# Envisat RA-2 Individual Echoes: A Unique Dataset for a Better Understanding of Inland Water
## 2 Theoretical Model of Specular Returns from Inland Waters
### The Theoretical Complex Radar Altimeter Echo, Z(n,r)  
All current radar altimeters are nadir pointing and ping the Earth at regular intervals. Envisat and other \"older generation\" radars pinged at about 2 KHz. More recent altimeter systems are using \(\sim\)20 KHz PRF. The echo of each radar ping is range-gated in all cases. Figure 1 illustrates a satellite in orbit and the Earth illuminated by a single radar pulse. \(S_{0}\) is the annulus of the range bin at the initial contact, \(S_{r}\) is the annulus of a generic range bin at a further range from nadir. The thickness of an annulus corresponds to one radar range bin. The radar nadir point sweeps along a ground track. With Envisat's PRF this means a ping every 3.8 m along the ground track. The successive series of IEs are designated by echo number \(n\). Figure 1: Range annuli for a single echo. The pattern moves along the satellite’s nadir track line with each successive radar echo. Let \(z\)(\(n\),\(r\)) designate the observed \(n\)th complex radar echo in the range bin \(r\), corresponding to range annulus \(S_{r}\). Here, we describe the model \(\mathcal{Z}(n,r)\) that would exactly match an observed \(z\)(\(n\),\(r\)) if all model assumptions are satisfied. Our model assumptions are the following. * There is zero reflectivity from land: all reflectivity is from water surfaces. * Water surfaces are flat: there is no wind roughening. * The water surface area is known exactly: location and extent. In this work, we analyze the electromagnetic behavior of the water surface under the stated strong assumptions and we will show that results from practical cases are in good agreement with the expectations, even under the strongest simplifying hypothesis. At the \(n\)th ech,\(o\) the radar altimeter is at location \(\overrightarrow{p_{n}}\), and the range from altimeter to a location on the Earth rigid surface at \(\overrightarrow{x}\) is \(R\left(\overrightarrow{x}|\xi\right)=\left|\left|\overrightarrow{x}+\mathbf{ k}\cdot\xi-\overrightarrow{p_{n}}\right|\right|\), where \(\xi\) is the water level with respect to a reference height, and \(\mathbf{k}\) is the unit vector perpendicular to the geoid. The phase of the round trip echo from one specular elemental scatterer at this location is \(\frac{4\pi}{\lambda}R\left(\overrightarrow{x}\right)\). The complete complex return \(z\)(\(n\),\(r\)) is a sum of elemental scatterers within the radar footprint ([PERSON], 1951), weighted appropriately by the radar's range point spread function. The model for \(z\)(\(n\),\(r\)) is thus:  
\[\mathcal{Z}(n,r|\xi)\ \propto\ \int\frac{\sin(\pi\left(\left(r\Delta-R\left( \overrightarrow{x}\right)\right)/\Delta\right)}{\pi(\left(r\Delta-R\left( \overrightarrow{x}\right)\right)/\Delta}M\left(\overrightarrow{x}\right)e^{ \left(-i\frac{4\pi}{\lambda}R\left(\overrightarrow{x}\right)\right)}\ dS \tag{1}\]  
where, \(\Delta\) is the radar range bin width; \(M\left(\overrightarrow{x}\right)\) is a map of the Earth surface, having a value of 1 where there is water, and 0 otherwise; and \(\lambda\) is the radar wavelength. There are three terms within the area integral. The \(e^{\left(-i\frac{4\pi}{\lambda}R\left(\overrightarrow{x}\right)\right)}\) is the complex radar echo of one elemental surface, with a phase shift due to the round trip distance between radar and surface. The \(M\left(\overrightarrow{x}\right)\) limits the integration to the scatterers within the physical boundaries of water surfaces. The \(sin(x)\)/\(x\) term is the radar in-range response, weighing each elemental scatterer according to the distance from the range bin center. The average weight on a scatterer located within the \(r\)th annulus is the result of integrating the power in Equation (1) in the interval [\(-1/2\)\(\dots\)\(1/2\)] about the bin center. For scatterers that physically fall in the adjacent range bin (adjacent annulus) the average weight is lower but not negligible; two bins away it is expected to be even lower, and so forth.