2.4.2.1 Model-based retrackers

In model-based retrackers, the parameters of interest (range between satellite and sea surface, significant wave height, and backscatter coefficient) are estimated by fitting the waveform to a certain model via a maximum likelihood estimation (MLE) approach. What really counts is the choice of the model.

Figure 2. Schematic representation of the waveform classes within the PISTACH processing [15].

Although the Brown [18] or Hayne [19] model is pretty successful over open ocean surface, there is little theoretic echo model elsewhere. Maybe the most universal coastal waveform model is the one proposed by Enjolras et al.:

$$P\_r(t) = A \left[ \left. \int\_{S \in water} S\_r \left( t - \frac{2h}{c} \right) \frac{G^2(\theta)}{h^4} \sigma\_{water}(\theta) d\mathcal{S} + \iint\_{S \in land} S\_r \left( t - \frac{2h}{c} \right) \frac{G^2(\theta)}{h^4} \sigma\_{lamd}(\theta) d\mathcal{S} \right] \right] \tag{1}$$

In the model the return power is expressed as the weighting average of the water surface echo and land surface echo. This model is not very practical: the geometry of the coastline, the relief, the nature of the terrain, etc., in a word, all characteristics of the coast that are extremely diverse all over the world. Therefore, it is difficult to determine the a priori parameters in the model.

There are also a couple of models, initially developed for ice surface, which are popularly used in coastal retrackers, such as the so-called 5β model [20] and E model [21], with the following expressions:

$$P\_r(t) = \beta\_1 + \beta\_2 \Phi(\frac{t - \beta\_3}{\beta\_4})(1 + \beta\_5 Q) \text{ 5\% model} \tag{2}$$

$$P\_r(t) = \beta\_1 + \beta\_2 \Phi\left(\frac{t - \beta\_3}{\beta\_4}\right) \exp\left(-\beta\_5 Q\right) \to \text{model} \tag{3}$$

where:

$$Q = \begin{cases} 0 & \text{t} < \beta\_3 + \beta\_4/2 \\ \text{t} - \beta\_3 - \beta\_4/2 & \text{t} > \beta\_3 + \beta\_4/2 \end{cases} \tag{4}$$

The two models are both reduced forms of Brown model, except permitting a negative mispointing angle. When β5Q < < 1, the two models are essentially the same. The official Envisat ICE2 retracker is based on a simplified version of 5β model, where β<sup>5</sup> is always set to 0. Obviously, their physical mechanisms are based on open ocean surface and are often not suitable for coastal waveforms.

Hamili et al. [22] proposed a "Brown + Gaussian peaky (BGP)" model for the surface where a strong land scatter is presented in a Brown background. This model is suitable for the coastal zone with vertical structures behaving like corner reflectors.

As have been noted, the most cumbersome problem in coastal waveform is the contamination of land in the radar footprint. Fortunately, usually the contamination does not present in the entire waveform. One can retrack a portion of the waveform bins which are free from land contamination (this portion is called sub-waveform, e.g. [23–25]). After rejecting the contaminated bins one can still retrieve useful information.

In applying the sub-waveform technology, the most important issue is to determine the extent of a sub-waveform. The algorithms designed by PISTACH (RED3 and ICE3 [15]) define a sub-waveform with a fixed range: from the 22nd to the 52nd bins. Therefore no more than 1/3 of the 104 bins (for Jason-2) or 128 bins for (Envisat and HY-2A) are included in the retracker. Consequently, for the land-free waveforms, the precision is worse than the traditional retrackers such as MLE3 and MLE4. ALES can be regarded as an improved version of RED3, in which the

### Coastal Altimetry: A Promising Technology for the Coastal Oceanography Community DOI: http://dx.doi.org/10.5772/intechopen.89373

sub-waveform extent is dependent on the significant wave height (SWH). As a consequence, under high sea state conditions, the sub-waveform can cover almost the entire waveform and is difficult to eliminate the contaminated bins.

In Xu et al. [26], a new strategy was proposed in which the extent of the subwaveform was identified by the "differential spectrum". For the Brown waveforms, the neighboring bins are unlikely to change rapidly except for the leading edge, so if there are land-contaminated bins, the corresponding differential spectrum will show a peaky pattern. The neighboring bins can thus be flagged as invalid ones in the retracking procedure. This strategy has been tested on HY-2A altimeter waveforms and has made significant improvement.

## 2.4.2.2 Model-free retrackers

The model-free retrackers (or empirical retrackers) do not assume a priori surface features. It can provide robust estimator regardless of physical background. The most famous model-free retracker is the OCOG (Offset Center Of Gravity) retracker that was proposed by Wingham et al. [27]. The idea is to approximate the waveform envelope to a rectangle shape, to find the center of gravity of the rectangle, and to subtract the half of the rectangle width (Figure 3) from the center of gravity:

$$t\_{\rm COG} = \frac{\sum\_{i=1}^{N} iV\_i^2}{\sum\_{i=1}^{N} V\_i^2} \text{ A} = \sqrt{\frac{\sum\_{i=1}^{N} V\_i^4}{\sum\_{i=1}^{N} V\_i^2}} \text{ W} = \frac{\left[\sum\_{i=1}^{N} V\_i^2\right]^2}{\sum\_{i=1}^{N} V\_i^4} \text{ t\_0} = t\_{\rm COG} - \text{W/2} \tag{5}$$

Another retracker is the simple threshold retracker. Finding the bin with the maximum power, say, M, and finding the first bin whose power exceeds M\*p%, where p% is a threshold percentage.

OCOG retracker is the most robust retracker, but it has been shown that OCOG retracker usually has relatively large bias (even larger than the simple threshold retracker). On the other hand, the threshold retracker can generate unexplainable results occasionally. The modified threshold retracker adopts the advantages of both retrackers. It uses A in Eq. (5) rather than M as the maximum bin power. The modified threshold retracker is the most widely used model-free retracker. The official Envisat ICE1 retracker and PISTACH ICE1 and ICE3 retrackers all belong to modified threshold retrackers.

Although it is pretty easy to implement, the most troublesome issue of the (modified) threshold retracker is to determine the value of p%. Apparently 50% is reasonable, but various studies gave different values. Tseng et al. [29] pointed that

Figure 3. Principle of an OCOG algorithm (from [28]).

20% is better, and the PISTACH ICE retrackers preferred p = 30% [15]. It seems that the threshold somewhat depends on the characteristics of the study area.

Another model-free algorithm is the curve spline interpolation. It has been mentioned briefly by a couple of authors without details. The idea is to interpolate between neighboring bins when implementing the threshold retracker. It can unlikely bring significant difference from the threshold retracker.

#### 2.4.2.3 Retracking strategy

Scientists have been debating for many years on the retracking strategy. Someone insisted that applying one single algorithm for all kinds of waveforms is a better choice. For instance, ALES developers use their retracker even over open ocean surface, and their analysis shows that the precision of ALES is not substantially worse than the official MLE4 over open ocean surface. On the other hand, PISTACH carried out a waveform classification before retracking. The classification is primarily dependent on the waveform pattern and secondarily on auxiliary information such as land cover model. Different retrackers are implemented for different waveform classes.

Our opinion is that a classification may be preferable because the waveforms have a large variety. A specialized algorithm for a certain waveform would improve the retracker precision significantly. There may be bias between different retrackers and inconsistency in the retracker transition. If the bias between retrackers is not compensated, there might be unexplained jumps in the sea level measurements, and this is the reason why some researchers are inclined to use one unique retracker. We hold that this problem can be solved either by simulative analysis or by calibration.

### 2.5 Geophysical corrections at the coast

The geophysical corrections near the coast also need specific considerations. Many terms of the geophysical corrections have larger uncertainty at the coast than over the open ocean.

#### 2.5.1 Atmospheric propagation corrections

The most uncertain error source comes from the wet tropospheric correction because the onboard radiometer suffers severely from land contamination in the coastal area. A simple but effective approach is to extrapolate a model-based correction (using, for example, atmospheric reanalysis data from the European Center for Medium-Range Weather Forecasts, ECMWF), but the corresponding spatial resolution is relatively low for coastal applications. Other approaches include an improved radiometer-based correction accounting for the land contamination effect [30], or the computation of GNSS-derived Path Delay (GPD, Fernandes et al. [31]).

Concerning the ionospheric correction, the imperfect coastal altimeter range measurements lead to significant errors, generating outliers in the correction values. The median + MAD (mean absolute bias) criterion is more preferable than the mean + standard deviation criterion, because the outliers are easier to detect and remove. The along-track profile of ionospheric corrections is further spatially low-pass filtered with a cutoff frequency of 100 km.

#### 2.5.2 Sea state bias

Another important correction is the sea state bias (SSB). The SSB depends on the retracking algorithm, because it contains the tracker bias. A careful analysis showed

## Coastal Altimetry: A Promising Technology for the Coastal Oceanography Community DOI: http://dx.doi.org/10.5772/intechopen.89373

that for Jason-2 GDR products, the SLAs obtained from MLE3 and MLE4 retrackers have large bias. From a statistical analysis using cycles 1–238 for a couple of altimeter passes over the open ocean, we obtained: SLAMLE3SLAMLE4 = +2.3 cm. Near the coast, this bias appeared to be even larger and even more critical as it is not constant. Figure 4 shows both MLE3 and MLE4 SSB corrections as a function of SWH for an arbitrary pass (cycle 16, pass #153). MLE3 SSB has a clear bias (+3 cm) relative to MLE4 SSB. Moreover, MLE3 SSB seems to have more outliers, in particular near the coast. The bias observed between MLE3 and MLE4 sea level estimates mainly corresponds to a bias in the SSB corrections.

Deeper investigation showed that the MLE3 SSB outliers are often related to large altimeter waveform-derived off-nadir angle estimation values [32], which probably suffer from errors given the good attitude control of Jason-2 satellite. For this reason, we adopted the MLE4 SSB in the computation of all SLAs, resulting in a relative bias <1 cm for all retrackers.

## 2.5.3 Ocean tide and DAC

The coastal ocean tide corrections, provided by global models, are also far from accurate. There are two families of tide solutions in most altimetry products: the family of the Goddard Ocean Tide (GOT) models developed by Ray et al. [33], and the family of the Finite Element Solution (FES) models developed by Lyard et al. [34].

Ray compared different tide solutions against 196 shelf-water tide gauges and 56 coastal tide gauges. Their accuracy was characterized by the RSS (root sum square) error of the eight main tidal constituents (Q1, O1, P1, K1, N2, M2, S2, and K2). For the shelf-water gauges, the accuracy of GOT4.8 was 7.04 cm (European coasts) or 6.11 cm (elsewhere), while the accuracy of FES2012 was 4.82 cm (European coasts) or 4.96 cm (elsewhere). For the coastal tide gauges, the accuracy of GOT4.8 and FES2012 was 8.46 and 7.50 cm, respectively [35]. In comparison, the accuracy of GOT4.7 and FES2004 in shelf-water was 7.77 and 10.15 cm, respectively. These results illustrate the significant improvement in coastal ocean tide solution during the last decade.

Figure 4. SSB difference with respect to the significant wave height (SWH).
