Estimation of Point Rainfall Frequencies

Me Éireann Irish Meeorologial Servie Tehnial Noe 6 Esimaion of Poin Rainfall requenies D.L. izgerald Me Éireann, Glasnevin Hill, Dublin 9, Ireland UDC: 55.577.37 45 Oober, 2007 ISSN 393-905X

ESTIMATION O POINT RAINALL REQUENCIES WorkPakage.2, lood Sudies Updae A sudy underaken by Me Éireann for he Offie of Publi Works OPW

Table Of Conens Exeuive Summary. Page iii Inroduion.. DD Model... 2 2. Rainfall Daa.... 5 3. Mapping...... 4. Reliabiliy and Auray 6 5. Effes of Climae Change..... 7 6. Comparisons wih TN40. 8 7. Referenes.... 2 Appendix A...... 22 Developmen and Implemenaion of he Deph- Duraion-requeny relaionships Appendix B...... 3 Esimaion of he parameers of he log-logisi disribuion for lef-ensored samples Appendix C...... 34 Cheks and Confidene Inervals for he gridded rainfall esimaes Appendix D..... 42 Langbein s ormula Average Reurrene Inervals Appendix E...... 44 Duraions less han 5 mins Appendix...... 45 Glossary of erms used Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 ii

Exeuive Summary A deph duraion frequeny model is developed whih allows for he esimaion of poin rainfall frequenies for a range of duraions for any loaion in Ireland. The model onsiss of an index median rainfall and a log-logisi growh urve whih provides a muliplier of he index rainfall. Rainfall saion daa were analysed and an index rainfall exraed, inerpolaed and mapped on a 2km grid. The model was fied o series of annual maxima and he growh urve parameers were deermined; hese were also inerpolaed and mapped on a 2km grid. Compuer appliaions were wrien o apply he model and produe gridded oupus of he reurn period rainfalls whih an easily be mapped; an appliaion for deriving rariy esimaes was also developed. An aoun is also given of he reliabiliy and probable auray of he model and he probable effes of Climae Change on exreme rainfalls. Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 iii

Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 iv

Design Rainfall Esimaes for he Irish lood Sudies Updae Proje Inroduion The work was underaken by Me Éireann using funding provided by he Offie of Publi Works OPW and is a module of he lood Sudies Updae SU Proje. The requiremen was o a produe a gridded se of parameer values summarising he rainfall Deph-Duraion-requeny DD relaionship, and hereby enable he produion of onsisen esimaes of poin rainfall frequenies over duraions ranging from 5 minues o 25 days. The esimaes were o supersede hose provided in Logue 975 in whih he mehods of SR lood Sudies Repor,975 were adaped o Irish ondiions. All he design rainfall oupus are for sliding duraions e.g. an 8- day esimae is for 92 onseuive hours and may sar a any hour of day; his onrass wih he raw daa whih are mosly for fixed duraions e.g. daily values read a 0900UTC. The body of his doumen desribes he model, he daa, he onversion of daily daa from fixed o sliding duraions, he mehods of spaial inerpolaion, and onains an assessmen of reliabiliy levels and onfidene inervals for he gridded esimaes. More ehnial desripions of some of hese opis are given in he appendies. Possible effes of limae hange and omparisons wih he esimaes from Logue 975 are also inluded. The model developed enables he esimaion of rainfall frequenies a any loaion. As series of annual maxima were employed hroughou, he rainfall frequenies are expressed in erms of reurn periods; his and oher ehnial erms will be explained. Some guidane going beyond he requiremens e.g. esimaion of 5-minue and 0-minue reurn period rainfalls is given and he maer of onvering reurn periods o average reurrene inervals is reaed. An underlying assumpion is ha he 94-2004 daa used in his sudy will reasonably represen he upoming rainfall regime. Given he onsensus view ha we are in a period of global warming his is no a safe assumpion even in he medium erm. General indiaors of he effes of global warming on he preipiaion regime are available bu are heavily dependen on he pariular parameerisaions used in he general irulaion model. The indiaions of he laes assessmen of he Irish limae modelling group, C4I Communiy Climae Change Consorium for Ireland, are given. However, appropriae adjusmens are no inluded in he esimaes of he reurn period rainfalls as i appears ha for quie a number of years ino he fuure he indiaions of he effes of global warming on preipiaion regime will hange from assessmen o assessmen. The laes advie on he probable effes of limae hange on exreme rainfalls should be sough. Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007

. The Deph-Duraion-requeny DD Model A DD model onsiss of:. An index rainfall value e.g. he median or mean 2. A growh urve whih provides a muliplier of he index rainfall The model developed here applies o a single loaion and is mahemaial form enables he esimaion of reurn period rainfalls over a range of duraions D and reurn periods T. Appendix A gives he jusifiaion for using he log-logisi disribuion as he growh urve and he median of he series of annual maxima as he index rainfall. The growh urve is of form: R T, D = T, T = R 2, D, T > where is he umulaive disribuion funion. RT,D is he rainfall of duraion D wih reurn period T, where T is he average number of years beween years wih one or more rainfalls exeeding he value RT,D. I is imporan o noe ha T is no he average reurrene inerval ARI beween rainfalls exeeding RT,D. Analysis of annual maxima leads naurally o he expression of ime inervals in erms of reurn periods. Analysis of parial duraion series PDS, also ermed peak-over-hreshold POT analysis, leads o average reurrene inervals. How o onver a given ARI ino a value of T so ha rainfall for a given ARI may be esimaed from annual series is disussed in he nex seion. R2,D is he populaion median a duraion D i.e. half he annual falls exeed R2,D sine T = 2 orresponds o = 0.5 and is he index rainfall ; i as as a saling faor in he DD model. A plausible form for he variaion of he median wih duraion is see Appendix A R s 2, D = R2, D 2 where D = is a suiably hosen uni duraion whih is 24 hours d for boh he d o 25d rainfalls D >= and for duraions less han 24-hours D <. Thus he d median rainfall, R 2,, plays a pivoal role in he DD model. The full DD model ombines and 2 and is: R T, D s = R2, D T, T> 3 The exponen s in 2 deermines he muliplier of R2, yielding R2,D as duraion, D, varies. The exponen in is he shape parameer of he log-logisi growh urve see Appendix A and deermines he muliplier of R2,D whih yields RT, D, he rainfall a reurn period T and duraion D. Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 2

The final forms of he models onsised of 3 wih 24 hours as he uni duraion and wih he exponens of form:. o 25 days: D = a +b lnd, D ranging o 25 4 s D = e + f lnd 5 Here ln is he naural logarihm. Noe ha wih D =, = a = 24, he shape parameer a he -day 24-hour duraion. Boh exponens are aken o be funions of D and no of T..2 24hours o 5 minues: D = 24 + h ln D, D < 6 s = s 7 Here D = 0.75 a 8 hours, D = 0.5 a 2 hours, D = 0.25 a 6 hours and so on. The shape parameer is, again, a funion of duraion bu he duraion exponen s is no. Only wo parameers, h and s need be deermined as 24 is available from he work on o 25 days.. The reasons for hese hoies are disussed in Appendix A..3 Implemening he DD model The wo DD models were fied o he saion daa by a mehod desribed in Appendix A. Geosaisial mehods Kianidis, 997 were used o inerpolae he saion values of he parameers o he 2km grid. Two esimaes are required:. RT,D he reurn period rainfall a duraion D, given reurn period T and duraion D and 2. T, he reurn period given duraion D and rainfall amoun RT,D Esimaing RT,D and/or T a grid poins is hen sraighforward. The mehod of inerpolaion beween grid poins is desribed a he end of Appendix A where he maer of he mos appropriae parameer values o aah o he represenaive poin of a ahmen is briefly disussed. Programs were wrien o esimae RT,D over he 2km grid and also o alulae eiher RT,D or T a any loaion..4 Conversion aors for Parial Duraion Series PDS By definiion annual maximum series AMS onsis of he highes fall for eah year; he seond highes fall is ignored wheher or no i exeeds he highes fall in oher years. Parial duraion series PDS onsis of all falls exeeding a erain hreshold ogeher wih heir imes of ourrene. The reurn period T is bes hough of as he inverse of a probabiliy e.g. he rainfall orresponding o T = 50 has a probabiliy of 0.02 of being exeeded nex year. Risk an also be expressed in erms of reurn period rainfalls. The rainfall orresponding o T = 238 years has a probabiliy of 0.9 of no being exeeded during he nex 25 years Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 3

Zuhini & Adamson, 989 i.e. he risk r is 0.. Assuming ha he annual maxima are saisially independen and all are drawn from he same disribuion n T = / r 8 where he design horizon is n years and he risk is r. The analysis of PDS gives he average period beween rainfall evens ha exeed a pariular value and is ofen ermed he average reurrene inerval ARI for a given duraion D. or high values of T, values of ARI and T are nearly equal bu for T less han 20 years he differene an be signifian. Langbein 949 provides a formula see Appendix D for he relaionship beween T and ARI whih yields: T =.6 for ARI =/2 T =.58 for ARI = T = 2.54 for ARI = 2 The approximaion T = ARI + 0.5 improves as ARI inreases. Thus he PDS rainfall for ARI = 2 is he reurn period rainfall for T =2.54 years. As annual series were used in his sudy we an only esimae growh urves for reurn period rainfalls bu he Langbein formula enables heir onversion ino PDS rainfalls wih a known ARI. Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 4

2. Rainfall daa for he lood Sudies Updae 2. Irish Saions Noe: In his seion Irish daa is aken o mean daa from rainfall saions in he Republi of Ireland 2.. Daily0900-0900UTC daa: The requiremen was o form series of annual maxima for a leas six duraions ranging from one o weny five days. On assessing he amoun of qualiy onrol needed o rea eah duraion i was deided ha he produion of omplee daily series was he beer opion. The Me Éireann arhive for periods wihin 94-2004 was used as his had already undergone exensive qualiy onrol. However, dry monhs and daily falls in exess of 75.0mm were re-examined and some fauly daa orreed before forming he iniial able of daily rainfall. High-qualiy saions were piked ou by examining he number of aumulaed oals or missing days in eah monh of reord. or hese saions aumulaions were broken up ino daily values and missing days esimaed o give a omplee se of daily values. or mos saions esimaions were made by using up o six neighbouring saions wih similar wihin abou 0% average annual rainfall AAR; hese were ranked in order of preferene and he firs o have a omplee reord for a period requiring esimaion used as his had he pereived advanage of using a oal aually reorded in he general area raher han a weighed mean. If he AAR of he saion requiring esimaion differed onsiderably from is neares neighbours hen hree neighbours wih omplee daily reords were hosen and he global monhly raios of arge saion over neighbour deermined. or missing monhs he oal was aken as a weighed mean of he hree esimaed oals and hen redued o daily values by referene o he daily falls a he neares neighbour. Monhs wih a oal bu no daily values were reaed by forming he weighed mean of he hree neighbours and giving i equal weigh o he arge saion oal; he agreemen beween he wo was usually good. Again daily values were found from he esimaed monhly oal by referene o he neares neighbour. The remaining missing or aumulaed days were reaed by muliplying he daily values a he neares available neighbour by he monhly raio. Annual maxima exraed from he original and reaed daa showed ha he differenes were usually small as he qualiy of he hosen saions was high. Using hese mehods daily values for 474 saions were exraed; heir average period of reord was 4.2 years, wih a range of 20 o 64 years. Annual maxima for, 2, 3, 4, 6, 8, 0, 2, 6, 20 and 25 days ogeher wih he saring dae were pu ino a able of 24,98 rows. The loaion of hese saions plus 03 saions from Norhern Ireland are shown in igure. Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 5

2..2 Shor-duraion falls Daa for nine sliding duraions beween 5 minues and 24 hours are available from 39 saions for periods ranging from 5 o 55 years bu 37 of he saions have 30 or more years of reord. The loaions of he 39 saions used are shown in igure 2. Mosly he daa were exraed from Dines reorder hars bu sine he mid-nineies hese have been replaed by ipping buke reorders TBR a some saions. In he periods of overlap he differenes beween he Dines and TBRs were found o be generally small and no adjusmen for he ransiion was made. The maxima of all falls aaining or exeeding a leas one of a se of hresholds were exraed. The hresholds are: Duraion 5m 30m h 2h 3h 4h 6h 2h 24h Threshold mm 4.0 5.0 6.0 0.0 2.5 5.0 20.0 25.0 30.0 The dae assigned is ha of he day on whih mos, or all, of he oal for he rainfall even was reorded. Qualiy onrol onsised of:. Cheking he daes of ourrene and values of he 2. 0900-0900UTC oals of 30mm or more agains hose of he 24h values. 3. Examining he rows of he able for onsiseny. Doubful or missing values deeed were esimaed by referene o he neares saions. or he 6 synopi saions a more exensive qualiy onrol was possible as all lok- hour o 24 lok-hour oals reahing or exeeding he hresholds and heir daes of ourrene were heked agains he orresponding absolue maximum values. To exra annual series we mus deal wih he problem of years wih no value exeeding he hreshold. In he ase of synopi saions good esimaes an be made from he lok-hour values, if desired. The sraegy applied o all saions was o rus in he qualiy of he daa and assume ha he missing values were below he appropriae hresholds and hene unimporan for fiing a probabiliy disribuion log-logisi o exremes. The sample ould be regarded as ensored and mehods ha ake aoun of his developed o deermine he parameers Appendix B. In he even, only he ordered se of values greaer han or equal o he sample median was needed o fi he disribuion; forunaely, a all saions hese series were omplee. 2.2 Daa from Norhern Ireland Through he good offies of he UK Me Offie and he Cenre for Eology & Hydrology Wallingford he daily and hourly series of annual maxima from Norhern Ireland used in he lood Esimaion Handbook 999 were made available. Of he daily saions 03 wih a leas 20 years of reord a duraions of, 2, 4 and 8 days were used in his sudy. The hourly daa onsised of 8 saions wih lenghs of reord beween and 9 years and wih daa a duraions of, 2, 4, 6, 2, 8 and 24 hours. Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 6

Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 7

2.3 Conversion from fixed o sliding duraions The daa available were he -day, 2-day..25-day annual maxima derived from daily oals read a 0900UTC. However, he aim was o esimae he 24-hour, 48- hour 600-hour reurn period rainfalls where 24-hour is he annual maximum for any 24-hour period wihin he year. How o do his from he 0900-0900UTC daa involves onversion faors from fixed o sliding duraions. The maer was examined in wo ways:. A he 4 longer-erm synopi saions he log-logisi disribuion was fied o he, 2, 3, 4, 6, 8, 0, 2, 6, 20 and 25-day annual maxima of he 0900-0900UTC rainfall oals and also o heir 24, 48, 72, 96, 44, 92, 240, 288, 384, 480 and 600 lok-hour oals. Reurn period rainfalls of 2, 5, 0, 20, 50, 00, 250, 500 and 000 years were alulaed for eah and he raios examined. No allowane was made for he ransiion from lok-hour values o absolue values as inspeion showed ha, even for 24 hours, he faor was very lose o one. Use of he year January-Deember revealed signifian endof-year effes espeially a he longer duraions. As a hek he April-Marh period was used and i was found ha he values of he reurn period rainfalls were lile hanged and so April-Marh was used as he rainfall year. 2. or he eleven 0900-0900UTC duraions, 2, 3, 4, 6, 8, 0, 2, 6, 20 and 25- days hresholds of 23, 30, 35, 40, 48, 55, 60, 65, 70, 75 and 80mm were se and he n-day exeedanes ompared wih he orresponding lok-hour values; orresponding was aken as saring wihin a erain inerval ha had he saring dae of he 0900-0900UTC aumulaions as a fairly enral value. Various inervals were ried and he general effe of narrowing he inerval n xi was o slighly derease he raios. Sample average raios = were n i= yi derived from he individual evens, his in preferene o he raio of he sample avg xi averages = whih gave slighly lower values. As poined ou in avg y i Dwyer & Reed 995 he raio of he sample averages is equivalen o a weighed sum of he individual raios, wih greaer weigh given o he larger evens. The resuls of fiing he log-logisi disribuion, presened below in Table A, sugges ha for he median rainfall he onversion faors should be : d 2d 3d 4d 6d 8d 0d 2d 6d 20d 25d RP =2.5.08.06.05.04.04.03.03.02.02.0 The work on evens gives : Evens..06.04.04.03.03.02.02.0.0.0 Noe ha he faors based on means of aual evens are generally lower. This may be aribued in fair measure o he fa ha for evens he imes of ommenemen of he 0900-0900UTC and hourly aumulaions are fairly lose while, espeially for he 24 and 48-hour duraions, he ime of sar of he 0900-0900UTC and hourly annual maxima may, on oasion, be separaed by monhs i.e. here is an exra soure of Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 8

variaion. If you exra he mean of. as a haraerisi muliplier for he -day evens i is from a posiively skewed disribuion wih a median of.05 bu wih 32 of a oal 2285 ases in exess of.25 and a range from.0 o 2.0. In Table A he range of he RP2 median faor is from.086 o.257 wih a mean of.53, a median.48 and a sandard deviaion of abou 0.03. Sine i is he index rainfall, he values for he 2-year reurn period rainfall are mos imporan bu i is of ineres o see how he faors vary wih inreasing reurn period. Table A gives he means over foureen synopi saions of he raios of he lok-hour esimaes of reurn period rainfalls o he orresponding fixed-duraion values for eah of he eleven duraions. In general he raios derease wih reurn period bu are nearly onsan for he one-day rainfalls and for duraions of 6 days or more, while he 2- day falls inrease wih reurn period. As expeed he derease of he raio wih duraion is onsisen over all reurn periods. Table A ixed v Sliding Duraions Mean of Adjusmen aors Days RP 2 RP5 RP0 RP20 RP50 RP00 RP250 RP500 RP000 Evens.53.48.47.46.45.46.48.49.52.0 2.076.064.058.05.044.038.032.026.022.060 3.062.048.040.032.023.07.009.002 0.996.044 4.044.036.032.028.023.020.06.03.0.037 6.04.029.022.05.007.002 0.994 0.989 0.984.027 8.042.032.026.02.05.00.004.000 0.996.024 0.034.027.022.08.03.00.005.002 0.999.02 2.026.03.034.037.040.043.048.05.054.07 6.023.022.02.02.02.02.02.02.02.04 20.07.05.05.04.03.02.02.0.0.0 25.06.0.008.006.003.00 0.999 0.995 0.993.00 Sine he onversion from -day 0900-0900UTC o 24 hour median values is ruial o he shor-duraion model, he medians of annual maximum series a he 39 saions for whih absolue 24-hour maxima were available were ompared wih he medians of he 0900-0900UTC annual maximum series over he same years. The quarile summary of he 39 raios is: Minimum Q Median Mean Q3 Maximum.0.09.23.24.64.33 Only 3 of he 39 values exeeded.20 and only exeeded.25. The inerquarile range yields an esimae of abou 0.05 for he sandard deviaion. Linear regression gives.28 as he raio. The ompeing mean value is.5, he onversion faor for he log-logisi esimae of he median raio and his was adoped as i ompares wih.6 used in EH 999 and.4 used in he New Zealand HIRDS sysem. Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 9

2.4 Noaion When i is neessary o disinguish beween he wo, he sliding-duraion rainfalls obained by muliplying he fixed-duraion falls by he appropriae onversion faor will be referred o as d, 2d, 3d 25d rainfalls; he fixed duraion falls will be labelled d09, 2d09,.25d09. Sub-daily duraions are always sliding duraions. The Irish shor-duraion daa onsis of absolue maxima exraed from rainfall evens for duraions up o and inluding 24 hours. In wha follows he laer value is referred o as abs24. The d value is a lose approximaion o abs24 and has he advanage of being muh more widely available. The d rainfall will be labelled slide24 when onsidering duraions of 24h o 5m i.e. i is being used as a subsiue for abs24. Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 0

3. Mapping 3. Mapping he Index Rainfall The pivoal value is he median one-day rainfall, RMEDd whih is losely approximaed by: RMED d =.5RMEDd 09 9 The number of values of RMEDd available was 577 of whih 03 were in Norhern Ireland. This did no seem adequae o produe values on a 2km grid. orunaely, here is a srong linear relaionship beween AAR average annual rainfall and RMEDd. In he ase of he 96-990 averages AAR690 he relaion RMED d = 0.0366 AAR690 0 has a oeffiien of deerminaion R 2 = 0. 978 ha an be inreased o 0.993 by he addiion of loaion o-ordinaes e.g. easing and norhing in meres for eah saion: RMEDd = 0.0333AAR + 0.000062easing - 0.0000353norhing To exploi he srong relaionships 0 or values of AAR690 on a grid were needed. The saions for whih AAR690 were available numbered 946 of whih 242 were in Norhern Ireland. Drawing on onsiderable experiene of esimaing AAR, 0 values were added in daa-sparse areas. The 047 daa poins were used o produe values of AAR690 on a 2km grid using geosaisial mehods Kianidis, 997 and he R pakage, geor. The gross dependene of AAR on elevaion and loaion was removed by linear regression and he residuals inerpolaed o he grid using ordinary kriging wih a moving neighbourhood i.e. weighed linear ombinaions of nearby values. The values of he regression equaion a eah grid poin were added o he gridded residuals o produe he final resul. Comparison wih previously mapped values of AAR690 showed very good agreemen. Kriging was again used o produe gridded values of RMEDd. The number of saions having values of boh RMEDd and AAR690 was 468. The residuals from he regression of RMEDd on AAR690 were inerpolaed o he 2km grid and he regression esimaes added o produe he final mapping, igure 3. 3.. Comparison of reorded and inerpolaed values of he median rainfall There were 09 saions having a value of RMEDd bu no value of AAR690. Using gridded RMEDd as he daa values, he 09 kriging esimaes KMEDd of RMEDd were obained ogeher wih heir kriging sandard errors ksd. RMEDd The raio is approximaely normally disribued wih mean 0.995 and KMEDd sandard error of 0.084. The perenage differenes are less han 5% in 02 of he 09 ases, 66 are less han 7.5%, 50 less han 5% and 30 less han 2.5%. I indiaes ha KMEDd is an appropriae esimaor of RMEDd and jusifies he assumpion ha he mapped value is a good esimaor of he aual median rainfall a a sie. The kriging sandard error, ksd,of KMEDd refles he variabiliy of he esimaes whih are weighed means of he surrounding grid-poin values and so he sandard Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007

error will be higher where he median rainfall hanges more rapidly wih disane e.g. in and near mounainous areas. I is reasonably well approximaed by: 2 2 ksd = 0.00 KMEDd, R = 0.8 2 This kriging sandard error inreases monoonially wih KMEDd and ranges from abou mm o 9mm as KMEDd varies from 3 o 94 mm. As he higher values of KMEDd our in he mounains where he densiy of he raingauge nework is lowes, he unerainy aahing o he inerpolaed values is highes here. The kriging sandard error an be used o esimae he error of inerpolaion and hene is of ineres as a measure of he reliabiliy of KMEDd. Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 2

3.2 Mapping he model parameers 3.2. d o 25d sliding duraions No useful relaionships were found beween he four model parameers and median rainfall, AAR or easing and norhing. Hene, all parameers were inerpolaed o he 2km grid using ordinary kriging wih a moving neighbourhood i.e. a weighed mean wih weigh dependen on disane. 3.2.2 24h o 5m sliding duraions The model adoped mean ha no new mappings were required. The gridded d shape parameer = 24 and he grid of he median one-day rainfall were needed bu hese had been mapped as par of he work on he d o 25d model. The wo model parameers h and s for he shor-duraions falls an hen be alulaed a any poin by means of he sheme oulined in Appendix A. 3.2.3 Enforing Consiseny on he model parameers or he 4-parameer model D = a + b lnd and s D = e + f lnd 3 To mee he requiremen ha, for he same reurn period T, he esimae should inrease wih duraion we mus have: e + 2f lnd + b lnt- > 0 4 Near 000 years and near day his beomes e + 7b > 0 or a given duraion, o have a posiive rae of hange wih inreasing reurn period T requires: a + b lnd > 0 5 A abou 30 days his requires a + 3.5 b > 0 3.2.4 d o 25d Irish model In nearly 7% of ases here ours he problem ha a high reurn periods ~000years ha he 2-day esimae may be slighly less han he -day esimae. The basi reason is ha he shape parameer is dereasing oo sharply. This an be simply orreed by enforing he ondiion : e + 7b > 0 6 or a given duraion he reurn period rainfall esimaes mus inrease wih T bu his presened no problem. An example of he oupu is shown in igure 4. 3.2.5 Consiseny of he model for he Norhern Ireland daa The parameer values for he Irish daa for d o 8d had similar quariles o he d o 25d daa see Table, Appendix A. Hene, i was deided o use he Norhern Ireland daa as if i were for periods up o 25d, a onvenien bu deidedly risky assumpion. The ondiion e+7b > 0 had o be imposed in nearly 7% of ases and his adjusmen also remedied he failure of 8% of he saions o mee he requiremen a +3.5 b >0. Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 3

Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 4

3.2.6 Model for 24 hours o 5 minues Reall ha he model is of form: s 24 + h ln D = R2, D T 7 R T, D A a given duraion he esimae inreases wih reurn period T if: + h ln D 0 8 24 > A a given reurn period he esimae inreases wih duraion if: s + hln T > 0 9 These ondiions were kep in mind when assigning values o h and so no onsiseny problems arose wih he shor-duraion model as 0.03 h 0, ln D 0 and s 0.33. An example of he model oupu is shown in igure 5. 3.3 alls of duraions less han 5 minues Esimaes of rainfall dephs for duraions less han 5min are onsidered in Appendix E. The onlusion is ha i is probably bes o employ 5-minue esimaes and apply he given formula for he mean fraion as a funion of he fraional duraion. In he ase of en and five minue duraions he fraions of he fifeen minue dephs are 0.85 and 0.6 respeively. Esimaion for inervals of less han 5 minues duraion should be regarded as highly speulaive. Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 5

4. Reliabiliy and Probable Auray of he Reurn Period Rainfalls These maers are disussed in more deail in Appendix C where he enaive onlusions are:. The 24-hour o 600-hour esimaes may be used wih fair onfidene for reurn periods up o abou 500 years 2. The esimaes for duraions of less han 24 hours may be used wih fair onfidene for reurn periods up o abou 250 years. The saisial analysis was done on he assumpion ha he daa are represenaive of he fuure rainfall regime. Given onerns abou probable hanges in preipiaion limae in he shor o medium erm due o global warming his is by no means assured. How, or even if, o adjus he esimaes is no a quesion o whih here is a good answer a presen bu some general, perhaps even useful, observaions are made in he nex seion. Aeping he urren onsensus on he high likelihood of hanges in he preipiaion limae, here seems o be lile sense in esimaing 500-year reurn period rainfalls. However, equaion 8 shows ha he very praial maer of esimaing a 0% risk for a ime horizon of 50 years requires a reurn period rainfall for 475 years. Also in Appendix C a mehod is given for aahing a sandard error o any esimaed reurn-period rainfall from knowledge of he values of he median rainfall and he shape parameer plus a very large assumpion abou he effeive sample size. This saisial measure of spread gives an idea of he probable auray of he value. I has o be regarded as merely a rough guide. Informaion on he rainfall exremes of he las 00 years or so may be had in Rohan 986 or Hand e al.2004. Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 6

5. Probable Effes of Climae Change on Exreme Rainfalls The mos reen IPCC repor on regional limae projeions Chrisensen e al., 2007 saes ha over Norhern Europe beween 980-999 and 2080-2099 he median hange in preipiaion was a 5% inrease for he monhs of DJ, 2% for MAM, 2% for JJA and 8% for SON. Relaive o he wees period in 980-999 here was a 43% inrease in we evens in DJ. However hese were over a large area and he model resoluion a ~ 200km was raher oarse. The laes assessmen from C4I Communiy Climae Change Consorium for Ireland saes ha over Ireland by mid-enury here may well be:. An inrease of abou 5% in winer rainfall amouns 2. Drier summers wih 20% lower preipiaion in some areas, mos likely he eas and souheas. 3. A 20% inrease in he wo-day exreme rainfalls, espeially in norhern areas and smaller inreases in he one and five day exremes. 4. More frequen rainfall exremes in auumn. All his would sugges an inrease in exreme rainfalls for duraions of 24 hours or more, espeially in auumn-winer. Drier summers sugges an inrease in he frequeny of droughs. The breakdown of droughs is someimes he oasion of heavy shor-duraion rainfalls. The general suggesion of mos of he senarios is ha safey faors of maybe as muh as 20% on rainfall deph migh be inorporaed as an aemp a a no regres poliy in he fae of unerainy. A purely saisial exerise in izgerald 2005 omes up wih safey faors for - day rainfalls a Phoenix ParkDublin of abou % for a 20-year reurn period rainfall, 9% for he 00-year value and 33% for 000-year rainfall, his based on 22 years of daily daa. Given he wide variaion in prediions from assessmen o assessmen or beween saisial exerises i would be wise always o seek he laes advie on he probable effes of limae hange on exremes of preipiaion before onsidering an adjusmen o he model esimaes of reurn period rainfall. Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 7

6. Comparisons wih he esimaes of Tehnial Noe 40 TN40 Sine 975 he mehods disussed in TN40 Logue,975 have been used o supply design rainfalls in Ireland. The mehods were hose of he Briish lood Sudies ReporSR,975 adaped o Irish ondiions. The mehodology was o fi a se of growh urves based on he generalised exreme value disribuion Hosking & Wallis, 997 o series of annual maxima. The index rainfalls were he 2-day 0900-0900UTC rainfall wih a five year reurn period and he -hour rainfall wih five year reurn period. Growh urves were defined mosly in erms of duraion and average annual rainfall AAR. The presen sudy uses he log-logisi disribuion as he growh urve and he median as he index rainfall. or an AAR of 00mm he ables of TN40 an be used o haraerize is growh urves in erms of he shape parameer of he log-logisi disribuion i.e. he values of he 50-year reurn period rainfall over he median rainfall exraed from Table III of TN40 were insered in he equaion M 50 / M 2 = T = 49. Sine 00mm is near he naional average rainfall hese values are aken are roughly orresponding o he mean values of he parameers of he DD model i.e. he mean value of 0.9 for 24 was used ogeher wih rae of hange parameers wih he logarihm of duraion of -0.028 for 24hours or longer and of -0.00 for duraions of less han 24 hours o yield he following: Table C Comparison of TN40 and SU in erms of heir Mean log-logisi Shape Parameers Sudy 5m 30m h 2h 4 h 6 h 2h d 2 d 4 d 8 d 6d 25d TN40 0.240 0.230 0.220 0.200 0.85 0.80 0.70 0.60 0.55 0.30 0.20 0.0 0.05 SU 0.235 0.230 0.220 0.25 0.20 0.200 0.95 0.90 0.70 0.50 0.30 0.0 0.00 TN40 employed alendar monh values and hese were aken as equivalen o he SU 25d falls. Agreemen is surprisingly good wih he main differenes enered on he 24-hour mean. More signifian are he differenes in he ranges of he shape parameers e.g. he 24-hour shape parameer has a range from 0.9 o 0.0 in TN40 and he -hour shape parameer ranges 0.26 o 0.6, he highes values in areas of low AAR and he lowes values in mounainous areas. In he SU sudy he ranges of he shape parameers are wider e.g. he d shape parameer varies beween 0.3 and 0. and igure 6 shows ha quie a number of he lowes values are in lowland areas. This helps o explain he wide ranges of he spo value perenages for he 24-hour duraion in igure 7 and for he -hour duraion in igure 8 The main poin o noe is ha mos of he values are in he range 85% o 25% in boh ases. Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 8

Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 9

Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 20

7. Referenes Abramowiz, M. & Segun I. A. 965, Handbook of Mahemaial unions, Dover, New York Chrisensen e al. 2007, Regional Climae Change Projeions, Climae Change 2007: The Physial Siene Basis, Conribuions of Working Group o he ourh Assessmen Repor of he Inergovernmenal Panel on Climae Change [ Solomon e al. eds.], CUP C4I Commumiy Climae Change Consorium for Ireland hp://www.4i.ie David, H. A. 98, Order Saisis, Wiley, New York Dwyer, I.J. & Reed, D.W. 995, Allowane for disreizaion in hydrologial and environmenal risk esimaion, IH Repor No. 23, Cenre for Eology and Hydrology, Wallingford EH 999, lood Esimaion Handbook, Vol. 2 Rainfall requeny Esimaion, Cenre for Eology & Hydrology, Wallingford eller, W. 968, An Inroduion o Probabiliy Theory and is Appliaions, Vol., Wiley, New York eller, W. 97, An Inroduion o Probabiliy Theory and is Appliaions, Vol. 2, Wiley, New York izgerald, D. L. 2005, Analysis of exreme rainfall using he log logisi disribuion, Sohasi Environmenal Researh and Risk Assessmen SERRA, Vol. 9, No. 4, pp 249-257, Springer, Berlin SR 975, Unied Kingdom lood Sudies Repor, Vol. 2 Meeorologial Sudies, Naural Environmen Researh Counil NERC, London Hand W. H. e al. 2004, A ase sudy of wenieh-enury exreme rainfall evens in he Unied Kingdom wih impliaions for foreasing, Meeor. Appl., Vol., pp5-3 HIRDS 2002 High Inensiy Rainfall Design Sysem, Naional Insiue of Waer and Amospheri Researh NIWA, Wellingon, NZ HMSO 98 Handbook of Meeorologial Insrumens, Her Majesy s Saionery Offie, London Hosking, J. R. & Wallis, J. R. 997, Regional requeny Analysis, CUP Kianidis, P. K. 997 An Inroduion o Geosaisis, CUP Kuosoyiannis, D. e al. 998, A mahemaial framework for sudying rainfall inensiy-duraion-frequeny relaionships, Jnl. of Hydrol., Vol. 206, pp 8-35, Elsevier Langbein W. B. 949, Annual floods and he parial duraion mehod, Trans. Amer. Geophys. Union, Vol. 30, pp 879-88 Logue, J. J. 975, Exreme Rainfalls in Ireland, Tehnial Noe No. 40, Me Eireann, Dublin NOAA 2004, Preipiaion requeny Alas 4, Vol. 2 R Developmen Core Team 2006 R: A Language and Environmen for Saisial Compuing Ribeiro, P. J. & Diggle, P. J. 2006, geor version.6-9, A pakage for Geosaisial Analysis Rohan P.K. 986, Climae of Ireland, Governmen Sales Publiaion Offie, Dublin Wang, Q. J. 990, Esimaion of he GEV Disribuion from ensored samples by he mehod of Parial Probabiliy Weighed Momens, Jnl. of Hydrol., Vol. 20, pp 03-4 Zuhini, W. & Adamson, P.T. 989, Boosrap onfidene inervals for design sorms from exeedane series, Hydrologial Sienes Jnl., Vol. 34, pp 4-48 Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 2

Appendix A Developmen and Implemenaion of he Deph-Duraion-requenyDD Relaionships The DD model applied is based on he median, R2,D, as he index rainfall; his makes he log-logisi disribuion izgerald,2005 a naural hoie for he growh urve sine is usual 3-parameer form in erms of, he umulaive disribuion funion, is: R T, D g = = T, a > 0, > 0, g >= 0 A a RT,D is he rainfall a reurn period T and duraion D, wih T =2 = 0.5 being he median R2,D. Hene a + g = R2,D i.e. he sum of he loaion and sale parameers is he median. or posiive random variables, suh as rainfall, he loaion parameer may be aken as he minimum. There are wo ses of annual maxima o onsider:. 0900-0900 UTC rainfall aumulaions for duraions beween and 25 days 2. he shor-duraion series for 9 duraions ranging 5 minues o one day, all being sliding duraions. Unlike he 0900-0900UTC daily series, he shor-duraion daa may have missing years as values were reorded only if above prese hresholds. or analysis of individual duraions, mehods of parameer esimaion for lef-ensored samples were developed. Deails are given in Appendix B. Iniially, he 3-parameer log-logisi disribuion was fied o he daily 0900-0900UTC series of annual maxima using he mehods oulined in izgerald2005. This work revealed ha for boh daases he probabiliy weighed momen PWM and maximum likelihood ML esimaes of he log-logisi parameers exhibied oo irregular a variaion over he ranges of duraion o be suiable for direly fiing a DD relaion a mos saions. or he shor-duraion daase he average over some 40 saions showed he mean of he log-logisi shape parameer inreasing, albei somewha unseadily, from 24 hours o a maximum a hour, wih slighly lower values a 30 and 5 minues. The sale parameer expressed as a fraion of he median showed lile paern beyond a endeny for he lowes values o our in he o 4-hour duraions. Anoher feaure was ha he values of he shape parameer were someimes undesirably high >= 0.4 beause here were high growh raes a low values of he sale parameer. or he 474 daily saions he average shape parameer dereased slowly from day or 2 days o 25 days. The sale parameer expressed as a fraion of he median showed is highes values a he long duraions, wih he lowes values in he o 3-day duraions bu, overall, he variaion wih duraion was errai. Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 22

The firs aemp o overome he problems was o pu he log-logisi disribuion in growh-urve form: R T, D a D = + [ T ] D, T > A2 R2, D R2, D a D The parameers K D = and D were expressed as funions of duraion, R2, D ypially s D = a + bln D and K D = onsan, e + f D and D. The sample median was used for R2, D. Maxima greaer han or equal o he median R T, D were exraed from he full daases, pu in he form and ordered samples R2, D formed. The median ploing posiion lood Sudies Repor, Vol. 2, 975 was used for T, wih T = 2 for he firs row. A able wih duraion varying aross he rows and reurn period T 2 down he olumns was hen formed. All series of maxima greaer han or equal o he median were omplee and here was no furher need of mehods of parameer esimaion for ensored samples. As suggesed by Kuosoyiannis e al. 998 he parameers were esimaed simulaneously. In general here was good agreemen beween he PWM or ML soluions for individual duraions and he pormaneau soluion, wih D = a + blnd and K D = e + fd. However, as wih he PWM and ML soluions, he values of D and K D fluuaed markedly beween neighbouring saions making i diffiul o dee any paern. I was deided use a growh urve wih he median as sale parameer i.e. o ry loglinear relaions of form: R T, D s = D T R2,, T > A3 where D = is a suiably hosen uni duraion or variaion aross he rows his implies R T, D s = D R T, A4 and for variaion down he olumns a log-logisi median growh urve: R T, D = T R2, D A5 inal form of he o 25-day model or he shape parameer he earlier work had indiaed a slow variaion wih duraion ; his was assumed o be of form D = a + b ln D A6 Assuming s is onsan in equaion A4 in onjunion wih A5 and A6 gave good resuls bu here was suffiien reduion in he range of he residuals o make i worhwhile o assume s o also be of form s D = e + f lnd A7 Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 23

The four parameers were esimaed simulaneously using he R rouine lm and he 2 resuls were exellen wih a oeffiien of deerminaion R in exess of 0.99. In addiion agreemen wih he log-logisi PWM/ ML quanile esimaes and hose derived from A3, A6 and A7 were esed for reurn periods up o 250 years and found good. All four parameers were highly signifian in he sense of being muh greaer han heir sandard errors. Noe han for D = we ge = a in equaion A6 and s = e in A7. Use of he daily 0900-0900UTC daa for Norhern Ireland Daily daa for, 2, 4 and 8 days were available and 03 saions wih more han 20 years of reord were used. Table A ompares he NI -day and s values wih hose derived for he Me Eireann daily saions, using a fiing of equaion A3 o,2, 3,4,6,8 days and,2,3,4,6,8,0,2,6,20 and 25days, day being he uni duraion. Table A Values of he -day parameers of he DD model N I,2,4,8 Irl,2,3,4,6,8 Irl o 25 days s s s Minimum 0.23 0.253 0.08 0.69 0.079 0. 76 Q 0.7 0.362 0.54 0.333 0.56 0.3 Median 0.98 0.4 0.80 0.370 0.83 0.348 Mean 0.200 0.409 0.8 0.372 0.84 0.347 Q3 0.227 0.453 0.200 0.40 0.205 0.377 Maximum 0.35 0.554 0.34 0.56 0.347 0.508 I is a mild surprise ha he enral values for NI are he higher. However, he sandard deviaions of and s are roughly 0.04 and 0.07 respeively, he duraions and periods of reord differen and so he differenes are no highly signifian. I is ineresing ha he -day parameer esimaes for he Irish o 8 and o 25-day daases are so similar. This lends some jusifiaion o using he NI esimaes for he 4-parameer model on he same basis as hose of he 4-parameer model derived from he o 25-day daase bu, learly, esimaes of reurn period rainfalls for duraions longer han abou 0 or 2 days for loaions in NI should be reaed wih auion. Effes on he parameer esimaes of onvering from fixed o sliding duraions Table A is based on fixed-duraion 0900-0900UTC daa. On shifing o sliding duraions by muliplying he 0900-0900UTC daa by he appropriae onversion faors and refiing he DD model he shape parameer of he growh urve is unhanged and he duraion exponen s dereases by 0.09. Implemening he DD model a grid poins for sliding duraions of d o 25d As desribed in he main ex, saion values of he four parameers required by equaions A6 and A7 and also saion values of he median d rainfall were exrapolaed o a 2km grid. A grid poins equaion A3 ould hen be used o derive he reurn period rainfalls for any duraion beween d and 25d. Noe ha all model oupus are for sliding duraions. If fixed-duraion design rainfalls are needed i is neessary o divide he sliding-duraion rainfall by he appropriae faor. The faors are: d 2d 3d 4d 6d 8d 0d 2d 6d 20d 25d.5.08.06.05.04.04.03.03.02.02.0 Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 24

inal form of he 24-hour o 5-minue model The firs maer o resolve onerned he pivoal 24-hour sliding value, slide24 whih is merely a relabelling of he median d rainfall. This hange of noaion is emphasise he near equivalene of he median d rainfall o he median of he 24-hour maximum, abs24, available a he shor-duraion saions. To es he effe of using slide24 insead of abs24, he enire olumn of he ordered 24-hour values and heir median ploing posiions available a he 39 shor-duraion saions were exraed from he able. The se for omparison was generaed using equaion A5 wih:. duraion D = 2. slide24 insead of abs24 as R2,, he median rainfall 3. d or 24-hour shape parameer = = 24 orunaely, inspeion showed ha agreemen was good. Tesing for one o one orrespondene over all he 24-hour daa a all he shor-duraion saions he mean regression oeffiien was.037 and he oeffiien of deerminaion R 2 = 0. 994. The larger differenes were usually a he higher reurn periods, wih he generaed values exeeding he reorded values. The residuals were: Minimum s quarile Median 3 rd Quarile Maximum -7.2-2.3-0.2 +2. 28.6 Slide24 and he assoiaed shape parameer 24 were available a all he daily saions. Slide24 was used insead of abs24 even a he shor-duraion saions. However, in seing up he shor-duraion model abs24 was used as a hek on he resuls obained using slide24. or he 24-hour abs24 o 5-minue daase, assuming D = a +b lnd and also s D = d + e lnd again gave he highes oeffiien of deerminaion in exess of 0.97 and ofen over 0.99. Unforunaely, a some saions, b and/or e were no well deermined. However, assuming or s or boh onsan yielded only slighly lower 2 R. Indeed, 2, 3 and 4-parameer models all performed well in omparisons wih quanile esimaes from A or from he PWM/ML esimaes for individual duraions. Esimaing from he daa was o ignore he parameer 24 obained from he daily series; 24 had proved suessful in generaing he series suessfully subsiued for he reorded values abs24. Using he oeffiien of deerminaion as a measure of fi, i was found ha nohing was los by:. Using 24 = he -day shape parameer from he daily daa, as he saring value of he shape parameer a he uni 24-hour duraion. 2. The subsiuion of slide24, he median sliding 24-hour value, for he median absolue value abs24 3. Assuming he duraion exponen s onsan i.e. no a funion of duraion giving a DD model : R T, D s 24 + h ln D = R 2, D T A8 Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 25

s Equaion A8 an be regarded as going aross he op row using R 2, D = R2, D and hen down he olumn o RT,D wih shape parameer 24 + hln D applying a s h ln D s+ h ln T duraion D. Sine D T = D i is fully equivalen o : 24 s+ h ln T s+ h ln T R T, D = R2, T D = R T, D A9 Equaion A9 orresponds o going down he olumn D = o he appropriae value of T and hen aross he row o ge RT,D. Boh equaions aoun for nearly all of 2 he variane of he shor-duraion daa R in he range 0.974 o 0.998 wih a median value of 0.995. The parameer s was well-deermined and indeed he parameer h was saisially signifian a mos saions. Over he ounry s was well-defined and varied wih he value of slide24; here did no seem o be any paern o he variaion of h. Linearising equaion A9 he log-log regression was performed wih:. The slide24 series in he uni duraion olumn and he reorded values for he eigh duraions 2 hours o 5 minues. 2. The reorded values in all 9 olumns i.e. abs24 as he median a uni duraion. 3. Using he slide24 series a he uni duraion and jus he hree shores duraions hour, 30 minues and 5 minues; his was o examine he abiliy of he model o express he shorer duraions in erms of he 24-hour. Naurally, his oeffiien of deerminaion was lowes bu sill subsanial, wih a range over he saions from 0.89 o 0.93. The parameer s The resuls were : Log-log regression esimaes of parameer s minimum s quarile Median Mean 3 rd quarile maximum slide24 0.35 0.374 0.409 0.42 0.443 0.50 abs24 0.305 0.376 0.402 0.407 0.432 0.496 <= hr 0.296 0.369 0.39 0.398 0.425 0.486 The loseness of he hree ses of values is highly saisfaory. The mean values also agree well wih he exponen derived from averages of he shor-duraion daa. Considering all he daa and he values greaer han or equal o he median we ge: Average fraion of 24-hour oal for 8 duraions 24h = uni duraion 2h 6h 4h 3h 2h h 0.5h 0.25h All daa 0.828 0.648 0.526 0.448 0.36 0.239 0.90 0.58 >= median 0.82 0.63 0.57 0.454 0.380 0.275 0.203 0.50 Assuming fraion = D, D <, we ge = 0.4 wih R 2 = 0. 993 for all he daa. or he daa greaer han or equal o he median, = 0.403 wih R 2 = 0. 998. The parameer h A feaure of he shor duraion daa is ha a saions suh as Birr, having a low median 24-hour rainfall 32mm, he median -hour fall is abou one hird of he 24- hour value; a high reurn periods his inreases o nearly one half implying ha he -hour fall as a fraion of he 24-hour inreases wih reurn period i.e. in A9 h is Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 26

negaive; from A8 his also implies ha he shape parameer inreases as duraion dereases. This is a key feaure o whih he model mus give expression. A saions suh as Valenia Observaory median 24-hour = 5mm he median one hour fall was abou one quarer of he median 24-hour bu was somewha less han one quarer a he higher reurn periods. This an be aered for by having values of h whih are posiive. However, onsiseny would be very diffiul o mainain if posiive values of h were allowed. I is preferred o impose a value of zero in suh ases. In effe his is o regard he highes hour values as oo low for he orresponding values of heir reurn periods or o onsider he growh rae of he 24- hour maxima as high, as may be seen from he raio of he sample maximum o he sample median for he full range of duraions: Sample Maximum/Sample Median a Valenia Observaory 24hrs 2hrs 6hrs 4hrs 3hrs 2hrs hr 0.5hr 0.25hr 2.3 2.0.8.9 2.0 2..9 2.3 2.7 The value of s a Valenia is 0.43 and a Birr 0.3. Pursuing suh onsideraions he ounry ould be readily divided aording o ranges of median 24-hour values, eah wih a ypial value of s. Geing a orresponding ypial value of h proved diffiul as i ranged from posiive o negaive wihin eah s-aegory and was generally small, ranging from +0.034 o -0.058. However, as he muliplier lnd an be as high as 4.6 is value is imporan in deermining he fraion of he 24-hr rainfall ha applies a lower duraions: RP s h 2hr hr 5min 50 years 0.38-0.02 0.8 0.38 0.27 50 years 0.38 0.0 0.77 0.30 0.75 The summary saisis for h over all he shor-duraion saions using boh slide24 and abs24 values a he uni duraion plus he resuls of using only he 24-hour, -hour, 30-minue and 5-minue duraions insead of he usual 9 are: Log-log regression esimaes of he parameer h 24h minimum s quarile Median Mean 3 rd quarile maximum slide24-0.053-0.08-0.005-0.007 +0.005 +0.034 abs24-0.058-0.02-0.009-0.0-0.002 +0.07 <= hr -0.05-0.024-0.0-0.04-0.005 +0.04 The sandard deviaion is abou 0.05 in all hree ases and so he means are unsable. or individual duraions, values of h an be found from he raios of wo reurn period rainfalls by applying eiher A8 or A9. The reurn period rainfall values used were RP5 RP0 RP0, and sine hese ould be esimaed wih reasonable auray RP2 RP2 RP5 from he daa. Boh he mehod of quarile means Logue, 975 and he median ploing posiion were used o deermine he values. The resuls showed ha only for duraions of hour or less was h onsisenly negaive. Mean and median values of h inreased slowly wih duraion beween 2 and 2 hours. However, he mos srongly Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 27

negaive values of h over all duraions ourred a 6, 4 and 3 hours respeively. or duraions of hour or less he summary saisis are: Summary saisis of he parameer h for duraions of hour or less Mehod minimum s quarile Median Mean 3 rd quarile Maximum Quarile -0.053-0.023-0.00-0.0 +0.005 +0.030 means Ploing posiion -0.090-0.024-0.009-0.0 +0.004 +0.044 The value of h applying is fuzzy raher han risp bu a se of values giving saisfaory mappings has been derived using he following guidelines:. h depends on median rainfall, being mosly zero or posiive a he higher values. However here were some negaive values of h in his aegory >= 47mm. Posiive values are no allowed in he model for alulaing h a grid poins as hey imply a derease in growh rae as duraion dereases. 2. Wihin eah aegory of median rainfall he value of h depends on 24, being low when he shape parameer is high. There was one exepion. 3. h also depends on he deails of he onveive or fronal/onveive aiviy a he saion, espeially if here is a marked jump beween he highes and seond highes rainfalls reorded a he shorer duraions bu no a 24 hours. Examinaion of individual saions revealed his as he faor ausing he more exreme negaive values of h. 4. As s +h lnt- mus be greaer han zero, he rule s > 0h was applied. 5. High values of he shape parameer >= 0.40 should be avoided. Aemps o express he dependenies and 2 by regression equaions were no suessful and he rieria for fixing h were evenually hosen by inspeion of mappings of he parameers h, s and 24. In hoosing he final values of h he shor-duraion saions were divided aording o ranges of values of he median 24-hour rainfall: >= 60mm, >= 47mm and < 60mm, >= 35mm and < 47mm and < 35mm. Eah of hese aegories had a haraerisi value of he parameer s. Bes values of h were hen seleed by examining is values a saions wihin he aegory, keeping in mind boh he guidelines and he resuls in he above ables. The final sheme is: slide24 >= 60mm : s =0.48 if 24 < 0.5 h = -0.0, oherwise h = 0 slide24 >= 47mm & slide24 < 60mm: s = 0.43 if 24 < 0.6 h = -0.05, oherwise h = 0 Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 28

slide24 >= 35mm & slide24 < 47mm: s = 0.375 if 24 >= 0.25h = -0.0 if 24 > 0.6 & 24 < 0.25 h = -0.05 if 24 <= 0.6 h = -0.023 slide24 < 35mm s = 0.33 if 24 >= 0.25 h = -0.05 if 24 > 0.6 & 24 < 0.25 h = -0.023 if 24 <= 0.6 h = -0.030 Implemening he DD model for duraions of 24 hours o 5 minues The DD model is given by equaion A8 i.e. s 24 + h ln D R T, D = R 2, D T, wih R2, = slide24 As grids of 24 and slide24 were available, grids of reurn period rainfalls for any duraion ould be generaed using he above sheme o supply values of s and h. The mappings produed in his way were mosly saisfaory bu, espeially a duraions of hour or less, here were a few anomalous spos indiaing sudden jumps in he rainfall values. Boh he number and exen of hese anomalies were small bu o deal wih hem furher smoohing was used. The simple sheme applied a eah grid poin was o add in he values of he four neares neighbours, weigh all five equally, and ake he mean as he grid poin value. While his may seem like over-smoohing, espeially a he longer duraions, he only visible effe on he mappings was o remove he anomalous spos. In view of his he maer of he degree of smoohing was no pursued furher. Implemenaion of he DD model a any loaion for duraions of 25d o 5m The form of he model onsidered was as in equaion A5: R T, D D = R2, D T where R 2, D = R2, D s or d o 25d duraions s, like, is a funion of duraion bu for sub-daily duraions i is no. A grid poins R2,D and D an be found for any duraion. Two asks were addressed:. Given duraion D and reurn period T, esimae RT,D, he reurn period rainfall 2. Given duraion D and amoun RT,D, esimae he reurn period T or a loaion on an easing or norhing gridline he wo neares grid poins were used, while for a poin wihin a grid box, he four neares grid poins were used in he inerpolaions. Weighs for he grid poins were aording o he inverse of he square of heir disanes from he loaion. A a grid poin and for duraions >= d only he grid poin iself was used. or subdaily duraions he grid poin and is four neares neighbours were used on an equal fooing for he reason indiaed in he paragraph above. Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 29

To ge he esimae of RT,D or T a a loaion wo mehods of inerpolaion were examined:. Esimae RT,D or T a eah grid poin and ake a weighed mean 2. Esimae R2,D and D a eah grid poin and ake he weighed means, Then esimae RT,D or T from equaion A5 The seond opion proved he more saisfaory. Esimaes of RT,D were praially he same for boh bu he esimae of T provided by he seond opion was he more saisfaory and is he sheme underlying he programs. EH999 disusses he idea of a represenaive poin rainfall for a ahmen whih raises he quesion of he mos appropriae DD parameers R2,D and D. Some averaging of heir values over grid poins in he ahmen is obviously required bu wheher he averaging should be simple or weighed would depend on he hydrologial onex. Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 30

Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 3 Appendix B Esimaion of he Parameers of he log-logisi disribuion for lef-ensored samples or shor duraion falls only values above prese hresholds were reorded. As a resul here was no value in some years. To esimae he parameers i is hen neessary o onsider ha over n years n-r annual maxima were reorded and r were known o be below he hreshold. Maximum likelihoodml heory lends iself quie well o his ype of ensoring and he mehod of parial probabiliy weighed momens Wang,990 an also be applied. or he log-logisi disribuion ake he usual 3-parameer form izgerald, 2005. or onveniene wrie he probabiliy disribuion funion in erms of he umulaive disribuion funion CD i.e. CD where, 2 = = + = + x x x a a g x a g x a x f B Maximum Likelihood Soluion or n years of reord wih n-r values above he hreshold 0 he likelihood L is: i r n i i r r n r n i i r a f L + = = = 0 0 Now i a g = and so L l ln = an be wrien: + + = a g r n a r n r l i i i ln ln ln ln ln ln 0 Using a f g ; a a + = = = = ; ln We ge from 0 = a l : i i i r n i i a g a g r r n 0, 2 + = = = B2 rom 0 = l = = a g r n r - a g r n r n i i i 0 ln ln 2 0 B3 rom 0 = g l 2 0 0 0 g r g i r n i i + = = B4

Equaions B3 and B4 an boh be used o deermine he shape parameer, raising he possibiliy ha, on oasion, i may no be possible o reonile hem i.e. he ML soluion may no onverge. This proved o be he ase and he mehod of parial probabiliy weighed momens provided a more robus mehod of parameer esimaion. Parial Probabiliy Weighed MomenPPWM Soluion: Consider he ordered sample of n-r values greaer han he hreshold 0 from n years of reord. x 0 The runaed umulaive disribuion funion is G x x > 0 =. 0 s E xg x is a regular probabiliy weighed momen Hosking & Wallis, 997 of he runaed disribuion for a sample of size n-r. In erms of he daa: p s + = E G s i i 2... i s n r i i= s+ n r n r... n r s G = s s Bu by definiion E G = G dg G = 0 Subsiuing for and G: 0 0 0 g + a s p s + = d B5 s+ In his form ps+ is ermed a parial probabiliy weighed momen Wang, 990. Taking 0 as known, he PPWMs are expressed in erms of inomplee bea funions Abramowiz & Segun,965. Considering d p q p q p q p q p q = p q = p p + q d and aking on boh sides we ge: u p q u u = pb p, q, u p + q B p +, q, u B6 rom B5 and B6 we an now ge from he expressions for p, p2 and p3: a p g B +,, 0 = 0 0 p g K = 0 2 2 2 + 0 0 p p g + 2 = 0 0 0 K where K = 2 p2 p 2 and 0 0 0 0 Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 32

K 2 = 6 0 p3 4 3 0 p2 22 3 0 2 0 0 0 p 2 0 0 The hree equaions give: 2 0 = K 2 K + /2 p2 p 0 B7 g = p-k/ B8 p g 0 a = B +,, B9 0 Seing 0 = 0 gives he l-momen soluion Hosking & Wallis,997 in PWM form. Seing 0 = 2 implies g =0, a = 0 = smed, he sample median. The equaion for he shape parameer redues o: 3p3 p2 p + smed / 2 = B0 2 p2 p Serving as heks we also have: p2 smed / 2 = and p p smed = B +,,/ 2 Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 33

Appendix C Cheks and Confidene Inervals for he gridded rainfall esimaes General Here he gridded esimaes of reurn period rainfall are heked agains esimaes provided by fiing he log-logisi disribuion o he daa. urher, a rude mehod of esimaing he sandard error of quanile esimaes for he gridded daa is esed. d09 o 25d09 Please noe ha in his seion daa are fixed duraion e.g. daily 0900-0900UTC. I is quie feasible o wrie down expressions for he sandard errors of quaniles of he 4-parameer model. The orrelaion marix is remarkably onsan over he saions. However, he values were obained using R 2,, he median -day rainfall, as if i were a known onsan in: R T, D e+ f ln D 24 + b ln D = D T C R2, where 24 is he shape parameer a 24hours i.e. D = Sine he sample median was used, esimaes of he sandard error of R 2, D should allow for he sandard error of R 2, and he ovariane of R 2, wih eah of he four parameers above e.g. ov R 2,, 24 = 0. 038 while ov R 2,, e =+0.053 over he 577 saions. Wha his means for a single loaion is no quie lear bu i is assumed ha he sign of he iner-saion ovariane applies. The ovarianes are small bu so are he erms based on var a and var e and so srings of small erms are obained. Having low onfidene in his approah a rude bu dire mehod was adoped. The basi assumpion made is ha sine he model produes quanile esimaes similar o hose obained by fiing he log-logisi disribuion o individual duraions, he model sandard error for a given duraion is also similar o ha derived from loglogisi heory for he individual duraion. Now, resuls of boosrapping exerises and hose of maximum likelihood ML heory izgerald, 2005 an be used. or a given duraion rewrie C as: R T, D = R2, D T D C2 To sale down R2,D, he sliding-duraion gridded median o is fixed-duraion value he following faors from Table A were used: d 2d 3d 4d 6d 8d 0d 2d 6d 20d 25d.5.08.06.05.04.04.03.03.02.02.0 Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 34

ML heory gives: var R T, D R T, D var R2, D R2, D 2 = + var ln T 2 2 D and ov R T, D, = 0 D The negaive value of ov R 2,, 24 indiaes ha ov R T, D, D may well be negaive, making he assumpion of a zero value aepable in ha i inreases he variane. urher for any duraion D: 2 9D varr2,d var = and 2 2 π + 3 n R2, D D = 3 D n 2 C3 Plunging on, i is now assumed ha despie he very differen mehods of arrival a he values of R 2, D and D, he above ML formulae apply o he gridded values wih n, he number of years of reord, se a 4 years i.e. he average over he 577 saions. Now he sandard errors seml of he following ases an be examined: a. The 2p/3p l-momen soluion for he series of annual maxima for individual duraions wih he loaion parameer regarded as fixed bu iniially unknown izgerald, 2005 b. The soluion based on he annual maxima >= he sample median 4-p daa ha uses he daa for he eleven duraions o simulaneously esimae he four parameers.. The soluion based on inerpolaed gridded values of he four parameers, assuming 4 years of reord 4p-grid Noe ha for ases b and he loaion parameer is zero and he sale parameer is he median. In ase a he sum of he sale and loaion parameers is he median. or he Phoenix Park 22 years of daily daa were available and his gave he opporuniy o ompare i wih he period 94-2004 used in his sudy. rom he annual maximum series and from he grids of he parameers we ge: Case Shape parameer Sale parameer Loaion parameer 2p/3p 88-2002 0.274 22.2 2.0 2p/3p 94-2004 0.300 20.0 4. 4p-daa 0.224 34.6 0.0 4-p grid 0.239 33.0 0.0 The following able, C, gives he reurn period rainfalls and heir sandard errors derived from annual maximum series AMS of -day 0900-0900UTC daa for he period 94-2004. a Phoenix Park. The uni of rainfall is he millimere. Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 35

Table C -day Reurn Period Rainfalls and heir sandard errors Model 2p/3p Model Model RP Daa 94-2004 4p-daa 94-2004 4p-gridded years Rainfall seml Rainfall seml Rainfall seml 2 34..3 34.6.7 33.0 2. 5 44.3 2.4 47.0 2.8 45.9 3.5 0 52.8 3.7 53.6 4.0 55.7 5.2 20 62.5 5.5 66.9 5.6 66.6 7.4 50 78.4 8.9 82.8 8.6 83.5.4 00 93.5 2.5 96.9.4 98.7 5.4 250 8.8 9.3 9. 6.4 23.0 22.5 500 43. 26.4 39.2 2.4 45.2 29.5 000 72.9 35.8 62.7 27.5 7.4 38.4 The agreemen beween he hree esimaes of he reurn period rainfalls is exellen. or he2p/3p and 4-p models he number of years of reord is 64 while for he 4pgrid i is aken as 4. Many of he surrounding saions are full period and so in he ase of Phoenix Park n = 4 is likely a bi low. Noneheless he esimae of he sandard error by inerpolaion from he gridded parameer values is reasonable. To ge an idea of how realisi he quaniles and ML esimaes of he sandard error in Table C are he nex able ompares boosrap esimaes of he sandard error wih seml, boh derived from 22 years of daa. Table C2 Model 2p/3p -day 0900-0900UTC daa 88-2002 RP years Rainfall mm sdboosrap seml 2 34.3.0.0 5 44.4.7.7 0 52.4 2.5 2.5 20 6.4 3.6 3.6 50 75.8 5.7 5.7 00 89.2 8.2 8.2 250.3 3.0 2. 500 32. 8.2 6.3 000 57.2 25. 2.7 The agreemen beween he boosrap and maximum likelihood esimaes of he sandard error is good. The indiaions from Table C2 are ha even he 500 and 000- year reurn period rainfalls derived from he gridded values of he parameers in Table are OK even if i does ake some alimiisaion o hink of 7 ± 38 as being onsisen wih 57 ± 25. The value of he sandard error is ha i gives some noion of he loal variaion e.g. for he gridded values in Table C, making he usual normal disribuion assumpion, you migh regard he 000-year reurn period rainfall for -day even as being beween 33mm and 209mm in 66% of ases. Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 36

A 0 days agreemen beween he hree quanile esimaes oninues o be very good, wih values of he 000-year reurn period rainfalls of 283 ± 45, 260 ± 32 and 28± 39 for he hree esimaes. However, beween 2 and 20 days and espeially a 6 days he individual 2p/3p parameer esimaes appear somewha anomalous. I was deided hen o es he 4-day esimaes. or he 4-day 0900-0900UTC falls during he period 94-2004 a Phoenix Park he parameer values are: Case Shape parameer Sale parameer Loaion parameer 2p/3p 94-2004 0.26 55.3 38.0 4p-daa 94-2004 0.59 94.3 0.0 4-p gridded 0.75 92.7 0.0 The reurn period rainfalls and heir sandard error esimaes are: Table C3 4-day Reurn Period Rainfalls and heir sandard errors Model Model Model RP 2p/3p-daa 94-2004 4p-daa94-2004 4p-gridded years RRmm seb srap seml RRmm seml RRmm seml 2 93.3 3.0 2.6 94.3 3.2 92.7 4.4 5 2.6 4.4 4.2 7.6 4.9 8.2 6.7 0 26.9 5.9 6.0 33.8 6.7 36.2 9.4 20 42.5 8.0 8.4 50.7 9.0 55.2 2.7 50 66.3 2. 2.6 75.3 2.9 83.3 8.4 00 87.3 6.6 6.7 96. 6.5 207.3 23.8 250 220.3 25.0 23.7 227. 22.3 243.7 32.7 500 249.8 33.7 30.6 253.7 27.7 275.2 4. 000 284. 44.9 39. 283.4 33.5 30.8 5. A he higher reurn periods he gridded esimaes of boh he reurn period rainfalls and sandard errors appear a lile high bu are nearly all wihin one sandard error of he oher esimaes. or he 25-day values we have: Case Shape parameer Sale parameer Loaion parameer 2p/3p 94-2004 0.82 98.6 29. 4p-daa 0.44 22.8 0.0 4-p grid 0.6 25.0 0.0 Noe ha he l-momen soluion for 25-days has a lower loaion parameer han ha of 4-days. As he loaion parameer of he log-logisi disribuion bears he inerpreaion of he absolue minimum, an annual 25-day value of some 29mm appears unrealisially low. The daa used so far has been based on a year of April- Marh. Using January-Deember annual 25-day maxima we ge quie differen as he values of he parameers: 0.224, 75.0 and 5.8 respeively. However, Table C4 shows ha even he 500-year reurn period rainfall 2p/3p and grid esimaes are no seriously dissimilar given he size of he sandard errors. Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 37

The ses of parameer values give he following, Table C4, wih he Jan-De figures in brakes: Table C4 25-day0900-0900UTC reurn period rainfalls and heir sandard errors Model Model Model RP 2p/3p-daa 94-2004 4p-daa 94-2004 4p-grid years RRmm seb srap seml RRmm seml RRmm seml 2 27.726.8 4.6 4.4 3.9 3.6 22.8 3.8 25.0 5.4 5 56.054. 6.2 6.3 6.0 6.0 49.9 5.7 56.3 8.2 0 76.274.5 7.7 7.8 8.4 8.7 68.5 7.7 78..3 20 97.696.9 9.7 9.7.52.2 87.6 0.3 200.8 5.2 50 229.323.2 3.9 3.6 6.88.5 25. 4.4 233.9 2.6 00 256.726.9 8.4 8.2 25.824.8 238.0 8.2 26.9 27.7 250 298.230. 27.0 27.3 30.235.7 27.8 24.3 303.9 37.6 500 334.5353.6 35.0 37.2 38.246.3 300.4 29.9 339.9 46.7 000 375.7404.5 47.4 50.5 47.659.6 332.0 36.3 380.0 63.7 Now he 4p-daa model seems o yield low values of boh reurn period rainfalls and of he sandard error. The gridded values are in good agreemen wih he 2p/3p l- momen esimaes, espeially hose derived from he April-Marh daa. Conlusions for duraions of d09 o 25d09 or duraions of one day or greaer he grids of he median -day rainfall and of he parameer values an provide esimaes of he reurn period rainfalls ogeher wih rough esimaes of he sandard error; he laer is o provide onfidene inervals. Making he usual normal-ype assumpion you migh hopefully say ha he 25-day rainfall wih a reurn period of 500 years is 340 ± 47mm i.e. has abou a 66% hane of lying beween 293mm and 387mm. Given he broad assumpions made in deriving is values i would hardly be wise o go beyond one sandard error in his ase. ollowing esablished saisial praie any quanile esimae ha was no, say, five even four imes he sandard error should be reaed wih reserve. All onlusions apply equally well o he sliding-duraion esimaes and sandard errors i.e. d o 25d esimaes. Taken as wha hey are i.e. annual exeedane probabiliies based on he assumpion ha he annual series 94-2004 adequaely represen he fuure, he esimaes may be used wih reasonable onfidene for reurn periods of up o abou 500 years. Climae hange onsideraions migh be deal wih by adjusing he amouns using safey faors. Sliding Duraions less han 24 hours The same ideas an be applied o shor-duraion falls for whih he DD model is:, 2, 24 + h ln D R T D = R D T, D <, h <= 0, 24 hln D D = + s R 2, D = R2,24h D, 24 hours is he uni duraion; R2,24h = median 24 - hour fall. or 2-hour falls D = 0. 5 and so on. The assumpion ha h is zero or negaive implies ha he rae of growh inreases as duraion dereases. Imposing his ondiion in Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 38

deriving he parameers s and h over he grid see Appendix A mean ha he usual praie of following he daa as losely as possible had, in some ases, o be violaed. This, aken ogeher wih he srong loal variabiliy a shor duraions Rohan, 986 & Hand e al. 2004 makes he esimaes of he sandard errors arrived a for duraions shorer han 24 hours even more enaive han hose for duraions of o 25 days. The daa onsised of 39 Irish saions wih a mean reord lengh of n = 44 years and his value was used in deermining sandard errors sine i did no differ muh from he value of n = 4 for he daily saions. The eigh shor-duraion saions available from Norhern Ireland had muh shorer periods of reord han he Irish saions and were used mainly o hek on he parameer values obained from he Irish daa. The effe of using grid values is shown in he following hree ases for Mullingar, WaerfordTyor and Claremorris: Mullingar A Mullingar he sheme making he parameers h and s funions of he median 24- hour rainfall had an unusually srong effe on boh he shape parameer D and on he exponen s and is he mos exreme ase enounered. This is illusraed by omparing he grid esimaes of he shape and sale parameers for hour wih hose esimaed from he daa i.e. PPWM or PWM soluionsee Appendix B for he annual series of -hour maxima and he log-log regression soluion of he DD model based on daa >= he median rainfall for duraions beween 24 hours and 5 minues. 24 median h s 24 h sale loaion DDgrid 0.224 33.9-0.023 0.330 0.297.9 0.0 DDdaa 0.234 34.3-0.00 0.367 0.237 0.7 0.0 PPWM ------- -------- --------- ------- 0.298 06.4 4.6 The grid has high values of boh he shape parameer h and of he sale parameer, R2,h, he median -hour fall. The hree ses of esimaes are: -hour Reurn Period Rainfalls a Mullingar mm RPyears PPWM DDdaa DDGrid Grid seml 2.0 0.7.9 0.9 5 4.3 4.9 8.0.7 0 6.9 8.0 22.9 2.6 20 9.8 2.5 28.5 3.8 50 24.9 26.9 37.8 6.2 00 29.7 3.8 46.6 8.8 250 37.5 39.6 6.3 3.5 500 43.4 42.8 75.3 6.6 A he 00-year reurn period he PPWM and DD-daa esimaes agree well and he grid esimae appears high. In 48 years of reording shor-duraion rainfalls he -hour maximum a Mullingar was 34.5mm for whih he median ploing posiion is over Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 39

68 years. While he 00-year reurn period rainfall esimae of 46.6mm is high he oher wo esimaes appear a bi low. Also, if he rule of humb of being hary of esimaes ha are no a leas five imes as large as he esimae of he sandard error, hen he grid esimae of 6mm for 250 years is suspe. Nowihsanding, i is more omforable o onsider a 66% hane of he 250-year reurn period rainfall for a sliding duraion of hour as lying beween 48mm and 75mm han he 30.5mm o 44.5mm suggesed by he PPWM esimae. WaerfordTyor 24 median h s 24 h sale loaion DDgrid 0.80 44.4-0.05 0.375 0.228 3.5 0.0 DDdaa 0.85 45.7-0.025 0.433 0.265.5 0.0 PPWM ------- -------- --------- ------- 0.286 0.3.4 The grid value of he -hour median rainfall is high bu, by way of ompensaion, he shape parameer is low and brings he quanile esimaes ino line wih hose based direly on shor-duraion daa, as may be seen from he following able: -hour Reurn Period Rainfalls a Tyor mm RPyears PPWM DD-daa DD-Grid Grid seml 2.7.5 3.5 0.8 5 6.7 6.6 8.5.3 0 20.7 20.6 22.3 2.3 20 25.3 25. 26.4 2.7 50 32.7 32.3 32.8 4.2 00 39.7 38.9 38.5 5.6 250 5.3 49.6 47.5 8.0 500 62.3 59.7 55.7 0.5 The esimae from he gridded values of he DD parameers of 5.6 as he sandard error of he 00-year reurn period rainfall appears low bu hen he PPWM esimae is only 7.4 while he DD-daa esimae is 6.5. Claremorris 24 median h s 24 h sale loaion DDgrid 0..200 37.5-0.05 0.375 0.248.4 0.0 DDdaa 0.202 38.8-0.009 0.394 0.23. 0.0 PWM ------- -------- --------- ------- 0.369 5.2 6. Claremorris is ineresing beause he PPWM soluion has a high value of he shape parameer. The -hour daa is omplee for he 47 years of reord and so he soluion is a PWM and a no a PPWM soluion. The effe of he grids has been o inrease boh he median -hour rainfall and he shape parameer over he values suggesed by he DD-daa model. Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 40

The able of esimaes is: -hour Reurn Period Rainfalls a Claremorris mm RPyears PWM DD daa Grid Grid seml 2.3..4 0.7 0.5 5 4.8 5.3 6.8.3.0 0 8.0 8.4 9.7.9.6 20 2.5 2.9 23.7 2.7 2.4 50 28.0 27.3 29.9 4. 4.3 00 34.4 32. 35.6 5.6 6.4 250 45.9 39.7 44.8 8.3 0.8 500 57.6 46.6 53.2 0.9 5.8 The wo highes values reorded a Claremorris were 34.6mm and 24.9mm wih median reurn periods of 68.4 years and 28 years respeively. The agreemen beween he esimaes is quie good. Again, using he rule of humb for he sandard error on he gridded values, he 500-year year value is suspe even hough he agreemen beween he esimaes is good. In brakes is he boosrap esimae of he sandard error from he -hour daa whih looks he more realisi a he 250 and 500-year reurn periods. Conlusions for duraions beween 24 hours and 5 minues: Grid esimaes of shor duraion falls may be used wih reasonable onfidene up o abou he 250-year reurn period rainfall. However, i does no seem sensible o onsider an amoun ha should our on average one in he 250 years when he preipiaion limae 50 years hene is unknown. I makes more sense o inerpre he 250-year value as he amoun ha has nearly a 0% hane of being exeeded a leas one during he nex 25 years. The sandard error esimaes mus be onsidered enaive. Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 4

Appendix D Langbein s ormula or rainfall evens above a erain high hreshold assume ha:. hey our on average one in n years i.e. he average reurrene inerval ARI is n 2. he proess of ourrenes is Poisson eller, 968 Then he probabiliy of a leas one even in a given year = e Average inerval beween years wih a leas one even = e where T is he reurn period. Rewriing in erms of ARI we ge: T = exp ARI D ARI =0 is equivalen o T =0.5 and so when only series of annual maxima are available we an approximae he parial duraion series rainfall for ARI = 0 by he reurn period rainfall for T = 0.5. To ge an idea of he size of he effes of using equaion D assume ha he growh urve of he reurn period rainfall is log-logisi and hen we have: R T, D = T,T = reurn period,t = 2 = median, D = duraion R2, D The exponen is posiive and he range of values enounered inreases as duraion dereases. Choosing a ypial value for eah duraion we ge: Table D n n = T aor onvering Reurn Period rainfall o PDS rainfall wih ARI = T Duraion Typial T = Reurn Periodyears Growh Curve Exponen 2 5 0 25 50 00 25d 0.00.044.02.006.002.00.00 8d 0.30.058.06.007.003.00.00 4d 0.50.067.08.008.003.002.00 d 0.90.085.023.00.004.002.002 6hr 0.200.090.025.0.004.002.002 hr 0.220.00.027.02.005.002.002 /4hr 0.235.07.029.03.005.002.002 Some redene is len o hese raher speulaive ompuaions by he resuls in NOAA Alas 4, Vol. 2 whih deals wih he Ohio River basin. Analysing boh PDS and AMS daa separaely hey find values for he 24-hour duraion ha are quie lose o hose in he able. Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 42

Shor Average Reurrene InervalsARI or AMS he median orresponds o T = 2; values of T beween and 2 years are no learly meaningful. However, ARI values even hose less han one year are meaningful. As noed above, Langbein s formula an be used for fraional ARI values and provides a means of esimaing PDS rainfalls for ARI less han 2. When AMS daa are available Jenkinson s quarile mehod SR, 975, Vol.2 provides anoher mehod of esimaing PDS rainfall for low ARI. ARI = ½ orresponds o he mean of firs quarilemean of he lowes 25% of he ordered annual series, while he mean of he seond quarile 25% o 50% serves for ARI =. The geomeri mean is usually employed in preferene o he arihmei mean in quarile analysis of rainfall Logue, 975. Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 43

Appendix alls of duraions less han 5 minues Dines reorders provided he bulk of he daa for falls of duraions lower han 24 hours. These reording rain gauges are no aurae below 5 minues Logue, 975. Noneheless, esimaes of he daily maximum raes over a period of abou 5 minues have been exraed a Irish synopi saions for periods of 30 o 50 years. Maximum values a he saions range 8mm o 2mm. The shor-duraion model properly applies o duraions beween 24 hours and 5 minues. I an be exrapolaed o lower duraions bu esimaes for duraions shorer han 5 minues migh well be unrealisi. Indeed, as he shape parameer onrolling he rae of growh inreases as duraion dereases here is a need o hek ha he 0- minue and 5-minues values are realisi. Employing he hydrologial onep of a rainfall profile of given duraion EH999, Vol. 2,h. 4, regard he 5-minue and 0-minue falls as sub-periods of a 5-minue rainfall even wih oal rainfall se o one i.e. ake our 5-minue esimae as orre. Saisis an hen supply reasonable esimaes for he expeed maxima of sub-periods by way of he disribuion of he maxima of n random divisions of he uni inerval eller, 97; David, 980. The enral values of his disribuion are: Table E Mean and Median of he disribuion of maxima of n divisions of a uni inerval N 2 3 4 5 6 8 0 2 5 Median 0.75 0.59 0.50 0.44 0.39 0.32 0.28 0.25 0.2 Mean 0.75 0.6 0.52 0.46 0.4 0.34 0.29 0.26 0.22 The mode of he disribuion is less han he median and he sandard errors for n = 2, 8 and 2 are 0.4, 0.09 and 0.07 respeively. The equaion 0.37550.0683ln D mean = D, D = fraion of uni duraion is almos exa. or omparison, NOAA Alas4, Vol. 2, Table 4..4 gives figures for sub-periods of 60-minue aumulaions, based on daa wih a resoluion of 5 minues, ha agree well wih he heoreial values above. or unimodal evens he equaion provides a profile e.g. for a 5-minue even wih oal fall =, he mean maximum 0-minue fall D= 0.666 is 0.85 and he 5- minue D = 0.333 is 0.6 and is independen of reurn period. Also he maer of where in he uni period he maxima are loaed is sidesepped. By way of omparison, direly using he DD model o generae summary saisis for reurn periods beween 2 and 250 years gives mean values of 0.856 o 0.875 for he 0-minue fall and 0.66 o 0.70 for he 5-minue fall. The mean of he DD model values on he 2km grid a he 50-year reurn period is 2.mm while he 5-minue mean muliplied by 0.6 is 0.6mm whih more in aord wih he 8 o 2mm maxima reorded over 30 o 50 years. Hene, for duraions of less han 5 minues he indiaion is ha i is probably beer o employ 5-minue esimaes and apply he formula for he mean fraion as a funion of he fraional duraion. Esimaion for inervals of less han 5 minues duraion should be regarded as highly speulaive. Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 44

Appendix Glossary of erms used α-level onfidene inerval Bounds wihin whih an esimae lies wih perenage probabiliy of α %. AAR - Average annual rainfall oal; he period 96-990AAR690 was used in his sudy. AEP Annual exeedane probabiliy for a given duraion i is he probabiliy of exeeding a given rainfall amoun a leas one in any year. AMS Annual Maximum Series Time series onaining he larges value in eah year 2-monh period of reord for a pariular duraion. ARIAverage Reurrene Inerval for a given duraion i is he average ime span beween exeedanes of a prese rainfall oal. CD Cumulaive Disribuion union - The CD, x, is he probabiliy of a value of he random variable X being less han or equal o a number x. Conveive aiviy- In meeorologial erms i refers o he generaion of showers in air aused o rise by hea ransfer from he surfae of he earh. C4I Communiy Climae Change Consorium for Ireland based in he headquarers of Me Eireann, he Irish Naional Meeorologial Servie, his proje may be aessed a: hp://www.4i.ie Dines Rainfall Reorder A iling siphon rainfall reorder of Briish Meeorologial Offie design ha produes a reord of rainfall over ime. DDDeph-Duraion-requeny Model Mehod of esimaion of a rainfall amoun as a funion of duraion and of frequeny; frequeny is usually expressed in erms of reurn period T see below. The basi omponens of a DD model are an index rainfall and a growh urve see below ha provides he esimae as some muliple of he index rainfall. Easing and norhing o-ordinaes of a loaion expressed as he disane easwards and he disane norhwards from a fixed daum. ixed-duraion Rainfall - Rainfall aumulaion beween fixed hours e.g. 24-hour oal read a 0900UTC eah day. ronal Aiviy- Rainfall aused by air moions indued by onrass aross zones dividing air masses of differing haraerisis. General Cirulaion Model Compuer model of he ineraions of he amosphere wih he earh-sun sysem used for he prediion of limae hange. h Geomeri mean - n roo of he produ of a sample of n values of a posiive variable suh as rainfall. Growh Curve A formula giving he inrease of rainfall wih reurn period see below for a known duraion. I provides he faor by whih he index rainfall is muliplied in DD relaionship. Index Rainfall A suiably hosen value, usually a enral value suh as he median or he mean rainfall, ha is muliplied by a growh faor in a DD relaionship. Inerpolaion- Any mehod of ompuing new daa poins from a se of exising daa poins. Kriging mehod of inferring a value a a loaion wih no daa as a disaneweighed average of he daa a neighbouring loaions. Me Eireann- Irish Naional Meeorologial Servie hp://www.me.ie Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 45

NOAA- Hydrologial Design Sudies Cenre hp://www.nws.noaa.gov/oh/hds OPW- Offie of Publi Works hp://www.opw.ie HIRDS- High Inensiy Rainfall Design Sysem New Zealand www.niwa.o.nz/n/ools/hirds L-momens- momens ompued from linear ombinaions of he ordered sample values ha provide summary saisis suh as dispersion and skewness and are ofen more effiien han ordinary momens in parameer esimaion of probabiliy disribuions. L-momens are inimaely onneed o probabiliy weighed momens see below. Loaion parameer - A value subraed from or added o he variable x o ranslae he graph of is probabiliy disribuion along he x-axis. PDSParial Duraion Series - or a given duraion i is he series of all evens during he period of reord ha exeed a prese hreshold ogeher wih heir imes of ourrene. Also known as POT Peak over hreshold series. Poisson proess- The pariular proess of ourrenes for whih he ourrene of an even does no affe he probabiliy of ourrene of he nex even. POT Peak over Threshold Analysis Saisial analysis of parial duraion series Probabiliy disribuion funion- or a oninuous random variable x i yields he relaive frequeny or probabiliy of ourrene of x over all subses of is range of values. PWMsProbabiliy weighed momens - Cerain weighed linear funions of he ordered sample daa ha saisial heory shows as boh useful and effiien for parameer esimaion of probabiliy disribuions. L-momens see above are a developmen of PWM heory. Parial probabiliy weighed momensppwms are anoher developmen of PWMs and are used for ensored samples. Quaniles - Values aken a regular inervals from he CDsee above of a oninuous random variable or an ordered sample x, x,... x he f-quanile is he daa value x wih he 2 n fraion f of he daa less han or equal o x; i may orrespond o a sample value x i or i may be inerpolaed beween wo suessive sample values. An f-quanile divides an ordered sample ino f approximaely equal ses of values. Quariles or an ordered sample hey are he quaniles for whih f assumes values 25%, 50% he sample median and 75% i.e. hey are he hree poins dividing he sample values ino four groups, wih 25% of he observaions in eah group. The inerquarile range is he differene beween he values of he 3 rd quarile and he s quarile and is a measure of dispersion. Residual Observed value minus he value esimaed by a model. Reurn Period T - Average number of years beween years wih rainfalls exeeding a erain value. I is he inverse of he AEP defined above e.g. a 50-year reurn period orresponds o an AEP of 0.02. Is imporane sems from is being a basi omponen of he DD model used o alulae he reurn period rainfall. Reurn Period Rainfall The esimae supplied by he D relaion when he reurn period T, he duraion D and he model parameers are supplied. I is, perhaps, bes hough of in risk erms e.g. he 00-year reurn period rainfall has nearly a 0% hane of being exeeded in a 0-year period. 2 R Coeffiien of Deerminaion - The proporion of he sum of squares variabiliy aouned for by a regression model. Sale parameer- Divisor onrolling how spread ou he disribuion is e.g. he median is he sale parameer of a wo-parameer log-logisi disribuion. Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 46

Shape parameer - Power law exponen whose variaion hanges he shape of a disribuion e.g. for he log-logisi disribuion he shape parameer deermines he posiion of he peak probabiliy and he skewness. Skewness A measure of he deparure from symmery of a disribuion Sliding duraion rainfall Term used for he rainfall oal for a given duraion when exraing maxima from effeively oninuous rainfall reords; i ofen exeeds and anno be less han he orresponding fixed duraion rainfall, as for he laer he hour of reading is prese. Sandard deviaion- Measure of dispersion or spread of values abou heir mean Sandard error Esimaed sandard deviaion of a sample saisi suh as he mean i.e. he sandard deviaion of he sampling disribuion of he mean Tipping Buke Reorder- A ylinder funnels preipiaion on o one of wo balaned bukes of fixed apaiy. On reahing apaiy he reeiving buke ips, sending a signal o a reorder while he seond buke beomes he reeiver. Unimodal- Having one maximum on is probabiliy disribuion urve i.e a single peak UTC - Coordinaed Universal Time or Universal Time Coordinaed UTC is he inernaional ime sandard. I is he urren erm for wha was ommonly referred o as Greenwih Meridian Time GMT. Esimaion of Poin Rainfall requenies. Me Éireann, Oober 2007 47

Lis of Me Éireann Tehnial Noes 44. Lynh, Peer, 984: DYNAMO---A one dimensional primiive equaion model. 45. Lynh, Peer, 984: Iniializaion using Laplae Transforms. 46. Lynh, Peer, 984: Iniializaion of a baroropi limied area model using he Laplae Transform ehnique. 47. Leeh, L~S, 985: A provisional assessmen of he rereaional qualiy of weaher in summer, in erms of hermal omfor and he adverse effe of rainfall. 48. Lynh, Peer, 986: Numerial foreasing using Laplae ransforms: Theory and appliaion o daa assimilaion. 49. Hamilon, J, P Lennon and B O'Donnell, 987: Objeive analysis of limaologial fields of emperaure, sunshine and rainfall. 50. Lynh, Peer, 987: The Slow Equaions. Par I: Derivaion and Properies of he Sysem. Par II: Appliaion o Coninuous Daa Assimilaion. rainfall. 5. Logue, J J, 989: The esimaion of exreme wind speeds over sandard errain in Ireland. 52. Lynh, Peer, 99: ilered Equaions and ilering Inegraion Shemes. 53. MDonald, A, 99: Semi-Lagrangian mehods. 54. Lynh, Peer, 996: The elasi pendulum: a simple mehanial model for amospheri balane. 55. MDonald, A, 998: The origin of noise in semi-lagrangian inegraions. 56. Lynh, Peer, 999: Resonan Moions of he Swinging Spring. 57. MDonald, A, 200: A sep oward ransparen boundary ondiions for meeorologial models. 58. Lynh, Peer, 200: Resonan Rossby Wave Triads and he Swinging Spring. 59. Sheridan, Tom, 200: Analysis of rends a some Irish rainfall saions. 60. MDonald, A, 2004: Transparen laeral boundary ondiions for barolini waves I: he foundaions. 6. izgerald, D L, 2007: Esimaion of Poin Rainfall requenies.