# Centimeter-Level Precise Orbit Determination for the Luojia-1A Satellite Using BeiDou Observations
## 3 BeiDou Multipath Modeling with Luojia-1A Observations
### Modeling the Multipath of BeiDou Observations  
The pseudorange observations of BeiDou are reported as suffering from the elevation-dependent multipath biases, which may lead to an adversary impact on positioning accuracy [24]. Currently, the elevation-dependent code variations of BDS-2 IGSO and MEO satellites are based on the data from ground stations. The ground stations are not able to capture the elevation-dependent biases for the GEO satellites, due to a stable geometry relationship. The LEO onboard BDS observations can be used to capture the geometric change of the GEO satellites, which can be used as a complementary approach for the multipath biases modeling. The multipath error can be estimated with the linear combination of code and carrier-phase observations, which can be expressed as follows [25, 26]:  
\[MP_{i}=C_{i}-\frac{f_{i}^{2}+f_{j}^{2}}{f_{i}^{2}-f_{j}^{2}}\lambda_{i}\varphi_ {i}+\frac{2f_{j}^{2}}{f_{i}^{2}-f_{j}^{2}}\lambda_{j}\varphi_{j}+\frac{f_{i}^{2 }+f_{j}^{2}}{f_{i}^{2}-f_{j}^{2}}\lambda_{i}N_{i}-\frac{2f_{j}^{2}}{f_{i}^{2}-f _{j}^{2}}\lambda_{j}N_{j}-D_{c} \tag{1}\]  
where \(C\) and \(\varphi\) represent the code and carrier-phase observables, respectively. The \(f\) is the frequency, \(\lambda\) is the wavelength, and \(N\) represents the carrier phase ambiguity. The subscripts \(i,j\) (\(i\
eq j\)) are used to specify different frequencies; \(D_{c}\) is the constant part of hardware delays. The BDS code bias was analyzed based on Luojia-1A onboard measurements. In this section, the MP1 and MP2 represent the code multipath of B1 and B2 frequency. The MP1 and the elevation of several BDS satellites are shown in Figure 5, which indicates that most BeiDou satellites present systematic elevation-dependent variations in multipath combination. According to the GEO satellites, such as the C02 and C05, it is obvious that the multipath is decreased as the elevation increases. As for the IGSO satellite, e.g., C07, the trend is similar to the GEO satellites but with a more dramatic variation. Another IGSO satellite, the C10 satellite, has a low elevation, so it has a larger multipath bias without an obvious trend. Considering the MEO satellites, the multipath is linearly increasing as the elevation angle decreases, which is shown in panel (c). The multipath error of the C14 satellite is generally smaller than 2 m without significant elevation-dependent variation. Figure 4: The sky view of the BeiDou satellites from the ARF of Luojia-1A satellite. The red, green, and blue lines represent the geostationary earth orbit (GEO), inclined geostationary (IGSO), and medium earth orbit (MEO) satellites, respectively. To mitigate the impact of multipath biases, an elevation-dependent model is established with the BeiDou observations from the Luojia-1A satellite. The elevation-dependent modeling method proposed by [PERSON] and [PERSON] [24] is adopted in this study for the multipath (MP) series. To define the piecewise linear models for MP series, the elevations interval, which varies from 0\({}^{\circ}\) to 90\({}^{\circ}\), is split by 10 nodes with the spacing of 10\({}^{\circ}\). Therefore, the values of 10 nodes are estimated for the elevation-dependent model, and the restriction that the average of model values is zero. The MP observables with the elevation \(e\) between \(e_{k}\) and \(e_{k+1}\) can be expressed as follows:  
\[MP_{f}^{s}(e)=\Bigg{(}\frac{e-e_{k}}{e_{k+1}-e_{k}}\Bigg{)}p_{f,k+1}^{s}+ \Bigg{(}1-\frac{e-e_{k}}{e_{k+1}-e_{k}}\Bigg{)}p_{f,k}^{s}+v_{i} \tag{2}\]  
where \(f\) represents the frequency and \(s\) is the type of the BeiDou satellites. There is a 10\({}^{\circ}\) difference between \(e_{k}\) and \(e_{k+1}\). \(P_{f,k}^{s}\) and \(P_{f,k+1}^{s}\) are the MP values of the contiguous grid point, which are estimated using the methods of least square estimation. Figure 6 shows the elevation-dependent MP models, which are built respectively for the GEO, IGSO, and MEO satellites for both B1 and B2 frequencies.