## 3 Results
### SWOT Data Filtering  
The Laplacian diffusion filter was applied to remove the noise and, thus, reduce the difference between the spectrum obtained from SWOT estimates with and without noise (Figure 6). Given the results obtained from the non-filtered data spectra, \(\lambda_{c}\) is first chosen to be 60 km. We then choose smaller \(\lambda_{c}\)s (50, 40, 30 and 15 km) to see how much lower we can go with this filter. We go down to 15 km, which is the expected wavelength at which SWOT will measure SSH. For comparison, we also  
Figure 6: (**Top**) Spectra of the data before filtering, from cycle 1 to 122, corresponding to box 1 (pass 15), (**left**), and to box 2 (pass 168), (**right**). Error bars denote 95% confidence intervals. (**Bottom**) Corresponding Signal to Noise Ratio (SNR), with a horizontal black line indicating where the noise is more than 15% of the signal, and the vertical line the corresponding wavelength. choose \(\lambda_{\rm c}=200\) km, which is the wavelength resolved by present-day altimeter constellation fields [11]. We lastly choose \(\lambda_{\rm c}=100\) km as an intermediate value between 60 and 200 km. In Figures 7 and 8, we show the effect of the filter on SSH at different values of \(\lambda_{\rm c}s\). For Figure 7, the effect of the filter is mainly seen in the pseudo-SWOT data, especially in the northern part where smaller structures are present. In Figure 8, as there are more, smaller structures, we can see more differences between the filtered outputs with respect to the model interpolated and pseudo-SWOT data. These differences are not only in the shape of the structures that are present, but also in their intensity. On the top row of Figures 7 and 8, the original model is also included to show that differences do also arise from the interpolation onto the SWOT grid. In Figure 7, especially for the 200 km \(\lambda_{\rm c}\), we can observe that the original structure present is significantly altered. This emphasizes the importance for development of interpolation techniques to fill the gap between the two swaths with the help of the nadir altimeter data. In Figures 7 and 8, the SSH images for the different filters look very similar, but the differences are amplified when the first derivatives (Figures 9 and 10) and second derivatives (Figures 11 and 12) are calculated. After applying the Laplacian diffusion filter, we can now retrieve the structures present in the pseudo-SWOT SSH in the absolute geostrophic velocity plots (Figures 9 and 10). With a 15 km \(\lambda_{\rm c}\) filter, the effect of the noise can be still clearly observed, especially for box 2 where smaller structures are present. As a result, although the main structures are recovered after filtering, their shapes are not accurately retrieved. Even if spurious structures remain, with a 30 km \(\lambda_{\rm c}\) there is a large improvement with respect to the 15 km \(\lambda_{\rm c}\). This improvement seems greater for box 1 than box 2, as the noise seems to have a greater effect within box 2 than box 1. For a 40 km \(\lambda_{\rm c}\), in box 1, we can no longer qualitatively see any remaining noise, but we can see some in box 2. For \(\lambda_{\rm c}s\) greater than or equal to 50 km, the effect of the noise is no longer observed in either box 1 (Figure 9) or box 2 (Figure 10). On the other hand, we observe a large decrease of the magnitude of the velocities from the 15 to the 200 km \(\lambda_{\rm c}\). With this filtering method, the intensity of the structures present, and thus the signal, decreases with the increase of \(\lambda_{\rm c}\). In the relative vorticity plots, the loss of signal with the increase of \(\lambda_{\rm c}\) is even more evident. With no filtering, the relative vorticity of box 1 ranges from \(-1.82\)f to \(1.66\)f for SSH\({}_{model}\) and from \(-15.22\)f to \(18.16\)f for SSH\({}_{obs}\) (Figure 4). With a 200 km \(\lambda_{\rm c}\), this reduces to \(-0.23\)f to \(0.14\)f for both SSH\({}_{model}\) and SSH\({}_{obs}\) (Figure 11).