5--0 Stochastc Hydrology Hydrologcal statstcs and extremes Marc F.P. Berkens Professor of Hydrology Faculty of Geoscences Hydrologcal statstcs Mostly concernes wth the statstcal analyss of hydrologcal tme seres n relaton to extremes,,.e. floods and droughts.
5--0 Example: Rhne at Lobth Daly averaged dscharge (m 3 /s) 90-00 00 Example: Rhne at Lobth Flow duraton curve of Rhne dscharge
5--0 Extreme events Want to know: What s the probablty dstrbuton that a flood of gven sze occurs? What s the sze of a flood that belongs to a gven desgn frequency? Frst queston that must be answered s: What consttutes a flood? wo methods:. he largest dscharge per year (Maxmum values). All dscharge values above a certan threshold (Peak over hreshold (PO) data or partal duraton data) Maxmum values of Rhne at Lobth 4000 000 0000 Q max (m 3 /s) 8000 6000 4000 000 0 90 9 9 93 94 95 96 97 98 99 00 Year Maxmum daly averaged dscharge (m 3 /s) for each year n 90-00 00 3
5--0 Assumptons about maxmum values. he maxmum values are realsatons of ndependent random varables.. here s no trend n tme. 3. he maxmum values are dentcally dstrbuted. In ths case a sngle probablty dstrbuton can be assumed for the maxmum values. Probablty and recurrence tmes Cumulatve probablty dstrbuton: Exceedence probablty: Return perod or recurrence tme: F( y) Pr( Y y) P( y) Pr( Y y) ( y) P( y) F( y) Recurrence tme: the average number years between two consecutve flood events of a gven sze Note: : the actual number of years between flood events of a gven sze s tself random. 4
5--0 Recurrence tmes from data Fˆ ( y) n Pˆ( y) ˆ( y) Analyss of maxmum values of Rhne dscharge at Lobth for recurrence tme Y Rank F(y) P(y) (y) 790 0.0097 0.9909.0098 800 0.094 0.98058.098 905 3 0.093 0.97087.0300 306 4 0.03883 0.967.0404 30 5 0.04854 0.9546.050 3444 6 0.0585 0.9475.069 3459 7 0.06796 0.9304.079.......... Fˆ ( y) 940 937 90 9 0.87379 0.8930 0.6 0.0680 7.93 9.3636 9300 943 9 93 0.88350 0.909 0.650 0.09709 8.5833 0.3000 950 94 0.96 0.08738.4444 Fˆ ( y) 9707 95 0.933 0.07767.8750 9785 96 0.9304 0.06796 4.743 9850 97 0.9475 0.0585 7.667 074 98 0.9546 0.04854 0.6000 00 99 0.967 0.03883 5.7500 365 00 0.97087 0.093 34.3333 93 0 0.98058 0.094 5.5000 80 0 0.9909 0.0097 03.0000 Large recurrence tmes From a seres of n maxmum values: largest recurrence tme to be assessed: n+ years! For desgn purposes: y o ths end: for large s necessary!. Ft a probablty dstrbuton (whch( one?). Use t to extrapolate to large values of y (maxmum values y wth F(y) ) close to ). 5
5--0 he Gumbel dstrbuton It can be proven (see( secton 4... Syllabus) that maxmum values of the followng dstrbutons follow a Gumbel dstrbuton: - Exponental - Gaussan - loggaussan - Gamma - Logstc - Gumbel pdf: f Y ( y) be exp( e b( ya) b( ya) ) cpdf: F ( z) exp( e Y b( ya) ) Fttng the Gumbel dstrbuton ypes of methods: graphcal usng Gumbel paper and lnear regresson method of moments maxmum lkelhood estmaton 6
5--0 akng the double logarthm of the cpdf: ln[ F ln{ ln[ F y y Y ( y)] ln[(exp( e Y b( ya) ( y)]} b( y a) a ln{ ln[ FY ( y)]} b ( y) a ln{ ln[ ]} b ( y) )] e Plot maxmum y vs ln{ ln[ ]} n Gumbel paper b( y a) Plot maxmum y vs (y ) on specal Gumbel (double logarthmc) ) paper Gumbel paper. Plot maxmum y vs ln{ ln[ ]} n or,, plot maxmum y vs (y ) on specal Gumbel (double logarthmc) ) paper. Ft a straght lne by eye or by regresson to determna a and b. 7
5--0 Rhne data set Gumbel paper aˆ 5604, bˆ 0.0005877 Method of moments Mean and varance of a Gumbel varate Y: Y 0.577 a Y b s Y 6b ) Estmate mean m Y and varance ) Equate these to the above expressons 3) Solve for a and b: bˆ 6 s Y 0.577 aˆ my bˆ 8
5--0 0000 8000 6000 Method of moments aˆ 56, bˆ 0.0006674 Rhne data set Q max (m 3 /s) 4000 000 0000 8000 6000 4000 000 0 0 00 000 0000 Recurrence tme (Years) Estmaton of the -year event ˆ y aˆ ln( ln( )) bˆ In case of the Rhne data set: MoM parameters: 3 y50 56 6 ln( ln(0.999)) 78 m / s. wth parameters from regresson: 7736 m 3 /s 9
5--0 Estmaton of confdence lmts In case of regresson: yˆ t ( ) ˆ 95s ˆ y y t95s ˆ ( ) Y t 95 : the 95-pont of the student s t-dstrbuton and the standard error of predcton ( ) estmated as: Y s Y ˆ N Yˆ ( ) ( y yˆ ) N ( x( ) x) N s ( x( ) x) wth x( ) ln[ ln(( x N N x( ) ) / )] Estmaton of confdence lmts In case of method of moments: (. 0.5x 0.6x ) Vâr( yˆ ) bˆ N wth x( ) ln[ ln(( ) / )] Assumng a Gaussan estmaton error: yˆ.96 Vâr( y ) y yˆ.96 Vâr( y ) 0
5--0 he number -year events n a gven perod he expected number of -year events n N years: Np N / he actual number n of -year events n N years s a random varable obeyng a bnomal dstrbuton: Pr( n events y n N years) N p n p n N n ( ) 0 9 Examples: Pr(events y n 0 years) 0.0 ( 0.0) 0. 094 00 Pr(one or more flood events occur n 0 years) = -Pr(no events) = 0.99 0 0. 0956. he tme untl the next -year event he expected number of years n untl the next -year event: he actual number of years n untl the next -year event s a random varable obeyng a geometrc dstrbuton (wth( p=/ =/): Pr( m years untl event y ) ( p) m p
5--0 Other extreme value dstrbutons Generalzed extreme value dstrbutons ype II ype (Gumbel) ype III (Webull) Other extreme value dstrbutons Other maxmum value dstrbutons used for maxmum values: log-normal normal, log-pearson Pearson-type III. 0000 8000 Lognormal 6000 4000 Gumbel Q max (m 3 /s) 000 0000 8000 6000 4000 000 0 0 00 000 0000 Recurrence tme (Years)
5--0 Mnmum values (e.g. low flows) ake -Z or /Z or use Webull dstrbuton on Z mn estng the assumptons Independence: : Von Neuman s Q Q n n ( Y Y ) ( Y Y ) Lower crtcal area. If larger than crtcal value: no evdence that data are dependent! Rhne data set: Q=.07 ; upper crtcal value at =0.05:.68 -> no evdence for dependence 3
5--0 estng the assumptons rends: Mann-Kendall test n j sgn( Y Y j ) ' 8 /[ n( n )(n 5)] For n>40 has standard Gaussan dstrbuton wth two sded crtcal area (.e. sgnfcant at 95% (=0.05)( =0.05) accuracy trend s < -.96 or > >.96); otherwse no evdence of a trend. Rhne data set: =.99 -> > sgnfcant trend at 95% accuracy. Yearly maxma of average dayly runoff of the Rhne at Lobth 4000 000 y =.68x - 805 R = 0.036 0000 Q max (m 3 /s) 8000 6000 4000 000 0 900 90 940 960 980 000 Year 4
5--0 estng the assumptons estng for some dstrbuton: test. Defne a m classes (as n a hstogram) and assgn the n data values. Count the number of data fallng n each each class : n 3. Ft the proposed dstrbuton functon F(Y) ) to the data 4. Calculate the expected number of data fallng nto each class as: 5. he followng test statstc s calculated: e n n FY ( yup) FY ( ylow ) e ( n n e n m ) estng the assumptons estng for some dstrbuton: test m X follows a ch-squared dstrbuton wth m- degress of freedom: here s an upper crtcal area for the 0-hypothess 0 that the data follow the proposed dstrbuton. Rhne data and proposed Gumbel dstrbuton and 0 classes: 3.70 Rhne data and proposed lognormal dstrbuton and 0 classes: 4.36 Lower boundary crtcal area for m- = 9 degrees of freedom and =0.05: 30.44 -> both dstrbutons cannot be dscarded! 5