## 2 Data and Methods
### Optimal Interpolation Method
Assuming that the observations are isotropic and homogeneous, then the covariance of the observations at points i and j would only depend on the distance \(d_{ij}\) between the two points:  
\[\mathbf{A}=\left(\langle\sigma^{2}\rangle C\left(d_{ij}\right)+\langle \varepsilon_{i}\varepsilon_{j}\rangle\right)_{ij=1,N}\mathbf{C}_{r}=\left( \langle\sigma^{2}\rangle C\left(d_{ij}\right)\right)_{j=1,N} \tag{7}\]  
where \(\sigma^{2}\) represents the a priori MDT variance, and \(\varepsilon_{i}\) denotes the error on the observation located at i. This study assumed that the errors at different locations were uncorrelated with each other, and that the errors of the observations were uncorrelated with the observations. \(C(r)\) denotes the a priori covariance function of MDT, which was suggested by [PERSON] and [PERSON] (2018):  
\[C(r)=\left(1+r+\frac{1}{6}r^{2}-\frac{1}{6}r^{3}\right)e^{-r} \tag{8}\]where \(\mathbf{r}=\sqrt{\left(\frac{x}{x_{0}}\right)^{2}+\left(\frac{y}{y_{0}}\right)^{2}}\), and \(x_{0}\) and \(y_{0}\) denote the zonal and meridional correlation radii of the MDT, respectively. Consequently, the errors of the estimated MDT could be expressed as:  
\[\varepsilon(\mathbf{r})=\sigma^{2}-\sum_{i=1}^{N}\sum_{j=1}^{N}\mathbf{A}_{ij} ^{-1}\mathbf{C}_{r,j}\mathbf{C}_{r,j} \tag{9}\]  
Theoretically, the mean of the values to be estimated should be zero when applying the OIM, but it is difficult to satisfy this condition in practice. Therefore, in this study, a large-scale prior MDT was removed from the observation, which could be obtained by applying a Gaussian filter on the raw MDT; that is, the difference between the selected MSS and GGM, so that the mean of the remaining residuals was close to zero, and the residuals were then treated as the observations in MDT modeling. The large-scale prior MDT depends on the filtering radius of Gaussian filtering. The appropriate filtering radius can be estimated by the trial and error method. In this study, the filtering radius was set to 400 km. The MDT value of each grid point was estimated based on the procedure of its surrounding points using the OIM for each grid point, where the key parameters are the error of the observations, the a priori variance of the observations, and the covariance between the observations. There are two estimation methods for the error of the observations, one of which can estimate the error of the MDT by using the error information of the MSS and GGM according to the error propagation theory [30]. However, the calculations of this method will increase as the maximum d/o of the GGM increases. In this study, an informal error was introduced for the replacement of the error of the GGM, where several higher-order GGMs like GECO, EGM2008, SGG-UGM-1, and XME2019e_2159 [31; 32; 33; 34] were used to calculate the geoid of the study area. Then, the mean of these geoids was calculated and the difference between the geoid calculated by the GGM used in this study and the mean value was taken as the error of the GGM [35]. Notably, this informal error did not consider factors such as the omission error of GGM, which may lead to a smaller result than the true values. Second, in order to derive an accurate estimation of the observation error, an a priori MDT (e.g., DTU13 MDT) was used to compare with the raw MDT, and the RMS differences between the two models could be computed in a \(1^{\circ}\) grid as the error of the observation. However, this error information can be overestimated due to the variability of the mean circulation by the a priori MDT. Therefore, in this study, the two methods were combined, and for each grid point, the observation error was set to the larger one in the two methods. The a priori variance and correlation function of the MDT can be calculated by using the a priori MDT (e.g., CNES-CLS18 MDT). Since the large-scale signal of the MDT was removed in this experiment, only the covariance of the residual part of the MDT needed to be estimated.