## 2 Methodology
### Particle Filter  
Particle filters, similar to the EnKF, are sequential Monte Carlo methods that calculate the posterior distribution of states and parameters by a set of random samples. The advantage of the PF, in comparison to the EnKF, is that it relaxes the assumption of a Gaussian error structure, which allows the PF to more accurately predict the posterior distribution in the presence of skewed distributions [27; 28]. In detail, consider _ne_ independent and identically distributed random variables \(x_{t,i}\sim p(x_{t}|y_{1:t})\) for \(i=1,2,\ldots,ne\), the posterior density, based on the sequential importance sampling (SIS) method, can be approximated as a discrete function:  
\[p(x_{t}|y_{1:t})=\sum_{i=1}^{n\epsilon}w_{t,i}\delta(x_{t}-x_{t,i}) \tag{13}\]  
where \(w_{t,i}\) is the posterior (updated) normalized weight of the \(i\)th particle drawn from the proposed distribution and \(\delta\) is the Dirac delta function. Assume the system state to be a Markov process, and apply the Bayesian recursive expression to the filtering problem. The updated expression for the importance weights (not normalized) can be expressed as:  
\[w_{t,i}^{a}=w_{t,i}^{f}\cdot\frac{p(y_{t}|x_{t,i}^{f},\theta_{t,i}^{f})p(x_{t,i}^{f}|x_{t-1,i}^{f},\theta_{t,i}^{f})}{q(x_{t,i}^{f}|x_{1:t-1,i}^{f},\theta_ {t,i}^{f},y_{1:t}^{f})} \tag{14}\]  
Equation (14) provides the mechanism to sequentially update the importance weights, given an appropriate choice of the proposal distribution \(q(x_{t,i}^{f}|x_{1:t-1,i}^{f},\theta_{t,i}^{f},y_{1:t}^{f})\). Consequently, the expression of the proposal distribution will significantly affect the efficiency and complexity of the PF method. A common choice of the proposal distribution is the transition prior function [25]:  
\[q(x_{t,i}^{f}|x_{1:t-1,i}^{f},\theta_{t,i}^{f},y_{1:t}^{f})=p(x_{t,i}^{f}|x_{t -1,i}^{f},\theta_{t,i}^{f}) \tag{15}\]  
When the transition prior is chosen as the proposal distribution, Equation (14) can be simplified as:  
\[w_{t,i}^{a}=w_{t,i}^{f}p(y_{t}|x_{t,i}^{f},\theta_{t,i}^{f}) \tag{16}\]  
Therefore, the normalized updating weight can then be obtained via the following equation:\[w_{t,i}^{a}=\frac{w_{t,i}^{f}p(y_{t}|x_{t,i}^{f},\theta_{t,i}^{f})}{\sum\limits_{ i=1}^{ne}w_{t,i}^{f}p(y_{t}|x_{t,i}^{f},\theta_{t,i}^{f})} \tag{17}\]  
where \(p(y_{t}|x_{t,i}^{f},\theta_{t,i}^{f})\) can be obtained from the posterior likelihood equation:  
\[p(y_{t}|x_{t,i}^{f},\theta_{t,i}^{f})=\frac{L(y_{t}|x_{t,i}^{f},\theta_{t,i}^{ f})}{\sum\limits_{i=1}^{ne}L(y_{t}|x_{t,i}^{f},\theta_{t,i}^{f})} \tag{18}\]  
\(L(y_{t}|x_{t,i}^{f},\theta_{t,i}^{f})\) is the posterior likelihood function. If a Gaussian model error is to be used, the likelihood may be approximated via a parametric likelihood function:  
\[L(y_{t}|x_{t,i}^{f},\theta_{t,i}^{f})=\frac{1}{\sqrt{2\pi R_{t}}}\exp(-\frac{ 1}{2R_{t}}[y_{t}-y_{t,i}^{f}]^{2}) \tag{19}\]  
For the particle filter through SIS, a serious limitation is the depletion of the particle set, which means that, after a few iterations (time steps), all the particles except one are discarded because their importance weights are insignificant [27]. To address the above issue, sampling importance resampling (SIR) algorithms are usually applied to eliminate the particles with small importance weights, and replace them with the particles with large importance weights. In our study, the sampling importance resampling (SIR) particle filter is implemented as elaborated by [PERSON] et al. [28], in which \(w_{t,i}^{f}\) are set equal to \(1/ne\) before moving on to the next time step, and the posterior weight \(w_{t,i}^{a}\) is equal to \(p(y_{t}|x_{t,i}^{f},\theta_{t,i}^{f})\).