The impact of the time series resolution on the reliability of the maximum precipitation models

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1 E3S Wb of Cofrcs 7, 38 (7) DOI:.5/ 3scof/7738 EKO-DOK 7 Th mpact of th tm srs rsoluto o th rlablty of th mamum prcptato modls Bartosz Kaźmrczak,*, Katarzya Wartalska, ad Marc Wdowkowsk Wrocław Uvrsty of Scc ad Tchology, Faculty of Evromtal Egrg, Wybrzż Wyspańskgo 7, 5-37 Wrocław, Polad Isttut of Mtorology ad Watr Maagmt Natoal Rsarch Isttut, Podlśa Strt 6, -673 Warsaw, Polad Abstract. I th papr, two mamum prcptato modls for gca wr dvlopd. For ths purpos archval pluvographc rcords from th tm spa 96 wr usd. Th frst modl was dvlopd o th bass of rafall data of duratos ragg from 5 muts to 6 days. I th scod modl, rafall data wth duratos ragg from m to 6 days wr usd, ad prcptato amouts for 5 muts durato wr trapolatd. Gralzd potal dstrbuto was usd to dvlop th modls. Both modls wr compard wth masurmt data usg rlatv rsdual ma squar rror. Itroducto Th basc form of quattatv rafall dscrpto ar modls of dpdc: tsty I ( mm/m), ut tsty q ( dm 3 /s ha) or th hght h ( mm) of prcptato o ts durato t ad th probablty of cdac p []. Th rlatoshp btw prcptato tsty (ut tsty or hght) ad ts durato s most oft prstd trms of tsty-durato-frqucy (IDF) or dpthdurato-frqucy (DDF) curvs, for dffrt probablts p of prcptato occurrc. Ths curvs rprst th hyprbolc famly gv by th gral quato [, ]: a I ( t b) whch: a, b, c ad ar th mprcal coffcts, dpdt o th occurrc probablty of a gv rafall ad o th clmatc ad physographc factors of th catchmt. A srs of homogous obsrvato for dcads s rqurd for thr dtrmato []. Authortatv for dsgg of aras draag systms ar both short-trm prcptato wth a hgh-tsty ad log-lastg prcptato wth a sgfcat trrtoral scop ad hgh ffccy. Rcommdd by th Europa stadard PN-EN 75:8 modlg of stg or wly dsgd swrag systms coutrs Polad a barrr of a lack of c () * Corrspodg author: bartosz.kazmrczak@pwr.du.pl Th Authors, publshd by EDP Sccs. Ths s a op accss artcl dstrbutd udr th trms of th Cratv Commos Attrbuto cs 4. (

2 E3S Wb of Cofrcs 7, 38 (7) DOI:.5/ 3scof/7738 EKO-DOK 7 accss by dsgrs to approprat ad rlabl rafall databass. At th trac to hydrodyamc modls ar most commoly rqurd storm hytographs wth a 5 muts tm rsoluto, such as Eulr s typ II modl prcptato. Th bass for th dvlopmt of modl prcptato ar th modls of mamum rafall th form of IDF or DDF curvs [3 ]. I Polad, accss to th prcptato sourc data s a mattr for th Isttut of Mtorology ad Watr Maagmt (IMWM), th owr of th largst umbr of mtorologcal statos th coutry. By, prcptato th IMWM wr rcordd o papr pluvographs, whch volvd a vry tm-cosumg dvlopmt of masurmt data. Howvr, du to th umbr of th statos ad th lgth of th masurmt srs (oft dcads) papr pluvographc strps ar a trmly valuabl rsarch matral o prcptato Polad. Wth approprat commtmt, formato about rafall hght ca b rad from th papr strps wth a rsoluto tm of 5 muts. Basd o ths form of aalogu but cotous rcords of rafall vts, may modls of mamum prcptato hav b dvlopd ad currtly appld grg practc [,, ]. Sc 7, th IMWM bga to rcord prcptato by dgtal ra gaugs, tally paralll wth tradtoal dvcs. Tm srs ar currtly rcordd wth a -mut tm rsoluto. Th assumptv rsoluto from th pot of vw of urba hydrology s a mattr of cocr bcaus th swag systms dsgg ad modlg t s dd formato about shortr prcptato, lastg 5 muts []. Th am of th study s to valuat th mpact of tm srs rsoluto o th rlablty of th mamum prcptato modls. Th aalyss, wr coductdusg two mamum prcptato modls basd o archval pluvographs from gca from th tm spa 96. Th frst modl was basd o rafall data of duratos ragg from 5 muts to 6 days. I th scod modl, rafall data wth duratos ragg from muts to 6 days wr usd, ad th amouts of prcptato wth th durato of 5 muts wr trapolatd. Both modls wr compard wth masurmt data. Matrals ad mthods Archvd pluvographs from mtorologcal stato of IMWM gca from th tm spa 96 wr usd as rsarch data. Masurg stato gca, as part of a atoal masurmt ad obsrvato twork at hydrologcal ad mtorologcal srvc, s a syoptc stato whch s partcpatg th tratoal wathr motorg program (Wathr World Watch) as part of th World Mtorologcal Orgazato (WMO), of whch Polad s a mmbr. Th stato buldg s locatd o th south-astr outskrts of th cty of gca, at a lvato of m abov th sa lvl. Th prdomat lad us both th mucpalty ad rural ara aroud th stato ar flds ad wastlad []. I ordr to dtrm th rlatoshp btw th amout of rafall from durato ad probablty of cdac h(t,p), thr must b do a slcto of data o whch th rlatoshp wll b dvlopd. Elaboratg archval pluvographs authors lmtd prod of aalyss to moths from May to Octobr (V X) [,, ]. For th purpos of ths papr, basd o th Pak-Ovr-Thrshold (POT) mthod [3 9] thr wr solatd from th tstd 5-yars prod top 5 mamum amouts (h, mm) of rafall for ach of th followg rafall duratos: t = 5,,, 3, 4, 5, 6, 9,, 8, 36, 7, 8, 44, 6, 88, 43, 576, 7 ad 864 muts. I th frst plac, th top 5 amouts of rafall wr ordrd dcrasg. Th thr wr succssvly assgd to t th mprcal probablty of cdac accordg to () from p =. (for th hghst valu) to p =.98 (for th lowst valu) [, ]:

3 , N m N m p () whr m s th squc umbr wth a dcrasg ordrd strg of th umbr of N. Th amout of rafall rcordd for slctd valus of mprcal probablty ar show Tabl. Tabl. Th amout of rafall (h, mm) rcordd for slctd valus of mprcal probablty (m =, 5,, 5 ad 5). t, m p t, m p To dscrb th masurmt data gralzd potal dstrbuto (GED) was usd. klhood fucto of ths dstrbuto dscrbs a quato: l l l ),, ( l (3) whr α, λ ad μ ar paramtrs of th dstrbuto. Paramtrs ca b dtrmd by th mamzato of th lklhood fucto or by solvg th systm of ormal quatos [ 4]: l (4) Quatls of a radom varabl for th GED dscrbd by th quato ) ( l p h (5) Th cocdc of thortcal dstrbutos wth masurd data was amd usg th Adrso-Darlg tst for statstcs [5, 6]: X F X F A l l (6) DOI:.5/, 38 (7) scof/ E3S Wb of Cofrcs 38 3 EKO-DOK 7

4 E3S Wb of Cofrcs 7, 38 (7) DOI:.5/ 3scof/7738 EKO-DOK 7 whr: -th valu th dcrasg ordrd radom sampl, F() cumulatv dstrbuto fucto for th thortcal dstrbuto. Th ull hypothss H (wh th masurmt data wr sutabl for tstd thortcal dstrbuto), wr tak o a sgfcac lvl of.5 f th A tst statstc was lss tha th crtcal valu A kr. Th altratv hypothss was tak othrws. Th crtcal valus ca b rad from th statstcal tabls or obtad by Mot Carlo mthod [5, 7]. For GED dstrbuto ad N = 5 crtcal valu A kr =.73. Rlatv rsdual ma squar rror (R RMSE ) was usd to valuat th apttud of vstgatd dstrbutos ad to dscrb th masurmt data h t, hm, R RMSE % (7) hm, whr: h t th thortcal amout of rafall (mm), h m amout of rafall from masurmts (mm). 3 Rsults Calculato rsults of partcular paramtrs of GED wr prstd Tabl. Th paramtr stmats wr dtrmd by umrcal mamzato of th log-lklhood fucto (3). Th calculatos wr carrd out for ach of duratos of mamum prcptato amouts aalyzd th papr. GED dstrbuto fulfls th complac crtro A for ach of th aalyzd rafall duratos. Thr wr also calculatd rlatv rsdual ma squar rror statstcs, covrg th tr rag of data all duratos. I ths cas R RMSE = 3.3%. Tabl. Calculato rsults of paramtrs of GED. t, λ, γ, α A t, λ, γ, R m /mm mm RMSE α m /mm mm A R RMSE Followg th quato (5) ad th paramtrs lstd Tabl, th prcptato amout wth ay cdac probablty ad slctd rafall durato ragg from 5 to 864 muts ca b calculatd. I ths rspct, t should b otd that th currt -mut tm stp of rafall rgstrato wll ot allow th futur to stmat paramtrs for prcptato wth th durato of 5 muts. So ths paramtrs wll hav to b trapolatd. Th paramtrs λ ad γ hav a clarly markd trd, whl th paramtr α s dprvd of a trd. Sc thr s o dpdcy trd α(t), th ma valu of ᾱ =.837 was assumd th calculatos. Basd o th calculatd GED dstrbuto paramtrs thr wr prpard plots (Fg. ) showg thr dpdc o th rafall durato (from to 864 muts). 4

5 E3S Wb of Cofrcs 7, 38 (7) DOI:.5/ 3scof/7738 EKO-DOK , /mm , mm. 5. t, m Fg.. Th dpdc of th paramtrs λ ad γ of th rafall durato. Th rlatoshp of paramtrs λ ad γ of th rafall durato ar dscrbd as fuctos: t (8) t (9) for th coffct of dtrmato of.99 ad.994, rspctvly. Fally, a modl dscrbg th dpdc of th amout of rafall o ts durato ad a spcfd cdac probablty, basd o th quatls of GED dstrbuto (5), taks th form of: h.56.47,95 5.6t,66 t l( p) Basd o th obtad formula, prcptato amouts for th 5 m durato ad th cdac probablty p =.,.98,.96,.49 ad.98 wr computd. Th rcvd rsults wr compard wth th masurmt data (Tabl ) ad th prcptato amout calculatd usg formula (5) wth th paramtrs from Tabl. Th rsults ar show Tabl 3. Tabl 3. Calculato rsults of paramtrs of GED. p h h m, mm t, mm h R t, mm (5) RMSE () R RMSE % % % % % % % % % % Thr wr also calculatd rlatv rsdual ma squar rror statstcs, covrg th tr rag of data (t = 5 m). I ths cas R RMSE =.9% for h calculatd by (5) ad R RMSE = 4.5% for h calculatd by (). Th ft qualty of th quatos for rafall data from gca s show th h-h plot (Fg. ). () 5

6 E3S Wb of Cofrcs 7, 38 (7) DOI:.5/ 3scof/7738 EKO-DOK h by (5) h by () h t, mm h m, mm Fg.. Th h-h plot for th GED dstrbuto. Th aalyss of th rsults clarly dcats that th trapolato of th GED dstrbuto paramtrs dos ot produc accptabl rsults. Calculatd accordg to () rafall hghts of a t = 5 m durato do ot corrlat wth th rsults of masurmts. 4 Coclusos I ths papr, two mamum prcptato modls for gca wr dvlopd. For ths purpos archval pluvographc rcords from th tm spa 96 wr usd. Th frst modl was dvlopd o th bass of rafall data of duratos ragg from 5 muts to 6 days. I th scod modl, rafall data wth duratos ragg from m to 6 days wr usd, ad prcptato amouts for 5 muts durato wr trapolatd. Gralzd potal dstrbuto (GED) was usd to dvlop th modls. Both modls wr compard wth masurmt data usg rlatv rsdual ma squar rror. Th coductd aalyss allowd to draw th followg coclusos: Th mamum prcptato modl (5) wth th paramtrs show Tabl accuratly maps th masurmt rsults - R RMSE = 3.3%. I partcular, for rafall wth a durato of t = 5 m, th dffrcs btw th prdctd ad masurd prcptato amouts at th lvl of R RMSE =.9% wr otd. I th cas of usg th modl () to dtrm prcptato hghts wth a durato of t = 5 m (paramtrs trapolato) obtad rsults wr ot corrlatd wth th masurmt rsults - R RMSE = 4.5%. Thrfor, ths mthod should b rgardd as approprat for stmatg prcptato amouts for th durato of t = 5 m. Prcptato amouts wth a durato of t = 5 m ca b stmatd wth som appromato basd o th prcptato hghts data for th durato of t = m. Th aalyss of masurmt data dcats that durg th prod cosdrd, th mamum prcptato wth a durato of t = 5 m rprstd o avrag 68% of mamum rafall hghts wth a durato of t = m. Howvr, ths stmato should b tratd oly dcatvly, sc ths varablty ragd th cas of th largst 5 prcptatos from 55% to 75%. I ordr to draw up rlabl coclusos, furthr rsarch ths fld volvg othr masurmt statos s dd. I th lght of th obtad rsults, t s rqustd to cras th tmporal rsoluto of rcordd by IMWM rafall for 5-mut or shortr trvals. 6

7 E3S Wb of Cofrcs 7, 38 (7) DOI:.5/ 3scof/7738 EKO-DOK 7 Th work was ralzd wth th allocato No. 4/69/6 awardd for Faculty of Evromtal Egrg Wroclaw Uvrsty of Scc ad Tchology by Mstry of Scc ad Hghr Educato yars 6 7. Rfrcs. A. Kotowsk, B. Kaźmrczak, A. Dacwcz, Th modlg of prcptatos for th dmsog of swr systms (Polsh Acadmy of Sccs, Warsaw, ). A. Kotowsk, Th prcpls of saf dmsog of swr systms (Sdl-Przywck, Warsaw, 5) 3. E. Burszta-Adamak, M. Mrowc, Watr Sc Tchol 68, (3) 4. A. Kotowsk, K. Wartalska, M. Nowakowska, Ochroa Środowska 38, (6) 5. B. Kowalska, D. Kowalsk, G. Łagód, M. Wdomsk, Modllg of Hydraulcs ad Pollutats Trasport Swr Systms wth Emplary Calculatos SWMM (ubl Uvrsty of Tchology, ubl, 3) 6. S. Y. Park, K. W., I. H. Park, S. R. Ha, Dsalato 6, 3 (8) 7. T. G. Schmtt, Kommtar zum Arbtsblatt A 8 Hydraulsch Bmssug ud Nachws vo Etwässrugssystm (DWA, Hf, ) 8. T. G. Schmtt, M. Thomas, KA Wassrwrtschaft, Abwassr, Abfall 47, () 9. D. Słyś, A. Stc, Evromt Protcto Egrg 38, 4 (). A. Wałęga, G. Kaczor, B. Stęplwsk, Pol J Evro Stud 5, 5 (6). E. Bogdaowcz, J. Stachý, Mamum rafall Polad. Dsg charactrstcs (Th Publshg Hous of th Isttut of Mtorology ad Watr Maagmt, Warsaw, 998). B. Kaźmrczak, M. Wdowkowsk, Prodca Polytchca Cvl Egrg 6, (6) 3. W. Jakubowsk, Mtorology Hydrology ad Watr Maagmt 3, (5) 4. A. Malhota, S. achac-cloutrb, G. Talbota, A.C. Favrc, J Hydrol 476, 7 (3) 5. S. Cols, A troducto to statstcal modlg of trm valus (Sprgr Srs Statstcs, Sprgr-Vrlag, odo, ) 6. C. Oyutha, Joural of Urba ad Evromtal Egrg 6, () 7. W.. Shy, N. Ismal, A. A. Jma, Watr Rsour Maag 8, (4) 8. S. W, B. J. Valdés, S. Stschdr, T. W. Km, Stoch Ev Rs Rsk A 3, (6) 9. A. G. Ylmaz, I. Hossa, B. J. C. Prra, Hydrol Earth Syst Sc 8 (4). R. D. Gupta, D. Kudu, J Stat Pla Ifr 37, (7). R. D. Gupta, D. Kudu, J Stat Comput Sm, 69 (). B. Kaźmrczak, A. Kotowsk, J Hydrol 55 (5) 3. F. ao, G. D Baldassarr, A. Motaar, Watr Rsour Rs 45, 7 (9) 4. S. H., S. J. Mag, Irrgat Dra 5 (3) 5. R. D Agosto, M. A. Stphs, Goodss of Ft Tchqus (Marcl Dkkr, Nw York, 986) 6. S. Hogjoo, J. Youghu, J. Chagsam, H. Ju-Hag, Stoch Ev Rs Rsk A 6, () 7. A. S. Hassa, ItrStat Elctroc J (5) 7