## 3 Results
### SWOT Data Filtering
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). (blue) curves is much smaller than it is between the model-interp (blue) and pseudo-SWOT (red) curves (top left panel of Figures 13 and 14). Moreover, the pseudo-SWOT filtered and the model-interp. curves separate at smaller wavelengths. If we look at the dashed line of the top right panel of Figures 13 and 14, we can more accurately determine this wavelength separation value by looking at where the dashed curve starts decreasing. This corresponds to 30 and 40 km wavelengths for box 1 and 2, respectively. Between wavelengths of 60 and 30 km (40 for box 2), the SNR (dashed line) is greater than 1, indicating some over-filtering/smoothing, but this value is very low (1.064 for box 1 and 1.088 for box 2). The pattern of the SNR of the filtered model-interp. over the filtered pseudo-SWOT data (solid line of top right panel of Figures 13 and 14) is similar to that of Figure 6. The noise gains importance as the wavelengths reduce from 60 km, but the SNR values of the 30 km \(\lambda_{\rm c}\) are lower than the non-filtered ones. On the other hand, this indicates that, given that the filtered model-interp. and pseudo-SWOT spectra are still quite different, a larger \(\lambda_{\rm c}\) is necessary. In both the left and right panels of the 30 km \(\lambda_{\rm c}\) of Figures 13 and 14, anomalous patterns are observed below the 10 km wavelength. Not only are these spatial scales very small, but, if we look at the spectra values, they are about \(10^{-12}\) m\({}^{2}\)/km or lower. We consider these values to be too low for discussion. With a 60 km \(\lambda_{\rm c}\), the noise is further reduced, but we lose more signal too. If we look at the continuous line of the right panel of Figures 13 and 14, we see that it remains approximately constant at 1. For box 1, it reaches a minimum SNR of 0.715 and for box 2 of 0.697. Looking at the dashed line, the SNR becomes larger than 1 at wavelengths lower than 80 km for both boxes. This means that, although we eliminate all the noise, we are also eliminating part of the signal that was initially present. At wavelengths greater than 10 km, the power spectra of the filtered pseudo-SWOT reach a maximum difference nearly two orders of magnitude smaller than the original model-interp. spectra. Lastly, with a 200 km \(\lambda_{\rm c}\) filtering, the model-interp. and pseudo-SWOT spectra curves are identical, and the SNR (solid line) is approximately 1 (Figures 13 and 14, bottom row): however, as observed in Figures 7-12, we lose a lot of signal. In the 20-80 km wavelength range, we can see how the SNR curves (dashed line) rapidly increase and the values are greater with a 200 km \(\lambda_{\rm c}\) than with a 60 km \(\lambda_{\rm c}\). At wavelengths lower than 80 km, on average, there is about one order of magnitude difference between the filtered spectra and the original SSH\({}_{model}\). It is also interesting to note that the purple and yellow curves separate from the red and blue at 80 km instead of 60, showing that this cut-off exceeds that necessary to remove the noise. This also emphasizes how with SWOT a major advancement could be made as lower cut-off wavelengths will be possible, and thus the observation of smaller scale structures than with contemporary satellites. On the one hand, this result is expected thanks to the 2D swath instead of only 1D nadir data, but, on the other hand, it is important to remember that these are simulated from expected errors and that not all errors are implemented (see Appendix B). To further quantify the differences observed between the different \(\lambda_{\rm c}\)s in Figures 7-12, the RMSE between the interpolated model and pseudo-SWOT data shown in these Figures were calculated as described in Section 2.3.3. In Figure 15, it is interesting to focus on the minimum points of the RMSE curves.