# Performance Analysis of Undifferenced NRTK Considering Time-Varying Characteristics of Atmosphere
## 2 Positioning Model of BDS Undifferenced NRTK
### User Error Correction and Positioning
With  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Type** & **Receiver Errors** & **Satellite Errors** & **Atmospheric Errors** \\ \hline Single reference station & \(dt_{A}\) & \(\delta_{A,j}\) & \(dt^{s}\) & \(\delta_{j}^{s}\) & \(T_{A}^{s}\) & \(\delta_{A}^{s}\) \\ Multiple reference stations & \(\alpha_{A}\cdot dt_{A}+\alpha_{B}\cdot dt_{B}+\alpha_{C}\cdot dt_{C}\) & & \(dt^{s}\) & \(\delta_{j}^{s}\) & \(\alpha_{A}\cdot T_{A}^{s}+\alpha_{B}\cdot T_{B}^{s}+\alpha_{C}\cdot T_{C}^{s}\) \\  & \(\alpha_{A}\cdot\delta_{A,j}+\alpha_{B}\cdot\delta_{B,j}+\alpha_{C}\cdot\delta_ {C,j}\) & & \(\alpha_{A}\cdot\delta_{A,j}^{s}+\alpha_{B}\cdot\delta_{B,j}^{s}+\alpha_{C}\cdot \delta_{C,j}^{s}\) \\ User station & \(dt_{r}\) & \(\delta_{r,j}\) & \(three reference stations fitting the atmospheric error of the user, higher accuracy and faster convergence times can be achieved in long baselines. After the observation errors of the user are corrected, the undifferenced NRTK algorithm only needs a solution for the integer ambiguity of the user. Due to the elimination or significant weakening of the user error, the integer ambiguity resolution of user stations is the same as in conventional RTK. The estimation of user position parameters is mainly achieved by four steps:  
1. The user error is corrected by the UD error correction value of the reference station; the corrected user is not affected by atmospheric delay error and can accurately fix the integer ambiguity. 2. The float solutions of all parameters are solved using the least squares estimation method, and irrelevant parameters are eliminated using the parameter elimination method, leaving only coordinate and ambiguity unknown parameters. The coordinate unknown parameters and ambiguity unknown parameters (using \"\"\(\bullet\)\" symbol representation) and variance covariance matrix by least squares estimation are as follows: \[ \begin{bmatrix}\hat{x}_{k}\\ \Delta\tilde{N}_{k}\end{bmatrix} , \begin{bmatrix}Q_{\hat{x}_{k}}&Q_{\hat{x}_{k }\Delta\tilde{N}_{k}}\\ Q_{\Delta\tilde{N}_{k}\hat{x}_{k}}&Q_{\Delta\tilde{N}_{k}}\end{bmatrix} \] (18)
3. The floating solution \(\left(\Delta\tilde{N}_{k},Q_{\Delta\tilde{N}_{k}}\right)\) is the input, and the integer ambiguity is searched and fixed by the LAMBDA method, and the integer ambiguity value is: \[\Delta\tilde{N}_{k}=\xi(\Delta\tilde{N}_{k})\] (19) where \(\xi\) is the integer mapping function: \((\xi:R^{n}\to Z^{n})\)
4. By using standard least squares adjustment, finally the solution parameters with fixed ambiguity are obtained. The coordinate parameters and the variance covariance matrix with fixed ambiguity are: \[\left\{ \begin{array}{l}\widetilde{x}_{k}=\hat{x}_{k}-Q_{\hat{x}_{k}\Delta \tilde{N}_{k}}Q_{\Delta\tilde{N}_{k}}^{-1}\left(\Delta\tilde{N}_{k}-\Delta \tilde{N}_{k}\right)\\ Q_{\widetilde{x}_{k}}=Q_{\hat{x}_{k}}-Q_{\hat{x}_{k}\Delta\tilde{N}_{k}}Q_{ \Delta\tilde{N}_{k}}^{-1}Q_{\Delta\tilde{N}_{k}\hat{x}_{k}}\end{array} \right.\] (20)  
Ambiguity Fixing of Reference Stations under Constraints of Atmospheric Time-Varying Characteristics  
If the integer ambiguity of the reference station is correctly fixed, so that the correctness and reliability of the error correction information can be guaranteed, then centimeter-level accuracy of the user station can be achieved. The ambiguity fixing of the reference station is the basis for achieving high-accuracy positioning. Therefore, this section mainly studies how to apply appropriate constraints on atmospheric parameters to improve the efficiency of ambiguity fixing.