## 2 Methods
### INS Model  
In this paper, the mechanization is conducted in the \(e\)-frame (earth-centered earth-fixed frame) for easily integrating the state of INS with the GNSS observables. Then, the dynamic equation of INS can be described as  
\[ \begin{pmatrix}\dot{x}^{e}_{\text{INS}}\\ \dot{\psi}^{e}_{\text{INS}}\\ \dot{C}_{b}\end{pmatrix} = \begin{pmatrix}\mathbf{c}^{e}_{\text{INS}}\\ \mathbf{c}^{e}_{\text{b}}\end{pmatrix} = \begin{pmatrix}\mathbf{c}^{e}_{\text{INS}}\\ \mathbf{c}^{e}_{\text{b}}\end{pmatrix} -2\mathbf{\omega}^{e}_{\text{ie}}\times\mathbf{v}^{ e}_{\text{INS}}+\mathbf{g}^{e}\\ \mathbf{c}^{e}_{\text{b}} \begin{pmatrix}\mathbf{\omega}^{b}_{\text{e}\mathbf{b}}\times \end{pmatrix}  \tag{6}\]  
where \(\mathbf{x}^{e}_{\text{INS}}\) is the position vector in the \(e\)-frame, respectively; \(\mathbf{v}^{e}_{\text{INS}}\) is the velocity vector in the \(e\)-frame; \(\mathbf{C}^{e}_{\text{b}}\) represents the rotation matrix from the \(b\)-frame (body frame) to the \(e\)-frame; \(\mathbf{f}^{b}\) is the specific force vector generated by the accelerometers in the \(b\)-frame; \(\mathbf{\omega}^{e}_{\text{ie}}\) is the earth rotation vector of the \(e\)-frame against the \(i\)-frame (inertial frame) in the \(e\)-frame; \(\mathbf{g}^{e}\) denotes the local gravity vector in the \(e\)-frame; \(\mathbf{\omega}^{b}_{\text{e}\mathbf{b}}\) denotes the rotation rate vector of the \(b\)-frame against the \(e\)-frame projected to the \(b\)-frame; and \([\cdot\times]\) denotes the skew-symmetric matrix. By using the Phi-angle error model, the INS error model can be written as [42]  
\[ \begin{pmatrix}\delta\dot{x}^{e}_{\text{INS}}\\ \delta\dot{v}^{e}_{\text{INS}}\\ \dot{\mathbf{\phi}}\end{pmatrix} = \begin{pmatrix}\delta\mathbf{v}^{e}_{\text{INS}}\\ -2\mathbf{\omega}^{e}_{\text{ie}}\times\delta\mathbf{v}^{e}_{\text{INS}}+\mathbf{C}^{e}_{ \text{b}}\mathbf{f}^{b}\times\mathbf{\phi}+\mathbf{C}^{e}_{\text{b}}\delta\mathbf{f}^{b}+\delta \mathbf{g}^{e}\\ -\mathbf{\omega}^{e}_{\text{ie}}\times\mathbf{\phi}-\mathbf{C}^{e}_{\text{b}}\delta\mathbf{ \omega}^{b}_{\text{b}}\end{pmatrix}  \tag{7}\]  
in which \(\mathbf{\phi}\) indicates the correction vector of attitude; \(\delta\mathbf{g}^{e}\) represents the gravity error in the e-frame; and \(\delta\mathbf{f}^{b}\) and \(\delta\mathbf{\omega}^{b}_{\text{i}\mathbf{b}}\) are the sensor errors of the accelerometer and gyroscope, respectively. Bias and scale factor errors along with white noise can be used to model the sensor error [42], which can be expressed as  
\[ \begin{cases}\delta\dot{\mathbf{f}}^{b}=\mathbf{B}_{a}+diag\begin{pmatrix}\mathbf{f}^{b} \end{pmatrix}\mathbf{S}_{a}+\mathbf{w}_{\mathbf{v}}\\ \delta\mathbf{\omega}^{b}_{\text{i}\mathbf{b}}=\mathbf{B}_{g}+diag\begin{pmatrix}\mathbf{\omega }^{b}_{\text{i}\mathbf{b}}\end{pmatrix}\mathbf{S}_{g}+\mathbf{w}_{\mathbf{\phi}}\end{cases}  \tag{8}\]  
in which, \(\mathbf{B}_{a}\) and \(\mathbf{S}_{a}\) indicate the bias and scale factor errors of the accelerometer, respectively; \(diag\) denotes the diagonal matrix; \(\mathbf{B}_{g}\) and \(\mathbf{S}_{g}\) indicate the bias and scale factor errors of the gyroscope, respectively; and \(\mathbf{w}_{\mathbf{v}}\) and \(\mathbf{w}_{\mathbf{\phi}}\) indicate the corresponding random white noise. Bias and scale factor errors can be modeled as first-order Gauss-Markov processes and expressed as [42]  
\[ \begin{cases}\begin{pmatrix}\dot{\mathbf{B}}_{a}\\ \dot{\mathbf{B}}_{g}\end{pmatrix}=\begin{pmatrix}\frac{-1}{\tau_{b}}\mathbf{B}_{a}\\ \frac{-1}{\tau_{b}}\mathbf{B}_{g}\end{pmatrix}+\begin{pmatrix}\mathbf{w}_{ba}\\ \mathbf{w}_{bg}\end{pmatrix}\\ \begin{pmatrix}\dot{\mathbf{S}}_{a}\\ \dot{\mathbf{S}}_{g}\end{pmatrix}=\begin{pmatrix}\frac{-1}{\tau_{a}}\mathbf{S}_{a}\\ \frac{-1}{\tau_{a}}\mathbf{S}_{g}\end{pmatrix}+\begin{pmatrix}\mathbf{w}_{sa}\\ \mathbf{w}_{bg}\end{pmatrix}\end{cases}  \tag{9}\]  
where \(\tau_{(\bullet)}\) and \(\mathbf{w}_{(\bullet)}\) (denotes the subscript \(ba\), \(bg\), \(sa\), or \(sg\)) denote the corresponding correlation time and driving white noise, respectively.