Assessment of extreme discharges of the Vltava River in Prague

Transcription

1 Flood Recovery, Iovatio ad Respose I 05 Assessmet of extreme discharges of the Vltava River i Prague M. Holický, K. Jug & M. Sýkora Kloker Istitute, Czech Techical Uiversity i Prague, Czech Republic Abstract Damage of costructio works due to floodig i 997 ad 00 i the Czech Republic iitiated ivestigatios of structural failures ad reassessmet of available data for discharge extremes. I this study, hydrological data for 66 aual maximum discharges of the Vltava River i Prague sice 87 are aalysed usig various statistical methods. Momet characteristics of the measuremets the mea, stadard deviatio ad skewess are estimated ad the ehacig effect of a exceptioal observatio i 00 is detected. The aual maxima are described by two- or three-parameter logormal distributios ad the extreme value distributios of the type I ad II. Stadard statistical Kolmogorov ad chi-square tests are applied to assess goodess of fit of the theoretical models. It appears that a two-parameter logormal distributio may be the most suitable theoretical model. Assumig this distributio, extreme discharges correspodig to characteristic ad desig values are estimated. It is show that the partial safety factor estimated from the measuremets sigificatly differs from the recommeded value of.5. The discharge i 00 correspods to a exceptioally log retur period. It is cocluded that statistical methods provide a valuable backgroud for evaluatio ad predictio of discharges. However, the preseted aalysis should be further improved to iclude o-statistical aspects that ifluece discharges such as the effects of water maagemet ad deforestatio. Keywords: discharge, probabilistic assessmet, extremes, failure, statistical methods. doi:0.495/friar0800

2 06 Flood Recovery, Iovatio ad Respose I Itroductio A umber of structures i the Czech Republic were affected by the floodig i July 997 i Moravia ad i August 00 i Bohemia. I particular damage ad destructio caused to structures i the historic city of Prague i 00 was o a uprecedeted scale. Mai observed causes of structural damage have bee subdivided ito geotechical ad structural aspects. The geotechical causes iclude: - Isufficiet foudatio (depth, width), - Udergroud trasport of sedimets ad ma-made groud (propagatio of cavers), - Icreased earth pressure due to elevated udergroud water. The major structural causes cover: - Isufficiet structural robustess (o rig beams as idicated i Figure ), - Use of iadequate costructio materials (ufired masory uits), - Material property chages caused by moisture (volume, stregth). Figure : Failure of a structure with isufficiet robustess. Water levels recorded i Prague ad its surroudigs durig the floodig seem to be exceptioally high. However, was the floodig really so exceptioal ad upredictable? What was the actual retur period correspodig to the measured amout of water? Aual maximum discharges of the Vltava River i Prague are assessed to aswer these questios. Available measuremets are aalysed usig various statistical methods with a particular focus o ifluece of the measuremet Q 00. Characteristic ad desig values of discharges are the determied o the basis of extreme discharges correspodig to specified probabilities. Partial safety

3 Flood Recovery, Iovatio ad Respose I 07 factors are the derived as the ratio of the desig value over the characteristic value, EN 990 []. Fially a retur period correspodig to the discharge Q 00 is estimated. Statistical evaluatio of aual maximum discharges Aual maximum discharges Q i of the Vltava River i Prague recorded by the Czech Hydrometeorological Istitute sice 87 are aalysed usig basic statistical methods provided by Ag ad Tag []. Statistical characteristics of the discharges are iitially estimated by the classical method of momets for which prior iformatio o the type of a uderlyig distributio is ot eeded. The resultig characteristics are give i Table for the samples without ad with the observatio Q 00. Table : Sample characteristics of the aual maxima i m 3 /s (sample size = 65 or 66). Sample characteristic Mea Stadard deviatio CoV Coefficiet of skewess Formula used i the aalysis s = m = Q i i= i= w = ( )( ) s ( Q i m) Without Q With Q 00 Eh. factor s v = m 3 i= 3 ( Q i m) It appears that the sample mea, stadard deviatio ad coefficiet of variatio are iflueced by the discharge Q 00 rather isigificatly (the ehacig factor varies from.0 up to.07). However, the coefficiet of skewess seems to be cosiderably affected by Q 00 (the ehacig factor is.). 3 Probabilistic distributios The characteristics provided i Table idicate that the aual maxima might be well described by a two-parameter logormal distributio havig the lower boud at the origi (LN0) or more uiversal three-parameter logormal distributio (LN) havig the lower boud (for a positive skewess) geerally differet from zero. Other possible theoretical models are extreme value distributios: the type II called also the Fréchet distributio (F) or type I, a popular Gumbel distributio (G) with the costat skewess of.4.

4 08 Flood Recovery, Iovatio ad Respose I Relative frequecy LN0 LN F G March August Aual maximum discharges i m 3 /s Figure : Histogram of the aual maxima ad the selected probabilistic distributios. Probability desity fuctios of the cosidered theoretical models ad a histogram of the aalysed measuremets are show i Figure. It follows that the logormal distributio LN0 fits the ivestigated sample very well. To compare goodess of fit of the cosidered distributios, Kolmogorov-Smirov ad χ - tests are applied. A hypothesis that a theoretical distributio fits well the sample distributio should be accepted uder the coditio: r K r = K 0 / K p ( χ = χ 0 / χ p ) () Otherwise the hypothesis should be rejected. I eq. () K 0 deotes a test value; K p critical value ad K r relative test value of the Kolmogorov-Smirov test. Aalogous symbols are used for the chi-square test. Relative test values are listed i Table for the samples without ad with the discharge Q 00. Table : Results of the Kolmogorov-Smirov ad chi-square tests. Probabilistic distributio Without Q 00 With Q 00 Logormal distributio LN Logormal distributio LN Fréchet distributio F Gumbel distributio G K r χ r K r χ r

5 Flood Recovery, Iovatio ad Respose I 09 It follows that all the applied distributios meet the coditio () i accordace with the Kolmogorov-Smirov test. However, the chi-square test idicates that the measured frequecies sigificatly differ from theoretical values for all the cosidered distributios. It appears that the logormal distributio LN0 is the most suitable model. Less favourable test results are observed for the three-parameter logormal LN ad Fréchet distributio F, ad the worst test results are obtaied for the Gumbel distributio (the fixed skewess of.4 may be rather low). If the discharge Q 00 is ivolved, the tests provide more favourable results for all the distributios, except for the Gumbel distributio. It should be oted that the test results are idicative oly. Suitable models should ot be solely selected o the basis of the statistical tests oly, but also takig ito accout experiece with discharges measured at other localities. Experiece of the Czech Hydrometeorological Istitute idicates that the logormal distributio LN0 could be a suitable model. Therefore, this distributio is further cosidered i estimatio of extreme discharges. 4 Parameter estimatio The method of momets applied i Sectio to estimate the sample characteristics is ofte cosidered to be rather iefficiet. Assumig that the uderlyig distributio of the sample is the logormal distributio LN0, the sample characteristics ca be improved by the maximum-likelihood method, which is cosidered as the most efficiet method for parameter estimatio, particularly for large samples. The maximum-likelihood estimators qˆ of ukow parameters θ of the distributio (here mea m ad stadard deviatio s) are obtaied maximizig the logarithm of a likelihood fuctio: maxl L q Q ˆ () q [ ( )] q where Q = (Q,,Q ) is the sample, q realizatio of the vector of the parameters θ ad L(q Q) is the likelihood fuctio: L ( q Q) = f ( Qi q) (3) i where f( ) deotes the probability desity fuctio of the uderlyig distributio. Here the o-liear cojugate-gradiet method implemeted i the software package Mathcad is applied. Compariso of the distributio parameters estimated by the method of momets ad the maximum-likelihood method is idicated i Table 3. Table 3: Estimated parameters i m 3 /s. Method Without Q 00 With Q 00 m s m s Momets Maximum-likelihood

6 0 Flood Recovery, Iovatio ad Respose I It appears that the estimates of the mea are early idepedet of the applied method (differeces about %). However, the stadard deviatios estimated by the maximum-likelihood method are systematically greater tha those obtaied by the method of momets (differeces about 0%). 5 Estimatio of extreme values Upper fractiles Q p of the logormal distributio LN0 are further estimated usig the classical coverage method for the give cofidece level γ, see e.g. ISO 49 [3]: P(Q p,cov > Q p ) = γ (4) I accordace with EN 990 [], the characteristic value Q k is obtaied as the 0.98 fractile of aual maxima while the desig value Q d is the fractile of the life-time maxima correspodig to the probability: p d = - Φ(α E β) = - Φ( ) = (5) where Φ deotes the cumulative distributio fuctio of the stadardised ormal distributio, α E is the FORM sesitivity factor (cosiderig the recommeded value of 0.7 for the leadig actio) ad β is the reliability idex equal to 3.8 for the referece period of 50 years. Assumig statistical idepedece of the aual maxima, the desig value is estimated as follows: ( qˆ ) F 50 d Q d = p (6) where F - ( ) deotes the iverse cumulative distributio fuctio of the uderlyig distributio of the aual maxima. Partial safety factor γ Q for ufavourable effects of a variable actio is cosequetly obtaied as the ratio Q d / Q k. Estimated extreme discharges ad partial factors are summarized i Table 4. Table 4: Estimated extreme discharges i m 3 /s. Method Desig value (p d ) Partial factor γ Q Without Q 00 With Q 00 Method Expect. γ = 0.75 Expect. γ = 0.75 Characteristic value (p k ) Momets Maximumlikelihood Momets Maximumlikelihood

7 Flood Recovery, Iovatio ad Respose I It is idicated that the extreme values predicted from the available data icludig the discharge Q 00 are greater tha those estimated without this discharge (by about 9% for the method of momets ad 4% for the maximumlikelihood method). It also appears that the extreme discharges predicted by the maximum-likelihood method are greater tha those obtaied by the method of momets (by about 6 9% for the characteristic value ad 9% for the desig value). Furthermore, the upper fractiles estimated cosiderig the commoly accepted 0.75 cofidece level are greater tha the expected upper fractiles (by about 6% i case of the characteristic values ad % i case of the desig values). The partial safety factor γ Q 3.0 derived from the data seems to be sigificatly greater tha the recommeded value.5. 6 Retur period of the discharge Q 00 Expected retur periods T correspodig to the discharge Q 00 are derived usig the relatioship: T = / [ F(Q 00 qˆ )] (7) where F( ) deotes the cumulative distributio fuctio of the uderlyig distributio. Expected retur periods are listed i Table 5. Table 5: Expected retur periods i years. Method Without Q 00 With Q 00 Momets Maximum-likelihood 0 80 It appears that the retur period is cosiderably affected by the fact whether the discharge Q 00 is take ito accout or ot. I additio the estimates based o the maximum-likelihood method are sigificatly lower tha those obtaied usig the method of momets. Cosiderig the data without Q 00 ad the maximum-likelihood method, the observed discharge Q 00 = 550 m 3 /s correspods to the exceptioally log retur period of 0 years. Obviously the discharge Q 00 could have bee hardly expected. Note that estimates of the retur period may also eormously vary with a type of the applied distributio as idicated by Holicky ad Sykora [4]. It is emphasized that the preseted aalysis is based o statistical methods oly. More detailed aalysis should also cosider o-statistical iflueces that may have evolved durig the period covered by the measuremets (sice 87). I particular discharges may be strogly depedet o a river maagemet icludig modificatios of depth, width ad roughess of a river chael ad removal of vegetatio. Effects of deforestatio ad other ma-made itervetios i eviromet should be also take ito accout. 7 Coclusios Ivestigatio of structural failures due to floodig i Moravia (997) ad Bohemia (00) idicates that mai causes of structural damage may be

8 Flood Recovery, Iovatio ad Respose I subdivided ito geotechical ad structural reasos. Statistical aalysis of available data for aual discharge maxima shows that: - Discharges may be well described by a two-parameter logormal distributio LN0. - The characteristic ad desig discharges predicted usig data icludig the discharge i 00 are greater tha those estimated without cosiderig the discharge i 00 (by about 5%). - Extreme discharges predicted by the maximum-likelihood method are greater tha those by the method of momets (by about 0%). - The recommeded value of the partial safety factor γ Q =.5 is cosiderably lower tha the value derived from the available data (γ Q 3.0). - The discharge observed i 00 correspods to a exceptioally log retur period ad, therefore, could have bee hardly expected. It appears that statistical methods provide a valuable backgroud for evaluatio ad predictio of discharges. It is further oted that the preseted results are idicative oly sice purely statistical methods are used i the assessmet. Effects of water maagemet ad ma-made itervetios i eviromet should be also take ito accout. Ackowledgemets This study has bee coducted at the Kloker Istitute, Czech Techical Uiversity i Prague, with fiacial support of the project GACR 03/06/5 Reliability ad risk assessmet of structures i extreme coditios. Refereces [] EN 990 Eurocode: Basis of structural desig, CEN, 00. [] Ag, A.H.-S. & Tag, W.H., Probabilistic cocepts i egieerig plaig ad desig, Joh Wiley ad Sos: New York, 975. [3] ISO 49 Statistical methods for quality cotrol of buildig materials ad compoets, ISO, 997. [4] Holicky, M. & Sykora, M, Probabilistic Evaluatio ad Predictio of Discharges o the Vltava River i Prague, CTU Reports (Vol. 8, No. 3), Proc 3rd Czech/Slovak Symposium Theoretical ad Experimetal Research i Structural Egieerig, CTU Publishig House: Prague, p. 47 5, 004.