A Review and Some New Issues on the Theory of the H/V Technique for Ambient Vibrations
Abstract
In spite of the HorizontaltoVertical Spectral Ratio (HVSR or H/V) technique obtained by the ambient vibrations is a very popular tool, a full theoretical explanation of it has been not reached yet. A short excursus is here presented on the theoretical models explaining the H/V spectral ratio that have been development in last decades. It leads to the present two main research lines: one aims at describing the H/V curve by taking in account the whole ambientvibration wavefield, and another just studies the Rayleigh ellipticity. For the first theoretical branch, a comparison between the most recent two models of the ambientvibration wavefield is presented, which are the Distributed Surface Sources (DSS) one and the Diffuse Field Approach (DFA). A mention is done of the current developments of these models and of the use of the DSS for comparing the H/V spectral ratio definitions present in literature. For the second research branch, some insights about the connection between the socalled osculation points of the Rayleigh dispersion curves and the behaviour of the H/V curve are discussed.
Keywords
Rayleigh Wave Spectral Ratio Love Wave Body Wave Ambient Vibration15.1 Introduction
The HorizontaltoVertical Spectral Ratio (HVSR or H/V) technique is a way to retrieve information about the shallowsubsoil seismic properties (which are of engineering interest) by singlestation measurements carried out on the Earth’s surface. This method is widely used in seismic exploration as a tool for a quick detection and evaluation of seismicamplification effects in terms of Swave resonance frequency as well as for constraining the elastic properties of the shallow geological structure (usually under the assumption of horizontally layered medium). Nevertheless, some controversial aspects about the exact physical interpretation of the outcome provided by this technique (the H/V curve) remain. Most of them are related with the nature of the ambientvibration wavefield and of its sources. These differences in the H/V curve modelling might have consequences in the results of inversion procedures used to infer the subsoil stratigraphical profile from experimental measurements.
From the experimental point of view, this technique requires a threecomponent groundmotion acquisition and consists in performing the ratio between its horizontal and vertical Fourier spectrum, properly averaged on an adequate sample. This ratio, which is a function of the frequency, is called the H/V (or HVSR) curve (or function). The ratio is usually computed by using groundmotion velocity spectra, but displacement or acceleration spectra can be used as well. The two horizontal motion components can be combined in different ways (vide infra).
In order to fully exploit the H/V curve to constrain subsoil seismicproperties, some theoretical model is necessary to link the H/V pattern to the mechanical properties of geological bodies under the measuring site. As the H/V refers to ambient vibrations, any model of H/V is also a model, explicit or tacit, of the ambientvibration wavefield, and thus it should be consistent with the other findings about the ambientvibration wavefield, and not just gives a plausible way to reproduce the H/V curve only.
In next section, a short excursus on the history of the H/V theoretical explanations is presented. The most part of the proposed models, which are the sole ones considered in this review, describes the Earth as 1D medium, i.e., a stack of homogeneous and isotropic horizontal layers overlying an halfspace with the same characteristics. They are the models widely used, while 2D and 3D ones, which are very cumbersome under many aspects, have been playing, so far, a minor role, being their use limited to specific problems (see, e.g., BonnefoyClaudet et al. 2004).
15.2 A Short Review on the H/V Theory
Kanai and Tanaka (1961) use the ambientvibration horizontalmotion spectra to infer seismic subsoil properties, even if they already recognize that the ambientvibration features depend on both site mechanicalproperties and ambientvibration sources’ characteristics. Other authors, however, note that ambientvibration spectra often reflect more the sources’ characteristics rather than the subsoil ones (cf. Tokimatsu 1997).
15.2.1 The H/V Origins: BodyWave Based Theories
The one described above is, de facto, the first theoretical explanation of the H/V curve, whose the most important implication is probably that the peak frequency and amplitude of the function HV correspond to the Swave resonance frequency and amplification factor of the site, respectively. Although this description is probably inadequate, it marked a turning point and made the fortune of the H/V technique. Indeed, while the statement about the amplitude has been proved to be almost always false, the correspondence between Swave resonance and H/Vpeak frequency has been always confirmed since then, in innumerable field experiments as well as by numerical simulations. This is by far the most useful and the most used feature of the H/V curve, but, surprisingly, it has not find a suitable complete theoretical explanation yet. Just in the particular case that surface waves are considered only (vide infra), the analytical formulae of Malischewsky and Scherbaum (2004) for the Rayleigh ellipticity demonstrate that the implication concerning the peak frequency is correct, in so far the impedance contrast is high enough. It is worth noting that Nakamura’s theory explains the H/V curve just around its main peak frequency, and any extension to the whole H/V curve requests further assumptions (cf. Bard 1998).
15.2.2 The Role of the Surface Waves
The fact that the H/V can be described in term of body waves travelling along particular patterns only is not at all obvious. In fact, the composition of ambient vibrations in term of the different seismic phases is not clearly understood till today, but all authors share the opinion that them are composed by all seismic phases travelling in the subsoil, although in not univocally defined proportions: the key and controversial aspect is the relative contribution of these seismic phases (see, e.g., SESAME 2004). In fact, contrasting results exist both in field experiments and in numerical simulations, and it seems likely that the content in different seismic phases can drastically change in dependence on the subsoil stratigraphy and on sources’ characteristics as well as in different frequency ranges.
So, as a sort of “counterparty” of the theories relied on body waves, theories based on surfacewave dominance have been developed. Already Nogoshi and Igarashi (1971) compare H/V curves from ambient vibrations with the ellipticity pattern of Rayleigh fundamentalmode, reckoning from the possibility of this comparison that this seismic phase plays the main role in the ambient vibrations. Subsequently, several other authors (e.g., Lanchet and Bard 1994, 1995; Tokimatsu 1997; Konno and Ohmachi 1998; Wathelet et al. 2004) have been agreeing on the close relation existing between the H/V spectral ratio and the ellipticity of Rayleigh waves, which is reckoned as a consequence of their energetic predominance. In particular, Arai et al. (1996) and Tokimatsu (1997), like Nogoshi and Igarashi (1971), explain the ambientvibration H/V curve by the ellipticity of the first mode of Rayleigh waves, and consider the feasibility of this explanation a suggestion of the surfacewave dominance. Surfacewave based is also the interpretation given by Konno and Ohmachi (1998), who point out that the H/V peak by ambient vibrations could be explained by the ellipticity of the fundamental Raylegh mode as well as by the Airy phase of the fundamental Love mode, and also examine the role of the first higher Rayleigh mode. Moreover, in numerical simulations performed by these Authors, the H/Vpeak amplitude roughly approximates the Swave amplification factor, providing that a specific proportion between Rayleigh and Love waves exists; this mimics the Nakamura’s statement, but in terms of surface waves instead that of body waves.
15.2.3 The Sources’ Role and the FullWavefield
The abovementioned theories give an explanation of the possible origin of the H/V curve, especially around its lowerfrequency peak, but do not insert this explanation in a theory of the ambientvibration wavefield. In other words, they say nothing about the origin of the H/Vcurve overall shape, since they are not models for the ambientvibration wavefield. In order to construct such a model, besides the composition of the ambientvibration wavefield in terms of different seismic phases, another key element is its dependence on the subsoil properties. Without this piece of information, no possibility exists of using any experimental datum to estimate subsoil characteristics. For the models based on the hypothesis that just vertically incident P and S waves are important to describe the H/V curves, this aspect is simply exhausted by computing the propagation of these phases in a stratified model, as is the case of the abovementioned Herak’s approach. When the characteristics of free Rayleigh waves are needed, classical algorithms to compute them in a stratified medium can be applied. Besides these simple cases, some models have been developed in last couple of decades that manage this aspect by means of more detailed analysis of the ambientvibration wavefield.
Lanchet and Bard (1994, 1995) consider that the ambientvibration wavefield cannot be described in a deterministic way, because the greatest number of its sources are randomly located on the Earth’s surface. So, they carry out a numerical simulation of the ambientvibration wavefield by arranging a number of sources of different kinds acting in aleatory ways inside a given horizontal circle surrounding the receiver. For computational reasons, sources are located at depth of 2 m, while the receiver is on the Earth’s surface. In this simulation, ambientvibration displacement is the sum, in the time domain, of the ones produced by these sources in a fixed lapse of time and the H/V curve is the ratio between their horizontal and vertical Fourier amplitudespectra. This is a purely numerical way to simulate the H/V curve, which has been used many times since then. By means of this model, Lanchet and Bard show the correspondence between the H/V peakfrequency and the Swave resonancefrequency. Moreover, they also show that the H/V peakfrequency corresponds to the ellipticity peakfrequency of the Rayleigh fundamental mode as well as to the firstpeak position of the ratio between horizontal and vertical groundmotion produced by S waves incident from a range of angles. Finally, they suggest that the overall shape of the H/V is determined by all seismic phases, and check that the peak amplitude, depending on many variables, does not correspond to the site amplification factor.
About in the same period, Field and Jacob (1993) propose a theoretical way to connect ambientvibration displacement powerspectrum to the Green’s function of the ground. They assume that the ambient vibrations are generated by an infinitude of uncorrelated pointlike sources, uniformly located on the Earth’s surface. The H/V curve in a point of the Earth’s surface is obtained as the square root of the ratio between the horizontal and the vertical total power, computed, for any subsoil profile, as a finite sum of the contribution, in the frequency domain, given by the sources in a succession of annuli centred on the receiver and with increasing radii. Differently from the Lanchet and Bard’s model, which is purely numerical, this is an analytical model, although the sums have to be computed numerically. A decade later, Arai and Tokimatsu (2000, 2004) specialize this model to surface waves generated by sources with independent phases, which are approximated as continuously distributed on the Earth’s surface. In this way, the total average spectral power is given by an integration on the horizontal plain, which can be carried out analytically. A sourcefree area around the receiver also exists in this model, with a radius equal to one wavelength of each propagation mode, in order to guarantee the surfacewave dominance and the possibility of describing these waves as plane waves. In order to make the powerintegrals converging, these Authors insert a exponential damping factor originated by the “scattering” of the considered waves in the subsoil. A slightly modified version of this model was proposed by Lunedei and Albarello (2009), in which the damping originates by the material viscosity and the sourcefree area dimension does not more depend on each single propagation mode and can be done independent from the frequency too.
Fäh et al. (2001) use two ways to generate H/V synthetic curves. The first one is a numerical simulation made by a finite difference technique: these Authors agree that ambientvibration sources are superficial, but they also introduce buried sources to describe scattering and wave conversion due to lateral heterogeneities. A large number of sources, with positions, depths and timedependences chosen randomly, are distributed around a receiver. The second technique is a mode summation (Landisman et al. 1970). They particularly focus on the Rayleigh wave ellipticity of fundamental and higher modes, to explain the H/Vpeak frequency, which they regard as the only trustworthy element, in that its amplitude and other features of the H/V curve also depend on other variables besides the Swave velocity profile. Moreover, they identify stable parts of the H/V ratio, which are independent of the sources’ distance and are dominated by the ellipticity of the fundamental Rayleigh mode, in the frequency band between the H/Vpeak frequency, which they check to be close to the site Swave fundamental resonancefrequency, and the first minimum of the H/V curve.
In their very important series of papers, BonnefoyClaudet et al. (2004, 2006, 2008) carry out a systematic study of the H/V curve by numerical simulations, in which the ambient vibrations are generated by a multitude of pointlike forces, randomly oriented in the space and located relatively near to the observation point. They take advantage by a code developed by Hisada (1994, 1995) to compute the full displacement wavefield produced by these sources at some receivers, which are located on the Earth’s surface. The total displacement at each receiver is computed by summing up, in the time domain, the one due to each sources. The H/V curve at each receiver is then computed as ratio between the average horizontal and the vertical Fouriertransform of this total displacement. In BonnefoyClaudet et al. (2006) the quasiindependence of the H/V curve from the specific sources’ timedependence has been confirmed. A dependency on the spatial horizontal distribution of nearsurface sources as well as on the depth of buried sources has instead been observed, which however only slightly concerns the mainpeak frequency. It instead shows relevant effects on H/Vpeak amplitude and on the appearance of secondary peaks. By using surface sources and several simple stratigraphical profiles, BonnefoyClaudet et al. (2008) check the good correspondence between the H/Vpeak frequency and the Swave resonance one. They also conclude that the H/V peakfrequency could be explained, depending on the stratigraphical situation, by Rayleigh ellipticity, Love Airy phase, Swave resonance or a mix of them. In particular, the possibility of explaining the H/V main peak in term of Rayleigh ellipticity seems limited to profile with high impedance contrast (more than 4). An interesting result of this work is the coming out of the significant role of Love waves in the H/V curve and, more in general, in composing the ambientvibration horizontal groundmotion. Moreover, the importance of taking into account all seismic phases propagating in the subsoil in constructing a suitable H/V model as well as the key role of the impedance contrast in controlling the origin of the H/V peak have been pointed out.

Lowfrequencies (below the Swave resonance frequency, f _{ S }), where ambientvibration spectralpowers are relatively low; in this range, the shallow layer acts as a highpass filter, with an effect as more pronounced as sharper the impedance contrast is; both near sources and body waves dominate the wavefield; power spectra and H/V curves are significantly affected by sourcefree area dimension, V _{ P } /V _{ S } ratio and impedancecontrast strength at the bottom of the shallow layer;

Highfrequencies (above max{f _{ P }, 2f _{ S }}, where f _{ P } is the Pwave resonance frequency), where surface waves (both Love and Rayleigh, in their fundamental and higher modes) dominate the wavefield; in this range, spectral powers smoothly decrease with frequency as an effect of material damping, which also results in the fact that relatively near sources mostly contribute to ambient vibrations, as more as the frequency increases; H/V curves are almost unaffected by subsoil configuration and source/receiver distances;

Intermediate frequencies, where the most of the ambientvibration energy concentrates; in this range, sharp peaks in the horizontal and vertical spectral powers are revealed around its left and right bounds; irrespective of the subsoil structure and sourcefree area considered, horizontal ground motion is dominated by surface waves, with a varying combination of Love (in the fundamental mode) and Rayleigh waves that depends on the shallowlayer Poisson’s ratio (Lovewave contribution increases with it) and, to a minor extent, on the strength of the impedance contrast; in the vertical component, Rayleigh and other phases play different roles, both depending on the sourcefree area dimension and of V _{ P } and V _{ S } profiles.
Stratigraphical profiles used in the numerical experiments
M2  
h (m)  V _{ S } (m/s)  ν  ρ (g/cm^{3})  D _{ P }  D _{ S } 
25  200  0.333  1.9  0.001  0.001 
5,000  1,000  0.333  2.5  0.001  0.001 
∞  2,000  0.257  2.5  0.001  0.001 
M2*  
h (m)  V _{ S } (m/s)  ν  ρ (g/cm^{3})  D _{ P }  D _{ S } 
25  200  0.01–0.\( \overline{49} \)  1.9  0.001  0.001 
5,000  228–1,520  0.333  2.5  0.001  0.001 
∞  2,000  0.257  2.5  0.001  0.001 
M3  
h (m)  V _{ S } (m/s)  V _{ P } (m/s)  ρ/ρ _{ 4 }  D _{ P }  D _{ S } 
5  30  500  1  0.001  0.001 
25  100  500  1  0.001  0.001 
50  150  500  1  0.001  0.001 
∞  500  1,500  1  0.001  0.001 
15.2.4 A Different Point of View: The Diffuse Wavefield
The model proposed more recently, named DFA (Diffuse Field Approach), significantly differs from the other ones, because it assumes that ambient vibrations constitute a diffuse wavefield. This means that seismic waves propagate in every (threedimensional) spatial direction in a uniform and isotropic way and that a specific energetic proportion between P and S waves exists, which is the same whenever and wherever. This theory, initially developed in a fullspace (SánchezSesma and Campillo 2006), has been afterwards applied to an halfspace and to a layered halfspace (SánchezSesma et al. 2011; Kawase et al. 2011). The link between the H/V curve and the subsoil configuration is simply given by the Green’s function, computed for source and receiver located in the same position: its imaginary part, in the spectral domain, is proportional to the average spectralpower of the ambientvibration groundmotion. A key element in the DFA theory, which is implied in the diffuse character of the wavefield, is the loss of any trace of the sources’ characteristics, so no link between displacement and its sources is involved in this theory, ergo, no description of ambientvibration sources is necessary. The model can describe the ambientvibration fullwavefield as well as its surfacewave component only, depending on whether the fullwavefield Green’s function or its surfacewave component is used.
15.2.5 Current Research Branches

The branch that studies the ambientvibration wavefield as a whole; in this case, the theory aims to explain the H/V curve as it is measured in field, with all its components in terms of different seismic phases; this theory has to face the problems about the role of body and surface waves as well as about the role of the sources;

The branch that should be better named “ellipticity theory” or “Rayleighwave H/V”; the subject is, in this case, just the Rayleigh ellipticity, both in theory and in experiments; as it chooses, a priori, to take into account the Rayleigh ellipticity only, the relative theory does not need to deal neither with body waves nor with the wavefield sources, while experiments are devoted to extract Rayleigh waves from the recorded signal (e.g., Fäh et al. 2001).
Currently, the first theory is essentially represented by models that consider surface sources, in all possible variants (the purely numerical one or the semianalytical DSS) and the DFA: in next section a comparison between the DSS and the DFA is summarized, while in the subsequent a mention to new developments in these models is done. Afterwards, a section is devoted to the ellipticity theory.
15.3 Comparison Between the DSS and the DFA Models
In last years, some conference notes (GarcíaJerez et al. 2011, 2012a, b, c) were presented to compare the most recent two models of the H/V spectral ratio: the Distributed Surface Sources (DSS) and the Diffuse Field Approach (DFA). Each of them is a complete theory of the ambient vibrations and has solid theoretical foundations. Through this section, which summarizes the salient elements of these comparisons, G _{ij}(x _{ A }, x _{ B }, ω) is the frequencydomain displacement Green’s function for the considered Earth’s model at the point x _{ A } on the free surface along the ith Cartesian axis due to a pointlike force located at the point x _{ B } and directed along the jth Cartesian axis. The three Cartesian spatial directions are marked by subscripts 1, 2 (for the horizontal plane) and 3 (for the vertical direction), while r and θ are the polar coordinates on the horizontal plane.
15.3.1 The DSS Model
15.3.2 The DFA Model
The DFA model assumes that the relative power of each seismic phase is prescribed by the energy equipartition principle. Under this hypothesis, proportionality exists between the Fouriertransformed autocorrelation (power spectrum), at any point of the medium, and the imaginary part of the Green’s function computed when source location corresponds to the one of the receiver (SánchezSesma et al. 2011). The assumption of a major role of multiple scattering involving all possible wavelengths is behind this formulation.
15.3.3 Comparison
The differences between DSS and DFA model are shown in a more explicit form if their versions for surface waves are compared (Eqs. 15.9, 15.10 or 15.11, 15.12 vs 15.14, 15.15). The formulae have a similar structure, but the contributions of each wavetype and mode to the total power differ. Indeed, they depend on A _{• m } in the DFA formulation and on \( {\left(\frac{A_{\bullet m}}{k_{\bullet m}}\right)}^2 \) or \( \frac{{\left({A}_{\bullet m}\right)}^2}{k_{\bullet m}{\alpha}_{\bullet m}}\cdot \exp \left(2{\upalpha}_{\bullet m}{x}_{\min}\right) \) in the DSS one, where “•” indicates or Love or Rayleigh waves. The square operator in the last model is a consequence of the power computation; in the DFA model, the correct physical dimension is guaranteed by an appropriate factor that multiplies the imaginary part of the Green’s function. So, while in the DSS the energy repartition among contributing waves depends on the energy of each wave (expressed by its square amplitude), in the DFA this repartition is established by the Green’s function for coincident source and receiver. This is a very important physical difference between the two models. The common inverse wavenumber 1/k _{• m } in the DSS formulae represents an effect of the longrange wave propagation from the generic source to the receiver, while the other 1/k _{• m } factor or the correspondent 1/α _{• m } is the effect of the integration on the horizontal distance to compute the total source distribution effect. Both these elements are obviously absent in the DFA. In both the considered models, the function HV restricted to surfacewaves tends to the ellipticity of (nondispersive) Rayleigh waves over a halfspace and depend on the characteristics of the deeper medium, as ω → 0.
In order to study the differences and similarities of these two models, a set of synthetic tests was performed (see notes quoted at the beginning of the section): results relative to stratigraphic profiles listed in Table 15.1 are here shown. The group of profiles M2* is generated by varying the profile M2, and all these profiles basically consist of a layer overlying an halfspace (although a intermediate thick buffer layer exists, which prevents from sharply unrealistic truncation of surfacewave higher modes in the range of frequency of interest). The profile M3, instead, presents two major and a weak impedance contrasts. For the DSS model, σ_{1} ^{2} = σ_{2} ^{2} = σ_{3} ^{2} = 1/3 was set.
The results obtained indicate that both the DSS and the DFA provide reasonable fullwavefield and surfacewave synthetics of H/V spectral ratios. In spite of the rather different underlying hypotheses, DFA and DSS lead to similar H/V curves for stratigraphic profiles with a dominant impedance contrast (M2*). Relative H/V main peaks match the first Swave resonance frequency (f _{S}) in a very good way. Nevertheless, peak amplitudes may differ and show nontrivial dependence on impedance contrast and Poisson’s ratio. Results relative to DSS also depend on the source distribution around the receiver. For both models, surface waves represent the dominant contribution at high enough frequencies, whereas body waves play an important role around and below f _{S}. For a stratigraphy with more impedance contrasts, some variability occurs in the overall shape of the H/V curve in the fullwavefield DSS when sources are present or absent near the receiver. Whenever near sources are eliminated from the DSS computation (so surface waves are playing the major role), both DFA and DSS provide very similar results, and this seems suggest that, although physical bases are different, surfacewave behaviour described by DFA and DSS is very similar. In any case, the differences in the overall H/V curve features make clear that further investigations on the relationships between DFA and DSS are still necessary.
15.4 A Mention to the Most Recent Results in H/V Modelling
To overcome some limits of the fullwavefield DSS model, a new version of it has been very recently proposed by Lunedei and Albarello (2014, 2015). This new theory bases on describing the ambientvibration groundmotion displacement and its generating force fields as threevariate, threedimensional stochastic processes stationary both in time and space. In this frame, the displacement power can be linked with the source filed power via the Green’s function, which, in turn, depends on the subsoil configuration.
About the DFA model, very recently GarcíaJerez et al. (2013) have shown some consequence, at low and high frequencies, of its application to a simple crustal model. The most recent development of this model is its application to a case where a lateral variation exists, by Matsushima et al. (2014).
 1.
No combination, that is, two H/V curves are computed by considering separately the two directions,
 2.
Arithmetic mean, \( H\equiv \left({H}_N+{H}_E\right)/2, \)
 3.
Geometric mean, \( H\equiv \sqrt{H_N\cdot {H}_E}, \)
 4.
Vector summation, \( H\equiv \sqrt{H_N^2+{H}_E^2}, \)
 5.
Quadratic mean, \( H\equiv \sqrt{\left({H}_N^2+{H}_E^2\right)/2}, \)
 6.
Maximum horizontal value, \( H\equiv \max \left\{{H}_N,{H}_E\right\}, \)
 (a)
The square root of the ratio between the arithmetic mean of the spectral powers on the L timewindows,
 (b)
The arithmetic mean of the H/V ratios computed in each of the L timewindows.
15.5 Rayleigh Ellipticity Theory
In this research branch, the H/V curve is identified a priori and by definition with the ellipticity of Rayleigh waves, which is the subject of the study. A short summary on this topic can be found, e.g., in SESAME (2004). Moreover, a part of the popular Geopsy software (http://www.geopsy.org/) is focused on the ellipticity. Fäh et al. (2001) propose a way to extract Rayleigh ellipticity experimentally and to compare it with a theoretical model. Malischewsky and Scherbaum (2004) investigate some important properties of H/V on the basis of Rayleigh waves by reanalysing an old formula of Love, and obtaining essential results to apply the H/V method. Later, the theory for the ellipticity of Rayleigh waves was carefully studied by Tran (2009)and Tran et al. (2011) with particular regard to applications for the H/V method.
15.5.1 Osculation Points

if \( {{\overline{\nu}}_1}^{(1)}<{\nu}_1<{{\overline{\nu}}_1}^{(2)} \) the H/V curve has two peaks,

if \( {{\overline{\nu}}_1}^{(2)}<{\nu}_1<0.5 \) the H/V curve has one peak and one zeropoint,
It turns out that the osculation point is, for LOH again, the point where the H/V curve changes its behaviour dramatically. The practical consequences of this behaviour are discussed for models in Israel and Mexico in Malischewsky et al. (2010).
15.6 Conclusions
This short excursus on the way to construct a theory able to explain the H/V curve features shows that, in spite of the strongly different hypothesis underlying the various proposed theories, the key element of the H/V curve, i.e., the main peak frequency, is reproduced in a more than acceptable way by all of them. Even thought, in order to be able to profoundly understand the relative role of the model and of the stratigraphy in affecting the synthetic H/V curves, a big systematic comparative work would be necessary, the capability of different theories of giving realistic features of this quantity reinforces the idea that the H/V curve, and in particular its main peak frequency, express intrinsic properties of the subsoil, i.e., that it is eminently determined by the stratigraphical profile, ergo it gives a true piece of information about the subsoil seismic properties. By a phrase, the H/V seems to resist theories!
Notes
Acknowledgments
Authors are grateful to Prof. Dario Albarello for useful suggestions about the subject of this paper.
References
 Albarello D, Lunedei E (2011) Structure of an ambient vibration wavefield in the frequency range of engineering interest ([0.5, 20] Hz): insights from numerical modelling. Near Surface Geophys 9:543–559. doi: 10.3997/18730604.2011017 CrossRefGoogle Scholar
 Albarello D, Lunedei E (2013) Combining horizontal ambient vibration components for H/V spectral ratio estimates. Geophys J Int 194:936–951. doi: 10.1093/gji/ggt130 CrossRefGoogle Scholar
 Arai H, Tokimatsu K, Abe A (1996) Comparison of local amplifications estimated from microtremor fk spectrum analysis with earthquake records. In: Proceedings of the 11th world conferences on earthquake engineering (WCEE), Acapulco. http://www.nicee.org/wcee/
 Arai H, Tokimatsu K (2000) Effect of Rayleigh and Love waves on microtremor H/V spectra. In: Proceedings of the 12th world conferences on earthquake engineering (WCEE), Auckland. http://www.nicee.org/wcee/
 Arai H, Tokimatsu K (2004) Swave velocity profiling by inversion of microtremor H/V spectrum. Bull Seismol Soc Am 94(1):53–63CrossRefGoogle Scholar
 Bard PY (1998) Microtremor measurements: a tool for site effect estimation? In: Proceedings of the 2nd international symposium on the effects of surface geology on seismic motion, Yokohama, pp 1251–1279Google Scholar
 BonnefoyClaudet S, Cornou C, Kristek J, Ohrnberger M, Wathelet M, Bard PY, Moczo P, Fäh D, Cotton F (2004) Simulation of seismic ambient noise: I. Results of H/V and array techniques on canonocal models. In: Proceedings of the 13th world conferences on earthquake engineering (WCEE), Vancouver. http://www.nicee.org/wcee/
 BonnefoyClaudet S, Cornou C, Bard PY, Cotton F, Moczo P, Kristek J, Fäh D (2006) H/V ratio: a tool for site effects evaluation. Results form 1D noise simulation. Geophys J Int 167:827–837. doi: 10.1111/j.1365246X.2006.03154.x CrossRefGoogle Scholar
 BonnefoyClaudet S, Köhler A, Cornou C, Wathelet M, Bard PY (2008) Effects of Love waves on microtremor H/V ratio. Bull Seismol Soc Am 98(1):288–300. doi: 10.1785/0120070063 CrossRefGoogle Scholar
 Fäh D, Kind F, Giardini D (2001) A theoretical investigation of average H/V ratios. Geophys J Int 145:535–549CrossRefGoogle Scholar
 Field E, Jacob K (1993) The theoretical response of sedimentary layers to ambient seismic noise. Geophys Res Lett 20(24):2925–2928CrossRefGoogle Scholar
 Forbriger T (2003) Inversion of shallowseismic wavefields: I. Wavefield transformation. Geophys J Int 153:735–752CrossRefGoogle Scholar
 GarcíaJerez A, Luzón F, SánchezSesma FJ, Santoyo MA, Albarello D, Lunedei E, Campillo M, IturraránViveros U (2011) Comparison between two methods for forward calculation of ambient noise H/V spectral ratios. AGU Fall Meeting 2011, 5–9 Dec 2011, San Francisco. http://abstractsearch.agu.org/meetings/2011/FM/sections/S/sessions/S23A/abstracts/S23A2230.html
 GarcíaJerez A, Luzón F, Albarello D, Lunedei E, SánchezSesma FJ, Santoyo MA (2012a) Comparison between ambient vibration H/V synthetics obtained from the Diffuse Field Approach and from the Distributed Surface Load method. In: Proceedings of the XXIII general assembly of the European seismological commission (ESC 2012), 25–30 Aug 2012, Moscow, pp 412–413Google Scholar
 GarcíaJerez A, Luzón F, Albarello D, Lunedei E, Santoyo MA, Margerin L, SánchezSesma FJ (2012b) Comparison between ambient vibrations H/V obtained from the diffuse field and distributed surface source models. In: Proceedings of the 15th world conferences on earthquake engineering (WCEE), 24–28 Sept 2012, Lisbon. http://www.nicee.org/wcee/
 GarcíaJerez A, Luzón F, Lunedei E, Albarello D, Santoyo MA, Margerin L, SánchezSesma FJ (2012c) Confronto fra le curve H/V da vibrazioni ambientali prodotte dai modelli di distribuzione superficiale di sorgenti e di campo diffuso, Atti del XXXI Convegno Nazionale del Gruppo Nazionale di Geofisica della Terra Solida, 20–22 Nov 2012, Potenza, pp 148–157. http://www2.ogs.trieste.it/gngts/ (Sessione 2, Tema 2) (in Italian)
 GarcíaJerez A, Luzón F, SánchezSesma FJ, Lunedei E, Albarello D, Santoyo MA, Almendros J (2013) Diffuse elastic wavefield within a simple crustal model. Some consequences for low and high frequencies. J Geophys Res 118:1–19. doi: 10.1002/2013JB010107 Google Scholar
 Harkrider DG (1964) Surface waves in multilayered elastic media. Part 1. Bull Seismol Soc Am 54:627–679Google Scholar
 Herak M (2008) ModelHVSR – a Matlab® tool to model horizontaltovertical spectral ratio of ambient noise. Comput Geosci 34(11):1514–1526. doi: 10.1016/j.cageo.2007.07.009 CrossRefGoogle Scholar
 Hisada Y (1994) An efficient method for computing Green’s functions for a layered halfspace with sources and receivers at close depths. Bull Seismol Soc Am 84(5):1456–1472Google Scholar
 Hisada Y (1995) An efficient method for computing Green’s functions for a layered halfspace with sources and receivers at close depths (part 2). Bull Seismol Soc Am 85(4):1080–1093Google Scholar
 Kanai K, Tanaka T (1961) On microtremors. VIII. Bull Earthq Res Inst 39:97–114Google Scholar
 Kausel E, Malischewsky P, Barbosa J (2015) Osculations of spectral lines in a layered medium. Wave Motion (in press). doi: 10.1016/j.wavemoti.2015.01.004, (http://dx.doi.org/10.1016/j.wavemoti.2015.01.004)
 Kawase H, SánchezSesma FJ, Matsushima S (2011) The optimal use of horizontaltovertical spectral ratios of earthquake motions for velocity inversions based on diffusefield theory for plane waves. Bull Seismol Soc Am 101(5):2001–2014. doi: 10.1785/0120100263 CrossRefGoogle Scholar
 Konno K, Ohmachi T (1998) Groundmotion characteristics estimated from spectral ratio between horizontal and vertical components of microtremors. Bull Seismol Soc Am 88(1):228–241Google Scholar
 Lanchet C, Bard PY (1994) Numerical and theoretical investigations on the possibilities and limitations of Nakamura’s technique. J Phys Earth 42:377–397CrossRefGoogle Scholar
 Lanchet C, Bard PY (1995) Theoretical investigations on the Nakamura’s technique. In: Proceedings of the 3rd international conference on recent advanced in geotechnical earthquake engineering and soil dynamics, 2–7 Apr 1995, St. Louis (Missouri), vol IIGoogle Scholar
 Landisman L, Usami T, Sato Y, Massè R (1970) Contributions of theoretical seismograms to the study of modes, rays, and the earth. Rev Geophys Space Phys 8(3):533–589CrossRefGoogle Scholar
 Lunedei E, Albarello D (2009) On the seismic noise wavefield in a weakly dissipative layered Earth. Geophys J Int 177:1001–1014. doi: 10.1111/j.1365246X.2008.04062.x (Erratum: Geophys J Int 179:670. doi: 10.1111/j.1365246X.2009.04344.x)
 Lunedei E, Albarello D (2010) Theoretical HVSR curves from full wavefield modelling of ambient vibrations in a weakly dissipative layered Earth. Geophys J Int 181:1093–1108. doi: 10.1111/j.1365246X.2010.04560.x (Erratum: Geophys J Int 192:1342. doi: 10.1093/gji/ggs047)
 Lunedei E, Albarello D (2014) Complete wavefield modelling of ambient vibrations from a distribution of correlated aleatory surface sources: computation of HVSR. Special Session “Ambient Noise for soil and building studies” of the Second European Conference on Earthquake Engineering and Seismology (2ECEES), 24–29 Aug 2014, IstanbulGoogle Scholar
 Lunedei E, Albarello D (2015) Horizontaltovertical spectral ratios from a fullwavefield model of ambient vibrations generated by a distribution of spatially correlated surface sources. Geophys J Int 201:1140–1153. doi: 10.1093/gji/ggv046 CrossRefGoogle Scholar
 Malischewsky PG, Scherbaum F (2004) Love’s formula and H/Vratio (ellipticity) of Rayleigh waves. Wave Motion 40:57–67CrossRefGoogle Scholar
 Malischewsky PG, Zaslavsky Y, Gorstein M, Pinsky V, Tran TT, Scherbaum F, Flores Estrella H (2010) Some new theoretical considerations about the ellipticity of Rayleigh waves in the light of siteeffect studies in Israel and Mexico. Geofisica Int 49:141–151Google Scholar
 Matsushima S, Hirokawa T, De Martin F, Kawase H, SánchezSesma FJ (2014) The effect of lateral heterogeneity on horizontaltovertical spectral ratio of microtremors inferred from observation and synthetics. Bull Seismol Soc Am 104(1):381–393. doi: 10.1785/0120120321 CrossRefGoogle Scholar
 Nakamura Y, Ueno M (1986) A simple estimation method of dynamic characteristics of subsoil. In: Proceedings of the 7th Japan earthquake engineering symposium, Tokyo, pp 265–270 (in Japanese)Google Scholar
 Nakamura Y (1989) A method for dynamic characteristics estimation of subsurface using microtremor on the ground surface. Q Rep Railw Tech Res Inst 30(1):25–30Google Scholar
 Nakamura Y (2000) Clear identification of fundamental idea of Nakamura’s technique and its applications. In: Proceedings of the 12th world conference on earthquake engineering (WCEE), Auckland. http://www.nicee.org/wcee/
 Nakamura Y (2008) On the H/V spectrum. In: Proceedings of the 14th world conference on earthquake engineering (WCEE), Beijing. http://www.nicee.org/wcee/
 Nogoshi M, Igarashi T (1971) On the amplitude characteristics of microtremor (part 2). J Seismol Soc Jpn 24:26–40 (in Japanese with English abstract)Google Scholar
 SánchezSesma FJ, Campillo M (2006) Retrieval of the Green’s function from cross correlation: the canonical elastic problem. Bull Seismol Soc Am 96(3):1182–1191. doi: 10.1785/0120050181 CrossRefGoogle Scholar
 SánchezSesma FJ, Rodríguez M, IturraránViveros U, Luzón F, Campillo M, Margerin L, GarcíaJerez A, Suarez M, Santoyo MA, RodríguezCastellanos A (2011) A theory for microtremor H/V spectral ratio: application for a layered medium. Geophys J Int 186:221–225CrossRefGoogle Scholar
 SESAME (2004) European Research Project, WP12–deliverable D23.12. Guidelines for the implementation of the H/V spectral ratio technique on ambient vibrations: measurements, processing and interpretation (see Bard PY et al (2004) The SESAME Project: an overview and main results. In: 13th world conference on earthquake engineering (WCEE), Vancouver, 1–6 Aug 2004, paper no. 2207, http://www.nicee.org/wcee)
 Tokimatsu K (1997) Geotechnical site characterization using surface waves. In: Ishihara K (ed) Earthquake geotechnical engineering: proceedings of ISTokyo ’95, the first international conference on earthquake geotechnical engineering, Tokyo, 14–16 Nov 1995, vol 3. A A Balkema Publishers, Rotterdam, pp 1333–1368Google Scholar
 Tsai NC (1970) A note on the steadystate response of an elastic halfspace. Bull Seismol Soc Am 60:795–808Google Scholar
 Tran TT (2009) The ellipticity (H/Vratio) of Rayleigh surface waves. PhD dissertation, FriedrichSchillerUniversity, JenaGoogle Scholar
 Tuan TT, Scherbaum F, Malischewsky PG (2011) On the relationship of peaks and troughs of the ellipticity (H/V) of Rayleigh waves and the transmission response of single layer over halfspace models. Geophys J Int 184:793–800CrossRefGoogle Scholar
 Wathelet M, Jongmans D, Ohrnberger M (2004) Surfacewave inversion using a direct search algorithm and its application to ambient vibration measurements. Near Surface Geophys 2:211–221CrossRefGoogle Scholar
Copyright information
Open Access This chapter is distributed under the terms of the Creative Commons Attribution Noncommercial License, which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.