# Performance Analyses of a RAIM Algorithm for Kalman Filter with GPS and NavIC Constellations
## 4 \(\boldsymbol{\
### Uncertain \(a_{\ell q}\)  
Uncertainties in the knowledge of \(a_{\ell q}\) arise because actual values of the parameters needed to calculate it are not known. Suppose the actual value of \(\tau_{\text{tr}}\) for a satellite is \(\varrho_{\text{tr}}\) times 25 h (modeled value), where \(\varrho_{\text{tr, min}}\ \leq\ \varrho_{\text{tr}}\ \leq\ 1\). Similarly, the scale factor for \(\tau_{\text{un}}\) is \(\varrho_{\text{un}}\), which lies between \(\varrho_{\text{un, min}}\) and one. Actual \(\sigma_{\text{tr}}^{2}\), \(\sigma_{\text{un}}^{2}\) and \((\sigma_{\text{ml}}^{2}+\sigma_{\text{n}}^{2})\) also take on values below their respective modeled values. Their actual values can be obtained by multiplying scale factors \(\mathcal{K}_{\text{tr}}\), \(\mathcal{K}_{\text{un}}\), and \(\mathcal{K}_{\text{ml-n}}\), respectively, to the respective modeled values. The lower limits of these scale factors are \(\mathcal{K}_{\text{tr, min}}\), \(\mathcal{K}_{\text{un, min}}\), and \(\mathcal{K}_{\text{ml-n, min}}\), respectively, and the upper limits are one. All lower limits should be determined by prior analysis. A possible approach would be to analyze several months of data, following [32; 33] for the tropospheric error and URA, respectively. The autocorrelation functions, power spectral densities, and error distributions of the data can be formed to determine both upper and lower bounds of the time constants and variances. The ratios of the lower to upper bounds will give the lower limits of the scale factors, and the upper bounds will be the modeled values discussed earlier. Although not considered in this paper, modeled values may change across satellites. Similar analyses may be done for multipath and receiver noise, but this will be application specific. Equation (17) is now re-written as  
\[ \begin{array}{ll}E\Big{(}\underline{\boldsymbol{\eta}}^{(\ell)}(i)\underline{ \boldsymbol{\eta}}^{(\ell+q)}(i)\Big{)}&=(\sigma^{(\ell)}(i))^{2}\exp^{(- \beta q\Delta t)}=\text{mp}^{(\ell)}\big{(}\theta^{(\ell)}(i)\big{)}\text{mp} ^{(\ell+q)}\big{(}\theta^{(\ell+q)}(i)\big{)}\mathcal{K}_{\text{tr}}\sigma_{ \text{tr}}^{2}\exp^{(-q\Delta t/(\sigma_{\text{tr}}\cdot\boldsymbol{\eta}_{ \text{tr}}))}+\\ &\mathcal{K}_{\text{ura}}\sigma_{\text{ura}}^{2}\exp^{(-q\Delta t/(\sigma_{ \text{ura}}\cdot\boldsymbol{\eta}_{\text{tra}}))}+\mathcal{K}_{\text{ml-n}}(( \sigma_{\text{ml, }i}^{(\ell)})^{2}+(\sigma_{\text{n, }i}^{\ell})^{2})\exp^{(-q\Delta t/\tau_{\text{ml-n}})}\end{array}  \tag{22}\]  
In this work, no uncertainty in the time constant of multipath and noise is assumed. However, this can be relaxed if needed. Using Equation (22), modified \(\beta_{\ell q}\), \(\beta_{\ell q}^{\text{mod}}\) is  
\[\beta_{\ell q}^{\text{mod}}=-\log\Bigg{(}\frac{E\Big{(}\underline{\boldsymbol{ \eta}}^{(\ell)}(i)\underline{\boldsymbol{\eta}}^{(\ell+q)}(i)\Big{)}}{\Big{(} \sigma^{(\ell)}(i)\Big{)}^{2}}\Bigg{)}\Big{/}(q\Delta t) \tag{23}\]  
Comparing Equations (17) and (22), it is evident that \(\exp^{(-\beta_{\ell q}^{\text{mod}}q\Delta t)}\leq\exp^{(-\beta_{\ell q}q \Delta t)}\).