Table 1
The P-III distribution parameter estimation results and the distribution fitting test results for AMF volumes
Station
|
Period
|
|
|
|
χ2 test
|
K-S test
|
Yingchang Station
|
Period 1
|
5.67
|
2.07
|
1.37
|
2.070(5.991)
|
0.004(0.189)
|
Period 2
|
5.28
|
2.41
|
1.10
|
0.158(5.991)
|
0.009(0.349)
|
Chenglingji Station
|
Period 1
|
2.78
|
1.84
|
0.76
|
0.419(5.991)
|
0.004(0.189)
|
Period 2
|
3.25
|
3.08
|
0.53
|
1.082(5.991)
|
0.028(0.349)
|
Notes: The critical values are in parentheses.
|
Table 2
The mixed von Mises distributions parameter estimation results and the distribution fitting test results for AMF durations and interval times
Variable
|
Station
|
Period
|
|
|
|
χ2 test
|
K-S test
|
AMF duration
|
Yichang Station
|
Period 1
|
0.29
|
1.80
|
16.74
|
4.063(12.592)
|
0.023(0.189)
|
0.71
|
2.72
|
2.51
|
Period 2
|
0.25
|
1.21
|
37.97
|
0.214(5.991)
|
0.011(0.349)
|
0.75
|
2.36
|
2.61
|
Chenglingji Station
|
Period 1
|
0.40
|
0.78
|
16.29
|
3.791(9.488)
|
0.016(0.189)
|
0.51
|
2.09
|
2.13
|
0.09
|
5.21
|
30.99
|
Period 2
|
0.28
|
0.37
|
700.87
|
1.345(3.841)
|
0.040(0.349)
|
0.51
|
1.60
|
8.43
|
0.21
|
2.58
|
26.44
|
AMF interval time
|
Between the two stations
|
Period 1
|
0.45
|
2.60
|
9.39
|
0.449(7.815)
|
0.005(0.189)
|
0.55
|
3.52
|
1.68
|
Period 2
|
1
|
3.35
|
1.35
|
0.002(9.488)
|
0.015(0.349)
|
Notes: The critical values are in parentheses.
|
Table 3
Vine copula structures of the five-dimensional joint distributions of AMF hydrograph coincidence for the two studied periods
|
Period 1 (R-Vine)
|
Period 2 (R-Vine)
|
Variable
|
Copula function
|
Parameter
|
Variable
|
Copula function
|
Parameter
|
Tree1
|
Td, d2
|
Independence
|
Td, d2
|
Gaussian
|
par = −0.44
|
d2, d1
|
Tawn type 2
|
par 1= 20.00
par 2= 0.08
|
d2, V2
|
Tawn type 1
|
par 1= 20.00
par 2= 0.31
|
d1, V1
|
Frank
|
par = 6.20
|
d2, V1
|
Clayton
|
par = 0.80
|
d2, V2
|
Joe–Frank
|
par 1 = 3.60
par 2 = 0.93
|
V1, d1
|
Survival Clayton
|
par = 2.04
|
Tree2
|
Td, d1; d2
|
Independence
|
Td, V2; d2
|
Rotated Tawn type 1 180 degrees
|
par 1 = 20.00
par 2 = 0.20
|
d2, V1; d1
|
Independence
|
Td, V1; d2
|
Independence
|
d1, V2; d2
|
Rotated Tawn type 1 90 degrees
|
par 1 = −3.86
par 2 = 0.10
|
d2, d1; V1
|
Tawn type 2
|
par 1 = 20.00
par 2 = 0.31
|
Tree3
|
Td, V1; d2, d1
|
Rotated Tawn type 1 90 degrees
|
par 1 = −6.68
par 2 = 0.11
|
V2, V1; Td, d2
|
Rotated Tawn type 1 90 degrees
|
par 1 = −3.21
par 2 = 0.42
|
V1, V2; d1, d2
|
Student t
|
par 1 = 0.39
par 2 = 2.96
|
Td, d1; d2, V1
|
Independence
|
Tree4
|
Td, V2; V1, d1, d2
|
Tawn type 2
|
par 1 = 20.00
par 2 = 0.01
|
V2, d1; Td, d2, V1
|
Rotated Tawn type 1 90 degrees
|
par 1 = −2.70
par 2 = 0.58
|
Table 4
AMF hydrograph coincidence risk probability for different return period combinations
Risk probability
|
Station
|
Yichang
|
station
|
R02
|
Period\R01
|
1000
|
500
|
100
|
50
|
10
|
5
|
Chenglingji
|
1000
|
Period 1
|
3.00 × 10−5
|
3.00 × 10−5
|
4.00 × 10−5
|
6.00 × 10−5
|
1.10 × 10−4
|
1.70 × 10−4
|
Period 2
|
<1.00 × 10−5
|
<1.00 × 10−5
|
<1.00 × 10−5
|
<1.00 × 10−5
|
3.00 × 10−5
|
7.00 × 10−5
|
500
|
Period 1
|
3.00 × 10−5
|
4.00 × 10−5
|
8.00 × 10−5
|
1.40 × 10−4
|
2.00 × 10−4
|
2.90 × 10−4
|
Period 2
|
<1.00 × 10−5
|
<1.00 × 10−5
|
<1.00 × 10−5
|
<1.00 × 10−5
|
5.00 × 10−5
|
1.10 × 10−4
|
100
|
Period 1
|
7.00 × 10−5
|
1.00 × 10−4
|
2.70 × 10−4
|
4.50 × 10−4
|
1.05 × 10−3
|
1.59 × 10−3
|
Period 2
|
<1.00 × 10−5
|
<1.00 × 10−5
|
2.00 × 10−5
|
4.00 × 10−5
|
2.60 × 10−4
|
4.60 × 10−4
|
50
|
Period 1
|
7.00 × 10−5
|
1.00 × 10−4
|
3.50 × 10−4
|
5.80 × 10−4
|
1.75 × 10−3
|
2.73 × 10−3
|
Period 2
|
<1.00 × 10−5
|
<1.00 × 10−5
|
5.00 × 10−5
|
7.00 × 10−5
|
4.80 × 10−4
|
8.50 × 10−4
|
10
|
Period 1
|
1.10 × 10−4
|
1.70 × 10−4
|
7.50 × 10−4
|
1.31 × 10−3
|
5.26 × 10−3
|
9.35 × 10−3
|
Period 2
|
3.00 × 10−5
|
5.00 × 10−5
|
2.70 × 10−4
|
3.80 × 10−4
|
1.81 × 10−3
|
3.36 × 10−3
|
5
|
Period 1
|
1.20 × 10−4
|
2.20 × 10−4
|
1.11 × 10−3
|
1.89 × 10−3
|
7.68 × 10−3
|
1.41 × 10−2
|
Period 2
|
7.00 × 10−5
|
1.10 × 10−4
|
4.30 × 10−4
|
7.00 × 10−4
|
3.17 × 10−3
|
5.70 × 10−3
|
Table 5
AMF hydrograph coincidence volumes for different return period combinations
Volume (m3)
|
Station
|
Yichang
|
Station
|
R02
|
Period\R01
|
1000
|
500
|
100
|
50
|
10
|
5
|
Chenglingji
|
1000
|
Period 1
|
1.57 × 1011
|
1.52 × 1011
|
1.41 × 1011
|
1.36 × 1011
|
1.23 × 1011
|
1.16 × 1011
|
Period 2
|
1.18 × 1011
|
1.14 × 1011
|
1.04 × 1011
|
9.98 × 1010
|
8.89 × 1010
|
8.35 × 1010
|
500
|
Period 1
|
1.52 × 1011
|
1.48 × 1011
|
1.36 × 1011
|
1.31 × 1011
|
1.18 × 1011
|
1.12 × 1011
|
Period 2
|
1.15 × 1011
|
1.11 × 1011
|
1.01 × 1011
|
9.71 × 1010
|
8.62 × 1010
|
8.08 × 1010
|
100
|
Period 1
|
1.42 × 1011
|
1.37 × 1011
|
1.26 × 1011
|
1.21 × 1011
|
1.08 × 1011
|
1.01 × 1011
|
Period 2
|
1.08 × 1011
|
1.04 × 1011
|
9.48 × 1010
|
9.05 × 1010
|
7.96 × 1010
|
7.42 × 1010
|
50
|
Period 1
|
1.37 × 1011
|
1.32 × 1011
|
1.21 × 1011
|
1.16 × 1011
|
1.03 × 1011
|
9.64 × 1010
|
Period 2
|
1.05 × 1011
|
1.01 × 1011
|
9.18 × 1010
|
8.75 × 1010
|
7.66 × 1010
|
7.12 × 1010
|
10
|
Period 1
|
1.25 × 1011
|
1.21 × 1011
|
1.09 × 1011
|
1.04 × 1011
|
9.13 × 1010
|
8.48 × 1010
|
Period 2
|
9.80 × 1010
|
9.40 × 1010
|
8.45 × 1010
|
8.02 × 1010
|
6.93 × 1010
|
6.39 × 1010
|
5
|
Period 1
|
1.20 × 1011
|
1.15 × 1011
|
1.04 × 1011
|
9.88 × 1010
|
8.58 × 1010
|
7.93 × 1010
|
Period 2
|
9.46 × 1010
|
9.06 × 1010
|
8.11 × 1010
|
7.68 × 1010
|
6.59 × 1010
|
6.05 × 1010
|