ABE 5 Rabi H. Mohtar Hydrologic Frequency Analysis Return Period and Probability year: if time is very large the average time between events is years. he epected number of occurrences of a year event in N years is N/. hese are only statistical epectations and are not certainties. Recurrence Interval: time between occurrences. he average value of recurrence interval is. Average time of a year event recurrence is years \ Probability of a year P in any year is /. P / Assumptions: Flows are independent from year to year. May not be true since weather pattern is cyclic in most cases! Statistical properties of the system are not changing, i.e., no changes in watershed as an operator. his is also not totally true, but is adjusted with new data about the system. Weather rainfall input Input Watershed System Runoff event Output Risk Analysis Probability of k occurrences of Q * Q * > Q in a year is governed by the binomial distribution: n! k n k f k;p, n P P n k! k! where n! nnn ; and 0! Eample: Probability of occurrences of a 0year event in 0 years P 0.05 0 0! 8 f,0.05, 0 0.05 0.95 0.6 8!! Note: All references to equations and tables are from the tet book Design Hydrology and
his probability is independent of the event Q. his means that 6% of large 0year records will contain eactly peaks eceeding Q. Probability of not eceeding + probability at least one eceedence n f P, n where fp,n is the probability of year will be eceeded at least once in an nyear, if n, then fp,0.6 \ a structure built on a year, probability that flow will be eceeded is 0.6 at least once. If you want to be 90% sure not to eceed a design capacity of a 5year period fp,5 would be 0.90. \0. 5 \8 years What does failure mean? Flow eceeded! Danger! Human life or inconvenience? Consequences Return periods using economic analysis. Danger: interest rate sensitivity See Figure. in the tet book! Frequency determination Approach depends on quantity, quality, and type of hydrologic data. Long flow record at the site the only one considered Any statistics involve these terms: Population Sample Population statistics: o Mean µ o Standard deviation s, s ªvariance o C v coefficient of variation ª dimensionless o Skewness ýªmeasure of symmetry, for normal distribution γ0 Sample statistics: X S i n X o o S o Ĉ v o C s i n n Note: All references to equations and tables are from the tet book Design Hydrology and
C C s v s n i n n s n sample size If we assume that the set sample is independent and represents the population with no trend \ we can have frequency analysis Probability plotting Look at data in able. in the tetbook. 6 events out of eceeded 0 cfs \0% probability of eceeding 0 \5year flood is 0 his intuitive approach has a limit though. Systematic approach is needed:. Rank data from largest to smallest m. Calculate plotting position P, where ntotal number of records and mrank n +. Plot the observation on a log normal or probability paper with P along the probability scale and the magnitude along the variable scale P represents fraction of data greater or equal to the corresponding value. See the curvature with or without probability paper! Fit a line to the data. Probability distributions Probability density function pdf vs Cumulative density function cdf P P X Used for probability of random event P P d Bell shape normal pdf Prob a <bx < b P X b P X a Mean and s on normal distribution Used to calculate probability of event less than a certain value Prob Xb< Normal distribution: P ep s π Prob X < m s Note: All references to equations and tables are from the tet book Design Hydrology and
Log normal distribution: ln P ep m Y σ Y π s Y µ Y and s Y are mean and s.d. of lnx Etreme value type : β β P ep ep α α α,b Λ Λ S 6 α β 0.45S π Pearson type III: P Po e α / δ / α a/ δ with the mode at Xa and lower bound at X0, ddifference between mean and mode PoP X a. Log Pearson \ take log of events and apply Pearson type III Mode at a d difference in the mode and mean Distribution Skewness Normal 0 Lognormal C v + C v Etreme value.9 Log Pearson III Any value Standardized variable: X X + C K v X magnitude of X at return period X mean, C v coef of variation K frequency factor dependent on distribution For normal distribution Prob Qb< 500 Prob ªProb Zb< 0.896 500 59 9 Z < 006 Note: All references to equations and tables are from the tet book Design Hydrology and
µ Prob Q <b Prob Z < σ hese are in tables to simplify calculations Eample Frequency Analysis: able. data µ 599 σ 006 C v 0.69 Log Normal: use X X + C K v t K : able.5 Etreme Value type I Use X K : able.6 Log Pearson III K : able.7 Use procedure for X in book Steps for LP Probability Plotting:.ransform observations to logarithm Yi log X i.compute mean logarithm, Y.Compute standard deviation of the logarithm S Y 4.Compute C s : skewness 5.Compute Y Y + S k 6.Calculate Y X antilogy Please refer to chapter of the tet for further details on the subject. Note: All references to equations and tables are from the tet book Design Hydrology and