Altimeter measurements are corrected for several geophysical parameters in order to access ocean signals of interest, like mesoscale or sub-mesoscale variability. The ocean tide is one of the most critical corrections due to the amplitude of the tidal elevations and to the aliasing phenomena of high-frequency signals into the lower-frequency band, but the internal-tide signatures at the ocean surface are not yet corrected globally.

Internal tides can have a signature of several centimeters at the surface with wavelengths of about 50–250 km for the first mode and even smaller scales for higher-order modes. The goals of the upcoming Surface Water Ocean Topography (SWOT) mission and other high-resolution ocean measurements make the correction of these small-scale signals a challenge, as the correction of all tidal variability becomes mandatory to access accurate measurements of other oceanic signals.

In this context, several scientific teams are working on the development of new internal-tide models, taking advantage of the very long altimeter time series now available, which represent an unprecedented and valuable global ocean database. The internal-tide models presented here focus on the coherent internal-tide signal and they are of three types: empirical models based upon analysis of existing altimeter missions, an assimilative model and a three-dimensional hydrodynamic model.

A detailed comparison and validation of these internal-tide models is
proposed using existing satellite altimeter databases. The analysis focuses
on the four main tidal constituents: M

Since the early 1990s, several altimeter missions have been monitoring sea level at a global scale, nowadays offering a long and very accurate time series of measurements. This altimetry database is nearly homogeneous over the entire ocean, allowing the sampling of many regions that were poorly sampled or not sampled at all before the satellite era. Thanks to its current accuracy and maturity, altimetry is now regarded as a fully operational observing system dedicated to ocean and climate applications (Escudier et al., 2017).

The main difficulty encountered when using altimeter datasets for ocean
studies is related to the long revisit time of the satellites, which results
in the aliasing of high-frequency ocean signals into a much lower-frequency
band. Concerning tidal frequencies, the 9.9156 d cycle of
TOPEX/Poseidon and Jason altimeter series induces the aliasing of the
semidiurnal M

The upcoming Surface Water Ocean Topography (SWOT) mission, led by NASA, CNES, and the UK and Canadian space agencies, is planned for 2021 and will measure sea surface height with a spatial resolution never proposed before, thus raising the importance of the correction of the internal-tide surface signature. Internal tides (denoted ITs) are generated by an incoming barotropic tidal flow on a bathymetric pattern within a stratified ocean and can have amplitudes of several tens of meters at the thermocline level and a signature of several centimeters at the surface, with wavelengths ranging approximately between 30 and 250 km for the lowest three modes of variability (Chelton et al., 1998). From the perspective of the SWOT mission and of high-resolution ocean measurements in general, removing these small-scale surface signals is a challenge because we need to be able to separate all tidal signals to access other oceanic variability of interest such as mesoscale, sub-mesoscale or climate signals.

A large part of the internal-tide signal remains coherent over long times, with large stable propagation patterns across ocean basins, such as the North Pacific and many other regions (Dushaw et al., 2011). The amplitude of the coherent signal appears to be greatly diminished in the equatorial regions, which may be caused by the direct disrupting effect of the rapid equatorial wave variations (Buijsman et al., 2017) or merely masked by the background noise. The seasonal variability of the ocean medium and the interaction with mesoscale eddies and currents may also disrupt the coherence of the internal tide in many other areas, which makes the non-coherent internal tides' variability more complex to observe and model (Shriver et al., 2014).

In this context, and since conventional satellite altimetry has already shown its ability to detect small-scale internal-tide surface signatures (Ray and Mitchum, 1997; Dushaw 2002; Carrere et al., 2004), several scientific teams have developed new internal-tide models, taking advantage of the very long altimeter time series now available. These internal-tide models are of three types: empirical models based upon analysis of existing altimeter missions, usually more than one; assimilative models based upon assimilating altimeter data into a reduced-gravity model; and three-dimensional hydrodynamic models, which embed internal tides into an eddying general circulation model. In the present paper, the analysis is focused on seven models that yield a coherent internal-tide solution: Dushaw (2015), Egbert and Erofeeva (2014), Ray and Zaron (2016), Shriver et al. (2014), Clément Ubelmann (personal communication, 2017), Zaron (2019), and Zhao et al. (2016, 2019a).

The objective of this paper is to present a detailed comparison and a
validation assessment of these internal-tide models using satellite
altimetry. The present analysis focuses on the coherent internal-tide signal
for the main tidal constituents: M

After a brief description of the participating models (Sect. 2), an analysis of the differences between internal-tide models is presented in Sect. 3. Section 4 describes the altimeter dataset used, the method of comparison and the validation strategy. The validation results of the different internal-tide corrections versus altimetry databases are described in Sects. 5 and 6. Finally, a discussion and concluding remarks are gathered in Sect. 7.

This section gives a brief overview of the internal-tide models evaluated in this study. We considered five purely empirical models involving data merging, one data assimilative model and also one pure hydrodynamic model simulating tides and internal tides using the gravitational forcing and a high spatial resolution but without any internal-tide data constraint. The list of participating IT models is given in Table 1.

List of the participating IT models. Most of the models are global models except one that is currently available in only two areas (Hawaii and Azores, noted in italic). E – empirical model; A – assimilative model; H – hydrodynamic model. Abbreviations used for altimeter missions: TP – TOPEX/Poseidon; J1 – Jason-1; J2 – Jason-2; EN – Envisat; GFO – GEOSAT Follow-On; C2 – CryoSat-2; AL – AltiKa.

The purely empirical models are based upon the analysis of existing conventional altimeter missions, usually more than one. The five empirical models used in the present study are briefly described below.

This global model was computed using a frequency–wavenumber tidal analysis (Dushaw et al., 2011; Dushaw, 2015). The internal tides were assumed to be composed of narrow-band spectra of traveling waves, and these waves are fitted to the altimeter data in both time and position. A tidal analysis of a time series allows extracting accurate tidal estimates from noisy or irregular data under the assumptions that the signal is temporally coherent and described by a few known frequencies. The frequency–wavenumber analysis generalizes such an analysis to include the spatial dimension, making the strong assumptions that both time and spatial wave variations are coherent. In addition to the known tidal constituent frequencies, the solution also requires accurate values for the local intrinsic wavelengths of low-mode internal waves. Internal-tide properties, which depend on inertial frequency, stratification and depth, were derived using the 2009 World Ocean Atlas (Antonov et al., 2010; Locarnini et al., 2010) and Smith and Sandwell global seafloor topography (Smith and Sandwell, 1997). The solution is a spectral model with no inherent grid resolution; tidal quantities of interest derived from the solution are both inherently consistent with the data employed and influenced by non-local data.

The fit used M

The RAY model provides a global chart of surface elevations associated with the
stationary M

The internal-tide solution is obtained from all altimetry satellites in the
period 1990–2013, except for the CryoSat-2 mission. The method relies on a
simultaneous estimation of the mesoscales and coherent M

The method is further described in Ubelmann et al. (2020). Further improvements are expected after introducing additional tide components in the same inversion and after considering slow (or seasonal) variation in the phases.

The High Resolution Empirical Tide (HRET) model provides an empirical
estimate for the baroclinic tides at the M

HRET was initially developed to evaluate plausible spatial models for the baroclinic tides, seeking ways to improve on some previous models (Zhao et al., 2012; Ray and Zaron, 2016). It uses a local representation of the wave field as a sum of waves modulated by an amplitude envelope consisting of a second-order polynomial, thus generalizing the spatial signal model used in previous plane wave fitting (Ray and Mitchum, 1997; Zhao et al., 2016). The details of the implementation in HRET differ in additional ways from previous approaches. Specifically, the wavenumber modulus and direction of each wave component are determined by local two-dimensional Fourier analysis of the along-track data, and the coefficients in the spatial model are determined by weighted least-squares fitting to along-track slope data – the latter removes the need for rather arbitrary along-track high-pass filtering used in other estimates. Hence, the model is fully empirical in the sense that it does not use an a priori wavenumber dispersion relation.

The above-described approach to building local models for the baroclinic
waves is applied to overlapping patches of the ocean, which are then blended
and smoothly interpolated on a uniform latitude–longitude grid. Using the
standard error estimates from the original harmonic analysis and
goodness-of-fit information from the spatial models, a mask is prepared
which smoothly damps the model fields to zero in regions where the estimate
is believed to be too noisy to be useful. These are generally regions near
the coastline where the number of data used are reduced or regions in
western boundary currents or the Southern Ocean where the baroclinic tides
cannot be distinguished from the continuum of energetic mesoscale
variability. HRET version 7.0 was provided for the present validation
analysis. Note that the model is still being refined and version 8.1 is
available at present: it has improved O

This model is constructed by a two-dimensional plane wave fit method (Zhao
et al., 2016). In this method, internal tidal waves are extracted by fitting
plane waves using SSH (sea surface height) measurements in individual fitting windows (160 km by
160 km for M

Gary D. Egbert and Svetlana Y. Erofeeva have developed a reduced-gravity (RG) data
assimilation scheme for mapping low-mode coherent internal tides (Egbert and
Erofeeva, 2014) and applied this to a multi-mission dataset to produce
global first mode M

Within the RG scheme used, the vertical-mode coupling terms are dropped to
obtain independent equations for the propagation of each mode with spatially
variable reduced water depth, which are determined from local bathymetry and
stratification. These simplified equations are identical to the linear
shallow-water equations used in the OSU Tidal Inversion Software (OTIS,

This OTIS-RG assimilation scheme has been applied to construct global maps
of first mode temporally coherent internal-tide elevations. Available exact
repeat mission data, except GFO, were assimilated
(TP-Jason, ERS/Envisat), with the AVISO weekly gridded SSH product used to
reduce mesoscale variations before harmonic analysis. Solutions are computed
in overlapping patches (

The hydrodynamic internal-tide solution is provided by the three-dimensional
ocean model HYCOM (HYbrid Coordinate Ocean Model), which embeds tides and
internal tides into an eddying general circulation model (Shriver et al., 2014). A free simulation, i.e., without any data assimilation, is used for
the present study; this run used an augmented state ensemble Kalman filter
(ASEnKF) to correct the forcing and reduce the M

A first analysis of the model differences consists in visualizing the patterns of IT models' amplitude in the regions of interest defined in Fig. 1. These seven regions are characterized by a well-known and nearly permanent internal-tide signal, already pointed out by previous studies (Egbert et al., 2000; Carrere et al., 2004; Nugroho, 2017). From the seven regions of interest, the North Pacific area (NPAC) and Luzon regions were selected for the comparison hereafter because they are more energetic regions; moreover, all tested models are available in the NPAC region and the Luzon area is characterized by strong semidiurnal and diurnal baroclinic tides.

Localization of the internal-tide regions studied in the present paper.

Figure 2 shows the M

M

Amplitude of the IT models for the M

Amplitude of the IT models for the M

Figure 4 shows the amplitude of the three IT solutions available for the K

Concerning diurnal tides in the global ocean, the ZARON solution is not
defined over large regions of the world ocean, including latitudes poleward
of the diurnal-tide critical latitude and regions where the IT amplitude is
negligible and/or not separable from background ocean variability. The ZHAO
solution stops at the diurnal critical latitude, while the EGBERT solution
is defined over a wider range of latitudes (until 60

Amplitude of the IT models for the K

Following Stammer et al. (2014), the standard deviation (SD) of all the IT
models listed in Table 1 was computed for each tidal constituent with
respect to elevation

The computation of the SD

Global maps shown in Figs. 5 and 6 illustrate the mean amplitude and the
standard deviation of the M

The map of M

Global maps of mean amplitude of the M

Global maps of standard deviation of the M

The mean standard deviation value is computed over the different regions
studied. In order to eliminate any residual barotropic variability likely
existing in the empirical IT models in shallow waters, only data located in
the deep ocean are used to compute the standard deviation; values are gathered
in Table 2. Over all regions, the standard deviation is stronger for M

The diurnal K

The altimeter measurements used correspond to the level-2 altimeter product
L2P, with 1 Hz along-track resolution (LRM), produced and distributed by
Aviso

The altimeter period from 1993 onwards is sampled by 12 altimeter
missions available on different ground tracks (

Jason-2 (denoted J2 in the text and figures) is a reference mission flying in the reference TP track with a 10 d cycle and sampling latitudes between

CryoSat-2 (denoted C2 hereafter) is characterized by a drifting polar orbit sampling all polar seas and it has a nearly repetitive sub-cycle of about 29 d.

The mission's time series and the number of cycles used for the present study are listed in Table 3. It is worth pointing out that much of the T/P and Jason data have been used in most of the IT empirical solutions tested (see Table 1), but all models are independent of CryoSat-2 mission data.

Due to sub-optimal time sampling, altimeters alias the tidal signal to
much longer periods than the actual tidal period. The aliased frequencies of
the four main tidal waves studied are listed in Table 3 for the two orbits used.
It is noticeable that the diurnal tide K

Spatial-mean SD (cm) of the M

Description of the altimeter database for the validation study, along with the associated aliasing periods for the main tidal components.

The altimeter sea surface height (SSH) is defined as the difference between
orbit and range, corrected from several instrumental and geophysical
corrections as expressed below:

tide includes the geocentric barotropic tide, the solid Earth tide and the
pole tide corrections. The geocentric barotropic tide correction was updated
compared to the altimetry standards listed in Pujol et al. (2016) and comes
from the FES2014b tidal model (

IT is the internal-tide correction, taken one by one from each model studied in this paper;

other_corr includes the dynamic atmospheric correction, the wet tropospheric correction, the dry tropospheric correction, the ionospheric correction, the sea state bias correction and complementary instrumental corrections when needed, as described in Pujol et al. (2016).

The sea level anomaly (SLA) is defined by the difference between the SSH and
a mean profile (MP) for repetitive orbits or a mean sea surface (MSS) for
drifting orbits. Mean profiles computed for TOPEX or the Jason orbit for the
reference period of 20 years (1993–2012), have been used within the present
study for the J2 mission (Pujol et al., 2016), and the MSS_CNES_CLS_11 also referenced on the same 20
years period was used for the C2 drifting orbit mission (

Satellite altimetry databases can be used to evaluate many geophysical corrections and particularly global barotropic tidal models as already examined by other authors (Stammer et al., 2014; Carrere et al., 2016a, b; Lyard et al., 2006; Carrere, 2003). We propose using a similar approach to validate the concurrent IT models listed in Table 1.

First, we generate the corresponding IT correction for each along-track altimeter measurement, computed from the interpolation of each IT atlas onto the satellites' ground tracks and the use of a tidal prediction algorithm. Each tidal component is considered separately for the clarity of the analysis, keeping in mind that the various IT models do not all contain the same waves.

Second, the altimeter SSH using IT corrections from each model tested can then be computed, and the differences in the sea level contents are analyzed for different time and spatial scales. In particular, considering several altimeters allows the study of different temporal periods. As the missions considered, J2 and C2, have different ground tracks and different orbit (cycle) characteristics, several aliasing characteristics are tested.

Third, the impact of each IT model on SSH can be estimated for short
temporal scales (time lags lower than 10 d), which are the main concern
here as we consider the main high-frequency tidal components M

Fourth, the variance of SSH differences at crossover points is computed on
boxes of 4

Fifth, along-track SLA statistics can be calculated from 1 Hz altimetric
measurements and allow for a higher spatial resolution in the analysis. The
maps of the variance difference of SLA using either the IT correction tested
or the reference ZERO correction are computed on boxes of 2

Sixth, the mean of these variance reduction estimations at crossovers and for along-track SLA is computed for each studied region, which allows an easier analysis and comparison of the performances of the IT model tested.

Finally, in order to quantify the impact of each IT model on the SLA variance reduction in terms of spatial scales, a spectral analysis of J2 SLA is performed in the different regions of interest, and details are given in Sect. 6.

This section gathers the validation results of each IT model using the
satellite altimetry databases described previously. For the clarity of the
analysis, each IT correction is compared to a reference correction using a
ZERO correction. For the ZERO correction, no IT correction is applied, as in
the actual altimeter geophysical data records version-D and version-E (GDR-D and GDR-E) processing (Pujol et al., 2016; Taburet
et al., 2019). The complete diagnostics and analysis are presented hereafter
for the largest semidiurnal (M

To investigate and quantify the regional impact of the M

Figure 8 displays the maps of along-track J2 SLA variance differences using
the M

Maps of SSH variance differences at crossovers using either the M

Maps of SLA variance differences using either the M

One should note that those J2 results might be biased in favor of the empirical models, as J2 data are used in all of them except for the DUSHAW model (see Table 1). To check these results, similar diagnostics are computed using the C2 altimeter database, as described in Sect. 4.1, which is an independent database for all models. Validation results are given in Figs. 9 and 10 for C2 SSH crossover differences and C2 SLA, respectively.

Validations with the C2 database show similar results as for J2, with a
significant variance reduction in the C2 SSH differences and SLA for most
models in all IT regions; variance gain patterns are generally similar but
more widely spread and stronger in C2 SSH maps compared to J2 particularly in the
Atlantic Ocean and in the west Pacific. The pattern is different for the
UBELMANN model in the NATL region, likely due to some inclusion of J2
errors or signal or larger-scale signals in the model (see Sect. 6). The
ground track pattern of the C2 orbit explains the lack of crossover data at
0

Maps of SSH variance differences at crossovers using either the M

Maps of SLA variance differences using either the M

Mean values for C2 data, averaged over the strong IT regions, are also given
in Table 4. Mean C2 SLA variance gains are comparable to J2 mission on all
IT regions. C2 validation results for the M

Mean variance reduction for the J2 and C2 altimeter databases, within
each IT region, when using the different M

The maps of K

Maps of SSH variance differences at crossovers using either the K

The maps of SLA variance differences using the EGBERT, ZARON and ZHAO K

Maps of SLA variance differences using either the K

The mean statistics of altimeter variance reduction, over the regions
defined in Fig. 1, are given in Table 5 for the SLA and the SSH
differences of J2 and C2 missions and for the different regions studied;
note that we focus on Luzon, Tahiti, Hawaii, Madagascar and global areas
because mean K

Mean variance reduction for J2 and C2 altimeter databases, within
each IT region, when using the different K

In order to quantify the impact of each IT model on the altimeter SLA
variance reduction as a function of spatial scales, a spectral analysis of
J2 along-track SLA is performed. This analysis is not conducted for other
missions because the duration of the C2 mission time series used is too
short to allow a proper spectral estimation at the aliasing frequency of M

The J2 SLA spectral analysis is performed for each of the IT regions
described in Fig. 1. For each area, a frequency–wavenumber spectrum is
computed for the along-track SLA and for the SLA corrected from each IT
solution; the spectral density at a 62 d frequency, which is the aliasing
frequency band of the M

Results for the different regions are gathered in Fig. 13 and show that
all empirical models generally manage to remove an important amount of
coherent IT energy for the first mode (wavelengths of about 150 km):
the reduction in energy reaches about 50 %–80 % depending on the area. Some
empirical models also perform well for shorter scales. The DUSHAW model is
generally less efficient in the different regions except in the Gulf of
Guinea where it is as efficient as others for the first mode. In the Tahiti,
Luzon, Gulf of Guinea and NATL regions, ZARON is the most efficient model
with a very significant reduction in the energy for the first and the second
IT modes: the ZARON model removes 80 % of the energy at the M

Normalized difference of the power spectral density of J2 SLA as a function of wavelength and for each IT region studied. Blue line – DUSHAW model; green – EGBERT model; red – HYCOM model; light blue – RAY model; purple – ZARON model; light green – ZHAO model; black – UBELMANN model.

The black curves show the performances of the UBELMANN model in the NATL and NPAC regions: it is very efficient in NPAC with a similar energy reduction as the ZARON model for the first and second modes, and it also removes some signal at shorter scales. In the NATL area, the UBELMANN model seems to be more efficient than all other models for all wavelengths and also for large scales, which likely indicates that the model also includes some large-scale signals which are not internal tides but rather some residual barotropic tide signals or even some non-tidal ocean signal aliasing.

The assimilative model, EGBERT, has performances comparable to the purely empirical models for the first mode, but it does not have enough energy for the shorter IT modes except for two regions: for the Madagascar region the EGBERT model reduces the SLA energy for scales of 60–70 km, and for the Gulf of Guinea region it reduces energy in shorter modes compared to other models (scales shorter than 60 km).

It is also interesting to point out that the pure hydrodynamic model, HYCOM, removes energy for the three first IT modes in some of the regions studied: although weaker than for the empirical models, the HYCOM gain reaches 55 % for the first mode, 40 % for the second mode and 15 % for the third mode in the Tahiti area. The gain is weak but noticeable in the NATL, NPAC, Luzon and Madagascar regions, but the local rise of energy in some regions also indicates that the hydrodynamic model still has some uncertainties, particularly in the Gulf of Guinea region and for short IT scales in the Madagascar region.

Seven models of the coherent IT surface signature have been extensively compared within the present study: Dushaw (2015), Egbert and Erofeeva (2014), Ray and Zaron (2016), Shriver et al. (2014), Clément Ubelmann, personal communication, Zaron (2019), and Zhao et al. (2016). They are of three types: empirical models based upon analysis of existing altimeter missions, an assimilative model and a three-dimensional hydrodynamic model.

Recently updated Jason-2 and CryoSat-2 altimeter databases have been used to
validate these new models of coherent internal tides over the global ocean,
focusing on the four main IT frequencies: M

All empirical models display generally good performance for M

The assimilative model (EGBERT) has performances comparable to the empirical models, but it also removes some variability in regions of strong currents, likely due to some remaining mesoscale variability in the assimilated data.

The hydrodynamic solution, computed from a HYCOM simulation, is also able to reduce some of the internal-tide variability in most of the IT regions studied, which is a very encouraging result. However, the analysis indicates that it is not yet mature enough to be compared to empirical models. The HYCOM solution has stronger amplitudes compared to the other models, which is likely due to the effects of the relatively short HYCOM time series duration (1 year) on the IT estimation (see Ansong et al., 2015). Indeed, some tests showed that using a reduction coefficient (Buijsman et al., 2020) that accounts for the short duration of the time series used in the analysis slightly improves the performance of the HYCOM hydrodynamic solution. Ongoing work is testing whether operational HYCOM simulations, which assimilate altimeter measurements of mesoscale eddies and improve the underlying stratification relative to observations (e.g., Luecke et al., 2017), will yield improvements in the skill of the predicted internal tides in HYCOM.

The results described here and for which we provide a scientific
justification, have been also presented at the last OSTST (Ocean Surface
Topography Science Team) meetings of Ponta Delgada Miguel (2018; program available at

In addition, the impact of using the ZARON IT correction has also been estimated for the level-4 (L4) altimeter products, which are global gridded data. A significant improvement was detected in all the regions of interest, and it was demonstrated that this new correction reduces the remaining IT signal in the L4 AVISO/CMEMS (Copernicus Marine Environment Monitoring Service) products (Faugère et al., 2019; Zaron and Ray, 2018). Accordingly, this IT correction will be used to compute the SLA for the next Duacs reprocessing product Duacs-2021, which is currently being undertaken. Moreover, the implementation of this new IT correction is planned in the future CMEMS L3 and L4 altimeter product version coming in 2021.

The present study indicates that the use of the altimetry database is a valuable tool to validate models of IT surface signature in the global ocean. It particularly complements the in situ validation processes which are generally more localized in space or time due to the availability of in situ datasets (Dushaw et al., 1995, 2017; Dushaw, 2006, 2015; Zaron and Ray, 2018).

Within the SWOT mission preparation, several teams pursue ongoing efforts concerning the better understanding and modeling of IT in the global ocean, and the work presented here could help validate the new model solutions produced. The perspectives of improvement of IT models concern the coherent internal tides through the inclusion of higher IT modes and more tidal frequencies. Many initiatives are also being conducted to try to better understand and model the non-stationary component of the internal tides. Work is progressing on the modeling of the seasonal and interannual internal-tide variability: Zhao (2019b), Zaron (2019), Richard D. Ray (personal communication, 2019) and Clément Ubelmann (personal communication, 2020). And within the SWOT Science Team and other projects, several teams are also working on 3D simulations using different general circulation models such as HYCOM, MITgcm, NEMO (CMEMS–Mercator–Ocean, project in progress) or even a specific spectral approach (Barbot et al., 2020).

Amplitudes of O

In this region, the S

Global maps shown in Figs. A3 and A4 illustrate the mean IT amplitude and
the standard deviation of the IT models for the O

Amplitude of the IT models for the O

Spatial-mean SD of models for S

For O

Amplitude of the IT models for the S

Global maps of mean amplitude of the O

Global maps of standard deviation of the O

The maps of SLA and crossover variance differences using each of the three
different O

The three models remove a significant amount of J2 SLA variance mostly in
the Luzon Strait or west Pacific region where the amplitude of the O

Mean variance reduction for J2 and C2 altimeter databases, in the
Luzon region, when using the different O

The mean statistics of altimeter variance reduction for O

Maps of SLA variance differences using either the O

Maps of SSH variance differences at crossovers using either the O

The maps of SLA and crossover variance differences using each of the three
different S

Using the three models for S

The mean statistics of the altimeter variance reduction are gathered in
table C3 for the SLA and the SSH differences of the J2 and C2 missions and
for the different regions studied; the analysis focuses on Tahiti, Hawaii,
NPAC, Madagascar and the Luzon areas because mean S

Mean variance reduction for J2 and C2 altimeter databases, within
each IT region, when using the different S

Maps of SLA variance differences using either the S

Maps of SSH variance differences at crossovers using either the S

Level 2P (L2P) altimetry products, with 1 Hz along-track resolution (LRM), are produced and distributed by Aviso

The supplement related to this article is available online at:

BKA, BD, GE, SE, RDR, CU, EZ, ZZ, JFS and MCB provided the different IT models that have been used for the study. LC designed the experiments and carried them out with the help of the CLS team. LC analyzed the results with contributions from all co-authors. LC prepared the paper with contributions from all co-authors.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Developments in the science and history of tides (OS/ACP/HGSS/NPG/SE inter-journal SI)”. It is not associated with a conference.

This work has been performed within the framework of the SWOT-ADT (Algorithm Definition Team) and funded by CNES. We thank Romain Baghi for his help in the processing.

This paper was edited by Mattias Green and reviewed by C. K. Shum and one anonymous referee.