reproduction of temporal scaling by a rectangular pulses rainfall model

HYDROLOGICAL PROCESSESHydrol. Process. 16, 611–630 (2002)DOI: 10.1002/hyp.307

Reproduction of temporal scaling by a rectangular pulsesrainfall model

Jonas Olsson,1* and Paolo Burlando2

1 Institute of Environmental Systems, Kyushu University, Fukuoka, Japan2 Institute of Hydromechanics and Water Resources Management, ETH Zurich, Switzerland

Abstract:

The presence of scaling statistical properties in temporal rainfall has been well established in many empiricalinvestigations during the latest decade. These properties have more and more come to be regarded as a fundamentalfeature of the rainfall process. How to best use the scaling properties for applied modelling remains to be assessed,however, particularly in the case of continuous rainfall time-series. One therefore is forced to use conventional time-series modelling, e.g. based on point process theory, which does not explicitly take scaling into account. In light ofthis, there is a need to investigate the degree to which point-process models are able to ‘unintentionally’ reproducethe empirical scaling properties. In the present study, four 25-year series of 20-min rainfall intensities observed inArno River basin, Italy, were investigated. A Neyman–Scott rectangular pulses (NSRP) model was fitted to theseseries, so enabling the generation of synthetic time-series suitable for investigation. A multifractal scaling behaviourwas found to characterize the raw data within a range of time-scales between approximately 20 min and 1 week. Themain features of this behaviour were surprisingly well reproduced in the simulated data, although some differenceswere observed, particularly at small scales below the typical duration of a rain cell. This suggests the possibility ofa combined use of the NSRP model and a scaling approach, in order to extend the NSRP range of applicability forsimulation purposes. Copyright 2002 John Wiley & Sons, Ltd.

KEY WORDS temporal rainfall; Neyman–Scott rectangular pulses model; scaling properties

INTRODUCTION

At present, the field of stochastic rainfall time-series modelling is dominated by two conceptually differentapproaches. On one hand, the representation of storm occurrence as a point process and the internal stormintensity structure as a cluster of rectangular pulses mimicking rain cells has proved to be an elegant andphysically realistic way to describe temporal rainfall. The theoretical foundations of the approach were laid byRodriguez-Iturbe et al. (1987a, 1988) and it has been developed and tested further using observed rainfall fromvarious climates (e.g. Rodriguez-Iturbe et al., 1987b; Entekhabi et al., 1989; Islam et al., 1990; Cowpertwait,1991; Burlando and Rosso, 1993; Onof and Wheater, 1993; Bo et al., 1994; Cowpertwait et al., 1996a,b). Twomain versions of this modelling approach exist, that is the Bartlett–Lewis and the Neyman–Scott clusteringscheme, respectively, essentially differing in the way pulses are distributed relative to the storm origin (e.g.Rodriguez-Iturbe et al., 1987a). Both models have five main parameters in their first formulation (Rodriguez-Iturbe et al., 1987a), which have been shown to reflect well the seasonal and climatological features of therainfall-generating mechanisms (e.g. Rodriguez-Iturbe et al., 1987a; Burlando, 1989; Islam et al., 1990).

On the other hand, building models on scaling properties of temporal rainfall has emerged as a second mainapproach in recent years. Originating from fractal theory (Mandelbrot, 1982), the assumption underlying thisconcept is the absence of characteristic scales where statistical properties distinctly change, but the presence

* Correspondence to: Dr Jonas Olsson, Swedish Meteorological and Hydrological Institute, SE-603 57 Nowkoping, Sweden.E-mail: [email protected]

Received 12 June 2000Copyright 2002 John Wiley & Sons, Ltd. Accepted 21 February 2001

612 J. OLSSON AND P. BURLANDO

of a symmetry linking rainfall statistics at different temporal aggregation levels. Scaling properties, generallymultifractal, manifested in a power-law variation of some statistical quantity versus time-scale, have beenobserved in many data sets (e.g. Hubert et al., 1993; Olsson, 1995; Burlando and Rosso, 1996; Harris et al.,1996; de Lima and Grasman, 1999). Random cascade processes basing their theoretical foundation in fractalgeometry thus have been advocated to be suitable for modelling the scaling behaviour (e.g. Schertzer andLovejoy, 1987; Gupta and Waymire, 1990) and a number of cascade models for temporal rainfall have beendeveloped (e.g. Menabde et al., 1997; Olsson, 1998; Deidda et al., 1999).

Although scaling-based rainfall models are very promising, they are still in their infancy. Althoughreproducing the observed scaling, the ability of the models to also reproduce well the other fundamentalfeatures of continuous rainfall, such as the clustering and the presence of dry periods, is still not clear(e.g. Olsson, 1998). Therefore rectangular pulses modelling is likely to remain the most robust and practicalapproach to simulate continuous rainfall time-series for the time being. In light of this, it is of interestto investigate the degree to which scaling properties in the historical data used for model calibration arereproduced in model simulated data. Even if rectangular pulses models do not explicitly include any scalingbehaviour in their construction but are based on characteristic scales, e.g. the typical lifetime of a rain cell, thisdoes not exclude that they may approximately reproduce the observed scaling over certain ranges. Rectangularpulses models have been shown in several studies to reproduce well the conventional rainfall statistics overa range of aggregation levels, normally between 1 h and 1 day (e.g. Rodriguez-Iturbe et al., 1987a, 1988;Entekhabi et al., 1989; Islam et al., 1990; Burlando and Rosso, 1993). For example, a model calibrated using,for example, 12- and 24-h statistics, has been shown also to reproduce statistics reasonably well at a 1- or 2-hscale (e.g. Rodriguez-Iturbe et al., 1987b; Bo et al., 1994). This suggests that the model is in some respectsable to transfer statistics through scales. Bo et al. (1994) argue that this ability is related to the approximatelyscaling structure of the Bartlett–Lewis clustering model power spectrum. It should be mentioned further thatthe same model for some parameter values may lead to asymptotic second-order self-similarity, althoughsuch values seldom appear to be encountered in practice (e.g. Rodriguez-Iturbe et al., 1988; Entekhabi et al.,1989).

The aim of the present study is to investigate the scaling behaviour of synthetic rainfall data simulated by aNeyman–Scott rectangular pulses model. The issue of scaling in rectangular pulses models has been tackledpreviously, notably by Onof et al. (1996), but there is a clear need for more comprehensive evaluations. Thefirst issue is whether scaling is at all present in the simulated data; the second is to what accuracy are scalingparameters reproduced in the historical data to which the model has been fitted (provided both data sets doexhibit scaling). For this assessment, three indicators of scaling—empirical probability distribution function,power spectrum and variation of statistical moments with scale—are studied for both historical and simulateddata. Included in this analysis is a thorough investigation of scaling in a rainfall data base from the Arno Riverbasin in central Italy, data that have been used previously in connection with rectangular pulses modelling(e.g. Islam et al., 1990; Burlando and Rosso, 1993). The scaling parameters are related to both geographicaland seasonal climatological characteristics.

EXPERIMENTAL SET-UP

Study region and historical data

The historical data set that has been used to carry out the present analysis includes four stations locatedin the Arno River basin, which is located in central Italy, approximately between latitudes 43° and 44°N andlongitudes 10° and 11°E (see Figure 1). It drains a total of 8168 km2 in Pisa, where the Arno River flows intothe Tyrrenian Sea, being characterized in its lower reaches by gentle hills and, mostly, completely flat areas.The basin area is bounded by the Appennine Mountains forming an arc from north to east with an averageelevation of 1000 m a.s.l. and a maximum elevation of about 2000 m a.s.l. The southern part of the watershedis bounded by the Chianti Hills, with elevations that do not exceed 900 m a.s.l. The climatic characterization

Copyright 2002 John Wiley & Sons, Ltd. Hydrol. Process. 16, 611–630 (2002)

RAINFALL MODELLING 613

Camaldoli

VallombrosaLivorno

FirenzeThyrrenian

Sea

0 40 Km20

Figure 1. Sketch of the Arno River basin and location of the rainfall stations investigated

of the basin is representative of the high variability of the Mediterranean climate, with a total yearly amountof rainfall ranging between about 700 and 1700 mm, heavy precipitation during autumn, and dry summerswith convective thunderstorms. A data set consisting of 25 years (1962–1986) of 20-min rainfall data for 14stations is available for the catchment and the neighbouring areas.

Four stations out of that data set (Camaldoli, Firenze, Livorno and Vallombrosa) have been selected for thepresent analysis. These are reasonably representative of the different conditions within the catchment, rangingfrom a coastal storm-dominated energy-rich regime for stations located at sea level, to a more orographicallycontrolled regime for inland stations at higher altitudes (Figure 1). Table I summarizes information aboutthe stations, including a few geographical characteristics that may be meaningful with respect to resultinterpretation. There are remarkable differences in mean rainfall amounts and the proportion of dry periodsbetween the two stations located at higher elevation (probably influenced by orographic effects), and the othertwo located on the coast (Livorno) and in the main valley of the Arno River basin (Firenze). Table II reportssome such descriptive statistics of the historical data, relevant to the following analysis.

In the analyses, the time-series were investigated both in their entirety, representing the lumped rainfallprocess, and split into seasons. The period January to March was assumed representative for winter, April toJune for spring, July to September for summer, and October to December for autumn.

NSRP model simulated data

Among the various stochastic rainfall models proposed in the literature, the Neyman–Scott rectangularpulses (NSRP) model has attracted considerable attention. Over many years the model has proved capable of

Table I. Characteristics of the investigated stations in the Arno River basin

Station Latitude Longitude Distance tosea (km)

Elevation(m a.s.l.)

Resolution(min)

Data recordlength

Camaldoli 43°48’N 11°49’E c. 125 1111 20 1962–1986Firenze Ximeniano 43°47’N 11°15’E c. 80 51 20 1962–1986Livorno 43°33’N 10°18’E 0 5 20 1962–1986Vallombrosa 43°44’N 11°33’E c. 100 955 20 1962–1986



Table II. Relevant descriptive statistics of the historical (Hist) and simulated (NSRP) data at the original temporal aggregation(20 min). Parameters given are zero-depth probability P�X D 0�, conditional (i.e. for X > 0) mean Mnc�X� and coefficient

of variation CVc�X�, and 10-year (lumped process) or 10-season (winter and summer) maximum value Max10

Camaldoli Firenze Livorno Vallombrosa

Hist NSRP Hist NSRP Hist NSRP Hist NSRP

Lumped P�X D 0� 0Ð86 0Ð95 0Ð94 0Ð97 0Ð94 0Ð98 0Ð88 0Ð96Mnc�X� 0Ð445 0Ð64 0Ð50 0Ð61 0Ð51 0Ð83 0Ð39 0Ð61CVc�X� 2Ð14 1Ð46 2Ð00 1Ð73 2Ð695 1Ð95 2Ð30 1Ð49Max10 68Ð6 18Ð7 31Ð1 26Ð4 74Ð8 32Ð5 49Ð5 15Ð8

Winter P[X D 0] 0Ð81 0Ð93 0Ð92 0Ð94 0Ð92 0Ð96 0Ð84 0Ð94Mnc�X� 0Ð37 0Ð49 0Ð37 0Ð40 0Ð37 0Ð44 0Ð33 0Ð43CVc�X� 1Ð65 1Ð28 1Ð44 1Ð31 1Ð65 1Ð32 1Ð775 1Ð31Max10 12Ð8 6Ð2 7Ð3 4Ð9 9Ð1 6Ð0 13Ð3 5Ð6

Summer P�X D 0� 0Ð92 0Ð98 0Ð97 0Ð99 0Ð97 0Ð99 0Ð93 0Ð98Mnc�X� 0Ð53 1Ð18 0Ð92 1Ð59 0Ð73 1Ð92 0Ð48 0Ð985CVc�X� 3Ð02 1Ð49 2Ð24 1Ð52 2Ð86 1Ð46 2Ð62 1Ð40Max10 67Ð2 18Ð7 40Ð0 26Ð7 28Ð6 26Ð2 24Ð0 15Ð0

0 t1tj

t1(1)

t2(1)t3

(1)

t2(1)

tc1

tci = duration of the i-th pulse

ici = intensity of the i-th pulse

X(t)

0

t1(1)

t1(1) = displacement of the j-th cell from the cluster center

Wj+1W1

ic

Wj

Wj = interarrival time of events (pulses)

W2

ttime

rain

fall

inte

nsity

Figure 2. Sketch of the Neyman–Scott rectangular pulses model

capturing statistical properties of both the continuous and the extremal process (e.g. Burlando, 1989; Burlandoand Rosso, 1993; Cowpertwait, 1994, Cowpertwait et al., 1996a,b). The model, illustrated in Figure 2, is basedon Poisson arrivals of storms, and associated with each arrival is a cluster of rectangular pulses of randomheight and duration, displaced randomly from the cluster origin. The superposition of these pulses providesthe description of the storm profile. In the earliest model formulation it is assumed that both the intensity andthe duration of a pulse are independent and identically distributed, following an exponential distribution. Thepulses are displaced from the cluster origin according to an exponential distribution, and the number of cells



Table III. Comparison between historical (Hist), analytically fitted (NSRP, f), and NSRP simulated (NSRP, s) statisticsfor July, station Firenze, at aggregation levels 1, 6 and 24 h. Parameters given are zero-depth probability P�X D 0�, mean

Mn�X�, variance Var�X�, and lag-1 autocorrelation r(1)

1 h 6 h 24 h

P�X D 0� E�X� Var�X� r(1) P�X D 0� E�X� Var�X� r(1) P�X D 0� E�X� Var�X� r(1)

Hist 0Ð982 0Ð0377 0Ð319 0Ð2205 0Ð952 0Ð226 2Ð94 0Ð114 0Ð872 0Ð905 13Ð1 0Ð0528NSRP, f — 0Ð0377 0Ð319 0Ð2205 — 0Ð226 2Ð94 0Ð114 — 0Ð905 14Ð0 0Ð0342NSRP, s 0Ð985 0Ð0379 0Ð3345 0Ð197 0Ð948 0Ð222 3Ð03 0Ð125 0Ð856 0Ð835 14Ð1 0Ð0281

are distributed geometrically. Such a structure yields a five-parameter model, which must be calibrated byfitting observed statistics to analytical expressions of the second-order properties, as reported by, for example,Burlando (1989). Most literature contributions recognized the need for seasonal parameter estimation, whichenables the model to preserve statistics of the observed process throughout different scales of aggregationand across the year. In several cases this leads up to monthly based parameter estimates (see, e.g. Burlandoand Rosso, 1993), such as in the present case. Table III shows an example of the agreement between somehistorical, analytically fitted, and simulated parameters at three aggregation levels (1, 6 and 24 h) for July atstation Firenze. The parameters used in the model fitting were the mean value at 1 h together with varianceand lag-1 autocorrelation at 1 and 6 h (for further details, see Burlando and Rosso, 1993).

Based on the above parameterization, 20-min series corresponding to 160 years were simulated andinvestigated extensively, as described hereinafter. Descriptive statistics of the simulated data are given inTable II. From a qualitative point of view, it may be concluded that the model captures well both the seasonaland the interstation parameter variations of the historical data. However, it is clear that the model systematicallyoverestimates the zero-depth probability somewhat, which also leads to a corresponding overestimation of theconditional mean (i.e. rainfall amounts during non-zero periods). Further, variability (CV) as well as maximumvalues are systematically somewhat underestimated by the model, and generally the model accuracy is lowerfor 20-min than for 1-h values (compare Tables II and III). This decrease in model accuracy for time-scalesbelow the mean lifetime of a rain cell (typically 0Ð5–1 h) generally is known (e.g. Burlando and Rosso,1993), but overall the model also provides a reasonably robust representation of the rainfall process at a20-min time-scale.

Scaling analyses

The three most commonly applied indicators of temporal scaling in rainfall are the empirical probabilitydistribution function (PDF), the power spectrum and the variation of statistical moments with scale. In allcases, a power-law behaviour qualitatively indicates the presence of scaling and the power-law exponent (i.e.slope) provides a quantitative measure. The connections of these indicators with scaling theories and theirapplication to rainfall time-series have been covered extensively elsewhere (e.g. Ladoy et al., 1991; Fraedrichand Larnder, 1993; Olsson, 1995; Harris et al., 1996; Tessier et al., 1996), and here the method are onlybriefly reviewed.

Let XT denote the temporal rainfall intensity process at a certain location, sampled at a time resolution T.If the tail of the empirical PDF of intensities XT follows a power-law form

Pr �XT > x� / x�qcr �1�

for high threshold intensities x, the series is characterized by hyperbolic intermittency, an intrinsic property offractal and multifractal scaling processes (e.g. Fraedrich and Larnder, 1993; Harris et al., 1996; Tessier et al.,1996). Hyberbolic intermittency is often contrasted with an exponential tail behaviour, and both types havebeen found in rainfall time-series (e.g. Zawadzki, 1987; Georgakakos et al., 1994; Olsson, 1995; Svensson



et al., 1996; de Lima and Grasman, 1999). The issue is both directly and highly relevant to engineering design,because return periods of extreme intensities may differ widely, being lower in the hyperbolic case. Theambiguous findings obtained to date indicate that temporal rainfall may in fact exhibit both a hyperbolic andan exponential behaviour, presumably primarily depending on the rainfall-producing atmospheric mechanismand thus secondarily related to, for example, season and geographical location. This possibility is supportedparticularly by Georgakakos et al. (1994), who found both types of behaviour for isolated storms observedby the very same gauge. In cases where a hyperbolic tail has been suggested, the value of qcr was found tovary widely within the range 1 to 10, as reported by the literature referred to above.

The power spectrum E(f) is an indicator of scaling in its most basic form, i.e. absence of characteristic,symmetry-breaking scales. This requires that the spectrum takes the form

E�f� / f�ˇ �2�

where f is the frequency and ˇ is a characteristic exponent. Thus variability at different frequencies (orscales) are linked by a unique relationship, a basic requirement underlying every scaling-based descriptionand modelling approach (e.g. Harris et al., 1996). Contrary to the empirical PDF, spectral analyses of observedrainfall data have indicated a rather unambiguous scaling behaviour, albeit restricted to certain scale ranges.With few exceptions, analyses have supported the following general pattern, where s is the time scale: ˇ ³ 0for s > 1–2 weeks, 0 < ˇ < 1 for 1–2 weeks > s > 1–2 h, 1 < ˇ < 2 for s < 1–2 h (e.g. Fraedrich andLarnder, 1993; Georgakakos et al., 1994; Fabry, 1996; Harris et al., 1996; Tessier et al., 1996). The lowerfrequency range has been proposed to relate to the random nature of weather systems and cyclone tracks,the middle frequency range to baroclinic forcing and frontal disturbances, and the higher frequency range tointernal turbulence of rain-producing systems (e.g. Fraedrich and Larnder, 1993; Fabry, 1996). Within thisgeneral pattern, the value of ˇ has been shown to vary with altitude and storm type (e.g. Georgakakos et al.,1994; Harris et al., 1996), but scaling has not been rejected in a qualitative sense. Even so, the above patterndoes suggest that scaling models, as generally defined today, will have upper and lower time-scale limits ofapplicability when used for temporal rainfall.

A more comprehensive indicator of scaling is the variation of the average coarse-grained moments hXq�Ti,

where q denotes moment order, with coarse-graining scale parameter �, defined as the scale under studydivided by the smallest scale available (thus � ½1 with � D 1 corresponding to 20 min). In this context, ascaling behaviour manifests in the relationship

hXq�Ti / ��Kq �3�

where Kq is a characteristic function describing the growth curve of scaling exponents with respect to momentorder. For details of the coarse-graining moment analysis procedure, see, for example Tessier et al. (1993),Olsson (1995) and Harris et al. (1996). The value K0

1�D dKq/dq at q D 1� is a measure of the intermittency ofthe mean process, and often is denoted C1 (e.g. Tessier et al., 1993, 1996; Davis et al., 1994). For time-seriesC1 is bounded theoretically by 0 and 1, and for rainfall it has been found to vary from slightly larger than 0 upto c. 0Ð6 (e.g. Hubert et al., 1993; Olsson, 1995; Harris et al., 1996; Tessier et al., 1996; de Lima and Grasman,1999). The moment scaling behaviour is related to both the empirical PDF and the power spectrum exponent.The exponent, qcr, defines a critical order above which moments diverge and Kq becomes linear (e.g. Man-delbrot, 1974). If the exponent, ˇ, exceeds 1, Equation (3) does not necessarily hold (Menabde et al., 1997).

If the Kq function is convex, as normally is the case for rainfall time-series, the data set is termedmultiscaling or multifractal (e.g. Frisch and Parisi, 1985; Gupta and Waymire, 1990). If, on the other hand, Kq

is linear the data set is termed simple scaling or monofractal. Often a visual inspection suffices to determinethe type of scaling, but because Kq for the present data showed in some cases to be weakly curved but nearlylinear an objective determination procedure was required. For this purpose, it is useful to express Kq as

Kq D C1�q � 1�ϕq �4�



where ϕq is a function quantifying the departure of Kq from a linear, simple scaling behaviour (Burlandoand Rosso, 1996). The 95% confidence interval of Kq was calculated and compared with the simple scalingcurve Kq D C1(q � 1). If the latter is located within the confidence interval, a simple scaling hypothesiscannot be rejected. Further, a measure of the curvature of Kq was desired in order to compare the ‘degree ofmultiscaling’ between different data sets, and ϕ4 was used for this purpose.

Concerning rainfall time-series, moment scaling has been investigated mainly for the total rainfall process,either as separate events or continuous series with dry periods included. However, Burlando and Rosso (1996)showed also that the annual maximum process may exhibit scaling properties. On the basis of these findings,the authors developed a framework for depth–duration–frequency curves encompassing both a simple scalingand a multiscaling behaviour, and demonstrated an increased accuracy and robustness as compared withconventional approaches. A similar approach, restricted to simple scaling, was made recently by Menabdeet al. (1999).

Denote by H�T the annual maximum rainfall intensity process obtained by a moving average window oflength �T, where T is the sampling time resolution (see Burlando and Rosso (1996) for details of the process).A scaling behaviour of the qth order moments hHq

�Ti may be expressed as

hHq�Ti / ��˛q �5�

where ˛q is a characteristic moment scaling function with properties similar to the function Kq in the caseof total process coarse-grained moments. To quantify departures from simple scaling, ˛q may be written inthe form

˛q D ˛1qϕq �6�

where ˛1 is a key parameter in scaling models of depth–duration–frequency curves (Burlando and Rosso,1996; Menabde et al., 1999). Similar to the total process moment analysis, 95% confidence intervals of ˛q

were used to test the hypothesis of simple scaling (i.e. ˛q D ˛1q) versus multiscaling, and ϕ4 was used tospecify the degree of multiscaling.

RESULTS AND DISCUSSION

Empirical PDF

The distribution of rainfall intensities simulated by the NSRP model described previously, by con-struction belongs to the exponential family. A hyperbolic behaviour similar to the one often exhibitedby observed data therefore can not be reproduced. The comparison between observed and simulateddata for the rainfall series analysed in this study essentially confirms this. Figure 3 shows the empiricalPDFs of the lumped time-series from the four stations, plotted as both log–log (Figure 3a) and semilog(Figure 3b) diagrams. For all stations, a hyperbolic tail behaviour appears to be the best approximation,especially for Camaldoli and Livorno. For Vallombrosa and particularly Firenze, the situation is some-what less clear, with tails approaching an exponential shape (Figure 3b). From these findings it may beexpected that the difference in the PDFs of observed and model simulated data is smaller for Firenze

Table IV. Empirical PDF exponent qcr for the historical data

Lumped Winter Spring Summer Autumn

Camaldoli 2Ð26 2Ð95 2Ð59 1Ð84 2Ð27Firenze 2Ð125 2Ð92 1Ð99 1Ð44 3Ð03Livorno 2Ð16 2Ð17 1Ð81 1Ð23 2Ð24Vallombrosa 2Ð37 3Ð43 2Ð33 2Ð04 2Ð06



−6

−5

−4

−3

−2

−1

00 1 2 4 5 6 7 83

Log[x]

Log[

Pr(

XT>

x)]

Camaldoli

Firenze

Livorno

Vallombrosa

a)

1.E−06

1.E−05

1.E−04

1.E−03

1.E−02

1.E−01

1.E+000 20 40 60 80 100 120

xP

r(X

T>

x)

CamadoliFirenzeLivornoVallombrosa

b)

Figure 3. Historical empirical probability distribution functions for all stations, lumped process, plotted as log–log (a) and semilog(b) diagrams. The curves have been horizontally offset, thus the x-axis is only relative. The straight lines in (a) have been fitted by

regression to the filled symbols

a) Camaldoli

−7

−6

−5

−4

−3

−2

−1

0−3 −2 −1 0 1 2

Log[x ]

Log[

Pr(

XT>

x)]

Historical

NSRP simulated

b) Firenze

−7

−6

−5

−4

−3

−2

−1

0−3 −2 −1 0 1 2

Log[x]

Log[

Pr(

XT>

x)]

Historical

NSRP simulated

Figure 4. Historical and NSRP simulated empirical probability distribution functions for Camaldoli (a) and Firenze (b), lumped process

than the other stations. This is confirmed in Figure 4 comparing PDFs for Camaldoli and Firenze. ForCamaldoli (Figure 4a), the difference is striking qualitatively as well as quantitatively; the probabilityof extreme intensities (corresponding to c. 200–250 mm/h) is more than two orders of magnitude toolow in the simulated data. For Firenze (Figure 4b), on the other hand, the tails’ shape agrees ratherwell, although the probability of high intensities also is here underestimated systematically in the simu-lated data.

The PDFs with hyperbolic tails have been further investigated. The exponents, qcr, of a linear regressionmodel fitted to the PDF tails as illustrated in Figure 3a have been estimated and are reported in Table IV.The difference between stations is small, with all values being in the vicinity of 2Ð2. One may associatethe slightly lower values observed for Firenze and Livorno with the more erratic occurrence of convectiveevents, as compared with the rainfall regimes of Camaldoli and Vallombrosa, which are more influenced byorographic conditioning. The values of qcr are in the lower range of results from previous investigations of



continuous time-series (see earlier section), implying a comparably modest increase in return periods withincreasing intensity.

The seasonal PDF analyses of observed data revealed a rather distinct pattern, with spring to autumngenerally displaying a clear hyperbolic behaviour, whereas in winter exponential tails generally weresuggested. In consequence of the above, the NSRP model performed at least reasonably well during winterfor all stations, with fits generally similar to the situation in Figure 4b. For spring, summer and autumn,however, in about half the cases the fit was poor, resembling the situation in Figure 4a. Table IV shows theseasonal variation of qcr (for comparison, qcr also was estimated for the winter PDFs not displaying clearhyperbolicity). A clear cyclicity is present, with considerably lower values during the summer than duringthe winter half-year. The results reflect the significant presence of convective events over a large part of theyear and the high frequency of extreme intensities during summer, as compared with the more homogeneous,frontal-dominated winter rainfall.

a) Camaldoli

4

5

6

7

8

−7 −6 −5 −4 −3 −2 −1 0

Log[f ]

Log[

E(f

)]

Historical

NSRP simulated

1 week

6 hours

b) Firenze

4

5

6

7

8

−7 −6 −5 −4 −3 −2 −1 0

Log[f]

Log[

E(f

)]

Historical

NSRP simulated

1 week

2 hours

d) Vallombrosa

4

5

6

7

8

−7 −6 −5 −4 −3 −2 −1 0

Log[f ]

Log[

E(f

)]

Historical

NSRP simulated

1 week

6 hours

c) Livorno

4

5

6

7

8

−7 −6 −5 −4 −3 −2 −1 0Log[f ]

Log[

E(f

)]

1 week

4 hours

Historical

NSRP simulated

Figure 5. Historical and NSRP simulated power spectra for all stations, lumped process



Power spectrum

Figure 5 shows the observed and model simulated power spectra for all stations, lumped process, averagedover logarithmically spaced frequency intervals to reduce statistical scatter. For the spectra observed, a patternconsistent across the stations is evident, showing an extended power-law section with ˇ < 1 from a time-scale of 2–6 h up to around 1 week or more, up to 1 month in the case of Vallombrosa (Figure 5d). At largerscales the spectra flatten approaching the zero-ˇ characteristic of white noise. At scales below 2–6 h a secondpower-law section is exhibited with ˇ > 1, also agreeing well with general patterns described in the literature(see earlier section). The break at a few hours is sometimes pronounced, as is the case for Firenze and Livorno,whereas for Camaldoli it is only weakly suggested and the power-law section may in fact be extended down tothe smallest scales. In some previous investigations, this break has been hypothesized as spurious and relatedto an insufficient intensity resolution of the measuring device (e.g. Fraedrich and Larnder, 1993; Olsson,1995). As noted by Harris et al. (1997), the conditional mean is the parameter relevant to instrument-relatedbreaks; the larger the mean, the greater the accuracy. This does not seem to explain the present breaks because,for example, the value at Camaldoli is similar to and even slightly lower than Firenze (see Table II), whichshould imply a clearer break at the former station.

The model-simulated spectra are overall similar to those observed, with three differences. Figure 5 shows(i) the absence of a pronounced break at a time-scale of a few hours, (ii) a weak oscillation within thepower-law section, similar in structure for all stations, and (iii) an upward displacement as compared withthe historical spectra. The first difference is expected owing to the expected approximate power-law shape ofthe spectra of rectangular pulses models below some upper limit (e.g. Bo et al., 1994), corresponding hereto a few weeks in agreement with the observed spectra (see Figure 5). The oscillating behaviour presumablycan be attributed to weak characteristic scales associated with the model construction and parameter values,preventing a perfect power-law shape (similar features were indeed found by Bo et al., 1994). The overallincreased energy level of the simulated spectra probably is related to the overestimated zero-depth probabilityand conditional mean (Table II), indicating a temporal structure of many short but intense events.

The values of the power spectrum exponent ˇ are given in Table V. The section between the break at2–6 h and the start of the transition to ˇ ³ 0 at around 1 week found for the historical data has been usedin the estimation of ˇ (for consistency, the same range also was used for the simulated data from the samestation). The values for the historical data vary between 0Ð36 and 0Ð57, i.e. clearly below 1, in agreement withliterature values for the same range of frequencies (see earlier section). The highest value is at Camaldoli,highlighting the more correlated nature of the temporal rainfall structure in this region as compared to theothers, particularly Livorno. The interstation variation of ˇ for the model-simulated spectra is similar to theobserved, but the value generally is c. 0Ð04 too high, indicating a somewhat too smooth representation ofrainfall in the NSRP model. This result is expected considering the simplification of rainfall events intoclusters of rectangular pulses on which the model is based.

A picture consistent with these findings also is offered by the analysis of the seasonal process spec-tra, which generally agree well with the pattern displayed in Figure 5: a scaling behaviour up to1–2 weeks, weakly oscillating for the model-simulated data and with a weak break at a few hours

Table V. Power spectrum exponent ˇ for the historical (Hist) and simulated (NSRP) data

Lumped Winter Spring Summer Autumn

Hist NSRP Hist NSRP Hist NSRP Hist NSRP Hist NSRP

Camaldoli 0Ð57 0Ð63 0Ð68 0Ð74 0Ð56 0Ð58 0Ð315 0Ð35 0Ð67 0Ð74Firenze 0Ð49 0Ð47 0Ð69 0Ð79 0Ð50 0Ð49 0Ð28 0Ð285 0Ð65 0Ð60Livorno 0Ð36 0Ð38 0Ð55 0Ð57 0Ð34 0Ð36 0Ð24 0Ð31 0Ð42 0Ð42Vallombrosa 0Ð46 0Ð50 0Ð54 0Ð65 0Ð40 0Ð46 0Ð28 0Ð30 0Ð54 0Ð535



for the historical data. Table V shows the seasonal values of the exponent ˇ, calculated using fre-quency ranges determined similar to the lumped process. A seasonal variation of ˇ is shown, withthe lowest value in summer and the highest value, about twice the summer value, during the winterhalf-year. This pattern corresponds well to the climatology, characterized by long, frontal rains duringwinter producing a smoother, more correlated structure than the convective summer showers. The ˇ val-ues from model-simulated data reproduce well the observed seasonal variation, but are on average c. 0Ð04too high.

Statistical moments

Total process. For historical data, lumped process, moment scaling is well represented for all stationsin the range 20 min up to 1–2 weeks (the scaling regime), and a typical example is shown in Figure 6,station Vallombrosa. Slight deviations from perfect scaling sometimes occur for the smallest time-scales(weak convexity for q < 1 and weak concavity for q > 1), in agreement with some previous investigations ofrainfall time-series (e.g. Olsson, 1995; de Lima and Grasman, 1999). This possibly is related to the generalinability of rainfall gauges to fully resolve the very lowest and highest intensities; effectively the former(emphasized when q < 1) becomes overestimated and the latter (q > 1) underestimated. The accuracy of alinear approximation may be quantified by R2 of moment-fitting straight regression lines in the scaling regime.The values typically are very close to 1, but even a reduction of the order of 0Ð005 implies a clearly visibleincrease in the curvature. For the historical data, R2 ³ 0Ð994 for 0 q 4 with little difference betweenstations.

The above characteristics are overall well reproduced by the moment curves of the NSRP simu-lated data, as illustrated by Figure 6. Two differences are, however, evident. First, the convexity atsmall scales of the historical curve corresponding to q D 0 is not present in the simulated data. Thissuggests that model calibration is little influenced by low-intensity inaccuracies in the historical data.Second, in the scaling regime moments of order q > 1 are overestimated consistently in the simu-lated data, the only exceptions being for the highest order moments at the smallest scales. This isin line with the systematically overestimated conditional mean (dry periods do not contribute to themoments) in the simulated data, as illustrated already by Table II. It may be noted that the resultsare in striking contrast to the results of Onof et al. (1996), who found a pronounced underestimation

−10

−5

0

5

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15

20

25

0 5 10 15 20 25

Log λ

Log

<XλT

q >

Historical

NSRP simulated

1 week

Figure 6. Historical and NSRP simulated moment scaling curves for Vallombrosa, lumped process, moment orders 0 q 4 (bottom totop)



a) Winter

−10

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20

30

0 5 10 15 20

Log λ

Historical

NSRP simulated

1 week

Log

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q>

Log λ

b) Summer

−10

0

10

20

30

0 5 10 15 20

Historical

NSRP simulated

1 week

Log

<XλT

q>

Figure 7. Historical and NSRP simulated moment scaling curves for Camaldoli, winter (a) and summer (b), moment orders 0 q 4(bottom to top)

of high-order moments by the rectangular pulses model. The absence of small-scale curvature in thesimulated data is reflected in a higher R2 of the regression, being on average 0Ð997 in the scalingregime.

The seasonal process moment curves exhibit some notable differences as compared with the lumped processcurves described above. For winter, the curvature of the moment curves of orders q 6D 1 in the scaling regime ismore pronounced (Figure 7a; R2 ³ 0Ð990), whereas for summer nearly perfect scaling is obtained (Figure 7b;R2 ³ 0Ð995). The spring curves generally resemble summer, and the autumn curves winter. We note that thecurvature thus increases with decreasing relative measurement accuracy, represented by the conditional mean(Table II). The seasonal variation in moment curve shape agrees well with the findings of Onof et al. (1996)for English data.

The agreement between historical and simulated curves varies with season, winter and summer beingextreme cases with spring and autumn in between. For winter, the simulated curves exhibit a pronouncedconcavity for all q 6D 1, whereas for summer near-perfect scaling is obtained, as shown for Camaldoliin Figure 7a and b. The model thus manages to reproduce the qualitative differences in moment scalingwith season in the historical data for positive moments. This also is evident from the seasonal varia-tion of R2 averaged over all stations, which equals 0Ð989 in winter, 0Ð996 in spring, 0Ð999 in summerand 0Ð991 in autumn. For q < 1, there is a distinct underestimation of the simulated moments at shorttime-scales. This probably is a combined effect of low-intensity inaccuracies in the historical data andthe imperfect representation of rainfall intensities in the NSRP model for short temporal aggregationlevels mentioned earlier. In winter, positive moments in the scaling regime are strongly overestimatedby the model (Figure 7a). As for the lumped process, this possibly could be explained by the overes-timated conditional mean (Table II). However, although the overestimation of conditional mean is evenmore significant for summer, the moments are far less overestimated and even in some cases underesti-mated (Figure 7b). Besides the above-mentioned imprecision of the model at short time-scales, this canbe explained by considering the reproduction of extreme values. In summer, the overestimated condi-tional mean is compensated for by an underestimation of the very largest values (Table II), the influenceof which increases with increasing moment order. In winter, when moments are dominated far less bya few extreme values, the underestimation does not have the compensatory effect observed for sum-mer data.



−1

0

1

2

3

0 1 2 3 4q

Kq

CamaldoliFirenzeLivornoVallombrosa

Figure 8. Historical Kq functions for all stations, lumped process

Moving over to the characteristic Kq function, results from the historical data, lumped process, are shownin Figure 8. All functions are clearly convex, as evidenced by the values of ϕ4 clearly exceeding 1 (Table VI),which implies multiscaling, with small differences between stations. From somewhere in the range 2 < q < 3the functions become linear for higher-order moments, in good agreement with the values of qcr found inthe PDF analysis (Table IV). The intermittency parameter C1 is shown to be of the order 0Ð4–0Ð5, withthe highest value for Livorno and the lowest for Camaldoli, as reported by Table VII. This could suggest astructured dependence on the distance to the sea and related effects of the rainfall regime, such as orography.The values of C1 agree well with previous investigations of continuous rainfall time-series in southern Europe(e.g. Hubert et al., 1993; de Lima and Grasman, 1999).

The Kq functions from the simulated data match well the location of the observed functions, but the formerare markedly less curved, as seen in Figure 9a, which shows the functions for Livorno. The simulated Kq

function actually is close to linear, indicating a simple scaling behaviour. As shown in Figure 9b, however, thesimple scaling function (C1(q � 1)) is outside the 95% confidence interval of Kq and thus the simulated dataalso must be regarded as multifractal. This is the case for all stations, lumped process. The underestimateddegree of multiscaling is illustrated by the curvature parameter ϕ4, being on average 0Ð35 lower for thesimulated data, as reported by Table VI. The variation of C1 between stations in the simulated data is identicalto the historical data, but the simulated values are constantly c. 0Ð1 too large (see Table VII). This points to anexcessive intermittency (e.g. Davis et al., 1994; Harris et al., 1996), which is in line with the overestimatedzero-depth probability as already discussed with reference to values reported by Table II.

Table VI. Degree of multiscaling, as expressed by ϕ4, for the historical (Hist) and simulated (NSRP) data. The suffix ‘m’denotes multiscaling and ‘s’ that simple scaling could not be rejected

Lumped Winter Spring Summer Autumn Maximum

Hist NSRP Hist NSRP Hist NSRP Hist NSRP Hist NSRP Hist NSRP

Camaldoli 1Ð74 m 1Ð16 m 1Ð31 m 1Ð07 s 1Ð35 m 1Ð09 m 1Ð55 m 1Ð11 m 1Ð64 m 1Ð07 s 1Ð13 m 1Ð01 sFirenze 1Ð31 m 1Ð23 m 1Ð20 m 1Ð09 s 1Ð20 m 1Ð16 m 1Ð22 m 1Ð07 m 1Ð23 m 1Ð08 s 1Ð00 s 1Ð04 sLivorno 1Ð50 m 1Ð16 m 1Ð26 m 1Ð08 s 1Ð46 m 1Ð14 m 1Ð20 m 1Ð08 m 1Ð39 m 1Ð05 s 1Ð07 m 1Ð01 sVallombrosa 1Ð49 m 1Ð13 m 1Ð37 m 1Ð04 s 1Ð52 m 1Ð14 m 1Ð24 m 1Ð08 m 1Ð60 m 1Ð13 m 1Ð05 m 1Ð01 s



Table VII. Moment scaling exponents C1 (total process, lumped and seasonal) and ˛1 (annual maximum process), for thehistorical (Hist) and simulated (NSRP) data

Lumped Winter Spring Summer Autumn Maximum

Hist NSRP Hist NSRP Hist NSRP Hist NSRP Hist NSRP Hist NSRP

Camaldoli 0Ð39 0Ð48 0Ð34 0Ð42 0Ð42 0Ð50 0Ð52 0Ð625 0Ð34 0Ð44 0Ð665 0Ð63Firenze 0Ð50 0Ð57 0Ð44 0Ð50 0Ð51 0Ð59 0Ð64 0Ð71 0Ð445 0Ð525 0Ð74 0Ð75Livorno 0Ð52 0Ð62 0Ð46 0Ð53 0Ð54 0Ð66 0Ð64 0Ð71 0Ð50 0Ð61 0Ð75 0Ð77Vallombrosa 0Ð43 0Ð53 0Ð375 0Ð46 0Ð45 0Ð55 0Ð56 0Ð65 0Ð40 0Ð52 0Ð69 0Ð665

a)−1

0

1

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3

0 1 2 3 4q

Kq

Historical

NSRP simulated

0 1 2 3 4q

Kq

b)−1

0

1

2

3

NSRP simulated

95% confidence interval

Simple Scaling

Figure 9. Historical and NSRP simulated Kq functions for Livorno, lumped process (a) and assessment of multiscaling by comparing the95% confidence interval with the simple scaling function C1(q � 1) (b)

From the seasonal analysis we find that the summer Kq function defines, in general, an upper limit (forq > 1) and the winter function a lower limit, with the remaining curves in between, as shown by Figure 10displaying the seasonal variation for the station Camaldoli. All historical seasonal functions are multiscalingaccording to the 95% confidence interval criterion. The degree of multiscaling, as expressed by ϕ4, does notshow any unambiguous seasonal pattern (Table VI), which contrasts with the higher degree of multiscalingduring summer found by Onof et al. (1996) for English rainfall. It is interesting to note that the lumpedprocess ϕ4 generally is similar to, or sometimes even larger than, the maximum seasonal ϕ4. Thus, the‘most multiscaling’ season fixes the character of the lumped process, as argued by, for example, Harris et al.(1997). The parameter C1, however, exhibits a clear and consistent seasonal pattern, with the highest value forsummer and the lowest for winter (Table VII). This is not surprising, being in agreement with the higher levelof intermittency resulting from short-term summer showers than from the lengthy frontal passages prevailingin winter.

Generally, the agreement between historical and simulated seasonal process Kq functions is similar tothe case of lumped process functions, i.e. accurate location but too weakly multiscaling, as evidenced by theunderestimated ϕ4. Table VI shows in fact that in autumn and winter the hypothesis of simple scaling generallycould not be rejected for the simulated data, whereas in spring and summer the model qualitatively reproducedthe multiscaling behaviour of the historical data. The seasonal variation of C1 in the historical data is exactlyreproduced by the model, but as for the lumped process the simulated values are c. 0Ð1 larger (Table VII).



−1

0

1

2

3

0 1 2 3 4q

Kq

TotalWinterSpringSummerAutumn

Figure 10. Historical seasonal Kq functions for Camaldoli

Annual maximum process. The annual maxima of all the historical data analysed exhibit nearly perfectscaling of moments in the range 20 min to 1 week, as exemplified by the curves for Firenze in Figure 11a.For all stations, the mean R2 of the fitted linear regression model estimated for the range 0 q 4 exceeds0Ð997. As expected from the tendency of the model to underestimate extreme values shown in Table II, thesimulated moment curves are located below the historical curves, as shown in Figure 11b displaying the caseof Livorno, q D 2. The moment scaling, however, also is well represented by the simulated data, even thoughthe station mean R2 is systematically slightly lower than the historical value.

As in the Kq functions for the total rainfall process, the historical maximum process ˛q functions generallyare weakly convex. Figure 12a shows the historical function of Camaldoli, which is the most curved, as

a)−10

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0 1 2 3 4 5 6 7 8 9

Log λ

q=0

q=1

q=2

q=3

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<XλT

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Log λ

b)−6

−4

−2

0

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4

6

8

10

Log

<XλT

q>

Historical

NSRP simulated

Figure 11. Historical moment scaling curves for Firenze, annual maximum process (a) and comparison between historical and NSRPsimulated curves for Livorno q=2 (b)



0

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4

0 1 2 3 4q

α q

a)

Historical


Simple Scaling

0 1 2 3 4α q

0

1

2

3

4

q

NSRP simulated


Simple Scaling

b)

Figure 12. Historical (a) and NSRP simulated (b) ˛q function for Camaldoli, annual maximum process, and assessment of multiscaling bycomparing the 95% confidence interval with the simple scaling function ˛1q

0.30

0.40

0.50

0.60

0.70

0.60 0.65 0.70 0.75 0.80

C1

Historical

NSRP simulated

α1

Figure 13. Relationship between C1 and ˛1 for the four stations, historical and NSRP simulated data. Lines were fitted by regression

indicated by the largest value of ϕ4 (see Table VI) and simple scaling is clearly rejected. Also Livornoand Vallombrosa are (weakly) multiscaling, whereas for Firenze simple scaling could not be rejected(Table VI). One can note that the difference in the scaling character of the maximum process betweenstations is qualitatively identical to the total (lumped) process; Camaldoli is the ‘most’ and Firenze the‘least’ multiscaling. In other words, the scaling of the total process appears in a weaker version for themaximum process. As reported by Table VI, the maximum process ϕ4 is for all stations c. 0Ð45 lower thanthe corresponding value for the total (lumped) process. In Table VII, the maximum process moment scalingexponents ˛1 are given. The value is c. 0Ð7 throughout, which is similar to results obtained for rainfall datafrom Australia and South Africa (Menabde et al., 1999). As shown in Figure 13, ˛1 is linearly correlatedremarkably well to the total process exponent, C1, from the same station, emphasizing the link between thescaling of the total and annual maximum rainfall process, respectively.



The model-simulated ˛q functions are all visually similar to the historical functions, but generally witha slightly underestimated degree of multifractality (Table VI). For one station, Firenze, the scaling type ofthe historical data was reproduced correctly, whereas for the other simple scaling of the simulated data issuggested, contrary to the observed multiscaling (see Figure 12b). However, as the multiscaling character ofhistorical data is constantly weak, the simulated simple scaling behaviour provides a reasonable approximation.The values of ϕ4 are only slightly underestimated (Table VI). Concerning the exponent ˛1, the simulated valuesare all close to the historical (Table VII) and the linear relationship with C1 found for the historical data isreproduced rather well by the model (Figure 13). Overall, the scaling nature of the historical annual maximumprocess is replicated very well by the model, despite the underestimation of the actual maxima apparent fromTable II.

SUMMARY AND CONCLUSIONS

An extensive and comprehensive analysis was carried out to evaluate the capability of a popular stochas-tic model of temporal rainfall, the Neyman–Scott rectangular pulses (NSRP) model, to reproduce thescaling behaviour of 20-min historical data sets to which it was fitted. Concerning the empirical proba-bility distribution function (PDF), this is by construction exponential in the model, which therefore failedto reproduce the hyperbolic PDFs characterizing the historical data at all seasons except winter. Model-simulated power spectra reproduced well the power-law shape of the historical spectra, but the breakat a few hours suggested in the latter was not present in the model spectra. Variations of the powerspectrum exponent ˇ in the historical data were reproduced by the model, but the values were slightlyoverestimated, indicating an overly smooth process. The scaling behaviour of statistical moments foundbetween 20 min and 1–2 weeks in the historical data for both the total and the annual maximum processwas well reproduced by the model, as were seasonal variations. Considering the total process, for springand summer, as well as for the lumped process, the historical multiscaling was qualitatively reproducedby the model, although of a weaker character. For autumn and winter, also multiscaling in the histor-ical data, the hypothesis of simple scaling, however, could not be rejected for the simulated data. Themodel reproduced the historical variations of the exponent C1, but somewhat overestimated its value, indi-cating an overly intermittent process. Considering the maximum process, generally the model generatedsimple scaling contrary to the historical (weak) multiscaling, but the exponent ˛1 was reproduced accu-rately.

The remarkably well reproduced scaling of power spectra and statistical moments elucidates the previouslyfound ability of the NSRP model to represent rainfall over a range of aggregation levels. Another noteworthymodel feature was the ability of producing both a simple scaling and a multiscaling behaviour. Themain limitation of the model is the inability of accommodating a hyperbolic distribution of the rainfallintensities, which can lead to an underestimation of extreme values. Some authors (e.g. Burlando andRosso, 1993), however, have noticed such problems to occur essentially for durations smaller than the meanlifetime of a cell, reflecting the above-mentioned structural problems of the model. The underestimation ofextremes is in turn probably causing the constantly too weak degree of multiscaling in the simulated data,sometimes making the type of scaling qualitatively different from the historical data. One could speculateabout the possibility to include in the model hyperbolically distributed rainfall intensities, which would,however, substantially increase the complexity of the model and possibly lead to overparameterization withaccompanying calibration difficulties. Given the proven satisfactory behaviour of the model for temporalranges of technical interest (see, e.g. Cowpertwait, 1991; Burlando and Rosso, 1993), it appears practicable tofirst investigate systematically the degree to which scaling properties are reproduced for other climates beforestarting to modify the model.

In addition to the question of scaling in the model-simulated data, the analysis of historical data offeredthe opportunity to shed some light on two unsettled and sometimes disputed issues. First, the shape of



the empirical PDFs estimated from data indicates that temporal rainfall is not hyperbolic or exponential,but that the rainfall-generating mechanisms may produce both types of behaviour, as well as intermediatevariants. Generally, a hyperbolic behaviour seemed associated with rainfall triggered by convective processes,whereas for frontal rainfall exponentiality was suggested. This makes great demands upon rainfall models,which should be flexible enough to allow both distribution types. A potentially successful approach wouldbe to represent ‘pure convective’ and ‘pure frontal’ rainfall by two different stochastic processes, with thefinal time-series being a mixture of the two. Models that can account for different cell types are proposedin the literature (e.g. Cowpertwait, 1994), but calibration difficulties, for example, related to the separationof convective and frontal rainfall in observed time-series (see, e.g. Linderson et al., 1993), may limit anyexpected improvement in performance and thus the practical applicability. The second issue concerns a scalingbreak at a time-scale of a few hours. The present analyses, particularly the spectral, generally supportedthe presence of such a break, although instrumental effects could not be ruled out as causing the break.If real, the break would complicate the use of scaling-based models for temporal rainfall, although thedistribution of events and their internal structure, could in principle be represented by two separate scalingmodels.

The analyses of seasonal time-series showed that the scaling behaviour and parameters well reflect wellthe physical properties of the dominant rainfall-generating processes, generally supporting previous similarfindings (e.g. Over and Gupta, 1994; Harris et al., 1996; Svensson et al., 1996). The clear seasonal variation ofscaling exponents also underlined the necessity of taking seasonality into account in scaling-based modellingapproaches.

A new and potentially significant finding in the historical data is the systematic relationship between themoment scaling behaviour of the total and the annual maximum process. If confirmed by further analyses,such a relationship would in principle allow the estimation of extreme values for stations with an insufficientrecord length for analyses of annual maxima. From moment scaling analyses of the total rainfall process,the scaling behaviour of annual maxima could be estimated and then, for example, applied to derivedepth–duration–frequency curves by the method proposed by Burlando and Rosso (1996).

Closing on a general note, the present results add to previous findings of scaling and fractal-like features inoutput from general models not explicitly incorporating such features (e.g. Crawford et al., 1998). It appearsthat well-designed statistical models based on a sound geometrical representation of the actual physical processcan ‘unintentionally’ reproduce general scaling properties of the latter. Thus, rather than simply disqualifyingapproaches not based explicitly on scaling (as sometimes done in the non-linear community), they should firstbe tested for approximate reproduction of scaling. Further, coupling conventional and scaling-based methodsis potentially fruitful. In the present context, this could mean, for example using the NSRP model down to ascale of typical event durations, and using a scaling model to disaggregate the actual storm profile (see also,e.g. Koutsoyiannis and Foufoula-Georgiou, 1993). Such experiments are underway.

ACKNOWLEDGEMENTS

JO gratefully acknowledges funding from Axel and Margaret Ax:son Johnsons Foundation during theinitialization of this study, and the European Commission S&T Fellowship Programme in Japan during thefinalization. The authors are grateful to Renzo Rosso (Politecnico di Milano, Italy) for his support.

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reproduction of temporal scaling by a rectangular pulses rainfall model

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