EXTREME VALUE DISTRIBUTION TO DETERMINE MAXIMUM PRECIPITATION FOR DIFFERENT RETURN PERIODS By Jonathan Anthony D’Cruz Guided by Prof. Shreenivas N. Londhe Professor In Civil Engineering And Dean (Academics)
Extreme value distribution to predict maximum precipitation
Dec 07, 2014
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EXTREME VALUE DISTRIBUTION TO DETERMINE MAXIMUM PRECIPITATION FOR DIFFERENT RETURN PERIODS
By Jonathan Anthony D’CruzGuided by
Prof. Shreenivas N. LondheProfessor In Civil Engineering And Dean (Academics)
INTRODUCTION On the 26th of July, 2005, a record 944 mm of rainfall fell over
the city of Mumbai, causing large scale flooding, claiming several lives and causing large scale property damage.
Such events forced scientists to take a strong view of flood frequency analysis, which is the use of historical flow records to produce guidance about the expected behavior of future flows.
An attempt has been made to compute the return period, which is, an estimate of the interval of time between events like an earthquake, flood or river discharge flow of a certain intensity or size.
The Hershfield technique and Gumbel’s theory of extreme values were used for the estimation of maximum one day rainfall.
OBJECTIVE In recent years, heavy precipitation events have
resulted in several damaging floods in India. The consecutive flash floods over three major metro cities in the same year i.e. Mumbai in July 2005, Chennai in October 2005 and again in December 2005 and Bangalore in October 2005 caused heavy damages to the economy, life and property.
Therefore a detailed study on extreme weather events mostly rainfall was urgently needed to have a clear idea about the impact of climate change on the extreme weather events of the country. As they are essential for planning and design of structures such as buildings, check dams, storage reservoirs, etc.
MOTIVATION Flood Analysis is a vast topic which encompasses
numerous variables and parameters. An attempt has been made to obtain the most rudimentary results which may be used as a reference and benchmark for further studies.
Mathematical modeling of natural phenomena is, at best unpredictable. Which is reflected by Dingman [1994, p. 141]:
“The concepts of PMP and Flood Analysis are controversial. Can we really specify an upper bound to the amount of rain that can fall in a given time? We must recognize that the plotted values are only those that have been observed historically at the infinitesimal fraction of the earth covered by rain gages, and higher amounts must have fallen at ungauged locations at other times and places. And, conceptually, we can always imagine that a few more molecules of water could fall beyond any specified limit.”
STUDY AREA The study area is Colaba, Mumbai. It is located at latitude 18° 54′ 36″ N and longitude
72° 48′ 36″ E. Its elevation is 11 meters above mean sea level. Average yearly temperature is 27.2 °C Rainfall available is for the past 35 years.
MAXIMUM RAINFALL OVER 35 YEAR PERIOD
PROBABLE MAXIMUM PRECIPITATION
Hershfield TechniquesChow suggested the following formula for most of the frequency distributions applicable to hydrologic analysisXm = ( Ẋ+ Kmσ)X= Value of the event.Ẋ = Mean of the annual maximum series of N years. σ = Standard deviation of the series of annual maximum.Xm = Estimate of 1 Day Probable Maximum Precipitation.
Km = (XL – XN-1)/ σn-1
XL = Largest value of annual maximum series omitting the largest value of the series.
XN-1= Mean of annual maximum series omitting the largest value of the series.
σn-1 =Standard Deviation of annual maximum series omitting the largest value of the series.
Km= (544.3- 224.8676471)/ 101.6399693 = 3.142782855X= 234.8885714 + (3.142782855 * 117.6662473)
=604.688036 mmTherefore, the probable maximum precipitation for Colaba, Mumbai based on the maximum precipitation data for the past 35 years is 604.688036 mm.
CALCULATION OF PMPPROBABLE MAXIMUM PRECIPITATION
RANK RAINFALL (x) XN-1 STANDARD DEVIATION XN-1 - XN-1(avg) (XN-1 - XN-1(avg))^21 575.6 2 544.3 544.3 319.4323529 102037.02813 477.6 477.6 252.7323529 63873.642224 421.2 421.2 196.3323529 38546.392815 417.2 417.2 192.3323529 36991.733996 345.5 345.5 120.6323529 14552.164587 279.4 279.4 54.53235294 2973.7775178 261.9 261.9 37.03235294 1371.3951649 249.7 249.7 24.83235294 616.645752610 244.2 244.2 19.33235294 373.739870211 243.7 243.7 18.83235294 354.657517312 241.6 241.6 16.73235294 279.971634913 233 233 8.132352941 66.1351643614 217 217 -7.867647059 61.8998702415 213.3 213.3 -11.56764706 133.810458516 206.9 206.9 -17.96764706 322.836340817 206.2 206.2 -18.66764706 348.481046718 184.9 184.9 -39.96764706 1597.41281119 184.4 184.4 -40.46764706 1637.63045820 183.3 183.3 -41.56764706 1727.86928221 180.9 180.9 -43.96764706 1933.15398822 175.9 175.9 -48.96764706 2397.83045823 175.7 175.7 -49.16764706 2417.45751724 174.4 174.4 -50.46764706 2546.983425 165.4 165.4 -59.46764706 3536.40104726 162.8 162.8 -62.06764706 3852.39281127 153.9 153.9 -70.96764706 5036.40692928 150.3 150.3 -74.56764706 5560.33398829 148.6 148.6 -76.26764706 5816.75398830 147.7 147.7 -77.16764706 5954.84575331 138.3 138.3 -86.56764706 7493.95751732 138.2 138.2 -86.66764706 7511.28104733 128.7 128.7 -96.16764706 9248.21634134 125.9 125.9 -98.96764706 9794.59516435 123.5 123.5 -101.3676471 10275.39987
Average 234.8885714 224.8676471 351243.2344 Variance 10330.68337 Standard Deviation 101.6399693
RETURN PERIOD & RISK ANALYSIS A return period also known as a recurrence interval is an
estimate of the interval of time between events like an earthquake, flood or river discharge flow of a certain intensity or size.
Recurrence interval = (n+1)/mn is number of years on record;m is the rank of the event being considered
Risk analysis gives the likelihood of at least one event that exceeds design limits during the expected life of the structure.1/T=m/(n+1)R=1-(1-(1/T))^n
RISK ANALYSISRAIN FALL RISK ANALYSIS
RANK
RAINFALL (x) R.I. (T) RETURN PERIOD (1/T) RISK ANALYSIS
(m/(n+1)) (1-(1/T))^n1-((1-(1/T))^n)
Percent Terms
1 575.6 36 0.0277777780.37307317
7 0.626926823 62.69268231
2 544.3 18 0.0555555560.13526161
5 0.864738385 86.47383847
3 477.6 12 0.0833333330.04757736
3 0.952422637 95.24226368
4 421.2 9 0.1111111110.01620547
3 0.983794527 98.379452695 417.2 7.2 0.138888889 0.00533423 0.99466577 99.46657698
6 345.5 6 0.1666666670.00169299
8 0.998307002 99.83070022
7 279.45.14285714
3 0.1944444440.00051682
4 0.999483176 99.94831763
8 261.9 4.5 0.2222222220.00015133
6 0.999848664 99.98486635
9 249.7 4 0.254.23784E-
05 0.999957622 99.99576216
10 244.2 3.6 0.2777777781.13104E-
05 0.99998869 99.99886896
11 243.73.27272727
3 0.3055555562.86624E-
06 0.999997134 99.99971338
12 241.6 3 0.3333333336.86761E-
07 0.999999313 99.99993132
13 2332.76923076
9 0.3611111111.54841E-
07 0.999999845 99.99998452
14 2172.57142857
1 0.3888888893.26743E-
08 0.999999967 99.99999673
15 213.3 2.4 0.4166666676.41339E-
09 0.999999994 99.99999936
16 206.9 2.25 0.4444444441.16269E-
09 0.999999999 99.99999988
17 206.22.11764705
9 0.4722222221.93103E-
10 1 99.99999998
18 184.9 2 0.52.91038E-
11 1 100
19 184.41.89473684
2 0.5277777783.93663E-
12 1 100
20 183.3 1.8 0.5555555564.71641E-
13 1 100
21 180.91.71428571
4 0.5833333334.92727E-
14 1 100
22 175.91.63636363
6 0.6111111114.40447E-
15 1 100
23 175.71.56521739
1 0.6388888893.29177E-
16 1 100
24 174.4 1.5 0.6666666671.99874E-
17 1 100
25 165.4 1.44 0.6944444449.50947E-
19 1 100
26 162.81.38461538
5 0.7222222223.38386E-
20 1 100
27 153.91.33333333
3 0.758.47033E-
22 1 100
28 150.31.28571428
6 0.7777777781.37266E-
23 1 100
29 148.6 1.24137931 0.8055555561.28187E-
25 1 10030 147.7 1.2 0.833333333 5.8171E-28 1 100
31 138.31.16129032
3 0.8611111119.84833E-
31 1 100
32 138.2 1.125 0.8888888893.99496E-
34 1 100
33 128.71.09090909
1 0.916666667 1.693E-38 1 100
34 125.91.05882352
9 0.9444444441.16269E-
44 1 100
35 123.51.02857142
9 0.9722222223.38386E-
55 1 100
ESTIMATED MAXIMUM ONE DAY RAINFALL FOR DIFFERENT RETURN
PERIODS Gumbel’s Method was used to estimate the maximum one day
rainfall for different return periods which is explained below. XT = (Ẋ + σKT) KT = - (1.00+1.75(log (log (T/ (T-1))))) Ẋ = Mean of the annual maximum series of N years. T= Return Period XT= Maximum rainfall corresponding to Time period.
Estimation Of Maximum Rainfall (Model 1)- Maximum Of 35 YearsRANK RAINFALL (x) STANDARD DEVIATION RETURN PERIOD FREQUENCY FACTOR MAX.RAINFALL
1 575.6 340.7114286 116084.2776 2 -0.087567102 224.58487922 417.2 182.3114286 33237.45699 5 0.773854859 325.94516873 123.5 -111.3885714 12407.41384 10 1.344191152 393.05449994 184.4 -50.48857143 2549.095845 15 1.665970069 430.91701755 175.7 -59.18857143 3503.286988 20 1.891271417 457.42738176 206.2 -28.68857143 823.0341306 25 2.06481257 477.84731797 125.9 -108.9885714 11878.5087 30 2.205993197 494.45951248 241.6 6.711428571 45.04327347 35 2.325007656 508.46349719 180.9 -53.98857143 2914.765845 40 2.427881777 520.568309
10 174.4 -60.48857143 3658.867273 45 2.51847549 531.228131111 544.3 309.4114286 95735.43213 50 2.599410704 540.751474112 345.5 110.6114286 12234.88813 55 2.67255001 549.357501813 128.7 -106.1885714 11276.0127 60 2.739264255 557.207516614 153.9 -80.98857143 6559.148702 65 2.800591732 564.423690615 138.2 -96.68857143 9348.679845 70 2.857337751 571.100781816 183.3 -51.58857143 2661.380702 75 2.910139521 577.313767917 421.2 186.3114286 34711.94842 80 2.959509909 583.122996218 477.6 242.7114286 58908.83756 85 3.005867826 588.577758319 175.9 -58.98857143 3479.651559 90 3.04955986 593.71883620 206.9 -27.98857143 783.3601306 100 3.130061449 603.191155921 148.6 -86.28857143 7445.717559 22 162.8 -72.08857143 5196.762131 23 165.4 -69.48857143 4828.661559 24 244.2 9.311428571 86.70270204 25 261.9 27.01142857 729.6172735 26 233 -1.888571429 3.566702041 27 243.7 8.811428571 77.64127347 28 184.9 -49.98857143 2498.857273 29 138.3 -96.58857143 9329.352131 30 147.7 -87.18857143 7601.846988 31 150.3 -84.58857143 7155.226416 32 217 -17.88857143 320.0009878 33 213.3 -21.58857143 466.0664163 34 279.4 44.51142857 1981.267273 35 249.7 14.81142857 219.3784163
Average 234.8885714 470741.7554 Variance 13845.34575 Standard Deviation 117.6662473
Estimation Of Maximum Rainfall (Model 2)- Month Of June For 35 YearsRANK RAINFALL (x) STANDARD DEVIATION RETURN PERIOD FREQUENCY FACTOR MAX. RAINFALL
1 36 -113.6542857 12917.29666 2 -0.087567102 140.79340912 69.8 -79.85428571 6376.706947 5 0.773854859 227.96031923 97.5 -52.15428571 2720.069518 10 1.344191152 285.67239834 91.1 -58.55428571 3428.604376 15 1.665970069 318.23306575 175.7 26.04571429 678.3792327 20 1.891271417 341.03121026 90.2 -59.45428571 3534.81209 25 2.06481257 358.59176337 86 -63.65428571 4051.86809 30 2.205993197 372.87776898 134.8 -14.85428571 220.6498041 35 2.325007656 384.92078989 180.9 31.24571429 976.2946612 40 2.427881777 395.33057710 174.4 24.74571429 612.3503755 45 2.51847549 404.49771511 133.3 -16.35428571 267.4626612 50 2.599410704 412.687513812 345.5 195.8457143 38355.5438 55 2.67255001 420.088448113 110.9 -38.75428571 1501.894661 60 2.739264255 426.839233214 78.2 -71.45428571 5105.714947 65 2.800591732 433.044933715 102.2 -47.45428571 2251.909233 70 2.857337751 438.787038416 111.9 -37.75428571 1425.38609 75 2.910139521 444.130026417 421.2 271.5457143 73737.07495 80 2.959509909 449.125794318 477.6 327.9457143 107548.3915 85 3.005867826 453.816731519 47 -102.6542857 10537.90238 90 3.04955986 458.237909320 118.4 -31.25428571 976.8303755 100 3.130061449 466.383829721 99.7 -49.95428571 2495.430661 22 36.6 -113.0542857 12781.27152 23 62.4 -87.25428571 7613.310376 24 122.8 -26.85428571 721.1526612 25 248.6 98.94571429 9790.254376 26 233 83.34571429 6946.50809 27 126 -23.65428571 559.5252327 28 122.4 -27.25428571 742.7960898 29 96.4 -53.25428571 2836.018947 30 147.7 -1.954285714 3.819232653 31 82.2 -67.45428571 4550.080661 32 209.7 60.04571429 3605.487804 33 118 -31.65428571 1001.993804 34 279.4 129.7457143 16833.95038 35 170.4 20.74571429 430.3846612
Average 149.6542857 348137.1269 Variance 10239.32726 Standard Deviation 101.189561
ESTIMATION OF MAXIMUM RAINFALL (MODEL 2)- MONTH OF JUNE FOR 35 YEARS
Estimation Of Maximum Rainfall (Model 3)- Month Of July For 35 YearsRANK RAINFALL (x) STANDARD DEVIATION RETURN PERIOD FREQUENCY FACTOR MAX. RAINFALL
1 575.6 412.0628571 169795.7982 2 -0.087567102 152.86306542 417.2 253.6628571 64344.84509 5 0.773854859 257.86693783 123.5 -40.03714286 1602.972808 10 1.344191152 327.3886384 184.4 20.86285714 435.2588082 15 1.665970069 366.61219385 83.2 -80.33714286 6454.056522 20 1.891271417 394.0755226 119.7 -43.83714286 1921.695094 25 2.06481257 415.22949067 125.9 -37.63714286 1416.554522 30 2.205993197 432.43884118 98.3 -65.23714286 4255.884808 35 2.325007656 446.9462259 135 -28.53714286 814.3685224 40 2.427881777 459.486166610 130.4 -33.13714286 1098.070237 45 2.51847549 470.529175811 544.3 380.7628571 144980.3534 50 2.599410704 480.394852912 125.8 -37.73714286 1424.091951 55 2.67255001 489.310240213 24.4 -139.1371429 19359.14452 60 2.739264255 497.442438314 153.9 -9.637142857 92.87452245 65 2.800591732 504.918011115 100.5 -63.03714286 3973.68138 70 2.857337751 511.835122616 183.3 19.76285714 390.5705224 75 2.910139521 518.27144617 77.8 -85.73714286 7350.857665 80 2.959509909 524.289497818 181.1 17.56285714 308.453951 85 3.005867826 529.940341319 175.9 12.36285714 152.8402367 90 3.04955986 535.266224620 161.1 -2.437142857 5.939665306 100 3.130061449 545.079044521 148.6 -14.93714286 223.1182367 22 51.6 -111.9371429 12529.92395 23 144.9 -18.63714286 347.3430939 24 102.3 -61.23714286 3749.987665 25 117.8 -45.73714286 2091.886237 26 79.2 -84.33714286 7112.753665 27 243.7 80.16285714 6426.083665 28 184.9 21.36285714 456.3716653 29 19.8 -143.7371429 20660.36624 30 95.6 -67.93714286 4615.45538 31 132.4 -31.13714286 969.5216653 32 103.9 -59.63714286 3556.588808 33 213.3 49.76285714 2476.341951 34 114.8 -48.73714286 2375.309094 35 249.7 86.16285714 7424.037951
Average 163.5371429 505193.4017 Variance 14858.62946 Standard Deviation 121.895978
ESTIMATION OF MAXIMUM RAINFALL (MODEL 2)- MONTH OF JULY FOR 35 YEARS
Estimation Of Maximum Rainfall (Model 4)- Month Of August For 35 YearsRANK RAINFALL (x) STANDARD DEVIATION RETURN PERIOD FREQUENCY FACTOR MAX. RAINFALL
1 184.3 68.14571429 4643.838376 2 -0.087567102 110.62635962 107.8 -8.354285714 69.7940898 5 0.773854859 165.00610833 87 -29.15428571 849.9723755 10 1.344191152 201.01023384 72.4 -43.75428571 1914.437518 15 1.665970069 221.32345715 35.7 -80.45428571 6472.89209 20 1.891271417 235.54625536 206.2 90.04571429 8108.230661 25 2.06481257 246.50154177 110.3 -5.854285714 34.27266122 30 2.205993197 255.41397648 48.7 -67.45428571 4550.080661 35 2.325007656 262.92710789 88.3 -27.85428571 775.8612327 40 2.427881777 269.421333810 173.4 57.24571429 3277.071804 45 2.51847549 275.140323511 21.5 -94.65428571 8959.433804 50 2.599410704 280.249592612 79.7 -36.45428571 1328.914947 55 2.67255001 284.866722613 128.7 12.54571429 157.3949469 60 2.739264255 289.078251914 100.3 -15.85428571 251.3583755 65 2.800591732 292.94972615 86.6 -29.55428571 873.4558041 70 2.857337751 296.531982316 71 -45.15428571 2038.909518 75 2.910139521 299.865246517 259 142.8457143 20404.89809 80 2.959509909 302.981894818 33.7 -82.45428571 6798.709233 85 3.005867826 305.908372219 174.9 58.74571429 3451.058947 90 3.04955986 308.66655820 60.9 -55.25428571 3053.03609 100 3.130061449 313.748453321 98.1 -18.05428571 325.9572327 22 70 -46.15428571 2130.21809 23 101.1 -15.05428571 226.6315184 24 244.2 128.0457143 16395.70495 25 261.9 145.7457143 21241.81323 26 85 -31.15428571 970.5895184 27 116.5 0.345714286 0.119518367 28 72.2 -43.95428571 1931.979233 29 138.3 22.14571429 490.4326612 30 65.8 -50.35428571 2535.55409 31 150.3 34.14571429 1165.929804 32 161 44.84571429 2011.13809 33 191.6 75.44571429 5692.055804 34 111.2 -4.954285714 24.54494694 35 67.8 -48.35428571 2338.136947
Average 116.1542857 135494.4269 Variance 3985.130202 Standard Deviation 63.12788767
ESTIMATION OF MAXIMUM RAINFALL (MODEL 2)- MONTH OF AUG. FOR 35 YEARS
Estimation Of Maximum Rainfall (Model 5)- Month Of September For 35 YearsRANK RAINFALL (x) STANDARD DEVIATION RETURN PERIOD FREQUENCY FACTOR MAX. RAINFALL
1 27.5 -66.54857143 4428.712359 2 -0.087567102 88.45664562 175.3 81.25142857 6601.794645 5 0.773854859 143.4659773 88.8 -5.248571429 27.54750204 10 1.344191152 179.8869414 181.9 87.85142857 7717.873502 15 1.665970069 200.43534115 78 -16.04857143 257.5566449 20 1.891271417 214.8228046 65.1 -28.94857143 838.0197878 25 2.06481257 225.90492557 23 -71.04857143 5047.899502 30 2.205993197 234.92054418 241.6 147.5514286 21771.42407 35 2.325007656 242.52065899 76.8 -17.24857143 297.5132163 40 2.427881777 249.0900719
10 115.1 21.05142857 443.1626449 45 2.51847549 254.875273411 148.4 54.35142857 2954.077788 50 2.599410704 260.043695212 29.5 -64.54857143 4166.518073 55 2.67255001 264.7142813 30.7 -63.34857143 4013.041502 60 2.739264255 268.974568414 6.3 -87.74857143 7699.811788 65 2.800591732 272.890864515 138.2 44.15142857 1949.348645 70 2.857337751 276.514594516 37.2 -56.84857143 3231.760073 75 2.910139521 279.886449717 97.8 3.751428571 14.07321633 80 2.959509909 283.03918118 21.9 -72.14857143 5205.416359 85 3.005867826 285.999539719 97.5 3.451428571 11.91235918 90 3.04955986 288.789658520 206.9 112.8514286 12735.44493 100 3.130061449 293.930389521 91.5 -2.548571429 6.495216327 22 162.8 68.75142857 4726.758931 23 165.4 71.35142857 5091.026359 24 132 37.95142857 1440.310931 25 59.8 -34.24857143 1172.964645 26 134 39.95142857 1596.116645 27 45.1 -48.94857143 2395.962645 28 19 -75.04857143 5632.288073 29 38.9 -55.14857143 3041.364931 30 77.1 -16.94857143 287.2540735 31 32.6 -61.44857143 3775.926931 32 217 122.9514286 15117.05379 33 38.3 -55.74857143 3107.903216 34 65 -29.04857143 843.819502 35 125.7 31.65142857 1001.812931
Average 94.04857143 138649.9674 Variance 4077.940218 Standard Deviation 63.85875209
ESTIMATION OF MAXIMUM RAINFALL (MODEL 2)- MONTH OF SEPT. FOR 35 YEARS
CONCLUSION
Co-relating the Probable Maximum Rainfall (PMP) and one day maximum rainfall for different return periods (For maximum of 35 years), both obtained by different techniques, it is observed that the values lie very close to each other viz. PMP being 604.68mm and One day maximum rainfall for 100 year return period being 603.1911559mm. The ratio being 1.00246.
The graph between one day rainfalls for different return periods is plotted. This figure would be used for obtaining the maximum 1-day rainfall values for return period up to 1000 years. In view of the linearity exhibited by the curves it is possible to use these figures for extrapolation for the periods beyond 1000 years also. The trend analysis for the estimated one day rainfall for different return period was carried out and it is found that the logarithmic trend line gives the better coefficient of correlation (R2=0.99).
Hence the model is accurate. And is able to predict future events.
RETURN PERIOD (YEAR) VS. ONE DAY MAXIMUM RAINFALL (MM)
THANK YOU
GUMBEL’S DISTRIBUTION
VALUES ASSUMED
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