GEOPHYSICAL RESEARCH LETTERS, VOL. 20, NO. 10, PAGES 915-918, MAY 21, 1993
A TEST OF THE GREAT CIRCLE
APPROXIMATION
IN THE ANALYSIS
OF SURFACE
WAVF.•S
Dolors Alsina and Roel Snieder
Departmentof TheoreticalGeophysics,
UtrechtUniversity
ValerieMaupin
Institutede Physiquedu Globede Strasbourg
Abstract.A new way of reconstructing
wavefrontsis developedfor testingthe validityof thegreatcircleapproximation in dispersionanalysisof surfacewaves.Phaseand time
delaysbetweenpairs of stationsof an array are usedto reconstructthe wave fronts or surfacesof equal phase.This
tech•que is applied to fundamentalmode Rayleighwaves
recordedat the NARS stationsdudlag the ILIHA (Iberian
LithosphereHeterogeneityand Anisotropy)project.The resultingwave fronts are comparedwith the theoreticalones,
i.e. calculatedassumingthat the waveshavetraveledalong
the greatcirclepath with a constantphasevelocity,andthe
angle betweenboth fronts is calculated.Waves traveling
mosfiythrougha laterallyhomogeneous
oceanicpathbefore
arrivingto the Iberian Peninsulaare analyzedfor the frequencyrangeof 10 to 60 mHz. Theseshowanglesup to 8
degreesbetween the theoreticalwave fronts and the wave
frontsreconstructed
usingthe delaydata. This impliesthat
we makea relativeerrorlessthan 1% whencalculating
pathaveragedphasevelocitiesignoringthe deviationof the arrival of thewavefrom thegreatcircledirection.
usedas a criterionof validity of the greatcircle approximation and as an indicationof ray bendingeffects.Earlier
works have alreadybeendone to computephasevelocities
andthe approachazimuthof the wave train for a givenperiod, by meansof a triangularnetwork [Evernden,1954]. Althoughthe tripartitephasevelocity methodwas designedto
handle surface waves incident upon the array from any azimuth, the resultsof Knopoff et al.[1967] showedthat when
thedirectionof propagationdoesnot lie alongoneof thelegs
of the network,the phasevelocitiescomputedmay be in significanterror.SchwabandKausel[ 1976]proposed
theextensionto quadripartitenetworksto solvethis limitation.With
only 3 stationsof a tripartitearray one must assumeplane
wave fronts; in order to allow for curvature of the
wavefronts,a fourth stationis includedin the array.In the 4
stafion'smethod,the wave front is approximatedby the arc
of a circledivergingfrom, or convergingtowarda fixedpoint
on the free surface.In the methodproposedin thispaper,the
form of thereconstructed
wavefront is not fixeda priori. The
difference
in curvature
between
the reconstructed
and the
theoreticalwave fronts gives us information about lateral
heterogeneities
in the lithosphereand uppermanfiebothbelow the arrayandalongthe wholepathalongwhichthe wave
Introduction
Oneof thegoalsof modemseismology
is to mapthelat- has traveled.
eralheterogeneity
in the Earth.The analysisof surfacewave
Reconstruction of wavefronts: method
datais particularlyvaluablefor studyinglateralvariationsof
the crustanduppermantle.StandardsurfacewavetomograIn the geometricaloptics approximationthe wavefield
phyrelieson thegreatcircleapproximation
whichstatesthat canbe expressed
as u(r,co)=A(r,co)e
i•*(r'•) in thefresurfacewavesare only influencedby theintegralof thelocal quencydomain. Wave fronts are surfacesof equal phase
phaseor groupvelocityover the source-receiver
minor arc. • = cox r(r, co).TheEikonalequation
IVrl2= 1/ca deterHowever,the great circle approximationcannotbe used if
lateralvariationshaveproducedstrongdeviationsof the surface wave ray path and if the structureis not smoothon a
scaleof a wavelength.
The goalsof thisresearchare (1) to testwhetherthe observedwavefieldarrivesin the greatcircle direction,and (2)
to evaluatethe biasin phasevelocitycalculations
in casea
deviationis observed.
For thispurposea newway of calculatingwavefrontsis developed.This is appliedto datafrom
the NARS network,recordedduringthe ILIHA project.An
minesthe wave fronts,sinceit givesa constrainton the function r(r, co).In the right-handside of the Eikonal equation
for dispersivesurfacewaves,c is the localphasevelocity.
To reconstructthe wave frontswe first measurethe phases of the fundamentalRayleigh mode as a functionof the
frequency.The phasesare measuredusingthe residualdispersion analysis method describedby Dziewonski et al.
[1972]. The travel time function for each station is calculat-
ed by dividing the measuredphasesby co. For each event
the stationwith the shortestepicentraldistanceis takenas a
array enablesthe measurementof the azimuth of the direction of arrival of the wave. The phasevelocityCm• measuredalongthe greatcircle can be relatedto the real local
oneC•l by the expression
Cme,= C•l/cos a, a beingthe deviationof the ray pathfrom the greatcircledirection.Phase
delaysfor the fundamentalmode of the Rayleighwavesare
calculatedbetweenthe stationsfor differentfrequencybands.
The wave frontsare represented
by lines connectingequal
phasedelays.The anglea betweenthe observedandthe theoretical wave fronts (correspondingto waves traveling
througha homogeneous
medium)is calculated.This angle
providesinformationaboutthe deviationof the path of the
reference, and the difference between its travel time and the
true wave from the source-receiver minor arc; it can thus be
equation
IVrl2= 1/c.2,,
with Cinthe phasevelocitycorre-
travel time of the other available stations is calculated. After-
wardsthe time delaysare averagedover frequencybandsof
6 or 8 mHz, between 10 and 60 mHz.
Next we performweightedfitting of the phaseplaneby
bothforcingthe delaytime functionto satisfythe delaytime
data while being as close as possibleto a solutionof the
Eikonalequation.Our null hypothesis
is thatwavescomein
alongthe greatcircleandtravelthroughthe regionof study
with a fixed phasevelocityqn. From all the functionsr(r)
thatfit the measureddelaytime data,we wantto find the one
closestto our initial hypothesis,
i.e. satisfyingthe Eikonal
sponding
to a realisticlaterallyhomogeneous
model.We apply thusa regularizafion
basedon the Eikonalequation:this
criterion allows the reconstructed wave fronts to deviate from
the onesin the laterallyhomogeneous
referencemodelonly
whenthis is requiredby the data. The Mercatortransforma-
Copyright1993by the AmericanGeophysical
Union.
tion is used
Papernumber93GL00774
y = In(tan(O/2)) ,
0094-8534/93/93 GL-00774 $03.00
915
x=½
(1)
916
Alsinaetal.:Testof theGreatCircleApproximation
with 0 and• the colatitudeandlongitude.An optimumfit is
achievedby minimizing the sum:
We have selected earthquakes for which the surface
wavetrain has been recorded at a minimum of 4 stations. The
frequency-band
of the analysisis 10 to 60 mHz, butfor most
events the data are available in a more narrow band. This is
S= 5 • (di- z'(ri))
2+
wheredi representsthe delay time data, ,(ri) the delaytime
afterinversionandCinthe inputphasevelocityor the theoretical phasevelocity we assumefor a homogeneousstructure
of the regionof study;7 and/• are scalingfactorssuchthat
both termsin the expression(2) are of orderunity. All the
stationshave the same weight. As initial phasevelocity we
usedvaluesdeterminedby Payo [1970] in the Iberian shield
by meansof a Long Period TriangularArray. The second
term incorporatesall our a priori notionsaboutthe solution
into r(ri). 0 < •: < 1 weightsthe relativeimportanceof the
two terms:if e is closeto 0, the phasefunctionis a solution
of the Eikonalequationbutwith a relativelybadfit to thedata, and vice versa if e is close to 1. As (2) is non-linearin
•:(ri) an iterative approachis needed.The minimizationis
performedby usinga conjugategradientsmethod. The summarion in the function S is over the pairs of stations.The
gradientof thephasefunctionis parameterized
asfollows:
* = 12
Tn(x)Tm(y)
n =0 .....
*=
N, m =0 .....
M (3)
bnm
Tn(x)Tin(y)
whereTn(x) representsthe Chebyschevpolynomialof order
n and direction
x. The condition
ay(ax .r)= ax(•y'r)
(4)
links the coefficientsa•mand the bnm,so that the coefficients
barnwith n>O can be expressedin the a=n.
The wave fronts are representedby lines connectingequal
time delays.Note that in the 2nd term of the functionS, the
regularizationterm, there is no informationaboutthe direction of the gradientof the phase,but only aboutits norm.In
other words, with this method we can obtain information
aboutthe directionof propagationof the waveswithoutan a
priori assumptionfor this direction.The area comprisedbetween 36 ø N and 44 ø N and between 8 ø W and 2 ø E is the
due eitherto a decreaseof signalover noiseratio insidethe
frequencyband of interest,eitherto interferenceof different
wavetrainswhich prohibit the measurementof a reliable
phaseat a numberof frequencies.
Among the earthquakes
thatwereinitially selected,the onesthatgavevery irregular
phasevelocitieswere rejectedso that only the nine earthquakesin figure1 havebeenused.
Discussion of results
With the phase data available we reconstructedwave
frontsfor theeventslistedin table1, for frequencies
between
10 and 50 mHz and for frequencies
up to 62 mhz for the
Azoresevent (nr.1). We tried both 2nd (N = 2, M = 2) and
4th (N = 4, M = 4) order polynomialsto adaptthe phase
functionby iteration.For the numberof pointsavailable(we
had a maximumof 7 pairsof stationsper event),not much
differenceis observedafter bothinversions,
usingdifferent
valuesof e in the rangebetween0.25 and 0.75. The results
are givenfor the 4th orderpolynomialinversionandfor e =
0.5. In figures2 and 3 two examplesare shownfor events
with a north-west(eventnr. 4) and north-east(eventnr. 2)
back-azimuthrespectively.All the eventsstudiedshow a
reasonable
agreement
betweenthetheoretical
andtheexperimentalwave fronts,i.e. to a greatextentit seemsthatwaves
arrivein the greatcircledirection.We shouldpointout that
changeson the startingvelocityc• of about0.1 km/s gave
variationsof lessthan2 degreeson themeanvalueof theangle.Two kindsof deviations
fromtheshapeof wavefrontsin
an homogeneous
model are observedin the reconstruction
of
the wave fronts: a difference in the direction of arrival of the
wave, and a difference in the curvature of the fronts. Waves
crossingthe IberianPeninsulacarryinformationbothabout
thestructure
belowIberiaandaboutthepathfollowedbefore
reachingthis region.A differencein the directionof arrival
indicatesa deviationof the path of the true wavefrom the
source-receiver
minorarc,possiblydueto refraction,scattering or/andmultipathingeffectsbeforearrivingto Iberia;a
difference
in curvature
is anevidence
of lateralheterogeneity
bothbelowIberiaandalongtheentirepath.
On averagingthe angleswe have mixed thesetwo ef-
arearepresented
in figure2 and3. However,the inversionis
fects.The valueof thestandard
deviation
canbe helpfulto
only performedin the region coveredby the stationsfrom
distinguishoneeffectfrom the other:a relativehigh standard
whichthe dataare used. We calculatethe anglea i(xi, Yi) bedeviationimplies a differenceof curvaturebetweenthe two
tween the theoreticalwave fronts (i.e. computedassuming kinds of fronts,which may correspondto any value of the
the waveshadtraveledthrougha homogeneous
mediumwith
averageangle; a low standarddeviationindicatesthat the
a phasevelocity c• and the experimentalwave fronts (i.e.
curvatureof the wave front is similarto the geometricalone,
calculatedusingphasedifferences).For thiswe usethe computedphasegradientvectorsfor both wavefields.The values
obtainedin the region coveredby the stations,are averaged
date
N
to give a meanvalueof the angle:
nr
ai (xi, Yi)
6r=l•
n i=l
(5)
where n is the numberof grid points(xi, Yi) usedfor the inversion.In our inversiona grid of 20 x 20 pointshas been
used. By definition,a clockwiserotationof the experimental
wave fronts with respectto the theoreticalones gives negativeangles,while an anticlockwiserotationimpliespositive
values. The standard deviation of the values is calculated as
O'a
=(1
-I1 i=l
• (ai(xi,
Yi)- a)2)
«
(6)
Data
The datausedin this studyare the fundamentalmodeof
the Rayleighwavesrecordedby the NARS stationsduring
theILIHA project[Paulssen,
1990].
1
22-07-88
8
2
06-08-88
5
3
08-08-88
7
4
11-08-88
6
5
14-08-08
7
6
22-O9-88
6
7
01-11-88
4
8
13-12-88
4
13-02-89
7
Fig. 1. GreatCirclepathsfor the eventsused.Circlesrepresent the epicenters;the trianglethe locationof the seismologicalnetwork.N is the numberof stationsused.
Alsinaet al.' Testof theGreatCircleApproximation
who indicated that these differences
880811, fm=30.3 mhz, c=3.85 km /s
42N
40N
38N
•1/•••_
• +/R
ef'NE
26
zz zz• •
z
•/•••• ••••
zz z z ,
, •-•/•
zz
••'•
•-•-
•
•
•
•17
129.3
•19
113.6 113.8
129.5
•25
193.7 193.6
8W
6W
4W
2W
0
and 8) and to the South (Coast of West Africa and Alboran
2E
Fig. 2. Recons•ction of •e wavefrontsfor the event4, for
a center•equency of •0.• mHz. Solid lines give •e wave
frontsobt•ned •om the inversionof the meas•ed phasedelays; dashed•nes give •e wave fronts co•espondingto a
homogeneous
modelwith a referencephasevel•ity of •.8•
•/s. •e •version has only beenperformedin •e region
coveredby the s•tions. The me• v•ue for the •gle in •s
regionis •. 8 ß •. 2 deuces. •e cffclesrepresentthe l•ation of •e stationsused.Only the •ection •d the spacing
of •e l•es •e telever, since •ey have been recons•ucted
usingph•e &fiefences.
880806, frn=21.8 mhz, c=3.90 km /s
ß ',•x••e•-•
xx
•
/ ReftNE25
'
in the behaviour of the
curvesmay be due to differencesin the structurealongseismic paths. For the eventswith a north and south backazimuth(events3,7,8 in figure2), a differencein curvatureis
dominantfor frequenciesup to 30 mHz whereasfor higher
frequenciesa cleardeviationfrom thegreatcirclepathis observed:in generalhigher angle valuesand smallerstandard
deviationsare observedfor higher frequencies.FOr these
eventsthe waveshave crossedseveralcontinentalmarginsto
the North (North of Scoffandand Bay of Biscay,eventsnr.3
•22
179.1
178.8
v• = 3.2 ø
36N
917
•17
NE21
152.8
99.1
98.2
152.9
NE26
128.9
128.8
NE27
191.6
192.1
o, =0.•
Fig. 3. SameasFigttre2 for theevent2.
Sea, eventnr.7). This last regionis characterizedby a complex lithosphericstructuredue to the active tectonichistory
[Paulssen,1990]. The fact that for some eventsthe deviation
increasesat lower frequenciesmay come from less precise
phasemeasurements.
For the Chile event for which waves
have traveledthroughthe South Americanplate and along
the Arianfie Ocean, an irregularpatternis observed:for frequenciesaround25 and 35 mHz it seemsthat thereis a clear
deviationfrom the great circle direction,with high valuesfor
the averageangle and low standarddeviation;for lower and
higher frequenciesa differencein curvaturebetweenthe experimentaland the theoreticalwave fronts is observed.For
the Thailand event (see figure 3) the reconstruction
of the
wave fronts suggeststhat the waveshave traveledalong anotherpath than the greatcircle. This eventgivesthe highest
value (about8 degrees)for frequenciesbetween10 and 25
mHz. This eventhascomplexwaveforms;this is not surprising sincethe waveshavetraveledthroughcomplexcontinental structures;this
notwithstanding
it is surprisingthat simple
wave fronts are obtained.Only frequenciesup to 25 mHz
couldbe usedfor this event,due to the difficultyin obtaining
reliablephasemeasurements
for higherfrequencies.
From the curveson figure 4 we see that the anglesare
small,up to 8 degrees,giving a relativeerror of lessthan 1%
in thepath-averaged
valueof thephasevelocity:
=I
-
Creal
I x100%
= 1-COScosta
x100%
• 1%
(7)
Or
Anglebetween
observed
andtheoretical
wavefronts
.
,
and suggestsa systematicdeviationof the reconstructed
wave fronts due to a different arrival direction.Figure 2
showsa typicalexampleof wavefront curvaturedifference,
whereasfigure3 showsan exampleof a largerdifferencein
directionof arrival.Figure 4 showthe anglesmeasuredas a
functionof frequency.In figure4(a) resultsfor eventswith a
•, (1)
[]
x
o
(4)
(6)
(9)
pureoceanicgreatcirclepatharepresented,
whereasfigure
4(b) showsthe resultsfor eventswith a greatcirclepathpartly oceanicandparfly continental.Event2 is the only event
with a purecontinentalpath.
For the events1,4,6,9 in figure4(a) a differencein curvatureis the dominantfeaturefor all frequencybands,with values of the averageanglesmuch closerto zero than for the
eventsin figure4(b). In thiscasewaveshavebeentraveling
along a relative short homogeneous
oceanicpath before
reachingthe Peninsula.This small deviationof the wave
frontsfrom thegreatcircledirectionin therangeof frequenciesused,wasakeadyobservedby Payo[ 1970] andBadalet
al.[1990] and explainedas an effect of the continentalmargin.For theeventnr. 9, a generalincreaseof thevalueof the
angleand a decreaseof the standarddeviationwith the frequencyis observed.This meansthat for higherfrequencies
the deviationfrom the greatcirclepathbeeamesmoreimportant. This is what we in generalwould expect,sincehigher
frequenciessampleshallowerdepthswith a more complex
[]
(2)
AO
(3)
(5)
x
o
10.
20.
30.
40.
(7)
(8)
50.
frequency(mhz)
structure. However, this is not what we observe for all the
events.For the Azores event (nr.1), for which a wider fre-
Fig. 4. Anglesbetweenwave frontsaveragedin the region
quencybandhasbeenstudied,the behaviourof the curveis
quite irregular.This was alreadyobservedby Payo [1970]
used in the inversion. Vertical bars indicate the standard de-
viation.In thelegendthenumberrefersto Figure 1.
918
Alsinaet al.:Testof theGreatCircleApproximation
The observed deviation of the direction of the wave front can
in principalbe relatedto horizontalgradientof the phasevelocity usingthe theoryof Woodhouse
andWong [1986]. If
the great circle approximationis valid, the directionof the
observedwave fronts shouldmatch the polarizationof the
Acknowledgments.
This researchwas supported
by the
SpanishMinistry of Educationand Sciencethrougha PhD
grant (FPI), and by the NetherlandsOrganizationfor Scientific Research (NWO) through the Pionier project PGS
76-144. The EEC is gratefullyacknowledged
for theirfinancial supportto the ILIHA project.
surface waves [Paulssenet a1.,1990; Levshin et al.,1992].
The small relativeerror obtainedimpliesthat for the studied
References
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can be usedfor dispersionmeasurements.
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that a goodagreementbetweenthe reconstructed
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we find smoothwave frontssatisfyingthe phasereasonably
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smooth. Having more wavelengthsinsidethe network,i.e.
workingat higherfrequencies,we would expectmore complex wave fronts,giving moredetailedinformationaboutthe
structurebelow Iberia. The problemis to find phasedata at
thesefrequencies;at periodssmallerthan 20s for oceanic
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aspossible.
Conclusions
In this studyit has beenfound that for surfacewaves
travelingmostlythroughoceanicpathsbeforereachingthe
regionof study,andfor frequenciesbetween10 and60 mHz,
a goodagreementis achievedbetweenthedirectionof arrival
of the fundamentalmodeof theRayleighwavesandthegreat
circle.Anglesup to 8 degreeshavebeenobtainedbetween
the irue and the theoreticalwavefront,whichrepresentsa relative error of lessthan 1% in the path-averaged
value of the
phasevelocity. This very good fit of the phasedata with
simplewavefronts(seefigure2 and 3) showsthatthe traditional methodsfor measuringphase velocitiesof surface
wavesare ratherrobust.One shouldnot forgetthat few data
have been used: only low frequenciesfor waves traveling
mostly throughan oceanicpath beforereachingthe Iberian
Peninsula.
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D. Alsina and R. Snieder,Departmentof TheoreticalGeophysics,Instituteof EarthSciences,UtrechtUniversity,P.O.
Box 80.021, 3508 TA Utrecht,The Netherlands
V. Maupin,Institutde Physiquedu Globede Strasbourg,
5 rue Descartes,67084 Strasbourg
Cedex,France.
(Received:January25, 1993;
revised: March 22, 1993;
accepted:March 22, 1993.)