stochastic modelling of daily rainfall: the impact of adjoining wet days on the distribution of...

Environmental Modelling & Software 13 (1998) 317–324

Stochastic modelling of daily rainfall: the impact of adjoining wetdays on the distribution of rainfall amounts

Tom ChapmanSchool of Civil and Environmental Engineering, University of New South Wales, Australia

Abstract

Daily rainfall data for each month of the year have been classified according to the number of adjoining wet days (0, 1 or 2).The data sets used were long-term records for 14 stations in Australia, 6 in South Africa, and 24 in North America, and medium-term ( < 20 years) records for 22 island stations in the Western Pacific. For all regions, a nonparametric test showed a lowprobability that the data in the different classes were from the same distribution, at least for some months of the year. Stochasticmodels, which treat the classes separately, generally resulted in a better fit than currently used models which group the data together.The magnitude of the ratios of the class means to the overall mean daily rainfall shows that serious errors may result from modelswhich do not take account of these differences, either explicitly by separate modelling of the classes, or implicitly by a multi-statetransition probability matrix for rainfall amounts. This work was supported by author-developed software for the statistical analysisof historical rainfall data, parameter estimation by maximum likelihood for a range of models, comparison of model fitting by theAkaike Information Criterion, and daily rainfall simulation. 1998 Elsevier Science Ltd. All rights reserved.

Keywords:Rainfall; Stochastic modelling

Software and data availability

Program title: dayrain.f

Developer: E/Prof Tom Chapman

Contact address: PO Box 106, Crows Nest2065, Australia; phone/fax:1 61 2 9929 7701; email:[email protected]

First available: 1998

Hardware required: Power Macintosh or PCwith at least 500K freememory

Software required: Fortran 77 compiler

Program language: Fortran 77 (comprises 40subprograms and 3840lines of code)

1364-8152/98/$—See front matter 1998 Elsevier Science Ltd. All rights reserved.PII: S1364-8152 (98)00036-X

Availability and cost: Source code available atno charge; author will runprogram on user’s data atcost to be arranged.

Data: The daily rainfall dataused in this paper can bemade available forresearch purposes only,subject toacknowledgement ofsources.

1. Introduction

Simulated sequences of rainfall are a useful tool inthe design and operational management of hydraulicstructures, roof-tank water supplies, irrigation schemesand dryland farming enterprises, as they provide a rangeof scenarios which may differ markedly from the detailsof the historical record, while retaining that record’s stat-

318 T. Chapman/Environmental Modelling & Software 13 (1998) 317–324

istical properties. In recent years, these applications havebeen extended to providing the inputs to models ofenvironmentally sensitive structures such as tailingsdams and landfills, where the quality of the outflow isa predominant consideration. For some applications,monthly or even annual sequences may be adequate, butmost effort has been given to developing models for thetime interval at which rainfall is usually observed, thatis, daily.

Stochastic models of daily rainfall have generallybeen divided into two parts, a model of rainfall occur-rence, which provides a sequence of dry and wet days,and a model of rainfall amounts, which simulates theamount of rainfall occurring on each wet day.

Models of rainfall occurrence are themselves of twomain types, those based on Markov chains and thosebased on alternating renewal processes.

Markov chain models specify the state of each day as‘wet’ or ‘dry’, and develop a relation between the stateof the current day and the states of preceding days. Thenumber of preceding days taken into account is the orderof the Markov chain. Most Markov chain modelsreferred to in the literature are first order (e.g. Buishand,1978; Roldan and Woolhiser, 1982; Zucchini and Adam-son, 1984). Models of second and higher orders havealso been studied (Chin, 1977; Coe and Stern, 1982;Singh et al., 1981).

In alternating process renewal models, distributionsare assumed for the lengths of dry and wet spells. It isassumed that all intervals are independent, and that thedistributions may be different between dry and wetspells. The distributions most used have been the trunc-ated negative binomial distribution (Buishand, 1978) andthe truncated geometric distribution (Roldan andWoolhiser, 1982).

Models used for rainfall amounts include the two-parameter gamma distribution (Coe and Stern, 1982;Richardson and Wright, 1984), the exponential distri-bution (Allan and Haan, 1975; Woolhiser and Roldan,1982), the Weibull distribution (Zucchini and Adamson,1984), the mixed exponential distribution (Woolhiserand Roldan, 1982), and a skewed normal distribution(Nicks and Lane, 1989).

As a result of seasonal variations in rainfall, it is notsurprising that the parameters for these models of rainfalloccurrences and rainfall amounts vary through the year.The usual method of handling this variation is to deriveseparate parameter sets for each month (or sometimesfor each season) of the year, giving rise to a very largenumber of parameters for the overall model. Attemptshave been made to reduce the number of parameters byfitting the parameter variation through the year to a poly-nomial (Coe and Stern, 1982) or a Fourier series(Woolhiser and Roldan, 1982; Zucchini and Adamson,1984).

Srikanthan and McMahon (1985), in a study covering

the main climatic regions of Australia, followed Allanand Haan (1975) by extending the Markov chain conceptto a multi-state model or transition probability matrix,in which the daily rainfalls are grouped into up to sevenclasses of given magnitude ranges, and the probabilitiesare calculated for transition from each class to any other.The lowest class gives the occurrences of dry days, thetop class is modelled by a skewed normal distribution(requiring estimation of three parameters), and inter-mediate classes are modelled by a linear distribution.Separate parameter values are calculated for each month,so that when all seven classes are used, the total numberof parameters is 123 (7 3 6 1 3) 5 540.

A similar approach has more recently been used byGregory et al. (1993) in a study of area-average dailyrainfalls for 3-month seasonal periods in Britain. Theygrouped the daily rainfalls into 10 classes, the lowestbeing for dry days, while the limits for the other classeswere selected to give approximately equal numbers ofrain days in each class. The gamma distribution was usedfor the top class and to interpolate between the limits ofthe intermediate classes. They showed that the multi-state model was more successful in simulating the vari-ances of seasonal rainfall than either a first or secondorder two-state model.

A comprehensive review of approaches to modelingdaily rainfall can be found in Woolhiser (1992).

As noted above, stochastic rainfall models havebecome an integral part of models in which the overallemphasis is on water quality as much as quantity.Examples include the US Environment Protection Auth-ority HELP (Hydrologic Evaluation of LandfillPerformance) model, and the US Agricultural ResearchService’s models SPUR (Simulation of Production andUtilization of Rangelands), EPIC (Erosion ProductivityImpact Calculator), SWRRB (Simulation of WaterResources from Rural Basins) and WEPP (Water Ero-sion Prediction Project). The rainfall input for thesemodels is provided by one of three stochastic daily rain-fall models: WGEN (Richardson and Wright, 1984),CLIGEN (Nicks and Lane, 1989), or USCLIMAT.BAS(Hanson et al., 1994). These models use a first orderMarkov chain for rainfall occurrences, and a gamma,skewed normal, or mixed exponential distribution,respectively, for rainfall amounts. Monthly parametershave been calculated for a large number of locations inthe US (e.g. 180 cities for the HELP model), and areavailable as part of the program packages.

In these models, the rainfall amounts on wet days havebeen taken as belonging to one probability distribution.However, Buishand (1978) found significant differencesbetween mean rainfalls on wet days grouped accordingto the number of adjoining wet days, and Cole and Sher-riff (1972) applied separate models to rainfalls for a soli-tary wet day, the first day of a wet spell, and the otherdays of a wet spell.

319T. Chapman/Environmental Modelling & Software 13 (1998) 317–324

Table 1Summary of daily rainfall data sets

Region No. of stations Mean record Mean annuallength (years) rainfall (mm)

Australia 14 74 254–1629South Africa 6 95 215–1020North America 24 97 214–1279Pacific islands 22 20 860–3810

The purpose of this paper is to determine whether thedistributions for rainfall amounts classified according tothe number of adjoining wet days are significantly differ-ent, to examine how these differences may be incorpor-ated in model structures, and to quantify the errors thatmay ensue from neglecting these differences.

2. Data sets

For Australia, South Africa and North America, long-term daily rainfall records were selected for sites cover-ing a range of annual rainfalls and seasonal distributionsof rainfall, while for the Pacific islands, 22 sites with 181 years of records were downloaded from the Compre-hensive Pacific Rainfall Data Base (Morrissey andShafer, 1995). A summary of the data sets is given inTable 1, while the site locations are shown in Figs. 1–4. Where more than a few days of data were missing in agiven year, the whole year was excluded, and the recordtreated as continuous.

To examine the effect of record length, data for 20,50 and 100 years were analysed separately, using themost recent part of the record in each case.

Fig. 1. Location of Australian rainfall stations.

Fig. 2. Location of South African rainfall stations.

3. Nonparametric tests

For each station, the rainfall data for each month wereclassified according to the number of adjoining wet days(0, 1 or 2). Thus, Class 0 comprises solitary day rainfalls,Class 1 comprises rainfalls on days at the beginning orend of a wet spell (of at least 2 days duration), and Class2 comprises rainfalls in the interior of a wet spell (whichis therefore of at least 3 days duration).

Typical empirical frequency distributions for theseclasses are shown in Fig. 5. The Kolmogorov–Smirnovtwo-tailed test (Conover, 1980) was used to determinethe probability that the data from pairs of classes werederived from the same empirical distribution. The prob-abilities were calculated using the exact algorithm givenby Kim and Jennrich (1973). Table 2 shows the percent-age of months in which these probabilities are below0.05, for nominal record lengths of 20, 50 and 100 years.In every case, the number of significant differences


Fig. 3. Location of North American rainfall stations.

Fig. 4. Location of rainfall stations in the western Pacific ocean.

Fig. 5. Typical frequency distributions of daily rainfall (Durban, Jan-uary 1974–93).

between classes increases with record length. For a givenrecord length, the greatest number of differences occursbetween Classes 0 and 2, and the least between Classes0 and 1.

Table 2 also demonstrates notable differences in thiseffect between the regions studied, with the Australian

and Pacific Islands data having the largest percentage ofoccurrences and the North American data least.

4. Model comparisons

This section examines the effect of separate treatmentof these rainfall classes on selection of a best fit model.For each rainfall record, four rainfall occurrence modelswere tested with separate parameter values for eachmonth. They were Markov chains of orders 1 and 2(denoted as MC1 and MC2), the truncated negativebinomial distribution (TNBD), and the truncated geo-metric distribution (TGD). Three of these models (MC1,TNBD and TGD) were also tested with parameters vary-ing smoothly through the year according to a Fourierseries. Tests were made with 0, 1 and 2 harmonics (asthe 2-harmonic model was not selected for any data set,tests with higher order harmonics were not undertaken).All fitting was by maximum likelihood (with the dataclassified by the ordinal day of the year for the Fourierseries models), and the ‘best’ of the 13 models wasselected by the Akaike Information Criterion (AIC)(Akaike, 1974).


Table 2Percentage of months showing significant (p , 0.05) differences between distributions of rainfall classified according to number of adjoiningwet days

Location: Australia South Africa North America Islands

Record length 20 50 100 20 50 100 20 50 100 20(years):

Classes 0 and 1 20 37 85 7 22 31 13 21 31 21Classes 1 and 2 42 64 92 28 29 64 20 34 47 54Classes 0 and 2 61 77 98 39 51 78 29 49 58 67

Using the selected occurrence model, five modelswere tested for the distribution of rainfall amounts: theWeibull, mixed exponential, two-parameter gamma,skewed normal, and two-parameter kappa distributions.The skewed normal distribution (three parameters) wasobtained by fitting a normal distribution to data normal-ised by the Box–Cox transformation. The parameter forthis transformation was determined by minimising thevariance of the transformed variable, using theexpression for the log likelihood given by Hipel et al.(1977). The models for rainfall amounts used separateparameters for each month; Fourier fittings were notattempted in this case.

Each of these models was applied to three sets of data:all rainfall classes grouped together, solitary wet days(Class 0) fitted separately from the other wet days(Classes 1 and 2), and each class of data fitted separately.In addition, the Srikanthan–McMahon model was fittedto each data set, and the AIC calculated by the methoddescribed in Chapman (1995, 1997). For the resultsgiven here, the last 100 years of records (20 for thePacific islands) were used where available; otherwise thewhole record was used.

The results for the North American and South Africandata are shown in Tables 3 and 4; similar presentationsfor the Australian and Western Pacific data are in Chap-man (1994) and Chapman (1995, 1997), respectively.While it is not the purpose of this paper to dwell oncomparisons between different probability distributions,it may be noted that all but one of the North Americanstations were best fitted by a rainfall occurrence modelof type MC2 or TNBD, with the parameters fittedmonthly. There is considerably more variation in theoccurrence models selected for the other three data sets.For the rainfall amount models, only the mixedexponential and the skewed normal distributions wereselected for any of the data sets.

The bold cells in Tables 3 and 4 show which of themodels differentiated by rainfall classes resulted in theminimum AIC. The model where all the classes werefitted separately was selected for the large majority ofthe North American and South African data, and all theAustralian and Pacific islands data. It may also be notedthat the Srikanthan–McMahon model performed better

than the other models for 53 out of 66 station records,including all but one of the North American rainfall sta-tions.

5. Magnitude of the effect

The general nature of the differences between the dis-tributions of daily rainfalls, classified according to thenumber of adjoining wet days, is that the mean dailyrainfalls for Classes 0 and 2 are respectively lower andhigher than the overall mean daily rainfall. This variesfrom month to month, and the potential impact on mode-ling has been evaluated by considering the lowestmonthly ratio of Class 0 rainfall to mean daily rainfall,and the highest monthly ratio of Class 2 rainfall to meandaily rainfall. Means and standard deviations of thesequantities (Table 5) show that the effect is least for theNorth American stations and greatest for the Pacificislands, where on average the rainfall on solitary wetdays would be overestimated by a factor of nearly 4 bya model which grouped all the data together. For theAustralian stations, the rainfall on solitary wet dayswould on average be overestimated by a factor of 2.5,while the rainfall in the interior of wet spells would beunderestimated by a factor of 2.3.

For the North American stations, Fig. 6 shows thatthere is an indication of an increasing impact with longi-tude west, so that models which group the data can beapplied with less error in the east than in the westernStates.

For the North American and South African stations,the Class 1 daily rainfalls are generally within a few percent of the overall daily mean, but for the Australian andPacific island stations they average 15–20% below theoverall daily mean.

These differences between the rainfall classes can bemodelled effectively by fitting probability distributionswith different parameters. However, Fig. 7 shows thatthe Srikanthan–McMahon model gives a close simul-ation of each class, even though the classes are notexplicitly identified. It has also been shown (Srikanthanand McMahon, 1985) that this model has the additionaladvantage of preserving the variances of monthly andannual totals.


Table 3Comparison of daily rainfall models for North American stations

Station Spell model All classes together Solitary separately All classes separately S-M model

Model AIC Model AIC Model AIC AIC

Albany M-MC2 ME 113 100 ME 113 085 ME 113 056 112 621Baltimore M-MC2 ME 116 129 ME 116 143 ME 116 134 115 473Big Rapids M-TNBD SN 102 009 SN 102 010 SN 102 007 101 694Brookings M-MC2 SN 78 743 SN 78 734 SN 78 738 78 374Clarinda M-TNBD SN 93 068 SN 93 064 ME/SN 93 052 92 624Corvallis M-MC2 ME 120 462 ME/SN 120 024 ME/SN 119 234 118 031Crookston M-TNBD SN 84 164 SN 84 175 SN 84 136 83 799Dialville A1-TGD ME 83 762 ME 83 795 ME 83 804 83 567Fairbury M-TNBD SN 90 591 SN 90 547 ME/SN 90 516 90 010Fayetteville M-MC2 ME 105 144 ME 105 138 ME 105 125 104 524Helena M-TNBD SN 78 035 SN 77 791 SN 77 641 77 347Little Rock M-MC2 ME 108 763 ME 108 673 ME 108 586 108 107Mesa M-MC2 SN 34 635 ME/SN 34 590 –* – 34 516Minn-StPaul M-TNBD SN 102 117 SN 101 996 ME/SN 101 906 101 038Montreal M-MC2 ME 96 790 ME 96 733 ME 96 754 95 823Moscow M-TNBD SN 97 690 SN 97 682 SN 97 621 97 329Newitt M-TNBD ME 78 613 ME 78 621 ME 78 631 78 504New York M-MC2 ME 120 298 ME 120 259 ME 120 237 119 526Olga M-TNBD SN 110 686 SN 110 672 SN 110 610 110 844Raleigh M-MC2 ME 112 102 ME 112 120 ME 112 125 111 631Sacramento M-MC2 ME 59 097 ME/SN 58 961 –* – 58 564Spokane M-TNBD SN 90 276 SN 90 136 SN 90 065 89 323Tucson M-MC2 SN 50 141 SN 50 125 SN 50 123 50 051Vancouver M-TNBD SN 109 233 SN 109 020 SN 108 619 108 391

ME 5 mixed exponential; SN5 skewed normal; S-M5 Srikanthan–McMahon; M-5 monthly parameters; A1-5 annual parameters with 1harmonic; other symbols defined in the text. Bold cells show the lowest AIC for each station.*Insufficient data in at least one month.

Table 4Comparison of daily rainfall models for South African stations

Station Spell model All classes together Solitary separately All classes separately S-M model

Model AIC Model AIC Model AIC AIC

Bredasdorp M-MC1 SN 73 247 SN 73 138 SN 73 045 70 043Cullinan M-MC2 SN 54 013 ME/SN 54 042 –* – 54 210Dewetsdorp M-MC2 SN 47 742 SN 47 703 –* – 48 013Durban M-TNBD SN 102 079 SN 101 978 ME/SN 101 747 101 350Springbok M-MC2 SN 36 743 SN 36 695 –* – 36 864Versamelhoek M-TNBD ME 54 686 ME/SN 54 683 ME/SN 54 603 54 164

ME 5 mixed exponential; SN5 skewed normal; S-M5 Srikanthan–McMahon; M-5 monthly parameters; other symbols defined in the text.Bold cells show the lowest AIC for each station.*Insufficient data in at least one month.

6. Conclusions

It has been shown that, when daily rainfalls are classi-fied according to the number (0, 1 or 2) of adjoiningwet days, the sets of data usually belong to differentprobability distributions, with the mean of each set typi-cally increasing with the class number.

The difference between Classes 1 and 2 may reason-ably be attributed to systematic differences in the dur-ation of rainfall; on average, the duration of rainfall on

the first or last day of a wet spell will be less than inthe intervening days.

The difference between Class 0 and the other classesmay be partly due to rainfall duration, but may also bean indication of the causative mechanism for the rainfall,with Class 0 rainfalls being typically convectional, whilethe other classes are typically frontal or cyclonic.

It has also been shown that most current daily rainfallsimulation models, which neglect this effect, will resultin significant overestimation of Class 0 rainfall and


Table 5Means and standard deviations of ratios of class mean to overall meandaily rainfall, for lowest monthly value in Class 0 and highest monthlyvalue in Class 2

Data set Class 0 Class 2

Mean SD Mean SD

Australia 0.40 0.16 2.33 0.66South Africa 0.67 0.07 2.20 0.50North America 0.70 0.13 1.58 0.24Pacific islands 0.29 0.10 1.55 0.26

Fig. 6. Ratio of annual mean of Class 0 rainfall to overall mean dailyrainfall for US stations, plotted against longitude.

Fig. 7. Mean daily rainfall per wet day, for each class of rainfall, atHelena, Montana. The solid lines show the historical record, and thedashed lines are for a 1000-year simulation using the Srikanthan–McMahon model.

underestimation of Class 2 rainfall. Where these modelsare applied to calculation of daily water balances or sur-face erosion, these errors will be carried through the cal-culations, with unpredictable results.

For the data sets studied, the magnitude of the differ-ences between rainfall classes is least in the eastern

United States, increasing westwards, and is greatest inAustralia and the islands of the western Pacific.

The Srikanthan–McMahon model, which is largelybased on a transition probability matrix for classes ofrainfall amounts on successive days, is effective inmodeling rainfalls classified according to the number ofadjoining wet days.

Acknowledgements

This paper would not have been possible without thegenerous provision of data from several sources. Dr R.Srikanthan of the Australian Bureau of Meteorology pro-vided the Australian data. The South African data weremade available by Dr M. Dent from the records held atthe Computing Centre for Water Research, University ofNatal. The Pacific islands data were downloaded fromthe Comprehensive Pacific Rainfall Data Base main-tained by the Oklahoma Climatological Survey. Most ofthe North American records were provided by Dr BenHarding, of Hydrosphere Resource Consultants,Boulder, CO. The records for Corvallis and Vancouver,Oregon, were given by Jinfan Duan of Oregon State Uni-versity, from records maintained by the US Forest Ser-vice, while the data for Montreal were provided by DrFred Fabry. Computing facilities and Internet connec-tions were provided by the School of Civil Engineeringof the University of New South Wales. Figs. 1–4 wereprepared with the assistance of the generic mapping toolssoftware (Wessel and Smith, 1991).

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