GEOPHYSICS, VOL. 50, NO.4 (APRIL 1985); P. 582-595, 31 FIGS.

Scattering characteristics of elastic waves by an elastic heterogeneity

R. Wu* and K. Akit

ABSTRACT

Elastic wave scattering by a general elastic heterogen­eity having slightly different density and elastic con­stants from the surrounding medium is formulatedusing the equivalent source method and Born approxi­mation. In the low-frequency range (Rayleigh scattering)the scattered field by an arbitrary heterogeneity havingan arbitrary variation of density and elastic constantscan be equated to a radiation field from a point sourcecomposed of a unidirectional force proportional to thedensity contrast between the heterogeneity and themedium, and a force moment tensor proportional to thecontrasts of elastic constant. It is also shown that thescattered field can be decomposed into an "impedance­type" field, which has a main lobe in the backscatteringdirection and no scattering in the exact forward direc­tion, and a "velocity type" scattered field, which has amain lobe in the forward scattering direction and no

INTRODUCTION

Elastic wave scattering has become a topic of current interestin both general and exploration geophysics, because of its closerelation with various kinds of heterogeneities in elastic media.The Earth has been found to be laterally inhomogeneous inevery scale, and elastic wave scattering could be the mosteffective tool to examine these inhomogeneities. In generalgeophysics, seismic wave scattering by the heterogeneities nearthe mantle-core boundary of the Earth was proposed byHaddon and Cleary (1974) to interpret the precursors ofPKIKP in seismograms (see Doornbos, 1976). The phase andamplitude fluctuation across a large seismic array (such asLASA or NORSAR) were used to estimate the parameters ofvelocity inhomogeneities under the arrays based on Chernov'stheory of wave scattering in random media (Aki, 1973; Capon,1974; Berteussen et al., 1975). Seismic coda waves from localearthquakes were attributed to backscattering of seismic waves(Aki, 1969; Aki and Chouet, 1975) and attempts have beenmade to infer the properties of local small-scale heterogeneities

scattering in the exact backward direction. For Miescattering we show that the scattered far field is a prod­uct of two factors: (1) elastic Rayleigh scattering of aunit volume, and (2) a scalar wave scattering factor forthe parameter variation function of the heterogeneitywhich we call" volume factor." For the latter we derivethe analytic expressions for a uniform sphere and for aGaussian heterogeneity. We show the relations betweenvolume factors and the 3-D Fourier transform (or I-DFourier transform in the case of spherical symmetry) ofthe parameter variations of the heterogeneity. The scat­tering spatial pattern varies depending upon variouscombinations of density and elastic-constant pertur­bations. Some examples of scattering pattern are givento show the general characteristics of the elastic wavescattering.

from their studies (Aki, 1982; Sato, 1982b; Wu and Aki, 1984).Apparent attenuation caused by seismic wave scattering and itsrelation with the intrinsic absorption were also discussed (Aki,1980a, 1980b, 1982;Sato, 1981, 1982a;Wu, 1980, 1982a,1982b;Richards and Menke, 1983) and the problem is still open. Inmany of the problems mentioned above the solution is ob­tained by a scalar wave scattering theory without guarantee ofcorrectness. We need to develop full elastic wave treatments forthese problems. In exploration geophysics, following the intro­duction of shear wave sources and three-component geo­phones, the need for applying elastic wave scattering to thecomplex object and structural exploration has become appar­ent and pressing. Especially in the case of vertical seismicprofiling (VSP), where the source and receiver arrangements arefavorable for receiving wide-angle reflected or scattered wavesand the targets are often complicated, seismic wave scatteringhas vast possibilities of application. Some VSP experimentshave been done to examine the explosion-formed fracturevolume and hydrofractures (Turpening, 1984; Turpening andBlackway, 1984).

Therefore, for both general and exploration geophysics, we

Presented at the 53rd Annual International SEG Meeting, September 12, 1983, in Las Vegas. Manuscript received by the Editor January 31, 1984;revised manuscript received October 31, 1984.*Presently Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139; Institute ofGeophysical and Geochemical Prospecting, Baiwanzhuang, Beijing, China.tDepartment of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute ofTechnology, Cambridge, MA02139.C 1985 Society of ExplorationGeophysicists. Allrights reserved.

582

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Scauering by an Elastic Heterogeneity 583

RAYLEIGH SCATTERING OF ELASTIC WAVES

(2)

(1)

(4)

(3)

(5)

F·=fQ dV~W20pVUO(x)1 I """"-' I 0',

pIx) = Po + 6p(x),

f,(X) = Ao + 6A(X),

Suppose an arbitrary heterogeneity with parameters

and

We write the displacement field V as the sum of "primarywave" V O and "scattered wave" V' :

and the force moment tensor of the equivalent point source

u, = JQiXk dV(x)

- 6ik 6A VV . U()(xo)

- 611 V[Ufdxo) + U~.i(XO)J,

The scattered field can be obtained as an integral repre­sentation using the body force equivalent method and the Bornapproximation (Miles, 1960; Haddon and Cleary, 1974; Guber­nat is et al., 1977a, b: Aki and Richards, 1980, chap. 13). How­ever. for Rayleigh scattering the scattered field can be equatedto the radiation field of an equivalent point source. Since thesize of the inclusion is small compared with the wavelengthinvolved, phase differences between the radiation far fields ofequivalent body force from different parts of the inclusion canbe neglected. Using integration by parts, we get the total singleforce

I Here subscripts before comma denote Cartesian components andsubscripts after comma. differentiation with respect to that coordi­nate. Repeated subscripts imply summation over those subscripts.

is situated in an isotropic. homogeneous medium with parame­ters Po' f.() . and 110' where the following conditions are assumedto be satisfied

where Qi is the equivalent body force, V is the volume of theinclusion, lip, 6/e, and 611 are the average excesses of its densityand Lame constants, and &ik is the Kronecker delta.' Fordetails see Wu (1984).

From equation (5) it can be seen that the force momenttensor due to 6/e V has only diagonal elements and all theelements have the same strength, which corresponds to anexplosion-type point source; on the other hand, the momenttensor due to 611 V can have both diagonal and off-diagonalelements. Because of the symmetry with respect to i and k,l'vt ik = 'Vlkl • the diagonal elements correspond to on-line forcecouples. while the ofT-diagonal-element pairs correspond to

tering patterns for a uniform elastic sphere and for a sphericallysymmetric heterogeneity having Gaussian parameter vari­ations. We also show that the scattered field can be decom­posed into an "impedance-type" and a "velocity-type" scat­tered field with quite different scattering characteristics.

need to advance the elastic wave scattering theory in order tomodel and understand better the scattering phenomena andalso to develop some effective algorithms. Elastic wave scatter­ing by a single inclusion in a homogeneous elastic medium isthe basis of more advanced scattering theory. though the topicitself is a rather old one. Elastic wave scattering by an elasticspherical inclusion has been studied by several authors Yingand Truell (1956) and Yamakawa (1962) treated scattering forplane P-wave incidence and Einspruch et al. (1960) derived theformulas for plane S-wave incidence. (Gubernatis et al., 1977bcorrected some errors in earlier works.) In their method theelastic wave equations (a scalar wave equation and a vectorwave equation) were decomposed into three scalar wave equa­tions for scalar potentials, which were solved in spherical coor­dinates, and the boundary conditions were matched to deter­mine the unknown coefficients. The solutions are Infinite series.which converge slowly for large inclusions compared with thewavelength. In addition, for a general elastic sphere, a matrixequation must be solved in order to obtain the expandingcoefficients for each term of the series. Therefore. there are noexplicit expressions for scattered fields and the general charac­teristics of the spatial scattering pattern have not been exposed.However, in Yamakawa's paper (1962), some results of Ray­leigh scattering (when the radius of the sphere is much smallerthan the wavelength) for plane P-wave incidence were shownexplicitly. Knopoff (1959a, b) also showed some results for arigid and unmovable sphere. Because of the infinite rigidity andthe infinite density of the sphere in his treatment. the scatteringpattern derived there is much simpler and not representative forelastic wave scattering.

Besides the complexity of calculation, the eigenfunction ex­pansion method only works for spherical or circular-cylindricalinclusion (see Morse and Feshbach, 1953; Pao and Mao. 1973).For a general case of arbitrary shape and arbitrary parametervariation of the inclusion, only approximate methods are ex­pected.

Instead of solving the partial differential equations. Miles(1960) formulated the scattering problem into an equivalentintegral equation using the elastodynamic Green's functionderived by Stokes (1849; see also Love, 1944. section 212 andRayleigh, 1896, section 378) and obtained explicit expressionsin the case of Rayleigh scattering using Born approximation.This approach was also used by Haddon and Cleary (ILJ74) forthe P-wave scattering near the mantle-core boundary and byHudson (1977) for the scattered waves in the coda of P Guber­natis et al. also formulated the scattering problem of a homoge­neous inclusion into an integral equation (1977a) and obtainedformulas for the case of Born approximation in the wholefrequency range (I 977b). In this paper, we follow the approachof integral equation and Born approximation. and we gener­alize the formulation to an elastic heterogeneity of arbitraryshape and arbitrary variation of parameters. We derive theformulas for Rayleigh scattering in terms of equivalent sourceshaving different force and force moments corresponding to thedifferent parameter perturbations. This representation IS prob­ably more familiar to geophysicists. In the Mie scattering range(when the wavelength is comparable to the size of inclusion),the scattering pattern can be decomposed into two factors. Oneis the Rayleigh scattering, the other is the "volume factor."which is a generalization of the "shape factor" of Gubernatis etal. (1977a, b) for uniform inclusions. To demonstrate the scat­tering characteristics, we give numerical results and plot scat-

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584 Wu and Aki

torsion-type double couples (we show this In detail in thefollowing). V co

2 (a6) {8P. (~o) 8~ }= - 2" 2" - Sin e+ - - sin 2841t ao ~o Po ao ~o

(6)

r

where the subscript r stands for the r-component, and merstands for meridian component. Because of the symmetry of theproblem with respect to the polar axis, there is no latitudinalcomponent of the S-wave, i.e., pUfat = O. This result was ob­tained by Gubernatis et al. (1977a) by direct integration of theintegral representation of the scattered field.

(10)xPlane P-wave incidence

When a plane P-wave

u7 = exp [ -ico(t - xdao)],

U~ = U~ = 0,

is incident on the inclusion along the x direction (we takeXl = X, X2 = y, and X3 = z) where a o is the P-wave velocity inthe medium, we can calculate the equivalent point forces fromequations (4) and (5)as

S-waves

scattered

)(

P-waV88

(b)

scattered

Elastic - wave Rayleigh scattering

equivalent

forces

y

Y, = Cos 8

z Yz =Sin 8 Cos ep

Y3 =Sin8Sinep

(a)

P-wave incidence

(7)

(9)

o

o

co l8A + 28~M = -i - V 0 8A- a o

o

V co2

{8P 8A 28~ }= - - - cos 8 - - cos? 841t a6 Po Ao + 2~o Ao + 2~o

1 .x - e-IW(t-r/rzo)

r '

3

PU; = L UfYii= 1

and

F I = co 2 Sp V exp (- icot),

F 2 = F 3 = 0,

Here we took the center of the inclusion at the origin of thecoordinates. ~ is the moment tensor of the equivalent pointsource. Therefore, for P-wave incidence the equivalent forcesconsist of a single force in the incident direction, contributedfrom the density contrast, and a moment tensor contributedfrom the contrast of elastic constants. We can see in this casethat 8Acontributes an isotropic explosion-type source, while 8~

functions as an on-line force couple (Figure 1b).Knowing the equivalent point source, we can derive the

scattered far field (see Aki and Richards, 1980, chap. 4; or Wu,1984) as

o, = Fj *Gij + M j k * Gij.k

co 2 V 1= e-iro(t-r/.o)

41ta6 r

{8P ~ 2~ 2}

X Po YiYl ~ Ao + 2~o Yi - Ao + 2~o YiYI

co 2 V 1+ e- iro(t-r/Ilo)

41t~6 r

x l8P

(8il - Yiytl- 2(~O) 81l (8i1 YI - YiYi)]. (8)Po ao ~o

If we take spherical coordinates having their polar axis in theincident direction XI (i.e., in the direction of particle motion)(Figure Ia), we can write the scattered P-wave PU; and S-wavePU~er as

and

PU~e< = - U~ sin 8 + U~ cos ecos <D + U~ cos 8 sin <DFIG. l. (a) Spherical coordinate system for P-wave incidence,

and (b) the scattering patterns for different equivalent forces.

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and (11)

the scattering will have only a main lobe in the forward direc­tion for P-waves and no backscattering. We called this..velocity-type" scattering.

In general, if 8A/tco = 811/110 and Ao = 110, we can decomposethe scattered field into an impedance-type field and a velocity­type scattered field. Since

585

(12)

(13)

{OZ, [. 8 (~o). ]x Zso SIO - a

osin 28

{OZP ( 1 2 )x -- cos 8 - - - - cos? 0ZPo 3 3

oa ( 1 2 ')}- - cos 8 + - + - cos· 8 .ao 3 3

Op oZp oa----

Po ZPo a o

V (02 e iw(t - r/ilo)

PU P- - - ----r - 4rr a~ r

where Zp is the impedance for P-waves, then

In a similar way.

O~ [ . (~o). ]}- ~o sm 8 + ao

SIO 28 •

where Z, is the S-wave impedance, Z, = p~

op 1 (OP OA + 2(11)= Po + 2: Po + Ao + 2110

o(pa) op So--=-+­poao Po ao

Scattering by an Elastic Heterogeneity

For acoustic scattering, 11 = 0, and equation (9) agrees withRayleigh's result (Rayleigh, 1896, section 296, p. 152). However,the existence of shear rigidity increases greatly the complexityof the scattering pattern. Figure 1 gives the spatial patterns ofPU; and PU~er for opV, oAY, and oilV, respectively. Since thedecomposition of the spatial pattern of PU; is not unique, thisresult is equivalent to that of Miles (1960), but different inrepresentation. In his case, the scattering pattern by oilV wasexpressed as a quadripole plus an isotropic part, while in ourcase it is expressed as a dipole. We believe our representation ismore natural.

The resultant scattering pattern of an inclusion will exhibitquite different appearances depending upon various combi­nations of Sp, OA, and oil. Figures 2 and 3 are examples. Noticethat when OA, oil, and op have the same sign, indicating theinclusion is harder and heavier (or softer and lighter), the p-p

scattering always has its maximum in the backscattering direc­tion (Figure 2), because all the scattered field due to OA, 011, andop are in-phase in backward direction, as seen from equation(9). Also note that, when op/Po = 8A./Ao = Oil! 110 . indicatingthere is no velocity contrast between the inclusion and the

medium, i.e. oa/ao = Hop/po + (OA + 2(11);0'0 + 2110)]' therewill be only one main lobe in the back scattering direction andno forward scattering (Figure 3). We call this .. impedance-type"scattering. When the density perturbation 8p has the oppositesign to that of the elastic constants OA and 611. i.e., when theinclusion is lighter and harder than the medium (or hea vier andsofter), the whole scattering pattern will be turned over In sucha way as to swap the forward and backward directions Figure3 also shows the case of OA/Ao = 011/110 = - 0P/Po Note that.when there is no impedance contrast between the inclusion andthe medium, i.e., when the following quantity vanishes,

0," Sp--=1/2­fJ-o Po

150·

160·

140· 130· 120· 110· 100· 90· 80· 70· 60· 50 0 40·------::r----~/----

8>- ~2[e~ i'» Po

13',

'0

30·

20·

150·

160 0

140· 130· 120· 110· 100· 90· 80· 70· 60· 50· 40·~---;--\ \ I I ;-;-

VELOCITY 8"\ 8/" 8pI TYPE T: = ~=-~

IMPEDANCE 8, 8/" 8pTYPE \;, • ;;;, , Po

30·

20·

10·

20·

30·

.. 10·-,,

0.5 \~---++---t-L---t----+---+---+ --+---f- 0°

--',,~-,-,

\\\III

/

/" ...... _--"'?

_J "_L__ \.120· 110· 100· 90· 80· 70· 60· 50· 40·

p.- s

p-p

1700.

170·'

180·

16o- . _:".: f\:- ,100

O·

40·140· 130· 120· 110"ibo· 90· 80· 70· 60· 50·

170· .

180·

160·

150·

FIG. 2. Scattering patterns of Rayleigh scattering for planeP-wave incidence. The upper half is of P-P scattering, the lowerhalf is of P-S scattering. All the patterns are axially symmetricabout the x-axis.

FIG. 3.Same as Figurc 2. The scattering pattern of velocity-typeand impedance-type scattering.

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586 Wu and Aki

and

v? = V~ = 0

is incident upon the inclusion along the x-direction and havingits particle motion in the y-direction (Figure 4a), the equivalentforces can be derived from equations (4)and (5)as

F I = F3 = 0,

FZ = ooz op V exp (- ioot),

7"2 =Cos 8

7"1 = Sin 0 Sin ¢

7"3 =Sin 8 Cos¢

x

(a)

Elastic-wave Rayleigh scattering

y

(14)

When a plane S-wave

V~ = exp [ -ioo(t - Xl/~O)]

Plane S-wave incidence

and

S-WBve Incidence

S-W8vee

scatteredP-waV88

ecattered

force I

equivalent

namely a single force FZ in the direction of particle motion ofthe incident wave due to density contrast and a double-coupleforce in the polarizarion plane due to the contrast of shearrigidity.

The scattered far field can be obtained as

(b)FIG. 4. (a) Spherical coordinate system for S-wave incidence,

and (b) the scattering patterns for different equivalent forces.

Vi = Fj *Gij + M jk *Gij. k

= oozop V {YiYZ ~ e-iw(t-rloo)

4npo a~ r

(YiYZ - 0iZ) 1 -iwl,-r/Po)}- - e~~ r

+ ooZ oilV {-2Yi YlYZ ~ e-iwlt-r/oo)

~o 4npo a~ r

+ (2YiYIYZ - Oi;YZ - 0iZYl) ~ e-iW(,-r/Po1}. (16)~o r

We take the direction of particle motion of the incident field(y-axis) as the polar axis of the spherical coordinates (Figure 4).The scattered P-wave sV; and the scattered S-wave sV~er andsUfalcan be written as

SVP ~ vP V 002

{op 2 (~o) oil } 1 -iw«-r/oo)r = L.. i Yi = - 2 - Yz - - - YIYZ - e

i=1 4n a o Po a o 110 r

= ~ 00: {Op cos 8 _ (~o) oil sin 28 sin <I>} ~ e-iw(t-r/oo),

4n a o Po a o 110 r

sV~er = -V2 sin 8 + V 3 cos <I> cos 8 + VI sin <I> cos 8

= ~ 00: (ao)2{_ op sin 8 + oil (sirr' 8 _ cos" 8) sin <I>} ~ e-iwl,-r/Po)

4n a o ~o Po 110 r

= _ ~ 00: (ao)Z {Op sin 8 + oil cos 28 sin <I>} ~ e-iw(t-r/ Po),

4n a o ~o Po 110 r

(17)

(18)

and

. V 002(a )Z Oil I.SVS = - V sin <I> + V cos <I> = - - -.£ - cos 8 cos <I> - e-1W(t-r/Po)lat 3 I 4n a~ ~O 110 r .

(19)

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587

10·

20·

30·

30·

ZO°

30·

30·

O'

20·

'. 10·

- 10·

. 20·

, 10'

40'

40·

o Coming out

X Going in

01'- oPfio=fio

§jJ-'I/Z§ffLo Po

§jJ-,s§ffLo Po

SU~erin X- Y Plane

14>'!90·, No U~" I

~'~'2~Ao!-Lo Po

SUS...'in X-Z Plene

14>'0·, 180·, No U~,,)

60· SO' 40·

"­\

__-+---+---+1_.J.\+I---+--+_--1

--+-----+--+ -----f--

No Urin X-Z Plane

'Q-,

.>,\

\\\\

----\7I

81. ~ 1/2~//>::-j-Lo- Po

8>- Of. [E //\0 fLo Po

130· IZO· 110· 100· 90· 80· 70· 60· 50·

s- s

140·

140·

140 130 IZO

150·

160·

170·

s-s180· __+_---l-___--+ ___+------+-_++-_

s-p,

1700

Tf'160·Y

ISO·

ISO·

160

170·

180·

FIG, 5. Scattering pattern of Rayleigh scattering in x-y plane forplane S-wave incidence. The upper half is of S-S scattering, thelower half is of Sop scattering. Note that in x-y plane there areonly meridian components of scattered S-waves.

(20)

{OZsI (~o) -Jx Zso Lcos 0 - ao

sin 20 sin ¢

- ~~ LCos 0 + (~:) sin 20 sin ¢lI'

SuP = ~ co2~ e-iwit-rloo)

r 4][ a~ r

ScaUering by an Elastic Heterogeneity

Note that the scattered S-wave due to &11 V is attributed to anequivalent double force couple. For each single couple. if wetake the polar axis of the spherical coordinates parallel to theforce direction, there will be only meridian components ofS-waves. Therefore we can plot the scattering patterns contrib­uted from different equivalent forces as in Figure 4, where wedecomposed the pattern due to &11 V into two parts, each ofwhich corresponds to a pattern due to a single couple. Theresultant scattering pattern will be the vector sum of those threepatterns. Figures 5 to 8 present examples. The S-wave scatter­ing depends only on the contrasts of densities and shear modu­lus that are physically expected. As in the case of P-waveincidence, the S-S scattering patterns have main lobes in thebackscattering direction, if both the density and shear modulushave the same sign. Otherwise the main lobes will turn over tothe forward direction (as in Figure 8), In the same manner as forP-wave incidence, there will be no scattering in the forwarddirection when &p/Po = &11/110' i.e., when the S-wave velocitycontrast between the inclusion and the medium vanishes. Onthe other hand, when &p/Po = - 011/110' or in other words whenthe shear wave impedance of the inclusion matches that of themedium, there will be no scattering in the backward direction.In x-z plane (y = 0) (Figure 6) the meridian component U~er isthe only nonzero component of the scattered waves. The lati­tudinal component Ulalcomes solely from the transverse (withrespect to the particle motion of the incident wave) equivalentforce couple M 12' It has two lobes having maxima along they-axis(Figure 7).

Using equations (12) and (13), the scattered field can bedecomposed into an impedance-type field and a velocity-typefield as follows,

FIG,6. Same as Figure 5, but in x-z plane.

{OZ

x -' (sin e + cos 28 sin ¢)Z,O

{(OZ, O~) }x -' + - cos e cos <!> ' (22)Z,O ~o

For both P-wave and S-wave incidences, the convertedwaves (P-S or SoP) have only sidelobes (with respect to theincident direction), while the common-mode scattered wavesalways have main lobes along the incident direction (either inthe forward or backward direction),

Regarding the scattering strength, the scattered S-waves arealways stronger than the scattered P-waves (suppose ole ~ 011)

140· ,130· 120·

30·

10·

O'

30·

20·

.20·

, 10·

40·

40·

~'2§ffLo Po

8/LIBP-';;;"2Pa

8/LBP1r=P;

o Coming out

X Going in

//

130· IZO' 110· 100· 90· 80· 70· 60' SO'

FIG.7. Same as Figure 5, but in y-z plane.

140·

160·

170·

ISO·

s-s180· -T f

Sop

170· #/~'-y

160· 1"-z

ISO·(21)- ~~ (sin 8 - cos 28 sin <!»},

and

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588 Wu and Aki

x f [OP(~) _ OA(~)t.t, A + 2 t,

v Po 0 110

dV is the equivalent force moment tensor for the scattering ofthe elementary volume dV(~) with elastic constant pertur­bations bA(~) and bll(~) in the heterogeneity. Therefore equation(24) implies that the scattered field is a superposition of thescattered fields by all the volume elements of the heterogeneity,each of which is of Rayleigh scattering type. Taking the Fraun­hofer approximation to Gij(~) (see Aki and Richards, 1980),equation (24) can be further simplified. For plane P-wave inci­dence, we have

40·

130· 120· 110· 100· 90· 80· 70· 60· 50· 40·140·

140· 130· 120· 110· 100· 90· 80· 70· 60· 50·

150·SU~er

30·in X-Y Plane

(4):: t 90~ No UCat)

160· 20·»<

-c,<,

-:/

170· I 10·f s-s

180· o·S-P SU~

170· f· 10·

TVELOCITY --~L~160· TYPE iLo Pc 20·

IMPEDANCE --- 8iL 8p150· TYPE fL;=P; 30·

FIG. 8. Same as Figure 5, the scattering pattern of velocity typeand impedance type scattering.

(25)

and

(23)

and

(OZ ISUS(x) = e- iw(t - r / 1lo1

, 47t~6 r

(29)

bp(~) = bpo P(~).

OA@ = OAoP(~),

bll(~) = bllo P(~),

x f [oP(~) (Oil - YiYI)" Po

( ~o) 201l@ ]- - -- (Oil - YiYI)YIUo 110

x ei(WI>O)~l ~(iw/Ilo)(x'~) dV(~). (26)

In the case of plane S-wave incidence,

(OZ ISUP(x) = e-iw(t-r/>o)

, 4nu6 r

xf[OP(~) YiYZ - (~o) 2bll(~) YiY1YZ]Po Uo 110

x exp [i ~ ~1 - i~ (x • ~)J dV(~), (27)~o Uo

x f lbP(~) (OiZ - YiYZ) - bll(~)v Po 110

X (OilYZ + 0iZYI - 2YiYIYZ)]

x exp [i ~ ~I - i~ (x . ~)] dV@. (28)~o ~o

Suppose Sp, OA, and oil have the same form of spatial vari­ation. We introduce the parameter variation function P(~) suchthat

and

In Mie scattering (when the wavelength is comparable to thesize of inclusion) the equivalent source of scattering by aninclusion can no longer be regarded as a point source. Thephase differences of the incident field at different parts of theinclusion and of the scattered field from different parts of theinclusion can no longer be ignored. Nevertheless, if the totalscattered field is still much weaker than the incident field, theBorn approximation can still be a useful tool for calculating thescattered field and deriving the scattering characteristics.

Here we formulate the problem for a general arbitrary elasticheterogeneity. The scattered waves can be written using theBorn approximation as

ELASTIC WAVE SCATTERING OF ANARBITRARY ELASTIC HETEROGENEITY

fi]lUo

U.(x) = - bp __J * G.. dVI r t)(2 IJ

[see equations (17), (18), (9) and (10) or Figures 2 and 5, etc.].Therefore, after a large distance of propagation and scattering,the scattered waves will be composed mostly of S-waves.

For an elastic sphere of volume V = (4/3)na3, the amplitude

of the scattered field has the familiar dependence of (OZa3,

similar to that for acoustic scattering.

-f [Ojk OA(V . UO)

+ bll(UY, k + UUJ * Gij. k dV. (24)

Compared with equations (4) and (5), we recognize that theterm inside the square brackets of equation (24) with the factor

Ui(X) = JQj(~) * Gij(x, ~) dV(~),

where U i (x) is the scattered field at point x, Gij is the elastody­namic Green's function, and Qi is the equivalent body force (seeAki and Richards, 1980, chap. 4); the integration is over thewhole volume. Substituting the expression of Qi into equation(23) and integrating out the terms with the gradients of elasticconstant using integration by parts, we have

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Scattering by an Elastic Heterogeneity 589

(30)

where oPo , 01.,0' and 0110 are the parameter perturbations at thecenter of the heterogeneity and satisfy

oPo Iv P(~) dV(~) = op V, etc.

Putting equation (29) into equation (25) yields

(02 I .PuP(x) = __ - e- Ilo( t - r/aol

I 47l:U~ r

x f P(~) exp [i ~ ~I - i~ (x· ~)l dV(~). (31)" ao ao J

We recognize that the formula is the same as the Rayleighscattering with the volume V replaced by a factor

0\(x) = f P(~) exp [i ~ ~I - i~ (x . ~)j dV(~) (32)t' ao ao

We call 0\(x) the "volume factor." In the case of a uniforminclusion P(~) = I, it becomes the "shape factor" of Gubernatiset al. (1977b). In this way we can rewrite equations (25) through(28)as

PUr(x) = ~ e ;W(I r'O)E\(x)0I(x ),

r

PU7(X) = ~ e iwU r/llolE2(X)02(X),r

(33)

and

SU7(X) = ~ e iW(I-r/llolE4(X)04(X).r

where E1-E4 are the elastic wave Rayleigh scattering factors fora unit volume, and 0\-04 are the corresponding volume fac­tors which are the scalar wave scattering patterns

and

I 8S4=-2sin-,

~o 2

where x1 IS the unit vector in x (-direction, i.e., the incidentdirection, SI S4 are the exchange slowness vectors (Figure 9),and 8 is the scattering angle. Note that equation (34) is in theform of a 3-0 spatial Fourier transform. Putting K, = coSn , wehave

On (x) = 1P(~) exp (iK n • ~) dV(~) = P3 (Kn), (36)

where 1\(K n ) is the 3-0 Fourier transform of P@. If theparameter's spatial variation is spherically symmetrical, i.e.,P(~) is only the function of I~ I = r r- we introduce the sphericalcoordinates v.. 01. <pd having the polar axis along the S,direction. Then we can integrate equation (36) with respect toHI and <PI to obtain

/

FIG. 9. The coordinate system for the calculation of volumefactors.

In equation 34 we have

J( 1)2 (1)2 7S2 = 1S21 = - + A - ~ cos 8,a o f-'o Uo f-'O

I _ I_s =-x --x

3 ~o 1 ao'

(35)

150· ~

160· .. 05

/

1700 I

{180· --t-- t-+-o-

170· \\

160· \150·

140·

30·

I _ I_s =-x --x

4 ~o 1 ~o'FIC;. 10. Volume factors of S-wave scattering OS for a uniform

sphere for different frequencies.

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590 Wu and Aki

180·f--+-+-+-+--+--+--+--+---+-~--+--+---+---+---+---+---+---+--;---'-l' O·

50· 40·

(37)

150·

160·

170·

170·

160·

150·

140·

140·

130· 120· 110· 100· 90· 80· 70· 60·

130· 120· 110· 100· 90· 80· 70· 60· 50· 40·

30·

20·

10·

10·

20·

30·

0 n(9) = ~ (~(rl) sin (coSnrl)r[ dr,coSn Jo-4n 1 0

=-----coSn 2 o(coSn)

x f":(rd cos (coSnrd dr,

2n 0 ~

= - coSn

o(coSn)

P(coSn),

where P(coSn) is the 1-0 spatial Fourier transform of the param­eter variation P(r).

We now show two examples, a uniform sphere and a localGaussian type spherical heterogeneity to demonstrate the gen­eral scattering characteristics.

FIG. 11.0 s, when (co/~o)a = 2n. A uniform sphere

For a uniform sphere, P(r) is a boxcar function. The resultingvolume factor is

(38)

(39)

4na3

[sin coSna l0 n(6) = ---2 S - cos coSna ,(coSna) co na

where a is the radius of the sphere. Note that

sin (coSna) I----"-- - cos coSn a::::: - (coSnaf,

coSna 3

0 n (9) ::::: V, when coSn a ~ 1.

In Rayleigh scattering, the phase differences between thevolume elements are neglected, so the volume factor is equal tothe real volume. For Mie scattering, the volume factor is gener­ally smaller than the volume because of the interference be­tween the fields from different volume elements. Figures 10-12and 13-15 give the volume factors for S-wave scattering 0 s =

0 4 and for wave converting 0 c = O 2 = 0 3 , respectively. Inplotting, we normalized the values with the volume of thesphere. It can be seen that for P-P or S-S scattering the volumefactor 0 always has the main lobe in the forward scatteringdirection, since all the scattered waves from different volumeelements are always in-phase along the incident direction.When the volume becomes larger, small back lobes and sidelobes start to appear (Figures 11 and 12), but most of thescattered energy for the common-mode scattering (P-P or S-S)

will be contained in the main forward lobe. However, becausethe scattered waves have different velocities from the incidentwaves, destructive interference also occurs in the forward direc­tion. Therefore the volume factor 0 c along the incident direc­tion will reduce its value and will become oscillatory withincreasing frequency. The scattering pattern then becomesmore complex, and the converted energy diverges to all direc­tions (e.g.,Figure 15).

The general scattering pattern is a combination of E(x) and0(6). Figures 16 through 26 show examples of scattering pat­terns. Note that for P-P or S-Smer scattering the main lobebecomes sharper and much bigger than the side lobes whenfrequency goes higher (Figure 17 and 22), while the convertedwaves and cross-coupled waves (SUra,) diverge into many smalllobes in all directions and become much smaller compared withthe main lobe (Figures 17, 22, and 26). Also remarkable is the

30·

30·

10·

10·

20·

20·

40·

40·

40·

~aolO

FIG. 12.0 s, when (como)a = 10.

130· 120· 110· 100· 90· 80· 70· 60· 50·

130· 120· 110· 100· 90· 80· 70· 60· 50·

130· 120· 110· 100· 90· 80· 70· 60· 50·

140·

140·

140·

160·

170·

170·

150·

180·f--+-+-+--+--+--+--+---+---+-~O-+---+---+---+---+--+---+-+--+---'-JO·

160·

140· 40·

€'l150· 30·

160· 20·

170· w 10·-a=7T1%

180· O'

170· - 10·

160· 20·

150· 30·

150·

FIG. 13. Volume factors of converted wave 0 c for a uniformsphere for different frequencies.

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ScaUering by an Elastic Heterogeneity 591

150'

140' 130' 120' 110· 100' 90· 80· 70' 60· 50· 40';-~/@-cl

130 ' 150'

140' 130' 120' 110' 100' 90' 80' 70' 60'".---~~

(~~~~2~ IAo fLo Po

50' 40'

30'

160' 160' 20'

170· _

1800 -------t-+------- +---+--+--+-+---j+-k

170' -

160'

---~,~I~.

+-_+~-+_4 +-?'~ 0

j10·

170·

180'

170'

160'

P~P

t- +- --+---+_ ........

P -s

- 10'

-dlri"",,-<--+--+- 0.;.5."".+-__+--+--+--+'o·_3~C

10'

20'

150· 150' 30·

140· 130' 120· 110· 100' 90' 80' 70' 60· 50' 40'/

140' 130· 120' 110· 100' 90' 80' 70· 60· 50' 40'

FIG. 14.e C, when ((j)/~ola = 2rr. FIG. 17.Same as Figure 16 for (m/uola = 10.

30·

20'

'0·

30·

20'

40'

40·

50'

130' 120· 110· 100· 90· 80· 70' 60· 50·140'

160' _

//

170' - rI,,

180' -r t-- -+-- +0.2

170· I-- ~x160"

I I

I,,

'50·, I\ I

\ II

140' 130' 120· 110' 100"'1l9' 80' 70· 60'

150'

130' 120· 110' 100' 90· 80' 70' 60· 50'"r::':

140'

170·

160·

170· --

150·

160'

180'

150'

FIG. 15.e c, when ((j)/~ola = 10. FIG. 18. Same as Figure 16, but for OA/Ao = 01-1/1-10 = Sp/Po, i.e.,the impedance type scattering.

FIG. 19. Same as Figure 18, when (m/uola = 10.

160·

170'pop

180· +---< +-----+-

P-S

170· - ~x160'

150' 30·

10'

20'

50· 40·

20·

30·

~a =:100 0

"IMPEDANCE ( ~ ~ ~ ~ !!E)TYPE" x, fL. Po

10·

l2~o"=~C+=--+- +--+--+--;--+--+--;--10'

0.2 0.5

130·140'

150'

10'

FIG. 16. Scattering patterns of a uniform sphere for P-waveincidence for different frequencies. When OA(Ao = OI-l!l-Io =2(op/Pol and Ao = 1-10' The upper half is of P-P scattering, thelower half is of P-S scattering. All the patterns are axiallysymmetric about the x-axis.

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592 Wu and Aki

140' 50' 40' 140' 130' 120' 110' 100' 90' 80' 70' 60' 50' 40'

40'130' I?D' 110' 100' 90' 80' 70' 60' 50'140'

150'SUS /(~o)

Mer .80

30'"IMPEDANCE (~,~)

TYPE" ~ Po160' 20'

170' 10'

S-S Jt2MP.

180' O'S-P

0.2SU~

0.5

170'

f'10'

----tr -- ~o=l

160 e, 20'Y ----- ~ 0= 1r

f3.150' 30'30'

20'

10'

40'130' 120' 110' 100' 90' 80' 70' 60' 50'

p-s

P-P

"VELOCITY (~, ~ , _ £e )TYPE" A. 1". P.

140'

170'

170'

150'

160'

160'

180·'f-.-+--+--+--+--+-f--+-+--+--I-;;;;;.=-<-f--+--+-+'-+--+--+----:c O'

150'

FIG. 20. Same as Figure 16, but for 8A/A = 8!l/!l0 = - 8p/Po,i.e., the velocity type scattering.

FIG. 23. Same as Figure 21 but for 8!l/!l0 = 8p/po, i.e., theimpedance type scattering.

140' 130' 120' 110' 100' 90' 80' 70' 60' 50' 40'140' 130' 120' 110' 100' 90' 80' 70' 60' 50' 40'

150' 150' "IMPEDANCE (~, £e)TYPE" 1". p.

30'

20'

20'

10'

30'

S-S

S-P

170'

160'

170'

160'

150'

10'

112~O180'f--+-+-+--+--+--+--+--+~~~F==:;:::::::+--+--+--+--+--+---+----j O'

10'

20'

10'

20'

30'

w------ - 0"'..".

f3.~'y

s-s

s-p

170'

180·1--+--+--+--+---+--jL,-+--+¥--+-~~-+'+-I____+--+--+-+-+--+----.j0·

160'

170'

150'

160'

140' 130' 120' 110' 100' 90' 80' 70' 60' 50' 40' 140' 130' 120' 110' 100' 90' 80' 70' 60' 50' 40'

FIG.21. Scattering patterns of a uniform sphere in the x-y planefor S-wave incidence for different frequencies when O!l/!lo =2(op/Pol. The upper half is of S-S scattering, the lower half is ofS-P scattering. Note that there exists only the meridian compo­nents of scattered S-waves.

FIG. 24. Same as Figure 23 for ((J)/~o)a = 10.

140' 40'

40'130' 120' 110' 100' 90' 80' 70' 60' 50'140'

150'"Vi~~~"TY(~,-~)

sus lI.!!!o)230'Mer /3

0

Ii Po

160' 20'

170' 10'

S-S

180' O'

S-P0.5

170' sU Pf(!o'o)210'

T ...'!!... 0 ,7Tr f3.

~' f30160' 20'

Y

150' 30'

40'

40130' 120 110 100 90 80 70 60 50

130' 120' 110' 100' 90' 80' 70' 60' 50'

140'

140'\ \

· (~'2~ IsU~er/ 10 30

fio Po

· 20

· ~a=IO 10'e;S-S

~ ---- I":':.· 't~ 0.5 1 1.0S-P

~o=IO SU~'f-~

f'f3. 10'

+· 20

Y

· 30

/ . . . . . . . . .

160

170

180

170

150

160

150

FIG. 22.Same as Figure 21 for ((J)/~o)a = 10.FIG. 25. Same as Figure 21 but for O!l/!lo = -op/Po, i.e., the

velocity type scattering.

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Scauering by an Elastic Heterogeneity 593

140' 130' 120' liD' 100' 90' 80' 70' 60' 50' 40' 140 0 130' 120' uo- 100' 90' 50' 70' 60' 50' 40'

l30'~Oo=O.5ISO' SU~at [500 ®P 3D'

~a:l(3.

160' 20' 160' 20'

170 10' 170' 10'

S-Slot ~, ~.

180' .- D' 180 0 D·0.5

S -SlotSU~ot/?

170' 10' 170 0 10'

w160' 73, 0

7T 20' 160' - 20'

_~ __~__j 30'

8 t

ISO' 150 0

~ °0=0.5 3D'

140' 130' 120' no- 100' 90' 80' 70' 60' 50' 40' 140 0 130' 120' 1100 100° 90° 80· 70' 60' 50' 40'

FIG. 26. Scattering patterns of scattered latitudinal componentof a uniform sphere for S-wave incidence for different fre­quencies. The plane has a 45 degree angle to the x-y plane witha common x-axis.

FIG. 27. Volume factors for a Gaussian heterogeneity. Theupper half-plane is the case of P-P, the lower half-plane is forP-S.

impedance-type scattering which does not have a main lobe inthe forward direction and has features similar to the convertedor cross-coupled waves (Figures 18, 19,23 and 24).

Since there is no latitudinal component of scattered S-wavesin the x-y plane [see equation (22)], we took a plane with thecommon x-axis having 45 degree angle with the x-r plane(Figure 26) to present the changes of the scattering pattern withincrease of frequency. Figure 26 gives the patterns in that planefor three frequencies. Note that there will be four small lobes oflatitudinal component bending close to the x-axis when thewavelength becomes shorter, though the x-axis is a node. Thesefour small lobes give rise to the scattered cross-component(here the z-component) in the nearly forward direction. whichmight be very useful in detecting the existence of scatteringinclusions (Turpening, 1984).

A Gaussian spherical heterogeneity

Suppose the parameter variation of a spherical heterogeneityhas a Gaussian shape, i.e.,

P(r) = exp (r2/a~), (40)

where ao is the characteristic length of the heterogeneity. Fromequation (37) we have

en = (fiao)3 exp [-(wSnao)24l (41)

In Figure 27, eP are shown on the upper half-plane and eC onthe lower half-plane. Compared with the case of a uniformsphere, the patterns for high-frequencies are much simpler.though the low-frequency patterns are similar. Because of thesmoothness of the parameter variation, small side lobes do notappear in the high-frequency range. The scattered energy grad­ually concentrates to the forward lobe and becomes narrowerand narrower without splitting when frequency goes higher andhigher. Therefore the total scattering patterns also hecomesimpler for high frequencies. Figures 28 and 29 show the pat-

terns of impedance type and velocity type for P-wave incidence.Figure 30 shows the patterns for S-wave incidence only for(w/~o)ao = 10 to compare with the case of uniform sphere ofFigure 24. The patterns for cross-coupled component SUral areshown in Figure 31 and are presented in the same manner as inFigure 26. The four lobes gradually bend toward the forwarddirection without splitting when frequency increases.

ACKNOWLEDGMENT

We are grateful to Dr. H. Sato for his helpful discussionsabout the use of the Born approximation to the elastic wavescattering problem when he visited MIT during 1982. Dis­cussions with Dr. R. Turpening about the possibility of apply­ing the crosscoupling effect by scattering to exploration werealso very useful. Dr. K. Burrhus helped design and plot the 3-Dscattering patterns. We thank both of them. We are also grate­ful to the reviewers and the associate editor who pointed outthe works of Gubernatis et al., so that we could revise our paperappropriately. This research is supported partly by the VSPproject of Earth Resource Laboratory of Massachusetts Insti­tute of Technology (MIT) funded by CGG (CompagnieGenerale de Geophysique). Part of this paper is (the treatmentabout the uniform sphere) taken from the report "Verticalseismic profiling research" of the Earth Resources Laboratoryof MIT (Wu, 1983). We express here our gratitude to MIT forallowing us to publish the results in the report

REFERENCES

Aki, K" 1969, Analysis of the seismic coda of local earthquakes asscattered waves: 1. Geophys. Res., 74,615-63 L

- 1973. Scattering of P waves under the Montana Lasa: 1.Geophys. Res" 78. 1334. 1346.

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594 Wu and Aki

140' 130' 1200 110' 100' 90' 80' 70' 60' 50' 40' 140' 130' 120' 110· 100" 90° 80· 70' 60' 50' 40'

150'pur

30' 150' iOo : 1O 30'w "impedance typeLa;Oo~05 "i mpedonce type

/--- ------160" - , , 20' 160· SU~.r 20',",

170"I

10' 170' 10'I 5-5

p-p I10I

180'~:~:; tr

180' O'p-s 0.' 0'

S-P

170' PU;er 10' 170 0 10'\~ SU~ (ofter x100),

160' , 20' 160' 20',,,,

30'150" , 150',,

140' 130' 120· 110' 100 90' 80' 70' 60' 50' 40' 140' 130' 120' "0' 1000 90' 80' 70' 60' 50' 40'

FIG. 28. Scattering patterns of a Gaussian heterogeneity ofimpedance type for different frequencies. The upper half-planeis for P-P, the lower half-plane is for P-S.

FIG. 30. Same as Figure 28, for (oo/po)ao = 10, the upper half­plane is for S-S, the lower half-plane is for SoP.

10'

20'

20'

10'

30'

30'

10

40'

40'

77"....... ---- .. ,- \\I

II

.:0.'

130' 120' 110' 100' 90' 60' 70' 60' 50'140'

140'

180·!----''-----------.:'+''il'O.;;===--+-------jO·I

170'

160'

150'

170'

160 0

140" I~OO [20 0 II O' 100' 90' 80' 70' 60' 50' 40'

150' pu~/(*aot -5; °0 =0.530'

0

velocity type --,------160' -, 20'

" -,

-rv>> ,17Q" -<, v

10-,

p-p \

180'p-s

, ,17QO P S (W )2 , , 0.5

Um• r / a;oa ,-,

I'160'

,,,\

"150'\ 30'0.5 \1 \\, \

140· 130' 120· 110· 100' 90' 80' 70' 60' 50' 40'

FIG.29. Same as Figure 28, for velocity type heterogeneity. FIG. 31. Scattering patterns of scattered latitudinal componentof a Gaussian heterogeneity for S-wave incidence. The graphicplane has 45 degree angle to the polarization plane of theincident wave (x-y plane) with a common x-axis.

--- 1980a, Attenuation of shear waves in the lithosphere for fre­quencies from 0.05 to 25 Hz: Phys. Earth Planet. Inter., 21,50--60.

--- 1980b, Scattering and attenuation of shear waves in the litho­sphere: 1. Geophys. Res., 85, 6496-6504.-- 1982, Scattering and attenuation: Bul!. Seis. Soc. Am., 72,

5319-5330.Aki, K. and Chouet, B. 1975, Origin of coda waves: source, attenuation

and scattering effects: J. Geophys. Res., 80, 3322-3342.Aki, K. and Richards, P., 1980, Quantitative seismology, Ch. 13: W. H.

Freeman and Co.Berteussen, K. A., Christofferson, A., Husebye, E. S. and Dable, A.

1975, Wave scattering theory in analysis of P wave anomalies atNORSAR and LASA: Geophys. 1. Roy Astr. Soc., 42, 403-417.

Capon, 1., 1974, Characterization of crust and upper mantle structureunder Lasa as a random medium: Bull. Seis. Soc. Am., 64, 235-266.

Doornbos, D. J., 1976, Characteristics of lower mantle inhomogeneitiesfrom scattered waves: Geophys. 1. Roy Astr. Soc., 44, 447-470.

Einspruch, N. G., Witterholt, E. J. and Trull, R. 1960, Scattering of aplane transverse wave by a spherical obstacle in an elastic medium:1. Appl, Phys., 31,806-818.

Gubernatis, J. E., Domany, E., and Krumhansl, 1. A. 1977a, Formalaspects of the theory of the scattering of ultrasound by flows inelastic materials: J. App!. Phys., 48, 2804-2811.

Gubernatis, 1. E., Domany, E. Krumhansl, J. A. and Huberman, M.1977b, The Born approximation in the theory of the scattering ofelastic waves by flows: 1. Appl. Phys., 48, 2812-2819.

Haddon, R. A. W. and Cleary, 1. R. 1974, Evidence for scattering ofseismic PKP waves near the mantle-core boundary: Phys. Earth andPlanet. Int., 8, 211-234.

Hudson, J. A., 1977, Scattered waves in the coda of P: J. Geophys., 43,359-374.

Knopoff, L., 1959a, Scattering of compressional waves by sphericalobstacles: Geophysics, 24, 30--39.

--- 1959b, Scattering of shear waves by spherical obstacles: Geo­physics, 24, 209-219.

Love, A. E. H., 1944, A treatise on the mathematical theory of elasticity,4th ed: Dover Publications, Inc.

Miles, J. W., 1960, Scattering of elastic waves by small inhomoge­neities: Geophysics, 25, 642-648.

Morse, P. M. and Feshbach, H., 1953, Methods of theoretical physics:McGraw-Hill Book Co.

Pao, Y. H., and Mao, C C, 1973, Diffraction of elastic waves anddynamic stress concentrations: Crane Russak Co. Inc.

Rayleigh, 1. W. S., 1896. The theory of sound, V. II: Dover Pub­lications, Inc. (1945 edition).

Richards, T. G. and Menke, W., 1983, The apparent attenuation of

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Ying, C. F. and Truell. R. 1956, Scattering of a plane longitudinal waveby a spherical ohstacle in an isotropically elastic solid: J. Appl.Phys., 27, 1087-1097.

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