## 3 Methodology
### Retrackers
#### 3.2.3 Two-Step Physical-Based Retracker  
The in-house isardSAT SAR ocean retracker in [65] is integrated in a two-step fitting procedure to operate over inland waters, providing robust surface height estimation with a minimal modification to the SAR ocean retracker model. The waveforms reflected from an inland water body might exhibit ocean-like shapes (winds blowing over lakes) or, most commonly, a peakier shape (smoother surfaces) for small water bodies. To adapt to the latter ones, the MSS, describing the sea surface roughness [66], was used instead of the SWH as a fitting parameter; this approach is initially considered in [59], when exploiting the SAMOSA model over Tibetan plateau lakes. The SAR ocean retracker in [65] considers a scattering model of rough surface with Gaussian statistics, and so the normalized backscattering coefficient (surface radiation pattern) as function of the incidence angle \(\theta_{inc}\) can be modeled as [67]:  
\[\sigma^{0}(\theta_{inc})=\sigma^{0}(0)\cdot\exp\left(-\frac{(\tan\theta_{inc} )^{2}}{MSS}\right) \tag{4}\]  
where the MSS represents the variance of the surface slopes. From the relationship in Equation (4), it can determined that, when the variance of the surface slopes (MSS) is large, a broad surface radiation pattern is obtained (rough surfaces), while, when MSS tends to small values, the pattern is more directed as expected for specular returns (smooth surfaces). Therefore, the MSS can be accordingly adjusted as a fitting parameter to properly fit specular returns. With the objective of exploiting the SAR ocean retracker [65] with minimal changes over inland waters, the same retracker model can be used, but the SWH is fixed (to a value close to zero), while the MSS or surface roughness is fitted. A similar approach with the SAMOSA model is exploited in [89] to fit waveform returns from leads. To ensure that the different typology of waveforms (ocean- and specular-like) can be fitted, a two-step retracking is implemented. In a first run, the normal operation of the physical-based retracker in [65] was considered, i.e., the SWH was one of the fitting parameters (keeping MSS constant). Then, a second run of the fitting was implemented but this time the SWH was fixed (to a very small value around 1e-5) and the MSS was fitted so that the radiation pattern of the rough surface was adjusted. Then, the estimate of the surface height was extracted from the fit with the best correlation coefficient between the measured waveform and the modeled waveform. In this way, it was ensured that peaky-like waveforms could be properly fitted using the MSS as a fitting parameter, since the normal SAR ocean retracker would fail to adjust the model. An example of the operation of the two-step retracker is shown in Figure 3 over a peaky waveform. The in-house isardSAT SAR ocean retracker integrated in the two-step procedure [65] is an implementation based on the SAR ocean model proposed by [PERSON] et al. [53]. It is a physically-based model of the altimetric power waveform over the ocean received by the altimetric sensor after delay-Doppler (or SAR) processing. Details regarding the definition and implementation of this retracker can be found in [65]. In SAR or delay-Doppler altimetry, different Doppler beams, also known as looks (\(l\)), are synthesized to focus on a given surface on the ground [65]. Thus, averaging of all these Doppler power beams would lead to the final multilook power waveform to be retracked.