# An Efficient BDS-3 Long-Range Undifferenced Network RTK Positioning Algorithm
## 2 Long-Range URTK Algorithm
### Reference Station Network Double-Difference Integer AR
Although the double-difference observation equation can eliminate most of the errors (e.g., satellite and receiver clock errors, orbital errors, hardware delays at the satellite and receiver, and observation noise), for long-range RTK, if reasonable measures are not used to process the atmospheric delay, the fixed speed and correctness of the double-difference ambiguity will be significantly affected. Therefore, in order to prevent the rank defect problem caused by the estimation of atmospheric delay parameters in each epoch, and to consider the time-varying property of atmospheric delay, this study processes the ionospheric and tropospheric parameters using a random walk model-based method, considering the constraint between epochs. The random walk process used for the troposphere and ionosphere is defined as follows [30; 31]:  
\[ \begin{cases}RZTD_{t}-RZTD_{t+1}=\omega_{T},\omega_{T}\sim N(0,\sigma_{T}^{2} )\\ I_{t}-I_{t+1}=\omega_{I},\omega_{I}\sim N(0,\sigma_{I}^{2})\end{cases}  \tag{5}\]  
where \(RZTD_{t}\) and \(RZTD_{t+1}\) are the relative zenith tropospheric delays of epochs \(t\) and \((t+1)\), respectively; \(I_{t}\) and \(I_{t+1}\) are the ionospheric delays of the two epochs \(t\) and \((t+1)\), respectively; \(\omega_{T}\) and \(\omega_{I}\) represent the zenith tropospheric and ionospheric parameters of the epoch difference, respectively. Equation (5) indicates that the atmospheric parameter variations at different observation times can be expressed with a random walk process. The zenith tropospheric parameters satisfy a normal distribution, with a mean of zero and a variance of \(\sigma_{T}^{2}\). In addition, the ionospheric parameters also satisfy a normal distribution, with a mean value of zero and a variance of \(\sigma_{T}^{2}\). As long as an appropriate variance can be obtained, the corresponding constraints on atmospheric parameters can be imposed. In this study, the tropospheric delay power spectral density is \(1\) cm\(/\sqrt{h}\), and the ionospheric delay power spectral density is \(1\) m\(/\sqrt{h}\)[32]. After the power spectral density is given, the corresponding variance can be calculated, and the weight of the constraint equation can be calculated using the variance. The specific expression of the virtual observation equation of the atmospheric parameter random walk process is as follows:  
\[\left\{ \begin{aligned} v_{T}=RZTD_{t}-RZTD_{t+1},\beta_{T}& =\frac{\sigma_{0}^{2}}{\sigma_{T}^{2}}=\frac{\sigma_{0}^{2}}{\sigma_{T}^{2} \cdot\Delta t}\\ v_{I}=I_{t}-I_{t+1},\beta_{I}&=\frac{\sigma_{0}^{2}}{ \sigma_{T}^{2}}=\frac{\sigma_{0}^{2}}{\sigma_{T}^{2}\cdot\Delta t}\end{aligned} \right. \tag{6}\]  
where \(\beta_{T}\) and \(\beta_{I}\) represent the weights of the tropospheric delay and ionospheric current  
ened only through constraining the B1I or B1C frequency between epochs. Therefore, the multi-frequency data of the BDS are beneficial to the constraint and solution of ionospheric parameters. The observation equation and constraint conditions are combined, and a parameter elimination method is used in order to eliminate the parameters of the normal equation. The least squares (LSQ) algorithm is employed in order to solve the unknown parameters of Equation (7). Further, the least square ambiguity decorrelation adjustment (LAMBDA) method is applied to the AR. Finally, the double-difference integer ambiguity algebraic sum between the closed reference stations of the same satellite is zero, and \(ratio>3\) is used to test the integer ambiguity results, which improves the reliability of the integer AR results.