Climatic data Models applied to mean monthly data

F. Meza, E. Varas Agricultural and Forest Meteorology 100 2000 231–241 233 Furthermore, if the heat flow towards the soil is neglected, one can find the ratio of sensible heat to latent heat or Bowen ratio, on a daily basis Campbell, 1977. Sensible heat is responsible for temperature variations, so it is possible to obtain a relationship between temperature differences and solar radiation, being temperature a reflection of radiation balance. Using this argument, Bristow and Campbell 1984, suggested the following relationship for daily R G , as a function of daily R A and the difference between maximum and minimum temperatures 1T, ◦ C: R G R A = A h 1 − exp−B1T C i 3 Athough coefficients A, B and C are empirical, they have some physical meaning. Coefficient A represents the maximum radiation that can be expected on a clear day. Coefficients B and C control the rate at which A is approached as the temperature difference increases. Values most frequently reported for these coefficients are 0.7 for A, the range 0.004 to 0.010 for B and 2.4 for C. Since clear days present large temperature differ- ences A tends to be the ratio between global solar radi- ation and Angot radiation, hence the sum of Angström coefficients a and b tends to be similar to A. 2.3. Allen model, 1997 Allen 1997, suggested the use of a self-calibrating model to estimate mean monthly global solar radiation following the work of Hargreaves and Samani 1982. He suggested that the mean daily R G can be estimated as a function of R A , mean monthly maximumT M , ◦ C and minimum temperatures T m , ◦ C. R G R A = K r T M − T m 0.5 4 Previously, Allen 1995, had expressed the empiri- cal coefficient K r as a function of the ratio of atmo- spheric pressure at the site P, kPa and at sea level P , 101.3 kPa as follows: K r = K ra P P 0.5 5 In his work, Allen suggested values of 0.17 for interior regions and 0.20 for coastal regions for the empirical coefficient K ra .

3. Climatic data

In order to compare the behavior of the different models, monthly climatological data of 21 stations representing different climatic regions of Chile were collected. Data ranged from Arica latitude 18.3 ◦ S to Punta Arenas 53.1 ◦ S and was registered between the years 1971 and 1992. Selected meteorological variables were T M , T m , P, mean monthly degree of cloud cover x and R G . For the locations mentioned in Table 1, monthly val- ues of maximum and minimum temperatures, cloud cover and atmospheric pressure for each year in the period 1971 to 1992, were available. Unfortunately, for global solar radiation only the average value for each month in that period was available and monthly radiation values for each year were impossible to ob- tain from Dirección Meteorológica de Chile. In addition to the above, data from La Paloma sta- tion was collected to compare the behaviour of models based on temperature differences when they are ap- plied to estimate monthly global solar radiation. The selected meteorological variables in this case were T M , T m , P, and R G between the years 1971 and 1978. Finally, data from Santiago station was used to compare the behaviour of Bristow–Campbell and Allen models when they are applied to estimate daily global solar radiation. The meteorological variables were daily T M , T m , P, and R G .

4. Models applied to mean monthly data

The extension of the reviewed models to apply them to monthly averages requires some explanation. The Angström model was originally derived for daily so- lar radiation and hours of sunshine. Nonetheless, be- ing a linear function it can be readily applied to mean monthly data since the expected value of a sum is equal to the summation of the expected values. Allen’s model was derived for monthly data so it can readily be used. However, the Bristow–Campbell model is de- fined for daily data and has no evident extrapolation to mean monthly values. For this reason, one can ex- pect to find a new set of coefficients when the same expression is applied to monthly data. With the values of temperature, atmospheric pres- sure and sunshine hours, mean monthly global solar 234 F. Meza, E. Varas Agricultural and Forest Meteorology 100 2000 231–241 radiation was calculated at each site, using the ex- pressions and empirical coefficients suggested by Angström 1924, Bristow and Campbell 1984, and Allen 1995. Results show that models using the coefficients proposed in the literature do not esti- mate correctly the historical average in each location. The slope of the relationship between calculated and observed radiation is significantly different from unity. This is especially notorious in the case of the Bristow–Campbell model, although this result was ex- pected since the coefficients suggested by the authors are applicable to daily data. Given the results, it was necessary to change the Allen and Bristow–Campbell model coefficients to obtain a better fit, following the idea suggested by Castillo and Santibáñez 1981 for the Angström model. Least squares coefficients, which minimize the sum of square errors for each location were calculated and included in Table 2. Due to the fact that monthly solar radiation values, were not available for each year, as mentioned in the section about climatic data, the A and C coefficients of Bristow–Campbell model were assumed fixed and the B coefficient was adjusted to minimize the square Table 2 Adjusted coefficients K ra and B of Allen and Bristow–Campbell models Locality K ra B Arica 0.3354 0.01354 Iquique 0.2854 0.01619 Antofagasta 0.4717 0.01944 Copiap´o 0.2577 0.00203 Vallenar 0.3457 0.00200 La Serena 0.2697 0.00677 La Paloma 0.1589 0.00347 Quintero 0.2731 0.00589 Valpara´ıso 0.0114 0.01144 Santiago 0.2593 0.00202 Curic´o 0.4348 0.00152 Constituci´on 0.2423 0.00555 Chillan 0.2316 0.00159 Concepci´on 0.3402 0.00242 Temuco 0.2583 0.00154 Osorno 0.3756 0.00150 Puerto Montt 0.3252 0.00290 Ancud 0.2820 0.00493 Puerto Ays´en 0.2870 0.00463 Balmaceda 0.3058 0.00348 Punta Arenas 0.3471 0.00389 Table 3 Regression between calculated and observed mean monthly global solar radiation using adjusted parameters of 20 Chilean localities Model Slope Upper Lower R 2 limit. 95 limit. 95 Angström 0.959 0.970 0.939 0.892 Allen 0.999 1.010 0.990 0.961 Bristow–Campbell 1.152 1.170 1.138 0.928 errors. The available data made it impossible to study the contribution of coefficients A and C. However, A represents the maximum radiation on a clear day and its value represents the observed data reasonably well. Moreover, a change in coefficient C does not affect significantly the calculated global solar radiation. Observed and calculated values for different locations and models are shown in Fig. 1. In this figure the improvement in the relationships when us- ing locally calibrated coefficients can be appreciated. The Angström model results using the coefficients proposed by Castillo and Santibáñez 1981 are also included for comparison. Slopes of the different mod- els and the coefficients of determination are given in Table 3. Allen’s model presents the best relationship. It has a higher coefficient of determination and the slope is equal to unity with 90 confidence interval. The Bristow–Campbell model tends to under-estimate global solar radiation but explains a large proportion of sample variance. The Angström model fit the data poorer than the other two. 4.1. Models applied to monthly data. Since the available data of global solar radiation for most stations is only the average value for each month, it was necessary to examine if the relationships with the adjusted coefficients represent accurately the monthly values for each year. One station available with monthly global solar radiation data, is La Paloma. In this case the models with the adjusted coefficients derived with the average monthly values were used to estimate monthly global solar radiation for each year. A comparison between estimated monthly values for La Paloma, compared to observed monthly values is shown in Table 4. Results show that monthly global radiation for each year can be adequately estimated with the derived F. Meza, E. Varas Agricultural and Forest Meteorology 100 2000 231–241 235 Fig. 1. a Comparison between observed and measured mean monthly global solar radiation using Angström parameters from the literature see text; b Comparison between observed and measured mean monthly global solar radiation using Angström adjusted parameters; c Comparison between observed and measured mean monthly global solar radiation using Allen parameters from the literature; d Comparison between observed and measured mean monthly global solar radiation using Allen adjusted parameters; e Comparison between observed and measured mean monthly global solar radiation using Bristow-Campbell parameters from the literature; and f Comparison between observed and measured mean monthly global solar radiation using Bristow–Campbell adjusted parameters. 236 F. Meza, E. Varas Agricultural and Forest Meteorology 100 2000 231–241 Fig. 1 Continued. F. Meza, E. Varas Agricultural and Forest Meteorology 100 2000 231–241 237 Table 4 Regression between calculated and observed monthly global solar radiation at La Paloma station Model Slope Upper Lower R 2 limit. 95 limit. 95 Allen 1.000 1.010 0.990 0.97 Bristow–Campbell 0.994 1.006 0.982 0.96 models. Allen’s model presents the best relationship between observed and calculated monthly solar global radiation because it explains a large proportion of the sample variance. In both models the slope is equal to unity with 90 confidence interval. This verifies that the models can be used to estimate monthly values for different years. 4.2. Global solar radiation distribution functions A probability distribution function for global solar radiation was obtained as a derived distribution, when radiation is expressed as a function of temperature dif- ferences and temperature differences are expressed as a Fourier series with a random component. This ran- dom error was found to be a random variable with nor- mal distribution. This hypothesis was tested in both for the Bristow–Campbell and the Allen models using the Anderson–Darling test for normal distribution. Once a distribution model for solar radiation is calculated, confidence intervals for estimates can be computed. 4.3. Temperature difference modelling. Temperature has a marked seasonal variation due to periodicity in the earth’s orbit about the sun. For this reason temperature variations can be represented us- ing mathematical cyclic functions. In this paper, dif- ferences between maximum and minimum tempera- tures were modelled using a Fourier series once the stationary component was removed, as suggested by Van Wijk and De Vries 1966 and Campbell and Nor- man 1997. These authors applied Fourier series with one term to represent air temperatures. The 1T in location i and month j 1T ij , ◦ C can be expressed as a function of mean annual 1T in location i 1T i , ◦ C, Fourier series coefficients at location i C i , D i and an error or residual in location i and month j E ij , ◦ C as follows: Table 5 Average temperatures 1T i and Fourier series coefficients C i and D i of 20 Chilean localities Locality 1 T i C i D i Arica 06.344 0.722 1.412 Iquique 05.629 0.629 1.162 Antofagasta 06.489 0.269 0.945 Copiap´o 14.545 0.640 − 0.292 Vallenar 13.193 1.264 0.177 La Serena 07.856 0.220 − 0.044 La Paloma 14.143 0.330 0.345 Quintero 08.366 0.670 0.495 Valpara´ıso 05.549 0.804 0.330 Santiago 13.917 2.539 1.830 Curic´o 14.612 4.235 2.851 Constituci´on 08.397 0.723 0.188 Chillan 13.802 3.991 2.910 Concepci´on 10.073 2.134 1.365 Temuco 11.494 3.052 2.111 Osorno 11.031 3.140 1.592 Puerto Montt 08.592 1.862 0.773 Ancud 07.255 1.636 1.051 Puerto Ays´en 06.823 1.427 0.345 Balmaceda 09.078 2.130 1.151 Punta Arenas 07.019 1.829 0.665 1T ij = 1T i + C i cos 2πj 12 + D i sin 2πj 12 + E ij 6 The coefficients C i and D i are given in Table 5 for the sites used in this work. 4.4. Probability distribution functions If X is a continuous random variable with a proba- bility density function fx and Y is a monotonic func- tion of X, then the probability function of Y can be obtained multiplying the inverse function by the abso- lute value of the Jacobian of the transformation J or determinant of the first derivative of wy with respect to X Walpole and Myers, 1992: gy = f [wy]|J | 7 Using this procedure probability density and proba- bility distribution functions for R G estimated by Allen and Bristow–Campbell models were derived. 238 F. Meza, E. Varas Agricultural and Forest Meteorology 100 2000 231–241 4.5. Bristow–Campbell model In this case the distribution function is calculated using Eq. 3 and replacing 1T ij for its expression in terms of annual 1T in each location and the corre- sponding Fourier series coefficients. Combining both the expressions, an equation for the residuals is ob- tained. Residuals were found to be well represented by a normal distribution model, so the probability dis- tribution of the errors was assumed known. The distri- bution hypothesis was tested using Anderson–Darling test. The probability density function for solar radiation following Eq. 7, is equal to the product of the normal density function evaluated at the residuals for location i and month j and the absolute value of the transfor- mation Jacobian Eq. 8. The residuals are given in this case by Eq. 9 and the first derivative by Eq. 10. gR Gij = [J ]f R Gij 8 The residuals are given by the following equation ex- pressed as a function of terms already defined: E ij = − ln 1 − R Gij 0.7R Aij B i 12.4 − 1T i − C i cos 2πj 12 − D i sin 2πj 12 9 The first derivative is: | J | = 1 B i 12.4 1 6.8 − ln 1 − R Gij 0.7R Aij − 1.42.4 × 1 1 − R Gij 0.7R Aij 1 R Aij 10 The cumulative distribution function CDF is ob- tained by integrating the probability density function. The CDF was evaluated numerically, using very small intervals and the trapezoidal integration method, to define confidence intervals for global solar radia- tion. Results for two locations Arica and Vallenar are shown graphically in Fig. 2 a,b. 4.6. Allen’s model Similarly, for Allen’s model, 1997, the probability density function is obtained using Eq. 4 and replac- ing 1T ij for its expression in terms of annual 1T in each location and the corresponding Fourier series co- efficients Eq. 6. Residuals in this case were also found to be well represented by a normal distribution model, so the probability distribution of errors was assumed known. The probability density function for solar radiation is shown in Eq. 8.The residuals are given in this case by Eq. 11 and the first derivative by Eq. 12: E ij = R Gij R Aij K rai P P 0.5 − 1T i − C i cos 2πj 12 − D i sin 2πj 12 11 The Jacobian is: | J | = 2R Gij P K 2 ra i R Aij 2 P 12 The CDF is obtained integrating the probability den- sity function. It was evaluated numerically to define confidence intervals for global solar radiation. Re- sults for Arica and Vallenar are shown graphically in Fig. 2c,d. The expected value for global solar radiation given by the CDF using Allen’s model are higher than the Angot radiation because the limits of integration derived in this case were zero and infinite. On the other hand, the CDF using Bristow-Campbell model have clear and defined limits which are zero and A times the Angot radiation. For this reason the CDF obtained with Bristow–Campbell model is more accurate and has smaller confidence intervals.

5. Models applied to daily data