scattering of elastic waves by a general anisotropic basin. part 2: a 3d model

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2003; 32:653–670 (DOI: 10.1002/eqe.233)

Scattering of elastic waves by a general anisotropic basin.Part 2: a 3D model

Marijan Dravinski∗;†

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

SUMMARY

Scattering of plane harmonic waves by a three-dimensional basin of arbitrary shape embedded withinelastic half-space is investigated by using an indirect boundary integral equation approach. The materialsof the basin and the half-space are assumed to be the most general anisotropic, homogeneous, linearlyelastic solids without any material symmetry (i.e. triclinic).

The unknown scattered waves are expressed in terms of three-dimensional triclinic time harmonicfull-space Green’s functions. The results have been tested by comparing the surface response of semispherical isotropic and transversely isotropic basins for which the numerical solutions are available.

Surface displacements are presented for a semicircular basin subjected to a vertical incident planeharmonic pseudo-P-, SV -, or SH -wave. These results are compared with the motion obtained for thecorresponding equivalent isotropic models. The results show that presence of the basin may causesigni�cant ampli�cation of ground motion when compared to the free-�eld displacements. The peakamplitude of the predominant component of surface motion is smaller for the anisotropic basin than forthe corresponding isotropic one. Anisotropic response may be asymmetric even for symmetric geometryand incidence. Anisotropic surface displacement generally includes all three components of motionwhich may not be the case for the isotropic results. Furthermore, anisotropic response strongly dependsupon the nature of the incident wave, degree of material anisotropy and the azimuthal orientation ofthe observation station.

These results clearly demonstrate the importance of anisotropy in ampli�cation of surface groundmotion. Copyright ? 2003 John Wiley & Sons, Ltd.

KEY WORDS: three-dimensional anisotropic basin; site response

1. INTRODUCTION

Damage analysis after large earthquakes often shows that strong ground motion can be highlylocalized. Numerous studies have con�rmed that alluvial valleys and sedimentary basins aregenerally exposed to larger surface motion ampli�cations and longer duration of shaking than

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

†E-mail: [email protected] 11 February 2002

Revised 17 May 2002Copyright ? 2003 John Wiley & Sons, Ltd. Accepted 11 July 2002

654 M. DRAVINSKI

the sites on crystalline rocks [1; 2]. While the response of isotropic basins has been studiedin great detail [3], the corresponding anisotropic problem is much more di�cult to solve.Most of the earth materials are intrinsically anisotropic. One of the earliest investigations

on the role of anisotropy in surface layer of the Earth is due to Stonley [4]. He demonstratedthat presence of transversely isotropic materials may result in signi�cant di�erences in wavepropagation compared to the motion in isotropic materials.Elastodynamic response of anisotropic linearly elastic solids of arbitrary geometry must

be solved numerically. For scattering problems involving large characteristic lengths one ofthe most e�ective method is the boundary integral equation approach. This technique requiresdiscretization only along the boundary of the model, thus greatly reducing the size of the prob-lem, and the radiation conditions in the far �eld are satis�ed exactly [5]. However, boundaryintegral equation approach in elastodynamics requires evaluation of the corresponding Green’sfunctions, i.e. the displacement and stress �elds in an unbounded solid subjected to a pointforce with the same material properties as in the original problem. The Green’s functions arethe kernels in the boundary integral equations and they must be computed repeatedly manytimes in order to solve the discretized system of equations. Consequently, it is essential to de-velop e�cient algorithms for evaluation of the Green’s functions. For isotropic media this canbe achieved with standard numerical techniques [6], however anisotropic Green’s functionsare much more di�cult to calculate. Recently, Wang and Achenbach [7] derived anisotropic3D Green’s functions using the Radon transform. The solution was obtained in terms of inte-grals over a unit sphere and a unit circle which can be accurately evaluated. However, theseintegrals involve the Kelvin–Cristho�el matrix, its inverse, and it derivative of the inverse,as well as the corresponding eigenvalues and eigenvectors [8]. Since for triclinic media theKelvin–Christo�el matrix includes 24 independent variables (21 material constants and threecomponents of the propagation vector) the Green’s function kernels are too complex to besolved analytically and then incorporated into a computer code. Using a computer algebrasystem within the programming environment Dravinski and Niu [8] eliminated this problementirely . In addition, the e�ciency of the Green’s functions calculations has been signi�cantlyimproved through the elastodynamic state vector approach [9] which eliminates reevaluationsof the common parts of the integrands and thus resulting in a much faster algorithm.These recent advances in evaluation of the 3D anisotropic Green’s functions provide the

necessary tools for the boundary integral equation formulation for scattering of waves in themost general anisotropic media. Two di�erent boundary integral approaches have been per-suaded. Niu and Dravinski [10] used a direct boundary integral formulation to study scatteringof elastic waves by a three-dimensional cavity embedded within a triclinic full space subjectedto a plane harmonic wave. The numerical evaluation of the free terms based on the propertiesof the Green’s functions and the computation of the integration constants which arise in directformulation were considered in detail.On the other hand the present paper deals with a more general problem of surface ground

motion ampli�cation by a basin of arbitrary shape embedded in a half-space when subjectedto plane harmonic incident waves using an indirect boundary integral equation approach. Thisformulation facilitated the analysis of the problem involving the most general anisotropicmaterials with a manageable amount of computational e�ort.This research is a generalization of the works by Dravinski and Wilson [11] and Zheng

and Dravinski [12]. The former considered the corresponding 2D triclinic problem while thelatter investigated the three-dimensional problem limited to transversely isotropic materials.

Copyright ? 2003 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2003; 32:653–670

SCATTERING OF ELASTIC WAVES BY A GENERAL ANISOTROPIC BASIN 655

The model discussed here is fully three dimensional involving the most general geometry andanisotropic properties of the materials.

2. STATEMENT OF PROBLEM

Geometry of the problem is depicted by Figure 1. The model consists of a half-space with athree-dimensional basin of arbitrary shape. The domains of the half-space and the basin aredenoted by D1 and D2, respectively. The interface S between the basin and the half-spaceis assumed to be su�ciently smooth. Materials are taken to be linearly elastic, homogeneousand triclinic exhibiting no symmetry [13].The steady-state equations of motion for the basin and the half-space can be written as

�(J )ij; j + �(J )!2u(J )i =0; (); j ≡ @

@xj; J =1; 2 (1)

and the linearized stress–strain relation is given by

�(J )ij = c(J )ijpqe(J )pq ; J =1; 2 (2)

where e(J )pq ; �(J )ij ; �

(J ); !, and c(J )ijpq are the strain tensor, the stress tensor, the mass density, thecircular frequency, and the tensor of elastic constants, respectively. Throughout the paper thesuperscript (J ) refers to the domain DJ and the summation over repeated indices is understood.Summation over repeated superscripts is being suppressed. Unless stated di�erently, the rangeof indices is assumed to be from 1 to 3. The strain tensor components are de�ned by

e(J )pq =12(u

(J )p; q + u

(J )q;p) (3)

where u(J )p denotes the displacement vector. The symmetry of the stress and strain tensors, andthe existence of the positive strain energy density function imply the symmetry and positivede�niteness of c(J )ijpq

c(J )ijpq= c(J )jipq= c

(J )ijqp= c

(J )pqij

c(J )ijpqe(J )ij e

(J )pq ¿0

(4)

for any non-zero real valued tensor e(J )ij . Thus, in general, out of 81 components of c(J )ijpq only

21 are independent. In particular, the constitutive equations for a triclinic material can bewritten as [13]

�11�22�33�23�13�12

=

C11 C12 C13 C14 C15 C16C22 C23 C24 C25 C26

C33 C34 C35 C36* C44 C45 C46

C55 C56C66

e11e22e332e232e132e12

(5)


656 M. DRAVINSKI

x1

x 3

x2

O

SD2

D1

incident wave

γ n inc

Figure 1. Problem geometry.

where ∗ denotes the symmetric part of the matrix. In order to relate the material propertiesdescribed by the fourth-order tensor cijpq to a second-order tensor Ckl (k; l=1; 2; : : : ; 6) the fol-lowing subscript contracting convention has been used: 1→ 11; 2→ 22; 3→ 33; 4→ 23; 5→ 13;and 6→ 12. Thus, for example, c1122 is equivalent to C12 and so on.The stress-free boundary conditions are given by

�(J )3i =0; x3 = 0; x ∈ DJ ; J =1; 2 (6)

Perfect bonding between the half-space and the basin can be stated as

u(1)i = u(2)i ; x∈S (7)

t (1)i = t (2)i ; x∈S (8)

where t (J )i =�(J )ij �j are components of the traction vector and � denotes unit normal vector onS (Figure 1).The incident �eld is assumed to be a plane harmonic wave. As the incident wave strikes the

basin it generates scattered waves. These interact with the free-�eld resulting in ampli�cation(constructive interference) or reduction (destructive interference) of motion. The objective ofthe paper is to determine the unknown scattered waves and subsequently the total displacementand stress �elds throughout the elastic media.

3. SOLUTION OF PROBLEM

As the incident wave strikes the interface S, it is partially transmitted into the basin andpartially re�ected back into the half-space. Consequently, the waves in the half-space consistof the free-�eld and the scattered wave �eld, while the motion inside the basin comprises ofthe scattered waves only, or

u(1)i = u�i + u(1)si ; x∈D1

u(2)i = u(2)si ; x∈D2 (9)

where the superscripts s and � denote the scattered and free-�elds, respectively.



3.1. Free-�eld

The incident wave, assumed to be propagating in the x1x3-plane, is of the following form:

uinc =AincUincei(kincninc · x−!t) (10)

k inc =!vinc

(11)

ninc = [sin �0; 0;− cos �0]T (12)

vinc ∈{v1; v2; v3} (12′)

Uinc∈{U(1);U(2);U(3)} (13)

where Ainc and Uinc denote the amplitude factor and the polarization vector, respectively,ninc is the direction of propagation, �0 is the angle of incidence measured clockwise fromthe x3-axis, vinc is the velocity of propagation, superscript T indicates the transpose, and k inc

denotes the wavenumber. In addition, vk and U(k); k=1 : 3 are, respectively, the eigenvaluesand eigenvectors of the matrix

�ip=1�ci�p�n�n�; �; �=1; 3 (14)

so that

(�ip − v2�ip)U (k)p =0 (15)

For convenience a unit amplitude factor, Ainc, has been assumed.It should be noted that for any choice of incident direction [ninc1 ; 0; n

inc3 ]

T there exist threepositive eigenvalues {v21 ; v22; v23} and three orthogonal eigenvectors {U(1);U(2);U(3)}. Therefore,both the polarization vector and the speed of the incident wave depend upon its direction ofpropagation ninc.If the eigenvalues {vm; m=1; 2; 3} are arranged in the ascending order then m is called the

incidence index. Thus the smallest velocity v1 corresponds to an incident pseudo-SH -wave(pSH), the intermediate velocity v2 is associated with a pseudo-SV -wave (pSV ) and themaximum velocity v3 implies a pseudo-P-wave (pP).For a choice of the incident wave sometimes it may be convenient to scale the corresponding

polarization vector according to the following procedure. First, the polarization scaling index, s,is de�ned based on the type of the incident wave. This index corresponds to the ‘predominant’component of motion associated with P-, SV -, and SH -polarizations for an incident directionbeing along the negative x3-axis in a co-ordinate system {x1; x2; x3}. Therefore, for incidentpP-, pSV -, and pSH -waves the polarization scaling index s is assigned values 3; 1; and 2;respectively. Consequently, the scaled polarization vectors are of the form

[Umi ]→

[Umi

Ums

](16)

The use of the scaled polarization vectors for the anisotropic free-�eld allows directcomparison with the corresponding isotropic results which are available in literature (e.g.Reference [14]).


658 M. DRAVINSKI

As the incident waves strikes the surface of the half-space it generates the re�ected wave�eld. In general, there will be three possible re�ected waves [13; 15] and the re�ected wave�eld can be written as

uref =3∑m=1A(m)V(m)ei(k

(m)n(m) · x−!t) (17)

where A(m); V(m); k(m), and n(m) denote the amplitude factor, polarization vector, wavenumber,and the direction of propagation, respectively. Using the Snell’s law, the stress-free boundaryconditions along the surface of the half-space, and the radiation conditions it is possible todetermine the unknown re�ected wave �eld. Then, the resulting free-�eld is calculated by

u� = uinc + uref (18)

This completes the free-�eld discussion. The unknown scattered waves are considered next.

3.2. Scattered wave �eld

For the indirect integral equation approach considered in this paper, the scattered wave �eldis assumed to be expressed in terms of single layer potentials [16; 17]

u(J )p (x; !)=∫S (J )q(J )k (y)g

(J )pk (x; y; !) dy; x∈DJ ; J =1; 2 (19)

where q(J )k are the unknown density functions and S (J ) are auxiliary surfaces de�ned insideand outside of the interface S [12]. In addition, g(J )pk (x; y; !) are components of the full-spaceGreen’s functions which satisfy the following equations of motion:

{�(J )ip + �(J )!2�ip}g(J )pk (x; y; !)= − �ik�(x − y) (20)

where

�(J )ip = c(J )ijpq@@xj

@@xq

(21)

and the factor e−i!t is understood. Corresponding stress Green’s functions are given by

h(J )ijk (x; y; !)= c(J )ijpqg

(J )pk; q(x; y; !) (22)

Detailed discussion on evaluation of the displacement and stress full-space triclinic Green’sfunctions can be found in the paper by Dravinski and Niu [8].If the unknown density functions are assumed in the form of discrete point sources then

the scattered wave �eld becomes [12]

u(1)p = amg(1)p1 (x;xm;!) + bmg

(1)p2 (x;xm;!) + cmg

(1)p3 (x;xm;!)

u(2)p =d‘g(2)p1 (x;x‘; !) + e‘g

(2)p2 (x;x‘; !) + f‘g

(2)p3 (x;x‘; !)

m=1; : : : ; M ; ‘=1; : : : ; L (23)



where the coe�cients am to f‘ are still to be determined and M and L denote the numberof sources along the auxiliary surfaces S (1) and S (2), respectively. These coe�cients will beevaluated through the stress-free boundary conditions (6) and the continuity conditions (7)and (8).Suppose that the stress-free conditions (6) are imposed at P locations at the surface

of the half-space {xp∈D1 | x3 = 0; p=1; : : : ; P} and at Q locations along the surface ofthe valley {xq∈D2 | x3 = 0; q=1; : : : ; Q}: Furthermore, let the continuity conditions (7) and(8) be imposed at N points along the interface S; i.e. at {xn∈S; n=1; : : : ; N}. Then boththe stress-free boundary conditions and the continuity conditions can be writtenas

Gg=w (24)

where g is a 3(M + L) vector of unknown density coe�cients de�ned by

g=(a1; : : : ; aM ; b1; : : : ; bM ; c1; : : : ; cM ; d1; : : : ; dL; e1; : : : ; eL; f1; : : : ; fL)T (25)

while G and w denote known complex 3(P+Q+2N )× 3(M +L) matrix and 3(P+Q+2N )vector, respectively [12]. Therefore, system (24) represents 3(P + Q + 2N ) equations in3(M + L) unknowns, which is solved in the least-squares sense by using QR-decomposition.Detailed convergence analysis of the method can be found in the paper by Zheng andDravinski [12].Once the density coe�cients a; b; c; and d are known, the scattered waves can be evaluated

according to Equation (23). This completes the discussion of the scattered wave �eld. Thenumerical results are considered next.

4. NUMERICAL RESULTS

The choice of auxiliary surfaces, S (J ); the number of sources, M and L, and the num-ber of collocation points, N , are chosen based on the extensive parametric study of Zhengand Dravinski [12]. Therefore, the next section deals with the testing of themethod.

4.1. Testing of the method

Since for the problem at hand no numerical solution is available, the testing of the method isdone by considering two limit cases: an isotropic and a transversely isotropic problems. Forthese two models the numerical solutions are available in literature.

4.1.1. Isotropic limit test. This problem consists of a semispherical basin of unit radius sub-jected to a vertically incident plane harmonic P-wave. The problem, originally solved bySanchez-Sesma [18], was tested by several researchers including [19; 20].


660 M. DRAVINSKI

Based on the material properties of the half-space and the basin given in [18], it followsthat

C1 =

3 1 1 0 0 03 1 0 0 0

3 0 0 0* 1 0 0

1 01

; �1 = 1 (26)

C2 =

1:2 0:6 0:6 0 0 01:2 0:6 0 0 0

1:2 0 0 0* 0:3 0 0

0:3 00:3

; �2 = 0:6 (27)

where ∗ denotes the symmetric part of the matrix. The frequency of the incident wave ischosen to be !=2:721=s which corresponds to a dimensionless frequency of one half [18].The isotropic test results are depicted by Figure 2.Apparently the isotropic limit results of this study agree very well with the results of

Sanchez-Sesma [18].

4.1.2. Transversely isotropic limit test. In this test, the response of a semispherical trans-versely isotropic basin is compared with that obtained by Zheng and Dravinski [12]. Thematerial constants for the basin and the half-space are assumed to be

C1 =

4:13 1:47 1:01 0 0 04:13 1:01 0 0 0

3:62 0 0 0* 1 0 0

1 04:13−1:47

2

; �1 = 1 (28)

C2 =

1:38 0:49 0:337 0 0 01:38 0:337 0 0 0

1:260 0 0 00:33 0 0

0:33 01:38−0:49

2

; �2 = 0:67 (29)



Figure 2. Isotropic limit test. Surface response of a semispherical basin of unit radius subjected to avertically incident plane harmonic P-wave. Solid and dash lines denote the results obtained in this work

while the stars and open circles denote the results of Sanchez-Sessma [18].

The results of the comparison are depicted by Figure 3. It is evident from Figure 3 thatthe two results agree very well.Therefore, the two limit tests verify the numerical accuracy of the solution proposed in the

present study. The response of a triclinic basin is considered next.

4.2. Triclinic results

4.2.1. Material properties. A general triclinic material requires 21 di�erent elastic constants.In order to simplify this choice it is useful to observe that an isotropic material with ahigher degree of symmetry may appear triclinic if it is appropriately rotated. For example,transversely isotropic material constants cijkl de�ned in a co-ordinate system {xi} and triclinicmaterial constants c′′ijkl in a co-ordinate system {x′′i } can be related through

c′′ijmn= ‘(3)ip ‘

(3)jq ‘

(3)mr ‘

(3)ns ‘

(2)pt ‘

(2)qu ‘

(2)rv ‘

(2)sw ctuvw (30)


662 M. DRAVINSKI

Figure 3. Transversely isotropic limit test. Response of a semispherical basin subjected to a verticalplane harmonic pP-wave. Solid, dash, and dash-dot lines denote the results of this study while open

circles, stars, and triangles represent the results obtained by Zheng and Dravinski [12].

where ‘(2)ij denotes the rotation about the x2-axis for a positive angle � and ‘(3)ij represents the

rotation about the x′3-axis for the same angle �: Therefore, it is possible to generate triclinicproperties by starting with a transversely isotropic material. For that purpose, the propertiesof transversely isotropic materials for the half-space and the basin are assumed to be theone de�ned by Equations (28) and (29). Then for the rotation angle �=10◦, the triclinicproperties for the half-space and the basin follow to be

C′′1 =

4:0660 1:4562 1:0595 −0:0122 0:1801 0:00554:1280 1:0250 −0:0332 0:0772 0:0058

3:5848 0:0166 −0:0939 −0:00631:0112 −0:0070 0:0553

* 1:0494 −0:00831:3201

(31)



and

C′′2 =

1:3582 0:4854 0:3555 −0:0040 0:0613 0:0019

1:3793 0:3420 −0:0114 0:0257 0:0020

1:2447 0:0072 −0:0410 −0:00240:3339 −0:0026 0:0193

0:3484 −0:00290:4415

(32)

These material properties will be used to simulate the motion for a triclinic model.It should be pointed out that there is no restriction on the value of the rotation angle �. To

illustrate that point the numerical results will include a case of the triclinic materials obtainedthrough the rotation angle �=20◦ as well.In order to make meaningful comparison of anisotropic and isotropic results it is neces-

sary to construct the appropriate reference isotropic material. As suggested by Fedorov [21]this can be accomplished by comparing the Kelvin–Christo�el tensors for anisotropic andisotropic materials. For the sake of completeness a brief summary of the Fedorov’s approachis presented here.The Kelvin–Christo�el tensor for an anisotropic material is de�ned by

�ip(n)= cijpqnjnq (33)

while for an isotropic material it is given by

��kl = ��kl + ( �+ ��)nknl (34)

where � and �� are the Lam�e constants and n denotes the propagation direction. One can com-pare the Kelvin–Christo�el tensors of the two materials by forming the di�erencematrix

�=�− �� (35)

The isotropic material most similar to the given anisotropic solid is the one for which thedi�erence between � and �� is minimal. This condition is satis�ed if the sum of the squaresof all components of � (i.e., the Frobenius norm ‖�‖22) is minimized. However, due tothe symmetry of � and �� it follows that

‖�‖22 =∑k; l(�kl)2 = tr(�

2) (36)

where tr denotes the trace operation. Since both � and �� depend on the direction of the wavepropagation n, to eliminate this and obtain a relation-dependent solely on the properties of thematerials, one must average the di�erence matrix � (35) over all directions of n. Therefore,


664 M. DRAVINSKI

Figure 4. Surface response along the x= x1-axis of a semispherical triclinic valley of unit ra-dius subjected to a vertically incident pP-wave with Ainc = 1 and the unscaled polarization vec-tor Uinc = (−0:0370; 0:0065; 0:9993). Corresponding response for an equivalent isotropic model (EI)

subjected to a vertical incident P-wave is included as well.

the problem leads to minimization of the quantity

〈F〉= tr〈�2〉= tr〈(�− ��)2〉= min (37)

where the averages are de�ned by

〈F〉≡ 14

∫ 2

0d�

∫

0F(�; �) sin � d� (38)

By performing the minimization procedure the Lam�e constants for the most similar isotropicmaterial are derived to be [21]

��= 130(3cikik − ciikk); �= 1

30(3ciikk + cikik)− �� (39)

These properties will be used to simulate the response of an equivalent isotropic basin. Thus,for the material properties de�ned by Equations (31) and (32), corresponding Lam�e constants



Figure 5. Surface response along the x= x1-axis of a semispherical triclinic valley (�=20◦) of unitradius subjected to a vertically incident pP-wave with Ainc = 1 and the unscaled polarization vec-tor Uinc = (−0:0528; 0:0192; 0:9984). Corresponding response for an equivalent isotropic model (EI)

subjected to a vertically incident P-wave is included as well.

are calculated to be

��1 = 1:2253; �1 = 1:2082 (40)

��2 = 0:4114; �1 = 0:4077 (41)

and the resulting elasticity tensors become

�C1 =

3:6589 1:2082 1:2082 0 0 03:6589 1:2082 0 0 0

3:6589 0 0 0* 1:2253 0 0

1:2253 01:2253

; ��1 = 1 (42)


666 M. DRAVINSKI

Figure 6. Surface response along the x= x1-axis of a semispherical triclinic valley of unit ra-dius subjected to a vertically incident pSV -wave with Ainc = 1 and the unscaled polarization vec-tor Uinc = (0:9841;−0:1735; 0:0376). Corresponding response for an equivalent isotropic model (EI)

subjected to a vertical incident SV -wave is included as well.

and

�C2 =

1:2305 0:4077 0:4077 0 0 01:2305 0:4077 0 0 0

1:2305 0 0 0* 0:4114 0 0

0:4114 00:4114

; ��2 = 0:67 (43)

This concludes the analysis of the material properties. Triclinic response of the basin isconsidered next.

4.2.2. Incident pP-wave. Surface response of a semispherical valley subjected to a verticallyincident pP-wave is depicted by Figures 4 and 5 for two di�erent triclinic materials (�=10◦

and 20◦). Results for both triclinic and equivalent isotropic models are presented.



Figure 7. Surface response along the y= x2-axis of a semispherical triclinic valley of unit ra-dius subjected to a vertically incident pSV -wave with Ainc = 1 and the unscaled polarization vec-tor Uinc = (0:9841;−0:1735; 0:0376). Corresponding response for an equivalent isotropic model (EI)

subjected to a vertical incident SV -wave is included as well.

Several observations can be made based on these results:

• Presence of the basin caused signi�cant ampli�cation of surface motion when comparedto the free-�eld results.

• The predominant component of motion, U3, for the anisotropic model exhibits smallerpeak amplitude then for the equivalent isotropic case.

• The anisotropic response may be non-symmetric even for symmetric geometry and inci-dent wave.

• The surface motion depends upon the degree of anisotropy of the materials, and• The motion of the anisotropic basin in general includes all three displacement componentswhile the corresponding isotropic response may not.

4.2.3. Incident pSV-wave. Surface response for a pSV -incidence is depicted by Figures 6and 7. Again both anisotropic and corresponding isotropic response are presented.


668 M. DRAVINSKI

Figure 8. Surface response along the x= x1-axis of a triclinic semispherical valley of unitradius subjected to a vertically incident pSH -wave with Ainc = 1 and the unscaled polarization vectorUinc = (01736; 0:9848; 0:0000). Corresponding response for an equivalent isotropic model (EI) subjected

to a vertical incident SH -wave is included as well.

The results can be summarized exactly the same way as in the case of pP-incidenceexcept that here the predominant component of motion is the x1-direction. In addition, it canbe seen that the anisotropic response strongly depends upon the azimuthal orientation of theobservation station relative to the incident wave (x1-axis versus x2-axis).

4.2.4. Incident pSH-wave. Surface motion for this incidence is depicted by Figures 8 and 9.The results for the pSH -incidence exhibit the same features as in the case of pP- and

pSV -incidence except that the predominant component of motion is in the x2-direction.

5. SUMMARY AND CONCLUSIONS

Ampli�cation of surface ground motion has been considered for a 3D elastic basin of arbitraryshape embedded within a half-space and subjected to an incident plane harmonic pseudo-P-; SV -, or SH -wave. The materials were assumed to be generally anisotropic without any



Figure 9. Surface response along the y= x2-axis of a semispherical triclinic valley of unit ra-dius subjected to a vertically incident pSH -wave with Ainc = 1 and the unscaled polarization vectorUinc = (01736; 0:9848; 0:0000). Corresponding response for an equivalent isotropic model (EI) subjected

to a vertical incident SH -wave is included as well.

symmetry, i.e. triclinic. The resulting motion has been obtained by using an indirect boundaryintegral equation method. The method is based on the full-space time harmonic Green’sfunctions.Testing of the method has been done for isotropic and transversely isotropic semispherical

basins. The numerical results were presented for a semispherical triclinic valley subjected to avertically incident plane harmonic pP-, pSV -, or pSH -waves. The results can be summarizedas follows:

1. Presence of the basin may cause signi�cant ampli�cation of surface ground motion whencompared to the free-�eld displacements.

2. The peak amplitude of the predominant component of surface motion is smaller for theanisotropic basin than for the corresponding isotropic one.

3. Anisotropic response may be asymmetric even for symmetric geometry and incidence.4. Anisotropic surface displacement generally includes all three components of motionwhich may not be the case for the isotropic results.


670 M. DRAVINSKI

5. Anisotropic response strongly depends upon the nature of the incident wave, degree ofmaterial anisotropy and the azimuthal orientation of the observation station.

The results clearly demonstrate the importance of anisotropy in ampli�cation of surfaceground motion.

ACKNOWLEDGEMENTS

This research was made possible in part through the sabbatical leave granted to the author by the Schoolof Engineering at USC.The author would like to thank the two anonymous reviewers for their constructive comments.

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scattering of elastic waves by a general anisotropic basin. part 2: a 3d model

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