# An Effective Surrogate Ensemble Modeling Method for Satellite Coverage Traffic Volume Prediction
## 3 Satellite Coverage Traffic Volume Modeling and Prediction Approximation
### Satellite Coverage Traffic Volume Modeling  
Given any ground site \((u,w)\), where \(u\) and \(w\) are the geographical longitude and latitude, the ground traffic density can be statistically obtained according to the historical data, denoted as \(D(u,w)\). For the SCTV calculation, the ground traffic density in the satellite coverage region \(S\) around the site \((u,w)\) should all be considered. First, the area \(a_{S}\) of the coverage region can be calculated by  
\[\begin{array}{l}a_{S}=2\pi R_{e}^{2}(1-\cos\theta)\\ \theta=\arcsin\Big{(}\frac{h+R_{e}}{R_{e}}\sin\alpha-\alpha\Big{)},\end{array} \tag{17}\]  
where \(R_{e}\) is the radius of the earth, \(h\) and \(\alpha\) are the altitude and half-beam angle of the satellite with the corresponding nadir point site \((u,w)\). \(\theta\) is the geocentric half-cone-angle of the coverage region. The diagram of the satellite coverage region on the earth's surface is presented in Figure 1. The SCTV \(y(u,w)\) is defined as  
\[y(u,w)=\frac{\iint_{(x_{1},x_{2})\in S}D(x_{1},x_{2})dx_{1}dx_{2}}{a_{S}}, \tag{18}\]  
where \((x_{1},x_{2})\) are the ground sites which belong to the satellite coverage region \(S\) around the nadir point site \((u,w)\). Detailed solution process of SCTV can be found in the literature [25]. Notice that for most ground business, there would be large geographical distribution variances of SCTV. Take air traffic monitoring as an example, the air traffic SCTV data is downloaded from TianTuo-3 (National University of Defense Technology, Changsha, China) as shown in Figure 2 (National University of Defense Technology developed and launched TianTuo-3 micro-satellite in May 2014, which achieves worldwide collection of the air traffic SCTV data [26]). It can be seen that aircraft traffic distribution is dense in population agglomeration while scarce in vast ocean regions.  
To predict SCTV, traditionally a global SCTV distribution data table is first statistically constructed on the ground according to historical data and uploaded to the satellite. Then SCTV is predicted onboard by the data table lookup. When the SCTV distribution is scarce, the satellite payload is preferably kept at low power or shut down to save onboard resources. However, with the dense SCTV distribution, the payload should be maintained at high power (with full capacity) for better reception of the real-time signals. To update the date table with dynamically accumulated data, the SCTV data table should be uploaded each time the satellite passes over the ground station. Moreover, for better SCTV prediction, more data is preferred to construct a table with fine resolution. Then large data transmission and storage are necessitated, which is prohibitive for satellite communication and onboard data handling. To solve these problems, this paper proposed to distill the data into a surrogate model to be uploaded to satellite. Through sampling, a small amount of training points \(\mathbf{x}=(u,w)\) with the corresponding SCTV responses \(y(\mathbf{x})\), the surrogate \(\hat{y}(\mathbf{x})\) can be constructed on the ground. Then the surrogate \(\hat{y}(\mathbf{x})\) is uploaded to the satellite instead of the SCTV data table. When the satellite passes over the ground station, only a few parameters of the surrogate are demanded to update. In this way, the valuable communication link resource would be saved, and the SCTV prediction accuracy could be improved compared to table lookup. Moreover, for satellite missions of collecting the ground traffic business, such as air traffic monitoring and ship trajectory identification, the SCTV is only related to the ground traffic. Thus, the proposed method and SCTV prediction results can be directly generalized to more complex satellite constellations.  
Figure 1: Diagram of the coverage region on the earth’s surface.  
Figure 2: Aircraft traffic density distribution.