## 2 Problem Statement and Preliminaries  
The underwater positioning error model is denoted in Equation (1) [37]. In this paper, the positions calculated by high-accuracy reference navigation system are reference information. \[\left\{ \begin{array}{l}\delta\dot{L}=\frac{1}{{{R_{M}}+h}}\delta V_{N}-\frac {{V_{N}}}{{{({R_{M}}+h)}^{2}}}\delta h+\omega_{L}\\ \delta\dot{\lambda}=\frac{{\delta V_{\text{F}}\sec L}}{{{R_{N}}+h}}+\frac{{{V_ {\text{F}}\sec L}\tan L}}{{{R_{N}}+h}}\delta L+\frac{{{V_{\text{F}}\sec L}}}{{{ ({R_{N}}+h)}^{2}}}\delta h\\ \delta h=\delta V_{U}+\omega_{h}\end{array} \right. \tag{1}\]  
where \(\lambda\), \(L\), and \(h\) indicate the longitude, latitude, and height of the AUV, respectively; \(V_{N}\), \(V_{E}\), and \(V_{U}\) are the velocities of the AUV along the north, east, and upward directions, respectively; \(\delta L=L_{INS}-L_{\text{Ref}}\), \(\delta\lambda=\lambda_{INS}-\lambda_{\text{Ref}}\), and \(\delta h=h_{INS}-h_{\text{Ref}}\) are the latitude, longitude, and height errors, respectively; \(L_{INS}\) and \(L_{\text{Ref}}\) denote the latitudes of the AUV measured by the INS and the reference system, respectively; \(\lambda_{INS}\) and \(\lambda_{\text{Ref}}\) are the longitudes of the AUV measured by the INS and the reference system, respectively; \(h_{INS}\) and \(h_{\text{Ref}}\) denote the heights of the AUV measured by the INS and the reference system, respectively; \(\delta V_{E}=V_{E}^{INS}-V_{E}^{\text{Ref}}\), \(\delta V_{N}=V_{N}^{INS}-V_{N}^{\text{Ref}}\), and \(\delta V_{U}=V_{U}^{INS}-V_{U}^{\text{Ref}}\) are the velocity errors in three directions; \(V_{E}^{INS}\) and \(V_{E}^{\text{Ref}}\) are the east velocities of the AUV measured by the INS and the reference system, respectively; \(V_{N}^{INS}\) and \(V_{N}^{\text{Ref}}\) are the north velocities of the AUV measured by the INS and the reference system, respectively; \(V_{U}^{INS}\) and \(V_{U}^{\text{Ref}}\) are the upward velocities of the AUV measured by the INS and the reference system, respectively; \(R_{N}\) and \(R_{M}\) are the long and short axis radii of the earth; \(\omega_{L}\), \(\omega_{\lambda}\), and \(\omega_{h}\) are the process noises on three axes. Equation (1) can be rewritten as the state-space model, shown in Equation (2). \[\left[ \begin{array}{l}\delta\dot{L}\\ \delta\dot{\lambda}\\ \delta h\end{array} \right]=\mathbf{FX}+\mathbf{u}+\boldsymbol{\omega}=\left[ \begin{array}{ccc}0&0&-\frac{V_{N}^{\text{IS}}}{{{({R_{M}}+h)}^{2}}}\\ \frac{V_{E}^{\text{ref}}\sec L}{{{R_{N}}+h}}&0&\frac{V_{E}^{\text{ref}}\sec L}{{ {({R_{N}}+h)}^{2}}}\delta h\\ 0  
where \(\mathbf{X}=\left[ \begin{array}{cc}\delta L&\delta\lambda&\delta h\end{array}  \right]^{T}\) is the state vector; \(\mathbf{u}=\left[ \begin{array}{cc}\frac{\delta V_{N}}{R_{M}+\hbar}&\frac{ \delta V_{F}\sec L}{R_{N}+\hbar}&\delta V_{U}\end{array} \right]^{T}\) is the input information; \(\mathbf{\omega}=\left[ \begin{array}{cc}\omega_{L}&\omega_{\lambda}&\omega_{h }\end{array} \right]^{T}\) is the process noise; \(\mathbf{Q}\) is the process noise covariance matrix; the process noise is assumed to follow the Gaussian distribution with zero means and the covariance matrix \(\mathbf{Q}\); \(k\) is the time of \(k\). The state transition matrix \(\mathbf{F}_{k|k-1}\) is approximated as follows:  
\[\mathbf{F}_{k|k-1}\approx\mathbf{I}_{n}+\mathbf{F}\Delta t \tag{3}\]  
where \(n\) denotes the dimension of the state vector; \(\mathbf{I}_{n}\) is the identity matrix of the \(n\) dimensions; \(\Delta t\) is the time interval. The measurement function is shown as follows:  
\[\mathbf{Z}=\mathbf{H}\mathbf{X}+\mathbf{v}=\left[ \begin{array}{cc}L^{INS}-L ^{\text{Ref}}\\ \lambda^{INS}-\lambda^{\text{Ref}}\\ h^{INS}-h^{\text{Ref}}\end{array} \right]=\left[ \begin{array}{cc}1&0&0\\ 0&1&0\\ 0&0&1\end{array} \right]\left[ \begin{array}{c}\delta L\\ \delta\lambda\\ \delta h\end{array} \right]+\left[ \begin{array}{c}\text{v}_{L}\\ \text{v}_{\lambda}\\ \text{v}_{h}\end{array}