## 4 Ground Track Repeat Period of GPS, GLONASS, and BDS Satellites
### Theory of Ground Track Repeat Period Model  
Before calculating the ground track repeat period, the satellite operation period is introduced first. The satellite operation period is the period of the satellite that rounds the orbit of itself one time. If the earth is stationary, the satellite operation period is totally same as the ground track repeat period for the static observation station. However, it is well known that the earth exists by rotation and revolution instead of remaining stationary. Thus, the satellite operation period (denoted as \(T_{\text{sop}}\)) is different from the satellite ground track repeat period.  
Considering that only an approximation ground track repeat period with units in day needs to be determined in this experiment instead of an accurate period with units in seconds, the simplest fraction method is adopted [21]. The simplest fraction method can be expressed as follows:  
\[NT_{sop}=DT_{sd} \tag{6}\]  
where \(N\) and \(D\) mean that after \(D\) solar days, the satellite operations \(N\) rounds are in their own orbit. \(T_{sd}\) denotes the sidereal day, which is 86,164 s. \(T_{sop}\) is the satellite operation period with units in seconds. \(T_{sop}\) can be calculated by two methods; one is based on the [PERSON]'s third law, which is expressed as follows [22]:  
\[T_{sop}=2\pi\sqrt{\left(R+H\right)^{3}/GM} \tag{7}\]  
where \(GM\) = 3.986005 \(\times\) 10\({}^{14}\) m\({}^{3}\)/s\({}^{2}\) is the earth's gravitational constant; \(R\) and \(H\) denote the mean radius of the earth and the height of the satellite from the ground, respectively. Based on Equation (7), \(T_{sop}\) can be calculated by the broadcast ephemeris. More information can be found in [23; 24]. Another method is based on the information of the satellite's orbit, which is given by the GNSS operation official government sector. The relevant information is summarized in Table 2.  
Figure 3: Latitude and longitude of the two floods. The map denotes Henan Province, China.  
It should be noted that the \(T_{sp}\) calculated by the broadcast ephemeris and [PERSON]'s third law is more accurate than that by the given orbital repetition information. However, for the GLONASS satellite, the broadcast ephemeris only includes the satellite coordinates instead of the root parameter of the satellite's orbit. Thus, only the given orbital repetition information can be used to calculate \(T_{sp}\). Since only an approximation period with units in days is needed in this experiment, the GLONASS satellite period calculated by official information can be used in subsequent analyses.  
After estimating parameter \(T_{sp}\), the \(N\) and \(D\) can be determined by the search and test process. For example, the \(T_{sp}\) of the GLONASS satellite can be calculated by Table 2, which is 40,560 s. Thus, Equation (6) can be converted to:  
\[\frac{N}{D}=\frac{T_{sd}}{T_{sp}}=\frac{86,164}{40,560}=2.21435 \tag{8}\]  
where \(T_{sd}\) is the sidereal day.  
Then, by using the search and test method, it can be found that only the simplest fraction (\(17/8\)) = 2.215 can approximately satisfy the equation. This indicates that the GLONASS satellite runs around the earth seventeen times about every eight solar days. Thus, in order to analyze the influence of flood on GLONASS satellite, data set collected on eight day ago before flood should be used. The repetition period of BDS MEO satellite can also be calculated by the above process.