Weather and Climate 21, 35-46 (2001)

Comparing the rainfall-producing models in stochastic weathergenerators

AbstractWeather generators can be used to simulate daily time series o f weather elements such as rainfall,maximum and minimum temperature, and solar radiation. Here we focus on the rainfall component,and assess the accuracy o f three different models in simulating monthly means and variances o fprecipitation amount and frequency, and in simulating the length of wet and dry spells, at three NewZealand sites. The first and simplest model assumes two states (wet and dry days), where today's statedepends only on what occurred the previous day. The second model simulates alternating sequencesof wet and dry spells rather than each day separately. The third model is a modification of the first,where transition probabilities and rainfall distributions depend on how wet was the previous month.The first and third models proved to be superior to the second model at reproducing the observations.The first model, however, has the advantage o f simplicity, which makes i t easier to modify thestatistical parameters to cope with future changes in rainfall climate.

1. Introduction

C.S. Thompson and A.B. Mullan

National Institute of Water and Atmospheric Research,Wellington, New Zealand

The synthetic generation o f daily weather elements is frequently used to supplementobservations o f climatological data and to provide a way to simulate the impacts o fweather variability on a wide range of management decisions. These so-called "weathergenerators" have typically been developed along the lines o f the model proposed byRichardson (1981), i n which related variables including maximum and minimumtemperatures and solar radiation are conditioned on the daily occurrence o r non-occurrence o f precipitation. That is, different distributions are used for these othervariables according to whether the day is "wet" or "dry". The potential o f weathergenerators to synthesise long records o f climatological data has led to a number o fapplications. Developed with widely different research objectives in mind, weathergenerators can provide weather records for water quality and other hydrologic studies(precipitation), o r they may produce input for crop-growth simulations and impacts(precipitation, maximum and minimum temperatures and solar radiation). They may alsoprovide the means for extending the synthesis of climate data to locations where recordsare not available (Hutchinson, 1995; Semenov and Brooks, 1999). A third area o fapplication has arisen from climate change studies in which weather generators canproduce site-specific scenarios at the daily time-step (Wilks, 1992, 1999a).

Corresponding author: Craig Thompson, NIWA, P.O. Box 14-901, Kilbirnie, Wellington, New Zealand.Email: c.thompson@niwa.cri.nz

36 W e a t h e r and Climate 21

Within New Zealand, weather generators are being used in a scientific programme "toenhance the understanding o f the sensitivity o f New Zealand's natural and managedenvironments t o climate variability and change" (Kenny et al., 2000). C a l l e dCLIMPACTS, the programme, which began i n 1993, is an on-going collaborativeresearch e f fo r t between t w o Universities and f ive C rown Research Institutes.CLIMPACTS i n v o l v e s t h e d e s i g n , deve lopmen t a n d e v a l u a t i o n o fagricultural/horticultural/pastoral impact models, weather generators, along with toolsto evaluate climate risk and sensitivity, and scenarios of climate change. Since many ofthe impact models developed by the CLIMPACTS collaborators require daily weatherdata, there is a need to generate such synthetic time series.

Several weather generators have been developed for CLIMPACTS (Thompson andMullan, 1995, 1997), based on techniques described in the literature (Richardson, 1981,Racsko et al., 1991, Wilks, 1989). These models were also assessed in terms o f theirability to simulate important meteorological quantities such as the distributions o ftemperature exceedances and spells o f wet and dry days, as well as the monthly andannual variability o f rainfall and temperature. (Thompson, 1997). T h e weathergenerators have been incorporated into CLIMPACTS, and are used by the various impactmodels to generate daily time series o f weather elements to assess crop responses. A nimportant consideration within CLIMPACTS is the capability to apply the weathergenerators to the simulation o f future climates, such as might occur under variousscenarios of greenhouse gas increases (Mullan et al., 2001a). This requires adjusting theparameters of the generator (e.g., as in Wilks, 1992), and argues for as simple a weathergenerator as possible, subject to an adequate simulation of current climate.

In this presentation we focus on the precipitation-producing processes within theweather generator and assess their performance. Within CLIMPACTS daily time seriesof precipitation, maximum and minimum temperatures and solar radiation are requiredfor weather generators, and high quality data from 15 sites nationally were prepared(Porteous, 1997, Mullan et al., 2001b). We illustrate the model performance using dailyhistorical data for Ohakea (241m1 northwest o f Palmerston North, on the ManawatuPlains), Ruakura, (near Hamilton) and Lincoln (17 km southwest o f Christchurch),covering the periods 1954-1990, 1972-1995 and 1950-1991 respectively (where start dayis determined by availability of daily solar radiation data). The three sites, selected fromthe CLIMPACTS network of stations, are situated in different rainfall climate regions ofNew Zealand. On average, Ruakura receives about 1200 mm o f rainfall annually, andOhakea about 920 mm, and both sites have pronounced winter-time rainfall maxima.Lincoln is much drier wi th about 660 m m and has a relatively uniform rainfalldistribution throughout the year.

2. Structure of Weather Generators used in CLIMPACTSAll weather generators consider precipitation as the primary weather element, with other

Thompson & MuIlan: Rainfall models in stochastic weather generators 3 7

weather variables on any day being conditioned on whether the day is wet or dry.Basically, the modelling o f precipitation involves two component processes: (a) theoccurrence process (i.e., the sequence of wet or dry days), and (b) the intensity process(i.e., precipitation amounts on wet days). Details o f precipitation occurrence processeswill be presented shortly. Precipitat ion intensities o n wet days are stochasticallysimulated from a statistical distribution that is usually, but not exclusively, characterisedby a gamma distribution. Such a distribution has a long "tail", so that low precipitationtotals occur frequently, and high totals only occasionally. The precipitation amount onany day is completely independent of the amount of precipitation (no rainfall included)for the previous day (Katz, 1977). Separate gamma distributions are fitted to each monthof data, and their seasonal cycle is represented by a Fourier series analysis using annualand semi-annual cycles.

The weather variables o f maximum temperature, minimum temperature and solarradiation are represented as a first order tri-variate autoregressive model (Richardson,1981). S ince these variables are correlated, they can not be simulated individuallywithout risking the simulation o f non-physical events, such as the precipitation fromclear skies. The autoregressive model is;

x(t) = [A] x(t-1) + [13] E (t)where the (3 x 3) parameter matrices [A] and P31 reflect the serial and cross-correlationsof the three variables, the s's are independent normal variates w i th a N(0,0,2)distribution. T h e x's are normalised residuals (i.e. a N(0,1) distribution) conditional onwhether the day is wet or dry according to:

xk = (X - k i ) / Gkj k = 1,2,3; j = 0,1where X is the actual daily value of the weather variable in question. Fo r each of thek = 1,2,3 weather variables, separate means and standard deviations are used for dry(j 0 ) and wet (j = 1) days.

The [A] and [B] matrices are determined from matrices o f the lagged and unlaggedcorrelations among the three elements of x (e.g. Matalas, 1967; Richardson, 1981), and areassumed to be the same for wet and dry days. Elements o f [A] primarily represent timedependence (i.e. auto-correlation) in the autoregressive model, while [3] serves mainly toproduce appropriate contemporary correlations among the simulated variables (Wilks, 1989).

Weather generator model parameters are fitted to observed daily climate data, for wetand dry days, with their seasonal cycle being represented by annual and semi-annualcycles from a Fourier series analysis.

2.1 Precipitation occurrence processes

In CLIMPACTS, there are several different stochastic processes that have been proposedto model the characteristics o f daily precipitation occurrence. T h r e e componentprocesses used by the authors are presented.

Markov chain-dependent process: This is a relatively simple approach to themodelling o f the precipitation occurrence made popular by Gabriel and Neumann

38 W e a t h e r and Climate 21

(1962), Katz (1977) and Richardson (1984 Precipitation occurrence is represented as atwo-state first-order Markov process. That is, the day is either wet or dry, and theprobability of precipitation depends only on the precipitation state of the previous day.This model can be expressed in terms o f transition probabilities poi (probability a wetday follows a dry day) and p i 1 (the probability a wet day follows a wet day). Twoalternative parameters frequently used are:

d P H P o i which is the lag-1 autocorrelation (i.e., i ts persistence) f o r theprecipitation occurrences, and

= Pol/(1 — which is the long-term climatological probability of a wet day. Theprobabilities, pij, are estimated by counting the number o f transitions from State i toState j, and dividing by the number of these transitions originating in State i. T h e pijconstitute a 2x2 matrix, so that all four transition probabilities can be specified from justtwo o f them. T h i s representation o f the precipitation process is to f i t all the dataunconditionally (Wilks, 1989): model parameters have been estimated using all historicalweather data for a given time period (e.g., month). As an example, the mean annualaverages o f the Ohakea parameters are: d = 0.3039 and Tu = 0.4267 (alternatively,Poi — 0.2970 and PH = 0.6009), for the 1954-1990 data period.

Rainfall renewal process: Precipitation occurrences are simulated using a "renewalprocess" (Wilby et al., 1998) in which the lengths or alternating wet and dry spells (i.e.,runs of one or more consecutive wet or dry days) are represented by "mixed geometric"distributions (Racsko et al., 1991; Semenov and Porter, 1995). This model was developedbecause the Markov chain process tends to simulate long spells of wet or dry weathertoo infrequently, and this limitation could be critical for crop-simulation and othersimilar modelling studies. With a change in distribution at 8 days, the mixed distributionhas a probability o f 1—p for short spells and a probability p for longer spells. Havingrandomly chosen a short (up to 8 days) or long spell, another random number decidesthe exact length within this spell, in the subsequent simulation phase. Parameters of themixed distribution are estimated separately for wet and dry spells using the entirehistorical weather record. By definition each wet period is followed by a dry one. A sdeveloped by Racsko et al., (1991) the renewal model has six parameters, three each torepresent the wet and dry spells (for the annual mean and each Fourier coefficient). I ntheir non-standard notation, a t Ohakea f o r example, the geometric distributionparameter for short spells o f wet (dry) days is 0.4217 (0.3558). T h e probability p (therun length will be larger than 8 days) is 0.0197 (0.0692), and the mean length of the longspells of wet (dry) days is 8 + 1.73 (3.81) days.

Conditional Markov chain-dependent process: The above two processes areexamples o f unconditionally fitting all the historical weather data to derive parametersfor precipitation occurrence. W i l ks (1989) developed a rather different approach toimproving the simulation o f seasonal variability by "conditioning" synthesised dailyrainfalls on prior monthly rainfall amounts. In a conditional rainfall model, precipitation

Thompson & Mullan: Rainfall models in stochastic weather generators 3 9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1Jan F e b M a r A p r M a y J u n J u l A u g S e p O c t N o v D e c

Months

---E—Normal—e---Dry—x - - Unconditional

Figure 1. Transition matrix probabilities for daily precipitation occurrences at Ohakea,conditional on the tercile category of total monthly precipitation and unconditional for aprobability of a wet day following a wet day (pa).

is stratified into subsets o f months according to which tercile (i.e., dividing the recordinto thirds) category they fall into. For a two-state first order Markov process, the dailytransition probabilities, PH, for the wet (top third), near-normal (middle-third), and dry(lower-third) monthly precipitation terciles are shown in Figure 1 for Ohakea. Transitionprobabilities, fitted to the upper tercile o f monthly precipitation distributions (i.e., wetmonths), are greater than those derived for the near-normal category, which in turn aregreater than those in the dry category. The respective unconditional probabilities are ingeneral quite close to those estimated using only data from the central third o f themonthly precipitation distributions. Similar patterns are also seen in the poi transitions.In addition to these stratified daily transition probabilities, the monthly transitionprobabilities (e.g., wet tercile to wet tercile, wet tercile to near-normal tercile, etc) are alsoestimated from the observations. Then, in the simulation phase, a month is first assignedto a tercile category, and the appropriate transition probabilities used. This conditionalmodel has rather more parameters (9) than the other two process models considered.

The precipitation intensity process for a conditional Markov process is modelledfrom a gamma distribution having a common shape parameter but having different scale(or intensity) parameters for each tercile category. T h e synthesis of daily precipitationtotals is complex. Ful l details can be found in Wilks (1989), and proceeds along the linesof using one of three possible precipitation outcomes (i.e., dry, wet, or normal) for thefollowing month, each made on the basis of the synthesised total for the current month.This transition occurrence process constitutes a three-state first-order Markov process andis represented by a matrix of observed monthly transition probabilities. For example, atOhakea, the probability of a wet month followed by another wet month is 0.38, a wet monthfollowed by a dry month is 0.30, and a dry month followed by another dry one is 0.36.

40 W e a t h e r and Climate 21

Averages Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecObserved 65 58 68 67 86 85 86 76 69 76 64 89Markov 78 57 72 66 78 86 84 77 71 70 82 74tprob 0.10 0.84 0.61 0.92 0.30 0.85 0.78 0.92 0.83 0.33 0.01 0.08Renewal 89 76 74 78 84 93 86 79 70 70 85 96tprob 0.00 0.01 0.41 0.07 0.83 0.28 0.97 0.61 0.91 0.31 0.00 0.39Conditional 67 63 67 67 89 83 91 80 72 72 75 75tprob 0.85 0.48 0.97 0.94 1.00 0.79 0.57 0.62 0.62 0.53 0.11 0.12

Standard deviationsObserved 35 29 46 29 37 43 67 32 37 35 29 48Markov 40 37 42 36 38 35 36 34 39 33 38 43F bpro 0.38 0.11 0.54 0.15 0.99 0.11 0.05 0.70 0.80 0.60 0.07 0.40Renewal 36 37 38 34 32 40 34 31 30 28 33 40Fprob 0.92 0.13 0.14 0.29 0.24 0.59 0.02 0.72 0.08 0.07 0.39 0.12Conditional 39 44 41 34 41 31 39 38 31 33 36 37Fprob 0.49 0.01 0.39 0.28 0.52 0.01 0.14 0.27 0.12 0.61 0.15 0.05

b. Rainday frequency (days of at least 0.1 mm)

Averages Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecObserved 10 9 11 12 15 15 15 15 14 15 13 12Markov 10 8 10 11 14 15 15 15 14 14 13 11tprob 0.79 0.43 0.17 0.24 0.17 0.98 0.42 0.18 0.54 0.20 0.89 0.03Renewal 13 11 12 12 15 15 15 16 15 15 15 14tprob 0.00 0.00 0.21 0.31 0.92 0.76 0.58 0.55 0.56 0.54 0.00 0.00Conditional 10 9 11 12 14 14 15 14 14 14 13 12tprob 0.88 0.98 0.58 0.86 0.12 0.75 0.94 0.24 0.68 0.20 0.52 0.45

Observed 4 3 4 4 4 4 5 4 4 5 3 4Markov 3 3 4 3 4 4 4 4 4 3 4 3Fprob 0.56 0.75 0.83 0.35 0.28 0.25 0.51 0.95 0.83 0.02 0.19 0.64Renewal 4 3 4 4 4 4 45 4 4 5 3 4Fprob 0.55 0.52 0.63 0.42 0.54 0.18 0.35 0.42 0.80 0.09 0.05 0.82Conditional 4 4 4 3 4 4 4 5 4 4 4 3Fprob 0.77 0.31 0.94 0.29 0.66 0.78 0.47 0.06 0.23 0.33 0.19 0.41

3. Model performanceBefore generating synthetic time series of daily data, parameter files of the precipitationoccurrence and intensity were f i rs t prepared f o r a l l three sites f r o m the dailyobservations. Then we synthesised 100 years o f daily precipitation for each site and foreach o f the rainfall generation models described. I n order t o produce statisticalproperties o f the synthetic data that are close to the "true" distributions o f the rainfallgenerator, long time series are required (Semenov et al., 1998). The longer the time series,

Table 1. Statistical evaluation o f monthly averages and standard deviations betweenobserved and simulated times series for each of the three rainfall generators at Ohakea, of(a) rainfall totals and (b) rainday frequency. Test statistics, either a t-test or F-test, significantat the 0.05 probability level are highlighted in the table..

a. Rainfall totals (mm)

Standard deviations

Thompson & Mullan: Rainfall models in stochastic weather generators 4 1

Rainfall total RaindaysLocation/ Average standard Average StandardRainfall Generator deviation deviationOhakeaMarkov Chain 1 1 1 1

Renewal Process 3 1 4 1Conditional Markov Chain 3RuakuraMarkov Chain 1 1

Renewal Process 1 1 4 1Conditional Markov Chain 1 1LincolnMarkov Chain 1 1Renewal Process 5 4 5Conditional Markov Chain 1 1 1

the more likely that an outcome from statistical testing will provide a significant resultwhen there is a difference between the historical and synthesised data.

Statistical tests o f simulated precipitation amount and frequency were based oncomparisons with historical data. Tests o f monthly and annual means and varianceswere performed using t and F tests, respectively. Monthly and annual rainfall data usuallyshow a small degree o f skewness, but we have not transformed the data to make thedistributions "more normal" since we are assessing the relative performance o f eachrainfall generator. Sequences of wet and dry spells were tested with a X2 test. For thesethree statistical tests, the significance level is calculated. A small probability (e.g. 0.05 or0.10) indicates a significant difference existing between the historical and synthetic timeseries, and by implication, the simulation is poor.

An evaluation o f the generators, comparing historical and simulated rainfall fo rOhakea, is given in Table 1 for the monthly averages and standard deviations of rainfallaccumulations and raindays. The two statistical analyses (t and F tests) were performedto test whether there were significant differences between the observations and thesimulations. Table 1 also shows the probability of achieving the test statistic by chance.Note that even for a "correct" simulation of climate, we would expect a p-value below0.05 for one in twenty tests. For the three rainfall generators, the comparison shows thatsignificant differences do exist between the historical and simulated records This can beseen more readily in the summary Table 2 (for all three sites). T h e renewal process

Table 2. Frequency of significant differences (at the 0.05 probability level) between historicaland synthetic time series in mean (Av) and standard deviation (std) o f the monthly andannual values of rainfall totals and raindays. Results shown for each of the three rainfallgenerators for a 100-year simulation at Ohakea, Ruakura and Lincoln. A non-occurrence inany column indicates a zero occurrence.

42 W e a t h e r and Climate 21

1200

1000

o>, 800co 6000-

400u_

200

1200

1000

800a) 6000-co 400

200

Figure 2a. Frequency distributions of spells o f consecutive days of no rainfall for thesimulated record by the three rainfall generators and for the historical record at Ohakea.

0

Dry day spells at Ohakea

2 3 4

Wet day spells at Ohakea

Lim

M i n E•tr i mmari —5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5

Spell length (days)

_In Observed• Markov0 Renewal10 Conditiona_l_h—

ri Observed• Markov10 Renewal —13 Conditional —

E r r r i r . -2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5

Spell length (days)

Figure 2b. Frequency distributions o f spells o f consecutive raindays for the simulatedrecord by the three rainfall generators and for the historical record at Ohakea.

model does not perform as well as the other two. Thompson (1997) also noted thiswhen simulating a 500-year record. There is little difference between the conditionalMarkov model and the "traditional" Markov chain.

The length of wet and dry spells in the precipitation record is an important aspect inimpact studies in the CLIMPACTS programme, such as in the assessment of very dryepisodes in the water balances of soils in pasture and crop yields. Figure 2 presents thefrequency distributions o f dry day runs, and wet day runs, for each model and thehistorical record. In order to make comparisons the simulated record has been scaled bythe ratio of the number of years in the historical and synthesised records (e.g., by 37/100for Ohakea). Table 3 provides a summary o f how well each rainfall generator did inproducing the historical sequences o f wet and dry spells. For dry weather spells bothMarkov chain models have a superior performance to the renewal process, which failed

Table 3. Success in simulating observed spells of wet and dry-days for each of the threerainfall generators for a 100-year simulation at Ohakea, Ruakura and Lincoln. The thresholdfor success is set at the 0.05 probability level with a X2 test.

Thompson 8c MuIlan: Rainfall models in stochastic weather generators 43

Location/Rainfall generator Dry spells Wet spellsOhakeaMarkov Chain V(0.43) x(0.04)Renewal Process x(0.00) x(0.00)Conditional Markov Chain V(0.51) x(0.05)RuakuraMarkov Chain V(0.13) x(0.01)Renewal Process x(0.00) x(0.00)Conditional Markov Chain x(0.02) V (0.70)LincolnMarkov Chain V(0.87) V(0.07)Renewal Process x(0.00) x(0.00)Conditional Markov Chain V(0.49) V (0.20)

the Z2 test at the 0.05 probability level at all sites. There is a large contribution to the teststatistic in the very short duration (i.e., less than four days) runs in that too many shortspells are simulated by the renewal process model when compared with the historicalrecord. This is also evident to a lesser extent in the run day distributions for the Markovmodels. The lack of f i t in the models may be related to having an inadequate model ofthe occurrence process that may be resolved by using more complex and higher ordermodels. A t longer durations (i.e., more than 8 days) all the rainfall generators producedistributions o f spells o f dry weather that are similar to the historical record. Similarpatterns can also be seen in the distributions of the spells of wet days.

180 O h a k e a CLO160140

-Es. 120 -E100

ce 4600

20

80

0Jan

MonthsFeb M a r A p M a y J u n J u A u g S e p O c N o v D e c

Markovo Renewal13 Conditional

Figure 3. Monthly rainfall totals for the 90-percentile level at Ohakea for the historical andsynthesised records.

44 W e a t h e r and Climate 21

Quantile values (0%, 10%, 90%, and 100%) were evaluated fo r mean monthlyprecipitation in order to compare extremes derived by simulation with the historical data.Quantiles can provide useful information on the available water resource and itsvariability at a particular site or region. Figure 3 provides the monthly 90 percentile levelsfor Ohakea. Over all months the bias is relatively small for each model. Some monthsthe 90 percentile is overestimated; in others i t is underestimated. O v e r the quantilerange, the bias in general increases from the Oc'/0 quandle to the maximum monthly value(100% quantile) where the models consistently overestimate the maximum monthlyprecipitation. O f the three generators, the renewal process model replicates thehistorical record more accurately than the two Markov rainfall processes.

4. Discussion and ConclusionsThe three rainfall generators have a similar structure in that they use observed data to fitparameters f o r the daily distributions o f the rainfall occurrence process and theprecipitation intensity. A l l models analyse wet and dry days separately and include amechanism for selecting the precipitation state o f each day in the synthesised record.They are a constituent component of the class of weather generators that can synthesiselong records o f daily weather elements. T h e models have been shown capable o fsynthesising daily precipitation data, which also reproduce features observed in thehistorical record. The conditional Markov rainfall occurrence model does not appear tooffer a superior performance to the usual Markov chain approach, although i t doesincorporate some aspects o f longer-period precipitation variations through i tsconditioning process (Wilks, 1989). Although the simulations do reflect the monthly andannual means and (rainday) totals, there is a tendency in all the models to overestimatethe frequencies of the short durations of wet and dry spells. The cause of this is unclear,and it may be related to having an inadequate model of the occurrence process that maybe resolved by more complex models.

Comparisons of weather generators have been carried out for Europe and NorthAmerica (Hayhoe, 2000, Semenov et al., 1998, Wilks, 1999b). Results show that the bestperforming generator varies with the location, and there is not yet any agreement on theoptimum design. Thus it was necessary to carry out a similar comparison for the NewZealand climate. Our finding is that the simplest first-order Markov model performsadequately for the three key sites tested. Wilks (1999b) found that the Markov model wasalso adequate for simulating daily rainfall in the central and eastern United States, but wasinferior to more complex alternative models for western U.S. sites.

Although the conditional Markov model in the New Zealand situation performssimilarly to the simpler first-order Markov process, i t does has the disadvantage o futilising many more parameters. T h i s makes i t difficult to apply to future climates —other than trivial offsets to monthly means, assuming all the other interrelationshipsremain unchanged from the current climate. O n the other hand, Wilks (1992) haspublished a methodology by which the first-order Markov parameters (means and

Thompson & Mullan: Rainfall models in stochastic weather generators 4 5

variances), and also the shape and intensity parameters, can be adjusted for future climatescenarios given down-scaled climate information. T h i s methodology has beenimplemented in the CLIMPACTS programme (Thompson and MuHan, 1999, 2001,Mullan et al., 2001a).

A widely recognised deficiency in weather generators is their general inability tosimulate sufficient interannual variability in rainfall. This deficiency shows up in ourresults in the failure of the weather generators to match the observed monthly standarddeviations o f rainfall totals and rainday frequencies. O f the 19 failures (or 18% of 108months compared over all 3 sites and 3 models, Table 2), 16 o f these are fo r asignificantly smaller standard deviation than the observations indicate. This characteristicis known as "overdispersion" (Cox, 1983), and in spite o f a lot of research attention tothis problem, it has not yet been resolved. To correct the problem, it may be necessaryto improve the component rainfall process model, to relax the assumption o f climatestationarity, or to explicitly parameterise the lower frequency (e.g., seasonal) correlationstructure of the data. The effect of overdispersion when applying the simulated weatherdata to crop models is to produce too many years where the yield is "near normal", withless year to year variation than observed (Alistair Hall, pers. comm.). Further research onthis issue is needed.

Current weather generators only discuss temporal issues for a single site. Recentdevelopments have seen weather generators capable of the simultaneous simulation ofrainfall and temperature over both time and space (see for example Wilks, 1999c),

AcknowledgementThis wo rk has been funded b y The Foundation

Technology, under Contract number C01612.

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