We present a new gridded sea surface height and current dataset
produced by combining observations from nadir altimeters and drifting buoys.
This product is based on a multiscale and multivariate mapping approach
that offers the possibility to improve the physical content of gridded
products by combining the data from various platforms and resolving a
broader spectrum of ocean surface dynamic than in the current operational
mapping system. The dataset covers the entire global ocean and spans from
1 July 2016 to 30 June 2020. The multiscale approach
decomposes the observed signal into different physical contributions. In the
present study, we simultaneously estimate the mesoscale ocean circulations
as well as part of the equatorial wave dynamics (e.g. tropical instability
and Poincaré waves). The multivariate approach is able to exploit the
geostrophic signature resulting from the synergy of altimetry and drifter
observations. Sea-level observations in Arctic leads are also used in the
merging to improve the surface circulation in this poorly mapped region. A
quality assessment of this new product is proposed with regard to an
operational product distributed in the Copernicus Marine Service. We show
that the multiscale and multivariate mapping approach offers promising
perspectives for reconstructing the ocean surface circulation:
observations of leads contribute to improvement of the coverage in delivering gap-free maps
in the Arctic and observations of drifters help to refine the mapping in regions
of intense dynamics where the temporal sampling must be accurate enough to
properly map the rapid mesoscale dynamics. Overall, the geostrophic
circulation is better mapped in the new product, with mapping errors
significantly reduced in regions of high variability and in the equatorial
band. The resolved scales of this new product are therefore between 5 %
and 10 % finer than the Copernicus product (https://doi.org/10.48670/moi-00148, Pujol et al., 2022b).
Introduction
Several oceanographic applications (e.g. operational oceanography, marine
weather, and climate monitoring) rely on high-quality
observational datasets. The European Union (EU) Copernicus Marine and
Climate Change Services provide operational services and indicators on the
observed state of the climate. Sea-level and surface currents are, among
others, key variables distributed by the services. They are also listed as
Essential Climate Variables (ECVs) for the detection of climate change and
the characterization of climate system variability (Bojinski et al., 2014).
As part of the Copernicus Services, the Sea Level Thematic Assembly Centre
(SL-TAC) delivers near-real-time and delayed-time sea-level and surface
current products (along-track Level-3 and gridded Level-4 products) that
are used by the ocean science community to study, understand, and monitor the
evolution of the ocean system. These products do not resolve the entire
spectrum of the ocean surface variability; they have resolution limits of
about 60 km for the along-track products (Dufau et al., 2016) and
>200 km × 20 d for the gridded products (Ballarotta et al.,
2019), but recent nadir altimetry instruments, such as the new Sentinel-3A
and 3B SAR missions, or future missions based on large swath technologies
(e.g. the upcoming Surface Water and Ocean Topography (SWOT) mission) offer,
for example, the possibility of observing finer ocean structures (Morrow et
al., 2019), which could be used to provide better gridded product resolution.
In addition, the growing needs to develop observing systems or methods with
finer spatial scales and higher frequencies have been identified by the ocean
scientific community and the Copernicus Services as research and development priorities to
serve Copernicus marine users and decision-makers (see, for example, International Altimetry Team, 2021, or the “Copernicus Marine Service Evolution Strategy: R&D
priorities – Version 5 June 30, 2021” document, https://marine.copernicus.eu/sites/default/files/media/pdf/2021-09/CMEMS Service_evolution_strategy_RD_priorities_v5-June-2021.pdf, last access: 1 December 2022).
Therefore, with the support of the French Space Agency (CNES), the
development of new experimental products has been undertaken, aiming at
improving the resolution of the current Level-3 and Level-4 sea-level
products (Mulet et al., 2021a; Ballarotta et al., 2020; Ubelmann et al.,
2022; Prandi et al., 2021) and preparing operational systems for the SWOT
era (Ubelmann et al., 2015, 2021; Le Guillou et al., 2021;
Beauchamp et al., 2020).
The present study focuses on the development and assessment of experimental
global gridded products based on a recent multiscale and multivariate
mapping approach (Ubelmann et al., 2021, 2022) and applied to real Earth
observations. Here we investigate the possibility of improving the content
of gridded products in combining the data from various platforms (in situ
and satellite) and in resolving a larger spectrum of the ocean surface
dynamic than in current operational products.
The paper is structured as follows: the data sources and merging methods
used in this study are described in Sect. 2. Section 3 presents the
experiments and validation metrics. The quality assessment of the new
products is proposed in Sect. 4. The key results are then summarized in
Sect. 5.
Data and methodsData sources
The mapping method used in this study takes input data from remote sensing
and in situ observations, which are summarized in Table 1 and described
below.
List of observation datasets used in this study.
Product typeGlobal altimeter SLA productsArctic leads' altimeter SLA productsDrifters' geostrophic velocity productProduct ref.SEALEVEL_GLO_PHY_L3_REP_OBSERVATIONS_008_062ExperimentalAOMLSpatial coverage[90∘ S–90∘ N], [0∘ E–360∘ E]>60∘ N[90∘ S–90∘ N], [0∘ E–360∘ E]PeriodFrom 15 January 2016 to 30 June 2020From 15 January 2016 to 30 June 2020From 15 January 2016 to 30 June 2020Sea-level anomaly products
The global ocean sea surface height (SSH) observations are from the
(delayed time, DT) Level-3 altimeter satellite along-track data, reprocessed
in 2021 and distributed by the EU Copernicus Marine Service (product
reference SEALEVEL_GLO_PHY_L3_MY_008_062,
https://doi.org/10.48670/moi-00146, Pujol et al., 2022a). These data cover the period from
1 January 1993 to 31 December 2020 over the world ocean (excluding ice-covered areas; see, for example, Fig. 1) and are available at a sampling rate of 1 Hz
(∼ 7 km spatial spacing). Homogenization and cross-validation
are applied to the dataset to remove any residual orbit error,
long-wavelength error (lwe), large-scale biases, and discrepancies between
different data streams. The list of geophysical and environmental
corrections applied to the datasets is described in the quality information
document (Taburet et al., 2021) and summarized below in Eq. (1). In
this study, unfiltered sea-level anomalies (SLAs) corrected with dynamic
atmospheric correction (dac), ocean tide, and lwe corrections are considered
in the multiscale and multivariate mapping.
SLA=orbit-range-∑environmental
corrections-∑geophysical
corrections-mean sea surface,
where ∑(environmental corrections)=wet tropospheric+dry
tropospheric+ionospheric+sea-state bias, ∑(geophysical
corrections)=solid earth tide+load tide+ocean tide+pole tide+dynamic atmospheric correction (see Taburet et al., 2021, for
the references associated with each mission correction). The mean sea surface
used here is the CNES-CLS18 (Mulet et al., 2021b).
Example of sea-level altimetry coverage for a 7 d period (from
1 July 2019 to 7 July 2019). Colour scale represents the sea-level anomaly
amplitude in metres. For this time interval, data originate from six
altimeters: Jason-3, Sentinel-3A, Sentinel-3B, SARAL/AltiKa, Cryosat-2,
and Haiyang-2A.
Sea-level anomaly products in Arctic leads
In the polar regions, satellite sea-level observations are limited by the sea
ice. Thanks to dedicated processing, sea level can however be estimated
within fractures in the ice (leads). The echoes from the altimeters over the
ice-covered region are classified to identify peaky waveforms corresponding
to lead echoes. Range estimation is then made with specific retracking
methods, and it is corrected from instrumental and geophysical corrections
to obtain a sea-level anomaly (Prandi et al., 2021). To ensure continuity with the
open ocean, the corrections are derived from the global ocean Level-3
along-track processing (Taburet et al., 2021) when possible. The noticeable
exceptions concern (1) the wet tropospheric correction that comes from the
European Centre for Medium-Range Weather Forecasts (ECMWF) model since
onboard radiometer estimates are not reliable over ice, (2) the sea-state
bias correction which is not applied since waves and winds are considered small
over leads, and (3) orbit error corrections which are not applied as they are difficult
to compute over this small region. Then a constant bias of ∼ 8 cm is applied for each mission to ensure continuity with the
SEALEVEL_GLO_PHY_L3_MY_008_062 open-ocean SLA
previously described. These products cover the Arctic region (up to
88∘ N) at a sampling rate of 20 Hz (∼ 350 m)
for three altimetry missions: SARAL/AltiKa, Sentinel-3A, and CryoSat-2 (Fig. 2
and Table 2).
Example of Arctic leads' sea-level altimetry coverage for a 7 d
period (from 1 July 2019 to 7 July 2019). Colour scale represents the sea-level anomaly amplitude in metres.
Geostrophic current anomaly products
To further constrain the surface circulation, we used delayed-time
horizontal surface velocities from the NOAA's Atlantic Oceanographic and
Meteorological Laboratory (AOML) Surface Velocity Program (SVP; Lumpkin and
Centurioni, 2019). The data cover the entire world ocean and are available
at a 6 h frequency. The SVP drifters are designed to follow the 15 m depth
circulation, which is the centre depth of their drogues. When the drogue is
lost, they follow the surface current but are also under the direct
influence of the wind. AOML distributes a flag to indicate whether the
drogue is lost or not (Lumpkin et al., 2013). These data are also distributed
by the IN SITU Thematic Assembly Centre of the EU Copernicus Marine Service
(see Product User Manual;
http://marine.copernicus.eu/documents/PUM/CMEMS-INS-PUM-013-044.pdf, last access: 9 January 2023) with an
additional wind slippage correction for undrogued buoys derived from the Rio (2012) methodology. For the study, the undrogued and drogued drifters are
selected over the global ocean and the period from 1 June 2016 to 31 July 2020.
Note that for specific experiments described hereafter, we excluded
drifters' trajectories between -10∘ S and 10∘ N (e.g.
Fig. 3) to isolate and evaluate only the impact of the equatorial wave's
mode in this region. As in Mulet et al. (2021a), we computed the geostrophic
velocity anomaly components, which are defined as
2Uanom=Ubuoy-Uekman-Ustokes-Uinertial-Utidal-Uahf-Uslip-Umdt3Vanom=Vbuoy-Vekman-Vstokes-Vinertial-Vtidal-Vahf-Vslip-Vmdt,
where Ubuoy (Vbuoy) is the
drifter's zonal (meridional) velocity. Each component is corrected as follows.
The wind-driven component Uekman
(Vekman) uses an update of the model used in Mulet
et al. (2021a) and described in Etienne (2021a). The Ekman component is not
available in the Mediterranean basin, so there is no drifter used in this
region for the study. In this recent version, ERA5 wind stress (Hersbach et
al., 2018) replaces the ERA Interim data, and the equatorial symmetry of the
wind driven parameters is removed.
The Stokes drift Ustokes
(Vstokes) from ERA5 reanalysis (Hersbach et al., 2018) is also removed from the surface drifter velocity (undrogued
drifters). No Stokes drift is removed from the 15 m depth velocity, as this
component is supposed to mostly vanish in the first 2–4 m.
The wind slippage is the direct effect of the wind on the buoy
Uslip (Vslip). This correction
is significant only in the case of drogue loss (Etienne et al., 2021), when
the drifters are advected by the surface current.
Then the data are filtered from the tidal and inertial
velocities Uinertial+Utidal
(Vinertial+Vtidal) as well
as the residual high-frequency ageostrophic
signal Uahf (Vahf). Finally,
the mean geostrophic velocity (CNES-CLS2018; Mulet et al., 2021b)
Umdt (Vmdt) is subtracted to
obtain the geostrophic velocity anomaly.
Example of the drifter's trajectory coverage for the period 1 January 2019 to
31 December 2019. Colour scale represents the velocity amplitude in
metres per second (m s-1).
Methods
Two mapping methods are compared in this study: the operational DUACS (Data
Unification and Altimeter Combination System) mapping approach and the
multiscale and multivariate MIOST (Multiscale Inversion of Ocean Surface
Topography) mapping approach. Each method is described in detail in
reference articles, such as Le Traon et al. (1998, 2003), Ducet et al. (2000),
or Pujol et al. (2016) for the DUACS method and Ubelmann et al. (2021,
2022) for the MIOST method. A description of the methods is given in
Appendix A, and we propose hereafter to focus on the specific developments
and processes that are considered in this study.
It is important to mention that DUACS maps are constrained by a single-scale
covariance function (Arhan and Colin de Verdière, 1985; Le Traon et al.,
1998) and focus mainly on the geostrophic circulation (i.e. processes with
typical space and timescales >100 km, 10 d). Consequently,
they do not resolve the full spectrum of ocean surface variability. It is,
for example, the case for the equatorial surface dynamics (see, for example, Fig. 7). While slow Rossby waves are already resolved within geostrophy in DUACS
maps, faster equatorial waves such as Poincaré waves are filtered out,
even though the space–time coverage of altimetry data allows for sampling of
large-scale waves with periods of 4–10 d and more (Farrar and Durland,
2022). The multiscale approach proposed by the MIOST method offers the
possibility to solve some of the missing surface variabilities in DUACS,
accounting for the covariances of various surface processes in a single
inversion. The covariance functions in the MIOST system are expressed as
wavelet modes, and the inversion is performed in this space using a
variational approach (Ubelmann et al., 2021). In the following, we focus on
the main components that have been tested in this study with the MIOST
method: the geostrophy component already investigated in Ubelmann et al. (2021) and two new components associated with the equatorial wave's dynamic.
The geostrophy component follows the same formulation provided in Ubelmann
et al. (2021) (see their Sect. 2.3.2.1, where the analytical formula of
the ensemble of wavelet elements is given) and is also reported in Appendix A. The covariance function associated with the geostrophy component is plotted
for a given point (5∘ N, 210∘ E) in Fig. 4a and c,
shown as a function of space (bottom left panel) and as a function of time
(top left panel). This covariance function is similar to what is currently
used for altimetry mapping with DUACS.
Example of spatio-temporal covariance models at (5∘ N, 210∘ E) for (a, c) the geostrophy component and for (b, d) a westward-propagating wave component, e.g. TIW.
In the present study, we simultaneously estimate the surface signatures of
the geostrophy and equatorial tropical instability waves (TIWs) and
Poincaré waves. As for the geostrophy component, the equatorial wave
covariances are expressed as a reduced wavelet basis, with typical wavelength
and propagation speed given in the literature (e.g. Shinoda et al., 2009;
Farrar, 2008, 2011; Farrar and Durland, 2012; Tanaka and Hibiya, 2019). For
Poincaré waves, we built an ensemble of wavelets between 10∘ S
and 10∘ N which follow the dispersion relation (Matsuno, 1966):
ω=k2⋅c2+β⋅c⋅2⋅n+1,
where ω is the time frequency, c=±2.8 m s-1 is the
Poincaré wave propagation speed (considered a constant here), k is
the spatial wavenumber, and n is a positive integer defining the wave mode.
The wavelets are localized with a Hamming window with half-widths of 1000 km in the zonal direction, 300 km in the meridional direction, and 5 d in
the temporal direction. For the TIW component, we also built an ensemble of
wavelets between 10∘ S and 10∘ N which follow the
dispersion relation (Matsuno, 1966):
ω=c⋅k,
where ω is the time frequency, c=-0.5 m s-1 is the TIW
propagation speed (considered a constant here), and k is the spatial
wavenumber. The wavelets are localized here with a Hamming window with
half-widths of 500 km in the zonal direction, 300 km in the meridional
direction, and 20 d in the temporal direction. The covariance function for
a westward-propagation-wave-like TIW is illustrated in Fig. 4b and d for
a given point (5∘ N, 210∘ E), shown as a function of
space (bottom right panel) and as a function of time (top right panel). Note
that for Poincaré waves, both eastward and westward propagation is
considered. A more detailed description of the equatorial wave's components
implemented in MIOST is provided in Appendix A.
Experiments and validation metricsExperiments
We produced 4 years (from 1 July 2016 to 30 June 2020) of SSH maps using the
MIOST multiscale and multivariate approach by combining the Level-3
altimeter dataset from SARAL/AltiKa, Envisat, Jason-1, Jason-2, Jason-3,
Cryosat-2, Haiyang-2A, Haiyang-2B, Sentinel-3A, and Sentinel-3B missions, the
Level-3 Arctic lead sea-level anomaly products from SARAL/AltiKa,
Sentinel-3A, and CryoSat-2 missions, and geostrophic current anomaly
data from the AOML drifter database. These MIOST products are available on the
AVISO + (Archivage, Validation et Interprétation des données des
Satellites Océanographiques) website (see Sect. 5, “Data availability”,
for more details).
Specific maps were also made to quantitatively assess the quality of these
MIOST products. Table 3 summarizes the list of experiments conducted in this
study, indicating the input data used in the mapping and the physical
content of the maps.
DUACS allsat-1 and MIOST allsat-1 experiments focus on the geostrophic variability. These SSH maps were
produced from six altimeters (Jason-3, Cryosat-2, Sentinel-3A, Sentinel-3B,
Haiyang-2A, Haiyang-2B) for the period 1 January 2019 to 31 December 2019, excluding
one altimeter (Saral/AltiKa, over open-ocean region) from the mapping to perform independent
assessments. The MIOST allsat-1 80 % drifters + equatorial waves + L3 Arctic experiment focuses on the geostrophic and equatorial
wave variabilities. This experiment is based on (1) 80 % of the drifter
data, (2) the six altimeters previously mentioned over ocean, and (3) lead
altimeter observations. The Saral/AltiKa dataset (over open-ocean region) and the remaining 20 %
of the drifter trajectories were here excluded from the mapping to perform
independent assessments. Note that for these specific maps, drifter
trajectories between -10∘ S and 10∘ N (e.g. Fig. 3)
were also excluded to evaluate only the impact of the equatorial wave's mode
in this region.
List of mapping experiments with the input data and physical
content considered.
The validation metrics are based on statistical and spectral analysis.
One quantitative assessment is based on the comparison between SSH maps and
independent SSH along-track data. This diagnostic follows three main steps: (1)
the SSH gridded data are interpolated to the locations of the independent SSH
along-track, geo-referenced by their longitude, latitude, and time; (2) the
difference SSHerror=SSHmap-SSHalong-track is calculated;
and (3) a statistical analysis on the SSHerror is performed in
1∘× 1∘ longitude × latitude boxes.
Prior to the statistical analysis, a filtering operation can be applied to
isolate the spatial scales of interest. For example, the analysis can be
performed over the spatial range [65–500 km] typically representative of
the medium mesoscale ocean signal. This excludes the noisy part of the
reference signal (along-track) as well as possible large-scale biases (scale >500 km). In the study, the validation metric is based on the
error variance scores in 1∘× 1∘ longitude × latitude
boxes (or averaged over a specific region of interest), defined as
σerrx,y=∑t=1NSSHerrorx,y,t-SSHerrorx,y,t‾2N.
The similar statistical analysis can also be performed on the geostrophic
velocity errors Uerror=Umap-Udrifter for the zonal
component and Verror=Vmap-Vdrifter for the
meridional component.
The comparison of the error variance score between two experiments informs
about the gain or reduction Δ of the mapping error; for example,
Δ=100⋅σerrEXP2-σerrEXP1σerrEXP1
The previous diagnosis is undertaken in physical space (space–time space).
For a more descriptive assessment by wavelength and to avoid spatio-temporal
filtering of independent and study datasets, diagnostics can be performed in
frequency space, using spectral analysis of SSH altimetry and gridded
datasets. More specifically, a spectral analysis can be applied to altimetry
data to estimate the effective resolution of gridded SSH products. It is
described, for example, in Ballarotta et al. (2019). Here, we recall the main
processing steps for the estimation of the effective resolution: (1) the
SSHmap data are interpolated to the locations of independent
SSHalong-track data, (2) the along-track and interpolated data are
divided into overlapping segments of 1500 km length every 300 km, (3) each
segment is stored in a database and referenced by its median coordinates
(longitude, latitude), and (4) finally, between latitudes 90∘ N–90∘ S and longitudes 0∘–360∘ E, we consider
10∘× 10∘ longitude × latitude boxes
for the global products every 1∘ increment. All available segments
referenced in the 10∘× 10∘ box are selected to
compute the mean power spectral densities of the independent signal
(SSHalong-track) and the mapping error (SSHmap-SSHalong-track). Before the spectral calculation, the signals are
detrended, and a Hanning window is applied. The signal-to-noise (SNR) ratio
(Eq. 8) is then derived from the power spectral density (PSD) of the along-track SSH (SSHalong-track) and the PSD of the
error (SSHmap-SSHalong-track). As in Ballarotta et al. (2019), the
effective resolution is then given by the wavelength λs where the
SNR(λs) is 2 (Eq. 9), i.e. the wavelength where the
SSHerror is 2 times lower than the signal SSHalong-track.
8SNRλ=PSD(SSHalong-track)(λ))PSDSSHerrorλ9SNRλs=2
ResultsQualitative assessment
Here we qualitatively assess the gridded products from the DUACS allsat-1 and MIOST allsat-1 80 % drifters + equatorial waves + L3 Arctic experiments.
The SLA maps from the DUACS and MIOST mapping approaches are relatively
similar in the subpolar region, as illustrated in Fig. 5 by an example of
SLA reconstruction on 15 February 2019 for (a) the DUACS mapping approach and (b)
the MIOST mapping approach. More significant differences take place in the
Arctic basin: in contrast to the DUACS products, the use of Arctic lead
observations in MIOST offers the possibility to extend sea-level mapping
into ice-covered area and thus to deliver gap-free maps to end users
(Fig. 5b).
Example of sea-level anomaly maps on 15 February 2019 over the Arctic
region constructed with the DUACS mapping approach (a) and with the MIOST
mapping approach (b). The black line contour indicates the 15 % sea-ice
concentration from the OSI-SAF product.
From a global perspective, the MIOST maps are slightly more energetic than
the DUACS maps as illustrated in Fig. 6 with the variance maps and their
differences. The difference between MIOST and DUACS variance maps (Fig. 6c) indicates regions of higher variability in the MIOST maps (>10 %) than in the DUACS maps, such as in the equatorial band, regions of
low variability at mid-latitudes, and coastal and polar regions. Tropical ocean
regions are prone to lower SSH variability (10 %) in the MIOST maps than
in the DUACS maps.
Variance (in m2) of sea-level anomaly maps constructed with
(a) the DUACS approach, (b) the MIOST approach, and (c) the difference between the
MIOST and DUACS variance maps expressed in percent.
The large SSH variability in the equatorial band of the MIOST maps is mainly
associated with the equatorial wave components. The zonal
wavenumber–frequency spectrum of SSH in the Pacific has been investigated
in several studies (e.g. Shinoda et al., 2009; Farrar, 2008, 2011) to examine
the SSH variability associated with tropical and equatorial waves. Figure 7
shows contours of the base 10 logarithm of power in the wavenumber–frequency
space calculated from SSH in the equatorial Pacific (region [10∘ S–10∘ N], [180–280∘ E]) for the period 2008
to 2018, for (a) DUACS, (b) MIOST with equatorial wave modes, and (c) the
GLORYS12V1 reanalysis (Lellouche et al., 2018). The rapid equatorial wave
dynamics are resolved in the GLORYS12v1 ocean numerical simulation (Fig. 7c): the zonal wavenumber–frequency spectrum of the SSH in the Pacific
reveals significant spectral peaks at periods close to 4, 5, and 7 d for a wavelength >20∘ in longitude. These peaks
are associated with inertia-gravity (Poincaré) waves. These SSH
variabilities for timescales smaller than 10 d are filtered in the DUACS
mapping approach (Fig. 7a). In contrast, the MIOST multiscale mapping
approach (MIOST allsat-1 80 % drifters + equatorial waves + L3 Arctic) resolves spectral peaks near 4, 5, and 7 d
for wavelengths >20∘ in longitude (Fig. 7b). We show
in the next section that these equatorial wave modes in MIOST also
contribute to a significant reduction of the mapping error in this region. For
timescales >10 d, each dataset has relatively similar
spectral contents, particularly the energetic westward propagation of
equatorial Rossby waves for negative wavenumbers.
Zonal wavenumber–frequency spectrum of SLA in the equatorial Pacific computed for (a) DUACS, (b) MIOST with equatorial wave modes, and (c) the GLORYS12V1 reanalysis. White lines represent the theoretical dispersion
relation curves for equatorial waves corresponding to the Kelvin [1], Yanai
[2], Rossby [3], and Poincaré [4] waves.
The first assessment is a comparison of the DUACS allsat-1 and MIOST allsat-1 experiments. Both
experiments aim to map the mesoscale circulation from altimetry data only.
The SARAL/AltiKa altimeter and drifter sensors are not included in the
mapping but are used as independent validation.
Sea-level anomaly quality
The largest SSH mapping error σerr in DUACS allsat-1 reaches 50–100 cm2 in
the western boundary surface current and over the continental plateaus
(Fig. 8a and b). In the offshore low-variability region, the error
variance is <10 cm2. Figure 8c and d show the difference in
mapping error between the MIOST allsat-1 and DUACS allsat-1 experiments for all spatial scales and the spatial
scale between 65 and 500 km, respectively. A blue (red) pattern means a
reduction (increase) of the mapping error in MIOST compared to DUACS. For all
spatial scales considered, MIOST mapping errors are smaller than those of
DUACS, especially at mid-latitudes, with an average reduction in mapping error
between 5 % and 10 %. The largest reduction in mapping error
(∼ 10 %) is found in regions of high variability. In the
intertropical region, MIOST and DUACS have similar scores. For spatial
scales between 65 and 500 km, MIOST mapping errors are reduced by
∼ 10 % compared to DUACS in high-variability regions at
mid-latitudes. In low-variability regions, the mapping error is between 3 and
4 % smaller with MIOST than with DUACS, but the mapping errors are locally
larger with MIOST than with DUACS: for example, in the Argentine Sea, the
Siberian plateau, and the New Zealand plateau. Table 4 summarizes the results of
the comparison over different regions of interest (Arctic, Antarctic,
equatorial band, low-variability region, and high-variability region).
Overall, the geostrophic flows in the MIOST SSH maps are closer to the
independent SARAL/AltiKa observations than those in DUACS maps.
Variance of the difference SSHmap-SSHalong-track computed
for the DUACS allsat-1 experiment and in considering (a) all spatial scales
and (b) the spatial scale between 65 and 500 km. Gain/loss of the mapping
error variance of SLA in the MIOST allsat-1 experiment relative to the DUACS
allsat-1 mapping error variance for (c) all spatial scales and (d) the scale
between 65 and 500 km. Blue colour means a reduction of error variance in
MIOST.
Regionally averaged mapping error variance and gain/reduction of
error variance on the SSH variable between MIOST and DUACS.
Figure 9a and b show the validation against the independent drifter
velocity data in terms of mapping error σerr for the zonal and
meridional velocities. The largest mapping error σerr in DUACS
reaches 300 to 400 cm2 s-2 in the western boundary surface current
(e.g. the Gulf Stream and the Kuroshio, Mozambique, and Agulhas currents). In
offshore low-variability regions, the error variance is <80 cm2 s-2. The differences in mapping error between MIOST and DUACS
are shown in Fig. 9c and d for zonal and meridional velocities,
respectively. Mapping errors are smaller in MIOST than in DUACS, mainly in
the core of the ocean gyres. In the intertropical region, the DUACS maps
appear to be closer to the independent drifter velocities than MIOST. Table 5 summarizes the results of the comparison over different regions of
interest (Arctic, Antarctic, equatorial band, low-variability region, and
high-variability region). Overall, MIOST surface velocities are slightly
closer to drifter velocities than the DUACS surface velocities.
Variance of the difference Umap-Udrifter computed for
the DUACS allsat-1 experiment and in considering (a) the zonal velocity
component and (b) the meridional velocity component. Gain/loss of the mapping
error variance of currents in the MIOST allsat-1 experiment relative to the
DUACS allsat-1 mapping error variance for (c) the zonal velocity component
and (d) the meridional velocity component. Blue colour means a reduction of
error variance in MIOST.
Regionally averaged mapping error variance and gain/reduction of
error variance on the surface currents between MIOST and DUACS.
The comparison of the MIOST allsat-1 80 % drifters + equatorial waves + L3 Arctic experiment with MIOST allsat-1 examines the impact of the
equatorial waves' mode and the drifters' observations in the MIOST mapping
approach.
Sea-level anomaly quality
The differences in mapping error between MIOST allsat-1 80 % drifters + equatorial waves + L3 Arctic and MIOST allsat-1 are shown in Fig. 10a
and b for all spatial scales and the spatial scale between 65 and 500 km,
respectively. For all spatial scales considered, we observe that the
equatorial wave modes locally reduce the mapping error in the equatorial
band by more than 10 %. However, coastal equatorial regions (e.g.
Indonesian Archipelago, western and eastern parts of Africa, and South
America) are prone to deterioration. This suggests that the equatorial wave
mapping is not adapted in these coastal regions where different ocean
processes are at play. In extra-equatorial regions, we evaluate the impact
of drifter observations in MIOST. This impact is moderate on the SLA mapping
(a few percent of difference in the mapping error variance), with a reduction
of error variance mainly in the high-variability regions. For a spatial scale
between 65 and 500 km (Fig. 10b), the equatorial wave modes deteriorate
the mapping solution in the western and central equatorial Pacific Ocean and in
the Indian Ocean, while a reduced mapping error is found in the eastern
equatorial Pacific and the equatorial Atlantic. In the extra-equatorial
region, the impact of drifter observations remains moderate (with 1.5 %
error variance reduction in the high-variability region). Overall, the
drifters reduce the mapping errors primarily in regions of intense dynamics
where the temporal sampling must be sufficiently accurate to properly map
the rapid mesoscale dynamics. Table 6 summarizes the results of the
comparison over different regions of interest (Arctic, Antarctic, equatorial
band, low-variability region, and high-variability region).
Gain/loss of the mapping error variance of SLA in the MIOST allsat-1
80 % drifters + equatorial waves + L3 Arctic experiment relative to
the MIOST allsat-1 mapping error variance for (a) all spatial scales and (b)
the scale between 65 and 500 km. Blue colour means a reduction of error
variance in MIOST when drifters are included in the mapping and with
equatorial wave parametrization.
Regionally averaged mapping error variance and gain/reduction of
error variance on the SSH variable between MIOST allsat-1 80 % drifters + equatorial waves + L3 Arctic and MIOST allsat-1.
The differences in mapping error of surface geostrophic currents between
MIOST allsat-1 80 % drifters + equatorial waves + L3 Arctic and MIOST allsat-1 are shown in Fig. 11a and b for the zonal component and the
meridional component of the velocity, respectively. It is difficult to draw
conclusions from this diagnosis: the mapping errors are reduced with MIOST
in some regions in the tropics (such as the Bay of Bengal) and in the Kuroshio
extension. Overall, the contribution of drifters remains moderate for the
restitution of geostrophic currents (only a few percent improvement in the open
ocean), as summarized in Table 7.
Gain/loss of the mapping error variance of currents in the MIOST
allsat-1 80 % drifters + equatorial waves + L3 Arctic experiment
relative to the MIOST allsat-1 mapping error variance for (c) the zonal
velocity component and (d) the meridional velocity component. Blue colour
means a reduction of error in MIOST when drifters are included in the
mapping and with equatorial wave parametrization.
Regionally averaged mapping error variance and gain/reduction of
error variance on the surface currents between MIOST allsat-1 and MIOST allsat-1 80 % drifters + equatorial waves + L3 Arctic.
The comparison of the MIOST allsat-1 80 % drifters + equatorial waves + L3 Arctic and DUACS allsat-1 experiments allows the
complete MIOST product distributed to users to be evaluated against the DUACS method.
Sea-level anomaly quality
The differences in mapping error between MIOST allsat-1 80 % drifters + equatorial waves + L3 Arctic and DUACS allsat-1 are shown in Fig. 12a
and b for all spatial scales and the spatial scale between 65 and 500 km,
respectively. We have the same pattern as found in the previous sections:
for all spatial scales considered (Fig. 12a), the equatorial wave modes
help to reduce the mapping error variance in the equatorial band by more
than 20 % locally. At mid-latitudes, the mapping error is between 5 %
and 10 % smaller with MIOST than with DUACS. For spatial scales between 65
and 500 km, MIOST and DUACS solutions are globally equivalent, except in the
high-variability region where the mapping error is between 10 % and 20 %
smaller with MIOST than with DUACS. The mapping errors are locally larger
with MIOST than with DUACS in regions where the circulation interacts with
bathymetry features such as in the Argentine Sea and near the Siberian plateau
and New Zealand plateau. Table 8 summarizes the results of the comparison
over different regions of interest: mapping errors are ∼ 11 % smaller in high-variability regions in MIOST than in DUACS. In other
regions, the errors are ∼ 3 %–6 % smaller.
Gain/loss of the mapping error variance of SLA in the MIOST allsat-1
80 % drifters + equatorial waves + L3 Arctic experiment relative to
the DUACS allsat-1 mapping error variance for (a) all spatial scales and (b)
the scale between 65 and 500 km. Blue colour means a reduction of error
variance in MIOST.
Regionally averaged mapping error variance and gain/reduction of
error variance on the SSH variable between MIOST allsat-1 80 % drifters + equatorial waves + L3 Arctic and DUACS allsat-1.
The differences in mapping error of surface geostrophic currents between
MIOST allsat-1 80 % drifters + equatorial waves + L3 Arctic and DUACS allsat-1 are shown in Fig. 13a and b for the zonal component and the
meridional component of the velocity, respectively. The mapping errors are
globally smaller in MIOST than in DUACS, particularly in the high-variability regions. In the tropical regions, DUACS outperforms MIOST for
reconstructing the surface geostrophic velocities. Overall, the mapping
errors are on average between ∼ 2 % and 5 % smaller with
MIOST than with DUACS (Table 9).
Gain/loss of the mapping error variance of currents in the MIOST
allsat-1 80 % drifters + equatorial waves + L3 Arctic experiment
relative to the DUACS allsat-1 mapping error variance for (c) the zonal
velocity component and (d) the meridional velocity component. Blue colour
means a reduction of error in MIOST.
Maps of effective spatial resolution (in km) for (a) the DUACS
allsat-1 and (b) MIOST allsat-1 80 % drifters + equatorial waves + L3
Arctic experiments and (c) gain/loss of effective resolution (in %)
between MIOST and DUACS. Blue means finer resolution in MIOST than in DUACS.
Regionally averaged mapping error variance and gain/reduction of
error variance on the surface currents between MIOST allsat-1 80 % drifters + equatorial waves + L3 Arctic and DUACS allsat-1.
The effective spatial resolution quantifies the minimum spatial scale
resolved in the maps (Ballarotta et al., 2019). Maps of the effective
spatial resolution (expressed in kilometres) are presented in Fig. 14a and
b for DUACS allsat-1 and MIOST allsat-1 80 % drifters + equatorial waves + L3 Arctic, respectively. For each experiment, the effective
spatial resolution varies from ∼ 500 km at the Equator to
∼ 100 km at high latitudes and a mean value at mid-latitudes
close to 200 km. The difference in effective spatial resolution between the
two experiments is shown in Fig. 14c. The resolution of the SLA maps of
the MIOST experiment is overall finer than in the SLA maps of the DUACS
experiment. It is between 5 % and 10 % finer than the DUACS maps in
regions of high variability (the Gulf Stream, Kuroshio, and Agulhas regions), in
the Atlantic and equatorial Pacific, and in the Norwegian and Greenland
seas. Some regions (e.g. tropical regions, coastal regions, the East China
Sea, the New Zealand Shelf, or the Argentine Sea) are subject to a coarser
effective resolution in MIOST maps than in DUACS maps. These regions will
require further investigation in the near future.
Data availability
The MIOST gridded products are hosted
on the AVISO+ (Archivage, Validation et Interprétation
des données des Satellites Océanographiques) website (https://doi.org/10.24400/527896/a01-2022.009, Ballarotta et al., 2022).
The reference DUACS maps are hosted on the EU Copernicus Marine Service
portal (https://doi.org/10.48670/moi-00148, Pujol et al., 2022b). The multiscale and multivariate products are distributed on a regular grid:
the spatial grid extends from 0 to 360∘ E in
longitude and 80∘ S to 90∘ N in latitude, with a grid
spacing of 0.1∘; the temporal grid covers the period 1 July 2016
to 30 June 2020 with a time step of 1 d. The dataset is distributed in
netCDF4 format. Each netCDF file contains six variables: sla, adt, ugosa,vgosa, ugos, and vgos (see the list of variables available in the MIOST product in Fig. 15).
List of variables available in the multiscale and multivariate product.
Summary and conclusions
Ubelmann et al. (2021, 2022) evaluated the multiscale and multivariate
mapping approach in the Observing System Simulation Experiment (OSSE) and
the Observing System Experiment (OSE) for the simultaneous mapping of mesoscale
circulation, coherent internal tides, and surface geostrophic and ageostrophic
velocities. Here, we extend the application of the MIOST solution to the
simultaneous mapping of equatorial waves and mesoscale circulation from real
observations. Furthermore, we investigate the levels of mapping improvement
by enhancing the sampling of the ocean surface state with in situ data and
altimetry data in the Arctic sea-ice regions. We found that the Arctic lead
SSH observations allow the monitoring coverage in
this remote region to be significantly improved. The gap-free maps, proposed with MIOST, hence offer the opportunity to
end users to study the Arctic surface circulation and
its connections to the subpolar and mid-latitude regions. It is important to
mention that this polar mapping will need to be validated against
independent data in the near future. Drifters' observations have a moderate
impact in the mapping. They mainly contribute to reduce mapping errors in
regions of intense dynamics where the temporal sampling must be accurate
enough to properly map the rapid mesoscale dynamics. It is important note
that drifter observations can potentially improve surface circulation in
areas not or poorly sampled by altimeters. Therefore, their impact on the
sea-level reconstruction may be larger over periods of weak altimeter
sampling.
The ocean surface circulation involves a superposition of processes acting
at widely different spatial and temporal scales, from the geostrophic
large-scale and slow-varying flow to the mesoscale turbulent eddies and, at
even smaller scales, the mixing generated by the internal wave field. It is
also important to mention that the DUACS maps are constructed from altimetry
data using an interpolation method optimized for mapping mesoscale
variability. Consequently, some ocean surface variabilities are not or
poorly represented in these DUACS maps: equatorial wave dynamics is thus
part of the filtered ocean signals in DUACS. The multiscale approach allows
the observed SSH to be decomposed into various physical contributions. Here, we
explored and validated the possibility of improving the content of altimetry
maps by simultaneously estimating the ocean mesoscale circulations as well
as the equatorial wave dynamics associated with the tropical instability waves
and Poincaré waves. We show that mapping these ocean surface
variabilities from altimeter observations broadens the spectrum of mappable
space–timescales and reduces mapping errors by almost 20 % locally
relative to independent data, primarily in the equatorial Pacific and
Atlantic basins. This is possible because the spatio-temporal coverage of
the altimeter data allows large-scale waves of 4 d periods and
longer to be sampled. At global scale, we also found that, compared to the operational
DUACS mapping approach, MIOST approach improves the surface mesoscale
circulations in regions of high variability. Consequently, the effective
resolution of the maps produced by the multiscale approach is finer than the
DUACS maps, particularly in the western boundary currents and in the
equatorial band.
This experimental product is currently available on the AVISO + (Archivage,
Validation et Interprétation des données des Satellites
Océanographiques) website (see the “Data availability” section for more
details), but our results suggest that the multiscale and multivariate
mapping approach is very promising for use in an operational context. It is
also worth mentioning that several other global gridded products exist as an
alternative to the DUACS/MIOST products which provide only the geostrophic
part of the surface current. Examples of these other products that provide a
broader spectrum of ocean surface current variability (e.g. the total
surface currents) include (1) the Copernicus GLORYS12v1 global ocean
reanalysis (Lellouche et al., 2018; 10.48670/moi-00021, Drévillon et al., 2022), (2)
the Copernicus GLOBCURRENT product (Rio et al., 2014;
10.48670/moi-00050, Etienne, 2021b), or (3) the OSCAR product (Dohan, 2021;
10.5067/OSCAR-25F20 Dohan, 2021) distributed by the NASA-JPL Distributed
Physical Oceanography Active Archive Center (PO. DAAC).
To conclude, these results pave the way for the exploration of new types of
ocean signals that may eventually be mapped with MIOST from remote sensing
and in situ observations. Future work could consist of enriching the MIOST
components in considering oceanic signals missing in the maps and yet
captured by observing systems: for example, in mapping high-frequency
signals such as the near-inertial oscillation from drifter observations, in
using SSH lead products in the Southern Ocean (Auger et al., 2022), or by
enhancing the SLA map content with a dynamical model approach (Ubelmann et
al., 2015) or artificial intelligence methods (Beauchamp et al., 2020).
Mapping approaches tested in this studyThe optimal interpolation (DUACS mapping approach)
The DUACS mapping approach constructs a SSH field on a regular grid by
combining measurements from various altimeters. It is based on a global
suboptimal space–time objective analysis that considers along-track
correlated errors as described, for instance, in Ducet et al. (2000) or Le
Traon et al. (2003). The mathematical formulation, known as optimal
interpolation, is described hereafter.
We assume a state to estimate, denoted x, and partial
observations, denoted y, which can be related to the state by a
linear operator H such as
y=Hx+ϵ,
where ϵ is an independent signal (e.g. observation error) not
related to the state. If we define B the covariance matrix of
x and R the covariance matrix of ϵ, both variables
being assumed Gaussian, then the linear estimate is written as
xa=BHTHBHT-R-1y.
The observation vector y represents the SLA observations. The state vector
x is the gridded SLA. The operator H (formally a trilinear
interpolator transforming the gridded state SLA to the equivalent
along-track SLA) is not considered explicitly. The matrices
BHT and HBHT, representing the covariance of the
signal in the (grid, obs) and (obs, obs) spaces, are directly written with
the analytical formula of the Arhan and Colin de Verdière (1985)
covariance model as described in Ducet et al. (2000), Le Traon et al. (2003), or Pujol et al. (2016):
A3Cx,y,t=1+ar+16(ar)2-16(ar)3e-are-tLt2A4r=x-CpxtLx2+y-CpytLy2,
where, x, y, and t correspond to the zonal, meridional, and temporal position; Lx,
Ly, and Lt are the zonal, meridional, and temporal decorrelation scale;
Cpx and Cpy denote the phase speed; and
a is a constant (3.337).
This covariance model is mainly optimized for mesoscale signal
reconstruction. The R matrix represents the representativity and
instrumental errors. Since the covariance of mesoscale SLA is assumed to
vanish beyond a few hundreds of kilometres in space and beyond 10–20 d
in time (Le Traon and Dibarboure, 2002), separate inversions are performed
locally, selecting observations over time and space windows adjusted to these
values. In practice, since the number of observations is limited to less
than 1000 (Le Traon et al., 1998), the inversion in observation space is
computationally manageable. More details on the map production are given in
Pujol et al. (2016).
In DUACS, the geostrophic current (Ug, Vg) is then directly
derived from the mapped SSH:
A5Ugx,y=-gfc∂SSHx,y∂yA6Vgx,y=gfc∂SSHx,y∂x,
where g is the gravity, and fc is the Coriolis frequency, which is a function
of latitude.
A multiscale and multivariate mapping approach
The optimal interpolation requires the inversion of a matrix of the same
size as the observation vector y. When the number of observations exceeds
the size of the state to resolve, it can be interesting to use an equivalent
formulation given by the Sherman–Morrison–Woodbury transformation, allowing for
an inversion in state space, with a matrix of the size of the state vector
x:
xa=(HTR-1H+B-1)HTR-1y.
The formulation of the multiscale and multivariate mapping algorithm is
detailed in Ubelmann et al. (2022). Here we recall the main principle. We
consider an extended state vector × composed by N physical components that
will later be assumed independent. In this study N=3 for (1) geostrophy and
equatorial waves, (2) tropical instability waves (TIWs), and (3) Poincaré
waves:
x=x1T,…,xNTT.
Each component xk represents the state of the surface topography and
surface current to be resolved in the grid space, denoted xk=hkT,ukT,vkTT. The key aspect of the method
is a rank reduction of the state vector, through a subcomponent
decomposition, such as xk, which can be written as
xk=Γk,hΓk,uΓk,vηk=Γkηk,
where ηk is the reduced state vector for component k, and Γk,h, Γk,u, and Γk,v are the subcomponent matrices
expressed in topography and currents, respectively. Note that for some
components, one of the blocks can be set to zeros (e.g. if the geostrophy
component is considered to have zero contribution to SSH, which is the case for
the equatorial wave components). Their concatenation is called Γk,
which is the matrix transforming the reduced state vector in the grid space
for topography and currents. In practice, Γk will be a wavelet
decomposition of the time–space domain, with elements of appropriate
temporal and spatial scales to represent the component k. These wavelet
scales, and their specified variance set with a diagonal matrix noted
Qk, will define the equivalent covariance model Bk in the grid
space for component k:
Bk=ΓkQkΓkT.
The observation vector y is also extended to the observed surface topography
and surface current noted y=hoT,uroTT. Then,
if Hk is the observation operator for component k (from grid space to
observation space), we note Gk=HkΓk, the
subcomponent matrix expressed in observation space. In these conditions, the
observation vector y is the sum of all component contributions plus the
unexplained signal ϵ (instrument error and representativity):
y=∑k=1NGkηk+ϵ.
If we use the notation η=η1T,…,ηkTT for the concatenation of the subcomponent state
vectors, and G=(G1,…,GN), then we have
y=Gη+ϵ.
Applying the same transformation from Eqs. (A1), (A2), and (A7) to the reduced state vector η, the global solution is written as
ηa=GTR-1G+Q-1-1GTR-1y,
where Q is the covariance matrix of η, expressed as the
concatenation of the diagonal matrices Qk for each component. Finally,
the solution in the reduced-space projects into the grid space with the
following relation:
xa=Γηa.
In practice, to solve Eq. (A13), each block of G is directly filled
from the analytical expression of the reduced-space elements constituting
the columns of the matrix. Also, in many situations, the GTR-1G+Q-1 matrix, noted A hereafter, would be
too large to be inverted (as required by Eq. A13 explicitly). We use a
preconditioned conjugate gradient method to solve η=A-1z,
where z=GTR-1y is computed initially from G and the
observation vector y. The algorithm involves many iterations of Aη
computations for updated η until convergence is reached (when Aη
approaches z). Note that if A is too large to be written explicitly, the
result Aη can still be computed in two steps from a matrix
multiplication of G and then of GT. Once the solution η is
obtained, the projection in physical grid space given by Eq. (A14) is
applied sequentially, by summing the analytical expression of the ripples
applied to grid coordinates (the columns of Γ), separately for each
component k. As in any inversion based on linear analysis, the result
strongly relies on the choice of covariance models, here defined by the
reduced elements of each component.
Geostrophy component
Geostrophy is the component that has a signature on both topography and
currents and on which some synergy between altimetry and drifter
observations can be expected. Following the formulation provided in Ubelmann
et al. (2021), here we define the gridded variable H1 to resolve, and
the corresponding gridded geostrophic current field (U1, V1) is written
U1=-gfc∂H1∂yV1=gfc∂H1dx.
The proposed reduced state for geostrophy is based on an element
decomposition of H1, expressed by Γ1,h with wavelets of
various wavelength and temporal extensions. This will allow
the standard covariance models used in altimetry mapping to be approximated, accounting for
specific variations with wavelength and time. A given p element of the
decomposition Γ1,h is expressed as follows:
Γ1,hi,p=coskx,pxi-xp+ky,pyi-yp+Φp×ftapxi-xpLxp,yi-ypLyp,ti-tpLtp,
where the ith line of the matrix stands for a given grid index of
coordinates (xi, yi, ti). For the ensemble of p, Φp is alternatively 0 and π/2, such that all subcomponents are
defined by pairs of sine and cosine functions to allow for the phase degree of
freedom. kx,p and ky,p are zonal and meridional wavenumbers
respectively, set to vary in the mappable mesoscale range (between 80 and
900 km with a spacing inversely proportional to the wavelet extensions,
allowing a signal of any intermediate wavelength to be represented). (xp,
yp, tp) are the coordinates of a space–time pavement. The function
ftap localizes the subcomponent in time and space (at scales
Ltp , Lxp, and Lyp, respectively) as geostrophy has a local extension of covariances. It is expressed as
ftapδx,δy,δt=cosπ2δxcosπ2δycosπ2δy,forδx,δy,δy<1,1,10,elsewhere.
In practice, Lxp and Lyp will be set to 1.5 the wavelength
of element p and Ltp to the decorrelation timescale. Then, the same
element p of the decomposition also has an expression in the geostrophic current
(through the geostrophic relation Eq. A15) written in the Γ1,u and Γ1,v matrices:
Γ1,ui,p=-gfc∂Γ1,hi,p∂yiΓ1,vi,p=gfc∂Γ1,hi,p∂xi.
The whole time–space domain is paved with similar subcomponents, along
coordinates (xp, yp, tp) for wavelengths between 80 and
900 km spanning in all directions of the plan. The ensemble can be seen as a
wavelet basis. Finally, each subcomponent p is assigned an expected variance
in the Q1 matrix, consistent with the power spectrum observed from
altimetry at the corresponding wavelength with isotropy assumption.
Equatorial wave component
Here we define the gridded variables H2 and H3 to resolve TIW and
Poincaré waves, respectively, and we consider no contributions of the
equatorial wave components to the geostrophic currents; therefore the
corresponding gridded geostrophic current fields (U2, V2) and
(U3, V3) are written U2=U3=0, V2=V3=0. The
reduced state is represented in the time–space domain by the following
Γ2,h and Γ3,h matrix:
A19Γ2,hi,p=cosω2,t,pti-tp-k2,x,pxi-xp×ftapxi-xpL2,xp,yi-ypL2,yp,ti-tpL2,tpA20Γ3,hi,p=cosω3,t,pti-tp-k3,x,pxi-xp×ftapxi-xpL3,xp,yi-ypL3,yp,ti-tpL3,tp,
where k2,x,p and k3,x,p refer to the zonal wavenumber, and
ω2,t,p and ω3,t,p are the frequency which
satisfies the dispersion relation of the wave component (Matsuno, 1966);
for example,
ω2,t,p=c2⋅k2,x,pfor the TIW,c2=-0.5ms-1ω3,t,p=k3,k,p2⋅c32+β⋅c3⋅2⋅n+1for the Poincaré
waves,c3=±2.8ms-1,
where c2 and c3 denote the wave propagation speed (the sign indicating
the direction of propagation, negative for westward and positive for eastward),
β is the meridional gradient of the Coriolis frequency fc, and
n=1,2,3…
In the present study, we chose L2,tp=20 d, L2,xp=500 km, and L2,yp=300 km for the TIW component and
L3,tp=5 d, L3,xp=1000 km, and L3,yp=300 km for the equatorial Poincaré wave component. As for the
geostrophy component, the function ftap localizes the subcomponent in
time and space (at scales Ltp , Lxp, and Lyp,
respectively).
Author contributions
CU implemented the multiscale and multivariate algorithm. PV and PP prepared the Arctic lead sea-level dataset, and HE and SM prepared the drifter dataset. MB carried out the experiments and prepared the manuscript and figures. YF, GD, RM and NP participated in the discussion and interpretation of the results. All authors checked and provided related comments for this work.
Competing interests
The contact author has declared that none of the authors has any competing interests.
Disclaimer
Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Acknowledgements
The authors would like to thank the AVISO + (Archivage, Validation et Interprétation des données des Satellites
Océanographiques) team for their support and expertise in the
distribution of the dataset. We are grateful to the three anonymous
reviewers for their comments and suggestions to improve the manuscript.
Financial support
The work presented here was carried out in
the framework of the DUACS-R&D project funded by CNES.
Review statement
This paper was edited by Giuseppe M. R. Manzella and reviewed by three anonymous referees.
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