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: M2, K1, O1 and S2. The validation
process is based on a statistical analysis of multi-mission altimetry
including Jason-2 and Cryosphere Satellite-2 data. The results show a
significant altimeter variance reduction when using internal-tide
corrections in all ocean regions where internal tides are
generating or propagating. A complementary spectral analysis also gives some
estimation of the performance of each model as a function of wavelength and
some insight into the residual non-stationary part of internal tides in the
different regions of interest. This work led to the implementation of a new
internal-tide correction (ZARON'one) in the next geophysical data records version-F (GDR-F) standards.
Introduction
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 M2 lunar tide into a 62 d period, and the diurnal K1 tide is
aliased into a 173 d period, the latter of which is very close to the
semiannual frequency and raises complex separation problems. The long
duration of the global ocean altimeter database available has allowed the
community to overcome this separation problem, and new global ocean
barotropic tidal solutions (Stammer et al., 2014) have been produced taking
advantage of altimeter data: among them the last Goddard/Grenoble Ocean Tide
model (denoted GOT: Ray, 2013) and the last finite-element solution for ocean
tide (denoted FES2014: Carrere et al., 2016a; Lyard et al., 2020), which are commonly
used as reference for the barotropic tide correction in actual altimeter
geophysical data records (denoted GDRs). Moreover this altimeter database has
been used in numerous studies to validate new instrumental and geophysical
corrections used in altimetry, thanks to the analysis of their impact on the
sea level estimation at climate scales, as well as at lower temporal scales
like mesoscale signals; in particular, it has proven its efficiency for
validating global ocean models (Shum 1997; Stammer et al., 2014; Carrere et al., 2016b; Quartly et al., 2017).
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: M2, S2, K1 and O1. The validation process
is based on a statistical analysis and on a comparison to multi-mission
altimetry including Jason-2 (denoted J2 hereafter) and Cryosphere Satellite-2
(also named CryoSat-2 or C2 hereafter) LRM data (low-resolution mode). For
the sake of clarity, only results for the main tidal components M2 and K1
are presented in the core of this paper, and O1 and S2 validation results
are gathered in the Appendix.
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.
Presentation of participating internal-tide models
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.
Model nameType ofGrid resolutionConstituents testedAltimeter data usedAuthorsmodelprovidedDUSHAWE0.05∘M2TP + J1Dushaw (2015)EGBERTA0.03∘M2, K1, O1, S2ERS-EN + TP-J1-J2Egbert and Erofeeva (2014, 2002)HYCOMH0.08∘M2No data assimilatedShriver et al. (2014)RAYE0.05∘M2GFO + ERS-EN + TP-J1-J2Ray and Zaron (2016)UBELMANNE0.1∘M2All except C2Ubelmann et al. (2020)ZARON (HRET)E0.05∘M2, K1, O1, S2TP-J1-J2 + ERS-EN-AL + GFOZaron (2019)ZHAOE0.1∘M2, K1, O1, S2GFO + ERS-EN + TP-J1-J2Zhao et al. (2016)Empirical models
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.
DUSHAW
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 M2, S2, N2, K2, O1 and K1 constituents, with spectral bands
for barotropic, mode-1 and mode-2 wavenumbers. Data from the TOPEX/Poseidon (TP) and Jason-1
altimetry programs were employed. These data had the barotropic tides
removed, but the fit allowed for residual barotropic variations. Employing
all constituents and wavelengths simultaneously in a single fit minimized
the chance that the solution for a particular constituent was influenced by
noise from nearby tidal constituents. To account for regional variations in
the internal-tide characteristics (and reduce computational cost)
independent fits were made in 11∘× 11∘
overlapping regions. The global solution was obtained by merging the
regional solutions together using a cosine taper over a 1∘
interval; the solution is therefore sometimes discontinuous within these
overlapping zones. For this study, global maps of the harmonic constants for
the two first baroclinic modes of the largest semidiurnal tidal constituent
M2 were computed on a regular 1/20∘ grid (Dushaw, 2015; the
complete solution is available from http://www.apl.washington.edu/project/project.php?id=tm_1-15, last access: 17 December 2020). This global M2 solution was tested against
pointwise, along-track estimates for the internal tide, with satisfactory
comparisons in the Atlantic and Pacific oceans. Comparisons were also made
to in situ measurements by ocean acoustic tomography in the Pacific and
Atlantic, showing a good predictability in both amplitude and phase. By
comparisons to the tomography data, internal tides within the Philippine Sea
(Dushaw, 2015) or Canary Basin (Dushaw et al., 2017) were less predictable.
Some of these comparisons found good agreement between hindcasts and time
series recorded in the western North Atlantic about a decade before the
altimetry data were available, which is consistent with the extraordinary
temporal coherence of this IT signal in many regions of the world's oceans.
RAY
The RAY model provides a global chart of surface elevations associated with the
stationary M2 internal-tide signal. This map is empirically constructed from
multi-mission satellite altimeter data, including GFO (GEOSAT Follow-On), ERS (European Remote Sensing satellite), Envisat,
TOPEX/Poseidon, and the J1 and J2 missions. Although the present-day altimeter
coverage is not entirely adequate to support a direct mapping of very
short-wavelength features such as surface internal-tide signatures, using
an empirical mapping approach produces a model that is independent of any
assumption about ocean wave dynamics. The along-track data from each
satellite mission were subjected to tidal analysis, and the M2 fields were
high-pass-filtered to remove residual noise from barotropic and other
long-wavelength modeling errors. Filtered data from all mission tracks were
then interpolated to a regular grid. The complete description of the
methodology is described in Ray and Zaron (2016, Sect. 3). Validation
using some independent data from CryoSat-2 showed a positive variance
reduction in most areas except in regions of large mesoscale variability,
due to some contamination from non-tidal ocean variability in these last
regions (Ray and Zaron, 2016). In the model version used in the present study,
those regions have been masked with a taper to give zero elevation. The
model grid has a 1/20∘ resolution and it is defined over the
50∘ S–60∘ N latitude band.
UBELMANN
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 M2 internal tides.
Indeed, the mesoscale signal is known to introduce errors in the tidal
estimation (non-zero harmonics in a finite time window). To mitigate that
issue, most existing methods subtract the low-frequency altimetry field from
AVISO (Archiving, Validation and Interpretation of Satellite Oceanographic data) as a proxy for mesoscales (e.g., Ray and Zaron, 2016). However, the
estimate of the mesoscale is itself contaminated by internal tides (e.g.,
Zaron and Ray, 2018) aliased into a low frequency, which also introduces
errors. For these reasons, Clément Ubelmann (personnal communication, 2017) proposed here a simultaneous estimation,
accounting for the covariances of mesoscales and internal tides in a single
inversion. In practice, these covariances are expressed as a reduced wavelet
basis (local in time and space) for mesoscales and as a plane wave basis
(local in space only) for internal tides. The plane wave wavelength and
phase speed rely on the first and second Rossby radii of deformation climatology
by Chelton et al. (1998). Although the inversion cannot be done explicitly
(because of the long time window extending the basis size for the mesoscale), a
variational minimization allows for a converged solution after about 100
iterations (typical degree of freedom for the problem). For this study, only
the M2 internal-tide solution (for mode 1 and mode 2) is considered, but the
mesoscale solution is also of interest because the internal-tide
contamination should be minimized compared to the standard AVISO processing.
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.
ZARON
The High Resolution Empirical Tide (HRET) model provides an empirical
estimate for the baroclinic tides at the M2, S2, K1 and O1 frequencies, as well
as the annual modulations of M2, denoted MA2 and MB2. The development of
HRET begins with assembling time series of essentially all the exact-repeat
mission altimetry along the reference and interleaved orbit ground tracks of
the TOPEX/Poseidon–Jason missions, the ERS–Envisat–AltiKa missions and
the GEOSAT Follow-On mission (Zaron, 2019). Standard atmospheric path delay
and environmental corrections are applied to the data, including the removal of
the barotropic tide using the GOT4.10c model and the removal of an estimate for
the mesoscale sea level anomaly using a purpose-filtered version of the
Ssalto/Duacs multi-mission L4 sea level anomaly product (Zaron and Ray,
2018). Conventional harmonic analysis is then used to compute harmonic
constants at each point along the nominal 1 Hz ground tracks (Carrere et
al., 2004), and these data are used as inputs for subsequent steps.
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 O1 relative to the results shown here and made minor changes to the other constituents.
ZHAO
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 M2). Prerequisite wavenumbers are calculated using climatological
ocean stratification profiles. For each window, the amplitude and phase of
one plane wave in each compass direction (angular increment 1∘)
are determined by the least-squares fit. When the fitted amplitudes are
plotted as a function of direction in polar coordinates, an internal tidal
wave appears to be a lobe. The largest lobe gives the amplitude and
direction of one internal tidal wave. The signal of the determined wave is
predicted and removed from the initial SSH measurements. This procedure can
be repeated to extract an arbitrary number of waves (three waves here). Four
tidal constituents M2, S2, O1 and K1 are mapped separately using their
respective parameters and are used in the present paper (model version
Zhao16). This mapping technique dynamically interpolates internal tidal
waves at off-track sites using neighboring on-track measurements, overcoming
the difficulty posed by widely spaced ground tracks. There are a large
number of independent SSH measurements in each fitting window, compared to a
single time series of SSH measurements used by pointwise harmonic analysis.
As a result, non-tidal noise caused by tidal aliasing can be efficiently
suppressed. This technique resolves multiple waves of different propagation
directions; therefore, the decomposed internal-tide fields may provide more
information on generation and propagation.
Assimilative model
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 M2 and K1 solutions. This scheme is based on the
Boussinesq linear equations for flow over arbitrary topography with a free
surface and horizontally uniform stratification. As in Tailleux and
McWilliams (2001) and Griffiths and Grimshaw (2007), vertical dependence of
the flow variables are described using flat-bottom modes (which depend on
the local depth H(x,y)), yielding a coupled system of (two-dimensional) partial differential equations (PDEs) for the modal coefficients for surface elevation and horizontal velocity.
Equations for each mode are coupled through interaction coefficients, which
can be given in terms of the vertical-mode eigenvalues following the
approach of Griffiths and Grimshaw (2007). Modes are decoupled wherever
bathymetric gradients are zero, and for a flat bottom the system reduces to
the usual single-mode RG shallow-water equations.
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,
https://www.tpxo.net/otis, last access: 17 December 2020; Egbert and Erofeeva, 2002), thus
allowing the use of the assimilation system to map internal tides by simply
modifying depth and fitting along-track harmonic constants as a sum over a
small number of modes. With some extensions to OTIS, coupling terms for the
first few modes can be included in the dynamics.
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 (∼ 20∘× 30∘) and then merged
(via weighted average on overlaps) into a global solution. It may be noted that adjacent solutions almost always match quite well even without this
explicit tapering.
Hydrodynamic model
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 M2 barotropic tidal error to
about 2.6 cm (Ngodock et al., 2016). The value of such a simulation is to
provide some information about the interaction of internal tides with
mesoscales and other oceanic signals in addition to the internal-tide
signal itself, which means that it can give access to the non-coherent
internal-tide signal too. For the present study, a 1-year simulation
(simulation no. 102 on year 2014) has been run and a harmonic
analysis of the steric 1 h SSH allowed the extraction of the M2 internal-tide
signal which remains coherent in this period. The non-assimilative quality
of the simulation makes it entirely independent from the altimeter database
used for the validation. The spatial resolution of the native grid is
1/24∘, but data have been interpolated on a 1/12.5∘ grid
to provide the tidal atlas for the present analysis.
Comparison of internal-tide modelsQualitative comparison of IT elevations
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 M2 IT amplitude of each model in the NPAC located around the Hawaiian islands. In this region, all models have
similar amplitudes and similar beam patterns demonstrating northeastward
propagation with one clear northward beam; amplitudes are often greater than
2 cm. The amplitude's pattern varies along IT beams with short spatial
scales, indicating that most of the models capture a part of the higher-order IT modes: typical 70 km patterns are visible corresponding to the
second M2 IT mode wavelength in this region. The ZHAO solution shows
cleaner and smoother patterns likely due to the theoretical plane wave
approximation used for the estimation. RAY, ZHAO and EGBERT propagate until
150∘ W, while ZARON propagates farther to the east and EGBERT has
the most attenuated amplitudes in the region. The UBELMANN and DUSHAW models
show similar patterns but both maps are noisier compared to other solutions.
HYCOM also shows similar beams but with clearly stronger amplitudes, and
some noise is also noticeable in the maps.
M2 IT amplitudes in the Luzon region are plotted in Fig. 3. Only six models
are plotted as UBELMANN is not defined for this area. The models have an M2
amplitude greater than 2 cm in the Luzon region, and HYCOM has stronger
amplitudes than the other models. The IT propagation pattern also shows
small spatial scales (of the order of 100 km eastward of the strait)
indicating that higher IT modes are also enhanced at the semidiurnal
frequency, but the models do not agree on a clear common pattern: DUSHAW has
a rather noisy structure and a discontinuity appears along longitude
125∘ E due to the effect of the different computational patches
used to estimate the global solution. All other models show a strong M2
amplitude across the Luzon Strait; on the east side of the strait, two beams
northward and southward along Taiwan and the Philippines, respectively, are visible, and a wide eastward beam is visible in the ZARON, ZHAO
and HYCOM maps. The patterns are noisier for the EGBERT and RAY solutions.
The ZARON and HYCOM solutions are close to zero in shallow waters, while
RAY, ZHAO and EGBERT are not defined; DUSHAW is defined in shallow waters
showing some propagation patterns, but one must be careful as an empirical
model might have difficulties separating IT surface signatures from small
scales of barotropic tides occurring in these areas. At the strait itself
the main wave propagation is expected to be predominantly in the west and/or
east directions, which is challenging for empirical techniques to recover
owing to the primarily north–south altimeter track orientations. The problem
was discussed in some detail by Ray and Zaron (2016), and their model does indeed have very little eastward-propagating energy from the strait (see also Zhao,
2019a). Plots of the M2 IT for other regions defined in Fig. 1 are provided
as a Supplement.
Amplitude of the IT models for the M2 tide component in the NPAC region (north Hawaii).
Amplitude of the IT models for the M2 tide component in the Luzon area.
Figure 4 shows the amplitude of the three IT solutions available for the K1 wave in the Luzon region, where amplitudes of the diurnal IT are the most important.
Models show large-scale (about 200 km or more) patterns on both sides of the
Luzon Strait. The K1 scales are greater than the M2 scales as expected from
theoretical wavelengths. The K1 amplitude reaches 2 cm on the west side,
while patterns and amplitudes of the models differ on the east side of the
strait: ZHAO has weaker amplitudes and some different spatial patterns,
while ZARON and EGBERT have the solutions that lie closest to one to each
other. For these three models, the amplitude of K1 becomes zero at about
24∘ N when getting close to the K1 critical latitude.
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 K1 tide component, in the Luzon
area.
Quantitative comparison of IT models
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 ηj=ξje-iσt, where ξj is
the time-independent amplitude of a tide component at a wet grid point j,
σ is tidal frequency and i=-1. First, the mean elevation
of each tidal constituent across models taken into account (N) is computed
at every grid point according to
ηmean=1N∑j=1Nξje-iσt=HmeancosGmean+isinGmeane-iσt,
where ξj=HjcosGj+isinGj with Hj the
amplitude and Gj the Greenwich phase lag of the tide considered. Then
the SD between all involved models (N) can be computed for each constituent
at each grid point according to
SDtide=1N∑n=1N1T∫0TRe(ηn-ηmean)2dt12=(1N∑n=1N12[(HncosGn-HmeancosGmean)2+HnsinGn-HmeansinGmean2])1/2,
where Hn and Gn are the amplitude and the Greenwich phase lag of a
constituent given by each model, respectively, and Hmean and Gmean
are the mean amplitude and Greenwich phase lag computed from all models from
Eq. (1).
The computation of the SDtide was performed for the four tidal
constituents M2, S2, K1 and O1, after re-gridding bilinearly the models to a
common 1/20∘ grid. The maps of SD are computed over the global
ocean. Note that the DUSHAW model was not included in this SD calculation,
as it increases too much the SD value over the global ocean due to noisier
patterns in wide regions and makes the results difficult to analyze.
Global maps shown in Figs. 5 and 6 illustrate the mean amplitude and the
standard deviation of the M2 and K1 IT models, respectively. Near-coastal
regions, shallow-water regions and regions of low signal to noise are
masked-out in the maps as they are not defined in most of the studied
models. The mean M2 amplitudes reach more than 2 cm in all the known
generation sites – in the Pacific, the Indian Ocean around Madagascar, the
Indonesian Seas and in the Atlantic offshore of Amazonia. K1 has a
significant mean amplitude above 1.5 cm in the Luzon Strait region, in the
Philippine Sea and east of Palau and about 0.5–0.7 cm in some regions of
the Indian and Pacific oceans.
The map of M2 SD shows small values, generally below 1 cm for M2,
indicating good agreement of the IT models in all IT regions defined in
Fig. 1 for the M2 wave; the ratio SD / mean amplitude for the M2 wave reaches
only 0.2–0.3 around IT generation regions with some clear beam patterns
indicating that models agree with each other in those areas. Some larger SD
values are found around Luzon Strait, above Madagascar and in the Indonesian
seas. For the diurnal wave K1, IT models provide coherent information in the
Luzon region, in Tahiti and Hawaii and in the Madagascar region.
Global maps of mean amplitude of the M2(a) and K1(b) IT
models (cm).
Global maps of standard deviation of the M2(a) and K1(b) IT models (cm).
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 M2,
consistent with the fact that M2 is the most important IT component in the
global ocean. The standard deviation is largest in the Luzon and Madagascar
regions, where models give rather different solutions as already seen in the
previous section.
The diurnal K1 tide takes on the largest standard deviation value, of 0.25 cm, in the Luzon region, where this diurnal component has the most
significant amplitudes.
Presentation of the altimeter database and the method of comparisonThe altimeter database
The altimeter measurements used correspond to the level-2 altimeter product
L2P, with 1 Hz along-track resolution (LRM), produced and distributed by
Aviso+ (https://www.aviso.altimetry.fr/en/data/, last access: 17 December 2020
AVISO), as part of the Ssalto ground processing segment. The version of the
products considered is nearly homogeneous with the DT-2014 standards
described in Pujol et al. (2016), except for the tide correction as described
below.
The altimeter period from 1993 onwards is sampled by 12 altimeter
missions available on different ground tracks (https://www.aviso.altimetry.fr/en/missions.html, last access: 17 December 2020). For the purpose of the
present study, we use the databases for two different missions:
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
± 66∘; the entire mission time span in the reference track can
be used for the study which represents nearly 8 years of data;
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 K1 is the most difficult to observe
with satellite altimetry as it is aliased to the semiannual period by the J2
orbit and to a nearly 4-year period by the C2 satellite orbits. C2 aliasing
periods are very long compared to Jason's ones.
Spatial-mean SD (cm) of the M2 and K1 IT models for each studied
region.
RegionSD M2 (cm)SD K1 (cm)Tahiti0.360.07Hawaii0.330.07Madagascar0.460.10Gulf of Guinea0.210.07Luzon0.540.25NATL0.15–NPAC0.20–
Description of the altimeter database for the validation study,
along with the associated aliasing periods for the main tidal components.
MissionJ2C2Repeat period (d)9.9156sub-cycle of 28.941Cycles used1–288 (8 years)14–77 (5 years)Time period12 Jul 2008–6 May 201628 Jan 2011–22 Feb 2016Darwin nameAliasing (d)Aliasing (d)O145.7294.4K1L173.21430M262.1370.7S258.7245.2
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:
SSH=orbit-range-tide-IT-other_corr,
where
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 (https://www.aviso.altimetry.fr/en/data/, last access: 17 December 2020; Carrere et al., 2016a; Lyard et al., 2020);
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 (https://www.aviso.altimetry.fr/en/data/products/auxiliary-products/mss.html, last access: 17 December 2020; Schaeffer et al., 2012; Pujol et al., 2016; Appendix A).
Method of comparison
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 M2, K1, O1 and S2.
Moreover, these short temporal scales also impact climate studies since high
temporal-frequency errors increase the formal estimation error of
long-timescale signals (Ablain et al., 2016; Carrere et al., 2016b). The
impact of using each of the studied corrections on the SSH performances is
estimated by computing the SSH differences between ascending and descending
tracks at crossovers of each altimeter. Crossover points with time lags
shorter than 10 d within one cycle are selected in order to minimize the
contribution of the ocean variability at each crossover location. For the
purpose here, we avoid all strong assumptions about internal tide and assume
coherent internal tides have short autocorrelation scales.
Fourth, the variance of SSH differences at crossover points is computed on
boxes of 4∘× 4∘ holding all measurements within the time
span of the mission considered according to
DiffVarSSH=Var(ΔSSHITi)-Var(ΔSSHITzero),
where ΔSSHITzero is the SSH differences at crossovers using a
zero IT correction within a 4∘× 4∘ box for the period
considered and ΔSSHITi is the SSH differences at
crossovers using one of the IT models listed in Table 1 within the same box
and period. The resulting maps give information on the spatiotemporal
variance of the SSH differences within each box. As SSH differences are
considered, this variance estimation is twice the variance difference of
SLA. A reduction in this diagnostic indicates an internal consistency of sea
level between ascending and descending passes within a 10 d window and
thus characterizes a more accurate estimate of SSH for high frequencies.
However, the spatial resolution of this diagnostic is limited due to the
localization of crossovers and the 4∘ resolution of the grid.
Particularly for C2, the mission ground tracks' pattern induces a
non-homogeneous spread of crossovers over the global ocean, with no
crossovers around latitudes 0∘ and ± 50∘. For J2,
all latitudes are covered with crossovers but the number of points is not
homogeneous over the ocean: it is limited at the Equator and increases
towards the poles.
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∘× 2∘ according to
DiffVarSLA=Var(SLAITi)-Var(SLAITzero),
where SLAITi (or SLAITzero) are the SLA computed using one of
the IT corrections listed in Table 1 (or using the zero IT correction) in the period considered and within one 2∘× 2∘ box.
Although high-frequency signals are aliased in the lower-frequency band
following the application of the Nyquist theory to each altimeter sampling,
SLA time series contain the entire ocean variability spectrum. The SLA
variance reduction diagnostic shows an improvement of the studied IT
correction, on the condition that the correction is decorrelated from the
sea level.
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.
Variance reduction analysis using satellite altimeter data
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 (M2) and diurnal (K1) components; results for
the second-largest semidiurnal (S2) and diurnal (O1) IT are gathered in the
Appendix of the paper.
M2 component
To investigate and quantify the regional impact of the M2 IT corrections,
the maps of SSH variance difference at crossovers using either IT correction
from each model or a ZERO reference correction, are plotted for
the J2 mission in Fig. 7. Note that the quantification and the regional
analysis of the M2 IT correction can be performed for the seven IT models
participating in the present study. Most of the IT models reduce the
altimeter SSH variance in all IT regions. The RAY and ZARON models are the
most efficient, with a variance reduction reaching more than 5 cm2 in many areas. The HYCOM and DUSHAW models reduce SSH
variance in some locations but also raise the variance locally: mostly in
large deep ocean regions where IT signal can be weak in other models for
HYCOM, while the DUSHAW model raises variance mostly in areas of strong
currents. Mean values, averaged over the strong IT regions shown in Fig. 1, are listed in Table 4: the more energetic areas for the M2 IT seem to be the
Luzon Strait and Hawaii regions with a mean SSH variance reduction
greater than 2 cm2 for the ZARON model. The ZARON model is
the most efficient in all areas except in the North Atlantic (NATL) region where the UBELMANN
model reduces slightly more variance. Over the global ocean, the EGBERT,
ZARON, ZHAO and RAY models have similar mean performances, but RAY reduces a
bit more the J2 variance globally (0.34 cm2).
Figure 8 displays the maps of along-track J2 SLA variance differences using
the M2 IT correction from each model and a ZERO reference
correction. Spatial patterns are similar to those in Fig. 7. However,
using the along-track SLA allows for a better spatial resolution in the
output variance maps. In addition, regions of strong IT and regions of
strong ocean currents are more clearly identified. The DUSHAW model raises
SLA variance in several mesoscale regions (Gulf Stream, Agulhas current,
Argentine basin and Kuroshio currents), likely indicating that the model
does not properly separate IT and other oceanic signals in these strong-current areas; the ZHAO model also raises the variance in those
regions slightly, while EGBERT reduces the SLA variance in the Gulf Stream and Agulhas
regions. HYCOM raises the variance over wider regions in the three oceans
than the empirical and assimilative models do: this is likely due to its
intrinsic characteristic of the free hydrodynamic model which may induce more
phase errors compared to constrained or empirical models and also due to the
short HYCOM time series duration used to extract the IT atlas, which
induces stronger IT amplitudes (see Ansong et al., 2015; Buijsman et al., 2020). These maps also indicate that the four models RAY, EGBERT, ZARON and
ZHAO, reduce the SLA variance in some additional IT areas which are not
specifically investigated in the present study: the Indonesia seas and south
of Java, north of Sumatra, between the Solomon Islands and New Zealand in
Pacific, off the Amazonian shelf, and in many regions of the Atlantic Ocean.
Mean values, averaged over the strong IT regions identified in Fig. 1, are
given in Table 4: mean J2 SLA variance reductions are weaker than the
crossover difference variances by construction, but they indicate similar
conclusions as for J2 crossover differences: the ZARON model is the most
efficient to reduce the SLA variance in all IT regions, except in NPAC and
NATL, where the UBELMANN model is slightly more efficient. Mean values over
the global ocean are close for the four models EGBERT, ZHAO, ZARON and RAY,
with the two last ones showing a slightly better performance than others.
Maps of SSH variance differences at crossovers using either the M2 IT
correction from each model or a ZERO reference correction in
the SSH calculation for the J2 mission (cm2). J2 cycles 1–288
have been used.
Maps of SLA variance differences using either the M2 IT correction
from each model or a ZERO reference correction in the SLA
calculation for the J2 mission (cm2). J2 cycles 1–288 have been
used.
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∘ and ± 50∘ latitudes bands. C2 SLA variance maps
have similar patterns compared to J2, and some additional IT regions are
pointed out, which corroborates the quality of the different IT models
tested. Over both C2 SSH and SLA, the HYCOM and DUSHAW models show a
significant addition of variance in some regions, similarly as for J2
results.
Maps of SSH variance differences at crossovers using either the M2 IT
correction from each model or a ZERO reference correction in
the SSH calculation for the C2 mission (cm2). C2 cycles 14–77
have been used.
Maps of SLA variance differences using either the M2 IT correction
from each model or a ZERO reference correction in the SLA
calculation for the C2 mission (cm2). C2 cycles 14–77 have been
used.
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 M2 IT component show that the ZARON
model performs better than other models in most IT regions studied, with a
maximum reduction in SSH difference variance of 3.2 cm2 in
Luzon and 2.2 cm2 in the Madagascar area. RAY reduces variance a bit more in the Tahiti region; on average over the global ocean, the ZARON
and RAY models are the most efficient.
Mean variance reduction for the J2 and C2 altimeter databases, within
each IT region, when using the different M2 internal-tide models and
compared to the ZERO correction case; variance reduction in altimeter SLA
(white lines) and for altimeters crossover differences (italic lines) for
each mission, in square centimeters. For each IT region, the maximum
variance reduction across the different models is in bold.
M2RAYZHAOZARONEGBERTHYCOMDUSHAWUBELMean variance reduction for J2 database (cm2) SLATahiti-0.68-0.55-0.73-0.63-0.39-0.58Hawaii-0.65-0.58-0.74-0.62-0.30-0.55Madagascar-0.61-0.51-0.68-0.66-0.10-0.41Gulf of Guinea-0.13-0.12-0.14-0.10-0.02-0.05Luzon-1.37-1.22-1.73-1.51-1.04-0.66NATL-0.15-0.13-0.18-0.16-0.08-0.09-0.20NPAC-0.29-0.28-0.35-0.30-0.13-0.25-0.36global-0.23-0.20-0.26-0.24-0.05-0.11CrossoversTahiti-1.45-1.23-1.52-1.31-0.84-1.30Hawaii-1.93-1.92-2.17-1.92-1.25-1.90Madagascar-0.74-0.69-0.79-0.81+0.50-0.45Gulf of Guinea-0.16-0.25-0.26-0.12-0.05-0.24Luzon-1.83-1.75-2.16-1.24+0.73-0.69NATL-0.11-0.11-0.09-0.09+0.25+0.09-0.13NPAC-1.05-1.01-1.20-1.10-0.39-1.02-1.12global-0.36-0.31-0.36-0.33+0.12-0.18Mean variance reduction for C2 database (cm2) SLATahiti-0.70-0.54-0.68-0.63-0.44-0.46Hawaii-0.56-0.47-0.60-0.58-0.30-0.37Madagascar-0.55-0.45-0.55-0.49-0.17-0.13Gulf of Guinea-0.09-0.07-0.12-0.08-0.01-0.02Luzon-1.32-1.25-1.56-1.19-1.16-0.23NATL-0.14-0.13-0.16-0.14-0.11-0.04-0.11NPAC-0.25-0.24-0.29-0.28-0.13-0.18-0.28global-0.23-0.16-0.21-0.19-0.07-0.07CrossoversTahiti-1.78-1.27-1.68-1.42-1.28-1.17Hawaii-1.34-1.10-1.39-1.25-0.77-0.66Madagascar-2.08-1.55-2.21-1.90-0.45-0.92Gulf of Guinea–––––––Luzon-3.07-2.51-3.22-2.39-2.61-0.80NATL-0.22-0.15-0.24-0.20-0.14+0.02-0.11NPAC-0.39-0.39-0.47-0.42-0.12-0.29-0.45global-0.60-0.45-0.59-0.55-0.22-0.06K1 component
The maps of K1 SSH variance difference at crossovers using the K1 IT correction
from EGBERT, ZARON and ZHAO models are plotted in Fig. 11
for the J2 and C2 missions. Note that unlike the M2 wave analysis, the
quantification and the regional analysis of the K1 IT correction can be
performed for only three IT models participating in the present study that provide a K1 solution, as the diurnal tides are more difficult to detect and
sort out by altimetry. The K1 IT solutions are compared to a ZERO reference
correction. The three models have different approaches to take into account the
diurnal tides' critical latitude and regions where amplitude of K1 IT is
negligible and/or not separable from background ocean variability (cf Sects. 2 and 3.1), which explains the large non-defined regions in ZARON and ZHAO
maps compared to EGBERT. Results show that the three IT models all reduce
the J2 SSH variance strongly in the west Pacific or Luzon and Indonesian
regions (more than 2 cm2), while a weaker variance reduction
is visible in the central Indian and central Pacific areas (0.5–1 cm2). The reduction is also important for C2 SSH in the east
Pacific or Luzon area and south of Java, and results are noisier in the other
oceans where diurnal IT is weak, but C2 data are likely less efficient for
testing the K1 tide due its very long alias compared to M2 tide (see Table 3).
The ZARON model reduces slightly more C2 variance in the southern part of
the Indian Ocean.
Maps of SSH variance differences at crossovers using either the K1
IT correction from each model or a reference ZERO correction in
the SSH calculation for the J2 and C2 missions (cm2). J2 cycles
1–288 have been used; C2 cycles 14–77 have been used.
The maps of SLA variance differences using the EGBERT, ZARON and ZHAO K1 IT
models are plotted in Fig. 12 for the J2 and C2 missions. Spatial SLA
patterns are consistent with the SSH maps of Fig. 11 and allow a better
spatial resolution compared to SSH maps as also noted for M2 results: using
the EGBERT model allows a significant reduction in the J2 SLA variance mostly in
the Luzon Strait or west Pacific region and the northern Indonesian seas, where
the amplitude of the K1 IT is the most important; a weak variance gain is
also visible in the IT regions around Tahiti, Hawaii and north of Madagascar
but also in some large ocean current regions, in the central Indian Ocean and
east of Australia. The other maps indicate that ZHAO is less efficient than
the two others in the Luzon region, while ZARON reduces slightly more
variance for the C2 mission in the west Pacific area.
Maps of SLA variance differences using either the K1 IT correction
from each model or a reference ZERO correction in the SLA
calculation for the J2 and C2 missions (cm2). J2 cycles 1–288
have been used; C2 cycles 14–77 have been used.
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 K1 statistics are not significant in the other regions of large
semidiurnal tides defined in Fig. 1. The values in Table 5 indicate a
significant variance reduction mainly in the Luzon region as expected from
the analysis of global maps. The ZARON and EGBERT models are the most
efficient IT solutions in the Luzon region, with similar variance gains for
both models at C2 crossovers. ZARON shows a significant variance gain
compared to the ZERO correction for both missions tested, reaching 3
and 2.4 cm2, respectively, for J2 crossovers
and C2 crossovers.
Mean variance reduction for J2 and C2 altimeter databases, within
each IT region, when using the different K1 internal-tide models and
compared to the ZERO correction case; variance reduction in altimeter SLA
(white lines) and for altimeter crossover differences (italic lines) for
each mission, in square centimeters. For each IT region, the maximum
variance reduction across the different models is highlighted in bold.
K1ZHAOZARONEGBERTMean variance reduction for J2 database (cm2) SLATahiti-0.04-0.04-0.06Hawaii-0.03-0.05-0.05Madagascar-0.06-0.05-0.07Luzon-0.53-1.03-1.09global-0.05-0.05-0.06CrossoversTahiti-0.08-0.14-0.10Hawaii-0.05-0.15-0.10Madagascar-0.09-0.14-0.14Luzon-1.82-3.01-2.85global-0.17-0.21-0.12Mean variance reduction for C2 database (cm2) SLATahiti-0.03-0.03-0.03Hawaii-0.02-0.03-0.02Madagascar-0.03-0.04-0.05Luzon-0.51-0.86-0.80global-0.04-0.04-0.03CrossoversTahiti-0.02-0.04-0.09Hawaii-0.20-0.12-0.09Madagascar-0.03-0.07-0.04Luzon-1.37-2.41-2.41global-0.10-0.12-0.08Wavelength analysis for the M2 wave
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 M2
(cycle duration is 370 d for C2). Moreover, this diagnostic only focuses
on the main M2 IT because the K1 aliasing frequency by J2 sampling is 173 d (see Table 2), which makes it barely separable from the semiannual
ocean signal.
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 M2 tidal component by Jason's orbit, is extracted in
both cases and then the normalized difference of the spectral density is
computed and plotted as a function of wavelength. This computation gives an
estimation of the percentage of energy removed at the M2 frequency thanks to
each IT model correction, as a function of wavelength and for the different
regions studied.
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 M2 frequency
for the first internal-tide mode and 70 % for the second mode in the
Tahiti region. With respect to the first mode, the ZARON model removes
nearly 80 % of the energy in the Gulf of Guinea, 60 % in the Luzon,
Madagascar and NPAC regions, and 50 % in the NATL region. We speculate
that the regions for which ZARON removes less variance may be regions with
stronger IT non-stationarity (Zaron, 2017). In the Madagascar region, ZARON,
EGBERT, RAY and ZHAO perform similarly for the first mode. Only a few models
manage to reduce the IT energy for the second and the third modes: RAY and
ZARON reduce more than 60 % of the second mode energy around Tahiti and up
to 30 % in other regions except in NATL where they only reduce about 15 % of the second mode energy. Aside from the fact that models are not
perfect, these results corroborate the fact that the non-stationary IT part
is even more significant for higher IT modes (Shriver et al., 2014; Rainville
and Pinkel, 2006). Around Tahiti, the curves indicate that the RAY model also
reduces the SLA energy for a third mode of IT (∼ 20 %). The
ZHAO model also removes some energy at short scales in the Madagascar and
Luzon regions.
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.
Discussion
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: M2, K1, O1 and S2. First, the
analysis shows clearly the value of using such a complete altimeter database
to validate IT models. The great quality of the database allows
the investigation of small-amplitude signals over the entire ocean, and the
different sampling characteristics of the various missions complement each
other well. The results point to a significant altimeter variance reduction
when using the new IT correction models over all ocean regions where
internal tides are generated and propagating. Moreover, the spectral
approach quantifies the efficiency of the variance reduction potential of
each model as a function of horizontal wavelengths – the latter is
particularly valuable information for the SWOT mission, which will focus as
never before on short wavelength phenomena.
All empirical models display generally good performance for M2, K1, O1 and
S2, but the DUSHAW solution performs slightly less well. The ZARON and RAY
models have similar results for the first three IT modes, but the ZARON
model removes more variability than all other models over most of the strong
IT regions analyzed. It is also noticeable that some models (DUSHAW and
ZHAO) still remove some variability in areas of strong currents, likely due
to some residual leakage of the mesoscale variability. The UBELMANN solution
appears to also remove some large-scale, likely residual barotropic tide
signal in the northeast of the Azores area.
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 https://meetings.aviso.altimetry.fr/programs/2018-ostst-complete-program.html, last access: 18 January 2021) and Chicago (2019; program available at https://meetings.aviso.altimetry.fr/programs/2019-ostst-complete-program.html, last access: 18 January 2021): in the light of these findings, the recommendation came to use
an internal-tide model for the correction of all along-track nadir altimeter
databases as well as the upcoming high-resolution SWOT wide-swath altimeter
mission. Consequently, the ZARON model is being implemented in the next
version of the altimeter GDRs (GDR-F-standard: https://www.aviso.altimetry.fr/en/data/, last access: 17 December 2020),
which will be available on AVISO.
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).
Comparing internal-tide models for O1 and S2 waves
Amplitudes of O1 and S2 tide components for each IT model are plotted in
Figs. A1 and A2, respectively, in the Luzon area. Concerning the O1 tide, ZHAO and ZARON show a similar south-west pattern on the west side of the Luzon Strait, with an amplitude reaching more than 2 cm for ZARON and only the
half for ZHAO's solution. On the east side of the strait, the three models are
quite different: ZHAO has the weaker amplitudes, ZARON has strong large-scale patterns propagating far eastward (1.5 cm amplitude with 200 km wide
features) and decaying to zero above 22∘ N; and EGBERT shows a
third very different pattern with zero amplitude along latitudes
15 and 21∘ N and also east of the Philippines and
amplitudes reaching about 1 cm at 22–23∘ N.
In this region, the S2 IT amplitude shows smaller spatial scales than O1 and
close to the M2 ones as expected. The EGBERT S2 solution is very different from
others and mostly shows a noisy pattern in this Luzon area. ZHAO and ZARON
show similar features of about 1 cm amplitude and with a clear eastward
propagation in the Pacific Ocean and a northwestward direction west of the
strait; ZARON has stronger amplitudes.
Global maps shown in Figs. A3 and A4 illustrate the mean IT amplitude and
the standard deviation of the IT models for the O1 and S2 tidal component, respectively. S2 mean amplitudes show similar patterns as M2 with weaker
amplitudes as expected (below 1 cm); the main S2 generation sites are visible
around Hawaii in the Pacific Ocean, off Amazonia, around Madagascar, north of
Sumatra, south of Lombok, in the Banda and Celeb seas, around the Solomon Islands,
in the Luzon area, and on the Saipan ridge. O1 IT has similar patterns as K1 but with
weaker mean amplitudes.
Amplitude of the IT models for the O1 tide component in the Luzon area.
Spatial-mean SD of models for S2 and O1 tide components for each
studied region (in centimeters).
RegionSD S2 (cm)SD O1 (cm)Tahiti0.080.06Hawaii0.110.06Madagascar0.150.08Gulf of Guinea0.080.06Luzon0.160.18
For O1, the Luzon Strait region mainly stands out with stronger standard
deviation values in the Luzon Strait and eastwards in the Philippine Sea (values
around 0.4 cm). The S2 standard deviation reaches 0.1–0.5 cm in the Hawaii,
Madagascar and Luzon regions, where the amplitude of the S2 IT signal is more
important. The mean standard deviation is computed in the different regions
studied, using only data located in the deep ocean, and values are gathered in
Table A1. The S2 mean standard deviation is at least 3 times smaller compared to
M2, which is coherent with the fact that S2 IT has a smaller amplitude than
M2; stronger values occur in the Luzon and Madagascar regions, where mean S2 IT
amplitude is maximal. O1 IT has the strongest standard deviation (0.18 cm)
in the same Luzon area as K1, where diurnal internal tides have the most
significant amplitudes in the ocean, which indicates that O1 IT models have
some uncertainties in this region.
Amplitude of the IT models for the S2 tide component in the Luzon area.
Global maps of mean amplitude of the O1(a) and S2(b) IT
models (cm).
Global maps of standard deviation of the O1(a) and S2(b) IT models (cm).
Validation results for O1 IT models
The maps of SLA and crossover variance differences using each of the three
different O1 IT models are plotted in Figs. B1 and B2, respectively, for
both J2 and C2 missions; the O1 IT solutions are compared to a ZERO
reference correction. First, it is noticeable that as for K1, the ZARON O1
solution is not defined in large ocean regions, mostly taking into account
the diurnal critical latitude and regions where the O1 IT amplitude is
negligible and/or not separable from background ocean variability. The ZHAO
O1 solution is not defined beyond the diurnal-tide critical latitude, while the EGBERT solution is defined on a wider range of latitudes.
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 O1 IT is the
most important in the ocean; the variance reduction reaches 1–2 cm2 in this area. The EGBERT model removes some C2 variability
(0.5 cm2 on C2 SLA) in the middle of the Indian Ocean around
latitude 20∘ S, but maps are noisier for the two other models in
this region; some C2 SLA variance reduction occurs west of Luzon Strait and
north of the Indonesians seas, but in the Philippine Sea the three models both
reduce and raise the C2 SLA variance in the 10–25∘ N
latitude band with a zonal band pattern; the variance raise is minimal with the EGBERT model. This zonal effect, only visible in C2 SLA data, might be
explained by some residual TP-Jason errors or even oceanic variability in
the O1 IT models in this area. The maps of the variance differences at
crossovers are consistent with SLA results for both missions, and they
indicate a significant J2 variance reduction in the Indonesian and
Philippine areas; the C2 crossover maps indicate a weaker and noisier
impact compared to J2 data.
Mean variance reduction for J2 and C2 altimeter databases, in the
Luzon region, when using the different O1 internal-tide models and compared
to the ZERO correction case; variance reduction in altimeter SLA (white
lines) and for altimeters crossover differences (italic lines) for each
mission, in square centimeters. The maximum variance reduction across the
different models is highlighted in bold.
Mean variance reduction on Luzon regionZHAOZARONEGBERTMean variance reduction for J2 database (cm2) SLA-0.30-0.41-0.41crossovers-1.15-1.53-1.14Mean variance reduction for C2 database (cm2) SLA-0.35-0.13-0.46crossovers-0.18-0.11-0.12
The mean statistics of altimeter variance reduction for O1 IT are given in Table B1 for the SLA and the SSH crossover differences of J2 and C2
missions; notice that only the Luzon region is presented because the O1 amplitude is
not significant elsewhere. The figures show that the ZARON O1 model reduces J2 variance more than other models (1.5 cm2 of SSH crossover
variance in the area), but the EGBERT and ZHAO solutions are a bit more
efficient when considering mean C2 SLA values. Mean C2 crossover variance
differences are very weak, reflecting the noisy corresponding variance maps
in the region as seen in Figs. B1 and figure B2. These weaker or noisier
results noted with C2 crossovers for O1 frequencies can likely be explained
by the fact that the C2 temporal series are shorter than J2 ones, which make
the analysis noisier particularly for such a small-amplitude signal, in
addition to the fact that crossovers statistics are smoothed in larger boxes
compared to SLAs.
Maps of SLA variance differences using either the O1 IT correction
from each model or a reference ZERO correction in the SLA
calculation for the J2 and C2 missions (cm2). J2 cycles 1–288
have been used; C2 cycles 14–77 have been used.
Maps of SSH variance differences at crossovers using either the O1
IT correction from each model or a reference ZERO correction in
the SSH calculation for the J2 and C2 missions (cm2). J2 cycles
1–288 have been used; C2 cycles 14–77 have been used.
Validation results for S2 IT models
The maps of SLA and crossover variance differences using each of the three
different S2 IT models are plotted in Figs. C1 and C2, respectively, for both J2
and C2 missions; the S2 IT solutions are compared to a ZERO reference
correction. The ZARON S2 solution is not defined in large deep-ocean regions
(white areas in the maps) where the S2 IT amplitude is negligible and/or not
separable from background ocean variability.
Using the three models for S2 IT correction allows a small but well-detected
reduction in the J2 and C2 SLA variances in the same regions of the ocean as
for the main semidiurnal IT (see Figs. 7–10 for M2 IT): variance reduction
is maximal (about 0.5–1 cm2) west of the Hawaii region, north of
Madagascar, in the Luzon Strait or west Pacific region and also in the
Indonesian islands, north of Sumatra, and between the Solomon Islands and New
Zealand. The C2 maps show similar reduction patterns, but the variance gain
is weaker than for J2. Both the EGBERT and ZHAO models remove some variance
south of Africa in the Agulhas currents, while ZARON does not; the ZHAO model
clearly impacts the altimeter variance in most of the great ocean currents
areas, which likely indicates that the model might contain some residual
oceanic signal and/or some J2 error and not only IT. The patterns of the
crossover variance differences are consistent for J2 missions but with
weaker values than for the SLA; for the C2 mission, the crossover maps indicate
a weaker and noisier impact than for SLA as already noted for the O1 frequency.
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 S2 statistics are not
significant elsewhere. The figures show weak SLA variance reductions with
stronger values in the Luzon, Madagascar and Hawaii regions, where the amplitude of the S2 IT is the most important: if looking at J2 SLA, the three models are equivalent in Hawaii and Madagascar, but EGBERT and ZARON are
more efficient at reducing variance in the Luzon area (0.28 and
0.25 cm2, respectively). Looking at C2 SLA, the three models give similar results in Madagascar, EGBERT and ZHAO reduce more variance in the Luzon region, and EGBERT is more efficient in the Hawaii region. Unlike the results obtained
for the M2 and K1 waves and described in previous sections, the variance
reduction for crossover differences is weaker than for the SLA for the S2 wave
and mean values are hardly useful; this is likely explained by the weak S2
IT signal in the ocean in addition to the fact that crossover statistics are
performed in large boxes, which tends to smooth them even more. This analysis
results suggest that the EGBERT and ZARON S2 IT solutions are the most efficient
in the different regions of interest.
Mean variance reduction for J2 and C2 altimeter databases, within
each IT region, when using the different S2 internal-tide models and
compared to the ZERO correction case; variance reduction in altimeter SLA
(white lines) and for altimeter crossover differences (italic lines) for
each mission, in square centimeters (0 is for |value| < 0.005 cm2). For each IT region, the maximum
variance reduction across the different models is highlighted in bold.
Mean varianceZHAOZARONEGBERTreduction (cm2)Mean variance reduction for J2 database (cm2) SLATahiti-0.02-0.02-0.05Hawaii-0.14-0.15-0.17Madagascar-0.14-0.14-0.15Luzon-0.15-0.25-0.28NPAC-0.05-0.06-0.07global-0.04-0.04-0.06CrossoversTahiti-0.04-0.04-0.05Hawaii-0.12-0.11-0.11Madagascar-0.02-0.02-0.03Luzon0-0.01-0.05NPAC-0.07-0.06-0.07global0-0.02-0.02Mean variance reduction for C2 database SLATahiti-0.01-0.01-0.02Hawaii-0.09-0.08-0.15Madagascar-0.08-0.06-0.09Luzon-0.16-0.09-0.16NPAC-0.04-0.04-0.07global-0.01-0.02-0.02CrossoversTahiti00+0.01Hawaii-0.01-0.010Madagascar-0.02+0.020Luzon+0.03+0.02+0.02NPAC000global000
Maps of SLA variance differences using either the S2 IT correction
from each model or a reference ZERO correction in the SLA
calculation for the J2 and C2 missions (cm2). J2 cycles 1–288
have been used; C2 cycles 14–77 have been used.
Maps of SSH variance differences at crossovers using either the S2 IT correction from each model or a reference ZERO correction in
the SSH calculation for the J2 and C2 missions (cm2). J2 cycles
1–288 have been used; C2 cycles 14–77 have been used.
Data availability
Level 2P (L2P) altimetry products, with 1 Hz along-track resolution (LRM), are produced and distributed by Aviso+, as part of the Ssalto ground processing segment (https://www.aviso.altimetry.fr/en/data/, last access: 8 January 2021). The version of the L2P products used in the present study corresponds to the DT-2014 standards described in Pujol et al. (2016), except for the tide correction. The FES2014b global tidal atlas is available at the Aviso+ web page (https://www.aviso.altimetry.fr/en/data/products/auxiliary-products/global-tide-fes.html, last access: 8 January 2021). MSS_CNES_CLS_11 referenced on the 20-year altimetric period is also available at the Aviso+ web page (https://www.aviso.altimetry.fr/en/data/products/auxiliary-products/mss.html, last access: 8 January 2021). Brian Dushaw's baroclinic tide model can be downloaded at https://apl.uw.edu/project/project.php?id=tm_1-15 (last access: 8 January 2021). The latest version of Edward Zaron's baroclinic tide model can be downloaded at https://ezaron.hopto.org/~ezaron/downloads.html (last access: 8 January 2021); HRET version 7.0 was used for the present study. Other baroclinic tide models have been provided specifically by the authors for the present study and are not publicly available, but they might be provided by their respective authors on demand.
The supplement related to this article is available online at: https://doi.org/10.5194/os-17-147-2021-supplement.
Author contributions
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.
Competing interests
The authors declare that they have no conflict of interest.
Special issue statement
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.
Acknowledgements
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.
Review statement
This paper was edited by Mattias Green and reviewed by C. K. Shum and one anonymous referee.
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