Assignment 3 .max. Likelihood And Nash Model.xlsx
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Extreme Value ype 1 ( Gumbel ) with maximum likeli Submitted by : Hamse Hussein Adam ID: P Department of Hydraulic and Water Resourc
Annual maximum flows in sugar Creek at Cr Year Xi 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988
log(X) 7860 2730 6480 12800 26300 15100 14600 7300 8580 15100 15100 21800 6200 2130 11100 14300 11200 6670 5440 9370 6900 9680 6810 7730 5290 12200 9750 7390 13100 7190 8850 6290 18800 9740 2990 6950
Fi=(i-0.35)/N
Rank(i) 3.895 3.436 3.812 4.107 4.420 4.179 4.164 3.863 3.933 4.179 4.179 4.338 3.792 3.328 4.045 4.155 4.049 3.824 3.736 3.972 3.839 3.986 3.833 3.888 3.723 4.086 3.989 3.869 4.117 3.857 3.947 3.799 4.274 3.989 3.476 3.842
17 2 8 29 39 33 32 14 18 33 33 38 6 1 25 31 26 9 5 20 11 22 10 16 4 27 24 15 30 13 19 7 36 23 3 12
XiFi 0.427 0.042 0.196 0.735 0.991 0.837 0.812 0.350 0.453 0.837 0.837 0.965 0.145 0.017 0.632 0.786 0.658 0.222 0.119 0.504 0.273 0.555 0.247 0.401 0.094 0.683 0.606 0.376 0.760 0.324 0.478 0.171 0.914 0.581 0.068 0.299
XiFi2 3355.615 115.500 1271.077 9403.077 26063.974 12641.410 11848.462 2555.000 3883.000 12641.410 12641.410 21045.385 898.205 35.500 7015.769 11238.333 7366.154 1479.372 648.615 4721.038 1884.231 5373.641 1685.038 3101.910 495.090 8336.667 5912.500 2775.987 9959.359 2332.141 4232.115 1072.526 17185.128 5656.692 203.167 2076.090
1432.589645 4.886538462 249.3266272 6907.64497 25830.0669 10583.12936 9615.482249 894.25 1757.30641 10583.12936 10583.12936 20316.89053 130.1245891 0.591666667 4434.325937 8832.17735 4844.662722 328.1170776 77.33491124 2378.677071 514.5399408 2983.059698 416.9390039 1244.740911 46.33532216 5696.722222 3585.400641 1042.774671 7571.666502 756.4508711 2023.81928 182.8793721 15708.96976 3285.23284 13.80491453 620.1652696
1989 1990 1991 39
9390 12400 21200 402810.000 10328.462 M100 βo
3.973 4.093 4.326 154.315 3.957
N mean(M1) sta(S) cv1
L1 L2 L3 L4
21 28 37
0.529 0.709 0.940
39 10328.462 5320.598 0.515
mean(M1y) sta(Sy) cv(M2y
bo = 2b1-bo = 6b2- 6b1+bo= 20b3-30b2+12b1-bo=
4971.885 2632.549162 8791.282 6232.793557 19922.564 18722.10191 256836.321 193064.789 6585.547 4950.379 M110 M120 β1 β2 3.957 0.235 0.059 10328.462 2842.632 517.457 387.550
maximum likelihood parameter estimator ein Adam ID: PRAMIT/1833/11 nd Water Resources Engineering
ar Creek at Crawfordsville, Indiana. XiFi3
1-Fi 611.606 0.207 48.906 5074.462 25598.259 8859.979 7803.334 312.988 795.294 8859.979 8859.979 19613.614 18.851 0.010 2802.721 6941.186 3186.297 72.775 9.221 1198.487 140.509 1655.981 103.166 499.492 4.337 3892.760 2174.224 391.709 5756.408 245.362 967.801 31.183 14359.610 1907.962 0.938 185.254
Xi(1-Fi) 0.573 0.958 0.804 0.265 0.009 0.163 0.188 0.650 0.547 0.163 0.163 0.035 0.855 0.983 0.368 0.214 0.342 0.778 0.881 0.496 0.727 0.445 0.753 0.599 0.906 0.317 0.394 0.624 0.240 0.676 0.522 0.829 0.086 0.419 0.932 0.701
4504.385 2614.500 5208.923 3396.923 236.026 2458.590 2751.538 4745.000 4697.000 2458.590 2458.590 754.615 5301.795 2094.500 4084.231 3061.667 3833.846 5190.628 4791.385 4648.962 5015.769 4306.359 5124.962 4628.090 4794.910 3863.333 3837.500 4614.013 3140.641 4857.859 4617.885 5217.474 1614.872 4083.308 2786.833 4873.910
Xi(1-Fi)2 2581.359 2503.887 4187.173 901.491 2.118 400.309 518.559 3084.250 2571.306 400.309 400.309 26.121 4533.714 2059.592 1502.787 655.511 1312.355 4039.373 4220.104 2306.600 3646.078 1915.778 3856.862 2770.920 4346.156 1223.389 1510.401 2880.800 752.949 3282.169 2409.589 4327.828 138.713 1711.848 2597.472 3417.986
Xi(1-Fi)3 1479.317 2397.953 3365.843 239.242 0.019 65.178 97.728 2004.763 1407.625 65.178 65.178 0.904 3876.907 2025.265 552.949 140.347 449.229 3143.461 3716.938 1144.429 2650.419 852.275 2902.536 1659.000 3939.400 387.406 594.478 1798.654 180.515 2217.568 1257.311 3589.878 11.915 717.659 2420.977 2396.972
1393.901 4418.891 17593.975 156391.615 4010.041
0.471 0.291 0.060 19.427
M130
4418.115 2078.780 978.093 3608.718 1050.229 305.644 1277.436 76.974 4.638 145973.679 82202.148 55103.791 3742.915 2107.747 1412.918 M101 M102 M103
β3
m s P W U X100
Maximum Likelihood estimator 3.957 0.2437807 0.01 3.0348543 2.3267853 33420.844
C0 C1 C2
2.515517 0.802853 0.010328
d1 d2 d3
1.432788 0.189269 0.001308
Instantaneous Unit Hydrograph (IUH) Nash M
Submitted by : Hamse Hussein Adam ID: PRAMIT/18 Department of Hydraulic and Water Resources Enginee GIVEN DATAS Drainage area= 36km^2 Abstractions= 0.5cm/hr Base flow= 5m^3/s Table
6hr period Rainfall in (c stream flow ( Reqired
1 1.5 15
2 3.5 75
3 2.5 170
4 1.5 185
Detemine the peak of the IUH Using Nash model
Solution
Calculation of M1Q & M2Q 1 Base flow
0 5 5 5 5 5 5 5 5 5
stream flow (m3/sec.)
time
0 15 75 170 185 147 84 43 18 8
0 6 12 18 24 30 36 42 48 54
2
3
ord DOR Avg DOR
0 10 70 165 180 142 79 38 13 3
0 5 35 82.5 90 71 39.5 19 6.5 1.5
sum
M1Q = M2Q =
where
25.251428571 737.28
Calculation of M1h & M2h 1
2
3
time
DOR
interval Δt
4
5
inc.areamoment arm
0 6 12 18 24
0 1 3 2 1
0 6 6 6 6
0 1 3 2 1
sum
0 3 9 15 21
7
M1h= M2h=
11.57143 181.2857
calculate the value of n & K nk = M1Q – M1h nk =
13.68
M2Q – M2h = n(n + 1)k2 + 2 *nk* M1h M2Q – M2h = 555.9943 2 *nk* M1h= 316.5943 (nk)2= 187.1424 nk2 ? From eqn (1)
M2Q – M2h = n(n + 1)k2 + 2 *nk* M1h 555.9943= 555.9943=
(nk)^2+nk^2+316.5943 187.1424 +nk^2+316.5943
nk2= 52.2576 But, nk=13.68 from thus
n=13.68/k
(13.68/k)*k^2=52.2576 k= n=
3.82 3.581152
time(hr)
t/3
e-t/3
(t/3)3
t 1 0 6 12 18 24 30 36
2 0 2 4 6 8 10 12
3 1 0.135164 0.018269 0.002469 0.000334 4.51E-05 6.1E-06
4 0 8 64 216 512 1000 1728
42 48 54 60 66 72 78 84 90
14 16 18 20 22 24 26 28 30
time(hr)
runoff
0 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90
time(hr)
0 6 12 18 24 30 36 42 48 54 60 66
8.24E-07 1.11E-07 1.51E-08 2.04E-09 2.75E-10 3.72E-11 5.03E-12 6.79E-13 9.18E-14
2744 4096 5832 8000 10648 13824 17576 21952 27000
IUHO
0 18.03633218 19.502954347 8.8968521465 2.8504588448 0.7525008693 0.1757571234 0.0377238055 0.0076112069 0.001464783 0.0002715864 4.885941E-05 8.573854E-06 1.473413E-06 2.487372E-07 4.135164E-08
cumec Avg.
0 9.01816609 18.769643263 14.199903247 5.8736554957 1.8014798571 0.4641289964 0.1067404645 0.0226675062 0.004537995 0.0008681847 0.0001602229
6hr UGO
20 18 16 14 12 10 8 6 4 2 0
0 10 20 30 40 50 60
72 78 84 90
2.871663E-05 5.023634E-06 8.610753E-07 0.000000145
8 6 4 2 0
0 10 20 30 40 50 60
ograph (IUH) Nash Model
Adam ID: PRAMIT/1833/11 Water Resources Engineering
5
6
7
8
9
147
84
43
18
8
Nash model
4
5
6
7
8
9
interval Δt
inc.area
mom't arm
1st momt
2nd momt_a
2nd momt_b
0 6 6 6 6 6 6 6 6 6
0 5 35 82.5 90 71 39.5 19 6.5 1.5 350
0 3 9 12 15 18 21 24 27 30
0 30 420 1485 2160 2130 1422 798 312 81
0 180 5040 26730 51840 63900 51192 33516 14976 4374
0 90 630 1485 1620 1278 711 342 117 27
8838
251748
6300
M1Q = sum 1st momt sum inc.area
and
6
7
8
1st momt
2nd momt_a
2nd momt_b
M2Q = (sum 2st momt a+sum 2st momt b) sum inc.area
AbstractioRain fall
2
0 3 27 30 21
0 9 243 450 441
0 18 54 36 18
81
1143
126
IUHO
6hr UGO
EQ(1)
+ 2 *nk* M1h
k^2+316.5943 +nk^2+316.5943
us
n=13.68/k
~3 ~3
u(t) cm/hr u(t) cumec by averaging (3)*(4)/6 (5)*2.78*36 5 6 7 0 0 0 0.1802191465 18.0363321799 9.01816609 0.1948736446 19.5029543468 18.7696432634 0.0888974035 8.8968521465 14.1999032466 0.028481803 2.8504588448 5.8736554957 0.0075189935 0.7525008693 1.8014798571 0.0017561663 0.1757571234 0.4641289964
0 0.5 0.5 0.5 0.5
0 1.5 3.5 2.5 1.5
0.0003769365 7.605122836E-05 1.463612104E-05 2.713692635E-06 4.882035165E-07 8.567000218E-08 1.47223551E-08 2.485384186E-09 4.131858048E-10
0.0377238055 0.0076112069 0.001464783 0.0002715864 4.8859408E-05 8.5738538E-06 1.4734133E-06 2.4873725E-07 4.1351635E-08
0.1067404645 0.0226675062 0.004537995 0.0008681847 0.0001602229 2.87166309E-05 5.02363356E-06 8.61075274E-07 0.000000145
25
20
15
10
5
0 0
10 20 30 40 50 60 70 80 90 100
20 18 16 14 12 10 8 6 4 2 0
IUH
0 10 20 30 40 50 60 70 80 90 100
8 6 4 2 0 0 10 20 30 40 50 60 70 80 90 100
2st momt b)
Annual maximum flows in sugar Creek at Cr Year Xi 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988
log(X) 7860 2730 6480 12800 26300 15100 14600 7300 8580 15100 15100 21800 6200 2130 11100 14300 11200 6670 5440 9370 6900 9680 6810 7730 5290 12200 9750 7390 13100 7190 8850 6290 18800 9740 2990 6950
Fi=(i-0.35)/N
Rank(i) 3.895 3.436 3.812 4.107 4.420 4.179 4.164 3.863 3.933 4.179 4.179 4.338 3.792 3.328 4.045 4.155 4.049 3.824 3.736 3.972 3.839 3.986 3.833 3.888 3.723 4.086 3.989 3.869 4.117 3.857 3.947 3.799 4.274 3.989 3.476 3.842
17 2 8 29 39 33 32 14 18 33 33 38 6 1 25 31 26 9 5 20 11 22 10 16 4 27 24 15 30 13 19 7 36 23 3 12
XiFi 0.427 0.042 0.196 0.735 0.991 0.837 0.812 0.350 0.453 0.837 0.837 0.965 0.145 0.017 0.632 0.786 0.658 0.222 0.119 0.504 0.273 0.555 0.247 0.401 0.094 0.683 0.606 0.376 0.760 0.324 0.478 0.171 0.914 0.581 0.068 0.299
XiFi2 3355.615 115.500 1271.077 9403.077 26063.974 12641.410 11848.462 2555.000 3883.000 12641.410 12641.410 21045.385 898.205 35.500 7015.769 11238.333 7366.154 1479.372 648.615 4721.038 1884.231 5373.641 1685.038 3101.910 495.090 8336.667 5912.500 2775.987 9959.359 2332.141 4232.115 1072.526 17185.128 5656.692 203.167 2076.090
1432.589645 4.886538462 249.3266272 6907.64497 25830.0669 10583.12936 9615.482249 894.25 1757.30641 10583.12936 10583.12936 20316.89053 130.1245891 0.591666667 4434.325937 8832.17735 4844.662722 328.1170776 77.33491124 2378.677071 514.5399408 2983.059698 416.9390039 1244.740911 46.33532216 5696.722222 3585.400641 1042.774671 7571.666502 756.4508711 2023.81928 182.8793721 15708.96976 3285.23284 13.80491453 620.1652696
1989 1990 1991 39
9390 12400 21200 402810.000 10328.462 M100 βo
3.973 4.093 4.326 154.315 3.957
N mean(M1) sta(S) cv1
L1 L2 L3 L4
21 28 37
0.529 0.709 0.940
39 10328.462 5320.598 0.515
mean(M1y) sta(Sy) cv(M2y
bo = 2b1-bo = 6b2- 6b1+bo= 20b3-30b2+12b1-bo=
4971.885 2632.549162 8791.282 6232.793557 19922.564 18722.10191 256836.321 193064.789 6585.547 4950.379 M110 M120 β1 β2 3.957 0.235 0.059 10328.462 2842.632 517.457 387.550
maximum likelihood parameter estimator ein Adam ID: PRAMIT/1833/11 nd Water Resources Engineering
ar Creek at Crawfordsville, Indiana. XiFi3
1-Fi 611.606 0.207 48.906 5074.462 25598.259 8859.979 7803.334 312.988 795.294 8859.979 8859.979 19613.614 18.851 0.010 2802.721 6941.186 3186.297 72.775 9.221 1198.487 140.509 1655.981 103.166 499.492 4.337 3892.760 2174.224 391.709 5756.408 245.362 967.801 31.183 14359.610 1907.962 0.938 185.254
Xi(1-Fi) 0.573 0.958 0.804 0.265 0.009 0.163 0.188 0.650 0.547 0.163 0.163 0.035 0.855 0.983 0.368 0.214 0.342 0.778 0.881 0.496 0.727 0.445 0.753 0.599 0.906 0.317 0.394 0.624 0.240 0.676 0.522 0.829 0.086 0.419 0.932 0.701
4504.385 2614.500 5208.923 3396.923 236.026 2458.590 2751.538 4745.000 4697.000 2458.590 2458.590 754.615 5301.795 2094.500 4084.231 3061.667 3833.846 5190.628 4791.385 4648.962 5015.769 4306.359 5124.962 4628.090 4794.910 3863.333 3837.500 4614.013 3140.641 4857.859 4617.885 5217.474 1614.872 4083.308 2786.833 4873.910
Xi(1-Fi)2 2581.359 2503.887 4187.173 901.491 2.118 400.309 518.559 3084.250 2571.306 400.309 400.309 26.121 4533.714 2059.592 1502.787 655.511 1312.355 4039.373 4220.104 2306.600 3646.078 1915.778 3856.862 2770.920 4346.156 1223.389 1510.401 2880.800 752.949 3282.169 2409.589 4327.828 138.713 1711.848 2597.472 3417.986
Xi(1-Fi)3 1479.317 2397.953 3365.843 239.242 0.019 65.178 97.728 2004.763 1407.625 65.178 65.178 0.904 3876.907 2025.265 552.949 140.347 449.229 3143.461 3716.938 1144.429 2650.419 852.275 2902.536 1659.000 3939.400 387.406 594.478 1798.654 180.515 2217.568 1257.311 3589.878 11.915 717.659 2420.977 2396.972
1393.901 4418.891 17593.975 156391.615 4010.041
0.471 0.291 0.060 19.427
M130
4418.115 2078.780 978.093 3608.718 1050.229 305.644 1277.436 76.974 4.638 145973.679 82202.148 55103.791 3742.915 2107.747 1412.918 M101 M102 M103
β3
m s P W U X100
Maximum Likelihood estimator 3.957 0.2437807 0.01 3.0348543 2.3267853 33420.844
C0 C1 C2
2.515517 0.802853 0.010328
d1 d2 d3
1.432788 0.189269 0.001308
Instantaneous Unit Hydrograph (IUH) Nash M
Submitted by : Hamse Hussein Adam ID: PRAMIT/18 Department of Hydraulic and Water Resources Enginee GIVEN DATAS Drainage area= 36km^2 Abstractions= 0.5cm/hr Base flow= 5m^3/s Table
6hr period Rainfall in (c stream flow ( Reqired
1 1.5 15
2 3.5 75
3 2.5 170
4 1.5 185
Detemine the peak of the IUH Using Nash model
Solution
Calculation of M1Q & M2Q 1 Base flow
0 5 5 5 5 5 5 5 5 5
stream flow (m3/sec.)
time
0 15 75 170 185 147 84 43 18 8
0 6 12 18 24 30 36 42 48 54
2
3
ord DOR Avg DOR
0 10 70 165 180 142 79 38 13 3
0 5 35 82.5 90 71 39.5 19 6.5 1.5
sum
M1Q = M2Q =
where
25.251428571 737.28
Calculation of M1h & M2h 1
2
3
time
DOR
interval Δt
4
5
inc.areamoment arm
0 6 12 18 24
0 1 3 2 1
0 6 6 6 6
0 1 3 2 1
sum
0 3 9 15 21
7
M1h= M2h=
11.57143 181.2857
calculate the value of n & K nk = M1Q – M1h nk =
13.68
M2Q – M2h = n(n + 1)k2 + 2 *nk* M1h M2Q – M2h = 555.9943 2 *nk* M1h= 316.5943 (nk)2= 187.1424 nk2 ? From eqn (1)
M2Q – M2h = n(n + 1)k2 + 2 *nk* M1h 555.9943= 555.9943=
(nk)^2+nk^2+316.5943 187.1424 +nk^2+316.5943
nk2= 52.2576 But, nk=13.68 from thus
n=13.68/k
(13.68/k)*k^2=52.2576 k= n=
3.82 3.581152
time(hr)
t/3
e-t/3
(t/3)3
t 1 0 6 12 18 24 30 36
2 0 2 4 6 8 10 12
3 1 0.135164 0.018269 0.002469 0.000334 4.51E-05 6.1E-06
4 0 8 64 216 512 1000 1728
42 48 54 60 66 72 78 84 90
14 16 18 20 22 24 26 28 30
time(hr)
runoff
0 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90
time(hr)
0 6 12 18 24 30 36 42 48 54 60 66
8.24E-07 1.11E-07 1.51E-08 2.04E-09 2.75E-10 3.72E-11 5.03E-12 6.79E-13 9.18E-14
2744 4096 5832 8000 10648 13824 17576 21952 27000
IUHO
0 18.03633218 19.502954347 8.8968521465 2.8504588448 0.7525008693 0.1757571234 0.0377238055 0.0076112069 0.001464783 0.0002715864 4.885941E-05 8.573854E-06 1.473413E-06 2.487372E-07 4.135164E-08
cumec Avg.
0 9.01816609 18.769643263 14.199903247 5.8736554957 1.8014798571 0.4641289964 0.1067404645 0.0226675062 0.004537995 0.0008681847 0.0001602229
6hr UGO
20 18 16 14 12 10 8 6 4 2 0
0 10 20 30 40 50 60
72 78 84 90
2.871663E-05 5.023634E-06 8.610753E-07 0.000000145
8 6 4 2 0
0 10 20 30 40 50 60
ograph (IUH) Nash Model
Adam ID: PRAMIT/1833/11 Water Resources Engineering
5
6
7
8
9
147
84
43
18
8
Nash model
4
5
6
7
8
9
interval Δt
inc.area
mom't arm
1st momt
2nd momt_a
2nd momt_b
0 6 6 6 6 6 6 6 6 6
0 5 35 82.5 90 71 39.5 19 6.5 1.5 350
0 3 9 12 15 18 21 24 27 30
0 30 420 1485 2160 2130 1422 798 312 81
0 180 5040 26730 51840 63900 51192 33516 14976 4374
0 90 630 1485 1620 1278 711 342 117 27
8838
251748
6300
M1Q = sum 1st momt sum inc.area
and
6
7
8
1st momt
2nd momt_a
2nd momt_b
M2Q = (sum 2st momt a+sum 2st momt b) sum inc.area
AbstractioRain fall
2
0 3 27 30 21
0 9 243 450 441
0 18 54 36 18
81
1143
126
IUHO
6hr UGO
EQ(1)
+ 2 *nk* M1h
k^2+316.5943 +nk^2+316.5943
us
n=13.68/k
~3 ~3
u(t) cm/hr u(t) cumec by averaging (3)*(4)/6 (5)*2.78*36 5 6 7 0 0 0 0.1802191465 18.0363321799 9.01816609 0.1948736446 19.5029543468 18.7696432634 0.0888974035 8.8968521465 14.1999032466 0.028481803 2.8504588448 5.8736554957 0.0075189935 0.7525008693 1.8014798571 0.0017561663 0.1757571234 0.4641289964
0 0.5 0.5 0.5 0.5
0 1.5 3.5 2.5 1.5
0.0003769365 7.605122836E-05 1.463612104E-05 2.713692635E-06 4.882035165E-07 8.567000218E-08 1.47223551E-08 2.485384186E-09 4.131858048E-10
0.0377238055 0.0076112069 0.001464783 0.0002715864 4.8859408E-05 8.5738538E-06 1.4734133E-06 2.4873725E-07 4.1351635E-08
0.1067404645 0.0226675062 0.004537995 0.0008681847 0.0001602229 2.87166309E-05 5.02363356E-06 8.61075274E-07 0.000000145
25
20
15
10
5
0 0
10 20 30 40 50 60 70 80 90 100
20 18 16 14 12 10 8 6 4 2 0
IUH
0 10 20 30 40 50 60 70 80 90 100
8 6 4 2 0 0 10 20 30 40 50 60 70 80 90 100
2st momt b)
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