## 3 Results
### SWOT Data Filtering
For box 2, with no filtering, the relative vorticity ranges from \(-0.71\)f to \(1.50\)f for SSH\({}_{model}\) and from \(-17.39\)f to \(17.71\)f for SSH\({}_{obs}\) (Figure 5). For 200 km, it reduces to\(-0.07\)f to \(0.10\)f for SSH\({}_{model}\) and from \(-0.09\)f to \(0.06\)f for SSH\({}_{obs}\) (Figure 12). There is approximately two orders of magnitude difference between the vorticity calculated from the original data, and that filtered at \(\lambda_{\rm c}=200\) km. For box 1, the velocity appears to contain no further noise with \(\lambda_{\rm c}=40\) km, but this filtering is not sufficient to properly reconstruct the relative vorticity. With a 50 km \(\lambda_{\rm c}\), some noise is still present, and with 60 km \(\lambda_{\rm c}\), there appears to be no remaining noise. For box 2, the velocity appears to have no further noise with a 50 km \(\lambda_{\rm c}\), and, similarly to box 1, we use a 60 km \(\lambda_{\rm c}\) to qualitatively remove remaining noise in the relative vorticity plots. The relative vorticity fields present unrealistic small-scale structures at larger \(\lambda_{\rm c}s\) values than SSH and velocity. This is expected as the noise effects increase as higher order derivatives are reached. Nevertheless, the larger structures present in the images are recovered from the non-filtered image with a 60 km \(\lambda_{\rm c}\) filter for both box 1 and 2. Not as much signal is lost with a 60 as with a 200 km \(\lambda_{\rm c}\) but some is still lost. For the mesoscale, given the relative vorticity and structures observed in Figures 11 and 12, this does not seem to have a large impact. However, there may be an impact when wanting to observe finer scales as we retrieve normalized relative vorticity much lower than 1. Spectra were computed for \(\lambda_{\rm c}s\) of 30, 60 and 200 km to visualize these effects. The corresponding SNR is also calculated in two different ways by using two references. One is by dividing the filtered model-interpolated data by the filtered pseudo-SWOT data, and the other by dividing the non-filtered model-interpolated data by the filtered pseudo-SWOT data. This is shown in Figures 13 and 14. Figure 7: S8H (m) on 23 January 2009 corresponding to pass 15 of cycle 2. Figure 8: SSH (m) on 3 February 2009 corresponding to pass 168 of cycle 2. Figure 9: Absolute geostrophic velocity (m/s) on 23 January 2009 corresponding to pass 15 of cycle 2. Figure 10: Absolute geostrophic velocity (m/s) on 3 February 2009 corresponding to pass 168 of cycle 2. Figure 11: Relative vorticity normalized by f on 23 January 2009 corresponding to pass 15 of cycle 2. Figure 12: Relative vorticity normalized by f on 3 February 2009 corresponding to pass 168 of cycle 2. Figure 13: Box 1 region (pass 15) mean of cycles 1 to 122. (**Left**) Spectra of SSH\({}_{model}\) (blue) and SSH\({}_{\text{obs}}\) (red) before filtering and after applying the different cut-off wavelengths shown in the different rows (30, 60 and 200 km) in purple and orange, respectively. Error bars denote 95% confidence intervals. (**Right**) SNR of SSH\({}_{model}\) and SSH\({}_{obs}\), both filtered (solid line) and of SSH\({}_{model}\) non-filtered and filtered SSH\({}_{obs}\) (dashed line). As a consequence of the application of the filter, the separation of the spectral curves of SSH\({}_{model}\) (model-interp.) and SSH\({}_{obs}\) (pseudo-SWOT) is reduced. As seen in Figure 6, with no filter, the model-interp. (blue) and pseudo-SWOT (red) curves separate at a wavelength around 60 km. With a 30 km \(\lambda_{c}\), the noise level is still high, as observed in Figures 7-12, but the power spectra difference at wavelengths smaller than 60 km between the pseudo-SWOT filtered (yellow) and the model-interp. Figure 14: Box 2 region (pass 168) mean of cycles 1 to 122. **(Left)** Spectra of SSH\({}_{model}\) (blue) and SSH\({}_{obs}\) (red) before filtering and after applying the different cut-off wavelengths shown in the different rows (30, 60 and 200 km) in purple and orange, respectively.