## 2 Data and Methods
### Optimal Interpolation Method  
The MDT is expressed by the difference between the MSS and geoid, which is given by:  
\[\mathrm{h}=\overline{\mathrm{\eta}}-\mathrm{N} \tag{1}\]  
with the MDT h, the MSS \(\overline{\mathrm{\eta}}\), and the geoid N, which should be combined under the same coordinate system and tide system. Considering the GOCE-derived GGM and the MSS models used in this study, the models were transformed to the GRS80 and zero-tide system. In order to mitigate the signal distortion and leakage problems induced by the traditional filtering methods, this study adopted the OIM (also known as multivariate objective analysis) for the MDT modeling, which was first applied to oceanographic parameter estimation by [PERSON] [28]. The basic principle of this method can be understood as a linear weighted average method for the observations, thus it is essential for the determination of the variance/covariance information. Compared to the traditional Gaussian filtering method, the OIM retains more detailed signals in the combination and helps derive a more \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Ocean Numerical Model** & **Spatial Resolution** & **Time Scale** & **References** \\ \hline SODA3 & \(1/4^{\circ}\) & 1980–2017 & [PERSON] et al. [22] \\ ORA55 & \(1/4^{\circ}\) & 1979–2018 & [PERSON] et al. [23] \\ Copernicus & \(1/4^{\circ}\) & 1993–2019 & Product User Guide [24] \\ CNES-CLS13 MDT & \(1/4^{\circ}\) & 1993–2012 & [PERSON] et al. [25] \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Details of the ocean numerical models. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Geostrophic Velocity** & **Max** & **Mean** & **RMS** \\ \hline Zonal component & 823 & 4.2 & 14.3 \\ Meridional component & 445.8 & 6.9 & 11.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Statistics of the errors of the drifter geostrophic velocity (units: mm/s). reasonable MDT solution. Supposing there are N points used for MDT modeling, the weights for each point are expressed as the following:  
\[\eta_{i}=\sum_{j=1}^{N}A_{ij}^{-1}\varphi_{j} \tag{2}\]  
where \(A_{ij}\) denotes the covariance between point i and j. For each estimated point r and the observation point i, the estimated MDT value \(\langle h\rangle(r)\) could be expressed as:  
\[\langle h\rangle(r)=\sum_{i=1}^{N}C_{ri}\eta_{i} \tag{3}\]  
where \(C_{ri}\) denotes the covariance between r and i. Considering a linear weighted estimator, the estimated MDT \(\langle h\rangle(r)\) could be rewritten as:  
\[\langle h\rangle(r)=\sum_{j=1}^{N}\alpha_{ij}\varphi_{j} \tag{4}\]  
where \(\alpha_{ij}\) denotes the weighting factors. Considering the difference between the estimated MDT \(\langle h\rangle(r)\) and the true MDT \(h(r)\), the error variance between them is expressed as follows:  
\[ \begin{split}\overline{\left(h(r)-\langle h\rangle(r)\right)^{2}} &=\overline{\left(h(r)-\sum\limits_{j=1}^{N}\alpha_{ij}\varphi_{j} \right)^{2}}\\ &=\mathbf{C}_{rr}-2\sum\limits_{j=1}^{N}\alpha_{ij}C_{ij}\\ &+\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N}\alpha_{ri}\alpha_{ri} A_{ij}\\ &=\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N}(\alpha_{ri}-\sum \limits_{i=1}^{N}\mathbf{C}_{rij}A_{ij}^{-1})(\alpha_{ij}-\sum\limits_{j=1}^{N }\mathbf{C}_{rij}A_{ij}^{-1})\mathbf{A}_{ij}+\mathbf{C}_{rr}\\ &-\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N}C_{ri}C_{ij}A_{ij}^{-1 }\end{split}  \tag{5}\]  
To minimize the error variance in Equation (5), let \(\alpha_{ri}=\sum_{i=1}^{n}C_{ri}A_{ii}^{-1}\), and then Equation (4) can be rewritten as:  
\[\langle h\rangle(r)=\sum_{i=1}^{N}\alpha_{i}O(r_{i}),\alpha_{i}=\sum_{j=1}^{ N}A_{ij}^{-1}C_{rj} \tag{6}\]  
with the estimated MDT value \(\langle h\rangle(r)\) at point r, raw observation \(O(r_{i})\), which is the difference between MSS and geoid; the covariance matrix of the observation \(\mathbf{A}\); and covariance vector \(\mathbf{C}\) between the observed and estimated MDT.