Surveys in Geophysics (2006) 27:1–17 DOI 10.1007/s10712-005-7261-3
Springer 2006
ESTIMATION OF ELASTIC MODULI IN A COMPRESSIBLE GIBSON HALF-SPACE BY INVERTING RAYLEIGH-WAVE PHASE VELOCITY JIANGHAI XIA1, YIXIAN XU2, RICHARD D. MILLER1 and CHAO CHEN2 1
Kansas Geological Survey, The University of Kansas, 1930 Constant Avenue, Lawrence, Kansas, 66047-3726, U.S.A E-mail:
[email protected]; 2 Institute of Geophysics and Geomatics, China University of Geosciences, Wuhan, Hubei, 430074, P.R. China
(Received 14 December 2004; Accepted 28 April 2005)
Abstract. A Gibson half-space model (a non-layered Earth model) has the shear modulus varying linearly with depth in an inhomogeneous elastic half-space. In a half-space of sedimentary granular soil under a geostatic state of initial stress, the density and the Poisson’s ratio do not vary considerably with depth. In such an Earth body, the dynamic shear modulus is the parameter that mainly affects the dispersion of propagating waves. We have estimated shear-wave velocities in the compressible Gibson half-space by inverting Rayleigh-wave phase velocities. An analytical dispersion law of Rayleigh-type waves in a compressible Gibson halfspace is given in an algebraic form, which makes our inversion process extremely simple and fast. The convergence of the weighted damping solution is guaranteed through selection of the damping factor using the Levenberg-Marquardt method. Calculation efficiency is achieved by reconstructing a weighted damping solution using singular value decomposition techniques. The main advantage of this algorithm is that only three parameters define the compressible Gibson half-space model. Theoretically, to determine the model by the inversion, only three Rayleigh-wave phase velocities at different frequencies are required. This is useful in practice where Rayleigh-wave energy is only developed in a limited frequency range or at certain frequencies as data acquired at manmade structures such as dams and levees. Two real examples are presented and verified by borehole S-wave velocity measurements. The results of these real examples are also compared with the results of the layered-Earth model. Keywords: dispersion curve, Gibson half-space, inversion, phase velocity, Rayleigh wave, shear-wave velocity
1. Introduction The shear (S)-wave velocities of near-surface materials (such as soils) are of fundamental interest in many environmental and engineering studies. There are three basic ways to perform shear tests in situ in soil mechanics: in situ shear box, shear vane, and penetration (e.g., Whitlow, 1995). During these tests, either a soil sample needs to be carefully cut or remolded (the first method) or invasive penetrations need to be performed (the second and third
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methods). Analysis of surface-wave dispersion offers an alternative way to evaluate shear strength rapidly, non-invasively, and cost-effectively. Near-surface S-wave velocities are important to construction safety and environmental study. For example,V30 s , the average S-wave velocity of the upper 30 m of the Earth, plays a key role in soil classification and is one of the critical parameters in the construction industry and for earthquake safety.V30 s was accepted in the U.S.A. for the UBC (Uniform Building Code) for site classification in 1997 (Dobry et al., 2000). Wills et al. (2000) utilized the UBC classification for site-condition mapping in California. The nearsurface S-wave velocity of a construction site is also a fundamental parameter in the new provisions of Eurocode-8 (Sabetta and Bommer, 2002). S-wave velocity is also a critical parameter in slope-stability analysis (Hack, 2000). The correlation between N-value (Craig, 1992), an index value of formation hardness used in soil mechanics and foundation engineering, and S-wave velocity (Imai and Tonouchi, 1982) shows a bright future for non-invasive technologies in defining near-surface S-wave velocities. Rayleigh waves are surface waves that travel along or near the surface of the ground. They are usually characterized by relatively low velocity, low frequency, and high amplitude (Sheriff, 1991). Rayleigh waves are the result of interfering P and Sv waves, Particle motions for the fundamental mode of Rayleigh waves in a homogeneous medium moving from left to right are elliptical in a counter-clockwise (retrograde) direction along the free surface and are still elliptical when reaching sufficient depth. As depth increases, the particle motion becomes prograde at a depth of 0.2 k (where k is the wavelength) (Aki and Richards, 1980, p. 162). The motion is constrained to a vertical plane consistent with the direction of wave propagation. For the case of a solid homogeneous half-space, the Rayleigh wave is not dispersive. It travels at a velocity of approximately 0.9194 v when Poisson’s ratio is equal to 0.25 and where v is the S-wave velocity in the half space (Sheriff and Geldart, 1985, p. 49). Stokoe and Nazarian (1983) and Nazarian et al. (1983) presented a surface-wave method, called Spectral Analysis of Surface Waves (SASW), which analyzes Rayleigh waves generated from impact acoustic sources to produce near-surface S-wave velocity profiles. SASW has attracted the attention of engineering scientists and has been widely applied to many engineering projects (e.g., Sanchez-Salinero et al., 1987; Sheu et al., 1988; Stokoe et al., 1989; Song et al., 1989; Gucunski and Wood, 1991; Hiltunen, 1991; Stokoe et al., 1994). A technique (Park et al., 1999; Xia et al., 1999) utilizing a multichannel recording system to estimate near-surface S-wave velocity from highfrequency (‡2 Hz) Rayleigh waves (Multichannel Analysis of Surface Waves – MASW) has been applied to more and more near-surface problems (e.g., Xia et al., 1998, 2002a, b, and 2004a; Miller et al., 1999; Ivanov et al., 2000; Calderon-Macias and Luke, 2002; Yilmaz and Eser, 2002). Xia et al. (2002a)
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reported that the differences between MASW results and direct borehole measurements are 15% or less and random. Studies on the MASW method have been extended to areas of utilization of higher modes (Beaty et al., 2002; Beaty and Schmitt, 2003; Xia et al., 2003), determination of near-surface quality factors (Q) by inverting the attenuation coefficients of Rayleigh waves (Xia et al., 2002c), optimization of field parameters of data acquisition (Zhang et al., 2004), and deployment of autojuggie (Steeples et al., 1999) with the MASW method (Tian et al., 2003a, 2003b). Discussion on the multichannel analysis of surface-wave dispersion can also be found in Lin and Chang (2004). MASW applications in near-surface geophysics can be found in numerous other publications (e.g., Park et al., 1996, 1999; Ivanov et al., 2000; Calderon-Macias and Luke, 2002; Xia et al., 2002c; and Lin et al., 2004). Understanding the resolving power of MASW techniques and improving the resolution of S-wave velocity results were discussed by Xia et al. (2005). An overall review of high-frequency surface wave techniques, mainly developed at the Kansas Geological Survey, and their applications to near-surface geophysics can be found in Xia et al., (2004b). Inversion of the shallow seismic wavefield which utilized all signals in a shot gather, including surface waves and body waves, was discussed by Forbriger (2003a, b). The researches mentioned above are all focused on a layered-Earth model. For a non-layered medium, surface waves are always dispersive due to the inhomogeneity of the elastic properties in real soil bodies (e.g., Richart et al. 1970). In a half-space of sedimentary, especially very shallow subsurface, granular soil/sand under geostatic state of initial stress, the density and the Poisson’s ratio do not vary considerably with depth (e.g., Bachrach et al., 1998). In such an Earth body, the dynamic shear modulus is the parameter that mainly affects the dispersion of propagating waves (Vardoulakis and Verttos, 1988). Gibson (1967) introduced an assumption of shear modulus distribution linearly with depth in an inhomogeneous elastic half-space, which is called a compressible Gibson half-space. Vardoulakis and Verttos (1988) derived an analytical dispersion law of Rayleigh-type waves in the compressible Gibson half-space. The dispersion law has an algebraic form, which makes our inversion process extremely simple and fast. Based on the dispersion law, we have proved that an iterative solution technique to a weighted damping equation using the Levenberg-Marquardt (L-M) method (Marquardt, 1963) and the singular value decomposition (SVD) techniques (Golub and Reinsch, 1970) is very effective in determining the elastic moduli of the Gibson half-space. Reconstruction of S-wave velocities by the inversion of Rayleigh-wave phase velocities could be achieved in real time in situ. In particular, only three Rayleigh-wave phase velocities are needed to define the Gibson half-space model. This could dramatically reduce field efforts of data acquisition, especially in some difficult environments such as a shallow marine environment (Luke et al., 1996; Luke and Stokoe, 1998; Park et al.,
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2000) or a noisy environment (Xia et al., 2004a), and make inversion possible with as few as three data points.
2. The method In this section, we first list the dispersion law of the Gibson half-space and its partial derivatives, and then formulate an inversion algorithm by the L-M method and the SVD technique. 2.1. Gibson half-space and dispersion law A Gibson half-space (Gibson, 1967) is defined as an inhomogeneous elastic half-space z ‡ 0, with a constant density q and Poisson’s ratio m, and a dynamic shear modulus G increasing linearly with depth (Figure 1). The shear modulus variation in the Gibson half-space is given by: G ¼ G0 ð1 þ mzÞ;
ð1Þ
where G0 > 0 is the shear modulus at the free surface, m is a measure of inhomogeneity which has the dimensions of inverse length. The limiting value m=0 corresponds to the homogeneous elastic half-space, where Rayleigh waves do not exhibit dispersion.
Figure 1. A Gibson half-space (Gibson, 1967).
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Vardoulakis and Verttos (1988) derived an approximate algebraic form of the dispersion law for the fundamental mode of the Rayleigh wave in the Gibson half-space (Eq. 1). 1 þ Cffi Xv
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 ; þ 2 Xv 0:35ð3:6 mÞ
ð0:25 m 0:5Þ
ð2Þ
where C ¼ c=vs0 ; Xv ¼
ð3Þ
0:56ð3:6 mÞX ; 1:5 þ m
ð4Þ
c is the Rayleigh-wave phase velocity, vs0 is the S-wave velocity at the free surface, pffiffiffiffiffiffiffiffiffiffiffiffi ð5Þ vs0 ¼ G0 =q; X ¼ 2pf=mvs0 ; and
ð6Þ
f is frequency in Hz, and q is an average density of the Gibson half-space. C and W are called dimensionless velocity and dimensionless frequency, respectively. Vardoulakis and Verttos (1988) explained that the relative error induced by making the approximation (Equation 2) is 1–3 percent.Based on Eq. 1, the S-wave velocity at depth z can be written as pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð7Þ vs ðZÞ ¼ G=q ¼ G0 ð1 þ mzÞ=q ¼ vs0 ð1 þ mzÞ: If the density of the Gibson half-space q can be determined by another means, the shear-modulus, Young’s modulus, and the bulk modulus at depth z can be estimated by S-wave velocities and Poisson’s ratio. 2.2. Partial derivatives of dispersion law Equation 2 shows that the fundamental mode of the Rayleigh wave in the Gibson half-space, c, is a function of the S-wave velocity at the surface vs0, the measure of inhomogeneity m, and Poission’s ratio m. The partial derivatives of Equation 2 are as follows:
@c
2 þ @vs0 ¼ Xv
!1=2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 1 1 1 þ þ þ ; X2v 0:35ð3:6 mÞ X2m X2v 0:35ð3:6 mÞ
ð8Þ
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2
@c 1 4 ¼ Xv þ @m X2v
!1=2 3 1 1 5 vs0 ; and þ 2 0:35ð3:6 vÞ m Xv
ð9Þ
8 !1=2 " #9 < = 0 @c 1 1 1 2Xm 1 þ ¼ 2 X0m þ 0:5 2 þ vs0 ; @m : Xv Xv 0:35ð3:6 vÞ X3v 0:35ð3:6 mÞ2 ; ð10Þ where X0v
0:56 3:6 m ¼ 1þ X ð1:5 þ mÞ 1:5 þ m
ð11Þ
is the partial derivative of Wv with respect to Poisson’s ratio m. As indicated by Equation 2, Poisson’s ratio is in the range of 0.25–0.5. This constraint is replaced by following variable substitutions: m ¼ 0:5ðb2 þ b1 Þ þ ðb2 b1 Þðarctan pÞ=p; 0:25 b1 b2 0:5; and 1
