## 2 A Novel Filtering Strategy for Reliable Navigation
### Adaptation of GNSS Covariance to the Changing Signal Conditions
Firstly, the Bhattacharyya distance (DB) for two univariate normal acceleration distributions is given as:  
\[BD=\frac{1}{4}\frac{(a_{IMU}-a_{GNSS})^{2}}{\sigma_{IMU}^{2}+\sigma_{GNSS}^{2 }}+\frac{1}{2}log\left(\frac{\sigma_{IMU}^{2}+\sigma_{GNSS}^{2}}{2\sqrt{ \sigma_{IMU}^{2}\sigma_{GNSS}^{2}}}\right) \tag{2}\]  
Then, \(BC\) is obtained as:  
\[BC=e^{-BD}*100\% \tag{3}\]  
Figure 1: Block diagram of adaptive scheme. Figure 2: Consistency between accelerations. Generally, in open sky conditions, the consistency between these two accelerations is high. However, in areas where the GNSS measurement quality degrades, the consistency between the two accelerations decreases since the errors in the GNSS derived acceleration distributions are no longer centered on zero. Thus, in areas with a good consistency, a higher weight (lower variance) is assigned to GNSS measurement updates while in areas with poor consistency, a lower weight is used. Then a scale factor based on the consistency between the accelerations is computed, which in turn is used to scale the _a priori_ variance as shown in the block diagram of Figure 1. Figure 3 shows the proposed algorithm flowchart to select an adaptive _a priori_ variance for GNSS measurements based on the consistency of the acceleration values obtained using the two sensors. As with other non-adaptive filters, initially an assumption is made about the _a priori_ variance for the GNSS measurements. The goal of the above algorithm is then to find a value to scale this _a priori_ variance so that the assumed measurement noise distribution closely matches the true error characteristics. As mentioned in Appendix A, the condition of the minimum number of Doppler measurements required for obtaining the user acceleration must however be met before the consistency between the accelerations can be computed. The method described in Appendix A can be considered a loosely coupled consistency check as the comparison is done after the user accelerations are made available by the two systems. However, the comparison could also be done using a tightly coupled approach. With the knowledge of the orientation of the IMU and the direction cosines of the satellites, the accelerations on the IMU could be mapped to the line of sight acceleration from the Doppler derivative of each satellite. This method would obviate the condition of having a certain minimum number of Doppler measurements. Figure 3: GNSS _a priori_ variance computation. In order to mitigate the effect of noise in the Doppler measurements, the consistency is computed as a moving average of the consistency values at three GNSS epochs, each two typically separated by 50 ms. A buffer to store the consistency values of three consecutive epochs is thus maintained. This introduces a latency of two epochs at the beginning before the _a priori_ variance scale factor can be computed. Moreover, if enough Doppler measurements are not available somewhere during the processing, then the buffer is reset as it has to hold the consistency values of three consecutive epochs. Once the buffer is full, the current value of consistency is repeatedly computed by scaling the initial _a priori_ variance with several _a priori_ variance scale factors. The final _a priori_ variance is chosen such that the moving average of the consistency values is maximized.