Introduction

Shear wave velocity (Vs) is a key parameter for site characterization in geotechnical engineering. Vs value is sensitive to ground compactness and rigidity variations and Vs imaging techniques allow the delineation of geological boundaries in the subsurface (Hun- ter et al., 2002), as well as ground densification (Kim and Park, 1999) or decompaction resulting from landslide effects (Jongmans et al., 2009). In earthquake engineering, shear wave velocity is also used in charts for liquefaction potential assessment during an ear- thquake (Finn, 2000). In the same field, numerous studies have shown that the ground motion characteristics (amplitude, duration) are amplified at sites where soft soil layers cover firm bedrock, and that contrasts in Vs values strongly control the dynamic site response and the resulting damage (see, for instance,Bard and Riepl, 1999;Sommerville and Graves, 2003). They are therefore required for evaluating site effects in seismic hazard assessment.

Shear wave velocity (Vs) can be in-situ measured by using various methods including borehole tests, shear wave refraction and reflection studies, and surface-wave techniques (Jongmans, 1992; Dasios et al., 1999; Hunter et al., 2002; Boore, 2006). In recent years, surface waves have been increasingly used for deriving Vs as a function of depth, taking advantage of the dispersion properties of these waves (Socco and Jongmans, 2004, for a review, see). Surface wave methods are divided in two main categories based on the kind of sources that generate the observed signals, i.e. active and passive methods. The first ones record vibrations generated by an artificial source, the frequency band of which is generally above 2 Hz (Tokimatsu, 1997). Their penetration depths are usually limited

3.2. Inversion strategy 87 to a few tens of meters (Jongmans and Demanet, 1993; Park et al., 1999; Socco and Strobbia, 2004). On the contrary, ambient vibrations or microtremors are also produced by sources of lower frequency (Aki, 1957; Satoh et al., 2001; Okada, 2003), making both methods complementary for defining the dispersion curve over a wide frequency range (Nguyen et al., 2004; Wathelet et al., 2004;Park et al., 2007; Socco et al., 2008)). Surface wave dispersion depends essentially on the Vs profile (Vs values and thicknesses of the different layers), and to a lesser degree, on the P-wave velocity and density profiles (Xia et al., 1999). Starting from seismic records, Vs profiles are thus estimated in two steps : 1) deriving the dispersion curve by transforming the recorded ground motion from the time- space domain to the velocity-frequency domain, and 2) inverting the dispersion curve to retrieve the shear wave velocity structure.

Numerous transforming methods have been proposed to process surface waves and estimate their dispersion curve, depending on the sensor layout (linear or two-dimensional arrays), the surface wave type (Love or Rayleigh) and the considered velocity (group or phase). An overview of the different techniques can be found inSocco and Strobbia (2004) and in Wathelet et al. (2008). Three main methods can be combined to determine the dispersion curve on a wide frequency range : the frequency-wavenumber method (Lacoss et al., 1969), the high-resolution frequency-wavenumber method (Capon, 1969) and the modified spatial auto-correlation technique (Aki, 1957; Bettig et al., 2001). In the second step, the phase velocity dispersion curve is usually inverted using a classical linearized algorithm (Herrmann, 1987;Satoh et al., 2001) or direct search techniques like the Monte Carlo approach (Edwards, 1992;Mosegaard and Tarantola, 1995;Socco and Boiero, 2008), the neighborhood algorithm (Sambridge, 1999; Wathelet et al., 2004; Wathelet, 2008) or the genetic algorithm (Yamanaka and Ishida, 1996;Kind et al., 2005;Parolai et al., 2005;

Dal Moro et al., 2007).

A classical assumption in surface wave inversion studies is the predominance of the fundamental mode for the vertical motion. Indeed, several blind studies reported a good agreement between Vs profiles derived from surface wave inversion with fundamental mode assumption and independent geological and seismic data (Asten et al., 2005; Asten and Boore, 2005; Boore and Asten, 2008). Other studies showed that the measured disper- sion curve may be influenced by higher modes (Socco and Strobbia, 2004; Cornou et al., 2006). In the literature, this influence was reported in case of deep sources (Keilis-Borok, 1986), inversely dispersive media (Gucunski and Wood, 1991;Tokimatsu et al., 1992; Pa- rolai et al., 2006), or 2D/3D lateral variations that act as deep secondary sources (Schlue and Hostettler, 1987; Uebayashi, 2003). For considering higher modes in the inversion scheme, Tokimatsu et al. (1992) proposed a formulation relating the apparent phase ve- locity dispersion curve to the relative energy of the multiple Rayleigh modes, which was used by Arai and Tokimatsu (2005); Parolai et al. (2005). This formulation is based on the assumption of Rayleigh waves excited vertically at the surface of an elastic medium, and does not take into account the geometry of the array layout. Another approach was developed by Lai (1998) who match the experimental dispersion curve with the effective phase velocity curve, which accounts for multi-mode Rayleigh wave propagation.

These two approaches help in addressing the multiple mode content of the measured dispersion curve that is encountered in some cases. However, they do not provide any gui- delines regarding the inversion procedure itself, which suffers the major draw back of the

88 3. Inversion des ondes de surface en milieu 1D non-uniqueness of the solution. This problem, which exists even when the whole disper- sion curve is perfectly known in the appropriate frequency range, is accentuated with the number of parameters to invert. This issue will be discussed in the following section. In practice, the dispersion curve is also affected by uncertainty and is determined in a limited frequency band, owing to the signal energy, the sensor characteristics and the high-pass filter effect of the soil layers (Scherbaum et al., 2003;Wathelet et al., 2008). This data qua- lity degradation makes the non-uniqueness problem worse. To limit these effects, a special attention must be paid to the data acquisition (Socco and Strobbia, 2004) with the aims of lowering uncertainty and measuring the dispersion curve on the widest possible frequency band. Other ways to reduce the non-uniqueness problem by constraining the final model are to jointly invert surface wave dispersion curves and other data, like refraction or reflec- tion travel times (Dal Moro and Pipan, 2007;Dal Moro, 2008) and Horizontal-to-Vertical (H/V) Spectrum data (Scherbaum et al., 2003;Arai and Tokimatsu, 2005;Parolai et al., 2005). Luo et al. (2007) recently showed that the joint inversion of the fundamental and higher modes could also reduce the non-uniqueness problem, if the different modes can be identified. Finally, K¨ohler et al. (2007) extended the modified spatial autocorrelation technique to three-component analysis and determined both Love and Rayleigh waves.

Joint inversion of the two dispersion curves was successfully tested at one site where exis- ting shear-wave profile was available. All these strategies however require additional data, which are not acquired in current and present day practice.

The aim of the present study is to investigate the influence of parameterization on the dispersion curve inversion and the possibility to extract further constraints on the ground model from the inversion process itself. In our study, we focus on the cases where the fundamental mode is the most energetic and is derived over a wide frequency range. This limiting assumption will be discussed in the application section.

The relation between non-uniqueness and parameterization is first highlighted and a short review of the usual practice in parameterization for surface wave inversion is pre- sented. We then analyze Vs profiles measured at twenty strong-motion sites and selected in the framework of the ongoing European project NERIES (Network of Research Infra- structures for European Seismology ; http ://www.neries-eu.org/). A part of this project is indeed dedicated to the comparison of the different methods available for deriving Vs pro- files on sites where shallow soil layers (at most 200 m) are covering firm bedrock. Seismic passive (Ambient Vibrations measurements) and active (MASW) tests were performed with the same equipment at all sites, resulting in the determination of dispersion curves over a wide frequency range (NERIES-JRA4, 2008a, 2009b). First, we analyzed the Vs profiles provided by borehole test data previously acquired at these twenty sites. Repre- sentative Vs profiles were defined for such geological structures with soil layers overlying bedrock, that can be in most cases approximated with 1 or 2 linearly increasing velocity layers over half space. However, we show that inverting with parameterizations allowing linear velocity laws usually results in smoothed models and that introducing information on the bedrock depth significantly helps guiding the inversion towards the right solution.

Three synthetic models were designed from the preliminary study on borehole Vs profiles.

They are used to understand the influence of parameterization on the dispersion curve inversion and to propose a two-step inversion scheme for sites with a strong Vs contrast at the bedrock top. This strategy consists in first estimating the bedrock depth range, using

3.2. Inversion strategy 89 parameterizations with uniform layers, and then introducing this information to constrain the inversion with linearly increasing velocity layers. The last part of the paper presents the application of the proposed procedure at two sites where active seismic and ambient vibration methods were applied to derive dispersion curves over a wide frequency range.