Lambeck, K., 1973. Geodetic methods. In: Plate Tectonics, (X. Le Pichon, J. Francheteau and J. Bonnin, Eds), Elsevier, Amsterdam, 55-62.
Developmentsin Geotectonics6
PLATE,TE,CTONTICS
XAVIER LE PICHON,JEAN FRANCHETEAUandJEAN BONNIN
Cente National pour I'Exploitation des Océans,
Centre Océanologique de Bretagne, Plouzané (France)
ELSEVIERSCIENTIFICPUBLISHINGCOMPAI.IY
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I
Preface
The origin of this book is a reviewpaperthat ProfessorLeon Knopoff suggested
be
prepared for the final upper Mantle symposium held in Moscow in 1971. we have
greatly enlargedthe scopeof this proposedpaper.
The book is an attempt to give a broad expositionof the plate-tectonicshypothesis.
We feel that, at a time when plate tectonicsis often usedto justify wild extrapolations
from poor data with little rigor, our approachmay have some value. Accepting plate
tectonicsas a valid working hypothesis,we try to presentin a logical fashion the main
underlying conceptsand somerelatedapplications.The emphasisis placedon the tight
constraintsthat the hypothesis imposeson any ifiterpretation made within its framework. The dynamics of the plates and the origin of the motions are not discussed.
There is not yet a satisfactory answerto theseproblems,one of the difficulties being
that the rigid lithosphere is an efficient screenbetween us and tåe asthenosphere.
In its first five yearsof life, plate tectonicshas beenresponsiblefor an extraordinarily profuseliterature.It has not been our intention to provide a comprehensivebibliography of this most recent literature. We have selectedabout 600 referencesfrom
articlesand books availableto us by early 1972.Most of thesewere chosenbecausewe
felt that they were either significant or representativeof trends in research.There are
no doubt somegrievousomissions.
It isnot possibleat the presenttime to cover adequâtelythe implications of the still
evolving plate'tectonics hypothesis upon the different fields of earth science:Our
position on many problems is controversial and partly biased. The choice of the
problemsis itself biased,becausewe have put the emphasison those with which we, as
marine geophysicists, are most familiar. After a brief introduction in Chapter l, we
define plate tectonics (Chapter 2). In Chapter 3, we describethe rheologicalstratification of the upper layers of the earth, defìning lithosphere and asthenosphere.In
Chapter4, we discussthe kinematicsof relativemotions, instantaneousand fìnite, on a
plane and on a sphere.In chapter 5, we cohsider"absolutemotions,', that is motions
within a reference frame external to the plates. Chapter 6 is concernedwith processes
at accretingplate boundaries and Chapter 7 with processesat consuming plate boundaries.
The book is largely the result of the close collaboration of two of the authors(X.
L,ePichon and J. Francheteau). The third author (J. Bonnin) provided a fi¡st version of
chapter 7 and contributed to the generalorganizationof the work. In many places,we
have used the clear expositions of various problems that have been given by Dan
McKenzie. His analytical solutions, in particular, are very convenientfor discussion.
We iistr also to thank Jason Morgan who commented on the first four parts of the
manuscript and made many valuable suggestions.
PREFACE
Several colleagues critically read parts of the manuscript at different stages and
made constructive suggestions,particularly J. Brune, J. Cann, J. Coulomb, K.
Lambeck, J.L. Le Mouel, L. Lliboutry, D.P. McKenzie,H.D. Needham,A.R. Ritsema
and E. Thellier. A. weill helped in some of the computations.K. Lambeckwrote part
of Chapter 4. Many colleagueskindly gave us permission to use their illustrations and
sent us papersin advanceof publication. We thankYvette Potard and Nicole Uchard
for typing severalversions of the manuscript with skitl and style, and Daniel Carré and
SergeMonti for drafting assistance.A.R. Ritsema arrangedfor us to deleteour review
paper from the Final Upper Mantle Symposium specialvolume and encouragedus to
publistr an expandedversion.
This work was supportedby the Centre Nationalpour I'Exploitation desOcéans,and
was undertaken at the Centre océanologique de Bretagne.we are grateful to its
Director, René Chauvin, for his friendship and support. Revisionsof part of this work
were done by the first author while working at the Institute of Geophysics and
Planetary Physicsin La Jolla with a Cecil and Ida Green scholarship.
Pointe du Diable,
June 1972
Foreword
This book marks the transition from the era of the Upper Mantle Project
Project(from I97I-1977).
(1963-1970) to that of the Geodynamics
The conclusionof the activeperiod of the Upper MantleProjectwasceiebratedin
August l97I by a fìve-daysreviewsymposiumduring the XVth GeneralAssemblyof
the InternationalUnion of Geodesyand Geophysicsin Moscow.One of the invited
speakersat the time was Xavier Le Pichon,who presenteda reviewpaperon Plate
Tectonics,compiledtogetherwith his co-workersJeanBonninand JeanFrancheteau.
The broad set-upof their report madeinclusionof this major reviewpaperin the
therefore,publishedin
proceedingsof the symposiumimpossible.Theseproceedings
1972 as a specialissueof Tectonophysicsentitled The UpperMantle, and also as
did not contain this paper on
in Geotectonics,
Volume 4 in the seriesDevelopments
PlateTectonics.
The developmentof the conceptof PlateTectonicsand the finding of supporting
evidencefor the hypothesisare amongthe major resuitsof the work executedduring
- and for a part under the auspicesof - the UMP.The presentVolume therefore,
made up-to-dateas to December1972 with the data from literatu¡e as well as the
may properlybe consideredasa
newestresultsof the work of the authorsthemselves,
direct continuation- andin effectpart - of The UpperMantle.
During the UMP a great deal of scientifìc ingenuity was directed to the quasi
new
aspectsof the earth'scrust and upper mantle.In rapid succession,
steady-state
layers
in
basicfacts were discoveredabout the prominenceand extent of low-velocity
the upper mantle and crust, and about the existenceand delineationof important
in the uppermantle.But alsomore dynamicalaspectsemerged,
lateralinhomogeneities
such as the really astonishingdegreeof relativemotion betweencrustalblocks of all
sizesat leastduring the past few 100 m.y. It is this dynamicalaspectthat forms the
subjectof the presentVolume and that in largemeasuredid lead to the initiation of
Project.
the Geodynamics
The problem of the driving forcesfor the important relativemotion, and for the
creationand consumptionof greaterand smallerplatesof the earth'slithospherehas
not yet been solved conclusively.Ttris publication therefore, is also an interim
document on the presentstate of the art of Plate Tectonics.As such it may be
expectedto sewe as the inspiringbasefor further studiesin the frameworkof the
Project.
InternationalGeodynamics
Januarv1973
A. ReinierRitsema
of the
editor of the proceedings
final UMP symposium
MEASUREMENTSOF INSTANTANEOUSIVIOVEMENTS
55
variation.
to eliminatethis statistical
motion,as muchas 1,000yearsmay be necessary
However,the resultsof Daviesand Brune (197l) show a fair quantitativeagreement
between seismicdip-slip rates obtainedalong consumingplate boundarieswith the
Brune method, using the last seventyyearsof data, and rates derivedfrom those
piateboundaries
by the Vine and Matthewsmethod.
obtainedat accreting
Over continentaltransform faults, the method has given ratesin agreementwith
geodeticratesusinga width Ws of 20 km (about5-6 cm/yearfor the SanAndreas
fault and I 1 cm/year basedon the seismicityof the last 31 yearsfor the Anatolian
fault). Overoceanict¡ansformfaults,if the motionoccursentirelyby dislocation,Wç
et a1.,1970).lf we
is muchsmailerand of the orderof 5 km (Brune,1968;Northrop
can assumeas a first approximationthat the thicknessof the brittle lithosphereinby thermal
creaseswith age away iiom the accretingplate boundary, as suggested
modeis,oneshouldmakethe estimationof the averagelVo asa function of the average
age of the crust along the transformfauit. For exampleBrune'scalculationstbr the
Romanchefiacture zone actuallyusethe lengthsof three fracturezones,Romanche
rate of I.7
(550 km). Usinga spreading
(880 km), Chain(330 km) and St.-Paul's
cm/year(Le Pichon,1968),the meanageof the seafloor alongthesethreefiacture
zonesis 9.5 m.y. For the SouthPacificEltaninfracturezonecompiex,Herron(1972)
rate
indicatesthree transformfäuits which givea meanageof 5 m.y. usinga spreading
thicknessof I.2 km found for the Eltaninfracturezone
of 4.5 cm/year.The average
thicknessof
6.5kmtor
average
of 5 m.y.whereasthe
woulcithusapplyto an average
ageof 9.5 m.y. Similarreasoningcanbe appliedto
the Romancheappliesto an average
the othertransformfaultsstudiedby Brune(1968)andNorthropet al. (1970)and the
thickeningof the brittle lithoof a progressive
resultsseemto confirm the existence
age.
spherewith increasing
To summarize,the Brune method is, with the geodeticmethod on the continents,
the only direct method availableat presentto measurerelativemotion at consuming
plate boundaries.It is much lessprecisethan the Vine and Matthewsmethod. An
unknown systematicerror may exist if the seismicityof the period usedis not repreovertimes of the orderof a miilion
sentativeof the "steadystate" seismicity(averaged
years).The main imprecisionresultsfrom the uncertaintyof the empiricalrelations
magnitude.However,this
giving the seismicmoment as a funçtion of surface-wave
by determiningthe moment directly by studyof individual
difficulty may be lessened
when possible,rather than by assuminga simplemagnitudeversusmoearthquakes,
A secondsourceof error is the estimationof the areaof brittle
ment relationship.
faulting. The method cannot be appliedto oceanictransform faults to obtain the
relativevelocity of plates but may be the best way to measurethe variationof the
thicknessof the brittle lithospherenearthe crest.
GeodeticmethodsL
of motion betweenparts of the earth'scrustis essentially
The direct measurement
I
This section has been written by Kurt .Lambeck,GRGS, Observatoirede Meudon, Meudon,
France.
KINEMATICS OF RELATIVE MOVEMENTS
one of repeatedlymeasuringthe positionsof well-definedpoints. As the expected
motions are of the order of a few cmfyear(ignoringthe catastrophicdisplacements
in both the
occurringnear earthquakeepicenters)the highestprecisionis necessary
are
and in the definitionof the pointsbetweenwhich the measurements
measurements
made. Also, as we do not know if the movementsare continuousand gradual,or
sporadicand abrupt, we must be ableto obtain thq highly accuratepositionswithin
be
strort time intervals.The elasticityof the platesrequiresthat the measurements
made betweenpoints that are at considerabledistanceson either side of the plate
deformationin localizedareas
the instantaneous
boundary.Otherwiseone is measuring
oververy long time periodswill be requiredto obtainthe
and'continuousobservations
relativemotions of the platesasa whole.
of the positionsof points on the earth'ssurfacecanbe made
Direct measurements
usingvariousterrestrial,astronomicalor spatialmethods.The first onesareof limited
applications,however.They requireintervisibilitybetweenstationsand the methodis
ümited to measuringmotions acrossplate boundarieslocated on continents.These
boundariesare in generalvery complexand deformedover a wide regionso that an
elaboratenetwork has to be constructedto ensurethat at leastsomeof the pointslie
geodeticnets
on the "rigid" partsof the plates.Unfortunately,itisacharacteristicof
that the more extensivethe net the lessprecisethe results,as many circumstantial
factorsbecomeimportant. Spatialgeodeticmethodscircumventthis problembecause
by severalthousandsof kilometers.
one can measurethe positionsof points separated
That is, the points can be selectedto iie far from the piate boundariesand motions
acrossoceanicpiate boundariescan be measured.The spatialmethodshaveonly been
for
developedin recentyearsand they do not yet providethe high accuracynecessary
they are very promisingand will
measuringthe tectonic movements.Nevertheless
probably give the requiredresultsin the nearfuture. Terrestrialand spatialgeodetic
are complementarylthe former can providethe motions occurringimmeasurements
mediatelyacrossthe boundariesand the latter canprovidethe motionsof the plateas
are requiredfor a completeunderstanding
a whole. Thesetri,o types of dispiacements
of the tectonicmotions.
In satellitegeodesyone often speaksof reiativeand absolutepositiondetermina"absolute"in that they refer to an inertialreference
tion. Positionsare considered
frame havingfor origin the earth'scenterof mass.This inertial frame can be defined
by a z-axisparallelto the earth'smeanaxisof rotation,an x-axisin the pianeof the
meanequatoraiongthe vernalequinoxat an adoptedepocha5rda y-¿¡¡saiongz x x.
Givenpoints on the earth'ssurfacecan be relatedto this frameif the motion of a
frame attachedto the earth'scrust about this inertiaiframe is known. That is, if
precession,
nutation,polar motion and variationsin the earth'srate of rotationare
with respectto this inertialframe
known. The motion of an earthsatelliteis described
for example(seebelow),satellite
and, with the dynamicmethodof satellitegeodesy
trackingstationsare determinedin the inertialsystem.Thesestationpositionsare of
coursetime-dependent
due to the earth'smotion and thev can be relatedto some
MEASUREMENTSOF INSTANTANEOUSMOVEMENTS
terrestrialframe only if the abovementionedmotions of the earth are known. Any
variationin the positionof the stationdue to tectonicmotion manifestsitself as a
changein the coordinatesof the point with respect to the terrestrialframe. The
difficulty is of coursetwo-fold: the motionsof the earth arenot known with sufficient
accuracyand will have to be improvedwith the samespatialtechniquesas usedfor
measuringthe tectonic motions, and the terrestrialframe can only be pragmatically
in one
defined by the coordinatesof the stationsthemseives.A small displacement
frame
and
of
terrestrial
any
concept
absolute
will
the
defìnition
of
the
change
station
positionsbecomes
meaningiess
at the levelof accuracysoughthere.
The spatiai methodsthat are likely to be of interest in the near future are the
to the moon and
measurements
precisetrackingof ciose-earth
sateilites,
iaser-range
these
long-baseiine
intertèrometry.In view of the abovediscussion
radio-teiescope
and
methodswill giverelativemotionsonly. They are in many casescompiementary
campaign.
togetherin any future observing
will probablyhaveto be considered
from severai
stationsit is
simuitaneously
to sateilites
aremeasured
If the distances
possibleto determinethe relativestationpositionsby a purely geometricmethod,
without recourseto any orbital theory. This method is usuailyreferredto as the
usedin the pastfbr
and hasbeensuccessfully
geometricmethodof satellitegeodesy
and for simultaneousdirection and distance
simultaneousdirection measurements
obtainedhavegenerallybeen of the sameorder as the
The accuracies
measurements.
only
data;about 5-i0 m for directionmeasurements
accuracyof the observational
(Gaposchkinand Lambeck, 197l) and about 2-5 m for the directionand distance
and Dargnies,797l).Relativepositionsof
(Lambeck,1968;Cazenave
rneasurements
points separated
by severalthousandkilometerscanbe determinedin this way.
observations
beyondthis levelof accuracyonly the simultaneous
For improvements
of laserdistanceswill provide the means.The method has not yet been fully used
becausemany stationsare requiredto givepreciseand unambiguousresuits.For measuringthe motions of someof the largeplates,as many as six stations,threeon each
plate,arerequiredto givethe completemotionand to ensurethat any relativemotions
betweenthe points on the platecanalsobe detected.It is not, of course,necessary
that all six stationsobservethe satelliteat the sametime. Simultaneousobservations
from subgroupsof any four stationsat a time will provideimportantinformation.For
measuringthe motion of smallplatesrelativeto big plates,four stationswill suffice,
three on the main plateand the fourth on the smallplate, providedthereis no motion
betweenthe stationson the principalplate.With the geometricmethodbeingbasedon
very simple hypotheses,the station positionscan be determinedwith an accuracy
particularlyas the station-satelliteconfiguracomparableto that of the observations
tion canbe optimized.
Station positions can also be determinedby the so-calleddynamic method of
satellitegeodesy.The forcesacting on the satelliteare assumedknown or partially
known so that the satellite'spositionscan be computedat any instantof obierv¿tion
the equationsof thesatellite's
Beeause
asa function of known andunknownparameters.
,'.
58
KINEMATICS
oF RELATIVE
MoVEMENTd$
.î
motion are refèrred to a well-definedinertial frame centeredon the earth's center of.,,
mass,these satellitepositionsalso refer to this inertial frame.The observationsof the, ri
satellite positions and the computedgeocentricpositionsare related to the unknown
orpartially known geocentricstation positions,giving a set of equationswhosesolution yields, amongstother information, the correctionto the station'sposition.The
difficulty with this rnethod is that we need to know all the forces acting on the
satellite and that we have to know very preciselyìhe motion of a frame attachedto
the earth's crust relativeto the inertial frame. At present,an accuracyof between5
and 10 m has been obtainedusing observationaldata of about 10-15 m accuracy
(Gaposchkinand Lambeck, 197l). The advantageof the dynamicapproachis that the
satellitedoesnot haveto be simultaneously
visiblefrom severalstationsat a time. This
meansthat the separationbetweenstationscan be very much greaterthan for the
geometricapproachand that fewer stationsare requiredto completelymeasurethe
motionsof the majorplates.
The immediategoal in satellitegeodesyis for observational
accuracies
of about20
cm in station-satellitedistanceand for this w.ecanusethe existingsatellites,
particularly GEOS i and GEOS2. Accuraciesof about 20-50 cm in stationposition could
be achievedwithin about three yearsfrom now (1972) if a suitableobservationcampaignis organized.Theseaccuracies
are still not very interestingfor measuring
tectonic
motions but they are most valuablefor other interactionsbetweensatellitegeodesy
and geodynamics.
Beyondthis level of accuracywe needto know the earth'smotion
relative to an inertial frame with accuraciesthat are higher than the presentlyused
methods can provide.To achieveaccuraciesbetter than about i0 or 15 cm, special
satelliteswill be requiredto ensurethat we can establishthe point to which we make
the measurementand to minimize, for the dynamicmethod, the non-gravitational
forces.Such a satellitehas been proposedby Weiffenbach(1970) and is now being
studiedin detail (Weiffenbach
and Hoffinan, 1970)for a possiblelaunchdaten 1974.
The altitude of the orbit will be between3,000 and 4,000 km and the optimum
distancebetweenlaserstationsfor the seometricsolutionis of ttris order.
Lunør laser ranging.Distancemeasurements
betweenlaserstationson the earth and
retroflectorson the moon are of the sameorderof accuracyasachievedby rangingto
close-earthsatellites;an accuracyof about 10 cm is envisagedin the near future.
Instrumentalcomplexity is, however,very considerablyincreased,as the moon is so
group (Alley et al.,
much further away. For example,the McDonald Observato.ry
1970) is usinga2.7 m telescope
for transmittingand receivingthe laserbeam,whereas
for close-earthsatellitetracking a 40 cm transmittingopticsis quite adequate(Lehr
andPearlman,797O).
The reflectorson the moon couid be used in a geometricmannerexactlyas discussedabove for close-earth
satellites.However,the geometryis considerablypoorer
now, and reliable resultscan only be obtained if the terrestrial stationsare well
distributed around the entire globe. Benderet al. (1968), for*example,haveinvesti-
MEASUREMENTSOF INSTANTANEOUSMOVEMENTS
59
gated the geometry for very high satellites(maximum distance 110,000km) and
of about
concludethat with 12 globally distributedstationsandwith rangeaccuracies
of about40 cm.
with accuracies
15 cm, it is possibleto measureinterstationdistances
For the moon these results will be poorer. An alternativemethod, similar to the
dynamicmethodof sateilitegeodesy,is to usethe moon'sorbit asreferencefor a short
time period. Alley and Bender(1968) discussthis approachfor measuringrelative
longitudesbetweentwo stations.The methodis to observedistancesto the reflectorat
throughthe
times about equallydistributedabout the momentwhen the moon passes
gives
the time at
measure
of
a
local meridian.The differenceof thesemeasurements
from a secondstationgivesthe
occurs.Repeatingthe measurements
which this passage
differencein longitudesof the two stations.We needto know, however,the precise
variationsin the motion of the reflectoron the moon for the period of observation
(usualiy about 8 hours) to obtain a good geometry.It is not at all clearwith what
accuracywe will be able to predict this motion in the future. We needalsoto know
the'polar motion, the variationsin the earth'srate of rotation and the earthandmoon
tides. Separatingthesevariousphenomenawill againrequire many well distributed
stations.It wouid seemthat the lunar laserrangingmethods are most suitablefor
studying the motions of the rnoon and perhapsthe earth's rotation. For tectonic
motions,the useof laserrangingto artificialsatelliteswould be more appropriate.
One of the limiting factors in laserrangingto objectsoutsideof the earth'satmosphereis the retardation of the laserpulseby the atmosphere.The effect can be
correctedfor, but uncertaintiesof a few cm may remain.Studiesby Hopfield(1972),
however,are very encouragingin this respect:comparisonsof refractioncorrections
with correctionbased
and surfacemeasurements
computedfrom model atmospheres
to betterthan I cm nearthe zenith.
on atmosphericsoundingsindicateagreement
radio interferometry(Y.L.B.I.).In the classicalinterferometricmethods
Long-baseline
at two antennasseparatedby a known bæea radio signalis receivedsimultaneously
line. Becauseof the different path lengthsbetweenthesetwo terminalsand the radio'
signalsource,the two signalsreceivedwill exhibit a phasedifferencethat is a measure
If the baseiineis known the orientation
of the path lengttrdifferenceof thesesigrrals.
of the sourcerelativeto the baselinecanbe determined.
To measurethe phasedifferencethe receivedsignalshaveto be comparedagainsta
standardoscillator.In the past,whenthe local oscillatorswere not very stable,the two
antennasrvereconnectedelectrically and the signalscomparedto the samefrequency
standard.This need for a direct connectionlimited very much the maximum length of
baselinethat couid be achievedbecauseof electronicdelaysin the land line. This need
alsomeant a limit to the resoiutionthat can be obtainedsincethe longerthe baseline
the larger the phase difference and the more precisethe determination. Since the
developmentof stableatomic clocks theseproblemscan now be overcome.The receivedsignalat each terminal is noù recordedon tape, togetherwith the time coÍitrol
derivedfrom the frequcncy standa¡d.At somelater date, the two tapescanbe brought
60
KINEMATICSOF RELATTVEMOVEMENTS
together and analyædfor the phasedifference.Now the length of the baselineis only
limited by the fact that the source must be simultaneouslyvisible from the two
appliedin resolvingstellarsources;
antennas.The methodhas beenmost successfully
that is in measuringdifferencein angles.Cohenet at.(1968),in a summaryof results,
report resolutionsof better than 0.b01 sec of arc for a baselineof 6,300 km. The
inverseof those resultswould be that we can obtait the baselinewith a comparabie
are known and if we can overcomeseveralother
accuracyif the source
difficulties.
Generallywe will not know the sourcepositionsto solvefor baselineand source
positionsat the sametime. Also, we haveto improvethe stability of the clocks.For
that the local oscillatorsare
it is only necessary
the sourceresolutionmeasurements,
stablefor the period of observationand drifts from one period to the next areunimportant as we are makingreiativemeasurements.
For the baselinedetermination,we
frequencystandards
phase
Even
control.
hydrogen-maser
requiremuch more stringent
may not be adequatefor centimeteraccuracies.
Atmosphericrefractionalsois a probat radio frequencies
both the
because
lem, more seriousthan with lasermeasurements,
troposphericwater vapor content and electron densitiesin the ionospherebecome
important. If presentavailableatmosphericmodelsare used,the uncertaintyin the
refractioncorrectionis of the order of 10-20 cm and is dependenton the frequency
of the ¡adio-source.
Improvementscanbe expectedif the atmosphericdensitiescanbe
meazuredat the time of observation(Dickinsonet al., 1970).
'
Up to now the resultsfor baseünedeterminationhaveprovidedaccuracies
that are
similar to thoseobtainedby satellitegeodesymethods.Recentresultsby Cohenand
Shaffer (1971) between stationsin Australia and on the west coast of the United
Statescomparevery favorably with resultsobtainedby Gaposchkinand Lambeck
(197I) for the coordinatesof nearby satellitetracking stationsthat havebeenconnectedto the corresponding
radio-telescopes
by groundsurvey.
So far we have not said anything about the nature of the radio-sources.
Several
possibilitiesexist. Stellarsourcesare most convenientbecausethey are alreadythere
and becausemany of them lie outsidethe galaxyand are not expectedto exhibit
proper motions(relativemotions of the stars).Thusthey form an ideal
measurable
inertial referenceframe with respect to which the earth'smotion in spacecan be
measured.The disadvantage
of natural sourcesis that the signalsreceivedon earthare
quite weak requiringvery largetelescopes
in order to observethem. This meansnot
only that there may be somedifficulty in relatingthe eiectroniccenterof the instrupointson the "rigid" earthwith betterthan centimeteraccuracy,
ment to well-defined
but it alsomeansthat the useof V.L.B.l. techniquewill be a costlyway of measuring
tectonicmotions;particularlyasseveralstationswill be requiredon eachplatein order
to separateout the variousother geophysical
phenomena
introducingvariationsin
stationpositions.Artificial radio-sources
canalsobe used.Micheliniand Grossi(1972)
for examplehave attemptedto use radio-signals
from the geostationary
satellite
placed
ATS-5.Radiosourceshavebeen
on the moonbv someof the Apollomissions.
MEASUREMENTSOF INSTANTANEOUSMOVEMENTS
6l
The practicaladvantageof these approachesis that the signalsreceivedare much
can be usedand that the analysis
strongerso that very much smallerradio-telescopes
of data is simpierand lesscostly than it is for stellarsourceobservation.On the other
hand,we wiil haveto know preciselythe forcesacti!g on the satellite.
Terrestrialgeodefic measurements.^ simple technique of measuringdisplacements
alongfault zonesis to establisha line of surveymarksacrossthe fault and to measure
the offsetsof the line. This can readilybe donewith an accuracyof a few millimeters
so that it offersa very simpleand accuratemethod of measuringfault-creepslippage
faults (Tocher, 1960;Nasonand Tocher,
along the SanAndreasfault and associated
1970). The latter, for example,report motions of about 1 mm/month acrossthe
with the SanAndreassystem.Continuousrecording
fault, a fault associated
Calaveras
are now being made at some20
of úe movementsis possible,and measurements
points alongvarioussectionsof the Californianfault system.However,fault zonesare
not alwaysconfinedto very narrowzonesandwe will often requiremore sophisticated
methods.
Classicaltriangulationmethods,usingtheodolitedirection and invar tape distance
havebeenusedin the pastto detecttectonicmotionsalongfauit zones.
measurements,
An areamuch investigatedin this manneris the San Andreasfault (Whitten, 1960,
1970;Meade,1966).Meade,for example,givesresultsfor a surveymadein 1944utd
along the fault are quite convincingiyestabresurveyedin 1966. The dispiacements
lisiredand in good agreementwith fault-creepslippageratesknown to occur on the
fault; the differencein positionsfor points on either sideof the fault amountedto 60
cm in 20 years.Thesemethods give convincingresults only when the time interval is
can
about 20-30 years.More precisemethodsare requiredso that the displacements
be determinedover shorter time intervals,in order to obtain a cleareridea as'to
whetherthesemotions are continuousor sporadic.Electronicdistancemeasurements
fot example,giveaccuracies
providea meansof doing this. Conventionalgeodimeters,
of about 2-3 parts in 106, or about 4-6 cm in distancesof about20 km. A geo'
the SanAndreasfault systemfor more than 600 km has
dimetertraversecriss-crossing
(Hoffmann,1968).
by the CaliforniaDepartmentof WaterResources
beenestablished
severaltimesduringthe last l0years and a
Partsof the traversehave beenremeasured
generalmovementof about 2-4 cmlyearhas been found aiong the fault although
occur.
displacements
many anomalous
Further improvements are possible with laser instruments and many tests have
indicatedthat a precisionapproaching1 in 107 is possible,but that, asfor the spatial
the atmosphericrefraction limits the accuracyto below this value.The
measruements,
problem is not insurmountableif the atmosphericdensity alongthe ray path can be
observationsaretaken horizontally through an extremelyvariableatmosphere.But the
problem is not insurmountableif ttre atnospheric density along the ray path cán be
measured,either by discretesamplingor by measuringthe integrateddensity alongthe
62
KINEMATICS OF RELATTVE MOVEMENTS
path. The latter canbe observedby measuringthe distancesat two optical frequencies,
With thesetechniques,accusincethe refraction correctionis frequency-dependent.
raciesof I in 107 can be realized(Owens,1967).For a distanceof 30 or 40 km, this
representsdistanceaccuracybetter than I cm.
Astronomical obsert¡ationsof latinde- Observationsof astronomic latitude can, in
principle,alsoprovidea meansof testingthe plate-tectonics
hypothesis.The astronomical latitude of a station is defined asthe complementof the anglebetweenthe earth's
instantaneous
axis of rotation and the station'svertical,and is observed,for example,
.measuring
the zenith distanceof a known star as it passesthrough the station's
by
meridian.Variationsin the astronomicallatitudearethereforecausedby polarmotion,
by earth tides and by tectonicmotionsof the stations.The difficulty liesin separating
thesevariouscomponents.The polar motion hasperiodicparts as well as a possible
secuiardrift, and the former canbe eliminatedby suitablyaveragingthe latitude data.
The earth tide effectscan usuallybe adequatelymodelled,as far as the short-period
terms areconcerned,but this cannotbe saidfor the long periods,suchasthe 18.6-year
period, as the earth's elasticresponseto long-perioddeformationsis poorly understood.
Latitude observationshave been made for some70 yearsby five stationsof the
becauseof
International Latitude Servicebut the resultsare quite inhomogeneous
Thelatitudedataat eachstationare
variationsin methodsof reductionand observation.
averagedover a 6-yearintervalto eliminatethe annualand l4-month Chandlerperiod.
The interpretationof the variationsin the meanvaluesfor eachof the stationsdepends
on the hypothesismade.Markowitz (1968), for example,analyzedthe dataassuming
that the stationsarefixed with respectto eachother and found a seculaîdrift of about
0.003"/year along longitude 65'W. Whitten (1970) on fhe other hand, interpreted
Markowitz'sresultsassumingthere to be no secularmotion of the pole. He found that
it was possibleto explainthe variationsin meanpole positionif North Americahad
turnedabout5" clockwiseandEurasiaabout5o anticlockwise
duringthelast107years.
Theseresultsare almostidenticalto thosefound by Le Pichon(1968) from an altogetherdifferent method.
over a year is about 0.02" (or 0.6 m)
The accuracyfor a mean latitude averaged
accordingto Markowitz(1968). Thus at least50 yearsof good data is requiredto
observeany drift betweentwo stationson different plates.For a separationof the
tectonicmotionsfrom the secularmotion of the pole,manymo¡e stationsarerequired
than now participatein the InternationalLatitudeServiceobservingprogram.
Length of sinkingzone method
Isackset al. (1968) demonstrated
that the seismicactivitywithin deepand intermediate seismiczonesshowsa well-definedrelativemaximumin somedepth rangein the
upper mantle. They suggested
that the length of the seismiczone,from the surfaceto
its maximum in activity, is a measureof the amountof underth¡ustingduringthe past
MEASUREMENTSOF INSTANTANEOUS MOVEMENTS
63
10 m.y. The suggestionwas basedon the observationthat the time which hasbeen
necessary
to thrust into the mantlea lengthof plateequalto the presentlengthof the
seismiczone,usingthe calculatedslip ratesof Le Pichon(1968), did not varywith the
calculated
slipratebut remainedcloseto 10 m.y. (Fig.19).
Fiji x
=
¡,.topon-Monchurio
:<
asourh amerìco
/
Mortonas+R,tukyu
-
,t
N e w H e b r i d e sO
f 20s -1705
I n d o n e s loo
l¡J
Tonqoo
z
t\¡
=
I
//
,/
./
P h ¡ l i p p r n e+s
P ol o u
xDeepesl Events
¡lKurite-Komchotko
New Zeolond
,/
'
x S p o n i s hD e e p
Eorthquoke
-
*/
V
,/ Kerñodec
)/a
t!
q,
'
New Zeolond)
North lsloîd
,/
/'/
,/
:c
,/
F(9
a s. arosko+ E. Areuf ions
tD C,entrot Amerrco
Mexrc¡f-anto^mo1
1r9
z
Souln sonowrcn
UJ
,/
.
Z e o l o n d- S o u t h l s . + M o c q u o r i e R i d g e
JNew
, A l e u t i o n sn e o r 1 7 O oE
Zc-tr À1".1^ .
A.Ç¡rrrrhle¡n
.Chi¡o,
I
CALCULATEOSLIP RATE
I ARC (CM,/YEAR)
Fig.19. Calculatedratesof underthrusti¡g (Le Pichon, 1968) versuslength of seismiczones.Crosses
indicate unusual deep events.Note the approximately linea¡ increasein the length of the zone with
respectto the calculatedslip rate. (After Isackset at., 1968.)
The most remarkablecorrelationbetweenlengthof seisrniczoneandpredictedslip
rate existsfor plate bounda¡iesaiongwhich the distanceto the pole of relativemotion
variesgreatly. An examplestrikingly illustratedby McKenzieis the Pacific-India plate
boundary from Macquarie Island to Tonga where the length of the seismiczone
increasesfrom less than 60 to 800 km as the predicted rate increasesfrom 0 to 8
cmfyear(McKenzie,I969a; seeFig.73). Another exampleis the Pacific-Philippine
plate boundary from the Palau-Yap trenchesto the Izu-Bonin trench (Katsumata
and Sykes,1969).
Isacks et al. suggestedtwo possibleexplanationsfor this correlation. Either the
present'seismiczoneswere createdduring the most recent 10 m.y. old episodeof
spreading,or 10 m.y. was the thermal time constant of the plate within the seismic
zone. The first explanation can now be ruled out, as the JOIDES results(Maxwellet
al., 1970) havedemonstratedthat there was no worldwide pauseof spreadingduiing
the last 70 m.y. McKenzie (1969a) has shown that _!hesecondexplanationis possible
64
KINEMATICSOF RELATIVEMOVEMENTS
(seeChapter7: Thermalregimeof sinkinglithosphericplate,pp.212-215). If one
that thereis a temperaturelimit ?nbeyondwhich the materialdoesnot behave
assumes
anymore as a brittle solid, the plate will loseits-identity within the mantle onceit has
reachedthis temperature.The time necessaryfor the plate to rcactrT is independent
of the consumingrate, provided thät the mantle is at a constanttemperatureand that
the changein pressureof the plate as it sinks does-notproduce a changein temperaarenot validduein particularto pressureture (McKenzie,1969a).Theseassumptions
approximaand adiabaticheating,but they leadto a reasonable
inducedphasechanges
In addition,the thicknessof the
tion if one reasonsin termsof potentialtemperatures.
plate as it entersthe consumingzonemay be quite variabledependingon itsageand
the time necessaryto heat it will vary roughly as the squareof its thickness(\il.J.
Morgan, personal communication, 1972). In conclusion,this explanation would
for a plateto reachthe temperatureT is about 10 m.y.
assumethat the time necessary
The difficultiesin usingthis methodarenumerous.First, it is basedon an empirical
heating
correlationwhich can be explained,but not yet predicted,by the progressive
arepoorly known or unknown:
of the piate as it sinks.Too many variableparameters
distribuphasechanges,
actualgeometryof plate,thermalconstants,pressure-ilcluded
tion of temperaturewithin the mantle. Second,one hasto assumethat the plate did
not begin to be thrust into the mantle iater than 10 m.y. ago.This is not necessarily
true. Third, the length of the zone should be measuredaiong the directionof slip,
which shouldbe estimatedfrom fault planesolutions.Fourth, Fig.19 showsthat the
empiricalreiationshipbetweenrate of slip and [engthof zoneis poorly defined.
This method should,in fact, be usedto obtain a better knowledgeof the thermal
time constant,and consequentlythe thickness,of the different deepseismiczones.
Sucha study is possiblenow that we havemuch more detaileddataon the geometry
of the deep seismiczones(Isacksand Molnar,1'97I) and better calculatedratesof slip
(Morgan, 197Ib). In this study, one strould take the age of the lithosphereinto
that the Japanseismiczone,in which the lithoaccount.It is alreadyvery suggestive
sphereis very old p 200 m.y.), is the longestand the MiddleAmericaseismiczone,in
which the lithosphereis very young (a few million years),is among the shortest
(Molnar and Sykes,1969).Clearly,this studywould setimportantneededconstraints
on the thermalmodelsof the sinkingpiate.
of direction of relativemotion
Methods of meøsurement
The transþrm-føult method
Transformfaults a¡e plate boundariesaiongwhich surfaceis conservedand consequently lie alongthe directionof relativemotion betweenplates.A knowledgeof their
geometry uniquely definesthe di¡ection of motion. This methôd is very simple to
apply and quite powerful. Heezenand Tharp (1965) were probably the first who
that the motionsof the continentson either sideof a mid-oceanridgemay
suggested
be inferred from the directionof fracturezones.Morgan(1968) and Le Pichon(1968)
MEASUREMENTSOF INSTANTANEOUSMOVEMENTS
65
uSedtransformfaults to obtain the directionsof relativemotion betweenplatesin a
quantitativeway.
Ridge-ridgeand ridge-arctransformfaultshavethe property that a residualinactive
traceexistsin their prolongation,beyondthe accretingplate margin.Thus the active
part givesthe presentdirection of relativemotion while the fossilpart givesa geologi
cal record of the past relativemotions of the two platesthrough time. This type of
transformfault generallylies underwaterin the oceanand hasa very cieartopographic
expression.Menard (1955) first describedthesetopographicfeaturesin the Pacific
Oceanand calledthem fracturezones,the namebeing appliedto both the activeand
fossil parts of transform faults. Menardand Chase(1970) defìnethem as "long and
naffow bandsof grossiyirregulartopographycharacteÀzed
by voicanoes,
linearridges
and scarps,and typically separatingdistinctivetopograph-ic
provinceswith different
regionaldepths". Fracturezoneshavenow beendiscoveredaiongthe wholelengthof
the accretingplateboundaries.
Their length,their size,and the offset of the ridgecrest
alongthem are highly variable.In general,the sizeof the topographicfeatureincreases
with the length of the offset and this is understandable
on mechanicaland thermal
considerations.
The lengthof the fracturezoneshowssomecorrelationwith the length
of the offset.This resultsfrom the fact that transformfaultswith largeoffsetsarevery
stable,whereastransformfaults with smalloffsetsare not and may disappeardue to
second-order
modificationof the geometryof the accretingplate boundary.However,
no ciearcorrelationhasbeenfound betweenrelativevelocity alongthe boundaryand
sizeof the fracturezone.
Typically,fracturezonesconsistof a lineartrougha few km (- 10 km) wide anda
few hundred metersdeep.This trough is often obscuredor filled by sedimentsand
may be difficult to map without a seismicreflection survey. The trough may be
borderedby one or two basementridgesor scarps,which may havea heightof a few
kilometersand are easyto map. The total width of a largefracture zone is about 30
km. However,the structureis often complexand many exceptionsto the preceding
descriptionexist, as shownby Fig.20.The ma'gneticfield is generallyfeatureless
over
the fracturezone and this has beenexplainedby Cann and Vine (1966) by a demagnetizationof the rocks due to shearmetamorphism.
On eachsideof the zoneof quiet
magneticfield, the Vine-Matthewsmagneticlineationsare offset by a distanceequalto
the offset of the ridge crest at the time they werecreated.This providesprobablythe
surestway to identify and map a fracture zone,especiallyif the offset is not large.The
preciselocationsof epicenterscan also be used to determinethe geometryof the
aqtivepart of afracturezone(Sykes,
1963,1965,1966a,L967;SykesandLandisman,
1Þ64).
The major difficulty in surveyinga fracture zone is that it is possibleto go from one
fracture zone to a parallelone, from one crossingto the other. This is especiallytrue if
there are numerouscloæly spacedfracture zonesof equalimportanceand whorethey
are disruptedby an important charr-ge
in trend. Oncethe fracture zone is mapped,it is
necessÍrryto determine the exact line of slip. Yet-' it is not clear which part of the
66
KINEMATICSOF RELATIVEMOVEMENTS
¡,Dle
¡-e
LINE of VOLCANOES
@
TROUGH
SIMPLE,SHALLOW
@
rRouGHs
GREAT
( lor)
l r r
@
' r r r r r r r r r
SCARp
-æ'e
I
r r l r r r r r
I
t
I
(high)
<1
:ll:;tJ:3""93rtff.iåÈ3
--
G R E A TR I D G E S
æ
COMPLEX TROUGH
o n d G R E A TR I D G E
characteristic
of fracturezones.(AfterMenardandChase,
FîÉ.zO.Stytizedtypesof topography
1970.)
fracture zone can be taken as the exact witnessof the path of a flow line and how
much the fracture zone can deviate locally from a flow line. Francheteauand Le
Pichon (unpublished)have shown that, in the Gibbs(or Charlie)fracturezone,near
52oN in the Atlantic Ocean,the axis of the trough follows the samesmallcircle to
whereasthe scarpson eachsidemay deviatefrom a
within 3 km overlong distances,
small circle by as muchas 10 km locally (Fig.z1).In general,the fracturezonesdefine
flow lines to whithin 3-5 km. The effects of the smalldeviationsfrom latitude ci¡cles
are probably absorbedby the elasticityof the plates.Thesedeviations,however,put a
limit to the resolutionof the direction of relativemotion one canget from a fracture
zone.
Theoretically, the informationscontainedin the geometryof the flow line followed
by a transform fault is sufficient to determine uniquely the location of the pole of
relative motion. This results from the fact that the curvatureof a small circle varies
with the dista¡ce 0 to the pole of rotation as l/R sin 0 whereR is the radius of the
spherical earth. Thus, the variation of the curvatureobeys the samelaw as the
variation of the rate of accretionand containsthe sameinformation as fa¡ as the
In practice,it is very difficult to definepreciselythe
location of the poleis concerned.
curyature of the iatitude circle along the small portion of an active transform fault,
one is generallyonly
unlessone is very closeto the pole of rotation. Consequently,
able to definea locusof the pole,which is the greatcircleperpendicularto the portion
of the active transform fault. The only useful information then is the azimuth of the
transform fault, which can be consideredas the derivativeof the different mapped
positions of the fracture zone. Locally, this aeimuthmay be greatly in error and a
Many authors(Morgan,1968 ; Le Pichon,1968)have
smoothingmethod is necessary.
useda visualsmoothingmethod.It is probablybest to do the smoothingnumerically
portion of
over the different positionstaken at equalintervalalongthe corresponding
more completelylater in this chapter(p. 115).
a transformfault. This will be discussed
with accretingplate boundariesare easüyidentifìed,
If transformfaults associated
this is not so in the caseof consumingplate boundaries,whereit is not possibleto
recognizewhether a topographictrench is due to thrusting of one plate below the
3
r
Þ
(/)
c
H
rd
z
È
(t)
'11
z
U)
È
Þ
ztsl
z
frj
c
(t)
Fìg.21. Topography of the Gibbs (or Charlie) fracture zone a¡rdcomputed srnallcircle adjustedto
the points shown by black dots. (After Francheteauand Le Pichon,unpr.rblished
wo¡k.)
E
3
z
Fl
(A
;j
68
KTNEMATTcs
oF RELATTvEMovEMrNrs.ì
.,:i
'other or to pure strike-slip.Thus, along conzumingplate boundaries,earthquakefault-:l
:
plane solutionsand geodeticmeasurementsarethe only possiblemethods.
- ..;:
'
Fault-ptanesolution method
Along a plate boundary, the movement on the fault plane is mostly the result of
successivedislocations which occur each time the .accumulatedstrain exceedsthe
plate's elastic limit. The opposite sides of the f¿rult then return to a position ol
equilibrium, releasingin elastic wavesa gteat part of the accumulatedenergy.For an
earthquakeof magnitudelarger than 5.5-6, thesewavescan be recordedover the
whole earth. The first motion of a compressionalP-wavecan be either a comprôssion
or a rarefaction. The distribution of these first motions, as recorded at different
stationssurroundingthe epicenter,can be usedto obtain the senseand type of displacementwhich occurredon the fault plane.The "fault plane" or "focal mechanism"
solution is a powerful way to determinethe direction of relative movementbetween
plates, in spite of the fact that the precision of a given solution rarely exceeds
10'-15" and that there is an inherent ambiguityof 90' which must be raisedon the
basis of other considerations.The techniquewas perfectedin the 1960's (Stauder,
1962; Wickens and Hodgson, 1967) and the resultsshowedthat all well recorded
earthquakescould be satisfactorily explained by the equivaientdouble-couplepoint
source (Honda, 1962). The technique was first applied to plate teotonics by Sykes
(1967) who demonstratedits remarkable efficiency with the uæ of the long-period
seismographrecords of the World Wide StandarduedSeismographNetwork which
commencedoperation n 1962. Previously,solutionsslroweda large proportion of
inconsistentreadings,due to the non-homogeneityof the network. In additionto the
P-waveonset method, fault-planesoiutionscan be obtainedfrom the di¡ectionand
polarizationof S-waveonset(e.g.,Stauder,L962),from surfacewaves(Brune, 1961)
and from the amplitudeof free oscillationsdata(Giibertand McDonald,1961).These
last two methodsare more diffìcult to useandlessreliablethan the methodsbasedon
onsetsof P- and S-waves.In the following, we wili briefly discussthe P-waveonset
method,makingextensiveuseof avery cleardiscussion
by McKenzie(1971b).
Let us considera purely horizontal motion on a verticalplane,the senseof motion
being.indicatedby the arrowson Fig.22.Intuition suggests
that points in front of the
aüows are being pustredwhile points behind them arebeingpulled. This is confirmed
by dislocationtheory. In directionssuchas.B and E,the initial motion is awayfrom
the focus of the earthquake(compression)while in directionssuch as.4 and F it is
toward the source(rarefactionor dilatation). The radiation fìeld is divided by two
perpendicularpianesinto dilatationaland compressional
quadrSnts.
Theseplanes,called
nodal planes,are planesalongwhich in theory the motion ofP is null. This property,
actually,heipsto determinewhether the stationwherethe wavesarerecordedis close
to a nodal plane.One of the nodal planesis the fault plane,the other is the auxiliary
plane which is perpendicularto the slip vector. It is not'possibleto decidefrom the
wave-radiationpattern only which plane is the fault-plane.This ambiguity is funda-
MEASUREMENTSOF INSTANTANEOUSMOVEMENTS
ó
o¡totot¡on
.,,rft
(2) compression
1,",,.,;{_øry"
(3)Dilototion
Fig.22.Theradiationfield of a strike-slip
earthquake.
Thea¡rowson the raysmarktheinitial
direction
of motionof theground.Notethattheauxiliary
planeisperpendicular
to theslipplane.
(AfterMcKenzíe,
197Ib.)
mental to the double-couplesourcemechanism.The decisionmust be madeon the
basisof other informations:the earthquakemay havebeenaccompanied
by a surface
breakand the strike of the breakaswell as the senseof motion alongit mustagreewith
one of the two possiblesolutions.The distributionof aftershocks,the ellipticity of
isoseismal
linesand the directionof propagation
of the initial breakareotherpossible
waysof removingthis ambiguity.
Let us definethe normal to the auxilary plane(calledits pole) by the unit vector
u1,the normalto the fault planeby u, and the unit vectoralongthe verticalby ar.
The slip vectorin the fault planeis parailelto the pole of the auxiliaryplanezl. The
horizontalprojectionof the slip vectorthen is uh = n t - (u, . ar) s, and consequently, the strike of uh may be easilyobtainedby adding90o to the strike of the auxiliary
plane.If the auxiiiary planeis vertical,as in Fig.22,the strikeof u¡ is the sameasthe
strike of the fault plane.But this resultis not true otherwiseand the strikesof u¡ and
of the fault plane in the horizontal plane will in generalbe different. McKenzieand
Parker(1967) showedthat, alonga givenpiate boundary,one can use the horizontal
projection of the slip vector to raisethe ambiguity betweenfault plane and auxiliary
piane. This results from the fact that u¿ should have a consistent direction as one
movesalongthe plate boundary: the rigrdity of platesimposescontinuity in direction
of relativemovementbut the fault plane itself may haveany direction. Thus a choice
betweenthe two planesmay be madeon the basisof the necesssary
continuity of a¿
alongthe plate boundary.The intersectionof the two planes,calledthe null vectoror
.B-axisis in the direction ü1 x rr2. The P-(or Pressure)axis lies in the dilatational
quadrant, is normal to the null vector and bisectsthe two pianes whiie the Z- (or
Tension) axis similarly bisects the compressionalquadrant and is normal to the null
vector; written as unit vectors they lie along (ø1 +u2) I,t2 and (21 -u) l,[2.
\ryhithin an homogeneousmaterial the triaxial stress-field applied to the material
slrouldcorrespondto P, B and T, whereP is the axis of maximum compressivestress,.B
the axis of intermediateand ?trof minimum compressivestress.It has beenshownby
Isacksand Molnar (1971) that tlis is true within the plates sinking in the mantlé in
which earthquakesoccur by failure within an- homogeneousmaterial. However,
70
KINEMATICSOF RELATTVEMOVEMENTS
producedby
McKenzie (1969b) has pointed out that this is not true of earthquakes
plane
is alreadya predifferential motion between plates. This is becausethe fault
involvedin shallowearthquakesare
existing plane of weaknessand that shearstresses
at least an order of magnitudetoo small to proã.rct'fracturewithin an homogeneous
failure may occur atananfle differentof 45"
fault-freematerial.Thus,in thesecä,ses,
from the axis of greateststressand the P,.8, I-axeshaveno simplerelationto the
triaxial stressfield.
In practice,to obtain a fault-planesoiution on the basisof recordsof the radiation
wave pattern at stations far away from the earthquake,it is necessaryto know the
di¡ilctions in which the rays left the focus. But the path of the ray dependson the
veiocity structureof the earth along its path, on the angulardistancebetweenthe
sourceand the receiverand on the hypocentraldepth. Thus, the calculatedanglei
between the rays and the downward vertical may be affected by sizeableerrors,
especiallyfor crustalearthquakeswhich occur in regionsof largevelocity gradientsand
if the structure is not radially homogeneous.This will be speciallytrue if the nodal
planes are not steeply dipping as in a strike-slipsolution.Valuesof i for a given
distanceand hypocentraldepth can be .obtainedfrom tables (seeRitsema,1958;
Sykes,1967).Allowancefor a velocity in'the focal regionlessthan 7.8 km/sec(crustal
é =r + ' , Ð = e z ' e
O
9
"t
#o
x SP-
MAY
t9,
1963
Fig.23. Example of a nearly pure strike-slip mechanism for an event on a No¡ttt Atlantic ffacture
open circles:dilatations;crosses:wave
zone (event 5 of Sykes, 1968). Solid circles: compressions;
cha¡acter on seismogramindicates station is near nodal plane. Smallersymbols representpoorer
data. ó and 6 are strike and dip of the nodal planes.A¡rows indicate senseof sheardisplacementoR
the plane that was chosenas the fault plane. (After Sykes,1968.)
MEASUREMENTSOFINSTANTANEOUSMOVEMENTS
II
is madevery seldom,althoughit could easilybe done,usingfor example
earthquakes)
Ritsema'scurves.
Knowing the anglei, a convenientway to representthe radiationpatternin two
dimensionsis to imaginea small spherecenteredon the focus of the earthquakeon
onto
which the first-motiondirectionsareplotted and to project the lower hemisphere
the
iatter
case
projection.
In
equal-area
or an
a horizontalplaneusingI stereographic
for example,the two coordinatesused to describea station are its azimuth at the
epicenterand its radial distanceà whereR=,f 2 sin (i/2), usinga sphereof ra<iius
unity. Thus, the intersectionof the spherewith the plane of projectionis the circle
of tàuit-planesolutions
R = 1. Fig.23-25 show three examplesof representations
It
solutions.
normaifaultingand thrust-faulting
for dominantlystrike-slip,
respectiveiy
is easyto seethat the strike of a nodalplanein the horizontalpianeis obtainedby
the angieon the circleof radiusR = I ciockwisefrom the north.The strike
measuring
i
of ø¿ is obtainedby adding90o to the strikeof the auxiliaryplane.The complement
the distanceR = ',f 2 sin (i/2). For a
of the dip of a pianeis obtainedby measuring
"pure" strike-slipsolutionwith a verticalfault plane,the circleis dividedinto four
É =3 2 : s =s 2 ' E
I
I
I
I
¡r
t/
p^o
۾:
õ
o
I
I
I
I
d
o
q
Do
X O
l
I
I
I
I
MAR.20.
1966
I
Fig.24. Example of a nearly pure normal-tàulting solution of an East Atiican earthquake (event l3
of Sykes, 1968). Note that the two nodal planeshave approximately the same strike and that
consequently,the horizontal projectioh of the slip vector is uniquely detìned (with a possibleerror
of 18"1;C and T correspondto the P and T-axes.n = J2 sin [(90'-6 )/2]. (After Sykes,1968.)
72
KINEMATICS OF RELATTVE MOVEMENTS
rì
*\:353'. ô=5e'w
.zs: 8.
I
8'E
\
\
I
I
I
.¡.1
O
o
aa
r¡
þ"\
? ¡t'rt¡
:
I
ì
o
I
I
,l
I
I
I
ìEPT. t?, t964
Fig.25. Example of a nearly pure thrust-faulting solution of an earthquakeon MacquarieRidge
(event17 of Sykes,1968).(After Sykes,1968.)
equal quadrantsand the motion along a nodal plane is such that it goesfrom a
diiatationalto a compressional
quadrant.For a purelynormal-faulting
solution,the
centerof the circleis occupiedby an ovalfilied with a dilatationalquadrant.For pure
thrust-faultingthe oval is filled with a compressional
quadrant.For theseiast two
cases,there is no ambiguityin determiningthe strikeof u¡ as the two nodalplanes
have the samestrike. However,a combinationof strike-slipwith normal-faultineand
thrust-faultingis possible.
This method is the most generalmethod yet devisedto study motions between
platesas it appliesto any plate boundaries.SinceSykes(1967),it hasbeenwidely
appliedby many authorsand first by McKenzieandParker(1967) alonga consumìng
plate boundaryand by Isackset al. (1958) overthe wholeearth.The precisionobtained with it is routinely of the order of 20" for an earthquakeof magnitude6 or
greater,and could normally be improved considd¡ablyif a proper distribution of
recordingstationsexistedandif the upperstructureat the epicenterwerewell known.
However,the distributionof recordingstationson the focal sphereis most often very
(DaviesandMcKenzie, 1969)
inhomogeneous
There is an interestingapplicationof this methodto micro-earthquake
studiesusing
MEASUREMENTSOF INSTANTANEOUS MOVEMENTS
t3
local recordingnetworks.It would be of greatinterestto comparethe resultsobtained
to largeearthquakes.
to thosecorresponding
relativeøngularvelocity betweenplatesand estimationof
Calculationof instøntaneous
eftors
Introduction
relativeangularvelocitybetweentwo platescan be represented
The instantaneous
co.The problemof determiningthe relativemotion betweentwo
by a pseudo-vector
the
piateson the earth'ssurfacethen reducesto the calculationof th¡ee parameters,
coordinatesof {rl, and their probableerrors.Alternativeiy,the determinationof o¡ can
be obtained in two steps.One first computesk, the unit vector along or (entireiy
determinedby two parameters,i.e., the ratios of two cartesiancoordinateswith
respectto the third one). This is equivaientto finding the location at which the
(latitude@
whichis defìnedby two parameters
rotationaxispiercesthe earth'ssurface,
and longitudeÀ). Then one has to obtain the third parameterc..r,the magnitudeof or.
In practice,to make this determrnation,we havea population1 of pointsalongthe
commonboundaryof the two platesat which the magnitudev and (or) the direction
vlv of the relativevelocity vector v at a point r havebeenmeasured.We will define
v and u, are
u=vlv.lt is important to note that thesetwo setsof measurements,
independentand are not obtainedin generalat the samepoints. Thus there are two
setsof errors,thoseon u and u andnot simplyone,the errorson v. In addition,these
errors are often difficult to appreciateand systematicerrors may be present.For
example,fault-planesolutionsare often affectedby unknown asymmetryaroundthe
earthquakefocus, especiallyif the fault planeis not vertical.Finally, as the data are
generailydiscontinuousand inhomogeneous,
it is not easyto test their internalconsistency.
Very little systematicwork has been done up to now on the problem of determining c¡ from the y anda measuredfor ^Iand evaluatingits probableerror.McKenzie
and Parker(1967) apparentlydeterminedthe America/Pacificunit rotation vectork
by constructionof great circlesperpendicularto the horizontal projectionsof slip
vectorson a giobe.Morgan(1968) obtainedo from a bootstrapoperationcombininga
to
on distances
similarbut computerizedgeometricalconstructionwith considerations
polesbasedon the law of variationof v. He first obtainedthe location of the eulerian
to u at point r
pole (À, @)by geometrical
constructionof greatcirclesperpendicular
(Fig.9,upperpart). The greatcirclestend to bundlein an areawhich is elongatedalong
the generaldirectionof the great circles.Then a visualinspectionof the variationof p
with distanceto the chosenpole (Fig.9, lower part) led him to choosea "best"
position for the eulerianpole within this elongatedarea.This then fìxed the valueof
both /c and<,r.
Le Pichon (1968) used insteaä two computerized search methods. In the first
method, the k chosenwas suchthat the function.{ = Ð(g- rlù' be minimized,where0