Homepage: https://jwim.ut.ac.ir/
University of Tehran Press
Frequency Analysis and Rainfall-Runoff Simulation Based on the Tree Sequence of Vine Copula
Mohammad Nazeri Tahroudi1 | Rasoul Mirabbasi Najafabadi2
1. Department of Water Engineering, Shahrekord University, Shahrekord, Iran. E-mail: [email protected] 2. Corresponding Author, Department of Water Engineering, Shahrekord University, Shahrekord, Iran. E-mail:
Article Info ABSTRACT
Article type:
Research Article
Article history:
Received: November 01, 2022 Received in revised form:
November 17, 2022 Accepted: January 23, 2023 Published online: April 14, 2023
Keywords:
Canonical Structure, Conditional Density, Four-Dimensional copula, Tree Structure,
Zayandeh-Rood.
Rainfall-runoff simulation is one of the most important challenges in the management of water resources in each basin, considering the climatic changes and the increase of extreme values in recent years, especially in different regions of Iran. In this study, the rainfall and runoff values in the statistical period of 1993-2019 were used regarding the rainfall-runoff simulation in the Qale Shahrokh sub-basin in the Zayandeh-Rood Dam basin. In this study, the Vine copula-based simulation approach was used to simulate and joint frequency analysis the flow discharge in Qale Shahrokh station given by the rainfall in Chelgerd, Meyheh and Marghmalek stations. By analyzing the tree sequence of vine copulas and using internal rotated copulas, D-vine copula was selected as the best copula based on the studied statistics. By using the best copula and the conditional relationship c(u4|u1,u2,u3), the flow discharge simulation was done given by rainfall of the upstream stations. The simulation results showed an error rate equal to 18.57 (m3/s) based on the RMSE statistic and the efficiency of the model was 0.86 based on the NSE statistic. The results of the simulation of flow discharge values given by rainfall in the upstream stations showed that the proposed approach has simulated the average of observed values with high accuracy. By considering the condition of rainfall occurrence in the frequency analysis of flow discharge in Qale Shahrokh station, the joint return period curve and the conditional probability of occurrence in this sub-basin were obtained.
Using this curve, it is possible to estimate the flow discharge values of the studied station with high accuracy along with different return periods and different probability of occurrence. Considering the fact that the proposed method is based on the condition of rainfall in the region as well as the marginal distribution according to the data, it has no implementation limitations and is somehow specific to the studied region.
Cite this article: Nazeri Tahroudi, M., & Mirabbasi Najafabadi, R. (2023). Frequency Analysis and Rainfall-Runoff Simulation Based on the Tree Sequence of Vine Copula. Journal of Water and Irrigation Management, 13 (1), 259-274.
DOI: https://doi.org/ 10.22059/jwim.2023.350666.1027
© The Author(s). Publisher: University of Tehran Press.
DOI: https://doi.org/ 10.22059/jwim.2023.350666.1027
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>
` a :B : #
O=
8 (
. . .
1 1
1
2 2|1 2 1
1
| 1,...,1 1 1
: 0;1 , 1,...,
: :
: |
: | ,...,
i i d j
d d d d d
First Sample w U j d
Then u w
u C w u
u C w u u
M
= 9K
>>
P
#K S= #K S 4
Cj|j-1,...,1, j = 1…,d
& 67 Q<L :;<
>
Q . P [ K I ?^
=
S= #K <6 =h
#K S= K S 4 W # g
6B = &
#K
= I #f
6 =
# ]# 9
)
Nazeri Tahroudi et al., 2022b
= .(
I ^t4 C>=
K
Czado
) (2019
?=
>
>
.
2 . 3 . ) 2
qL
>
>
P ) O7 I 9=
(RMSE
c R C
- ) M>G K (NSE
9>
I ^t4
Akaike (AIC)
>9
) P D>= I ^t4 (BIC
&
_
= 6%&
6 = N6 P
>? _ 6 % + :;<
<6 -
# )
Nash and Sutcliffe, 1970; Zhang and Singh, 2006
.(
O=
9 (
2 1
1 ( )
N
i i
i
RM SE p p
N
O=
10 (
2 2 ln( ) A IC m L
O=
11 (
2 1
ln 1 ( ) ln( )
N
i i
i
BIC N p p m N
N
O=
12 (
2 1
2 1
( )
1
( )
N
i i
i N
i ave
i
p p
N SE
p p
pi
pi
= c>K K
>? + = = = K & +
. =
9K N
&
9Km
& 6 (
= " F6B S= K + • B L
.Q "
pave
K & + e #6 = = = D>
:B . =
` a C& '(
= I #f : { = ft7 )
4 ( . =
Q b = & 6% @ = L = &
↓
← 9* O # & >J6 Q<L 6%?F& =
↓
← ] = K = :;< N6 P &:;< =
= & >9
↓
← G7 &:;< = c 6 67 * #K N6
↓
← >? (c(u4|u1,u2,u3)) >J6 5 4 * 5 =
@ = -
Figure 4. Flowchart of research steps
3 . 4 5
>? Q L 9* O P E :>GXK
T&
R#B SH U & 9GH R#B L 7 L =
6% @ = &
Q b = &
1398 - 1372 >* I >>JK #F . <6
&
= # =
I #f :
) 5 ( & >J6 P>= 6%?F& . = 9* O #
= ( = F&
` #6%>&
&
>
D
= I #f :
) 6 ( : = L#K = . = )
6 ( & >J6 P>= 6%?F& #L #K
9* O #
> |K ` #6%>& . &
9* O # D>
& >J6 ( S #K Q& ? &
9* O #
. =
Figure 5. Observed values of rainfall (mm) and flow (m3/s) in the studied stations
Figure 6. Correlation, scatter cloud and histogram of the initial values of rainfall (mm) and flow discharge
(m3/s) in the studied stations
6% @ = + = [GF\ > G5 &
= c>K K o
R_Chel R_Mey
R_Mor
o U & 9GH 6% L = + =
Q_Qal
<6 " L & 67 9* O P . #
T^ P :;< MG6N
R C
= # D
:;< 67 & 67 .QE H Q* B x &
&
= MG6N
] =
= & >9
AIC BIC Log Likelihood
N6 c 67 67
:;< MG6N & 67 = = . #
D-vine
c R =
AIC=-743.7 BIC=-716.7
Log
Likelihood=377.8
= # ^ :;< 67 = _ 6 . N6 K = :;<
D-vine
:>GXK 5 9=
+O 9* O # = D>
" L { )
1 ( : ) 7 ( 67 = _ 6 . g
D-vine
<6 = &
:= H 6%?F& " Q7 #L#
Q7 ?K C DE = #L & >J6 Q<L P>= *#?H P &
€<B 6 K 6%?F&
" #g K 6%?F& + = 6 = . # :;< G$% [ D>E
D-vine
N6
? F& . = " L #4
) 1 ( #6 t :;< Q iN 180
[ E L
= # ^ :;<
&
N6 G7 #6 t :;< .
180 #6 t # 7 :;< L C #( = L#K = =
& >J6 Q<L 6%=
:= H MG6N I L
D> :;< S= #K # P . = .Q q(
C DE P #=b =
Q7 " #g K + C DE " = F6^ D> Kb = &
. &
T&
P> 5
_ 6
] =
& >9
AIC BIC Log-Likelihood
Q7 FK
&
= # :;<
:;< cN6 #6 t G7 &
180 [ E L
= # ^ :;< <6 . N6 K = :;<
&
G7
" QH = I L FK " #g K + #K
>?
C DE &
. &
:;< <6 = :B Q
D-vine
:;<
g G7 &
" L )
1 ( . =
Table 1. The structure of D-vine in the correlation analysis of the studied values Par Tau
Copula Edge
Tree
0.16 0.14
Clyton-180 R_Mey, Q_Qal
T1 R_Mor, R_Mey Frank 0.67 5.58
1.02 0.61
Clyton-180 R_Chel, R_Mor
0.06 0.57
Clyton-180 R_Mor, Q_Qal | R_Mey
T2
2.39 0.61
Frank R_Chel, R_Mey | R_Mor
0.09 0.40
Clyton-180 R_Chel, Q_Qal | R_Mor, R_Mey
T3
Figure 7. The tree structure of the selected D-vine in the 4-D analysis of the studied values
3 . 1 .
:;< QH = Q L
>? = ` H cN6 &
+ <6 = U & 9GH 6% L = +
6% @ = Q b = &
6 ( G;< ] >+ = # + :;< S= #K 4 * 5 <6 = .
>? ( L = ) 6%=
= 9= .
> |K :;<
9= 5 :;< 4 * 5 = = cN6 &
L = + <6 = 6= . 67 (
c (u4|u1,u2,u3)
C>( = ` H L = + P>FNK >=
d|6
= L = + H Q= d = 6= #W P = . = # & 6 ( +
c (u4|u1,u2,u3)
+
C>= . ? X :;<
#X + P K
= L = W6 # + = = =x
L = L + . #
? X
@ = I >>JK = j 6 L = 4 Q* B &
:;< S= #K <6 = 6 P . =
6% @ = Y#H Z = L = + = #
>? Q b = &
= _ 6 : I #f
) &
8 (
) 9 ( ) : = L#K = . g 8
(
>? " & #K
= :;< = 6?
Q 6% #K =#7
I >>JK
= " P > >= >F Z + . & C #( K & &
>? =#7 P>= 8t67
>? K & + #L
>? " .
= :;< = 6?
>? @ = + " F^ :>*
&
= B Z + g =#7 _ 6 Q 6% #K ) : = L#K = . & C #( =#7
9 D> ( & #K
>? O7 D>
B :;< = 6?
57 / 18
> d = c9 6 D> " c R . =
>?
B 9* O # D = R#B L = >J6 5 86
/ 0 >F4 B = L#K = . = 95
>? f -
) : 9
( T # 5 & #K C>= # P 5 >FNK
>? >FNK = +
L . #L @ = + Y#H Z =
6% L = + e #6 9* O #
B 56 / 47 + = ^ P #= > d = c9 6
>?
B 80 / 47
>? + >F .Q = > d = c9 6
= #
= = B K & + e #6
>= = :;< S= #K 4 * 5 = r( . =
" > P>= 6%= 67 &
= # =
#W ? X
= . <6 L = Q = &
+O @ = = c 6 L = 4 Q =
9* O #
` I &
10 20 50 100 *
Q = Q . 67 (
T&
Q = L = +
&
K 100 { = = *
: ) 10 ( g .
Figure 8. Simulation and observation values of flow discharge given by the rainfall in upstream stations
Figure 9. 95% confidence intervals of simulation and observation values of flow discharge given by the rainfall
values in upstream stations
) : = L#K = 10
( Q = = & #K L = 4 Y#H MG6N Ib F6B MG6N &
= U & 9GH 6%
#L
Q = = .
K <f 9* O # :X L = + MG6N &
300
>J6 > d = c9 6
>J6 [K Q = I <K . = >J6 5 Q = =
T&
Q* B Q P
[K
= ^ [ >J6 E 9 Q = # ^
* B # Q* B
T&
>67 MG6N Ib F6B = = [
H = &
= #K Q = = L = P>FNK = G Q* B # ^ g MG6N " F6B MG6N &
) : = L#K = . 10
( K L = + 4 Y#H " F6B * Q = = & #K
T K 50 C& f #L " & * Q = Y#H W6 a . =
& #K
C& L = + Y#H " F6B C DE = " F6B K . =
50 L = + C& * D f
r( #= L#K := H * Q = +K
p ? = Q= d e = L#K = _ 6 . >
9* O # +O = T B C>= L = @ = W
50
= > d = c9 6 9* O # +O e #6 #4
I >>JK > MG6N I 9* O = L#K = .Q%> W6
@ = ( S #K #&
) &
Khalili et al., 2016;
Tahroudi et al., 2019
DE > % ?L (
= uO C
@ = #L :>*
= B &
T&
#7
: #= & #7 QN >%= ~ 7 F67 S= Q ` D* #L# e = L#K = q* . & #7
] %B C>( C>= XO ) : . #
10 Ib F6B = L = K >>JK Q = & ( MG6N
e K &
T&
>J6 5
q(
= .Q Q = " • #4 10
" F6B = * 99
/ 0
K 09 / 0 I 9* O Q D P . = L = #K
. et al Nazeri Tahroudi
)
2022 a , b
. & D> (
Figure 10. The return period of the flow discharge given by the occurrence of rainfall corresponding to the
conditional probabilities
4 . 7 8
>? 9* O P E :>GXK
T&
6% @ = L = U & 9GH R#B Q b = &
1398 K 1372 :;< <6 = :;< = PFR #W P = .QE q( I #f P # 7 &
9= 5 &
67 * #K P # 7 ] = &
& >9
AIC BIC Log-Likelihood
:;< & 67
&
C-vine
D-vine R-vine
T&
Q* B P> 5
&
:+6%
& >9 = L#K = .QE H = # &
g :;< G$% [ D>E
D-vine
= c :;< # ^ :;< 67 67 = _ 6 . N6 K
&
:;< = #
D-vine
?* N6 = G$% [ D>E €<B P>^
>J6 Q<L 6%?F& K = &
K &
€<B Q7 P 7 T& .
:;< E 9 N6 = P> 5 Q* B #6 t &
180 6%= 67 L
I >>JK :
>? Q .QE H = # &
6% @ = + = L#K = L = + &
<6 = Q b = :;<
D-vine
:;<
#6 t G7 &
180
>? _ 6 .QE I #f [ E L
>= L = + 86
O7 D> >( " f 57
/ 18
>? . #= > d = c9 6
Q>+E#
67 67 w>H N6 = G>* SH & 6 ( D>
:;<
D-vine
wE# " P . =
I >>JK e #6
>? b = QH = &
>F4 B 95
Y#R# P D> f
> |K 8 & .
E :>GXK Q L 4 * 5 S= #K <6 C& '( P Gf T&
Z = L = 4 Y#H " F6B
6% @ = Y#H Q b = &
4 * 5 <6 = =
c (Q_Qal| R_Chel R_Mey, R_Mor)
= = . = g MG6N Ib F6B = L#K = " P >
. P>FNK L = + #K
>J6 5 = L#K = >(
H = #=
>? b = Q>9OH Y#H Z L = Y#H " F6B =
+O & = iN & #F g = P .Q #7 = C DE = L#K = t> Q #K
@ =
= . = >< 6E H <6 # B &
P P :;< L 9* O
:;< #K D> #K #K &
>•>( <6 GB &
C DE 9= 5 Q* B P :;< = Q?% #K #K :;<
@ . =
>? >(
@ = -
= S #K <6 :>*
T& &
J6 Q<L = c 6 67 *#K P> 5 & >
. F>GH >E JL Q X h>&
5 .
! &: ;<
C& '( Q FB v f Q FB QXK d P ) # E
F { 4 6E = (INSF
"
4005011
"
.Q ` a
6 .
>
+ # ) 1. Dependency model
2. Archimedean functions 3. Non-Archimedean function 4. Plackett
5. Cascade of simple building blocks 6. Chain law
7. Drawable vine trees 8. Canonical vine trees 9. Nested trees
7 .
* @ 2
&
>
h
#
# e #K SE ‚ 9K
% . #L
8 . *
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