## 2 Materials and Methods
### 3D/4D-Var Techniques  
The goal of any variational DA technique is to find the optimal state \(x\), henceforth called \"analysis\", that minimizes a cost function [24]. For the case of the 3D-Var method, the main differentiation being that it is a time-independent method that is less computationally demanding and requires less data but may suffer from accurately capturing fast evolving weather systems. However, it can provide accurate results for more slowly time-paced variables and is still useful for many practical applications due to the low computational cost. The method can be summarized in minimizing functional J with respect to the Analysis state \(x\) in the following cost function [12]:  
\[J(x)=\frac{1}{2}\Big{[}\left(x-x^{b}\right)^{T}B^{-1}\left(x-x^{b}\right)+ \left(y-H(x)\right)^{T}R^{-1}(y-H(x))\Big{]}, \tag{1}\]  
where superscripts T, \(-1\) are the standard notation for transpose and inverse operators; \(x^{b}\) is the background state or field (previous forecast or initialization); and \(y\) is the observation vector. \(H(x)\) is the observation operator that maps the model variable to the observation space, introducing additional representation errors. Additionally, this method requires the estimation of the background and observation error covariance matrices \((B,R)\), respectively. The computation of the observation and most importantly the background error covariance matrices is a heavily discussed and researched topic (see, e.g., [13; 20; 24; 25]). The approach for estimating matrix \(B\) for the case study considered here is discussed in the next sections. The extension of 3D-Var is named 4D-Var, which similarly to 3D-Var solves an optimization problem, with the main difference being that it considers data over specific time bins, which allows it to account for the temporal evolution of the atmosphere, in contrast to the 3D-Var that is not time dependent, and it considers a single analysis time by assimilating observations from that specific time into the model. This difference makes the 4D-Var more computationally intensive due to the use of the fully nonlinear model and consideration of the entire assimilation window. Hence, it makes it more suitable for research and high-performance computing environments, whereas the 3D-Var can be easier implemented for operational purposes [12; 26], and it can be used to correct the initial and boundary conditions of the outer domain in an operational setting. In equation form, 4D-Var considers the splitting of the time dimension in discretized n time-bins, and the observations \(\mathbf{y}_{i}\) that belong to each time bin are used. So, Equation (1) is transformed into  
\[J(x)=\frac{1}{2}\Bigg{[}(x-x^{b})^{T}B^{-1}\left(x-x^{b}\right)+\sum_{i}^{n}(y _{i}-H_{i}(x_{i}))^{T}R^{-1}(y_{i}-H_{i}(x_{i}))\Bigg{]}, \tag{2}\]  
where the observational last term on the right hand side is a sum over all time bins [14]. Variational methods offer an advantage over sequential techniques by achieving a state x that effectively assimilates all available observations simultaneously through the minimization of J. However, the use of variational data assimilation comes with certain drawbacks, most notably its reliance on several assumptions, with a main one being that the error statistics do not propagate in time [26]. This leads to a very strong reliance of the assimilation results on the background error calculation, which is often hard to accurately quantify. The precision of prescribed errors becomes pivotal for the accuracy of the resulting analysis, given that imperfect observations and prior information are integral inputs in the assimilation process [25]. The last limitation is shared with sequential DA techniques. Additionally, while the variational approach allows for the incorporation of linearized dynamical and physical processes, it is crucial to acknowledge that real errors within the Numerical Weather Prediction (NWP) system can exhibit highly nonlinear behavior.