# Wave Orbital Velocity Effects on Radar Doppler
## 2 Materials and Methods  
DDA is similar in concept to conventional radar altimetry, in that a power/time curve (in the following: \"waveform\") of a reflected radio wave pulse is produced, and by analyzing such a curve, an accurate monitoring can be carried out of both SSH and SWH. In order to clarify the subject of  
Figure 1: Experimental waveforms both in delay Doppler altimeter (DDA) mode (blue line) and conventional altimeter mode (red line) for various swell events detected by CryoSat-2 altimeter (track 681, cycle 030) as reported in [11]. the present paper, it is necessary to briefly recall main principles of DDA and its basic definitions, according to current literature [21; 22; 23; 24]. DDA provides a decisive improvement of the resolution along the direction of the flight track (also known as along-track or azimuth direction) by taking into account the Doppler shift caused by the relative velocity of the satellite--or aircraft--borne antenna. The waveform is formed, in the same way as in conventional altimetry, by the variation in time of the illuminated sea surface (see Figure 2), but in DDA, the return signal is split into \(\Delta_{\text{dy}}\) wide strips perpendicular to the along-track direction Y. The resolution along the Y direction is given by the Doppler shift resolution \(\text{f}_{\text{res}}\), i.e., by the accuracy with which the altimeter can separate the return beam frequencies. Following [21; 22], the Doppler frequency \(\text{f}_{\text{d}}\) and the distance along Y from the Nadir are linked by the following two equations:  
\[\text{f}_{\text{d}}=\frac{2\text{V}_{\text{S}}}{\lambda}\frac{\text{Y}}{\text {h}}, \tag{1}\]  
\[\text{Y}=\frac{\lambda\text{h}}{2\text{V}_{\text{S}}}\text{f}_{\text{d}}, \tag{2}\]  
where Vs is the Earth/satellite relative velocity, h is the distance from the ground to the antenna, and \(\lambda\) is the electromagnetic wavelength. It is easy to see that the effect is governed by the projection Vc of Vs along the connection from the sea surface to the antenna, which can be taken to be approximately equal to V\({}_{\text{S}}\)Y/h. A point (or a surface element on the sea) gives a contribution to the reflected intensity (and is therefore considered in the measurement procedure) only if its Doppler shift frequency \(\text{f}_{\text{d}}\) is comprised between \(\text{f}_{\text{res}}\)/2 and \(-\text{f}_{\text{res}}\)/2, and therefore its velocity component Vc is comprised between \(\text{Vs}\Delta_{\text{dy}}\)/(2h) and \(-\text{Vs}\Delta_{\text{dy}}\)/(2h), \(\Delta_{\text{dy}}\) being the nominal width of the resolution along Y (see Figure 3). We have thus:  
\[\Delta\text{d}_{\text{y}}=\frac{\lambda\text{h}\text{f}_{\text{res}}}{2\text {V}_{\text{S}}}. \tag{3}\]  
In a typical DDA altimeter, the footprint corresponds to an elongated strip on the surface, with an across-track resolution of 5-6 km and an along-track resolution of around 300 m. In presence of waves, the water surface is not still, and each water particle is affected--among others--by a vertical orbital velocity component VZ; such a component algebraically adds to the Vc velocity due to the  
Figure 2: Doppler resolution strip along-track (from ESA CryoSat-2 Product Handbook-Baseline D 1.0 -C2-LI-ACS-ESL-5319). antenna relative movement, thus introducing a further Doppler shift \(\rm f_{R}\), indicated in the following as \"R-effect\" [14; 16]. It is worth noting that a similar disturbance effect, sometimes known as \"velocity bunching\", on imaging SAR radars has been discovered as far back as in the 1980s [25], and has been considered and studied ever since [26]. It constitutes a limit to the applications of SAR, but it was recently found that it can been used to extract further information about the wind and the sea state (cut-off effect, [27; 28]).