There are five major spectral transmittances for the atmospheric components: Rayleigh scattering, aerosol extinction, ozone absorption, water vapor absorption and permanent gas absorption, according to the work by Leckner (1978). Spectral transmittance of these different components was written both as a function of wavelength and air mass. Yang et al. (2001) suggested a new form so-called hybrid model which considered the spectral and temporal physical processes and still preserved the simplicity of Angstrom correlation. In their model they used Eq. (5.8) to obtain the spectrally averaged transmittance for each irradiative transfer process. All these broadband transmittance values can be calculated using the measured or calculated values of some atmospheric-climatic and geographic parameters of the site of interest as outlined in Yang et al. (2001). The values for the clear sky beam and diffuse components of the solar irradiation on a horizontal surface are then calculated with the integral forms:

Hb Iq Tb, clearSinhdt (5.10a)

Hd Iq Td, clearSinhdt (5.10b)

with

Tb, clear ~ max(°, TQzTwTgTrTa °.°l-3) (5.11a)

Td, clear ~ max{0, [TQZTwTg(1 – TrTa)+0.013]} (5.11b)

where I0 is the integral of the solar spectrum for all the wavelengths outside the atmosphere and h is sun altitude. As it is clear, subscripts oz, w, g, r, and a stand to indicate that various transmittances are for ozone absorption, water vapor absorption, permanent gas absorption, Rayleigh scattering and aerosol extinction, respectively.

In the first version of the hybrid model (Yang et al. 2001), the relation between monthly-mean daily solar irradiation and sunshine duration was established as:

H = (a + bn/N)Hb + (c + dn/N)Hd (5.12)

in order to find four regression coefficients a, b, c and d, between H and n/N. In this expression, extraterrestrial daily solar irradiation on horizontal surface, namely H0, above the location of interest outside the atmosphere and the inclusion of the diffuse component of the site are embedded into the delicate calculation of the beam and diffuse components. Yang and co-workers used the measured H and n values together with the calculated Hb and Hd values using the data of 16 stations of Japan in 1995 and obtained the coefficients as: a = 0.391, b = 0.518, c = 0.308, d = 0.320 for n/N > 0, and a = 0.222 and c = 0.199 for n/N = 0.

In the model of Yang et al. it is possible to obtain the radiation values in any preferred time interval since the time integrals equation (5.10) gave this flexibility. For example, from the hourly values, daily sums can be obtained. They used 14 different stations from Japan to validate their model from different locations at different latitudes and altitudes. They also compared the model with the model proposed by Gopinathan (1988) and concluded that their model showed better performances.

In fact, to start from hourly values might be significant as the bright sunshine periods at different times within a day have different contributions to the amount of daily radiation as mentioned in section 2. Monthly averages of hourly values of bright sunshine and hourly solar irradiation at the same intervals correlate quite well for a one year data set of a specific location (Akinoglu et al. 2000). A further similar analysis can be carried out using a larger and reliable data set from different locations in which the hourly values of these variables at the same average air mass may also be considered.

Gueymard (2003a), compared 21 broadband spectral models including the above model of Yang et al. (2001) for the predictions of direct solar transmittance and irradiance, and as the result of a detailed investigation recommended four of them, one of which is the Yang and co-worker’s model. As stated in their recent article of Yang et al. (2006) which presents a similar but modified version of the model, some other investigators (Paulescu and Schlett (2003); Madkour et al. (2006)) also verified the high performance of this broadband model (Yang et al. 2001).

In updated version of the hybrid model, in addition to some corrections of the first version, Yang et al. (2006) showed that the relation between global solar irradiation and sunshine duration for monthly-mean scale can be easily extended to daily scale and even to hourly scale. Also, this version introduced global aerosol and ozone data sets to improve the accuracy of radiation estimation.

In the modified version, global solar irradiation is written as:

R = Tc j (тЬ, clear + Td, clear)I0dt (5.13)

AT

where AT is any preferred time interval. Bright sunshine hour in this new version is now introduced into the formula through Tc = R/Rciear, which was thought to be a function of n/N, that is Tc = f (n/N), a newly defined parameter which is the ratio of the surface solar irradiation R to the surface solar irradiation under clear sky, Rclear. This parameter is of course similar to the quantity which Angstrom used in obtaining his simple correlation between solar radiation and bright sunshine hours, H/Hc of Eq. (5.1). Tc can be obtained by regression analysis using hourly, daily or monthly bright sunshine records. Yang et al. used a two-step procedure to obtain tc: in step one, they regressed the daily data of 67 stations at the year 1995, in Japan. In step two using the result of pass one, excluded the data with RMSE>2.0MJm-2 (Yang et al. 2006) (RMSE is the root mean square error which is the square-root of average of the square of calculated minus measured values of a quantity and explained briefly in section 5). Remaining data was then used to obtain the function f (n/N) for daily and monthly mean daily solar irradiation, respectively as:

Tc = 0.2505 + 1.1468n/N – 0.3974(n/N )2 (5.14)

Tc = 0.2777 + 0.8636n/N – 0.1413(n/N )2.

A limitation for these equations was for n/N = 1, Tc = 1 must be used since the radiation in this case is the clear sky value Rclear.

They tested the model for the seven sites in China, seven in USA and twelve in Saudi Arabia comparing the results with the estimations of two different models and with Angstrom-Prescott type quadratic correlations that they obtained directly by regression analysis for the same data set they used in obtaining Eq. (5.14). Their results showed that the new formalism which started with spectral considerations has better performance and hence universally applicable.

It may be questioned that why the quadratic forms, Eq. (5.14), are chosen in obtaining the function f (n/N). They used this form due to some reasons (Yang 2007): as their experience on humid regions (Japan) showed that the quadratic form gives better correlations and also following the conclusion reached in a recent article (Suehrcke 2000) that those non-linear forms are better than linear. They also followed the work of Iqbal (1979) who proposed quadratic relations between the fractional monthly averages of daily diffuse and beam components and bright sunshine hours. It is rather interesting that Suehrcke (2000) reached a non-linear form using a physical formalism he developed, which starts from bi-modal character of instantaneous or short-term irradiation. This characteristic is due to the fact that the cloud within a time interval may intermittently obscure the sun rays in rather short times so that significantly reduces the irradiation without changing the bright sunshine records of the instrument.

A physical model which starts by defining various spectrally averaged physical properties (reflectance, transmittance etc.) of the atmosphere within the entire solar spectrum, measurable by the instruments such as pyranometers and pyrheliometers, may give quadratic forms rather than linear as will be discussed in the following section. Such instruments measure the total integrated instantaneous solar global and beam irradiation over the whole spectrum.

The FORTRAN code of the model of Yang et al. is included in the CD-ROM. In the readme. doc file within modelYangetal. zip, some explanations of how to use the programs are given. Clear-sky, hourly, daily and monthly models are in the zip file and each model has an application as an example.