#### 3.2. Pre-Filtering Analysis of Simulator Outputs

In this study, we focus on the analysis of spatial scales of individual passes. Due to the irregular time sampling of the SWOT data, future studies will be devoted to temporal interpolation of passes. Moreover, prior swath filtering is necessary to determine the quality of the dynamical variables that can be derived from SWOT data, and how it can be improved before combining different swaths for temporal interpolation. As an illustration, we focus on the treatment of two 2° × 2° boxes. Box 1 is within pass 15 and was chosen close to the north African coast as it is a region where anticyclonic eddies are shed from the Algerian Current [

25,

35]. For example, in the snapshot shown in

Figure 4, part of an anticyclonic eddy is present on the eastern part of the domain. Box 2 is within pass 168, and this subdomain south of the Balearic island of Menorca was chosen because it contains smaller structures than in box 1 (see

Figure 2). In

Figure 5, filament-like structures and smaller eddies can be observed, especially at the northern part of the domain.

The effect of the filter is assessed for SSH and its derived dynamical variables: absolute geostrophic velocity and relative vorticity. These were calculated as explained in

Section 2.3.1.

As observed in the first and middle columns (model and model interpolated onto SWOT grid data, respectively) of

Figure 4 and

Figure 5, SSH and its derived variables reveal fine scale features, but the noise level masks the signal of these features when derived variables are obtained from pseudo-SWOT SSH. We can also see how the effect of the noise is lower in regions with high SSH gradients. If we compare the velocity derived from pseudo-SWOT data of box 1 and 2, for box 1, the region with high values can still be appreciated as they reach 0.9 m/s, but not for box 2 as they only reach 0.4 m/s.

To have information on the spatial scales resolved and the effect of the noise, spatial Fourier power spectra for each filter were calculated as described in

Section 2.3.2. The spectra were calculated for each individual cycle, and then averaged over the 122 cycles in which both passes 15 and 168 are available (cycle 123 stops at pass 132).

Figure 6 compares the spectra of model data interpolated onto the SWOT grid and the pseudo-SWOT data. The SWOT noise starts to dominate at wavelengths lower than 60 km. In the top panel of

Figure 6, the red and blue curves separate at around 60 km for both boxes. If we look at the zoom inset, we see that for pass 15 the lines separate at slightly higher wavelengths than for pass 168.

Nevertheless, there does not seem to be a significant difference between the mean spectra of both passes. Note also how the power spectral energy level of SSH

${}_{obs}$ at wavelengths lower than 20 km stabilizes around 3.7 × 10

${}^{-9}$ for both passes, whilst the energy level of SSH

${}_{model}$ reduces until it reaches the grid scale. If we look at the signal to noise ratio (SNR), we find that below 50 km wavelength the energy of the noise is significant with respect to that of the signal (SNR values below 15 dB at wavelengths smaller than 47.6 km, i.e., the energy of the noise accounts for more than 15% of the energy of the signal for these scales). Such low SNR values make particularly challenging the denoising issue for scales below 50 km [

36]. Consequently, we expect that the best filter parametrization will be one corresponding to

${\lambda}_{\mathrm{c}}$ between 47.6 and 60 km.

#### 3.3. 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,

${\lambda}_{\mathrm{c}}$ is first chosen to be 60 km. We then choose smaller

${\lambda}_{\mathrm{c}}s$ (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 choose

${\lambda}_{\mathrm{c}}$ = 200 km, which is the wavelength resolved by present-day altimeter constellation fields [

11]. We lastly choose

${\lambda}_{\mathrm{c}}$ = 100 km as an intermediate value between 60 and 200 km.

In

Figure 7 and

Figure 8, we show the effect of the filter on SSH at different values of

${\lambda}_{\mathrm{c}}s$. 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

Figure 7 and

Figure 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

${\lambda}_{\mathrm{c}}$, 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

Figure 7 and

Figure 8, the SSH images for the different filters look very similar, but the differences are amplified when the first derivatives (

Figure 9 and

Figure 10) and second derivatives (

Figure 11 and

Figure 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 (

Figure 9 and

Figure 10). With a 15 km

${\lambda}_{\mathrm{c}}$ 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

${\lambda}_{\mathrm{c}}$ there is a large improvement with respect to the 15 km

${\lambda}_{\mathrm{c}}$. 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

${\lambda}_{\mathrm{c}}$, in box 1, we can no longer qualitatively see any remaining noise, but we can see some in box 2. For

${\lambda}_{\mathrm{c}}s$ 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

${\lambda}_{\mathrm{c}}$. With this filtering method, the intensity of the structures present, and thus the signal, decreases with the increase of

${\lambda}_{\mathrm{c}}$.

In the relative vorticity plots, the loss of signal with the increase of

${\lambda}_{\mathrm{c}}$ is even more evident. With no filtering, the relative vorticity of box 1 ranges from −1.82f to 1.66f for SSH

${}_{model}$ and from −15.22f to 18.16f for SSH

${}_{obs}$ (

Figure 4). With a 200 km

${\lambda}_{\mathrm{c}},$ this reduces to −0.23f to 0.14f for both SSH

${}_{model}$ and SSH

${}_{obs}$ (

Figure 11). For box 2, with no filtering, the relative vorticity ranges from −0.71f to 1.50f for SSH

${}_{model}$ and from −17.39f to 17.71f for SSH

${}_{obs}$ (

Figure 5). For 200 km, it reduces to−0.07f to 0.10f for SSH

${}_{model}$ and from −0.09f to 0.06f 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

${\lambda}_{\mathrm{c}}$ = 200 km. For box 1, the velocity appears to contain no further noise with

${\lambda}_{\mathrm{c}}$ = 40 km, but this filtering is not sufficient to properly reconstruct the relative vorticity. With a 50 km

${\lambda}_{\mathrm{c}}$, some noise is still present, and with 60 km

${\lambda}_{\mathrm{c}},$ there appears to be no remaining noise. For box 2, the velocity appears to have no further noise with a 50 km

${\lambda}_{\mathrm{c}}$, and, similarly to box 1, we use a 60 km

${\lambda}_{\mathrm{c}}$ to qualitatively remove remaining noise in the relative vorticity plots. The relative vorticity fields present unrealistic small-scale structures at larger

${\lambda}_{\mathrm{c}}s$ 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

${\lambda}_{\mathrm{c}}$ filter for both box 1 and 2. Not as much signal is lost with a 60 as with a 200 km

${\lambda}_{\mathrm{c}}$, but some is still lost. For the mesoscale, given the relative vorticity and structures observed in

Figure 11 and

Figure 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

${\lambda}_{\mathrm{c}}s$ 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

Figure 13 and

Figure 14.

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

${\lambda}_{\mathrm{c}}$, the noise level is still high, as observed in

Figure 7,

Figure 8,

Figure 9,

Figure 10,

Figure 11 and

Figure 12, but the power spectra difference at wavelengths smaller than 60 km between the pseudo-SWOT filtered (yellow) and the model-interp. (blue) curves is much smaller than it is between the model-interp (blue) and pseudo-SWOT (red) curves (top left panel of

Figure 13 and

Figure 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

Figure 13 and

Figure 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

Figure 13 and

Figure 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

${\lambda}_{\mathrm{c}}$ 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

${\lambda}_{\mathrm{c}}$ is necessary. In both the left and right panels of the 30 km

${\lambda}_{\mathrm{c}}$ of

Figure 13 and

Figure 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

${\lambda}_{\mathrm{c}},$ the noise is further reduced, but we lose more signal too. If we look at the continuous line of the right panel of

Figure 13 and

Figure 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

${\lambda}_{\mathrm{c}}$ filtering, the model-interp. and pseudo-SWOT spectra curves are identical, and the SNR (solid line) is approximately 1 (

Figure 13 and

Figure 14, bottom row): however, as observed in

Figure 7,

Figure 8,

Figure 9,

Figure 10,

Figure 11 and

Figure 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

${\lambda}_{\mathrm{c}}$ than with a 60 km

${\lambda}_{\mathrm{c}}$. 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).

In

Figure 15, it is interesting to focus on the minimum points of the RMSE curves. Looking at the

${\lambda}_{\mathrm{c}}$ corresponding to the different minimum points, for both boxes, the minimum of the curve for SSH and the absolute geostrophic velocity (Vg) is found for a 30 km (29)

${\lambda}_{\mathrm{c}}$. It is slightly higher, 40 km (41), for relative vorticity. This directly relates to the amplification of fine scale structures, and thus the effect of the noise, in the computation of second-order derivatives. It is also in accordance with what is found in the SNR in

Figure 6, which shows that we cannot recover the signal at wavelengths lower than 40–50 km. With the qualitative (

Figure 7,

Figure 8,

Figure 9,

Figure 10,

Figure 11 and

Figure 12) and the spectra (

Figure 13 and

Figure 14) plots, we saw that, for box 2, as the signal is lower than in box 1, the effect of the noise is greater, and larger

${\lambda}_{\mathrm{c}}s$ are necessary. The RMSE plots show us another point of view. As the signal is not as intense in box 2 as in box 1, the over-smoothing (signal lost) due to the Laplacian diffusion filter is lower, and thus we observe lower RMSE values in

Figure 15. Moreover, the improvement of the RMSE values is greater for box 2 than box 1. For SSH, the RMSE reduces by 0.05 and 0.07 m from the no filter (0

${\lambda}_{\mathrm{c}}$) to the minimum RMSE, for box 1 and box 2, respectively. For Vg, the RMSE reduces by 0.42 and 0.455 m/s, and for

$\zeta $/f by 3.2 and 3.35.

For SSH, we could say that with a 40 km

${\lambda}_{\mathrm{c}}$ it is sufficient, but if we do not want to see the effect of the noise in vorticity, we need a greater

${\lambda}_{\mathrm{c}}$ of 60 km. Therefore, together with what is observed in

Figure 4,

Figure 5,

Figure 6,

Figure 7,

Figure 8,

Figure 9,

Figure 10,

Figure 11,

Figure 12,

Figure 13,

Figure 14 and

Figure 15, we consider that, with this filtering technique applied to this region, SWOT will be able to resolve wavelengths down to a 40–60 km wavelength range. This is the

${\lambda}_{\mathrm{c}}$ range where we found that there is a compromise between filtering out the noise of SSH and its derived variables (Vg and

$\zeta $/f), and over-smoothing the original image as little as possible. It is important to note that

${\lambda}_{\mathrm{c}}$ depends on the signal-to-noise ratio between SSH signals and instrument noise at fine scales. As such, it can be expected to change from region to region (and from season to season) depending on the energy levels at fine scales and on the noise level.