METHODOLOGY 5

METHODOLOGY 5.1 GENERAL The analyses of flood frequency are carried out by different methodology. Different statistical methods and with the help of software HEC-SSP the analyses has been done to find out maximum probable flood for different return period. 5.2 METHODS OF FLOOD FREQUENCY ANALYSIS The Various methods of finding flood of different frequencies can be classified in two main types i.e., I) Empirical Methods, II) Statistical Methods and Statistical Software Package. 5.2.1 Empirical Methods Empirical Methods are listed below, 1) Fullers Formula 2) Creagers Equation 3) Correlation with Mean Flow – R. S. Varshneys Relation 4) Percentage Increase Method of Design Flood. The empirical formulae to determine peak flow should be employed only when there is insufficient available hydrologic information for performing the detailed and precise analysis. The main in drawback of the method is due to the uncertainty involved in the subjective decision of the hydrologist on the factor of safety to be adopted. Moreover, no indication is available regarding the probable frequency of the maximum recorded floods or maximum historical flood on the stream under consideration. The empirical methods will not be discussed here, as these methods involve various constants which are difficult to predict. The results of these methods are also quite approximate. 5.2.2 Statistical Methods If the entire flood peaks of all the years of records of a basin are arranged in the order of their magnitude, they are found to constitute a statistical arrangement, which may be arranged in a frequency curve. Most of the distributions encountered in hydrology are pronounced skew. The flood frequency curves are also skew and therefore an additional parameter is involved in their calculations. Various techniques have been evolved to develop graphical or mathematical means of solving this difficulty of skewness and to fit the observed data to smooth frequency curves. The means of deriving the frequency curves are explained without going into details of theoretical consideration involved. Statistical Methods are listed below, Return Period About 100 Years 1) Normal Frequency Evaluation Graphical Method 2) California Methods B) Large Return Periods 3) Fosters Method Type-I III 4) Hazens Method 5) Gumbels Method (Using Frequency Factors) 6) Gumbles Method (Statistical Approach) 7) Powels Method 8) Ven Te Chows Method 9) Log-Normal Method 10) Log Pearson Type-III Method 5.3 NORMAL FREQUENCY EVALUATION GRAPHICAL METHOD (RETURN PERIOD LESS SAY ABOUT 100 YEARS) This is the simplest and easiest method of drawing frequency curve. According to law of probabilities, the possible percentage of future floods that will equal or exceed a given flood Q may be obtained by the following relation. QUOTE (5.3.1) Where, p Calculated possibilities of future flood i.e. in 100 year how many times this magnitude of flood will be exceeded. m the number of years during the period of record that a flood was equaled or exceeded. n total no of years of record. After calculating p, QUOTE (5.3.2) Where p Calculated possibilities of future flood i.e. in 100 year how many times this magnitude of flood will be exceeded. m no of years the discharge Q was exceeded. I Future flood frequency i.e. after how many years Q will be exceeded. An estimate of the probable frequency of flood flows of the Sabarmati River at Dharoi Dam has been made in Table 5.1 and frequency curve is shown in Fig 5.1. According to Normal Frequency Evaluation Graphical Method the lowest magnitude of flood 118 is came in every year and calculated possibilities of it is 100 and the highest magnitude of flood 13186 is came once in 21 year and calculated percentage possibilities of it is 3.45. Table 5.1 Calculations for Probability Plotting of Sabarmati River by Normal Frequency Evaluation Graphical Method from 29 year data at Dharoi Dam (1978 to 2006). Normal Frequency Evaluation Graphical MethodSr. No.Peak Flow in cumecNo of OccurrencesSummation of Occurrencesp100m/nI100/p1118129100.001.00224412896.551.04334712793.101.07439312689.661.12561112586.211.16665912482.761.21768412379.311.26870912275.861.32973212172.411.381096212068.971.4511103911965.521.5312104511862.071.6113117911758.621.7114140011655.171.8115151011551.721.9316158511448.282.0717169911344.832.2318204511241.382.4219394111137.932.6420518811034.482.902158911931.033.222259461827.593.632361611724.144.142470481620.694.832573311517.245.802678341413.797.252778741310.349.67287881126.9014.502913186113.4529.00 Fig. 5.1 Frequency curve for Normal Frequency Evaluation Graphical Method. 5.4 CALIFORNIA METHODS 5.4.1 Modified California Method The flood frequency f in100 years is based upon the following equation. QUOTE (5.4.1) 5.4.2 Modern California Method Here the equation for the flood frequency is given by the equation. QUOTE (5.4.2) Where m rank of the flood n total no of years of record f in 100 years no. of times the flood is likely to exceed y. According to California Method the value of f indicates that in 100 years no of times the flood is likely to exceed. The lowest magnitude of flood 118 is come in 98.2759 in 100 years by Modified California Method and by Modern California Method it is 96.6667. The highest magnitude of flood 13186 is come in 1.7241 in 100 years by Modified California Method and by Modern California Method it is 3.3333. The probable flood frequency has been estimated for the Sabarmati River. The calculations are shown in Table 5.2 and frequency curve is shown in Fig 5.2 and 5.3. Fig 5.2 Frequency Curve for Modified California Method Fig. 5.3 Frequency Curve for Modern California Method 5.5. FOSTERS METHOD So long as the problem under consideration deals with return periods considerably less than the period of record, one may obtain a reasonable trust worthy estimate of flood frequency by interpolation. When floods approaching or transcending the maximum of record must be considered, it becomes extremely important to know the exact nature of statistical distribution with which we are dealing. H. A. Foster (1924) suggested the case of Pearsons skew function for fitting observed flood data. Pearson adopted the general differential equation, QUOTE (5.5.1) Where x is the deviation of the variable X, J the frequency corresponding to x, a is a constant f(x) which is a function of x. Standard Deviation QUOTE (5.5.2) And the coefficient of skew, QUOTE (5.5.3) The coefficient of skew must be adjusted for the size of sample by, QUOTE (5.5.4) And QUOTE (5.5.5) Equations 5.5.4 and 5.5.5 are used for Foster Type III ( Hazen) and Foster Type I respectively. Factor K corresponding to various percentage frequencies and the computed coefficient of skew can be selected from Table 5.3 and 5.4 for foster type III and type I respectively. These factors K multiplies by the coefficient of variation and added to the mean of X define the frequency curve. Table 5.3 Skew curve Factors for Foster Type III Curve CsFrequency Percent9995805020510.10.010.0010.00010.0-2.33-1.64-0.840.000.841.642.333.093.734.274.760.2-2.18-1.58-0.85-0.030.831.692.483.084.164.845.480.4-2.03-1.51-0.85-0.060.821.742.623.674.605.426.240.6-1.88-1.45-0.86-0.090.801.792.773.965.046.017.020.8-1.74-1.38-0.86-0.130.781.832.904.255.486.617.821.0-1.59-1.31-0.86-0.160.761.873.034.545.927.228.631.2-1.45-1.25-0.85-0.190.741.903.154.826.377.859.451.4-1.32-1.18-0.84-0.220.711.933.285.116.828.5010.281.6-1.19-1.11-0.82-0.250.681.963.405.397.289.1711.121.8-1.08-1.03-0.80-0.280.641.983.505.667.759.8411.962.0-0.99-0.95-0.78-0.310.612.003.605.918.2110.5112.812.2-0.90-0.89-0.75-0.330.582.013.706.20—2.4-0.83-0.82-0.71-0.350.542.013.786.47—2.6-0.77-0.76-0.68-0.370.512.013.876.73—2.8-0.71-0.71-0.65-0.380.472.023.956.99—3.0-0.67-0.66-0.62-0.400.422.024.027.25— Table 5.4 Skew curve Factors for Foster Type I Curve CsFrequency Percent9995805020510.10.010.0010.00010.0-2.08-1.64-0.920.000.921.642.082.392.532.592.620.2-1.91-1.56-0.93-0.050.891.722.252.662.832.943.000.4-1.75-1.47-0.93-0.090.871.792.422.953.183.553.440.6-1.59-1.38-0.92-0.130.851.852.853.243.593.803.920.8-1.44-1.30-0.91-0.170.831.902.753.554.004.274.431.0-1.30-1.21-0.89-0.210.801.952.923.854.424.754.431.2-1.17-1.12-0.86-0.250.771.993.094.154.835.255.501.4-1.06-1.03-0.83-0.290.732.033.254.455.255.756.051.6-0.96-0.95-0.80-0.320.692.073.404.755.255.756.051.8-0.87-0.87-0.76-0.350.642.103.545.056.086.757.202.0-0.80-0.79-0.71-0.370.582.133.675.356.507.257.80 The calculation for the mean, standard deviation and the coefficient of skew are shown Table 5.5. Table 5.5 Frequency analysis computations for Sabarmati River by Fosters Method Type I and III Standard Deviation QUOTE And the coefficient of skew QUOTE Adjusted coefficient of skew For Foster Type III ( Hazen), Factors corresponding to various percentage frequencies and the computed coefficient of skew 1.084432 are taken from Table 5.4. These factors multiplied by the standard deviation and added to the mean value define the frequency curve. Similarly, The adjusted coefficient of skew for foster type I curve is given by, For Foster Type I, And the factor K is taken from Table 5.4 Table 5.7 Calculation for the flood frequencies Fosters Type I curve Fig. 5.4 Frequency Curve for Fosters Type III Method Fig. 5.5 Frequency Curve for Fosters Type I Method The calculation for the mean standard deviation and the coefficient of skew are the same as in the Fosters method (Type III). The calculation for the flood frequencies by Hazens method are shown in Table 5.9.Fig 5.6 shows the curve fitting for the expected flood peaks for the Hazens Method. Table 5.8 Skew Curve Factors According to Hazens Method the maximum flood of magnitude 13186 cumec recorded which was occurred in 1993 may occur once in 69.61 years. Table 5.9 Calculation for flood frequencies Hazens Method Fig. 5.6 Frequency Curve for Hazens Method 5.7 GUMBLES METHOD (STATISTICAL APPROACH) The peak discharge (x) is arranged in the ascending order of magnitude. The next column indicates the probability Fm according to Gumble. The values of F1 the probability of the lowest flood is given by Where n is the number of years of data available. QUOTE (5.7.1) The value of QUOTE , the probability of the highest flood is given by QUOTE (5.7.2) The probabilities of the intermediate floods such as F2, F3, and F4.F30 are worked out from the relation QUOTE (5.7.3) The reduced variate y corresponding to any value of probability is given by QUOTE (5.7.4) Then the straight-line equation between the probability of any flood of magnitude x and the corresponding reduced variate y is given as QUOTE (5.7.5) The calculation for determining the various parameters in the Gumbles statistical approach are given in Table5.10. Table 5.10 Calculation for Gumbles Method (Statistical Approach)Sr. NoYear Flood X(cumec)FmReduced Variate YXYY2X2(X-Y)(X-Y)2119991180.0345-1.2141-143.26501.474113924119.2141114212.004220042440.0678-0.9902-241.61670.980659536244.9902360020.214319793470.1010-0.8296-287.86630.6882120409347.82959120985.42419873930.1343-0.6970-273.91450.4858154449393.69698154997.31519826110.1676-0.5802-354.48610.3366373321611.58017374030.31619806590.2008-0.4733-311.88380.2240434281659.47327434904.99720036840.2341-0.3729-255.06530.1391467856684.3729468366.27819787090.2674-0.2769-196.33470.0767502681709.27692503073.75919987320.3007-0.1838-134.54330.0338535824732.1838536093.121019959620.3339-0.0924-88.90370.0085925444962.09242925621.8211198110390.3672-0.0018-1.90990.000010795211039.00181079524.812198610450.40050.088792.71060.007910920251044.91131091839.613199611790.43370.1800212.18760.032413900411178.821389616.714200014000.46700.2726381.66330.074319600001399.72741959236.715200215100.50030.3674554.70420.134922801001509.63262278990.716198515850.53360.4649736.91840.216225122251584.53512510751.417198916990.56680.5662961.95080.320628866011698.43382884677.418200120450.60010.67211374.39910.451741820252044.32794179276.719199039410.63340.78383088.80010.6143155314813940.21621552530420199251880.66670.90274683.02460.8148269153445187.09732690597921199458910.69991.03066071.40981.0622347038815889.96943469173922198859460.73321.17016957.28551.3691353549165944.82993534100323200561610.76651.32448159.62401.7540379579216159.67563794160424198370480.79971.498510561.30892.2455496743047046.50154965318425199173310.83301.699912461.74022.8896537435617329.30015371864026198478340.86631.941115206.75913.7680613715567832.05896134114627199778740.89962.245717682.59335.0431619998767871.75436196451628200678810.93282.665921010.21277.1072621101617878.334162068148291993131860.96613.367144399.185311.337717387059613182.633173781809Total9524215.5294152306.688443.690663420386095226.471633899290Average32840.53555251.9548Where c and m are statistical constant to be derived from data x and y available, as under, QUOTE QUOTE Hence QUOTE Where, m indicates the Slope of the line, y indicates the reduced variate corresponding to Fn, c indicates the intercept of the straight line, X indicates the expected peak discharge, X and Y indicate average values. The values of the expected peak discharge X are computed in Table 5.11 and Fig 5.7 5.8 shows the curve fitting for the expected flood peaks for the Gumbles Method (Statistical Approach). Table 5.11 Calculation for flood peaks by Gumbles Statistical Approach. Gumbles Method (Statistical Approach)Return Period (year)Probability FnReduced VariateExpected peakProbability in 50.81871.60936359.192520.0000100.90482.30228343.562010.0000150.93552.70799505.58526.7000200.95122.995110327.99075.0000300.96723.400711489.49193.3000500.98023.912112954.00392.00001000.994.600114924.44701.00002000.9955.295816916.66730.50005000.9986.213619545.02370.2000 According to Gumbles Method (statistical approach) the maximum flood of magnitude 13186 cumec recorded which was occurred in 1993 may occur once in 55.88 years. Fig. 5.7 Calculated Flood for Gumbles Method(Statistical Approach) Fig. 5.8 Flood Frequency Curve for Gumbles Statistical Method s standard deviation of recorded floods K frequency factor Table 5.12 Frequency factors for Gumbles method Years of RecordReturn Period – Years51025501001000150.9671.7032.6323.3214.0056.465200.9191.6252.5173.1793.8366.006250.8881.5752.4443.0263.7295.842300.8661.5412.3933.0263.653350.8511.5162.3542.9793.5985.576400.8381.4952.3262.9433.5545.576450.8291.4782.3032.9133.52500.821.4662.2332.8893.4915.478550.8121.4662.2672.8693.467600.8071.4462.2532.8523.446650.8011.4372.2412.8373.429700.7971.432.232.8243.4135.359750.7921.4232.222.8123.4800.7881.4172.2122.8023.387850.7851.4132.2052.7933.376900.7821.4092.1982.7853.367950.781.4052.1932.7773.3571000.7791.4012.1872.773.3495.261 Thus using these values and the values of K from Table 5.12 in equation 5.8.1 the values of flood of several return periods are calculated in Table 5.14 andFig 5.9 shows the curve fitting for the expected flood peaks for Gumbles Method (Frequency Factor). Table 5.14 Calculation for Flood Frequencies by Gumbles Method. (Using Frequency Factors) According to Gumbles method (Using Frequency Factors) the maximum flood of magnitude 13186 cumec recorded which was occurred in 1993 may occur once in 45.85 years. Fig. 5.9 Flood Frequency Curve for Gumbles Frequency Factor Method 5.9 POWELS METHOD A modification in the value of k was made by R. W. Powel (1943) in formula. QUOTE (5.9.1) Where QUOTE Where from The mean and the standard deviation are calculated as the same in previous methods. QUOTE Now values of the factor K is calculated for the various values of Return period T and then substituted in the equation 5.9.3 for finding out the corresponding expected flood peaks. These calculations are shown in Table 5.15 andFig 5.10 shows the curve fitting for the expected flood peaks for Powels Method. Table 5.15 Calculation of flood frequencies by Powels Method Fig. 5.10 Flood Frequency Curve for Powels Method According to Powels Method the maximum flood of magnitude 13186 cumec recorded which was occurred in 1993 may occur once in 80.33 years. 5.10 VEN TE CHOWS METHOD Another modification in Gumbles method was introduced by Ven Te Chow, in which use was made of frequency factor. The equation was given as below QUOTE (5.10.1) Where, QUOTE (5.10.2) Or QUOTE (5.10.3) a and b are the parameters estimated by the method of moments from observed data. Table 5.16 Calculation for determining a and b coefficient for Ven Te Chows Method Ven Te Chows MethodRank (k)Annual Peak flow X cumecreturn period XTYXTXT2T (n 1)/m11318630.0000-1.8320-24156.70073.35622788115.0000-1.5234-12006.00582.32083787410.0000-1.3395-10547.52061.7944478347.5000-1.2066-9452.29681.4558573316.0000-1.1014-8074.19971.2130670485.0000-1.0136-7144.07371.0274761614.2857-0.9378-5777.90260.8795859463.7500-0.8706-5176.80760.7580958913.3333-0.8099-4771.37470.65601051883.0000-0.7543-3913.11230.56891139412.7273-0.7025-2768.66740.49351220452.5000-0.6539-1337.31350.42761316992.3077-0.6079-1032.78740.36951415852.1429-0.5638-893.67900.31791515102.0000-0.5214-787.29920.27181614001.8750-0.4802-672.25330.23061711791.7647-0.4399-518.61920.19351810451.6667-0.4002-418.19060.16011910391.5789-0.3608-374.85450.1302209621.5000-0.3214-309.15910.1033217321.4286-0.2816-206.13050.0793227091.3636-0.2411-170.91470.0581236841.3043-0.1993-136.29860.0397246591.2500-0.1555-102.50180.0242256111.2000-0.1089-66.55990.0119263931.1538-0.0580-22.77890.0034273471.11110.00000.00000.0000282441.07140.070417.18760.0050291181.03450.169419.99110.0287Sum95242-17.2457-100800.823416.9784 In this method a plotting position T has been assigned for each value of Y arranger in descending order of magnitude. For example if rank of Y is m, its plotting position i.e. its return period is QUOTE where n is the total number of years of observation. With the use of Equation 5.10.2 or 5.10.3 we can calculate QUOTE for the Corresponding value of T as explained in Table 5.16 From the calculation performed in Table 5.16 the value of, Now the values of (a) and (b) can be deduced from the two equation given below, which is derived on the basis of least square method. 95242 QUOTE Making coefficient (a) equal we get And QUOTE Now using these values in equation 5.10.1 we can calculate the probable floods YT for various return periods T, as shown in Table 5.17 andFig 5.11 shows the curve fitting for the expected flood peaks for Ven Te Chows Method. Table 5.17 Calculation for flood frequencies by Ven Te Chows Method Ven Te Chows methodReturn PeriodXTEstimated Flow X in Cumec1.25-0.1555415758.49774372-0.52139023003.4053715-1.01363136023.87707310-1.33953788023.69215920-1.65215519941.96280250-2.056806112424.96738100-2.360035114285.62991500-3.06075118585.336951000-3.361998520433.84016 Fig. 5.11 Flood Frequency Curve for Ven Te Chows Method According to Ven Te Chows Method the maximum flood of magnitude13186 cumec recorded which was occurred in 1993 may occur once in 70.45 years. 5.11 THE LOG NORMAL METHOD This is based on the log normal probability law and assumes that the flood values are such that their natural logarithms are normally distributed. Carrying out the following steps derives the frequency curve. Mean and Standard deviation. Mean QUOTE cumecs Standard Deviation QUOTE The values of QUOTE is found out to be 321409428.61 which is calculated in Table 10.1 The coefficient of variation is given by, Adjust the coefficient of variation Compute the value of K.s for each value of probability. And then find X for the corresponding recurrence interval. Table 5.18 calculation of mean and standard deviation. Log Normal MethodSr. No.YearFlood in cumecDeviation from meanX2119931318699029804960422006788145972113240931997787445902106810041984783445502070250051991733140471637820961983704837641416769672005616128778277129819885946266270862449199458912607679644910199251881904362521611199039416574316491220012045-123915351211319891699-158525122251419851585-169928866011520021510-177431470761620001400-188435494561719961179-210544310251819861045-223950131211919811039-22455040025201995962-23225391684211998732-25526512704221978709-25756630625232003684-26006760000241980659-26256890625251982611-26737144929261987393-28918357881271979347-29378625969282004244-30409241600291999118-316610023556Sum95242321409428Average3284.207 Now from Table 5.19 select the factor K corresponding to the computed value of Cs and selected probability values. Interpolation may be necessary. Table 5.19 Chow frequency Factors Skew Coeff CsProb. at meanProbability in equal to or greater than the given variateCorresponding Coeff. of Variation99958015020510.010.001—-0502.331.650.8400.841.642.333.093.7200.149.32.251.620.850.020.841.672.43.223.950.0330.248.72.181.590.850.040.831.72.473.394.180.0670.3482.111.560.850.060.821.722.553.564.420.10.447.32.041.530.850.070.811.752.620.724.70.1360.546.71.981.490.850.090.81.772.73.884.960.1660.646.11.911.460.850.10.791.792.774.055.240.1970.745.51.851.430.850.110.781.812.844.215.240.230.844.91.791.40.840.130.771.822.94.375.810.2620.944.21.741.370.840.140.761.842.974.556.110.292143.71.681.340.840.150.751.853.034.726.40.3241.143.21.631.310.830.160.731.863.094.876.710.6511.242.71.581.290.820.170.721.873.155.047.020.3811.342.21.541.260.820.180.711.883.215.197.310.4091.441.71.491.230.810.190.691.883.265.357.620.4361.541.31.451.210.810.20.681.893.315.517.920.4621.640.81.411.180.80.210.671.893.365.668.260.491.740.41.341.140.780.220.651.893.45.88.580.5171.8401.341.140.780.220.641.893.445.968.880.5441.939.61.311.120.780.230.631.893.486.19.20.57239.21.281.10.770.240.611.893.526.259.510.5962.138.81.251.080.760.240.61.893.556.399.790.622.238.41.221.060.760.250.591.893.596.5110.120.6432.338.11.21.040.50.250.581.893.596.6510.430.6672.437.71.171.020.740.260.571.883.656.7710.720.6912.537.41.1510.740.260.561.883.676.910.950.7132.637.110120.990.730.260.551.873.77.0211.250.7342.736.81.10.970.720.270.541.873.727.1311.550.7552.836.81.080.960.720.270.531.863.747.2511.80.7762.936.31.060.950.710.270.521.863.767.3612.10.7963361.040.930.710.280.511.53.787.4712.360.8183.235.51.010.90.690.080.491.843.87.6512.850.8573.435.10.980.880.680.290.471.833.847.8413.360.9663.634.70.950.860.670.290.461.813.87813.830.933.834.20.920.840.660.290.441.83.898.1614.230.966433.90.90.820.650.290.421.783.918.314.714.5330.840.780.630.30.391.753.938.615.621.081532.30.80.740.620.30.371.713.958.8616.451.155 Compute the value of K.s for each value of probability (i.e. corresponding return period). And then find X (expected flood) x K.s for the corresponding recurrence interval. The required calculation for flood frequency are shown in Table 5.20 andFig 5.12 shows the curve fitting for the expected flood peaks for Log Normal Method. Table 5.20 Calculations for Flood Frequencies by Log Normal Method Log Normal MethodProbabilityX meanStd.dev.K from TableKsX x Ks in CumecsRec. interval9932843388.05-0.8772-2972.0010311.99903121.019532843388.05-0.8048-2726.7059557.29414081.058032843388.05-0.6424-2176.48591107.514111.255032843388.05-0.2938-995.41032288.58973522032843388.050.40681378.26044662.2603675532843388.051.76865992.11239276.11230420132843388.053.197610833.641514117.641471000.132843388.058.41428507.086431791.0863610000.0132843388.0515.049650988.857554272.8574810000 Fig. 5.12 Flood Frequency Curve for Log Normal MethodAccording to Log Normal Method the maximum flood of magnitude 13186 cumec recorded which was occurred in 1993 may occur once in 84.60 years. 5.12 THE LOG PEARSON TYPE III DISTRIBUTION In this method expected flood magnitude for the desired recurrence interval is given by, QUOTE (5.12.1) Where, Q is the expected flood magnitudes QUOTE is the mean of the logarithmic values of the annual flood K is the frequency factor taken from Table 5.22 a or 5.22 b corresponding to the skew coefficient Cs of the logarithmic values of flood and the corresponding years of return period and Sy is the standard deviation of the logarithms. The necessary calculation for finding out these parameters are shown in Table 5.21 from which we get, QUOTE , QUOTE , QUOTE , Hence, mean of logarithms, Table 5.21 Computation of logarithmic Mean, Standard deviation and skewness Coefficient of the data. Standard Deviation, Now the unbiased Skewness Coefficient is given by, Now using the value of the unbiased skew coefficient Cs, the values of the factor K are found out from the Table 5.22a or 5.22b by interpolating, corresponding to the various recurrence interval. Table 5.22 a K Values for Log Pearson Type-III Distribution Skew Coeff CsRecurrence Interval in Years1.01011.05261.11111.2525102550100200Percent Chance999590805020104210.5Positive Skew3-0.667-0.665-0.66-0.636-0.960.421.182.2783.1524.0514.972.9-0.69-0.668-0.681-0.651-0.390.441.1952.2773.1344.0134.9092.8-0.714-0.711-0.702-0.666-0.3840.461.212.2753.1143.9734.8472.7-0.74-0.736-0.724-0.6810.3760.4791.2242.2723.0933.9324.7832.6-0.769-0.762-0.747-0.696-0.3680.4991.2382.2673.0713.8894.7182.5-0.796-0.79-0.771-0.711-0.360.5181.252.2623.0483.8454.6522.4-0.832-0.819-0.795-0.725-0.3510.5371.2622.2563.0233.84.5842.3-0.867-0.85-0.819-0.739-0.3410.5551.2742.2482.9973.7534.5152.2-0.905-0.882-0.844-0.752-0.330.5741.2842.242.973.7054.4442.1-0.946-0.914-0.869-0.3765-0.3190.5921.2942.232.9423.6564.3722-0.99-0.949-0.92-0.788-0.3070.6271.3022.2192.9123.6054.391.9-1.037-0.984-0.92-0.788-0.2940.6271.312.2072.8813.5534.2231.8-1.087-1.02-0.97-0.808-2820.6431.3182.1932.8483.4994.1471.7-1.14-1.056-0.97-0.808-0.2680.661.3242.1792.8153.4444.0691.6-1.197-1.093-0.994-0.817-0.2540.6751.3292.1632.783.3883.991.5-1.256-1.131-1.018-0.825-0.240.691.3332.1462.7433.333.911.4-1.318-1.169-1.168-0.832-0.2250.7051.3372.1282.7063.2713.8281.3-1.383-1.206-1.064-0.838-0.210.7191.3392.1082.6663.2113.7451.2-1.449-1.243-1.086-0.844-0.1950.7321.342.0872.6263.1493.6611.1-1.518-1.28-1.107-0.848-0.180.7451.3412.0662.5853.0873.5751-1.588-1.317-1.128-0.852-0.1640.7581.342.0432.5423.0223.4890.9-1.66-1.353-1.147-0.854-0.1480.7691.3392.0182.4982.9573.4010.8-1.733-1.388-1.166-0.856-0.1320.781.3361.9932.4532.8913.3120.7-1.806-1.423-1.183-0.857-0.0990.791.3331.9672.4072.8243.2230.6-1.88-1.458-1.2-0.857-0.0990.81.3281.9392.3592.7553.1320.5-1.955-1.491-1.216-0.856-0.0830.8081.3231.912.3112.6863.0410.4-2.029-1.523-1.231-0.855-0.0660.8161.3171.882.2612.6152.9490.3-2.104-1.155-1.245-0.853-0.050.8241.3091.8492.2112.5442.8560.2-1.218-1.586-1.258-0.85-0.0330.831.3011.8182.1592.4722.7630.1-2.252-1.616-1.27-0.846-0.0170.8361.291.7852.1072.42.670-2.362-1.645-1.282-0.84200.8421.2821.7512.0542.3262.576 Table 5.22 b K Values for Pearson Type III Distribution Negative Skew-0.1-2.1-1.673-2.58-0.836-0.0170.8461.271.71622.5252.482-0.2-2.472-1.7-1.301-0.830.0330.851.2581.681.9452.1782.388-0.3-2.544-1.726-1.309-0.8240.050.8531.2451.6431.892.1042.294-0.4-2.615-1.75-1.317-0.8160.0660.8551.2311.6061.8342.0292.201-0.5-2.686-1.77-1.323-0.8080.0830.8561.2161.5671.7171.9552.108-0.6-2.755-1.797-1.328-0.80.0990.8571.21.5281.721.882.016-0.7-2.824-1.819-1.333-0.790.1160.8571.1831.4881.6631.8061.926-0.8-2.891-1.839-1.336-0.780.1320.8561.1661.4481.6061.7331.837-0.9-2.957-1.839-1.339-0.7690.1480.8541.1471.4071.5491.661.749-1-3.022-1.877-1.34-0.7580.1640.8521.1281.3661.4921.5881.664-1.1-3.087-1.894-1.341-0.7460.180.8481.1071.3241.4351.5181.581-1.2-3.149-1.91-1.339-0.7320.1950.8411.0861.2821.3791.4491.501-1.3-3.211-1.925-1.337-0.7190.210.8381.0641.241.3241.3831.424-1.4-3.271-1.937-1.333-0.7050.2250.8321.0411.1981.271.3181.351-1.5-3.33-1.951-1.329-0.690.240.8241.0181.1571.2171.2561.282-1.6-3.388-1.962-1.318-0.6750.2540.8170.9941.1161.1661.141.155-1.7-3.444-1.972-1.31-0.660.2680.8080.971.0751.1161.1971.216-1.8-3.499-1.981-1.302-0.6430.280.7990.9451.0351.0691.0871.097-1.9-3.553-1.989-1.294-0.6270.2940.7880.920.9961.0231.0371.044-2-3.605-1.996-1.302-0.6090.3070.7770.8950.9590.980.990.995-2.1-3.656-2.001-1.294-0.5920.3190.7650.8690.9230.9390.9460.949-2.2-3.705-2.006-1.284-0.5740.330.7520.8440.8880.90.9050.907-2.3-3.753-2.009-1.274-0.5550.3410.7390.8190.8550.8640.8670.869-2.4-3.8-2.011-1.262-0.5370.3510.7250.7950.8230.830.8320.833-2.5-3.845-2.012-1.25-0.5180.360.7110.7710.7930.7980.7990.8-2.6-3.889-2.013-1.238-0.4990.3680.6960.7470.7640.7680.7690.769-2.7-3.973-2.012-1.224-0.4790.3760.6810.7241.7380.740.740.741-2.8-3.973-2.01-1.21-0.460.3840.6660.7020.7120.7140.7140.714-2.9-4.013-2.007-1.195-1.440.390.6510.6810.6830.6890.690.69-3-4.051-2.003-1.18-0.420.960.3960.9360.6660.6660.6670.667 These values of K are multiplies with the standard deviation S and added to the mean of the logarithm y, which in turn gives the logarithmic value of the expected flood. Hence the antilogy of this value gives the magnitude of the expected flood for the corresponding value of the recurrence interval. This calculation wherein the flood frequencies are computed are shown in Table 5.23 andFig 5.13 shows the curve fitting for the expected flood peaks for Log Pearson type III Method. Table 5.23 Calculation for flood frequencies by Log Pearson Type III Method Log Pearson Type-IIIReturn PeriodProb.CsKSyK SylogX YKSPeak Flood X In cumec1.0199-0.1802-2.39760.5377-1.28911.954790.08601.0595-0.1802-1.69460.5377-0.91122.3326215.10391.1190-0.1802-1.55680.5377-0.83712.4067255.11791.2580-0.1802-0.83120.5377-0.44692.7969626.4456250-0.18020.02300.53770.01243.25621803.7314520-0.18020.84920.53770.45663.70045016.55081010-0.18021.26040.53770.67773.92158346.4210254-0.18021.68720.53770.90724.151014157.3907502-0.18021.95600.53771.05174.295519747.50791001-0.18022.24740.53771.20844.452228326.50552000.5-0.18022.40680.53771.29414.537934506.3963 Fig. 5.13 Flood Frequency Curve for Log Pearsons Type-III Method According to Log Pearson Type III Methodthe maximum flood of magnitude 13186 cumec recorded which was occurred in 1993 may occur once in 22.49 years. 5.13 STATISTICAL SOFTWARE PACKAGE For the analysis of Flood frequencies for the Tapi River, HEC-SSP software is used. The HEC-SSP executable code and documentation are public domain and were developed by the hydrologic engineering center for the U. S. Army corps of Engineers. The software developed with United States Federal Government resources, and is therefore in the public domain. This software can be downloaded for free from the HEC internet site (HYPERLINK http//www.hec.usace.army.milwww.hec.usace.army.mil). HEC does not provide technical support for this software to non-corps users. 5.13.1 HEC-SSP HEC-SSP is an integrated system of software, designed for interactive use in a multi-tasking environment. The system is comprised of a graphical user interface (GUI), separate statistical analysis components, data storage and management capabilities, mapping, graphics, and reporting tools. Over a period of many years, the Hydrologic Engineering Center has supported a variety of statistical packages that perform frequency analysis and other statistical computations. Historically, the programs that received the most use within the Corps of Engineers were HEC FFA (Flood Frequency Analysis) and STATS (Statistical Analysis of Time Series Data). EFA incorporates Bulletin 17B procedures that have been adopted by the Corps for flow frequency analysis. The STATS software package is used for statistical analysis of time series data. STATS can provide either analytical or graphical frequency analysis, specified by the user. STATS has the capability of computing monthly and annual maximum, minimum, and mean values along with computing a volume-duration analysis. Two other packages that have received a lot of use within the Corps of Engineers are REGFRQ (Regional Frequency Computation) and MLRP (Multiple Linear Regression Program). REGFRQ performs regional frequency analysis and MLRP is a multiple linear regression analysis tool. 5.13.2 Overview of Program Capabilities HEC-SSP is designed to perform statistical analyses of hydrologic data. The following is a description of the major capabilities of HEC-SSP. User Interface The user interacts with HEC-SSP through a graphical user interface (GUI). The main focus in the design of the interface was to make it easy to use the software, while still maintaining a high level of efficiency for the user. The interface provides for the following functions File management Data entry, importing, and editing. Statistical analyses. Tabulation and graphical displays of results. Reporting facilities Statistical Analysis Components Flow Frequency Analysis (Bulletin 178) – This component of the software allows the user to perform annual peak flow frequency analyses. The software implements procedures in Bulletin 17B, Guidelines for Determining Flood Flow Frequency, by the Interagency Advisory Committee on Water Data. General Frequency Analysis – This component of the software allows the user to perform annual peak flow frequency analyses by various methods. Additionally the user can perform frequency analysis of variables other than peak flows, such as stage and precipitation data. Volume Frequency Analysis – This component of the software allows the user to perform a volume frequency analyses on daily flow or stage data. Duration Analysis – This component of the software allows the user to perform a duration analysis on any type of data recorded at regular intervals. The duration analysis can be used to show the percent of time that a hydrologic variable is likely to equal or exceed some specific value of interest. Coincident Frequency analysis – This component of the software assists the user in computing the exceedance frequency relationship for a variable that is a function of two other variables. Calculation Here, General Frequency Analysis and bulletin 17 B plot for FFA is done. Expected Peak flow is tabulated in Table 5.24, Fig 5.14 shows the tabular form of general frequency analysis and, 5.15 and 5.16 shows the curve fitting for the expected flood peaks for HEC SSP. Fig. 5.14 Shows the tabular form of Frequency curve of Sabarmati river at Dharoi Dam. Fig. 5.15 shows the general frequency graphical plot for Sabarmati project at Dharoi Dam Fig. 5.16 shows the General Frequency Analytical plot for Sabarmati project at Dharoi Dam Table 5.24 Calculation for flood frequencies analysis by HEC-SSP with the help of Log Pearson Type III Method LabelsObserved Events (Median plotting positions)UnitsPercentCUMECTypePROBLOG PEARSON III12.3811318625.782788139.1847874412.5857834515.9867331619.3887048722.7896161826.1905946929.59258911032.99351881136.39539411239.79620451343.19716991446.59915851550.00015101653.40114001756.80311791860.20410451963.60510392067.0079622170.4087322273.8107092377.2116842480.6126592584.0146112687.4153932790.8163472894.2182442997.619118 Here, Fig 5.17 and 5.18 shows the bulletin 17 B plot for flood frequency analysis curve fitting for the expected flood peaks for HEC SSP. Fig. 5.17 shows the magnitude of scatter peak inflow for the Dharoi Dam. Fig. 5.18 shows the bulletin 17 B plot for FFA of the Dharoi Dam. PAGE MERGEFORMAT 45
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