# The Potential of LEO Satellite-Based Opportunistic Navigation for High Dynamic Applications
## 2 The Navigation Filter
### Observation Model
#### 2.2.1 Doppler Shift Model  
The Doppler shift measurement is subjected to many error sources. The Multipath error is negligible for a vehicle travelling in an open sky. The Doppler shift induced by the ionosphere delay variation is higher than that of GNSS due to the faster variation of the ionosphere pierce point of the LEO SVs. Nevertheless, simulations we carried out using ionosphere grid maps showed that the ionosphere-induced Doppler shift stays largely below 1 Hz at 1.626 GHz. The tropospheric delay variation is dominated by the fast height variation of the vehicle rather than the elevation variation of the satellite. The usage of a tropospheric delay correction model (for instance, the MOPS [19]) is known to compensate accurately for this error. The expected residual is therefore small enough compared to theDoppler shift measurement noise. In this study, we will ignore the Doppler shift induced by the atmosphere and by the multipath propagation. The \(k\)th-satellite range rate measurement, \(\hat{\rho}^{k}\) (in m/s), is given by the opposite sign of the Doppler frequency shift \(F_{d}\) multiplied by the carrier wavelength \(\lambda\). It can be written as the relative velocity between the receiver's antenna and the transmitter's antenna projected onto the line of sight (LOS) unit vector \(\mathbf{u}^{k}\) between them, plus the receiver clock drift (converted in m/s), \(c\dot{\delta}t\), minus the satellite clock drift, \(c\dot{\delta}t^{k}\) (converted in m/s), and an additive Gaussian thermal noise \(\eta_{\hat{\rho}}\)[20]  
\[\dot{\rho}^{k}=-\lambda F_{d}\cong\left(\mathbf{u}^{k}\right)^{T}\left(\mathbf{v}-\mathbf{ v}^{k}\right)+c\dot{\delta}t-c\dot{\delta}t^{k}+\eta_{\hat{\rho}} \tag{4}\]  
The LOS unit vector \(\mathbf{u}^{k}\) is written as:  
\[\mathbf{u}^{k}=\frac{\mathbf{r}-\mathbf{r}^{k}}{\|\mathbf{r}-\mathbf{r}^{k}\|} \tag{5}\]  
where \(\mathbf{r}^{k}=\mathbf{r}^{k}\left(t-\tau^{k}\right)\) and \(\mathbf{v}^{k}=\mathbf{v}^{k}\left(t-\tau^{k}\right)\) are, respectively, the position and the velocity of the \(k\)th-satellite at the time of burst transmission \(t-\tau^{k}\), where \(t\) is the burst reception time and \(\tau^{k}\) is the propagation delay of the signal between the satellite antenna and the receiver antenna. Usually, if the satellite clock drift \(\delta t^{k}\) is known, it is used to correct the range rate measurement \(\hat{\rho}^{k}\). In our case, it is unavailable. The predicted range rate computed at the burst reception time can be written as:  
\[\hat{\rho}^{k}=\left(\frac{\hat{\mathbf{r}}-\hat{\mathbf{r}}^{k}\left(\hat{t}-\tau^{k} \right)}{\|\hat{\mathbf{r}}-\hat{\mathbf{r}}^{k}\left(\hat{t}-\tau^{k}\right)\|} \right)^{T}\left(\hat{\rho}-\mathbf{v}^{k}\left(\hat{t}-\tau^{k}\right)\right)+c \hat{\delta}t \tag{6}\]  
In (6), we assumed that the reconstructed propagation time is close to the actual propagation time once the filter has converged, i.e., \(\tau^{k}\approx\tau^{k}\). In order to linearize the range rate model, we define the following perturbations of the user position, user velocity, user time and user clock drift, respectively:  
\[ \begin{array}{l}\hat{\mathbf{r}}=\mathbf{r}+\delta\mathbf{r}\\ \hat{\mathbf{v}}=\mathbf{v}+\delta\mathbf{v}\\ \hat{t}=t+\delta t\\ \hat{\delta}t=\dot{\delta}t+\delta\left(\dot{\delta}t\right)\end{array}