Constructing a non-linear relationship between the incoming solar radiation and bright sunshine duration

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1 INTERNATIONAL JOURNAL OF CLIMATOLOGY Int. J. Climatol. 3: (2) Published online 28 October 29 in Wiley Online Library (wileyonlinelibrary.com) DOI:.2/joc.232 Constructing a non-linear relationship between the incoming solar radiation and bright sunshine duration Khil-Ha Lee* Research Scientist, KORDI, 27 Sa2-dong, Sangrok-gu, Ahansan-si, Kyunggi-do, , S. Korea ABSTRACT: This paper reports the application of a non-linear relationship between the incoming shortwave solar radiation and bright sunshine duration. The newly suggested equation is a modified form of the existing Angstrom equation. Measurements of solar radiation and sunshine radiation from 997 to 26 at 2 meteorological stations were used to calibrate and validate the suggested equation. The model parameters required to specify the nature of the relationship between solar radiation and sunshine duration were determined by automatically minimizing the difference between the modelled and measured solar radiation. At the 2 meteorological stations, the absolute error (AE) is in the range of MJm 2 day for the original Angstrom equation, while it is in the range of MJm 2 day for the modified equation. The root mean square error (RMSE) is also improved by 7 8% for the modified method. The results show that the newly suggested equation generally provides better performance than the existing Angstrom equation. Copyright 29 Royal Meteorological Society KEY WORDS Angstrom; modified Angstrom; solar radiation; sunshine duration Received 3 May 29; Revised 25 August 29; Accepted September 29. Introduction Solar radiation (R s ) provides information on how much of the sun s energy strikes the surface at a place on earth during a certain time period. The measured data are obviously the best form of R s information. There are very few meteorological stations that measure R s, especially in developing countries, and this measurement is costly and time consuming. Efforts have been made to estimate R s in various ways, and a common practice has been undertaken to assess the R s using appropriate correlations for those places where no measured values are available. The Angstrom equation (Angstrom, 924; Prescott, 94) has been a dominant tool in use for a long time as a basic approach to estimate the R s. The Angstrom equation has been very conveniently used in a large number of locations (Gopinathan, 988; Annear and Wells, 27), and many scientists have presented slightly different model parameters for different locations (Doorenbos and Pruitt, 977). Allen (997) recommended the values for Angstrom parameters when estimating R s, when there is no available data on sunshine duration and direct measurements of R s. R s is defined as the radiation of the Sun disk that is absorbed and scattered in the atmosphere column by gases, aerosols and clouds, passing through a unit plane area in unit time. Hence, atmospheric constituents, such * Correspondence to: Khil-Ha Lee, Research Scientist, KORDI, 27 Sa2-dong, Sangrok-gu, Ahansan-si, Kyunggi-do, , S. Korea. khil ha@yahoo.com as molecules, aerosols and clouds, can affect solar radiation; and these atmospheric constituents can be included in building the relationship between R s and bright sunshine hours (n). Consequently, other attempts have been made to modify the Angstrom equation, including more meteorological parameters, such as surface albedo, latitude, ambient temperature, total precipitation, humidity, elevation, the amount of cloud cover, etc. (Hay, 979; Hargreaves et al., 985; Supit and Van Kappel, 988; Dorvlo and Ampratwum, 2). The main purpose of this study is to suggest a modified form of the existing Angstrom equation to improve both its accuracy and fitness, introducing non-linearity between R s and n. The newly suggested equation was facilitated at the 2 meteorological stations on the Korea Peninsula and tested against the measured data. The new equation shows better performance compared to the existing Angstrom equation. 2. A modified Angstrom equation Angstrom (924) and Prescott (94) suggest a correlation in a linear form as follows: R ( s n ). = () where R s is the total incoming shortwave solar radiation (MJm 2 day ), R a is the extraterrestrial radiation (MJm 2 day ), a and b are the model parameters, n is the bright sunshine duration (hour) and N is the total day length (hour). The following equations were used to Copyright 29 Royal Meteorological Society

2 A RELATIONSHIP BETWEEN RADIATION AND SUNSHINE DURATION 885 compute the extraterrestrial radiation R a and total day length N: R a = 5.392d r (ω s sin ϕ sin δ + cos φ cos δ sin ω s ) (2) where R a = extraterrestrial radiation (mm/day), d r = relative distance earth sun, ω s = sunset hour angle (radians), ϕ = latitude of site (+ for Northern Hemisphere, for Southern Hemisphere) (radians) and δ = solar declination (radians). N = 24 π ω s (3) d r = +.33 cos(2π J/365) (4) ω s = arccos( tan φ tan δ) (5) δ =.493 sin(2πj/365.45) (6) where J = Julian day As a matter of fact, Equation () is the most widely used relationship to estimate the average daily global radiation on a horizontal surface. In Equation (), n/n denotes the cloudiness fraction, while R s /R a is affected by various atmospheric conditions, as stated earlier. The input of extraterrestrial shortwave radiation R a is absorbed by atmospheric gases, particularly water vapour and ozone, and is scattered by air molecules and aerosol particles in clear sky conditions, and additionally by clouds when they are present (Maidment, 993). The global solar radiation reaching the earth s surface, R s, consists of two components: direct and diffuse radiation. Direct radiation is the part of solar radiation that directly travels through space and the atmosphere to the surface. There is extinction of the direct radiation in the atmosphere by gases and aerosols. Diffuse radiation is that part of solar radiation which gets scattered by atmospheric constituents, such as molecules, aerosols and clouds. Direct radiation causes shadows, and diffuse radiation is responsible for skylight (Hunt, 979). In this study, Equation () is generalized in an attempt to improve accuracy and performance as follows and is called the modified Angstrom equation : R ( s n ) c = a + b (7) An additional parameter c, which describes the influence of the impeding factor and non-linearity between R s /R a and n/n, is introduced in Equation (7). Consequently the Angstrom Equation () is a special form of Equation (7), giving c =.. 3. Materials 3.. The study region and data used for the study Study sites were selected on the basis of data completeness and reliability (Figure ). The days on which observations were not available were averaged and filled with neighbouring values. The measured weather data were checked for integrity, quality and reasonableness. Data quality and integrity check were made for all locations, following the recommendations of previous studies (Allen, 996; Temegsen et al., 999; Irmak et al., 23;). To check the integrity of R s, clear sky envelopes (Allen, 995, 997) were calculated. There was some mismatch between the measured and the clear sky radiation envelope because some points never reached a clear sky. These points needed further scrutiny; adjustment multipliers were applied to force the upper surface of the measured R s to reach computed clear-sky radiation envelops, as shown in Figure 2. This adjustment was based on the assumption that there were some clear-sky days at each location and that a single factor could produce suitable calibration correction for the measurement (Allen, 996). The Korean Peninsula has a moderate climate, characterized by distinct wet and dry seasons. The dry season coincides with the north-west wind, which is predominant from November to March. The wet season results from the south-east wind, which brings moisture-laden air from the Pacific Ocean; this season lasts from May to October, accounting for 7% of the peninsula s annual precipitation. July August is usually the wettest season. All sites exhibit typical daily and seasonal variations in moderate temperature trends, ranging between a seasonal maximum from April to October to a seasonal minimum from November to March. Coastal sites show less variability in temperature with a diurnal temperature range of approximately 7 C. The radiation data used for this study corresponded to the period of (2 months) and consisted of 3652 carefully screened daily values. The Korea Meteorological Administration (KMA) provided the meteorological data. All stations were also close to the reference condition (Allen, 996). A summary of site information, including daily manual observations, such as mean temperature, relative humidity, wind speed and R s at the 2 stations (2 inland and 9 coastal, including 3 islands) used for this investigation is given in Table I. Solar radiation is measured with pyranometers of the CMP2, and sunshine duration with the MS-93. The expected daily accuracy of the pyranometers is ±2%, and the integration error of the sunshine duration metres is less than min/day. The data used for the study are daily averages on a horizontal surface. Table I also presents a summary of the weather stations, including their geographical coordinates, elevation and measurement heights. Stations 3 and 6 are located in a mountainous area and are influenced by the orographic effect. The logarithmic wind profile equation (Brustaert, 99) was used to adjust the measured wind speed at each height to a reference height of 2 m Determining the parameters for the Angstrom equation One of the most significant issues in the modelling approach is how to determine the model parameters; accuracy strongly depends on the model parameters. Copyright 29 Royal Meteorological Society Int. J. Climatol. 3: (2)

3 886 K.-H. LEE Figure. Study region and meteorological stations. The present study uses a single-criterion optimization technique to minimize the difference between the time series of measured and modelled solar radiation and explore the feasibility of estimating the values of two or three parameters in the Angstrom equation. In general, a numerical model might have n parameters to be calibrated, using m observations. The distance between the m model-simulated responses and the m observations is defined by an objective function (O), such as the RMSE between the modelled responses and observations. The goal of a model calibration is then to find the preferred value for the n parameters within the feasible set of parameters that minimize O. The Shuffled Complex Evolution algorithm (SCE, Duan et al., 993, 994) is a general-purpose global optimization method designed to handle many of the response problems encountered in the calibration of nonlinear simulation models. The SCE randomly samples the feasible parameter space, which is prescribed within reasonable values ( a 3, b 3, c 3 for this study) to select a population of points; then the population is partitioned into several complexes, each of which Copyright 29 Royal Meteorological Society Int. J. Climatol. 3: (2)

4 A RELATIONSHIP BETWEEN RADIATION AND SUNSHINE DURATION 887 (a) 4 Rs (MJ m 2 d ) 3 2 (b) 4 Rs (MJ m 2 d ) 3 2 Jan Feb Mar Apr May Jun Jul Month Aug Sep Oct Nov Dec Figure 2. An example of the adjustment multipliers (=.5) to force the upper surface of measured R s to reach computed clear-sky radiation envelopes at Hugsando station (year 999): (a) before adjustment, (b) after adjustment. evolves independently in a manner based on the downhill simplex algorithm (Nelder and Mead, 965). The reader can refer to Duan et al. (993, 994) for more details Evaluating the efficiency of fit To quantify the efficiency of fit, the Nash Sutcliffe coefficient of efficiency (NSC) (97) and RMSE were used as follows: NSC = m (R s.est R s.obs ) 2 i= (8) m (R s.obs R s.obs ) 2 i= m (R s.est R s.obs ) 2 i= RMSE = m (9) where R s.est and R s.obs are simulated and observed values of the incoming shortwave solar radiation respectively and R s.obs is the mean observed, incoming shortwave solar radiation. m is the number of data points. NSC has a maximum perfect score of. and no minimum, with values greater than indicating satisfactorily results. Physically, NSC is minus the ratio of the mean square error to the variance of the observed data (Nash and Sutcliffe, 97). The normal distribution of the error structure is assumed to determine the confidence intervals of the regression model. 4. Outcomes 4.. Calibration of the solar radiation for each station Calibration of both methods (Equations () and (7)) is performed to find the preferred parameter set at the individual meteorological station. The simple linear coefficients a and b of the observed daily data for the original Angstrom equation are shown in Table II. The simple linear coefficients a and b are in the range of.5.2 and.5.6 respectively. The corresponding AE(RMSE) is in the range of.26.58( )MJm 2 day, while the correlation coefficient (r) is in the range of However, the modified Angstrom equation shows a better fit, as shown in Table III. The coefficients a, b, and c of Equation (7) are in the range of.8.46,.5.62 and.68.74, respectively. The corresponding AE-(RMSE) is in the range of.89.54( )MJm 2 day, while the correlation coefficient is in the range of Figure 3 visually presents a relative comparison of the basic statistics for both methods (RMSE in Figure 3(a) and AE in Figure 3(b)) Validation of the modified Angstrom equation The data are divided into two groups: one for calibration (8 stations and called the first group) and the other for validation (three stations and called the second group). The first group is applied to both methods to find the preferred parameter, using the single-criterion optimization technique mentioned in the earlier section. Copyright 29 Royal Meteorological Society Int. J. Climatol. 3: (2)

5 888 K.-H. LEE Table I. Summary of weather stations used for the study. Station index Station name Ele. Lat. Lon. Ht Hw Mean Std Rs Sunhr WS RH T Rs Sunhr WS RH T Inland Andong Cheongju Chupungnyeong Deagu Daejeon Daegwallyeong Gwangju Jeonju Jinju Seoul Suwon Wonju Coast 3 Busan Gangneung Hugsando Incheon Jeju Jejugosan Mokpo Pohang Seosan Ele., elevation (m); Lat., latitude (degree); Lon., longitude(degree); Ht, height of thermometer above the ground (m); Hw, height of anemometer above the ground (m); Rs, Incoming solar radiation (MJm 2 day ); Sunhr, bright sunshine duration (hours); WS, wind speed(m/sec); RH, relative humidity(%); Tmean, daily mean temperature( C). The mean and standard deviation values for all variables are for the period of the study (997 26). Copyright 29 Royal Meteorological Society Int. J. Climatol. 3: (2)

6 A RELATIONSHIP BETWEEN RADIATION AND SUNSHINE DURATION 889 Table II. Simple linear coefficients a and b of the observed daily data and the corresponding basic statistics for the original Angstrom equation (MJm 2 day for RMSE and AE). Station index Station name a b c RMSE AE r Inland Andong Cheongju Chupungnyeong Deagu Daejeon Daegwallyeong Gwangju Jeonju Jinju Seoul Suwon Wonju Coast 3 Busan Gangneung Hugsando Incheon Jeju Jejugosan Mokpo Pohang Seosan Table III. Model parameter a, b and c fitted by the observed daily data and the corresponding basic statistics for the modified Angstrom equation (MJm 2 day for RMSE and AE). Station index Station name a b c RMSE AE r Inland Andong Cheongju Chupungnyeong Deagu Daejeon Daegwallyeong Gwangju Jeonju Jinju Seoul Suwon Wonju Coast 3 Busan Gangneung Hugsando Incheon Jeju Jejugosan Mokpo Pohang Seosan The corresponding results are as follows: R ( s n ). = () R ( s n ).649 = () Then, the second group (Jeonju, Jinju, and Mokpo stations) is used to compare the accuracy and performance for each method. Comparisons between measured and estimated daily solar radiation by Equations (8) and (9) at three stations are presented in Figure 4 and the corresponding basic statistics are shown in Table IV. Copyright 29 Royal Meteorological Society Int. J. Climatol. 3: (2)

7 89 K.-H. LEE AE (MJ m 2 day ) (a) Inland Coast RMSE (MJ m 2 day ) (b) Original Modified Station No. Figure 3. Relative comparison of basic statistics for both methods. Rs / Ra (a) Rs / Ra = (n/n) Rs / Ra (b) Rs / Ra = (n/n) Rs /Ra (c) (d) (e) Rs / Ra = (n/n) Rs / Ra = (n/n) (f) Rs / Ra = (n/n) n/n Rs / Ra = (n/n) n/n Figure 4. Solar radiation equations fitted to both the functions against the observed data. Red line indicates the nonsimultaneous 95% CI associated with the observed solar radiation. Figure 4 shows the Rs fitted to both functions against the observed data. The preferred parameter set for a, b Copyright 29 Royal Meteorological Society and c are.77,.552 and. for the original function and.28,.556 and.649 for the modified function. Int. J. Climatol. 3: (2)

8 A RELATIONSHIP BETWEEN RADIATION AND SUNSHINE DURATION 89 Table IV. Basic statistics of the estimated daily solar radiation using the preferred parameter set at three stations. Index name Angstrom Modified RMSE NSC AE r RMSE NSC AE r 8 Jeonju Jinju Mokpo MJm 2 d for RMSE and AE. (a) Angstrom Modified (b) Probability of exceedance (c) Confidence level Figure 5. A statistical evaluation of the estimated R s against the observed R s. The figure shows the EP of upper/lower quantiles of the estimated R s for a range of confidence level probability. The blue line indicates the best-fitted R s using the preferred parameter set and the red line indicates the nonsimultaneous 95% confidence intervals (CI) associated with the observed R s (95%CI =±. in Figure 4(a); 95%CI =±. in Figure 4(b) and 95%CI =±. in Figure 4(c)). For the Jeonju station, the NSC and AE(RMSE) is.88 and.49(3.34)mjm 2 day, respectively, for the modified function and.85 and.7(4.24)mjm 2 day, respectively, for the original function. The NSC of the Jinju and Mokpo stations is.92 and.9, respectively, for the modified equation and.9 and.88, respectively, for the original function. The AE(RMSE) of the Jinju and Mokpo stations is.38 and.68(2.67 and 3.29)MJm 2 day, respectively, for the modified function and.58 and.6(3.6 and 3.98)MJm 2 day, respectively, for the original function. A statistical evaluation of the estimated R s against the observed R s is presented in Figure 5. Figure 5 shows the exceedance probability (EP) of upper/lower quantiles Copyright 29 Royal Meteorological Society Int. J. Climatol. 3: (2)

9 892 K.-H. LEE of the estimated R s for a range of confidence level probability. Three stations showed a similar level of statistical accuracies for the suggested method. All three validation sites associated with high EP values, ranging from near % at a high value of CI (.) to 3% at.6 of CI. It is obvious that the modified Angstrom equation provided better performance than the original Angstrom equation at each station. Eventually, all the 2 meteorological stations were combined to find the best parameter set for the modified Angstrom equation on the Korea Peninsula. The final form is given as follows: R ( s n ).649 = (2) The modified Angstrom model is advantageous in that the solar radiation brightness duration relationship can be derived from easily obtainable climate variables and needs only one more parameter compared to the existing Angstrom equation. 5. Summary and conclusions This study suggests a non-linear relationship between the incoming shortwave solar radiation and bright sunshine duration on the basis of the existing Angstrom equation. Measurements of solar radiation and sunshine radiation during the period from 997 to 26 at 2 meteorological stations were used to calibrate and validate the suggested equation. The model parameter was determined by automatically minimizing the difference between modelled and measured solar radiation. The primary conclusions of the present study are as follows: At the 2 meteorological stations, the AE is in the range of MJm 2 day for the original Angstrom equation, while it is in the range of MJm 2 day for the modified equation. For the three validation stations, the NSC is.88,.92 and.9, respectively, for the suggested equation and.85,.9 and.88 for the original Angstrom equation. The RMSE is less by 7 8% for the suggested revised equation. Assuming normal distribution to determine confidence intervals, the EP is high for the suggested equation fitted to the observed data. In general, the suggested equation provides better accuracy. Acknowledgements The author would like to thank Korea Meteorological Administration (KMA) for their kind cooperation and data provision. References Allen RG Evaluation of Procedure for Estimating Mean Monthly Solar Radiation from Air Temperature. Report. United Nations Food and Agricultural Organizaion (FAO): Rome, Italy. Allen RG Assessing integrity of weather data for reference evapotranspiration estimation. Journal of Irrigation and Drainage Engineering: ASCE 22(2): Allen RG Self-calibrating method for estimating solar radiation from air temperature. Journal of Hydrologic Engineering 2(2): Angstrom A Solar and terrestrial radiation. Quarterly Journal of Royal Meteorological Society 5: Annear RL, Wells SA. 27. A comparison of five models for estimating clear-sky solar radiation. Water Resources Research 43: W45, DOI:.29/26WR555. Brustaert W. 99. Evaporation into the Atmosphere, Theory, History and Application. Kluwer academic publishers: Dordrecht, The Netherland. Doorenbos J, Pruitt WO Guideline for Predicting Crop Water Requirements. FAO Irrigation and Drainage Paper 24. FAO: Rome, Italy, 56. Dorvlo ASS, Ampratwum DB. 2. Harmonic analysis of global irradiation. Renewable Energy 2: Duan QY, Gupta VK, Sorooshian S Shuffled complex evolution approach for effective and efficient global minimization. Journal of Optimization Theory and Applications 76: Duan QY, Sorooshian S, Gupta VK Optimal use of the SCE- UA global optimization method for calibrating watershed models. Journal of Hydrology 58: Gopinathan KK A general formula for computing the coefficients of the correlation connecting global solar radiation to sunshine duration. Solar energy 4(6): Hargreaves GL, Hargreaves GH, Riley P Irrigation water requirement for Senegal River Basin. Journal of Irrigation and Drainage Engineering: ASCE : Hay JE Calculation of monthly mean solar radiation for horizontal and inclined surfaces. Solar Energy 23(4): Hunt VD Energy Dictionary. Van Nostrand Reinhold Company Inc.: New York. Irmak S, Allen RG, Whitty EB. 23. Daily grass and alfalfareference evapotranpiration calculations as part of the ASCE standardization effort. Journal of Irrigation and Drainage Engineering: ASCE 29(5): Maidment DR Handbook of Hydrology. McGraw-Hill: New York. Nash JE, Sutcliffe JV. 97. River flow forecasting through conceptual models, I-A discussion of principles. Journal of Hydrology : Nelder JA, Mead RA A simplex method for function minimization. The Computer Journal 7: Prescott J. 94. Evaporation from a water surface in relation to solar radiation. Transactions of the Royal Sociey of South Australia 64: 4 8. Supit I, Van Kappel RR A simple method to estimate global radiation. Solar Energy 63: Temegsen B, Allen RG, Jensen DT Adjusting temperature parameters to reflect well-watered conditions. Journal of Irrigation and Drainage Engineering: ASCE 25(): Copyright 29 Royal Meteorological Society Int. J. Climatol. 3: (2)