GeneralizedBremmerserieswith rationalapproximationfor thescatteringof waves

in inhomogeneousmedia

MATTHEUS J.N. VAN STRALEN�, MAARTEN V. DE HOOP

�AND

HANS BLOK�

June9, 1998

Abstract

The Bremmerseriessolutionof the wave equationin generallyinhomogeneousmedia,requiresthe introductionof pseudo-differential operators. In this paper, sparsematrixrepresentationsof thesepseudo-differentialoperatorsarederived. We focuson designingsparsematrices,keepingtheaccuracy high at thecostof ignoringany critical scattering-anglephenomena.Suchmatrix representationsfollow from rationalapproximationsofthe vertical slownessand the transverseLaplaceoperatorsymbols,and of the verticalderivative,asthey appearin theparabolicequationmethod.Sparsematrix representationsleadto a fastalgorithm. An optimizationprocedureis followed to minimize the errors,in the high frequency limit, for a given discretizationrate. The Bremmerseriessolverconsistsof threesteps:directionaldecompositioninto up-anddowngoingwaves,one-waypropagation,andinteractionof thecounter-propagatingconstituents.Eachof thesestepsisrepresentedby asparsematrixequation.Theresultingalgorithmprovidesanimprovementof theparabolicequationmethod,in particularfor transientwavephenomena,andextendsthelattermethod,systematically, for backscatteredwaves.

PACScodes: 43.20.Bi,43.20.Fn,41.10.H,42.20,42.82.-m,91.30.Fn.

�Laboratoryof ElectromagneticResearch, Facultyof ElectricalEngineering,

Delft Universityof Technology, P.O.Box5031,2600GADelft, theNetherlands.�Centerfor WavePhenomena,ColoradoSchoolof Mines,GoldenCO 80401-1887,USA.

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I. INTRODUCTION

Directionalwavefield decompositionis a tool for analyzingandcomputingthepropagationofwavesin configurationswith a certaindirectionality, suchasthewaveguidingstructurein Fig-ure1. Themethodconsistsof threemainsteps:(i) decomposingthefield into two constituents,propagatingupwardor downwardalongapreferreddirection,(ii) computingtheinteractionofthecounterpropagatingconstituentsand(iii) recomposingtheconstituentsinto observablesatthepositionsof interest.Themethodis beneficialbecauseit canbecomputationallyefficientandit canbeusedto separatedifferentpropagationphenomena,which is of importancein theinterpretationandinversionof measurements.

In thefrequency-domainBremmerseriesapproachto modelling,weencounterpseudo-differential operatorsin the directional (de)composition,in the downward and up-ward propagationor continuation,andin the reflectionsandtransmissionsdueto variationsin mediumpropertiesin the preferreddirection(De Hoop 1). For the numericalimplemen-tation,we employ a total rational-approximationapproachto find, upondiscretization,sparsematrix representationsof thesepseudo-differentialoperators.Therationalapproximationhasits roots in the parabolicequation(PE) method(Claerbout2, and Tappert3), and hasbeenextendedandexploredby Ma 4, Greene5, HalpernandTrefethen6, andCollins 7. Theratio-nal approximationshouldbecarriedout in a delicateway, to ensureconservationof acousticpower flow, seealsoCollins andWestwood 8. The Bremmerseriesgeneratessystematicallythebackscatteredfield, a topic investigatedby Collins andEvans9, andCollins 10. For fixedsamplingrates,allowing the numericalgrid to be coarse,we consideroptimizationsof thematrix representationsfor thethreesteps,(de)composition,propagation,andinteraction,suchthatthenumericaldispersionis minimized.Theideaof optimizationwasexploitedby Collins11, andCederberg et al. 12; thenumericaldispersionwascarefullyanalyzedby Trefethen13,Beaumontetal. 14, andHolberg 15� 16.

The improvementin accuracy and efficiency, and extensionsof rational approximationtechniquesremainto get significantattention. Recentadvancesin the applicationto explo-rationseismicscanbefoundin GravesandClayton17, andRuhl et al. 18. They controlledtheerrorsat largescatteringanglesby initiating thepropagationwith a transverselyhomogeneousbackgroundphaseshift. In the field of integratedoptics, the PE methodwasintroducedbyFleck, Morris andFeit 19. Variousextensionsof the PE-stylemethodhave beendevelopedsincethen.Thesearenow known asBeamPropagationMethods(BPMs).Examplesof presentBPMs arethe Methodof Lines 20, the Mode ExpansionMethod21 andwide-anglemethodsbaseduponhigher-orderrationalapproximations22� 23� 24. For a recentoverview we refer toHoekstra25. For thedevelopmentsin oceanacoustics,we refer the readerto Collins 7 � 26. Arecentoverview of PEmethodsin underwateracousticsis givenby LeeandPierce27.

The discretizationof the ‘one-way’ wave equation,the propagationstep,is basedon the

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third-orderThiele-typecontinued-fractionapproximationof the left vertical wave-slownesssymbol. We enforcethe associatedvertical slownessoperatorto be self-adjoint (‘energy-conserving’),astheexactoneis self-adjointin thereal ��� . This implies thatwe departfromtheprincipalsymbolanalysisof DeHoop1. Theone-waywaveequationis thusapproximatedby a partial differentialequation.By discretizingthe transversederivatives(the Laplaceop-erator)accordingto a rationalapproximationof its spectrum,thepartialdifferentialequationis transformedinto a systemof ordinarydifferentialequations.Thesolutionof this systemisformally written asa productintegral. Theexponentin thediscretizedversionof theproductintegral is thenreplacedby its ��� �� -Pade approximation.Sucha procedureguaranteesnu-mericalstability. We will pay mostattentionto the ��������� -Pade approximation,which yieldstheCrank-Nicholsonimplicit finite-differenceschemein thepreferreddirection.Theresultingalgebraicequations,which now involve sparsematrices,canbesolvedrapidly with standardproceduresavailablein varioussoftwarelibraries.In two dimensions,directmatrix inversionsarecarriedout; in threedimensions,iterative techniquesshouldbeapplied.

Theverticalphaseandgroupslownessesassociatedwith theultimatesystemof algebraicequationscanbeevaluatedandareusedto analyzethe numericalartifactsintroducedby thevariousrationalapproximations.Theaccuracy of theverticalgroupslownessascomparedwiththe exactverticalslownessis indicative for thenumericalanisotropy; the differencebetweentheverticalphaseandgroupslownessesis indicative for thenumericaldissipation.To arriveat an optimal systemof algebraicequations,fixing the bandwidthof the wave field andthesamplingrate,a simultaneousoptimizationof the Thiele-typeapproximation,the transversefinite-difference,andtheverticalfinite-differencerepresentationsis carriedout. Sincetheop-timizationwill dependon themedium’s wave speed,theoptimalparametersetwill vary withfrequency andpositionaswell. Theoptimizationprocedureis repeatedfor the composition,decomposition,reflectionandtransmissionoperators.With thevariousmatrix representations,theBremmercouplingseriessolutionof thewave equationcannow becomputed.For typicalmediumfluctuations,asa rule of thumb,oneneedsto considerthreetermsof this seriestoobtainanaccuraterepresentation.A general– a priori – rule for how many termsoneshouldcomputeis hardto give,however. In this respect,notethatin theFourierdomaintheBremmerseriesdoesnotnecessarilyconverge.

The outline of the paperis asfollows. In the next section,a summaryof the methodofdirectionaldecomposition,leadingto a coupledsystemof one-way wave equationsis given.In SectionIII, the conceptof generalizedslownesssurfaceis introduced,which is usedinrepresentationsof the Green’s functionsof the one-way wave equations.In SectionIV, theBremmercouplingseriesandits numericalimplementationarediscussed.The remainderofthepaperis dedicatedto thederivationandoptimizationof approximate,sparse,matrix repre-sentations.In SectionV, theone-waywaveequationsarediscretizedfor thepurposeof solving

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for theone-wayGreen’s functions.In SectionVI, transverse,transparentboundariesareintro-duced. In SectionVII, theoptimizationprocedurefor the one-way propagationis explained.SectionsVIII andIX containthediscretizationsandoptimizationsof the(de)compositionandinteractionoperators.Finally, our algorithmis illustratedin SectionX by variousnumericalexamples.

II. DIRECTIONAL WAVE FIELD DECOMPOSITION

For thedetailsonthederivationof theBremmercouplingseriessolutionof theacousticswaveequation,we refer the readerto De Hoop 1. Here,we restrictourselvesto a summaryof themethod.Ourconfigurationis two-dimensional.

Let ��� acousticpressure[Pa], ����� particlevelocity [m/s], ��� volumedensityof mass[kg/m� ], � � compressibility[Pa!#" ], $%� volumesourcedensityof injection rate[s!#" ], and&�' � volumesourcedensityof force[N/m � ]. We assumethatthecoefficient � is smooth,andthat � is constant,for thepurposeof wave field decomposition.Furthermore,we assumethatthesefunctionsareconstantoutsideacompactdomain.Thisprovisionenablesusto formulatethe acousticwave propagation,when necessary, as a scatteringproblemin a homogeneousembedding.The smoothnessentailsthat the singularitiesof the wave field (in particulartheonesin theneighborhoodof thewavearrival) arisefrom theonesin thesignaturesof thesourcedistributions. The formationof caustics,associatedwith multi pathingof characterictics,iscapturedin thenumericalproceduredevelopedin thispaper.

We carryout our analysisin the time-Fourierdomain. To show thenotation,we give theexpressionfor theacousticpressure(�)�+*#,-�/.0�1�3254687:9<;>=@? �BA i .DC/�E�)�+*#,F� C/�HG@C0�JILKNM�.PORQTSVU (1)

Underthis transformation,assumingzeroinitial conditions,we have W 6YX i . . In theFourierdomain,theacousticwavefield satisfiesthesystemof first-orderequationsW ' (�[Z i .\� (� ' � (&�' � (2)

i .0� (�RZ]W�� (�^�_� ($VU (3)

The changeof the wave field in spacealonga directionof preferencecannow be expressedin termsof the changesof the wave field in the direction transverseto it. The directionofpreferenceis taken along the * � -axis (or ‘vertical’ axis) and the remaining(‘transverse’or‘horizontal’) coordinateis denotedby * " .

Sincewe allow themediumto vary with all coordinatesandhencealsowith thepreferreddirection,weareforcedtocarryoutthewavefielddecompositionfrom thesystemof first-orderequationsratherthanthesecond-orderscalarHelmholtzequation.

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A. The reduced system of equations

Theprocedurerequiresaseparatehandlingof thehorizontalcomponentof theparticlevelocity.FromEqs.(2)and(3) we obtain (� " � i � !#" . !#" �`W " (�5A (& " �a� (4)

leaving, uponsubstitution,thematrixdifferentialequation�`W �cbedgf h Z i . (i dgf h � (j h � (k d � (i dlf h � (i dgf h �+* " �gm "ln * � �a�Jm "�o i. W " � (5)

in which theelementsof theacousticfield matrixaregivenby(j " � (��� (j � � (� � � (6)

theelementsof theacousticsystem’soperatormatrixby(i "Lfp" � (i � f � �qSV� (7)(i "Lf � �r�s� (i � fp" �tAum " �� !#" m " ��Zv�w� (8)

andtheelementsof thenotionalsourcematrixby(k " � (& � � (k � �qm " �+� !#" (& " ��Z ($VU (9)

It is observedthattheright-handsideof Eq.(4)and(i dgf h containthespatialderivative m " with

respectto the horizontalcoordinateonly. m " hasthe interpretationof horizontal slownessoperator. Further, it is notedthat

(i "Lf � is simplyamultiplicativeoperator.To ensurethat the mediumis smooth,we employ equivalent mediumaveragingat any

point over a box that is twice the spatialsamplesize. (This procedureis derived from theoneby CoatesandSchoenberg 28 for smoothingdiscontinuitiesmakinguseof theequivalentmediumaveragingof Schoenberg andSen29).

B. The coupled system of one-way wave equations

To distinguishup- anddowngoingconstituentsin thewave field, we shallconstructanappro-priatelinearoperator

(� dlf h with (j d � (� dlf h (x h � (10)

that,with theaidof thecommutationrelation �`W � (� dgf h �D�zy{W � � (� dgf h�| , transformsEq.(5)into(� dgf h �`W �lb�h f } Z i . (~ h f } � (x } ��A��`W � (� dlf h � (x h Z (k d � (11)

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asto make(~ h f } , satisfying (i dgf h (� hgf } � (� dgf h (~ hgf } � (12)

a diagonalmatrix of operators.We denote(� dlf h asthe compositionoperatorand

(x } asthewavematrix. Theexpressionin parentheseson theleft-handsideof Eq.(11)representsthetwoso-calledone-waywaveoperators.Thefirst termon theright-handsideof Eq.(11)is represen-tativefor thescatteringdueto variationsof themediumpropertiesin theverticaldirection.Thescatteringdueto variationsof themediumpropertiesin thehorizontaldirectionis containedin(~ h f } and,implicitly, in

(� dlf h .To investigatewhethersolutionsof Eq.(12)exist,weintroducethecolumnmatrixoperators(�0���#�d accordingto (� ���#�d � (� dlfp" � (� � ! �d � (� dgf � U (13)

Uponwriting thediagonalentriesof(~ h f } as(~ "Lfp" � (� ���#� � (~ � f � � (� � ! � � (14)

Eq.(12)decomposesinto thetwo systemsof equations(i dgf h (�0���#�h � (�a���#�d (� ���#� U (15)

By analogywith the casewherethe mediumis translationallyinvariantin the horizontaldi-rection, we shall denote

(� ���#� as the vertical slownessoperators. Notice that the operators(�0���#�" composetheacousticpressureandthat the operators(�0���#�� composetheverticalparticle

velocity, whereastheelementsof(x } maybephysicallynon-observable.

In De Hoop1 anAnsatzprocedurehasbeenfollowedto solve thegeneralizedeigenvalue-eigenvectorproblem(15) in operatorsense:in theacoustic-pressure normalizationanalogwehave y (�0���#�� � (i � fp" (i "Lf � | �qSVU (16)

In thisnormalizationwefind theverticalslownessoperatorto be(� ���#� ��A (� � ! � � (� � (� "`��� � (� � (i � fp" (i "Lf � � (17)

while thegeneralizedeigenvectorsconstitutethecompositionoperator(����� (i "Lf � (i "Lf �(� A (��� U (18)

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Notethatwe have decomposedthepressure(up to a multiplicationby density)viz. accordingto(j " � (j �" Z (j !" ,

(j �" � (i "Lf � (x " and(j !" � (i "Lf � (x � . In termsof theinverseverticalslowness

operator,(� !#"�� (� !#"`��� , thedecompositionoperatorthenfollowsas(� !#" � �� � (i !#""Lf � (� !#"(i !#""Lf � A (� !#" � U (19)

Usingthedecompositionoperator, Eq.(11)transformsinto�`W �cb>dlf } Z i . (~ dlf } � (x } ��A[� (� !#" � dgf } �W � (� }uf � � (x � Z�� (� !#" � dgf } (k } � (20)

which canbe interpretedasa coupledsystemof one-way wave equations.Thecouplingbe-tweenthe counter-propagatingcomponents,

(x } , is apparentin the first source-like term ontheright-handside,whichcanbewrittenasA (� !#" �`W � (�\�D� � (� (�(� (� � � (21)

in which(�

and(�

representthe transmissionand reflectionoperators,respectively. In theacoustic-pressurenormalizationanalog,for constantdensity, we thusfind(� ��A (� � "� (� !#" �W � (� �aU (22)

III. THE ONE-WAY WAVE PROPAGATOR

To arrive at a coupledsystemof integral equationsthat is equivalent to Eq.(20) and thatcan be solved in termsof a Neumannexpansion,we have to invert the operatoroccurringon the left-handside. We set

(� ���#� � �`W � Z i . (� ���#� �g!#" . The one-sidedelementarykernels(� ���#� �+* " � * ��n *#� " � *#�� � associatedwith theseoperatorsare the so-calledone-way Green’s func-tions.They satisfytheequationsW � (� ���#� Z i . (� ���#� (� ���#� � b �E* " A�* � " � b �E* � A�* �� �a� (23)

togetherwith theconditionof causality.Now, consider

(� � (� ���#� , (� � (� ���#� and(� � (� ���#� . Theoperator

(�actsona testfield

(� as� (� (� ���E* " � * � ��� 2��g��� � 2�¡l¢ £��e� � (� �E* " � * ��n * � " �l¤¥� (� �E* � " �l¤¥�HGH* � " G#¤5U (24)

Let usdefinetheinitial-valueproblemof determininga function(¦ �+* " � * ��n ¤¥� satisfying�`W � Z i . (� � (¦ �qS for * �-§ ¤#� (¦ �E* " �l¤ n ¤¥��� (� �E* " �l¤¥�YU (25)

Thenit is observedthat � (� (� �e�+* " � * � �D��2 ¡>¨� 7 ! 4 (¦ �+* " � * ��n ¤@�HG:¤5U (26)

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A. The product integral

We observe that the vertical slownessoperatorsat different levels of * � do not necessarilycommutewith oneanotherdueto theheterogeneityof themedium.Thuswearriveata ‘time’-orderedproductintegralrepresentation30 of theone-sidedpropagators(cf. Eq.(25))associatedwith theone-waywaveequations,where‘time’ refersto theverticalcoordinate* � ,(¦ ���#� �BU©� * ��n * � � ���3ª�«��­¬5yp* � � A®* �g| �°¯± ² ¡>¨³� 7 ¡ ¢ ¨ ;>=@? y´A i . (� ���#� ��U8�l¤¥�HG#¤ | µ ¶· (� �BU©�/* �� �>U (27)

In this expression,theoperatororderingis initiatedby ;>=@? y�A i . (� �BU©�/* �� �HG#¤ | actingon(� �BU©� * �� �

followedby applying ;>=@? y�A i . (� ��U8�l¤¥�HG#¤ | to theresult,successively for increasing¤ .If the mediumin the interval y *#�� � * �g| wereweakly varying in the vertical direction, the

Trotterproductformulacanbeappliedto the productintegral in Eq.(27). This resultsin theHamiltonianpathintegral representationsfor theGreen’s functions,(� ���#� �E* " � * ��n * � " �/* � � ���3ª¸«��­¬5yp* � � A�* �g| �#2�¹»º��E* � �" �l¼ � �" �;c=H?¾½ A i .s2 ¡>¨� 7 ¡ ¢ ¨ G#¤:M�¼ � �" �`¿ � * � �" ��Z (À ���#� �+* � �" �l¤#�l¼ � �" �lO�ÁT� (28)Â

beinga setof paths �+* � �" �`¤¥�>�g¼ � �" �`¤¥� � in (horizontal)phasespacesatisfying* � �" �`¤°�t* �� �Ã�Ä* � " ,*#� �" �¤[�r* � �\�Å* " . In Eq.(28),(À ���#� is theso-calledleft symbolof

(� ���#� , i.e.(� ���#� �+* " �lm "cn * � � ;c=H? �BA i .Y¼ " * " ��� (À ���#� �+* " � * � �l¼ " � ;>=@? ��A i .Y¼ " * " �0U (29)

Thepathintegral in Eq.(28)is to beinterpretedasthelatticemulti-variateintegral(� ���#� �+* " �/* ��n * � " � * � � �D�3ª�«��`¬°y * �� A�* �g| ��Æ©ÇÈK} É 4 2 }³Ê 7 " �E.0Ë �^Ì ��G#¼�� Ê �" }R!#"³Í 7 " GH*Î� Í �";>=@?Ͻ A i . }Ð' 7 " M�¼�� ' �" �E*Î� ' �" A�*Î� ' !#" �" ��Z (À ���#� �+*Î� ' �" �g¤ ' A "��Ñ !#"/Ò * � �l¼�� ' �" � Ñ !#"/Ò * � O�ÁT� (30)

with * � 9 �" �Å* � " �Î* � } �" �r* " , and Ò * � �r* � AÓ* �� . All theintegrationsaretakenovertheinterval��AuÔq�cÔT� , Ñ !#" Ò * � is the stepsizein ¤ , and �E* � Í �" �g¼ � Í �" � arethe coordinatesof a pathat thediscretevalues¤ Í of ¤ asÕV�t����Ö�Ö�Öe� Ñ . If Ò * � is sufficiently small,thelatticeintegralreducesto thecaseÑ ��� : (� ���#� �E* " � * ��n * � " � * � � �D×Ī¸«��`¬°y * �� A�* �g| � 2 �E.0Ë ��Ì ��G#¼ � �";>=@? y�A i .PM�¼ � �" �+* " A®* � " ��Z (À ���#� �E* " � * � A "� Ò * � �l¼ � �" � Ò * � O | U (31)

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B. The generalized slowness surface

In theacousticpressurenormalizationanalog,thecharacteristicdifferentialoperator(cf. Eq.(17))is givenby(� ��A-m " m " A�� !#" �m " �¥�:m " Z]� !¥� �`m " �¥���m " �¥��A®� !#" �`m " m " �¥��Zv�:�s� (32)

with left symbol(Ø � (Ø � Z (Ø " Z (Ø 9 , in which(Ø � ��A-¼ �" Zv�:�°� (Ø " ��AF� !#" �`m " �¥�:¼ " � (Ø 9 ��AF� !#" �`m �" �¥�)ZÙ� !¥� �`m " �¥� � U (33)

In our configuration,we have(Ø " � (Ø 9 ��S , sincewe assumethedensityto beconstant.In

(Ø �we cansubstituteÚÃ�Û�­�:�¥� !#"`��� , thewavespeedin themedium.

Usingthecompositionrulefor symbolsof pseudo-differentialoperators,theoperatorequa-tion

(� �V� (� (cf. Eq.(17))is transformedinto a characteristicequationfor thecorrespondingleft symbols,A��E.0Ë ��Ì � 2e¡ ¢ £ �e� ��2�Ü ¢ £ �e� � (34)(À �+* " �g¼ � " � ;c=H? y i .u�E* " A�* � " ���`¼ " A�¼ � " � | (À �+* � " �g¼ " �HG:¼ � " GH* � " Z (Ø �E* " �l¼ " �D��SÓUThis equationdefinesthe generalizedslownesssurfaceandhassolutions

(À ���#� , which appearin the thin-slabone-way Green’s functions(31). Thetwo branchesare

(À ���#� �E* " �l¼ " � suchthatILKÝM (À ���#� �+* " �l¼ " �cO3Þ S and ILKNM (À � ! � �+* " �l¼ " �cO § S . Thesechoicesare consistentwith theconditionthat

(� ���#� areboth causal. Due to the up/down symmetryof the mediumwe have(À ���#� ��A (À � ! � . Notethatas . X Ô thecompositionof symbolstendsto anordinarymultipli-cation,andthesolutionof Eq.(34)reducesto theprincipalpartsof thesymbols.Theprincipalpartof theverticalslownesssymbolcorrespondsto theverticalcomponentof gradientof traveltime, in accordancewith theeikonalequation(whichcanbeobtainedfrom thehigh-frequencyapproximationof thepathintegral,seeDeHoop1)

IV. THE GENERALIZED BREMMER COUPLING SERIES

A. The coupled system of integral equations

Applying theoperatorswith kernelsEq.(28)to Eq.(20)weobtaina coupledsystemof integralequations.In operatorform, they aregivenby� b>dlf h A (ß dgf h � (x h � (x � 9 �d � (35)

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in which(x � 9 � denotestheincidentfield. In ourconfigurationthedomainof heterogeneitywill

berestrictedto theslab �`SH�/*#à+á/â ã� | , seeFigure1, andtheexcitationof thewaveswill bespecifiedthroughaninitial conditionat thelevel * � �rS , viz.(x � 9 �" �+* " � * � �D� 2�¡ ¢ £ �e� � (� ���#� �+* " �/* ��n * � " �lSä� (x " �E* � " �lSä�HGH* � " � (x � 9 �� �E* " � * � �D��SÓ� (36)

in therangeof interest,* �-å yæS@� * àEá â ã� | ; thesecondequationreflectstheassumptionthatthereisnoexcitationbelow theheterogeneousslab. Theintegraloperatorsin Eq.(35)aregivenby� (ß "Lfp" (x " ���E* " � * � ��� 2 ¡>¨� 7:9 2�¡ ¢ £ �e� � (� ���#� �+* " �/* ��n * � " �l¤¥��� (� (x " ���+* � " �g¤@�HG@* � " G#¤°� (37)� (ß "Lf � (x � ���E* " � * � ��� 2 ¡>¨� 7:9 2 ¡l¢ £ �e� � (� ���#� �+* " �/* ��n * � " �l¤¥��� (� (x � ���+* � " �l¤¥�HGH* � " G#¤°� (38)� (ß � fp" (x " ���E* " � * � ��� 2 ¡cç´è`é ê¨� 7 ¡>¨ 2 ¡l¢ £L�e� � (� � ! � �+* " � * ��n * � " �l¤¥��� (� (x " ���+* � " �g¤@�HG@* � " G#¤°� (39)� (ß � f � (x � ���E* " � * � ���32 ¡cç´è`é ê¨� 7 ¡>¨ 2 ¡l¢ £L�e� � (� � ! � �+* " � * ��n * � " �l¤¥��� (� (x � �e�+* � " �l¤¥�HGH* � " G:¤°U (40)

They describetheinteractionbetweenthecounter-propagatingconstituentwaves.Wecanrepresenttheactionof theone-sidedGreen’s kernelsby productintegrals,viz.,(x � 9 �" ��U8� * � �D� ¯± ² ¡>¨³� ¢ 7:9 ;>=@? y�A i . (� ���#� ��U8�l¤ � �HG#¤ � | µ ¶· (x " ��U8�lSä�0� (41)

while � (ß "Lfp" (x " ���BU©�/* � �D�q2 ¡>¨� 7:9 ¯± ² ¡>¨³� ¢ 7 � ;>=@? y´A i . (� ���#� ��U8�l¤ � �HG#¤ � | µ ¶· � (� (x " �e��U8�l¤¥�HG#¤5� (42)

andsoon.

B. Bremmer series

If .��ëA i ì (and ¼ " � i ¼îí" å i I ï ) with ì realandsufficiently large, theNeumannexpansioncanbe employed to invert � bedgf h A (ß dgf h � in Eq.(35). Sucha procedureleadsto the Bremmercouplingseries,(x d � 4ÐÍ 7:9 (x � Í �d � in which

(x � Í �d � (ß dgf h (x � Í !#" �h for Õ § �¸� (43)

canbe interpretedasthe Õ -timesreflectedor scatteredwave. This equationindicatesthat thesolutionof Eq.(35)canbe found with the aid of an iterative scheme.We will consideronlya finite andsmallnumberof termsof theseriesin Eq.(43)andin fact take the frequencies.closeto real-valued.

10

C. An iterative scheme

To arriveat aniterativeschemefor thesolutionof Eq.(35),considerthe Õ -timesreflectedcon-stituentwave. Wesplit theinterval y{SH� *#àEá â ã� | into Ñ thin slabswith thicknessÒ * � . Set(x � Í �d ��U8�cð Ò * � �D� (ñ � Í �dgfp" ��U8�cð:��Z (ñ � Í �dgf � ��U8�cð:�0� (44)ÕV���ò� � ��Ö�Ö�Ö and ð°�qS@������Ö�Ö�Ö>� Ñ , where(cf. Eq.(43))(ñ � Í �dgfp" �BU©�cðH�ó� � (ß dgfp" (x � Í !#" �" �e��U8�cð Ò * � �0� (45)(ñ � Í �dgf � �BU©�cðH�ó� � (ß dgf � (x � Í !#" �� �e��U8�cð Ò * � �0U (46)

Uponcomparisonwith Eq.(42)wefind that(ñ � Í �"Lfp" ��U8�cð:�D� 2 'gô ¡ ¨� 7:9 ¯± ² ' ô ¡>¨³� ¢ 7 �1;>=@? y�A i . (� ���#� ��U8�l¤ � �HG:¤ � | µ ¶· (õ � Í �"Lfp" �BU©�g¤@�HG:¤5� (47)

with(õ � Í �"Lfp" �BU©�g¤@���Û� (� (x � Í !#" �" ����U8�l¤¥� . Similarly,(ñ � Í �"Lf � ��U8�cð:�D� 2 'gô ¡>¨� 7:9 ¯± ² 'gô ¡>¨³� ¢ 7 �1;c=H? y�A i . (� ���#� �BU©�l¤ � �HG#¤ � |/µ ¶· (õ � Í �"Lf � ��U8�l¤¥�HG#¤Ó� (48)(ñ � Í �� f h ��U8�cð:����A¾2 ¡ ç´è­é ê¨� 7 ' ô ¡>¨°¯± ² 'gô ¡ ¨³� ¢ 7 � ;>=@? y´A i . (� � ! � �BU©�g¤ � �HG#¤ � | µ ¶· (õ � Í �� f h �BU©�l¤¥�HG#¤Ó� (49)

with(õ � Í �"Lf � ��U8�l¤¥�1�Û� (� (x � Í !#" �� ���BU©�g¤@� , (õ � Í �� fp" �BU©�l¤¥����� (� (x � Í !#" �" �e��U8�l¤¥� , (õ � Í �� f � ��U8�l¤¥�1�Û� (� (x � Í !#" �� ���BU©�l¤¥� .

In aconfigurationwith constantdensity, wehave(� ��A (� andhence(õ � Í �� fp" ��A (õ � Í �"Lfp" and(õ � Í �"Lf � �tA (õ � Í �� f � U

To constructtheiterationscheme,we carryout thefollowing steps.Let(Â

denotethethin slabpropagator (cf. Eq.(31))( ��U8�cð:�D� ¯± ² 'gô ¡>¨³� ¢ 7 � ' !#" � ô ¡>¨ ;c=@? y´A i . (� ���#� �BU©�l¤ � �HG#¤ � | µ ¶· U (50)

Then,usingthesemi-groupproperty,(ñ � Í �"Lfp" �BU©�cðH�\� ¯± ² 'gô ¡ ¨³� ¢ 7 � ' !#" � ô ¡ ¨ ;>=@? y�A i . (� ���#� ��U8�l¤ � �HG#¤ � | µ ¶·2 � ' !#" � ô ¡>¨� 7:9 ¯± ² � ' !#" � ô ¡>¨³� ¢ 7 � ;>=@? y�A i . (� ���#� ��U8�l¤ � �HG#¤ � |/µ ¶· (õ � Í �"Lfp" ��U8�l¤¥�HG#¤Z 2 'gô ¡ ¨� 7 � ' !#" � ô ¡>¨s¯± ²'gô ¡>¨³� ¢ 7 �1;c=@? y´A i . (� ���#� �BU©�l¤ � �HG#¤ � |/µ ¶· (õ � Í �"Lfp" ��U8�l¤¥�HG#¤V� (51)

11

whichcanbewrittenas(ñ � Í �"Lfp" ��U8�cð:�D� ( �BU©�cðH� (ñ � Í �"Lfp" �BU©�lðVAT����Z (ö � Í �"Lfp" ��U8�cð:�a� (52)

where (ö � Í �"Lfp" ��U8�cð:���32 'gô ¡>¨� 7 � ' !#" � ô ¡>¨s¯± ²'gô ¡>¨³� ¢ 7 � ;>=@? y�A i . (� ���#� ��U8�l¤ � �HG#¤ � | µ ¶· (õ � Í �"Lfp" ��U8�l¤¥�HG#¤5U (53)

Recursionrelationssimilar to theonein Eq.(52)canbefoundfor theotherelementsof(ñ, viz.,(ñ � Í �"Lf h ��U8�cð:�÷� ( ��U8�cð:� (ñ � Í �"Lf h ��U8�cð[Aø�^��Z (ö � Í �"Lf h �BU©�cðH� for ð°�t��� � ��Ö�Ö�Ö>� Ñ �(ñ � Í �� f h ��U8�cð:�÷� ( ��U8�cð�Zr��� (ñ � Í �� f h �BU©�lð¸ZÅ�^��Z (ö � Í �� f h �BU©�cðH� for ð°� Ñ AT��� Ñ A � ��Ö�Ö�Öe�gS U(54)

Here (ö � Í �"Lf h �BU©�cðH��� 2 'gô ¡>¨� 7 � ' !#" � ô ¡>¨s¯± ²'gô ¡>¨³� ¢ 7 ��;>=@? y´A i . (� ���#� ��U8�l¤ � �HG#¤ � | µ ¶· (õ � Í �"Lf h �BU©�l¤¥�HG#¤V� (55)(ö � Í �� f h �BU©�cðH���32 'gô ¡>¨� 7 � ' � " � ô ¡>¨s¯± ²�³� ¢ 7 'gô ¡>¨�;>=@? y i . (� � ! � �BU©�g¤ � �HG#¤ � | µ ¶· (õ � Í �� f h �BU©�l¤¥�HG#¤ÓU (56)

Theinitial valuesfor therecursivescheme(54)aregivenby(ñ � Í �"Lf h �E* " �lSä�ù� SÓ� (57)(ñ � Í �� f h �E* " � Ñ �÷� SÓ� (58)

again,for Õ[����� � ��Ö�Ö�Ö .D. Numerical issues

The implementationof the iterative schemeis asfollows. It is initiatedby the calculationoftheincidentfield,

(x � 9 �" , accordingto(x � 9 �" �BU©�cð Ò * � ��� ( �BU©�lð:� (x � 9 �" ��U8�^�­ð�AT��� Ò * � � for ðÓ����� � ��Ö�Ö�Öe� Ñ � (59)

seeFigure2, with initial condition(x � 9 �" �BU©�gSä��� (x " ��U8�lSä�a� (60)

12

accordingto Eq.(36). During the forward propagation,at eachof the discretelevels,(õ � " �h fp"

arecomputedandstored;(õ � " �h f � aresetto zero. The procedureis continuedby the backward

propagationdefinedby the secondrecursionin Eq.(54). At eachof the discretelevels,(x � " ��

is computed(Eq.(44))andusedto calculate(õ � � �h f � ; the latterquantityis storedasbefore. The

schemecontinuesto switchfrom backwardto forwardpropagationbasedonthefirst recursionin Eq.(54),andsoon.

To evaluatetheelementsof(ö � Í � , weapplythetrapezoidalrule. Then(ö � Í �"Lf h ��U8�cð:�D× "� Ò * �ûú (õ � Í �"Lf h �BU©�cð Ò * � ��Z ( �BU©�lð:� (õ � Í �"Lf h ��U8�^�`ðVAv�^� Ò * � �Lü%� (61)(ö � Í �� f h ��U8�cð:�D×zA "� Ò * �%ú (õ � Í �� f h ��U8�cð Ò * � ��Z ( ��U8�cð�Zr��� (õ � Í �� f h ��U8�^�­ð�ZÅ�^� Ò * � �LüÏU (62)

which formulasareaccurateýÝ�/� Ò * � ���>� .Thewholenumericalschemeis elucidatedin a flow chart,Figure3. In the remainderof

thispaper, wewill deriveapproximate,sparsematrix representationsfor(�,(Â,(� !#" and

(�.

V. SPARSE MATRIX REPRESENTATION OF THE PROPAGATOR

Considerthehomogeneousone-waywaveequation(cf. Eq.(20))W � (x � 9 �" ZÙÇ8. (� (x � 9 �" ��SÓ�þ* �uå �`SH� * àEá/â ã� | � (63)

which is satisfiedby the leading-orderterm of the Bremmerseries. In this section,we willderive a sparsematrix representationfor its propagator,

( ��U8�cð:� . In the following approxima-tions,wewill maintainthehyperbolicityof thetime-domain-equivalentequations,but wewilldeformthepost-criticalregime.

A. Approximating the vertical slowness leading to a partial differential equation

Theprincipalpartof theleft verticalwave-slownesssymbolis givenby (cf. Eq.(34))(À ÿgÿ �+* " � * � �g¼ " �D�Ûy{Ú !¥� �+* " � * � ��AÙ¼ � " | "`��� U (64)

In the upperleft cornerof Figure4, Ú (À ÿgÿ is shown. The interval Ú>¼ "¾å yæSH��� | correspondsto wave propagationunderan angleof 0 to 90 degreesand is called the pre-criticalregion.The interval Ú>¼ "Vå �����cÔT� correspondsto the evanescentwave constituentsandis calledthepost-criticalregion.

For our numericalscheme,we considerThiele’s third-ordercontinuedfraction’s approxi-mationof theleft symbol(seeSerafiniandDeHoop31 andDeHoopandDeHoop32)(À dBd�dÿgÿ � �Ú Z � �PA�� � Ú � ¼ � "�� !#" � A�� " Ú>¼ � " Z�� � Ú � ¼� "�� U (65)

13

Thewavefront associatedwith thisapproximationis freefrom artificial singularitiesin aconeof propagationanglesaboutvertical,unlike thesecond-orderapproximationthatis commonlyused.Theoperatorassociatedwith this left symbolis notsymmetricnorself-adjoint.To createa symmetricThiele approximation,we extract the local wave speedÚ from the squarerootexpression;theverticalslownessoperatorthenbecomes(�� Ú !#"`��� ���YZ �D� "`��� Ú !#"`��� � (66)

where � o A-ÚYm �" �`ÚPÖ��aU (67)

This approximationis valid for smallvaluesof thecommutatory{��lm "L| , which will bethecasefor mediavaryingsmoothlyon thescaleof thedominantwavelength.

The third-orderThiele approximationis now appliedto the symbolof the operator ���uZ�D� "`��� . This leadsto a symmetricoperator(� dBd�d , theprincipalsymbolof which still equalsthe

expressionin Eq.(65),(� dBd�d �qÚ !#"`����� �YZqy��YZ�� � � | !#" y�� " ��Z�� � � � |�� Ú !#"`��� U (68)

Theinternalstructureof(� d�dBd is suchthatits symbolcapturessomeof thecontributionsbeyond

theprincipalpart.Fromnow onwewill freelyomit thesuperscriptdBd�d .Accordingto Thiele’s formula, we have � " � �^Ë � , � � � �^Ë�� , � � � ��Ë � . The param-

eters, � " , � � and � � , however, can be adjustedby minimizing the differencebetweentheexactspectral-domainverticalslownessandits continued-fractionapproximation,definedbyEq.(65),with respectto the � � -norm,over all the propagatingmodes(i.e., the real slownesssurface).In this minimization,we mustbeawareof thefactthatthepole(at �¼� " �B�P�ÄÚ�!¥��� !#"� )introducedby theThiele-typeapproximationwill lie outsidethepre-criticalregion of the ¼ " -plane.Theoptimizationprocedurecanbeviewedasa replacementof Thiele’scontinuedfrac-tion by Newton interpolation.Usinganoptimizationroutinebasedon thesimplex method33,thevalues� " �qSHU�� ��� , � � �qS@U�� ��� , � � ��SHU�� � � areobtained.

In casethepropagationanglesappearingin thewave field arerestricted,theoptimizationshouldbecarriedout accordingly. This typeof optimizationwasalsoconsideredby LeeandSuh34 andfollowedby Bunks35.

Figure 4 shows the exact and approximatedvertical slownesseswith � " , � � and � � setto Thiele’s valuesandset to optimal values,respectively. In the secondrow of figuresthedifferenceswith theexactslownessaredrawn. In thebottomrow therealpartof theverticalslownessesasa functionof complex-valued Úe¼ " is shown. In thesepictureswe recognizethesingularitiesin theverticalslownesses:a branchcut from Úe¼ " ��� to Z�Ô in theexactvertical

14

slownessandpolesat � � and ��U8��S for thetwo approximatedverticalslownesses,respectively.Sinceno singularitiesarepresenton the imaginaryaxis, the approximationis accuratefor alargerangeof imaginaryvaluesof Úe¼ " .

Fundamentaldifferencesbetweentheapproximateandexactverticalslownessesoccurnearthe singularitiesin the horizontalslownessplane. Sinceit is impossibleto approximateabranchcutwith afinite setof polesfrom arationalapproximation,wehaveto restrictourselvesto approximatingthepre-criticalwave propagationascompleteaspossible.Theinfluencesoftheartificial post-criticalwavepropagationmustthenbesuppressed.A dissipationtrick canbedesignedfor thatpurpose.

B. The dissipation trick

To suppressartifacts,andaliasing(whichmayarisefrom thediscretizationto becarriedout inthehorizontaldirection),from thelargehorizontalwavenumbercomponents,wemayreplacetherealfrequency . by acomplex one,. � �Å.%���ÃA i �P�Ã�just in theexpressionfor i . (� (andlater in

(�and

(� � , seeequations(80)). In a homogeneousmedium,it impliesperFouriercomponentor planewave, w�+* � �l¼ " �/.0� ;c=H? �BA i .Y¼ " * " � , anam-plification factorof theform! w�+* � Z Ò * � �l¼ " �/.0� Ë" w�+* � �g¼ " ��.0� ! � ;c=@? ½ A . Ò * �� Ú$#V�¼ " ��.F� i �P� ÁTU (69)

Theamplificationfactoris attenuative in nature.The # -factoris givenby#[�`¼ " �/.F� i �P�\� �Im M � Ú (À dBd�d �`¼ "cn .u�B�ÃA i �P� �lO � (70)

and is plotted in Figure5. Here, � is assumedto be very small ( �&% � ). This particularcomplexification resultsin a local bandlimitationfilter, suppressingthe artifactsassociatedwith thepost-criticallypropagatingmodes.

C. The comoving frame of reference

To reducediscretizationartifacts in the vertical derivative, the numericalcomputationsaredonein a comoving frameof reference,traveling in thedirectionof preference.Theaccuracyis improved,sincethis transformationenablesus to take into accountthe leadingphaseshiftexactly. Thechangeof frameyields(seeClaerbout2)(' �+* ' �/.0��� ;c=H? y i .$(��+* ' � | (x � 9 �" �E* ' �/.0�0� (71)

15

in which (��+* ' �D��2 ¡>¨� 7:9 Ú !#" �+* " �l¤¥�HG#¤V� (72)

is thelocal verticaltravel time. SubstitutingEq.(71)into Eq.(63),leadstoW � (' Z i . ú ;>=@? � i .$(<� (� ;>=@? ��A i .$(#��AÙÚ !#" ü (' �qSVU (73)

D. Discretization in the transverse direction

First,weintroducethediscretizationin thetransversedirection,of thepartialdifferentiationW#�" .(In three-dimensionalwave propagationthis wouldbethetwo-dimensionalLaplaceoperator.)Our startingpoint is a low-order, implicit, finite-differenceapproximation.The spectrumorsymbolof the resultingoperatoris a rational function in horizontalslowness. The mediumhasto besmoothon the scaleof thesizeof the operator. A uniform grid is employed in thehorizontaldirection.Thehorizontalsamplinginterval is denotedas Ò * " .

The discretizationof the one-dimensionalLaplaceoperatoris formulatedin termsof re-cursive filters basedon nearest-neighboursinteraction(seeMitchell and Griffiths 36). Ourrecursivefilter actingona function

('is definedthrough�B�0Z�) �lb �" �+*­W �" ('-, �z� Ò * " � !¥� ) "/b �" (' � (74)

where b �" (' � (' �+* " Z Ò * " ��A � (' �E* " ��Z (' �+* " A Ò * " �0U (75)

In Eq.(74), � denotestheidentityand *­W �" ('-, �.*­W �" , (' representstheapproximateLaplaceoper-atorhaving actedon

('. Usinga Taylorseriesexpansionof

('at * " ª Ò * " about* " , thevalues) " �t� , ) � ���^Ë@� � arefound(seeMitchell andGriffiths36). Then *­W �" , �qW �" Z%ýN� � Ò * " � � � . For

a finite-bandwidthsolutiongeneratedwith a givensamplinginterval, however, we leave thisorderestimate,anduse) � asaparameterto improvetheoverallaccuracy. Requiringthatin thelimit Ò * "0/ S thespectrumof *`W<�" , tendsto thespectrumof theLaplaceoperatorupthelowestorder, leadsnecessarilyto ) " �ë� . Theparameter) � is thendeterminedby minimizing non-linearly the differencebetweenthe approximateandthe exact Laplaceoperatorspectrawithrespectto the � � -normover theNyquistinterval. Thusamoreaccurateapproximatespectrumis obtainedover thespatialbandwidthasa whole. Usinganoptimizationroutinebasedon thesimplex method37, thevalue ) � �ÛSHU8�1��S is obtained.It is noted,however, that if it is knownbeforehandthat theactualspatialbandwidthof thewave field to beextrapolatedis limited bya horizontalwave numberlessthe Nyquist wave number ð "Lf 24365 � Ì Ë Ò * " , the optimizationshouldbecarriedout over thissub-bandonly.

16

Figure6 illustratesthespectraof theexactandtheapproximateLaplaceoperatorswith op-timal valuesandTaylorvaluesfor ) " and ) � , respectively. TheapproximatedLaplaceoperatorhaspolesat ¼ " �q¼87 " �Û�E. Ò * " � !#" � ª Ø�9�:+:<;>= ú �PA "� ) !#"� ü Z ��? Ì � � (76)

(? ��SH�cªV���cª � ��Ö�Ö�Ö ). In practice) � Q3� , andthesepolesaresituatedin thecomplex ¼ " -plane

atReM�¼ " O � Ì Ë:� . Ò * " � (determinedby theNyquisttheorem),faraway from therealaxis.Thepolesof theapproximateLaplaceoperatorcarryoverin theapproximationof theverti-

cal slownessoperator. Theverticalslownesssymbolbecomesperiodicin accordancewith theNyquisttheorem,andtheLaplacepolesareslightly shifted:¼07 " X �E. Ò * " � !#"@� ª Ø�9�:<:+;>= ú �PA "� �A) !#"� Z�. Ò * " � � �Lü1Z �"? Ì � � (77)

seeFigure7. For example,taking ) � �3S@U©�1�òS , � � �qS@U�� � � , anda samplingrateof 8 pointsperwavelength,thepolesarelocatedat¼ � " �3ªV��U8��S � �FZ�� ? and ¼87 " �3ª5� � A � U � SH��S i ��Z�� ? U (78)

E. The system of ordinary differential equations

Substituting*­W#�" , , i.e.,Eq.(74)into Eq.(67)andtheresultinto theone-waywaveequation(73)with verticalslowness(68),weobtainasystemof ordinarydifferentialequationsof theform(� W � (' Z Ç8. (� � (B !#" (' �qSVU (79)

Theoperatorsinvolvedare(� � (� �E* � �/. n * " � b �" � Ò * " � , (� � � (� � �E* � �/. n * " � b �" � Ò * " � and(B � (B �+* � �/. n * " � b �" � Ò * " � ; they aresparse,viz., 3, 5 and3 bandsrespectively, andaregivenby(� Ö � ���\Z ) �cb �" ���`Ú�!#"`��� ;c=@? ��A i .$(<�DÖ��÷Z � � �DCgÚ�!¥� b �" �`Úc�B��� ;>=@? �BA i .$(<��Öä� �(� �HÖz� � " �DClÚ�!¥�î���YZ ) �cb �" ��� b �" �)Öò� �÷Z � � � �C Ú�! � b �" �`Úc� b �" ��Öä�/� � (80)

while (B Ö@� ;c=H? �+Ç8.$(#�)Ú !#"`��� ���YZ ) �cb �" �\Ö<� (81)

with horizontalsamplingrateparameter�DCÃ���E. Ò * " ËòÚe� !¥� U (82)

Thequantity��Ì �DC equalsthenumberof grid pointsperwavelengthin thelocalmedium.

17

F. Integration of the vertical derivative

The solution of equation(79) can formally be written as a product integral. This productintegral will have to beevaluated,recursively, for every propagationstepwith size Ò * � . Theproductintegral is givenby (seealsoEq.(27))(' ��U8� * � Z Ò * � �D� ¡>¨ � ô ¡>¨³� 7 ¡>¨v;c=@? ú A i .%� (� !#" (� � (B !#" ����U8�l¤¥�HG#¤òü (' ��U8� * � �aU (83)

It is appoximatedby(' ��U8� * � Z Ò * � ��× ;>=@? ú A i . Ò * � � (� !#" (� � (B !#" ����U8� * � Z "� Ò * � �Lü (' ��U8� * � �0U (84)

To speedupthecomputations,theexponentis approximatedby thesolutionof aunitarymatrixequation.The �+��g� -Pade approximantscanbeusedfor this purpose.They aregivenby (seeNumericalRecipes38)

;>=@? �FE»��× ¯GGGGGG± GGGGGG²�YZ��DHIE�PAJ�DHIE � lowestorder:(1,1) �� �YZ�� "L" E�ÃA�� "L" E � � �\ZK� " 9 E�FAL� " 9 E � � higherorder:(2,2) � (85)

where E o Ò * � W �>MMMM * � Z "� Ò * � � (86)

is thederivative thatappearsin Eq.(79).Thehighertheorderof theapproximation,thelargerthe stepsizeÒ * � canbe. For stability considerationsof the associatedhigher-orderimplicitfinite-differenceschemes,seeWidlund 39.

According to Pade’s formula, we have �DH�� "� . Insteadof taking this exact value, welet �DH be a free parameterfor an optimizationprocedure.Minimizing the differencefor thepre-criticalwaveconstituents,gives �DHP�ÄSHU�N �>� for 2 pointsperwavelength,�DHP�ÄSHU�� � S for 5pointsperwavelength,�DH0��S@UO�@��S for 10pointsperwavelength.The(1,1)-Padeapproximationof theexponentyieldsthewell-known Crank-Nicholsonscheme(seeRichtmeyer andMorton40), in ournotationgivenby� (� (B ���E* � Z "� Ò * � ��� b"P (' � �+* � � �D���A i �DH:. Ò * � � (� � �E* � Z "� Ò * � ��� (' � �+* � ��Z (' � �+* � Z Ò * � � �0� (87)

with � b P (' � ���E* � ��� (' � �+* � Z Ò * � ��A (' � �E* � �a� (88)

18

where (' � �`¤¥��� (B !#" �+* � Z "� Ò * � � (' �`¤¥� for ¤ å �E* � � * � Z Ò * �g| U (89)

TheschemeEq.(87)shouldbeinterpretedasacentereddifferenceapproximationof theverticalderivative,andhenceis accurateup to ýÝ�/� Ò * � ���l� .G. Propagator matrix

To arrive at an explicit matrix representationfor the one-way propagator, we introducethearray, (' Ê �E* � �/.0��� (' �+* " Ê � * � �/.0�0� (90)

where * " Ê �RQ Ò * " , and Q»� ����Ö�Ö�Ö>� k ¡ £ labelsthe samplesin the tranversedirection. In thisnotation,with the aid of Eqs.(87)and(89), the approximateone-way wave propagationcanfinally bewritten in thematrix form(' Ê �+* � Z Ò * � �/.0��� (B ÊpÍ � (S !#" � Í ' (S � 'UT � (B !#" � T , MMM * � Z "� Ò * � (' ,»�+* � �/.0�\� (91)

wherethematrices(S,(S � and

(Barefunctionsof * � and. , andareconstructedfromtheoperators(�

,(� � and

(B: (S � (� (B Z i �DH:. Ò * � (� � � (92)(S � � (� (B A i �DH:. Ò * � (� � U (93)

Thematrices(S

and(S � contain5 non-vanishingbands,while

(Bhas3 non-vanishingbandsonly.

Observe that (S � �E.0��� (S �BAP.0�0� (94)

thus(S �� .0��� (S �E.0� if . is real-valued( �ø��S ).

Thenumericalschemefollowing Eq.(91),involvestwo matrix-vectormultiplicationsandtwicesolvinga matrixequationperpropagationstep:(B T , (' �, �+* � ��.0� � (' T �E* � �/.0�0� (95)(' � �' �+* � ��.0� � (S � 'UT (' � T �+* � �/.0�0� (96)(S Í ' (' � ' �E* � Z Ò * � ��.0� � (' � �Í �E* � �/.0�Y� (97)(' Ê �E* � Z Ò * � ��.0� � (B Ê{Í (' �Í �+* � Z Ò * � �/.0�0U (98)

19

Thelaststepadjustsfor thechangeto thecomoving frameof reference.Introducethediagonalmatrix V , representingthechangeof frame,with diagonalelements(cf. Eq.(71))V\,),»�+* � �D� ;c=H? y i .$( �+* " ,-� * � � | U (99)

Thenthepropagatormatrix (cf. Eqs.(50)and(91)) is givenby* ( �`ð�Zr�^� , Ê , �WVu�+* � Z Ò * � � Ê�Ê ¢ (B Ê ¢ Í � (S !#" � Í ' (S � 'UT � (B !#" � T , MMM * � Z "� Ò * � U (100)

AppendixA containsa discussionon absorbingboundaryconditionsto which the matrix(S

is subjected.Several techniquesexist for carryingout thematrix inversions41. Theinversionof the tridiagonalmatrix

(Bin Eq.(95) is carriedout by forward/backward substitution(see

NumericalRecipes38).

VI. THE VERTICAL PHASE AND GROUP SLOWNESSES

Let(À4XIY and

(À-Z denotethe vertical phaseandvertical groupslownessassociatedwith a real-valued plane wave, respectively. We will analysethe accuracy of our numericalschemewith the aid of thesequantities. The analysisof the accuracy of finite-differencepropaga-tion schemeswith phaseandgroupvelocitiescanbefoundin Trefethen13. Beaumontetal. 14

andHolberg 15� 16 assessedandimprovedtheaccuracy of suchschemes.

A. General considerations

Substituteaplanewaveconstituentinto Eqs.(71)and(98):(x � 9 �" �E* ' �/.0��� ;>=@? ��A i . (À X[Y Ò * � � ;>=@? y´A i .Y¼ " Ò * "B| � (101)

or, in themoving frameof reference,(' �E* ' �/.0��� ;c=@? � i .$(<� ;>=@? ��A i . (À X[Y Ò * � � ;>=@? y�A i .Y¼ " Ò * "B| U (102)

We thenobtainthe amplificationfactorof the finite-differencescheme(seeRichtmeyer andMorton 40), \ !#" \ � � ;c=@? y´A i . (À X[Y Ò * �g| ;>=@? y i .YÚ !#" Ò * �g| � (103)

in which(À X[Y , \ and

\ � aretheleft symbolsof thenumericalrepresentationfor(�, and

Sand

S � ,respectively. For ahomogeneousmedium(constantÚ ), wefind\ � S Ê�Ê Z � S ÊäÊ � " :+;>= �E.Y¼ " Ò * " ��Z � S Ê�Ê � � :<;�= � � .Y¼ " Ò * " �a� (104)

20

independentof Q , togetherwith a similar expressionfor

\ � , replacingS

byS � . For real-valued. , we have

\ �+�E.0�u� \ ��AÃ.0� , thus! \ !#" \ � ! ��� in a homogeneousmedium.Thesymbolof the

finite differenceschemeis theverticalphaseslowness(À X[Y givenby(À X[Y �¼ " �/.0���rÚ !#" A Æ ;>] �\ !#" \ � �

i . Ò * � � (105)

which reduces,if . is real-valued,to(À X[Y �¼ " ��.0����Ú !#" Z �. Ò * � Ø�96] �\ �aU (106)

Fromtheverticalphaseslownesstheverticalgroupslowness(À^Z canbederived,viz.,(À Z �rW�_�� . (À XIY �\� (À XIY Z�.u�W�_ (À X[Y �aU (107)

Theterm Ú1�`W�_ (À`X[Y � canbeidentifiedwith thedelayof theenvelopeperwavelength.Togetherwith Eq.(103),this leadsto(À Z �`¼ " �/.0�1�qÚ !#" A i � Ò * � � !#" M \ !#" W>_ \ Ar� \ � � !#" W>_ \ � O�� (108)

andreduces,if . is real-valued,to(À Z �`¼ " �/.0�1��Ú !#" A � � Ò * � � !#" ILKÝM \ !#" W>_ \ O[U (109)

It is observed that(À X[Y and

(À Z arefunctionsof the horizontalandvertical samplingratesperwavelengths, Ú�ËH�E. Ò * " � and Ú�Ë:� . Ò * � � . Note that

(À-Z A (À is indicative for the numericalanisotropy, whereas

(À X[Y A (À Z is indicative for thenumericaldissipation.

B. Optimization

Theverticalphaseandgroupslownessesarefunctionsof ourparametersMa� " �U� � �U� � �b) � ���DH��b�uOarisingfrom the variousapproximationsmadeto arrive at a sparsematrix representationforthepropagator. Sofar, we have foundtwo distinctparametersets:onearisingfrom consistentrationalexpansions,andonearising from step-wiseoptimization. For a samplingrateof 5pointsperwavelengthin boththehorizontalandverticaldirections( Ò * " � Ò * � ��SHU �>c ) thephaseandgroupslownessesfor thefirst parametersetareshown in Figure8. Theexactverticalslownessis shown in thesamegraph.Significantdeviationsareobservedfor largepropagationangles.Theparametersetcorrespondingto thestep-wiseoptimizationleadsto a betterresult(Figure9). In thestep-wiseoptimization,however, it is hardto control themovementof thepolesin the complex horizontalslownessplane,introducedby the approximations.Onecan

21

control this movementin an optimizationschemeof the phaseandgroupslownessesfor allparameterstogether;thenoneexpectsamoreaccurateresultaswell.

Theoptimizationof thephaseandgroupslownessesresultsin aminimizationof numericalanisotropy andnumericaldissipationfor thepre-criticalwaveconstituents.In theoptimizationprocedurespecialattentionmustbepaidto thecontraintthatthepolesof

\stayoutsidethepre-

critical region in the ¼ " -plane.Theoveralloptimizationis carriedout for anobjectivefunctiongivenby thesumof thesquareddifferencesbetween

(À^Z and(À , and

(À4XIY and(À^Z . Theresultsare

illustratedin Figure10, asbeforefor 5 pointsperwavelength.Fromnumericaltests,we findthat the accuracy remainsmoreor lessthe samewhenwe vary the transversesamplingratefrom 5 down to 2.5 pointsperwavelength,keepingthe verticalsamplingratethesame.Theprocedureis evenlesssensitive to theverticalsamplingrate.

Sofar, theoptimizationprocedurewasfocussedon therealslownesssurface.However, aswehaveseenin thepreceedingsections,theapproximationsleadto artifactsin thepost-criticalregime. In fact, the post-criticalconstituentshave beenmappedon propagatingconstituents(slow waves).Wehavedesignedadissipationtrick to attenuatetheseconstituents.Besidesthistrick, someof theartificial constituentsareforcedto leave thecomputationaldomainthroughthe transparentboundaryconditions. The dissipationtrick inducesa complexification of theparametersusedin the optimization,andhenceit seemsto be naturalto complexify the op-timization procedureaccordingly(seealso Collins 47). Pursuingthis complexification, theoptimizationprocedureis extendedwith the constraintthat the post-criticalpower is attenu-atedstronglywhile theattenuation/amplificationin thepre-criticalregionis keptminimal(lessthan �>d ). Theresultingparametersetis shown in thefourth columnof Table1. Thefirst col-umnof this tablerepresentstheexpansionvalues,andthesecondcolumnrepresentsthevaluesobtainedby step-wiseoptimization.Theaccuracy of thecomplex parametersetis illustratedin Figure11.

We note,however, that thecomplexification of parametersis not alwaysdesired.For ex-ample, in the applicationof our numericalschemeto long-rangepropagationin low-lossywaveguides,theaccumulativepower lossis of key importance.In suchconfigurations,we set�T�rS andkeeptheparametersreal.Theoptimizationleadsto thethird columnof Table1.

Thepolesarisingin verticalphaseslowness,for thedifferentparametersets,aregiveninTable2.

C. A numerical example

We illustratethe final, optimized,one-way propagationby computingthe wave field excitedby a line sourcein a homogeneousmedium. Thewave speedof themediumis chosento beÚ5���1e�S�S m/s. Thenumericalgrid consistsof 99 pointsalongthe * " -directionand60 pointsalong the * � -direction. The discretizationstepis � � m in both directions. Given a source-

22

signaturewith trapezoidalspectrumwith cornerfrequencies��S , � S , � � , and� S Hz, we en-

countersamplingratesof 2.97 to 11.9 pointsper wavelength. The line sourceis locatedat* " �f��SòS m, * � ��S m, andwe will show snapshotsof thepressurefield at SHU�� s. Thesourceisa verticalforce.For thisconfigurationwehaveanalyticexpressions.

In Figure12 the snapshotis shown usingthe first parameterset,while in Figure13 thesnapshotfor thefourth parametersetis shown. Theleft partsof thefigures( SNQ�* " Qg�òS�S m)representthe numericalsolutions,while the right parts( ��S�SøQ * " Q � � S�S m) representtheexactsolution.Theerrorsoccurringin Figure13areillustratedin Figure14,theabsoluteonesfor SÓQ]* " Qh�òS�S m andtherelativeonesfor ��S�SÓQ]* " Qq� � S�S m. Fromthisfigurewefind thattherelative errorat thenumericalwave front is lessthan2.5%up to anglesof approximately65degrees.Repeatingthisaccuracy analysisfor theotherparametersets,weconcludethattheapproximationsareacceptableup to 20 degrees,40 degrees,and80 degreesfor the standardparabolicapproximation,parameterset1, andparameterset3, respectively. Thethird param-etersetis moreaccuratefor higheranglesthanthefourthparameterset;however, post-criticalartifactsremainapparentin thenumericalresultsobtainedwith this real-valuedparameterset.

VII. SPARSE MATRIX REPRESENTATIONS OF THE(DE)COMPOSITION OPERATORS

Thecomputationof thegeneralizedBremmercouplingseriesstartswith thedecompositionofthe initial field into up- anddowngoingconstituentsin accordancewith Eq.(19). Uponcom-pletingthecalculationof a sufficiently largenumberof termsof theBremmercouplingseries,theconstituentsarerecomposedinto observableswith theaid of Eq.(18).In theseprocedures,theoperator

(�andits inversemustbecomputed.Here,usingtheresultsof SectionV, we will

derivesparsematrix representationsfor theseoperators.

A. The composition operator

The compositionoperatorcontainsthe vertical slownessoperator. To find a sparsematrixrepresentationfor this operator, we usethe sameapproximationas in Eq.(68) for one-waywavepropagation.Wewrite, however, thefractionin aslightly differentway,(� dBdBd ��Ú !#"`���i� �YZ�� � ��Zqy��YZ�� � � | !#" �Djb� � Ú !#"`��� � (110)

with � � � � �� � � �Dja�k� " A � �� � U (111)

23

We set (� d�dBd � (� dBdBd" Z (� dBdBd� � (112)

with (� d�dBd" ��Ú !#"`��� M �YZ�� � �0O�Ú !#"`��� � (� dBdBd� �qÚ !#"`��� M@y��\Z�� � � | !#" �Dj[�0O1Ú !#"`��� U (113)

SubstitutingdiscretizationEq.(74)into Eq.(67)andtheresultinto theseoperators,yields���YZ ) �cb �" �ml Ú !#"`��� * (� d�dBd" ('^,6n � ú �B�0Z�) �cb �" ��Ú !¥� ZK� � �DClÚ !¥� b �" üol Ú "`��� ('-n �ú ���YZ�) �gb �" ��Ú !¥� Z�� � �DCgÚ !¥� b �" üol�Ú �B��� * (� d�dBd� ('^,6n � �DjU�DCcÚ !¥� b �" lgÚ "`��� ('pn U (114)

The two equationscontaintridiagonalmatricesonly. Hence,they canbesolvedrapidly withtheforward/backwardsubstitutionprocedure(seeNumericalRecipes38).

B. The decomposition operator

Thedecompositionoperatoressentiallycontainstheinverseof theverticalslownessoperator.Usingapproximation(68)asbefore,wefind thatÚ "`��� (� dBd�d Ú "`��� �q� � y´�YZK� � � | !#" yr� !#"� ZK� !#"� �s� " Z�� � �t�ûZ�� � | U (115)

Factoringtheoperatorbetweenbrackets,yields � !#"� Zu� !#"� �s� " Zv� � �6��Zw� � �zyr�Dx Ay� | yr�-z Ay� | .Thenit is straightforwardto invert theverticalslownessoperator, viz.,� (� dBdBd � !#" ��Ú "`���i� � !#"� yr�-zDAK� | !#" yr�Dx\A�� | !#" y��YZ�� � � | � Ú "`��� U (116)

Observe that the symbolof this operatorhastwo poleson both sidesof the origin (the zerocrossingsof

(À dBdBd ).Usingthefactorization,wedecomposetheinverseverticalslownessinto threeoperators,� (� dBd�d � !#" � ({ !#"� ({ !#"� ({ " � (117)

with ({ " �zy��0ZK� � � | Ú "`��� � ({ � �Ûyr�DxYA�� | � ({ � �Ûyr�-z�AK� | � � Ú !#"`��� U (118)

Theinverseverticalslownessoperatoris thenappliedin two steps,({ � l ({ � � � (� dBdBd � !#" (' � n � ({ " (' �24

whichinvolvestwicesolvinganequation.SubstitutingEq.(67),with theaidof Eq.(74)thetwoequationsarediscretizedú �Dx^���YZ�) �cb �" ��A��DClÚ !¥� b �" Ú � ü|l Ú !#" * ({ � �/� (� dBd�d � !#" (' � , n � ú ���YZ ) �cb �" ��Z�� � �DClÚ !¥� b �" Ú � ü|l Ú !#"`��� (' n �� � ú �-z����YZ ) �cb �" ��A��DClÚ !¥� b �" Ú � üol Ú !¥�B��� *g� (� dBdBd � !#" ('^, n �Û�B�YZ ) �lb �" � l Ú !#" * ({ � � � (� dBdBd � !#" (' � , n U

(119)

Theassociatedtridiagonalmatricesareinvertedusingtheforward/backwardsubstitutionpro-cedure(seeNumericalRecipes38).

C. A numerical example

At the computationalboundarieswe apply, asbefore,the Robin boundaryconditions(Sec-tion VI). We alsoemploy an optimizationprocedurefor the (de)compositionoperators,con-sideringtheparametersubsetMa� " �U� � �U� � �b) � �b�uO . To avoid instabilities,we have to move thepolesarisingin the symbolsof the approximateoperatorsinto the complex horizontalslow-nessplane. Table3 containstheoutcomeof the optimizationfor a samplingrateof 5 pointsperwavelength.Figure15 shows thesymbolsof thecompositionoperatorfor differentvaluesof � .

To demonstratetheeffectof wavefield decomposition,wehavecomputedthepressureex-citedby aline explosionsource.Theconfigurationwasotherwisethesameasin Figures12-13.Figure16showsasnapshotat SHU�� s. Notethattheartificial headwaveenforcestheapproximatewave front to be vertical at the level of the source. The radiationpatternbecomesclosetoisotropicasit should.

VIII. SPARSE MATRIX REPRESENTATIONS OF THEREFLECTION/TRANSMISSION OPERATORS

Up to principalparts,we cansimplify the reflectionandtransmissionoperators(cf. Eq.(22))sincethen W � (� × "� (� !#" �/�`W � Ú !¥� � Ö��0U (120)

Substitutingthisapproximationinto Eq.(22),yields(� × "� (� !#" �/�`W � Ú !¥� � Ö�� n (121)

norationalapproximationis requiredto arriveatasparsematrixrepresentation,andits validityextendsinto thepost-criticalregime.To stayconceptuallycloseto theoriginalexpression(22)

25

we replaceW � Ú�!¥� by� Ú�!#"/W � Ú�!#" . This replacementis justifiedonly prior to applyingthefinite-

differenceapproximation.In thisapproximation,wehave

"� ÚÎW � Ú !#" F.D.A X �Ò * � Ú�!#"��+* � Z "� Ò * � ��AÙÚ�!#"e�+* � A "� Ò * � �Ú !#" �+* � Z "� Ò * � ��Z]Ú !#" �E* � A "� Ò * � � � (122)

which resemblestheexactreflectioncoefficientatnormalincidence.

On principal symbol level, the operatororderingcanbe further interchanged.Thus, thereflectionoperatorcanbewritten in thesymmetricform(� × "� Ú !#"`��� �W � Ú !#" � "`��� (� !#" �Ú !#"`��� �W � Ú !#" � "`��� Öä�aU (123)

This form guaranteesthat thereflection/transmisionoperatorsvanishin any region wherethemediumis * � -invariant.Upondiscretizingthisequation,usingEqs.(32)and(74),weget� ú ���\Z�) �cb �" ��Ú !¥� Z��DClÚ !¥� b �" üol �`W � Ú !#" � !#"`��� Ú "`��� * (� ('^, n �Û�B�\Z ) �cb �" � l Ú !#"`��� �`W � Ú !#" � "`��� (' n U (124)

It is notedthat thefreechangeof orderof operatorsin thefrequency domainis accurateonlynearthediagonalof theintegralkernelof theircomposition.

A. A numerical example

The discretizedreflectionoperatorcontainsonly two freeparameters:) � and � , through �DC .To optimizefor theseparameters,we would have to considerevery possiblechangein wavespeed.It turnsout,however, thattheoptimizationis ratherinsensitive to themediumchanges,so that we canrestrictourselvesto usinga singleparameterset, ) � � �^˥� � , � ��SHU8� . Thevalueof � is takenpositive,to movethepoleat90degreespropagationangleinto thecomplexhorizontalslownessplane.

Figure17 shows thesymbolof thediscretizedreflectionoperatorfor ) � � �^Ë@� � and �t�SHU S (theplane-wave reflectioncoefficient). In this figurearealsoshown thesymbolsfor �t�SHU S}� and �r�ÄSHU©� , aswell astheexactreflectioncoefficient. In Figure18 we show a snapshotof thepressurefield correspondingto thefirst two termsof theBremmerseries,excitedby avertical line forcein a two-layermedium.Thenumericalgrid, andthepositionof thesource,arethesameasin Figures12-13. Theuppermediumhasa wave speedÚR���~e�S�S m/sandthelowermediumhasa wave speedÚ-���>��SòS m/s. Theinterfaceis locatedat * � � � SòS m, andthesnapshottime is SHU�� s, asbefore.Thereflectionandtransmissionareaccuratelymodelled;theheadwave,however, is only mimicedby our numericalscheme(for details,seeDe HoopandDe Hoop48).

26

IX. NUMERICAL SIMULATIONS

In this section,we show examplesof wave field modelling in variousconfigurations,usingour Bremmercouplingseriesalgorithm. First, we considertwo examplesthatoriginatefromintegratedoptics.Here,wehavetranslatedthoseconfigurationsinto theiracousticcounterparts.Second,we consideranexamplefrom explorationseismics,viz., a saltstructureembeddedinsparsesystemof sedimentarylayers.

Ifk _ denotesthe numberof frequenciesrelevant for the wave field computations,

k ¡ £denotesthenumberof discretizationpointsof theconfigurationin thehorizontaldirectionasbefore,Ñ denotesthenumberof thin slabsof theconfigurationi.e. thenumberof discretiza-tion pointsin the preferreddirectionasbefore,thenthe computationaleffort per term in theBremmerseriesis of theorder

� k _|� k ¡ £ Ñ (thefactor�

accountsfor thepropagatorontheonehandandthereflectionoperatoron theotherhand;thefactor � reflectsthemaximumnumberof non-vanishingbandsin thematricesencounteredin our numericalscheme).For waveguid-ing problems,

k _ is eitheroneor small; for transientphenomenak _ will beproportionalto

thenumberof time samplesto bemodelled.Ouroptimizationapproachallowsusto employ acoarsecomputationalgrid; typicallywesamplethewavefield by

� UO�ÎA�� pointsperwavelength.

A. Waveguide

Our first exampleis, in fact,a benchmark test(seeNolting 49). It considersthepropagationof a singlemodethrougha waveguide(see,for example,Vassallo50), orientedunderdifferentangleswith respectto the numericalgrid. Thedetailsof this testconfigurationcanbe foundin Nolting 49. Thewave speedis approximately�1� � � m/s insidethe waveguideand �1e�S�S m/soutside.Thewaveguidehasa thicknessof approximately�~��S m. Thefrequency is �~e�S Hz. Wetake a numericalgrid with 779 pointsalong the * " -directionand1360pointsalongthe * � -direction. The discretizationstepis approximately��U�� m in both directions(about6.7 pointsperwavelength).Thetotal lengthof thewaveguideis

� U S � km. In this simulation,we usedthethird (real-valued)parameterset.In thiswaveguidethereexist only 11discretemodes.

Two of thewaveguidingmodes(thefundamentalandthetenthmode)arelaunchedinto thewaveguide. In thefirst experimentthewall of thewaveguideis parallelto the * � -axis. In thesecondexperiment,thewaveguideis tilted with respectto thecomputationalcoordinatesystemby 20 degrees.Thefield at theoutputlevel planeis comparedwith original field at the inputlevel plane. Theoretically, thesefields mustbe identical. Figure19 shows the fundamentalmodeat theendof theverticalwaveguide,while Figure20showsthefundamentalmodeat theendof thetilted waveguide.Figures21and22arethesame,but thenfor thetenth-ordermode.For theverticalwaveguide,thesymbolsaccountfor all the interactions;for the tilted waveg-uide,aninterplaybetweentheBremmercouplingandthesymbolinteractionstakesplace.Up

27

to thetenth-ordermodetheresultsaresatisfactory. It is conjecturedthatsomeof theradiatingmodescanbeproperlymodelledwith ourschemeaswell.

B. Block

In the secondexample,we investigatethe time-harmonicplanewave interactionin a homo-geneousmedium( Ú»� �1e�SòS m/s)with a squareblock (sized ��S�S m by ��S�S m) with significantlysmallerwave speed( ÚÓ� �1� �>� m/s). The frequency is �~e Hz. Thenumericalgrid contains99pointsalongthe * " -directionand60pointsalongthe * � -direction.Weplotonly 59pointsalongthe * " -direction.Thediscretizationstepis � m in bothdirections(14 pointsperwavelengthin-side the heterogeneity).Full, accurate,numericalresponseshave beenpresentedby Martinet al. 51. Our results,shown in Figure23, arevery similar to theirs. (The right half of thefigure is theBremmerseriessolution,the left half is the full solutionobtainedby an iterativeintegral-equationtechniquebaseduponminimizing anintegratedsquareerrorcriterion.) Fig-ure24correspondsto thesecondtermof theBremmercouplingseries,while Figure25showsthefield correspondingto theleadingtermof theBremmercouplingseries.Thehigher-ordertermsin theBremmerseriesbecomerapidly smaller. For thecalculationswe usedparameterset4; in thereflectionoperator, we set �v�qSHU S � .C. Geological dome

As a third example,we considerthesaltmodelof Figure26. This modelcontainsthreesed-imentarylayers. Thewave speedis constantin the upperandbottomlayers,while the wavespeedvariesin theleft partof themiddlelayer. An approximate,transientverticalline forceisappliedat * " ��� � S�S m, * � �rS m. Figures27-29show snapshotsat SHUON�� s. For thecalculationswe usedparameterset4. Thenumericalgrid contains199pointsin the * " -directionand156pointsin the * � -direction. Thediscretizationstepis � � m in bothdirections.We usedparam-eterset4. In Figure27 we have calculatedthefirst six termsof theBremmercouplingseries.Figures28and29show snapshotsof theleadingtermandthesecondtermof theseries.

Thisexampleillustratesthatour approximateschememodelsthetransmittedandreflectedbodywavesaccuratelyup to largescatteringangles.Also theline diffractionat themajorkinkof the seconddeepestinterfaceis capturedby the method. Someheadwave (post-critical)energy is mimicedby theartifactsof ourapproximations,but is inaccurate.

X. DISCUSSION OF THE RESULTS

We have developeda fastnumericalimplementationof the Bremmercouplingseriesin twodimensions,for scatteringanglesup to critical. Thekey ingredientwasa rationalapproxima-

28

tion of the vertical slownessin termsof the horizontalslowness. Our schemehasaddedtothevariousexisting one-way wave propagationandBPM schemesin thefollowing ways: weconsideredthe third-orderThieleapproximationratherthanthefirst- or second-orderone,toenhancethe accuracy andto remove artificial body wavesaroundthe vertical direction; weenhancedthe accuracy by optimizing the phaseandgroupslownesssurfacesfor any desiredsamplingrates;wehaveimprovedthetransformationto amoving frameof reference;weintro-duced(de)compositionoperatorsto incorporateany desiredsource-or receiver-typewith theappropriateradiationcharacteristics;wehave takencareof thebackscatteredfield with theaidof theBremmercouplingseries.Thebackscatteredfield includesphenomenalike turningraywaves. Thoughwe have shown resultsin two dimensionsonly, the algorithmcanbe readilyextendedto threedimensionsby employing a hexagonalgrid in thehorizontaldirections(forreference,seeMersereau52, andPetersonandMiddleton53).

We have illustratedthegeneralizedBremmerserieswith rationalapproximationwith var-ious examples.Theexampleshave beentaken from the differentfields of applicationof themethod.With a view to explorationgeophysics,the waveguideis associatedwith coalbeds.Figure16 representsthewave-equationmigration-operatorresponse,thetangentsto thefrontdefiningthedips. In migration,theBremmerseriesapproachis particularlyusefulif multiplearrivalsplaya rolecreatingaproperimage.

The efficiency of the methoddependson the application. The methodis designedin thefrequency domain,andhenceallows for a relatively coarsesamplingof the field; the lowerthe frequency, the coarserthe grid canbe. If only a small numberof frequencieshasto beevaluated,theBremmerseriesapproachhasa particularadvantageover full-wave modellingschemes.Thoughtherangeof scatteringanglestreatedaccuratelyis limited justbelow critical,themediumcanberelatively stronglyheterogeneous.If themediumis weaklyheterogeneous,however, competingapproachesexist to calculatethepropagator. We mentionthephase-shift(GazdagandSguazzero54), the split-stepFourier (Stoffa et al. 55), the McClellan transform(Hale56), andthephase-screen(Wu 57) representations.If theconfigurationconsistsof weaklyandstronglyheterogeneousregions,the differentapproachescanbe combinedinto a hybridscheme,matchingthemthroughtheverticalphaseslownesses.

In casecritical anglephenomenaare important,a more precisesymbol for the verticalslownessoperatorhasto beused.A candidateis theuniformexpansiondevelopedby FishmanandGautesen58. Thenthethin slabpropagatoris directlyevaluatedusingFouriertransforms.

To incorporatevariabledensityin theproceduredevelopedin thispaper, while keepingthenumericalanalysisthesame,we referto De Hoop1. Thekey is to port thewave-fielddecom-positionprocedurefrom the acoustic-pressurenormalizationanalogto the vertical-acoustic-power-flux normalizationanalog.Thestructureof theoperator

(�remainsthesame(Laplacian

plusmediumslownesssquared)but is subjectedto a transformationof the wave speedfunc-

29

tion. The complicationis hiddenin the (de)compositionoperators,which now dependon afractionalpower

(� !#"`� � . We cansubjectalsothis power to thesparsificationprocedurebasedonacontinued-fractionexpansionasdiscussedin themainbodyof thispaper.

ACKNOWLEDGMENTS

Theresearchreportedin this paperhasbeenfinanciallysupportedthrougha SpecialResearchFundof the Executive Boardof the Delft University of Technology, Delft, the Netherlandsandthroughresearchgrantsfrom theStichtingFundfor Science,TechnologyandResearch(acompanionorganizationto theSchlumbergerFoundationin theU.S.A.).Wealsoappreciatethefinancialsupportby thesponsorsof theConsortiumProjectattheCenterfor WavePhenomena.

30

APPENDIX A: TRANSVERSE TRANSPARENT BOUNDARIES

Ourdiscretizationcoexistswith applyingperiodicboundaryconditionsin thetransversedirec-tion. It is standardpracticeto make theboundariesof thecomputationaldomainabsorbing,tosimulateanunboundedconfigurationinstead.Severalapproacheshavebeendevelopedfor thisadjustment;we mentionthework of ClaytonandEngquist42 basedon theparabolicequationin the transversedirections.Competingapproachesarethe TransparentBoundaryConditionby Hadley 43, andtheoneby Berenger44 basedon a perfectlymatchedlayerfor absorptionattheboundary. Wefollow themethodologydescribedby Arai etal. 45, whichis ageneralizationof Hadley’s approach.Arai et al. arrivedat a linearboundaryconditionandmadeit adaptive.We employ its simplestform, theso-calledRobinconditions.

TheRobinboundaryconditionemploystheone-waywaveequations(asin Eq.(63))in bothtransversedirectionsat theedgesof thecomputationaldomain,W " (' Z i .�� ���#� (' ��SÓ� at * " � ! �� S andat * " ���#�� � k ¡ £ Zr�^� Ò * " U (A1)

By choosing� ���#� adaptively, wewill show thattheseboundaryconditionscanbemadehighlytransparent.

Thekey matricesin our numericalschemeconsistof 5 bands.Thus,we requireestimatesof thefield at two discretizationpointsoutsidethecomputationaldomain,in accordancewithtransparentboundaryconditions.Theestimateof thefield at thefurthestsamplepointswill bebasedon theadditionalconstraint,W �" (' Z i .�� ���#� W " (' � invariant � at * " � ! ��ùS andat * " ���#�� � k ¡ £ Zr��� Ò * " U (A2)

We will illustratetheestimationprocedureat the left boundaryat * " � Ò * " . Thenthefieldsamples

(' 9 and(' !#" have to bedetermined.In termsof a lowest-orderfinite-differencerepre-

sentation,Eq.(A1)becomes, (' " A (' 9 Z i . Ò * " � � ! � (' 9 �rS5U (A3)

This leadsto (' 9 �z�B�PA i . Ò * " � � ! � � !#" (' " U (A4)

Theconstraint(A2) is usedto construct(' !#" , with thefinite-differencerepresentation(' " A � (' 9 Z (' !#"� Ò * " � � Z i .�� � ! � (' 9 A (' !#"Ò * " � (' � A � (' " Z (' 9� Ò * " � � Z i .�� � ! � (' " A (' 9Ò * " � (A5)

31

with thesolution(' !#" � (' � Z (' "0ú i . Ò * " � � ! � A��FZq���PA i . Ò * " � � ! � �g!#"e�A� A � i . Ò * " � � ! � �Lü�PA i . Ò * " � � ! � U (A6)

Thenumericalschemeassociatedwith theRobinboundaryconditionis� neutrallystableif ILKwMa� � ! � O ��S ,� stableif ILKwMa� � ! � O�QøS ,� unstableif ILKwMa� � ! � O��TS .Theboundaryconditionsreduceto theNeumannboundaryconditionfor � � ! � ��S , andto theDirichlet boundaryconditionfor � � ! � X Ô .

Weadaptthevalueof � ���#� by assumingthatthefield behaveslikea planewave( � ;c=H? �BA i .�� ���#� * " � ) nearthecomputationalboundaries.At thepreviouspropagationstep,thehorizontalslownessassociatedwith this planewave canbeestimated;at thecurrentstep,thatvalueof � ���#� canthenbeapplied.At theleft boundaries,with Eq.(A3),weget

� � ! � × � i . Ò * " � !#" �H�PA (' �(' " � U (A7)

If theplanewave wereto travel inward( ï ; Ma� � ! � O�� S at the left boundary),� ���#� is resettoits imaginarypart ILKwMa� ���#� O . In this way, inward travelling wavesareattenuated.Theeffec-tivenessof thetransparentboundaryconditionscanbedeterminedfrom thegraphof reflectioncoefficient associatedwith a plane-wave hitting the boundary, seeClaytonandEngquist42.Keys 46 extendedtheone-wayabsorbingboundaryequationto thecasewheretwo distinguish-ableplanewaveshit thecomputationalboundarysimultaneously.

In ournumericalschemethetransparentboundaryconditionsaretranslatedinto anadjust-mentof theouterelementsof thematrices

S � andS

. For example,theleft boundaryelementsofS

become(cf. Eqs.(A4)-(A6))S "L" X S "L" Zq���PA i . Ò * " � � ! � � !#" y S " 9 Z (A8)� i . Ò * " � � ! � AJ�-Z����PA i . Ò * " � � ! � � !#" �A� A � i . Ò * " � � ! � � � S "�!#"B| �S "� X S "� Zq���PA i . Ò * " � � ! � � !#" S "�!#" � (A9)S � " X S � " Zq���PA i . Ò * " � � ! � � !#" S � 9 U (A10)

32

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33

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34

� C R.T. Coatesand M. Schoenberg, “Finite-differencemodelingof faults and fractures”,Geoph.60, 1514-1526(1995).� H M. Schoenberg andP.N. Sen,“Propertiesof a periodicallystratifiedacoustichalf-spaceandits relationto aBiot fluid”, J. Acoust.Soc.Am.73, 61-67(1983).� 9 C. DeWitt-Morette, A. Maheshwari andB. Nelson,“Path integration in non-relativisticquantummechanics”,PhysicsReports50, 255-372(1979).� " H.C. SerafiniandM.V. de Hoop, “Even-versus-oddorderThiele approximationsof theone-way operatorin thespace-timedomain”,61stAnn. Mtg. Soc.Explor. Geophys.,Ex-pandedAbstracts, 1580-1583(1991).�L� M.V. deHoopandA.T. deHoop,“Scalarspace-timewavesin their spectral-domainfirst-andsecond-orderThieleapproximations”,WaveMotion15, 229-265(1992).�L� We employed the routine E04CCFof the NAG FORTRAN Library Manual Mark 15(1991).� � M.W. Lee andS.Y. Suh,“Optimization of one-way wave equations”,Geoph.50, 1634-1637(1985).� j C.Bunks,“Optimizationof paraxialwaveequationoperatorcoefficients”,62ndAnn.Mtg.Soc.Explor. Geophys.,ExpandedAbstracts, 897-900(1992).� x A.R. Mitchell andD.F. Griffiths,Thefinite differencemethodin partial differential equa-tions, Wiley, Chichester(1985).� z We employed the routine E04CCEof the NAG FORTRAN Library Manual Mark 15(1991).� C W.H. Press,B.P. Flannery, S.A. Teukolsky andW.T. Vetterling,NumericalRecipes, Cam-bridgeUniversityPress,Cambridge(1986).� H O.B. Widlund, “On thestability of parabolicdifferenceschemes”,Math. Comp.19, 1-13(1965).� 9 R.D. Richtmeyer andK.W. Morton, Differencemethodsfor intial-valueproblems, Wiley,New York (1967).� " We employedtheroutinesF07BRFandF07BSFof theNAG FORTRAN Library ManualMark 15 (1991).

35

� � R.W. ClaytonandB. Engquist,“Absorbingboundaryconditionsfor wave-equationmigra-tion”, Geoph.45, 895-904(1980).� � G.R.Hadley, “Transparantboundaryconditionfor thebeampropagationmethod”,IEEEJournalof QuantumElectronics28, 363-370(1992).��� J.-P. Berenger, “A perfectlymatchedlayer for theabsorptionof electromagneticwaves”,J. Comp.Phys.114, 185-200(1994).� j Y. Arai, A. MarutaandM.Matsuhara,“Transparentboundaryfor thefinite-elementbeam-propagationmethod”,Opt.Lett.18, 765-766(1993).� x R.G. Keys, “Absorbingboundaryconditionsfor acousticmedia”, Geoph.50, 892-902(1985).� z M.D. Collins,“Higher-orderPadeapproximationsfor accurateandstableelasticparabolicequationswith applicationto interfacewave propagation”,J. Acoust.Soc.Am.89, 1050-1057(1991).� C M.V. deHoopandA.T. deHoop,“Interfacereflectionsof sphericalacousticwavesin thefirst- andsecond-orderrationalparabolicapproximationsandtheir artifacts”, J. Acoust.Soc.Am.93, 22-35(1993).� H H.-P. Nolting andR. Marz,“Resultsof benchmarktestsfor differentnumericalBPM al-gorithms”,IEEEJournalof LightwaveTechnology13, 216-224(1995).j 9 C. Vassallo,Opticalwaveguideconcepts, Elsevier, Amsterdam(1991).j " O.J.F. Martin, A. DereuxandC. Girard,“Iterativeschemefor computingexactly thetotalfield propagatingin dielectricstructuresof arbitraryshape”,J. Opt.Soc.Am.A 11, 1073-1080(1994).j � R.M. Mersereau,“The processingof hexagonally sampledtwo-dimensionalsignals”,Proc.IEEE67, 930-949(1979).j � D.P. PetersenandD. Middleton, “Samplingandreconstructionof wave-number-limitedfunctions in

k-dimensionalEuclideanspaces”,Information and Control 5, 279-323

(1962).j � J.GazdagandP. Sguazzero,“Migration of seismicdataby phaseshift plusinterpolation”,Geoph.49, 124-131(1984).

36

j�j P.L. Stoffa, J.T. Fokkema,R.M. de Luna FreireandW.P. Kessinger, “Split-stepFouriermigration”,Geoph.55, 410-421(1990).j�x D. Hale,“3-D depthmigrationvia McClellantransformations”,60thAnn. Mtg. Soc.Ex-plor. Geophys.,ExpandedAbstracts, 1325-1328(1990).j�z R.-S.Wu, “Wide-angleelasticwave one-way propagationin heterogeneousmediaandanelasticwavecomplex-screenmethod”,J. Geophys.Res.99, 751-766(1994).j�C L. Fishmanand A.K. Gautesen,“An exact, well-posed,one-way reformulationof theHelmholtzequationwith applicationto direct and inversewave propagationmodeling,”in: New perspectivesonproblemsin classicalandquantumphysics, editedby A.W. SaenzandP.P. Delsanto,GordonandBreach,Newark,to appearin 1996.

37

FIGURES AND TABLES

FIG. 1. A sampleconfiguration. Betweenthe entranceand the exit levels, the configura-tion’s mediummay vary in the vertical andhorizontaldirections. Outsidethat regionthemediumis constant.

FIG. 2. Thelevelsatwhich thefield is calculated.

FIG. 3. Flow chartof thenumericalimplementationof theBremmercouplingseries.

FIG. 4. Theexactverticalslowness(left column)andits Thieleapproximantswith � " �z�^Ë � ,� � �ù�^Ë�� , � � �÷�^Ë � (middle) andwith � " �ëS@UO���@� , � � � SHU�� ��� , � � �ëSHU�� � � (right).In theupperleft figure thedottedline correspondsto the imaginarypartof theverticalslowness,while in the two otherupperfiguresthe dottedlines correspondto the realpartof theexactverticalslowness.Thefiguresin thesecondrow show thedifferencesbetweenthe approximantsandthe exact expression.In the bottomrow, the real partsof the vertical slownessesas functionsof complex ¼ " are shown. The contourlinescorrespondto thevaluesS@U©���gSHU � ��Ö�Ö�Öc� � U S .

FIG. 5. The # -factor(left) andthecorrespondingamplificationfactor(right) associatedwiththedissipationtrick in thethird-orderThieleapproximation;� " �qSHU�� ��� , � � ��S@U�� ��� and� � �JS@U�� � � . The relative imaginarycircular frequency � � S@U S � . Note that the notchoccursbeyondthecritical angle.

FIG. 6. Thespectraor symbolsof theapproximateW<�" operatorswith ) � ���^Ë@� � (upperleft)and ) � �ëS@U©�1�òS (upperright). The bottomfiguresshow the correspondingdifferenceswith theexactspectra,which arealsodottedin theupperfigures.We set .Å� � s!#" andÒ * " � Ì m.

FIG. 7. The third-orderThiele approximationof the vertical slownesssymbol including thespectrumof the discretizedLaplaceoperator( ) � � SHU©�~��S , � " �óSHUO� ��� , � � � S@U�� ��� ,� � � SHU�� � � , anda samplingrateof 8 pointsperwavelength).Thepolesarelocatedat¼ " ×Û��U8��S � � � U��>e>�H�be@U©��S � ��Ö�Ö�Ö .

FIG. 8. Thephaseslowness(upperleft) andthegroupslowness(upperright) associatedwiththe discretized,approximate,one-way wave equationwith � " ���^Ë � , � � ���^Ë�� , � � ��^Ë � , ) � � �^Ë@� � , �DH��ë��Ë � , anda samplingrateof 5 pointsperwavelength.Theexactverticalwave-slownesssymbolis dotted. In thebottomrow thedifferencebetweenthephaseslownessandthegroupslowness,andthedifferencebetweenthegroupslownessandtheexactslownessareplotted.

38

FIG. 9. Thephaseslowness(upperleft) andthegroupslowness(upperright) associatedwiththe discretized,approximate,one-way wave equationwith � " � SHUO� ��� , � � � S@U�� ��� ,� � � S@U�� � � , ) � = 0.0888,�DH�� SHUO� � e , anda samplingrateof 5 pointsperwavelength.The exact vertical wave-slownesssymbol is dotted. In the bottomrow the differencebetweenthe phaseslownessand the group slowness,and the differencebetweenthegroupslownessandtheexactslownessareplotted.

FIG. 10. Thephaseslowness(upperleft) andthegroupslowness(upperright) associatedwiththe discretized,approximate,one-way wave equationwith � " � SHU � � � , � � � S@U�� � e ,� � � SHU�� � � , ) � �JSHU8��� � , �DH[� SHU�� � e , anda samplingrateof 5 pointsperwavelength.The exact vertical wave-slownesssymbol is dotted. In the bottomrow the differencebetweenthe phaseslownessand the group slowness,and the differencebetweenthegroupslownessandtheexactslownessareplotted.

FIG. 11. Thedifferencebetweentherealpartsof thephaseslownessandthegroupslowness(upperleft), andthedifferencebetweentherealpartsof thegroupslownessandtheexactslowness(upperright). Themiddlerow showstheimaginarypartsof thephaseslowness(left) andthegroupslowness(right). Theparametersetis givenin column4 of Table1.Two samplingratesareshown: 5 pointsperwave length(solid line) and10 pointsperwave length (dottedline). The bottomfigure shows the imaginarypartsof the phaseslowness(solid line) andthegroupslowness(dottedline ona largerscale.

FIG. 12. Snapshotat 0.3sof a vertical line forcesourceresponsein a homogeneousmedium( Ú�� �1e�SòS m/s). Theright-handsideis obtainedanalytically, while theleft-handsideisobtainedwith ournumericalscheme.Here,weusedparameterset1 of Table1.

FIG. 13. Snapshotat 0.3sof a vertical line forcesourceresponsein a homogeneousmedium( Ú�� �1e�SòS m/s). Theright-handsideis obtainedanalytically, while theleft-handsideisobtainedwith ournumericalscheme.Here,weusedparameterset4 of Table1.

FIG. 14. Theabsoluteerror( SwQÄ* " Q���S�S m) andtherelative error( �òS�SwQÄ* " Q � � S�S m) ofthesnapshotshown in thepreviousfigure.Therelativeerroris in %.

FIG. 15. Therealpartof thesymbolof theverticalslowness(composition)operator(upperfig-ure)andits differencewith theexactsymbol(lowerfigure).Thedottedline correspondsto therealpartof theexactslownesssymbol;theotherlinescorrespondto � " �zS@UO� ��� ,� � �qSHU�� ��� , � � �rSHU�� � � and ) � ��SHU8�1��S , while � equalsS (solid line), SHU SH� (dashedline)and S@U©� (dashdotline).

39

FIG. 16. Snapshotof an explosion line sourcein a homogeneousmedium( ÚÏ� �~e�S�S m/s).Theright-handsideis obtainedanalytically, while theleft-handsideis obtainedwith ournumericalscheme.

FIG. 17. Therealpartof thesymbolsof thediscretizedandtheexact (dottedline) reflectionoperators(plane-wave reflectioncoefficients). The differencesbetweenthe discretizedandexactsymbolsareplottedin thelowerfigure.Thewave speedequals� aboveand

�below theinterface.Weused) � ����Ë@� � , and �T��SH�gSHU©�ò�lSHU S}� (solid,dashedanddashdotlines,respectively).

FIG. 18. Reflectionandtransmissionataninterface.

FIG. 19. The fundamentalwaveguiding modeat the entrancelevel (upperfigure) andat theexit level (bottomfigure). The waveguide is not tilted with respectto the coordinatesystem.

FIG. 20. The fundamentalwaveguiding modeat the entrancelevel (upperfigure) andat theexit level (bottom figure). The waveguide is tilted with an angleof

� S degreeswithrespectto thecoordinatesystem.(Leadingtermof theBremmercouplingseriesonly.)

FIG. 21. The tenth-orderwaveguidemodeat theentrancelevel (upperfigure)andat theexitlevel (bottomfigure).Thewaveguideis not tilted with respectto thecoordinatesystem.

FIG. 22. The tenth-orderwaveguidemodeat theentrancelevel (upperfigure)andat theexitlevel (bottomfigure).Thewaveguideis tilted with anangleof

� S degreeswith respecttothecoordinatesystem.(Leadingtermof theBremmercouplingseriesonly.)

FIG. 23. Planewave(from thetop)responsein aconfigurationwith ablockin ahomogeneousembedding.

FIG. 24. Planewave(from thetop)responsein aconfigurationwith ablockin ahomogeneousembedding.Secondtermof theBremmercouplingseries.

FIG. 25. Planewave(from thetop)responsein aconfigurationwith ablockin ahomogeneousembedding.Leadingtermof theBremmercouplingseries.

FIG. 26. Wavespeedsof a two-dimensionalsaltmodel.

FIG. 27. Snapshotat Cs�óSHU�N>�òS s of the acousticpressuredueto a vertical line force in themodelof Figure26. Six termsof theBremmercouplingseries.

40

FIG. 28. Snapshotat Cs�óSHU�N>�òS s of the acousticpressuredueto a vertical line force in themodelof Figure26. Leadingtermof theBremmercouplingseries.

FIG. 29. Snapshotat Cs�óSHU�N>�òS s of the acousticpressuredueto a vertical line force in themodelof Figure26. Secondtermof theBremmercouplingseries.

41

horizontal direction

(x1)

(x)

2

vert

ical

dire

ctio

n

x2= x

2

exit

x2

= 0ENTRANCE

EXIT

FIG. 1.

42

SÒ * �� Ò * �ð Ò * �

Ñ Ò * ����

���

FIG. 2.

43

Incidentfield:j " �`S@� C/�D�Ù�)�SH� C/� or

j � �SH� C/����� � �SH� C/� is given

FourierTransformA X (j âr�U�+âr�gà���ã ��Ö©�lSH�/.0�For every .

decomposition:(x d �Û� (�0!#"/� dlf h (j h

For every term Õ of BremmerCouplingSeries, (ÕV�rSH����� � ��Ö�Ö�Ö )(ñ � Í �"Lf h ��Ö©�lSä�1�qS , (ñ � 9 �"Lf h �BÖÈ�lS���� (x " ��ÖÈ�gSä�For everydownwardpropagationstep ð , ( ð5����� � ��Ö�Ö�Öe� Ñ )(ñ � Í �"Lf h ��ÖÈ�lð:��� ( ��Ö©�cð:� ñ � Í �"Lf h �BÖÈ�cð[Aø�^��Z (ö � Í �"Lf h ��ÖÈ�lð:�(x � Í �" ��ÖÈ�lð Ò * � �\� (ñ � Í �"Lfp" �BÖÈ�cðH��Z (ñ � Í �"Lf � ��ÖÈ�lð:�

Calculate(õ � Í � " �"Lf h ��ÖÈ�lð Ò * � �

compositionA X (j � Í �" and(j � Í ��

addingto previousfields+ storage(ñ � Í �� f h ��Ö©� Ñ ���qSFor everyupwardpropagationstepð , ( ð°� Ñ AT�ò��Ö�Ö�Öe�gS )(ñ � Í �� f h ��ÖÈ�lð:��� ( ��Ö©�cð�Zr�^� ñ � Í �� f h �BÖÈ�cð�ZÅ�^��Z (ö � Í �� f h ��Ö©�cð:�(x � Í �� ��ÖÈ�lð Ò * � �\� (ñ � Í �� fp" �BÖÈ�cðH��Z (ñ � Í �� f � ��ÖÈ�lð:�

Calculate(õ � Í � " �� f h ��ÖÈ�lð Ò * � �

compositionA X (j � Í �" and(j � Í ��

addingto previousfields+ storage

InverseFourierTransformA X �)��Ö©� * � � C/� and � � �BÖÈ� * � � C/� at thepositionsof interest

FIG. 3.

44

0 1 2−1

0

1

2Exact

0 1 2

Thiele

0 1 2

Optimised

0 0.5 1

−0.2

0

0 0.5 1

0 1 20

2

0 1 20

2

cα1

cγ

cα1

cγIII

cα1c

γIII

cα1

cγIII

−c

γ

cα1

cγIII

−c

γ

Re{cα1}

Im{c

α1}

Re{cα1}

Im{c

α1}

Re{cα1}

Im{c

α1}

0 1 20

2

FIG. 4.

45

0 1 2 3 4 510

−1

100

101

102

103

104

105

Q−

fact

or

c α1 c α1

0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Am

plifi

catio

n fa

ctor

FIG. 5.

46

−1

−0.8

−0.6

−0.4

−0.2

0

spec

trum

<∂ 1

>

0 0.2 0.4 0.6 0.8 1−0.1

0

0.1

0.2

0.3

0.4

sp

ectr

um <

∂ 1>

+α 12

α1 α1

0 0.2 0.4 0.6 0.8 1

FIG. 6.

47

c α1

cγ

0 2 4 6 8 10 12 14 16−4

−3

−2

−1

0

1

2

3

4

FIG. 7.

48

0 0.5 1 1.5 2−0.5

0

0.5

1

1.5

0 0.5 1 1.5 2−0.1

0

0.1

0 0.5 1 1.5 2−0.1

0

0.1

c γph

c α1

c γph

− c

γgr

c α1

c γgr

− c

γc

γgr

0 0.5 1 1.5 2−0.5

0

0.5

1

1.5

FIG. 8.

49

0 0.5 1 1.5 2−0.5

0

0.5

1

1.5

0 0.5 1 1.5 2−0.1

0

0.1

0 0.5 1 1.5 2−0.1

0

0.1

c γph

c α1

c γph

− c

γgr

c α1

c γgr

− c

γc

γgr

0 0.5 1 1.5 2−0.5

0

0.5

1

1.5

FIG. 9.

50

0 0.5 1 1.5 2−0.5

0

0.5

1

1.5

0 0.5 1 1.5 2−0.1

0

0.1

0 0.5 1 1.5 2−0.1

0

0.1

c γph

c α1

c γph

− c

γgr

c α1

c γgr

− c

γc

γgr

0 0.5 1 1.5 2−0.5

0

0.5

1

1.5

FIG. 10.

51

0 0.2 0.4 0.6 0.8 1−0.02

−0.01

0

0.01

0.02

0 0.2 0.4 0.6 0.8 1−0.02

−0.01

0

0.01

0.02

0 0.2 0.4 0.6 0.8 1−0.01

−0.005

0

0.005

0.01

0 0.2 0.4 0.6 0.8 1−0.01

−0.005

0

0.005

0.01

cα1

c R

e{γph

− γ

gr}

cα1

c Im

{γph

}

cα1

c Im

{γgr

}

cα1

c R

e{γph

− γ

}

cα1

c Im

{γph

}

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

−0.2

0

0.2

FIG. 11.

52

−1 −0.5 0 0.5 1

x

x

1

2

400 800 1200

0

400

800

FIG. 12.

53

−1 −0.5 0 0.5 1

x

2

1

x

400 800 1200

0

400

800

FIG. 13.

54

0

400

800

800400 1200x

x

1

2

0−1.3 1.3 right

left0 0.05 0.1 0.15 0.2

FIG. 14.

55

0 0.5 1 1.5 2−0.5

0

0.5

1

1.5

c γ

c α1

c γex

act −

c γ

num

0 0.5 1 1.5 2−0.1

0

0.1

FIG. 15.

56

−2500 −2000 −1500 −1000 −500 0 500 1000 1500 2000 2500

x

x

1

2

400 800 1200

0

400

800

FIG. 16.

57

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

α1

Re{

R num

exac

t }R

e{R n

um }

−R

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−5

0

5

FIG. 17.

58

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8

x

x 2

1

400 800 1200

0

400

800

FIG. 18.

59

0 200 400 600 800 10000

0.05

0.1

x1

abs(

pres

sure

)

x1

abs(

pres

sure

)

0 200 400 600 800 10000

0.05

0.1

FIG. 19.

60

0 200 400 600 800 10000

0.05

0.1

x1

abs(

pres

sure

)

x1

abs(

pres

sure

)

0 200 400 600 800 10000

0.05

0.1

FIG. 20.

61

0 200 400 600 800 10000

0.02

0.04

0.06

0.08

0.1

x1

abs(

pres

sure

)

x1

abs(

pres

sure

)

0 200 400 600 800 10000

0.02

0.04

0.06

0.08

0.1

FIG. 21.

62

0 200 400 600 800 10000

0.02

0.04

0.06

0.08

0.1

x1

abs(

pres

sure

)

x1

abs(

pres

sure

)

0 200 400 600 800 10000

0.02

0.04

0.06

0.08

0.1

FIG. 22.

63

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x1

x 2

x1

x 2

x1

x 2

x1

x 2

x1

x 2

x1

x 2

x1

x 2

x1

x 2

50 100 150 200 250

0

50

100

150

200

250

300

FIG. 23.

64

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x1

x 2

x1

x 2

x1

x 2

x1

x 2

x1

x 2

x1

x 2

x1

x 2

x1

x 2

x1

x 2

x1

x 2

x1

x 2

x1

x 2

50 100 150 200 250

0

50

100

150

200

250

300

FIG. 24.

65

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x1

x 2

x1

x 2

x1

x 2

x1

x 2

x1

x 2

x1

x 2

x1

x 2

x1

x 2

x1

x 2

x1

x 2

x1

x 2

50 100 150 200 250

0

50

100

150

200

250

300

FIG. 25.

66

500 1000 1500 2000 2500 3000

0

500

1000

1500

2000

x 2

x1

c

2000 2500 3000 3500 4000 4500

FIG. 26.

67

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

x1

x 2

500 1000 1500 2000 2500 3000

0

500

1000

1500

2000

FIG. 27.

68

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

x1

x 2

500 1000 1500 2000 2500 3000

0

500

1000

1500

2000

FIG. 28.

69

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

x1

x 2

500 1000 1500 2000 2500 3000

0

500

1000

1500

2000

FIG. 29.

70

TABLE 1. Theparametersetsfor propagation.

parameterset 1 2 3 4���@�U�>�0.2 0.2 0.2 0.2���-�+�>�0.2 0.2 0.2 0.2

Re�a� �[� 1/2 0.526 0.486 0.5104Im �a� �b� 0 0 0 -0.0340Re�a� �~� 1/8 0.364 0.349 0.2207Im �a� �<� 0 0 0 -0.0131Re�a�D� � 1/2 0.825 0.841 0.6685Im �a�D� � 0 0 0 -0.0310Re�"� �1� 1/12 0.089 0.114 0.1207Im �"� �~� 0 0 0 0.0063Re�a�D� � 1/2 0.540 0.529 0.4679Im �a�D� � 0 0 0 -0.0066�

0 0 0 0.0406Figures 8 9 10 11

12 1318

TABLE 2. Thepolesof thenumericalverticalphaseslowness.

parameterset pole1 pole2

1 �>���} �¡>¡£¢ ¤`�O¥�¡�¤}¦ i ���O¦`�~§} |¢q�>��§>¤}¦"� i2 �>�¨�1¤} ��ª©K¤«��¤>¤��>¤ i ¥4�O¥�¡�¬>¬£¢q�>��¦> �§�¤ i3 �>��¤}¦�­}¥®©K¤«��¤}¥>¦4� i ����§>­�¡>­£¢q�>�¨�a¥`�<� i4 �>�� �¬}¦��ª©h�>�¨�~�}¤�­ i �>�¯�~¡«�1­°©K¤«�¨�a¥`�>� i

71

TABLE 3. Theparametersetfor (de)composition.

parameterset 2’���@�����0.2���-�I���0.2

Re�a� �[� 0.526Im �a� �U� 0Re�a� �~� 0.364Im �a� �+� 0Re�a�D� � 0.825Im �a�D� � 0Re�"� �~� 0.13Im �"� �<� 0�

0.01

72