The Seventh Asia-Pacific Conference on
Wind Engineering, November 8-12, 2009,
Taipei, Taiwan
AN EMPIRICAL JOINT PROBABILITY DENSITY FUNCTION OF
WIND SPEED AND DIRECTION
1
Jun Chen1 and Xiaoqin Zhang2
Associate Professor, Department of Building Engineering & State Key Laboratory of
Disaster Reduction in Civil Engineering, Tongji University ,Shanghai, China
cejchen@tongji.edu.cn
2
Postgraduate, Department of Building Engineering, Tongji University
Shanghai, China, zhangxiaoqin0808@163.com
ABSTRACT
An empirical joint probability density function (JPDF) of mean wind speed and direction is presented in this
paper. The proposed JPDF model is built up by marginal distributions of wind speed and wind direction that are
assumed as an Extreme-Value equation. Details of the JPDF model are first discussed with focus on application
procedure for both unimodal and bimodal wind data and approaches for determining the modal parameters. It is
then applied to field measured joint probability density of wind sample from parent population and extreme
value data as well. Design wind prediction for given return period using the proposed JPDF is also briefly
discussed. The performance of the proposed JPDF model is assessed by goodness-of-fit criteria. It is concluded
from the results that the proposed JPDF model can represent quite well the joint distribution properties of wind
speed and direction from different data source.
KEYWORDS: JOINT PROBABILITY DENSITY FUNCTION, WIND SPEED, WIND DIRECTION
Introduction
Field measurement of wind turbulent properties and analysis is a long-term and
fundamental task for structural wind engineering since uncertainties of wind load is in fact the
key factor that affects the analysis accuracy of wind-resistant structures. Among many
turbulent wind parameters as wind spectrum, wind profile, turbulent intensity and so on, the
joint probability density function of mean wind speed and direction (short as JPDF hereafter)
is an very important but less addressed property. The importance of wind directionality for
design wind prediction, wind-induced fatigue analysis and wind-energy assessment has been
underlined and emphasized by many researchers. Without considering wind directionality, as
pointed out by Moriarty [Moriarty 1983], the design extreme wind speed might be
overestimated. In the current non-directional wind-induced fatigue analysis procedure, it
assumes that the wind blows with constant direction during the whole life of the structure
concerned, which in turn leads to in general conservative prediction [Repetto and Solari, 2004,
Xu and Chen 2008]. In the evaluation of wind-power resources available at a given site, the
knowledge of joint probability density function is crucial for positioning the wind turbines to
maximize the capturable energy [Carta 2008].
Several JPDF models have been proposed by researchers in the past decades that can
be broadly classified into continuous function and discrete function. McWilliams
[McWilliams 1979] suggested an isotropic Gaussian model based on the assumptions that the
wind speed component along the prevailing wind direction follows normal distribution with
non-zero mean and a given variance, while the wind speed component along the direction
orthogonal to the prevailing direction is independent and normally distributed with zero mean
The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan
and same variance. Weber [Weber 1991] extended McWilliams model to the anisotropic
Gaussian model in which the same variance limitation was released. Angular-linear
distribution function is often adopted to model the JPDF, and normally the wind speed is
modeled as Gaussian distribution and wind direction is modeled as von-Mises distribution
[Marida 2000]. More recently, Carta [Carta 2008] proposed a new JPDF model based on the
angular-linear distribution originally suggested by Jonhson [Jonhson and Wehrly 1978 ]. This
model is derived using marginal PDF of wind speed and wind direction, which are assumed as
normal Weibull mixture distribution and von Mises distribution respectively. As for discrete
JPDF model, the basic idea is divide the whole circle into several sub-sections, in which the
probability density of wind speed is assumed as Weibull, Gumbel or other functions and some
measures are introduced to account for the correlation between each section [Gu 1999, Ge and
Xiang 2002, Matsui 2002, Xu and Chen 2008].
None of the above JPDF models, however, enjoy universal acceptance nowadays.
More attempts are necessary to archive a better JPDF model. In this connection, an empirical
JPDF model is presented in this paper based on the classical directional statistical theory on
angular-linear distribution. The proposed JPDF model is actually built up by marginal
distributions of wind speed and wind direction whose distribution is assumed as an ExtremeValue equation. The framework of the proposed JPDF model is introduced in the following
section. The performance of the model is then assessed by applying to field measured JPDFs
from different sources.
Proposed Empirical JPDF model
Basic functions
There has been extensive research carried out over the last forty more years to address
separately the probability distribution of mean wind speed or wind direction. Therefore, the
most distinct way to form the joint probability density function of wind speed and direction is
using the information of the marginal distributions. In wind-induced structural fatigue
analysis, for instance, the general approach is to divide the wind direction into several
subsections and apply identical probability function say Weibull distribution for wind speed
in each subsection and assume statistical independence among subsections [Ge 2002]. For
angular-linear distribution, Johnson and Wehrly [Johnson 1978] suggested to construct the
joint probability density using marginal distribution as
fV Θ ( v,θ ) = 2π g (ζ ) fV ( v ) fΘ (θ ) ; 0 ≤ θ ≤ 2π ; − ∞ ≤ v ≤ ∞
(1)
where fV ( v ) , fΘ (θ ) is probability density function for mean wind and direction respectively;
ζ is a circular variable and g (ζ ) is its probability density function, and g (ζ ) can be
derived from cumulative probability distribution of mean wind speed and direction.
Inspired by Eq.1 the following empirical JPDF model is suggested based on the
following two assumptions: (1) the existence of one or multi- prevailing wind directions, and
the whole circle [0~2 π ] section can be accordingly divided into several sub-section (usually
two) each possessing only one prevailing wind direction; (2) in each sub-section the JPDF
model is identical as:
fΩi (U ,θ ) = a * fU ( v, b, c ) + d * fΘ (θ , e, f ) + g * fU (v, b, c)* fΘ (θ , e, f )
(2)
where Ωi is the ith sub-section, a,b,c,d,e,f, g are seven unknown model parameters to be
determined; function fU ( v, b, c ) , fΘ (θ , e, f ) are the extreme-value equation as
The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan
⎛
⎛ v −b ⎞ ⎛ v −b ⎞ ⎞
fU ( v, b, c ) = exp ⎜ − exp ⎜ −
(3)
⎟ −⎜
⎟ + 1⎟
⎝ c ⎠ ⎝ c ⎠ ⎠
⎝
⎛
⎛ θ −e ⎞ ⎛θ −e ⎞ ⎞
(4)
f (θ , e, f ) = exp ⎜ − exp ⎜ −
0 < θ ≤ 2π
⎟−⎜
⎟ + 1⎟
f ⎠ ⎝ f ⎠ ⎠
⎝
⎝
Given the above JPDF, the marginal PDF for wind direction/wind speed can then be
calculated by integrating over θ / U .
Estimation of model parameters
Two technical issues associated with the proposed JPDF model are definition of each
sub-section and determination of modal parameters. Taking the most common situation of
two prevailing wind directions as example, suppose θ1 is the wind direction having the lowest
occurrence frequency of wind data and θ 2 is another wind direction having the lowest
occurrence frequency of wind data in the range of [θ1 + π − π / 4, θ1 + π + π / 4] . The two subsections can then be defined as
Ω1 = {θ ∈ (θ1,θ2 ) mod 2π } Ω2 = {θ ∈ (θ2 ,θ1 ) mod 2π }
,
The prevailing wind direction in each sub-section can then be calculated as
θΩ1 =
( f0 − f−1 ) βk + βk −1 ( f0 − f+1 ) = β
2 f0 − f −1 − f +1
k −1
+
f0 − f −1
( βk − βk −1 )
2 f0 − f +1 − f −1
(6)
(7)
where ( βk −1, βk ] is the upper and lower bound of the wind direction interval having the
maximum wind occurrence; f0 is the wind occurrence in this interval, f −1 and f +1 are the
wind occurrence value in the adjacent interval [Mardia 2000]. Finally, the unknown modal
parameters can be identified by fitting the model to field measured data using non-linear least
square algorithm.
Prediction of design wind speed
A simplified procedure is used in this paper to predict the design wind speed at a
certain wind direction zone ⎡⎣θ j − Δβ / 2,θ + Δβ / 2⎤⎦ using the proposed JPDF. By this way,
the value obtained can be directly compared with that determined by other discrete JPDF
model. In particular, the design wind speed at certain wind direction zone
⎡⎣θ j − Δβ / 2,θ + Δβ / 2⎤⎦ in return period R (year) can be determined by the following
equation
Δβ
2
Δβ
θj−
2
∫
θj+
∫
U max
0
P ( u , θ ) dudθ = 1 −
1
R
(8)
where θ j Δβ and U max is, respectively, wind direction, interval and design wind speed.
Application and Discussion
Various types of field measured wind data are employed in this section to assess the
applicability of the proposed model. In specific, unimodal and bimodal wind data and wind
The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan
sample from parent population and extreme value data are adopted. The verification
procedure is as follows: (1) compute the discrete joint probability density function using field
measured mean wind speed and direction; (2) fit the proposed model to the measured one to
determine the modal parameters. The performance of the proposed model is judged by
goodness-of-fit criteria.
Data Source
The proposed JPDF is applied to three field measured discrete JPDFs, denoted as Case
1 to 3. In Case 1, the measured JPDF was computed from hourly mean wind speed and wind
direction of a city for 100 year, which is considered as bimodal sample from parent
population [Chen 2009]. In Case 2, the measured JPDF was computed from weekly extreme
value of wind speed and direction of a city for more than thirty year, which is considered as
bimodal sample from extreme value [Yang 2002].. In Case 3, the measured JPDF was
calculated from 5-year records of hourly mean wind speed and direction from a 50 m high
mast [Chen 2007], which is considered as unimodal sample from parent population.
For each case, the whole circle is divided into 16 sections each of 22.5 degree interval,
the number of wind speed falling in each section are counted and the occurrence rate is
calculated accordingly through Eq.9, where N j , N is the number of wind data in the section
and total wind data. The measured JPDF (wind occurrence rate) for Case 1, Case2 and Case 3
are given in Table 1, Table 2 and Table 3 respectively.
Pij =
N ij
(9)
N * ΔU * Δθ
Table 1: Measured JPDF of Case 1 (%)
m/s
N
NNE
NE
ENE
E
ESE
SE
SSE
S
SSW
SW
WSW
W
WNW
NW
NNW
0～2
1.8026
1.6270
1.8458
1.1105
1.7163
4.8378
5.2799
1.4097
0.9750
0.9929
0.8395
0.4451
0.1370
0.0655
0.5716
1.6002
2～4
4.9092
3.8122
2.4531
1.1745
2.3638
7.7166
6.4052
2.3072
1.9872
1.8845
1.9991
0.6803
0.2620
0.0893
0.8395
3.7615
4～6
3.6246
2.0795
0.6371
0.3855
1.5079
3.7318
1.8964
1.2042
1.1372
1.4603
0.9795
0.2947
0.0729
0.0372
0.4495
1.9664
6～8
1.4692
0.5210
0.1503
0.0997
0.6698
1.0479
0.4525
0.4421
0.4882
0.5686
0.2679
0.0774
0.0268
0.0164
0.1131
0.6505
8～10
0.5002
0.1563
0.0461
0.0283
0.2858
0.3126
0.1727
0.1221
0.1637
0.2114
0.1012
0.0357
0.0149
0.0045
0.0387
0.2263
10～12
0.1370
0.0164
0.0119
0.0104
0.1250
0.1131
0.0670
0.0342
0.0298
0.0432
0.0104
0.0119
0.0015
0.0015
0.0089
0.0432
12～14
0.0417
0.0074
0.0015
0.0000
0.0566
0.0536
0.0238
0.0134
0.0089
0.0074
0.0074
0.0060
0.0030
0.0000
0.0060
0.0164
14～16
0.0089
0.0030
0.0015
0.0000
0.0298
0.0119
0.0134
0.0074
0.0074
0.0104
0.0000
0.0000
0.0000
0.0000
0.0015
0.0045
16～18
0.0074
0.0000
0.0030
0.0000
0.0149
0.0179
0.0045
0.0000
0.0060
0.0015
0.0015
0.0015
0.0030
0.0000
0.0030
0.0045
18～20
0.0030
0.0000
0.0015
0.0030
0.0060
0.0089
0.0000
0.0000
0.0000
0.0000
0.0000
0.0015
0.0000
0.0000
0.0000
0.0015
20～22
0.0000
0.0000
0.0000
0.0030
0.0000
0.0030
0.0030
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
22～24
0.0015
0.0000
0.0000
0.0000
0.0015
0.0000
0.0015
0.0015
0.0000
0.0000
0.0015
0.0000
0.0000
0.0000
0.0000
0.0000
24～26
0.0000
0.0000
0.0015
0.0000
0.0030
0.0000
0.0000
0.0000
0.0015
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
26～28
0.0000
0.0000
0.0000
0.0000
0.0015
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
28～30
0.0000
0.0000
0.0000
0.0000
0.0015
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0015
0.0000
30～32
0.0000
0.0015
0.0000
0.0000
0.0015
0.0015
0.0015
0.0000
0.0015
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
32～34
0.0000
0.0000
0.0000
0.0015
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
34～36
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0015
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
36～38
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
38～40
0.0000
0.0000
0.0000
0.0000
0.0000
0.0015
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
40～42
0.0000
0.0000
0.0000
0.0000
0.0015
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
Total
12.5052 8.2242
5.1533
2.8163
6.7863 17.8580 14.3227 5.5418
4.8065
5.1801
4.2081
1.5540
0.5210
0.2143
2.0334
8.2748
Comparison of Measured and Calculated JPDF
The field measured JPDFs for case 1 to 3 are depicted on left side of Fig. 1, and the
proposed JPDFs are shown on right side of Fig. 1 accordingly. The identified modal
parameters and the Chi-square error for the three cases are given in Table 4. It is observed
The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan
from Fig.1 that the proposed JPDF model can represent the field measurement results quite
well. The Chi-square error for Case 1, Case 2 and Case 3 is larger than 0.94, the minimum
value is 0.942 for case 3 which indicates a very good fit for all cases. Furthermore,
introducing the identified seven unknown parameters into Eq.5, the calculated value is unity
for each cases which also indicates a proper PDF model.
Table 2: Measured JPDF of Case 2
m/s
0-2
2-4
4-6
6-8
8-10
10-12
12-14
14-16
16-18
18-20
20-22
22-24
24-26
26-28
N
0
0.876
2.378
1.314
0.313
0.125
0
0
0
0
0
0
0
0
NNE
0
1.001
3.004
1.377
0.313
0.063
0.063
0.063
0
0
0
0
0
0
NE
0
0.814
3.191
2.253
0.626
0.063
0
0
0
0.125
0
0
0
0
ENE
0
1.502
5.882
2.879
0.751
0.125
0
0
0
0
0
0.063
0
0
E
0
2.315
6.821
2.128
0.501
0.125
0
0
0
0
0
0
0
0.063
N
0.081
0.465
0.126
0.152
0.081
0.071
0.076
0.106
0.071
0.035
0.02
0.005
0.01
0
0.005
0
NNE
0.101
0.753
0.647
0.647
0.44
0.324
0.314
0.197
0.142
0.066
0.046
0.015
0
0
0
0
NE
0.086
0.87
0.728
0.389
0.288
0.197
0.167
0.025
0.01
0.015
0.005
0
0
0
0
0
ENE
0.142
3.075
4.42
3.363
1.29
0.435
0.177
0.278
0.263
0.147
0.056
0.046
0.01
0.01
0
0.005
E
0.243
3.798
6.144
6.528
2.19
0.501
0.329
0.253
0.152
0.116
0.116
0.046
0.02
0.005
0
0
ESE
0
1.252
4.38
2.128
0.501
0.125
0.063
0.063
0
0
0
0
0
0
SE
0
1.189
3.88
2.065
0.688
0.188
0
0.063
0
0
0
0
0
0
SSE
0
1.377
4.13
2.441
0.751
0.125
0.063
0.063
0
0
0
0
0
0
S
0
0.751
1.877
0.814
0.188
0
0.125
0
0
0
0
0
0
0
SSW
0
0.501
0.939
0.25
0.063
0.063
0
0
0
0
0
0
0
0
SW
0
0.375
0.563
0.063
0.125
0
0
0
0
0
0
0
0
0
WSW
0
0.501
0.688
0.25
0.063
0
0.063
0
0
0
0
0
0
0
W
0
0.751
1.377
0.563
0.188
0.125
0.063
0
0
0
0
0
0
0
WNW
0
1.502
2.879
2.128
1.064
0.438
0.125
0
0.063
0
0
0
0
0
NW
0
1.252
3.379
2.003
1.001
0.25
0.063
0.063
0
0
0
0
0
0
NNW
0
1.064
2.566
1.627
0.438
0.063
0.063
0
0
0
0
0
0
0
WSW
0.101
0.278
0.455
0.329
0.066
0.03
0.01
0.005
0.005
0
0
0
0
0
0
0
W
0.101
0.602
0.637
0.607
0.182
0.071
0.015
0.005
0
0
0
0
0
0
0
0
WNW
0.091
0.475
0.435
0.303
0.061
0.015
0.01
0
0
0
0
0
0
0
0
0
NW
0.051
0.379
0.142
0.147
0.051
0.035
0
0
0
0.005
0
0
0
0
0
0
NNW
0.308
1.102
0.622
0.389
0.223
0.131
0.025
0.035
0.03
0.005
0.01
0
0.005
0
0
0
Table 3: Measured JPDF of Case 3
m/s
0-2
2-4
4-6
6-8
8-10
10-12
12-14
14-16
16-18
18-20
20-22
22-24
24-26
26-28
28-30
30-32
ESE
0.253
3.004
4.733
4.243
1.446
0.47
0.217
0.137
0.056
0.02
0.005
0.01
0.01
0
0
0
SE
0.293
2.908
3.965
2.655
0.86
0.43
0.142
0.051
0.04
0.025
0.005
0.01
0
0
0
0.005
SSE
0.172
1.76
2.528
2.215
0.753
0.354
0.121
0.091
0.051
0.01
0.01
0.005
0.005
0
0
0
S
0.283
1.35
2.336
2.599
0.925
0.465
0.228
0.086
0.025
0.025
0.005
0.015
0.01
0.005
0
0
SSW
0.187
0.91
1.163
1.077
0.531
0.142
0.167
0.137
0.106
0.025
0
0.005
0
0
0
0
SW
0.071
0.531
0.855
0.753
0.314
0.051
0.015
0.005
0.005
0
0.005
0
0
0
0
0
Table 4: Modal Parameters and Chi-square Error
Case
1
2
3
Ω1
a
0.979
b
2.273
Modal parameters
c
d
e
f
1.531 0.007 117.001 20.117
g
7.759
Chi-square
error γ 2
0.951
Ω2
0.128
2.811
1.711
-0.005 290.241 28.594
4.980
0.967
Ω1
-7.178
4.759
1.438
-0.001
83.238 115.357 12.607
0.946
Ω2
Ω
0.532
4.925
1.795
-0.005 310.465 33.401
2.876
0.976
0.584
4.934
2.227
-0.023
5.884
0.942
Sect
ion
94.119
33.042
Determination of Design Wind Speed
Taking Case 2 as an example, the design wind speed for 100y return period is calculated
and given in Table 5. In [Ge 2002, Yang 2002] the joint distribution probability model is
suggested by assuming identical probability model with different model parameters for each
wind direction section. The prediction of design wind speed for 100y for Gumbel, Frechet,
The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan
Weibull distribution are also given in Table 5 for comparison. Figure 2 plots the design wind
speed at each wind direction. It is seen from Fig.2 that some kind of ‘smoothing effect’ can be
observed for values obtained by the proposed JPDF model, since the value for relatively small
occurrence wind direction become large while those for relatively large occurrence wind
direction become small. Generally, the value obtained by the proposed JPDF mode is small
than that from separated Gumbel/Frechet/Weibull distribution. Further work is necessary to
explore the meaning of directional design wind speed.
8
6
Frequency
4
6
4
2
2
2
00
90 45 )
4
8
10
12
14
16
(m
/s)
/s)
(m
ree
180
225
(Deg
270
ction
e
315
ir
dD
360
Win
135
6
d
ee
Sp
d
ee
Sp
8
10
12
14
ind
ind
6
W
W
4
2
360
315
225
180
135
0
0
90 45
(D
ction
Dire
d
in
W
270
Probability Dens
ity
8
e)
egre
(a) Bimodal wind sample from parent population
ty (%)
Probability Densi
6
4
4
(m
/s
Wind 100 150
200
Dire
ction
250
300
(Deg
ree)
6
4
2
2
0
0
45
Win 90135
dD
180
irec 225
tion 270
(De 315
gre
e) 360
ed
8
50
Sp
e
0
0
16
14
12
10
)
2
W
in
d
ty(%)
Probability Densi
6
10
8
6
4
2
in
W
16
14
s)
12
m/
e
pe
dS
d(
7
6
6
2
2
1
(m
)
0
0 45
16
14
12
10
90
135
Win
180
d Dir
ectio 225 270
n (D
egre 315 360
e)
6
4
2
(c) Unimodal wind sample from parent population
Figure 1: Filed measured JPDF (left) and Computed JPDF (right)
8
W
ind
4
d
8
6
Sp
ee
0
0 45
Wind 90 135
180
Dire
ction 225
(Deg 270
ree) 315 360
16
14
12
10
/s
1
2
2
3
2
3
4
)
4
5
(m
/s
5
Sp
ee
d
ty(%)
Probability Densi
7
W
in
d
Probability Dens
ity
(b) Bimodal wind sample of extreme value
The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan
337.5
0.0
28
22.5
24
315.0
45.0
20
292.5
67.5
16
12
8270.0
8
90.0
2
6 247.5
112.5
0
4
225.0
8
135.0
202.5
180.0
157.5
Gumbel
Frechet
Weibull
JPDF
Figure 2: Comparison of design wind speed for Case 2
Table 5: Design wind speed prediction at returen period 100Y
Direction
Gumbel
distribution
Frechet
distribution
Weibull
distribution
JPDF
N
NNE
NE
ENE
E
ESE
SE
SSE
S
SSW
SW
WSW
W
WNW
NW
NNW
W
15.22*
17.81
23.91
22.48
26.79
17.65
16.75
17.73
17.3
12.8
12.98
15
17.07
20.44
18.6
15.21
15.22
12.38
14.99
18.06
18.84
21.37
14.98
14.5
15.02
13.79
11.22
10.27
12.76
13.97
16.89
15.41
13.11
12.38
13.21
15.43
18.86
17.32
18.19
15.93
15.89
15.47
15.38
11.84
12.91
13.76
17.02
17.95
16.57
14.46
13.21
13.75
14.19
14.5
14.56
14.62
14.52
14.45
14.21
12.74
11.12
10.93
12.81
14.11
13.72
13.67
13.73
13.75
*Unit: m/sec
Concluding Remarks
An empirical joint probability density function (JPDF) of mean wind speed and
direction is suggested in the paper. The proposed JPDF model is built up by marginal
distributions of wind speed and wind direction that are assumed as an Extreme-Value
equation. Different marginal distribution can be adopted for wind speed and wind direction
thereby forming different JPDF model. The proposed JPDF model is applied to various field
measured wind sample of extreme value or from parent population with unimodal or bimodal
distribution. It is concluded from the results that the proposed JPDF model can represent quite
well the joint distribution properties of wind speed and direction from different data source.
There are still some problems remains for the JPDF model and needs further work. For
The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan
instance, the value of JPDF may less than zero according to the definition, and the meaning of
the design wind speed and the joint distribution function in the angular-liner expression is not
very clear.
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Acknowledgement
Financial support from The 11th Young Teacher Research Fund by Fok Ying Tung
Education Foundation through project No 111077 ‘ Field measurement and modeling of joint
probability density function of wind speed and direction’ to the first author is highly
appreciated. The wind data in case 2 are collected from open source for academic research
purpose. Any opinion and conclusions presented in this paper are entirely those of the authors.