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AGROCLIMATOLOGY

Evaluation of Solar Radiation Prediction Models in North America

^a Dep. of Plant Sci., Univ. of Saskatchewan, 51 Campus Dr., Saskatoon, SK S7N 5A8, Canada
^b Dep. of Crop, Soil, and Environ. Sci., Univ. of Arkansas, 1366 W. Altheimer Dr., Fayetteville, AR 72704
^c Dep. of Geogr., Univ. of Saskatchewan, 9 Campus Dr., Saskatoon, SK S7N 5A5, Canada

^* Corresponding author (lpurcell{at}uark.edu).

Received for publication February 11, 2003.

	ABSTRACT

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
Solar radiation data at the earth's surface (R_s, MJ m^–2 d^–1) are not typically recorded at weather stations, but they may be predicted from other meteorological measurements. For one location, Keiser, AR, we developed an empirical equation for predicting R_s. The mechanistic models of Hargreaves–Samani (HS) and two forms of the Bristow–Campbell model, described by Thornton and Running (TR) and Weiss et al. (WS), were also evaluated for predicting R_s at 13 sites, covering a 23° range in latitude and a 42° range in longitude. For the HS, TR, and WS models, we used coefficients as they were originally published, and for the HS model, a site-specific coefficient (HS-SS) was derived and evaluated for each site. Regression of predicted vs. observed R_s values using the empirical equation for Keiser gave r² values (0.77) similar to the best of the mechanistic models. The HS-SS model had the lowest root mean square error of 3.50 MJ m^–2 d^–1, followed by the TR (3.56), the HS (3.86), and the WS (4.33) models. Predicted vs. observed values gave r² values ranging from 0.72 (TR model) to 0.56 (WS model). There was a slight superiority of the TR model over the HS-SS and HS models. Similar fits (r² > 0.87) and errors were found among the TR, HS-SS, and HS models when R_s values were averaged over a 7-d period, and it was concluded that these three models provided accurate and precise R_s estimations for our sites without further model modification.

Abbreviations: DOY, day of year • HS, Hargreaves–Samani model • HS-SS, site-specific Hargreaves–Samani model • k_Rs, Hargreaves–Samani coefficient • MAE, mean absolute error • R_a, total extraterrestrial radiation • R_s, total solar radiation at the earth's surface • RMSE, root mean square error • T_max, maximum air temperature • T_min, minimum air temperature • T_tmax, maximum transmittance that occurs for a given site under clear-sky conditions • TR, Thornton–Running model • WS, Weiss et al. model

	INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
SOLAR RADIATION at the earth's surface (R_s, MJ m^–2 d^–1) is an important variable used in agricultural sciences, particularly for crop modeling and estimating crop evapotranspiration, but it is also important for hydrology, meteorology, and soil physics. When compiling historical data sets for analysis, the addition of R_s data is frequently necessary, but historic weather data typically do not include R_s. For example, the ratio of weather stations collecting R_s data relative to those collecting temperature data in the USA is approximately 1:100, and worldwide the estimate is approximately 1:500 (NCDC, 1995, as cited by Thornton and Running, 1999).

The utility of weather data sets is greatly expanded by including R_s. Radiation estimates for historical weather can be obtained by predicting R_s using either a site-specific radiation model or a mechanistic prediction model. A site-specific model relies on empirical relationships of R_s with commonly recorded weather station variables. Although a site-specific equation requires a data set with actual R_s data for determining appropriate coefficients, this approach is frequently simpler to compute and may be more accurate than complicated mechanistic models. These simple, site-specific equations, therefore, may be very useful to those interested in sites near to where these models are developed.

The strength of a mechanistic model is that once coefficients are developed, they should be applicable to the geographical region to which they were originally developed. Ideally, a mechanistic model would be accurate, precise, and require no further calibration to determine coefficients; the inputs for the model would be simple and easily obtained; and the model would be applicable globally.

One common approach to predict R_s that is used by several mechanistic models is to first determine extraterrestrial radiation (R_a, MJ m^–2 d^–1), which may be calculated using standard geometric methods (Sellers, 1965) for any given day of year (DOY) based upon latitude and the solar constant (Allen et al., 1998). By modifying R_a with an estimate of atmospheric transmissivity (T_t), R_s is predicted, as shown in Eq. [1]:

[1]

Different models use different approaches for determining T_t that are based on empirical coefficients, temperature differentials, cloudiness, or daily sunshine hours (e.g., Annandale et al., 2002; Richardson, 1985; Supit and van Kappel, 1998).

Hargreaves and Samani (1982) developed a simple mechanistic model for estimating R_s, which they used for evapotranspiration calculations. They proposed that T_t for a given day was proportional to the square root of the differences between maximum temperature (T_max) and minimum temperature (T_min). Annandale et al. (2002) modified the model slightly to include a correction for altitude (Alt, m) such that:

[2]

In this equation, k_Rs is an empirical coefficient with values set at 0.16 for inland sites and 0.19 for coastal sites.

An alternate method for estimating T_t was developed by Bristow and Campbell (1984), who divided T_t into two components:

[3]

In Eq. [3], T_tmax is the maximum transmittance that occurs under clear-sky conditions for a given location, and T_tf is the fraction of T_tmax that is realized on a specific day. The expression for T_tf is an exponential relationship requiring site-specific calibration for the coefficients b and c:

[4]

The T_min value in the Bristow–Campbell model (Bristow and Campbell, 1984) was a 2-d average of T_min from the day in question and of the next day's. This analysis model was derived from data obtained from three sites in the northwestern USA. Although their model provided relatively accurate and unbiased predictions, the authors recognized that further work was required to extend this approach over a wide geographic area.

Weiss et al. (2001) evaluated 14 variations of the Bristow–Campbell model (Bristow and Campbell, 1984) using 10 yr of data from Mead, NE. Their evaluations included modifications to the Bristow–Campbell model (Bristow and Campbell, 1984) that were suggested by Ndlovu (1994). Although one of the models evaluated by Weiss et al. (2001) was used successfully to predict radiation at several independent sites after including a bias correction factor, the applicability of this approach to other regions is doubtful. The empirical and site-specific determinations of values for T_tmax and for coefficients in Eq. [4] are not trivial and require large amounts of actual data with solar radiation (McVicar and Jupp, 1999).

Thornton and Running (1999) and Thornton et al. (2000) refined the Bristow–Campbell model (Bristow and Campbell, 1984) by evaluating R_s over a wide geographic area and eliminating the need for site-specific calibration of coefficients. Their model calculates a unique T_tmax for each day based upon R_a, atmospheric pressure, vapor pressure, and zenith angle. The additional requirement for vapor pressure as an input is essentially eliminated by algorithms that estimate vapor pressure from inputs of T_max and T_min. Although this model is considerably more complex than other modifications of the Bristow–Campbell model (Bristow and Campbell, 1984), their approach has greatly extended the general approach originally formulated by the Bristow–Campbell model (Bristow and Campbell, 1984).

Our aim was to evaluate the performance of three currently used mechanistic models for predicting R_s (HS, TR, and WS models) and two site-specific methods. The specific objectives of our paper were to (i) evaluate methods to predict R_s by use of a minimal set of weather variables; (ii) determine the prediction accuracy from site-specific and non-site-specific equations for 13 sites varying widely in climate, latitude, and longitude; and (iii) assess the ability of mechanistic models to predict R_s without recalibrating model coefficients.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
Keiser Site-Specific Model
A site-specific model to predict radiation, R_s, was made for Keiser, AR, using a multiple-regression equation. The data set used to generate the model contained daily R_s, T_max, T_min, and precipitation for the years of 1997 and 1998. The total number of daily observations, N, was 464. A data set containing the same variables from 1991, 1993, and 1999 to 2002 (N = 1708) was reserved for testing the model.

The model was constructed using a quadratic-response surface in the General Linear Model Procedure of SAS (v. 8.2, SAS Inst., Cary, NC). The input variables were DOY, precipitation (mm), T_max, and T_min. Terms were made for each variable as linear, quadratic, and the cross-products between all variables. Nonsignificant terms were eliminated based on type III sums of squares where P > 0.10. Regression was performed twice until all terms remaining were significant at P < 0.10. The final model, in units of in MJ m^–2 d^–1, had the form

[5]

where the coefficient ß₀ was the intercept and ß₁ to ß₁₂ were coefficients for variables X₁ to X₁₂, respectively.

Extraterrestrial Radiation
Daily values for R_a were calculated using standard geometric procedures. The calculation of R_a that we used was adapted from a larger model that calculates hourly R_a in addition to direct and diffuse radiation (S.K. Carey, unpublished results, 2003). Although these and similar equations are published elsewhere (e.g., Allen et al., 1998; Annandale et al., 2002), their utility is hampered by their unfamiliarity to those outside climatology and meteorology. We created a simple program in GWBASIC and Microsoft Excel that calculates R_a for any location for a normal year (DOY 1 to 365), requiring only that the user provides longitude and latitude. These programs are freely available upon request.

Modified Hargreaves–Samani Model
We used a modified form of the Hargreaves–Samani equation (Hargreaves and Samani, 1982) (referred to as the HS model in this paper) that includes a correction for altitude (Annandale et al., 2002). Nominal values for k_Rs of 0.16 and 0.19 were suggested for inland and coastal regions, respectively (Hargreaves and Samani, 1982; Allen et al., 1998; Annandale et al., 2002), and these values were used in our evaluations. Hargreaves and Samani (1982) defined an inland region as one with weather patterns dominated by a large landmass whereas a coastal region was one with weather patterns dominated by close proximity to a large body of water. We evaluated both inland and coastal forms of the HS model for the 13 locations in our data set, regardless of a site's proximity to a large body of water.

Site-Specific Coefficients for Modified Hargreaves–Samani Model
Instead of using the nominal k_Rs values of 0.16 or 0.19 (Hargreaves and Samani, 1982; Allen et al., 1998), we derived k_Rs for individual sites. This method (subsequently referred to as the HS-SS model) requires a data set containing actual measurements of R_s. For each site, data were divided into two sets for model creation and model testing. Equations [1] and [2] were combined and rearranged to solve for k_Rs such that:

[6]

To determine k_Rs, the numerator of Eq. [6] was regressed against the denominator with the intercept being forced through the origin. The slope of this simple linear regression is equivalent to a site-specific k_Rs term.

Weiss et al. Model
We chose in our evaluations the most simple of the 14 modified Bristow–Campbell algorithms presented by Weiss et al. (2001) (Algorithm 5 from their Table 2; referred to as the WS model in this paper). This model is similar to a yearly model presented by Goodin et al. (1999). The specific value of T_tmax was 0.75, and the b and c coefficients (Eq. [4]) were 0.226 and 2, respectively. A 2-d moving average for T_max and T_min was used to smooth the temperature data, and R_s was normalized by R_a such that:

[7]

Thornton–Running Model
The calculation of R_s using the TR model is complex, and each day for which R_s is predicted requires hourly calculations and summations to determine that day's T_tmax (Thornton and Running, 1999; Thornton et al., 2000). We used a software package developed by P.E. Thornton (Mtclim, v. 4.3) that automates these calculations. The Mtclim software is available online (http://www.ntsg.umt.edu/bioclimatology/mtclim/; verified 22 Dec. 2003). The software requires specific information of latitude, average annual rainfall, and altitude for each site and daily data for T_max, T_min, and precipitation.

Model Testing and Assessment
The observed R_s and predicted values of R_s for each site were first evaluated as daily observations, and then a second set of variables were calculated by use of a 7-d centered-moving average (PROC EXPAND, SAS version 8.2, SAS Inst., Cary, NC) on observed and predicted values of R_s. For the Keiser site-specific model and the HS-SS model, the test data sets were not part of the data sets used in model creation or coefficient determination. The Keiser site-specific model was tested only on Keiser data but for different data than those used to create the empirical model.

The mechanistic models were tested on 13 sites covering a range of 23° latitude and 42° longitude (Table 1). Data sets for all sites had no missing data and were seamless, except for Davis, Keiser, Lincoln, and Stoneville. Davis had data missing for 59 d in 1998 and 106 d in 2000. Keiser was the most fragmented data set, having data missing for 21 d in 1991, 181 d in 1997, and 29 d in 1998. Lincoln had data missing for 8 d in 1998 and for 99 d in 2001. Stoneville had missing data for 6 d in 1964, 10 d in 1965, 4 d in 1967, and 5 d in 1969 and had no observations for years 1994 and 1995.

	RESULTS AND DISCUSSION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
Coefficients for the Keiser site-specific model are listed in Table 2. The multiple-regression model had a R² of 0.81 and a RMSE of 3.42. Radiation predicted from this empirical equation against observed R_s values in an independent data set (N = 1709) from the same location resulted in linear relationships for both daily observations (r² = 0.77) and the 7-d moving average (r² = 0.91). The model using daily observations had a RMSE of 3.34 MJ m^–2 d^–1 and a bias error of 0.59 MJ m^–2 d^–1; the 7-d moving average lowered the RMSE to 1.82 MJ m^–2 d^–1, but the bias remained unchanged. Mean absolute errors were 2.84 MJ m^–2 d^–1 and 1.51 MJ m^–2 d^–1 for predictions using daily observations and 7-d moving averages, respectively.

Modified Hargreaves–Samani Model
The nominal coefficients for k_Rs that were originally proposed by Hargreaves and Samani (1982) for inland (0.16) and coastal sites (0.19) were both initially evaluated.

The HS coastal model had higher values of bias and RMSE for each site (data not shown), including the sites on the Florida peninsula (Alachua, Hastings, Lake Alfred, and Ocklawaha), which were 50 to 100 km from the ocean. The coastal form of the HS model overpredicted R_s by about 2 MJ m^–2 d^–1 more than the HS inland form (data not shown), and we did not include the HS coastal model in further evaluations and comparisons. Subsequent discussion of the HS model pertains only to the inland form.

For all sites combined, R_s prediction from daily values using the HS inland equation resulted in slightly lower RMSE and bias compared with the TR and WS models (Table 3). Combined over sites, the HS model overpredicted R_s by 0.56 MJ m^–2 d^–1 with a RMSE of 3.86 MJ m^–2 d^–1. For individual sites, the HS model tended to overpredict R_s daily values. Bias for model performance ranged from 2.86 MJ m^–2 d^–1 for Alachua to underprediction at two sites, Saskatoon (–1.86 MJ m^–2 d^–1) and Stoneville (–0.15 MJ m^–2 d^–1), and RMSE ranged from 3.43 (Hastings, Tucson) to 4.63 (Lincoln) MJ m^–2 d^–1. The range of r² values was from 0.87 (Davis) to 0.52 (Stoneville).

Recalibrating k_Rs for a specific site improved prediction performance over the HS model. Regression of predicted vs. observed values of R_s for the HS-SS model had equivalent or slightly improved r² values compared with the fits obtained using the HS model (Table 3). For each site, the RMSE of the HS-SS model was lowered by approximately 0.35 MJ m^–2 d^–1 relative to the HS model, and the overprediction encountered with the HS model was reduced by 1 MJ m^–2 d^–1 or less. The RMSE ranged from 2.49 to 3.80 MJ m^–2 d^–1. Bias values less than 0.5 in magnitude were observed for Alachua, Hastings, Keiser, Lexington, Lincoln, Odessa, and Stoneville. Bias values of magnitude 0.5 to 1.5 were found for Corvallis, Davis, Lake Alfred, Ocklawaha, Saskatoon, and Tucson. The greatest underprediction was for Saskatoon (bias of –1.28 MJ m^–2 d^–1; RMSE of 3.80 MJ m^–2 d^–1). The greatest overprediction occurred for Tucson (bias of 1.01 MJ m^–2 d^–1; RMSE of 2.79 MJ m^–2 d^–1) and Davis (bias of 1.08 MJ m^–2 d^–1; RMSE of 2.49 MJ m^–2 d^–1).

Combining R_s data from all sites also indicated that the HS-SS model was superior to the HS model (Table 3). The RMSE for the HS-SS model (3.50 MJ m^–2 d^–1) was lower than the RMSE for the HS model (3.86 MJ m^–2 d^–1). Regression of predicted vs. observed values increased the adjusted r² from 0.65 to 0.68 for the HS and HS-SS models, and bias was decreased from 0.56 to –0.02 MJ m^–2 d^–1, respectively.

Therefore, if researchers have several years of R_s data for a site, a site-specific k_Rs may increase predictive accuracy compared with the nominal k_Rs value of 0.16. For three sites (Lexington, Stoneville, and Tucson), however, our data indicated that the nominal k_Rs value of 0.16 performed nearly as well as the site-specific k_Rs. Only minor reductions in RMSE (about 0.10 MJ m^–2 d^–1) for the HS-SS model compared with the HS model were seen at these three sites, and bias actually increased at Tucson. Generally, at locations where no R_s data are available for site-specific estimation of k_Rs, the nominal k_Rs value of 0.16 appears to be well suited.

Weiss et al. Model
For each of the 13 sites, the WS model had RMSE values for predicted vs. observed R_s that were considerably higher than for the other models tested, ranging from 2.84 (Tucson) to 4.13 (Stoneville) MJ m^–2 d^–1 (Table 3). Bias ranged from –3.36 (Stoneville) to 1.11 (Alachua) MJ m^–2 d^–1. Adjusted r² values for the WS model were also generally lower than for other models and ranged from 0.28 (Stoneville) to 0.83 (Odessa). Because the coefficients that were used to derive the WS model were originally from data from Mead, NE (Weiss et al., 2001), it is not surprising that the WS model poorly predicted R_s in other regions. Given the close proximity of Lincoln, NE, to Mead, NE (66 km), it is somewhat surprising that WS model performance at Lincoln, NE, was also poor relative to the performance of the other models. Nevertheless, the inability of the WS model to be extended to other areas without coefficient calibration is clearly evident.

Thornton–Running Model
The more complicated approach of the Bristow–Campbell model used in the TR model was successful in eliminating the need for site-specific calibration of coefficients, as required by other forms of the Bristow–Campbell model. Regression of predicted vs. observed R_s for each site individually and for all sites combined resulted in the highest r² values for the TR model compared with the other models that we evaluated (Table 3).

For most sites, there was a negative bias for the TR model resulting in predicted R_s values lower than the actual seasonal mean and lower than the HS-SS model (Table 3). Over all sites combined, RMSE for the TR model was 3.56 MJ m^–2 d^–1, bias was –1.60 MJ m^–2 d^–1, and MAE was 3.45 MJ m^–2 d^–1. For individual sites, RMSE ranged from 2.52 (Ocklawaha) to 3.92 (Stoneville) MJ m^–2 d^–1, and bias ranged from 0.09 (Lexington) to –3.44 (Tucson) MJ m^–2 d^–1. Thornton and Running (1999) found the weighted mean of the MAE for 40 sites was 2.39 MJ m^–2 d^–1 with a bias of 0.51 MJ m^–2 d^–1.

Model Performance with Seven-Day Moving Averages
Compared with regressions of predicted R_s vs. observed R_s on a daily basis, performance of all models was improved by regressing a 7-d moving average of predicted R_s against the 7-d moving average of observed R_s (Table 4). Reducing the amplitude of day-to-day variation with a 7-d moving average was an effective way of lowering RMSE while bias remained unchanged. The moving average lowered RMSE by 1 to 2 MJ m^–2 d^–1 compared with model performances using daily averages (Table 3). When using the 7-d moving averages, adjusted r² values for the predicted vs. observed relationship were >0.84 for the HS, HS-SS, and TR models and >0.53 for the WS model. For many purposes, such as crop modeling and irrigation scheduling, the ability to predict the average R_s over a weekly period may be more important than the absolute R_s value on a given day. From this standpoint, the HS and TR models gave accurate results over a 7-d time frame and required no site-specific calibration.

	CONCLUSIONS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

For users interested primarily in predicting R_s at a specific site, an empirical equation can effectively predict R_s, as demonstrated by the empirical equation developed for Keiser (Table 2). The precision of this equation was equivalent or better than the other models evaluated for this site (Table 3). This empirical site-specific equation had a simple input data set requiring only four variables (DOY, precipitation, T_min, and T_max), which were used to generate a proficient R_s prediction model from less than 2 yr of daily observations. Predicting R_s for only one or two sites may be easier with a simple empirical model, such as the one illustrated for Keiser, than a complex mechanistic model requiring coefficient determination and R_a calculation.

For some sites, the HS-SS model using a site-specific coefficient for k_Rs may increase the accuracy and precision of R_s predictions, but the nominal k_Rs value of 0.16 appears to be acceptable in the absence of additional R_s data for k_Rs determination. The HS and TR models were both robust, over the sites we evaluated, and accurately predicted R_s for locations varying by up to 23° latitude and 42° longitude. The TR model for most locations was slightly superior to the HS model but is considerably more complicated to use than is the HS model.

	ACKNOWLEDGMENTS

 
The authors gratefully acknowledge the following people in helping us identify solar radiation data sets that were used in these evaluations: B. Bugbee, R.F. Denison, D.B. Egli, J.D. Ray, E.A. Ripley, P.J. Sexton, T.R. Sinclair, J.E. Specht, and E.D. Vories. Appreciation is also extended to P.E. Thornton for helpful suggestions on using Mtclim software.

	REFERENCES

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 

• Allen, R.G. 1997. Self-calibrating method for estimating solar radiation from air temperature. ASCE J. Hydrol. Eng. 2:56–67.
• Allen, R.G., L.S. Pereira, D. Raes, and M. Smith. 1998. Crop evapotranspiration: Guidelines for computing crop water requirements. FAO Irrig. and Drainage Paper 56. FAO, Rome.
• Annandale, J.G., N.Z. Jovanic, N. Benade, and R.G. Allen. 2002. Software for missing data error analysis of Penman–Monteith reference evapotranspiration. Irrig. Sci. 21:57–67.
• Bristow, K.L., and G.S. Campbell. 1984. On the relationship between incoming solar radiation and daily maximum and minimum temperature. Agric. For. Meteorol. 31:159–166.
• Goodin, D.G., J.M.S. Hutchinson, R.L. Vanderlip, and M.C. Knapp. 1999. Estimating solar irradiance for crop modeling using daily air temperature data. Agron. J. 91:845–851.[Abstract/Free Full Text]
• Hargreaves, G.H., and Z.A. Samani. 1982. Estimating potential evapotranspiration. J. Irrig. Drain. Eng. 108:225–230.
• McVicar, T.R., and D.L.B. Jupp. 1999. Estimating one-time-of-day meteorological data as inputs to thermal remote sensing based energy balance models. Agric. For. Meteorol. 96:219–238.
• Ndlovu, L.S. 1994. Generating data for crop simulation model. Ph.D. diss. Biol. Syst. Eng. Dep., Washington State Univ., Pullman.
• Richardson, C.W. 1985. Weather simulation for crop management models. Trans. ASAE 28:462–470.
• Sellers, W.K. 1965. Physical climatology. Univ. of Chicago Press, Chicago.
• Supit, I., and R.R. van Kappel. 1998. A simple method to estimate global radiation. Sol. Energy 63:147–160.
• Thornton, P.E., H. Hasenauer, and M.J. White. 2000. Simultaneous estimation of daily solar radiation and humidity from observed temperature and precipitation: An application over complex terrain in Austria. Agric. For. Meteorol. 104:255–271.
• Thornton, P.E., and S.W. Running. 1999. An improved algorithm for estimating incident daily solar radiation from measurements of temperature, humidity, and precipitation. Agric. For. Meteorol. 93:211–228.
• Weiss, A., C.J. Hays, Q. Hu, and W.E. Easterling. 2001. Incorporating bias error in calculating solar irradiance: Implications for crop yield simulations. Agron. J. 93:1321–1326.[Abstract/Free Full Text]

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