## 2 Algorithm Principle
### The Principle of Barometric Height Measurement and Its Auxiliary Positioning Algorithm
#### 2.1.1 The Principle of Barometric Pressure Measurement
The improvement principle of VDOP by elimination of clock bias  
When the clock bias between the receiver on the moving carrier and the LEO satellites is eliminated by exploiting the two-way communication link [6], we assume that the coordinates of each LEO satellite in the ECEF coordinate system are \((x_{i},y_{i},z_{i})\), and \(\rho_{iLEO}=\sqrt{(x_{i}-x_{c})^{2}+(y_{i}-y_{c})^{2}+(z_{i}-z_{c})^{2}}\) represents the pseudorange measurement value to the \(i\)-th satellite. We omit the complicated derivation and directly give the corresponding geometric matrix as follows:  
\[H^{\ddagger}=\left[ \begin{array}{ccc}\frac{\partial\rho_{iLEO}}{\partial x_{ c}}&\frac{\partial\rho_{iLEO}}{\partial y_{c}}&\frac{\partial\rho_{iLEO}}{ \partial z_{c}}\\ \frac{\partial\rho_{iLEO}}{\partial x_{c}}&\frac{\partial\rho_{iLEO}}{\partial y _{c}}&\frac{\partial\rho_{iLEO}}{\partial z_{c}}\\ \frac{\partial\rho_{iLEO}}{\partial x_{c}}&\frac{\partial\rho_{iLEO}}{\partial y _{c}}&\frac{\partial\rho_{iLEO}}{\partial z_{c}}\\ \end{array} \right], \tag{12}\]  
where there are:  
\[\left\{ \begin{array}{c}\frac{\partial\rho_{iLEO}}{\partial x_{c}}=\frac{(x_ {c}-x_{i})}{\rho_{iLEO}}|x_{c}=x_{\Theta}\\ \frac{\partial\rho_{iLEO}}{\partial y_{c}}=\frac{(y_{c}-y_{i})}{\rho_{iLEO}}|y _{c}=y_{\Theta}\\ \frac{\partial\rho_{iLEO}}{\partial z_{c}}=\frac{(x_{c}-z_{i})}{\rho_{iLEO}}| z_{c}=z_{\Theta}\\ \end{array} \right., \tag{13}\]  
and where \((x_{\Theta},y_{\Theta},z_{\Theta})\) is the nominal position solution of \((x_{c},y_{c},z_{c})\). The covariance matrix of the position becomes:  
\[\text{cov}(Pos)=\left[ \begin{array}{ccc}\sigma_{x_{c}}^{2}&\sigma_{x_{c}y_{ c}}^{2}&\sigma_{x_{c}z_{c}}^{2}\\ \sigma_{x_{c}y_{c}}^{2}&\sigma_{y_{c}}^{2}&\sigma_{y_{c}z_{c}}^{2}\\ \sigma_{x_{c}z_{c}}^{2}&\sigma_{y_{c}z_{c}}^{2}&\sigma_{z_{c}}^{2}\\ \end{array} \right]. \tag{14}\]Let \(H^{\Xi}=\left[\left(H^{\dagger}\right)^{T}H^{\dagger}\right]^{-1}\), then:  
\[\text{cov}(Pos)=\left[ \begin{array}{ccc}H_{xx}^{\Xi}&H_{xy}^{\Xi}&H_{xz}^{\Xi} \\ H_{xy}^{\Xi}&H_{yy}^{\Xi}&H_{yz}^{\Xi}\\ H_{xz}^{\Xi}&H_{yz}^{\Xi}&H_{zz}^{\Xi}\end{array} \right]\sigma_{URE}^{2}, \tag{15}\]  
where \(\sigma_{URE}^{2}\) is the variance of the user ranging error (URE), and \(\sigma_{x,}^{2}\), \(\sigma_{y,}^{2}\), and \(\sigma_{z}^{2}\) are the variances of the corresponding components of each positioning error, respectively. Finally, VDOP can be expressed as:  
\[VDOP=\frac{\sigma_{z}^{2}}{\sigma_{URE}^{2}}=\sqrt{H_{zz}^{\Xi}}. \tag{16}\]  
It can be seen that when the clock bias is eliminated, the covariance matrix is independent of the clock state; at this time, VDOP will become smaller, since the precision factor can be regarded as a linear mapping from the measurement error in the observation to the state estimation error. In the case of the same measurement error, a smaller precision factor will make the error of the state estimation smaller; this is important for improving the VDOP value, because the receiver cannot receive evenly distributed signals from the satellite at the zenith position and it is impossible to receive signals from the other side of the earth.