Badsar, SA

Degrande, G

Haegeman, Wim

Schevenels, M

English

Ghent University Academic Bibliography

IntroductionThe Spectral Analysis of Surface Waves (SASW) method is widely used to determine the dynamic shearmodulus and the material damping ratio of soils. It is based on an in situ experiment where wavesare generated by means of an impact hammer, a falling weight, or a hydraulic shaker. The resultingwave field is recorded by a number of sensors at the soil’s surface and used to determine dispersion andattenuation curves. An inverse problem is solved to identify the shear wave velocity and the materialdamping ratio profiles: the (theoretical) dispersion and attenuation curves are calculated for a given soilprofile and compared to the (experimental) corresponding curves derived from the surface wave test.The soil profile is subsequently adjusted in order to minimize the distance between the experimental andthe theoretical curves.This paper focuses on the determination of the material damping ratio of the soil by means of the SASWmethod. Two new methods to determine the material damping ratio profile of the soil are presented,based on (1) the half-power bandwidth method and (2) the spatial decay of the Arias intensity. Bothmethods are applied to a test site in Heverlee (Belgium).The paper consists of three parts, addressing (1) the in situ experiment, (2) the determination of dis-persion curve, attenuation curve, and Arias intensity from the measurement data, and (3) the inverseproblem where the soil profile is identified.The in situ testThe SASW experiment is usually performed using a relatively small number of receivers (about 10),located at exponentially increasing distances from the source. However, there is a trend to use a largernumber of receivers (about 50-100) to capture the spatial variation of the wave field in a more accurateway. This is important for the determination of the attenuation curve. The attenuation curve is deter-mined from the spatial decay of the wave field, which is more accurately measured in the case with alarge number of receivers.In the experiment performed in Heverlee, surface waves are generated by means of a hammer impact ona small aluminum foundation. The response has been measured at 90 equidistant locations from 1m to90m from the source.The wave field recorded in the SASW test is contaminated by noise. To mitigate the influence of thenoise, the test is repeated a number of times (i.e. multiple hammer impacts are recorded) and the averagetransfer function HˆEzz(r, ω) from the hammer force (channel i) to the free field response at a distance r(channel j) is computed. To this end, the H1 estimator [2] is used:HˆEzz(rj, ω) =Sˆji(ω)Sˆii(ω)with Sˆij(ω) =1NN∑k=1xˆki (ω)xˆk∗j (ω) (1)where xˆki (ω) is the frequency content of the signal recorded in channel i for impact k, and xˆk∗j (ω) isthe complex conjugate of xˆkj (ω). The index i refers to the force channel, the index j refers to the j-threceiver, located at a distance rj from the source.The dispersion and attenuation curvesIn the past two decades, much attention has been paid to the determination of the dispersion curve fromthe data recorded in the SASW test, resulting in several well established methods. The determination ofthe attenuation curve has been tackled only recently [5, 7]. Existing methods to determine the attenuationcurve are based on the spatial decay of the Rayleigh wave and rely on the knowledge of the geometricdamping. These methods may lead to incorrect results (1) if the geometric damping is calculated basedNear Surface 2010 – 16th European Meeting of Environmental and Engineering Geophysics6 – 8 September 2010, Zurich, Switzerlandon an inaccurate shear wave velocity profile and (2) when higher modes contribute to the wave field inthe soil.(a) (b)0 20 40 60 80 1000100200300Frequency [Hz]Phase velocity [m/s](c)0 20 40 60 80 10000.10.20.30.4frequency [Hz]Attenuation coefficient [1/m]Figure 1 Experimental (a) f–k spectrum H˜(kr, ω), (b) dispersion curve CER(ω) and (c) attenuationcurve αER(ω) for the site in Heverlee.Two alternative methods to determine the material damping ratio have therefore been developed. Thefirst new method is based on the half-power bandwidth method and proceeds as follows. First, thefrequency-wavenumber content H˜Ezz(kr, ω) of the transfer function HˆEzz(rj , ω) is computed by meansof a discrete approximation to the Hankel transformation [3]. Figure 1(a) shows the f–k spectrumH˜Ezz(kr, ω) for the site in Heverlee. This spectrum has been normalized for each individual frequency. Itexhibits peaks corresponding to the Rayleigh modes. The peak corresponding to the dominant Rayleighwave is identified and the dispersion curve CER(ω) is derived from the peak’s position. The resultingdispersion curve is shown in figure 1(b).Next, the attenuation curve αER(ω) is derived from the peak’s width, using the half-power bandwidthmethod. The half-power bandwidth method has originally been developed in the field of structural dy-namics to determine the modal damping ratio of a structure from the width of the peaks in the structure’sfrequency response function. It is verified numerically that the half-power bandwidth method is alsoapplicable to the wavenumber content of the soil’s response, resulting in the attenuation coefficient ofthe surface waves. The attenuation curve αER(ω) for the site in Heverlee is shown in figure 1(c). In thenext section, the curve αER(ω) is inverted to determine the material damping ratio profile of the soil.In this alternative approach, the experimental attenuation curve is derived directly from the experimentaldata, avoiding the use of a (possibly incorrect) estimate of the soil’s shear wave velocity. Moreover, theoccurrence of multiple Rayleigh modes does not affect the attenuation curve of either the fundamental orthe dominant Rayleigh wave, as long as modes occur as separate, non-interfering peaks in the frequency-wavenumber spectrum. When modes interfere, it is not always possible to identify the mode numberand estimate the degree of their participation, however.The second new method for the determination of the material damping ratio does not involve the deter-mination of the attenuation curve; it is based on the decay of the Arias intensity at the soil’s surface. Inthis method, an experimental intensity curve IEzz(rj), which gives the spatial decay of the Arias inten-sity, is calculated. Figure 2 shows the Arias intensity IEzz(rj) at different receivers at the surface at thesite Heverlee. In the next section, the curve IEzz(rj) is inverted to determine the material damping ratioprofile of the soil.In this method, there is no need to identify mode numbers. The occurrence and interference of higherRayleigh modes does not affect the resulting material damping ratio, as the contribution of all modes isproperly taken into account in the computation of the theoretical Arias intensity ITzz. Compared to thefirst new method, this approach involves a slightly higher computational effort.Near Surface 2010 – 16th European Meeting of Environmental and Engineering Geophysics6 – 8 September 2010, Zurich, Switzerland0 20 40 60 80 900102030Distance [m]Arias intensity [m/s]Figure 2 The experimental Arias intensity curve IEzz(rj) calculated at different offsets at the site inHeverlee.The inverse problemThe soil profile is finally determined from the experimental dispersion, attenuation, and intensity curvesthrough the solution of an inverse problem. The direct stiffness method [4, 10] is used to calculatethe theoretical dispersion, attenuation, and intensity curves of a soil with a given stiffness and damp-ing profile. The profile is iteratively adjusted in order to minimize a misfit function that measures thedistance between the theoretical and the experimental dispersion, attenuation, and intensity curves. Theminimization problem is solved with a gradient based local optimization method.The results of the inversion procedure are presented in figure 3. Figure 3(a) shows the shear wave ve-locity profile determined from the inversion of the dispersion curve. This profile is used in the inversionof both the attenuation and the intensity curve. Figure 3(b) compares the material damping ratio pro-files determined using the two proposed methods. These damping profiles do not agree well. A directassessment of the accuracy of these soil profiles is impossible due to the lack of a reference soil profilefor the test site in Heverlee. A comparison is therefore performed where the theoretical transfer func-tion HˆTzz(rj , ω) is computed for both soil profiles and compared to the experimental transfer functionHˆEzz(rj , ω). The results are shown in figure 4 for three different offsets from the source.(a)0 100 200 300 40005101520Shear wave velocity [m/s]Depth [m](b)0 0.02 0.04 0.06 0.08 0.105101520Material damping ratio [−]Depth [m]Figure 3 (a) The shear wave velocity profile as determined by the SASW method and (b) the materialdamping ratio profile as determined by the half-power bandwidth (red line) and the intensity curve (blueline) methods for the site in Heverlee.Figure 4(a) compares the experimental and the theoretical displacement transfer functions at 2 m fromthe source. It is observed that both proposed methods lead to a good correspondence with the experimen-tal results at small distances from the source, where the influence of the material damping on the responseis small. At larger distances, the second method leads to a theoretical transfer function HˆTzz(rj, ω) thatagrees very well with the experimental data, as shown in figures 4(b) and 4(c). The theoretical transferfunction HˆTzz(rj , ω) obtained with the first method is slightly worse but still acceptable. While bothmaterial damping ratio profiles are completely different, the resulting transfer functions HˆTzz(rj , ω) arequite similar. This is explained by the non-uniqueness in the inverse problem of the SASW method: alarge variety of soil profiles can be found that correspond well to the experimental data. Schevenels etal. [8, 9] performed a detailed study of this non-uniqueness.Near Surface 2010 – 16th European Meeting of Environmental and Engineering Geophysics6 – 8 September 2010, Zurich, Switzerland(a) 0 20 40 60 80 10010−410−2100Frequency [Hz]Normalized transfer function [−]2 m(b) 0 20 40 60 80 10010−410−2100Frequency [Hz]Normalized transfer function [−]20 m(c) 0 20 40 60 80 10010−410−2100Frequency [Hz]Normalized transfer function [−]80 mFigure 4 Comparison between the experimental (black line) results and the SASW with the half-powerbandwidth method (red line) and SASW with the intensity curve (blue line) method at offsets (a) 2 m (b)20 m and (c) 80 m for the site in Heverlee.ConclusionThis paper focuses on the determination of the material damping in the soil by means of the SASWtest. Two new methods are proposed. The first method is based on the half-power bandwidth methodapplied to the width of the peaks in the f–k spectrum to determine the attenuation of Rayleigh waves.The second method uses the spatial decay of the Arias intensity at the surface to determine the materialdamping ratio. Both methods are used to determine the material damping ratio in the soil at a site inBelgium. The identified soil profiles are used to simulate the wave field registered in the SASW test,which is confronted with the experimental data. Good results are obtained with both methods, the secondmethod performing slightly better than the first.References[1] G. Degrande, S.A. Badsar, M. Schevenels, G. Lombaert, and A. Teughels. Application of the Coupled LocalMinimizers method to the optimization problem in the Spectral Analysis of Surface Waves method. Journalof Geotechnical and Geoenvironmental Engineering, Proceedings of the ASCE, 134(10):1541–1553, 2008.[2] D.J. Ewins. Modal testing: theory and practice. Research Studies Press Ltd., Letchworth, UK, 1984.[3] T. Forbriger. Inversion of shallow-seismic wavefields: I. Wavefield transformation. Geophysical JournalInternational, 153(3):719–734, 2003.[4] E. Kausel and J.M. Roe¨sset. Stiffness matrices for layered soils. Bulletin of the Seismological Society ofAmerica, 71(6):1743–1761, 1981.[5] C.G. Lai. Simultaneous inversion of Rayleigh phase velocity and attenuation for near-surface site character-ization. PhD thesis, Georgia Institute of Technology, 1998.[6] S. Nazarian and M.R. Desai. Automated surface wave method: field testing. Journal of Geotechnical Engi-neering, Proceedings of the ASCE, 119(7):1094–1111, 1993.[7] G.J. Rix, C.G. Lai, and A.W. Spang Jr. In situ measurement of damping ratio using surface waves. Journal ofGeotechnical and Geoenvironmental Engineering, Proceedings of the ASCE, 126(5):472–480, 2000.[8] M. Schevenels, G. Lombaert, G. Degrande, and S. Franc¸ois. A probabilistic assessment of resolution inthe SASW test and its impact on the prediction of ground vibrations. Geophysical Journal International,172(1):262–275, 2008.[9] M. Schevenels, S.A. Badsar, G. Lombaert, and G. Degrande. The robustness of ground vibration predictionsbased on SASW tests. Proceedings of Leuven Symposium on Applied Mechanics in Engineering, Leuven,Belgium, CD-ROM, 2008.[10] M. Schevenels, S. Franc¸ois, and G. Degrande. EDT: An ElastoDynamics Toolbox for MATLAB. Computers& Geosciences, 35(8):1752–1754, 2009.Near Surface 2010 – 16th European Meeting of Environmental and Engineering Geophysics6 – 8 September 2010, Zurich, Switzerland

Determination of the damping ratio in the soil from SASW test using the half-power bandwidth method and Arias intensity

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Determination of the damping ratio in the soil from SASW test using the half-power bandwidth method and Arias intensity

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