THE DETERMINATION OF THE DAMPING PARAMETER OF SOILS WITH THE SASW METHOD

Transcription

1 COMPDYN 9 ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering M. Papadrakakis, N.D. Lagaros, M. Fragiadakis (eds.) Rhodes, Greece, - June 9 THE DETERMINATION OF THE DAMPING PARAMETER OF SOILS WITH THE SASW METHOD S. A. Badsar, M. Schevenels, and G. Degrande Department of Civil Engineering, K.U.Leuven Kasteelpark Arenberg, B- Leuven, Belgium sayedali.badsar@bwk.kuleuven.be Keywords: Material damping Attenuation Rayleigh waves Half-power bandwidth Insitu tests Shear modulus. Abstract. This paper presents a technique for the determination of the material damping ratio of shallow soil layers. It is based on the spectral analysis of surface waves (SASW) test. The technique is an alternative to existing methods, where the damping ratio is determined from the spatial decay of the Rayleigh wave. These methods are based on the hypothesis that the response of the soil in the SASW test is due to a single mode surface wave. Therefore, the resulting attenuation curve can become erroneous when higher modes contribute to the soil s response. Moreover, in these methods, the estimate of the attenuation curve is based on an estimate of the geometric spreading function. The latter is computed using the shear wave velocity of the soil, which is determined by inversion of the experimental dispersion curve. Errors in the experimental dispersion curve and in the inversion procedure lead to an erroneous geometric spreading factor and, consequently, experimental attenuation curve. In the proposed technique, the f-k method is used to determine the experimental dispersion and attenuation curves and, subsequently, the soil profile. The dispersion curve is derived from the peak s position, while the attenuation curve is derived from its width, using the half-power bandwidth method. In this method there is no need for calculating the geometric damping. The occurrence of higher Rayleigh modes does not affect the attenuation curve associated with the fundamental Rayleigh wave, as higher modes appear as separate peaks in the f-k spectrum and do not interfere with the peak corresponding to the fundamental Rayleigh wave. Synthetic examples are used to validate the outlined technique; the results are compared with those obtained using two state-of-the-art methods. It is demonstrated that all three methods work well when applied to a regular soil profile, where the stiffness of the soil increases with depth. In case of soil profiles with soft layers trapped between stiffer layers, or profiles including a lot of thin layers that are approximated by a few thick layers, the proposed technique leads to accurate results.

2 INTRODUCTION The Spectral Analysis of Surface Waves (SASW) method aims to determine the dynamic shear modulus and the material damping ratio of shallow soil layers. It is based on an in situ experiment where Rayleigh waves are generated by means of an impact hammer, a falling weight, or a hydraulic shaker. The resulting wave field is recorded by a number of sensors at the soil s surface and used to determine the dispersion and attenuation curves of the soil. An inverse problem is solved to identify the corresponding soil profile. The SASW method has been utilized in different applications over the past couple of decades. It has been used to investigate pavement systems [], to assess the quality of ground improvement [], to determine the thickness of waste deposits [8], and to identify the dynamic shear modulus [,, 6,, ] and the material damping ratio [9,, ] of shallow soil layers. The shear wave velocity and material damping ratio are important for calculation of vibrations in the free field or in buildings due road or rail traffic, industrial machinery, and construction activities. The present study is performed in the context of this category of applications, with emphasis on determination of the material damping ratio. In order to determine dynamic soil properties, both in situ and laboratory methods have been used in the literature. Laboratory measurements like resonant column or torsional shear tests are often used to determine properties of cohesive soils, but for non-cohesive soils, there is a risk of sample disturbance. Although laboratory tests are useful for parametric studies of soil properties [], in situ tests preserve the natural status of the soil and avoid sample disturbance. Moreover, a larger volume of the soil is examined when performing in situ tests, avoiding a bias in the results due to local variations of the properties. Borehole techniques are well known for in situ determination of soil properties. These techniques have a good resolution at large depths compared to surface wave methods. However in the frame of the current research, only the information of the shallow soil layers are of interest. Moreover, surface wave methods have the same range of operating frequencies as the frequency range of interest in the ground vibration analysis, and they are non-invasive tests to save time and money [8]. The surface wave methods established by Lai [9] and Rix et al. [], and extended by Foti [], to determine the material damping ratio are discussed in this paper. These methods are based on the hypothesis that the response of the soil in the SASW test is due to a single surface wave []. If multiple surface waves contribute to the response (e.g. due to the high contrast or inclusion of a softer layer), this assumption does not hold and the resulting attenuation curves are affected. Moreover, in these methods, the estimate of the attenuation curve is based on an estimate of the geometric spreading function. The latter is computed using the shear wave velocity of the soil, which is determined by inversion of the experimental dispersion curve. Errors in the experimental dispersion curve and in the inversion procedure (e.g. due to the nonuniqueness of the problem) lead to an incorrect estimate of the geometric spreading factor and, consequently, the experimental attenuation curve. In this paper, an alternative method for the determination of the material damping ratio is proposed, using the half-power bandwidth method. The half-power bandwidth method has originally been developed in the field of structural dynamics to determine the modal damping ratio of a structure from the width of the peaks in the structure s frequency response function. It has been applied to the frequency-wavenumber content of the soil s response in this research. The occurrence of multiple Rayleigh modes does not affect the attenuation curve of either the fundamental or the dominant Rayleigh wave, as all modes appear as separate, non-interfering

3 Figure : Shear wave velocity profile corresponding to the regular, the irregular, and the multilayer soil profile. peaks in the frequency-wavenumber spectrum. Moreover, the experimental attenuation curve is derived directly from the experimental data, avoiding the use of a (possibly incorrect) estimate of the soil s shear wave velocity. This paper is organized as follows: section presents three synthetic examples including a regular, an irregular, and a multilayer soil profile. These examples are used in sections to to validate the proposed method and compare it with existing techniques. In these sections, the presented technique as well as two state-of-the-art methods are elaborated and the objective functions used in the inversion procedures are explained. SYNTHETIC EXAMPLES The presented technique as well as two state-of-the-art methods are elaborated and compared in the following sections. The comparison is performed based on three different synthetic examples: a regular soil profile where the stiffness of the layers increases with depth, an irregular soil profile in which a soft layer is trapped between two stiffer layers, and a multilayer soil profile with a smooth change of the properties between different layers are considered. For each of these profiles, a Spectral Analysis of Surface Waves (SASW) experiment is simulated. To simulate an SASW test, the response of the soil due to a transient point force is calculated using the direct stiffness method [7]. The response is subsequently transformed to the frequency-space domain and the frequency-wavenumber domain. In the following sections, the results are processed using different methods in order to determine the experimental dispersion and the attenuation curves. The first soil profile consists of a layer with a thickness of m on a halfspace. The shear wave velocity C s is m/s in the layer and m/s in the halfspace (figure a). The Poisson s ratio ν is / throughout the medium, resulting in a dilatational wave velocity C p of m/s in the layer and m/s in the halfspace. The material damping ratio β in both shear and volumetric deformation is equal to. in the layer and. in the halfspace (figure a). The density ρ is 9 kg/m in both the layer and the halfspace. The second soil profile is an irregular soil profile with a soft layer trapped between two stiffer layers. This reverse profile results in participation of the higher modes in the soil s response. The top layer has a thickness of m and the second layer has a thickness of m. The shear wave velocity in the top layer is equal to m/s, in the second layer m/s, and in the halfspace m/s (figure b). Poisson s ratio ν equals / throughout the medium, resulting in a dilatational wave velocity of m/s in the top layer, m/s in the second layer, and 8 m/s in the halfspace. The material damping ratio β in both shear and volumetric deformation is. in the top layer,. in the second layer, and. in the halfspace (figure b). The density ρ is 9 kg/m everywhere.

4 Figure : Material damping ratio profile corresponding to the regular, the irregular, and the multilayer soil profile. Force [N].. Force [N] x..... Time [s] Time [s] Figure : Time history and frequency content of a Ricker wavelet with a characteristic period T D =. s and a time shift t s =. s. The third soil profile can be considered as a profile consisting of layers on a halfspace where the material properties vary smoothly within and between the layers. This profile is chosen in an attempt to mimic a real soil, where the material properties do not vary abruptly. The profile is modeled using layers with a thickness of. m on top of a homogeneous halfspace (figure ). The shear wave velocity C s gradually increases from m/s at the surface to m/s at a depth of. m. The material damping ratio β decreases with depth, from. at the surface to. at a depth of. m (figure c). Poisson s ratio ν equals / and the density ρ is equal to 9 kg/m throughout the medium. In an in situ experiment, the response of the soil due to an impulsive force applied by means of an impact hammer, a falling weight, or a hydraulic shaker is considered. For the present simulations, the force is represented by a point load Fz E (t) corresponding to a Ricker wavelet, defined as: F E z (t) = ( ) ( ) π(t ts ) π(t ts ) exp () T D with T D =. s the characteristic period and t s =. s a time shift (figure ). The resulting vertical displacement u E z (r, t) at the soil s surface is calculated at every m up to m (figure ), using the ElastoDynamics Toolbox (EDT) [7] for Matlab, which is based on the direct stiffness method [7]. When the arrival times of the ground vibrations are compared, it is observed that wave propagation in the soil delays the time signals for an increasing distance from the source. As all traces are scaled with respect to their peak value, the effect of attenuation is not visible. The loading function Fz E (t) is transformed from the time to the frequency domain by means of a forward Fourier transformation: ˆF E z (ω) = T D F E z (t)e iωt dt ()

5 Figure : Time history of the vertical displacement u E z (r, t) at the surface of the regular, the irregular, and the multilayer soil profile. Figure : Transfer function ĤE zz (r, ω) for the regular, the irregular, and the multilayer soil profile. where ω is the circular frequency and a hat above a variable denotes its representation in the frequency domain. Similarly, the displacement u E z (r, t) is transformed from the time-space domain to the frequency-space domain: û E z (r, ω) = u E z (r, t)e iωt dt () The transfer function ĤE zz (r, ω) from the force ˆF z E(ω) to the displacement ûe z (r, ω) is defined as: Ĥzz(r, E ω) = ûe z (r, ω) () ˆF z E (ω) Figure shows the transfer function ĤE zz(r, ω). Due to geometric and material damping in the soil, the transfer function ĤE zz (r, ω) generally decreases as the distance r increases. In comparison with figures b and c, figure a exhibits a relatively smooth transfer function for the regular soil profile. It is observed in figure that peaks are clearly distinguishable in the three cases, implying that the spatial sampling is good enough to obtain the modal dispersion curves rather than an apparent (effective) dispersion curve. This has been verified by choosing finer spatial sampling in the synthetic data set. The transfer function ĤE zz (r, ω) is transformed to the frequency-wavenumber domain according to the procedure proposed by Forbriger []. This procedure is similar to a classical slant stack analysis [6], but the Fourier transformation is replaced by a Hankel transformation. In this way, the cylindrical symmetry of the problem is properly accounted for. The f-k spectrum of the response is defined as: H E zz(k r, ω) = Ĥ E zz(r, ω)j (k r r)r dr () where a tilde above a variable denotes its representation in the frequency-wavenumber domain and J (k r r) is the zeroth order Bessel function of the first kind. Equation is approximated by

6 Figure 6: Transfer function H E zz(c R, ω) for the regular, the irregular, and the multilayer soil profile. truncation of the integral at r = r M (the position of the farthest receiver) and the discretization of the integration domain between r = and r = r M. In order to mitigate the effect of the truncation of the integration domain, an exponential window ŵ(r, ω) is applied to the data in the frequency-space domain. The application of an exponential window can be considered as the introduction of artificial damping, resulting in a higher spatial attenuation of the surface waves. The window ŵ(r, ω) is applied for each frequency ω independently and defined as: The exponent Âart(ω) is chosen so that: ŵ(r, ω) = e Âart(ω)r (6) ŵ(r max, ω)ĥe zz (r max, ω) ŵ(r min, ω)ĥe zz(r min, ω) = q (7) where r min and r max denote the positions of the nearest and the farthest receiver, respectively. For frequencies ω where equation 7 results in a negative exponent Âart(ω), this exponent is set equal to zero. The application of the window ensures that the amplitude ratio of the response at the farthest and the nearest receiver does not exceed a value q. A value q = has been used here to avoid overestimation of the attenuation coefficient at the low frequencies. Following Forbriger [], the Bessel function J (k r r) in equation () is replaced by a zeroth order Hankel function H () (k r r)/ to account for the fact that the wave field only consists of outgoing waves. The following approximation is thus obtained: H E zz (k r, ω) = M j= Ĥ E zz (r j, ω)h () (k r r j )ŵ(r j, ω)r j r j (8) where M is the number of receivers, r j is the source-receiver distance for receiver j, and r j = r j r j. The transfer function ĤE zz (r, ω) is normalized at each frequency to the corresponding maximum value. The resulting frequency-wavenumber spectrum H zz E (k r, ω) is shown in figure 6. The spectrum is plotted in terms of the phase velocity C r (ω) = ω/k r (ω) instead of the wavenumber k r. The maxima in the frequency-wavenumber spectrum correspond to the dispersion curves of the soil. The occurrence of higher modes is clearly visible in figures 6b and 6c. In the following sections, the transfer function in the frequency-space domain ĤE zz(r, ω) and in the frequency-wavenumber domain H zz E (k r, ω) are used to determine the experimental dispersion and attenuation curves by means of three different methods. In order to asses the results, they are compared to the target dispersion and attenuation curves, which are directly computed with EDT, and shown in figures 7 and 8. 6

7 Figure 7: Target fundamental dispersion curves of the regular, the irregular, and the multilayer soil profile. Attenuation coefficient [/m] Attenuation coefficient [/m] Attenuation coefficient [/m] Figure 8: Target attenuation curves of the regular, the irregular, and the multilayer soil profile. Each of the computed fundamental dispersion curves shown in figure 7 starts from the Rayleigh wave velocity corresponding to the halfspace and ends at the phase velocity corresponding to the top layer. The regular profile has a descending dispersion curve (figure 7a) that covers a relatively small range of phase velocities, stating the low contrast between the top layer and the halfspace. Figure 7b shows the dispersion curve corresponding to the irregular soil profile. The curve ascends in the higher frequencies, revealing the presence of a stiffer top layer. The dispersion curve corresponding to the multilayer soil profile is descending (figure 7c). It covers a relatively wide band of phase velocities, showing a relatively high contrast between the top layer and the halfspace. The Rayleigh wave attenuation curves (figure 8) start from zero and continue with a positive slope. The slope is equal to the horizontal wavenumber k r (ω) at each frequency. The initial slope is a measure for the value of the material damping ratio at large depths (large wavelengths), while the slope of the curve at the higher frequencies gives an estimate to the material damping ratio of the top layer(s). Comparing figures 8a and 8b with figure 8c, it can be seen that sharp changes in the layer stiffness and/or material damping ratio result in sharp variations of the slope of the corresponding attenuation curve. METHOD : AMPLITUDE AND PHASE REGRESSION IN THE FREQUENCY- SPACE DOMAIN In the first method, the experimental dispersion curve CR(ω) E and the attenuation curve αr(ω) E are determined by means of an amplitude and phase regression of the transfer function ĤE zz (r, ω) in the frequency-space domain [9, ]. It is assumed that the response is due to a single surface wave and can be expressed as a product of three factors: ( ) ĥ E ω zz(r, ω) = ζ(r, ω)exp i CR(ω) r exp ( αr(ω)r ) E (9) E 7

8 The factor ζ(r, ω) accounts for the wave decay due to the geometric spreading of the wave fronts over an increasing area. This factor is equal to the wave decay in a soil without material damping and depends on the stratification of the soil. As this stratification is initially unknown, the geometric spreading factor is initially assumed to be equal to the factor in a homogeneous halfspace, i.e. ζ(r, ω) = / r. The second factor exp( iωr/cr E (ω)) is a harmonic function that depends on the phase velocity CR E(ω) of the surface wave. The third factor exp( αe R (ω)r) is an exponentially decaying function that accounts for the wave decay due to material damping. This function depends on the attenuation coefficient αr E(ω). For each frequency, the phase velocity CR(ω) E and the attenuation coefficient αr(ω) E are determined by fitting the function ĥe zz (r, ω) to the experimental transfer function ĤE zz (r, ω). The fitting procedure is performed in two steps. In the first step, the moduli of the functions ĥe zz (r, ω) and ĤE zz(r, ω), calculated at M offset points, are fitted in order to determine the attenuation coefficient αr E(ω): M α E R (ω) j= αr E (ω) = arg min ĤE zz (r j, ω) ζ(r j, ω)exp( αr E (ω)r j) The values taken by αr E (ω) during the iterations are constrained by upper and lower bounds to avoid physically meaningless solutions. In the second step, the attenuation coefficient αr E(ω) is kept fixed and the complex valued functions ĥe zz(r, ω) and ĤE zz(r, ω), calculated at M offset points, are fitted in order to determine the phase velocity CR E(ω): C E R(ω) = arg min M CR E(ω) j= ( ) ω zz(r j, ω) ζ(r j, ω)exp i ĤE CR E(ω)r j exp( αr(ω)r E j ) Next, an inverse problem is solved to determine the shear wave velocity profile corresponding to the experimental dispersion curve CR(ω). E This profile is used to compute a new estimate of the geometric spreading factor ζ(r, ω) and the procedure is repeated until convergence is reached. Finally, an inverse problem is solved to determine the material damping ratio of the soil corresponding to the experimental attenuation curve αr(ω). E Method is used to identify the dispersion and attenuation curves for each example soil profile. The identified curves are compared to the target curves and used to determine the soil profile. The dispersion and attenuation curves corresponding to the resulting soil profile are also computed. The latter curves may differ from the target curves. The difference is due to errors in the determination procedure of the attenuation curve and errors in inversion procedures to identify the shear wave velocity and material damping ratio. First, the regular soil profile is considered (figure a). Figure 9a compares the target and the identified dispersion curve. The difference between these curves can be explained by errors in the phase regression process to identify the phase velocity CR E (ω). This leads to a difference between the target and the identified shear wave velocity profile (figure a). The target and the identified attenuation curve agree well (figure a). At the higher frequencies, better correspondence can be attained by choosing a finer spatial sampling. The identified and the target material damping ratio profile agree reasonably well (figure a). Second, the irregular soil profile is considered (figure b). Figure 9b shows that the identified and the target dispersion curves agree well. The identified shear wave velocity profile (figure b) and the corresponding dispersion curve are also in agreement with their targets. The identified attenuation curve does not match very well with the target curve (figure b). The () () 8

9 Figure 9: Target dispersion curve (gray line), identified dispersion curve (black line), and dispersion curve corresponding to the identified soil profile calculated by method (dashed line) for the regular, the irregular, and the multilayer soil profile. mismatch is due to errors in the amplitude regression procedure to determine the attenuation curve αr E (ω). The response of the soil in the SASW test is assumed to be due to a single mode surface wave. When multiple surface waves contribute to the response due to the inclusion of a softer layer, this assumption is no longer valid and the resulting attenuation curves may be incorrect. The latter is the case for the current irregular soil profile. As is shown in figure 6, contributions of higher modes to the response begin from 6 Hz; the frequency at which the identified attenuation curve starts large deviations from the target curve (figure b). Finally, the multilayer soil profile is considered (figure c). In this case, the soil properties vary gradually with depth. However, in the inversion procedure, the soil is modelled as a horizontally layered medium. In order to mimic real practice, the number of layers is kept as small as possible, but sufficiently large to allow for a good fit of the experimental and the theoretical curves. This approach results in a shear wave velocity profile consisting of layers on a halfspace and a material damping ratio profile of layers on a halfspace. Figure 9c shows that the identified and the target dispersion curves are slightly different. The figure also reveals a minor difference between the target dispersion curve and the dispersion curve corresponding to the identified shear wave velocity profile, which is shown in figure c. The target and the identified attenuation curves are compared in figure c. The large difference between these curves can be explained by errors in the calculation of the geometric spreading factor ζ(r, ω). The function ζ(r, ω) is computed using the shear wave velocity of the soil, which is determined by inversion of the identified dispersion curve CR E (ω). Errors in the identified dispersion curve and errors from the inversion procedure (e.g. due to the non-uniqueness of the problem) may lead to an incorrect estimate of the shear wave velocity profile and geometric spreading factor ζ(r, ω) and, consequently, the identified attenuation curve αr E (ω). Figure shows the geometric spreading factor ζ(r, ω) for the multilayer soil profile (figure c) and its equivalent layer soil profile (figure c). Comparing figures a and b, it can be noticed that the target (figure a) is different from the calculated (figure b) geometric spreading factor, resulting in the different identified attenuation curve (figure c). METHOD : f -k ANALYSIS AND AMPLITUDE REGRESSION In the second method, the experimental transfer function ĤE zz (r, ω) is transformed to the frequency-wavenumber domain []. The surface waves occur as peaks in the resulting f-k spectrum H zz(k E r, ω). The dispersion curve of the dominant Rayleigh wave is determined from the position of the highest peak in the f-k spectrum. Next, an inverse problem is solved to determine the shear wave velocity profile corresponding to the experimental dispersion curve CR E (ω). This profile is used to compute the geometric spreading factor ζ(r, ω). The moduli 9

10 Attenuation coefficient [/m] Attenuation coefficient [/m] Attenuation coefficient [/m] Figure : Target attenuation curve (gray line), identified attenuation curve (black line), and attenuation curve corresponding to the identified soil profile calculated by method (dashed line) for the regular, the irregular, and the multilayer soil profile. Figure : Target (gray line) and identified (dashed line) shear wave velocity profile calculated by method for the regular, the irregular, and the multilayer soil profile. Figure : Geometric damping ζ(r, ω) for the multilayer soil profile and a layer soil profile approximating the multilayer soil profile Figure : Target (gray line) and identified (dashed line) material damping ratio profile calculated by method for the regular, the irregular, and the multilayer soil profile.

11 frequency [Hz] Figure : Target dispersion curve (gray line), identified dispersion curve (black line), and dispersion curve corresponding to the identified soil profile calculated by method (dashed line) for the regular, the irregular, and the multilayer soil profile. Attenuation coefficient [/m] Attenuation coefficient [/m] Attenuation coefficient [/m] Figure : Target attenuation curve (gray line), identified attenuation curve (black line), and attenuation curve corresponding to the identified soil profile calculated by method (dashed line) for the regular, the irregular, and the multilayer soil profile. of the functions ĥe zz(r, ω) and ĤE zz(r, ω) are subsequently fitted in a similar way as in the first method in order to determine the attenuation curve αr E (ω). No iteration is required in this method since the geometric spreading factor ζ(r, ω) is calculated in a direct way and does not need to be updated. Finally, an inverse problem is solved to determine the shear wave velocity and the material damping ratio of the soil corresponding to the experimental dispersion curve CR(ω) E and the attenuation curve αr(ω) E [, ]. Method is applied to the three soil profiles. It is demonstrated in the following that in comparison with the phase regression, the f-k analysis leads to more accurate results for the identified phase velocities. The calculated geometric spreading function is more accurate accordingly. For the regular soil profile, figure a shows a very good correspondence between the target and the identified dispersion curve, revealing the advantage of using the f-k analysis for the identification of the phase velocity. Also, there is a good correspondence between the target and the identified shear wave velocity profile (figure 6a). The target and the identified attenuation curve agree fairly well as shown in figure a. This results in a material damping ratio profile that corresponds well to the target profile (figure 7a). For the irregular soil profile, the identified and the target dispersion curves agree very well as shown in figure b. This results in a correct estimation of the identified shear wave velocity profile (figure 6b), and accordingly, a good estimation of the geometric spreading factor ζ(r, ω). There is still a major difference between the target and identified attenuation curve (figure b). This is due to participation of higher modes in the response, which affects the identified material damping ratio profile (figure 7b). For the multilayer soil profile, the identified and the target dispersion curve agree well (figure c). This results in an estimate of the shear wave velocity profile as shown in figure 6c. This

12 Figure 6: Target (gray line) and identified (dashed line) shear wave velocity profile calculated by method for the regular, the irregular, and the multilayer soil profile Figure 7: Target (gray line) and identified (dashed line) material damping ratio profile calculated by method for the regular, the irregular, and the multilayer soil profile. estimate leads to a different geomtric damping function, resulting in a large difference between the identified and the target attenuation curve (figure c). The identified material damping ratio profile and the target (figure 7c) are therefore quite different. METHOD : f -k ANALYSIS USING THE HALF-POWER BANDWIDTH METHOD An alternative method to determine the dispersion and attenuation curves is presented. This method is based on the peaks in the f-k spectrum H zz(k E r, ω). First, either the peak corresponding to the dominant (the highest peak) or the fundamental (the first peak) Rayleigh wave is identified. The dispersion curve is derived from the peak s position in a similar way as in the second method discussed in the previous section. The attenuation coefficient αr E (ω) is derived from the peak s width, using the half-power bandwidth method. The half-power bandwidth method has originally been developed in the field of structural dynamics to determine the modal damping ratio ξ of a structure from the width of the peaks in the structure s frequency response function. The half-power bandwidth is defined as the frequency bandwidth ω where the magnitude of the frequency response function is / times the peak value []. For a weakly damped single degree of freedom system, the half-power bandwidth equals (figure 8): ω = ξω res () where ω res is the resonance frequency. This procedure can be generalized to multi-degree of freedom systems with widely spaced resonance frequencies. The half power bandwidth method is also applicable to the representation of the response in the frequency-wavenumber domain. At every frequency ω, the half-power bandwidth k(ω), defined as the difference k (ω) k (ω) between the wavenumbers k (ω) and k (ω), is equal to (figure 9): k(ω) = ξ(ω)k res (ω) ()

13 ξ γξ Ĥ E zz(ω) ω ω ω res ω Figure 8: half-power bandwidth method. ξ γξ HE zz (k) k k k res k Figure 9: half-power bandwidth method in the wavenumber domain. with ξ(ω) a frequency dependent damping coefficient and k res (ω) the wavenumber where the response is maximum: k res (ω) = k (ω) + k (ω) () Introducing the linear relation αr(ω) E = ξ(ω)k res (ω) between the damping coefficient ξ(ω) and the attenuation coefficient αr E (ω) into equation (), the following expression is obtained for the attenuation coefficient αr E(ω): αr E (ω) = k (ω) k (ω) () In order to avoid mixing of adjacent peaks, it is suggested to use a smaller scale factor γ than to ensure that the calculated attenuation coefficient α E R (ω) corresponds to the picked peak: α E R (ω) = k (ω) k (ω) γ (6)

14 Attenuation coefficient [/m] Attenuation coefficient [/m] Attenuation coefficient [/m] Figure : Target attenuation curve (gray line), identified attenuation curve (black line), and attenuation curve corresponding to the identified soil profile calculated by method (dashed line) for the regular, the irregular, and the multilayer soil profile Figure : Target (gray line) and identified (dashed line) material damping ratio profile calculated by method for the regular, the irregular, and the multilayer soil profile. A value of γ =. is used in this paper. In this alternative approach, the occurrence of multiple Rayleigh modes does not affect the attenuation curve of either the fundamental or the dominant Rayleigh wave, as all modes occur as separate, non-interfering peaks in the frequency-wavenumber spectrum. Moreover, the identified attenuation curve is derived directly from the experimental data, avoiding the use of a (possibly incorrect) estimate of the soil s shear wave velocity. This method is fast and easy as it does not need iterative refinements of the geometric damping function. Method is applied to soil profiles,, and. The identified dispersion curves as well as the identified shear wave velocity profiles are the same as in the previous section. Hence, only the attenuation curves and their corresponding material damping ratio profiles are discussed in the following. It is demonstrated in figure that the method works when applied to the wavenumber content of the response. In this figure, the target attenuation curves are calculated by the direct stiffness method, while the identified curves have been determined using the halfpower bandwidth method. For the regular soil profile, a perfect match is observed in figure a between the target and the identified attenuation curve. This results in an excellent agreement between the target and the identified material damping ratio profile as shown in figure a. For the irregular soil profile, figure b shows a good match between the target and the identified attenuation curve. The identified material damping ratio profile (figure b) has a slight difference with the target profile, which can be explained by errors in the inversion procedure. For the multilayer soil profile, the identified and the target attenuation curve agree well (figure c), with minor differences at low frequencies. These are related to the maximum receiver offset, which limits the maximum wavelength. Good quality data in the low frequency range demand information of corresponding long wavelengths. The identified material damping ratio profile can be considered as an approximation to the target profile (figure c).

15 Methods and use an amplitude regression scheme to determine the attenuation coefficients. The more the smoothness of the transfer function, the more accurate the identified attenuation coefficient. This can be seen by comparing figures corresponding to the regular soil profile for the transfer function (figure a) and for the resulting identified attenuation curve (figures a and a) with the corresponding figures for the irregular and multilayer soil profile (figures b,c, b,c, b,c). It is also visible from figures b,c, b, and c that the amplitude regression to determine the attenuation curve may lead to inaccurate results when higher modes dominate the soil s response and/or when a multilayer soil profile is modeled as a soil profile with a few soil layers. 6 CONCLUSION The half-power bandwidth method is successfully applied to the frequency-wavenumber content of the soil s response. The width of the peaks in the f-k spectrum are used to determine the experimental attenuation coefficients, corresponding to an experimental phase velocity. The phase velocity is identified using the information of the position of the peaks. This results in a consistent dispersion and attenuation curve. The technique has been successfully validated using synthetic data obtained from the numerical simulation of an SASW experiment. The technique is compared with two state of the art methods; amplitude and phase regression in the frequency-space domain and f-k analysis and amplitude regression. It is demonstrated that all the three methods have a good performance when applied to a regular soil profile. The third method leads to more robust results when applied to irregular and multilayer soil profiles, as it directly identifies the attenuation curve. The results are not affected by the errors in calculating the shear wave velocity profile and the geometric damping. The proposed technique is not limited to shallow soil layers and can be extended to a wide range of surface wave test applications ACKNOWLEDGMENTS The results presented in this paper have been obtained within the frame of the project G.9.6 In situ determination of material damping in the soil at small deformation ratios, funded by the Research Foundation - Flanders and the project OT// A generic methodology for inverse modeling of dynamic problems in civil and environmental engineering, funded by the Research Council of K.U.Leuven. The second author is a postdoctoral fellow of the Research Foundation - Flanders. REFERENCES [] M.O. Al-Hunaidi. Analysis of dispersed multi-mode signals of the SASW method using the multiple filter/crosscorrelation technique. Soil Dynamics and Earthquake Engineering, :, 99. [] A.K. Chopra. Dynamics of structures. Pearson Prentice Hall, Upper Saddle River, NJ, 7. [] V. Cuellar and J. Valerio. Use of the SASW method to evaluate soil improvement techniques. In Proceedings of the th international conference soil mechanics and foundation engineering, pages 6 6, 997.

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