Bull Earthquake EngDOI 10.1007/s10518-013-9543-3

ORIGINAL RESEARCH PAPER

Simplified seismic soil classification: the Vfz matrix

Silvia Castellaro · Francesco Mulargia

Received: 18 May 2012 / Accepted: 16 October 2013© Springer Science+Business Media Dordrecht 2013

Abstract Site effect assessment studies aim at predicting the effect of seismic shaking onstructures by modeling the subsoil as an oscillator coupled to another oscillator representingthe construction. The resulting amplification functions and response spectra depend on somany strong assumptions and parameters that, in the standard engineering practice, simplifiedseismic classifications appear preferable to complex modeling procedures which can onlyoffer an illusory better accuracy. Since stratigraphic seismic amplification is not properlyrelated to the absolute rigidity of subsoil but to impedance contrasts, the standard simplifiedapproaches based on the ‘average’ rigidity of subsoil in the first few meters (e.g. Vs30) canhardly be effective. Here it is proposed a simplified soil classification approach that takesinto account the basic Physics of seismic amplification and its parameters, i.e. the averageshear wave velocity of the cover layer, the resonance frequency and the impedance contrastbetween the cover and the bedrock, which we summarize as VfZ. A possible classificationapproach is illustrated through a set of examples.

Keywords Stratigraphic seismic amplification · VfZ · Passive seismic methods ·Site effects

1 Introduction

The assessment of seismic site effects at the scale of urban planning (shake maps, seis-mic microzonation) or at the scale of the single construction (building codes) requires theknowledge of the mechanical properties of subsoil down to the bedrock and of the ‘charac-teristic’ bedrock motion expected at the site. As output, it normally provides the SH-wavebedrock-to-surface amplification function—that is the ratio between the Fourier spectra ofthe accelerogram on the surface and on the bedrock—and the response spectrum—that isthe expected maximum acceleration/velocity/displacement on a single degree of freedom

S. Castellaro (B)· F. MulargiaDipartimento di Fisica e Astronomia, Università di Bologna, V.le Carlo B. Pichat 8,40127 Bologna, Italye-mail: silvia.castellaro@unibo.it

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oscillator of a specified damping and eigen-period, which mimics the behavior of a build-ing. The first function mostly depends on the subsoil properties, while the latter is mostlydependent on the input ground motion (see also “Appendix”).

Determining the input needed for this kind of analysis requires the measurement of a largenumber of parameters (P and S-wave velocity profile down to the bedrock, density profile,depth of the water table, shear modulus dependence with strain, etc.) and the analysis itselfis based on several assumptions, such as the existence of a characteristic earthquake and thedominance of vertically propagating SH-waves, which are in many cases contrary to availableevidence.

As a consequence, taking into account all uncertainties associated to the inputs revealsthe huge uncertainty associated to the output of the current numerical modelling procedures.Some of the latter will be quantified in the following section and show that the results of the1D numerical seismic response modelling can only be used to provide first-approximationestimates of the bedrock-to-surface amplification function and of the response spectrum andthat the earth motion values accurate to several significant digits required by some seismiccodes is just illusory.

Furthermore, since the uncertainty associated to the modeling procedures increases withthe number of parameters and since the global uncertainty is ruled by the variables withthe largest error (which are the same in 1D or 3D procedures), it follows that even the3D approaches deal with comparatively large errors. A nice example of this is the lack ofconvergence of the 3D seismic modeling algorithms to the same input data presented at theESG4 symposium (Bard 2011).

Clearly, this does not mean that research on seismic response modeling should be haltedbut that the results of neither 3D nor 1D algorithms can be simply incorporated in the dailyengineering practice. The need for quick simplified alternatives to seismic soil responseassessment for the standard daily practice appears scientifically justified.

Despite a number of approaches presented in the literature (Dobry et al. 2000; Rodriguez-Marek et al. 2001; Stewart et al. 2003; Zhao et al. 2006), at present, the best known simplifiedprocedure is based on Vs30, that is the shear-wave velocity of a homogeneous layer equiv-alent to the first 30 m depth, which is used as a proxy to the SH amplification factor, i.e. themaximum of the amplification function (Borcherdt 1994). This approach, which was devel-oped on a purely empirical basis, has been shown to suffer from statistical (Castellaro et al.2008) and physical problems (Lee and Trifunac 2010).

In this paper we try to cast the basis for an alternative simplified approach by first assess-ing what are the minimum physical parameters necessary to quantify seismic stratigraphicamplification. Then we show that the same effort currently used to measure Vs30 can producesubsoil classifications based on more physically meaningful parameters.

2 Methodological incongruences in the current approach

1. Use of full accelerograms in the numerical codes (1D linear equivalent analysis and sim-ilar) Despite the growing interest in this branch of research, using complete real earthquakerecordings as input functions for standard 1D equivalent-linear codes is not completely jus-tified. These codes assume that the seismic input is a SH wave vertically propagating fromthe bedrock to the surface, where it is reflected down to the bedrock and up again, with themaximum amplitude at the resonance frequency of the system.

However, unless the ground motion is induced by a very deep earthquake—usually irrel-evant in engineering terms—the dominant part of the seismogram for both amplitude and

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Fig. 1 Left Vs profiles of the 3 subsoil models. Right accelerograms of the 40 input motions used in the model.All the recordings are normalized to an amplitude of 0.25 g and are divided into 4 groups of 10 recordings(black close to the source, blue, cyan and green far from the source)

duration is surface waves (NMSOP 2002), that is waves propagating in a way which is notmodeled by the standard 1D numerical approaches. This is also clear by noting that no mag-nitude type is defined on S-waves (MS , the name of which could be misleading, is actuallycalculated on surface waves), thus suggesting that the maximum amplitude motion is notlinked to body waves. We are therefore facing the paradox that we use codes modeling SHwaves but we use input recordings which are only in minimum part SH waves. To quantifythe consequences of this, let us consider 3 subsoils described in terms of their Vs profile(Fig. 1, left) and 40 input motions, each normalized to a maximum acceleration of 0.25 g(Fig. 1, right). The input motion are the wavetrains generated by the same source recordedat different distances. As it can be seen, by moving far away from the source (bottom part ofthe figure), surface waves become dominant and body waves can hardly be distinguished.

Let us divide the 40 input motions into 4 groups of 10 recordings and for each group (andfor each subsoil) let us calculate the average response spectrum (and standard deviation). Asexpected (Fig. 2), a strong shift in the frequency peak of the response spectra is observed: bymoving far from the source, the dominant period in the response spectrum becomes longer.

There is little new in this since response spectra are strongly dependent on the input motioncharacteristics, and by moving far from the source the low frequencies prevail.

What is more interesting is that in the near field, where body waves play a role, the responsespectra are completely different from spectra in the far field, which tend to be more similar.However, in the latter the dominant waves are surface waves and should not be modeled withthe 1D numerical codes used both here and in the standard practice.

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Fig. 2 Response spectra expected on the 3 subsoils of Fig. 1 (left) and for the input motions of Fig.1 (right).The 40 input motions are divided into 4 groups of 10 recordings (same colors as in Fig. 1 (right) black 1–10,blue 11–20, cyan 21–30, green 31–40). Each thick line is the average response spectrum for each group whilethe standard deviation is given by the dashed lines

This stands for the fact that full accelerograms should not be used as input for 1D numericalcodes. While SH waves are certainly important for the resonance effects, the present modelingprocedures are still deficient because they ignore the role (both in terms of amplitude andduration) of surface waves.

2. PGA0 (peak ground acceleration at the bedrock) PGA0 is the basic parameter that character-izes the seismic input at a site in almost all applications. It is defined from the Ground MotionPrediction Equations (GMPE), i.e. relations that describe the attenuation of log(PGA0) as afunction of magnitude and source distance. The effects of the uncertainty associated to thisvariable seem to be rarely taken into account. A glance at the published GMPE that report thisuncertainty shows that the 1σ deviation on the log(PGA0) (�log(PGA0)) is always largerthan 0.2 (Campbell 1981; Boore et al. 1993; Kramer 2000), which implies an uncertainty�PGA0 = 10log PGA0�log(PGA0) > PGA0 × 0.2. Such an evidence is often ignored inboth research and common practice, resulting in building codes imposing the use of 3 or 4significant digits for PGA0 (cfr. the debate in Stucchi et al. 2011; Mucciarelli and Albarello2012).

3. Vs profiles Errors associated to the estimate of the Vs profiles are, to be optimistic, ofthe order of 20 % (Asten and Boore 2005; Mulargia and Castellaro 2009). Unfortunately,the analysis of the uncertainty associated to each input parameter used in the numericalsimulation and their propagation to the results is not requested by the regulations of somecountries in the standard cases, where the Vs30 approach is accepted, and as a consequence itis not performed in the standard engineering practice in those countries. Preliminary studies(e.g. Boaga et al. 2011) show that this effect can be very large.

3 What controls stratigraphic seismic amplification?

Seismic amplification has several causes (Anderson 2007), the most important of which isstratigraphic amplification. This is due to the existence of impedance contrasts (Z = ρ V,density × seismic wave velocity) in the subsoil (Aki e Richards, 1980), which rule seismicwave reflection and transmission at the interfaces, determining wave interference and ‘guidedwave’ effects.

To a first approximation, the non-resonant amplification produced from the energy con-servation of waves travelling through materials with gradually changing velocities is givenby

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A(f) = (Z0/<Zi>)1/2 (1)

where the index 0 indicates the property at the source and i at the site. At a specified frequency,amplification would theoretically be given by the ratio between the seismic impedance atthe source depth Z0 and the impedance averaged over the quarter-wavelength depth <Zi>

(Joyner et al. 1981; Day 1996). Equation 1 makes it clear that it is not the absolute Vs valuethat controls seismic amplification.

4 The VfZ approach

In order to characterize how seismic amplification is related to impedance contrasts, the 1Dequivalent linear response of a dataset of subsoils (30 layers per model) with the followingproperties is studied:

1) Layer 1: The surface layer is characterized by:

a. Vs ranging from 100 to 600 m/s (step 100 m/s),b. thickness of 3, 5, 7, 10, 15, 20, 40, 50, 60, 80, 100, 200, 300 m,

2) Layer 2: underlies Layer 1, characterized by increasing Vs (200, 300, 400, 500, 600,700, 1,000, 1,200, 1,600, 2,000 m/s). When Layer 1 has Vs = 200 m/s, Layer 2 hasVs increasing from 300 to 2,000 m/s, when Layer 1 has Vs = 300 m/s, Layer 2 has Vsincreasing from 400 to 2,000 m/s and so on,

3) Layer 3 to 30: the velocity profile increases in a exponentially decaying way down to thebedrock, which is located at 2 km depth, together with the source.

In the above models, the maximum impedance contrast Z always occurs between Layer 1 andLayer 2, giving resonance frequencies between 0.08 and 50 Hz. For each Layer 1 thickness,there are 45 different Vs profiles and the total amount of subsoil models investigated is 13(thicknesses of Layer 1) × 45 (impedance contrasts between Layer 1 and Layer 2) = 585. Oneexample of subsoil profile used is given in Fig. 3a. Density is 2 × 103 kg/m3 for all layersand no water table is assumed. The shear modulus and damping versus strain curves usedfor all models are illustrated in Fig. 3b, c for the cover and the bedrock. The specific choiceof these curves clearly affects the fundamental mode amplitude and the amplitude decay ofhigher modes of the amplification function but the absolute values are not of primary interestin a methodological paper like the present one. They should instead be taken into accountfor a possible future practical application of the method, which should be tuned on specificgeological conditions.

To avoid the incongruence of using whole accelerograms to model input motion, and inorder to minimize the number of variables, the input motion function (the earthquake) is keptas simple as possible assuming it as a Ricker wavelet (Curtis 1975; Hosken 1988; Ryan 1994)with frequency of 1 and 0.5 Hz. This represents the onset of the SH-wave of intermediate-small and intermediate-large earthquakes, respectively. PGA0 is set equal to 0.35 g. Rickerwavelets have been used in the literature to define the subsoil amplification function (e.g.Kawase and Aki 1989) and to study topographic effects (e.g. Pagliaroli et al. 2011). Theiruse is justified in this context because the amplification function is strongly dependent on thesubsoil property and little sensitive to the specific input motion. The use of Ricker waveletsto estimate response spectra, which on the opposite are strongly sensitive to the input motiondetails, needs a separate discussion which is outlined in “Appendix”, together with someother limitations of the present procedure.

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Fig. 3 a Example of subsoil model used in the analysis (zoom in the first 500 m depth. The real model reaches2 km depth), which represents 40 m cover Vs = 200 m/s overlying a semi-rigid layer (Vs0 > 700 m/s). b andc shear modulus and damping ratio versus shear strain used for the cover and the bedrock layer (Seed et al.1970)

Fig. 4 SH wave amplification functions for the models characterized by 80 m thickness cover layer with dif-ferent Vs (black 100 m/s, blue 200 m/s, green 300 m/s, magenta 400 m/s, cyan 500 m/s, red 600 m/s) overlyinga stiffer layer with increasing impedance contrast (see the legends in Fig. 5 for each set of Vs). The maxima ofthe amplification function—for each model and for each tested pseudo-bedrock depth—are plotted in Fig. 5,grouped according to the Vs of the cover. Peaks at low frequency appear sharp due to the frequency step ofcalculation (0.1 Hz). Before sampling the maxima, the curves have been fitted with exponential functions toget less biased estimates of the maxima

The 1D equivalent-linear site response simulations for the 585 models is run by usingthe computer code for equivalent linear earthquake site response analysis of layered soils byBardet et al. (2000). As an example, Fig. 4 shows the amplification functions obtained fromthe models with the maximum impedance contrast at 80 m depth.

Let us now plot only the maximum amplification (the main peaks in Fig. 4) for eachtested Vs of Layer 1 as a function of its frequency of occurrence, which depends on thebedrock depth and we obtain the plots shown in Fig. 5. Each line in these plots connects the

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Fig. 5 Amplification factors for the SH wave expected at the resonance frequency as a function of Vs ofthe cover layer and of the impedance contrasts Z (for simplicity, Z is simply the ratio between the Vs of twolayers, being density the same for all layers). The input motion used for the calculation is a Ricker waveletwith 1 s period. The Fa values must be interpreted only as relative (not in absolute sense) because they dependon many other variables and assumptions

points characterized by the same impedance contrast between Layer 1 and Layer 2. Theseare sections of a 4D function of the type:

Fa = f(<Vs>, f0, Z) (2)

This function, only graphically defined, allows to get a quick estimate of the maximumSH-wave amplification factor Fa from the average <Vs> of the cover layer, its resonancefrequency f0 and the impedance contrast Z between Layer 1 and Layer 2.

<Vs>, f0 and Z (or VfZ) constitute the minimal physical basis for a simplified classifi-cation of the stratigraphic site amplification potential.

It is to be noted that the absolute values of Fa depend on many other variables not explicitlyconsidered in the modeling. Figure 5 is therefore to be read only in relative sense (high orlow amplification).

4.1 VfZ versus Vs30

It is not our intention to set boundaries between new site classes because this procedure—ifrigidly instead of statistically interpreted—inevitably adds problems (Mulargia and Castellaro2009). However, let us discuss what would potentially be the benefits of a subsoil classificationbased on <Vs>, f0 and Z rather than on Vs30.

4.1.1 VfZ

We group our 585 soil modes in terms of the expected amplification at low (<1 Hz) orhigh frequency (≥ 1 Hz) and in terms of low (<1.5), intermediate (1.5–2) and high (>2)

amplification. We name these classes C1, C2, . . .C6 as shown in Fig. 6.In Fig. 7 we plot the first 100 m of some of our subsoil models divided according to their

amplification potential into the C1, C2, …, C6 classes. As expected, the classes characterized

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Fig. 6 Example of seismic site classes defined on the basis of the expected amplification frequency (oddclasses: f0 < 1 Hz; even classes: f0 ≥ 1 Hz) and value (Fa)

Fig. 7 Example of subsoil models classified according to the subsoil classes defined in Fig. 6 (zoom of thefirst 100 m depth)

by f0 < 1 Hz (C1, C3, C5) are related to subsoils with strong impedance contrasts atlarger depths than classes (C2, C4, C6). However, several different models give the sameamplification factors and a simple description in terms of Vs ‘averaged’ within a certaindepth (like Vs30 or VsH) is not sufficient to achieve this classification.

The effects of the proposed classification (Fig. 6) on the response spectra for an inputground motion Ricker wavelet with frequency 1 Hz are given in Fig. 8. We find that the

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Fig. 8 Average response spectra for C1–C6 site classes for (left) input ground motion which is a Rickerwavelet with 1 Hz frequency and (right) input ground motion which is a Ricker wavelet with 0.5 Hz frequency

maximum acceleration in the response spectrum is expected on soils with Fa ≥ 2 andf0 ≥ 1 Hz, which is intuitive. The minimum acceleration is expected on soils with Fa <1.5 andf0 < 1 Hz.

For the 0.5 Hz input motion, the maximum acceleration is expected on soil classes withFa ≥ 2 and f0 < 1 Hz, which is again intuitive. The minimum acceleration is expected onsoils with Fa <1.5 and f0 ≥ 1 Hz.

4.1.2 Vs30

The VfZ matrix and the related subsoil classes (Fig. 6) are conceived to distinguish amplifi-cation factors and frequencies and, as it has just been shown, this implies a ‘predictive power’on the response spectra, as a function of the earthquake magnitude.

Approaches based on Vs30/VsH do not have the same capability because they do notexplicitly take into account the main reason for stratigraphic amplification, that is the existenceof an impedance contrast. As a further verification, we group the amplification maxima (andrelated frequencies) of the 585 models as a function of their Vs30 soil class as defined bythe Italian Building Code (NTC 2008) and summarized in Table 1. Note that the Vs30 soilclasses are very similar worldwide, what changes is mostly the class-labeling (A, B, C. . .).

Results are illustrated in Fig. 9 and show that Vs30 cannot effectively discriminate neitherdifferent soil amplifications, nor different frequencies of amplification. Subsoils classified asB, C and D give completely overlapped amplification levels and frequencies of amplification.Class B and Class E results are largely overlapped, too. Additionally, soils classified as Bor C can result in any size of amplification at any frequency of engineering interest. Sincethe bedrock-to-surface amplification function is little sensitive to the specific ground motioninput, this result does not change with the specific input.

Class A, representing subsoils with maximum 5 m cover overlying a stiff bedrock (Vs0 >

800 m/s), obviously leads to amplification at high frequency only. However, it is still largelyoverlapped to soils E and B.

We now analyze the average response spectra derived from our models grouped in theirVs30 site class. Differently from the bedrock-to-surface amplification functions, responsespectra are strongly sensitive to the specific input ground motion used (cfr. also “Appendix”).In Fig. 10a we show the results when the input motion is the 1 Hz Ricker wavelet. We seethat at short periods the highest accelerations are expected for buildings on soil classes C

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Table 1 Vs30-based site classes according to the Italian building code (NTC 2008)

Class Vs30 requirements Other specifications

A >800 m/s Outcropping bedrockB 360–800 m/s Soft rocks, very compacted coarse grained sediments

(NSPT,30 > 50) or very stiff fine grained sediments(cu,30 > 250 kPa), thicker than 30 m and characterized by agradual improvement of the mechanical properties with depth

C 180–360 m/s Compacted coarse grained sediments (15 < NSPT,30 < 50) orstiff fine grained sediments (70 < cu,30 < 250 kPa), thickerthan 30 m and characterized by a gradual improvement of themechanical properties with depth

D <180 m/s Poorly compacted coarse grained sediments (NSPT,30 < 15) orloose fine grained sediments (cu,30 < 70 kPa), thicker than30 m and characterized by a gradual improvement of themechanical properties with depth

E Bedrock (Vs > 800 m/s)within the first 20 m. Vs0 asfor class C or D.

Fig. 9 SH wave amplification factors expected at the resonance frequency for the 585 subsoil models, groupedaccording to their Vs30 site class (Table 1). The complete overlap of class B, C, D and, in large part, E, canbe observed

and E while at long periods there is no significant difference between the classes. When theinput motion is a Ricker wavelet with a lower frequency (e.g. 0.5 Hz in Fig. 10b), the patternchanges at long periods, where the D and C class (those having the bedrock at higher depths)show the maximum response spectra. The frequency band of the maxima shifts from shorterto longer periods, consistently with the input motion dominant period.

The low class D normalized acceleration values at short periods might appear unexpectedwhile they are not since D classes are associated, according to our models, to very deepbedrocks, therefore damping plays a major role at short periods and is responsible for lowacceleration values.

The simplified numerical-modeling approach suggests that the Vs30 parameter is not anideal site response proxy even when response spectra are considered, because the latter arevery sensitive to the specific frequency content of the input motion compared to the subsoileigen-frequency, information which is not included in the Vs30 parameter.

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Fig. 10 Average response spectra for the Vs30 site classes. a Input ground motion is a Ricker wavelet with1 Hz frequency, b input ground motion is a Ricker wavelet with 0.5 Hz frequency

In summary, a simplified subsoil classification has to rely on the minimum physical infor-mation needed to predict stratigraphic amplification, that is on <Vs>, f0, Z or similar com-binations. Note that Vs30 classes, are not unambiguously related neither to Fa, nor to f0.

Since the final goal of site effect assessment studies is to predict the behavior of anoscillator (the structure) founded on another oscillator (the subsoil) and since such behaviorsare both a function of frequency, shifting the reasoning from a depth-dependent approach(Vs30/VsH) to a frequency dependent approach (f0) appears very natural.

4.2 Other simplified seismic soil classification experiences

In Italy, simplified soil classification schemes trying to overcome some limits of the Vs30approach are already applied and are in some way similar to the VfZ approach. Following,e.g., the national guidelines for seismic microzonation (ICMS 2008), a first-approximationSH-wave amplification factor can be derived from specific tables where the entry parametersare the bedrock (Vs0>800 m/s) depth, the average cover <Vs> value, the material type(clay, sand, etc.), the expected Vs gradient of the subsoil (linear, exponential, etc.) and PGA0.Compared to this method, the VfZ approach relies on less inputs and does not require to seta threshold Vs value to define what a bedrock is.

Alternative simplified approaches existed even before at regional scale (e.g. Atto di indi-rizzo per la MZS in Emilia Romagna 2007) that provided a first order estimate of Fa basedon the thickness of the cover layer (H), its ‘average’ <Vs> and the bedrock stiffness (higheror lower than 800 m/s), thus introducing the fundamental concept of impedance contrast.

Further approaches suggesting to exploit Vs from the dispersion curve and f0 from H/Vto estimate Fa have also been recently presented (Cadet et al. 2011).

4.3 VfZ in Practice

There exist a number of ways to measure the basic parameters <Vs>, f0 and Z in the range ofengineering interest ∼[0.1–20] Hz, which corresponds approximately to 1 km to 1 m depth.

f0: A reasonably good estimate of f0 is provided by the microtremor Horizontal to Vertical(H/V) spectral ratio (Nakamura 1989). It has been shown both empirically and theoreticallythat the H/V curve shows peaks at frequencies which are good proxies of the SH-resonancefrequency of the subsoil (SESAME 2004 and references therein). In turn, since the resonancefrequencies are generated by impedance contrasts, the H/V peaks are indicative of the pres-ence of impedance contrasts Z in the subsoil, which is what matters to seismic amplification.

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In principle, it is possible to estimate f0 from the Vs profile through numerical modelingprocedures but this implies that the subsoil model is known down to the seismic bedrock(easily at depth much higher than 50 m for f0 ≤ 2 Hz, which is a depth well beyond thecapabilities of standard geophysical techniques) and the propagation of the uncertainty in allthe parameters used for the model is taken into account. Considering that H/V measurementsare inexpensive in time and money, estimating f0 from H/V rather than from modeling appearspreferable.

<Vs>: There exist a plethora of active and passive multichannel techniques to measurethis parameter. In general—and for different reasons—all these suffer from a limited pene-tration capability in respect to the need of characterizing the subsoil down to 0.1–1 Hz (i.e.depth >100 m). In recent years the combined approach of surface wave based multichanneltechniques (ReMiTM, ESAC, MASW, etc.) and H/V techniques is preferred and allows toget reasonable estimates of Vs ‘down to f0’. Under favorable circumstancies an estimate ofVs can be obtained from the H/V curve alone but only in presence of a constraint (Castellaroand Mulargia 2009).

Z: for low f0, getting the Z that generates that resonance can be hard for the same reasonsdescribed above about Vs. It is widely accepted that microtremors are composed essentiallyof surface waves (mostly Rayleigh with variable amounts of Love waves, Bonnefoy-Claudetet al. 2008; Endrun 2011), therefore the H/V peak amplitude A0 is not linearly related tothe SH-amplification function. Nevertheless, under the assumption above, there still exists afirst approximation relation between the H/V peak amplitude A0 and the impedance contrastthat generates it. In the case of a wavefield composed of 80 % Rayleigh and 20 % Lovewaves (Castellaro and Mulargia 2009), this is shown in Fig. 11. This helps in getting a first-approximation estimate of Z and its uncertainty from the H/V curve. As already pointed out inibid., since smoothing is necessary but affects the H/V peaks amplitude, this should be kept toa minimum standardized value (5–10 %). Another parameter that may affect the experimentalH/V peaks amplitude is a 2D structure of the subsoil. However, since we are dealing withsimplified soil classifications, our discussion applies by definition only to subsoils which canbe approximated as 1D.

We note that reasoning in the frequency domain rather than in the depth domain is verynatural when the VfZ parameters are determined from single station and/or multichannelsurface wave based techniques since these work in the frequency domain. On the opposite,the Vs30/VsH approach requires to shift from the frequency domain (experimental data)to the depth domain (inversion procedures) to the frequency domain again (amplificationfunction and response spectrum).

Fig. 11 Relation between the H/V peak amplitude and the impedance contrast that generates it for differentPoisson’s ratios ν

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Fig. 12 Case 1: a sharp impedance contrast. a experimental (red mean + standard deviation) and theoretical(blue) H/V curve for the model in panel c, b Rayleigh wave phase velocity spectra with the theoretical (blue)dispersion curve for the model in panel c recorded at a site with c about 20 m cover overlying a stiff layer

4.3.1 Case 1: a sharp impedance contrast

Let us consider the case illustrated in Fig. 12, where the subsoil Vs model was obtained bythe joint fit of H/V and the dispersion curve of Rayleigh wave (vertical component). TheH/V curve shows a peak with f0 = 3–4 Hz, A0 = 6 ± 0.5 due to the resonance of about 20 msoft cover (<Vs> = 240 m/s) on a rigid layer (Vs0 ≈ 700 m/s). The impedance contrastamong the cover and the bedrock-like layer is Z ≈ 3.

Since Vs, f0 and Z are known, we can enter the VfZ matrix (Fig. 5) which provides Fa≈ 2.5 at f0 = 4 Hz. To validate the results of the simplified VfZ approach, we perform a1D equivalent linear modeling and get the SH-amplification function illustrated in Fig. 13,which suggests the same Fa.

4.3.2 Case 2: smooth impedance contrasts

Let us now consider a case where Vs increases gently with depth. Figure 11 suggests thata significant H/V peak can be observed only for Z > 1.5. In the same way, significantamplification of SH-waves is expected for Z > 1.5 (Fig. 5).

It follows that a rather flat H/V curve indicates a subsoil with low Z ad will result in lowFa, that can be estimated by following the low impedance contrast curves for each <Vs> ofthe cover in Fig. 5.

We now consider (Fig. 14) the case of a clayey subsoil showing a modest H/V peak atf0 = 0.8 Hz, the existence of which is however acknowledged in the area by a number ofother surveys. The joint fit of the H/V and dispersion curve gives the Vs profile illustratedin panel C. The dispersion curve provides information down to the maximum depth of 20 m(the maximum explored wavelength is about 40 m) and the Vs model below this depth isprovided by the fit of the H/V curve.

The fundamental site frequency (f0 = 0.8 Hz) is traced back to a bedrock-like layer atabout 70 m depth. We have <Vs> = 230 m/s and Z = 1.7. According to the simplified VfZ

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Fig. 13 SH-amplification function for the subsoil of Fig. 12 from a 1D equivalent-linear site response analysis.This gives Fa(SH) ≈ 2.5, which is the same result obtained from the simplified VfZ approach for <Vs> =240 m/s, f0 = 3–4 Hz and Z = 3

Fig. 14 Case 2: smooth impedance contrasts. a experimental (red mean + standard deviation) and theoretical(blue) H/V curve for the model in panel c, b Rayleigh wave phase velocity spectrum (blue theoretical dispersioncurve for the model in panel c), c Vs model for the subsoil obtained from the joint fit of the two surveys. Thesurvey in b provides information down to 5 Hz only, the rest of the model below this frequency was obtainedfrom the H/V modeling

approach, the expected amplification Fa = 1.8 (Fig. 5). To validate the results of the simplifiedVfZ approach, we perform a 1D equivalent linear modeling and get the SH-amplificationfunction illustrated in Fig. 15, which suggests Fa = 1.6 at the resonance frequency.

4.3.3 Case 3: several impedance contrasts

Let us now see how a subsoil with more than one relevant impedance contrast might beclassified according to the simplified approach. We consider the case of Fig. 16, where two

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Fig. 15 SH-amplification function for the subsoil of Fig. 14 from a 1D equivalent-linear site response analysis.This gives Fa(SH) ≈1.6, which is close to the 1.8 obtained from the simplified VfZ approach for <Vs> =230m/s, f0 = 0.8 Hz and Z = 1.7

Fig. 16 Case 3: several impedance contrasts. a experimental (red mean + standard deviation) and theoretical(blue) H/V curve for the model in panel c, b Rayleigh wave phase velocity spectrum (blue theoretical dispersioncurve for the model in panel c), c Vs model for the subsoil obtained from the joint fit of the two surveys. Thesurvey in b provides information down to 5 Hz only, the rest of the model below this frequency was obtainedfrom the H/V modeling

H/V peaks appear at 0.5 and 5 Hz. The latter is related to an overconsolidated clay layer atabout 10 m depth, while the low frequency peak is related to the local bedrock at about 200 mdepth.

The joint fit of the H/V and dispersion curve (Fig. 16a, b) gives the Vs model of Fig. 16c.The 1D equivalent-linear modeling suggests Fa = 1.7 at f0 = 0.5 Hz (Fig. 17) and thesimplified VfZ approach for <Vs> = 300 m/s and Z = 2, gives Fa = 1.8.

By considering f0 = 5 Hz as the relevant frequency, we would have <Vs> =100 m/s, Z < 2 and a negligible Fa (Fig. 5).

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Fig. 17 SH-amplification function for the subsoil of Fig. 16 from a 1D equivalent-linear site response analysis.This gives Fa(SH) ≈ 1.7–1.8 at f0, which is the same result obtained from the simplified VfZ approach for<Vs> = 300 m/s, f0 = 0.4 Hz and Z = 2.5

5 Discussion and conclusions

Since the final goal of site effect assessment studies is to predict the behavior of an oscillator(the structure) coupled to another oscillator (the subsoil), it is convenient to shift the reasoningfrom a depth-dependent approach (Vs30) to a frequency dependent approach (f0).

By observing that the main cause for stratigraphic seismic amplification is the existence ofimpedance contrasts in the subsoil, we propose a simplified seismic site classification schemebased on three parameters: <Vs>, f0 and Z (in short, VfZ), that is the average velocity ofthe cover layer, the resonance frequency of the subsoil and the impedance contrast betweenthe cover and the bedrock or pseudobedrock.

The 1D numerical analysis of seismic response of several subsoils (all characterized byincreasing Vs with depth) allows to generate a 4D function relating the amplification factorexpected for the SH-wave, Fa, to (<Vs>, f0, Z).

VfZ constitutes the minimum physical basis for a first-order approximation of stratigraphicamplification and can be measured with the same instrumental effort required to measureVs30. Among the several techniques available, the joint fit of surface-wave based multichan-nel techniques and H/V or—under favourable circumstances—the H/V alone accompaniedby an adequate knowledge of the site stratigraphy appear to offer the best cost effective per-formance. In particular, the H/V provides a sufficiently reliable estimate of f0 while the jointfit of dispersion curves and H/V provide an adequate estimate of <Vs> down to f0. Z canbe inferred from the H/V peak amplitude or from the Vs profile.

We have shown that the proposed simplified classification scheme VfZ can be used alsoon sites where no specific resonances are measured (due to the absence of sharp impedancecontrasts) and on soils presenting several resonances.

A common objection to the proposed method is that, once the VfZ parameters are knownfor a site, one could—with a modest additional effort—perform a complete 1D numericalmodeling rather than applying the simplified procedure.

Actually, the application of numerical models require the study of seismic response withseveral (3–9, depending on the specific regulation) different inputs (earthquakes), in order toprovide an average value and its uncertainty (which is often ignored because not requested

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by law) and also requires the user to operate several other input choices (water table position,shear modulus and damping curves as a function of strain rate for all layers, position of thesource, choice of the characteristic earthquake, etc.). All this makes the 1D numerical proce-dure more cumbersome and give results which offer just an illusory accuracy, as discussed inthe introduction. The use of full accelerograms in 1D numerical models conceived to modelvertically incident SH waves is not fully justified and is the reason why we have based ourmodels on simple Ricker wavelets.

The VfZ matrix helps also in the interpretation of the H/V curves. In fact, it allows arough estimate of the expected SH-wave amplitude from the H/V peak amplitude and allowsto understand the amplification potential of the observed peaks.

It is important to emphasize that the Fa values provided by the VfZ matrix (and in generalby any simplified procedure) should only be interpreted as relative values (low or high ampli-fication) because they depend on a very large number of other parameters and assumptionsnot explicitly taken into account by the models.

Final note The aim of this work is not to propose a further series of tables to estimate Fabut only to indicate what should be the minimum physical parameters required to achievea simplified seismic soil classification and to show why it is preferable to adopt simplifiedbut rational classifications rather than more articulated—but only illusively more accurate—classifications.

Acknowledgments We thank the two very competent anonymous reviewers for their accurate observations.

Appendix

We have used the 1D equivalent linear (1D-EQL) analysis, which gives reasonable estimatesof ground vibration under a seismic event (Idriss 1990) and is generally conservative overthe results of nonlinear methods or real records from downhole acceleration arrays. In thesecond case, the difference between the 1D-EQL approach and experimental observationscan be due to both non-linear behaviours but also to the fact that many input parameters of themodels are measured on very small laboratory samples compared to the scale of the physicalphenomenon. As a result, damping, stiffness versus shear strain curves and other parametersmeasured in the laboratory are usually conservative compared to what one would measureon the field, where the volume of investigation includes layering, fractures and many moreheterogeneities than those existing at the laboratory scale.

It has been reported in the literature (Sugito et al. 1994; Yoshida et al. 2002; Kausel andAssimaki 2002) that at very low periods the 1D-EQL approach under predicts the motionas an effect of high-frequency overdamping and that results are unrealistic in the case ofhigh shear strain levels (>0.1 %) induced by seismic excitation. On the contrary, Kwok et al.(2007) have shown that soil models with low modal frequency (approximately <1 Hz) areexpected to result in lower amplification values.

Given such limits for the 1D-EQL approach, our models appear usable at strain levelsapproximately <0.1 %.

Additionally, as input motion for our models we used Ricker wavelets rather than realearthquake recordings. It is well known that the amplification transfer function dependsstrictly on the subsoil and not on the excitation function. This can clearly be seen in Fig. 18,where the amplification transfer function obtained by using a 1 Hz and a 0.5 Hz Ricker wavelet(red) is compared to the average (±2σ) transfer function obtained from 10 real accelerograms(PGA0 = 0.35 g) on 3 ‘typical’ subsoil models, i.e.:

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Fig. 18 Surface-to bedrock transfer functions obtained from a set of 10 real accelerograms (average ±2sigma interval in black, solid ± dashed lines) compared to the usage of 2 Ricker wavelets as input motion(1 and 0.5 Hz, respectively, in red). The subsoil is represented by (left) 30 m cover (Vs = 200 m/s) overlyingthe bedrock, (centre) 100 m cover (Vs = 200 m/s) overlying the bedrock, (right) 200 m cover (Vs = 500 m/s)overlying the bedrock. In all cases the bedrock is represented by a 800 m/s half space

Fig. 19 Response spectra obtained from a set of 10 real accelerograms (average ± 2 sigma interval in black,solid ± dashed lines) compared to the usage of 2 Ricker wavelets as input motion (1 and 0.5 Hz, respectively,in solid and dashed red, respectively). The subsoil is represented by (left) 30 m cover (Vs = 200 m/s) overlyingthe bedrock, (centre) 100 m cover (Vs = 200 m/s) overlying the bedrock, (right) 200 m cover (Vs = 500 m/s)overlying the bedrock. In all cases the bedrock is represented by a 800 m/s half space

1) Left: 30 m (Vs = 200 m/s, f0 = 1.7 Hz) cover overlying the bedrock,2) Centre: 100 m (Vs = 200 m/s, f0 = 0.5 Hz) cover overlying the bedrock,3) Right: 200 m (Vs = 500 m/s, f0 = 0.6 Hz) cover overlying the bedrock.

In all cases the bedrock is characterized by Vs = 800 m/s and no difference is observedbetween the results produced by different input motions.

On the other hand, the response spectrum is mostly dependent on the input motion, so thatwhen we use as excitation function a Ricker wavelet with a specific frequency, we expectto get the maximum of the response function more or less around that frequency. This canclearly be seen in Fig. 19 where we used 1 Hz (red) and 0.5 Hz (dashed red) Ricker waveletsas input motions and we had the maximum of the response spectrum at approximately 1 and2 s. The subsoil specific properties certainly affect also this function but to a minor extent.

The models also suggest that the Ricker wavelet is a good proxy for the average responseto real accelerograms for periods around the Ricker wavelet proper period. Therefore, underthe linear elastic assumption, a Ricker wavelet with frequency 1 Hz is adequate to provideresponse spectra at periods ≤1 s, a Ricker wavelet with frequency 0.5 Hz is adequate toprovide response spectra at about 1.5–2.5 s etc.

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