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Analysis of effects of active sources on observed phase velocity based on the thin layer method
Journal of Applied Geophysics 73 (2011) 49–58
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Journal of Applied Geophysics j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / j a p p g e o
Analysis of effects of active sources on observed phase velocity based on the thin layer method Hua-You Chai a,⁎, Kok-Kwang Phoon b, Chang-Fu Wei a, Ying-Fa Lu c a b c
State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan 430071, China Department of Civil Engineering, National University of Singapore, Singapore 117576, Singapore Key Laboratory of Geological Hazards on Three Gorges Reservoir Area, Ministry of Education, China Three Gorges University, Yichang 443002, Hubei, China
a r t i c l e
i n f o
Article history: Received 10 November 2009 Accepted 13 November 2010 Available online 21 November 2010 Keywords: Near-field effects Surface waves Mode Thin layer stiffness method Effective phase velocity
a b s t r a c t In the wave field induced by active sources, the observed phase velocity of surface waves is influenced by both mode incompatibility (i.e. non-planar spread of surface waves is idealized as plane waves) and body waves. Effects of sources are usually investigated based on numerical simulations and physical models. Several methods have been proposed to mitigate the effects. In application, however, these methods may also have difficulties since the energy of the body waves depends on soil stratification and parameters. There are multiple modes of surface waves in layered media, among which the higher modes dominate the wave field for soils with the irregular shear velocity profiles. Considering the mode incompatibility and the higher modes, we derive analytical expressions for the effective phase velocity of the surface waves based on the thin layer stiffness method, and investigate the effects of the body waves on the observed phase velocity through the phase analysis of the vibrations of both the surface waves and the body waves. The results indicate that the effective phase velocity of the surface waves in layered media varies with the frequency and the spread distance, and is underestimated compared to that of the plane surface waves in the spread range less than about one wavelength. The oscillations that appeared in the observed phase velocity are due to the involvement of the body waves. The mode incompatibility can be ignored in the range beyond one wavelength, while the influence range of the body waves is far beyond one wavelength. The body waves have a significant influence on the observed phase velocity of the surface waves in soils with a soft layer trapped between the first and the second layers because of strong reflections. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The propagation behavior of surface waves (i.e. Rayleigh waves in this paper) in horizontally layered media is often studied under the assumption that the surface waves at the free state (i.e. no source) travel in the form of plane wavefronts. The modes of the surface waves at the free state are usually termed as the normal modes. Under this assumption, the phase velocity of the normal modes depends only upon the frequency (or wavelength) and the penetration depth. The dispersions of surface waves can be obtained by performing surface wave testing. The surface waves can be generated by either active or passive sources (Tokimatsu et al., 1992a,b; Morikawa et al., 2004; Park et al., 2005; Malovichko et al., 2005; Roberts and Asten, 2006; Yoon and Rix, 2009). The active sources are those artificially generated in the wave testing by, for example, a sledge hammer, a falling weight or an explosive, whereas the other sources are termed as passive sources, such as nearby traffic loading and microtremors induced by sea waves. Dispersions over
⁎ Corresponding author. E-mail address: [email protected] (H.-Y. Chai). 0926-9851/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jappgeo.2010.11.005
a wide range of wavelengths can be obtained by combining active and passive tests. In general, the wave fields induced by passive sources are omni-directional. Two-dimensional arrays (Tokimatsu et al., 1992a; Morikawa et al., 2004; Roberts and Asten, 2006) are used in these tests. The phase velocity of the superposed vibrations of various modes is imaged by frequency–wavenumber spectra or extracted by the spatial autocorrelation (SPAC) method (Tokimatsu et al., 1992a; Morikawa et al., 2004; Roberts and Asten, 2006). Active sources are often used to obtain the properties of shallow soils. Two-station measurements are originally used in the surface wave testing, in which dispersion curves are obtained by spectral analysis of surface waves (SASW) (Gucunski and Woods, 1992; Nazarian and Desai, 1993; Al-Hunaidi, 1993; Foti, 2000). For a wave field with multiple modes, the surface vibration is the superposition of all the modes at a given frequency. The phase velocity of the superposed vibration is termed as the effective phase velocity (Lai and Rix, 1999; Foti, 2000; Strobbia, 2003; O'Neill et al., 2003; O'Neill and Matsuoka, 2005) or the apparent phase velocity of the surface waves (Tokimatsu et al., 1992a,b). In this paper, the term “effective phase velocity” is used. In SASW, since the phase velocity is calculated from the phase lag of the superposed vibration, the spectral analysis
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H.-Y. Chai et al. / Journal of Applied Geophysics 73 (2011) 49–58
technique cannot be used to determine the phase velocity of a single mode (Al-Hunaidi, 1993; Lu and Zhang, 2004). To account for higher modes, the multi-station measurement techniques have been developed. The multichannel analysis of surface waves (MASW), such as frequency–wavenumber (f–k) or frequency–slowness (f–p) transforms, is used to obtain the dispersion images of the modes based on the plane-wave assumption (Al-Hunaidi, 1996; Park et al., 1998; Foti, 2000; O'Neill et al., 2003; Strobbia, 2003; Xia et al., 2003; Lu and Zhang, 2004; Yoon, 2005; Yoon and Rix, 2009). When the spatial aperture of the measurement and the number of the receivers are large, the dispersions of some higher modes can be obtained over limited frequency ranges (Foti, 2000; O'Neill et al., 2003; Strobbia, 2003). If the spatial aperture of the measurement is small, however, the resolution of the wavenumber is low. In this case, all the modes are superposed together, the modes cannot be identified and separated in such a case, and thus only the effective phase velocity can be obtained (Foti, 2000; Strobbia, 2003; Lu and Zhang, 2004). Based on the assumption of plane surface waves, the dispersions of modes can be calculated by the propagator matrix model (Foti, 2000; Strobbia, 2003) and the effective phase velocity can be analyzed using the models presented by Tokimatsu et al. (1992a,b) and Lai and Rix (1999) for passive and active sources, respectively. If a source exists, however, the assumption of a pure and plane surface wave field is no longer applicable. Effects of the source on the analysis of surface waves can be attributed to two factors: one is the mode incompatibility (Zywicki and Rix, 2005), and the other is the presence of the body waves including the direct body waves and the reflections from the interfaces of layers. The wave field under the influence of these two factors is called the near-field, and all others are termed as the far-field. Tokimatsu et al. (1992a,b) and Lai and Rix (1999) have proposed the expressions of the effective phase velocity, which take into account the higher modes but ignore the near-field effects. The near-field effect can be investigated via analytical methods, numerical methods, physical modeling and field testing (Zywicki and Rix, 2005; Strobbia and Foti, 2006; Bodet et al., 2009; Yoon and Rix, 2009). It has been recognized that the near-field effect can be mitigated by using the cylindrical beamformer and multi-offset phase analyses, or by increasing the total number of receivers and the array center distance in an array-based testing (Zywicki and Rix, 2005; Strobbia and Foti, 2006; Yoon and Rix, 2009). These proposals are helpful for mitigating the near-field effects and analyzing the observed phase velocity. However, validation and generalization of results from numerical and physical models need to be supported using theoretical analysis. The mode incompatibility and the body waves have different influences on the observed phase velocity, which must be investigated separately. In this paper, an analytical expression of the effective phase velocity is derived to account for both the mode incompatibility and the higher modes, based on the thin layer stiffness method. Effects of the mode incompatibility and the body waves on the observed phase velocity are investigated separately. Because the pattern and the energy of the direct body waves and the reflections are related to Poisson's ratio and soil stratifications, we shall study the near-field effect in homogenous half space media with different Poisson's ratios and layered half space media with regular or irregular soil profiles (Tokimatsu et al., 1992b). 2. Thin layer stiffness matrix method To determine the dispersions of the normal modes and the displacement responses of modes under harmonic loadings, two procedures can be adopted, namely, the Thomson–Haskell propagator matrix method (Tokimatsu et al., 1992b; Foti, 2000; Strobbia, 2003; Lu and Zhang, 2004) and the stiffness matrix method (Kausel and Roësset, 1981; Kausel and Peek, 1982; Ganji et al., 1998; Foinquinos and Roësset, 2001). In applying the propagator matrix algorithm, however,
there exist some inconveniences, including: (a) matrices and their derivatives appear in the displacement response of modes (Tokimatsu et al., 1992b), (b) when the material damping is taken into account, the elements of the propagator matrix are complex, the solution of the complex propagator matrix is difficult and time-consuming, and (c) the contour integral technique is needed in resolving the responses of soils to point or disk loadings since the temporal and spatial Fourier transform is used (Kausel and Peek, 1982). There are also some inconveniences in the evaluation of the stiffness matrix because of the transcendental or trigonometric functions in the matrix. To resolve the aforementioned issues, the soil layers are divided into “thin” layers in the sense that the layer thickness is smaller than the wavelength, and as a consequence, the transcendental or trigonometric functions in the stiffness matrix are reduced to simple algebraic functions and the explicit expression of response of soils to loadings can be obtained. The availability of an explicit analytical expression is very useful for deriving the effective phase velocity of the surface waves and studying the effects of the body waves. When the layer thicknesses are less than 1/8 to 1/10 of the wavelength, the matrix stiffness of the jth thin layer for P–SV waves can be expressed as (Kausel and Roësset, 1981; Kausel and Peek, 1982; Foinquinos and Roësset, 2001): 2
ð1Þ
Kj = Aj k + Bj k + Cj
where Cj = Gj − ω2Mj, and k and ω are the wavenumber and the circular frequency, respectively. If the displacement vector is rearranged first by grouping all horizontal (x or r) interface displacements followed by all vertical (z) interface displacements (rather than arranged by alternating horizontal and vertical displacements of interfaces), the coefficient matrices Aj, Bj, and Cj in Eq. (1) can be presented in the following forms: 2 2 λj + 2μ j λj + 2μ j 6 6 hj 6 2 λj + 2μ j 6 λj + 2μ j Aj = 6 66 6 4
3
7 " # 7 7 Axj 0 7 7= 7 0 Azj 2μ j μ j 7 5 μ j 2μ j 3 2 λj −μ j − λj + μ j 6 7 6 7 2 6 7 0 λj + μ j − λj −μ j 7 16 7=4 Bj = 6 7 T 26 6 7 λj + μ j λj −μ j Bxzj 6 7 4 5 − λj + μ j − λj −μ j 2 3 μj −μ j 6 7 6 −μ j 7 μj 7 16 7 6 Gj = 6 λj + 2μ j − λj + 2μ j 7 7 hj 6 6 7 4 5 − λj + 2μ j λj + 2μ j 2 3 2 1 6 7 7 ρhj 6 1 2 6 7 Mj = 7 6 6 2 1 4 5 1 2
2
μ j ρj hj ω2 − 6 6 hj 3 6 6 6 6 μ j ρj hj ω2 6− − 6 h 6 6 j 6 Cj = 6 6 6 6 6 6 6 6 6 6 4 " =
Cxj
0
0
Czj
#
−
hj
2
−
3 5
0
3
μ j ρj hj ω2 − hj 6
μj
Bxzj
ρj h j ω 3
λj + 2μ j hj
2
−
− λj + 2μ j hj
ρj hj ω 3
2
−
ρj h j ω 6
7 7 7 7 7 7 7 7 7 7 7 7 27 − λj + 2μj ρj h j ω 7 − 7 hj 6 7 7 7 7 7 2 λj + 2μj ρj hj ω 5 − hj 3
ð2Þ
H.-Y. Chai et al. / Journal of Applied Geophysics 73 (2011) 49–58
where hj, λj and μj are the thickness and Lamé's constants of the jth layer, respectively. Except for Bxzj, the submatrices Axj, Azj, Cxj and Czj are symmetrical. The elements of Axj, Azj and Bxzj depend on the material parameters of the jth layer, while those of Cxj and Czj are related to both material parameters and frequency. 2.1. Layered media with a rigid base For layered media with a rigid base, since the horizontal and the vertical displacements are zero at the base, the global matrix can be obtained by assembling the matrices of the thin layers as follows: 2 ˜ =4 K
˜ x + C˜ x k2A
kB˜ xz
˜ xz kB
˜ z + C˜ z k2A
T
3 5
ð3Þ
˜ x; A ˜ z ; C˜ x ; C˜ z and B˜ xz are assemblies of the corresponding where A matrices of thin layers. ⇀ ⇀ ˜ϕ At a free state, we have K = 0 , that is, 2 4
3 ⇀ ⇀ 5 ϕ⇀x = 0 2˜ ϕz k Az + C˜ z
2˜ ˜ kA x + Cx
kB˜ xz
T kB˜ xz
ð4Þ
h
ϕ = ux;1 ; ux;2 ; ⋯; ux;N−1 ; ux;N ; uz;1 ; uz;2 ; ⋯; uz;N−1 ; uz;N ⇀
iT
⇀ ϕ = ⇀x : ϕz
ð5Þ
⇀
N is the number of the thin layers; ϕx and ϕz are the horizontal and the vertical displacement vectors, respectively. The free surface is numbered as 1 (see Fig. 1). Eq. (4) can be rewritten as ("
˜x A 0
B˜ xz ˜z A
#
" #) ⇀ ⇀ 1 C˜ x 0 k ϕx = 0: + 2 ⇀ k B ϕz ˜ Txz C˜ z
ð6Þ
Accordingly, the solution to Eq. (4) can be converted into an eigenvalue problem. 2.2. Layered media with an underlying half space The stiffness matrix of a half space layer for P–SV waves is given as (Kausel and Roësset, 1981; Wolf, 1985): 8 <
1−rβ2
r α K0 = 2kμ 1 :2 1−r r α β
1 0 − rβ 1
where vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 u u ω ω 2 rα = t1− ; rβ = 1− kcp0 kcs0
ð8Þ
cp0 and cs0 are velocities of the compression and the shear waves, respectively. The above matrix is not in the form of Eq. (1). Thus, the global matrix is not in the form of Eq. (6) and layered media with an underlying half space are not an eigenvalue problem. For the waves with a small wavenumber k, however, the propagating directions of the waves are not far from the vertical direction [also known as paraxial waves, Kausel, 2001], and Eq. (7) can be expanded using Taylor's series. Keeping the terms up to k2, the matrix of a half space layer can be reduced to (Ganji et al., 1998; Kausel, 2001) 2
ð9Þ
K0 = A0 k + B0 k + C0 where 2
where the displacement vector is defined as follows: ⇀
51
9 = 1 0 ;
ð7Þ
3 2−α " 0 7 − 6 α 1 cs0 6 1−2α 0 7 A0 = iμ 6 7; B0 = μ 2 ω4 α 1 1−2α 5 0 α3 " # 1 0 C0 = iωρcs0 ; α = cs0 = cp0 0 1= α
1
#
0
ð10Þ
pffiffiffiffiffiffiffiffi i = −1. The elements of matrix B0 are real, while those of matrices A0 and C0 are imaginary. These observations imply that, for the paraxial waves, the half space layer can be modeled by an absorbing boundary containing elastic and damping components (see Fig. 1). It can be deduced from Eq. (9) that for layered media with an underlying half space, the global matrix is in the form of Eq. (3). The wavenumbers and the displacements of modes can be solved in a way similar to the solution of Eq. (6). The material damping can be considered by replacing the real Lamé's constants with imaginary ones (Wolf, 1985): λ j + 2μ j = λj + 2μ j 1 + 2iξpj ; μ j = μ j 1 + 2iξsj
ð11Þ
where ξpj and ξsj are the damping ratios of the compression and the shear waves of the jth thin layer, respectively. In general, ξpj ≠ ξsj. Here, ξpj = ξsj = ξj is assumed.
Fig. 1. Artificial boundaries and non-uniform thin layers for a layered half space.
52
H.-Y. Chai et al. / Journal of Applied Geophysics 73 (2011) 49–58
2.3. Discretization of media with a half space layer
3.1. Observed phase velocity
If a half space layer is truncated by an artificial boundary, such as a rigid base and the absorbing boundary expressed by Eq. (9), the behavior of P–SV waves can be studied using the eigenvalue method. In this case, part of the energy is reflected from the base or the boundary. For Rayleigh waves, since the vibration displacement of the particle attenuates exponentially with the depth in the half space layer, the vibrations are negligible before the boundary, if the truncated thickness of the half space layer is about 2 times of the analyzed wavelength. If the body waves are induced by point sources, they spread out in the form of hemispherical wavefronts when the spread distance (i.e. r) is larger than the wavelength. The geometric attenuation of the displacement amplitudes with spread distance is proportional to r− 1 (Graff, 1975). If the thickness of the half space layer is large enough compared to the wavelength of the body waves, the energy of the reflections from the artificial boundary can be ignored due to the geometrical and the material attenuations of the waves. It is found that at least 5 times of the analyzed wavelength should be retained for the body waves (Ganji et al., 1998). In order to compare the results with others, the material damping ratio of 0.05% is assumed in the following (see also Gucunski and Woods, 1992). At least ten thin layers are necessary over one wavelength. The thicknesses of the thin layers above the half space layer are uniform, and increase with the depth in the half space layer, as shown in Fig. 1.
Because the time term, exp(iωt), is the same for all modes of the harmonic waves, it is omitted for brevity in the following displacement expressions. For layered media acting by a harmonic loading uniformly distributed on a circular area, the vertical displacement of the nth interface of the thin layers at the distance r from the source center is given by (Kausel and Peek, 1982): n
uz ðr; ωÞ = −
iqRπ N nl 2 ð2Þ ∑ ϕ J1 ðkl RÞH0 ðkl r Þ = kl r≥R 2 l=1 z
ð12Þ
where R is the radius of disk loading, q is the intensity of the loading, and N is the number of the modes of the generalized Rayleigh waves for a given frequency; N = 2N for the rigid base and N = 2ðN + 1Þ for the absorbing boundary; J1(klR) is the Bessel function of the first kind of the first order; H(2) 0 (klr) is the Hankel function of the second kind of the zero order; and ϕnl z represents the normalized vertical displacement of the lth mode at the nth thin layer interface. kl is the wavenumber of the lth mode. At the surface, n = 1. Omitting the superscript “n” in Eq. (12), the vertical displacement at the surface can be rewritten as uz ðr; ωÞ = −
iqRπ qRπ ARz −iAIz = 2 2
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ARz + AIz exp½−iðφz + π = 2Þ ð13Þ
2.4. Dispersions of modes of surface waves
where
For a given frequency, the matrices C˜ x and C˜ z can be obtained from Eq. (2). A series of eigenvalues (1/k2) can be determined by solving the linear eigenvalue problem, i.e., Eq. (6). There are 4 N wavenumbers for media with the rigid base and 4(N + 1) wavenumbers for media with the absorbing boundary. These wavenumbers can be real or complex (if the material damping is taken into account, all of them are complex). Half of them correspond to the generalized Rayleigh waves radiating from the source. For a complex wavenumber, the negative imaginary part implies that the waves attenuate in the horizontal propagation direction. Thus, we choose the waves with the negative imaginary part, if the wavenumbers are complex, or the waves with the positive real part, if the wavenumbers are real (Kausel and Peek, 1982). Among the modes of the generalized Rayleigh waves, some modes attenuate exponentially with the depth in the truncated half space layer and approach zero before the artificial boundary, implying that the effects of the artificial boundary on the behavior of these waves can be ignored. These waves can be regarded as the real Rayleigh waves because the amplitudes of the real Rayleigh waves in a layered half space also approach to zero at infinity depth. Thus, the wavenumbers corresponding to the modes of the Rayleigh waves can be identified according to the attenuation characteristics of the waves, and the dispersions are calculated from the wavenumbers and the frequency.
φz = arctan AIz = ARz
3. Analysis of effective phase velocity The wave field induced by active sources includes the surface waves and the body waves. Thus, the observed phase velocity is that of the total vibrations of the full waves. By comparing the observed phase velocity to the effective phase velocity of the surface waves, the effects of the body waves can be investigated. The effects of the mode incompatibility can be studied by comparing the effective phase velocities of the surface waves induced by active sources and those of the surface waves under assumption of the plane waves. The expressions of the observed phase velocity and the effective phase velocity of the surface waves are derived in Sections 3.1 and 3.2, respectively.
"
ARz
ð14aÞ
# ð2Þ 2 H0 ðkl r Þ 1l = Re ∑ ϕz J1 ðkl RÞ kl l=1 N
"
ð14bÞ
# ð2Þ 2 H0 ðkl r Þ 1l AIz = −Im ∑ ϕz J1 ðkl RÞ : ð14cÞ kl l=1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 2 ARz + AIz accounts for the geometrical and the material The term attenuations and φz is the phase angle. Let kˆz ðr; ωÞ denote the effective N
(or apparent) wavenumber of the surface response of the full waves, i.e., ∂φz ðr; ωÞ kˆz ðr; ωÞ = : ∂r
ð15Þ
Eq. (15) can be calculated using the finite difference method: φ ðr + Δr; ωÞ−φz ðr; ωÞ : kˆz ðr; ωÞ = lim z Δr→0 Δr
ð16Þ
Thus, the observed phase velocity is given as cz ðr; ωÞ = ω = kˆz ðr; ωÞ = lim
Δr→0 φz ðr
ωΔr : + Δr; ωÞ−φz ðr; ωÞ
ð17Þ
3.2. Effective phase velocity of surface waves For a given frequency, assuming that there are M modes of real Rayleigh waves in N modes of the generalized Rayleigh waves, the total vertical displacement of the surface waves is given as n
uzR ðr; z; ωÞ = −
iqRπ M nl 2 ð2Þ ∑ ϕzR J1 ðkl RÞH0 ðkl r Þ = kl r≥R 2 l=1
ð18Þ
where ϕnl zR represents the normalized vertical displacement of the lth normal mode of the surface waves at the nth interface of the thin
H.-Y. Chai et al. / Journal of Applied Geophysics 73 (2011) 49–58 nl layers. If ϕnl z and N are replaced by ϕzR and M, respectively, Eqs. (13) to (14c) still hold for the surface waves as well. From Eq. (18), the displacement of the lth mode is given as
iqRπ nl 2 ð2Þ ϕzR J1 ðkl RÞH0 ðkl r Þ = kl uzR;l ðr; z; ωÞ = − 2
ð19Þ
from which it is clear that the mode propagates in the form of a Hankel function instead of propagating as plane wavefronts in the radial direction. If the material damping is ignored, the wavenumbers and the displacements of the modes of the surface waves ϕnl zR are real. The mth order Hankel functions of the first and the second kind can be rewritten as
The derivatives of Bessel functions and Hankel functions are related as (Wolf, 1985): h i dJ0 ðxÞ dY0 ðxÞ ð2Þ ð2Þ = − J1 ðxÞ; = −Y1 ðxÞ; H0 ðxÞ = −H1 ðxÞ ;x dx dx h i d d ð2Þ ð2Þ ð2Þ ½xJ1 ðxÞ = x J0 ðxÞ; ½xY1 ðxÞ = xY0 ðxÞ; H1 ðxÞ = H0 ðxÞ−H1 ðxÞ = x: ;x dx dx
ð25Þ Based on these equations, the effective wavenumber can be derived, i.e., M
kˆzR ðr; ωÞ = ð1Þ Hm ðkl r Þ
= Jm ðkl r Þ + iYm ðkl r Þ
ð2Þ
Hm ðkl r Þ = Jm ðkl r Þ−iYm ðkl rÞ
ð20aÞ
ð20bÞ
53
M
∑ ∑ Bˆlm km ½Y0 ðkl r ÞJ1 ðkm r Þ−J0 ðkl r ÞY1 ðkm r Þ
l=1 m=1 M M
:
l=1 m=1
Accordingly, the effective phase velocity of the surface waves is expressed as M
where Jm(klr) and Ym(klr) are the mth order Bessel functions of the first and the second kind. In the far field where klr NN 1, Jm(klr) and Ym(klr) can be approximated as
czR ðr; ωÞ =
ω = kˆzR ðr; ωÞ
M
ω ∑ ∑ Bˆlm ½Y0 ðkl rÞY0 ðkm r Þ + J0 ðkl r ÞJ0 ðkm r Þ l=1 m=1 M M
∑ ∑ Bˆlm km ½Y0 ðkl rÞJ1 ðkm rÞ−J0 ðkl rÞY1 ðkm rÞ
l=1 m=1
ð27Þ
sffiffiffiffiffiffiffiffiffi 2 π m Jm ðkl r Þ≈ cos kl r− − π πkl r 4 2
ð21aÞ
sffiffiffiffiffiffiffiffiffi 2 π m sin kl r− − π : Ym ðkl r Þ≈ πkl r 4 2
ð21bÞ
2 −1 1m where Bˆlm = ϕ1l ðkl km Þ J1 ðkl RÞJ1 ðkm RÞ. zR ϕzR In the far field, with Eqs. (21a) and (21b), it follows from Eq. (27) that M
M
ω ∑ ∑ Blm cos½ðkl −km Þr m=1 czR ðr; ωÞ≈ Ml = 1 M =
It can be derived from the above equations that
∑ ∑ Blm kl cos½ðkl −km Þr
l=1 m=1
rffiffiffiffiffiffiffi rffiffiffi qR π 1 π nl 2 uzR;l ðr; z; ωÞ≈ ϕzR J1 ðkl RÞ exp−i kl r + kl 2kl r 4
M
M
∑ ∑ Blm cos½ðkl −km Þr
l=1 m=1 M M
∑ ∑ Blm c−1 l cos½ðkl −km Þr
l=1 m=1
ð28Þ ð22Þ
Clearly, Eq. (22) is an approximation of Eq. (19) in the far field, and it can be seen that the mode excited by active sources travels approximately in the form of cylindrical wavefronts and the wavenumber approaches to kl (the wavenumber of the lth normal mode). The effective wavenumber of surface waves at the distance r from the source center can be obtained from Eq. (13), namely, h i ∂ arctan AIz = ARz ð Þ ∂φ r; ω z kˆzR ðr; ωÞ = = ∂r ∂r 1 A A − A A = Iz Rz Rz Iz 2 2 ;r ;r ARz + AIz
ð26Þ
∑ ∑ Bˆlm ½Y0 ðkl r ÞY0 ðkm r Þ + J0 ðkl r ÞJ0 ðkm r Þ
ð23Þ
2 −3 = 2 −1 = 2 ˆ 1m where Blm = ϕ1l J1 ðkl RÞJ1 ðkm RÞðkl km Þ = ðkl km Þ Blm , cl zR ϕzR is the phase velocity of the lth normal mode. In the far field, the cylindrical surface waves also approach to the plane ones. Eq. (28) holds for both plane and cylindrical surface waves. The term (kl − km)r in Eq. (28) represents the phase difference of the lth and the mth modes at the distance r. When (kl − km) r → nπ(n = 0, 1, 2, ⋯), cos[(kl − km)r] → ±1. The lth and the mth modes π interfere constructively. When ðkl −km Þr→ + nπ(n = 0, 1, 2, ⋯), 2 cos[(kl − km)r] → 0. The lth and the mth modes interfere destructively. The term (kl − km)r in Eq. (28) implies that the effective phase velocity is influenced by the interference of the modes. There is only a single non-dispersive mode of the surface waves in a half space. As a special case, the effective phase velocity of surface waves can be obtained from Eq. (27):
czR ðr; ωÞ = cR By using Eqs. (20a) and (20b), Eqs. (14b) and (14c) can be reduced
½Y0 ðkrÞ2 + ½ J0 ðkrÞ2 Y0 ðkrÞJ1 ðkrÞ−J0 ðkrÞY1 ðkrÞ
ð29Þ
to:
ARz
M J ðk r Þ 1l 2 = ∑ ϕzR J1 ðkl RÞ 0 l kl l=1
ð24aÞ
2 Y ðk rÞ 1l ϕzR J1 ðkl RÞ 0 l kl
ð24bÞ
M
AIz = ∑
l=1
where cR is the phase velocity of the normal mode. Using Eqs. (21a), (21b) and (29), it can be shown that czR(r, ω) → cR in the far field. From Eq. (26) or (29), the wavenumber of the surface waves in a half space can be written as Y ðkrÞJ1 ðkr Þ−J0 ðkrÞY1 ðkr Þ kˆzR ðr; ωÞ = k 0 ½Y0 ðkr Þ2 + ½ J0 ðkr Þ2
ð30Þ
54
H.-Y. Chai et al. / Journal of Applied Geophysics 73 (2011) 49–58
where k is the wavenumber of the plane surface waves. It follows that ∂kˆzR ∂kˆzR ∂k ∂kˆzR 1 = = ∂ω ∂k ∂ω ∂k cR 2krff½Y0 ðkrÞ2 −½ J0 ðkrÞ2 gY1 ðkrÞJ1 ðkrÞ−f½Y1 ðkrÞ2 −½ J1 ðkrÞ2 gY0 ðkrÞJ0 ðkrÞg 1 : cR f½Y0 ðkrÞ2 + ½ J0 ðkr Þ2 g2
=
ð31Þ Thus, the group velocity of the surface waves in a half space is ∂ω czg ðr; ωÞ = ∂kˆ
zR 2
= cR
2 2
f½Y0 ðkrÞ + ½J0 ðkrÞ g : 2krff½Y0 ðkrÞ2 −½J0 ðkrÞ2 gY1 ðkr ÞJ1 ðkrÞ−f½Y1 ðkrÞ2 −½J1 ðkrÞ2 gY0 ðkr ÞJ0 ðkrÞg
ð32Þ It can be seen from Eqs. (21a), (21b) and (32) that czg(r, ω) → cR in the far field. For a homogenous half space, the group and the phase velocities are equal under the assumption of plane surface waves. By comparing Eq. (29) with Eq. (32), it can be seen that the group and the phase velocities are different, under the influence of the mode incompatibility. Although Eqs. (27) and (28) are derived by ignoring the material damping, they are applicable for the soils with the material damping because the material damping has little influence on the phase velocity of the normal modes and the effective phase velocity of the surface waves (Chai et al., submitted for publication). 4. Effects of active sources 4.1. Homogenous half space 4.1.1. Phase velocity of plane surface waves In a homogenous elastic half space, the regression relationship between the phase velocities of the plane surface waves is given as follows (Graff, 1975): 0:87 + 1:12ν cR = cs ≈ 1+ν
ð33Þ
where ν is Poisson's ratio. 4.1.2. Effects of mode incompatibility For a given shear wave velocity, the phase velocity of the surface waves under active sources can be calculated from Eq. (27). The theoretical phase velocity cR can be determined using Eq. (33). The variation of the dimensionless velocity (c/cR) with the dimensionless distance (r/λR) is shown in Fig. 2 for ν = 0.0, 0.25, 0.4 and 0.45. The
effect of Poisson's ratio on the normalized effective phase velocity of surface waves is minimal. It can be seen from Fig. 2 that the effective phase velocity of the surface waves is smaller than cR for r/λR b 1; the shorter the dimensionless distance, the smaller the phase velocity. The phase velocity approaches to cR when the spread length is greater than one wavelength. This implies that the effect of the mode incompatibility can be ignored beyond one wavelength. 4.1.3. Effects of body waves The energy of the direct body waves depends on Poisson's ratio. The observed phase velocity can be calculated from Eqs. (13) to (17). The loading parameters R and qRπ/2 in Eq. (13) are assumed to be 0.01 m and 1 N/m, respectively, for both homogenous and layered media. The observed phase velocities for ν = 0.0, 0.25, 0.4 and 0.45 are in agreement with those obtained by Foinquinos and Roësset (2001) (the differences are too small to be shown at the scale of Fig. 2). Thus, if the thicknesses of the thin layers are significantly smaller than the studied wavelength, the observed phase velocity can be evaluated very accurately using the thin layer stiffness method. Comparing the observed phase velocities and the phase velocity of the surface waves, it is found that oscillations appear in the observed phase velocity due to the influence of the body waves. It can be seen in Fig. 2 that a larger oscillation corresponds to a higher Poisson's ratio. To analyze the energy distribution pattern of the direct waves in the surface wave field, the ratio of the displacement amplitude of the full waves to that of the surface waves (in the vertical direction) as a function of the dimensionless distance r/λR is given in Fig. 3 for different Poisson's ratios. The results in Fig. 3 show that the direct body waves have stronger energy for larger Poisson's ratios, implying that the stronger the body waves, the stronger the oscillations in the observed phase velocity. It is also found that the influence range of the body waves is far beyond one wavelength. 4.2. Layered media 4.2.1. Effective phase velocity of plane surface waves In layered media, there are multiple modes in the wave field. Three layered media used by Tokimatsu et al. (1992b) and Yoon and Rix (2009) are selected for illustration: (a) The shear wave velocity of the layers increases with the depth in Case I. (b) There is a soft layer trapped between stiff layers 1 and 3 in Case II. (c) A stiff layer is trapped between soft layers 1 and 3 in Case III.
1.4 1.1
ν=0.0 ν=0.25 ν=0.4 ν=0.45
1.3
Effect of mode incompatibility
1.2
1.05
Surface waves 1.1
u/uzR
c/cR
1 0.95 0.9
Surface waves Observed (ν =0.00) Observed (ν =0.25) Observed (ν =0.40) Observed (ν =0.45)
Oscillations induced by the body waves
0.85 0.8
0
1
2
3
4
5
6
7
8
r / λR Fig. 2. Variation of the dimensionless phase velocity with the dimensionless distance in half spaces.
1 0.9 0.8 Full waves
0.7 0.6
0
1
2
3
4
5
6
7
8
r / λR Fig. 3. Variation of the dimensionless vertical displacement with the dimensionless distance in half spaces.
H.-Y. Chai et al. / Journal of Applied Geophysics 73 (2011) 49–58
a
Vertical displacement φz
1l
100
0
1
2
3
4
5x10-4
Mode 4
Frequency(Hz)
The material parameters for these media are listed in Table 1 (reproduced from those used by Tokimatsu et al., 1992b). The vertical displacements of the first several normal modes are shown in Fig. 4 for the aforementioned three cases. It can be seen from Fig. 4 that (a) the fundamental mode dominates the surface wave field over most of the frequency range in Case I, (b) the higher modes are predominant in the frequency range greater than 20 Hz in Case II, and (c) the higher modes play a significant role in the narrow frequency range around 10 Hz in Case III. It can be seen from Eq. (28) that if the mode incompatibility is ignored, the effective phase velocity of the surface waves czR varies with the dimensionless distance kr. The effective wavelength is defined as the ratio of effective phase velocity to frequency. Assuming that λR is the average effective wavelength over the studied spread length, we can replace the dimensionless distance kr with r = λR . We can show that the effective phase velocity varies with r = λR , and the variations are different at different frequencies. A typical result is shown in Fig. 5, where subscript R20 denotes the Rayleigh waves at a frequency of 20 Hz. Although the fundamental normal mode dominates the surface wave field over most of the frequency range for the regular profile Case I, the effective phase velocity is different from the phase velocity of the fundamental mode, due to the influence of the higher modes.
55
Mode 2 Mode 3 10 Mode 1
Case I 2
b
Vertical displacement φz
1l
100
0
0.5
1.5x10-4
1 Mode 5 Mode 4 Mode 3
4.2.3. Effects of body waves It can be seen from Fig. 7 that due to the influence of the body waves, some oscillations appear in the observed phase velocity. These oscillations do not die out as they do in half spaces. For Case II, oscillations are even stronger, implying that the body waves have
1 2 3 4
Shear wave velocity (m/s) Case I
Case II
Case III
80 120 180 360
180 120 180 360
80 180 120 360
Frequency(Hz)
10
Mode 1
Case II 2
Vertical displacement φz
c
1l
100
0
1
2
Mode 4
3
4
5x10-4
Mode 2 Mode 3 Mode 1
10
Case III
2 Fig. 4. Normalized vertical displacements of the first several normal modes: (a) Case I; (b) Case II; and (c) Case III.
Table 1 Material parameters for three typical layered media. Layer number
Mode 2
Frequency(Hz)
4.2.2. Effects of mode incompatibility It can be seen from Eq. (22) that in the far field, the geometric decay of the displacement is nearly in the form of r− 1/2. The variation of the vertical displacement of the fundamental mode with the dimensionless distance r/λR20 at a frequency of 20 Hz is shown in Fig. 6(a) for Case I. It can be seen that the effect of the mode incompatibility on the displacement can be ignored when the spread length is greater than one average wavelength. Because of the superposition of various modes, some oscillations appear in the displacement, as shown in Fig. 6(b). The effect of the mode incompatibility on the effective phase velocity of the surface waves is shown in Fig. 7. It can be seen that the effect is also limited within the spread length less than one average effective wavelength. In this range, the smaller the ratio of the spread length to the wavelength, the greater the effective phase velocity is underestimated. The variations of the effective phase velocity of the surface waves with frequency at positions 1 m and 5 m from the source are shown in Fig. 8. In order to compare the effective phase velocity with the dispersions of the normal modes, the dispersions of the first several modes are also shown. Clearly, the variations of the effective phase velocity with frequency at different positions are different, and the phase velocity is underestimated due to the mode incompatibility. At significantly lower frequencies, the discrepancy of the effective phase velocity at 1 m is more significant than that at 5 m, due to the stronger effect of mode incompatibility.
Thickness (m)
Poisson's ratio
Density (kg/m3)
2 4 8 ∞
0.35 0.35 0.35 0.35
1800 1800 1800 1800
more significant influence on the observed phase velocity. The ratios of displacements of the full waves to that of the surface waves are given in Fig. 9, showing that the body waves for Case II have stronger energy than those for Case I. This can be due to the stronger reflections from the interfaces of layers for media where a soft layer is trapped at a shallow depth.
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H.-Y. Chai et al. / Journal of Applied Geophysics 73 (2011) 49–58
120
a Fundamental normal mode
1.2 Case I
100 1.1
Effect of body waves
cR20 (m/s)
80
c/cR20
1 60
Superposition of normal modes
40
0.9 Effect of mode incompatibility
Observed
0.8
Surface waves
20 0.7 0
0
1
2
3
4
5
6
7
8
0.6
r / λ R 20
0
1
2
3
4
5
6
7
8
r / λ R 20
Fig. 5. Variation of the effective phase velocity with the dimensionless distance at a frequency of 20 Hz in Case I.
b
1.2
Case II
5. Conclusions
Effect of body waves 1.1
a
0.9 Effect of mode incompatibility 0.8
Real Imaginary Magnitude
20 Effect of mode incompatibility
Surface waves
0
1
2
3
4
5
6
7
8
r / λ R 20
c
15
Observed
0.7 0.6
25
Displacement (×10-9 mm)
1
c/cR20
Discrete analytical expressions for the effective phase velocity of the surface waves, which account for the mode incompatibility and higher modes, are derived based on the thin layer stiffness matrix method. An analytical expression of the effective phase velocity of the surface waves in the far field is also developed. The field within one average effective wavelength is influenced by both the mode incompatibility and the body waves. In this range, the effective phase velocity is underestimated in comparison with that of
1.2
Case III I
10
Effect of body waves
1.1
Decay inr -1/2
5 1
c/cR20
0 -5 -10
1
2
3
4
5
6
7
Surface waves
0.6 Real Imaginary Magnitude
5
Displacement (×10-7 mm)
Observed
8 0.7
6
4
0
1
2
3
4
5
6
7
8
r / λ R 20 Fig. 7. Variation of the dimensionless phase velocity with the dimensionless distance at a frequency of 20 Hz: (a) Case I; (b) Case II; and (c) Case III.
3 Effect of mode incompatibility
2 1 0 -1 -2
Effect of mode incompatibility
0.8 0
r / λR 20
b
0.9
0
1
2
3
4
5
6
7
8
r / λ R 20 Fig. 6. Variation of the vertical displacement with the dimensionless distance at a frequency of 20 Hz for Case I: (a) fundamental mode; and (b) superposition of modes.
the plane surface waves; the smaller the ratio of the spread length to the wavelength, the more the effective phase velocity is underestimated. The mode incompatibility will result in the underestimation of the observed phase velocity, while the body waves may induce some oscillations. The effects of the mode incompatibility can be ignored in the field where the spread length is beyond one average effective wavelength. The observed phase velocity has some oscillations due to the influence of the body waves. The influence range is far beyond one average effective wavelength. In layered media, especially in Case II where a soft layer is trapped between the first and the third layers, the body waves are strongly reflected at the interfaces of layers.
H.-Y. Chai et al. / Journal of Applied Geophysics 73 (2011) 49–58
a
a
400
2 Case I
Case I 1m 5m
Mode 4
300
1.5
u/uzR
Phase velocity (m/s)
Mode 3
57
Mode 1
200
Surface waves
Mode 2 1 100 Full waves Mode incompatibility 0
0
50
Rayleigh waves of surface layer 100
150
0.5
200
0
1
2
3
Frequency (Hz)
b
Phase velocity (m/s)
Mode 3 Mode 5 300
7
8
7
8
Case II
5m
Mode 4
1.5 Rayleigh waves of surface layer
200
Surface waves
1 100
0
Mode 2 Full waves 0.5
0
50
100
150
200
0
1
2
3
4
5
6
r / λ R 20
Frequency (Hz)
Phase velocity (m/s)
6
1m
Mode 1 Mode incompatibility
c
5
2
Case II
400
u/uzR
b
4
r / λ R 20
Fig. 9. Variation of the dimensionless displacement with the dimensionless distance at a frequency of 20 Hz: (a) Case I; and (b) Case II.
400
Case III 1m
Mode 3
5m
300 Mode 1 200 Mode 4
Mode 2
can take the near-field effects into account (Ganji et al., 1998; O'Neill et al., 2003). In the wave field with multiple modes, the effective phase velocity of the surface waves is dependent on both the frequency and the spread distance. In the far field where the effects of mode incompatibility and body waves are insignificant, the effective phase velocity can be analyzed using Eq. (28).
100 Mode incompatibility 0
0
50
Rayleigh waves of surface layer 100
150
200
Frequency (Hz)
Acknowledgments This research is supported by the one hundred talents program of the Chinese Academy of Sciences.
Fig. 8. Effective phase velocity curves of the surface waves at positions 1 m and 5 m and dispersion curves of the first several normal modes: (a) Case I; (b) Case II; and (c) Case III.
References
The analysis of the observed phase velocity based on the assumption of the plane waves is complicated by the near-field effects. Because the influence of the body waves is strongly related to soil stratification, it is difficult to identify an approximate range of the influence. Although mode incompatibility can be mitigated in the wave field beyond one average effective wavelength, the effect of body waves can persist over longer distances in complicated soil layers. Conventional criteria proposed to mitigate the near-field effects in wave monitoring may not hold for these complicated cases. It has been suggested that robust full-waveform inversion techniques
Al-Hunaidi, M.O., 1993. Insight on the SASW nondestructive testing method. Canadian Journal of Civil Engineering 20 (6), 940–950. Al-Hunaidi, M.O., 1996. Nondestructive evaluation of pavements using spectral analysis of surface waves in frequency wave-number domain. Journal of Nondestructive Evaluation 15 (2), 71–82. Bodet, L., Abraham, O., Clorennec, D., 2009. Near-offset effects on Rayleigh-wave dispersion measurements: physical modeling. Journal of Applied Geophysics 68 (1), 95–103. Chai, H.Y., Cui, Y.J., Wei, C.F., submitted for publication. A parametric study of the apparent phase velocity of surface waves in layered media. Soil dynamics and earthquake engineering. Foinquinos, R., Roësset, J.M., 2001. Elastic layered half-spaces subjected to dynamic surface loads. In: Kausel, E., Manolis, G. (Eds.), Chapter 4 in Wave Motion in Earthquake Engineering. WIT Press, Southampton, U K. Foti, S., 2000. Multistation methods for geotechnical characterization using surface waves. Ph.D. thesis, Politecnico di Torino.
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Ganji, V., Gucunski, N., Nazarian, S., 1998. Automated inversion procedure for spectral analysis of surface waves. Journal of Geotechnical and Geoenvironmental Engineering 124 (8), 757–770. Graff, K.F., 1975. Wave Motion in Elastic Solids. Ohio State University Press, USA. Gucunski, N., Woods, R.D., 1992. Numerical simulation of the SASW test. Soil Dynamics and Earthquake Engineering 11 (4), 213–227. Kausel, E., 2001. The thin-layer method in seismology and earthquake engineering. In: Kausel, E., Manolis, G. (Eds.), Chapter 5 in Wave Motion in Earthquake Engineering. WIT Press, Southampton, U K. Kausel, E., Peek, R., 1982. Dynamic loads in the interior of a layered stratum: an explicit solution. Bulletin. Seismological Society of America 72 (5), 1459–1481. Kausel, E., Roësset, J.M., 1981. Stiffness matrices for layered soils. Bulletin. Seismological Society of America 71 (6), 1743–1746. Lai, C.G., Rix, G.J., 1999. Inversion of multi-mode effective dispersion curves. In: Jamiolkowski, M., Lancellotta, R., Lo Presti, D. (Eds.), Pre-failure Deformation Characteristics of Geomaterials, Vol. 1. Balkema, Rotterdam, pp. 411–418. Lu, L.Y., Zhang, B.X., 2004. Analysis of dispersion curves of Rayleigh waves in the frequency–wavenumber domain. Canadian Geotechnical Journal 41 (4), 583–598. Malovichko, A.A., Anderson, N.L., Malovichko, D.A., Shylakov, D.Y., Butirin, P.G., 2005. Active–passive array surface wave inversion and comparison to borehole logs in southeast Missouri. Journal of Environmental and Engineering Geophysics 10 (3), 243–250. Morikawa, H., Sawada, S., Akamatsu, J., 2004. A method to estimate phase velocities of Rayleigh waves using microseisms simultaneously observed at two sites. Bulletin. Seismological Society of America 94 (3), 961–976. Nazarian, S., Desai, M.R., 1993. Automated surface wave method: field testing. Journal of Geotechnical Engineering 119 (7), 1094–1111. O'Neill, A., Dentith, M., List, R., 2003. Full-waveform P–SV reflectivity inversion of surface waves for shallow engineering applications. Exploration Geophysics 34 (3), 158–173. O'Neill, A., Matsuoka, T., 2005. Dominant higher surface-wave modes and possible inversion pitfalls. Journal of Environmental and Engineering Geophysics 10 (2), 185–201.
Park, C.B., Miller, R.D., Xia, J.H., 1998. Imaging dispersion curves of surface waves on multi-channel record. The 68th Annual Meeting of the Society of Exploration Geophysicists, New Orleans, Louisiana, pp. 1377–1380. Park, C.B., Miller, R.D., Ryden, N., Xia, J., Ivanov, J., 2005. Combined use of active and passive surface waves. Journal of Environmental and Engineering Geophysics 10 (3), 323–334. Roberts, J., Asten, M., 2006. Investigation of near source effects in array-based (SPAC) microtremor surveys. Third International Symposium on the Effects of Surface Geology on Seismic Motion, Grenoble, France, Paper Number: 97. Strobbia, C., 2003. Surface wave methods — acquisition, processing and inversion. Ph.D. thesis, Politecnico di Torino. Strobbia, C., Foti, S., 2006. Multi-offset phase analysis of surface wave data (MOPA). Journal of Applied Geophysics 59, 300–313. Tokimatsu, K., Shinzawa, K., Kuwayama, K.S., 1992a. Use of short period microtremors for Vs profiling. Journal of Geotechnical Engineering 118 (10), 1544–1558. Tokimatsu, K., Tamura, S., Kojima, H., 1992b. Effects of multiple modes on Rayleigh wave dispersion characteristic. Journal of Geotechnical Engineering 118 (10), 1529–1543. Wolf, J.P., 1985. Dynamic Soil–Structure Interaction. Prentice-Hall, Inc., Englewood Cliffs. Xia, J.H., Miller, R.D., Park, C.B., Tianb, G., 2003. Inversion of high frequency surface waves with fundamental and higher modes. Journal of Applied Geophysics 52 (1), 45–57. Yoon, S.S., 2005. Array-based measurements of surface wave dispersion and attenuation using frequency–wavenumber analysis. Ph.D. thesis, The Georgia Institute of Technology. Yoon, S.S., Rix, G.J., 2009. Near-field effects on array-based surface wave methods with active sources. Journal of Geotechnical and Geoenvironmental Engineering 135 (3), 399–406. Zywicki, D.J., Rix, G.J., 2005. Mitigation of near-field effects for seismic surface wave velocity estimation with cylindrical beamformers. Journal of Geotechnical and Geoenvironmental Engineering 131 (8), 970–977.
Contents lists available at ScienceDirect
Journal of Applied Geophysics j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / j a p p g e o
Analysis of effects of active sources on observed phase velocity based on the thin layer method Hua-You Chai a,⁎, Kok-Kwang Phoon b, Chang-Fu Wei a, Ying-Fa Lu c a b c
State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan 430071, China Department of Civil Engineering, National University of Singapore, Singapore 117576, Singapore Key Laboratory of Geological Hazards on Three Gorges Reservoir Area, Ministry of Education, China Three Gorges University, Yichang 443002, Hubei, China
a r t i c l e
i n f o
Article history: Received 10 November 2009 Accepted 13 November 2010 Available online 21 November 2010 Keywords: Near-field effects Surface waves Mode Thin layer stiffness method Effective phase velocity
a b s t r a c t In the wave field induced by active sources, the observed phase velocity of surface waves is influenced by both mode incompatibility (i.e. non-planar spread of surface waves is idealized as plane waves) and body waves. Effects of sources are usually investigated based on numerical simulations and physical models. Several methods have been proposed to mitigate the effects. In application, however, these methods may also have difficulties since the energy of the body waves depends on soil stratification and parameters. There are multiple modes of surface waves in layered media, among which the higher modes dominate the wave field for soils with the irregular shear velocity profiles. Considering the mode incompatibility and the higher modes, we derive analytical expressions for the effective phase velocity of the surface waves based on the thin layer stiffness method, and investigate the effects of the body waves on the observed phase velocity through the phase analysis of the vibrations of both the surface waves and the body waves. The results indicate that the effective phase velocity of the surface waves in layered media varies with the frequency and the spread distance, and is underestimated compared to that of the plane surface waves in the spread range less than about one wavelength. The oscillations that appeared in the observed phase velocity are due to the involvement of the body waves. The mode incompatibility can be ignored in the range beyond one wavelength, while the influence range of the body waves is far beyond one wavelength. The body waves have a significant influence on the observed phase velocity of the surface waves in soils with a soft layer trapped between the first and the second layers because of strong reflections. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The propagation behavior of surface waves (i.e. Rayleigh waves in this paper) in horizontally layered media is often studied under the assumption that the surface waves at the free state (i.e. no source) travel in the form of plane wavefronts. The modes of the surface waves at the free state are usually termed as the normal modes. Under this assumption, the phase velocity of the normal modes depends only upon the frequency (or wavelength) and the penetration depth. The dispersions of surface waves can be obtained by performing surface wave testing. The surface waves can be generated by either active or passive sources (Tokimatsu et al., 1992a,b; Morikawa et al., 2004; Park et al., 2005; Malovichko et al., 2005; Roberts and Asten, 2006; Yoon and Rix, 2009). The active sources are those artificially generated in the wave testing by, for example, a sledge hammer, a falling weight or an explosive, whereas the other sources are termed as passive sources, such as nearby traffic loading and microtremors induced by sea waves. Dispersions over
⁎ Corresponding author. E-mail address: [email protected] (H.-Y. Chai). 0926-9851/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jappgeo.2010.11.005
a wide range of wavelengths can be obtained by combining active and passive tests. In general, the wave fields induced by passive sources are omni-directional. Two-dimensional arrays (Tokimatsu et al., 1992a; Morikawa et al., 2004; Roberts and Asten, 2006) are used in these tests. The phase velocity of the superposed vibrations of various modes is imaged by frequency–wavenumber spectra or extracted by the spatial autocorrelation (SPAC) method (Tokimatsu et al., 1992a; Morikawa et al., 2004; Roberts and Asten, 2006). Active sources are often used to obtain the properties of shallow soils. Two-station measurements are originally used in the surface wave testing, in which dispersion curves are obtained by spectral analysis of surface waves (SASW) (Gucunski and Woods, 1992; Nazarian and Desai, 1993; Al-Hunaidi, 1993; Foti, 2000). For a wave field with multiple modes, the surface vibration is the superposition of all the modes at a given frequency. The phase velocity of the superposed vibration is termed as the effective phase velocity (Lai and Rix, 1999; Foti, 2000; Strobbia, 2003; O'Neill et al., 2003; O'Neill and Matsuoka, 2005) or the apparent phase velocity of the surface waves (Tokimatsu et al., 1992a,b). In this paper, the term “effective phase velocity” is used. In SASW, since the phase velocity is calculated from the phase lag of the superposed vibration, the spectral analysis
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H.-Y. Chai et al. / Journal of Applied Geophysics 73 (2011) 49–58
technique cannot be used to determine the phase velocity of a single mode (Al-Hunaidi, 1993; Lu and Zhang, 2004). To account for higher modes, the multi-station measurement techniques have been developed. The multichannel analysis of surface waves (MASW), such as frequency–wavenumber (f–k) or frequency–slowness (f–p) transforms, is used to obtain the dispersion images of the modes based on the plane-wave assumption (Al-Hunaidi, 1996; Park et al., 1998; Foti, 2000; O'Neill et al., 2003; Strobbia, 2003; Xia et al., 2003; Lu and Zhang, 2004; Yoon, 2005; Yoon and Rix, 2009). When the spatial aperture of the measurement and the number of the receivers are large, the dispersions of some higher modes can be obtained over limited frequency ranges (Foti, 2000; O'Neill et al., 2003; Strobbia, 2003). If the spatial aperture of the measurement is small, however, the resolution of the wavenumber is low. In this case, all the modes are superposed together, the modes cannot be identified and separated in such a case, and thus only the effective phase velocity can be obtained (Foti, 2000; Strobbia, 2003; Lu and Zhang, 2004). Based on the assumption of plane surface waves, the dispersions of modes can be calculated by the propagator matrix model (Foti, 2000; Strobbia, 2003) and the effective phase velocity can be analyzed using the models presented by Tokimatsu et al. (1992a,b) and Lai and Rix (1999) for passive and active sources, respectively. If a source exists, however, the assumption of a pure and plane surface wave field is no longer applicable. Effects of the source on the analysis of surface waves can be attributed to two factors: one is the mode incompatibility (Zywicki and Rix, 2005), and the other is the presence of the body waves including the direct body waves and the reflections from the interfaces of layers. The wave field under the influence of these two factors is called the near-field, and all others are termed as the far-field. Tokimatsu et al. (1992a,b) and Lai and Rix (1999) have proposed the expressions of the effective phase velocity, which take into account the higher modes but ignore the near-field effects. The near-field effect can be investigated via analytical methods, numerical methods, physical modeling and field testing (Zywicki and Rix, 2005; Strobbia and Foti, 2006; Bodet et al., 2009; Yoon and Rix, 2009). It has been recognized that the near-field effect can be mitigated by using the cylindrical beamformer and multi-offset phase analyses, or by increasing the total number of receivers and the array center distance in an array-based testing (Zywicki and Rix, 2005; Strobbia and Foti, 2006; Yoon and Rix, 2009). These proposals are helpful for mitigating the near-field effects and analyzing the observed phase velocity. However, validation and generalization of results from numerical and physical models need to be supported using theoretical analysis. The mode incompatibility and the body waves have different influences on the observed phase velocity, which must be investigated separately. In this paper, an analytical expression of the effective phase velocity is derived to account for both the mode incompatibility and the higher modes, based on the thin layer stiffness method. Effects of the mode incompatibility and the body waves on the observed phase velocity are investigated separately. Because the pattern and the energy of the direct body waves and the reflections are related to Poisson's ratio and soil stratifications, we shall study the near-field effect in homogenous half space media with different Poisson's ratios and layered half space media with regular or irregular soil profiles (Tokimatsu et al., 1992b). 2. Thin layer stiffness matrix method To determine the dispersions of the normal modes and the displacement responses of modes under harmonic loadings, two procedures can be adopted, namely, the Thomson–Haskell propagator matrix method (Tokimatsu et al., 1992b; Foti, 2000; Strobbia, 2003; Lu and Zhang, 2004) and the stiffness matrix method (Kausel and Roësset, 1981; Kausel and Peek, 1982; Ganji et al., 1998; Foinquinos and Roësset, 2001). In applying the propagator matrix algorithm, however,
there exist some inconveniences, including: (a) matrices and their derivatives appear in the displacement response of modes (Tokimatsu et al., 1992b), (b) when the material damping is taken into account, the elements of the propagator matrix are complex, the solution of the complex propagator matrix is difficult and time-consuming, and (c) the contour integral technique is needed in resolving the responses of soils to point or disk loadings since the temporal and spatial Fourier transform is used (Kausel and Peek, 1982). There are also some inconveniences in the evaluation of the stiffness matrix because of the transcendental or trigonometric functions in the matrix. To resolve the aforementioned issues, the soil layers are divided into “thin” layers in the sense that the layer thickness is smaller than the wavelength, and as a consequence, the transcendental or trigonometric functions in the stiffness matrix are reduced to simple algebraic functions and the explicit expression of response of soils to loadings can be obtained. The availability of an explicit analytical expression is very useful for deriving the effective phase velocity of the surface waves and studying the effects of the body waves. When the layer thicknesses are less than 1/8 to 1/10 of the wavelength, the matrix stiffness of the jth thin layer for P–SV waves can be expressed as (Kausel and Roësset, 1981; Kausel and Peek, 1982; Foinquinos and Roësset, 2001): 2
ð1Þ
Kj = Aj k + Bj k + Cj
where Cj = Gj − ω2Mj, and k and ω are the wavenumber and the circular frequency, respectively. If the displacement vector is rearranged first by grouping all horizontal (x or r) interface displacements followed by all vertical (z) interface displacements (rather than arranged by alternating horizontal and vertical displacements of interfaces), the coefficient matrices Aj, Bj, and Cj in Eq. (1) can be presented in the following forms: 2 2 λj + 2μ j λj + 2μ j 6 6 hj 6 2 λj + 2μ j 6 λj + 2μ j Aj = 6 66 6 4
3
7 " # 7 7 Axj 0 7 7= 7 0 Azj 2μ j μ j 7 5 μ j 2μ j 3 2 λj −μ j − λj + μ j 6 7 6 7 2 6 7 0 λj + μ j − λj −μ j 7 16 7=4 Bj = 6 7 T 26 6 7 λj + μ j λj −μ j Bxzj 6 7 4 5 − λj + μ j − λj −μ j 2 3 μj −μ j 6 7 6 −μ j 7 μj 7 16 7 6 Gj = 6 λj + 2μ j − λj + 2μ j 7 7 hj 6 6 7 4 5 − λj + 2μ j λj + 2μ j 2 3 2 1 6 7 7 ρhj 6 1 2 6 7 Mj = 7 6 6 2 1 4 5 1 2
2
μ j ρj hj ω2 − 6 6 hj 3 6 6 6 6 μ j ρj hj ω2 6− − 6 h 6 6 j 6 Cj = 6 6 6 6 6 6 6 6 6 6 4 " =
Cxj
0
0
Czj
#
−
hj
2
−
3 5
0
3
μ j ρj hj ω2 − hj 6
μj
Bxzj
ρj h j ω 3
λj + 2μ j hj
2
−
− λj + 2μ j hj
ρj hj ω 3
2
−
ρj h j ω 6
7 7 7 7 7 7 7 7 7 7 7 7 27 − λj + 2μj ρj h j ω 7 − 7 hj 6 7 7 7 7 7 2 λj + 2μj ρj hj ω 5 − hj 3
ð2Þ
H.-Y. Chai et al. / Journal of Applied Geophysics 73 (2011) 49–58
where hj, λj and μj are the thickness and Lamé's constants of the jth layer, respectively. Except for Bxzj, the submatrices Axj, Azj, Cxj and Czj are symmetrical. The elements of Axj, Azj and Bxzj depend on the material parameters of the jth layer, while those of Cxj and Czj are related to both material parameters and frequency. 2.1. Layered media with a rigid base For layered media with a rigid base, since the horizontal and the vertical displacements are zero at the base, the global matrix can be obtained by assembling the matrices of the thin layers as follows: 2 ˜ =4 K
˜ x + C˜ x k2A
kB˜ xz
˜ xz kB
˜ z + C˜ z k2A
T
3 5
ð3Þ
˜ x; A ˜ z ; C˜ x ; C˜ z and B˜ xz are assemblies of the corresponding where A matrices of thin layers. ⇀ ⇀ ˜ϕ At a free state, we have K = 0 , that is, 2 4
3 ⇀ ⇀ 5 ϕ⇀x = 0 2˜ ϕz k Az + C˜ z
2˜ ˜ kA x + Cx
kB˜ xz
T kB˜ xz
ð4Þ
h
ϕ = ux;1 ; ux;2 ; ⋯; ux;N−1 ; ux;N ; uz;1 ; uz;2 ; ⋯; uz;N−1 ; uz;N ⇀
iT
⇀ ϕ = ⇀x : ϕz
ð5Þ
⇀
N is the number of the thin layers; ϕx and ϕz are the horizontal and the vertical displacement vectors, respectively. The free surface is numbered as 1 (see Fig. 1). Eq. (4) can be rewritten as ("
˜x A 0
B˜ xz ˜z A
#
" #) ⇀ ⇀ 1 C˜ x 0 k ϕx = 0: + 2 ⇀ k B ϕz ˜ Txz C˜ z
ð6Þ
Accordingly, the solution to Eq. (4) can be converted into an eigenvalue problem. 2.2. Layered media with an underlying half space The stiffness matrix of a half space layer for P–SV waves is given as (Kausel and Roësset, 1981; Wolf, 1985): 8 <
1−rβ2
r α K0 = 2kμ 1 :2 1−r r α β
1 0 − rβ 1
where vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 u u ω ω 2 rα = t1− ; rβ = 1− kcp0 kcs0
ð8Þ
cp0 and cs0 are velocities of the compression and the shear waves, respectively. The above matrix is not in the form of Eq. (1). Thus, the global matrix is not in the form of Eq. (6) and layered media with an underlying half space are not an eigenvalue problem. For the waves with a small wavenumber k, however, the propagating directions of the waves are not far from the vertical direction [also known as paraxial waves, Kausel, 2001], and Eq. (7) can be expanded using Taylor's series. Keeping the terms up to k2, the matrix of a half space layer can be reduced to (Ganji et al., 1998; Kausel, 2001) 2
ð9Þ
K0 = A0 k + B0 k + C0 where 2
where the displacement vector is defined as follows: ⇀
51
9 = 1 0 ;
ð7Þ
3 2−α " 0 7 − 6 α 1 cs0 6 1−2α 0 7 A0 = iμ 6 7; B0 = μ 2 ω4 α 1 1−2α 5 0 α3 " # 1 0 C0 = iωρcs0 ; α = cs0 = cp0 0 1= α
1
#
0
ð10Þ
pffiffiffiffiffiffiffiffi i = −1. The elements of matrix B0 are real, while those of matrices A0 and C0 are imaginary. These observations imply that, for the paraxial waves, the half space layer can be modeled by an absorbing boundary containing elastic and damping components (see Fig. 1). It can be deduced from Eq. (9) that for layered media with an underlying half space, the global matrix is in the form of Eq. (3). The wavenumbers and the displacements of modes can be solved in a way similar to the solution of Eq. (6). The material damping can be considered by replacing the real Lamé's constants with imaginary ones (Wolf, 1985): λ j + 2μ j = λj + 2μ j 1 + 2iξpj ; μ j = μ j 1 + 2iξsj
ð11Þ
where ξpj and ξsj are the damping ratios of the compression and the shear waves of the jth thin layer, respectively. In general, ξpj ≠ ξsj. Here, ξpj = ξsj = ξj is assumed.
Fig. 1. Artificial boundaries and non-uniform thin layers for a layered half space.
52
H.-Y. Chai et al. / Journal of Applied Geophysics 73 (2011) 49–58
2.3. Discretization of media with a half space layer
3.1. Observed phase velocity
If a half space layer is truncated by an artificial boundary, such as a rigid base and the absorbing boundary expressed by Eq. (9), the behavior of P–SV waves can be studied using the eigenvalue method. In this case, part of the energy is reflected from the base or the boundary. For Rayleigh waves, since the vibration displacement of the particle attenuates exponentially with the depth in the half space layer, the vibrations are negligible before the boundary, if the truncated thickness of the half space layer is about 2 times of the analyzed wavelength. If the body waves are induced by point sources, they spread out in the form of hemispherical wavefronts when the spread distance (i.e. r) is larger than the wavelength. The geometric attenuation of the displacement amplitudes with spread distance is proportional to r− 1 (Graff, 1975). If the thickness of the half space layer is large enough compared to the wavelength of the body waves, the energy of the reflections from the artificial boundary can be ignored due to the geometrical and the material attenuations of the waves. It is found that at least 5 times of the analyzed wavelength should be retained for the body waves (Ganji et al., 1998). In order to compare the results with others, the material damping ratio of 0.05% is assumed in the following (see also Gucunski and Woods, 1992). At least ten thin layers are necessary over one wavelength. The thicknesses of the thin layers above the half space layer are uniform, and increase with the depth in the half space layer, as shown in Fig. 1.
Because the time term, exp(iωt), is the same for all modes of the harmonic waves, it is omitted for brevity in the following displacement expressions. For layered media acting by a harmonic loading uniformly distributed on a circular area, the vertical displacement of the nth interface of the thin layers at the distance r from the source center is given by (Kausel and Peek, 1982): n
uz ðr; ωÞ = −
iqRπ N nl 2 ð2Þ ∑ ϕ J1 ðkl RÞH0 ðkl r Þ = kl r≥R 2 l=1 z
ð12Þ
where R is the radius of disk loading, q is the intensity of the loading, and N is the number of the modes of the generalized Rayleigh waves for a given frequency; N = 2N for the rigid base and N = 2ðN + 1Þ for the absorbing boundary; J1(klR) is the Bessel function of the first kind of the first order; H(2) 0 (klr) is the Hankel function of the second kind of the zero order; and ϕnl z represents the normalized vertical displacement of the lth mode at the nth thin layer interface. kl is the wavenumber of the lth mode. At the surface, n = 1. Omitting the superscript “n” in Eq. (12), the vertical displacement at the surface can be rewritten as uz ðr; ωÞ = −
iqRπ qRπ ARz −iAIz = 2 2
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ARz + AIz exp½−iðφz + π = 2Þ ð13Þ
2.4. Dispersions of modes of surface waves
where
For a given frequency, the matrices C˜ x and C˜ z can be obtained from Eq. (2). A series of eigenvalues (1/k2) can be determined by solving the linear eigenvalue problem, i.e., Eq. (6). There are 4 N wavenumbers for media with the rigid base and 4(N + 1) wavenumbers for media with the absorbing boundary. These wavenumbers can be real or complex (if the material damping is taken into account, all of them are complex). Half of them correspond to the generalized Rayleigh waves radiating from the source. For a complex wavenumber, the negative imaginary part implies that the waves attenuate in the horizontal propagation direction. Thus, we choose the waves with the negative imaginary part, if the wavenumbers are complex, or the waves with the positive real part, if the wavenumbers are real (Kausel and Peek, 1982). Among the modes of the generalized Rayleigh waves, some modes attenuate exponentially with the depth in the truncated half space layer and approach zero before the artificial boundary, implying that the effects of the artificial boundary on the behavior of these waves can be ignored. These waves can be regarded as the real Rayleigh waves because the amplitudes of the real Rayleigh waves in a layered half space also approach to zero at infinity depth. Thus, the wavenumbers corresponding to the modes of the Rayleigh waves can be identified according to the attenuation characteristics of the waves, and the dispersions are calculated from the wavenumbers and the frequency.
φz = arctan AIz = ARz
3. Analysis of effective phase velocity The wave field induced by active sources includes the surface waves and the body waves. Thus, the observed phase velocity is that of the total vibrations of the full waves. By comparing the observed phase velocity to the effective phase velocity of the surface waves, the effects of the body waves can be investigated. The effects of the mode incompatibility can be studied by comparing the effective phase velocities of the surface waves induced by active sources and those of the surface waves under assumption of the plane waves. The expressions of the observed phase velocity and the effective phase velocity of the surface waves are derived in Sections 3.1 and 3.2, respectively.
"
ARz
ð14aÞ
# ð2Þ 2 H0 ðkl r Þ 1l = Re ∑ ϕz J1 ðkl RÞ kl l=1 N
"
ð14bÞ
# ð2Þ 2 H0 ðkl r Þ 1l AIz = −Im ∑ ϕz J1 ðkl RÞ : ð14cÞ kl l=1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 2 ARz + AIz accounts for the geometrical and the material The term attenuations and φz is the phase angle. Let kˆz ðr; ωÞ denote the effective N
(or apparent) wavenumber of the surface response of the full waves, i.e., ∂φz ðr; ωÞ kˆz ðr; ωÞ = : ∂r
ð15Þ
Eq. (15) can be calculated using the finite difference method: φ ðr + Δr; ωÞ−φz ðr; ωÞ : kˆz ðr; ωÞ = lim z Δr→0 Δr
ð16Þ
Thus, the observed phase velocity is given as cz ðr; ωÞ = ω = kˆz ðr; ωÞ = lim
Δr→0 φz ðr
ωΔr : + Δr; ωÞ−φz ðr; ωÞ
ð17Þ
3.2. Effective phase velocity of surface waves For a given frequency, assuming that there are M modes of real Rayleigh waves in N modes of the generalized Rayleigh waves, the total vertical displacement of the surface waves is given as n
uzR ðr; z; ωÞ = −
iqRπ M nl 2 ð2Þ ∑ ϕzR J1 ðkl RÞH0 ðkl r Þ = kl r≥R 2 l=1
ð18Þ
where ϕnl zR represents the normalized vertical displacement of the lth normal mode of the surface waves at the nth interface of the thin
H.-Y. Chai et al. / Journal of Applied Geophysics 73 (2011) 49–58 nl layers. If ϕnl z and N are replaced by ϕzR and M, respectively, Eqs. (13) to (14c) still hold for the surface waves as well. From Eq. (18), the displacement of the lth mode is given as
iqRπ nl 2 ð2Þ ϕzR J1 ðkl RÞH0 ðkl r Þ = kl uzR;l ðr; z; ωÞ = − 2
ð19Þ
from which it is clear that the mode propagates in the form of a Hankel function instead of propagating as plane wavefronts in the radial direction. If the material damping is ignored, the wavenumbers and the displacements of the modes of the surface waves ϕnl zR are real. The mth order Hankel functions of the first and the second kind can be rewritten as
The derivatives of Bessel functions and Hankel functions are related as (Wolf, 1985): h i dJ0 ðxÞ dY0 ðxÞ ð2Þ ð2Þ = − J1 ðxÞ; = −Y1 ðxÞ; H0 ðxÞ = −H1 ðxÞ ;x dx dx h i d d ð2Þ ð2Þ ð2Þ ½xJ1 ðxÞ = x J0 ðxÞ; ½xY1 ðxÞ = xY0 ðxÞ; H1 ðxÞ = H0 ðxÞ−H1 ðxÞ = x: ;x dx dx
ð25Þ Based on these equations, the effective wavenumber can be derived, i.e., M
kˆzR ðr; ωÞ = ð1Þ Hm ðkl r Þ
= Jm ðkl r Þ + iYm ðkl r Þ
ð2Þ
Hm ðkl r Þ = Jm ðkl r Þ−iYm ðkl rÞ
ð20aÞ
ð20bÞ
53
M
∑ ∑ Bˆlm km ½Y0 ðkl r ÞJ1 ðkm r Þ−J0 ðkl r ÞY1 ðkm r Þ
l=1 m=1 M M
:
l=1 m=1
Accordingly, the effective phase velocity of the surface waves is expressed as M
where Jm(klr) and Ym(klr) are the mth order Bessel functions of the first and the second kind. In the far field where klr NN 1, Jm(klr) and Ym(klr) can be approximated as
czR ðr; ωÞ =
ω = kˆzR ðr; ωÞ
M
ω ∑ ∑ Bˆlm ½Y0 ðkl rÞY0 ðkm r Þ + J0 ðkl r ÞJ0 ðkm r Þ l=1 m=1 M M
∑ ∑ Bˆlm km ½Y0 ðkl rÞJ1 ðkm rÞ−J0 ðkl rÞY1 ðkm rÞ
l=1 m=1
ð27Þ
sffiffiffiffiffiffiffiffiffi 2 π m Jm ðkl r Þ≈ cos kl r− − π πkl r 4 2
ð21aÞ
sffiffiffiffiffiffiffiffiffi 2 π m sin kl r− − π : Ym ðkl r Þ≈ πkl r 4 2
ð21bÞ
2 −1 1m where Bˆlm = ϕ1l ðkl km Þ J1 ðkl RÞJ1 ðkm RÞ. zR ϕzR In the far field, with Eqs. (21a) and (21b), it follows from Eq. (27) that M
M
ω ∑ ∑ Blm cos½ðkl −km Þr m=1 czR ðr; ωÞ≈ Ml = 1 M =
It can be derived from the above equations that
∑ ∑ Blm kl cos½ðkl −km Þr
l=1 m=1
rffiffiffiffiffiffiffi rffiffiffi qR π 1 π nl 2 uzR;l ðr; z; ωÞ≈ ϕzR J1 ðkl RÞ exp−i kl r + kl 2kl r 4
M
M
∑ ∑ Blm cos½ðkl −km Þr
l=1 m=1 M M
∑ ∑ Blm c−1 l cos½ðkl −km Þr
l=1 m=1
ð28Þ ð22Þ
Clearly, Eq. (22) is an approximation of Eq. (19) in the far field, and it can be seen that the mode excited by active sources travels approximately in the form of cylindrical wavefronts and the wavenumber approaches to kl (the wavenumber of the lth normal mode). The effective wavenumber of surface waves at the distance r from the source center can be obtained from Eq. (13), namely, h i ∂ arctan AIz = ARz ð Þ ∂φ r; ω z kˆzR ðr; ωÞ = = ∂r ∂r 1 A A − A A = Iz Rz Rz Iz 2 2 ;r ;r ARz + AIz
ð26Þ
∑ ∑ Bˆlm ½Y0 ðkl r ÞY0 ðkm r Þ + J0 ðkl r ÞJ0 ðkm r Þ
ð23Þ
2 −3 = 2 −1 = 2 ˆ 1m where Blm = ϕ1l J1 ðkl RÞJ1 ðkm RÞðkl km Þ = ðkl km Þ Blm , cl zR ϕzR is the phase velocity of the lth normal mode. In the far field, the cylindrical surface waves also approach to the plane ones. Eq. (28) holds for both plane and cylindrical surface waves. The term (kl − km)r in Eq. (28) represents the phase difference of the lth and the mth modes at the distance r. When (kl − km) r → nπ(n = 0, 1, 2, ⋯), cos[(kl − km)r] → ±1. The lth and the mth modes π interfere constructively. When ðkl −km Þr→ + nπ(n = 0, 1, 2, ⋯), 2 cos[(kl − km)r] → 0. The lth and the mth modes interfere destructively. The term (kl − km)r in Eq. (28) implies that the effective phase velocity is influenced by the interference of the modes. There is only a single non-dispersive mode of the surface waves in a half space. As a special case, the effective phase velocity of surface waves can be obtained from Eq. (27):
czR ðr; ωÞ = cR By using Eqs. (20a) and (20b), Eqs. (14b) and (14c) can be reduced
½Y0 ðkrÞ2 + ½ J0 ðkrÞ2 Y0 ðkrÞJ1 ðkrÞ−J0 ðkrÞY1 ðkrÞ
ð29Þ
to:
ARz
M J ðk r Þ 1l 2 = ∑ ϕzR J1 ðkl RÞ 0 l kl l=1
ð24aÞ
2 Y ðk rÞ 1l ϕzR J1 ðkl RÞ 0 l kl
ð24bÞ
M
AIz = ∑
l=1
where cR is the phase velocity of the normal mode. Using Eqs. (21a), (21b) and (29), it can be shown that czR(r, ω) → cR in the far field. From Eq. (26) or (29), the wavenumber of the surface waves in a half space can be written as Y ðkrÞJ1 ðkr Þ−J0 ðkrÞY1 ðkr Þ kˆzR ðr; ωÞ = k 0 ½Y0 ðkr Þ2 + ½ J0 ðkr Þ2
ð30Þ
54
H.-Y. Chai et al. / Journal of Applied Geophysics 73 (2011) 49–58
where k is the wavenumber of the plane surface waves. It follows that ∂kˆzR ∂kˆzR ∂k ∂kˆzR 1 = = ∂ω ∂k ∂ω ∂k cR 2krff½Y0 ðkrÞ2 −½ J0 ðkrÞ2 gY1 ðkrÞJ1 ðkrÞ−f½Y1 ðkrÞ2 −½ J1 ðkrÞ2 gY0 ðkrÞJ0 ðkrÞg 1 : cR f½Y0 ðkrÞ2 + ½ J0 ðkr Þ2 g2
=
ð31Þ Thus, the group velocity of the surface waves in a half space is ∂ω czg ðr; ωÞ = ∂kˆ
zR 2
= cR
2 2
f½Y0 ðkrÞ + ½J0 ðkrÞ g : 2krff½Y0 ðkrÞ2 −½J0 ðkrÞ2 gY1 ðkr ÞJ1 ðkrÞ−f½Y1 ðkrÞ2 −½J1 ðkrÞ2 gY0 ðkr ÞJ0 ðkrÞg
ð32Þ It can be seen from Eqs. (21a), (21b) and (32) that czg(r, ω) → cR in the far field. For a homogenous half space, the group and the phase velocities are equal under the assumption of plane surface waves. By comparing Eq. (29) with Eq. (32), it can be seen that the group and the phase velocities are different, under the influence of the mode incompatibility. Although Eqs. (27) and (28) are derived by ignoring the material damping, they are applicable for the soils with the material damping because the material damping has little influence on the phase velocity of the normal modes and the effective phase velocity of the surface waves (Chai et al., submitted for publication). 4. Effects of active sources 4.1. Homogenous half space 4.1.1. Phase velocity of plane surface waves In a homogenous elastic half space, the regression relationship between the phase velocities of the plane surface waves is given as follows (Graff, 1975): 0:87 + 1:12ν cR = cs ≈ 1+ν
ð33Þ
where ν is Poisson's ratio. 4.1.2. Effects of mode incompatibility For a given shear wave velocity, the phase velocity of the surface waves under active sources can be calculated from Eq. (27). The theoretical phase velocity cR can be determined using Eq. (33). The variation of the dimensionless velocity (c/cR) with the dimensionless distance (r/λR) is shown in Fig. 2 for ν = 0.0, 0.25, 0.4 and 0.45. The
effect of Poisson's ratio on the normalized effective phase velocity of surface waves is minimal. It can be seen from Fig. 2 that the effective phase velocity of the surface waves is smaller than cR for r/λR b 1; the shorter the dimensionless distance, the smaller the phase velocity. The phase velocity approaches to cR when the spread length is greater than one wavelength. This implies that the effect of the mode incompatibility can be ignored beyond one wavelength. 4.1.3. Effects of body waves The energy of the direct body waves depends on Poisson's ratio. The observed phase velocity can be calculated from Eqs. (13) to (17). The loading parameters R and qRπ/2 in Eq. (13) are assumed to be 0.01 m and 1 N/m, respectively, for both homogenous and layered media. The observed phase velocities for ν = 0.0, 0.25, 0.4 and 0.45 are in agreement with those obtained by Foinquinos and Roësset (2001) (the differences are too small to be shown at the scale of Fig. 2). Thus, if the thicknesses of the thin layers are significantly smaller than the studied wavelength, the observed phase velocity can be evaluated very accurately using the thin layer stiffness method. Comparing the observed phase velocities and the phase velocity of the surface waves, it is found that oscillations appear in the observed phase velocity due to the influence of the body waves. It can be seen in Fig. 2 that a larger oscillation corresponds to a higher Poisson's ratio. To analyze the energy distribution pattern of the direct waves in the surface wave field, the ratio of the displacement amplitude of the full waves to that of the surface waves (in the vertical direction) as a function of the dimensionless distance r/λR is given in Fig. 3 for different Poisson's ratios. The results in Fig. 3 show that the direct body waves have stronger energy for larger Poisson's ratios, implying that the stronger the body waves, the stronger the oscillations in the observed phase velocity. It is also found that the influence range of the body waves is far beyond one wavelength. 4.2. Layered media 4.2.1. Effective phase velocity of plane surface waves In layered media, there are multiple modes in the wave field. Three layered media used by Tokimatsu et al. (1992b) and Yoon and Rix (2009) are selected for illustration: (a) The shear wave velocity of the layers increases with the depth in Case I. (b) There is a soft layer trapped between stiff layers 1 and 3 in Case II. (c) A stiff layer is trapped between soft layers 1 and 3 in Case III.
1.4 1.1
ν=0.0 ν=0.25 ν=0.4 ν=0.45
1.3
Effect of mode incompatibility
1.2
1.05
Surface waves 1.1
u/uzR
c/cR
1 0.95 0.9
Surface waves Observed (ν =0.00) Observed (ν =0.25) Observed (ν =0.40) Observed (ν =0.45)
Oscillations induced by the body waves
0.85 0.8
0
1
2
3
4
5
6
7
8
r / λR Fig. 2. Variation of the dimensionless phase velocity with the dimensionless distance in half spaces.
1 0.9 0.8 Full waves
0.7 0.6
0
1
2
3
4
5
6
7
8
r / λR Fig. 3. Variation of the dimensionless vertical displacement with the dimensionless distance in half spaces.
H.-Y. Chai et al. / Journal of Applied Geophysics 73 (2011) 49–58
a
Vertical displacement φz
1l
100
0
1
2
3
4
5x10-4
Mode 4
Frequency(Hz)
The material parameters for these media are listed in Table 1 (reproduced from those used by Tokimatsu et al., 1992b). The vertical displacements of the first several normal modes are shown in Fig. 4 for the aforementioned three cases. It can be seen from Fig. 4 that (a) the fundamental mode dominates the surface wave field over most of the frequency range in Case I, (b) the higher modes are predominant in the frequency range greater than 20 Hz in Case II, and (c) the higher modes play a significant role in the narrow frequency range around 10 Hz in Case III. It can be seen from Eq. (28) that if the mode incompatibility is ignored, the effective phase velocity of the surface waves czR varies with the dimensionless distance kr. The effective wavelength is defined as the ratio of effective phase velocity to frequency. Assuming that λR is the average effective wavelength over the studied spread length, we can replace the dimensionless distance kr with r = λR . We can show that the effective phase velocity varies with r = λR , and the variations are different at different frequencies. A typical result is shown in Fig. 5, where subscript R20 denotes the Rayleigh waves at a frequency of 20 Hz. Although the fundamental normal mode dominates the surface wave field over most of the frequency range for the regular profile Case I, the effective phase velocity is different from the phase velocity of the fundamental mode, due to the influence of the higher modes.
55
Mode 2 Mode 3 10 Mode 1
Case I 2
b
Vertical displacement φz
1l
100
0
0.5
1.5x10-4
1 Mode 5 Mode 4 Mode 3
4.2.3. Effects of body waves It can be seen from Fig. 7 that due to the influence of the body waves, some oscillations appear in the observed phase velocity. These oscillations do not die out as they do in half spaces. For Case II, oscillations are even stronger, implying that the body waves have
1 2 3 4
Shear wave velocity (m/s) Case I
Case II
Case III
80 120 180 360
180 120 180 360
80 180 120 360
Frequency(Hz)
10
Mode 1
Case II 2
Vertical displacement φz
c
1l
100
0
1
2
Mode 4
3
4
5x10-4
Mode 2 Mode 3 Mode 1
10
Case III
2 Fig. 4. Normalized vertical displacements of the first several normal modes: (a) Case I; (b) Case II; and (c) Case III.
Table 1 Material parameters for three typical layered media. Layer number
Mode 2
Frequency(Hz)
4.2.2. Effects of mode incompatibility It can be seen from Eq. (22) that in the far field, the geometric decay of the displacement is nearly in the form of r− 1/2. The variation of the vertical displacement of the fundamental mode with the dimensionless distance r/λR20 at a frequency of 20 Hz is shown in Fig. 6(a) for Case I. It can be seen that the effect of the mode incompatibility on the displacement can be ignored when the spread length is greater than one average wavelength. Because of the superposition of various modes, some oscillations appear in the displacement, as shown in Fig. 6(b). The effect of the mode incompatibility on the effective phase velocity of the surface waves is shown in Fig. 7. It can be seen that the effect is also limited within the spread length less than one average effective wavelength. In this range, the smaller the ratio of the spread length to the wavelength, the greater the effective phase velocity is underestimated. The variations of the effective phase velocity of the surface waves with frequency at positions 1 m and 5 m from the source are shown in Fig. 8. In order to compare the effective phase velocity with the dispersions of the normal modes, the dispersions of the first several modes are also shown. Clearly, the variations of the effective phase velocity with frequency at different positions are different, and the phase velocity is underestimated due to the mode incompatibility. At significantly lower frequencies, the discrepancy of the effective phase velocity at 1 m is more significant than that at 5 m, due to the stronger effect of mode incompatibility.
Thickness (m)
Poisson's ratio
Density (kg/m3)
2 4 8 ∞
0.35 0.35 0.35 0.35
1800 1800 1800 1800
more significant influence on the observed phase velocity. The ratios of displacements of the full waves to that of the surface waves are given in Fig. 9, showing that the body waves for Case II have stronger energy than those for Case I. This can be due to the stronger reflections from the interfaces of layers for media where a soft layer is trapped at a shallow depth.
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H.-Y. Chai et al. / Journal of Applied Geophysics 73 (2011) 49–58
120
a Fundamental normal mode
1.2 Case I
100 1.1
Effect of body waves
cR20 (m/s)
80
c/cR20
1 60
Superposition of normal modes
40
0.9 Effect of mode incompatibility
Observed
0.8
Surface waves
20 0.7 0
0
1
2
3
4
5
6
7
8
0.6
r / λ R 20
0
1
2
3
4
5
6
7
8
r / λ R 20
Fig. 5. Variation of the effective phase velocity with the dimensionless distance at a frequency of 20 Hz in Case I.
b
1.2
Case II
5. Conclusions
Effect of body waves 1.1
a
0.9 Effect of mode incompatibility 0.8
Real Imaginary Magnitude
20 Effect of mode incompatibility
Surface waves
0
1
2
3
4
5
6
7
8
r / λ R 20
c
15
Observed
0.7 0.6
25
Displacement (×10-9 mm)
1
c/cR20
Discrete analytical expressions for the effective phase velocity of the surface waves, which account for the mode incompatibility and higher modes, are derived based on the thin layer stiffness matrix method. An analytical expression of the effective phase velocity of the surface waves in the far field is also developed. The field within one average effective wavelength is influenced by both the mode incompatibility and the body waves. In this range, the effective phase velocity is underestimated in comparison with that of
1.2
Case III I
10
Effect of body waves
1.1
Decay inr -1/2
5 1
c/cR20
0 -5 -10
1
2
3
4
5
6
7
Surface waves
0.6 Real Imaginary Magnitude
5
Displacement (×10-7 mm)
Observed
8 0.7
6
4
0
1
2
3
4
5
6
7
8
r / λ R 20 Fig. 7. Variation of the dimensionless phase velocity with the dimensionless distance at a frequency of 20 Hz: (a) Case I; (b) Case II; and (c) Case III.
3 Effect of mode incompatibility
2 1 0 -1 -2
Effect of mode incompatibility
0.8 0
r / λR 20
b
0.9
0
1
2
3
4
5
6
7
8
r / λ R 20 Fig. 6. Variation of the vertical displacement with the dimensionless distance at a frequency of 20 Hz for Case I: (a) fundamental mode; and (b) superposition of modes.
the plane surface waves; the smaller the ratio of the spread length to the wavelength, the more the effective phase velocity is underestimated. The mode incompatibility will result in the underestimation of the observed phase velocity, while the body waves may induce some oscillations. The effects of the mode incompatibility can be ignored in the field where the spread length is beyond one average effective wavelength. The observed phase velocity has some oscillations due to the influence of the body waves. The influence range is far beyond one average effective wavelength. In layered media, especially in Case II where a soft layer is trapped between the first and the third layers, the body waves are strongly reflected at the interfaces of layers.
H.-Y. Chai et al. / Journal of Applied Geophysics 73 (2011) 49–58
a
a
400
2 Case I
Case I 1m 5m
Mode 4
300
1.5
u/uzR
Phase velocity (m/s)
Mode 3
57
Mode 1
200
Surface waves
Mode 2 1 100 Full waves Mode incompatibility 0
0
50
Rayleigh waves of surface layer 100
150
0.5
200
0
1
2
3
Frequency (Hz)
b
Phase velocity (m/s)
Mode 3 Mode 5 300
7
8
7
8
Case II
5m
Mode 4
1.5 Rayleigh waves of surface layer
200
Surface waves
1 100
0
Mode 2 Full waves 0.5
0
50
100
150
200
0
1
2
3
4
5
6
r / λ R 20
Frequency (Hz)
Phase velocity (m/s)
6
1m
Mode 1 Mode incompatibility
c
5
2
Case II
400
u/uzR
b
4
r / λ R 20
Fig. 9. Variation of the dimensionless displacement with the dimensionless distance at a frequency of 20 Hz: (a) Case I; and (b) Case II.
400
Case III 1m
Mode 3
5m
300 Mode 1 200 Mode 4
Mode 2
can take the near-field effects into account (Ganji et al., 1998; O'Neill et al., 2003). In the wave field with multiple modes, the effective phase velocity of the surface waves is dependent on both the frequency and the spread distance. In the far field where the effects of mode incompatibility and body waves are insignificant, the effective phase velocity can be analyzed using Eq. (28).
100 Mode incompatibility 0
0
50
Rayleigh waves of surface layer 100
150
200
Frequency (Hz)
Acknowledgments This research is supported by the one hundred talents program of the Chinese Academy of Sciences.
Fig. 8. Effective phase velocity curves of the surface waves at positions 1 m and 5 m and dispersion curves of the first several normal modes: (a) Case I; (b) Case II; and (c) Case III.
References
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