The centerless circular array (CCA) method can be used to estimate phase velocities of Rayleigh waves from microtremor records obtained using an array of seismic sensors placed around a circumference. The method analyzes the power spectral densities of synthesized waveforms to obtain a spectral ratio, from which phase velocities can be inferred. Noise limits the resolution of the CCA method at long wavelengths. A new method is proposed to quantitatively evaluate the noise-to-signal ratio using the coherence between array seismograms, allowing estimation of the longest resolvable wavelength. Analysis of field data using larger arrays confirms the applicability and high performance of the CCA method into surprisingly long wavelengths.

1. Centerless circular array method: Inferring phase
velocities of Rayleigh waves in broad wavelength
ranges using microtremor records
Ikuo Cho,1
Taku Tada,2
and Yuzo Shinozaki2
Received 20 December 2005; revised 8 May 2006; accepted 21 June 2006; published 29 September 2006.
[1] The centerless circular array (CCA) method, proposed by ourselves in an earlier work,
is an algorithm of microtremor exploration which can be used to estimate phase velocities
of Rayleigh waves by analyzing vertical component records of microtremors that are
obtained with an array of three or five seismic sensors placed around a circumference. We
have confirmed, through field tests, the applicability of our CCA method to arrays on the
order of several to several hundred meters in radii and have revealed its remarkably high
performance in long-wavelength ranges, the upper resolution limit extending as far as
several 10 times the array radius. We have also invented a mathematical model that
enables to evaluate signal-to-noise ratios in a given microtremor field. Scrutiny of field
data has borne out our hypothesis that noise is the principal factor that biases the analysis
results of the CCA method in long-wavelength ranges and that its longest resolvable
wavelength is determined by the signal-to-noise ratio. Combined use of the CCA method
and our new method of signal-to-noise ratio analysis provides a powerful methodological
tool that allows one to extract maximal information from microtremor records obtained
with a simple seismic array.
Citation: Cho, I., T. Tada, and Y. Shinozaki (2006), Centerless circular array method: Inferring phase velocities of Rayleigh waves in
broad wavelength ranges using microtremor records, J. Geophys. Res., 111, B09315, doi:10.1029/2005JB004235.
1. Introduction
[2] We have invented and published [Cho et al., 2004] an
algorithm that enables to estimate phase velocities of
Rayleigh waves by analyzing vertical component records
of microtremors that are obtained with an array of three or
five seismic sensors placed around a circumference. In this
method, all information on the field of vertical component
microtremors is integrated into a single quantity, called the
‘‘spectral ratio,’’ which contains information on the phase
velocities alone. Since this method does not separate indi-
vidual plane wave components with different arrival azi-
muths, it possesses higher resolution in long-wavelength
ranges than the frequency-wave number spectral method
[Capon, 1969]. Theory predicts that the resolution of our
method in long-wavelength ranges depends upon the signal-
to-noise (SN) ratio, or the ratio of the power of propagating
plane wave components to that of nonpropagating compo-
nents (incoherent noise) contained in the array seismograms
(in the following, we refer to incoherent noise simply as
‘‘noise’’ unless otherwise stated). We have confirmed,
through field data analysis, that our method is applicable
to small-sized arrays on the order of 5–15 m in radii, and
that analysis of sufficient resolution and stability is possible
in broad wavelength ranges extending from a little less than
four times up to several 10 times the array radius [e.g., Cho
et al., 2004].
[3] In the present study, we refer to the above cited
algorithm as a centerless circular array (CCA) method and
demonstrate its remarkably broad applicability that is not
limited to small-sized arrays as well as its high performance
that continues into surprisingly long wavelengths. We also
propose a new method to quantitatively evaluate signal-
to-noise ratios in microtremor records, and discuss how the
presence of noise restricts the efficacy of the CCA method
in long-wavelength ranges.
[4] We first corroborate, by way of real data analysis, the
applicability of our CCA method to arrays of larger sizes.
We deploy seismic arrays of several to several hundred
meters in radius at a test site where the subsurface structure
is known from geophysical exploration, and apply the CCA
method to microtremor records to estimate the phase veloc-
ities of Rayleigh waves. The estimates are then checked
against ‘‘reference’’ phase velocities that have been inferred
using other established methods. We next analyze the noise-
to-signal (NS) ratios (reciprocals of the SN ratio) of the
array seismograms using our new algorithm. On the basis of
the estimated NS ratios and the reference phase velocities,
we calculate the apparent values of phase velocities which
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 111, B09315, doi:10.1029/2005JB004235, 2006
1
Geological Survey of Japan, National Institute of Advanced Industrial
Science and Technology, Tsukuba, Japan.
2
Department of Architecture, Faculty of Engineering, Tokyo University
of Science, Tokyo, Japan.
Copyright 2006 by the American Geophysical Union.
0148-0227/06/2005JB004235$09.00
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2. theory expects should be obtained by the CCA method in
long-wavelength ranges, and compare them with the results
of real data analysis. This allows us to check the validity of
our hypothesis that noise limits the resolution of the CCA
method in long-wavelength ranges.
2. Method
2.1. Determining the Phase Velocity
[5] In the present section we briefly review the algo-
rithm of the CCA method, which was first presented by
Cho et al. [2004] and was later reformulated in a broader
context by Cho et al. [2006] as part of their general theory
of microtremor exploration methods using circular arrays.
The theoretical background is fairly complicated and
lengthy to describe, so we restrict ourselves to citing its
very gist in the present paper and refer the interested
reader to Cho et al. [2006] for more details. To help
understanding, we shall make reference, wherever appro-
priate, to specific equations appearing in the earlier works
of Cho et al. [2004, 2006].
[6] Suppose we deploy a circular seismic array of
radius r in a field of microtremors, whose vertical
component we denote by z(t, r, ). If we synthesize the
complex waveforms
Z0 t; rð Þ ¼
Z
À
z t; r; ð Þd ð1Þ
Z1 t; rð Þ ¼
Z
À
z t; r; ð Þ exp ið Þd ð2Þ
(compare equations (11), (12) and (20) of Cho et al.
[2004] and equation (48) of Cho et al. [2006]) by
integrating the seismograms with regard to azimuth, their
power spectral densities, which we denote by GZ0Z0(r, r; !)
and GZ1Z1(r, r; !), respectively (! being the angular
frequency), can each be represented in the following way:
GZ0Z0 r; r; !ð Þ ¼ 42
XM
i¼1
fi !ð ÞJ2
0 rki !ð Þð Þ ð3Þ
GZ1Z1 r; r; !ð Þ ¼ 42
XM
i¼1
fi !ð ÞJ2
1 rki !ð Þð Þ ð4Þ
(compare equation (58) of Cho et al. [2006]), where M is the
number of Rayleigh wave modes present, fi(!) is the intensity
of the ith mode, J0(Á) and J1(Á) are the zeroth- and first-order
Bessel functions of the first kind respectively, and ki(!) stands
for the wave number of the ith mode. By taking their mutual
ratio, we have
GZ0Z0 r; r; !ð Þ
GZ1Z1 r; r; !ð Þ
¼
PM
i¼1 i !ð ÞJ2
0 rki !ð Þð Þ
PM
i¼1 i !ð ÞJ2
1 rki !ð Þð Þ
ð5Þ
(compare equation (28) of Cho et al. [2004] and equation (64)
of Cho et al. [2006]), where
i !ð Þ ¼ fi !ð Þ=f !ð Þ; f !ð Þ ¼
XM
i¼1
fi !ð Þ ð6Þ
is the power partition ratio for the ith mode.
[7] When it can be safely assumed that the fundamental
Rayleigh wave mode dominates the vertical component of
the microtremor field, equation (5) simplifies to
GZ0Z0 r; r; !ð Þ
GZ1Z1 r; r; !ð Þ
¼
J2
0 rk !ð Þð Þ
J2
1 rk !ð Þð Þ
ð7Þ
(compare equation (30) of Cho et al. [2004] and equation
(74) of Cho et al. [2006]). The function on the right-hand
side is plotted against rk in Figure 1. Once the spectral ratio
on the left-hand side is known from measurement records, it
is possible to estimate rk by inverting the above equation for
each frequency !. Since r is known, one can obtain the
wave number k, and finally the phase velocity c = !/k. The
analysis is limited to the range 0 rk 2.405, where there
is a one-to-one correspondence between the value and the
argument of the function J0
2
(Á)/J1
2
(Á). For the sake of
simplicity, we conduct, in the present study, measurements
at a test site where the fundamental mode is expected to
dominate, and make use of equation (7) in the analysis of
data.
[8] Equation (7) holds in noise-free situations. When
noise is present, we have, under the assumption that the
power n(!) of noise contaminating the records is identical
for all sensors in the seismic array,
GZ0Z0 r; r; !ð Þ ¼ 42
XM
i¼1
fi !ð ÞJ2
0 rki !ð Þð Þ þ n !ð Þ=N
h i
ð8Þ
GZ1Z1 r; r; !ð Þ ¼ 42
XM
i¼1
fi !ð ÞJ2
1 rki !ð Þð Þ þ n !ð Þ=N
h i
ð9Þ
(compare equations (108) and (123) of Cho et al. [2006]),
where N is the number of sensors around the circumference.
Figure 1. Graph of the theoretical spectral ratio function
on the right-hand side of equation (7). The horizontal bar
denotes the argument range where phase velocity analysis is
feasible.
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3. In this case, equation (7), for the case of a single mode
dominating, must be replaced with
GZ0Z0 r; r; !ð Þ
GZ1Z1 r; r; !ð Þ
¼
J2
0 rk !ð Þð Þ þ !ð Þ=N
J2
1 rk !ð Þð Þ þ !ð Þ=N
; ð10Þ
where
!ð Þ ¼ n !ð Þ=f !ð Þ ð11Þ
is the noise-to-signal (NS) ratio of the array seismograms, or
the ratio of the power of incoherent noise to that of coherent
signals.
[9] It should be noted that in the equations we have cited
above, we have omitted terms representing the effects of
directional aliasing which become significant in short-
wavelength ranges [Cho et al., 2004, 2006], where direc-
tional aliasing refers to systematic errors that are caused by
incomplete identification of the directional characteristics of
the microtremor field because of the finite number of
sensors constituting the seismic array.
2.2. Determining the NS Ratio
[10] Henstridge [1979] pointed out that the average of the
seismograms recorded on a circle and the seismogram at its
center could each be regarded as the output and input of a
linear filter, and that their mutual coherence, which he
named the ‘‘circle coherence,’’ could be used as an indicator
of the extent to which noise and multiple phase velocities
prevail in those records. In order to substantiate his intuitive
observation, we define, in the present study, the squared
circle coherence by
coh2
!ð Þ ¼
jGZ0Z0 0; r; !ð Þj2
GZ0Z0 r; r; !ð ÞGZ0Z0 0; 0; !ð Þ
; ð12Þ
where GZ0Z0(0, r; !) represents the cross-spectral density of
the waveforms Z0(t, 0) and Z0(t, r) defined by equation (1),
and GZ0Z0(0, 0; !) is the power spectral density of the
waveform Z0(t, 0). According to Cho et al. [2006],
GZ0Z0 0; r; !ð Þ ¼ 42
XM
i¼1
fi !ð ÞJ0 rki !ð Þð Þ ð13Þ
GZ0Z0 0; 0; !ð Þ ¼ 42
XM
i¼1
fi !ð Þ þ n !ð Þ
h i
; ð14Þ
where we have omitted directional aliasing terms as in
section 2.1. Substituting equations (8), (13), and (14) into
equation (12) yields
coh2
¼
J2
J2 þ D þ =Nð Þ 1 þ ð Þ
ð15Þ
where
J ¼
XM
i¼1
iJ0 rkið Þ; D ¼
XM
i¼1
iJ2
0 rkið Þ À J2
ð16Þ
are the weighted mean and variance of J0(rki), respectively.
The relationship D ! 0 holds because
XM
i¼1
i J0 rkið Þ À
XM
j¼1
jJ0 rkj
À Áh i2
¼
XM
i¼1
iJ2
0 rkið Þ À
XM
i¼1
iJ0 rkið Þ
h i2
! 0 ð17Þ
Since D equals zero when and only when a single mode
dominates the field, it can be regarded as an indicator of the
extent to which multiple wave modes prevail. The squared
circle coherence coh2
takes values between zero and unity,
and equals unity when and only when noise is absent ( = 0)
and a single mode dominates (D = 0).
[11] In the meantime, it follows from equations (13) and
(14) that the spectral ratio
¼ GZ0Z0 0; r; !ð Þ=GZ0Z0 0; 0; !ð Þ ð18Þ
which is equivalent to the spectral ratio used in the classical
spatial autocorrelation (SPAC) method of Aki [1957] for the
purpose of phase velocity analysis [Cho et al., 2006], can be
written as
¼ J= 1 þ ð Þ ð19Þ
[12] By eliminating J from equations (15) and (19), we
obtain the following relationship between D and :
D ¼ A2
þ B þ C ð20Þ
A ¼ À2
; B ¼
2
coh2
À 22
À
1
N
; C ¼ 2 1
coh2
À 1
ð21Þ
Since D ! 0 must hold, takes the maximum value
max ¼ ÀB À
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
B2 À 4AC
p
=2A ð22Þ
when D = 0. Once we have estimated coh2
and from
measurement records, max is immediately derived from
equation (22); we can regard max as the estimate for itself
when the dominance of the fundamental mode (D = 0) can
be assumed.
[13] Even when it is not possible to assume the domi-
nance of the fundamental mode, rki(!) tends to zero for all
wave modes in the long-wavelength limit, so J0(rki) tends to
unity, and the variance D tends to zero. For this reason, max
is not expected to differ much from the actual value of . It
is also possible to pose, by way of quantitative theoretical
considerations, a certain limit on the possible range of the
value of (see Appendix A).
3. Data Acquisition and Analysis
[14] We conducted array measurements of microtremors
in Kasukabe City in the Kanto Plain, Japan, situated on a
floodplain about 40 km landward to the north of Tokyo Bay.
The Kanto Plain is a structural basin characterized by the
presence of three major stratigraphic features overlying the
seismic bedrock: in descending order, the Shimosa Group
(middle to late Pleistocene), the Kazusa Group (late Plio-
cene to early Pleistocene), and sedimentary layers of diverse
nature known collectively as the Miura Group (Miocene to
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4. middle Pleistocene) [e.g., Suzuki, 2002]. The site we have
chosen is located near one of the deepest depressions of the
pre-Neogene bedrock, and the bottom of the Miura Group
lies at a depth of approximately 3 km. The thicknesses of
the Shimosa and Kazusa groups are about 200 and 800 m
beneath our site, respectively, and are known to have little
lateral variations in its neighborhood. The surface of our site
is covered by soft soil overlying the Shimosa Group, which
is about 30 m thick and is composed of fluvial and
Holocene deposits.
[15] We have selected, as the base of our survey, a park
lying in a quiet part of the city, sized about 100 m per side.
The city center, formed around a railroad station, is located
about 1 km to the northeast of this park. An expressway
runs about 6 km to the southwest. The city lies in a
relatively quiet residential district where traffic congestion
rarely occurs.
[16] We deployed circular seismic arrays of 5, 25, 50,
100, 200, 300, and 600 m in radius centered on the park,
and measured the vertical component of microtremors. We
installed five sensors, for the 5 m array, and three sensors,
for all other arrays, both equidistantly on the circumference
(Figure 2). We also installed a sensor at the center of each
circle for the purpose of analyzing the NS ratio. All sensors
fell on the premises of the park for arrays sized 50 m or less
in radius. For larger arrays, the sensors on the circum-
ferences were installed at appropriate locations on paved
roads outside the park.
[17] Measurements were carried out on different occa-
sions for different array radii; the 5 m array was operated on
16 August 2001 (day A in the following), and the arrays
sized from 25 to 600 m were operated on 26 November
2002 (day B) under windy conditions. The 25 and 50 m
arrays were again put in operation on 18 December 2002
(day C) under windless conditions. All seismograms were
recorded in the daytime over a duration of 40 min. There
was almost no human traffic inside the park throughout all
measurement sessions, but at some seismic stations outside
the park, there were occasional or frequent passages of
automobiles nearby.
[18] The 5 m array was composed of six VSE-15D servo
velocity seismometers and an SPC-51 data recorder, both
manufactured by Tokyo Sokushin Corporation. The outputs
of the sensors were mutually synchronized by cable trans-
mission to a single recorder, and were digitized into 16-bit
data at a sampling rate of 100 Hz. For arrays sized 25 m or
larger, we used GPL-6A3P portable recording systems
manufactured by Akashi Corporation, composed of a
built-in accelerometer and a built-in data logger, which
make use of the Global Positioning System for automatic
time correction. For the 25 and 50 m arrays, the ground
motion was preamplified at a gain of 500, low-pass-filtered
at 50 Hz, and digitized into 24-bit data at a sampling rate of
100 Hz. For larger arrays, the preamplifier gain was set at
1000 and the low-pass filter cutoff frequency was set at
5 Hz. No measure was taken to protect the sensors from the
effects of wind.
[19] Prior to the measurements, we clustered all sensors at
an identical location and confirmed the mutual consistency
of the sensor outputs over the frequency range of interest
(huddle tests). Figure 3 shows the results of a huddle test
conducted on day C using GPL-6A3P seismometers. The
intersensor phase differences stayed within ±10° in the
frequency range of 0.2–7 Hz, while the magnitude-squared
coherences stayed above 0.95 in the frequency range of
0.2–3 Hz. The troughs in the power spectral density plots
near 0.1 and 0.7 Hz seem to correspond to local depressions
in the magnitude-squared coherence plots and to local
bulges in the phase difference plots.
[20] Power spectral densities were estimated with the
techniques of both segment averaging and smoothing in
the frequency domain [Bendat and Piersol, 1971, section 9].
From each set of array seismograms, we extracted from 10
Figure 2. Configuration of the seismic arrays deployed.
Figure 3. Results of the huddle tests conducted on day C. (a) Power spectral densities, plotted for all
four sensors. (b) Intersensor phase differences, plotted for all combinations of the four sensors.
(c) Intersensor magnitude-squared coherences, same as Figure 3b.
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5. to 48 data segments with a duration of either 20.48 or
40.96 s so as not to include parts where nonstationarity is
obvious, and calculated their power spectral densities by
fast Fourier transform. The power spectral densities were
then smoothed with a Parzen window of a bandwidth of
either 0.2 or 0.05 Hz, and were averaged for each frequency
before being used for estimating the spectral ratio (7), coh2
and . A grid search method was used in the inverse
analysis of equation (7).
4. Results
[21] Figure 4 plots the phase velocities of Rayleigh waves
estimated with our CCA method, together with the reference
phase velocity dispersion curve synthesized in the following
way: below 1.0 Hz, phase velocities inferred by Matsuoka
and Shiraishi [2002] with the SPAC method of Aki [1957]
using microtremor records; between 1.0 and 3.5 Hz, phase
velocities inferred by ourselves with the SPAC method
using microtremor records; above 3.5 Hz, phase velocities
of the fundamental mode Rayleigh waves calculated theo-
retically for a one-dimensional soil profile model (Figure 5),
which we have built on the basis of PS logging data
available to a depth of 160 m at a nearby drilling site.
The observed dispersion curves, drawn below 3.5 Hz,
connected smoothly with the theoretical dispersion curve
drawn above 3.5 Hz; in fact, geological data described in
section 3 imply that a horizontally layered soil model gives
a good enough approximation, while theoretical calculations
[Tokimatsu et al., 1992] using our soil model predict that the
fundamental mode dominates the field of Rayleigh waves at
all frequencies upward of 3.5 Hz.
Figure 4. Estimated phase velocities of Rayleigh waves. (a) Simultaneous plot of the estimates for all
array radii, together with the reference phase velocities shown in a thick gray curve. For the radii of 25
and 50 m, the solid and dotted black curves are for the data of day B, while the color curves are for the
data of day C. (b) Estimates for the 25 m array. (c) Estimates for the 100 and 200 m arrays. (d) Estimates
for the 300 and 600 m arrays. The horizontal bars in Figures 4b, 4c, and 4d denote resolvable frequency
ranges (solid part) and frequency ranges of limited resolution (dotted part).
Figure 5. Profile model for the P and S wave velocities
beneath the test site, compiled on the basis of PS logging
data at a nearby drilling site.
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6. [22] Simultaneous plot of the results for all array radii
(Figure 4a) demonstrates a good general agreement between
the phase velocities inferred by our CCA method and the
reference phase velocities. In the following, we refer to the
frequency range where the two phase velocities agree well
with each other as the ‘‘resolvable frequency range.’’ The
longest resolvable wavelength is obtained as the wavelength
corresponding to the phase velocity value at the lowest
resolvable frequency.
[23] For the array radii of 100 and 200 m, the agreement
between the estimated and reference phase velocity disper-
sion curves is poor over all frequencies, but over certain
frequency ranges, the estimated curve appears to be running
parallel to the reference curve while staying lower than the
latter (Figure 4c). Similar behavior is recognized for the 25
and 50 m arrays of day C and for the 600 m array, on the
lower-frequency side of the resolvable frequency range
where the agreement is fine. In the following, we refer to
frequency ranges where such behavior is recognized as
‘‘frequency ranges of limited resolution.’’
[24] Table 1 summarizes the resolvable frequency range,
the longest resolvable wavelength, and the frequency range
of limited resolution if there exists one, identified in the
analysis results for each measurement session. It should be
noted that for the 25 m array of day C, the estimated
dispersion curve, which falls away from the reference curve
below the resolvable frequency range, seems to again
approach the reference curve at around 0.4–0.5 Hz
(Figure 4b). This corresponds to a wavelength of about
2100 m or 84r.
[25] The results summarized in Table 1 reveal that no
simple proportionality holds between the longest resolvable
wavelength and the array radius. For the radius of 300 m,
the longest resolvable wavelength even takes a larger value
than for the radius twice larger. Also, for the array radii of
25 and 50 m, the measurements conducted on days B and C
produced longest resolvable wavelengths that were consid-
erably different. It also remains a question why, for the array
radii of 100 and 200 m, the estimated dispersion curve ran
parallel to but stayed lower than the reference curve even at
the best of their mutual agreement. We shall demonstrate in
section 5 that all these facts can be accounted for in a
unified and rational way by taking noise into consideration.
5. Interpretation of the Longest-Wavelength
Resolution Limit
[26] Figure 6 shows the phase velocities of Rayleigh
waves and the NS ratios, both estimated from records of
the 300 m array. The curve represents the values of max
defined by equation (22), whereas c stands for the upper
limit on the NS ratio that is required to keep the relative
error in the phase velocity estimate below a threshold value
of a (in the present study we set a = 0.05). When noise is
present, the spectral ratio GZ0Z0/GZ1Z1, represented by the
right-hand side of equation (10), takes a smaller value in the
range rk 1.4347 and a larger value in the range rk
1.4347 than the spectral ratio of the noise-free case repre-
sented by the right-hand side of equation (7) (Figure 7).
This means that as long as one uses the noise-free
equation (7) when noise is present in reality, one over-
estimates rk and underestimates the phase velocity in the
range rk 1.4347, while one underestimates rk and over-
estimates the phase velocity in the range rk 1.4347. On the
basis of the above considerations, we calculated c by
J2
0 rkð Þ þ c rkð Þ=N
J2
1 rkð Þ þ c rkð Þ=N
¼
J2
0 rk0
ð Þ
J2
1 rk0ð Þ
ð23Þ
where
rk0
¼
rk= 1 À að Þ for rk 1:4347
rk= 1 þ að Þ for rk 1:4347
8
:
ð24Þ
[27] Figure 6 shows that the longest-wavelength resolu-
tion limit of the CCA method roughly coincides with the
frequency (0.2 Hz) at which the NS ratio first exceeds c.
Figure 6 (bottom) shows both the values of c calculated
using the phase velocity estimates of the CCA method and
the values of c calculated using the reference phase
velocities, but their discrepancy remains very small
throughout the resolvable frequency range. This implies
that even at sites where no a priori data are available on the
subsurface structure, it is still possible to evaluate the
longest-wavelength resolution limit of the CCA method if
one calculates c using the phase velocity estimates of the
CCA method and then compares them with the estimates.
[28] Figure 8 summarizes the analysis results for the array
radii of 5, 25, 300 and 600 m; wavelengths corresponding to
the phase velocity estimates, normalized by the array radius r,
are plotted against frequency in Figure 8 (top). One can
observe that in all cases, the shortest resolvable wavelength
roughly corresponds to a little more than 3r, while the longest
resolvable wavelength takes varying values between 9r and
84r.
[29] The crosses in Figure 8 (top) and the arrows in
Figure 8 (middle) indicate the longest-wavelength (lowest-
Table 1. Summary of Field Data Analysis Results
Date Array Radius r, m
Resolvable Frequency
Range, Hz
Longest Resolvable
Wavelength
Frequency Range of
Limited Resolution, Hz
16 Aug 2001 (day A; breezy) 5 1.2–5 40r (200 m) —
26 Nov 2002 (day B; windy) 25 1.1–1.6 9r (225 m) —
50 1.0–1.2 5.6r (280 m) —
100 — — (0.3–0.8)
200 — — (0.25–0.65)
300 0.18–0.6 42r (12600 m) —
600 0.25–0.45 9r (5400 m) (0.18–0.25)
18 Dec 2002 (day C; windless) 25 0.9–1.6 14r (350 m) (0.4–0.9)
50 0.8–1.2 11r (550 m) (0.4–0.8)
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7. frequency) resolution limits. Figure 8 (bottom) compares,
for each array radius, the NS ratios (solid curve), estimat-
ed from field records, with the upper limit c (thin curve) on
the NS ratio that is required to keep the relative error in the
phase velocity estimate below 5%. In all cases, the longest-
wavelength resolution limit roughly coincides with the
frequency where first exceeds c (arrows in Figure 8
(bottom)).
[30] As we pointed out in section 4, the phase velocity
dispersion curve of Rayleigh waves, estimated for the array
radius of 25 m (day C), remains parallel to but falls lower
than the reference curve for frequencies below 0.9 Hz, but
again approaches the reference curve in the neighborhood of
0.4 Hz. As for the estimated NS ratio c, it surpasses the
curve near 0.7 Hz and stays above it over a certain
frequency interval, but again becomes comparable to near
0.4 Hz, a behavior that is starkly similar to the phase
velocity analysis results. The fact that the longest resolvable
wavelength turned out larger for the array radius of 300 m
than for 600 m also appears concordant with the observation
that the frequency, at which first exceeds c, is not much
different for both radii. The considerable difference in the
longest resolvable wavelengths on the windy day B and the
windless day C for the array radii of both 25 and 50 m has
also been explained rationally from the viewpoint of differ-
ent NS ratios.
[31] Figure 9 (top) shows the noise-free spectral ratios (7),
theoretically calculated using the reference phase velocities,
as well as the noise-inclusive spectral ratios (10), calculated
using both the reference phase velocities and the NS ratios
Figure 6. (top) Phase velocity estimates and (bottom) NS ratio estimates for the 300 m array.
Figure 7. Theoretical spectral ratio curves for the noise-
free case (equation (7)) and the noise-inclusive case
(equation (10)).
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8. inferred from field data, and compares them with the spectral
ratios of the measurement records. The case of the 25 m array
of day C is shown. The fall of the measured spectral ratios,
relative to the theoretical noise-free spectral ratio, for fre-
quencies below 0.4 Hz is mostly accounted for by considering
the effects of noise, and the same can be said of the phase
velocities of Rayleigh waves (Figure 9, bottom) that are
estimated by inverse analysis of the spectral ratios. This
observation corroborates our hypothesis that noise is the
principal factor that biases the analysis results of the CCA
method in long-wavelength ranges.
[32] We have conducted similar analysis for the array
radii of 600 and 100 m, and the results are shown in Figures 10
and 11. The fact that the resolvable frequency range extended
only as far down as about 0.25 Hz for the 600 m array, and the
fact that the estimated phase velocity dispersion curve ran
parallel to but stayed lower than the reference curve in the
frequency range between 0.3 and 0.8 Hz for the 100 m array,
are both accounted for satisfactorily by theoretical consider-
ation of the effects of noise.
6. Discussions
[33] In practical implementation of the CCA method, the
use of different types of smoothing in the process of
estimating power spectral densities may yield different
analysis results. According to our experience in real data
analysis, the very use of a spectral window tends to give
underestimated values for the spectral ratio on the left-hand
side of equation (7) in long-wavelength ranges; this empirical
finding is also supported by some theoretical reasoning which
is described in Appendix B. Underestimation of the spectral
ratio results in the overestimation of the parameter rk and,
consequently, in the underestimation of the phase velocity. To
avoid such biases in the phase velocity estimates and at the
same time maximize the resolving power of the CCA method
in long-wavelength ranges, it appears most desirable to avoid
Figure 8. Analysis results for the 5 m, 25 m (day C), 300 m, and 600 m arrays. (top) Wavelengths
corresponding to the phase velocity estimates, normalized by the array radius. Crosses denote longest-
wavelength resolution limits. (middle) Phase velocity estimates. Arrows denote lowest-frequency
resolution limits. (bottom) NS ratio estimates (solid curves), and upper bounds c on the NS ratios (thin
curves) that are required to keep the relative errors in the phase velocity estimates below 5%. Arrows
indicate frequencies at which first exceeds c.
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9. the use of a spectral window at all, or to use one with as narrow
a bandwidth as possible, and to stabilize the spectral density
estimates by taking as many data segments as possible.
[34] To maximize the amount of data used for analysis,
we overlapped, in the present study, consecutive data seg-
ments by half their duration, and applied a Hanning (cosine)
data window to each segment to taper off both ends [Carter
et al., 1973]. Prior to analysis, we calculated the root mean
square of the data in each segment, and discarded segments
for which it deviated significantly from normal values. We
also evaluated NS ratios of the data contained in different
parts (worth several segments) of the time series seismo-
grams, and discarded parts for which the NS ratios were
anomalously large. Such an automated method of data
selection makes it relatively easy to extract a large number
of segments from data of a finite time duration. It should be
borne in mind, however, that increasing the number of data
segments used in the analysis simply helps to suppress
statistical errors in the spectral density estimates but does
not in any way ameliorate the intrinsic NS ratio, or the ratio
of power between noise (nonpropagating components) and
signals (propagating components).
[35] It is possible, for simplicity’s sake, to classify noise
according to whether it originates in the interior or the
exterior of the recording system. Interior origins include
electrical noise, while exterior origins are thought to include
vehicular and human traffic in the neighborhood of indi-
vidual sensors, vibrations of machinery, wind, and the
vibrations of trees and buildings which wind may cause.
In addition, as the wavelength decreases and approaches the
array dimension, it becomes increasingly difficult to prop-
erly identify the directional characteristics of propagating
wave components (signals) with a finite number of sensors,
and to distinguish the apparent behavior of such signals
from that of nonpropagating components (noise). It is
convenient to include such cases in the ‘‘noise of exterior
origin’’ as we define it here, even if they do not fall into that
category in the strict sense of the word.
[36] The NS ratio int, corresponding to noise of internal
origin alone, can be estimated by the following formula from
Figure 9. (top) Spectral ratio estimates and (bottom) phase
velocity estimates for the 25 m array (day C), compared
with theoretical curves for both the noise-free and noise-
inclusive cases.
Figure 10. Same as in Figure 9 (600 m array).
Figure 11. Same as in Figure 9 (100 m array).
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10. the square root coh of the magnitude-squared coherence that
is inferred from huddle test records [Carter et al., 1973]:
int ¼ 1 À cohð Þ=coh ð25Þ
Meanwhile, the NS ratio , estimated from array seismo-
grams by the algorithm described in section 2.2, is thought
to include noise of both internal and external origins.
[37] Figure 12 shows NS ratios , inferred from seismo-
grams of the 25 and 50 m arrays of the windless day C;
power spectral densities of the records of each sensor; and
power spectral densities for noise of both internal and
external origins combined, estimated by multiplying the
total power spectral densities by /(1 + ). Figure 12 also
shows NS ratios int, inferred by equation (25) from the
huddle test records of the same date; and power spectral
densities for noise of internal origin alone, estimated by
multiplying the total power spectral densities by int/(1 + int) =
1 À coh. In the frequency range of 0.1–0.7 Hz, the
estimates of are, for the array radius of 25 m, 0.5–2 times
the estimates of int in magnitude and, for the array radius of
50 m, 1–10 times the latter. In either case, the corresponding
power of noise shows little dependence on frequency. This is
a reasonable result, because noise, whether it be of internal
or external origin, is supposed to originate from a large
number of unspecified and mutually incoherent vibration
sources, which are thought to have diverse peak frequencies.
[38] Figure 13 shows NS ratios and power spectral
densities of noise, inferred from records of the windy
day B, by the same analysis procedure as in the making
of Figure 12. In the frequency range of 0.1–0.7 Hz, the
estimates of are, for the array radius of 25 m, 2–10 times
larger than the estimates of int and, for the array radius
of 50 m, 2–50 times larger. Besides, the estimates are
10–100 times larger than their day C counterparts for
frequencies upward of 0.2 Hz. These findings imply that
strong winds augmented the noise of external origin.
Figure 12. Power spectral densities of the microtremors (left, solid color curves), NS ratio estimates
(right, color curves), and power spectral density estimates for the noise of both internal and external
origins combined (left, dotted color curves), for records of the 25 m (green) and 50 m (red) arrays day C.
Also shown are the NS ratio estimates (right, black curves) and power spectral density estimates (left,
black curves) for the noise of internal origin alone, estimated from huddle test records of the same date.
The power spectral density curves are drawn for all four sensors.
Figure 13. Same as in Figure 12, for arrays of radii 25 to 600 m, day B. For legibility, the power
spectral densities, corresponding to each array radius, are shown for the center sensor alone.
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11. [39] As a whole, the int estimates showed no significant
difference for days B and C, but in certain limited frequency
ranges, they were up to about 10 times larger on day C than on
day B. This may possibly be due to the wind directly shaking
the measurement instruments or rattling their hooks and lids
which were insufficiently fixed. The power of noise showed
as little dependence on frequency on day B as on day C.
[40] Incidentally, the total power of noise, estimated from
the array seismograms of day B, turned out to be higher for
the arrays of radii 100 m and upward, deployed in the
afternoon when the wind was stronger, than for the arrays of
radii 25 and 50 m, deployed in the morning when the wind
was less strong.
[41] The total NS ratio is expected to increase with array
radius as long as the whole array falls within the bounds of
the park. To explain this, let us assume that vibration
sources of microtremors are distributed in urban areas at
an approximately homogeneous density except in the park’s
interior. All sensors can be regarded as equally remote from
vibration sources when a small-sized array is deployed near
the center of the park. As the array size increases, sensors on
the circumference approaches vibration sources outside the
park, and accordingly, the ratio of wave components that do
not arrive coherently at all sensors increases. This reasoning
seems to account for the fact that in Figure 12, the
estimates are larger for the array radius of 50 m than for
25 m. However, the estimates of day B show no
recognizable difference for different array radii (Figure 13).
The increase in the noise of external origin, caused by
strong winds, may possibly have covered up any array size
dependence of that may have been present.
[42] If we rely on the above reasoning, the effects of noise
of external origin are expected to be small for small-sized
arrays on the order of several meters in radius unless there
are strong winds, so the CCA method alone is expected to
be sufficient for the purpose of exploring shallow subsur-
face structures, as long as we can minimize noise of internal
origin by using high-precision recording systems. In fact,
Cho et al. [2004] successfully applied the CCA method to
seismic arrays of radii 5 and 15 m, even in a park in an
urban area where traffic was fairly heavy. It is possible to
place the sensors at uneven intervals if site conditions make
it difficult to place them equidistantly around the circum-
ference [Cho et al., 2004, 2006].
[43] On the other hand, if the array radius is larger than a
certain level or if the measurements are conducted under
windy conditions, noise of external origin, beyond our
control, is expected to be large, so it is desirable to estimate
NS ratios using the method we have proposed in the present
study, and use those estimates as auxiliary data that help to
identify the longest-frequency resolution limit of the phase
velocity estimates. For the purpose of estimating NS ratios,
it is not necessary to install a sensor at the center of every
circular array deployed; it suffices to install a sensor at the
center of just one circular array with a relatively small
radius, because our method of NS ratio analysis has no
intrinsic resolution limit on the long-wavelength side.
7. Conclusion
[44] We have demonstrated that the Centerless Circular
Array (CCA) method of Cho et al. [2004] remains basically
applicable to real microtremor records when the seismic
array is on the order of several to several hundred meters in
radius. We have also defined a quantity, called the circle
coherence, which is a function of both the noise-to-signal
(NS) ratio of the array seismograms and the extent to which
multiple wave modes prevail. By making use of the circle
coherence, it is possible to evaluate NS ratios if only we
install a sensor at the center of a circular seismic array.
Results of real data analysis have borne out our hypothesis
that noise limits the resolution of the CCA method in long-
wavelength ranges. Ours is the first study, in the genealogy
of the spatial autocorrelation method [Aki, 1957] and other
methods of microtremor exploration using circular array
data [Cho et al., 2006], to propose a mathematical model
that is able to evaluate quantitatively both the NS ratio and
the extent to which multiple wave modes prevail.
[45] Scrutiny of the NS ratio estimates, obtained by circle
coherence analysis, allows us to evaluate theoretically the
extent to which phase velocity estimates of the CCA
method are biased by noise in long-wavelength ranges. A
method to analyze phase velocities of Rayleigh waves,
which uses simple seismic arrays and remains valid over
broad wavelength ranges, and a quantitative indicator of the
method’s longest-wavelength resolution limit: combined,
they are expected to provide a powerful methodological
tool that allows one to extract maximal information from
microtremor records.
Appendix A: Lower Limit for
[46] The weighted variance D of J0(rki) samples, defined
by equation (16), is expected to take a largest value when
part of the J0(rki) samples equals the maximum Jmax = 1.0
of the J0(Á) function and all the rest equals the minimum
Jmin = À0.4028. Since the weighted mean of the J0(rki)
samples should lie at J, it follows that the partition ratio
for the samples equaling Jmax should in this case be x =
(J À Jmin)/(Jmax À Jmin), so that the upper limit for D is
Dmax ¼ x Jmax À Jð Þ2
þ 1 À xð Þ Jmin À Jð Þ2
¼ Jmax À Jð Þ J À Jminð Þ ðA1Þ
By solving the quadratic inequation
0 A2
þ B þ C Dmax ! 0ð Þ ðA2Þ
we obtain the lower limit for the NS ratio as
min ¼
ÀB À
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
B2 À 4A C À Dmaxð Þ
p
2A
0 Dmax Cð Þ
0 Dmax ! Cð Þ
8
:
ðA3Þ
Appendix B: Effects of Spectral Windowing
[47] When the fundamental mode Rayleigh waves dom-
inate, equations (3) and (4) can each be rewritten, in the
long-wavelength limit rk ! 0, asymptotically as
GZ0Z0 r; r; !ð Þ ’ 42
f !ð Þ 1 À 1=2ð Þ rk !ð Þð Þ2
h i
ðB1Þ
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12. GZ1Z1 r; r; !ð Þ ’ 42
f !ð Þ Á 1=4ð Þ rk !ð Þð Þ2
ðB2Þ
This indicates that as long as f(!) and c(!) = !/k(!) both
vary slowly enough with frequency, GZ0Z0(r, r; !) and
GZ1Z1(r, r; !) are a concave and a convex function of !,
respectively. It then follows that if we artificially smooth
power spectral density estimates by applying a spectral
window of a finite width, we get power spectral density
estimates that are smaller than the true values for GZ0Z0(r, r; !)
and larger than the true values for GZ1Z1(r, r; !). This explains
why the use of a spectral window tends to give under-
estimated values for their mutual quotient GZ0Z0(r, r; !)/
GZ1Z1(r, r; !) in long-wavelength ranges.
[48] Acknowledgments. The authors are grateful to Tatsuro
Matsuoka for advice on the selection of the test site and to Takuya Shiga,
Takehito Tokunaga, and Michio Fukuyo for their help in the field measure-
ments. Comments by Edoardo Del Pezzo, an anonymous reviewer, and
Associate Editor Rodolfo Console have helped to improve the manuscript.
Part of the recording systems were borrowed from the National Research
Institute of Fire and Disaster courtesy of Shinsaku Zama and Ken
Hatayama. The present work was partially supported by grants-in-aid for
scientific research of the Japan Society for the Promotion of Science (JSPS)
and of the Japanese Ministry of Education, Culture, Sports, Science and
Technology (MEXT).
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ÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀ
I. Cho, Geological Survey of Japan, National Institute of Advanced
Industrial Science and Technology, Tsukuba Central 7, 1-1-1 Higashi,
Tsukuba, 305-8567, Japan. (ikuo-chou@aist.go.jp)
Y. Shinozaki and T. Tada, Department of Architecture, Faculty of
Engineering, Tokyo University of Science, 1-14-6 Kudan Kita, Chiyoda-ku,
Tokyo 102-0073, Japan. (sinozaki@rs.kagu.tus.ac.jp; kogutek@rs.kagu.
tus.ac.jp)
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