## 2 Elevation-Dependent Modeling of Satellite-Induced Code Bias  
### Formulation of MP Combinations  
Following the standard BDS measurement model, the code observations and the carrier phase observations on the \(i\)th frequency can be modeled as:  
\[P_{i}=\rho+c\mathrm{d}t_{r}-c\mathrm{d}t+I\cdot\gamma_{i}+T+\mu_{r,i}+\mu_{i}+ M_{i}+\varepsilon_{i}+\beta_{i} \tag{1}\]  
\[L_{i}=\rho+c\mathrm{d}t_{r}-c\mathrm{d}t-I\cdot\gamma_{i}+T+N_{i}+b_{r,i}+b_{ i}+m_{i}+\xi_{i} \tag{2}\]  
with \(\gamma_{i}=f_{1}^{2}/f_{i}^{2}\), where \(P\) is the measured pseudorange, \(L\) is the measured carrier phase, \(\rho\) is the geometric distance between the phase centers of the receiver and the satellite antennas, \(c\mathrm{d}t_{r}\) is the receiver clock offset, \(c\mathrm{d}t\) is the satellite clock offset, \(I\) is the slant ionospheric delay on the B1 frequency, \(\gamma\) is the ionospheric delay factor with respect to the \(I,f\) is the signal frequency, \(T\) is the tropospheric path delay, \(N\) is the integer phase ambiguity, \(\mu_{r}\) and \(\mu\) are the receiver- and satellite-dependent code hardware delays, respectively, \(b_{r}\) and \(b\) are the phase hardware delays at the receiver and satellite ends, respectively, \(M\) and \(m\) are the multipath effects in the pseudorange and carrier phase observations, respectively, \(\varepsilon\) and \(\xi\) are the receiver code and phase measurement noises, respectively, and \(\beta\) is the elevation-dependent SICB in BDS code observations, which is the focus of this study. In order to analyze the SICB \(\beta\), the effects of the other terms shown in Equation (1) should be eliminated as much as possible. The linear combination of the code and carrier phase observations is an effective way to remove the common terms in the code and carrier phase observation equations, namely, Equations (1) and (2). The frequency-independent terms, including the geometric distance, receiver clock offset, satellite clock offset and the tropospheric delay, can be removed when the sum of the combination coefficients is equal to zero. In addition, the frequency-dependent ionospheric delay can also be removed by forming the ionospheric-free combination based on its dispersive nature. According to the above statements, we actually need a geometry-free and ionospheric-free (GFIF) combination. In addition, the GFIF combination should only involve the code observations on a single frequency so that we can model the SICB for each frequency. The MP combination, which is usually used to analyze the characteristics of GNSS code observations, is an optimal case for the required GFIF combination. The dual-frequency carrier phase observations and single-frequency code observations can be used to form the MP combinations. The MP combinations for the code observations on the \(i\)th frequency can be described as shown below [21]:  
\[ \begin{array}{ll}\text{MP}_{i}&=P_{i}+(a_{ijq}-1)\cdot L_{j}-a_{ijq}\cdot L_ {q}\\ &=N_{e,ijq}+\left[M_{i}+\varepsilon_{i}+(a_{ijq}-1)\cdot(m_{j}+\xi_{j})-a_{ijq }\cdot(m_{q}+\xi_{q})\right]+\beta_{i}\end{array}  \tag{3}\]  
with  
\[\left\{ \begin{array}{l}a_{ijq}=(\lambda_{i}^{2}+\lambda_{j}^{2})/(\lambda_{ j}^{2}-\lambda_{q}^{2})\\ N_{e,ijq}=\mu_{r,i}+\mu_{i}+(a_{ijq}-1)\cdot(N_{j}+b_{r,j}+b_{j})-a_{ijq}\cdot( N_{q}+b_{r,q}+b_{q})\end{array} \right. \tag{4}\]  
where \(i,j\) and \(q\) denote three carrier frequencies, \(a\) denotes a linear factor, \(\lambda\) denotes the wavelength, \(N_{e}\) denotes the float phase ambiguity grouped with the constant code and phase hardware delays at both receiver and satellite ends, which is usually considered to be stable when there are no cycle slips, and the bracket describes the effects of multipath and noise errors in the MP combinations. For convenience, one of the carrier phase observations usually adopts the same frequency as the code observations in most practical applications, namely \(i=j\) or \(i=q\).