## 3 RMNCE-Based Satellite Selection
### RMNCE Based Method  
As described in Section 3.1, the RMNCE is employed to estimate the measurement noise variances of the pseudo-range and pseudo-range rate. Assuming that the number of satellites at epoch \(k\) is \(S_{N}\), the estimate of \(\mathbf{R}_{k}\) can be expressed in the following form:  
\[\mathbf{\hat{R}}_{k}^{RMNCE}=diag\Big{\{}\partial_{\rho,1}^{2},\partial_{\rho, 2}^{2},\cdots,\partial_{\rho,S_{N}}^{2},\partial_{\rho,1}^{2},\partial_{\rho, 2}^{2},\cdots,\partial_{\rho,S_{N}}^{2}\Big{\}} \tag{18}\]  
where \(\hat{\sigma}_{\rho,l}^{2}\) and \(\hat{\sigma}_{\rho,l}^{2}\) respectively denote the estimated variance of the pseudo-range and pseudo-range rate measurement noise of the \(l\)-th satellite.  
Since pseudo range can be measured from INS and GNSS simultaneously, the GNSS measurement noises are able to be evaluated by RMNCE without the interference of state estimation error coming from IAE or RAE. At each epoch \(\hat{\sigma}_{\rho}^{2}(k)\) is updated, then the selection operations can be expressed as follows:  
\[\Xi_{1}=\Big{\{}S_{l}\Big{|}\left|\rho_{GNSS}^{l}(k)\right.-\left.\rho_{INSS}^ {l}(k)\right|\leq\left.\Delta\rho_{thrd}\text{ and }\hat{\sigma}_{S_{l}}^{2}(k)\right.\leq \left.\hat{\sigma}_{thd}^{2},S_{l}\in S(k)\right\} \tag{19}\]  
\[\mathbf{W}=\left\{w_{l}\middle|\frac{1}{\hat{\sigma}_{S_{l}}^{2}},S_{l}\in\Xi_ {1}\right.\right\} \tag{20}\]  
\[\Xi_{2}=\left\{S_{l}^{\prime}|\operatorname{GDOP}_{W}=\min,S_{l}^{\prime}\in \Xi_{1}\right\} \tag{21}\]  
where \(S(k)\) is the set of observable satellites at epoch \(k\), \(\Delta\rho_{thrd}\) is the threshold of pseudo range difference between INS and GPS, \(\hat{\sigma}_{thd}^{2}\) is the threshold of variance, \(\Xi_{1}\) is the candidate set discarding the abnormal observations, and \(\Xi_{2}\) is the selection set.  
In Equation (19), the pseudo range difference between INS and GNSS is taken as an index to reject abnormal pseudo-range errors mainly caused by the multipath effect [37]. However, this index is biased because it is affected by the INS positioning error, which may cause the wrong detection. To cope with this problem, as shown in Equation (19), satellite measurement noise variances estimated by RMNCE are considered with the pseudo range difference together to detect the abnormal measurements. Moreover, as illustrated in Equations (20) and (21), the weighting matrix \(\mathbf{W}\) which is calculated by the estimated noise variances \(\left\{\hat{\sigma}_{S_{l}}^{2}\right\}\) is used to derive the weighted GDOP; thus, both satellite geometry and measurement noise characteristic have been taken into consideration in the satellite selection procedure.  
Besides, five satellites, when available, are selected to improve the positive accuracy in this research. In order to speed up processing, all the observable satellites are sorted in descending order according to elevation angle, and the first two as mentioned in [33] are selected. Then the remained satellites can be determined according to GDOP\({}_{W}\). The main procedure of the proposed satellite selection algorithm is shown in Figure 1.  
**Remark 1**.: _In Equation (15), the receiver clock offset delay is unimportant when only implementing the RMNCE method, because this term cancels in the SOMD. However, the satellite exclusion step described by Equation (19) requires that \(\rho_{GNSS}\) and \(\rho_{INS}\) should be comparable. Hence, the compensated \(\rho_{GNSS}\) not the direct measurement is used in Equation (19)._