# Estimating BDS-3 Satellite Differential Code Biases with the Single-Frequency Uncombined PPP Model
## 2 Methods
### Ionospheric Modeling  
The distribution of electron density in the ionosphere is known to be non-uniform. To simplify the study of the ionosphere, a common simplification is made by assuming that all electrons in the ionosphere are concentrated on an infinitesimally thin layer at a specific height. This simplified representation, referred to as the thin-layer ionospheric model, is depicted in Figure 1.  
The point at which the line-of-sight direction from the receiver intersects with the thin layer of the ionosphere is known as the ionospheric penetration point (IPP). The height of the thin layer ionosphere is typically set within the range of 350-450 km in ionospheric modeling. Under the assumption of the thin layer ionospheric model, the mapping function \(MF(z)\) facilitates the conversion of the ionospheric STEC along the line of sight between the satellite and the penetration point into the ionospheric VTEC at the penetration point. This conversion can be represented as follows:  
\[MF(z)=\frac{STEC}{VTEC}=[1-\frac{R_{e}^{2}\cos^{2}E}{\left(R_{e}+H_{ion}\right) ^{2}}]^{-\frac{1}{2}} \tag{13}\]  
where \(R_{e}\) denotes the mean Earth radius, which is set to 6371 km; \(E\) is the elevation angle of the satellite with respect to the receiver; and \(H_{ion}\) represents the ionospheric thin layer height which is taken as 450 km [19; 20].  
The utilization of the generalized triangular series function model enables simulation of the diurnal variation in local ionosphere total electron content (TEC) [12]. By employing this model to represent single-station ionosphere VTEC and applying a zero-mean condition, it becomes possible to separate the ionospheric VTEC and satellite DCBs from the ionospheric observations (Equation (8)). This separation can be resolved by solving the following equation:  
\[\left\{\begin{array}{l}\begin{array}{l}VTEC(\varphi_{IPIP},h_{IPIP})=\sum \limits_{a=0}^{2}\sum\limits_{b=0}^{2}\left[E_{ab}(\varphi_{IPIP}-\varphi_{R})^ {a}\cdot h_{IPIP}{}^{b}\right]+\sum\limits_{k=1}^{4}\left[C_{k}\cos(k\cdot h_{ IPP})+S_{k}\sin(k\cdot h_{IPP})\right]\\ \frac{h_{IPIP}=\frac{2\pi(t_{i}-14)}{24}}{\overline{ION}_{r,1}=\frac{A}{f_{1}^ {2}}\times MF(z)\times VTEC(\varphi_{IPIP},h_{IPP})-\frac{f_{2}^{2}}{f_{1}^{2} -f_{2}^{2}}DCB_{2,1}^{s}+\overline{d}t_{r}(1)\\ \sum\limits_{i=1}^{N}DCB^{i}=0\end{array}\right.\end{array}\right. \tag{14}\]  
In the equation, \(\varphi_{IPP}\) and \(\varphi_{R}\) denote the geomagnetic latitude of the IPP and the receiver, respectively; \(h_{IPP}\) is a function related to the local time \(t_{i}\) of the IPP, \(E_{ab}\), \(C_{k}\) and \(S_{k}\) are the coefficients of the generalized triangular series function model to be estimated, and \(N\) is the total number of satellites observed by the receiver.  
The specific procedure for estimating satellite DCBs using the SF uncombined PPP model is depicted in Figure 2. The process begins with the estimation of ionospheric observables utilizing the SF uncombined PPP approach. This involves using the SF BDS B1 observations, along with external precise ephemeris and clock offset products, as well as the high-precision priori known station coordinates. Subsequently, ionospheric observables are utilized in conjunction with a thin-layer ionosphere model and the zero-mean condition to jointly estimate the satellite DCBs and VTEC parameters.  
Figure 1: Schematic of thin-layer ionosphere model.