The ocean's sea surface height (SSH) field is a complex mix of motions in
geostrophic balance and unbalanced motions including high-frequency tides,
internal tides, and internal gravity waves. Barotropic tides are well
estimated for altimetric SSH in the open ocean, but the SSH signals of
internal tides remain. The transition scale,

The spatial scale of the transition from geostrophically balanced to unbalanced motions is estimated regionally from satellite altimetry data for the first time.

Results agree with in situ observations and predictions from high-resolution models including tidal forcing.

Global maps of multi-mission satellite altimetry sea surface height (SSH) are widely used in the ocean community, resolving the larger mesoscale dynamic scales greater than 150–200 km in wavelength (Chelton et al., 2011; Ballarotta et al., 2019). Our understanding of upper-ocean dynamics in the smaller mesoscale to submesoscale wavelength range (roughly 15–200 km) has seen great improvement in recent years due to the combined use of in situ measurements and state-of-the-art high-resolution ocean models (Sasaki et al 2014; Rocha et al., 2016a, b; Qiu et al., 2017, 2018; Klein et al., 2019). Processes at these spatial scales are essential in determining the upper-ocean energy budget through the kinetic energy cascade and energy dissipation (e.g., Ferrari and Wunsch, 2009; McWilliams, 2016; Rocha et al., 2016a). Additionally, they play a critical role in connecting the surface ocean with the interior through the modulation of the mixed layer seasonality and heat transfer (Capet et al., 2008; Klein et al., 2008; Thomas et al., 2008; Su et al., 2020; Siegelman, 2020).

Kinetic energy and SSH variance at these 15–200 km spatial scales are
partitioned between balanced (geostrophic) and unbalanced (ageostrophic)
motions. Quantifying the relative importance of each component of the flow
across the ocean is crucial for the correct calculation of geostrophic
currents from SSH for all satellite altimetry missions, including the
upcoming Surface Water and Ocean Topography (SWOT) high-resolution altimetry
mission. Barotropic tides are well estimated for altimetric SSH in the open
ocean, but the SSH signals of other ageostrophic high-frequency motions
remain. Recent results show that, depending on the location and season, the
energy and SSH signature associated with unbalanced motions (including
near-inertial flows, internal tides, and inertia-gravity waves) can overcome
that of the balanced motions at smaller scales (Rocha et al., 2016b; Qiu et
al., 2018; Chereskin et al., 2019), imposing a wavelength boundary beyond
which SSH measurements provided by satellite altimetry can no longer be used
to infer upper-ocean geostrophic flows. Documenting the spatial scale at
which this occurs (the so-called transition scale,

Tackling this problem needs high-resolution ocean data, ideally in space and
time. To date, progress on documenting

Along-track altimeter data have a finer spatial resolution than the mapped
data, and recent reprocessing now allows us to access oceanic scales down to
50–70 km for Jason-class altimeters and 35–50 km for SARAL/AltiKa
(Dufau
et al., 2016; Vergara et al., 2019; Lawrence and Callies, 2022). Most of the
unbalanced internal tide energy, and some of the internal gravity wave
energy, occurs at scales larger than 40 km wavelength and can be observed
with the latest along-track altimetry data (Zaron, 2019). Using along-track
SSH data from recent altimetric missions and a statistical approach based on
wavenumber spectral analysis, this paper will document the global
distribution of

Our satellite altimetry wavenumber spectral

Along-track SSH data from two missions with different technologies (Jason-3 – J3 conventional nadir altimetry; Sentinel-3A – S3 synthetic aperture radar nadir altimetry) are analyzed at a global scale. The time period analyzed spans their common 4-year period from March 2015 to March 2019.

Along-track SSH observations are maintained at their original 1 Hz observational position with 7 km spacing and are corrected for all instrumental, environmental, and geophysical corrections (Taburet et al., 2019). Only time-dependent variations of along-track SSH measurements are considered, following Stammer (1997), Le Traon et al. (2008), and Xu and Fu (2011, 2012). Since S3 is on a new repeat track, sea level anomalies (SLAs) are computed for both missions by subtracting the mean sea surface model CNES_CLS_2015 (Schaeffer et al., 2016; Pujol et al., 2018) from the along-track SSH measurements.

In order to obtain regionally varying spectral estimates, we apply the
methodology described in Vergara et al. (2019). We sample the along-track SSH
measurements inside a

For each average spectrum, we estimate the 1 Hz error level by fitting a straight flat line to the SLA power spectral density (PSD) level for wavelengths between 15 and 30 km; a similar technique was applied by Xu and Fu (2011), Dufau et al. (2016), and Vergara et al. (2019). This straight line fit is horizontal for J2 and S3 (white noise). The spectrum shape of S3 shows a slight slope over the 15 to 30 km wavelength range (red-type noise), which is a characteristic effect of the wind wave field on the synthetic aperture radar (SAR) measurements (Moreau et al., 2018). The differences in the unbiased spectrum and our methodology when applying either a red noise or white noise fit to the 15–30 km band of S3 data are explored in Appendix A.

The spatial patterns of the noise levels for Jason-3 and Sentinel-3A (Figs. 1a, 2a) approximately follow the spatial distribution of significant wave height (Dufau et al., 2016), with peaks in the regions of high sea state in the North Atlantic, Southern Ocean, and off the coast of southern Africa. The increased SSH noise of the current generation of satellite altimeters due to surface waves is well documented for both radar and SAR altimeters (Tran et al., 2002; Moreau et al., 2018, 2021). The latitudinal trend (Figs. 1c, 2c) shows an increase in the noise levels from the Equator towards the poles, in agreement with previous studies (Dufau et al., 2016; Vergara et al. 2019). Annual-mean Jason-3 wavenumber spectra noise levels range from 1.8 cm root mean square (rms) at the Equator to 2.8 cm rms in the Southern Ocean, whereas the Sentinel-3 SAR noise floor is smaller (1.4 cm rms at the Equator and 2.3 cm rms in the Southern Ocean). In general, noise levels observed for both satellites indeed show local maxima in the vicinity of the Gulf Stream, Kuroshio extension, and the ACC, related to local geophysical effects such as rain cells and more importantly the local wind wave field. Despite the relatively higher noise levels observed in these regions, the mesoscale signal is also strong, and therefore the signal-to-noise ratio remains favorable over these highly energetic regions (Figs. 1b and 2b).

Same as Fig. 1, but for Sentinel-3A data.

This computed flat noise spectral level is then subtracted from the PSD estimates over the entire wavenumber range, which provides an unbiased estimate of the regionally averaged spectrum (Xu and Fu, 2012). We then analyze the unbiased spectrum in order to determine two spectral slopes, taking into account the variations of spectral slope values in the fit. The mesoscale spectral slope is calculated within a geographically variable wavelength range: the maximum mesoscale wavelength is where the spectral shape significantly (at 95 %) departs from the observed mesoscale spectral slope (usually occurring at wavelengths larger than 500 km), and the minimum regional wavelength limit is based on the local eddy length scale, as in Vergara et al. (2019). Where possible, a second smaller-scale spectral slope is determined at wavenumbers between 30 km wavelength and the lower mesoscale spectral slope limit.

In order to analyze the two slopes from the regional unbiased spectrum
shape, we least-square-fit a linear model to the average spectrum obtained
from observations in the logarithmic space, defined as

The fitting algorithm is initialized using a first guess for the spectral slopes across the wavelength range: for the mesoscale spectral slope we follow Vergara et al. (2019) using the change in spectral slope at low wavenumber and the local eddy length scale as the wavelength bounds. The small-scale fit is initially computed as a linear fit between the spectral signal at 30 km wavelength and the signal at the local eddy length scale. These values are used for the first iteration and are recomputed at each step of the least-square minimization procedure to best resolve the double fit, maintaining the end points of the 30 km wavelength and the slope change at low wavenumber boundaries. The minimization process then adjusts the mesoscale and small-scale intercept to better capture the overall shape of the unbiased PSD.

An example of the results of this two-slope methodology is presented in
Fig. 3a for a region in the North Pacific Ocean in comparison to the
single-mesoscale-slope fit of Vergara et al. (2019) in Fig. 3b. The
one-slope mesoscale fit in Fig. 3b follows the wavenumber curve well
within the defined mesoscale range (vertical dashed lines) with a slope of

The observable wavelength (OWL) is defined as the threshold wavelength
where the SSH spectral signal exceeds this flat noise level (i.e., SNR

In addition to the fitting parameters for the model described by Eq. (1), we compute the uncertainty associated with the least-square fitting, related to each parameter. This helps us in the interpretation of the results by allowing us to estimate the validity of the spectral slope values for the large and small wavelength ranges and also their intercept.

The uncertainty (or error) emerges from the confidence interval envelope obtained when computing the regional average spectrum (gray shaded area in Fig. 3a and b). On this log scale, the 95 % envelope of the average PSD appears to grow considerably towards high wavenumbers. This is a consequence of the denoising method: the 95 % envelope is impacted by the subtraction of the noise plateau computed between 15 and 30 km wavelength, and the effect will become more evident in the high-wavenumber part of the spectrum given that the smaller amplitude of the PSD values is closer to the noise level.

Using the uncertainty estimates from the mesoscale and small-scale spectral
slopes, we can determine the error associated with their intercept by
propagating the uncertainty as

In this section, we analyze the temporal mean geographical distribution of
the mesoscale and small-scale spectral slopes computed using the model
described by Eq. (1). Additionally, we estimate (when possible) the
intercept wavelength between the two slopes in the spectral space. This
characteristic intercept wavelength may be considered a first-order
approach to the transition scale (

The methodology used to analyze the spectral shape does not explicitly separate the geostrophically balanced mesoscale regimes from small-scale regions of the spectrum. However, it allows us to infer their associated contributions to the observed PSD by means of the change in spectral slopes and their intercept. We will refer to the results of either part of the bilinear model as mesoscale and small-scale spectral slopes.

The geographical distribution of the mesoscale spectral slope values is very
close for the two missions analyzed (Figs. 4a and 5a), showing larger
slope values over the western boundary current systems, as well as the
Antarctic Circumpolar Current, indicative of the energetic mesoscale
circulation that is observed in both regions. The observed spectral slope
values in these highly energetic regions vary between

Same as Fig. 4 for Sentinel-3:

In some regions, the mesoscale spectral slope values obtained are slightly
lower (flatter slopes) for Sentinel-3A than for Jason-3. This could be
related to the white noise level model used for S3A. Red noise is often
observed in the 1 Hz SAR data at small wavelengths related to wave and swell
impacts on the signal processing (Moreau et al., 2018). Our methodology of
removing a white noise level computed over the 15–30 km wavelength range
could impact the unbiased PSD up to

Indeed, the zonal annual-mean distribution of the mesoscale spectral slope
values is the same for both missions (Figs. 4c and 5c, in blue), showing a
profile that is nearly symmetrical around the Equator, with values
increasing poleward, ranging between

The originality of this method is to estimate the second small-scale slope
from the wavenumber spectra when possible. The wavelength range for this
small-scale slope varies geographically but is generally between 30 and

On average, the linearly fitted small-scale spectral slopes vary between

Note that the optimal fitting algorithm is not able to separate the
contribution of two different spectral slopes in the intertropical band
equatorward of

The zonal-mean values of the valid small-scale slopes (Figs. 4c and 5c) show
a

The intercept of the fitted mesoscale and small-scale spectral slopes results can be used to obtain a characteristic wavelength for the change in their dynamical regime. If we consider that the mesoscale slopes reflect the balanced (s)QG dynamics at midlatitudes, and the valid small-scale slope values reflect the wave-like motions from internal tides or IGWs, the intercept wavelength therefore indicates the boundary at which the SSH variability would be mainly driven by either dynamical regime. This wavelength scale could be considered an approximation of the so-called transition scale from balanced to unbalanced motions (Qiu et al., 2017, 2018), which indicates the boundary between the circulation dominated by either geostrophically balanced or unbalanced motions (in terms of SSH variability). While Qiu et al. (2018) calculate the transition scale by explicitly filtering the contribution of balanced and unbalanced motions from the PSD in the wavenumber and frequency space, we compute the spatial scale directly from the observed PSD, assuming that its shape captures both dynamical regimes.

Considering the limitations inherent to satellite altimetry observations
(e.g., noise level, residual errors from corrections, and observability
wavelength), we also compute the uncertainty associated with our estimates
for the intercept wavelength. We then exclude any results that have the
following criteria: (1) uncertainty in the mesoscale and/or small-scale
spectral slope higher than 40 % and (2) an intercept wavelength value
less
than the local observable wavelength (OWL). Following these criteria, the
intercept wavelengths we can interpret from along-track altimetry are reduced
to a fraction of the world ocean. Nevertheless, within these constraints,
intercept wavelength spatial distribution shows higher values in the
tropical band and lower values towards the poles for the two missions
considered (Fig. 6). Both J3 and S3A show intercept wavelengths around
100 km on average in midlatitudes from 25–45

Intercept scale (in km) between the large- and small-scale
spectral slopes for J3

Using a state-of-the-art global ocean simulation, Qiu et al. (2018) recently explored the geographical and seasonal variations of the balanced to unbalanced transition scale, highlighting that the highly energetic western boundary current systems have relatively short transition scales, with the largest transition-scale values occurring in the relatively low-energy regions bounded by the intertropical and subpolar bands. Their modeled regional distribution reflected the local levels of mesoscale variability and the energy levels of unbalanced motions (near-inertial flows, internal tides and inertia-gravity waves). Large transition-scale values are also observed where prominent bathymetric features exist (Qiu et al., 2018) such as the North Atlantic Ridge and the western equatorial Pacific.

The observed J3 and S3A intercept values across each basin are similar to
recent modeling results of global estimates for the balanced to unbalanced
motion transition scale, with shorter transition wavelengths located in the
energetic western boundary current regions and longer values in the eastern
basins. Using the points that can be defined from our estimates, the
Kuroshio extension has intercept values of around 90–100 km for both J3 and
S3, with the Gulf Stream having values of around 60 km (Fig. 7), whereas the
eastern North Pacific intercept wavelengths reach values of 120–140 km. We
note that S3-A with lower noise has more geographical coverage of

Meridional average of the intercept wavelength presented in Fig. 4
over the North Pacific (130

Equatorward of 25

In this study we have explored the capability of currently available along-track data to capture the changes in the circulation variability at wavelengths shorter than 150 km. We used a statistical approach consisting of analyzing the shape of the SSH power spectral density, which can be indicative of the underlying circulation dynamics. In addition to the mesoscale spectral slope, we compute a secondary spectral slope at smaller spatial scales in a wavelength region that is characterized by a regime change from geostrophically balanced mesoscale motions to a potentially non-geostrophic regime. The methodology used here is based on an unbiased slope estimate after removal of white-type instrument noise. However, the least-squares fit takes into account the variance of the errors associated with the instrumental noise, which grows towards the high-wavenumber part of the SSH spectrum as the signal amplitude decreases and therefore increases the uncertainty of the estimated parameters towards short wavelengths.

A second outcome was to compute the intercept of the mesoscale and small-scale spectral slopes estimated in order to obtain a characteristic transition wavelength. We interpret this wavelength as a proxy for the transition scale that marks the boundary between the geostrophically balanced and unbalanced motions in the SSH signal. Despite the limited number of intercept wavelengths that pass our rigorous selection criteria, these intercept wavelengths show a distinctive geographical pattern. Higher spectral slope transition values are located around the tropics, sometimes exceeding 200 km in wavelength, and towards the eastern ocean boundaries (between 100–150 km), in agreement with circulation patterns with important wave-like variability at the mesoscale wavelength range (Pollmann, 2020; Tchilibou et al., 2018) compared to the local eddy field. The shortest intercept wavelengths are, on the other hand, observed over the western boundaries and towards high latitudes, where the circulation variability is dominated by an energetic mesoscale eddy field.

In our two-slope methodology, the larger mesoscale spectral slope estimate starts from a first guess based on the geographically variable wavelength range specified in Vergara et al. (2019). Then, the least-squares minimization of the two-slope fit allows this minimum wavelength range to be adjusted. As opposed to previous studies, we also include in our spectral slope estimates the inherent uncertainty that is contained in the altimetric observations. We have compared the results of the mesoscale spectral slopes from the bilinear solution against the observed spectrum (Fig. 8). For consistency, we used the locally variable wavelength range proposed in Vergara et al. (2019) to compute the mesoscale spectral slopes. Overall, we observe that the differences in spectral slopes when diagnosing either the optimal fit solution or the observed spectra are small (of the order of 0.1) across the world ocean. The differences are slightly higher in the equatorial regions (from both datasets) between 0.3 to 0.5 (Fig. 8) but remain smaller than the average uncertainty associated with the mesoscale spectral slope for the bilinear fit for these latitudes (Figs. 4 and 5).

Difference in the mesoscale spectral slope estimates computed over
the fitted solution (from Eq. 1) and the observed spectrum using the simple
linear fit approach of Vergara et al. (2019).

At wavelengths shorter than 150 km, the analysis of the SSH spectral shape becomes increasingly sensitive to the observation errors (i.e., instrumental error, accuracy of the altimetric corrections), and therefore the interpretation of the results at high wavenumber needs to account for the increased uncertainty compared to the mesoscale wavelength range. Our methodology for the analysis of the unbiased spectral shape assumes a white noise plateau for the J3 and S3A 1 Hz observations, which is removed to reveal the SSH spectrum free of instrumental errors. Using this first-order approximation for the instrumental error significantly increases the uncertainty of the spectral estimates towards high wavenumbers in comparison to their weak amplitude (i.e., the 95 % confidence interval envelope grows as we move towards the small-scale part of the spectrum). Thus, the uncertainty in the estimates compared to the signal is more important in the small-scale part of the spectrum than in the mesoscale wavelength range. This uncertainty also affects the estimates of the intercept wavelength in Eq. (2).

In addition, in regions where the first baroclinic Rossby radius is small
(e.g., high latitudes) and/or the mesoscale energy is intense, the mesoscale
spectral slope dominates the double fit and extends down to small
wavelengths. In this case, there is not much wavelength range above 30 km to
perform a second slope fit, and this combines with the larger error variance
at small scales. We therefore observe an inverse relationship between the
error associated with the small-scale spectral slope and the wavelength
range used to perform the small-scale fit (Fig. 9) (i.e., the range between
the intercept wavelength and the 30 km wavelength, the limit for computing
the noise plateau). We note that Jason-3 (Fig. 9a) has higher error variance
than S3A (Fig. 9b), as expected, and this larger error extends over a longer
wavelength range, whereas the S3A ratio of the small-scale spectral slope to error
tends to be confined to smaller wavelength ranges. Note that values of
small-scale slope error ratio

Error in the determination of the small-scale spectral slope
plotted against the wavelength range between the computed intercept scale
and 30 km wavelength (noise level limit) for Jason-3

At these smaller spatial scales, the observed variability may result from
different sources, both geophysical and instrument- or platform-related, and the
diagnosed small-scale spectral slope is potentially a combination of such
elements. Among the dynamical contributions, it has been shown that a
significant part of the SSH PSD spectrum at wavelengths shorter than 150 km
is related to phase-locked and non-phase-locked internal tides (Ray and
Zaron, 2016). An important cascade of energy is apparent in the SSH spectrum
around the tropical latitudes (Tchilibou et al., 2018), with an increased
high-frequency variability of tidal origin (mainly non-phase-locked) for
wavelengths shorter than 70 km (Tchilibou et al., 2022). Using a
high-resolution global ocean general circulation model (OGCM), Qiu et al. (2018) explored the influence of the
internal tides on their estimates of

In addition to the dynamical contributions, the altimetry-based small-scale
spectral slope estimates may also be influenced by errors in the altimetric
measurements used. One source of error that has been characterized at
wavelengths ranging from 30 to

The seasonal variability of the spectral characteristics derived from altimetric observations has been documented in recent literature (e.g., Dufau et al., 2016; Vergara et al., 2019; Lawrence and Callies, 2022), highlighting the fact that the variations of the spectral shape are related to changes in both the underlying circulation and surface ocean stratification as expected, but also to variations of the altimetric noise levels throughout the year. The methodology used in the present paper also reveals a marked seasonal change of the spectral slopes, with variations in the mesoscale wavelength range showing sharper values during summer months than during winter as a by-product of the interaction between the higher noise levels during winter and the presence of small-scale turbulence that is generated through vigorous vertical mixing (Sasaki et al., 2014; Callies et al., 2015). This small-scale variability is therefore partially masked by the increased noise levels (and increased uncertainty in our slope estimates) during winter months. On the other hand, during summer months the instrumental noise levels drop, and hence the SSH observability spans a large wavelength range with favorable signal-to-noise ratio. During summer months, higher spectral slope values are consistent with interior QG dynamics, suggesting that large eddies are formed through baroclinic instabilities in the thermocline and very little energy cascades to smaller scales (e.g., Callies et al., 2015). The combination of favorable conditions for the generation of eddies at mesoscales (larger than 100 km) and lower noise levels provides an ideal altimetric observability scenario during summer months. The seasonal variability of the mesoscale and small-scale spectral slopes is documented in Appendix B.

The intercept wavelength also modulates seasonally, suggesting that
information about the SSH variability in the sub-100 km wavelength range is
effectively reflected by this parameter computed from along-track altimetry
observations (Figs. 10 and 11). Changes in the upper-ocean stratification
will significantly modulate the energy levels of balanced and unbalanced
motions that collectively contribute to the SSH variability observed for
wavelengths ranging from 15 to 200 km. During summer months, a shallow mixed
layer with a sharp density gradient at its base works to enhance the surface
unbalanced motion kinetic energy (Rocha et al., 2016b) that surpasses the
energy levels of the geostrophic turbulence at

Intercept scale (in km) averaged during

Same as Fig. 10, but for Sentinel-3.

We note that our zonally averaged results are only calculated in the
non-masked areas and have partial coverage, but this first altimeter-based
seasonal

Although our analyses show the possibility to partially diagnose the small-scale part of the SSH spectrum, a thorough diagnosis of the impact of the instrumental noise levels on the methodology presented in this paper should also be carried out. This could be built around a series of observing system simulation experiments that simulate the along-track observations and also isolate the different contribution to the SSH energy spectrum. This is planned for future work.

There are two major implications for these spectral analysis results. The
first is that the observable wavelength of all SSH signals above the
instrument noise is limited to 60–70 km for Jason-3 and 50–70 km for S3-A
(Figs. 1 and 2). So, at present, any along-track altimetric studies
addressing either balanced ocean dynamics or internal tides or internal
gravity waves will be limited to these spatial scales by this instrument
noise level. Recent improvements in high-resolution

The performance of the upcoming SWOT mission, representing a new generation of altimeter technology, anticipates more than 1 order of magnitude of noise level reduction compared to current 1 Hz Jason observations (Fu and Ubelmann, 2014). Refined estimates for the SWOT SNR, including realistic wind-wave effects on the interferometric technology, anticipate an observability ranging from around 15 km in wavelength at low latitudes to around 30–45 km in wavelength towards the poles, with an important longitudinal dependence (Wang et al., 2019). This will greatly extend our capacity to estimate the smaller-spectral slope, especially at middle to high latitudes.

The second implication is for the

Our results indicate that at low latitudes, the intercept wavelength remains large (100–150 km), suggesting that the changes in the spectral slope will be well observed in two dimensions by the future SWOT mission with its reduced noise level. The estimated observability at high latitudes, particularly in the ACC, could still be a challenge for diagnosing the transition from geostrophic to non-geostrophic circulation regimes from SWOT observations alone unless noise reduction techniques are also applied to SWOT data. Wang et al. (2019) estimate the observed wavelengths for SWOT in the ACC to be between 30 and 45 km, with an important longitudinal dependency. Our very limited along-track estimates in this region indicate that the spectral slope break should occur in the 40–60 km wavelength range on average, as do the modeling estimates from Qiu et al. (2018). In situ observations from the Drake Passage report that half of the near-surface kinetic energy between 10 and 40 km wavelength is accounted for by ageostrophic motions (Rocha et al., 2016a), likely dominated by inertia-gravity waves. Our estimates also reveal an inherent geographical variability of the intercept wavelength, suggesting a localized dependence of the different dynamical regimes around the ACC that was also observed by Wang et al. (2019) for the region. This implies that the observability in the ACC will not be a constant threshold but rather a pattern dominated by localized and seasonal variability.

The altimetry data used in the present paper are fully
available at

OV, RM, and MIP conceived the methodology used to analyze the sea surface height spectra and their shape changes. The analyses presented in this paper are the collective effort of all the authors. OV wrote the paper in close collaboration with RM. MIP, GD, and CU provided altimetry data expertise and critical advice on the methodology.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The Jason-3 and Sentinel-3A data were obtained from the Geophysical Data
Record (GDR) available on the AVISO website
(

This research has been supported by the Centre National d'Etudes Spatiales (TOSCA program grant).

This paper was edited by Anne Marie Treguier and reviewed by two anonymous referees.