The document analyzes extreme rainfall events in Mumbai, India to determine maximum probable precipitation amounts for different return periods. It describes using the Hershfield technique and Gumbel's extreme value theory to analyze 35 years of rainfall data for Mumbai's Colaba region. Calculations determine the probable maximum precipitation is 604.688036 mm based on past data, and estimate maximum one-day rainfall amounts for return periods from 2 to 100 years.

1. 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)

2. INTRODUCTION On the 26 th 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.

3. 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.

4. 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.”

5. 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.

6. MAXIMUM RAINFALL OVER 35 YEAR PERIOD

7. PROBABLE MAXIMUM PRECIPITATION Hershfield Techniques Chow suggested the following formula for most of the frequency distributions applicable to hydrologic analysis X m = ( Ẋ+ K m σ) X= Value of the event. Ẋ = Mean of the annual maximum series of N years. σ = Standard deviation of the series of annual maximum. X m = Estimate of 1 Day Probable Maximum Precipitation. K m = (X L – X N-1 )/ σ n-1 X L = Largest value of annual maximum series omitting the largest value of the series. X N-1 = Mean of annual maximum series omitting the largest value of the series. σ n-1 =Standard Deviation of annual maximum series omitting the largest value of the series. K m = (544.3- 224.8676471)/ 101.6399693 = 3.142782855 X= 234.8885714 + (3.142782855 * 117.6662473) =604.688036 mm Therefore, the probable maximum precipitation for Colaba, Mumbai based on the maximum precipitation data for the past 35 years is 604.688036 mm.

8. CALCULATION OF PMP PROBABLE MAXIMUM PRECIPITATION RANK RAINFALL (x) XN-1 STANDARD DEVIATION XN-1 - XN-1(avg) (XN-1 - XN-1(avg))^2 1 575.6 2 544.3 544.3 319.4323529 102037.0281 3 477.6 477.6 252.7323529 63873.64222 4 421.2 421.2 196.3323529 38546.39281 5 417.2 417.2 192.3323529 36991.73399 6 345.5 345.5 120.6323529 14552.16458 7 279.4 279.4 54.53235294 2973.777517 8 261.9 261.9 37.03235294 1371.395164 9 249.7 249.7 24.83235294 616.6457526 10 244.2 244.2 19.33235294 373.7398702 11 243.7 243.7 18.83235294 354.6575173 12 241.6 241.6 16.73235294 279.9716349 13 233 233 8.132352941 66.13516436 14 217 217 -7.867647059 61.89987024 15 213.3 213.3 -11.56764706 133.8104585 16 206.9 206.9 -17.96764706 322.8363408 17 206.2 206.2 -18.66764706 348.4810467 18 184.9 184.9 -39.96764706 1597.412811 19 184.4 184.4 -40.46764706 1637.630458 20 183.3 183.3 -41.56764706 1727.869282 21 180.9 180.9 -43.96764706 1933.153988 22 175.9 175.9 -48.96764706 2397.830458 23 175.7 175.7 -49.16764706 2417.457517 24 174.4 174.4 -50.46764706 2546.9834 25 165.4 165.4 -59.46764706 3536.401047 26 162.8 162.8 -62.06764706 3852.392811 27 153.9 153.9 -70.96764706 5036.406929 28 150.3 150.3 -74.56764706 5560.333988 29 148.6 148.6 -76.26764706 5816.753988 30 147.7 147.7 -77.16764706 5954.845753 31 138.3 138.3 -86.56764706 7493.957517 32 138.2 138.2 -86.66764706 7511.281047 33 128.7 128.7 -96.16764706 9248.216341 34 125.9 125.9 -98.96764706 9794.595164 35 123.5 123.5 -101.3676471 10275.39987 Average 234.8885714 224.8676471 351243.2344 Variance 10330.68337 Standard Deviation 101.6399693

9. 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)/m n 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

10. RISK ANALYSIS RAIN FALL RISK ANALYSIS RANK RAINFALL (x) R.I. (T) RETURN PERIOD (1/T) RISK ANALYSIS (m/(n+1)) (1-(1/T))^n 1-((1-(1/T))^n) Percent Terms 1 575.6 36 0.027777778 0.373073177 0.626926823 62.69268231 2 544.3 18 0.055555556 0.135261615 0.864738385 86.47383847 3 477.6 12 0.083333333 0.047577363 0.952422637 95.24226368 4 421.2 9 0.111111111 0.016205473 0.983794527 98.37945269 5 417.2 7.2 0.138888889 0.00533423 0.99466577 99.46657698 6 345.5 6 0.166666667 0.001692998 0.998307002 99.83070022 7 279.4 5.142857143 0.194444444 0.000516824 0.999483176 99.94831763 8 261.9 4.5 0.222222222 0.000151336 0.999848664 99.98486635 9 249.7 4 0.25 4.23784E-05 0.999957622 99.99576216 10 244.2 3.6 0.277777778 1.13104E-05 0.99998869 99.99886896 11 243.7 3.272727273 0.305555556 2.86624E-06 0.999997134 99.99971338 12 241.6 3 0.333333333 6.86761E-07 0.999999313 99.99993132 13 233 2.769230769 0.361111111 1.54841E-07 0.999999845 99.99998452 14 217 2.571428571 0.388888889 3.26743E-08 0.999999967 99.99999673 15 213.3 2.4 0.416666667 6.41339E-09 0.999999994 99.99999936 16 206.9 2.25 0.444444444 1.16269E-09 0.999999999 99.99999988 17 206.2 2.117647059 0.472222222 1.93103E-10 1 99.99999998 18 184.9 2 0.5 2.91038E-11 1 100 19 184.4 1.894736842 0.527777778 3.93663E-12 1 100 20 183.3 1.8 0.555555556 4.71641E-13 1 100 21 180.9 1.714285714 0.583333333 4.92727E-14 1 100 22 175.9 1.636363636 0.611111111 4.40447E-15 1 100 23 175.7 1.565217391 0.638888889 3.29177E-16 1 100 24 174.4 1.5 0.666666667 1.99874E-17 1 100 25 165.4 1.44 0.694444444 9.50947E-19 1 100 26 162.8 1.384615385 0.722222222 3.38386E-20 1 100 27 153.9 1.333333333 0.75 8.47033E-22 1 100 28 150.3 1.285714286 0.777777778 1.37266E-23 1 100 29 148.6 1.24137931 0.805555556 1.28187E-25 1 100 30 147.7 1.2 0.833333333 5.8171E-28 1 100 31 138.3 1.161290323 0.861111111 9.84833E-31 1 100 32 138.2 1.125 0.888888889 3.99496E-34 1 100 33 128.7 1.090909091 0.916666667 1.693E-38 1 100 34 125.9 1.058823529 0.944444444 1.16269E-44 1 100 35 123.5 1.028571429 0.972222222 3.38386E-55 1 100

11. 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. X T = ( Ẋ + σK T ) K T = - (1.00+1.75(log (log (T/ (T-1))))) Ẋ = Mean of the annual maximum series of N years. T= Return Period X T = Maximum rainfall corresponding to Time period.

12. Estimation Of Maximum Rainfall (Model 1)- Maximum Of 35 Years RANK RAINFALL (x) STANDARD DEVIATION RETURN PERIOD FREQUENCY FACTOR MAX.RAINFALL 1 575.6 340.7114286 116084.2776 2 -0.087567102 224.5848792 2 417.2 182.3114286 33237.45699 5 0.773854859 325.9451687 3 123.5 -111.3885714 12407.41384 10 1.344191152 393.0544999 4 184.4 -50.48857143 2549.095845 15 1.665970069 430.9170175 5 175.7 -59.18857143 3503.286988 20 1.891271417 457.4273817 6 206.2 -28.68857143 823.0341306 25 2.06481257 477.8473179 7 125.9 -108.9885714 11878.5087 30 2.205993197 494.4595124 8 241.6 6.711428571 45.04327347 35 2.325007656 508.4634971 9 180.9 -53.98857143 2914.765845 40 2.427881777 520.568309 10 174.4 -60.48857143 3658.867273 45 2.51847549 531.2281311 11 544.3 309.4114286 95735.43213 50 2.599410704 540.7514741 12 345.5 110.6114286 12234.88813 55 2.67255001 549.3575018 13 128.7 -106.1885714 11276.0127 60 2.739264255 557.2075166 14 153.9 -80.98857143 6559.148702 65 2.800591732 564.4236906 15 138.2 -96.68857143 9348.679845 70 2.857337751 571.1007818 16 183.3 -51.58857143 2661.380702 75 2.910139521 577.3137679 17 421.2 186.3114286 34711.94842 80 2.959509909 583.1229962 18 477.6 242.7114286 58908.83756 85 3.005867826 588.5777583 19 175.9 -58.98857143 3479.651559 90 3.04955986 593.718836 20 206.9 -27.98857143 783.3601306 100 3.130061449 603.1911559 21 148.6 -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

13. Estimation Of Maximum Rainfall (Model 2)- Month Of June For 35 Years RANK RAINFALL (x) STANDARD DEVIATION RETURN PERIOD FREQUENCY FACTOR MAX. RAINFALL 1 36 -113.6542857 12917.29666 2 -0.087567102 140.7934091 2 69.8 -79.85428571 6376.706947 5 0.773854859 227.9603192 3 97.5 -52.15428571 2720.069518 10 1.344191152 285.6723983 4 91.1 -58.55428571 3428.604376 15 1.665970069 318.2330657 5 175.7 26.04571429 678.3792327 20 1.891271417 341.0312102 6 90.2 -59.45428571 3534.81209 25 2.06481257 358.5917633 7 86 -63.65428571 4051.86809 30 2.205993197 372.8777689 8 134.8 -14.85428571 220.6498041 35 2.325007656 384.9207898 9 180.9 31.24571429 976.2946612 40 2.427881777 395.330577 10 174.4 24.74571429 612.3503755 45 2.51847549 404.497715 11 133.3 -16.35428571 267.4626612 50 2.599410704 412.6875138 12 345.5 195.8457143 38355.5438 55 2.67255001 420.0884481 13 110.9 -38.75428571 1501.894661 60 2.739264255 426.8392332 14 78.2 -71.45428571 5105.714947 65 2.800591732 433.0449337 15 102.2 -47.45428571 2251.909233 70 2.857337751 438.7870384 16 111.9 -37.75428571 1425.38609 75 2.910139521 444.1300264 17 421.2 271.5457143 73737.07495 80 2.959509909 449.1257943 18 477.6 327.9457143 107548.3915 85 3.005867826 453.8167315 19 47 -102.6542857 10537.90238 90 3.04955986 458.2379093 20 118.4 -31.25428571 976.8303755 100 3.130061449 466.3838297 21 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

14. ESTIMATION OF MAXIMUM RAINFALL (MODEL 2)- MONTH OF JUNE FOR 35 YEARS

15. Estimation Of Maximum Rainfall (Model 3)- Month Of July For 35 Years RANK RAINFALL (x) STANDARD DEVIATION RETURN PERIOD FREQUENCY FACTOR MAX. RAINFALL 1 575.6 412.0628571 169795.7982 2 -0.087567102 152.8630654 2 417.2 253.6628571 64344.84509 5 0.773854859 257.8669378 3 123.5 -40.03714286 1602.972808 10 1.344191152 327.388638 4 184.4 20.86285714 435.2588082 15 1.665970069 366.6121938 5 83.2 -80.33714286 6454.056522 20 1.891271417 394.075522 6 119.7 -43.83714286 1921.695094 25 2.06481257 415.2294906 7 125.9 -37.63714286 1416.554522 30 2.205993197 432.4388411 8 98.3 -65.23714286 4255.884808 35 2.325007656 446.946225 9 135 -28.53714286 814.3685224 40 2.427881777 459.4861666 10 130.4 -33.13714286 1098.070237 45 2.51847549 470.5291758 11 544.3 380.7628571 144980.3534 50 2.599410704 480.3948529 12 125.8 -37.73714286 1424.091951 55 2.67255001 489.3102402 13 24.4 -139.1371429 19359.14452 60 2.739264255 497.4424383 14 153.9 -9.637142857 92.87452245 65 2.800591732 504.9180111 15 100.5 -63.03714286 3973.68138 70 2.857337751 511.8351226 16 183.3 19.76285714 390.5705224 75 2.910139521 518.271446 17 77.8 -85.73714286 7350.857665 80 2.959509909 524.2894978 18 181.1 17.56285714 308.453951 85 3.005867826 529.9403413 19 175.9 12.36285714 152.8402367 90 3.04955986 535.2662246 20 161.1 -2.437142857 5.939665306 100 3.130061449 545.0790445 21 148.6 -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

16. ESTIMATION OF MAXIMUM RAINFALL (MODEL 2)- MONTH OF JULY FOR 35 YEARS

17. Estimation Of Maximum Rainfall (Model 4)- Month Of August For 35 Years RANK RAINFALL (x) STANDARD DEVIATION RETURN PERIOD FREQUENCY FACTOR MAX. RAINFALL 1 184.3 68.14571429 4643.838376 2 -0.087567102 110.6263596 2 107.8 -8.354285714 69.7940898 5 0.773854859 165.0061083 3 87 -29.15428571 849.9723755 10 1.344191152 201.0102338 4 72.4 -43.75428571 1914.437518 15 1.665970069 221.3234571 5 35.7 -80.45428571 6472.89209 20 1.891271417 235.5462553 6 206.2 90.04571429 8108.230661 25 2.06481257 246.5015417 7 110.3 -5.854285714 34.27266122 30 2.205993197 255.4139764 8 48.7 -67.45428571 4550.080661 35 2.325007656 262.9271078 9 88.3 -27.85428571 775.8612327 40 2.427881777 269.4213338 10 173.4 57.24571429 3277.071804 45 2.51847549 275.1403235 11 21.5 -94.65428571 8959.433804 50 2.599410704 280.2495926 12 79.7 -36.45428571 1328.914947 55 2.67255001 284.8667226 13 128.7 12.54571429 157.3949469 60 2.739264255 289.0782519 14 100.3 -15.85428571 251.3583755 65 2.800591732 292.949726 15 86.6 -29.55428571 873.4558041 70 2.857337751 296.5319823 16 71 -45.15428571 2038.909518 75 2.910139521 299.8652465 17 259 142.8457143 20404.89809 80 2.959509909 302.9818948 18 33.7 -82.45428571 6798.709233 85 3.005867826 305.9083722 19 174.9 58.74571429 3451.058947 90 3.04955986 308.666558 20 60.9 -55.25428571 3053.03609 100 3.130061449 313.7484533 21 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

18. ESTIMATION OF MAXIMUM RAINFALL (MODEL 2)- MONTH OF AUG. FOR 35 YEARS

19. Estimation Of Maximum Rainfall (Model 5)- Month Of September For 35 Years RANK RAINFALL (x) STANDARD DEVIATION RETURN PERIOD FREQUENCY FACTOR MAX. RAINFALL 1 27.5 -66.54857143 4428.712359 2 -0.087567102 88.4566456 2 175.3 81.25142857 6601.794645 5 0.773854859 143.465977 3 88.8 -5.248571429 27.54750204 10 1.344191152 179.886941 4 181.9 87.85142857 7717.873502 15 1.665970069 200.4353411 5 78 -16.04857143 257.5566449 20 1.891271417 214.822804 6 65.1 -28.94857143 838.0197878 25 2.06481257 225.9049255 7 23 -71.04857143 5047.899502 30 2.205993197 234.9205441 8 241.6 147.5514286 21771.42407 35 2.325007656 242.5206589 9 76.8 -17.24857143 297.5132163 40 2.427881777 249.0900719 10 115.1 21.05142857 443.1626449 45 2.51847549 254.8752734 11 148.4 54.35142857 2954.077788 50 2.599410704 260.0436952 12 29.5 -64.54857143 4166.518073 55 2.67255001 264.71428 13 30.7 -63.34857143 4013.041502 60 2.739264255 268.9745684 14 6.3 -87.74857143 7699.811788 65 2.800591732 272.8908645 15 138.2 44.15142857 1949.348645 70 2.857337751 276.5145945 16 37.2 -56.84857143 3231.760073 75 2.910139521 279.8864497 17 97.8 3.751428571 14.07321633 80 2.959509909 283.039181 18 21.9 -72.14857143 5205.416359 85 3.005867826 285.9995397 19 97.5 3.451428571 11.91235918 90 3.04955986 288.7896585 20 206.9 112.8514286 12735.44493 100 3.130061449 293.9303895 21 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

20. ESTIMATION OF MAXIMUM RAINFALL (MODEL 2)- MONTH OF SEPT. FOR 35 YEARS

21. 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 (R 2 =0.99). Hence the model is accurate. And is able to predict future events.

22. RETURN PERIOD (YEAR) VS. ONE DAY MAXIMUM RAINFALL (MM)

23. THANK YOU

24. GUMBEL’S DISTRIBUTION

25. VALUES ASSUMED