Bulletin of the Seismological Society of America, Vol. 98, No. 1, pp. 265–279, February 2008, doi: 10.1785/0120060249
The Localized Boundary Integral Equation–Discrete Wavenumber Method for Simulating P-SV Wave Scattering by an Irregular Topography by Hong Zhou* and Xiaofei Chen†
Abstract
In this study, we present a new method, the local boundary integral equation–discrete wavenumber method (loBIE-DWM), for simulating the scattered P-SV waves by a 2D irregular surface topography. This method is rigorously derived from the basic formulation of Bouchon and Campillo’s BIE-DWM, which can provide accurate enough solutions for most problems, while the expensive computation cost, especially for the high-frequency problem, restricted its application. In this new algorithm we propose, the dimension of the inverse matrix involved is only proportional to the sampling number within the corrugated part of the surface. Therefore, its computation efficiency is increased dramatically while keeping the same accuracy as BIE-DWM, particularly for the problem in which the corrugated part of the topography is highly localized. Comparisons with previously existing validated results demonstrated the validity of the loBIE-DWM and further showed that its computational efficiency is much superior to the BIE-DWM. Finally, with this new method, we investigated the influences of the topography on the propagation of Rayleigh wave.
Introduction 2002; Komatitsch et al., 2004). The most advanced domain methods may accurately solve the problem of the scattering of seismic waves by an irregular topography, though at higher computation costs. The third category, the boundarytype numerical method, has been the most active and fruitful since Aki and Larner’s (1970) pioneered work. The boundary-type numerical methods, built upon the representation of the integral equation over the irregular surface, are generally more efficient and accurate for the problem of seismic wave scattering by an irregular topography because they are physically more suitable to the topographic model. The Aki–Larner method (AL) is the first effort in this direction (Aki and Larner, 1970), and it is quite successful; it was followed up by a number of extensions to various cases (Bouchon, 1973; Bard and Bouchon, 1980a, b, 1985; Takenaka et al., 1996). Although AL is limited by Rayleigh ansatz, it works well for relatively smoother and weaker topographic models. Inspired by AL, Chen proposed the global generalized reflection-transmission matrices method based on T-matrix theory (Chen, 1990, 1995, 1999, 2007), and it works very well for arbitrarily irregular topography because it was rigorously built upon the elastodynamic equation. The majority of the boundary-type methods belong to boundary integral equation (BIE) methods, which are often sorted to the direct BIE (e.g., Wong and Jennings, 1975;
Studies on seismic wave scattering by irregular topography have been an important subject in seismology and earthquake engineering for their wide applications in seismic hazards. During the past few decades, numerous advances have been made in this field. Roughly speaking, this progress can be sorted into three categories: the analytical methods, the domain-type numerical methods, and the boundary-type numerical methods. Although the analytical methods may provide accurate solutions, they are only available to a few special cases with higher symmetrical property, such as the semicylindrical canyon (Trifunac, 1971, 1973; Lee and Karl, 1992, 1993; Liang et al., 2001). The domain-type methods include the finite-difference method (Boore, 1972; Hestholm and Ruud, 1994, Pitarka and Irikura, 1996; Robertsson, 1996; Ohminato and Chouet, 1997; Hestholm and Ruud, 1998; Wang and Zhang, 2004; Zhang and Chen, 2006), the finite-element method (Smith, 1975; Bao et al., 1998; Koketsu et al., 2004; Zhang et al., 2005; Ding et al., 2006), and the spectral-element method (Komatitsch and Vilotte, 1998; Komatitsch, 2000; Komatitsch and Tromp, * Present address: Institute of Geophysics, China Earthquake Administration, Beijing 100081, China. † Present address: School of Earth and Space Science, University of Science and Technology, Hefei 230026, China.
265
266 Sánchez-Sesma and Rosenblueth, 1979; Wong, 1982; Dravinski, 1983; Dravinski and Mossessian, 1987; Ge et al., 2005; Ge and Chen, 2007) and the indirect BIE (e.g., Sanchez-Sesma and Campillo, 1991, 1993; Yokoi and Sánchez-Sesma, 1998). Properly treating the singularities of Green’s function on the boundary has been the primary issue of BIE. The direct BIE avoids singularity by setting a number of fictitious sources outside the interesting region closely along the irregular boundary, thus its accuracy heavily depends on the normal distances between the locations of the fictitious sources, and the boundary though its numerical implementation is relatively easier. In contrast, using singlelayer integral representation, the indirect BIE contains only weak singularities, which are treated analytically based on an approximate expansion of the Hankel function (SanchezSesma and Campillo, 1991). In addition to these, Bouchon and Campillo proposed a hybrid of BIE and a discrete wavenumber method (abbreviated to BIE-DWM hereafter), which can avoid the singularity issue thoroughly (Bouchon, 1985; Campillo and Bouchon, 1985; Bouchon, 1989; Gaffet and Bouchon, 1989). The BIE-DWM is merited for its stability and robustness in numerical computation and its suitability to arbitrary topography. However, its scheme of discretization with equal intervals involves more sampling points for relatively higher frequency problems, and so results in a large computation demand. This disadvantage limits its application and extension to a 3D problem. Thus, in order to increase its efficiency, some relevant ideas have been proposed. One is to reduce the unknown number. Campillo (1987), Gaffet and Bouchon (1989), and Kawase and Aki (1989, 1990) used a half-space Green’s function instead of using the full-space Green’s function, to limit the unknown forces to the irregular part of the interface only. However, a half-space Green is not suitable for application to arbitrary topography (i.e., mountaintype topography) (Kawase and Aki, 1990). Another idea to improve computational efficiency is to directly reduce the computation amount by an approximate technique (Bouchon et al., 1996). Such improvement, however, is achieved at the cost of reducing accuracy; hence, it should be applied carefully (Yokoi and Sánchez-Sesma, 1998). Different from the previously described methods, Zhou and Chen (2006a,b) proposed a new method to improve the efficiency of the BIE-DWM for SH wave scattering by an irregular topography. This method is several tens to hundreds of times faster than the original BIE-DWM because it limits the unknowns to the irregular part of topography while still using the simple and analytic full-space Green’s function. Besides the significant improvement to efficiency, this method can provide identical results with BIE-DWM for the same problem. In this study, we extended this idea to the problem of scattering of P-SV waves by an irregular topography. In what follows, we shall present the theoretical development first, then validate it, and finally apply it to investigate the influences of topography to the propagation of body and Rayleigh waves.
H. Zhou and X. Chen
Theoretical Development Formulation of BIE-DWM for P-SV Topographic Problems The problem considered here is seismic wave scattering due to an irregular topography subjected to incident P-SV waves in a homogeneous isotropic medium (Fig. 1). In this problem, the total seismic wave field can be written as ux; z; ω uin x; z; ω usc x; z; ω;
(1)
where u denotes the displacement in the frequency domain; and the subscripts in and sc denote the incident and scattered wave fields, respectively. According to the Huygens principle, the scattered waves can be represented by a Kirchhoff integral over the diffracting surface (see Bouchon, 1985; Campillo and Bouchon, 1985; Bouchon, 1989), Z usc x; z; ω
Gx x0 ; z ξx0 qx0 dsx0 ;
(2)
S
where G is the tensor of Green’s function for P-SV waves in a 2D elastic full space, qx0 is the distribution of the unknown force, and the integration domain S is the irregular topographic surface. It is noted that the Green’s tensor and force distribution are also the functions of ω, though it is not explicitly indicated here for compactness in the mathematical presentation. To efficiently solve this topographic scattering problem, we adopt the discrete wavenumber (abbreviated as DWN) method (see, e.g., Bouchon and Aki, 1977; Bouchon, 2003). With this method, the source-medium configuration is designed to be periodic along the horizontal dimension (Fig. 2); accordingly, the scattered wave field at an arbitrary discrete horizontal grid is reduced to the following finite summation over all the discrete Huygens source points (Bouchon, 1985; Campillo and Bouchon, 1985; Bouchon, 1989):
a
b
q
q
F
q
C
Figure 1.
L
*
Figure 2.
Ordinary topography model.
L
*
F
L
L
*
*
Periodic source configuration of DWN method.
Localized Boundary Integral Equation–Discrete Wavenumber Method for Simulating P-SV Wave Scattering
usc xn ; z; ω
N X L GDWN xn xm ; z 2N 1 mN
ξxm qxm ;
Formulation of the Localized BIE-DWM for P-SV Topographic Problems
(3)
where G is the discrete wavenumber representation of Green’s tensor G, fxm ; ξxm ; m 0; 1; 2; …; Ng are the locations of the Huygens point source along the irregular topographic surface, and qxm denotes the strength of the mth Huygens point source at the surface. L is the length of periodicity of the source-medium configuration, and 2N 1 is the total grids with an equal horizontal interval within one periodicity; thus, fxm mΔx; m 0; 1; 2; …; Ng and Δx L=2N 1. Moreover, at the horizontal grids, GDWN can be further expressed as DWN
GDWN xn ; z
N X
~ m ; z expixn km ; Gk
267
Recently, we have proposed a follow-up method to the BIE-DWM for the SH wave scattering problem caused by an
irregular surface topography. In that method, the dimension of the matrix being inversed is only proportional to the sampling number within the corrugated part of the surface (Zhou and Chen, 2006a). Therefore, it is particularly efficient for the problem in which the corrugated part of the surface is localized; thus, our method is named the localized BIE-DWM and is abbreviated loBIE-DWM. In this study, we shall extend the loBIE-DWM to the P-SV topographic problem. Similar to the SH topographic problem (Zhou and Chen, 2006a), the first step for such an extension is an orthogonal decomposition of the unknown source strength vector fqg; that is, fqg fqf g⊕fqc g;
(4)
(6)
mN
where kn 2πn=L denotes the discrete horizontal wave~ denotes the discrete Fourier transform of number, G GDWN , and its explicit expression is given in Appendix A. The irregular surface is a free surface on which traction should be vanished; that is,
tin xn ; zn
N X
fHDWN xn xm ; zn ξxm gqxm 0;
mN
(5) where zn ξxn and n 0; 1; 2; …; N; tin and HDWN are the tractions associated with uin and GDWN , respectively. Notice that equation (5) provides exactly the number of linear equations as that of the unknowns fqxm ; m 0; 1 2; …; Ng; thus, they could be directly applied to solve the unknowns after a minor modification on the collocation elements of the coefficient matrix (Bouchon, 1985; Campillo and Bouchon, 1985; Bouchon, 1989). This is the basic idea of the BIE-DWM. Its validity and the applicability to practical problems have been demonstrated in a number of studies (Bouchon, 1985; Campillo and Bouchon, 1985; Bouchon, 1989; Bouchon and Coutant, 1994; Bouchon, 1996). It is obvious, however, that the computation cost of BIE-DWM directly depends on the total sampling number within one spatial periodicity (i.e., x∈ L=2; L=2). On one hand, the dimension of the coefficient matrix involved in this linear equation system is proportional to the total sampling number; on the other hand, a proper estimation on the high frequency problem requires dense spatial sampling. Consequently, the dimension of the matrix is increased and so is the computation cost for highfrequency problems.
where, as shown in Figure 3, fqc g denotes the unknown source strengths of those sampling points within the corrugated part of the surface, while fqf g denotes the unknowns of the flat part of the surface: qf xm θ1 xm qxm ;
qc xm θ2 xm qxm ;
where θ1 and θ2 are two orthogonal window functions (Zhou and Chen, 2006a). Substituting equation (6) in equation (5) yields N X
HDWN xn xm ; zn qf xm
mN
Nb X
HDWN xn xm ; zn ξxm qc xm
(7)
mN a
tin xn ; zn 0; where N a and N b are the sampling point number corresponding to the irregular part of the topography in Figure 1. For an arbitrary sampling point over the flat part of the surface (i.e.,
L
Figure 3. The cartoon of the orthogonal decomposition of the unknown source strength vector fqg.
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H. Zhou and X. Chen
zn 0). Equation (7) becomes N X
with kp
HDWN xn xm ; 0qf xm
2πp ; L
for p 0; 1; 2; …; N:
mN Nb X
HDWN xn xm ; 0 ξxm qc xm
(8)
mN a
~ p ; 01 q~ f kp Hk
tin xn ; 0 0:
tun xn
H
DWN
non-zero quantity; 0;
xn ∈Ωc : xn ∈Ωf
qf xl
Nb X N X
~ p ; zn expikp xn ~ 0 kp 1 Hk H
nN a pN
(9)
xl qc xn
N X
~ 0 kp 1 ~t0in kp expikp xl H
pN
xn xm ; 0qf xm
HDWN xn xm ; 0 zm qc xm
Nb X
mN a
N X 1 ~ k 1 expikp xl H 2N 1 pN 0 p
xm tun xm
(10)
mN a
(13a)
tin xn ; 0 tun xn ;
for xl ∈Ωf , and
for xn ϵL=2; L=2: Then equation (10) can be executed by the discrete Fourier transform (DFT) with respect to fxn g over L=2; L=2, and becomes ~ p ; 0q~ f kp Hk
Nb X
~ p ; zm qc xm expikp xm Hk
mN a
~tin kp
(12)
Applying inverse DFT to equation (12) and exchanging summation order yields
mN Nb X
~ p ; zm qc xm Hk
tun xm ~ expikp xm tin kp ; 0 : 2N 1
With tun xn , equation (8) can be extended to the whole assembly of horizontal sampling within one spatial periodicity; that is, N X
X Nb mN a
It is noted that equation (8) holds only for xn ∈Ωf . For the range of xn ∈Ωc (in Fig. 3), equation (8) does not hold any more; the right-hand side of equation (8) is equal to a nonzero quantity whose value is unknown yet at this stage. We denote this unknown quantity by tun xn , and it obviously has following property:
Equation (11) can be solved as
Nb X 1 t x expikp xm ; 2N 1 mN un m
0
Nb X N X
~ p ; zn expikp xn ~ 0 kp 1 Hk H
nN a pN
xl qc xn
Nb X mN a
N X
~ 0 kp 1 ~t0in kp expikp xl H
pN N X 1 ~ k 1 expikp xl H 2N 1 pN 0 p
xm tun xm
a
(11)
(13b)
~ p ; z is a DFT of Hx ~ n ; z, and Hx ~ n ; z is given where Hk in Appendix B; further,
~ 0 kp Hk ~ p ; 0 and for xl ∈Ωc. Here, we denote H ~t0in kp ~tin kp ; 0 for simplicity. In compact form, equations (13a) and (13b) are expressed as
q~ f kp
~tin kp
N X 1 q x expikp xm ; 2N 1 mN f m
N X
1 t x ; 0 expikp xm ; 2N 1 mN in m
Qf Af Qc Sf Bf Tun ;
(14a)
Ac Qc Sc Bc Tun ;
(14b)
where Af , Ac , Bf , Sf , and Sc are the sub-blocks of matrices A and B and vector S as defined next:
Localized Boundary Integral Equation–Discrete Wavenumber Method for Simulating P-SV Wave Scattering Af
A1 ; A3
Bc B2 ;
Ac A2 ; Sf
S1 ; S3
Bf
B1 ; B3
Ef nm HDWN xn xm ; ξxn ;
(15)
Sc S2 ;
1 A1 A @ A 2 A; A3
0
1 B1 B @ B2 A and B3
0
1 S1 S @ S2 A: (16) S3
The dimensions of A and B and vector S are 22N 1 × 22N 1 and 22N 1 × 1, and their block elements are defined by N X
Amn
~ p ; zn expikp xn xm ; ~ 0 kp 1 Hk H
pN
(17a) N X
Bmn
~ 0 kp 1 expikp xn xm ; H
(17b)
~ 0 kp 1 ~t0in kp expikp xm ; H
(17c)
pN
Sm
N X pN
where n; m 0; 1; 2; …; N. The unknowns in equations (14a) and (14b) are 2
3 tun xNa 6 tun xN a 1 7 6 7 Tun 6 7; .. 4 5 . tun xNb Q Qf Qf;1 ; 2
Qf;1
f;2
2
3 qc xN a 6 qc xN a 1 7 6 7 Qc 6 7; .. 4 5 . qc xN b
and
Ec nm H
DWN
xm ∈Ωf ; xn xm ; ξxn ξxm ; Tin n tin xn ; ξxn ;
for xn ∈Ωc :
Substituting equation (19) in (20) yields a linear equation system for determining the unknown Qc as follows: fEf Bf Bc 1 Ac Af Ec gQc Tin Ef Sf Bf Bc 1 Sc :
With this linear equation, Qc can be determined; thus, Qf can be consequently determined via equation (19). Finally, substituting the solved Qc and Qf in equation (3), we can calculate the P-SV wavefield scattered by the irregular surface. Compared with the BIE-DWM proposed by Campillo and Bouchon (1985), the loBIE-DWM for the topographic problem of P-SV waves proposed here is more efficient in terms of computation. This is because, according to equation (21), the dimension of the matrices to be inversed in this method is proportional to the dimension of Ωc , which is only a fraction of the dimension of ΩL , while the dimension of the matrix to be inversed in BIE-DWM is proportional to the dimension of ΩL . Obviously, the more localized the corrugated part of the ground surface is, the more efficiently this new method performs. The previous description does not limit Ωc to a continuum region. This means the formulas are suitable to the topography with several separated irregular regions. In that case, Qc is composed with the total forces distributed on all irregular regions, while Qf is composed with the total forces on all flat regions.
2
and
Qf;2
3
qf xNb 1 6 qf xNb 2 7 6 7 6 7: .. 4 5 . qf xN
It can be seen from the preceding section that the loBIEDWM is rigorously derived from the basic formulation of Bouchon and Campillo’s BIE-DWM. Therefore, in terms of accuracy, the results of the loBIE-DWM should be identical to those of BIE-DWM. Indeed, this is exactly what we have
(18) By combining equations (14a) and (14b) and eliminating the unknown vector Tun, we obtain a linear relationship between Qf and Qc as follows: Qf Sf Bf Bc 1 Sc fBf Bc 1 Ac Af gQc : (19) Uin
To determine Qc , we recast equation (5) only on the corrugated part of the surface and obtain Ef Qf Ec Qc Tin ; where
(21)
Numerical Validations
3
qf xN 6 qf x1N 7 6 7 6 7; .. 4 5 . qf xN a 1
for xn ∈Ωc
for xn ; xm ∈Ωc ;
and 0
269
θ
(20) Figure 4.
Semicylindrical canyon model.
270
H. Zhou and X. Chen
2
Amplitude(V−COMP)
3 2.5 2
1 1.5
0.5
1 0.5 −2 1.5
Amplitude(H−COMP)
1.5
(a1) −1
0
2
0.5
1 (b1) −1
0 (1)x/a
Figure 5.
Amplitude(V−COMP)
2
1
0 −2
Amplitude(H−COMP)
1
(a2) 0 −2 3
1
2
0 −2
−1
0
1
2
(b2) −1
0 (2)x/a
1
2
Comparison between our results and BC due to vertical plane P and SV (η 1).
2
1.5
1.5
1
1 0.5
0.5 0 −2 2
(a1) −1
0
1
2
0 −2 2
1.5
1.5
1
1
0.5
0.5
0 −2
(b1) −1
Figure 6.
0 (1)x/a
1
2
0 −2
(a2) −1
0
1
−1
0
1
2
(b2) 2
(2)x/a
Comparison between our results and BC. P with 30° angle and SV with 15° angle (η 1).
seen in the comparisons between the numerical results in the model in Figure 4 and the two methods shown in Figures 5– 9. In these comparisons, the topographic model is a semicylindrical canyon with radius of a shown in Figure 4. Figures 5–8 display the amplitudes of the displacements
on the ground surface due to a vertically incident P or SV plane wave for dimensionless frequencies η 1 and 2, respectively, where the dimensionless frequency is defined by η ωa=2πβ. The computational parameters used in these comparisons are L 20a, Δx 0:2a and ωI
Amplitude(H−COMP)
Amplitude(V−COMP)
Amplitude(H−COMP)
Amplitude(V−COMP)
Localized Boundary Integral Equation–Discrete Wavenumber Method for Simulating P-SV Wave Scattering
4
2
3
1.5
2
1
1
0.5 (a1)
0 −2 3
−1
0
1
2
271
(a2) 0 −2 4
−1
0
1
2
3
2
2 1 0 −2
1 1
(b1) 2
(b2) 0 −2
−1
0 (1)x/a
Figure 7.
Comparison between our results and BC due to vertical plane P and SV (η 2).
3
1.5
2
1
1
0.5
0 −2 2
(a1) −1
0
1
2
1.5
1
1
0.5
0.5 (b1) −1
Figure 8.
0 (1)x/a
1
2
0 (2)x/a
1
2
(a2) 0 −2 2
1.5
0 −2
−1
0 −2
−1
0
1
2
(b2) −1
0 (2)x/a
1
2
Comparison between our results and BC. P with 30° angle and SV with 15° angle (η 2).
2πβ=L. Figure 9 displays the results from oblique incident waves (θ 30°, θ 15°), which once again show identical results with those of BIE-DWM. In addition to having an identical precision to that of BIE-DWM, the loBIE-DWM performs more efficiently. Fig-
ure 10 shows the comparison of the computational times consumed by the two methods for the semicylindrical canyon’s scattering problem for taking different ratios of L=a in the computation, where the Δx=a is fixed as 0.05 for all cases, and the dimensionless frequency η is set to be 1. The
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H. Zhou and X. Chen
Amplitude(V−COMP)
6
2
4
1.5 1
2
0.5
Amplitude(H−COMP) depth(km)
2.5
0 −2 3
(a1) −1
0
1
2
(a2) −1
0
1
2
1.5
2
1
1
0 −2 1 0.5 0 −0.5 −2
Figure 9.
0.5
−1
0
1
−1
0 (1)Position
1
(b1) 2
0 −2 1 0.5 0 −0.5 −2 2
−1
0
1
2
0
1
2
Topographic Influences on the Propagation of Rayleigh Wave
1000 800 600 400 200
Figure 10.
−1
Comparison between our results and BC for Gauss canyon. (1) P plane wave 30° angle. (2) Plane SV with 30° angle (η 1).
1200
0 0
(b2)
(2)Position
1400
CPUBIE−DWN/CPUour
0 −2 2
10
20 L/a
30
40
Ratio of CPU time between BC and our again L/a.
computed results shown here were executed on a personal computer with a P4 CPU with 2.8 GHz clock speed. It is obvious that the loBIE-DWM becomes more efficient than the BIE-DWM as the ratio of L=a increases (e.g., it is about 1000 times faster when L=a 40).
The classic Rayleigh wave is a prominent seismic phase in an elastic half-space with a flat ground surface. Its mechanism of generation and the characteristics of propagation have been well studied since it was discovered more than a century ago (Lamb, 1904). In many practical problems, topographic irregularity of the ground surface cannot be avoided; thus, its influences on the propagation of Rayleigh waves must be carefully taken into account. Until now, however, there has been a lack of a systematic investigation on such influences, though some studies have been carried out in the past three decades (Wong, 1982; Dravinski and Mossessian, 1987; Kawase, 1988; Sánchez-Sesma and Campillo, 1991, 1993; Chen, 1999). In this study, we investigate this issue through numerical simulations utilizing the efficient loBIE-DWM that we have presented previously in this article. Consider a homogeneous half-space covered by an arbitrarily irregular free surface. In this model, the velocities of S and P waves are respectively 1 and 1:73 km=sec, and an explosive source is buried 0.5 km deep and distanced 15 km away from the nearest edge of the corrugated part of the topography (see Fig. 11). Such an model is designed for
Localized Boundary Integral Equation–Discrete Wavenumber Method for Simulating P-SV Wave Scattering
273
20 19 18
position(km)
17 16 15 14 13 12 11 −1 −0.5
0
0.50
(A)topography(km)
Figure 11.
5
10
15
20
25
(B)time(s),horizontal component
300
5
10
15
20
25
30
(c)time(s),vertical coponent
Part (a) shows the model used here. Parts (b) and (c) show seismic responses in the concave canyon caused by an explosion source 15 km away from the canyon center with a 1-Hz center frequency. The left-hand side shows the horizontal components, and the righthand side shows the vertical components on the surface. Parts (d) and (e) show corresponding snapshots of the vertical and horizontal components. Line a is 1.7617 sec; line b is 3.229 sec; line c is 4.6967 sec; line d is 6.1644 sec; line e is 7.6321 sec; line f is 8.8063 sec; line g is 9.9804 sec; line h is 11.1546 sec; line i is 12.3288 sec; line j is 13.5029 sec; line k is 14.6771 sec; line l is 15.8513 sec; line m is 16.7319 sec; line n is 17.319 sec; line3 o is 20.5479 sec.
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H. Zhou and X. Chen
investigating the influence of corrugated topography on the propagation of P wave and Rayleigh wave because in this model, the Rayleigh wave not only has been formed prior to its propagating through the corrugation of topography but also can be separated clearly from the direct P wave in time series as seen in the next section. Two kinds of topography, the valleylike and mountainlike models, are considered here. Valleylike Topography: Arc, Semicircular, and Gaussian Valleys Figure 11b,c shows the time responses of the ground motion in the vicinity of and on the arc valley (Fig. 11a). It shows that on the source side (the lower side of the corner), the incoming Rayleigh wave and its reflection coexist and propagate in opposite directions. In valley area P wave, Rayleigh waves and creeping waves are moving from the lower corner to upper corner of the valley, and their reflections from the lower corner can be observed, although they are relatively weak. Figure 11d,e is a snapshot of wave propagation in the same topography model as Figure 11a. It displays complex features of seismic waves (P and Rayleigh waves and their conversions) and reveals the detailed conversion process among different seismic phases caused by the valley. Figures 12 and 13 are the theoretical seismograms on the ground for the Gaussian and semicylindrical canyons, re-
spectively. The semicylindrical canyon has a very abrupt slope change at the corners (x 59, 61); the Gaussian canyon has a smooth slope change at the corners, while the arc valley shown in Figure 11 has a moderate slope change at the corners (x 58, 62). The most notable discrepancy among the ground responses to these three canyons is the backward scatterings (or reflections) by their corners. It can be seen that the richness of energy of the reflected (or backward scattered) waves is proportional to the degree of changes in topographic slope. For instance, reflections or backward scatterings occurred in the semicylindrical canyon are strongest, and those in Gaussian canyon are almost negligible. Mountainlike Topography: Arc, Semicylindrical, and Gaussian Mountains Figures 14–16 are the theoretical seismograms on the ground for the Gaussian, arc, and semicylindrical mountains, respectively. Compared with those for valleylike topography, we find that the seismograms on the surface for mountainlike topography show different, while somewhat complementary, characteristics. A predominant feature that can be seen is that the lower corner (the nearest corner to the source) of the irregular topography generates stronger forward scattering bit weaker backward scattering than those of valleylike topography, especially for the cases of arc and semicylindrical mountains shown in Figures 15 and 16. Another notable
65 64 63
position(km)
62 61 60 59 58 57 56 −1 −0.5 0 0.50 topography(km)
5 10 15 20 25 (b)time(s),horizontal component
Figure 12.
300
5
10 15 20 25 (c)time(s),vertical coponent
Displacement on the ground of the Gaussian valley.
30
Localized Boundary Integral Equation–Discrete Wavenumber Method for Simulating P-SV Wave Scattering
275
64 63
position(km)
62 61 60 59 58 57 −1 −0.5 0 0.50 (a)topography(km)
5 10 15 20 (b) time(s),horizontal component
Figure 13.
250
5 10 15 20 (c) time(s),vertical coponent
25
Displacement on the ground of the semicylindrical valley.
64 63
position
62 61 60 59 58 57 56 −0.5 0 0.5 1 0 (a)topography(km)
5 10 15 20 25 (b)time(s),horizontal component
Figure 14.
300
5
10 15 20 25 (c)time(s),vertical coponent
Displacement on the ground of the Gaussian mountain.
30
276
H. Zhou and X. Chen
64 63
position
62 61 60 59 58 57 56 −0.5 0 0.5 1 0 (a)topography(km)
5 10 15 20 25 (b)time(s),horizontal component
Figure 15.
300
5
10 15 20 25 (c)time(s),vertical coponent
30
Displacement on the ground of the arc mountain.
64 63
position
62 61 60 59 58 57 56 −0.5 0 0.5 1 0 (a)topography(km)
5 10 15 20 25 (b)time(s),horizontal component
Figure 16.
300
5
10 15 20 25 (c)time(s),vertical coponent
Displacement on the ground of the semicylindrical mountain.
30
Localized Boundary Integral Equation–Discrete Wavenumber Method for Simulating P-SV Wave Scattering feature of the results is the stronger and richer reverberation recorded on the surface of the mountain, especially for the Rayleigh wave, as shown in Figure 16. The most predominant feature of the results for mountainlike topography is that there are more Rayleigh waves passing through the mountainlike topography. This means that the mountainlike topography does not prevent the effective passage of Rayleigh wave’s energy, unlike the valleylike topography. For Rayleigh wave propagation, in some senses, the mountainlike topography behaves likes an accelerator, while the valleylike topography behaves like a highly dissipated trap. These understandings are useful to the study and survey of seismic hazards.
Conclusions In this study we presented a new method, the loBIEDWM, for simulating the scattered P-SV waves by a 2D irregular surface topography. This method is rigorously derived from the basic formulation of Bouchon and Campillo’s BIE-DWM. The dimension of inverse matrix involved in this new algorithm is only proportional to the effective sampling number within the corrugated part of the surface. Therefore, its computation efficiency is increased dramatically while it maintains the same accuracy as BIE-DWM, particularly for the problem in which the corrugated part of the topography is highly localized. For instance, it is about 1000 times faster than BIE-DWM when L=a 40. With this efficient algorithm, we investigated topographic influences on the propagation of Rayleigh wave by applying to the valleylike and mountainlike topographic models, and we arrived at the following conclusions: (1) The most sensitive factor affecting on the propagation of Rayleigh wave is the change rate of topographic slope, and the larger the change, the stronger the scatterings (including forward and/or backward scatterings). (2) A valleylike topography usually dissipates the energy of propagating Rayleigh wave, while a mountainlike topography does not only dissipate the energy of propagating Rayleigh wave, but it also generates some scattered Rayleigh waves. In summary, the algorithm we developed in this study (i.e., loBIE-DWM) provides a powerful tool for investigating the topographic influences on seismic ground motion.
Acknowledgments This study was supported by the National Natural Science Foundation of China (Grant Numbers 40474011, 40521002, and 90715020) and by the Institute of Geophysics, China Earthquake Administration (Grant Number DQJB07B06).
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Appendix A
Green’s Function for Displacement GDWN
GDWN xx GDWN zx
GDWN xz ; GDWN zz
where GDWN xx x; z
N 2 X kp i E1 γ 2 E2 ; 2Lk2β μ pN γ 1
GDWN x; z xz
N isgnz zc X kp E1 E2 ; 2 2Lkβ μ pN
GDWN ; GDWN zx xz
GDWN x; z zz
N X k2p i γ E1 E2 : γ2 2Lk2β μ pN 1
Localized Boundary Integral Equation–Discrete Wavenumber Method for Simulating P-SV Wave Scattering
Appendix B
HDWN x; z zz
Green’s Function for Stress HDWN x; z
H DWN xx H DWN zx
H DWN xz ; H DWN zz
N X 1 2kp γ 1 E1 2Lk2β μ pN kp 2 2 2γ 2 kβ E2 n1 γ2
sgnz
where
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zc 2k2p
k2β E1
2k2p E2 n2
with s ω2 kp ; γ2 β2
N X kp 1 HDWN x; z 2γ n1 k2β E1 xx 2Lk2β μ pN γ 1 2kp γ 2 E2 n1
sgnz zc 2k2p E1 2γ 22 k2β E2 n2 ; N X 1 x; z H DWN sgnz zc 2k2p E1 xz 2Lk2β μ pN
2γ 22 k2β E2 n1 kp 2k2p k2β E1 2kp γ 2 E2 n2 ; γ1
H DWN x; z zx
N X 1 sgnz 2Lk2β μ pN zc 2γ 2n k2β E1 2k2p E2 n1
kp 2kp γ 1 E1 2γ 22 k2p E2 n2 ; γ1
s ω2 kp ; γ1 α2
Im γ 2 ≤ 0;
Im γ 1 ≤ 0;
E1 eiγ1 jzzc j eikp xxc ; E2 eiγ2 jzzc j eikp xxc : α and β are the velocity of P and S waves, respectively, and n1 and n2 are the x and z components, respectively, of the normal direction on a surface.
Laboratory of Computational Geodynamics, School of Earth and Space Science Peking University Beijing 100871, China
[email protected]
Manuscript received 10 June 2006
