Department de Medi Ambient i CieÁncies del SoÁl, Universitat de Lleida, Lleida, Spain

A new simple method for estimating monthly and daily solarradiation. Performance and comparison with other methodsat Lleida (NE Spain); a semiarid climate

F. Castellvi

With 3 Figures

Received May 23, 2000Revised February 14, 2001

Summary

A simple new method is proposed for estimating relativesunshine duration and solar irradiance on both a monthly anddaily basis. The method requires the occurrence of precip-itation, and maximum and minimum air temperatures asinput. Comparisons with two other published simple meth-ods: Bristow and Campbell (1984); and Hargreaves et al.(1985), and a modi®ed version of the method proposed byMcCaskill (1990), were conducted for a weather station atLleida (NE Spain), which has a continental semiarid climate.The new method performed slightly better than the othersmentioned above, mainly as a result of a reduction in errorswhen estimating cases of high solar radiation.

1. Introduction

The aim of this work was to ®nd a suitable, simplemethod for estimating solar radiation data toaccompany generated weather series for dailyprecipitation, and maximum and minimum airtemperatures in dry climates. The method mayalso be used for estimating missing data fromactual records.

Over the last two decades, increased interestin modeling radiation-mediated processes hasresulted in a greater demand for solar radiationdata. This demand has been re¯ected in thenumerous methods, developed for suitable radia-tion data, that have already been reported. Thesemethods range from simple empirical formulas to

complex numerical models that depend on theavailability of input data. Simple relationships forestimating sunshine duration and solar radiation,involving such factors as cloud cover, percentageof speci®c cloud types, evaporation, humidity,number of days with dust or smoke, air tempera-ture, precipitation, latitude and elevation, havebeen widely reported (Linacre, 1992). Simplerelationships are practical because in remote areasweather data is scarce and furthermore fullyinstrumented meteorological stations are costlyto set up and maintain. References comparing avariety of simple methods can be found inGoldberg et al. (1979), Meinke et al. (1995),Hunt et al. (1998), X. Yin (1999a and 1999b) andMeza and Varas (2000).

Good estimates of solar irradiance using cloudcover have been achieved that correlate well withsky transmissivity for example, Schei®nger andKromp-Kolb (2000). Records of cloud cover havebeen available for many years though for fewerweather stations than data for precipitation and airtemperature. The observation of cloudiness israther subjective, so cloud cover related data tendsto be less attractive than that of other weathervariables (Linacre, 1992). Relationships involvingair temperature and precipitation data, apart fromother site-speci®c geographical parameters, areboth valuable and attractive because thermo-

Theor. Appl. Climatol. 69, 231±238 (2001)

pluviomentric weather stations are common andgenerally possess records going back for severaldecades.

Some models, known as weather generators,normally require as input monthly means andstandard deviations for the occurrence and amountof precipitation, maximum and minimum airtemperatures and solar radiation in order to sto-chastically generate daily series of precipita-tion, maximum and minimum air temperaturesand solar radiation (see among many others,Richardson and Wright, 1984; Geng et al., 1986and Wallis and Grif®ths, 1995). When solar radia-tion data is not available or records are too short,solar radiation data may accompany daily pre-cipitation and air temperature generated data byimplementing a suitable simple method to esti-mate daily solar radiation. Weather generators areuseful for bypassing problems deriving from lackof data, records taken over too short a time spanand poor quality weather variables, when con-ducting long-term analyses or taking planningdecisions. Weather generators are also particularlyuseful for providing input data for other modelswhich are related to hydrological and agriculturalprocesses (Johnson et al., 1996).

Hunt et al. (1998), following a comparison ofseveral equations for estimating daily solar radi-ation in a humid climate (Canada), showed thatsmall errors result when the amount of precipita-tion and daily air temperature amplitude are takeninto account.

Equations involving the daily amount of precip-itation were not introduced in the present workdue to the dif®culties associated with reliablyreproducing the amount of precipitation assignedto a given wet event. The reliability of a generatedseries of daily precipitation data depends on aparticular method's performance when generatingwet and dry events and on the amount of precip-itation assigned to each given wet day. Further-more, the results obtained by Hunt et al. (1998)cannot be extrapolated to dry climates: whenprecipitation events are scarce, this variable hasreduced statistical weight.

Hence, the aim of this work was to proposea simple new method for estimating daily solarradiation, over ¯at terrain, from the daily occur-rence of precipitation and maximum and mini-mum air temperatures. The proposed method wascompared with another method that requires the

same input data, a modi®ed version of the Tamsimmethod (McCaskill, 1990), and two other meth-ods; the Bristow and Campbell method (Bristowand Campbell, 1984) and Hargreaves method(Hargreaves et al., 1985) that require daily maxi-mum and minimum air temperatures as input data.

The Lleida weather station was selected to testthe methods. Lleida, with a continental semi-aridclimate, is located in the river Ebro basin withinan important agricultural area in Catalonia,northeast Spain (42�N, 0.6� E) and at 120 mabove sea level.

2. Theory

2.1 Proposed method

Any monthly mean weather variable, �X�m, cal-culated from the corresponding daily values canbe determined as follows;

�X�m � fdry�Xdry�m � fwet�Xwet�mwith fdry � fwet � 1 �1�

where �Xdry�m and �Xwet�m are respectively themonthly mean weather variables for dry and wetdays in month m and, fdry and fwet, the corre-sponding monthly frequencies of dry and wetdays. When fdry�Xdry�m and fwet�Xwet�m are corre-lated, fdry�Xdry�m � a fwet�Xwet�m. Using Eq. 1 themonthly mean weather variable for dry and wetdays can be expressed as follows:

�Xdry�m � b�1ÿ fwet�ÿ1�X�m �2��Xwet�m � �1ÿ b�� fwet�ÿ1�X�m �3�with b a site-speci®c coef®cient to be determined.Note that when fwet is close to zero: equal or lessthan �1ÿ b�, physical inconsistencies may arisefrom Eq. 3. When fwet is zero, the monthly meanweather variable for wet days is zero. Thus, analgorithm has to be introduced in Eq. 3 to accountfor the range 0< fwet � �1ÿ b�.

The following approximation may be used,�1ÿ fwet�ÿ1 � �1� fwet�, for dry climates as thefrequency of wet days is low in most months. It isthen proposed to estimate monthly mean valuesfor dry and wet days using the following twoexpressions:

�Xdry�m �a1�1� fwet��X�m if fwet > 0

�X�m if fwet � 0

��4�

232 F. Castellvi

�Xwet�m �1ÿ a1�1ÿ � fwet�2�h i

� fwet�ÿ1�X�m if fwet > �1ÿ a1�a2�X�m � a3 if 0< fwet � �1ÿ a1�undefined if fwet � 0

8><>:�5�

where, a1, a2 and a3 are site-speci®c coef®cientsto be determined.

The reason for proposing Eq. 4 instead of Eq. 2was that Eq. 4 expresses a linear relationship withthe frequency of wet days. Eq. 4 may suit all thedata pairs better than Eq. 2 when determiningcoef®cient a1 in Eq. 4 and b in Eq. 2 using simplelinear ®t. Note that months with a relatively highfrequency of wet days may introduce a largerdeparture from the linear tendency when deter-mining coef®cient b than coef®cient a1. Further-more, the relative error, RE, for the approximationused is RE � � fwet�2 and thus for dry climates therelative error introduced is acceptable for mostmonths.

The ®rst algorithm in Eq. 5 was obtained bycombining Eq. 1 and Eq. 4. The second algorithmin Eq. 5 was proposed for simplicity. In principle,the range of fwet used for this algorithm is verynarrow. The drier the climate, the closer the coef-®cient a1 will be to unity and consequently thenarrower will be its range of monthly frequency ofwet days.

The proposed method estimates daily solarradiation according to the following three steps:

1. An equation to estimate actual bright sun-shine hours over length of day or relative brightsunshine duration, BSD, on a monthly basis isproposed as follows:

�BSD�m � A� B��Tm�b � C� fwet�c �6�where �Tm is the monthly air temperature ampli-tude for month m and A, B, C, b and c are sitespeci®c coef®cients to be determined.

2. A set of two equations, similar to Eq. 4 andEq. 5, are proposed to estimate the daily BSD forwet and dry days respectively as follows,

Dry day;

BSDk � a1�1� fwet��BSD�m;k if fwet > 0

�BSD�m;k if fwet � 0

��7�

Wet day;

BSDk �b1ÿ a1�1ÿ � fwet�2c� fwet�ÿ1�BSD�m;k if fwet > �1ÿ a1�a2�BSD�m;k � a3 if 0< fwet � �1ÿ a1�0 if fwet � 0

8<:�8�

where the sub-index k denotes the day of the year,fwet is the frequency of wet days in month m andBSDm;k is a daily value obtained from Eq. 6 byintroducing daily air temperature amplitude, �Tk,instead of monthly air temperature, �Tm. Itshould be noted that Eq. 7 and Eq. 8 depend ondaily air temperature amplitude, which correlatespositively with solar radiation and relative brightsunshine duration. In general, cloudy days willhave a higher minimum air temperature and alower maximum air temperature than their re-spective monthly mean values and vice-versa.

3. Daily solar radiation, Rs in MJ mÿ2, is ob-tained using the AngstroÈm expression (Martinez-Lozano et al., 1984),

Rsk � �a� b BSDk�Rak �9�where a and b are site-speci®c coef®cients (alsoknown as AngstroÈm coef®cients) and Rak, in MJmÿ2, the extra-terrestrial radiation (also known asAngot radiation) for day of year k that can bedetermined using the following expression:

Rak � 37:6�1� 0:033 cos�0:0172 k��� �! sin� sin � � sin! cos� cos �� �10�

where � is the solar declination in radians whichcan be calculated from the expression, � � 0:409sin �0:0172 k ÿ 1:39�; � is the latitude in radiansand ! is the sunset hour angle in radians whichcan be calculated from the expression, ! � arccos�ÿtan � tan ��.

2.2 Modi®ed Tamsim method

The Tamsim method (McCaskill, 1990) can beused to estimate any primary weather variableusing Eq. 11 (see below) without the last term: bysetting the coef®cient i equal to zero. Here,following Hunt et al. (1998), daily air temperatureamplitude, �Tk was introduced and a modi®edversion of the Tamsim method resulted:

BSDk � a� b cos �� c sin �� d cos 2�

� e sin 2�� f Pkÿ1 � g Pk

� h Pk�1 � i �Tk �11�where, a, b, c, d, e, f, g, h and i are empiricalcoef®cients, � is the day k of year converted toradian form �� � k 2�=365�;Pkÿ1;Pk, and Pk�1

are precipitation converted to binary form (0 fordry days and 1 for wet days) for the previous,

A new simple method for estimating monthly and daily solar radiation 233

current and following day, respectively. Solarradiation is obtained from Eq. 9.

Note that BSDk values determined using boththe method proposed and the modi®ed Tamsimmethod may sometimes be negative or greaterthan unity although this is not normally the case.Desired boundary values can be imposed: here,they were set to zero and unity respectively.

2.3 Bristow and Campbell method

Bristow and Campbell (1984) proposed a relation-ship linking daily atmospheric transmissivity andthe difference between daily maximum air tem-perature, Tx, and average minimum air tempera-ture, Tn, for the current and following day, DTk �Txk ÿ 0:5�Tnk � Tnk�1�. Daily solar radiation isestimated according to the following expression,

Rsk � � �1ÿ exp�b DTck �Tÿ1

m ��Rak �12�where � is the clear sky transmissivity to be esti-mated, b and c are empirical coef®cients, and�Tm is monthly air temperature amplitude asde®ned in Eq. 6.

Note that in the original Bristow and Campbellmethod (1984) the monthly air temperature ampli-tude was embedded in coef®cient b proposing anexponential relationship dependency. Though notshown, previous research was conducted at Lleidaand obtained similar or slightly better estimationsusing the dependence shown in Eq. 12 rather thaneither the original form suggested by Bristow andCampbell (1984) or another derived-form pro-posed by Donatelli and Campbell (1998).

The best ®t procedure for Eq. 12 is as follows(Donatelli and Campbell, 1998); the slope forestimated versus actual solar radiation should beas close to unity as possible and the root meansquare error and coef®cient of residual massshould be as low as possible (see details in Loagueand Green, 1991).

2.4 Hargreaves method

Hargreaves et al. (1985) proposed the followingrelationship between daily solar radiation anddaily air temperature amplitude,

Rsk � a� b Rak �T0:5k �13�

where a and b are two empirical coef®cients.

3. Climate, database and comparisonprocedure

The main relevant climatic features at Lleida areas follows: Consecutive days of persistent fog orovercast skies during winter and early spring dueto the stagnation of cold air (total of 58 days peryear on average); front associated rainfall duringlate spring and autumn, mainly falling in May,October and November and convection rainfallmainly in July and August. The driest and mosthumid months are February and October with18.7 mm and 39.5 mm respectively. July and Mayare the months with the minimum and maximummonthly frequencies of wet days: 0.08 and0.22 respectively. The mean annual amount ofprecipitation is 338.5 mm. The maximum andminimum monthly values for solar radiationwere recorded in July and December with25:4 MJ mÿ2 dayÿ1 and 4:9 MJ mÿ2 dayÿ1 respec-tively.

At Lleida station daily solar radiation data wereavailable for a six year period (1994 to 1999) anddaily precipitation, maximum and minimum airtemperatures and relative bright sunshine durationdata for a thirty year period (1970 to 1999). Theruns test (E. Linacre, 1992) was applied to analyzethe homogeneity of the monthly climate series forthe amount of precipitation, mean and absolutevalues for the maximum and minimum air tem-peratures and mean relative bright sunshine dura-tion. All the 30-year series data passed the runstest.

Calculated daily solar radiation obtained usingEq. 9 and measured relative bright sunshineduration were used for the ten year period (1970to 1979) to determine the empirical coef®cientsinvolved in all methods. Following Jagannathanet al. (1967) ten years is, on average, long enoughto obtain stable climate statistics for cloud cover atlocations similar to Lleida. Thus assuming noclimate change (none was detected by runs test)the relative bright sunshine duration and solarradiation variability were fully captured from thisdata set.

The next ten year period (1980 to 1989) wasused for validation and performance. Generalstatistics derived from linear ®t, root mean squareerror, t-test and F-test were determined on anannual basis. Kolmogorov-Smirnoff tests for dis-tribution functions on an annual, seasonal (con-

234 F. Castellvi

sidered as three consecutive months) and monthlybasis were also used for comparisons. All testswere conducted at 5% level of signi®cance.

4. Results

A day was considered wet when the amount ofprecipitation was greater than 0.2 mm (rain gaugemeasure error). Simulated annealing procedure wasused (Goffe et al., 1994), to optimize the coef-®cients in Eq. 6. Regression technique was usedfor all the other equations.

Note that any equation for estimating monthlymean relative bright sunshine duration can be usedin the ®rst step of the method proposed. The thirdstep may be avoided if an equation for estimatingmonthly mean solar radiation is used in the ®rststep. Following Meza and Varas (2000), theHargreaves and Bristow and Campbell equationswere calibrated on a monthly basis, but their per-formance on a monthly time basis gave poorerresults than combining the proposed equations,Eq. 6 and Eq. 9 (not shown).

4.1 Calibration of algorithms

The linear ®t through the origin to determine thecoef®cient a1 in Eq. 4 was a1 � 0:94, R2 � 0:93and SEE � 0:05. Coef®cients a2 and a3 in Eq. 5where determined from data for seven monthswhich failed in the range assigned to this algo-rithm. From the data set used for calibration fourmonths of July and three of August had monthlyfrequencies of wet days within the range0< fwet < 0:06. The regression ®t produced thefollowing results: a2 � 5:00 and a3 � ÿ3:41,R2 � 0:35 and SEE � 0:03.

The coef®cients obtained for Eq. 6 were asfollows: A � 13:54, B � ÿ14:41, C � ÿ0:145,b � ÿ0:045 and c � 0:635. The root mean squareerror of the estimate, RMSE, was RMSE � 0:02.Note that the values determined for coef®cients Band b were physically consistent: the greater thethermal amplitude, the greater the relative valuefor bright sunshine hours. The term involving themonthly frequency of wet days modi®es the in-dependent term A month by month. Coef®cients Cand c were also physically consistent: the greaterthe monthly frequency of wet days, the lesser thevalue for monthly relative bright sunshine dura-tion for a given monthly air temperature ampli-tude.

The period 1994 to 1999 was used to determineEq. 9. The coef®cients obtained from the linear ®twere a � 0:23 and b � 0:51 with a correlationcoef®cient of R2 � 0:91 and a standard error ofthe estimate of SEE � 2:52 MJ mÿ2. The coef-®cients were close to the values recommendedby Allen et al. (1994) for a variety of climates,a � 0:25 and b � 0:5.

4.2 Validation of algorithms

Figures 1 to 3 show agreement for actual andestimated monthly relative bright sunshine dura-tion (Eq. 6), and for the corresponding estimatesfor dry and wet days (Eq. 4 and Eq. 5 respec-tively) using Eq. 6 to estimate the monthly valuefor each year. Linear ®ts through the origin forthese equations are shown in Table 1. Performance

Fig. 1. Actual versus estimated (Eq. 6) monthly relativebright sunshine duration

Fig. 2. Actual versus estimated (Eq. 4) monthly relativebright sunshine duration for dry days

A new simple method for estimating monthly and daily solar radiation 235

was generally very good, although for wet daysthe variance captured was low, R2 � 0:44.

Table 2 shows the results obtained from linear®t through the origin, the root mean square error,and the actual and estimated annual mean andstandard deviation for daily solar radiation, foreach method. The results obtained for eachmethod were similar and performance was good.The method proposed and the modi®ed Tamsimmethod did not pass the t-test for the annual mean.The F-test applied for annual standard deviationwas rejected for all except the proposed method.

Table 3 shows the actual daily solar radiationdistribution function for intervals of 5MJ mÿ2 andthe absolute difference between the actual andestimated frequency by each method and interval.Also shown is the interval at which each methodfailed the Kolmogorov-Smirnoff test. Only theproposed method passed this test. Table 4 showsthe distribution function for each season using theproposed method. Only the third and fourthseasons passed the test: the method proposedwas unable to reproduce daily variability duringmost of winter, spring and early summer. Thecorresponding tests applied on a seasonal basis forthe other methods all resulted in rejection (not

Fig. 3. Actual versus estimated (Eq. 5) monthly relativebright sunshine duration for wet days

Table 1. Statistics obtained from the linear ®t through theorigin and the root mean square error (RMSE) obtainedwhen Eq. 6 was used to determine the monthly brightsunshine duration for dry and wet events (Eqs. 4 and 5respectively). The same statistics are shown for Eq. 6

Slope R2 RMSE

Eq. 4 1.06 0.82 0.08Eq. 5 0.96 0.44 0.09Eq. 6 1.06 0.91 0.06

Table 2. Statistics obtained from the linear ®t through the origin and the root mean square error (RMSE) obtained for actualversus estimated daily solar radiation. Also shown are the actual and estimated annual mean and standard deviation (STD) andthe methods that did not pass the t-test and F-test for means and standard deviations respectively (�)Method a R2 RMSE (MJ mÿ2) Mean (15.42 MJ mÿ2) STD (9.23 MJ mÿ2)

Proposed 1.01 0.91 2.55 15.06� 9.33Tamsim modi®ed 1.00 0.90 2.50 15.92� 8.73�Bristow and Campbell 0.97 0.90 2.54 15.36 8.93�Hargreaves 0.97 0.90 2.49 15.37 8.83�

Table 3. Actual daily solar radiation distribution function for intervals of 5 MJ mÿ2 and absolute differences in frequencybetween actual and estimated solar radiation by each method and interval. Also shown (�) is the lowest interval at which eachmethod failed the Kolmogorov-Smirnoff test

Interval Distribution Proposed Mod. Tamsim Bristow-Campbell Hargreavesfunction

0 0 ± ± ± ±5 0.175 0.015 0.053� 0.036� 0.00910 0.369 0.001 0.029 0.038 0.00615 0.526 0.001 0.005 0.008 0.00620 0.675 0.008 0.019 0.001 0.00925 0.835 0.016 0.017 0.035 0.036�30 0.998 0.009 0.010 0.026 0.03435 1 0.017 0.001 0.002 0.001

236 F. Castellvi

shown), except for the Hargreaves method thatpassed the test for the fourth season. Hence, theHargreaves method was only able to perform forvariability during autumn and early winter. Badperformances were obtained for all methods on amonthly time basis: all Kolmogorov-Smirnoff testswere rejected except in November and Decemberusing the proposed method (not shown).

5. Summary and conclusions

A new method was proposed for estimatingmonthly and daily bright sunshine duration andsolar radiation from precipitation occurrence andmaximum and minimum air temperatures for dryclimates. The method proposed performed wellon a monthly time basis when applied for acontinental semi-arid climate. Any equation canbe used for estimating the monthly mean relativebright sunshine duration in the ®rst step of themethod proposed. On the other hand, equationsfor estimating monthly mean solar radiation mayalso be used in the ®rst step for estimating dailysolar radiation directly from Eq. 7 and Eq. 8 (i.e.,avoiding the third step). Note that to calibrate theproposed method, whatever the method used inthe ®rst step, only actual monthly data is requiredfor estimating daily solar radiation. Good qualitydaily data is required for calibrating simplemethods for estimating daily solar radiation andin most weather generators (standard deviation isrequired as input).

The performance of four methods for estimat-ing the daily solar radiation was tested: the pro-posed method; a modi®ed version of the Tamsimmethod; the Bristow and Campbell method; and

the Hargreaves method. No method was clearlysuperior to the others. However, in general, theproposed method performed slightly better thanthe others. The Hargreaves method performedbetter than either the modi®ed Tamsim method orthat of Bristow and Campbell. However, it tendedto increase errors in cases with high solar radiation(in our case, following Table 3, Rs> 20 MJ mÿ2).For cases with high solar radiation, the methodproposed performed better than the Hargreavesmethod. Hence, for Lleida, the proposed methodwas more convenient for agriculture purposes.

Acknowledgments

This work is a preliminary study and forms part of theprojects, InterregII (2.3.GC. Code 82, Centre TecnoloÁgicForestal de Catalunya) and the CICYT projects HID96-1295-C04-03 and HID97-397, that provided ®nancial support inorder to obtain the database.

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Author's address: F. Castellvi, Department of MediAmbient i CieÁncies del SoÁl. University of Lleida, C. RoviraRoure, 177, Lleida 25198, Spain (E-mail: F-Castellvi@macs.udl.es).

238 F. Castellvi: A new simple method for estimating monthly and daily solar radiation