<figure><img src="image_33.png" /><figcaption>**Fig. 28.** Maps of the magnitudes of 4-day averages of surface velocity computed geostrophically from error-free and noisy simulated SWOT measurements of SSH with isotropic 2-dimensional smoothing using a Parzen smoother with a half-power filter cutoff wavelength of 25 km: Column (a) The 4-day average over the full model domain computed geostrophically from error-free model SSH fields at a time step of 0.5 day over the 4-day period; Column (b) The 4-day average over the full model domain computed geostrophically from model SSH fields at a time step of 0.5 day over the 4-day period with simulated uncorrelated SSH measurement errors with a standard deviation of $\sigma_h = 2.74$ cm; Column (c) The 4-day average computed geostrophically from simulated SWOT swath sampling of error-free model SSH fields at the times and locations of each satellite observation over the 4-day period; and Column (d) The 4-day average computed geostrophically from simulated SWOT swath sampling of model SSH fields at the times and locations of each satellite observation over the 4-day period with simulated uncorrelated SSH measurement errors with a standard deviation of $\sigma_h = 2.74$ cm. The bottom panels are the error maps computed by subtracting the error-free map in Column (a) from the error-contaminated maps in the top row of the respective Columns (b)–(d).</figcaption></figure>

as noted previously by about 10 days in the 14-day average result in misinterpretation of temporal variability as spatial variability in the centered difference estimates of the derivatives of the SSH field at these boundaries. These sampling issues in maps of geostrophically computed velocity constructed from simulated SWOT data have been discussed previously by Gaultier et al. (2016, see their Fig. 5c).

Aside from time averaging that was shown above to be ineffective, the effects of sampling and measurement errors can be suppressed by increasing the spatial smoothing of the SSH field prior to computing the velocity component fields using the geostrophic approximation. The S/N standard deviation ratios as functions of the half-power filter cutoff wavelength are shown in Figs. 30a and b for, respectively, 4-day and 14-day averages of geostrophically computed SWOT estimates of velocity magnitude. The solid lines are the S/N ratios for combined measurement and sampling errors, which represents the total errors in SWOT estimates of velocity. The dotted and dashed lines are, respectively, the S/N ratios for measurement and sampling errors alone, which can be isolated in our simulated SWOT data but of course could not be distinguished in real observational data.

The S/N ratio for measurement errors alone (the dotted lines in Figs. 30a and b) improves rapidly with increased smoothing, intersecting the threshold of 3.16 at a half-power filter cutoff wavelength of

about 20 km for 4-day averages and 14 km for 14-day averages. If measurement errors were the only issue in SWOT estimates of surface velocity, these would be the resolution capabilities of 4-day and 14-day averages according to our criterion summarized in Section 5. But as noted above from the maps in Figs. 28 and 29, SWOT estimates of surface velocity are more severely affected by sampling errors than measurement errors. This is evident from the dashed lines in Figs. 30a and b by the slow improvement of the S/N ratio for sampling errors alone when the smoothing is increased. The S/N ratio of the total errors (the solid lines in Figs. 30a and b) quickly converges toward the S/N ratio of the sampling errors alone, thus showing the dominance of the effects of sampling errors in the total errors of SWOT estimates of surface velocity.

A peculiarity of Fig. 30a is that the S/N ratios for sampling errors alone and for total errors in estimates of the magnitudes of 4-day averaged velocity asymptote with increased smoothing to a value that is coincidentally just slightly higher than 3.16 for filter cutoff wavelengths longer than about 50 km. This indicates that the signal variance and sampling error variance both decrease at the same rate with high spatial smoothing, resulting in an approximate constant S/N ratio. There is thus no benefit to smoothing with a filter cutoff wavelength longer than about 50 km. According to our criterion, the resolution capability of
SWOT 14-Day Average Geostrophic Velocity with Filter Cutoff Wavelength 25 km

<figure><img src="image_34.png" /><figcaption>**Fig. 29.** The same as Fig. 28, except the magnitudes of 14-day averages of surface velocity computed geostrophically from simulated SWOT measurements of SSH with isotropic 2-dimensional smoothing using a Parzen smoother with a half-power filter cutoff wavelength of 25 km.</figcaption></figure>

SWOT estimates of 4-day averaged surface velocity is about 52 km, which corresponds to the wavelength at which the S/N ratio for the total errors (and for sampling errors alone) intersects the threshold of 3.16. The resolution capability worsens to about 132 km in 14-day averages because of the increased sampling errors.

Because of the flatness of the S/N curves in Figs. 30a and b, the resolution capability inferred from the S/N ratio is very sensitive to the choice of threshold value of the S/N standard deviation ratio. Lowering the threshold from our recommended value of 3.16 to a value of 2, for example, would improve the estimated resolution capability from 52 km to 28 km in SWOT estimates of 4-day averaged velocity fields (see Fig. 51 in Section 13.5 below).

The results of this analysis indicate that reducing the standard deviation of the uncorrelated measurement errors of SWOT estimates of SSH from the value of $\sigma_h = 2.74$ cm assumed for our calculations would have negligible effect on the resolution capability of maps of geostrophically computed estimates of the time-averaged velocity field constructed from SWOT data. Moreover, time averaging does not improve the S/N ratios of SWOT estimates of surface velocity because the sampling errors from inadequate sampling of rapidly evolving submesoscale velocity structures worsens when increasing the averaging period from 4 days to 14 days. The S/N ratio can be improved by spatial smoothing, but only to a limited degree.

It can thus be concluded that geostrophically computed SWOT estimates of surface velocity will be most useful on a swath-by-swath basis with no time averaging. The resolution capability is then the value of about 32 km deduced in Section 6 from Fig. 18, except only within the SWOT measurement swaths rather than for the artificial swaths across the full CCS domain considered in Section 6.

As noted in Section 6, the 50-km widths of each of the two parallel SWOT measurement swaths will result in some contamination of the swath-by-swath estimates of geostrophically computed velocity from edge effects of the ~30-km smoothing. It may be possible to utilize the nadir altimeter on the SWOT satellite to help interpolate across the 20-km nadir gap, thus resulting in 30-km smoothed maps of SSH across the full 120-km swath. The different resolution and error characteristics of the nadir altimeter will have to be taken into consideration in order to assess the benefits of this merging of the KaRIN and nadir altimeter data from the SWOT satellite. We have not investigated this in the present study.

The results for WaCM estimates of surface velocity are much more encouraging than for the SWOT estimates considered above. With the factor-of-11 improvement in sampling for the 1200-km swath width considered in this section, sampling errors are much less of an issue for WaCM. The total errors are dominated by measurement errors and time averaging is much more effective at improving the S/N ratio.

The dominance of the effects of measurement errors is readily apparent from the 4-day averaged WaCM estimates of velocity shown in Fig. 31 for the case of spatial smoothing with the same half-power filter cutoff wavelength of 25 km used in Fig. 28 for SWOT estimates of velocity. In contrast to the error maps for SWOT estimates of velocity shown in Fig. 28, the sampling errors shown for WaCM in the bottom panel of Fig. 31c are mostly smaller than the measurement errors shown in the bottom panel of Fig. 31b. The total errors in the bottom panel of Fig. 31d are thus dominated by measurement errors.

The patterns of the total error field in Fig. 31d are similar to the patterns of measurement errors alone in Fig. 31b but the magnitudes of
<figure><img src="image_35.png" /><figcaption>**Fig. 30.** The S/N standard deviation ratios for the full CCS region for the magnitude of the vector-averaged velocity (referred to for expedience as “velocity”) computed from simulated satellite estimates of time-averaged SSH (for SWOT) and surface velocity (for WaCM) as functions of the half-power filter cutoff wavelength of 2-dimensional isotropic smoothing using a Parzen smoother: (a) Geostrophically computed SWOT estimates of 4-day averaged velocity; (b) Geostrophically computed SWOT estimates of 14-day averaged velocity; (c) WaCM estimates of 4-day averaged velocity; and (d) WaCM estimates of 14-day averaged velocity. The dotted lines correspond to estimates over the full model domain with simulated uncorrelated measurement errors ($\sigma_h$ = 2.74 cm for SWOT and $\sigma_{spd}$ = 0.50 m s⁻¹ for WaCM). The dashed lines correspond to estimates from simulated swath sampling of error-free fields. The thick solid lines correspond to estimates from simulated swath sampling with uncorrelated measurement errors ($\sigma_h$ = 2.74 cm for SWOT and $\sigma_{spd}$ = 0.50 m s⁻¹ for WaCM). The S/N ratios for WaCM are based on a swath width of 1200 km. The improved S/N ratios for WaCM with a swath width of 1800 km are shown later in Figs. 43a and b. The gray area in each panel indicates S/N standard deviation ratios less than 3.16, which corresponds to a S/N variance ratio of 10. The vertical dashed line in each panel indicates the wavelength above which the S/N standard deviation ratios exceed a value of 3.16 for the case of combined measurement and sampling errors.</figcaption></figure>

the total errors are somewhat larger. This is because the measurement errors alone in the bottom panel of Fig. 31b are more effectively suppressed by the averaging over the eight individual maps at 0.5-day intervals during the 4-day averaging period. As essentially all of the grid points in the CCS model domain are sampled only four times during each 4-day orbit repeat period (see the top panel of Fig. 23), i.e., half as many as the number of twice-daily maps averaged for the bottom panel of Fig. 31b, the residual measurement errors are about $\sqrt{2}$ times larger in the sampled field in the bottom panel of Fig. 31d.

Because sampling errors are less of an issue for WaCM estimates of surface velocity, the total errors can be reduced much more effectively than for SWOT by increasing the averaging period. This is visually evident from the maps of 14-day averages shown in Fig. 32.

The speckled appearance of the velocity estimates in the 4-day average with 25-km smoothing in the top panel of Fig. 31d renders WaCM estimates of surface velocity with this amount of smoothing too noisy for most applications. The S/N standard deviation ratio in this case is only 1.63, which is well below our recommended threshold of 3.16. The velocity estimates in the 14-day average with 25-km smoothing in the top panel of Fig. 32d are still somewhat speckled, but

the measurement errors have been reduced enough that the velocity estimates may be useful for many applications. The S/N ratio in this case is 2.89, which is close to our threshold of 3.16.

The effects of noise reduction from spatial smoothing are shown in Figs. 30c and d from the S/N standard deviation ratios as functions of the half-power filter cutoff wavelength for, respectively, 4-day and 14-day averages of WaCM estimates of velocity. The S/N ratios for sampling errors alone (the dashed lines) exceed the threshold of 3.16 for all choices of smoothing, thus showing the secondary importance of sampling errors to the total errors in WaCM estimates of surface velocity. With increased smoothing, the S/N ratios for the case of total errors (the solid lines) parallel the S/N ratios for the case of measurement errors alone (the dotted lines). The small offsets between this pair of lines in both Figs. 30c and d represent the relatively small but not negligible contributions of sampling errors to the total errors. According to our criterion, the resolution capabilities of WaCM estimates of 4-day and 14-day averaged surface velocity are about 60 km and 28 km, respectively. These are significant improvements over the 130-km resolution deduced in Section 6 for WaCM estimates of instantaneous velocity fields. It should be kept in mind, however, that the
<figure><img src="image_36.png" /><figcaption>**Fig. 31.** Maps of the magnitudes of 4-day averages of surface velocity from error-free and noisy simulated WaCM measurements of surface velocity with a swath width of 1200 km with isotropic 2-dimensional smoothing using a Parzen smoother with a half-power filter cutoff wavelength of 25 km: Column (a) The 4-day average over the full model domain computed from error-free model surface velocity fields at a time step of 0.5 day over the 4-day period; Column (b) The 4-day average over the full model domain computed from model surface velocity fields at a time step of 0.5 day over the 4-day period with simulated uncorrelated speed measurement errors with a standard deviation of $\sigma_{\text{spd}} = 0.50 \text{ m s}^{-1}$; Column (c) The 4-day average computed from simulated WaCM swath sampling of error-free model surface velocity fields at the times and locations of each satellite observation over the 4-day period; and Column (d) The 4-day average computed from simulated WaCM swath sampling of model surface velocity fields at the times and locations of each satellite observation over the 4-day period with simulated uncorrelated speed measurement errors with a standard deviation of $\sigma_{\text{spd}} = 0.50 \text{ m s}^{-1}$. The bottom panels are the error maps computed by subtracting the error-free map in Column (a) from the error-contaminated maps in the top row of the respective Columns (b)–(d).</figcaption></figure>

signal variance is reduced at short wavelengths in the time-averaged fields (see Figs. 24 and 25).

## 8.5. The resolution capabilities of time-averaged surface vorticity fields

The procedure summarized in Section 8.2 and applied in Section 8.4 to assess the resolution capabilities of SWOT and WaCM estimates of surface velocity is applied in this section to assess the resolution capabilities of vorticity estimated from space-time smoothed SWOT and WaCM data. Differentiation of the velocity fields to estimate vorticity amplifies the errors in the velocity fields. It can therefore be anticipated that the resolution capabilities will be worse for vorticity than for velocity. As in Section 8.4, the simulated SWOT estimates of vorticity derived from geostrophically computed velocity in this section neglect the limitations of the geostrophic approximation discussed in Section 3.

Maps of geostrophically computed SWOT estimates of vorticity are shown for 4-day and 14-day averages in Figs. 33 and 34, respectively, for the case of spatial smoothing with a half-power filter cutoff wavelength of 50 km. The larger smoothing compared with the 25-km smoothing used for the velocity fields in Figs. 28 and 29 is commensurate with the noise amplification from the differentiation to compute vorticity from the geostrophically computed velocity field. Qualitatively, the characteristics of the maps of vorticity errors in the

bottom panels of Figs. 33 and 34 are similar to the characteristics of the maps of velocity errors in the bottom panels of Figs. 28 and 29. The total errors in geostrophically computed SWOT estimates of vorticity are even more dominated by sampling errors than was the case in geostrophically computed SWOT estimates of velocity. This is to be expected because small-scale variability is much more energetic and evolves much more quickly in the vorticity field than in the velocity field. The errors in the SWOT estimates of time-averaged vorticity in the top panels of Figs. 33d and 34d are too large for the SWOT estimates of vorticity to be useful for most applications.

The S/N standard deviation ratios as functions of the half-power filter cutoff wavelength are shown in Figs. 35a and b for, respectively, 4-day and 14-day averages of geostrophically computed SWOT estimates of vorticity. As in the case of SWOT estimates of velocity in Figs. 30a and b, the S/N ratios for the effects of measurement errors alone (the dotted lines) on SWOT estimates of vorticity improve rapidly with increased smoothing, intersecting the threshold of 3.16 at half-power filter cutoff wavelengths of about 38 km and 32 km for 4-day and 14-day averages, respectively. SWOT estimates of vorticity are thus again dominated by sampling errors. In this case, however, the S/N ratios asymptote with increased smoothing to a value of 2 in 4-day averages. In 14-day averages, the S/N ratios increase more gradually to about the same value of 2 for a filter cutoff wavelength of 150 km. The
## WaCM 14-Day Average Velocity with Filter Cutoff Wavelength 25 km

<figure><img src="image_37.png" /><figcaption>Fig. 32. The same as Fig. 31, except the magnitudes of 14-day averages of surface velocity from WaCM with isotropic 2-dimensional smoothing using a Parzen smoother with a half-power filter cutoff wavelength of 25 km.</figcaption></figure>

S/N ratios never reach our recommended threshold of 3.16 in either the 4-day or 14-day averages.

Like the geostrophically computed SWOT estimates of velocity considered in Section 8.4, we conclude that geostrophically computed SWOT estimates of vorticity will be most useful on a swath-by-swath basis. The resolution capability is then the value of about 54 km deduced in Section 6 from Fig. 18, except only within the SWOT measurement swaths. Since the two parallel SWOT measurement swaths each have a width of 50-km, the 54-km half-power filter cutoff wavelength required to achieve a S/N ratio of 3.16 is comparable to the swath width. As discussed in Section 6, swath-by-swath estimates of vorticity will therefore be even more contaminated by edge effects of the smoothing than will be the case for swath-by-swath SWOT estimates of surface velocity.

Because of its factor-of-11 improvement in sampling for the 1200-km swath width considered in this section, WaCM estimates of surface vorticity are again much more encouraging than the SWOT estimates considered above. Maps of 4-day and 14-day averages of WaCM estimates of vorticity are shown in Figs. 36 and 37, respectively, for the case of spatial smoothing with the same half-power filter cutoff wavelength of 50 km considered for SWOT in Figs. 33 and 34. The error characteristics for WaCM estimates of vorticity are qualitatively very similar to those of WaCM estimates of velocity in Figs. 31 and 32. Overall, the total errors are dominated by measurement errors (compare the bottom panels of Figs. 36b and d for the case of 4-day averages and the bottom panels of Figs. 37b and d for the case of 14-day averages). Sampling errors are comparable in magnitude to measurement errors in the regions of strongest meandering large-scale

currents where submesoscale variability is most energetic. Elsewhere, measurement errors are much larger than sampling errors.

The S/N standard deviation ratios as functions of the half-power filter cutoff wavelength are shown in Figs. 35c and d for, respectively, 4-day and 14-day WaCM estimates of vorticity. The S/N ratios for sampling errors alone (the dashed lines) exceed the threshold of 3.16 for filter cutoff wavelengths of about 25 km in 4-day averages and about 10 km in 14-day averages. Sampling errors are thus of secondary concern and the total errors are dominated by measurement errors for all choices of spatial smoothing. As in Figs. 30c and d for space-time smoothed WaCM estimates of velocity, the small offsets between the S/N ratios for the case of total errors (the solid lines) and for the case of measurement errors alone (the dotted lines) in both Figs. 35c and d indicate that the effects of sampling errors on the resolution capabilities of space-time smoothed maps of vorticity constructed from WaCM data are small but not negligible.

By our criterion, the resolution capabilities of WaCM estimates of 4-day and 14-day averaged vorticity with combined measurement and sampling errors are about 87 km and 62 km, respectively. These are significant improvements over the 130-km resolution deduced in Section 6 for WaCM estimates of instantaneous vorticity fields. Note again, however, that the signal variance is significantly reduced at short wavelengths in the time averaged fields (see Figs. 24 and 25).

## 9. Sensitivity of estimated resolution capabilities to the choice of smoother

The assessments of resolution capability in Sections 6, 8.4 and 8.5 were based on the S/N standard deviation ratios computed from error-free and noise-contaminated velocity and vorticity fields smoothed
<figure><img src="image_38.png" /><figcaption>**Fig. 33.** Maps of 4-day averages of normalized surface vorticity $\zeta_s/f$ computed geostrophically from error-free and noisy simulated SWOT measurements of SSH with isotropic 2-dimensional smoothing using a Parzen smoother with a half-power filter cutoff wavelength of 50 km. The various combinations of measurement and sampling errors are the same as for Columns (a)–(d) of Figs. 26 and 28. The bottom panels are the error maps computed by subtracting the error-free map in Column (a) from the error-contaminated maps in the top row of the respective Columns (b)–(d). In order to see the relatively small effects of measurement errors alone, the color bar for the bottom panel of Column b differs from the color bar for the bottom panels of Columns c and d.</figcaption></figure>

spatially using the Parzen smoother that is summarized in Appendix A. The choice of the Parzen smoother for these calculations was motivated in part by the fact that it will be used in the onboard pre-processing of SWOT data. The Parzen smoother is also very efficient computationally. The degree to which our assessments of resolution capabilities are sensitive to the choice of smoother is investigated in this section from the S/N ratios computed from velocity and vorticity fields smoothed using the quadratic loess smoother that is summarized in Appendix C.

Computationally, the 2-dimensional loess smoother consists of a weighted least squares fit to a quadratic surface (Cleveland and Devlin, 1988). The primary advantages of the loess smoother are that its filter transfer function has smaller sidelobes and rolls off more steeply through the half-power filter cutoff wavelength than does the filter transfer function of the Parzen smoother (see Schlax and Chelton, 1992; see also Fig. C.3 in Appendix C). The loess smoother is therefore more effective at attenuating small-scale noise. (It also attenuates more of the small-scale signal variability.)

A disadvantage of the loess smoother is that the weighted least squares fits at each individual grid point are much more computationally intensive than the Parzen smoother. This computational effort increases as the product of the filter cutoff wavelengths in each dimension. The difference in computing time for the set of filter cutoffs considered in this study is more than a factor of 50, which was a significant issue because a total of 1200 smoothed maps were required to generate the signal-to-noise graphs in Figs. 18, 30, 35 and 43.

Another disadvantage of the loess smoother is that its total span for a given half-power filter cutoff wavelength is 83.5% larger than

the span of the Parzen smoother (see Appendix C). This results in potentially more problems with edge-effect artifacts in the smoothed fields.

The S/N standard deviation ratios for SWOT and WaCM estimates of instantaneous snapshots of velocity and vorticity smoothed spatially using the loess smoother are shown as functions of the half-power filter cutoff wavelength in Fig. 38. These S/N graphs are qualitatively similar to those in Fig. 18 based on the Parzen smoother but with somewhat larger values of the S/N standard deviation ratio at all filter cutoff wavelengths. The resolution capabilities inferred from our threshold S/N standard deviation ratio of 3.16 therefore improve, as summarized in Table 3.

The improved resolution capabilities inferred from loess smoothing are especially impressive for WaCM estimates of instantaneous snapshots of velocity and vorticity. Because of the flatness of the S/N curves, small increases in the S/N values result in large improvements in the resolution capability inferred from any specified threshold value of S/N ratio. As noted previously from Fig. 18 in Section 6.2, the flatness of the S/N curves also implies that the estimated resolution capability is very sensitive to the choice of the threshold value of the S/N standard deviation ratio used to define resolution capability. Lowering the threshold from our recommended value of 3.16 to a value of 2, for example, would improve the loess-based estimates of resolution capabilities from 99 km to 47 km for velocity and from 89 km to 59 km for vorticity.

It is important to bear in mind that the estimates of resolution capability in Fig. 38 assume that the measurement swaths sample the
SWOT 14-Day Average Vorticity/f with Filter Cutoff Wavelength 50 km

<figure><img src="image_39.png" /><figcaption>**Fig. 34.** The same as Fig. 33, except for 14-day averages of normalized vorticity $\zeta_3/f$ computed geostrophically from simulated SWOT measurements of SSH with isotropic 2-dimensional smoothing using a Parzen smoother with a half-power filter cutoff wavelength of 50 km.</figcaption></figure>

entire CCS model domain on each satellite overpass. While this is very nearly achieved by just one side of the two-sided measurement swaths of WaCM, complete coverage of the CCS model domain is a highly idealized representation of SWOT sampling. Because its sampling consists of two parallel swaths with 50-km width separated by a 20-km nadir gap, in-swath estimates of velocity and vorticity from SWOT measurements of SSH are subject to edge-effect contamination in the smoothed fields. The resolution capabilities suggested from Fig. 38a are therefore optimistic estimates of what will be achieved from in-swath estimates of actual SWOT data.

The S/N standard deviation ratios for 4-day and 14-day averaged estimates of velocity fields computed from simulated SWOT and WaCM data smoothed spatially using the loess smoother are shown as functions of the half-power filter cutoff wavelength in Fig. 39. The results are qualitatively very similar to those obtained in Fig. 30 using the Parzen smoother. The dependencies of the S/N ratios for measurement errors alone, sampling errors alone and combined measurement and sampling errors have very similar characteristics for both smoothing algorithms. The only significant differences are that the S/N ratios increase somewhat faster with increased smoothing and thus intersect our recommended threshold value of 3.16 at somewhat shorter half-power filter cutoff wavelengths. The resolution capabilities of SWOT and WaCM estimates of velocity fields inferred from the loess smoother are consequently somewhat better than those inferred from Fig. 30 based on the Parzen smoother. The resolution capability improves by about 30% and 20% for geostrophically computed SWOT estimates of, respectively, 4-day and 14-day averaged velocity fields, and by about

12% for WaCM estimates of velocity fields for both choices of time averaging (see Table 3).

The S/N standard deviation ratios for 4-day and 14-day averaged SWOT and WaCM estimates of vorticity based on the loess smoother are shown as functions of the half-power filter cutoff wavelength in Fig. 40. As in the case of the space–time smoothed velocity fields considered above, the results are qualitatively very similar to those obtained in Fig. 35 using the Parzen smoother. Recall from Figs. 35a and b that the S/N ratios for SWOT estimates of vorticity inferred from Parzen smoothing never reached the threshold of 3.16. The S/N ratios for sampling errors and combined measurement and sampling errors in SWOT estimates of 4-day averaged vorticity using the loess smoother asymptote with increased smoothing to a value that is coincidentally just slightly higher than 3.16. Because of the flatness of the S/N curve, the resolution capability inferred from this figure is sensitive to the choice of threshold value of the S/N standard deviation ratio. By our criterion of a threshold value of 3.16, the resolution capability inferred from loess smoothing is 100 km for 4-day averages (Fig. 40a). The resolution capability worsens to 143 km for 14-day averaged SWOT estimates of velocity (Fig. 40b). As discussed previously, the degradation of resolution capability with increased temporal averaging is from increased sampling errors because of the space–time sampling pattern of the SWOT 21-day exact-repeat orbit that samples the full CCS model domain in two 4-day subcycles that are interleaved spatially and separated by 6-day gaps.

For WaCM, the faster increases of the S/N standard deviation ratios with increased smoothing using the loess smoother result in about 19% improvements in the resolution capabilities of WaCM estimates of both
<figure><img src="image_40.png" /><figcaption>Fig. 35. The S/N standard deviation ratios for the full CCS region from estimates of time-averaged surface vorticity computed from simulated satellite estimates of time-averaged SSH (for SWOT) and surface velocity (for WaCM) as functions of the half-power filter cutoff wavelength of 2-dimensional isotropic smoothing using a Parzen smoother: (a) Geostrophically computed SWOT estimates of 4-day averaged vorticity; (b) Geostrophically computed SWOT estimates of 14-day averaged vorticity; (c) WaCM estimates of 4-day averaged total vorticity; and (d) WaCM estimates of 14-day averaged total vorticity. The dotted, dashed and thick solid lines correspond to the same combinations of signal and errors as in Fig. 30 and the gray area in each panel indicates S/N standard deviation ratios less than 3.16. The S/N ratios for SWOT in the top panels never exceed the threshold value of 3.16. The vertical dashed lines for WaCM in the bottom panels indicate the wavelengths above which the S/N standard deviation ratios exceed a value of 3.16 for the case of combined measurement and sampling errors based on a swath width of 1200 km and a standard deviation of $\sigma_{spd} = 0.50 \text{ m s}^{-1}$ for the speed measurement noise. The improved S/N ratios for WaCM with a swath width of 1800 km are shown in Figs. 43c and d.</figcaption></figure>

4-day and 14-day averaged vorticity fields (see Figs. 40c and d and Table 3).

The comparisons in Table 3 of the resolution capabilities of SWOT and WaCM estimates of velocity and vorticity based on the Parzen and loess smoothers provide a measure of how rigorously the resolution capabilities reported here should be interpreted. In addition to the subjectivity of our recommended choice of 3.16 as the minimum threshold value for the S/N standard deviation ratio, the half-power filter cutoff wavelength above which the S/N ratios exceed this threshold changes by 10%–40% as summarized in Table 3, depending on the variable of interest, the averaging time, and the choice of smoother. Importantly, however, the relative rankings of resolution capabilities for the two instruments and for the three choices of time averages considered in Figs. 38–40 are insensitive to the details of the smoothing algorithm.

The Parzen smoother and the loess smoother were both applied for all of the calculations in this study. The resolution capabilities will be

discussed based on the more conservative estimates obtained from the Parzen smoother.

## 10. The benefits of improved sampling and measurement accuracy for WaCM

A conclusion from Section 8 is that measurement errors are a minor concern in space-time smoothed maps of surface velocity and vorticity computed geostrophically from SWOT data. The total errors in these fields are dominated by sampling errors owing to the narrow swath width of SWOT measurements of SSH. Reducing the SWOT measurement errors would therefore have little effect on the resolution capabilities of space-time smoothed maps of the velocity and vorticity fields computed geostrophically from SWOT data over a domain the size of the CCS. Aside from multiple SWOT instruments operating simultaneously on different satellites in coordinated orbits, the only way to improve the resolution capability would be to reduce the sampling
WaCM 4-Day Average Vorticity/f with Filter Cutoff Wavelength 50 km

<figure><img src="image_41.png" /><figcaption>**Fig. 36.** Maps of 4-day averages of normalized total surface vorticity $\zeta/f$ computed from error-free and noisy simulated WaCM measurements of surface velocity with a swath width of 1200 km and a standard deviation of $\sigma_{spd} = 0.50 \ \text{m s}^{-1}$ for the speed measurement noise with isotropic 2-dimensional smoothing using a Parzen smoother with a half-power filter cutoff wavelength of 50 km. The various combinations of measurement and sampling errors are the same as for Columns (a)–(d) of Fig. 31. The bottom panels are the error maps computed by subtracting the error-free map in Column (a) from the error-contaminated maps in the top row of the respective Columns (b)–(d).</figcaption></figure>

errors by increasing the width of the measurement swath. This might be achievable to a limited degree at increased cost by increasing the power of the transmitted signals. But altimetry is fundamentally restricted to measurements at relatively small incidence angles, and hence narrow swaths, compared with the Doppler scatterometer measurements by WaCM.

In contrast to SWOT, the dominance of the measurement error contribution to the total errors in maps of surface velocity and vorticity constructed from WaCM data implies that the accuracy and resolution capability can be improved by reducing the measurement noise in the WaCM data. For the calculations in Section 8, it was assumed that the uncorrelated measurement errors will have a standard deviation of $\sigma_{u,v} = 0.354 \ \text{m s}^{-1}$ for each velocity component, which corresponds to a standard deviation of $\sigma_{spd} = 0.50 \ \text{m s}^{-1}$ in speed errors. It may be technologically possible to reduce the WaCM measurement errors, thus improving the resolution capabilities for maps of velocity and vorticity fields constructed from WaCM data.

While sampling errors were shown in Section 8 to be secondary to measurement errors in determining the resolution capabilities of space–time smoothed maps of velocity and vorticity constructed from WaCM data, they are not negligible. This was evident in Figs. 30c and d and Figs. 35c and d from the small offsets between the S/N ratios for the cases of total errors and measurement errors alone (see also the analogous figures with loess smoothing in Figs. 39c and d and Figs. 40c and d). The resolution capabilities of space–time smoothed maps of velocity and vorticity constructed from WaCM data can thus also be

improved by reducing the sampling errors, which can be achieved by increasing the swath width.

The improvements of the resolution capabilities from increasing the swath width and reducing the uncorrelated measurement errors are investigated individually and in combination in this section.

10.1. The benefits of increasing the WaCM measurement swath to 1800 km

Recent engineering studies have concluded that it may be possible to extend the swath width of WaCM from the 1200-km swath considered for the simulations in Section 8 to 1800 km. With the nadir gap of 100 km, WaCM sampling would then consist of a pair of parallel measurement swaths with 850-km widths. The full CCS model domain is more than completely sampled by just one of the parallel measurement swaths on ascending orbits (see the left panel of Fig. 41) and nearly the entire CCS model domain can be sampled with an appropriate choice of orbit ground tracks by just one of the parallel measurement swaths on descending orbits (see the right panel of Fig. 41).

The implications of the improved sampling afforded by the wider swath width of 1800 km are shown by the histograms of WaCM sampling of the CCS model domain in Fig. 42. For the 4-day period shown in the top panel, the average number of samples is about 6.5. For comparison, the average number of samples per 4-day repeat was only about 4 for the 1200-km swath considered in Section 8 (see the top panel of Fig. 23). For the 14-day period shown in the bottom panel of Fig. 42, the average number of samples for a swath width of 1800 km is
WaCM 14-Day Average Vorticity/f with Filter Cutoff Wavelength 50 km

<figure><img src="image_42.png" /><figcaption>Fig. 37. The same as Fig. 36, except for 14-day averages of normalized surface vorticity $\zeta/f$ computed from simulated WaCM measurements of surface velocity with 2-dimensional isotropic smoothing using a Parzen smoother with a half-power filter cutoff wavelength of 50 km.</figcaption></figure>

<figure><img src="image_43.png" /><figcaption>Fig. 38. The same as Fig. 18, except the S/N standard deviation ratios from simulated satellite estimates of instantaneous snapshots of surface velocity as functions of the half-power filter cutoff wavelength of 2-dimensional isotropic smoothing using a loess smoother rather than the Parzen smoother used for Fig. 18.</figcaption></figure>

about 23, compared with only about 13.5 for a swath width of 1200 km (see the bottom panel of Fig. 23).

The increased number of WaCM samples during each 4-day and 14-day averaging period reduces the sampling errors because of improved sampling of the rapidly evolving submesoscale variability compared

with the simulations in Section 8. It also reduces the effects of uncorrelated measurement errors somewhat through the time-averaging of a larger number of observations at each grid location over the course of the averaging period. The net effects of the smaller sampling and measurement errors on the S/N standard deviation ratios in space-time smoothed maps of velocity and vorticity with a swath width
<figure><img src="image_44.png" /><figcaption>**Fig. 39.** The same as Fig. 30, except the S/N standard deviation ratios from simulated satellite estimates of time-averaged surface velocity as functions of the half-power filter cutoff wavelength of 2-dimensional isotropic smoothing using a loess smoother rather than the Parzen smoother used for Fig. 30.</figcaption></figure>

of 1800 km are shown in Fig. 43. By our criterion, the resolution capabilities of WaCM estimates of 4-day and 14-day averaged surface velocity (Figs. 43a and b) are about 44 km and 21 km, respectively. These are substantial improvements over the resolutions capabilities of 60 km and 28 km deduced in Section 8.4 for a swath width of 1200 km (Figs. 30c and d). The resolution capabilities of WaCM estimates of 4-day and 14-day averaged vorticity with a swath width of 1800 km (Fig. 43c and d) are about 70 km and 51 km, respectively, which are again substantial improvements over the resolutions of 87 km and 62 km deduced in Section 8.5 for a swath width of 1200 km (Figs. 35c and d).

A noteworthy feature of Fig. 43 is that the offsets between the S/N ratios for the cases of total errors (the solid lines) and measurement errors alone (the dotted lines) are much smaller for the 1800-km swath width than for the 1200-km swath shown in Figs. 30c and d and Figs. 35c and d. In fact, these two curves in each panel of Fig. 43 are so similar that little would be gained by increasing the WaCM swath width beyond 1800 km. Further improvements in the resolution capabilities of space-time smoothed maps of velocity and vorticity constructed from WaCM data are more effectively achieved by reducing the uncorrelated measurement errors, which is considered in Section 10.2.

## 10.2. The benefits of reducing the WaCM measurement noise

The resolution capabilities of WaCM estimates of velocity and vorticity that were deduced in Section 6 for instantaneous maps and in Sections 8.4, 8.5 and 10.1 for time-averaged maps were based on the baseline standard deviation of $\sigma_{spd} = 0.50 \text{ m s}^{-1}$ for the speeds of the uncorrelated errors of WaCM measurements of surface velocity. For the simulations throughout this study, these speed uncertainties are assumed to be equally partitioned between each velocity component. Measurement errors could be reduced by increasing the transmit power of the WaCM antenna or by increasing the antenna size. The added cost of these options must be traded off against the scientific benefits of higher resolution maps of velocity and vorticity. To assess the scientific benefits of higher resolution maps, we extended the analyses of Sections 6, 8.4, 8.5 and 10.1 to determine the resolution capabilities of maps of velocity and vorticity constructed from simulated WaCM data with smaller uncorrelated measurement errors.

The effects of reducing the uncorrelated errors in WaCM measurements of surface velocity can be inferred from the derivations in Appendix 1.4 of the effects of measurement errors on the wavenumber spectral contents of smoothed WaCM estimates of velocity component and vorticity fields. It is shown from Eqs. (I.48a) and (I.52), with the
<figure><img src="image_45.png" /><figcaption>**Fig. 40.** The same as Fig. 35, except the S/N standard deviation ratios from simulated satellite estimates of time-averaged surface vorticity as functions of the half-power filter cutoff wavelength of 2-dimensional isotropic smoothing using a loess smoother rather than the Parzen smoother used for Fig. 35.</figcaption></figure>

latter partitioned as (I.54a) and (I.55a), that the wavenumber spectral values of the errors of smoothed WaCM estimates of, respectively, velocity component and vorticity fields both depend linearly on the variances $\sigma_u^2$ and $\sigma_v^2$ of the velocity component measurement errors. Since the variance of the smoothed fields is equal to the integral of their wavenumber spectrum according to Parseval's Theorem, the variances of the errors of the smoothed fields decrease linearly with decreasing variance of the velocity component measurement errors.

The S/N standard deviation ratios for smoothed estimates of velocity component and vorticity fields for any particular choice of half-power filter cutoff wavelength depends inversely on the measurement error standard deviation. The resolution capability defined as in previous sections to be the half-power filter cutoff wavelength above which the S/N standard deviation ratios exceed a value of 3.16 therefore decreases systematically with decreasing measurement error standard deviation, though not necessarily linearly since the signal contribution to the S/N standard deviation ratios does not necessarily vary linearly with filter cutoff wavelength.

We first consider the case of instantaneous snapshot maps of velocity and vorticity that could be obtained from a single overpass of the CCS model domain. The benefits of improving the WaCM measurement accuracy were quantified by repeating the calculations in Section 6 of the S/N standard deviation ratios for spatially smoothed WaCM maps

of instantaneous velocity and vorticity based on standard deviations of the uncorrelated speed measurement errors ranging from $\sigma_{spd} = 0.10 \text{ m s}^{-1}$ to $0.50 \text{ m s}^{-1}$, which we continue to assume are equally partitioned between the two orthogonal velocity components. As in Section 6, we isolate the effects of measurement errors by considering the idealized scenario of complete sampling of the full CCS model domain on each overpass of the WaCM satellite. This is nearly achieved for both ascending and descending orbits for a swath width of 1200 km (Fig. 20) and is essentially achieved for a swath width of 1800 km (Fig. 41). The variances of the error-free signals for each choice of half-power filter cutoff wavelength are unchanged from Section 6 in these calculations, but the variances of the errors of the smoothed fields decrease linearly with the measurement error variance as summarized above.

The dependencies of the resolution capabilities of smoothed maps of WaCM estimates of instantaneous velocity and vorticity on the standard deviation of the speed measurement errors were deduced from graphs of the S/N standard deviation ratios like those shown in Fig. 18b. The results are shown in Fig. 44 for both Parzen and loess smoothing. As discussed in Section 9, the filtering efficiency of the loess smoother improves the resolution capability compared with the Parzen smoother, albeit with higher computational effort. For discussion purposes, we will continue to adopt the more conservative resolution capabilities inferred from the Parzen smoother.
<figure><img src="image_46.png" /><figcaption>Fig. 41. The same as the bottom two panels of Fig. 20, except examples of the measurement swaths for single ascending and descending overpasses of WaCM for the case of a swath width of 1800 km. The ground tracks are for illustrative purposes and could be adjusted longitudinally to optimize the sampling of any specific region of the world ocean. The details of the sampling of the CCS region would change accordingly.</figcaption></figure>

<caption>**Table 3**</caption>

The resolution capabilities deduced in Sections 6, 8 and 9 for SWOT and WaCM estimates of the magnitude of surface ocean velocity and vorticity in an instantaneous snapshot and in 4-day and 14-day averages based on spatial smoothing using the Parzen and loess smoothers for measurement error standard deviations of $\sigma_h = 2.74$ cm for SWOT and $\sigma_{spd} = 0.50$ m s$^{-1}$ equally partitioned between the two velocity components for WaCM. A horizontal line indicates that the S/N ratio did not exceed the threshold of 3.16 for any of the choices of half-power filter cutoff wavelength considered in Section 8. The last column shows the percent improvement of the resolution capability inferred from the loess smoother compared with the Parzen smoother. The results for WaCM are based on a swath width of 1200 km. The improved resolution capabilities if the WaCM swath can be extended to 1800 km are listed in Table 4 based on the Parzen smoother. The resolutions based on Parzen smoothing that are listed in this table are shown graphically in Fig. 51 in Section 13.5 below.

<table><thead><tr><th>Instrument</th><th>Averaging period</th><th>Resolution from Parzen</th><th>Resolution from Loess</th><th>Percent improvement</th></tr></thead><tbody><tr><td><strong>Velocity</strong></td><td></td><td></td><td></td><td></td></tr><tr><td>SWOT</td><td>Snapshot</td><td>32 km</td><td>23 km</td><td>28%</td></tr><tr><td>SWOT</td><td>4 days</td><td>52 km</td><td>36 km</td><td>31%</td></tr><tr><td>SWOT</td><td>14 days</td><td>132 km</td><td>107 km</td><td>19%</td></tr><tr><td>WaCM</td><td>Snapshot</td><td>130 km</td><td>99 km</td><td>24%</td></tr><tr><td>WaCM</td><td>4 days</td><td>60 km</td><td>52 km</td><td>13%</td></tr><tr><td>WaCM</td><td>14 days</td><td>28 km</td><td>25 km</td><td>11%</td></tr><tr><td><strong>Vorticity</strong></td><td></td><td></td><td></td><td></td></tr><tr><td>SWOT</td><td>Snapshot</td><td>54 km</td><td>34 km</td><td>37%</td></tr><tr><td>SWOT</td><td>4 days</td><td>-</td><td>100 km</td><td>-</td></tr><tr><td>SWOT</td><td>14 days</td><td>-</td><td>143 km</td><td>-</td></tr><tr><td>WaCM</td><td>Snapshot</td><td>130 km</td><td>89 km</td><td>32%</td></tr><tr><td>WaCM</td><td>4 days</td><td>87 km</td><td>71 km</td><td>18%</td></tr><tr><td>WaCM</td><td>14 days</td><td>62 km</td><td>50 km</td><td>19%</td></tr></tbody></table>

For the sake of specificity in the discussion that follows, the resolution capabilities for the baseline standard deviation of $\sigma_{spd} = 0.50$ m s$^{-1}$ for the uncorrelated speed measurement errors will be compared with

<figure><img src="image_47.png" /><figcaption>Fig. 42. The same as the thick lines in Fig. 23, except histograms of the number of samples by WaCM during 4 days and 14 days based on a swath width of 1800 km rather than 1200 km.</figcaption></figure>

the resolutions that could be achieved by halving the noise standard deviation to $\sigma_{spd} = 0.25$ m s$^{-1}$. The numerical values of the resolution
<figure><img src="image_48.png" /><figcaption>Fig. 43. The same as the bottom two panels of Fig. 30 and the bottom two panels of Fig. 35, except the S/N standard deviation ratios for WaCM estimates of 4-day and 14-day averaged velocity and vorticity fields for the baseline standard deviation of $\sigma_{spd} = 0.50 \ \text{m s}^{-1}$ for the speed measurement noise and a swath width of 1800 km rather than 1200 km.</figcaption></figure>

capabilities are listed in Table 4a and b. The resolution capabilities for other choices of $\sigma_{spd}$ can be determined graphically from Fig. 44.

Reducing the speed measurement noise to $\sigma_{spd} = 0.25 \ \text{m s}^{-1}$ would dramatically improve the resolution capability of instantaneous snapshots of velocity from 130 km to 50 km (see Fig. 44a and Table 4a). The resolution capability of instantaneous snapshots of vorticity would improve from 130 km to 74 km (see Fig. 44b and Table 4b). The benefits of the higher resolution with the smaller measurement noise of $0.25 \ \text{m s}^{-1}$ for instantaneous snapshots of velocity and vorticity can be seen visually in Fig. 50 below.

The effects of improving the WaCM measurement accuracy on space-time smoothed maps of velocity and vorticity were similarly quantified by repeating the calculations in Sections 8.4, 8.5 and 10.1 of the S/N standard deviation ratios like those shown in Figs. 30, 35, 39, 40 and 43 for standard deviations of uncorrelated speed measurement errors ranging from $\sigma_{spd} = 0.10 \ \text{m s}^{-1}$ to $0.50 \ \text{m s}^{-1}$. For space-time smoothed maps, it was shown in Section 10.1 that the swath width of the WaCM measurements is also a factor in determining the resolution capability. We therefore estimated the resolution capabilities for both of the swath widths that were considered in Sections 8.4, 8.5 and 10.1. The results for swath widths of 1200 km and 1800 km are shown, respectively, by the blue and red lines in Fig. 45. The solid and dashed lines are the resolution capabilities inferred based on smoothing

using the Parzen and loess smoothers, respectively. For discussion purposes, we again consider the more conservative resolution capabilities inferred from the Parzen smoother.

The dependencies of the resolution capabilities of 4-day averaged WaCM estimates of the surface velocity field on the standard deviation of the speed measurement errors are shown in Fig. 45a. For the baseline standard deviation of $\sigma_{spd} = 0.50 \ \text{m s}^{-1}$ assumed in Sections 8.4 and 10.1 for the uncorrelated speed measurement errors, it was shown from Figs. 30c and 43a that increasing the swath width from 1200 km to 1800 km would improve the resolution capability for 4-day averaged velocity fields from 60 km to 44 km. Reducing the standard deviation of the measurement noise to $\sigma_{spd} = 0.25 \ \text{m s}^{-1}$ would further improve this resolution capability to 18 km (see Fig. 45a and Table 4c).

For 14-day averaged velocity fields, it was shown from Figs. 30d and 43b that the resolution capability for a speed measurement noise of $\sigma_{spd} = 0.50 \ \text{m s}^{-1}$ would improve from 28 km for a swath width of 1200 km to 21 km for a swath width of 1800 km. Reducing the standard deviation of the speed measurement noise to $\sigma_{spd} = 0.25 \ \text{m s}^{-1}$ would further improve the 14-day averaged velocity resolution capability to 10 km (see Fig. 45b and Table 4e), which is the inherent resolution of the pre-processed estimates of surface velocity with a footprint diameter of 5 km for the simulations in this study.
<figure><img src="image_49.png" /><figcaption>**Fig. 44.** The resolution capabilities of WaCM estimates of instantaneous snapshot maps of the full CCS region (which are essentially achieved on each satellite overpass with a swath width of 1800 km — see Fig. 41) as functions of the standard deviation $\sigma_{\text{spd}}$ of the uncorrelated errors of WaCM measurements of surface velocity: (a) maps of velocity; and (b) maps of vorticity. The solid and dashed lines in each panel correspond to the half-power filter cutoff wavelengths above which the S/N standard deviation ratios exceed a value of 3.16 using Parzen and loess smoothers, respectively. For reference, the vertical dashed line in each panel indicates a standard deviation of $\sigma_{\text{spd}} = 0.25 \text{ m s}^{-1}$ for the uncorrelated speed measurement errors, which corresponds to half of the baseline value of $\sigma_{\text{spd}} = 0.50 \text{ m s}^{-1}$.</figcaption></figure>

<caption>**Table 4**</caption>

The resolution capabilities deduced in Sections 6, 8 and 10 for maps of the surface ocean velocity and vorticity based on spatial smoothing of WaCM data using the Parzen smoother for: (a) instantaneous snapshots of velocity; (b) instantaneous snapshots of vorticity; (c) 4-day averages of velocity; (d) 4-day averages of vorticity; (e) 14-day averages of velocity; and (f) 14-day averages of vorticity. The results are listed for combinations of 1200 km and 1800 km swath widths and standard deviations of $\sigma_{\text{spd}} = 0.50 \text{ m s}^{-1}$ and $0.25 \text{ m s}^{-1}$ for the uncorrelated errors of WaCM measurements of current speed. The swath width for instantaneous maps of velocity and vorticity in (a) and (b) is assumed to be the full CCS model domain, which is essentially achieved on every satellite overpass with a swath width of 1800 km (see Fig. 41). Example maps with the resolutions listed in Tables a–d are shown in Figs. 48–50. The resolutions listed in this table are shown graphically in Fig. 51 in Section 13.5 below.

<table><thead><tr><th colspan="3">Swath width Measurement noise σ<sub>spd</sub></th><th colspan="3">Swath width Measurement noise σ<sub>spd</sub></th></tr><tr><th colspan="3">0.50 m s<sup>-1</sup> 0.25 m s<sup>-1</sup></th><th colspan="3">0.50 m s<sup>-1</sup> 0.25 m s<sup>-1</sup></th></tr></thead><tbody><tr><th colspan="3">(a) Instantaneous velocity maps</th><th colspan="3">(b) Instantaneous vorticity maps</th></tr><tr><td>CCS Model</td><td>130 km</td><td>50 km</td><td>CCS Model</td><td>130 km</td><td>74 km</td></tr><tr><th colspan="3">(c) 4-Day Averaged velocity maps</th><th colspan="3">(d) 4-Day Averaged vorticity maps</th></tr><tr><td>1200 km</td><td>60 km</td><td>28 km</td><td>1200 km</td><td>87 km</td><td>59 km</td></tr><tr><td>1800 km</td><td>44 km</td><td>18 km</td><td>1800 km</td><td>70 km</td><td>45 km</td></tr><tr><th colspan="3">(e) 14-Day Averaged velocity maps</th><th colspan="3">(f) 14-Day Averaged vorticity maps</th></tr><tr><td>1200 km</td><td>28 km</td><td>11 km</td><td>1200 km</td><td>62 km</td><td>42 km</td></tr><tr><td>1800 km</td><td>21 km</td><td>10 km</td><td>1800 km</td><td>51 km</td><td>34 km</td></tr></tbody></table>

The dependencies of the resolution capabilities of 4-day averaged WaCM estimates of the vorticity field on the speed measurement error standard deviation are shown Fig. 45c. For the baseline standard deviation of $\sigma_{\text{spd}} = 0.50 \text{ m s}^{-1}$ for the uncorrelated speed measurement errors, it was shown from Figs. 35c and 43c that increasing the swath width from 1200 km to 1800 km would improve the resolution capability for 4-day averaged vorticity fields from 87 km to 70 km. Reducing the standard deviation of the speed measurement errors to $\sigma_{\text{spd}} = 0.25 \text{ m s}^{-1}$ would further improve the resolution capability to 45 km (see Fig. 45c and Table 4d).

For 14-day averaged vorticity fields, it was shown from Figs. 35d and 43d that the resolution capability for a speed measurement noise of $\sigma_{\text{spd}} = 0.50 \text{ m s}^{-1}$ would improve from 62 km for a swath width of 1200 km to 51 km for a swath width of 1800 km. Reducing the standard deviation of the speed measurement noise to $\sigma_{\text{spd}} = 0.25 \text{ m s}^{-1}$ would further improve the 14-day averaged vorticity resolution capability to 34 km (see Fig. 45d and Table 4f).

While the improved spatial resolution capabilities summarized above for space-time smoothed maps of velocity and vorticity estimated from WaCM data for an increased swath width of 1800 km and a reduction of the standard deviation of uncorrelated speed measurement errors to $0.25 \text{ m s}^{-1}$ are impressive, we note again that much of the small-scale velocity and vorticity signals that are of interest are attenuated in 4-day and 14-day averaged fields, especially the latter (see Figs. 24 and 25). Depending on the application, the coarser spatial resolution of instantaneous snapshot maps of velocity and vorticity constructed from WaCM data (Fig. 44) may be preferable to the higher spatial resolution of the time-averaged maps (see Figs. 48–50).

## 11. Discussion

The preceding analysis has assessed the resolution capabilities of surface ocean velocity and vorticity fields estimated from simulated interferometric altimeter measurements of SSH by SWOT and Doppler radar measurements of surface velocity by WaCM. The simulated satellite data for this study were constructed based on the high-resolution model of the California Current System summarized in Section 2 with uncorrelated measurement errors added to the SSH and surface velocity fields with the standard deviations in the baseline designs of the SWOT and WaCM instruments.

For SWOT, our analysis includes a derivation of the baseline science requirement for the standard deviation of the uncorrelated errors of the SSH measurements, a number that cannot be found unambiguously in the SWOT documentation. This derivation (see Appendix F) concludes that the requirement for the uncorrelated errors of SWOT measurements of SSH, as specified in the SWOT Science Requirements Document (Rodríguez and Callahan, 2016) in terms of the wavenumber spectrum after smoothing in ground-based post-processing, corresponds to a standard deviation of $\sigma_h = 2.74 \text{ cm}$ for pre-processed SWOT estimates of SSH with a footprint size of 1 km.

Our error analysis includes derivations of analytical expressions for the variances (Appendix G) and the wavenumber spectra (Appendix I) of errors of the estimates of the derived variables (geostrophically computed velocity and vorticity for SWOT and vorticity for WaCM), with and without spatial smoothing. The errors are shown as functions of the half-power filter cutoff wavelength of the smoothing in Fig. 12. The wavenumber spectra shown in Figs. 13–15 quantify the scale dependencies of these errors. The equations for the variances and wavenumber spectra of the errors of the derived variables all depend explicitly on the variance of the uncorrelated measurement errors and are thus applicable to arbitrary specification of measurement noise. This allows an assessment of the degree of improvement in the accuracies of the derived variables that would be achieved by improving the baseline measurement accuracies.

The calculations presented in Sections 6, 8, 9 and 10.1 are based on the above standard deviation of $\sigma_h = 2.74 \text{ cm}$ for the uncorrelated
<figure><img src="image_50.png" /><figcaption>Fig. 45. The resolution capabilities of WaCM estimates of 4-day and 14-day averages (left and right columns) of surface velocity and vorticity (top and bottom rows) as functions of the standard deviation $\sigma_{spd}$ of the uncorrelated speed errors of WaCM measurements of surface velocity for swath widths of 1200 km and 1800 km (blue and red lines, respectively). The solid and dashed lines in each panel correspond to the half-power filter cutoff wavelengths above which the S/N standard deviation ratios exceed a value of 3.16 using Parzen and loess smoothers, respectively. For reference, the vertical dashed line in each panel indicates a standard deviation of $\sigma_{spd} = 0.25 \ \text{m s}^{-1}$ for the uncorrelated speed measurement errors, which corresponds to half of the baseline value of $\sigma_{spd} = 0.50 \ \text{m s}^{-1}$. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

errors of SWOT measurements of SSH with a footprint size of 1 km, and a standard deviation of $\sigma_{spd} = 0.50 \ \text{m s}^{-1}$ for the uncorrelated errors of WaCM measurements of surface ocean velocity with a footprint size of 5 km. The benefits of reducing the measurement noise are investigated for WaCM in Section 10.2. Improved measurement accuracy is not considered for SWOT because the total errors in maps of SWOT estimates of velocity and vorticity are so strongly dominated by sampling errors that reducing the measurement errors would offer little benefit.

For WaCM, we have assumed that the uncorrelated speed errors are equally partitioned between the two orthogonal velocity components, thus resulting in a baseline standard deviation of $\sigma_{\mu\nu} = 0.354 \ \text{m s}^{-1}$ for each component. In reality, the errors of each velocity component will vary differently across the measurement swaths because of the geometrical transformation issues discussed in Section 4.2. For approximation of the derivatives of the velocity components that are required to estimate vorticity from WaCM data, we have further assumed that the velocity estimates with 5 km footprint size are oversampled on a 1 km × 1 km grid. It is shown in Appendix H that 3-point centered differences on this grid retain more of the small-scale signal in the vorticity field without having to resort to centered difference estimates of the derivatives with a wider stencil width on the 5 km × 5 km grid on which the WaCM measurement errors are uncorrelated.

The measurement error standard deviations of $\sigma_h = 2.74 \ \text{cm}$ for SWOT and $\sigma_{spd} = 0.50 \ \text{m s}^{-1}$ for WaCM are swath-averaged values. It is shown in Fig. 1 of Appendix F that the SWOT measurement errors will increase toward each edge of the measurement swaths and with

increasing significant wave height (SWH). WaCM measurement errors also increase toward the edges of the measurement swaths and will depend on SWH (Rodríguez, 2018). In this study, we have considered the SWOT and WaCM measurement errors to have uniform standard deviations across the full widths of their measurement swaths. Because we have neglected the increases of measurement errors toward the swath edges, our analysis in this study likely underestimates the overall effects of measurement errors on space-time smoothed estimates of velocity and vorticity fields constructed from SWOT and WaCM data. The results presented in Sections 6 and 8–10 should therefore be considered optimistic estimates of the resolution capabilities of actual SWOT and WaCM data.

The CCS model used for our assessments of the resolution capabilities of SWOT and WaCM estimates of surface velocity and vorticity has some limitations that should be kept in mind. As summarized in Section 2, the model was forced with seasonally varying winds, heat flux and freshwater flux, and thus omits the significant effects of synoptic atmospheric forcing. Inertial motions are therefore poorly represented in the model. They will be considered unwanted “noise” in most applications of SWOT and WaCM data. While they are not a concern in this study because of their weak signal levels in the model, their effects on SSH and surface velocity will have to be addressed in the analysis of actual SWOT and WaCM data. We note that mesoscale and submesoscale variability may also be underrepresented in the model because of the seasonal atmospheric forcing. The model also has high dissipation and lacks tidal forcing. It is therefore likely that
internal wave variability is considerably underestimated in the model. In spite of these limitations, we believe that the model provides useful comparisons of the resolution capabilities of maps of surface velocity and vorticity constructed from SWOT and WaCM data.

The importance of the grid resolution of the model for proper representation of the effects of submesoscale variability is evident from Fig. 1. Submesoscale variability increases dramatically by increasing the grid resolution from 4 km to the 0.5 km grid spacing of the model used for our simulations. The choice of grid resolution is clearly an important consideration in choosing a model from which to assess the resolution capabilities of SWOT and WaCM estimates of surface velocity and vorticity. For example, Qiu et al. (2016) evaluated the resolution capability of maps of vorticity constructed from simulated SWOT data based on a model of the Kuroshio Extension with a grid spacing of 1/30° (approximately 3 km). The much less energetic submesoscale variability in that model compared with the CCS model used in this study is partly attributable to the coarseness of their model grid. The weaker submesoscale variability in the model output used in the Qiu et al. (2016) study is compounded by the fact that their analysis was based on daily averages rather than instantaneous snapshots of the model output.

The extent to which accurate model representation of submesoscale variability is important in model-based simulations of SWOT and WaCM data depends on the amount of smoothing that must be applied to the simulated satellite data to achieve adequate signal-to-noise ratio in maps of the variables of interest. Given the coarse resolution capabilities inferred from the analysis in Sections 6 and 8–10 and summarized in Tables 3 and 4, it may not be necessary to use a model with resolution as high as the 0.5-km grid of the CCS model used in this study. It nonetheless seems prudent to use as high a resolution model as possible in order to be sure that aliasing issues from discrete sampling at satellite overpass times are adequately accounted for. To this end, the daily averages analyzed by Qiu et al. (2016) are likely an issue, even for the coarse resolution capabilities of SWOT estimates of geostrophically computed velocity and vorticity. The twice-daily snapshots analyzed in this study are also less than ideal; evidence of aliasing from inadequate sampling of the rapidly evolving submesoscale field was readily apparent from the 4-day and 14-day averages of error-free vorticity in the bottom panels of Fig. 24.

In addition to analysis of the effects of uncorrelated measurement and sampling errors, we also investigated limitations of the geostrophic approximation that must be used to estimate surface velocity and vorticity from SWOT measurements of SSH (Section 3). Because of the above-noted underestimation of internal gravity wave energy in the CCS model used in this study, we were not able to use this model to assess the effects of the SSH signatures of internal waves on velocity estimates computed geostrophically from SSH. We did, however, use the CCS model to investigate the effects of ageostrophic variability from cyclostrophic motion, which results in geostrophically computed velocities that overestimate the positive vorticity of cyclonic features and underestimate the magnitude of the negative vorticity of anticyclonic features. The former are more prevalent, presumably because of the tendency for anticyclonic features to become inertially unstable. It was shown in Section 3 that the errors from cyclostrophic motion may often exceed 0.1 m s⁻¹ but can be reduced to less than 0.05 m s⁻¹ by smoothing with a half-power filter cutoff wavelength of 50 km or more.

Contamination of geostrophically computed SWOT estimates of surface velocity by internal gravity waves is an important issue in need of rigorous analysis that is beyond the scope of this study. Some insight into the effects of internal waves on the interpretation of SWOT data was provided in Section 3 based on two high-resolution simulations of the ocean circulation southeast of the Gulf Stream in the western North Atlantic (see Figs. 10 and 11). A simulation forced by low-frequency winds and without tides results in ageostrophic submesoscale features similar to the cyclostrophic features found in the CCS model used in this study. The same North Atlantic simulation forced by hourly winds and

ocean tides results in a dramatically different geostrophically computed velocity field consisting of internal wave structures that overwhelm the submesoscale velocity features that are of interest.

The apparent velocity structures computed geostrophically from the SSH signatures of internal waves are a gross misrepresentation of the surface manifestation of the actual internal wave velocities. SWOT data will have to be judiciously smoothed spatially and/or temporally to mitigate contamination of the geostrophically computed velocity estimates by internal gravity waves.

While contamination of SWOT estimates of surface velocity and vorticity by ageostrophic processes is clearly an issue, it should be kept in mind that SWOT was not designed to determine the surface velocity and vorticity fields that are the primary focus of this study. However, many users of SWOT data intend to investigate ocean dynamics from SWOT estimates of velocity and vorticity. The analysis presented in this study provides a useful framework for understanding the limitations of SWOT data for such applications.

It should also be noted that some of the same ageostrophic signals that contaminate SWOT estimates of velocity and vorticity will be considered unwanted “noise” in the context of many applications of the total velocity measured by WaCM. In particular, the surface velocity signatures of internal gravity waves and inertial motions could obscure some of the velocity signals of the mesoscale and submesoscale currents that will be the primary interest in many studies. A sense of the magnitudes of these ageostrophic velocities can be inferred from the modeling result shown in the bottom left panel of Fig. 10. Values exceeding 0.5 m s⁻¹ are common. The partitioning of this ageostrophic velocity between inertial motions, internal waves and other ageostrophic processes is presently not well understood.

A point worth emphasizing is that the “contamination” of WaCM data by ageostrophic processes differs fundamentally from the contamination of SWOT data by the same ageostrophic processes. The velocity signatures of inertial motions and internal waves are true velocities. In contrast, the apparent surface velocity field computed geostrophically from the SSH signatures of ageostrophic processes are fictitious velocities that do not exist in nature.

Although the spectral characteristics of SSH at high wavenumbers (short wavelengths) were not a primary focus of this investigation, useful insight into the nature of SSH variance at poorly observed small scales can be obtained from the CCS model used throughout this study and from the above two North Atlantic models considered briefly in Section 3 and Appendix F. It is shown in Figs. F.3 and F.4 that the rolloff at high wavenumbers in the SSH spectra from all three of these models is steeper than the approximate $l^{-5/2}$ dependence on wavenumber $l$ that is assumed in the 68th-percentile global average SSH spectrum that is the basis for defining the science requirements for SWOT measurement accuracy and resolution (Rodríguez and Callahan, 2016). This 68th-percentile wavenumber spectrum is based on extrapolation of along-track SSH spectra from nadir altimetry that resolves variability only down to a wavelength scale of about 70 km (Xu and Fu, 2012). The validity of this extrapolation cannot be determined from presently available observational data and must therefore be inferred from high-resolution ocean circulation models.

A consequence of the steeper rolloff of the wavenumber spectra from the three models considered in this study is that the 68th-percentile spectrum assumed in the SWOT requirements document may overestimate the variance at high wavenumbers, in which case the 15-km resolution of the SWOT data as defined spectrally in the science requirements would be overly optimistic. In particular, the 15-km smoothed white noise spectrum for the SWOT requirement of the standard deviation of $\sigma_h = 2.74$ cm for the uncorrelated measurement errors that is derived for a footprint size of 1 km in Appendix F may intersect the true SSH signal spectrum at a wavelength longer than 15 km. Achieving the science goal of resolving the signal spectrum down to a wavelength of 15 km would then require a higher measurement accuracy in the pre-processed SWOT estimates of SSH, i.e., a smaller standard deviation of the uncorrelated measurement errors.
Since little is known about the spectral characteristics of SSH at wavelengths shorter than 70 km, it is possible that the CCS model used in this study and the two North Atlantic models considered briefly in Section 3 and Appendix F underestimate the variance at high wavenumbers. As discussed at the end of Appendix F, the wavenumber spectrum of SSH at wavelengths shorter than 70 km is flatter in the MITgcm model (Rocha et al., 2016; Savage et al., 2017) and thus has higher variance at these high wavenumbers. The differences between the spectral characteristics of SSH at high wavenumbers from different models underscores how little is known about ocean variability on scales smaller than 70 km. One of the primary contributions of the SWOT mission will be to elucidate the nature of spatial variability of SSH at these smaller scales that have heretofore not been observable from space. This will enable testing of the various model representations of small-scale variability of SSH.

Notwithstanding the issues summarized above, the analysis presented in Sections 6 and 8–10 provides useful insight into the resolution that can be expected in maps of the surface velocity and vorticity fields constructed from SWOT and WaCM data. From simulated SWOT measurements of SSH and WaCM measurements of surface velocity based on the CCS model used in this study, it is shown in Appendix G and Section 4 (see also Tables 1 and 2 and the bottom left panels of Figs. 17b, 17c, 19a and 19b) that the standard deviations of the uncorrelated measurement errors assumed in our analysis result in velocity and vorticity errors that are too large for the SWOT and WaCM data to be useful for oceanographic applications at the full resolutions of the pre-processed data (footprint sizes of 1 km for SWOT and 5 km for WaCM). It will therefore be necessary to smooth the pre-processed satellite data in ground-based post-processing to reduce the effects of uncorrelated measurement errors and improve the signal-to-noise ratios in SWOT and WaCM estimates of velocity and vorticity.

The standard deviations of the errors of the variables of interest decrease rapidly with increased smoothing (see Fig. 12). The signal standard deviations also decrease with increased smoothing, but usually less rapidly than the error variances. The signal-to-noise (S/N) standard deviation ratios therefore generally improve with increased smoothing. From visual inspection of many maps such as the examples shown in Fig. 16 for S/N standard deviation ratios of 1, 2 and 3.16, we recommend a ratio of 3.16 (which corresponds to a S/N variance ratio of 10) as a guideline minimum threshold for maps of velocity and vorticity constructed from SWOT and WaCM data. While subjective, this S/N ratio is sufficiently high to distinguish the signal from the noise unambiguously. For readers willing to accept a lower S/N ratio, the corresponding resolution capabilities can be inferred from the graphs in Sections 6, 8.4, 8.5, 9 and 10.1, specifically Figs. 18, 30, 35, 38–40 and 43. The results are compared graphically in Fig. 51 in Section 13.5 below for the resolution capabilities of SWOT and WaCM estimates of velocity and vorticity for the cases of threshold S/N standard deviation ratios of 3.16 and 2.00.

In addition to the subjective choice of a threshold S/N standard deviation ratio of 3.16 adopted in this study to define resolution capability, the discussion in Section 9 (see also Figs. 44 and 45 in Section 10.2) shows that the precise values of the resolution capabilities of SWOT and WaCM estimates of surface velocity and vorticity depend to some extent on the details of the algorithm used to smooth the noisy satellite data. For example, it was shown in Section 9 that the resolution capabilities of instantaneous snapshots and space-time smoothed maps of velocity and vorticity that would be inferred based on the loess smoother are better than those inferred from the Parzen smoother (see Table 3) but the relative rankings of resolution capabilities for the two instruments and for the choices of time averages considered in Section 9 (instantaneous snapshots and time averages over 4 and 14 days) are insensitive to the details of the smoothing algorithm.

A disadvantage of the loess smoother is that it is more than a factor of 50 more computationally intensive than the Parzen smoother for the set of filter cutoffs considered in this study. This was a significant issue

since a total of 1200 smoothed maps were required to generate the S/N graphs in Figs. 18, 30, 35 and 43 that are based on Parzen smoothing. Another disadvantage of the loess smoother and other smoothers that attenuate small-scale noise more effectively than the Parzen smoother is that the desirable characteristics of their filter transfer functions (small side lobes and a steep rolloff through the half-power filter cutoff wavenumber) require wider spans of data for each smoothed estimate. Such smoothers are therefore subject to greater edge-effect contamination of smoothed estimates near the edges of the measurement swaths. The assessments of resolution capabilities in this study are based on the more conservative estimates obtained using the Parzen smoother.

We first investigated the effects of measurement errors alone on the resolution capabilities of SWOT and WaCM estimates of surface velocity and vorticity. The analysis in Section 6 considered the idealized case of measurement swaths that span the entire CCS model domain on each satellite overpass. This is a highly unrealistic representation of SWOT sampling, which consists of a 120-km swath with a 20-km nadir gap (see the top panels of Fig. 20). For WaCM, however, most of the CCS model domain is sampled on each overpass by just one of the two parallel measurement swaths for the case of a total swath width of 1200 km (see the bottom panels of Fig. 20). If the total swath width can be extended to 1800 km, essentially all of the CCS model domain would be sampled on each overpass by just one of the two parallel measurement swaths (see Fig. 41).

Based on uncorrelated measurement errors alone without consideration of the limitations of swath width, the conclusion of Section 6 is that the resolution capabilities of SWOT estimates of surface velocity and vorticity are substantially better than those of estimates obtained from WaCM with the baseline standard deviation of 0.50 m s⁻¹ for the uncorrelated measurement errors (see Table 3). If the WaCM measurement noise can be reduced to 0.25 m s⁻¹, however, the resolution capabilities approach those of SWOT estimates of velocity and vorticity (see Section 10.2 and Table 4a and b), with the distinct advantage of a factor-of-17 wider measurement swath.

Mapping of the velocity and vorticity fields over domains larger than individual measurement swaths requires combining measurements from multiple satellite overpasses. Space-time smoothing of the satellite data to construct such maps introduces sampling errors in addition to the measurement errors considered in isolation in Section 6. As discussed in detail in Section 7, sampling errors arise predominantly from inadequate sampling of rapidly evolving submesoscale features in the surface velocity and vorticity fields. To the extent that the CCS model used in this study may underestimate the SSH variance at high wavenumbers as discussed above, our assessment of resolution capabilities for SWOT estimates of velocity and vorticity may be overly optimistic. Because small-scale features generally evolve more rapidly than large-scale features, higher spectral variance at high wavenumbers would likely increase the aliasing artifacts in time-averaged maps of velocity and vorticity constructed from SWOT. This would decrease the signal-to-noise ratio in the time-averaged maps, thus worsening the resolution capabilities.

It is evident from the figures in Sections 8.4 and 8.5 that the sampling errors summarized in Section 7 are much worse for SWOT than for WaCM because of the narrow measurement swath of the SWOT instrument and the space-time sampling pattern of the SWOT orbit; during each 21-day exact-repeat orbit, the full CCS domain considered in this study is observed during two 4-day subcycles that are interleaved spatially and separated temporally by 6-day gaps (see Figs. 21–23). Spatially adjacent swaths are separated in time by 10 days. Sampling errors are consequently the primary concern in maps of geostrophically computed velocity and vorticity constructed from SWOT data. The sampling errors are so large that our threshold S/N standard deviation ratio of 3.16 is not achieved in either 4-day or 14-day averaged vorticity fields using the Parzen smoother, even with the maximum half-power filter cutoff wavelength of 150 km considered in this study. Moreover, the resolution capability of SWOT estimates of 14-day averages of
velocity is substantially worse than in 4-day averages because of the increased sampling errors in the longer time average.

We thus conclude that geostrophically computed SWOT estimates of velocity and vorticity will be useful primarily within individual measurement swaths. But even then, there are likely to be issues from the edge effects of spatial smoothing with a span that is comparable to the 50-km width of each of the two parallel measurement swaths. These edge effects will be exacerbated by the increased measurement errors toward the inner and outer edges of each measurement swath (see Fig. F.1) that have not been taken into consideration in this study. To mitigate the multi-faceted effects of sampling errors, mapping of the surface velocity and vorticity fields from SWOT data will likely have to be done with the aid of dynamic interpolation from a data assimilation model, as suggested, for example, by Ubelmann et al. (2015).

## 12. Space-Time Smoothed SSH Fields from SWOT Data

Because of amplification of errors by the centered-difference approximations of the derivatives required to compute geostrophic velocity and vorticity from SSH, SWOT estimates of the derivative quantities are extremely sensitive to the standard deviation of $\sigma_h = 2.74$ cm of the uncorrelated errors in the SSH measurements on a 1 km × 1 km grid (see Table 1). As pointed out in Section 11, however, SWOT was not designed to map the surface velocity and vorticity fields that are the focus of this study. It was specifically designed to map the global SSH field with significantly higher resolution than can be achieved from the present constellation of nadir altimeters.

The scientific value of the improved resolution and accuracy of space-time smoothed SSH fields constructed from SWOT data should not be underestimated. Pujol et al. (2012), Fu and Ubelmann (2014) and Gaultier et al. (2016) have previously shown that maps of SSH from SWOT will be superior to the SSH maps from presently available nadir altimeter data. We show this here from simulated altimeter data based on the CCS model used in this study.

Space-time smoothed maps of SSH were constructed from simulated sampling by SWOT and from simulated sampling by nadir altimeters processed in a manner similar to that of the “two-sat” SSH fields produced and archived by CLS/AVISO from the merged measurements from two simultaneously operating altimeters. Our intent is not to reproduce all of the details of the CLS/AVISO mapping procedure (see Pujol et al., 2016, and references therein). In view of the conclusion of Section 8 that SWOT measurement errors are negligible compared with sampling errors and the fact that sampling errors are much worse for the sparse sampling by two nadir altimeters than for the wide-swath SWOT altimeter, measurement errors were not included in our simplified form of the CLS/AVISO smoothing procedure. Instead, we consider only the effects of sampling errors by constructing a simulated CLS/AVISO SSH map by the following procedure.

The twice-daily SSH fields in the 30-day simulation of the CCS model used in this study were linearly interpolated in space and time to the locations of actual TOPEX/Poseidon and ERS-1 altimeter observations during the period 22 May to 21 June 1993, which corresponds to the same time period as the climatologically forced CCS model simulation. These simulated error-free TOPEX/Poseidon and ERS-1 data were smoothed along the satellite ground tracks with a half-power filter cutoff wavelength of 37 km. The along-track smoothed SSH values were then subsampled at every third point, corresponding to a spacing of about 18 km.

The resolution capability of the CLS/AVISO SSH fields is limited to wavelengths longer than about 200 km and time scales longer than about a month (see Appendix A.3 of Chelton et al., 2011). The subsampled and smoothed SSH values along the TOPEX/Poseidon and ERS-1 ground tracks were therefore smoothed and interpolated to the CCS model grid using a 2-dimensional objective analysis procedure with a half-power filter cutoff wavelength of 200 km in each dimension. To mimic the monthly smoothing in the CLS/AVISO SSH fields, all of

the smoothed and subsampled simulated TOPEX/Poseidon and ERS-1 SSH observations within the 30-day period were weighted equally in the 2-dimensional smoothing.

The effects of sampling errors on the 200-km by 30-day smoothed SSH fields constructed from simulated SWOT and nadir altimeter data are shown in Fig. 46. The “true” space-time smoothed SSH field constructed from twice-daily error-free model output is shown in Fig. 46a. The space-time smoothed maps constructed from simulated CLS/AVISO processing of error-free nadir altimeter data and from simulated error-free SWOT data are shown in the top panels of Figs. 46b and c, respectively. The bottom panels show the sampling errors in the maps constructed from the simulated altimeter data.

The superiority of SWOT sampling is visually apparent in Fig. 46 from the smaller sampling errors. The standard deviation of 0.34 cm for the sampling errors in the SWOT map is a factor of 3.2 smaller than the standard deviation of 1.09 cm in the nadir altimeter map. Moreover, the dynamic range of the errors is nearly a factor of four smaller in the SWOT map: the extrema of the sampling errors range from only −1.3 cm to +1.3 cm in the SWOT map compared with a range of −6.5 cm to +3.7 cm in the nadir altimeter map. The present resolution of 200 km by one month will thus be achieved from SWOT data with much smaller sampling errors than in the presently available CLS/AVISO SSH fields.

It is noteworthy that the CCS region considered in our analysis coincidentally lies within the latitude range of most favorable space-time sampling by SWOT. The mapping errors increase somewhat at higher and lower latitudes, although they are still much smaller than the errors in maps constructed from nadir altimeter data (Pujol et al., 2012; see their Fig. 1 and their Tables 2 and 3).

The simulations in Fig. 46 do not include the effects of uncorrelated measurement errors. As noted above, measurement errors have negligible effect for the 200-km smoothing applied for the simulated SWOT map of SSH Fig. 46c. Uncorrelated measurement errors are a more significant consideration in the simulated “two-sat” CLS/AVISO map of SSH in Fig. 46b because the point-to-point noise in the nadir altimeter measurements at intervals of 6 km along the satellite ground track has a standard deviation of about 3 cm. This is slightly higher than the $\sigma_h = 2.74$ cm standard deviation of the uncorrelated measurement errors in the pre-processed SWOT estimates of SSH with a smaller footprint size of 1 km. Smoothing the SWOT data 2-dimensionally to a resolution of 6 km (i.e., with a half-power filter cutoff wavelength of 12 km in each dimension — see Appendix B) to be commensurate with the 6-km along-track resolution of nadir altimeter data would reduce the SWOT noise standard deviation by about a factor of six (see Appendix D). The resulting standard deviation of about 0.46 cm for the noise in the 12-km smoothed SWOT data (see Fig. 12a) is more than a factor of six smaller than the 3-cm noise of the nadir altimeter data from which CLS/AVISO maps are constructed. Accounting for the effects of uncorrelated measurement errors in the simulations in Fig. 46 would thus worsen the errors in our simulated CLS/AVISO SSH field compared with the simulated SWOT SSH field.

A complete and rigorous analysis of the improved accuracy and resolution of SSH fields that will be achieved from SWOT data, including consideration of the effects of uncorrelated measurement errors mentioned above as well as orbit errors and other long-wavelength measurement errors, is beyond the scope of this study. Fig. 47 offers a sense of what can be expected from sampling errors alone. The two curves show the standard deviation of the sampling errors from spatially smoothed 30-day and 21-day averages of simulated SWOT data. The 1.09 cm standard deviation from our simulation of the sampling errors in CLS/AVISO SSH fields with 200-km by 30-day smoothing shown in Fig. 46b can be achieved from SWOT data with only 100-km by 21-day smoothing. One application that will especially benefit from the improved accuracy and improved spatial and temporal resolution of SSH fields constructed from SWOT data is the identification and tracking of mesoscale eddies (e.g., Chelton et al., 2011), and more generally, the study of mesoscale ocean dynamics. The smaller sampling errors from SWOT will significantly reduce the mislocation of eddies in presently available SSH fields (see, for example, Fig. 3 of Pascual et al., 2006).
<figure><img src="image_51.png" /><figcaption>**Fig. 46.** Maps of 30-day averaged SSH smoothed 2-dimensionally with a half-power filter cutoff wavelength of 200 km in each dimension: Column (a) The “true” space–time smoothed field computed from error-free model SSH fields at a time step of 0.5 day over the 30-day period; Column (b) The space–time smoothed field computed from simulated TOPEX/Poseidon and ERS-1 nadir altimeter sampling of error-free model SSH fields at the times and locations of each altimeter observation over the 30-day period; and Column (c) The space–time smoothed field computed from simulated SWOT swath sampling of error-free model SSH fields at the times and locations of each SWOT observation over the 30-day period. The bottom panels are maps of the sampling errors computed by subtracting the error-free map in Column (a) from the simulated satellite-sampled maps in the top row of the respective columns (b) and (c).</figcaption></figure>

## 13. Summary and conclusions

High-resolution global satellite estimates of surface ocean velocity and vorticity would be of broad interest to a wide range of physical, biological and chemical oceanography applications. In this study, we conducted an exhaustive investigation the effects of uncorrelated measurement errors and sampling errors on the resolution capabilities of instantaneous and space–time smoothed maps of velocity and vorticity fields derived from simulated measurements by two different techniques: (1) Altimetric measurements of SSH by interferometry from the Surface Water and Ocean Topography (SWOT) mission with a planned launch in 2021; and (2) Doppler radar measurements of surface ocean velocity from a future scatterometer mission that we have referred to as a Winds and Currents Mission (WaCM). WaCM is one of seven recommended NASA Earth Systems Explorer missions in the recent

decadal survey by the National Academies of Science, Engineering and Medicine (NASEM, 2018).

The analysis procedures used in this study and the results and conclusions of our assessments of the resolution capabilities of SWOT and WaCM estimates of surface ocean velocity and vorticity are summarized below.

### 13.1. *The analysis procedures*

Our strategy for assessing resolution capability is summarized in detail in Section 5. Simulated SWOT and WaCM data were generated based on, respectively, the SSH and surface velocity fields from 30 days of twice-daily output from the CCS model summarized in Section 2. Uncorrelated measurement errors with the baseline standard deviations were added to the model simulations of SSH and velocity. For SWOT, we used the error standard deviation of $\sigma_h = 2.74$ cm that is derived
<figure><img src="image_52.png" /><figcaption>**Fig. 47.** The standard deviations of the sampling errors in 2-dimensionally smoothed 30-day and 21-day averaged SSH fields constructed from simulated SWOT data as functions of half-power filter cutoff wavelength. The horizontal dashed line corresponds to the 1.09-cm standard deviation of the sampling errors in the simulated CLS/AVISO SSH field in the bottom panel of Fig. 46b constructed from error-free model SSH fields subsampled at the times and locations of the TOPEX/Poseidon and ERS-1 nadir altimeter observations over a 30-day period.</figcaption></figure>

for a footprint size of 1 km in Appendix F. For WaCM, we used a baseline standard deviation of $\sigma_{spd} = 0.50 \text{ m s}^{-1}$ for a footprint size of 5 km but we also considered the benefits of smaller measurement errors in Section 10.2. Noise-free and noisy SSH and velocity fields on the $0.5 \text{ km} \times 0.5 \text{ km}$ model grid were processed in manners that mimic the pre-processing that will be applied to the SWOT and WaCM data to achieve footprint sizes of 1 km and 5 km, respectively. Maps of noise-free and noisy SSH and velocity fields with these footprint sizes were then constructed on a $1 \text{ km} \times 1 \text{ km}$ grid over the full model domain and on subsampled grids within the measurement swaths at the times and locations of the overpasses of each of the two satellite instruments.

The errors of surface velocity and vorticity fields estimated from the simulated pre-processed SWOT and WaCM data are too large for the estimates to be useful scientifically (see Tables 1 and 2 and the bottom left panels of Figs. 17b, 17c, 19a and 19b). It will be necessary to smooth the pre-processed SWOT and WaCM data in ground-based post-processing to reduce the effects of measurement noise in the maps of the variables of interest in this study. Four combinations of SSH and velocity fields (noise-free on the full grid, noise-free on the subsampled grid, noisy on the full grid and noisy on the subsampled grid) were successively smoothed with half-power filter cutoff wavelengths ranging from 10 km to 150 km. We also investigated the added benefits of time averaging to reduce the effects of measurement and sampling errors.

For each filter cutoff and each choice of time averaging, we computed the spatial standard deviations of the smoothed noise-free fields and the error fields for the cases of measurement errors alone, sampling errors alone and combined measurement and sampling errors. Our definition of resolution capability is based on the signal-to-noise (S/N) ratios of the standard deviations of the smoothed noise-free fields and the smoothed error fields. The S/N ratios for the cases of measurement errors alone and sampling errors alone allow an assessment of the relative contribution of each type of error to the total errors in maps of velocity and vorticity constructed from the simulated satellite data.

With increased spatial smoothing, the noise standard deviations usually decrease more rapidly than the signal standard deviations. The S/N ratios therefore improve with increased smoothing. Based on visual comparisons of many maps of smoothed noise-free and noisy velocity and vorticity fields constructed from the simulated satellite data (see, for example, Fig. 16), we defined the resolution capability as the half-power filter cutoff wavelength above which the S/N standard deviation ratios exceed a threshold value of 3.16, which corresponds to a S/N variance ratio of 10. The improvements in the resolution capabilities

inferred based on a more liberal S/N standard deviation ratio of 2 are shown in Fig. 51 in Section 13.5 below.

The sensitivity of our assessment of resolution capability to the details of the smoothing algorithm was investigated in Section 9 (see also Figs. 44 and 45 in Section 10.2) from consideration of two different smoothers. The resolution capability inferred based on the loess smoother is better than that inferred based on the Parzen smoother (see Table 3). This is attributable to the more efficient attenuation of small-scale variability because of the better filtering properties of the loess smoother, albeit with more than 50 times more computational effort for the set of calculations in this study. The relative rankings of resolution capabilities for the two instruments and for the choices of time averaging considered in this study (instantaneous snapshots and time averages over 4 and 14 days) are insensitive to the details of the smoothing algorithm. We applied both the Parzen smoother and the loess smoother for the analysis presented here. Our assessments of resolution capabilities are based on the more conservative estimates inferred from the Parzen smoother.

## 13.2. Instantaneous snapshot estimates of velocity and vorticity

Based on our criterion of a minimum S/N standard deviation ratio of 3.16 and the Parzen smoother, we showed in Section 6 that the in-swath resolution capabilities (i.e., the resolution capabilities when measurement errors are the only consideration) for instantaneous snapshot maps of geostrophically computed SWOT estimates of velocity and vorticity would be 32 km and 54 km, respectively, based on the spatial variances of the signals in the CCS model considered in this study (see Fig. 18a). These numbers represent the wavelengths of the features that can be resolved with a S/N standard deviation ratio of 3.16. The corresponding radii of resolvable features are about four times smaller than the wavelength resolution (see Appendix B). The above resolution capabilities for instantaneous maps of SWOT estimates of velocity and vorticity correspond to the upper range of submesoscale variability.

In comparison, the in-swath resolution capabilities for instantaneous maps of velocity and vorticity constructed from WaCM data with the baseline standard deviation of $\sigma_{spd} = 0.50 \text{ m s}^{-1}$ for the uncorrelated measurement errors, equally partitioned between the two velocity components, would be 130 km for both variables (see Fig. 18b). Reducing the WaCM measurement errors to $\sigma_{spd} = 0.25 \text{ m s}^{-1}$ would improve these resolution capabilities to 50 km and 74 km for instantaneous maps of velocity and vorticity, respectively (see Fig. 44 and Table 4a and b).

In terms of measurement errors alone, and aside from the limitations of the geostrophic approximation discussed in Section 3, velocity and vorticity would thus be estimated in instantaneous snapshots with higher resolution from SWOT than from WaCM, although only modestly so if the WaCM measurement noise can be reduced to $\sigma_{spd} = 0.25 \text{ m s}^{-1}$.

An important qualification of the conclusion that the resolution capabilities of in-swath estimates of velocity and vorticity are better for SWOT than for WaCM is that the analysis of the effects of measurement errors alone in Section 6 considers the case of measurement swaths that sample the entire CCS model domain and thus neglects any edge effects of the smoothing near the edges of the measurement swaths. With its 120-km swath width and a 20-km nadir gap, it can be seen from the top panels of Fig. 20 that complete coverage of the CCS model domain on a single satellite overpass is a highly idealized representation of SWOT sampling. In addition to its sparse coverage, edge effects will be problematic because the half-power filter cutoff wavelengths of 32 km and 54 km required to achieve a S/N standard deviation ratio of 3.16 for in-swath SWOT estimates of velocity and vorticity are comparable to the width of each parallel measurement swath. The above numbers must therefore be considered the most optimistic possible outlook for the resolution capabilities of velocity and vorticity fields constructed from SWOT data within the confines of a single measurement swath.

In contrast, the idealized complete coverage of the CCS model domain considered for the analysis in Section 6 would be very nearly
achieved on each satellite overpass by WaCM for a region of this size. With a total swath width of 1200 km, WaCM coverage would be nearly complete with just one side of the two parallel measurement swaths (see the bottom panels of Fig. 20). WaCM coverage would be essentially complete on each overpass if the swath width can be extended to 1800 km (Fig. 41).

### 13.3. Space-Time Smoothed Estimates of Velocity and Vorticity

Mapping of the velocity and vorticity fields over a region larger than an individual measurement swath requires space-time smoothing of the satellite data from multiple overpasses of the region of interest. This is especially needed for SWOT because of its narrow 120-km swath width. For both SWOT and WaCM, time averaging can reduce the effects of the measurement errors considered in Section 6 but it also introduces the three types of sampling errors that are summarized in Section 7. We investigated the effects of combined measurement and sampling errors in Sections 8–10 for the cases of 4-day and 14-day averages. These choices of time averages correspond to the two subcycles of each repeat of the 21-day SWOT orbit (see Figs. 22 and 23). The 4-day averaging time also corresponds to the exact-repeat period of the WaCM orbit assumed in this study.

The resolution capabilities inferred from the analysis in Section 8 for SWOT and WaCM estimates of space-time smoothed velocity and vorticity fields are summarized in Table 3 for a swath width of 1200 km in the case of WaCM. Whereas SWOT outperformed WaCM from the consideration of measurement errors alone in Section 6, the relative performance of the two instruments is not so simple for space-time smoothed velocity and vorticity fields. Because of the narrow swath width of SWOT and the space-time sampling pattern of the SWOT 21-day exact-repeat orbit that samples the full CCS model domain in two 4-day subcycles that are interleaved and separated by 6-day gaps, the total errors in maps of geostrophically computed velocity and vorticity constructed from SWOT data are dominated by sampling errors. Improving the measurement accuracy would have very little effect on the total errors in SWOT estimates of space-time smoothed maps of velocity and vorticity.

For velocity, the resolution capability of SWOT estimates of 4-day averages is 52 km based on the Parzen smoother (see Fig. 30a), but with large gaps in the maps because of the sparse SWOT sampling of the CCS model domain during a 4-day period (see Fig. 28). Because spatially adjacent swaths are separated by 10 days, sampling errors are even worse in SWOT estimates of time-averaged velocity fields over the 14-day period required for SWOT to map the entire CCS model domain; the resolution capability degrades to 132 km (see Fig. 30b). For SWOT estimates of space-time smoothed vorticity fields, sampling errors are so large that the S/N standard deviation ratio never reaches our threshold minimum of 3.16 for either choice of averaging period based on the Parzen smoother (see Figs. 35a and b).

A conclusion of this study is thus that SWOT will be of limited value for spatially complete mapping of velocity and vorticity fields without the use of a data assimilation model. Its primary contribution will be the SSH mapping capabilities for which it was designed. SWOT will map the global SSH field with accuracy and resolution significantly better than are presently achieved from the merging of SSH measurements from multiple nadir altimetry missions (see Section 12 and Figs. 46 and 47). Among other applications, this will greatly benefit studies of mesoscale eddy dynamics.

For spatially complete global mapping of the surface velocity and vorticity fields, the Doppler radar measurements from a WaCM mission would be far superior to SWOT. With its wider measurement swath and the 4-day orbit repeat period assumed for the simulations of WaCM data in this study, sampling errors are much less of an issue than for SWOT. The effects of measurement errors in instantaneous snapshot maps of the velocity and vorticity can therefore be reduced with time averaging. By our criterion, the resolution capabilities for WaCM estimates of 4-day averaged maps of velocity and vorticity based on a swath width

of 1200 km and a standard deviation of $\sigma_{spd} = 0.50 \text{ m s}^{-1}$ for the uncorrelated measurement errors (see Figs. 30c and 35c) are 60 km and 87 km, respectively, which correspond to the lower range of mesoscale variability. Increasing the swath width to 1800 km would improve these resolution capabilities to 44 km and 70 km (see Section 10.1 and Figs. 43a and c).

For both swath widths, the spatial resolutions of WaCM estimates of velocity and vorticity could be further improved by increasing the averaging period, e.g., to the 14-day averaging period considered in Sections 8–10. It is not clear, however, how much of an advantage the improved spatial resolution in these longer time averages would be since increasing the averaging time also attenuates the smaller-scale velocity and vorticity signals that are of interest (see Figs. 24 and 25).

The fact that measurement noise is the primary source of errors in WaCM maps of the space-time smoothed velocity and vorticity fields means that the resolution capabilities would be improved if the standard deviation of uncorrelated WaCM measurement errors can be reduced from the baseline value of $\sigma_{spd} = 0.50 \text{ m s}^{-1}$ assumed in Sections 6, 8.4, 8.5, 9 and 10.1. The benefits of improving the WaCM measurement accuracy were investigated in Section 10.2 (see Fig. 45). If the measurement errors can be reduced to $\sigma_{spd} = 0.25 \text{ m s}^{-1}$, for example, with the same footprint size of 5 km, the resolution capabilities for WaCM estimates of 4-day averaged fields based on a swath width of 1800 km would improve from the above-noted 44 km and 70 km to 18 km and 45 km for velocity and vorticity, respectively (see Figs. 45a and c).

The resolution capabilities summarized above for WaCM estimates of 4-day averaged maps of surface velocity and vorticity constructed using the Parzen smoother are listed in Table 4c and d for the cases of 1200-km and 1800-km swath widths and standard deviations of $\sigma_{spd} = 0.50 \text{ m s}^{-1}$ and $0.25 \text{ m s}^{-1}$ for the uncorrelated speed measurement errors. For comparisons, the resolution capabilities of WaCM estimates of velocity and vorticity are also listed for the cases of instantaneous snapshots (Table 4a and b) and 14-day averages (Table 4e and f).

### 13.4. Visual assessments of the WaCM resolution capabilities

The improvements in the resolution of velocity and vorticity maps that would be achieved from WaCM over the present resolution capability of about 200 km by one month in geostrophically computed maps constructed from multi-mission nadir altimeter data are visually apparent from the 4-day averages shown in Figs. 48 and 49 for the CCS region considered in this study. For the purposes of this comparison, we have adopted the more conservative resolution capabilities inferred from Fig. 45 based on Parzen smoothing.

Although lacking the highly energetic submesoscale variability that is evident in the instantaneous maps in Fig. 3, the resolutions of about 60 km and 90 km that are attainable in 4-day averaged maps of, respectively, velocity and vorticity (Figs. 48b and 49b) with the baseline standard deviation of $\sigma_{spd} = 0.50 \text{ m s}^{-1}$ for the uncorrelated measurement noise and a swath width of 1200 km are a dramatic improvement over the present capabilities shown in Figs. 48a and 49a. The increased resolutions of about 45 km and 70 km that would be achieved in 4-day averaged maps of velocity and vorticity with a wider swath width of 1800 km are shown in Figs. 48c and 49c. The further improvements to resolutions of about 20 km and 45 km that would be achieved in 4-day averaged maps of velocity and vorticity from WaCM if the swath width can be increased to 1800 km and the standard deviation of the WaCM uncorrelated measurement noise can be reduced to $\sigma_{spd} = 0.25 \text{ m s}^{-1}$ are shown in Figs. 48e and 49e.

While the 4-day averaging in Figs. 48 and 49 reduces the measurement noise and improves the spatial resolution compared with instantaneous snapshots, the time averaging attenuates the small-scale signal variability that is of interest, as noted above (Fig. 24). Depending on the application, the coarser spatial resolution capability of the snapshots may be preferable to the 4-day averages. This is visually apparent
<figure><img src="image_53.png" /><figcaption>**Fig. 48.** Space-time smoothed maps of the magnitudes of vector-averaged error-free surface velocity with the wavelength resolutions that can be achieved from presently available altimeter data and from 4-day averages of WaCM data with the combinations of measurement noise $\sigma_{\text{spd}}$ and swath widths listed in Table 4c: (a) The resolution of 200 km by 30 days that can be achieved with the geostrophic approximation from SSH maps constructed from presently available merged nadir altimetry data; (b) A resolution of 60 km that could be achieved in 4-day averages of WaCM data for a swath width of 1200 km and $\sigma_{\text{spd}} = 0.50 \text{ m s}^{-1}$; (c) A resolution of 45 km that could be achieved in 4-day averages of WaCM data for a swath width of 1800 km and $\sigma_{\text{spd}} = 0.50 \text{ m s}^{-1}$; (d) A resolution of 25 km that could be achieved in 4-day averages of WaCM data for a swath width of 1200 km and $\sigma_{\text{spd}} = 0.25 \text{ m s}^{-1}$; and (e) A resolution of 20 km that could be achieved in 4-day averages of WaCM data for a swath width of 1800 km and $\sigma_{\text{spd}} = 0.25 \text{ m s}^{-1}$.</figcaption></figure>

from Fig. 50, which shows instantaneous snapshot maps of velocity and vorticity with the resolution capabilities that are achievable from WaCM data with measurement error standard deviations of $\sigma_{\text{spd}} = 0.50 \text{ m s}^{-1}$ and $0.25 \text{ m s}^{-1}$.

### 13.5. Final comments

A qualification of the analysis in Sections 6 and 8–10 is that we have neglected the effects of correlated measurement errors. Because of
<figure><img src="image_54.png" /><figcaption>**Fig. 49.** Space-time smoothed maps of error-free vorticity scaled by the local Coriolis parameter $f$ with the wavelength resolutions that can be achieved from presently available altimeter data and from 4-day averages of WaCM data with the combinations of measurement noise $\sigma_{\text{spd}}$ and swath widths listed in Table 4d: (a) The resolution of 200 km by 30-days that can be achieved with the geostrophic approximation from SSH maps constructed from presently available merged nadir altimetry data; (b) A resolution of 90 km that could be achieved in 4-day averages of WaCM data for a swath width of 1200 km and $\sigma_{\text{spd}} = 0.50 \text{ m s}^{-1}$; (c) A resolution of 70 km that could be achieved in 4-day averages of WaCM data for a swath width of 1800 km and $\sigma_{\text{spd}} = 0.50 \text{ m s}^{-1}$; (d) A resolution of 60 km that could be achieved in 4-day averages of WaCM data for a swath width of 1200 km and $\sigma_{\text{spd}} = 0.25 \text{ m s}^{-1}$; and (e) A resolution of 45 km that could be achieved in 4-day averages of WaCM data for a swath width of 1800 km and $\sigma_{\text{spd}} = 0.25 \text{ m s}^{-1}$.</figcaption></figure>

the spatial high-pass filtering property of the derivative operator, these long-wavelength errors are secondary to the effects of uncorrelated errors on estimates of the surface ocean velocity and vorticity fields that are the primary interest in this study. A more important limitation is that we considered the uncorrelated measurement errors to be spatially homogeneous across the SWOT and WaCM measurement swaths. In

reality, the measurement errors for both instruments increase toward the inner and outer edges of each swath.

Another caveat to our analysis is that we have assumed that the measurement errors of WaCM estimates of surface velocity are equally partitioned between the two orthogonal velocity components. We have thus neglected the geometrical transformation issues associated with the limited azimuthal diversity of the Doppler measurements of radial
<figure><img src="image_55.png" /><figcaption>**Fig. 50.** Spatially smoothed maps of the magnitudes of vector-averaged error-free surface velocity (top row) and error-free vorticity scaled by the local Coriolis parameter *f* (bottom row) with the wavelength resolutions that can be achieved from presently available altimeter data and from instantaneous snapshot maps of WaCM data on a single satellite overpass with the combinations of measurement noise *σ*<sub>spd</sub> and swath widths listed in Table 4a and b: (a) The resolution of 200 km by 30 days that can be achieved for velocity computed geostrophically from presently available merged nadir altimetry data (the same as Fig. 48a); (b) A resolution of 130 km that could be achieved for velocity from a snapshot of WaCM data with *σ*<sub>spd</sub> = 0.50 m s<sup>-1</sup>; (c) A resolution of 50 km that could be achieved for velocity from a snapshot of WaCM data with *σ*<sub>spd</sub> = 0.25 m s<sup>-1</sup>; (d) The resolution of 200 km by 30 days that can be achieved for vorticity from presently available merged nadir altimetry data (the same as Fig. 49a); (e) A resolution of 130 km that could be achieved from a snapshot of WaCM data with *σ*<sub>spd</sub> = 0.50 m s<sup>-1</sup>; and (f) A resolution of 130 km that could be achieved from a snapshot of WaCM data with *σ*<sub>spd</sub> = 0.25 m s<sup>-1</sup>.</figcaption></figure>

velocity near the swath edges. Errors of the cross-track velocity compo-
nent increase toward the inner edges of the two parallel measurement
swaths and errors of the along-track component increase toward the
outer edges of the measurement swaths.

The resolution capabilities inferred from our analysis are based
on a specified minimum S/N standard deviation ratio. Some readers
may feel that our recommended threshold value of 3.16 is overly

conservative. The resolution capabilities inferred based on lower S/N
ratios can be estimated graphically from Figs. 18, 30, 35, 38–40 and
43. To illustrate the sensitivities of inferred resolution capabilities to
the choice of threshold S/N standard deviation ratio, the results are
shown graphically in Fig. 51 for snapshots, 4-day averages and 14-day
averages of velocity and vorticity based on our recommended S/N ratio
of 3.16 (the red lines) and on a more liberal ratio of 2.00 (the green
<figure><img src="image_56.png" /><figcaption>**Fig. 51.** Graphical representation of the resolution capabilities for snapshots, 4-day averages and 14-day averages of SWOT and WaCM estimates of velocity and vorticity based on Parzen smoothing. Four different combinations of swath width and measurement noise are shown for WaCM. Note that swath width is irrelevant to the resolutions shown for snapshots in the top panels (see the discussion in Section 6). The red lines are the resolution capabilities listed in Tables 3 and 4 for the S/N standard deviation ratio threshold of 3.16 recommended in this study. The green lines are the resolution capabilities inferred from Figs. 18, 30, 35, 38–40 and 43 based on a more liberal S/N standard deviation ratio threshold of 2.00. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

lines) that we feel is the smallest value that should be considered (see Fig. 16).

Because of our incomplete treatment of measurement errors, the assessments of resolution capabilities deduced from the analysis in Sections 6 and 8–10 and summarized graphically in Fig. 51 are likely to be somewhat optimistic.

We also emphasize that our assessments of resolution capabilities are specific to the CCS region during the summertime, and in particular, to the representation of the CCS surface currents in the ROMS model with 0.5 km × 0.5 km grid resolution used here to simulate SWOT and WaCM data. Since the S/N ratios for WaCM estimates of the velocity and vorticity fields depend predominantly on measurement errors that

are relatively homogeneous geographically, the resolution capabilities inferred from this study will improve in regions or at times of more energetic spatial variability of velocity and vorticity. Likewise, the resolution capabilities for WaCM will be degraded when the velocity and vorticity variability is less energetic.

It is less clear how the resolution capabilities will change geographically and seasonally for SWOT because the S/N ratio depends predominantly on sampling errors owing to the narrow measurement swaths. While the signal variance increases when the variability is more energetic, so does the variance of the sampling errors. Depending on the relative increases in signal and sampling error variances, the resolution capabilities may actually worsen for SWOT when the variability is more
energetic. Likewise, the sampling error variance may decrease more than the signal variance when the variability is less energetic, thereby improving the resolution capabilities for SWOT. The results for SWOT are thus likely to be regionally and seasonally specific.

The immunity of WaCM measurements of surface velocity to errors from the geostrophic approximation that must be used for SWOT estimates of surface velocity is an important advantage of WaCM. In principle, surface divergence and hence vertical velocity can be estimated from WaCM measurements of surface velocity. This cannot be done from SWOT data since geostrophically computed velocity is non-divergent.

In practice, estimation of surface divergence from WaCM data will be challenging since the velocity field is nearly geostrophic on the scales that will be resolvable by WaCM. The non-divergent part of the velocity signal on these scales is therefore weak. In particular, the divergence is much weaker than the vorticity that has been a focus of this study. Divergence and vorticity both involve first derivatives of velocity. The residual noise variance for a given space–time smoothing of divergence will therefore be about the same as the residual noise variance for space–time smoothed vorticity. It can thus be anticipated that the signal-to-noise ratio that is the basis for defining the resolution capability in this study will be smaller for divergence than for vorticity. The resolution capability of WaCM estimates of space–time smoothed surface divergence will therefore be coarser than the resolution capabilities determined for vorticity in Sections 8–10.

We close by reiterating the important point that WaCM would provide simultaneous measurements of vector winds collocated with the measurements of surface ocean velocity that were considered in this study. The WaCM measurements of vector winds will have the same footprint size of 5 km as the surface ocean velocity measurements, which is about a factor-of-5 smaller than the footprint size of QuikSCAT measurements of vector winds. The WaCM technology for measurements of both wind velocity and surface current velocity is mature and has been demonstrated from an aircraft version of a Doppler scatterometer that has been built by NASA and flown in several field campaigns over the past year and additional field campaigns are planned in the future (Rodríguez et al., 2018).

## Acknowledgments

We thank Ralph Milliff, Bo Qiu, Ernesto Rodríguez and Lee-Lueng Fu for many helpful editorial comments and suggestions that improved the manuscript. This research was funded by NASA Grants NNX13AD78G, NNX14AM72G, NNX13AE32G, NNX14AM66G, NNX16AH76G, NNX14AM71G and NNX17AH54G. The two North Atlantic Ocean simulations in this study were performed using HPC resources from GENCI-TGCC with support from Grant 2017-A0010107638 for Jonathan Gula.

## Appendices

To provide a road map of the unusually large number of appendices (nine total) in this paper and to clarify the logic of the ordering of the appendices, we include here a brief overview of the content of each appendix.

**Appendix A** provides a detailed description of the Parzen smoother that is the primary filter used in this study to simulate the smoothing applied in the pre-processing of simulated SWOT and WaCM data and the additional smoothing applied in simulated ground-based post-processing of simulated SWOT and WaCM data to reduce the effects of measurement errors in estimates of velocity and vorticity. The filter transfer function for the Parzen smoother that is given by (A.11) and the relationship (A.12b) between the span of the Parzen smoother and the corresponding half-power filter cutoff wavelength are used extensively throughout later appendices.

**Appendix B** shows that the footprint diameter that is achieved by smoothing a field of uncorrelated white noise with a specified

half-power filter cutoff wavelength of $\lambda_c$ can be characterized as approximately $\lambda_c/2$. The science requirement for SWOT measurements of SSH with a footprint size (diameter) of 1 km can thus be achieved with isotropic 2-dimensional smoothing using a Parzen smoother with a half-power filter cutoff wavelength of 2 km. The errors of these pre-processed SWOT estimates of SSH are shown to be uncorrelated when sampled on a 1 km × 1 km grid.

The baseline design assumed here for WaCM measurements of surface velocity components with a footprint size of 5 km can be achieved with isotropic 2-dimensional smoothing using a Parzen smoother with a half-power filter cutoff wavelength of 10 km. While the errors of these pre-processed WaCM estimates of velocity are uncorrelated when sampled on a 5 km × 5 km grid, it is shown later in Appendix H that it is advantageous to oversample the WaCM data on a 1 km × 1 km grid because the response function for standard centered difference estimates of derivatives on the finer grid retains more small-scale variability in the vorticity signal.

**Appendix C** shows that the filter transfer function of the Parzen smoother is almost identical to that of the Gaussian-weighted smoother with an appropriately chosen e-folding scale. A feature of isotropic Gaussian smoothing in two dimensions, i.e., with Gaussian weighting dependent only on the radial distance of the simulated observations from the location of the smoothed estimate, is that it can be applied either as a single 2-dimensional weighted average or as separate 1-dimensional weighted averages in each of two orthogonal dimensions. The close similarity of the Parzen smoother to the Gaussian smoother thus implies that smoothing successively using a 1-dimensional Parzen smoother with the same half-power filter cutoff wavelength in each dimension is essentially equivalent to isotropic 2-dimensional smoothing.

While the Parzen smoothing that was applied to the simulated SWOT and WaCM data in Sections 6 and 8–10 was implemented truly isotropically, the derivations of analytical expressions for the wavenumber spectral content of 2-dimensionally smoothed fields in Appendices E, F, H and I and for the theoretical error spectra in Figs. 13–15 in Section 4.4, are greatly simplified if the Parzen smoothing is applied 1-dimensionally in each dimension.

**Appendix C** also includes a comparison of the filter transfer function of the Parzen smoother with that of the quadratic loess smoother. The latter has the desirable property that its filter transfer function rolls off more steeply through the half-power filter cutoff wavelength, but it is much more computationally intensive. It is shown in Sections 9 and 10.2 that the more effective attenuation of small-scale noise that is achieved with the loess smoother or other smoothers with similar filter transfer function properties improves our assessment of the resolution capability somewhat (see Table 3 and Figs. 44 and 45).

**Appendix D** determines the reduction of the variance of uncorrelated errors that is achieved by 2-dimensional smoothing. For isotropic smoothing with a half-power filter cutoff wavelength of $\lambda_c$, it is shown from (D.14c) that the error variance is reduced by the factor $4\Delta x \Delta y \lambda_c^{-2}$ that is the same as would be achieved by block averaging of measurement errors on a $\Delta x \times \Delta y$ grid with a span of $\lambda_c/2$ in each dimension. This variance reduction factor, which is equivalent to a standard deviation reduction factor of $2(\Delta x \Delta y)^{1/2} \lambda_c^{-1}$, is used extensively in the derivations in Appendices E–G and I.

**Appendix E** quantifies the effects of 2-dimensional smoothing on the 1-dimensional wavenumber spectrum of 2-dimensional uncorrelated white noise. As in the case of low-pass filtering of a 1-dimensional dataset, 2-dimensional smoothing of a 2-dimensional dataset attenuates the spectral content of the 1-dimensional spectrum at wavenumbers higher than the half-power filter cutoff wavelength $\lambda_c$. For uncorrelated noise, the spectral values are also attenuated at all wavenumbers (not just at high wavenumbers) by the constant multiplicative factor of $2\Delta x/\lambda_c$ in (E.11b), where $\Delta x$ and $\lambda_c$ are the grid spacing and half-power filter cutoff wavelength in the dimension perpendicular to the dimension of the 1-dimensional spectrum.
The results of Appendix E are used in the derivations of 1-dimensional spectra of smoothed fields of uncorrelated errors in Appendices F and I. In the context of the analysis presented in this paper, the interest is in 1-dimensional spectra in the alongshore dimension. In the context of actual satellite data, the interest is in 1-dimensional spectra in the along-track dimension.

Appendix F determines the variance of the uncorrelated errors of pre-processed SWOT estimates of SSH based on the science requirements for SWOT that are expressed in terms of the intersection of an hypothesized wavenumber spectrum of the SSH signal and the wavenumber spectrum of smoothed uncorrelated noise. The analysis concludes that the error standard deviation for the baseline design is $\sigma_h = 2.74$ cm.

Our derived error standard deviation of $\sigma_h = 2.74$ cm is used throughout this study, including in the propagation-of-error analysis in Appendix G to determine the standard deviations of errors of geostrophically computed SWOT estimates of velocity and vorticity. The derivations of the analytical formulas for the 1-dimensional wavenumber spectra of errors in SWOT estimates of SSH and geostrophically computed velocity and vorticity in Appendix I are also based on an error standard deviation of $\sigma_h = 2.74$ cm. Likewise, the error fields for the simulated SWOT data used in Sections 4, 6, 8 and 9 are predicated on an error standard deviation of $\sigma_h = 2.74$ cm.

Appendix G applies standard propagation-of-error analysis to derive analytical expressions for the variances of the errors of derivative quantities estimated from the pre-processed SWOT and WaCM data. In the case of SWOT, the velocity components are computed geostrophically from derivatives of SSH. The vorticity is then estimated from derivatives of the geostrophically computed velocity components, i.e., from second derivatives of SSH. In the case of WaCM, the velocity is measured directly and vorticity is obtained from derivatives of the velocity components. The derivatives in this study are all estimated using standard centered differences (see Appendix H). This finite differencing amplifies the errors in the measured variables.

The equations that are derived in Appendix G are all expressed in terms of the variances of the uncorrelated errors of the pre-processed SWOT and WaCM data. Throughout this study, the error variance of $\sigma_h^2 = (2.74 \text{ cm})^2$ that is derived in Appendix F is used for SWOT estimates of SSH and an error variance of $\sigma_{u,v}^2 = (0.354 \text{ m s}^{-1})^2$ is used for the variances $\sigma_u^2$ and $\sigma_v^2$ of the errors of WaCM estimates of the two orthogonal velocity components u and v. The equations for the variances of the errors of the derivative quantities (the two geostrophically computed velocity components and vorticity for SWOT and vorticity for WaCM) are applicable to any specified values of $\sigma_h^2$ or $\sigma_u^2$ and $\sigma_v^2$.

Appendix H examines the wavenumber spectral characteristics of the response functions of various finite-difference estimates of derivatives applied to the pre-processed SWOT and WaCM data and in combination with smoothing in ground-based post-processing.

For the amount of smoothing that will be required to achieve scientifically useful signal-to-noise ratios in geostrophically computed SWOT estimates of velocity and vorticity obtained from pre-processed SSH on a 1 km × 1 km grid, it is shown that standard centered differencing with a 3-point stencil width provides adequate estimates of derivatives.

For the 5 km × 5 km grid on which the errors of pre-processed WaCM estimates of velocity are uncorrelated, derivatives estimated from centered differences with a 9-point stencil width are significantly better than from standard centered differences with a 3-point stencil width. If the pre-processed WaCM data are oversampled on a 1 km × 1 km grid, however, it is shown that it is not necessary to estimate derivatives with a 9-point stencil for the amount of smoothing that will be required to achieve scientifically useful signal-to-noise ratios in WaCM estimates of vorticity. This is because a 3-point stencil retains considerably more small-scale signal variability in vorticity estimates when computed on an oversampled 1 km × 1 km grid rather than on a 5 km × 5 km grid. Although a 9-point stencil on a 5 km × 5 km grid

retains even more small-scale variability, that portion of the vorticity wavenumber spectrum is attenuated by the spatial smoothing that is required to achieve adequate signal-to-noise ratio. For the resolution capability of WaCM estimates of vorticity deduced from this study, any advantage of the 9-point stencil is therefore lost in WaCM estimates of vorticity computed on an oversampled 1 km × 1 km grid.

Throughout this study, we therefore assume that the pre-processed WaCM data will be available on a 1 km × 1 km grid and derivatives are estimated with a standard 3-point stencil. This oversampling leads to some complications in the derivations of analytical expressions for the standard deviations and wavenumber spectra of errors of the WaCM estimates of velocity and vorticity in Appendices G and I.

Appendix I derives analytical expressions for the 1-dimensional wavenumber spectra of the errors of all of the variables that are of interest in this study. The analysis begins with derivations in Appendix I.1 of the spectral characteristics of the errors of SSH and geostrophically computed velocity components and vorticity estimated from the pre-processed SWOT data. The effects on the spectral characteristics of the errors after isotropic 2-dimensional smoothing in ground-based post-processing of SWOT data are then derived in Appendix I.2. The analogous derivations for velocity components and vorticity estimated from pre-processed WaCM data and after smoothing in ground-based post-processing are presented in Appendices I.3 and I.4, respectively. The theoretical error spectra derived in Appendix I are shown in Figs. 13–15 to agree very well with spectra computed empirically from the simulated SWOT and WaCM error fields generated for the analysis in Sections 6 and 8–10.

As in Appendix G, the equations derived in Appendix I are all expressed in terms of the variances of the uncorrelated measurement errors and the filter transfer function for whatever smoother is applied in the ground-based post-processing. While this study assumes error variances of $\sigma_h^2 = (2.74 \text{ cm})^2$ and $\sigma_u^2 = \sigma_v^2 = (0.354 \text{ m s}^{-1})^2$ for SWOT and WaCM measurements of SSH and the two velocity components, respectively, the equations for the wavenumber spectra of the errors of the derivative quantities (the geostrophically computed velocity components and vorticity for SWOT and the vorticity for WaCM) are applicable to any specified values of $\sigma_h^2$ or $\sigma_u^2$ and $\sigma_v^2$.

## Appendix A. The Parzen smoother

Parzen smoothing is used extensively throughout this study. It is applied in the first step of our analysis of simulated satellite data to mimic the onboard pre-processing of the raw radar measurements to achieve footprint sizes of 1 km and 5 km for SWOT and WaCM, respectively (see Appendix B). It is subsequently applied to the maps of SSH for SWOT and surface velocity components for WaCM to mimic smoothing in ground-based post-processing to reduce the effects of measurement errors to acceptable levels for the variables of interest in this study (see Appendices C and D and Sections 6 and 8–10). To understand the filtering implied at both of these stages of smoothing, and to quantify what is meant by the “footprint size” of pre-processed data (Appendix B) and by “feature resolution” in smoothed fields constructed in ground-based post-processing (Appendix C), the filtering properties of the Parzen smoother are derived in this appendix.

The filtering properties of any smoother can be understood by considering the case of a 1-dimensional spatial series that is continuous. The results can be extended straightforwardly to the case of two dimensions. The results can also be extended to discrete sampling, but the equations become unnecessarily cumbersome for present purposes of characterizing the wavenumber content of the filtered output.

Any linear, space-invariant filter applied to a continuous spatial series $z(x)$ can be written as the convolution

$$ \tilde{z}(x) = w(x) * z(x) \equiv \int_{-\infty}^{\infty} w(x-s) z(s) ds, \qquad (A.1) $$

where the asterisk is short-hand notation for the convolution integral on the right side of the equation and $w(x)$ is the weighting function of the
<figure><img src="image_57.png" /><figcaption>Fig. A.1. The weighting functions (top panels) and log-linear plots of the squared filter transfer functions (bottom panels) for uniformly weighted running-average smoothers with spans of 1 km and 0.5 km (thin lines in panels a and b, respectively) and for a Parzen weighted running average smoother with a span of 1 km (thick line in panel b).</figcaption></figure>

smoother. The overbar distinguishes the smoothed value $\tilde{z}(x)$ from the unfiltered value $z(x)$. The Fourier transform of the weighting function $w(x)$, which is called the filter transfer function of the smoother, is given by

$$W(k) = \int_{-\infty}^{\infty} w(x) e^{-i2\pi kx} dx, \qquad (A.2)$$

where k is wavenumber in the x dimension.

The wavenumber contents of the unfiltered and smoothed data are defined by their Fourier transforms, which can be denoted as $Z(k)$ and $\bar{Z}(k)$, respectively. By the Convolution Theorem, the convolution integral (A.1) in the space domain can be expressed as multiplication in the wavenumber domain. The wavenumber content $\bar{Z}(k)$ of the smoothed output $\tilde{z}(x)$ is therefore related to the wavenumber content $Z(k)$ of the unfiltered data $z(x)$ by

$$\bar{Z}(k) = W(k)Z(k). \qquad (A.3)$$

The filter transfer function $W(k)$ thus defines the wavenumber content of the smoothed output.

In one dimension, any smoother can be characterized by its filter cutoff wavenumber $k_c$, which is defined here to be the wavenumber at which the squared value of the filter transfer function passes through a value of 0.5. The ideal filter has a filter transfer function that consists of values of 1 for wavenumbers $|k| \le k_c$ and 0 for higher wavenumbers. In practice, the filter transfer functions of real smoothers decrease gradually with increasing wavenumber across the low-wavenumber pass band and roll off steeply through the filter cutoff wavenumber $k_c$. Most real smoothers also have undesirable sidelobes at wavenumbers higher than $k_c$.

For the simple block-average smoother with uniform weighting over a span of $L_1$ centered on the estimation location, the weighting function is

$$w_1(x) = \frac{1}{L_1} \Pi\left(\frac{x}{L_1}\right). \qquad (A.4a)$$

The subscript 1 is used here to distinguish this uniform-weighted average from the Parzen weighted average considered below and $\Pi(x/L_1)$ is shorthand notation for the rectangle function defined by

$$\Pi\left(\frac{x}{L_1}\right) \equiv \begin{cases} 1 & \text{if } -L_1/2 \le x \le L_1/2 \\ 0 & \text{otherwise.} \end{cases} \qquad (A.4b)$$

The output (A.1) of the uniform-weighted average smoother with weighting function (A.4) is thus

$$\tilde{z}_1(x) = \frac{1}{L_1} \int_{-\infty}^{\infty} \Pi\left(\frac{x-s}{L_1}\right) z(s) ds = \frac{1}{L_1} \int_{x-L_1/2}^{x+L_1/2} z(s) ds. \qquad (A.5)$$

For discretely sampled values of $z(x)$, the integral on the right side of this equation becomes a discrete sum.

The filter transfer function (A.2) of the uniform-weighted average smoother is

$$W_1(k) = \int_{-\infty}^{\infty} w_1(x) e^{-i2\pi kx} dx = \frac{1}{L_1} \int_{-L_1/2}^{L_1/2} e^{-i2\pi kx} dx = \text{sinc}(kL_1), \qquad (A.6a)$$

where $\text{sinc}(kL_1)$ is standard shorthand notation for

$$\text{sinc}(kL_1) \equiv \frac{\sin(\pi k L_1)}{\pi k L_1}. \qquad (A.6b)$$

The half-power filter cutoff wavenumber for the uniform-weighted average smoother is the value $k_c$ of the wavenumber k at which the squared value of the filter transfer function (A.6) has a value of 0.5. The solution to this transcendental equation is

$$W_1^2(k_c) = \text{sinc}^2(k_c L_1) = 0.5 \qquad (A.7) \\ \Rightarrow \quad k_c = \frac{0.4430}{L_1}.$$

The weighting function (A.4) and the squared value of the filter transfer function (A.6) of the uniform-weighted average smoother are shown for the case of $L_1 = 1$ km in Fig. A.1a. A highly undesirable feature of the simple uniform-weighted average smoother is that its filter transfer function has large sidelobes outside of the main lobe that defines the range of wavenumbers $|k| \le k_c$ that are of interest. According to (A.3), these sidelobes contaminate the frequency content of the smoothed values $\tilde{z}(x)$ by admitting considerable variance at wavenumbers higher than the desired half-power filter cutoff wavenumber $k_c$.

The sidelobes of the uniform-weighted average smoother can be suppressed by replacing the uniform weighting with a tapered weighting. A simple approach to tapering is to apply a succession of uniform-weighted averages. For example, the weighting function for two successive applications of a uniform-weighted average with a span of $L_1$ can be shown to be equivalent to a single triangular weighted average with a span of $L_2 = 2L_1$. More generally, p successive applications of a
uniform-weighted average with a span of $L_1$ are equivalent to a single weighted average with a span of $L_p = pL_1$ and a weighting function that consists of a piecewise continuous polynomial of order $p-1$.

Mathematically, the weighting function $w_p(x)$ of p successive applications of a uniform-weighted average smoother with a span of $L_1$ consists of p self-convolutions of the rectangle weighting function (A.4) of the uniform-weighted average. The Fourier transform of this smoother (i.e., its filter transfer function) is therefore easily determined from the Convolution Theorem to be the multiplicative product of p of the filter transfer functions (A.6) of the uniform-weighted average,

$$W_p(k) = \text{sinc}^p(kL_1). \qquad (A.8)$$

The extremum of the jth sidelobe of this filter transfer function has a value of $\text{sinc}^p(k_j L_1)$, where $k_j = (j + 1/2)/L_1$, $j = 1, 2, 3, \dots$. Since $\text{sinc}^p(k_j L_1) = (-1)^j / [(j + 1/2)\pi]^p$, it has a magnitude less than 1 for all j and the sidelobe extrema decrease rapidly in magnitude, both with increasing j and with increasing p. For the case of the dominant j = 1 sidelobe, the sinc function has a value of $\text{sinc}(1.5) = \sin(1.5\pi)/(1.5\pi) = -0.212$. The dominant sidelobe of the filter transfer function for p successive applications of the uniform-weighted average thus has a magnitude of $0.212^p$ and a squared magnitude of $0.212^{2p}$.

There is no practical advantage to smoothing with more than p = 4 successive applications of a uniform-weighted average since the extremum of the dominant sidelobe of its squared filter transfer function is $4.08 \times 10^{-6}$, i.e., an attenuation factor of more than 50 dB. The quadruple pass of the uniform-weighted average is called the Parzen smoother. Mathematically, the weighting function of the Parzen smoother can be written as a quadruple convolution of the weighting function (A.4) of the uniform-weighted average,

$$w_4(x) = \frac{1}{L_1^4} \left[ \Pi\left(\frac{x}{L_1}\right) * \Pi\left(\frac{x}{L_1}\right) * \Pi\left(\frac{x}{L_1}\right) * \Pi\left(\frac{x}{L_1}\right) \right], \qquad (A.9)$$

where the subscript 4 signifies that the Parzen smoother is equivalent to 4 successive applications of a uniform-weighted average. The output of the Parzen smoother is the convolution

$$\bar{z}_4(x) = w_4(x) * z(x) = \int_{-\infty}^{\infty} w_4(x-s) z(s) ds. \qquad (A.10)$$

The filter transfer function (A.8) with p = 4 for the Parzen weighting function is

$$W_4(k) = \text{sinc}^4(kL_1) = \text{sinc}^4\left(\frac{kL_4}{4}\right), \qquad (A.11)$$

where $L_4 = 4L_1$ and $L_1$ is the span of each of the four successive applications of the uniform-weighted average. The motivation for introducing $L_4$ will become apparent below. The half-power filter cutoff wavenumber of the Parzen smoother is defined as in (A.7) to be the value $k_c$ of the wavenumber k at which the squared value of the filter transfer function (A.11) has a value of 0.5. The solution to this transcendental equation is

$$W_4^2(k_c) = \text{sinc}^8(k_c L_1) = \text{sinc}^8(k_c L_4/4) = 0.5 \\ \Rightarrow \quad k_c = \frac{0.2276}{L_1} = \frac{0.9104}{L_4}. \qquad (A.12a)$$

The value of $L_4$ for a Parzen filter that has a half-power filter cutoff wavelength of $\lambda_c = k_c^{-1}$ is thus

$$L_4 = 0.9104 \lambda_c. \qquad (A.12b)$$

While it would be very tedious to calculate the weights of the Parzen smoother from the quadruple convolution (A.9), it is straightforward with the help of integral tables to determine these weights from the inverse Fourier transform of the filter transfer function (A.11). The result is the piecewise cubic polynomial defined by

$$w_4(x) = \begin{cases} \frac{8}{3L_4} \left( 1 - \frac{24x^2}{L_4^2} + \frac{48|x|^3}{L_4^3} \right) & \text{if } 0 \le |x| \le L_4/4 \\ \frac{8}{3L_4} \left( 2 - \frac{12|x|}{L_4} + \frac{24x^2}{L_4^2} - \frac{16|x|^3}{L_4^3} \right) & \text{if } L_4/4 \le |x| \le L_4/2 \\ 0 & \text{if } |x| > L_4/2. \end{cases}$$

(A.13)

The Parzen smoother thus has a full span of $L_4$ that is defined above to be four times larger than the span $L_1$ of each of the four successive applications of the uniform-weighted smoother upon which it is based.

In principle, the Parzen smoother (A.10) can be implemented either as a succession of four applications of the uniform-weighted average (A.5) with span $L_1$ or as a single pass of (A.10) with the piecewise cubic weighting (A.13) and a span of $L_4 = 4L_1$. In practice, however, application to discretely sampled data results in small differences in the filtered output by the two procedures. When applied to data with a uniformly spaced sample interval of $\Delta x$, for example, an arbitrarily specified value of $L_4$ can result in a value of $L_1 = L_4/4$ that is a non-integer multiple of $\Delta x$. Moreover, it is desirable for the full span of the smoothing to be an odd multiple of $\Delta x$ so that the weighting function is symmetric. This assures that there is no phase shift at any of the wavenumbers in the filtered output. It is clearly not possible for both $L_1$ and $L_4$ to be odd integer multiples of $\Delta x$. The preferred implementation of the Parzen smoother with uniformly spaced data is therefore as a single pass of (A.10) with the piecewise cubic weighting (A.13).

The weighting function (A.13) and the square of the filter transfer function (A.11) of the Parzen smoother are shown for the case of $L_4 = 1$ km by the thick lines in Fig. A.1b. This is equivalent to a span of $L_1 = L_4/4 = 0.25$ km for each of the four successive applications of the uniform-weighted average (A.5).

For a given value of $L_1$, it can be noted from (A.7) and (A.12a) that the filter cutoff wavenumber $k_c$ is a factor of $0.228/0.443 = 0.515 \approx 0.5$ smaller for the Parzen smoother than for the single-pass uniform-weighted average smoother. To achieve the same half-power filter cutoff wavenumber $k_c$ with both the uniform-weighted average smoother and the Parzen smoother, the spans of the two smoothers can be adjusted accordingly. A uniformly weighted average with a span of $L_1 = 0.5$ km, for example, has a half-power filter cutoff wavenumber of $k_c = 0.443/0.5 = 0.886$ cpkm (cycles per km). As shown by the thin line in the bottom panel of Fig. A.1b, this is very close to the half-power filter cutoff wavenumber of $k_c = 0.910$ cpkm for the Parzen smoother with a span of $L_4 = 1$ km shown by the thick line.

The much smaller sidelobes of the filter transfer function of the Parzen smoother compared with those of the uniform-weighted average smoother (see Fig. A.1b) are highly desirable attributes of the Parzen smoother. This improved sidelobe suppression comes at the price of somewhat more gradual rolloff through the filter cutoff wavenumber $k_c$ of the filter transfer function of the Parzen smoother (see the bottom panel of Fig. A.1b).

## Appendix B. Footprint size and pre-processing of SWOT and WaCM data

### B.1. SWOT pre-processing

The raw radar measurements by the Ka-band Radar Interferometer (KaRIn) instrument on the SWOT satellite in its low-resolution mode over the ocean have a footprint size of about 100 m. The plan as presently summarized in the SWOT onboard processing document (Peral, 2016) is to smooth the raw measurements of SSH onboard the satellite to achieve the science requirement of a footprint size (diameter) of 1 km for ocean observations. The SWOT Algorithm Development Team has recently recommended changing the onboard processing to a smaller footprint size of 0.5 km posted on a 0.25 km × 0.25 km grid. The motivation for this increased resolution is to allow improved detection and removal of noisy outliers in the SWOT estimates of SSH.

The rescoping of the SWOT science requirements to a smaller footprint size of 0.5 km with a concomitant factor-of-two increase of the standard deviation of the uncorrelated measurement errors in the onboard estimates of SSH would change some of the details of the simulations of SWOT data in this study. As shown from the figures
in Sections 6, 8 and 9, however, even the science requirement of a footprint size of 1 km that was used for our simulations far exceeds the signal resolution capability for geostrophically computed SWOT estimates of the variables of interest in this study (surface velocity and vorticity). The difference between 1 km and 0.5 km footprint sizes in the onboard pre-processing of SWOT data is therefore irrelevant to the analysis presented in Sections 6, 8 and 9. The SWOT Algorithm Development Team acknowledges that the 0.5 km footprint size is unnecessarily fine for most applications of SWOT data. The resulting SSH measurements are likely to be smoothed to a wavelength resolution of 2 km × 2 km in ground-based post-processing for the SWOT dataset that is provided to the general science community. It is shown below that this is equivalent to the wavelength resolution of measurements with a footprint size of 1 km.

The onboard smoothing of the raw SWOT data will be achieved using a Parzen smoother (see Appendix A) in the cross-track direction and a Blackman–Harris smoother in the along-track direction (Peral, 2016). The parameters of each of these smoothers were carefully chosen to have spatial autocorrelation values of 0.5 at the same lag, which is essentially equivalent to having the same half-power filter cutoff wavenumber for both smoothers.

The rationale for applying a different smoother in each dimension in the onboard pre-processing of SWOT data is that the filter transfer function of the Blackman–Harris smoother has slightly better sidelobe suppression than the Parzen smoother [see Fig. 37 in Peral (2016); see also Harris (1978)]. But for technical reasons related to the hardware implementation of the onboard pre-processing, the Blackman–Harris smoother cannot be applied in the cross-track direction. The better sidelobe suppression of the Blackman–Harris smoother is not enough to make any significant, or even detectable, difference in the smoothed SSH values. For all intents and purposes, the 2-dimensional smoothing of the raw SWOT data could thus be achieved equally well by smoothing with a Parzen smoother in both dimensions. The filtering properties of the onboard pre-processed estimates of SSH are therefore discussed here in the context of 2-dimensional smoothing with the Parzen smoother that was examined in detail in Appendix A.

The resolution of smoothed estimates of SSH is defined by Peral (2016) as summarized below based on the lagged autocorrelation associated with the smoother applied to the raw data. An expression for this spatial autocorrelation can be derived by noting that the square of the filter transfer function of the smoother is proportional to the wavenumber power spectral density of the low-pass filtered output of the smoother applied to data consisting of uncorrelated (white) noise with unit standard deviation. The autocovariance of this filtered output can be obtained as the inverse Fourier transform of its power spectral density. For white noise input data with unit standard deviation, this autocovariance is equivalent to the autocorrelation.

For a Parzen smoother with a span of $L_4$ in the x dimension, the autocorrelation is thus the inverse Fourier transform of the square of the filter transfer function (A.11),

$$ \rho(x) = \int_{-\infty}^{\infty} \mathrm{sinc}^8\left(\frac{kL_4}{4}\right) e^{i2\pi kx} \, dk = 2 \int_{0}^{\infty} \mathrm{sinc}^8\left(\frac{kL_4}{4}\right) \cos(2\pi kx) \, dk. \quad (B.1) $$

This can be evaluated with the help of integral tables. The result is a piecewise continuous 7th-order polynomial that is symmetric about lag $x=0$. This is shown for positive lags $x$ in the bottom panel of Fig. B.1a for a span of $L_4 = 2$ km. The associated weighting function and squared filter transfer function are shown, respectively, in the top and middle panels of Fig. B.1a.

The minimum scale of features that can be resolved in the smoothed data is defined by Peral (2016) to be the lag at which the autocorrelation decays to a value of 0.5. Although somewhat subjective, this is a reasonable definition. For the Parzen smoother with a span of $L_4 = 2$ km shown in Fig. B.1a, the autocorrelation (B.1) decays to a value of 0.5 at a lag of $x = 0.495$ km (see the bottom panel of Fig. B.1a).

Since the autocorrelation is symmetric about zero lag, this can be interpreted as the radius of features that can be resolved. The feature diameter resolution after filtering with a Parzen smoother with a span of $L_4 = 2$ km is therefore approximately 1 km. This can be considered the footprint diameter of SWOT measurements of SSH smoothed with a half-power filter cutoff wavelength (A.12) of $\lambda_c = k_c^{-1} = L_4/0.910 \approx 2$ km. The autocorrelation for the Parzen smoother with $L_4 = 2$ km decays to a very small value of 0.050 at a radial lag of 1 km (see the bottom panel of Fig. B.1a). The resulting smoothed estimates of SSH are therefore essentially uncorrelated when posted on a 1 km × 1 km grid.

The present official requirement for the pre-processing of SWOT data in the SWOT onboard processing document (Peral, 2016) is for SSH estimates on a 1 km × 1 km grid to be statistically uncorrelated. From the preceding discussion, this can be achieved using a 2-dimensional Parzen smoother with a span of $L_4 = 2$ km in each dimension. The half-power filter cutoff wavenumber (A.12a) for this span is $k_c = 0.455 \approx 0.5$ cpkm (see the middle panel of Fig. B.1a). A wavenumber of 0.5 cpkm is also the Nyquist wavenumber for a sample interval of $\Delta x = 1$ km. The resolution and sampling requirements for the pre-processed SWOT estimates of SSH as stated in the onboard processing document (Peral, 2016) can thus be achieved using a Parzen smoother with a span of $L_4 = 2$ km in each dimension, and posting the smoothed SSH estimates on a 1 km × 1 km grid. The estimates of SSH obtained by smoothing with a half-power filter cutoff wavelength of 2 km could be posted on a finer grid, but the resulting gridded values would no longer be statistically uncorrelated.

As discussed at the beginning of this appendix, the onboard processing for SWOT estimates of SSH is likely to change to a smaller footprint size of 0.5 km. Based on the preceding discussion, this can be achieved by smoothing with a half-power filter cutoff wavelength of 1 km. The analysis throughout this study is based on the science requirements for a footprint size of 1 km, which is achieved by smoothing with a half-power filter cutoff wavelength of 2 km as summarized above. None of the conclusions of this study would change if the analysis were based on the smaller footprint size of 0.5 km since the results of the additional smoothing in ground-based post-processing that will be needed to achieve an adequate signal-to-noise ratio would be essentially the same for either footprint size.

## B.2. WaCM pre-processing

The WaCM mission is still in the early stages of planning. To reduce the measurement errors, the present plan is to smooth the pooled raw measurements of radial velocity from multiple antenna look angles into 5 km × 5 km areas from which two orthogonal components of velocity (e.g., along-track and cross-track components) are estimated. The feature resolution of these velocity estimates will thus effectively have a footprint diameter of 5 km. In analogy with the discussion in Appendix B.1 for pre-processing of SWOT data, this can be achieved using a 2-dimensional Parzen smoother with a half-power filter cutoff wavelength of 10 km. This corresponds to a span of approximately $L_4 = 10$ km (see Fig. B.1b). The associated autocorrelation decays to a value of 0.5 at a lag of $x = 2.476$ km, thus corresponding to a feature diameter resolution of approximately 5 km. The autocorrelation for the Parzen smoother with $L_4 = 10$ km decays to a very small value of 0.050 at a radial lag of 5 km (see the bottom panel of Fig. B.1b). The resulting smoothed estimates of surface velocity are therefore essentially uncorrelated when posted on a 5 km × 5 km grid.

The variables that are of primary interest in this study are the surface velocity field itself and the surface vorticity, which is computed from derivatives of the velocity components. For centered-difference estimates of these derivatives, it is shown later in Appendix H that it is advantageous to post the pre-processed WaCM estimates of surface velocity on a grid that is finer than 5 km × 5 km. For the analysis in this study, the smoothed velocity estimates were posted on a 1 km × 1 km. The measurement errors are not uncorrelated on this finer grid.
<figure><img src="image_58.png" /><figcaption>Fig. B.1. The weighting functions (top panels), log-linear plots of the squared filter transfer functions (middle panels) and lagged autocorrelation functions (bottom panels) for Parzen smoothers with spans of 2 km (Column a) and 10 km (Column b) that are used in this study to simulate pre-filtering of SWOT data and WaCM data, respectively. The half-power filter cutoff wavelengths of these Parzen smoothers (the reciprocals of the filter cutoff wavenumbers shown by the dotted lines in the middle panels) are approximately 2 km and 10 km. The dotted lines in the bottom panels indicate the lags at which the autocorrelation functions decay to values of 0.5 and 0.05.</figcaption></figure>

## Appendix C. Feature resolution

Maps of the variables of interest in this study (surface velocity and vorticity) constructed from SWOT and WaCM data pre-processed as summarized in Appendix B are too noisy to be useful for most applications (see the unsmoothed fields in the bottom left panels of Figs. 17a–17c and Figs. 19a–19b). For these variables, it will be necessary to smooth the pre-processed data in ground-based post-processing to mitigate the effects of the uncorrelated measurement errors. The variance of the errors can be reduced to any desired level with sufficient 2-dimensional smoothing. The error reduction factor is derived later in Appendix D. The purpose of this appendix is to quantify the filter cutoff wavelength and the implied feature resolution scale for a given amount of smoothing.

The procedure used in this study to mitigate the effects of measurement errors and increase the signal-to-noise ratio is to apply 2-dimensional smoothing to the noisy estimates of SSH from SWOT and surface velocity components from WaCM. This smoothing can be applied anisotropically, i.e., with different filter cutoff wavenumbers in each of two orthogonal dimensions. For the purposes of the analysis in Sections 6 and 8–10, however, the 2-dimensional smoothing was applied isotropically. The 2-dimensional weighting function for isotropic smoothing can be expressed as

$$w^{2d}(x, y) = w^{2d}(r), \qquad (C.1)$$

where $r = (x^2 + y^2)^{1/2}$ is the radial distance from the estimation location to each data point within the 2-dimensional span of the smoother. It is shown below that 2-dimensional smoothing with the Parzen smoother used in this study is essentially equivalent to separate 1-dimensional smoothing in each of two orthogonal dimensions. The effects of the 2-dimensional smoothing can thus be understood from consideration of the filtering properties of successive 1-dimensional smoothing in each dimension. This simplifies the derivation of the wavenumber spectral characteristics of 2-dimensionally smoothed fields presented later in Appendix I.

There are many possible choices of smoothers to reduce errors. While the details of the filter transfer functions differ for the various smoothers, any smoother applied in, say, the x dimension can be characterized by the half-power filter cutoff wavenumber $k_c$, or equivalently the half-power filter cutoff wavelength $\lambda_c = k_c^{-1}$, that is associated with the particular choice of the parameters of the smoother. For example, the values of $k_c$ for the uniform-weighted average and the Parzen smoother considered in Appendix A are related to the parameters $L_1$ and $L_4$ that define the spans of each of these smoothers by (A.7) and (A.12a), respectively. Other smoothers have analogous relations between $k_c$ and the parameters of the smoother; see, for example, the Gaussian and loess smoothers summarized later in this section. The primary distinctions between different smoothers are the steepness of
<figure><img src="image_59.png" /><figcaption>Fig. C.1. Log-log plots of the squared filter transfer functions of Parzen smoothers with half-power filter cutoff wavelengths of 20, 50 and 80 km (thin solid, thick solid and dashed lines, respectively). The horizontal dotted line corresponds to an ordinate value of 0.5. The half-power filter cutoff wavelengths for the Parzen smoothers are shown by the three vertical dotted lines, which are the wavenumbers at which each squared filter transfer function intersects the horizontal dotted line.</figcaption></figure>

<figure><img src="image_60.png" /><figcaption>Fig. C.2. Log-log plots of the squared filter transfer functions with a half-power filter cutoff wavelength of 50 km for the Parzen smoother (thick line, which is the same as the thick line in Fig. C.1) and a Gaussian-weighted smoother (thin line). The horizontal dotted line corresponds to an ordinate value of 0.5 and the vertical dotted line is the wavenumber at which the two squared filter transfer functions intersect the horizontal dotted line.</figcaption></figure>

the rolloff of the squared filter transfer function through $k_c$ and the amplitudes of the sidelobes at wavenumbers higher than $k_c$.

The analyses of resolution capability in Sections 6 and 8–10 determine the signal-to-noise ratio as a function of isotropic filter cutoff wavelength $\lambda_c$ for each of the variables of interest (the two components of velocity and vorticity) computed from the simulated satellite data (SSH for SWOT and the two velocity components for WaCM) smoothed with 19 different values of $\lambda_c$ ranging from 10 km to 200 km. These smoothed fields were generated for instantaneous maps, 4-day averages and 14-day averages of SSH for SWOT and the two velocity components for WaCM for fields constructed from error-free model output, measurement errors alone, sampling errors alone and combined measurement and sampling errors. Together with the pre-processed SWOT and WaCM data smoothed with $\lambda_c = 2$ km and 10 km, respectively, 20 different smoothed maps were computed for each measured variable, each choice of time averaging, each combination of measurement and sampling errors, and two choices of swath width for WaCM (1200 km and 1800 km). A total of 1200 smoothed maps of the simulated satellite data were thus required to generate the signal-to-noise graphs in Figs. 18, 30, 35, 38–40 and 43.

Because of the computational effort required for these calculations, the standard smoother used throughout this study was the

<figure><img src="image_61.png" /><figcaption>Fig. C.3. Log-log plots of the squared filter transfer functions with a half-power filter cutoff wavelength of 50 km for the Parzen smoother (thick line, which is the same as the thick line in Fig. C.1), the loess smoother (thick dashed line) and the uniformly weighted running average smoother (thin dashed line). The horizontal and vertical dotted lines are the same as in Fig. C.2.</figcaption></figure>

Parzen smoother that is discussed in detail in Appendix A. The Parzen smoother is computationally efficient and very easy to implement as a simple weighted average with weights defined by (A.13). For a given choice of half-power filter cutoff wavenumber $k_c$ and associated wavelength $\lambda_c = k_c^{-1}$, the span of a 1-dimensional Parzen smoother inferred from (A.12b) is

$$ \text{Parzen Smoother : } L_4 = 0.910 \lambda_c \approx \lambda_c. \qquad (C.2) $$

The squared filter transfer function for the Parzen smoother is shown in Fig. C.1 in log-log format (the standard format used for the wavenumber spectra shown throughout this paper) for filter cutoff wavelengths of $\lambda_c = 20, 50$ and 80 km, corresponding to weighted averages with spans of $L_4 = 18.2, 45.5$ and $72.8$ km.

It is useful to compare the Parzen smoother with the Gaussian-weighted smoother for which the weighting function in one dimension is

$$ w_G(x) = \frac{1}{\sqrt{\pi} L_G} e^{-x^2/L_G^2}, \qquad (C.3) $$

where $L_G$ defines the e-folding scale of the Gaussian weighting. The normalization factor $\sqrt{\pi} L_G$ constrains the weighting function (C.3) to have unit area, as required for unbiased smoothing. As discussed in Appendix A, the filter transfer function of any smoother is defined to be the Fourier transform (A.2) of the weighting function of the smoother. For the Gaussian smoother (C.3), this is

$$ W_G(k) = \int_{-\infty}^{\infty} w(x) e^{-i2\pi kx} dx = e^{-(\pi L_G k)^2}. \qquad (C.4) $$

As defined in Appendix A, the half-power filter cutoff wavenumber of the Gaussian smoother is the wavenumber $k_c$ at which the squared value of the filter transfer function (C.4) has a value of 0.5,

$$ W_G^2(k_c) = e^{-2(\pi L_G k_c)^2} = 0.5. $$

The parameter $L_G$ of the Gaussian smoother (C.3) is thus related to the desired half-power filter cutoff wavenumber $k_c$ and associated wavelength $\lambda_c = k_c^{-1}$ by

$$ L_G = \left( -\frac{\ln(0.5)}{2\pi^2} \right)^{1/2} \frac{1}{k_c} = 0.187 \lambda_c. \qquad (C.5) $$

The squared filter transfer function of the Gaussian smoother is shown in log-log format by the thin line in Fig. C.2 for the case of a half-power filter cutoff wavelength of $\lambda_c = 50$ km. For comparison, the thick line is the squared filter transfer function of the Parzen smoother from Fig. C.1 for the same half-power filter cutoff wavelength
of $\lambda_c = 50$ km. It is evident that the filtering properties of the Parzen smoother are indistinguishable from those of the Gaussian smoother for wavenumbers smaller than about $2k_c$. The two squared filter transfer functions differ somewhat at higher wavenumbers, but the variability at these wavenumbers is attenuated by more than 15 dB relative to the variability at the lower wavenumbers within the pass band of the smoother.

A useful feature of isotropic Gaussian smoothing in two dimensions is that it can be applied either as a single 2-dimensional weighted average or as separate 1-dimensional Gaussian smoothing with the weighting function (C.3) in each of two orthogonal dimensions. This can be seen by noting that the 2-dimensional weighting function (C.1) for the isotropic Gaussian smoother is

$$w_G^{2d}(x, y) = w_G^{2d}(r) = \frac{1}{\pi L_G^2} e^{-r^2/L_G^2} = \frac{1}{\pi L_G^2} e^{-(x^2+y^2)/L_G^2}. \quad (C.6a)$$

The normalization $\pi L_G^2$ constrains the 2-dimensional weighting function $w_G^{2d}(r)$ to have unit volume, as required for unbiased smoothing. The right side of (C.6a) can be rewritten as

$$w_G(x, y) = w_G(x) w_G(y), \quad (C.6b)$$

where $w_G(x)$ is the 1-dimensional Gaussian weighting function (C.3) in the $x$ dimension and $w_G(y)$ is an analogous expression in the $y$ dimension. At least in principle, the isotropic 2-dimensional Gaussian smoother can thus be applied as separate 1-dimensional Gaussian smoothers in each orthogonal dimension. It is of course not possible to extend the Gaussian weighting to $x, y \to \pm\infty$. The separability (C.6b) is therefore not precisely valid in practice.

Because of the very close similarity between the squared filter transfer functions of the Parzen and Gaussian smoothers (see Fig. C.2), and therefore between the weighting functions of the two smoothers that can be obtained from the inverse Fourier transforms of the filter transfer functions, smoothing isotropically with a 2-dimensional Parzen smoother shares the separability property (C.6b) for all intents and purposes. In other words, isotropic 2-dimensional Parzen smoothing can be implemented as separate 1-dimensional Parzen smoothing in each of two orthogonal dimensions. This property is used in Appendices D, E and I (see also Section 4) to derive theoretical expressions for the effects of smoothing on the magnitudes and wavenumber spectral characteristics of residual uncorrelated measurement errors after smoothing SWOT and WaCM data in simulated ground-based post-processing.

The very small sidelobes of the filter transfer function of the Parzen smoother, and the lack of sidelobes in the filter transfer function of the Gaussian smoother, are desirable characteristics of any smoother. An undesirable feature of these smoothers is that the rolloffs of their squared filter transfer functions through $k_c$ are gradual compared with some other smoothers. The assessment of resolution capability of SWOT and WaCM data deduced from smoothed fields in Sections 6 and 8–10 depends to some degree on the details of the filtering properties of the smoother that is used to suppress the errors. This is apparent in Sections 9 and 10.2 from the resolution capabilities inferred from the quadratic loess smoother compared with the Parzen smoother. The quadratic loess smoother introduced by Cleveland and Devlin (1988) has better filtering properties than the Gaussian and Parzen smoothers (Schlax and Chelton, 1992).

Mathematically, the quadratic loess smoother is a locally weighted least squares fit of the data to a quadratic function over a span of $L_q$. The standard weighting for the quadratic loess smoother is the tricubic function used by Cleveland and Devlin (1988). For a given choice of half-power filter cutoff wavelength $\lambda_c$, it can be shown that the span for this quadratic loess smoother is

$$\text{Quadratic Loess Smoother: } L_q \approx 1.67 \lambda_c. \quad (C.7)$$

This is 83.5% larger than the span (C.2) of the Parzen smoother.

The squared filter transfer function for a 1-dimensional quadratic loess smoother is shown in log-log format by the thick dashed line

in Fig. C.3 for the case of a half-power filter cutoff wavelength of $\lambda_c = 50$ km (i.e., a span of $L_q = 83.5$ km). For comparison, the thick solid line is the squared filter transfer function of the Parzen smoother from Figs. C.1 and C.2 for the same half-power filter cutoff wavelength of $\lambda_c = 50$ km. While the sidelobes of the quadratic loess smoother with tricubic weighting function are larger than the sidelobes of the Parzen smoother, they occur at wavenumbers closer to the filter cutoff wavenumber $k_c$ and the rolloff of the filter transfer function has much better characteristics for the loess smoother. Specifically, the squared filter transfer function of the loess smoother is somewhat flatter in the pass band and rolls off much more steeply through $k_c$. At a wavelength of 30 km, for example, i.e., a wavenumber of $k = 0.0333$ cpkm = 1.67 $k_c$, the squared filter transfer function for the loess smoother is 27.3 times smaller than that of the Parzen smoother (i.e., an attenuation factor of more than 14 dB). Even with its larger-amplitude sidelobes, the magnitude of the squared filter transfer function of the loess smoother is smaller than that of the Parzen smoother at all wavenumbers less than about 0.07 cpkm (a wavelength of about 14 km), by which point both squared filter transfer functions are down by about 50 dB from the values within the pass band of the smoother so that sidelobe contamination is no longer an issue. The inferiority of simple block averages (i.e., uniformly weighted running averages) is readily apparent from the numerous large sidelobes of its squared filter transfer function shown by the thin dashed line in Fig. C.3.

The quadratic loess smoother is thus seen to be more effective than the Parzen smoother for attenuating high-wavenumber variability. In addition to the large span (C.7) required to achieve a filter cutoff wavelength $\lambda_c$, a disadvantage of the loess smoother is that the least squares fitting procedure is computationally intensive on the 1 km × 1 km grid used in this study for simulated SWOT and WaCM data, especially for large spans $L_q$ in two dimensions. It is apparent from the figures in Sections 9 and 10.2, however, that the better noise suppression properties of the loess smoother improve the assessment of resolution capability (see Table 3).

The precise estimates of the resolution capabilities of SWOT and WaCM data thus depend somewhat on the specific details of the smoother used to suppress the effects of measurement errors. However, the relative resolution capabilities of SWOT compared with WaCM do not depend on the choice of smoother used in the analysis.

The formalism developed in Appendix B to define the footprint sizes of the pre-processed SWOT and WaCM data can be used to characterize the feature resolution in smoothed fields constructed in ground-based post-processing with any desired half-power filter cutoff wavenumbers $k_c$ and $l_c$ in the $x$ and $y$ dimensions. For the case of the Parzen smoother, it was shown in Fig. B.1 that smoothing in the $x$ dimension with spans of $L_4 = 2$ km for SWOT and $L_4 = 10$ km for WaCM results in an autocorrelations of 0.5 at lags of about 0.5 km and 2.5 km, respectively, and therefore feature diameter resolution scales of about 1 km for SWOT and 5 km for WaCM according to the criterion adopted in Appendix B.

More generally, the feature diameter resolution scale for a Parzen smoother applied in the $x$ dimension with an arbitrary span of $L_4$ in ground-based post-processing is

$$\text{Feature Diameter Resolution} \approx \frac{L_4}{2}. \quad (C.8a)$$

Since the half-power filter cutoff wavenumber (A.12a) for a Parzen smoother with a span of $L_4$ is $k_c \approx L_4^{-1}$, the feature diameter resolution scale (C.8a) can be expressed equivalently as

$$\text{Feature Diameter Resolution} \approx \frac{1}{2k_c}. \quad (C.8b)$$

In terms of half-power filter cutoff wavelength $\lambda_c = k_c^{-1}$, this is

$$\text{Feature Diameter Resolution} \approx \frac{\lambda_c}{2}. \quad (C.8c)$$

Regardless of the specific formulation of the smoothing procedure, the relationships (C.8b) and (C.8c) between feature diameter resolution
and filter cutoff wavenumber $k_c$ or wavelength $\lambda_c$ developed above for the Parzen smoother are approximately applicable to any smoother.

The preceding analysis clarifies the relation between the autocorrelation-based definition (C.8) of feature diameter resolution and the half-power filter cutoff wavelength $\lambda_c = k_c^{-1}$ that is often used by oceanographers to characterize the resolution of filtered fields. From (C.8c), the feature diameter resolution of fields smoothed with a half-power filter cutoff wavenumber $k_c$ is about $\lambda_c/2$. This definition of resolution, which is based on the lag at which the autocorrelation of the smoothed fields decreases to a value of 0.5 as suggested by Peral (2016), is coarser by 25% than the feature diameter resolution of $\lambda_c/2.5$ as defined somewhat subjectively in Appendix A of Chelton et al. (2011) using a different criterion. For consistency with the SWOT documentation, we adopt the Peral (2016) definition in this study.

## Appendix D. Error reduction by smoothing

Smoothing of the pre-processed SWOT estimates of SSH and WaCM estimates of surface velocity in ground-based post-processing as summarized in Appendix C clearly reduces the variance of the uncorrelated errors. The objective of this appendix is to quantify the reduction that can be achieved with 2-dimensional smoothing for given choices of the half-power filter cutoff wavenumbers $k_c$ and $l_c$ in the x and y dimensions, respectively. The analysis that follows is based on the Parzen smoother, but the results are at least qualitatively applicable to any smoother with parameters calibrated to the same filter cutoff wavenumber in each dimension.

The amount by which filtering 2-dimensionally with the Parzen smoother reduces the variance of uncorrelated errors compared with block averaging, i.e., compared with the uniform-weighted average (A.4), can be derived from consideration of smoothing in one dimension and then extending the results to two dimensions. Consider a 1-dimensional discretely sampled spatial series $\epsilon(x_m)$ of errors at M locations $x_m = m\Delta x$, $m = 0, 1, ..., M - 1$. It will be assumed that these errors are uncorrelated at the sample spacing of $\Delta x$. It was shown in Appendix B that SWOT measurements of SSH smoothed in pre-processing with a half-power filter cutoff wavelength of 2 km are uncorrelated with a spacing of $\Delta x = 1$ km. For WaCM, smoothing of the surface velocity measurements in pre-processing with a half-power filter cutoff wavelength of 10 km are uncorrelated with a spacing of $\Delta x = 5$ km.

The error reduction by smoothing in ground-based post-processing of the pre-processed SWOT and WaCM data can be quantified from the wavenumber spectral characteristics of the filtered errors. Parseval's Theorem for the sample variance $\sigma_e^2$ of the uncorrelated errors in one dimension without smoothing can be expressed as

$$ \sigma_e^2 = \sum_{i=1}^{M/2} S_e^{1d}(k_i) \Delta k, \qquad (D.1) $$

where $\Delta k = (M \Delta x)^{-1}$ is the Fourier wavenumber interval between independent spectral estimates for a record length of $M \Delta x$ and $S_e^{1d}(k_i)$ is the 1-sided, 1-dimensional power spectral density of the errors at wavenumber $k_i = i \Delta k$. The zero wavenumber corresponding to the index $i = 0$ is excluded from the sum (D.1) because the power spectral density at zero wavenumber is equivalent to the square of the sample mean value, which does not contribute to the sample variance. The wavenumber $k_{M/2} = (M/2)\Delta k$ at the upper range of the summation in (D.1) corresponds to the Nyquist wavenumber $k_{N'} = (2\Delta x)^{-1}$ that is the highest resolvable wavenumber for the discrete sample interval $\Delta x$. The 1-sided, 1-dimensional power spectral density $S_e^{1d}(k_i)$ on the right side of (D.1) doubles the power at all wavenumbers except the zero and Nyquist wavenumbers.

Since the errors $\epsilon(x_m)$ are uncorrelated, the wavenumber power spectral density $S_e^{1d}(k_i)$ is white, i.e., constant at all wavenumbers.<fnref n="5" />

The spectral values at wavenumbers $k_i = 0$ and the Nyquist wavenumber $k_i = M/2$ are not doubled in the 1-sided, 1-dimensional spectrum. Since

$$ S_e^{1d}(k_i) = \frac{\sigma_e^2}{(M/2)\Delta k} = \frac{\sigma_{N'}^2}{k_{N'}}, \quad i = 1, \dots, M/2. \qquad (D.2) $$

The wavenumber power spectral density after smoothing of the uncorrelated errors $\epsilon(x_m)$ in ground-based post-processing using a linear, space-invariant filter with a filter transfer function $W_{k_c}(k)$ that has a half-power filter cutoff wavenumber $k_c$ can be expressed in terms of the wavenumber spectrum (D.2) of the unfiltered errors by

$$ \bar{S}_e^{1d}(k_i) = |W_{k_c}(k_i)|^2 S_e^{1d}(k_i). \qquad (D.3a) $$

The overbar distinguishes the spectrum $\bar{S}_e^{1d}(k_i)$ of the smoothed errors from the spectrum $S_e^{1d}(k_i)$ of errors in the unfiltered pre-processed data. The filter transfer functions of the uniform-weighted average and the Parzen smoother considered below are given by (A.6) and (A.11), respectively. Since these filter transfer functions are both real (as is the case for any symmetric, linear filter), the absolute value can be omitted from (D.3a). After substituting (D.2) for $S_e^{1d}(k_i)$, the spectrum (D.3a) of the smoothed white noise can be written as

$$ \bar{S}_e^{1d}(k_i) = \frac{\sigma_e^2}{k_{N'}} W_{k_c}^2(k_i). \qquad (D.3b) $$

Parseval's Theorem for the sample variance of the residual uncorrelated errors after smoothing can be expressed analogous to (D.1) in terms of the 1-sided, 1-dimensional power spectral density (D.3) of the smoothed errors as

$$ \bar{\sigma}_e^2 = \sum_{i=1}^{M/2} \bar{S}_e^{1d}(k_i) \Delta k. \qquad (D.4a) $$

The overbar distinguishes the residual variance $\bar{\sigma}_e^2$ of the smoothed errors from the variance $\sigma_e^2$ of the errors of the unsmoothed pre-processed data that is given by (D.1). Substituting (D.3b) into the right side of (D.4a) expresses the variance of the smoothed errors in terms of the variance of the unsmoothed errors as

$$ \bar{\sigma}_e^2 = \frac{\sigma_e^2}{k_{N'}} \sum_{i=1}^{M/2} W_{k_c}^2(k_i) \Delta k. \qquad (D.4b) $$

Note again that the wavenumber $k_{M/2}$ at the upper range of the summation is the Nyquist wavenumber $k_{N'} = (2\Delta x)^{-1}$, a point that is important below.

Eq. (D.4b) can be interpreted as a discretized statement that the sample variance of smoothed white-noise errors is equal to the integrated area under the squared filter transfer function. This is seen by considering the limit as the record length $M \Delta x$ approaches infinity for fixed sample interval $\Delta x$. The wavenumber interval $\Delta k = (M \Delta x)^{-1}$ then becomes an infinitesimally small value $dk$ and the discrete wavenumbers $k_i = i \Delta k$ become continuous. The subscript $i$ can then be dropped and the discrete summation becomes an integral so that (D.4b) can be written as

$$ \bar{\sigma}_e^2 = \alpha \sigma_e^2, \qquad (D.5a) $$

where

$$ \alpha = \frac{1}{k_{N'}} \int_{0}^{k_{N'}} W_{k_c}^2(k) dk = 2\Delta x \int_{0}^{k_{N'}} W_{k_c}^2(k) dk. \qquad (D.5b) $$

The Nyquist wavenumber that defines the upper bound of the integral is imposed by the discrete sample interval $\Delta x$, regardless of the record length $M \Delta x$. [As noted above, the zero wavenumber at the lower bound of the integral in (D.5b) must be excluded from the discretized

the measurement errors are assumed to have a mean of zero, this detail is unimportant for $k_i = 0$. The non-doubling at $k_i = M/2$ will be ignored to avoid unnecessary lack of clarity in the analysis that follows.

<footnote n="5">⁵ The spectral values at wavenumbers $k_i = 0$ and the Nyquist wavenumber $k_i = M/2$ are not doubled in the 1-sided, 1-dimensional spectrum. Since</footnote>
form (D.4b) of this integral since it does not contribute to the sample variance.]

It is well known that 1-dimensional block averaging of L uncorrelated errors $\epsilon(x_m)$ in uniform-weighted averages with a span of $L_1 = L\Delta x$ on a sample grid spacing of $\Delta x$ on which the errors are statistically uncorrelated reduces the uncorrelated error variance $\sigma_c^2$ by a multiplicative factor of

$$\alpha_1 = \frac{1}{L}. \qquad (D.6)$$

The subscript 1 indicates that this value for $\alpha$ in the error variance reduction formula (D.5a) is specific to the uniform-weighted average. The integral on the right side of (D.5b) after substituting (D.6) on the left side of the equation is

$$\int_{0}^{k_N} W_1^2(k) dk = \frac{1}{2L\Delta x}, \qquad (D.7)$$

where $W_1(k)$ is the filter transfer function (A.6) of the uniform-weighted average with a span of $L_1 = L\Delta x$, which has a half-power filter cutoff wavenumber of $k_c \approx 0.5L_1^{-1}$ according to (A.7). The explicit dependence of the filter transfer function $W_{k_c}(k)$ on the half-power filter cutoff wavenumber $k_c$ has been dropped in (D.7) in favor of the subscript 1 in order to distinguish this filter transfer function from that of the Parzen smoother considered below.

It can be anticipated from the close agreement of the half-power filter cutoff wavenumbers of the two filter transfer functions in Fig. A.1b that the error reduction factor that is achieved using a 1-dimensional Parzen smoother with a span of $L_p = 2L_1 = 2L\Delta x$, i.e., twice the span of the uniform-weighted average smoother, is approximately the same as the error variance reduction factor (D.6) for a 1-dimensional block average of L uncorrelated errors over a span of $L_1 = L\Delta x$. This can be shown to be the case as follows.

An approximate analytical expression for the ratio of the variances $\bar{\sigma}_1^2$ and $\bar{\sigma}_4^2$ of, respectively, uniform-weighted averages with a span of $L_1 = L\Delta x$ and Parzen-weighted averages with a span of $L_4 = 4L_1 = 4L\Delta x$ applied to uncorrelated white noise can be derived from (D.5) by considering the ratio

$$R_L = \frac{\frac{\bar{\sigma}_4^2}{\bar{\sigma}_1^2} = \frac{\int_0^{k_N} W_4^2(k) dk}{\int_0^{k_N} W_1^2(k) dk} = \frac{\int_0^{k_N} \text{sinc}^8(kL\Delta x) dk}{\int_0^{k_N} \text{sinc}^2(kL\Delta x) dk}}{\frac{\bar{\sigma}_1^2}{\bar{\sigma}_4^2} = \frac{\int_0^{k_N} W_4^2(k) dk}{\int_0^{k_N} W_1^2(k) dk} = \frac{\int_0^{k_N} \text{sinc}^8(kL\Delta x) dk}{\int_0^{k_N} \text{sinc}^2(kL\Delta x) dk}} \qquad (D.8)$$

where $W_4(k)$ is the filter transfer function (A.11) of the Parzen smoother with a span of $L_4 = 4L\Delta x$, which yields $\text{sinc}^4(kL\Delta x)$ on the right side of (A.11) and has a filter cutoff wavenumber of $k_c \approx L_4^{-1}$ according to (A.12a). Exact solutions for the two integrals on the right side of (D.8) cannot be obtained analytically. However, solutions can be found in integral tables if the upper bounds of the two integrals in (D.8) are both $\infty$ rather than the Nyquist wavenumber $k_N$. The ratio (D.8) can thus be rewritten as

$$R_L = \frac{\frac{I_8 - \int_{k_N}^{\infty} \text{sinc}^8(kL\Delta x) dk}{I_2 - \int_{k_N}^{\infty} \text{sinc}^2(kL\Delta x) dk}}{\frac{I_2 - \int_{k_N}^{\infty} \text{sinc}^8(kL\Delta x) dk}{I_8 - \int_{k_N}^{\infty} \text{sinc}^2(kL\Delta x) dk}} \qquad (D.9a)$$

where

$$I_2 = \int_0^\infty \text{sinc}^2(kL\Delta x) dk = \frac{1}{2L} \qquad (D.9b)$$

$$I_8 = \int_0^\infty \text{sinc}^8(kL\Delta x) dk = \frac{151}{630L} \qquad (D.9c)$$

The solutions (D.9b) and (D.9c) for $I_2$ and $I_8$ are derived from the general solution obtained from integral tables for the integral from 0 to $\infty$ of $\text{sinc}^p(kL\Delta x)$ with $p = 2$ and 8.

Because the sidelobes of $\text{sinc}^8(kL\Delta x)$ decay so much faster than the sidelobes of $\text{sinc}^2(kL\Delta x)$ (see Figs. A.1b and C.3), the correction term in the numerator of (D.9a) is much smaller than the correction

<figure><img src="image_62.png" /><figcaption>Fig. D.1. The ratio $R_L = \bar{\sigma}_4^2/\bar{\sigma}_1^2$ of the residual variance $\bar{\sigma}_4^2$ after filtering uncorrelated white noise using a Parzen smoother with a span of $L_4 = 4L$ and the residual variance $\bar{\sigma}_1^2$ after filtering with a uniform-weighted running average with a span of $L_1 = L$. The dots correspond to the analytical approximation given by Eq. (D.11) and the solid line is the exact solution obtained by numerical integration of Eq. (D.8).</figcaption></figure>

term in the denominator. The ratio of variances of uncorrelated errors smoothed with the Parzen smoother and the uniform-weighted average can therefore be approximated as

$$R_L \approx \frac{I_8}{I_2 - \int_{k_N}^{\infty} \text{sinc}^2(kL\Delta x) dk} \qquad (D.10)$$

By approximating each of the sidelobes of $\text{sinc}^2(kL\Delta x)$ as rectangles, it can be shown with some effort that the correction term in the denominator of this equation is approximately $\pi^{-2}L^{-2}$ for sufficiently large $L$. The meaning of “sufficiently large” is quantified below from Fig. D.1. The ratio (D.10) of the variances of Parzen-weighted and uniform-weighted averages with spans of $4L\Delta x$ and $L\Delta x$, respectively, then becomes

$$R_L \approx \frac{I_8}{I_2 - \pi^{-2}L^{-2}} \qquad (D.11a)$$

where $I_2$ and $I_8$ are defined in terms of $L$ by (D.9b) and (D.9c).

The analytical approximation (D.11a) for $R_L$ is shown by the dots in Fig. D.1 for values of $L$ ranging from 2 to 25. For comparison, the solid line corresponds to the exact solution obtained by numerical integrations of the numerator and denominator of (D.8) with $k_N = (2\Delta x)^{-1}$. The analytical approximation is almost imperceptibly too large for $L = 2$ and becomes indistinguishable from the numerical solution for $L > 2$. It is thus seen that the large $L$ approximation (D.11a) is applicable to all practical choices of smoothing of the pre-processed SWOT and WaCM data. The ratio $R_L$ in Fig. D.1 has a value of 0.5 for $L = 5$. It is slightly larger for $L < 5$ and slightly smaller for $L > 5$. Over the range 20 km $\le L \le 200$ km considered in this study, the numerical value of (D.11a) decreases from 0.484 to 0.480 and can thus be approximated as

$$R_L \approx 0.482 \qquad (D.11b)$$

For practical purposes, $R_L$ can be considered to have a value of 0.5. In other words, the residual variance $\bar{\sigma}_4^2$ of white-noise errors smoothed using a Parzen smoother with a span of $L_4 = 4L\Delta x$ is approximately half as large as the residual variance $\bar{\sigma}_1^2$ of white-noise errors smoothed using a uniform-weighted average with a span of $L_1 = L\Delta x$. From (D.7)
and (D.8) with $R_L \approx 0.5$, the integral in (D.5b) for the Parzen smoother with a span of $L_4$ is thus

$$ \int_{0}^{k_{N}} W_{4}^{2}(k) dk = R_{L} \int_{0}^{k_{N}} W_{1}^{2}(k) dk \approx \frac{1}{4L\Delta x} = \frac{1}{L_{4}}. \qquad (D.12a) $$

Since the half-power filter cutoff wavenumber for the Parzen smoother with a span of $L_4$ is $k_c = 0.910 L_4^{-1}$ according to (A.12a), this can be written alternatively as

$$ \int_{0}^{k_{N}} W_{4}^{2}(k) dk = \frac{k_{c}}{0.910} \approx k_{c}. \qquad (D.12b) $$

From the form (D.12a) for the integral on the right side of (D.5b), it is apparent that the multiplicative error variance reduction factor (D.5b) for a 1-dimensional Parzen smoother with a span of $L_4 = 4L\Delta x$ is

$$ \alpha_4 \approx \frac{1}{2L} = \frac{2\Delta x}{L_4}. \qquad (D.13) $$

This verifies the relationship between the error variance reduction factor for the 1-dimensional Parzen smoother and error variance reduction factor (D.6) for a 1-dimensional block average with a span of $L_1 = L\Delta x$ that was anticipated above prior to considering the ratio $R_L$ defined by (D.8). Namely, smoothing using a Parzen smoother with a span of $L_p = 2L\Delta x$ yields an error variance reduction factor of $\alpha_4 \approx L^{-1}$, which is obtained by substituting $L_p$ for $L_4$ in (D.13). This is the same as the value (D.6) for $\alpha_1$ that is obtained from smoothing with a uniform-weighted running average with a span of $L_1 = L\Delta x$.

The above analysis in one dimension can be extended straightforwardly to the case of filtering in two dimensions in the form of separate 1-dimensional smoothing in each of two orthogonal dimensions for a sample grid spacing of $\Delta x \times \Delta y$ on which the errors are statistically uncorrelated. The multiplicative error variance reduction factor $\alpha$ in (D.5a) that is achieved with 2-dimensional smoothing in this manner using Parzen smoothers with spans of $L_4(x)$ in the x dimension and $L_4(y)$ in the y dimension is then just the product of factors of the form (D.13) in each dimension,

$$ \alpha_4 \approx \frac{4\Delta x \Delta y}{L_4(x) L_4(y)}. \qquad (D.14a) $$

The filtering of the SSH and surface velocity fields from the CCS model with simulated SWOT and WaCM uncorrelated measurement errors in Sections 6 and 8–10 was applied isotropically with a 2-dimensional weighting function of the form (C.1) that depends only on the radial distance between the estimation location and each data point. In general, this form of 2-dimensional smoothing differs from successive applications of 1-dimensional smoothing in two orthogonal dimensions. It was shown in Appendix C, however, that the two approaches are essentially equivalent for the Parzen smoother with spans $L_4(x) = L_4(y)$ because of its close similarity to Gaussian smoothing. The advantage of smoothing separately in each dimension is that the separability allows the use of the factor (D.14a) to determine the error reduction from 2-dimensional smoothing. Because of the essential equivalence of the two forms of isotropic 2-dimensional smoothing with the Parzen smoother, the error reduction factor (D.14a) is also applicable to the error fields smoothed in Sections 6 and 8–10 with an isotropic 2-dimensional Parzen weighting function.

To illustrate the effects of this error reduction with a specific example, consider the case of an isotropic Parzen smoother with spans of $L_4(x) = L_4(y) = 14$ km applied to uncorrelated errors on a 1 km × 1 km grid. Then $L_4(x) = 14\Delta x$ and $L_4(y) = 14\Delta y$. The error variance reduction factor (D.14a) is therefore $\alpha_4 \approx 4/14^2 = 1/49$, i.e., the same as would be achieved by uniform-weighted block averaging over a 7 km × 7 km area. Recall, however, that the Parzen smoother has much better filtering properties than the uniform-weighted average smoother (see Figs. A.1b and C.3).

Because the half-power filter cutoff wavenumber (A.12a) for the Parzen smoother is $k_c \approx L_4^{-1}(x)$ in the x dimension and $l_c \approx L_4^{-1}(y)$ in

the y dimension, a 2-dimensional span of $L_4(x) \times L_4(y)$ can be expressed alternatively as smoothing with half-power filter cutoff wavenumbers of approximately $k_c^{-1} \times l_c^{-1}$. The expression (D.14a) for the multiplicative reduction factor $\alpha$ in the relation (D.5) for the residual error variance after smoothing the pre-processed estimates with these half-power filter cutoff wavenumbers can thus be written alternatively by substituting $L_4(x) = k_c^{-1}$ and $L_4(y) = l_c^{-1}$ into (D.14a) to get

$$ \alpha_4 \approx 4 \Delta x \Delta y k_c l_c. \qquad (D.14b) $$

For the case of isotropic smoothing with a half-power filter wavelength of $\lambda_c = k_c^{-1} = l_c^{-1}$ in each dimension, this becomes

$$ \alpha_4 \approx \frac{4 \Delta x \Delta y}{\lambda_c^2}. \qquad (D.14c) $$

The subscript 4 on the multiplicative error variance reduction factor (D.14) to be used in the error reduction formula (D.5) is a reminder that $\alpha_4$ was derived specifically for the Parzen smoother. However, the form (D.14b) provides an approximate characterization of the reduction of uncorrelated error variance for other smoothers with parameters chosen to give the same filter cutoff wavenumbers $k_c$ and $l_c$. For isotropic smoothing, the error reduction for any smoother with a half-power filter cutoff wavelength $\lambda_c$ is given approximately by (D.14c).

## Appendix E. One-dimensional wavenumber spectra of 2-dimensionally smoothed uncorrelated error fields

The procedure followed in Sections 6 and 8–10 to improve the signal-to-noise ratio for SWOT and WaCM estimates of the variables considered in this study (surface velocity and vorticity) is to reduce the error variance by isotropic 2-dimensional smoothing of the variable measured by each instrument (SSH for SWOT and velocity components for WaCM). It was shown in Appendix C that the smoothing applied in Sections 6 and 8–10 using a truly isotropic 2-dimensional Parzen weighting function could be achieved essentially equivalently as successive applications of a 1-dimensional Parzen smoother in each of two orthogonal dimensions. This equivalence allowed the derivation of the error variance reduction factor (D.14b) for separate smoothing with half-power filter cutoff wavenumbers of $k_c$ and $l_c$ in the x and y dimensions.

The equivalence of the two methods of 2-dimensional smoothing is used in this appendix to quantify the effects of isotropic 2-dimensional smoothing on the wavenumber spectrum of residual measurement errors. Of particular interest is the effect of 2-dimensional smoothing on the 1-dimensional wavenumber spectrum<fnref n="6" /> of 2-dimensional uncorrelated white noise, which differs in an important way from what would be achieved with a 1-dimensional dataset. The difference is not necessarily intuitive but is needed for the derivation in Appendix F of the standard deviation of uncorrelated errors from the spectral specification of the science requirement for SWOT measurement accuracy. The results are also essential to the interpretation of the spectral analyses of the effects of smoothing of the measurement errors in ground-based post-processing that are derived for both SWOT and WaCM in Appendix I (see also Section 4.4). We therefore summarize the effects of 2-dimensional smoothing on the 1-dimensional wavenumber spectrum in this appendix.

Consider a 2-dimensional spatial field $\epsilon(x, y)$ of uncorrelated measurement errors (e.g., measurements of SSH by SWOT or measurements of a velocity component by WaCM) at discrete locations $(x_m, y_n)$ for $x_m = m\Delta x$, $m = 0, 1, \dots, M-1$ and $y_n = n\Delta y$, $n = 0, 1, \dots, N-1$, where $\Delta x$ and $\Delta y$ are the sample intervals in the x and y dimensions. It will be assumed that the errors $\epsilon(x_m, y_n)$ are uncorrelated on this

<footnote n="6">⁶ The direction of the 1-dimensional spectrum could be defined to be along the satellite ground track for the case of actual satellite data or in the alongshore direction for the case of the spectral analysis of the model output considered in Appendix I and Section 4.4.</footnote>