A comparison of alternatives for daily to sub-daily rainfall disaggregation

Journal of Hydrology 470–471 (2012) 138–157

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Journal of Hydrology

journal homepage: www.elsevier .com/locate / jhydrol

A comparison of alternatives for daily to sub-daily rainfall disaggregation

Alexander Pui a, Ashish Sharma a,⇑, Rajeshwar Mehrotra a, Bellie Sivakumar a,b, Erwin Jeremiah a

a School of Civil and Environmental Engineering, The University of New South Wales, Sydney, NSW 2052, Australiab Department of Land, Air and Water Resources, University of California, Davis, CA 95616, USA

a r t i c l e i n f o

Article history:Received 7 November 2011Received in revised form 31 July 2012Accepted 18 August 2012Available online 29 August 2012This manuscript was handled byKonstantine P. Georgakakos, Editor-in-Chief,with the assistance of Ana P. Barros,Associate Editor

Keywords:Rainfall disaggregationRandom multiplicative cascadesRandomized Bartlett–Lewis modelMethod of fragments resamplingAustralia

0022-1694/$ - see front matter � 2012 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.jhydrol.2012.08.041

⇑ Corresponding author.E-mail addresses: [email protected] (A. Pui),

Sharma), [email protected] (R. Mehrotra), s.bemar), [email protected] (E. Jeremiah).

s u m m a r y

This paper evaluates three distinct approaches for disaggregating daily rainfall to sub-daily sequences:(1) random multiplicative cascades (microcanonical and canonical versions), (2) point process (random-ized Bartlett–Lewis model – RBLM), and (3) resampling (method of fragments). These methods are usedto perform disaggregation of daily rainfall to hourly rainfall at four point locations across Australia (Syd-ney, Perth, Cairns, and Hobart), which are associated with different climatic regimes. The methods areevaluated based on parameter estimation procedures applied (including introduction of the sequentialMonte Carlo sampler in RBLM), the capability of the resulting sequences to reproduce standard validationstatistics, and the representation of observed rainfall variability and intermittency, within-day wet spells,and extreme rainfall percentiles. The results generally indicate that the method of fragments outperformsthe other models. While all the models are found to simulate reasonably well the commonly used statis-tical measures (e.g. mean and dry proportions) of rainfall at the hourly timestep, the microcanonicalmodel is found to significantly overestimate the hourly rainfall variance. With respect to extreme valuecharacteristics, the resampling approach is found to match well the observed intensity–frequency rela-tionship at an hourly scale, with the cascade models underestimating (canonical) and overestimating(microcanonical) extreme rainfall. The point process model’s performance is poor in Cairns but reason-ably good at other locations. An analysis of the empirical within-day wet- and dry-spell distributions fur-ther reveals that the cascade-based models are not robust for observed wet and dry spells.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

A wide range of applications involving planning, design, andmanagement of small water resources systems relies on rainfalldata at fine (or high) temporal resolutions, such as hourly. How-ever, fine-resolution rainfall data are generally scarce, especiallyfor rural and remote regions, because their measurements arecostly and time-consuming. One way to overcome this problemis to derive fine-resolution data from the widely available coar-ser-resolution (e.g. daily) data through data transformation proce-dures, generally referred to as rainfall disaggregation. The purposeof such modeling tools is to represent a range of scenarios thatcould possibly occur at fine resolutions, given the complex andscaling nature of the rainfall process and of the governing mecha-nisms. To this end, numerous models have been proposed in theliterature (e.g. Hershenhorn and Woolhiser, 1987; Cowpertwait,1991; Glasbey et al., 1995), which are also based on many differentconcepts. Popular among these include: (1) the Bartlett–Lewis/

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[email protected] ([email protected] (B. Sivaku-

Neyman–Scott rectangular pulse models based on point processtheory (Rodriguez-Iturbe et al., 1987; Onof and Wheater, 1993;Khaliq and Cunnane, 1996); and (2) the random cascade modelsbased on scale-invariance theory (Gupta and Waymire, 1993; Cars-teanu and Foufoula-Georgiou, 1996; Molnar and Burlando, 2005).

Given that there is a host of disaggregation methods with differ-ent theoretical underpinnings, it is highly useful, from a practicalperspective, to apply at least a few of them to assess their perfor-mances, towards their suitability and effectiveness. Several studieshave attempted this for studying temporal and/or spatial rainfall(see a recent review in Sharma and Mehrotra (2010)), and the com-parisons include: stochastic versus deterministic models (e.g. Hin-gray and Ben Haha, 2005), multiplicative cascade versus pulse-typemodels (e.g. Ferraris et al., 2003), and canonical versus microca-nonical cascade models (e.g. Hingray and Ben Haha, 2005; Molnarand Burlando, 2005; Serinaldi, 2010; Licznar et al., 2011). Whilemodel comparisons no doubt offer useful insights, a general con-clusion as to the better/best model is impossible to draw, sincerainfall properties significantly differ for different climatic regionsand generating mechanisms. Consequently, reliable comparisonsbetween/among these studies performed for different regions aregenerally hard to achieve.

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A. Pui et al. / Journal of Hydrology 470–471 (2012) 138–157 139

Considering the influence of geographic and climatic conditionsas well as the rainfall generating mechanisms on rainfall propertiesand its disaggregation, the present study focuses on the evaluationof the utility and suitability of three different types of disaggrega-tion models for Australian conditions: (1) both the canonical (CA-NON) and micro-canonical (MICRO) versions of the discretemultiplicative random cascades (Gupta and Waymire, 1993; Mol-nar and Burlando, 2005); (2) the randomized Bartlett–Lewis model(RBLM) (Koutsoyiannis and Onof, 2001); and (3) the non-paramet-ric resampling approach based on ‘method of fragments’ (MOF)(Sharma and Srikanthan, 2006). Rainfall data from four differentcities in Australia are studied: (1) Sydney (south east Australia;temperate to Mediterranean climate); (2) Perth (Western Austra-lia; arid/semi-arid climate); (3) Cairns (north east Australia; tropi-cal climate); and (4) Hobart (south east Australia, temperateclimate). These models are assessed in the context of their abilityto reproduce a number of important daily to sub-daily rainfall sta-tistics estimated from the observed records.

Our reasons for choosing particular versions of the above mod-els are as follows:

(a) The cascade-based models are chosen in light of the encour-aging results reported by earlier studies (Koutsoyiannis andMamassis, 2001; Molnar and Burlando, 2005) as well as fortheir appeal from a practical viewpoint because they are rel-atively parameter parsimonious (Sivakumar and Sharma,2008).

Table 1Details of selected rainfall stations for this study.

Station number Station name Latitude

009021 Perth Airport 31.93�S031011 Cairns Airport 16.87�S066037 Sydney Airport 33.95�S094008 Hobart Airport 42.83�S

Fig. 1. Locations of study sites across Austr

(b) The RBLM, being a widely-applied stochastic rainfall gener-ator, has also formed the basis for the development of othervariants, such as the Bartlett–Lewis-based hybrid model(Gyasi-Agyei and Willgoose, 1997) as well as power lawtailed–autocorrelation downscaling method applied in con-junction with the RBLM (Marani and Zanetti, 2007). As such,any biases or issues pertaining to its model structure maypermeate through to its more complex variants, and shouldbe duly noted.

(c) The MOF is included as it is relatively new within the pointtemporal rainfall disaggregation literature (with recentextensions of the methodology to ungauged locationsreported in Westra et al. (2012) and Mehrotra et al.(2012)), and we aim to test the performance of a non-para-metric model against its parametric counterparts.

It is helpful to note that while the random cascade models andthe method of fragments models have been used exclusively as‘disaggregators,’ the RBLM was initially formulated strictly as arainfall simulator. The RBLM has since been modified and accordedan appropriate ‘adjusting procedure’ to enable it to be applied as arainfall disaggregator (Koutsoyiannis and Onof, 2001). Since ourprimary objective here is to assess and compare each model’sability to obtain realistic fine-resolution rainfall (especially forthe purpose of design flood estimation), we are particularlyinterested in statistics that are pertinent to this objective:i.e. how realistically the models simulate rainfall variability and

Longitude Start/end year Years used

115.98�E 1964/2005 42145.75�E 1944/2006 63151.17�E 1964/2005 42147.50�E 1964/2005 42

alia: Sydney, Perth, Cairns, and Hobart.

Fig. 2. The RBLM with six parameters. The schematic above gives an indication ofhow they operate within the RBLM generation process. Storm arrivals followPoisson process with mean of 1/k, with generation duration of each stormexponentially distributed with mean 1/c. First cell is fixed to occur at storm originwhile later cells within a storm arrive by a second Poisson process with expectedvalue of b. Duration of each cell is exponentially distributed with mean of 1/g. Eachcell’s intensity is exponentially distributed with a mean of lx.

140 A. Pui et al. / Journal of Hydrology 470–471 (2012) 138–157

intermittency, rainfall extremes, within-day wet spells, alongwith low-order moment and autocorrelation characteristics. Inthis study, we perform disaggregation from the daily timestepdown to the hourly timestep (target timestep); however, we alsofocus on a number of intermediary timesteps (i.e. 6 h and 12 h)to examine re-aggregation statistics. The hourly target timestepis chosen, as it is deemed to be sufficiently fine for design floodestimation purposes in rural catchments; it also enables theassessment of all models from a common platform (since theRBLM disaggregation model applied here does not disaggregatebeyond the hourly timestep). In addition, we also discuss, whereappropriate, the impact of sensitivity of estimated parameters,to assess if future improvements (i.e. potential developments,such as modifications to parameter estimation techniques orregionalization of parameters) to the parametric models ispossible.

The rest of the paper is organized as follows. In Section 2, a briefdescription of the study area is presented, including climate andrainfall data characteristics. The model structure and methodologyof each rainfall disaggregation model is discussed in Section 3. Sec-tion 4 discusses the parameter estimation procedures and perfor-mance of rainfall disaggregation models; in particular, ithighlights the similarities and differences in model results, includ-ing inferences as to how model performance may be impacted byparameter estimation processes adopted and/or intrinsic biasestied to particular model structures. Conclusions are reported inSection 5.

Table 2Parameter values for the CANON and MICRO. Note that while CANON has constant paramlevels 1, 2, 3, 4, and 5 here correspond to timescales of 1440–720 min, 720–360 min, 360

Sydney PerthCANON b = 0.358, r2 = 0.0905 b = 0.349, r2 = 0.02

MICRO Level Dry probability ‘p’

Sydney Perth Cairns H

1 0.669 0.595 0.541 02 0.515 0.478 0.507 03 0.501 0.472 0.488 04 0.387 0.396 0.420 05 0.368 0.403 0.423 0

Fig. 3. Moment scaling relationships (left) and the MKP function (right) for observed dadivergence/convergence of moments with scale.

2. Study area and data

In this study, rainfall data from four different, and representa-tive, cities across Australia are studied: Sydney, Perth, Cairns, andHobart (see Fig. 1). As seen from Fig. 1, these four locations have

eters, the MICRO parameters are timescale dependent. For example, disaggregation–180 min, 180–90 min, and 90–45 min.

Cairns Hobart23 b = 0.364, r2 = 0.0114 b = 0.417, r2 = 0.0469

Variability parameter ‘a’

obart Sydney Perth Cairns Hobart

.670 0.862 0.985 0.875 0.835

.600 0.907 1.071 0.870 0.878

.562 1.072 1.187 0.982 1.093

.459 1.163 1.260 1.040 1.178

.430 1.399 1.316 1.163 1.527

ta at Sydney. Slope of lines on plot of Moment scaling relationships shows rate of


certain similarities among themselves in terms of their proximityto the coast (Sydney, Cairns, and Hobart on the east/southeastcoast; Perth on the southern west coast). At the same time, how-ever, they also have substantial differences in terms of climateand rainfall variability. At a basic level, according to the Köppenclassification scheme (i.e., based on annual average temperatureand rainfall), the four locations are hydroclimatologically diverse.Perth is classified as a dominant winter rainfall region with annualrainfall largely falling during the winter months of May–October,while the dry summer period occurs from November to April.Cairns is in the tropical region and experiences dominant summerrainfall (November–April), with dry periods occurring over thewinter period from May to October. Although Sydney’s climate istemperate to Mediterranean and Hobart lies in the temperate re-gion, both locations exhibit fairly uniform monthly rainfall, withlittle intra-annual variability.

For each of the above four cities, rainfall data are available atmany raingauge stations. However, since long rainfall records aredesirable for a reliable comparison of the disaggregation ap-proaches, only data from the station with the longest record fromeach city is considered in this study. Table 1 presents the locationsof these stations, along with the period and length of records.Although the shortest timestep used to assess model performancein this study is 1 h, we use continuous 6-min pluviograph rainfalldata maintained by the Australian Bureau of Meteorology (BOM)at all locations, with negligible missing data for our prescribedtimespan of each historical record (see Table 1). The availabilityof observed rainfall data at a scale finer than the target disaggrega-

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tion timestep is advantageous, as it allows for better parameteriza-tion of some of the models (i.e. when estimating moments across arange of timesteps as well as examining the probability distribu-tion of the breakdown coefficients for the cascade-based model).

3. Model descriptions

3.1. Random multiplicative cascades (RMC)

Random cascade models originate from turbulence theory (Ya-glom, 1966; Mandelbrot, 1974; Meneveau and Sreenivasan, 1987),and have been widely used across different climatic regions forrainfall modeling purposes (e.g. Gupta and Waymire, 1993; Carste-anu and Foufoula-Georgiou, 1996; Menabde et al., 1997; Molnarand Burlando, 2005). Although these models are purely phenome-nological, as the exact relationship between turbulence and rainfallremains unclear, their ability to reproduce the structures in ob-served rainfall and its statistical properties has justified their con-tinued application (Menabde et al., 1997). In the present study, twoversions of a larger group of models, which operate based on thetheory of scale invariance, are applied: (1) canonical cascade model(hereafter referred as CANON); and (2) micro-canonical cascademodel (hereafter referred as MICRO). These models are chosen inlight of the promising results reported in Molnar and Burlando(2005) when applied to temperate climate of Zurich, Switzerland(Sydney and Hobart have a temperate climate). A brief account ofthese two models is presented next. For extensive details, the

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d seasonal basis at (a) Sydney, (b) Perth, (c) Cairns, and (d) Hobart.

Cascade Weights

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(f) 60 − 30 min

Fig. 5. Probability density plot of breakdown coefficients (which are essentially observed cascade weights (0 < W < 1) from cascade level n to n + 1) (shown in red) andgenerated weights from the Beta distribution (shown in blue) for Sydney. Cascade levels shown are (a) 1920–960 min, (b) 960–480 min, (c) 480–240 min, (d) 240–120 min,(e) 120–60 min, and (f) 60–30 min. Note decreasing variance from higher to lower cascade levels (a)–(f). (For interpretation of the references to colour in this figure legend,the reader is referred to the web version of this article.)

Table 3Upper and lower limits of constraints applied to parameters (daily scale) of the RBLM.

Parameter Lower bound Upper bound

k 0.001 4.320j 0.010 10.000u 0.001 2.000a 1.000 40.000N 0.001 10.000lX(mm/d) 1.000 500.000


reader is directed to, for example, Gupta and Waymire (1993), Overand Gupta (1996), and Molnar and Burlando (2005).

3.1.1. Canonical cascade model (CANON)The CANON model distributes rainfall between successive sub-

divisions with b as the branching number (see Molnar and Burlan-do (2005) for a schematic representation of the model structure).As such, the ith interval after n levels of sub-division is denotedas Di

n. The dimensionless scale is defined as kn = b�n. The distribu-tion of mass then occurs via a multiplicative process through alllevels of the cascade (i.e. 1, . . .,n), such that the mass, ln, in sub-division Di

n is:

lnðDinÞ ¼ r0knP

nj¼1WjðiÞ for i ¼ 1;2; . . . bn ð1Þ

where r0 is the initial rainfall depth (in our case, daily rainfall) atn = 0, and Wj(i) (hereafter denoted as just W for ‘weights’) is a rangeof weights that essentially forms the cascade generator (Molnar andBurlando, 2005). The weight W is treated as an independent andidentically distributed (iid) random variable, subject to the condi-tion E(W) = 1, so that the mass is conserved on average throughall levels of the cascade. The properties of W can be estimated fromthe moment scaling function behavior across all scales of interest. In

particular, we need to obtain the rate of divergence/convergence ofmoments with scale. For a random cascade, the ensemble momentshave been shown to be a log–log linear function of the scale of res-olution. The slope of this scaling relationship is known as the Mah-ane Kahane Pierre (MKP) function (Mandelbrot, 1974; Kahane andPeyriere, 1976). Having identified the moment scaling function, anappropriate probability distribution for the weights can be specified(Over and Gupta, 1996; Molnar and Burlando, 2005). The intermit-tent lognormal �bCANON model weights generator can be expressedas W = BY, where B is a function of bCANON (the parameter whichcontrols intermittency) in rainfall and Y controls rainfall amounts.In order to divide rainfall into rainy and non-rainy periods, theintermittent section of the model is based on the following proba-bilities, derived according to the Bernoulli distribution:

PðB ¼ 0Þ ¼ 1� b�bCANON and PðB ¼ bbCANON Þ ¼ b�bCANON ð2Þ

where bCANON is the intermittency controlling parameter, B is theintermittency factor, and E(B) = 1. Variability in the positive partof the generator (i.e. where B > 0) is derived from the lognormal dis-tribution, and is expressed as:

Y ¼ b�r2 ln b

2 þrX� �

ð3Þ

where X is a normal N(0,1) random variable, r2 is the variance of Y,with E(Y) = 1. Note that a branching number b = 2 is used in thisstudy, leading to a partial simulation of the observed autocorrela-tion structure in the disaggregated rainfall field through the induc-tion of dependence from the coarser level (daily timestep) to thefiner target cascade level (hourly timestep) (Molnar and Burlando,2005).

3.1.2. Microcanonical cascade model (MICRO)The MICRO model, by definition, conserves mass exactly at each

cascade level. As such, it uses a branching order (b) of two by

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Fig. 6. Intermediate distribution densities of lx of Cairns at various stages of the SMC algorithm. The captions show densities at stages 1, 50, 100, 200, 300, and 640 (thetargeted posterior distribution).


default, and the MICRO weights (M) must either equal to a combi-nation of [0,1] when intermittency arises (hereafter ‘intermittentpart’), or [M, 1 �M], where 0 < M < 1 when there is no intermit-tency (hereafter ‘variability part’). The MICRO model also differsfrom its canonical counterpart because its weights (M) are scale-dependent and not independent and identically distributed (iid).Due to this property, it represents a version of bounded randomcascades, where the cascade generator is a function of scale. Theintermittency parameter (p.dry) and the variability parameter (a)of MICRO are estimated from the breakdown coefficients, whichare defined as the ratio of rainfall of a random field averaged overdifferent scales, to account for decrease in variance with decreasein timescales. Specifically, p.dry represents the probability thatone of the intervals in disaggregation is dry for scales between nand n + 1 and, therefore, is simply estimated from historical rainfallat each pre-defined cascade level (Molnar and Burlando, 2005). Toobtain a, non-zero cascade weights M (also referred to as ‘break-down coefficients’) are first calculated from non-overlapping adja-cent pairs of rainfall measurements as:

Mj;s ¼Rj;s

Rj;s þ Rjþ1;sj ¼ 1;3;5; . . . ;Ns � 1 ð4Þ

where Rj,s is the rainfall depth recorded over the time interval oflength s at position j in the time series and Ns is the total numberof records at timescale s. These weights are then modeled by thesingle parameter Beta distribution (Molnar and Burlando, 2005)(see also Section 4 for further details). To operate MICRO, pairs ofnon-zero weights are generated utilizing a unique property of thisBeta distribution. Recall that since b = 2, a pair of weights on bothsides for a given cascade level (Mleft and Mright) can then be simu-lated by drawing two independent realizations, x1 and x2, from agamma distribution with same parameter a, and taking ratios x1/(x1 + x2) and x2/(x1 + x2) which sum to unity, thus preserving rainfallamounts exactly (Koutsoyiannis and Xanthopoulos, 1990).

3.2. Randomized Bartlett–Lewis model (RBLM)

The Bartlett–Lewis model is a cluster-based model, originallydeveloped by Rodriguez-Iturbe et al. (1987), that represents rain-fall events as clusters of rain cells, where each cell is considereda pulse with a random duration and random intensity. The originalBartlett–Lewis process is characterized by five parameters (k, bRBLM,c, g, and lx) with the following description: The first parameter (k)characterizes the expected value of the Poisson-distributed stormgeneration process. Each storm origin will generate a variablenumber of ‘‘storm cells’’ until a certain time from the storm originis exceeded, at which time cell generation ceases (see Fig. 2). Thesecond parameter (bRBLM) represents the reciprocal of the expectedvalue of another Poisson process representing the generation ofadditional cells in the storm (the first cell always coincides withthe start of the storm). The third parameter (c) represents the reci-procal of the expected value of the storm generation durationwhich is characterized by an exponential distribution. In the origi-nal BLM, this time from the storm origin, often called the ‘‘genera-tion duration,’’ is taken to be exponentially distributed with anexpected value of 1/c, where c = u � g, where u is a dimensionlessparameter. The duration of each cell’s contribution is taken to beexponentially distributed with expected value 1/g. Although theg value is a constant in the original BLM, in the randomized version(RBLM), g are defined as independent and random for distinctstorms (hence the name ‘‘randomized’’) following a gamma distri-bution with shape a and scale 1/m (Rodriguez-Iturbe et al., 1988).This modification has been reported to improve the simulation ofdry spell lengths (Rodriguez-Iturbe et al., 1988), and thus is usedin this study. There are indeed more complex versions of theBartlett–Lewis model in the literature, such as the Bartlett–Lewisrectangular pulse gamma model (Onof and Wheater, 1994) andhybrid based models (Gyasi-Agyei and Willgoose, 1997). Suchmodels are not considered in this study, however, since our pur-pose here is to focus on differences in results (where present) that

Table 4Observed and model generated standard statistics for 1-, 6-, 12-, and 24-h timesteps across all study sites. Standard statistics computed from model disaggregated rainfall are computed from average of 10 realizations, with variationsbetween model runs shown in Figs. 7–10. All models are structured to reproduce the means, whereas MICRO is expected to reproduce the observed dry proportions, MOF lag-1, and RBLM variance, lag-11 and dry proportion statistics inthe simulations.

Hourly statistics 6 h statistics 12 h statistics 24 h statistics

OBS CANON MICRO RBLM MOF OBS CANON MICRO RBLM MOF OBS CANON MICRO RBLM MOF OBS CANON MICRO RBLM MOF

MeanSydney 0.113 0.113 0.113 0.114 0.113 0.680 0.677 0.680 0.683 0.680 1.359 1.353 1.359 1.365 1.359 2.719 2.707 2.719 2.731 2.719Perth 0.078 0.078 0.078 0.079 0.078 0.469 0.469 0.469 0.472 0.468 0.938 0.938 0.938 0.944 0.937 1.876 1.876 1.876 1.888 1.874Cairns 0.209 0.210 0.209 0.207 0.208 1.252 1.258 1.252 1.244 1.251 2.503 2.517 2.503 2.488 2.501 5.007 5.033 5.007 4.976 5.003Hobart 0.057 0.056 0.057 0.057 0.057 0.340 0.339 0.340 0.341 0.339 0.679 0.677 0.679 0.683 0.679 1.358 1.355 1.358 1.365 1.358

VarianceSydney 0.756 0.549 1.470 0.599 0.867 12.38 13.488 18.90 12.78 13.18 34.64 42.51 43.84 36.09 36.00 94.81 131.4 94.81 95.20 94.81Perth 0.288 0.180 0.463 0.338 0.318 4.420 4.432 5.932 4.900 4.560 12.17 13.96 14.02 12.67 12.30 31.04 43.39 31.04 31.37 31.01Cairns 2.285 1.767 4.700 1.492 2.439 37.84 43.09 57.35 36.78 39.34 107.6 137.1 132.6 108.6 110.1 301.0 428.3 301.1 300.9 300.9Hobart 0.202 0.161 0.386 0.228 0.204 3.212 3.634 4.460 3.534 3.318 8.839 11.03 9.976 9.056 8.695 21.36 32.37 21.36 21.54 21.36

Lag 1 autocorrelationSydney 0.505 0.672 0.336 0.709 0.453 0.395 0.419 0.116 0.406 0.328 0.36 0.356 0.122 0.287 0.308 0.307 0.215 0.307 0.302 0.306Perth 0.446 0.122 0.335 0.495 0.400 0.365 1.904 0.126 0.286 0.315 0.305 5.012 0.126 0.248 0.272 0.277 8.756 0.277 0.277 0.277Cairns 0.508 0.677 0.312 0.804 0.481 0.424 0.462 0.133 0.490 0.358 0.413 0.401 0.187 0.37 0.354 0.424 0.312 0.424 0.421 0.424Hobart 0.490 0.629 0.295 0.518 0.487 0.339 0.365 0.083 0.270 0.284 0.239 0.281 0.083 0.175 0.222 0.179 0.117 0.179 0.179 0.179

Dry proportionSydney 0.927 0.899 0.915 0.940 0.931 0.865 0.846 0.865 0.884 0.869 0.818 0.803 0.818 0.828 0.822 0.727 0.748 0.727 0.727 0.730Perth 0.929 0.907 0.917 0.941 0.932 0.869 0.860 0.869 0.880 0.872 0.828 0.822 0.828 0.833 0.831 0.755 0.773 0.755 0.755 0.757Cairns 0.895 0.867 0.876 0.906 0.898 0.801 0.796 0.801 0.833 0.803 0.733 0.738 0.733 0.761 0.735 0.634 0.663 0.634 0.636 0.636Hobart 0.925 0.897 0.910 0.954 0.929 0.845 0.832 0.845 0.882 0.849 0.779 0.777 0.779 0.805 0.783 0.667 0.703 0.667 0.668 0.672

144A

.Puiet

al./Journalof

Hydrology

470–471

(2012)138–

157

CANON MICRO RBLM MOF

0.10

80.

112

0.11

6 sydneyR

ainf

all (

mm

)a(i)


0.07

60.

079

perth

Rai

nfal

l (m

m)

a(ii)


0.20

50.

209

cairns

Rai

nfal

l (m

m)

a(iii)


0.05

550.

0575

hobart

Rai

nfal

l (m

m)

a(iv)


0.65

0.67

0.69

sydney

Rai

nfal

l (m

m)

b(i)


0.45

50.

475

perth

Rai

nfal

l (m

m)

b(ii)


1.23

1.25

1.27

cairns

Rai

nfal

l (m

m)

b(iii)


0.33

50.

345

hobart

Rai

nfal

l (m

m)

b(iv)


1.30

1.34

1.38

sydney

Rai

nfal

l (m

m)

c(i)


0.91

0.94

0.97

perthR

ainf

all (

mm

)

c(ii)


2.46

2.50

2.54

cairns

Rai

nfal

l (m

m)

c(iii)


0.66

50.

685

hobart

Rai

nfal

l (m

m)

c(iv)


2.60

2.70

sydney

Rai

nfal

l (m

m)

d(i)


1.82

1.88

1.94

perth

Rai

nfal

l (m

m)

d(ii)


4.95

5.05

cairns

Rai

nfal

l (m

m)

d(iii)


1.33

1.36

1.39

hobart

Rai

nfal

l (m

m)

d(iv)

Fig. 7. Mean for 1, 6, 12, and 24 h from rows (a–d) respectively, with columns denoting locations, dashed red line denotes observed mean. All models are expected toreproduce the observed means at all timescales. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)


0.6

1.0

1.4

sydney

Varia

nce

a(i)


0.20

0.30

0.40

perth

Varia

nce

a(ii)


1.5

2.5

3.5

4.5

cairns

Varia

nce

a(iii)


0.15

0.25

0.35

hobart

Varia

nce

a(iv)


1214

1618

20

sydney

Varia

nce

b(i)


4.0

4.5

5.0

5.5

6.0

perth

Varia

nce

b(ii)


3540

4550

55

cairns

Varia

nce

b(iii)


3.0

3.5

4.0

4.5

hobartVa

rianc

e

b(iv)


3438

4246

sydney

Varia

nce

c(i)


1213

1415

perth

Varia

nce

c(ii)


110

120

130

140

cairns

Varia

nce

c(iii)


910

1112

hobart

Varia

nce

c(iv)


100

120

140

sydney

Varia

nce

d(i)


3540

4550

perth

Varia

nce

d(ii)


300

350

400

450

cairns

Varia

nce

d(iii)


2530

35

hobart

Varia

nce

d(iv)

Fig. 8. Variance for 1, 6, 12, and 24 h from rows (a–d) respectively, with columns denoting locations, dashed red line denotes observed mean. This statistic is not used in theparameter estimation stage by any of the models considered, except RBLM. (For interpretation of the references to colour in this figure legend, the reader is referred to theweb version of this article.)


may be attributed to fundamental differences in model structuresbetween the cascades, point process, and resampling methods.

Two versions of RBLM calibrated to operate in ‘disaggregationmode’ in the literature are based on conditional simulation

(Koutsoyannis and Onof, 2001) and proportional adjusting (Glas-bey et al., 1995). While the logic of both these methods is similar,we use the adjusting procedure model, in view of its good perfor-mance reported recently (e.g. Debele et al., 2007) and also compu-


0.3

0.5

0.7

sydney

Lag−

1−Au

toco

rr.

a(i)


0.1

0.3

0.5

perth

Lag−

1−Au

toco

rr.

a(ii)


0.3

0.5

0.7

cairns

Lag−

1−Au

toco

rr.

a(iii)


0.3

0.5

hobart

Lag−

1−Au

toco

rr.

a(iv)


0.10

0.25

0.40

sydney

Lag−

1−Au

toco

rr.

b(i)


0.5

1.5

perth

Lag−

1−Au

toco

rr.

b(ii)


0.2

0.4

cairns

Lag−

1−Au

toco

rr.

b(iii)


0.1

0.3

hobart

Lag−

1−Au

toco

rr.

b(iv)


0.10

0.25

0.40

sydney

Lag−

1−Au

toco

rr.

c(i)


02

46

perth

Lag−

1−Au

toco

rr.

c(ii)


0.20

0.30

0.40

cairns

Lag−

1−Au

toco

rr.

c(iii)


0.05

0.20

0.35

hobart

Lag−

1−Au

toco

rr.

c(iv)


0.18

0.24

0.30

sydney

Lag−

1−Au

toco

rr.

d(i)


02

46

8

perth

Lag−

1−Au

toco

rr.

d(ii)


0.30

0.36

0.42

cairns

Lag−

1−Au

toco

rr.d(iii)


0.10

0.14

0.18

hobart

Lag−

1−Au

toco

rr.

d(iv)

Fig. 9. Lag-1-autocorrelation for 1, 6, 12, and 24 h from rows (a–d) respectively, with columns denoting locations, dashed red line denotes observed mean. This statistic is notused in the parameter estimation stage by any of the models considered, except RBLM. (For interpretation of the references to colour in this figure legend, the reader isreferred to the web version of this article.)


0.90

0.92

0.94

sydney

Prop

ortio

n D

ry

a(i)


0.91

0.93

perth

Prop

ortio

n D

ry

a(ii)


0.87

0.89

cairns

Prop

ortio

n D

ry

a(iii)


0.90

0.93

hobart

Prop

ortio

n D

rya(iv)


0.85

0.87

sydney

Prop

ortio

n D

ry

b(i)


0.86

00.

875

perth

Prop

ortio

n D

ry

b(ii)


0.80

0.82

cairns

Prop

ortio

n D

ry

b(iii)


0.83

0.86

hobart

Prop

ortio

n D

ry

b(iv)


0.80

00.

815

sydney

Prop

ortio

n D

ry

c(i)


0.82

00.

828

perth

Prop

ortio

n D

ry

c(ii)


0.73

00.

745

0.76

0 cairns

Prop

ortio

n D

ry

c(iii)


0.77

50.

790 0

.805

hobart

Prop

ortio

n D

ry

c(iv)


0.73

00.

745

sydney

Prop

ortio

n D

ry

d(i)


0.75

50.

765

0.77

5 perth

Prop

ortio

n D

ry

d(ii)


0.63

50.

650

0.66

5 cairns

Prop

ortio

n D

ry

d(iii)


0.67

0.69

hobart

Prop

ortio

n D

ry

d(iv)

Fig. 10. Dry proportion for 1, 6, 12, and 24 h from rows (a–d) respectively, with columns denoting locations, dashed red line denotes observed mean. This statistic is not usedin the parameter estimation by any of the models considered, except MICRO and RBLM. (For interpretation of the references to colour in this figure legend, the reader isreferred to the web version of this article.)


tational ease. The adjusting procedure modifies the initially gener-ated values ðfXsÞ at a lower timestep from a RBLM to obtain the ad-justed values (Xs):

Xs ¼ eXszPk

j¼1~Xj

!ð5Þ

where Z is the coarser-level variable (i.e. observed daily rainfall inthis case) and k is the number of finer-level variables within onecoarser-level period, which, for example, may typically be dailyrainfall, or a wet spell that continues for several days (Koutsoyian-nis and Onof, 2001). Given that rainfall depths in rainy intervals canbe assumed to be approximately gamma-distributed, the propor-tional adjusting procedure is deemed appropriate. Nevertheless,


some bias may be introduced. This bias has been addressed by usinga repetitive scheme (e.g. Koutsoyiannis and Onof, 2001), generatingBartlett–Lewis sequences until the difference between simulatedand observed daily totals is less than a pre-set value of d as definedbelow:

d ¼XL

i¼1

ln2 Zi þ c~Zi þ c

� � !0:5

ð6Þ

With reference to Eq. (6), the pre-set value, d, helps reduce biasthrough employing a logarithmic distance, such that extreme valuesare not associated with undue weight; similarly, the constant c isintroduced to avoid domination by very low values. The parameterL is the sequence of wet days generated, while Z values are the his-torical (numerator) and generated (denominator) daily depths ofday i. In this study, the value of d is set to 0.1 to avoid unacceptablebias. The detailed algorithm for coupling the RBLM with the adjust-ing procedure is set out in Koutsoyiannis and Onof (2001). Furtherdetails on the parameter estimation for RBLM are discussed later.

3.3. Resampling based on method of fragments (MOF)

This method stands in contrast to the cascade-based model andRBLM, as it is non-parametric. As such, it makes no major assump-

Fig. 11. Empirical intensity–frequency curves derived from observed and model disagfrequency curves for 1, 6, 12, and 24 h respectively, with black lines denoting observedparameter estimation by any of the models considered.

tions about the nature of the relationship between continuous andaggregate rainfall. The MOF generates sequences of rainfall that ex-hibit persistence attributes similar to those observed by maintain-ing temporal dependence at a daily timescale, and then using anon-parametric disaggregation logic to impart dependence tosub-daily timesteps (Sharma and Srikanthan, 2006). The MOFmethodology is based on resampling a vector of fragments repre-senting the ratio of the sub-daily to daily rainfall. The term ‘frag-ment’ here specifically represents the ratio of sub-daily rainfall todaily rainfall for a particular timestep of choice. For example, ifthe time series is made up of hourly rainfall, and assuming rainfallis uniform across the 24 h, then every fragment is equally weightedat 1/24. These weights will shift according to the distribution ofrainfall simulated for a given day. Resampling is performed via amodified k-nearest neighbor algorithm (Lall and Sharma, 1996),and can be described as follows:

Step 1: Form daily rainfall Ri ¼P

mXi;m� �

and sub-daily fragment(fi,m = Xi,m/Ri) time series, i denoting day and m a sub-daily time-step. Here, notation R is used to define daily rainfall, X to sub-daily rainfall, and f is used to define the sub-daily fragments.Step 2: To disaggregate daily rainfall Rt for a particular day t, forma moving window of length l days centered around day t. Formthe following four rainfall classes of the historical daily data:

gregated rainfall for 1, 6, 12, and 24 h at Sydney. Rows (a)–(d) denote intensity–intensity–frequency behavior. This is a validation statistic as it is not used in the


CLASS 1 : Rj P 0jðRj�1 ¼ 0;Rjþ1 ¼ 0ÞCLASS 2 : Rj P 0jðRj�1 P 0;Rjþ1 ¼ 0ÞCLASS 3 : Rj P 0jðRj�1 ¼ 0;Rjþ1 P 0ÞCLASS 4 : Rj P 0jðRj�1 P 0;Rjþ1 P 0Þ

ð7Þ

where time j represents a day falling within the moving windowcentered on the current day t.

Step 3: Identify the class, ct (ct e (1–4)) corresponding to thedaily rainfall Rt that is to be disaggregated.Step 4: Identify the k nearest neighbors of the conditioning vec-tor (Rt) as the days corresponding to the k lowest absolute dif-ferences |Rj � Rt| and satisfying the condition cj � ct. Specifyk ¼

ffiffiffinp

(Lall and Sharma, 1996), where n represents the samplesize of the class members falling within the moving window.The ranked daily rainfall from the lowest absolute departureis then given as Rj, j = 1,2, . . .,k. Sample a jth neighbor usingthe following conditional probability distribution (Lall andSharma, 1996; Mehrotra and Sharma, 2006):

Fig. 12. Empirical intensity–frequency curves derived from observed and model disaggrecurves for 1, 6, 12, and 24 h respectively, with black lines denoting observed intensity–estimation by any of the models considered.

pj ¼1=jPki¼11=i

ð8Þ

where pj represents the probability of selecting neighbor j, with j = 1for the neighbor having the smallest absolute departure. Using auniformly distributed random number (0,1), select a neighbor (Rj)using the probabilities in Eq. (8). The fragments used to disaggre-gate can be specified as fj,m = Xj,m/Rj.

Step 4: Use increments of t and repeat steps (2) to (4) until dis-aggregation is completed.

In the current study, the sampling window spans over 15 dayson either side of our day of interest. This would increase the sam-ple size available for sampling purposes and also account for ef-fects of seasonality. For example, if we want to disaggregatedaily rainfall on 16th January of our daily time series, we look intothe entire month of January (1–31) across the full observed record(minus the current year, if one disaggregates using historical rain-fall) to find n nearest neighbors closest to the daily rainfall total

gated rainfall for 1, 6, 12, and 24 h at Perth. Rows (a)–(d) denote intensity–frequencyfrequency behavior. This is a validation statistic as it is not used in the parameter


being disaggregated. This criterion of matching up daily rainfalltotals (i.e. selecting the n closest neighbors in terms of daily rainfalltotals from the historical record) is especially important because ithelps to identify whether our ‘wet’ day of interest lies at the start,the end, or in the middle of a storm. As such, our ‘selected’ dayfrom history is expected to be accorded with a more realistic dis-tribution, compatible with our ‘wet’ day of interest. Here, we fixn = 144, and hence k = 12, with the MOF reducing to resamplingfrom the marginal distribution when k < 12 to avoid excessive biaswhen simulating at the extremes. The MOF further aims to appro-priately represent the characteristics in the generated sub-daily se-quences by ensuring that disaggregated sub-daily rainfall frommodel output sum up to the daily rainfall and that the model issensitive to various sub-daily temporal patterns that shift withtime of the year and also the magnitude of daily rainfall. The aboveconditions are reflected by its generation procedure, which oper-ates under the assumption that daily rainfall values to be disaggre-gated are representative of daily rainfall formed by disaggregatingobserved continuous rainfall time series, and that sub-daily frac-tions represent temporal patterns applicable for the specific siteof interest (Sharma and Srikanthan, 2006).

Please note that all the approaches mentioned above do nothave an explicit mechanism of maintaining the correlation


between the 24th hour rainfall of the previous day and 1st hourrainfall of the current day and, as such, if a storm crosses over aday, there is a discontinuity in the disaggregated hourly rainfall val-ues. While it is possible to modify the MOF structure to incorporatethis dependence (Sharma and O’Niell, 2002), it would result inadded complexity in one model structure and, therefore, is not at-tempted in this paper. It should be pointed out, however, that theuse of wetness states allows representation of dependence acrossdays to a better extent, in contrast to what would be achieved hadan unconditional selection of the days to be resampled performed.

4. Parameter estimation, results, and discussion

In this section, first we discuss the parameter estimation meth-ods employed for the RMC models and the RBLM. In particular, wereport on the suitability of different approaches to parameteriza-tion, the fit of parameters to the data, as well as the stability ofthe parameters – including the proposal of a new approach toparameterization of the RBLM to account for parameter uncer-tainty. Next, we assess the model disaggregation results that com-prise of several ‘standard’ rainfall model validation statistics,followed by an extreme value analysis and empirical wet spellanalysis. Standard statistics are used here to describe those

gregated rainfall for 1, 6, 12, and 24 h at Cairns. Rows (a)–(d) denote intensity–intensity–frequency behavior. This is a validation statistic as it is not used in the


statistics that are considered to be most important – and are arequirement if the model is to be considered to be adequate forwater resource studies. The standard statistics that have been com-monly used in the literature include mean, standard deviation, lag-one autocorrelation of rainfall, and dry proportions (e.g. Onof andWheater, 1993). The intensity–frequency curves are produced toexamine the ability of the models to reproduce the distributionof annual extremes. The intensity–frequency curves are derivedfrom the maximum intensity in each year for a given duration.Empirical dry and wet spells are further used as evidence of eachmodel’s ability to reproduce the observed rainfall characteristics.

Since the aim here is to reproduce observed historical rainfallstatistics, rather than to perform forecasting, the entire historicalrainfall record at each of the study sites is used to calibrate theparametric models, to take advantage of the length of availabledata. This is conventionally the case with most publications on sto-chastic generation, with validation deferred to a set of attributesthat are not directly modeled. Historical daily rainfall (aggregatedfrom observed sub-daily data) of those same records are thendisaggregated to hourly rainfall, with a total of 10 realizations per-formed. As the cascade-based models, by virtue of their branchingnumber b = 2, sub-divide from 1440 min to 45 min after five


successive disaggregation levels, we perform linear interpolationfrom 45 min to the nearest 1-h timestep. The procedure for theRBLM is performed using the HYETOS program (Koutsoyiannisand Onof, 2000).

4.1. Parameter estimation

4.1.1. Parameter estimation of RMC modelsParameter estimation of the canonical model is straightforward,

as only two parameters (bCANON and r2) are required. First, mo-ments of order 0 to 4 are obtained for rainfall aggregated to time-steps of 30 min to 1920 min (We consider order 0–4 to reproducemodels as closely as practicable to those widely reported in the lit-erature, although other limitations/guidelines are also reported inthe literature; see, for example, Licznar et al. (2011)). A relation-ship between log(q), where q is the order of the moment (or mo-ment order) versus log (timescale) is then developed. Thegradient of each fitted line is obtained via linear regression. Theseestimated gradients, s(q), are subsequently plotted against mo-ment order. To estimate the parameters, bCANON and r2, the MKPfunction of W for the beta-lognormal model is then applied (Overand Gupta, 1996):

gregated rainfall for 1, 6, 12, and 24 h at Hobart. Rows (a)–(d) denote intensity–intensity–frequency behavior. This is a validation statistic as it is not used in the


sðqÞ ¼ ðbCANON � 1Þðq� 1Þ þ r2 ln b2ðq2 � qÞ ð9Þ

Optimization for the function in Eq. (9) uses a relative-offset con-vergence criterion that compares the numerical imprecision at thecurrent parameter estimates to the residual sum-of-squares to ob-tain nonlinear least-squares estimates of the parameters bCANON

and r2. An example of the moment scaling relationship and shapeof the MKP function for rainfall is illustrated in Fig. 3, which pre-sents the results obtained for Sydney rainfall data. Table 2 presentsthe CANON model parameter values for rainfall from all the fourlocations studied herein. It should be noted that, while the use oftwo parameters to characterize the CANON is appealing from the

1−2 3−4 5−6 7−8 9−10 11−12

Sydney (1−12hrs)

Spell Length (hrs)

Den

sity

0.0

0.1

0.2

0.3

0.4

0.5

a(i).

13−14 17−18 21−22

Sydney (13−24hrs)

Spell Length (hrs)

Den

sity

0.00

00.

005

0.01

00.

015

0.02

00.

025 OBS

MOFCANONMICRORBLM

a(ii).

1−2 3−4 5−6 7−8 9−10 11−12

Cairns (1−12hrs)

Spell Length (hrs)

Den

sity

0.0

0.1

0.2

0.3

0.4

0.5

c(i).

13−14 17−18 21−22

Cairns (13−24hrs)

Spell Length (hrs)

Den

sity

0.00

00.

010

0.02

00.

030

OBSMOFCANONMICRORBLM

c(ii).

Fig. 16. Distributions of empirical dry spell lengths for (a) Sydney, (b) Perth, (c) Cairnestimation by any of the models considered.

1−2 3−4 5−6 7−8 9−10 11−12

Sydney (1−12hrs)

Spell Length (hrs)

Den

sity

0.0

0.1

0.2

0.3

0.4

a(i).

13−14 17−18 21−22

Sydney (13−24hrs)

Spell Length (hrs)

Den

sity

0.00

00.

005

0.01

00.

015

OBSMOFCANONMICRORBLM

a(ii).

1−2 3−4 5−6 7−8 9−10 11−12

Cairns (1−12hrs)

Spell Length (hrs)

Den

sity

0.0

0.1

0.2

0.3

0.4

0.5

c(i).

13−14 17−18 21−22

Cairns(13−24hrs)

Spell Length (hrs)

Den

sity

0.00

00.

005

0.01

00.

015

0.02

0 OBSMOFCANONMICRORBLM

c(ii).

Fig. 15. Distributions of empirical wet spell lengths for (a) Sydney, (b) Perth, (c) Cairnestimation by any of the models considered.

viewpoint of simplicity, its parameters exhibit significant inter-an-nual variability as well as intra-annual (seasonal) variability (seeFig. 4). The single values of parameters shown in Table 2 are com-puted using the entire historical record, meaning that bCANON andr2 are estimated from the probability distributions derived fromrainfall data across the whole record instead of a subset of the re-cord (i.e. the annual and seasonal values).

The seasonal bCANON values at Sydney and Perth indicate in-creased intermittency in rainfall during summer compared to win-ter. Seasonal bCANON values at Cairns are also representative of moreintermittent rainfall during the dry season compared to less inter-mittency during its wet season. This finding is in agreement withthat of a similar study conducted on rainfall in the Swiss Alpine

1−2 3−4 5−6 7−8 9−10 11−12

Perth (1−12hrs)

Spell Length (hrs)

Den

sity

0.0

0.1

0.2

0.3

0.4

0.5

0.6

b(i).

13−14 17−18 21−22

Perth (13−24hrs)

Spell Length (hrs)

Den

sity

0.00

00.

004

0.00

80.

012 OBS

MOFCANONMICRORBLM

b(ii).

1−2 3−4 5−6 7−8 9−10 11−12

Hobart (1−12hrs)

Spell Length (hrs)

Den

sity

0.0

0.1

0.2

0.3

0.4

0.5

d(i).

13−14 17−18 21−22

Hobart (13−24hrs)

Spell Length (hrs)

Den

sity

0.00

00.

005

0.01

00.

015

0.02

00.

025 OBS

MOFCANONMICRORBLM

d(ii).

s, and (d) Hobart. This is a validation statistic as it is not used in the parameter

1−2 3−4 5−6 7−8 9−10 11−12

Perth (1−12hrs)

Spell Length (hrs)

Den

sity

0.0

0.1

0.2

0.3

0.4

0.5

b(i).

13−14 17−18 21−22

Perth (13−24hrs)

Spell Length (hrs)

Den

sity

0.00

00.

002

0.00

40.

006

0.00


b(ii)

1−2 3−4 5−6 7−8 9−10 11−12

Hobart (1−12hrs)

Spell Length (hrs)

Den

sity

0.0

0.1

0.2

0.3

0.4

0.5

0.6

d(i).

13−14 17−18 21−22

Hobart(13−24hrs)

Spell Length (hrs)

Den

sity

0.00

00.

002

0.00

40.

006

0.00


d(ii).

s, and (d) Hobart. This is a validation statistic as it is not used in the parameter


region, where it was reported that parameter variations to the CA-NON model may be large and dependent on seasonality as well aslocal climatology and orographic influences (Molnar and Burlando,2008). It should be noted that there may also be significant depen-dency between rainfall intensity and cascade model parameters,both in space (e.g. Over and Gupta, 1996; Veneziano et al., 2006;Rupp et al., 2012) and in time (e.g. Molnar and Burlando, 2005,2008; Veneziano et al., 2006; Serinaldi, 2010; Licznar et al.,2011). However, we do not address this dependency here, sinceit is a more complex problem and, thus, a deviation from the mainobjective of this study. We will investigate this issue in the future.

For the MICRO model, previous studies have shown that thevariability parameter a is dependent on successive timescalesand becomes smoother with increasing temporal resolution (e.g.Menabde and Sivapalan, 2000; Molnar and Burlando, 2005). As aresult, we choose to fit f(M) by a symmetric beta distribution foreach doubling timestep from 45 min to 1440 min (daily level), rep-resentative of the five levels of disaggregation. Fitting a function(e.g. power law) to establish a relationship between the parametervalue to the timestep (e.g. Menabde and Sivapalan, 2000; Molnarand Burlando, 2005; Paulson and Baxter, 2007; Rupp et al., 2009;Licznar et al., 2011) would also allow for: (1) a reduction in modelparameter values; and (2) the ability to compare parameter valuesacross studies. We perform this according to steps taken in Molnarand Burlando (2005), but, for sake of brevity, graphs showingparameter value to timescale are not shown here. The interestedreaders may obtain such additional details from the authors.

Here, we use the symmetric beta distribution with meanE[M] = 0.5 and apply the method of moments for parameter esti-mation through the relationship a = (1/(8Var[M]) � 0.5), where arepresents the model variability parameter. The dry probabilityparameter, p.dry, is also estimated for the same levels. The resultsobtained for these scale-dependent parameters at all locations areshown in Table 2. An assessment of the fit of the beta distributionto observed cascade weights, M, reveals that it does not properlycapture the decreasing variance of the observed weights at finertimescales at all locations. An example of the poor fit of the betadistribution to M is illustrated in Fig. 5, which presents the caseof Sydney rainfall data from 30 min to 1920 min. This may trans-late into the problem of over-simulating variance at finer stepsby the MICRO model, and will be discussed later.

4.1.2. Parameter estimation of RBLMThe parameters for RBLM have traditionally been estimated on

a monthly basis, assuming stationarity for the month (Bo et al.,1994), using the method of moments as the basis for estimation(see Appendix A). The vector of parameters to be estimated is de-noted as h, where h = {k, lx, j, u, a, m}. The second-order propertiesof the accumulated process over the time interval, T, which repre-sent selected statistical attributes of the rainfall time series (Rodri-guez-Iturbe et al., 1987), are estimated over varying levels ofaggregation (1–48 h). In this study, twelve statistical attributes(mean, variance, lag-1 auto-correlation, and dry proportion at the1-h, 24-h, and 48-h aggregation scales) are used to ascertainparameters, as recommended by Koutsoyiannis and Onof (2000),using an objective function that seeks to minimize the weightedsum of squared deviation of observed and modeled attributes.

Problems of parameter instability of RBLM during the optimiza-tion process have been well documented in the literature (e.g. Onofand Wheater, 1993; Khaliq and Cunnane, 1996). The instability ofparameters that control storm generation duration (u, a, and m),in particular, often results in the frequent generation of storms ofexcessively long durations, which are physically implausible.While the literature contains suggestions of alternate optimizationalgorithms to circumvent this difficulty (e.g. Onof and Wheater,1993), we address this issue through:

(a) Constraining parameters based on limits previously reportedin the literature.

(b) Introducing a Sequential Monte Carlo (SMC) sampler toascertain optimal outcomes.

The setting of parameter constraints is due to the fact thatalthough the mean squared error for the objective function is oftenminimal (i.e. less than 0.2), it is still possible to simulate physicallyimplausible storms. As such, we constrain a number of parametersto provide more realistic rainfall sequences (see Table 3) across alllocations. In particular, to ensure that more than a solitary cell isproduced within a storm, the lower boundary for j is set to 0.01(so that j does not become too small), and the upper boundaryfor u is set to 2.0 to prevent too large a value of u. An upperboundary on k of 4.32 is also fixed such that only a maximum offive storms per hour could be generated (see Table 3). It shouldbe noted that the constraints imposed are not overly restrictive,as they are not intended to reflect the rainfall characteristics at aparticular location but simply to prevent the trawling of redundantparameter space. Importantly, the constraints are flexible enoughto allow for the possibility of a varying number of acceptableuncertainty that can be implemented.

In this study, we use the Sequential Monte Carlo (SMC) samplerto ascertain the optimal parameter choices. The SMC sampler usesdata assimilation techniques, while executing a combination ofsampling and resampling processes (Jeremiah et al., 2011). By ini-tializing the simulation with multiple parallel sample runs (eachsample representing a fully functioning set of model parameters),the SMC method has an added advantage in better representingthe complete joint posterior distribution of the parameter vector.As such, the SMC sampler is able to identify both local and globaloptima and effectively explore the parameter space. The SMCmethod starts by sampling the posterior state from a prior distri-bution, p0 (initial knowledge and, in this article, a set of uniformdistribution is used) and gradually moves the population along atransition path, {bSMC} (Jeremiah et al., 2011), until finally con-verges with the targeted posterior distribution, denoted as bSMCs

.The symbol s denotes the number of iterations required to com-plete the simulation and observes the condition, s = 0, . . ., S, andstops when bSMCs

¼ 1. The intermediate distribution, ps, is calcu-lated at each stage of the algorithm, where additional informationof the data is gradually being added (see Eq. (10) below). The grad-ual increase of data (correction of the sampler from the prior to theposterior distribution) creates a surge of new local optima andmoves the population from one point to another. The update ofthe intermediate distribution is given as:

ps / p1�bSMCs0 pðhÞbSMCs ð10Þ

where h is the parameter vector h = {k, j, /, a, m, lx}.The example given in Fig. 6 (for Cairns, for the month of March)

shows the intermediate distributions of parameter lx, calculated atiterations 1, 50, 100, 200, 300, and 640. In total, the SMC samplerrequires 640 iterations to complete the simulation and convergethe particle population to the targeted posterior distribution. Thesampler starts by sampling each parameter from a uniform distri-bution and gradually moves the population as more data are in-cluded into the simulation. Out of the four locations studied,Cairns (wet months) is considered the most volatile, frequentlycausing the SMC sampler to collapse when new information isadded into the framework. The term ‘collapse’ used with the SMC(Arulampalam et al., 2002; Moradkhani et al., 2005) refers to thesituation when only a few samples with important weights domi-nate the sampling mechanism, thus preventing the populationfrom continuously giving a good representation of the posteriordistribution. An analysis of the SMC runs revealed that the final


converged parameters are overwhelmingly located within the pre-fixed thresholds. Results for the mean of 1000 sets of monthlyparameters identified at each station by the SMC are presentedin Appendix B, with a range of mean squared errors from 0.008to 2.46.

It should be noted that the use of a weighted objective function,ascertained through estimated moments as detailed before, repre-sents a quasi-likelihood and does not conform to the definition of aproper likelihood function. As a result, the ascertained optimalparameter sets do not represent the joint posterior distributionof the parameters, but rather the multiple optimal solutions thatcharacterize the parameter space. In this study, a total of 10monthly parameter sets from each group of 1000 are selected atrandom to account for the uncertainty in the realizations resultingfrom the RBLM.

4.2. Assessment of disaggregated rainfall

Assessing three stochastic generation models of varying levelsof complexity and parameterisation is a difficult task beset by anumber of challenges. Using a part of the sample for calibrationand the other part for validation (an often used approach especiallyfor estimation-type models) is not recommended here, given thedifficulty in using part of the sample in characterizing the full pop-ulation, and the added complexity of needing to maintain depen-dence that is not an option in approaches such as cross-validation.

It is for this reason that elaborate measures are used in validat-ing stochastic generation models. A seminal paper on what consti-tutes such measures (Stedinger and Taylor, 1982) suggests thatvalidation should be performed using the full data and focusingon attributes that are not aimed to be modeled directly but reflectthe inherent ability of the model structure to mimic the underlyingprocess being simulated. This is the approach we adopt in the cur-rent study. Statistics such as the variance across multiple time-scales as well as the extremes and their representation using IFDrelationships are all characteristics that neither model directlyaims to simulate.

The only other alternative for validation is to assume an under-lying population and then fit all methods to this sample, withassessment being done over the underlying population being as-sumed. While this approach has been adopted in past studies(e.g. Tarboton et al., 1998), it suffers from the need to makeassumptions about the population, which can unintentionally af-fect the outcome from the exercise. We avoid pursuing this path,given the complexity involved and the uncertainty it creates whenassessing the alternate methods.

4.2.1. 1-, 6-, 12- and 24-h standard validation statistics4.2.1.1. Mean. Reproduction of the mean is generally done well byall models at all study sites (see Table 4 and Fig. 7). The MICRO

Table 5Observed and model simulated average number of wet spells per wet day and within-dareproduce these statistics explicitly in the simulations.

Average number of wet spells per wet day

OBS CANON MICRO RBLM MO

Sydney 1.24 2.35 2.08 1.25 1.42Perth 1.85 2.38 2.25 1.90 1.92Cairns 2.03 2.34 2.33 1.26 2.29Hobart 1.68 2.19 2.04 1.24 1.61

Average number of dry spells per wet daySydney 0.66 1.42 1.08 0.26 0.66Perth 0.93 1.45 1.25 0.92 0.95Cairns 1.03 1.42 1.32 0.27 1.06Hobart 0.63 1.29 1.04 0.25 0.61

results replicate that of the observed data at all timesteps of inter-est, which is to be expected due to its exact conservation of mass ateach disaggregation level. The MOF also achieves the same resultas it is based on resampling of observed rainfall. The slight discrep-ancy between the observed and simulated mean by the RBLM atthe 24-h timestep may be attributed to its disaggregation method-ology. Unlike the other models used, HYETOS does not disaggre-gate based on solitary wet days one at a time, but does so onwet-day sequences (i.e. consecutive rainy days preceded and fol-lowed by at least one dry day). As a result, the mean computedfrom the aggregated rainfall for the RBLM from hourly to dailytimestep does not exactly match the observed 24-h mean. The CA-NON model slightly underestimates the mean at all timesteps forSydney, Perth, and Hobart, but overestimates the mean for Cairns.The scatter in the mean for different realizations may be attributedto its preservation of mass on average, but not so at exactly eachdisaggregation timestep.

4.2.1.2. Variance. The MOF and RBLM models are able to simulateobserved variance reasonably well, but not the RMC models. TheMICRO model, in particular, performs poorly at all sites, by consis-tently returning inflated variance at the target hourly disaggrega-tion timestep at all sites, although this result needs to be verifiedin the context of dependency between rainfall intensity and modelparameters for higher complexity MICRO model, which we do notattempt here. Recalling the poor fit of the Beta distribution (seeFig. 5) to observed cascade weights (breakdown coefficients) atthe sites, it is likely that the weights generated using the variabilityparameter a fail to adequately reflect the decreasing variance withscale of the observed cascade weights. As a result, the MICRO-sim-ulated variance deviates further from the observed variance forsuccessive cascades or disaggregation levels (see Fig. 8). It is rele-vant to note that a similar problem in reproducing the variancewith this version of MICRO has also been reported by Rupp et al.(2009), with an overestimation in their case. The other RMC coun-terpart, CANON, produces results that are converse to that of MI-CRO; it underestimates the hourly variance, but increasinglyoverestimates the variance at successive re-aggregation timesteps.The CANON model is also the only model where the variance doesnot converge to that of the observed one at 24 h (see Fig. 8). A pos-sible explanation for this may be the non-containment of weights,which may result in physically unrealistic rainfall (i.e. Re-aggre-gated rainfall sums larger than observed, as the sum of cascadeweights may be >1). Please note that the CANON parameters areestimated using the log-moment scaling and, therefore, it is ex-pected that CANON would not preserve the scaling of the secondmoment. Likewise, there is also the possibility of simulating twozero weights for a rainy period, further contributing to the poorsimulation of variance at the 24-h timestep. It is possible to reducethe likelihood of two zero weights being generated during a heavy

y wet spell length for all locations of study. Models considered are not structured to

Average within day wet spell length (h)

F OBS CANON MICRO RBLM MOF

4.23 3.77 3.57 4.20 4.003.18 3.81 3.63 2.96 3.483.38 3.73 3.48 4.90 3.293.30 3.39 3.16 2.63 3.21

Average within dry wet spell length (h)3.67 3.08 3.21 4.01 3.663.23 3.03 3.11 2.53 3.244.29 3.07 3.26 4.28 4.28

1 3.49 3.26 3.48 4.32 3.41


rainfall period through making beta a function of rainfall intensity;this, however, is not attempted here, but will be done in a futurestudy with more complex models. It should also be noted thatwhile we apply a combination of parameters, as identified by theSMC to the RBLM to account for uncertainty, the narrow widthsof the box plots shown in Figs. 7 and 8 (and also Figs. 9 and 10;see below) suggest that there is little variability across differentrealizations and that this may be due to model structure that per-forms repetitive matching of daily rainfall amounts, thus produc-ing combinations of storm/cell arrangements.

4.2.1.3. Lag-1 autocorrelation. The lag-1 autocorrelation is overallwell reproduced by the MOF and also to some extent by RBLM(with the exception for RBLM at hourly timestep at Perth and Ho-bart), but not the RMC models (see Fig. 9). In particular, MICRO sig-nificantly underestimates this statistic at about all sites, whileCANON produces consistently larger estimates at about all studysites. This result suggests that the RMC models do not outperformthe point process- and resampling-type models from the perspec-tive of simulating realistic temporal rainfall persistence structures;a key feature of the RBLM process is its ability to simulate thestorm inter-arrival times (via the Poisson process).

4.2.1.4. Dry proportion. Reproduction of observed within-day dryproportions is done reasonably well by all the models at all sites(see Fig. 10). It should be noted, however, that CANON underesti-mates dry proportions at all sites for the 1-, 6-, and 12-h timesteps,and overestimates at the daily level. In addition, RBLM appears tobe biased towards simulating greater number of dry spells thanthat observed at all sites, especially for 1-, 6-, and 12-h timesteps.

4.2.2. Extreme value analysisThere are a number of methods to examine extreme value infor-

mation from available rainfall time series, including the widely-used approach of extracting the annual maxima. Frequency analy-sis of the resulting annual maximum series then provides an aver-age range of application in terms of the return period from 1 to1000 years (Verhoest et al., 1997). Here, intensity–frequencycurves are constructed based on empirical cumulative distributionof the rainfall series from observed and disaggregated rainfall ser-ies. In this procedure, annual maximum rainfall is ranked from thehighest to the lowest, and empirical estimates of the annual excee-dence probability (AEP) are estimated according to:

AEPðmÞ ¼ ðm� 0:4Þ=ðN þ 0:2Þ ð11Þ

where m is the rank and N is the length of record (Pilgrim and Dor-an, 1987). These AEPs are then plotted against the correspondinglog-transformed rainfall intensities representing the intensity–fre-quency relationship for each of the models at each study site. Con-fidence bands for model-simulated intensity–frequency curves areformed using results from 10 realizations. The intensity–frequencycurves are presented for each model versus observed values at eachstudy site for the 1-, 6-, 12-, and 24-h aggregation levels (seeFigs. 11–14). While the observed values at all sites fall within thesimulated bounds (even at low AEPs) at the 6-, 12-, and 24-h levels,there are also some notable exceptions at the hourly target disag-gregation timestep. For Sydney, MICRO significantly overestimatesthe hourly empirical intensity–frequency. This result is consistentwith the problem of ‘inflated variance’ (as mentioned earlier). Con-versely, RBLM significantly underestimates the hourly empiricalintensity–frequency. The CANON model slightly underestimatesthe intensity–frequency at higher return periods at 1-h and 6-htimesteps, but reproduces the lower AEP values well (seeFig. 11a). For Perth, RBLM and MOF perform well, with observedvalues generally falling within simulated bounds of these models.However, CANON results in an underestimation of the hourly and

6-h intensity–frequency, while the MICRO model overestimatesthe hourly intensity–frequency by a significant margin (seeFig. 12a and b). For Cairns, both CANON and MOF perform reason-ably well, with the exception of a slight underestimation of high re-turn periods by CANON at the 6-h timestep, and by MOF for lowreturn periods at the hourly timestep. The RBLM performs poorlyin Cairns, with a significant underestimation of the hourly inten-sity–frequency (see Fig. 13). For Hobart, all the models are able toreproduce the observed intensity–frequency curves reasonablywell, with again the notable exception of MICRO at the hourly time-step (see Fig. 14). The results also suggest that the CANON modelslightly overestimates the 12-h and 24-h rainfall intensities. Theseresults are consistent with the ones reported for rainfall data fromtemperate climate data elsewhere around the world; for instance,Molnar and Burlando (2005) reported overestimation of longerduration rainfall extremes using the CANON model This limitationcan be attributed to the correlation structure of the CANON model(see Molnar and Burlando (2005) for details).

In view of these results, it may be inferred that CANON consis-tently underestimates the middle to lower tails of the extreme va-lue distribution at the sites, while MICRO returns a significantoverestimation. The RBLM appears to be better suited for semi-aridand temperate climate regions – performing reasonably well inPerth and Hobart, but poorly in Cairns; the version employed inthis study is not able to simulate intense storm bursts well. TheMOF reliably mimics the observed intensity–frequency, aided byits resampling logic using the observed data.

4.2.3. Empirical dry and wet spell analysisFrom a design flood perspective, generation of realistic dry and

wet spells is important, because rainfall intermittency and persis-tence need to be quantified to gauge antecedent catchment soilconditions, especially in rural, ungauged basins. This cannot beachieved simply by undertaking an intensity–frequency analysisof extreme values, and additional information is necessary. To thisend, the ability of CANON, MICRO, RBLM, and MOF to reproducedry and wet spells that are similar to that of observed rainfall arealso investigated here. For this purpose, a within-day wet spell isdefined by consecutive hours of rainfall within a rainy day, inaccordance with some previous studies (Menabde and Sivapalan,2000; Llasat, 2001). Intuitively, we further define a within-daydry spell as simply the consecutive dry hours that intersperse rainyevents, since we are interested in attributes related to antecedentproperties of within-day rainfall produced by the disaggregationmodels. The distribution of empirical dry and wet spells for the ob-served and the disaggregated rainfall at all study sites is shown inFigs. 15 and 16; for better visualization and comparison of the re-sults, separate plots are presented for 1–12 h (denoted as (i) foreach city) and 13–24 h (denoted as (ii) for each city). While all ofthe models simulate the shorter wet spells well (i.e. spells of lengthless than 8 h) across different locations, the RMC models are unableto simulate long wet spells of greater than 12 h in length. TheRBLM, although reasonably robust at other stations, performspoorly in Cairns, overestimating the frequency of long wet spells(see Fig. 15c(ii)). This is evident as the model average within wetspell length of 4.90 h is significantly larger than the observed valueof 3.38 h (see Table 5). It also appears that the RMC models are notrobust in terms of reproducing both variability in average numberof wet spells per wet day and average within-day wet-spell lengthsat the sites, as it consistently returns similar values for these statis-tics (see Table 5). With respect to dry spells, the RMC models tendto overestimate while the RBLM underestimates the occurrence ofwithin-day dry spells at all the locations, with the exception ofgood performance of RBLM in Perth (see Table 5). In contrast, theRMC models slightly underestimate the within-day dry spelllengths and display less inter-station spell length variability


compared to observed dry spell lengths. The RBLM produces longerthan observed dry spell lengths at Sydney and Hobart, which sug-gest that the inter-storm parameter k may be underestimated atthese locations (thus resulting in fewer storm arrivals per wetday) (see Table 5).

5. Summary

This paper has evaluated the performance of a number of dailyto sub-daily rainfall disaggregators, such as two versions of ran-dom cascade models (canonical – CANON; microcanonical – MI-CRO), the randomized Bartlet–Lewis model (RBLM), and themethod of fragments (MOF), using continuous rainfall data at fourcities in Australia (Sydney, Perth, Cairns, and Hobart). In particular,the parameter estimation procedures of each parametric modelwere assessed, as was the performance of the models in terms ofsimulation of standard rainfall statistics and a host of validationstatistics, such as extreme values, intensity–frequency relation-ships, and empirical wet spells. In general, the MOF model outper-forms the other models. This is not entirely unexpected given thatthe MOF logic operates based on resampling of observed rainfallfractions (at sub-daily timescale) and is, therefore, expected to pro-duce statistics that bear the closest resemblance to the observeddata. The strength of this model, however, may also serve as a lim-itation because its applicability is confined to areas where contin-uous sub-daily data are available. Also, the MOF model may not beflexible enough to simulate events beyond the limits of the histor-ical record. As a result, without conditioning the MOF model onsynthetic rainfall data representative of a future climate, it maybe limited by its assumption of climate stationarity. In the samevein, this argument holds true for the other models since theirparameters were estimated based on observed historical rainfallonly, followed by the assumption that these parameters remain va-lid for the future.

On the whole, the RBLM performs better than the cascade mod-els, albeit with a slightly inflated simulation of dry proportions atthe hourly scale as well as an underestimation of extreme rainfallat low return periods at Sydney and Cairns. The underestimation ofextreme rainfall has also been reported by Verhoest et al. (1997). Itshould be noted that the parameterization of the model is complex,and the choice of statistics (and the timescales at which these sta-tistics are gauged) remains subjective. As a result, we circum-vented this problem by choosing to implement parameterizationusing a Sequential Monte Carlo (SMC) sampler to identify regionsof likely optima. Another concern is that the model parametersmay not necessarily represent actual physical quantities becausethey depend on the timescale that is chosen for fitting (Foufoula-Georgiou and Guttorp, 1986), resulting in the use of an intuitiveparameter constraining procedure to ensure that somewhat realis-tic quantities are produced. The selection of the value for the pre-set value d for the adjusting procedure is also somewhat arbitrary,with lower d values vastly increasing the number of iterations re-quired to obtain close enough rainfall cell amounts matching ob-served rainfall. This results in increased computation time, andoccasionally creates ‘blocks’ (computer running out of memory)whereby the maximum number of iterations are reached but HYE-TOS is still unable to match up daily totals between disaggregatedand observed data.

The RMC models, in particular MICRO, did not outperform itssimpler counterpart, CANON. Since CANON is parameter parsimo-nious, its performance is judged to be relatively satisfactory con-sidering its simplicity, particularly if the exact conservation ofrainfall is not required. Perhaps estimating CANON parameterson a seasonal basis (in locations where seasonality is pronounced)may improve its performance, particularly from the viewpoint of

extreme value simulation. The poor performance of the CANONmodel has also been noted by Hingary and Ben Haha (2005), whoreported overestimation of a majority of statistics, especially thoserelated to extremes. Use of either a theoretical distribution otherthan log-normal, such as log-stable (e.g. Serinaldi, 2010; Ruppet al., 2012), or an empirical distribution (Hingary and Ben Haha,2005) may also improve the CANON model performance. The MI-CRO performed poorly by generating inflated hourly rainfall vari-ance and, consequently, overestimating the extreme rainfall atthe same timestep. This may be primarily attributed to the inap-propriate choice of the Beta distribution to Australian rainfall,therefore highlighting the importance of selecting an appropriateprobability distribution considering that the model is completelyreliant on cascade weights. However, consideration of dependencybetween rainfall intensity and model parameter values may alsocontribute to this problem, which we will address in our futureresearch.

It may be noted that, although this study makes use of data ofonly four locations in Australia, the model comparison measure,being based on the validation statistics, forms a generic indicatorof the strengths and drawbacks of the various models evaluated.Therefore, the present study (and similar ones) that discern the rel-ative limitations and strengths of each model, is important, asthere will be greater possibility in the future of wider applicationsof the disaggregation methods to downscaled daily rainfall forproper assessment of the changes in the occurrence and frequencyof extreme rainfall events. As such, information regarding the per-formance of the models is expected to guide hydrologists in apply-ing daily to sub-daily ‘disaggregators’ with greater insight.

Acknowledgements

We would like to thank Christian Onof and Demetris Koutsoy-iannis for providing assistance with the RBLM. We would also liketo acknowledge Ana Barros for her assistance during the revisionprocess, and the two anonymous reviewers for their constructivecomments and useful suggestions to improve the quality of themanuscript. This study was financially supported by the AustralianResearch Council.

Appendix A. Equations used for the modeled statistics for RBLMparameter estimation procedure

mean ¼ klxlcm

a� 1T lc ¼ 1þ j

/

variance ¼ 2m2�a

a� 2k1 �

k2

/

� �� 2m3�a

ða� 2Þða� 3Þ k1 �k2

u2

� �þ 2ða� 2Þða� 3Þ k1ðT þ mÞ3�a � k2

/2 ð/T þ mÞ3�a� �

k1 ¼ 2klcl2x þ

klcl2x/

/2 � 1

� �ma

a� 1

� �

k2 ¼klcl2

xj/2 � 1

� �ma

a� 1

� �

Autocovarianceðlag � sÞ ¼ k1

ða� 2Þða� 3Þ fTðs� 1Þ þ mÞ3�a

þ ½Tðsþ 1Þ þ m�3�a � 2ðTsþ mÞ3�ag

þ k2

/2ða� 2Þða� 3Þf2ð/Tsþ mÞ3�a � ½/Tðs� 1Þ þ m�3�a

� ½/Tðsþ 1Þ þ m�3�ag


Prðzero� rainÞ ¼ expð�kT � f1 þ f2 þ f3Þ

f1 ¼km

/ða�1Þ 1þ/ jþ/2

� ��1

4/ðjþ/Þðjþ2/Þþ/ðjþ/Þð4j2þ27j/þ36/2Þ

72

" #

f2 ¼km

ðjþ /Þða� 1Þ 1� j� /þ 32

/jþ /2 þ j2

2

� �

f3 ¼ki

ðjþ /Þða� 1Þm

mþ Tðjþ /Þ

� �a�1

� j/

1� j� /32j/þ /2 þ j2

2

� �

Appendix B

B.1. Optimal values of RBLM parameters estimated for Sydney usingSMC method

Month k j = b/g u = c/g a N lX(mm/d)

January 0.213 0.182 0.192 3.424 0.265 71.846February 0.207 0.125 0.573 3.858 0.770 56.401March 0.269 0.155 0.263 3.684 0.406 59.183April 0.200 0.174 0.156 4.519 0.602 47.057May 0.200 0.798 0.327 3.848 0.467 25.205June 0.205 0.173 0.489 3.921 1.116 33.204July 0.221 1.219 0.074 2.456 0.018 37.171August 0.144 2.528 0.636 2.011 0.062 41.012September 0.116 0.886 0.038 4.728 0.029 83.376October 0.207 0.190 0.026 2.398 0.014 122.768November 0.277 0.472 0.045 2.960 0.013 123.421December 0.203 0.203 0.017 2.861 0.007 197.054

B.2. Optimal values of RBLM parameters estimated for Perth using theSMC method

Month
k j = b/g
u = c/g

a
N lX(mm/
d)

January
0.017 2.451 0.183 14.445 0.468 34.201 February 0.039 0.764 1.053 24.312 2.201 70.653 March 0.065 0.167 0.092 31.043 3.306 23.776 April 0.108 0.453 0.061 30.966 0.715 54.831 May 0.248 0.217 0.026 19.712 0.293 79.716 June 0.362 0.184 0.023 24.147 0.517 67.505 July 0.428 0.298 0.042 26.264 0.526 60.347 August 0.408 0.455 0.046 38.50 0.526 56.356 September 0.341 0.294 0.042 39.778 0.601 55.650 October 0.153 0.117 0.035 29.228 1.619 34.658 November 0.111 1.076 0.176 24.447 1.795 12.261 December 0.055 3.005 1.719 37.54 8.045 7.616
B.3. Optimal values of RBLM parameters estimated for Cairns using theSMC method

Month
k j = b/g
u = c/g

a
N lX(mm/
d)

January
B.4. Optimal values of RBLM parameters estimated for Hobart usingthe SMC method

Month
k j = b/g
u = c/g

a
N lX(mm/d)
January
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A comparison of alternatives for daily to sub-daily rainfall disaggregation

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