## 3 Results and Discussion
### Validation  
Compared to the Gaussian filtering method, one of the advantages is that the OIM can give the error distribution of the MDT based on the observations. Figure 8 shows the estimated error of the MDT calculated by the OIM, and the error was larger near the Kuroshio current. The RMS of the errors of DTU18 DIRR4-MDT, DTU18 DIRR5-MDT, and DTU18 DIRR6-MDT was 6.1 mm, 6.0 mm, and 5.8 mm, respectively, which reflects the contribution of the improved GGM. Furthermore, in order to compare the OIM and Gaussian filter method, the MDTs based on DTU18 MSS and GOCE-derived GGMs were derived using a Gaussian filter method with a filter radius of 70 km [39], and the results were denoted as GF-R4-MDT, GF-R5-MDT, and GF-R6-MDT, respectively. The in situ geostrophic \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**MSS Models** & **Min** & **Max** & **Mean** & **RMS** \\ \hline DTU15 MSS & 14.7 & 61.0 & 21.9 & 22.2 \\ DTU18 MSS & 13.5 & 48.0 & 20.7 & 21.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Statistics of the errors of the MSS models (units: mm). \begin{table} \begin{tabular}{c c c c} \hline \hline
**MDT Solutions** & **Min** & **Max** & **RMS** \\ \hline DTU15 DIRR6-MDT & \(-\)381.5 & 276.2 & 77.7 \\ DTU18 DIRR6-MDT & \(-\)383.3 & 270.2 & 77.0 \\ \hline \hline \end{tabular}  Note: the mean values of the misfit between the estimated MDTs and the reference MDT model are removed. Figure 7: The MDTs of the (**a**) DTU15 DIRR6-MDT and (**b**) DTU18 DIRR6-MDT, and the corresponding differences between them and the reference MDT (**c,d**). ocean current velocity profiles were used for validation, in which the reference time of the MDT solution; i.e., the DTU18 DIRR4-MDT, DTU18 DIRR5-MDT, DTU18 DIRR6-MDT, GF-R4-MDT, GF-R5-MDT, and GF-R6-MDT, was unified to the same time period with the buoy data (1993-2016) using SLA provided by Aviso. Then, the associated geostrophic ocean current velocity was calculated using Equation (13). It was notable that the exportation of the geostrophic ocean current velocity from the MDT led to an error amplification [38], so a spatial filter was used to reduce the noise; the filter radius was selected as 60 km through experiments. In addition, the geostrophic ocean current velocity exported from the MDT was interpolated to the point of the buoy data, and the differences between them are shown in Figures 9 and 10, where the differences in the zonal and meridional directions are shown, respectively. The geostrophic velocities derived from the OIM method had smaller misfits against the buoy data than those of the Gaussian filter method, especially in coastal areas. The statistical results of the difference are shown in Table 8, in which it can be seen that the RMS of the differences between the DTU18 DIRR6-MDT-derived zonal (meridional) components and the in situ data were 122.6 mm/s (93.7 mm/s), which were 1.2 mm/s (1.8 mm/s) lower than that of the DTU18 DIRR5-MDT, and 2.2 mm/s (3.5 mm/s) lower than that of the DTU18 DIRR4-MDT. In addition, when making the comparison between the OIM solution and the Gaussian filtered solution, the RMS of the difference between the zonal (meridional) geostrophic ocean current velocity derived from the Gaussian filtered solution against the in situ data were about 10 mm/s (17 mm/s) larger than those of the OIM solution, which indicated that the OIM was better than the Gaussian filtering method. The comparison between Tables 4 and 8 shows that the RMS of the difference between the geostrophic ocean current velocity of the MDT models and the in situ data was about 10 mm/s lower than that of the estimated MDT, which indicated that the accuracy of the MDT models was similar to the estimated MDT. Figure 10: The differences in meridional geostrophic velocities between buoy data and information deduced from the MDT solutions: **(a)** DTU18 DIRR4-MDT; **(b)** DTU18 DIRR5-MDT; **(c)** DTU18 DIRR6-MDT; **(d)** GF-R4-MDT; **(e)** GF-R5-MDT; **(f)** GF-R6-MDT.