## Appendix A
\begin{array}{l}\mathbf{e}_{t}^{v_{t}}=\{(\mathbf{\dot{r}}_{t}^{i},\mathbf{v}_{t}^{i}), i=1,\ldots,n_{e}\}\\ \mathbf{e}_{t}^{d}=\{\mathbf{\dot{d}}_{t}^{i},i=1,\ldots,n_{e}\}\end{array} \right\}\text{ for }t=0,\ldots,T\]  
and the accumulated ensemble set defined as  
\[\mathbf{E}_{0:t}^{v_{t}}=\{(\mathbf{\dot{r}}_{0:t}^{i},\mathbf{v}_{0:t}^{i}),i=1,\ldots,n_{e }\}\quad\text{for }t=0,\ldots,T\]  
The \((2n+m)\)-vector \([\mathbf{\dot{r}}_{t},\mathbf{v}_{t},\mathbf{d}_{t}]\) has covariance matrix \(\mathbf{\Sigma}_{rvd}^{t}= \begin{bmatrix}\mathbf{\Sigma}_{rvd}^{t}&\mathbf{\Gamma}_{rvd}^{ t}\\ \mathbf{\Gamma}_{d,rv}^{t}&\mathbf{\Sigma}_{d}^{t}\end{bmatrix} \)  
The \((2n(t+1)+m)\)-vector \([\mathbf{\ddot{r}}_{0:t},\mathbf{v}_{0:t},\mathbf{d}_{t}]\) has covariance matrix \(\mathbf{\Sigma}_{rvd}^{0:t}= \begin{bmatrix}\mathbf{\Sigma}_{rvd}^{0:t}&\mathbf{\Gamma}_{rvd }^{0:t}\\ \mathbf{\Gamma}_{d,rv}^{0:t}&\mathbf{\Sigma}_{d}^{t}\end{bmatrix} \)  
1. **Initiate**
2. \(n_{e}=\text{No. of ensemble members}\)
3. Generate \( \begin{bmatrix}\mathbf{\ddot{r}}_{0:t}^{a(i)}\\ \mathbf{v}_{0}^{a(i)}\end{bmatrix} \sim f(\mathbf{\ddot{r}}_{0},\mathbf{v})\), \(i=1,...,n_{e}\)
4. \(\mathbf{E}_{0:0}^{i\
u}=\{(\mathbf{\dot{r}}_{0}^{a(i)},\mathbf{v}_{0}^{a(i)}),i=1,\ldots,n _{e}\}\)
5. Generate \(\mathbf{e}_{0:t}^{a(i)}\sim\mathcal{U}_{m}[0,1]\), \(i=1,...,n_{e}\) iid
6. \(\mathbf{d}_{0}^{i}=\mathbf{\dot{\psi}}_{0:t}(\mathbf{\ddot{r}}_{0:t}^{a(i)}),i=1,...,n_{e}\)
7. \(\mathbf{e}_{0}^{d}=\{\mathbf{d}_{0}^{i},i=1,\ldots,n_{e}\}\)
8. Estimate \(\mathbf{\Sigma}_{rvd}^{0}\) from \(\mathbf{E}_{0:0}^{\
u},\mathbf{e}_{0}^{d}\longrightarrow\mathbf{\dot{\Sigma}}_{rvd}^{0}\)
9. \( \begin{bmatrix}\mathbf{r}_{0}^{c(i)}\\ \mathbf{v}_{0}^{c(i)}\end{bmatrix} = \begin{bmatrix}\mathbf{\ddot{r}}_{0:t}^{u(i)}\\ \mathbf{v}_{0}^{u(i)}\end{bmatrix} +\mathbf{\ddot{\Gamma}}_{rvd}^{0}\mathbf{\dot{\Sigma}}_{ d}^{0-1}(\mathbf{d}_{0}-\mathbf{d}_{0}^{i}),\quad i=1,...,n_{e}\)
10. **Iterate**\(t=1,\ldots,T\):
11. Forwarding
12. Generate \(\mathbf{e}_{t}^{(i)}\sim\mathcal{U}_{n}[0,1]\), \(i=1,...,n_{e}\)
13. \( \begin{bmatrix}\mathbf{\ddot{r}}_{t}^{u(i)}\\ \mathbf{v}_{t}^{u(i)}\end{bmatrix} = \begin{bmatrix}\mathbf{\omega}_{t}(\mathbf{\ddot{r}}_{t -1:t}^{c(i)}\mathbf{\dot{r}}_{t}^{t(i)})\\ \mathbf{v}_{t-1}^{c(i)}\end{bmatrix} \), \(i=1,...,n_{e}\)
14. \(\mathbf{E}_{0:t}^{v_{t}}=\{\mathbf{E}_{0:t-1}^{v_{t}},(\mathbf{\ddot{r}}_{t}^{u(i)},\mathbf{v} _{t}^{u(i)}),i=1,\ldots,n_{e}\}\)
15. Generate \(\mathbf{e}_{t}^{d(i)}\sim\mathcal{U}_{m}[0,1]\), \(i=1,...,n_{e}\) iid
16. \(\mathbf{d}_{t}^{i}=\mathbf{\dot{\psi}}_{0:t}(\mathbf{\ddot{r}}_{t}^{a(i)},\mathbf{e}_{t}^{a(i)})\), \(i=1,...,n_{e}\)
17. \(\mathbf{e}_{t}^{d}=\{\mathbf{d}_{t}^{i},i=1,\ldots,n_{e}\}\)
18. Estimate \(\mathbf{\Sigma}_{rvd}^{0:t}\) from \(\mathbf{E}_{0:t}^{v_{t}},\mathbf{e}_{t}^{d}\longrightarrow\mathbf{\dot{\Sigma}}_{rvd}^{0:t}\)
19. \( \begin{bmatrix}\mathbf{\ddot{r}}_{0:t}^{c(i)}\\ \mathbf{v}_{0:t}^{c(i)}\end{bmatrix} = \begin{bmatrix}\mathbf{\ddot{r}}_{0:t}^{u(i)}\\ \mathbf{v}_{0:t}^{u(i)}\end{bmatrix} +\mathbf{\ddot{r}}_{rvd}^{0:t}\mathbf{\dot{s}}_{t}^{t-1 }(\mathbf{d}_{t}-\mathbf{d}_{t}^{i}),\quad i=1,...,n_{e}\)
20. **End iterate**
21. \(\mathbf{E}_{0:t}^{S}=\{(\mathbf{\ddot{r}}_{0:T}^{c(i)},\mathbf{\
u}_{0:T}^{c(i)}),i=1, \ldots,n_{e}\}\)
22. Select 23. For arbitrary \(t\in[0,T]\), select corresponding ensemble \(\mathbf{e}_{t}^{S}\) from \(\mathbf{E}_{0:T}^{S}\)
24. Estimate \(\mathbf{\mu}_{r_{\
u}}^{t}\mathbf{\Sigma}_{r_{\
u}}^{t}\) from \(\mathbf{e}_{t}^{S}\longrightarrow\mathbf{\dot{\mu}}_{r_{\
u}}^{t}\mathbf{\dot{\Sigma}}_{r_{ \
u}}^{t}\)
25. Assess 26. \(\hat{f}(\mathbf{r}_{t}|\mathbf{d}_{0:T})= \begin{bmatrix}\Phi_{n}(\mathbf{A};\mathbf{\dot{\mu}}_{ \
u}^{t},\mathbf{\dot{\Sigma}}_{\
u}^{t})\end{bmatrix} ^{-1}\times\Phi_{n}(\mathbf{A}; \mathbf{\dot{\mu}}_{\
u}^{t}+\mathbf{\ddot{\Gamma}}_{\
u|F}^{t}(\mathbf{r}-\mathbf{\dot{\mu}}_{ \
u}^{t}),\mathbf{\dot{\Sigma}}_{\
u|F}^{t})\times\varphi_{n}(\mathbf{r};\mathbf{\dot{\mu}}_{ \
u}^{t},\mathbf{\dot{\Sigma}}_{\
u}^{t})\)
27.