GEOPHYSICAL
RESEARCH LETTERS, VOL. 21, NO. 24, PAGES 2617-2620, DECEMBER
1, 1994
Findingsetsof acceptablesolutionswith a geneticalgorithm
withapplicationto surfacewavegroupdispersionin Europe
AnthonyLomaxandRoel Snieder
Department
of Theoretical
Geophysics,
Utrecht
University,
TheNetherlands
Abstract. We discussthe useof a geneticalgorithm(GA)
to invertdata for many acceptablesolutions,in contrastto
inversionfor a single, "optimum"solution. The GA is a
directedsearchmethod which does not need linearization of
the forwardproblemor a startingmodel, and it can be
In general this goal cannot be achieved with direct,
calculusbasedinversions.Thesemethodsrequireknowledgeof an adequatestartingsolution,and that perturbations of the model are linearly relatedto changesin the
data [Aki and Richards,I980]. They operatethrougha
applied
witha verylargemode!-space.
Consequently,
fewer singleor an iterativeperturbationof the solutionusing
assumptions
arerequiredanda greaterrangeof solutions
is locally determinedgradientsof the misfit surface;this
examined
than with many otherinversionmethods.
requiresa smoothanddifferentiablemisfitfunction.Only
We applytheGA to fundamental
Rayleighgroupdisper- a singlefinal solutionis identified,and, in general,this
sionestimates
for pathsacrossCentralEuropeand across solutionwill be in the neighbourhood
of and strongly
theEastEuropeanPlatformto determine"average",
layered dependent
on the startingsolution[Goldberg,1989].
S velocitymodelsseparatelyfor eachregion. The useof
Searchtechniquessuchas Monte Carlo and hedgehog
the GA allows an identicalmodel parameterizationand
allow a large modelspaceto be exploredand produce
broadparametersearchrangeto be usedfor bothregions.
multiple solutions[Keilis-Borokand Yanovskaya,!967].
The scatterof acceptable
solutionsshowsvelocity-depth
Howeverthesetechniques
becomeinefficientor impractitrade-offs
aroundthe Moho, indicatesthe depthresolution
cal
in
very
large
model
spaces.
The geneticalgorithm
of theinversion,andshowstheuncertaintyin uppermantle,
(GA)
method
[Goldberg,
1989]
is
one of a number of
S velocity estimates. The results indicate that a thicker
newer
techniques
that
give
a
more
efficient
samplingof a
crustandup to 0.3 km/sec(7%) higherS wavevelocitiesin
large
model
space;
it
has
been
used
recently
in geophysics
the upper 100 km of the mantle under the older East
European
PlatformthanunderCentralEuropeexplainmost [e.g., Stoffa and Sen, 1991; Sen and Stoffa, 1992;
Sambridgeand Drijkoningen,1992; Jin and Madariaga,
of the differences in the data sets.
1993; King, 1993].
The GA method is an iterative, non-local, controlled
search,which operateswith populationsof trial solutions
Introduction
to find new solutionswith lower "misfit", where the misfit
A goalof geophysical
inversionis to find all earthmodels is givenby a comparison
of predictionsfrom the solutions
which,whenoperateduponby someforwardmethod, with data. Beginningwith a randominitial populationof
produce
synthetic
datathatgivesan acceptable
agreement solutionsand corresponding
misfits,succeeding
populawithobserved
data[Keilis-Borok
andYanovskaya,
1967;Aki tionsarecreatedby threestochasticprocesses'1) selection
andRichards,
1980].However,forpractical
reasons,
most - saving those solutionswith smaller misfit, 2) crossover
inversions
include
various
assumptions,
simplifications
and - combining parts of two solutions to form new trial
restrictionswhich limit the allowable solutionsto a small solutions,and3) mutation- makingsmallchangesto the
regionof the physically
plausiblemodel-space.
For solutions. New populationsare createduntil a converexample,
directinversions
canfind solutions
onlyin the gencecriteriais reached.The GA searchproducesa large
neighbourhood
of somereference
model,and many set of solutionswhich give an estimate of the misfit
techniques
require
a simpleparameterization
of themodel surfacein the model space.
where
important
elements,
suchascrustal
velocityand
thickness,
arefixed. Thereis,however,
thepossibilityGeneticAlgorithmsfor AcceptableSolutions
thatthelimitedmodel-space
excludes
somereasonable
Unfortunately,
manyinverseproblems
in geophysics
are
non-linear
and
poorly
constrained.
Such
problems
may
addition,
manymethods
producea singlesolution,
complexregionsof
although
geophysical
problems
areoften
non-unique,
and havemultiple,broador topologically
space.Ourexperience
and
manysolutions,
perhapswithin differentlocal minima, minimummisfitin thesolution
willproduce
anacceptable
fit to thedata. Formany otherwork [e.g.,Goldberg,1989;Stoffaand Sen, 1991]
problems,
asa complement
to thesemorelimitedinver- indicatethatwith thistypeof problem,eachrunof a GA
sions,
it isuseful
toexplore
aslargea model
space
as tends to convergeto a single, local minima, and in
solutions
for a givendatasetandforwardmethod.In
possible,
retaining
largenumbers
of acceptable
•olutions.differentrunsdifferentminimamaybefounddepending
Copyright
!994
bytheAmerican
Geophysical
Union.
on the parameterscontrollingthe GA.
Many GA applications
convergeto a local minima
because
theyare configured
for rapidconvergence
with
the goalof findingan "optimum"solution. Here,theGA
Paper
number
94GL02635
0'094-8534/94/94GL-02635
$3.00
2617
is configured
to attempt
to findmanyacceptable
solutions
- solutions
representing
allregions
of themodelspace
that
give a misfit with the data below somelevel (Figure 1).
We begin with a GA similar to that described by
Sambridgeand Drijkoningen [1992], but set the ratesof
crossover and mutation relatively low so that many
solutionspassunchanged
to the followinggenerations,
we
flip many bits in eachmutatedstring,andthe bestsolution
of eachgenerationis not explicitly saved. In addition,we
define a minimum
50N
misfit value and reset lower misfits to
this value; this reducesthe stalling of the GA in deep
minima that are much lower than the acceptablelevel.
These adjustmentstend to producea smaller but more
stableand diverseset of acceptablemodelsrelativeto a
GA configuredfor rapid convergence.
Application to Group Velocity Dispersion
The older crust of the Precambrian East European
Platform (EEP) adjoinsthe youngercrust of Palaeozoic
Central Europe (CE) along the geologically distinct
Tornquist-TeisseyreZone (TTZ, Figure 2). Body and
surfacewave studieshave indicateda contrastof up to
10% in S wave velocityat the top of the mantleacrossthe
TTZ, with higher velocitiesunderthe EEP [Snieder,1988;
Zielhuis and Nolet, 1994]. Body wave studies also
indicateincreasedP wave velocity underthe EEP [Hurtig
et al., 1979; Spakmartet al., 1993].
In this study, the GA is usedto determinetwo setsof
layered velocity-modelswith group velocity dispersion
curvesthat matchgroupvelocityestimatesfor wavepaths
on each side of the TTZ. The observationsare digital
seismogramsfrom eventslocatedprimarily to the southeast of Europe recordedat distancesof about 10-30ø at
EuropeanNARS and GDSN stations(Figure 2).
Groupdispersionfor the fundamentalRayleighmodein
the period range of 7-300 sec is estimatedfrom the
observedwaveformsusing multiple-filter analysis(MFA)
[Dziewonskiet al., 1969]. Peakson the envelopesof the
narrow-bandseismogramsform group-velocity- period
data points for the GA inversion (Figure 3). Smooth
40N
Figure2. Map showing
Tornquist-Teisseyre
Zone(TTZ)
andgreatcircle pathsbetweensourcesandreceivers
used
in
thisstudyfor CE (solidlines)andthe EEP (dashed
lines).
When dividedby region,the group-velocity
dispersion
estimates
form two groupswhich,despitetheiroverlap
andscatter,havedistinctlydifferentcharacter
(Figure
3).
This groupingindicates
that inversionof thisdatamay
resolve significant differencesin "average"crustaland
uppermantle structurebetweenthe EEP and CE.
Model
Parameterization
The GA techniquepermits any model parameterization
that is compatiblewith the employedforwardalgorithm.
Sincethe GA techniquecan efficientlysearcha verylarge
numberof models,this parameterizationcan be liberally
defined,with few assumptionsand restrictions.
In this study the same model parameterization
is used
for the inversion
of the data from each side of the TTZ to
allow a relatively unbiasedestimateof the differences
in
structurebetweenthe two regions. This modelis defined
by 4 crustal and 14 mantle velocity-depthnodesanda
variableMoho depthbetween15 and70 km. Thedeepest
crustalnode is locatedat the Moho depth. The mantle
dispersioncurvesare not estimatedbecausethe fitting of
syntheticdispersioncurvesto the scatteredpeak valuesin nodesare spacedfrom the Moho to the bottomof the
increasing
the inversionproducesan effectivesmoothingof thedata, modelat 2371 km depth,with thenodespacing
in proportion
to depth.The location
oftwo
and because the scatter gives a frequency-dependent approximately
nodesat the Moho depth allows a step discontinuity
weightingto the curve fits.
betweenthe crust and mantle. The S velocityat each
misfit
surface
• •
.......•......a.
cceptablemisfitlevel....•;i
':-.'.•
.•...•...
%'i...L•:.½!½:
I1)
'•;':'"'•:•
'•.....
2
misfit surface
•acceptable
misfit
level
..•
.
==============================================================================
:.•::::::.•.::::.
:.::x::::::•:x:::::_.,,•
Central
Europe
,E:s,t
,••ropean
,elatf•••r•
Figure 1. a) A standardGA may find solutionslocated
near only one of the minima, 1, 2 or 3 in the data ntisfit
10
100
surface. But a goal of geophysicalinversionis to find all
Period (sec)
solutionswith misfit below someacceptancelevel (e.g. data
mode,Rayleigh
g•oup-velocity
variance). b) The identificationof a representativesample Figure3. Fundamental
of all acceptablesolutionsis a practicalway to achievethis estimates
asa function
of period
fromMFAanalysis
for
wavepathsin CE and in the EEP.
goal of geophysicalinversion.
node
canvarywithinapproximately
_+1km/sec(about constrainton the solutionsat depth primarily due to the
•0%) of theS velocity
fromtheiasp91model[Kennett lack of dispersiondata at greaterthan 300 secperiod.
There is increasedscatteron the dispersioncurves at
andEngdahl,
1991]at thecorresponding
depth.TheP
velocity
is determined
from the S velocityusinga shortestperiods;the correspondingscatterin the models
Poisson's
ratio of 0.25, the densityprofile is fixed occurs in the upper crust. This scatter indicatespoor
constrainton velocity at shallow depth due to the lack of
separately
for the crustandmantleandcorresponds
approximately
tothevalues
in iasp91.TheMohodepth shorterperiod data. There is also increasedscatterand
andS velocityparametersare discretizedwith a very diversity in the models near the Moho discontinuity
small
stepsize,givinganeffectively
continuous
variation
because a truncated set of low-order
surface-wave
modes
between
thelimitingvalues.This parameterization
gives cannot uniquely resolve a discontinuity. Some models
about
1045
possible
models,thoughthenumberof signifi- have a strong velocity contrast across a deep Moho
canfly
different
models
istheorder
of 10•ø.(Forcompari-discontinuityanda distinct,high-velocitymantlelid, while
son,in a similarstudywith phasevelocities,Calcagnile other models have a weaker Moho discontinuity at
andPanza[1978]usea "hedge-hog"
grid-searchfor about shallower depth and no high-velocity lid.
Figure 5 showsthe I c•and2(• distributionof acceptable
2X10
4possible
crust/mantle
models.)
A centrally-weighted,
5 point smoothingof velocityis models obtained with the modified GA for both CE and
applied
separately
to thecrustandmantlesections
of the the EEP paths. (For completeness,the resultsof threeGA
nodemodelto suppress
rapidoscillationof the solution. runsfor eachregionare shown,thougha singlerun of the
Thissmoothedmodel is convertedto a constant-velocity GA asconfiguredherecan indicatemost of the significant
layered
modelto calculate
synthetic
dispersion
curves.A variations in acceptablesolutions.) The resultsfor CE
starting
population
of modelsfor theGA searchis shown show a sharpincreasein velocity acrossa Moho discontinuity at about 25 km depth, and a well defined upper
inFigure4a; thesearchrangeis indicatedin Figure5.
mantle velocity structureto a depth of about 150 kin.
Below this depththe increasingspreadindicatesa lack of
Inversion Results
constraint on the solutions; the scatter in models for the
Figures4b and 4c showsthe resultsfrom the GA
appliedto the EEP data. The distributionof acceptable
models(solid lines) and correspondingdispersioncurves
showsthe mappingof uncertaintybetweenthe data space
andthemodelspace. Here, acceptablemodelsare defined
as thosemodelsgiving predictedgroup velocity values
with an r.m.s misfit in group velocity less than 0.9Ed,
whereEd is the r.m.s of the differencesin groupvelocity
between each of the data values and all the other data
valuesat eachperiod. This definitionof acceptance
level
is chosenso that the scatterof acceptabledispersion
curvesfalls within the averagescatterof the data.
The scatterof acceptablemodels is lowest from just
belowtheMoho to about350 km depth. This indicates
thedepthrangewherethe velocityis bestconstrained
by
the dispersiondata. Below about 350 km the models
begintofanoutandspantherangeof testedmodels(light
lines). This increase in scatter shows the loss of
EEP does not increasesignificantlyuntil below 300 km.
There is a difference in the maximum depth constrained
in the two regionsbecausethe longestperiod data available for CE is only about 150 sec, while the data for the
EEP extends to around 300 sec.
Because of the lack of
constraint,the depthextentof the low velocityzoneunder
CE cannot be determined
from this inversion.
The resultsin Figure 5 indicatethat significantchanges
in crustalthicknessand in uppermantle S velocity across
the TTZ can explain the main differencesbetween the
dispersioncurves for the two regions. The velocity
structuresin the upper 100 km of the mantle are well
constrainedin both regionsand show up to 7% higher
averageS wave velocitiesunder the EEP than underCE.
Taking in to accountthe differencesin data sets,model
parameterizationand inversionmethods,this result is in
agreementwith the contrastin upper mantle S velocity
acrossthe TTZ determinedby Zielhuis and Nolet [1994].
....•.:
{ .'...-.i½
•;.•
.:..4...
?:--:.
t!",
$
\
',
\
b)
' ,•-*.%\
x,
'::
'
.'
½)
.\
ß
1_l
30
4.0
5.0
6.0
,
,,
1o
S Velocity(km/s)
i
Iøo
Period (sec)
,
.
i
3.0
4.0
5.0
6.0
S Velocity (km/s)
Figure
4. Results
fromoneGAinversion
fortheEEPdata.a)Initialpopulation
of layered
earthmodels.
b)Observed
dispersion.data
(dots)
andrepresentative
setofsynthetic
dispersion
curves.
c)Representative
setot EarthmodelsAcceptable
results
aredrawnwithsolidblacklines,a sample
of all testedresults
areplotted
ingrey.About
4500models
weresampled
froma model
space
withabout
1045
members
using
a population
sizeof 60 and200 generations.
References
Aki, K. and P.G. Richards,
Quantitative
seistnology,
932
pp.,W.H. Freeman,San Francisco,1980.
Calcagnile,
G. andG.F. Panza,Crustanduppermantle
structure
undertheBalticShieldandBarents
Seafrom
thedispersion
of Rayleigh
waves,
Tectonophysics,
47,
59-71, 1978.
Dziewonski,
A.,BlochS.& Landisman,
M., A technique
(
.............
Ccmm!
-
1
' '•".,..•:E.'.½.
'\ /
fortheanalysis
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seismic
signals,
BullSeisin.
Soc. Am., 59, 427-444, 1969.
Goldberg,
D.E., Genetic
Algorithms
in Search,
Optimiz-
,.".e.•,:,'.,.:.E,
...
ationand MachineLearning,Addison-Wesley,
Reading, MA, 1989.
Hurtig,E., S. Gr•isslandR.-P. Oesburg,Velocityvariationsin theuppermantlebeneath
CentralEurope
and
the East EuropeanPlatform, Tectonophysics,
56,
EastEur
133-144, 1979.
3.0
4.0
5.0
S Velocity (km/s)
Jin, S. and R. Madariaga,Backgroundvelocityinversion
with a geneticalgorithm,Geophys.Res.Lett.,20,9396, 1993.
Figure 5. Upper450 km of resultsfrom 3 GA runsfor the Keilis-Borok,V.I. andT.B. Yanovskaya,
Inverseproblems
EEP andCE showingthe _+1
o (closelyspacedlines)andthe
in seismology(structuralreview), Geophys.J. R. Astr.
Soc., 13, 223-234, 1967.
+2(5 (outer lines) spreadin S velocityof acceptablemodels
and the search limits (long dashed lines). The spread Kennett, B.L.N. and Engdahl, E.R., Travel timesfor
indicates little resolution below about 150 km tbr CE.
global earthquakelocation and phase identification,
Geophys.d. b•t., 105, 429-466, 199I.
King, S., The geneticsof mantleviscosity(abstract),
Terra
Nova, 5, 581, 1993.
Sambridge,M. and G. Drijkoningen,Geneticalgorithms
Discussion
in seismic waveform inversion, Geophys.J. Int., 109,
323-342, 1992.
Two families of dispersioncurvesfrom Eurasiawere
inverted with a GA configuredto find large sets of Sen, M.K. and P.L. Stoffa, Rapid samplingof model
acceptable
solutions.The scatterin the setof acceptable spaceusinggeneticalgorithms:examplesfromseismic
models shows how the variance and the resolution of the
waveform inversion, Geophys. d. lnt., 108, 281492,
1992.
dispersion
datamapsbetweendataandthe modelspaces.
Snieder,
R.K., Large-scalewaveforminversionsof surface
This scattershowstrade-offsbetweenMoho velocity
contrast and depth, and between layer velocities and
wavesfor lateralheterogeneity2. applicationto surface
waves in Europe and the Mediterranean,J. Geophys.
thicknesses,
and indicatesthe maximumdepthresolution
Res., 93, 12067-12080, 1988.
of the inversionand the uncertaintyin the conclusion
that
Spakman,
W., S. van der Lee, and R. van der Hilst,
the uppermantle S velocitiesvary acrossthe TTZ.
Travel-timetomographyof theEuropean-Mediterranean
The GA presentedhere was configuredto improvethe
mantle down to 1400 km, Phys. Earth Planet.Inter.,
stability of the inversion, producing more consistent
79, 3-74, 1993.
imagesof the better-fittingregionsof the modelspace. Stoffa, P.L., and Sen, M.K., Nonlinear multiparameter
This stabilityis obtainedat the expenseof slowerconveroptimization
usinggeneticalgorithms:
Inversion
of
genceandsomewhatpoorer-fittingbestsolutionsthanwith
a more "standard" GA
and the results still show some
plane-wave
seismograms,
Geophysics,
56, 1794-1810,
1991.
dependenceon the GA parameters. More significant Zielhuis,A. andG. Nolet, Shear-wavevelocityvariations
modificationsto the GA, or perhapssome other search
in theuppermantlebeneath
centralEurope,
Geophys.
method,
mayberequired
toaaequately
define
thecomplex
topology of the acceptablemisfit region of the solution
spacefor many geophysicalproblems.
J. Int., ! 17, 695-715, 1994.
Anthony
LomaxandRoel Snieder,
Department
of
Theoretical
Geophysics,
UtrechtUniversity,
PO Box
Acknowledgements.We thank G. Laske,J.J. L6vSqueand
M. Sambridgefor their helpful reviews. This researchwas
supportedby the Netherlands Organization tbr Scientific
Research(NWO) throughthe Pionier projectPGS 76-144.
80.021, 3508 TA Utrecht, The Netherlands.
(Received
April 14, 1994,revisedAugust17,1994;
acceptedSeptember19, 1994.)