This study proposes an approach to estimate the ocean sea surface height signature of coherent internal tides from a 25-year along-track altimetry record, with a single inversion over time, resolving both internal tide contributions and mesoscale eddy variability. The inversion is performed on a reduced-order basis of topography and practically achieved with a conjugate gradient. The particularity of this approach is to mitigate the potential aliasing effects between mesoscales and internal tide estimation from the uneven altimetry sampling (observing the sum of these components) by accounting for their statistics simultaneously, while other methods generally use a prior mesoscale. The four major tidal components are considered (M2, K1, S2, O1) over the period 1992–2017 on a global configuration. From the solution, we use altimetry data after 2017 for independent validation in order to evaluate the performance of the simultaneous inversion and compare it with an existing model.

Ocean internal gravity waves forced by barotropic tidal currents, known as internal tides, have signatures on sea surface height (SSH) at scales below 200 km (e.g.,

For the sake of improving the empirical approaches, in this study we propose to tackle the issue of mesoscale aliasing in internal tide estimation. Indeed, the mesoscale signal is overall an order of magnitude above internal tides (the typical amplitudes of the latter being as small as

In the first section, we present the principle of the simultaneous estimation, with illustrations from idealized one-dimensional synthetic signals. Then, in Sect. 2, the application to altimetry data through the mode decomposition is detailed, followed by the results and validation with independent data. Finally, we will discuss its potential, in particular to handle non-stationary internal tides with future wide-swath altimetry data.

The estimation of pure harmonics from partial observations with errors or additional signals is generally performed with harmonic analysis. However, this inversion does not account for particular spatial covariances of the errors or additional signals. When additional errors are white random noise, the impact is minimized and harmonic analysis remains most optimal. However, when we have the knowledge of covariances for the additional signals, especially if they are non-zero between observation coordinates, the estimation can be improved with a full consideration of the signals in the inversion.

Here, we present the general inversion formula of a sum of

We assume

If we note

If we assume that the

The formulation presented above can be easily tested on a one-dimensional synthetic case, in particular to evaluate the gain of the simultaneous inversion as presented, rather than separate inversions.
Let us consider a one-dimensional signal, constituted by the sum of a broadband component following the spectrum shown in black in Fig.

Now let us assume partial observations of this total signal, as represented by the black dots in the figure, here at random times, with typical occurrences of 3 to 5 days and random observation errors of 0.01.

From these observations, we will illustrate the simultaneous inversion method applied to the two components (

For the broadband part,

For the narrowband part,

We used Eq. (

In parallel, we run the so-called “separate estimations”, where the inversions are applied separately for

We also run the so-called “sequential estimations”, consisting of separate estimations with a modified vector of observation, whom the separate estimation of the unconsidered component has been removed. Note that this sequential estimation is similar to what is generally done for internal tide estimations with previous removal of mesoscale signal (e.g., CMEMS/AVISO mesoscale maps).

The results of the different solutions are shown in Fig.

For the case of coherent internal tides, the waves span over 30 years of satellite altimetry, so a single inversion over time is not as trivial as in the illustration presented above. Even with suited spatial localization, the number of observations would exceed the limit of matrix invisibility (and even matrix storage). To overcome this issue, one can define covariances through the construction of a reduced basis as presented in

We used the along-track sea level anomalies (SLAs) from CMEMS labeled “GLOBAL OCEAN ALONG-TRACK L3 SEA SURFACE HEIGHTS REPROCESSED (1993-ONGOING) TAILORED FOR DATA ASSIMILATION” provided at 1 Hz resolution (approximately 6 km along-track posting). We then processed super observations at 0.33 Hz (approximately 18 km) by simply averaging groups of three consecutive points. This second step allowed us to significantly reduce the computing cost, without affecting the results (sensitivity tests, not shown, suggested an impact beyond averaging four consecutive points). The data from all missions between 1 January 1993 and 31 August 2017 were considered. Note that the data from 1 September 2017 will be used in the study for external validation: they are not used in the computation of the solutions.

This paragraph is a summary of the method presented in detail in

Illustration of one element of the basis for the mesoscale broadband signal

Then, if we note

The equivalent covariance models are

These covariance models will rely on the structure of elements in

The basis for mesoscales is the same as presented in

Illustration of the equivalent representer at a given location, for the mesoscale basis

The basis for internal tides presents some similarities in the construction, but accounting for specific aspects of the internal tide dynamics, in particular the forcing frequency and the horizontal dispersion relation.

For the M2 and K1 forcing frequencies, the contribution of the first two vertical modes are considered, because they have a major contribution to SSH at the horizontal scales where altimetry observations are accurate (limited to 50–80 km along-track;

For mode-1 (of any forcing frequency), we built a decomposition of plane waves following the dispersion relation between the time frequency

Using such a basis with constant terms for the

For the second baroclinic mode where only M2 and K1 forcing frequencies are considered, the bases are constructed exactly in the same way, with a second vertical baroclinic phase speed approximated by

Since the size of the parameter space is much smaller than the size of observation space, it is interesting to consider the Sherman–Woodbury transformation of Eq. (

The problem is solved on 15 by 15

In each tile, the first step consists of filling the

The second step is the resolution of Eq. (

To overcome the large computational and storage demands of the

Finally, from the solutions on the overlapping 15 by 15

Illustration of the same elements as presented in Fig.

Snapshots of the reconstructed solutions in sea surface height (m) for 20 May 2017 at 00:00 UTC in the North Atlantic basin, for the mesoscale component

Both mesoscale and coherent internal tide estimations are provided by the inversion. The mesoscale solution, supposedly less affected by internal tide aliasing, is not analyzed in this paper, which is focused on the internal tide solution. The illustration in Fig.

As mentioned above, the internal tides are resolved simultaneously but separately for each component (tidal frequencies and vertical modes). The illustration of all these components is particularly interesting in some regions where several components contribute, such as in the Philippine Sea, as shown in Fig.

Snapshots of the reconstructed internal tide solutions in the Philippine Sea for M2

In the following, we use the internal tide solution (written at the reference date) in a prediction mode assuming the stationary assumption persists over the period September 2017–December 2019, where we use independent altimetry data. These data also combine all satellite missions, and feature the same processing (barotropic tidal model in particular) as the main dataset used to compute the solution. To assess the quality of the reconstruction, we use a classical metric computing the variance difference before and after applying the prediction (e.g.,

Same as Fig.

Difference of sea level anomaly variance (cm

Variance reduction for Cryosat-2 crossovers differences (cycles 96–124) when using the new MIOST internal tide solution and either
a zero IT correction or the Zaron (2019) correction for M2 and K1 frequencies, in cm

Before analyzing the global solution, we propose to look at dedicated experiments where the estimation of internal tides is not simultaneous with mesoscales (only internal tide components are considered in the

The main simultaneous inversion experiment that featured the expected mitigation of aliasing issues over independent or sequential estimations is now validated globally and compared with an existing internal tide solution from

Another interesting diagnostic is the variance reduction at ground-track crossover differences, because it naturally suppresses some of the large timescale variability of mesoscales, but keeps the energy of internal tides. The diagnostic was run and integrated over different region, as detailed in

The simultaneous inversion of mesoscale variability and internal tide signatures on sea surface height has been successfully implemented globally, constituting in particular an additional empirical internal tide solution to the existing ones. It was designed to minimize aliasing effects, by accounting for the statistics of mesoscales and internal tides in a global linear analysis framework.

The computation of uncertainties could be an interesting next step. Indeed, it would be possible to provide estimations of errors, with respect to the covariance model prescribed through the mode decompositions for mesoscales and internal tides (first given in the parameter space, but then projectable in physical space). This would be a future improvement, considering that the successful quantitative validation with independent data was the most important milestone for this new internal tide model.

The methodology applied in this study could also be extended to internal tides with phases varying seasonally, as already implemented in

The internal tide solution from the MIOST analysis described in this article is available at

CU designed the methodology, coded the numerical implementation and wrote Sects. 1, 2, 3.1 and 3.2.1. LC provided expertise on the internal tide during the methodology design, supervised the global validation (Sect. 3.2.2), managed the data availability and wrote the data user handbook. CD ran the global validation. GD provided altimetry data expertise and critical advice on the methodology. YF helped with data management; MB and FB provided support during computational implementation. Finally, FL provided support on theoretical aspects of internal tides.

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

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

We thank the French National Centre for Space Studies (Centre National d'Etudes Spatiales) for providing financial support. We also thank Edward Zaron, Xongxiang Zhao and a third anonymous reviewer for their very helpful and constructive comments.

This research has been supported by the Centre National d'Etudes Spatiales (grant no. DUACS).

This paper was edited by Anna Rubio and reviewed by Edward Zaron and one anonymous referee.