperpositionofnine elemen- iseasier.Inalaterallyhomogeneousmedium,for
tary force systems, with each component of the example, radiation mustbe computed for only a
tensorgivingthestrength(moment)ofoneforce single azimuth. (3) The moment tensor is more
system.ThediagonalcomponentsM ,M ,and general than the DC representation. It includes
11 22
M correspond to linear dipoles that exert no DCs as special cases, but has two more free
33
torque, and the off-diagonal elements M , M , parametersthanaDC(sixvs.four),whichenable
12 13
M ,M ,M ,andM correspondtoforcecou- ittorepresentsourcesinvolvingvolumechanges
21 23 31 32
ples.Itisusuallyassumedthatthemomenttensor andmoregeneraltypesofshear.
is symmetric, (M = M , M = M ,
12 21 13 31
M = M ), so that the force couples exert no Higher-RankMomentTensors
23 32
net torque (see above), in which case only six As Eq. 6 shows, “the” moment tensor described
moment-tensor components are independent. In aboveisonlyoneofaninfinitesequenceofspa-
this case, the off-diagonal components corre- tialmomentsthatappearintheexpansionofthe
spond to three pairs of force couples, each Earth’s response to an earthquake. The later
exertingnonettorque(“doublecouples”). terms involve higher-rank moment tensors,
1570 Non-Double-CoupleEarthquakes
which contain information about the spatial and discontinuity in the x direction. This ambiguity
3
temporaldistributionoffailureinanearthquake, between “conjugate” faults is an example of the
andhavegreatpotentialvalueforstudyingsource fundamental limitations on the information that
finitenessandrupturepropagation. canbededucedfromequivalentforcesystems.
SurfaceSources(Faults) TensileFaults
A fault is a surface across which there is a dis- For a tensile fault in a homogeneous isotropic
continuityindisplacement.Theequivalentforce medium, with the fault lying in the x -x plane
1 2
distribution,f,foragenerallyorientedfaultinan andopeninginthex direction,n = n = 0and
3 1 2
elasticmediumis,fromEq.2, [u ] = [u ] = 0.Equation8givesamomentten-
1 2
sor with three non-zero components:
ðð
@ M ¼M ¼lAu and M ¼ðlþ2mÞAu,
11 22 33
fkð(cid:2),tÞ¼(cid:2) ½uiðx,tÞ(cid:4)cijkinj@(cid:2)idð(cid:2)(cid:2)xÞdA; corresponding to two dipoles in the fault plane
A
withmomentsoflAu,andathirddipoleoriented
(10)
normal to the fault and with a moment of
ðlþ2mÞAu.
where(cid:2) isthepositionwheretheforceisevalu-
ated,xisthepositionoftheelementofareadA,
VolumeSources
andtheintegrationextendsoverthefaultsurface
Somepossiblenon-DCsourceprocesses,suchas
(AkiandRichards2002,Eq.3.5).Theunitvector
polymorphic phase transformation, occur
normaltothefaultsurfaceisn(x)and[u(x,t)]is
throughout a finite volume rather than on a sur-
thedisplacementdiscontinuityacrossthefaultin
face. The equivalent force system often can be
thedirectionofn.Thecomponentsoftheelastic
expressed in terms of the “stress-free strain”
modulus tensor are c and d(x) is the three-
ijkl D e ,whichisthestrainthatwouldoccurinthe
dimensionalDiracdeltafunction. s ij
source volume if the tractions on its boundary
SubstitutingtheforcedistributionfromEq.10
wereheldconstantbyexternallyappliedartificial
into Eq. 8 gives the moment tensor of a general
forces (It might more accurately be called
fault,
the “fixed-stress strain.”). By reasoning that
involves a sequence of imaginary cutting,
Mij ¼(cid:2)cijkiAnk½ui(cid:4); (11) straining, and welding operations (e.g., Aki and
Richards 2002, Sect. 3.4), the moment tensor of
where A is the total fault area and the overbar suchavolumesourceisfoundtobe
indicatestheaveragevalueoverthefault.
ððð
M ¼ c D e dV: (12)
ShearFaults ij ijki s ki
V
For a planar shear fault (with normal n in the x
3
direction and displacement discontinuity [u] Forastress-freevolumechangeD Vinanisotro-
s
in the x direction, say, so that n = n = 0 and picmedium,forexample,
1 1 2
[u ] = [u ] = 0) in a homogeneous isotropic
2 3
medium (c ¼ld d d þ2md d ), Eq. 8 M ¼KD Vd ; (13)
ijkl ij km lm ik jl ij s ij
givesamomenttensorwithtwononzerocompo-
nentsM ¼M ¼mAu,where uistheaverage whereK = l + (2/3)misthebulkmodulus.
13 31
slip.Thiscorrespondstoapairofforcecouples, The stress-free strain is not, in general, the
one with forces in the x direction and moment strain that actually occurs in a seismic event
1
arm in the x direction, and the other with these (RichardsandKim2005).Becausethesourceis
3
directionsinterchanged. imbeddedintheEarth,itsdeformationisresisted
WegetthesameDCmomenttensorforafault by the stiffness of the surrounding medium,
withnormalinthex directionanddisplacement making the true strain changes smaller than the
1