Frequency Factor - Stochastic Hydrology - Lecture Notes, Study notes of Mathematical Statistics

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Frequency factor for Normal Distribution: 3   Frequency Analysis T T T T x K xK µ σ µ σ = + − = 1 2 2 1lnw p ⎡ ⎤⎛ ⎞ = ⎢ ⎥⎜ ⎟ ⎝ ⎠⎣ ⎦ 0 < p < 0.5 2 2 3 2.515517 0.802853 0.010328 1 1.432788 0.189269 0.001308T w wK w w w w + + = − + + + Docsity.com Consider the annual maximum discharge Q in cumec, of a river for 45 years : 4   Example – 1 Year Q Year Q Year Q Year Q 1950 804 1961 507 1972 1651 1983 1254 1951 1090 1962 1303 1973 716 1984 430 1952 1580 1963 197 1974 286 1985 260 1953 487 1964 583 1975 671 1986 276 1954 719 1965 377 1976 3069 1987 1657 1955 140 1966 348 1977 306 1988 937 1956 1583 1967 804 1978 116 1989 714 1957 1642 1968 328 1979 162 1990 855 1958 1586 1969 245 1980 425 1991 399 1959 218 1970 140 1981 1982 1992 1543 1960 623 1971 49 1982 277 1993 360 1994 348 Docsity.com cumec 7   Example – 1 (Contd.) 2 20 2 3 2 2 3 2.515517 0.802853 0.01032 1 1.432788 0.189269 0.001308 2.515517 0.802853 2.45 0.01032 2.452.45 1 1.432788 2.45 0.189269 2.45 0.001308 2.45 1.648 w wK w w w w + + = − + + + + × + × = − + × + × + × = 20 20 756.6 1.648 639.5 1810.5 x x K s= + = + × = Docsity.com Frequency factor for Extreme Value Type I (EV I) Distribution: •  To express T in terms of KT, 8   Frequency Analysis 6 0.5772 ln ln 1T TK Tπ ⎧ ⎫⎡ ⎤⎛ ⎞= − +⎨ ⎬⎜ ⎟⎢ ⎥−⎝ ⎠⎣ ⎦⎩ ⎭ 1 1 exp exp 0.5772 6 T T Kπ = ⎧ ⎫⎡ ⎤⎛ ⎞⎪ ⎪ − − − +⎨ ⎬⎢ ⎥⎜ ⎟ ⎝ ⎠⎪ ⎪⎣ ⎦⎩ ⎭ Ref: Applied Hydrology by V.T.Chow, D.R.Maidment, L.W.Mays, McGraw-Hill 1998 Docsity.com When xT = µ in the equation ; KT = 0 Substituting KT = 0, i.e., the return period of mean of a EV I is 2.33 years 9   Frequency Analysis T T xK µ σ − = 1 01 exp exp 0.5772 6 2.33 T π = ⎧ ⎫⎡ ⎤×⎛ ⎞⎪ ⎪ − − − +⎨ ⎬⎢ ⎥⎜ ⎟ ⎝ ⎠⎪ ⎪⎣ ⎦⎩ ⎭ = years T Tx Kµ σ= + Docsity.com Frequency factor for Log Pearson Type III Distribution: The PDF is where y = log x •  The data is converted to the logarithmic series by {y} = log {x}. 12   Frequency Analysis ( ) ( ) ( ) ( ) 1 yy e f x x β λ εβλ ε β − − −− = Γ log x ε≥ Docsity.com •  The mean , standard deviation sy, and the coefficient of skewness Cs are calculated for the converted logarithmic series {y} •  The frequency factor for the log Pearson Type III distribution depends on the return period and coefficient of skewness 13   Frequency Analysis y Docsity.com •  When Cs = 0, the frequency factor is equal to the standard normal deviate z and is calculated as in case of Normal distribution. •  When Cs ≠ 0, KT is calculated by (Kite, 1977) where k = Cs / 6 14   Frequency Analysis ( ) ( ) ( )2 3 2 2 3 4 5 11 6 1 3 1 3 TK z z k z z k z k zk k = + − + − − − + + Ref: Kite, G. W., Frequency and Risk Analysis in Hydrology, Water Resources Publications, Fort Collins, Colorado, 1977 Docsity.com The mean, = 2.725 cumec Standard deviation, s = 0.388 cumec Coefficient of skewness Cs = -0.2664 T = 20 years 17   Example – 3 (Contd.) y 1 2 2 1 2 2 1ln 1ln 0.05 2.45 w p ⎡ ⎤⎛ ⎞ = ⎢ ⎥⎜ ⎟ ⎝ ⎠⎣ ⎦ ⎡ ⎤⎛ ⎞= ⎜ ⎟⎢ ⎥ ⎝ ⎠⎣ ⎦ = Docsity.com 18   Example – 3 (Contd.) 2 2 3 2 2 3 2.515517 0.802853 0.01032 1 1.432788 0.189269 0.001308 2.515517 0.802853 2.45 0.01032 2.452.45 1 1.432788 2.45 0.189269 2.45 0.001308 2.45 1.648 w wz w w w w + + = − + + + + × + × = − + × + × + × = k = Cs / 6 = -0.2664/6 = -0.0377 Docsity.com = 1.581 19   Example – 3 (Contd.) 756.6 1.581 639.5 1767.6 T Tx x K s= + = + × = cumec ( )( ) ( )( ) ( )( ) ( ) ( ) 22 3 3 4 52 11.648 1.648 1 0.0377 1.648 6 1.648 0.0377 3 11.648 1 0.0377 1.648 0.0377 0.0377 3 TK = + − − + − × − − − − + × − + − Docsity.com i.e., if a CDF of a set of data plots a straight line on arithmetic paper, the data follows uniform distribution. •  The probability paper for a given distribution can be developed so that the cumulative distribution plots as a straight line on the paper. 22   Probability Plotting Docsity.com •  Constructing probability paper is a process of transforming the arithmetic scale to the probability scale so that the resulting cumulative distribution plot is a straight line •  The plot is prepared with exceedence probability or the return period ‘T’ on abscissa and the magnitude of the event on ordinate. 23   Probability Plotting Docsity.com Construction of Probability paper: •  Mathematical construction. •  Graphical construction 24   Probability Plotting Docsity.com Construct probability paper for exponential distribution with λ = 1/3 Soln: 1.  The values of F(x) are assumed and corresponding x, Y and Z values are calculated. 2.  Y is plotted against Z and the Y axis is labeled with the corresponding value of F(x) and the Z axis with corresponding value of x. 27   Example – 4 Docsity.com 28   Example – 4 (Contd.) F(x) Y x = Z 0.01 0.010 0.030 0.05 0.051 0.154 0.1 0.105 0.316 0.2 0.223 0.669 0.3 0.357 1.070 0.4 0.511 1.532 0.5 0.693 2.079 0.6 0.916 2.749 0.7 1.204 3.612 0.8 1.609 4.828 0.9 2.303 6.908 0.95 2.996 8.987 0.99 4.605 13.816 Docsity.com Y is plotted against Z : 29   Example – 4 (Contd.) 0   1   2   3   4   5   0   3   6   9   12   15   Y   Z   Docsity.com Probability paper for exponential distribution: 32   Example – 4 (Contd.) 0.99   0.7   0.8   0.9   0.95   0.5   0.1   F(x)   X   0   9   12   15  3   6   λ1 λ2 Docsity.com •  Any exponential distribution data will plot as a straight line . •  The slope of the line will change as λ changes. •  Slope of the line gives the λ value. •  For many probability distributions, the same graph paper may be used for all values of the parameters of the distribution. •  For some distributions like gamma, a separate graph paper is required for different values of the parameters. •  Many types of probability papers are commercially available. 33   Probability Plotting Docsity.com Graphical construction: •  Graphical construction is done by transforming the arithmetic scale to probability scale so that a straight line is obtained when cumulative distribution function is plotted •  The transformation technique is explained with the normal distribution. •  Consider the coordinates from the standardized normal distribution table. 34   Probability Plotting Docsity.com Arithmetic scale plot: 37   Probability Plotting -­‐3   -­‐2   -­‐1   0   1   2   3   0   0.1   0.2   0.3   0.4   0.5   0.6   0.7   0.8   0.9   1   Z   F(Z)   Docsity.com 0.1   0.2   0.4   0.5   0.6  0.05   0.7   0.8   0.95  0.9   0.99  0.01   Transformation plot: 38   Probability Plotting 0.3   F(Z) Docsity.com Normal probability paper (probabilities in percentage) : 39   Probability Plotting 99.9 99.0 95.0 90.0 80.0 70.0 50.0 30.0 10.0 5.0 1.0 1.0 50.0 99.9 Redrawn from source: http://www.weibull.com/GPaper/ Docsity.com •  First entry as 1, second as 2 etc. •  Let ‘n’ is the total no. of values to be plotted and ‘m’ is the rank of a value, the exceedence probability (p) of the mth largest value is obtained by various formulae. •  The return period (T) of the event is calculated by T = 1/p •  Compute T for all the events •  Plot T verses the magnitude of event on semi log or log log paper 42   Plotting Position Docsity.com Formulae for exceedence probability: California Method: Limitations –  Produces a probability of 100% for m = n 43   Plotting Position ( )m mP X x n ≥ = Docsity.com Modification to California Method: Limitations –  Formula does not produce 100% probability –  If m = 1, probability is zero 44   Plotting Position ( ) 1m mP X x n − ≥ = Docsity.com Most plotting position formulae are represented by: Where b is a parameter –  E.g., b = 0.5 for Hazen’s formula, b = 0.5 for Chegodayev’s formula, b = 0 for Weibull’s formula –  b = 3/8 0.5 for Blom’s formula –  b = 1/3 0.5 for Tukey’s formula –  b = 0.44 0.5 for Gringorten’s formula 47   Plotting Position ( ) 1 2m m bP X x n b − ≥ = + − Docsity.com –  Cunnane (1978) studied the various available plotting position methods based on unbiasedness and minimum variance criteria. –  If large number of equally sized samples are plotted, the average of the plotted points foe each value of m lie on the theoretical distribution line. –  Minimum variance plotting minimizes the variance of the plotted points about the theoretical line. –  Cunnane concluded that the Weibull’s formula is biased and plots the largest values of a sample at too small a return period. 48   Plotting Position Docsity.com –  For normally distributed data, the best formula is Blom’s plotting position formula (b = 3/8). –  For Extreme Value Type I distribution, the Gringorten formula (b = 0.44) is the best. 49   Plotting Position Docsity.com 52   Example – 3 (Contd.) Year Annual Max. Q Arranged data Rank (m) P(X > xm) T 1950 804 3069 1 0.021739 46 1951 1090 1982 2 0.043478 23 1952 1580 1657 3 0.065217 15.33333 1953 487 1651 4 0.086957 11.5 1954 719 1642 5 0.108696 9.2 1955 140 1586 6 0.130435 7.666667 1956 1583 1583 7 0.152174 6.571429 1957 1642 1580 8 0.173913 5.75 1958 1586 1543 9 0.195652 5.111111 1959 218 1303 10 0.217391 4.6 1960 623 1254 11 0.23913 4.181818 Docsity.com 53   Example – 3 (Contd.) 15   150   1500   1   10   100   An nu al  M ax im um  d is ch ar ge  (Q )   Return  period  (T)   Docsity.com