Introduction Model description Directory UMM :Data Elmu:jurnal:A:Agricultural & Forest Meterology:Vol100.Issue2-3.Febr2000:

Agricultural and Forest Meteorology 100 2000 231–241 Estimation of mean monthly solar global radiation as a function of temperature Francisco Meza a,∗ , Eduardo Varas b,1 a Departamento de Ciencias de Recursos Naturales, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile b Departamento de Ingenier´ıa Hidráulica y Ambiental, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile Received 14 December 1998; received in revised form 11 August 1999; accepted 13 August 1999 Abstract Solar radiation is the primary energy source for all physical and biochemical processes that take place on earth. Energy balances are a key feature of processes such as temperature changes, snow melt, carbon fixation through photosynthesis in plants, evaporation, wind intensity and other biophysical processes. Solar radiation level is sometimes recorded, but generally it needs to be estimated by empirical models based on frequently available meteorological records such as hours of sunshine or temperature. This paper evaluates the behavior of two empirical models based on the difference between maximum and minimum temperatures and compares results with a model based on sunshine hours. This work concludes that empirical models based on temperature have a larger coefficient of determination than the model based on cloud cover for the normal conditions of Chile. These models are easy to use in any location if the parameters are correctly adjusted. In addition, probability distribution functions and confidence intervals for solar radiation estimates using stochastic modeling of temperature differences were calculated. ©2000 Published by Elsevier Science B.V. All rights reserved. Keywords: Solar radiation; Temperature; Random variable; Fourier series

1. Introduction

In some cases a record of global solar radiation R G using instruments such as pyranometers or actinome- ters is available, however, there are many meteorolog- ical stations which do not measure solar radiation, but do register other variables such as precipitation, pres- sure and temperature. For this reason, this paper eval- ∗ Corresponding author. Fax: +56-2-553-92-31. E-mail addresses: fmezapuc.cl F. Meza, evarasing.puc.cl E. Varas. 1 Fax +56-2-686-58-76. uates proposed mathematical models to estimate so- lar radiation as a function of temperature differences and compares their performance with models based on sunshine hours. Solar radiation is the principal energy source for physical, biological and chemical processes, such as, snow melt, plant photosynthesis, evaporation, crop growth and is also a variable needed for biophysical models to evaluate risk of forest fires, hydrological simulation models and mathematical models of natu- ral processes. Hence, in many occasions, a record of observed solar radiation or an estimate of radiation is required. 0168-192300 – see front matter ©2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 1 9 2 3 9 9 0 0 0 9 0 - 8 232 F. Meza, E. Varas Agricultural and Forest Meteorology 100 2000 231–241

2. Model description

Extra-terrestrial solar radiation, also known as An- got radiation R A , MJ m − 2 day − 1 can be calculated as a function of the distance from the sun to earth d, km, the mean distance sun–earth d m , km, latitude φ, rad, solar declination δ, rad and solar angle at sunrise sunset H s , rad using the following expres- sion Romo and Arteaga, 1983: R A = 864001360 π d m d 2 × [H s sinφsinδ + cosφcosδsinH s ] 1 Using the preceding relationship, solar radiation can be calculated for any point in the earth’s outer atmo- sphere for each day of the year as a function of latitude and solar declination. However, gases and clouds in- troduce changes to both magnitude and spectral com- position of solar radiation. 2.1. Angström model, 1924 Since the beginning of the century, efforts have been made to estimate solar radiation as a function of extra-terrestrial solar radiation and the state of the atmosphere Castillo and Santibáñez, 1981. The pa- rameter most commonly used is hours of sunshine. Usually, the ratio of global solar radiation to Angot ra- diation is correlated to the ratio of effective sunshine hours to total possible sunshine hours. Effective sunshine hours n are measured with a heliograph Mart´ınez-Lozano et al., 1984. Although this instrument has a threshold, under which sunshine is not recorded, this error is not significant when esti- mating daily solar radiation. Angström 1924, suggested a simple linear re- lationship to estimate global solar radiation R G , MJ m − 2 day − 1 as a function of Angot radiation, actual sunshine hours n and potential or theoretical sunshine hours N. R G R A = a + b n N 2 Angström suggested values of 0.2 and 0.5 for empirical coefficients a and b respectively. Other authors, such as Bennett 1962, Davies 1965, Table 1 Angström coefficients a and b recommended for Chilean locali- ties. Castillo and Santib´añez, 1981 Locality a b Latitude Longitude Altitude ◦ S ◦ W m Arica 0.28 0.57 18.29 70.19 035 Iquique 0.23 0.47 20.13 70.09 008 Antofagasta 0.23 0.47 23.28 70.20 122 Copiap´o 0.26 0.51 27.21 70.20 283 Vallenar 0.22 0.46 28.35 70.46 469 La Serena 0.29 0.57 29.54 71.15 032 La Paloma 0.22 0.46 30.41 71.02 320 Quintero 0.22 0.45 32.47 71.32 002 Valparaiso 0.22 0.55 33.01 70.38 041 Santiago 0.22 0.44 33.27 70.42 520 Curic´o 0.23 0.47 34.58 71.13 227 Constituci´on 0.22 0.45 35.20 72.26 007 Chillan 0.23 0.47 36.36 72.02 124 Concepci´on 0.26 0.51 36.47 73.07 009 Temuco 0.23 0.47 38.46 72.39 114 Osorno 0.23 0.47 40.35 73.09 027 Puerto Montt 0.26 0.51 41.28 72.56 110 Ancud 0.26 0.51 41.54 73.48 020 Puerto Ays´en 0.26 0.51 45.24 72.42 010 Balmaceda 0.26 0.51 45.54 71.43 520 Punta Arenas 0.26 0.52 53.10 70.54 008 Monteith 1966, Penman 1948, and Turc 1961 have calibrated this expression for different places. Coefficients can vary significantly as Doorenbos and Pruitt 1975 show. In Chile, Castillo and San- tibáñez 1981, have recommended the values given in Table 1. 2.2. Bristow–Campbell model, 1984 Incoming solar radiation is determined by the state of the atmosphere. However, the dynamics of the atmosphere is very difficult to predict. Considering transformations experienced by solar radiation, one can expect to find a relationship to express solar radiation as a function of meteorological variables commonly registered at climatological stations. When solar radiations reaches the earth surface, part of it is reflected and part is absorbed. The same occurs with long-wave radiation that each body emits as a function of its temperature. As Chang 1968, reports, there is usually a good relation between net radiation and global solar radiation, since the latter one is the principal source of energy. 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