Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | , and 3 Sciences Discussions Earth System Hydrology and , A. Seed 2 10 mm) is investigated for erent estimation techniques. > ff , F. H. S. Chiew 1 ¨ oppen climate zone (strongly related to lat- 3190 3189 , L. Siriwardena 1,* cient resolution is available only at a limited number of locations ffi processes and, in particular, sediment entrainment (e.g. Dodov and ciency of prediction of the highest rainfall intensities (average intensity , B. Anderson ff 1 ffi erent fitting methods are compared for di ff 4 ¨ oschl This discussion paper is/has beenSystem under Sciences review for (HESS). the Please journal refer to Hydrology the and corresponding Earth final paper in HESS if available. Department of Infrastructure Engineering, TheCSIRO University Land of and Melbourne, Water, Australia Canberra,Centre Australia for Australian Weather andInstitute Climate of Research, Hydraulic Melbourne, Engineering Australia and Water Resources Management Vienna University now at: Water Division, Bureau of Meteorology, Melbourne, Australia step. The overall goal ofday this statistical research is distribution to of establish a rainfall means intensity of given estimating the the within- daily rainfall depth and other Rainfall data at high temporal resolutionof are surface required runo to accurately modelFoufoula-Georgiou, the 2005; dynamics Kandel et al.,respond 2005; to Mertens rainfall et intensity al., 2002).rainfall variations intensity These over at processes short su intervals.across However, Australia. measurement of On thetime other step, consequently hand many there models is used good to inform coverage water of managers rainfall use data a at daily a time daily also investigated and trends withitude) latitude, and K daily rainfall amountto were a identified. changing mix The of latitudinal rainfall trends generation are mechanisms likely across related the Australian continent. 1 Introduction Root Mean Square Error (RMSE)tion; between and the the fitted e andof observed the within-day 5 distribu- highest intensitytion was intervals). clearly The the best three-parameterGamma, performer. Generalised Good and Pareto results distribu- two-parameter were also Generalizedparameter obtained Pareto from functions, distributions, Exponential, which each may ofResults be which advantageous of when are di predicting two The parameter behaviour values. of the statistical properties of the within-day intensity distributions was The statistical behaviourwithin and the distribution wet of42 part high-resolution stations across (6 of min) Australia.tions rainy rainfall (TDFs) This in intensity representing days paper these data. compares (total Two goodness-of-fit nine rainfall statistics theoretical are reported: depth distribution the func- Abstract 1 2 3 4 of Technology, Austria * Received: 5 March 2011 – Accepted:Correspondence 7 to: March 2011 A. – W. Western Published: ([email protected]) 31 MarchPublished 2011 by Copernicus Publications on behalf of the European Geosciences Union. Hydrol. Earth Syst. Sci.www.hydrol-earth-syst-sci-discuss.net/8/3189/2011/ Discuss., 8, 3189–3231, 2011 doi:10.5194/hessd-8-3189-2011 © Author(s) 2011. CC Attribution 3.0 License. G. Bl The within-day behaviour of 6rainfall minute intensity in Australia A. W. Western 5 20 25 15 10 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | rate by ff erent distributions. ff ective parameter approach ff ects of short timescale rainfall intensity ff ect of the short time scale processes but ff erent theoretical distributions were fitted to ff 3192 3191 erent statistical distributions. ff -intensity relationship that can be updated on a daily basis ff or erosion). Second, nine di ff generation and erosion, that is the high intensities. The principal aspects of the Does the “best” TDFand vary how with do characteristics location of around the Australia distribution (i.e. relate to with climatic climate characteristics? zone) How well does eacheach of other? the TDFs perform and howWhich do approach they to rank parameter estimation withmoments, L-moments, shows respect LH-moments, the to or greatest Least skill: Squares the (LS)? method of ff – – – The intention of this paper is to examine how to best represent the cdf of 6 min rainfall There are several ways of capturing the e filtered to ensure data qualityfor and to runo exclude days of smallthe rainfall measured depth cumulative (not density of function interest for (CDF) estimating of the rainfall distribution intensity.functions Multiple parameter were methods values employed were to employed.Data assess processing Third, and the two analysis goodnessFortran-90. objective was of Each principally stage fit achieved is of via described custom in the more routines di detail written in in the following sections. 2 Data and methods High resolution rainfall dataand from a pluviograph detailed stations analysisThere across conducted were Australia to three was explore stages the obtained to distribution the of analysis. within-day intensities. First, the raw rainfall intensity records were It also aims todistributions examine variation between in locations. the statistical(6 To min) behaviour address of rainfall these the data aims within-dayaround recorded we intensity Australia. at analysed It high 42 israinfall resolution important Bureau disaggregation to method of note as Meteorology DF that models the pluviometer do paper not installations is require not an aiming explicit to time sequence. develop a new problem that are addressed by this work include: requirement of such models is the cdf of rainfall intensityusing within wet the and day. drytion fractions, of coupled with rainfall intensities antreatment appropriate during of continuous the the distribution wet func- TDF fraction.rainfall selection intensity In distributions problem, the in this Australia. absenceto paper Specifically, of quantify aims the a how aim to comprehensive of well filltensity this a the data investigation range was and, gap of in for available particular,runo within-day TDFs fit fit the the characteristics of measured rainfall within-day that rainfall are in- most relevant to depending on the catchment wetness orwork other has states shown such that, as from surface a cover.intensity water Point-scale quality/erosion across perspective, a the distribution day of andtime rainfall the sequence total of daily intensityBruijnzeel volume is (2004) are of reached of similar secondary primary conclusions value importance, for events. (Kandel while The et the key al., meteorological input 2005). Van Dijk and large catchments is impractical.calibrated) Alternatively model in parameters an can attemptwith be to modified a (e.g. capture long the (sayis e daily) not well model suited timefunction to step; (DF) nonlinear however, processes. approach thisshort in Another e approach time which is step the toKandel (say cumulative use et 6 probability the min) al., distribution density rainfall 2005).a intensity function This typically is (cdf) non-linear function runo input is of then (Van modified Dijk to and produce Bruijnzeel, a cdf 2004; of runo makes a first stepintensity in and its that representation by by examining di the within-day statisticalvariability in behaviour catchment of modelling. rainfall Thein rainfall time a series can short be time explicitly represented step model; however, running short time step distributed models on readily available hydrometeorological data (e.g. temperature, pressure). This paper 5 5 20 15 10 25 15 20 10 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | and 1 − valid day cient rain ffi ) equalled or . On average, P 1 − on any day need to be min R (0.1 mm/6 min) were considered 1 − discretisation), the practical need to 1 − ¨ oppen climate zones for Australia (Peel et ) of 1 mm h 3194 3193 min R ). Prior to the early 1990s the BOM pluviometer network used Dines is a reasonable compromise given the discretisation inherent in tipping 1 ¨ oppen climate zones present in continental Australia are represented and − ) of 6-min intervals where the intensity exceeded and found the results were insensitive to the exact level of the threshold. n 1 − 10 mm) and that on these days 88.2% of the total rainfall depth was received. In ≥ ) exceeded a threshold minimum ( Rainfall was also censored if 6-min intensity was less than 1 mm h The results of this censorship regime in terms of the number of rainy days on the Records of rainfall intensity measured using tipping bucket technology incur errors P R remove the artefact of single tips being spread over many time increments in the data accumulation is less than 10in mm. the Over data all after censorship stations, wasreasonable the 74.7% given average of our rainfall the interest depth total in retained rainfall processes depth, which sensitive was to considered large events. this accounted for 5.5% of thefrom rainfall 1.8% depth to at 9.2%. each station, We2 with mm undertook h this sensitivity proportion testing varying usingThe thresholds 1 of mm h 1 mm h bucket rain gauges (which is typically a 2 mm h record and the percentageis of summarised the in rainfall Table 2.tigation, depth The that showing bottom fell for line within example( describes the that the various data in categories analysed Darwincontrast, by almost 12.3% this half of inves- of valid Melbourne’s rainfall days depth had is su delivered on days where the total valid day pluviograph recordsdata from were consideration. censored First, inexceeded only 10 two mm days were where ways considered. the to( Second, total only eliminate rainfall those depth low 6-min ( intervals intensity in where fitting intensity the CDF. Finally, innumber order ( to numerically resolve theat higher least order four. moments, the for this analysis was defined240 as values, including one zeroes, for starting which at the 09:00 pluviograph h). record isat complete very (i.e. low rainthis rates work due to was resolution concerned problems with (see review the by upper Nystuen, end 1999). of As the rainfall intensity spectrum, the record where data was missing were not used rather than being in-filled. A 09:00 h (as per the Bureauday standard). characteristics; This therefore investigation inter-day was relationships concerned could only be with neglected intra- and periods of Pluviographs which recorded via a papermechanism. chart and pen Since connected that tobeen a time, float used tipping and siphon bucket and rainthese the gauges types time of with records of a are providedshowed individual 0.2 by mm that the tips tip BOM the recorded as size short 6similar. have (Srikanthan min time data. et This interval Srikanthan data is et al., al.tle from consistent 2002). (2002) dependence these with on two the Both gauge measurement conclusionsgauges. types characteristics of For this are (e.g. Fankhouser analysis, statistically (1998), bucket a size) who day was for found designated tipping lit- as bucket the period starting and finishing at 2.2 Quality control and censoring Rainfall intensity data forincrement each with station each was 24 suppliedpluviograph h at data period the divided BOM into standard 240 6-min intervals time (hereinafter referred to as 42 meteorological stations used.a separate This study) set provides ofspan a sites at broad (identified least spatial by 20 coverage Lu yrto across and 2439 and mm Australia, Yu, the at record 2002, mean Koombooloomba. lengths of for annual the Site rainfall ten elevations ranges K range fromthere from 196 is mm sea a at level selection tonon-seasonal rainfall of 760 m, regimes. sites nine from each of winter dominated, summer dominated and Pluviograph records were obtained fromfrom the the Australian 42 Bureau of sitesal., Meteorology shown 2007) (BOM) in are Fig.classification also 1. was shown. checked The with K Where site stations data. are Table very 1 close shows to pertinent a properties zone of boundary the the 2.1 Data 5 5 25 15 20 10 20 25 15 10 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | . The selection of TDFs 1 − threshold. Note that this yields 1 − 3196 3195 erent theoretical distribution functions (TDFs) (Table 3) were fitted to the data ff The utility of Wang’s (1997) LH-moment method (LH4 moments in this case) was Parameter values for each of the distributions were computed from the pluviograph The final three TDFs in Table 3a and b are Extreme Value Distributions (EVDs). entire range of intensityone values RMSE above value the per rain 1 mm h day analysed. A low RMSE value indicates that the fitted Consequently, the results presented in thisother paper three examine parameter only estimation the techniques. relative merit of the 2.5 Assessment of fit Two measures of goodness-of-fitFirst, were the selected Root to Mean quantifypared Square the with Error fit the (RMSE of – observed thewell defined rainfall distributions. the by intensity Eq. shape data 1) was of of computed. each the fitted TDF RMSE TDF quantifies matches com- how the recorded within-day data considering the also examined using thepursued GEV distribution even as though athe it test case. L-moment yielded estimates This a fitting becausepluviograph better method stations) the fit was inferior LH4 not to to estimations those the produced were upper by (for product tail a moment of large and the LS majority methods. distribution of than data in three ways:second first via by the the method-of-moments computation (productthird of using moments, L-moments a denoted least (Stedinger PM); squares et estimationmented (denoted al., an LS) 1993) technique. automatic (denoted The pattern LS LM);Monro, algorithm search 1971) and imple- with optimisation the method objective tonext (Hooke section) minimise and between the the Jeeves, Root measured Mean 1961; rainfallfor Square intensity Error the CDF (RMSE and first – the iteration see fittedinitialised of TDF. using Note the that values calculated LS via algorithm the the product moment parameter method. values of the fitted TDF were to question as the rainfallvalue intensity data data to set, which they atrecent are analysis least being of fitted heavy using is rainfall traditional not byis Wilson an in ways and extreme fact of Toumi (2005) “heavy shows thinking tailed” that – about the a distribution rainfall. characteristic feature of However, EVDs. a observation drawn from a large sample. The validity of including these EVDs is open These have been derived specifically to represent the distribution of the largest literature. The mathematicaltechniques formulation employed, of followed each the TDF,identified and in methods the the presented right-most parameter column in offor estimation Table Stedinger 3. each Table et 3b of gives al. the theparameters) mathematical (1993) distributions. have formulae been as used Other to describe distributiondistribution, rainfall (e.g. functions Rossi the et with two-component al., extreme greater value 1984);parameter flexibility however, given (more values that we for aim distributions subsequentlydistributions from to to predict those daily the that meteorological have observations, three or we less limited parameters. Nine di for the wet fraction ofintervals the in day. The the wet day fractionwas with is populated rainfall calculated as intensity with the exceeding proportion distributions 1 mm of h well 6-min known in the meteorological and hydrological whether a local (analysisshould by be individual taken. site) or Thereof a are regional enabling advantages (all of a sites both.sites better together) A while approach understanding local a of approach has regionaldue local the to analysis behaviour advantage the will and inclusion providemore contrasts of interested a between more in more understanding data. those the robustbetween site Here sites. relationship level we behaviour over and chose a in a exploring region local the variation approach2.4 because we are Theoretical distribution functions intensities). 2.3 General approach With any analysis of information from multiple stations a decision must be made as to and our primary interest in intensities that are significant to surface processes (i.e. high 5 5 25 15 20 10 20 25 15 10 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | (1) erent ), was ff 1 − ( mm h ’th probability of HI j I (larger where necessary to 30 000 following quality con- erent rainfall rates has been 1 8). Inspection of fitting results ff > − . 0 > of di 2 erences between the fit of di r ff duration 3198 3197 and the is the number of 6-min intervals during the wet frac- n 240 wf). Note that for the LS fitting method, the objective = n volume 2  j I is the number of remaining data points. For these bins, ranges were − value for each rain day in the record ( n r j ˆ I captures 30 min of rainfall in total but not necessarily from consecutive are the fitted and measured rainfall intensity at the n HI  I I HI 1 I n = P j and v u u u t ˆ I , where = 4 . 0 r n cient (PPCC) test (Filliben, 1975), the Kolmogorov-Smirnov test and Chi-squared The Chi-square test requires continuous data to be discretised into bins and it is Formal statistical testing of distribution fits was also considered. Several alterna- Given the ultimate aim of providing input to erosion models, where the highest inten- ffi trol). In ordertwo to summary quantify statistics the were goodness-of-fit computed: over mCOE all and RMSE90. the rain-days at a given station, the distribution contributed more thanof 50% of days the (except chi-square lognormal statisticupper – on 50% 70–75 tail. percent of Given days)distinguishing our and candidate that greater distributions. the interest statistic in was the insensitive upper to tail,2.5.1 the this testing Summary was statistics not for useful each for station The results of fittingvalue at and a one given pluviograph station are reported herein by one RMSE less than five datauntil points, all bins the had number atwere of least selected bins five for observation testing, was points. which reduced(i.e. were Only 30 and generally days observation ranges days that points). with met recalculated Thisrejected more the testing on than above about indicated criteria 3 half that hours of each the of candidate days rainfall tested. distribution was Subsequent analysis showed the lower half of suitable. Thus we usedHandbook (2011) the recommendations. Chi-squared test and followed the Engineeringrecommended that Statistics there be atupper least limit 5 of data the pointsensure first in that bin each it was contained bin at set and leastto arbitrarily at 5 2 to least data points). 1.5 5 mm The bins. h numberallocated The of on subsequent an bins was equal set probability basis using the fitted distribution. If bins existed with (Gamma, GEV, Weibull andter Gumbel) 1.3.5.14, tests (Engineering Heobe Statistics et fully Handbook, al., specified Chap- 2008) forChapter The the 1.3.5.16). Kolmogorov-Smirnov critical requires values Becauseneeded to the we be to distribution wanted valid estimate to to (Engineering the test Statistics parameter Handbook, all values the from distributions the consistently data, and these three tests were not distributions for the Anderson-Darling (Lognormal, Exponential and Weibull) and PPCC RMSE that a good fitachieved. to both the TDF provides a good approximation to the shape of the rainfall intensity CDF; showing e goodness of fit test. Of these, critical values only exist for a subset of the candidate over-emphasizing errors in theaveraging fit over of long the periodsTDFs. highest tended The one to average or reduce of two the di selected intensity as five providing highest values) a intensity reasonable but periods, balancebe that between designated noted these that competing factors. It should intervals. tives exist for testing whether ainclude sample the comes Anderson-Darling from test a (Stephens, hypothesised 1974), distribution. the These probability plot correlation co- the fit to the upperered, tail including of the the maximum rainfall 6-min distribution.intensity intensity; A 6 min the number periods; average of and of alternatives the thethese were 80th 2, measures consid- and 3, were 90th 5 percentile highly intensities. and cross-correlatedfor 10 Of (i.e. Melbourne course, highest and many of Darwin showed that some degree of averaging was useful (to avoid where: exceedance respectively; and tion (wf) of the day (thatfunction is: is to minimise the RMSE. sities are the most important, a second goodness-of-fit statistic was used to quantify 5 5 25 15 20 10 25 20 15 10 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | (2) erent 15%, ff ± ciency (Nash and ffi ciency (mCOE) (one ffi is the number of rain days erent TDFs (LGN3, GAMA, S cient of E ff ffi cient of E ffi (from the fitted TDF) and the observed HI I 3200 3199 erence. This is largely controlled by the skew- ff erence between the quality of fit for the various TDFs, ff 15.4) clearly inferior compared with the other two TDFs

= k HI HI I ˆ I is the mean value of the set; and − − k k HI I HI HI I I

1 are the fitted and measured mean intensities of the 5 highest intensity 1 = S = S P P k HI k I − e, 1970), but instead of squaring the error between measured and ob- 0 (top) and RMSE (bottom). A number of additional statistics are provided ff and . 1 HI HI ˆ I I = RMSE (i.e. 90% ofHerein this RMSE statistic values is denoted are as less RMSE90. than or equal to this RMSE value). The goodness-of-fit between the fitted data was quantifiedvalue using of the mCOE ModifiedmCOE per Coe is station) essentially similar as toSutcli defined the well by known Legates Coe served and data McCabe (which gives (1999). extraerror weight is The to computed outliers), the insteadderivation absolute and (refer magnitude discussion). to of the Legates andThe McCabe, range 1999 of for RMSE a values thorough at a station was summarised by the 90th percentile – – More quantitative results of the fitting for Melbourne and Darwin are shown in Figs. 4 ues associated with thethe lognormal TDF lognormal in fit Darwin (RMSE90 vary over a much wider range, with 3.1 Fit results for various TDFs In Fig. 4 thelognormal amount fit of than scatter either around theMelbourne the gamma and line-of-perfect or agreement Darwin. GEV is distributions,normal greater The and statistic for this mCOE more the is statistics than the 10%of support case lower a this for than similar both observation, either magnitude other withwith for TDF. the the The Melbourne; log- variation lognormal that in is TDF RMSE the at is 90th the percentile upper RMSE end is of 2.8 this range. In contrast, the RMSE val- show results for Darwin.parameter They estimation compare methods the (LM, fitting PMwhen and skill an LS) additional achieved degree and by of also the freedomIn show three is summary, the available: di improvement these i.e. in GPT3 two fit (right)(ii) figures versus location; GPT2 (iii) show (left). fitting that method; fitting and skill (iv) number varies of as TDF a parameters. function of: (i) TDF; also appears to be aDarwin. wider These range in figures observed give a distribution shapes qualitative idea at Melbourne of the thanand at range in 5. fit quality. Theserved charts are paired (referredwith to these as plots “chart-pairs”), asfacilitate showing described comparison fitted in of versus detail ob- fittingand by skill GEV) each using at figure PM two heading. for locations: three Melbourne The di (left) charts and in Darwin Fig. (right). 4 The charts in Fig. 5 tribution which used PM).at It is Melbourne) clear there that iswhile for little for some di others days there (for is exampleness a of 1 significant the February rainfall di 1970 intensitydistributions distribution being on more the flexible particular in terms day, with of the matching GPT2 the and variations GPT3 in skewness. There station. These distributions were fitted using the LS method (except for the Gamma dis- Fits of Exponential, Gamma, andbourne Generalised and Pareto 2 Darwin and are 3 shown in parameter Figs. TDFs 2 for Mel- and 3 for nine randomly selected days at each 3 Illustrative results: Melbourne and Darwin where: intervals of the day; in the pluviometer record for that station. mCOE Legates and McCabe, 1999): The meaning of theseare introduced two in statistics the will next section. become The clearer equation as used some to compute illustrative mCOE results was (as per 5 5 25 15 20 10 20 15 10 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | and 1.91 ff = ) so as to HI I in Melbourne). 1 − ). Given this result, LS erences in fitting skill 0.928 compared with ff = for higher observed values of HI I ) that are readily comprehended (for erence is most likely due to the fact 1 ff − values are used as the basis of compari- HI ). It is interesting to note that the middle-right I 3202 3201 0.891 compared to 0.741, and RMSE90 LS should not be ruled out at this point as the fit is = – putting result (ii) into context). in Darwin compared to 9.5 mm h LS 1 1 − − has more physical meaning than a normalised RMSE 1 − 1.58 compared with 2.03). The optimisation provided by the = is 30.2 mm h HI I exceeds 20 mm h HI I 7.6 and 8.3). This suggests that location-related di = ) has a very similar fit to the bottom-left chart-pair (GPT2 combination clearly provides the best fit of the combinations shown in Fig. 5 ered by the third parameter is less significant (mCOE ff PM LS . This is indicated by the negative bias and the position of the dashed regression A final point to note from the fitted versus observed plots in Fig. 4 is that both the Given that the elevated RMSE values for Darwin are driven by the higher rainfall 30 only marginally poorer butanalysis is it achieved with is one notable less possible model than to parameter. maximising decide In whether thework the fewer potential that model present goodness-of-fit, parameters follows this are thisvalues will more from TDF indeed desir- daily selection climate be study measurements).the a (i.e. However, selection question it attempting process is for in to an thatalso important predict TDFs it consideration TDF with is in two parameter important rather to than choose three not parameters. only the best fitting TDF but LS process narrows the gapextent between that the the GPT2 value andGPT3 of GPT3 goodness-of-fit the to third such parameter(and an must in be fact later questioned. figuresHowever, To the show summarise: this combination to of the be GPT2 the case across all the pluviograph stations). (GPT3 two conflicting conclusions can beof freedom drawn available regarding to GPT3 the over valueevident GPT2. of in The the advantage the of additional product the degree far moment third better parameter fits than is (middle most those chart-pairs), ofcompared GPT2 with (mCOE with the 2.56). fit However, statistics lookingment for at o GPT3 the bottom chart-pairs0.884, (LS and fit), the RMSE90 improve- Figure 5 illustrates two trends inthan fitting L-moments skill: while first, LS product is moments thedom are best available more of to successful the GPT3 three; noticeably and improvesthe second, the chart-pair the fitting at extra indices. the degree bottom of The right free- best (GPT3 fit is shown by 3.2 Impact of fitting method and number of TDF parameters experiences events having far higher intensitythe than observed Melbourne (i.e. many events where GAMA and GEV TDFs tendI to slightly underestimate lines being consistently below theerosion line-of-perfect-agreement. predictions Consequently, runo using thewith fitted the TDFs observed would data. tend to be underestimated compared of indicating the error magnitudeexample: in RMSE units of ( mm 1.0 h mmof h 0.1). Thus, fromGEV in the Darwin RMSE and data Melbourneues in have computed Fig. superior for 4 performance Darwin it to are can LGN3; more be (ii) than concluded RMSE90 double val- that: those (i) in GAMA Melbourne; and and (iii) Darwin in Darwin than inLGN3 Melbourne, fits while get the poorer LGN3 as value rainfall intensity is increases fivefold in higher general). intensity (suggesting of that monsoonal events,facilitate should comparison the between stations datasults)? (i.e. be It RMSE normalised is calculated (e.g.work the for was by to author’s nondimensional examine opinion TDF re- fits thatbased at on this each station unscaled was not rainfall not between intensity stations. necessary data For as this was task the suitable, RMSE and objective has of the this added advantage may be important.that In Darwin receives fact much the heavierheavier source rainfall if than of the Melbourne; the median approximately di son three or (e.g. times 90th median percentile Indeed, the RMSE90 values for the GAMA and GEV distributions are threefold larger (RMSE90 5 5 25 15 20 10 25 15 20 10 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | erent structural rela- ff 3204 3203 ort required to implement an LS solution was not justifiable. ff The desire to identify multiple candidate TDFs, as well as the TDF with the best fit, On the basis of these observations it is clear that the LS method represents theCandidate best TDFs should be first fit by PM and then optimised by LS to obtain the The superiority of the LS fitting is founded on the success of the PM fit in that the Note that LS fitting to GAMA and LGN3 caused technical problems and hence re- is that the parameter valuesothers. of some One TDFs reason mayidentifiable be (highlighted (i.e. more able earlier) to amenable be is tois predicted) prediction that than that than a a the three-parameter two-parameter parameters TDF.another A TDF TDF. of second For may possibility one example, be it TDF maypredicted more may be than that be the the more two twotionships identifiable parameters parameters between of than of the EXP TDF the GPT2, are parametersvariance, parameters due more and skewness, readily of perhaps the and kurtosis). statistics to of di the distribution (i.e. mean, In this section the focuswas is to reduce on the ranking number the ofwith fit candidates the from provided ultimate nine aim by down to the to thenof nine the use these TDFs. best these TDFs three in The from or a objective daily four subsequent TDFs, climate study variables. to predict the parameters PM approach (see especially GEV and GPT2 results –fitting top method, of followed Fig. by 6). PMthis and study then draws the is LM with method. regard Thus, to the fitting first method: conclusionhighest that fitting skill as measured by mCOE and RMSE90. 4.2 Ranking the TDFs PM parameter values were usedshow to substantially higher initialise mCOE the values LS thanvalues. the optimisation. Lower LM RMSE90 The fits, and values PM also are fitswas show to to in lower be minimise RMSE90 Fig. RMSE expected 6 values. as It theimproved is objective by noteworthy of the that the mCOE LS values LS process are algorithm also by substantially comparison with the mCOE values achieved using the achieved using other fitting methods. 4.1 Ranking the fitting methods The trends previously observed for individualin pluviographs Fig. are 6. reinforced by These the(top) show results that and the smaller LS method RMSE90Furthermore, produces in values consistently all (bottom) higher cases mCOE than bothvalues values is the either smaller. magnitude of The of reduction the thegreater in statistic extent range other than is implies fitting the better that better and LS methods. fits, the improves with the range the poorest of fit fits at by a all stations an improvement over those wasn’t robust, with theconditions. denominator of This problem the could algorithm belocation tending avoided parameter. toward by However, imposing given zero the a under poor numberL-moment some fitting of and performance constraints of product on LGN3 moment the obtainedselected estimation, with and it hence was the felt e that the TDF was unlikely to be fitting method (see definitionsthe in primary the tool figure for legend). ranking the The fitting results methods shown andsults TDFs in for Fig. 6 these were casesthese do cases not are as appear follows.an in For iterative Fig. GAMA numerical 6. an solutionwhen analytic The LS was CDF was impediments is required. attempted unavailable to using andsolution the LS Computation pattern so to calculation search times instead the algorithm in GAMA became coupled CDF. with In excessive the the numerical case of LGN3, estimation of the location parameter Figures 4 and 5 lookedmeaning at specific of results the for goodness-of-fit twothese pluviograph indices goodness-of-fit stations (mCOE results and for and illustrate all the RMSE90).RMSE90 42 bottom). stations Figure using 6 two Three summarises sets boxes of are box plots shown (mCOE for top, each of the nine TDFs, one for each 4 Results for all stations 5 5 25 15 20 10 25 15 20 10 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | ) PM and GAMA LS erent TDFs being ff is unequivocally the LS , EXP LS erence between the high ff cient flexibility to accommo- ffi , GPT2 LS 3206 3205 clearly the first choice distribution for fitting within-day LS erent locations, suggests that GPT3 has su ff Which TDF fits best at each pluviograph station? Can spatial trends in the goodness-of-fit statistics be discerned? – – While GPT3 provides the best fit across almost all the stations, the next question is The fact that GPT3 consistently provides the best fit, rather than di concentrated in the North-East anda lower similar values spatial in the pattern,and South albeit low and with values, South-West. to a This the is largerhigher pattern proportional magnitude of di of mean wet rainfallspect period intensity to rainfall Melbourne in intensities and the Darwin andquantified, earlier). it North spatial To reflects investigate of statistics the whether were Australia this employed. clustering (as could discussed be with re- constructed. Symbol size on thesediameters maps indicates indicating the a goodness of poorer fit,structed fit with for larger (i.e. circle the low four mCOE TDFsand not or from yet high eliminated a RMSE-90). (GPT3 qualitative,each Maps TDF. visual One were pattern inspection con- observed the by pattern the authors of was circle that larger sizes RMSE-90 looked values were similar for better at di date a range of within-dayrainfall rainfall CDFs intensity does distributions, and not perhaps varytion that strongly of the with both shape location. factors, of with Therainfall GPT3 answer intensity. is probably a combina- whether the level of fitting skillsibility, maps varies showing systematically the with spatial location. variation To of examine mCOE this and pos- RMSE-90 such as Fig. 7 were est mCOE and, independently,results the are lowest summarised RMSE90 in werebest Table identified. 4 fitting and TDF, The with demonstrate aggregated that GPT2at GPT3 and only 3 WEBL and distributions 1 providing pluviometer a stations lower respectively. RMSE90 result fit. Thus, on a station-by-station basis the TDF and fitting method showing the high- other TDFs, it is possible that at particular stations one of the other TDFs yields a better One factor that cannot betion. discerned To from understand Fig. how 6 much of is an whether influence fitting location skill has varies two with questions loca- were asked: The results shown in Fig.ever because 6 suggest the that range GPT3 of provides mCOE the and best RMSE90 fit values to overlaps the with data. the How- box plots of that there is a strong enoughwithin-day case rainfall to is consider not selecting an ataken EVD, classic to given the exclude extreme concern GEV, value GMBL that distribution. and WEBL Hence, distributions the from decision4.2.3 further was consideration. Variability with location 4.2.2 Elimination: extreme value distributions The performance of the three EVDsfor (GEV, GMBL fitting and by WEBL) LS). is The mixed (considerskill box results plots of for the GEV and GMBLand WEBL is are median second not RMSE90 only as is tosupports good GPT3, greater the while as than notion the the 1.0). that other EVDsdistributions. The are TDFs suitable good However, (median for results the representing mCOE forEVDs’ skill within-day is fit rainfall the shown is less intensity inferior GEV cannot to than and GPT3 be 0.9 (at WEBL all considered but exceptional one station). in Thus, that on balance the it is not considered The first two eliminations aredistributions straight forward. are The clearly skill shown poorereliminated by than as the candidate the LOGN distributions. and other LGN3 TDFs. Hence, LOGN and LGN3 were 4.2.1 Elimination: lognormal TDFs 5 5 25 15 20 10 20 15 10 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | , LS ) at a given i ) i RMSE90 (best fit) -th TDF (TDF i − mCOE (TDF ) i result is followed by GPT2 − LS is the best-fit benchmark, followed LS RMSE90 (TDF = 3208 3207 mCOE (best fit) i = i erence between the best performed TDF and the ff ¨ oppen climate zone. Similar figures were drawn for intensity threshold. The data are discussed in terms of . 1 − LS is zero or close to zero in both the upper and lower charts , are considered. To understand how these parameters vary LS , which exhibit the next lowest (and very similar) marginal error ¨ oppen classes and also with each of the explanatory variables. HI I LS ¨ oppen class first and then the explanatory variable and also ac- ¨ oppen climate zone, annual rainfall depth, annual rain days, mean – likely to have been ranked third if LS fitting could have been done, but and then EXP – clearly the best fit; – the next best fit and with only 2 parameters; – placed here because of the low marginal mCOE values; and PM and EXP PM LS LS LS LS GAMA placed fourth as it has significantly higher marginal mCOE values. marginal error in RMSE90 for TDF GPT3 GPT2 EXP marginal error in mCOE for TDF erences between K – – Figure 9a shows box plots of daily mean wet period intensity. Latitude and sta- Given these results, the authors suggest the following ranking of TDFs based on 1. 2. 3. 4. ff All the examplesshowed shown the use strongest latitude relationship as with the the explanatory rainfall variable behaviour. as it There consistently are, however, within-day statistics and daily rainfall amount was also examined. tion numbers are shownto on north the and x-axis.each are of coloured Boxes the by areboth explanatory K organised according variables by to and latitude K eachcording from to of south the the explanatory statistics.di variable (as Box in order Fig. was 9). varied This enabled assessment both of clarity of interpretation. Inas addition characterised the by behaviour ofbetween rainfall the stations highest an intensities exploratoryrelationships in analysis with the was K day, undertaken andrain the day rainfall existence depth, of elevation, and latitude considered. The relationship between the 5 Overview of within day intensity behaviour The TDFs are essentiallywithin day representing 6-min rainfall three intensity distributions: aspectsness. the of mean; In standard the deviation; addition, and statistical the skew- the day distribution wet exceeding fraction of the 1 parameter mm represents h these standard the statistical duration parameters of rather rainfall than within the GPT3 distribution parameters for choosing GPT2, exponential or gammabe distributions incorrect is to only interpret small.utility; their ranking Therefore, in below it point GPT3 would ofquite as attractive fact a options. because recommendation against they their rely on only two parameters they are viewed as Although the GPT3 distribution provided clearly the best fit, the performance penalty for then GAMA their performance as measured by the goodness-of-fit statistics mCOE and RMSE90: that the error forbecause GPT3 at most stations itFig. 8). gives the These results best reinforce fit. theby fact GPT2 Consider that GPT3 the mCOEmagnitudes. results first Turning (top to of the lower box plot, the GPT3 TDF of interest on apluviograph station-by-station station: basis. That is, for the The box plots shown inall the Fig. pluviometer 8 stations depict with the the TDFs range fitted of by both each product marginal moments error and statistic LS. Note across To rank the remaining four TDFs (EXP,GAMA,tics GPT2 and were GPT3) calculated some additional which statis- TDF focus over another, on rather the than simply marginalMarginal looking error error at associated the is magnitude with defined of selecting mCOE as and one the RMSE90. di 4.2.4 Final ranking 5 5 20 25 15 10 20 15 10 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | - ffi the Pacific ff . It can be concluded HI ects even approaching I ff set each other in terms of daily S. Very similar patterns of be- ff ◦ ering daily rainfall accumulations. ff ¨ oppen zones (Aw and BSh – see Fig. 1) ¨ oppen classes become evident. It is also ¨ oppen zones BWh, Bsh, Cfa and Cwa). The 3210 3209 20 m) over the 15 km east (i.e. upwind) of the ∼ cient of variation of within day wet period 6-min inten- (Fig. 9c). ffi S. A similar but noisier pattern was observed with wet HI ◦ I erences between K ff ects acting on the prevailing easterly winds blowing o ff ¨ oppen climate classes. Individual boxes represent all days within a cient of variation shows interesting behaviour with daily accumulation, first ffi Figure 10 shows how the within day statistics of rainfall vary with daily rainfall amount Taken together the changes in within-day statistical behaviour of rainfall intensity Figure 9b shows that the coe The one site that is a consistent and significant exception to the above trends is Figure 9a shows a trend of increasing rainfall intensity towards the equator, particu- ¨ ¨ oppen class. The inter-quartile range in CV remains approximately constant across all oppen class, so attributing the behaviour to a particular explanatory variable is di wet fraction (Fig. 10d) shows an almost linear growth with daily rainfall accumulation, show the highest meanintensity intensities for and a also given theCsa, daily Csb, greatest rainfall Cfb) show inter-day accumulation, the variability while lowesttrends intensity in the and in mean most inter-day intensity variability. southerly with This indicates latitude zonesThe that are (BSk, the coe not just dueincreasing, to then di reaching afrom plateau the or changes in beginning standard deviation, tobut which decrease. increases tends with to This daily rainfall asymptote behaviour accumulation shows towards results similar constant patterns behaviour to standard at deviation large but the daily changes rainfalls. are less Skewness pronounced. The 10 mm range inamount daily increases rainfall, there is beginningdeviation an with increase and the in skewness mean 10–20 (not mmto intensity shown) result range. (Fig. in for 10a) a and all As proportionallythe also climate daily greater rain standard zones. increase day rainfall in Fig. These the 10c). highest combine intensities The together observed most during northerly K fall systems towards therange) of equator. the wet The fractionintermediate possibly changes reflect latitudes, a in with mix inter-day of increasingAustralia frontal variability dominance and and (inter-quartile convective of systems convective one in systems the should frontal in be systems northern remembered in that southern Australia.than they 10 mm, only In which reflect interpreting accounts days for these with most data of rainfall the it accumulations rainfall greater for at the these sites. various K opposing trends in intensity andrainfall accumulation, wet although period there partially is an o towards increasing the trend in equator. daily rainfall accumulation probably reflect a shift in domination from frontal rainfall systems to convective rain- the intermediate latitudes considered (K tends to decrease towards the equator but has a slightly higher inter-quartile range in this magnitude. sity grows smoothly withnent latitude, is although smaller than the for proportionalfrom any this of change trend the across mean, that the standard thethe conti- deviation equator. standard or Again, deviation weaker grows patterns more wereK quickly observed with than mean the wet-day mean rainfallstations. and towards with Skewness (not shown) waswith observed an inter-quartile to range be from very about consistent 1.1 between to stations 2.4 and a median of 1.7. The wet fraction Cairns, Queensland. This iswith an Orographic area e with extremelyOcean high and rainfall up gradients the associated escarpmentthe of terrain the rises Great from Dividingsite. near Range. No sea The other level site sites ( is in at the 760 data m, set and are subject to orographic e of colours in Fig.clear 9a from di the rapidthe expansion between-day in variability in inter-quartile within-day range intensitythe distributions compared equator, becomes with larger especially the towards belowhaviour median were a that evident latitude for of6-min the intensities about within-day and wet 30 also for period standard deviation (notKoombooloomba shown) (31083). of This site is located on the Great Dividing Range near K cult. larly for latitudes lessday than mean 30 rainfall depth (annual rainfall/annual rain days). By considering the groups significant correlations between the explanatory variables, most notably latitude and 5 5 25 15 20 10 25 15 20 10 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | , where the intervals HI I cient of variation of wet pe- ). Mean intensity, standard ffi HI I 3212 3211 Implications of distribution function ranking: the relatively poor fit of the lognormal lesser degree the Gumbel distribution)of provided a fits quality to the thatGiven within-day approaches that rainfall but data the does extremetage, not value coupled exceed distributions with that the provide doubt of norainfall over the clear the data, validity GPT3 performance it of advan- distribution. is usingthis them recommended purpose. to that describe extreme within-day value distributions not be used for Variability of fit performancea theoretical with distribution location: functiontion the was (GPT3) being found importance consistently to of identifiedever, be the as location root small best mean in performing square with error between fitting theincreased. statistic sites. was same noted How- to distribu- increase as rainfall intensity (2 and 3 parameter) distributionbasis function for suggest modelling that it within-day rainfall should patterns. not be usedExtreme as value distributions: the the skill of the GEV and Weibull distributions (and to a tion to minimise root mean square error (Eq. 1). Parameter estimation methods: the utility ofusing fitting theoretical L-moment distribution functions methods wasproduct moment found method. to The be bestvalues by consistently fit product was inferior moments, achieved to then by improving first the the estimating standard fit parameter performance using a optimisa- – – – – The analysis has also provided insight into the within-day statistical behaviour of rain- In addition to these specific conclusions, the study provides a range of other more tions closer to the equator.of Skewness rainfall remained during approximately rain constant. days The tended duration to decrease towards the equator. Trends with daily excellent means to summarise2 the or distribution 3 of TDF within-daymodel). parameter data values (240 plus points) the by wet only fraction statistic (givingfall a and 3 the or inter-day 4 variationacross parameter in the this continent behaviour. towardsproperties the Clear including equator) trends the were with mean, identified standard latituderiod deviation for (increasing and 6-min key coe within-day intensity statistical variationdeviation and and maximum maximum intensities intensities also ( became more variable between days for loca- It is important tothe note measured rainfall that intensity in data absolute is terms very the high. quality This of suggests the that calibrated the TDF TDFs fits are an to general insights into thefitting nature distribution of functions within-day to rainfall it. intensity data and information on distributions. The ranking wasobjective made functions: on the root the mean basis squarepared of error to of performance the the with measured fitted theoretical respect within-daythe distribution measured pluviograph to com- 5 data; two highest and 6-min the rainfalldid intensities fitted not across versus have the to the day, be mean consecutive of 6 Summary and conclusions This study was conductedof as whether a within-day precursor rainfall to characteristicsfrom a and daily detailed intensity measurements investigation distributions of into canprimarily climatic the be variables. inferred question on Given identifying thiswhich the context to the most represent study appropriate within-day focussed demonstrated theoretical rainfall that intensities. distribution the function(s) three-parameter Inbest with Generalised fit, respect Pareto followed of by Distribution this the provides aim, two-parameter the the Generalised analysis Pareto, Exponential and Gamma the mean intensity anddaily wet rainfall accumulation fraction is together,intensity. due it to is growing clear rainfall that duration rather most than of increases the in increase in as does the inter-day variability (inter-quartile range) in wet fraction. Considering both 5 5 25 15 20 10 25 15 20 10 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | ff http://www. ¨ oppen-Geiger erent Rainfall Con- ff doi:10.5194/hess-11-1633- , Eur. J. Soil Sci., 55, 299–316, ff erence in daily accumulation is cient of variation and skewness ff ffi and soil loss from bench terraces, 1. An , 2005. ff cient Test for Normality, Technometrics, 17, ffi 3214 3213 , 2008. , 2011. e, J. V.: River flow forecasting through conceptual models part I – a cient test statistics from several probability distributions, J. Hydrol., 355, ff doi:10.1029/2005GL022465 ffi This work was supported by an Australian Research Council Discov- and infiltration into soil profiles, Hydrol. Process., 16, 731–739, 2002. ff doi:10.1016/j.jhydrol.2008.01.027 , 2007. event-based model of rainfall2004. infiltration and surface runo 2841–2848, 1997. Res. Lett., 32, 1–4, Events, in: Handbook of Hydrology, edited by: Maidment, D.Assoc., R., 69, McGraw-Hill, Inc., 730–737, 1993. 1974. bucket rain gauge under64 field pp., conditions: 2002. Final report, Bureau offrequency Meteorology, analysis, Melbourne, Water Resour. Res., 20, 847–856, 1984. ditions, J. Atmos. Ocean. Tech., 16, 1025–1043, 1999. climate classification, Hydrol. Earth2007 Syst. Sci., 11, 1633–1644, simulating runo application., National Oceanic AtmosphericDept. of Administration, Commerce, National NOAA, Weather1971. Silver Service, Spring, US MDTech. Memo. NWS HYDRO-12, 52,discussion of 91 principles, pp., J. Hydrol., 10, 282–290, 1970. Res., 40, 887–901, 2002. Assoc. Comput. Mach., 8, 212–229, 1961. to daily timesteps:doi:02010.01029/02004WR003380, 2005. a distribution function approach, Water Resour. Res., 41, W02003, logic and hydroclimatic model validation, Water Resour. Res., 35, 233–241, 1999. plot correlation coe 1–15, Resour., 28, 711–728, 2005. itl.nist.gov/div898/handbook/ rainfall data, Water Sci. Technol., 37, 121–129, 1998. 111–117, 1975. fall and basin geomorphology into nonlinear analyses of streamflow dynamics, Adv. Water Wilson, P. S. and Toumi, R.: A fundamental probability distribution for heavy rainfall, Geophys. Van Dijk, A. I. J. M. and Bruijnzeel, L.Wang, A.: Q. Runo J.: LH moments for statistical analysis of extreme events, Water Resour. Res., 33, Stephens, M. A.: EDF Statistics for Goodness of Fit and some Comparisons, J. Am. Stat. Stedinger, J. R., Vogel, R. M., and Foufoula-Georgiou, E.: Frequency Analysis of Extreme Rossi, F., Fiorentino, M., and Versace, P.: Two component extreme value distribution for flood Srikanthan, R., James, R. A., and Matheson, M. J.:. Evaluation of the performance of the tipping Nystuen, J. A.: Relative Performance of Automatic RainPeel, Gauges M. under C., Di Finlayson, B. L., and McMahon, T. A.: Updated world map of the K Nash, J. E. and Sutcli Monro, J. C.: Direct search optimisation in mathematical modelling and a watershed model Mertens, J., Raes, D., and Feyen, J.: Incorporating rainfall intensity into daily rainfall records for Lu, H. and Yu, B.: Spatial and seasonal distribution of rainfall erosivity in Australia, Aust. J. Soil Kandel, D. D., Western, A. W., and Grayson, R. B.: Scaling from process timescales Legates, D. R. and McCabe, G. J.: Evaluating the use of “goodness of fit” measures in hydro- Hooke, R. and Jeeves, T. A.: “Direct Search” solution of numerical and statistical problems, J. Heo, J. H., Kho, Y. W., Shin, H., Kim, S., and Kim, T.: Regression equations of probability Filliben, J. J.: Probability Plot Correlation Coe Engineering Statistics Handbook: e-Handbook of Statistical Methods, available at: Fankhouser, R.: Influence of systematic erros from tipping bucket rain gauges on recorded Dodov, B. and Foufoula-Georgiou, E.: Incorporating the spatio-temporal distribution of rain- Acknowledgements. ery Project GrantPEOPLE (DP-0557799) programme). and the AWWExchange WARECALC Scheme was project fellowship supported from of the by Australianand the Academy a Steve of European Wealands Marie Sciences. for Union We their Curie thank (FP7- constructive Murray International advice Peel and Research technical Sta guidance. References mum intensities, more complex behaviourand for strongly the increasing coe rainfall duration.due Most to of the duration di ratherbehaviour than are intensity changes. believed to Therainfall be spatial mechanisms linked across trends the to in continent. within-day a rainfall shift in dominance of frontal and convective rainfall accumulation demonstrated increases in mean, standard deviation and maxi- 5 5 30 25 20 15 10 25 20 15 10 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | ) and ). The P 1 − Class Rainfall Rainfall Days 3216 3215 ) where rainfall intensity exceeded the threshold (1 mm h n 30.333.9 153.135.3 151.2 1960–200535.2 149.2 1962–200534.2 147.5 1937–2005 538.1 142.1 1945-2005 6 Cfa 57837.8 147.1 1953–2005 Cfa Cfb 21241.5 145.0 1953–2005 Cfa 5042.8 147.2 1873–2005 1704 BSk 147.5 1938–2005 5 1106 35 1960–2005 630 Cfb 170 242.4 Cfb Cfb 583 124.3 Mar 4 294 65.8 Jun Cfb 60.5 Oct 617 657 63.6 31.5 Oct 684 62.7 Sep Oct 62.6 39.9 67.3 510 Sep 77.8 Nov 37.1 Jun Oct Aug 18.8 87 56.7 Feb 96 Mar 40.5 Dec 47.5 72 38.5 Feb 73 Feb Mar 29.3 45 Jun 91 100 93 85 18.217.9 127.720.4 122.2 1955–200524.9 118.6 1948–2005 42226.6 113.7 1953–2005 BSh28.8 118.5 1956–2005 731.9 114.7 1953–2005 6 BSh34.9 116.0 1953–2005 BWh 4 51733.8 546 117.8 1961–2005 BWh BWh 3330.8 121.9 1965–2005 152.6 15 Csa 59325.0 121.5 1969–2005 313 68 Csa12.4 128.3 1939–2005 Jan 234 235 25 Csb12.3 130.9 1956–2005 365 177 Csb19.6 98.5 136.8 1953–2005 598 BSh 46723.8 Jan 49.1 134.2 34.6 1966–2005 BWh Feb 795 2.2 3031.2 133.9 1969–2005 Jun 804 Jun 52 Aw27.6 Aug 136.8 107 1951–2005 376 172.6 625 Aw32.1 271 135.4 1955–2005 1.5 546 BSh 123.2 Jun 0.9 273 Jun35.0 133.7 1961–2004 BWh 167 Sep 2.1 4.9 98.537.7 Jul Sep 138.5 47 1954–2005 31.4 117 BWh 171513.8 140.8 48.8 Dec 1967–2005 Sep BWh 1430 Jul Jun 5.7 1510.6 435 143.1 9.2 1942–2005 428.5 Feb 35 282 BSk 23.417.7 142.2 20 1967–2002 Dec 284.8 6 Jan Jan 192 119.6 6320.7 141.1 1961–1993 25 29 Csb 161 Feb 17.4 14.4 Feb 176 Csb17.8 139.5 44.4 1964–1999 10.1 Feb Aw Dec 5819.2 Sep 145.6 20.8 1967–2005 60 304 Feb Aug 87 Aw21.1 146.8 28.9 1960–2005 8 340 May 82 1 4.223.4 455 149.2 1953–2005 Aw 760 BSh 1.6 Feb 707 41.1 91 23.4 1192 144.3 40 1959–2005 Sep Jul Cfa 32 Aug27.4 150.5 11.9 1964–2005 8 9 1746 Jul 308.5 30 99.326.4 63 153.1 1939–2005 Aw 192 Apr Sep Cwa31.5 443 146.3 1949–2000 418.6 Jan 919 BSh 56 9 Jul 94 Jul 2739 1031.0 37 145.8 1953–2005 11.9 Jan Aug 102.9 Cfa 150.3 1962–2005 259.5 4 303 481.7 28 Jan 1144 30 1606 1946–2005 Cfa Jan 260 BSh 25.6 0.9 18.2 Jan 455 Mar BSh 307 292.7 316.9 Feb 22 Sep 3.5 Jan Cfa 819 57 Feb Feb 81.1 Sep 3.8 1185 85.6 1.7 493 119 141.2 Feb 415 75 Jun 79 Oct Aug 171.7 10.7 16.4 Feb 643 72.2 84 Feb 49.9 Sep Sep Jan 9.6 138 37 44 90.9 23.5 Jan Sep 34.9 Jan Sep 65 97 20.3 Sep Aug 24 33 36.4 62 Jun 91 Aug 44 46 60 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − ce ce ffi ffi ce ffi s Harbour MO 4 700 (4.7) 13 626 (52.8) 1894 (12.3) 61 156 (88.2) 30 418 (5.2) 813 781 (74.7) 4 2 (0.01) 42 (0.16) 7 (0.05) 85 (0.1) 59 (0.01) 841 (0.08) Summary of data divided into categories based on daily total rainfall depth ( ff 10 4056 (27) 12 118 (47) 1886 (12.2) 8069 (11.6) 81 416 (13.9) 274 371 (25.2) Properties of the 42 study sites. ≥ ,n < ,n 2 mm 10 269 (68.3) 11 664 (75.5) 473 196 (80.9) P < . 10 10 0 ≤ ≥ ≥ 2 . Station:CriteriaP < 0 DaysP P (%) Melbourne Depth (%) Days (%) Depth (%) Darwin Days (%) Depth (%) All Stations 6603770014 AMO 72150 76031 Wagga Wagga AMO 85072 86071 East Sale Airport 91104 Melbourne Regional O 94008 20123003 Halls Creek4032 Airport Broome Airport6011 Port Hedland7045 Airport Carnarvon Airport8051 Meekatharra Airport9021 Geraldton Airport9741 Perth Airport9789 Albany Airport12038 Esperance 13017 Kalgoorlie-Boulder Airport 14015 Giles Meteorological O 14508 Darwin Airport 15135 15590 16 001 17043 Woomera Aerodrome 18012 Oodnadatta Airport 23034 Ceduna AMO 26021 27006 Mount Gambier AERO 27022 29041 Thursday Island MO 29127 Normanton Post O 31083 Mount Isa AERO 32040 Koombooloomba Dam 33119 Townsville AERO 36031 Mackay MO 39083 Longreach AERO 40223 Rockhampton AERO 44021 Brisbane AERO 48027 Charleville AERO 55024 Cobar MO 59040 Gunnedah SCS Co StationCode Station Name Lati- Longi- tude Period tude Elev Koppen Annual Max (m) Max Climate Rainfall Monthly Min Month Min Monthly Month Annual Rain the number of 6 min periodsnumber ( of days and daily rainfallindividually, depth and (mm) for are all listed stations for the combined. stations in Melbourne and Darwin Table 2. Table 1. Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | κ / α + ξ κ 0 / ≤ α 1 x x < ≤ ≤ x   0 : 0 0 ,ξ κ κ α α ≤ 0 0 + + , β < x > x > ξ ξ 0 κ > 0 ) κ > x ,x > 0; for ,x < κ > 0 ) ln( 0 ; for x = x > ∞ ∞ y κ < κ > ln( ; for 0 : 0 and = ∞ x < y β > ≤ 0 α > when when x < where < x < β 0; ,ξ ≤ 0  0 2 for where , x > i −∞ 0; for 0 y ): gamma function κ <  . µ ( 2 − ) i Γ α > y x > ξ for κ < ξ y σ − µ  x − y ) for σ /κ ln( x 1 i h } i ln( ) ) ) 2 1 ξ ) 3218 3217 /κ h ξ ξ  − α 1 − − 1 2 ) x α βx i κ − x ( α ) ( −  ( /κ −  ξ κ x 1 − Γ ( x α − α    x − exp( β ( x α location parameter (mean)scale parameter (std. 18.14–15 dev.) location parameter (mean)scale parameter (std. 18.15–16 dev.) location parameter inverse scale parameterlocation parameter 18.19–21 shape parameterinverse scale parameter scale parametershape parameter 18.19–21 scale parametershape parameter location parameter 18.22 scale 18.22 parametershape parameter location parameter scale parametershape parameter 18.17–19 scale parameterlocation parameter 18.19 18.16–17 exp 1 1 )] and parameter validity conditions for nine distributions − − κ y κ − h  { exp σ x exp ) α − − y ( ) ξ − µ β α α κ α κ α α σ µ σ ξ ξ β α κ ξ ξ κ ξ σ − 1 ) − 1 F h  1 x x exp h exp  ( ( 1 βx π π ( − − − − | 2 2 exp 1 1 exp 1 1 √ √ β | ======) ) ) ) ) ) LGN3 GPT3 ) ) ) x x x x x x x x x ( ( ( ( ( ( ( ( ( F F f F F F f F f )] and/or CDF[ x ( f Extreme value distributions Generalized GEV WeibullGumbel WEBL GMBL TDFNameLognormal Short LOGN Name ParametersExponential EXP GammaGeneralized Page Reference(s) GAMA Pareto GPT2 Stedinger et al. (1993) Extreme Value PDF[ Theoretical Distribution Functions tested for skill in fitting the CDF of within-day rainfall Gumbel Weibull DistributionLognormal (2-par) PDF and/or CDF and parameter validity conditions Generalised Pareto (3-par) Generalised Extreme Value Exponential Gamma Generalised Pareto (2-par) Lognormal (3-par) Details for all TDFs can also be found in: Table 18.1.2 and Table 18.2.1 (Stedinger et al., 1993). Table 3b. (Stedinger et al., 1993). 1 Table 3. intensity. This table alsoof indicates the the short distribution name andrelationships assigned used their to in function each the (scale, TDF, fitting lists process shape, the (pages or parameters from location), Stedinger et and al., indicates 1993). the source for Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | 3220 3219 GPT2 LS 7.0% WEBL LS 2.5% Fit statisticmCOE TDF FitRMSE90 method GPT3 % Stations GPT3 LS LS 100% 90.5% Combination of TDF and fitting method with the highest fitting skill for mCOE and Map showing the location of the selected 42 pluviograph stations together with the ¨ oppen climate zones (Peel et al., 2007) for Australia. Fig. 1. K Table 4. RMSE90 statistics, indicatingbest. the percentage of stations for which each combination is the Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper |

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Fitted cumulative density functions (CDFs) of EXP, GPT2, GPT3 and GAMA for nine Fitted cumulative density functions (CDFs) of EXP, GPT2, GPT3 and GAMA for nine

4040 7070 0 7070 0 5050 00 00 00 00 00 00 00 40 4

50 5 00 55 00 88 66 44 22 00 00 66 44 22 00 00 00 00 00 00 00 00 00 55 00 55 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 88 66 44 22

77 66 55 44 33 22 11 11 11 11 11 11 88 33 22 22 11

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) ) r r h h / / mm mm ( ( y y t t i i s s en en t t n n i i ll ll a a f f n n i i a a R R ) ) r r h h / / mm mm ( ( y y t t i i s s en en t t n n i i ll ll a a f f n n i i a a R R events representative of varying dailytechnique. rainfall depths for Darwin. TDFs were fitted using the LS Fig. 3. events representative of varying dailyLS rainfall technique. depths for Melbourne. TDFs were fitted using the Fig. 2. Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | ), as well 2 r th th th th th th 90 50 90 50 90 50 cient ( ffi 120 120 120 9.31% > 16 100 100 100 0.74% > 16 mm/hr 0.63% > 16 mm/hr 80 80 80 [mm/hr] [mm/hr] [mm/hr] HI HI HI I I I th th th 90 90 62.0 90 60 60 60 Measured Measured Measured 40 40 40 th TDF: Lognormal (3 param) TDF: Gamma TDF: Generalized Extreme Value th th 50 50 30.2 50 STATION: Darwin Airport Darwin STATION: 20 20 20 = 0.991 = 0.995 = 0.935 2 2 2 mCOE = 0.859 bias = -1.90 mm/hr r mCOE = 0.840 bias = -2.10 mm/hr r mCOE = 0.659 bias = 4.23 mm/hr r 0 0 0 5 0 0 5 0 5 0 0 0 80 60 40 20 15 10 80 60 40 20 50 15 10 15 10 140 120 100 140 120 100 250 200 150 100 7.6 3.0 8.3 3.4 4.3 15.4 th th th th th th 90 50 90 50 90 50 3224 3223 100 100 100 FITTING: Product Moment Method Product Moment FITTING: 80 80 80 [mm/hr] [mm/hr] [mm/hr] HI HI HI 0.14% > 16 mm/hr 0.14% > 16 mm/hr 0.18% > 16 mm/hr I I I 60 60 60 40 40 40 Measured Measured Measured and the RMSE for each event (both plotted versus the measured th th th 30 90 90 22.56 TDF: Generalized Extreme Value TDF: Lognormal (3 param) TDF: Gamma 20 90 20 20 I = 0.994 th th = 0.971 = 0.993 values, and also indicate the percentage of RMSE values greater than th 2 2 2 mCOE = 0.893 r bias = -0.39 mm/hr STATION: Melbourne Regional Office Melbourne STATION: bias = 0.906 mm/hr mCOE = 0.770 r mCOE = 0.878 bias = -0.60 mm/hr r 50 50 9.5 50 0 30 0 0 0 0 I 5 0 5 0 5 0 0 80 60 40 20 80 60 40 20 80 60 40 20 10 15 15 10 15 10 120 100 120 100 120 100 2.9 0.6 2.4 0.6 3.2 0.7

HI HI HI I I I RMSE RMSE RMSE [mm/hr] [mm/hr] [mm/hr] [mm/hr] [mm/hr] [mm/hr] Fitted Fitted Fitted and are hence outside the vertical scale of the plot. 1 − Six paired scatter plots are shown to illustrate the skill of three selected TDF’s: three Caption on next page. charts. The RMSE plots indicate the 50th (solid line) and 90th (dashed line) percentile 30 I ). Values for mCOE, bias and the square of Pearson’s correlation coe 30 16 mm h RMSE and measured as the line-of-perfect-agreement (solid)the and the linear regression line (dashed) are printed on Fig. 4. parameter Lognormal (top); Gamma (middle);data and is the for Generalized two Extreme pluviograph Value stations: (lower).in Melbourne pairs The on showing the left the and fitted DarwinI on the right. The plots are Fig. 4. Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | 1 − charts. 30 I ). Values for 30 I th th th th th th 90 50 90 50 90 50 120 120 120 ), as well as the line-of- 100 100 100 0.15% > 16 mm/hr 1.63% > 16 mm/hr 0.42% > 16 mm/hr 2 r 80 80 80 [mm/hr] [mm/hr] [mm/hr] HI HI HI I I I th th th 62.0 90 90 90 60 60 60 cient ( Measured Measured Measured ffi 40 40 40 th th th Product Moment Fit Least Squares Fit L-Moment Fit 30.2 50 50 50 20 20 20 = 0.963 = 0.994 = 0.996 2 2 2 bias = -2.98 mm/hr bias = -1.18 mm/hr bias = -0.46 mm/hr r mCOE = 0.760 r mCOE = 0.891 r mCOE = 0.929 TDF: Generalized Pareto (3 parameter) Pareto (3 TDF: Generalized 0 0 0 5 0 0 0 0 5 0 5 0 15 10 80 60 40 20 80 60 40 20 80 60 40 20 15 10 15 10 100 140 120 140 120 100 140 120 100 5.8 2.4 7.2 2.9 9.3 3.3 th th th th th th 90 50 90 50 90 50 3226 3225 erent fitting schemes: L-moments (top); Product ff 120 120 120 STATION: Darwin Aiport Darwin STATION: 100 100 100 1.48% > 16 mm/hr 0.58% > 16 mm/hr 3.59% > 16 mm/hr 80 80 80 [mm/hr] [mm/hr] [mm/hr] HI HI HI I I I th th th 62.0 90 90 90 60 60 60 Measured Measured Measured 40 40 40 th th th 30.2 50 50 50 Least Squares Fit Product Moment Fit L-Moment Fit 20 20 20 = 0.992 = 0.976 = 0.961 2 2 2 bias = -3.77 mm/hr bias = -1.03 mm/hr bias = -3.80 mm/hr mCOE = 0.884 r mCOE = 0.713 r mCOE = 0.741 r TDF: Generalized Pareto (2 parameter) Pareto (2 TDF: Generalized 0 0 0 0 0 0 5 0 5 0 5 0 80 60 40 20 80 60 40 20 80 60 40 20 15 10 15 10 15 10 140 120 100 140 120 100 140 120 100 7.6 2.7 9.2 3.3 10.8 3.6

HI HI HI I I I RMSE RMSE RMSE [mm/hr] [mm/hr] [mm/hr] [mm/hr] [mm/hr] [mm/hr] Fitted Fitted Fitted values, and also show the percentage of RMSE values greater than 16 mm h 30 I and the RMSE for each event (both plotted versus the measured Six paired scatter plots based on data from Darwin Airport are shown to illustrate first Caption on next page. 30 I perfect-agreement (solid) and the linearThe regression RMSE line plots (dashed) are indicatemeasured printed the on the 50th (solid line) and 90th (dashed line) percentile RMSE and Fig. 5. mCOE, bias and the square of Pearson’s correlation coe and are hence outside the vertical scale of the plot. the relative skillside) of and TDF’s having second two the (GPT2 success – of left three side) di or three parameters (GPT3 – right Moments (middle); and Leastfitted Squares Estimation (lower). The plots are in pairs showing the Fig. 5. Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper |

WEBL WEBL

LS not available not LS available not LS LGN3 LGN3 percentile percentile th th LOGN LOGN - 75 - median - 25 - whisker < 1.5 x interquartile range - outlier > 1.5 x interquartile range GPT3 GPT3 GPT2 GPT2 Fit by: Least Squares 3228 3227 GMBL GMBL

GEV GEV

LS not available not LS available not LS Fit by: Product Moments GAMA GAMA EXP EXP Fit by: L-Moments 1 5 0

25 20 15 10

0.8 0.7 0.6 0.5 0.4 0.9 (90th Percentile) (90th

modified Coefficient of Efficiency of Coefficient modified Error Square Mean Root Legend: Maps indicating the spatial variation of mCOE (left) and 90th percentile RMSE (right) Box plots showing the spread of mCOE (upper chart) and RMSE90 (lower chart) values Fig. 7. for GPT3 fitted usingfit least (i.e. squares maximum estimation. mCOE and Note minimum that RMSE). smaller circle sizes indicate a better Fig. 6. across the 42 pluviographeach stations. of The the results ninetechnique indicate TDFs the was spread and not of the ableskill values three is to associated indicated fitting be with by methods employed mCOEwhile (note values for outliers close are that the to identified the GAMA 1.0 above,analysis. this least and and does by squares LGN3 not RMSE90 estimation imply values distributions). data close was to High removed zero. from fitting Note any that subsequent Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper |

Cfb Cfa Cwa Csb Csa BSk BSh BWh Aw

27022 6 −10.

14508 3 −12.

14015 4 −12.

27006 8 −13.

29041 7 −17.

31083 8 −17.

3003 9 −17.

2012 2 −18.

32040 2 −19.

15135 6 −19.

4032 4 −20.

29127 7 −20.

33119 1 −21.

39083 4 −23.

36031 4 −23.

15590 8

GPT3 GPT3 −23.

6011 9 −24.

13017 −25

44021 4 −26.

7045 6 −26.

40223 4 −27.

17043 6 −27.

8051 8 −28.

59040 3 −30.

12038 8 percentile percentile −30.

55024 th th −31

16001 2 −31.

48027 5 −31.

9021 9 - 75 - median - 25 - whisker < 1.5 x interquartile range - outlier > 1.5 x interquartile range −31.

18012 1 −32.

9789 8 −33.

66037 9 −33.

76031 2 −34.

9741 9 −34.

GPT2 GPT2 23034 −35

72150 2 −35.

70014 3 −35.

26021 7 −37.

86071 8 −37.

85072 1 −38.

91104 5 −41.

94008 8 −42. (d) (b)

0

8 7 6 5 4 3 2 1 2 1 0 5 5 5

LS not available not LS LS not available not LS 0. 0. 0. 0. 0. 0. 0. 0.

2. 1. 0.

Wet Fraction Wet Coe cient of Variation of Coe cient

27022 6 −10.

3230 3229 14508 3 −12.

14015 4 −12. , and wet fraction. Boxes are arranged from highest

27006 8 −13.

29041 7

−17. HI

GAMA GAMA I 31083 8 −17.

3003 9 −17.

2012 2 −18.

32040 2 −19.

15135 6 −19.

4032 4 Fit by: Least Squares −20.

29127 7 −20.

33119 1 −21.

39083 4 −23.

36031 4 −23.

15590 8 −23.

6011 9 −24.

13017 −25

44021 4 −26.

7045 6 −26.

40223 4 −27.

17043 6 EXP EXP −27.

8051 8 −28.

59040 3 −30.

12038 8 −30.

55024 −31

16001 2 −31.

48027 5 −31.

9021 9 −31.

18012 1 −32.

9789 8 −33.

66037 9

Fit by: Product Moments −33.

76031 2 −34.

9741 9 −34.

23034 0 −35

4 3 2 1 0

0.2 0.1 72150 2 −35.

70014 3 −35.

26021 7 −37. RMSE90 - RMSE90(best) - RMSE90

mCOE(best)-mCOE Legend: 86071 8 −37.

85072 1 −38.

91104 5

Box plots showing the marginal error associated with choosing an alternate TDF and −41.

¨ 94008 8 −42. oppen climate classification of the stations (see Fig. 1). All intensity statistics use 6-min Box plots showing variation in daily mean wet period intensity, daily wet period intensity (c) (a) cient of variation, extreme intensity 0

5 0

50 40 30 20 10 80 70 60

10 40 35 30 25 20 15 Extreme Intensity (mm/h) Intensity Extreme Mean Intensity (mm/h) Intensity Mean ffi data. Boxes show theand notches inter-quartile show range, confidence whiskers limits extend on 1.5 the median. times the inter-quartile range Fig. 9. coe (most southerly) to lowestshow latitude K and are labelled with station number and latitude. Colours Fig. 8. fitting method than the bestGPT3/MLE available combination combination at as each at pluviograph.(therefore most zero Error error). is locations very The this low upperplots combination for box the plot error exhibits indicates the in the highest spread RMSE90.displaying of fitting low error skill values The for in mCOE most andabove, both attractive this the the does lower TDF upper not and and imply fitting lower data plots. was method removed combinations Note from that any are subsequent while those analysis. outliers are identified Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper |

Cfb Cfa Cwa Csb Csa BSk BSh BWh Aw

85 85

65 65

15 15

175 175

145 145

95 95

55 55

15 15

125 125

105 105

55 55

15 15

65 65

15 15

55 55

15 15

45 45

15 15

125 125

105 105

55 55

15 15

105 105

85 85

55 55

15 15

165 165

105 105

55 55

15 15 (b) (d) 2 1 0 1 0 5 5 5 8 6 4 2

2. 1. 0. 0. 0. 0. 0.

n Variatio of Coe cient n Fractio Wet

85

85 3231 65 65

, and wet fraction with daily rainfall accumulation for

15

15

175

175 HI

I

145 145

95 95

55 55

15 15

125 125

105 105

55 55

15 15

65 65

15 15

55 55

15 15

45 45

15 15

125 125

105 105

55 55

15

15

105 105

85 85

55 55

15 15

165 165

105 105

¨ oppen climate zones. Daily rainfall has been categorised into 10 mm bins with a lower

55 Box plots showing variation in daily mean wet period intensity, daily wet period intensity, 55

15 15 (a) (c) cient of variation, extreme intensity 5 0 0

10 80 60 40 20 40 35 30 25 20 15

ffi Mean Intensity (mm/h) Intensity Mean 140 120 (mm/h) 100 Intensity Extreme limit of 10 mmmaintain i.e. clarity bin and 1 labels includeswhere represent daily at the rainfalls middle least of ofaccumulation 10 10–20 the days mm. amount bin in fall Only some range. in cases somebox Note, and the due bins boxes some to observation are are observations this. only exist labelled bins, above drawn to bins the maximum are plotted missing for the second highest various K Fig. 10. coe