Generating Synthetic Rainfall on Various Timescales - Daily,Monthly and Yearly

Piantadosi, J., J.W. Boland and P.G. Howlett

Centre for Industrial and Applied MathematicsUniversity of South Australia, Mawson Lakes Campus, Adelaide SA 5095, Australia.

E-Mail: julia.piantadosi@unisa.edu.au

Keywords: Daily, monthly, yearly rainfall, correlation, copulas

EXTENDED ABSTRACT

The main objective of this paper is to present a modelfor generating synthetic rainfall totals on varioustimescales to be applicable for a variety of uses. Manylarge-scale ecological and water resources modelsrequire daily, monthly and yearly rainfall data asinput to the model. As historical data provides onlyone realisation, synthetic generated rainfall totals areneeded to assess the impact of rainfall variability onwater resources systems (Srikanthan 2005). Thus ourpreferred model should simulate rainfall for yearly,monthly, and daily periods. We believe that for watersupply issues no higher resolution is needed althoughhigher resolution would be useful in models designedto measure the risk of local flooding. The criticalfactors are daily, monthly and yearly totals and daily,monthly and yearly variation.

A model for generating yearly totals will be describedusing traditional time series methods. This model,along with a similarly constructed daily generationmodel by, Piantadosi et al. (2007b), will be cascadedto start with a synthetic yearly total, then generatea synthetic sequence of monthly totals (throughselection from a large number of realisations) thatmatch the yearly total, and subsequently performa similar operation for sequences of daily totals tomatch the required monthly totals.

We present a new model for the generation ofsynthetic monthly rainfall data which we demonstratefor Parafield in Adelaide, South Australia. Therainfall for each month of the year is modelledas a non-negative random variable from a mixeddistribution with either a zero outcome or a strictlypositive outcome. We use maximum likelihoodto find parameters for both the probability of azero outcome and the gamma distribution that bestmatches the observed probability density for thestrictly positive outcomes. We describe a new modelthat generates correlated monthly rainfall totals usinga diagonal band copula with a single parameter togenerate lag-1 correlated random numbers. Ourmodel preserves the marginal monthly distributionsand hence also preserves the monthly and yearly

means. We show that for the particular exampleof Parafield the correlation between rainfall totalsfor successive months is not significant and so it isreasonable to assume independence. This is howevernot true for daily rainfall. The correlation betweenrainfall on successive days is certainly small but it isreasonable as suggested by Katz and Parlange (1998)to conclude that the assumption of independence isthe primary reason for the low simulated monthlystandard deviations. We describe a new modelthat generates correlated daily rainfall totals usinga diagonal band copula with a single parameter togenerate lag-1 correlated random numbers.

The City of Salisbury supplies recycled storm-water to local businesses on a commercial basisand it is important that they understand the fullimplications of the likely distribution of rainfall andthe consequent impact on their ability to managethe capture, treatment and supply to consumers ofrecycled water. Recent work by Piantadosi (2004)used Matrix Analytic Methods to model the capture,storage and practical management of urban storm-water with stochastic input and regular demand.Howlett et al. (2005) describe subsequent researchon the application of these methods to the optimalcontrol of storm-water management systems. Thepractical value of this work has been the developmentby Piantadosi (2004) of computer simulations thatmodel a sequence of small capture and storage damson a suburban storm-water course. Because the realsystems are complicated by non-essential physicalfeatures and fixed infrastructure the most effectiveway to study their behaviour is by constructingand running an accurate computer simulation drivenby synthetic daily rainfall data. To underline theimportance of the simulated input we note thatsuburban storage systems are relatively small andwill overflow during sustained wet periods and willempty during sustained dry periods. It is importantthat water managers understand the level of riskassociated with such occurrences. These risks aremost easily calculated using repeated simulationsdriven by a realistic stochastic rainfall generator.Historical records of existing rainfall sequences arethought to be not sufficiently comprehensive.

2910

1 INTRODUCTION

Many large-scale ecological and water resourcesmodels require daily, monthly and yearly rainfall dataas input to the model. As historical data providesonly one realisation, stochastically generated rainfalltotals are necessary to assess the impact of rainfallvariability on water resources systems. Syntheticrainfall data provides alternative realisations that areequally likely but have not necessarily occurred in thepast (Srikanthan 2005). As a result the modelling ofrainfall records has become well established over thepast thirty years. In particular the gamma distributionhas been used many times to model rainfall totalson wet days. Valuable general reviews on weathergenerators are published by Wilks and Wilby (1999)and Srikanthan and McMahon (2001).

The performance of our models is assessed bycomparing the average of each characteristic from allthe synthetic generations with that of the observedhistorical data. Different characteristics are importantfor different applications, for example, the syntheticdaily totals should match the observed marginal dailydistributions and other key medium term statisticssuch as the monthly means and standard deviations.Current models generally tend to perform better at thetime step on which they are based, for example, a dailysynthetic rainfall model will tend to simulate the dailycharacteristics better than the monthly and yearlycharacteristics, such as monthly standard deviation,(Chiew et al. 2005). To overcome this problemwe present a rainfall generator (rGen) model thatsimulates rainfall at the time step on which they arebased and then the models will be cascaded to startwith a synthetic yearly total, then generate a syntheticsequence of monthly totals (through selection froma large number of realisations) that match the yearlytotal, and subsequently perform a similar operationfor sequences of daily totals to match the requiredmonthly totals.

The City of Salisbury in South Australia hasconstructed a diverse network of local storm-waterstorage systems which are being used to restoredegraded wetlands and supplement existing watersupplies. Salisbury lies in suburban Adelaideapproximately 15 km north of the CBD. Adelaideis the capital city of South Australia. The Parafieldsystem modelled by Piantadosi (2004) is a typicalexample of a storm-water storage and recyclingsystem within the Salisbury precinct. Since thecatchment is relatively small and consists mostly ofpaved surfaces the response time for run-off fromlocal rainfall is at most a few hours. Thus themonthly intake of storm-water is very strongly linkedto the daily rainfall. Our work is part of a broadresearch program to develop mathematical modelsthat can be used to simulate the operation of such

systems. Synthetic data from a suitable daily rainfallmodel can be used to drive the simulated operation ofspecific storm-water systems and hence assist in thedevelopment of optimal management strategies andrisk assessment.

2 METHODS AND STUDY AREA

2.1 The rainfall generator model rGen

We present a rainfall generator (rGen) algorithm thatwill use yearly rainfall generated using the methods ofDick and Bowden (1973) and a new daily generationmodel by Piantadosi et al. (2007b) together with asimilar model to generate synthetic monthly totals.Figure 1 shows a flowchart of the rainfall generator(rGen) algorithm. The algorithm proceeds as follows.Firstly, we estimate the parameters µ1, σ2

1, µ2, σ2

2and

p of the compound normal distribution to generatea yearly rainfall total. The choice of distribution isexplained in Section 2.7. Secondly, a large numberof synthetic monthly totals are generated using a 2-parameter gamma distribution. The two parameters,α and β, used to describe the gamma distribution arefound using maximum likelihood estimation. Section2.3 describes a new model for generating monthlyrainfall totals. We select a sequence of monthlytotals that best match the generated yearly total.The monthly rainfall totals preserve the monthlycharacteristics and sum to obtain the yearly total.Finally, we generate sequences of synthetic dailyrainfall totals using a model developed by Piantadosiet al. (2007b). The rainfall for each day of theyear is modelled as a non-negative random variablefrom a mixed distribution with either a zero outcomeor a strictly positive outcome. We use maximumlikelihood to find parameters for both the probabilityof a zero outcome and the gamma distribution thatbest matches the observed probability density for thestrictly positive outcomes. We generate sequencesof daily rainfall totals to match the required monthlytotals. The synthetic daily totals preserve the dailycharacteristics and sum to obtain the monthly totals.

2.2 Data

Daily rainfall time series data from two locationsare used for this case study. We have available 90years of official daily rainfall records supplied by theAustralian Bureau of Meteorology for Parafield, inSouth Australia, during the period 1901-1990. Therainfall records are measured in millimetres (mm). Weuse 30 years of daily rainfall records obtained fromthe Malaysian Meteorological Department for Mesing(1971-2000) to test our models. A summary of theavailable rainfall data is given in Table 1. The Tableshows that the mean annual rainfall in Parafield is

2911

Yearly rainfall model:compound normal distributionparameters: µ1, σ2

1, µ2, σ2

2and p

Output: yearly rainfall totals(preserves yearly statistics)

?

Monthly rainfall model:Gamma distributionparameters: α and β

Output: monthly rainfall totals(preserves monthly statistics & from which we select

a sequence matching the desired yearly total)

?

Daily rainfall model:

totals: non-negative random variable from amixed distribution

zero outcomerandom # ≤ Prob zero

total = 0

non-zero outcomerandom # > Prob zero

total: Gamma dist.parameters: α and β

Output: daily rainfall totals(preserves daily statistics & from which we selecta sequence matching the desired monthly totals)

Figure 1. Flowchart of the rainfall generator rGenalgorithm.

Table 1. Summary of daily rainfall data.

Location Record length (years) Mean annualrainfall (mm)

Parafield 90 (1901-1990) 495.46Mesing 30 (1971-2000) 2674.94

approximately 495 mm compared to the mean annualrainfall for Mesing, approximately 2675 mm. Figure2 shows a time series plot of the yearly totals atParafield from 1901-1990. The plot shows a veryslight decrease in yearly rainfall totals but this is notsignificant (with a P-value 0.793) and will not beconsidered in our models. Figure 3 shows the monthlyaverage rainfall for Parafield. It can be seen thatFebruary has the lowest average monthly rainfall of 19mm and June has the highest of 66 mm. Table 2 showsthat June has the highest variability. From Figure 4,taken from Australian Bureau of Meteorology, we seethe average daily rainfall for each month.

1910 1920 1930 1940 1950 1960 1970 1980 1990

200

300

400

500

600

700

800

Year

Rain

fall

(mm

)

Yearly rainfall totals for Parafield (1901−1990)

y = − 0.1245x + 501.59

Figure 2. Time series of yearly rainfall at Parafield.

1 2 3 4 5 6 7 8 9 10 11 120

20

40

60

80

100

120

Month

Rain

fall

(mm

)

Monthly rainfall averages for Parafield

Figure 3. Monthly averages of rainfall data for Parafield; theheights of the bars are the mean monthly totals and the linesrepresent two standard deviations of monthly totals.

2.3 Modelling Monthly Rainfall

The Gamma distribution has been widely used fordescribing rainfall data, Stern and Coe (1984), Katzand Parlange (1998), Wilks and Wilby (1999), andRosenberg et al. (2004). Wilks and Wilby (1999)suggest the reason for this is due to the flexiblerepresentation involving only two parameters. Anextended model is explained for situations when theprobability of a zero monthly total is non-zero. Themonthly rainfall totals for Parafield will be modelledas non-negative random variables. For each montht of the year there is a chance that no rain will falland so it is necessary to consider a model that allowspositive probability for a zero total. To begin wedivide the data into two groups; zero records and non-zero records. The probability

p0 = p0[t] = P [X [t] = 0]

of a zero total on month t is estimated by

p0 =k

n

where we use k = k[t] to denote the number of zerorainfall records and n to denote the total number ofrecords. The gamma distribution is used to model thestrictly positive component of the monthly rainfall.

2912

0 10 20 300

1

2

3

4

Day

Dai

ly r

ainf

all (

mm

)

January

0 10 20 300

1

2

3

4

Day

Dai

ly r

ainf

all (

mm

)

February

0 10 20 300

1

2

3

4

Day

Dai

ly r

ainf

all (

mm

)

March

0 10 20 300

1

2

3

4

Day

Dai

ly r

ainf

all (

mm

)

April

0 10 20 300

1

2

3

4

Day

Dai

ly r

ainf

all (

mm

)

May

0 10 20 300

1

2

3

4

Day

Dai

ly r

ainf

all (

mm

)

June

0 10 20 300

1

2

3

4

Day

Dai

ly r

ainf

all (

mm

)

July

0 10 20 300

1

2

3

4

Day

Dai

ly r

ainf

all (

mm

)

August

0 10 20 300

1

2

3

4

Day

Dai

ly r

ainf

all (

mm

)

September

0 10 20 300

1

2

3

4

Day

Dai

ly r

ainf

all (

mm

)

October

0 10 20 300

1

2

3

4

Day

Dai

ly r

ainf

all (

mm

)

November

0 10 20 300

1

2

3

4

Day

Dai

ly r

ainf

all (

mm

)

December

Figure 4. Daily averages of rainfall for Parafield.

The gamma distribution is defined on (0,∞) by thedensity function

p[α, β](x) =xα−1e−x/β

βαΓ(α)

where α > 0 and β > 0 are parameters. Theparameters α = α[t] and β = β[t] for month t willbe determined from the observed non-zero recordsby the method of maximum likelihood. The generaldistribution of rainfall on the interval [0,∞) for montht can now be modelled with a cumulative distributionfunction (CDF) in the form

F [p0, α, β](x) = p0 + (1 − p0)

∫ x

0

p[α, β](ξ)dξ.

Let xa = xa[t] and xg = xg [t] be the arithmeticand geometric means of the observed non-zero values

Table 2. Monthly means and standard deviations forParafield

mean standard deviationJanuary 19.68 19.88February 19.83 23.24March 22.74 25.40April 40.54 31.83May 61.49 35.22June 64.96 40.19July 61.58 26.78August 57.27 25.89September 48.97 25.50October 43.29 25.92November 29.23 21.86December 26.37 20.09

xi = xi[t] for month t in which case the maximumlikelihood equations can be written in the form

ψ(α) − loge(α) + loge

(

xa

xg

)

= 0 (1)

andαβ = xa (2)

where the digamma function ψ is defined by theformula

ψ(α)4=

Γ′(α)

Γ(α)

for each α > 0. The above equations can be solvedeasily using MATLAB or EXCEL.

2.4 Estimating the key parameters

Standard MATLAB functions were used to calculatethe maximum likelihood estimates α and β for theyears of monthly rainfall records at Parafield andMesing. Guenni et al. (1996) present a statisticalmethodology to estimate the parameters of a rainfallmodel. The α and β values found using maximumlikelihood estimation for Parafield and Mesing areshown in Table 3.

2.5 Synthetic rainfall generated by independentrandom numbers

Spearman’s correlation coefficient is calculated for the90 years of rainfall records at Parafield, to test if anytwo months are correlated. Table 4 shows the P-valuesfor the Spearman’s correlation between successivemonths (Dec, Jan), (Jan, Feb) and so on. The resultssuggest it is reasonable to assume independence, atthe α = 0.05 significance level, as only Septemberis significant. We obtain similar results for Mesing.In our monthly model, we generate synthetic monthlytotals using independent random variables. This isnot the case for daily rainfall totals. The assumption

2913

Table 3. Parameter estimates for monthly rainfallrecords at Parafield and Mesing.

Parafield MesingMonth α β α βJan 1.26 16.73 1.03 297.96Feb 0.88 24.35 0.96 123.98Mar 0.86 28.38 1.38 95.79Apr 1.28 31.61 2.20 60.56May 2.33 26.43 5.72 23.67Jun 2.18 29.75 6.05 24.78Jul 4.91 12.53 9.03 16.89Aug 4.26 13.45 8.65 20.98Sep 3.22 15.20 2.95 53.37Oct 2.21 19.59 6.03 30.12Nov 1.69 17.92 5.60 68.06Dec 1.50 18.02 2.53 255.77

of independence for daily rainfall totals is incorrectso a modified model is described in Subsection 2.6to generate correlated daily rainfall. To generate asequence {x[t]} of synthetic monthly rainfall we firstgenerate realisations {r[t]} of a sequence {R[t]} ofindependent random numbers, each one uniformlydistributed on the unit interval [0, 1], and then usestandard MATLAB functions to solve the equation

F [p0[t], α[t], β[t]](x) = r[t]

to find the corresponding monthly rainfall denoted byx = x[t]. If r[t] < p0[t] then x[t] = 0. The parametersp0[t], α[t] and β[t] are defined by the maximumlikelihood estimates from the observed monthly data.

Table 4. Spearman’s correlation between successivemonths at Parafield

Month P-value Month P-valueJan 0.177 Jul 0.143Feb 0.337 Aug 0.070Mar 0.711 Sep 0.044Apr 0.330 Oct 0.605May 0.861 Nov 0.218Jun 0.269 Dec 0.227

Figure 5 shows a histogram of the observed monthlytotals versus the generated monthly totals using the2-parameter gamma distribution for January and July.Similar results were obtained for Mesing.

2.6 Modelling Daily Rainfall

2.6.1 Synthetic rainfall generated by correlatedrandom numbers using a single parameter

Piantadosi et al. (2007b) show that synthetic rainfallgenerated by independent daily random variables

0 13 26 40 53 66 80 93 106 More0

5

10

15

20

25

30

35

40

45January

ObservedGenerated

14 28 42 56 70 85 99 113 127 More0

5

10

15

20

25July

ObservedGenerated

Figure 5. Histogram of observed versus generatedmonthly totals for January and July.

on successive days significantly underestimatesthe observed monthly standard deviations. Thecorrelation between rainfall on successive days iscertainly small but it is reasonable as suggestedby Katz and Parlange (1998) to conclude thatthe assumption of independence is the primaryreason for the low simulated monthly standarddeviations. Piantadosi et al. (2007a) developed anextended model using a doubly stochastic matrix todefine a copula that preserves the given marginaldistributions and matches a known grade correlationcoefficient so that the entropy of the doubly stochasticmatrix is maximized. Cooke and Waij (1986) andLewandowski (2005) describe a joint distribution ofprobability mass on the unit square [0, 1] × [0, 1],which we shall describe generically as a diagonalband copula. Copulas represent a useful approachto modelling of dependent random variables whilepreserving the known marginal distributions (Nelson1999). The diagonal band copula provides a simpleformula that allows the joint probability to concentratealong one or other of the diagonals while preservinga uniform marginal distribution in each individualvariable. This type of copula can be used to specifya degree of positive or negative correlation betweenthe two variables simply by specifying the width anddirection of the diagonal band and at the same timethe known marginal distributions can be preserved.Piantadosi et al. (2007b) show how this idea can beused to correlate the rainfall on different days whilestill allowing the non-zero totals to be modelled by atime-varying gamma distribution. We show that thegenerated synthetic daily data matches the observeddaily and monthly statistics.

2914

2.7 Modelling Yearly Rainfall

Preliminary data analysis indicated that yearly totalsmay be well described by a mixture of two normaldistributions, one for dry years and one for wetyears. We are concerned with the estimation of theparameters µ1, σ2

1, µ2, σ2

2and p of the compound

normal distribution with density

f(y) = pf1(y) + (1 − p)f2(y)

where

fi(y) = (2πσ2

i )−1/2 exp[

−(y − µi)2/2σ2

i

]

i = 1, 2

when additional information is available from eitherf1(y) or f2(y). Suppose that n1 observations aretaken from the normal density f1(y), say, yj , j =1, 2, . . . , n2. Newton’s method is a process yieldingsolutions to the likelihood function L(θ) where θ =(µ1, µ2, σ

2

1, σ2

2, p). Estimation by the method of

moments was used to obtain trial values for theNewton iteration procedure when independent sampleinformation is available from one of the populations.We refer the reader to Dick and Bowden (1973) forfurther details. The estimated parameters for Parafieldare µ1 = 424.30, σ2

1= 4551.30, µ2 = 609.14, σ2

2=

7342.45 and p = 0.398. Figure 6 shows a histogramof the observed yearly totals versus the generatedyearly totals using a compound normal distribution.

300 400 500 600 700 800 More0

2

4

6

8

10

12Yearly totals

ObservedGenerated

Figure 6. Histogram of observed yearly totals versusthe generated yearly totals using compound normaldistribution.

3 RESULTS AND DISCUSSION

We compare the performance of our models forgenerating monthly totals using data from twolocations, Parafield rainfall data and Mesing rainfalldata. The Kolmogorov-Smirnov statistic D is aparticularly simple measure. It is defined as themaximum value of the absolute difference betweentwo cumulative distribution functions (the absolute

value of the area between them). Thus, for comparingone data set SN(x) to a known cumulative distributionfunction P (x), the K − S statistic is

D = max∞<x<∞

|SN (x) − P (x)|

where N is the number of data points. We refer thereader to Press et al. (1992) for further details. Table 5shows that the K-S Test D (greatest distance betweenthe two cumulative distributions) is not significantfor each month at Parafield and Mesing. We cansee that D for each month is less than the criticalvalue 0.143 for Parafield and 0.248 for Mesing. Ifthe K-S Test D is greater than the critical value thedifference is significant. We generate any number of

Table 5. Kolmogorov-Smirnov statistic D of eachmonth for Parafield and Mesing respectively.

Parafield MesingK-S Test D K-S Test D

January 0.083 0.157February 0.133 0.088March 0.091 0.076April 0.062 0.169May 0.088 0.106June 0.081 0.100July 0.058 0.136August 0.075 0.147September 0.068 0.153October 0.087 0.113November 0.063 0.099December 0.052 0.090

synthetic yearly totals. For each year, a large numberof monthly sequences are generated and summed.The total that best matches the yearly total is chosenas the synthetic monthly sequence. Figure 7 showsthe generation of two different monthly realisationswhich match the yearly total 478 mm. It is shownthat the generated monthly realisations with the sameyearly total are quite different. The methodologyfor generating monthly totals preserves the monthlystatistics and matches the yearly total.

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec0

10

20

30

40

50

60

70

80

90

100

110Monthly totals for 2 realisations

Month

Rain

fall

(mm

)

Yearly total = 478Yearly total = 478

Figure 7. Generated monthly realisations matched to a generatedyearly total.

2915

A similar methodology is used to generate dailyrainfall totals that match the monthly totals. Foreach month, a large number of daily rainfall totalsare generated and summed. The daily totals that bestmatch the monthly total is selected as the syntheticdaily realisation. Figure 8 shows the generation of twodifferent daily realisations which match the monthlytotals for February (19 mm) and July (59 mm). Onceagain we can see the generated realisations are quitedifferent although the monthly totals are the same.This methodology for generating daily rainfall totalspreserves the daily statistics and matches the monthlytotals. Figures 7 and 8 illustrate the principle under

5 10 15 20 250

2

4

6

8

10

12February

Day

Rai

nfal

l (m

m)

Monthly total = 19Monthly total = 19

5 10 15 20 25 300

2

4

6

8

10

12

14

16

18July

Day

Rai

nfal

l (m

m)

Monthly total = 59Monthly total = 59

Figure 8. Generated daily realisations matched to a generatedmonthly total (February and July) .

which we base our procedure. Any sequence ofmonthly totals that give the same yearly total areequally likely. Thus we can select any of theserealisations as a possible sequence for the syntheticyear. The same principle is utilised for selecting asequence of daily totals to match a pre-determinedmonthly total. In this way, we ensure that the long-term statistics are preserved at all three time scales.

4 REFERENCES

Australian Bureau of Meteorology (2004), Bureau ofMeteorology http://www.bom.gov.au.

Chiew, F.H.S., R. Srikanthan, A.J. Frost and E.G.IPayne (2005), Reliability of daily and annualstochastic rainfall data generated from differ-ent data lengths and data characteristics, MOD-SIM2005, Melbourne, Dec. 2005, 1223–1229.

Cooke, R. M. and R. Waij (1986), Monte Carlo sam-pling for generalized knowledge dependence withapplication to human reliability, Risk Analysis, 6,335–343.

Dick, N.P. and D.C. Bowden (1973), Maximum like-lihood estimation for mixtures of two normal dis-tributions, Biometrics, 29, 781–790.

Guenni, L., M.F. Hutchinson, W. Hogarth, C.W. Roseand R. Braddock (1996), A model for seasonalvariation of rainfall at Adelaide and Turen, Eco-logical Modelling, 85, 203–217.

Howlett, P.G., J. Piantadosi, and C.E.M. Pearce(2005), Analysis of a practical control policy forwater storage in two connected dams, Cont. opt..V. Jeyakumar and A. Rubinov (eds.), Springer-Verlag, NY, 99, 433–450.

Katz, R.W. and M.B. Parlange (1998), Overdispersionphenomenon in stochastic modelling of precipita-tion, J. Climate, 11, 591–601.

Lewandowski, D. (2005), Generalized diagonal bandcopulas, Insurance Math. and Econ., 37, 49–67.

Malaysian Meteorological Department (2007), Ja-batan Perkhidmatan Kajicuaca Malaysia.

Nelson, R.D. (1999), An introduction to copulas,Springer Lecture Notes in Statistics, New York.

Press, W.H., B.P. Flannery, S.A. Teukolsky and W.T.Vettering (1992), Numerical recipes in C: The artof scientific computing, Cambridge Press, NY.

Piantadosi, J. (2004), Optimal policies for storageof urban stormwater, PhD Thesis, University ofSouth Australia.

Piantadosi, J., P.G. Howlett and J.W. Boland (2007a),Matching the grade correlation coefficient usinga copula with maximum disorder, J.I.M.O., 3(2),305–312.

Piantadosi, J., P.G. Howlett and J.W. Boland (2007b),A new model for correlated daily rainfall, (inpreparation).

Rosenberg, K., J.W. Boland and P.G. Howlett (2004),Simulation of monthly rainfall totals, ANZIAM J.,46(E), E85–E104.

Srikanthan, R. (2005), Stochastic Generation of DailyRainfall Data, MODSIM2005, Melbourne, 12-15Dec. 2005, 1915–1921.

Srikanthan, R. and T. A. McMahon (2001), Stochasticgeneration of annual, monthly and daily climatedata: A review, Hydr. and Earth Sys. Sci., 5(4),633–670.

Stern, R. D. and R. Coe (1984), A model fitting anal-ysis of daily rainfall, J. Roy. Statist. Soc. A, 147,Part 1, 1–34.

Wilks, D.S. and R.L.Wilby (1999), The weather gen-eration game: a review of stochastic weather mod-els, Prog. Phys. Geog., 23(3), 329–357.

2916