## 2 Algorithm Principle
### The Principle of Barometric Height Measurement and Its Auxiliary Positioning Algorithm  
#### 2.1.1 The Principle of Barometric Pressure Measurement  
Generally, the information obtained by measuring air pressure can assist the GNSS navigation and positioning system to increase the availability of the corresponding system [22]. The positioning accuracy of GNSS system in the vertical direction is relatively poor, especially in areas with large topographic relief, such as a winding road section or for high building structures, however, with the help of an altimeter, we can measure the relative height of the moving carrier, and effectively improve the accuracy of the system in the elevation direction. Common altimeters include a barometric altimeter and a radar altimeter; here, we use a barometric altimeter for analysis. In practical engineering applications, the barometric altimeter measures the height of the moving carrier, the change of local atmospheric pressure, and the relative height that changes due to geographical changes based on the law of atmospheric pressure changes. We know that the air pressure will decrease as the height increases. According to this law, we can use the following formula to determine the height information of the moving carrier [23]:  
\[H_{k}=\frac{T_{t}}{\
abla_{t}}[(\frac{P_{0}}{P_{s}})^{-(\frac{R_{d}V_{t}}{k_{ 0}})}-1] \tag{1}\]  
where \(T_{t}\) is the surface temperature, \(\
abla_{t}\) = 6.5 \(\times\) 10\({}^{-3}\) K m\({}^{-1}\) is the atmospheric temperature gradient, \(P_{0}\) is the atmospheric pressure at height z, \(P_{s}\) = 1013, 25 hPa, \(R_{d}\) = 278.1 \(J\cdot\text{kg}^{-1}\cdot\text{K}^{-1}\) is the gas constant (under dry air conditions), and \(g_{0}\) = 9.80665 m \(\cdot\text{s}^{-2}\). 2.1.2 Clock Bias Elimination and Altimeter-Assisted Positioning Algorithm Based on Similar Ellipsoid  
(1) Elevation information model based on similar ellipsoid  
According to the WGS84 coordinate system, we regard the earth as an ellipsoid, and we assume that the moving carrier is located on an ellipsoid that coincides with the center of the earth model, the semi-major axis (equatorial radius), semi-minor axis (polar radius), and the radii of the unitary circle coincide, and the eccentricity is equal; we call the earth (small ellipsoid) and the large ellipsoid similar ellipsoids. Suppose that the true position of the ECEF coordinate of the carrier on the large ellipsoid is \((x_{c},y_{c},z_{c})\), and the reference ECEF coordinate position on the earth (small ellipsoid) is \((x^{\prime}_{c},y^{\prime}_{c^{\prime}},z^{\prime}_{c})\); to use the related properties of similar ellipsoids, we first prove that the two ellipsoids are similar. **Definition 1**. : _Ellipsoids with coincident centers, long semi axis (equatorial radius), short semi axis (polar radius), overlapping radius of prime unitary circle and equal eccentricity are similar ellipsoids._  
**Theorem 1**. : _Assume that the semi-major axes of the small ellipsoid (Earth) and the large ellipsoid are: \(R_{e}\), \(R_{E}\), and the semi-minor axes are: \(R_{p}\), \(R_{p}\), and the radii of the circle at the same latitude are: \(R_{S}\), \(R_{L}\), which is shown in Figure 1. If a large ellipsoid is similar to a small ellipsoid, it should have:_  
\[\frac{R_{E}}{R_{e}}=\frac{R_{P}}{R_{p}}=\frac{R_{L}}{R_{S}} \tag{2}\]  
**Proof:** Let the spatial geodetic coordinate system (WGS-84) be attached to the reference ellipsoid. Since the reference ellipsoid is symmetrical around the \(z\)-axis, we can convert the three-dimensional ellipsoid into a two-dimensional plane rectangular coordinate system for analysis.