scattering of elastic waves by a general anisotropic basin. part 1: a 2d model

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2001; 30:675–689

Scattering of elastic waves by a general anisotropic basin.Part 1: a 2D model

Marijan Dravinski∗ and Mark S. Wilson

Department of Aerospace and Mechanical Engineering; University of Southern California; Olin Hall 430;Los Angeles; California 90089-1453; U.S.A.

SUMMARY

Scattering of incident plane harmonic pseudo P-, SH-, and SV-waves by a two-dimensional basin ofarbitrary shape is investigated by using an indirect boundary integral equation approach. The basin andsurrounding half-space are assumed to be generally anisotropic, homogeneous, linearly elastic solids. Nomaterial symmetries are assumed. The unknown scattered waves are expressed as linear combinationsof full-space time-harmonic two-dimensional Green functions. Using the Radon transform, the Greenfunctions are obtained in the form of �nite integrals over a unit circle. An algorithm for the accurateand e�cient numerical evaluation of the Green functions is discussed. A detailed convergence andparametric analysis of the problem is presented. Excellent agreement is obtained with isotropic resultsavailable in the literature.Steady-state surface ground motion is presented for semi-circular basins with generally anisotropic

material properties. The results show that surface motion strongly depends upon the material propertiesof the basin as well as the angle of incidence and frequency of the incident wave. Signi�cant modeconversion can be observed for general triclinic materials which are not present in isotropic models.Comparison with an isotropic basin response demonstrates that anisotropy is very important for assessingthe nature of surface motion atop basins. Copyright ? 2001 John Wiley & Sons, Ltd.

KEY WORDS: two-dimensional triclinic basin; scattering; site response

1. INTRODUCTION

Anisotropy is wide-spread in the Earth’s structure [1] and anisotropic materials are increas-ingly used in engineering applications of composites [2]. Consequently, scattering of elasticwaves by two-dimensional isotropic basins has been the subject of many studies aimed atthe prediction of earthquake-induced ground motion atop a sedimentary basin [3]. These stud-ies provide reasonable understanding of wave scattering and site ampli�cation for isotropicmodels. However, wave scattering problems for anisotropic media are characterized by greatercomplexity than the isotropic models.

∗ Correspondence to: Marijan Dravinski, Department of Aerospace and Mechanical Engineering, University ofSouthern California, Olin Hall 430, Los Angeles, California 90089-1453, U.S.A.

Received 30 March 2000Revised 11 August 2000

Copyright ? 2001 John Wiley & Sons, Ltd. Accepted 24 August 2000

676 M. DRAVINSKI AND M. S. WILSON

Figure 1. Geometry of the problem.

The problem of scattering of elastic waves by a basin embedded in an elastic half-space maybe solved using an boundary integral equation approach. This method has advantage over the�nite element technique that discretization takes place only along the boundary of the scatterersand the radiation condition in the far �eld is satis�ed exactly. The approximate boundaryintegral equation method considered in this paper originates in the works of Kupradze [4] andUrsell [5]. An extensive literature review of applications of the boundary integral method toelastodynamics can be found in the papers by Kobayashi [6] and Beskos [7].Modelling the scattered waves in anisotropic media leads to di�culties associated with

accurate and e�cient evaluation of the Green functions. However, a recent advance in theanalysis of full-space Green functions by Wang and Achenbach [8] simpli�es greatly thenumerical computation. They have obtained a new representation of the Green functions in ananisotropic medium by using the Radon transform [9]. While the standard Fourier transformtechnique requires evaluation of multiple improper integrals, application of the Radon transformresults in an integration over a unit circle.Recently, Zheng and Dravinski [10; 11] considered ampli�cation of pseudo-P-, Rayleigh,

SH-, and SV-waves by an orthotropic basin of arbitrary shape. In the present paper, theanalysis is extended to include generally anisotropic materials.

2. PROBLEM STATEMENT

The model consists of a sedimentary basin of arbitrary shape without corners perfectly embed-ded in a half-space. The basin is subjected to an incident plane harmonic wave. The problemis to determine the displacement �eld throughout the medium. A typical model geometry isdepicted in Figure 1. The smooth interface � divides the medium into two regions, the half-space D1 and the basin D2. The basin and half-space are assumed to be generally anisotropic,homogeneous, linearly elastic solids.Let xi, t be right-handed Cartesian co-ordinates and time, respectively. Latin subscripts take

on the values 1–3 while Greek subscripts assume only the values 1 and 3. Tensor indices willbe extended by ; k to indicate di�erentiation with respect to xk . The n-dimensional real andcomplex vector spaces are denoted Rn and Cn, respectively. All �eld variables are assumedto be independent of the coordinate x2; thus, x=(x1; 0; x3)∈R3.

Copyright ? 2001 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2001; 30:675–689

SCATTERING OF ELASTIC WAVES BY A GENERAL ANISOTROPIC BASIN 677

For a general anisotropic medium, the steady-state equations of motion are given by [2]

�i�; � + �!2ui= − fi (1)

where ui and �ij are the displacement and stress �elds, respectively, fi is the body force (forceper unit volume), � is the constant mass density, and ! is the circular frequency. Throughout,the time dependence factor exp(−i!t) is suppressed. Unless stated otherwise, summation overrepeated indices is understood.Hooke’s law determines each stress component as a linear combination of the strains epq

�ij= cijpqepq (2)

where the strain tensor is related to the displacements by

epq= 12(up;q + uq;p) (3)

and cijpq is the elastic tensor (sti�ness). Since �ij and eij are symmetric and the elastic tensorsatis�es the strict inequality

cijpqeijepq¿0 (4)

the following symmetry properties hold [2]:

cijpq= cjipq= cijqp= cpqij (5)

For zero body force, Equations (1)–(3) and (5) imply

ci�p�up; �� + �!2ui=0 (6)

Associated with each domain DJ is a displacement �eld u(J )i , a stress �eld �

(J )ij , an elastic

tensor c(J )ijpq, and a constant mass density �(J ), J ∈{1; 2}.

Since the surface of the half-space is stress-free, the boundary conditions are

�(J )i3 = 0 for x∈ @DJ ; J ∈{1; 2} (7)

where @D1 is the free surface of the half-space and @D2 is the free surface of the basin alongx3 = 0 (see Figure 1).If the basin and half-space are perfectly bonded along the interface �, then the displacements

and tractions are continuous across �, i.e.

u(1) = u(2)

t(1) = t(2)x∈� (8)

where the traction �elds are given by

t(J )i =�(J )ij j; J ∈{1; 2} (9)

and designates the outward unit normal to � (see Figure 1).The general incident plane harmonic wave for an anisotropic solid is represented by [2]

u(1)I =A0U0 exp(i!s0 · x) (10)

with amplitude A0, polarization vector U0, and slowness vector s0= n0=v0 with unit propagationdirection n0 and phase speed v0. As it strikes the basin, the incident wave is scattered. Theunknown scattered wave �eld is discussed next.



Figure 2. Displacement spectral amplitude for a full space {x1; x2; x3} along x3 = 0. A distributed harmonicload of unit magnitude acting in the x1 direction over the region {|x1|6 1; x3 = 0} is applied (after Barrosand Neto [14]). The material is orthotropic in the coordinate system {x′i} with material parametersc′1111 = 6; c′1133 = 6; c′3333 = 9; c′2323 = 1, and �=1. Dimensionless frequency is chosen !

√�c′2323 = 1.

The calculations were done with respect to a coordinate system {xi}, obtained from {x′i} through arotation about x′2-axes for an angle − �

12 . Solid and dash lines are results of this study. Stars and opencircles denote the results of [14].

3. PROBLEM SOLUTION

3.1. Scattered wave �eld

Upon reaching the basin, the incident wave is partially re ected and partially transmitted bythe interface �. Therefore, the total wave �eld for the half-space consists of both free- andscattered wave �elds. The motion within the basin, on the other hand, consists of scatteredwaves only. Thus,

u(1) = u(1)� + u(1) s; x∈D1u(2) = u(2) s; x∈D2

(11)

with corresponding tractions

t(1) = t(1)� + t(1) s; x∈D1t(2) = t(2) s; x∈D2

(12)

where the superscript s and � designate the unknown scattered waves and free-�eld, respec-tively. The free-�eld is the sum of the incident and the re ected waves in the absence of thevalley [12].In the indirect boundary element method, the scattered wave �eld is expressed in terms of

single-layer potentials [4; 5]

u(J )si (x)=∫�(J )q(J )k (x0)g

(J )ik (x;x0) dx0 J ∈{1; 2} (13)

where �(1) and �(2) are auxiliary curves de�ned inside and outside the basin (see Figure 2),q(J )k are unknown density functions, and g(J )ik are the two-dimensional time-harmonic



full-space Green functions de�ned in the appendix. If the functions q(J )k are assumed to belinear combinations of discrete line sources [10; 11]

(q1; q2; q3)(1)(x0)=M∑m=1(am; bm; cm)�(x0 − xm) xm ∈SM

(q1; q2; q3)(2)(x0)=L∑‘=1(d‘; e‘; f‘)�(x0 − x‘) x‘ ∈SL

(14)

where M and L are positive integers while SM ⊂�(1) and SL⊂�(2) are sets of M and Lpoints, respectively. Substitution of Equations (14) into the integral (13) yields the followingexpressions for the scattered wave �elds

u(1)si (x) = amg(1)i1 (x;xm) + bmg

(1)i2 (x;xm) + cmg

(1)i3 (x;xm) xm ∈SM (15)

u(2)si (x) = d‘g(2)i1 (x;x‘) + e‘g

(2)i2 (x;x‘) + f‘g

(2)i3 (x;x‘) x‘ ∈SL (16)

where summation over repeated indices m and ‘ is understood.The interface � has been augmented in the integral equation (13) by the auxiliary curves

�(J ), J ∈{1; 2}. In general, �(1) and �(2) are de�ned inside and outside of the interface �,respectively [10; 11]. The sources are distributed on �(1) and �(2) in such a way as to gen-erate the unknown scattered wave �elds. 3(M + L) density coe�cients (am, bm, cm, d‘, e‘,f‘ m=1; : : : ; M ; ‘=1; : : : ; L) are needed to approximate the scattered waves in the half-spaceand the basin. The next section discusses the evaluation of the density coe�cients.

3.2. Linear system of equations

Let N , P, and Q be positive integers and let a set of N points, SN ⊂�, approximate theshape of the interface �. Likewise, let SP ⊂ @D1 and SQ ⊂ @D2 be sets of P and Q points,respectively, which discretize the free-surface. By requiring the displacement �elds (11) andtraction �elds (12) to satisfy the continuity condition (8), it follows that:

u(1) s(x)− u(2) s(x)=−u� (x)t(1) s(x)− t(2) s(x)=−t� (x) x∈SN (17)

Furthermore, by requiring the stress �elds generated by the displacements (11) to satisfy thestress-free boundary conditions (7), it follows that:

�(1)i3 (x)= c(1)i3p�u

(1) sp; � (x)=0 x∈SP

�(2)i3 (x)= c(2)i3p�u

(2) sp; � (x)=0 x∈SQ

(18)

The free-�eld already satis�es the free-surface condition and therefore does not appear in (18).However, since the scattered wave �elds (15) and (16) are linear combinations of the full-spaceGreen functions, g(J )ik , they do not automatically satisfy the stress-free boundary conditions.By collecting the 2N vector equations (17) and the P + Q vector equations (18), the

following linear system is obtained:

Aa= b (19)



Here a∈C3(M+L) is the vector of unknown densities coe�cientsa=(a1; : : : ; aM ; b1; : : : ; bM ; c1; : : : ; cM ; d1; : : : ; dL; e1; : : : ; eL; f1; : : : ; fL)T: (20)

The right-hand side b∈C3(2N+P+Q) and 3(2N+P+Q)× 3(M+L) complex matrix A are givenby

A=

[H(1)ij ] 03P×3L

[G(1)ij ] [−G(2)ij ][T (1)ij ] [−T (2)ij ]03Q×3M [H(2)

ij ]

; b=

0 3P

[−u� (xn)][−t� (xn)]03Q

xn ∈SN (21)

where

G(1)ij = [g(1)ij (xn;xm)]∈CN×M ; G(2)ij =[g(2)ij (xn;x‘)]∈CN×L

H(1)ij = [h(1)i3j (xp;xm)]∈CP×M ; H(2)

ij =[h(2)i3j (xq;x‘)]∈CQ×L

T(1)ij = [h(1)ikj (xn;xm) k]∈CN×M ; T(2)ij =[h(2)ikj (xn;x‘) k]∈CN×L

x‘ ∈SL; xm ∈SM ; xn ∈SN ; xp ∈SP; xq ∈SQHere hijk = cijp�gpk; � are the stress Green functions (Appendix). 0n denotes the zero vectorof Cn. Similarly, 0m×n denotes the zero matrix of Cm×n (the set of m× n matrices over thecomplex �eld). The linear system (19) may be solved by the QR method [13] to determinethe solution vector a and, thus, the unknown scattered wave �elds (15) and (16). Once thescattered waves are known the displacement �eld throughout the medium can be calculatedaccording to Equation (11).Before solving system (19), it is necessary to evaluate the Green functions, locate the

auxiliary curves, and designate the collocation and source points. This is considered in thenext section.

4. NUMERICAL PROCEDURE

This section deals with the speci�c numerical issues which arise in the indirect boundaryintegral equation approach used in this paper. The numerical evaluation of the Green functionsfor anisotropic media is dealt with �rst, followed by a discussion of the auxiliary curves,collocation points, and sources. Finally, the convergence criteria are listed and the results ofa parametric analysis are presented.

4.1. Green functions

4.1.1. Integral representations of regular and singular parts. The primary numerical di�cultyin the indirect boundary integral equation approach is the accurate computation of the Greenfunctions. The standard Fourier transform approach to evaluate the Green functions requiresevaluation of improper integrals which is di�cult to accomplish numerically [14]. Recently,the Green functions have been derived for anisotropic media by using the Radon transform [8].



The key feature of the two-dimensional Radon transform approach is that the Green functionscan be described as integrals over a unit circle. This technique has been used for evaluatingthe Green functions in the present paper.The Green functions are de�ned in the Appendix. It can be shown that the displacement

Green functions can be written in the form

gpk =14�2

∫|n|=1n·y¿0

3∑m=1�mpk�(k

mn · y) dn (22)

where �mpk and � are de�ned in Equations (A4) and (A5), respectively. It should be notedthat both �mpk and k

m depend on n. Further modi�cations of Equation (22) are achieved asfollows: for a non-zero vector y=x−x0 and a unit vector n in the (x1; x3)-plane, let � be thecounter-clockwise angle between y and n. If y denotes the magnitude of y, then

n=y−1(y1 cos �+ y3 sin �; 0; y3 cos �− y1 sin �) (23)

and n · y=y cos �. Since n · y¿0 for �∈ (− �2 ;�2 ), Equation (22) becomes

gpk =14�2

∫ �=2

−�=2

3∑m=1�mpk�(k

my cos �) d� (24)

It can be shown [15] that �(kmy cos ��)+2 ln(1− 2� |�|) is a continuous function for �∈ (− �

2 ;�2 )

with �nite left (right) limit at �2 (− �2 ). Therefore, Equation (24) may be written as

gpk = gRpk − gSpk (25)

where

gRpk =14�2

∫ �=2

−�=2

3∑m=1�mpk[�(k

my cos �) + 2 ln(1− 2� |�|)] d� (26)

gSpk =12�2

∫ �=2

−�=2

3∑m=1�mpk ln(1− 2

� |�|) d�

=12�2

∫ �=2

0

3∑m=1[�mpk(�) + �

mpk(−�)] ln(1− 2

��) d� (27)

Equation (26) involves a regular integral which may be computed accurately by an openquadrature formula. On the other hand, Equation (27) incorporates an improper integral. If thesubstitution z=1− 2

�� is introduced into Equation (27), this yields

gSpk =�∫ 1

0

3∑m=1[�mpk(

�2 (1− z)) + �mpk( �2 (z − 1))]ln(z) d z (28)

Equation (28) is now in a form suitable for treatment by a weighted Gaussian quadraturescheme [16].It can be shown that the stress Green functions are given by (see the Appendix)

hijk =cijp�4�2

∫|n| = 1n·y¿0

3∑m=1kmn��mpk�

′(kmn · y) dn (29)



where prime denotes di�erentiation with respect to the argument. If the de�nition for n givenby (23) is introduced into Equation (29), the following representation is obtained:

hijk =cijp�4�2

∫ �=2

−�=2

3∑m=1kmn��mpk�

′(kmy cos �) d� (30)

It can be shown [15] that �′(kmy cos ��)+2(kmy cos �)−1 is a continuous function for �∈ (− �2 ;�2 )

with �nite left (right) limit at �2 (− �2 ). Therefore, Equation (30) may be written as

hijk = hRijk − hSijk (31)

where

hRijk =cijp�4�2

∫ �=2

−�=2

3∑m=1kmn��mpk

[�′(kmy cos �) +

2kmy cos �

]d� (32)

hSijk =cijp�2�2

∫ �=2

−�=2

3∑m=1n��mpk

1y cos �

d�

=cijp�2�2

∫ �=2

0

3∑m=1[n�(�)�mpk(�) + n�(−�)�mpk(−�)]

1y cos �

d� (33)

Equation (32) is a regular integral and may be computed accurately by an open quadratureformula. The integrand of Equation (33) has an apparent singularity when �= �

2 . However,since n�(− �

2 )=−n�( �2 ) and �mpk(− �2 )=�

mpk(

�2 ), the singularity is removable. Hence, the singular

part of the stress Green functions may also be computed by an open quadrature formula.Due to symmetry, there are six displacement Green functions and eighteen stress Green

functions to calculate. Dravinski and Zheng [17] proposed a method to simultaneously evaluatethe Green functions by utilizing the elastodynamic state vector approach. The key feature ofthis procedure is that it avoids repeated calculations of the functions needed to perform thenumerical integration. This method is used to compute the elements of the linear system (19).

4.1.2. Veri�cation of Green functions. The exact full-space time-harmonic two-dimensionalGreen functions for an isotropic solid are given by Kobayashi [6]. If the material propertiesin Equations (22) and (29) are taken to be isotropic, the method for calculating the Greenfunctions described in Section 4.1.1 may be compared to the exact results. Excellent agree-ment was found for various arguments (x), circular frequencies (!), and isotropic materialproperties.The numerical method for computation of the Green functions was veri�ed for an anisotropic

solid as well. For this purpose a full-space loaded uniformly by a harmonic load over the region{ | x1 |61; x3 = 0} was considered. The load acts in the x1-direction and has unit intensity. Forthis problem, Barros and Neto [14] calculated the displacement �eld for a orthotropic solidde�ned in the co-ordinate system x′i . The co-ordinates x

′i and xi and the corresponding sti�ness

tensors are related by

xi = ‘ijx′j (34)cijmn = ‘ip‘jq‘mr‘nsc′pqrs (35)



Figure 3. Auxiliary curves, sources (solid circles) and collocation points (open circles) of the problem.

where ‘ij denotes a counter-clockwise rotation of − �12 about x

′2-axis. It should be noted that the

angle of rotation in Figure 3 of Reference [14] should have been − �12 instead of

�12 (Barros,

1999: private communication).In Figure 2, the results of [14] are compared with the integration method reported in this

paper. Excellent agreement can be observed between the two results con�rming the accuracyof the proposed algorithms used for evaluation of the Green functions. Furthermore, as a resultof these tests, the optimum integration parameters may be speci�ed as well. For the Gauss–Legandre quadrature method, the order of quadrature is mgl = 10 and the number of panelsis ngl = 10. For the weighted Gaussian scheme which handles logarithmic singularities [16],mln = 20. These values will be adopted throughout the paper.

4.2. Convergence and parametric analysis

Solution of system (19) for the 3(M + L) source density coe�cients de�nes the scatteredwave �elds in the basin and the half-space. Equations (11) will then specify the total dis-placement throughout the medium. However, the choice of auxiliary curves, source location,and collocation points (Figure 3) necessary for evaluation of the scattered wave �eld has tobe determined through careful parametric study of the problem. In this work the proceduresuggested by Zheng and Dravinski [18] is adopted to determine these key parameters whichare:• �1; �2;�1;�2; �—size and position of the auxiliary curves, where �1¡1 and �2¿1 arescaling parameters used to generate the auxiliary surfaces �(1) and �(2) from the interface� [18] while �1;�2; and � are vertical o�sets de�ned by Figure 3;

• M , L1 and L2—number of source points along the auxiliary curves �(1) and �(2);• N—number of collocation points along the interface � between the half-space and the basinwhere the continuity of displacement and traction �elds is imposed;

• P, Q—the number of collocation points along the half-space surface, outside and inside thebasin, respectively, where the stress-free boundary conditions are imposed;

• size of the half-space surface where discretization by P occurs.



The lack of analytical solutions for the problem at hand requires an indirect veri�cation ofthe numerical calculations. This is accomplished by the following tests:

(1) Isotropic transparency: The elastic constants of the half-space and basin are chosento be identical and isotropic. The parameters are varied in order to produce the free-�eldresponse in an isotropic half-space. This provides the initial choice of the parameters neededfor evaluation of the scattered wave �eld.(2) Anisotropic transparency: The elastic constants of the half-space and basin are assumed

to be identical and anisotropic. The parameters are chosen to produce the free-�eld responsein an anisotropic half-space.(3) Norm test: For a half-space and basin of general anisotropic nature, the parameters are

chosen to minimize the norm ‖Aa − b‖.Test 1 is used to determine the initial estimate of the key parameters of the problem while

Tests 2 and 3 are used to determine the �nal numerical results.In this study, the optimal parameters are found to be (Figure 3)

�1 = 0:3; �2 = 1:7; �1 =0:3; �2 =0:7; �=2; �=4

L1 = M; P=Q; 106L2615; 3=2¡(2N + P +Q) : (M + L1 + L2)¡3(36)

This concludes the convergence analysis of the problem. The steady-state results are con-sidered next.

5. RESULTS

Using the optimized parameters identi�ed above, this section presents steady-state surfacedisplacement for various anisotropic materials.

5.1. Models

In order to study the role of anisotropy on the basin’s response it is necessary �rst to choosethe material properties for the half-space and the basin. The case of a semi-cylindrical basin ofunit radius subjected to incident plane harmonic pseudo P-, SH -, and SV -waves is considered.As for the isotropic response, the elastic constants of the half-space and basin are chosen

to correspond to isotropic models in the literature.

5.2. Role of general anisotropy

In order to assess the importance of anisotropy on surface ground motion the response ofa semi-circular basin of unit radius are examined for incident plane harmonic waves. Threemodels are considered. Model A involves isotropic materials while Models B and C incorpo-rate general anisotropic ones. The properties for the anisotropic models are generated in thefollowing way. Suppose an orthotropic material with elastic tensor c′′ijmn is described in righthanded Cartesian co-ordinate system {x′′i }: If the coordinate system {x′′i } is rotated �rst aboutthe x′′2 -axes for a positive angle � and then about the new x′3-axes for the same angle �, thisresults in a coordinate system {xi} in which the material is triclinic with the sti�ness tensor



de�ned by [19]

cijmn= ‘(2)ip ‘

(2)jq ‘

(2)mr ‘

(2)ns ‘

(1)pt ‘

(1)qu ‘

(1)rv ‘

(1)sw c

′′tuvw; i : : : w=1; 2; 3 (37)

Here ‘(1)ij and ‘(2)ij denote the rotation about the x′′2 - and x′3-axes, respectively. The components

of the elastic tensor for the orthotropic half-space are chosen to be

C(1)′′=

3:8 1:2 1:6 0 0 0· 3:6 2:1 0 0 0· · 4:6 0 0 0· · · 1:2 0 0· · · · 1:05 0· · · · · 0:75

(38)

where · denotes the symmetric elements of the upper triangular part of C(1)′′ : For the valleythe elastic tensor is given by

C(2)′′=

0:65 0:3 0:2 0 0 0· 0:9 0:5 0 0 0· · 0:45 0 0 0· · · 0:2 0 0· · · · 0:11 0· · · · · 0:19

(39)

The conversion convention from the fourth order to the second-order elastic tensors is assumedto be: 11→ 1; 22→ 2; 33→ 3; 23→ 4; 13→ 5; and 12→ 6: The material density of the half-spaceand the valley are chosen to be 1 and 2=3; respectively. For Model B, the triclinic materialis obtained by choosing the double rotation angle �=5◦, and for Model C, the rotation angleis chosen to be �=20◦: Therefore, the sti�ness tensor components for three di�erent modelsare given by Table I.The surface response for a semi-circular basin of unit radius subjected to incident pseudo

P-waves is depicted in Figure 4 and for incident pseudo SH -waves in Figure 5. The ampli-tude of the incident waves are chosen in such a way that as the materials become isotropic,the free-�eld reduces to the corresponding isotropic free-�eld as given in references [3] and[20]. The results for incident pseudo SV-waves are omitted in order to reduce the number of�gures.Presented results clearly demonstrate that anisotropy may play very important role in ampli-

�cation of surface ground motion of the basin. In particular, signi�cant mode conversions maytake place in anisotropic models that are not present in the isotropic ones. These are mani-fested through the presence all three components of motion. Corresponding isotropic modelsinvolve either two components of motion (incident P- and SV -waves) or one component ofmotion (incident SH -wave). In addition, the results show that presence of valley may producelarge ampli�cation of ground motion when compared to the free-�eld one.



Table I. Material properties for Models A (isotropic), B (triclinic), and C (triclinic).Subscripts 1 and 2 refer to the half-space and the valley, respectively.

A1 B1 C1 A2 B2 C2

c1111 4 3.7986 3.8066 2=3 0.6471 0.6258c1133 2 1.6113 1.7524 1=3 0.2042 0.2561c1123 0 −0:0075 −0:0883 0 −0:0019 −0:0187c1113 0 0.0075 −0:0293 0 0.0194 0.0455c1112 0 0.0004 0.0154 0 0.0032 0.0374c3333 4 4.5864 4.4031 2=3 0.4496 0.4465c2333 0 0.0068 0.0861 0 0.0002 0c1333 0 −0:0772 −0:2365 0 −0:0024 0.0001c1233 0 0.0429 0.1350 0 0.0257 0.0803c2323 1 1.1955 1.1481 1=6 0.1993 0.1916c1323 0 0.0121 −0:0019 0 0.0076 0.0199c1223 0 −0:0384 −0:1118 0 −0:0006 0.0118c1313 1 1.0586 1.1526 1=6 0.1126 0.1441c1213 0 −0:0025 −0:0168 0 −0:0033 −0:0396c1212 1 0.7518 0.7641 1=6 0.1915 0.2003

Figure 4. Surface displacement amplitudes for models A–C subjected to incident pseudo-P-waves with!=�=2 sec−1. Amplitudes: A0 = 1 for vertical incidence and A0 = − sin(�) otherwise. Displacementcomponents: u1 (solid line), u2 (dash–dot line), and u3 (dash line). Asterisks and circles correspond to

results obtained of Reference [3].



Figure 5. Surface displacement amplitudes for models A–C subjected to incident pseudo-SH-waves with!=�=2 sec−1. Amplitudes: A0 = 1 for all incident waves. Displacement components: u1 (solid line),u2 (dash–dot line), and u3 (dash line). Algorithmic parameters: N =60, L2 = 15; M =30, and P=30.

Asterisks correspond to results obtained of Reference [20].

6. CONCLUSION

Scattering of elastic waves by a two-dimensional basin embedded within an elastic half-spacehas been considered by using an indirect boundary integral equation approach. The materialswere assumed to be generally anisotropic, while the incident waves were plane harmonicpseudo-P-, SH-, or SV-waves.The unknown scattered waves were described in terms of full-space Green functions for

anisotropic media. The unknown scattered wave �eld is expressed in terms of single layerpotentials which involve full-space Green functions and unknown density functions.The scattered waves were determined by satisfying the stress-free boundary conditions along

the half-space surface and the continuity conditions between the basin and the half-space in aleast squares sense.Numerical results for steady-state surface response were presented for semi-circular basins

for three models: One involves isotropic materials and two incorporate triclinic materials.The surface ground motion spectra are presented for a range of parameters, and they demon-

strate the following: Presence of the basin may cause locally very large ampli�cation of thesurface motion. That motion depends in a complex manner upon a number of physical pa-rameters present in the problem, such as, frequency and nature of the incident wave, locationof the observation point at the surface of the half-space, and component of the motion whichis being observed. In addition, the presented results demonstrate the importance of anisotropy



upon surface motion. Signi�cant mode conversion has observed in anisotropic models whichdoes not exist in corresponding isotropic models.

APPENDIX: FULL-SPACE TIME-HARMONIC TWO-DIMENSIONAL GREENFUNCTIONS FOR ANISOTROPIC SOLIDS

An impulsive line load in the xk direction, applied on a line parallel to the x2-axis throughthe point x0, is given by

fi(x)=A�(x − x0)�ik (A1)

where �ik is the Kronecker delta and A is unity with dimensions of force and will be suppressedhereafter. The displacement Green functions, denoted gpk(x;x0), are a solution to Equation (6)with body force (A1):

ci�p�gpk; ��(x;x0) + �!2gik(x;x0)= − �ik�(x − x0): (A2)

The derivation of the two-dimensional time-harmonic full-space Green functions in the paperby Wang and Achenbach [8]. By using a two-dimensional Radon transform, de�ned by thefollowing pair of equations [9]:

f̂(s; n) = R[f(x)]=∫f(x)�(s− n · x) dx

f(x) =14�2

∫|n|=1

[∫@sf̂(s; n)n · x − s d s

]dn

they showed that the following integral representation for gpk is valid:

gpk(x;x0)=18�2

∫|n|=1

3∑m=1�mpk(n)�(k

m|n · y|) dn (A3)

where y=x − x0; km=!(�m=�)−1=2,

�mpk =EpmEkm�m

; m ∈ {1; 2; 3} (A4)

�(z)= i� exp(iz)− 2[cos(z)Ci(z) + sin(z)si(z)]: (A5)

In Equation (A5), si(z)=Si(z)− �2 and Ci and Si denote the cosine integral and sine integrals,

respectively [15]. If �ip= ci�p�n�n� (a symmetric, positive de�nite matrix), then the eigenvalues�m and eigenvectors Epm satisfy

�ip(n)Epm= �mEim; m∈{1; 2; 3}It should be noted that both �m and Epm are functions of n. An analytical solution for thethree eigenvalues follows from Cardano’s formula [15]. It can be shown [21] that if �m is adistinct eigenvalue then the product of eigenvectors appearing in Equation (A4) becomes

EpmEkm=AmpkAmqq



where Ampk is the classical adjoint of the matrix �ip − �m�ip and there is no summation of m.Likewise, if �2 = �3 is an eigenvalue of multiplicity two, then

Ep2Ek2 = �pk −A1pkA1qq

if all the eigenvalues are equal, EpmEkm= �pk (no summation over m).The stress Green functions are de�ned by hijk = cijp�gpk;� and substitution of Equation (A3)

yields

hijk(x;x0)=cijp�8�2

∫|n|=1

3∑m=1kmn��mpk(n)�

′(km|n · y|) dn (A6)

where

�′(z)= − � exp(iz) + 2[sin(z)Ci(z)− cos(z)Si(z)− z−1] (A7)

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scattering of elastic waves by a general anisotropic basin. part 1: a 2d model

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