## 4 Results and Analyses
### Pedestrian Data in Natural Canyon  
As discussed in Section 3.2, the natural canyon data set includes GNSS data collected using two different receivers, namely an OEM6 and u-blox6. The data collected using these two receivers were integrated separately with the IMU in order to obtain the integrated filters for the three cases: Standard, VB and VB Adaptive filter. Using data from different GNSS receivers further validates the analysis of the performance of the proposed algorithm.  
Firstly, as with the urban canyon data, the range errors were computed using a similar technique as that for the urban data using the GNSS data from the OEM6 receiver. These range errors plotted in Figure 12a show that a significant portion of the data is affected by biased measurements, which at some epochs go as high as nearly 120 m.  
Moreover, a cumulative density function (CDF) of absolute values of the range errors was also plotted along with the CDF of absolute values of the normal fit to the range errors. It can be observed in Figure 12b that there are not only errors with significant magnitude but there is also a high non-conformity between the two CDFs. This indicates that assuming a normal distribution for such errors is likely to lead to a sub-optimal navigation solution.  
The C/N\({}_{\mathrm{o}}\) values are plotted in Figure 13; these values are often below 40 dB. The position errors were computed as described earlier, neglecting the static LOS data at the beginning and end of the canyon. Moreover, the data with very high PDOP values (PDOP \(>\) 10) were also discarded during the position error computation.  
The horizontal and vertical errors obtained using GNSS data from the u-blox6 receiver integrated with the IMU data are shown in Figure 14a,b. As before, the error values obtained for GPS are also included along with the PDOP values.  
Figure 12: (**a**) Range errors in natural canyon (OEM6) (**b**) Range error distribution in natural canyon (OEM6).  
It is observed that there is a significant improvement in accuracy using the proposed scheme as compared to the standard filter. The maximum errors are also found to decrease dramatically with the proposed scheme. The reliability values for the same data set, as depicted in Figure 15, again shows that there is an improvement in the reliability of the navigation solution with the proposed adaptive VB filter.  
The absolute values of position errors and reliability values were re-calculated by processing the GNSS data collected using OEM6 receiver tightly integrated with the IMU data. The obtained results are tabulated in Table 4.  
In this data set, although the accuracy improves only slightly, there is a significant improvement in reliability with the proposed scheme. This again indicates proper characterization of GNSS measurement errors through the use of adaptive \(t\) distribution for the GNSS measurement errors.  
As for the urban canyon data, the availability of integrity information was also computed for the pedestrian data collected in natural canyon. The results tabulated in Table 5 indicate improvement in the availability of integrity information on using the proposed scheme.  
Figure 14: (**a**) Horizontal errors (u-blox6 + IMU)—natural canyon (**b**) Vertical errors (u-blox6 + IMU)—natural canyon.  
Figure 13: C/N\({}_{\mathrm{o}}\) values of measurements in natural canyon (OEM6).