## 1 Introduction
Furthermore, the \(R_{nl}\) cannot be obtained directly with this method, and its relationship with \(R_{nD}\) was not easy to model [11]. Hence, it is not ideal to apply sinusoidal model with polar-orbiting satellite data within mid-low latitudes where there are few available satellite data [11]. With the rapid development of machine learning methods and its wide application in remote sensing, several algorithmswere developed to estimate \(R_{nd}\) directly from instantaneous satellite observations at the top of the atmosphere (TOA) with machine learning methods [9; 11; 15]. Compared to the sinusoidal model, this kind of algorithm is easily to be implemented and its outputs are satisfactory, although they lack mechanisms and are heavily dependent on the quality and quantity of the samples for modeling [9; 11]. An empirical parameter referred as \(C_{d}\) is the alternative temporal expansion method, which was first proposed by [PERSON] and [PERSON] [16], was thought to be a general model to expand \(R_{ni}\) into \(R_{nd}\)[16; 17; 18]. \(C_{d}\) is defined as the ratio between \(R_{nd}\) and \(R_{ni}\):  
\[C_{d}=\frac{R_{nd}}{R_{ni}} \tag{2}\]  
Hence, if the value of \(C_{d}\) in a certain region at a certain time was determined, then the corresponding \(R_{nd}\) could be easily obtained from \(R_{ni}\) at that time by multiplying \(C_{d}\). \(C_{d}\) was first proposed based on the analysis results of the ground measurements during the summer midday in the Avignon region of France [16]. The authors found that the ratio between \(R_{nd}\) and \(R_{ni}\) at a certain time and region under a clear sky was very close to a constant, they thus defined \(C_{d}\) as 0.3 \(\pm\) 0.03. Afterwards, \(C_{d}\) was further studied and was found to have different values under different cases. Thereby, \(C_{d}\) was further revised as a function of various parameters (e.g., time, the day of year <_day_>, and latitude), such as the linear function [19] and quadratic function [17]. Because of its simple format and being easy to calculate, \(C_{d}\) has been widely used in practical applications [17; 18]. Lately, the definition of \(C_{d}\) has been applied with satellite data by [PERSON] [20]. Being different from the previous models, the calculation of \(C_{d}\) in this study was not from a model with a specific format but by the calculated \(R_{ni}\) and \(R_{nd}\) from satellite data using parameterization models; hence, this \(C_{d}\) contains some physical meanings of the temporal expansion of \(R_{n}\). Additionally, the authors claimed that this \(C_{d}\) could be the same for one image from any satellites, so it can be used generally. However, this \(C_{d}\) is difficult to be widely applied because of its unspecific format, with its accuracy highly dependent on the estimated \(R_{ni}\) and \(R_{nd}\). Generally, the previous \(C_{d}\) calculation models were usually developed using the ground measurements within a small region, but their performance in other regions has not been thoroughly evaluated. Therefore, there is still a great development space for \(C_{d}\) estimation for its application with satellite data. In this study, based on comprehensive clear-sky ground measurements, a new \(C_{d}\) model for clear sky that could be applied at any moment around the globe during 9:30-12:00 hours was proposed. Against the ground measurements, this model was evaluated by validating and inter-comparing it with other two existing models, and then it was applied in \(R_{nd}\) estimation with MODIS data. This paper is organized as follows: Section 2 introduces the data and methods, results and analysis are provided in Section 3, and the discussion and conclusions are given in Section 4.