Smiles an Frows by Migaration

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G EOP HYSI CS, VOL . 63, NO. 4 (JULY-AU G UST 1998); P. 1200–1209, 17 FIG S.

Tutorial

Smiles and frowns in migration/velocity analysis

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

ABSTRACT

R eliable seismic depth migrations require an a ccurate

input velocity model. Inaccurate velocity estimates will

distort point diffractors into smiles or frow ns on a depth

section. For both poststack and prestack migrated sec-

tions, high velocities cause deep smiles while low veloci-

ties cause shallow frowns on migrated gathers. H owever,

for prestack images in the offset domain, high veloci-

ties cause deep frowns while low velocities cause shallow

smiles. If the velocity is correct, there will be no varia-

tion in the depth migration as a function of o ffset and no

smiles or frow ns in the off set doma in. We explain migra-

tion responses both mathematically and graphically and

thereby provide the basis for d epth migration velocity

analysis.

INTRODUCTION

D epth migration is the process of positioning reflected seis-

mic arrivals at their proper subsurface locat ions. The a ccurate

depth migration of seismic reflectors requires accurate velocity

estimations. Inaccurate velocity estimates will cause moveout

artifactssuch as smilesand frownsto appear on depth-migrated

images. The elimination of these moveout features by adjust-

ing seismic velocities allows depth migration to be used as a

powerful velocity analysis tool. It ca n be argued on the b asis of

model studies (Lines et al., 1993) and with rea l dat a exa mples

(Whitmore and G aring, 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 b oth post-

stack and prestack depth migration o f point diffractors, both

mathematically and geometrically. Point diffractors are used

for their simplicity and because reflected wavefields can be

Published on G eophysicsOnline May 6,1998. Manuscript received by the E ditor November 26, 1997; revised manuscript received November 26, 1997.∗Formerly Memorial U niversity, D epartment of E arth Sciences, St. John’s,Newfo undland, Ca nada A1B 3X5; presently G X Technology Co rporation,5847 San Felipe, Su ite 3500, H oust on, Texa s 77057. E -mail: jzhu@gxt .com .‡Formerly Memorial U niversity, D epartment of E arth Sciences, St. John’s, Newfoundland, C anada A1B 3X5; presently The U niversity of C algary,D epart ment of G eology a nd G eophysics, 2500 U niversity D rive, N.W., Ca lgary, Albert a, C ana da T2N 1N4. E-mail: [email protected].∗∗Amoco C anad a P etroleum Compa ny, P.O. Bo x 200, Station M, C algary, Alberta, Ca nada T2P 2H8. E-mail: [email protected] 1998 Society of Exploration G eophysicists. All rights reserved.

considered a superposition of point diffra ctor arrivals, accord-

ing to H uygens’s principle. We show the effects o f velocity on

the depth migration of point diffraction arrivals so that onecan estab lish criteria fo r velocity a nalysis. We do this for bo th

zero-offset (poststack) and nonzero-offset (prestack) record-

ing configurations.

SMILES AND FROWNSIN POSTSTACK MIGRATION

In understanding the case of poststack migration, consider

two point diffractors in the middle of a uniform medium with

velocity v = 4000m/s. The depths of the diff ract ors are 600 and

800 m, respectively. We now examine the migration of a point

diffraction seismogram from thismo del recorded by coincident

source-receiver positions. The zero-offset section is shown in

Figure 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 t oo high (Figure 2c). If the velocity is too low, the poststack

migration images are shallow frow ns. If the velocity is correct,

the images are concentrated blobs at the proper depth. If the

migration velocity is high, the images are deep smiles. These

are described by data examples in Yilmaz (1987). The shal-

low frowns are described as undermigration, while the deeper

smiles are described as overmigrat ion.

The phenomenon of undermigrations and overmigrations

for poststack data can be described by considering the case

of coincident source-receiver positions. For simplicity, we con-

sider the case of a point diffractor for a coincident source-

receiver (or zero-offset) recording confi guration. 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 of

a Cartesian system ( x, z). The diffra ctor at depth z is vertically

below the origin. Consider a coincident source-receiver pair

S / 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 to

P b e t 0; let the total traveltime from S / R t o P b e t ; and let the

medium velocity be v. The one-way traveltime for an arrival

traveling from S / R to P is given by using the Pythagorean

theorem and the distance relationship betw een the sides of the

right triangle in Figure 3. That is,

x2 + v2t 20 = v 2t 2 (1)

or

t =

t 20 +

x2

v2. (2)

If w e wish to use the two-way reflection times (2t and 2t 0),

which are the arrival times for a reflection experiment, we can

consider the same d istance relationship by using the medium’s

half-velocity, v/2. Then the mo del in Figure 3 is essentially t he

exploding reflector model of Loewenthal et al. (1976). In this

model, 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 of

the medium. Except fo r a few pathological cases (Claerbout,

1985; Yilmaz , 1987), this exploding refl ector m ode l can be co n-sidered a suitable model for poststack data . Figure 4is a record

of such an explosion experiment from the diffractor model.

Let us consider the migration of a diffraction arrival of a

trace at o ffset x from the origin. Suppose the arrival time is t .

To migrate t his arrival, we use the principle of aplana tic sur-

faces (Sheriff, 1991). An aplana tic surface defi nes the locus of

possible reflection depth points that could exist for a given one-

way traveltime, t , and a migration velocity, vm . For a coincident

source-receiver position a nd a constant migra tion velocity, the

aplana tic surface is defi ned by the following equation of a circle

giving all possible locations of reflection po ints:

( x − xm)2 + z2m = v 2

m t 2, (3)

FIG. 1. Synthetic zero-offset seismograms of tw o point diffra c-tors.The diffracto rs are at depthsof 600and 800m, respectively,in the middle of the model.

where ( xm , zm) defi nes the migrated domain. If we substitute

the expression fo r t from eq uation (2), we obta in

z2m = v 2

m

t 20 +

x2

v2

− ( xm − x)2. (4)

Note that the observed reflection time in equation (2) is

expressed in terms of the actual velocity, v, which we generally

do 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 frow ns.(b) The true velocity collapses the hyperbolae to concentratedblobs. (c) A larger migra tion 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 in

Figure 4 is essentially the superposition of all these aplanatic

surfaces, one for each source-receiver pair. We hope to find

geometric criteria corresponding to casesw here vm < v, vm = v,

vm > v, which will allow us to find cases where the velocity is

correct.

In migration, we a re dealing with the superposition of wa ve-

field amplitudes that are distributed along aplanatic surfaces.

For the poststack situation, this is the w avefront superposition

method described b y R obinson a nd Treitel (1980). The mi-

grated image represents a summation of those amplitudes that

are in pha se such that they w ill interfere constructively. Math-

ematically, this constructive interference can be described by

the metho d o f stat ionary phase (Scales, 1995). The ba sic idea

of stationary phase is that highly oscillatory time sequences

tend to cancel upon migration except where the phase function

has a stationary point. This stationary point occurs where the

fi rst derivative of the phase function eq uals zero. In migration,

the phase function is the phase difference between the migra-

tion curve and the diffraction signatures of the data; locations

where the migration curve ista ngent to diffraction or reflection

events in the data give the stationary phase contributions to the

migration.

In migration, an a lternative kinematic description of the am-

plitude summation a long aplana tic surfaces is given by the en-

velope curves for the aplanatic surfaces. If a set of aplanatic

curves 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 aplanatic

curves defined by equation (4), as shown by Maeland (1989).

E ssentially the envelope is the solution of the system consist-

ing of equation (4) and its tangent curve (Sneddon, 1957). The

tangent curve of equation (4) is given by differentiating equa-

tion (4) with respect t o t he source-receiver position x ,

v2m

v2 x − ( x − xm) = 0. (5)

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

Eq uivalently, if we define β = vm/v,

(1 − β2) x = xm . (6)

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

have

x = xm

1 − β

2. (7)

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

z2m

v2m t

20

− x2m

v2 − v2m

t 20

= 1. (8)

Now let’s consider t hree specific cases.

1) Migrationvelocitysmallerthanthemediumvelocity, i.e.,v m < v .—In this case, equation (8) represents a hyperbola. Its

vertexison thedepth axiswith coordinate of vmt 0 below the ori-

gin. The apex is thus above the diffracto r position as vmt 0 < vt 0.

Obviously, it opens downward because the center of the hy-

perbola is just on the coordinate origin. Thus, the poststack

migration of the diffraction curve will be a shallow hyperbolic

frown when the migration velocity is too small, so that un-

dermigration partially collapses the original hyperbola into a

second, better focused hyperbola. This observation forms the

basis of residual migration and cascaded migration.

In fact, the formation of such shallow frowns can also be

well illustrated geometrically. Figure 5 illustrates this migration

case. The cyan circle is the diffra ctor point. The red curve ist he

original record we simulated for such an explosion. The blue

curves are the migration a planatics that finally superpose to

form the envelope of another hyperbola (in green) that isla ter-

ally much narrowed. This essentially indicates undermigration.

2) Migrationvelocitygreaterthanthemediumvelocity, i.e.,v m > v .—In this case, equation (8) can effectively be reformu-

lated as

z2m

v2m t

20

+ x2m

v2m − v2

t 20

= 1. (9)

This is the equa tion of a semiellipse with the center at the coor-

dinate origin. The vertex on t he depth z-axis is still vmt 0 (>vt 0),

which is now above the diffractor point P . The mouth of the

ellipse is toward the negative axis of depth, as we are only

interested in the positive z-values. Therefore, the migration of

the diff raction curve in Figure 4 will be deep elliptic smiles on

the migrated section when the migration velocity is too large.

Such a migration procedure is a lso geometrically repre-

sented in Figure 6. Most of the curves in Figure 6 are just thesame a s in Figure 5. Figure 6 is an excellent example showing

that when the migration velocity is too high, the superposition

of t he individual aplana tics forms an elliptic smile (in green) in

the migrated section.

3) Migration velocity equal to the medium velocity, i.e.,v m =v .—In thiscase, we haveto start fromequations(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 = vt 0 = z. This implies that when the




FIG. 5. Poststack migration of the diffractor model when a ve-locity smaller ( 3 km/s) tha n the true ve locity (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 large r (5 km/s) tha n the true velocity (4 km/s) is used.The superposition of all the wavefronts in blue results in anelliptic smile in green in t he fi nal migrat ion section.

FIG. 7. Poststack migration of the diffractor model when thetrue velo city (4 km/s) is used for m igra tion. The green cro ss isthe result of the constructivesuperposition of all the wavefrontsin blue. This indicates a perfect recovery of the diffra ctor pointin the migration.

migration velocity is correct, the superposition of the migra-

tion aplanatics finally collapses the recorded diffractions to

the correct spatial position, ( xm, zm) = (0, z). Parallel to such

mathema tical d evelopment, Figure 7 expresses geometrically

the procedure of reconstructing the true diffraction point by

simple superposition of aplanatic curves.

SMILES AND FROWNSIN THE PRESTACK

MIGRATED 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 mathematically

and geometrically. Since the mathema tical development is very

similar to that in the poststack migration case, we will focus on

the geo metrical a spects.

Let us first consider the same point diffractor P on the verti-

cal axis of a C artesian system. Now a ssume there is a source at

S and a receiver at R on the earth’s surface, with x -coordinates

of xs a nd xr , respectively (see Figure 8). If the medium velocity

is v , the total traveltime from the source S to the diffractor P

and b ack up to the receiver R can thus be given by the so-called

double squa re root relationship (Claerbo ut, 1985),

t =1

v

x2s + z2 +

x2r + z2

. (10)

Migration of a single diffraction arrival at R attributable to the

source at S can stillbe described by the concept of aplanaticsur-

faces. The aplanatic surface for the source-receiver pair shown

in Figure 8, analogous to equation (3), can be represented by

( xs − xm)2 + z2

m +

( xr − xm)2 + z2

m = vm t . (11)

Substituting equation (10) into the above, we have

F ( xm, zm; xs, xr ) = β

x2

s + 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) for

this special case o f constant velocity.

The fina l migration of all these arrivals recorded by different

receivers from many sources is the envelope of the individual

ellipses. The envelope of these ellipsesis essentially the solution

of eq uation (12) and its derivative equa tions (Sneddon, 1957),

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

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



1204 Zhu et al.

a nd

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

Following the same procedures as in the last section, we can

develop the same conclusions as those in t he poststack migra-

tion, although the mathematical derivations will be much more

complicated. Figure 9 schematically shows four shot gathers

from the diffraction mo del in Figure 8. Figure 10 geometricallysummarizes the migration procedure when a velocity that is

too small is used for prestack migration. Just as expected, the

superposition of all individual migration ellipses results in a

shrunken hyperbola (in green). Notice, however, that not all

of t he ellipses are ta ngent to the envelope. Figure 11illustrates

that the correct velocity allows the migration to reconstruct the

point diffractor model almost perfectly as long as the recording

coverage is sufficiently wide and dense. In contrast to Figure 11,

Figure 12d emonstrates that w hen a velocity that is too large is

used for prestack migration (vm > v), the superposition of in-

dividual elliptical aplana tics (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 tha n the true medium velocity. A low migration

velocity (vm = 3000m/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 migr at ion

results in smiling images (b). In either of these two cases, the

migrated images of the diffractors are not properly focused.

A smaller velocity results in image shallowing, while a larger

velocity leads to image deepening. Only when the true velocity

is used will the diffractions completely collapse to their true

positions (c). Thus, the final migrated section exhibits smile

and 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 diffra ctor lies at 0.8 km in depth in the middle o fthe model.

FIG. 10. Prestack migration of the diffractor mod el when a ve-locity sma ller (3.3 km/s) than t he true ve locity (4 km/s) is used.The cyan circle represents the diffracto r 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 the

true velo city (4 km/s) is used f or m igrat ion. The gre en cross isthe result of the constructivesuperposition of all the wavefrontsin blue. This indicates a perfect recovery of the diffractor pointin the migration.

FIG. 12. Prestack migration of the diffractor mod el when a ve-locity large r (5 km/s) tha n th e true velocity (4 km/s) is used.The superposition of all the wavefronts in blue results in anelliptic smile in green in the fi nal migratio n section.




poststack or prestack seismic data whenever errors exist in the

migration velocity.

These undermigrationa nd overmigration featuresare gener-

ally observed in both poststackand prestack migration sections.

However, smiles and frowns are often very difficult to observe

on the migration sections because individual diffracto rs are of-

ten just an element of the reflecting interface. G enerally, the

migration moveout effects on common image gathers (CIG s)

from prestack migrations are q uite pronounced and allow for

effective velocity analysis. Interestingly enough, the moveout

characteristics on CIG s are very different from those in the

migrated stacked sections.

PRESTACK DEPTH MIGRATION MOVEOUT

Many recent studiesdemo nstrate that prestack depth migra-

tion can accurately image reflections and diffractions without

dip restriction if a reasonable approximation of the velocity

field 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 for

velocity analysis (Versteeg, 1993; Lines et al., 1993). Here we

will 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 CIG s (Al-Yahya, 1989). Our analysis, however, will

no longer depend on the assumption of a la yered-earth model.

Co nsider the genera l subsurface structure and recording ge-

ometry as shown in Figure 14. P denotes the arbitrary scatter-

ing point in the eart h’s interior. S a nd R are a source-receiver

pair illuminating P . D is the surface image of P . Assuming

that the average velocity above P is v̄ and tha t the diffraction

received at R (because of a source wavelet from S and then

diffracted at P) never travels beneath P , then its arrival time

can be expressed as

t =1

v̄(SP + R P). (15)

When an incorrect average velocity v̄m is used for migration,

the diffraction signal received from P will be migrated to an

incorrect point P . P generally has both vertical and lateral

displacements from the true position P . We deno te t hese dis-

placementsw ith x = P Q a nd z = QP . In this case, the trav-

eltime will be

t =1

v̄m[SP + P R] =

1

v̄m[(SP − P1P) + ( RP − P2P)]

=1

v̄m[(SP − Q1P) + ( RP − Q2P)]

+1

v̄m(P2Q2 − P1Q1). (16)

FIG. 14. Migrat ion depth/velocity relat ionship diagra m in ageneral subsurface structure. (a) A wrong velocity migratesthe reflection to a position P , which has a latera l displacement x in addition to a vertical displacement z. (b) An enlargedview of the lower part of (a).



1206 Zhu et al.

From Figure 14b, the following rela tionship holds in the tri-

angle P QO :

sin αs

P O=

sin αr

QO=

sin θ

x. (17)

Thus, we have

P2Q2 − P1Q1 QO − P

O

= x

sin θ sin αr −

x

sin θ sin αs

= x

sin θ (sin αr − sin αs)

= x

cos(θ/2) sin

αr − αs

2 . (18)

In the derivation of the last equation ab ove, we have used the

relationships αr + αs + θ = 180◦, sin θ = 2sin(θ/2) cos(θ/2), and

sin αr − sin αs = 2cos[(αr + αs)/2]sin[(αr − αs)/2]. Therefore,

the distance part of the last term of equation (16) would be

an o rder smaller than x as long as |sin(αs − αr )/2)| < 0.10

a nd θ is not close to 180

◦

. The latt er condition generally holds,a s αs a nd αr would not be zero for most cases. The fi rst condition

is equivalent to |αs − αr | < 12◦, i.e., the difference betw een the

two illuminating angles being less than 12◦. Since |αs −αr | = 2α,

where α isthestructuraldipat P , thea boveinequality thusonly

holds for structures of gentle dip. In such cases, equation (16)

can be properly approximated by

t =1

v̄m[(SP − Q1P) + ( RP − Q2P)]. (19)

This equation is equivalent physically to the assumption that

the lateral displacement x is negligible compared to t he ver-

tical one. E liminating t from eq uations (15) and (19), we obta in

(1 − β)(SP + R P) = Q1P + Q2P, (20)

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

tions hold:

SP = z/cos αs; Q1P = − z co s αs;

RP = z/cos αr ; Q2P = − z co s αr .

Substituting these relations into equa tion (20) leads to

z = (β − 1) z

cos αs co s αr

. (21)

In t he case of a zero-offset source-receiver pair just a t D, αs =

αr = 0, we obtain

z = (β − 1) z. (22)

This implies the migra tion depth z will be shallower than the

true depth z if a velocity smaller than the true velocity (v̄m < v̄)

is used fo r migration, while it w ill be deeper if a higher velocity

(v̄m > v̄) isused.O nly when β = 1,i.e., the true medium velocity

is used for migration, will the diffractor be located properly. By

denoting z0 = (β − 1) z, equation (21) can be rewritten as

z(αs, αr ) = z0

cos αs co s αr

. (23)

Now let us consider the fo llowing three cat egories.

1) Migrationvelocity lessthan the truevelocity (v̄ m < v̄ ).—In this case, β < 1, z0 < 0, and z(αs, αr ) < z0. G enerally

the following relation,

z(αs + 1, αr + 2) < z(αs, αr ), (24)

also holds for any αs, αr a nd small nonnegative values of 1, 2.This relation indicates the migration image of P will form a

smile that curves upward on a CIG , which is a display of mi-

gration traces versus the source-receiver offset corresponding

to a fixed surface point.

2) Migration velocity greater than the true velocity (v̄ m >

v̄ ).—In this case, β > 1, z0 > 0, and z(αs, αr ) > z0. Similar

to equation (24), we have

z(αs + 1, αr + 2) > z(αs, αr ). (25)

This relation indicates the migration image o f P forms a frown

that curves downward on a C IG .

3) Migrationvelocityequal tothetruevelocity(v̄ m =v̄ ).—In

this special case, β = 1, z0 = 0. Thus

z(αs, αr ) = 0 (26)

for any source-receiver pair. This simply means that when the

true 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 horizontal

segment on the CIG displays.

To consider prestack migration velocity ana lysis in terms o f

offset and common midpoints (CM Ps), consider again Figure 8for a point diffractor at (0, z). The midpoint can be d enoted b y

X = ( xr + xs)/2 and the offset denoted by 2h so that x s = X −h

a nd xr = X + h . Equation (10) gives the total traveltime for

a particular point diffractor, but it can be embedded into an

equation for a correct migration ellipse:

t 1 = t 1( X ) =1

v(

( X − h − x)2 + z2

+

( X + h − x)2 + z2). (27)

In terms of migration velocity and migration coordinates, we

can also write the traveltime expression as

t 2 = t 2( X ) = 1vm

( X − h − xm)2 + z2

m

+

( X + h − xm)2 + z2

m

. (28)

When vm = v, the migration ellipse (28) is identical to the el-

lipse (27) for the true velocity. If t he velocity a nalysis location

xm coincides with the diffractor location x = 0, we then have

zm = z for all h . That is, the migrat ion depth equa ls the correct

depth of the diffractor regardless of the offset value when the

velocity iscorrect. When the velocity is not correct (vm = v), we




need to evaluate the envelope of migration ellipses for all val-

ues of the midpoints X . That is, we need to set the derivat ives

of t 1( X ) equal to the derivatives of t 2( X ) for all X . If, again, we

perform the velocity analysis at the actual diffractor location,

then t hese derivatives give, respectively, the slopes of t he cor-

rect and the incorrect migration ellipses at xm = x = 0. These

slopes are different unless both ellipses are flat at the analysis

location, i.e., unless X = 0 also. So we set X = xm = x = 0 into

the expressions for t 1( X ) and t 2( X ) to obtain

h2 + z2

v2 =

h2 + z2m

v2m

(29)

or

z2m = (β2 − 1) x2 + β2 z2, (30)

where β = vm/v.

In the case of vm < v, we have β < 1. E quation (30) can be

rewritten as

z2m + (1 − β2) x2 = β 2 z2, (31)

which is an ellipse. So this is a special case o f eq uation (24).I f vm > v, then β > 1 and eq uation (30) is essentially the hy -

perbolic equation

z2m − (β2 − 1) x2 = β 2 z2. (32)

This is a hyperbolic frow n in the half-plane o f positive depth.

I f vm = v, then β = 1 and equation (30) simplifies to zm = z .

This represents a horizontal line segment. Only in this special

case of using the correct velocity for migration will the final

stack of the migrated common reflecting point (CRP) gather

reach its highest energy in the migration section.

The above procedure of migrating a CRP gather also can be

represented geometrically. Figure 15shows a CR P gather from

the same diffra ctor model of Figure 8. This gather is essentiallyequivalent to t hat o f CM P ga ther in this special case of a single

diffracto r point in a constant-velocity medium. Figure 16shows

FIG. 15. CR P gather from a diffractor point model. The diffrac-tor lies at 0.8 km depth in the midd le of the model.

the migrated CR P gathersfor thisdiffractor model correspond-

ingto different migrationvelocities. B ecause the migrated CR P

gather is just the same a s the so-called CI G gathers, Figure 16

equivalently illustrates the CI G s corresponding to different mi-

gration velocities. The recorded hyperbola in the CRP gather

will appear as elliptical smiles when a smaller velocity is used

for migration, while a hyperbolic frown appears when a larger

velocity 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 CIG s. We can use the same point diffractor model

to illustrate these theo retical predictions. E ighty-one shot pro-

files are theoretically simulated, with each record consisting

of 60 traces. Figure 17 shows the CIG s for surface position

x = 1.0 km when different velocities are used in migration. It

clearly demonstrates that only when the true velocity is used

FIG. 16. CIG s when different migration velocities are used.Migra tion velocity is 3 (a), 4 (b), a nd 5 km/s (c), respectively.The circle in cyan represents the diffractor position. The red isthe scaled CR P gather.



1208 Zhu et al.

will the migration images of diffractors be independent of the

source-receiver offset. When the velocity is lower tha n the true

velocity (vm = 3000 m/s), the diff ract or ima ges form smiles at a

depth shallower tha n the true d epth. This is total in a greement

with equations (24) and (31). In contrast, when the velocity is

higher than the true velocity (vm = 5000m/s),t he diffractors are

expressed as frowns on a CI G at a depth greater than the true

depth. This is just what has been predicted by mathematical

expressions (25) a nd (32).

Thus, the migration velocity error is documented well on

both the fina l migration sections (Figure 13) and the CI G s(Fig-

ure 17). Interestingly, the CI G s show shallow smilesf or low ve-

locities and deep frowns for high velocities (Lines et al., 1993),

whereasthe final depth migration sectionsshow shallow frowns

for low velocities and deep smiles for high velocities (Yilmaz,

1987).

If we take a careful further look at Figures 13 and 17, we

see that the curvatures of the shallow smiles/frowns a re gen-

erally larger than the deep ones. This indicates that velocity

errors are more pronounced on shallow reflections and thus

are easier to correct. This also agrees with the general observa-

tion tha t sufficient of fset/depth rat io should be kept to analyze

properly the velocity errors (Lines, 1993). Luckily, we often

have more constraints available on the shallow parts of the

earth, such as w ell logs and geological exposures. We can also

more effectively use techniques such as fi rst-break tomogra phy

to constrain our near-surface velocity estimation. With respect

to the deeper structure, we generally have to accept that it is

coarsely defined and more ambiguous.

Though the diffractor model is oversimplified, it is of vital

significance in migration and velocity analysis theory because

any complicated structures can be considered as a continuum

of diffractors. Thisis especially suitab le for moveout ana lysis on

a C IG that corresponds to a single surface point. The smile and

FIG. 17. CI G s for a model with two point diffra ctors.

frown patterns on CI G s can be effectively used to q ualitatively

and quantitatively analyze the migration velocity (Al-Yahya,

1989).

CONCLUSIONS

D epth migration is very sensitive to errors in the velocity

model. Migration moveout features such as smiles and frowns

have previously been reported (Yilmaz, 1987; Lines et al.,

1993). We ha ve shown both mathema tically and geometrically

these smiles and f rowns on migra tion sections and C IG s. U s-

ing the simple diffractors model, we demonstrated tha t a fter

migration, either using the stacked data or the prestack shot

gathers, the diffractions migrate to shallow frow ns when the

migration velocity is too small and deep smiles when the ve-

locity is too large. Only when the migration velocity is exactly

the medium velocity will both the poststack and prestack mi-

gration produce concentrated blobs in the migration section.

However, the CIG s produced in the prestack depth migra-

tion provide an effective migration velocity analysis domain.

Our study starting from the very general subsurface structure

showed that the migration moveout in the CIG s is interestingly

different from t he patterns in the migration sections. In CI G s, a

lower velocity produces shallow smiles, while a higher velocity

results in deeper frowns. When the migration velocity is cor-

rect, the CIG s present horizontal segments at the exact diffrac-

tor depths, which indicate that the migration image of the

point diffra ctor is independent o f source-receiver offset. These

moveout patt erns of smiles and frowns can serve as both q ual-

itative and q uantitat ive criteria for migrat ion velocity analysis.

ACKNOWLEDGMENTS

We thank Jianhua Pan for his technical assistance. The au-

thors also thank Sven Treitel for providing us a copy of the

Maeland paper.




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Smiles an Frows by Migaration

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