Geophys. J. lnt. (1996) 125, 584-598

.Two-dimensional multivalued traveltime and amplitude maps by

uniform sampling of a Tay field

G.Lambaré,1P.S.Lucio1andA.Hanyga21ÉcoledesMinesdeParis,CentredeRecherche~nGéophysique,35rueSaintHonoré,77305Fontainebleau,France

2 lnstitutefor Solid Earth Physics, University of Bergen, Allegaten 41,5007 Bergen, Norway

Accepted 1996 January 8. Received 1995 December 18; in original form 1994 July 28

SUMMARyAnalgorithmforcomputingmultivaluedmarsfortra~eltime,amplitudeoranyother

ray-related variable is presented. It is based on a wavefront construction method, wherethe Tay field is decomposed into elementary cens defined by adjacent rays and

wavefronts. AsamplingcriterionforTaydensityinthephasespaceissuggested.ItisdemonstratedthatthisnewcriterionensuresuniformTaydensityovertheentireTayfield inc1uding caustics. The .method is applied to complex models.Keywords:Green'sfunctions,numericaltechniques,Taytheory,Taytracing,traveltime,wave propagation.

1 INTRODUCTION

The current interest in 3-D imaging has increased the impor-tanceofasymptoticmethodsofwavefieldcomputationbasedon Tay tracing. Migration and i.nversion requîtes repeatedevaluation of the wavefield in the scattering region from several

sources. Asymptotic methods offer a reasonable compromisebetweenaccuracyandcomputationalefficiency(Hanyga&Helle 1995). ln 3-D inversion/migration, the task is formidableandasymptoticmethodsbecomeunavoidable.Theefficientnumerical calculation of traveltimes and amplitudes is a majorchallengeinimagingbyasymptoticmethods.AsignificantbreakthroughwasachievedthroughthefinÎte-difference(FD)calculation of the first-arrival traveltime (Vidale 1988; Podvin&Lecomte1991).However,FDcomputationoftraveltimesleadstopoorimagingincomplexmedia(Geoltrain&Brac1991),owingtounreliableamplitudeinformation.Thesimul-taneous computation of traveltimes and amplitudes is possible

by dynamic Tay tracing of a densely and uniformly sampled

Tay field, and evaluation of traveltimes and asymptotic Taytheory(ART)amplitudesatagivenpointbyinterpolation(Lambaré et al. 1992; Vinje et al. 1993a,b; Sun 1992; Forgueset al. 1994).TheTaydensityhastobecontrolledinaIdertoensureaccuracy as weIl as computational efficiency of the algorithm.Raydensitycanbecheckedatsomeselectedplaces,suchasat a sequence of flat horizons (Lambaré et al. 1992) or wave-fronts(Vinjeetal.1993a,b;Sun1992).Thelatterapproachisfollowedinthispaper.lnaIdertokeeptheTaYdensityaboveaminimumlevelitisnecessarytoinsertadditionalraysastheTayfieldspreadsoutfromthesource.AnadditionalTayisgenerated by interpolation of initial data and subsequenttracing of the Tay.

584

TwocriteriaforTaydensityhavepreviouslybeensuggested,namelythemetricdistancebetweenadjacentrays(Lambaréetal.1992;Vinjeetal.1993a,b)andtheirangulardistance(Sun 1992). Instead of directly addressing the problem of the

precision of the ray-field sampling, these criteria attempt tocontraiTaydensitybyalooselyrelatedinputparameter.Asthe contrai parameter is not directly related to the complexÎtyoftheTayfield,eachparticularcomputationrequîtesatleastsomevisualizationoftheTayfieldinaIdertocheckthequalityoftheresult.Theshortcomingsoftheabove-mentionedalgorithms are particularly apparent in caustic regions.Weproposeamethodforcalculatingmultivaluedtraveltimeand amplitude maps in 2-D smooth heterogeneous velocity

fields. Our approach is based on the wavefront constructionmethodinitiallyproposedbyVinje,Iversen&Gj0ystdal(1992)andontheHamiltonianformulationofTaytracing.TheHamiltonianformulation(Goldstein1980)isaveryconvenienttoolinseismicTaytheory(seeBurridge1976;Hanyga,Lenartowicz&Pajchel1985;Chapman1985;Cerveny1989;Virieux&Farra1991).In;;IheHamiltonianapproach,rays in the configuration space (x)'are replaced by bicharacter-istic curves in the phase space (x, p), where p denotes theslownessvector.Thesetrajectories,definedastheintegralcurves of the Hamiltonian equations for a fixed source, do notintersect at caustics. ln the space (x, p), the bicharacteristicsspanaregularLagrangiansubmanifoldA(Maslov1972;Hanyga1984;Hanygaetal.1985).ThesubmanifoldAcanbeparametrizedbytwogloballydefinedcoordinates,namelythetraveltime 0" and the take-off angle (J. The ray-related variablesaresmoothfunctionsofthecoordinates(0",(J)andconsequentlyitispossibletointerp~latethemonA.ThetangentplanestoAcanbedeterminedateverypointbyparaxialTaytracing.WeproposetosampletheTayfieldalongraysbyamethod

(Ç)1996RAS

ofwavefront construction. The ray field is paved by elementary

quadrangular cells defined by adjacent rays and wavefronts.The cell sizes are checked to ensure a uniform precision of theinterpolationofthesubmanifoldAalongthesampledwave-fronts. First-order interpolation is used and an upper limit onthelocalmisfitofthesubmanifoldAwithitstangentplaneisused to cons train the ray density. Each elementary cell isprojectedandinterpolatedoveraregulargridin(x,z).Wepresent applications of our algorithm to Saille complex models.

2HAMILTONIANFORMULATIONOFRAYTHEORYTheHamiltonianforanisotropiemediumcalibewritteninthe form

1

(p2

)H(x, p) = 2: U2(X) - l ,wherexistherayposition,ptheslownessvectorassociatedwiththeray,ŒtheinternaIcoordinateoneveryray(Œhasthedimension of time) and u(x) the slowness. Bicharacteristics(x, p)(Œ) in the phase space satisfy a system of canonical or ray

parabolic paraxia! wavefront

Xo~

"

,,al"" paraxl ray

.... ... ... ... ... ... ..."'... :....Figure1.Paraxialapproximationaroundacentralray.Theparabolicparaxialwavefrontisshown.

Px

x

Phase Space

z

",,'

- - - - - - propagated isochrontopFigure2.ThepavingoftheLagrangianmanifold.TheLagrangianmanifoldispavedwithquadrangularcellsdefinedintermsof

bicharacteristics and isochrons.

(ê) 199/1 RAS. (;.TT 125. SR4-S9R

Traveltime and amplitude maps 585

differential equations:

{

OXOŒ=VpH=pIU2(X),op 1 p2 2OŒ=-VxH=2:U4(X)VU(x).(2)TheeikonalequationissatisfiediftheHamiltonianvanishesall

along the bicharacteristics: H(x, p) =O.SilicetheHamiltonian

does not depend on Œit is constant along any bicharacteristicand it is sufficient to ensure that it vanishes at the source point.

Traveltime is given by the integration of slowness along thetrajectory of the rays:

(1)

ru oxT(X(Œ»=T(X(Œo»+1p'OŒdŒ=T(X(Œo»+Œ-Œo,0

(3)

where . denotesthescalarproduct,whenceŒcoïncideswithtraveltime, up to an additive constant.

3PARAXIALRAYEQUATIONSLinearinterpolationofray-fieldvariablesaswellasamplitude

computation is based on paraxial ray tracing (Chapman 1985;Farra&Madariaga1987).Aperturbationofthebicharacter-istic (lix, lip) around a central bicharacteristic (Fig. 1) satisfies,in the first-order approximation, a linear system of ordinary

base of the ceg

L

(a)x distance

< dxVinje?

CONFIGURATIONSPACE

z

~'

P,%PHASESPACE

(b)

x and p distances< dxSun?

< dpSun?

Figure 3. Vinje's and Sun's criteria for checking the size of the cells.(a)Vinje'scriterion,wherethexdistanceofthetopofthecellmustnot exceed the value dXVinje.(b) Sun's criterion, where the x and pdistances of the top of the cell must not exceed the values dxsunand dpsun.

586G.Lambaré,P.S.LucioandA.HanygaP\JL

Pz

misfit

< dxmax?

< dpmax?top of the cellFigure4.Ournewcriterionforcheckingthesizeofthecells.The

misfit between the tangent plane (defined by the paraxial approxi-mation) and the exact manifold must not exceed a given value indistanceandslowness.Anewbicharacteristicisinterpolatedatthebase of the cell if the misfit exceeds (dxmax> dpmax).

+

Lz

+ +

Figure 5. Linear interpolation in the configuration space. Each cellon the Lagrangian manifold is projected on the configuration space.Itissubdividedintotwotriangles.Adenseregulargridisdefinedineach triangle. Values of the ray-field variables at the grid points aredetermined by linear interpolation from the values at the vertices

of triangles.

Ao

0"8

0 0

A2

0"180

maximumerror

Figure 6. Accuracy of the linear interpolation. Paraxial ray theory

provides an estimation for the error associated with our linearinterpolation.

differential equations:

obx

a; = VxVpH °bx+ VpVpHobp

P 2 1= -~

)(Vu(x)obx)+---z-

( )bp,

u (x u x

obpa; = -VxVxH°bx-VpVxH°bp

1 p2 p2=-~

( )VVu2(x)bx-~

( )VU2(X)(VU2(X)0bx)

2u x u x

(4)

1 2

+ U4(X) Vu (x)(p ° bp).Theeikonalequationimpliesthatthefirst-orderperturbationbHoftheHamiltonianvanishesalongtheperturbedraytrajectory:bH(x,p)=VxHobx+VpHobp

1 p2 1= ---Vu2(x)obx+-pobp=0.

2 U4(X) U2(X)(5)Thelineardifferentialsystemofequations(4)Ganbesolvedbythepropagatormatrixmethod(Aki&Richards1980).ThesolutionisgiveninterrnsofapropagatormatrixP,suchthat

(;;) (cr)= P(cr,cro)(;;) (cro).(6)

This matrix is defined as the 4 x 4 Jacobian matrix

o(x, p)

P(cr,cro)= o(xo,Po) (al 02

)Pl P2 '(7)whereabO2,PlandP2are2x2matrices(Farra&Madariaga

1987). It satisfies the linear system of ordinary differentialequations

oP = ( VxVpHocr-VxVxHVpVpH )p,

-VpVxH

(8)

where P(cro, cro)= 1 (initial condition) and the determinant ofP is equal to 1 (Liouville's theorem, Goldstein 1980).4RAYFIELDANDLAGRANGIANMANIFOLDA2-Drayfieldisnaturallyparametrizedbyanycoordinate8specifying the initial direction of the rays (the lake-off angle inthe case of a point-source radiation) and an internaI coordinatecr of the rays.

ln the phase space (x, p) the bicharacteristics parametrizedby(8,cr)spanasubmanifoldA(Pham1992).ThemanifoldAislocallyrepresentedbygraphsofsmoothfunctionssuchasVT(x),andconsequentlyisLagrangianandregular(Guillemin&Sternberg1977;Weinstein1979).AGaningeneralfoldoverthe configuration space (x). Caus tics of the ray field are imagesofthefoldsofAundertheprojectionmapping(x,p)-+(x)(Hanygaetal.1985).Consequently,atthecausticsthepro-jectionofA:(8,cr)-+(x,z)issingular(non-invertible),andmanyfunctionsassociatedwiththerayfieldGannatbeexpressedassmooth functions of the configuration space coordinates (x, z).ThetangentplanesofAinthephasespacearegivenbythe@1996RAS,GJI125,584-598

partialderivativesofthecoordinates(x,p)withrespectto8and(J.Theycanbeexpressedintermsofparaxialquantities8(x,p)j8(J and 8(x,p)j88, and, consequently, in terms of theirinitial values and the propagator matrices:

{

:(JG)(8,(J)=P((J,(Jo):(JG)(8,(Jo),:8G)(8,(J)=P((J,(Jo):8G)(8,(Jo),wherethepartialderivativeswithrespectto(Jareglvenby eq. (2).lnviewofLiouville'stheorem,regularityofAisensuredbythe condition that the derivatives of (x, p) with respect to8

and (Jare independent at any fixed value of (Jo, for example atthe source.Usingeq.(9)itispossibletodetermineabeamofparaxialrays around a central Tay. ln order to de scribe the variationofraysinthebeamweintroducethevector

8(x,p)

J((J,(Jo)= ao((J) = P((J, (Jo)J((Jo, (Jo),

-4.99

8-1.66...........

[IJ

.......

00

~1.66~

.....

N

p.. 4.99

-0.10

8~~1.37

.....

..r:1+-'p..

Q)

't:1 2.10

distanceinkm

-0.09 0.63 1.36

distanceinkm

-0.09 0.63 1.36

0.63

Traveltime and amplitude maps 587

where for a point source

J((Jo,(Jo)=

(

~

)

.

PzO-PxO

(11)

(9)ln the zero-order approximation of the Tay theory, theamplitudeAisgivenintermsofthegeometricalspreading

z

1 [8(X,Z)

JI}' = u (x) det8((J,8)by the equation

J}'((Jo)A((J)=A((Jo)}'((J)'Wenowdescribeamethodtoestimatethevaluesoftheparameters (x, p) and the traveltime T(x) in the vicinity of acentralTayforagivenTayfield.Supposeweknowthevalues

ofthevectorJandofthetraveltimeTtatthepoint

Al=(Xl'pdEA.WewanttoestimatethevaluesoftheslownessvectorPzandofthetraveltimeTzatapointAzEA(12)

px in .0001 sim

-5.00 -1.67 1.67 5.00

(10)

2.09

-4.99

8-1.66...........

[IJ

.......

0

g 1.66

~

.....

N

p.. 4.99

2.09

-0.10

s~~1.37

.....

..r:1+-'p..

Q)

't:1 2.10

px in .0001 sim

-5.00 -1.67 1.67 5.00

0.63Figure7.TheLagrangianmanifoldAforaconstant-velocityfield(v=2000ms-').ThepavinginthecensofAisshownforvarious2-Dprojectionsofthephasespace[(x,z),(x,PxJ,(Px,z),(Px,Pz)].Thesourcepointisatx=110m,z=110m.Thetraveltimestepis0.03s,andtheray-density criterion is dxmax=10manddPmax=10X10-6sm-1.lntheplane(x,z)theraysarestraightlinesandthewavefrontsarecircles.~1996RAS,GJI125,584-598

588G.Lambaré,P.S.LucioandA.HanygacorrespondingtothepointXzinthevicinityofAl'lnthevicinityofAbwehavethefollowingrelationbetweentheperturbedparameters(bX,bp)andtheperturbation(150",150):G:)=:0"(:)00"+:0(:)dO,

(13)wherethepartialderivativesareestimatedatAl'Weestimate00" and 00 froID the linear system consisting of eq. (13) and thedefinitionofox:ox=Xz-Xl'WethenobtainoptothefirstaIder froID eq. (13) and the traveltime to the second aIderfroID the relation

112=Tl+Pl.ox+-op.ox.2

(14)

5UNIFORMSAMPLINGOFTHELAGRANGIANMANIFOLDA

It remains to find a criterion for sampling the LagrangianmanifoldAinthephasespace(x,z,Px,Pz)'DrawingonthewavefrontconstructionmethodproposedbyVinjeetal.(1992,1993a)andSun(1992),wedividetheLagrangianmanifoldAintocurvilinearquadrilateralcells.Thisapproachhastheadvantage of an easy numerical implementation and of apossibleextensiontothreedimensions(Vinjeetal.1993b;

-4.77

S-1. 71""-..

[IJ, ,aa01.36

q.~N

Po< 4.42

-0.10

S~

q 1.36.~

..r:1

+'Po<

Q)

"d 2.08

distance inkm

-0.10 0.63 1.36

distanceinkm

-0.10 0.63 1.36

0.63

Lucio,Lambaré&Hanyga1995).Acellisdefinedasthesectionofaraytubeboundedbytwoisochrons.Itcanbespecified in terms of its four vertices (Fig. 2).ThecellsareconstructedproceedingalongtheraysfroIDaninitial set of cells. Saille criterion must be chosen to contraithecellsize.Anacceptablediscretizationofthecoordinate0"can be based on a periodic sampling. Any other sampling, for

example a constant step in z, could be chosen, but a constant

traveltime step has a more direct physical meaning.Thesamplingof0musttakeintoaccountthelocalray-fieldcomplexity.ArecentlyproposedcriterionbyVinjeetal.(1993a, b) is based on the maximum distance between rays on

successive wavefronts. Consider the top of a cell defined bythetwopointsAl(XbPl)andAz(xz,Pz)andassociatedwiththe bicharacteristics 01 and Oz. Vinje's criterion imposes(Fig. 3a)

Ixz - xli < dxVinje' (15)Thiscriterionhastheadvantageofbeingsimple,butitdoesnot guarantee uniform accuracy of interpolation. ln particular,the ray density is drastically underestimated in the vicinity ofcaustics.Asaresult,Vinjeetal.(1993a)havetoignoreSailleparts of the wavefrant adjacent to caustics.

ln aIder to overcome this limitation, Sun (1992) intraduced

an angular distance in addition to the met rie distance between

2.09

-4.77

S-1. 71""-..

[IJ

......

aaa 1.36

q.~N

Po< 4.42

2.09

-0.10

S~

q 1.36......

..r:1+'Po<Q)

"d 2.08

px ln .0001 sim

-4.43 -1.48 1.48 '4.43

px ln .0001 sim

-4.43 -1.48 1.48 4.43

0.63Figure8.TheLagrangianmanifoldAforaconstant-gradientvelocityfield.Velocityincreaseslinearlywithdepth,from2083mS-latz=0mto3833mS-latz=2000m.Thetraveltimestepis0.03s,andtheray-densitycntenonisdxmax=10manddPmax=10X10-6sm-1.lnthe(x,y)plane, the cens are defined by circulaI rays and wavefronts. Note the p-caustic at the bottom of the (x, Pz) projection.iÇ)1996RAS,GJl125,584-598

rays. With respect to Vinje's criterion, this criterion has theadvantageoftakingintoaccountinsomewaythedistancebetween the rays in terms of slowness (even if it considers onlythe direction of the slowness). ln this paper, a slightly alteredversionofSun'scriterionwasusedfortests,wheretheoriginalangulardistancewasreplacedbyanequivalenttestonslownessvectors (Fig. 3b):

{

Ixz- xii < dxsun,

Ipz- Pli < dpsun.(16)Thischoicewasmotivatedbythesimplicityoftheexpressionsandbythefactthatitretainstheunderlyingphilosphy.With

respect to Vinje's criterion, Sun's one gives a better raycoverage in the vicinity of caustic cusps but no significantimprovementinthecaseofsimplecaustics,wheretheslow-ness variation is orteil smaller. Numerical specification ofSun's criterion is still difficult, however, and in general ray

density has to be overestimated to obtain an overall satisfyingnumerical precision.Acorrectwaytoproceedshouldinvolveacriterionthatensuresauniformprecisionoftheinterpolation.Siliceweuseafirst-orderinterpolation,thelocalsamplingdensityoftheLagrangianmanifoldAshouldberelatedtothecurvatureofA.Agoodcriterionforestimatingthecurvatureofthesurfaceis the mismatch between a bicharacteristic and its paraxialapproximationfrOIDanadjacentbicharacteristic.TwobicharacteristicsRI'RzarecloseenoughiftheestimateofRzobtained by linear approximation based on paraxialmatricescomputedforRIdoesnotdeviatetoomuchfroID

Rz(Fig.4).Considertwopoints(Xl>pdand(xz,Pz)ontwodifferentbicharacteristics and suppose that the corresponding values

0.00

0.415

~0.80~

.1"'1

.d 1.915~

P-4

Q)

't:I I.BO

distanceinkm

0.00 0.415 0.90

Traveltime and amplitude maps 589

of (J,O are (Jl,Ol and (Jz, Oz. For a given perturbation of(jO=Oz-Olofthebicharacteristic01weGancomputeaparaxialestimateofthebicharacteristicOz.lnthevicinityofpointAlthecoordinates(x,p)Ganbeapproximatedintermsof(jOand(j(Jbythelinearequation(

DX

) (V'pH

)Dp=JIDO+-V'xH1(j(J, (17)whereJIdenotesthevalueofJat(Xl>pd.WecompareapointAzontheisochron(D(J=0)throughAlwiththelinearapproximation given by eq. (17). The above criterion amountsto an upper limit on the error in distance and slowness:

{

1(jX+Xl-xzl:o;dxmax,l(jp + Pl - Pzl:O; dPmax.

(18)Aswewillseelater,thiscriterionisrelatedtothecurvatureoftheLagrangianmanifoldAalongtheisochrons.FollowingtheideaofSun(1992)andVinjeetal.(1993a,b),a new bicharacteristic is traced froID the base of the cell whentwoendpointsofthebicharacteristicsturnouttobetoofarapartatthetopofthecell.Theparametersofthenewbicharacteristic are interpolated in the phase space at the baseofthecell(Fig.4).CubicHermiteinterpolation(Farin1993)in 0 is used for x, p and traveltime. ln order to ensure theembeddingofthecellsinthespace(x),,thenewinterpolatedpointhastobeonthebaseofthecell.AnabscissaCI:isintroduced such that x(CI:) =Xl+(xz-Xl)CI:,andderivativeswithrespecttoCI:ofparametersaregivenbycombinationsofthe paraxial parameters.

1.915 I.BO

1.100

1.7215

2.9150

2.9715

9.800

COMPLEXVELOCITYFIELDFigure9.Acomplexvelocityfield.25x25knotpointsspacedby100mareusedfortheB-splinerepresentationofthevelocityfield.ThevelocityrangesfroID1093mS-lto3800mS-1.(Ç)1996RAS,GJI125,584-598

590G.Lambaré,P.S.LucioandA.Hanygadistanceinkm

-4.96

6-080

~

[j)

.,-i

0003.36

~.~

~752 --1

-0.10

6~

.5 1.37

~

,0..Q)

-c 2.10

-0.10 0.64 1.37 2.10

-4.96

6-0.80~

[j)

.,-i

0003.36

~.~N

0..7.52

2.10

-0.10

6~

.5 1.37

~

,0..

Q)

-c 2.10

px ln .0001 sim

-4.72 -059 3.53 766

distanceinkm

-0.10 064 1.37

px ln .0001 sim

-4.72 -059 3.53 7.66

063 0.63Figure10.TheLagrangianmanifoldAinacomplexvelocityfield.Thecriterion(18)hasbeenapplied.Thesourceisatx=110m,Z=110m.Thetraveltimestepis0.03s,andtheraydensityisspecifiedbycriterion(18)withdxmax=10manddPmax=10X10-6sm-1.Therayfieldexhibitsmanycaustics,andprojectionsinvolvemanyoverlappingcells.Thedensityofraysintheconfigurationspaceisincreasedintheregionsofstrongcurvature of the rays.

6 INTERPOLATION INSIDE THE CELLS

For interpolating the results obtained at cell vertices on adensergridofpointsweusearobustfirst-orderscheme.Eachcellisdividedintotwotrianglesaccordingtoacriterionofthesmallestmisfitinpositionandslownessbetweentheoppositeedgesofthecell.Adensegridisdefinedineachtriangle.Foreachgridpoint,thevaluesof(p,8,T,J)areevaluated by linear interpolation froID the values at the verticesofthetriangle(Fig.5).TheprecisionoflinearinterpolationontherayfieldGanbethenestimatedwiththehelpofparaxial theory.

Suppose that on a given triangle Ao(80, 0"0), A1(81, 0"0)andA2(80,0"1),Aisapproximatedbyaquadraticexpression:{

x(8, 0-) =Xo+axL1O"+bxM+cAL1O")l+dxL10"L18+eAL18)Z,

p(8, 0")= Po + apL1O"+ bpL18 + cp(L1O")Z+ dpL10"L18+ ep(L18)Z,

(19)whereL10"=0"-0"0andL18=8-80,Linear interpolation of x, P on the triangle yields the

following(Fig.6)estimates:{

Xz-XoXl-Xo

xint(8, o")=xo+ -L10"+ -L18,0"1-0"0 81-80

P . (8 r) = P+Pz-PoL1+Pl-PoA

8mt' 0 0"8 8

L!..0"1-0"0 1- 0

(20)ThisresultGanbecomparedwiththesecond-orderapproxi-mation(19).Anestimateofthemaximumerroroftheapproximation (20) is restricted by the inequalities

1

1

iPx

1

1

1

iPx

1

1

1

azx

1Ixint-xl<8 arl L1O"î+8 a8l L18î+4 a8aO" L10"1L181

and

(21)

1

1

azp

1

z 11

alp

1

2 11

alp

1IPint-pl<8 aO"z L10"1+8 a8z L181+4 a8aO" L10"1L181

(22)

(1°1 denotes here the vector norm, L10"1= 10"1- 0"01 andL181=181-801).Thepartialderivativeswithrespectto0"aregivenbyeqs(2)and(4).Thefirst-orderderivativeswithrespect<1.:11996RAS,GJI125,584-598

S1.00~~.....

.ci 1.60

.....

Po.Q)

'U 2.00

0.60

Figure 11. Maps for traveltime obtained for the ray-field sampling

presentedinFig.10:(a)bythealgorithmofPodvin&Lecomte(1991);(b)byouralgorithmforthefirst-arrivaltraveltime;(c)byouralgorithmforthestrongest-arrivaltraveltime.Theverticalandhorizontalstepsare10m.~1996RAS,GJI125,584-598

(a)

~1.00~.....

..r:: 1.110

op)

Po.Q)

'C 2.00

(b)

~1.00~.....

..r:: 1.110op)

p..

Q)

'C 2.00

Traveltimeandamplitudemaps

591distanceinkm0.00 0.150 1.00 1.150 2.00

0.00

0.150distanceinkm0.00 0.150 1.00 1.150 2.00

0.00

.&.

o.

0.150 1.

2.

8.

Figure 12. Maps for amplitude obtained for the ray-field sampling

presented in Fig. 10: (a) the first-arrival amplitude; (b) the strongest-

arrivaiamplitude.Theverticalandhorizontalstepsare10m.toearegivenbytheparaxialparameters.Finally,

a2x

a(J2 ='lx'lpH'lpH-'lp'lpH'lxH,

a2pa(J2 = - 'lx 'lxH'lpH + 'lp 'lxH'lxH,

(23)a2x a

a(Jae = ae 'lpH,

a2p- aa(Jae = - ae 'lxH.Asweshallseelater,theassociatederrorisboundedbythevaluesofourray-densitycriterion.Theaccuracydependson

the constant traveltime step Ll(J chosen initially for samplingAandonthevaluesofdxmaxanddPmax.Theprecisionofinterpolation in the e direction is uniformly bounded, sincethe ray-density criterion (18) implies that

1

1

a2x

l

1

1

a2p

I2 ae2 Lleî<dxmax, 2 aeî Lleî< dPmax. (24)

Unfortunately, paraxial ray theory cannot pro vide anyinformation about the precision of the interpolated amplitude.

(a) distanceinkm0.00 0.60 1.00 1.60 2.00

1 1 1 1 10.00

0.60

S1.00

.....

.ci 1.60op)p..Q)

'U 2.00

(b) distanceinkm0.00 0.60 1.00 1.60 2.00

0.00

[00'0.60 iro.60o

s 1.00 II!III-1.000

.....

.ci 1.608-1.600

.....Po.Q)

'U 2.00 --2.000

(c)distanceinkm

0.00 0.60 1.00 1.60 2.00

1 1 1 1 10.00

592 G.Lambaré,P.S.LucioandA.HanygaNotethatsecond-aIderpartialderivativesin(Jcharacterizethe curvature of the wavefront (0"= constant) in the phasespace.NotealsothattheprecisionoflinearinterpolationofAisboundedforaninterpolationwithrespecttothecoordinates((J,0"),but flot with respect to (x, z), Silice the Jacobian matrix8(x, z)/8((J, 0")is flot invertible at caus tics.

7APPLICATIONSTherayprolongationschemepresentedaboveassumesasmoothvelocityfield.Thisrestrictionismotivatedbytheapplications in linear inversion, where ray tracing is muchfasterinasmoothbackground(Thierry&Lambaré1995;Ettrich&Gajewski1995).Theimplementationofcellraytracinginvelocitymodelswithinterfaceshasalreadybeen

(a) distanceinkm

0.00 1.33 1.780.44 0.89

0.75

0.92

fi]1.08

~

'1'"1

IV

e.1'"1+J-

IV

:>l\SM

+J

1.25

1.42

(b) distanceinkm0.00 0.44 0.89 1.33 1.78

0.00

0.38

0.78

IV"CJ~

+J.1'"1-0..el\S

1.13 +!+:

1.51

Figure 13. Multivalued traveltimes (a) and amplitudes (b) for avertical line of receivers at x = 2000mwithaverticalstepof10mobtainedbytheray-fieldsamplingpresentedinFig.10.Notenumeroustriplications and caustics.

developed(Vinjeetal.1993a;Pajchel&Moser1995)andapplied in inversion (Moser 1995).lnthemostgeneralcase,asmoothvelocityfieldGanbe specified in terms of cardinal cubic B-splines (de Boor1978).ThisensuressmoothnessuptothesecondaIder,which suffices to guarantee the continuity of the paraxialparameters. Integration of eqs (2) and (4) is implemented byafourth-orderRunge-Kuttaschemewithaconstantstep.Theschemerequiresfurtherimprovements.lnparticular,sampling in the ray direction should also be controlled andlocally adjusted by appropriate cell subdivision. Localreductions of ray density would prevent ray oversampling andimprove performance.

RayTheory

0

1 1.2 1.4time in s

Figure 14. Seismogram obtained froID the traveltime and amplitude

given in Fig. 12. The phase shift associated with crossing caustics hasbeentakenintoaccount.Thesourcesignatureisd2

s(t) = 2 exp[ -(t/0.005)2].dt (Ç)1996RAS,GJI125,584-598

000C\2

00CD

..-1

0

0

CO

..-1

S 00

..-1.

0

..Q

0

.......-C\2

p.,..-1

Q)

0

'\j00..-1

Q)

P.0

.

0

Q)

CD

U

Q) 0

0

CO

0

0

0

0

C\2

(a)

0.00

0.67

s~~

.....

,.q+>p"

Q)'U

1.33

2.00

(b)

0.00

0.67

s~~

.....

,.q+>Po

Q)'U

1.33

2.00

(c)

0.00

0.67

s~~

.....

,.q+>PoQ)'U

1.33

2.00

distance inkm

0.00 0.67 1.33 2.00

distance inkm

0.00 0.67 1.33 2.00

distanceinkm

0.00 0.67 2.001.33

(Ç)1996RAS,GJI125,584-598

Traveltime and amplitude maps 593

7.1 Choice ofthe values for dxmax and dpmaxThetolerancesdxmaxanddPmaxarerelatedtothedesiredaccuracyoftheLagrangianmanifoldsampling.Wehavesetdxmaxequaltothexandzstepinthemaps,whiledPmaxcorresponds to a maximum error in the direction of rays ofabout 1 to 2 degrees.

7.2 Constant velocity field

lnthefirstplaceweconsideraconstantvelocityfield

(v=2000mS-l).Thesourcepointisatx=110m,z=110m.Fig.7showsthepartitioningoftheLagrangianmanifoldAfor varions 2-D projections of the phase space [(x, z), (x, Px),(Px,z),(Px,Pz)].Thetraveltimestepis0.03s,andtheray-densitycriteriondxmax=10m,dPmax=10X10-6sm-1.Overthe (x, z) plane, the rays are straight lines and the wavefrontsarecircles.Therayfielddoesflotexhibitmultiplecoverageinthe configuration space.

7.3 Constant gradient ofvelocitylnthenextexample,velocityincreaseslinearlywithdepth,from2083mS-latz=0mto3833mS-latz=2000m.Fig.8showsvarionsprojectionsoftheLagrangianmanifoldA.Thetraveltimestepis0.03s,andtheray-densitycriterionisdxmax =10m,dPmax=10X10-6sm-1.ThecellsaredefinedbycirculaIraysandwavefronts.ThisfigurecaTIbecomparedwiththepreviousone.Asraysspread'apart,newraysareinsertedandthecellsaresubdividedtomeettheray-densitycriterion.Nocausticappearsintheconfigurationspace(x,z).OnecaTIsecafoldingofAovertheplane(x,Pz)atthebottomofthepicture.Anasymptoticsolutioninthe(x,Pz)domainexhibitsacausticinthisregion(calledap-caustic).Ap-causticcaTIalsobecharacterizedbyaninfinite(x,pz)-domainamplitude.

It is a general property of Lagrangian submanifolds that,ateverypointofaLagrangiansubmanifoldA,atleastoneofthefourprojectionsofAontheplanes(x,z),(x,Px),(Px,z)or(Px,Pz)isregular(Hanyga1984).TheFouriertransformoftheasymptoticsolutionwithrespecttothecorrespondingconjugate pairs of variables (z, Pz), (x, Px) or both involvesanon-vanishing'rayspreading'JacobianGeX,pz)jo(O,CT),o(Px, z)jo(O, CT), o(Px, pz)jo(O, CT). For a caustic in the con-figurationspaceonecaTIfindaregularasymptoticsolutionbysumming regular asymptotic 'plane' waves corresponding tothefixedvalueofPx[inthe(Px,z)domain]orPz[inthe(x,Pz)domain] (Maslov 1972).7.4AcomplexvelocityfieldLetusnowconsideramorecomplexvelocityfield.Thevelocityfield is the superposition of strong Gaussian heterogeneitiesFigure15.Comparisonofray-densitycriteriaforraytracingonthecomplex velocity field presented in Fig. 9. The source is at x =1000m,z=110m.Thetimestepis0.03s.Therayfieldissampledaccordingto:(a)Vinje'scriterion;(b)Sun'scriterion;(c)ouruniformsamplingcriterion.lnordertocompareequivalentresultsintermsofcom-putationalcost,wechoosethevaluesoftheray-densitycriteriainsuchawayastohavethesaillenumberofcens(about2305ineach case).

594G.Lambaré,P.S.LucioandA.Hanygasuperposed on a constant background (v =2000ms-1)(Fig.9).25x25knotpointsspacedby100mareusedfortheB-splinerepresentationofthevelocityfield.Thevelocityrangesfrom1093mS-1to3800mS-1.Fig.10showsthefourprojectionsoftheLagrangianmanifold.Thesourceisatx=110mandz=110m.Thetraveltimestepis0.03sandtheraydensityisspecifiedbythecriterion(18)withdxmax=10m,dpmax=1.0X10-5sm-1.Therayfieldexhibitsmatircaustics,andprojectionsinvolvematiroverlappingcens.Thedensityof rays in the configuration space is increased in the regionsof strong curvature of the rays.

TraveltimemapsareshowninFig.11forthesameparametersasinFig.10.Themapsarecomparedwiththeresult obtained for the first-arrival traveltime given by thefinite-differencetraveltimealgorithmofPodvin&Lecomte(1991).TheCPUtimeforthealgorithmofPodvin&Lecomtewas0.4s,ascomparedwith3.90sforourray-tracingcode(CPUtimesaregivenforaSparc10workstation).ThelatterCPUtimebreaksdowninto1.07sofray-fieldsamplingand 2.79 s of linear interpolation of the required parameters

(traveltime, amplitude, take-off angle, paraxial ray parametersandtheslownessvector).ThisCPUcostisfairlymodestascomparedwiththeCPUcostofthecodeofPodvin&Lecomte (1991), taking into account that the latter yields onlythefirst-arrivaltraveltimewhileourcodecomputesuptonineparameters and gives an the arrivais.lnourtests,thenumberofcenswas4400andthenumberofpointsinthemapswas201x201withastepof10m.AmplitudemapsareshowninFig.12.lnordertodemonstrate

s~

I=i.....

,.q.;.)P-a)

"d

distanceinkm

0.00 0.33

theabilityofouralgorithmtocopewithmultiplearrivais,Fig.13showsthemultipletraveltimesandamplitudesforaverticalline of receivers located at x=2000m.Manycausticscali be seen. Infinite amplitudes at caus tics are due to the factthatwehaveappliedasymptoticraytheory(ART)toevaluatethefield(Fig.14).ThesesingularitiescalibeeliminatedbysummationoverA,asexplainedinHanygaetal.(1995a).lnFig.15,comparisonoftherayfieldsampiedwiththecriteriaofVinjeetal.(1993a)(eq.15)andSun(1992)(eq.16)is shown. ln order to compare equivalent results in term ofcomputationalcost,wechoosethevaluesoftheray-densitycriteriainsuchawayastohavethesamenumberofcens(about2305)ineachcase.Thesourceisatx=1000m,z=110m.Thetimestepis0.03s.ThecomplexrayfieldsampledaccordingtoVinje'scriterion(eq.15)(dXVinje=100m)isshowninFig.15(a),whileFig.15(b)showstherayfieldsampledbySun'scriterion(eq.16)withdxsun=142m,dPsun=92X10-6sm-1.Fig.15(c)showstheresultofsamplingaccordingtocriterion(18)withdxmax=20m,dPmax=25X10-6sm-1.

Some attention must be paid to the caus tic zones. Vinje'scriterion leads to a drastic undersampling in the vicinity ofcaustics.Sun'scriterionbringssomeimprovementatthecausticcusp, but in a neighbourhood of simple caus tics the ray fieldis stin drasticany undersampled.Wealsocomparedtheerrorsoftheray-fieldinterpolationfortherayfieldssampiedwiththethreecriteria.Asareferenceforerrorestimation,weusedaverydenselysampiedrayfield(Fig.16)obtainedwithourray-tracingcodewithatraveltime

0.67 1.331.00 1.67 2.00

0.00

0.33

0.67

1.00

1.33

1.67

2.00Figure16.Referenceray-fieldsarnplingfortestingtheaccuracyoftheray-fieldinterpolation.Thetraveltirnestepis0.01s,dxmax=1rn,dpmax=10-6 s rn-I. 37516 cells were generated. l1J1996RAS,GJI125,584-598

stepof0.01sanddxmax=1.0m,dPmax=10-6sm-1.37516cellsweregenerated.Figs17and18showcomparisonsbetweenthe three criteria, for the erraIS IXin! - xrefl, and IPin!- Prefl asfunctionsofthecoordinateseand()"inaregionofA.Ouruniformsamplingcriterionappearstoensureabetterprecision,especiallyinthecausticzones.TheerraIismoreuniformlydistributed and has smaller extremal values. Sun's criteriongivesalittleimpravementwithrespecttoVinje'scriterion,butstill exhibits large erraIS localized in caus tic zones.

(a)

0.00.~

a:i

00

= 0.80

.S.

~0.80

....

=id

b 0.80

(b)

0.00

.:Ja:i00= 0.80

.El

.110.80

~

=id

b 0.80

(c)

.~

80= 0.80

.El

..g 0.80

~

=id

b 0.80

angleiDdegree871.U.100. 1115.

angleiDdegree871.U.100.

11&.

0.0

.<,,'

.,!

11.7

",8&.0

88.8angleiDdegree871.U.100.

11&.

0.00Figure17.ComparisonoftheinterpolationoftheTayfieldforthevariousray-densitycriteria.ThisfigurepresentstheerrorIXint-xœflassociatedwiththelinearinterpolationofx(B,cr)fortheray-fieldsamplings presented in Fig. 15 in respect to the reference samplingshowninFig.16.ErrorsaregiveninmetTesfor:(a)Vinje'scriterion(maximumerror64.9m);(b)Sun'scriterion(maximumerror65.0m);(c)ouruniformsamplingcriterion(maximumerror8.2m).@1996RAS,GJI125,584-598

Traveltime and amplitude maps 595

7.5 The Marmousi velocity fieldlnthelastexampleweappliedouralgorithmtoasmoothedsectionoftheMarmousimodel(Fig.19).TheB-splinerep-resentation of the velocity field is based on 33 x 34 knot pointsspacedat198m.Thesourceisatx=150m,Z=1550m.Thewavefront spacing is 0.03 s and the ray-density criterion isdxmax=1Om,dPmax=1Oxl0-6sm-1.Theextremecom-plexity of the wavefronts cali be seen in Fig. 20. Figs 21 and22 show the mars of traveltimes and amplitudes correspondingrespectivelytothefirstarrivaIandtothemostenergeticarrivaI.

0.00000oo

0.0000117

0.0000888

0.00008&0

0.00

i00:: 0.80

.a.

.g 0.80~

=id

b 0.80

Figure 18. Comparison of the interpolation of the Tay field for thevarious ray-density criteria. This figure presents the error Ipint - Proflassociatedwiththelinearinterpolationofp(0,T)fortheTayfieldsamplingspresentedinFig.15inrespecttothereferencesamplingshowninFig.16.ErrorsaregiveninsecondspeTmetTesfor:(a)Vinje'scriterion (maximum error 63.3 x 10-6 s rn-'); (b) Sun's criterion

(maximum error 63.4 x 10-6 sm-'); (c) our uniform sampling criterion

(maximum error 9.9 x 10-6 sm-').

(a) angleiDdegree871. 815. 100. 11&.

0.00.oua:i00::0.80

.a.0.80

.....=b 0.80

(b) angleiDdell'ee871. 815. 100. 1&.

0.00

.:1t:I00= 0.80

.a.

0.80.....=b 0.80

(c) angleiDdell'ee871. 815. 100. 11&.

596G.Lambaré,P.S.LucioandA.Hanyga0.00

0.715

~1.150

s:=

.,...

,.Q 2.215~

Pt

CI)

'C

9.00

distanceinkm

0.00 0.715 1.150 8.215 8.00

1.1500

~m~ 2.1500

9.1500

4..1500

15.1500

MARMOUSIVELOCITY FIELDFigure 19. Smoothed section of the Marmousi mode!. The B-spline representation of the velocity field is based on 33 x 34 knot points spaced

at198m.

s~~

.,...,

~

+'

~Q)

'"d

distance inkm

0.00 0.50 1.00 1.50 2.00 2.50 3.00

0.00

0.50

1.00

1.50

2.00

2.50

3.00Figure20.Ray-field'samplingintheMarmousimodelpresentedinFig.19.Thesourceisatx=150m,Z=1550m.Thewavefrontspacingis0.03 s and the ray-density criterion is dxmax=10manddpmax=10X10-6sm-1.Thereare15616cens.

... @1996RAS,GJI125,584-598

lntermsofcomputationaltimewehave3.34sforsamplingthe ray field (for 15616 cells) and 9.29 s for interpolatingamplitudes and traveltimes (334 x 334 points) on a Sparc10workstation.Theextracomputationaltimeforadditionalinterpolation of the take-off angle of the ray, slowness vectorand paraxial ray parameters is around 3 s.

8 CONCLUSIONTheHamiltonianapproachallowsustodealfairlyefficientlywiththecomplexityoftherayfield:thesingularitiesappearingin the configuration space are unfolded in the phase spaceyieldingaregular2-DmanifoldA.ParaxialraytheoryallowsacontraiofthesamplingoftherayfieldonA.AreliablecriterionforuniformsamplingoftheLagrangianmanifoldAhas been found. It corresponds to a uniform precision of thelinearinterpolationalongselectedwavefronts.Onthebasisoftheseconsiderationswehavedevelopedafastandrobustalgorithm for calculating multivalued maps of traveltimes and

amplitudes.Thealgorithmisparticularlyusefulformigrationandinver-sionofseismicdata.ItGannaturallybeextendedto3-D0.000

0.400

0.800

1.800

1.800Figure21.Traveltimemapsobtainedbytheray-fieldsamplingshawninFig.20ontheMarmousimode!.Thereare334x334pointsspacedby10minxandz.(a)ThefirsttraveltimearrivaI;(b)thestrongesta~riva!.~1996RAS,GJI125,584-598

Traveltime and amplitude maps 597

problems, for which the requirement offast modelling methodsisstillmorecrucial.The2-Dversionwasdevelopedfortestingtheconceptsthatwillbeappliedinthreedimension(Lucioetal.1995).Someextensionsandapplicationsbasedonthe2-D version are under development, namely, time-domain

Maslov integrals at caus tics and caus tic cusps (Hanyga,Lambaré&Lucio1995a)aswellasapplicationto2-Dasymptoticinversion(Thierry&Lambaré1995).ACKNOWLEDGMENTSThisworkwaspartirfundedbytheEuropeanCommissionand the Norwegian Research Council in the framework of theJOULEproject'3-DAsymptoticSeismicImaging'.PauloLuciowassponsoredbyagrantfroIDCNPq(Brazil).WeareindebtedtoPatrickComptefroIDIFPforkindlyprovidingasmoothed Marmousi velocity field, and to Roland Lehoucq(CEA)andMichelLégerfordiscussionsconcerningtopological(a)

~1.150~

.....

..Q 8.815

~~

CI.)

"CS8.00

(b)

~1.150~

.....

..Q 8.815

~~

CI.)

"CS8.00distanceinkm

0.00 0.'1'11 1.110 8.815 8.00

0.00

0.'1'11

distanceinkm

0.00 0.'1'11 1.150 8.815 8.00

0.00

8.000

0.000

0.'1'15 Il''''0.'1'110

1.1500

8.8150Figure22.Amplitudemapsobtainedbytheray-fieldsamplingshawninFig.21ontheMarmousimode!.Thereare334x334pointsspacedby10minxandz.(a)ThefirsttraveltimearrivaI;(b)thestrongestarriva!.ThedifferencebetweenthemapsagreeswiththeconclusionofGeoltrain&Brac(1991),concerningtheopportunityofusingorl.lytheminimumtraveltimearrivaIforimagingfroIDtheMarmousidata set.

(a) distance in km

0.00 0.'1'11 1.150 8.815 8.00

1 1 1 1 10.00

0.'1'15

1.110

.1"4

..Q 8.811

CI.)

"CS 8.00

(b) distance in km

0.00 0.'1'11 1.110 8.815 8.00

1 1 1 1 10.00

0.'1'11

1.110

!:I....

..1:18.811

CI.)

"'d 8.00

598G.Lambaré,P.S.LucioandA.Hanygacriteriainnon-convexspacemappings.Wearegratefultooneof the referees for valuable comments and to Pascal Podvin

for discussion and for providing the algorithm for calculatingfirst-arrival traveltime by finite differences.REFERENCESAki,K.&Richards,P.,1980.Quantitativeseismology:Theoryandmethods,W.H.Freeman&Co,SanFrancisco,CA.Burridge,R,1976.Somemathematicaltopicsinseismology,CourantInstitutofMathematicalSciences,NewYorkUniversity,NY.Cerveny, V., 1989. Ray tracing in factorized anisotropic inhomogeneous

media, Geophys. J. Int., 99, 91-100.Chapman,C.R.,1985.Raytheoryanditsextensions:WKBJandMaslovseismogram,J.Geophys.,58,27-43.de Boor, c., 1978. A practical guide to splines, Springer-Verlag,

Berlin, Germany.Ettrich,N.&Gajewski,D.,1995.EfficientprestackKirchhoffmigrationusingwavefrontconstruction,extendedabstract57thEAEGMeeting,Glasgow,A023,EAEG,Zeist,Netherlands.Farin, G., 1993. Curves and surfaces for computer aided geometricdesign:apracticalguide,3rdedn,AcademicPress,SanDiego,CA.Farra,V.&Madariaga,R,Seismicwaveformmodelinginhetero-geneousmediabyrayperturbationtheory,J.geophys.Res.,92,

2697-2712.Forgues,E.,Lambaré,G.,deBeukelaar,P.,Coppens,F.&Richard,V.,1994.AnapplicationofRay+Borninversiononrealdata,inSEGAnn.Meetg,ExpandedAbstracts,pp.1004-1007,LosAngeles,SEG,Tulsa,OK.Geoltrain,S.&Brac,J,1991.Caliweimagecomplexstructureswithfinite-d!fferencetraveltimes?,in61thSEGAnn.Meetg,ExpandedAbstracts,pp.1110-1113,SEG,Tulsa,OK.Goldstein,H.,1980.ClassicalMechanics,(lstedn1950)Addison-Wesley,Reading,MA.Guillemin,V.&Sternberg,S.,1977.GeometrieAsymptotics,AmericanMathematicalSoéiety,Providence,RI.Hanyga,A.,1984.DynamicraytracingonLagrangianmanifolds,Geophys.J.R.astr.Soc.,79,51-63.Hanyga,A.&Helle,H.B.,1995.SyntheticseismogramsfroIDgeneralized ray tracing, Geophys. Prospect., 43, 51-76.Hanyga,A.,Lenartowicz,E.&Pajchel,J.,1985.SeismicWavesintheEarth, Elsevier, Amsterdam.Hanyga,A.,Lambaré,G.&Lucio,P.S.,1995a.AsymptoticGreen's

.

functionsbyMaslovsummation,SEGAnn.Meetg,ExtendedAbstracts,pp.1297-1300,SEG,Tulsa,OK.Hanyga,A.,Thierry,P.,Lambaré,G.&Lucio,P.,1995b.2Dand3Dasymptotic Green's functions for linear inversion, in MathematicalMethods in Geophysical Imaging III, pp. 123-137, ed. Hassanzadeh,S.,Proc.SPIE,2571,SPIE,Bellingham,WA.Lambaré,G.,Virieux,J.,Madariaga,R&Jin,S.,1992.Iterativeasymptotic inversion in the acoustic approximation, Geophysics, 57,1138-1154.Lucio,P.S.,Lambaré,G.&Hanyga,A.,1995.3Dmultivaluedtraveltimeandamplitudemap,in57thEAEGMeetg,ExpandedAbstracts,p.147,EAEG,Zeist,Netherlands.Maslov,V.P.,1972.Théoriedesperturbationsetméthodesasympto-tiques,DunodetGauthier-Villars,Paris.Moser,T.J.,1995.Inversioninanonsmoothbackground,57thEAEGMeetg,ExpandedAbstracts,E036,EAEG,Zeist,Netherlands.Pajchel,J.,Moser,TJ.,1995.Recurcivecellraytracing,57thEAEGMeetg,ExpandedAbstracts,PO92,EAEG,Zeist,Netherlands.Pham,F.,1992.Géométrieetcalculdifférentielsurlesvariétés:Cours,études et exercices pour la maîtrise de mathématiques, InterÉditions,Paris.Podvin,P.&Lecomte,1.,1991.Finitedifferencecomputationoftraveltimesinverycontrastedvelocitymodels:amassivelyparallelapproach and its associated tools, Geophys. J. Int., 105,271-284.Sun,Y.,1992.Computationof2Dmultiplearrivaitraveltimefieldsbyaninterpolativeshootingmethod,in62thSEGAnn.Meetg,ExpandedAbstracts,pp.1320-1323,SEG,Tulsa,OK.Thierry,P.&Lambaré,G.,1995.2.5Dtrueamplitudemigrationonaworkstation,SEGAnn.Meetg,ExtendedAbstracts,pp.156-159,SEG,Tulsa,OK.

Vidale, J, 1988. Finite-difference calculation of travel time, Bull. seism.

Soc. Am., 78, 2062-2076. 'Vinje,V.,Iversen,E.&Gj0ystdal,H.,1992.Traveltimeandamplitudeestimationusingwavefrontconstruction,inEAEGAnn.Meetg,Abstract,pp.504-505,EAEG,Zeist,Netherlands.Vinje,V.,Iversen,E.&Gj0ystdal,H.,1993a.Traveltimeandamplitudeestimation using wavefront construction, Geophysics, 58, 1157-1166.Vinje,V.,Iversen,E.,Gj0ystdal,H.&Àsteb01,K.1993b.Estimationofmultivaluedarrivaisin3Dmodelsusingwavefrontconstruction,EAEGAnn.Meetg.Abstract,BOI9,EAEG,Zeist,Netherlands.Virieux,J&Farra,V.,1991.Raytracingin3Dcomplexisotropicmedia:ananalysisoftheproblem,Geophysics,56,2057-2069.Weinstein,A.,1979.Lecturesonsymplecticmanifold,AMS,Providence,RI.<Ç)1996RAS,GJI125,584-598