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Point process models of rainfall: Development of pulses within cells

Paul Cowpertwait

1

, Valerie Isham

2

, & Christian Onof

3

1 Institute of Information & Mathematical Sciences Massey University

Auckland, New Zealand

2 Department of Statistical Science University College

London, UK

3 Department of Civil Engineering Imperial College of Science and Technology

London, UK

Abstract

A conceptual stochastic model of rainfall is proposed in which storm origins occur in a Poisson process, where each storm origin generates a random number of raincells during the storm lifetime and each raincell has a random lifetime during which random instantaneous depths (or ’pulses’) occur in a Poisson process. A key motivation behind the model formulation is to account for the variability in rainfall data over small (e.g. five-minutes) and larger time intervals. Time series properties are derived to enable the model to be fitted to rainfall data. These include moments up to third order, the probability that an interval is dry, and the autocovariance function. To allow for distinct storm types (e.g. convective and statiform) more than one process may be superposed. Using the derived properties, a model consisting of two storm types is fitted to sixty years of five-minute rainfall data taken from a site near Wellington, New Zealand, using sample estimates taken at five-minute, one-hour, six-hours, and the daily levels of aggregation. The model is found to fit moments of the depth distribution up to third order very well over these time intervals. Using the fitted model, five minute series are simulated, and annual maxima extracted and compared with equivalent values taken from the historical record. At the five minute aggregation level, the simulated values underestimate the equivalent historical values. A good fit in the extremes is found at the 1h and 24h levels of aggregation.

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1 Introduction

In the literature on stochastic models of rainfall based on Neyman-Scott or Bartlett-Lewis processes (Rodriguez-Iturbe et al. (1987)), it is usual to find rainfall intensity treated as a constant variable throughout the lifetime of a rain cell (Cowpertwait (1994); Verhoest et al. (1997); Koutsoyiannis and Onof (2001)). Whilst such a model formulation has provided a good fit to rainfall series sampled at one hour time intervals, it has usually proved necessary to disaggregate the series when data at a finer resolution are required (Cowpertwait et al. (1996)).

In this paper, the model formulation is extended to produce more realistic rainfall profiles and also remove the need for disaggregation. We use a Bartlett-Lewis occurrence process of cell starting times. However, the methodology could be applied to the Neyman-Scott model, or to mixed processes.

2 Model Formulation

Storm origins{Ti}arrive in a Poisson process with rateλ. Each storm lasts for a duration {Di}, where Di ∼exp(γ). A Poisson process with rate β of starting times {Tij} for cells occurs during the storm duration, i.e. Ti < Tij < Ti+Di;{Tij}is aBartlett-Lewis process.

Each cell has durationLij ∼exp(η), provided this is less than the storm duration. During each time interval (Tij, Tij+Lij) cell pulses occur at times{Tijk}in a Poisson process with rate ξ, the process of pulses terminating when the cell or storm terminates. Associated with each pulse is an independent randomX representing the rainfall depth generated by the pulse.

The pulse arrival times{Tijk}form a one-dimensional point process, the process of rainfall depths forming a marked process (Cox and Isham (1980)). The rainfall at time t is X(t)dN(t), where dN(t) is unity if there exists a pulse at time t or zero otherwise, and X(t) is the magnitude of the pulse at time t. (dN(t) is the limiting form of δN(t), the number of points in a time interval (t, t+δ).) The expected number of pulses per storm isµp =βξ{γ(γ+η)}−1, and the expected total rainfall depth per storm is µpµX. In the fitting procedure (Section 7),X will be an independent exponential random variable with parameterµX =θ.

3 Covariance Structure of Pulse Arrival Times

The covariance structure of a point process is used in the derivation of a number of key mathematical properties. In Section 4, we shall use the covariance of the pulse arrivals

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to derive model properties for aggregated series, which is the usual form rainfall data are stored.

Two points (pulses) can occur at timesx1 and x2 (x2 =x1+u) when: (i) they come from the same cluster; (ii) they come from the same storm, but different clusters; (iii) they come from different storms. Therefore,

P{δN(x1) =δN(x2) = 1}δ−1 = (λµp)22 Z

t=0

λe−γ(t+u) Z t

s=0

e−η(s+u)β ds dt +ξ2

Z

t=0

λe−γ(t+u) Z t

s=0

e−ηsβ ds Z t+u

v=0

e−ηvβdv dt (1) After a little algebra, the following is obtained:

Cov{δN(x), δN(x+u)}δ−1 = λξ2β{γ(γ+η)}−1e−(γ+η)u +λβ2ξ2{ηγ(γ+η)}−1e−γu

−λβ2ξ2{η(2η+γ)(γ+η)}−1e−(γ+η)u

= Ae−γu+Be−(γ+η)u (2)

4 Second-Order Properties

LetYi(h) be the total rainfall in a time interval [(i−1)h, ih]. Then, Yi(h) =

Z ih

(i−1)h

X(t)dN(t) (3)

where X(t) is the pulse depth at time t and dN(t) is unity if there is a pulse at time t (zero otherwise). Hence,

Eh Yi(h)i

= Z h

0

E[X(t)] E[dN(t)]

= λµxµph (4)

Using (2) above, the aggregated variance and autocovariance for the Bartlett-Lewis Pulse (BLP) model follow as:

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V ar h

Yi(h) i

= Z h

0

Z h

0

Cov[X(s)dN(s), X(t)dN(t)]

= Z h

0

E(X2)E[dN(t)] + Z Z

s6=t

µ2x Cov[dN(s), dN(t)]

= λµpE(X2)h+ 2µ2x−2(γh+e−γh−1) +2µ2xB(γ+η)−2

(γ+η)h+e−(γ+η)h−1

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Covh

Yi(h), Yi+k(h)i

= Cov

"

Z h

0

X(s)dN(s),

Z (k+1)h

kh

X(t) dN(t)

#

=

Z (k+1)h

kh

Z h

0

Cov[X(s) dN(s), X(t)dN(t)]

=

Z (k+1)h

kh

Z h

0

µ2xCov[dN(s), dN(t)]

= µ2x−2e−γ(k−1)h(e−γh−1)22xB(γ+η)−2e−(γ+η)(k−1)h

(e−(γ+η)h−1)2 (6) To obtain more generality, the depths of pulses within the same cell can be treated as correlated with joint expectationE(X1X2), in which case the properties follow as:

V arh Yi(h)i

= λµpE(X2)h+ 2µ2x−2(γh+e−γh−1)

+2E(X1X2)B1(γ+η)−2[(γ +η)h+e−(γ+η)h−1]

−2µ2xB2(γ+η)−2[(γ+η)h+e−(γ+η)h−1] (7)

Covh

Yi(h), Yi+k(h)i

= µ2x−2e−γ(k−1)h(1−e−γh)2

+E(X1X2)B1(γ+η)−2e−(γ+η)(k−1)h

(1−e−(γ+η)h)2

−µ2xB2(γ+η)−2e−(γ+η)(k−1)h

(1−e−(γ+η)h)2 (8) whereB1 = etc.

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PD is given by:

−λ−1lnP(Y(h) = 0) = Z

0

dx[e−γx−f(h, x)−γ Z h

0

duf(u, x)]

+ Z h

0

dx[1−g(x)−γ Z x

0

dug(u)] (9)

where:

−lnf(u, x) = γ(x+u) + βξ

η(η+ξ)(1−e−ηx)(1−e−(η+ξ)u) + βuξ

η+ξ − βξ

(η+ξ)2(1−e−(η+ξ)u) (10) and,

−lng(u) = γu+ βuξ

η+ξ − βξ

(η+ξ)2(1−e−(η+ξ)u) (11)

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5 Third Moment

E((Y(h))3) λβξ3

= 6

(η+γ)2

E(X1X2X3)

γ(η+γ) + E(X1X2Xβ γ(η+γ)

1

η + λ

γ(η+γ) − γ η(2η+γ)

− µ3Xβ2 η(η+γ)(2η+γ)

1

η + λ

γ(η+γ) h− 2

(η+γ)+ 2e−(η+γ)h

(η+γ) +he−(η+γ)h

+ 6

(η+γ)(2η+γ)

− 2E(X1X2Xβ

η(η+γ)(2η+γ)+ µ3Xβ2

η2(2η+γ)(3η+γ)

×

h− 3η+ 2γ

(η+γ)(2η+γ)+ (2η+γ)e−(η+γ)h

η(η+γ) − (η+γ)e−(2η+γ)h η(2η+γ)

+ 6µ3Xβ2 ηγ3(η+γ)

1

η + λ

γ(η+γ) h− 2

γ + 2e−γh

γ +he−γh

+ 6

γ(η+γ)

2E(X1X2Xβ

ηγ(η+γ) − µ3Xβ2

η2(η+γ)(2η+γ) h− η+ 2γ

γ(η+γ)+ (η+γ)e−γh

ηγ − γe−(η+γ)h η(η+γ)

+ 12

η+γ

E(X1X2Xλβ

γ2(η+γ)2 − µ3Xλβ2 ηγ(η+γ)2(2η+γ)

1

2h2− h

η+γ + 1

(η+γ)2 − e−(η+γ)h (η+γ)2

+ 12µ3Xλβ2 ηγ3(η+γ)2

1

2h2− h γ + 1

γ2 − e−γh γ2

+ µ3Xλ2β2h3 γ3(η+γ)3 + 6E(X12X2)

ξγ(η+γ)2

h− 1

η+γ + e−(η+γ)h η+γ

+ 6E(X2Xβ ξηγ2(η+γ)

h− 1

γ +e−γh

γ − γ2

(η+γ)(2η+γ)

h− 1

η+γ +e−(η+γ)h η+γ

+ 3E(X2Xλβh2

ξγ2(η+γ)2 + E(X3)h ξ2γ(η+γ)

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6 Superposition of BLP Models

In previous work, it has been noted that distinct types of precipitation occur, for example convective and stratiform rain, which can be modelled using point processes that contain different cell types (e.g. Cowpertwait (1994)). Alternatively, the cell duration parameter (η) can be made random between storms, so that different distributions of cell dura- tion occur for different storms (Rodriguez-Iturbe et al. (1988)). Another approach uses superposed mixed independent processes of different storm types (Cowpertwait (2004)).

An advantage of the last approach is that properties from independent processes can be aggregated to give the properties of the superposed mixed process, thus making use of the derived properties of the independent processes. To illustrate, suppose there are n independent BLP processes, denoted BLPi, with parameters λi, βi, ξi, γi, ηi, and θi (i = 1, . . . , n). If {Yij(h) : j = 1,2, . . .} is the aggregated rainfall series sampled over an interval of widthh due to aBLPi process, then the total rainfall in the jth time interval, due to the superposition of thenindependent BLP processes, is the sumYj(h) =Pn

i=0Yij(h). Properties (up to order 3) of the superposed process are obtained as the sum of the equivalent properties from each independent BLPi process, i.e. Eh

{Yj(h)−µj(h)}ki

= Pn

i=0Eh

{Yij(h)−µij(h)}ki

(k ≤3), where µj(h) =E[Yj(h)] and µij(h) = E[Yij(h)].

Superposing processes can provide considerable flexibility. However, this can be at the expense of introducing large numbers of parameters into the model. Consequently, the number of superposed processes is usually restricted apriori. In the fitting procedure that follows we taken = 2, which might provide a suitable approximation for the occurrence of convective and stratiform storms.

7 Fitted Model

One motivation for developing the BLP model is to achieve a good fit to series of rainfall depths sampled over a range of time scales, from fine resolutions, e.g. five-minutes, to higher aggregation levels, such as daily. In this section we aim to verify that the model can achieve a good fit to properties sampled over a range of scales, including moments up to third order.

The National Institute of Water and Atmospheric Research (NIWA) provided us with a 60-year record (1945-2004) of rainfall data recorded at a site in Kelburn (near Wellington, New Zealand). The data were available at five-minute time intervals, and were therefore suitable for assessing the fit of the model at fine resolutions.

A superposed model consisting of two independent BLP processes has a total of 12 pa- rameters: λi, βi, ξi, γi, ηi, and θi (i = 1,2). Some of these parameters can be treated as

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equal for different storm types, thus reducing the total number. One convenient choice is the mean pulse intensityθi, as this can be estimated directly from the mean rainfall (see (16) below). We therefore takeθi =θ (i= 1,2), resulting in an 11-parameter model. All the parameters, except θ, can be estimated from the dimensionless functions:

Coefficient of variation, ν(h) = E h

{Yj(h)−E(Yj(h))}2i12

/E(Yj(h)) (12) Autocorrelation (lag 1), ρ(h) = Cor(Yj(h), Yj+1(h)) (13)

Coefficient of skewness, κ(h) = Eh

{Yj(h)−E(Yj(h))}3i /Eh

{Yj(h)−E(Yj(h))}2i32 (14)

As the above functions contain 10 parameters, at least 10 sampled properties are needed to fit the model to data. We chose the following twelve properties: the coefficient of variation, lag 1 autocorrelation, and coefficient of skewness, each taken at aggregation levels 5-minutes, 1-hour, 6-hours, and 24-hours, to cover a range of scales over which the model parameters might be identified. The sample estimates of these properties, denoted ˆ

ν(h), ˆρ(h), and ˆκ(h) (for h = 1/12,1,6, and 24h), were calculated for each calendar month in the series, by pooling all available data for the month over the 60-year record for Kelburn. For each month j, the estimates ˆλij, ˆβij, ˆξij, ˆγij, ˆηij (i= 1,2;j = 1, . . . ,12) were obtained by minimising the sum of squares:

X

h=1/12,1,6,24

n

1− ν(h)ν(h)ˆ 2

+

1− ν(h)ˆν(h)2

+

1− ρ(h)ρ(h)ˆ 2

+

1−ρ(h)ρ(h)ˆ 2

+

1− ˆκ(h)κ(h)2

+

1−κ(h)κ(h)ˆ 2o

(15)

In (15), the model function is divided by the sample estimate to ensure that large sample values do not dominate the minimisation procedure, so that each sample value receives an equal weight. In addition, the sum contains two terms per model function, one which contains the model function divided by the sample value, and the other which contains the reciprocal term. This helps to ensure that the optimal solution is not biased from above or below the sample values, when an exact fit to the sample value is not obtained.

The sum of squared terms (15) was minimised using the simplex method (Nelder and Mead (1965)); the resulting parameter estimates are given in Table 1. In Table 1, the scale parameterθj for the jth month has been estimated using the sample mean:

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θˆj = x¯j

"

λˆ1,jβˆ1,jξˆ1,j

ˆ

γ1,j(ˆγ1,j+ ˆη1,j)+

λˆ2,jβˆ2,jξˆ2,j

ˆ

γ2,j(ˆγ2,j+ ˆη2,j)

#−1

(16)

where ¯xj is the estimated mean 1hrainfall for thejth month (j = 1, . . . ,12). The sample and fitted values are shown in Figures 1–4.

An almost exact fit is obtained over all time intervals to properties up to third order (Figures 1–4). A possible exception is the fit to the five-minute lag 1 autocorrelation shown in Figure 4(a). However, the actual differences in the correlations are of the order 0.01, which is unlikely to be of any practical consequence.

Comparing the parameter estimates for the different superposed processes, type 1 storms tend to be longer than type 2 storms (γ1,j < γ2,j), and contain longer less intense cells (η1,j < η2,j; ξ1,j < ξ2,j; Table 1). Broadly speaking, type 2 storms may be interpreted as rare heavy convective storms, whilst type 1 storms may correspond to prevailing frontal weather systems. The estimates for the pulse depths are very small, generally θ <ˆ 0.01mm, which suggests the pulses may possibly correspond to small-scale droplets of rain.

Some evidence of seasonal variation is present in the parameter estimates (Table 1). For example, over the summer months of January and February ˆλ takes a lower value when compared with the other months reflecting the lower occurrence rate of summer storms.

The other estimates show less seasonal variation, which could be due to irregular changes in the sample estimates between adjacent months, especially in the autocorrelation (Fig- ure 4).

8 Extreme Values

Before the design of a hydrological system is implemented, it is usually essential to quantify the performance of the system under extreme conditions. For certain systems, e.g. urban sewerage networks, this can be achieved using a stochastic model of rainfall, such as the model presented in this paper, for simulation of series that can be feed into a hydraulic flow simulation model. It is therefore useful to assess the performance of a stochastic model of rainfall with respect to extreme values. Furthermore, as extreme values are unusual they form a stringent test when assessing model performance, especially if they are not directly used in the fitting procedure. In this section, extreme values generated by the model are compared to those in the 60-year historical record from Kelburn.

Using the fitted model (Table 1), twenty samples of five minute series, each of sixty years duration, were simulated. For each sample, the annual maximum over a five minute

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interval was extracted, ordered, and plotted against the reduced Gumbel variate. The results, with the equivalent value from the sixty year historical record, are shown in Figure 5. In addition, each generated sample of five minute data was aggregated to the 1h and 24h levels, and the annual maxima extracted for the aggregated series (Figures 6 and 7).

At the higher levels of aggregation there is good agreement; the historical values generally fall within the range of simulated sample values (Figures 6 and 7). However, for the five minute aggregation level there is evidence of underestimation for return periods in excess of about 3 years (Figure 5). There are many possible reasons for this, one of which is that there is probably some bias in the sample estimate of skewness, which is likely to be more significant at smaller aggregation levels, such as five minutes, due to high autocorrelation (e.g. compare Figures 4a and 4d). This requires further detailed investigation, and shall be reported in due course, along with a more extensive empirical analysis.

9 Summary and Conclusions

A stochastic process of pulses has been introduced into the cell structure of the Bartlett- Lewis rainfall model (Rodriguez-Iturbe et al. (1987)). Properties up to third order were derived and used to fit the model to five-minute data. The fitted values corresponded almost exactly with the observed values, indicating that the model was able to fit data sampled over a range of time scales from five-minutes upwards. This suggests that the model has potential application in a number of areas, for example urban drainage catch- ment studies, which usually require five-minute rainfall series.

The simulated extreme values at the 1hand 24hlevels of aggregation showed good agree- ment to the equivalent values taken from the historical record. The five minute extremes from the historical series were underestimated for return periods greater than about 3 years. This could be due to bias in the estimation of skewness, which could be more significant at the five minute aggregation level, compared to the higher levels, because of the larger values of autocorrelation. A reparametrisation of the model, e.g. usual a longer tailed distribution for pulse depth, may also improve the fit of simulated extremes at the five minute level, whilst retaining the existing good fit at the higher levels. Some further research would be required to assess these ideas.

The analysis of extremes based on annual maxima is a stringent test of model performance, and it is possible that the existing model, without further modification, will be sufficient for many practical applications. This largely depends on the intended application. The model would require further tests, appropriate and specific to any intended application, before being applied in practice.

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Acknowledgements

The National Institute of Water and Atmospheric Research are gratefully acknowledged for providing the data.

References

Cowpertwait, P. (1994). A generalized point process model of rainfall. Proceedings of the Royal Society of London, Series A, 447:23–37.

Cowpertwait, P. (2004). Mixed rectangular pulses models of rainfall.Hydrology and Earth System Sciences, 8:993–1000.

Cowpertwait, P., O’Connell, P., Metcalfe, A., and Mawdsley, J. (1996). Stochastic point process modelling of rainfall: I. Single-site fitting and validation; II. Regionalisation and disaggregation. Journal of Hydrology, 175:17–65.

Cox, D. and Isham, V. (1980). Point Processes. Chapman & Hall.

Koutsoyiannis, D. and Onof, C. (2001). Rainfall disaggregation using adjusting procedures on a Poisson cluster model. J.Hydrol, 246:109–122.

Nelder, J. and Mead, R. (1965). A simplex algorithm for function minimization.Computer Journal, 7:308–313.

Rodriguez-Iturbe, I., Cox, D., and Isham, V. (1987). Some models for rainfall based on stochastic point processes. Proc. R. Soc. Lond. A, 410:269–288.

Rodriguez-Iturbe, I., Cox, D., and Isham, V. (1988). A point process model for rainfall:

further developments. Proc. R. Soc. Lond. A, 417:283–298.

Verhoest, N., Troch, P., and Troch, F. D. (1997). On the applicability of bartlett-lewis rectangular pulses models in the modelling of design storms at a point. J. Hydrol., 202.

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Table 1: Fitted Model: Parameter Estimates for Kelburn Series.

month j λˆ1,j βˆ1,j ξˆ1,j γˆ2,j ηˆ1,j λˆ2,j βˆ2,j ξˆ2,j γˆ2,j ηˆ2,j θˆj 1 0.00641 0.133 509 0.0401 0.733 0.000273 1.43 4620 0.178 7.96 0.00676 2 0.00791 0.548 347 0.0988 1.23 0.000721 3.92 1810 0.269 7.73 0.0076 3 0.0116 0.359 446 0.107 0.822 0.00104 3.02 3250 0.401 7.12 0.00531 4 0.0109 0.137 684 0.0491 0.641 0.00123 1.53 4980 0.172 5.98 0.00333 5 0.0149 0.0654 320 0.0346 0.347 0.0015 0.773 2840 0.132 3.81 0.00537 6 0.0168 0.051 116 0.0299 0.228 0.00161 0.488 1510 0.179 2.8 0.0127 7 0.0101 0.0831 814 0.0245 0.294 0.00295 2.39 6210 0.197 5.94 0.00144 8 0.0166 0.909 661 0.103 1.38 0.00153 3.60 4350 0.271 6.53 0.00206 9 0.0153 0.398 211 0.0911 0.907 0.00146 2.85 1200 0.388 5.58 0.00824 10 0.0151 0.38 200 0.0923 0.897 0.000305 2.11 1620 0.236 6.17 0.0111 11 0.0122 0.259 389 0.0676 0.823 0.00103 3.74 3590 0.482 8.44 0.00547 12 0.0145 0.744 370 0.169 1.43 0.000218 2.46 2880 0.158 6.32 0.00705

The units are h−1 for all estimates except θwhich has unitsmm.

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1 2 3 4 5 6 7 8 9 10 11 12

0.100.120.140.160.18

month

1h mean / mm

x x

x x

x x

x

x

x x

x

x

Figure 1: The mean 1hrainfall against month. The sample estimates are shown as crosses (×) and the fitted values as lines.

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1 2 3 4 5 6 7 8 9 10 11 12

5.05.56.06.57.07.58.0

month

5min CV

x x

x x

x

x x x

x x

x x

1 2 3 4 5 6 7 8 9 10 11 12

4.04.55.05.56.0

month

1h CV

x x

x

x

x

x x

x

x x

x x

(a) (b)

1 2 3 4 5 6 7 8 9 10 11 12

3.03.54.04.5

month

6h CV

x x

x

x

x

x x x

x x

x x

1 2 3 4 5 6 7 8 9 10 11 12

2.02.22.42.62.83.0

month

24h CV

x x

x x

x

x x x

x x

x x

(c) (d)

Figure 2: Coefficient of variation against month for aggregation levels: (a) five-minute;

(b) 1-hour; (c) 6-hour; and (d) 24-hour. The sample estimates are shown as crosses (×) and the fitted values as lines.

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1 2 3 4 5 6 7 8 9 10 11 12

121416182022

month

5min skewness

x

x x

x

x x

x x

x x

x

x

1 2 3 4 5 6 7 8 9 10 11 12

89101112

month

1h skewness

x x

x

x x

x

x x

x x

x x

(a) (b)

1 2 3 4 5 6 7 8 9 10 11 12

5678

month

6h skewness

x x

x

x

x x

x

x x x

x x

1 2 3 4 5 6 7 8 9 10 11 12

4567

month

24h skewness

x x

x x

x

x x

x x

x x

x

(c) (d)

Figure 3: Coefficient of skewness against month for aggregation levels: (a) five-minute;

(b) 1-hour; (c) 6-hour; and (d) 24-hour. The sample estimates are shown as crosses (×) and the fitted values as lines.

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1 2 3 4 5 6 7 8 9 10 11 12

0.810.820.830.840.850.86

month

5min autocorrelation

x x

x

x x

x

x x

x x

x x

1 2 3 4 5 6 7 8 9 10 11 12

0.600.620.640.66

month

1h autocorrelation

x x

x

x x

x x

x

x x

x x

(a) (b)

1 2 3 4 5 6 7 8 9 10 11 12

0.340.360.380.400.420.440.46

month

6h autocorrelation

x x

x x

x x

x x

x x

x x

1 2 3 4 5 6 7 8 9 10 11 12

0.150.200.25

month

24h autocorrelation

x

x

x x

x

x x

x x

x x

x

(c) (d)

Figure 4: Lag 1 autocorrelation against month for aggregation levels: (a) five-minute; (b) 1-hour; (c) 6-hour; and (d) 24-hour. The sample estimates are shown as crosses (×) and the fitted values as lines.

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−1 0 1 2 3 4 5

0246810

Reduced Variate

5min annual maximum rain / mm

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− −− −

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− − −

− −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− − −

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2 5 10 20 50 100

T

Figure 5: Ordered annual maxima taken over five minute time intervals plotted against the reduced Gumbel variate. The values from the historical series are shown as (), and are connected with lines. The simulated values as shown as bars (–). For the simulated series there are 20 samples, each of 60 years duration.

(18)

−1 0 1 2 3 4 5

01020304050

Reduced Variate

1h annual maximum rain / mm

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T

Figure 6: Ordered annual maxima taken over one hour time intervals plotted against the reduced Gumbel variate. The values from the historical series are shown as (), and are connected with lines. The simulated values as shown as bars (–). For the simulated series there are 20 samples, each of 60 years duration.

(19)

−1 0 1 2 3 4 5

050100150200

Reduced Variate

24h annual maximum rain / mm − −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− −

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T

Figure 7: Ordered annual maxima taken over daily time intervals plotted against the reduced Gumbel variate. The values from the historical series are shown as () and are connected with lines. The simulated values as shown as bars (–). For the simulated series there are 20 samples, each of 60 years duration.

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