High frequency edge diffraction theoryP.F. Daley1 and E.S. Krebes2

Department of Geoscience, University of Calgary, Calgary, Alberta, Canada

INTRODUCTION

The work of Klem-Musatov (1980) dealing with elastic wavesdiffracted by the linear edges of seismic interfaces, using anextension of asymptotic ray theory (ART) (Červený andRavindra, 1970 and Červený, 2001, among others) and subse-quently dynamic ray tracing (DRT) (Červený and Hron, 1980)were employed to obtain a high frequency approximation toelastodynamic waves diffracted by linear edges. The methodpresented uses modifications of ART and DRT incorporatingthe boundary layer method. This theory is applicable to a threedimensional case of rays, emanating from a point source, prop-agating in a geological model with homogeneous layers sepa-rated by curved interfaces. The Russian text by Klem-Musatov(1980), once relatively inaccessible may be found now in theEnglish translation, Klem-Musatov (1995) as well as the transla-tion of a more recent work, Klem-Musatov et al. (2007). Adissertation based on the original text (1980) is the topic of thePh.D. thesis of Chan (1986), where many of the subjectscontained in that book are discussed. The papers by Bakker(1990), Hron and Chan (1995) and Gallop and Hron (1997)pursue some aspects of the theory presented in the works ofKlem-Musatov. The work of Bakker (1990) approaches the edgediffraction problem using a paraxial (DRT) approximationsimilar to that in Červený and Hron (1980) while Hron andChan (1995) arrive at essentially the same results using a moreconventional ART modified by the inclusion of boundary layertheory. As stated above, it is the intent of this paper to engage ina minimal amount of theoretical development and employpreviously derived formulae to introduce concepts in a formintended for numerical implementation.

Formulae for use in the computation of numerical resultswould ideally be such that they would (a) allow for simplephysical interpretation, (b) provide a practical means for theefficient calculation of the diffracted field and (c) in light of (b),also maintain a reasonable degree of accuracy when describingthe diffracted wavefield. A relatively inaccessible report byHron and Covey (1988) contains a more programmer friendlytreatment of edge diffraction theory. The ART approximation tothe edge diffracted arrival consists of the zero order asymptoticexpression for the incident wavefield at the diffractor, due to apoint source, multiplied by what was termed a diffraction coef-ficient. The resulting diffracted wavefield was obtained byconsidering the diffractor as a secondary source of seismicenergy and tracing its path to receivers in accordance with amodified form of Snell’s Law and zero order ART. If the modelwere truly three dimensional, the individual points of diffrac-tion must be replaced by diffraction edges. The theoryemployed here may be referred to as 2.5 D due to the fact thatout of plane geometrical spreading is included.

The accuracy of a method similar in development to the onedescribed above was determined in the work of Chan (1986)where an approximate method for SH waves was compared

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CANADIAN JOURNAL of EXPLORATION GEOPHYSlCSVOL. 36, NO. 1 (June 2011), P. 37-46

1 Correspondence to: P.F. Daley, Department of Geoscience, University ofCalgary, 2500 University Drive, Northwest, Calgary, Alberta, CanadaT2N 1N4; E-mail: pdaley@ucalgary.ca; Telephone: (403) – 220 – 8340.

2 krebes@ucalgary.ca; Telephone: (403) – 220 – 5028.

Manuscript received March 2011; revision accepted June 2011.

ABSTRACT

Diffractions make significant contributions to seismicamplitudes. While they are often not included in raytrace models due to their perceived computational diffi-culty and expense, this paper describes a tractablemethod for computing diffractions which could benefitseismic modeling in exporation geophysics. We presentformulae related to the diffraction of seismic waves bylinear edges in elastodynamic media, based on an exten-sion of the high frequency, zero order Asymptotic RayTheory (ART) formulation. Theoretical development hasbeen kept to a minimum. Rather, schematics indicatingrelevant details such as shadow boundaries andboundary rays have been included. The identification ofthe above is required as the argument of the diffractioncoefficient is dependent on the angle between the shadowboundary ray and the diffracted ray or equivalently thedifference between the diffracted and reflected arrivaltimes. More will be said about this in the text. Thereflected geometrical arrival does not exist in the shadowregion and its travel time is required to determine theargument of the diffraction coefficient. Using analyticcontinuation, within the aforementioned limits ofminimal theoretical discussion, this topic is also consid-ered. Klem-Musatov (1980) derived a very comprehen-sive set of formulae for the diffracted amplitude due to alinear edge in a three dimensional medium. This workwas the basis of the PhD. Thesis of Chan (1986) where thetheory, along with numerous numerical experimentsinvolving the comparison of the modified ART solutionwith “exact” solution methods, was presented.

A basic problem will first be considered to give a briefoverview of the theory of edge diffractions. The geometryof this problem involves a wedge embedded in a halfspaceand is such that both a source and receivers are located atthe surface of the halfspace. Only reflected rays andprimary diffractions are included in the solution. Themotivation for this is to provide some initial insight for theextension of edge diffraction theory to more complicatedand realistic geological structures. As an ART type solu-tion is used, the geometrical and diffracted wavefields arecomputed separately and the total wavefield is the sum ofthe two. This concept can be extended to combining anumber of diffracting edge models and additively obtainthe total wavefield.

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with the highly numerically accurateresults obtained by the AlekseevMikhailenko method (AMM), a pseudo-spectral method (see for example,Mikhailenko, 1985 and 1988).

As the treatment of this problemcontained in Klem-Musatov (1995 and2007) and Chan (1986) are quite compre-hensive, minimal theoretical aspects ofthe problem will be dealt with in thiswork. For clarity, the problem will bedefined in the next section along withsome comments and a restatement ofthe final formulae required for compu-tation of the diffracted wavefield due toan edge diffractor will be presented.Other rather inaccessible notes (Hron,1986) might be of some use for thoseinterested in the theoretical develop-ment of this problem but these are fairlymathematically intense and of minimalattraction to the majority of readers.

The two models that will be consideredin this work are shown in Figures 1 and2. They each consist of a wedge that isinfinite in the y direction with ray prop-agation constrained to the (r, z) plane.The receivers in both geometries arealong a surface receiver profile. Model Iwill be briefly considered first, and thenused, with minor modifications, forModel II. Numerical results for thecombination of Models I and II, termedModel III, are also presented.

THEORY

Consider a three dimensional Cartesiancoordinate system (r, y, z), in a halfspacez > 0 with an explosive point source ofcompressional (P) waves located at theorigin, (Figure 1). A semi-infinite wedgeis assumed to be located at a depth zDbelow the surface, occupying the threedimensional space

(zD£z<∞ ,0£r£rD,–∞<y<∞) .Receivers are placed at the surface in the(r, z) plane containing the source. Forthis case the reflected rays from the topof the wedge and the diffracted P wavefrom the edge at D = (rD, zD) are the onlyarrivals considered in the resultingsynthetic seismograms. The compres-sional (P) wave velocities in the halfplane, a, and in the wedge strip, a1, arechosen such that a <a1. The shear wave(SV) velocities are defined by the rela-tion b = a/@ –3 and the densities areobtained from Gardner’s Law. This will

P.F. Daley and E.S. Krebes

Figure 2. Geometry of Model II for a point source and a surface array of receivers. The wedge (dark gray)occupies the 3D space (rD£ r<∞ , zD£ z< ∞ , –∞<y<∞). As in Model I, the point of edge diffraction is inthe (r,z) plane at (rD,zD). In this model a reflected arrival off the flank of the wedge is not considered, onlythe ray reflected from the top of the wedge is of interest here. The boundary ray, which separates the illu-minated, WI, region from the shadow, WS (light gray) region is shown. In this instance, reflected arrivalsfrom the top of the wedge are only seen at a surface receiver, r, such that (r£2rD= rB). As for Model I thediffracted arrival is seen at all offsets.

Figure 1. Geometry of Model I for a point source and a surface array of receivers. The wedge (darkgray) occupies the 3D space (zD£ z<∞ , 0£ r£ rD, –∞<y<∞). The point of edge diffraction is in the (r,z)plane at (rD,zD). Ray propagation is assumed to lie in this plane. The shadow region WS (light gray) forthe ray reflected from the top of the wedge is defined by the boundary ray. Only reflected arrivals at asurface receiver, r, such that (rB=2rD£ r ) are seen on the synthetic traces. The diffracted arrival appearsat all offsets.

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be referred to as Model I. A modification of Model I will beconsidered, denoted as Model II and shown schematically inFigure 2. This second model also consists of a wedge embeddedin a halfspace with the wedge being in the three dimensionalspace (zD£z<∞ ,rD£ r<∞ ,–∞<y<∞) . All other parameters aresimilar to those for Model I, except for the elastic parameters inthe wedge, defined such that a <a2.

When considering diffraction from the wedge in Model I thereare two regions to be considered in the half plane, the illumi-nated (WI) and the shadow (WS) regions (Figure 1). The illumi-nated and shadow regions are separated from one another bywhat is termed the boundary ray. In the simple situations beingconsidered here the reflected arrival only appears on thesynthetic traces in the illuminated regions while the diffractedarrival exists in both regions.

The Fourier time transformed vector particle displacement ofan incident compressional wave generated at the point sourceat the origin and recorded at the surface receiver arrays may bewritten in terms of the zero order ART approximation as

(1)

where the subscript “G” (geometrical) indicates the reflectedPP arrival, r is the generally three dimensional position vectorof some point in the halfspace that may, when convenient bedenoted as M, w is the circular frequency, tG(r) [tG(M)] is thetravel time along the reflected ray from the source to a reflectionpoint on the wedge top, to some point r defining a surfacereceiver, LG(r) [LG(M)], is the 3D geometric spreading of the raybetween the source and the point r which for the reflectedarrival is given by

(2)

The reflected ray arrival time may be written

(3)

The term P in equation (1) is the product of all reflection andtransmission coefficients encountered along the geometrical rayfrom source to receiver. In the reflected ray case, P is the elasticPP reflection coefficient from the top of the wedge. The vectorCs = [CS

(V),CS(H)]T is the surface conversion coefficient vector

which partitions the incident particledisplacement at the free surface receiverinto its constituent vertical and hori-zontal components. F(w) is the Fouriertime transform of the band limitedsource wavelet, f(t), t being time. Aspherically symmetric radiation patternof this source function is assumed.

For the diffracted arrival the travel timefrom source to receiver consists of twoparts; the time it takes the ray to travelform the source to the diffraction edgein the (r,z) plane at (rD,zD) plus the timetaken for the ray to progress from thediffraction point (D) to the receiver atM. This infers that a diffracted arrivalgenerally has a different ray parameter(horizontal component of the slownessvector) on either side of the edgediffraction point.

From field data observations and resultsobtained from numerical modeling withfinite difference methods a number ofattributes of edge diffracted arrivalsmay be realized: (a) the diffracted wave-field is frequency dependent and (b)angular dependent measured at someunit distance about the diffraction pointon the diffracting edge and (c) the pointon the diffracting edge appears to act asa secondary source. For this reason asolution is assumed that may be writtenin terms of the reflected arrival at someminimal distance about the point D. Thediffracted arrival solution results bymultiplying the reflected arrival at the

High frequency edge diffraction theory

Figure 3. Analytic continuation of the geometrical wavefield into the shadow region to obtain some simu-lated measure of the geometrical arrival time in this region. A more complete discussion may be found inAppendix A.

UG(r,w) =F(w)Pexp[iwtG(r)]CSLG(r)

1 = 1LG (M) [ r2 + (2zD)2 ] 1/2 .M

tG (M) = [ r2 + (2zD)2 ] 1/2 .a

M

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point D by a radiation characteristic function, I(w,y). This func-tion is assumed to be a frequency (w) and angular (y)dependent in the (r,z) plane. The actual determination of thisfunction is yet to be addressed. However, based on empiricalobservations, the diffracted arrival may be written as

(4)

The three dimensional geometrical spreading LD(r)=LD(M) is thesum of the spreading from the source to the point of diffractionplus the addition of the spreading from this point to M. Thisquantity and the diffracted travel time will be given later.

It is convenient at this point to introduce another Cartesiancoordinate system, (u,v), whose origin is at D, together with arelated polar coordinate system, (r,y). Here, y is positive/nega-tive in the shadow/illuminated regions which are defined bythe boundary ray. The quantity rM is the distance from the pointof diffraction at D to the observation point M fully defined bythe additional coordinate, yM within the context of theCartesian system (u,v).

In light of the above Cartesian and polar coordinate systemsdefinitions, the radiation characteristic function I(w,y) will bereplaced in Equation (4) by a more general related function,W(z) with z=z(w,w(w,rM,yM)), termed the diffraction coeffi-cient. This coefficient is derived in a number of the previouslycited works to which interested readers are referred as its inclu-sion here is, apart from introducing unnecessary length, is dealtwith quite rigorously in the citations listed in the Introduction.

The vector particle displacement at the point M, which iscontained in the surface receiver array at the coordinates(rM,yM) may then be written as

(5)

where z=z(w,wM) and wM=w(w,rM,yM), with w being thecircular frequency and

(6)

with the “–” and “+” signs associated respectively with the illu-minated and shadow regions, and W(z) is the scaled comple-mentary error function. The asymptotic expansion for W(z) forlarge values of Íz˜, Íarg(z)˜<3p/4, is

(7)

indicating that the amplitude of the diffracted arrival is of theorder O(1/@ –w) for large values of the argument of z. Themeaning of the quantity w is found by studying the worksdealing with solution method for edge diffracted waves and isdefined as

(8)

Equation (8.b) for w2 results from the fact that at points at thesurface no reflected arrival exists in the shadow zone(rD£ r<∞ ). Equation (8.b) was obtained using analytic continua-tion. Appendix A contains a discussion of its derivation in asimplified manner avoiding excessive mathematical rigor.Figure 3 depicts schematically what is discussed in Appendix A.

The geometrical spreading LD(M) and arrival time tD(M) for thediffracted P arrival for Model I may now be written as

(9)

(10)

respectively. As before, the compressional wave velocity in thehalfspace, except for the wedge, is a. All other quantities havebeen previously defined.

Another simple geological model, depicted schematically inFigure 2 and denoted as Model II, will now be examined. It issimilar to Model I with the exception that the wedge now occu-pies the space (rD£r<∞ ,zD£ z<∞ ,–∞<y<∞ ) in the halfplane(z>0). The expressions for the reflected and diffracted wavesamplitudes and travel times are given by Equations (1) and (5),and (3) and (10) still hold, provided the coordinate system (r,y)is properly defined.

An additional model is constructed by overlaying Models I andII, and referred to as Model III. Numerical results will bepresented for this situation.

NUMERICAL RESULTS

The geological parameters defining the models mentionedearlier in this report are given in Table 1. Additional quantitiesrequired for model definition are the distance from the surfaceto the tops of the wedges, h = 400m, the horizontal distance fromthe source location to the points of diffraction, rD = 1000m andthe horizontal distance from the source to the points theboundary ray in Models I and II where surface receivers areboth at rB = 2000m. The offsets of all of the synthetic seismo-grams presented run from 0.0m to 3000m in steps of 50m. Thesetogether with a time scale and a brief description of what isshown in a given figure appear on all of the synthetic seismo-grams. A Gabor wavelet is used when producing the synthetictraces. The function W(z) is a standard function available inmost mathematical libraries such as IMSL, where it is denotedas ZERFE (Gautschi, 1979a and 1979b).

Schematics of the models used are given in Figures 1 and 2where additional information may be found in the captions.The synthetic seismograms presented include both the verticaland horizontal components of displacement at the surface foran elastic halfspace with an embedded wedge or wedges. Largevelocity contrasts between the halfspace and the wedges wereemployed to emphasize the reflected arrivals.

Figures 4 through 7 are associated with Models I and II withFigures 8.a and 8.b displaying the vertical and horizontalcomponents of displacement of Model III – a combination ofModels I and II. There are 3 panels in each of the Figures 4 – 7showing: (a) the reflected arrival, (b) the diffracted arrival and

P.F. Daley and E.S. Krebes

UD(r,w) =F(w)I(w,ϕ)P

exp[iwtD(r)]CS .LD(r)

UD(M,w) =F(w)W(z)P

exp[iwtD(M)]CS .LD(M)

W(z)=±e–z2 erf c(–zi) , with z= e w/2 wip/4

W(z)≈i/ p (e p/2w) = e / pw(tD – tR) ,ip/4 ip/4

w2=(a). 2w (tD(M)–tG(M)) – illuminated zonep

w2 = (b).2w (rM (1–cosϕM)) – shadow zonep a

{

LD(M) = [ z2 + r2 ] 1/2 + rMD D

tD(M) =[ z2 + r2 ] 1/2

+rM

a aD D

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High frequency edge diffraction theory

Figure 4. Vertical component of displacements of the (a) direct, (b) diffracted and (c) combined, forModel I with receivers located at the model surface.

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P.F. Daley and E.S. Krebes

Figure 5. Horizontal component of displacements of the (a) direct, (b) diffracted and (c) combined,for with receivers located at the model surface.

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High frequency edge diffraction theory

Figure 6. Vertical component of displacements of the (a) direct, (b) diffracted and (c) combined, forModel II with receivers located at the model surface.

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P.F. Daley and E.S. Krebes

Figure 7. Horizontal component of displacements of the (a) direct, (b) diffracted and (c) combined,for Model II with receivers located at the model surface.

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(c) the combination of the two. Only the total traces (reflecteddiffracted) are shown for Model III in Figures 8.a and 8.b. Thevertical and horizontal components are the arithmetic sum ofthe corresponding traces for Model I and Model II.

CONCLUSIONS

Two simple geological models have been considered in whichsome of the basic concepts of diffraction theory in elastic mediawere presented. This was done within the framework of asymp-totic ray theory (ART) and certain extensions thereof describedin previously cited works such as Klem-Musatov (1995), Chan(1986) and Klem-Musatov et al. (2007). ART produces resultsequivalent to those derived using the high frequency geomet-rical optics solution method and it has been shown in previousnumerical experiments to be reasonably accurate whencompared to “exact” methods (see for example, Chan, 1986).

The models were designed to investigate properties of edgediffraction from a 3D wedge. Specifically, the introduction ofthe boundary ray for different geometries and ray types wereconsidered. This ray is such that it defines the illuminated andshadow zones for the reflected arrival. The diffracted ray existsin both regions while the reflected ray exists only in the illumi-nated region. The formulae for the edge diffracted arrival wereobtained from other works cited in the Introduction. Thesmooth transition of the diffracted arrival across the boundaryray from the illuminated region to the shadow region was takenas an indication that the formulae being used satisfied thisconstraint. Comparison of the modified ART solution for thisproblem has been checked for more complex media using finitedifference and related methods and as such has not beenincluded here. The diffraction coefficient is a function of thedifference of the diffracted and direct travel times. As the directarrival does not exist in the shadow zone it was necessary to

introduce the concept of analytic continua-tion to provide an appropriate value for thisquantity. What has been presented here isnot a definitive source of all theory that isrequired for the introduction of edgediffracted arrivals into synthetic traces forcomplex structures, but rather a simpleintroduction to the topic to provide a basisfor the numerical implementation of diffrac-tion theory.

ACKNOWLEDGEMENTS

The support of the sponsors of CREWES isduly noted. The first author also receivesassistance from NSERC through Operatingand Discovery Grants held by Professors E.S.Krebes (7991-2006), L.R. Lines (216825-2008)and G. F. Margrave (217032-2008).

REFERENCESAbramowitz, M. and I.A. Stegun, 1980, Handbookof mathematical functions: U.S. GovernmentPrinting Office, Washington.

Bakker. P.M., 1990, Theory of edge diffraction interms of dynamic ray tracing: Geophysical JournalInternational, 102, 177-189.

Červený, V., 2001, Seismic Ray Theory: CambridgeUniversity Press, Cambridge.

Červený, V. and R. Ravindra, 1970, Theory ofseismic head waves: University of Toronto Press,Toronto.

Červený, V. and F. Hron, 1980, The ray seriesmethod and dynamic ray tracing for three dimen-sional inhomogeneous media: Bulletin of theSeismological Society of America, 70, 47-77.

Chan, G.H., 1986, Seismic diffraction fromwedges: Ph.D. Thesis, University of Alberta,Edmonton.

Gallop, J.B. and F. Hron, 1997, Diffractions andboundary conditions in asymptotic ray theory:Geophysical Journal International, 133, 413-418.

Hron, F., 1986, Unpublished notes on diffractiontheory for seismic applications.

High frequency edge diffraction theory

Figure 8. Vertical (a) and horizontal (b) components of displacement for the combination of reflectedand diffracted arrivals, for Models I and II with receivers located at the model surface. The displace-ment of the two models is just the additive sum of the constituent parts.

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Hron, F. and E.R. Kanasewich, 1971, Synthetic seismograms for deepseismic sounding studies: Bulletin of the Seismological Society ofAmerica, 61, 1169-1200.

Hron, F. and J.D. Covey, 1988: Theory of seismic waves diffracted by thelinear edges of seismic interfaces: Technical Report F88-E-13, AMOCOResearch Centre, Tulsa.

Hron, F. and G.H. Chan, 1995, Tutorial on the numerical modeling ofedge diffracted eaves by the ray method: Studia Geophysica etGeodætica, 39, 103-137.

Gautschi, W., 1979a, A computational procedure for the incompletegamma function: ACM Transactions on Mathematical Software, 5,466-481.

Gautschi, W., 1979b, Algorithm 542: Incomplete gamma functions:ACM Transactions on Mathematical Software, 5, 482-489.

Klem-Musatov, K.D., 1980, Theory of edge waves and their use in seis-mology: NAUKA (in Russian).

Klem-Musatov, K.D., 1995, Theory of edge waves and their use in seis-mology: SEG Publications, F. Hron and L.R. Lines, Editors.

Klem-Musatov, K.D., A.M. Aizenberg, J. Pajchel and H.B. Helle, 2007,Edge and tip diffractions: theory and applications in seismicprospecting: SEG Publications, L.R. Lines and P.F. Daley, Editors.

Mikhailenko, B.G., 1985, Numerical experiment in seismic investiga-tion: Journal of Geophysics, 58, 101-124.

Mikhailenko, B.G., 1988, Seismic fields in complex media: Academy ofSciences of the U.S.S. R., Siberian Division, (in Russian).

Table 1. Elastic parameters for Models I and II.

P Velocity (km/s) S Velocity (km/s) Density (gm/cm3)

Wedges I & II 2.50/1.60 1.44/0.92 2.20/1.50

Halfspace 2.00 1.15 1.80

P.F. Daley and E.S. Krebes

APPENDIX A

Assume a point source of compressional (P) waves located atthe origin of an isotropic 3 dimensional Cartesian half space inwhich there is embedded a wedge of infinite dimension in they direction (Figure 3). At a time after an impulsive excitation ofthe point source the direct wavefront will have progressed todefine the wavefront surface at tG. Wavefronts resulting fromreflection from the top of the wedge and transmission throughit have not been considered. However, the diffracted wave-front from the impinging of the wavefront originating at theorigin on point D is included. This diffracted wavefrontdefined by tD will be assumed to be propagating in the half-space. The ray associated with the point source at the originwhich is such that it passes some small distance e : e Æ 0 fromthe point D will be called the boundary ray and denoted RB.This ray separates the illuminated zone (WI) from the shadow(WS) zone for the direct wavefront which originates from apoint source at some arbitrary origin, O.

In the shadow zone the direct geometrical arrival does notexist even though its travel time in this region is required todetermine the argument of the diffraction coefficient, W(z),where

z = eip/4 w/2 w (A.1)

and w is defined (tentatively) by the relation

. (A.2)

In the shadow region the travel time of the geometrical arrivalto the point Mmust be determined by analytic continuation oftG(M0) from the shadow/illuminated boundary into theshadow region. As ST is the tangent plane to tG(M0) on the

boundary ray RB it may be interpreted as the local representa-tion of tG(M0) there, which is to say that from the view point ofseismic energy partitioning due to encounters of the ray withinterfaces it is identical to a seismic plane wave at that point.From a mathematical point of view the analytic continuationof the travel time along the plane representation of the wavefront ST requires that the travel time is the same along theplane wave front as it is at M0. The same argument may beused to infer that the amplitude along the plane wave front STmust also be the same as its value at M0.

Using the coordinate system (r,y) defined in the text which iscentered at the diffraction point D, assuming that all rays areconstrained to lie in the (x,z) plane of the (x,y,z) Cartesiansystem initially assumed here, the travel time of the directgeometrical arrival on the seismic boundary ray , is and as aresult of the preceding argument

t G(M)=t G(M 0) where M∈ST . (A.3)

If t1 is the time taken for the direct geometrical arrival to travelfrom the point source to the edge diffraction point D, then itmay be seen from Figure 3 that

tD(M)=t 1+rM/a (A.4)

and

t G(M)=t 1+rM cosϕM/a (A.5)

where a was defined in the text as the wave velocity in thehalf space. Substituting equations (A.4) and (A.5) into (A.2) thequantity z expressed in terms of w may be obtained from

(A.6)

w2 = [ 2w (tD(M)–tG(M))]p

w2 = [ 2w (tD(M)–tG(M))] = [(2w)(rM)(1–cosϕM)].p p a