# Simulation-informed deep learning for enhanced swot observations of fine-scale ocean dynamics

Eugenio Cutolo, Carlos Granero-Belinchon, Ptashanna Thiraux, Jinbo Wang, Ronan Fablet

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Eugenio Cutolo, Carlos Granero-Belinchon, Ptashanna Thiraux, Jinbo Wang, Ronan Fablet. Simulation-informed deep learning for enhanced swot observations of fine-scale ocean dynamics. 2025. [hal-05004872v2](https://hal.science/hal-05004872v2)

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Distributed under a Creative Commons Attribution 4.0 International License
SIMULATION-INFORMED DEEP LEARNING FOR ENHANCED
SWOT OBSERVATIONS OF FINE-SCALE OCEAN DYNAMICS

E. Cutolo<fnref n="1" />,<fnref n="2" />, C. Granero-Belinchon<fnref n="1" />,<fnref n="2" />, P. Thiraux<fnref n="1" />,<fnref n="2" />, J. Wang<fnref n="4" />, R. Fablet<fnref n="1" />,<fnref n="2" />

ABSTRACT

Fine-scale ocean dynamics at tens-of-kilometers scales and days-to-weeks timescales remain hard to observe cleanly. The Surface Water and Ocean Topography (SWOT) mission samples sea-surface height (SSH) at unprecedented resolution, but KaRIn measurements contain scale-dependent, sea-state-modulated noise that can obscure fronts, filaments, and small eddies. We introduce SIMPGEN, a simulation-informed approach that uses high-resolution ocean model output as a physically grounded reference to learn a prior that reconstructs SSH from SWOT without noisy/clean training pairs. SIMPGEN jointly learns a scale- and orientation-aware realism metric with respect to a reference dataset and a neural inversion operator that suppresses quasi-isotropic high-wavenumber contamination while preserving anisotropic submesoscale features. Applied to SWOT KaRIn observations exhibit higher 24 h phase coherence in the 50–10 km band than standard denoising baselines. The framework provides a ‘third way’ between classical data assimilation and supervised methods, learning directly from the abundant outputs of numerical models and global observations.

**Keywords** Satellite altimetry · Deep learning · SWOT · KaRIn noise · Submesoscale ocean dynamics

# 1 Introduction

Oceanic processes at scales of 5–150 km, evolving over days to weeks, fundamentally influence the ocean’s physical and ecological state by redistributing heat and nutrients, driving water mass formation, and modulating air–sea exchanges [1, 2, 3]. Their energetic nature also shapes broader circulation patterns, underscoring the need to observe and model submesoscale phenomena [4].

However, capturing fine-scale features remains challenging. In situ platforms (e.g., moorings, Argo floats, gliders) provide detailed but sparse snapshots [5], while nadir along-track satellite sensors balance resolution, accuracy, and revisit frequency, often limiting their ability to observe small-scale variability [6, 7]. The Surface Water and Ocean Topography (SWOT) mission and its Ka-band Radar Interferometer (KaRIn) [6, 8] provide unprecedented high-resolution SSH observations. However, they involve complex noise processes [9], potentially obscuring small-scale SSH signals.

High-resolution numerical modeling [10] and data assimilation [11] fill observational gaps by integrating measurements into ocean models, yet remain prone to biases from sub-grid parameterizations, limited and irregularly-sampled data and observation noise [5, 3]. Deep learning (DL) offers a complementary solution for estimating ocean variables from incomplete observations [12, 13, 14, 15, 16]. Networks trained on high-resolution simulations can predict quantities like subsurface temperature [17], sea surface height (SSH) [18, 19], or velocity fields [20], but often struggle when confronted with sparse, noisy, and real-world data [6, 21]. Domain adaptation methods help but require extensive, well-characterized observational datasets [15, 22]. As a result, the operational SWOT SSH denoising [9] based on a supervised learning from simulation data cannot fully reveal the potential of SWOT SSH observations, especially regarding fine-scale patterns.

<footnote n="1">¹IMT Atlantique, Lab-STICC, UMR 6285, 29238, CNRS, Brest, France</footnote>

<footnote n="2">²ODYSSEY Team-Project, INRIA Ifremer IMT-Atl., 35042, CNRS, Brest, France</footnote>

<footnote n="4">⁴Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, USA
eugenio.cutolo@pm.me,</footnote>
Unsupervised learning provides an additional pathway for exploiting real measurements lacking ground-truth labels [23]. Among these methods, Generative Adversarial Networks (GANs) [24] have been used in oceanography for gap filling and super-resolution [25, 26, 14]. Building on these ideas, we introduce an adversarial learning framework for unsupervised processing of SWOT SSH observations referred to as SIMPGEN (Simulation Informed Metric and Prior for Generative Ensemble Network). Unlike standard GAN approaches that rely on a binary discriminator, we train a metric used as a prior in a variational cost; the generator is an inversion operator mapping noisy observations to physically consistent fields. The metric is trained with a triplet objective so that realistic model states are closer to each other than to generated outputs. SIMPGEN pairs a neural inversion operator generating noise-free, dynamically plausible reconstructions with an adversarial discriminator that enforces realism. We show that SIMPGEN mitigates SWOT KaRIn noise and preserves submesoscale structures more effectively than operational SWOT denoising [9]. The paper is organized as follows: section 2 presents the SWOT mission, KaRIn noise characteristics, and the synthetic and real datasets. section 3 details the SIMPGEN framework. section 4 illustrates its performance in reconstructing SWOT SSH fields. Finally, section 5 discusses implications for ocean state estimation and potential synergies with data assimilation.

## 2 Context and Case study: SWOT mission and the KaRIn noise

Launched in December 2022 by NASA, CNES, the Canadian Space Agency, and the UK Space Agency, the Surface Water and Ocean Topography (SWOT) mission addresses the need for higher-resolution observations of global ocean dynamics [6, 8]. Its Ka-band Radar Interferometer (KaRIn) resolves wavelengths as short as ≈10–15 km an order of magnitude improvement over the ≈150 km typical of previous nadir altimeters [27, 28]. This capability opens a window onto submesoscale processes and strongly nonlinear internal waves that shape ocean kinetic energy and circulation. However, KaRIn observations remain prone to wave-modulated noise, particularly at finer spatial scales, which limits accurate estimates of geostrophic velocities, vorticity, and strain rates. Conventional filters (e.g., median, Lanczos) risk oversmoothing small-scale features, and direct supervised training of machine-learning denoisers is hampered by the lack of paired noisy and ground truth [9].

In this study, noise denotes components we do not aim to retain in a balanced mesoscale/submesoscale SSH analysis of SWOT Level-2 KaRIn data. These include instrumental/processing errors (e.g., speckle/thermal noise, residual roll and baseline-phase errors, geolocation jitter, striping, and the effects of 2 km pixel averaging), whose statistics depend on sea state and swath geometry. More generally, because our method maps the content of raw scenes to the "style" of a numerical reference composed of daily SSH snapshots, it should suppress variability absent from that reference, including unbalanced high-frequency motions; among them note that the reference models used here do not include tidal forcing.

We evaluate the approach in two complementary experiments: a controlled synthetic benchmark with known truth, and a real-data study that also probes sensitivity to the choice of numerical reference.

### 2.1 Synthetic experiment.

We use the eddy-resolving eNATL60 model of the North Atlantic (grid spacing ≈1.7–1.9 km) [29] as the reference field. Synthetic SWOT scenes are generated by sampling model SSH onto the swath-aligned 2 km grid and adding KaRIn-like, wave-modulated errors parameterized by significant wave height and distance to nadir. Although on-orbit analyses indicate lower noise than prelaunch budgets (by roughly a factor of four) [30], these controlled scenes provide a benchmark with known truth.

### 2.2 Real-data experiment and reference sensitivity.

For real observations, we use the SWOT L3LR_SSH product, derived from the KaRIn Low-Rate (L2) ocean data (NASA/JPL-CNES) and distributed free of charge by the AVISO+ and DUACS teams. In particular, we analyze the fast-sampling phase (one-day repeat orbit, ≈3 months), which provides frequent ≈2 km native KaRIn sampling across the swath [31]. We use the unfiltered sea-surface height anomalies (SSHA) while the CLS U-Net-filtered product serves as our baseline for comparison [9]. To assess sensitivity to the numerical reference used to learn the prior/metric, we train two otherwise identical SIMPGEN variants: SIMPGEN/eNATL60, trained on eNATL60 snapshots (1/60°), and SIMPGEN/MITgcm, trained on the ECCO MITgcm LLC4320 reference (1/48°). Aside from swapping the reference, all settings are identical (architecture, patch scales, preprocessing, and training schedule).
## 2.3 Diagnostics used in this paper.

Because no ground truth exists for real SWOT, we evaluate denoising with three complementary diagnostics:

(i) Along-track PSDs. We compute one-dimensional PSDs along the across-track direction on the native 2 km grid. For each pass, linear detrending is applied to each cross-track line, Welch spectra are averaged across lines within a swath, and then across passes; panel A reports the median, with shaded envelopes indicating day-to-day variability at each wavelength.

(ii) 24 h day-offset relative variation (upper-bound error). To quantify phase-coherent fidelity without truth, we compare each method SSH<sub>denoised</sub> at day D to an independent raw SWOT pass at D ± 1 (about 24 h apart) and normalize by the variance of the day-D raw field:

$$RV_{\text{SSH}_{\text{denoised}}}(k) = 100 \times \frac{\text{PSD}(\text{SSH}_{\text{denoised}}(D) - \text{SSH}_{\text{raw}}(D \pm 1))}{\text{PSD}(\text{SSH}_{\text{raw}}(D))}, \quad (1)$$

with $X \in \{\text{raw}, \text{CLS}, \text{SIMPGEN/eNATL60}, \text{SIMPGEN/MITgcm}\}$. Lower $RV_X(k)$ indicates better preservation of structures that persist over 24 h. Results are summarized over the 50–10 km and > 50 km bands.

(iii) Directional wavelet analysis. To diagnose scale-orientation behavior, we compute orientation-binned wavelet energies on 50 km × 50 km tiles at nominal scales 4, 8, and 16 km. For each tile we form the relative directional energy

$$R_{\theta}(s) = \frac{E_{\theta}(s)}{\sum_{\phi} E_{\phi}(s)}, \quad (2)$$

where $E_{\theta}(s)$ is the wavelet energy at scale s and orientation $\theta$; this normalization highlights anisotropy while removing scene-to-scene amplitude differences. Tiles are then classified by their dominant orientation: along-track dominated if the direction of maximum $R_{\theta}$ lies within a 22.5° fixed sector around the along-track axis, otherwise across-track dominated. We average $R_{\theta}$ separately within these two classes to reveal energy aligned or unaligned with the dominant direction, which helps discriminate physically coherent features from quasi-isotropic small-scale noise. For a detailed explanation of how the polar plots of wavelet coefficients correspond to the SSH fields, please refer to the supporting information S2.

These diagnostics are used consistently in Sections section 4 to compare raw SWOT, CLS, and the two SIMPGEN variants, and to document robustness to the choice of reference dataset.

# 3 Simulation-Informed Metric and Prior for Generative Ensemble Network

We introduce an unsupervised, adversarial deep-learning framework to reconstruct physically consistent sea-surface height (SSH) fields from noisy SWOT observations, leveraging numerical simulations as reference data (Figure 1). Unlike supervised approaches that require paired noisy/clean examples, we operate with unpaired datasets: $\mathcal{O}$, a collection of noisy observations, and $\mathcal{S}$, a reference set of high-resolution numerical simulation outputs. We posit that $\mathcal{S}$ approximates the distribution of true but unobserved ocean states corresponding to $\mathcal{O}$.

We start from the canonical variational data assimilation (DA) objective for estimating the state x from an observation $y \in \mathcal{O}$ and repurpose it as a DA-inspired training loss for our neural networks, rather than solving it with a fixed 3D-Var optimizer:

$$J(\mathbf{x}; \mathbf{y}) = J_{\text{prior}}(\mathbf{x}) + \frac{1}{2}(\mathbf{H}(\mathbf{x}) - \mathbf{y})^{\top}\mathbf{R}^{-1}(\mathbf{H}(\mathbf{x}) - \mathbf{y}), \quad (3)$$

where $\mathbf{H}$ maps state to observation space (here, identity plus swath masking) and $\mathbf{R}$ collects observation-error weights (diagonal in practice). The term $J_{\text{prior}}(\mathbf{x})$ encodes prior knowledge about the state. In the widely used 3D-Var approach, the prior typically relies on Gaussian assumptions [11] and is defined as:

$$J_{\text{prior}}(\mathbf{x}) = \frac{1}{2}(\mathbf{x} - \mathbf{x}_b)^{\top}\mathbf{B}^{-1}(\mathbf{x} - \mathbf{x}_b), \quad (4)$$

where $\mathbf{x}_b$ is the background state and $\mathbf{B}$ the background error covariance matrix. This linear-quadratic formulation is convenient, but a Gaussian prior in $\mathbf{B}$ cannot represent the complex, non-linear, and anisotropic mesoscale dynamics of ocean states [32]. Instead of prescribing the prior through the matrix $\mathbf{B}$, we learn a metric $M_{\Phi}$ that maps SSH fields into a latent space where Euclidean distances reflect multi-scale, orientation-aware similarity to the reference set $\mathcal{S}$. The prior then penalizes the distance to the reference distribution using an ensemble of states $\mathbf{x}_s \in \mathcal{S}$:

$$J_{\text{prior}}(\mathbf{x}) = \mathbb{E}_{\mathbf{x}_s \in \mathcal{S}} \| M_{\Phi}(\mathbf{x}) - M_{\Phi}(\mathbf{x}_s) \|^2. \quad (5)$$
<figure><img src="image_2.png" /><figcaption>Figure 1: SIMPGEN schematic. Numerical-model snapshots supply anchor (A) and positive (P) examples; SWOT observations provide the input and a negative (N) example. A metric maps tiles (patchify → wavelet energies → MLP) to an embedding, while a U-Net-like denoiser produces the denoised field (D). Training alternates two goals: the triplet loss makes the embeddings close for (A,P) and far from N $||M_\Phi(A) - M_\Phi(P)||_\downarrow$, $||M_\Phi(A) - M_\Phi(N)||_\uparrow$; the denoiser is optimized so that $M_\Phi(D)$ moves onto the model manifold (toward A/P) while remaining consistent with the observation.</figcaption></figure>

Rather than performing per-swath optimization of (3), we train a neural inversion operator $G_\theta$ that maps $(\mathbf{y}, \mathbf{R}) \mapsto \hat{\mathbf{x}}$ and is optimized to minimize the expected objective:

$$ \hat{\theta} = \arg \min_{\theta} \mathbb{E}_{\mathbf{y} \in \mathcal{O}} [J(\hat{\mathbf{x}}, \mathbf{y})] \quad \text{with} \quad \hat{\mathbf{x}} = G_{\theta}(\mathbf{y}, \mathbf{R}). \qquad (6) $$

This generator provides fast reconstructions and, during training, supplies challenging (“hard”) examples for the metric. To make $M_\Phi$ meaningful, we then optimize its parameters with a triplet loss that uses the current generator to define hard negatives:

$$ \hat{\Phi} = \arg \min_{\Phi} \mathbb{E} \left[ \| M_{\Phi}(\mathbf{x}_a) - M_{\Phi}(\mathbf{x}_p) \|^2 - \| M_{\Phi}(\mathbf{x}_a) - M_{\Phi}(\mathbf{x}_n) \|^2 + \text{margin} \right]_+, \qquad (7) $$

with $(\mathbf{x}_a, \mathbf{x}_p) \in S$ and the negative $\mathbf{x}_n$ drawn from a mixture of sources: (i) purely noisy SSH fields $\mathbf{y} \in \mathcal{O}$ (projected to state space on the swath via identity plus masking), and (ii) generator outputs $\hat{\mathbf{x}} = G_\theta(\mathbf{y}, \mathbf{R})$; the hinge operator is $[\cdot]_+$. Using both kinds of negatives is crucial: noisy-only negatives make the task too easy (the metric degenerates into a noise detector), whereas including generator-based negatives forces $M_\Phi$ to separate realistic-looking but out-of-manifold reconstructions from true simulation states, tightening the prior and discouraging trivial generation. In practice, we sample the two negative types in roughly equal proportion (results are not sensitive to this ratio).

In summary, training alternates two steps until convergence: (i) with $M_\Phi$ frozen, minimize (6) to update $G_\theta$; (ii) with $G_\theta$ frozen, minimize (7) to update $M_\Phi$. This adversarial procedure jointly learns a generator-based solver for the DA-inspired objective (3) and a simulation-informed metric prior (5) shaped by (7), enabling learning from unpaired $\mathcal{O}$ and $S$ while preserving fidelity to the observations at large scales. Architectural details, wavelet settings, and SWOT-specific preprocessing (swath geometry, masking, and $\mathbf{R}$ construction) are provided in S1–S4.

# 4 Results and Discussion

We present the results to illustrate the effectiveness of the proposed approach for denoising SWOT observations in two different and independent scenarios: one using purely simulated data, where both the reference dataset and the noisy
dataset (with added KaRIn-like noise) were available, and another combining simulated data as a reference with real SWOT observations.

## 4.1 Synthetic SWOT Observations

We first consider an application to the SWOT denoising benchmark based on synthetic noisy SWOT observations derived from high-resolution numerical simulations. In this controlled setting, where the ground truth is available, we compare our method against the state-of-the-art baseline, a U-Net trained in a supervised manner (similar to [9]). Specifically, the U-Net is provided with paired noisy and true SSH fields during training. In contrast, our adversarial learning scheme is trained without direct supervision, accessing only the noisy fields alongside the uncoupled reference SSH. The used dataset consists of the entire data-challenge SWOT orbital cycle 13, which includes 270 passes. In particular, we used the left swath for training and tested on the right swath after flipping it to preserve the symmetric nadir-distance-dependent error.

Figure 2 provides a comprehensive visual comparison between the ground truth (Panel A), the noisy dataset (Panel B), and the reconstructions obtained using both the baseline U-Net (Panel C) and our method (Panel D). While the supervised U-Net produces a smoother representation, the neural inversion operator preserves small-scale structures, particularly in regions with high spatial gradients. The RMSE averaged over the entire test dataset (Panel M) further supports this advantage: our trained neural inversion operator consistently outperforms the baseline U-Net denoiser, particularly at smaller scales (from 0.03 to 0.02 at 10 km). In contrast, when smaller scales are smoothed out, the RMSE difference between the two methods becomes less pronounced.

The results suggest that training the U-Net with an RMSE-based loss leads to excessive smoothing, whereas our method effectively balances noise removal and detail preservation by leveraging the adversarial learning framework. Naturally, more sophisticated training strategies for the baseline such as incorporating additional terms in the loss function could improve performance. However, such modifications would inevitably increase training complexity and the risk of overfitting to the training dataset.

Panels E-L present the average wavelet coefficients for the different datasets, normalized by the maximum scale energy, for an along-track-dominated average (left column) and an across-track-dominated average (right column). The noiseless reference (Panels E, F) exhibits coherent energy distributions aligned with the dominant spatial directions. Conversely, in the noisy cases (Panels G, H), the wavelet coefficients display a significant spread of energy across multiple directions and scales, revealing the impact of the simulated KaRIn noise. Although the simulated KaRIn noise has an inherent anisotropy due to its dependence on cross-swath distance and the significant wave height (SWH), we observe that, on average, this anisotropy does not manifest clearly in the wavelet coefficients.

Both the baseline U-Net denoiser (Panels I, J) and SIMPGEN (Panels K, L) yield nearly indistinguishable results in restoring the original anisotropy, demonstrating comparable effectiveness in noise reduction while maintaining the expected energy distributions.

In the average along-track spectra shown in Panel N, we show that SIMPGEN achieves a spectral energy distribution that preserves small-scale wavelengths, closely aligning with the noiseless reference. By contrast, the baseline U-Net denoiser underestimates small-scale spectral energy, in agreement with the oversmoothing effect discussed earlier.

In summary, our neural inversion operator effectively removes isotropic noise while preserving anisotropy associated with small-scale features and maintaining the physical coherence of the SSH field across dominant scales and directions. Notably, our method achieves these results without relying on a supervised dataset that in the following experiment, the real SWOT observations, is not available.

## 4.2 Demonstration with Real SWOT Observations

We evaluate denoising on real SWOT sea-surface height (SSH) during the fast-sampling phase (approximately 24 h revisit) in three regimes: the Mediterranean, the Gulf Stream, and the Southern Ocean (ground tracks in panel G of Figure 3), all the analysis are made on average of these 3 areas. To verify that performance is not tied to a particular numerical reference, we trained two otherwise identical SIMPGEN variants that differ only by the dataset used to learn the prior/metric: eNATL60 (1/60°) and MITgcm/ECCO LLC4320 (1/48°), hereafter SIMPGEN/eNATL60 and SIMPGEN/MITgcm. Even here as a baseline we used the CLS supervised unet.

In Panel A we show across-track power spectral densities (medians over passes; shaded variation in ±24 h). The raw SWOT spectrum follows the mesoscale $k^{-11/3}$ slope at large scales but flattens below 50 km. All denoisers steepen this tail toward the expected $k^{-5}$ decay. Both SIMPGEN variants align most closely with these theoretical slopes preserving variance in the 50–10 km submesoscale band while suppressing excess small-scale energy whereas CLS also reduces
<figure><img src="image_3.png" /></figure>

Figure 2: (A–D) Example snapshot of SSH fields: (A) noiseless ground truth, (B) noisy field, (C) supervised U-Net denoised field, and (D) SIMPGEN denoised field. (E–L) Polar plots of the average wavelet coefficients normalized by the scale energy for two cases: across-track dominated snapshots (right column) and along-track dominated snapshots (left column). Panels (E,F) represent noiseless fields, (G, H) noisy fields, (I, J) supervised U-Net denoised fields, and (K, L) SIMPGEN denoised fields. (M) RMSE as a function of spatial scale, where the neural inversion operator outperforms the supervised model, particularly at smaller scales. (N) Power spectral density comparison across wavelengths.

noise but remains systematically higher (and more variable) than SIMPGEN across 50–10 km. At wavelengths >50–100 km the curves collapse, indicating large-scale energy conservation. The MITgcm-trained variant rolls off slightly earlier below ~15–20 km than the eNATL60-trained one, consistent with the coarser effective resolution of LLC4320.

The shaded envelopes in panel A quantify the inter-day spread at each wavelength, which grows toward the smallest wavelengths (of order 10–15% near 10 km) and decays to a few percent by 50–100 km. In the 50–10 km band, CLS exhibits not only a higher median PSD but also a larger inter-day spread than SIMPGEN/NATL60, consistent with greater residual noise; when averaged over this band, the CLS spread is higher by roughly a few percent (approximately 3–6%) relative to SIMPGEN.

In panel B, both SIMPGEN variants lie below CLS throughout 50–10 km, corresponding to a band-averaged reduction of a few percent (approximately 3–6%), and remain comparable to CLS and raw at wavelengths >50 km, indicating no large-scale adjustments.

Directional wavelet diagnostics showed in panel C-F are only showed for raw data and SIMPGEN/NATL60 are computed on 50 km × 50 km tiles at nominal scales of 4, 8, and 16 km. After denoising the energy increases along the dominant direction and decreases in orthogonal directions, most clearly at 4–8 km, consistent with suppression of quasi-isotropic small-scale noise and retention of balanced finescale variability. This redistribution occurs within the high-wavenumber band; the large-scale background remains unchanged.

Finally, we illustrate a Mediterranean crossover region (pink box in Panel G), rich in submesoscale structure that is also apparent in chlorophyll concentration [33] (H). The raw SWOT SSH swath (I) is denoised by SIMPGEN/eNATL60 (L) and by CLS (M): SIMPGEN preserves sharp fronts and filaments maintaining variance in the 50–10 km band and
co-locating with CHL features, whereas CLS produces a smoother field with reduced small-scale contrast. The boxed areas mark locations where submesoscale preservation is most evident. Other examples relative to the 3 other regions considered (showed in panel G) are in the Supplementary Information S6.

<figure><img src="image_4.png" /><figcaption>Figure 3: (A) Across-track power spectral density (PSD) for raw SWOT (blue), CLS (red), SIMPGEN/eNATL60 (green), and SIMPGEN/MITgcm (orange). Curves are medians over passes; shaded bands denote the 24 h variability at each wavelength. (B) Twenty-four-hour relative variation with respect to the day-D raw variance. (C-F) Directional wavelet energy on 50 km × 50 km tiles at nominal scales 4, 8, and 16 km, averaged separately over tiles classified as along-track-dominated and across-track-dominated, to highlight energy aligned or unaligned with the dominant direction. (G) Locator map. (H-M). Example Mediterranean crossover region the 12th of June 2023: CHL image (H); raw SSH (I); denoised SSH from SIMPGEN (L) and CLS (M). The Mediterranean test region corresponds to area B in the global map, red boxes highlight submesoscale rich areas. Additional MITgcm examples and Southern Ocean results are provided in the Supplementary Information.</figcaption></figure>

# 5 Conclusion

We presented SIMPGEN, a simulation-informed, unsupervised framework that reconstructs physically coherent sea-surface height (SSH) fields from SWOT KaRIn observations without requiring noisy-clean training pairs. The approach learns a scale- and orientation-aware metric from high-resolution numerical scenes and uses it as a prior within a variational reconstruction. In controlled synthetic tests based on eNATL60, SIMPGEN preserves finescale variability more faithfully than a supervised U-Net baseline, reducing error while avoiding the oversmoothing typically associated with supervised denoisers.

Applied to real SWOT Level-2 KaRIn SSH during the fast-sampling phase, SIMPGEN strongly attenuates high-wavenumber noise yet preserves energy in the 50-10 km band. A 24 h day-offset diagnostic shows that SIMPGEN reduces the upper-bound error proxy throughout 50-10 km compared with both raw and the CLS denoiser, by on the order of a few percent when averaged over the band (approximately 3-6% median across passes; panel B). At wavelengths > 50 km, the relative variation remains comparable to raw and CLS.

Furthermore because our prior/metric is built from physically meaningful, orientation-binned wavelet energies, rather than opaque features, SIMPGEN is not a black box: the same scale-orientation descriptors that guide learning also diagnose what the network does. Directional-wavelet analyses show that denoising redistributes energy toward the dominant flow orientation at 4-8-16 km while reducing it in orthogonal sectors, a fingerprint consistent with suppression of quasi-isotropic KaRIn noise and retention of balanced finescale structure.

To assess sensitivity to the choice of numerical reference used to learn the prior, we trained otherwise identical models on two datasets: eNATL60 (1/60°) and LLC4320 (1/48°). The two SIMPGEN variants exhibit nearly indistinguishable 24 h relative-variation performance in 50-10 km, and both improve over CLS, while the MITgcm-trained variant shows a slightly earlier roll-off below ~15-20 km, consistent with its coarser effective resolution and simulator smoothing. This robustness suggests that SIMPGEN's gains are not tied to a single model. At the same time, the small but
systematic differences between references imply a clear path for further improvement: as higher-fidelity and better regionally tuned numerical references become available (finer grids, richer submesoscale content, improved physics, or mixtures of priors across regimes), the learned prior should better match local dynamics and further enhance finescale recovery without compromising large-scale fidelity.

Because our target product prioritizes balanced mesoscale/submesoscale variability, part of the unbalanced signal (e.g., internal gravity waves) is attenuated; users seeking those components should prefer the raw Level-2 product or specialized filters. Beyond denoising, the same directional-wavelet machinery we use to characterize noise anisotropy could be repurposed to help disentangle balanced finescale variability from unbalanced motions (e.g., internal waves), a direction we leave for future work.

Looking ahead, SIMPGEN's simulation-informed prior can be extended to other variables (e.g., surface velocity, temperature) and coupled with data assimilation systems to improve finescale state estimates. More broadly, the approach offers a third way to exploit the expanding archives of numerical models and global observations: rather than prescribing parametric covariances as in classical DA, or requiring paired noisy/clean examples as in supervised ML, we learn a physically grounded prior/metric from model ensembles and use it to regularize reconstructions that remain faithful to the observations. Its unsupervised design and demonstrated robustness across two distinct references make it a practical tool for forthcoming high-resolution missions and data-sparse regimes, with a clear path to incremental gains in ocean understanding as numerical references improve.

## Open Research Section

The synthetic SWOT observations and ground-truth data used in this study are available at https://2024b-dc-swotdenoising.readthedocs.io/en/latest/1_getstarted/getstarted_data.html. The SWOT Level-3 raw and filtered products were obtained from https://tds-odatis.aviso.altimetry.fr/thredds/catalog/dataset-13-swot-karin-nadir-validated/13_lr_ssl/v2_0_1/catalog.html. For the reference-robustness experiment, we additionally used the PO.DAAC MITgcm/ECCO LLC4320 KaRIn-simulated Level-2 product [34]. For the chlorophyll-a overlays, we used the Copernicus Marine Service (CMEMS) near-real-time OLCI daily Level-3 product [33]; access requires a free CMEMS account. For visualization we composited CHL within ±12-24 h of each SWOT pass, applied the provider quality flags, and bilinearly mapped the fields onto the 2 km SWOT swath grid (details in SI). The SIMPGEN code, together with scripts for data download and preprocessing, will be made publicly accessible upon acceptance of this paper.

## Conflict of Interest Statement

The authors have no conflicts of interest to disclose.

## Acknowledgments

This work was supported by the French National Research Agency (ANR-21-CE46-0011-01), within the program *Appel à projets générique 2021*. This work was supported by the Brittany Region Government (22006858), within the program *Stratégie d'Attractivité Durable 2022*. This work was supported by the Finistère Department Government, within the program *Aide aux Projets de Recherche 2022*. J. Wang is supported by the NASA Physical Oceanography Program grant 80NSSC25K7643. E. Cutolo would like to extend his sincerest gratitude to S. Bolumar, whose unwavering support, patience, and encouragement were invaluable throughout the course of this work.

## Appendices

* **Appendix A.** Unified methods: adversarial scheme; directional wavelets; $G_\theta$ details; $M_\Phi$ training; ensemble inference.

* **Appendix B.** Characterizing noise with the neural metric.

* **Appendix C.** Regional examples (Gulf Stream, Mediterranean, Southern Ocean).

This Supporting Information (SI) complements the GRL paper "Simulation-Informed Deep Learning for Enhanced SWOT Observations of Fine-Scale Ocean Dynamics" by providing implementation details, diagnostics, and regional examples that could not fit in the main text. **Appendix A** gathers all methodological material into a single coherent
narrative: we first state the variational objective and adversarial training loop that couple a neural inversion operator with a simulation-informed metric; we then motivate the use of directional wavelets; next we detail the architectures and training strategies for $G_\theta$ and $M_\Phi$; finally we describe the ensemble inference routine. **Appendix B** analyzes the learned metric as a tool to characterize noise. **Appendix C** presents additional regional examples. Figures A1, B1, and C1–C2 illustrate the synthetic wavelet analysis, metric sensitivity, and multi-region SWOT snapshots, respectively.

# A Unified methods: objective, wavelet representation, networks, and inference

## A.1 Problem setup and adversarial learning scheme

We adopt an *unsupervised, adversarial deep-learning* formulation with two unpaired datasets: $\mathcal{O}$ (noisy SWOT observations) and $\mathcal{S}$ (reference snapshots from a high-resolution numerical model). Variational data assimilation (DA) provides the organizing principle, but we use a *DA-style objective as a training loss* rather than solving it per swath. This section specifies the objective and the alternating optimization that jointly learn a neural inversion operator $G_\theta$ (generator) and a simulation-informed metric $M_\Phi$.

**Neural inversion operator training.** Let $\mathbf{H}$ map model states to SWOT observation space (identity on observed pixels, masking elsewhere), and let $\mathbf{R}$ be a positive-definite observation-error weighting (diagonal in practice). Given $\mathbf{y} \in \mathcal{O}$, the neural inversion operator predicts $\hat{\mathbf{x}} = G_\theta(\mathbf{y}, \mathbf{R})$. We train $G_\theta$ as a learned solver by minimizing a DA-inspired objective with a learned, simulation-informed prior:

$$ \hat{\theta} = \arg \min_{\theta} \mathbb{E}_{(\mathbf{y}, \mathbf{R}) \in \mathcal{O}} \left[ \frac{1}{2} \| \mathbf{H}\hat{\mathbf{x}} - \mathbf{y} \|_{\mathbf{R}^{-1}}^2 + \lambda_{\text{prior}} \mathbb{E}_{\mathbf{x}_s \in \mathcal{S}} \| M_{\Phi}(\hat{\mathbf{x}}) - M_{\Phi}(\mathbf{x}_s) \|^2 \right]. \quad (\text{A1}) $$

Here $M_\Phi$ maps fields to a latent (metric) space in which Euclidean distances reflect multi-scale, orientation-aware similarity to $\mathcal{S}$. Expectations are implemented as mini-batch averages over $\mathcal{O}$ and $\mathcal{S}$.

**Metric training.** With $G_\theta$ fixed, the metric network $M_\Phi$ is optimized so that reference snapshots are close and out-of-manifold states are far apart:

$$ \hat{\Phi} = \arg \min_{\Phi} \mathbb{E} \left[ \| M_{\Phi}(\mathbf{x}_a) - M_{\Phi}(\mathbf{x}_p) \|^2 - \| M_{\Phi}(\mathbf{x}_a) - M_{\Phi}(\mathbf{x}_n) \|^2 + \text{margin} \right]_+, \quad (\text{A2}) $$

where $(\mathbf{x}_a, \mathbf{x}_p) \in \mathcal{S}$, and $\mathbf{x}_n$ are negatives drawn from a mixture of (i) purely noisy SSH and (ii) current $G_\theta$ outputs, which stabilizes training by diversifying out-of-manifold examples. We set $\lambda_{\text{prior}}=1$ and margin=1 (no particular sensitivity observed).

**Adversarial training loop.** We alternate two updates until convergence: (i) with $G_\theta$ frozen, minimize (A2) to update $M_\Phi$; (ii) with $M_\Phi$ frozen, minimize (A1) to update $G_\theta$. This coupling yields a fast, generator-based solve of the DA-inspired objective while $M_\Phi$ learns a simulation-informed regularity notion.

## A.2 Directional wavelets and the simulation-informed metric ($M_\Phi$)

The metric $M_\Phi$ replaces restrictive Gaussian priors by mapping SSH fields into a latent space where Euclidean distances reflect physical similarity across scales and directions. For interpretability and physics awareness, we feed $M_\Phi$ with *first-order directional wavelet (scattering) coefficients* computed with *kymatio*.

**Directional wavelet representation.** Directional wavelets provide a multi-scale, multi-orientation analysis that preserves spatial localization and reveals anisotropy, complementing azimuthally averaged Fourier spectra. For a field $f(x, y)$,

$$ W_f(s, \theta, x, y) = \iint f(x', y') \psi_{s,\theta}(x-x', y-y') dx' dy', $$

with $\psi_{s,\theta}$ a scaled/rotated mother wavelet. We use Morlet-type wavelets with dyadic scales $2^j$, $j = 0, \dots, J-1$, and $L$ orientations $\theta_\ell = \ell\pi/L$, $\ell = 0, \dots, L-1$. Each pair $(j, \ell)$ targets size $\sim 2^j$ and orientation $\theta_\ell$.

**Synthetic motivation.** Figure A1 illustrates: (i) synthetic plane waves at various orientations/wavelengths and their polar wavelet-energy portraits; (ii) synthetic power-law fields ($E(k) \propto k^{-11/3}$) showing how isotropic vs. anisotropic organization across scales is recovered. This motivates directional wavelets for SWOT's 2D observations with strong directional variability and informs the representation used by $M_\Phi$.
**Configuration and patching.** Let $\phi_j^{(\ell)}(\mathbf{x})$ denote the modulus (first-order scattering) coefficients at scale $2^j$ and orientation $\theta_\ell$. We use $J = 3$ scales (approximately 4, 8, 16 km given a ~2 km pixel) and $L = 8$ orientations in $[0, \pi)$. Inputs are split into $26 \times 26$ patches (~52 x 52 km), matching SWOT's ~120 km swath (with ~20 km nadir gap) while maintaining local stationarity; overlapping tiles are averaged when reassembled.

**Metric network and training.** An encoder–decoder produces a compact embedding $\mathbf{z}$ from wavelet features:

$$ \mathbf{z} = \mathrm{Enc}(\{\phi_j^{(\ell)}(\mathbf{x})\}; \mathbf{\Phi}_{\mathrm{enc}}), \quad \hat{\phi}_j^{(\ell)}(\mathbf{x}) = \mathrm{Dec}(\mathbf{z}; \mathbf{\Phi}_{\mathrm{dec}}). \qquad (\mathrm{A3}) $$

We pretrain the autoencoder with an $\ell_2$ reconstruction term plus an $\ell_1$ sparsity penalty on encoder parameters,

$$ \mathcal{J}_{\mathrm{AE}} = \sum_{j, \ell} \|\phi_j^{(\ell)}(\mathbf{x}) - \hat{\phi}_j^{(\ell)}(\mathbf{x})\|_2^2 + \lambda \|\mathbf{\Phi}_{\mathrm{enc}}\|_1, \quad \lambda = \frac{1}{2}, \qquad (\mathrm{A4}) $$

which encourages disentangled, interpretable directions in feature space. After pretraining, $M_\Phi$ is optimized with the triplet loss (A2): reference snapshots $(\mathbf{x}_a, \mathbf{x}_p) \in \mathcal{S}$ are pulled together in latent space, while negatives $\mathbf{x}_n$—drawn from a mixture of purely noisy SSH and current $G_\theta$ outputs—are pushed away. These distances are precisely those used by the prior term in (A1).

**Scalar score for diagnostics.** While distances in (A2) are computed in the latent space, for diagnostics we report a scalar score $m(\mathbf{x})$ obtained by a shallow head applied to $\mathbf{z}$; this is what appears in violin plots and sensitivity visualizations.

### A.3 Neural inversion operator ($G_\theta$)

We instantiate $G_\theta$ as a U-Net with attention gates to emphasize dynamically relevant patterns and suppress noise-dominated regions. Inputs are the noisy SSH $\mathbf{y}$, an optional observation-mask channel, and a scalarized observation-error covariance $\mathbf{R}$.

**Randomized observation-error weighting (training).** We expose the model to a range of noise levels by sampling

$$ \mathbf{R} = \epsilon \mathbf{I}, \quad \epsilon \sim \mathcal{N}(\mu, \sigma^2), \qquad (\mathrm{A5}) $$

with $\mu = 0.5$ and $\sigma = 0.1$ (std. $\approx 20\%$ of the mean). This effectively modulates the balance between the data term and the metric prior in (A1).

**Ensemble inference (test time).** At inference, we randomize $\mathbf{R}$ by resampling $\epsilon$ to generate $E=30$ realizations $\{\hat{\mathbf{x}}^{(e)}\}_{e=1}^E$ and average them:

$$ \bar{\mathbf{x}} = \frac{1}{E} \sum_{e=1}^{E} \hat{\mathbf{x}}^{(e)}, \quad \mathrm{Var}[\mathbf{x}] = \frac{1}{E-1} \sum_{e=1}^{E} (\hat{\mathbf{x}}^{(e)} - \bar{\mathbf{x}})^2, $$

which broadens the range of addressed noise scenarios and yields robust SSH estimates; the variance serves as a spread diagnostic (reported on unobserved pixels).

## B Characterizing noise with the neural metric

The scalar score $m(\cdot)$ enables quantitative comparisons across datasets. Figure B1A shows distributions over multiple sets. As expected, the noiseless synthetic reference remains centered near zero; noisy synthetic SWOT broadens; real noisy SWOT widens further, indicating additional instrument/sampling effects. Among denoisers, a supervised baseline (CLS) reduces dispersion relative to raw inputs, while our adversarial approach further narrows the distribution and lowers the median, particularly on real SWOT.

To interpret the score, Figure B1B visualizes sparse autoencoder coefficients in polar scale–orientation coordinates. Positive weights (red) elevate the score; negative (blue) reduce it. The structure highlights sensitivity to energy spread across orthogonal directions at 4–8 km, consistent with KaRIn noise being predominantly across-track and directionally diffuse compared to coherent oceanic structures.
**C Regional examples (Gulf Stream, Mediterranean, Southern Ocean)**

**Atlantic Ocean — Gulf Stream.** Energetic mesoscale eddies and strong fronts yield high signal-to-noise. High-frequency contamination in raw SSH is substantially reduced after denoising, with sharper eddy edges and fronts retained. CLS and our approach both improve small-scale structure; our method avoids excess smoothing of larger scales.

**Mediterranean Sea.** Lower-energy dynamics and a small first baroclinic Rossby radius (5–15 km) produce compact, relatively isotropic structures at the 50 × 50 km patch scale. Denoising lowers high-frequency energy while preserving compact eddies/filaments. In the 20–30 km band, preserving energy is important; our approach suppresses the smallest-scale noise while maintaining that band.

**Southern Ocean.** Intense eddy activity associated with the ACC yields large gradients and anisotropic along-track structures. Denoising restores frontal sharpness and eddy definition and reduces small-scale temporal variability. Residual energy below ~10 km is hardest to model; methods trained elsewhere may underperform. The directional, scale-aware constraints help mitigate this.
<figure><img src="image_5.png" /><figcaption>Figure A1: Directional wavelet analysis on synthetic fields. Left: plane waves (different orientations and wavelengths) and their polar energy portraits (radius = scale, angle = orientation). Right: power-law correlated fields ($k^{-11/3}$) and corresponding portraits, highlighting scale-dependent anisotropy.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Figure B1: (A) Violin plots of the neural score across datasets (noiseless synthetic reference, noisy synthetic SWOT, noisy real SWOT, supervised CLS denoiser, and our approach). (B) Polar visualization of sparse autoencoder coefficients (radius = scale, angle = orientation); red/blue indicate positive/negative contributions to the score.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure C1: Snapshots from 15 April 2023 for different SWOT processing and three regions: Mediterranean Sea (first row), Gulf Stream (second row), Southern Ocean (third row). Columns: raw SWOT SSH, CLS-denoised SSH, and our denoised SSH.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Figure C2: Same layout as Figure C1 for 03 June 2023.</figcaption></figure>
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