## 3 Observing System Simulation Experimental Framework
### WRFDA-4 DVar Assimilation  
WRFDA was used in the GEO-MW OSSE to assimilate the synthetic GEO-MW observed brightness temperature data from GMR, GeoSTAR, and GIMS. WRFDA is a data assimilation system built within the WRF software framework and used for application in both research and operational environments. The WRFDA was developed and maintained by the Mesoscale and Microscale Meteorology (MMM) Laboratory of NCAR and is freely available to the general community, together with further documentation, test results, etc. WRFDA web page ([[http://www2.mmm.ucar.edu/wrf/users/wrfda/index.html](http://www2.mmm.ucar.edu/wrf/users/wrfda/index.html)]([http://www2.mmm.ucar.edu/wrf/users/wrfda/index.html](http://www2.mmm.ucar.edu/wrf/users/wrfda/index.html)) (accessed on 21 March 2021)). WRFDA can assimilate various observation data that include satellite brightness temperature and is also suitable for use in a broad range of applications across scales ranging from kilometers for regional and mesoscale modeling to thousands of kilometers for global-scale modeling. Various assimilation schemes are supported in WRFDA, including 3 DVar, 4 DVar, and hybrid ensemble-variational (EnVar).  
4 DVar assimilation is an extension of 3 DVar in the time dimension, considering that the characteristics of observation data change with time. 4 DVar essentially solves the analysis field that best matches all effective observation data in the assimilation time window. Prior to 2006, the global 4D-Var data assimilation system was in operation in the leading global numerical weather forecasting centers [39; 40; 41]. In this study, 4 DVar was employed because it assimilates all observation data within the assimilation time window at a fixed time step and is more suitable for assimilation of GEO-MW observation data with a high-temporal resolution. The data assimilation problem is solved by minimizing the cost function in the 4 DVar scheme:  
\[J(x)=\frac{1}{2}(x-x_{b})^{T}B^{-1}(x-x_{b})+\frac{1}{2}\sum_{i=0}^{n}(H_{i}(M _{0\to i}(x)-y_{i})^{T}R_{i}^{-1}(H_{i}(M_{0\to i}(x)-y_{i})+J_{c} \tag{1}\]  
where \(x\) is the atmospheric state to be solved, \(x_{b}\) is the background state; \(B\) is the error covariance matrix of \(x_{b}\); \(y_{i}\) is the observation at time \(i\); \(H\) is the observation operator; \(R_{i}\) is the observational error covariance matrix at time \(i\); \(M_{0\to i}\) is the model integration from the analysis time to time \(i\); and \(J_{c}\) is the weak constraint term based on the digital filter.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Physical Process Type** & **Parametric Scheme** & **Parameter** \\ \hline Microphysical process & [PERSON] & mp\_physics = 2 \\ Longwave radiation process & RRTM & ra\_hw\_physics = 1 \\ Shortwave radiation process & Dudhia & ra\_sw\_physics = 1 \\ Land surface model & [PERSON] & sf\_surface\_physics = 2 \\ Boundary layer & YSU & bl\_pbl\_physics = 1 \\ parameterization process & & \\ Cumulus convection & & \\ parameterization scheme & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: WRF physical parameterization scheme.  
The system framework of WRFDA-4 DVar is shown in Figure 2. The WRFDA system iteratively updates the background state \(x_{b}\) and finally obtains the analysis field \(x_{a}\), which initializes the new prediction. It also supports updating boundary conditions, including side boundary updating and lower boundary updating.