## 2 Methodology
### Model of INS Measurement  
For inertial navigation, the following three types of coordinate systems are commonly used: the inertial frame (I-frame), the Earth-centered Earth-fixed frame (E-frame), and the body frame (B-frame). In terms of the dead-reckoning algorithm, it can be expressed as follows:  
\[\begin{array}{c}\dot{r}^{e}_{INS}\\ \left(\dot{\dot{v}}^{\prime}_{INS}\right)=\left(\begin{array}{c}C^{e}_{b}f^{b }_{ib}-2\Omega^{e}_{ie}\varphi^{e}_{INS}+g^{e}\\ C^{e}_{b}\Omega^{b}_{eb}\end{array}\right)\\ \end{array} \tag{7}\]  
In Equation (7), the variables \(r^{e}_{INS}\) and \(\
u^{e}_{INS}\) stand for the position and velocity vectors of the INS within the E-frame reference frame. The value \(C^{e}_{INS}\) corresponds to the direction cosine matrix, signifying the transformation from the B-frame to the Earth-fixed reference frame. The term \(f^{b}_{ib}\) signifies the specific force vector experienced by the system in the Earth-fixed reference frame concerning the I-frame reference frame. Additionally, \(\Omega^{e}_{ie}\) represents the skew-symmetric matrix associated with the Earth's rotation vector \(\omega^{e}_{ie}\) in relation to the inertial reference frame, but described within the Earth-fixed frame. The notation \(g^{e}\) designates the gravity vector experienced in the Earth-fixed reference frame.  
The symbol \(\Omega^{b}_{eb}\) is indicative of the skew-symmetric matrix, while \(\omega^{b}_{eb^{\prime}}\) and \(\omega^{b}_{eb}\) can be expressed as follows:  
\[\omega^{b}_{eb}=\omega^{b}_{ei}+\omega^{b}_{ib}=\omega^{b}_{ib}-\omega^{b}_{ ie}=\omega^{b}_{ib}-C^{b}_{e}\omega^{e}_{ie} \tag{8}\]  
where \(\omega^{b}_{ib}\) denotes the gyroscope angular velocity and \(C^{b}_{e}\) denotes the inverse matrix of \(C^{e}_{b}\).  
By using the dynamic equation, the error state model of INS can be expressed as follows [38]:  
\[\begin{array}{c}\dot{\delta r}^{e}_{INS}\\ \left(\dot{\delta\dot{v}}^{\prime}_{INS}\right)=\left(\begin{array}{c}-2 \Omega^{e}_{ie}\delta v^{e}_{INS}+\left[\left(C^{e}_{b}f^{b}_{ib}\right)\times \right]\varphi^{e}-C^{e}_{b}\delta f^{b}_{ib}\\ \dot{\varphi}^{e}\end{array}\right)\\ -\Omega^{e}_{ie}\varphi^{e}+C^{e}_{b}\delta\omega^{b}_{ib}\end{array} \tag{9}\]  
where \(\left[\left(C^{e}_{b}f^{b}_{ib}\right)\times\right]\) denotes the skew-symmetric matrix of \((C^{e}_{b}f^{b}_{ib})\). The errors of the specific force and gyroscope angular velocity can be written as follows:  
\[f^{b}_{ib}=\delta b_{a}+\varepsilon_{a} \tag{10}\]  
\[\omega^{b}_{ib}=\delta b_{g}+\varepsilon_{g} \tag{11}\]  
As a result, the linearized state equation of the INS can be written as follows:  
\[\begin{bmatrix}\dot{\delta r}^{e}_{INS}\\ \dot{\delta v}^{e}_{INS}\\ \dot{\varphi}^{e}\\ \dot{\delta b}_{a}\\ \dot{\delta b}_{g}\end{bmatrix}=\begin{bmatrix}0&I&0&0&0\\ 0&-2\Omega^{e}_{ie}&\left[\left(C^{e}_{b}f^{b}_{ib}\right)\times\right]&-C^{e }_{b}&0\\ 0&0&-\Omega^{e}_{ie}&0&C^{e}_{b}\\ 0&0&0&0\\ 0&0&0&0\end{bmatrix}\begin{bmatrix}\delta r^{e}_{INS}\\ \delta v^{e}_{INS}\\ \varphi^{e}\\ \delta b_{a}\\ \delta b_{g}\end{bmatrix}+\begin{bmatrix}0&0\\ -C^{e}_{b}&0\\ 0&C^{e}_{b}\\ 0&0\\ 0&0\end{bmatrix}\begin{bmatrix}\varepsilon_{a}\\ \varepsilon_{b}\\ 0&0\\ 0&0\end{bmatrix}\begin{bmatrix}\varepsilon_{b}\\ 0&0\\ 0&0\\ 0&  
The ratio test threshold \(T\) is adapted based on the C/N\({}_{0}\) and the observed multipath effects. Equation (13) defines the adaptive threshold \(T\), as follows:  
\[T=k_{0}+k_{1}\cdot\left(\frac{C/N0}{M}\right) \tag{13}\]  
where \(k_{0}\) and \(k_{1}\) are empirical constants, C/\(N_{0}\) is the carrier-to-noise density ratio, and \(M\) is a measure of the multipath effect derived from the signal quality indicators.