JOURNAL
OF GEOPHYSICAL
RESEARCH,
VOL. 93, NO. B10, PAGES
12,067-12,080, OCTOBER
10, 1988
Large-ScaleWaveformInversionsof SurfaceWaves for Lateral Heterogeneity
2. Applicationto SurfaceWaves in Europeandthe Mediterranean
ROEL SNIEDER
Departmentof TheoreticalGeophysics,
Universityof Utrecht,TheNetherlands
Linear surfacewave scatteringtheory is usedto reconstructthe lateral heterogeneityunder Europeand the
Mediterranean
usingsurfacewave datarecordedwith the Networkof Autonomously
RecordingSeismographs
(NARS). The waveforminversionof the phaseandthe amplitudeof the directsurfacewaveleadsto a variance
reductionof approximately
40% andresultsin phasevelocitymapsin the periodranges30-40 s, 40-60 s and
60-100 s. A resolutionanalysisis performedin order to establishthe lateral resolutionof theseinversions.
Using the phasevelocity perturbations
of the threeperiodbands,a two-layermodel for the S velocityunder
Europeand the Mediterraneanis constructed.
The S velocityperturbations
in the deepestlayer (100-200 km)
aremuchmorepronounced
thanin thetoplayer (0-100 kin), whichconfirmsthatthelow-velocityzoneexhibits
pronounced
lateralvariations.In bothlayersthe S velocityis low underthe westernMediterranean,
whiletheS
velocityis high underthe Scandinavian
shield.In the deepestlayer a highS velocityregionextendsfrom Greece
under the Adriatic to northem Italy. Several interestingsmaller features, such as the Massif Central, are
reconstructed.One of the spectacular
featuresof the reconstructed
modelsis a sharptransitionin the layer
between100 and 200 km nearthe Tomquist-Tesseyre
zone.This would indicatethat thereis a sharptransition
at depthbetweenCentralEuropeandthe EastEuropeanplatform. The waveforminversionof the surfacewave
coda leads to good waveform fits, but the reconstructedmodels are chaotic.This is due both to a lack of
sufficientdatafor a goodimagingof the surfacewaveenergyon the heterogeneities
andto an appreciable
noise
componentin the surfacewavecoda.
INTRODUCTION
One of the main tasks of modem seismologyis to map the
lateral heterogeneityin the Earth. Low-order spectralmodelsof
lateral heterogeneityhave been constructedusing P wave delay
times [Dziewonski,1984], surfacewave dispersiondata [Nataf et
al., 1986], or surface waveforms [Woodhouse and Dziewonski,
1984; Tanimoto,1987]. Thesestudiesproducedextremelysmooth
Earth models because of the low-order expansion of the
heterogeneityin sphericalharmonics.However,recentlarge-scale
tomographicinversionsof P wave delay times have shown that
lateralheterogeneity
existsdownto depthsof at least500 km on a
horizontalscaleof a few hundredkilometers[Spakman,1986a,b].
Lateral variations in the P velocity on this scale can be
analyzedaccuratelyusing delay time tomography.In principle,
tomographicinversionscould also be applied to S wave delay
times.In practice,this is not so simplebecausethe presenceof the
low-velocitylayer renderstheS wave tomographyproblemhighly
nonlinear. In fact, it is shown by Chapman [1987] that the
tomographicinversionproblemis illposedif a low-velocitylayer
is present.The fact that the low-velocity layer exhibits strong
lateral variations [York and Helmberger, 1973; Souriau, 1981;
Paulssen,1987] posesan additionalcomplication.
measurements,
cannotbe usedin that case.Up to this point, this
fact hasbeenconsequentlyignored.
The breakdown of ray theory means that scattering and
multipathingeffects can be important.In the companionpaper
[Snieder, this issue; (hereafterreferred to as "paper 1")], linear
surfacewave scatteringtheoryis presented.It is shownin paper 1
how this theorycan be usedto map the lateralvariationsof the S
velocity in the Earth. With this method,completewaveformsof
surfacewave data can be inverted,so that not only the phasebut
alsothe amplitudecanbe usedfor inversion.Unfortunately,there
is at this point only linear theory for surfacewave scatteringin
three dimensions [Snieder, 1986a, b; Snieder and Nolet, 1987;
Snieder and Romanowicz, 1988; Romanowiczand Snieder, 1988]
which limits the applicability of this method. Large-scale
inversionof both the phase and the amplitudeof surfacewave
data has also been performedby Yomogidaand Aki [1987], who
applieda scatteringformalismto the Rytov field of surfacewaves.
However, their methodis basedon the assumptionthat surface
waves satisfy the two-dimensionalwave equation, which has
neverbeenshown(andwhichis probablynot true).
In this paper, large-scalewaveform inversionsusing linear
scatteringtheory, as presentedin paper 1, are appliedto surface
wave data recorded with the Network of Autonomously
RecordingSeismographs
(NARS) [Dost et al., 1984; Nolet et al.,
One could use surface wave data instead because Love waves
1986]
for
events
in
southern
Europe.The inversionswith linear
and Rayleigh waves are stronglyinfluencedby the S velocity.
scattering
theory,
which
will
be
called the "Born inversion," are
However, fundamentalmode surface wave data (which are most
easilymeasuredandidentified)that penetrateas deepas 200 km, appliedboth to the surfacewave coda and to the direct surface
have a horizontalwavelengthof the order of 300 km. This means wave. Details of the inversionmethodwith numericalexamples
that lateral heterogeneities
on a scaleof a few hundredkilometers are shownin paper1.
The Born inversionis first appliedto the surfacewave coda.To
are no longersmoothon a scaleof a wavelengthof thesewaves.
Thereforeray theory, which forms the basis of all dispersion this end the natureof the surfacewave coda is investigatedin
section 2, and the conditionsfor the validity of the Born
approximation for the surface wave coda are established.In
inversionsof the completewaveform,parameters
like the source
Copyright1988by theAmericanGeophysical
Union.
mechanism,station amplification,etc., should be specified
correctly.The proceduresthat are followed in this study are
Papernumber7B6094.
0148-0227/88/007B- 6094505.00
reported in section 3. As shown in paper 1, it may be
12,067
12,068
Source-reciever
SNIEDER:
LARGE-SALE
INVERSION
OFSURFACE
WAVEDATA,2
minor arcs.
Event 5300 (Algeria), stationNE03, depth=10 km.
i
i
i
i
i
/•A
T>30
s.
A•A^
T>25
s.
,
i
i
600
800
i
1000
1200
1400
time (s.)
Fig. 2. Low passedseismograms
for an Algerianeventrecordedin NE03
(Denmark).
Fig. 1. Source-receiverminor arcs for the seismogramsused in the
inversions.The dashedminorarcscorrespond
to the seismograms
usedin
the computation
of the spectrafor the westernMediterranean,
while the
minor arcsusedfor the spectrafor the pathsin Europeare dotted. The
seismograms
for all minor arcs(dashed,dotted,and solid) are usedin the
than30 s. (This is confirmedlaterby Figure4b.) Thereforeonly
periods larger than 30 s have been used in the inversions
presented in this paper. It is verified that for all recorded
waveform inversions.
seismograms
low passfilmring at 30 s leadsto a codalevel which
is low enoughtojustify theBorn approximation.
This conditionfor the validity of the Born approximationmay
advantageous
to performa nonlinearinversionfirst for a smooth
be
overconservative.
In a field experiment,surfacewavesreflected
referencemodel in order to renderthe problemmore linear. The
from a concretedam on a tidal flat have beenusedsuccessfullyto
results of this nonlinear inversion are shown in section 4. Section
the locationof thisdamusingBorn inversion[Snieder,
5 features waveform inversions of the surface wave coda, while in reconstruct
1987a]. Due to the very large contrastposedby this dam, the
section 6 the Born inversions of the direct wave are shown. The
direct
surface
wave
and
the
reflected
surface
wave
had
reliability of theseresultsis investigatedin section7, where a
resolutionanalysisis presented.A two-layer model of the S approximatelythe samestrength,andBorn inversionwas strictly
velocity underEuropeand the Mediterraneanis finally presented not justified. Nevertheless,an accuratereconstructionof the
locationof this dam was achieved.The reasonfor this discrepancy
in section 8.
is that the geometry of the heterogeneityprecluded multiple
scattering.In such a situation, linear inversion gives at least
2. NATURE OF THeESURFACE WAVE CODA
qualitativelygoodresults.
Before proceedingwith the inversion,it is instructiveto study
the surfacewave coda in somemore detail. In this study,surface
wave datarecordedby theNARS array[Dost et al., 1984;Nolet et
Event3082 (Greece), stationNE03, depth=13km.
al., 1986] are used for shallow events around the Mediterranean
and a deeper event in Rumania. Figure 1 shows the source
receiver minor arcs for the seismogramsused in this study.
Becausethe inversionis linear, it is importantto establishfirst the
conditionsfor the validity of the Born approximationfor the
i
i
i
i
i
ß
surface wave coda.
In Figure 2 a seismogramis shown for an event in Algeria
recorded at station NE03 in Denmark, low passed at several
differentperiods. For the seismogram
low passedat 16 s, the
codahasapproximately
the samestrengthasthedirectwave.This
meansthatthe Born approximation
cannotbe usedto describethe
surfacewave coda at theseperiods.However,for periodslarger
ß
ß
than 20 s the coda is much weaker than the direct wave, which
ß
justifiesthe Born approximationfor theseperiods. Low passed
seismogramsrecordedin the same stationfor a Greek event at
approximatelythe samedepth are shownin Figure 3. For this
I
I
I
I
,
I
600
800
1000
1200
1400
event in the seismogramlow passedat 25 s, there is still an
time (s.)
appreciablesecondarywave train (around700 s) just after the
arrival of the direct wave, and the Born approximationfor the Fig. 3. Low passedseismograms
for a Greekeventrecordedin NE03
surfacewave coda is thereforeonly justified for periodslarger (Denmark).
SNIEDER:
LARGE-SALE
INVERSION
OFSURFACE
WAVE DATA,2
be defined by subtractinga constantnoise level from the coda
spectrumandby divisionby the spectrumof the directwave. This
normalizedcoda level is approximatelyequal to the interaction
coefficients,seeequations(1) and(5) of paper1. (One shouldbe a
Paths through the western Mediterranean
'
i
'
i
--
direct wave
........
coda
,
i
,
_
0.01
0.02
0.03
12,069
0.04
0.05
Frequency (Hz)
Fig. 4a. Spectraof the directwave, surfacewave coda,and background
noisefor wavepathsthroughthe westemMediterranean.
The examplesshownin the Figures2 and 3 showthat the coda
level is very different for the differentwave paths.This is verified
by dividing the seismogramsin two groups.One groupconsistsof
seismograms for with wave paths through the western
Mediterranean (the dashed lines in Figure 1), while the other
groupis for the wave pathsthrougheasternand centralEurope
(the dottedlines in Figure 1). For eachgroupthe spectrumof the
direct surfacewave is determined,as well as the spectumof the
coda (definedby groupvelocitiesbetween1.6 and 2.9 km/s), and
the spectrumof the signalbefore the arrival of the direct wave.
The spectrumof the signalbeforethe arrival of the direct wave is
consideredto give an estimateof the backgroundnoise level.
From an academicpoint of view this is acceptablebecausebody
waves andhigher-modesurfacewavesare noisefor our purposes.
On the otherhand, thisproceduremay give an overestimateof the
bit careful with this identification because an organized
distributionof scatterersleads to extra frequency dependent
factors;see Snieder [1986a] for an exampleof scatteringof
surfacewavesby a quarterspace.)
The normalizedcoda levels are comparedwith the interaction
termsfor differentheterogeneities
the Figures5a and5b. (In this
examplethe absolutevalue of the interactiontermsis averaged
over all scatteringangles.)Theseinhomogeneities
havea constant
relative shearwave velocity perturbationdown to the indicated
depth,while the densityis unperturbed;
furthermore,õk=õg. For
the wave pathsthrougheasternandcentralEuropethesecurvesin
Figure 5b can only be comparedwith the interactionterms for
periodslongerthan30 s, becausetheconditionof linearitybreaks
down for shorter periods. All shown heterogeneitiesfit the
normalizedcoda level within the accuracyof the measurements.
Also, it follows from Figures la and b of paper 1 that these
heterogeneities
have approximately
the sameradiationpattern.
This meansthatfor thesewavepathsit is virtuallyimpossible
to
determinethe depthof the heterogeneity
from the surfacewave
coda. For the paths through the western Mediterraneanthis
situationis differentbecauseit canbe seenfrom Figure5a that a
shallowheterogeneity(or topography)fits the normalizedcoda
spectrum
betterthana deeperinhomogeneity.
Thisis an indication
that the lateral heterogeneityin easternand centralEuropeis
presentat greaterdepthsthan in the westernMediterranean.
3. PROCEDURESFOR T}m INVERSION OF SURFACE
WAVE SEISMOGRAMS
In orderto performwaveformfits of surfacewave data,several
parametersand proceduresneed to be specified.All inversions
presentedin this paper have been performedwith the model
shownin Figure6. This modelis equalto the M7 modelof Nolet
[1977],exceptthattheS velocityin the top 170 km is 2% lower
background
noise.
than
intheM7model.
This
compensates
forthefactthat
theM7
Thespectra
forthepaths
through
thewestern
Mediterranean
are model
isfortheScandinavian
shield
which
hasananomalously
shown
inFigure
4a.Note
thattheaverage
coda
level
ismuchhighS velocity.
Theanelastic
damping
of thePREM
model
weaker than the strengthof the direct surface wave, even for
periodsas short as 16 s (0.06 Hz). For periodslarger than 25 s
(0.04 Hz) the noiselevel has aboutthe samestrengthas the coda
level. The meansthat inversionsusingthe surfacewave codafor
these periods can only give meaningfulresults if there is an
abundanceof data, which leadsto a systemof linear equations
o
whichis sufficiently
overdetermined
to average
out the
d
•.
Paths through Europe
i
direct wave
........
coda
before direct wave
•
contaminatinginfluenceof noise. For the wave paths through
easternand central Europe (Figure 4b) the situationis different.
For all frequenciesthe coda level lies above the noise level,
althoughfor periodslonger than 50 s (0.02 Hz) this differenceis
marginal. For these seismogramsthe coda energy increases c• o
rapidly as a function of frequency;for periodsshorterthan 22 s
• '
(0.045 Hz) the codaspectrumis evenhigherthanthe spectrumof co
the directwave. This meansthat thereis only a relativelynarrow
frequencyband where the Born approximationis valid and where
•
the coda stands out well above the noise level.
The fact that the codalevel for the pathsthrougheasternand
I
[
I
[
I
[
I
[
I
,
centralEuropeincreases
morerapidlyas a functionof frequency
0.01
0.02
0.03
0.04
0.05
than for the paths through the western Mediterranean has
Frequency(Hz)
implicationsfor the depthof the heterogeneities
that generatethe Fig. 4b. Spectraof the direct wave, surfacewave coda,and background
coda.In orderto quantifythisnotiona normalizedcodalevel can noiseforwavepaths
through
eastern
andcentral
Europe.
_•o
12,070
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LARGE-SALE
INVERSION
OFSURFACE
WAVEDATA,2
Reference model for inversions
Pathsthroughthe western Mediterranean
'
i
'
i
'
i
,
i
,
i
,
i
S-velocity
........ density
o
z=33
km.
.........
topography
,
g•.
.
_• o
,,
//
o
z
w.-
-
o
,
,
I
0.01
0.02
0.03
0.04
I
50
0.05
•
I
100
,
I
150
•
I
200
•
I
,
250
I
300
,
I
350
,
400
Depth (km)
Frequency (Hz)
Fig. 5a. Normalizedcodalevel for the •vavepathsthroughthe westem
Mediterraneanand the (normalized)integratedradiationfor different
inhomogeneities
asdefinedin section3.
Fig. 6. Startingmodel for the inversions.
in this studyare, in general,ratherweak (ms=5-6),so that the
reportedsourcemechanismsare not always reliable. All
seismograms
with a strongdifference
in waveshapebetween
the
[Dziewonski
andAnderson,
1981]
isassumed;
thisdamping
isnot dataandthe(initial)synthetics
havenotbeenusedin the
varied
intheinversions.
For the sourcemechanismsand the event locationand depth,
the centroid moment tensor solutions, as reported in the
InternationalSeismological
Centrebulletins,are usedwhenever
available.For the remainingeventsthe sourceparametersfrom
the PreliminaryDeterminationof Epicentersbulletinsare used.
inversion.
Surface
waverecordings
thattriggered
onthesurface
wave have also been discarded.
The stationmagnifications
are not includedin the inversions
presented
here.The stationmagnification
includesnot only the
instrumental
magnification
but alsothe magnification
effectsof
the local environment of the station. As a check, the inversions
For the Rumania event the source mechanism as determined from
presentedhere have also beenperformedwith a simultaneous
GEOSCOPE data (B. Romanowicz, personal communication, inversion for the station amplifications.Even though this
1986) is employed.No inversionfor the sourcemechanismis produced
stationmagnifications
aslargeas 10%,theresultin the
performed
because
theNARSstations
provideonlya limited models
forlateral
heterogeneity
wasminimal.
azimuthalcoverage,whichmeansthatthesourcemechanisms
are
In paper1, the waveforminversionis formulatedas the least
poorly constrained
by the data.The sourcestrengthis usually squares
solutionof a largematrixequation.The rowsandcolumns
rather inaccurate;this parameteris determinedby fitting the of this matrix can be scaled at will [Van der Sluis and Van der
envelopes
of thesynthetics
to thedataenvelopes.The eventsused Vorst, 1987]. It has been shownby Tanimoto[1987] that it is
importantthateachseismogram
getsmoreor lessthesameweight
in the inversion. Similarly, the different frequency components
within each seismogramshouldhave a comparableweight.For
the shalloweventsusedhere, the low frequenciesare not excited
Paths throughEurope
'
i
,
i
,
I
,
i
,
/i
'
!
,!
very well. In orderto compensate
for this,dataandsynthetics
are
scaled in the frequency domain with a factor 1/co. In the
,
to_
data
'-
z=170
km.
.......
cv o
-cd
'
O
O
Z
to
z:110
km.
kin.
z=33
i: [
inversions
for thesurface
wavecoda,a scaling
withMo/sxt•-•nA
is
/'
/
-
applied,whereMo is the strengthof the momenttensorandA is
the epicentral distance.This factor corrects for the different
strengthsand geometrical spreadingfactors of the different
events.In the inversionfor the direct wave the seismogramsare
scaledin sucha way thatmaximumamplitudein thetime domain
is equalized.
,,/,,,,,,//,/,;/,//,
Last, in the Born inversions both for the coda and the direct
wave,eachseismogram
is scaledwith a factorl/x/1+ E/<E>. In
this expression,E is the energy of the data residualof the
seismogram
underconsideration,
while <E > denotesthe average
of this quantityfor all seismograms.
This weight factorensures
thatthe seismograms
with appreciable
misfitsgetmoreor lessthe
sameweightin the inversion,so that the contaminating
influence
0.01
0.02
0.03
0.04
0.05
Frequency(Hz)
of outliersis reduced.In the meantime,seismograms
with a good
initial fit havea low weightin the inversion;thispreventsa small
Fig. 5b. Normalizedcodalevel for the wavepathsthrougheasternand
centralEuropeand the (normalized)integratedradiationfor different amount of spurious noise in these seismogramsgetting an
excessiveweight in the inversion.
inhomogeneities
asdefinedin section3.
,
,
SNIEDER:LARGE-SALEINVERSIONOF SURFACEWAVE DATA,2
4. NONLINEAR INVERSION OF Tim DtRECT WAVE
As mentionedin paper 1, it is advantageousto perform a
-5%L
12,071
+5%
(•c/c after constrained nonlinear inversion.
noldinear inversion of the direct wave first because this renders
the Born inversionmore linear. For this inversionthe procedures
describedin paper 1 are usedfor determiningthe phasevelocity
perturbationof a smoothreferencemodel. In this inversionthe
phasevelocityis determinedon a rectangulargrid of 12x12 points
in the domain shown in Figure 1 and is interpolated at
intermediatelocationsusing bicubic splines.The relative phase
velocityperturbationõc/c is assumed
to be constantin theperiod
bands 30-40 s, 40-60 s, and 60-100 s.
In Figure 7a the phase velocity perturbation for periods
between 60 s and 100 s is shown for the unconstrainedcase, i.e.,
T=0 in equation(21) of paper 1. Note that the phasevelocity
perturbations
arenot confinedto thevicinity of the sourcereceiver
paths.This is an artifact of the bicubic splineparameterization,
whichhasan oscillatorynaturenearplaceswherethe interpolated
function changesrapidly. These artifacts can be removed by
switchingon the regularizationparameter? in expression(21) of
paper 1. The constrainedsolution(?>0) is shownin Figure 7b.
This regularizationgoesat the expenseof the waveformfit, andit
is subjectivehow much regularizationone wants to impose on
thesesolution.However,in thisstudythenonlinearinversionfor a Fig.7b. Relative
phase
velocity
perturbation
(5cIc ) forperiods
between
smoothreferencemodel is only the first step in the complete 60 and 100 s determinedfrom constrainednonlinear waveform fitting
waveform inversion, so that there is no need to obtain the (7>0.).
maximuminformationfrom this nonlinearinversion.For periods
largerthan 30 s the resultingreferencemodelsfor the employed
5. BORN INVERSION OF TH2ESURFACE WAVE CODA
value of ? (see, for example,Figure7b) producea waveformfit
which is sufficientlygood to warrant a linear inversionfor the
The Born inversioncan be applied both for inversionof the
remainingdataresidual.(This meansthat the phaseshift between direct surface wave, as well as for the coda. In this section the
the data and syntheticsis at the most 45ø, and that the amplitude waveform inversion of the surface wave coda is discussed. The
mismatchis not largerthan30%.) Thesereferencemodelsfor the surfacewave coda is extractedfrom the full seismogramwith a
phase velocity perturbationsare used in the subsequentBorn time window that allows groupvelocitiesbetween1.74 and 2.90
inversions.Waveformfits of thenonlinearinversionarepresented km/s. At both endsthiswindow is taperedwith a cosinetaperover
in section 6.
a lengthof 100 s.
In this inversionthe depthdependence
of the heterogeneityis
prescribedto consistof a constantrelativeS velocityperturbation
-5%
+5%
•5[•/[•downto a depthof 170km, while thedensityis unperturbed.
after unconstrained nonlinear inversion.
The perturbationson the Lain6 parametersare equal. In the
inversiona model of 100xl00 cells is determined(with a cell size
of 35x35km2), so that 10,000unknownsare determined
in the
inversion.The 42 seismograms
produce2520 data points,where
the real and imaginary parts of each spectral component are
countedas independentvariables.This meansthat the resulting
systemof linear equationsis underdetermined.
Increasingthe cell
sizehasthe disadvantage
thatthe scatteringintegral(5) of paper1
is not discretizedaccurately.Imposing a smoothness
constraint
alsois no option, sincescatteredsurfacewaves are most sensitive
to abrupt lateral changes of the heterogeneity. As argued in
section2, it is difficult to obtain a good depthresolutionfor this
kind of inversion. This, and the consideration that for a fixed
depth dependenceof the heterogeneitythe resulting system of
linear equationis alreadyunderdetermined,
makesit unjustifiable
to perform an inversion with more degreesof freedom with
respectto the depthdependence
of the heterogeneity.
The result of the Born inversion of the surface wave coda for
periodsbetween30 and 100 s is shownin Figure 8. For this
inversion,threeiterationshavebeenperformed;accordingto the
resultsof paper1 thisis sufficientto imagetheheterogeneity.
The
Fig.
7a.Relative
phase
velocity
perturbation
(•5c
Ic) forperiods
between
reconstructed
model
hasamessy
appearance
andisdominated
by
60 and100s determined
fromunconstrained
nonlinear
waveform
fitting ellipsoidalstructures.These structuresare reminiscentof the
(T=0).
"smiles"thatoccurin improperlymigratedsections
in exploration
12,072
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•qYERSION
OFSURFACE
WAVEDATA,
2
-5O/o
h--
+5%
for z<170 km., 30. s.<T<100s.
Envelopeof modelfor pointscatterers.
Fig.8. RelativeS velocityperturbation
([•/•) fromtheinversion
of the Fig. 10. Filteredenvelopeof the modeldeterminedfrom a waveform
surface
wavecoda.Theheterogeneity
extends
downtoa depthof 170km. inversionof synthetics
computedfor positive(circles)and negative
(triangles)
pointscatterers
with[•/• constant
down
toa depth
of 170km.
seismics[Berkhout,1984]. Note the oscillatorynatureof the
solution,
whichis a consequence
of thefactthatthisimageis better
resolution
alongthese
ellipsoidal
stripes.
Thedirection
of
reconstructed
essentially
witha correlation
method
(paper1). It these
ellipsoidal
stripes
isthusdetermined
bythegeometry
ofthe
can be seenin Figure1 that the majorityof thepathsin central eventsandthestations;
it doesnotnecessarily
reflectthestructure
Europerunsin thesoutheast-northwest
direction.
The ellipsoidal of theinhomogeneity.
structures
in easternEuropeandbetweenSicily and Franceare
This directivityof the oscillationsin the reconstructed
model
centeredaroundthesepaths.This meansthat just as in the can be removedby computingthe two-dimensional
spatial
syntheticexampleof section8 of paper 1, the shapeof these envelopeof the solutionin Figure8. Subtracting
a smoothed
ellipsoidalstripesdoesnotnecessarily
coincidewith thelocations versionof this envelopefrom the envelopeitself enhances
the
of thescatterers
andthatsomesmearing
of thetrueinhomogeneitycontrastsin the final solution.The resultof this procedureis
alongthesestripes(smiles)hastakenplacein theinversion.More shownin Figure9. Oneshouldalwaysbe carefulin applying
this
data,and especiallymore crossingpathsare neededto obtaina kindof imageprocessing
techniques
because
it mayintroduce
an
unwanteddegreeof subjectivity
in theresultingpatterns.
On the
otherhand,thesemethodsmay help to extractsomeorderout of
0.5% ..........
',1111',1111111•
3%
an apparent
chaos.Unfortunately,
in theresulting
model(Figure
Envelope of model from coda inversion.
9) thisgoalis onlypartlyreached.Someof the heterogeneities
could be relatedto familiar geologicalstructuressuch as the
Tomquist-Tesseyre
zone and the northernedgeof the African
continent,while otherheterogeneities
appearto be distributed
at
random.
In orderto establishthe significance
of theseresults,the same
inversion
is performed
withsynthetic
datafora modelconsisting
of ninepositive
andeightnegative
pointscatterers.
Theenvelope
of theresulting
modelis shownin Figure10.Theheterogeneity
is
reconstructed
nearmostpointscatterers
but alsoin areasaway
fromthesepointscatterers
(Spain,southwest
France,
Aegean
Sea,
etc.). This meansthat even if the datawere noisefree, thereare
insufficientdata to constrainthe resultingmodel.For noisecorrupteddatathiseffectis aggravated.
Nevertheless,
a reasonable
goodfit of thesurfacewavecodais
achieved,with a variancereductionof 25%. It may appear
surprising
thatthevariance
reduction
is only25%,despite
thefact
thatthesystemof linearequations
is underdetermined.
However,
the linearequationsare not only underdetermined
but alsoselfcontradictory.
Thiscanbe seenfor examplein Figures
4a and4b,
wheretheaverage
amplitude
spectra
for thecodashowa jagged
Fig. 9. Filteredenvelopeof themodelin Figure8.
appearance.(This is even more pronouncedfor the individual
SNIEDER:
LAR6E-SALE
INVERSION
OFSURFACE
WAVEDATA,2
Event 3218 (Greece), stationNE02.
Fit of (coda)dataand synthetics,for T>50 s.
I
--
I
12,073
I
'
I
I
I
'
'
I
data
data
........ synthetics
........ synthetics
5300NE15
5300NE04
3218NE12
o-3218NE02
- _.. . .__
__•
I
I
600
I
800
I
1000
1200
time (s)
I
I
400
Fig. 1la. Examplesof the waveformfit of the surfacewave coda,low
passedat a comerperiodof 50 s.
I
600
800
[
I
t
1000
I
1200
time (s)
Fig. 12. Full waveformfit of the surfacewave coda,high passedat a
comerperiodof 50 s.
spectra.) The scattering theory leads, in general, to much
smootherspectra, so that it is impossibleto fit all spectral
Inversionsfor heterogeneities
whichextendto a depthdifferent
componentsperfectly, thus producing an imperfect variance from 170 km producealmostthe samemodel;only the strengths
reduction.
of theseheterogeneities
differsfrom themodelshownin Figure8.
Some waveform fits of the surface wave coda are shown in
An inversionfor data band passedbetween30 and 40 s gives
Figures11a and 1lb both low passedandhigh passedat a comer virtually the same result as Figure 8, which confirmsthat for
period of 50 s. For periodslarger than 50 s (Figure 11a) the longer periods the surfacewave coda containsa large noise
andnotmuchscattered
surfacewaves.
waveformfit is poor.This is consistent
with the resultsof section component
These resultsdo not imply that mappinglateralheterogeneity
2, whereit was arguedthat the codalevel doesnot standout very
well abovethe noise level. However, for the higher frequencies using the surfacewave coda is impossible.In fact, it has been
imagingof
(periodsfrom 30 to 50 s, seeFigure 1lb) a reasonablewaveform shownin a controlledfield experimentthat successful
fit is obtained. Most of the beats in the surface wave coda are
the surfacewave codais possible[Snieder,1987a]. However,the
reproducedin the synthetics.It shouldbe rememberedthat the surfacewave coda data at our disposalare currentlytoo sparseto
seismograms
in thesefiguresonly showthe surfacewavecoda.In produce an accuratereconstructionof the lateral heterogeneity.
orderto seethesedatain their properperspective,a fit of the coda This is exacerbatedby the fact that the noiselevel in the surface
with
(band passedfor periodsbetween30 s and 50 s) is shown in wave codais relativelyhigh, which can only be compensated
Figure 12 togetherwith the direct wave. Unfortunately,the fact a redundantdataset.Largenetworksof digitalseismicstations,as
that good waveform fits are achieveddoes not establishthe formulated in the ORFEUS [Nolet et al., 1985] and PASSCAL
reliability of the resultingmodelsbecausethe linear systemof proposals,are necessaryto achievethis goal. Alternatively,the
Born inversionof the surfacewave codamightbe usedin regional
equationsis underdetermined.
studies where one wishes to study tectonic features such as
continental margins or the boundaries of major geological
Fit of (coda)dataand synthetics,for 30 s.<T<50 s.
formations.A systemof portabledigital seismographs
would be
data
very usefulfor thiskind of investigations.
....... synthetics
The directivityof the solutionin Figure 8 is related to the
source-receivergeometry.This means that apart from a denser
networkof stations,we shouldaim to employthesestationin such
a way that the wavepathsof the recordedwavescoverthe domain
.
5300NE04
of interestin a more or lesshomogeneous
fashion.Furthermore,
three-component
data might help to constrainthe locationof the
inhomogeneitybecause the horizontal componentsimplicitly
'
I
'
I
'
I
'
I
contain information
3218NE12
,•
---'...
. - -,,
o
3218NE02
I
I
600
800
•
on the backazimuth
from the receiver
to the
scatterer.Numerical experiments,similar to the experimentthat
producedFigure 10, could be used to determinewhich station
configurationshouldbe used to image a particularstructure.In
this way, the designof networkscan be related to a particular
geophysicalor geologicalproblem.
I
1000
I
1200
6. BORN INVERSION OF TI-tE DIRECT SURFACE WWAVE
time (s)
Fig. 1lb. Examples
of the waveformfit of the surfacewavecoda,high
passed
ata comerperiod
,of50 s.
/
Linear scatteringtheory can also be used to describethe
distortionof the direct wave [Snieder, 1987b]. This distortioncan
12,074
SNIEDER:
LARGE-SALE
INVERSION
OFSURFACE
WAVE DATA,2
(Note that in contrast to the case for the surface wave coda, the
•c/c for 30 s.<T<40
s. after Born inversion.
directwave standsout well abovethe noiselevel for all employed
frequencies;seeFigures4a and4b.) A separateBorn inversionis
performedfor each of thesefrequencybands,so that the phase
velocity perturbation is determined independently for each
frequency band. In order to justify the isotropic approximation
(paper 1), a time window is used to extractthe direct wave from
the completeseismograms.
The Born inversionspresentedhere are performedfor a model
of 100x100 cells with a cell size of 35x35 km 2. In the Born
inversion of the surface wave coda in section 5, no a priori
smoothnessconstraintwas imposed becausescatteredsurface
waves are most efficiently generated by sharp lateral
heterogeneities.
This led to an underdetermined
systemof linear
equations.For the Born inversionof the directwave the available
data set also producesan underdeterminedsystem of linear
equations.One alternativewould be to increasethe cell size, but
accordingto the exampleof Figure7 in paper1, rathersmallcells
are neededto producethe requiredfocusing/defocusing.
Instead
of this,the smoothingoperatorof equation(26) in paper1 is used
in this inversion to constrain the solution.
In these inversions the
values ct•-0.66 and N=4 are used, which implies an effective
correlationlengthof 140 km. The Born inversionsareperformed
Fig.13a.Relative
phase
velocity
perturbation
(•c/c ) forperiods
between in threeiterations(seealsopaper1); it hasbeencheckedthatmore
30 and 40 s as determinedfrom nonlinearwaveforminversionplus a
subsequentBorn inversion.The dominantwavelengthof the employed iterationsdo not changetheresultingmodelsverymuch.
The phasevelocityperturbationfor the threefrequencybands
wavesis shownfor comparison.
are shownin Figure 13. The phasevelocitiesare the resultof both
the nonlinear inversion for the smooth reference medium and the
subsequent
Born inversion.See Figure 7b as an exampleof the
eitherbe due to ray geometricaleffectsor to multipathingeffects
that are not accountedfor by ray theory.In the Born inversion
presentedin this section, the isotropic approximationis used
(paper 1). This means that the relative phase velocity
perturbations
are retrievedfrom the linearwaveforminversionof
the directwave.This quantityis assumedto be constantwithin the
frequencybandsemployed(30-40 s, 40-60 s and 60-100 s).
-5%
(5c/c for 40 s.<T<60
+5O/o
s. after Born inversion.
contribution
of the nonlinear
inversion
for the smooth reference
mediumto thephasevelocitymodelof Figure13c.
Note that the resulting phase velocity patterns vary
considerably
on a scaleof onehorizontalwavelength.This means
that ray theory cannot be used to model the effects of these
heterogeneifies,
while surfacewavescattering
theorytakeseffects
suchasmultipathinginto account.Surprisingly,
Figure13cfor the
phasevelocity determinedfrom Born inversionis not too different
from Figure 7a for the unconstrained
nonlinearinversionusing
-5%h-=(5c/c for 60 s.<T<100
+5O/o
s. after Born inversion.
for T=75 s.
Fig.13b.AsFigure13a,butforperiods
between
40 and60 s.
Fig.13c.AsFigure13a,butforperiods
between
60and100s.
$NIEDER:
LARGE-SALE
•¾ERSION
OFSURFACE
WAVEDATA,2
Event5300 (Algeria),stakonNE15, 40 s.<T<60 s.
Event3218 (Greece),stationNE12, 30 s.<T<40 s.
I
'
I
'
I
data
12,075
I
'
'
I
I
I
I
data
,,,
......
synthetics
:"/•/"• ::/•
lat.
hom.
',,;
'""•,
I
,
I
500
I
600
I
700
northnear
fit
,
400
800
500
600
700
hme (s)
time (s)
Fig. 16. As Figure 14 for an Algerianevent recordedin NE15
Fig. 14. Waveform
fit for periods
between
30 and40 s for thelaterally (Netherlands)
for periodsbetween40 and60 s.
homogeneous
starting
model(top),afterthe nonlinear
inversion
for a
smoothreference
model(middle),andafterBorninversion(bottom),for a
Greekeventrecordedin NE12 (Spain).
Waveform fits after the nonlinear inversion for the smooth
referencemodel and after the subsequent
Born inversion(the
"finalfit") are shownin Figures14-19. In Figure14, resultsfor
stationNE12 nearMadrid are shown.The amplitudeof the direct
raytheory.
Thesmaller-scale
features
of Figure13careabsent
in
Figure7a because
the splineinterpolation
doesnot allowthese surfacewaveis changedconsiderably
in the inversion.
Note that
small-scalefeatures(nor does ray theory). Nevertheless,the
the waveform fit has slightly deterioratedin the nonlinear
overallpatternin theseFigures
isthesame.Apparently,
raytheory inversion. The reason for this is that the 42 seismogramsare
is rather robust to violations of the requirement that the
invertedsimultaneously,
so that it is possiblethat the fit of one
heterogeneity
is smooth.This may explainthe success
of seismogram
is improvedat the expense
of anotherseismogram.
dispersion
measurements
in situations
whereray theoryis not In Figure15 an exampleis shownfor a seismogram
recorded
at
justified.Most of the information
on the S velocitystructure NE01 (Gothenborg).
Forthisseismogram
theamplitude
is already
underEuropein the crustanduppermantleis determined
from quitegoodfor the laterallyhomogeneous
starting
model,butthe
surface
wavedispersion
measurements.
Forexample,
Panzaet al. phaseis adjustedin the Born inversion.In the preceding
[1982] delineateda heterogeneity
betweenCorsicaandnorthern examplesthe Born inversionrealizedthe fit betweendata and
Italy with a scaleof approximately
250 km, fromanomalouslysynthetics.
Thisis notthecasefor all seismograms.
In Figure16 a
low Rayleighwavephasevelocities
between
40 and60 s. Their seismogram
for an event in Algeria recordedat NE15
resultsare thereforeinconsistentwith the (ray) theory that they (Netherlands)is shown. For this seismogram
the nonlinear
employed.
Nevertheless,
thislow phasevelocityanomaly
is also inversionperformedmostof thewaveformfit.
visiblein Figure13b, whichis constructed
usingsurfacewave
By superposing
theseismograms
forthethreefrequency
bands,
scatteringtheory.
seismogramsfor the full bandwidth (30-100 s) can be
Event4042 (Greece),stationNE01,30 s.<T<40s.
I
I
-
I
data .
'
I
'
I
'
m
' '" '""i'",
"%
I
,
I
,
600
I
'
I
'
I
'
........synthetics /r• :,,• .
I
700
,,:
final
fit
I
•
800
time (s)
Fig.15. AsFigure14,fora Greekeventrecorded
inNE01(Oothenborg)
for periodsbetween30 and40 s.
'
data
ear
fit
., "", ' ' ! i
500
Event3218 (Greece),stationNE12, 30 s.<T<100s.
I
:',
500
600
time (s)
700
800
Fig.17. AsFigure14forthefullbandwidth
(30-100s).
12,076
SNIEDER:
LARGE-SALE
•WERSION
OFSURFACE
WAVE
DATA,
2
Event5300 (Algeria),stationNE15, 30 s.<T<100s.
'
i
TABLE 1, VarianceReductions
for the WaveformInversions
,
data
Period, s
30-40
40-60
60-100
nonlinear
Nonlinear
15%
37%
21%
Bom
20%
27%
25%
Nonlinear +Bom
31%
54%
41%
fit
smoothreferencemedium the solutionis rather heavily
constrained
(compare
Figures7a and7b) sothatlargervariance
reductions could be achieved with the nonlinear inversion. The
smallest
variance
reduction
occurs
in theperiodrangefrom30 to
ß
40 s. This is not surprising
becausethesesurfacewaveshavethe
.
shallowest
penetration
depthsandaretherefore
moststrongly
subjected
to lateralheterogeneity
andthereforemostdifficultto
,
I
I
I
,
fit. Surprisingly,
thevariancereduction
for periods
between
40to
I
60 s is largerthanfor60-100s.Thereason
forthismightbethat
surfacewavesbetween60-100s. are influenced
by the lowvelocityzone,whichisreported
toexhibitstrong
lateralvariations
Fig.18. AsFigure14forthefullbandwidth
(30-100s).
[YorkandHelmberger,
1973;Paulssen,
1987].Thetotalvariance
reductionis largerthan the variancereduction
obtainedby
constructed.
Figure17 displays
theseismogram
of Figure14,but Yomogida
andAki [1987]for surface
waveswhichpropagated
400
500
600
700
time (s)
now for the full employedbandwidth.The final fit betweenthe
throughthe Pacific. (They obtaineda variancereductionof
dataandthesynthetic
is extremely
good.Notethatthetail of the approximately
30%.) However,it is difficultto compare
these
directwave (around725 s) is fittedquitewell afterthe Born resultsbecause
on theonehandthepathsof propagation
of the
inversion.For therecording
of theAlgerianeventin NE15,the surface
wavesthattheyusedaremuchlongerthanin thisstudy
full bandwidth
dataareshownin Figure18.The troughin the butontheotherhandEuropeandtheMediterranean
ismuchmore
waveformaround420 s hasbeenadjusted
well in thenonlinear heterogeneous
thanthe Pacific.
inversion,
whereas
thefit of thestartof thesignal
(around
400s)
is improvedconsiderably
in the subsequent
Born inversion.
Unfortunately,
theimprovement
in thewaveformfitsis notfor all
7. A RESOLUTIONANALYSISOFTIlE INVERSION
FOR THE DmECT WAVE
seismograms
asdramatic
asin thepreceding
examples.
Figure19
Justaswiththeinversion
forthesurface
wavecoda,
thequality
(Denmark).
Thephaseof thesignalis slightlyimproved
in the of the waveform fit is no measure of the resolution of the
this issue,synthetics
havebeen
nonlinear
inversion,
butthefinalwaveform
fit isnotimpressive. inversion.In orderto address
Thequalityof thewaveform
fitsis expressed
by thevariance
featuresthe waveform fit for a Greek event recordedat NE02
reductionsshownin Table 1. Both in the nonlinearinversionand
in the subsequent
Born inversionthevariancereductionis of the
orderof 25%, although
thisdiffersconsiderably
between
the
different frequencybands.In the nonlinearinversionfor the
-s%h--
+s%
6c/c modelfor resolution
experiment.
Event3218 (Greece),stationNE02, 30 s.<T<100s.
'
--
I
,
i
,
i
,
data
........
syni•eti•i
.../f•
),,/)
,,"'-..'"'
_.
ß
.
final fit
5OO
6OO
7OO
time (s)
Fig. 20. Synthetic
modelof the relativephasevelocity
perturbation
Fig.19. AsFigure14fora Greek
eventrecorded
inNE02(Denmark)
for (•5c/c) fortheresolution
experiment
of section
7. Thesource-receiver
thefull bandwidth(30-100 s).
minorarcsaresuperposed.
SNIEDER:
LARGE-SALE
INWERSION
OFSURFACE
WAVEDATA,2
,5c/c model after nonlinear inversion, 60 s.<T<100 s.
12,077
wavepathsrunsin a bundlefrom Greeceto northwestern
Europe
and encountersa suite of positiveand negativeanomalies.This
leadsto a smearingof the solutionunderGermanyandDenmark
in
the
northwest-southeast direction and
a
subsequent
underestimateof the true inhomogeneity.A similar smearingin
the
northwest-southeast
direction
is visible
in
the northern
Adriatic; this area also suffersfrom a lack of crossingray paths.
One of the most conspicuousfeaturesin Figure 13 is the high
phasevelocitiesunderGreece.This is no artifactof the inversion
because this feature is not present in the results from the
resolutionanalysis(Figure22).
In conclusion,the reconstructedphase velocity models are
meaningless
outsidethe dottedline in Figures24 and25. In the
area enclosedby this line, lateral smearingin the northwestsoutheastdirection occurs under Denmark, Germany, and the
northern Adriatic, while there is an east-westsmearing in the
southern Mediterranean.
8. A MODEL FOR TIDES VELOCITY UNDER EUROPE
AND Tim MEDiTERRANEAN
The phasevelocityperturbations
presented
in section6 canbe
convertedto a depthmodelusingthe phasevelocityinformation
Fig. 21. Reconstruction
of the modelof Figure20 afterthe constrai•.ed of the differentfrequencybands.However,thesephasevelocities
nonlinearinversionfor periodsbetween60 and100s.
are influencednot only by the composition
of the crustandupper
mantle but also by the crustalthickness.The crustalthickness
is knownfrom refraction
computedusing asymptoticray theory [Woodhouse
and Wong, underEuropeand the Mediterranean
studies,
and
it
is
therefore
possible
to
correctfor the varying
1986] for the phasevelocitymodel shownin Figure 20. For
convenience,the minor arcsof the usedsource-receiver
pairs are crustalthickness. The referencemodel shown in Figure 6 has a
also shown in this Figure. The resultingsyntheticshave been crustalthicknessof 33 km. By determiningthe phasevelocityfor
subjected
to the sametwo-stepinversionasthe surfacewavedata the same model, but with a different crustal thickness, the
of theeffectof crustalthickness
from section6. As a representative
example,the resultsfor the followinglinearparameterization
periodbandbetween60 and100 s arepresented
in thissection.In on the fundamentalRayleigh mode phase velocity has been
determined:
Figure21 the modelasderivedin the nonlinearinversionfor the
5c
smooth reference model is shown. The thin lines show the model
of Figure20; in the ideal casethe inversionwouldreproducethis
model. Since only the directwave is usedin this inversion,the
solutionis only nonzeroin the vicinity of the source-receiver
minor arcs. Apart from the positive anomaly in the northern
Adriatic, the heterogeneitiesare placed more or less at their
•
c
= F ( z - 33 km ) (%)
(1)
6c/c model after Born inversion, 60 s.<T<100 s.
correct location. The reconstructedmodel after the subsequent
Born inversionis depictedin Figure22.
The strengthof the model afterBorn inversionis closerto the
true model than after the nonlinear inversion alone. However, the
magnitudeof the reconstructed
heterogeneityis still much less
than the magnitudeof the input model. The physicalreasonfor
this is that the model
used in this resolution
test consists of
alternatingpositiveandnegativeanomalies.A smearingof these
anomalies leads to a reduction of the magnitude of these
anomalies.In the true Earth an alternationbetween positive and
negativephase velocity anomaliesmay also occur, so that the
reconstructedmodels (Figure 13) may underestimatethe true
phasevelocityperturbations.
Surprisingly,
the heterogeneities
arebetterpositioned
afterthe
Born inversion (Figure 22) than after the nonlinear inversion
alone (Figure 21). The reason for this is that with the ray
geometricalnonlinear inversion one basically measurescertain
path integralsover the sourcereceiverminor arc (see equations
(10a)-(10d) of paper 1), whereasin the Born inversionmore
completewave informationis used.
It follows from Figure 22 that the east-westresolutionin the Fig. 22. Reconstruction
of ,he model of Figure20 after the constrained
southern
Mediterranean
is ratherpoor.Thisis dueto thefactthat nonlinear
inversion
andthesubsequent
Borninversion
forperiods
between
there are no crossingrays in that region. A large portionof the 60 and100s.
12,078
SNIEDER:
LARGE-SALE
•WERSIONOFSURFACE
WAVE DATA,2
Smoothed crustal thickness in domain.
Fig. 23. Smoothedcrustalthicknessusedin the correctionfor the varying
Moho depth.
same smoothingis applied to the crustal thickness,as for the
reconstructions
shownin Figure 13. The variationsin the crustal
thicknessare as large as 25 km in the area of interest.For the
shortest-periodband this leads to a phasevelocity perturbationof
4.5%, which is of the same order of magnitude as the
perturbationsas determinedfrom the Born inversion(Figure 13a).
After correcting for the crustal thickness,a standardlinear
inversion [Nolet, 1981] leads to the S velocity perturbationsfor
depthsbetween 0 and 100 km, and between 100 and 200 km. A
simpleresolutionanalysisshowsthat incorporatinga third layer is
unjustified.The resultingS velocity perturbationsare shownin
Figures24 and25. A biasin the S velocityof thereferencemodel
would show up as a dominanceof either positive or negative
velocity perturbations.Likewise, a bias in the attenuationwould
show up in structuresthat would for the majority of the paths
produce a marked focusingor defocusing.As can be seenfrom
Figures24 and 25, and from the wave pathsin Figure 1, neither
effect seemsto be present.
The S velocitymodelsin theseFigurescan be comparedwith
mapsof the S velocity as compiledsubjectivelyfrom a wide range
of surface wave and body wave data [Panza et al., 1980;
Calcagnile and Scarpa, 1985]. In general, there is a
correspondence
of the large-scalefeatures.The velocityis highin
the Scandinavian shield, which can be seen in the northern area of
in thisexpression,z is the crustalthicknessin kilometers.
The parameterF is for the differentfrequencybandsgivenby
F=-0.180(%/km)
for
30s<T<40s
F=-0.113
for
40s<T<60s
for
60s<T<100s
(%/km)
F=-0.076(%/km)
(2)
The crustal thicknessused in this study is adoptedfrom
Meissner[1986] andStokoet al. [1987], and is shownin Figure
23. In the area outsidethe dottedline in Figures24 and 25 the
default value is assumed(33 km). For consistencyreasons,the
-s%
inversionof Figures24 and 25. Under the westernMediterranean
the velocity is low [Marillier and Mueller, 1985], whereasthe
Adriatic is characterizedby a S velocity higher than in the
adjacentregions.This high velocity under the Adriatic is more
pronouncedin the lowestlayer (Figure 25) than in the top layer
(Figure 24). Note that the Alps do not showup in Figures24 and
25, whereasPanza et al. [1980] and Calcagnile and Scarpa
[1985] report large anomaliesboth in the westernAlps and the
eastern Alps. A reason for this discrepancymight be that the
depth-averaged
structureof the Alps deviatesnot very muchfrom
the rest of Europe,so that the surfacewaves are not perturbed
strongly.
+s%
for 0 km.<z<100km.
-5%
+5O/o
(513/13
for 100 km.<z<200km.
Fig.24. Relative
Svelocity
perturbation
([•/•) between
thesurface
anda
Fig.25. Relative
S velocity
perturbation
([•/•) fordepths
between
100
depthof 100km.
and200 km.
$NIEDER:LARGE-SALE
INVERSION
OF$U•ACE WAVE DATA,2
Early tomographic studies using P wave delay times
[Romanowicz,1980; Hovland et al., 1981; Hovland and Husebey,
1982; Babuska et al., 1984] producedrather different resultsfor
the P velocityunderEurope. The only consistentfeaturesof these
studiesare the low velocity in the Pannonianbasin and the high
velocity under the Bohemian massif for the upper layer (0-100
km). Both featurescan also be seen in Figure 24. (In Figure 23
the Pannonianbasin shows up as a region with a thin crust,
whereasthe Bohemianmassifcan be identifiedby its thick crust.)
A more recent tomographicinversionwith a much larger data set
producedmore detailedresults [Spakmart,1986a,b; Spakmartet
al., 1988]. In his studiesthe subductionof Africa under Europe
has been imaged spectacularly.The subductionof Africa of the
African slab under Europe can also be seen in Figure 25 as a
positive velocity anomaly in the deepestlayer (100-200 km)
under the Adriatic and northern Italy. Panza et al. [1982]
observedrelatively low velocitiesin the lid betweenCorsicaand
Italy; the sameanomalyis visible in Figure24.
In Figures24 and 25 the Rhine Grabenshowsup as a zone of
relatively low velocitiesextendingtoward the southeastfrom the
Netherlands.Most wave pathsin thisregion are in the southeastnorthwestdirection.This may explainwhy in the top layer (Figure
24) only the northern part of the Rhine Graben can be seen,
whereasthe southernpart of the Rhine Graben (which trendsin
the north-southdirection) is not delineated. The variationsin the
S velocity for the deepestlayer (Figure 25) reflect the lateral
12,079
9. CONCLUSION
Linear inversionof a large set of surfacewave data is feasible
with present-daycomputers.The Born inversionfor the surface
wave coda (using42 seismograms)
ran in roughly one night on a
superminicomputer. The inversionof the direct surfacewave for
the threefrequencybandstakesapproximatelythe sametime. The
nonlinear
inversion
for
the
smooth
reference
model
is
comparativelyfast andtakesabout3 hoursfor the threefrequency
bands.With the presentgrowth in computerpower, larger data
sets can soon be inverted with the same method,possiblyon a
global scale.
Reasonable
waveform
fits of the surface
wave
coda can be
obtained,leadingto a variancereductionof approximately25%
for the surface wave coda. However, with the data set used in this
studymany artifactsare introducedin the inversion(the analogue
of the smiles in explorationseismics). The fact that the surface
wave codacontainsa relatively largenoisecomponentis an extra
complication.A larger (redundant)data set is neededto perform
an accurateimagingof the inhomogeneityin the Earth usingthe
surfacewave coda. It would be interestingto set up controlled
experimentsto probe tectonicstructureslike continentalmargins
or the Tornquist-Tesseyre
zonewith scatteredsurfacewaves.
Applicationof Born inversionto the direct surfacewavesleads
to detailed S velocity models on a scale conaparableto the
wavelength of the surface waves used. With the data set
employed,a lateral resolutionof approximately300 km can be
variationsof the low-velocity zone. It is noted by York and
Helmberger [1973] and Paulssen [1987] that strong velocity achieved in some regions (Italy, France, Alps, western
variationsof the low-velocityzone exist, which is confirmedby Mediterranean),while in other areas,smearingalong the wave
Europe,the
Figure 25. Under the Massif Central the positive anomalyin the paths occurs(southernMediterranean,northeastern
Adriatic).
More
data
are
needed
to
achieve
a
more
evenly
top layer (Figure24) andthe negativeanomalyin thebottomlayer
distributed
resolution.
Only
a
limited
depth
resolution
can
be
(Figure 25) indicate a pronouncedlow-velocity zone, which is
obtained.
consistentwith the resultsof Souriau [1981].
The fact that a modelof theheterogeneity
is constructed
with a
The featurewhich showsthe power of the inversionmethodof
horizontal
length
scale
comparable
to
the
wavelength
of
the
used
this paper most spectacularlyis the high-velocity anomaly in
surfacewavesimpliesthat scatteringandmultipathingeffectsare
easternEuropein the deepestlayer (Figure25). (It is possiblethat
this area of high velocitiesextendsfarthereastward,but this area operative. This means that for this situation, dispersion
measurements
are not justified.Nevertheless,the resultingmodel
is not sampledby the data.) From the waveform point of view,
for theS velocitybearscloseresemblance
to theS velocitymodels
this anomalyis neededto producethe focusingneededto fit the
constructed
by
Panza
et
al.
[1980]
and
Calcagnile and Sca•;pa
amplitudesof seismograms
recordedin the northernstationof the
[1985],
which
are
largely
based
on
surface
wave dispersion
NARS array. For this reason,this anomalyis locatedaway from
measurements.Apparently, the phaseas deducedfrom ray theory
(but closeto) the sourcereceiverminor arcs.This zoneof a highS
is relativelyrobustfor structuresthat are not smoothon scaleof a
velocitycloselymarkstheTomquist-Tesseyre
zone,theboundary
wavelength.
betweencentralEuropeand the eastEuropeanplatform.Note that
Linear waveform inversionis a powerful and rigorousmethod
this transition zone is not visible in the upper layer of the Sto fit surfacewave data. Presently,the main limitation is imposed
velocity model (Figure24). This is consistentwith the findingsof
Hurtig et al. [1979], who showedby fitting travel time curvesthat by the availability of high-qualitydigital surfacewave data. A
network of seismometers,as describedin the ORFEUS [Nolet et
below 100 km the eastern European platform has higher P
al., 1985] or PASSCAL proposals,will increasethe resolutionand
velocities than central Europe. According to Figure 25 this
reliability of the resultingmodels.A datadistributioncenterlike
transition at depth between Central Europe and the eastern
ODC (ORFEUS Data Center) provides access to digital
Europeanplatformis very sharp.
seismologicaldata at low costs.Born inversionfor surfacewaves,
The modelspresentedproducea wealth of interestingfeatures.
applied to these data, may help to constructaccurateS velocity
However, at this point one shouldbe extremely careful with a
models of the Earth.
physical interpretation of these models. As shown by the
resolution analysis of section 7, parts of these models are
Acknowledgments.I thank GuustNolet for his stimulatinginterestand
subjected to strong lateral smearing, which could produce advicefor this research.His nonlinearwaveformfitting programformed
unwantedartifacts.Furthermore,the numberof seismogramsthat the backboneof the nonlinearwaveforminversionpresentedin this study.
contributesto the reconstructionof a particularinhomogeneityis Wire Spakmanprovidedme kindly with his tomographyprogram,which
relatively small. This meansthat errorsin individualseismograms formed the basisfor the softwarefor the Bom inversion.The help of
Gordon Shudofsky, Bemard Dost, and Hanneke Paulssen was
can distortthe reconstructed
images. Larger data setswith a more indispensable
for handlingthe NARS data. The constructive
commentsof
even path distributionare neededto producemodelswhich are Brian Mitchell and an anonymousreviewerare greatlyappreciated.The
lesslikely to containartifactsand which are more robustto errors NARS projecthasbeenfundedby AWON, the EarthSciencebranchof the
NetherlandsOrganizationfor the Advancementof PureResearch(ZWO).
in individual seismograms.
12,080
SNIEDER:
LARGE-SALE
INVERSION
OFSURFACE
WAVEDATA,2
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(ReceivedOctober22, 1987;
revisedApril 21, 1988;
accepted
April 25, 1988.)