CLIMAK: a Stochastic Model
For Weather data generation
Francesco
Danuso and Giuliano Castelli
Dipartimento
di Produzione Vegetale e Tecnologie Agrarie, University of Udine
Via
delle Scienze 208, 33100 Udine(Italy)
Fax +39 432 558603
E-mail:
danuso@dpvta.uniud.it
ABSTRACT
Background. Many human activities and
ecological processes are affected by climatic conditions. Despite the
difficulties in long term meteorological forecasting, a statistical description
of climate is possible and used for planning purposes and strategic decisions.
With this aim, stochastic models for the generation of daily weather data have
been developed (Climak). The generated meteorological data can be used: a) to
perform Monte Carlo simulations with deterministic models (e.g., crop growth
and ecological models, models for climate risk assessment); b) to better
classify the climates; c) to assess environmental scenarios for the effects of
climatic changes by "what if" procedures; d) to spatially interpolate
the climate parameters, so obtaining data for location not covered by
meteorological stations. These data can be used as input for agro-ecological
models, mainly when a probabilistic evaluation of climatic uncertainty and risk
is of interest. Climak has been developed to directly take into account the
among-years variability.
Methods.Climak generates, as first, the occurrence of rain and the rainfall amount for rainy days. After rainfall generation, minimum and maximum air temperatures are generated, with different parameters for rainy and dry days. Solar radiation is obtained from the astronomical photoperiod and from the daily thermal excursion. Evapotranspiration (reference or potential) is generated from solar radiation, if available; if not, it is obtained from photoperiod and daily maximum temperature.
Results. The model was evaluated using
meteorological data coming from different locations of northern Italy and
comparing its capability to reproduce the climatic properties with that of
another weather generator (Wxgen). The behaviour of Climak was quite
satisfactorily even if some minor problems have been highlighted. The minimum
number of years of data to correctly estimate the climatic parameters was also
determined.
Conclusions. Climak was shown to be
satisfactorily accurate in generating meteorological data representing the
climate of a site, even when compared with a well known weather generator. At
present the model still has some limitations and it has not been tested in
climates other then the temperate ones. These drawbacks will be improved in the
next versions of Climak.
Keywords: climate models,
weather generators, software, meteorological data, planning
RIASSUNTO
Scopo. Molte attività umane che
coinvolgono decisioni strategiche e di progettazione dipendono dalle condizioni
climatiche. I modelli stocastici del clima consentono la generazione di dati
meteorologici per esegue simulazioni Monte Carlo con modelli di deterministici
(ad esempio, modelli colturali), classificare i climi; valutare gli scenari
ambientali dovuti ai cambiamenti climatici con procedure "what if" e
spazializzare i parametri climatici per aree lontane dalle stazioni
meteorologiche.
Viene presentato un modello stocastico
del clima (Climak), sviluppato per generare sequenze giornaliere di dati meteorologici
da impiegare come input per i modelli agro-ecologici, e, in particolare, per le
analisi probabilistiche e di rischio.
Metodo. Climak ha una struttura simile ad
altri “weather generators” ma tiene conto direttamente della variabilità fra
gli anni. Climak genera per primo l'evento piovoso (giorno secco o piovoso) e
la quantità di pioggia per i giorni piovosi. Successivamente vengono generate
la temperatura minima e massima dell’aria, con parametri diversi per i giorni
piovosi o secchi. La radiazione solare è ottenuta dal fotoperiodo astronomico e
dalla escursione termica giornaliera. L’evapotraspirazione (di riferimento o
potenziale) è generata dalla radiazione solare se disponibile; se no, viene
ottenuta dal fotoperiodo e dalla temperatura massima giornaliera.
Risultati. Il modello è stato
valutato sui dati meteorologici ottenuti in alcune località del nord Italia, in
confronto con il modello Wxgen, per la capacità di riprodurre le proprietà
climatiche. Il comportamento di Climak è risultato soddisfacente. E’ stata
eseguita anche una valutazione del numero minimo di annate meteorologiche
necessarie a stimare correttamente i parametri climatici.
Conclusioni. Climak è risultato
accurato nella generazione di dati meteorologici di una determinata località,
anche se confrontato con un generatore climatico ben noto ed usato da tempo. Al
momento il modello presenta alcune limitazioni: per esempio, non è stato
sottoposto a test in climi diversi da quelli temperati. Altri aspetti saranno
migliorati nelle prossime versioni del programma.
Parole chiave: clima, modelli
stocastici, generatore climatico, software, dati meteorologici.
INTRODUCTION
The climate of a site is the resultant of
variations in some relevant climatic factors such as air temperature, precipitation,
solar radiation and evaporative demand of the atmosphere. Over time, the
weather factors show periodical trends (daily, annual, multi-annual) and
stochastic variability. The randomness level of climatic variables can be very
different; for example, the rain event can be considered as a purely stochastic
process while the daylight duration (photoperiod) can be considered as a
completely systematic factor. The weather factors also show correlation and
dependence on the previous values of the same and other variables.
Many human activities and ecological
processes are controlled by the climatic conditions, and so great efforts have
been devoted to weather forecasting. Intrinsic limits prevent the long term
forecasting but a statistical description of the climate is possible. This
knowledge allows climatic stochastic models for the generation of weather
data to be developed (Jones et
al., 1970; Richardson, 1981; Larsen and Pense, 1982; Wgen, Richardson and
Wright, 1984; Shu Geng et al., 1985;
Wxgen, Richardson and Nicks, 1990; USClimate, Hanson et al., 1994; MTCLIM, Thornton et
al., 1997; ClimGen, Stockle et al.,
1998; Skills and Richardson, 1998; Parlange
and Katz, 2000; GEM, Johnson et al.,
2001). The generated meteorological data can be used to perform Monte Carlo
simulations with deterministic models (e.g., models of crop growth and of
ecological processes, models for the climate risk assessment). The climatic
models also permit the classification of different climates, the generation of
meteorological data for sites not covered by meteorological stations and the
study of the effects of climatic
changes by "what if" methods (Wilks, 1992).
A model (Climak; Danuso and Della Mea,
1994) for the generation of daily
values for rainfall (Prec, mm/d), minimum
air temperature (Tn, °C), maximum air
temperature (Tx, °C), integral of
solar radiation (Rad, MJ/m²/d) and
evapotranspirative demand of the atmosphere (ETr, mm/d) is described. The weather generation procedure involves
the estimation of climatic parameters from the historical meteorological
recordings and the weather generation from deterministic models and residuals
sampled from probability distributions. The scheme of the whole generation
procedure is reported in figure 1.
Climak was applied to the regional scale
(Danuso et al., 1997) by creating a
grid of climatic parameters for the plains of Friuli-Venezia Giulia to be used
to perform Monte Carlo simulations with agro-ecological models.
MODEL
Rainfall
Rain is generated first because it is
required for the generation of the other variables. Two aspects are considered:
the occurrence of the rain event (if a day is rainy or not) and, for the rainy
days, the amount of rainfall.
a) Rain
occurrence
Model. The rain event
is assumed to be a stochastic process, represented by a first order Markov
chain (Larsen and Pense, 1982;
Richardson, 1985). The state of each day (rainy or dry) is obtained from the
state of the previous day and from the dry to dry (Pdd) and rainy to dry (Prd)
transition probabilities. The complementary
dry to rainy (Pdr) and rainy
to rainy (Prr) probabilities are
computed as:
Pdr = 1 - Pdd and Prr = 1 - Prd
The transition probabilities are considered
on a monthly base and then the model requires 24 parameters for the rain event
generation (12 for Pdd and 12 for Prd).
Parameter estimation. The transition
probabilities are calculated on a monthly base and are estimated, for a
location, on all the available recordings in the data set as:
Pdd=Ndd/Nd and Prd=Nrd/Nr
where:
Ndd number of dry days in the month, preceded by a dry day;
Nd total number of dry days in the data set, for the month;
Nrd number of dry days in the month preceded by a rainy day;
Nr total number of rainy days in the data set, for the month;
b) Rainfall
amount
Model: The rainfall
amount distribution typically follows a negative exponential-like distribution,
with parameters varying during the course of the year. All these distributions
are limited to positive values and allow also the generation of rare intense precipitation.
The most used probability density function (pdf) for the rainfall amount
generation is the Gamma distribution (Jones et
al., 1970; Larsen and Pense, 1982) but exponential and mixed-exponential
distributions are also used.
In Climak the precipitation amount is
generated by sampling from a two parameters Gamma pdf:
if x>0 and Ag>0 and Bg>0;
otherwise Gamma(x)=0
where G(Ag) is the gamma function. Ag
and Bg are specific parameters for
each month. Moreover, because the rain gages cannot detect rainfall below a
sensitivity threshold (Sthr, about
0.2 mm), the rainfall data are corrected for this threshold.
The total number of parameters needed to
describe the rainfall amount is 25 (Sthr;
Ag and Bg for each month).
Parameter estimation. Before
estimating the parameters of the Gamma distribution, the sensitivity threshold
of the instrument (Sthr) is
subtracted from rainfall data, in order to obtain a distribution starting from
0. Then the Ag and Bg parameters are estimated, on a monthly
base, by the method of the moments (Ag=M²/V;
Bg=V/M, where M is the mean and V the variance of the daily rainfall amounts). Often, for the
estimation of the Gamma pdf parameters, the polynomial approximation of maximum
likelihood from Greenwood and Durand (1960) has been used (Larsen and Pense,
1982; Shu et al, 1985). However, the
method of the moments was here adopted because, for our data, it showed
slightly better in reproducing the original climate statistics.
Air temperature
Model: Air temperature
has a strong dependence on the rainfall status of the day and then it is
normally generated with different parameters for rainy and dry days. Moreover
the yearly trend of minimum and maximum temperatures show, in high latitudes, a
quite regular sinusoidal shape. The approach to calculate daily minimum and
maximum temperatures uses the following steps: 1) minimum and maximum air
temperature, both in rainy and dry days, are considered separately; 2)
depending on the rain status of the day, the mean value of minimum temperature
for that day, is calculated from the proper trend; 3) a random residual is
added to the value of 2); this residual
is sampled from a bivariate normal distribution, autocorrelated with the minimum
temperature residuals of the previous day (first-order autoregressive process);
4) depending on the rain status of the day, the mean value of maximum
temperature for that day is calculated, from the proper trend;
5) a random residual is added to the value
of 4); this residual is sampled from a
bivariate normal distribution of minimum and maximum temperature residuals,
taking into account the correlation between the residuals of minimum and
maximum temperature for the same day.
The minimum (Tn) and maximum (Tx) air
temperatures are obtained as:
Tn=Tnd+Rn and Tx=Txd+Rx if
the day is dry,
Tn=Tnr+Rn and Tx=Txr+Rx if
the day is rainy,
where Tnd
is the average daily minimum temperature for the dry days, obtained as a
function of the date, by interpolating:
Tnd=a+b·sin(Date·2p/365)-c·cos(Date·2p/365)+d·sin(Date·4p/365)-e·cos(Date·4p/365)
This expression
(Carlini, 2003; SIPEAA 2003) is equal to the second order Fourier series
of the type:
Tnd=A+B·sin[(Date-C)·2p/365] + D·sin[(Date-E)·2p/182.5]
by putting:
a=A A=a
b=B·cos(2pC/365) B=(b2+c2)1/2
c=-B·sin(2pC/365) D=(d2+e2)1/2
d=D·cos(4pE/365) C=365/(2p)·acos(b/B)
e=-D·sin(4pE/365) E=365/(4p)·acos(d/D)
where:
A annual average minimum temperature of
the dry days (°C);
B semi-amplitude of the first term (°C);
C phase shift for the first term (days),
assumed as a constant for a location;
D semi-amplitude of the second term (°C);
E phase shift for the second term
(days), assumed as a constant for a location;
Date date as day of the year (from 1 to 365).
A, B and D are obtained by sampling, at the beginning of each year, from the
normal distributions N(MA,SA), N(MB,SB), N(MD,SD),
so determining the specific trend for each year. Since statistics of measured data do not show correlations among
parameters, independent samplings are used.
Analogous calculations, with the specific
parameters, are made for the trends of minimum temperatures for rainy days (Tnr), the maximum temperatures for dry
days (Txd) and maximum temperature
for rainy days (Txr).
Rn is a random residual sampled from a
bivariate normal distribution N2(0,n) with
0
vector of means and n covariance matrix of residuals from the minimum
temperature
of the previous day and the same of the present day. The dispersion
matrix
corresponds to:
where:
SRn standard
deviation of the residuals from the trends of the minimum
temperature, both in dry and rainy days;
RRnn autocorrelation
coefficient for minimum temperature residuals, with
time lag of 1 day.
Rx is a random residual sampled from a
bivariate normal distribution N2(0,x),
where
the dispersion matrix x is:
where:
SRx standard
deviation of the residuals from the trends of the maximum
temperature,
both in dry and rainy days;
RRnx correlation
coefficient between minimum temperature residuals and
maximum
temperature residuals of the same day.
The model requires 80 parameters to describe
the air temperature. Of these, 32 parameters are used for the Fourier series (C, E,
MA, MB, MD, SA, SB
and SD of the four different
temperature trends Tnd, Tnr, Txd,
Txr), while 48 parameters are used to
characterise the monthly distributions of the residuals (SRn, SRx, RRnn, RRnx).
Parameter estimation. The yearly
trends of the four combinations of temperature are obtained by the following
procedure:
1) The values of a,
b, c, d and e are estimated using linear regression, C
and E are calculated and are assumed to be constant among years, for a
specific site.
2) Year by year, the values of A, B
and D are estimated using linear
regression and assuming C and E as constants.
3) The parameters for the normal
distributions of A, B and D are then calculated. The mean values for A, B and D are obtained as:
MA=Aj/n, MB=Bj/n and MD=Dj/n
where: j= index of the year: j=1,..., n
n= number of years in the data set
The standard deviations for A (SA),
B (SB) and D (SD) are obtained as:
, 
and 
4) By using the values of A, B
and D from linear regression on the
data set of each year, the trends of the temperatures are obtained and then the
residuals from minimum (Rn) and
maximum temperature (Rx) are
calculated. While four temperature trends are calculated (depending on the rain
status of the day) for the residual distributions no distinction is made
between rainy and dry days .
5) Subsequently the residuals are evaluated
on a monthly base. It is assumed that there is a normal distribution of
residuals and no difference between the residual distributions of dry and rainy
days. For each month, the procedure computes the standard deviation of Rn and Rx (SRn, SRx), the autocorrelation of Rn residuals with those of the previous
day (RRnn) and the correlation
between Rn and Rx (RRnx). This specific
correlation, instead of others (e.g.,
autocorrelation of Tx or
cross-correlation) was selected because it showed the highest values.
Given that the residuals are obtained from
the specific trend of each year, the among year variability of the Rn and Rx distributions has not been considered. The variability of the
trends is taken into account by calculating the variability among years of A, B
and D.
Solar radiation
Model. The maximum
(potential) daily integral of solar radiation (Rmax) strictly depends on photoperiod (Ph), while the actual radiation at soil level (Rad) also depends on the cloudiness of the day. Cloud data are not,
in general, available; to evaluate the atmosphere transmittance (related to
cloudiness) the effect of cloudiness on daily thermal excursion (Es=Tx-Tn)
is considered. Normally, on cloudy days the maximum temperature is low because
sun radiation is strongly intercepted and the minimum temperature is high
because the long wave irradiation from the soil during the night is reduced by
the moisture of the atmosphere, so reducing Es.
Radiation is modelled in two steps: the first
calculates the annual trend of the maximum daily radiation (Rmax). This was found to be well
linearly related to the duration of the photoperiod (figure 2); this relation
is considered to be the same for all the years of each location. The second
step evaluates the distribution of the ratio between the daily radiation and Rmax. This ratio (Rr=Rad/Rmax), ranges from 0 to 1 and is
linearly related to the daily thermal excursion (Es). Moreover, the distribution of Rr values for the different Es
classes was found to be a Beta pdf (figure 3). The equation for this distribution
is:
if 0£x£1 and Ab>0 and Bb>0;
otherwise Beta(x)=0
The Beta pdf, being defined for real values
ranging from 0 to 1, fits well the different shapes of the Rr distributions at varying Es.
Ab and Bb are parameters, calculated respectively
with an exponential and an hyperbolic regression above the specific values for
each of the twenty Es classes adopted.
The model needs
six parameters for the generation of the solar radiation: two for the
calculation of Rmax from the photoperiod, two for the best exponential
fit on the twenty values of Ab, and two for the best hyperbolic fit on the
twenty values of Bb.
Parameter estimation A preliminary
step for the estimation of the radiation parameters is the calculation of the
daily astronomical photoperiod depending on latitude and the day of the year.
This is performed with the method described in Keisling (1982). The parameters
of the linear relation between Rmax
and Ph are obtained by selecting only
the maximum values of the solar radiation
in ten-day periods of the year: Rmax
= b1·Ph + b0
Subsequently the program calculates the ratio
between the daily radiation Rad and
the estimated function for Rmax (Rr). The radiation data are then divided
into twenty classes, depending on twenty classes of daily air temperature
excursion. For each class and from the ratio Rr the two parameters of the Beta distribution are evaluated, using
the moments:
Ab=M²·(1-M)/V - M
Bb=Ab·(1-M)/M
where M
and V are the mean and variance of Rr, for each excursion class.
Then an exponential fit is adapted on the
twenty Ab values and two best fit parameters are obtained. In a similar way a
hyperbolic fit is adapted on the twenty Bb values and two other best fit
parameters are obtained.
Evapotranspiration
Model. The reference
evapotranspiration always shows a good linear relation with the radiation
(Doorembos and Pruitt, 1977); the dependence on maximum air temperature and
photoperiod is less good. Given that radiation data are often not available in
the historical meteorological series, two different approaches are adopted,
depending on availability of radiation data:
a) if solar radiation data are available, the
daily reference evapotranspiration is obtained as a linear function of the
daily radiation, plus a residual from a normal distribution (unique for all the
months) with standard deviation Setr: ETr=a1·Rad + a0 + N(0,Setr)
b) if radiation data are not available, ETr is modelled as a function of maximum
air temperature and photoperiod. The following relation was found to give a
good fit:
ETr= c0
+ c1·Tx·Ph² /1000+N(0,Setp)
where Setp
is the standard deviation of the residuals, related to the photoperiod by a
linear function: Setp=d1·Ph + d0. In
this case the standard deviation of the residuals is not considered as a
constant, but linearly depending on the photoperiod.
Three parameters are then needed for the
generation of the reference evapotranspiration if radiation data are available
and 4 in the other case.
Parameter estimation: the estimation
of the parameters a1 and a0 is simply made by linear
regression of ETr vs. Rad. Then, the standard deviation of the
residuals (Setr), obtained
subtracting the predicted data from the measured ones, is evaluated.
With no available radiation data, c1 and c0 come from a linear regression of ETr vs. Tx·Ph². The
residuals of this regression show a distribution with a variance increasing
with the photoperiod. Then the standard deviation of residuals is calculated
separately for each of 12 photoperiod classes. These values are then regressed
against photoperiod to give the d0
and d1 parameters.
It should be noted that the procedure for
evapotranspiration can be applied to simulate every variable representing the
evapotranspirative demand of the atmosphere (e.g., without limitation due to
water shortage). So, data of “reference evapotranspiration”, “potential
evapotranspiration”, “evapotranspiration from class A pan” and others can be
generated, depending on the data set used to estimate the parameters.
Procedure for weather data generation
Climak generates first the occurrence of
rainy or dry day and the rainfall amount if the day is rainy (figure 1). After
rainfall generation, minimum and maximum air temperature are generated,
separately, for rainy and dry days. Solar radiation is obtained from the
astronomical photoperiod and from the daily thermal excursion. If radiation
data are available, the evapotranspiration is generated from solar radiation;
if not, it is obtained from photoperiod and maximum temperature.
The rain
event is generated by sampling, for each day to be generated, a random
value from the uniform distribution U(0,1).
If the current day is dry, the Pdd
probability is used; if the sampled value U(0,1)
is less than Pdd the following day is
set to "dry", otherwise it is set to "rainy". The same
procedure is adopted with the Prd
transition probability if the present day is rainy. The considered probability
is that specific for each month.
On the rainy days, the rainfall amount (Prec) is
obtained by sampling a value from a G(Ag,Bg)
pdf and adding the threshold value for the instrumental sensitivity:
Prec=G(Ag,Bg)+Sthr
where the parameters Ag and Bg are specific
for each month.
The sampling routine for the Gamma
distribution was implemented merging two algorithms: the first allows the
generation from a Gamma pdf when the parameter Ag is less than 1 (RGS algorithm; Ahrens et al., 1972; Tadikamalla, 1981; Bratley et al. 1984); the second is used when Ag is greater than or equal to 1 (G3A algorithm; Fishman, 1978).
In general, the rainfall amount distribution
has the form of a Gamma pdf with Ag<1
(negative exponential-like); the complete routine has been implemented because
it is used also for the generation of the solar radiation.
The generating method for the air temperature values is strictly
related to the procedure for the estimation of the parameters. After rainfall
generation, the minimum and maximum temperatures are generated separately,
depending on the status of the day (rainy or dry).
At the beginning of a new year the program
samples a value for the parameters A,
B and D from the normal probability distributions N(MA,SA), N(MB,SB)
and N(MD,SD), for each
temperature combination, in order to obtain the annual trends of temperatures
by the Fourier series. C and E are considered to be constant for all
the years while means and standard deviations of A, B and D are different in relation to the year
and for the minimum/maximum and rainy/dry combinations.
Using these parameters, the annual trend of
minimum and maximum air temperature on dry and rainy days is generated, for
each year; then, month by month, each temperature is obtained by adding the
residuals to the trends. The residuals are obtained as follows:
- The minimum temperature residual (Rn) is sampled from the autocorrelated
normal distribution with 0 mean, SRn
standard deviation and RRnn
autocorrelation:
where:
Rn minimum temperature residual of the
current day;
R1n residual of minimum temperature of the
previous day, already generated;
N(0,1) value sampled from a normal distribution with
0 mean and 1 standard deviation;
- The maximum temperature residual (Rx) is sampled from the bivariate normal
distribution of parameters 0 mean, SRx
standard deviation and RRnx
correlation coefficient, depending on the value Rn:
The generation of radiation data is based on air temperature excursion and so a
previous generation of air temperature is required. As in the estimation phase,
Climak calculates first the daily photoperiod (Ph). From Ph, the maximum
daily radiation Rmax is obtained as Rmax=b1·Ph+b0, for each day of the year.
After that, for each day, a value of Rr is generated by sampling from a Beta
probability distribution: Rr=B(Ab,Bb),
where Ab and Bb are parameters of the Beta pdf, chosen in relation to the
temperature excursion of the day. For the Beta generation, a well-known
property of this distribution is used:
B(Ab,Bb)=G(Ab,1)/(G(Ab,1)+G(Bb,1))
where B(Ab,Bb) is a value extracted from the
Beta pdf and G(Ab,1) and G(Bb,1) are values from the Gamma
distribution. The generation of a Beta pdf is made by using the algorithm for
the Gamma generation twice and setting Ab
or Bb to 1.
Finally, the daily solar radiation is
calculated as Rad = Rmax × Rr.
If radiation parameters are available, the
daily reference evapotranspiration is
obtained, after the generation of radiation, by generating a value from the
normal distribution N(0,Setr) as:
ETr= a0
+ a1·Rad +N(0,Setr)
When the radiation parameters are missing,
Climak calculates ETr from maximum
temperature and photoperiod. The standard deviation of the ETr residuals (Setp) is,
in this case, obtained as a function of the photoperiod as: Setp
= d1·Ph+d0
The residuals are then generated from a
normal distribution N(0,Setp) and the daily reference
evapotranspiration is obtained from maximum air temperature, photoperiod and
residuals as:
ETr= c0
+ c1·Tx·Ph²/1000 + N(0,Setp)
MODEL IMPLEMENTATION
Program use
The current version of Climak (2.0, 4
November 2003) is a GUI (Graphical Using Interface) program that is able to
estimate parameters (text files with cmk
name extension) from historical data (text files with met extension) and to generate meteorological data (text files with
gen extension) from the parameter
file (figure 4).
The generation of meteorological data is
performed by using the climate parameters contained in the parameter files.
Only the variables for which the related parameters are in the parameter file
are generated: that is, if some values are missing, the corresponding climate
variables are not generated. As a minimum, Climak can generates only the
rainfall. The generation of temperature requires rainfall; radiation requires
temperature and evapotranspiration requires radiation or temperature (figure
1).
Structure of the meteorological data files
The meteorological data files of Climak are
text files with name extension met (recorded
data) and gen (generated data). All
the required columns have to be present, even if data are missing. Missing
values have to be indicated by a full stop (.). The meteorological data files
have seven columns (variables) which must follow the order: year code, day of
the year, precipitation, minimum air temperature, maximum air temperature,
solar radiation, reference evapotranspiration. If a variable is not available,
its column has to be filled with full stops. The column of year and day of the
year cannot contain missing values, while the other variables can.
The minimum set of variables required for the
estimation of a parameter set is formed by the year code, the day of the year
and the rainfall; in this case only the rainfall parameters can be estimated.
To estimate also the temperature parameters, rainfall and both minimum and
maximum temperatures have to be supplied. The estimation of the radiation
parameters needs the temperature variables. The estimation of the
evapotranspiration parameters requires the radiation variable or the
temperature variables; when the radiation variable is present, by default, the
parameters of ETr are obtained from
radiation.
Structure of the parameter file
The generation of meteorological data is
performed by using the climatic parameters contained in the parameter files.
Climatic parameters are estimated by the Climak program from recorded
meteorological data.
The parameter files are text files with cmk extension and 148 rows of numeric
data. Climak recognises as comment each line starting with an asterisk.
The file of estimated parameters has one
column of 148 elements. Each parameter has a definite position in the column.
Parameters that cannot be estimated are indicated by a full stop (missing
value).
A complete set of parameters is of 148 items.
Of these, 6 are used for the general site description, 49 for the rainfall
generation, 80 for the temperatures, 6 for the radiation and 3 or 4 for the
reference evapotranspiration (depending on the generation method).
Only the variables for which the related
parameters are in the parameter file are generated: that is, if some parameters
are missing, the corresponding climate variable is not generated.
Latitude (as degrees and decimals) is used
for the photoperiod calculations and then its value has an effect on the
generation of radiation and on evapotranspiration. Longitude and altitude serve
only for documentation or when a spatial interpolation has to be performed.
The structure of the parameter file is
reported in tables 1, 2, 3 and 4.
MODEL EVALUATION
Number of years needed to estimate Climak parameters
The goodness of the generated weather data
depends on the model but also on the quality of the estimate of parameters.
This depends, in turn, on the estimation method and on the number of year of
data available for the estimation.
An evaluation of the minimum sample size to
estimate the Climak parameters was carried out. If the meteorological data
comes with too short a time period, the estimated values may not correctly
represent the statistical properties of the climate of the site and so the
generated meteorological data should be used with caution.
The minimum number of years needed to give a
correct estimate of the parameters depends on the degree of variability of the
climate itself and on the specific parameters. For example, rainfall values are
known to present higher variability than temperature values.
Using the historical meteorological data
series of Udine (from 1960 to 1989) the estimation of parameters has been
performed, with the specific Climak routine, using a sample of years increasing
size from 5 to 30 years. In figure 5 the estimated values for Bg (Gamma parameter for the generation
of rainfall amount), for each month, of different series length are reported.
For the transition probability which controls the rain events, Pdd requires, in the average of months,
less year data to obtain a stable estimate than Prd (15 and 20 years, respectively). The Pdd and Prd values are
more easily estimated for the winter and spring months (about 10 years are
enough). More years are progressively required for the summer and fall months
(up to 20 years).
The estimation of the annual temperature
trend parameters (MA, MB, MD,
C, E) were good with just 10-15 years, for all the Tn-Tx
and dry-rainy combinations.
Performance of Climak and Wxgen
The aim of the weather generators is to
generate meteorological sequences having the same statistical properties as the
actual ones. Thus, there are two ways to evaluate the goodness of the generated
data: the first involves the statistical comparison of parameters of the
observed and generated data sets (centrality and dispersion parameters,
distributions). Another, pragmatic, way considers the generated data as
acceptable when a dynamic simulation model, requiring meteorological data as
input, gives the same results with both recorded and generated data.
Climak has already been evaluated and
compared with other weather generators also by other authors. Acutis et al. (1998) evaluated
Climak as compared with ClimGen (Stockle et
al., 1998) and with Wxgen-Vxparm (Berndt and Williams, 1991) using
long-period series from different localities: Akron (USA), Haarveg (NL), Modena
(IT), Tolouse (FR) and Udine (IT). The behaviour of the models was judged
satisfactorily.
In the present paper, a comparison between
recorded and generated data has been made. The behaviour of Climak seems quite
satisfactory, particularly for the generation of temperature and radiation.
Table 5 reports a comparison between annual statistics derived from historical
data recorded at Padua (North-East Italy) and those generated by Climak and by
another weather stochastic generator (Wxgen, Richardson and Nicks, 1990). The
climatic parameters for Wxgen have been estimated by the program Wxparm (Berndt
and Williams, 1991). Climak gives quite satisfactory results, particularly for
the temperature and radiation generation, even considering the monthly
statistics of generated values. Both the models give very similar annual
statistics that are not statistically different from those of the original data
(figures 6, 7, 8 and 9).
While the normal rainfalls are well
reproduced by both generators (figure 9), they are not able (Climak less than
Wxgen) to reproduce the extreme values of observed rainfalls. So, other
generation methods should be applied to generate those values (for example,
sampling from a Poisson pdf).
CONCLUSIONS
Climak was found
to be sufficiently accurate in generating meteorological data that represents
the climate of a site, even when compared with a well known weather generator.
The minimum
number of years needed to give a correct estimate of the parameters depends on
the degree of variability of the climate itself and on the specific parameters.
For example, rainfall required more years of data than temperature in order to
be correctly estimated.
In a previous evaluation (Acutis et al., 1999) it was found that Climak
did not reproduce the climate as well as Climgen, when using short series to
estimate parameters. This is due to the characteristics of Climak, as it was
especially designed to cope with the year-to-year variability. In effect, when
using long series to estimate the parameters, Climak is better at reproducing
the climate itself (Acutis et al.,
1998).
The goodness of a weather model basically
depends on its model structure, on methods and algorithms for parameter
estimation and on algorithms for data generation (sampling from pdf). The
algorithms implementation in the computer program plays a major role in model
quality, particularly for the sampling phase. So the weather generator was
evaluated for the real performance of the software. At present the model has
some limitations. It has not been tested in climates other then the temperate
ones. So care should be taken when using the weather generator in such
situations.
The model is “unit-free” for rainfall,
radiation and evapotranspiration but temperature must be in Celsius degrees.
This is because absolute values of thermal excursion are used in radiation
calculation. Instead, the units or rainfall, radiation and evapotranspiration
can be other than those used in this study. This aspect will be fixed in the
next versions. Spatial co-ordinates (latitude and longitude) are also required
as degrees and decimals because latitude is used for the photoperiod
calculation.
In general, the behaviour of the model is
considered satisfactory but some aspects could be improved and inserted into
the next versions of Climak: a) generation of other variables such as dewpoint
temperature, wind speed, etc.; b) the variables (especially, temperature and
rainfall) at a higher time resolution (hourly values, within-storm rainfall
intensity, etc.); c) account for spatial correlation in order to generate
realistic data on a regional or watershed bases; d) developing methods to
spatialise the climatic parameters in hill or mountain conditions; e) a link to
global circulation models (GCM) which can supply parameters for the weather
generator derived from climate scenario forecasting; f) the use in the model of
multi annual trends, related, for example, to the solar spot cycle or to the El
Niño Southern Oscillation (ENSO, Woolhiser et
al., 1993) to obtain the climatic parameters, thus providing also a sort of
climate forecasting; g) the creation of regional georeferenced database of
climate parameters of Climak, particularly for hill conditions.
ACKNOWLEDGEMENTS
Work supported by the Italian Ministry of the
University and of the Scientific and Technological Research (MURST - 60%). The
authors wishes to thank Vincenzo Della Mea for his help in developing and
programming the first version of Climak, and Laura Carlini for the temperature
function linearization. The Istituto di Agronomia e Coltivazioni erbacee of the
University of Padua and the Ufficio Idrografico del Magistrato alle Acque are
also gratefully acknowledged for supplying the meteorological data used for the
model evaluation.
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