# Wave Orbital Velocity Effects on Radar Doppler
## 2 Materials and Methods
Since the radial direction from the water surface to the antenna is nearly vertical, \(\rm f_{R}\) is given by following Equation (4) (Figure 4):  
\[\rm f_{R}=\frac{2V_{Z}}{\lambda}. \tag{4}\]  
The nearly vertical apparent velocity of a water particle with respect to the antenna is therefore \(\rm V_{C}+V_{S}Y/h\), and its total Doppler frequency shift \(\rm f_{t}\) is given by Equation (5):  
\[\rm f_{t}=f_{d}+f_{R}=\frac{2V_{S}Y}{\lambda h}+\frac{2V_{Z}}{\lambda}=\frac{2 }{\lambda}\bigg{(}\frac{V_{S}Y}{h}+V_{Z}\bigg{)}. \tag{5}\]  
Figure 4: Effect of particle vertical velocity on the apparent position of an elementary surface area. The red point, positioned in 1, has a positive vertical velocity component \(\rm V_{Z1}\), which adds to its \(\rm V_{C}\); it therefore appears to be farther (position 1) from the nadir; the green point, positioned in 2, has a negative \(\rm V_{Z2}\); it therefore looks closer (position 2). Figure 3: Along-track resolution as determined by antenna velocity relative to the Earth. If \(\rm f_{t}\) for a given point (red point in Figure 4) is greater than \(\rm f_{res}\) /2 or less than \(\rm-f_{res}\)/2, the point will be not taken into account, and the value of its instantaneous elevation \(\rm\eta\) will not be considered by the algorithms, even if its geometrical position (position 1 in Figure 4) is located within the nominal width \(\rm\Delta_{dy}\); conversely, a point located outside the nominal position (green point in Figure 4) will appear, due to the R-effect, to be located inside (position 2' in Figure 4), and will be considered in all the statistics. It is easy to see that a potential bias does exist, and it has to be evaluated; it remains to be seen whether the phenomenon is significant enough, i.e., if the order of magnitude of the R-effect is the same as that of the Vs induced Doppler, and if the error averages out in practical application. Considering, for instance, the parameters of the CryoSat-2 satellite operating in SAR mode (see Table 1), and assuming, e.g., a 2 meter-high sinusoidal 225 meter-long wave in deep water (T \(\approx\) 12 s), the vertical orbital velocity component \(\rm V_{Z}\) varies between \(\rm-0.52\) m/s and \(\rm+0.52\) m/s, while for a shorter and higher wave (H = 4 m; L = 125 m; T \(\approx\) 9 s) the effect is more important (\(\rm V_{Z}\) varies between \(\rm-1.41\) m/s and \(\rm+1.41\) m/s). Figure 5 shows, graphically, some comparisons between the orbital velocity induced Doppler frequency shift \(\rm f_{R}\) and the satellite velocity Doppler effect \(\rm f_{d}\). \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Satellite Height h & 700,000 m \\ Satellite Velocity \(\rm V_{S}\) & 7000 m/s \\ Electromagnetic Wavelength \(\rm\lambda\) & 0.0221 m \\ Frequency Resolution \(\rm f_{res}\) & 312.5 Hz \\ Nominal Space Resolution \(\rm\Delta_{dy}\) & 360.1 m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main parameters of CryoSat-2 altimeter in DDA mode. A more accurate assessment is thus necessary, and some numerical experiments described in the next paragraph help clarify the issue.