International audienceThe evaluation of the natural frequency of random Vs profiles before analyzing the fundamental Rayleigh-wave dispersion characteristics is proposed in this paper. The inclusion of this parameter optimizes the effectiveness of random inverse searching to estimate Vs profiles. To demonstrate this method, a numerical test was performed using the "experimental" Rayleigh-wave dispersion curve obtained for a fictitious TEST site

Cuadra, C.

Karkee, M.

Tokeshi, K.

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Estimation of Vs profile using its natural frequency andRayleigh-wave dispersion characteristicsK. Tokeshi, M. Karkee, C. CuadraTo cite this version:K. Tokeshi, M. Karkee, C. Cuadra. Estimation of Vs profile using its natural frequencyand Rayleigh-wave dispersion characteristics. Advances in Geosciences, European GeosciencesUnion, 2008, 14, pp.75-77. <hal-00297048>HAL Id: hal-00297048https://hal.archives-ouvertes.fr/hal-00297048Submitted on 2 Jan 2008HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Adv. Geosci., 14, 75–77, 2008www.adv-geosci.net/14/75/2008/© Author(s) 2008. This work is licensedunder a Creative Commons License.Advances inGeosciencesEstimation of Vs profile using its natural frequency andRayleigh-wave dispersion characteristicsK. Tokeshi, M. Karkee, and C. CuadraDepartment of Architecture and Environment Systems, Akita Prefectural University, Yuri-Honjo, 015-0055 Akita, JapanReceived: 30 May 2007 – Revised: 29 August 2007 – Accepted: 3 September 2007 – Published: 2 January 2008Abstract. The evaluation of the natural frequency of randomVs profiles before analyzing the fundamental Rayleigh-wavedispersion characteristics is proposed in this paper. The in-clusion of this parameter optimizes the effectiveness of ran-dom inverse searching to estimate Vs profiles. To demon-strate this method, a numerical test was performed using the“experimental” Rayleigh-wave dispersion curve obtained fora fictitious TEST site.1 IntroductionHighly populated cities in numerous developing countriesare facing a high seismic risk. In many cases, the damagescaused by earthquakes have been the consequence of the in-teraction between ground motion and soil-structure interac-tion; the so-called “site effect”. In that respect one well-known example is the 1985 Mexico City earthquake thatcaused 60 000 casualties. The characteristics of the groundstructure, in particular the Vs profile: shear wave velocityand thickness of each layer, are very important for earthquakedisaster prevention, and the role of geophysical prospectingis preponderant. These geophysical prospecting techniquesshould be economic, fast and reliable.One of the most recent techniques for estimating the soilprofile is using the characteristics of Rayleigh-wave disper-sion. The Rayleigh-wave is a type of surface wave, whichis dispersive due to that the phase velocity of Rayleigh-wavevaries with frequency. Haskell (1953) showed the basis forcalculating the theoretical Rayleigh-wave dispersion on amultilayered media. Nowadays, there are several methodsfor estimating the Rayleigh-wave dispersion characteristicsfrom ambient vibrations. For example, the high resolutionf -k spectral method (Capon, 1969), the SAC method (Toki-Correspondence to: K. Tokeshi(tokeshi@akita-pu.ac.jp)matsu, 1995), and the SASW method (Nazarian and Stokoe,1984), among other methods, are some of the most usedtechniques. Whichever method is used, the obtained disper-sion curve has some variations of phase velocity at each fre-quency, and this fact should be considered during the inverseanalysis process.Concerning to the inverse analysis of Rayleigh-wave dis-persion characteristics, a full systematic search using wideranges of values for shear-wave velocity and thickness woulddemand huge computation times, due to the large total num-ber of combinations that increases with the number of lay-ers in the performed analysis. Nowadays, there are sev-eral methods, e.g., the Neighbourhood Algorithm (NA) andthe Genetic Algorithm (GA) (Sen and Stoffa, 1995). Oneof fastest methods would be the NA method proposed bySambridge (1999), which uses a stochastic direct techniquefor non-linear geophysical inverse problems through multi-dimensional parameter spaces. However the NA code is notavailable for personal computers. The random inverse analy-sis used in this research could be considered as the first gen-eration in the GA to find new and better sets of ranges for thenext generation of searching.2 Experimental Rayleigh-wave dispersion at fictitiousTEST siteTo show the effectiveness of the here presented method, anumerical test was performed on a fictitious TEST site. Thephysical parameters of each layer are shown in Table 1. The“experimental dispersion”, which is actually the theoreticalRayleigh-wave dispersion curve, was calculated using the de-fined physical parameters of Table 1 (Fig. 1). Only 14 pointsfrom a frequency band of 3.4∼7.0 Hz of this experimentaldispersion curve (black points in Fig. 1) were selected forfaster evaluation. The criteria used for minimum and max-imum limits of the frequency band were (1) that the phasePublished by Copernicus Publications on behalf of the European Geosciences Union.76 K. Tokeshi et al.: Estimation of Vs profileTable 1. Physical parameters assumed for TEST site.Layer Density (ton/m3) VP (m/s) Vs (m/s) H (m)1 1.32 330 150 52 2.06 1950 240 103 2.06 1950 380 64 2.19 2490 930 125 2.28 2910 1470 406 2.38 3500 1850 307 2.50 3800 2500 ∞ 㪇㪉㪇㪇㪋㪇㪇㪍㪇㪇㪏㪇㪇㪈㪇㪇㪇㪈㪉㪇㪇㪈㪋㪇㪇㪈㪍㪇㪇㪈㪏㪇㪇㪉㪇㪇㪇㪉 㪊 㪋 㪌 㪍 㪎F㫉㪼㫈㫌㪼㫅㪺㫐 㩿H㫑㪀P㪿㪸㫊㪼 V㪼㫃㫆㪺㫀㫋㫐 㩿㫄㪆㫊㪀 F㫌㫅㪻㪸㫄㪼㫅㫋㪸㫃 㫄㫆㪻㪼 ±㪈㪇% 㫊㫋㪸㫅㪻㪸㫉㪻 㪻㪼㫍㫀㪸㫋㫀㫆㫅2.9 ～3.4 Hz           Fig. 1. Experimental Rayleigh-wave dispersion characteristics usedas target for estimating Vs profile at fictitious TEST site.velocity for frequencies lower than 3.4 Hz showed high un-certainty and (2) that the values of phase velocity for fre-quencies higher than 7.0Hz may be affected by higher modesof Rayleigh-waves. A standard deviation (σ=±10%) of thephase velocity at each frequency (dash lines in Fig. 1) wasconsidered due to the variability of actual Rayleigh-wave dis-persion curve.The random searching for possible Vs profiles was carriedout through multidimensional parametric spaces. Values ofall parameters for all layers were assigned simultaneouslywith random values within their respective ranges. Table 2shows the ranges used for Poisson ratio (ν), S-wave velocity(Vs ), and thickness (H ) in each layer during inverse analy-sis. The shear wave velocity and the thickness have stronginfluence on the characteristics of Rayleigh-wave dispersion.The P-wave velocity (VP ) in each layer was calculated byEq. (1).VP = Vs(1− ν0.5− ν)1/2(1)And the density (ρ) in each layer was determined byEq. (2), which is the empirical relation proposed by Gard-ner et al. (1974).ρ = 0.31V 0.25P (2)where, ρ is in (t/m3) and VP is in (m/s).Table 2. Ranges of parameters used for random inverse analysis.Layer ν Vs (m/s) H (m)1 0.25∼0.49 50∼300 1∼302 100∼500 1∼503 200∼1000 1∼504 200∼2000 1∼1005 200∼3000 1∼1006 200∼4000 1∼100A huge number of trials (1.26×1023 combinations) wouldobtained if the increments of 10 m/s for S-wave velocity and1m for thickness are applied in respective parameter rangesof Table 2. So, the full systematic calculation would take ahuge computer consuming time.As one key for optimizing computation time, the valueof natural frequency of each random Vs profile was calcu-lated before assessing its respective Rayleigh-wave disper-sion. Only the Vs models whose values for the natural fre-quency are within a certain frequency band and lower thana certain frequency where the value of phase velocity be-comes asymptotic were taken into account for calculating oftheoretical fundamental Rayleigh-wave dispersion. The fre-quency range of 2.9∼3.4 Hz was chosen as appropriate fre-quency band for checking the natural frequency of randomVs profiles at TEST site (see Fig. 1). In this sense, Fa¨h etal. (2003) reported that stable parts of the H/V ratio of am-bient vibrations are dominated by the ellipticity of the fun-damental mode of Rayleigh-wave in the frequency band be-tween the natural frequency of unconsolidated or “soft” sed-iments and the first minimum of the average H/V ratio. Fur-thermore, Bonnefoy-Claudet et al. (2006) concluded that thepredominant frequency of the H/V ratio from synthetic am-bient vibrations when sources are on the surface is related tothe fundamental mode of Rayleigh-waves. The authors haveverified this relationship using their own synthetic ambientvibrations. A detailed description of their numerical simula-tion is given in Tokeshi et al. (2000).3 Results from random inverse analysisThe total number of trials used for random inverse analy-sis was 10 millions, which took 1 day to finish calculation.After satisfying the two conditions, the value of fundamen-tal Rayleigh-wave velocity within±10% of the experimentalphase velocity at each frequency and the value of natural fre-quency of Vs profile within a frequency band lower than theasymptotic frequency of Rayleigh-wave dispersion, all but62 solutions were rejected and interpreted as possible solu-tions for Vs profiles. Figure 2 shows the 10 best solutions(gray lines), the former Vs profile (dashed line), and, the av-erage of the 10 solutions (solid line). The average of Vs pro-Adv. Geosci., 14, 75–77, 2008 www.adv-geosci.net/14/75/2008/K. Tokeshi et al.: Estimation of Vs profile 77                 㪇㪈㪇㪉㪇㪊㪇㪋㪇㪌㪇㪍㪇㪎㪇㪏㪇㪐㪇㪈㪇㪇㪈㪈㪇㪈㪉㪇㪇 㪌㪇㪇 㪈㪇㪇㪇 㪈㪌㪇㪇 㪉㪇㪇㪇 㪉㪌㪇㪇 㪊㪇㪇㪇V㫊 㩿㫄㪆㫊㪀D㪼㫇㫋㪿 㩿㫄㪀 㪈㪇 㪹㪼㫊㫋 㫊㫆㫃㫌㫋㫀㫆㫅㫊 F㫆㫉㫄㪼㫉 㫄㫆㪻㪼㫃 A㫍㪼㫉㪸㪾㪼 㫆㪽 㫊㫆㫃㫌㫋㫀㫆㫅㫊Fig. 2. Vs profiles obtained from random inverse analysis of exper-imental Rayleigh-wave dispersion characteristics.file was obtained as the average of corresponding values forthicknesses and shear wave velocities of each layer. It is clearthat the variation of possible solutions for Vs profile increaseswith depth, but the fitting of average with former Vs profileis acceptable.4 ConclusionsFor optimizing random inverse analysis, the previous evalu-ation of the natural frequency of each random Vs profile wasincluded, just before calculation of fundamental Rayleigh-wave dispersion characteristics. One test on the proposedoptimization was carried out on a fictitious TEST site, whosephysical parameters of each layer were defined beforehand.The theoretical or experimental fundamental Rayleigh-wavedispersion curve obtained at this TEST site was used as thetarget during random inverse searching.The conclusions are as follows:– The inclusion of a previous evaluation of the natural fre-quency of each random Vs profile is an effective tool foroptimizing the random inverse searching.– The variation of Vs profiles from possible solutions in-creases with depth, but the average of possible solutionsis acceptable when comparing with the former Vs pro-file model.Acknowledgements. The authors thank the support of the JapanSociety for the Promotion of Science (JSPS): Grants-in-Aid for theScientific Research, Project No. 18560556.Edited by: P. FabianReviewed by: two anonymous refereesReferencesBonnefoy-Claudet, A., Cornou C., Bard, P. Y., Cotton, F., Moczo,P., Kristek, J., and Fah, D.: H/V ratio: a tool for site effectsevaluation. Results from 1-D noise simulations, Geophys. J. Int.,167, 827–837, 2006.Capon, J.: High-resolution frequency-wavenumber spectrum anal-ysis, Proc. IEEE, 57(8), 1408–1418, 1969.Fa¨h, D., Kind, F., and Giardini D.: Inversion of local S-wave ve-locity structures from average H/V ratios, and their use for theestimation of site-effects, J. Seismol., 7, 449–467, 2003.Gardner, G. H. F., Gardner, L. W., and Gregory, A. R.: Formationvelocity and density; the diagnostic basics for stratigraphic traps,Geophysics, 39, 770–780, 1974.Haskell, N. A.: The dispersion of surface waves in multilayeredmedia, BSSA, 43, 17–34, 1953.Nazarian, S. and Stokoe, K. H.: Use of surface waves in pave-ment evaluation, Transportation Research Record 1070, TRB,National Research Council, 132–144, 1984.Sambridge, M.: Geophysical inversion with a neighbourhood al-gorithm I. Searching a parameter space, Geophys. J. Int., 103,4839–4878, 1999.Sen, M. K. and Stoffa, P. L.: Global Optimization Methods in Geo-physical Inversion, Advances in Exploration Geophysics, 4, El-sevier Science Ltd., 1995.Tokeshi, J. C., Sugimura, Y., and Karkee, M. B.: Parametric studyon simualted microtremors and its application to interpretationof microtremor records, J. Struct. Constr. Eng., AIJ, 535, 69–78,2000.Tokimatsu, K.: Geotechnical site characterization using surfacewaves, Proc. of the First Int. Conf. on Earthquake Geotech. Eng.IS-Tokyo’95, Tokyo, 1333–1368, 1995.www.adv-geosci.net/14/75/2008/ Adv. Geosci., 14, 75–77, 2008

Estimation of <i>Vs</i> profile using its natural frequency and Rayleigh-wave dispersion characteristics

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Estimation of <i>Vs</i> profile using its natural frequency and Rayleigh-wave dispersion characteristics

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