Experiences of Hydrological
Frequency Analysis in Taiwan
- Coping with Extraordinary Extremes
and Climate Change
Prof. Ke-Sheng Cheng
Department of Bioenvironmental Systems Engineering,
National Taiwan University
Outline
• Introduction
• Improving accuracy of frequency analysis
– Presence of outliers (extraordinary rainfall extremes)
– Spatial correlation (non-Gaussian random field
simulation)
• Frequency analysis of multi-site rainfall
extremes
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Introduction
• Frequent occurrences of disasters induced by
heavy rainfalls in Taiwan
– Landslides
– Debris flows
– Flooding and urban inundation
• Increasing occurrences of rainfall extremes
(some of them are record breaking) in recent
years
• Almost all of the long-duration rainfall
extremes (longer than 12 hours) were
produced by typhoons.
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Examples of annual max events in Taiwan
Design
durations
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Proportion
of rainfall
stations
observing
annual max
rainfalls.
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• A large number of rain gauges maintained by
CWB and WRA. Some of them have record
length longer than 50 years. Most of them
have less than 30 years record length .
• Design rainfalls play a key role in studies
related to climate change and disaster
mitigation.
• Problems in rainfall frequency analysis
– Short record length (less than 30 years) (small
sample size)
– Record breaking rainfall extremes (presence of
extreme outliers)
• Typhoon Morakot
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• Typhoon Morakot (2009)
 World record
2467
1825 2361
1624
0805-0810
accumulated rainfall
 Morakot
8
• Catastrophic storm rainfalls (or extraordinary
rainfalls) often are considered as extreme
outliers. Whether or not such rainfalls should
be included in site-specific frequency analysis is
disputable.
– 24-hr annual max. rainfalls (Morakot) of 2009
•
•
•
•
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甲仙
泰武
大湖山
阿禮
1077 mm
1747 mm
1329 mm
1237 mm
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• Frequency analysis of 24-hr annual maximum
rainfalls (AMR) at Jia-Sien station using 50
years of historical data
– 1040mm/24 hours (by Morakot) excluding
Morakot – 901 years return period
– Return period inclusive of Morakot – 171 years
The same amount (1040mm/24 hours) was found to be
associated with a return period of more than 2000
years by another study which used 25 years of annual
maximum rainfalls.
06/25/2013
2013 AOGS Conference
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Morakot rainfalls were not included in the frequency analysis.
Rainfall
(mm)
1/14/2016
Return
period
Rainfall
(mm)
Return
period
Rainfall
(mm)
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Return
period
Total
Rainfall
Record
length
11
• Concurrent occurrences of extraordinary
rainfalls at different rain gauges
– Several stations had 24-hr rainfalls exceeding 100-yr
return period.
– Site-specific events of 100-yr return period.
– What is the return period of the event of multi-site
100-yr return period?
• (100)^4 = 100,000,000 years (4 sites), assuming
independence
– Redefining extreme events
• multi-site extreme events w.r.t. specified durations and
rainfall thresholds
• Spatial covariation of rainfall extremes
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• Spatial covariation of rainfall extremes
– By using site-specific annual maximum rainfall
series for frequency analysis implies a significant
loss of valuable information.
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Station A
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Station B
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• Coping with outliers
– Regional Frequency Analysis
– Stochastic simulation of multi-site event-maximum
rainfalls
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Fundamental concept of regional frequency
analysis
• For areas with short record length or without
rainfall or flow measurements, hydrological
frequency analysis needs to be conducted
using data from sites of similar hydrological
characteristics.
• Data observed at different sites within a
“homogeneous region” can be combined and
used in regional frequency analysis.
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General procedures of regional frequency
analysis (Hosking and Wallis, 1997)
1. Data screening
– Correctness check
– Data should be stationary over time.
2. Identifying homogeneous regions
– A set of characteristic variables should be chosen
and used for delineation of homogeneous regions.
– Characteristic variables may include geographic
and hydrological variables.
– Homogeneous regions are often determined by
cluster analysis.
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3. Choice of an appropriate regional frequency
distribution
– GOF test using rescaled samples from different
sites within the same homogeneous region.
– The chosen distribution not only should fit the
data well but also yield quantile estimates that are
robust to physically plausible deviations of the
true frequency distribution from the chosen
frequency distribution.
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4. Parameter estimation of the regional
frequency distribution
– Estimating parameters of the site-specific
frequency distribution
– Estimating parameters of the regional frequency
distribution using record-length weighted average.
5. Calculating various quantiles of the
hydrological variable under investigation.
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An experimental stochastic simulation
• We generated 7 random samples (sample size
n = 40) of a gamma distribution with
mean=600 and std dev =346.
• The data set is considered equivalent to annual
max rainfalls (record length = 40) at 7 stations.
• Outliers were detected in four of the seven
series.
• Return period of the maximum value of each
individual AMR series was calculated.
Max value in AMR 1797.758 1159.976 1567.84 2066.767 1326.539 1288.869 1624.943
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2012 AOGS Conference
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Study area and rainfall stations
25 rainfall stations in southern
Taiwan. (1951 – 2010)
Not all stations have the same
record length.
Annual maximum rainfalls (AMR)
of various durations (1, 2, 6, 12,
18, 24, 48, 72 hours)
06/25/2013
2013 AOGS Conference
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Event-maximum rainfalls of various
design durations
• Event-max 1, 2, 6, 12, 18, 24, 48, 72-hr
typhoon rainfalls at individual sites.
• Approximately 120 events
• Complete series
24
Delineation of homogeneous regions
(K-mean cluster analysis)
• 24 classification features (8 design
durations x 3 parameters – mean, std
dev, skewness)
– Normalization of individual features
• Two homogeneous regions (25 stations)
25
Regional frequency analysis
• Delineating homogeneous regions
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2012 AOGS Conference
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Hot spots for occurrences of extreme rainfalls
1992 – 2010
Number of extreme
typhoon events
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2012 AOGS Conference
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Rescaled variables for regional
frequency analysis – Frequency
factors
Kijk 
xijk  ik
 ik
29
Goodness-of-fit test and parameters
estimation
L-moment ratio diagrams (LMRD) for goodness-of-fit
test
 Pearson Type 3 distribution
Parameters estimation
 Method of L-moments
 Record-length-weighted regional parameters
30
L-moment-ratio diagram GOF test
甲仙（2）， t = 24hr
PT3
Gaussian
EV-1
31
32
Covariance structure of the random
field
• Covariance matrices are semi-positive definite.
• Experimental covariance matrices often do not
satisfy the semi-positive definite condition.
• Modeling the covariance structure of
frequency factors (event-max rainfalls) by
variogram modeling.
33
Relationship between semivariogram
and covariance function
1
 h   Var Z x  h   Z x 
2
 C 0   C h 
C xi , x j   C h 
γ(h)
Sill
C(h)
Influence range
sill  C 0   1
34
Semi-variogram modeling (24-hr
EMR)

 h 
 a 
 h    1  exp  

  C 0  1
35
Semi-variograms of EMRs of other
durations
36
Stochastic Simulation of
Multi-site Event-Max Rainfalls
• Pearson type III (Non-Gaussian) random field
simulation
• Covariance Transformation Approach
– Covariance matrix of multivariate PT3 distribution
– Covariance matrix of multivariate standard Gaussian
distribution
– Multivariate Gaussian simulation
– Transforming simulated multivariate Gaussian
realizations to PT3 realizations
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 XY   AX AY  3 AX CY  3C X AY  9C X CY UV
 2 BX BY 
2
UV
 
AX  1   X 
 6 
4
 Y 
AY  1   
 6
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 6C X CY 
3
UV
BX 
4
BY 
3
1 
CX   X 
3 6 
3
1  Y 
CY   
3 6 
X
X 
Y
 Y 
 
6  6 
 
6  6
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2
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• Each simulation run generated one sample of
t-hr multi-site event maximum rainfalls.
• Simulated samples preserved the spatial
covariation of multi-site EMRs as well as the
marginal distributions.
• 10,000 samples were generated.
– Multi-site t-hr EMRs of 10,000 typhoon events.
• The number of typhoons vary from one year to
another.
– Annual count of typhoons is a random variable.
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• Determination of t-hr rainfall of T-yr return
period
– Average number of typhoons per year, m = 2.43
– Return period, T=100 years
– Exceedance probability of the event-max rainfalls,
pE=1/(100*2.43)
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Return period, T = 50yr
Duration, t = 24hr
2500
EMR
AMR
雨量
2000
1500
1000
500
0
41
Return Period, T = 200yr
Duration, t = 24hr
4000
EMR
AMR
3500
雨量
3000
2500
2000
1500
1000
500
0
42
24hr雨量(mm)
24hr重現期(年)
48hr雨量(mm)
48hr重現期(年)
72hr雨量(mm)
72hr重現期(年)
北港(2)
358
11
433
11
460
12
大埔
720
53
921
57
1069
118
沙坑
607
32
759
31
888
52
中坑(3)
497
24
636
26
744
42
溪口(3)
324
15
403
16
450
24
六溪
791
43
1024
55
1122
59
木柵
778
102
1037
54
1188
80
阿蓮(2)
630
41
868
42
924
41
旗山(4)
519
17
760
23
819
22
美濃(2)
338
4
509
6
569
6
新豐
908
63
1179
51
1195
36
屏東(5)
676
40
878
45
938
45
南和
614
15
988
39
1100
62
嘉義
526
22
646
17
697
24
台南
535
61
709
53
736
42
高雄
538
40
755
57
801
53
恆春
528
27
715
34
730
28
關子嶺(2)
1098
92
1490
126
1683
221
甲仙(2)
1040
65
1614
177
1915
204
紹家
818
110
1176
141
1333
130
三地門
825
18
1109
25
1235
29
大湖山
1329
48
1958
89
2533
306
樟腦寮(2)
868
22
1395
55
1846
141
泰武(1)
1747
47
2938
121
3417
171
阿禮
1237
23
1908
38
2335
66
43
IDF Curve
44
Estimating return period of multisite Morakot
rainfall extremes
• Four rain gauges within the Kaoping River
Basin recorded 24-hr rainfalls close to
1,000mm (908, 1040, 825, 1237 mm).
• Define a multi-site extreme event
– over 1,000 mm 24-hr rainfalls at all four sites
– Among the 10,000 simulated events, only 8 events
satisfied the above requirement.
– Average number of typhoons per year, m=2.43
– Multi-site extreme event return period T = 514
years.
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• Further look at the simulation results
– Preserving the spatial pattern of rainfall extremes
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Type A
47
48
Type B
49
50
Type C
51
52
Type D
53
54
Summary
• We developed a stochastic approach for
simulation of multi-site event-max rainfalls to
cope with the problems of outliers and short
record length in hydrological frequency
analysis.
• By increasing the sample size and considering
the spatial covariation of EMRs, the return
periods of site-specific and multi-site rainfall
extremes can be better estimated.
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Rationale of BVG simulation using
frequency factor
• From the view point of random number
generation, the frequency factor can be
considered as a random variable K, and KT is a
value of K with exceedence probability 1/T.
• Frequency factor of the Pearson type III
distribution can be approximated by
Standard
normal
deviate
X
1 3
X 
KT  z  z  1  z  6 z  
6 3
 6 
2
X 
X  1X 
 z  1   z     
 6 
 6  3 6 
3
4
5
2
[A]
2
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• General equation for hydrological
frequency analysis
X T   X  KT X
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• The gamma distribution is a special case
of the Pearson type III distribution with a
zero location parameter. Therefore, it
seems plausible to generate random
samples of a bivariate gamma
distribution based on two jointly
distributed frequency factors.
X
1 3
X 
KT  z  z  1  z  6 z  
6 3
 6 
2
X 
1X 
2
 z  1   z     
 6 
 6  3 6 
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3
X 
4
5
2
[A]
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Gamma density
1 x
f X ( x ; ,  ) 
 
( )   


0

2
2
     0
 
 1
e
x /
, 0  x  
       0

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
2
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
 6
KT  z  z 2  1
X

1 3
 
 
  1 
z  6 z  X   z 2  1  X   z X    X 
3
 6 
 6 
 6  3 6 


2


3
4
5
• Assume two gamma random variables X and
Y are jointly distributed.
• The two random variables are respectively
associated with their frequency factors KX
and KY .
• Equation (A) indicates that the frequency
factor KX of a random variable X with gamma
density is approximated by a function of the
standard normal deviate and the coefficient
of skewness of the gamma density.
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
 6
KT  z  z 2  1
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X

1 3
 
 
  1 
z  6 z  X   z 2  1  X   z X    X 
3
 6 
 6 
 6  3 6 


2


3
4
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5
• Thus, random number generation of the
second frequency factor KY must take into
consideration the correlation between KX
and KY which stems from the correlation
between U and V.
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Conditional normal density
• Given a random number of U, say u, the
conditional density of V is expressed by
the following conditional normal density
V |U (v | U  u) 
2

1
 1  v  UV u  
 
 exp  
2
2
2 (1  UV
)
 2  1  UV  


2

u
1


with mean UV and variance
UV .
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V |U (v | U  u) 
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2

1
 1  v  UV u  
 
 exp  
2
2
2 (1  UV )
 2  1  UV  


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X   X  K X
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Flowchart of BVG simulation (1/2)
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Flowchart of BVG simulation (2/2)
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 XY ~ UV Conversion
 XY   AX AY  3 AX CY  3C X AY  9C X CY UV
2
3
 2 BX BY UV
 6C X CY UV
X 
AX  1   
 6 
4
 Y 
AY  1   
 6
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4
[B]
X
3
1X 
CX   
3 6 
Y
3
1 
CY   Y 
3 6 
X 
BX 
 
6  6 
 Y 
BY 
 
6  6
2
2
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• Frequency factors KX and KY can be
respectively approximated by

6
K X  U  U 2 1
X




3
4
5
1 3
 Y 
 Y 
 Y  1  Y 
2
2
KY  V  V  1
 V  6V    V  1    V     
6 3
 6
 6
 6  3 6 


Y
1
 
 
  1 
 U 3  6U  X   U 2  1  X   U  X    X 
3
 6 
 6 
 6  3 6 
2


2


3
4
5
where U and V both are random variables
with standard normal density and are
UV
correlated with correlation
coefficient
.
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• Correlation coefficient of KX and KY can be
derived as follows:
K
X KY
 Cov( K X , KY )  EK X KY 
2
3
4
5


1



1

 X
 X
 X
 X 
2
3
2
X
 U  6U    U  1    U       
U  U  1
6 3
 6 
 6 
 6  3  6   

 E

2
3
4
5


Y 1 3
 Y 
 Y 
 Y  1  Y  
2
2
V

V

1

V

6
V

V

1

V













6
3
6
6
6
3
6

















1/14/2016






Laboratory for Remote Sensing Hydrology and Spatial Modeling,
Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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3
2
5
   X 4 


  
1
  1 
K X  U 1      U 2  1  X   X    U 3  6U   X    X 
 6  3 6 
  6  
 6  6   3
 AXU  BX U 2  1  CX U 3  6U   DX
3
2
5
   Y 4 




1

1

 
 
 
KY  V 1      V 2  1  Y   Y    V 3  6V   Y    Y 
 6  3 6 
  6  
 6  6   3
 AYV  BY V 2  1  CY V 3  6V   DY

1 
1 
 
 
AX  1   X  , BX  X   X  , C X   X  , DX    X 
6  6 
3 6 
3 6 
 6 
4
3
2
Y  Y 
1  Y 
1  Y 
 Y 
AY  1    , BY     , CY    , DY    
6 6
3 6 
3 6 
6
4
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3
2
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
5
5
74
EK X KY 
 AX AYUV  AX BYU V 2  1  AX CYU V 3  6V 



2
2
2
 AX DYU  BX AYV U  1  BX BY U  1V  1 
 B C U 2  1V 3  6V   B D U 2  1

X Y
X Y


E
 C X AY U 3  6U V  C X BY U 3  6U V 2  1



3
3
3
 C X CY U  6U V  6V   C X DY U  6U 



2
3
 DX AYV  DX BY V  1  DX CY V  6V   DX DY 
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EK X KY 







 AX AYUV  AX CYU V 3  6V  BX BY U 2  1 V 2  1

 E

3
3
3
 C X AY U  6U V  C X CY U  6U V  6V  DX DY 









E[ K X ]  E AX U  BX U 2  1  C X U 3  6U  DX  DX
Since KX and KY are distributed with zero
means, it follows that
DX  DY  0
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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K
 EK X KY 
X KY






 A A UV  A C U V 3  6V  B B U 2  1 V 2  1 
X Y
X Y
X Y

 E
 C X AY U 3  6U V  C X CY U 3  6U V 3  6V




 


 C A E U V   6  
 C C E U V   6 E UV   6 E U V   36  

 
 AX AY UV  AX CY E UV 3  6UV  BX BY E U 2V 2  1
3
X
Y
UV
3
X
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Y
3
3
3
UV
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• It can also be shown that
E U V   6
 
E U V   E UV   3
2
E U 2V 2  2UV
1
3
3
3
3
UV
 9UV
3
UV
Thus,
 XY   K
X KY
 2 BX BY 
2
UV
1/14/2016
  AX AY  3 AX CY  3C X AY  9C X CY UV
 6C X CY 
3
UV
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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 XY ~ UV Single - Value Relationsh ip
We have also proved that Eq. (B) is indeed a single-value function.
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Proof of Eq. (B) as a single-value function
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Laboratory for Remote Sensing Hydrology and Spatial Modeling,
Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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• Therefore,
AX AY  3 AX CY  3CX AY  9CX CY
2
2 
2
2
  2  2


 X
  X     Y 
 Y  
    1        1    
 6 
 6    6 
 6 










1/14/2016
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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1/14/2016
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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• The above equation indicates  XY
increases with increasing UV , and thus
Eq. (B) is a single-value function.
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Conceptual description of a gamma
random field simulation approach
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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