2656

CLEAR IDENTIFICATION OF FUNDAMENTAL IDEA OF NAKAMURA'STECHNIQUE AND ITS APPLICATIONS
Yutaka NAKAMURA1

SUMMARY
A method which employs microtremor has been introduced for estimating dynamic characteristics of surface
layers, in early 1950. Then usage of this method has received lots of criticism considering uncertainty aboutsource of microtremor. After an introduction of the Nakamura’s technique (H/V or QTS technique;
Nakamura, 1989), many people have paid a renewed great attention for estimating dynamic characteristics ofground and structures using microtremor, since clear and reliable information was provided by very simple
and inexpensive noise measurements.
In recent years, although several researchers claimed that theoretical ground of this technique is not clear andconsensus based on experiment couldn’t be reached, there have been many successful experimental studies
based on these technique. Many theoretical studies have been performed, for explaining the amount of types
of waves included in microtremor and checking the applicability of the QTS technique. And some of them
are suggested that the peak on H/V ratio can be explained with the fundamental peak of Rayleigh waves.
From the output of these researches, explanation of microtremor with Rayleigh waves caused some confusion
between users and the author decided to clear out this problem.
The basic idea and the main goal of QTS technique are tried to be re-explained in present paper. The author’s
explanation about the effects of contents of Rayleigh waves in microtremor is also given. Other possible
usage of products from QTS technique (predominant frequency and amplification factor) for hazard
estimation is also given. As it is well known, occurrence of earthquake damage depends upon strength,
period and duration of seismic motions. And these parameters are strongly influenced by seismic response
characteristics of surface ground and structures. This reality makes investigation of vulnerability of groundand structures an important issue, before the earthquake occurs. For this purpose, vulnerability indices called
K values were proposed by Nakamura (1996). K values are simply derived from strains of ground andstructures. Formulation of K values for ground (Kg) and some application examples are also given in present
paper. These new values give a chance to estimate vulnerabilities of all types of structures and ground, before
the real damage occurs.

INTRODUCTION
Damages caused by the recent earthquakes are concluded as a direct result of local geological conditions
affecting the ground motion. Best approach for understanding ground conditions is through direct observation ofseismic ground motion, but such studies are restricted to areas with relatively high rates of seismicity. Becauseof these restrictions in other methods, such as high rates of seismicity and the availability of an adequate
reference site, non-reference site methods have been applied to site response studies. Microtremor is a very
convenient tool to estimate the effect of surface geology on seismic motion without needing other geologicalinformation.
H/V (or QTS, Quasi-Transfer Spectra) technique fits very well to this description and it has received great
attention from all over the world with its simplicity together with quick information about dynamic
characteristics of ground and structures. Although several researchers claimed that theoretical background of
this technique is not clear, there have been many successful experimental studies performed. Method is
attractive since it gives the ease of data collection and it can be applied in areas of low or even no seismicity.

1

System and Data Research Co. Ltd., 3-25-3 Fujimidai, Kunitachi-shi, Tokyo, Japan yutaka@sdr.co.jp

H/V technique was developed by author with relating bore hole investigations together with strong motion
records analysis, on the various geological site conditions. It was hypothesized that the vertical componentof ambient noise keeps the characteristics of source to sediments surface ground, is relatively influenced byRayleigh wave on the sediments and can therefore be used to remove both of the source and the Rayleighwave effects from the horizontal components. It is effective to identify the fundamental resonant frequency
of a sedimentary layer, with implied amplification factors that are more realistic than those obtained fromsediment to rock site ratios. It has been shown by many researchers (ex. Ohmachi et. al., 1991; Lermo et. al.,
1992; Field and Jacob, 1993, 1995) that how such H/V ratio of noise can be used to identify the fundamental
resonant frequency and amplification factor of sediments.
Looking to the examples in the study of Nogoshi and Igarashi (1971) which compared H/V of Rayleigh wave
with that of microtremors and concluded that microtremor was mostly composed of Rayleigh wave, some oftheoretical studies (Lachet and Bard, 1994; Konno and Ohmachi, 1998; Bard, 1998) suggested that the peak on
H/V can be explained with the fundamental mode of Rayleigh waves. If we think that this approach is true,
microtremor should be considered to consist of only Rayleigh wave. On the other hand, if we check theexamples given on Nogoshi and Igarashi (1971) carefully (which will be discussed later), we can clearly see
that, at the peak frequency of H/V of Rayleigh wave, the energy of Rayleigh wave is very small, nearly close
to the zero. Rayleigh wave has its maximum energy at near trough frequency of H/V. Because of this, peakof H/V cannot be explained by Rayleigh wave energy. As it is explained by Nakamura(1989), H/V ofmicrotremor at peak frequency range can be explained with vertical incident SH wave.
2. WAVES IN MICROTREMOR AND CHARACTERISTICS OF STRONG GROUND MOTION
Many observations and experiences on microtremor records show that microtremor consists of both body andsurface waves, but there is no established theory concerning what kinds of wave motions the microtremor ismade up from. In Nakamura (1989), the purpose of the author was the estimation of the amplification factor
caused by multiply refracted vertical incident SH waves. For this purpose, Rayleigh wave contained by
microtremor was considered as noise and eliminated inside the H/V process.

Strong Motion Observation Site

Izumikoshi Tunnel
Geological

Atamai
Izumikoshi

Atamai
Izumikoshi

Atami / Izumikoshi Ha / Hi

Fresh Rock

NS
EW
UD

Maximum Acceleration of H Component

Weathered RockMax. Acc. of Atami (Surface)

Seismic experiences shows that strong motion at the rock site in horizontal and vertical components does notshow much differences. Also if we take the H/V ratio on the record measured simultaneously at the groundsurface and basement, we can see that maximum acceleration ratio is consistent as shown in Figure 1.

Cross-section

Max. Acc. of Izumikoshi (Basement)

Maximum Acceleration of V Component

Atami Ha/Va

Figure 1. Relation of Maximum accelerations on surface and basement (Nakamura and Saito, 1983)

Figure 2 shows the difference in records caused by difference in earthquakes and observation sites. As we canfollow, records in different stations even for the same earthquake are different because of the different sitecharacteristics. On the other hand, for different earthquakes same site gives almost same type of records. In
other words, it may be said that, affecting dynamic characteristics, the effect of surface layer is most criticalamong the other factors.
For the investigation of dynamic characteristics, boring is one of the most accurate method, but it is costly andtime consuming and is not available all the time. If we measure both at surface Hf and basement Hb and takethe ratio we can get the transfer function. Method which employs microtremor is well known and there may
be some cases that transfer function calculated from Hf/Hb ratio and from velocity model of boring data show
some differences. The reason for this is basically the evidence of surface waves. In practice Rayleigh wave
should be eliminated to estimate transfer function correctly. H/V method (Nakamura, 1989) was introduced2

for this purpose and verified with both strong ground motion and microtremor data.

Figure 2.

Difference in Accelerogram due to difference in earthquake and observation stations(Nakamura, 1989)

3. RELATION BETWEEN H/V SPECTRUM OF MICROTREMOR AND H/V OF RAYLEIGH WAVES

Nogoshi and Igarashi (1971) gives examples on
different sites. Figure 3 shows two of theseexamples for the microtremor records in
Hakodate, Japan. It can be clearly seen fromthese figures that, energy of Rayleigh wave does
not appear on the peak of H/V of Rayleigh wave.
We can easily see that there is no energy aroundthe peak frequency of H/V and amplitude isalmost zero for horizontal and zero for verticalhvofmt0.5
0.4
HV

Nakamura (1989), gives theoretical definition of
the H/V technique with multiple refraction of SH
waves. On the other hand, there is a group of
researchers who try to explain the peak of H/Vratio with the evidence of Rayleigh waves. As
it is mentioned before, microtremor consists ofmany kinds of waves. Link to the surface
waves is based on the assumption that noise
predominantly consists of surface waves.
Under this assumption Bard (1998) explains that,many researchers are agree on following two
arguments; First, H/V is basically related to theellipticity of Rayleigh waves because of the
predominance of Rayleigh waves in verticalcomponent.
Second, this ellipticity isfrequency dependent and exhibits a sharp peakaround the fundamental frequency for sites
displaying a high enough impedance contrast
between the surface and deep materials. Thisapproach, simply comes from the similarity of
the figures of H/V ratio of microtremor and H/Vof fundamental mode of Rayleigh waves, but justlooking to this similarity this conclusion cannot
be reached.

h
v

0.3
0.2
0.1
0
0

5

10
freq in Hz

15

20

Å
This figure, Fourier Spectrum of
H and V with linear frequency
axes, is derived from the above.
Å-----Æ
H and V ofRayleighwave

0-Level of Amplitude

Å0-lineÆ

0-Level of Amplitude

Figure 3. Relation of H, V and H/V for microtremorand for Rayleigh wave (Nogoshi & Igarashi, 1971)3

velocity of Rayleigh waves. For trough and minimum group velocity(for almost all impedance ratio values) frequency changes between 1.52. On the other hand, only for the peak frequency varies in a wider
range for different impedance ratio. The energy of Rayleigh wave isalmost zero at peak frequency of H/V, and at the trough frequency of
H/V the energy becomes to maximum. When the impedance ratio isgreater than 2.5, Rayleigh wave does not affect to the H/V peak ofground motion . And on the other hand, when the impedance ratio isless than 2.5, Rayleigh wave affects to the H/V peak of ground motion.
It is consistent with the experience of many researchers measuring
microtremor. Figure 7 shows the change of H/V ratio of peak andtrough of Rayleigh wave versus impedance ratio. In this figure, forimpedance ratio around two, there is not a big difference in the H/V ofRayleigh wave. For the impedance ratio bigger than two, H/V ratio
for peak of Rayleigh wave increases immediately. It seems impossible
to estimate the impedance ratio from the H/V peak of Rayleigh wave.
All these examples prove that, as it was explained by Nakamura(1989),
H/V peak is mostly related with SH wave and cannot be explained with
the Rayleigh waves.
Relationship between Characteristics ofRayleigh Wave and Impedance Ratio
Peak, Trough of H/V of Rayleigh Wave

Peak and Trough Frequencies of H/V,
Frequency of Minimum Group Velocity
in unit Cs/4h

1

0.1
0.1

1

10

Frequency in unit Cs/4h
Figure 4.

H/V of Rayleigh wave
for two layered ground

1

0
1

2

3

Frequency in unit Cs/4h
Figure 5.

Group velocity ofRayleigh waves

1 0 .03
trough frequency of H/Vpeak frequency of H/V
freq. of min. group velocity

2.5

H/V of Rayleigh wave

For different wave-lengths, Ohta (1962) calculated H/V and phasevelocity of Rayleigh waves for two layers model for various impedance
ratio (varying between 1.2-4.5) and Poisson’s ratio (varying between
0.25-0.49) both in sedimentary and basement layers.
By using thesecalculation results, H/V of Rayleigh wave versus frequency is drawn in
Figure 4 to show the relations between impedance, peak and frequency.
And frequency in this figure is normalized with Cs/4h, Cs is S wave
velocity and h is depth of sediment. Group velocities calculated fromphase velocities are normalized with S wave velocity and drawn withthe frequency normalized with Cs/4h. Figure 5 shows the distributionof group velocity versus frequency. As we can follow from this figure,
if impedance ratio is less than two, energy of Rayleigh wave distributes
to the lower frequencies. Figure 6 shows the change of impedance
ratio for frequencies of trough and peak of H/V and minimum group

10Group Velocity in unit Cs

components of Rayleigh waves. On the other hand, Rayleigh wave
energy gets its maximum on later frequencies at minimum groupvelocity of Rayleigh wave and this is nearly equal to trough frequency
which is almost two times of the H/V peak frequency.

2
1.5
1
0.5
0

1 .0

Peak of H/V
Trough of H/V

0 .1

0

1

2
3
Impedance Ratio

4

5

1

10
I m p e d a n c e R a tio

Figure 6. Relationship between characteristics ofRayleigh wave and impedance ratio.
4

Figure 7. Relationship between peak andtrough of H/V of Rayleigh wave
and impedance ratio

5. EXPLANATION ON H/V RATIO (QTS) TECHNIQUE
Figure 8, shows the typical geological structure of
sedimentary basin. Definition of ground motions and theirspectra at different places are defined in following lines.
Here microtremor is divided into two parts considering it
contains Rayleigh wave and other waves. Then, horizontal
and vertical spectra on the surface ground of the sedimentarybasin (Hf, Vf) can be written as follows.
H f = Ah * H b + H s
Th =

H f

V f = Av * V b + V s
Tv =

Hb

Outcrop of RockHr, Vr = Hb, Vb
Hf, VfBasement
GroundSurface Ground(1)

Hb, VbFigure 8.

Vf

(2)

Vb

Typical geological structure of
sedimentary basin

where Ah and Av are amplification factor of horizontal and vertical motions of vertically incident body wave.
Hb and Vb are spectra of horizontal and vertical motion in the basement under the basin (outcropped basin). Hs
and Vs are spectra of horizontal and vertical directions of Rayleigh waves. Th and Tv are amplification factor ofhorizontal and vertical motion of surface sedimentary ground based on seismic motion on the exposed rockground near the basin. In general, P wave velocity is more than three-four times of S wave velocity. In such
sedimentary layer, vertical component cannot be amplified (Av=1) around the frequency range where horizontal
component receives large amplification. If there is no effect of Rayleigh waves, Vf ≅ Vb. On the other hand, if
Vf is larger than Vb, it is considered as the effect of surface waves. Then estimating the effect of Rayleighwaves by Vf/Vb (=Tv), horizontal amplification can be written as,Hs  Ah + 
Vf
Hb 
T
QTS 
*
Th = h =
=
=
Hb
Tv Hb

V
Av + s 

Vb
Vb
Vb 

Hf


Hs 
Ah + H f Ah * Hb + Hs Hb 
Hb 
=
= ⋅
QTS=
Vf
Av *Vb +Vs
Vb 
Vs 
Av + Vb 
where ,

(3)

(4)

In equation (4), Hb/Vb ≅ 1. Hs/Hb and Vs/Vb are related with the route of energy of Rayleigh waves. If there isno influence of Rayleigh wave, QTS = Ah/Av. If amount of Rayleigh wave is high, then second term in above
formulation gets dominant and QTS = Hs/Vs and the lowest peak frequency of Hs/Vs is nearly equal to the lowest
proper frequency Fo of Ah (see, Figure 6). In the range of Fo, Av=1. QTS shows stable peak at frequency Fo.
Even when influence of Rayleigh wave is large, Vs become small (which results in a peak of Hs/Vs) around thefirst order proper frequency due to the multiple reflection of horizontal motions. And QTS = Ah, if
microtremors of the basement Vb is relatively large comparing to the Rayleigh wave. Briefly, QTS represents
the first order proper frequency due to multiple reflection of SH wave in the surface ground layer and resulted
amplification factor, regardless of the influence degree of Rayleigh waves.

If influence of Rayleigh wave becomeslarger, QTS<1 in the wide range of
frequency and if influence is small, QTS
locally expected to be smaller than one, inthe narrow frequency range at frequency
several times higher than Fo, because of theinfluence of vertical motion.

18
16
Amplitude/Amplification Factor

Figure 9, shows a schematic comparison of
Horizontal (Hf), Vertical (Vf), Hf/Hb (spectral
ratio of sediment site to the reference site)and Hf/Vf (H/V technique). As it can befollowed QTS is smaller than the theoreticaltransfer function. Since Hf includes theeffects of Rayleigh waves, Hf/Hb is bigger
than the theoretical transfer function.

Vf=Av*Vb+Vs

14
12
10
8
Hf=Ah*Hb+Hs, HH=Hf/Hb
6
Ah: Transfer Fanction of SH

4
2

Hb

QTS=Hf/Vf0
0

1

Figure 9.

23Frequency in unit Fo

45

Schematic comparison of Horizontal (Hf),
Vertical (Vf), Hf/Hb (spectral ratio of
sediment site to the reference site)

Main waves consisting of microtremors are either body waves or Rayleigh waves, or depending on the locationand other conditions can be mixture of both waves. For such microtremors, if we calculate QTS, first orderproper frequency by multiple reflections of SH wave in the surface layer and its amplification factor can becalculated correctly. More explanation regarding Rayleigh waves can be found in Nakamura(1996).
5

6. BASEMENT DEPTH RELATED WITH QTS
From above discussions we can now
conclude that peak of QTS is caused bymultiple refraction of S waves andQTS(Quasi Transfer Spectrum) represents themeaning coming from its name. On theother hand, here it should also be addressed
the depth of the basement related with QTS.
The frequency Fo related with QTS is;

Cs
(5)
4h
and the amplification factor A for thisfrequency is related with impedance ratio.
If densities for basement and surface layer aresame then,

A0 =

Cb
Cs

Depth in km

F0 =

(6)

and depth of basement h is,h=Cb
4 A0 ⋅ F0(7)

Cb is S wave velocity of basement.
estimated depth by QTS method
meacured depth of basement groundFigure 10, shows the depth of basement alongaShinkansen
line
estimatedfrommicrotremor for the case of basement velocity
Cb=600m/sec. The comparisons of thesecalculated values are compared with boringdata and results showed that, alluviumdiluvium contact line is the basement of QTS.

Distance along the Shinkansen in km
Figure 10. Estimated depth from microtremor
measurement along a Shinkansen(Nakamura et. al., 1990)

7. H/V OF EARTHQUAKE DATA
Figure 11(a), shows H/V of strong ground motion of two earthquake different earthquake recorded at the same
station. Although waveforms look different, when we take H/V, they look similar. Another example can befound also in a recent study of Okuma et. al. (1999) which is given in Figure 11(b). Results again shows that
NS

H

V

H/V

H/V (QTS) of Strong Motion Acceleration

NS

Freaqency(a) Osaki (Nakamura et. al., 1989)(b) Miyazaki (Okuma et. al. , 1999)

Figure 11. H/V of strong ground motion for different earthquakes recorded at the same station.
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H/V gives similar characteristics for the different earthquakes recorded at the same stations. If we think that H/Vof earthquake record can be explained by Rayleigh wave, we should see same characteristics for all directions of
the station, but this is out of practice and it is proper to think that QTS of earthquake motion indicate the transfer
function of local surface ground, as same as QTS of microtremor.
8. VULNERABILITY INDEXES, K-VALUES FOR SURFACE GROUND
K values have been proposed by Nakamura(1996) for accurately
estimating earthquake damage of surface ground and structures.
Here in this paper, only the formulation of Kg, K value forground, will be given as an example of application of outputs
from QTS method. For calculating Kg, shear strain of theground is considered. Figure 12 simply shows the shear
deformation of surface ground.

γ = Ag ⋅dh

Ag x d

H
Vs

(8)

d

where Ag is amplification factor of surface layer, h is thicknessof surface layer and d is seismic displacement of the basement.
Putting the S wave velocities of the basement and surface layer(Cb and Cf) natural frequency F of the surface layer can be
expressed as,

Vb
4 Ag ⋅ h(9)

Acceleration α in the basement can be written asγ =Ag ⋅ α b(2πF )

2⋅ 4 Ag ⋅

g

Fg

(10)

Cb

α
=
⋅ 2b
Fg π C b= c ⋅ K gα
αb = (2πFg) d and shear strain γ is expressed as follows,
where,
2

;

Kg =

Ag

Marina District in San Francisco, U.S.A.
500
Reclaimed Land: Severe Damage

100
50

Ag2

1
c = 2
π ⋅ vbFigure 12 . Surface ground deformation

Kg-value for Surface GroundFg =Basement Ground Vb

2Hill side Area:
10
5

1
0. 5

No Damage
Damaged Area
0

500

100

1500 mDistance from Coast line

Fg

Figure 13.

Kg values calculated for Lomac is expected to be almost constant for various sites.
Prieta earthquake (Nakamura et.
Effective shear strain defined by e % of equation (8)al., 1990)
becomes to nearly equal to the product of Kg and αb,under the assumptions of e = 60% and Cb = 600m/s. Kg is a value corresponding to the site and can beconsidered as a vulnerability index of the site, which might be useful to select weak points of ground.
K values for various types of structures and some application examples can be found in Nakamura (1996),
Nakamura (1997), Nakamura and Gurler (1999).
9. CONCLUDING REMARKS
It is proved in the present paper that peak of H/V ratio (QTS), either for microtremor of for earthquakes cannot be
explained with Rayleigh waves, since Rayleigh wave energy is very small for the peak frequency but high on thetrough of H/V ratio. Author’s explanation in Nakamura (1989) is correct for explaining this peak with SH waves.
Vulnerability index, K value for ground which gives a chance to estimate the earthquake damage before theearthquake occurs is also briefly explained710. REFERENCES
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