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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:
[email protected]
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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
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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
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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
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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.
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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
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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.
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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
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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.
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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
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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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