# Performance Analyses of a RAIM Algorithm for Kalman Filter with GPS and NavIC Constellations
## 3 Modified Pseudorange Error Models and \(\mathbf{\
abla}\)  
In this section, first, the adopted error models of the carrier-smoothed pseudorange measurements with different time constants are explained. Following this, it is briefly discussed how they are incorporated in the SKF navigation filter. Finally, necessary modifications of \(\mathbf{\
abla}^{i}_{\text{sub}}\) for satellite \(i\) are presented. It is assumed that the Ionospheric error is eliminated with dual frequency measurements. The following models are adopted for other error sources with dual frequency correction terms, when needed. Residual zenith tropospheric delay after applying corrections of the MOPS model [31] has a latitude- and longitude-dependent bias, as noted in [32]. It can be corrected in the pseudorange measurements using a lookup table as a function of latitude and longitude. The remaining zenith error, \(\underline{\mathbf{\eta}}_{\text{tr}}(i)\), is a zero-mean FOGM process with a standard deviation (\(\sigma_{\text{tr}}\)) of 0.2 m and time constant (\(\tau_{\text{tr}}\)) of 25 h for any visible satellite \(i\). The broadcast ephemeris and satellite clock error, which is overbounded by the user range accuracy (URA), is denoted by \(\underline{\mathbf{\eta}}_{\text{cc}}(i)\). It is modeled as an FOGM process, having a zero mean, unit standard deviation (\(\sigma_{\text{ura}}\)), and 20 h time constant (\(\tau_{\text{ura}}\)). The time between two effectively independent range error samples for satellite clock and orbit error is considered 50 h [33], which is taken as approximately 2.5 \(\times\) the time constant. It is justified in [34] that common errors of code and carrier measurements remain unchanged in the carrier-smoothed code, thus allowing the same models to be used. As multipath and receiver noise are application specific, they are modeled for safety critical aviation applications in this paper. Their standard deviations, denoted as \(\sigma_{\text{ml, }i}\) and \(\sigma_{\text{n, }i}\), respectively, are obtained from [27] for airborne users. In addition, the constant term of the receiver noise standard deviation is inflated further to bound the autocorrelation function of carrier-smoothed pseudoranges due to multipath and noise derived in [35] for airborne users. The inflated value is 0.23 m in place of 0.15 m. Finally, multipath and receiver noise together (\(\underline{\mathbf{\eta}}_{\text{ml-n}}(i)\)) are represented as a zero-mean FOGM process with a time constant (\(\tau_{\text{ml-n}}\)) of 100 s and standard deviation \(\sqrt{\sigma_{\text{ml, }i}^{2}+\sigma_{\text{n, }i}^{2}}\). In the current work, NavIC is considered the second constellation. The same models as those of GPS are adopted for it, but with frequency bands appropriately changed. The approach described in the paper also holds for other constellations with model parameters suitably changed. The SKF navigation filter estimates error in the a priori position, velocity and acceleration of the user in ECEF coordinates, and receiver clock offsets and drifts for GPS and NavIC from respective truths. The states of the aforementioned FOGM processes for each visible satellite are the \"consider parameters\" of the SKF. The zenith tropospheric error is projected onto the line of sight (LOS) using the mapping function of the MOPS model in the measurement model matrix. In the simulation studies, the position error of the SKF is compared with that of an EKF that estimates all FOGM states to depict comparable performance. Since the SKF takes into account the effects of the \"consider parameters\" in its estimates, no additional terms other than those in Equation (14) are needed for PL calculation. With modified error models, the \(a_{\text{min}}\) and \(a_{\text{max}}\) mentioned after Equation (6) for test statistics are re-calculated as follows.