# Prospects for future satellite estimation of small-scale variability of ocean surface velocity and vorticity

Dudley B. Chelton, Michael G. Schlax, Roger M. Samelson, J. Thomas Farrar, M. Jeroen Molemaker, James C. Mcwilliams, Jonathan Gula

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Dudley B. Chelton, Michael G. Schlax, Roger M. Samelson, J. Thomas Farrar, M. Jeroen Molemaker, et al.. Prospects for future satellite estimation of small-scale variability of ocean surface velocity and vorticity. Progress in Oceanography, 2019, 173, pp.256-350. 10.1016/j.pocean.2018.10.012. insu-03683196

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# Prospects for future satellite estimation of small-scale variability of ocean surface velocity and vorticity

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Dudley B. Chelton<sup>a,∗</sup>, Michael G. Schlach<sup>a</sup>, Roger M. Samelson<sup>a</sup>, J. Thomas Farrar<sup>b</sup>, M. Jeroen Molemaker<sup>c</sup>, James C. McWilliams<sup>c</sup>, Jonathan Gula<sup>d</sup>

## ARTICLE INFO

**Keywords:**

Satellite oceanography

Wide-swath altimetry

Doppler scatterometry

Submesoscale variability

Mesoscale variability

Sea level measurement

Surface current measurement

Relative vorticity

Geostrophic approximation

## ABSTRACT

Recent technological developments have resulted in two techniques for estimating surface velocity with higher resolution than can be achieved from presently available nadir altimeter data: (1) Geostrophically computed estimates from high-resolution sea surface height (SSH) measured interferometrically by the wide-swath altimeter on the Surface Water and Ocean Topography (SWOT) Mission with a planned launch in 2021; and (2) Measurements of ocean surface velocity from a Doppler scatterometer mission that is in the early planning stages, referred to here as a Winds and Currents Mission (WaCM). In this study, we conduct an analysis of the effects of uncorrelated measurement errors and sampling errors on the errors of the measured and derived variables of interest (SSH and geostrophically computed velocity and vorticity for SWOT, and surface velocity and vorticity for WaCM). Our analysis includes derivations of analytical expressions for the variances and wavenumber spectra of the errors of the derived variables, which will be useful to other studies based on simulated SWOT and WaCM estimates of velocity and vorticity. We also discuss limitations of the geostrophic approximation that must be used for SWOT estimates of velocity.

The errors of SWOT and WaCM estimates of velocity and vorticity at the full resolutions of the measured variables are too large for the unsmoothed estimates to be scientifically useful. It will be necessary to smooth the data to reduce the noise variance. We assess the resolution capabilities of smoothed estimates of velocity and vorticity from simulated noisy SWOT and WaCM data based on a high-resolution model of the California Current System (CCS). By our suggested minimum threshold signal-to-noise (S/N) variance ratio of 10 (a standard deviation ratio of 3.16), we conclude that the wavelength resolution capabilities of maps of velocity and vorticity constructed from WaCM data with a swath width of 1200 km are, respectively, about 60 km and 90 km in 4-day averages. For context, the radii of resolvable features are about four times smaller than these mesoscale wavelength resolutions. If the swath width can be increased to 1800 km, the wavelength resolution capabilities of 4-day average maps of surface velocity and vorticity would improve to about 45 km and 70 km, respectively. Reducing the standard deviation of the uncorrelated measurement errors from the baseline value of $\sigma_{spd} = 0.50 \text{ m s}^{-1}$ to a value of $0.25 \text{ m s}^{-1}$ would further improve these resolution capabilities to about 20 km and 45 km.

SWOT data will allow mapping of the SSH field with far greater accuracy and space-time resolution than are presently achieved by merging the data from multiple nadir altimeter missions. However, because of its much narrower 120-km measurement swath compared with WaCM and the nature of the space-time evolution of the sampling pattern during each 21-day repeat of the SWOT orbit, maps of geostrophically computed velocity and vorticity averaged over the 14-day period that is required for SWOT to observe the full CCS model domain are contaminated by sampling errors that are too large for the estimates to be useful for any amount of smoothing considered here. Reducing the SSH measurement errors would do little to improve SWOT maps of velocity and vorticity. SWOT estimates of these variables are likely to be useful only within individual measurement swaths or with the help of dynamic interpolation from a data assimilation model. By our criterion, in-swath SWOT estimates of velocity and vorticity have wavelength resolution capabilities of about 30 km and 55 km, respectively.

<footnote><sup>a</sup> College of Earth, Ocean and Atmospheric Sciences, Oregon State University, Corvallis, OR, United States</footnote>

<footnote><sup>b</sup> Woods Hole Oceanographic Institution, Woods Hole, MA, United States</footnote>

<footnote><sup>c</sup> Institute of Geophysics and Planetary Physics, Department of Atmospheric and Oceanic Sciences, University of California, Los Angeles, CA, United States</footnote>

<footnote><sup>d</sup> Université de Brest, CNRS, IRD, Ifremer, Laboratoire d'Océanographie Physique et Spatiale, LOPS, IUEM, Brest, France</footnote>

<footnote>\* Corresponding author.</footnote>

<footnote>E-mail address: chelton@coas.oregonstate.edu (D.B. Chelton).</footnote>
In comparison, in-swath estimates of velocity and vorticity from WaCM data with $\sigma_{spd} = 0.50 \text{ m s}^{-1}$ have a wavelength resolution capability of about 130 km for both variables. Reducing the WaCM measurement errors to $\sigma_{spd} = 0.25 \text{ m s}^{-1}$ would improve the resolution capabilities to about 50 km and 75 km for velocity and vorticity, respectively. These resolutions are somewhat coarser than the in-swath estimates from SWOT data, but the swath width is more than an order of magnitude wider for WaCM. Instantaneous maps of velocity and vorticity constructed in-swath from WaCM data will therefore be much less prone to edge effect problems in the spatially smoothed fields.

Depending on the precise value of the threshold adopted for the minimum S/N ratio and on the details of the filter used to smooth the SWOT and WaCM data, the resolution capabilities summarized above may be somewhat pessimistic. On the other hand, aspects of measurement errors and sampling errors that have not been accounted for in this study will worsen the resolution capabilities presented here. Another caveat to keep in mind is that the resolution capabilities deduced here from simulations of the CCS region during summertime may differ somewhat at other times of year and in other geographical regions where the signal variances and wavenumber spectra of the variables of interest differ from the CCS model used in this study. Our analysis nonetheless provides useful guidelines for the resolutions that can be expected from SWOT and WaCM.

# 1. Introduction

Presently available global sea-surface height (SSH) fields constructed from satellite altimeter data by Collecte Localis Satellites (CLS) and archived by Archivage, Validation, Interprétation des données des Satellite Océanographiques (AVISO) are able to resolve time scales of about a month and wavelength scales of about 200 km, corresponding to feature radius scales of about 50 km (see Appendix A.3 of Chelton et al., 2011). The 25+ year CLS/AVISO record of SSH fields (Pujol et al., 2016) has been extremely useful for studies of mesoscale eddies, large-scale ocean circulation variability and sea level rise (e.g., Fu et al., 2010; Lee et al., 2010; Willis et al., 2010; Chelton et al., 2011). Notwithstanding the many successful applications of this dataset, it is clear from high-resolution numerical simulations and satellite infrared and visible observations of sea surface temperature (SST) and ocean color that energetic variability exists on smaller scales than can be resolved by the CLS/AVISO SSH fields. Of particular interest are wavelength scales smaller than ~50 km (radius scales less than ~10 km), which is a commonly used definition for submesoscale variability. Numerical models have shown that submesoscale variability is important to the physics and biology of the ocean (e.g., Capet et al., 2008; Klein and Lapeyre, 2009; Lévy et al., 2001). There is thus a strong interest in high-resolution satellite observations of surface velocity and its associated vorticity to complement the modeling. The objective of this study is to investigate the prospects for future high-resolution satellite measurements of small-scale velocity and vorticity variability.

In addition to the interest in submesoscale variability, much remains to be learned about mesoscale ocean dynamics. Velocity and vorticity can be estimated geostrophically from the CLS/AVISO SSH fields, but with a very coarse spatial resolution imposed by the ~200 km wavelength resolution limitation of the SSH fields. The improved resolutions that are anticipated from the two satellite technologies considered in this study are likely to advance understanding of mesoscale variability in a manner similar to the advances achieved from the CLS/AVISO multi-altimeter merged SSH dataset compared with the single-altimeter observations obtained from TOPEX/Poseidon (see, for example, Fig. 1 of Chelton et al., 2011).

Our particular interest here in satellite estimation of surface velocity and vorticity is motivated in part by their effect on local wind-driven vertical velocities (Ekman pumping). It is well known that surface currents contribute to Ekman pumping in mesoscale eddies through their effects on the relative wind and hence the surface stress. The surface velocity within a rotating eddy generates a curl of the surface stress with sign opposite that of the vorticity of the eddy, thus attenuating the eddy and generating an Ekman pumping velocity that is often larger on mesoscales and smaller scales than that from the curl of the large-scale background wind stress field (e.g., Dewar and Flierl, 1987; Martin and Richards, 2001; McGillicuddy et al., 2007; Gaube et al., 2015). Mesoscale eddies also generate Ekman pumping to an

even greater degree through their effects on the gradient of the total vorticity (planetary plus relative vorticity) that generates horizontal divergences of Ekman transport and hence vertical velocities (Stern, 1965; Mahadevan et al., 2008; McGillicuddy et al., 2008; Gaube et al., 2015). This vorticity-gradient-induced Ekman pumping is sometimes referred to as nonlinear Ekman pumping. The two effects of surface currents on Ekman pumping increase rapidly with decreasing scale and increasing Rossby number (see Fig. 7 of Gaube et al., 2015). A small-scale cutoff for this increase presumably exists, but both effects are likely important into at least the upper range of submesoscale variability.

Determination of Ekman pumping on mesoscale and smaller scales thus depends critically on accurate knowledge of the surface ocean velocity at these scales. Recent technology developments have resulted in two techniques for estimating ocean surface velocity that promise higher resolution than is presently achieved from the CLS/AVISO SSH fields. One of these is the Surface Water and Ocean Topography (SWOT) Mission (Fu and Ferrari, 2008; Durand et al., 2010) that is slated for launch in 2021. SWOT will measure SSH altimetrically by radar interferometry (Fu and Rodriguez, 2004) with a footprint size of about 1 km. Surface velocity can then be estimated geostrophically. The other technology for estimating ocean surface velocity is based on the Doppler shift of radar returns from the moving sea surface (Chapron et al., 2005; Rodríguez et al., 2018; Rodríguez, 2018). An important distinction of Doppler radar systems is that they provide direct measurements of ocean surface velocity, rather than the indirect estimates computed geostrophically from measurements of SSH as in the case of SWOT.

Satellite-based Doppler radar systems for measuring surface ocean currents are in the early stages of development, both in Europe and in the U.S. The European Doppler scatterometer mission concept summarized by Chapron et al. (2005) has evolved to a proposed Doppler radar system called the Sea surface Kinematics Multiscale (SKIM) mission (Ardhuin et al., 2018) that has been designed to measure surface currents, ice drift and ocean waves across a swath width of 320 km. A primary goal of SKIM is to measure the wave spectrum. The incidence angles of the SKIM radar measurements have therefore been chosen to be 6° and 12° in order to maximize the sensitivity to surface wave tilt while minimizing the sensitivity to winds. The SKIM radar thus cannot measure surface vector winds.

The Doppler radar in the mission under development in the U.S. will have a much wider measurement swath than SKIM (at least 1200 km, and possibly as much as 1800 km) and will measure radar backscatter at much higher incidence angles that will provide collocated measurements of surface currents and vector winds, but not the wave spectrum. Wind speed and direction are inferred from the roughness of the sea surface by conventional scatterometry (see, for example, Sec. 2 of Chelton and Freilich, 2005, for a summary of the QuikSCAT scatterometer). An aircraft version called DopplerScatt has been built by the National Aeronautics and Space Administration Jet Propulsion
<figure><img src="image_3.png" /><figcaption>**Fig. 1.** Snapshots for a region off the central California coast from ROMS models of the California Current System (CCS) with grid resolutions of 4, 1.5 and 0.5 km (left to right): Row (a) Sea surface height; Row (b) Sea surface temperature; and Row (c) Normalized vorticity $\zeta/f$, where f is the local Coriolis parameter at each grid point.</figcaption></figure>

Laboratory (NASA/JPL) and has been flown in several field campaigns (Rodríguez et al., 2018) as a proof of concept for a future satellite Doppler scatterometer mission that we refer to in this study as a Winds and Currents Mission (WaCM). In the recent decadal survey of recommended Earth-observing satellite missions by the National Academies of Sciences, Engineering, and Medicine (NASEM, 2018), WaCM is one of seven recommended NASA Earth System Explorer missions. The baseline performance of WaCM that is assumed in this study (see Section 4.2) is supported by the actual performance of the airborne DopplerScatt instrument (Rodríguez et al., 2018).

The collocated WaCM measurements of surface currents and vector winds are expected to have footprint sizes of about 5 km. For the wind measurements, this is much smaller than the footprint sizes of about 25 km for the QuikSCAT and ASCAT scatterometers. The smaller footprint size for WaCM will be achieved with a larger antenna size and the use of a Ka-band radar (35.8 GHz), rather than the Ku-band radar (13.4 GHz) used for QuikSCAT or the C-band radar (5.3 GHz) used for ASCAT. This study considers only the surface ocean velocity estimates. Our baseline simulations of WaCM measurements of surface currents

assume a swath width of 1200 km. Recent engineering studies suggest that it may be possible to broaden the swath width to 1800 km, which would greatly improve the sampling. The benefits of the wider swath width are investigated as part of this study.

The viability of SWOT and WaCM estimates of surface velocity for investigation of small-scale variability clearly depends on the signal-to-noise ratios of the measurements (SSH for the case of SWOT and surface velocity for the case of WaCM). For SWOT, the utility of the data also depends on the validity of the geostrophic approximation. The objectives of this study are to investigate these issues and assess the resolution capabilities for mapping of the surface velocity and vorticity fields for both SWOT and WaCM. The approach is based on simulated SWOT and WaCM sampling of, respectively, the SSH and surface velocity fields from 30 days of twice-daily snapshots from a high-resolution model of the California Current System (CCS).

Our focus is on the limitations imposed by the baseline science requirements for the cross-swath averages of the standard deviations of uncorrelated instrumental measurement errors. In reality, the instrumental errors for both SWOT and WaCM vary across the measurement
<figure><img src="image_4.png" /><figcaption>**Fig. 2.** A representative summertime snapshot of sea surface temperature (SST) from the ROMS model of the CCS on 5 June (left panel) in latitude–longitude coordinates at the full 0.5 km × 0.5 km grid resolution of the model. The model was forced by the seasonal cycle wind stress derived from QuikSCAT scatterometer data, which is shown for the month of June in the right panel. The box in each panel delineates the truncation of the full model domain to mitigate edge effects in the analyses of spatially smoothed fields considered in this study.</figcaption></figure>

swaths (see Fig. F.1 in Appendix F for the case of SWOT), with smallest errors near the centers of the swaths and increasing errors toward both edges of the swaths. The measurement errors also depend on the significant wave height (see again Fig. F.1 in Appendix F for the case of SWOT). The significant wave height (SWH) dependence and cross-track variations of the instrumental errors are not considered in the simulations presented in this study. Rather, we consider the cross-swath average instrumental measurement errors for the conditions of 2-m significant wave height that are specified for the baseline design requirements of the measurement errors.

We note that the error characteristics of SWOT data could have been simulated more accurately and completely for this investigation by using the simulator software available from the SWOT Project Office (Gaultier et al., 2017; see also Qiu et al., 2016; Gaultier et al., 2016). Simulator software is not yet available for WaCM. Moreover, the dependencies of WaCM measurement errors on swath location and SWH have not yet been fully quantified, although it is known that WaCM measurement errors will increase toward the edges of the measurement swaths in a manner similar to the cross-track variations of SWOT measurement errors (Rodríguez, 2018). For consistency in our treatments of SWOT and WaCM measurement errors, we have therefore chosen to use our simpler simulations of SWOT measurement errors to be consistent with our simulations of WaCM measurement errors.

Measurements from both SWOT and WaCM are also subject to larger-scale errors from geophysical corrections for a variety of environmental effects. While important, these larger-scale errors are not addressed in the analysis presented here as they are generally secondary to the effects of uncorrelated instrumental errors for the purposes of estimating surface velocity and vorticity on the mesoscales and submesoscales that are of primary interest in this study.

We are also not able to address the importance of internal gravity waves, which are underrepresented in the CCS model used to simulate SWOT and WaCM data in this study (see Sections 2 and 3). This internal wave variability is one of several ageostrophic processes that affect the accuracy of SWOT estimates of surface velocity and vorticity. The contributions of ageostrophic processes to contamination of geostrophically computed surface velocity and vorticity from SWOT measurements of SSH cannot be fully assessed from the CCS model used in this study since it was forced with seasonally varying winds,

heat fluxes and freshwater fluxes and lacks tidal forcing. This issue is addressed to a limited degree in Section 3 and Appendix F (see Figs. 10, 11 and F.4) from a pair of simulations of the Gulf Stream region off the southeastern seaboard of the U.S.

It is noteworthy that the seasonally forced CCS model that is used in this study may also misrepresent larger-scale geostrophic processes. Mesoscale and submesoscale variability might be stronger with more realistic wind forcing that includes synoptic atmospheric variability. The analysis presented here is based on model simulations for a 30-day period in early summertime. Since weather systems over the CCS are usually not very energetic at this time of year, inclusion of synoptic atmospheric forcing may have only modest effects on the model simulation for the geographical location and time of year considered here. This potential limitation of the model should nonetheless be kept in mind in the interpretation of the results presented in this study.

Because of the various aspects of measurement errors summarized above that are not taken into consideration in this study, the conclusions about the effects of measurement errors presented here are likely somewhat optimistic assessments of the resolution capabilities of SWOT and WaCM. On the other hand, if the uncorrelated instrumental errors can be reduced from the baseline science requirement values used in our simulations, the effects of instrumental errors in the simulations presented here may prove to be somewhat pessimistic assessments of the resolution capabilities for the baseline consideration of 2-m significant wave height.

In addition to the effects of internal gravity waves and other small-scale ageostrophic processes, the ability to map the space–time evolution of the surface velocity field depends on the swath width over which the measurements are made. For SWOT, the swath width will be 120 km with a nadir gap of 20 km. For the analysis in Section 8, we have assumed that velocity estimates will be obtained by WaCM across a swath width of 1200 km with a nadir gap of 100 km. The sampling coverage in each satellite overpass of a given region will thus be more than an order of magnitude better for WaCM than for SWOT. The SWOT orbit will have an exact repeat period of 21 days. For the simulations in this study, we have assumed that WaCM will have the same 4-day repeat orbit as QuikSCAT. The net effect of the different swath widths and orbit repeat periods for SWOT and WaCM is that a given location within the CCS region considered here is sampled on average about
<figure><img src="image_5.png" /><figcaption>**Fig. 3.** Snapshots from the ROMS model for the same time as the SST map in Fig. 2 at the full 0.5 km × 0.5 km grid resolution of the model: Column (a) the speed of the total surface velocity; and Column (b) the normalized vorticity $\zeta/f$ computed from the total surface velocity, where $f$ is the local Coriolis parameter at each grid point. The maps in the top panels are for the truncated domain delineated by the boxes in Fig. 2 in the model x, y coordinate system that is rotated by a polar angle of 24° relative to latitude–longitude coordinates. The bottom panels are enlargements of the Central California Current System (CCCS) region delineated by a box in each of the top panels.</figcaption></figure>

once a day by WaCM with a 1200-km swath width and about once a week by SWOT (see Section 7). Extending the WaCM swath width to 1800 km with the same nadir gap of 100 km would increase the sampling to an average of more than 1.5 samples per day over the CCS model domain. Our analysis includes an assessment of the impact of this improved sampling on the resolution capability of space–time smoothed velocity and vorticity fields constructed from WaCM data.

The resolution limitations of maps of geostrophically computed surface velocity and vorticity constructed from simulated SWOT data have previously been considered by Fu and Ubelmann (2014), Qiu et al. (2016) and Gaultier et al. (2016). In this study, we extend these previous investigations of the resolution capabilities of simulated SWOT data and compare the results with the resolution capabilities of maps of velocity and vorticity constructed from simulated WaCM data. Our approach is more systematic than has been used in past studies. In particular, we propose a specific criterion for defining resolution capability and we partition the mapping errors between instrumental measurement errors and sampling errors. By sampling errors, we mean the errors in mapped fields of surface velocity and vorticity that are imposed by the limited swath widths of the SWOT and WaCM measurements and by the discrete and irregular temporal sampling of the rapidly evolving submesoscale features in the velocity and vorticity fields at a given location. The ability to distinguish between

<figure><img src="image_6.png" /><figcaption>**Fig. 4.** (a) The scale dependencies of the 80th, 90th, 95th and 99th percentile points of the distributions of absolute values of the normalized vorticity $\zeta/f$ as functions of half-power filter cutoff wavelength; the dashed line corresponds to the root-mean squared value of $\zeta/f$. (b) The scale dependencies of selected percentage points symmetric about the median (i.e., the 50th percentile point) in the distributions of $\zeta/f$ as functions of half-power filter cutoff wavelength; the dashed line corresponds to the mean value of $\zeta/f$, which is indistinguishable from zero. For both panels, $\zeta$ was computed from error-free model fields of total surface velocity for the complete CCS model domain at the full 0.5 km × 0.5 km grid resolution of the model after 2-dimensional isotropic smoothing with the half-power filter cutoff wavelengths indicated along the abscissa.</figcaption></figure>

measurement and sampling errors in maps constructed from simulated satellite data provides insight into the relative benefits of reducing the measurement noise versus improving the sampling, either by increasing the measurement swath widths or by combining the measurements from multiple satellites.

This paper is organized as follows. The CCS model from which simulated SWOT and WaCM data are derived is summarized in Section 2. The limitations of the geostrophic approximation that must be used for SWOT estimates of surface velocity and vorticity are discussed in Section 3. The uncorrelated measurement errors for SWOT and WaCM and their effects on the errors of the derived quantities (geostrophically computed velocity and vorticity for SWOT and vorticity for WaCM) are examined in detail in Section 4, including determination of the wavenumber spectral characteristics of the errors.

The strategy adopted in this study to assess the resolution capabilities for estimates of surface velocity and vorticity from noisy SWOT and WaCM data is presented in Section 5. The procedure is first applied in Section 6 to instantaneous maps of the velocity and vorticity fields constructed from SWOT and WaCM data for the idealized case of measurement errors alone without consideration of sampling errors from
<figure><img src="image_7.png" /><figcaption>**Fig. 5.** The same as Fig. 3, except snapshots from the ROMS model for: Column (a) SSH; Column (b) the magnitude of geostrophically computed velocity; and Column (c) normalized geostrophically computed vorticity $\zeta_g/f$.</figcaption></figure>

the limited swath widths of the SWOT and WaCM measurements. This analysis of the effects of measurement errors alone can be interpreted as the best possible resolution capabilities within a single measurement swath.

For mapping of the velocity and vorticity fields over regions larger than the width of an individual measurement swath, sampling errors can be as important as, or even more important than, measurement errors to the overall accuracies of the mapped variables. The space–time sampling characteristics of SWOT and WaCM data are summarized in Section 7. The combined effects of measurement and sampling errors on the resolution capabilities of maps of surface ocean velocity and vorticity estimated from space–time smoothed SWOT and WaCM data are then investigated in Sections 8–10.

The conclusion of Section 8 is that the resolutions of maps of geostrophically computed velocity and vorticity constructed from SWOT data are limited almost totally by sampling errors. Improving the SWOT measurement accuracy would therefore have very little effect on the resolution capabilities of the variables considered in this study. The resolutions of maps of velocity and vorticity constructed from WaCM data are primarily limited by measurement errors but sampling errors are not negligible. The benefits of increasing the WaCM swath width (thus reducing the sampling errors) and reducing the WaCM

measurement noise are investigated separately and in combination in Section 10.

Supporting technical details for the calculations in Sections 4, 6 and 8–10 are provided in a series of nine appendices.

The error analysis in Section 4 merits special mention since the results will be useful to other studies. Analytical expressions for the variances of the errors of SWOT estimates of velocity and vorticity computed geostrophically from SSH are derived in Appendix G.1 in terms of the variance $\sigma_h^2$ of the uncorrelated errors of pre-processed SWOT estimates of SSH. By “pre-processed”, we mean the satellite data that will be distributed to general users after smoothing of the raw data onboard the satellite, possibly with additional ground-based smoothing to achieve a specified footprint size. The numerical results presented in this study are based on the SSH error standard deviation of $\sigma_h = 2.74$ cm that is derived in Appendix F for the case of pre-processed data with a footprint size of 1 km. It would be straightforward to obtain numerical results from the analytical expressions for any specified value of $\sigma_h$.

An analytical expression for the variance of errors of WaCM estimates of vorticity is similarly derived in Appendix G.2 in terms of the variances $\sigma_u^2$ and $\sigma_v^2$ of the uncorrelated errors of WaCM estimates of each orthogonal velocity component $u$ and $v$. The result is therefore valid for any application of WaCM data, given specifications of the velocity component error variances. The analysis in this study assumes
<figure><img src="image_8.png" /><figcaption>**Fig. 6.** Column (a) Maps of the ageostrophic velocity vectors defined to be the vector differences of the total surface velocity minus the geostrophically computed velocity with the magnitudes of the differences shown in color. Note that the left half of the color bar is not used in this figure. Column (b) The large-scale ageostrophic velocity defined to be the vector differences and their magnitudes in (a) smoothed isotropically with a half-power filter cutoff wavelength of 150 km, referred to here as the Ekman ageostrophic velocity field. Column (c) The magnitudes of the non-Ekman ageostrophic velocity vectors defined to be the total ageostrophic velocities in Column (a) minus the large-scale ageostrophic velocities in Column (b). The bottom panels are enlargements of the CCCS region delineated by a box in each of the top panels. The vectors in Columns (a) and (b) were subsampled on a 15 km × 15 km grid in the top panels and a 7.5 km × 7.5 km grid in the bottom panels.</figcaption></figure>

an equal error standard deviation of $\sigma_{u,v} = 0.354 \text{ m s}^{-1}$ for each velocity component. If the errors of the two velocity components are uncorrelated with each other, this corresponds to current speed errors with a standard deviation of $\sigma_{\text{spd}} = 0.50 \text{ m s}^{-1}$, which is the present baseline design for WaCM.

Our analysis includes determinations of the wavenumber spectral characteristics of errors of velocity and vorticity computed geostrophically from pre-processed SWOT measurements of SSH, as well as the wavenumber spectral characteristics of velocity components and vorticity estimated from pre-processed WaCM data. Analytical expressions for these wavenumber spectra of the errors are derived for SWOT and WaCM in Appendices I.1 and I.3, respectively. Analytical expressions are also derived in Appendices I.2 and I.4 for the wavenumber spectra of the errors of all of the variables of interest after 2-dimensional smoothing is applied in simulated ground-based post-processing. As with the equations for the error variances discussed above, the equations derived in Appendix I for the error spectra are all expressed in terms of the measurement error variances $\sigma_h^2$ for SWOT and $\sigma_u^2$ and $\sigma_v^2$ for WaCM. The results are therefore valid for any specification of

the uncorrelated errors of SWOT and WaCM measurements of SSH and surface velocity.

Throughout this study, the surface vorticity fields are computed from the components of the total velocity fields from simulated WaCM data and from the components of surface velocity fields computed geostrophically from simulated SWOT measurements of SSH. We are also interested in the surface ocean velocity fields themselves, which are characterized here by the magnitude of the total or geostrophically computed vector surface velocity field, rather than by the vector component fields. For the 4-day and 14-day time averages considered in Sections 8–10, the analysis is based on the magnitudes of the vector averaged velocity fields. While it would be straightforward to consider each velocity component separately constructed from instantaneous maps and time averages of simulated SWOT measurements of SSH and WaCM measurements of surface velocity, the results are more difficult to interpret than the scalar velocity magnitude field because of the highly anisotropic and geographically inhomogeneous nature of the velocity field in the CCS region. Because the velocity errors are random in each component, direction errors will generally decrease
as the signal-to-noise ratio of satellite estimates of velocity magnitude increases.

## 2. The CCS model

A detailed description and validation of the CCS model used in this study can be found in Molemaker et al. (2015). Since the model representation of submesoscale variability is an important consideration for simulating SWOT and WaCM data in this investigation, a summary of the model is provided here. The computational code was ROMS, the Regional Oceanic Modeling System (Shchepetkin and McWilliams, 2005, 2009), which solves the hydrostatic primitive equations for the velocity, potential temperature, and salinity with a seawater equation of state. To simulate local flows in a realistic large-scale environment, the model was configured for the CCS utilizing open boundary conditions and a sequence of three nested subdomains (Marchesiello et al., 2003; Penven et al., 2006).

Because the primary target is submesoscale currents with horizontal scales of <10 km near the eastern topographic slope, an aggressive approach to nesting was taken with successively finer resolution in a sequence of steps where each “child” grid utilizes “parent” grid data at the open boundaries of the regional domains (Mason et al., 2010). The procedure consists of off-line, one-way nesting from larger to finer scales without feedback from the child grid solution onto the parent grid. It is implicitly assumed that a numerical “zoom” around a specific phenomenon is valid when it has an essentially local dynamical behavior, albeit with important influences from its environment of basin and regional circulation.

Bottom topography was defined from the SRTM30-plus bathymetry dataset based on the 1-minute Smith and Sandwell (1997) global dataset but using higher resolution data where available. A Gaussian smoothing kernel with a width of 4 times the topographic grid spacing was applied to avoid aliasing wherever the topographic data are available at higher resolution than the computational grid. The maximum depth for all grids was set to 6000 m, which is not a serious distortion for the U.S. west coast region. Models formulated with a terrain-following coordinate such as ROMS have computational restrictions with regard to the steepness and roughness of the topography (Beckmann and Haidvogel, 1993). Where the steepness of the topography exceeded these criteria, additional local smoothing was applied. The above procedure results in topography that is increasingly well resolved in the nested grids with successively higher resolution but may differ significantly from the original data in the coarser grids. Following Mason et al. (2010), the topography near the boundaries of the nested domains is matched with the parent topography.

The largest-scale simulation used in this study covers the full Pacific basin (see Fig. 1 of Molemaker et al., 2015). This grid, as well as the nested grids described below, is orthogonal based on an oblique Mercator projection and designed to have nearly uniform spacing in both horizontal dimensions. For the Pacific basin, the grid spacing varies from 12.5 km at the central latitude of the grid to 8.5 km at the north and south extremes of the grid near 40°S and 55°N; this is comparable to what is used in global mesoscale eddy-resolving models.

The simulation was forced at the surface by the QuikSCAT-based mean-monthly Scatterometer Climatology of Ocean Winds (SCOW; Risien and Chelton, 2008), and monthly heat and freshwater fluxes from the Comprehensive Ocean-Atmosphere Data Set (COADS; Da Silva et al., 1994), using a weak feedback from SST (Barnier et al., 1995). The open boundary information and initial state were taken from the Simple Ocean Data Assimilation monthly climatology (SODA; Carton and Giese, 2008). The model SST fields have both gyre-scale contrasts and mesoscale eddy fluctuations visible in regions of high SST gradients (see Figs. 1 and 2 in Molemaker et al., 2015; see also Figs. 1 and 2). It should be noted that inertial motions are poorly represented in our model because of the lack of high-frequency atmospheric forcing.

The Pacific basin model was spun-up from interpolated SODA data for 2 years, by which time an approximate statistical equilibrium was

reached for kinetic energy. The model was then run for an additional 10 years. The mean-monthly climatology over this 10 years was used to force the first nested grid along the U.S. west coast (see Fig. 2 of Molemaker et al., 2015) at its open boundaries. The climatological monthly boundary information from SODA forced the outermost Pacific basin model with a seasonal cycle. Mesoscale eddy activity was passed through boundary conditions to the first nested grid with an update time scale of five days. The successive nested models were boundary forced at increasingly shorter time scales, with boundary updates every 12 h for the CCS nested model analyzed in this study (see below). This was verified by comparing maps of surface eddy kinetic energy with altimetry-derived eddy kinetic energy. With open boundary conditions in nested grids, it is important to avoid computational artifacts associated with boundary-trapped features (e.g., rim currents) and noisy fields. In the ROMS-to-ROMS nesting interface, these artifacts are largely avoided, even for realistic flows with high mesoscale activity (Mason et al., 2010).

The three nested grids along the U.S. west coast are rotated by a polar angle of 24° so that the x and y axes are aligned approximately cross-shore and alongshore, respectively. As with the grid for the full Pacific basin, each of the three nested grids is discretely orthogonal, and they vary even less in their spacing over their relatively smaller domains. The first nested subdomain has a grid spacing that varies from 4 km to 3.97 km. The grids for the two smaller domains have average horizontal spacings of 1.5 km, and 0.5 km, respectively. The successive levels of grid refinement spontaneously exhibit an increasing amount of submesoscale variability (c.f., Capet et al., 2008).

The importance of the grid resolution for model representation of the submesoscale variability in the inner nested grid is readily apparent from Fig. 1. Submesoscale variability in the SST and vorticity fields is visibly much more energetic with a grid spacing of 0.5 km, even compared with a grid spacing of 1.5 km. The choice of model grid resolution is clearly an important consideration in the assessment of the resolution capabilities of SWOT and WaCM estimates of surface velocity and vorticity. This issue is discussed further in Section 11.

In all of the grids, there were 40 stretched vertical levels with a resolution of a few meters near the surface. The time step was 1600 s for the full Pacific basin model with ~10 km grid spacing and decreased to 600 s, 240 s and 90 s for the nested models with successively smaller grid spacings of 4 km, 1.5 km and 0.5 km, respectively. Dissipation was imposed in the model by a hyper viscous term, which is a by-product of the 3rd-order upwind advection. The effective hyper viscosity scales with the local velocity and the grid scale. The stepping procedure also includes a damping term that is much smaller than the hyper viscous term.

An important point for this study is that the high dissipation in the model significantly attenuates internal gravity waves compared with observations. While the internal gravity wave energy increases with the increased resolution of each nested grid, it is much weaker than in the real ocean, even at the highest grid resolution. In addition to the high dissipation, the lack of high-frequency atmospheric forcing and the absence of ocean tidal forcing contribute to the weak internal gravity wave energy in the CCS model used in this study. The effects of higher internal gravity wave energy are briefly considered in Section 3 from a pair of models of the Gulf Stream region off the southeastern seaboard of the U.S., one with weak internal waves like the CCS model used here and the other with energetic internal waves forced by high-frequency winds and tides (see Figs. 10 and 11).

For the investigation of the resolution capabilities of SWOT and WaCM in this study, we consider only the inner nested model, which extends from Point Conception in the south to approximately the Oregon/California border in the north. There are 1200 × 1800 grid points. With the 0.5-km grid spacing of this inner nested model, this corresponds to 600 km in the cross-shore dimension by 900 km in the alongshore dimension. For the analysis in this study, we consider only the 30-day period from day 141 to day 171, which corresponds
<figure><img src="image_9.png" /><figcaption>**Fig. 7.** Maps showing the scale dependence of the magnitudes of the non-Ekman ageostrophic velocities on the 0.5 km × 0.5 km model grid for the CCCS region after 2-dimensional isotropic smoothing with half-power filter cutoff wavelengths of 10, 20, 30, 40, 60 and 80 km. Note the different color bars for the upper and lower rows of panels and that the left half of the color bars are not used in either set of panels.</figcaption></figure>

to the early summertime period from 21 May through 20 June when submesoscale variability is fully developed. The climatological average wind stress field for the month of June is shown in the right panel of Fig. 2. For this study, the model output was subsampled at intervals of 0.5 days. It will be seen from Fig. 24 below that this sample interval is too coarse to fully resolve the rapidly evolving submesoscale variability. Since the time interval between successive measurements by both SWOT and WaCM is longer than this (see Section 7), the rapid evolution of submesoscale variability is even less well resolved in the satellite data.

A representative map of the model SST field on 5 June is shown in the left panel of Fig. 2 for the full domain of the 0.5-km inner nested model in latitude–longitude coordinates. The map reveals a rich distribution of scales of variability. The cold water near the coast is associated with wind-driven upwelling forced by the summertime equatorward winds. The meandering ribbon of upwelled cold water that separates from the coast at the northern corner of the model domain is the core of the equatorward flowing California Current. Submesoscale features are evident along SST fronts throughout the model domain.

The speed and vorticity of the surface currents associated with the SST field in Fig. 2 are shown in the rotated model grid coordinates in Fig. 3. The relative vorticity $\zeta = \partial v/\partial x - \partial u/\partial y$ (referred to hereafter as just the vorticity) was computed from the cross-shore ($u$, positive onshore) and alongshore ($v$, positive poleward) velocity components

using centered differences on the 0.5 km × 0.5 km model grid. Except in the alongshore wavenumber spectra in Figs. 13–15, the vorticity is normalized throughout this study by the local Coriolis parameter $f$ at each grid point. A mesoscale anticyclonic eddy can be seen in the north-west corner of the Central California Current System (CCCS) region in the bottom panels of Fig. 3. An additional mesoscale anticyclone and a mesoscale cyclone can be seen in the northwest corner of the full CCS model domain in the top panels. Most of the rest of the model domain is dominated by submesoscale variability, the most energetic of which is associated with the core of the meandering equatorward-flowing California Current that separates from the coast near Cape Blanco, just to the north of the model domain. Energetic submesoscale features are also associated with two offshore jets, one near the center of the model domain and the other near the southern boundary of the model.

The highly energetic vorticity field at submesoscales in Fig. 3b can be quantified by characterizing the scale dependence of the distribution of the magnitude of the normalized vorticity $\zeta/f$, which can be interpreted as a Rossby number. The scale dependence of $|\zeta|/f$ was determined by isotropic smoothing of the map in the top panel of Fig. 3b using a Parzen smoother (see Appendix A) with successively longer half-power filter cutoff wavelengths ranging from 0 to 150 km. To reduce edge effects in the smoothed fields along the northern, southern and offshore boundaries of the model domain, the analysis throughout the remainder of this study was restricted to the region delineated by the black lines in both panels of Fig. 2.
<figure><img src="image_10.png" /><figcaption>**Fig. 8.** The scale dependencies of the 80th, 90th, 95th and 99th percentile points of the distributions of the magnitudes of the non-Ekman ageostrophic velocities on the full 0.5 km × 0.5 km model grid for the full CCS model domain after 2-dimensional isotropic smoothing with the half-power filter cutoff wavelengths indicated along the abscissa. The dashed line corresponds to the root-mean squared (RMS) value of the non-Ekman ageostrophic velocities.</figcaption></figure>

The distributions of $|\zeta|/f$ from the smoothed fields of normalized vorticity are shown in Fig. 4a as a function of the half-power filter cutoff wavelength of the smoothing. The four solid lines correspond to the 80th, 90th, 95th and 99th-percentile points of the cumulative probability distribution of $|\zeta|/f$. The root mean squared (RMS) value of $|\zeta|/f$ is about 0.5 in the unsmoothed field and decreases to about 0.1 at large scales (see the dashed line in Fig. 4a) and coincides approximately with the 80th percentile point of the distribution at all scales. It would correspond to the 68th-percentile if the distribution were Gaussian. The distribution of Rossby numbers $|\zeta|/f$ in the unsmoothed fields is thus long-tailed toward large magnitudes with values exceeding 0.85 at about 5% of the grid points and exceeding 1.7 at about 1% of the grid points. The high values of $|\zeta|/f$ are concentrated geographically in the regions of strongest ocean velocity (see Fig. 3). Much of the small-scale variability in these regions is thus highly nonlinear and ageostrophic. The long-tailed nature of the distributions of $|\zeta|/f$ decreases rapidly with increasing scale. All of the percentile points and the RMS decrease very slowly for filter cutoff wavelengths longer than about 100 km.

From a close inspection of the vorticity maps in Fig. 3b, it can be seen that small-scale features with positive vorticity are more intense than their counterparts with negative vorticity. This is most easily seen from the enlargement in the bottom panel of Fig. 3b. This asymmetry of the vorticity distribution is quantified as a function of scale in Fig. 4b. At the full resolution of the model grid, the vorticity is skewed toward positive values. This skewness decreases with increasing scale, becoming very small for half-power filter cutoff wavelengths longer than about 50 km. The average values of $\zeta/f$ are very close to zero at all scales (see the dashed line in Fig. 4b).

### 3. Limitations of geostrophically computed velocity

The high incidence of large Rossby numbers $|\zeta|/f$ at small scales in Fig. 4a is indicative of limitations of the validity of the geostrophic approximation. This is investigated in this section from comparisons of the surface currents and vorticity from the total velocity field (as in Fig. 3) with the surface currents and vorticity computed geostrophically from the model SSH fields.<fnref n="1" /> The derivatives for the geostrophically computed velocity components $u_g = -g\bar{f}^{-1}\partial h/\partial y$ and $v_g = g\bar{f}^{-1}\partial h/\partial x$

were approximated using centered differences of sea surface height $h$ on the 0.5 km × 0.5 km model grid. The geostrophically computed vorticity $\zeta_g = \partial u_g/\partial x - \partial v_g/\partial y$ was then approximated using centered differences of the geostrophically computed velocity components. Maps of SSH, the magnitudes of geostrophically computed surface velocity and normalized geostrophically computed vorticity are shown in Fig. 5.

To the untrained eye, the existence of energetic submesoscale variability is not easily discerned in the SSH map in Fig. 5a. The preponderance of highly energetic submesoscale variability evident in the map of normalized geostrophically computed vorticity in Fig. 5c attests to the extreme spatial high-pass filtering effects of double differentiation of the SSH fields. Irrespective of the validity of the geostrophic approximation, the ability to estimate vorticity from SWOT measurements of SSH clearly depends critically on the magnitudes of small-scale errors in the SSH measurements. This is investigated in Section 6 from consideration of uncorrelated measurement errors alone. The added effects of sampling errors from the limited swath width and discrete overpass times of the SWOT satellite on maps of the spatially and temporally evolving velocity and vorticity fields computed geostrophically from space-time smoothed simulated SWOT data are considered in Section 8. In this section, we consider only error-free SSH fields and their spatial derivatives.

Qualitatively, the geostrophically computed surface current speed and the normalized geostrophically computed vorticity in Figs. 5b and c look very similar to the surface current speed and normalized vorticity in Figs. 3a and b that were computed from the total velocity. The vector differences between the total and geostrophically computed surface velocity (referred to hereafter as the ageostrophic velocity<fnref n="2" />), are shown in Fig. 6. From the unsmoothed velocity differences in Fig. 6a it can be seen that the ageostrophic velocity field on large scales is dominated by the expected wind-driven surface Ekman velocity associated with the climatological wind stress used to force the model. The ageostrophic velocity field smoothed using a Parzen smoother with a half-power filter cutoff wavelength of 150 km is shown in Fig. 6b. These large-scale ageostrophic velocity vectors are predominantly aligned approximately 52° to the right of the equatorward alongshore direction, which is at least qualitatively consistent with expectations from the equatorward alongshore winds during the early summertime period considered in this study (see the right panel of Fig. 2). The magnitudes of these large-scale ageostrophic currents are typically about 0.1 m s⁻<fnref n="1" />. For the purposes of the analysis in this section, we will define the 150-km smoothed ageostrophic velocity field in Fig. 6b to be the wind-driven Ekman velocity.

The small-scale features in the ageostrophic velocity field in Fig. 6a that are superimposed on the large-scale wind-driven Ekman velocity field in Fig. 6b coincide with the regions of energetic submesoscale variability noted above from Figs. 3 and 5. This becomes apparent after subtracting the 150-km smoothed ageostrophic velocity field in Fig. 6b from the unsmoothed ageostrophic velocity field in Fig. 6a. The magnitudes of the resulting “non-Ekman ageostrophic velocities” (Fig. 6c) often exceed 0.05 m s⁻<fnref n="1" /> and differences in excess of 0.15 m s⁻<fnref n="1" /> are common in regions of the CCS with energetic submesoscale variability.

The scale dependence of the non-Ekman ageostrophic velocities is shown in Fig. 7 for the CCCS region after isotropic smoothing using a Parzen smoother with selected half-power filter cutoff wavelengths ranging from 10 to 80 km. The non-Ekman ageostrophic features diminish in magnitude rather slowly with increasing scale because of the tight

<footnote n="1">¹ For estimation of surface velocity from SWOT data, it is important to distinguish the velocity computed from SSH data by the geostrophic equations from the truly geostrophic velocity that is valid only for small Rossby number. To clarify this subtle but pedagogically important point, we use the somewhat cumbersome terminology “geostrophically computed velocity” rather than “geostrophic velocity”.</footnote>

<footnote n="2">² The attribution “ageostrophic velocity” is not strictly correct. Some of the apparent velocity structures computed geostrophically from SSH are not representative of actual features in the velocity field. For example, the geostrophically computed velocity field associated with a long-crested internal gravity wave consists of opposing parallel velocity jets that straddle the wave crests and troughs. This is a glaring misrepresentation of the surface manifestation of internal wave velocities. This contamination of SWOT estimates of the surface velocity field is discussed in more detail at the end of this section.</footnote>
clustering of small-scale features. The residual non-Ekman ageostrophic velocities exceed 0.1 m s⁻¹ in many small-scale features, even after smoothing with a half-power filter cutoff wavelength of 40 km. Several patches of non-Ekman ageostrophic velocities of 0.03 m s⁻¹ still exist with the maximum smoothing of 80 km shown in the bottom right panel of Fig. 7. Bearing in mind the gradual rolloff of the filter transfer function of the Parzen smoother (see Appendix A), the magnitudes of the band-pass filtered non-Ekman ageostrophic velocities in Fig. 7 must be considered lower-bound estimates of the true values. The imperfections of the band-pass filtering become progressively more of an issue as the filter cutoff wavelength approaches the 150 km filter cutoff used to define the Ekman ageostrophic velocity in Fig. 6b from which the smoothed non-Ekman ageostrophic velocity magnitudes in Fig. 7 were derived by band-pass filtering.

The statistics of the scale dependence of the magnitudes of the non-Ekman ageostrophic velocities are summarized in Fig. 8 from the 80th, 90th, 95th and 99th-percentile points in the distributions of the magnitudes of the non-Ekman ageostrophic velocities over the full CCS model domain as functions of half-power filter cutoff wavelength. On all scales, the RMS value corresponds very closely to the 80th-percentile point in the distributions. The distributions of smoothed residual non-Ekman velocities are thus long-tailed toward large magnitudes, especially at small scales. In the unsmoothed fields, the RMS value of the magnitudes of the non-Ekman ageostrophic velocities is about 0.05 m s⁻¹ and values in excess of 0.1 m s⁻¹ occur at about 5% of the grid points over the full CCS model domain. It is evident from Fig. 6c that high non-Ekman ageostrophic velocities are highly inhomogeneous geographically, occurring over much higher percentages of the grid points within the regions of most energetic submesoscale variability.

As expected from the maps in Fig. 7, the distribution points of the magnitudes of residual non-Ekman ageostrophic velocities decrease rather slowly with increasing scale. Even after smoothing with a half-power filter cutoff wavelength of 50 km, for example, values in excess of 0.02 m s⁻¹ and 0.05 m s⁻¹ occur at, respectively, about 20% and 2% of the grid points over the full CCS model domain and at much higher percentages of the grid points within the localized regions of energetic submesoscale variability.

The submesoscale features in the ageostrophic velocity field in Figs. 6 and 7 are attributable mostly to contributions of cyclostrophic motion to the total velocity field in the CCS model used for this study. This can be shown by considering the force balance between geostrophy and centripetal acceleration in the steady-state momentum equation, referred to in meteorology as the gradient wind balance. The cyclostrophic momentum balance for axially symmetric flow in cylindrical coordinates is

$$-fV - \frac{V^2}{R} = -\frac{1}{\rho} \frac{\partial P}{\partial r}, \quad (1)$$

where r is radial distance, R is the radius to the center of curvature, ρ is the water density, P is pressure, and V is the angular velocity, which is defined to be positive for counterclockwise rotation and negative for clockwise rotation. For the northern hemisphere considered here, the angular velocity is thus positive in cyclones and negative in anticyclones.

The momentum equation (1) can be rearranged into the form

$$V_g = V + \frac{V^2}{fR}, \quad (2a)$$

where $V_g = (\rho f)^{-1} \partial P / \partial r$ is the geostrophically computed angular velocity. Since the second term on the right side of (2a) is always positive, it is evident that $V_g > V$ for any nonzero cyclostrophic velocity V. The angular velocity $V_g$ computed geostrophically is thus an overestimate of the positive cyclostrophic velocity in cyclones and an underestimate of the magnitude of the negative cyclostrophic velocity in anticyclones. As a consequence, the vorticity computed geostrophically is overestimated in cyclones and underestimated in anticyclones.

The positive bias of geostrophically computed vorticity can be shown mathematically by rewriting (2a) as

$$V_g = (1 + \epsilon_R) V, \quad (2b)$$

where

$$\epsilon_R = \frac{V}{fR} \quad (3)$$

is the Rossby number of the rotational flow. The average relative vorticity within the circular area of the assumed axially symmetric flow with radius R can be obtained from Green's Theorem, which gives

$$\bar{\xi} = \frac{\oint_C \bar{v} \cdot d\bar{l}}{\int_{A_C} dA} = \frac{V 2\pi R}{\pi R^2} = \frac{2V}{R}. \quad (4)$$

It is thus apparent that (3) can be expressed alternatively as

$$\epsilon_R = \frac{1}{2} \frac{\bar{\xi}}{f}. \quad (5)$$

Multiplication of (2b) by 2/R and substitution of (4) for 2V/R gives

$$\bar{\xi}_g = (1 + \epsilon_R) \bar{\xi} = \bar{\xi} + \frac{1}{2} \frac{\bar{\xi}^2}{f}, \quad (6)$$

where $\bar{\xi}_g = 2V_g/R$ is the average vorticity computed geostrophically within the circular area of axially symmetric flow with radius R. Since the second term on the right side of (6) is positive, the geostrophically computed vorticity $\bar{\xi}_g$ is larger than the true vorticity $\bar{\xi}$, thus confirming that the geostrophically computed vorticity overestimates the positive vorticity in cyclones and underestimates the magnitude of negative vorticity in anticyclones.

After division of both sides by f, (6) can be rearranged into the form

$$\frac{\bar{\xi}_g}{f} - \frac{\bar{\xi}}{f} = \frac{1}{2} \left( \frac{\bar{\xi}}{f} \right)^2, \quad (7)$$

For cyclostrophic flow, the differences between the normalized geostrophically computed vorticity $\bar{\xi}_g/f$ and the normalized vorticity $\bar{\xi}/f$ are thus related quadratically to the latter. The validity of the cyclostrophic relation (7) is shown in the left panels of Fig. 9. The top left panel is a map of $(\bar{\xi}_g - \bar{\xi})/f$ in the CCCS region computed from the bottom panels of Figs. 5c and 3b after a small amount of smoothing with a half-power filter cutoff wavelength of 20 km. The bottom left panel shows binned averages of the normalized differences in the top left panel as a function of the normalized vorticity $\bar{\xi}/f$. The curve in the bottom left panel corresponds to the quadratic relation (7). The quadratic fit is quite good. The discrepancies of the binned averages from the theoretical quadratic relation (7) are presumably attributable to deviations of the flow field from the axially symmetric flow assumed in the derivation of (7).

It is noteworthy that the features in the top left panel of Fig. 9 that are most compact and have the largest magnitudes are all positive. This is also evident from the asymmetry of the distribution of adequately sampled bins in the binned scatter plot in the bottom left panel of Fig. 9. The most energetic ageostrophic features in the vorticity field are thus cyclonic and the positive vorticities $\bar{\xi}$ of these features are overestimated by the geostrophically computed vorticities $\bar{\xi}_g$. The lack of compact anticyclonic features is likely because small-scale anticyclones are susceptible to inertial instability (e.g., Rayleigh, 1916; Flierl, 1988; Kloosterziel and van Heijst, 1991; Kloosterziel et al., 2007). The most intense small-scale features in the ageostrophic vorticity field are therefore associated almost exclusively with submesoscale cyclones.

The relevance of cyclostrophic flow can be further investigated from the ratios $\bar{\xi}_g/\bar{\xi}$. For the axially symmetric flow assumed in the derivation above, the ratio obtained from (6) is

$$\frac{\bar{\xi}_g}{\bar{\xi}} = 1 + \frac{1}{2} \frac{\bar{\xi}}{f}. \quad (8)$$
<figure><img src="image_11.png" /><figcaption>**Fig. 9.** Maps of the differences (left) and ratios (right) of the normalized geostrophically computed vorticity $\zeta_g/f$ in Fig. 5c and the normalized total vorticity $\zeta/f$ in Fig. 3b after smoothing with a half-power filter cutoff wavelength of 20 km. The bottom panels show binned averages of the gridded values in the maps in the top panels as functions of $\zeta/f$. The vertical bar on each binned average represents the ±1 standard deviation of the data values within the bin and the smooth lines correspond to the theoretical solutions derived in the text for cyclostrophic motion.</figcaption></figure>

The validity of this cyclostrophic relation is shown in the right panels of Fig. 9. The top right panel is a map of $\zeta_g/\zeta$ in the CCCS region computed from the bottom panels of Figs. 5c and 3b after a small amount of smoothing with a half-power filter cutoff wavelength of 20 km. The bottom right panel shows binned averages of the ratios in the top right panel as a function of the normalized vorticity $\zeta/f$. Note again the predominance of positive values of $\zeta$. The line in the bottom right panel corresponds to the linear relation (8). The discrepancies of the binned averages from the theoretical linear relation (8) are again presumably attributable to deviations of the flow field from the axially symmetric flow assumed in the derivation of (8). The differences are especially evident in the regions of large-scale flow where $|\zeta|/f$ is small.

The conclusion of the preceding analysis is that the errors from the effects of cyclostrophic motion on surface velocities computed geostrophically from SWOT data may often exceed 0.1 m s⁻¹ in the regions of high velocity where small-scale vorticity magnitudes are strongest. These non-Ekman ageostrophic velocities can mostly be reduced to less than 0.05 m s⁻¹ by smoothing with a half-power filter cutoff wavelength of 50 km or more. The small-scale cyclostrophic motions that are responsible for these residual errors in the geostrophic approximation result in geostrophically computed velocities that overestimate the positive vorticity of cyclonic features and underestimate the magnitudes of the negative vorticity of anticyclonic features. The former occurs more commonly than the latter because of the tendency for anticyclonic features to become unstable.

In addition to errors from the existence of small-scale cyclostrophic motions, velocity estimates computed geostrophically from SWOT measurements of SSH will be contaminated by the SSH signatures of ageostrophic internal gravity waves. Hints of such features are evident

from close inspection of Fig. 5c in the form of subtle wavy striations in the geostrophically computed vorticity. These are most clearly seen in the central northern area of weak submesoscale variability in the bottom panel. As discussed in Section 2, internal gravity waves are underestimated in the CCS model used in this study. Artifacts from internal waves are therefore likely to be worse in actual geostrophically computed SWOT estimates of velocity than is suggested from Fig. 5c.

Present understanding of the sea-surface height signatures of the oceanic internal gravity wave field is limited, and consequently the extent of this contamination of estimates of surface velocity and vorticity computed geostrophically from SWOT data is not addressed systematically here. Its potential importance can be illustrated, however, through comparison of the ageostrophic velocity fields from two high-resolution ROMS simulations of the Gulf Stream region off the southeastern seaboard of the U.S., both with a grid spacing of 0.75 km × 0.75 km. Similar to the CCS model used in this study, one of the North Atlantic simulations (referred to here as the LF simulation) was forced at low frequencies with monthly smoothed winds and does not include ocean tides (Gula et al., 2015). The other (referred to here as the HF simulation) was forced at high frequencies with hourly winds and includes boundary forcing by ten ocean tidal constituents. Although the region of the North Atlantic model domain is very different dynamically from the CCS region considered in this study, the availability of the HF and LF simulations provides useful insight into the contamination of SWOT data that can be expected from internal waves forced by high-frequency winds and ocean tides.

Snapshots of the magnitudes of the total surface velocities, the geostrophically computed surface velocities and their differences (the “ageostrophic velocities”) over the full model domain are shown in
<figure><img src="image_12.png" /><figcaption>**Fig. 10.** Snapshot maps of the magnitudes of the total velocity, the geostrophically computed velocity and the ageostrophic velocity (top to bottom) for the HF and LF simulations (left and right columns, respectively) of a region of the North Atlantic Ocean off the southeastern seaboard of the U.S. The grid resolution was 0.75 km × 0.75 km in both simulations. The HF simulation was forced with high-frequency (hourly) winds and tides. The LF simulation was forced with monthly winds and did not include tides. The ribbons of high velocity in the western and northern regions of each map are the Gulf Stream as it flows northward off the Florida coast and then turns northeastward along the continental slope to Cape Hatteras at the northern corner of the model domain. The x and y axis labels are longitude in degrees east and latitude in degrees north and the box in each panel is the area over which the non-Ekman ageostrophic velocities were computed for Fig. 11. Note that the left half of the color bar is not used in the bottom panels.</figcaption></figure>

**Fig. 10.** As in the CCS model used for this study (Figs. 3a and 5b), the geostrophically computed velocity is qualitatively very similar to the total surface velocity in each of the two North Atlantic simulations. Moreover, the total and geostrophically computed velocity fields are qualitatively similar between the two North Atlantic simulations. But the ageostrophic velocity fields are dramatically different in the HF and LF simulations. Features resembling the cyclostrophic motions discussed above from the CCS model are evident in the ageostrophic

velocity field from the LF model, especially in the immediate vicinity of the Gulf Stream. Such features are swamped in the HF model by striated internal gravity wave structures. The geostrophically computed velocities associated with these internal waves are a gross misrepresentation of the actual surface velocities associated with internal waves. We nonetheless refer to these striated features as “ageostrophic velocities” since they would be misinterpreted as velocity structures in geostrophically computed SWOT estimates of the velocity field.
<figure><img src="image_13.png" /><figcaption>**Fig. 11.** Maps showing the scale dependence of the magnitudes of smoothed non-Ekman ageostrophic velocities (defined as in Fig. 6c) from the subregions of the two North Atlantic simulations indicated by the boxes in Fig. 10. The left and right columns correspond to the results for the HF and LF models, respectively, after 2-dimensional isotropic smoothing with half-power filter cutoff wavelengths of 20, 40, 60 and 80 km (top to bottom). The x and y axis labels are distance in kilometers. Note that the left half of the color bar is not used in this figure.</figcaption></figure>

To assess the effects of internal gravity waves on SWOT estimates of the surface velocity fields, the “non-Ekman ageostrophic velocity field” was computed from each North Atlantic simulation in the same manner as in Fig. 6c, i.e., as the differences of each of the total ageostrophic

velocity fields from their respective 150-km smoothed ageostrophic field (not shown here). The residual velocity fields were then smoothed with various half-power filter cutoff wavelengths as in Fig. 7. The magnitudes of the resulting smoothed non-Ekman ageostrophic velocity
fields are shown in Fig. 11 for the region southeast of the Gulf Stream indicated by the boxes in Fig. 10. Features suggestive of internal gravity waves can be faintly seen in the LF simulation with 20-km smoothing but are not readily apparent with higher smoothing. In contrast, internal waves are highly energetic in the HF simulation and decrease in magnitude rather slowly with increased spatial smoothing. Fig. 11 suggests that significant contamination of the geostrophically computed velocity field by the SSH signatures of internal waves persist in some regions even after smoothing with a half-power filter cutoff wavelength of 80 km.

In principle, the internal wave contamination that is evident in Figs. 10 and 11 can be reduced by time averaging as in the simulations of geostrophically computed SWOT estimates of space–time smoothed velocity and vorticity in Section 8. In practice, the interleaved sampling pattern during each repeat of the 21-day orbit of SWOT (see Figs. 21–23) restricts the geographical distribution of locations with repeat sampling, thus limiting the benefits of time averaging to mitigate contamination by internal gravity waves. Qiu et al. (2018) note that it may be possible to remove some of the internal wave contamination based on model estimates of the stationary part of diurnal and semidiurnal tidal signals. Contamination by nonstationary internal gravity waves is much more problematic.

A thorough analysis of the effects of internal gravity waves on interpretation of geostrophically computed SWOT estimates of space–time smoothed velocity and vorticity is beyond the scope of this study. For present purposes, the contrasting velocity fields from the two North Atlantic simulations in Figs. 10 and 11 serve as a cautionary warning that the assessments of the resolution capabilities for SWOT estimates of velocity and vorticity in Sections 6, 8 and 9 may be somewhat optimistic since our simulations are based on a CCS model with unrealistically weak internal gravity waves. A rigorous analysis of the effects of contamination of geostrophically computed velocity estimates by internal gravity waves is needed to fully understand the limitations of future use and interpretation of SWOT data.

It should be noted that other effects of synoptic-scale wind forcing besides internal gravity waves may also not be adequately represented in the CCS model used in this study since it was forced by seasonally varying winds. High-frequency wind forcing can also affect mesoscale and submesoscale variability. For this and other reasons, the resolution capabilities of actual SWOT and WaCM data are likely to be somewhat worse than the estimates presented here.

## 4. The error characteristics of measured and derived variables from SWOT and WaCM data

For our investigation of the resolution capabilities of SWOT and WaCM estimates of surface velocity and vorticity, the measurement characteristics that are of interest are the footprint sizes of the data that will be distributed to general users (referred to here as pre-processed data), the standard deviations of the uncorrelated measurement errors for the specified footprint size, and the space–time sampling patterns that are imposed by the measurement swath width and the orbit configuration for each instrument. The effects of footprint size and uncorrelated measurement errors on SWOT and WaCM estimates of velocity and vorticity are summarized in this section. Discussion of the equally important consideration of sampling patterns on the ability to map the velocity and vorticity fields (especially an issue for SWOT because of its much narrower swath width) is deferred to Section 7.

### 4.1. SWOT errors

The measurement characteristics for the SWOT altimeter are documented in a suite of project reports published by the Jet Propulsion Laboratory and available online at https://swot.jpl.nasa.gov/documents.htm. The most germane of these reports are the Science Requirements Document (Rodríguez and Callahan, 2016), the Onboard Processing

and Algorithm Theoretical Basis Document (Peral, 2016), and the Mission Performance and Error Budget Document (Esteban Fernandez, 2017). While the versions of these documents that are listed in the references may undergo revisions between now and the planned launch date in 2021, no major changes are expected in the instrumental measurement errors and orbit configuration that are crucial to the simulations of SWOT data used in this study.

The baseline science requirements for SWOT are for SSH measurements with a footprint size of 1 km on a 1 km × 1 km grid across the measurement swath. SWOT will also include a nadir altimeter, but that is not considered in this study because it has a very different footprint size and measurement accuracy compared with the Ka-band Radar Interferometer (KaRIn). KaRIn will measure SSH with unprecedented resolution and spatial coverage across two parallel measurement swaths with 50-km widths separated by a 20-km gap centered on the satellite ground track (see Section 7).

In addition to uncorrelated errors from instrument noise, the KaRIn estimates of SSH will be contaminated by spatially correlated (long-wavelength) errors from orbit errors and various environmental effects, including significant wave height (SWH) and the effects of dry gases, water vapor and ionospheric free electrons on atmospheric refraction. Of particular note, the validity of the so-called inverted barometer correction of 1 cm hPa⁻¹ for sea level pressure effects on SSH is unknown on the small scales measurable by KaRIN. The uncertainties of the sea level pressure fields from the atmospheric model that will be used as a basis for the inverted barometer correction are also unknown on these small scales.

Consideration of the complete error budget for SWOT is beyond the scope of this study. Readers are referred to Rodríguez and Callahan (2016) for a discussion of the present understanding of the various aspects of SWOT measurement errors. Our focus is on SWOT estimates of small-scale surface velocity and vorticity variability, both of which are spatially high-pass filtered versions of the SSH field (see Fig. 5) since they are computed from spatial derivatives of SSH. With the possible exception of the inverted barometer correction mentioned above, most of the spatially correlated errors have relatively large scale and are thus expected to be of secondary importance. We therefore consider only the effects of uncorrelated measurement errors in this study.

In its low-resolution mode over the ocean, the raw radar measurements by the KaRIn instrument will have a footprint size of about 100 m. To reduce the measurement errors, while at the same time reducing the unnecessarily large data volume over the ocean, the present official plan as summarized by Peral (2016) and Esteban Fernandez (2017) is to smooth the raw measurements of SSH in an onboard processor to achieve the SWOT science requirements for a footprint size of 1 km for ocean observations.<fnref n="3" /> It is shown in Appendix B.1 that this footprint size can be achieved by smoothing the raw measurements using a 2-dimensional Parzen smoother with a half-power filter cutoff wavelength of 2 km. It is further shown in Appendix B.1 that pre-processed SWOT estimates of SSH obtained in this manner are spatially uncorrelated on a 1 km × 1 km grid.

For the simulations of SWOT data in this study, a critically important characteristic of the pre-processed SWOT estimates of SSH is the

<footnote n="3">³ 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. This higher resolution increases the standard deviation of the uncorrelated measurement errors in the onboard estimates of SSH by a factor of two compared with a footprint size of 1 km. It will be shown in Section 6 that even the science requirement of 1-km footprint size used for the simulations in this study exceeds the signal resolution capability by more than an order of magnitude for in-swath SWOT estimates of surface velocity and vorticity computed geostrophically from SSH. The results of the additional smoothing in ground-based post-processing that will be needed to achieve an adequate signal-to-noise ratio would therefore be essentially the same for either footprint size.</footnote>
standard deviation of the uncorrelated instrumental errors. As discussed in detail in Appendix F, it is not possible to determine this unambiguously from the SWOT documentation. In part, this is because the actual science requirement is specified in terms of the wavenumber spectrum of SSH after 2-dimensional smoothing in ground-based post-processing, rather than in terms of the measurement accuracy itself in units of SSH (see Appendix F). On the few occasions where a standard deviation of the uncorrelated measurement errors is stated, inconsistent values are given (see Appendix F), perhaps because the stated value is for an evolving “present best estimate” of what will actually be achieved from SWOT on-orbit, rather than for the baseline science requirement that is considered in this study. The former is slightly better than the science requirement and may continue to improve somewhat as the techniques for processing the KaRIn data are refined between now and the launch date.

The wavenumber spectral specification of the measurement errors in the SWOT science requirements is “reverse engineered” in Appendix F to determine the corresponding standard deviation of the uncorrelated errors of the pre-processed SWOT estimates of SSH. The derived value for the swath-averaged science requirement for conditions of 2-m SWH and a footprint size of 1 km is shown to be $\sigma_h = 2.74$ cm.

The smallest value cited in the SWOT documentation for the swath-averaged standard deviation of uncorrelated errors is $\sigma_h = 2.4$ cm for 2-m SWH, which is somewhat better than the value of 2.74 cm derived in Appendix F. The dependencies of the present projected estimate of SWOT measurement errors on swath location and SWH are shown in Fig. F.1 in Appendix F. Our assumed swath-averaged error standard deviation of $\sigma_h = 2.74$ cm appears to be more representative of the projected estimate for 4-m SWH.

The difference between 2.74 cm and 2.4 cm is a minor distinction for this investigation. A more significant issue is that the analysis presented in this study does not simulate the cross-track variation of the SWOT measurement errors. As shown in Fig. F.1, the measurement errors increases toward both edges of the measurement swath, resulting in cross-track variations that are much larger than the difference between 2.74 cm and 2.4 cm. For 2-m SWH, for example, the estimates of SSH errors in Fig. F.1 range from 1.9 cm at the center of the swath to 3.0 cm at the inner edge of the swath and more than 4 cm approaching the outer edge of the swath. It will be seen in Sections 6, 8 and 9 that geostrophically computed SWOT estimates of velocity and vorticity must be smoothed considerably to achieve adequate signal-to-noise ratio. This smoothing leads to edge effects that would be exacerbated by the larger measurement errors near the swath edges.

As noted in the introduction, the error analysis in this study could have been carried out using the simulator software available from the SWOT Project Office (Gaultier et al., 2017). Analogous simulator software is not yet available for WaCM (Rodríguez, 2018). For consistency in our analysis of the effects of SWOT and WaCM measurement errors, we have therefore chosen to simulate the error standard deviation for simulated SWOT data as spatially constant with a swath-averaged value of $\sigma_h = 2.74$ cm. Because our simulations do not account for the higher measurement errors near the edges of the measurement swaths, the analysis presented here likely underestimates the overall effects of measurement errors on space–time smoothed fields of velocity and vorticity constructed geostrophically from SWOT measurements of SSH. The results presented in Sections 6, 8 and 9 should therefore be considered optimistic assessments of the resolution capabilities of actual SWOT data.

For SWOT, the velocity and vorticity fields that are the focus of this investigation must be estimated geostrophically, which requires differentiation of SSH for velocity and double differentiation of SSH for vorticity. The spatial derivatives are estimated in this study by centered differences of the discrete SSH values. The effects of the uncorrelated errors of SWOT measurements of SSH on geostrophically computed velocity and vorticity are derived in Appendix G.1 using propagation-of-error analysis. The results are summarized in the first column of

<caption>Table 1</caption>

The standard deviations of errors of SWOT estimates of sea surface height $h$, the geostrophically computed cross-shore and alongshore velocity components $u_g = -g f^{-1} \partial h / \partial y$ and $v_g = g f^{-1} \partial h / \partial x$, the magnitude $(u_g^2 + v_g^2)^{1/2}$ of the geostrophically computed velocity and the geostrophically computed vorticity $\zeta_g = \partial v_g / \partial x - \partial u_g / \partial y$ normalized by the Coriolis parameter $f$. The calculations are all based on the value of $f = 8.75 \times 10^{-5} \text{ s}^{-1}$ at the central latitude 37°N of the California Current model domain shown in Fig. 2. The results for the pre-processed SWOT estimates of SSH with 1-km footprint on a 1 km × 1 km grid and the science requirement of $\sigma_h = 2.74$ cm for the standard deviation of the uncorrelated errors of the SSH measurements are listed in the first column. The reduction of error that can be achieved by smoothing with half-power filter cutoff wavelengths of $\lambda_c = 10$ km and 50 km are listed in the second and third columns. The error estimates in the first and second columns are derived in Appendices G.1 and G.3 by propagation-of-error analysis. The error estimates in the third column were obtained by integrating the analytical expressions for the error spectra derived in Appendix 1.2 with $\lambda_c = 50$ km.

<table><thead><tr><th>Variable</th><th>SWOT with<br>1-km footprint</th><th>SWOT with<br>5-km footprint<br>(10-km smoothing)</th><th>SWOT with<br>50-km smoothing</th></tr></thead><tbody><tr><td>SSH</td><td>2.74 cm</td><td>0.55 cm</td><td>0.11 cm</td></tr><tr><td>$u_g$, $v_g$</td><td>2.17 m s<sup>-1</sup></td><td>0.26 m s<sup>-1</sup></td><td>0.013 m s<sup>-1</sup></td></tr><tr><td>Speed</td><td>3.07 m s<sup>-1</sup></td><td>0.37 m s<sup>-1</sup></td><td>0.018 m s<sup>-1</sup></td></tr><tr><td>$\zeta_g / f$</td><td>39.1</td><td>3.2</td><td>0.045</td></tr></tbody></table>

Table 1, which shows that the $\sigma_h = 2.74$-cm standard deviation of SSH measurement errors for a footprint size of 1 km results in a large standard deviation of $2.17 \text{ m s}^{-1}$ for the errors of SWOT estimates of each geostrophically computed velocity component at the central latitude 37°N of the CCS model domain where the Coriolis parameter is $f = 8.75 \times 10^{-5} \text{ s}^{-1}$. This error increases by about 11% at the 32.5°N southern corner of the CCS model domain where $f = 7.81 \times 10^{-5} \text{ s}^{-1}$ and decreases by about 11% at the 42°N northern corner of the model domain where $f = 9.73 \times 10^{-5} \text{ s}^{-1}$. The $\sigma_h = 2.74$-cm SSH measurement errors result in an extraordinarily large standard deviation of 39.1 f for the errors of geostrophically computed SWOT estimates of vorticity at 37°N.

Because of these large errors, SWOT estimates of velocity and vorticity computed geostrophically from the unsmoothed pre-processed data are likely of little value for most oceanographic applications. The errors listed in the first column of Table 1 are much larger than the errors summarized for WaCM estimates of velocity and vorticity in Section 4.2, but the footprint size of 5 km for pre-processed WaCM estimates of surface velocity is a factor-of-5 coarser than the 1-km footprint size for SWOT. The large errors of SWOT estimates of velocity and vorticity obtained from the pre-processed SWOT data can be reduced by smoothing commensurately with the half-power filter cutoff wavelength of 10-km that corresponds to the 5-km footprint size of pre-processed WaCM estimates of surface velocity (see Appendix B.2).

The effects of smoothing the pre-processed SWOT estimates of SSH isotropically with a half-power filter cutoff wavelength of 10 km in ground-based post-processing, i.e., the same smoothing applied in pre-processing of WaCM estimates of velocity, are derived in Appendix G.3 using propagation-of-error analysis. The results are summarized in the second column of Table 1. The standard deviations of errors of geostrophically computed SWOT estimates of velocity and vorticity are reduced by factors of about 8 and 12, respectively. The resulting uncertainties are comparable to the uncertainties of velocity and vorticity derived below in Section 4.2 based on pre-processed WaCM data with the same footprint size of 5 km (see Table 2).

Eqs. (G.2), (G.5), (G.8), (G.17a) and (G.20) in Appendices G.1 and G.3 from which the error estimates in the first two columns of Table 1 were computed are all expressed in terms of the variance $\sigma_h^2$ of the errors of the pre-processed SWOT estimates of SSH for a footprint size of 1 km. It is therefore straightforward to recompute the standard deviations of the errors of geostrophically computed velocity components and vorticity based on any specified value of $\sigma_h^2$.
<caption>**Table 2**</caption>

The standard deviations of errors of WaCM estimates the cross-shore and alongshore velocity components *u* and *v*, the velocity magnitude (*u*² + *v*²)¹/² and the vorticity *ζ* = *∂v*/∂x - *∂u*/∂y normalized by the Coriolis parameter *f* at the central latitude 37°N of the California Current model domain shown in Fig. 2. The results for the pre-processed WaCM estimates of the velocity components with 5-km footprint size on a 5 km × 5 km grid and the baseline standard deviation of *σ<sub>spd</sub>* = 0.50 m s⁻¹ for uncorrelated speed errors, which corresponds to *σ<sub>u,v</sub>* = 0.354 m s⁻¹ for the uncorrelated measurement errors of each orthogonal component of velocity, are listed in the first column. The results for the same footprint size of 5 km oversampled on a 1 km × 1 km grid are listed in the second column. The reduction of error that can be achieved by smoothing with a half-power filter cutoff wavelength of *λ*<sub>c</sub> = 50 km are listed in the third column. The error estimates in the first and second columns are derived in Appendix G.2 by propagation-of-error analysis. The error estimates in the third column were obtained by integrating the analytical expressions for the error spectra derived in Appendix I.4 with *λ*<sub>c</sub> = 50 km.

<table><thead><tr><th>Variable</th><th>WaCM with<br>5-km footprint<br>and 5-km grid</th><th>WaCM with<br>5-km footprint<br>and 1-km grid</th><th>WaCM with<br>50-km smoothing<br>and 1-km grid</th></tr></thead><tbody><tr><td><i>u, v</i></td><td>0.35 m s<sup>-1</sup></td><td>0.35 m s<sup>-1</sup></td><td>0.071 m s<sup>-1</sup></td></tr><tr><td>Speed</td><td>0.50 m s<sup>-1</sup></td><td>0.50 m s<sup>-1</sup></td><td>0.10 m s<sup>-1</sup></td></tr><tr><td>ζ/f</td><td>0.81</td><td>2.4</td><td>0.12</td></tr></tbody></table>

## 4.2. WaCM errors

As WaCM is not yet a proposed NASA mission, there is no formal documentation of the details of the Doppler measurements of surface velocity. The technical aspects of WaCM are summarized by Rodríguez (2018). A Doppler radar measures the radial component of velocity parallel to the antenna pointing angle. Estimation of vector velocity requires measurements from multiple look angles. This is achieved using a spinning pencil-beam antenna that measures a given location on the sea surface from several different look angles over a short period of time as the satellite moves along its orbit.

Issues with the geometrical transformations of pooled measurements of radial velocity from multiple look angles to estimate two orthogonal components of velocity are not addressed in our simulations of WaCM data. We have assumed that the two orthogonal velocity components have already been estimated in onboard and ground-based pre-processing of the measurements pooled over 5 km × 5 km areas and that the uncorrelated errors of surface current speed in these pooled estimates have a standard deviation of *σ<sub>spd</sub>* = 0.50 m s⁻¹ in the baseline design for WaCM. In reality, the errors will differ for the two velocity components (see Rodríguez, 2018). Because of limited azimuthal diversity of the Doppler measurements of radial velocity near the swath edges, the errors of the cross-track velocity component increase toward the inner edges of the two parallel measurement swaths. Likewise, the errors of the along-track component increase toward the outer edges of the measurement swaths.

It is shown in Appendix B.2 that an effective footprint size of 5 km can be achieved by smoothing high-resolution data using a 2-dimensional Parzen smoother with a half-power filter cutoff wavelength of 10 km. The estimates of surface velocity that we refer to here as pre-processed WaCM data were therefore simulated from the CCS model output based on isotropic 2-dimensional smoothing of noisy velocity components on the full 0.5 km × 0.5 km resolution of the model grid with a half-power filter cutoff wavelength of 10 km.

Since velocity will be estimated directly by WaCM, the only derived quantity is the vorticity, which is computed from spatial derivatives of the velocity components. We assume throughout this study that the speed measurement uncertainties are equally partitioned between orthogonal velocity components and that the errors of each velocity component are uncorrelated with each other. The baseline standard deviation of *σ<sub>spd</sub>* = 0.50 m s⁻¹ for the speed measurement errors therefore corresponds to a standard deviation of *σ<sub>u,v</sub>* = 0.354 m s⁻¹ for the uncorrelated errors of pre-processed WaCM estimates for each orthogonal velocity component *u* and *v*.

For a footprint size of 5 km, pre-processed WaCM estimates of surface velocity are spatially uncorrelated on a 5 km × 5 km grid (see Appendix B.2). The effects of the uncorrelated velocity component errors on WaCM estimates of vorticity computed on a 5 km × 5 km grid are derived in Appendix G.2 using propagation-of-error analysis. The results are summarized in the first column of Table 2, which shows that the standard deviation of *σ<sub>u,v</sub>* = 0.354 m s⁻¹ for the velocity component errors results in a standard deviation of 0.807*f* for the errors of WaCM estimates of vorticity at the central latitude 37°N of the CCS model domain.

It is advantageous to oversample the WaCM estimates of surface velocity on a 1 km × 1 km grid in order to retain more of the high-wavenumber variability of the vorticity signal in the centered differences used here to estimate the derivatives in the definition of vorticity (see Appendix H). For the simulations in this study, oversampling the WaCM data on a 1 km × 1 km grid has the additional practical advantage of giving a spatial gridding that is equivalent to the 1 km × 1 km gridding assumed here for the pre-processed SWOT data. The footprint size from the filtering in the pre-processing of WaCM data is still 5 km. The standard deviation of the errors of WaCM estimates of each velocity component is therefore still *σ<sub>u,v</sub>* = 0.354 m s⁻¹, but the velocity component errors on a 1 km × 1 km grid are spatially correlated.

The effects of correlation of the measurement errors on the vorticity estimated from WaCM data on a 1 km × 1 km grid are incorporated in the propagation-of-error analysis in Appendix G.2. The results are summarized in the second column of Table 2. The oversampling increases the standard deviation of the vorticity errors by about a factor of 3 to a value of about 2.4*f* at 37°N. While this increase of the errors is undesirable, the standard deviation of the vorticity signal also increases in the centered difference estimates of the derivatives on the oversampled grid. It is shown in Appendix H that this retains more of the short-wavelength vorticity variability where much of the signal variance lies (see Figs. 3b and 4). The net effect of oversampling the WaCM data on a 1 km × 1 km grid is to improve the signal-to-noise ratio, even though the noise standard deviation is higher on the 1 km × 1 km grid.

The standard deviation of the errors of WaCM estimates of velocity with 5-km footprint (i.e., with smoothing with a half-power filter cutoff wavelength of 10 km) in the first two columns of Table 2 is somewhat larger than the standard deviation of geostrophically computed SWOT estimates of velocity with 10-km smoothing listed in the second column of Table 1. On the other hand, the errors of WaCM estimates of vorticity on a 1 km × 1 km grid are somewhat smaller than the errors of geostrophically computed SWOT estimates of vorticity with 10-km smoothing to achieve a footprint size of 5 km (compare the bottom elements in the second columns of Tables 1 and 2). The apparent contradiction of higher velocity errors but lower vorticity errors for WaCM is attributable to the different wavenumber spectral characteristics of the errors of SWOT and WaCM estimates of the velocity components and vorticity (see Section 4.4) that arise from the different orders of differentiation required to obtain the estimates.

Eqs. (G.10), (G.12) and (G.15) in Appendix G.2 from which the error estimates in the first two columns of Table 2 were computed are all expressed in terms of the variances *σ*<sub>u</sub><sup>2</sup> and *σ*<sub>v</sub><sup>2</sup> of the errors of the pre-processed WaCM estimates of the velocity components. It is therefore straightforward to recompute the standard deviations of the vorticity errors based on any specified values of *σ*<sub>u</sub><sup>2</sup> and *σ*<sub>v</sub><sup>2</sup>.

## 4.3. Error reductions from smoothing in ground-based post-processing

The errors of SWOT and WaCM estimates of velocity and vorticity listed in the first two columns of Tables 1 and 2 are too large for the data to be useful for most applications, even with the 10-km smoothing to achieve a footprint size of 5 km. SWOT and WaCM data will have to be further smoothed in ground-based post-processing to
reduce the effects of measurement errors. The reductions of the errors of the measured and derived variables that are achieved with this additional smoothing could be derived analytically, albeit tediously, by the propagation-of-error analysis procedure used in Appendix G.3 for the case of smoothing of the SWOT data with a half-power filter cutoff wavelength of 10 km. For SWOT, this would require determination of the autocorrelations of smoothed SSH errors at the three spatial lags of $2\lambda_x$, $2\sqrt{2}\lambda_x$ and $4\lambda_x$ that appear in Eqs. (G.17a) and (G.21) in Appendix G for each choice of half-power filter cutoff wavelength $\lambda_c$. A similar propagation-of-error analysis could be carried out for WaCM estimates of smoothed velocity and vorticity.

The variances of the residual errors after isotropic 2-dimensional smoothing with a given $\lambda_c$ are much more easily determined for each variable of interest from the analytical expressions for the wavenumber spectra of the measured and derived variables that are derived for SWOT and WaCM in Appendices I.2 and I.4, respectively.

The procedure for determining the standard deviation of residual errors from the wavenumber spectrum after smoothing in ground-based post-processing is based on Parseval's Theorem, which relates the variance of a variable to the integral of the power spectral density of the variable. For finite record length, this integral becomes the discrete sum (D.1) in Appendix D. The residual error variance after isotropic 2-dimensional smoothing was obtained by integrating the analytical expression for the wavenumber spectra of the residual errors for half-power filter cutoff wavelengths $\lambda_c$ ranging from 10 km to 150 km. Examples of these wavenumber spectra are presented and discussed below in Section 4.4 for $\lambda_c = 20$ km, 50 km and 80 km. The standard deviation of the residual errors is the square root of the residual error variance computed from the integrated spectrum for each choice of $\lambda_c$.

The residual error standard deviations obtained in this manner are shown in Fig. 12 for SSH from SWOT (panel a) and for velocity and vorticity (panels b and c, respectively) estimated from both SWOT and WaCM data on $1 \text{ km} \times 1 \text{ km}$ grids. These estimates assume complete instantaneous sampling of the full CCS model domain, i.e., without the sampling errors discussed in Sections 7–10. The errors initially drop very quickly with increased smoothing and then more gradually as the smoothing is further increased. In the case of SSH, the residual errors in Fig. 12a decrease as $\lambda_c^{-1}$. The velocity errors in Fig. 12b decrease as $\lambda_c^{-2}$ and $\lambda_c^{-1}$ for SWOT and WaCM, respectively. The corresponding vorticity errors in Fig. 12c decrease as $\lambda_c^{-3}$ and $\lambda_c^{-2}$. The more rapid decreases for SWOT than for WaCM are because of attenuation of the variance of the geostrophically computed velocity component errors from the band-pass filtering operation of the response function for centered difference estimates of the geostrophic derivatives (see Appendices H and I and the discussion in Section 4.4 of the wavenumber spectral characteristics of the residual errors of the smoothed variables).

To provide some quantitative numbers for the error reductions achieved with isotropic 2-dimensional smoothing in ground-based post-processing, the values of the standard deviations for the case of a filter cutoff wavelength of $\lambda_c = 50$ km are listed for SWOT and WaCM in the third columns of Tables 1 and 2, respectively. The errors of SWOT estimates of SSH and geostrophically computed velocity components and vorticity are reduced by factors of 25, 167 and 875, respectively, compared with estimates from the pre-processed SWOT data. Compared with the 10-km smoothing applied to SWOT data in Section 4.1 to match the 5-km footprint size of WaCM data (see the second column of Table 1), the errors of geostrophically computed SWOT estimates of velocity and vorticity with 50-km smoothing are further reduced by factors of 21 and 81, respectively. The errors of WaCM estimates of velocity components and vorticity on a $1 \text{ km} \times 1 \text{ km}$ grid with 5-km footprint, the baseline speed noise of $\sigma_{\text{spd}} = 0.50 \text{ m s}^{-1}$ and 50-km smoothing are reduced by more modest factors of 5 and 20, respectively, compared with the 10-km smoothing in the pre-processing (see the second and third columns of Table 2).

### 4.4. Wavenumber spectral characteristics of the errors

Alongshore wavenumber spectra of the signals and errors in simulated SWOT and WaCM estimates of the velocity components and vorticity from the CCS model provide insight into the significant challenges in mapping small-scale variability from the satellite data. In the case of SWOT, velocity and vorticity are computed geostrophically and the alongshore wavenumber spectra of SSH signal and errors are also considered. The error spectra provide wavenumber decompositions of the error standard deviations presented in Sections 4.1–4.3. As noted previously, we consider only the effects of uncorrelated measurement errors in this study since they are the primary limitation for SWOT and WaCM estimates of velocity and vorticity. The error spectra presented below assume a standard deviation of $\sigma_h = 2.74 \text{ cm}$ for the errors of pre-processed SWOT estimates of SSH and a standard deviation of $\sigma_{u,v} = 0.354 \text{ m s}^{-1}$ for the errors of pre-processed WaCM estimates of each velocity component. The latter corresponds to a standard deviation of $\sigma_{\text{spd}} = 0.50 \text{ m s}^{-1}$ for speed errors equally partitioned between the two velocity components.

A minor caveat to the general applicability of the analytical equations in Appendix I for the theoretical wavenumber spectra of the errors of velocity components and vorticity computed geostrophically from SSH is that they are based on a specified constant value of the Coriolis parameter $f$. For the calculations in this section, we have used the value of $f$ at the central latitude $37^\circ\text{N}$ of the CCS model domain. At the northern extent of the model domain, the Coriolis parameter is 11% larger than its value at $37^\circ\text{N}$; at the southern extent, it is 11% smaller.

The use of a constant value of $f$ in the analytical expressions for the geostrophically computed velocity and vorticity error variances is a minor issue in this study. This is confirmed in Figs. 13a, b and 14 from the very close agreement between the spectra of the errors computed theoretically based on the analytical expressions in Appendix I with constant $f$ (the green lines in each panel of the above-noted figures) and the spectra determined empirically from errors of velocity components and vorticity computed geostrophically from maps of simulated SWOT SSH measurement errors using the local value of $f$ at each grid point (the blue lines in each panel of the figures).

Before presenting the alongshore wavenumber spectra of the variables of interest, we pause to point out one other very minor issue in the error spectra presented for both SWOT and WaCM in Figs. 13–15. The 2-dimensional smoothing applied in simulated ground-based post-processing of the SWOT and WaCM error fields from which the error spectra were computed empirically (the blue lines in Figs. 13–15) was applied truly isotropically using a 2-dimensional Parzen weighting function (see Appendix A) that depends only on the radial distance of each data point from the estimation location. It is shown in Appendix C that essentially equivalent isotropic 2-dimensional smoothing can be achieved through sequential 1-dimensional smoothing using the Parzen filter in each of two orthogonal dimensions. The advantage of sequential 1-dimensional smoothing is that it facilitates the derivations in Appendix I of the analytical expressions for the wavenumber spectral characteristics of residual errors in smoothed fields that are shown by the green lines in Figs. 13–15. The essential equivalence of true isotropic 2-dimensional smoothing and sequential 1-dimensional smoothing can be inferred from these figures by the very close agreement between the theoretical spectra computed from the analytical expressions in Appendix I and the spectra computed empirically from the simulated error fields.

#### 4.4.1. Error spectra from pre-processed SWOT data

The alongshore wavenumber spectra of error-free SSH from the CCS model and uncorrelated SSH measurement errors from unsmoothed simulated pre-processed SWOT estimates of SSH are shown in the top panel of Fig. 13a. The spectrum of uncorrelated SSH measurement errors computed empirically from the simulated SWOT data is shown by the blue line. The theoretical spectrum of SSH measurement errors
<figure><img src="image_14.png" /><figcaption>**Fig. 12.** The standard deviations of residual errors as functions of half-power filter cutoff wavelength with 2-dimensional isotropic Parzen smoothing for footprint sizes of 1 km and 5 km for SWOT and WaCM, respectively: (a) SWOT estimates of SSH; (b) Geostrophically computed SWOT estimates of each of the components of velocity (thin lines, which are indistinguishable for the two components), and WaCM estimates of each component of the total velocity (thick lines, which are again indistinguishable for the two components); and (c) geostrophically computed SWOT estimates of vorticity (thin line) and WaCM estimates of the total vorticity (thick lines), normalized by the Coriolis parameter *f* at the central latitude 37°N of the CCS model domain. The thick solid lines in (b) and (c) are for WaCM with the baseline speed noise of $\sigma_{\text{spd}} = 0.50 \text{ m s}^{-1}$ and the thick dashed lines are for the smaller speed noise of $\sigma_{\text{spd}} = 0.25 \text{ m s}^{-1}$ considered in Section 10. The residual standard deviations in these figures were computed from the error variances determined for SWOT and WaCM, respectively, by integrating the analytical formulas in Appendix I.3 and I.4 for each filter cutoff wavelength $\lambda_c$. The power-law dependencies on $\lambda_c$ that are labeled on each curve were determined from log-log versions of each panel (not shown here).</figcaption></figure>

given by the analytical expression (I.2) derived in Appendix I.1 is shown by the green line, which agrees very well with the empirical error spectrum. These error spectra intersect the SSH signal spectrum at a wavenumber of about 0.022 cycle per km (cpkm), which corresponds to a wavelength of about 45 km. The relevance of this intersection of the signal and noise spectra to the resolution capability of SWOT estimates of SSH is discussed later in the context of the interpretation of Fig. 1b.

The alongshore wavenumber spectra of signals and errors for SWOT estimates of cross-shore and alongshore velocity components $u_g$ and $v_g$ computed geostrophically from centered difference estimates of derivatives of simulated pre-processed SWOT estimates of SSH are shown in, respectively, the second and third panels of Fig. 13a. The error spectra computed empirically from the simulated SWOT data based on the local value of the Coriolis parameter *f* at each grid point are again shown by blue lines in these panels. The analytical expressions (I.5) and (I.10) derived in Appendix I.1 for the error spectra are shown by the green lines, which again agree very well with the empirical error spectra. A notable feature of the $v_g$ spectra is that the spectral power of $v_g$ errors exceeds the spectral power of the $v_g$ signal at all wavenumbers. The $v_g$ signal is thus undetectable from unsmoothed pre-processed SWOT data. In the case of the $u_g$ spectra, the signal power exceeds the error power for wavenumbers below about 0.02 cpkm but this signal variance will be masked in maps of $u_g$ by the order-of-magnitude higher power of the errors at higher wavenumbers.

The dramatic differences between the wavenumber characteristics of the alongshore wavenumber spectra of the errors of $u_g$ and $v_g$ merit some discussion. The spectrum is “white” (constant with alongshore wavenumber) for $v_g$ errors but “blue” (dominated by high alongshore wavenumber variability) for $u_g$ errors. The spatial high-pass filtering of $u_g$ errors implied by this blueness occurs because the white noise SSH is differentiated in the alongshore dimension, i.e., in the same dimension as the wavenumber spectrum. (The dropoff of spectral power at the highest wavenumbers arises because of the response function for the centered difference approximation of the geostrophic derivatives; see Fig. H.1 in Appendix H.)

The $v_g$ errors are similarly spatially high-pass filtered, but from differentiation of SSH in the cross-shore dimension. This cross-shore high-pass filtering would be evident in the cross-shore wavenumber spectrum of $v_g$, but is not manifest in its alongshore wavenumber spectrum because the centered difference estimates of the derivative of SSH in the geostrophic equation for $v_g$ are perpendicular to the alongshore dimension. It is shown in Appendix I, however, that the

blueness of the cross-shore spectrum of $v_g$ nonetheless has a large effect on the spectra of error fields smoothed 2-dimensionally in ground-based post-processing. (The cross-shore spectrum of $u_g$ is white for the same reason that the alongshore spectrum of $v_g$ is white, namely that the alongshore centered difference estimates of the derivative of SSH in the geostrophic equation for $u_g$ are perpendicular to the cross-shore dimension.)

The alongshore wavenumber spectra of signal and errors for SWOT estimates of vorticity $\zeta_g$ computed geostrophically from second derivatives of simulated pre-processed SWOT estimates of SSH are shown in the bottom panel of Fig. 13a. The spectrum of $\zeta_g$ errors computed empirically from the simulated SWOT data (the blue line) again agrees very well with the theoretical spectrum of $\zeta_g$ errors (the green line) computed by the analytical expression (I.24) in Appendix I based on the constant value of $f = 8.75 \times 10^{-5} \text{ s}^{-1}$ at the central latitude 37°N of the CCS model domain. In this case, the error spectra are nearly three orders of magnitude more energetic than the $\zeta_g$ signal spectrum at all wavenumbers. The vorticity signal estimated from unsmoothed pre-processed SWOT data is thus completely swamped by the errors at all wavenumbers.

For comparison with the wavenumber spectra of errors of estimates of velocity components and vorticity computed from pre-processed WaCM data that are discussed below, the alongshore wavenumber spectra of signals and errors for SWOT estimates of SSH, $u_g$, $v_g$ and $\zeta_g$ are shown in Fig. 13b for isotropic 2-dimensional smoothing with a half-power filter cutoff wavelength of $\lambda_c = 10 \text{ km}$ in simulated ground-based post-processing of the pre-processed SWOT data. This 10-km smoothing is the same as the smoothing applied in this study to simulate the 5-km footprint size in pre-processing of WaCM data (see Appendix B.2). The error spectra computed empirically from the smoothed SWOT data (the blue line in each panel) again agree very well with the theoretical spectra (the green lines) that were computed by the analytical expressions (I.29), (I.31), (I.33) and (I.35) in Appendix I.2 for isotropic 2-dimensional smoothing with a filter cutoff wavelength $\lambda_c = 10 \text{ km}$.

Since the wavenumber spectra of error-free SSH, $u_g$, $v_g$ and $\zeta_g$ signals are red, nearby values of each of these variables are correlated. The power of the signal spectra after 2-dimensional smoothing is therefore attenuated only at wavenumbers higher than $\lambda_c^{-1}$. (The small attenuation at wavenumbers smaller than $\lambda_c^{-1}$ is from imperfections of the filter transfer function of the Parzen smoother used here to smooth the 2-dimensional fields of interest.) The spectral power of the errors
<figure><img src="image_15.png" /><figcaption>**Fig. 13.** Alongshore wavenumber power spectral densities of simulated satellite estimates of the signals and errors for: Column (a) SWOT estimates of (top-to-bottom) SSH and geostrophically computed cross-shore and alongshore velocity and vorticity obtained from simulated pre-processed SWOT data; Column (b) the same as Column (a), except after isotropic smoothing using a Parzen smoother with the same half-power filter cutoff wavelength of 10 km used in the pre-processing of WaCM data; and Column (c) WaCM estimates of (top-to-bottom) cross-shore velocity, alongshore velocity and vorticity obtained from simulated pre-processed WaCM data. The red lines are the signal spectra computed from the model after applying the pre-filtering for SWOT (columns a and b, with additional 10-km smoothing in the latter) and WaCM (column c). The blue lines are the spectra computed empirically from the simulated error fields, which were computed geostrophically based on the local value of the Coriolis parameter *f* at each grid point in the case of SWOT. The green lines are the theoretical spectra of errors derived in Appendix I, which are based on the constant value of *f* at the central latitude 37°N of the CCS model domain in the case of SWOT. All of the spectra were smoothed by ensemble averaging over the individual spectra computed from alongshore grid lines that extend the full length of the model domain with a cross-shore spacing of 5 km. For reference, selected power-law rolloff dependencies on alongshore wavenumber *l* are labeled in the top panels of Columns a and b. Note that the rolloff of *l*⁻¹/² is not significantly different from the rolloff of *l*⁻¹¹/³ that was deduced from along-track altimeter data by Le Traon et al. (2008) and is consistent with the *l*⁻⁵/³ spectral rolloff of velocity in surface quasigeostrophic (SQG) theory (Held et al., 1995). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

of SWOT estimates of these variables is also attenuated at wavenumbers higher than $\lambda_c^{-1}$. Because the errors are spatially uncorrelated in the cross-shore direction, 2-dimensional smoothing also attenuates the power of the along-track wavenumber spectra of the errors by an additional factor of $2\Delta x/\lambda_c$ at all wavenumbers. This is discussed in detail in Appendix E [see Eq. (E.11)] and Appendix I. For the grid spacing of $\Delta x = 1$ km and filter cutoff wavelength of $\lambda_c = 10$ km,

this multiplicative factor is approximately $2\Delta x/\lambda_c = 0.2$. Note that the vorticity errors with 10-km smoothing in the bottom panel of Fig. 13b are still more than an order of magnitude larger than the signal at all wavenumbers.

Because of the overall reduction of the spectral power of the measurement errors at all wavenumbers when applying isotropic 2-dimensional smoothing, the intersection of the smoothed signal and
<figure><img src="image_16.png" /><figcaption>**Fig. 14.** Alongshore wavenumber power spectral densities of simulated satellite estimates of the signals and errors after 2-dimensional isotropic smoothing of the pre-processed SWOT data with half-power filter cutoff wavelengths of 20, 50 and 80 km (left, middle and right columns, respectively) for: Row (a) SSH; Row (b) geostrophically computed cross-shore velocity; Row (c) geostrophically computed alongshore velocity; and Row (d) geostrophically computed vorticity. The solid red lines are the signal spectra from the model after isotropic smoothing. The blue lines are the spectra computed empirically from the smoothed error fields constructed based on the local Coriolis parameter *f* at each grid point. The green lines are the theoretical spectra of smoothed SWOT error fields derived in Appendix I.2 based on the constant value of *f* at the central latitude 37°N of the CCS model domain. For reference, the dashed red and green lines are the signal spectra and theoretical error spectra in Fig. 13a based on the pre-processed SWOT data without additional smoothing in simulated ground-based post-processing. All of the spectra were smoothed by ensemble averaging as in Fig. 13. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

error spectra shifts to higher wavenumbers (shorter wavelengths) with increased smoothing. Recall from Fig. 13a that the unsmoothed SSH signal and error spectra intersect at a wavelength of about 45 km. For the smoothing with the 10-km half-power filter cutoff wavelength used for Fig. 13b, the SSH signal and error spectra intersect at a wavenumber of about 0.03 cpkm, which corresponds to a wavelength of about 33 km.

The intersection of the SSH signal and noise spectra at a wavelength of 33 km in Fig. 13b is surprising in view of the fact that the science requirement for SWOT measurement accuracy was chosen with the intent that the 68th-percentile of the global average SSH signal spectrum intersects the spectrum of uncorrelated measurement errors

at a wavelength of 15 km after isotropic 2-dimensional smoothing with a half-power filter cutoff wavelength of 15 km, which is only slightly more than the 10-km smoothing shown in Fig. 13b. With 15-km filtering, the signal spectrum from the CCS model used in this study intersects the noise spectrum at a wavelength of 30 km (see Fig. F.3 in Appendix F).

The longer 30-km wavelength of the intersection of the signal and noise spectra found in this study compared with the 15-km wavelength in the specification of the science requirement for the accuracy and resolution of SWOT measurements of SSH occurs because the SSH spectrum from the CCS model falls off as about $l^{-7/2}$ at alongshore wavenumbers *l* near 1/15 cpkm (see Figs. 13a and F.3), which is steeper
<figure><img src="image_17.png" /><figcaption>Fig. 15. Alongshore wavenumber power spectral densities of simulated satellite estimates of the signals and errors after 2-dimensional isotropic smoothing of the pre-processed WaCM data with half-power filter cutoff wavelengths of 20, 50 and 80 km (left, middle and right columns, respectively) for: Row (a) alongshore velocity; Row (b) cross-shore velocity; and Row (c) vorticity. The solid red lines are the signal spectra from the model after smoothing. The blue lines are the spectra computed empirically from the simulated error fields after smoothing and the green lines are the theoretical spectra of smoothed error fields derived in Appendix I.4 based on the constant value of f at the central latitude 37°N of the CCS model domain. For reference, the dashed red and green lines are the signal spectra and theoretical error spectra from Fig. 13c for the pre-processed WaCM data without additional smoothing in simulated ground-based post-processing. All of the spectra were smoothed by ensemble averaging as in Fig. 13. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

than the approximate $l^{-5/2}$ rolloff of the assumed 68th-percentile of the global-average SSH wavenumber spectrum upon which the SWOT science requirements are based. The steeper rolloff of the spectrum of SSH from the CCS model reflects both the specifics of the CCS dynamical regime and the absence from the model of other possible physical sources of SSH variance, including, for example, the SSH signatures of internal waves that would flatten the spectral rolloff of the SSH wavenumber spectrum at high wavenumbers.

The uncertainty of the rolloff of the SSH spectrum at wavelengths shorter than the approximate 70-km wavelength resolution limitation of nadir altimeter data (Xu and Fu, 2012) is an important issue for interpretation of SWOT data. Spectra of SSH are shown in Fig. F.4 in Appendix F for a region southeast of the Gulf Stream in the two ROMS simulations of the North Atlantic circulation considered in Section 3 (see Figs. 10 and 11), one with a weak internal wave field similar to the CCS model and the other with an energetic internal wave field from inclusion of high-frequency wind and tidal forcing. The SSH spectra in the models without and with high-frequency forcing have spectral rolloffs of about $l^{-5}$ and $l^{-3}$, respectively, at wavenumbers higher than about 1/50 cpkm. Because the flattening of the SSH spectrum from the model with high-frequency forcing occurs only at wavenumbers higher than about 1/50 cpkm, the intersection of the SSH signal spectrum and the noise spectrum for 15-km smoothed SWOT data occurs at only a slightly higher wavenumber (shorter wavelength) than for the model

without high-frequency forcing. As for the CCS model used in this study, the SSH spectra from both North Atlantic models intersect the noise spectrum at wavelengths longer than 15 km.

The issue of the uncertainty of the spectral characteristics of SSH at high wavenumbers and its implications for the resolution of SWOT measurements of SSH are discussed further in Appendix F. The conclusion is that the wavenumber spectral characteristics of SSH are not presently known from observations at wavelengths shorter than 70 km and must therefore be inferred from high-resolution models. However, different models give different results (see the discussion in Appendix F). SWOT will provide the first observational data from which model representations of small-scale SSH variability can be tested.

### 4.4.2. Error spectra from pre-processed WaCM data

The alongshore wavenumber spectra of signals and errors of WaCM estimates of velocity components *u* and *v* and vorticity *ζ* are shown in Fig. 13c for speed measurement errors with a standard deviation of $\sigma_{spd} = 0.50 \text{ m s}^{-1}$. As discussed in Section 4.2 and in more detail in Appendix I.3, we have assumed that the pre-processed WaCM velocity estimates will be isotropically smoothed with a half-power filter cutoff wavelength of 10 km and oversampled on a 1 km × 1 km grid. As for the case of SWOT considered above, the theoretical error spectra computed by the analytical expression (I.40) for *u* and *v* and the analytical expression (I.43) partitioned as (I.45b) and (I.46a) for *ζ* (the green
lines) agree very well with the error spectra computed empirically from the simulated WaCM data (the blue lines). The error spectra for $v$ and $\zeta$ are very similar in magnitude and wavenumber characteristics to the spectra of errors of $v_g$ and $\zeta_g$ from the simulated SWOT data with 10-km smoothing shown in Fig. 13b. The error spectra for $u$ from pre-processed WaCM data and for $u_g$ from SWOT data with 10-km smoothing applied in post-processing are very different because of the filtering of the first-difference operator used to calculate $u_g$ from SSH, as discussed previously.

### 4.4.3. Error spectra from post-processed SWOT and WaCM data

To illustrate the effects of smoothing in ground-based post-processing to improve the signal-to-noise ratio for scientific applications of SWOT and WaCM data, the wavenumber spectral characteristics of the signals and errors for isotropic 2-dimensional smoothing with half-power filter cutoff wavelengths of $\lambda_c = 20$ km, 50 km and 80 km are shown in Figs. 14 and 15 for SWOT and WaCM, respectively. The signal and error spectra are all attenuated at wavenumbers higher than $\lambda_c^{-1}$. The 2-dimensional smoothing additionally attenuates the error spectra at all wavenumbers by the factor of $24x/\lambda_c$ noted above in Section 4.4.1 in the discussion of the spectra of SWOT data smoothed with $\lambda_c = 10$ km in Fig. 13b.

An important thing to note from Figs. 14 and 15 is that the noise spectra are attenuated much more quickly than the signal spectra with increased smoothing. Smoothing therefore improves the signal-to-noise variance ratio. This is a key point in our approach to assessing the resolution capability of maps of velocity and vorticity constructed from SWOT and WaCM data.

The effects of isotropic 2-dimensional smoothing of pre-processed SWOT and WaCM data in simulated ground-based post-processing to reduce the effects of uncorrelated measurement errors are examined in Section 6 from instantaneous maps of the variables of interest with the same filter cutoff wavelengths of $\lambda_c = 20$ km, 50 km and 80 km used for the spectra on Figs. 14 and 15. The strategy used in this study to define the resolution capabilities from the resulting maps of signals and errors for these and other choices of $\lambda_c$ is summarized in the next section.

## 5. A strategy for assessing resolution capability from noisy satellite observations

The resolution capabilities of SWOT and WaCM estimates of velocity and vorticity are defined in this study by the amount of smoothing required to reduce the errors to achieve a specified signal-to-noise (S/N) ratio. The effects of uncorrelated measurement errors on estimates of surface velocity and vorticity are attenuated in Section 6 by isotropic 2-dimensional spatial smoothing of instantaneous snapshots of the simulated pre-processed SWOT and WaCM data. In a further effort to suppress noise, temporal averaging is applied in addition to 2-dimensional spatial smoothing of the simulated satellite data in Sections 8–10.

There is ambiguity in how to define the signal in the S/N metric. We define the S/N ratio based on the spatial variances of the residual signal and residual errors after smoothing. The signal variance is thus defined to be the spatial variance of the true value of the variable of interest after applying the same smoothing that is applied to reduce the errors. An arguably better choice is to compare the spatial variance of the smoothed errors with the spatial variance of the unsmoothed signal. From an observational perspective, however, this strategy would have tenuous relevance to the maps that can be constructed from the error-contaminated satellite observations since it is not possible to reduce the errors by smoothing the observed data without also smoothing the signal.

While the “moving target” of the systematic reduction of signal variance with increased smoothing may not be ideal, this is the only practical approach to assessing the resolution capability of the noisy satellite estimates; the smoothed signal is the best possible estimate of

the variable of interest that could be achieved from noisy satellite data after applying a specified amount of smoothing to reduce the effects of uncorrelated measurement errors. Because of spatial correlation of the signal variability as indicated by the redness of the signal wavenumber spectra in Figs. 13–15, the error variance usually decreases more rapidly than the signal variance, in which case the S/N ratio increases with increased smoothing. The preferential attenuation of the error variance compared with the signal variance is evident from the example wavenumber spectra shown in Figs. 14 and 15.

For the case of the uncorrelated measurement errors considered in this study, there is a direct relationship between the signal-to-noise (S/N) variance ratio and the cross correlation between the error-free and noisy fields. To see this, consider $\hat{z}(x, y)$ to be an estimate of a signal $z(x, y)$ contaminated by uncorrelated errors $\epsilon(x, y)$,

$$ \hat{z}(x, y) = z(x, y) + \epsilon(x, y). $$

The squared cross correlation between $\hat{z}(x, y)$ and $z(x, y)$ is

$$ \hat{\rho}_{zz}^2 = \frac{(\text{Cov}[(z + \epsilon), z])^2}{\text{Var}[z + \epsilon]\text{Var}[z]} = \frac{(\text{Cov}[z, z] + \text{Cov}[z, \epsilon])^2}{(\text{Var}[z] + 2 \text{Cov}[z, \epsilon] + \text{Var}[\epsilon])\text{Var}[z]}, \quad (9) $$

where Cov and Var are shorthand notations for the spatial cross covariance and variance. Since the errors $\epsilon$ are uncorrelated with the signal $z$, their cross covariance is $\text{Cov}[z, \epsilon] = 0$. And since $\text{Cov}[z, z] = \text{Var}[z]$, the squared cross correlation can be expressed as

$$ \hat{\rho}_{zz}^2 = \frac{(\text{Var}[z])^2}{(\text{Var}[z])^2 + \text{Var}[\epsilon]\text{Var}[z]} = \frac{1}{1 + \gamma^{-2}}, \quad (10a) $$

where

$$ \gamma^2 = \frac{\text{Var}[z]}{\text{Var}[\epsilon]} \quad (10b) $$

is the S/N variance ratio.

Specification of a threshold of minimum S/N variance ratio $\gamma^2$ for which the error-contaminated estimates are scientifically useful is somewhat subjective. Example maps are shown in Fig. 16 for $\gamma^2 = 1$, 4 and 10 for the case of normalized vorticity computed from noisy WaCM data, which is considered in detail later in Section 6.2. A choice of $\gamma^2 = 1$ (Fig. 16a) is clearly too liberal since any given small-scale feature is as likely to be attributable to noise as to real variability of the signal of interest. A mathematically convenient choice of threshold is $\gamma^2 = 10$, which corresponds to a S/N standard deviation ratio of $\gamma = 3.16$ (Fig. 16c). The signal standard deviation is then about three times larger than the error standard deviation, thus resulting in a clear distinction between the signal and errors in maps of the noisy fields $\hat{z}(x, y)$. For a value of $\gamma^2 = 10$, the squared cross correlation (10a) between the signal $z$ and the noisy estimate $\hat{z}$ is $\hat{\rho}_{zz}^2 = 1.10^{-1} = 0.91$ so that the cross correlation between $z$ and $\hat{z}$ is $\hat{\rho}_{zz} = 0.95$.

A less conservative choice of $\gamma^2 = 4$ for the threshold S/N variance ratio corresponds to a signal standard deviation that is twice as large as the error standard deviation (Fig. 16b). The squared cross correlation between the signal $z$ and the estimate $\hat{z}$ is then $\hat{\rho}_{zz}^2 = 1/1.25 = 0.80$ so that the cross correlation between $z$ and $\hat{z}$ is $\hat{\rho}_{zz} = 0.89$.

Our subjective assessment from visual inspection of maps with varying S/N ratios such as those shown in Fig. 16 is that a S/N standard deviation ratio of $\gamma = 2$ is insufficient to distinguish the signal from the errors unambiguously. We have therefore adopted the threshold S/N standard deviation ratio of $\gamma = 3.16$ to define the resolution capability in the analysis that follows. Readers can judge for themselves from the examples of error-contaminated maps in Fig. 16 and in the maps in Sections 6 and 8 whether they consider this threshold to be overly conservative. Since the amount of smoothing required to achieve a specified threshold S/N ratio increases with increasing value of the S/N ratio, lowering the threshold from our preferred choice of $\gamma = 3.16$ yields higher resolution assessments (see Fig. 51 in Section 13.5 below). The relative resolution capabilities of the spatially smoothed instantaneous fields constructed from SWOT versus WaCM data in
<figure><img src="image_18.png" /><figcaption>**Fig. 16.** Examples showing the characteristics of noisy data after isotropic 2-dimensional smoothing to achieve signal-to-noise standard deviation ratios of 1, 2 and 3.16 (rows a, b and c, respectively). For illustration purposes, these are maps of the normalized vorticity $\zeta/f$ constructed from simulated noisy WaCM data as in Section 6.2 below for the case of uncorrelated speed measurement errors with a standard deviation of $\sigma_{spd} = 0.50 \ \text{m s}^{-1}$. The left, middle and right columns show, respectively, the noisy estimates of $\zeta/f$, the error-free $\zeta/f$ signal and the residual errors after smoothing with the half-power filter cutoff wavelength $\lambda_c$ labeled to the left of each row.</figcaption></figure>

Section 6, or of the spatially smoothed time-averaged fields constructed by either instrument in Sections 8–10, would not change substantially by adopting a threshold different from $\gamma = 3.16$. In other words, changing the threshold value of $\gamma$ would not qualitatively alter our conclusions regarding the relative resolution capabilities of SWOT and WaCM.

## 6. The effects of measurement errors on estimates of instantaneous velocity and vorticity fields

The effects of uncorrelated measurement errors are investigated in this section in isolation from the additional effects of sampling errors by assessing the resolution capabilities of maps of velocity and vorticity constructed from simulated SWOT and WaCM data for the idealized scenario of complete sampling of the full CCS domain on each satellite overpass. In reality, of course, the variables of interest can only be estimated within the measurement swaths at the time of each overpass. The analysis of simulated noisy SWOT and WaCM observations in this section must therefore be viewed as the best possible resolution of instantaneous fields that could be achieved within the actual SWOT and WaCM measurement swaths. The effects of sampling errors from the limited swath widths and irregular revisit intervals are investigated separately and in combination with measurement errors in Sections 8–10 based on simulated time averages of noisy SWOT and WaCM observations.

For the reasons discussed at the end of the introduction, velocity estimates are characterized in this study in terms of the scalar magnitude of the vector-averaged velocity (which is equivalent to the current speed in the instantaneous maps considered in this section), rather than considering each velocity component separately. In the

case of SWOT, the adequacy of the geostrophic approximation was discussed in Section 3 and this issue is not considered any further in this study. The question addressed for SWOT in this section is thus how the uncorrelated errors of the SSH measurements affect the resolution capabilities of estimates of velocity and vorticity computed geostrophically from SWOT data, irrespective of the accuracy of the geostrophic approximation.

### 6.1. SWOT

The procedure for simulating pre-processed SWOT estimates of SSH for this study was to smooth the SSH fields on the $0.5 \ \text{km} \times 0.5 \ \text{km}$ grid of the CCS model using an isotropic Parzen smoother with a half-power filter cutoff wavelength of 2 km to mimic the footprint size of 1 km in the baseline science requirements for SWOT (see Appendix B.1). These smoothed SSH fields were then subsampled on a $1 \ \text{km} \times 1 \ \text{km}$ grid over the full CCS model domain. The measurement errors with 2-km smoothing are shown in Appendix B.1 to be uncorrelated for this sample spacing. The errors of SWOT measurements of SSH on this sample grid were simulated by adding Gaussian-distributed random errors with the standard deviation of $\sigma_h = 2.74 \ \text{cm}$ derived in Appendix F.

To assess the effects of errors in SWOT measurements of SSH, the estimates of velocity and vorticity that are of interest here were computed geostrophically from the error-free and noisy SSH fields. The results for error-free SSH and geostrophically computed surface velocity and vorticity are shown in the upper left panels of Figs. 17a–17c for the CCCS region of the full CCS model domain. As noted previously from Fig. 5, the rich distribution of small-scale variability in the vorticity field in the top left panel of Fig. 17c compared with the smooth SSH field in the top left panel of Fig. 17a from which it was computed
is a testament to the extreme spatial high-pass filtering operation of double differentiation to compute the vorticity geostrophically from SSH. Even the single differentiation of SSH to compute the velocity field geostrophically in the upper left panel of Fig. 17b results in considerable small-scale variability compared with the SSH field.

The effects of the $\sigma_h = 2.74$-cm SSH measurement errors on unsmoothed SWOT estimates of SSH and geostrophically computed velocity and vorticity are shown in the lower left panels of Figs. 17a–17c. The SSH field has a distinctly speckled appearance but most of the features in the error-free fields are recognizable in the error-contaminated field. In contrast, the geostrophically computed velocity and vorticity signals are completely swamped by the errors. The masking of the velocity and vorticity signals is not surprising in view of the large error standard deviations derived in Section 4.1 (see Table 1 and the signal and noise spectra in Fig. 13a).

It is clear that the noisy pre-processed SWOT estimates of SSH will have to be smoothed in ground-based post-processing to reduce the errors in geostrophically computed estimates of velocity and vorticity. Illustrative examples of error-free and noisy SSH fields are shown, respectively, in the top and bottom rows of Fig. 17a for smoothing with half-power filter cutoff wavelengths of 20, 50 and 80 km. The uncorrelated SSH measurement errors are quickly mitigated with increased smoothing (see Fig. 12a). A small amount of residual SSH measurement errors is visually apparent with 20-km smoothing but the errors become invisible with 50-km smoothing.

The reduction of errors in SWOT estimates of velocity requires more smoothing than for SSH (Fig. 17b). The residual errors of the velocity field computed geostrophically from the SSH field with 20-km smoothing for which the S/N standard deviation ratio is 1.5 would likely still render the velocity estimates unusable for most applications. The errors in the velocity estimates are essentially eliminated with 50-km smoothing for which the S/N standard deviation ratio is 7.70. Qualitatively, it thus appears that the resolution capability for SWOT estimates of velocity falls somewhere between 20 km and 50 km. This is quantified below.

Because of the amplification of measurement errors by differentiation, even more smoothing is required to obtain useful SWOT estimates of vorticity (Fig. 17c). The signal of interest is completely unrecognizable in the vorticity field computed geostrophically from the SSH field with 20-km smoothing. The noisy field with 50-km smoothing for which the S/N standard deviation ratio is 2.74 still has a somewhat mottled appearance compared with the 50-km smoothed error-free field. For example, the mesoscale anticyclone near the northwest corner of the CCCS region is broken up into multiple smaller-scale features in the noisy field. Numerous other artifacts of the 50-km smoothed measurement errors could easily be misinterpreted as small-scale signals. Qualitatively, it thus appears that the resolution capability for SWOT estimates of vorticity is somewhat coarser than 50 km.

It is noteworthy that the S/N standard deviation ratio of 2.74 for 50-km smoothing falls below our minimum threshold of 3.16, thus providing further rationale for why we advocated the value of 3.16 in Section 5 to define the resolution capability. The residual errors diminish very quickly with increased smoothing (see Fig. 12c). The S/N standard deviation ratio of 7.93 achieved by smoothing with a half-power filter cutoff wavelength of 80 km (see the right panels of Fig. 17c) is more than sufficient to eliminate essentially all of the residual errors in the geostrophically computed vorticity field.

To determine the resolution capability of in-swath SWOT estimates of velocity and vorticity, we smoothed the error-free and noisy SSH fields with half-power filter cutoff wavelengths ranging from 10 km to 150 km. We then computed the velocity and vorticity fields geostrophically from these smoothed error-free and noisy SSH fields. The resulting S/N standard deviation ratios are shown as functions of half-power filter cutoff wavelength in Fig. 18a. According to our criterion of a S/N threshold of 3.16, the wavelength resolution capability of in-swath SWOT estimates of instantaneous velocity is approximately 32 km,

which falls between the smoothing shown in the second and third columns of Fig. 17b. For in-swath SWOT estimates of vorticity, the S/N threshold of 3.16 implies a resolution capability of approximately 54 km, which is slightly more than the 50-km smoothing shown in the third column of Fig. 17c.

It is shown in Appendix C that the feature resolution capability is approximately half of the wavelength resolution. Fig. 18a thus suggests that SWOT will be able to detect velocity and vorticity features with diameter scales of approximately 16 km and 27 km, respectively, within each measurement swath. These resolution capabilities will be degraded somewhat by edge effects in the smoothing from the fact that the 50-km swath widths of the SWOT measurements (see Section 7) are comparable to the half-power filter cutoff wavelengths required to achieve the S/N threshold of 3.16. This is especially an issue for SWOT estimates of vorticity since they require more smoothing than for SWOT estimates of velocity.

## 6.2. WaCM

A similar procedure was followed to simulate the effects of measurement errors on WaCM estimates of velocity and vorticity. The 5-km footprint size anticipated for pre-processed WaCM estimates of surface velocity was achieved by isotropically smoothing the 0.5 km × 0.5 km gridded surface velocity fields from the CCS model with a half-power filter cutoff wavelength of 10 km (see Appendix B.2). In the baseline design, the measurement errors with this smoothing are uncorrelated with a speed standard deviation of $\sigma_{spd} = 0.50 \text{ m s}^{-1}$ on a 5 km × 5 km grid. Assuming that these speed uncertainties are equally partitioned between two orthogonal velocity components, this corresponds to an error standard deviation of $\sigma_{u,v} = 0.354 \text{ m s}^{-1}$ for each component.

As discussed previously, the simulated WaCM data in this study were oversampled on a 1 km × 1 km grid in order to improve the derivative estimates for calculation of the vorticity signal (see Appendix H). The 10-km smoothing and 1 km × 1 km oversampling were achieved by adding uncorrelated errors with a standard deviation of $3.54 \text{ m s}^{-1}$ to each unsmoothed velocity component on the 0.5 km × 0.5 km model grid prior to smoothing and subsampling. This 10-fold increase in noise level compared with the baseline standard deviation of $\sigma_{u,v} = 0.354 \text{ m s}^{-1}$ for the uncorrelated errors of pre-processed WaCM estimates of surface velocity components is commensurate with the 10-fold difference between the 0.5 km × 0.5 km model grid spacing and the 5-km footprint size of the WaCM data.

The effects of the velocity component measurement errors on 10-km smoothed estimates of velocity and vorticity from pre-processed WaCM data are shown in the lower left panels of Figs. 19a and 19b. The smoother appearances of the noisy velocity and vorticity fields compared with the lower left panels of Figs. 17b and 17c for SWOT estimates of velocity and vorticity are because of the 10-km smoothing applied in the pre-processing of WaCM data compared with only 2-km smoothing in pre-processing of SWOT data. The velocity signal of interest is vaguely detectable in the noisy field in the bottom left panel of Fig. 19a but the vorticity signal is completely masked in the bottom left panel of Fig. 19b.

The resolution capabilities of in-swath WaCM estimates of velocity and vorticity were assessed by applying the same smoothing to the error-free and noisy velocity component fields. Examples of the velocity and vorticity fields computed from smoothed velocity fields with and without simulated measurement errors are shown in Figs. 19a and 19b for half-power filter cutoff wavelengths of 20, 50 and 80 km. For in-swath velocity, the S/N ratio is only 2.30 with the 80-km smoothing shown in the last column of Fig. 19a. According to the S/N = 3.16 criterion adopted here, this amount of smoothing is insufficient. The inadequacies of 80-km smoothing are visually apparent from the discrepancies between the smoothed map of error-free velocity in the top right panel and the smoothed map of the velocity signal plus errors in the bottom right panel.
<figure><img src="image_19.png" /><figcaption>**Fig. 17a.** Maps of simulated SWOT estimates of SSH for the CCCS region shown in the bottom panel of Fig. 5a. The top panels are the error-free fields from the pre-processed SWOT data with no additional smoothing (left), and after isotropic 2-dimensional smoothing with half-power filter cutoff wavelengths of 20, 50 and 80 km (columns 2, 3 and 4, respectively). The bottom panels show the fields computed with the addition of simulated uncorrelated SSH measurement errors with a standard deviation of $\sigma_h$ = 2.74 cm but without sampling errors (i.e., as if the measurement swaths spanned the full CCS model domain on each satellite overpass). The same color bar is used for all of the panels.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>**Fig. 17b.** The same as Fig. 17a, except maps of simulated SWOT estimates of the magnitudes of geostrophically computed surface velocity for the CCCS region shown in the bottom panel of Fig. 5b. The panels in the top row were computed from error-free SSH fields and the panels in the bottom row were computed with the addition of simulated uncorrelated SSH measurement errors with a standard deviation of $\sigma_h$ = 2.74 cm. The same color bar is used for all of the panels and the signal-to-noise standard deviation ratio is labeled above each of the bottom panels.</figcaption></figure>
<figure><img src="image_21.png" /><figcaption>**Fig. 17c.** The same as Fig. 17a, except maps of simulated SWOT estimates of geostrophically computed and normalized vorticity $\zeta_x/f$ for the CCCS region shown in the bottom panel of Fig. 5c. The panels in the top row were computed from error-free SSH fields and the panels in the bottom row were computed with the addition of simulated uncorrelated SSH measurement errors with a standard deviation of $\sigma_h = 2.74$ cm. The same color bar is used for all of the panels and the signal-to-noise standard deviation ratio is labeled above each of the bottom panels.</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>**Fig. 18.** The scale dependencies of the ratios of the standard deviations of the signal and the uncorrelated measurement errors for a snapshot of the full CCS region from simulated (a) SWOT; and (b) WaCM estimates of instantaneous surface velocity (thin lines) and vorticity (thick lines) after isotropic 2-dimensional smoothing using a Parzen smoother with the half-power filter cutoff wavelengths indicated along the abscissas. The velocity and vorticity were computed geostrophically for SWOT. The gray areas correspond to signal-to-noise (S/N) standard deviation ratios less than 3.16, which is equivalent to a signal-to-noise variance ratio of 10. The vertical dashed lines indicate the wavelengths above which the S/N standard deviation ratios exceed 3.16. The dynamic range of the abscissa is larger in (b) to accommodate the lower S/N ratios of WaCM estimates of velocity and vorticity with the baseline speed standard deviation of $\sigma_{spd} = 0.50$ m s$^{-1}$ for the uncorrelated errors of WaCM measurements of velocity. The improved resolution capabilities that would be achieved from WaCM with smaller measurement errors are summarized later in Fig. 44 and Table 4a and b.</figcaption></figure>

For the case of in-swath vorticity, the S/N ratio is only 1.72 with the 80-km smoothing shown in the last column of Fig. 19b. There are numerous artifacts in the bottom right panel that would be misinterpreted as small-scale vorticity signals.

The S/N standard deviation ratios for velocity and vorticity are shown as functions of the half-power filter cutoff wavelength in Fig. 18b. By the S/N = 3.16 threshold adopted here, the wavelength resolutions

for in-swath WaCM estimates of velocity and vorticity are coincidentally both approximately 130 km. The more rapid convergence toward this threshold for vorticity compared with velocity is because of the noise attenuation at both high and low wavenumbers in the band-pass filtering operation of the response function for centered difference estimates of the derivatives in the calculation of vorticity (see Appendix H).
<figure><img src="image_23.png" /><figcaption>**Fig. 19a.** Maps of simulated WaCM estimates of the magnitudes of the total surface velocity for the CCCS region shown in the bottom panel of Fig. 3a. The top panels are the error-free fields from the pre-processed WaCM data with no additional smoothing (left), and after isotropic 2-dimensional smoothing with half-power filter cutoff wavelengths of 20, 50 and 80 km (columns 2, 3 and 4, respectively). The bottom panels show the fields computed with the addition of simulated uncorrelated velocity measurement errors with a speed standard deviation of $\sigma_{spd} = 0.50 \text{ m s}^{-1}$ equally partitioned between the two velocity components. The same color bar is used for all of the panels and the signal-to-noise standard deviation ratio is labeled above each of the bottom panels.</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>**Fig. 19b.** The same as Fig. 19a, except maps of simulated WaCM estimates of normalized vorticity $\zeta/f$ computed from the total surface velocity for the CCCS region shown in the bottom panel of Fig. 3b. The same color bar is used for all of the panels and the signal-to-noise standard deviation ratio is labeled above each of the bottom panels.</figcaption></figure>
A noteworthy feature of the S/N curves for WaCM estimates of velocity and vorticity in Fig. 18b is that they are much flatter than for the SWOT estimates in Fig. 18a. This flatness implies that the standard deviation of the residual errors decreases only slightly faster than the signal with increased smoothing. It also means that the resolution capabilities for WaCM are more sensitive to the choice of the threshold value of the S/N standard deviation ratio that is used to define resolution capability. Lowering the threshold value from our recommended value of 3.16 to a value of 2, for example (see Fig. 51 in Section 13.5 below), would correspond to much higher resolution capabilities of 63 km and 88 km for WaCM estimates of velocity and vorticity, respectively. For SWOT, the same choice of a threshold S/N value of 2 would only improve the resolution capabilities from 32 km to 24 km for velocity and from 54 km to 44 km for vorticity.

### 6.3. Discussion of the effects of measurement errors alone

It is seen from Fig. 18 that the resolution capability of in-swath SWOT estimates of instantaneous velocity and vorticity are superior to those from WaCM data with the baseline noise standard deviation of $\sigma_{spd} = 0.50 \text{ m s}^{-1}$ when considering the effects of measurement errors alone, i.e., by considering the unrealistic case of measurement swaths that sample the entire CCS model domain on each satellite overpass. However, in addition to the limitations of the geostrophic approximation discussed in Section 3, SWOT data are severely limited by the narrow measurement swaths (see the top panels of Fig. 20). It is shown in Section 8 that the sampling errors arising from these narrow measurement swaths are a more significant limitation than measurement errors for mapping the geostrophically computed velocity and vorticity fields over the full CCS model domain considered in this study. But even if SWOT estimates of these variables are considered only within the narrow measurement swaths, the 32-km and 54-km smoothing required to achieve a S/N ratio of 3.16 (see Fig. 18a) for geostrophically computed velocity and vorticity, respectively, will lead to artifacts from edge effects near the edges of the two parallel 50-km measurement swaths.

With its much wider measurement swaths (see the bottom panels of Fig. 20 for a swath width of 1200 km and Fig. 41 for a swath width of 1800 km), edge effects are less of an issue for WaCM than for SWOT. Measurement errors are therefore more limiting than sampling errors. This is quantified in Sections 8–10. The resolution capability of ~130 km for in-swath WaCM estimates of instantaneous snapshots of velocity and vorticity is coarse for investigations of small-scale variability. But it is much better than the present resolution capability of about 200 km by about 1 month (see Fig. 50).

The improved resolutions that could be achieved from WaCM for measurement noise standard deviations smaller than the baseline value of $\sigma_{spd} = 0.50 \text{ m s}^{-1}$ are investigated in Section 10.2. If the noise can be reduced to $\sigma_{spd} = 0.25 \text{ m s}^{-1}$, for example, it is shown that the resolution capabilities of instantaneous snapshot maps of velocity and vorticity constructed from WaCM data would be about 50 km and 74 km, respectively. These values are still somewhat coarser than can be achieved in SWOT estimates of velocity and vorticity, but over a swath width that is more than an order of magnitude wider for WaCM and is therefore less prone to edge effects in the smoothed fields.

Thus far, we have considered only the effects of spatial smoothing to mitigate the effects of measurement errors. At least in principle, the residual errors can also be reduced by applying temporal smoothing. In practice, this introduces the additional effects of the sampling errors that are discussed in Section 7. The effects of combined space-time smoothing are investigated in Sections 8–10 where it is shown that temporal smoothing is advantageous for WaCM but, perhaps counterintuitively, detrimental for SWOT.

## 7. SWOT and WaCM sampling characteristics

The sampling characteristics of the SWOT and WaCM radars are fundamentally different. SWOT will measure SSH across a swath width of 120 km with a 20-km gap centered on the satellite ground track, thus resulting in two parallel 50-km swaths. For the purposes of the analysis in Section 8, we have assumed that WaCM will have a swath width of 1200 km with a 100-km gap centered on the satellite ground track, thus resulting in two parallel 550-km swaths. Examples of SWOT and WaCM measurement swaths for single simulated satellite overpasses of the CCS model domain are overlaid on a map of the vorticity field in Fig. 20. It is shown in Sections 8 and 9 that the factor-of-11 better sampling coverage for WaCM compared with SWOT has a profound effect on the ability to map the space-time evolution of the surface velocity and vorticity fields.

Recent engineering developments indicate that it may be possible to increase the swath width of WaCM to 1800 km, which is a factor-of-17 better coverage than SWOT. The improved sampling afforded by this wider swath and the resulting improvements in the resolution capabilities for time-averaged maps of velocity and vorticity constructed from WaCM data are summarized in Section 10.

The limited swath widths and the discrete space-time sampling patterns result in the following three sources of sampling errors that occur independently of the effects of measurement errors considered in Sections 4 and 6:

1. One form of sampling errors arises from the need to synthesize the measurements from multiple satellite overpasses in order to map the variables of interest over regions larger than the measurement swaths from a single satellite overpass. The interest in this study is in mapping the velocity and vorticity fields over the full CCS model domain. Because of the rapid evolution of submesoscale features in these variables between the times of different satellite overpasses, spatial discontinuities can occur across the edges of neighboring or overlapping measurement swaths from the different orbits. For example, adjacent pairs of SWOT measurement swaths are sampled about 10 days apart (see the left panel of Fig. 21). Any temporal evolutions of the structure, location and intensity of small-scale features in the regions of overlap will be misinterpreted as spatial variability in time-averaged maps constructed from measurements along multiple satellite overpasses.

2. An equally important and related source of sampling errors arises from differences between the time average of the discretely sampled fields and the true time average. Because of the rapid evolution of submesoscale variability, the estimated time-average is corrupted by unresolved temporal variability between the discrete samples. This is similar to the classical problem of aliasing but this source of sampling errors is more complicated in maps constructed from the measurements from multiple satellite overpasses. The discrete sampling then occurs at different times in different regions of the mapping domain, resulting in a patchwork of different aliasing artifacts.

3. The two sources of sampling errors summarized above can be avoided if the satellite data are smoothed and analyzed on a swath-by-swath basis. But this introduces a third source of sampling errors in the form of edge effects that can occur from incomplete data within the span of the smoother. This type of sampling errors was neglected in Section 6 by considering the idealized case of SWOT and WaCM measurement swaths that sample the full CCS model domain on each satellite overpass. An estimate of the spatially smoothed field within an actual measurement swath is imperfect if the distance between the estimation location and a swath edge is less than half of the span of data that are included in the weighting function of the smoother. The imperfections worsen the closer the estimates
<figure><img src="image_25.png" /><figcaption>**Fig. 20.** Examples of the measurement swaths for single ascending and descending overpasses of SWOT (top panels) and WaCM with a swath width of 1200 km (bottom panels) overlaid on the snapshot of the normalized vorticity $\zeta/f$ from the top panel of Fig. 3b. The improved sampling coverage by WaCM with a swath width of 1800 km is shown in Fig. 41. The ground tracks for SWOT in this figure are the actual planned orbit. The ground tracks for WaCM 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>

are to the swath edge. This source of error is worse for some smoothing algorithms than for others and the magnitudes of the errors can depend on the detailed structure of the field near the edges of the measurement swaths. Edge effects are especially problematic when smoothing with a half-power filter cutoff wavelength that is comparable to or larger than the swath width.

All three of the above sources of sampling errors are exacerbated by a narrow swath width and are thus much bigger issues for SWOT than for WaCM. The magnitudes of the first two sources of sampling errors summarized above also depend strongly on the satellite repeat period. SWOT will be launched into a prograde orbit (ascending ground tracks from southwest to northeast) with an inclination of 77.6°, an altitude of 890.3 km and a 21-day exact repeat with 292 orbits per repeat period. As WaCM is in the early stages of planning, the exact orbit has not yet been determined. For the simulations in this study, we have assumed the same orbit configuration as the QuikSCAT satellite, which is a retrograde orbit (ascending ground tracks from southeast to northwest) with an inclination of 98.7°, an altitude of 802.7 km and a 4-day exact repeat with 57 orbits per repeat period.

The effects of the above orbit parameters on the space-time sampling patterns are shown in Fig. 21. The dots in each panel indicate the

longitudes and times of ascending ground tracks over a longitudinal range of 25° at a fixed reference latitude. The sampling patterns are similar for the descending ground tracks, except shifted in time by about half a day. The longitudes of the descending ground tracks relative to the longitudes of the ascending ground tracks vary with latitude and are therefore not displayed in Fig. 21.

Consider first the space-time sampling pattern for the WaCM orbit adopted for this study, which is simpler than the sampling pattern for SWOT. As noted above, we assume a swath width of 1200 km for the analysis in 8; the improved sampling for a swath width of 1800 km is considered in Section 10.1. As shown in the right panel of Fig. 21, ascending ground tracks cross the western and eastern portions of the CCS model domain (indicated by the vertical dashed lines) approximately one day apart. This is followed by a 2-day gap and then the same pair of ascending ground tracks are sampled again with the same 1-day time separation. During the 2-day gap, the outer portions of a 1200-km WaCM measurement swath from ascending ground tracks outside of the CCS model domain sample small portions of the model domain. In addition, the CCS model domain is sampled twice along descending ground tracks (not shown in Fig. 21) approximately half a day earlier than each ascending ground track. The net result is that nearly every grid point in the CCS model domain is sampled at least
Times and Longitudes of Ascending Ground Tracks

<figure><img src="image_26.png" /><figcaption>**Fig. 21.** Time and relative longitude plots of sequential ascending ground track overpasses of SWOT and WaCM measurements along a fixed latitude. There are analogous sampling patterns for the descending overpasses, except shifted approximately half a day in time. The sampling pattern shown for SWOT is based on the orbit parameters of the planned 21-day exact repeat mission. The sampling pattern shown for WaCM is based on the orbit parameters for the 4-day exact repeat QuickSCAT mission. For reference, the vertical dashed lines in both panels indicate the longitudinal extent of the western and eastern corners of the CCS model domain and the two pairs of horizontal dotted lines in both panels indicate the 3.5-day sampling periods during which the CCS model domain is sampled in the two 4-day subcycles of SWOT sampling during each 21-day exact repeat period of the SWOT orbit.</figcaption></figure>

<figure><img src="image_27.png" /><figcaption>**Fig. 22.** Illustration of the two 4-day subcycles of SWOT sampling of the CCS model domain over a 14-day interval of the 21-day exact repeat period (see the left panel of Fig. 21). The swaths from the first 4-day subcycle (days 3.5–7.0) are shown in blue in the left panel and the swaths from the second 4-day subcycle (days 13.5–17.0) are shown in red in the middle panel. The combined swaths from the two 4-day subcycles are overlaid in the right panel. The intersecting diagonal lines overlaid on the right panel are the ground tracks of the 10-day exact repeat orbit of the TOPEX, Jason-1, Jason-2 and Jason-3 altimeters. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

four times by WaCM during each 4-day repeat period (see Fig. 23 below).

The sampling pattern for SWOT is considerably more complicated than for WaCM. From the left panel of Fig. 21, it can be seen that the SWOT ground tracks will map out a grid with a westward-migrating longitude spacing of 2.466° during the first half of each 21-day repeat period. This track spacing is coarse relative to the 120-km span of the two parallel measurement swaths of the SWOT radar. The SWOT ground tracks then map out another grid with the same coarse westward migrating longitude spacing of 2.466° during the second half of the 21-day repeat period. The grid of ground track overpass times from the second half of each repeat period is interleaved spatially half

way between the grid from the first half, thus resulting in a longitude spacing of 1.233° of the ground tracks during each 21-day repeat period. Within the longitudinal extent of the CCS model domain, this interleaved pattern of ground tracks results in the two 4-day subcycles indicated by pairs of horizontal dotted lines in Fig. 21. The 4-day subcycles are separated by 6-day gaps in coverage.

In addition to the sampling summarized in the left panel of Fig. 21 for the ascending SWOT orbits, there is analogous sampling on descending orbits. These descending orbits occur about half a day later than the ascending orbits.

With the 120-km span of the two parallel measurement swaths of the SWOT radar and its 77.6° orbit inclination, the 1.233° longitudinal
<figure><img src="image_28.png" /><figcaption>**Fig. 23.** Histograms of the numbers of samples by SWOT and WaCM (thin and thick lines, respectively) during the first 4 days and the full 14 days (top and bottom panels, respectively) of the 14-day sampling period for the CCS model domain during each 21-day exact repeat period of the SWOT orbit. The histogram values are expressed as percentages of the total number of grid points in the CCS model domain. The histograms for WaCM are based on a swath width of 1200 km. The improved sampling by WaCM with a swath width of 1800 km is shown later in Fig. 42.</figcaption></figure>

spacing of ground tracks provides very nearly complete coverage of the Earth's surface between about 78°N and 78°S during each 21-day repeat period. The interleaved sampling of the CCS region in two 4-day subcycles separated by 6-day gaps is shown in the left and middle panels of Fig. 22. It can be seen from the right panel of Fig. 22 that there are small diamond-shaped regions that are never sampled by SWOT. These diamond-shaped regions account for about 3% of the area in the CCS region. For context, the thick lines in the right panel of Fig. 22 show the coarse ground track spacing of the 10-day repeat orbit that has been maintained since September 1992 by the TOPEX/Poseidon, Jason-1, Jason-2 and Jason-3 sequence of nadir altimeters. An important aspect of the SWOT sampling over the 21-day repeat period is the characteristic noted previously from the left panel of Fig. 21 that spatially adjacent parallel swaths are sampled about 10 days apart. It will be seen in Section 8 that this results in severe artifacts in maps of the variables of interest constructed from multiple satellite overpasses because of the effects of the first of the three sampling errors summarized above.

From the space-time sampling pattern of the SWOT orbit, there are two natural choices of time scale for SWOT sampling of the CCS model domain considered in this study. About 73% of the region is sampled in each of the two 4-day periods but approximately 14 days are required for SWOT to sample the full CCS region (excluding the above-noted small diamond-shaped regions that are never sampled). Histograms of SWOT and WaCM sampling of the CCS model domain are shown in Fig. 23 for 4-day and 14-day periods. For the case of the 4-day period shown in the top panel, 52% of the grid points in the CCS model are sampled once by SWOT and 27% are not sampled at all. The remaining 21% of the grid points that are sampled twice correspond to the regions of overlap of the measurement swaths from the intersecting ascending and descending ground tracks. During the same 4-day period, about

90% of the model grid points are sampled four times by WaCM with a 1200-km swath (see Section 10.1 for the case of an 1800-km swath), and about 9% of the grid points are sampled five times. About 1% of the grid points are sampled only three times by WaCM.

Histograms of the SWOT and WaCM sampling for the 14-day period are shown in the bottom panel of Fig. 23. About 53% of the CCS model grid points are sampled twice by SWOT during this 14-day period while about 26% and 17% of the grid points are sampled once and three times, respectively. The 3% of the grid points that are never sampled by SWOT correspond to the white diamond-shaped regions in Fig. 22. During the same 14-day period, about 88% of the model grid points are sampled either 13 or 14 times by WaCM. All but about 1% of the remaining grid points are sampled more than 14 times.

The effects of the sampling characteristics summarized above on maps of space-time smoothed velocity and vorticity constructed from simulated SWOT and WaCM data over the CCS model domain are investigated in Sections 8–10, both separately and in combination with the effects of the measurement errors that were previously considered in isolation in Section 6.

## 8. The effects of combined measurement and sampling errors on estimates of time-averaged velocity and vorticity fields

The improvements of the signal-to-noise ratios of SWOT and WaCM estimates of velocity and vorticity that can be achieved from spatial smoothing alone within a single measurement swath were assessed in Section 6. Mapping the variables of interest over a domain larger than a single swath of the instrument requires time averaging of measurements from multiple satellite orbits. While time averaging can mitigate the effects of uncorrelated measurement errors, it can also introduce artifacts from the three types of sampling errors discussed in Section 7. The question addressed in this section is whether the benefits of the attenuation of measurement errors from time averaging prevail over the added effects of sampling errors. It will be shown that sampling errors are the primary issue for SWOT and measurement errors are the primary issue for WaCM.

Because of the 4-day and 14-day subcycles in the 21-day repeat orbit of SWOT that were discussed in Section 7, two natural choices for temporal averaging period are 4 days and 14 days. These are the times required for, respectively, coarse and nearly complete SWOT coverage of the full CCS model domain (see Fig. 22). The 4-day averaging period matches the exact repeat period of the orbit assumed here for WaCM. With the swath width of 1200 km used for the simulations of WaCM data in this section, essentially all of the grid points in the CCS model domain are sampled by WaCM at least four times during each 4-day repeat period (see Fig. 23). In contrast, none of the grid points are sampled more than twice during a 4-day subcycle of the SWOT orbit; about half are sampled only once and about a quarter are not sampled at all. SWOT therefore benefits much less than WaCM from error reduction by time averaging. Moreover, the sampling errors discussed in Section 7 are much more severe for SWOT because its swath width is narrower by more than an order of magnitude than the 1200-km swath width of WaCM assumed for the analysis in this section. The benefits of increasing the WaCM swath width to 1800 km are summarized in Section 10.

### 8.1. *Time averages of error-free velocity and vorticity*

Before determining the S/N ratios of SWOT and WaCM estimates of velocity and vorticity from space-time smoothed fields of noisy SSH and velocity components, it is useful to provide context by looking at the effects of time averaging alone on the velocity and vorticity fields of interest. Maps of the magnitude of error-free surface velocity and vorticity from the CCS model at the full 0.5 km × 0.5 km grid resolution of the model with no spatial smoothing are shown in Fig. 24 for a snapshot and for 4-day and 14-day vector-averaged velocity centered on the time of the snapshot. The two time averages were computed from snapshots of the model output at intervals of 0.5 days.
<figure><img src="image_29.png" /><figcaption>**Fig. 24.** An illustration of the effects of time averaging on error-free surface velocity and vorticity fields centered on the same date as the snapshots in Fig. 3: Column (a) The instantaneous snapshots (the same as the top panels of Fig. 3a and b, repeated here for easy comparison with the other two panels of this figure); Column (b) Four-day average maps; and Column (c) Fourteen-day average maps. The averages were constructed from model output at intervals of 0.5 days. The velocity maps in the top panels of columns (b) and (c) are the magnitudes of the 4-day and 14-day vector-averaged velocity fields.</figcaption></figure>

The periodic structures with very small scales in the vorticity fields in the bottom panels of Figs. 24b and c that are not present in instantaneous maps such as that shown in the bottom panel of Fig. 24a are likely aliasing artifacts from discrete sampling of submesoscale features that are rapidly advected by the strong jet-like currents that are evident in the top panels. Such aliasing issues would not occur if the model output had been saved at a sufficiently fine sample interval, e.g., hourly rather than twice per day. The existence of this aliasing emphasizes how quickly the small-scale variability evolves, thus exposing one of the challenges in mapping submesoscale variability from satellite observations.

Notwithstanding the possible aliasing artifacts, it is visually apparent that the energetic small-scale variance in the instantaneous maps in Fig. 24a is dramatically reduced in the 4-day and 14-day averages

in Figs. 24b and c. This is quantified by the alongshore wavenumber spectra shown in Fig. 25. In the case of velocity, the variance at the smallest wavenumbers (longest wavelengths) is reduced by modest factors of about 2 and 3 in 4-day and 14-day averages, respectively. The amount of variance attenuation from time averaging alone increases monotonically with increasing wavenumber. At a wavenumber of 0.02 cpkm (a wavelength of 50 km), for example, the variance of the magnitude of the vector-averaged velocity is reduced by about a factor of 4 in the 4-day average and about a factor of 11 in the 14-day average. At a wavelength of 10 km, these reductions of variance increase to factors of about 9 and 49, respectively.
<figure><img src="image_30.png" /><figcaption>**Fig. 25.** Alongshore wavenumber spectra of velocity and normalized vorticity computed from the model surface velocity fields for the instantaneous snapshot, the 4-day average and the 14-day average (thin, medium and thick lines, respectively) shown in Fig. 24. The velocity spectra for 4-day and 14-day averages were computed from the magnitudes of the vector-averaged velocity fields over the respective time periods. The smoothed spectra were computed by ensemble averaging as in Figs. 13–15, except from the raw 0.5 km × 0.5 km grid with a cross-shore spacing of 5 km.</figcaption></figure>

The variance reductions from time averages alone are even greater for vorticity. The vorticity variance at the longest wavelengths is reduced by factors of about 4 and 8 in the 4-day and 14-day averages, respectively. These variance reductions increase to factors of about 7 and 29 at a wavelength of 50 km and factors of about 16 and 80 at a wavelength of 10 km.

The reductions of variance at the lowest wavenumbers in the time-averaged fields in both panels of Fig. 25 are perhaps surprising. This apparently occurs because the most energetic features with large scales in the alongshore dimension are generally narrow in the cross-shore dimension. The locations of these elongated features evolve quickly in time. Their contributions to the alongshore wavenumber spectrum are thus attenuated in the 4-day and 14-day averaged velocity and vorticity fields.

As summarized in Section 5, our metric for defining the resolution capability is based on the square root of the ratio of the spatial variance of the signal to the spatial variance of the total errors. In the space-time smoothed fields considered in Sections 8.4, 8.5, 9 and 10, the total errors are partitioned between measurement errors and sampling errors. In order for the S/N ratio to improve, the space-time smoothing must attenuate the total error variance more than the amount by which the signal variance is attenuated by the same space-time smoothing.

## 8.2. The data processing procedure for assessing resolution capability

The procedure for assessing the relative importance of measurement errors and sampling errors and determining the resolution capabilities for SWOT and WaCM estimates of space-time smoothed velocity and vorticity is the following:

Step 1. Twice-daily snapshots of SSH (for simulated SWOT data) and surface velocity (for simulated WaCM data) were constructed on the complete 0.5 km × 0.5 km model grid in two combinations:

(i) The error-free model output of SSH and surface velocity.

(ii) The model output of SSH and surface velocity with Gaussian-distributed uncorrelated measurement errors added. To simulate SWOT measurement errors, the uncorrelated errors added to the SSH fields had a standard deviation of 5.48 cm on the 0.5 km × 0.5 km model grid. To simulate WaCM measurement errors, the uncorrelated errors added to each velocity component had a standard deviation of 3.54 m s⁻¹ on the 0.5 km × 0.5 km model grid. When smoothed in step 2 below to achieve footprint diameters of 1 km and 5 km for SWOT and WaCM, respectively, this yields the desired measurement error standard deviations of σ<sub>h</sub> = 2.74 cm for simulated pre-processed SWOT estimates of SSH and σ<sub>u,v</sub> = 0.354 m s⁻¹ for simulated pre-processed WaCM estimates of each velocity component. The latter corresponds to the baseline standard deviation of σ<sub>spd</sub> = 0.50 m s⁻¹ for speed measurement errors equally partitioned between the two velocity components.

Step 2. To simulate pre-processing of SWOT data, the twice-daily error-free and noisy SSH fields from step 1 were smoothed isotropically on the 0.5 km × 0.5 km model grid with a half-power filter cutoff wavelength of 2 km to achieve the desired SWOT footprint diameter of 1 km. To simulate pre-processing of WaCM data, the twice-daily error-free and noisy surface velocity fields from step 1 were similarly smoothed isotropically on the 0.5 km × 0.5 km model grid with a half-power filter cutoff wavelength of 10 km to achieve the WaCM footprint diameter of 5 km. The smoothed SSH and surface velocity fields on the 0.5 km × 0.5 km model grid were then subsampled on a 1 km × 1 km grid to simulate the pre-processed SWOT and WaCM data used throughout the rest of this study.

Step 3. The twice-daily snapshots of simulated pre-processed SWOT estimates of SSH and WaCM estimates of surface velocity from step 2 were sampled on the complete grid and at only the times and locations of the in-swath measurements to obtain maps with four different combinations of signal and simulated measurement and sampling errors for each choice of averaging period (4 days and 14 days):

(i) Error-free SSH and surface velocity signals on the complete 1 km × 1 km grid.

(ii) SSH and surface velocity signals plus Gaussian distributed uncorrelated measurement errors on the complete 1 km × 1 km grid.

(iii) Error-free SSH and surface velocity signals sampled only at the 1 km × 1 km grid points within the measurement swaths at the time of each individual satellite overpass rounded to the nearest 0.5-day time step of the model.

(iv) SSH and surface velocity signals plus Gaussian distributed uncorrelated measurement errors sampled only at the 1 km × 1 km grid points within the measurement swaths at the time of each individual satellite overpass rounded to the nearest 0.5-day time step of the model.

Step 4. For each of the two averaging periods considered here (4 days and 14 days), the time-averaged error-free SSH and surface
<figure><img src="image_31.png" /><figcaption>**Fig. 26.** Maps of 4-day averaged SSH computed 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 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 from model SSH fields at a time step of 0.5 day over the 4-day period with simulated uncorrelated measurement errors with a standard deviation of $\sigma_h$ = 2.74 cm; Column (c) The 4-day average computed 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 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 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>

velocity component fields on the 1 km × 1 km grid (map i from step 3) and the time-averaged maps of SSH and surface velocity with the three combinations of errors in maps ii–iv from step 3 (measurement errors alone, sampling errors alone, and combined measurement and sampling errors) were additionally smoothed spatially with half-power filter cutoff wavelengths ranging from 10 to 150 km.

Step 5. Velocity and vorticity were computed geostrophically from each set of four space-time smoothed maps of SSH from step 4 using centered difference estimates of the spatial derivatives in order to assess the resolution capabilities for SWOT. Likewise, vorticity was computed from each set of space-time smoothed maps of surface velocity components from step 4 using centered difference estimates of the spatial derivatives to assess the resolution capabilities for WaCM.

Step 6. The errors for each set of three space-time smoothed fields with measurement and/or sampling errors from step 5 were computed for each variable of interest by subtracting the space-time smoothed error-free value on the complete 1 km × 1 km grid from the space-time smoothed error-contaminated value at each 1 km × 1 km grid point. The sets of three error fields as functions of the half-power filter cutoff wavelength of the spatial smoothing allow assessments of the

effects of measurement errors alone, sampling errors alone, and the combination of measurement and sampling errors.

Step 7. The spatial variances of each of the three error fields obtained in step 6 and the spatial variance of the space-time smoothed error-free signals from step 5 were computed from the 1 km × 1 km gridded values for each variable of interest for each of the half-power filter cutoff wavelengths of the spatial smoothing. In the case of SWOT, the 1 km × 1 km grid points outside of the sample swaths during the 4-day and 14-day averaging periods (see Fig. 22) were excluded from the variance calculations. There were no grid points with missing values in any of the WaCM fields for either time average.

Step 8. The resolution capabilities for SWOT and WaCM estimates of each variable of interest were defined as summarized in Section 5 to be the half-power filter cutoff wavelength above which the signal-to-noise variance ratios exceed a value of 10 (i.e., a signal-to-noise standard deviation ratio of 3.16). This half-power filter cutoff wavelength was estimated by linear interpolation between the discretely chosen filter cutoff wavelengths that were applied in step 4.
### 8.3. SWOT Estimates of Space-Time Smoothed Sea Surface Height

To provide context for understanding the combined effects of measurement and sampling errors on SWOT estimates of surface ocean velocity and vorticity that are the primary interest of this study, it is enlightening to look at time averages of the simulated SWOT measurements of SSH from which surface velocity and vorticity are computed geostrophically in Sections 8.4 and 8.5.

We first consider the 4-day subcycle of the SWOT orbit with the measurement swaths shown in the left panel of Fig. 22. The four maps of 4-day average SWOT measurements of SSH from step 4 of the procedure summarized in Section 8.2 are shown in the top four panels of Fig. 26 for the case of 2-dimensional spatial smoothing with a half-power filter cutoff wavelength of 25 km. The map in the top panel of Fig. 26d simulates the 4-day average SSH field that would be obtained from SWOT data based on the combination of SWOT sampling errors and measurement errors with a standard deviation of $\sigma_h = 2.74$ cm for a footprint diameter of 1 km. The maps in the top panels of Figs. 26b and c represent the SSH fields that would be obtained from SWOT data if it were possible to isolate, respectively, the measurement errors and sampling errors in real data. The bottom three panels are the differences of each of the three error-contaminated fields in the associated top panels minus the error-free field in Fig. 26a.

The background mottled appearance of the error field in the bottom panel of Fig. 26d represents the residual uncorrelated measurement errors after smoothing with the half-power filter cutoff wavelength of 25 km used for Fig. 26. The most important thing to note is that the largest errors with magnitudes exceeding 1 cm in the bottom panel of Fig. 26d coincide geographically with the largest errors in the bottom panel of Fig. 26c that includes only the effects of sampling errors. The total errors in Fig. 26d are thus seen to be dominated by sampling errors, at least in the regions of most energetic submesoscale variability in the eastern portion of the CCS model domain.

The 4-day average SSH maps in Figs. 26c and d are affected by two of the three sources of sampling errors discussed in Section 7. Some of the sampling errors in the bottom panels of Figs. 26c and d are edge-effect artifacts from incomplete data within the span of the smoother near the edges of the SWOT measurement swaths. But most are aliasing artifacts from inaccurate representation of the 4-day average of the SSH field. As discussed in Section 7 from the top panel of Fig. 23, 27% of the CCS model domain is not sampled at all during the 4-day period. These white areas in Figs. 26c and d are sampled over the second 4-day subcycle that begins 6 days later (see the middle panel of Fig. 22). Within the 73% of the CCS model domain that is measured by SWOT during the first 4-day subcycle, 52% of the region is sampled only once and 21% is sampled twice. The latter corresponds to the diamond-shaped areas formed by the intersections of ascending and descending measurement swaths. One or two samples are not adequate to resolve the rapidly evolving submesoscale features during the 4-day averaging period.

Because none of the grid points in the CCS model domain are sampled more than twice and about half are sampled only once, time averaging over the 4-day subcycle does little to reduce the measurement errors. Most of the reduction of measurement errors in Fig. 26d is therefore attributable to the 25-km spatial smoothing that reduces the $\sigma_h = 2.74$ cm standard deviation of the uncorrelated noise by about a factor of 12.5 (see Appendix D and Fig. 12a) and results in the background mottled appearance noted above.

It is noteworthy that the SSH errors in the bottom panel of Fig. 26b that includes only the effects of measurement errors are much smaller than the mottled background noise from residual measurement errors in the bottom panel of Fig. 26d, even in the regions of weak submesoscale variability where sampling errors are relatively small in Fig. 26d. This is because the measurement errors in Fig. 26b are much more effectively suppressed by the averaging of the eight individual SSH maps at 0.5-day intervals during the 4-day averaging period. From

the sampling characteristics noted above, only 21% of the CCS model domain benefits from temporal averaging of the measurement errors. But those regions are sampled only twice during the 4-day averaging period, thus reducing the measurement noise by half as much as the averaging of eight realizations in Fig. 26b. The remaining 79% of the CCS model domain is sampled only once (52%) or not at all (27%) during the 4-day period and thus do not benefit at all from temporal averaging of the measurement errors.

Intuitively, it may seem that the errors in Fig. 26d can be suppressed by increasing the averaging period to 14 days to increase the number of samples, thus also expanding the SWOT coverage to nearly all of the CCS model domain (see the right panel of Fig. 22). However, because the SWOT sampling consists of only two 4-day subcycles during the 14-day period, measurement errors are only slightly reduced in the longer 14-day average. This is because only about 17% and 1% of the region are sampled three and four times, respectively; about 53% of the CCS model domain is sampled twice during the 14-day period and about 26% is sampled only once.

More importantly than the small reduction of the effects of measurement errors from the modest increase in the numbers of samples at any given location, the longer averaging period of 14 days includes considerably more temporal evolution of the energetic submesoscale features compared with the 4-day averaging period for Fig. 26. The SWOT sampling during the 14-day period consists of two 4-day subcycles that are interleaved spatially and separated by a 6-day gap. The sampling errors are consequently dramatically worse in the 14-day average SSH field than in the 4-day average SSH field (compare the bottom panels of Figs. 27c and d with the bottom panels of Figs. 26c and d). In addition to the inadequate resolution of the rapidly evolving submesoscale features and the edge effects of smoothing that were both evident in the 4-day average maps in Fig. 26, the third source of sampling errors discussed in Section 7 arises in the 14-day average from discontinuities of SSH at the boundaries of spatially adjacent measurement swaths that are separated by about 10 days in time (see the left panel of Fig. 21). It will be seen in Sections 8.4 and 8.5 that the velocity and vorticity computed geostrophically from centered difference estimates of the derivatives of the SSH field at these boundaries misinterpret temporal variability as spatial variability.

It is clear from Fig. 26 and especially from Fig. 27 that errors from inadequate sampling of rapidly evolving submesoscale features will severely impact the accuracy of geostrophically computed velocity and vorticity fields that are defined in terms of first and second derivatives, respectively, that are approximated by centered differences of the SSH in space-time smoothed simulated noisy SWOT data. In contrast, the far superior sampling by WaCM and the fact that it measures velocity directly and thus requires only first derivatives of velocity to estimate vorticity, sampling errors are much less of an issue for WaCM than for SWOT. This is shown below from simulated SWOT and WaCM estimates of surface velocity and vorticity in Sections 8.4 and 8.5, respectively.

### 8.4. The resolution capabilities of time-averaged surface velocity fields

As in Section 6, the simulated SWOT estimates of velocity in this section neglect the limitations of the geostrophic approximation discussed in Section 3. The question addressed here for SWOT is thus how the combined SSH measurement errors and sampling errors affect the resolution capabilities of geostrophically computed SWOT estimates of velocity, irrespective of the accuracy of the geostrophic approximation. It will be seen that the resolutions that can be achieved in SWOT estimates of velocity are so coarse that the geostrophic approximation may not be a major limitation.

The four maps of the magnitudes of 4-day vector-averaged geostrophically computed SWOT estimates of velocity from step 5 of the procedure summarized in Section 8.2 are shown in the top four panels of Fig. 28 for the case of spatial smoothing with a half-power filter cutoff wavelength of 25 km. These are the magnitudes of the
<figure><img src="image_32.png" /><figcaption>Fig. 27. The same as Fig. 26, except for 14-day averages of SSH computed 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>

surface velocity field computed geostrophically from the simulated SWOT estimates of SSH with 25-km smoothing in the top four panels of Fig. 26. The map in the top panel of Fig. 28d simulates the magnitude of the velocity field that would be estimated geostrophically from SWOT data based on the combination of SWOT sampling and simulated measurement errors with a standard deviation of $\sigma_h = 2.74$ cm. The maps in the top panels of Figs. 28b and c represent the magnitudes of the velocity fields,<fnref n="4" /> that would be obtained from SWOT data contaminated by, respectively, measurement and sampling errors individually. With real observations, it is of course not possible to separate the two sources of errors in space–time smoothed maps. Analogous to Fig. 26, the bottom three panels are the differences of each of the three error-contaminated fields in the associated top panels minus the error-free field in Fig. 28a.

As in the case of the SSH mapping errors in Figs. 26c and d, the largest errors in the velocity fields in the bottom panel of Fig. 28d coincide geographically with the largest errors in the bottom panel of Fig. 28c that includes only the effects of sampling errors. The total errors in Fig. 28d are thus dominated by sampling errors, at least in the regions of most energetic submesoscale variability in the eastern portion of the CCS model domain. These sampling errors arise primarily from inaccurate representation of the 4-day average of the velocity field because of unresolved space–time variability of the rapidly evolving submesoscale features in the SSH field.

While the patterns of the total error field in the bottom panel of Fig. 28d are generally similar to the patterns of sampling errors alone in the bottom panel of Fig. 28c, there is considerably more small-scale variability in the total error field. The small-scale noise in the total error field in the bottom panel of Fig. 28d is much more energetic than the small-scale noise in the map of measurement errors alone in the bottom panel of Fig. 28b. As discussed in Section 8.3 from comparisons of the SSH errors in the bottom panels of Figs. 26b and d, this is because the simulated measurement errors in Fig. 28b have been much more effectively suppressed by the averaging of the eight individual maps at 0.5-day intervals during the 4-day averaging period. The effects of measurement errors are even more evident in the total error field for geostrophically computed velocity in the bottom panel of Fig. 28d than in the total error field for SSH in Fig. 26d because the relatively small mottled background noise in the 25-km smoothed SSH field is amplified in the centered difference estimates of the derivatives in the geostrophic equations. This amplification of noise in maps of geostrophically computed velocity constructed from simulated SWOT data has been discussed previously by Pujol et al. (2012) and Gaultier et al. (2016).

As in the case of the SSH fields considered in Section 8.3, increasing the averaging time from 4 days to 14 days does not improve the accuracy of SWOT estimates of the surface velocity field (Fig. 29). The sampling errors are even worse in the 14-day averaged geostrophically computed velocity field than in the 14-day average SSH field in Fig. 27. In addition to sampling errors from inadequate resolution of the rapidly evolving submesoscale velocity field and the edge effects of smoothing that were both evident in the 4-day average maps in Fig. 28, the sampling errors that arise from discontinuities at the boundaries of spatially adjacent measurement swaths that are separated temporally

<footnote n="4">⁴ For expediency, the magnitude of the estimated vector-averaged velocity fields will be referred to hereafter as velocity fields, with understanding that “velocity” implicitly refers to the magnitude of the time-averaged vector velocity field.</footnote>