Models applied to daily data

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

5.1. Allen’s model Allen’s model, 1997 includes a correction term for barometric pressure which in fact represents the alti- tude of the station above sea level, since the pressure as a function of elevation can be expressed in terms of the pressure at sea level, the temperature gradient, the temperature at station elevation and the Avogadro air constant. This correction term is small compared to the influence of the temperature difference on radiation. F. Meza, E. Varas Agricultural and Forest Meteorology 100 2000 231–241 239 Fig. 2. a Expected values and confidence limits 5 and 95 of daily mean global radiation using Bristow-Campbell model and Angot radiation for Arica; b Expected value and confidence limits 5 and 95 of daily mean global radiation using Bristow-Campbell model and Angot radiation for Vallenar; c Expected value and confidence limits 5 and 95 of daily mean global radiation using Allen model and Angot radiation for Arica; d Expected value and confidence limits 5 and 95 of daily mean global radiation using Allen model and Angot radiation for Vallenar. This model tends to over estimate global solar ra- diation in a daily basis, and frequently estimates radi- ation in excess of the extra-terrestrial radiation, since the condition expressed by Eq. 13 is fulfilled. This model does not have a limit for the estimated solar radiation. 1T P K ra 2 P 13 This condition is frequently true when the model is applied to points located in interior regions which usually experience large daily temperature variations. Even though Allen’s model has a larger coeffi- cient of determination, the slope is clearly less than unity, indicating that the model over-estimates solar radiation. 5.2. Bristow–Campbell model This model is defined solely in terms of temperature differences and is thus simpler to apply. The value for A coefficient is 0.7, which is a reasonable value for clear days. This type of day usually is associated to large temperature differences. 240 F. Meza, E. Varas Agricultural and Forest Meteorology 100 2000 231–241 Fig. 2 Continued. Table 6 Regression between calculated and observed daily global solar radiation at Santiago station Model Slope Upper Lower R 2 limit. 95 limit. 95 Allen 0.561 0.549 0.571 0.85 Bristow–Campbell 1.090 0.979 1.202 0.79 The behavior of the Bristow–Campbell model is more consistent and reliable, since it has an upper limit given by parameter A. The regression analysis shows that the Bristow–Campbell model performs better Table 6. On the other hand, Bristow–Campbell model gives consistently a better estimate when applied to daily data.

6. Conclusions