Doubly stochastic Poisson pulse model for fine scale rainfall

Transcription

1 Artcle Doubly stochastc Posson pulse model for fne scale ranfall Thayakaran, Rasah and Ramesh, N. I Avalable at Thayakaran, Rasah and Ramesh, N. I (2016) Doubly stochastc Posson pulse model for fnescale ranfall. Stochastc Envronmental Research and Rsk Assessment, 31 (3). pp ISSN It s advsable to refer to the publsher s verson f you ntend to cte from the work. For more nformaton about UCLan s research n ths area go to and search for <name of research Group>. For nformaton about Research generally at UCLan please go to All outputs n CLoK are protected by Intellectual Property Rghts law, ncludng Copyrght law. Copyrght, IPR and Moral Rghts for the works on ths ste are retaned by the ndvdual authors and/or other copyrght owners. Terms and condtons for use of ths materal are defned n the CLoK Central Lancashre onlne Knowledge

2 Doubly stochastc Posson pulse model for fne-scale ranfall R Thayakaran College of Health and Wellbeng, Unversty of Central Lancashre Preston, Lancashre, PR1 2HE, UK Tel.: +44 (0) rthayakaran@uclan.ac.uk and N I Ramesh Department of Mathematcal Scences, Unversty of Greenwch, Martme Greenwch Campus Old Royal Naval College, Park Row, Greenwch London SE10 9LS, UK May 12, 2016 Abstract Stochastc ranfall models are wdely used n hydrologcal studes because they provde a framework not only for dervng nformaton about the characterstcs of ranfall but also for generatng precptaton nputs to smulaton models whenever data are not avalable. A stochastc pont process model based on a class of doubly stochastc Posson processes s proposed to analyse fne-scale pont ranfall tme seres. In ths model, ran cells arrve accordng to a doubly stochastc Posson process whose arrval rate s determned by a fnte-state Markov chan. Each ran cell has a random lfetme. Durng the lfetme of each ran cell, nstantaneous random depths of ranfall bursts (pulses) occur accordng to a Posson process. The covarance structure of the pont process of pulse occurrences s studed. Moment propertes of the tme seres of accumulated ranfall n dscrete tme ntervals are derved to model 5-mnute ranfall data, over a perod of 69 years, from Germany. Second-moment as well as thrd-moment propertes of the ranfall are consdered. The results show that the proposed model s capable of reproducng ranfall propertes well at varous sub-hourly resolutons. Incorporaton of thrd-order moment propertes n estmaton showed a clear mprovement n fttng. A good ft to the extremes s found at larger resolutons, both at 12-hour and 24-hour levels, despte underestmaton at 5-mnute aggregaton. The proporton of dry ntervals s studed by comparng the proporton of tme ntervals, from the observed and smulated data, wth ranfall depth below small thresholds. A good agreement was found at 5-mnute aggregaton and for larger aggregaton levels a closer ft was obtaned when the threshold was ncreased. A smulaton study s presented to assess the performance of the estmaton method. Key Words: Moment propertes; Pont process; Precptaton; Ranfall generator; Stochastc models; Sub-hourly ranfall 1

3 1 Introducton An mportant challenge we face n envronmental or ecologcal mpact studes s to provde fast and realstc smulatons of atmospherc varables such as ranfall at varous temporal scales. Stochastc pont process models provde a bass for generatng synthetc ranfall nput to hydrologcal models where the observed data at the requred temporal scale are not avalable. They also enable us to assess the rsk assocated wth hydrologcal systems. There has been a number of stochastc pont process models developed by many authors over the years. Among these, the models based on Posson cluster processes (Rodrguez-Iturbe et al. (1987), Cowpertwat (1994), Onof and Wheater (1994) and Wheater et al. (2005)) utlzng ether the Neyman-Scott or Bartlett-Lews processes have receved much attenton, snce ther model structure reflects well the clmatologcal features of the ranfall generatng mechansm. A good revew of developments n modellng ranfall usng Posson cluster processes s provded by Onof et al. (2000). In addton, ranfall models based on Markov processes have also made a reasonable contrbuton to help tackle ths challenge. See for example, Smth and Karr (1983), Bardossy and Plate (1991), Ramesh (1998), Onof et al. (2002) and Ramesh and Onof (2014) amongst others. Much of the work on ths topc, however, has concentrated on modellng ranfall data recorded at hourly or longer aggregaton levels. Stochastc models for fne-scale ranfall are equally mportant, because n some hydrologcal applcatons there s a need to reproduce ranfall tme seres at fne temporal resolutons. For example, sub-hourly ranfall s requred as nput to urban dranage models and for small rural catchment studes. In addton, the study of clmate change mpacts on hydrology and water management ntatves requres the avalablty of data at fne temporal resolutons, whch s usually not avalable from general crculaton model (GCM) smulatons. The best avalable approach to generatng such ranfall currently les n the combnaton of an hourly stochastc ranfall smulator, together wth a dsaggregator makng use of downscalng technques. There has been some recent work on modellng fne-scale ranfall usng pont process models. Rather than attemptng to reproduce actual ranfall records at a fne-scale, usng downscalng technques or by other methods, these stochastc pont process models am to generate synthetc precptaton tme seres drectly from the proposed stochastc model. One good example of ths s provded by the work of Cowpertwat et al. (2007, 2011) who developed a Bartlett-Lews pulse model to study fne-scale ranfall structure. Ther model partcularly enables the reproducton of the fne tme-scale propertes of ranfall. A class of doubly stochastc Posson processes was employed by Ramesh et al. (2012, 2013) and Thayakaran and Ramesh (2013) to study fne-scale ranfall ntensty usng ranfall bucket tp tmes data. They utlsed maxmum lkelhood methods to estmate parameters of ther models. Our man objectve n ths paper s to develop a smple stochastc pont process model capable of reproducng fne-scale structure of the ranfall process. The other objectve s to provde a fast and effcent way of generatng synthetc fne-scale ranfall nput to hydrologcal models drectly from one stochastc model. In ths regard, and to take ths fne-scale ranfall modellng work further, we develop a smple pont process model based on a doubly stochastc Posson process, followng the Posson cluster pulse model approach of Cowpertwat et al. (2007). Our prelmnary work on ths (Ramesh and Thayakaran, 2012), analysng propertes of ranfall tme seres at sub-hourly resolutons, produced encouragng ntal results. In ths paper, we extend ths work further and accommodate thrd-order moments n estmaton. Mathematcal expressons for the moment propertes of the accumulated ranfall n dsjont ntervals are derved. The proposed stochastc model s ftted to 69 years of 5-mnute ranfall tme seres from Germany. The results of the analyss show that the proposed model s capable of re- 2

4 producng ranfall propertes well at varous sub-hourly resolutons. Furthermore, the analyss ncorporatng thrd-order moments produced better results than the one that used only up to second-order moments. Unlke Cowpertwat et al. (2007), who used superposton of two Bartlett-Lews pulse models to account for dfferent storm types, we use one smple model to reproduce sub-hourly ranfall structure. The novel contrbuton of ths study s the dervaton of the thrd-order moment propertes of the proposed model, as well as ther ncorporaton n estmaton, to reproduce fne-scale structure of ranfall more accurately. The proposed model provdes a sold framework to generate synthetc fne-scale ranfall nput to hydrologcal models drectly from one stochastc pont process model. In addton, the avalablty of a new stochastc model for the generaton of fne-scale ranfall, at varous sub-hourly scales, provdes scentsts wth a useful tool for envronmental or ecologcal mpact studes. The followng secton provdes a background to ths work, llustrates the model framework and then focuses on dervng moment propertes of varous component processes, such as the cell and pulse arrval processes. Propertes of the aggregated ranfall sequence are studed and mathematcal expressons for the thrd-order moment and the coeffcent of skewness are derved. Secton 3 presents the results of data analyss usng 5-mnute ranfall aggregatons and compares the results of two dfferent analyses that used second and thrd-order moments n estmaton. Extremes of the hstorcal data are compared wth the smulated extremes at varous resolutons. The proporton of dry ntervals s also studed. A smulaton study s carred out to evaluate the performance of the estmaton method. Conclusons and possble further work are summarsed n Secton 4. 2 Model framework and moment propertes 2.1 Background Doubly stochastc Posson processes (DSPP) provde a flexble set of pont process models for fne-scale ranfall. Ramesh et al. (2012, 2013) utlsed ths class of processes and developed stochastc models, for a sngle-ste and multple stes respectvely. These models were used to analyse data collected n the form of ranfall bucket tp-tme seres. One of the advantages of these models, over most other pont process models for ranfall, s that t s possble to wrte down ther lkelhood functon whch allows us to estmate the model parameters usng maxmum lkelhood methods. However, the ranfall bucket tp-tme seres s not usually avalable for a long perod of tme. Most of the longer seres of ranfall data are avalable n accumulated form, hourly or sub-hourly, rather than n tp-tme seres format. Moreover, the above DSPP models cannot be ftted drectly to data collected n aggregated form, such as hourly ranfall, usng the maxmum lkelhood method. Therefore t s useful to look for alternatve models, based on doubly stochastc processes, that can be used to model sub-hourly data collected n aggregated form. Motvated by the performance of the above class of doubly stochastc models, we seek to develop models wth the same structure that can be used to analyse accumulated ranfall sequences at fne tme scales. 2.2 Model formulaton The pont process model we propose here s constructed from a specal class of DSPP where the arrval rate of the pont process s governed by a fnte-state rreducble Markov chan. See 3

5 for example, Ramesh (1995, 1998) and Davson and Ramesh (1996). Suppose that the ran cells, at tme T say, arrve accordng to a two-state DSPP where the arrval rate s swtchng between the hgh ntensty (φ 2 ) and low ntensty (φ 1 ) states at random tmes controlled by the underlyng Markov chan. The transton rates of the Markov chan are λ (for 1 2) and µ (for 2 1) respectvely. Therefore, the parameters of the cell arrval process M(t) are specfed by the arrval rate matrx L of the cell occurrences and the nfntesmal generator Q of the underlyng Markov chan, where φ1 0 λ λ L = and Q =. 0 φ 2 µ µ Each ran cell generated by the process has a random lfetme of length L whch s taken to be exponentally dstrbuted wth parameter η and ndependent of the lfetme of other cells. A cell orgnatng at tme T wll be actve for a perod of L and termnates at tme T +L. When the cell at T s actve nstantaneous random pulses of ranfall occur, durng T, T + L ), at tmes T j accordng to a Posson process at rate ξ. Ths process of nstantaneous pulse arrval termnates wth the cell lfetme. Therefore, each cell of the pont process generates a seres of pulses durng ts lfetme, and assocated wth each pulse s a random ranfall depth, X j. As a result, the process {T j, X j } takes the form of a marked pont process (Cox and Isham, 1980). In our dervaton of model propertes later on n Secton 2.3, we treat the pulses n dstnct cells as ndependent but allow those wthn a sngle cell to be dependent. We shall refer to ths model as a doubly stochastc pulse (DSP) model. A dagram showng the structure of ths process s gven n Fgure 1. It s clear from ths that the formulaton of our model s very smlar to that of Cowpertwat et al. (2007), but the dfference les n the mechansm for the cell arrvals. Snce ths process operates n contnuous tme, we ntegrate the DSP process to obtan ranfall depths over dscrete dsjont tme ntervals and use ther moment propertes for model fttng and assessment. 2.3 Moment propertes of the pulse process It follows from the structure of the pont process that the moment propertes of the DSP process are functons of those of the cell arrval process. Therefore, we shall frst revew the propertes of the cell arrval process before movng onto derve the propertes of the pulse process. The second-order moment propertes of the cell arrval process M(t) can be obtaned as functons of the model parameters and these are gven below (see for example, Ramesh (1998)). The statonary dstrbuton π = (π 1, π 2 ) of the cell arrval process M(t) s obtaned by solvng πq = 0, where 0 = (0, 0), and s gven by π = (µ/(λ + µ), λ/(λ + µ)). Let 1 be a column vector of ones, then the mean arrval rate of the cell process s gven by E(M(t)) = m = πl1 = λφ 2 + µφ 1 λ + µ. The covarance densty of the cell arrval process M(t) can be obtaned as, for t > 0, c M (t) = λµ (λ + µ) 2 (φ 1 φ 2 ) 2 e (λ+µ)t = Ae (λ+µ)t (1) whch shows that ts covarance decays exponentally wth tme. 4

6 (a). φ 2 φ 1 T Tme (b). Tme T L T + L T +1 L +1 T +1 + L +1 (c). X 2 X 3 X 1 X j T T 1 T 2 T 3 T j T + L Tme Fgure 1: Schematc descrpton of the DSP model: (a) the cell arrval process based on a two-state DSPP. (b) the cell lfetmes of the two cells at tme T and T +1. (c) the pulse process n the cell that orgnates at tme T and termnates at tme T + L. 5

7 We shall now study the moment propertes of the pulse arrval process and focus our attenton on dervng an expresson for ts covarance densty. These moment propertes are requred to derve the statstcal propertes of the aggregated ranfall process later n Secton 2.4. The lfetmes L of the ran cells, under the DSP model framework, are assumed to be exponentally dstrbuted wth parameter η and hence we have E(L ) = 1/η. Let us take N(t) as the countng process of pulse occurrences from all cells generated by the process M(t). An actve cell generates a seres of nstantaneous pulses at Posson rate ξ durng ts lfetme and therefore the mean number of pulses per cell s ξ/η. As noted earler, the arrval rate of the cell process s m and hence the mean arrval rate of the pulse process s gven by E(N(t)) = mξ/η. It s well known that the covarance densty of a pont process can be expressed as a functon of ts product densty. As shown by Cox and Isham (1980), the product densty of the pont process N(t) at dstnct tme ponts t 1,t 2,...,t k for k=1,2,3,... can be wrtten as p k (t 1,..., t k )dt 1... dt k = P{dN(t 1 ) = dn(t 2 ) = = dn(t k ) = 1}. We consder two dstnct pulses at tme t and t + u (u > 0) whch may come from the same cell or dfferent cells generated by the process M(t). For pulses that come from the same cell, the contrbuton to the product densty p 2 (t, t + u) s gven by p 2 (t, t + u) = E(N(t))ξe ηu = (mξ 2 /η)e ηu. For two pulses at tme t and t + u that come from dfferent cells, wth ther cell orgns at t v and t + u w respectvely, the contrbuton to the product densty p 2 (t, t + u) of the pulse process N(t) s gven by u p 2 (t, t + u) = ξ 2 e ηv v=0 w=0 e ηw p M 2 (u + v w)dwdv. (2) From equaton (1), the product densty of the cell arrval process M(t) becomes p M 2 (u + v w) = c M (u + v w) + m 2 = Ae (λ+µ)(u+v w) + m 2. (3) Substtutng (3) n (2) and completng the ntegral shows that the contrbuton to the product densty by two pulses that come from dfferent cells s p 2 (t, t + u) = ( ) 2 mξ e (1 e ηu ) + ξ A (λ+µ)u 2 e ηu. (4) η η 2 (λ + µ) 2 The covarance densty (Cox and Isham, 1980) of the pulse process N(t) for u 0 can be wrtten n terms of ts product densty as c(u) = Cov{N(t), N(t + u)} = ( ) mξ δ(u) + p 2 (t, t + u) η ( ) 2 mξ, η where δ( ) s the Drac delta functon. Substtuton from (4) and rearrangng the terms n the above expresson gves, after some algebra, the covarance densty of ths DSP process N(t) for u 0 as ( ) mξ c(u) = δ(u) + A 1 e (λ+µ)u + B 2 B 1 e ηu, (5) η where A 1 = ξ 2 A/(η 2 (λ + µ) 2 ), B 1 = (ξm/η) 2 + ξ 2 A/ (η 2 (λ + µ) 2 ) and B 2 = (ξ 2 m/η). In the above expresson, A 1 and B 1 correspond to the contrbuton from pulses generated by dfferent cells whereas B 2 corresponds to the contrbuton from dfferent pulses wthn the same cell, where the depths of these pulses may be dependent. 6

8 2.4 Moment propertes of the aggregated ranfall In most applcatons, the ranfall data are usually avalable n aggregated form n equally spaced tme ntervals of fxed length. The DSP process we have developed, however, evolves n contnuous tme. We now, therefore, derve mathematcal expressons for the moment propertes of the aggregated ranfall arsng from the DSP process. These expressons are useful to descrbe the propertes of the accumulated ranfall and can be used for model fttng and assessment. Let Y (h) be the total amount of ranfall n dsjont tme ntervals of fxed length h, for = 1, 2,.... We can express ths as Y (h) = h ( 1)h X(t)dN(t), where X(t) s the depth of a pulse at tme t. Wthout assumng any dstrbuton for the pulse depth, let EX(t) = µ X be the mean depth of the pulses. Then the mean of the aggregated ranfall n the ntervals can be wrtten as E Y (h) = h ( 1)h E(X(t))dN(t) = ( ) mξ µ X h. (6) η Usng the well known Campbell s theorem from Daley and Vere-Jones (2007), and utlzng the covarance densty of the pulse arrval process gven n (5), we can now work out the varance and autocovarance functon of the aggregated ranfall process. Followng ths, we have Var Y (h) = = h h 0 h 0 0 Cov X(s)dN(s), X(t)dN(t) E(X 2 )E dn(t) + 2 h h 0 s E X(s)X(t) Cov dn(s), dn(t), where the double ntegral s separated nto two parts to account for the cases s = t and s t. In addton, we need to dstngush whether the pulses at tmes t and s belong to the same cell or come from dfferent cells. For contrbutons to Cov dn(s), dn(t), when pulses come from dfferent cells, the multpler E X(s)X(t) n the above expresson becomes µ 2 X. We shall wrte ths multpler as E X j X k when pulses come from the same cell. Ths wll allow us to accommodate some wthn-cell depth dependence. However, t s assumed that any two pulses wthn a cell, regardless of ther locaton wthn the cell, have the same expected product of depths. Under ths settng, the varance functon becomes Var Y (h) = E(X 2 ) ( mξ η ) h + 2 µ 2 XA 1 ψ 1 (λ + µ) + 2 E X j X k B 2 B 1 µ 2 X ψ1 (η), (7) where ψ 1 (λ + µ) = (λ + µ)h 1 + e (λ+µ)h /(λ + µ) 2 and ψ 1 (η) = ηh 1 + e ηh /η 2. Followng a smlar approach we can derve the autocovarance functon for the aggregated ranfall n two dstnct ntervals. Agan by dstngushng the contrbutons from pulses wthn the same cell, we derved an expresson gven as, for k 1, Cov Y (h), Y (h) +k = = (k+1)h h kh 0 (k+1)h h kh 0 Cov X(s)dN(s), X(t)dN(t) E X(s)X(t) Cov dn(s), dn(t) = µ 2 XA 1 ψ 2 (λ + µ) + E X j X k B 2 B 1 µ 2 X ψ2 (η), (8) 7

9 where ψ 2 (λ + µ) = e (λ+µ)(k 1)h 1 e (λ+µ)h 2 /(λ + µ) 2 and ψ 2 (η) = e η(k 1)h 1 e ηh 2 /η 2. When consderng the specal case where all pulse depths are ndependent E (X j X k ) can be replaced by µ 2 X n equatons (7) and (8). Although the second-order propertes capture the characterstcs of the process well n most pont process applcatons, hgher moments may provde mproved results n terms of reproducton of the propertes of nterest. To ths end, followng Cowpertwat et al. (2007), we shall ncorporate the thrd-order moment n our analyss. The dervaton of the thrd moment (Y (h) E ) 3 follows a smlar approach to that of the second moment, but becomes more complex algebracally. Therefore only an outlne of the dervaton s gven along wth the fnal expresson n Appendx A. The thrd moment about the mean µ (h) ( 3 = E Y (h) Y (h) µ (h) = E E 3 Y (h) 3E ) 3 Y (h) 3 s Var From ths the coeffcent of skewness of the aggregated ranfall process s shown to be ( ) 3 E Y (h) E Y (h) κ (h) µ (h) 3 = ( ) 2 3/2 = 3/2. (9) E Y (h) E Y (h) Var Y (h) Y (h) E Y (h) 3. 3 Model fttng and assessment We shall explore the applcaton of the proposed DSP model n the analyss of fne-scale ranfall data and assess how well t reproduces the propertes of the ranfall over a range of sub-hourly resolutons. We am to do ths usng 69 years ( ) of 5-mnute ranfall data from Bochum n Germany. In ths analyss, we consder the specal case where the pulse depths X j are ndependent random varables that follow an exponental dstrbuton wth parameter θ. Therefore, we have µ X = E X(t) = 1/θ. The data, recorded at the 5-mnute aggregaton level, allow us to ft the model over a range of sub-hourly accumulatons. We shall make use of the mathematcal expressons for the moment propertes of the accumulated ranfall n our model fttng. There are 7 parameters n our model and we estmate 6 of them by the method of moments approach usng the observed and theoretcal values of these propertes. The parameter µ X = 1/θ s estmated separately for each month from the sample mean of ranfall depth by usng the equaton, whch follows from equaton (6), ( ) η µ X = x, (10) mξ where x s the estmated average of hourly ranfall for each month. Although we employed the method of moments to estmate the other parameters, whch essentally equates the sample moments to theoretcal moments from the model, the estmaton can be done n dfferent ways. In ths applcaton, we constructed an objectve functon as the sum of squares of dfferences between the sample and theoretcal values of the moment propertes 8

10 at dfferent aggregaton levels and then mnmzed t usng standard optmsaton routnes. Ths s carred out separately for each month by consderng the data for a month as realsatons of a statonary pont process. Essentally the process of model fttng nvolves calculatng the emprcal mean, varance, correlaton, coeffcent of varaton and skewness from the observed data at each aggregaton level and matchng these wth the correspondng theoretcal values, calculated usng equatons (6) to (9), for a gven set of parameter values. The role of the objectve functon and the optmser employed was to fnd the best possble match usng a mnmum error crteron. We used the statstcal software envronment R for the optmsaton and for the smulaton of the process (R Development Core Team, 2011). A number of optons were avalable for the objectve functon dependng on the applcaton, but we used a form of weghted sum of squares. We employed the routnes optm and nlmnb n R for parameter estmaton n our analyss. The followng subsectons descrbe two dfferent methods used to estmate the frst 6 parameters of the model and dscuss the results produced. Once these parameter estmates were determned, equaton (10) was used to estmate the fnal parameter µ X. 3.1 Estmaton usng second-order moments The frst 6 parameters of the model λ, µ, φ 1, φ 2, η and ξ were estmated usng the followng dmensonless functons; the coeffcent of varaton ν(h) and the autocorrelaton ρ(h) at lag 1 of the aggregated ranfall process. Explctly, these are ν(h) = ( E Y (h) E Y (h) E(Y (h) ) ) 2 1/2, ρ(h) = Corr Y (h), Y (h) +1. (11) We need at least 6 sample propertes of the aggregated process to ft the model and we employed 8 propertes n our estmaton. They are ν(h) and ρ(h) at h=5, 20, 30 and 60 mnutes aggregaton levels. The estmates of the functons from the emprcal data, denoted by ˆν(h) and ˆρ(h) (for h= 1/12, 1/3, 1/2 and 1 hours), were calculated for each month usng 69 years of 5-mnute ranfall seres accumulated at approprate tme scales. Parameter estmates can be obtaned by usng an objectve functon constructed from the sum of squares of dfferences between the sample values and ther correspondng theoretcal values of the proposed model. The estmated values of parameters {ˆλ, ˆµ, ˆφ1, ˆφ2, ˆη and ˆξ} for each month were obtaned by mnmsng the weghted sum of squares of dmensonless functons, as gven below n equaton (12), usng standard optmsaton routnes: h= 1 12, 1 3, 1 2,1 1 Var(ν(h)) ˆ (ˆν(h) 1 ν(h))2 + (ˆρ(h) ρ(h))2. (12) Var(ρ(h)) ˆ In the above expresson, ν(h) and ρ(h) are theoretcal values as gven by equatons (6)-(11) whereas ˆν(h) and ˆρ(h) are calculated for each month from the 69 years of data. In addton, followng Jesus and Chandler (2011), the weghts n the objectve functon were taken as the recprocal of the varance of the emprcal values of the functons calculated separately for each of the 69 years. We also expermented wth other objectve functons but ths was found to gve better results. The above functon was mnmsed separately for each month to obtan estmates of the model parameters. We used the smplex algorthm of Nelder and Mead (1965) for the optmsaton, snce t does not requre the calculaton of dervatves. 9

11 The estmated parameters were then used to calculate the ftted values of the varous theoretcal propertes. The observed and ftted values of the mean, standard devaton, coeffcent of varaton and lag 1 autocorrelaton of the aggregated ranfall are dsplayed n Fgures 2-5. In almost all cases near perfect fts, and n some cases exact, were obtaned for all propertes wth the excepton of the lag 1 autocorrelaton. The mean gave an excellent ft at all tme scales. The standard devaton and coeffcent of varaton showed near perfect fts for most months, at all tme scales, wth small devatons from the perfect ft durng the summer months. The autocorrelaton was reproduced well at smaller aggregaton levels, however there appeared to be a slght underestmaton at larger aggregatons for all months. One pont to note here s that h=10 mnutes aggregaton was not used n the fttng but the model has certanly reproduced all the propertes well for ths tme-scale. Ths reveals that the model s capable of producng estmates of quanttes not used n the fttng whch adds strength to ths DSP modellng framework. Mean plot for 60 mnute levels of aggregaton Mean plot for 30 mnute levels of aggregaton Mean plot for 20 mnute levels of aggregaton Mean plot for 10 mnute levels of aggregaton Mean plot for 5 mnute levels of aggregaton FITTED OBSERVED Fgure 2: Observed and ftted values of the mean of the aggregated ranfall for the DSP model at h=5, 10, 20, 30, 60 mnutes aggregatons, usng second-order moments n estmaton. 10

12 SD plot for 60 mnute levels of aggregaton SD plot for 30 mnute levels of aggregaton SD plot for 20 mnute levels of aggregaton SD plot for 10 mnute levels of aggregaton SD plot for 5 mnute levels of aggregaton FITTED OBSERVED Fgure 3: Observed and ftted values of the standard devaton of the aggregated ranfall for the DSP model at h=5, 10, 20, 30, 60 mnutes aggregatons, usng second-order moments n estmaton. 11

13 CV plot for 60 mnute levels of aggregaton cv FITTED OBSERVED CV plot for 30 mnute levels of aggregaton cv CV plot for 20 mnute levels of aggregaton cv CV plot for 10 mnute levels of aggregaton cv CV plot for 5 mnute levels of aggregaton cv Fgure 4: Observed and ftted values of the coeffcent of varaton of the aggregated ranfall for the DSP model at h=5, 10, 20, 30, 60 mnutes aggregatons, usng second-order moments n estmaton. AC1 plot for 30 mnute levels of aggregaton correlaton AC1 plot for 20 mnute levels of aggregaton correlaton AC1 plot for 10 mnute levels of aggregaton correlaton AC1 plot for 5 mnute levels of aggregaton correlaton FITTED OBSERVED Fgure 5: Observed and ftted values of the autocorrelaton (lag one) of the aggregated ranfall for the DSP model at h=5, 10, 20, 30 mnutes aggregatons, usng second-order moments n estmaton. 12

14 3.2 Estmaton ncorporatng thrd-order moments We now analyse results whch ncorporated thrd-order moments n the fttng, snce ths has been found useful n modellng ranfall (Cowpertwat et al., 2007). Wth the coeffcent of skewness κ (h) gven n equaton (9), we now have three model functons to utlse n estmaton. Under the model framework, we have sx parameters to estmate usng the moment method and we consder nne propertes to nclude the coeffcent of skewness. These are the coeffcent of varaton ν(h), the autocorrelaton ρ(h) at lag 1 and the coeffcent of skewness κ(h), all at three dfferent aggregaton levels of h = 5, 10 and 20 mnutes. The followng objectve functon, whch ncorporates thrd-order moments n parameter estmaton, was used n our analyss: 1 Var(ν(h)) ˆ (ˆν(h) 1 ν(h))2 + Var(ρ(h)) ˆ (ˆρ(h) 1 ρ(h))2 + (ˆκ(h) κ(h))2. Var(κ(h)) ˆ h= 1 12, 1 6, 1 3 Ths objectve functon was mnmsed, separately for each month, to obtan estmates of the model parameters. These are gven n Table 1. Estmated parameters showed some varaton across months. Values of ˆµ were larger for summer months showng smaller mean sojourn tmes (1/µ) n the hgher ranfall ntensty state. The estmates ˆφ 2 were also hgher, n general, for the summer months and showed that the cell arrval rates n state 2 vary from about 2.4 to 7.8 cells per hour. The pulse arrval rate ˆξ also showed varaton across months from about 223 to 290 pulses per hour throughout the year. The mean duraton of cell lfetme (1/η) fell between 8 to 24 mnutes. It was notceable that the cell duratons were shorter n summer months (8-10 mnutes) when compared wth other months. Ths s consstent wth the nature of the summer ranfall, as they are mostly generated from thunder storms of hgh ntensty and shorter duraton. The estmates also showed that the mean depth of the pulses (µ x ) tend to be larger n summer months wth the hghest value n July. Table 1: Parameter estmates for the DSP model ncorporatng thrd-order moments n estmaton. Month ˆλ ˆµ ˆφ1 ˆφ2 ˆη ˆξ ˆµx JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Fgures 6 to 9 show the correspondng results when the thrd-order moments are ncorporated nto the parameter estmaton process. These clearly show an mprovement over the earler results of the method that used the moments up to the second-order. Although the same model s used here, the parameter estmates are obvously dfferent from the earler values due 13

15 to the fact that the coeffcent of skewness was used n estmaton for ths second method. In most cases a near perfect ft was obtaned and n some cases an exact ft was obtaned when the thrd-order moments were ncorporated. The mean, standard devaton and coeffcent of varaton have all been reproduced remarkably well at all aggregaton levels, ncludng those that were not used n fttng (h=30, h=60 mnutes). Comparng these wth earler results reveals that ncorporatng thrd-order moments n estmaton has certanly produced much better results for most of the propertes. The mean appears to be estmated equally well by both methods. The standard devaton and coeffcent of varaton were clearly n better agreement wth ther emprcal counterparts at all aggregatons when thrd-order moments are used. Ths s vsble n the plots, partcularly for the summer months where the frst method consstently showed a slght underestmaton. The autocorrelaton at lag one, however, was an excepton. Although t was estmated well at small aggregaton levels, the ft got worse at hgher levels of aggregaton, especally at those that were not used n fttng, showng consstent underestmaton. Nevertheless, the observed and ftted values of the coeffcent of skewness were n very good agreement n Fgure 10. Here agan the results showed a near perfect ft at all aggregaton levels. In addton to ths, as we wll see n Secton 3.3, ncorporaton of thrd-order moments n estmaton reproduced the extremes better than the earler method. Mean plot for 60 mnute levels of aggregaton Mean plot for 30 mnute levels of aggregaton Mean plot for 20 mnute levels of aggregaton Mean plot for 10 mnute levels of aggregaton Mean plot for 5 mnute levels of aggregaton FITTED OBSERVED Fgure 6: Observed and ftted values of the mean of the aggregated ranfall for the DSP model at h = 5, 10, 20, 30, 60 mnutes aggregatons, usng thrd-order moments n estmaton. 14

16 SD plot for 60 mnute levels of aggregaton SD plot for 30 mnute levels of aggregaton SD plot for 20 mnute levels of aggregaton SD plot for 10 mnute levels of aggregaton SD plot for 5 mnute levels of aggregaton FITTED OBSERVED Fgure 7: Observed and ftted values of the standard devaton of the aggregated ranfall for the DSP model at h = 5, 10, 20, 30, 60 mnutes aggregatons, usng thrd-order moments n estmaton. 15

17 CV plot for 60 mnute levels of aggregaton cv CV plot for 30 mnute levels of aggregaton cv CV plot for 20 mnute levels of aggregaton cv CV plot for 10 mnute levels of aggregaton cv 4 8 CV plot for 5 mnute levels of aggregaton cv FITTED OBSERVED Fgure 8: Observed and ftted values of the coeffcent of varaton of the aggregated ranfall for the DSP model at h = 5, 10, 20, 30, 60 mnutes aggregatons, usng thrd-order moments n estmaton. 16

18 AC1 plot for 30 mnute levels of aggregaton correlaton AC1 plot for 20 mnute levels of aggregaton correlaton AC1 plot for 10 mnute levels of aggregaton correlaton AC1 plot for 5 mnute levels of aggregaton correlaton FITTED OBSERVED Fgure 9: Observed and ftted values of the autocorrelaton (lag one) of the aggregated ranfall for the DSP model at h = 5, 10, 20, 30 mnutes aggregatons, usng thrd-order moments n estmaton. Skewness plot for 60 mnute levels of aggregaton skew FITTED OBSERVED Skewness plot for 30 mnute levels of aggregaton skew Skewness plot for 20 mnute levels of aggregaton skew Skewness plot for 10 mnute levels of aggregaton skew Skewness plot for 5 mnute levels of aggregaton skew Fgure 10: Observed and ftted values of the coeffcent of skewness of the aggregated ranfall for the DSP model at h = 5, 10, 20, 30, 60 mnutes aggregatons, usng thrd-order moments n estmaton. 17

19 3.3 Extremes and proporton of dry perods In many hydrologcal applcatons, more emphass s placed upon a stochastc model s ablty to reproduce the propertes of extreme ranfall rather than the usual moment propertes. One good example arses n urban dranage modellng wthn the context of flood estmaton. One can fnd many other examples of ths n analyses nvolvng envronmental data, see for example Ramesh and Davson (2002), Davson and Ramesh (2000) and Leva et al. (2016). In addton, knowledge of the extremes enables us to assess the rsk assocated wth hydrologcal systems. In vew of ths, we shall evaluate the performance of our proposed model n capturng the propertes of extreme ranfall. In ths regard, we compare the extreme values of the 69 years of observed ranfall data wth those generated by the proposed DSP model. The annual maxma of the emprcal data from the observed 69 year long hstorcal record were extracted, ordered and plotted aganst the correspondng Gumbel reduced varates at each aggregaton level. Ffty copes of the 5-mnute ranfall seres, each 69 years long, were then smulated from the ftted model. Each copy of the smulated data was subsequently aggregated to generate 1, 12 and 24 hour ranfall seres. The annual maxma of each of the 50 smulated seres, at each aggregaton level, were extracted and ordered to make up the nterval plots aganst the correspondng Gumbel reduced varates. These were supermposed on the correspondng Gumbel reduced varate plots of the emprcal data for comparson. Fgure 11 shows the results for h=5 mnutes (top panel) and h=60 mnutes (bottom panel) aggregatons whereas the results of h=12, 24 hours are dsplayed n Fgure 12. The red sold crcles show the mean of the maxma from the 50 smulatons and the blue squares connected by the sold lne show the emprcal annual maxmum values. At h=5 mnutes aggregaton level the model vastly underestmates the extremes. When h=60 mnutes there was evdence of over-estmaton at the lower end and underestmaton at the upper end of the reduced varates. Nevertheless, there was evdence of substantal mprovement at larger values of h. Clearly there was very good agreement at h=12, 24 hour aggregatons, as all of the emprcal values fell wthn the range of the smulated values. Furthermore, the mean of the smulated annual maxma fell reasonably close to the emprcal annual maxma, at h=12 and 24, for much of the perod. It s encouragng to note that ths was the case for the values correspondng to the return perods from about 5 years to 100 years. Hence, the proposed model, although underestmatng the extremes at sub-hourly aggregaton, appears to capture the extremes well at larger aggregatons. The estmaton of extreme values at smaller tme-scales s a common problem for most stochastc models for ranfall, and our results concur wth the fndngs of prevous publshed studes, see for example Cowpertwat et al. (2007) or Verhoest et al. (1997). In order to asses the performance of the two estmaton methods n reproducng extremes, we compared the ordered emprcal annual maxma wth those generated by the parameter estmates of the two methods. Table 2 shows the means of the ordered annual maxma from 50 smulatons for the two methods, together wth the ordered emprcal annual maxma, at 12-hour and 24-hour aggregaton levels. These are compared for a range of reduced Gumbel varates coverng the values that correspond to the return perod from 5 years to 100 years. Note that the nterval plots based on the smulatons, usng the method whch ncorporates thrdorder moments n estmaton (Method 2), are shown n Fgure 12 for the whole perod. Results presented n Table 2 show that Method 2 performed much better than the method that used second-order moments n estmaton (Method 1), at both aggregaton levels. The above result and Fgure 12 clearly show that Method 2 outperformed Method 1 and reproduced extremes well at hgher aggregatons. 18

20 Extreme value plot 5 mnute 5m annual maxma (mm) mean emprcal T Extreme value plot 60 mnute 60m annual maxma (mm) mean emprcal T Fgure 11: Ordered annual maxma of the 5-mnute (top panel) and one-hour (bottom panel) aggregated ranfall plotted aganst the reduced Gumbel varate. Emprcal annual maxmum values are shown as blue squares connected by a sold lne. Interval plots based on annual maxma of 50 smulatons, each 69 years long, are also shown. The red sold crcles are the mean of the 50 smulated maxma. Another property of nterest to hydrologsts s the proporton of ntervals wth lttle or no ran. Ths wll help to quantfy the proporton of dry perods. Very often the gauge ranfall data are recorded n a rounded form (to the nearest 0.1 mm) and hence, followng Cowpertwat et al. (2007), we calculate the proporton of ranfall below a small threshold (ˆp{Y h < δ} for δ > 0 ) nstead of the actual proporton of dry ntervals (δ = 0). Therefore by choosng smaller values of δ we can provde approxmate estmates of the proporton of dry ntervals. The proporton of ntervals below a gven threshold were calculated for each month, usng 50 samples of 69 years of smulated 5-mnute data, for each of the 5-mnute, 1 hour and 24 hour aggregatons (h = 1/12, 1, 24). The mean of the 50 values was calculated, at each aggregaton, for each calendar month. The observed proportons for the hstorcal data and the average of the smulated values, from the ftted model, for dfferent thresholds are gven n Table 3. In order to fnd a good estmate of the proporton of dry perods at h = 1/12 aggregaton level, a small threshold of δ = 0.05 mm was used. For the hourly ranfall, two threshold values of δ = 0.05, 0.1 mm were used. Hgher threshold values (δ = 0.5, 2 mm) were used for the daly ranfall, as occasonal lght ran durng the day may cause dscrepances between the observed and smulated proportons. The numercal results presented n Table 3 show that, n general, the model reproduces the observed proportons well at 5-mnute aggregaton, although t tends to over-estmate slghtly. However, the dfferences we observed are between 0.01 for February and for June. At the hourly aggregaton level, the model tended to consstently underestmate the proportons each month when δ = 0.05 mm. However, when the threshold was ncreased to 0.1 mm the observed 19

21 Extreme value plot 12 hour 12h annual maxma (mm) mean emprcal T Extreme value plot 24 hour 24h annual maxma (mm) mean emprcal T Fgure 12: Ordered annual maxma of the 12-hour (top panel) and 24-hour (bottom panel) aggregated ranfall plotted aganst the reduced Gumbel varate. Emprcal annual maxmum values are shown as blue squares connected by a sold lne. Interval plots based on annual maxma of 50 smulatons, each 69 years long, are also shown. The red sold crcles are the mean of the 50 smulated maxma. and smulated proportons were n good agreement, except for October, wth the summer month June agan showng the largest dfference of Fnally, at a daly level, there was consstent over-estmaton when the threshold was set at 0.5 mm. When the threshold ncreased to 2 mm the observed and smulated proportons became very close. Ths ndcates that the frequency of very lght ran events, over a perod of a day, was greater n the observed data. The dfferences tended to be larger n the summer months and smaller n the wnter months, wth June once agan showng the hghest dfference of and February showng the smallest dfference of

22 Table 2: Ordered emprcal annual maxma for the 12-hour and 24-hour aggregated ranfall and the correspondng ordered reduced Gumbel varates. The average annual maxma, based on 50 smulatons of 69-year long seres, for the two estmaton methods are also shown. 12-hour aggregaton 24-hour aggregaton Gumbel Emprcal Smulated maxma Emprcal Smulated maxma Reduced varate maxma Method 1 Method 2 maxma Method 1 Method Table 3: Proporton of ranfall below defned thresholds at dfferent tme scales. Here ˆp O and ˆp S represent the estmates of the proportons for the observed and smulated seres. Month p o p s p o p s p o p s p o p s p o p s JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC threshold δ(mm) scale(h) 1/12 1/

23 3.4 Smulaton study Snce the lkelhood functon of the model we proposed for the accumulated ranfall data was not avalable, we employed the moment method to estmate the parameters n our analyss. The method of moments smply equates sample moments from the observed data to the theoretcal moments of the model beng ftted to obtan estmates of the parameters. It s mportant to note that moment estmators, unlke maxmum lkelhood estmators, do not necessarly have the usual large sample propertes leadng to asymptotc results. Nevertheless, we carred out a smulaton study to evaluate the statstcal performance of the estmaton method we employed n our analyss. The observed data was a 69-year long accumulated ranfall seres n 5-mnute ntervals and our stochastc pont process model was ftted separately for each month. A month wth 31 days had observatons, although most of them were zero, especally for a summer month. The smulaton study was carred out for a typcal summer month, August, wth smulated ranfall data over a perod of length n l = 20, 40, 60, 69 years, as descrbed n the algorthm gven below. Smulate one hundred (n = 100) sample seres from the ftted values of the parameters (θ) for the month. Aggregate the smulated data n h = 5, 10, 20 mnutes ntervals. Calculate ther sample statstcs: coeffcent of varaton ν(h), lag 1 autocorrelaton ρ(h) and coeffcent of skewness κ(h) for each value of h. Compute the moment estmates (ˆθ) for the smulated samples, usng the objectve functon used n Secton 3.2. Compute the mean, bas and mean squared error of the moment estmates, separately for each of the parameters, usng the expressons Mean = 1 n n ˆθ, =1 Bas = 1 n n (ˆθ θ), =1 MSE = 1 n n (ˆθ θ) 2, =1 where n=100 and θ = λ, µ, φ 1, φ 2, η and ξ are the ftted parameter values of the emprcal ranfall data for the month under study and ˆθ s the correspondng estmate for the smulated data. The above steps were repeated wth n l = 20, 40, 60, 69 years of data to produce a table of average, bas, root mean squared error ( MSE) for each of the 6 parameters. The results are presented n Table 4 where the ftted parameter values for August are noted as the true values used n the smulaton. The means of the smulated estmates showed that these are generally close to the correspondng true values. The means of ˆµ and ˆξ were convergng towards ther true values as n l became large, whereas the means of ˆφ 2 and ˆη dd not seem to exhbt ths property. It s worth pontng out that the parameters φ 2 and η are postvely correlated and when φ 2 ncreased η also ncreased. The reason for ths s that much of the ran cells come from the hgh ntensty state 2 and, as we are equatng the ranfall moments n our estmaton, ncreased arrval rate φ 2 of ran cells results n smaller cell lfetmes of 1/η, and vce-versa. Means of ˆλ and ˆφ 1 do not seem to show much varaton. 22

24 Table 4: Smulaton study results: values of the mean, bas and root mean squared error ( MSE) for the moment estmates of the parameters based on 100 smulated seres of length n l years for the month of August. True values are the ftted parameter values for the observed ranfall data for August. Mean Bas n l ˆλ ˆµ ˆφ1 ˆφ2 ˆη ˆξ True values n l ˆλ ˆµ ˆφ1 ˆφ2 ˆη ˆξ E E E E MSE n l ˆλ ˆµ ˆφ1 ˆφ2 ˆη ˆξ E E E E The bas of ˆµ and ˆξ became smaller when n l became larger but both ˆφ 2 and ˆη seemed to show a slght negatve bas. As we can see from the values, ˆλ and ˆφ 1 both show very small negatve bas and agan do not show much varaton wth n l. The root mean squared errors seemed to stay more or less at the same level for most of the parameters, as n l ncreased, except for ˆφ 2 and ˆη whch showed a slght ncrease. One possble reason for that mght be that the moment estmates may be slghtly based because of the seral correlaton n the data, as noted by Cowpertwat et al. (2007), especally at sub-hourly aggregaton levels whch show hgh autocorrelaton. Ideally we would have lked the bas and MSE of the estmators to converge to zero asymptotcally for all parameters, as would those of the maxmum lkelhood estmators whch have good large sample statstcal propertes. However, the means of the smulated estmates were closer to ther true values and the relatve szes of the bas and root mean squared errors seemed to be reasonably small for the applcaton. 4 Conclusons and future work We have developed a doubly stochastc pont process model, usng nstantaneous pulses, to study the fne-scale structure of sub-hourly ranfall tme seres. Second and thrd-order moment propertes of the aggregated ranfall for the proposed DSP model were derved. The model was used to analyse 69 years of 5-mnute ranfall data. The emprcal propertes of the ranfall 23

25 accumulatons were shown to be n very good agreement wth the ftted theoretcal values over a range of sub-hourly tme scales, ncludng those that were not used n fttng. Although the use of second-order moments n estmaton produced very good results, ncorporaton of thrd-order moments showed a clear mprovement n fttng. Overall, the results of our analyss suggest that the proposed stochastc model s capable of reproducng the fne-scale structure of the ranfall process, and hence could be a useful tool n envronmental or ecologcal mpact studes. The smulated extreme values at daly and 12-hourly aggregatons are n very good agreement wth ther emprcal counterparts. However, although the model reproduces the moment propertes well, t underestmates the extremes at fne tme-scale. The results from the analyss of the proporton of dry perods, usng ntervals wth ranfall depths below approprate threshold levels, show that the model generally reproduces the observed proportons well. The smulaton study shows that the estmaton method used s capable of reproducng the estmates closer to the true values, although t may not have the desred large sample propertes whch maxmum lkelhood estmators exhbt. Overall, the above analyses ndcate that the proposed modellng approach s able to ft data over a range of sub-hourly tme scales and reproduce most of the propertes well. It has potental applcaton n many areas, as t provdes a fast and effcent way of generatng synthetc fne-scale ranfall nput to hydrologcal models drectly from one stochastc model. Despte ths, there s potental to develop the model further to employ a 3-state doubly stochastc model for cell arrvals and also to explore ts capablty to handle aggregatons at hgher levels. The 3-state model, along wth further developments to study other hydrologcal propertes of nterest, may be a frutful area for future work on. Although our model based on one doubly stochastc process has performed well, another possblty s to consder superposng two doubly stochastc pulse processes, as n Cowpertwat et al. (2007), to better account for dfferent types of precptaton, such as convectve and stratform. Acknowledgments The authors would lke to thank the Edtor, Assocate Edtors and two anonymous referees for ther constructve comments and suggestons on earler versons of the manuscrpt. Ther comments have greatly mproved the qualty of the manuscrpt. References Bardossy, A. and Plate, E. J. (1991). Modelng daly ranfall usng a sem-markov representaton of crculaton pattern occurrence. Journal of Hydrology, 122(1): Cowpertwat, P., Isham, V., and Onof, C. (2007). Pont process models of ranfall: developments for fne-scale structure. Proceedngs of the Royal Socety A: Mathematcal, Physcal and Engneerng Scence, 463(2086): Cowpertwat, P., Xe, G., Isham, V., Onof, C., and Walsh, D. (2011). A fne-scale pont process model of ranfall wth dependent pulse depths wthn cells. Hydrologcal Scences Journal, 56(7): Cowpertwat, P. S. (1994). A generalzed pont process model for ranfall. Proceedngs of the Royal Socety of London. Seres A: Mathematcal and Physcal Scences, 447(1929): Cox, D. R. and Isham, V. (1980). Pont processes, volume 12. CRC Press. 24