Global solar radiation in Central European lowlands estimated by

Agricultural and Forest Meteorology 131 (2005) 54–76
www.elsevier.com/locate/agrformet
Global solar radiation in Central European lowlands
estimated by various empirical formulae§
Miroslav Trnka a,*, Zdeněk Žalud a, Josef Eitzinger b, Martin Dubrovský c
a
Institute of Agrosystems and Bioclimatology, Mendel University of Agriculture and Forestry Brno,
Zemědělská 1, 613 00 Brno, Czech Republic
b
Institute for Meteorology, University of Natural Resources and Applied Life Sciences, Vienna, Austria
c
Institute of Atmospheric Physics, Czech Academy of Sciences, Prague, Czech Republic
Received 20 April 2005; received in revised form 23 April 2005; accepted 10 May 2005
Abstract
Seven methods for estimating daily global radiation, RG, were tested in the Central Europe case study area (lowlands of
Austria and the Czech Republic) assuming that no measured global radiation data for parameterisation are available, i.e. with all
empirical coefficients required by the selected methods being obtained from previously published studies. The variability
explained (R2), the root mean square error (RMSE) and mean bias error (MBE) indicated that the highest precision could be
expected when sunshine duration was used as predictor. The method generally known as the Ångström–Prescott method
explained 96% of the RG variability with the RMSE value (annual mean) equalling 1.6 MJ m2 day1 and MBE being
0.1 MJ m2 day1. It was found to be ultimately the best of all tested methods. Where there were no reliable estimates of the
empirical coefficients necessary for this equation, the multiple regression method between measured sunshine duration and RG,
was found to perform satisfactory from April to August. Where there were no sunshine duration data, the formula including
cloud term and daily temperature range, yielded a sufficiently precise estimates (R2 = 0.91; RMSE = 2.3 MJ m2 day1;
MBE = 0.2 MJ m2 day1). Where the cloud cover records were not available, the one of the methods employing the total daily
precipitation might be used (R2 = 0.86; RMSE = 3.1 MJ m2 day1; MBE = 0.2 MJ m2 day1). Where the precipitation data
are not available, the temperature-based method despite the relatively large deviations (R2 = 0.82; RMSE = 3.5 MJ m2 day1;
MBE = 0.3 MJ m2 day1) might be considered as an alternative. The missing RG data could also be substituted by the values
measured in a nearby station. The precision of the radiation estimated in this way decreased with increasing distance between the
two stations: R2 decreased from 0.95 to 0.60 as the distance increased from 17 to 369 km. When the annual mean RMSE was
studied it was found that it increased by approximately 0.15 MJ m2 day1 per 10 km in the study region and variability
explained decreased by approximately 1% for the same distance. The RG estimates based on temperature or combination of
temperature and precipitation were biased by about 10% during several months. The value of RMSE of these methods reached
up to 30%, and even the best estimates based on sunshine duration hours were loaded by RMSE to the extent of 10–20% during
the growing season. Therefore, any further application relying on these estimates, especially if they are based on literature
§
The nomenclature of the paper tries to follow the original one used by the authors of the tested methods.
* Corresponding author. Tel.: +420 5 4513 3083; fax: +420 5 4513 3083.
E-mail address: [email protected] (M. Trnka).
0168-1923/$ – see front matter # 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.agrformet.2005.05.002
M. Trnka et al. / Agricultural and Forest Meteorology 131 (2005) 54–76
55
derived coefficients, might be significantly distorted and error propagation analysis is strongly recommended to any user of
estimated RG data.
# 2005 Elsevier B.V. All rights reserved.
Keywords: Solar radiation; Model estimates; Accuracy; European lowlands
aA
aH
Ångstöm empirical coefficient (Eq. (1))
Hargreaves empirical coefficient
(Eq. (7))
aS
Supit empirical coefficient (Eq. (3))
bA
Ångstöm empirical coefficient (Eq. (1))
bD
empirical parameter (Eq. (6))
bH
Hargreaves empirical coefficient
(Eq. (7))
bS
Supit empirical coefficient (Eq. (3))
cS
Supit empirical coefficient (Eq. (3))
Cw
mean total cloud cover during daytime
observations (Eq. (3)) (octas)
d
distance to a nearby station providing
data (Eqs. (10) and (11)) (km)
D
day of the year (Eq. (2))
Dl
function correcting the effect of site
differences in day length (Eq. (4))
eS(Tmax) saturation vapour pressure at temperature Tmax (Eq. (4)) (kPa)
eS(Tmin) saturation vapour pressure at temperature Tmin (Eq. (4)) (kPa)
f(Tavg) function based on the daily mean
(Eq. (6))
f(Tmin) function based on the minimum temperature (Eq. (6))
MBE mean bias error (Eq. (9))
(MJ m2 day1)
n
actual sunshine duration (Eqs. (1) and
(2)) (h)
N
potential sunshine duration (Eqs. (1)
and (2)) (h)
Qest
estimated global radiation (Eqs. (9) and
(10)) (MJ m2 day1)
Qobs
observed global radiation (Eqs. (9) and
(10)) (MJ m2 day1)
2
R
coefficient of determination
RA
extraterrestrial radiation (MJ m2
day1)
RA,proxy extraterrestrial radiation at a nearby
station (Eq. (8)) (MJ m2 day1)
RG
global solar radiation (MJ m2 day1)
RG,proxy global radiation at a nearby station
(Eq. (8)) (MJ m2 day1)
Rmax
total radiation under a real atmosphere
for completely clear days
(MJ m2 day1)
RMSE root mean square error (Eq. (10))
(MJ m2 day1)
Tmax
maximum daily temperatures (8C)
Tmin
minimum daily temperatures (8C)
Tnc
empirical parameter (Eq. (6))
yMBE
mean bias error symbol used in Eq. (9)
(MJ m2 day1)
d
yRMSE function describing dependence of
RMSE in RG values on distance
(Eq. (12)) (MJ m2 day1)
yRMSE root mean square error symbol used in
Eq. (10) (MJ m2 day1)
Greek letters
b
Winslow empirical coefficient (Eq. (4))
iþ1
i
i
DT
calculated as Tmax
ðTmin
þ Tmin
Þ=2
where i stands for day of measurement
(Eq. (6)) (8C)
t
clear sky transmisivity (Eq. (6))
tcf
atmospheric transmittance function at
(Eq. (4))
tf max cloud correction factor (Eq. (5))
tt,max maximum (cloud-free) daily total transmittance at a location (Eq. (5))
1. Introduction
Daily global solar radiation (RG) is required by
most models that simulate crop growth, because the
growth is primarily based on photosynthetic processes, which involve the utilisation of radiation and
its conversion to chemical energy. Global solar
radiation is also an indispensable input for most
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M. Trnka et al. / Agricultural and Forest Meteorology 131 (2005) 54–76
methods of estimating potential and actual evapotranspiration, which are part not only of crop growth
models; but also of hydrological and soil water
balance models for various spatial scales. It has been
noted many times (Supit and van Kappel, 1998; Liu
and Scott, 2001) that continuous records of global
solar radiation measurements are relatively scarce,
because sufficiently precise observations demand a
high level of maintenance. The ratio between the
number of stations observing daily RG and those
measuring temperature and precipitation is highly
variable from less than 1:10 in Germany (Oesterle,
2001) or 1:20 in the Czech Republic and Austria to
1:500 on the global level (Thornton et al., 1997). By
contrast, the majority of the weather stations register
alternative meteorological variables such as sunshine
duration, cloud cover, air temperature or precipitation.
Therefore, some techniques are required to estimate
RG from other available data for the sites where RG is
not measured or is partly missing. Daily radiation data
at a given site might be either substituted by the data
measured at a nearby station (Nonhebel, 1993; Hunt
et al., 1998), estimated by remote sensing techniques
(Diak et al., 1996; Stewart et al., 1999; Marion and
George, 2001 or Wyser et al., 2002), or produced by
some other method. These methods include stochastic
weather generators (Richardson, 1981; Cooter and
Dhakhwa, 1996; Hansen, 1999), linear interpolation
(Soltani et al., 2003), use of higher order statistics (Safi
et al., 2002), application of neural networks method
(Reddy and Ranjan, 2003), and – the method that this
paper discusses – the application of various empirical
relationships established between daily global radiation and other more frequently measured meteorological parameters (Ångström, 1924; Klabzuba et al.,
1999; Winslow et al., 2001).
While the threshold distance for acceptable RG
data precision from a neighbouring station was
shown to be high in some regions (e.g. more than
385 km for Ontario province as reported by Hunt
et al. (1998)) the studies conducted under European
conditions seem to suggest smaller threshold
distances (depending on topography and other
factors), thus limiting the possibility for using data
from a nearby station (Nonhebel, 1993). For
example, the threshold distance for Central Europe
was estimated by Vanı́ček (1984) to be within 100–
200 km, depending on the station location and
season. In this case the threshold distance was based
on the known precision (5%) of the pyranometers
used by the meteorological services in the study area.
Remote sensing data are still scarce, with only
limited precision for a particular site (Wyser et al.,
2002) and, moreover, they are rarely available for
periods prior to 1980, which limits their possible use.
Stochastically generated data may be useful for
exploring possible model scenarios for an average
theoretical situation using long-term simulations.
However, data derived by this approach cannot be
used for model validation and simulation analysis
during a particular period of time as the method is
not capable of generating data that would match the
actual weather at particular time of interest (Liu and
Scott, 2001). Despite the fact that linearly interpolated RG data might prove to be a good substitute
for a few missing values at a particular location, they
cannot be applied to stations without RG measurement, as average monthly solar radiation values are
required. In spite of promising results reported by
Reddy and Ranjan (2003), the use of neural network
analysis is limited by the nature of the method,
which requires a relatively high number of input
variables and sufficient testing prior to its transfer to
the site distant from the region where the relationships were originally established.
As a result, the most frequently used approach has
been based on empirical relationships that require
the development of a set of equations to estimate
solar radiation from commonly measured meteorological variables. The number of such equations
that have been published and tested is relatively
high, which makes it difficult to choose the most
appropriate method for a particular purpose and site.
In order to assist in the selection process, seven of
these methods were chosen and compared, applying
data from Austrian and Czech stations that are
representative of lowlands in the Central Europe.
The tested methods were selected according to (1)
their data requirements (the selected methods utilise
only daily variables normally available at a majority
of weather stations) and (2) their practical applicability, which was judged mainly by availability of
the necessary empirical coefficients. All of the tested
methods are applicable without the need for
calibration by locally measured RG data, although
such a procedure might increase their precision in
M. Trnka et al. / Agricultural and Forest Meteorology 131 (2005) 54–76
some cases. In addition to the seven selected
methods the error introduced by substituting missing
radiation data by data measured at neighbouring
stations was also examined in order to quantify the
relationship between the estimated precision and
distance of the weather stations. The relationship
was then used to define the threshold distance, below
which such substitution would perform better than
other tested methods.
The underlying approach in most of the methods
currently used is to express solar radiation reaching
the earth’s surface (RG) as a fraction of daily total
extraterrestrial radiation (RA). This is based on the
attenuation of incoming radiation through the atmosphere. The physics involved in the interaction
between radiation and atmospheric constituents is
complex, but the relationship between atmospheric
transmittance and some weather variables can be
described empirically. The main input variables in
these relationships are sunshine duration (Ångström,
1924 modified by Prescott (1940) or Klabzuba et al.
(1999)), temperature (Hargreaves et al., 1985 or
Donatelli and Campbell, 1998), temperature in
combination with cloud cover (Supit and van Kappel,
1998) and temperature in combination with total daily
precipitation (Thorton and Running, 1999 or Winslow
et al., 2001).
One of the study objectives was to evaluate the
accuracy and applicability of seven selected models
for estimating daily RG values across the study area
for different data availability situations at 10
observational sites: (i) sunshine duration, (ii) cloud
cover and temperature, (iii) temperature and total
daily precipitation, or (iv) only temperature data.
The study compared the uncertainties in the selected
RG estimation methods in order to advise potential
users which methods gave the best results under
specific circumstances in the study region or in
comparable environments. The results of this
analysis could serve as a basis for selecting the
most suitable method for estimating missing radiation data. In the present analysis, the empirical
coefficients for all methods were based on literature
sources. This procedure might lead to a much larger
error than those referred to in recently published
studies, which in general operate with equations
parameterised with the help of solar radiation data
measured at the given site.
57
2. Materials and methods
2.1. Data collection
Fig. 1 shows the locations of the Czech and
Austrian stations used in the study. The climate of the
study area is influenced by mutual penetration and
mingling of the maritime and continental effects. It is
characterised by the prevailing westerly winds,
intensive cyclonal activity causing frequent alterations
of air masses and comparatively high precipitation.
The climate is influenced by the altitude and
geographical relief and especially in Austria the
orographical impact of the Alps on the spatial
distribution of cloud cover must also be taken into
account. However, the effect on cloud cover from the
Alps, which may have a significant impact on the
results of the tested methods in our study, decreases
with the distance from the mountain chain. Moreover,
our interest in an alternative estimation of site-specific
global solar radiation focused mainly on the intensively cultivated lowlands (less than 750 m above sea
level), which in Austria account for 50% of the
country. Out of the total country area, 40% is
cultivated, with arable land accounting for 18% and
permanent grassland covering the rest. The topography of the Czech Republic is rather different: more
than 92% of the land is below 750 m, which means
that over 54% of total area can be cultivated, with 72%
arable land. Owing to the fact that most of Central and
Eastern Europe (and virtually all agriculturally
important areas) are less than 750 mm above sea
level, only the stations within this altitude range were
included in the study. Table 1 lists the basic
characteristics of 10 sites with continuous RG
measurements, which were obtained from the databases of the Austrian Meteorological Service (ZAMG)
and Czech Hydrometeorological Institute (CHMI).
These sites have daily records for six years or more of
minimum temperature, maximum temperature, mean
temperature, precipitation, cloud cover, daily total
sunshine and global radiation.
In case of Austrian meteorological stations the
WMO first class Star pyranometer (produced by
PHILIPP SCHENK GmbH Wien & Co. KG, Vienna,
Austria) was used during the whole time period
included in the study. At the Czech stations the CM5
pyranometers (Kipp&Zonnen, Delft, The Netherlands)
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M. Trnka et al. / Agricultural and Forest Meteorology 131 (2005) 54–76
Fig. 1. Map of the study area with the location of the meteorological stations (solar radiation observatories) used.
Table 1
The solar radiation stations used in the studya
Station name
Latitude
Longitude
Altitude (m)
Tmean (8C)
Prec.b
(mm)
Data available in yearsd
Number
of years
Gmunden
Graz
Gross-Enzersdorf
Hradec Králové
Kocelovice
Kremsmünsterc
Kuchařovice
Langenlois
Ostrava-Poruba
Retzc
478550
468580
488120
508110
498280
488030
488530
488280
498480
488460
138550
158260
168340
158500
138500
148080
168050
158420
188150
158550
426
340
153
285
519
383
334
210
242
242
9.9
10.8
10.7
8.5
7.5
9.7
8.5
10.1
8.3
10.2
1261
822
546
612
591
1044
484
473
721
431
1984–1990; 1993–1995; 1997–2001
1990–2001
1996–2001
1984–2000
1985; 1987–2000
1989; 1990; 1993; 1996; 1997; 1999–2001
1984–1999
1992; 1993; 1995; 1996; 1998–2001
1985–1991; 1994; 1996; 1998; 1999
1981; 1982; 1994; 1995; 1997–2001
12
12
6
17
15
8
16
8
11
9
a
Values of Tmean, and Prec. were calculated with data including only those years indicated in the last column i.e. only years with complete
measured data sets (i.e. the period varies in length from 6 to 17 years depending on the station).
b
Mean annual sum of the precipitation at the given site.
c
At these stations the cloud cover has not been measured and therefore Supit and van Kappel method could not be applied.
d
Only those years without any missing data were used in the study—with exception of those stations (Footnote c).
M. Trnka et al. / Agricultural and Forest Meteorology 131 (2005) 54–76
were used until 1993/1994 when the CM11 sensors of
the same company replaced them. Only in case of the
Ostrava-Poruba station the CM5 sensor has been
retained until now due to its sufficient precision. At
both countries solar radiation stations are supervised
and follow vigorous calibration and maintenance
scheme. The sunshine duration hours were measured
by the Campbell–Stockes sunshine recorders at all
stations while the cloud cover was estimated three
times per day by highly trained observers. Thermometers and rain gauges used in the study correspond to
the WMO standards and are regularly calibrated and
properly maintained. Observations of air temperature,
cloud cover, sunshine and radiation either starts at
0700 h with three recording times and 7 h intervals or,
in those stations equipped with the automated
measurement systems, every 10 min, with daily
means being calculated from all readings. The data
records on precipitation relate to the 24-h period
starting at 0700 h. Although at least part of the rainfall
recorded for a given day might fall on the following
day, no adjustment was made. At the beginning of the
study 16 stations with all necessary data were
available. After careful verification of the completeness and reliability of data, the final list of 10 stations
was selected. Weather records from the 10 stations
were then scrutinised again and only those years
without missing or corrupt data were used. In total,
114 complete observation years (i.e. 41 640 observation days) were available for the calculations, with
exception of cloud cover, where only 97 years (i.e.
35 427 observation days) were available.
Daily total extraterrestrial radiation (RA) was
calculated as a function of latitude, day of the year,
solar angle and solar constant according the procedure
suggested by Allen et al. (1998) using the method of
Duffie and Beckman (1980). The procedure included
the relative distance of the Earth from the Sun. This
term, which varied over a range of 3.3%, was not
included by some authors of earlier studies (Bristow
and Campbell, 1984). The generally accepted solar
constant value was used, i.e. 1367 W m2, which
corresponds to 118.1 MJ m2 day1. This value
coincides with the solar constant recommended by
Allen et al. (1998) and was recently confirmed by
Gueymard (2004), who derived a solar constant of
1366.1 W m2 based on the 24 years of observation
data.
59
Following the data check, the distribution of
observed RG values was determined. Eleven intervals
with 3 MJ m2 day1 intervals were defined and the
number of daily records in each interval was plotted
(Fig. 2a). Most of the data was in the 0–12.0
MJ m2 day1 range. In order to assess the proportion
of days with high and low atmospheric transmisivity,
the RG versus RA ratio was also calculated for each day
(Fig. 2b). The closer the ratio to one, the higher was
Fig. 2. Number of the observational days belonging to the selected
categories: (a) distribution of observational days within the predefined interval of daily RG sums; (b) distribution of observational
days according to the RG vs. RA ratio. In both cases number of days
in each category is expressed as a percentage of all observational
days in the database i.e. 41 640.
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M. Trnka et al. / Agricultural and Forest Meteorology 131 (2005) 54–76
the atmospheric transmisivity on the given day. As
shown in Fig. 2b, the distribution of days in individual
categories according to the RG versus RA ratio was
more or less uniform with exception of the values
greater than 0.70.
2.2. Methods
2.2.1. Estimation of solar radiation using hours of
sunshine
Ångström (1924) developed a model for estimating
the solar radiation reaching the Earth’s surface (RG)
applying the relationship between the measured
sunshine duration and the maximum day length and
total radiation under a real atmosphere for completely
clear days Rmax. The original version of the formula is
not suitable for sites where no radiation data are
available, because of the need to know Rmax, which is
only possible with radiation measurements. Therefore,
Prescott (1940) proposed an improved version of the
original formula where RG is based on the fraction of
daily total atmospheric transmittance of the extraterrestrial solar radiation (RA), which is determined as
a fraction of actual (n) and potential sunshine duration
(N) during the day:
n
R G ¼ R A aA þ bA
N
(1)
where aA and bA are empirical coefficients determined for the particular site. Despite the fact that
the constants aA and bA have been derived for many
locations (Martı́nez-Lozano et al., 1984; Supit and
van Kappel, 1998) and various attempts have been
made to model these constants (Rietveld, 1978 or
Abdel-Wahab, 1993), the general applicability of
these methods remains geographically limited (Gueymard et al., 1995). As the present study was not aimed
at deriving the coefficients for the study region; but
rather at evaluating the performance of various RG
models without site-specific parameterisation, the
coefficients were derived from already published
studies. The Ångström–Prescott coefficients were
interpolated using maps available at www.supit.cistron.nl/start.htm (2004), which were produced within
the MARS project and originate from an extensive
database of 89 stations over Western and Central
Europe.
While the Ångström–Prescott method is based on
the linear relationship between the variables and
assumes certain physical meaning of the coefficients,
the second method tested in our study involved a
statistical analysis of the relationship between the
measured RG and the relative sunshine duration. The
equation is based on the observed RG in Hradec
Králové solar observatory from 1960 to 1979 and was
developed by Klabzuba et al. (1999) in the following
form:
n
n
þ 22:9
RG ¼ 7:19 þ 0:258 9:28 106
N
N
ðD 174:7Þ2
(2)
where D stands for the day of the year. The method
was intended for use in applications such as crop
growth models and climate models for the Czech
Republic and the main advantage is its simplicity.
On the other hand as it was derived for a single
location in the Czech Republic it can be used only
within similar climatic and orographical conditions,
i.e. in the region north of Alps. The authors of the
method also emphasised that the precision during
winter-months was low and that the method was
designed for application during the vegetation season
(i.e. March–October). The proposed equation is based
on the concept recently published by Almorox and
Hontoria (2004) who have found that a third degree
equation should be applied when the hours of sunshine
are used as the RG predictor.
2.2.2. Estimation of solar radiation using cloud
cover
One of the most important atmospheric phenomena limiting solar radiation at the Earth’s surface are
clouds and accompanying weather patterns (Supit
and van Kappel, 1998). A correlation between the
daily RG and cloud cover was produced as early as
1920s by Kimball (1928). Many studies to estimate
RG from observation of various cloud layer amounts
and cloud types have been carried out (Barker, 1992;
Kasten and Czeplak, 1980). The analysis of cloud
cover and RG showed that there was a non-linear
relationship and thus square root equations could be
used for relating global radiation to different cloud
amounts (Wörner, 1967). In this study the following
combination of Wörner (1967) and Hargreaves et al.
M. Trnka et al. / Agricultural and Forest Meteorology 131 (2005) 54–76
(1985) models proposed by Supit and van Kappel
(1998) was applied:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Cw
RG ¼ RA aS ðTmax Tmin Þ þ bS
1
8
þ cS
(3)
where Cw is the mean total cloud cover during daytime
observations (in octas), Tmax and Tmin stand for the
maximum and minimum daily temperatures and aS, bS
and cS symbols are empirical constants. These coefficients were determined in the same way as in Eq. (1),
i.e. by interpolation from available at www.supit.cistron.nl/start.htm (2004).
2.2.3. Estimation of solar radiation using daily
extreme temperatures and sum of precipitation
Most of methods applying these input parameters
were designed because of the general availability of
these inputs (De Jong and Stewart, 1993; Liu and
Scott, 2001) from weather stations. Many studies were
derived from Bristow and Campbell (1984) empirical
algorithm for estimating RG from daily maximum and
minimum temperatures.
Improvements in the original Bristow and Campbell
equation include the daily total precipitation. The
original model reduces the daily RA value by the fraction
lost due to clouds using the total atmospheric
transmisivity term (t), which can be calculated from
the daily temperature amplitude and site-specific
empirical coefficients. The simplicity of the equation
and its predictive accuracy make it attractive for use in
ecosystem simulations. However, it was designed for
use with coefficients determined from long-term sitespecific climatological data. Considerable effort has
therefore concentrated on improving the applicability of
the method (Running et al., 1987; Thorton and Running,
1999; Winslow et al., 2001). The method proposed by
Winslow et al. (2001) was designed as a globally
applicable model and the prediction equation is
bes ðTmin Þ
RG ¼ tcf Dl 1 (4)
RA
es ðTmax Þ
where eS(Tmin) and eS(Tmax) are saturation vapour
pressures at temperature Tmin and Tmax, respectively.
Variable tcf accounts for atmospheric transmittance
and is estimated from site latitude, elevation and mean
61
annual temperature. Function Dl corrects the effect of
site differences in day length, which causes a variation
between the time of maximum temperature (and minimum humidity) and sunset. The value b is a coefficient, which remains stable except in mountainous
regions with very large temperature amplitude. As the
elevation of the highest station was 519 m above sea
level (Table 1) the b value of 1.041 was used as
recommended by Winslow et al. (2001).
The method proposed by Thorton and Running
(1999) is also based on the Bristow and Campbell
(1984) study and is as follows:
RG ¼ RA tt;max t f max
(5)
where tt,max is the maximum (cloud-free) daily total
transmittance at a location with a given elevation and
depends on the near-surface water-vapour pressure on a
given day of the year, and tf max stands for the proportion of tt,max observed on a given day (cloud correction).
The original method was improved by Thornton et al.
(2000) by applying a snow pack model and subroutine
for correcting measurements at the stations with
obstructed horizons. The modified version of the
method including the snow pack model was included
in the present study, as it improves the estimated RG
precision during winter months. The radiation algorithm parameters were set according to the results,
which were derived for the conditions in Austria and
are generally applicable throughout the Central Europe
region (Thornton, 2003, personal communication).
2.2.4. Estimation of solar radiation using daily
Tmax, and Tmin only
Using only daily extremes of air temperatures to
estimate atmospheric transmisivity provides a strong
physical basis that in combination with good data
availability makes such methods very popular
(Hargreaves et al., 1985; Donatelli and Marletto,
1994; Donatelli and Campbell, 1998; Liu and Scott,
2001). Cloud cover reduces daily maximum temperature because of the smaller short wave radiation input.
Moreover, the presence of clouds at night increases the
minimum air temperature because of the greater
emissivity of clouds compared to a clear sky. These
phenomena were taken into account in the Bristow and
Campbell model (1984) that was improved by
Donatelli and Marletto (1994) in order to correctly
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M. Trnka et al. / Agricultural and Forest Meteorology 131 (2005) 54–76
estimate peak daily radiation values. The approach
was exploited further by Donatelli and Campbell
(1998) with the elimination of one calibration
parameter and the replacement of an exponential
function by the hyperbolic function of the daily mean
temperature. The daily global radiation (RG) is
estimated as RA multiplied by the transmisivity
coefficient of the atmosphere to solar radiation. The
final version of the method has the following form:
RG ¼ RA t½1 expðbD f ðTavg ÞDT 2 f ðTmin ÞÞ
(6)
where t stands for the clear sky transmisivity (set to be
0.75), DT, f(Tavg) and f(Tmin) are functions based on
the daily mean and minimum temperatures. The subsequent calculations also include two empirical parameters bD and Tnc that are listed in Table 2. The latter
is used in order to calculate the f(Tmin) parameter. Both
parameters can be parameterised as shown by Donatelli and Campbell (1998) and Ducco et al. (1998) for a
given site using readily available climatological data,
e.g. mean June temperature and mean annual temperature (excluding June).
Another relatively simple method of estimating
daily global radiation by relating the daily temperature
range (the difference between maximum and minimum temperatures) to global radiation was proposed
by Hargreaves et al. (1985):
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
RG ¼ RA aH ðTmax Tmin Þ þ bH
(7)
where aH and bH are the empirical constants derived
from maps published at www.supit.cistron.nl/start.htm
(2004). The model was originally validated for the
Senegal River Basin, i.e. for a completely different set
of climatic conditions. Since then it has been widely
used mainly because of the easily available inputs.
However, owing to the low accuracy of the model, its
application for locations in Europe is considered to be
limited (Choisnel et al., 1992).
2.2.5. Estimation of solar radiation using data
from another meteorological station
One of the most frequently used methods is the
substitution of the missing data by those from an
another nearby ‘‘representative’’ meteorological station. The applicability of the method is hampered by
the insufficient density of the station network and
relatively short length of RG records. It has been
investigated by number of authors (Nonhebel, 1993;
Hunt et al., 1998) and it was also proposed by Allen
et al. (1998) as a possible way of replacing missing
data when calculating the daily evapotranspiration. In
our study, we compare the performance of this method
with the empirical methods described earlier. As the
database does not contain records for all stations
during all seasons, the period 1996–2001, in which
most of the station records overlap, was chosen for the
assessment. The missing values were estimated for
five stations. Two of them (Graz and Ostrava-Poruba)
represent two most distant stations (situated furthest to
the south and north-west, respectively), while Kremsmünster, which is located only about 40 km north of
the Alps, represents the stations placed in the western
Table 2
Summary of coefficients for estimating global radiation from Eqs. (1), (3), (6) and (7)
Supit and van
Kappel (Eq. (3))a
Donatelli and
Campbell (Eq. (6)) b
Hargreaves
et al. (Eq. (7))a
Station name
Ångström–
Prescott (Eq. (1))a
aA
bA
aS
bS
cS
Tnc
bD
aH
bH
Gmunden
Graz
Gross-Enzersdorf
Hradec Králové
Kocelovice
Kremsmünster
Kuchařovice
Langenlois
Ostrava-Poruba
Retz
0.21
0.21
0.21
0.20
0.20
0.21
0.21
0.21
0.21
0.21
0.55
0.52
0.54
0.56
0.55
0.55
0.54
0.54
0.56
0.54
0.08
0.09
0.09
0.08
0.08
–
0.07
0.08
0.07
–
0.40
0.40
0.40
0.40
0.40
–
0.40
0.40
0.40
–
0.10
0.10
0.00
0.30
0.30
–
0.30
0.20
0.30
–
30.7
27.0
36.4
31.3
31.9
31.7
35.3
31.9
30.9
31.7
0.32
0.28
0.31
0.30
0.31
0.32
0.31
0.30
0.30
0.31
0.15
0.15
0.16
0.16
0.16
0.15
0.16
0.16
0.16
0.16
0.80
0.80
0.80
0.80
0.80
0.80
0.80
0.80
0.80
0.80
a
b
Values of coefficients were estimated using maps available at www.supit.cistron.nl/start.htm (2004).
The coefficient values were derived according to methodology described by Ducco et al. (1998).
M. Trnka et al. / Agricultural and Forest Meteorology 131 (2005) 54–76
part of the area of interest. The remaining two stations
(Retz and Kucharovice) were chosen to examine the
behaviour of the method at relatively small distances
(less than 40 km) over a uniform landscape. For each
of the five stations the measured RG values were
sequentially replaced by data measured at the
remaining nine stations and compared with the
measured data. Because of the relatively large latitude
effect, each daily value measured at the neighbouring
station was corrected by applying following formula
proposed by Allen et al. (1998):
RG ¼ RG;proxy
RA
RA;proxy
63
accounts for variability explained by the given
method) and slope of the regression line including
regression function forced through origin were provided for the whole year and individual months.
Besides daily data, monthly totals were also examined. All of these statistical methods were applied to
examine performance of Eqs. (1)–(7) during days of
specified daily RG totals (Fig. 2a) and during days with
particular RG/RA (Fig. 2b).
3. Results and discussion
(8)
where RG,proxy and RA,proxy are the measured global
radiation and extraterrestrial radiation at the neighbouring station, and RA represents the daily value of
the extraterrestrial radiation at the location, which was
estimated.
2.2.6. Performance indicators
The performance of each model was assessed on
the basis of three characteristics: the mean bias error
(MBE) as an indicator of systematic error, the root
mean square error (RMSE), which gives an idea of the
magnitude of the non-systematic error (Davies and
McKay, 1989). The relative value of MBE and RMSE
was determined as the ratio of the appropriate
characteristic and the mean for the given time period.
The MBE is calculated as
P
ðQobs Qest Þ
yMBE ¼
(9)
Nobs
where Qobs and Qest substitute observed and estimated
global radiation values (MJ m2 day1) and Nobs is the
number of observations.
The RMSE is calculated as
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P
ðQobs Qest Þ2
yRMSE ¼
(10)
Nobs
Both characteristics were calculated for each month
and for the whole year from the whole series of RG and
expressed in actual units, i.e. MJ m2 day1, as well
as in the relative terms, i.e. percentage of the mean for
a given month or for a whole year. In order to illustrate
relationship between the observed and calculated daily
total RG, the coefficient of determination (which
3.1. Estimation by the models of the solar
radiation variability
The performance of the seven models (Eqs. (1)–
(7)) is compared in Table 3. Of the two models which
use sunshine duration data (Eqs. (1) and (2)), the
Ångström–Prescott method explains the highest
portion of RG variability out of all tested methods
(Table 3) and the slope of the regression line forced
through the origin was closest to unity. Eq. (2)
(Klabzuba et al., 1999) was less accurate and in
contrast to Eq. (1) tended to overestimate the daily
global radiation totals. Eq. (3) (Supit and van Kappel,
1998) using daily extreme temperatures and mean
daily cloudiness performed quite well, and the R2
values were lower than those recorded for Eqs. (1) and
(2). Models using daily extreme temperatures and
daily precipitation total (Eqs. (4) and (5)) had similar
regression line slope values, but the variability
explained by Eq. (4) was much higher (85%) in
comparison to Eq. (5) (79%). Of the two models using
only daily extreme temperatures as inputs (Eqs. (6)
and (7)), the former (Donatelli and Campbell, 1998)
showed slightly higher R2 values, while there was
almost no difference between the methods in the
regression line slopes.
When plain linear regression was used instead of a
regression line forced through origin, the variability
increased for all seven tested methods and increased
significantly for Eqs. (5) and (7), both of which tended
to overestimate lower daily RG totals and to underestimate the higher ones. An analysis of the
performance of the models at the individual sites
shows that the site has no significant impact on the
value of R2 (Table 3). The highest site-to-site
64
M. Trnka et al. / Agricultural and Forest Meteorology 131 (2005) 54–76
Table 3
Coefficient of determination (R2) and slope of linear regressions forced through the origin of estimated against measured solar radiation
Station name
Ångström–
Prescott
(Eq. (1))
Klabzuba
et al. (Eq. (2))
Supit and
van Kappel
(Eq. (3))
Winslow
et al.
(Eq. (4))
Thornton
and Running
(Eq. (5))
Donatelli and
Campbell
(Eq. (6))
Hargreaves
et al.
(Eq. (7))
R2 a
Slopeb
R2
Slope
R2
Slope
R2
Slope
R2
Slope
R2
Slope
R2
Slope
Gmunden
Graz
Gross-Enzersdorf
Hradec Králové
Kocelovice
Kremsmünster
Kuchařovice
Langenlois
Ostrava-Poruba
Retz
0.96
0.96
0.97
0.97
0.97
0.96
0.97
0.96
0.96
0.95
0.99
1.00
0.95
1.01
0.98
0.98
1.00
0.98
0.99
0.98
0.94
0.92
0.95
0.94
0.94
0.94
0.94
0.93
0.94
0.93
1.02
1.04
1.02
1.04
1.04
1.02
1.06
1.03
1.08
1.04
0.90
0.91
0.91
0.92
0.91
–
0.92
0.89
0.91
–
1.02
1.01
1.01
0.95
0.96
–
0.94
1.06
1.04
–
0.86
0.86
0.84
0.88
0.88
0.86
0.88
0.85
0.86
0.82
0.97
0.99
0.89
1.03
0.98
0.89
0.98
1.03
1.11
0.93
0.85
0.81
0.84
0.86
0.85
0.82
0.86
0.83
0.85
0.82
0.95
1.04
0.88
0.96
0.94
0.97
0.96
0.95
1.02
0.95
0.83
0.83
0.82
0.86
0.86
0.83
0.86
0.83
0.84
0.80
0.99
1.03
0.91
1.05
1.00
0.91
0.99
1.05
1.13
0.94
0.81
0.82
0.82
0.84
0.84
0.81
0.84
0.82
0.83
0.79
0.98
1.06
1.02
1.02
0.97
0.91
0.97
1.01
1.09
0.94
All stations
0.96
0.99
0.94
1.03
0.91
0.99
0.86
0.97
0.82
0.92
0.82
0.99
0.83
0.99
a
b
Values of coefficients of determination of the linear regression line with the best fit.
Values of slope of the regression line forced through the origin.
variability of R2 was found for Eq. (6) and (7). As far
as the value of the slope of regression line is concerned
(i.e. the tendency of the models to overestimate or
underestimate the RG values), the influence of a
particular locality was more pronounced. The tendency to overestimate RG was higher in OstravaPoruba and Graz (the sites with the largest air
pollution potential), while the greatest tendency for
underestimation was found in Gross-Enzersdorf and
Kremsmünster (i.e. two rural sites). On the other hand,
the proximity of the station to Alps did not
significantly influence the performance of individual
methods, although it should be also remembered that
the altitude of all stations was less than 600 m above
the sea level.
The results are generally in accordance with the
previously published studies, which mostly concluded
that RG estimates based on the hours of sunshine were
superior to other methods. However, the variability
explained by Eq. (1) was significantly higher than the
results reported by Almorox and Hontoria (2004) for
Spain (82–88%). One of the largest studies for
European conditions conducted by Supit and van
Kappel (1998) found an R2 value of 0.95 (based on
data from 89 stations), which is almost identical to the
present results. Martı́nez-Lozano et al. (1984) found
that the coefficient of correlation for a great number of
stations worldwide was in the range of 0.83–0.95,
which corresponds to R2 ranging from 0.69 to 0.90.
Iziomon and Mayer (2001) found that after proper
parameterisation based on the measured RG values,
Eq. (1) was capable of explaining more than 99.5% of
the monthly global radiation variability under Central
European conditions. For hourly data, however, the
variability explained was significantly lower and
ranged from 80% to 88% depending on the site.
Eq. (3) derived by Supit and van Kappel (1998)
performed better than the methods based solely on the
cloud term, as the variability explained by this method
was found to be between 92% and 95% in our study
compared with 74% (Barr et al., 1996) or 80% (Supit
and van Kappel, 1998).
The methods using daily temperature extremes and
precipitation totals as predictors of RG performed
quite well compared with other regions of the world.
Liu and Scott (2001) found that the best of the tested
methods was capable of explaining only 79% of daily
RG variability at the 39 Australian stations. Results
achieved by De Jong and Stewart (1993) were even
less satisfactory with 57% of variability explained.
The variability explained by Eq. (4) was found to be
82–88% in Central European conditions, which is
consistent with the claim that in mid-latitudes Eq. (4)
can explain between 69% and 91% of the variability
(Winslow et al., 2001). With Eq. (5), 81–85% of the
variability could be explained, which is comparable
M. Trnka et al. / Agricultural and Forest Meteorology 131 (2005) 54–76
with results reported by Thorton and Running (1999)
and significantly better than the performance of the
method in a warm and moist environment (Almeida
and Landsberg, 2003).
Use of daily temperature extremes as RG predictors
yielded satisfactory results in the Central European
conditions, as Eqs. (6) and (7) explain on average 83%
and 82% of the daily RG variability (at some stations
even 86%). One of the largest studies including Eq. (7)
in European conditions found that the method was
capable of explaining on average 79–84% of the daily
variability (Supit and van Kappel, 1998). These results
are better than those reported for the frequently used
Bristow and Campbell (1984) method under similar
climatic conditions. This method explained 68% of the
daily variability in Australian conditions (Liu and Scott,
2001) and 79% in Chile (Meza and Varas, 2000).
Testing of Eq. (6) found that the method explained 68–
92% of the daily variability (Ducco et al., 1998;
Donatelli and Campbell, 1998). Eqs. (6) and (7) were
found to be superior to other methods based on the daily
temperature extremes cited in literature (Barr et al.,
1996; De Jong and Stewart, 1993).
3.2. Bias of the models in estimating solar
radiation
In Fig. 3a, the cumulative frequency function (CF)
for MBE for the Ångström–Prescott equation shows
that 83% of all daily values were estimated with a
deviation of less than 2.0 MJ m2 day1 and over
97% of these deviations lay within 4.0 MJ m2
day1 range. These results were partly mimicked only
by Eq. (3) when 68% and 90% of the values,
respectively, were in the same range. In the case of
Eq. (4)–(7), the CF of the daily bias error showed
slightly better results for Eqs. (5) and (6). The annual
MBE value (Table 4 and Fig. 4a) was within
0.12 MJ m2 day1 (Eq. (1)) and 0.70 MJ m2 day1
1 (Eq. (7)) when all stations were included and can
thus be ignored on an annual basis. However, careful
examination of Fig. 4a shows a clear annual cycle of
the relative MBE values for all methods. In the case of
Eq. (2) the MBE cycle agreed with the description
given by Klabzuba et al. (1999), as the method is
designed for use during the growing season and yields
relatively large errors during autumn and winter
months. Eqs. (1) and (3) showed a relatively small
65
change in MBE values during the year. Eq. (3) slightly
overestimated RG from April to October while Eq. (1)
overestimated RG mainly in November, December and
January (by 10–20%). The remaining four equations
underestimated RG during the cold half of the year
(October–March) and overestimated it during the
warm half. Comparison of the MBE at individual sites
yielded results similar to the slope of the regression
line. Methods based on the extreme temperatures
showed significantly higher annual MBE than average
only in the urban sites, i.e. Graz and especially at
Ostrava-Poruba. In the latter case, industrial pollution
from heavy industry and coal-burning power plants
probably explains why all methods except for Eq. (1)
overestimated RG at this site. The air pollution causes a
decrease in atmospheric transmittance during most of
the year, thus influencing both measured hours of
sunshine and RG. The precision of Eq. (1) is therefore
compromised much less than that of the other methods
that use other meteorological variables that are not
always influenced in the same way as RG. In the case of
Eq. (2) the unsatisfactory performance in the polluted
environment was inherent in the method itself because
of the fact that it was derived using datasets measured
at a single site in a relatively unpolluted though urban
environment. This explanation is supported by the fact
that the sites in the countryside (e.g. Kremsmünster or
Retz) slightly underestimate of RG in Eqs. (4)–(7).
The MBE values found in previous studies for the
Ångström–Prescott method (Eq. (1)) do not differ
from those found in this study. For example, Supit and
van Kappel (1998) found the average MBE value for
89 stations was 0.22 MJ m2 day1, which is almost
the same value as in our study. The MBE ranged
from 2.20 to +1.36 MJ m2 day1 for all stations
included in their study, whereas in our study the
MBE range was smaller (0.50 to +0.32 MJ m2
day1). This difference was probably caused by the
smaller number of stations in a climatologically more
homogeneous region. This finding is nevertheless
interesting because no parameterisation was carried
out in our study and both empirical coefficients were
interpolated from rather crude maps.
The relative MBE for Eq. (1) at 10 stations was
found to be 1.1% on average (ranging from 4.1% to
+3.1%) which is surprisingly lower than Iziomon and
Mayer (2002), who had found that the MBE value
related to Eq. (1) parameterised at the given station
66
M. Trnka et al. / Agricultural and Forest Meteorology 131 (2005) 54–76
Fig. 3. Comparison of cumulative frequency of (a) daily bias error and (b) daily root square error for the seven models across all sites.
was within 3.7–6.0% (when one set of parameters for
all 12 months was used). Performance of Eq. (2) is
comparable with the results reported by Oesterle
(2001), who applied a one-dimensional regression
equation and found the MBE value at seven German
stations to be 0.2 MJ m2 day1 on average. The same
method (one-dimensional regression equation) was
then applied on the daily mean cloudiness versus RG
relationship and yielded an MBE of 0.5 MJ m2
day1 (Oesterle, 2001). In general, the application of
cloud term as a single RG predictor yielded a higher
systematic error than a combination of cloud fraction
with another predictor such as temperature range
(Supit and van Kappel, 1998) or relative humidity
(Oesterle, 2001). The MBE value for Eq. (4) for
Central Europe (ranging from 1.21 to 1.56 MJ
m2 day1) compares well with the results reported by
Winslow et al. (2001) for five sites at which the
method was originally developed. Thorton and
Running (1999) reported an MBE value of 0.5 MJ
m2 day1 for sites where the method was calibrated
and tested and 1.26 to 0.68 MJ m2 day1 for 13
M. Trnka et al. / Agricultural and Forest Meteorology 131 (2005) 54–76
67
Table 4
Summary of performance of various methods given by annual mean bias error (MBE)
Station name
Gmunden
Graz
Gross-Enzersdorf
Hradec Králové
Kocelovice
Kremsmünster
Kuchařovice
Langenlois
Ostrava-Poruba
Retz
All stations
S.D.
a
b
Ångström–
Prescott
(Eq. (1))
Klabzuba
et al.
(Eq. (2))
Supit and
van Kappel
(Eq. (3))
Winslow
et al.
(Eq. (4))
Thornton
and Running
(Eq. (5))
Donatelli
and Campbell
(Eq. (6))
Hargreaves
et al.
(Eq. (7))
Actual a
Relativeb
Actual
Relative
Actual
Relative
Actual
Relative
Actual
Relative
Actual
Relative
Actual
Relative
0.29
0.28
0.50
0.32
0.04
0.04
0.19
0.10
0.28
0.07
3.05
2.46
4.14
3.10
0.39
0.35
1.68
0.89
2.78
0.58
0.36
0.51
0.17
0.70
0.57
0.13
0.78
0.38
1.09
0.53
3.70
4.46
1.40
6.60
5.30
1.10
7.04
3.41
10.87
4.25
0.60
0.49
0.14
0.40
0.09
–
0.50
1.11
0.90
–
6.20
4.30
1.20
3.50
0.80
–
4.50
10.03
8.10
–
0.17
0.22
1.21
0.70
0.05
0.87
0.03
0.66
1.56
0.59
1.75
1.93
10.10
6.70
0.50
7.61
0.28
5.98
15.60
4.73
0.37
1.21
0.92
0.30
0.10
0.32
0.23
0.17
1.03
0.07
3.86
10.48
7.60
2.90
0.82
2.80
2.04
1.56
10.25
0.57
0.28
0.66
1.10
0.87
0.10
0.81
0.06
0.84
1.75
0.61
2.94
5.71
9.30
8.20
0.91
7.12
0.54
7.59
17.48
4.87
0.79
1.46
0.80
0.90
0.32
0.29
0.26
0.85
1.71
0.08
8.22
12.66
6.60
8.80
2.90
2.55
2.39
7.73
17.10
0.65
0.12
0.25
1.1
2.23
0.57
0.27
5.20
2.72
0.18
0.55
1.7
5.03
0.19
0.77
1.70
7.09
0.33
0.55
3.00
4.94
0.32
0.83
2.90
7.65
0.70
0.60
6.32
5.68
Value of the actual annual mean bias error as defined in Eq. (8) expressed in MJ m2 day1.
Relative value of the annual mean bias error calculated as the fraction of the MBE abs. and annual mean of measured RG expressed in %.
lowland stations out of the 27 stations in the entire
Austrian database (Thornton et al., 2000). In the
present study, including seven previously untested
stations, the MBE ranged from 0.92 to
1.21 MJ m2 day1. In general Eq. (5) performs
significantly better in the Central Europe than in
Brazil, for example, where a systematic underestimation of 2.42 MJ m2 day1 was reported (Almeida and
Landsberg, 2003).
3.3. Random errors of the models in estimating
solar radiation
Cumulative frequency of daily root square error is
presented in Fig. 3b. It is clear that Eq. (1) is the best
performing model, followed by Eqs. (2)–(4). The
performance of Eqs. (5)–(7) is very similar. The figure
shows that 90% of the daily values estimated with
Eq. (1) had an error of less than 2.5 MJ m2 day1.
Eqs. (2) and (3) yielded 90% of the estimates with a
deviation of less than 3.6 MJ m2 day1, Eq. (4)
estimated 90% of the daily values with a deviation of
less than 5.0 MJ m2 day1, while the deviations of
the three remaining methods were 0.5 MJ m2 day1
larger.
The RMSE value averaged over the entire
observational period in the database was found to
be in the range of 1.57 MJ m2 day1, i.e. 14.5% for
Eq. (1), and 3.46 for Eqs. (6) and (7), i.e. 32.1%
(Table 5). The precision of the tested models
expressed in terms of the RMSE is consistent with
the previously examined parameters, i.e. R2 and MBE.
In comparison with Eq. (1), Eqs. (2) and (3) show a
29% and 42% increase in the RMSE value, which
suggests a higher overall deviation of the predicted
daily values by these two methods. However, when the
daily temperature extremes in combination with the
precipitation data were used as inputs (Eqs. (4) and
(5)), the RMSE value was more then double that of
Eq. (1). The error was even greater when Eqs. (6) and
(7), i.e. the methods applying only the daily minimum
and maximum temperatures, were use. The collation
of the performance of the individual methods at 10
observation sites (Table 5) also suggests that the interstation differences in the RMSE values for each of the
methods were minor with standard deviation ranging
from 0.11 to 0.27 MJ m2 day1, which corresponds
to 7.0–7.8% in terms of the coefficient of variance.
The distribution of the relative RMSE values over
individual months is shown in Fig. 4b, which
illustrates the general trend of all methods to greater
relative RMSE during period from October to March.
This phenomenon may be explained by the relatively
large significance of even small absolute deviations as
the daily global radiation totals during winter tend to
be small in high latitudes. In addition, some of the
tested methods, e.g. Eqs. (2) or (5), are less reliable in
the cold half of the year. Fig. 4b shows that even the
best method, i.e. Eq. (1), yields relative RMSE in the
range of 10.7% (August) and 31.8% (December).
68
M. Trnka et al. / Agricultural and Forest Meteorology 131 (2005) 54–76
Fig. 4. Annual and monthly values of (a) the relative mean bias error (MBE) and (b) of the relative root mean square error (RMSE) calculated for
each method and pooled together for all sites used in the study.
M. Trnka et al. / Agricultural and Forest Meteorology 131 (2005) 54–76
69
Table 5
Summary of performance of various methods given by annual root mean square error (RMSE)
Station name
Ångström–
Prescott
(Eq. (1))
Klabzuba
et al.
(Eq. (2))
Supit and
van Kappel
(Eq. (3))
Thornton
and Running
(Eq. (5))
Winslow
et al.
(Eq. (4))
Donatelli
and Campbell
(Eq. (6))
Hargreaves
et al.
(Eq. (7))
Actual a
Relative b
Actual
Relative
Actual
Relative
Actual
Relative
Actual
Relative
Actual
Relative
Actual
Relative
Gmunden
Graz
Gross-Enzersdorf
Hradec Králové
Kocelovice
Kremsmünster
Kuchařovice
Langenlois
Ostrava-Poruba
Retz
1.66
1.60
1.56
1.51
1.41
1.64
1.46
1.64
1.65
1.80
17.34
13.90
13.00
14.40
13.02
14.34
13.16
14.89
16.47
14.42
2.00
2.40
2.10
2.15
2.12
2.18
2.24
2.20
2.34
2.50
20.81
20.40
17.20
20.40
19.61
19.13
20.27
19.98
23.29
20.01
2.55
2.30
2.60
2.30
2.40
–
2.30
2.95
2.60
–
26.60
20.40
21.50
21.80
22.30
–
21.00
26.75
23.90
–
2.89
3.00
3.50
3.00
2.85
3.25
2.83
3.31
3.49
3.55
30.13
25.70
29.20
28.40
26.38
28.44
25.81
29.97
34.78
28.49
3.11
3.60
3.50
3.00
3.06
3.53
3.00
3.23
3.19
3.17
32.40
31.40
29.20
28.80
28.31
30.88
27.09
29.31
31.76
27.80
3.36
3.50
3.80
3.30
3.14
3.60
3.18
3.70
3.89
3.91
34.99
30.20
31.80
31.60
29.05
31.51
28.72
33.57
38.74
31.32
3.50
3.70
3.70
3.30
3.18
3.61
3.18
3.54
3.75
3.77
36.20
31.90
31.00
31.80
29.50
31.60
28.73
32.05
37.35
30.25
All stations
Standard deviation
1.57
0.11
14.50
1.37
2.20
0.14
20.39
1.43
2.28
0.21
24.71
2.31
3.09
0.27
28.60
2.53
3.21
0.21
29.74
1.71
3.46
0.27
32.03
2.83
3.46
0.22
32.05
2.60
a
b
Value of the actual annual root mean square error as defined in Eq. (9) expressed in MJ m2 day1.
Relative value of the annual root mean square error calculated as the fraction of the RMSE abs. and annual mean of measured RG expressed in %.
The actual RMSE values based on the results of all
114 observational years for Eq. (1) were 0.7 MJ
m2 day1 in December and 2.3 MJ m2 day1
during June. The relative RMSE values of Eq. (2)
were comparable with results attained by Eq. (1) only
in the period from April to August. In the remaining
months this method was highly unreliable in terms of
relative RMSE (e.g. 75.9% in December). Supit and
van Kappel’s method (Eq. (3)) clearly outperformed
the other four methods, but the relative RMSE range
was significantly higher in comparison with Eq. (1).
The December relative RMSE was 11.9% and in
August 7.6% higher than in Eq. (1). The two methods
using daily extreme temperatures and total precipitation as predictors of the daily global radiation total
estimated the RG with relative RMSE within 20.6%
and 71.6%, with Winslow’s method (Eq. (4)) showing
a more stable performance, especially during winter
months, than the Thornton and Running method
(Eq. (5)). As far as RMSE is concerned, the
introduction of precipitation as an additional parameter produced slightly better results than reliance
only on the daily extremes. However, the improvement transferred to the RMSE value decreased by only
0.3 MJ m2 day1 or 10% of the total RMSE error.
The non-systematic estimation error in Eq. (1)
expressed in terms of RMSE (Table 5) was found to be
between 1.4 and 1.8 MJ m2 day1 (i.e. 13.0–17.9%),
which is relatively high compared with 2.6% reported
by Iziomon and Mayer (2001). However, the method
was parameterised for the given location in their
analysis. On the other hand, the RMSE value was well
within the range reported by Supit and van Kappel
(1998), i.e. 1.4–5.0 MJ m2 day1. The relative
RMSE range for Eq. (2) was almost twice as high
as the values reported by Barr et al. (1996) during
November and December but very similar from April
to September. The annual mean RMSE found for
Eq. (2) corresponded well with findings reported by
Oesterle (2001). Eq. (3) performed within the range
published by Supit and van Kappel (1998) and
outperformed the methods examined by Barr et al.
(1996), Supit and van Kappel (1998) and Oesterle
(2001) based only on the cloud fraction term. RMSE
values of estimates made with help of Eqs. (4) and (5)
appear high when the mean value based on daily
estimates for all 10 stations was found to be
3.09 MJ m2 day1 (i.e. 28.6%), 3.21 MJ m2 day1
(29.7%), respectively. De Jong and Stewart (1993)
proposed a method based on the daily temperature
extremes and precipitation total that could be used to
estimate the daily value with relative RMSE of 10.7%
(in July) and 15.7% (in November). However, this
method requires parameterisation based on global
radiation measured at the given site and separate sets
of regression coefficients for each month. Liu and
Scott (2001) also proposed a method for Australian
conditions producing daily RG values with a smaller
non-systematic error than Eqs. (4) and (5), but the
difference is negligible as the RMSE ranged from
70
M. Trnka et al. / Agricultural and Forest Meteorology 131 (2005) 54–76
2.23 MJ m2 day1 to 3.49 MJ m2 day1 at 39
examined stations. Moreover the application of Liu
and Scott’s method in the study region would require
local parameterisation, which was necessary neither for
Eqs. (4) nor (5). The method proposed by Winslow et al.
(2001), i.e. Eq. (4), performed very well under Central
European conditions (Table 5) in comparison with the
five station where the method was developed. At these
sites the RMSE ranged from 2.46 to 4.41 MJ m2
day1. Despite the fact that according to some studies in
mid-latitudes (e.g. Barr et al., 1996) the daily
temperature range is considered to be a bad predictor
of RG (especially during winter months), it has been
found that Eqs. (6) and (7) performed fairly well
(Table 5). The RMSE values found for 11 stations over
the world that ranged from 2.49 to 5.02 MJ m2 day1
(Donatelli and Campbell, 1998) compare well with the
results reported in our study. Eq. (7) performed quite
well in the study region as the reported RMSE value was
smaller or equal to the non-systematic error reported by
Supit and van Kappel (1998), i.e. 3.61 MJ m2 day1
(average based on 89 stations) and by Hunt et al. (1998),
who found that RMSE for eight Ontario stations was
between 4.2 and 4.7 MJ m2 day1.
3.4. Performance of the tested models in
estimating high and low values of solar radiation
Fig. 5a examines the performance of the tested
methods for different daily RG totals using relative
MBE as an indicator. The chart clearly shows that all
the methods overestimate the RG values during days
with observed daily totals between 0.1 and
9.0 MJ m2 day1 and underestimate them on days
with a daily total higher than 24.1 MJ m2 day1. The
significance of this finding is all the greater since
49.6% of all the values in the database are for days
with RG of less than 0.1–9.0 MJ m2 day1 and 8.1%
are for days with RG higher than 24.1 MJ m2 day1.
It should be noted that Eqs. (5) and (7) are most
reliable for daily totals of RG in excess of 9.1 MJ m2
day1 and smaller than 24.0 MJ m2 day1. Eqs. (1)–
(4), (6) perform with relative MBE smaller than 10.0%
for days with RG between 6.1 and 30.0 MJ m2 day1
thus including over 62% of the database values.
However, all of the tested methods tended to perform
poorly for days with RG below 6.0 MJ m2 day1
(more than 37% of all observations) with relative
MBE values exceeding 88% (Eq. (5)) and relative
RMSE in some cases greater than 100%. Such an error
will clearly have an impact on subsequent analysis that
relies on estimated RG data. Further investigation
proved that the majority of RG values of less than
9.0 MJ m2 day1 (over 85%) were recorded between
October and March and should not therefore affect the
use of Eqs. (1)–(7) during the vegetation season in the
Central European conditions. More important in this
case is the tendency of all methods to underestimate
the RG during days with higher solar exposure
especially when Eqs. (3)–(7) are used.
Besides the evaluation of the tested method
performance for the predefined intervals of measured
RG, the performance of Eqs. (1)–(7) was assessed
under a whole range of atmospheric transmisivity
conditions (Fig. 5b). The ratio between the measured
RG and RA calculated for the given station and day was
used as an indicator of transmisivity. In general, low
ratios indicated cloudy conditions with low atmospheric transmisivity, while high values suggest that
the day was clear with relatively small fraction of RA
absorbed and reflected by the atmosphere or clouds.
All methods performed well (according to relative
MBE values) when the ratio was between 0.45 and
0.55. Both methods based on the hours of sunshine
(i.e. Eqs. (1) and (2)) performed quite well over a
much wider range (0.15 < RG versus RA < 0.80) in
comparison with the other five methods. For
‘‘extreme’’ values RG versus RA ratio even these
two methods yielded a relative MBE of more than
15%. Days when the RG versus RA ratio was either
0.80 or greater or 0.15 or smaller accounted for more
than 14% of all observation days in the database
(Fig. 2b). The reliability of Eqs. (3)–(7) in particular is
limited under these circumstances since the relative
MBE could be even higher than 100% (depending on
the method used). The highest bias of all methods was
found for Eq. (7). Eqs. (4)–(6) showed relatively low
precision for both high and low RG versus RA ratios; in
other words neither of these methods was fully reliable
with extremely low/high atmospheric transmisivity.
3.5. Use of data from another meteorological
station
The final part of the study explored the possible
replacement of missing RG data by measurements
M. Trnka et al. / Agricultural and Forest Meteorology 131 (2005) 54–76
71
Fig. 5. Values of the relative mean bias error (MBE) for (a) predefined intervals according to the measured daily RG (Fig. 2a) and (b) during
observational days with given RG vs. RA ratio (Fig. 2b).
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M. Trnka et al. / Agricultural and Forest Meteorology 131 (2005) 54–76
from a neighbouring station. The results obtained by
this method partially support conclusions of similar
studies carried out in the mid-latitude but under
similar climatic conditions. The variability explained
and RMSE between the proxy and measured RG values
decreased and increased as a function of the distance
in a curvilinear rather than linear manner (Fig. 6).
Functions that described the relationship were derived
in the following form:
R2 ¼ 1:2 107 d 2 1:39 104 d þ 0:97
(11)
ydRMSE ¼ 1:6 105 d 2 þ 1:54 102 d þ 1:94
(12)
where d is the distance between the site providing the
data and that receiving the data. These functions were
given correlation coefficients of 0.91 and 0.93. It was
found that the stations near the Alps (e.g. Kremsmünster or Graz) did not perform differently to the stations
where the influence of this mountain range was limited
(e.g. Kucharovice, Ostrava-Poruba). However, the
findings were of limited validity for the lowland
stations in the Alpine river valleys, where both the
significant mezoclimatic effect of the mountain chain
and horizon obstruction influence the relationship.
Studies presented by Nonhebel (1993) and Hunt
et al. (1998) revealed similar R2 (and RMSE) versus
distance function behaviour. The annual mean RMSE
value in Central European conditions increased by
approximately 0.15 MJ m2 day1 per 10 km of additional distance, while the explained variability
decreased by approximately 1.3% for the same distance. A comparison of these results with the estimates
based on the seven tested methods (Eqs. (1)–(7)) leads
to the conclusion that it is better to use Eq. (1)
provided that the hours of sunshine are available at
the station rather than RG measured at a nearby station,
even where such data are available from a station as
close as 17 km away.
3.6. Selection of an appropriate method of
estimating solar radiation
Fig. 6. Correlations (R2) between estimated and measured global
radiation (a) root mean square errors (RMSE) associated with the
estimation (b) dependence on the distance between the site providing the measured radiation and the site for which the radiation is
being estimated in Central Europe.
The final overview of the study results is presented in
Fig. 7, which offers a basic guide to the selection of a
suitable method for calculating missing daily values of
global solar radiation. The flowchart shows variability
explained and systematic and non-systematic errors for
the selected methods recommended for estimating daily
RG in Central European conditions. It should be noted
that these results were achieved without any further
calibration of any method, i.e. based solely on the
already available coefficients and constants derived
from previously published studies. This chart might
help to quantify the error arising from application of
these methods for further calculations.
The propagation of the error in RG estimates (e.g.
calculation of evapotranspiration or crop growth
models) has been already analysed in previous studies
(De Jong and Stewart, 1993; Lindsey and Farnsworth,
1997; Llasat and Snyder, 1998; or Xie et al., 2003).
Some of them (Lindsey and Farnsworth, 1997 or Xie
et al., 2003) concluded that the effect of error in the RG
daily estimate causes relatively insignificant changes
M. Trnka et al. / Agricultural and Forest Meteorology 131 (2005) 54–76
73
Fig. 7. Overview of the best performing methods tested in the present study. The chart is designed as a tool for selecting proper method for
estimating daily values of global solar radiation (if there is no direct measurement) based on the available data. Basic precision characteristics of
each method are given. Note. The chart applies for lowland stations located outside of the main Alpine mountain chain.
to the final result (potential evaporation or county crop
yield, etc.). In the case of potential evapotranspiration
the RG overestimation by 4% causes an error in the
potential evapotranspiration of between 1.6% and
3.6% depending on the time of the year (Llasat and
Snyder, 1998). Lindsey and Farnsworth (1997) found
that the use of cloud cover as a RG predictor usually led
to an underestimation of the potential evaporation
from a water surface by 14% on average and in some
cases by as much as 39%. The effect of the systematic
error in the RG estimates on the simulated spring wheat
yield was examined by Nonhebel (1994), who had
found that the overestimation of the RG values by 10%
led to a 5% increase in the potential yield, while the
underestimation by the same percentage would
simulated yield to be 9% lower. Similar findings
were also reported by Trnka (2002) and Trnka et al.
(2004) for spring barley and winter wheat. However,
the estimation error varies significantly during
individual months of the growing season (Fig. 4) thus
making the estimate of the final impact difficult. De
Jong and Stewart (1993) used estimated RG values
(with the relative RMSE during the growing season of
between 10.7% and 14.3%) in the WOFOST crop
model (van Diepen et al., 1988). The difference in the
wheat yield obtained with observed and estimated RG
expressed in terms of RMSE was on average 344–
643 kg ha1, depending on the location. ALMANAC,
another crop growth model (Kiniry et al., 1992),
showed significant variability in maize yields in eight
Texas counties (12% to +17%) when the RG value
was decreased by 11% during the growing season.
When the RG value was increased by 10% the yields
varied from between 8% and +3% of the original
values (Xie et al., 2003).
As the RG estimates based on Eqs. (4)–(7)
performed during several months with MBE show a
variability of nearly 10%, with RMSE reaching 30%,
74
M. Trnka et al. / Agricultural and Forest Meteorology 131 (2005) 54–76
caution should be exercised when estimated RG values
are applied instead of the measured ones. Even the
best estimates based on Eqs. (1)–(3) made during the
growing season with RMSE have a variability of 10–
20%, thus any calculations (e.g. evapotranspiration,
potential evaporation or crop model simulations)
based on the approximated data might be significantly
distorted.
4. Conclusions
Seven methods for estimating daily global radiation
were tested with no data for parameterisation
available, so that all empirical coefficients required
by the selected methods were derived from previously
published studies. Variability explained, root mean
square error (RMSE) and mean bias error (MBE)
indicated that the highest precision was reached when
sunshine duration was used as predictor. The method
of Ångström–Prescott (1940), i.e. Eq. (1), was found
to be the best of the tested methods. It explained 96%
of the RG variability with the RMSE value (annual
mean) equalling 1.6 MJ m2 day1 and MBE being
0.1 MJ m2 day1. If there are no reliable estimates of
coefficients available for Eq. (1), the method of
Klabzuba et al. (1999), i.e. Eq. (2), can be used from
April to August at least in the present study region. In
the remaining months the uncertainty of the estimate
increases considerably. If there are no reliable
sunshine duration data available at the site, the Supit
and van Kappel (1998) formula, i.e. Eq. (3), which
uses daily mean cloud cover and extreme air
temperatures as predictors, yields sufficiently precise
estimates (R2 = 0.91; annual mean RMSE = 2.3 MJ
m2 day1; annual mean MBE = 0.2 MJ m2 day1).
If cloud cover is not available, the method proposed by
Winslow et al. (2001), i.e. Eq. (4), which employs the
daily precipitation total, should be used (R2 = 0.86;
annual mean RMSE = 3.1 MJ m2 day1; annual
mean MBE = 0.2 MJ m2 day1). If precipitation is
not measured, then Eq. (6) proposed by Donatelli and
Campbell might be applied (R2 = 0.82; annual mean
RMSE = 3.5 MJ m2 day1; annual mean MBE = 0.3
MJ m2 day1). Where measured RG data are available at a neighbouring station, these measurements
might be used. If the distance between the stations is
less than 5 km, the accuracy of the surrogate radiation
data is greater than the Ångström–Prescott method
(Eq. (1)). As the distance between the stations
increases, the representativeness of the radiation data
from the nearby station decreases. The analysis of the
dependence of accuracy of the RG values taken from
the neighbouring station on the distance between the
stations has shown that annual mean RMSE value
increases approximately by 0.15 MJ m2 day1 per
10 km and variability explained decreases by approximately 1.3% for the same distance. The final choice of
the RG data source then depends on the availability of
input data used by individual methods and the distance
to the closest site with RG measurements, can be seen
in Fig. 7. However, it should be taken into account that
Eqs. (4)–(7) performed poorly for daily RG estimates
at least during several months of the year, and even the
best RG estimates (applying Eqs. (1) and (3)) are
loaded by RMSE ranging from 10% to 20% during the
growing season.
The results of the study suggest that the estimated RG
values have an inherent error, which might compromise
the precision of the subsequent applications. Therefore,
if the estimated RG data are used as an input for models
of daily evapotranspiration, crop models, etc.: (i) proper
analysis of the error propagation should be made or (ii)
methods that integrate sunshine duration or air
temperature values over the day and not only a daily
average (extremes) should be considered if possible.
However, applying the latter procedure would require
more detailed data input, which is still not readily
available in form of the digitised databases, especially
in the Czech Republic and other Central and Eastern
European countries and generally prior to 1990s when
automatic weather stations were introduced on a large
scale through the weather service networks of Austria
and the Czech Republic.
Acknowledgements
This study was conducted with support of projects
GACR 521/03/D059 and 521/02/0827. Authors would
like to thank to the Austrian Meteorological service
(ZAMG) and Czech Hydrometeorological Institute
(Ozone and Solar Observatory in Hradec Králové),
which generously provided data necessary for the
study and also to Jerome C. Winslow and Peter E.
Thornton for helpful consultations prior to the
M. Trnka et al. / Agricultural and Forest Meteorology 131 (2005) 54–76
application of their RG models. We would like to also
thank to Herbert Formayer and Daniela Semerádová
for technical support and to Asist Prof. Philip Weihs
for pre-review of the manuscript. We are grateful to
regional editor Dr. J.B. Stewart and to two anonymous
reviewers for helpful suggestions and clarification of
the final text. We dedicate this paper to Prof. Em. Inge
Dirmhirn who has been a leading global radiation
expert and above all an excellent teacher and an
inspiration to the authors.
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