## 3 Methodology
### Water Level Filtering Based on Occurrence  
Defining the \(\mathrm{WL}_{a}\) of the target in terms of the \(\mathrm{WL}_{i}\) of maximum occurrence allows one to filter out most of the waveforms that present significant contamination, i.e, the waveforms dominated by off-nadir scatterers whose maximum power is higher than the peak power of the target surface. Such an approach is valid under the assumption that the \(\mathrm{WL}_{i}\) of maximum occurrence around the target is indeed the water level of the target, which is typically the case.  
The \(\mathrm{WL}_{a}\) of the target for each satellite pass is estimated using an algorithm based on the density, or occurrence, of the observed \(\mathrm{WL}_{i}\) along the transect, \(\mathrm{\widetilde{WL}}_{i}\). Occurrence refers to the number of \(\mathrm{WL}_{i}\) observations within a WL interval, represented by a moving WL window. Mathematically, this can be expressed as\[\text{Occurrence}(\text{WL})=\sum_{i=1}^{N}f(\text{WL}_{i},\text{WL}) \tag{4}\]  
where  
\[f(\text{WL}_{i},\text{WL})=\begin{cases}1,&\text{if }\text{WL}_{i}^{\text{obs}} \in[\text{WL}-\frac{\omega}{2},\text{WL}+\frac{\omega}{2}]\\ 0,&\text{otherwise}\end{cases}\]  
with \(\omega\) defined as the width of the WL window. An occurrence graph is built by evaluating the occurrence at a sequence of WLs. Initially, the WL range of \(\vec{\text{WL}}_{i}\) is calculated, and then the range is extended by twice the window's width on each side. Subsequently, evenly spaced WL values are computed along this extended range.  
The algorithm iteratively computes the \(\text{WL}_{a}\) through several steps when multiple WLs are observed in one pass. Initially, an occurrence graph is constructed with the specified window width and step size. If the maximum occurrence in the graph is 1, \(\text{WL}_{a}\) is determined as the median of the observed \(\text{WL}_{a}\)\(\vec{\text{WL}}_{i}\). However, if there is only one point with the maximum occurrence, the corresponding \(\text{WL}_{i}\) is chosen as \(\text{WL}_{a}\).  
In case multiple points share the maximum occurrence, and there exists a single plateau of \(\text{WL}_{i}\)s with the maximum occurrence, the mean WL of the plateau is considered to be \(\text{WL}_{a}\). Plateaus are defined such that peaks separated by a distance below 5 cm are regarded as part of the same plateau. However, if there are multiple plateaus or peaks, accurate filtering of the WL becomes challenging, and \(\text{WL}_{a}\) is determined as the median of \(\vec{\text{WL}}_{i}\).  
The emergence of multiple plateaus can be attributed to several bright scatterers at different elevations, including puddles, pools, and tributaries. Despite their small size, under favorable conditions, these unexpected water surfaces can exhibit brightness comparable to or even surpassing that of the targeted reservoir.  
The inter-value distance was set to 0.1 cm, since it produces a finer WL spacing than the uncertainty imposed by geophysical corrections, which are examined in the next subsection. The moving window width was set to 20 cm for FFSAR and to 80 cm for UFSAR-derived \(\text{WL}_{a}\)s. The width of the moving window depends on the observed accuracy of the \(\text{WL}_{i}\)s, which is less than 5 cm for FFSAR and 20 cm for UFSAR-derived \(\text{WL}_{i}\)s. Considering that the error could be either positive or negative, these values are doubled to add a smoothing effect to the occurrence calculation. Otherwise, noise in the observations could lead to neighboring peaks of occurrence. Figure 5 shows an occurrence graph obtained over Itoiz Reservoir on 8 April 2023.  
Figure 5: Occurrence graph generated from the Sentinel-6 satellite pass over the Itoiz Reservoir on 8 April 2023. The \(\text{WL}_{i}\) measurements depicted in this graph are derived from FFSAR data. Additionally, for reference, the in situ \(\text{WL}_{t}\) and the \(\text{WL}_{a}\) derived from UFSAR using the OCOG and ocean retracker methods have been included as vertical lines.