# Solar Radiation Parameters for Assessing Temperature Distributions on Bridge Cross-Sections
## 2 State of Art of Estimation Models for Solar Radiation
### Estimation Models for Global Solar Radiation
#### 2.1.1 Estimation Models for Daily Global Solar Radiation  
The global solar radiation is mainly related to meteorological factors such as the duration of sunshine, extent of cloud cover, ambient temperature and so on. Accordingly, empirical regression analyses or artificial intelligence techniques are used by researchers to establish the estimation models of daily global solar radiation based on different meteorological factors. * Sunshine duration fraction models  
In 1924, a linear equation relating the clearness index and the sunshine duration fraction was proposed by [PERSON] [10], as shown below:  
\[H/H_{\rm C}=a+b(S/S_{0}) \tag{1}\]  
where \(H/H_{\rm C}\) is the clearness index, \(H\) is the daily global solar radiation (averaged over one month), \(H_{\rm C}\) is the average clear-day global solar radiation, \(S/S_{0}\) is the sunshine duration fraction, \(S\) is the daily sunshine duration (averaged over one month), \(S_{0}\) is the maximum daily sunshine duration (averaged over one month), and \(a\) and \(b\) are empirical coefficients. However, it is difficult to properly define the concept of \"clear day\" in the equation proposed by [PERSON]. Therefore, a daily extraterrestrial solar radiation on a horizontal surface (averaged over one month) (\(H_{0}\)) was proposed by [PERSON] to be used instead of \(H_{\rm C}\), resulting in the Angstrom-Page equation as follows [11]:  
\[H/H_{0}=a^{\prime}+b^{\prime}(S/S_{0}) \tag{2}\]  
where \(a^{\prime}\) and \(b^{\prime}\) are empirical coefficients in Angstrom-Page equation. The Angstrom-Page equation is one of the classical equations for estimation of the daily global solar radiation. Different mathematical expressions have been proposed by other researchers to improve its accuracy, including using quadratic [12], cubic [13], logarithmic [14], exponential [15] and power expressions [16]. However, the improvement in the accuracy of equations using these expressions is not obvious when comparing with the linear expression [17]. Therefore, the linear Angstrom-Page equation is generally used due its sufficient accuracy and lower computational effort. In China, research on global solar radiation models began in the 1960s, and was based on the Angstrom-Page equation with empirical coefficients for different geographic zones [18]. However, China is vast in territory, and environmental conditions in different regions are obviously different. Thus, it is difficult to establish a common set of empirical coefficients applicable to all regions. Some researchers have used other meteorological and geographic parameters such as the ambient temperature, altitude, longitude and latitude information, relative humidity, and cloud cover to improve the accuracy of the Angstrom-Page equation [19; 20; 21]. It is noted that sunshine duration has the greatest influence on the daily global solar radiation compared with other meteorological factors. Moreover, considering other meteorological factors may reduce the accuracy of the estimated values [22]. Non-sunshine duration models  
The ambient temperature was chosen by some researchers to establish an estimation model for global solar radiation. The difference between the daily maximum and minimum temperatures was used to predict the daily global solar radiation by [PERSON] and [PERSON] (H-S model) in 1982 [23]. To improve the accuracy of the H-S model, some meteorological factors such as the altitude [24], and precipitation [25; 26; 27] were considered. Based on the H-S model, an exponential equation considering the daily temperature range was proposed by [PERSON] and [PERSON] (B-C model) in 1984 [28]. [PERSON] et al. compared the results of these empirical models with measured data from the Mediterranean region and concluded that the H-S model had the highest accuracy [29]. However, [PERSON] et al. suggested using the B-C model after comparing various empirical models with measured data from 15 meteorological stations in Northern China due to better accuracy and ease of determining the empirical factors [30].