sample grid. Analogous to (D.1) for the 1-dimensional case considered in Appendix D, Parseval's Theorem for the relationship between the sample variance $\sigma_e^2$ of this error field and its 2-dimensional wavenumber sample power spectral density $S_e^{2d}(k_i, l_j)$ at wavenumbers $k_i = i\Delta k$ and $l_j = j\Delta l$ is

$$ \sigma_e^2 = \sum_{i=-M/2+1}^{M/2} \sum_{j=-N/2+1}^{N/2} S_e^{2d}(k_i, l_j) \Delta l \Delta k, \quad (E.1) $$

where $\Delta k = (M\Delta x)^{-1}$ and $\Delta l = (N\Delta y)^{-1}$ are the discrete Fourier wavenumber intervals that are defined by the record lengths $M\Delta x$ and $N\Delta y$ in the $x$ and $y$ dimensions. As in the 1-dimensional case (D.1), the zero wavenumbers corresponding to indices $i = j = 0$ are excluded from the sums in (E.1) because the spectral values at these wavenumbers correspond to the squared values of the sample mean in each dimension, which do not contribute to the sample variance $\sigma_e^2$. The indices $i = M/2$ and $j = N/2$ that define the limits of the summations in (E.1) correspond to the Nyquist wavenumbers $k_N = (2\Delta x)^{-1}$ and $l_N = (2\Delta y)^{-1}$ in the $x$ and $y$ dimensions, respectively

The 2-dimensional form (E.1) of Parseval's Theorem can be interpreted as a discretized statement that the sample variance is equal to the volume under the 2-dimensional wavenumber sample power spectral density. Analogous to the procedure followed in one dimension in Appendix D, this is seen by considering the limit as the record lengths $M\Delta x$ and $N\Delta y$ in the $x$ and $y$ dimensions approach infinity for fixed sample intervals $\Delta x$ and $\Delta y$. The wavenumber intervals $\Delta k$ and $\Delta l$ then become infinitesimally small values $dk$ and $dl$ and the discrete wavenumbers $k_i = i\Delta k$ and $l_j = j\Delta l$ become continuous. The subscripts $i$ and $j$ can then be dropped and the discrete summations become integrals so that (E.1) can be written in simpler form as<fnref n="7" />

$$ \sigma_e^2 = \int_{-k_N}^{k_N} \int_{-l_N}^{l_N} S_e^{2d}(k, l) dl dk. \quad (E.2) $$

The variance of the discretely sampled error field $\epsilon(x_m, y_n)$ with infinitely long record lengths $M\Delta x$ and $N\Delta y$ in the two dimensions is thus the integrated volume under the 2-dimensional sample power spectral density. A point that is important to the analysis below and in Appendix F is that the ranges of integration are finite. The Nyquist wavenumbers $k_N$ and $l_N$ that define the lower and upper bounds of the integrals are imposed by the discrete sample intervals $\Delta x$ and $\Delta y$, regardless of the record lengths $M\Delta x$ and $N\Delta y$.

The integral representation (E.2) of Parseval's Theorem greatly simplifies the notation in the analysis that follows. It should be kept in mind, however, that the observable resolutions $\Delta k$ and $\Delta l$ of the wavenumbers $k$ and $l$ in the $x$ and $y$ dimensions will be finite and are imposed by the finite record lengths $M\Delta x$ and $N\Delta y$.

The 2-dimensional wavenumber sample power spectral density $S_e^{2d}(k, l)$ of 2-dimensional uncorrelated errors is constant (i.e., white) with a value that can be defined to be $S_0$ at all wavenumbers $k$ and $l$ except $k = 0$ and $l = 0$ at which $S_e^{2d}(k, l) = 0$ for errors that have a sample mean value of zero. Parseval's Theorem (E.2) then reduces to

$$ \sigma_e^2 = 4k_N l_N S_0 = \frac{S_0}{\Delta x \Delta y}. $$

The constant 2-dimensional power spectral density for a specified white noise error variance $\sigma_e^2$ obtained by inverting this equation is

$$ S_e^{2d}(k, l) = S_0 = \Delta x \Delta y \sigma_e^2, \quad \text{for } -k_N < k \le k_N \text{ and } -l_N < l \le l_N. \quad (E.3) $$

For the case of errors $\epsilon(x, y)$ of SWOT estimates of SSH in units of cm, the 2-dimensional white-noise spectral value of $S_0$ has units of cm$^2$/cpkm$^2$. Similarly for the case of errors $\epsilon(x, y)$ of WaCM estimates of a velocity component in units of m s$^{-1}$, $S_0$ has units of (m s$^{-1}$)$^2$/cpkm$^2$.

The 1-dimensional wavenumber sample power spectral density in the $y$ dimension of a 2-dimensional field can be obtained from the 2-dimensional wavenumber sample power spectral density by integrating over all wavenumbers $k$ in the $x$ dimension. For the 1-sided, 1-dimensional spectrum that combines the power at positive and negative wavenumbers $l$, this relationship for the 1-dimensional white noise spectrum becomes,<fnref n="8" />

$$ S_e^{1d}(l) = 2 \int_{-k_N}^{k_N} S_e^{2d}(k, l) dk = 4k_N S_0 = \frac{2S_0}{\Delta x}, \quad \text{for } 0 < l \le l_N. \quad (E.4a) $$

From (E.3), this constant 1-sided, 1-dimensional white noise spectrum can be expressed in terms of the uncorrelated error variance $\sigma_e^2$ as

$$ S_e^{1d}(l) = 2\Delta y \sigma_e^2, \quad \text{for } 0 < l \le l_N. \quad (E.4b) $$

If the variance of the 2-dimensional white noise is suppressed by smoothing 2-dimensionally in the form of separate 1-dimensional smoothing with half-power filter cutoff wavenumbers of $k_c$ and $l_c$ in the $x$ and $y$ dimensions, respectively, the 2-dimensional spectrum $\bar{S}_e^{2d}(k, l)$ of the smoothed error fields is related to the 2-dimensional spectrum $S_e^{2d}(k, l)$ of the unsmoothed errors by

$$ \bar{S}_e^{2d}(k, l) = W_{k_c}^2(k) W_{l_c}^2(l) S_e^{2d}(k, l), \quad (E.5a) $$

where $W_{k_c}(k)$ and $W_{l_c}(l)$ are the filter transfer functions of the smoother in the $x$ and $y$ dimensions, respectively. It has been assumed in (E.5a) that the smoother is symmetric and linear so that the filter transfer functions are real and hence the wavenumber spectrum of the smoothed errors depends on the squares of the filter transfer functions rather than their complex inner products. If the filter cutoff wavenumbers $k_c$ and $l_c$ are the same, separate 1-dimensional smoothing in this manner with 1-dimensional Parzen smoothers in each dimension is essentially equivalent to isotropic smoothing with a 2-dimensional Parzen weighting function that depends only on the radial distance from the estimation location (see Appendix C).

From (E.3), the spectrum (E.5a) of the smoothed error fields can be expressed in terms of the uncorrelated error variance $\sigma_e^2$ as

$$ \bar{S}_e^{2d}(k, l) = \Delta x \Delta y \sigma_e^2 W_{k_c}^2(k) W_{l_c}^2(l). \quad (E.5b) $$

As in (E.4a), the 1-sided, 1-dimensional wavenumber spectrum of the smoothed error fields in the $y$ dimension is obtained from $\bar{S}_e^{2d}(k, l)$ by integrating over all wavenumbers $k$ in the $x$ dimension and doubling the power at each wavenumber except the zero and Nyquist wavenumbers,

$$ \bar{S}_e^{1d}(l) = 2 \int_{-k_N}^{k_N} \bar{S}_e^{2d}(k, l) dk. $$

From (E.5b), this can be expressed as

$$ \begin{aligned} \bar{S}_e^{1d}(l) &= 2\Delta x \Delta y \sigma_e^2 W_{l_c}^2(l) \int_{-k_N}^{k_N} W_{k_c}^2(k) dk \\ &= 4\Delta x \Delta y \sigma_e^2 W_{l_c}^2(l) \int_{0}^{k_N} W_{k_c}^2(k) dk. \end{aligned} \quad (E.6a) $$

The last expression follows from the fact that $W_{k_c}^2(k)$ is symmetric about $k = 0$.

For the Parzen smoother used in this study, the integral on the right side of (E.6a) was shown by (D.12b) in Appendix D to be approximately

<footnote n="7">⁷ The zero wavenumbers $k = l = 0$ that are included in the integral (E.2) must be excluded from the discretized form (E.1) of this integral since they do not contribute to the sample variance. If the errors have a mean of zero, however, inclusion of the zero wavenumbers in (E.2) is an irrelevant point since the spectral values are then zero at $k = 0$ and $l = 0$. The Nyquist wavenumbers $-k_N$ and $-l_N$ at the lower range of each of the integrals must also be excluded from (E.2). To simplify the discussion, this minor distinction between the integral and discrete forms of Parseval's Theorem will be ignored in the analysis that follows.</footnote>

<footnote n="8">⁸ As noted in footnote 5, the spectral values at wavenumbers $l = 0$ and $l_N$ are not doubled in the 1-sided, 1-dimensional spectrum in (E.4). As in Appendix D this technicality will be ignored to avoid unnecessary lack of clarity in the analysis that follows.</footnote>
equal to the half-power filter cutoff wavenumber $k_c$. This relation for the integral of the squared filter transfer function holds approximately for any other smoother with smoothing parameters calibrated to the same half-power filter cutoff wavenumber $k_c$. The 1-sided, 1-dimensional spectrum (E.6a) of smoothed errors therefore simplifies to

$$ \bar{S}_e^{1d}(l) \approx 4\Delta x \Delta y \sigma_c^2 k_c W_{lc}^2(l). \quad (E.6b) $$

It is illuminating to contrast the different effects of 2-dimensional smoothing on the 2-dimensional spectrum (E.5b) compared with the 1-sided, 1-dimensional spectrum (E.6b). The two spectra can be expressed in terms of the constant white-noise spectral value $S_0$ by substituting (E.3) into (E.5b) and (E.6b) to get

$$ \bar{S}_e^{2d}(k, l) = W_{kc}^2(k) W_{lc}^2(l) S_0 \quad (E.7) $$

$$ \bar{S}_e^{1d}(l) = 4 k_c W_{lc}^2(l) S_0. \quad (E.8) $$

The 2-dimensional spectral values $\bar{S}_e^{2d}(k, l)$ are thus equal to $S_0$ multiplied by the product of the squared values of the filter transfer functions in each dimension at the particular wavenumbers $k$ and $l$. In contrast, the 1-dimensional spectral values $\bar{S}_e^{1d}(l)$ are equal to $S_0$ multiplied by the squared value of the filter transfer function in only the $y$ dimension with an additional constant multiplicative factor of $4k_c$ at every wavenumber $l$. A factor of 2 can be accounted for by the doubling of the power in the 1-sided, 1-dimensional spectrum. The additional factor of $2k_c$ represents a constant attenuation that is applied at every wavenumber $l$. For smoothing in the $x$ dimension with a filter cutoff wavelength of $\lambda_c = k_c^{-1} = 50$ km, for example, this corresponds to an attenuation by a factor of 25 cpkm at every wavenumber.<fnref n="9" />

For the benefit of the analysis in Appendix F, and to illustrate the importance of the above distinction between the effects of 2-dimensional smoothing on the 2-dimensional and 1-dimensional spectra, each of these spectra can be compared with their counterparts for the unsmoothed error fields. From (E.5a), the 2-dimensional spectrum of the smoothed error fields is just the 2-dimensional spectrum of the unsmoothed fields multiplied by the squared values of the two filter transfer functions at each wavenumber $k$ and $l$. Consider an ideal filter with transfer functions given by

$$ W_{kc}(k) = \begin{cases} 1 & \text{if } |k| \le k_c \\ 0 & \text{otherwise} \end{cases} \quad (E.9a) $$

$$ W_{lc}(l) = \begin{cases} 1 & \text{if } |l| \le l_c \\ 0 & \text{otherwise.} \end{cases} \quad (E.9b) $$

The 2-dimensional spectral values of the smoothed error fields are then related to the 2-dimensional spectral values of the unsmoothed error fields by

$$ \bar{S}_e^{2d}(k, l) = \begin{cases} S_e^{2d}(k, l) & \text{if } |k| \le k_c \text{ and } |l| \le l_c \\ 0 & \text{otherwise.} \end{cases} \quad (E.10) $$

For the case of 1-dimensional spectra, it can be seen from (E.4b) and (E.6b) that the 1-dimensional spectrum of smoothed errors is related to the 1-dimensional spectrum of unsmoothed errors by

$$ \bar{S}_e^{1d}(l) = 2\Delta x k_c W_{lc}^2(l) S_e^{1d}(l). \quad (E.11a) $$

For a half-power filter cutoff wavelength of $\lambda_c$ in the $x$ dimension, this is

$$ \bar{S}_e^{1d}(l) = \frac{2\Delta x}{\lambda_c} W_{lc}^2(l) S_e^{1d}(l). \quad (E.11b) $$

For the ideal filter in the $x$ dimension considered above that has the filter transfer function (E.9a), the expression (E.11a) for the spectrum of smoothed errors becomes

$$ \bar{S}_e^{1d}(l) = \begin{cases} 2\Delta x k_c S_e^{1d}(l) & \text{for } 0 < l \le l_c \\ 0 & \text{otherwise.} \end{cases} \quad (E.12a) $$

For the case of a Parzen smoother, a filter cutoff wavenumber of $k_c$ can be achieved according to (A.12a) with a span of $L_4 \approx k_c^{-1}$. Neglecting the imperfections of the filter transfer function of the Parzen smoother for the sake of illustration, a span of $L_4 = L\Delta x$ expresses (E.12a) as

$$ \bar{S}_e^{1d}(l) = \begin{cases} 2L^{-1} S_e^{1d}(l) & \text{for } 0 < l \le l_c \\ 0 & \text{otherwise.} \end{cases} \quad (E.12b) $$

The 1-dimensional spectral values at all wavenumbers in the pass band would therefore be attenuated by a factor of $2L^{-1}$.

In contrast to the 2-dimensional spectrum of smoothed error fields that is unchanged in the low-wavenumber pass band $|k| \le k_c$ and $|l| \le l_c$ according to (E.10) for an ideal filter, the smoothing in the $x$ dimension reduces the 1-dimensional spectral values within the pass band $|l| \le l_c$ in the $y$ dimension by the multiplicative factor of $2\Delta x k_c$ in (E.12a) compared with the 1-dimensional spectrum (E.4b) of the unfiltered errors. This perhaps non-intuitive point is important to the derivation in Appendix F of the error variance of the pre-processed SWOT estimates of SSH from the science requirement specification in terms of the 1-sided, 1-dimensional power spectral density of SWOT data smoothed with the above hypothetical ideal filter.

## Appendix F. The standard deviation of uncorrelated errors in SWOT estimates of SSH

The generation of simulated noisy SWOT measurements of SSH from the CCS model for the analyses of resolution capabilities in Sections 6 and 8–10 and determination of the variances and wavenumber spectra of the errors of geostrophically computed SWOT estimates of velocity and vorticity in Section 4 from the equations in Appendices G and I, respectively, all require knowledge of the variance or standard deviation of the uncorrelated errors in SWOT estimates of SSH. It is difficult to find this information in the SWOT documentation. Several different values are given or indirectly implied in two of the SWOT documents. The Algorithm Theoretical Basis document for the SWOT onboard processor (Peral, 2016) states, in the caption of Fig. 47, that the swath-averaged standard deviation for a significant wave height (SWH) of 2 m is $\sigma_h = 2.54$ cm for the uncorrelated measurement errors of SWOT data<fnref n="10" /> smoothed 2-dimensionally onto a 1 km × 1 km grid with an isotropic half-power filter cutoff wavelength of 2 km as summarized in our Appendix B.1. The Mission Performance and Error Budget document (Esteban Fernandez, 2017) gives three different values for the standard deviation of uncorrelated measurement errors for 2-m SWH. The specification of the white-noise contribution to the total error spectrum in the unnumbered first equation in Sec. 5.1 [see also Eq. (F.1) below] can be combined with the unnumbered last equation in Sec. 5.1.1 for the 1-sided, 1-dimensional power spectral density of the uncorrelated SSH measurement errors to deduce a value of $\sigma_h = 2.74$ cm. The swath-averaged standard deviation for 2-m SWH is subsequently stated in Secs. 5.1.3 and 5.4.5 to be 2.5 cm and 2.4 cm, respectively.

<footnote n="9">⁹ The units on this attenuation factor are because $2k_c$ multiplies the constant 2-dimensional white noise spectral value $S_0$ on the right side of (E.8), thus assuring the proper units of the 1-dimensional spectrum on the left side of the equation.</footnote>

<footnote n="10">¹⁰ In the preceding appendices, the variance $\sigma_c^2$ of uncorrelated errors was denoted generically with the subscript $\epsilon$ in order to be applicable to either SWOT or WaCM, which measure different variables. Hereafter, the subscript $\epsilon$ is replaced with a symbol that is indicative of the variable of interest. As the focus in this appendix is on the errors of SWOT estimates of SSH, the subscript $h$ is used in place of $\epsilon$. Other symbols are used in Appendices G and I, depending on the variable of interest (see the discussion in the introduction of Appendix G).</footnote>
<figure><img src="image_63.png" /><figcaption>Fig. F.1. The current projected performance of the SWOT instrument. The graph shows the dependencies of the standard deviation of uncorrelated errors of SWOT measurements of SSH on significant wave height (SWH) and swath location relative to the satellite ground track. The seven solid lines correspond to SWH values increasing from 2 m to 8 m at increments of 1 m (bottom to top). The errors for 0-m SWH are essentially the same as those shown for 2-m SWH. This projected performance incorporates improvements over the baseline science requirements for SWOT. The horizontal dashed line is the estimate $\sigma_h = 2.74$ cm of the cross-swath average standard deviation of uncorrelated measurement errors derived in Appendix F from the baseline science requirements.</figcaption></figure>

Source: The solid lines in the figure are from Fig. 2.5 of Gaultier et al. (2017).

Four different values for the standard deviation $\sigma_h$ of the uncorrelated errors of SWOT measurements of SSH can thus be found in the SWOT documentation. Among these, the value of $\sigma_h = 2.74$ cm is derived below from the wavenumber spectral specification of the science requirements and thus represents the baseline accuracy requirement for uncorrelated errors in SWOT measurements of SSH with a footprint size of 1 km. Although not stated as such, the three slightly smaller values noted above may be projected estimates (at the time of writing of the various sections of the reports) of the accuracy that will actually be achieved from SWOT in orbit. Alternatively, they may consider only the effects of instrumental errors and neglect smaller sources of uncorrelated errors such as the effects of spacecraft pointing errors on the signal-to-noise ratio across the measurement swaths.

Since the baseline requirement for the standard deviation of the uncorrelated measurement errors is ambiguous in the SWOT documentation, and because this information is crucial for the simulated SWOT sampling of the CCS model output in this study and likely in other studies, an expression for the error standard deviation is derived in this appendix. As in the spectral specification of SWOT measurement accuracy requirements in Rodríguez and Callahan (2016), the standard deviation of uncorrelated measurement errors derived below is the global swath-averaged value for SWOT estimates of SSH with a footprint size of 1 km in conditions of 2-m SWH. In reality, the measurement errors increase toward each edge of the two parallel measurement swaths. They also increase with increasing SWH, which varies geographically and temporally. The most recent projected estimates of the dependencies of the SWOT measurement errors on swath location and SWH are shown in Fig. F.1 (see also Fig. 2.5 of Gaultier et al., 2017).

The required variance $\sigma_h^2$ of the uncorrelated errors in the pre-processed SWOT estimates of SSH with a footprint size of 1 km is determined here from the baseline spectral characterization of the SWOT measurement errors that is specified in the SWOT Science Requirements Document (Rodríguez and Callahan, 2016) in terms of the residual errors after smoothing of the pre-processed data with a half-power filter cutoff wavelength of $\lambda_c = 15$ km. Explanation of what exactly is meant by this smoothing is contradictory in the SWOT

<figure><img src="image_64.png" /><figcaption>Fig. F.2. The 1-sided, 1-dimensional along-track wavenumber power spectral density of the science requirements for measurement errors of SWOT data before and after (blue and red lines, respectively) smoothing in ground-based post-processing to eliminate variability with wavelengths shorter than a half-power filter cutoff wavelength of $\lambda_c = 15$ km (shown by the dotted black line). The dotted red line corresponds to the requirement for red noise from orbit errors and long-wavelength measurement errors. The dashed red line corresponds to the requirement for residual uncorrelated errors after smoothing either 2-dimensionally with an ideal filter that has a magnitude of 1 for wavelengths longer than $\lambda_c$ and 0 for shorter wavelengths or smoothing only 1-dimensionally in the cross-track dimension with any realizable low-pass filter that has a half-power filter cutoff wavelength of $\lambda_c$. The solid red line is the sum of the power spectral densities of the red noise and the low-pass filtered uncorrelated errors. The dashed blue line corresponds to the uncorrelated errors in pre-processed SWOT data for a footprint size of 1 km that is derived in Appendix F without the 15-km smoothing. The solid blue line is the sum of the power spectral densities of the red noise and the white noise in the pre-processed SWOT data. The thick black line is the 68th-percentile SSH signal power spectral density from the SWOT Science Requirements Document (Rodríguez and Callahan, 2016). A power-law rolloff of $l^{-5/2}$ for alongshore wavenumber $l$ is shown for reference as the thin black line. The dashed and solid green lines show, respectively, the spectra of the residual uncorrelated noise and the total noise that would be obtained if the SWOT data were smoothed 2-dimensionally using a realizable Parzen smoother with a half-power filter cutoff wavelength of $\lambda_c = 15$ km (For interpretation of the references to colour in this figure, the reader is referred to the web version of this article.).</figcaption></figure>

documentation. The requirement 2.7.2a in Sec. 2.7 of the Science Requirements Document (Rodríguez and Callahan, 2016) explicitly states that 15-km smoothing is to be performed 2-dimensionally. Consistent with this specification, the Mission Performance and Error Budget document (Esteban Fernandez, 2017) states in Sec. 4.1 that the objective of the SWOT mission is to characterize SSH variability at spatial scales of 15 km and larger, which implicitly implies 2-dimensional smoothing. But later in Sec. 5.1.1 of the document, it is stated that the smoothing is to be applied only 1-dimensionally in the cross-track dimension by “an ideal square filter” that is later clarified in Sec. 5.1.2 to be the simple running average filter with the uniform weighting function given by (A.4) in our Appendix A (which was shown in Figs. A.1 and C.3 to be far from ideal in the wavenumber domain).

A rationale for smoothing only in the cross-track dimension is given in the discussion following the last unnumbered equation in Sec. 5.1.1 of Esteban Fernandez (2017) based on consideration only of the along-track wavenumber spectrum of smoothed SSH, as opposed to the resolution of the 2-dimensional SSH fields themselves. While smoothing only in the cross-track dimension is justifiable if the interest is solely in the 1-dimensional along-track wavenumber spectrum, the smoothing would more generally be applied isotropically in two dimensions to achieve the resolution of 15 km. The analysis that follows therefore assumes that the 15-km smoothing is applied in this manner.
In the discussion that follows, we will consider both an idealized filter and a realizable filter that rolls off gradually through the half-power filter cutoff wavenumber. As expressed by (D.5a), smoothing the pre-processed estimates of SSH with half-power filter cutoff wavenumbers of $k_c$ and $l_c$ in the x and y dimensions, respectively, reduces the uncorrelated error variance $\sigma_h^2$ by the approximate multiplicative factor (D.14b) of $\alpha = 4\Delta x \Delta y k_c l_c$, where $\Delta x$ and $\Delta y$ are the sample intervals at which the measurement errors are uncorrelated. For isotropic 2-dimensional smoothing with $k_c = l_c = 1/15$ cpkm and a uniform sample interval of $\Delta x = \Delta y = 1$ km, this multiplicative factor is $\alpha = 4/15^2 = 1/56.25$. The science requirement for measurement accuracy could thus have been specified in terms of the variance $\bar{\sigma}_h^2$ of residual errors after smoothing isotropically with a filter cutoff wavelength of $\lambda_c = 15$ km. The variance of the unsmoothed pre-processed SWOT estimates of SSH that gives this value of $\bar{\sigma}_h^2$ could then be simply calculated as $\sigma_h^2 = \alpha^{-1} \bar{\sigma}_h^2 = 56.25 \bar{\sigma}_h^2$.

The baseline requirement for instrument noise is instead specified in the Science Requirements Document (Rodríguez and Callahan, 2016) in terms of the 1-sided, 1-dimensional (along-track) wavenumber power spectral density of the residual white-noise contribution to the total SSH measurement errors after smoothing as

$$ \bar{S}_h^{1d}(l) = 2 \text{ cm}^2/\text{cpkm}, \quad \text{for } 1/1000 \le l \le 1/15 \text{ cpkm}. \qquad (\text{F.1a}) $$

The overbar signifies the 15-km smoothing of the SWOT data. The instrumental errors are thus specified explicitly only for wavenumbers $l$ smaller than $l_c = 1/15$ cpkm, i.e., for wavelengths $\lambda = l^{-1}$ longer than the half-power filter cutoff wavelength of $\lambda_c = l_c^{-1} = 15$ km.

For 2-dimensional smoothing with an ideal isotropic 2-dimensional filter, the specification of the spectrum (F.1a) would be extended from the filter cutoff of $l_c = 1/15$ cpkm to the Nyquist wavenumber $l_N$ of the SWOT data as

$$ \bar{S}_h^{1d}(l) = 0, \quad \text{for } 1/15 < l \le l_N \text{ cpkm}. \qquad (\text{F.1b}) $$

For the present official planned footprint size of 1 km and sample grid spacings of $\Delta x = \Delta y = 1$ km for the pre-processed SWOT estimates of SSH as specified in the onboard processing document (Peral, 2016), the Nyquist wavenumber is $l_N = 0.5$ cpkm. The residual white noise error spectrum (F.1a) and (F.1b) after smoothing with the ideal isotropic 2-dimensional filter is shown by the dashed red line in Fig. F.2.

It was shown in Appendix E that the constant 1-sided, 1-dimensional along-track wavenumber spectrum of residual uncorrelated measurement errors after smoothing with half-power filter cutoff wavenumbers of $k_c$ and $l_c$ in the x and y dimensions is related to the variance $\sigma_h^2$ of the unsmoothed errors by (E.6b). For the ideal isotropic 2-dimensional filter specified by (F.1a) and (F.1b), the filter transfer function $W_{l_c}(l)$ on the right side of (E.6b) is given by (E.9b). Then (E.6b) simplifies to

$$ \bar{S}_h^{1d}(l) = \begin{cases} 4\Delta x \Delta y k_c \sigma_h^2 & \text{if } 0 < l \le l_c \\ 0 & \text{otherwise.} \end{cases} \qquad (\text{F.2}) $$

Comparing (F.2) with the science requirement (F.1) expresses the variance of the pre-processed SWOT estimates of SSH as

$$ \sigma_h^2 = \frac{1 \text{ cm}^2/\text{cpkm}}{2\Delta x \Delta y k_c}. \qquad (\text{F.3a}) $$

The units in the numerator of this equation, in combination with the units of the variables in the denominator, assure that the left side of (F.3a) has units of cm$^2$. For a sample interval of $\Delta x = \Delta y = 1$ km between uncorrelated errors in pre-processed SWOT data and the post-processing half-power filter cutoff wavenumber of $k_c = 1/15$ cpkm in the science requirement specification, this gives a variance of

$$ \sigma_h^2 = 7.5 \text{ cm}^2 = (2.74 \text{ cm})^2 \qquad (\text{F.3b}) $$

for the errors of the pre-processed SWOT estimates of SSH. The requirement for 1 km × 1 km pre-processed estimates of SSH with isotropic 2-dimensional smoothing with a half-power filter cutoff wavelength

of 2 km to achieve a footprint size of 1 km is therefore that the uncorrelated errors have a standard deviation of $\sigma_h = 2.74$ cm.

As recently recommended by the SWOT Algorithm Development Team (see Appendix B.1), a smaller footprint size of 0.5 km is likely to be adopted in the near future for the onboard pre-processed SWOT estimates of SSH. For pre-processing of the raw SWOT data with a smaller isotropic filter cutoff wavelength of 1 km in order to achieve the footprint size of 0.5 km, the standard deviation (F.3b) of allowed measurement errors would increase by a factor of two.

The overall average measurement error standard deviation of $\sigma_h = 2.74$ cm derived above from the baseline specification (F.1) in terms of the white noise spectrum of the instrumental errors for 2-m SWH is shown as the dashed line in Fig. F.1. It is visually apparent that the across-swath average of the most recent projected estimate of the measurement errors that will be achieved by SWOT for 2-m SWH is smaller than 2.74 cm. An overall average of 2.74 cm appears to be more representative of the projected errors for a SWH of about 4 m. If the projected estimates of the measurement error characteristics shown in Fig. F.1 are achieved, the results presented in Sections 6, 8 and 9 may be somewhat pessimistic for 2-m SWH. From the analysis in Section 8, however, space-time smoothed maps of the geostrophically computed velocity and vorticity fields are much more susceptible to sampling errors than measurement errors. The distinction between 2.74 cm and the slightly smaller values of 2.54 cm, 2.5 cm and 2.4 cm noted above from the various SWOT documents is therefore a minor issue in the analysis of resolution capability presented in this paper.

The effects of a $\sigma_h = 2.74$ cm standard deviation of errors in pre-processed SWOT data on the standard deviations of errors of geostrophically computed SWOT estimates of velocity and vorticity are quantified

<figure><img src="image_65.png" /><figcaption>Fig. F.3. The alongshore wavenumber power spectral density of SSH computed from the CCS model. The spectrum shown by the red line was smoothed by ensemble averaging over the individual spectra computed from alongshore grid lines that extend the full length of the model domain shown in Fig. 5a with a cross-shore spacing of 5 km. The blue line is the spectrum computed empirically from simulated SSH error fields with a footprint size of 1 km and a standard deviation of $\sigma_h = 2.74$ cm after smoothing 2-dimensionally using a Parzen smoother with a half-power filter cutoff wavelength of $\lambda_c = 15$ km. The green line is the theoretical spectrum of SSH errors derived in Appendix I.2 for the same filtering. The black line is the 68th-percentile SSH spectrum from Fig. F.2 that is the basis for defining the science requirements for SWOT. Power-law rolloffs of $l^{-5/2}$ and $l^{-7/2}$ for alongshore wavenumber $l$ are overlaid for reference as thin lines. As noted in the caption for Fig. 13, the rolloff of $l^{-7/2}$ 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 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>
<figure><img src="image_66.png" /><figcaption>Fig. 4. The wavenumber power spectral densities of SSH computed from the boxed region of the North Atlantic HF and LF simulations shown in Fig. 10 (blue and red lines, respectively). The spectra were smoothed by ensemble averaging over the individual spectra computed from “cross-shore” grid lines (southeast to northwest) over the region shown by the boxes in Fig. 10. The thick black line is the 68th-percentile SSH spectrum from Figs. F.2 and F.3 that defines the science requirements for SWOT and the green line is the theoretical spectrum of SSH errors derived in Appendix I.1 and shown previously in Fig. F.3 after smoothing 2-dimensionally using a Parzen smoother with a half-power filter cutoff wavelength of $\lambda_c = 15$ km. Power-law rolloffs of $l^{-5}$, $l^{-3}$ and $l^{-5/2}$ for wavenumber $l$ are overlaid for reference as thin lines (For interpretation of the references to colour in this figure, the reader is referred to the web version of this article.).</figcaption></figure>

in Appendix G using propagation-of-error analysis. The wavenumber spectral characteristics of the errors in SWOT estimates of SSH and geostrophically computed velocity components and vorticity estimated from the pre-processed SWOT data and after additional smoothing in ground-based post-processing are derived in Appendix I.

The uncorrelated measurement errors in the SWOT estimates of SSH are superimposed on spatially correlated (long-wavelength) noise from orbit errors and various environmental effects. The science requirement for these red-noise errors is that they have a 1-sided, 1-dimensional (along-track) power spectral density no larger than

$$S_{\text{red}}^{1d}(l) = 0.0125 l^{-2} \text{ cm}^2/\text{cpkm}, \quad \text{for } 1/1000 < l < 1/15 \text{ cpkm}, \quad (\text{F.4})$$

which is shown extrapolated from the wavenumber $l_c = 1/15$ cpkm to the Nyquist wavenumber $l_N = 0.5$ cpkm for $1 \text{ km} \times 1 \text{ km}$ sampling by the dotted red line in Fig. F.2. It can be seen that smoothing with an ideal isotropic 2-dimensional filter that has a half-power filter cutoff wavelength of $\lambda_c = l_c^{-1} = 15$ km would have very little attenuation effect on the red noise since it has so little power at wavenumbers $l > l_c$. The science requirement specification for the along-track power spectral density of the total measurement errors in the SWOT data is thus given approximately by the sum of the red-noise spectrum (F.4) and the spectrum (F.1) of the residual white noise after ideal low-pass filtering with a half-power filter cutoff wavelength of 15 km. This total measurement error spectrum is shown by the solid red line in Fig. F.2.

The analysis above based on isotropic 2-dimensional smoothing neglects the effects of imperfections of the filter transfer function of any realizable low-pass filter that could be applied in practice in ground-based post-processing of the pre-processed SWOT estimates of SSH. Post-launch verification of the SWOT instrument performance as specified spectrally by (F.1) for SWOT data smoothed 2-dimensionally

and isotropically as explicitly stated in the Science Requirements Document (Rodríguez and Callahan, 2016) would therefore be subject to imperfections of the filter transfer function. For example, the dashed green line in Fig. F.2 shows the 1-sided, 1-dimensional power spectral density of residual white noise errors after smoothing 2-dimensionally using a Parzen smoother with a half-power filter cutoff wavelength of $\lambda_c = l_c^{-1} = 15$ km. The solid green line is the sum of this realizable low-pass filtered white noise spectrum and the red-noise spectrum (F.4). Because of the gradual rolloff of the filter transfer function through the half-power filter cutoff wavelength $l_c$, the flatness of the residual error spectrum to the value specified by (F.1a) would not be detectable when the pre-processed SWOT data are smoothed 2-dimensionally with a realizable filter. The requirement for uncorrelated measurement errors in the SWOT estimates of SSH that are specified in the Science Requirements Document based on isotropic 2-dimensional smoothing therefore cannot be formally tested in the spectral form (F.1).

The issue of the untestability of the science requirement specification (F.1) in terms of an ideal 2-dimensional filter can be alternatively addressed by “reverse engineering” the science requirement to determine the required wavenumber spectral characteristics of the unfiltered errors of SWOT estimates of SSH. From (E.4b) in Appendix E and the variance (F.3) derived above from (F.1), the along-track white noise spectrum for the uncorrelated errors in the unfiltered pre-processed SWOT estimates of SSH in the baseline design is

$$S_h^{1d}(l) = 2\Delta y \sigma_h^2 = 15 \text{cm}^2/\text{cpkm} \quad \text{for } 0 < l \le l_N, \quad (\text{F.5})$$

where $l_N = (2\Delta y)^{-1}$ is again the Nyquist wavenumber for a sample interval $\Delta y$.

The spectrum (F.5) of unfiltered uncorrelated measurement errors is shown by the dashed blue line in Fig. F.2 for the case of a footprint size of 1 km and grid spacings of $\Delta x = \Delta y = 1$ km. Except at the short wavelengths over which the total SWOT measurement errors are dominated by the contribution from uncorrelated measurement errors, the wavenumber spectrum of long-wavelength measurement errors shown by the dotted red line in Fig. F.2 is approximately the same with and without smoothing with the half-power filter cutoff wavelength of 15 km considered above. The wavenumber spectrum of total errors in the pre-processed SWOT data without smoothing can thus be approximated by the sum of the dashed blue line and the dotted red line, which is shown as the solid blue line in Fig. F.2. In contrast to the solid red line that is untestable because it is based on a hypothetical ideal 2-dimensional low-pass filter, the requirement that the wavenumber spectrum of unsmoothed measurement errors be no greater than the solid blue line can be tested in post-launch verification. This will also avoid edge effect problems from 15-km smoothing in the cross-track dimension near the edges of the narrow 50-km swath on each side of the satellite ground track (see further discussion below). The specification (F.1) of the SWOT measurement errors based on idealized filtering can thus be tested indirectly from the along-track spectrum of unfiltered pre-processed SWOT data.

The preceding analysis is based on the science requirements as specified by Rodríguez and Callahan (2016) in terms of isotropic 2-dimensional smoothing with a half-power filter cutoff wavelength of 15 km in each dimension. As noted previously, Sec. 5.1.1 of the Mission Performance and Error Budget document (Esteban Fernandez, 2017) specifies smoothing only 1-dimensionally in the cross-track dimension. In that case, the wavenumber power spectral density would be attenuated uniformly at all wavenumbers by the factor $\alpha = 1/56.25$ derived above, without the gradual rolloff at high wavenumbers that would occur with along-track smoothing in addition to the cross-track smoothing. This is true regardless of the imperfections of the filtering in the cross-track dimension, i.e., perpendicular to the along-track dimension of the 1-dimensional wavenumber spectrum.

The along-track wavenumber spectrum of this cross-track 1-dimensionally smoothed SSH extends uniformly to the Nyquist
wavenumber $l_{\lambda'} = (2\Delta y)^{-1}$ cpkm with the value of $\bar{S}_h^{1d}(l) = 2 \text{ cm}^2/\text{cpkm}$ in the spectral specification (F.1) of the science requirements for the uncorrelated measurement errors. However, the specification in the unnumbered first equation of Sec. 5.1 of Esteban Fernandez (2017) [equivalent to our Eq. (F.1a)] considers only the portion of this attenuated spectrum for wavenumbers smaller than 1/15 cpkm. This is effectively equivalent to assuming idealized isotropic smoothing in both the cross-track and along-track dimensions, as in our analysis above, with the distinction that the SWOT performance on-orbit can be tested spectrally if only 1-dimensional cross-track smoothing is applied. This cross-track smoothing can only be applied across the narrow 50-km swath on each side of the satellite ground track. This is only 3.3 times the half-power filter cutoff wavelength of 15 km. The cross-track smoothing will therefore be susceptible to artifacts from edge effects of the smoothing. It is preferable to test the SWOT performance from the spectrum (F.5) of unfiltered pre-processed SWOT estimates of SSH, as noted above.

We note again that testability of the science requirement for SWOT measurement accuracy would be a moot issue if the uncorrelated measurement errors had been specified in terms of the variance $\sigma_h^2$ given by (F.3b) derived above for the pre-processed SWOT estimates of SSH with a footprint size of 1 km. It would then be straightforward to test the SWOT performance either from the noise in the SSH fields or spectrally from (F.5) without the need for ground-based post-processing to smooth either 2-dimensionally or 1-dimensionally (cross-track) with a half-power filter cutoff wavelength of $\lambda_c = 15$ km. This will avoid any of the edge effect problems mentioned above that could arise from smoothing with a half-power filter cutoff wavelength of 15 km across a swath width of only 50 km.

It is enlightening to compare the 68th-percentile SSH wavenumber spectrum upon which the science requirement for SWOT measurement accuracy is based with the alongshore wavenumber spectrum of SSH from the CCS model used throughout this study. The two spectra are shown in Fig. F.3 along with the empirical and theoretical spectra of residual SSH measurement errors after isotropic 2-dimensional smoothing with a half-power filter cutoff wavelength of $\lambda_c = 15$ km (see Appendix I.2 for the details of the calculation of the theoretical error spectrum). As discussed above, the science requirement is for the SSH signal and error spectra with 15-km smoothing to intersect at a wavenumber of 0.067 cpkm (a wavelength of 15 km), thus achieving a signal-to-noise ratio of 1 at the wavelength resolution goal of 15 km. It is evident from Fig. F.3 that the SSH signal spectrum from the CCS model used in this study intersects the error spectrum at a smaller wavenumber of about 0.033 cpkm, which corresponds to a wavelength of about 30 km. At this wavelength, the spectral power in the CCS model is about a factor of 5 less energetic than the spectral power in the assumed 68th-percentile spectrum.

The discrepancy between the intersections of the white-noise spectrum for SWOT measurement errors with 15-km isotropic 2-dimensional smoothing and the two SSH signal spectra shown in Fig. F.3 raises a question about whether the spectral specification of the science requirements for SWOT will be achieved for the actual SSH spectrum. The difference occurs because the CCS model spectrum rolls off more steeply toward high wavenumbers than does the 68th-percentile spectrum assumed in the Science Requirements Document. In the wavenumber range from about 0.003 cpkm to 0.1 cpkm (wavelengths from about 300 km to 10 km), the dependence of the CCS model spectrum on alongshore wavenumber $l$ is approximately $l^{-7/2}$ compared with a rolloff of about $l^{-5/2}$ for the assumed 68th-percentile SSH spectrum.

The flatter assumed 68th-percentile wavenumber spectrum is based on extrapolation of along-track SSH spectra from nadir altimetry that resolves variability only down to wavelength scales of about 70 km (Xu and Fu, 2012). The validity of this extrapolation is unknown since observational data are not presently available to determine the SSH spectrum on these scales. Velocity measurements by acoustic Doppler current profilers can be inverted by assuming geostrophy (see, for example,

Fig. 4 of Rocha et al., 2016). Such inversions omit the SSH signals from internal gravity waves and other small-scale ageostrophic processes. The spectral characteristics at wavelengths shorter than 70 km must therefore be inferred from high-resolution ocean circulation models.

The contributions of internal gravity waves and other small-scale ageostrophic processes to the spatial variability of SSH will flatten the spectrum (i.e., increase the variance) at high wavenumbers. Some insight into their effects on the wavenumber spectrum of SSH can be gained from the pair of North Atlantic simulations shown in Figs. 10 and 11 of Section 3. The wavenumber rolloff in the SSH spectrum computed from the model forced by monthly winds and no ocean tides (the LF simulation) is about $l^{-5}$ over all wavenumbers in the region southeast of the Gulf Stream (Fig. F.4). Up to a wavenumber of about 1/50 cpkm, the spectral rolloff is also about $l^{-5}$ in the model forced by high-frequency winds and ocean tides (the HF simulation). At higher wavenumbers, however, the spectral rolloff in the HF model flattens to about $l^{-3}$. As in the case of the spectrum from the CCS model shown in Fig. F.3, the spectra from both the HF model and the LF model intersect the error spectrum at wavelengths significantly longer than the resolution goal of 15 km.

On the other hand, Savage et al. (2017) found flatter spectral rolloffs between $l^{-2}$ and $l^{-1}$ at wavenumbers between 1/70 cpkm and 1/30 cpkm in high-resolution versions of the Massachusetts Institute of Technology general circulation model (MITgcm) for a wide range of dynamical regimes (see also Rocha et al., 2016). They attribute most of this flattening of the SSH spectrum to internal gravity waves. Depending on the signal power levels, a consequence of the flatter spectral rolloffs is that the SSH signal spectra from the MITgcm model can intersect the noise spectrum for 15-km smoothed SWOT data at shorter wavelengths than from the three ROMS models considered in Figs. F.3 and F.4. If the spectrum of SSH in the real ocean is similar to the SSH spectra from the MITgcm model, the wavelength resolution goal of 15 km will be achieved.

It is apparent from the discussion above that the details of the wavenumber spectral characteristics of SSH differ from one model to another and likely from one region to another. These differences are apparently attributable to a variety of technical details of the model formulations. One of the primary contributions of the SWOT mission will be to provide the first observational evidence from which the wavenumber spectral characteristics of SSH as represented in the various models can be tested.

## Appendix G. The standard deviations of the errors of SWOT and WaCM estimates of derivative quantities

The variables of interest in this study include derivatives of the basic measurements by SWOT and WaCM. For SWOT, the velocity must be computed geostrophically from derivatives of the noisy SSH measurements and the vorticity must be estimated from additional derivatives of the geostrophically computed velocity estimates, i.e., double differentiation of the noisy SSH measurements. For WaCM, the velocity will be measured directly and vorticity will be computed from derivatives of the noisy velocity component measurements. Estimation of these derivatives by finite differencing of the gridded fields of the variable measured by each instrument amplifies the errors in the measured variables. The effects of these error amplifications on the uncertainties of the SWOT and WaCM estimates of the derivative fields are quantified in this appendix using standard propagation-of-error analysis.

Throughout this appendix, errors are denoted by $\epsilon$ with a subscript symbol that is indicative of the variable of interest and the variances of these errors are denoted by $\sigma^2$ with the same subscript. As in Appendix F, the subscript $h$ is used in Appendix G.1 to indicate the errors of pre-processed SWOT estimates of SSH. The subscripts $u_g$, $v_g$ and $\zeta_g$ are used in Appendix G.1 to indicate the errors of geostrophically computed SWOT estimates of the two velocity components and vorticity. Similarly, the subscripts $u$, $v$ and $\zeta$ are used in Appendix G.2 to indicate
the errors of WaCM estimates of the two velocity components and vor-
ticity. As in Appendices E and F, overbars are used in Appendix G.3 to
indicate the variances of residual errors after 2-dimensional smoothing
of the pre-processed SWOT data.

An important point to note is that the expressions (G.2b), (G.5) and (G.8) derived below for the variances of the errors of geostrophically computed SWOT estimates of the x and y components of velocity and vorticity are all expressed in terms of the variance σ²_h of the pre-processed SWOT estimates of SSH. The value of σ²_h = (2.74 cm)² that is derived for a footprint size of 1 km in Appendix F is used in the applications of these equations in Section 4.1, but the same equations can be used for any specified value of σ²_h. Likewise, the expression (G.15) derived below for the variance of errors of WaCM estimates of vorticity is expressed in terms of the variances σ²_u and σ²_v of WaCM estimates of the x and y components of velocity. A value of σ²_u = σ²_v = (0.354 m s⁻¹)² for a footprint size of 5 km is used in the application of this equation in Section 4.2, but the same equation can be used for any specified values of σ²_u and σ²_v.

As noted previously, the uncorrelated measurement errors for both SWOT and WaCM will vary in the cross-track direction and will depend on significant wave height. The constant values of σ<sub>h</sub> = 2.74 cm and σ<sub>u</sub> = σ<sub>v</sub> = 0.354 m s<sup>-1</sup> used throughout this study are the average values across the measurement swaths. In actuality, these errors increase toward the edges of the swaths as shown in Fig. F.1 for the case of SWOT. Because we have neglected the cross-track variations of instrumental noise, the effects of uncorrelated errors on the resolution capabilities of SWOT and WaCM estimates of the variables of interest are likely to be underestimated in Sections 6 and 8–10. In other words, the resolution capabilities inferred from the analysis in this study are likely somewhat optimistic.

G.1. Geostrophically computed SWOT estimates of velocity and vorticity with 1 km footprint

The standard deviation of the uncorrelated instrumental noise con-
tribution to the pre-processed SWOT measurements of SSH was derived
in Appendix F from the wavenumber spectral specification (F.1) in the
baseline requirement for SWOT measurement errors. For measurements
with a footprint size of 1 km, the derived value of the standard devia-
tion for the baseline design is σ<sub>h</sub> = 2.74 cm. It is shown in the bottom
panel of Fig. B.1a that SWOT SSH measurement errors with a footprint
size of 1 km are essentially uncorrelated on a Δx × Δy = 1 km × 1 km
grid.

The error $\epsilon_{v_g}(x, y)$ of the alongshore component of geostrophically computed velocity $v = g f^{-1} \partial h / \partial x$ at grid location $(x, y)$ estimated from cross-shore centered differences of the height errors at the adjacent grid locations in the $x$ dimension is

$$
\epsilon_{v_g}(x, y) = \frac{g}{f} \frac{\epsilon_h(x + \Delta x, y) - \epsilon_h(x - \Delta x, y)}{2\Delta x}. \qquad (G.1)
$$

Since the height errors are assumed here to be spatially homogeneous,
the variance of these alongshore geostrophically computed velocity
errors obtained by propagation-of-error analysis is

$$
\sigma_{v_g}^2 = \frac{g^2}{f^2} \frac{\sigma_h^2}{2\Delta x^2} \left[ 1 - \rho_h(2\Delta x) \right], \qquad (G.2a)
$$

where $\rho_h(2\Delta x)$ is the autocorrelation of the SSH measurements at a
spatial lag of $2\Delta x$. This lagged autocorrelation is essentially zero since
the height errors for a footprint size of 1 km are essentially uncorrelated
on a 1 km × 1 km grid as noted above from the bottom panel of
Fig. B.1a. The variance of the alongshore geostrophically computed
velocity errors is therefore

$$
\sigma_{v_g}^2 = \frac{g^2}{f^2} \frac{\sigma_h^2}{2\Delta x^2}. \qquad (G.2b)
$$

The Coriolis parameter at the central latitude 37°N of the CCS model
domain is $f = 8.75 \times 10^{-5}$ s$^{-1}$. For gravitational acceleration $g =$

9.81 m s⁻², an SSH error variance of σ²_h = (2.74 cm)² and a grid spacing of Δx = 1 km, the standard deviation of the errors of the alongshore geostrophically computed velocity component is

$$
\sigma_{v_g}(37^\circ\text{N}) = 2.17 \text{ m s}^{-1}. \qquad (\text{G.3})
$$

The error $\epsilon_{u_g}(x, y)$ of the cross-shore component of geostrophically computed velocity $u = -g f^{-1} \partial h / \partial y$ at grid location $(x, y)$ estimated similarly from alongshore centered differences of the height errors at the adjacent grid locations in the $y$ dimension is

$$
\epsilon_{u_g}(x, y) = - \frac{g}{f} \frac{\epsilon_h(x, y + \Delta y) - \epsilon_h(x, y - \Delta y)}{2\Delta y}. \qquad (G.4)
$$

Analogous to the derivation of (G.2b), the variance of these cross-shore
geostrophically computed velocity errors on a 1 km × 1 km grid is

$$
\sigma_{u_g}^2 = \frac{g^2}{f^2} \frac{\sigma_h^2}{2\Delta y^2}. \qquad (G.5)
$$

For a uniform grid spacing Δx = Δy = 1 km, the cross-shore geostroph-
ically computed velocity errors $\epsilon_{u_g}(x, y)$ at the central latitude 37°N of
the CCS model domain have the same standard deviation (G.3) as the
alongshore velocity errors,

$$
\sigma_{u_g}(37^\circ\text{N}) = 2.17 \text{ m s}^{-1}. \qquad (\text{G.6})
$$

The error $\epsilon_{\zeta_g}(x, y)$ of the vorticity $\zeta_g = \partial v_g / \partial x - \partial u_g / \partial y$ at grid location $(x, y)$ calculated from centered differences of the geostrophically computed velocity component errors is

$$
\epsilon_{\zeta_g}(x, y) = \frac{\epsilon_{v_g}(x+\Delta x, y) - \epsilon_{v_g}(x-\Delta x, y)}{2\Delta x} - \frac{\epsilon_{u_g}(x, y+\Delta y) - \epsilon_{u_g}(x, y-\Delta y)}{2\Delta y}. \quad (\text{G.7a})
$$

From the expressions (G.1) and (G.4) for the geostrophically com-
puted velocity component errors calculated from centered differences
of height errors, (G.7a) can be written as the sum of four centered
differences of height errors with center points at (x + Δx, y), (x - Δx, y),
(x, y + Δy) and (x, y - Δy),

$$
\epsilon_{\zeta_g}(x, y) = \frac{g}{f} \frac{1}{2\Delta x} \left[ \frac{\epsilon_h(x + 2\Delta x, y) - \epsilon_h(x, y)}{2\Delta x} - \frac{\epsilon_h(x, y) - \epsilon_h(x - 2\Delta x, y)}{2\Delta x} \right] \\
+ \frac{g}{f} \frac{1}{2\Delta y} \left[ \frac{\epsilon_h(x, y + 2\Delta y) - \epsilon_h(x, y)}{2\Delta y} - \frac{\epsilon_h(x, y) - \epsilon_h(x, y - 2\Delta y)}{2\Delta y} \right]. \quad (G.7b)
$$

For a uniform grid spacing Δy = Δx, this simplifies to

$$
\epsilon_{\zeta_g}(x, y) = \frac{g}{f} \frac{1}{4\Delta x^2} \left[ \begin{aligned}
& \epsilon_h(x + 2\Delta x, y) \\
& + \epsilon_h(x - 2\Delta x, y) \\
& - 4\epsilon_h(x, y) \\
& + \epsilon_h(x, y + 2\Delta y) \\
& + \epsilon_h(x, y - 2\Delta y)
\end{aligned} \right]. \qquad (G.7c)
$$

For the 1 km × 1 km grid on which SWOT estimates of SSH are uncor-
related, the variance of the errors of SWOT estimates of $\zeta_g$ obtained by
propagation-of-error analysis is

$$
\sigma_{\zeta_g}^2 = \frac{g^2}{f^2} \frac{20 \sigma_h^2}{(4 \Delta x^2)^2}. \qquad (G.8)
$$

At the central latitude 37°N of the CCS model domain, the standard
deviation of the errors of geostrophically computed SWOT estimates of
vorticity is thus

$$
\sigma_{\zeta_g}^2 (37^\circ\text{N}) = 3.43 \times 10^{-3} \text{ s}^{-1}. \qquad (\text{G.9a})
$$

Compared with the planetary vorticity of *f* = 8.75 × 10⁻⁵ s⁻¹ at 37°N,
this standard deviation of the geostrophically computed vorticity errors
is

$$ \sigma_{\zeta_g}(37^\circ\text{N}) = 39.1f. \qquad (G.9b) $$

The uncertainties (G.3) and (G.6) of geostrophically computed SWOT estimates of velocity and the uncertainty (G.9) of geostrophically computed SWOT estimates of vorticity are very large compared with the oceanographic signals of interest. Clearly the SWOT data must be smoothed spatially and/or temporally to achieve scientifically useful estimates of geostrophically computed velocity and vorticity. This is discussed further in Appendix G.3 and Section 4.

## G.2. WaCM estimates of vorticity with 5 km footprint

In the baseline design, the standard deviation of speed uncertainties in WaCM estimates of surface velocity for a footprint size of 5 km (i.e., after isotropic smoothing of the raw WaCM measurements with a half-power filter cutoff wavelength of 10 km) is $\sigma_{spd} = 0.50 \text{ m s}^{-1}$. WaCM estimates of velocity with a footprint size of 5 km are essentially uncorrelated on a $\Delta x \times \Delta y = 5 \text{ km} \times 5 \text{ km}$ grid (see the bottom panel of Fig. B.1b). We will assume that the errors of the two velocity components are uncorrelated with each other.

In reality, the standard deviations $\sigma_u$ and $\sigma_v$ of the errors of the 10-km smoothed velocity estimates differ for the two orthogonal velocity components $u$ and $v$. In Sections 6 and 8–10, we have assumed that the speed uncertainties are equally partitioned between $u$ and $v$. The standard deviations $\sigma_u$ and $\sigma_v$ are then both equal to a value $\sigma_{u,v}$ that is related to the baseline design of $\sigma_{spd} = 0.50 \text{ m s}^{-1}$ by

$$ \sigma_{u,v} = \frac{\sigma_{spd}}{\sqrt{2}} = 0.354 \text{ m s}^{-1}. \qquad (G.10) $$

In the analysis that follows, we allow for the possibility of different error standard deviations for each velocity component.

The errors of WaCM estimates of vorticity $\zeta = \partial v/\partial x - \partial u/\partial y$ can be calculated from centered differences of the velocity errors,

$$ \epsilon_{\zeta(x,y)} = \frac{\epsilon_v(x + \Delta x, y) - \epsilon_v(x - \Delta x, y)}{2\Delta x} - \frac{\epsilon_u(x, y + \Delta y) - \epsilon_u(x, y - \Delta y)}{2\Delta y}. \qquad (G.11) $$

Since the $u$ and $v$ measurement errors are assumed to be spatially homogeneous, uncorrelated with each other, and individually uncorrelated spatially on the 5 km $\times$ 5 km grid, the variance of the errors of $\zeta$ obtained by propagation-of-error analysis for a grid spacing of $\Delta x \times \Delta y = 5 \text{ km} \times 5 \text{ km}$ on which the velocity component errors are uncorrelated is

$$ \sigma_{\zeta}^{2} = \frac{\sigma_{v}^{2}}{2\Delta x^{2}} + \frac{\sigma_{u}^{2}}{2\Delta y^{2}}. \qquad (G.12) $$

For equal partitioning of the speed uncertainties between the $u$ and $v$ velocity components with standard deviations given by (G.10), the standard deviation (G.12) of the errors of WaCM estimates of vorticity for a uniform 5 km $\times$ 5 km grid is

$$ \sigma_{\zeta} \Big|_{5\text{km}} = 7.08 \times 10^{-5} \text{ s}^{-1}. \qquad (G.13a) $$

Compared with the planetary vorticity of $f = 8.75 \times 10^{-5} \text{ s}^{-1}$ at the central latitude 37°N of the CCS model domain, this is

$$ \sigma_{\zeta}(37^\circ\text{N}) \Big|_{5\text{km}} = 0.807f. \qquad (G.13b) $$

It is shown in Appendix H that it is advantageous to oversample the WaCM data on a grid spacing of $\Delta x = \Delta y = 1 \text{ km}$ because the response function of centered differences on the finer grid retains more of the high wavenumber variability in the vorticity signal. The footprint size from the filtering in the pre-processing of WaCM data is still 5 km. The standard deviations of the errors of WaCM estimates of the velocity components are therefore still $\sigma_u^2$ and $\sigma_v^2$, but the velocity component errors on a 1 km $\times$ 1 km grid are spatially correlated. We will continue

to assume, however, that the errors of the two velocity components are uncorrelated with each other, even on the oversampled $\Delta x \times \Delta y = 1 \text{ km} \times 1 \text{ km}$ grid. The variance of the vorticity errors (G.11) then becomes

$$ \sigma_{\zeta}^{2} \Big|_{1\text{km}} = \frac{\sigma_{v}^{2}}{2\Delta x^{2}} \left[ 1 - \rho_{v}(2\Delta x) \right] + \frac{\sigma_{u}^{2}}{2\Delta y^{2}} \left[ 1 - \rho_{u}(2\Delta y) \right], \qquad (G.14) $$

where $\rho_v(2\Delta x)$ and $\rho_u(2\Delta y)$ are the autocorrelations of, respectively, the alongshore velocity errors $\epsilon_v$ at a spatial lag of $2\Delta x$ and the cross-shore velocity errors $\epsilon_u$ at a spatial lag of $2\Delta y$.

For the Parzen smoother with a half-power filter cutoff wavelength of 10 km in the baseline design for pre-processing of WaCM data, the autocorrelations of errors of $u$ and $v$ at a lag of 2 km are about 0.638 (see the bottom panel of Fig. B.1b). The variance (G.14) of the errors of WaCM estimates of vorticity on the oversampled 1 km $\times$ 1 km grid then becomes

$$ \sigma_{\zeta}^{2} \Big|_{1\text{km}} = \frac{0.362 \sigma_{v}^{2}}{2\Delta x^{2}} + \frac{0.362 \sigma_{u}^{2}}{2\Delta y^{2}}. \qquad (G.15) $$

Because of the smaller values of $\Delta x$ and $\Delta y$ in the denominators on the right side of (G.15) compared with their values in the denominators on the right side of (G.12), the standard deviation of vorticity errors is larger on the oversampled 1 km $\times$ 1 km grid. This is the case despite the smaller factor of 0.362 in the numerators of (G.15) that arises from the spatial correlations of the errors on the 1 km $\times$ 1 km grid.

For equal partitioning of the speed uncertainties between the $u$ and $v$ velocity components with standard deviations (G.10), the standard deviation of the errors of WaCM estimates of vorticity on the oversampled 1 km $\times$ 1 km grid is

$$ \sigma_{\zeta} \Big|_{1\text{km}} = 2.13 \times 10^{-4} \text{ s}^{-1}. \qquad (G.16a) $$

Compared with the planetary vorticity of $f = 8.75 \times 10^{-5} \text{ s}^{-1}$ at the central latitude 37°N of the CCS model domain, this is

$$ \sigma_{\zeta}(37^\circ\text{N}) \Big|_{1\text{km}} = 2.43f. \qquad (G.16b) $$

## G.3. Geostrophically computed SWOT estimates of velocity and vorticity with 5 km footprint

The standard deviation (G.10) of velocity component errors from WaCM is much smaller than the standard deviations (G.3) and (G.6) of the errors of geostrophically computed velocity components estimated from SWOT measurements of SSH. Likewise, the standard deviation (G.16) of the errors of WaCM estimates of vorticity on a 1 km $\times$ 1 km grid is much smaller than the standard deviation (G.9) of the errors of geostrophically computed vorticity estimated from SWOT data on a 1 km $\times$ 1 km grid. It must be kept in mind, however, that the footprint size is five times coarser for the pre-processed WaCM measurements of surface velocity than for the pre-processed SWOT measurements of SSH. In this section, we quantify the reduction of the errors of geostrophically computed SWOT estimates of velocity and vorticity that can be achieved in ground-based post-processing by smoothing in a manner that is commensurate with the smoothing with the same half-power filter cutoff wavelength of 10 km planned for the pre-processing of WaCM data.

Consider first the errors of SWOT estimates of the velocity component $v_g$ computed geostrophically from centered differences of SWOT estimates of SSH smoothed using the same Parzen smoother with a half-power filter cutoff wavelength of 10 km considered in Appendix G.2 for the pre-processing of WaCM data. The expression analogous to (G.2a) for the variance of the resulting smoothed estimates of $v_g$ is

$$ \bar{\sigma}_{v_g}^2 = \frac{g^2}{f^2} \frac{\bar{\sigma}_h^2}{2\Delta x^2} \left[ 1 - \bar{\rho}_h(2\Delta x) \right] = \frac{\bar{\sigma}_h^2}{\sigma_h^2} \left[ 1 - \bar{\rho}_h(2\Delta x) \right] \sigma_{v_g}^2, \qquad (G.17a) $$

where the overbar on $\bar{\sigma}_h^2$ distinguishes the variance of the smoothed SSH errors from the variance $\sigma_h^2$ of the pre-processed SWOT estimates
of SSH and $\sigma_{v_g}^2$ on the right side of (G.17a) is the variance (G.2b) of the errors of unsmoothed SWOT estimates of $v_g$. The term $\bar{\rho}_h(2\Delta x)$ in the square brackets is the autocorrelation of the smoothed estimates of SSH at a lag of $2\Delta x$.

Whereas the lagged autocorrelation was zero at nonzero lags for the case of unsmoothed SSH data considered in (G.2b), it is nonzero after smoothing with a half-power filter cutoff wavelength of 10 km. As noted in Appendix G.2 from the bottom panel of Fig. B.1b, this autocorrelation for the case of the Parzen smoother used here has a value of about 0.638 at a lag of $2\Delta x = 2$ km. And from (D.14c) in Appendix D, smoothing of the errors of the pre-processed estimates of SSH with a half-power filter cutoff wavelength of 10 km reduces the SSH error variance by approximately a factor of 25. Then $\bar{\sigma}_h^2/\sigma_h^2 \approx 0.040$ and (G.17a) becomes

$$ \bar{\sigma}_{v_g}^2 = 0.0145 \sigma_{v_g}^2. \qquad (G.17b) $$

From (G.2b), the variance of the errors of the unsmoothed SWOT estimates of $v_g$ is $\sigma_{v_g}^2 = (2.17 \text{ m s}^{-1})^2$ at the central latitude 37°N of the CCS model domain. The standard deviation (G.17b) of the errors of SWOT estimates of the velocity component $v_g$ computed geostrophically from smoothed SSH at 37°N is therefore

$$ \bar{\sigma}_{v_g}(37^\circ\text{N}) = 0.261 \text{ m s}^{-1}. \qquad (G.18) $$

The standard deviation $\bar{\sigma}_{v_g}$ of the errors of SWOT estimates of the velocity component $v_g$ computed geostrophically from smoothed SSH is the same as $\bar{\sigma}_{v_g}$. Smoothing the pre-processed SWOT estimates of SSH with the same half-power filter cutoff wavelength of 10 km that is planned for pre-processed WaCM estimates of surface velocity thus reduces the error standard deviations (G.3) and (G.6) by more than a factor of 8. Moreover, the standard deviations of the errors of $v_g$ and $v_g$ computed geostrophically from the smoothed SWOT estimates of SSH are somewhat smaller than the standard deviations (G.10) of pre-processed WaCM estimates of the two velocity components.

Derivation of the standard deviation of the errors of estimates of vorticity computed geostrophically from smoothed SWOT estimates of SSH is tedious. Analogous to (G.7c), the errors of the geostrophically computed vorticity estimated from smoothed SWOT data can be expressed as

$$ \bar{\epsilon}_{\zeta_g}(x, y) = \frac{g}{f} \frac{1}{4\Delta x^2} \left[ \begin{aligned} & \bar{\epsilon}_h(x + 2\Delta x, y) \\ & + \bar{\epsilon}_h(x - 2\Delta x, y) \\ & - 4\bar{\epsilon}_h(x, y) \\ & + \bar{\epsilon}_h(x, y + 2\Delta y) \\ & + \bar{\epsilon}_h(x, y - 2\Delta y) \end{aligned} \right], \qquad (G.19) $$

where the overbars indicate the errors after smoothing of the SWOT data with the same half-power filter cutoff wavelength of 10 km considered in Appendix G.2 for pre-processing of WaCM data. Whereas the errors $\epsilon_h$ of the pre-processed SWOT estimates of SSH were uncorrelated in (G.7c), the errors $\bar{\epsilon}_h$ of smoothed SSH are correlated.

Assuming that the error statistics are spatially homogeneous, the variance of the errors (G.19) of smoothed geostrophically computed vorticity after grouping of terms is

$$ \bar{\sigma}_{\zeta_g}^2 = \frac{g^2}{f^2} \frac{\bar{\sigma}_h^2}{(4\Delta x^2)^2} \left[ \begin{aligned} & 20 + \bar{\rho}_h(0, 4\Delta y) \\ & + 2\bar{\rho}_h(-2\Delta x, 2\Delta y) \\ & - 8\bar{\rho}_h(0, 2\Delta y) \\ & + 2\bar{\rho}_h(2\Delta x, 2\Delta y) \\ & + \bar{\rho}_h(-4\Delta x, 0) \\ & - 8\bar{\rho}_h(-2\Delta x, 0) \end{aligned} \right] $$

$$ \begin{aligned} & - 8 \bar{\rho}_h(2\Delta x, 0) \\ & + \bar{\rho}_h(4\Delta x, 0) \\ & + 2 \bar{\rho}_h(-2\Delta x, -2\Delta y) \\ & - 8 \bar{\rho}_h(0, -2\Delta y) \\ & + 2 \bar{\rho}_h(2\Delta x, -2\Delta y) \\ & + \bar{\rho}_h(0, -4\Delta y) \end{aligned} $$

where $\bar{\rho}_h(m\Delta x, n\Delta y)$ is the 2-dimensional lagged autocorrelation between the errors of smoothed SWOT estimates of SSH separated by $m\Delta x$ in the x dimension and $n\Delta y$ in the y dimension. The overbars on these lagged autocorrelations again indicate that they are based on the smoothed SWOT estimates of SSH. Because the 2-dimensional smoothing is applied isotropically, the lagged autocorrelations in the above expression are symmetric about zero lag and $\bar{\rho}_h(m\Delta x, n\Delta y)$ depends only on the radial distance $[(m\Delta x)^2 + (n\Delta y)^2]^{1/2}$. The variance of the smoothed geostrophically computed vorticity errors can then be written in terms of 1-dimensional lagged autocorrelations as

$$ \bar{\sigma}_{\zeta_g}^2 = \frac{g^2}{f^2} \frac{\bar{\sigma}_h^2}{(4\Delta x^2)^2} \left[ \begin{aligned} & 20 + 2\bar{\rho}_h(4\Delta x) \\ & - 16\bar{\rho}_h(2\Delta x) \\ & + 8\bar{\rho}_h(2\sqrt{\Delta x^2 + \Delta y^2}) \\ & - 16\bar{\rho}_h(2\Delta y) \\ & + 2\bar{\rho}_h(4\Delta y) \end{aligned} \right]. \qquad (G.20) $$

For a uniform grid spacing $\Delta y = \Delta x$, this simplifies to

$$ \begin{aligned} \bar{\sigma}_{\zeta_g}^2 &= \frac{g^2}{f^2} \frac{\bar{\sigma}_h^2}{(4\Delta x^2)^2} \left[ 20 + 4\bar{\rho}_h(4\Delta x) - 32\bar{\rho}_h(2\Delta x) + 8\bar{\rho}_h(2\sqrt{2\Delta x}) \right] \\ &= \frac{\bar{\sigma}_h^2}{\sigma_h^2} \left[ 1 + \frac{1}{5}\bar{\rho}_h(4\Delta x) - \frac{8}{5}\bar{\rho}_h(2\Delta x) + \frac{2}{5}\bar{\rho}_h(2\sqrt{2\Delta x}) \right] \sigma_{\zeta_g}^2. \end{aligned} \qquad (G.21) $$

where $\sigma_{\zeta_g}^2$ is the variance (G.8) of the errors of unsmoothed SWOT estimates of $\zeta_g$. As noted above in the derivation of (G.17b), $\bar{\sigma}_h^2/\sigma_h^2 \approx 0.040$ for Parzen smoothing of the errors with a filter cutoff wavelength of 10 km. For the grid spacing $\Delta x = \Delta y = 1$ km of SWOT data, the autocorrelations of smoothed SSH at lags of 2 km, $2\sqrt{2} = 2.828$ km and 4 km are about 0.638, 0.403 and 0.155, respectively (see the bottom panel of Fig. B.1b). The variance (G.21) of vorticity estimates computed geostrophically from SWOT estimates of SSH smoothed using a Parzen smoother with a half-power filter cutoff wavelength of 10 km is therefore

$$ \bar{\sigma}_{\zeta_g}^2 = 0.00684 \sigma_{\zeta_g}^2. \qquad (G.22) $$

From (G.9a), the variance of vorticity errors computed from unsmoothed SSH is $\sigma_{\zeta_g}^2 = (3.43 \times 10^{-3} \text{ s}^{-1})^2$. The standard deviation of the errors of SWOT estimates of vorticity computed geostrophically from 10-km smoothed SSH is therefore

$$ \bar{\sigma}_{\zeta_g} = 2.84 \times 10^{-4} \text{ s}^{-1}. \qquad (G.23a) $$

At the central latitude 37°N of the CCS model domain where $f = 8.75 \times 10^{-5} \text{ s}^{-1}$, this is

$$ \bar{\sigma}_{\zeta_g}(37^\circ\text{N}) = 3.23f. \qquad (G.23b) $$

Smoothing the pre-processed SWOT estimates of SSH with the same half-power filter cutoff wavelength of 10 km that is planned for pre-processed WaCM estimates of surface velocity thus reduces the standard deviation (G.9) of geostrophically computed vorticity estimates by more than a factor of 12. The standard deviation (G.23) of these geostrophically computed SWOT estimates of vorticity is only
<figure><img src="image_67.png" /><figcaption>Fig. H.1. Log–log plots of the squared filter transfer functions (also referred to as the response functions) for centered differences. The thick solid and dashed lines are for, respectively, 3-point and 9-point centered differences on a 1 km grid. The thin solid and dashed lines are for 3-point and 9-point centered differences on a 5 km grid assuming that the discretely sampled data on the 5 km grid have been filtered to attenuate variability at wavenumbers higher than the Nyquist wavenumber of 0.1 cpmk. The short dashed straight line in this log–log plot corresponds to the $(2\pi k)^2$ response function for derivatives of a continuous variable.</figcaption></figure>

slightly higher than the standard deviation (G.16) of WaCM estimates of vorticity computed from 10-km smoothed velocity estimates on a 1 km × 1 km grid.

Even with 10-km smoothing, the standard deviations of the errors of the estimates of velocity and vorticity from SWOT in (G.18) and (G.23) and from WaCM in (G.10) and (G.16) are much larger than the oceanographic signals of interest. Clearly, these errors must be further reduced with additional spatial and/or temporal smoothing. The effects of spatial smoothing on the variances of residual errors of SWOT and WaCM estimates of velocity components and vorticity are discussed in Section 4.3 (see Fig. 12).

## Appendix H. The filter transfer functions for smoothed finite difference estimates of derivatives

The quantities of interest in this study of the resolution capabilities of SWOT and WaCM include variables that require spatial differentiation of the variables measured by the two instruments. In the case of SWOT, the SSH measurements must be differentiated once to estimate velocity components geostrophically and a second time to estimate vorticity geostrophically. In the case of WaCM, the surface velocity measurements must be differentiated once to obtain surface vorticity. These derivatives must be approximated from the discretely sampled data using finite difference methods. The effects of finite differencing on the wavenumber spectral content of the estimated derivative fields are quantified in this appendix, both from the pre-processed data and with subsequent spatial smoothing in ground-based post-processing.

The standard procedure for estimating derivatives $z'(x)$ from grid-ded values of a 1-dimensional variable $z(x)$ is from centered differences of $z(x)$ at the adjacent grid locations,

$$z'_{3pt}(x) = \frac{1}{2\Delta x} \left[ z(x + \Delta x) - z(x - \Delta x) \right]. \qquad (H.1)$$

This finite difference spans across three adjacent points, referred to as the stencil width of the derivative estimate. To distinguish this standard centered difference estimate of derivatives from the estimate considered below based on a stencil width of nine points, (H.1) will be referred to in this appendix as a 3-point centered difference. As this is a linear filter, it can be written in the form (A.1) of a convolution when applied to a continuous spatial series $z(x)$. For discrete sampling at an interval $\Delta x$, this convolution integral can be written as a discrete convolution but the equations in the analysis that follows become more

cumbersome without providing any improved insight into the filtering properties of centered difference estimates of derivatives.

The filter weighting function in the convolution integral (A.1) for the 3-point centered difference filter (H.1) can be determined by the impulse response method (Bracewell, 1978), which gives

$$w_{3pt}(x) = \frac{1}{2\Delta x} \left[ \delta(x + \Delta x) - \delta(x - \Delta x) \right], \qquad (H.2)$$

where $\delta(x)$ is the Dirac delta, which can be visualized as having a value of infinity at $x = 0$ and zero elsewhere with unit area. The corresponding filter transfer function obtained analytically as the Fourier transform (A.2) of the filter weighting function (H.2) is

$$W_{3pt}(k) = \frac{1}{2\Delta x} \int_{-\infty}^{\infty} \left[ \delta(x + \Delta x) - \delta(x - \Delta x) \right] e^{-i2\pi kx} dx.$$

Using the sifting property of the Dirac delta reduces this to

$$W_{3pt}(k) = \frac{1}{2\Delta x} \left[ e^{+i2\pi k \Delta x} - e^{-i2\pi k \Delta x} \right] = i \frac{\sin(2\pi k \Delta x)}{\Delta x}. \qquad (H.3a)$$

Substitution of Euler's formula $i = e^{i\pi/2}$ expresses the response function for 3-point centered differences in the standard form of $A e^{i\phi}$,

$$W_{3pt}(k) = \frac{\sin(2\pi k \Delta x)}{\Delta x} e^{i\pi/2}. \qquad (H.3b)$$

The 3-point centered differencing thus introduces a quadrature phase shift of $\phi = \pi/2$ at each wavenumber $k$ with an amplitude of $A = \sin(2\pi k \Delta x)/\Delta x$. This filter transfer function is more commonly referred to in the literature as the response function of the finite difference estimate of the derivative. The squared magnitude of the complex response function (H.3) is

$$|W_{3pt}(k)|^2 = W_{3pt}^*(k) W_{3pt}(k) = \frac{\sin^2(2\pi k \Delta x)}{\Delta x^2}, \qquad (H.4)$$

where the superscript asterisk denotes the complex conjugate.

For the pre-processing of SWOT and WaCM data with half-power filter cutoff wavelengths of 2 km and 10 km, respectively, the minimum gridding of the pre-processed data to avoid aliasing of the resolved variability (wavelengths longer than 2 km for SWOT and 10 km for WaCM) is 1 km for SWOT and 5 km for WaCM. We will therefore consider centered differences for these two choices of $\Delta x$.

The squared response function (H.4) is shown in log–log format in Fig. H.1 by the thick and thin solid lines for the cases of $\Delta x = 1$ km and 5 km, respectively. It is assumed in this figure that there is no spectral power at wavenumbers higher than the Nyquist wavenumbers for the sample intervals of 1 km and 5 km, which are $k_{max} = 0.5$ cpmk and 0.1 cpmk, respectively. The response functions have therefore both been truncated at $k_{max}$ in Fig. H.1, thus eliminating irrelevant aliases of the response function. The squared response functions for both 3-point centered differences considered in Fig. H.1 rise monotonically to a maximum of $\Delta x^{-2}$ at a wavenumber of $k = (4\Delta x)^{-1}$ and then drop monotonically to zero at the Nyquist wavenumber $(2\Delta x)^{-1}$.

Although generally thought of as a high-pass filter, it can be seen from Fig. H.1 that the response function for 3-point centered differences is actually a band-pass filter. [Note that the logarithmic abscissa in Fig. H.1 distorts the $\sin^2$ dependence on wavenumber $k$ in (H.4)]. The attenuation of variability at low wavenumbers is the 3-point centered difference representation of the squared filter transfer function $(2\pi k)^2$ of high-pass filtering from differentiation of a continuous variable $z(x)$, which is shown as the short dashed line in Fig. H.1. The attenuation of the variability at the highest wavenumbers for 3-point centered difference estimates of the derivative arises from the smoothing implied by the differencing between grid points separated by $2\Delta x$. An undesirable characteristic of the 3-point centered difference filter is thus that the variability at the highest wavenumbers is attenuated compared with the squared filter transfer function $(2\pi k)^2$ of the true derivative. This high-wavenumber attenuation can be reduced and the overall accuracy of the derivative estimate can be improved by increasing the number
<figure><img src="image_68.png" /><figcaption>Fig. H.2. Log–log plots of the filtering from the combined effects of finite differencing and smoothing with a Parzen smoother with half-power filter cutoff wavelengths of $\lambda_c = 20, 50$ and $80$ km (top to bottom). This combined filtering consists of the product of the squared filter transfer functions for each operation that are shown in Figs. C.1 and H.1. The thick solid and dashed lines are for Parzen smoothing of, respectively, 3-point and 9-point centered differences on a 1 km grid. The thin solid and dashed lines are for Parzen smoothing of, respectively, 3-point and 9-point centered differences on a 5 km grid assuming that the discretely sampled data on the 5 km grid have been filtered to attenuate variability at wavenumbers higher than the Nyquist wavenumber of 0.1 cpkm. The thin dashed line becomes indistinguishable from the thick solid and thick dashed lines for smoothing with half-power filter cutoff wavelengths larger than $\lambda_c = 40$ km. All four lines are barely distinguishable for $\lambda_c = 80$ km. The short dashed line corresponds to the $(2\pi k)^2$ response function for derivatives of a continuous variable without smoothing.</figcaption></figure>

of grid points that are used in the estimate, i.e., the stencil width of the derivative estimate (see, for example, Arbic et al., 2012).

The general centered difference estimate with a stencil width of M points can be written as

$$z'_{Mpt}(x) = \sum_{m=1}^{(M-1)/2} a_m \left[ z(x + m\Delta x) - z(x - m\Delta x) \right], \quad (H.5)$$

where the weights $a_m$ differ for each choice of M. For the centered differences (H.1) with a 3-point stencil, i.e., for $M = 3$, there is only one term in the sum (H.5) and the weight is $a_1 = (2\Delta x)^{-1}$. Stencil widths greater than $M = 9$ are seldom used in practice. The weights for estimates of the derivatives with a 9-point stencil (referred to in this appendix as 9-point centered differences) can be shown to be

$$a_1 = \frac{4}{5\Delta x} \qquad (H.6a)$$

$$a_2 = -\frac{1}{5\Delta x} \qquad (H.6b)$$

$$a_3 = \frac{4}{105\Delta x} \qquad (H.6c)$$

$$a_4 = -\frac{1}{280\Delta x} \qquad (H.6d)$$

Analogous to the relationship between (H.1) and (H.3a) for the 3-point centered difference, the filter transfer function (response function) obtained by the impulse response method for the general M-point centered difference estimate (H.5) of the derivative is

$$W_{Mpt}(k) = 2i \sum_{m=1}^{(M-1)/2} a_m \sin(2\pi f m \Delta x). \qquad (H.7)$$

For the case of $M = 3$, this reduces to (H.3a). The square of the response function (H.7) for 9-point centered difference estimates of the derivative is shown in Fig. H.1 for the cases of $\Delta x = 1$ km and 5 km (the thick and thin dashed lines, respectively). Compared with the response functions for the standard 3-point centered differences, the 9-point stencils retain significantly more high-wavenumber variability of the true derivative shown by the short dashed line.

The disadvantage of stencils wider than $M = 3$ is that they have larger “edge effects”. Because of missing terms in the summation (H.5), centered differences cannot be computed closer than $(M - 1)/2$ points from the ends of the data record. This problem could be addressed by ratcheting down to successively narrower stencils at the ends of the data record, but the price paid for such a procedure is that the filter transfer function changes for each of the successively reduced stencils.

The degree to which it is advantageous to estimate derivatives using a wider stencil in this study depends on the subsequent processing of the derivative fields. As noted previously, the derivative variables of interest here are surface velocity and vorticity computed geostrophically from pre-processed SWOT estimates of SSH (the latter computed from second derivatives of SSH) and the vorticity computed from pre-processed WaCM estimates of surface velocity. It is shown in Sections 6 and 8–10 that maps of these variables must be smoothed in ground-based post-processing to mitigate the effects of measurement errors.

The combined effects of finite differencing and smoothing on the wavenumber spectral content $S_{z'}(k)$ of a derivative variable $z'(x)$ can be expressed in terms of the product of the square of the response function (H.7) of the finite differencing and the squared filter transfer function of the smoother,

$$\bar{S}_{z'}(k) = |W_{Mpt}(k)|^2 |W_{k_c}(k)|^2 S_z(k), \qquad (H.8a)$$

where $S_z(k)$ is the wavenumber spectrum of the pre-processed satellite data (SSH for SWOT and surface velocity components for WaCM), $W_{k_c}(k)$ is the filter transfer function of the particular choice of smoother with a half-power filter cutoff wavenumber $k_c$, and $\bar{S}_{z'}(k)$ is the wavenumber spectrum of centered differences of smoothed values of the variable $z(x)$. For the Parzen smoother considered in detail in Appendices A–C, the filter transfer function $W_{k_c}(k)$ is given by (A.11) with a span of $L_4 = 0.910 k_c^{-1} \approx k_c^{-1}$ according to (A.12a). Since the Parzen smoother is symmetric and real, its transfer function is also symmetric and real. But from (H.3) and (H.7), the response function $W_{Mpt}(k)$ for centered differencing is complex. Then (H.8a) can be written as

$$\bar{S}_{z'}(k) = |W_{Mpt}(k)|^2 W_{k_c}^2(k) S_z(k). \qquad (H.8b)$$
Examples of the filter transfer function $|W_{Mpr}(k)|^2 W_{k'}^2(k)$ for combined M-point centered differencing and spatial smoothing are shown in Fig. H.2 for Parzen smoothing with half-power filter cutoff wavelengths of $\lambda_c = 20, 50$ and $80$ km (corresponding to wavenumbers of $k_c = \lambda_c^{-1} = 0.05, 0.02$ and $0.0125$ cpmk, see Fig. C.1) with 3-point and 9-point centered difference estimates of the derivatives for grid spacings of $\Delta x = 1$ km and $5$ km (see Fig. H.1). Several conclusions can be drawn from Fig. H.2:

1. For the SWOT grid spacing of $\Delta x = 1$ km, the filter transfer functions for spatial smoothing combined with 3-point or 9-point centered differencing (the thick solid and thick dashed lines, respectively) are nearly indistinguishable for any of the choices of smoothing shown in Fig. H.2. Since the signal-to-noise ratios for SWOT estimates of any of the derivative quantities of interest in this study are inadequate for smoothing with filter cutoff wavelengths less than 20 km, it can be concluded that the 9-point centered differencing offers no advantage over 3-point centered differencing for SWOT data on a $1$ km $\times$ $1$ km grid.

2. For a grid spacing of $\Delta x = 5$ km relevant to WaCM, the filter transfer functions for spatial smoothing combined with 3-point or 9-point centered differencing (the thin solid and thin dashed lines, respectively) differ significantly for $\lambda_c = 20$ km. Centered differencing with a stencil width of 9 points on a 5 km grid thus retains significantly more of the high-wavenumber content of the derivatives and therefore gives improved estimates of the derivatives with this choice of smoothing on a 5 km grid with $\lambda_c = 20$ km. The 9-point stencil offers a very slight improvement for $\lambda_c = 50$ km and negligible improvement for $\lambda_c = 80$ km. It can be concluded that, for filter cutoff wavelengths $\lambda_c \lesssim 60$ km, it would be advantageous to use 9-point centered differencing if the pre-processed WaCM data are posted on a $5$ km $\times$ $5$ km grid.

3. Although gridding of pre-processed WaCM data on a $5$ km $\times$ $5$ km Nyquist-sampled grid is sufficient to avoid aliasing as discussed in Appendix B, the pre-processed data can be posted on a finer grid. While “oversampling” the WaCM data in this manner offers no advantage for mapping of the surface velocity components, it is evident from Fig. H.2 that it is advantageous for mapping of the velocity component derivative fields from which vorticity is computed. This is seen for the case of $\lambda_c = 20$ km in the top panel of Fig. H.2 from the fact that the combined effects of the filter transfer function of the smoother and the response functions for both 3-point and 9-point centered differences with $\Delta x = 1$ km (the thick solid and thick dashed lines) retain more of the high-wavenumber content of the derivatives than either 3-point or 9-point centered differences with $\Delta x = 5$ km (the thin solid and dashed lines). This implies that 3-point centered differencing of pre-processed WaCM data on an oversampled $1$ km $\times$ $1$ km grid (the thick solid line) gives better estimates of the derivatives than can be obtained even from 9-point centered differencing on the Nyquist-sampled $5$ km $\times$ $5$ km grid (the thin dashed line). For $\lambda_c \approx 40$ km (slightly shorter than the case of $\lambda_c = 50$ km shown in the middle panel of Fig. H.2), the filter transfer functions for 3-point centered differences with $\Delta x = 1$ km are still slightly better than for 3-point centered differences with $\Delta x = 5$ km but become nearly indistinguishable from 9-point centered differences with $\Delta x = 5$ km. For $\lambda_c \gtrsim 80$ km, all of the centered difference estimates of derivatives considered here give essentially equivalent results.

On the basis of the conclusions above deduced from Fig. H.2, it is preferable for the purposes of WaCM estimates of small-scale vorticity signal variability to grid the pre-processed WaCM data on the same $1$ km $\times$ $1$ km grid as SWOT rather than on the $5$ km $\times$ $5$ km grid that satisfies Nyquist sampling of pre-processed WaCM data. Moreover, for both WaCM and SWOT, a 3-point stencil width for centered differencing

on the $1$ km $\times$ $1$ km grid is adequate for the filter cutoff wavelengths of $\lambda_c \gtrsim 20$ km that are relevant to signal estimation in the assessments of the resolution capabilities of derivative fields in Sections 6 and 8–10.

The above conclusion that oversampling the WaCM data on a $1$ km $\times$ $1$ km grid is preferable to Nyquist sampling on a $5$ km $\times$ $5$ km grid was deduced from consideration of the effects of sampling and centered difference stencil width on preservation of the small-scale (high-wavenumber) signal contributions to the WaCM estimates of velocity component derivatives for calculation of vorticity. The same considerations apply to the error contributions to derivative estimates from WaCM data. Note, however, that the finer gridding accentuates the noise at high wavenumbers in finite-difference estimates of derivatives. For the case of white noise and red signal in the velocity measurements, it is preferable to attenuate the small-scale variability in the error field. The conflicting goals of signal preservation and noise suppression at high wavenumbers complicate the choice of the best gridding of the WaCM data for mapping of the vorticity field.

From simulations, we found that the noise suppression from $5$ km $\times$ $5$ km gridding is more beneficial than signal preservation in the signal-to-noise ratio of velocity component derivative fields smoothed with half-power filter cutoff wavelengths less than about 25 km. For larger smoothing scales, the advantage of signal preservation from $1$ km $\times$ $1$ km gridding is more beneficial. Since the analysis in Sections 6 and 8–10 concludes that wavelength scales smaller than about 25 km cannot be adequately resolved in vorticity maps constructed from WaCM measurements of surface velocity,<fnref n="11" /> oversampling on a $1$ km $\times$ $1$ km grid is preferable for the simulations presented in this study. A practical advantage of this oversampling is that the grid for simulated WaCM data then matches the $1$ km $\times$ $1$ km grid of the simulated SWOT data.

## Appendix I. The wavenumber spectral characteristics of the errors of SWOT and WaCM estimates of velocity and vorticity

Analytical expressions are derived in this appendix for the 1-dimensional wavenumber spectra of errors of estimates of each of the variables of interest in this study (SSH, two orthogonal velocity components and vorticity) based on the unsmoothed pre-processed estimates of SSH for SWOT and velocity for WaCM, as well as for the 1-dimensional wavenumber spectra of the variables of interest computed from SSH and velocity fields smoothed in ground-based post-processing. A noteworthy feature of the analysis is that the equations for each of the wavenumber spectra derived in this appendix are expressed in terms of the variance of the pre-processed data. The equations can therefore accommodate variances of the uncorrelated measurement errors (SSH for SWOT and surface velocity components for WaCM) that differ from the present baseline specifications.

In the case of SWOT, the relevant equations for the wavenumber spectra of unsmoothed fields are (I.2a), (I.5a), (I.10a) and (I.24a) for, respectively, the errors of SWOT estimates of SSH, the $x$ and $y$ components of geostrophically computed velocity and geostrophically computed vorticity. These equations are all expressed in terms of the variance $\sigma_h^2$ of the pre-processed SWOT estimates of SSH. The analogous equations for the wavenumber spectra of smoothed SWOT estimates of these variables are (I.29a) and (I.31a), (I.33a) and (I.35b). The value of $\sigma_h^2 = (2.74 \text{ cm})^2$ that is derived in Appendix F is used in the applications of these equations in Section 4.4, but the above-referenced equations can be used for any specified value of $\sigma_h^2$.

A limitation of the formulas derived below for the wavenumber spectra of errors of geostrophically computed SWOT estimates of velocity and vorticity is that they require the use of a constant value

<footnote n="11">¹¹ It can be seen from Figs. 44c and d that the resolution capability of vorticity maps constructed from WaCM data would exceed 25 km if the standard deviation of speed measurement errors could be reduced to about 0.1 m s⁻¹. At the present time, such measurement accuracy does not seem technologically feasible.</footnote>
of the Coriolis parameter *f*. The theoretical spectra for these errors that are shown in Figs. 13a, 13b and 14 are based on the value of *f* = 8.75 × 10⁻⁵ s⁻¹ at the central latitude 37°N of the CCS model domain. Over the latitude range of the CCS model (see the left panel of Fig. 2a), *f* varies from 7.81 × 10⁻⁵ s⁻¹ at the 32.5°N southern corner of the model domain to 9.73 × 10⁻⁵ s⁻¹ at the 42°N northern corner of the model domain. Compared with its value at 37°N, the Coriolis parameter is 11% larger at 42°N and 11% smaller at 32.5°N. This small variation of *f* and the fact that *f* varies approximately linearly over the latitudinal range of the CCS model domain result in negligibly small differences between the theoretical and empirical spectra that are based on the local value of *f* at each grid point (see Figs. 13a, 13b and 14).

In the case of WaCM, the relevant equations for the wavenumber spectra of the errors of unsmoothed velocity component and vorticity fields are, respectively, (I.40b) and (I.43), with the latter partitioned as (I.45b) and (I.46a). These equations are all expressed in terms of the variances σ_u² and σ_v² of the errors of WaCM estimates of the x and y components of velocity. The analogous equations for the wavenumber spectra of the errors of WaCM estimates of smoothed velocity component and vorticity fields are, respectively, (I.48a) and (I.52), with the latter partitioned as (I.54a) and (I.55a). A value of σ_u² = (0.354 m s⁻¹)² is used for both σ_u² and σ_v² in the application of these equations in Section 4.4, but the above-referenced equations can be used for any specified values of σ_u² and σ_v². This fact is used in Section 10.2 to determine how much the resolution capability of WaCM estimates of velocity and vorticity would improve if engineering efforts can reduce the measurement error variance σ_u².

## 1.1. Errors of SSH and geostrophically computed velocity and vorticity from unsmoothed SWOT data

The 2-dimensional wavenumber power spectral density of a spatially uncorrelated error field is constant at all wavenumbers *k* and *l* in the *x* and *y* dimensions, respectively, which correspond to the cross-shore and alongshore dimensions for the CCS model domain considered here but would correspond to cross-track and along-track in actual SWOT data. For the case of uncorrelated instrumental errors with variance σ_h² for SWOT estimates of SSH, this constant (white) power spectral density is related to σ_h² by Eq. (E.3) in Appendix E, which is repeated here for convenience,

$$S_h^{2d}(k, l) = \frac{\sigma_h^2}{4k_{N'}l_{N'}} = \Delta x \Delta y \sigma_h^2, \qquad (I.1)$$

where $k_{N'} = (2\Delta x)^{-1}$ and $l_{N'} = (2\Delta y)^{-1}$ are the Nyquist wavenumbers for the sample intervals of $\Delta x$ and $\Delta y$. The errors of pre-processed SWOT estimates of SSH obtained as summarized in Appendix B.1 by smoothing the raw SWOT data with a half-power filter cutoff wavelength of 2 km in each dimension are uncorrelated for a grid spacing of $\Delta x \times \Delta y = 1$ km $\times$ 1 km. The variance of the errors of these pre-processed SWOT estimates of SSH that is derived in Appendix F is $\sigma_h^2 = (2.74 \text{ cm})^2$.

The 1-sided, 1-dimensional alongshore power spectral density of the errors of the pre-processed SWOT estimates of SSH is given by (E.4), which can be expressed in terms of $\sigma_h^2$ as<fnref n="12" />

$$S_h^{1d}(l) = 2 \int_{-k_{N'}}^{k_{N'}} S_h^{2d}(k, l) dk = 4\Delta x \Delta y k_{N'} \sigma_h^2 = 2\Delta y \sigma_h^2. \qquad (I.2a)$$

For $\sigma_h^2 = (2.74 \text{ cm})^2$, the 1-dimensional alongshore spectrum (I.2a) of SSH measurement errors in the baseline design is

$$S_h^{1d}(l) = 15.0 \text{ cm}^2/\text{cpkm}. \qquad (I.2b)$$

The 2-dimensional wavenumber spectrum of the errors of cross-shore velocity $u_g$ computed geostrophically from centered differences of the errors of pre-processed SWOT estimates of SSH can be determined analytically from the 2-dimensional spectrum (I.1) of SWOT SSH measurement errors by

$$S_{u_g}^{2d}(k, l) = \frac{g^2}{f^2} |W_{3pt}(l)|^2 S_h^{2d}(k, l) = \frac{g^2}{f^2} \Delta x \Delta y \sigma_h^2 |W_{3pt}(l)|^2, \qquad (I.3)$$

where $W_{3pt}(l)$ is the response function for the alongshore derivative in the geostrophic relation $u_g = -g f^{-1} \partial h / \partial y$ estimated from alongshore centered differences with a 3-point stencil width. It is shown in Appendix H that $W_{3pt}(l)$ is given by an expression analogous to (H.3a), except with cross-shore grid spacing $\Delta x$ and wavenumber *k* replaced with their alongshore counterparts $\Delta y$ and *l*,

$$W_{3pt}(l) = i \frac{\sin(2\pi \Delta y l)}{\Delta y}. \qquad (I.4)$$

The 1-sided, 1-dimensional alongshore spectrum of $u_g$ errors obtained from the 2-dimensional spectrum (I.3) as in (I.2a) by integrating over all cross-shore wavenumbers *k* and doubling the power at each wavenumber *l* is

$$S_{u_g}^{1d}(l) = 2 \int_{-k_{N'}}^{k_{N'}} S_{u_g}^{2d}(k, l) dk = \frac{g^2}{f^2} 2\Delta y \sigma_h^2 |W_{3pt}(l)|^2 \qquad (I.5a)$$

$$= \frac{g^2}{f^2} |W_{3pt}(l)|^2 S_h^{1d}(l), \qquad (I.5b)$$

where $|W_{3pt}(l)|^2 = \Delta y^{-2} \sin^2(2\pi \Delta y l)$ is the squared magnitude of (I.4). The form (I.5b) for $S_h^{1d}(l)$ follows from (I.2a).

The wavenumber spectrum of the errors of alongshore velocity $v_g$ computed geostrophically from the errors of pre-processed SWOT estimates of SSH can be determined analytically following a procedure similar to that used above for the wavenumber spectrum of the errors of the cross-shore geostrophically computed velocity $u_g$. In this case, however, the cross-shore derivative in the geostrophic relation $v_g = g f^{-1} \partial h / \partial x$ is estimated by centered differences with a 3-point stencil in the cross-shore direction, rather than in the alongshore direction. The response function for these cross-shore centered differences is (H.3a), which is repeated here for convenience,

$$W_{3pt}(k) = i \frac{\sin(2\pi \Delta x k)}{\Delta x}. \qquad (I.6)$$

The 2-dimensional wavenumber spectrum of $v_g$ errors obtained from cross-shore centered differences of the pre-processed SSH errors is thus related to the 2-dimensional spectrum (I.1) of SWOT SSH measurement errors by

$$S_{v_g}^{2d}(k, l) = \frac{g^2}{f^2} |W_{3pt}(k)|^2 S_h^{2d}(k, l) = \frac{g^2}{f^2} \Delta x \Delta y \sigma_h^2 |W_{3pt}(k)|^2. \qquad (I.7)$$

The 1-sided, 1-dimensional alongshore spectrum of the errors of geostrophically computed alongshore velocity $v_g$ is

$$S_{v_g}^{1d}(l) = 2 \int_{-k_{N'}}^{k_{N'}} S_{v_g}^{2d}(k, l) dk = \frac{g^2}{f^2} 4\Delta x \Delta y \sigma_h^2 \int_0^{k_{N'}} |W_{3pt}(k)|^2 dk \qquad (I.8a)$$

$$= \frac{g^2}{f^2} 2\Delta x S_h^{1d}(l) \int_0^{k_{N'}} |W_{3pt}(k)|^2 dk. \qquad (I.8b)$$

The last expression follows from (I.2a) and both forms for $S_{v_g}^{1d}(l)$ use the fact that $|W_{3pt}(k)|^2$ is symmetric about $k = 0$. With the definition (I.6) of $W_{3pt}(k)$ and the transformation of variables $k' = 2\pi \Delta x k$, the integral on the right sides of (I.8a) and (I.8b) is

$$\int_0^{k_{N'}} |W_{3pt}(k)|^2 dk = \frac{1}{2\pi \Delta x^3} \int_0^{\pi} \sin^2(k') dk' = \frac{1}{4\Delta x^3}. \qquad (I.9)$$

Then (I.8a) and (I.8b) become

$$S_{v_g}^{1d}(l) = \frac{g^2}{f^2} \frac{\Delta y}{\Delta x^2} \sigma_h^2 \qquad (I.10a)$$

<footnote n="12">¹² As noted in footnotes 5 and 8, the spectral values at wavenumbers *l* = 0 and *l*_{N'} are not doubled in the 1-sided, 1-dimensional spectrum. In the interest of clarity in the analysis that follows, this technical detail will again be ignored as in Appendices D and E.</footnote>
$$ = \frac{g^2}{f^2} \frac{1}{2\Delta x^2} S_h^{1d}(l). \qquad (I.10b) $$

The expression (G.7b) in Appendix G for the errors of $\zeta_g$ in terms of finite differences of the uncorrelated errors of pre-processed SWOT estimates of SSH can be written as

$$ \epsilon_{\zeta_g}(x, y) = \epsilon_1(x, y) + \epsilon_2(x, y) + \epsilon_3(x, y) + \epsilon_4(x, y), \qquad (I.11) $$

where

$$ \epsilon_1(x, y) = \frac{g}{f} \frac{1}{2\Delta x} \frac{\epsilon_h(x + 2\Delta x, y) - \epsilon_h(x, y)}{2\Delta x} \qquad (I.12a) $$

$$ \epsilon_2(x, y) = -\frac{g}{f} \frac{1}{2\Delta x} \frac{\epsilon_h(x, y) - \epsilon_h(x - 2\Delta x, y)}{2\Delta x} \qquad (I.12b) $$

$$ \epsilon_3(x, y) = \frac{g}{f} \frac{1}{2\Delta y} \frac{\epsilon_h(x, y + 2\Delta y) - \epsilon_h(x, y)}{2\Delta y} \qquad (I.12c) $$

$$ \epsilon_4(x, y) = -\frac{g}{f} \frac{1}{2\Delta y} \frac{\epsilon_h(x, y) - \epsilon_h(x, y - 2\Delta y)}{2\Delta y}. \qquad (I.12d) $$

The contributions $\epsilon_1(x, y)$ and $\epsilon_2(x, y)$ correspond to centered cross-shore differences of SSH errors with center points at $(x + \Delta x, y)$ and $(x - \Delta x, y)$, respectively. Similarly, the contributions $\epsilon_3(x, y)$ and $\epsilon_4(x, y)$ correspond to centered alongshore differences of SSH errors with center points at $(x, y + \Delta y)$ and $(x, y - \Delta y)$, respectively.

It was shown in Appendix H that 1-dimensional centered differences can be written as convolution integrals. For the 2-dimensional fields considered here, centered cross-shore differences at $(x, y)$ can be written as the convolution

$$ \frac{\epsilon_h(x + \Delta x, y) - \epsilon_h(x - \Delta x, y)}{2\Delta x} = w_{3pt}(x) * \epsilon_h(x, y), \qquad (I.13a) $$

where, analogous to (H.2), the filter weighting function for a 2-dimensional field of SSH errors is

$$ w_{3pt}(x) = \frac{1}{2\Delta x} \left[ \delta(x + \Delta x, y) - \delta(x - \Delta x, y) \right]. \qquad (I.13b) $$

The similar expression for centered alongshore differences at $(x, y)$ is

$$ \frac{\epsilon_h(x, y + \Delta y) - \epsilon_h(x, y - \Delta y)}{2\Delta y} = w_{3pt}(y) * \epsilon_h(x, y), \qquad (I.14a) $$

where

$$ w_{3pt}(y) = \frac{1}{2\Delta y} \left[ \delta(x, y + \Delta y) - \delta(x, y - \Delta y) \right]. \qquad (I.14b) $$

The response functions (filter transfer functions) obtained as in (H.3a) as the Fourier transforms of (I.13b) and (I.14b) are given by (I.4) and (I.6), respectively,

The four contributions (I.12a)–(I.12d) to $\epsilon_{\zeta_g}(x, y)$ can be written as shifted convolution integrals similar to (I.13a) and (I.14a),

$$ \epsilon_1(x, y) = \frac{g}{f} \frac{1}{2\Delta x} w_{3pt}(x - \Delta x) * \epsilon_h(x, y) \qquad (I.15a) $$

$$ \epsilon_2(x, y) = -\frac{g}{f} \frac{1}{2\Delta x} w_{3pt}(x + \Delta x) * \epsilon_h(x, y) \qquad (I.15b) $$

$$ \epsilon_3(x, y) = \frac{g}{f} \frac{1}{2\Delta y} w_{3pt}(y - \Delta y) * \epsilon_h(x, y) \qquad (I.15c) $$

$$ \epsilon_4(x, y) = -\frac{g}{f} \frac{1}{2\Delta y} w_{3pt}(y + \Delta y) * \epsilon_h(x, y). \qquad (I.15d) $$

The weighting functions on the right sides of (I.15a) and (I.15b) are cross-shore-shifted versions of the weighting function (I.13b) of centered cross-shore differences at $(x, y)$. Likewise, the weighting functions on the right sides of (I.15c) and (I.15d) are alongshore-shifted versions of the weighting function (I.14b) of centered alongshore differences at $(x, y)$. By the Shift Theorem, the Fourier transforms of these shifted weighting functions are

$$ \mathcal{F}[w_{3pt}(x \pm \Delta x)] = e^{\pm 2\pi \Delta x k} W_{3pt}(k) \qquad (I.16) $$

$$ \mathcal{F}[w_{3pt}(y \pm \Delta y)] = e^{\pm 2\pi \Delta y l} W_{3pt}(l), \qquad (I.17) $$

where $\mathcal{F}$ indicates the Fourier transform and $W_{3pt}(k)$ and $W_{3pt}(l)$ are given by (I.6) and (I.4). The shifted weighting functions in the convolution integrals (I.15a)–(I.15d) thus result in phase rotations of the Fourier transforms $W_{3pt}(k)$ and $W_{3pt}(l)$.

The results above can be used to determine the Fourier transform of the vorticity errors (I.11). This Fourier transform can be written as the sum of the Fourier transforms of each of the four terms on the right side of (I.11),

$$ \mathcal{F}[\zeta_g(k, l)] = \mathcal{F}_1(k, l) + \mathcal{F}_2(k, l) + \mathcal{F}_3(k, l) + \mathcal{F}_4(k, l). \qquad (I.18) $$

By the Convolution Theorem and the response functions (I.16) and (I.17) of the shifted centered differences, the four Fourier transforms on the right side of (I.18) are

$$ \mathcal{F}_1(k, l) = \frac{g}{f} \frac{1}{2\Delta x} e^{i2\pi\Delta x k} W_{3pt}(k) \mathcal{F}[h(k, l)] \qquad (I.19a) $$

$$ \mathcal{F}_2(k, l) = -\frac{g}{f} \frac{1}{2\Delta x} e^{-i2\pi\Delta x k} W_{3pt}(k) \mathcal{F}[h(k, l)] \qquad (I.19b) $$

$$ \mathcal{F}_3(k, l) = \frac{g}{f} \frac{1}{2\Delta y} e^{i2\pi\Delta y l} W_{3pt}(l) \mathcal{F}[h(k, l)] \qquad (I.19c) $$

$$ \mathcal{F}_4(k, l) = -\frac{g}{f} \frac{1}{2\Delta y} e^{-i2\pi\Delta y l} W_{3pt}(l) \mathcal{F}[h(k, l)], \qquad (I.19d) $$

where $\mathcal{F}[h(k, l)]$ is the Fourier transform of the SSH errors. Then (I.18) becomes

$$ \begin{aligned} \mathcal{F}[\zeta_g(k, l)] &= \frac{g}{f} \mathcal{F}[h(k, l)] \left[ \left( \frac{e^{i2\pi\Delta x k} - e^{-i2\pi\Delta x k}}{2\Delta x} \right) W_{3pt}(k) \right. \\ & \qquad \left. + \left( \frac{e^{i2\pi\Delta y l} - e^{-i2\pi\Delta y l}}{2\Delta y} \right) W_{3pt}(l) \right] \\ &= i \frac{g}{f} \mathcal{F}[h(k, l)] \left[ \frac{\sin(2\pi\Delta x k)}{\Delta x} W_{3pt}(k) + \frac{\sin(2\pi\Delta y l)}{\Delta y} W_{3pt}(l) \right] \\ &= -\frac{g}{f} \mathcal{F}[h(k, l)] \left[ |W_{3pt}(k)|^2 + |W_{3pt}(l)|^2 \right]. \end{aligned} \qquad (I.20) $$

The last relation follows from (I.4) and (I.6).

The 2-dimensional power spectral density of the errors of SWOT estimates of vorticity obtained as the squared magnitude of the Fourier transform (I.20) is related to the power spectral density (I.1) of the SSH errors by

$$ \begin{aligned} S_{\zeta_g}^{2d}(k, l) &= \frac{g^2}{f^2} S_h^{2d}(k, l) \left[ |W_{3pt}(k)|^4 \right. \\ & \qquad \left. + 2 |W_{3pt}(k)|^2 |W_{3pt}(l)|^2 + |W_{3pt}(l)|^4 \right] \\ &= \frac{g^2}{f^2} \Delta x \Delta y \sigma_h^2 \left[ |W_{3pt}(k)|^4 \right. \\ & \qquad \left. + 2 |W_{3pt}(k)|^2 |W_{3pt}(l)|^2 + |W_{3pt}(l)|^4 \right]. \end{aligned} \qquad (I.21a) \\ (I.21b) $$

The 1-sided, 1-dimensional alongshore spectrum of the errors of SWOT estimates of $\zeta_g$ is

$$ \begin{aligned} S_{\zeta_g}^{1d}(l) &= 2 \int_{-k_N}^{k_N} S_{\zeta_g}^{2d}(k, l) dk \\ &= \frac{g^2}{f^2} 2\Delta x \Delta y \sigma_h^2 \left( 2 \int_0^{k_N} |W_{3pt}(k)|^4 dk \right. \\ & \qquad \left. + 4 |W_{3pt}(l)|^2 \int_0^{k_N} |W_{3pt}(k)|^2 dk + \frac{1}{\Delta x} |W_{3pt}(l)|^4 \right), \end{aligned} \qquad (I.22) $$

where $k_N = (2\Delta x)^{-1}$ is again the Nyquist wavenumber in the $x$ dimension. The expression (I.22) uses the fact that $|W_{3pt}(k)|^2$ and $|W_{3pt}(l)|^4$ are both symmetric about $k = 0$. The second term on the right side of (I.22) is the cross spectral term that arises from the fact that all four contributions (I.12) to the vorticity errors (I.11) are cross correlated since they share in common the SSH error $\epsilon_h(x, y)$.
The second integral on the right side of (1.22) is given by (I.9). With the same transformation of variables $k' = 2\pi\Delta x k$ that was used in (I.9), the first integral is

$$ \int_{0}^{k_N} |W_{3pt}(k)|^4 dk = \frac{1}{2\pi\Delta x^5} \int_{0}^{\pi} \sin^4(k') dk' = \frac{3}{16\Delta x^5}. \quad (I.23) $$

Then (1.22) becomes

$$ S_{sg}^{1d}(l) = \frac{g^2}{f^2} 2\Delta y \sigma_h^2 \left( \frac{3}{8\Delta x^4} + \frac{1}{\Delta x^2} |W_{3pt}(l)|^2 + |W_{3pt}(l)|^4 \right) \quad (I.24a) $$

$$ = \frac{g^2}{f^2} S_h^{1d}(l) \left( \frac{3}{8\Delta x^4} + \frac{1}{\Delta x^2} |W_{3pt}(l)|^2 + |W_{3pt}(l)|^4 \right), \quad (I.24b) $$

where $S_h^{1d}(l)$ is the 1-sided, 1-dimensional alongshore spectrum (I.2a) of the SSH errors.

The analytical expressions (I.2a), (I.5a), (I.10a) and (I.24a) for the 1-sided, 1-dimensional alongshore spectra of errors of the pre-processed SWOT estimates of SSH and the two velocity components and the vorticity computed geostrophically from the pre-processed SWOT data are shown for $\sigma_h^2 = (2.74 \text{ cm})^2$ by the solid green lines in Fig. 13a.

## 1.2. Errors of SSH and geostrophically computed velocity and vorticity from smoothed SWOT data

The errors of SWOT estimates of velocity and vorticity computed geostrophically from the pre-processed SWOT estimates of SSH (see Appendix G and Table 1 in Section 4 and the maps in the bottom left panels of Figs. 17b and 17c) are too large for the velocity and vorticity estimates to be scientifically useful. It will be necessary to estimate velocity and vorticity geostrophically from SWOT SSH fields that have been smoothed in ground-based post-processing.

Throughout this study, we use isotropic filtering with $k_c = l_c = \lambda_c^{-1}$, where $k_c$ and $l_c$ are the half-power filter cutoff wavenumbers in the x and y dimensions, respectively, and $\lambda_c$ is the desired isotropic half-power filter cutoff wavelength in km. It is shown in Appendix C that separate 1-dimensional filtering in each dimension using the Parzen smoother is essentially equivalent to isotropic smoothing with a 2-dimensional Parzen smoother that depends only on radial distance from the estimation location. This equivalence facilitates the derivation of the theoretical wavenumber spectra of errors in this appendix. For a desired filter cutoff wavelength of $\lambda_c$, the span of the Parzen smoother in each dimension as given by (A.12b) in Appendix A is $L_4 = 0.910 \lambda_c \approx \lambda_c$. The filter transfer function of the Parzen smoother in the x dimension, for example, is given by (A.11), which can be written in terms of the half-power filter cutoff wavelength $\lambda_c$ as

$$ W_{\lambda_c}(k) = \sin^4 \left( \frac{kL_4}{4} \right) \approx \sin^4 \left( \frac{k\lambda_c}{4} \right) = \left( \frac{\sin(\pi k \lambda_c / 4)}{\pi k \lambda_c / 4} \right)^4. \quad (I.25) $$

The 2-dimensional wavenumber spectrum of 2-dimensionally smoothed SSH measurement errors is obtained from the 2-dimensional wavenumber spectrum (I.1) of uncorrelated errors of the pre-processed SWOT estimates of SSH by

$$ \bar{S}_h^{2d}(k, l) = W_{\lambda_c}^2(k) W_{\lambda_c}^2(l) S_h^{2d}(k, l) = \Delta x \Delta y \sigma_h^2 W_{\lambda_c}^2(k) W_{\lambda_c}^2(l), \quad (I.26) $$

where $\sigma_h^2$ is the variance of the SSH measurement errors. The overbar distinguishes the 2-dimensional spectrum $\bar{S}_h^{2d}(k, l)$ of smoothed SSH measurement errors from the 2-dimensional spectrum (I.1) of unsmoothed errors. The 1-sided, 1-dimensional alongshore spectrum of the smoothed SSH measurement errors is

$$ \bar{S}_h^{1d}(l) = 2 \int_{-k_N}^{k_N} \bar{S}_h^{2d}(k, l) dk = 4\Delta x \Delta y \sigma_h^2 W_{\lambda_c}^2(l) \int_0^{k_N} W_{\lambda_c}^2(k) dk. \quad (I.27) $$

The right side of this equation uses the fact that $W_{\lambda_c}^2(k)$ is symmetric about $k = 0$.

For the Parzen smoother used in this study, it is shown in Appendix D that the integral of the associated squared filter transfer

function on the right side of (I.27) is given by (D.12), which can be written as

$$ \int_{0}^{k_N} W_{\lambda_c}^2(k) dk = \frac{1}{L_4} = \frac{1}{0.910 \lambda_c} \approx \lambda_c^{-1}. \quad (I.28) $$

The 1-dimensional wavenumber spectrum (I.27) of smoothed SSH measurement errors is therefore given approximately by

$$ \bar{S}_h^{1d}(l) = \frac{4\Delta x \Delta y \sigma_h^2}{\lambda_c} W_{\lambda_c}^2(l) \quad (I.29a) $$

$$ = \frac{2\Delta x}{\lambda_c} W_{\lambda_c}^2(l) S_h^{1d}(l), \quad (I.29b) $$

where $S_h^{1d}(l)$ is the 1-sided, 1-dimensional alongshore spectrum (I.2a) of the SSH errors. As discussed in Appendix E, the multiplication factor $W_{\lambda_c}^2(l)$ represents the low-pass filtering in the alongshore dimension that attenuates the spectral power $S_h^{1d}(l)$ at high wavenumbers and the multiplicative factor $2\Delta x \lambda_c^{-1}$ represents the additional attenuation of the spectral power $S_h^{1d}(l)$ at all wavenumbers from the low-pass filtering in the cross-shore dimension.

The 2-dimensional wavenumber spectrum of the errors of the geostrophically computed cross-shore velocity $u_g$ estimated from centered alongshore differences of the smoothed SSH errors is obtained analogously to (I.3), except from the 2-dimensional spectrum (I.26) of smoothed SSH,

$$ \bar{S}_{u_g}^{2d}(k, l) = \frac{g^2}{f^2} |W_{3pt}(l)|^2 \bar{S}_h^{2d}(k, l) \\ = \frac{g^2}{f^2} \Delta x \Delta y \sigma_h^2 W_{\lambda_c}^2(k) W_{\lambda_c}^2(l) |W_{3pt}(l)|^2. \quad (I.30) $$

The 1-sided, 1-dimensional alongshore spectrum of the errors of smoothed cross-shore geostrophically computed velocity $u_g$ is

$$ \bar{S}_{u_g}^{1d}(l) = 2 \int_{-k_N}^{k_N} \bar{S}_{u_g}^{2d}(k, l) dk \\ = \frac{g^2}{f^2} 4\Delta x \Delta y \sigma_h^2 W_{\lambda_c}^2(l) |W_{3pt}(l)|^2 \int_0^{k_N} W_{\lambda_c}^2(k) dk. $$

Substitution of (I.28) for the integral on the right side of this equation gives

$$ \bar{S}_{u_g}^{1d}(l) = \frac{g^2}{f^2} \frac{4\Delta x \Delta y \sigma_h^2}{\lambda_c} W_{\lambda_c}^2(l) |W_{3pt}(l)|^2. \quad (I.31a) $$

From (I.29a), this can be written alternatively in terms of the 1-dimensional alongshore spectrum $\bar{S}_h^{1d}(l)$ of smoothed SSH measurement errors,

$$ \bar{S}_{u_g}^{1d}(l) = \frac{g^2}{f^2} \bar{S}_h^{1d}(l) |W_{3pt}(l)|^2, \quad (I.31b) $$

where $\bar{S}_h^{1d}(l)$ can be expressed in terms of the 1-dimensional alongshore spectrum $S_h^{1d}(l)$ of the unsmoothed SSH measurement errors by (I.29b).

The 2-dimensional wavenumber spectrum of the errors of geostrophically computed alongshore velocity $v_g$ estimated from centered cross-shore differences of the smoothed SSH errors is obtained analogously to (I.7), except from the 2-dimensional spectrum (I.26) of smoothed SSH,

$$ \bar{S}_{v_g}^{2d}(k, l) = \frac{g^2}{f^2} |W_{3pt}(k)|^2 \bar{S}_h^{2d}(k, l) \\ = \frac{g^2}{f^2} \Delta x \Delta y \sigma_h^2 W_{\lambda_c}^2(k) W_{\lambda_c}^2(l) |W_{3pt}(k)|^2. \quad (I.32) $$

The 1-sided, 1-dimensional alongshore spectrum of the errors of smoothed alongshore geostrophically computed velocity $v_g$ is

$$ \bar{S}_{v_g}^{1d}(l) = 2 \int_{-k_N}^{k_N} \bar{S}_{v_g}^{2d}(k, l) dk \\ = \frac{g^2}{f^2} 4\Delta x \Delta y \sigma_h^2 W_{\lambda_c}^2(l) \int_0^{k_N} W_{\lambda_c}^2(k) |W_{3pt}(k)|^2 dk. \quad (I.33a) $$
From (I.29a), this can be written alternatively in terms of the 1-dimensional alongshore spectrum $\bar{S}_h^{1d}(l)$ of smoothed SSH measurement errors,

$$ \bar{S}_{v_g}^{1d}(l) = \frac{g^2 \lambda_c}{f^2} \bar{S}_h^{1d}(l) \int_0^{k_N} W_{\lambda_c}^2(k) |W_{3pt}(k)|^2 dk, \quad (I.33b) $$

where $\bar{S}_h^{1d}(l)$ can again be expressed in terms of the 1-dimensional alongshore spectrum $\bar{S}_h^{1d}(l)$ of the unsmoothed SSH measurement errors by (I.29b). With $W_{3pt}(k)$ and $W_{\lambda_c}(k)$ defined by (I.6) and (I.25), there does not appear to be an analytical solution for the integral on the right sides of (I.33a) and (I.33b). The integral must therefore be evaluated numerically. The result is shown as a function of the half-power filter cutoff wavelength $\lambda_c$ in Fig. 1.1a.

The 2-dimensional wavenumber spectrum of the errors of geostrophically computed vorticity estimated from finite differences of the smoothed SSH errors is obtained analogously to (I.21a), except from the 2-dimensional spectrum (I.26) of smoothed SSH,

$$ \bar{S}_{\xi_g}^{2d}(k, l) = \frac{g^2}{f^2} \bar{S}_h^{2d}(k, l) \left[ |W_{3pt}(k)|^4 + 2 |W_{3pt}(k)|^2 |W_{3pt}(l)|^2 + |W_{3pt}(l)|^4 \right] \quad (I.34a) $$

$$ = \frac{g^2}{f^2} \Delta x \Delta y \sigma_h^2 W_{\lambda_c}^2(k) W_{\lambda_c}^2(l) \left[ |W_{3pt}(k)|^4 + 2 |W_{3pt}(k)|^2 |W_{3pt}(l)|^2 + |W_{3pt}(l)|^4 \right]. \quad (I.34b) $$

The 1-sided, 1-dimensional alongshore spectrum of the errors of smoothed geostrophically computed vorticity errors is

$$ \bar{S}_{\xi_g}^{1d}(l) = 2 \int_{-k_N}^{k_N} \bar{S}_{\xi_g}^{2d}(k, l) dk \\ = \frac{g^2}{f^2} 2 \Delta x \Delta y \sigma_h^2 W_{\lambda_c}^2(l) \left[ 2 \int_0^{k_N} W_{\lambda_c}^2(k) |W_{3pt}(k)|^4 dk + 4 |W_{3pt}(l)|^2 \int_0^{k_N} W_{\lambda_c}^2(k) |W_{3pt}(k)|^2 dk + 2 |W_{3pt}(l)|^4 \int_0^{k_N} W_{\lambda_c}^2(k) dk \right]. \quad (I.35a) $$

The third integral on the right side of (I.35a) is approximately $\lambda_c^{-1}$ according to (I.28). Then (I.35a) can be expressed as

$$ \bar{S}_{\xi_g}^{1d}(l) = \frac{g^2}{f^2} \frac{4 \Delta x \Delta y \sigma_h^2}{\lambda_c} W_{\lambda_c}^2(l) \left[ \lambda_c \int_0^{k_N} W_{\lambda_c}^2(k) |W_{3pt}(k)|^4 dk + 2 \lambda_c |W_{3pt}(l)|^2 \int_0^{k_N} W_{\lambda_c}^2(k) |W_{3pt}(k)|^2 dk + |W_{3pt}(l)|^4 \right]. \quad (I.35b) $$

From (I.29a), this can be written alternatively in terms of the 1-dimensional alongshore spectrum $\bar{S}_h^{1d}(l)$ of smoothed SSH measurement errors,

$$ \bar{S}_{\xi_g}^{1d}(l) = \frac{g^2}{f^2} \bar{S}_h^{1d}(l) \left[ \lambda_c \int_0^{k_N} W_{\lambda_c}^2(k) |W_{3pt}(k)|^4 dk + 2 \lambda_c |W_{3pt}(l)|^2 \int_0^{k_N} W_{\lambda_c}^2(k) |W_{3pt}(k)|^2 dk + |W_{3pt}(l)|^4 \right], \quad (I.35c) $$

where $\bar{S}_h^{1d}(l)$ can again be expressed in terms of the 1-dimensional alongshore spectrum $\bar{S}_h^{1d}(l)$ of the unsmoothed SSH measurement errors by (I.29b). The second integral on the right sides of (I.33a) and (I.33b) is the same integral that appears on the right sides of (I.33a) and (I.33b) for which the solution obtained numerically is shown as a function of the half-power filter cutoff wavelength $\lambda_c$ in Fig. 1.1a. The numerical solution for the first integral on the right sides of (I.35b) and (I.35c) is shown as a function of $\lambda_c$ in Fig. 1.1b. It is apparent from Fig. 1.1a and 11b that the value of the first integral decreases with increasing $\lambda_c$ much more quickly than the value of the second integral.

The analytical expressions (I.29a), (I.31a), (I.33a) and (I.35b) for the 1-sided, 1-dimensional alongshore spectra of the residual errors of SWOT estimates of SSH and geostrophically computed velocity components and vorticity after smoothing in ground-based post-processing of the pre-processed SWOT data are applicable to any low-pass filter with a half-power filter cutoff wavelength of $\lambda_c$. The results based on the Parzen smoother are shown for $\sigma_h^2 = (2.74 \text{ cm})^2$ by the solid green lines in Fig. 13b for a half-power filter cutoff wavelength of $\lambda_c = 10 \text{ km}$ and in Fig. 14 for half-power filter cutoff wavelengths of $\lambda_c = 20, 50$ and $80 \text{ km}$. The theoretical spectra in these figures agree very well with the spectra shown in Figs. 13b and 14 by the solid blue lines that were computed empirically from the simulated SWOT error fields generated for the analysis in Sections 6, 8 and 9.

### I.3. Errors of pre-processed WaCM estimates of velocity and vorticity

The baseline design for the errors of the speeds of WaCM estimates of surface ocean velocity is for isotropic smoothing of the raw WaCM data in pre-processing with a half-power filter cutoff wavelength of 10 km to achieve a standard deviation of

$$ \sigma_{\text{spd}} = 0.50 \text{ m s}^{-1}. \quad (I.36) $$

The measurement errors with 10-km smoothing are essentially uncorrelated for a grid spacing of $\Delta x \times \Delta y = 5 \text{ km} \times 5 \text{ km}$ (see Appendix B.2).

If the 10-km smoothed WaCM data were subsampled on the 5 km × 5 km grid on which the errors are uncorrelated, the derivation of analytical expressions for the 1-dimensional alongshore spectra of the errors would be a straightforward modification of the derivation of the analytical expression (I.2a) for the 1-dimensional alongshore spectrum of the errors of pre-processed SWOT estimates of SSH. For WaCM estimates of vorticity $\zeta = \partial v/\partial x - \partial u/\partial y$, however, it is advantageous to grid the WaCM data on a finer grid with dimensions of $\Delta x \times \Delta y = 1 \text{ km} \times 1 \text{ km}$. The primary advantage is that centered difference estimates of the derivatives on the finer grid retain more of the short-wavelength variability in the vorticity signal (see Appendix H). For the comparisons with geostrophically computed velocity and vorticity estimated from simulated SWOT data, it is also convenient for the analysis in this study to grid the WaCM data onto the same 1 km × 1 km grid as the SWOT data.

With the 10-km smoothing assumed in the baseline design for pre-processing of WaCM data, the standard deviation of the errors of overlapping smoothed estimates of the speeds of WaCM measurements of velocity on the finer 1 km × 1 km grid is still (I.36). But the errors are no longer uncorrelated on the finer grid. As summarized below, this complicates the derivation of analytical expressions for the spectra of errors of the velocity components and vorticity estimated from pre-processed WaCM data.

The footprint size of the raw WaCM radar data will likely be of order 100 m. For the simulated WaCM data in this study, the smallest possible footprint size is the 0.5 km × 0.5 km grid spacing of the CCS model. This is a factor of 10 finer than the 5 km × 5 km grid on which the pre-processed data are uncorrelated with 10-km smoothing. To achieve the baseline design of (I.36) for the standard deviation of the speeds of 10-km smoothed errors of surface velocity measurements, the analysis in Appendix D shows that the variance of unsmoothed errors in simulated raw measurements with a footprint size of 0.5 km × 0.5 km is a factor of 100 times larger, hence the standard deviation is a factor-of-10 larger. This can be seen from the error variance reduction formula (D.5a) with the reduction factor $\alpha$ given by (D.14c) for the case of smoothing of the simulated raw measurements on a grid of $\delta x \times \delta y = 0.5 \text{ km} \times 0.5 \text{ km}$ using a Parzen smoother with an isotropic half-power filter cutoff wavelength of $\lambda_c = 10 \text{ km}$. The standard deviation of the speeds of the unsmoothed velocity errors on the simulated raw measurement grid is thus

$$ \sigma'_{\text{spd}} = 10 \sigma_{\text{spd}} = 5.0 \text{ m s}^{-1}, \quad (I.37) $$
<figure><img src="image_69.png" /><figcaption>Fig. 1.1. Numerical solutions as functions of half-power filter cutoff wavelength $λ_c$ for the integrals labeled in the lower left corner of each panel that appear in the equations in Appendix I: (a) The integral on the right sides of Eqs. (I.33a) and (I.33b), which also appears as the second integral on the right sides of Eqs. (I.35a), (I.35b), (I.35c) and (I.56) and as the integral on the right sides of Eqs. (I.46a) and (I.46b) with $λ_c = 10$ km; (b) The first integral on the right sides of Eqs. (I.35a), (I.35b) and (I.35c); (c) The integral on the right sides of Eqs. (I.48a) and (I.48b), which also appears as the integral on the right side of Eq. (I.54a) and in the denominator on the right side of Eq. (I.55b); and (d) The integral on the right side of Eq. (I.55a) and in the numerator on the right side of Eq. (I.55b). The dashed lines in panels (c) and (d) correspond to the approximate solutions obtained by replacing the term $W_{10\,km}^2(k)$ in the integrands with a value of 1 for all wavenumbers $k$, as in Eqs. (I.49) and (I.56), respectively.</figcaption></figure>

where the prime distinguishes this standard deviation of the errors of the simulated raw measurements from the standard deviation (I.36) of the speed errors after smoothing isotropically with a 10-km filter cutoff.

In general, the standard deviations $\sigma_u$ and $\sigma_v$ of the errors of the 10-km smoothed velocity estimates differ for the two orthogonal velocity components $u$ and $v$. Throughout Sections 6 and 8–10, however, we have assumed that the speed uncertainties in the pre-processed WaCM data are equally partitioned between $u$ and $v$. In this case, the standard deviations $\sigma_u$ and $\sigma_v$ are both equal to a value $\sigma_{u,v}$ that is related to the baseline design (I.36) by

$$\sigma_{u,v} = \frac{\sigma_{\text{spd}}}{\sqrt{2}} = 0.354 \, \text{m s}^{-1}.$$

The subscript $u, v$ is intended to indicate that the error standard deviation is the same for each velocity component. If the speed uncertainties are also equally partitioned between $u$ and $v$ on the simulated raw measurement grid, the standard deviations $\sigma'_u$ and $\sigma'_v$ of the uncorrelated errors of each velocity component in the simulated unsmoothed raw data are both equal to a value $\sigma'_{u,v}$ that is related to (I.37) by

$$\sigma'_{u,v} = \frac{\sigma'_{\text{spd}}}{\sqrt{2}} = 3.54 \, \text{m s}^{-1}.$$

In the derivations that follow, we allow for the possibility of different error standard deviations for each velocity component. Parseval's

Theorem for the 2-dimensional white noise wavenumber spectrum of the uncorrelated errors of the $u$ or $v$ velocity component on the $\delta x \times \delta y = 0.5 \, \text{km} \times 0.5 \, \text{km}$ simulated raw measurement grid can then be expressed as

$$S'_{u,v}^{2d}(k, l) = \frac{\sigma'_{u,v}^2}{4k'_{N}l'_{N}} = \delta x \delta y \sigma'_{u,v}^2 \quad (I.38)$$

where $k'_{N} = (2\delta x)^{-1}$ and $l'_{N} = (2\delta y)^{-1}$ are the Nyquist wavenumbers in the $x$ and $y$ dimensions. The subscript $u, v$ with the semicolon signifies the choice of a subscript of either $u$ or $v$ on both sides of (I.38), depending on which velocity component is of interest. The spectrum thus has the same form for each velocity component. The only difference is that $\sigma'_{u,v}^2$ on the right side of (I.38) is replaced with $\sigma'_u^2$ or $\sigma'_v^2$, depending on whether the interest is in the spectrum $S'_{u,v}^{2d}(k, l)$ of $u$ or the spectrum $S'_{v,u}^{2d}(k, l)$ of $v$ on the left side of (I.38).

The form (I.38) for the spectrum of velocity component errors from WaCM data is fundamentally different from the spectra (I.5) and (I.8) of the errors of geostrophically computed velocity components estimated from SWOT data. This is because of the different filtering imposed in the case of SWOT by the filter transfer functions (response functions) (I.4) and (I.6) for centered differences in orthogonal directions as summarized in Appendix I.1.

The 2-dimensional wavenumber spectrum of $u$ or $v$ after smoothing the simulated raw WaCM data with half-power filter cutoff wavelengths
of 10 km in each dimension (which is shown in Appendix C to be essentially equivalent to isotropic smoothing for the case of the Parzen smoother considered here) is

$$ \begin{aligned} \tilde{S}_{u;v}^{2d}(k, l) &= W_{10 \text{ km}}^2(k) W_{10 \text{ km}}^2(l) S'_{u;v}^{2d}(k, l) \\ &= \delta x \delta y \sigma'_{u;v}^2 W_{10 \text{ km}}^2(k) W_{10 \text{ km}}^2(l), \end{aligned} \quad (I.39a) $$

where $W_{10 \text{ km}}^2(k)$ and $W_{10 \text{ km}}^2(l)$ are the squared values of the filter transfer functions of smoothers with half-power filter cutoff wavelengths of 10 km in the x and y dimensions, respectively. For the case of the Parzen smoother used in this study, the filter transfer function in the x dimension with a half-power filter cutoff wavelength of 10 km is given by (A.11) in Appendix A with a span of $L_4 = 9.10$ km. The filter transfer function for Parzen smoothing in the y dimension is the same, except with wavenumber $k$ replaced with $l$. The tilde on the left side of (I.39a) is intended as a reminder that isotropic 2-dimensional smoothing with a half-power filter cutoff wavelength of 10 km has been applied to unsmoothed errors of the simulated raw measurements on the 0.5 km × 0.5 km grid.

With the 10-km smoothing applied to the raw WaCM data in pre-processing, the smoothed velocity component estimates could be subsampled onto a 5 km × 5 km grid with minimal aliasing that arises from the imperfect gradual rolloff of the filter transfer function through the half-power filter cutoff wavenumber of 0.1 cpkm (which is equivalent to the Nyquist wavenumber for $\Delta x = \Delta y = 5$ km). For the reasons discussed above, we assume that the raw data after 10-km smoothing are subsampled onto a $\Delta x \times \Delta y = 1$ km × 1 km grid. The 2-dimensional wavenumber spectrum of the smoothed measurement errors on this grid is the same as (I.39a), but is truncated at the Nyquist wavenumbers $k_N = (2\Delta x)^{-1} = 0.5$ cpkm and $l_N = (2\Delta y)^{-1} = 0.5$ cpkm that are associated with the 1 km × 1 km grid spacing. In addition to the motivations discussed previously, the subsampling onto this 1 km × 1 km grid eliminates essentially all aliasing by virtue of the fact that the Nyquist wavenumbers are five times higher than the half-power filter cutoff wavenumber of 0.1 cpkm for the 10-km smoothing applied in the pre-processing of WaCM data.

For the $\Delta x \times \Delta y = 1$ km × 1 km subsample grid, the grid spacings for our simulated raw WaCM data are related to $\Delta x$ and $\Delta y$ by $\delta x = 0.5\Delta x$ and $\delta y = 0.5\Delta y$. And since the variances $\sigma_u^2$ and $\sigma_v^2$ of the errors of these simulated raw and unsmoothed WaCM velocity component estimates are related to the variances $\sigma_u^2$ and $\sigma_v^2$ of the errors of the 10-km smoothed velocity component estimates by $\sigma'_{u;v} = 100\sigma_{u;v}^2$, the spectrum (I.39a) can be written alternatively as

$$ \tilde{S}_{u;v}^{2d}(k, l) = 25 \Delta x \Delta y \sigma_{u;v}^2 W_{10 \text{ km}}^2(k) W_{10 \text{ km}}^2(l). \quad (I.39b) $$

This expresses the 2-dimensional spectrum of the velocity component errors in terms of the grid spacing and variance after the 10-km smoothing and the $\Delta x \times \Delta y = 1$ km × 1 km subsampling that are applied in our simulated pre-processing of the WaCM data.

The 1-sided, 1-dimensional alongshore spectrum of the errors of WaCM estimates of velocity components obtained by integrating the 2-dimensional spectrum (I.39b) over all cross-shore wavenumbers $k$ and doubling the power at each alongshore wavenumber $l$ is

$$ \begin{aligned} \tilde{S}_{u;v}^{1d}(l) &= 2 \int_{-k_N}^{k_N} \tilde{S}_{u;v}^{2d}(k, l) dk \\ &= 100 \Delta x \Delta y \sigma_{u;v}^2 W_{10 \text{ km}}^2(l) \int_0^{k_N} W_{10 \text{ km}}^2(k) dk. \end{aligned} \quad (I.40a) $$

This expression uses the fact that $W_{10 \text{ km}}^2(k)$ is symmetric about $k = 0$. The integral on the right side of (I.40a) is equal to $(0.910 \lambda_c)^{-1} \approx \lambda_c^{-1}$ according to (D.12b). Since $\lambda_c = 10$ km = $10\Delta x$ for the smoothing applied in the pre-processing of WaCM data, the 1-dimensional alongshore spectrum (I.40a) of the errors of WaCM estimates of each velocity component on a $\Delta x \times \Delta y = 1$ km × 1 km grid is

$$ \tilde{S}_{u;v}^{1d}(l) = 10 \Delta y \sigma_{u;v}^2 W_{10 \text{ km}}^2(l). \quad (I.40b) $$

It should be noted that the multiplicative factors of 25, 100 and 10 in (I.39b), (I.40a) and (I.40b), respectively, are all specific to the choices of $\delta x = \delta y = 0.5$ km for the footprint size of the raw data as simulated in this study from the $\delta x \times \delta y = 0.5$ km × 0.5 km gridded velocity fields in the CCS model and the assumed 10-km smoothing and $\Delta x = \Delta y = 1$ km subsampling in the pre-processing of WaCM estimates of velocity. We retain $\Delta x$ and $\Delta y$ in these equations as a reminder of the need to keep track of units and to help facilitate adaptation of the equations to other choices of $\delta x$, $\delta y$, and $\Delta y$.

For example, consider the same footprint size of $\delta x = \delta y = 0.5$ km for the raw measurements but for subsampling of the 10-km smoothed raw data onto the grid with $\Delta x = \Delta y = 5$ km on which the pre-processed WaCM estimates of velocity components are uncorrelated. Then $\delta x = 0.1\Delta x$ and $\delta y = 0.1\Delta y$ rather than $0.5\Delta x$ and $0.5\Delta y$ as in the analysis above. The multiplicative factors in (I.39b), (I.40a) and (I.40b) would then become 1, 4 and 2, respectively. If not for the imperfections of the filter transfer function of the Parzen smoother, the values of $W_{10 \text{ km}}(k)$ and $W_{10 \text{ km}}(l)$ would be 1 at all wavenumbers $k < k_N$ and $l < l_N$ for the new Nyquist wavenumbers of $k_N = l_N = 0.1$ cpkm associated with the 5 km × 5 km grid. The imperfections of the filter transfer functions are nonetheless small for these ranges of $k$ and $l$. The resulting analytical expression (I.39a) for the 2-dimensional spectra of velocity component errors on a $\Delta x \times \Delta y = 5$ km × 5 km would then have the same form as the 2-dimensional spectrum (I.1) of errors of pre-processed SWOT estimates of SSH. Likewise, the expressions (I.40a) and (I.40b) for the 1-dimensional spectra of velocity component errors would have the same form as the 1-dimensional spectrum (I.2a) of errors of SWOT estimates of SSH.

The 2-dimensional wavenumber spectrum of the errors of vorticity $\zeta = \partial v / \partial x - \partial u / \partial y$ estimated from the errors of pre-processed WaCM estimates of the two velocity components is obtained straightforwardly from the 2-dimensional spectrum (I.39b). For centered difference estimates of the derivatives on the $\Delta x \times \Delta y = 1$ km × 1 km subsampling grid assumed in the derivation of (I.39b), the 2-dimensional spectrum of vorticity errors is

$$ \tilde{S}_{\zeta}^{2d}(k, l) = \tilde{S}_{\partial u / \partial y}^{2d}(k, l) + \tilde{S}_{\partial v / \partial x}^{2d}(k, l), \quad (I.41) $$

where

$$ \tilde{S}_{\partial u / \partial y}^{2d}(k, l) = |W_{3pt}(l)|^2 \tilde{S}_{u}^{2d}(k, l) \quad (I.42a) $$

$$ \tilde{S}_{\partial v / \partial x}^{2d}(k, l) = |W_{3pt}(k)|^2 \tilde{S}_{v}^{2d}(k, l). \quad (I.42b) $$

The tildes in the above equations again signify that 10-km smoothing has been applied in the pre-processing of WaCM data. There is no cross spectral term in (I.41) because the errors of WaCM estimates of $u$ and $v$ have been assumed to be uncorrelated. The response functions $W_{3pt}(k)$ and $W_{3pt}(l)$ of the centered difference estimates of the derivatives in the $x$ and $y$ dimensions, respectively, are given by (I.4) and (I.6) with $\Delta x = \Delta y = 1$ km.

The 1-sided, 1-dimensional alongshore spectrum of $\zeta$ errors obtained from (I.41) and (I.42) is

$$ \tilde{S}_{\zeta}^{1d}(l) = 2 \int_{-k_N}^{k_N} \tilde{S}_{\zeta}^{2d}(k, l) dk = \tilde{S}_{\partial u / \partial y}^{1d}(l) + \tilde{S}_{\partial v / \partial x}^{1d}(l), \quad (I.43) $$

where

$$ \tilde{S}_{\partial u / \partial y}^{1d}(l) = 2 \int_{-k_N}^{k_N} \tilde{S}_{\partial u / \partial y}^{2d}(k, l) dk = 2 |W_{3pt}(l)|^2 \int_{-k_N}^{k_N} \tilde{S}_{u}^{2d}(k, l) dk \quad (I.44a) $$

$$ \tilde{S}_{\partial v / \partial x}^{1d}(l) = 2 \int_{-k_N}^{k_N} \tilde{S}_{\partial v / \partial x}^{2d}(k, l) dk = 2 \int_{-k_N}^{k_N} |W_{3pt}(k)|^2 \tilde{S}_{v}^{2d}(k, l) dk. \quad (I.44b) $$

According to (I.40a), the integral on the right side of (I.44a) with the multiplicative factor of 2 is equal to the 1-dimensional alongshore spectrum $\tilde{S}_u^{1d}(l)$ of the cross-shore velocity component errors. The
contribution (I.44a) to the 1-dimensional alongshore spectrum (I.43) of vorticity errors is therefore

$$ \bar{S}_{\partial u/\partial y}^{1d}(l) = |W_{3pt}(l)|^2 \bar{S}_u^{1d}(l), \qquad (I.45a) $$

where $\bar{S}_u^{1d}(l)$ can be expressed alternatively in the form (I.40b) to get

$$ \bar{S}_{\partial u/\partial y}^{1d}(l) = 10 \Delta y \sigma_u^2 |W_{3pt}(l)|^2 W_{10 \text{ km}}^2(l). \qquad (I.45b) $$

From (I.39b), the contribution (I.44b) to the 1-dimensional wavenumber spectrum of vorticity errors can be written as

$$ \bar{S}_{\partial v/\partial x}^{1d}(l) = 100 \Delta x \Delta y \sigma_v^2 W_{10 \text{ km}}^2(l) \int_0^{k_N} |W_{3pt}(k)|^2 W_{10 \text{ km}}^2(k) dk. \qquad (I.46a) $$

This equation uses the fact that $|W_{3pt}(k)|^2$ and $W_{10 \text{ km}}^2(k)$ are both symmetric about $k = 0$. From (I.40b), this can be written alternatively in terms of the 1-dimensional wavenumber spectrum of alongshore velocity component errors,

$$ \bar{S}_{\partial v/\partial x}^{1d}(l) = 10 \Delta x \bar{S}_v^{1d}(l) \int_0^{k_N} |W_{3pt}(k)|^2 W_{10 \text{ km}}^2(k) dk. \qquad (I.46b) $$

The integral on the right sides of (I.46a) and (I.46b) is the same integral encountered in (I.33b) for the case of $\lambda_c = 10$ km. Numerical solutions for this integral are shown as a function of half-power filter cutoff wavelength $\lambda_c$ in Fig. 1.1a. For $\Delta x = \Delta y = 1$ km, the integral has a value of 0.0221 at $\lambda_c = 10$ km.

The analytical expressions (I.40) and (I.43), with the latter partitioned as (I.45a) and (I.46a), for the 1-sided, 1-dimensional alongshore spectra of errors of WaCM estimates of the two velocity components and the vorticity computed from the pre-processed WaCM data are applicable to any low-pass filter with a half-power filter cutoff wavelength of 10 km. The results based on the Parzen smoother with $\sigma_u = \sigma_v = \sigma_{u,v} = 0.354$ m s$^{-1}$ are shown by the solid green lines in Fig. 13c. These theoretical spectra agree very well with the spectra shown in Fig. 13c by the solid blue lines that were computed empirically from the simulated WaCM error fields generated for the analysis in Sections 6 and 8–10.

## 1.4. Errors of smoothed WaCM estimates of velocity and vorticity

The errors of WaCM estimates of velocity and vorticity computed from the pre-processed WaCM data (see Appendix G and Table 1 in Section 4 and the bottom left panels of Figs. 19a and 19b) are too large for the velocity and vorticity estimates to be useful for most applications. It will be necessary to smooth the pre-processed WaCM data in ground-based post-processing to reduce the effects of measurement errors. As in Appendix I.3, we assume here that the pre-processed WaCM estimates of velocity have been smoothed with a filter cutoff wavelength of 10 km and subsampled from the raw measurement grid (0.5 km × 0.5 km for the simulations in this study) onto a $\Delta x \times \Delta y = 1$ km × 1 km grid.

The 2-dimensional wavenumber spectrum of WaCM estimates of each velocity component after smoothing in each dimension with a filter cutoff wavelength of $\lambda_c$ in ground-based post-processing is obtained from the 2-dimensional wavenumber spectrum (I.39b) of the errors of the pre-processed estimates of the two velocity components by

$$ \begin{aligned} \bar{S}_{u,v}^{2d}(k,l) &= W_{\lambda_c}^2(k) W_{\lambda_c}^2(l) \bar{S}_{u,v}^{2d}(k,l) \\ &= 25 \Delta x \Delta y \sigma_{u,v}^2 W_{10 \text{ km}}^2(k) W_{10 \text{ km}}^2(l) W_{\lambda_c}^2(k) W_{\lambda_c}^2(l). \end{aligned} \qquad (I.47) $$

The subscript $u, v$ with the semicolon again indicates that the equation has the same form for both velocity components, differing only in the choice of a subscript $u$ or $v$ on both sides of (I.47). The combined tilde and overbar signify that the spectrum on the left side of (I.47) is based on WaCM velocity component estimates that have been 2-dimensionally smoothed twice, first in pre-processing with a half-power filter cutoff wavelength of 10 km in each dimension (signified as in Appendix I.3 by the tilde) and then in ground-based post-processing

with a half-power filter cutoff wavelength of $\lambda_c$ in each dimension (signified by the overbar).

The 1-sided, 1-dimensional alongshore spectrum of the smoothed velocity errors is

$$ \begin{aligned} \bar{S}_{u,v}^{1d}(l) &= 2 \int_{-k_N}^{k_N} \bar{S}_{u,v}^{2d}(k,l) dk \\ &= 100 \Delta x \Delta y \sigma_{u,v}^2 W_{10 \text{ km}}^2(l) W_{\lambda_c}^2(l) \int_0^{k_N} W_{10 \text{ km}}^2(k) W_{\lambda_c}^2(k) dk. \end{aligned} \qquad (I.48a) $$

From (I.40b), this can be written alternatively in terms of the 1-dimensional alongshore spectrum $\bar{S}_{u,v}^{1d}(l)$ of pre-processed WaCM estimates of the velocity components,

$$ \bar{S}_{u,v}^{1d}(l) = 10 \Delta x \bar{S}_{u,v}^{1d}(l) W_{\lambda_c}^2(l) \int_0^{k_N} W_{10 \text{ km}}^2(k) W_{\lambda_c}^2(k) dk. \qquad (I.48b) $$

When the filter cutoff wavelength $\lambda_c$ in the ground-based post-processing is large compared with the 10-km filter cutoff wavelength of the smoothing applied in pre-processing of WaCM data, the filter transfer function $W_{10 \text{ km}}(k)$ in the integrand on the right sides of (I.48a) and (I.48b) is approximately 1 across the low-wavenumber pass band of the filter transfer function $W_{\lambda_c}(k)$. In this case, the 10-km filtering in the pre-processing has relatively little effect on the integral on the right sides of (I.48a) and (I.48b). Then (I.48a), for example, reduces to the approximate form

$$ \bar{S}_{u,v}^{1d}(l) \approx 100 \Delta x \Delta y \sigma_{u,v}^2 W_{10 \text{ km}}^2(l) W_{\lambda_c}^2(l) \int_0^{k_N} W_{\lambda_c}^2(k) dk. $$

Substitution of (I.28) simplifies this to

$$ \bar{S}_{u,v}^{1d}(l) \approx \frac{100 \Delta x \Delta y}{\lambda_c} \sigma_{u,v}^2 W_{10 \text{ km}}^2(l) W_{\lambda_c}^2(l). \qquad (I.49) $$

More generally, the 1-dimensional alongshore spectra of velocity component errors smoothed with a half-power filter cutoff wavelength of $\lambda_c$ in each dimension must be computed from (I.48a) or (I.48b). Numerical solutions for the integral on the right sides of (I.48a) and (I.48b) are shown as a function of $\lambda_c$ by the solid line in Fig. 1.1c. The dashed line is the value of the integral when $W_{10 \text{ km}}^2(k)$ is replaced with a value of 1 for all wavenumbers $k$ as in (I.49). The two lines converge rapidly and the approximate solution (I.49) is accurate to better than 10% for ground-based post-processing with $\lambda_c \ge 22$ km.

The 2-dimensional wavenumber spectrum of the errors of vorticity estimated from smoothed errors of WaCM estimates of the velocity components is obtained analogously to (I.41) and (I.42), except with the 2-dimensional spectra (I.40) of the errors of pre-processed WaCM estimates of the two velocity components on the right sides of (I.42a) and (I.42b) replaced with the 2-dimensional spectra (I.47) of the two smoothed velocity component errors,

$$ \bar{S}_{\zeta}^{2d}(k,l) = \bar{S}_{\partial u/\partial y}^{2d}(k,l) + \bar{S}_{\partial v/\partial x}^{2d}(k,l), \qquad (I.50) $$

where

$$ \bar{S}_{\partial u/\partial y}^{2d}(k,l) = |W_{3pt}(l)|^2 \bar{S}_u^{2d}(k,l) \qquad (I.51a) $$

$$ \bar{S}_{\partial v/\partial x}^{2d}(k,l) = |W_{3pt}(k)|^2 \bar{S}_v^{2d}(k,l). \qquad (I.51b) $$

As in (I.47), the combined tilde and overbar signify spectra that are based on doubly smoothed WaCM velocity component estimates.

The 1-sided, 1-dimensional alongshore spectrum of the smoothed vorticity errors obtained from (I.50) and (I.51) is

$$ \bar{S}_{\zeta}^{1d}(l) = 2 \int_{-k_N}^{k_N} \bar{S}_{\zeta}^{2d}(k,l) dk = \bar{S}_{\partial u/\partial y}^{1d}(l) + \bar{S}_{\partial v/\partial x}^{1d}(l), \qquad (I.52) $$

where

$$ \bar{S}_{\partial u/\partial y}^{1d}(l) = 2 \int_{-k_N}^{k_N} \bar{S}_{\partial u/\partial y}^{2d}(k,l) dk = 2 |W_{3pt}(l)|^2 \int_{-k_N}^{k_N} \bar{S}_u^{2d}(k,l) dk \qquad (I.53a) $$
$$ \bar{S}_{\partial v/\partial x}^{1d}(l) = 2 \int_{-k_{\mathcal{N}'}^{k, \mathcal{N}'}} |\bar{S}_{\partial v/\partial x}^{2d}(k, l)|^2 \bar{S}_{v}^{2d}(k, l) dk. \qquad (I.53b) $$

From (I.47), the first contribution (I.53a) to the 1-dimensional alongshore spectrum (I.52) of vorticity errors can be written as

$$ \bar{S}_{\partial u/\partial y}^{1d}(l) = 100 \Delta x \Delta y \sigma_u^2 |W_{3pt}(l)|^2 W_{10 \text{ km}}^2(l) W_{\lambda_c}^2(l) \int_0^{k_{\mathcal{N}'}^k} W_{10 \text{ km}}^2(k) W_{\lambda_c}^2(k) dk. \qquad (I.54a) $$

The integral on the right side of this equation is the same as the integral on the right sides of (I.48a) and (I.48b). Numerical solutions for this integral are shown as a function of $\lambda_c$ in Fig. 1.1c.

With (I.48a), the spectrum (I.54a) can be expressed more compactly in terms of the 1-dimensional alongshore spectrum $\bar{S}_u^{1d}(l)$ of the u component of smoothed velocity errors,

$$ \bar{S}_{\partial u/\partial y}^{1d}(l) = |W_{3pt}(l)|^2 \bar{S}_u^{1d}(l). \qquad (I.54b) $$

This could be written in terms of the 1-dimensional spectrum $\bar{S}_u^{1d}(l)$ of pre-processed WaCM estimates of the u component of smoothed velocity errors by substituting (I.48b) for $\bar{S}_u^{1d}(l)$ on the right side of (I.54b).

A similar procedure can be followed for the second contribution (I.53b) to the 1-dimensional alongshore spectrum (I.52) of vorticity errors. From (I.47), the expression (I.53b) can be written as

$$ \bar{S}_{\partial v/\partial x}^{1d}(l) = 100 \Delta x \Delta y \sigma_v^2 W_{10 \text{ km}}^2(l) W_{\lambda_c}^2(l) \int_0^{k_{\mathcal{N}'}^k} |W_{3pt}(k)|^2 W_{10 \text{ km}}^2(k) W_{\lambda_c}^2(k) dk. \qquad (I.55a) $$

With (I.48a), this can be expressed in terms of the 1-dimensional alongshore spectrum $\bar{S}_v^{1d}(l)$ of the v component of smoothed velocity errors,

$$ \bar{S}_{\partial v/\partial x}^{1d}(l) = \bar{S}_v^{1d}(l) \frac{\int_0^{k_{\mathcal{N}'}^k} |W_{3pt}(k)|^2 W_{10 \text{ km}}^2(k) W_{\lambda_c}^2(k) dk}{\int_0^{k_{\mathcal{N}'}^k} W_{10 \text{ km}}^2(k) W_{\lambda_c}^2(k) dk}. \qquad (I.55b) $$

This could be written in terms of the 1-dimensional spectrum $\bar{S}_v^{1d}(l)$ of pre-processed WaCM estimates of the v component of velocity errors by substituting (I.48b) for $\bar{S}_v^{1d}(l)$ on the right side of (I.55b).

As in the case of the approximate expression (I.49) for the 1-dimensional spectrum of smoothed velocity component errors, the expression (I.55b) can be simplified when the filter cutoff wavelength $\lambda_c$ in the ground-based post-processing is large compared with the 10-km filter cutoff wavelength of the smoothing applied in the pre-processing of WaCM data. Then the filter transfer function $W_{10 \text{ km}}(k)$ in the integrands on the right sides of (I.55a) and (I.55b) is approximately 1 across the low-wavenumber pass band of the filter transfer function $W_{\lambda_c}(k)$. In this case, the 10-km filtering in the pre-processing has relatively little effect on the integral on the right side of (I.55a) and either of the two integrals on the right side of (I.55b). Then (I.55a), for example, reduces to the approximate form

$$ \bar{S}_{\partial v/\partial x}^{1d}(l) \approx 100 \Delta x \Delta y \sigma_v^2 W_{10 \text{ km}}^2(l) W_{\lambda_c}^2(l) \int_0^{k_{\mathcal{N}'}^k} |W_{3pt}(k)|^2 W_{\lambda_c}^2(k) dk. \qquad (I.56) $$

The integral on the right side of this equation is the same as the integral on the right side of (I.33). Numerical solutions for this integral were shown as a function of $\lambda_c$ in Fig. 1.1a.

More generally, the 1-dimensional alongshore spectra of vorticity errors smoothed with a half-power filter cutoff wavelength of $\lambda_c$ in each dimension must be computed from (I.55). Numerical solutions for the integral on the right side of (I.55a) and in the numerator of (I.55b) are

shown as a function of $\lambda_c$ by the solid line in Fig. 1.1d and numerical solutions for the integral in the denominator on the right side of (I.55b) was shown as a function of $\lambda_c$ by the solid line in Fig. 1.1c. The dashed lines are the values of the integrals when $W_{10 \text{ km}}^2(k)$ is replaced with a value of 1 for all wavenumbers $k$ as in (I.56). The approximate solution to the new integral in (I.55a) and the numerator of (I.55b) is accurate to better than 10% for ground-based post-processing with $\lambda_c \ge 37$ km.

The analytical expressions (I.48) and (I.52), with the latter partitioned as (I.54) and (I.55), for the 1-sided, 1-dimensional alongshore spectra of the residual errors of WaCM estimates of the two velocity components and vorticity after smoothing in ground-based post-processing of the pre-processed WaCM data are applicable to any low-pass filter with a half-power filter cutoff wavelength of $\lambda_c$. The results based on the Parzen smoother with $\sigma_u = \sigma_v = \sigma_{u,v} = 0.354 \text{ m s}^{-1}$ are shown by the solid green lines in Fig. 15 for filter cutoff wavelengths of $\lambda_c = 20, 50$ and $80 \text{ km}$. These theoretical spectra agree very well with the spectra shown by the solid blue lines that were computed empirically from the simulated WaCM error fields generated for the analysis in Sections 6 and 8–10.

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