The Seventh Asia-Pacific Conference on
Wind Engineering, November 8-12, 2009,
Taipei, Taiwan
CROSS-CORRELATION OF FLUCTUATING COMPONENTS OF
WIND SPEED BASED ON STRONG WIND MEASUREMENT
1
Mayumi Fujimura1 and Junji Maeda2
JSPS Research Fellow (Graduate), Graduate School of Human-Environment Studies,
Kyushu University, 6-10-1 Hakozaki, Higashi-ku, Fukuoka 812-8581, Japan,
[email protected]
2
Professor, Faculty of Human-Environment Studies, Kyushu University,
6-10-1 Hakozaki, Higashi-ku, Fukuoka 812-8581, Japan,
[email protected]
ABSTRACT
We report on the structure of atmospheric turbulence, focusing especially on the spatial correlations of three
components of fluctuating wind speed using our records taken from measurements at a transmission tower
located at the tip of a cape during Typhoon KIRK. Our results show that the non-dimensional cross-spectrum of
longitudinal component of fluctuating wind speed between two distantly-positioned points of the Kármán type
has the best fit to the measurement results, and the approximate expression to the Kármán type without special
functions can sufficiently explain several characteristics of the non-dimensional cross-spectrum of isotropic
turbulence. In addition, we show that the cross-correlation coefficient between longitudinal and vertical lateral
components of fluctuating wind speed is approximately expressed using the spatial correlation coefficient
between the components of fluctuating wind speed in the same direction.
KEYWORDS: TURBULENCE STRUCTURE, CROSS-SPECTRUM, CROSS-CORRELATION
Introduction
For the dynamic response analysis of a structure under strong winds, the spectral
response method in a frequency domain or the step-by-step integration of motion equation in
a time domain is used. The estimation of the wind response of a slender structure requires the
longitudinal and/or lateral components (u- and v-components) of fluctuating wind speed in a
horizontal direction, and the vertical lateral component (w-component) is rarely considered.
However, we must set up the wind field including w-component in the case of calculating the
response of a suspension-bridge or a long-span conductor which is strongly affected by wcomponent. The spectral response method postulates a linear deformation of a structure, and
so for the structure for which deformation is not linear, we generally have to perform the stepby-step integration of motion equation. For example, for a coupling system with cable, such
as a transmission tower, an increase in the gradient angle of the conductor-plane with the
wind speed enlarges not only the geometrical nonlinearity of the conductor’s rigidity but also
the effect of w-component of wind speed on the response as revealed by [Fujimura and Maeda
et al. (2007)]. In this case, it is necessary to generate the three-dimensional field of fluctuating
wind speed including the three components in a time domain. And then the nonlinear
response simulation of the coupled system of tower and conductor is performable using the
step-by-step integration method.
In order to generate the three-dimensional wind field with the three components of
fluctuating wind speed under strong winds, information about the spatial structure between
the three components is needed. When the gust loading factor method was introduced to the
The Seventh Asia Pacific Conference on Wind Engineering
November 8-12, 2009, Taipei, Taiwan
Photo 1: View of observed tower from sea.
Photo 2: Arrangement of anemometer.
U1 (214.5m)
U2 (203.5m)
U3 (183.5m)
U1~U8
: Ultrasonic anemometer
Observed tower
214.5m
U4 (148.0m)
U5 (110.0m)
: For three components
of wind speed
U6 (83.0m)
: For two components
of wind speed
U7 (53.0m)
20m
U8 (30.0m)
Figure 1: Relation between the observed tower
and the azimuth direction.
Figure 2: Position of anemometers.
wind load design of structures and buildings, many semi-empirical equations concerning the
spatial structure of wind were proposed in studies by [Davenport (1961) and Harris (1971)].
Currently, Kármán’s spectrum, which was derived from the theory of isotropic turbulence in
the work by [Kármán (1948)], is often used as the power spectrum of fluctuating wind speed.
This spectrum expression is in good agreement with the measurement results as reported by
[Maeda and Adachi (1983), Maeda and Makino (1988)]. [ESDU75001 (1975)] describes
Kármán’s cross-spectrum as being the best expression for non-dimensional cross-spectrum of
fluctuating wind speed between two distantly-positioned points, but the exponential
expression proposed in a study by [Davenport (1961)] as an approximate expression is used in
practice because it is a simpler and more convenient form of expression to use than Kármán’s
cross-spectrum. However, the exponential expression is very different from Kármán’s crossspectrum derived from the theory of isotropic turbulence in that its correlation value is
consistently 1 at the point where frequency is zero regardless of the distance between two
points, and that it does not have any negative values as pointed out by [Maeda and Makino
(1980)]. As for the cross-correlation between the longitudinal component (u-component),
horizontal lateral component (v-component) and vertical lateral component (w-component) of
fluctuating wind speed, [ESDU74031 (1974)] describes the co-variance uv and vw as being
small and which can be ignored, but this is not so for the uw co-variance near the ground, and
the formulated value of uw / σ uσ w as a function of height. However, there are few earlier
reports which describe the co-variances and cross-correlation coefficients between u- and wcomponents of fluctuating wind speed at two distantly-positioned points, and we cannot find
any later reports.
The Seventh Asia Pacific Conference on Wind Engineering
November 8-12, 2009, Taipei, Taiwan
Table 1: Mean characteristics of the measurement data.
Sample No. Anemometer No.
U4
No. 1
U6
U8
U4
No. 2
U6
U8
U4
No. 3
U6
U8
U (m/s) σ u (m/s) σ v (m/s) σ w (m/s)
26.6
2.6
2.4
1.6
25.2
2.6
2.5
1.7
23.4
2.3
3.0
1.6
28.9
3.2
2.9
2.0
26.8
3.2
3.1
2.0
25.3
2.8
3.4
1.9
22.2
2.5
3.0
1.6
20.6
2.5
2.8
1.5
18.8
2.3
2.9
1.3
x
L u (m)
149.9
143.2
75.9
195.9
166.4
121.0
203.9
241.7
191.4
x
L v (m)
72.5
89.5
81.1
61.6
73.2
133.0
112.1
104.8
145.2
x
L w (m)
43.1
42.6
23.0
36.8
39.2
34.9
42.1
33.3
16.3
Kyushu
Japan
Observed
tower
T9612
(KIRK)
Figure 3: Path of Typhoon KIRK.
In this paper, the authors estimate the non-dimensional cross-spectrum of fluctuating
wind speed between two distantly-positioned points using wind measurement data at a 200m
high transmission tower located at the tip of a cape during the passing of Typhoon KIRK
(1996) And the estimated values are compared with Davenport’s exponential expression and
an approximate expression to the Kármán type proposed by [Maeda and Makino (1980)]
without uncontrollable special functions. Additionally, the authors discuss the co-variance
and cross-correlation coefficients between u- and w-components of fluctuating wind speed
from their data and propose an approximate expression for the values, which is useful for the
three-dimensional simulation of a fluctuating wind field in a time domain.
Outline of Strong Wind Observation
Observed Tower and Terrain around Observed Tower
The observed tower was a 214.5m tall 500kV transmission tower at the tip of a cape at
a height of 50m above sea level, as shown in Photo 1. Figure 1 shows the relation between
the observed tower and the azimuth direction. Hills stood to the north of the tower, and it was
surrounded by sea on its south, east and west sides.
Outline of Measurement Data
Eight ultrasonic anemometers, four for three components of wind speed and four for
two components, were positioned on the tower for wind measurement, as shown in Figure 2.
Every anemometer was located several meters from a main post in a horizontal direction, as
shown in Photo 2. The sampling frequency of measurement was 120 Hz. The strong wind
observations were conducted during typhoons and non-typhoons from 1996 to 1997, and in
this paper we selected a total of three samples measured at U4, U6 and U8 when the mean
wind speed of U2 was more than 20m/s and the wind direction was from the west where the
observed tower was downwind during the passing of Typhoon KIRK (No.12 in 1996), whose
path is shown in Figure 3. Each sample was 600 seconds long. Table 1 shows the mean wind
The Seventh Asia Pacific Conference on Wind Engineering
November 8-12, 2009, Taipei, Taiwan
speed, U , the standard deviation, σ , and the longitudinal scale of turbulence, x L , at every
anemometer calculated using every sample, and Figures 4(a), 4(b) and 4(c) show the time
histories recorded at anemometers U4, U6 and U8 for sample 1, respectively. Wind records of
U1 at the top of the tower were not included in this study because measurements were
affected by a position light near anemometer U1.
Non-dimensional Cross-spectrum of Fluctuating Wind Speed
Proposed equations for non-dimensional cross-spectrum
The non-dimensional cross-spectrum of u-component of fluctuating wind speed
between vertically-distantly-positioned points A and B, as shown in Figure 5, is written as
follows:
~
Suc (ζ , n)
Suc (ζ , n) =
,
(1a)
SuA (n) SuB (n)
∞
S (ζ , n) = ∫ Ruc (ζ ,τ ) exp(− j 2πnτ )dτ .
c
u
(1b)
−∞
10
0
0
100 200 300 400 500 600
Time (sec)
v-component (m/s)
40
30
20
10
0
0
100 200 300 400 500 600
Time (sec)
40
30
20
10
0
0
100 200 300 400 500 600
Time (sec)
15
10
5
0
-5
-10
-15
15
10
5
0
-5
-10
-15
w-component (m/s)
20
0
100 200 300 400 500 600
Time (sec)
(a) U4
0
100 200 300 400 500 600
Time (sec)
(b) U6
0
w-component (m/s)
30
15
10
5
0
-5
-10
-15
100 200 300 400 500 600
Time (sec)
w-component (m/s)
v-component (m/s)
40
v-component (m/s)
u-component (m/s)
u-component (m/s)
u-component (m/s)
Where Ruc (ζ ,τ ) = E[u A (t )uB (t + τ )] , SuA (n) and SuB (n) are the power spectra at points A and B
respectively, ζ is the vertical distance between points A and B and n is the frequency. The
non-dimensional cross-spectrum of Davenport and Kármán types and an approximate
expression to the Kármán type proposed in the work study by [Maeda and Makino (1980)] are
written as follows, respectively:
| nζ | ⎞
~
⎛
Davenport: S uc = exp⎜ − k r
(2a)
⎟,
U ⎠
⎝
11/ 6
⎧⎪⎛ θ ⎞5 / 6
⎫⎪
~
⎛θ ⎞
(2b)
Kármán: Suc = 1.772⎨⎜ ⎟ K 5 / 6 (θ ) − ⎜ ⎟ K1/ 6 (θ )⎬ ,
⎪⎩⎝ 2 ⎠
⎪⎭
⎝2⎠
15
10
5
0
-5
-10
-15
15
10
5
0
-5
-10
-15
15
10
5
0
-5
-10
-15
0
100 200 300 400 500 600
Time (sec)
0
100 200 300 400 500 600
Time (sec)
0
100 200 300 400 500 600
Time (sec)
(c) U8
Figure 4: Time histories of wind speed for sample 1.
The Seventh Asia Pacific Conference on Wind Engineering
November 8-12, 2009, Taipei, Taiwan
~
Maeda and Makino: Suc = exp(−k1θ ) ⋅ (1 − k2θ 2 ) .
(2c)
0.4
2
2 1/ 2
x
Where kr = 13(ζ / zm ) , zm = 0.5( z A + z B ) , θ = {(0.747 ζ Lu ) + (2πnζ U ) } , k1 = 1.0, k2 = 0.2
and K µ (θ ) is the modified Bessel function of the second kind of order µ .
Measurement Results of Non-dimensional Cross-spectrum
Figure 6 shows the non-dimensional cross-spectra of every non-dimensional distance,
x
ζ Lu , estimated from the measurement data in the case of n=0. The spectra and crosscorrelation coefficients which are mentioned later are estimated using the Auto-regressive
Model and the AR order is decided determined by the minimum FPE criterion as introduced
by [Maeda and Makino (1981), Maeda and Makino (1992)]. In the figure, the expressions
written with Eqs. (2a), (2b) and (2c) have been added. As shown in the figure, the estimated
values of measurement results decrease with an increase in the non-dimensional distance.
Kármán’s expression is in good agreement with the measurement results, but Davenport’s
expression is not in agreement with them. And the expression proposed by Maeda and
Makino is in very good agreement with Kármán’s expression.
Figures 7(a) and 7(b) show the non-dimensional cross-spectra of the lateral
components (v- and w-components) of fluctuating wind speed estimated from the
measurement data in the case of n=0, respectively. The non-dimensional distance on the
horizontal axis is defined as ζ 0.5x Lu using the characteristic of the theory of isotropic
turbulence that is x Lu :x Lv :x Lw = 2 : 1 : 1 . The estimated values of measurement results of v- and wcomponents decrease with an increase in the non-dimensional distance and trace Kármán’s
expression as well as those of u-component.
Figures 8(a), 8(b) and 8(c) show the non-dimensional cross-spectra of u-component
of every non-dimensional frequency, n xLu U , in the cases that ζ x Lu ≅ 0.28, 0.44 and 0.82
wB z
uB
B
U
ζ
~
S uc (ζ ,0)
vB
wA
1.0
Measured
Davenport
Kármán
Maeda ζ / xL
u
y
vA
uA
A
0.0
x
0.0 1.0 2.0 3.0 4.0 5.0 6.0
Figure 5: Positions of points A and B.
Figure 6: Non dimensional cross-spectra of
u-component in the case of n=0.
~
S wc (ζ ,0)
~
S vc (ζ ,0)
1.0
1.0
Measured
Davenport
Kármán
Maedaζ / 0.5 x L
u
0.0
Measured
Davenport
Kármán
Maedaζ / 0.5 x Lu
0.0
0.0 1.0 2.0 3.0 4.0 5.0 6.0
0.0 1.0 2.0 3.0 4.0 5.0 6.0
(a) v-component.
(b) w-component.
Figure 7: Non dimensional cross-spectra of v- and w-components in the case of n=0.
~
S uc (ζ , n)
~
S uc (ζ , n)
Measured
1.0
Davenport
Kármán
Maeda
nζ / x L u
1.0
0.0
0.0
Measured
Davenport
Kármán
Maeda
nζ / x L u
0.0
0.2
0.4
0.6
0.8
(a) ζ / Lu ≅ 0.28
x
1.0
0.0
0.2
0.4
0.6
0.8
(b) ζ / Lu ≅ 0.44
x
1.0
~
S uc (ζ , n)
Measured
Davenport
Kármán
Maeda
1.0
nζ / x L u
0.0
0.0
0.2
0.4
0.6
0.8
1.0
(c) ζ / Lu ≅ 0.82
x
Figure 8: Non-dimensional cross-spectra of u-component for each non-dimensional distance.
The Seventh Asia Pacific Conference on Wind Engineering
November 8-12, 2009, Taipei, Taiwan
respectively. In the figures, the expressions written with Eqs. (2a), (2b) and (2c) have been
added. Regardless of non-dimensional distance, Kármán’s expression is in good agreement
with the measurement results, and especially in the case of ζ x Lu ≅ 0.44 when the
measurement results include negative values as shown in Figure 8(b), although it is clear that
Davenport’s cross-spectrum cannot express this characteristic. The expression proposed by
Maeda and Makino can sufficiently explain the characteristics of the measurement results.
Cross-correlation Coefficient between u- and w-components of Fluctuating Wind Speed
Definition of Cross-correlation Coefficient between u- and w-components
The cross-correlation coefficient between u- and w-components of fluctuating wind
speed at vertically-distantly-positioned points A and B, as shown in Figure 5, is written as
follows:
R c (ζ ,τ )
~c
Ruw
(ζ ,τ ) = uw
.
(3)
σ uAσ wB
Where R (ζ ,τ ) = E[u A (t )wB (t + τ )] and σ uA and σ uB are the standard deviations of u- and wcomponents at points A and B, respectively.
c
uw
Measurement Results of Cross-correlation Coefficient between u- and w-components
~
c
Figures 9(a), 9(b) and 9(c) show the values of uw / σ uσ w = Ruwc (0,0) , R~uw
(0,τ ) and
~c
Ruw (ζ ,0) estimated from the measured data, respectively. As shown in the figures, the values
of uw / σ uσ w are between -0.3 and -0.4 at every height, z. The values of R~uwc (0,τ ) are distributed
nearly symmetrically against the vertical axis of τU x Lu = 0 and close to zero with an increase
in absolute values of non-dimensional time, τU x Lu . The values of R~uwc (ζ ,0) are close to zero
with an increase in non-dimensional distances, ζ x Lu , and the absolute values of R~uwc (ζ ,0)
are smaller than 0.1 in the case of ζ x Lu > 0.5 .
In order to evaluate the statistical nature of R~uwc (0,τ ) and R~uwc (ζ ,0) , we define the ratio
~
of the cross-correlation coefficient to uw / σ uσ w , Ruwc* (ζ ,τ ) , and Figures 10(a) and 10 (b) show
~ c*
~ c*
the measurement results of Ruw (0,τ ) and Ruw (ζ ,0) respectively. In Figure 10(a), the averages of
~ c*
Ruw
(0,τ ) estimated from all the measurement data are discretely expressed. The longitudinal
0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
~
uw / σ u σ w
0
50
100
~c
R c (ζ ,0)
Ruw
(0, τ )
0.1 uw
0.1
0.0 0.2 0.4 0.6 0.8 1.0 1.2
150 200 -5 -4 -3 -2 -1 0 1 2 3 4 5
0.0
0.0
x
z (m)
ζ / x Lu
τU / Lu
-0.1
-0.1
Measured
-0.2
-0.2
-0.3
-0.3
Measured
Measured
-0.4
-0.4
-0.5
-0.5
~c
~c
(c) Ruw (ζ ,0)
(b) Ruw (0,τ )
(a) uw / σ uσ w
Figure 9: Co-variance and cross-correlation coefficients in the cases of
τ = 0 and ζ = 0 between u- and w-components.
1.0
0.0
~ c*
Ruw
(0, τ )
1.0
Measured
Equation (4)
Kármán
τU / x Lu
-5 -4 -3 -2 -1 0 1 2 3 4 5
~ c*
Ruw
(ζ ,0)
Measured
Equation (5)
Kármán
0.0
ζ / x Lu
0.0 0.2 0.4 0.6 0.8 1.0 1.2
~ c*
~ c*
(a) Ruw
(b) Ruw
(0,τ )
(ζ ,0)
Figure 10: Ratio of cross-correlation coefficient to uw / σ uσ w
in the cases of ζ = 0 and τ = 0 .
The Seventh Asia Pacific Conference on Wind Engineering
November 8-12, 2009, Taipei, Taiwan
spatial correlation coefficient, f (r ) , and the lateral spatial correlation coefficient, g (r ) ,
between the components of fluctuating wind speed in the same direction proposed by
[Kármán (1948)], which are written as Eqs. (4) and (5) respectively, have been added in
Figures 10(a) and 10(b) respectively:
f (r ) = f (U τ ) = α1 | a1U τ |1 / 3 K1 / 3 (| a1U τ |) ,
(4)
(5)
g (r ) = g (ζ ) = α1 | a1ζ |1/ 3 {K1/ 3 (| a1ζ |) − 0.5 | a1ζ | K − 2 / 3 (| a1ζ |)} ,
2/3
x
x
where α1 = 2 Γ(1 / 3) ≅ 0.5926 and a1 = π Γ(5 / 6) Γ(1 / 3) Lu ≅ 0.7468 Lu . As shown in Figure
10(a), Eq. (4) is in good agreement with the measurement results. And as shown in Figure
10(b), the measurement results vary but Eq. (5) traces the median measurement result.
Derivation of Approximate Expression for Measurement Result
[Hinze (1987)] showed that the cross-correlation coefficient between u- and wcomponents is defined using the spatial correlation coefficient between the components of
fluctuating wind speed in the same direction, f (r ) and g (r ) , as follows:
f (r1 ) − g (r1 )
~c
Ruw
(ζ ,τ ) =
⋅ |Uτ | ζ .
(6)
r12
Where r12 = (U τ ) 2 + ζ 2 . Eq. (6) was derived by assuming that the co-variance between the
mutually-perpendicular components of fluctuating wind speed is zero in the field of isotropic
turbulence. However, the measurement results of the co-variance between u and w
components, uw , are not zero, as shown in Figure 9(a). Additionally, in considering that
~c
~c
Ruw
(0,τ ) ≅ uw / σ uσ w ⋅ f (r ) and Ruw
(ζ ,0) ≅ uw / σ uσ w ⋅ g (r ) as shown in Figure 10, it is thought that
the approximate expression for the measurement results of the cross-correlation coefficient
between u- and w-components can be written as follows:
⎫ g (r ) ⎧
⎫
f (r ) ⎧
uw
uw
~c
Ruw
(ζ ,τ ) = 21 ⎨| U τ | ζ +
(U τ ) 2 ⎬ − 21 ⎨| U τ | ζ −
(ζ ) 2 ⎬
(7)
r1 ⎩
r
σ uσ w
σ
σ
1
u w
⎭
⎩
⎭
~
Figures 11(a) and 11(b) show the comparisons of the measurement results of Ruwc (ζ ,τ )
with Eq. (7) in the cases of ζ x Lu = 0.25 and ζ x Lu = 1.04, respectively. In the case of
ζ x Lu = 0.25, the approximate expression is in good agreement with the measurement results.
On the other hand, in the case of ζ x Lu = 1.04, the approximate expression includes the
positive values and is different from the measurement results. However, as shown previously,
there are few correlations between u and w components in the case of ζ x Lu > 0.5, and so it is
thought that this distinction is not important.
Conclusions
Some of our findings concerning the spatial correlation of three components of
fluctuating wind speed based on the wind speed measurement at the transmission tower
-5 -4 -3 -2 -1
0.0
0 1
2
3
4
τU / x Lu
Measured
-0.3
-5 -4 -3 -2 -1
0.0
0 1
2
3
4
5
τU / x Lu
-0.1
-0.1
-0.2
5
Equation (7)
~c
Ruw
(ζ ,τ )
(a) ζ / xLu = 0.25
-0.2
-0.3
Measured
Equation (7)
~c
Ruw
(ζ ,τ )
(b) ζ / xLu = 1.04
Figure 11: Comparison of measurement results of cross-correlation coefficients between
u- and w-components with Eq. (7) in the cases of ζ x Lu = 0.25 and ζ x Lu = 1.04 .
The Seventh Asia Pacific Conference on Wind Engineering
November 8-12, 2009, Taipei, Taiwan
located at the tip of a cape during Typhoon KIRK (1996) are summarized as follows.
(1) The measurement results of the non-dimensional cross-spectra are in good agreement with
Kármán’s expression derived from the theory of isotropic turbulence.
(2) The approximate expression to the Kármán type without special functions can sufficiently
explain several characteristics of the measurement results of the non-dimensional crossspectra.
(3) The cross-correlation coefficients between the longitudinal and vertical lateral
components of fluctuating wind speed are close to zero with an increase in absolute values
of non-dimensional time and non-dimensional distances.
(4) The cross-correlation coefficients between the longitudinal and vertical lateral
components of fluctuating wind speed are approximately expressed using the spatial
correlation between the components of fluctuating wind speed in the same direction.
The spatial correlation of the three components of fluctuating wind speed, as described
in this paper, can be useful for the three-dimensional simulation of a wind field in a time
domain.
Acknowledgement
This work was supported by Grant-in-Aid for JSPS Fellows, 21-4044, Japan Society
for the Promotion of Science.
References
Davenport, A. G. (1961), “The Spectrum of Horizontal Gustiness near the Ground in High Winds”, Q. J. Roy.
Met. Soc., 87, 194-211.
ESDU74031 (1974), “Characteristics of Atmospheric Turbulence near the Ground Part II: Single Point Data for
Strong Winds (neutral atmosphere)”, Engineering Sciences Data Unit, Item No. 74031.
ESDU75001 (1975), “Characteristics of Atmospheric Turbulence near the Ground Part III: Variations in Space
and Time for Strong Winds (neutral atmosphere)”, Item No. 75001.
Fujimura, M., Maeda, J., Morimoto, Y. and Ishida, N. (2007), “Aerodynamic Damping Properties of a
Transmission Tower Estimated Using a New Identification Method”, 12th International Conference on Wind
Engineering Preprints-Vol.1, 1047-1054.
Harris, R. J. (1976), “The Nature of the Wind”, Proceedings of CIRIA Seminar on the Modern Design of Windsensitive Structures, Construction Industry Research and Information Association, London.
Hinze, J. O. (1987), Turbulence, McGraw-Hill, 181-184.
Kármán, T. v. (1948), “Progress in the Statistical Theory of Turbulence”, Proceedings of Natl. Acad. Sci.,
Vol.34, 530-539.
Maeda, J. and Makino, M. (1980), “Classification of Customary Proposed Equations Related to the Component
of the Mean Wind Direction in the Structure of Atmospheric Turbulence and these Fundamental Properties”,
Trans. Architect. Inst. Japan, No.287, 77-87 (in Japanese) .
Maeda, J. and Makino, M. (1981), “On the Methods of Spectral Analyses of Turbulent Winds – Adaptation of
Auto-regressive Model”, Trans. Architect. Inst. Japan, 300, 19-29 (in Japanese) .
Maeda, J. and Makino, M. (1983), “On the Spatial Structures of Longitudinal Wind Velocities near the Ground
in Strong Winds”, Journal of Wind Engineering and Industrial Aerodynamics, 15, 197-207.
Maeda, J. and Makino, M. (1988), “Power Spectra of Longitudinal and Lateral Wind Speed near the Ground in
Strong Winds”, Journal of Wind Engineering and Industrial Aerodynamics, 28, 31-40.
Maeda, J. and Makino, M. (1992), “Characteristics of Gusty Winds Simulated by an A.R.M.A. Model”, Journal
of Wind Engineering and Industrial Aerodynamics, 41-44, 427-436.