DSpace at VNU: On the relationship of peaks and troughs of the ellipticity (H V) of Rayleigh waves and the transmission response of single layer over half-space models

Geophysical Journal International Geophys J Int (2011) 184, 793–800 doi: 10.1111/j.1365-246X.2010.04863.x On the relationship of peaks and troughs of the ellipticity (H/V) of Rayleigh waves and the transmission response of single layer over half-space models Tran Thanh Tuan,1 Frank Scherbaum2 and Peter G Malischewsky3 Hanoi University of Science, VNU, Vietnam E-mail: tuantt@vnu.edu.vn Potsdam, Germany E-mail: fs@geo.uni-potsdam.de Friedrich-Schiller University Jena, Germany E-mail: p.mali@uni-jena.de University SUMMARY One of the key challenges in the context of local site effect studies is the determination of frequencies where the shakeability of the ground is enhanced In this context, the H/V technique has become increasingly popular and peak frequencies of H/V spectral ratio are sometimes interpreted as resonance frequencies of the transmission response In the present study, assuming that Rayleigh surface wave is dominant in H/V spectral ratio, we analyse theoretically under which conditions this may be justified and when not We focus on ‘layer over half-space’ models which, although seemingly simple, capture many aspects of local site effects in real sedimentary structures Our starting point is the ellipticity of Rayleigh waves We use the exact formula of the H/V -ratio presented by Malischewsky & Scherbaum (2004) to investigate the main characteristics of peak and trough frequencies We present a simple formula illustrating if and where H/V -ratio curves have sharp peaks in dependence of model parameters In addition, we have constructed a map, which demonstrates the relation between the H/V -peak frequency and the peak frequency of the transmission response in the domain of the layer’s Poisson ratio and the impedance contrast Finally, we have derived maps showing the relationship between the H/V -peak and trough frequency and key parameters of the model such as impedance contrast These maps are seen as diagnostic tools, which can help to guide the interpretation of H/V spectral ratio diagrams in the context of site effect studies Key words: Site effects; Theoretical seismology; Wave propagation I N T RO D U C T I O N The analysis of ambient vibrations has become an increasingly popular tool for the estimation of local site effects and the characterization of shallow site structure This can be seen for example in several major recent research initiatives either being completely devoted to ambient vibrations such as SESAME (http://sesame-fp5.obs.ujfgrenoble.fr/index.htm) or having one or more subprojects dealing with ambient vibration related issues, such as HADU and NERIES (http://www.geotechnologien.de/forschung/forsch2.11k.html; http://www.neries-eu.org/) and the new monograph by Mucciarelli et al (2009) just to name a few In particular, the H/V spectral ratio technique, originally introduced by Nogoshi & Igarashi (1971), also known as Nakamura’s method Nakamura (1989, 2000, 2009), has become the primary tool of choice in many of the ambient vibration related studies Considering that the most dominant contributions to ambient vibrations are known to come from surface waves, although the exact composition may change depending on the particular site (cf the publications from the SESAME project referenced above), this means that it is the characteristics of the ellipticity of Rayleigh waves which is actually analysed However, the fundamenC 2010 The Authors Geophysical Journal International tals of the H/ V-technique are controversial [the history and different opinions are discussed e.g by Bonnefoy-Claudet et al (2006) and Petrosino (2006)] These different opinions even refer to the term H/V -technique itself In this paper Rayleigh-wave ellipticity is considered an essential part of H/V -technique, without excluding the important analysis of body waves Recently, Albarello & Lunedei (2009) found out that the body wave interpretation provides better results around the resonance frequency but not for higher frequencies We aware of the fact that the trough of the H/V -curve may be masked by higher modes, Love and body waves However, our intention in this paper is to theoretically analyse certain new relationships between parameters of interest by using fundamental mode Rayleigh waves alone While the amount of applications of ambient vibrations analysis in recent years is impressive, on the theoretical side numerous challenges remain What for example is the relationship between the H/V peak frequency and the peak frequency of the transmission response of a medium where the shakeability of the site would be expected to be enhanced? Under what conditions is it allowed to assume their approximate equivalence? This question is important, especially when the H/V -ratio is obtained from noise recordings only These questions turn out to be surprisingly 793 C 2010 RAS GJI Seismology Accepted 2010 October 21 Received 2010 July 22; in original form 2009 December 794 T T Tuan, F Scherbaum and P G Malischewsky challenging theoretically even for very simple model and they have only rarely been addressed in the literature (e.g Malischewsky & Scherbaum 2004; Malischewsky et al 2008 and Haghshenas et al 2008) In the present paper we are focusing on some of these issues, namely the properties of peaks and troughs of the ellipticity of Rayleigh waves and their relationships to the transmission response We derive a set of relationships, which might be used in practical applications to guide the interpretation of H/V spectral ratios, however it is not yet applied to its full potential in the framework of this paper with γ = − 2ν1 (1 − ν1 ) (2) For model ‘layer over half-space’, the H/V -ratio has a singularity if and only if rs < F(ν1 , ν2 , rd ) (3) and it has a zero-point if and only if rs < G(ν1 , ν2 , rd ) (4) The function F is given by F(ν1 , ν2 , rd ) = A(ν2 , rd ) arctan [B(ν2 , rd )(ν1 − 0.2026)] and the auxiliary functions A and B are defined by C H A R A C T E R O F H / V - R AT I O O F R AY L E I G H WAV E S A(ν2 , rd ) = 0.297 + 0.061rd − 0.058rd2 + 0.170ν2 − 0.589rd ν2 Malischewsky & Scherbaum (2004) presented an analytically exact formula of H/V for a 2-layer model of compressible media Later, Malischewsky et al (2008) used this formula together with the secular equation to investigate the region of prograde Rayleigh particle motion depending on material parameters These studies form the basis of the present investigation of two special features of the H/V -ratio: the singularity (or maximum) and the zero (or minimum) point An older paper by Suzuki (1933) analyses the surface amplitudes of Rayleigh waves in a stratified medium as well However, the formulas are much more difficult than the ones from the papers cited above and they are valid for Poissonian media only Earlier studies of the singularity and the zero point [see e.g SESAME H/V User Guidelines .] concluded that the singularity occurs at a frequency which is close (i.e less than per cent difference) to the fundamental resonance frequency for S-waves only if the S-wave impedance contrast exceeds a value of (BonnefoyClaudet et al 2008) For low contrast, the ellipticity ratio only exhibits maxima and minima at certain frequencies and no zeros or singularities In this case, the maxima occur at frequencies that may range between 0.5 to 1.5 times the S-wave fundamental resonance frequency It is also possible that H/V-curve of the fundamental mode may exhibit a peak at the frequency f p and a trough at a higher frequency f z Konno & Ohmachi (1998) reported a value of f z /f p equalling two for a limited set of velocity profiles Stephenson (2003) concludes that peak/trough structures with a frequency ratio around two witness both a high Poisson’s ratio in the surface soil and a high impedance contrast to the substrate One question addressed here is under which conditions the H/V ratio derived from the Rayleigh-wave ellipticity exhibits a singularity and a zero point, respectively, or if it only exhibits a maximum and a minimum point Let us denote the shear-wave velocities of the layer and the half-space by β1 and β2 , respectively and the corresponding densities of mass by ρ1 and ρ2 The shear-wave ratio β1 /β2 is denoted by rs and the density ratio ρ1 /ρ2 by rd One can show numerically that the character of H/V is relatively stable concerning changes of Poisson’s ratio ν2 of the half-space and of the densities of mass ρ1 and ρ2 However, it changes dramatically with Poisson’s ratio ν1 of the layer and the impedance contrast Malischewsky et al (2008) prove for the simple model ‘layer with fixed bottom’, which is a special case of the model ‘layer over halfspace’ when the impedance contrast is infinitive large or rs = 0, that ν1 = 0.2026 is the lower limit for the existence of a singularity in H/V and ν1 = 0.25 is the lower limit for existence of a zero point The value 0.2026 is the solution of equation √ π √ γ − γ sin =0 (5) (1) + 0.373rd2 ν2 − 0.284ν22 + 0.817rd ν22 − 0.551rd2 ν22 , B(ν2 , rd ) = 29.708 − 42.447rd + 23.852rd2 − 14.309ν2 + 75.204rd ν2 − 59.881rd2 ν2 + 121.370ν22 − 246.328rd ν22 + 170.027rd2 ν22 (6) The formulas (5)–(6) are the result of numerical calculations and can be applied with good accuracy (the error is often less than 1–2 per cent) for the intervals < ν2 < 0.5 and 0.3 < rd < 0.9, which cover the practically important cases For each pair of values (ν2 , rd ), the equation rs = F(ν1 , ν2 , rd ) represents a curve in the domain (ν1 , rs ) on which the H/V -ratio curve changes its property from having maximum points only to having singularities We are not able to present a similar formula for the function G It has to be determined numerically for each pair of values (ν2 , rd ) separately by determining the critical value of rs , for each value of ν1 , at which the H/V -ratio curve changes its property from having a zero point to having only a minimum point A careful numerical analysis of function F shows that the leading parameter is ν1 while there is almost no dependence on ν2 and only a weak dependence on rd : the maximum difference of F(ν1 ) on ν2 is only about 0.55 per cent and on rd about 3.3 per cent in the whole range of rd from 0.3 to 0.9 and of ν2 from to 0.5 By fixing ν2 and rd (ν2 = 0.3449, rd = 0.7391) we can use F(ν1 ) and G(ν1 ) to divide the region (ν1 , rs ) into four parts R1 , R2 , R3 , R4 with different character of H/V (see Fig 1) The blue curve AOB is the graph of the function rs = F(ν1 ) and the green curve COD, which is plotted numerically in this case, is the graph of the function rs = G(ν1 ) To the right of the segments COB (R1 ) we observe one peak and one zero point, which is the most interesting case The region R3 left from the segments AOD belongs to the case of one maximum and one minimum The other regions R2 and R4 are smaller and less important: two peaks occur within the segments AOC and two zeros within BOD The special point O is common to all four regions, but the investigation of its meaning is beyond the scope of this paper Because the relationships between the H/V peak and the trough (or the maximum and the minimum, respectively) and the peak frequency of the transmission response of the medium are essential in applying the H/V -method, we map them as contour lines in the key parameters (ν1 , rs )-plane in Figs 2(a) and (b), respectively We have constructed contour lines for f p /fr es in Fig 2(a), where f p is the position of the first peak (or maximum) and fr es is the S-wave resonance frequency of the medium and contour lines for f z /f p in Fig 2(b), where f z is the position of the zero (or minimum) These maps show all possible values of these two ratios in the key parameters domain for the fundamental mode H/V -curve of C 2010 The Authors, GJI, 184, 793–800 Geophysical Journal International C 2010 RAS Relationship of peaks and troughs of H/V-ratio 795 Figure Illustration of the four regions R1 , R2 , R3 , R4 of H/V with different character in dependence on ν1 and rs The four figures in the lower panel correspond to the four types of H/V -ratio curves as marked by the four stars in the upper panel The x-axis ¯f is non-dimensional frequency and is the ratio of thickness of the layer to wavelength of the S-wave in the layer Rayleigh surface waves In the region R4 , in which two zero points exist, the second one is represented The values for ν2 and rs are the same as in Fig We have already proven (Malischewsky et al 2008) that f p /fr es becomes for rs = (‘layer with fixed bottom’), that is, when the shear-wave contrast is very high, the frequency of the first peak is the S-wave resonance frequency of the layer The blue line in Fig 2(a) is the graph of the function rs = F(ν1 ) which becomes with our ν2 and rd rs = 0.291 arctan [18.147 (ν1 − 0.2026)] (7) It separates regions of H/V with at least one singularity (on the right—regions R1 and R2 ) from regions with a maximum (on the left—regions R3 and R4 ) The continuous red region is for the domain with at least one singularity of H/V while the dotted brown region corresponds to a maximum For the region R2 , with two singularities, the first one is plotted The value f p /fr es is close to with per cent deviation for high values of ν1 and of the shear-wave contrast This agrees well with other observations In the left (dotted brown) region, where maxima and minima of H/V occur, this value of f p /fr es is also observed We also note that the frequency of the peak adopts its maximum on the blue curve rs = F(ν1 ) This is in close connection with the remarkable curve in fig of Malischewsky & Scherbaum (2004), which presents the frequency C 2010 The Authors, GJI, 184, 793–800 Geophysical Journal International C 2010 RAS of the peak value of H/V in dependence on 1/rs By using (7) with the value ν1 = 0.4375 from this paper, we obtain for the maximum of the frequency of the peak value β2 /β1 = 2.5637, which is very close to 2.6 presented by Malischewsky & Scherbaum (2004) So this strange behaviour in Fig finds a simple explanation The ratio of the position of the zero or trough to the position of the peak or maximum (contour lines of f z /f p ) in dependence on ν1 and rs is presented in Fig 2(b) The green line rs = G(ν1 ) separates the region of zero(s) right (solid red contour lines-regions R1 and R4 ) from the region of trough(s) (minima) left (dotted brown contour lines-regions R2 and R3 ) For region R4 with two zero points, the second one is plotted The function √ rs = G(ν1 ) starts at ν1 = 0.25 for rs = with f z /f p = ≈ 1.73 which is in conformity with the considerations of Malischewsky et al (2008) The value of this ratio is observed in the down-left corner of the figure with high value of Poisson’s ratio and impedance contrast This is consistent with conclusion of Stephenson (2003) However, this value of ratio is also observed in the region with low Poisson’s ratio value where the H/V -ratio curve has only maximum and minimum points and it is almost unchanged with the impedance contrast Since the minimum points are generally not well distinct, they have not a great practical significance 796 T T Tuan, F Scherbaum and P G Malischewsky Figure (a) Contours of f p /fr es as a function of ν1 and rs (b) Contours of f z /f p as a function of ν1 and rs In regions with red continuous lines, the H/V -ratio curve has at least one singularity (Fig.2a) or at least one zero point (Fig 2b) In regions with brown dotted lines, it has only a maximum point (Fig 2a) and a minimum point (Fig 2b) In this figure, ν2 = 0.3449, rd = 0.7391 POTENTIAL PRACTICAL A P P L I C AT I O N S H/V -measurements yield positions of peaks and troughs, i.e f p and f z Let us assume that for a certain region the less important parameters ν2 and rd are known (ν2 = 0.3449, rd = 0.7391 in our case) Since we get only two data points from the H/V -ratio curve, we are able to derive two unknown parameters maximum The first example of the potential application is to get the resonance frequency of the medium from the map in Fig 2(a) if we know the two key parameters (Poisson’s ratio of the layer ν1 and the impedance contrast rs ) In this case, the thickness of the layer does not have an effect We can get the resonance frequency without knowing it Let us demonstrate this application by using a synthetic data set for a model with two layers over half-space of Lieg´e, Belgium (Wathelet 2005; see Table 1) and taking into account that this more complex model can be replaced with reasonable accuracy by the simple model one layer over half-space Note that ν2 and rd used for constructing Fig are different from the values of Table But usually these values are not known in advance and we have avoided C 2010 The Authors, GJI, 184, 793–800 Geophysical Journal International C 2010 RAS Relationship of peaks and troughs of H/V-ratio Table Parameters for the model two layers over half-space used for the synthetic data set Thickness (m) P-wave (m/s) S-wave (m/s) Density (g cm–3 ) 1st layer 2nd layer Half-space 7.8 20 – 310 1112 2961 193 694 2086 2 a recalculation of Fig The error in doing this is very small (see Section 2) The use of synthetic H/V -ratio data is quite common Făah et al (2003) used them to constrain the velocity solutions from the H/V ratio inversion and they found a good agreement with the observed data especially in the frequency range between the peak and the first trough in the H/V -ratio curve Fig shows how H/V -ratios’ dependence on frequency extracted from our synthetic data set The computation has been performed with GEOPSY (www.geopsy.org) and follows the SESAME H/V User Guidelines (2005) It is based on modal summation with all modes of Rayleigh and Love waves (i.e in practice 20 modes) by using Bob Hermann’s code The source distribution is spatially homogeneous and located on the surface (i.e no deep sources) The source orientations are randomly distributed The window length is 60 s with 50 per cent overlap, i.e 20 windows are used for the whole time interval of about 10 The frequencies of the peak and trough of H/V -curve observed in Fig are f p ≈ 5.1 Hz and f z ≈ 9.8 Hz To apply the method, we make the approximation that these peak and trough frequencies are those of Rayleigh waves This model is ‘two layers over half-space’ but we will treat it as a simple model ‘layer over half-space’ whose the new parameters of the layer are the average of those from the original model, as it is necessary in applying our methods As a starting point, we assume that we not know anything about this model except the synthetic H/V -ratio curve We now calculate the average parameters to be used in the method First, the new thickness is d = d1 + d2 = 27.8 m and the new shear velocity is β1 d1 + β2 d2 β¯ = = 553.43 m/s d (8) Here we have denoted the thicknesses of the two layers by d1 and d2 and the shear-wave velocities by β1 and β2 , respectively Note that the average shear-wave velocity can be alternatively taken by 797 conserving the traveltime in two layers as β¯ = d = 387.9 m/s d1 /β1 + d2 /β2 (9) but our investigation for approximating the model ‘two layers over half-space’ by the simple model ‘layer over half-space’ shows that the first averaging method gives more accurate results in peak and trough frequencies The new Poisson’s ratio is calculated to be 0.4579 in the same manner We now assume that we know nothing about this model except the Poisson’s ratio ν¯ = 0.4579 and the impedance ratio r¯s [= 553.43/2086] = 0.2653 From the map in Fig 2(a), by projecting, we can derive the ratio f p /fr es to be about 1.035 And from the H/V -ratio curve in Fig 3, we observe a clear peak at a frequency about f p ≈ 5.1 Hz in the H/V -ratio curve This implies that fr es = 4.9275 Hz Since the resonance frequency is calculated when the ratio of the thickness of the layer to the wavelength of the shear wave in the layer is one fourth (i.e λd¯ = d βf¯r es = 14 ), we can infer either the β1 thickness or the shear wave velocity if the other is known already For instance, if we assume d = 27.8 m is also known in advance, we can calculate the shear wave velocity as β¯ = 4d fr es = 547.94 m/s which is very close to the calculated average value of the model It should be mentioned that the theoretically obtained possible deviation of the H/V -peak from the S-wave resonance (up to 40 per cent, see Section 2) still has to be demonstrated with simulated H/V curves for simple models similar to the ones in Fig However, this is beyond the scope of the present paper A second potential application is to get the key parameters (Poisson’s ratio of the layer and the impedance contrast) from the other parameters and the peak and trough frequency in the H/V -ratio curve We assume that somehow we have the resonance frequency of the layer (in this example, it is calculated from the total thickness β¯ ] = 4.9769 Hz The and the average shear velocity) as fr es [= 4d other parameters are assumed to be unknown We can then map the peak frequency ratio f p /fr es as a function of rs and f z / f p (Fig 4a) and of ν1 and f z / f p (Fig 4b), respectively From these maps one can infer the average Poisson’s ratio of the layers ν1 and the average shear-wave contrast rs of any several-layer model from the H/V -measurements at one single station when it is treated as a simple model ‘layer over half-space’ These two maps Figure Synthetic H/V -curves versus frequency Each coloured curve corresponds to the result from one individual time window The black solid line is the average H/V (geometric mean of individual H/V curves) Dashed lines correspond to confidence intervals (±1 sigma for log normally distributed amplitude ratios; courtesy of M Ohrnberger) C 2010 The Authors, GJI, 184, 793–800 Geophysical Journal International C 2010 RAS 798 T T Tuan, F Scherbaum and P G Malischewsky Figure (a) Contours of f p /fr es as a function of rs and f z / f p in region R1 (b) Contours of f p /fr es as a function of ν1 and f z / f p in region R1 C 2010 The Authors, GJI, 184, 793–800 Geophysical Journal International C 2010 RAS Relationship of peaks and troughs of H/V-ratio refer to the region R1 of Fig 1(a) We omit here maps for the regions R2 , R3 and R4 as they are less important For example, if we know from the data that the peak occurs at the S-wave resonance frequency and the ratio between the trough and peak frequency is about 2, we obtain from Figs 4(a) and (b) that rs = 0.25 and ν1 = 0.46, respectively In our example, the ratio of trough to peak frequency and the ratio of the peak frequency to the resonance frequency of the layer can be obtained from the H/V -ratio curve as 9.8 Hz fz = 1.9216 and = fp 5.1 Hz fp 5.1 Hz = 1.0247 (10) = fr es 4.9769 Hz By using the contour line of f p / fr es = 1.0247 in Fig 4(a) together with the value of f z / f p = 1.9216 we obtain the shear-wave contrast of our simple model as 0.269; similarly, from Fig 4(b) we obtain Poisson’s ratio in the layer as 0.485 From the parameters of the actual model we have the average of the shear-wave contrast and of Poisson’s ratio as 0.2653 and 0.4579, respectively These values are close to the results obtained from the maps Figs 4(a) and (b) with a relative error of per cent for the shear-wave contrast and of 1.3 per cent for Poisson’s ratio One thing that should be noted here is the relative error of the results with the uncertainty of the data (frequencies of peak and trough) While the real data provides the reliable indication of the fundamental resonance frequency of the soils (see BonnefoyClaudet et al (2008)), the trough frequency is not reliable because the experimental H/V -ratio is related not only to the ellipticity of the fundamental mode of Rayleigh waves, but also to higher modes of Rayleigh waves, Love waves and body waves From the maps in Fig 4, we observe that the contour lines can be considered parallel in most cases with the inclined angle of α1 in the first map and of α2 in the second map where tan α1 ≈ 1.37 and tan α2 ≈ The relative errors of the results can be calculated as δrs = fz δ fz f p rs tan α1 and δν1 = fz δ fz f p ν1 tan α2 (11) where δrs = rrs s , δν1 = νν11 and δ f z = f zf z In this example, the relative errors of results caused by the uncertainty of data are δrs = 5.21 δ f z and δν1 = 3.96 δ f z , which are relatively big The third potential application of the method is for the case when only the shear wave velocity of the basin and the thickness of the layer are already known We will determine the shear wave and the Poisson’s ratio of the layer Since the resonance frequency is not known in this case, we have to construct another map to apply the method Fig is a modified map of the first map in Fig This map consists of contour lines of the peak frequency in the domain of the ratio f z / f p and β1 The procedure of the method in this case is as follows: first, from the new map we can infer the value of β1 to be 554 m/s by projecting the first peak as 5.1 Hz and the ratio of trough to peak frequency as 1.9216 This value of β1 is almost consistent with the average value of the model To get the value of ν1 we calculate the resonance frequency from the derived β1 and apply the second maps in Fig The relative error of the resulting β1 in this case due to the uncertainty of the trough frequency is calculated in the same manner as in example two and is δβ1 = 0.49 δ f z Finally, it should be noted that the method applied above is simple and easy to use but it is based on the relationship between peak/trough frequencies of the fundamental mode of the ellipticity of Rayleigh surface waves and the parameters for the simple model ‘layer over half-space’ For models of several layers over half-space, the average values of parameters in layers have to be used Hence, the method achieves accurate results only for models without big jumps in parameters, especially in S-wave velocity The method also requires at least the thickness or the S-wave velocity of the layer to be known a priori For some sites where both of them are not available, for example, when the sedimentary thickness cover is large, the method cannot be applied C O N C LU S I O N S The H/V -method has become increasingly popular over the last few decades as a convenient, practical and low cost tool used to Figure Contours of f p as a function of β1 and f z / f p in region R1 with β2 = 2086 m/s C 2010 The Authors, GJI, 184, 793–800 Geophysical Journal International C 2010 RAS 799 800 T T Tuan, F Scherbaum and P G Malischewsky determine subsurface site characteristics in urbanized areas.The physics behind the H/V -peak are not yet completely understood However, Bonnefoy-Claudet et al (2008) have shown on simulated microtremors that the H/V peak frequency astonishingly provides the resonance frequency of the site regardless of the underlying physics of the H/V peak is (Rayleigh waves for high impedance contrast, S-wave resonance and/or Love waves for moderate to low contrast) We studied Rayleigh waves and have presented a map, which shows the relationship between the H/V -peak frequency of the Rayleigh fundamental mode and the S-wave resonant frequency of the layer with dependence on other model parameters In addition, a general formula (function F) is presented showing under which conditions the H/V -ratio curve has a sharp peak or only a broad maximum It turns out that when Poisson’s ratio of the layer and the impedance contrast, which are the key parameters of the site, are located on or in the vicinity of the graph of F, the difference between the H/V -peak frequency and the S-wave resonance frequency of the layer is often very high (up to 40 per cent) Finally, we have presented some maps indicating values of the model parameters (e.g Poisson’s ratio of the layer and the impedance contrast) by using information about the H/V -peak and trough frequency These maps are meant as tools to aid in the interpretation of site characteristics from H/V measurements AC K N OW L E D G M E N T S We kindly acknowledge the support of Matthias Ohrnberger from the University of Potsdam in providing us with the synthetic data set and its processing This work was supported by the Deutsche Forschungsgemeinschaft (DFG) under Grant No MA 1520/6-2 Further, P.G.M gratefully acknowledges the support of Bundesministerium făur Bildung und Forschung (BMBF) in the framework of the joint project ‘WTZ Germany-Israel: System Earth’ under Grant No 03F0448A This work was also supported by Vietnam’s National Foundation for Science and Technology Development (NAFOSTED) in a project under Grant No 107.02-2010.07 REFERENCES Albarello, D & Lunedei, E., 2009 Alternative interpretations of horizontal to vertical spectral ratios of ambient vibrations: new insights from theoretical modelling, Bull Earthq Eng., 8, 519–534, doi:10.1007/s10518009-9110-0 Bonnefoy-Claudet, S., Cotton, F & Bard, P.-Y., 2006 The nature of noise wavefield and its applications for site effects studies A literature review Earth-Sci Rev., 79, 205–227 Bonnefoy-Claudet, S., Kăohler, A., Cornou, C., Wathelet, M & Bard P.-Y., 2008 Effects of Love waves on microtremor H/V ratio, Bull seism Soc Am., 98(1), 288300 Făah, D., Kind, F & Giardini D., 2003 Inversion of local S-wave velocity structures from average H/V ratios, and their use for the estimation of site-effects, J Seismol., 7, 449–467 Guidelines for the implementation of the H/ V spectral ratio technique on ambient vibration-measurements, processing and interpretations 2005 SESAME European research project, deliverable D23.12 Haghshenas, E., Bard, P.Y & Theodulilis, N., 2008 Empirical evaluation of microtremor H/V spectral ratio, Bull Earthq Eng., 6, 75–108 Konno, K & Ohmachi, T., 1998 Ground-motion characteristics estimated from spectral ratio between horizontal and vertical components of microtremor, Bull seism Soc Am., 88, 228–241 Malischewsky, P.G & Scherbaum, F., 2004 Love’s formula and H/V -ratio (ellipticity) of Rayleigh waves, Wave Motion, 40, 57–67 Malischewsky, P.G, Scherbaum, F., Lomnitz, C., Tran Thanh, T., Wuttke, F & Shamir, G., 2008 The domain of existence of prograde Rayleigh-wave particle motion for simple models, Wave Motion, 45, 556–564 Mucciarelli, M., Herak, M & Cassidy, J., (Ed.) 2009 Increasing Seismic Safety by Combining Engineering Technologies and Seismological Data, Springer, Dordrecht Nakamura, Y., 1989 A method for dynamic characteristics estimation of subsurface using microtremor on the ground surface Q Rep RTRI, 30(1), 25–33 Nakamura, Y., 2000 Clear identification of fundamental idea of Nakamura’s technique and its applications, Proc 12WCEE, No 2656, 177–402 Nakamura, Y., 2009 Basic Structure of QTS (HVSR) and Examples of Applications, in Increasing Seismic Safety by Combining Engineering Technologies and Seismological Data, pp 33–51, eds Mucciarelli, M., Herak, M & Cassidy, J., Springer, Berlin, doi:10.1007/978-1-4020-91964 Nogoshi, M & Igarashi, T., 1971 On the amplitude characteristics of microtremor (part 2), J seism Soc Japan, 24, 26–40 (In Japanese with English abstract) Stephenson, W.R, 2003 Factors bounding prograde Rayleigh-wave particle motion in a soft-soil layer, Pacific Conference on Earthquake Engineering, Christchurch, New Zealand Petrosino, S., 2006 Attenuation and velocity structure in the area of Pozzuoli-Solfatara (Campi Flegrei, Italy) for the estimate of local site response, PhD thesis, Universit`a degli Studi di Napoli “Federico II” Suzuki, T., 1933 Amplitude of Rayleigh waves on the surface of a stratified medium Bull Earthq Res Inst., 11, 187–195 Wathelet, M., 2005 Array recordings of ambient vibrations: surfacewave inversion, PhD thesis, University of Lige, Belgium Available at: http://marc.geopsy.org/publications.html (Last accessed 2008 June 10) C 2010 The Authors, GJI, 184, 793–800 Geophysical Journal International C 2010 RAS ... 2008 and Haghshenas et al 2008) In the present paper we are focusing on some of these issues, namely the properties of peaks and troughs of the ellipticity of Rayleigh waves and their relationships... International C 2010 RAS Relationship of peaks and troughs of H/V-ratio 795 Figure Illustration of the four regions R1 , R2 , R3 , R4 of H/V with different character in dependence on ν1 and rs The. .. regions, but the investigation of its meaning is beyond the scope of this paper Because the relationships between the H/V peak and the trough (or the maximum and the minimum, respectively) and the peak