The fundamental frequency of a site with the Nakamura’s method

The fundamental frequency of a site with the Nakamura’s method

The fundamental frequency of a site with the Nakamura’s method: The areas with a homogenous seismic behavior are identified with the methods of seismic micro-zoning which record, analyze and interpret the environmental seismic noise.

The HVSR (Horizontal to Vertical Spectral Ratio) method, named also Nakamura’s method, is one of the most used to determine the transfer function of a site, and then its resonant frequency; in fact, according to this method, the ratio between the Fourier amplitude spectra of horizontal (H) to vertical (V) components of the ambient noise vibrations assumes the maximum value at the resonant frequency.

The site effect is the ratio between the two horizontal components of seismic motion, at the surface (S) and that at the bedrock (B), depending on the frequency:

SE(ω) = HS(ω) / HB(ω)               (1)

The source spectral shape, instead, is defined as the ratio between the vertical components:

AS(ω) = VS(ω) / VB(ω)               (2)

So, the modified effect of site (the effect of site compensated with the spectral shape) results:

SM(ω) =SE(ω) /AS(ω) = [HS(ω) * VB(ω)]/[HB(ω) * VS(ω)]          (3)

Nakamura has experimentally obtained that, at the frequencies of interest,
HB/VB = 1, so the modified site effect is the ratio of horizontal to vertical components, both at the surface:

SM(ω) =HS(ω) /VS(ω)              (4)

Therefore, it’s possible to estimate the local seismic response without a reference site.

The Nakamura’s method assumes that:

  • the microtremors consist mainly of Rayleigh waves, which propagate in a surface layer overlaid on to a rigid substrate of rock;

  • the sources of microtremors are local (the contribution of sources in depth is neglected);

  • the vertical component isn’t affected by amplification of the surface layer;

  • the surface sources do not affect the characteristics of the motion at the top of the bedrock;

  • the amplification due to the site effects is caused by a sedimentary layer lying on an elastic half space.

The main advantages of this method are: no energizations, low costs, ability to obtain information by recording single station measurements of ambient vibration.

Criteria for a reliable H/V curve

A H/V curve is reliable if the following conditions are satisfied:

f0 = 10/lw             (5)

nc(f0) > 200             (6)

σA(f) < 2     per     0.5f0 < f < 2f0     se     f0 > 0.5 Hz             (7)

σA(f) < 3     per     0.5f0 < f < 2f0     se     f0 < 0.5 Hz                   

where: lw is the length of time windows without transients, expressed in seconds; f0 is the frequency of the peak; nc is the number of significant cycles, defined as: nc = lw * nw * f0 (nw is the number of selected windows); σA is the standard deviation of the amplitudes.

Equation 5 means that, at the frequency of interest, there be at least 10 significant cycles in each window; equation 6 establishes the minimum number of significant cycles; equation 7 assigns un upper limit to the standard deviation.

Further, a peak is clear if at least five of the following conditions are satisfied (clarity conditions):

Amplitude conditions:

∃ f ∈ [f0/4, f0] | AH/V(f) < A0/2            (8)

∃ f+ ∈ [f0, f0/4] | AH/V(f+) < A0/2            (9)

A0 > 2            (10)

Stability conditions:

fpeak[AH/V(f) ± σA(f)] = f0 ± 5%            (11)

σf < ε(f0)            (12)

σA(f0) < θ(f0)            (13)

where: AH/V(f) is the amplitude curve at the generic frequency f; A0 is the peak’s amplitude at the fundamental frequency f0; σf is the standard deviation of the peak’s frequency (f0 ± σf); ε(f0) and θ(f0) are the threshold values for stability conditions: in table 1 the recommended values are listed:

Tab. 1: recommended values for the thresholds of the stability conditions.

Frequency range [Hz] ε(f0)[Hz] θ(f0) [Hz]
< 0.20 0.25 f0 3.00
0.20 – 0.50 0.25 f0 2.50
0.50 – 1.00 0.15 f0 2.00
1.00 – 2.00 0.10 f0 1.78
> 2.00 0.05 f0 1.58 

Mode of data acquisition

During the measurements, some precautions are necessary. We list them below.

It’s advisable to note the occurrence of disturbance factors and the weather conditions on the measurement field sheet.

  • The sensors: the natural frequency of the sensor must be lower that the frequency of interest. All three channels must have tha same gain and must be syncronised.

  • The recording duration: the length of the recording must satisfie the reliability conditions (eq. 5, 6 and 7). Table 2 shows the acquisition’s times recommended.

Tab. 2: recommended values for the recording time.
f0 [Hz] Min. lw [s] Min. num. of significant cycles Min. num. of windows Min. duration of the useful signal (s) Min. recording duration [min]
0.2 50 200 10 1000 30
0.5 20 200 10 400 20
1 10 200 10 200 10
2 5 200 10 100 50
5 5 200 10 40 30
10 5 200 10 20 2
  • Soil-sensor coupling: the coupling between the sensor and the soil is very important to ensure that measurements are meaningful. In fact, the signal could be perturbed if the soil is soft or irregular, for example grass, mud, uncompacted snow, ice, gravel. Generally, concrete or asphalt does not affect the measurements. In case of snow, it’s appropriate to install the sensor in a metal box in order to avoid sensor tilting due to local melting under the sensor legs; the high grass should be removed, both in order to improve coupling and to avoid that the wind entirely invalidates the measurements. If the ground is sloping, the sensor must be placed in a metal box filled with sand.

  • Disturbances: in urban areas, subsurface structures, traffic areas and industries should be avoided, as far as possible; it could be very useful to record measurements for several days (also during the night). Near the highways (short-term high-amplitude sources), the acquisition must be carried out at least 15-20 meters; the most continuous sources (urban environment) alter the measures even at shorter distances, but since the signal is less impulsive, it is preferable to record in the city rather than near to highways.

  • Weather conditions: wind, heavy rain and pressure’s variations can strongly alter the signal, specially at low frequencies. Measurements during unfavorable weather conditions should be avoided.


Data processing

In the framework of the European research project SESAME (Site EffectS assessment using AMbient Excitations) the following processing sequence has been established:

  • selection of the most stationary time windows for all three components; an anti-triggering algorithm STA/LTA (short term average/long term average) detects transients and avoid the windows which contain them. Tha ratio STA/LTA should be lower than a small threshold (tipically around 1.5-2) over a long enough duration. Sometimes it’s useful to use a minimum threshold, in order to avoid ambient vibrations;


Fig. 1: subdivision of the acquisition time in stationary windows on all channels simultaneously.

  • computation of the amplitude spectra of any component for each window;

  • smoothing of the amplitude spectra for each time window: this step must be done with great care, because a very strong smoothing increases the numerical stability and make the results’ interpretation more easy, but it can hide peaks that must not be neglected;

  • computation of the quadratic mean of the two horizontal components of the spectrum, for each window:

SH(nw) = √(SNorth-South(nw)2 + SEast-West(nw)2)/2            (14)

  • computation of the H/V ratio for each window:

SH(nw)/SV(nw)            (15)

  • geometric mean of H/V ratios for all windows;

  • determination of the frequency of the peak;

  • curve’s interpretation.



If the measurements satisfie the reliability criteria abovementioned, the H/V curve is charaterized by a single peak, very pronounced.

His amplitude does not have any physical meaning; some researchers interpret the value of the amplitude as an estimate of the amplification of the site, but it’s very important keep in mind that the absence of any peak in the curve H/V does not imply absence of amplification.

Then, the presence of a peak can be seen as a sufficient but not necessary condition for establishing that at a given site there is amplification.

However, an amplitude value of about 4-5, implies that, at a given depth, there is a strong impedance contrast.

It’s necessary verify that the frequency of the peak is compatible with that of the sensor f > f0, and that the peak has not industrial origin (for example, by varying the smoothing: peak of non-industrial origin does not change).

Special cases:

  • industrial origin peak: the peak appears very pronounced in all components; it’s advisable to record measurements for several days or at different hours;

  • not very pronounced peak: in this case the first condition for the clarity (eq. 8) is not satisfied and, sometimes, even the second one (eq. 9). This occurs mainly at lowest frequencies: it’s rather difficult to determine the cause, as it may be:

  • a site with frequency very low;

  • an impedance contrast not very strong (<4);

  • measurements carried out during windy days;

  • poor coupling soil-sensor;

  • sensor unsuitable: f > f0;

  • low frequency sources close;

  • inappropriate smoothing.

In this case it’s recommended to check the site’s geology (if the site is rocky the frequency must be discarded), the weather conditions, the frequency of the sensor and the smoothing’s parameters; it’s recommended also to verify if the amplitude of the curve near to f=0 is >>2 (this effect is due to the wind, traffic, or the sensor used); if the fourth condition for the clarity (eq. 11) is not satisfied, the data must be processed again, using a larger time window.

  • broad peak: if the peak is broad also after varying the smoothing, probably there is a sloped interface.

  • multiple peaks: if the H/V curve has multiple peaks, it’s recommended to increase the bandwidth of the smoothing, or repeat the measurements. If there are two strong impedance contrasts at two different scales, the H/V curve can have two peaks that satisfie the abovementioned criteria. So, the geology must be check: if this is the case, both the frequencies are characteristic of the site, the lower is the fundamental and there may be amplification for frequencies between f0 e f1.

For a non-rocky site, if the H/V curve is characterized by a very pronounced, non-industrial origini peak, it’s possible to deduce that there is a strong impedance contrast that can amplifie the soil motion. The frequency of the peak is the fundamental frequency of the site. If the thickness h of the layer is known, it’s possible to estimate the propagation’s velocity of S waves, with the equation: VS = 4*f0*h. Conversely, knowing the velocity, the layer thickness is obtained.


Fig. 2: example of a reliable H/V curve with a clear peak: all the criteria for the reliability and the clarity are satisfied. The peak is very pronounced (his amplitude is about 6), the fundamental frequency F0 is 0.7 Hz. Depth of the bedrock: 196 m: type: gneiss.

chiara cocorullo

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