Proeedings of the 14 th Interntionl Conferene on Environmentl Siene nd Tehnology Athens, Greee, 3-5 Septemer 215 ANALYSIS AND MODELLING OF RAINFALL EVENTS IOANNIDIS K., KARAGRIGORIOU A. nd LEKKAS D.F. Anlysis nd Simultion of Environmentl Systems (ASES), University of the Aegen E-mil: ssm135@ss.egen.gr ABSTRACT The purpose of this study is the sttistil nlysis of rinfll events to explore ptterns nd dependenies tht would llow the generliztion in ses of missing or trunted dt. More speifilly, in this pper we estimte intensity durtion frequeny (IDF) urves, whih re widely used to model rinfll. We use dt from Eresos meteorologil sttion nd estimte the prmeters of the model for fixed return periods. Sensitivity nlysis is onduted to hek whether the estimtes re optiml. Finlly, more generl model is pplied tht llows for simultneous modelling of rinfll durtion, intensity nd frequeny (vi return periods). Keywords: rinfll modelling, intensity-durtion-frequeny urves, sensitivity nlysis, Monte Crlo simultion 1. Introdution It is ritil for hydrologil design nd wter engineering to develop modelstht explin the pttern under whih rinfll events our nd urtely predit thefuture inflow of rinflls. More speifilly, it is neessry to find reltionshipstht onnet rinfll intensity nd durtion while simultneously tking into ountthe frequeny of ourrene. Surprisingly, model proposed lmost entury go ybernrd (1932) is still one of the most widely used for rinfll modeling. A moregenerl formul for the sme purpose ws proposed nd thoroughly exminedmore reently y Koutsoyinnis et l. (1998). The tools used for the modelling of suh hrteristis re urves ommonly known s intensity rinfll frequeny (IDF) urves. An Intensity-Durtion-Frequeny urve (IDF urve) is grphil representtion of the proility tht rinfll of given durtion nd intensity will our. The prmeters tht mke up the xes of the grph re: Rinfll Durtion (in hrs) Rinfll Intensity (in mms/hr) Rinfll Frequeny (Return Period) (in yrs) Dt omes from Eresos (Lesvos, Greee) meteorologil sttion over period of 3 yers (17.11.29-16.12.212) nd onsists of 234 rinfll events. For eh event the following vriles re reorded: (R) Totl rinfll mount (in mms) (D) Event durtion (in hrs) (AI) Averge intensity (in mm/hrs) To hve preliminry piture of the dt, we provide in Tle 1 their sttistil hrteristis. The return period (T r ) is defined s T r = 1, where F is the umultive distriution funtion of the 1-F(r) totl rinfll mount. For every return period we wnt to find n IDF urve tht desries the reltion etween durtion (D) nd intensity (I) nd stisfies the following eqution: I= (D+) (1) where, nd re prmeters to e estimted. It is ommon for prmeters nd to hve similr vlues for every return period in given lotion. In suh ses, one n pply more
generl model tht simultneously onnets rinfll durtion, intensity nd frequeny (vi return periods). The generl model, whih inludes the return period (T r ) nd n dditionl prmeter K, is defined s: I= KT r d (D+) (2) Tle 1: Desriptive sttistis of rinfll events Sine it is nturl for rinfll durtion nd intensity to e inversely proportionl we hoose to pply the proposed model whih oviously stisfies this property. The stter plot for the Eresos dt set lerly onfirms inverse proportionlity of durtion nd intensity. Grph 1: Durtion-Intensity stter plot Similr nlyses hve een onduted in mny geogrphil res nd hve resulted in glol ppreition of the model. One n refer to Stern nd Coe (1984), Lee (25) nd Ro nd Ko (26) for relted results.
2. Model estimtion We pplied model (1) for 4 different return periods, tht is for rinflls tht pper one every 2, 5, nd 2 yers. The estimtes long with the 95% onfidene intervls for the prmeters re presented in the following tle. Tle 2: Prmeter estimtes with 95% onfidene intervls The resulting IDF urves in n eqution form re: Return period T r =2 yers I= Return period T r =5 yers I= Return period T r = yers I= Return period T r =2 yer I= 4.69 (D-.47).37 12.98 (D-.29).73 15.39 (D-.36).62 21.86 (D-.26).69s As mentioned in the previous setion, n IDF urve is grphil method with 3 xes, durtion, intensity nd return period. The ove equtions in ommon plot provide this grphil piture. 6 Intensity-Durtion-Frequeny urves Tr=2 yers Tr=5 yers Tr= yers Tr=2 yers 4 Rinfll intensity (in mm/h) 3 2 2 4 6 8 12 Rinfll durtion (in hrs) Grph 2: IDF urves for T r = 2, 5, nd 2 yers
Sine estimtes for nd were lose, we pplied the generl model, given in (2), whih inludes the return period (T r ) nd the extr prmeter K nd found the following results: Tle 3: Estimtes nd onfidene intervls for generl model A 3D plot ws reted tht grphilly depits the ove eqution. Generl model Intensity (in mm/hrs) 4 4 3 3 2 2 1 9 8 7 6 4 3 2 2 4 6 8 12 Return Period (in yers) Durtion (in hrs) Grph 3: IDF urve with durtion, intensity nd return period 3. Sensitivity nlysis In non-liner models, there is hne tht estimtion methods n result in su-optiml estimtes of the prmeters. This n hppen when the ojetive funtion we try to mximize (here R 2 ) gets stuk in lol optim or is not well defined for the seleted model. To hek whether the set of prmeters is the optiml (the glol mxim hs een rehed), n nlysis using Monte Crlo type method ws onduted to investigte prmeters ehviour. For eh return period T r, we rete rndom smples ( i, i, i ), where i, i, i ~Uniform for different hoies of intervls nd eh time we keep the prmeter-vetor with the lrgest R 2. The ove proedure ws repeted times. Results, s shown in the following grph, vlidted the optimlity of the prmeter estimtes. 4. Conlusions We pplied model tht hs een extensively used in the literture to model rinfll, IDF urves. The prmeters of the model were estimted nd grphil representtions of the IDF urves were reted. Sensitivity nlysis ws onduted on the prmeters to hek for their optimlity nd the estimtes were found optiml. This ws n initil prt of ongoing reserh with dt from only one meteorologil sttion nd n e further explored in two diretions:
Sptil generliztion of the model y using dt from nery sttions in the islnd of Lesvos nd possily the roder North-Estern Aegen re. Temporl generliztion of the model y ompring predited future dt with future inflows of tul dt to vlidte the ury of the model to predit rinfll ehvior..3533.3532.3533.3532.3533.3532.779.779.779.3531.353.3529.3531.353.3529.3531.353.3529.7788.7788.7788 -.5 -.45 -.4.35.4.45 T r =2 yers 4 4.5 5 -.4 -.3 -.2.7.8.9 T r =5 yers 15 1 1 1.9883.9882.9883.9882.9883.9882.9298.9298.9298.9881.9881.9881.9297 -.6 -.4 -.2.9297.4.6.8 T r = yers.9297 14 16 18.988 -.4 -.2.988 T r =2 yers.7.8.988 15 2 25 Grph 4: R 2 s funtion of model prmeters REFERENCES 1. Bernrd, M. (1932), Formuls for rinfll intensities of long durtion, Trnstions ofthe Amerin Soiety of Civil Engineers, 96, 592 624 2. Koutsoynnis D. et l. (1998), A mthemtil frmework for studying rinfll intensity-durtionfrequeny reltionships, Journl of Hydrology, 26, 118 135 3. Lee, C.-Y. (25), Applition of Rinfll Frequeny Anlysis on studying rinfll distriution hrteristis of Chi-Nn plin re in southern Tiwn, Crop, Environment & Bioinformtis, 2, 31-38 4. Ro, A.R nd Ko, S.C. (26), Sttistil Anlysis of Indin rinfll dt, Joint Trnsporttion Reserh Progrm, Finl Report 5. Stern, R.D. nd Coe, R. (1984), A model fitting nlysis of dily rinfll dt, Journl of the Royl Sttistil Soiety. Series A (Generl), 147, 1-34.