Question

Yearly peak flows Freqanaly River measured at a gaging site over a period of 35 years (ranked in the descending order) are given in the Table below. Calculate the magnitude of the 100-year flood using: Normal distribution Log-normal distribution Gumbel (Extreme-Value) distribution and Log-Pearson Ill distribution. 1 un 프 Rank Q (m/s) 99 2 86.1 3 80.7 4 76.4 5 75.7 6 74.1 7 72 8 71.3 9 71.1 10 69.9 11 67.4 12 67.1 13 66.2 14 64.7 64 16 59.9 17 58.2 18 57.8 19 57.5 10 55.7 21 49.6 22 49.6 23 48.3 24 47.2 25 42 26 40.6 27 37.5 28 33.8 29 33 30 32 31 31.2 32 31 33 30.1 34 27.3 35 19.3 36 15.4 37 7.6

Answer 1

It is assumed that data is following a normal distribution and the formulas of normal distribution is applied.

Whenever data follows log normal distribution we need to make the data into a normal data by applying natural logarithm to them.

Now, we can apply the normal dirstribution formulas and the obtained result is converted back into a result of lognormal by applying exponential to it.

G43 : =G41/37 F J K L M N P R A Rank 1 1 LV487777 2 3 4 2 3 4 5 6 7 8 9 5 6 7 8 10 11 12 13 14 15 16 17 9 10 11 12 13 Mean Standard Error Median Mode Standard Deviation Sample Variance Kurtosis Skewness Range Minimum Maximum Sum Count Confidence Level(95.0% 53.25135135 3.499840455 57.5 49.6 21.28869838 453 2086787 -0.55771025 -0.166681389 91.4 7.6 99 1970.3 37 7.09800543 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 B C D E Q[m 3/s) (x-xmean) (x-xmean) 2 (x-xmean) 3 99 45.74864865 2092.938853 95749.12424 86.1 32.84864865 1079.033718 35444.79948 80.7 27.44864865 753.4283126 20680.58904 76.4 23.14864865 535.8599343 12404.43334 75.7 22.44864865 503.9418262 11312.81299 74.1 20.84864865 434.6661505 9062.201851 72 18.74864865 351.5118262 6590.371724 71.3 18.04864865 325.753718 5879.414403 71.1 17.84864865 318.5742586 5686.12001 69.9 16.64864865 277.1775018 4614.630841 67.4 14.14864865 200.1842586 2832.33674 67.1 13.84864865 191.7850694 2655.964042 66.2 12.94864865 167.6675018 2171.067571 64.7 11.44864865 131.0715559 1500.592191 64 10.74864865 115.5334478 1241.828437 59.9 6.648648649 44.20452885 293.900381 58.2 4.948648649 24.48912345 121.1880677 57.8 4.548648649 20.69020453 94. 11247087 57.5 4.248648649 18.05101534 76.69242193 55.7 2.448648649 5.995880205 14.68180396 49.6 -3.651351351 13.33236669 -48.68115513 49.6 -3.651351351 13.33236669 -48.68115513 48.3 -4.951351351 24.5158802 -121.3867366 47.2 -6.051351351 36.61885318 -221.5935467 42 - 11.25135135 126.5929072 -1424.341278 40.6 -12.65135135 160.056691 -2024.933434 37.5 -15.75135135 248.1050694 -3907.99012 33.8 -19.45135135 378.3550694 -7359.51739 33 -20.25135135 410.1172316 -8305.428151 32 -21.25135135 451.6199343 -9597.5339 31.2 -22.05135135 486.2620964 -10722.73634 31 -22.25135135 495.122637 -11017.14776 30.1 -23.15135135 535.9850694 -12408.77866 27.3 -25.95135135 673.472637 -17477.52503 19.3 -33.95135135 1152.694259 -39135.52777 15.4 -37.85135135 1432.724799 -54230.56976 7.6 -45.65135135 2084.04588 -95139.51071 G Н. LN(Q) (y) (y-ymean) (y-ymean) 2 4.59512 0.7279724 0.529943753 4.455509 0.5883619 0.346169747 4.390739 0.5235911 0.274147622 4.335983 0.4688352 0.219806448 4.326778 0.4596307 0.211260351 4.305416 0.438268 0. 192078875 4.276666 0.4095186 0.167705505 4.266896 0.3997488 0.159799131 4.264087 0.3969398 0.15756124 4.247066 0.3799182 0.144337806 4.210645 0.3434975 0.11799055 4.206184 0.3390366 0.114945783 4.19268 0.325533 0.105971715 4.169761 0.3026137 0.091575057 4.158883 0.2917356 0.085109655 4.092677 0.225529 0.050863335 4.063885 0.1967379 0.038705786 4.056989 0.1898413 0.036039713 4.051785 0.1846375 0.03409099 4.01998 0.1528327 0.02335782 3.903991 0.0368433 0.001357432 3.903991 0.0368433 0.001357432 3.877432 0.0102841 0.000105762 3.854394 -0.012754 0.000162654 3.73767 -0.129478 0.01676452 3.703768 -0.163379 0.026692837 3.624341 -0.242807 0.058955025 3.520461 -0.346687 0.120191661 3.496508 -0.37064 0.137373959 3.465736 -0.401412 0.161131264 3.440418 -0.426729 0.182097979 3.433987 -0.43316 0.187627835 3.404525 -0.462622 0.214019412 3.306887 -0.560261 0.313892153 2.960105 -0.907042 0.82272591 2.734368 -1.13278 1.28319049 2.028148 -1.838999 3.381918225 Total 1970.3 Average 53.25135 Sheet1 Total Total 16315.51243 -54765.02084 Standard deviation 21.28869838 Sheet2 + Total 143.0845 Average 3.867147! Total 10.01102543 Standard deviation 0.527336742 ... READY

2 Q=1970-3 m/s = 33.25135 msec (Mean discharge). 1970.3 37 n 2(Q - Qrean) ² standard deviation n-1 16315-51243 37 -1 standard deviation=24-288698.38 m/s Q + kis s-standard deviation For Normal distribution (6 26 ) . . 3- ol. p Cazor) = 1-0.01 = 0.99. from 2-tables zrky = 2.325 Q1 = 53.25135 +(2-325) (21-288698.38) Q = 102.747 m/ (Normal distribution) Above solution following is based on assumption that dabas Normal distaibution. If data is following a log-Normal distribution. then 2 we need to apply logarithm to the data. (Now it follows normal distribution .. y-loge (O)

EY= 143.0845 2 143.0845 37 = 3.867147 sy = (y-7) n- to allo8543 Sy = 0.527 7336742 "T=100 = ý tkp sy = 3.867147+ (2.325) 6 5277336752) II o 3.094127793 YT-100 nga e 5.094/27783 e *T=160= 163-0615591 m/s (log-Noormal distribution) Gumbel distribution 0 5772 tln kiando Tf 100 klou -5772 c 5772-41600149 J6 TT -4.022949227 too kioo 3-136680644 (3136680644] [21-2236 Q108 = 130-62719 m/s. Chumbel distribution). 53-25135 +

log-Pearson Type II distribution CSE 0 (x-2)3 (n-1) (A-2) s} = 37 3G:-) :-54465 02084 5=2128869838 (5 = 37 (-54765.02084 (36) (35) (2128861833) --0-166681388 k= = -0.02778.231 용 wszlin. (13] / potato (0.01) 9.- 3.034854259 Za 2.516 + 0.80298 to 01033 It.1-432809 1 013136 +0.001518 23.034554 - که ان. 2.516+08029 (3-034854) +0.0103368-134851) i + 14328 (3 034) +0-19936-0348)2+ o gal316834) - 3-034854- (-51672.636 to 095143) (1+4+348337+0.174352 030617) Z = 9.126854

«т: 2+ (-)} + 4 (2247) - (-) + 2 + 4 5 а ) + (5-ла1)(на) - (-а). to us) ояту - (2 раза) (- 9-ого4 (an (2010), (- гнаў , 2-1263, e-04 23 0 00080 75541 xio1-2662x106 25.5158/09 К= 2 - 02 8243 . От = + 1 ч = 3-867)41 + (2023 243)(os 27333612) е 4:315 14 24 6тор От e 4-1384 247 131-52311 41 1/3 .

Chipboard Font Alignment Number Styles C1 (x-xmean) F J K L M 1 vunna/ Mean Standard Error Median Mode Standard Deviation Sample Variance Kurtosis Skewness Range Minimum Maximum Sum Count Confidence Level(95.0%4 53.2513513513513 3.49984045458245 57.5 49.6 21.288698379156 453.20867867868 -0.557710249775354 -0.166681388513445 91.4 7.6 99 1970.3 37 7.09800543032736 B 1 Rank Qim3s) 2 99 3 =42+1 86.1 4 = A3+1 80.7 5 = 44+1 76.4 6 = 45+1 75.7 7 = 46+1 74.1 8 = A7+1 72 9 =48+1 71.3 10 = A9+1 71.1 11 =A10+1 69.9 12 = A11+1 67.4 13 =A12+1 67.1 14 = A13+1 66.2 15 - A14+1 64.7 16 - A15+1 64 17 = 16+1 59.9 18 =417+1 58.2 19 = A18+1 57.8 20 =A19+1 57.5 21 =A20+1 55.7 22 =421+1 49.6 23 = A22+1 49.6 24 = A23+1 48.3 25 =A24+1 47.2 26 =A25+1 42 27 = A26+1 40.6 28 = A27+1 37.5 29 =A28+1 33.8 30 =429+1 33 31 =A30+1 32 32 =431+1 31.2 33 - A32+1 31 34 = A33+1 30.1 35 =434+1 27.3 36 = 435+1 19.3 37 = A36+1 15.4 38 = 437+1 7.6 39 40 Total 41 =SUM(B2:B38) 42 Average 43 =B4137 С D (x-xmean) lix-xmean) 2 =B2-$B$43 =C2^2 =B3-$B$43 =C32 =B4-$B$43 =C42 =B5-$B$43 -C52 =B6-$B$43 -C642 =B7-$B$43 =C72 =B8-$B$43 =C82 =B9-$B$43 C92 =B10-$B$43 =C102 =B11-$B$43 =C112 =B12-$B$43 =C122 =B13-$B$43 =C1342 =B14-$B$43 =C142 =B15-$B$43 -C1542 =B16-$B$43 =C1692 =B17-$B$43 =C172 =B18-$B$43 =C182 =B19-$B$43 =C1992 =B20-$B$43 =C20~2 =B21-$B$43 =C212 =B22-$B$43 =C222 =B23-$B$43 =C232 =B24-$B$43 =C242 =B25-$B$43 =C252 =B26-$B$43 =C2692 =B27-$B$43 =C272 =B28-$B$43 =C282 =B29-$B$43 =C292 =B30-$B$43 =C302 =B31-$B$43 -C312 =B32-$B$43 -C322 =B33-$B$43 =C3312 =B34-$B$43 =C342 =B35-$B$43 =C352 =B36-$B$43 =C36 2 =B37-$B$43 =C372 =B38-$B$43 =C382 E (x-xmean) 3 =C23 =C33 =C43 =C543 =C643 =C743 =C83 =C93 =C10^3 =C11 3 =C123 =C133 =C14 3 =C1543 =C1643 =C1743 =C1893 =C1993 =C203 =C213 =C223 =C233 =C243 =C2543 =C263 =C2743 =C2843 =C293 =C303 =C313 =C323 =C333 =C343 =C35 3 =C3643 =C3743 =C3843 G LN(Q) (9) ELN(B2) ELN(B3) ELN(B4) =LN(B5) =LN(B6) =LN[B7] =LN(B8) ELN(B9) =LN(B10) =LN(B11) =LN(B12) =LN(B13) =LN(B14) ELN(B15) =LN(B16) =LN(B17) ELN(B18) ELN(B19) =LN(B20) =LN(B21) =LN(B22 =LN(B23) =LN(B24) ELN(B25) =LN(B26) =LN(B27) =LN(B28) =LN(B29) =LN(B30) ELN(B31) =LN(B32) =LN(B33 ELN(B34) =LN(B35) =LN(B36) =LN(B37) =LN(B38) H (y-ymean) (y-ymean) 2 =G2-$G$43 =H22 =G3-$G$43 =H3 2 =G4-$G$43 =H42 =G5-$G$43 =H542 =G6-$G$43 =H62 =G7-$G$43 =H72 =G8-$G$43 =H82 =G9-$G$43 =H9^2 =G10-$G$43 =H102 =G11-$G$43 =H112 =G12-$G$43 =H12 2 =G13-$G$43 =H13^2 =G14-$G$43 =H142 =G15-$G$43 =H152 =G 16-$G$43 =H16 2 =G17-$G$43 =H172 =G18-$G$43 =H182 =G19-$G$43 =H1992 =G20-$G$43 =H202 =G21-$G$43 =H212 =G22-$G$43 =H222 =G23-$G$43 =H232 =G24-$G$43 =H242 =G25-$G$43 =H25-2 =G26-$G$43 =H2692 =G27-$G$43 =H2742 =G28-$G$43 =H282 =G29-$G$43 =H29 2 =G 30-$G$43 -H30*2 =G31-$G$43 =H312 =G32-$G$43 =H322 =G33-$G$43 =H332 =G34-$G$43 =H34-2 =G35-$G$43 =H35 2 =G36-$G$43 =H3642 =G37-$G$43 =H372 =G38-$G$43 =H382 Total =SUM(E2:E38) Total =SUM(G2 G38) Total =SUM(D2:D38) Standard deviation =SQRT(D4Y(37-11) Sheet2 + Total =SUM(12:138) Standard deviation ESORT(147(37-11) Average =G4737 Sheet1 * READY

Calculations are carried out in Ms excel for accuracy and for easy computating facility.

All the formulas are given in the spread sheet image.Kindly use them.

I hope you have understood.

All the Best.!

Please do mention your questions in the comments seciton.