1. Introduction

Presently available sea surface height (SSH) fields constructed from measurements by multiple nadir-viewing satellite altimeters are able to resolve midlatitude variability on wavelength scales down to about 200 km and monthly time scales (Chelton et al. 2011; Ballarotta et al. 2019). This resolution capability will soon be greatly improved by SSH measurements from the Surface Water and Ocean Topography (SWOT) satellite (Fu and Ferrari 2008; Durand et al. 2010; Fu and Ubelmann 2014) that has a target launch date in November 2022. The primary instrument on SWOT is the Ka-band Radar Interferometer (KaRIn) that will determine SSH from measurements at a microwave frequency of 35.75 GHz using radar interferometry and synthetic aperture radar technology (see appendix B; see also Fu and Rodríguez 2004). KaRIn will estimate SSH with a footprint diameter of order 1 km across a pair of swaths 50 km wide separated by a 20-km gap centered on the satellite ground track. The high accuracy and spatial resolution of the KaRIn data and the wide KaRIn measurement swath will significantly improve the space–time resolution capability of SSH fields. This will greatly benefit studies of mesoscale ocean dynamics. Among other applications, the smaller errors in maps of SSH constructed from SWOT data will reduce the mislocations of mesoscale eddies in presently available SSH fields (e.g., Pascual et al. 2006). The innovative measurements of SSH by KaRIn will also provide insight into ocean dynamics in the submesoscale regime, and the role of these small scales in influencing or connecting to the dynamics at larger scales.

While the primary goal of the SWOT mission is to improve the present SSH mapping capabilities, SWOT will also enable the first spaceborne estimates of instantaneous surface vector velocity and relative vorticity (referred to hereinafter as vorticity) that can be computed geostrophically from the gridded SSH fields within the two SWOT measurement swaths. Estimates of these variables from the SSH fields constructed from merged multimission altimeter data are limited by the above-noted 200-km wavelength and monthly time-scale resolution of these SSH fields. Surface vector velocity can be estimated geostrophically from a single nadir-viewing altimeter on the coarse grid of crossovers of ascending and descending ground tracks. However, in addition to the effects of SSH measurement noise on errors in the geostrophically computed across-track velocities, the accuracies of the vector velocity estimates at the crossovers are limited by sampling errors from nonsimultaneity of the ascending and descending overpasses and errors from the geometrical transformation of the two across-track velocity estimates that are in general nonorthogonal at the crossovers. The sampling errors and geometrical transformation errors both vary latitudinally because of latitudinal variations of the time separation and angle between ascending and descending ground tracks (Schlax and Chelton 2003).

The ability to estimate vector velocity and vorticity instantaneously by altimetry is thus unique to the gridded SSH estimates across the two parallel KaRIn measurement swaths. Although the SWOT mission was not designed to determine the surface velocity and vorticity fields, many researchers intend to investigate ocean dynamics from SWOT estimates of these variables. The equations from which the standard deviations and wavenumber spectra of noise can be computed for all of the variables of interest (SSH, geostrophically computed velocity components u_g and υ_g, and the vorticity ζ_g) are presented in this study for any choice of footprint diameter in the ground-based processing and subsequent additional spatial smoothing, referred to here as postprocessing.

Our reference to u_g and υ_g as “geostrophically computed velocity components” rather than “geostrophic velocity components” is deliberate. There are several reasons the pressure field might not be in geostrophic balance with the velocity field. For example, the geostrophic approximation is valid only for small Rossby number. Because of the high incidence of large Rossby numbers at small scales (see Figs. 4 and 5 of Chelton et al. 2019, referred to hereinafter as C19), the velocity components computed from the geostrophic equations (see appendix D) give velocity estimates that may not be truly geostrophic on all scales. A significant concern is the possibility of erroneous contributions to the computed velocity from gradients of the ageostrophic SSH signatures of internal tides and internal waves with short wavelengths that may be resolvable in the SWOT data. Because of the large degree of smoothing that will have to be applied in postprocessing to mitigate the effects of measurement noise on estimates of velocity and vorticity computed geostrophically from SWOT data (see sections 3–5), most of the ageostrophic signals will be attenuated by the filtering. We therefore neglect errors of the geostrophic approximation in the analysis that follows.

It is noteworthy that the surface velocity field on the large scales that are resolvable in SWOT data includes wind-driven Ekman currents. These large-scale ageostrophic contributions to velocity cannot be addressed from velocities computed geostrophically from SWOT data (Yu et al. 2021).

The SWOT satellite will also include a dual-frequency nadir altimeter that will measure SSH at Ku-band and C-band frequencies of 13.6 and 5.3 GHz with a footprint size and measurement precision that are very different from KaRIn. The footprint for 1-s averages of nadir altimeter measurements from the SWOT orbit altitude of 890 km for a significant wave height (SWH) of 2 m has across-track and along-track dimensions of approximately 3.5 km × 10 km (see Figs. 22 and 23 of Chelton et al. 2001). These footprint dimensions increase with increasing SWH. The nadir altimeter measurements are expected to have a noise standard deviation of 1.7 cm (see Table 5 of JPL 2017a). In comparison, the noise standard deviation of KaRIn measurements of SSH for a much smaller footprint diameter of 2 km is expected to be about 1.4 cm (see section 2 and appendix C). Because the precision and footprint size for KaRIn differ so much from those of the nadir altimeter, the latter is not considered in this study. Hereinafter, we will generally refer to SWOT estimates of SSH with the implicit understanding that we consider only the estimates from KaRIn.

This study also considers only the effects of uncorrelated measurement errors (which we will refer to interchangeably as measurement noise) in KaRIn estimates of SSH and how they affect spatially smoothed estimates of instantaneous SSH as well as SWOT estimates of surface velocity and vorticity computed from instantaneous SSH for a chosen footprint diameter, both without additional smoothing and with spatial smoothing applied in postprocessing. The effects of long-wavelength measurement errors are not addressed in this study. Because of the high-pass filtering operation of the derivative operator, long-wavelength measurement errors have relatively little effect on SWOT estimates of velocity and vorticity. We also do not consider the sampling errors that arise from space–time smoothing of SWOT data from multiple overpasses of SWOT, which are addressed in section 8 of C19 and discussed briefly here in section 6.

The analysis in this study draws heavily from the appendixes of C19. The standard deviation σ_h of the uncorrelated noise in SWOT estimates of SSH depends on the chosen footprint diameter. Although σ_h for a specified footprint diameter is the fundamental characterization of the uncorrelated measurement errors, this information is difficult to find and ambiguous in the current versions of the SWOT documents. Furthermore, the footprint diameter of the SWOT estimates of SSH in the data products that will be provided by the SWOT Project Office has evolved from the original plan that was to provide only a single product with a footprint diameter of 1 km. The new plan is to provide two data products, one with a footprint diameter of 0.5 km and the other with a footprint diameter of 2 km. To date, the only documentations of this change are the recent SWOT Product Description Document (JPL 2020) and revision B of the Onboard Processing and Algorithm Theoretical Basis Document (JPL 2021). To the enlightened reader, hints of the change to footprint diameters of 0.5 and 2 km can be inferred from sections 2.7.1a and 2.7.1b of the current version of the Science Requirements Document (JPL 2018). The current version of the Mission Performance and Error Budget Document (JPL 2017a) describes only the original planned footprint diameter of 1 km.

An objective of this study is to present the equations for the standard deviations and along-track wavenumber spectra of the noise of all of the variables of interest (SSH, geostrophically computed velocity components, and vorticity) in terms of the standard deviation σ_h of SSH noise for each of the three footprint diameters of 0.5, 1, and 2 km. These equations are presented in a series of appendixes, beginning with a summary in appendix A of the science requirements for KaRIn. The requirement for the uncorrelated errors in KaRIn measurements of SSH that are the focus of this study are specified in a nonconventional manner in terms of the swath-averaged along-track wavenumber spectrum of the noise after smoothing the data with a half-power filter cutoff wavelength of 15 km. After an overview of the onboard and ground-based processing of SWOT data in appendix B, the swath-averaged standard deviations of the measurement noise in the SSH estimates without 15-km smoothing are derived in appendix C from the spectral characterization for the three footprint diameters of 0.5, 1, and 2 km. Our analysis assumes that the spectral characteristics of uncorrelated measurement errors are “white” (i.e., constant) for all wavenumbers up to the Nyquist wavenumber of (2Δy)⁻¹ that is associated with an along-track sample grid spacing of Δy that is equal to the footprint diameter. Note, however, that the science requirements specify a white-noise spectrum only for wavenumbers up to 1/15 cycles per kilometer (cpkm), i.e., wavelengths longer than 15 km (see appendix A).

The equations for the swath-averaged standard deviations of the noise in velocity components computed from SSH by finite-difference approximations of the derivatives in the geostrophic equations, and of the noise in vorticity estimated from finite-difference approximations of the derivatives of the geostrophically computed velocity components, are presented in appendix D. The discussion in appendix D of the standard deviations of the noise in SWOT estimates of all three variables (SSH, velocity, and vorticity) is extended in appendix E to present equations for the along-track wavenumber spectral characteristics of the noise without 15-km smoothing.

As summarized in section 2, the conclusion from the analysis in appendixes D and E is that velocity and vorticity computed geostrophically from the SWOT SSH data products will be too noisy to be used directly for most applications, even for the coarsest footprint diameter of 2 km considered here. Most analyses of SWOT data over oceans will therefore be based on the 2-km SSH product provided by the SWOT Project, with additional smoothing applied in postprocessing to reduce the noise and improve the signal-to-noise ratio. The additional smoothing will be especially important for SWOT estimates of velocity and vorticity. The equations for the standard deviations and wavenumber spectra of the noise in smoothed SWOT estimates of all three variables are presented in sections 3 and 4.

Example maps of SSH and geostrophically computed velocity magnitude and vorticity constructed from simulated SWOT data are presented in section 5 for selected choices of smoothing in postprocessing. The simulated SWOT data include uncorrelated noise with the swath-averaged standard deviation σ_h derived in appendix C for the footprint diameter of 2 km that will likely be the SWOT data product preferred by most users. The purpose of presenting these maps is to provide a visual sense of what to expect from SWOT data with various amounts of smoothing applied in postprocessing. The maps also provide insight into the contamination that can be expected from edge effects of smoothing within the 50-km width of the two measurement swaths.

For easy application, the equations in appendixes C–E and sections 3 and 4 are all expressed in terms of the across-track and along-track grid spacings Δx and Δy and standard deviation σ_h of the SWOT data product with the specified footprint diameter. The table and figures presented in this study are based on the prelaunch estimates of σ_h that are derived in appendix C for the footprint diameters of 0.5, 1, and 2 km. The SSH noise standard deviation σ_h that appears in the equations from which the table and figures were generated can be replaced with the actual value determined after launch from in-orbit data for any specified footprint diameter, grid spacing of the data, swath location, and significant wave height. Moreover, the equations presented here will be applicable to future versions of the SWOT data products in which the measurement noise standard deviations σ_h are likely to decrease through improvements in the onboard and ground-based processing of SWOT data.

2. The effects of uncorrelated measurement noise on “unsmoothed” SWOT estimates of SSH, velocity, and vorticity

The errors in SWOT estimates of SSH, velocity, and vorticity have previously been investigated by C19 for the case of a footprint diameter of 1 km that was the original plan for the gridded SWOT data product. As noted in the introduction, the SWOT Project has recently replaced the 1-km data product with two gridded SSH products with footprint diameters of 0.5 and 2 km. The error analysis in C19 is extended here in appendixes C–E to derive the standard deviations and wavenumber spectra of the noise for these new footprint diameters. We refer to SWOT estimates of SSH, velocity, and vorticity with a chosen footprint diameter as “unsmoothed.” In actuality, they are produced by two-dimensional smoothing of the raw SWOT data that have a resolution of tens of meters (see appendix B) using a filter that has a half-power filter cutoff wavelength of approximately twice the footprint diameter (see appendix B.1 of C19). SWOT estimates of SSH, velocity, and vorticity generated from SWOT data with any of the three footprint diameters considered in this study are referred to as “smoothed” when they are subsequently filtered in postprocessing to improve the signal-to-noise ratio by reducing the small-scale noise.

Expressions for the standard deviations of the noise in unsmoothed SWOT estimates of SSH, the geostrophically computed velocity components u_g and υ_g, and the vorticity ζ_g computed from u_g and υ_g are presented in appendixes C and D for each of the three footprint diameters of 0.5, 1, and 2 km. The results are summarized in the top half of Table 1. The noise standard deviation for SWOT measurements of SSH with a footprint diameter of 2 km that is the data product likely to be preferred by most users is 1.37 cm. As noted in the introduction, this is smaller than the noise deviation of about 1.7 cm in SSH measurements from conventional nadir altimeters with their much larger footprint dimensions. The noise standard deviations for SWOT estimates of velocity components and vorticity are comparable to or larger than the signal standard deviations in most regions of the World Ocean for all three footprint diameters considered here. The challenges of distinguishing signal and noise are further underscored in appendix E by the wavenumber spectra of unsmoothed SWOT estimates of velocity and vorticity that are shown to be more energetic than the signal spectra over most wavenumbers (see Fig. E1).

Table 1

The swath-averaged standard deviations of the noise in SWOT estimates of SSH, geostrophically computed velocity components u_g and υ_g and relative vorticity ζ_g. The top half of the table lists the noise standard deviations for “unsmoothed” data with footprint diameters of 0.5, 1.0, and 2.0 km. Results for the case of 0.5-km footprint diameter are shown for a sample grid spacing of 0.5 km × 0.5 km on which the measurement errors are uncorrelated, and on an oversampled grid spacing of 0.25 km × 0.25 km. The bottom half of the table lists the residual noise standard deviations for the case of 2-km footprint diameter after additional smoothing in postprocessing using a Parzen smoother with selected half-power filter cutoff wavelengths. The geostrophic calculations are based on a Coriolis parameter of f_37°N = 8.75 × 10⁻⁵ s⁻¹ at the central latitude 37°N of the model of the California Current System used here (see Fig. 3). The noise in u_g and υ_g for other latitudes can be determined by multiplying the values in the fourth column by f_37°N/f, where f is the magnitude of the Coriolis parameter at the latitude of interest. The noise of ζ_g/f can be determined by multiplying the values in the last column by (f_37°N/f)².

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The conclusions from the analysis in appendixes C–E and the noise statistics summarized in the top half of Table 1 and in Fig. E1 are thus that SWOT data will have to be smoothed in postprocessing for most ocean applications in order to reduce the noise, especially for SWOT estimates of velocity and vorticity. Such smoothing attenuates the signal as well as the noise. The noise power is attenuated more than the signal power because the SSH, velocity, and vorticity signals are more dominated by spatially correlated low-wavenumber variability (see the thick black lines in Fig. E1). The signal-to-noise ratios therefore improve with increased smoothing. Estimation of ζ_g is the most challenging because the signal is less dominated by large-scale variability (see Fig. E1d) and is therefore more attenuated by the smoothing. The effects of smoothing on the standard deviation and wavenumber spectra of residual noise in SWOT estimates of SSH, velocity, and vorticity are quantified in sections 3 and 4.

3. The standard deviations of smoothed noise in SWOT estimates of SSH, velocity, and vorticity

The equations for the swath-averaged standard deviations of residual noise in SWOT estimates of SSH, u_g, υ_g, and ζ_g after the two-dimensional smoothing in the postprocessing that will be required for most ocean applications of the SWOT data that will be provided by the SWOT Project are derived in appendix G.3 of C19. Our simulated postprocessing is based on isotropic two-dimensional smoothing using the Parzen smoother that is discussed in detail in appendix A of C19. The results would be very similar for other choices of smoother, as long as the parameters of the smoother are calibrated to give the same half-power filter cutoff wavelength λ_c. A useful characteristic of the Parzen smoother is that its filter transfer function is almost identical to that of a Gaussian smoother (see Fig. C2 in appendix C of C19). A convenient property of the Gaussian smoother is that isotropic two-dimensional smoothing that depends only on the radial distance of each data point from the estimation location can be achieved equivalently through successive one-dimensional smoothing operations in each of two orthogonal dimensions. The similarity of the filter transfer functions of the Parzen and Gaussian smoothers implies that isotropic two-dimensional smoothing with the Parzen smoother can essentially be achieved equivalently with separate one-dimensional smoothing in two orthogonal dimensions. A distinct advantage of separate one-dimensional smoothing is that it facilitates the derivations of the analytical equations in section 4 below for the wavenumber spectra of residual noise in smoothed estimates of SSH, u_g, υ_g, and ζ_g. While the Parzen smoother nearly replicates the exact equivalence of isotropic two-dimensional smoothing and sequential one-dimensional smoothing in two orthogonal dimensions for the Gaussian smoother, this equivalence is only approximate for most other smoothing algorithms.

To provide context for the analysis in this section, it is useful to display the noise standard deviations graphically for unsmoothed SWOT estimates of SSH, velocity, and vorticity as functions of the half-power filter cutoff wavelengths associated with each of the footprint diameters of 0.5, 1, and 2 km. It is shown in appendix B.1 of C19 that SSH estimates with a footprint diameter of 1 km can be achieved with isotropic two-dimensional smoothing of raw SWOT data using a Parzen smoother (see, however, the footnote in appendix B of this paper) with a half-power filter cutoff wavelength of 2 km, i.e., twice the desired footprint diameter. It is also shown in appendix B.1 of C19 that the noise in these estimates of SSH is uncorrelated on a grid spacing of 1 km × 1 km. SWOT estimates of SSH, velocity, and vorticity with the footprint diameters of 0.5 and 2 km can correspondingly be obtained by two-dimensional smoothing of raw SWOT data using Parzen smoothers with half-power filter cutoff wavelengths of 1 and 4 km, respectively, and the noise for these footprints is uncorrelated on grids with spacings equal to the respective footprint diameters. (As discussed in appendixes B and C of this study, the SWOT Project may decide to smooth the SSH with a half-power filter cutoff wavelength somewhat longer than 4 km for the 2 km × 2 km data product; such increased smoothing would not significantly alter any of the conclusions of this study.)

The standard deviations listed in the top half of Table 1 for the SSH noise with the three footprint diameters of 0.5, 1, and 2 km and sample grid spacings equal to the footprint diameters (rows 2–4 in Table 1) are displayed at their associated filter cutoff wavelengths of 1, 2, and 4 km by the solid circles in Fig. 1a. The corresponding noise standard deviations for the geostrophically computed velocity components u_g and υ_g and the vorticity ζ_g are shown by the solid circles in Figs. 1b and 1c. The noise standard deviation for SSH with the footprint diameter of 0.5 km is the same on the oversampled 0.25 km × 0.25 km grid as on the 0.5 km × 0.5 km grid. However, the noise standard deviations for u_g, υ_g, and ζ_g with the footprint diameter of 0.5 km are larger on the oversampled 0.25 km × 0.25 km grid (see row 1 in Table 1 and the triangles in Figs. 1b and 1c). This is because the finite-difference approximations of the derivatives are computed on a smaller grid spacing, which increases the noise in accord with (D3), (D4), and (D7).

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The standard deviations of residual noise as functions of half-power filter cutoff wavelength λ_c for isotropic two-dimensional smoothing of SWOT data with a Parzen smoother: (a) SSH, (b) velocity component estimates computed geostrophically from SSH based on the Coriolis parameter f_37°N = 8.75 × 10⁻⁵ s⁻¹ for a latitude of 37°N, and (c) vorticity computed from the geostrophic estimates of velocity and normalized by f_37°N. The three solid circles correspond to the error standard deviations for footprint diameters of 0.5, 1, and 2 km (filter cutoff wavelengths of 1, 2, and 4 km). The solid lines are the standard deviations of residual noise after smoothing for footprint diameters of 1 km on a 1 km × 1 km grid (thin lines) and 2 km on a 2 km × 2 km grid (thick lines). The open triangles in (b) and (c) are the noise standard deviations for a footprint diameter of 0.5 km on an oversampled 0.25 km × 0.25 km grid. The open circles in (b) and (c) are the standard deviations for a footprint diameter of 1 km on a 1 km × 1 km grid smoothed with the half-power filter cutoff wavelength of 4 km; these noise standard deviations are larger than for the footprint diameter of 2 km on a 2 km × 2 km grid because the finite differences are computed on a smaller grid spacing.

Citation: Journal of Atmospheric and Oceanic Technology 39, 7; 10.1175/JTECH-D-21-0167.1

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Because of the excessively large noise standard deviations for all three variables with the footprint diameter of 0.5 km, this case is not considered further in the discussion that follows. The standard deviations of the residual noise in SWOT estimates of SSH, u_g, υ_g, and ζ_g after isotropic two-dimensional smoothing in postprocessing are determined in this section only for the footprint diameters of 1 and 2 km.

The right sides of the forms (3a) and (4a) for the variances of smoothed estimates of υ_g and u_g are the same as the right sides of the expressions (D3b) and (D4) for the variances of noise in unsmoothed υ_g and u_g, except with overbars to indicate the variances and lagged autocorrelations of smoothed estimates. The lagged autocorrelations for smoothed SSH inside the square brackets in (3) and (4) are always larger than their counterpart lagged autocorrelations for unsmoothed SSH inside the square brackets in (D3b) and (D4). The forms (3b) and (4b) for ${\overline{\sigma }}_{{\upsilon }_g}^2$ and ${\overline{\sigma }}_{u_g}^2$ allow calculation of the variances of the residual noise in smoothed geostrophically computed SWOT estimates of the velocity components for any choice of half-power filter cutoff wavelength λ_c from the variance ${\sigma }_h^2$ of the unsmoothed SSH noise for the footprint diameter of interest, the specified across-track and along-track grid spacings Δx and Δy, respectively, and the lagged autocorrelations ${\overline{\rho }}_h(2\mathrm{\Delta }x)$ and ${\overline{\rho }}_h(2\mathrm{\Delta }y)$ of the smoothed SSH for the particular choice of λ_c.

Some general comments can be made on the interpretations of (3) and (4). As discussed above from (1), the residual variance ${\overline{\sigma }}_h^2$ of the smoothed SSH noise is independent of footprint diameter for any specified filter cutoff wavelength λ_c when the grid spacings are chosen commensurately to be equal to the footprint diameter. In contrast, the variances of residual noise in u_g and υ_g depend on the footprint diameter through their dependencies on the variance ${\sigma }_h^2$ of unsmoothed SSH noise that itself depends on the footprint diameter. The factor-of-4 difference between the values of ${\sigma }_h^2$ for footprint diameters of 1 and 2 km on the right sides of (3b) and (4b) exists for any choice of filter cutoff wavelength λ_c. With increased smoothing, however, this factor-of-4 difference is moderated by the lagged autocorrelations inside the square brackets that increase with increasing λ_c. Because the lag distances 2Δx and 2Δy are twice as large for the commensurate grid spacings associated with the 2-km footprint diameter as for the 1-km footprint diameter, the autocorrelation values at these lags are always smaller for the larger footprint diameter. For small half-power filter cutoff wavelength λ_c, these differences in the lagged autocorrelation values are substantial.

Although not readily apparent, the terms in square brackets in (3) and (4) converge with increasing λ_c toward values that are a factor of 4 larger for the 2-km footprint diameter than for the 1-km footprint diameter. This factor-of-4 increase offsets the factor-of-4 smaller value of ${\sigma }_h^2$ for the 2-km footprint diameter. The net effect is that the variances of the residual noise in smoothed estimates of u_g and υ_g from the two footprint diameters converge with increasing λ_c. An intuitive explanation for the convergence of the variances of the residual noise in u_g and υ_g for the two footprint diameters with increasing λ_c is given later in section 4 from the equations for the wavenumber spectra of smoothed u_g and υ_g noise.

As in the cases of the variances (3) and (4) of the residual noise in smoothed estimates of u_g and υ_g, the variance ${\overline{\sigma }}_{{\zeta }_g}^2$ of the residual noise in smoothed ζ_g in the form (6b) can be calculated for any choice of half-power filter cutoff wavelength λ_c from the variance ${\sigma }_h^2$ of the unsmoothed SSH noise for the footprint diameter of interest, the specified grid spacings Δx and Δy [assumed to be the same in (6)] and the three lagged autocorrelations of smoothed SSH on the right side of (6). Whereas ${\overline{\sigma }}_{u_g}^2$ and ${\overline{\sigma }}_{{\upsilon }_g}^2$ depend on the autocorrelation of smoothed SSH at the single lag 2Δx, the variance ${\overline{\sigma }}_{{\zeta }_g}^2$ of the residual noise in smoothed estimates of ζ_g depends on the autocorrelations of smoothed SSH at lags of 2Δx, $2\sqrt{2}\mathrm{\Delta }x$ and 4Δx.

Numerical values of the standard deviations ${\overline{\sigma }}_h,\hspace{0.17em}{\overline{\sigma }}_{u_g},\hspace{0.17em}{\overline{\sigma }}_{{\upsilon }_g},\hspace{0.17em}{\overline{\sigma }}_{{\zeta }_g}$, and ${\overline{\sigma }}_{{\zeta }_g}/f$ of the residual noise in the smoothed variables are listed in the bottom half of Table 1 for four choices of the half-power filter cutoff wavelength λ_c. The case of λ_c = 15 km corresponds to the filtering in the specification of the science requirements for SWOT estimates of SSH that are discussed in appendix A. The cases of λ_c = 30, 50, and 70 km that are listed in the table include all of the choices of smoothing that are applied for the spectra in section 4 and for the maps in section 5.

The variances ${\overline{\sigma }}_h^2$, ${\overline{\sigma }}_{u_g}^2$, ${\overline{\sigma }}_{{\upsilon }_g}^2$, and ${\overline{\sigma }}_{{\zeta }_g}^2$ of the residual noise of the smoothed variables computed from (1), (3), (4), and (6) could have been computed alternatively from Parseval’s theorem (C2) in appendix C by integrating the wavenumber spectra of the residual noise after smoothing, as presented below in section 4. An advantage of that approach is that the expressions for the spectra isolate the separate filtering operations from the smoothing and the three-point centered differencing on the residual variances of the smoothed noise computed from SWOT data. For grid spacings Δx and Δy equal to the footprint diameter that is the minimum interval at which the SSH noise is uncorrelated, the products $\mathrm{\Delta }x\mathrm{\Delta }y {\sigma }_h^2$ that appear in the equations for the spectra of all of the smoothed variables (see section 4) are independent of footprint diameter, as discussed previously from (1). The only differences between the expressions for the footprint diameters of 1 and 2 km are thus the factors that involve the response function (E2) of the three-point centered difference estimates of the derivatives. The differences between the thin and thick lines in Figs. 1b and 1c are thus solely attributable to the filtering effects of finite-difference approximations of the derivatives on the different grids. This point is discussed further in section 4.

4. The wavenumber spectra of smoothed noise in SWOT estimates of SSH, velocity, and vorticity

The solution (12) for $c_{{\lambda }_c}$ can be understood qualitatively from consideration of smoothing with a simple running average. In one dimension with a span of MΔx for a grid spacing Δx on which the measurement noise is uncorrelated, i.e., a grid spacing equal to or larger than the footprint diameter (see appendix B.1 of C19), a block average of M uncorrelated observations reduces the noise variance by the well-known factor of M⁻¹. It is shown in appendix A of C19 that the half-power filter cutoff wavelength for running averages with a span of MΔx is given approximately by λ_c = 2MΔx. The number of points in the block average is thus related to the filter cutoff wavelength by M = λ_c/(2Δx). The noise variance reduction factor can therefore be expressed in terms of the filter cutoff wavelength as $M^{-1}=2\mathrm{\Delta }x{\lambda }_c^{-1}$. This is equivalent to the attenuation factor (12).

The expressions (13) and (14) for the spectra of u_g and υ_g noise after smoothing in postprocessing provide mathematical explanations for the convergence with increasing filter cutoff wavelength λ_c of the variances of the residual noise in u_g and υ_g for the footprint diameters of 1 and 2 km that was noted in section 3 from Fig. 1b. This can be understood intuitively by noting that the spectra (13) and (14) of smoothed estimates of u_g and υ_g noise involve products of the response function for three-point centered differences and the filter transfer function of the smoothing with a half-power filter cutoff wavelength of λ_c. The latter depends only on λ_c and not on the footprint diameter [see Eq. (10) for the case of the Parzen smoother used in this study], but the response function for three-point centered differences differs for the different grid spacings associated with the two footprint diameters. In particular, the bandpass filtering property of the sin² dependence on the square of the response function (E2) attenuates more of the high-wavenumber variability for the footprint diameter and grid spacing of 2 km compared with the footprint diameter and grid spacing of 1 km. This dependence on grid spacing is shown in Fig. H.1 of C19 in the context of finite differences on sample grids with grid spacings of 1 and 5 km.

For any particular choice of footprint diameter and grid spacing, the bandpass filtering from the response function for three-point centered differences is the same for any choice of spatial smoothing. As the half-power filter cutoff wavelength λ_c of the smoothing is increased, the filter transfer function of the smoothing attenuates progressively more of the variability at lower wavenumbers. This low-pass filtering eventually attenuates the variance within the wavenumber band over which the response functions for three-point centered differences differ on the sample grid spacings of 1 and 2 km for the respective footprint diameters of 1 and 2 km. The effects of this double filtering can be seen from Fig. H.2 of C19 in the context of finite differencing on sample grids with spacings of 1 and 5 km. The convergence of the standard deviations of the residual noise in Fig. 1b for the footprint diameters of 1 and 2 km thus occurs because the differences between the response functions for three-point centered differences on the two grid spacings become progressively more attenuated with increased spatial smoothing.

It can be noted that the first two terms in the spectrum (15) of smoothed estimates of ζ_g involve the product of the response function W_3pt(k, Δx) for three-point centered differences and the filter transfer function $W_{{\lambda }_c}(k)$ of the smoothing with a half-power filter cutoff wavelength of λ_c. As discussed above from (13) and (14), the differences between these products for the footprint diameters of 1 and 2 km decrease with increased filter cutoff wavelength λ_c. This explains the convergence of the standard deviations of the residual noise in Fig. 1c for the footprint diameters of 1 and 2 km.

The expressions (8), (13), (14), and (15) are applicable to any low-pass filter with half-power filter cutoff wavelength λ_c, with the caveat that the equivalence of isotropic two-dimensional smoothing and separate one-dimensional smoothing in two orthogonal dimensions is less valid for some choices of smoothing operation than for the Parzen smoother used in this study.

The spectra of noise in SSH, u_g, υ_g, and ζ_g computed from (8), (13), (14), and (15), respectively, are shown in Fig. 2 after applying isotropic two-dimensional Parzen smoothing in postprocessing of simulated SWOT estimates of SSH for footprint diameters of 1 and 2 km (thin and thick red lines, respectively) and half-power filter cutoff wavelengths of 15, 30, and 50 km. For reference, the noise spectra from Fig. E1 of unsmoothed variables for the footprint diameter of 2 km that will likely be the SWOT product preferred by most users are shown by the purple dashed lines in each panel. As discussed in appendix A, the choice of λ_c = 15 km in Fig. 2 corresponds to the filtering applied in the specification of the science requirements for SWOT estimates of SSH. The other two choices of λ_c in this figure illustrate the decrease with increasing λ_c of the spectral power of the noise at all wavenumbers in the SWOT estimates of each variable. The rapid convergence of the noise spectra of smoothed u_g, υ_g, and ζ_g for the two footprint diameters is also apparent, as was discussed above from (13), (14), and (15) and in section 3 from the standard deviations of the smoothed noise in Figs. 1b and 1c. Differences between the spectra of smoothed u_g and υ_g noise for the two footprint diameters can be seen for λ_c = 15 km but the two noise spectra are indistinguishable for the cases of λ_c = 30 and 50 km. The spectra of smoothed ζ_g noise for the two footprint diameters are barely distinguishable for λ_c = 30 km but become indistinguishable for the case of λ_c = 50 km.

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Swath-averaged wavenumber spectra of smoothed signals (thick black lines) computed in the alongshore dimension of the model of the California Current System (CCS) considered here (see section 5) for which descending swaths of SWOT are approximately parallel to the coast (see Fig. 3) and smoothed noise (red lines) in SWOT estimates of SSH, velocity components u_g and υ_g computed geostrophically from SSH based on the Coriolis parameter f_37°N = 8.75 × 10⁻⁵ s⁻¹ at the central latitude 37°N of the CCS model, and vorticity ζ_g computed from u_g and υ_g. The spectra of residual errors after smoothing the SWOT noise using a Parzen smoother with half-power filter cutoff wavelengths of (left to right) λ_c = 15, 30, and 50 km are shown for footprint diameters of 1 and 2 km (thin and thick red lines, respectively, which become less distinguishable with increasing λ_c). The horizontal and vertical red dashed and solid lines and the red dot in the top-left panel are as in Fig. E1. For reference, the purple dashed lines are the spectra of unsmoothed noise for the footprint diameter of 2 km and the thin black lines are the spectra of unsmoothed signals (see Fig. E1).

Citation: Journal of Atmospheric and Oceanic Technology 39, 7; 10.1175/JTECH-D-21-0167.1

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The distinctions between the wavenumber dependencies of the smoothed noise and the smoothed signals of SSH, u_g, υ_g and ζ_g from the model of the California Current System considered in this study (see section 5) can be seen from comparison of the spectra shown by the red and thick black lines, respectively. For reference, the thin black lines are the spectra of the unsmoothed signals. The two-dimensional smoothing attenuates both the signal and the noise spectra at wavenumbers higher than the filter cutoff wavelength of ${\lambda }_c^{-1}$. The noise spectra for all of the variables are further attenuated progressively more at lower wavenumbers with increasing λ_c. The ζ_g spectrum is also somewhat attenuated at lower wavenumbers with increasing λ_c, but much less so than the ζ_g noise spectrum. In all cases, the noise spectra are attenuated much more quickly than the signal spectra with increasing λ_c, thus improving the signal-to-noise ratio. For all four variables, smoothing with a half-power filter cutoff wavelength of at least λ_c = 50 km is required for the residual noise spectra to become less energetic than the smoothed signal spectra at all wavenumbers.

5. Example maps of simulated noisy SWOT estimates of SSH, velocity, and vorticity

A visual assessment of the effects of the statistical characterizations of the noise in sections 2–4 on SWOT estimates of SSH and geostrophically computed velocity and vorticity is presented in this section from maps constructed from simulated SWOT data based on the Regional Oceanic Modeling System (ROMS) model of the California Current System (CCS) that was considered by C19. The details of this model are summarized by Molemaker et al. (2015) and in section 2 of C19. The model does not include tides, was forced only by seasonal winds, has a high dissipation that attenuates internal wave variability, and has vertical and horizontal grid resolutions that are not adequate to resolve the full spectrum of internal wave variability. The small-scale features of tides and internal waves are therefore likely underestimated in the model used in this study. The conclusions from the analysis presented here may thus be an overly optimistic assessment of the capabilities for SWOT estimates of velocity and vorticity.

An instantaneous snapshot of SSH for the full model domain on the 0.5 km × 0.5 km grid of the model is shown in Fig. 3a. The speed of the associated geostrophically computed velocity based on the local Coriolis parameter at each grid location is shown in Fig. 3b and the geostrophically computed vorticity normalized by the local Coriolis parameter is shown in Fig. 3c. A descending pair of the parallel swaths from a single simulated overpass of SWOT is overlaid on each of these three maps. This simulated SWOT overpass samples the meandering core of the southward California Current. For reference, the full set of measurement swaths during each 21-day repeat of the SWOT orbit within the bounds of the map is shown in Fig. 3d. The different colors illustrate the 10-day subcycle of the SWOT orbit that is discussed in section 6; the swaths shown in red will be sampled about 10 days earlier and 10 days later than the swaths shown in blue.

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Maps of error-free instantaneous (a) SSH, (b) the magnitude of geostrophically computed velocity, and (c) the geostrophically computed vorticity for model day 27 May on the 0.5 km × 0.5 km grid of the model of the CCS considered here. (d) Simulated SWOT measurement swaths during two 5-day periods that are separated by 10 days in the 21-day exact repeat SWOT orbit (blue for days 3–8 and red for days 13–18). The diamond-shaped holes in the sampling pattern in (d) are never observed during the 21-day SWOT orbit. The shaded regions in (a)–(c) are the pair of simulated SWOT measurement swaths for the single descending overpass over the CCS model domain that is considered in Figs. 4 and 5. The large box in (d) is the full CCS model domain and the small box in each panel is the subregion of the CCS model that is shown in rotated coordinates in Figs. 4 and 5. The California calibration and validation site for SWOT is centered at 35.7°N, 125.4°W near the bottom-left corner of the small boxes.

Citation: Journal of Atmospheric and Oceanic Technology 39, 7; 10.1175/JTECH-D-21-0167.1

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To the untrained eye, the submesoscale features that the SWOT mission seeks to observe are difficult to identify in the error-free SSH field in Fig. 3a because the variance of SSH is so small on scales smaller than 50 km. They are evident as subtle yellow dimples imbedded in the orange, light red and green areas of the larger mesoscale features in the SSH field. The spatial high-pass filtering from differentiation of the SSH field emphasizes these submesoscale features in the geostrophically computed speed field shown in Fig. 3b. The submesoscale features are even more emphasized in the geostrophically computed vorticity field in Fig. 3c that was calculated from double differentiation of the SSH field. For the analysis in this study, the simulated SSH fields at the model grid points within the simulated SWOT measurement swaths were used directly without consideration of the differences in location and orientation of the model grid relative to the swath-based grids of the SSH estimates that will be provided by the SWOT Project.

To simulate SWOT data, the SSH field in Fig. 3a was smoothed isotropically with the half-power filter cutoff wavelength of 4 km that is the minimum filter cutoff for a footprint diameter of 2 km. (As discussed in appendixes B and C, the SWOT Project may smooth the SSH with a half-power filter cutoff wavelength somewhat longer than 4 km for the 2 km × 2 km data product.) The smoothed SSH field on the 0.5 km × 0.5 km grid of the model was then subsampled on a 2 km × 2 km grid within the two parallel swaths of the simulated SWOT overpass, thus mimicking the SWOT data product that will likely be preferred by most users. As in the previous sections, this simulated SWOT data product is referred to here as “unsmoothed” to distinguish it from the SSH field obtained by additional smoothing in the simulated postprocessing considered below. Spatially homogeneous uncorrelated SSH measurement errors with the swath-averaged standard deviation of σ_2km = 1.37 cm that is derived in appendix C for a footprint diameter of 2 km with 4-km smoothing were added to the gridded values of the model SSH field. The across-swath variations of the measurement noise shown in Fig. C1 and the long-wavelength measurement errors discussed in appendix A are not considered here. Simulated SWOT estimates of velocity and vorticity were computed geostrophically from the error-free and noisy SSH fields with the derivatives approximated as three-point centered differences.

To see the improvement of the signal-to-noise ratio in maps of smoothed SSH, velocity, and vorticity constructed from noisy simulated SWOT measurements of SSH, the unsmoothed error-free and noisy SSH fields within only the measurement swaths were smoothed two-dimensionally in simulated postprocessing using the Parzen smoother with selected choices of half-power filter cutoff wavelength. The residual effects of measurement noise on the smoothed SWOT estimates of SSH, velocity, and vorticity are assessed here from consideration of the subregion near the coast in the center of the CCS model domain that is indicated by the small rotated box in each panel of Fig. 3. For reference, the California calibration and validation site for SWOT is centered at 35.7°N, 125.4°W, which is near the bottom-left corner of the small rotated boxes. The SSH and geostrophically computed speed and vorticity within the small box are shown in the left column of Fig. 4 in rotated coordinates that are aligned with the model grid. These three error-free maps are the same as in the subdomain of the full-resolution maps in Fig. 3, except they have been smoothed and subsampled as summarized above to simulate the SWOT data product with a footprint diameter of 2 km. The edges of the pair of swaths in the simulated single descending overpass of SWOT that are shown in Figs. 3a–c are indicated by the parallel lines overlaid on each panel in the left column of Fig. 4.

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(left) Example maps of instantaneous error-free (top) SSH, (middle) the magnitude of geostrophically computed velocity, and (bottom) geostrophically computed vorticity for model day 27 May within the region of the CCS model shown by the small boxes in Fig. 3. The maps have been rotated by a polar angle of 24° relative to the latitude–longitude coordinates in Fig. 3 to align with the grid of the CCS model. The geostrophic calculations are based on the local Coriolis parameter f at each grid point and the parallel lines in the left panels are the swath edges for the simulated overpass of SWOT shown in Figs. 3a–c. (right four columns, top to bottom) SSH, geostrophically computed speed, and vorticity based only on simulated data within the measurement swaths of the single overpass of SWOT for a footprint diameter of 2 km on a 2 km × 2 km grid smoothed using a Parzen smoother with half-power filter cutoff wavelengths of λ_c = 0 km (i.e., unsmoothed), 15, 30, and 50 km. The left pair of swaths in each column shows the smoothed fields without measurement errors and the right pair of swaths includes the effects of spatially homogeneous uncorrelated SSH measurement errors with the standard deviation of σ_2km = 1.37 cm for the footprint diameter of 2 km.

Citation: Journal of Atmospheric and Oceanic Technology 39, 7; 10.1175/JTECH-D-21-0167.1

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The other four columns in Fig. 4 show the error-free and noisy fields (the left and right pair of swaths in each column) sampled only within the pair of measurement swaths from the simulated SWOT overpass. These four columns correspond to the fields constructed from the simulated SWOT data with smoothing applied based only on SSH observations within the measurement swaths with half-power filter cutoff wavelengths of λ_c = 0 km (i.e., no smoothing), 15, 30, and 50 km. The filter cutoff of 15 km corresponds to the smoothing in the specification of the science requirements for SWOT (see appendix A). According to the criteria proposed in section 5 of C19, the filter cutoffs of 30 and 50 km correspond approximately to the resolution capabilities of instantaneous maps of geostrophically computed velocity and vorticity, respectively (see Figs. 17b,c and 18 and the top panels of Fig. 51 of C19). Note, however, that the analysis of the resolution capability in instantaneous maps in section 6 of C19 did not take into consideration the edge-effect sampling errors that are discussed below.

Consider first the fields constructed from unsmoothed simulated SWOT data that are shown in the second column of Fig. 4. The noise in the SSH measurements is apparent from the speckles in the right pair of swaths. The effects of this visually subtle SSH measurement noise on SWOT estimates of velocity and vorticity completely swamp the submesoscale features in the unsmoothed maps of both of these derivative fields. The noise standard deviations of 0.54 m s⁻¹ in each velocity component and 4.9f in the vorticity estimates (see the fourth row in the top half of Table 1) are much more energetic because of the amplifications of the effects of SSH measurement noise discussed in appendix D that arise from the finite-difference approximations of the derivatives. It can be inferred from these maps that velocity and vorticity fields will not likely be detectable on any scales from unsmoothed SWOT data, at least in the CCS region considered here, and likely in most regions of the World Ocean. Additional smoothing in postprocessing will be required to attenuate the uncorrelated measurement errors and improve the signal-to-noise (S/N) ratio.

When the simulated SWOT data are smoothed within the measurement swaths with half-power filter cutoff wavelengths of 15, 30, and 50 km (third, fourth, and fifth columns of Fig. 4, respectively), the residual noise diminishes rapidly for all three variables, as shown previously in Fig. 1. The standard deviations of the residual noise in each variable are listed in the bottom half of Table 1. (Note that the residual effects of measurement errors in Table 1 and Fig. 4 do not include the edge-effect sampling errors discussed below that arise from smoothing within the narrow SWOT measurement swaths.) The signals are also attenuated by the smoothing, but not as rapidly as the noise from the effects of uncorrelated measurement errors (see Fig. 2). The S/N ratio thus improves with increased smoothing.

For the case of SSH, the residual noise in SWOT estimates of SSH can still be seen with 15-km smoothing from close inspection of the top panel in the third column of Fig. 4. There is no visual evidence of residual SSH noise with filter cutoffs of 30 or 50 km.

For SWOT estimates of velocity, the residual noise is readily apparent in the swath maps with 15-km smoothing shown in the middle panel of the third column of Fig. 4. The most energetic features in the error-free geostrophically computed velocity field with 15-km smoothing in the left pair of swaths can be detected in the noisy field with 15-km smoothing in the right pair of swaths. However, there are numerous features in the 15-km smoothed noisy field that could easily be misinterpreted as signals without the foreknowledge of the details of the error-free smoothed field in the left pair of swaths. The residual velocity noise becomes relatively subtle with the 30-km smoothing in the middle panel of the fourth column of Fig. 4. For the threshold S/N standard deviation ratio of 3.16 recommended by C19 (corresponding to a variance ratio of 10), the resolution capability of SWOT estimates of velocity is 32 km. The resolution capability for a more liberal threshold S/N standard deviation ratio of 2 (a variance ratio of 4) is 24 km. Note, however, that the C19 assessments of the resolution capability of instantaneous maps of geostrophically computed velocity do not include consideration of the edge-effect sampling errors that are discussed below.

The challenge of estimating vorticity from SWOT data is very apparent from the bottom panels of Fig. 4. Because of the large amplification of the effects of SSH measurement noise (see appendix D) from finite difference approximations of the double derivatives required to compute vorticity geostrophically from SSH, the signals are still swamped by the noise with 15-km smoothing. With foreknowledge of the smoothed error-free field in the left pair of swaths in the bottom panel of the fourth column of Fig. 4, the features in the 30-km smoothed error-free geostrophically computed vorticity field can be identified in the 30-km smoothed noisy field in the right pair of swaths. But there are numerous energetic artifacts in the 30-km smoothed noisy field that could easily be misinterpreted as submesoscale features. The residual noise becomes relatively subtle with the 50-km smoothing in the bottom-right panel of Fig. 4. The resolution capability of SWOT estimates of instantaneous vorticity deduced by C19 is 54 km based a threshold S/N standard deviation ratio of 3.16 and 44 km for a threshold S/N standard deviation ratio of 2. We note again that these assessments of resolution capability by C19 do not take into consideration edge-effect sampling errors.

The edge-effect sampling errors alluded to several times above become readily apparent from the spatial structures of the errors of the simulated SWOT estimates of SSH, velocity, and vorticity, rather than the maps of the signals without and with noise that are shown in Fig. 4. This can be achieved for the snapshot fields considered here by subtracting smoothed fields computed from the spatially complete error-free fields in the left column of Fig. 4 from smoothed estimates computed from simulated error-free and noisy SWOT data within just the measurement swaths. For each choice of half-power filter cutoff, these errors can be partitioned into a contribution from sampling errors and a contribution from residual measurement noise. The former corresponds to the errors of the smoothed fields that would be incurred if the smoothing were applied to error-free fields within just the SWOT measurement swaths. For the case considered here in which estimates are constructed from a pair of measurement swaths from a single simulated overpass of SWOT, sampling errors consist only of edge effects of smoothing from incomplete data within the span of the smoother near the edges of the measurement swaths. When the SSH fields are constructed from space–time smoothing of SWOT data from multiple overpasses, sampling errors are also incurred from the discrete and irregular sampling of rapidly evolving small-scale features. This is briefly discussed in section 6 below and is addressed in detail in section 8 of C19. The present analysis considers only the instantaneous fields constructed from simulated SWOT data within the measurement swaths of a single overpass.

Maps of the sampling errors and residual measurement noise in smoothed swath estimates of instantaneous SSH, velocity, and vorticity are shown in the middle two columns of Fig. 5 for the half-power filter cutoff wavelengths of λ_c = 30 and 50 km that were considered in Fig. 4. The right column shows the maps for a larger filter cutoff of λ_c = 70 km to illustrate the increased edge effects with increased smoothing beyond what was considered in Fig. 4. Note that different color bars are used for all of the error maps in Fig. 5 compared with the color bars for the maps in the left column of Fig. 5 and the color bars for all of the maps in Fig. 4.

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(left) The same example maps in rotated coordinates as in Fig. 4 showing (top) error-free SSH, (middle) the magnitude of geostrophically computed velocity, and (bottom) geostrophically computed vorticity. The geostrophic calculations are again based on the local Coriolis parameter f at each grid point and the parallel lines indicate the swath edges for the simulated overpass of SWOT shown in Figs. 3a–c. (right three columns, top to bottom) The errors of SSH, geostrophically computed speed, and geostrophically computed vorticity based only on simulated data within the single overpass of SWOT for a footprint diameter of 2 km on a 2 km × 2 km grid smoothed using a Parzen smoother with half-power filter cutoff wavelengths of λ_c = 30, 50, and 70 km. The left pairs of swaths in each column show the edge-effect sampling errors from mapping of error-free fields and the right pairs of swaths include the effects of spatially homogeneous uncorrelated SSH measurement errors with the standard deviation of σ_2km = 1.37 cm for the footprint diameter of 2 km.

Citation: Journal of Atmospheric and Oceanic Technology 39, 7; 10.1175/JTECH-D-21-0167.1

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We focus first on the sampling errors from edge effects in the smoothing of error-free data that are shown in the left pair of swaths in each of the right three columns of Fig. 5. The smoothing in these figures was two-dimensional and isotropic and was applied based only on data within the two measurement swaths. For estimates at any location across the measurement swath, complete data exist in the along-track dimension. Along the center of the swath, the data in the across-track dimension are symmetrically distributed about the estimation locations but the data distributions become more and more asymmetric as the estimation locations approach the swath edges; complete data exist toward the center of the swath unless the half span of the smoothing exceeds the distance to the far edge of the swath. Toward the near edge of the swath, however, no data are available beyond the edge of the swath. At the extreme estimation locations along the swath edges, data are available only in the half plane toward the center of the swath.

With large enough half-power filter cutoff wavelength, the estimates in one swath can include some data from the other swath on the opposite side of the 20-km nadir gap. SSH data along the ground track are also available from the nadir altimeter. As discussed in the introduction, the very different measurement noise and footprint diameter complicate the merging of nadir altimeter data with the KaRIn estimates of SSH across the two measurement swaths. Data from the nadir altimeter are not considered in this study.

Edge effects are visually apparent for all three variables and all three choices of filter cutoff λ_c in the maps in the left pair of swaths in each column of Fig. 5 that were constructed by smoothing error-free simulated data within just the two parallel measurement swaths. With increased smoothing, these edge effects increase in magnitude and extend farther toward the centers of the swaths. For the case of smoothing with a half-power filter cutoff wavelength of λ_c = 70 km, all but a narrow strip along the center of each swath is contaminated by edge effects.

The case of a radial distance of r₁ = λ_c/3, for example, gives a fractional volume of 0.9609. About 96% of the weighting of the Gaussian smoother thus lies within a radial distance of λ_c/3 from the estimation location. The exact value for the Parzen weighting determined by numerical integration, rather than the Gaussian weighting derived analytically above, is about 98%, thus verifying the close similarity between the Parzen and Gaussian weighting functions. Since virtually all of the weighting of the Parzen smoother is from data values within a radius of λ_c/3 it is apparent why the center half of each swath is not strongly contaminated by edge effects with a filter cutoff of λ_c = 50 km. For the case of 70-km smoothing, however, most of the swath is contaminated by edge effects.

The magnitude and extent of contamination by edge effects are very situation specific. They are most severe when there are features near the edges of the swath that have strong gradients in the across-track dimension. Such is the case for the swaths considered in Fig. 5 that straddle the meandering core of the southward California Current. Edge effects will be less pronounced in more quiescent regions where lateral gradients are weaker such as in the offshore region of the model domain shown in Fig. 3 where submesoscale variability is much less energetic.

The right pairs of swaths in each column in Fig. 5 show the combined effects of sampling errors and residual measurement noise. The latter is manifest as the mottled features in the smoothed error fields that diminish in magnitude with increasing filter cutoff wavelength λ_c. With 30-km smoothing, the total error field is dominated by residual measurement noise for all three variables. For the region considered in Fig. 5, sampling errors from edge effects become larger than the residual measurement noise with 50-km smoothing. The total error fields for all three variables are almost completely dominated by sampling errors from edge effects with 70-km smoothing.

We emphasize again that the maps in Figs. 4 and 5 were constructed from simulated SWOT data with homogeneous SSH measurement noise with the swath-averaged standard deviation of σ_2km = 1.37 cm. The errors near the edges of the swaths will be exacerbated by the larger measurement errors toward the swath edges (see Fig. C1) that have not been considered in this study. Edge-effect sampling errors will therefore be larger from actual SWOT data than inferred from the analysis presented here.

6. Summary and conclusions

The Ka-band Radar Interferometer (KaRIn) on the Surface Water and Ocean Topography (SWOT) satellite with a target launch date in November 2022 promises to revolutionize satellite altimetry by providing high-resolution and accurate measurements of SSH across a pair of 50-km swaths separated by a 20-km nadir gap. The primary oceanographic goal of the SWOT mission is to resolve SSH variability on wavelengths longer than about 15 km, which is nearly a factor of 5 better than can presently be achieved from along-track measurements by conventional nadir-viewing altimeters (Xu and Fu 2012), and more than an order of magnitude better than the resolution capability of SSH fields constructed from measurements by multiple nadir-viewing satellite altimeters (Chelton et al. 2011; Ballarotta et al. 2019). An especially exciting aspect of the SWOT mission is that the swath measurements by KaRIn will provide two-dimensional views of instantaneous SSH fields. This offers the first opportunity to obtain satellite-based estimates of instantaneous surface velocity and relative vorticity computed geostrophically from the gridded SSH measurements across the two parallel measurement swaths.

One of the goals of this study was to determine the standard deviations of the uncorrelated SSH measurement noise for footprint diameters of 0.5, 1, and 2 km (appendix C) from the spectral specification of the science requirement for KaRIn (appendix A), and to quantify the effects of the uncorrelated SSH measurement errors on the noise in geostrophically computed velocity and vorticity (appendix D). We did not address the errors in the geostrophic approximation (see, e.g., the discussion in section 3 of C19). Nor did we consider the effects of the long-wavelength measurement errors (see appendix A).

Our analysis also considered only the effects of swath-averaged uncorrelated errors in the SSH measurements for a significant wave height (SWH) of 2 m. We did not account for the increases of the measurement errors toward both edges of each measurement swath or the increase of measurement errors with increasing SWH (see Fig. C1). The conclusions of this study are therefore likely somewhat optimistic assessments of the resolution capabilities of SWOT estimates of SSH, velocity, and vorticity.

The equations in appendixes C–E and sections 3 and 4 for the standard deviations and wavenumber spectral characteristics of the noise in these three variables are all expressed in terms of the SSH standard deviation σ_h that is the fundamental characterization of the uncorrelated SWOT measurement errors for the chosen footprint diameter. They can therefore be applied for any location across the measurement swath and any SWH, based on the corresponding standard deviation of the SSH measurement noise at that location. Moreover, the prelaunch estimates of σ_h derived in appendix C for the three footprint sizes considered in this study can be replaced in the equations in appendixes C–E and sections 3 and 4 with actual values determined after launch from in-orbit data, or with smaller values that may be obtained in the future through improvements in the onboard and ground-based processing of SWOT data.

The analysis in appendix C first considered KaRIn data with the original planned footprint diameter of 1 km on a 1 km × 1 km grid and then considered the recently revised plan for KaRIn data products with footprint diameters of 0.5 km on an oversampled 0.25 km × 0.25 km grid and 2 km on a 2 km × 2 km grid (JPL 2020, 2021). The standard deviations of the SSH estimates with these three footprint diameters were derived from the science requirement for SWOT, which is expressed in a nonconventional form in terms of the wavenumber spectrum after smoothing with a half-power filter cutoff wavelength of 15 km (see appendix A). Because of the large 5.48-cm standard deviation of the SSH measurement noise with the footprint diameter of 0.5 km and the large data volume on the oversampled 0.25 km × 0.25 km grid, the vast majority of users will likely prefer the SWOT data product with a footprint diameter of 2 km on a 2 km × 2 km grid. The noise standard deviation of these SSH estimates are expected to be σ_2km = 1.37 cm if this data product is produced by smoothing the high-resolution SWOT data product with a half-power filter cutoff wavelength of 4 km (see the discussion in appendixes B and C). The smaller footprint size was not considered in our analysis of smoothed simulated SWOT estimates of SSH, velocity, and vorticity in sections 3–5.

The standard deviations of the noise in SWOT estimates of velocity and vorticity computed geostrophically from SSH with the three footprint diameters of 0.5, 1, and 2 km were derived by propagation-of-error analysis in appendix D. The SWOT estimates of SSH, velocity, and vorticity with a chosen footprint diameter are produced by smoothing the raw SWOT data with a half-power filter cutoff wavelength that is twice the footprint diameter. The resulting fields for any choice of footprint diameter are referred to here as “unsmoothed” to distinguish them from the estimates obtained in sections 3–5 with additional two-dimensional smoothing applied in postprocessing. Equations for the wavenumber spectral distributions of the noise variance of unsmoothed SWOT estimates of SSH, velocity, and vorticity are presented in appendix E. The effects of SSH noise are amplified by the finite difference approximations of the derivatives in the Eqs. (D1a) and (D1b) for geostrophically computed velocity and the Eq. (D2b) for vorticity. The amplification of the effects of SSH measurement noise is much greater for vorticity because it consists of the sum of two terms, both of which involve double differentiation of SSH. Mapping the geostrophically computed surface vorticity field from SWOT data will therefore be much more challenging than mapping the SSH and geostrophically computed surface velocity fields.

As summarized in section 2, the conclusion from the analysis in the appendixes is that the noise in instantaneous maps of all three variables constructed from unsmoothed SWOT data with any of the three footprint diameters considered here will be too large for the SWOT data products to be used directly for most applications. This is acknowledged in the science requirements for KaRIn that are based on smoothing with a half-power filter cutoff wavelength of λ_c = 15 km to improve the signal-to-noise (S/N) ratio in SWOT estimates of SSH. It was shown in section 3 that the residual noise decreases rapidly with increased smoothing. The noise standard deviations for SSH, velocity components, and vorticity decrease as approximately ${\lambda }_c^{-1}$, ${\lambda }_c^{-2}$, and ${\lambda }_c^{-3}$, respectively (see Fig. 1). Equations for the wavenumber spectral characteristics of the smoothed noise in each of the three variables are presented in section 4.

While smoothing with a half-power filter cutoff wavelength of λ_c = 15 km may yield an adequate S/N ratio for SSH, the velocity and vorticity computed geostrophically from in-swath SWOT data will likely have to be smoothed with filter cutoff wavelengths of about 30 and 50 km, respectively. Visual assessments of the effects of this smoothing were presented in section 5 from maps of instantaneous estimates of SSH, velocity, and vorticity constructed from simulated SWOT data based on a high-resolution numerical model of the California Current System. This simulated SWOT sampling provides insight into the contamination of smoothed estimates of the three variables from edge effects of the smoothing within the 50-km width of each measurement swath. These edge effects arise from incomplete data within the span of the smoother. They will be compounded by the larger measurement errors toward the swath edges that were not considered here (see Fig. C1). Our assessment of SWOT capabilities may therefore be somewhat optimistic. The measurement noise decreases with increased smoothing but the errors from edge effects worsen. Smoothing of the in-swath data will likely have to be limited to a half-power filter cutoff wavelength of about 50 km. Even then, edge effects will likely be too large except within approximately the center half of each 50-km swath (see Fig. 5). The estimates in this portion of the swath will nonetheless yield important new insight into global mesoscale variability, especially for the smoothed SWOT estimates of vorticity.

The analysis here considered only spatial smoothing of in-swath instantaneous SWOT data to improve the S/N ratio. Intuitively, edge effects and measurement errors can be reduced with space–time smoothing from multiple SWOT overpasses of the region of interest. In practice, however, the space–time sampling pattern of the SWOT orbit is not well suited to reducing the errors by time averaging in regions of energetic and rapidly evolving submesoscale variability. The 21-day exact-repeat orbit consists of two subcycles of about 10 days. During each 21-day period, the sampling consists of an interleaved pattern of measurement swaths in which adjacent swaths are sampled 10 days apart (see Fig. 3d; see also Figs. 21 and 22 of C19 and the related discussion in the text of that paper). This sampling pattern introduces a second source of sampling errors that arises from the unresolved rapid evolution of small-scale features during the 10-day time separation between adjacent swaths. While space–time smoothing will be useful for mapping mesoscale variability in SSH (see section 12 of C19), it is shown in section 8 of C19 that the resolution capabilities of SWOT estimates of velocity and vorticity actually worsen with time averaging because of sampling errors from the unresolved variability, at least for the time averages of 4 and 14 days considered in that study. Mapping of the surface velocity and vorticity fields from SWOT data will likely have to be done with the aid of dynamic interpolation from a data assimilation model, as suggested, for example, by Ubelmann et al. (2015).

The need to smooth SWOT data with a half-power filter cutoff wavelength of 30–50 km for estimates of velocity and vorticity will be disappointing to investigators interested in smaller submesoscale variability. It should be kept in mind that this is much better than the approximate 200-km-resolution limitation of the SSH fields constructed from measurements by multiple nadir-viewing altimeters. SWOT data will undoubtedly provide important new insight into ocean dynamics at the short end of the range of mesoscale variability and the long end of the range of submesoscale variability. It is also noteworthy that recent modeling and observational studies suggest that the variability of SSH from which velocity and vorticity will be computed geostrophically from SWOT data may be dominated by baroclinic tides and internal waves on wavelength scales up to 50 km or more (e.g., Rocha et al. 2016; Qiu et al. 2017, 2018; Callies and Wu 2019). SWOT data will undoubtedly provide new insight into the SSH structures of these ageostrophic processes. It may be challenging to separate baroclinic tides and internal wave variability from some of the geostrophic submesoscale signals of interest. The resolution of these other processes in SWOT data may therefore prove to be comparable to the resolution limitations that are imposed by the measurement noise that was the focus of this study.

Data availability statement.

The snapshot SSH field from the model of the California Current System that was analyzed for Fig. E1 and Figs. 2–5 was obtained from Jeroen Molemaker at the University of California, Los Angeles.