Environmental Modelling & Software 21 (2006) 1650e1661www.elsevier.com/locate/envsoft

Estimating average daytime and daily temperatureprofiles within Europe

Thomas A. Huld a,*, Marcel Suri a, Ewan D. Dunlop a, Fabio Micale b

a Institute for Environment and Sustainability, Renewable Energies Unit, TP 450, European Commission,DG Joint Research Centre, via Fermi 1, I-21020 Ispra (VA), Italy

b Institute for the Protection and Security of the Citizen, Agriculture and Fisheries Unit, TP 266, European Commission,

DG Joint Research Centre, via Fermi 1, I-21020 Ispra (VA), Italy

Received 9 February 2005; received in revised form 29 June 2005; accepted 29 July 2005

Available online 26 September 2005

Abstract

We present a methodology for estimating the average profiles of daytime and daily ambient temperature from a spatially-continuous databasefor any location within Europe. The primary database with 1-km grid resolution was developed by interpolation of monthly averages of 7 dailyvalues of temperature: minimum and maximum and 5 measurements at 3-h intervals from 6:00 to 18:00 hours Greenwich Mean Time. Witha little over 800 meteorological stations available, we obtained a cross-validation root mean square error of 1.0e1.2 �C, while the interpolationerror is lower, at 0.5e0.7 �C.

A polynomial fit was applied to estimate the daytime temperature profile (assuming only time from sunrise to sunset) from the interpolated 3-hmeasurements for each month. The curve fit coefficients make it possible to calculate a number of derived data, such as average daytime tem-perature, maximum daytime temperature and time of its occurrence within the region. An example demonstrates the coupling of the simulateddaytime temperature profile with a model for assessing the relative efficiency of electricity generation by crystalline silicon photovoltaic modules.

As an alternative to the polynomial fitting, a double-cosine method was applied to enable calculation of daily (24-h) temperature profiles for eachmonth using interpolated minimum and maximum temperatures. Compared to the polynomial curve-fitting, this method does not offer lower errors,but it provides data which are more suitable for estimation of solar thermal heating or calculation of degree days for building heating/cooling.� 2005 Elsevier Ltd. All rights reserved.

Keywords: Daytime and daily temperatures; Geographical information system; Interpolation; Temperature curve fits; Solar energy applications

Software availability

Developed standalone and GRASS GIS software: Free underthe GNU GPL license and can be obtained by contact-ing the corresponding author.

Hardware requirements: Any hardware running Linux or aUnix variant, web server (such as Apache) with PHPmodule, C and CCC compilers.

* Corresponding author. Tel.: C39 0332 785273; fax: C39 0332 789268.

E-mail addresses: thomas.huld@jrc.it (T.A. Huld), marcel.suri@jrc.it (M. Suri),

ewan.dunlop@cec.eu.int (E.D. Dunlop), fabio.micale@jrc.it (F. Micale).

1364-8152/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.envsoft.2005.07.010

1. Introduction

Estimates of temperature values at a specific time of day,from daytime and daily profiles, are needed for a number ofenvironmental, ecological, agricultural and technical applica-tions, ranging from natural hazards assessments (Camiaet al., 1999), crop growth forecasting (Zhang et al., 2002;Micale and Genovese, 2004) to design of solar energy systems(Al-Ajlan et al., 2003; Kenny et al., 2003; Garcıa and Balen-zategui, 2004). The pressure for spatially and temporallymore accurate simulations is a driving force for using dataat higher spatial resolution.

In this work we consider the average temperature profile asa curve (or a set of points) representing the typical variation of

1651T.A. Huld et al. / Environmental Modelling & Software 21 (2006) 1650e1661

temperature within a day for a given location. This is calculatedfrom temperature measurements at regular intervals during theday. The average of all data taken at the same time of day dur-ing the different days of a particular month forms one point onthe average temperature profile for that month. For instance,the average of all measurements taken at 6:00 hours GMT dur-ing days in June over the last 10 years would supply one pointfor the average temperature profile for June for the given loca-tion. The daily profile e the curve characterising the entire 24-hcycle of a day e is distinguished from the daytime profile,which is the curve representing only the period of time fromlocal sunrise to sunset. We also use an average minimum (max-imum) temperature that is the arithmetic average of all mea-sured daily minimums (maximums) for each month. Theaverage daily (daytime) temperature is understood as the aver-age of the entire 24-h temperature cycle (daytime cycle) of alldays in a month. Unlike the average temperature profile, this isa single number for each month.

The methods used for estimates of daily temperature vari-ability vary greatly in complexity and sophistication. Tradi-tionally, tabulated values for average temperatures (or theaverage of the minimum and maximum) have been publishedfor a given region or country. At the other end of the scale,some groups have analyzed detailed daily time series of tem-peratures to prepare synthetic data (see e.g. Aguiar et al.,1999; Knight et al., 1991). The time series capture not onlythe average daily or daytime temperature variation but alsoa statistically correct picture of fluctuations in temperatureduring a day, making it possible to simulate a ‘‘typical’’ dailytemperature profile. However, such investigations require datathat are more detailed than most meteorological ground sta-tions provide. For this reason, the data availability is ratherpoor, with measurements only for a few locations over largeregions.

For estimation of meteorological characteristics over largeregions, the data from ground stations are typically used asa basis for interpolation techniques. Spatial interpolationmakes it possible to estimate any meteorological characteristic(such as a maximum temperature) at locations away fromthose for which direct measurements exist. In this way, esti-mates can be made for scales up to continents and grid spatialresolution is typically in the order of several kilometres. Theinterpolation methods vary in complexity and accuracy, fromsimple Thiessen tessellation and inverse square distance(Goovaerts, 2000) to more complex methods such as TruncatedGaussian Filter (Jolly et al., 2005; Thornton et al., 1997), krig-ing and co-kriging methods (Jolly et al., 2005) andvariations of spline interpolation (Mitas and Mitasova, 1999;Jeffrey et al., 2001; Hofierka et al., 2002). The choice of meth-ods is partly determined by the speed of computation requiredand nature of the modelled phenomena; whereas methods suchas Thiessen polygon methods are very fast, kriging and multi-variate splines require more computational effort. The optimumcomputational environment is a Geographical Information Sys-tem (GIS) that integrates spatially-distributed data and ded-icated methods for their management, analysis, modellingand visualisation.

For estimation of a meteorological value at a time differentfrom the times at which the measurements are being made,a temporal interpolation (or extrapolation) has to be realizedusing the measured values at the discrete times determinedby the pointwise data. For estimating the daily temperaturevariation a number of different methods have been used,most often based on measured values of minimum and maxi-mum temperatures during a day; see for instance Parton andLogan (1981), Wann et al. (1985), ESRA (2000), and Feidaset al. (2002). Temperatures are assumed to oscillate betweenthese two extremes with a given functional dependence. Thetimes of the minimum and maximum may be assumed (Partonand Logan, 1981; ESRA, 2000) or they may be calculatedfrom more detailed data (Feidas et al., 2002). However, thismethod risks overestimating the daily temperature variationsince the average of daily maximum temperatures is not thesame as the temperature at the time of the day when the tem-perature on average is highest. If more detailed temperaturemeasurements are available it is advantageous to use theseto make a more accurate estimate of the average daily curve.In cases where only daytime temperatures are important,a number of daytime temperature measurements may beused to make an improved estimate of the temperatures attimes during the day when no measurements are available.

In this paper we present a high-resolution GIS database oftemperatures over Europe, created by spatial interpolation ofmonthly averages of 7 daily measurements available from ap-proximately 800 meteorological stations. Two methods havebeen implemented and tested for simulation of daytime anddaily profiles from the primary database e polynomialcurve-fitting and the double-cosine method presented in theEuropean Solar Radiation Atlas (ESRA). We then present ex-amples of the temperature-related products derived from theprofile simulation methods. Finally, the application of curve-fitting method for an assessment of the efficiency of photovol-taic modules has been demonstrated.

2. Geographical region

Geographically, the region covers the European subconti-nent, as well as Turkey, Iceland, and parts of North Africa.The map projection used is a Lambert Azimuthal equal-areawith the central point at 48 � N and 18 � E (near Bratislava,Slovakia). All the data were integrated within the open-sourcegeographical information system GRASS (http://grass.itc.it/).The size of the region is 5000 km in EasteWest directionand 4500 km in NortheSouth. The 1-km grid resolution ofthe GIS database is given by the digital elevation model(DEM) that was derived from the USGS SRTM-30 data(http://srtm.usgs.gov/). This choice has been made for compat-ibility with subsequent applications.

3. Input temperature data

The ambient temperature measurements are obtainedfrom the European Meteorological Monitoring Infrastructure

1652 T.A. Huld et al. / Environmental Modelling & Software 21 (2006) 1650e1661

(EMMI) maintained by the Agriculture and Fisheries Unit ofthe Joint Research Centre (Micale and Genovese, 2004). TheEMMI database contains a set of 30 meteorological character-istics measured in various years starting from 1933 for a totalof 5179 meteorological stations located in the region,though not all data are available for every station. In the data-base, the daily values of 7 temperature characteristics areavailable:

- Five daytime values, measured at time horizons 6:00, 9:00,12:00, 15:00 and 18:00 hours at Greenwich Mean Time(GMT);

- Daily minimum and maximum.

Since the region stretches from longitude of w30 � W tow60 � E, the local time is shifted �2 to C4 h relative to theGMT, though only few measuring stations are located at theextremes.

The availability of temperature data in EMMI varies stronglyfor each station. Most stations do not have any of the 5daytime temperature values. Many stations only have measure-ments for a couple of years. In order to obtain satisfactorystatistics and consistency, we applied the following criteria for in-cluding a station into the working data set:

- Data should be available for as long time as possible, toget reasonable statistics. Preferably, the same set of yearsshould be used for all stations, in order to avoid spuriousspatial variation. This criterion was relaxed for isolatedcoherent regions.

- For a given month in a given year, the station must have atleast 10 daily values for each of the 7 data fields (the 5daytime temperatures and the minimum and maximum).This is to ensure that each year in the period is adequatelyrepresented.

An analysis of the data availability found that for the period1995e2003 the criteria were fulfilled for approximately800 stations over the region (the number varies a littlebetween months). This time period was taken as a base forcalculations. However, for a few regions shorter periods hadto be selected. For Slovenia, Croatia, Macedonia and Serbiaand Montenegro, the data are available only for the period1998e2003, for Iceland it is the period 2000e2002 and forBosnia Herzegovina this database does not have any records.Some regions at the margin of our study area (East of theUral Mountains, Middle East) have no data at all. As we didnot have data for the sea, this area was explicitly masked outfrom the database. The same goes for regions far (O150 km)from the nearest measurement station. This criterion mainlyaffects the limits of the region in the Middle East and WesternSiberia.

For each month, the average value, representing period1995e2003 (in some regions shorter), was calculated fromeach of the 7 daily measurements (maximum and minimumand 5 daytime temperatures). These data were used in furtherprocessing, explained below.

4. Methods

4.1. Spatial interpolation of the temperature data

The monthly averages of the 7 daily measurements (5 daytime tempera-

tures, minimum and maximum) for the meteorological stations were spatially

interpolated using a multidimensional regularized spline with tension (RST).

The method has been implemented in GRASS GIS as the command s.vol.rst

(Neteler and Mitasova, 2004). The RST takes into account also the elevation

above sea level as the third dimension, thus capturing a more complex spa-

tially and temporally variable relation between temperature and the elevation

in the mountainous areas with representative distribution of point measure-

ments. The RST control parameters of tension, smoothing and vertical scaling

can be tuned by the embedded cross-validation procedure to minimize the

predictive error as documented in the work of Hofierka et al. (in press). In

a comprehensive comparison with other multiquadrics and kriging ap-

proaches, Hofierka et al. (2002) have demonstrated that RST is suitable for

applications requiring interpolation for multiple time periods using data at

the same locations.

The Root Mean Square Error (RMSE) from the interpolation for a given

time of the day and for a given month was calculated as follows:

Eðt;MÞZ 1

Ns

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXNs

nZ1

ðTnðt;MÞ � Tn;iðt;MÞÞ2vuut ð1Þ

where Tnðt;MÞ is the monthly average of temperature at station number n at

time t for month M. The interpolated temperature is denoted as Tn;iðt;MÞ.The total number of measurement stations is Ns.

As the interpolation RMSE does not provide information about the areas

between the input points, cross-validation was also used to estimate the predic-

tive accuracy of the interpolation method. The RMSE assessment by the cross-

validation is based on removing one input data point at a time, performing the

interpolation for the location of the removed point using the remaining sam-

ples and calculating the residual between the actual value of the removed

data point and its estimate. The procedure is repeated until every sample

has been, in turn, removed (Tomczak, 1998). As a result of the exten-

sive cross-validation of the selected data, the following interpolation

parameters were found as optimal: tension 20, smoothing 0.1 and vertical

scaling 200.

4.2. Polynomial fitting for simulating daytime profiles

The grid maps of 5 daytime points on the temperature profile for each

month were used for constructing the average daytime (i.e. from sunrise to

sunset) temperature profile using polynomial curves and least-squares fitting.

We have tested polynomials of 2nd and 3rd order that are well representing

the time of the day between sunrise and sunset. The knowledge of the daytime

dynamics of temperatures is needed in applications such as solar electricity

generation and solar heating.

We calculated the fits using a new GRASS module r.fitpoly, developed

for this project. The input into this program is a series of N daytime

temperature grid maps and the calculation outputs grid maps of the polynomial

fit coefficients. The coefficients can then be used to reconstruct the average

daytime temperature curve for each month at any point in the region.

The r.fitpoly program can use any order of polynomial for the fitting of the

data.

Once the grid maps of curve fit coefficients have been calculated

by r.fitpoly they can be used to calculate maps of the temperature at any

given time during the day by plugging the desired time into Eq. (2) (or the

equivalent 2nd order equation). However, the grid maps of curve fit coeffi-

cients can also be used to calculate a number of derived data sets, such as

maximum daytime temperature and time of its occurrence or average daytime

temperature.

The r.fitpoly program can switch between GMT and local solar time, so

that both the regional and continental aspects of the temperature variability

can be simulated.

1653T.A. Huld et al. / Environmental Modelling & Software 21 (2006) 1650e1661

4.2.1. Maximum daytime temperature and time ofmaximum temperature

The time of maximum temperature can be found by simple differentiation

of the polynomial. For the 2nd order polynomial the resulting equation is lin-

ear and solved trivially except in extreme cases such as the daytime tempera-

ture profile at locations north of the Polar Circle in winter, where there is

practically no consistent temperature profile. For the 3rd order polynomial:

TðtÞZc3;3t3Cc3;2t2Cc3;1tCc3;0 ð2Þ

(where the coefficient cj;i denotes the i-th order term coefficient of a polynomi-

al of order j ) the time of maximum daytime temperature tmax is found as one

of the solutions to the equation:

tmaxZ�c3;2G

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðc3;2Þ2�3c3;3c3;1

q3c3;3

ð3Þ

To find the right solution with a high level of confidence a reasonable restric-

tion of the time must be made, such as for instance 11 ! tmax ! 19. Alterna-

tively, the requirement can be made that the second derivative be negative

which entails:

3c3;3tmaxCc3;2!0 ð4Þ

Finally, the value of the maximum daytime temperature T(tmax) can be calcu-

lated by Eq. (2).

4.2.2. Average temperature over daytime

An estimate of the average temperature during the daytime can easily be

found by integration:

TZ1

ts � tr

Z ts

tr

TðtÞdt ð5Þ

where tr is the time of sunrise and ts is the time of sunset. For a 2nd order poly-

nomial this gives:

TZ1=3c2;2

�t2s CtstrCt2

r

�C1=2c2;1ðtsCtrÞCc2;0 ð6Þ

while for a 3rd order polynomial it becomes:

TZ1=4c3;3

�t3s Ct3

r Ctrt2s Ctst

2r

�C1=3c3;2

�t2s CtstrCt2

r

�C1=2c3;1ðtsCtrÞCc3;0 ð7Þ

The local solar sunrise tr and sunset ts times are calculated as:

trZ12� 12

parccos

�� sin f sin d

cos f cos d

�ð8Þ

tsZ24� tr ð9Þ

where f is the geographical latitude and d is the declination of the day in the

year. The formula is not valid in polar regions during periods when the Sun

does not rise or does not set. In these cases the calculated day length will

be less than 0 or greater than 24 h.

4.3. ESRA method for simulating daily profiles

Due to the absence of 3-h temperature measurements in the GMT night-

time in the EMMI database, the above mentioned approach can be only

used for simulating the daytime temperature profile. However, some applica-

tions (e.g. heating/cooling of buildings) need inputs from a simulation of

full daily (i.e. 24-h) cycle of temperatures. For simulating the daily cycle, we

have used a method presented in ESRA (2000, pp. 166e167). The ESRA method

estimates the dependence of the temperature on the time of the day (given as

the local solar time) from only two inputs e minimum and maximum daily

temperatures. It assumes that the temperature daily profile T(t) can be described

using three piecewise cosine functions, dividing the day into three periods: from

midnight to sunrise (tdawn), from tdawn to the time of peak temperature (tpeak), and

from tpeak to midnight.

TðtÞZTm � Ta cos�pðtdawn � tÞ=

�24Ctdawn � tpeak

��for 0!t%tdawn

TðtÞZTmCTa cos�p�tpeak � t

�=�tpeak � tdawn

��for tdawn!t%tpeak

TðtÞZTm � Ta cos�pð24Ctdawn � tÞ=

�24Ctdawn � tpeak

��for tpeak!t%24

ð10Þ

Here, Tm is the mean daily temperature, estimated simply as the average of the

minimum and maximum temperatures:

TmZðTminCTmaxÞ=2 ð11Þ

while Ta is the amplitude of daily temperature variation divided by 2:

TaZðTmax � TminÞ=2 ð12Þ

The peak daily temperature is here taken to occur at tpeak Z 15 h.

5. Results and discussion

5.1. Interpolation and cross-validation errors

The result of the spatial interpolation, as described inSection 4.1, is a set of 7 grid maps for each month integratedinto the GRASS GIS database, representing the monthly aver-ages of measured ambient temperature:

- Five daytime temperatures for 3-hourly intervals from6:00 to 18:00 hours GMT;

- Minimum and maximum daily temperature.

As an example of the database, Fig. 1 shows a grid map ofthe average temperature in May over the European subconti-nent at 12:00 hours GMT obtained by the interpolation of832 point values. Some features are clearly visible:

- Strong vertical temperature gradient is well resolved inregions like the Alps and the Carpathians. This showsthe strength of the multidimensional RST interpolation.

- In regions where distribution of meteorological stations isnot representative (missing stations at higher elevation),the RST interpolation fails to account for the verticalgradient. This can be seen in the Caucasus Mountains.

The values of the RMSE from interpolation at the 5 mea-surement times during the day are presented for four selectedmonths in Fig. 2. The error is calculated as the RMS of the dif-ference between monthly averages of measured temperaturesand the values obtained from the spatial interpolation. Forthe monthly values, the interpolation RMS error stays withinthe range of 0.5e1.0 �C while the cross-validation error issomewhat higher, within 1.0e1.3 �C (Fig. 3).

The cross-validation procedure tends to overestimate theerror since it contains less information than the full interpola-tion. Especially in mountains, the removal of one point canseriously impair the interpolation and overestimate the temper-ature at the missing point. As an example, Fig. 4 shows a scat-terplot of cross-validation errors versus elevation for eachmeasuring station in March, at 12:00 hours GMT. At highelevations, the errors are generally large and positive, especially

1654 T.A. Huld et al. / Environmental Modelling & Software 21 (2006) 1650e1661

Fig. 1. Map of average temperature at 12:00 hours GMT in May [ �C]. The red points indicate the position of the meteorological stations.

when closer to noon. This indicates the importance of havinga representative number and distribution of measurements inhigh mountains.

The real error at points between the meteorological stationsis likely to fall somewhere between the error values ofinterpolation and cross-validation though probably closer tothe cross-validation value. The interpolation error is stronglydependent on the chosen interpolation method and on theset-up of its control parameters (such as smoothing, tension,etc.). Many interpolation methods, including RST, can betuned to have no interpolation errors. However, such set-upmay result in anomalies and high errors away from the pointsused in the interpolation. Therefore, a proper balance betweeninterpolation and cross-validation errors has to be found.

5.2. Daytime temperature model

The contribution of the fitting procedure to the overall errorfrom simulation was tested for polynomial coefficients of 2ndand 3rd order calculated by the r.fitpoly program. The simulatedvalues (from interpolation and curve-fitting) for the points rep-resenting the meteorological stations were compared to themeasured temperature over all the sites for each month. Inthe same way, we used the output from the cross-validation ex-ercise and the fitting algorithm and the output was also com-pared to the original data. The results are shown in Figs. 5and 6 for both 2nd and 3rd order fits.

Comparing the curve-fitting RMSE to the interpolation andcross-validation errors it can be seen that the fitting procedure

does not add much to the overall error. When comparing withthe interpolation error it is seen that the additional error issmallest in summer. This is likely due to the fact that all 5 day-time measurements are well within the daytime. In winter thefit errors are larger, especially for the 2nd order fit. In this case,morning and evening (or, in extreme cases such as north of thePolar Circle, all) measurements of the 3-h cycle represent tem-perature before sunrise or after sunset. Especially the timearound sunrise is not well approximated by a low-orderpolynomial.

Fig. 2. Interpolation errors at the 5 measurement times for four selected

months. The error is calculated as the RMS of the difference between monthly

averages of measured temperatures and the values obtained from the spatial

interpolation [ �C].

1655T.A. Huld et al. / Environmental Modelling & Software 21 (2006) 1650e1661

The method described above involves performing the spa-tial interpolation of the temperature data for each of the 5 day-time temperature values and then fitting the interpolated datain time to find the coefficients of the polynomial for eachgrid point. However, we have also tested the opposite opera-tion, i.e. fitting the temperature values for each station toa polynomial and then interpolating the polynomial coeffi-cients over the region. We have found that the final outputsfor both alternatives are practically identical, with differencesbetween the results much smaller than the interpolation errorsengendered in each of the methods. On the other hand, thealternative approach has the advantage of being less computa-tionally demanding (it saves time of interpolation of 12 or 24data layers for 2nd and 3rd order polynomials, respectively)and hence also saves disk space.

At high latitudes (close and above the Polar Circle) thismethod will not work well in peak summer or winter. In win-ter, the day length is so short that the result has little meaningother than giving the noon temperature. In summer, the start

Fig. 3. Cross-validation and interpolation errors for 12 months. The graphs

show the average of the RMS errors at all 5 daytime values for the cross-

validation and the interpolation [ �C].

Fig. 4. Scatterplot of the cross-validation errors against elevation for all the

ground stations used for the interpolation of measurements at time 12:00 hours

GMT for March.

and end of the day are well outside the bounds of the dataused for the curve fit, and especially the 2nd order fit is likelyto underestimate the average daytime temperature.

5.3. Comparison of daytime and daily temperaturemodels

The monthly averages of the 5 daytime measurements wereused for assessing the accuracy obtained from simulations bythe polynomial curve-fitting (Section 4.2) and by the ESRAdaily model (Section 4.3) for each station.

It should be noted that the curve-fitting method requiresmore data (5 values) than the ESRA method (based only ontemperature minimum and maximum), so it would be expectedto be more accurate. That said, the comparison reveals that theRMSE is almost twice as large when using the ESRA dailytemperature model (Fig. 7). The reason is not a less accurateinterpolation of minimum and maximum temperatures, as

Fig. 5. Comparison of RMS errors at the measuring stations when fitting the

interpolated daytime temperatures to 2nd and 3rd order polynomials. The

two fitting procedures are compared to the errors resulting only from the inter-

polation. Values for each month are averages of the 5 daytime values [ �C].

Fig. 6. Error calculation as in Fig. 5, but for curve fits based on the cross-

validation values at each measuring station [ �C].

1656 T.A. Huld et al. / Environmental Modelling & Software 21 (2006) 1650e1661

the RMSE is in the range 0.6e1.0 �C, i.e. similar to the RMSEfrom the interpolation of daytime temperatures.

The larger RMSE of the ESRA method stems from the factthat the monthly average of the daily maximum temperaturemeasured at the meteorological stations is not a good indicatorof the maximum of the average daily temperature profile (thesame is true for the minimum). While the measured daily max-imum (minimum) can occur at any time during the day, thevalue calculated from the average daytime profile (Eq. (3))represents the specific time when temperature on average isat its peak. These are two different concepts.

In order to investigate this in more detail we used a grid mapof the time of the maximum daytime temperature tmax and fromthis we calculated a grid map of the maximum temperature atthis time T(tmax) (see Section 4.2.1). Subtracting the grid mapof T(tmax) from the interpolation of average measured maximumtemperatures we get a new grid map of the ‘‘overshoots’’.A graphical interpretation of the map is not very informativesince there is no clear geographical dependence of the over-shoots. Instead we have performed a probability analysis ofthe temperature overshoots. The results are shown in Fig. 8.For each of the four months in the graph, the curve shows theprobability of the temperature overshoot falling in each of 14‘‘bins’’ from �0.5 �C to C3 �C. Few points have less than0.5 �C deviation and the most probable deviation is in the range0.75e1.5 �C depending on the month. Only in December doa significant number of points show a deviation greater than2 �C.

The difference between the two methods is visible also inthe mean bias error (MBE). In Fig. 9a, we show for four dif-ferent months the mean bias error of the interpolated and fitted(3rd order) data relative to the actual station data. The MBE isplotted as a function of the time during the day. Values aregenerally in the range �0.1 ! MBE ! 0.1 �C except forDecember where MBE can reach absolute values of a littleless than 0.25 �C. These values are close to the accuracy of an-alogue temperature measurements at many of the stations. The

Fig. 7. Comparison of the RMS errors resulting from applying the ESRA dou-

ble-cosine simulation method for simulating daytime temperature profile with

polynomial curve-fitting procedure by r.fitpoly (assuming 2nd and 3rd order

fits) [ �C].

Fig. 8. Probability distribution of overshoots of averages of measured maxi-

mum temperatures (interpolated) over the maximum of the daytime tempera-

ture curve (interpolated and fitted to a 3rd order polynomial). The probability

distribution is the probability of values falling into bins 0.25 �C wide.

Fig. 9. Mean bias error (MBE) of simulated temperature against the monthly

averaged measured data as a function of time during the day for four different

months. The results of the two methods are compared: (a) MBE for the inter-

polated and 3rd order fitted data; (b) MBE for the ESRA daily temperature

model [ �C]. Note the different scales on the ordinate axes of the two plots.

1657T.A. Huld et al. / Environmental Modelling & Software 21 (2006) 1650e1661

higher MBE in December is due to the fact that the days areshort so a fit using values at 6:00 and 18:00 hours will usedata outside the daytime.

These low values of MBE are contrasted with the muchhigher values for the ESRA method shown in Fig. 9b (notea different scale). The MBE varies much more in this case,with a consistent variation from �1.5 �C in the morning toC2.5 �C in the afternoon. The afternoon results are causedby the overshoot in the maximum temperature. The deviationin the morning is due to the fact that the actual temperaturerise after sunrise is much steeper than indicated by a cosinefunction that has a minimum at sunrise. Thus the model func-tion does not react quickly enough and the temperature isunderestimated.

5.4. Derived temperature data

As an example of the application of the polynomial-fittingmethod, for the month of June we present a calculation of theaverage time when the maximum of the temperature profile,tmax occurs (Fig. 10a). The time represents the local solartime. The map reveals geographical variation of tmax withinEurope in the range of 13e16 h depending on the proximityto the sea and elevation. Fig. 10b shows the value of the max-imum temperature at this time T(tmax).

In general, we can state that the daily temperature curvecalculated from the maximum and minimum temperatures(Eq. (10)) results in significant overshoots. This is the casealso for more sophisticated methods than the one used here,such as the method used by Feidas et al. (2002), where thetime of maximum temperature is calculated from more de-tailed temperature data sets but the minimum and maximumtemperatures range is still used to calculate the amplitude ofthe daily temperature variation.

Both temperature models can be used for simulating dailyand daytime average temperature, see Eqs. (6), (7), and (11).An example in Fig. 11 shows the average daytime temperaturefor May in Europe that was calculated using the 3rd orderpolynomial (Eq. (7)).

6. Application: estimation of temperature effects onthe conversion efficiency of photovoltaic modules

The temperature database together with both simulation mod-els is generally applicable to many projects that need daily/daytime temperature profiles in Europe. However, it was origi-nally constructed within the PVGIS project (Suri et al., 2005)for the purpose of improving the prediction of the electricitygeneration from photovoltaic (PV) systems.

The PV systems convert solar radiation directly into electri-cal energy. Obviously, the amount of energy produced dependson solar radiation. However, the conversion efficiency dependson the instantaneous values of the irradiance and the tempera-ture of the PV modules. For PV modules made from crystal-line silicon (comprising the greater part of the current worldPV production) this dependence can be expressed as a functionfor the relative efficiency hrel (King et al., 1998):

hrelZð1CaiðTm� T0ÞÞ�

1Cc1 ln

�Gi

G0

�Cc2

�ln

�Gi

G0

��2

CbvðTm � T0Þ�

ð13Þ

where Gi is the in-plane irradiance, and Tm the module temper-ature; G0 Z 1000 W/m2 and T0 Z 25 �C are the values atStandard Test Conditions (STC), at which the referenceefficiency of a PV module is normally measured. The otherparameters were determined experimentally by Kenny et al.(2003) to have the values ai Z 1.20 ! 10�3 �C, bv Z�4.60 !10�3 �C, c1 Z 0.033 and c2 Z�0.0092. The module temperatureis estimated from the ambient temperature Tamb and irradi-ance Gi according to:

TmZðTNOCT � 20Þ Gi

800CTamb ð14Þ

where TNOCT is the nominal operating cell temperature.For modules in a free-standing configuration, an approximatevalue for TNOCT is 48 �C.

PVGIS is an online system for estimating solar radiation inthe European subcontinent based on the GRASS solar radia-tion model r.sun (Suri and Hofierka, 2004). Applying theabove equations together with the r.sun model, we calculatedthe relative efficiency hrel for PV modules oriented due Southand inclined 25 � from the horizontal (Fig. 12).

The efficiency of PV modules is shown as an overall valuefor the entire year and its values range from around 88% inSouthern Spain and Northern Africa up to 95% in Scotlandand Scandinavia. This effect works to slightly offset theadvantage of higher total irradiation in Southern Europe.The total irradiation in Europe varies by a factor of approxi-mately 2.5 from Northern Scandinavia to Southern Spainand Italy (Suri et al., 2005), and thus this effect seems slight.However, this should be compared to the overall rate of powerlosses in a PV system. An often-used rule of thumb states thatthe expected output from a PV system is 75% of what wouldbe obtained if the system always converted solar radiation toelectric power at the efficiency obtained under STC and ifthere were no other losses (losses in cables, in DC to AC con-version, etc.). Seen in this way, the loss of efficiency is equalto one-quarter to one-half of the total losses. Since other lossesare likely to be independent of geographical position, thisshows that a simple rule of thumb is insufficient to estimatethe expected real performance of a PV system. The informa-tion on the expected relative efficiency at different geograph-ical locations therefore helps PV system operators to assesswhether a given system is performing satisfactorily at the givenradiation input.

In addition to these considerations it should be noted thata difference in performance of about 7% relative to theexpected performance is not insignificant when consideringthe economic feasibility of a given installation.

1658 T.A. Huld et al. / Environmental Modelling & Software 21 (2006) 1650e1661

Fig. 10. Map of time of maximum daytime temperature in June (A) given as local solar time, and the corresponding temperature (B) at the maximum point [ �C].

7. Conclusions

We have presented a methodology for developing a GISdatabase and tools for estimation of the average daytime/daily

ambient temperature profiles for any point in large geograph-ical regions, applied to the European Subcontinent. The GISdatabase is created by spatial interpolation of monthly aver-ages of 5 daytime measurements at 3-h intervals from 6:00

1659T.A. Huld et al. / Environmental Modelling & Software 21 (2006) 1650e1661

Fig. 11. Average daytime temperature (assuming the period of day from sunrise to sunset) for Europe in June [ �C].

Fig. 12. Relative efficiency of crystalline silicon photovoltaic modules inclined 25 � from the horizontal position over the course of the year, varying due to geo-

graphic distribution of temperature and irradiance [%].

1660 T.A. Huld et al. / Environmental Modelling & Software 21 (2006) 1650e1661

to 18:00 hours GMT and two daily measurements of minimumand maximum temperatures from approximately 800 meteoro-logical stations. Polynomial fitting implemented as the r.fitpolyprogram within the GRASS GIS software is employed toestimate daytime (approximately from sunrise to sunset) tem-perature profiles between the 3-h values. The ESRA double-cosine method was implemented for estimating full 24-h dailyprofiles. The GIS database and both methods provide data toother models that take into account high-resolution spatialand time variability of the modelled phenomena.

Cross-validation of the interpolation method shows that theRMS errors are modest, with values in the range of 1e1.2 �C.The actual interpolation errors are significantly lower, in therange of 0.5e0.9 �C. The polynomial fit gives only a minorcontribution to the overall error. A reasonable spatial distribu-tion of the measurement stations is necessary to obtain a goodinterpolation results, mainly in mountains.

A comparison of r.fitpoly and the ESRA methods shows thatfor simulating daytime temperature profiles the first methodgives smaller errors. However, the r.fitpoly method is limitedin scope in that it does not predict nighttime temperatures.This deficiency could be addressed by including temperaturesmeasured at regular intervals during the all 24-h cycle. Forsuch an application, further investigation should be made tofind good fitting functions.

Once the grid maps of polynomial fit coefficients have beenconstructed they can be used for different purposes. We havepresented how an arithmetic manipulation can estimate thetime of maximum daytime temperatures and average daytimetemperatures. Here the use of the GRASS GIS software isbeneficial, since such manipulation is almost trivial usingthe built-in map algebra (r.mapcalc module). Another moreinvolved example is the application of the temperature datato estimate photovoltaic module performance in regions ofEurope.

The database is developed from a set of measuring stationswith average distances between stations much larger than thecomputational grid size (1 km). This inevitably means thatthere may be local phenomena affecting the temperaturewhich will not be resolved by the interpolation. At regionaland local scales, mainly in mountains, these could be terraingeometry (slope inclination and aspect) and terrain shadow-ing, as well as land cover. Using the digital elevation model,the shadowing dynamics and the influence of the terrain ge-ometry may be simulated using other GRASS GIS modulesr.shadow and r.sun. On the other hand, the proper modellingof these effects on temperature needs more experiments. An-other option is to supplement the data from ground stationswith data from satellites estimating the ground and surfacetemperatures (Carissimo et al., 2005; Doicu et al., 2005).This will depend on the availability of sufficient data to getgood long-term averages.

The use of open-source software makes it easier for othergroups to apply these methods to different geographical re-gions and data sets. Most of the software is already part ofthe GRASS software package. Additional utilities are avail-able upon request.

The extended work, including maps, graphs and animationscan be consulted at the web http://re.jrc.cec.eu.int/pvgis/pv/temper/.

The contemporary version of database represents onlymonthly averages for a period of 1995e2003. The automatingof the existing algorithms enables incorporation of new meas-urements every year. If needed (e.g. for prediction models) thetools can also be applied for processing the daily data.

Acknowledgements

This work has been carried out in the European CommissionJoint Research Centre under the Action No. 2324 SOLAREC,within the project PVGIS. The authors would like to thank thecolleagues Jaroslav Hofierka (University of Presov, Slovakia)and Tomas Cebecauer (Institute of Geography SAS, Slovakia)for discussions and helpful comments and Stefania Orlandi forher assistance in accessing the meteorological database.

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