1. WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful IslamDr. Akm Saiful Islam WFM 5201: Data Management and Statistical Analysis Akm Saiful Islam Lecture-8: Probabilistic Analysis June, 2008 Institute of Water and Flood Management (IWFM) Bangladesh University of Engineering and Technology (BUET)

2. WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful IslamDr. Akm Saiful Islam Frequency Analysis  Continuous Distributions Normal distribution Lognormal distribution Pearson Type III distribution Gumbel’s Extremal distribution  Confidence Interval

3. WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful IslamDr. Akm Saiful Islam Log-Normal Distribution The lognormal distribution (sometimes spelled out as the logarithmic normal distribution) of a random variable is one for which the logarithm of follows a normal or Gaussian distribution. Denote, then Y has a normal or Gaussian distribution given by:, (1)

4. WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful IslamDr. Akm Saiful Islam Derived distribution: Since, the distribution of X can be found as: (2) Note that equation (1) gives the distribution of Y as a normal distribution with mean and variance. Equation (2) gives the distribution of X as the lognormal distribution with parameters and.

5. WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful IslamDr. Akm Saiful Islam Estimation of parameters (, ) of lognormal distribution: Note:,, Chow (1954) Method: (1) (2) (3) (4)The mean and variance of the lognormal distribution are: (5) The coefficient of variation of the X s is: (6) The coefficient of skew of the X s is: (7) Thus the lognormal distribution is skewed to the right; the skewness increasing with increasing values of. and

6. WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful IslamDr. Akm Saiful Islam Example-1: Use the lognormal distribution and calculate the expected relative frequency for the third class interval on the discharge data in the next table

7. WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful IslamDr. Akm Saiful Islam Frequency of the discharge of a River ClassNumberObserved Relative Frequency 25,00020.03 35,00030.045 45,000100.152 55,00090.136 65,000110.167 75,000100.152 85,000120.182 95,00060.091 105,00000.000 115,00030.045

8. WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful IslamDr. Akm Saiful Islam Solution According to the lognormal distribution is

9. WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful IslamDr. Akm Saiful Islam So from the standard normal table we get The expected relative frequency according to the lognormal distribution is 0.145

10. WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful IslamDr. Akm Saiful Islam Example-2: Assume the data of previous table follow the lognormal distribution. Calculate the magnitude of the 100-year peak flood.

11. WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful IslamDr. Akm Saiful Islam Solution: The 100-year peak flow corresponds to a prob(X > x) of 0.01. X must be evaluated such that P x (x) = 0.99. This can accomplished by evaluating Z such that P z (z)=0.99 and then transforming to X. From the standard normal tables the value of Z corresponding to P z (Z) of 0.99 is 2.326. The values of S y and are given The 100-year peak flow according to the lognormal distribution is about 1,30,700 cfs.

12. WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful IslamDr. Akm Saiful Islam Extreme Value Distributions Many times interest exists in extreme events such as the maximum peak discharge of a stream or minimum daily flows. The probability distribution of a set of random variables is also a random variable. The probability distribution of this extreme value random variable will in general depend on the sample size and the parent distribution from which the sample was obtained.

13. WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful IslamDr. Akm Saiful Islam Extreme value type-I: Gumbel distribution Extreme Value Type I distribution, Chow (1953) derived the expression To express T in terms of, the above equation can be written as (3)

14. WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful IslamDr. Akm Saiful Islam Example-3: Gumble Determine the 5-year return period rainfall for Chicago using the frequency factor method and the annual maximum rainfall data given below. (Chow et al., 1988, p. 391) Year Rainfall (inch)Year Rainfall (inch)Year Rainfall (inch) 19130.4919260.6819380.52 19140.6619270.6119390.64 19150.3619280.8819400.34 19160.5819290.4919410.7 19170.4119300.3319420.57 19180.4719310.9619430.92 19200.7419320.9419440.66 19210.5319330.819450.65 19220.7619340.6219460.63 19230.5719350.7119470.6 19240.819361.11 19250.6619370.64

15. WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful IslamDr. Akm Saiful Islam Solution The mean and standard deviation of annual maximum rainfalls at Chicago are 0.67 inch and 0.177 inch, respectively. For, T=5, equation (3) gives

16. WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful IslamDr. Akm Saiful Islam Log Pearson Type III For this distribution, the first step is to take the logarithms of the hydrologic data,. Usually logarithms to base 10 are used. The mean, standard deviation, and coefficient of skewness, Cs are calculated for the logarithms of the data. The frequency factor depends on the return period and the coefficient of skewness. When, the frequency factor is equal to the standard normal variable z. When, is approximated by Kite (1977) as

17. WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful IslamDr. Akm Saiful Islam Example-4: Calculate the 5- and 50-year return period annual maximum discharges of the Gaudalupe River near Victoria, Texas, using the lognormal and log-pearson Type III distributions. The data in cfs from 1935 to 1978 are given below. (Chow et al., 1988, p. 393) Year1930 1940 1950 1960 1970 0 55900 13300 23700 9190 1 58000 12300 55800 9740 2 56000 28400 10800 58500 3 7710 11600 4100 33100 4 12300 8560 5720 25200 538500 22000 4950 15000 30200 6 17900 0 17900 1730 9790 14100 717200 46000 25300 70000 54500 825400 6970 58300 44300 12700 94940 20600 10100 15200

18. WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful IslamDr. Akm Saiful Islam Solution

19. WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful IslamDr. Akm Saiful Islam It can be seen that the effect of including the small negative coefficient of skewness in the calculations is to alter slightly the estimated flow with that effect being more pronounced at years than at years. Another feature of the results is that the 50-year return period estimates are about three times as large as the 5-year return period estimates; for this example, the increase in the estimated flood discharges is less than proportional to the increase in return period.

20. WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful IslamDr. Akm Saiful Islam Confidence Interval

21. WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful IslamDr. Akm Saiful Islam

22. WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful IslamDr. Akm Saiful Islam