GEOPHYSICS, VOL. 63, NO. 4 (JULY-AUGUST 1998); P. 1200–1209, 17 FIGS.

Tutorial

Smiles and frowns in migration/velocity analysis

Jinming Zhu∗, Larry Lines‡, and Sam Gray∗∗

ABSTRACT

Reliable seismic depth migrations require an accurateinput velocity model. Inaccurate velocity estimates willdistort point diffractors into smiles or frowns on a depthsection. For both poststack and prestack migrated sec-tions, high velocities cause deep smiles while low veloci-ties cause shallow frowns on migrated gathers. However,for prestack images in the offset domain, high veloci-ties cause deep frowns while low velocities cause shallowsmiles. If the velocity is correct, there will be no varia-tion in the depth migration as a function of offset and nosmiles or frowns in the offset domain. We explain migra-tion responses both mathematically and graphically andthereby provide the basis for depth migration velocityanalysis.

INTRODUCTION

Depth migration is the process of positioning reflected seis-mic arrivals at their proper subsurface locations. The accuratedepth migration of seismic reflectors requires accurate velocityestimations. Inaccurate velocity estimates will cause moveoutartifacts such as smiles and frowns to appear on depth-migratedimages. The elimination of these moveout features by adjust-ing seismic velocities allows depth migration to be used as apowerful velocity analysis tool. It can be argued on the basis ofmodel studies (Lines et al., 1993) and with real data examples(Whitmore and Garing, 1993) that iterative prestack depth mi-gration provides a very general velocity analysis method forstructurally complex media.

In this paper, we examine migration moveout for both post-stack and prestack depth migration of point diffractors, bothmathematically and geometrically. Point diffractors are usedfor their simplicity and because reflected wavefields can be

Published on Geophysics Online May 6, 1998. Manuscript received by the Editor November 26, 1997; revised manuscript received November 26, 1997.∗Formerly Memorial University, Department of Earth Sciences, St. John’s, Newfoundland, Canada A1B 3X5; presently GX Technology Corporation,5847 San Felipe, Suite 3500, Houston, Texas 77057. E-mail: jzhu@gxt.com.‡Formerly Memorial University, Department of Earth Sciences, St. John’s, Newfoundland, Canada A1B 3X5; presently The University of Calgary,Department of Geology and Geophysics, 2500 University Drive, N.W., Calgary, Alberta, Canada T2N 1N4. E-mail: lines@geo.ucalgary.ca.∗∗Amoco Canada Petroleum Company, P.O. Box 200, Station M, Calgary, Alberta, Canada T2P 2H8. E-mail: shgray@amoco.com.c© 1998 Society of Exploration Geophysicists. All rights reserved.

considered a superposition of point diffractor arrivals, accord-ing to Huygens’s principle. We show the effects of velocity onthe depth migration of point diffraction arrivals so that onecan establish criteria for velocity analysis. We do this for bothzero-offset (poststack) and nonzero-offset (prestack) record-ing configurations.

SMILES AND FROWNS IN POSTSTACK MIGRATION

In understanding the case of poststack migration, considertwo point diffractors in the middle of a uniform medium withvelocity v = 4000 m/s. The depths of the diffractors are 600 and800 m, respectively. We now examine the migration of a pointdiffraction seismogram from this model recorded by coincidentsource-receiver positions. The zero-offset section is shown inFigure 1. The images obtained by migrating the input diffrac-tion hyperbola in Figure 1 are compared for the cases wherethe velocity is too low (Figure 2a), exactly correct (Figure 2b),and too high (Figure 2c). If the velocity is too low, the poststackmigration images are shallow frowns. If the velocity is correct,the images are concentrated blobs at the proper depth. If themigration velocity is high, the images are deep smiles. Theseare described by data examples in Yilmaz (1987). The shal-low frowns are described as undermigration, while the deepersmiles are described as overmigration.

The phenomenon of undermigrations and overmigrationsfor poststack data can be described by considering the caseof coincident source-receiver positions. For simplicity, we con-sider the case of a point diffractor for a coincident source-receiver (or zero-offset) recording configuration. This record-ing configuration, as shown in Figure 3, is the situation whichwe simulate with stacked data.

Suppose there is a point diffractor P on the vertical axis ofa Cartesian system (x, z). The diffractor at depth z is verticallybelow the origin. Consider a coincident source-receiver pairS/R on the surface of the earth with lateral offset x from the

1200

Smiles and Frowns in Migration 1201

origin. Let the one-way vertical traveltime from the surface toP be t0; let the total traveltime from S/R to P be t ; and let themedium velocity be v. The one-way traveltime for an arrivaltraveling from S/R to P is given by using the Pythagoreantheorem and the distance relationship between the sides of theright triangle in Figure 3. That is,

x2 + v2t20 = v2t2 (1)

or

t =√

t20 +

x2

v2. (2)

If we wish to use the two-way reflection times (2t and 2t0),which are the arrival times for a reflection experiment, we canconsider the same distance relationship by using the medium’shalf-velocity, v/2. Then the model in Figure 3 is essentially theexploding reflector model of Loewenthal et al. (1976). In thismodel, reflection seismograms containing arrivals with two-way propagation are described by one-way propagation of ex-plosions that propagate to the surface with half the velocity ofthe medium. Except for a few pathological cases (Claerbout,1985; Yilmaz, 1987), this exploding reflector model can be con-sidered a suitable model for poststack data. Figure 4 is a recordof such an explosion experiment from the diffractor model.

Let us consider the migration of a diffraction arrival of atrace at offset x from the origin. Suppose the arrival time is t .To migrate this arrival, we use the principle of aplanatic sur-faces (Sheriff, 1991). An aplanatic surface defines the locus ofpossible reflection depth points that could exist for a given one-way traveltime, t , and a migration velocity, vm. For a coincidentsource-receiver position and a constant migration velocity, theaplanatic surface is defined by the following equation of a circlegiving all possible locations of reflection points:

(x − xm)2 + z2m = v2

mt2, (3)

FIG. 1. Synthetic zero-offset seismograms of two point diffrac-tors. The diffractors are at depths of 600 and 800 m, respectively,in the middle of the model.

where (xm, zm) defines the migrated domain. If we substitutethe expression for t from equation (2), we obtain

z2m = v2

m

(t20 +

x2

v2

)− (xm − x)2. (4)

Note that the observed reflection time in equation (2) isexpressed in terms of the actual velocity, v, which we generallydo not know but hope to determine, whereas the estimated

FIG. 2. Poststack migration of a model with two point diffrac-tors. (a) A smaller migration velocity produces shallow frowns.(b) The true velocity collapses the hyperbolae to concentratedblobs. (c) A larger migration velocity results in deep smiles.

FIG.3. The zero-offset (poststack) geometry for a point diffrac-tor at P.

1202 Zhu et al.

velocity used in migration is vm. Migration of the record inFigure 4 is essentially the superposition of all these aplanaticsurfaces, one for each source-receiver pair. We hope to findgeometric criteria corresponding to cases where vm<v, vm= v,vm>v, which will allow us to find cases where the velocity iscorrect.

In migration, we are dealing with the superposition of wave-field amplitudes that are distributed along aplanatic surfaces.For the poststack situation, this is the wavefront superpositionmethod described by Robinson and Treitel (1980). The mi-grated image represents a summation of those amplitudes thatare in phase such that they will interfere constructively. Math-ematically, this constructive interference can be described bythe method of stationary phase (Scales, 1995). The basic ideaof stationary phase is that highly oscillatory time sequencestend to cancel upon migration except where the phase functionhas a stationary point. This stationary point occurs where thefirst derivative of the phase function equals zero. In migration,the phase function is the phase difference between the migra-tion curve and the diffraction signatures of the data; locationswhere the migration curve is tangent to diffraction or reflectionevents in the data give the stationary phase contributions to themigration.

In migration, an alternative kinematic description of the am-plitude summation along aplanatic surfaces is given by the en-velope curves for the aplanatic surfaces. If a set of aplanaticcurves is described by F(x, z, t)= 0, then its envelope is de-fined by curves satisfying F(x, z, t)= 0 and d F/dx= 0.

For doing this, we need to find the envelope of the aplanaticcurves defined by equation (4), as shown by Maeland (1989).Essentially the envelope is the solution of the system consist-ing of equation (4) and its tangent curve (Sneddon, 1957). Thetangent curve of equation (4) is given by differentiating equa-tion (4) with respect to the source-receiver position x,

v2m

v2x − (x − xm) = 0. (5)

FIG. 4. Geometrical expression of a zero-offset section for apoint diffractor.

Equivalently, if we define β = vm/v,

(1− β2)x = xm. (6)

Now let’s consider the case of vm 6= v. For this case, we canhave

x = xm

1− β2. (7)

Substituting equation (7) into equation (4) leads to

z2m

v2mt2

0

− x2m(

v2 − v2m

)t20

= 1. (8)

Now let’s consider three specific cases.

1) Migration velocity smaller than the medium velocity, i.e.,vm < v.—In this case, equation (8) represents a hyperbola. Itsvertex is on the depth axis with coordinate of vmt0 below the ori-gin. The apex is thus above the diffractor position as vmt0 <vt0.Obviously, it opens downward because the center of the hy-perbola is just on the coordinate origin. Thus, the poststackmigration of the diffraction curve will be a shallow hyperbolicfrown when the migration velocity is too small, so that un-dermigration partially collapses the original hyperbola into asecond, better focused hyperbola. This observation forms thebasis of residual migration and cascaded migration.

In fact, the formation of such shallow frowns can also bewell illustrated geometrically. Figure 5 illustrates this migrationcase. The cyan circle is the diffractor point. The red curve is theoriginal record we simulated for such an explosion. The bluecurves are the migration aplanatics that finally superpose toform the envelope of another hyperbola (in green) that is later-ally much narrowed. This essentially indicates undermigration.

2) Migration velocity greater than the medium velocity, i.e.,vm > v.—In this case, equation (8) can effectively be reformu-lated as

z2m

v2mt2

0

+ x2m(

v2m − v2

)t20

= 1. (9)

This is the equation of a semiellipse with the center at the coor-dinate origin. The vertex on the depth z-axis is still vmt0 (>vt0),which is now above the diffractor point P. The mouth of theellipse is toward the negative axis of depth, as we are onlyinterested in the positive z-values. Therefore, the migration ofthe diffraction curve in Figure 4 will be deep elliptic smiles onthe migrated section when the migration velocity is too large.

Such a migration procedure is also geometrically repre-sented in Figure 6. Most of the curves in Figure 6 are just thesame as in Figure 5. Figure 6 is an excellent example showingthat when the migration velocity is too high, the superpositionof the individual aplanatics forms an elliptic smile (in green) inthe migrated section.

3) Migration velocity equal to the medium velocity, i.e.,vm = v.—In this case, we have to start from equations (4) and (6)because equation (8) is no longer valid. When vm= v, equa-tion (6) gives xm= 0. Substituting this value of xm into equa-tion (4), we obtain zm= vt0= z. This implies that when the

Smiles and Frowns in Migration 1203

FIG. 5. Poststack migration of the diffractor model when a ve-locity smaller (3 km/s) than the true velocity (4 km/s) is used.The cyan circle represents the diffractor position. The red isthe scaled recorded hyperbolic diffraction curve. The super-position of all the wavefronts in blue results in a shrunkenhyperbola in green in the final migration section.

FIG. 6. Poststack migration of the diffractor model when a ve-locity larger (5 km/s) than the true velocity (4 km/s) is used.The superposition of all the wavefronts in blue results in anelliptic smile in green in the final migration section.

FIG. 7. Poststack migration of the diffractor model when thetrue velocity (4 km/s) is used for migration. The green cross isthe result of the constructive superposition of all the wavefrontsin blue. This indicates a perfect recovery of the diffractor pointin the migration.

migration velocity is correct, the superposition of the migra-tion aplanatics finally collapses the recorded diffractions tothe correct spatial position, (xm, zm)= (0, z). Parallel to suchmathematical development, Figure 7 expresses geometricallythe procedure of reconstructing the true diffraction point bysimple superposition of aplanatic curves.

SMILES AND FROWNS IN THE PRESTACKMIGRATED SECTION

In this section, we will show that the smiles and frowns dis-cussed above are common to prestack migrated stacked sec-tions. They can also be verified easily, both mathematicallyand geometrically. Since the mathematical development is verysimilar to that in the poststack migration case, we will focus onthe geometrical aspects.

Let us first consider the same point diffractor P on the verti-cal axis of a Cartesian system. Now assume there is a source atSand a receiver at R on the earth’s surface, with x-coordinatesof xs and xr , respectively (see Figure 8). If the medium velocityis v, the total traveltime from the source S to the diffractor Pand back up to the receiver Rcan thus be given by the so-calleddouble square root relationship (Claerbout, 1985),

t = 1v

(√x2

s + z2 +√

x2r + z2

). (10)

Migration of a single diffraction arrival at R attributable to thesource at Scan still be described by the concept of aplanatic sur-faces. The aplanatic surface for the source-receiver pair shownin Figure 8, analogous to equation (3), can be represented by√

(xs − xm)2 + z2m +

√(xr − xm)2 + z2

m = vmt. (11)

Substituting equation (10) into the above, we have

F(xm, zm; xs, xr ) = β[√x2s + z2 +

√x2

r + z2]

− [√(xs − xm)2 + z2m +

√(xr − xm)2 + z2

m

] = 0. (12)

This is essentially an ellipse in the migrated space (xm, zm) forthis special case of constant velocity.

The final migration of all these arrivals recorded by differentreceivers from many sources is the envelope of the individualellipses. The envelope of these ellipses is essentially the solutionof equation (12) and its derivative equations (Sneddon, 1957),

Fxs(xm, zm; xs, xr ) = 0 (13)

FIG. 8. Prestack geometry for a point diffractor at P.

1204 Zhu et al.

and

Fxr (xm, zm; xs, xr ) = 0. (14)

Following the same procedures as in the last section, we candevelop the same conclusions as those in the poststack migra-tion, although the mathematical derivations will be much morecomplicated. Figure 9 schematically shows four shot gathersfrom the diffraction model in Figure 8. Figure 10 geometricallysummarizes the migration procedure when a velocity that istoo small is used for prestack migration. Just as expected, thesuperposition of all individual migration ellipses results in ashrunken hyperbola (in green). Notice, however, that not allof the ellipses are tangent to the envelope. Figure 11 illustratesthat the correct velocity allows the migration to reconstruct thepoint diffractor model almost perfectly as long as the recordingcoverage is sufficiently wide and dense. In contrast to Figure 11,Figure 12 demonstrates that when a velocity that is too large isused for prestack migration (vm>v), the superposition of in-dividual elliptical aplanatics (in blue) results in another ellipse(in green) in the final migrated section.

Figure 13 shows a numerical example of prestack depth mi-grations that correspond to cases of velocity lower than, equalto, and higher than the true medium velocity. A low migrationvelocity (vm= 3000 m/s) results in frowning migrated image (a),caused by an insufficient collapse of diffractions. In contrast,using a velocity that is too high (vm= 5000 m/s) in migrationresults in smiling images (b). In either of these two cases, themigrated images of the diffractors are not properly focused.A smaller velocity results in image shallowing, while a largervelocity leads to image deepening. Only when the true velocityis used will the diffractions completely collapse to their truepositions (c). Thus, the final migrated section exhibits smileand frown patterns whether the migration is performed on

FIG. 9. Some representative prestack recording events froma diffractor point model. Each event corresponds to a sourceposition. The diffractor lies at 0.8 km in depth in the middle ofthe model.

FIG. 10. Prestack migration of the diffractor model when a ve-locity smaller (3.3 km/s) than the true velocity (4 km/s) is used.The cyan circle represents the diffractor position. The super-position of all the wavefronts in blue results in a shrunkenhyperbola in green in the final migration section.

FIG. 11. Prestack migration of the diffractor model when thetrue velocity (4 km/s) is used for migration. The green cross isthe result of the constructive superposition of all the wavefrontsin blue. This indicates a perfect recovery of the diffractor pointin the migration.

FIG. 12. Prestack migration of the diffractor model when a ve-locity larger (5 km/s) than the true velocity (4 km/s) is used.The superposition of all the wavefronts in blue results in anelliptic smile in green in the final migration section.

Smiles and Frowns in Migration 1205

poststack or prestack seismic data whenever errors exist in themigration velocity.

These undermigration and overmigration features are gener-ally observed in both poststack and prestack migration sections.However, smiles and frowns are often very difficult to observeon the migration sections because individual diffractors are of-ten just an element of the reflecting interface. Generally, themigration moveout effects on common image gathers (CIGs)from prestack migrations are quite pronounced and allow foreffective velocity analysis. Interestingly enough, the moveoutcharacteristics on CIGs are very different from those in themigrated stacked sections.

PRESTACK DEPTH MIGRATION MOVEOUT

Many recent studies demonstrate that prestack depth migra-tion can accurately image reflections and diffractions withoutdip restriction if a reasonable approximation of the velocityfield is available (Versteeg, 1993; Lines et al., 1993). Neverthe-less, the velocity model is the key component in these migra-tions. Theoretically, there exist several alternative methods forvelocity analysis (Versteeg, 1993; Lines et al., 1993). Here wewill analyze the prestack migration moveout features that arefundamental to the basic theories in interval velocity analysis

FIG. 13. Prestack depth migration of a model with two pointdiffractors. A migration velocity smaller than, equal to, andlarger than the true velocity is used for images in (a), (b),and (c).

utilizing CIGs (Al-Yahya, 1989). Our analysis, however, willno longer depend on the assumption of a layered-earth model.

Consider the general subsurface structure and recording ge-ometry as shown in Figure 14. P denotes the arbitrary scatter-ing point in the earth’s interior. S and R are a source-receiverpair illuminating P. D is the surface image of P. Assumingthat the average velocity above P is v̄ and that the diffractionreceived at R (because of a source wavelet from S and thendiffracted at P) never travels beneath P, then its arrival timecan be expressed as

t = 1v̄

(SP+ RP). (15)

When an incorrect average velocity v̄m is used for migration,the diffraction signal received from P will be migrated to anincorrect point P′. P′ generally has both vertical and lateraldisplacements from the true position P. We denote these dis-placements with1x= P′Q and1z= Q P. In this case, the trav-eltime will be

t = 1v̄m

[SP′ + P′R] = 1v̄m

[(SP− P1 P)+ (RP− P2 P)]

= 1v̄m

[(SP− Q1 P)+ (RP− Q2 P)]

+ 1v̄m

(P2 Q2 − P1 Q1). (16)

FIG. 14. Migration depth/velocity relationship diagram in ageneral subsurface structure. (a) A wrong velocity migratesthe reflection to a position P′, which has a lateral displacement1x in addition to a vertical displacement 1z. (b) An enlargedview of the lower part of (a).

1206 Zhu et al.

From Figure 14b, the following relationship holds in the tri-angle 1P′QO:

sinαs

P′O= sinαr

QO= sin θ

1x. (17)

Thus, we have

P2 Q2 − P1 Q1 ' QO− P′O

= 1x

sin θsinαr − 1x

sin θsinαs

= 1x

sin θ(sinαr − sinαs)

= 1x

cos(θ/2)sin

αr − αs

2. (18)

In the derivation of the last equation above, we have used therelationships αr +αs+θ = 180◦, sin θ = 2 sin(θ/2) cos(θ/2), andsinαr − sinαs= 2 cos[(αr +αs)/2] sin[(αr −αs)/2]. Therefore,the distance part of the last term of equation (16) would bean order smaller than 1x as long as |sin(αs − αr )/2)| < 0.10and θ is not close to 180◦. The latter condition generally holds,asαs andαr would not be zero for most cases. The first conditionis equivalent to |αs − αr|< 12◦, i.e., the difference between thetwo illuminating angles being less than 12◦. Since |αs−αr| = 2α,whereα is the structural dip at P, the above inequality thus onlyholds for structures of gentle dip. In such cases, equation (16)can be properly approximated by

t = 1v̄m

[(SP− Q1 P)+ (RP− Q2 P)]. (19)

This equation is equivalent physically to the assumption thatthe lateral displacement 1x is negligible compared to the ver-tical one. Eliminating t from equations (15) and (19), we obtain

(1− β)(SP+ RP) = Q1 P + Q2 P, (20)

where β = v̄m/v̄. From Figure 14, the following general rela-tions hold:

SP= z/cosαs; Q1 P = −1zcosαs;RP= z/cosαr ; Q2 P = −1zcosαr .

Substituting these relations into equation (20) leads to

1z= (β − 1)z

cosαs cosαr. (21)

In the case of a zero-offset source-receiver pair just at D, αs =αr = 0, we obtain

1z= (β − 1)z. (22)

This implies the migration depth z will be shallower than thetrue depth z if a velocity smaller than the true velocity (v̄m < v̄)is used for migration, while it will be deeper if a higher velocity(v̄m> v̄) is used. Only whenβ = 1, i.e., the true medium velocityis used for migration, will the diffractor be located properly. Bydenoting 1z0 = (β − 1)z, equation (21) can be rewritten as

1z(αs, αr ) = 1z0

cosαs cosαr. (23)

Now let us consider the following three categories.

1) Migration velocity less than the true velocity (v̄m < v̄).—In this case, β < 1, 1z0 < 0, and 1z(αs, αr ) < 1z0. Generallythe following relation,

1z(αs + ε1, αr + ε2) < 1z(αs, αr ), (24)

also holds for any αs, αr and small nonnegative values of ε1, ε2.This relation indicates the migration image of P will form asmile that curves upward on a CIG, which is a display of mi-gration traces versus the source-receiver offset correspondingto a fixed surface point.

2) Migration velocity greater than the true velocity (v̄m >v̄).—In this case, β > 1, 1z0 > 0, and 1z(αs, αr )>1z0. Similarto equation (24), we have

1z(αs + ε1, αr + ε2) > 1z(αs, αr ). (25)

This relation indicates the migration image of P forms a frownthat curves downward on a CIG.

3) Migration velocity equal to the true velocity (v̄m = v̄).—Inthis special case, β = 1, 1z0 = 0. Thus

1z(αs, αr ) = 0 (26)

for any source-receiver pair. This simply means that when thetrue velocity is used for migration (v̄m = v̄), the migration im-ages of the diffractor point P will be at the exact depth, regard-less of source-receiver offset. So, its images form a horizontalsegment on the CIG displays.

To consider prestack migration velocity analysis in terms ofoffset and common midpoints (CMPs), consider again Figure 8for a point diffractor at (0, z). The midpoint can be denoted byX = (xr + xs)/2 and the offset denoted by 2h so that xs= X−hand xr = X + h. Equation (10) gives the total traveltime fora particular point diffractor, but it can be embedded into anequation for a correct migration ellipse:

t1 = t1(X) = 1v

(√

(X − h− x)2 + z2

+√

(X + h− x)2 + z2). (27)

In terms of migration velocity and migration coordinates, wecan also write the traveltime expression as

t2 = t2(X) = 1vm

(√(X − h− xm)2 + z2

m

+√

(X + h− xm)2 + z2m

). (28)

When vm= v, the migration ellipse (28) is identical to the el-lipse (27) for the true velocity. If the velocity analysis locationxm coincides with the diffractor location x= 0, we then havezm= z for all h. That is, the migration depth equals the correctdepth of the diffractor regardless of the offset value when thevelocity is correct. When the velocity is not correct (vm 6= v), we

Smiles and Frowns in Migration 1207

need to evaluate the envelope of migration ellipses for all val-ues of the midpoints X. That is, we need to set the derivativesof t1(X) equal to the derivatives of t2(X) for all X. If, again, weperform the velocity analysis at the actual diffractor location,then these derivatives give, respectively, the slopes of the cor-rect and the incorrect migration ellipses at xm= x = 0. Theseslopes are different unless both ellipses are flat at the analysislocation, i.e., unless X= 0 also. So we set X= xm= x= 0 intothe expressions for t1(X) and t2(X) to obtain

h2 + z2

v2= h2 + z2

m

v2m

(29)

or

z2m = (β2 − 1)x2 + β2z2, (30)

where β = vm/v.In the case of vm<v, we have β < 1. Equation (30) can be

rewritten as

z2m + (1− β2)x2 = β2z2, (31)

which is an ellipse. So this is a special case of equation (24).If vm>v, then β > 1 and equation (30) is essentially the hy-

perbolic equation

z2m − (β2 − 1)x2 = β2z2. (32)

This is a hyperbolic frown in the half-plane of positive depth.If vm= v, then β = 1 and equation (30) simplifies to zm = z.

This represents a horizontal line segment. Only in this specialcase of using the correct velocity for migration will the finalstack of the migrated common reflecting point (CRP) gatherreach its highest energy in the migration section.

The above procedure of migrating a CRP gather also can berepresented geometrically. Figure 15 shows a CRP gather fromthe same diffractor model of Figure 8. This gather is essentiallyequivalent to that of CMP gather in this special case of a singlediffractor point in a constant-velocity medium. Figure 16 shows

FIG. 15. CRP gather from a diffractor point model. The diffrac-tor lies at 0.8 km depth in the middle of the model.

the migrated CRP gathers for this diffractor model correspond-ing to different migration velocities. Because the migrated CRPgather is just the same as the so-called CIG gathers, Figure 16equivalently illustrates the CIGs corresponding to different mi-gration velocities. The recorded hyperbola in the CRP gatherwill appear as elliptical smiles when a smaller velocity is usedfor migration, while a hyperbolic frown appears when a largervelocity is used for migration. If the correct velocity is used,then the CIG gather will show a horizontally aligned segment.

Therefore, the velocity error in migration is very well ex-pressed on CIGs. We can use the same point diffractor modelto illustrate these theoretical predictions. Eighty-one shot pro-files are theoretically simulated, with each record consistingof 60 traces. Figure 17 shows the CIGs for surface positionx = 1.0 km when different velocities are used in migration. Itclearly demonstrates that only when the true velocity is used

FIG. 16. CIGs when different migration velocities are used.Migration velocity is 3 (a), 4 (b), and 5 km/s (c), respectively.The circle in cyan represents the diffractor position. The red isthe scaled CRP gather.

1208 Zhu et al.

will the migration images of diffractors be independent of thesource-receiver offset. When the velocity is lower than the truevelocity (vm= 3000 m/s), the diffractor images form smiles at adepth shallower than the true depth. This is total in agreementwith equations (24) and (31). In contrast, when the velocity ishigher than the true velocity (vm= 5000 m/s), the diffractors areexpressed as frowns on a CIG at a depth greater than the truedepth. This is just what has been predicted by mathematicalexpressions (25) and (32).

Thus, the migration velocity error is documented well onboth the final migration sections (Figure 13) and the CIGs (Fig-ure 17). Interestingly, the CIGs show shallow smiles for low ve-locities and deep frowns for high velocities (Lines et al., 1993),whereas the final depth migration sections show shallow frownsfor low velocities and deep smiles for high velocities (Yilmaz,1987).

If we take a careful further look at Figures 13 and 17, wesee that the curvatures of the shallow smiles/frowns are gen-erally larger than the deep ones. This indicates that velocityerrors are more pronounced on shallow reflections and thusare easier to correct. This also agrees with the general observa-tion that sufficient offset/depth ratio should be kept to analyzeproperly the velocity errors (Lines, 1993). Luckily, we oftenhave more constraints available on the shallow parts of theearth, such as well logs and geological exposures. We can alsomore effectively use techniques such as first-break tomographyto constrain our near-surface velocity estimation. With respectto the deeper structure, we generally have to accept that it iscoarsely defined and more ambiguous.

Though the diffractor model is oversimplified, it is of vitalsignificance in migration and velocity analysis theory becauseany complicated structures can be considered as a continuumof diffractors. This is especially suitable for moveout analysis ona CIG that corresponds to a single surface point. The smile and

FIG. 17. CIGs for a model with two point diffractors.

frown patterns on CIGs can be effectively used to qualitativelyand quantitatively analyze the migration velocity (Al-Yahya,1989).

CONCLUSIONS

Depth migration is very sensitive to errors in the velocitymodel. Migration moveout features such as smiles and frownshave previously been reported (Yilmaz, 1987; Lines et al.,1993). We have shown both mathematically and geometricallythese smiles and frowns on migration sections and CIGs. Us-ing the simple diffractors model, we demonstrated that aftermigration, either using the stacked data or the prestack shotgathers, the diffractions migrate to shallow frowns when themigration velocity is too small and deep smiles when the ve-locity is too large. Only when the migration velocity is exactlythe medium velocity will both the poststack and prestack mi-gration produce concentrated blobs in the migration section.However, the CIGs produced in the prestack depth migra-tion provide an effective migration velocity analysis domain.Our study starting from the very general subsurface structureshowed that the migration moveout in the CIGs is interestinglydifferent from the patterns in the migration sections. In CIGs, alower velocity produces shallow smiles, while a higher velocityresults in deeper frowns. When the migration velocity is cor-rect, the CIGs present horizontal segments at the exact diffrac-tor depths, which indicate that the migration image of thepoint diffractor is independent of source-receiver offset. Thesemoveout patterns of smiles and frowns can serve as both qual-itative and quantitative criteria for migration velocity analysis.

ACKNOWLEDGMENTS

We thank Jianhua Pan for his technical assistance. The au-thors also thank Sven Treitel for providing us a copy of theMaeland paper.

Smiles and Frowns in Migration 1209

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