# Envisat RA-2 Individual Echoes: A Unique Dataset for a Better Understanding of Inland Water
## 3 Range Determination for a Specular Target
### Simplified Algorithm
#### 3.2.3 Water Level  
In this section, we describe our proposed approach to complete the process for estimating the water level. First, we start with a moving 25-echo coherent summation as described above. Then, we choose a particular echo index, \(n\), corresponding to the CPA. The corresponding range waveform is P\({}_{r}\) (the n subscript is now discarded). A first approximation of the water level would be the range bin corresponding to the maximum of the echo power. In terms of mere quantization error, this would provide water level estimations with \(\mathrm{\,rmse}=\left(\frac{RGW}{2}\right)^{2}\,\sim 14\) cm, given the Envisat range gate width (\(RGW\)) about 0.47 m (Table 1). However, a much more accurate estimate is made possible by interpolating the peak samples. In particular, later in this paper, we will show \(\mathrm{\,rmse}<1\) cm with real world data.  
The expected range waveform for a specular echo can be approximated with a Gaussian function as follows (modified from [PERSON] et al. (2016), Equation (2)):  
\[(r)=\mathbf{P}_{0}e^{-(r-r_{0})\tau_{p})^{2}/2\sigma_{p}^{2}} \tag{10}\]  
with standard deviation \(\sigma_{p}=0.513\tau_{p}\), where \(\tau_{p}\,=3.125\) ns is the effective pulse length.  
This approximation is only valid at or near nadir; therefore, it is supposed to be sufficient within the span of the main lobe. The approximation should not be used to range water bodies located very far off the nadir (but that is not likely since the specular echo will not be detected anyway).  
In general, the interpolated range \(r_{0}\) and the peak power \(\mathbf{P_{0}}\) are found by minimizing the following cost function:  
\[\min_{r_{0},P_{0}}\sum_{i}|P_{r_{i}}-(r_{0}-r_{i})|^{2} \tag{11}\]  
where the sum is intended as limited to those waveform samples having significant power originated by the specular reflection; the rest are either noise or interference and should be avoided. In typical RA-2 signals, a specular echo extends over at most three range bins. In practice, two of the three are so much stronger than the third that the usage of only the two str ongest range bins, \(P_{L}\) and either \(P_{L-1}\) or \(P_{L+1}\), can be sufficient for the assessment of \(r_{0}\).  
With only two bins, the solution for range is simplified to the following:  
\[r_{o}=\ \frac{{r_{L}^{\prime}}^{2}\,-\ r_{L}^{2}\,+\ \frac{2\sigma_{p}^{2}}{ \tau_{p}^{2}}ln\left(\frac{P_{L}^{\prime}}{P_{L}}\right)}{2\left(r_{L}^{ \prime}-r_{L}\right)} \tag{12}\]  
where \(P_{L}^{\prime}=\ \left\{\begin{array}{ll}P_{L-1}\,\,if\ P_{L-1}>P_{L+1}&\\ P_{L+1}\,\,if\ P_{L+1}>P_{L-1}&\text{and}\ r_{L}^{\prime}\ \text{is the corresponding range bin}.\end{array}\right.\)  
It is important to notice that Equation (12) represents an analytical solution in closed form, unlike what commonly happens with altimetry waveforms, generally requiring some form of fitting or \"retracking\" exercise.  
Figure 8 is an example of a specular waveform from Envisat crossing over Rio Nahuapa (further discussed in Section 4.2) with coherent burst summing (Equation (5)) of 25 echoes. The example has been extracted from the dataset described in Section 4.1. The overlaid curve in Figure 8 is the best fit of Equation (10) to the data. There is a near perfect fit to all three top range bins, although only the highest two were used in the fitting. The interpolated range is approximately half way between bins 2 and 3.