# Assimilation of Dynamic Combined Finite Discrete Element Methods Using the Ensemble Kalman Filter
## 3 SHPB Experiment and HOSS Model Setup
### Ensemble Kalman Filter  
The ensemble Kalman Filter (EnKF) [41; 43] is an approximation to the Kalman filter [37] designed for non-linear models. The EnKF is widely implemented in several fields of science for its effectiveness in producing good results and the overall ease of implementation. The model mean and error covariance matrix is approximated through an ensemble of simulation results, which are used in the EnKF formulation. The observations are used to update the ensemble, through the Kalman filter formulas, and then evolved through the model between assimilation cycles. Details on the EnKF method and algorithm can be found in [43; 44]. The EnKF algorithm analysis equations are performed in the following way: For a vector of \(m\) observations \(\mathbf{y}^{o}\) and an ensemble of \(N\) model simulations valid at time \(t_{k}\), the EnKF analysis equations are given by:  
\[\mathbf{x}_{i}^{a} = \mathbf{x}_{i}^{f}+\mathbf{K}\Big{(}\mathbf{y}_{i}^{o}-\mathbf{H }\mathbf{x}_{i}^{f}\Big{)},\quad i=1,\ldots,N, \tag{3}\] \[\mathbf{K} = \mathbf{P}^{f}\mathbf{H}^{T}\Big{(}\mathbf{H}\mathbf{P}^{f} \mathbf{H}^{T}+\mathbf{R}\Big{)}^{-1}, \tag{4}\]  
where \(\mathbf{K}\in\mathbb{R}^{n\times m}\) is the Kalman gain matrix; \(\mathbf{x}_{i}^{a}\in\mathbb{R}^{n}\) is the analysis; \(\mathbf{P}^{f}\in\mathbb{R}^{n\times n}\) is the model covariance matrix; \(\mathbf{R}\in\mathbb{R}^{m\times m}\) is the covariance matrix of the observations; \(\mathbf{H}\in\mathbb{R}^{m\times n}\) is the observation operator used to map the model state to the observations; and \(\mathbf{y}_{i}^{o}\in\mathbb{R}^{m}\) is the observation vector with Gaussian white noise. The EnKF approximates the model covariance matrix using the ensemble simulation. There are a number of problems with this approximation, namely the introduction of artificial cross-correlations between model state variables, which can introduce errors when performing assimilation with the Kalman filter equations [45]. To solve this particular problem, the correlation matrix of the model (\(\mathbf{P}^{f}\)) is modified by removing the cross-correlation between parameters and variables that we know are not correlated. That is, the off-diagonal terms in \(\mathbf{P}^{f}\) that correspond to correlation between parameters and variables that we know are not correlated is set to zero. This is called localization of the covariance matrix. It is also well-known that using the EnKF to estimate model parameters can lead to ensemble collapse, that is, when the parameter estimate in all ensemble membersis essentially the same. In order to avoid ensemble collapse in the model parameters, an inflation technique is implemented. The inflation, as the name suggests, increases the difference between the ensemble parameter average and each individual ensemble parameter value. That is, the parameters are inflated as  
\[\hat{p}_{i}=\bar{p}+\lambda(p_{i}-\bar{p}), \tag{5}\]  
where \(p_{i}\) is the HOSS parameter value from ensemble \(i\), \(\bar{p}\) is the average, \(\hat{p}_{i}\) is the inflated parameter for ensemble \(i\), and the inflation factor is denoted by \(\lambda\). For our problem, \(\lambda\) is set to approximately 45% of the standard deviation for the initial parameter spread. For the HOSS model, the parameters considered cannot be changed mid-simulation. That is, the HOSS parameters are set at the start of the simulation, and once the simulation is underway, they cannot be modified. Due to this, the EnKF is implemented as a smoother [46], which considers the results of HOSS over time as a single state vector to be assimilated using the observations over time. Additionally, the EnKF is implemented in an iterative way to better approximate the HOSS parameter values. The full mathematical framework for the iterative EnKF can be found in [47; 48; 49]. EnKF algorithm implementation is done in the following steps:  
1.