(3) Grouping

The waveform series can be seen as being composed of two parts: The main waveform information and the noise information. Therefore, the subscript *D* = {1, ··· , *M*} of the elementary matrix **S***i* was divided into two disjoint subsets, namely *D*1 = {1, ··· , *l*}, 1 < *l* < *M* containing the first few leading components that describe the main waveform information, and *D*2 = *D* − *D*1 containing the residual components that describe the noise information. Then, the trajectory matrix S can be decomposed into two parts, **S***D*1 = **S**1 + **S**2 + ··· + **S***l* and **S***D*2 = **S***l*+<sup>1</sup> + **S***l*+<sup>2</sup> + ··· + **S***M*, namely:

$$\mathbf{S} = \mathbf{S}^{D\_1} + \mathbf{S}^{D\_2} \tag{3}$$

The parameter *l* denotes the leading components, which provides a good description of the main waveform information, and the lower M-*l* components represent the noise information. If *l* is too small, a part of the main waveform information will miss. Alternatively, if *l* is too large, a part of the main waveform information is approximated with the noise information. Currently, there is no clear standard for determining this parameter [13]. To properly choose the parameter *l*, the ratio calculated

in Equation (4) was used to estimate the contribution of the *i*-th candidate principal component corresponding to the raw waveform series.

$$\text{Ratio}\_{l} = \lambda\_{i} / \sum\_{i=1}^{M} \lambda\_{i} \tag{4}$$

where λ*i* is the *i*-th eigenvalue.

> (4) Reconstruction

Reconstruction was performed to restore **S***D*1 and **S***D*2 in Equation (3) to new series *XD*1 (*t*) and *XD*2 (*t*) of length *N*, respectively. Taking reconstruction **S***D*1 as an exa *M*∗ = min(*<sup>M</sup>*, *N* − *M* + 1) mple to describe its specific process and assuming that the element in **S***D*1 is *yi*,*j*, and *K*∗ = max(*<sup>M</sup>*, *N* − *M* + <sup>1</sup>), the reconstruction of **S***D*1 into the new series *XD*1 (*t*)*gt* : 1 < *t* < *N* can be obtained as follows:

$$\mathcal{g}\_t = \begin{cases} \frac{1}{t} \sum\_{i=1}^t \mathcal{Y}\_{i, t-i+1} & 1 \le t < M^\* \\\frac{1}{M^\*} \sum\_{i=1}^t \mathcal{Y}\_{i, t-i+1} & M^\* \le t \le K^\* \\\frac{1}{N-t+1} \sum\_{i=t-K^\*+1}^{N-K^\*+1} \mathcal{Y}\_{i, t-i+1} & K^\* < t \le N \end{cases} \tag{5}$$

#### 2.2.2. Waveform Retracking Method

Accurate range estimates were obtained using methods of waveform retracking (e.g., Ocean, Ice-2, Belta5, Threshold) [17]. These retracking methods and their applications to the coastal altimeter waveform have been reviewed in Gommenginger et al. [8]. Some studies have shown that the standard Ocean retracking method, Ice-2 retracking algorithm, and Belta5 retracker are not appropriate for complex coastal waveforms [6,8,10,17]. However, the threshold retracker is easy to implement and successful in producing valid SSHs from coastal altimeter waveform retracking [6,17–20]. Therefore, the threshold retracker was used in our study.

The threshold retracker was developed by Davis [21,22]. It is based upon the dimensions of the rectangle computed using the offset center of gravity (OCOG) retracking algorithm [6,8]. The threshold level is referenced to the OCOG amplitude as 25%, 50%, and 75% of the amplitude. The retracking gate estimate is determined by linearly interpolating between adjacent samples of a threshold crossing the steep part of the leading-edge slope of the waveform [8]. The selection of the appropriate threshold level (such as 25%, 50%, or 75% of the OCOG amplitude) is very important. However, a reasonable choice is very difficult and difficult to grasp [6,19]. For the coastal altimeter waveform, a 50% threshold level retracking can usually obtain better results [19]. The formula and explanations in the calculation of the 50% threshold retracker have been presented by Deng [6] and Gommenginger et al. [8].

#### 2.2.3. Estimating Sea Surface Height

The retracked gate was estimated from 50% threshold retracker to determine retracked range correction by [5]:

$$
\Delta R\_{\text{Ref}} = \frac{c}{2} \times \tau \times (G\_{\text{Ref}} - G\_{\text{Ref}}) \tag{6}
$$

where Δ*R*Ret is the range correction derived from the waveform retracking methods (m), *c* is the light velocity in vacuum, τ is the time interval for one gate Jason-1 satellite (3.125 ns), *G*Ret is the retracked gate estimated from the 50% threshold retracker, and *G*Ref is the reference tracking gate on-board observation of Jason-1 satellite (32, in unit of gates) [23]. Then, the precise SSH value of retracking waveform was calculated by Equation (7) [5,16]:

$$SSH\_{\text{corr}} = R\_{\text{Alt}} - \left(R + \Delta R\_{\text{ret}}\right) - corr \tag{7}$$

where *SSH*corr is the retracked SSH (m); *R*Alt is the satellite altitude (m); *R* is the range of satellite measurement (m); and *corr* is the error correction (m), which includes wet troposphere correction, dry troposphere correction, ionosphere correction, sea-state correction, dynamic atmospheric correction, ocean tide correction, solid Earth-tide correction, loading tide correction, and pole tide correction. All these error corrections can be found in Jason-1 SGDR products.
