## 2 Materials and Methods
### Robust Extended Kalman Filter Algorithm  
The solution of the standard Kalman filtering algorithm will satisfy the minimum unbiased variance only when the observation model and the state model are comparatively accurate, and the noise of the observation signal and the state noise satisfy Gaussian white noise, are uncorrelated with each other, and the mean value is zero. However, when the observation environment is in a more complex situation, it is usually difficult to meet the assumptions [18; 19]. For example, when there is a gross error in the observation data, or there is a missed period jump, the observation information will be contaminated, which will lead to jumping points in the localization results, and even malfunction of the filter and the phenomenon of filter divergence; therefore, the accuracy, as well as the reliability of the results, cannot be guaranteed [20; 21]. At this time, if the mean estimation based on the anti-divergence Kalman filter is applied for the solution, it detects the gross errors according to the observation residuals and downscales the observations with excessive residuals. This achieves the anti-divergence effect and then effectively suppresses the influence of the gross errors on the results, improves the reliability of the results, and guarantees its solution accuracy. The difference-resistant EKF introduces robust estimation theory into the parity model, which ensures that the valuation is near-optimal when the actual model matches the prior hypothesis model, ensures the valuation is less affected by the coarse difference when the actual model slightly deviates from the prior hypothesis model, and ensures that the estimate will not change significantly when the actual model deviates from the hypothesis model by a large margin [22]. The more commonly used robust estimation is the robust mean estimation, and the anti-variance extended Kalman filter introduces the IGG III model from the robust M estimation into the extended Kalman filter processing, where the IGG III model divides the observed residuals into three segments, constructs the anti-variance equivalent weights according to the adaptive downscaling factor, \(r_{i}\) and uses iterative computation to obtain the anti-variance solution [23]. The adaptive downscaling factor \(r_{i}\) in IGG III model is:  
\[r_{i}=\left\{ \begin{array}{ccc}1&,&|\overline{v}_{i}|\leq k_{1}\\ \frac{k_{1}}{|\overline{v}_{i}|}\left(\frac{k_{2}-|\overline{v}_{i}|}{k_{2}- k_{1}}\right)^{2}&,&k_{1}<|\overline{v}_{i}|<k_{2}\\ 0&,&|\overline{v}_{i}|\geq k_{2}\end{array} \right. \tag{13}\]  
where \(k_{1}=1.5\sim 2.5\), \(k_{2}=3.0\sim 5.0\), \(\overline{v}_{i}\) are the standardized residuals of the observations, and the calculation formula is:  
\[\overline{v}_{i}=\frac{V_{i}}{std(V)} \tag{14}\]  
where \(std()\) is the standard deviation function, and \(V\) is the vector of observed residuals. After obtaining the adaptive downscaling factors, the equivalent variance and covariance of the observations can be obtained as follows:  
\[\left\{ \begin{array}{ccc}\overline{Q}_{i,i}=\frac{1}{r_{ij}}Q_{i,i}\\ \overline{Q}_{ij}=\frac{1}{r_{ij}}Q_{ij}\\ \overline{Q}_{i,j}=\frac{1}{\sqrt{r_{i}r_{j}}}Q_{ij}\end{array} \right. \tag{15}\]  
where \(Q_{i,i}\) and \(Q_{ij,j}\) represent the variances of observations \(i\) and \(j\), and their equivalent variances are \(\overline{Q}_{i,i}\) and \(\overline{Q}_{j,i}\), respectively. \(Q_{i,j}\) represents the covariance of observations \(i\) and \(j\), and its equivalent covariance is \(\overline{Q}_{i,j}\).