Satellite altimeters routinely supply sea surface height (SSH)
measurements, which are key observations for monitoring ocean dynamics.
However, below a wavelength of about 70 km, along-track altimeter
measurements are often characterized by a dramatic drop in signal-to-noise
ratio (SNR), making it very challenging to fully exploit the available altimeter
observations to precisely analyze small mesoscale variations in
SSH. Although various approaches have been proposed and applied to identify
and filter noise from measurements, no distinct methodology has emerged for
systematic application in operational products. To best address this
unresolved issue, the Copernicus Marine Environment Monitoring Service
(CMEMS) actually provides simple band-pass filtered data to mitigate noise
contamination of along-track SSH signals. More innovative and suitable noise
filtering methods are thus left to users seeking to unveil small-scale
altimeter signals. As demonstrated here, a fully data-driven approach is
developed and applied successfully to provide robust estimates of noise-free
sea level anomaly (SLA) signals (Quilfen, 2021). The method combines empirical mode
decomposition (EMD), used to help analyze non-stationary and non-linear
processes, and an adaptive noise filtering technique inspired by discrete
wavelet transform (DWT) decompositions. It is found to best resolve the
distribution of SLA variability in the 30–120 km mesoscale wavelength band.
A practical uncertainty variable is attached to the denoised SLA estimates
that accounts for errors related to the local SNR but
also for uncertainties in the denoising process, which assumes that the SLA
variability results in part from a stochastic process. For the available
period, measurements from the Jason-3, Sentinel-3, and SARAL/AltiKa missions
are processed and analyzed, and their energy spectral and seasonal
distributions are characterized in the small mesoscale domain. In anticipation
of the upcoming SWOT (Surface Water and Ocean Topography) mission data, the
SASSA (Satellite Altimeter Short-scale Signals Analysis,

Satellite altimetry has supported studies related to ocean dynamics for more than 25 years, often looking to push the limits of these observations to capture ocean motions at ever smaller scales. New paradigms are thus emerging from this observational effort, among them the distinction between balanced and unbalanced motions that can lead to characteristic changes in sea surface height (SSH) signal variations and associated spectrum in the 30–200 km wavelength range (e.g., Fu, 1983; Le Traon et al., 2008; Dufau et al., 2016; Tchibilou et al., 2018) or the role of upper-ocean sub-mesoscale dynamics that is critical to the transport of heat between the ocean interior and the atmosphere (Su et al., 2018).

One of the main limitations is that altimetry measurements are often characterized by a low signal-to-noise ratio (SNR), which has a significant impact on geophysical analysis capability at spatial scales smaller than 120 km. The main sources of noise are induced by instrumental white noise, errors related to processing, including the retracking algorithm and corrections, and errors related to the intrinsic variability of radar echoes in the altimeter footprint that causes the notorious spectral hump in 20 and 1 Hz data (Sandwell and Smith, 2005; Dibarboure et al., 2014). Furthermore, because retrieved parameters are obtained from the same waveform retracking algorithm, they have highly correlated errors, i.e., the standard MLE4 processing produces four estimated parameters with correlated errors (SSH; significant wave height, SWH; sigma-0; and off-nadir angle). These errors directly limit the accuracy of the SSH measurement, requiring advanced denoising techniques (Quartly et al., 2021).

Analysis of fine-scale ocean dynamics therefore requires preliminary noise
filtering, and low-pass or smoothing filters (e.g., Lanczos, running mean,
or Loess filter) are frequently used. These filters effectively smoothen
altimeter signals, but result in the systematic loss of small-scale (

To overcome these difficulties, an adaptive noise removal approach for satellite altimeter measurements has been derived. It is based on the non-parametric empirical mode decomposition (EMD) method developed to analyze non-stationary and non-linear signals (Huang et al., 1998; Huang and Wu, 2008). EMD is a scale decomposition of a discrete signal into a limited number of amplitude- and frequency-modulated functions (AM/FM), among which the Gaussian noise distribution is predictable (Flandrin et al., 2004). Noise removal strategies can then be developed with results often superior to wavelet-based techniques (Kopsinis and McLaughin, 2009). An EMD-based technique was successfully applied to altimetry data to more precisely analyze along-track altimeter SWH measurements to map wave–current interactions (Quilfen et al., 2018; Quilfen and Chapron, 2019) known to predominate at scales smaller than 100 km. In particular, the method is suitable for processing non-stationary and non-linear signals, and thus for accurate and consistent recovery of strong gradients and extreme values. Building on local noise analysis, the denoising of small mesoscale signals is performed on an adaptive basis to the local SNR. A detailed description of the EMD denoising approach applied to satellite altimetry data is given in Quilfen and Chapron (2021).

In this paper, the method is extended to more thoroughly evaluate an experimental data set of denoised sea level anomaly (SLA) measurements, from three reference altimeters, the Jason-3, Sentinel-3, and SARAL/AltiKa, in order to capture short mesoscale information. Section 2 provides a description of the data sets used and Sect. 3 describes the denoising methodology main principles. In Sect. 4, which presents the results, examples of denoised SLA signals are given, and the energy spectral and seasonal distributions of denoised measurements are characterized in the small mesoscale domain for these three altimeters. Section 5 presents key features for comparison with other distributed data sets that make our approach more attractive. A discussion follows, analyzing the main results, and a summary is given. Appendices A and B provide details on the denoising scheme and power spectral density (PSD) calculation, respectively.

The Copernicus Marine Environment Service (CMEMS) is responsible for the dissemination of various satellite altimeter products, among which the level 3 along-track SSHs distributed in delayed mode (product identifier: SEALEVEL_GLO_PHY_L3_REP_OBSERVATIONS_008_062) are the state-of-the-art product that takes into account the various improvements proposed in the framework of the SSALTO/DUACS activities (Taburet et al., 2021). The input data quality control verifies that the system uses the best altimeter data. From these products, which include data from all altimetry missions, we use the “unfiltered SLA” variable to derive our analysis of SLA measurements.

The present study aims to provide research products, the SASSA (Satellite
Altimeter Short-scale Signals Analysis) data set, and innovative solutions
for better exploitation of the mesoscale mapping capabilities of
altimeters. The analysis is therefore limited to three current altimeter
missions, Jason-3, Sentinel-3, and SARAL/AltiKa, each carrying an instrument
with particular distinctive characteristics. The Jason-3 altimeter is the
reference dual-frequency Ku-C instrument and is used as the reference
mission for cross-calibration with other altimeters to provide consistent
products in the CMEMS framework. The Satellite with ARgos and ALtiKa (SARAL)
mission carries the AltiKa altimeter, which makes measurements at higher
effective resolution due to a smaller footprint obtained in the Ka band (8 km
diameter vs 20 km on Jason-3) and a higher pulse repetition rate. The
altimeter on board Sentinel-3 is a dual-frequency Ku-C altimeter that
differs from conventional pulse-limited altimeters in that it operates in
delay Doppler mode, also known as synthetic aperture radar mode (SARM). SARM
is the primary mode of operation, providing

Although only the quality-controlled CMEMS data are used as input in our analysis, ancillary data are useful in supporting the analysis of SLA data. Indeed, since some of the larger non-Gaussian SLA errors, correlated with high sea state conditions and rain or slick events, are expected to remain after the EMD analysis, SWH and radar cross-section (sigma-0) are also provided in the denoised SLA products to allow for further data analysis and editing. These are provided by the sea state Climate Change Initiative (CCI) products, developed by the European Space Agency (ESA) and processed by the Institut Français de Recherche pour l'Exploitation de la Mer (IFREMER, Dodet et al., 2020).

The proposed denoising technique essentially builds on the EMD technique (Huang et al., 1998; Wu and Huang, 2004; Huang and Wu, 2008) and its filter bank characteristics when applied to Gaussian noise (Flandrin et al., 2004). The technique was first adapted to process satellite altimeter SWH measurements (Quilfen et al., 2018; Quilfen and Chapron, 2019; Dodet et al., 2020), and the algorithm is described in detail in Quilfen and Chapron (2021). For the processing of the SLA data analyzed in this study, only limited modifications were made, and the algorithm is only briefly described below.

Three main elements characterize the properties of the denoising algorithm: (1) the EMD algorithm that adaptively splits the SLA signal on an orthogonal basis without having to conform to a particular mathematical framework; (2) the denoising algorithm that relies on high-frequency local noise recovery and analysis; (3) an ensemble-average approach to estimate a robust denoised SLA signal and its associated uncertainty.

EMD is a data-driven method, often used as an alternative to wavelets in denoising a wide variety of signals. EMD decomposes a 1D signal into a set of amplitude- and frequency-modulated components, called intrinsic modulation functions (IMFs), which satisfy the conditions of having zero mean and a number of extrema equal to (or different by one than) the number of zero crossings. IMFs are obtained through an iterative algorithm, called sifting, which extracts the high-frequency component by iteratively computing the average envelope from the extrema points of the input signal. The sifting algorithm is first applied to the input SLA signal to derive the first IMF, IMF1, which is removed from the SLA signal to obtain a new signal on which the process is repeated until it converges when the last calculated IMF no longer has a sufficient number of extrema. The original signal is exactly reconstructed by adding all the IMFs. Figure 1 shows two sets of IMF for two passages of SARAL over the Gulf Stream area. Panels (a) and (g) show the two SLA signals and the associated SWH signals for reference (red curves), and the other panels display the full set of IMF (six and four derived IMFs for these two cases, a number that can vary with signal length and observed wavenumber spectrum). Shown in panels (b) and (h), IMF1 genuinely maps the high-frequency noise in term of amplitude and phase, which can provide a direct approach to help remove high-frequency noise from the SLA signal. Local analysis of this high-frequency noise is used to predict and remove the lower-frequency noise embedded in other IMFs, as detailed below. In panel (b), IMF1 is also associated with high-frequency noise but shows non-stationary noise statistics that are related to changes in mean sea state conditions. As expected, the high-frequency noise of SLA increases with SWH. These two examples are general cases, but IMF1 can also contain geophysical information in cases where the SNR is locally very high, for example in the presence of very large geophysical gradients, or can show the signature of outliers related to the so-called spectral hump (rain, slicks, etc.). Indeed, depending on the SNR and in specific configurations for the numerical sifting algorithm, the type of large SLA gradient signature shown in IMF2 (panel i) can very well show up in IMF1, in case there are no detectable extrema between measurements number 2 and 8. Since IMF1 analysis is at the heart of the denoising strategy described below, careful preprocessing of IMF1 is necessary before denoising the full signal.

SARAL SLAs (

Flandrin at al. (2004) applied EMD to a Gaussian noise signal to
demonstrate that the IMF1 has the characteristics of a high-pass filter
while the higher-order modes behave similarly to a dyadic low-pass filter
bank, for which, as they move down the frequency scale, successive frequency
bands have half the width of their predecessors. Unlike Fourier or wavelet
decompositions for which the noise variance is independent of the scale, the
noise contained in each IMF is now “colored” with a different energy
level for each mode. Flandrin et al. (2004) deduced that the variance of the
Gaussian noise projected onto the IMF basis can be modeled as follows for
the low-pass filter bank:

Flandrin et al. (2004) then numerically derive, using Eq. (1) and for
different values of

Equation (2) therefore gives the expected noise energy in each IMF to determine
the different thresholds below which signal fluctuations can be associated
with noise. The threshold formulation introduces the constant factor

A detailed description of the entire denoising scheme can be found in Quilfen and Chapron (2021), and the main steps are given in Appendix A.

Figure 2 provides illustrations of the general approach taken to denoising
SLA signals. They show the PSD of SLA (black
curves), the associated IMFs (blue curves), and the IMFs of a white noise
(red curves) whose standard deviation has been adjusted to fit the SLA
background noise between 30 and 15 km wavelength. It is presented for the
Agulhas Current area, panel (a), and for the Gulf Stream area, panel (b).
For clarity, only the first three IMFs are shown. As expected, for white
noise, the EMD filter bank is composed of a high-pass filter with the IMF1,
and a low-pass filter bank with the higher ranked IMFs. A similar structure
is observed for the IMFs of the SLA signal with identical cut-off
wavelengths, which is the result of the noise shaping the frequency content
of the SLA signal. This similarity shows the consistency of separate
denoising of each IMF of rank

Mean PSD of the first three IMFs
(first

A is an important factor to adjust because it is directly related to the
improvement obtained in the SNR. Kopsinis and McLaughin (2009) perform the
optimization of the

The two cases (Fig. 1) show AltiKa SLA measurements in the Gulf Stream area, and Fig. 3 illustrates the EMD denoising principles for these examples. Described in Sect. 3.2, denoising a segment of SLA data after an initial expansion into an IMF set is a two-step process: (1) wavelet analysis of IMF1 to separate and evaluate the high-frequency part of the Gaussian noise and any geophysical information embedded in IMF1; (2) EMD denoising of a set of 20 realizations of reconstructed noisy SLA series to estimate a mean denoised SLA series and its uncertainty.

Pass 597, panels (g) to (k), is associated with rather low sea state conditions with little variability, and IMF1 (black curve, panel h) has little amplitude modulation, but rather large phase modulation due to the the high SNR in several portions of the segment (minor alternation of minima and maxima). Because of this relatively large phase modulation, a significant portion of the IMF1 is identified in the first step as “useful signal” by the wavelet analysis. In the second step however, this residual IMF1 signal will be almost completely removed. Indeed, it is well below the SNR prescribed by using the high-frequency noise jointly derived from the IMF1 wavelet processing and the threshold values set with Eqs. (A1), (2), and (3) (blue lines in Fig. 3). Only a small modulation between data records 80 and 90 therefore shows up in the denoised SLA signal. Figure 3i shows IMF2, and its associated threshold derived from the IMF1 threshold (i.e., Eq. 2), which maps the large SLA gradient in the Gulf Stream and mesoscale features near 70 km wavelength with some eddies appearing well above the threshold and other smaller amplitude oscillations that will be canceled in the second denoising step. As shown in Fig. 2, the SNR increases rapidly for IMF2 compared to IMF1. In this case, the uncertainty attached to the denoised SLA is almost constant below 1 cm, as shown in Fig. 3h.

AltiKa pass 53 crosses the Gulf Stream 19 d before, but this is a very
different situation. Quite frequently, such a case corresponds to high and
variable sea state conditions with abrupt changes in SWH, as shown in Fig. 1.
Strong westerly continental winds were present for several days before the
AltiKa passage, which turned to the northwest the day before. SWH was less
than 2 m near the coast between records 80 and 120, then a first large
increase occurred on the northern side of the Gulf Stream near record 60,
and a second on its southern side to reach sea state conditions with SWH

In cases where sigma-0 blooms or rain events corrupt limited portions of a data segment, and for which the data editing step was not performed, the impact is more difficult to analyze. It will depend on the magnitude and length of the associated errors which can vary greatly. However, the proposed EMD denoising process is not a data editing process and the results are certainly still affected by some of the largest errors. It should benefit from improvements in data editing procedures and retracking algorithms that will be used for future CMEMS products.

SARAL short data segments in the Gulf Stream area for cycle 106
and passes 53

For SLA measurements performed by a given altimeter instrument, the mean SNR
is expected to vary primarily with sea state. The mean SNR is then a
function of the climatological distribution of sea state conditions that are
dependent on ocean basins and seasons. The proposed denoising approach can
efficiently adapt to the local SNR, allowing for a single global value for
the control constant

The two-step analysis for each region then follows.

For each region, a set of discrete

An optimal value of

The problem to consider further is the presence, in limited portions of the processed data segments, of outliers associated with high waves or artifacts caused by sigma-0 blooms or rain events. These will likely appear in IMF1 and IMF2 series and the most energetic events will not be thresholded since the thresholds are calculated using the median absolute deviation from zero of IMF1. For this reason, processing IMF1 using wavelet analysis is an important step to separate, as much as possible, the possible useful geophysical signal in IMF1 from outliers, and to estimate the underlying Gaussian noise.

Mean PSD of SARAL SLA along-track
measurements: observed (thick black), best fit (thin black), best fit plus
WGN (red), retrieved (dashed red), and observed minus WGN
(green). WGN is estimated as the average in the range of 15–25 km of the mean
observed PSD (bold black curve). The PSD is the average of PSDs computed
over all data segments covering the years 2016–2018: the Gulf Stream (72–60

The reasoning used above to set the control parameter

In this simulation, the SNR is close to 1 on average near 50 km wavelength,
as shown in Fig. 5, but can be greater than 1 locally in a wavelength range
down to 30 km. Such small mesoscale geophysical information emerging from
the noise level can be retrieved from IMF1 using dedicated wavelet
denoising analysis. As shown in Fig. 5, the average PSD of IMF1 shows the
plateau of high-frequency noise, but also significant energy content over a
wider wavelength range associated with both lower-frequency noise and
geophysical information. The PSD curves of the IMF1 and the simulated SLA
intersect between 50 and 30 km wavelength. After wavelet decomposition of
the IMF1, the wavelet denoising scheme specifies the maximum level to be
retained for geophysical signal recovery. In the general case, only the
level containing the finest scales is systematically discarded, and Fig. 5
shows the PSD of the signal recovered from IMF1 after using the Huang and
Cressie (2000) denoising scheme (red dashed curve). The wavelet denoising
acts as a low-pass filter with a sharp cut-off near 25 km wavelength and a
significant amount of noise is also filtered out at longer wavelengths. In
this simulation, the processing results in the recovery (red curve) of the
full PSD (thin black curve) of the simulated signal because the

Mean PSD of SARAL SLA along-track
measurements: simulated noise-free SLAs (thin black), simulated noise-free
SLAs

The EMD denoising algorithm is then found to be robust and consistent in processing the AltiKa measurements. A workable rule can be defined to adjust the method to provide a global data set of denoised SLA measurements whose PSD are regionally consistent with the expected SLA geophysical signals. Such an approach is not easily applicable or numerically consistent for the Sentinel-3 and Jason-3 measurements. Their PSDs do not exhibit the expected white noise plateau in the 10–25 km wavelength range, Fig. 6. The red-type noise in Sentinel-3 measurements has already been discussed and analyzed in several studies, and has been shown to be mainly related to the effects of swell on SARM observations (Moreau et al., 2018; Rieu et al., 2021). The reason why the Jason-3 PSD also shows a tilted PSD in the high-frequency range is more puzzling. One possible explanation is that it results from poor data editing (especially the rain flag) for Jason-3. Indeed, while an effective rain flag was used for AltiKa so that rain has little influence on data quality (Verron et al., 2021), this is not the case for Jason-3, which is therefore likely to be more impacted by rain events. Associated errors may shape the noise distribution differently than white noise, as discussed in the previous section. Indeed, we found many more short segments of continuous measurements in the AltiKa data set than in the Jason-3 data set, both of which are distributed in the same CMEMS product, due to more efficient data editing. Therefore, the adjustment of the EMD denoising process for Jason-3 and Sentinel-3 was performed by using the AltiKa results as reference.

For Sentinel-3, Fig. 6 shows that its PSD for all analyzed regions is in
excellent agreement with AltiKa's PSD over the entire wavelength range down
to 25 km, which is a striking result, showing that the two altimeters have
similar average noise level and shape above 25 km wavelength. The same value
of

For Jason-3 and the 3-year data set analyzed, the control constant

For reference, Fig. 6 shows a

Mean PSD of observed and denoised SLA
along-track measurements for Jason-3 (blue), AltiKa (black), Sentinel-3
(red), and four regions: Gulf Stream (panel

Seasonal variations in SSH in the small mesoscale range, with a wavelength less than about 100 km, are very difficult to analyze because the noisy SLA spectral slopes are strongly shaped by errors related to the hump artifact, not dependent on the sea state, and by the instrumental and processing noise which is correlated with sea state conditions. Although the behavior of the resulting total noise is not well understood, it is genuinely postulated that the total SLA noise in the 1 Hz measurements can be considered to be WGN when a large amount of long segments is used to calculate the spectrum. This enabled the empirical study of seasonal variations in SSH by removing an average noise PSD from the average PSDs of altimeter measurements (e.g., Vergara et al., 2019; Chen and Qiu, 2021). However, the applicability of this assumption may not be verified at the regional level, as suggested by the results presented in Figs. 4 and 5. This certainly also depends on the performance of the data editing. Therefore, the adoption of an alternative approach based on the analysis of the along-track EMD-denoised SLA measurements, rather than on the denoising of the SLA spectrum, is likely more suited. It has also been shown that sea state-related errors are essentially removed by the EMD processing. Figure 6 shows that consistent spectral slopes of the denoised SLA are well obtained for the different altimeters. Hereafter, we only use AltiKa denoised measurements because the period covered is much longer for more consistent analysis of seasonal variations. The data used cover 6 years, from summer 2013 to winter 2018–2019, and the eight different regions shown in Fig. 7 are defined to cover various climatological sea state conditions and expected energy levels in the small mesoscale variability of SLA.

Yearly averaged SWH (m) computed over 2016–2018 from the Climate Change Initiative L4 products. Dashed black boxes define the eight areas analyzed in the section.

For each region, the average SLA spectrum is shown in Fig. 8 for the boreal
summer and winter, for the entire data set, and for a data set limited to
segments having more than 80 % of measurements with SWH

Distinct regions can be considered in Fig. 8. The intra-tropical regions,
2 and 3, show no seasonality, a result in agreement with previous studies
(Vergara et al., 2019; Chen and Qiu, 2021). In these regions, stable low
sea state conditions cannot introduce strong errors in the analyses. In the
rough southern oceans, regions 7 and 8 (the Drake Passage) show a small
apparent increase in small mesoscale energy in the austral winter,
disappearing when the high sea state threshold is applied. In the latter
case and for the Drake Passage, 130 and 73 AltiKa passes satisfy the
criterion and were used to estimate the mean spectrum for the boreal JJA and
DJF, respectively. This suggests the absence of seasonality, in agreement
with the results of Rocha et al. (2016), who used acoustic Doppler current profile (ADCP) measurements in the
Drake Passage, and disagrees with Vergara et al. (2019) and Chen and Qiu (2021), who used the standard approach applied to altimeter data. As
mentioned, high sea conditions make it difficult to assess the results
obtained by the different approaches, and better data editing and retracking
algorithms (Passaro et al., 2014; Thibaut et al., 2017; Moreau et al., 2021)
would improve the current analysis. The Agulhas region, number 5, which
experiences mixed sea conditions, shows no seasonality, which is in
agreement with the results of Chen and Qiu (2021). Three regions, 1, 4, and
6, show seasonality, insensitive to the filtering of high sea conditions.
Strong seasonality in mesoscale dynamics on scales of 1–100 km, driven by
turbulent scale interactions, are found in the Gulf Stream area using
numerical modeling experiments and in situ observations (Mensa et al., 2013;
Callies et al., 2015), confirming the present altimetry results. Chen and
Qiu (2021) also show this strong seasonality in the Gulf Stream region and
in region 6, west of Australia, as also obtained in our results. Conversely,
we find seasonality in the western tropical Atlantic, region 4, not reported
by the Chen and Qiu (2021) study. Overall, for regions showing seasonality in
the small 30–100 km mesoscale range that is apparently unaffected by high
sea states events, SLA variability is found to be greater in winter of each
hemisphere, consistent with stronger atmospheric being a source of enhanced
sub-mesoscale ocean dynamics (Mensa et al., 2013). For reference and
evaluation of the data, and although different dynamics may be at work, a

Mean PSD (m

Probability density function (PDF, %) of the absolute values of
IMF1 (m,

Three-year mean value of

For a processed data segment, the resulting denoised SLA segment is the
average of 20 realizations of the denoising process. An uncertainty

The probability density function (PDF) of IMF1, of the high-frequency noise, and
of

The uncertainty parameter

The spatial distribution of

In this study, we highlight key features that make our approach to denoising SLA data different and more attractive than the approaches currently used in other distributed products. It relies on (1) the EMD algorithm that adaptively splits the SLA signal into a set of empirical functions that share the same basic properties such as wavelets, but without having to conform to a particular mathematical framework; (2) a denoising algorithm that relies on a thorough and robust analysis of the local Gaussian noise affecting the SLA data over the entire wavenumber range; (3) an ensemble-average approach to estimate a robust denoised SLA signal and its associated uncertainty; (4) a calibration of the method to provide a realistic distribution of SLA variability by adjusting the mean level of the PSD function.

It is therefore useful to compare our approach with the CMEMS products, but
also with the Data Unification and Altimeter Combination System (DUACS)
experimental 5 Hz products distributed by the Aviso

For illustration, a selection of AltiKa passes in the Gulf Stream region is
shown in Fig. 11. Figure 11a shows (the same pass as in Figs. 1g and 3f) that EMD is best suited for analyzing strongly nonlinear signals in order to
accurately map the large SLA gradient (more than 40 cm in less than 50 km),
while CMEMS shows the expected limitations/artifacts due to low-pass
filtering, e.g., smoothing of gradients and poor localization of extrema.
Figure 11b and c show two passes for which small mesoscale features
(magnified in the insets) are recovered, and match well, for the SASSA and
DUACS products, while the

CMEMS unfiltered (dotted blue), SASSA (black), DUACS 5Hz (red),
and CMEMS filtered (green) SLAs (m) for different AltiKa
passes:

Figure 12 shows the PSD for the same AltiKa products and the Gulf Stream region. For the CMEMS and DUACS products, the low-pass filter applied at about 65 km (CMEMS) and 40 km (DUACS) wavelengths results in a sharp decrease in PSD with increasing wavenumber, whereas the SASSA PSD is in close agreement with the PSD obtained by removing WGN (computed as the average PSD between 15 and 30 km wavelength) from the unfiltered SLAs. The fact that SASSA products can provide a “realistic/physical” representation of the SLA variance distribution over the entire resolved wavenumber spectrum is a direct result of the chosen approach. For DUACS products, it is unclear whether the variance between 40 and 120 km wavelength corresponds to variance of unfiltered data or to variance of both unfiltered data and errors associated with HFA corrections resulting from the geophysical variability of SWH at these scales, as discussed above.

Mean PSD of SARAL SLA along-track
measurements: CMEMS unfiltered (blue), SASSA (black), DUACS 5 Hz (red), CMEMS
filtered (green), and CMEMS unfiltered minus a mean WGN
computed over 15–30 km wavelength. The PSDs are computed as the average of
PSDs obtained for all individual data segments covering the year 2017 and
the Gulf Stream region (72–60

The SASSA data set

The MATLAB code used to generate the SASSA data set is freely available on
the CERSAT website at

Satellite altimetry is certainly ideally suited to statistically characterize ocean mesoscale variability thanks to its global, repeat and long-term sampling of the ocean. In particular, the estimation of sea surface height wavenumber spectra is a key unique contribution of satellite altimetry. However, below a wavelength of about 100 km, along-track altimeter measurements can be affected by a dramatic drop in the SNR ratio. It thus becomes very challenging to fully exploit altimeter observations for analysis of SLA distributions, especially within the small mesoscale 30–120 km range.

To overcome these difficulties, and to extend previous efforts to characterize spectral distributions, an adaptive noise removal approach for satellite altimeter sea level measurements is proposed. It essentially builds on the non-parametric EMD method developed to analyze non-stationary and non-linear signals. It further exploit the fact that a Gaussian noise distribution becomes predictable after EMD. Each altimeter segment can then be analyzed to adjust the filtering process.

Applied, this data-driven approach is found to consistently resolve the distribution of the SLA variability in the 30–120 km wavelength band. A practical uncertainty variable is then attached to the denoised SLA estimates that takes into account errors in the altimeter observations as well as uncertainties in the denoising process.

Here, measurements from the Jason-3, Sentinel-3, and SARAL/AltiKa altimeters have been processed and analyzed, and their energy spectral and seasonal distributions more unambiguously characterized in the small mesoscale domain. In particular, the data-driven methodology helps to more consistently adjust the approach to local sea state conditions. Anticipating data from the upcoming Surface Water and Ocean Topography mission (e.g., Morrow et al., 2019), these denoised SLA measurements for three reference altimeter missions have already yielded valuable opportunities to assess global small mesoscale kinetic energy distributions, as well as to study possible correlation between SLA high-resolution measurements with sea state variability conditions.

Data denoising is performed on data segments of 128 continuous measurements
to limit large variations in noise statistics due to high sea state
conditions. No gap filling is performed for missing values. In addition to
the data editing performed for CMEMS products, additional outlier detection
is performed to remove the largest isolated SLA peak values. For each data
point in a segment, the difference in SLA with neighboring values is tested,
within a sliding window of five points, and its SLA value is replaced by the
average of neighboring values if the difference is greater than 4.5 times
the standard deviation of the IMF1 of the segment. For each data segment, a
reference high-frequency noise energy level,

The IMF1 processing used to estimate the high-frequency Gaussian noise is however
necessary and useful in two different parts of the denoising algorithm: (1) as discussed above, to compute

To schematize, the EMD-based denoising algorithm is applied to each data
segment as follows, with an iterative part of

perform an EMD expansion of the noisy signal

perform the IMF1 wavelet denoising to separate the IMF1 stochastic noise,

perform a reconstruction of signal

randomly modify the positions of the noise

perform an EMD expansion of

carry out the denoising of IMFs by hard thresholding with Eqs. (A1), (2),
and (3), and reconstruct a denoised signal

iterate

make an ensemble average of the

The SLA wavenumber spectra are calculated in regions of different sizes using fast Fourier transforms (FFTs), after detrending and applying a 50 % cosine taper window (Tukey window), for overlapping ground track segments of 128 continuous measurements. This corresponds to segment lengths of about 800 km or more, which is adequate for our study for which we focus primarily on the wavelength range below 120 km. Each mean spectrum is computed as the average of the individual spectra over at least the 2016–2018 period common to all three altimeters, which ensures that a sufficiently large number of segments are used.

YQ and BC conceived and organized the manuscript. YQ wrote the manuscript with inputs from all authors. YQ and JFP collected and processed the data. YQ implemented the EMD denoising method. JFP supervised the overall production of the SASSA data set.

The contact author has declared that neither they nor their co-authors have any competing interests.

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This study was conducted within the Ocean Surface Topography Science Team
(OSTST) activities. A grant was awarded to the SASSA project by the TOSCA
board in the framework of the CNES/EUMETSAT call CNES-DSP/OT 12-2118.
Altimeter data were provided by the CMEMS at

The authors are grateful to the two reviewers for their constructive comments.

This research has been supported by the Centre National d'Etudes Spatiales (grant no. CNES-DSP/OT 12-2118).

This paper was edited by Giuseppe M. R. Manzella and reviewed by two anonymous referees.