**3. Results**

#### *3.1. SSA-Denoised Waveform Series*

We experimented with the 20-Hz waveforms data of Jason-1 cycle 340 pass 153 from 13.80◦ N to 21.75◦ N (pass with purple color in Figure 1, hereafter called c340-p153 track). This track had a total of 3118 waveforms and contained the altimeter data moving from both land to ocean and ocean to land. The deepest part of the waters that this track moved was about 1730 m, and the shallowest part was about 7 m. The c340-p153 track had su fficient data volume and a certain representativeness. The waveforms of the c340-p153 track were connected end-to-end to construct a waveform series with length of *N* = 3118 × 104. Then, this waveform series was denoised with SSA.

The main purpose of the waveform retracker is to find the position of the leading-edge component with respect to the fixed nominal tracking point [6,8]. Therefore, it is critical to retain the slope information of the leading-edge component when performing noise reduction of the waveform series with SSA. It was assumed that a component with a ratio lower than 0.01% was considered as the noise information. That is, the ratio of 0.01% was used as a boundary value to distinguish the main waveform information and the noise information. The components with a ratio higher than 0.01% were classified as the main waveform information, and those lower than 0.01% were classified as the noise information. This means we can reconstruct the main waveform information from the components with a ratio higher than 0.01%.

Figure 3 presents the contribution ratio of the eigenvalues {λ*i*} *M i* corresponding to the raw waveform series. As can be seen from Figure 3, except for the ratio of the first eigenvalue reaching about 70.5%, the others were less than 10%, and the ratio of each eigenvalue starting from the 49th eigenvalue was within 0.01%. The ratio of the first 48 eigenvalues was more than 99%. Therefore, it was preferable to choose *l* = 48 in reconstruction to obtain the SSA-denoised waveform series. Thus, the main waveform information can be well preserved.

**Figure 3.** Ratio of the eigenvalues.

Figure 4a is the beginning part of the raw waveform series and the SSA-denoised waveform series, the latter corresponding to the main waveform information. Figure 4b is the beginning part of the residual series (i.e., noise information of the waveform series), defined as the difference between the raw and SSA-denoised waveform series. As can be seen from Figure 4, by comparison, the raw waveforms and the SSA-denoised waveforms mainly differed in the thermal noise component and amplitude, while the slope information of leading-edge component was well retained. This indicates that SSA allowed a noise reduction on Jason-1 waveforms. The larger amplitude of the residual series in Figure 4b was mainly concentrated on the junction of different waveforms. The reason is that the trailing edge component of the waveform was affected by the low-frequency signal of the thermal noise component of the following waveform. However, the leading edge was less affected.

**Figure 4.** Beginning part of the rawwaveform series, the SSA-denoisedwaveform series, and corresponding residual series defined as the difference between the raw and SSA-denoised waveform series. (**a**) Raw waveform series and SSA-denoised waveform series; (**b**) residual series.

#### *3.2. Comparison of Retracked SSHs*

The 20-Hz waveforms data of the c340-p153 track were denoised using the SSA algorithm described earlier to obtain the SSA-denoised waveforms. Both the raw and SSA-denoised waveforms were retracked by the 50% threshold retracker, respectively. Figure 5 compares the raw SSH, the retracked SSHs from the 50% threshold, and SSA + 50% threshold retracker, respectively, with referenced geoidal heights calculated by the EGM2008 model. Here, the SSA + 50% threshold retracker refers to using the 50% threshold retracker for the SSA-denoised waveforms. As can be seen from Figure 5, the deviation of SSH from geoidal height increased as the tracks approached the land. In the coastal region, the retracked SSH profile was smoother than the raw SSH profile, and the former was more similar to the geoidal height than the latter.

The success of the retracker in producing a better SSH estimate value was identified by computing the standard deviations of the difference between SSHs and geoid heights, and the improvement percentage (IMP). The calculation formula of IMP is as follows [18].

$$\text{IMPP} = \frac{\delta\_{\text{raw}} - \delta\_{\text{retracked}}}{\delta\_{\text{raw}}} \times 100\% \tag{8}$$

where, δraw and δretracked are the standard deviations of the differences between raw SSHs and geoidal heights, and retracked SSHs and geoidal heights, respectively. The geoidal heights are calculated by EGM2008 model [24] in the present study.

**Figure 5.** Comparison of raw SSH, retracked SSH, and referenced geoid height (calculated by EGM2008 model) along the track of Jason-1 cycle340 pass153.

The IMP was compared in three cases: (1) The entire c340-p153 track; (2) part of the c340-p153 track from land to ocean within 10 km from the coastline; and (3) part of the c340-p153 track from ocean to land within 10 km from the coastline. Table 2 shows the standard deviations (STDs) of the differences between raw SSH, retracked SSH, and geoidal heights and the IMP in these three cases. In Table 2, the STDs of differences between raw SSH and geoidal heights were smaller than that between the retracked SSH and geoidal heights. This means that waveform retracking iswas successful in improving the quality of altimeter data, especially in the coastal region. Whether in the open ocean or coastal region, the IMP values from the SSA + 50% threshold retracker were larger than those from the 50% threshold retracker. This indicates that SSA successfully improved the retracked SSHs estimate both in the open ocean and coastal region.


**Table 2.** Standard deviations of differences between raw SSH, retracked SSH, and geoidal heights and improvement percentage (IMP).

#### *3.3. Comparison of Retracked SSHs Discrepancies at Crossover Points*

SSA was used to reduce the noise information contained in the 20-Hz waveforms data of Jason-1 GM from cycle 500 to cycle 537 in the South China Sea to obtain the SSA-denoised waveforms, which was the same process as using SSA to denoise the waveforms of the c340-p153 track. Then, the 50% threshold retracker was used to retrack the raw and SSA-denoised waveforms.

There was an SSH difference at the crossover point between the ascending and descending tracks, e.g., Jason-1 GM ground tracks in Figure 1b. The crossover differences can be used to evaluate the quality of the retracked SSHs [10]. In order to compare the retracked SSHs from the 50% threshold retracker and that from the SSA + 50% threshold retracker quantitatively, the mean, STD, and root mean square (RMS) of the crossover differences were calculated (Table 3). The statistical results were categorized into two classes, according to the nearest distances from the crossovers to land: Distances less than 10 km and greater than 10 km. The number in the brackets in column one indicates the number of crossovers in these two classes.


**Table 3.** Statistical results of crossover differences between the retracked SSHs from the 50% threshold and SSA + 50% threshold retracker.

As can be seen from Table 3, the accuracy of the retracked SSHs in the coastal region (distances less than 10 km from the land) was lower than that in the open ocean (distances more than 10 km from the land). Regardless of whether the distance was more than 10 km or less than 10 km, both the STDs and RMSs from the SSA + 50% threshold retracker were smaller than those from the 50% threshold retracker, which shows that the retracked result of the SSA + 50% threshold retracker was better than that of the 50% threshold retracker. This indicates SSA can effectively improve the precision of the retracked SSHs, whether in the open ocean or coastal region. This conclusion is consistent with Section 3.2.

#### **4. MSSH Model and Validation**

#### *4.1. MSSH Model from SSA-Denoised Waveform Retracked SSHs*

All the 20-Hz waveforms data from Jason-1 SGDR products, including ERM1, ERM2, and GM in Table 1, were denoised with the SSA described earlier to obtain SSA-denoised waveforms, and the 50% threshold retracker was performed for these SSA-denoised waveforms to obtain the 20-Hz retracked SSHs. These 20-Hz retracked SSHs were compressed by linear regression to obtain 1-Hz retracked SSHs. In this linear regression, the SSHs over three-times larger than the STD were eliminated by an iterative outlier detection, and data with less than 10 points were also not considered. Then, these 1-Hz SSHs were used to establish an MSSH model over the South China Sea with grid of 2 × 2 (shows in Figure 6). The process of establishing MSSH model mainly includes data preprocessing, the removal of the temporal oceanic variability, crossover adjustment, and gridding, which has been detailed by Yuan et al. [25].
