# Evaluation of HY-2 Series Satellites Mapping Capability on Mesoscale Eddies
## 2 Materials and Methods
### Methods
#### 2.2.2 Optimal Interpolation  
The optimal interpolation (OI) [6] uses minimum variance estimation to organically integrate observations with error into the background field, in order to obtain an optimized estimate of the initial value of the model [33]. Specifically, the analyzed value at each spatial grid is obtained by adding the revised variance to the first guess, and the variance is weighted by the observations deviation from the background field at a certain range of grid points. The background error covariance matrix \(B\) we used was chosen as follows:  
\[B\big{[}x_{i}(t_{1}),x_{j}(t_{2})\big{]}=\sqrt{B_{ii}B_{j}}exp\Bigg{\{}-\frac{ d_{ij}^{2}}{L_{x}^{2}}-\frac{d_{t}^{2}}{L_{t}^{2}}\Bigg{\}} \tag{1}\]  
where \(i,j=1,2,\ldots,dim(x)\); \(d_{ij}\) is the physical distance between \(x_{i}\) and \(x_{j}\); \(d_{t}=|t_{1}-t_{2}|\); \(d_{t}\) is the time interval between moment \(t_{1}\) and \(t_{2}\); and \(C=exp\Big{\{}-d_{ij}^{2}/L_{x}^{2}-d_{t}^{2}/L_{t}^{2}\Big{\}}\) refers to the spatial-temporal correlation. The OI algorithm depends strongly on the smoothing scales. Thus, the parameters of spatial scalar \(L_{x}\) and temporal scale \(L_{t}\) are tuned so that the algorithm can produce the minimum RMSE between the gridded products and model data. Every group of smoothing scales for the reference group and evaluation group at high, middle and low latitudes is shown in Table 2.  
The multi-satellite mapping procedure used in DUACS is OI [25,34]. To confirm the data merging capability of the OI analysis, we selected January 2019 Kuroshio extension as the study target, with six altimeters available. The SLA generated by the mapping procedure was then compared to the DUACS DT2018 product. After sensitivity tests, the final parameters were chosen to be \(L_{t}\) = 21 days, \(L_{x}\) = 200 km, and \(r\) = 0.03, respectively.  
Detailed analysis results of the comparison are shown in Figure 2; the RMS values of the DUACS product (Figure 2a) and SLA synthetic product (Figure 2b) are 18.18 cm and 18.28 cm, respectively, and the RMSE value between two datasets is 2.2 cm. The difference (Figure 2c) between two products is around 0-2 cm on the whole, while it can be as intense as 4-6 cm at the transition regions between cold and warm eddies. However, these differences can be acceptable according to the error budget of DUACS products [2,26]. Generally, our mapping procedure can well reflect the SLA field, thus, we can infer that it is equivalent to the interpolation algorithm in the DUACS product.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Reference Group**} & \multicolumn{3}{c}{**Evaluation Group**} \\ \hline  & \begin{tabular}{c} **High** \\ **Latitudes** \\ \end{tabular} & \begin{tabular}{c} **Middle** \\ **Latitudes** \\ \end{tabular} & \begin{tabular}{c} **Low** \\ **Latitudes** \\ \end{tabular} & \begin{tabular}{c} **High** \\ **Latitudes** \\ \end{tabular} & \
\begin{tabular}{c} **Middle** \\ **Latitudes** \\ \end{tabular} &
\begin{tabular}{c} **Low** \\ **Latitudes** \\ \end{tabular} \\ \hline \(L_{x}\)/km & 80 & 100 & 120 & 70 & 90 & 120 \\ \(L_{t}\)/day & 7 & 7 & 3 & 3 & 3 & 3 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Tuned spatial scale \(L_{x}\) and temporal scale \(L_{t}\) for reference groups and evaluation groups at high, middle and low latitudes. Unit of \(L_{x}\) is km and unit of \(L_{t}\) is day.  
Figure 1: Workflow of the evaluation methodology.