## 3 Switched Variational Estimation Filtering in Underwater Navigation
### Proposed Optimal Variational Estimation in Navigation Information Acquisition
Set \(\kappa=\xi(t_{k})\); the logarithm of \(q(\kappa)\) in iteration \(l\) +1 is expressed as:\[\log q^{(l+1)}(\kappa)= -\tfrac{1}{2}\bigg{\{}\xi^{(l)}(t_{k})\Big{\{}\mathbf{Z}(t_{k})-h \Big{[}\mathbf{X}^{(l+1)}(t_{k})\Big{]}\Big{\}}^{T}\times\mathrm{E}^{(l+1)} \Big{(}\mathbf{R}_{k}^{-1}\Big{)}\Big{\{}\mathbf{Z}(t_{k})-h\Big{[}\mathbf{X}^{ (l+1)}(t_{k})\Big{]}\Big{\}}\Big{\}} \tag{43}\] \[-\tfrac{1}{2}\Big{[}1-\xi^{(l)}(t_{k})\Big{]}\mathbf{Z}^{T}(t_{k}) \mathrm{E}^{(l+1)}\Big{(}\mathbf{R}_{k}^{-1}\Big{)}\mathbf{Z}(t_{k})+\Big{[}1- \xi^{(l)}(t_{k})\Big{]}\log\Big{[}1-\tau^{(l)}(t_{k})\Big{]}\] \[+\xi^{(l)}(t_{k})\log\Big{[}\tau^{(l)}(t_{k})\Big{]}+C_{\xi}\]  
The expectation of \(\xi(t_{k})\) in iteration \(l+1\) is determined as:  
\[\mathbf{Z}_{k}^{**(l)}=\frac{1}{2n_{x}}\sum_{i=1}^{2n_{x}}h\Big{(}\chi_{i,k}^{** (l)}\Big{)} \tag{37}\]0  
where \(a\) and \(b\) are calculated in Equations (45) and (46), respectively. \[\mathbf{Z}_{k}^{**(l)}=\frac{1}{2n_{x}}\sum_{i=1}^{2n_{x}}h\Big{(}\chi_{i,k}^{** (l)}\Big{)} \tag{37}\]1  
\[\mathbf{Z}_{k}^{**(l)}=\frac{1}{2n_{x}}\sum_{i=1}^{2n_{x}}h\Big{(}\chi_{i,k}^{** (l)}\Big{)} \tag{37}\]2  
where \(\mathbf{D}_{k}^{(l+1)}\) and \(\mathbf{E}_{k}^{(l+1)}\) are estimated by Equations (47)-(52). **Case 1.3: \(\mathbf{D}_{k}^{(l+1)}\) and \(\mathbf{E}_{k}^{(l+1)}\) are written as:**  
\[\mathbf{Z}_{k}^{**(l)}=\frac{1}{2n_{x}}\sum_{i=1}^{2n_{x}}h\Big{(}\chi_{i,k}^{** (l)}\Big{)} \tag{37}\]3 \[\mathbf{Z}_{k}^{**(l)}=\frac{1}{2n_{x}}\sum_{i=1}^{2n_{x}}h\Big{(}\chi_{i,k}^{** (l)}\Big{)} \tag{37}\]4  
\[\mathbf{Z}_{k}^{**(l)}=\frac{1}{2n_{x}}\sum_{i=1}^{2n_{x}}h\Big{(}\chi_{i,k}^{** (l)}\Big{)} \tag{37}\]5  
**Case 2.3: \(\mathbf{D}_{k}^{(l+1)}\) and \(\mathbf{E}_{k}^{(l+1)}\) are written as follows. **  
The cubature points are selected by using Equation (49). \[\mathbf{Z}_{k}^{**(l)}=\frac{1}{2n_{x}}\sum_{i=1}^{2n_{x}}h\Big{(}\chi_{i,k}^{** (l)}\Big{)} \tag{37}\]6  
\[\mathbf{Z}_{k}^{**(l)}=\frac{1}{2n_{x}}\sum_{i=1}^{2n_{x}}h\Big{(}\chi_{i,k}^{** (l)}\Big{)} \tag{37}\]7  
\[\mathbf{Z}_{k}^{**(l)}=\frac{1}{2n_{x}}\sum_{i=1}^{2n_{x}}h\Big{(}\chi_{i,k}^{** (l)}\Big{)} \tag{37}\]8  
\[\mathbf{Z}_{k}^{**(l)}=\frac{1}{2n_{x}}\sum_{i=1}^{2n_{x}}h\Big{(}\chi_{i,k}^{** (l)}\Big{)} \tag{37}\]9  
Then, \(\mathbf{Z}_{k}^{***(l+1)}\) is calculated as Equation (52). \[\mathbf{P}_{xz}^{(l+1)}=\frac{\mathrm{E}^{(l)}[\xi(t_{k})]}{2n_{x}}\sum_{i=1}^ {2n_{x}}\Big{[}\mathbf{X}^{(l)}(t_{k})-\chi_{i,k}^{**(l)}\Big{]}\Big{[}\mathbf{ Z}_{k}^{**(l)}-h\Big{(}\chi_{i,k}^{**(l)}\Big{)}\Big{]}+\frac{\left(1-\mathrm{E}^{(l)} \right)[\xi(t_{k})]}{2n_{x}}\sum_{i=1}^{2n_{x}}\Big{[}\mathbf{X}^{(l)}(t_{k})- \chi_{i,k}^{**(l)}\Big{]}\Big{(}\mathbf{Z}_{k}^{**(l)}\Big{)}^{T} \tag{38}\]0  
**Remark 5**: _The expectations of \(\tau(t_{k})\) and \([1-\tau(t_{k})]\) in iteration \(l+1\) are calculated by using Equation (53)._  
\[\left\{ \begin{array}{l}\mathrm{E}^{(l+1)}\{\log[\ \tau(t_{k})]\}=\psi\Big{[} \alpha^{(l)}(t_{k})\Big{]}-\psi\Big{[}\alpha^{(l)}(t_{k})+\beta^{(l)}(t_{k}) \Big{]}\\ \mathrm{E}^{(l+1)}\{\log[\ 1-\tau(t_{k})]\}=\psi\Big{[}\beta^{(l)}(t_{k}) \Big{]}-\psi\Big{[}\alpha^{(l)}(t_{k})+\beta^{(l)}(t_{k})\Big{]}\end{array} \right. \tag{53}\]  
_where \(\psi(\bullet)\) is the digamma function._Set \(\kappa=\tau(t_{k})\); the update of \(q(\kappa)\) in iteration \(l+1\) can be expressed as:  
\[q^{(l+1)}(\kappa)=B\Big{(}\kappa;\alpha^{(l+1)}(t_{k}),\beta^{(l+1)}(t_{k})\Big{)} \tag{54}\]  
The parameters of \(\tau(t_{k})\) in iteration \(l+1\) are calculated by using Equation (55). \[\left\{ \begin{array}{l}\alpha^{(l+1)}(t_{k})=\alpha^{(l+1)}\big{(}t_{k}^{+} \big{)}-\mathrm{E}^{(l+1)}[\xi(t_{k})]+1\\ \beta^{(l+1)}(t_{k})=\beta^{(l+1)}\big{(}t_{k}^{+}\big{)}+\mathrm{E}^{(l+1)}[ \xi(t_{k})]\end{array} \right. \tag{55}\]  
The updates of \(\alpha^{(l+1)}\big{(}t_{k}^{+}\big{)}\) and \(\beta^{(l+1)}\big{(}t_{k}^{+}\big{)}\) are derived by:  
\[\left\{ \begin{array}{l}\alpha^{(l+1)}\big{(}t_{k}^{+}\big{)}=\rho\alpha^{(l )}(t_{k})\\ \beta^{(l+1)}\big{(}t_{k}^{+}\big{)}=\rho\beta^{(l)}(t_{k})\end{array} \right.