## 2 A Novel Filtering Strategy for Reliable Navigation
### Adaptation of GNSS Covariance to the Changing Signal Conditions  
The navigation filters used in personal navigation devices usually make an _a priori_ assumption of the measurement covariance of the sensors being used. For a relatively good MEMS IMU like Analog Device's ADIS16488 unit (Analog Devices, Inc., Norwood, MA, USA), a tactical grade MEMS IMU, the stochastic part of the measurement errors can be considered constant. However, the same may not apply for the case of GNSS measurements. For instance, the measurement covariance of non-line of sight (NLOS) GNSS measurements is most likely to increase as the user moves into a deep canyon. Hence, a fixed _a priori_ covariance for GNSS measurements in harsh environments will not correctly represent the true error characteristics of GNSS measurements. Hence, some adaptive technique must be used to adaptively tune the measurement covariance of GNSS measurements. Several techniques have been proposed in the literature to adapt the GNSS measurement covariance to the changing signal conditions. These techniques however come with several drawbacks, including filter instability, divergence, computational complexity or poor accuracy [7,8]. Hence, in order to adapt the GNSS measurement variance with the changing signal conditions and considering the drawbacks of the prevalent adaptive techniques, a new technique initially proposed by [9] is used in this work. This method scales the _a priori_ variance of the GNSS measurements before they are fed to the navigation filter and hence avoids various drawbacks of the earlier adaptive techniques. The proposed method computes the user acceleration from GNSS Doppler measurements and compares its consistency with measurements from accelerometers in the IMU. With a tactical grade MEMS IMU, the user acceleration obtained using accelerometers is fairly accurate and hence is taken as a reference to compare with that obtained with GNSS Doppler measurements. Thus, during good GNSS signal conditions, the two accelerations are highly consistent whereas in GNSS signal degraded environments, the consistency between the two accelerations degrades. Based on this characteristic, a scaling factor is computed to scale the covariance of GNSS measurements thus adapting it in accordance with the changing GNSS signal conditions. A top level block diagram of the proposed method is depicted in Figure 1.  
Firstly, the user accelerations are computed along three axes of the Earth Centered Earth Fixed (ECEF) frame using the GNSS Doppler measurements. The details on the computation of user acceleration from GNSS are given in Appendix A. The user accelerations are also detected by the accelerometers inside the IMU. These accelerations, however, are in the IMU sensor frame. Thus they are transformed into the ECEF frame using rotation matrices so that they can be directly compared with the accelerations obtained using GNSS measurements. The errors associated with each solution are assumed to have a normal distribution. In the case when the GNSS measurement errors are assumed to have \(t\) distributed errors, the GNSS variance adaptive part, however, still assumes normal distribution for Doppler measurements while computing user acceleration from Doppler measurements. If \(\alpha_{IMU}\) and \(\alpha_{GNSS}\) are the acceleration values along a certain axis obtained using the accelerometer and GNSS respectively with associated standard deviations \(\sigma_{IMU}\) and \(\sigma_{GNSS}\), then the consistency between the two accelerations can be expressed as the area overlap between the two acceleration distributions as shown in Figure 2.  
The overlap between the distributions can be calculated as a Bhattacharyya coefficient (BC) [10].