# Blending Sea Surface Winds from the HY-2 Satellite Scatterometers Based on a 2D-Var Method
## 2 Data and Methods
### 2D-Var Method  
Following the basic assumptions in meteorological variational data assimilation, a 2D-Var method is applied to blend the HSCATs observations with the ERA5 background wind field. Conceptually, given a set of scatterometer observations \(\mathbf{x}_{o}^{k}\) (\(\mathbf{x}\) denotes a vector, \(k\) denotes the \(k\)th satellite scatterometer) and a vector of background winds \(\mathbf{x}_{b}\), the analyzed wind vector \(\mathbf{x}\) is obtained by minimizing the following objective function:  
\[\min\left[\int\left(\mathbf{x}_{o}^{k},\mathbf{x},\mathbf{x}_{b}\right)\right] =J_{b}(\mathbf{x}_{b},\mathbf{x})+J_{o}\left(\mathbf{x}_{o}^{k}, \mathbf{x}\right)=\frac{1}{2}(\mathbf{x}-\mathbf{x}_{b})^{T}\mathbf{R}_{b}^{ -1}(\mathbf{x}-\mathbf{x}_{b})+\frac{1}{2}\left(\mathbf{x}-\mathbf{x}_{o}^{k} \right)^{T}\mathbf{R}_{o}^{-1}\left(\mathbf{x}-\mathbf{x}_{o}^{k}\right) \tag{1}\]  
where \(\mathbf{J}_{b}\) and \(\mathbf{J}_{o}\) denote the background (\(\mathbf{B}\)) and the observational (\(\mathbf{O}\)) terms, respectively. \(\mathbf{R}_{b}\) is the background error covariance matrix, and \(\mathbf{R}_{o}\) is the observation error covariance matrix. For computational convenience, the state vector x is replaced by the analysis increment \(\Delta\mathbf{x}=\mathbf{x}-\mathbf{x}_{b}\). In practice, an alternative formulation of the observational term, which was developed for the ambiguity removal of SeaWinds on QuikSCAT [18], is used in this study. That is, the observation term \(J_{o}\) is written as:  
\[J_{o}=\sum_{i,j}^{N_{1}N_{2}}\left\{\sum_{k}^{M_{ij}}\left[\frac{\left(\Delta u _{ij}-\Delta v_{ijk}^{(o)}\right)^{2}}{\sigma_{u}^{2}}+\frac{\left(\Delta u_{ ij}-\Delta v_{ijk}^{(o)}\right)^{2}}{\sigma_{v}^{2}}-2\ln p_{k}\right]^{- \lambda}\right\}^{-1/\lambda} \tag{2}\]  
with (_i,j_) the indices of the 2D-Var grid cell, \(N_{1}\) and \(N_{2}\) the number of 2D-Var grids along the zonal and the meridional direction, \(M_{ij}\) the number of HSCATs observations at point (_i,j_). Moreover, \(\Delta u_{ij}\) and \(\Delta v_{ij}\) stand for the analysis wind component increments at grid (_i,j_), \(\Delta u_{ij,k}^{(o)}\) and \(\Delta v_{ij,k}^{(o)}\) are the difference between the \(k\)th observation and the background wind components, \(\sigma_{u}\) and \(\sigma_{v}\) denote the errors in the observed wind fields. Note that, Equation (2) is similar to the cost function used in the scatterometer ambiguity removal scheme, such that \(\lambda=4\) is used without any adaption [19; 20]. Furthermore, note that the index \(k\) indicates the satellite number rather than the ambiguous solution. Consequently, the a priori probability of the \(k\)th observation being the \"true\" wind is the inverse of the number of observations within the grid (_i,j_), i.e., \(p_{k}=1/M_{ij}\), following the assumption that the three HSCATs contribute equally to the cost function. As shown in Figure 1, there is usually only one HSCAT observation in a grid cell (except for the grids at high latitudes), such that the term 2 ln\(p_{k}\) is usually equal to zero. The background wind term of Equation (1) is written as:  
\[J_{b}(\Delta\mathbf{x})=(\Delta\mathbf{x})^{T}B_{u,v}^{-1}(\Delta\mathbf{x}) \tag{3}\]  
where \(B_{u,v}^{-1}\) denotes the background error covariance matrix. In the numerical implementation, Equation (3) is transformed to the frequency domain and is expressed in terms of velocity potential (\(\chi\)) and stream function (\(\psi\)) increments, and the background error covariance is factorized into standard deviation (SD) errors and error correlations, which both can be expressed as diagonal matrices. As described in ref. [21], the 2D-Var analysis field can be optimized by adjusting the background error correlation function, and the background or observation error variances. Consequently, on the one hand, the SD errors of HSCATs winds are estimated for different collocation windows and then used to construct the observational error covariance matrix.