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Seismic migration:basic concepts and popular methods

Part 1By IRSHAD R. MUFTIMobil Research nd DevelopmentCorporationDallas, Texas

S ortly after the introduction of the seismic reflectionmethod, it was realized that the various events identi-fied in a seismic section do not, in general, rep resent thetrue locations of th e subsurface structures. The reflec tedenergy reaching the surface propagates in the form ofever-expanding wavefronts. In a nonuniform medium ,the variations in velocity tend to influence and modifythe direction of propagation and are accompanied bymutual interference of wavefron ts. In the zones of suddenchangesalong the geologic nterfaces such as faults, a por-tion of this energy undergoes diffraction. Conseq uently,the record o f events constituting a seismic section repre-sents a distorted im age of the subsurface reflectors, animage which has undergone a complicated proces s offocusing, defocusing, interference and diffraction. Anumerical procedure aimed at correcting for these prop-agation e ffects is known as migration.

The first comprehensive and practically meaningfulalgorithm for doing migration was put forward by J. G.Hagedoorn in 1954. He developed a method of reposi-tioning the various events identified in a time sectionsuch that the transformed section would correctly repre-sent the geometry of the subsurface reflectors. Hage-doorns meth od utilizes the concep ts of ray theory an dyields reasonably goo d results in the ab sence of stronglydipping events or lateral chang es in the sub surface velo-cities. Moreove r, it is quite efficient and can be easily ex-tended to three dimensions.In 1971, Jon Claerbout introduced a new method ofmigration which makes use of an approximate versionof the wave equation. Such an appro ach is directly basedon the distribution of en ergy of the seismic record , andit opens up the possibility of taking into account alltypes of phenomena associated with the propagation ofwave s such as diffraction and interference. T he im por-tance of C laerbouts work was readily recognized ; itfollowed a period of intense effort in research and de-velopment in the area of wave equation migration.

It is not the intention of this summa ry to review andevaluate every article written on this subject. Rather, itshall be restricted to the most common migration algo-rithms in current use. Some basic ideas and assumptionsunderlying seismic migration are a logical place tobegin.

The propagation of seismic energy into the subsurfaceis a very complicated phenomenon which d epends on alarge number of factors. Any attempt to migrate theseismic data observed at the surface that takes into ac-count all thes e factors becom es hopeles sly involved. Inorder to make things tractable, migration is accomplishedby introducing the following simp lifying assump tions:l The ea rth behaves like a fluid medium in whichvariations in density can be ignored. In th at cas e, we canuse the simplest form of the w ave equation for investi-gating the migration problem.9 The real earth can be treated as a two-dimensionalmedium . It is admittedly true that in some situations, in-cluding migration, this assum ption can lead to seriouserrors. On the other hand, the acquisition of field da ta

is usually done along linear profiles. More over, theassum ption of a two-dimensional earth is almost invari-ably inherent at all stag es of seismic data processin g.l Every subsurface reflector is made up of a set ofelementary diffractors. This m eans that the superpo sitionof th e wave fields genera ted by the individual diffractorsis approx imately equivalent to th e energy reflected bythe corresponding subsurface horizon.l A common depth point (C DP) seismic section is theproduc! of upward p ropagating waves only. In the pro-cessof stacking the observed seismic data, the m ultiplestend to cancel. A key assumption is that a seismic sec-

tion is comp letely free from multiples. Under these con-ditions, the seismic sourceswhich are actually located ator near the surface of the ground can be replaced by thesubsurface reflectors which explode simultaneously atan initial time t = 0. The energy emanating from thereflectors travels only in the upward direction withoutgenerating any multiples on its way to the surface. Thissimplification reduc es the traveltime by one-half. On aseismic section, it amou nts to reducing the value of thetime sampling interval by one-half.The concept of exploding reflectors is a rather crudeway of explaining things and it seem s o have no counter-part in other disciplines; however, it works reasonably

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well for m igrating the seismic sections.Befo re going into the details of migration, it would beuseful to examine how some simple structures appear ona seismic section.T e simplest example o f a seismic image is due to apoint diffractor. But since a point d iffractor can be usedas a building block to construct a reflector o f any arbi-trary geometry, this example is also the most important.Figure 1 shows a point diffractor P and a set of geo-phones G,, . . . . G5 located along the surface. The energyemanating from P prop agates upwards in the form ofan expanding wavefron t. Three different positions ofthe wavefront corresponding to times t,, t,, and t2 areindicated as partially drawn concentric circles. In ac cor-dance with th e well known Ferma ts principle, the trans-fer of energy from P to the various geophones will takeplace along a set of linear paths or rays as indicated inFigure 1. Let the location of the middle geophone G,correspond to the origin of coordinates (X = 0, z = 0)with positive t-axis pointed downward; z denotes thedepth of the point diffractor and x represents the hori-

zontal position of any of the geophones from the origin.The time (usually referred to as arrival time) needed forthe energy to travel from the point P to the various geo-phones is given byI = distance from the point diffractor to the geophone

velocity of the mediumwhe re the numerator of the fraction is equal to (9 + z):.If, using this relationship, t is plotted as a function ofx for different geop hone s, the resulting curve will ha vethe form of a hyperbola. Let us use this relation to com-pute the arrival times of the wavefron t to the variousgeophones by setting z = 2,000 m and velocity = 4,000m/set. The resu lts are shown in Figure 2; it representsthe seismic image of the point diffractor.It is obvious that as we increase the depth of the pointdiffractor, it will ta ke the wavefron t longer to reach thesurface, losing more and more of its curvature during itsupward path. A flatter front implies that the arrivaltimes to the various geoph ones will differ more slowlyand the corresponding seismic event will appear as abroader hyperbola., By the same argument, the shal-lower the depth of the diffractor, the narrower the eventbecomes.F.gure 3 shows a horizontal reflector approximated bya set of equidistant p oint diffractors PI, . . . P,. The cor-responding s eismic section consists of five iden tical bu tlaterally d isplaced hyperb olas. It is obvious that a pro perseismic representation 6f the reflector requires a muchlarger number of point diffractors. In that case, thevarious hyperbo las lose their identity due to destructiveinterference of the corresponding wavefronts and the re-sulting seism ic section is reduced to a co ntinuous line orthe envelope running tangentially along the apexesof thehyperbolas. But note that as we approach either end ofthe reflector, the effect of interference gradually re-duces. Consequently, the seismic image o f the reflectoris identical to the shape of the geologic model except fora gradual tapering of energy along the hyperbolic arcs.

Figure 1. Various stagesof evolution of an expanding wave-front originating at the point P.

00 7t

201 1

SEISMIC TRACES3

3.000 - 2.000 -1000 0 1.HORIZONTAL DISTANCE (M)

Figure 2. Se ismic mage of a point diffractor located at a depthof 2 ,000 m in a medium of velocity 4,000 m/set.

PROFILE AXIS G, G2 G, G, GS0 -.. _ - -_

2.0 I0 1 2 ; 4 5 6 7 8HORIZONTAL DISTANCE (KM)

Figure 3. Approximate representationof horizontal linear re-flector and the correspondingseismic mage.

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Figure 4 shows a reflector PIP2 dipping at an anglea = 30 degrees. We shall employ the above mentionedproce dure to construct its seismic section. Since it is alinear reflector, it would suffice to compute only twohyperbolas associated with the end points P, and P ,.Note that the hyperbola associated with the point P, isbroader than that for P1 due to differences in depth.The line P,P, which meets the two hyperbolas tangen-tially at P, and P 2I s the required envelope and repre-sents the image of the reflector.Most of the energy leaving the reflector P,P, will prop a-gate to the surface along the direction n ormal to the lineP,P,. Thus, the energy arriving at the geop hone G,willbe mostly due to the point source P, and will reach thesurface at the time which corresponds to the two-waytravel along the raypath between PI and G , Since on atime section, the traces are plotted at right angles to thedirection of the profile, the event assoc iated with P, willappear on the time trace of the geophone G,su ch thatPIG, = P,G, A simple way to determine P,s to draw acircular arc throug h P , with G,as its center; P,cor-

00 G, GIF f C$ Gzt -0

20s -> ,_ 140 1 2 3 4 5 6 3 8HORIZONTAL DISTANCE (KM)

Figure 4. A dipping linear reflector P, P, and its seismic mageP: P: obtained by ray-theory considerations.respon ds to the intersection of this arc and the verticalline passing through G, By a similar proced ure, we candetermine the image Pi for the point P,.Snc e every other structure can be visualized as madeup of linear segm ents dipping a t different angles, theimportance of the case under consideration cannot beoveremphasized.

Conclusions which can be drawn from this examina-tion of fundamen tal migration assum ptions are:l Every point belonging to a reflector in the geologic

depth mod el acts like a diffractor and gives rise to ahyperbola on the corresponding seismic section. Conse-quently, as long as we ignore multiples, a one-to-onecorrespondence exists between a given set of diffractorsand the corresponding set of hyperbolas. Except inzones of sudden chang es in reflectivity such as faultsand termination of reflectors, the individual hyperb olaslose their identity due to destructive interference.l Every diffractor belonging to the geologic modelrepresents the apex o f its hyperbola in the seismic sec-tion. The seismic mage of such a diffractor obtained bysimple considerations involving rays is one of th e pointsbelonging to this hyperbo la. Thus in Figu re 4, the P,of

the seismic section belongs to the hyperbola associatedwith the point P, of the geologic model.l For any given combination of velocity and depth ,the corresponding hyperbola representsa curve of max-imum convexity. This implies that the two hyperbolasshown in Figure 4 cannot be interchanged. Either ofthem is as convex as it could be for the respective depth.In the case of real data obtained in an area wherereliable velocity control exists, one can identify the dif-fraction patterns by measuring the curve of the hyper-bolic type features. Conversely, one can look for anduse such features for estimating subsurface velocities.l Formation of th e seismic section involves a rolling

down of the geologic depth points along their hyper-bolas. Thus in Figure 4, the point PI rolls down to thepoint P, The distance along which this rolling downtakes place increases with depth.l The seismic section can be regarded as a moreor less distorted and displaced image of the geologicmodel. and usually appears larger than the true geo-logic model. This ap parent inc rease in size amoun ts to a

PROFILE DIRECTION00'

05 :

G IO8Y6 15

I20 1 f

25 i 1 I I

I 14 15 16 17

Pb

Figure 5. A simple example of a synthetic seismicsection.loss of lateral resolution. More over, its displacem ent isapproximately along the downdip direction and takesplace in such a manner that it tends to overlie the object.Consequently, the seismic sectionof a salt dome isbroader than the dome and fully encloses t from above.Similarly, by visualizing a synclinal structure as mad eup of linear re flectors dipping at different angles, it canbe easily explained why its seismic section sometimesgives rise to the familiar bow-tie pattern.

Some of the basic features of the seismic section andhow they differ from the corresponding features in thegeologic model have now been reviewed. The variousmeth ods of transforming the seismic sections into thetrue geom etry of the corresponding geologic structures- the procedure of migration - will now be discussed.L et us use our newly acqu ired skill to compute a syn-thetic seism ic section showin g a dipping reflector under-lying a diffractor - for convenience we shall use thereflector in F igure 4. Th e results are shown in F igure 5.The seismic responseof the planar reflector appears as achain of events along th e dashed line connecting thepoints P, and P i. The seismic response of the diffrac-tor can be identified as a hyperbola with its apex P, on

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SEZSMKMIGRATZON1111)

There3gottabe an easier ownwardcontinuationf source nd eceiver.trace 5. For sub sequen t iscussion, num berof hyper-bolaspassinghrough he variouspointshavebeenaddedas well as somerandom noiseon trace 12 at time 1.95set and on trace 14 at time 1.75 sec.The dashe d yperboliccurvesshown n Figure 5 w ereobtainedby usinga velocity 4,000 m/set. Let us startwith the hyperbola passin g hrough the point P,. Theevent which falls on this curve constitutes he seismicresponse f the diffractor who se true location corre-spon ds o the point P,. Conseq uently, he processofsumm ing he energy associated ith this event and as-signing he sumvalue o the point whichcorrespondsothe apexof the hyperbolawill und o he effect of d iffrac-tion and the seismic esponsewill co llapse o the dif-fractor. But this s exactlywha t migration s expected oaccom plish. his procedure an alsobe used or m igrat-ing the plan ar reflector. We observe hat the curvepass-ing through PI touches he seismic vent of the planarreflector tangentiallyand passeshrough the responseon traces2, 3 and 4. Thus f we sumup the energyalongthis curve and assig n t to the point PI, we are able tomigrate the left end of the reflector from its apparentpos ition P,to the correct pos ition PI. A sim ilar pro-cedureenablesus to m igrate P,to P,. In this way, weare able to migrate the planar reflector from positionP,P2I o the correctpositionPIP,.The ingeniousmethoddescribed bovewas ntroducedby J. G. Hagedoo rn. n the absence f digitalcomputers,he useda large numberof p recomputed urves or dif-ferent velocitiesand depths.When it comes o d ealing

with the real size seismic ections, ucha procedure sextremely laboriousand is not worth the effort - evenat todays prices of oil and g as. Subs equentworkerswere able to automa teHage doorns procedureby mak-ing the assumptionhat every point on the seismic ec-tion can be treated as the apex of a hyperbola. In theabsenceof a reflector, the su m signal results n zeroenergy except for any residualenergy associatedwithnoise.The summationprocessusually yields good results.However, since t is basedon the ray theory approxima-tion, it fails to take into accountsmallchangesn phaseand cangive rise o a variety of co mpu tational o ise.Asan example, he sumsignalassignedo the point P, willalso include he noiseat the lower portion of trace 12,whereas oint P, m ay indicatea weak diffracting regiondue to the noisepresenton trace 14. Similarly, point Pawill derive ts energy rom the planarreflector below t.Theseare somesimpleexamples f the so-calledmigra-tion noise.The ideaof su mm ing p the diffractionenergy ecordedat the surface nd assigningt to the correspondingourceof diffraction is equivalent o deriving the strengthofthe wave field at the location of the source rom thevalueof the field alonga surfacenfluenced y the source.This invokes he possibility f doingmigrationby usingsomeof the well-know n esultsrom the theoryof optics.Considera wave field defined as a function of spaceand time over a closedsurface. Then the value of the

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iFigure 6. A hemispherical surface over which the Kirchhoffsummation is carried out.field a t any po int inside he surfaceand at t ime I = 0can be computedby meansof an expression ssociatedwith the nam e of Kirchhoff (seee.g., B aker and Cop-son, 1950). shallnot go into further detailswhich canbe obtainedelsewhere.t may be mentioned,however,that the resultsobtained in this manner remain unaf-fectedby the shapeof the closedsurfacewhich may bechosen o suit the problemunder nves tigation. or ourpurpos es, good choicewould be as shown n Figure6;it co nsists f a hemisph erical owl whose lat surfacerepresentshe zone of field measurem ents.et u s makethe radiusof the bowl infinitely large. In that case, he

curvedsurfacewou ld be too far away to be influencedby any subsurface eflectors. Since he field measu re-mentsare confinedwithin a finite area, we must ntro-ducean additionalassum ption that the value of thefield outside he surveyed reabecomes egligibly mall.It is not a good assumption, ut we cannotavoid it. Inany case, he effective portion of the closedsurface snow confined to a finite area where he field measure-mentsare available.Keeping n view the conceptof theexploding reflector, if a su bsurfacepoint with coor-dinates x, y, z) b elongs o a reflector, he value of thewave field at such point at time t = 0, will be anoma-louslystrongas compared o the pointswhereno reflec-torsare present. his s an elegant pproacho migrationand in p rinciple, it shou ld ead to superiorresultsascompared o the m igration methodsbasedon straightsummation. In practice, however, the quality ofmigrateddata obtainedby this method s often inferiorto that obtainableby other schem esn current use.Atleastsomeof the problems ssociated ith this methodmay be the directconsequencef too muchcompromisebetween h e theory and the practical imitations. Thetheory implies h at data to be migratedare basedon a3-D seismic urvey. n actualpractice, he field measu re-mentsare usuallyobtainedalong a profile rather thanover an area. This necessitatesurther comprom ise iththe theory. Sometime n the future, when 3-D acquisi-tion of field data has becomea commonpractice, hisapproximation ould be eliminated.Lli

(The concluding part of this article, discussing modern migra-tion techniques, will appear next month.)

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1.00 migration basic concepts

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