nonlinear inversion of the sh wave equation in a half-space for density and shear modulus...

Nonlinear inversion of the SH wave equation in a half-spacefor density and shear modulus determination

Wenhao ZhuDepartment of Engineering Science and Mechanics, The Pennsylvania State University, University Park,Pennsylvania 16802

Xu JunDepartment of Mechanics, Huazhong University of Science and Technology, Wuhan, 430074,People’s Republic of China

Joseph L. RoseDepartment of Engineering Science and Mechanics, The Pennsylvania State University, University Park,Pennsylvania 16802

~Received 27 February 1997; accepted for publication 8 October 1997!

This paper deals with the problem of multi-parameter nonlinear inversion of theSH wave equationin an elastic half-space. A numerical approach combining Born iteration and a regularizingtechnique is presented for simultaneously reconstructing 2-D distributions of density and shearmodulus in a scatterer embedded in a half-space. The well-to-well source–receiver schemecommonly used in geophysical exploration is considered in the model in which two incidentfrequencies are used to uncouple the parameters in frequency domain. The weighted residualmethod, along with bilinear interpolating functions, are used in the discretization procedure.Computer simulations have been conducted on several examples with different density and shearmodulus configurations. The numerical results show that the approach proposed has a uniformlyconvergence for the given objects and has a feature of treating the limited-source well-to-wellscheme that causes a more ill-conditioned equation in the inversion procedure. ©1998 AcousticalSociety of America.@S0001-4966~98!05102-9#

PACS numbers: 43.35.Cg, 43.35.Zc, 43.20.Bi@HEB#

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INTRODUCTION

The problem of inverse scattering has received mattention for its potential application in many practical fieldIntensive study has been performed on linear inverse plems, such as ray CT, diffraction tomography, Born or Rytinversion, etc.,1–5 for the situations of weak scattering ohigh-frequency approximation. Recently, many efforts habeen made to avoid the linearizing assumption and to invothe diffracting and multiple scattering effects in inverproblems. Newton6 generalized the Gelfand–LevitanMarchenko integral equation into three dimensions. Sianalytical solutions are hardly sought in this case, manymerical methods have been developed. Johnson and Tr7

used method of moment along with sinc basis functioTarantala8 applied an iterative method combined with an otimization procedure to investigate seismic inversion. Chet al.9 presented a distorted Born iterative method to sothe inverse scattering problem of the EM wave equationpermittivity reconstruction. However, most previous worwere restricted for single parameter inversion in an infinmedium. A more practical application is to consider the eltic wave inversion in a half-space as in seismic exploratand NDT, where the boundary surface effects cannot beglected.

In this paper, the multi-parameter nonlinear inversionSH waves in an elastic half-space is investigated numcally. In the problem, the source–receiver scheme isranged in a form of well-to-well format as in seismic explration ~see Fig. 1!. The point sources on the excitation arr

850 J. Acoust. Soc. Am. 103 (2), February 1998 0001-4966/98/1

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are assumed monochromatic and horizontally polarizedproduceSH waves. The forward scattered fields are mesured by the receiver array and then used in the inversprocedure. A numerical algorithm is developed that reforthe Born iterative procedure for inversion into a regularizoptimization procedure. Detailed numerical results with vaous types of density and shear modulus configurationspresented.

I. PROBLEM FORMULATION

The densityr and shear modulusm in an isotropic elas-tic half-space containing an inhomogeneous scattererV arewritten as follows:

r~r !5r0@11a~r !#, m~r !5m0@11b~r !#, ~1!

wherer0 , m0 are the density and shear modulus of the baground medium, respectively, anda, b are the correspondingdeviations fromr0 , m0 , which vanish outside the scattereV. The governing equation for theSH wave in frequencydomain yielded by a point source can be written as followomitting the time dependency of exp(2ivt):

m~r !¹2w~r ,r s!1“m~r !•“w~r ,r s!1v2r~r !w~r ,r s!

52s~v!d~r2r s! ~2!

with the boundary condition

]w

]yUy50

50, ~3!

85003(2)/850/8/$10.00 © 1998 Acoustical Society of America

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wherew(r ,r s) is the out of plane displacement field atr dueto a point source loaded atr s , ands(v) is the spectrum ofthe source. We can express the total fieldsw(r ,r s) as a sumof incident fields and the scattered fieldswtot(r ,r s)5win(r ,r s)1wsc(r ,r s) with the incident fieldwin(r ,r s) sat-isfying the following boundary value problem:

¹2win~r ,r s!1k02win~r ,r s!

52s~v!d~r2r s!/m0 ,]win

]y Uy50

50, ~4!

where k25v2r0 /m0 , k is the wave number of the background medium. From Eqs.~1!–~4!, we have

¹2wsc~r ,r s!1k2wsc~r ,r s!52$k2a~r !wtot~r ,r s!1“

•@b~r !“wtot~r ,r s!#%. ~5!

Applying Green’s theorem to Eq.~5!, integrating by parts thesecond term on the right side of Eq.~5!, and using the con-dition that b vanishes outside the scatterV, we obtain theintegral equation:

wsc~r ,r s!5E EV

@k2a~r0!G~r ,r0!wtot~r0 ,r s!

2b~r0!“G~r ,r0!•“wtot~r0 ,r s!#dr0 , ~6!

where r is on the receiver array,G(r ,r0) is the half-planeGreen’s function having the form

G~r ,r0!5i

4@H0

~1!~kur2r0u!1H0~1!~kur 82r0u!#, ~7!

and whereH0(1) is the zeroth-order Hankel function of firs

kind, r 8 is the mirror point ofr about the surface plane.Obviously, Eq.~6! is a nonlinear integral equation ina,

b, w, and some supplement equations are needed to enEq. ~6!. For the well-to-well scheme used here, each posource on the source array produces forward scattered fithat are measured by all of the receivers on the receivarray and then used as known data on the left side of Eq.~6!.When the point source is excited over the entire source arwe obtain a 2-D data set for one frequencyv1 . Repeatingthe above procedure for another frequencyv2 , we then ob-tain two equations fora, b:

FIG. 1. The geometry of the well-to-well scheme.

851 J. Acoust. Soc. Am., Vol. 103, No. 2, February 1998


setldsg

y,

wsc~r ,r s ,v i !5E EV

@ki2a~r0!G~r ,r0 ,v i !w

tot~r0 ,r s ,v i !

2b~r0!“G~r ,r0 ,v i !–“wtot~r0 ,r s ,v i !#dr0 ,

i 51,2. ~8!

Becausewtot(r ,r s ,v i) ( i 51,2) are unknown, we write thetotal fields forv i in the region ofV as

wtot~r ,r s ,v i !5win~r ,r s ,v i !

1E EV

@ki2a~r0!G~r ,r0 ,v i !w

tot

3~r0 ,r s ,v i !2b~r0!“G~r ,r0 ,v i !

–“wtot~r0 ,r s ,v i !#dr0 ,

rPV, i 51,2, ~9!

which are the direct scattering equations to be employedcalculating the fieldwtot in V. Combining Eqs.~8! and ~9!,we can solve numerically for the two 2-D functions:a, b,and the wave fieldswtot of two frequencies in the object. Aa part of the computer simulations in the paper, the forwscattered wave fields on the receiving line are calculatedEq. ~6! for the given objects.

II. DISCRETIZATION AND REGULARIZED ITERATIONALGORITHMS

Two inverse scattering Eqs.~8! for parametersa, b, andtwo direct scattering Eqs.~9! for wave fieldswtot of twofrequencies are therefore obtained. To implement themerical solutions to Eqs.~8! and ~9!, a weighted residuamethod and a Born iteration algorithm using a regularizatprocedure are used.

A. Discretization and interpolation

Consider a rectangular imaging regionD between thesource and receiving lines, which is large enough to conthe scatterer regionV. We discrete the regionD into N smallsquare elementsD5D1øD2•••øDN , and DiùD j50,when iÞ j . In each element, the unknown parametersa, b,and wave fieldw are interpolated by a set of bilinear bafunctions along with their values at the four nodes of telement:

f~r !5(i 51

4

f i f i~r !, ~10!

wheref~r ! is any ofa, b, wtot, f i is its value at thei th nodeof the element, andf i(r ) ( i 51,2,3,4) are bilinear interpolating functions about the spatial variablesx, y, which satisfyf i(r j )5d i j ( i , j 51,2,3,4), wherer j is the coordinate of thej th node of the element. The detailed forms off i and itsderivativesdi5] f i(r )/]x and t i5] f i(r )/]y are given in theAppendix.

Now the domain integral~on D! of Eq. ~8! can be dis-cretized into a sum of element integrals. In each of them,unknown functionsa, b, wtot are approximated by expan

851Zhu et al.: Nonlinear inversion of the SH wave equation


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sions like Eq.~10!. Then, the residual of such an approximtion compared with the left side of Eq.~8! are forced to be aminimum by simply using a set ofd functions as theweighted functions~point match!, where the matching pointare chosen on the receiving line. IfM number receivers areused in the problem, from Eq.~8! we have

wsc~rm ,r s ,v i !5(l 51

L

a lAml~wtot,v i !2(l 51

L

b lBml~wtot,v i !,

m51,...,M , i 51,2, ~11!

where

Aml~wtot,v!5k2(j 51

4 EDl j

E el j ~r !G~rm ,r ,v! f j

3S (i 51

4

Wl j ,itot f i D dr , ~12a!

Bml~wtot,v!5(j 51

4 EDl j

E el j ~r ! f jF ]G~rm ,r ,v!

]x

3(i 51

4

wl j ,itot di1

]G~rm ,r ,v!

]y

3(i 51

4

wl j ,itot t i Gdr

m51,...,M ,l 51,...,L, ~12b!

and whereL is the total number of nodes in the discretiztion, the subscriptl j is related to the j th element (j51,...,4) neighboring thel th node, andel j (r ) is a characterfunction defined as

el j ~r !

5 H1,0,

if j th element is connected tol th node,if j th element is not connected tol th node.

~13!

Similarly, the direct scattering Eq.~9! for wtot can be dis-cretized into the same form as Eq.~11!:

wtot~r k ,r s ,v i !5win~r k ,r s ,v i !1(l 51

L

wltotCkl~a,b,v i !,

k51,...,L, i 51,2, ~14!

where

Ckl~a,b,v!5k2(j 51

4 EDl j

E el j ~r !G~r k ,r ,v!

3 f j S (i 51

4

a l j ,i f i D dr2(j 51

4 EDl j

E el j ~r !

3F]G~r k ,r ,v!

]xdi1

]G~r k ,r ,v!

]yti G

3S (i 51

4

b l j ,i f i D dr k,l 51,...,L. ~15!



Finally, if S number of point sources are employed in thscheme, we can rewrite Eqs.~11! and ~14! as a couple ofmatrix equations:

Wsc5G~Wtot!–X, ~16a!

W in5T~X!–Wtot, ~16b!

where

G~Wtot!5FA~Wtot,v1!

A~Wtot,v2!

B~Wtot,v1!

B~Wtot,v2!G ,~17!

T~X!5F10 01G2FC~X,v1!

00

C~X,v2!G ,Wsc5@w1,1

sc ~v1!,...,wM ,1sc ~v1!,...,w1,S

sc ~v1!,...,

wM ,Ssc ~v1!,...,w1,S

sc ~v2!,...,wM ,Ssc ~v2!#T,

WI5@w1,1I ~v1!,...,wL,1

I ~v1!,...,w1,SI ~v1!,...,wL,S

I ~v1!,

...,w1,SI ~v2!,...,wL,S

I ~v2!#T, I 5 in,tot, ~18!

X5@a1 ,...,aL ,b1 ,...,bL#T,

and whereA, B are matrices ofM3S rows andL columnswith their elements defined by Eqs.~12a! and ~12b!, C is adiagonal matrix composed ofS number of sameL3L sub-matrices on the diagonal line with their elements definedEq. ~15!, ‘‘ T’’ denotes transpose. It is noted thatT(X) is asquare matrix whereasG(W tot) does not need to be squareas a regularization procedure will be applied to Eq.~16a!.

B. Regularizing procedure and Born iteration

It is known that the integral Eq.~6! for the parameters isa first kind Fredholm integral equation which is generaill-posed and thus the discretized equation is ill-conditionIn general, we should define a well-behaved problem withsolution as an acceptable approximation for the previousconditioned problem. To realize this, we adapt tTikhonov10 regularizing procedure, that is, instead of solvithe ill-posed matrix equation directly, we solve an optimiztion problem which minimizes the cost function given by

iG~Wtot!–X2Wsci21giH–Xi25min, ~19!

which leads to a normal equation,

Re@~G* ~Wtot!–G~Wtot!1gH* –H !–X#5Re@G* –Wsc#,~20!

whereH is a smoothing matrix,g is the regularizing parameter, and* denotes conjugation transpose. Generally,creasingg may decrease the ill-posed condition or increathe convergence rate, that may also decrease the accurathe solution. So there is a trade-off between accuracycomputing time. Estimates forg should be carried out duringthe numerical analysis. Furthermore, in most cases,smoothing matrixH is selected as the unit matrix or somdiagonal matrix. In the next section, Eq.~20! will be used to



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replace Eq.~16a! as the inverse scattering iterative equatioTo solve the nonlinear Eqs.~16b! and~20!, we make use

of the Born iterative method alternatively to the inverse adirect scattering procedures. This method is described aslows:

~1! First, substitute the total fieldWtot in the coefficient ma-trix of Eq. ~20! by the incident fieldW in, and solve Eq.~20! for X(1) as the first-order Born approximation ofX.

~2! Then, substitute the parameterX in the coefficient matrixof Eq. ~16b! by X(1), solve Eq.~16b! for W(2) as thesecond-order Born approximation ofWtot ~the first-orderBorn approximation ofWtot is W(1)5W in!.

~3! Next, substituteWtot in G(Wtot) of Eq. ~20! by W(2), andsolve the equation forX(2) as the second-order Born approximation ofX.

~4! Repeat steps~2! and~3! to obtain the higher-order Bornapproximations, and so on.

A diagram illustrating the cross iterative proceduregiven in Fig. 2.

There are two criteria to control the iterative proceduone is for the wave field~direct scattering phase! which isdefined as relative residual error~RRE!:

RRE5iWsc~ j !2Wsci2/iWsci2, ~21!

whereWsc is the measured or synthetic scattered field,Wsc(j )

is calculated from Eq.~6! with the total field in the integrandreplaced by thej th-order Born approximationW( j ). Theother is for the parameters~inverse scattering phase! which isdefined as the mean-square error~MSE!:

MSE15AED

~a~ j !2a!2 drY ED

a2 dr ,

~22!

MSE25AED

~b~ j !2b!2 drY ED

b2 dr ,

wherea ( j ), b ( j ) are j th-order Born approximations ofa, b.Since the aim of this paper is for the inverse scatter

for a, b, criterion ~22! is used in the numerical iterationpresented in the next section. However, it is noted that fopractical case, the real parametersa, b are unknown, so onlycriterion ~21! is available for controlling the iterative procedure.

III. NUMERICAL RESULTS AND CONCLUSION

Several examples with different density and shear molus configurations are considered and the computer simtions are conducted to evaluate the algorithms propoabove. The synthetic data of the scattered fields are prodby Eq. ~6! with the assumed various parameter functions

A. Computer simulations

The SH wave velocity in the background medium is sto beCT5Am0 /r054000 m/s. In examples 1–3, the imaing regionD is discretized into 11311 square elements wita side length ofh50.8 m. Six point sources~spaced equallywith 3.2-m interval! and 20 receivers~1.6-m interval! areused in these examples. This is an asymmetrical sour



.

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g

a

-a-ded

e–

receiver arrangement but is more practical because the fethe sources used, the less the experiments required. Theincident frequencies aref 151000 Hz andf 251500 Hz. Forexample 4, due to the complication of the parameter configrations, we use a net of 15315 elements~16316 nodes!along with eight point sources and thirty receivers.

1. Example 1

The parametersa,b are assumed to be ellipsoids with adistribution of the following form:

a~r !520.15A12S x210

2.4 D 2

2S y212

3.2 D 2

,~23!

b~r !520.1A12S x210

2.4 D 2

2S y212

2.4 D 2

.

This is shown in Fig. 3~a! and Fig. 4~a!. The reconstructedresults for the second, fourth, and ninth Born iterations agiven in Fig. 3~b!, ~c!, ~d! and Fig. 4~b!, ~c!, ~d! for a andb,respectively. In the iterative procedure, the unit matrixused as the smoothing matrix andg is selected in the regionof 1027– 1026. After the ninth iteration, the criteria of con-vergence are MSE150.039 and MSE250.127. It is difficultto obtain the two criteria below a satisfactory value at thsame time. Then, by changing the smoothing matrix bydiagonal matrix and recalculating the problem, it is realizethat MSE150.028 and MSE250.093 at the ninth iteration,which are somewhat better than the previous results. Timproved results are shown in Fig. 5~b!, ~c!, ~d!, and Fig.6~b!, ~c!, ~d!.

2. Example 2

The parametera is assumed to be a homogeneousquare cylinder with a constant value20.8 @shown in Fig.7~a!# andb is assumed to be an ellipsoid@shown in Fig. 8~a!#with the distribution as follows:

b~r !520.1A12S x210

3.2 D 2

2S y212

3.2 D 2

. ~24!

The results of the reconstruction for the second, fourth, asixth Born iterations are shown in Fig. 7~b!, ~c!, ~d! and Fig.8~b!, ~c!, ~d!. After the 6th iteration, we have MSE150.030 and MSE250.068. It converges faster than the example 1.

3. Example 3

The parametera is assumed to be a step function@atower as shown in Fig. 9~a!# with two constant values20.04and 20.08. b is assumed to be two homogeneous squacylinders with the same value of20.05, as shown in Fig.

FIG. 2. The cross iterative procedure.



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is

-

aFig.

FIG. 3. Reconstruction for density with an original distribution as~a!, at ~b!second,~c! fourth, ~d! ninth iterations, respectively.

FIG. 4. Reconstruction for shear modulus with an original distribution~a!, at ~b! second,~c! fourth, ~d! ninth iterations, respectively.



10~a!. The two incident frequencies are selected asf 1

51000 Hz andf 252000 Hz. The reconstructed results fthe first, third, and sixth Born iterations are given in Fi9~b!, ~c!, ~d! and Fig. 10~b!, ~c!, ~d!. After the sixth iteration,MSE150.021 and MSE250.078. The convergence ratefaster also.

4. Example 4The parametera is assumed to be for two similar ellip

soids with the same maximum20.05, as shown in Fig.

s

FIG. 5. Reconstruction for density with the same distribution as Fig. 3~a! byusing, a diagonal smoothing matrix at~a! second,~b! fourth, ~c! ninth itera-tions, respectively.

FIG. 6. Reconstruction for shear modulus with the same distribution as4~a! by using a diagonal smoothing matrix at~a! second,~b! fourth, ~c! ninthiterations, respectively.



a as

FIG. 7. Reconstruction for density with an original distribution as~a!, at ~b!second,~c! fourth, ~d! sixth iterations, respectively.

FIG. 8. Reconstruction for shear modulus with an original distribution~a!, at ~b! second,~c! fourth, ~d! sixth iterations, respectively.



s

FIG. 9. Reconstruction for density with an original distribution as~a!, at ~b!first, ~c! third, ~d! sixth iterations, respectively.

FIG. 10. Reconstruction for shear modulus with an original distribution~a!, at ~b! first, ~c! third, ~d! sixth iterations, respectively.



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11~a!. b is assumed to be for a homogeneous square cylinwith a value of20.05, shown in Fig. 12~a!. In this example,we use a net of 15315 elements with 16316 nodes and 8sources plus 30 receivers. The incident frequencies aresame as in example 3. The results of the reconstructionsthe first, fourth, and sixth Born iterations are shown in F11~b!, ~c!, ~d! and Fig. 12~b!, ~c!, ~d!. After the sixth itera-tion, MSE150.050 and MSE250.100.

B. Discussions

The four examples presented here represents four tof parameter distributions:~1! both a andb are continuous;~2! a has jumps whereasb is continuous;~3! both a andbhave jumps; and~4! b has jumps whereasa is continuous.All the examples in general show rapid convergence,both MSE1 and MSE2 fall below 0.1 within ten iterationThere is a trend in the results that the inversion accuracydensitya is better than that for shear modulusb. It is partlybecause in the discretization process the interpolation ofor a’s coefficients is higher than that forb’s @see Eqs.~12!and ~15!#. Example 1 also shows that ifa and b cannotconverge synchronically, we can try to use a nonunifodiagonal smoothing matrix. It is found after several tries tthe optimizing factorg for the regularization falls in the region of 1027– 1026 which gives a rapid convergence angood accuracy. Finally, the algorithm presented abovestable enough to treat the more ill-conditioned equationswe have met in the examples where asymmetrical sourreceiver schemes are used with fewer sources.

FIG. 11. Reconstruction for density with an original distribution as~a!, at~b! first, ~c! fourth, ~d! sixth iterations, respectively.



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IV. CONCLUSIONS

The problem considered in this paper represents a mpractical case that concerns two parameters in a bounmedium. We have presented the inversion algorithms baon Born iteration and a regularizing procedure and a serieexamples by numerical simulation to test the convergeand accuracy of the algorithms. The elementary resultssented here show promise that the approach can be exteto the case of elastic vectoral waves.

APPENDIX: THE INTERPOLATING FUNCTIONS ANDITS DERIVATIVES

Defining coordinate transform:

x51

4 (i 51

4

xi111

2hj, y5

1

4 (i 51

4

yi111

2hh, ~A1!

where (xil ,yil ) is the coordinate ofi th node ofl th element,and h is the side length of the element. Thus any elemwill be transformed into the standard element as shownFig. A1.

The interpolation functions used in the discretization agiven as follows:

f 15 14~11j!~11h!, f 25 1

4~11j!~12h!,~A2!

f 35 14~12j!~12h!, f 45 1

4~12j!~11h!,

and its derivatives as:

FIG. 12. Reconstruction for shear modulus with an original distribution~a!, at ~b! first, ~c! fourth, ~d! sixth iterations, respectively.



pl.

a-

] f i

]x5di , d15

1

2h~11h!, d25

1

2h~12h!,

d3521

2h~12h!, d45

21

2h~11h!,

~A3!

FIG. A1. The standard element in thej, h plane.



] f i

]y5t i , t15

1

2h~11j!, t25

21

2h~11j!,

~A4!

t3521

2h~12j!, t45

1

2h~12j!.

1A. C. Kak, Proc. IEEE67, 1245–1272~1979!.2A. J. Devaney, IEEE Trans. Geosci. Remote Sens.GRS-22, 3–13~1984!.3S. J. Norton and M. Linzer, IEEE Trans. Biomed. Eng.BME-28, 202–220~1984!.

4J. D. Achenback, D. A. Sotiropoulos, and H. Zhu, ASME Trans. J. ApMech.54, 754–760~1987!.

5W. H. Zhu, J. Acoust. Soc. Am.87, 2371–2375~1990!.6R. G. Newton, J. Math. Phys.21, 1689–1715~1980!.7S. A. Johnson and M. L. Tracy, Ultrason. Imaging5, 361–392~1983!.8A. Tarantala,Inverse Problem Theory~Elsevier, Amsterdam, 1987!.9W. C. Chew and Y. M. Wang, IEEE Trans. Med. ImagingMI-9 , 218–225~1990!.

10A. N. Tikhonov,On the Problems with Approximately Specified Informtion in Ill-posed Problems in the Natural Science, edited by A. N.Tikhonov and A. V. Goncharsky~MIR, Moscow, 1987!.



nonlinear inversion of the sh wave equation in a half-space for density and shear modulus...

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