The 1st Iranian Conference on Geotechnical Engineering, 22-23 October 2013
University of Mohaghegh Ardabili, Ardabil, Iran
OHN10110191115
Improve HVNR Technique by Using Wavelet Transform
H.Rahnema1, M.H.Bozorgi2
1- Assistant Professor, Department of Civil and Environmental Engineering, Shiraz University of
Technology, Shiraz, Iran
2- Ms Candidate of Earthquake Engineering, Department of Civil and Environmental Engineering,
Shiraz University of Technology, Shiraz, Iran
[email protected]
[email protected]
Abstract
The spectral ratio between horizontal and vertical components noise (H/V ratio) of microtremors has been
used to estimate fundamental resonance frequency of the site. In this study we investigate the application
of the wavelet analysis to improve the (HVNR) technique for site effect estimation. Wavelet analysis has
been widely used in the last decade for its capacity to localise the signal in both the time and frequency
domains. This property, which derived directly from multiresolution analysis provide us the ability to
decompose a signal in a well localized set of coefficients and identify the non-stationary portions of it.In
the present study we use the WT in order to eliminate the non-stationarities in microtremor signals before
we calculate the spectrum of each one using conventional FFT algorithms.
Keywords: HVNR, Wavelet transform, non-stationary, FFT algorithm.
1.
INTRODUCTION
During the last two decades Horizontal to Vertical Noise Ratio or Nakamura method (HVNR) has proved an
invaluable tool for estimating a site’s fundamental (f0) frequency using cheap, rapid and fair accurate
method. Since its first proposal by Nogoshi & Igarashi [1], and latest revision by Nakamura [2] the HVNR
method widely used by many researchers but without following a de-facto standard for the stationarity of the
selected microtremor signal. The use of HVNR method in urban areas and especially in densely populated areas
seems to be the most demanding case since there will be artificial non-stationary signals from unknown sources which
cannot be predicted. Nonstationary components contaminate the signal and reveals frequencies that are not
related with site’s characteristics. Nonstationary signal caused by daily human activities such as movement
of machinery in factories, motor cars, people walking and natural phenomena such as flow of water in rivers,
rain, wind, variation in atmospheric pressure and ocean waves.
In the present study we will estimate first the Horizontal to Vertical Noise Ratio (HVNR) using ordinary
methodology and then we calculate the HVSR using wavelet-filtered microtremor signals in order to identify
the variation of HVSR when the non-stationary components eliminated.
2.
THE NAKAMURA TECHNIQUE
A quick estimate of the surface geology effects on seismic motion is provided by the horizontal to vertical
noise spectral ratio technique (HVNR). This technique firstly introduced by Nogoshi and Igarashi [1], was
put into practice by Nakamura [2], and became in recent years widely used since it provides a reliable
estimate of the fundamental frequency of soft soil deposits( Lermo and Chavez-Garcia and Bard et al)[3,4].
The resulting curves often show a clear maximum. The frequency at which this peak occurs is empirically
found to correlate with the fundamental resonance frequency at the measurement site (e.g. Lachet and Bard;
Lermo and Chávez-García; Dravinski et al.)[5,6,7]. such method is widely used since it significantly reduces
field data acquisition time and costs. The evaluation of site response using the HVNR technique is largely
adopted since it requires only one mobile seismic station with no additional measurements at rock sites for
comparison. The principal two basic hypothesis explanations for using ambient noise are two. Nogoshi and
Igarashi [1] showed through their investigations that HVNR on microtremors are directly related to the
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University of Mohaghegh Ardabili, Ardabil, Iran
ellipticity curve of Rayleigh waves because of the predominance of Rayleigh waves in the vertical
component. Nakamura [2] instead, asserted that the H/V-ratio on microtremors gives a good estimate of the S
waves site response function. Dividing the horizontal component of surface ground motion by the vertical, a
removal of the source as well as the Rayleigh wave effects can be obtained. Rayleigh waves, which is
associated with an inversion of the direction of Rayleigh waves rotation (Nogoshi and Igarashi,; Lachet and
Bard)[1,3]. Thus, the ratio between the horizontal and vertical spectral components of motion can reveal the
fundamental resonance frequency of the site[9].
3.
SIGNAL PROCESSING
The Fourier analysis (Papoulis)[10], which decomposes a time series into orthogonal frequency components
or vice versa, has been widely used for geophysical signal processing. The fast Fourier transform (FFT)
algorithm has made the Fourier analysis very attractive for many data processing applications, mainly
because of the orthogonal properties of the Fourier series and of its simple expression. However, upon
inspection of the expression of FT.
F
(1)
f (t ) e it dt
The FT yields information on how much but not when (in time) the particular frequency components
exist. Such information is sufficient in a case of the stationary signals as the frequency content of such
signals does not change in time and all frequency components exist all the time. the FT provides just
information on the frequency content; however, the information on the frequency localization in time is
essentially lost in the process thus the frequency information cannot be associated with time.
The limitation of the Fourier transform, i.e. it gives only the global frequency content of a signal, is
overcome by the short time Fourier transform (STFT). The STFT is able to retrieve both frequency and time
information from a signal. The STFT calculates the Fourier transform of a windowed part of the original
signal, where the window shifts along the time axis. This transform which also called Gabor transform is
defined as [11] :
G
b,k
( )
f (t ) e it
g t b dt
k
(2)
Where g(t) is a Gaussian function :
g t 2
k
1
.k
e
t2
4k
(3)
which also called a window function. (Alternative formulations replaced the Gaussian function by other
window functions). Thus the Gabor transform localizes the FT of a signal f(t) at around t=b. The width of the
window is determined by the fixed positive constant k. In the majority of STFT methods there are
undesirable computational complexities when either narrowing of the window is required for better
localization or widening the window required to obtain more global picture [12].
The analysis of a non-stationary signal using the FT or the STFT does not give satisfactory results.
Better results can be obtained using wavelet analysis. One advantage of wavelet analysis is the ability to
perform local analysis. Wavelet analysis is able to reveal signal aspects that other analysis techniques miss,
such as trends, breakdown points, discontinuities, etc. In comparison to the STFT, wavelet analysis makes it
possible to perform a multiresolution analysis (MRA)[15]. By using MRA it is possible to analyze a signal at
different frequencies with different resolutions. The change in resolution is schematically displayed in Figure
1.
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University of Mohaghegh Ardabili, Ardabil, Iran
Figure1. Multiresolution time-frequency plane
For the resolution of Figure 1, it is assumed that low frequencies last for the entire duration of the signal,
whereas high frequencies appear from time to time as short burst. This is often the case in practical
applications. Wavelet transform can be accomplished in two ways: Continuous Wavelet Transform (CWT)
and Discrete Wavelet Transform (DWT).
The CWT is defined as follows :
w( a, b)
1
a
f (t ).g (
t b
)dt
a
(4)
Where g(t) is the wavelet function, and a and b are the scale and translation factors, respectively. The wavelet
function g(t) decays rapidly to zero with increasing t and has zero mean. The domain (range) of nonzero
values of the wavelet is called the support. The scale factor controls the dilation or compression of the
wavelet, whereas at lower scales, the wavelet is compressed and characterizes the rapidly changing details of
the signal. At higher scales, the wavelet is stretched over a greater time span and the slowly changing and
coarse features are better resolved [13].
In practical applications, the DWT is generally preferred because waveforms are recorded as discrete
time samples. DWT can be implemented quickly via the Mallat algorithm [14].
Figure2. Mallat algorithm
As shown in Figure 2, in discrete wavelet analysis, a signal(S) is segregated into an approximation (A) and a
detail (D) as Eq. (5):
S=A1+D1=A2+D1+D2=A3+D1+D2+D3.
(5)
The approximations are the high-scale, low-frequency components of the signal. The details are the lowscale, high-frequency components. The approximation is then itself split into a second-level approximation
and detail, and the process is repeated. For n-level decomposition, there are n+1 possible ways to decompose
or encode the signal.
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University of Mohaghegh Ardabili, Ardabil, Iran
4.
PROCESS OF ESTIMATION HVNR
According to the following algorithm, using the wavelet transform, the signal is preprocessed,, see Figure 3 .
Using by discrete wavelet
for remove region high-frequency
Using by continuous wavelet for
select the local time zone that has
a same frequency
Estimation
of hvnr
Figure3. Signal preprocessing using wavelets
Because, transient signals (non-stationary signals) should be removed from the original signal. So using
continuous wavelet diagram (time - frequency), which has a similar frequency to the local time zone is
selected. Then HVNR, is estimated, see Figure 5.
The horizontal to vertical spectral ratio (HVNR) is estimated from the three components of the
microtremor recordings (North-South, East-West and Vertical) through the following steps [11], see Figure 4.
Figure4. Estimation of hvnr algorithm
4.
RESULT AND DISCUSSION
According to set of three microtremor recordings by record time 1m5.536s and sampling rate 128µs is
presented below, see Figure 5.
Figure5. Horizontal (East-West) component, Horizontal (North-south) component, Vertical component
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University of Mohaghegh Ardabili, Ardabil, Iran
We estimate the HVNR following the steps defined in section 4. The calculated amplification factor over
frequency is depicted in Figure 6. We can identify amplification for frequencies around 1.25Hz by amplitude
1.44 and around 3.33Hz by amplitude 1.513 and around 11.85 by amplitude 2.176, see Figure 6.
Figure6. HVNR from unfiltered by wavelet microtremors
In Figure 6, The black curve represents H/V geometrically averaged over all colored individual H/V curves.
The two dashed lines represent the H/V standard deviation. The grey area, represent the averaged peak
frequency and its standard deviation. The frequency value is at the limit between the dark grey and light grey
areas.
By using CWT, select local time between 24s and 44s which has a similar frequency, for three-component,
see Figure 7.
Figure7. Continuous Wavelet Transform (CWT) for three-component
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In the zone time between 24 s to 44 s, estimate HVNR, see Figure 8.
Figure 8. HVNR from filtered by wavelet microtremors
The latter estimation of HVNR indicates that the dominant frequency is around 0.894 Hz by amplitude 1.917
and 5.046 Hz by amplitude 1.161 and 11.46 Hz by amplitude 2.25.
5.
CONCLUSIONS
The precision of the HVNR method for site effect investigations depends on the stationarity of the recorded
microtremors. This is not obtainable when measurements take place in urban areas. In the present study we
apply wavelet transform in order to identify non-stationarities in microtremors and then eliminate short
period transients using non-linear wavelet filtering. This approach does not nullify the commonly used
HVNR method but it is incorporated with it by adding one more procedure in order to achieve better and
trustworthy results.
11.
REFERENCES
1.
Nogoshi, M. and Igarashi, T. (1971). “On the Amplitude Characteristics of Microtremor (part 2) ”, Jour
Seism. Soc. Japan, 24, 26-40
2.
Nakamura, Y. (1989). “A method for dynamic characteristics estimation of subsurfaceusing microtremor
on the ground surface”. Q.R. Railway Tech. Res. Inst. Rept., 30, 25-33.
3.
Lermo J., Chavez-Garcia F. J. (1994). “Are microtremor useful in site response evaluation? ” Bull.
Seism. Soc. Am., 84: 1350-1364.
4.
Lermo, J, & Chavez-garcia, F. J. (1993). “Site effect evaluation using spectral ratio with only one
station. Bull”. Seism. Soc. Am., 83, 1574-1594.
5.
Dravinski M, Ding G, Wen KL (1996). “Analysis of spectral ratios for estimating ground motion in deep
basins”. Bull Seismol Soc Am 86(3):646–654
6.
Bard, P. Y. (1999). “ Microtremor measurements: a tool for site effect estimation? In The effects of
surface geology on seismic motion”. Irikura et al. (ed.), Balkema, Rotterdam, 1279-1251.
7.
Lachet C, Bard PY (1994). “Numerical and theoretical investigations on the possibilities and limitations
of Nakamura’s technique”. J Phys Earth 42:377–397.
8.
Brigitte Endrun (2011). “Love wave contribution to the ambient vibration H/V amplitude peak observed
with array measurements”. J Seismol 15:443–472 .DOI 10.1007/s10950-010-9191-x.
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9.
F. Panzera, G. Lombardo, S. D’Amico and P. Galea (2013). “Speedy Techniques to Evaluate Seismic
Site Effects in Particular Geomorphologic Conditions: Faults, Cavities, Landslides and Topographic
Irregularities”, Engineering Seismology, Geotechnical and Structural Earthquake Engineering, Dr
Sebastiano D'Amico (Ed.), ISBN: 978-953-51-1038-5, InTech, DOI: 10.5772/55439.
10. Papoulis, A., (1962). “The Fourier integral and its applications”. McGraw-Hill Press, New York.
11. G.Hloupis, F.Vallianatos, J.Stonham,(2004). “A Wavelet representation of HVSR technique”, Bulletin
of the Geological society of Greece, vol 36, pp.1269-1278.
12. Chui, C.K., (1992). “An introduction to wavelets”, Academic press, New York.
13. Aa . Zhang, H., Thurber, C. and Rowe, C. (2003). “Automatic P-wave Arrival Detection and Picking
with Multiscale Wavelet Analysis for Single-Component Recordings”, Bull. Seism.Soc. Am., 93, 19041912.
14. Mallat, S. (1989). “A Theory for Multi-Resolution Signal Decomposition: The Wavelet Representation,
IEEE Trans”, Pattern Anal.Machine Intelligence, 11, 674-693.
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