## 1. Introduction

Merging along-track ocean altimetry data into continuous maps in time and space is a challenging exercise that is extremely useful for research and applications. In particular, it provides synoptic views of the geostrophic currents and ocean dynamic content, not directly given by the along-track data. The most common gridded altimetry reconstructions (e.g., AVISO maps, http://www.aviso.altimetry.fr/duacs/, Aviso, 2015) from the available constellation of satellites are based on linear state analysis (following Bretherton, et al. 1976) with predefined time and space covariance models (e.g., Le Traon et al. 1998).

The mesoscale sea surface height (SSH) captured by altimetry satellites is principally dominated by quasigeostrophically balanced dynamics (outside of the equatorial zone) that evolve with typical time scales of a few weeks and spatial scales of a few hundreds of kilometers. The linear mapping analyses, as they have been performed so far, capture a significant part of these mesoscale dynamics (e.g., Ducet et al. 2000). These dynamics are known to be dominated, in large part, by the first baroclinic mode (e.g., Wunsch, 1997). Although well captured by the linear mapping, the actual evolution of the first baroclinic mode, especially at short spatial scales, is strongly nonlinear, which may limit the performance of the mapping. In Ubelmann et al. 2015, we have shown that accounting for the simplest nonlinear representation of the first baroclinic mode SSH evolution allows for significantly reducing interpolation errors between two fields of SSH, through the so-called dynamic interpolation. The fields of SSH were transported forward and backward by a nonlinear propagator conserving the potential vorticity expressed in the first baroclinic mode framework. This direct forward/backward approach, easy to implement, demonstrated the concept of dynamic interpolation, but was not directly applicable to observations unevenly sampled in time and space. In this present study, we propose to implement the use of this nonlinear propagator in an inverse approach, similar to the standard mapping analysis, allowing to process realistic along-track observations (of any distribution, with noise).

In the first section, we will briefly review the standard linear mapping analysis as commonly used for ocean altimetry data. Based on the same approach but introducing a correction term to the innovations, accounting for predictable nonlinear evolutions, we will propose in section 2 the implementation of a propagator in the analysis to perform dynamic mapping. The method will be finally tested with Observing System Simulation Experiments (OSSEs) in the Gulf Stream region, described and analyzed in section 3 globally and as a function of scale.

## 2. The standard mapping methods for altimetry data

The mapping methods commonly used for merging multisatellite altimetry data are based on a three-dimensional (time and space) linear analysis, applied to sea level anomaly (SLA) observations with respect to a background state (mean dynamic topography) and predefined time and space covariance functions for the departure from the background state.

### General formulation

*t*≫

*t*

_{0}, with a sharp cutoff for

*t*≪

*t*

_{0}. The westward propagation of eddies can also be accounted for by coupling

## 3. The dynamic mapping using a nonlinear propagator

### a. The state estimation problem

As described in Ubelmann et al. (2015), it is possible to use a very simple but nonlinear propagator to account for a predictable part in the time evolution of the SSH signal, and as a result reduce the errors of the state estimate between SSH fields given at different times. In this section, we propose defining the state estimate problem using the nonlinear propagator with an inverse approach similar to the standard mapping method summarized above. This new implementation will allow dynamic reconstructions from realistic along-track data unevenly sampled in time and space.

*t*around tref, noted

*t*) of the nonresolved dynamics. For

### b. Resolution in a reduced space using Green’s functions

### c. Computation of matrix

*j*of the

### d. Parameterization of matrix

## 4. The OSSEs

### a. Experiment setup

The outputs of an ocean general circulation model (OGCM) are used in the following as a reference truth from which an observation system is simulated, providing synthetic altimetric observations along virtual satellite tracks. The mapping methods presented above will be tested using these synthetic observations and the analyzed states will then be compared with the truth for performance estimation.

The OGCM fields come from a global MITgcm simulation at 1/16° horizontal resolution (Hill et al. 2007; Menemenlis et al. 2008) provided by the ECCO project. The Gulf Stream region has been chosen for this study, as shown in Fig. 1. The 6-hourly output fields over 1 year have been sampled by three virtual satellites, flying on Joint Altimetry Satellite Oceanography Network (Jason) orbits for the first two (shifted in time and space) and on Ka-band Altimeter (ALtiKa) orbit for the third one. The ground tracks are shown in Fig. 1. An instrument noise error has been added consistently with standard values of 3.3 cm at 1 Hz (~5.5-km ground spacing) (Xu and Fu 2011)

The reference mean state (a mean dynamic topography from which covariance matrices will be defined) is the time-mean SSH of the simulation outputs. The observations

### b. Implementation of the mapping algorithms

The standard mapping solution has been first computed following the description in section 1. The following parameters have been chosen for the covariance model Eq. (3): to = 15 days, *L* = 150 km, and *L* were revealed to be above optimal values, which happen to be close to those currently found in the literature (e.g., Le Traon et al. 1998). This could be optimized by considering regional and seasonal dependences in further studies. To avoid large matrix inversions, the analysis is performed locally in time and space: ±20 day time windows and a 400-km radius are chosen, knowing that correlation vanishes beyond a few hundreds of kilometers and beyond 20 days. In space, the local domains overlap every 100 km in zonal and meridional directions. Once computed, the local 2D solutions are linearly interpolated between the central locations of analysis. Finally, the matrix

The dynamic mapping solution has then been computed following the description in section 2. The reduced Fourier basis

### c. A look at the representer functions

The covariance between a particular grid point and the rest of the domain at different times is an interesting indicator of the mapping model characteristics. For the standard mapping, this correlation is directly given by the covariance model Eq. (2). For dynamic mapping, the covariance is

### d. Results and comparison with standard mapping

The analysis with standard and dynamic mapping has been performed on 1 years’ worth of data. The results are presented and analyzed in this section, from both direct diagnostics (error maps) and spectral diagnostics (mean power spectral densities and spectral coherences).

Figure 4 shows the maps of error variance between the reference run and the two sets of reconstructions over 1 year. The improvements of the dynamic solution over the linear solution are clear in the Gulf Stream current where the errors are reduced by almost 30% on average. However, the improvements are not significant along the continental shelf where the errors remain strong. In this area, the barotropic mode (not considered in the dynamic propagator) may be important. Further dynamic considerations would be therefore worth investigating, in order to account for other modes or dynamics in such specific coastal regions. It is nevertheless encouraging that the dynamic solution with only the first baroclinic mode is not worse than the linear solution on the continental shelf.

Figure 5 is an illustration of the results for a particular time, showing in the left column the SSH and in the right column the corresponding geostrophic velocity. The top panels are the reference fields, which are instant snapshots of the truth at time of analysis. The second row is the interpolated fields with the linear solution. As expected, from the comparison with the truth, the large scales (largest ~500-km eddies) are well resolved but important errors remain at smaller scales due to the sparse observation sampling in time and space. For example, the ~125-km eddy at 35°N, 289°E is not resolved. The third row is the interpolated fields with the dynamic solution, which is able to resolve smaller scales accurately. For example, the above-mentioned eddy is partially resolved. The error fields (the fourth and fifth rows) indicate significant improvements consistent with the error variance diagnostics presented above, for both SSH and its derivative geostrophic velocity.

Some scale-dependent diagnostics have been performed on the whole series of analyses with respect to the reference in order to better quantify the improvements as a function of scale. Indeed, given the steep SLA spectral slope (between −3 and −4), the error variance diagnostic mentioned above principally reflects the larger scale errors. It is therefore interesting to look at spectral content, with a particular attention on the phases indicating whether a given scale is phased or not between the reference and the interpolation reconstructions. The upper panel of Fig. 6 shows the power spectral density of the reference (black), the linear interpolation reconstruction (blue) and the dynamic interpolation reconstruction (red). The spectrum of linear interpolation starts dropping below the reference spectrum at scales above ~300 km, and has a steep drop below 140 km. However, the dynamic interpolation spectrum seems to better represent the energy below 300 km, and the firm drop only occurs at ~120 km. However, it is important to note that correct energy does not mean absence of errors, so comparing the phases is relevant to assess if a given scale is well resolved or not. To do so, the cross-power spectral densities have been computed between the interpolated fields and the reference fields. After normalization with the square root of the power spectral density of each field, we obtain the so-called spectral coherence represented in the lower panel. This diagnostic clearly reveals that the interpolated fields are well phased (spectral coherence close to 1) from long wavelengths down to a range of scales where coherence progressively drops toward zero. Here, it is clear that the dynamic interpolation outperforms the linear interpolation. The wavelength of 0.5 coherence, which can be considered as a threshold for assessing resolving capabilities, is located at ~145 km for linear interpolation and ~125 km for dynamic interpolation. This highlights the better resolving capabilities of dynamic interpolation over linear interpolation. We found that unlike the power spectral density, the spectral coherence was insensitive to the covariance parameters. The systematic gain in the spectral coherence, especially for scales between 125 and 250 km, seems inherent to the dynamic propagation that allows for better resolution of the phase of small eddies. These spectral coherences can be compared with those obtained by Ubelmann et al. (2015) from an idealized framework where interpolation was performed between two full images of SSH. While being significant, the relative gain from linear to dynamic interpolation is less pronounced here. We have two explanations for this. First, this more realistic case treats larger scales (because of the limited number of nadir satellites and the noise considered) and it was shown in Ubelmann et al. (2015) that the relative performances were decreasing as scales increase, because the nonlinearity is less important (larger scales are more linear). Second, these realistic data are distributed all over in time and space, whereas in the idealized case the observations were snapshots separated in time (with no observations in between). Since the introduction of dynamics allows reduction of the time decorrelation, its effect should be more pronounced when the observations suffer from time resolution like in the idealized case by construction. However, it is interesting to note that in this study simulating a realistic dataset of altimeter data, the dynamic interpolation method is still of interest in the strong western boundary currents.

## 5. Conclusions

This study presented a practical implementation of the dynamic interpolation introduced in Ubelmann et al. 2015. We followed here an inverse approach resolving a 3D (space + time) least squares problem similar to the standard optimal interpolation. However, unlike standard inversions following linear models, we proposed to use a locally linearized propagator with appropriate correction term for nonlinear evolutions of the local state in the innovations. Significant improvements from linear optimal mapping to nonlinear dynamic mapping have been shown, which encourages applying the method to process new maps of real altimetry data.

This method can be applied to any nonlinear propagator. Although this simple quasigeostrophic (QG) propagator performs efficiently in the Gulf Stream region, more sophisticated propagators may have to be developed for other regions. For instance, in coastal regions we may implement topography effects, for example, through adjustment of the Rossby radius and the addition of a barotropic mode if this latter is dominant. Also, in the equatorial regions, the quasigeostrophic (QG) propagator would clearly not be an appropriate choice. Other propagators (maybe simply linear waveguides) may be developed for these regions.

This dynamic interpolation method presents some specific features with respect to data assimilation in OGCMs. Unlike OGCM data assimilation, the solution is not a model trajectory (or a sequentially adjusted model trajectory). Here, the propagator is only used to determine a better covariance, but the analysis is performed separately from the mean sea surface height (MSSH) background, not from a model trajectory. For example, in the absence of observations, our mapping solution is the MSSH, not a free model run. The term “propagator” is meant to distinguish from an OGCM: the propagator only represents a deterministic SSH evolution, not necessarily a realistic evolution with input of energy, eddy generation, and other physics. In this sense, the dynamic interpolation keeps strong fidelity with the data. Beyond the deterministic evolution accounted for, all other dynamics are parameterized in a specific covariance matrix, like all dynamics are in standard interpolation.

Part of the research presented in the paper was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. Support from the SWOT project is acknowledged.

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