# Wave Orbital Velocity Effects on Radar Doppler
## 3 Simulation and Results
### Single Waveforms  
In the following, a few single WF results are reported for monochromatic waves, in order to highlight the basic aspects of the problem. The sea state is assumed to be made up of sine wave-trains in deep water, directed along the flight direction Y. The geometric (size and spatial resolution of the test area) computational parameters are reported in Table 2.  
Figure 6 provides a comparison between the waveforms for a single swell wave-train aligned with satellite track (Y direction), computed with, and without, R-effect.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters of the Simulations** & **Value** \\ \hline Size of test area in across-track (X) direction Lx & 6000 m \\ Size of test area in along-track (Y) direction Ly & 1000 m \\ Computational grid along the across-track (X) direction DX & 1 m \\ Computational grid along the track (Y) direction DY & 1 m \\ Resolution of distance between antenna and sea surface & 0.47 m \\ \hline \hline \end{tabular}
\end{table}
Table 2: Main parameters of the single measurement numerical simulationsThe difference between the two reconstructions is very small, to the point of being negligible: it is most likely that the positive effect (points geometrically outside the nominal Doppler strip moving into the resolution area) is balanced by the negative effect (points exiting the nominal strip).  
Things change, sometimes significantly, when the interaction between two wave trains is considered. A common situation is the superimposition of two wave trains, as represented in the following, by adding a wave train of height H\({}_{1}\) and a length L\({}_{1}\) of a few hundred metres--such as can be expected from a long ocean swell--to a shorter one with height H\({}_{2}\) and wavelength L\({}_{2}\). Figure 7 shows some results.  
Figure 6: Simulated waveform with (solid green line), and without (dashed red line), R-effect for two different single sine wave trains: (**a**) wave height H = 3 m, wavelength L = 300 m, wave period T \(\approx\) 14 s; (**b**) H = 4 m, L = 300 m, T \(\approx\) 14 s.  
Figure 7: Simulated waveform with (solid green line), and without (dashed red line), R-effect when a longer sinusoidal wave train (wave height H, wavelength L\({}_{1}\)) is superimposed to a shorter one (wave height H\({}_{2}\) and wavelength L\({}_{2}\)): (**a**) H\({}_{2}\) = 4 m, L\({}_{2}\) = 50 m; (**b**) H\({}_{2}\) = 6 m, L\({}_{2}\) = 50 m. (**c**) H\({}_{2}\) = 2 m, L\({}_{2}\) = 150 m; (**d**) H\({}_{2}\) = 6 m, L\({}_{2}\) = 150 m.  
The influence of the R-effect is clearly negligible in the case represented in Figure 7a, while it is relevant in the other examples. It is also useful to notice that some of the resulting waveforms bear a similarity to the shapes reported in [11; 13] and represented in Figure 1. Similar tests carried out by varying the wavelength and the height of the two components consistently provide a pseudo waveform well different from the classical shape of a Gaussian sea, thus opening new possibilities to use high frequency DDA data to monitor the presence of very long swells, as already suggested in [11; 12]. While it appears that the R-effect can, in certain circumstances, affect the formation of DDA response, it is however necessary, in order to get a more realistic appraisal, to take into account what happens when successive measurements are taken from different positions of the satellites, as well as when more realistic sea state is considered.