## 2 Materials and Methods
### Configuration Set-Up  
The weather forecasts were conducted using WRF (WRF-4.4), with data assimilation executed by the WRF Data Assimilation model (WRF-DA-4.4). WRF is capable of delivering atmospheric simulations appropriate for a wide array of applications, while WRF-DA generates the analysis--integrating ground-based observations to produce perturbed initial and boundary conditions for WRF. Both models were configured to offer a coupled system of high spatial resolution analysis and forecasts covering southern Greece. The WRF model was initialized at 00 UTC and ran daily with a ramp-up period of 24 h, forecasting the next 72 h. This will henceforth be referred to as the forecasting horizon. More details on similar WRF runs and the parametrizations made for the specific test case are presented in [21], but with different forcing data, so an additional validation is performed here to establish the baseline control case run. The nesting configuration is illustrated schematically in Figure 1.  
Concerning the WRF-DA preprocessing, the system requires the observations to be translated into littleR format, which is a specific ASCII-based file format ([[https://www2.mmm.ucaredu/wrf/users/wrfda/OnlineTutorial/Help/little.html](https://www2.mmm.ucaredu/wrf/users/wrfda/OnlineTutorial/Help/little.html)]([https://www2.mmm.ucaredu/wrf/users/wrfda/OnlineTutorial/Help/little.html](https://www2.mmm.ucaredu/wrf/users/wrfda/OnlineTutorial/Help/little.html)), last accessed online 28 October 2024). Then, the littleR files are processed through the WRF preprocessor, and the observational errors are calculated depending on the category of the sensor that is reported in the littleR file (SYNOP, BUOY, etc. ), as well as the prescribed height of measurement, elevation, and pressure. It is assumed that the errors of the field measurements are statistically independent of each other, so the observational covariance matrix R is diagonal, and its values represent each sensor's instrumental error [12], suggested by the WRF-DA suite. Quality control of the observations was carried out so that outliers, duplicates, and observations that are not in the analyzed space-time window were removed. A crucial part of the accuracy of any variational DA technique is the approximation of the covariance matrix B, as shown in Equations (1) and (2). There are two methods implemented within WRF-DA to approximate the matrix B: the Ensemble method (ENS) and the climatological approach known as the National Meteorological Center method (NMC), introduced by [PERSON] and [PERSON] [27]. The Ensemble method uses ensemble perturbations from a given set of forecasts. However, for regional studies, the NMC method is more commonly used. Since our model has been run operationally on a daily basis, forecasting four days in each run, the overlapping forecasted days from different simulations can be utilized to approximate the model error, denoted by \(\varepsilon_{b}\). This procedure assumes that background errors can be effectively represented by statistical data derived from averaged forecast differences, typically from a series of month-long forecasts with 36 h and 12 h validity at the same time. This concept is expressed in the following equation form:  
\[B=\overline{(\mathbf{x^{b}}-\mathbf{x^{t}})(\mathbf{x^{b}}-\mathbf{x^{t}})^{\mathsf{T}}}= \overline{\varepsilon_{b}\varepsilon_{b}\mathbf{z}^{\mathsf{T}}}\cong\overline{ (\mathbf{x^{i+36}}-\mathbf{x^{i+12}})(\mathbf{x^{i+36}}-\mathbf{x^{i+12}})^{\mathsf{T}}}, \tag{3}\]  
where \(\mathbf{x^{t}}\) is the unknown true atmospheric state, and the use of an overbar signifies the average over all samples that are collected beforehand for the estimation of the background error. The subtracted states \(\mathbf{x^{i+36}}\mathbf{x^{i+12}}\) are all the forecasted outputs that co-occur and originate from a specific WRF simulation that has been run 1 day apart [28].