The study area is located in the North Atlantic Ocean, in the Gulf Stream region (see Fig. ), bounded to the north and south by 33 and 43° N, and to the west and east by 63 and 52° W, respectively. This sector is characterized by one of the most energetic western boundary currents of the global ocean, with strong jets, meanders and eddies that generate intense temperature and salinity fronts over a wide range of spatial and temporal scales . Such conditions make it an ideal natural laboratory for testing downscaling methods that aim to reconstruct fine-scale structures from coarser information.

Within this domain, SSS, SST and SSH exhibit distinct patterns of spatial variability. As illustrated in Fig. , SSS and SST share sharp frontal features and filamentary structures associated with mesoscale and frontal activity. In the present GLORYS-based setup, SSH appears smoother than expected at the fine scale. The close resemblance between SSS and SST gradients suggests that SST is a particularly informative auxiliary variable for constraining the small-scale structure of SSS, whereas SSH primarily encodes the broader mesoscale circulation that organizes the positions and strengths of fronts and eddies.

This leads to the working hypothesis that underpins our modeling strategy: at the mesoscale and at the fine scales represented by GLORYS, SSS variability is tightly coupled to SST through processes such as mixed-layer dynamics and lateral advection, whereas SSH provides a complementary large-scale dynamical constraint. We therefore expect high-resolution SST to provide strong local guidance for reconstructing SSS gradients, with SSH acting as an additional constraint on the overall dynamical context. The diffusion model is designed to exploit these relationships by conditioning the generative process on both variables.

Our models were trained on GLORYS12V1 CMEMS global ocean reanalysis daily data available at 1/12° horizontal resolution on 50 vertical levels and produced in the framework of the Copernicus Marine Environment Monitoring Service . In this study, we only use the surface layer. The dataset used in this study consists of daily gridded fields of SSS, SST, and SSH over the Gulf Stream region for the period 1993–2020.

To emulate satellite-based salinity observations, we explicitly distinguish between a high-resolution reference field and a coarse-resolution observation field. The native GLORYS SSS at 1/12° is taken as the reference high-resolution SSS, while a corresponding coarse SSS field at 1/3° is obtained by applying a 4×4 spatial averaging and subsampling operator. This correction makes the degradation factor consistent with the ratio between 1/12 and 1/3°. The resulting grid spacing of about 35–40 km is intermediate between the effective footprint of present L-band radiometer missions (typically 40–100 km) and the nominal sampling of commonly used Level-4 multi-mission products, and thus provides a useful idealized proxy for current satellite SSS products . Only SSS is degraded in this way; SST and SSH are kept at their native 1/12° resolution and are used as idealized high-resolution auxiliary fields in the controlled experiments. The choice of 1/3° is not intrinsic to the method: in principle, the same workflow could be applied to another target resolution by adapting the degradation operator.

This idealized treatment of SSS, SST, and SSH should be kept in mind when considering satellite applications. The present SSS/SST/SSH conditioning experiments should therefore be interpreted as controlled tests rather than as direct operational satellite configurations.

Before training, all variables were linearly normalized to zero mean and unit variance over the training period. This normalization places SSS, SST and SSH on comparable numerical scales and stabilizes optimization. During training, square patches of 64×64 pixels (approximately 600km×600km) were randomly extracted from the domain and provided to the model (see Fig. ). Working with patches rather than full-domain images reduces the memory required per training sample and increases the number of training examples, which is critical given the high dimensionality of diffusion models. At the same time, the energetic nature of the Gulf Stream ensures that strong contrasts and eddy structures are frequently present, even within such smaller spatial windows. These patches therefore provide a rich set of examples for learning the multivariate relationships between SSS, SST and SSH that are exploited in the generative downscaling framework. In doing so, our methodology aims to obtain an SSS patch of 64×64 pixels at 1/12° resolution, independently of any geolocation information within the region of interest.

In this study, downscaling refers to the reconstruction of a high-resolution SSS field from a coarser SSS observation and high-resolution auxiliary variables. We follow the notation commonly used in inverse problems and data assimilation: X denotes the unknown high-resolution state, Y denotes the available observation, and H is the observation operator mapping the state space to the observation space. In the present case, X is the SSS field on the 1/12° grid and Y is the corresponding coarse SSS observation on the 1/3° grid. The two are linked by H, 1Y=HX+η, where H represents spatial averaging and sub-sampling from 1/12° to 1/3°, and η is an observation noise term. In our experiments based on GLORYS, Y is synthetically generated from X by applying H.

In addition to the coarse SSS Y, we assume that high-resolution auxiliary variables Z=(SST,SSH) are available on the same 1/12° grid as X. They can be used alone or together to help estimate the high-resolution state X. These fields share many of the fronts and filaments present in SSS and provide information on the underlying dynamics. From a learning perspective, the downscaling problem consists in estimating a mapping: 2X^=fθ(Y,Z), parametrized by the model weights θ, such that the prediction X^ is a high-resolution SSS field on the 1/12° grid, consistent with Y through the observation operator H and statistically similar to the reference high-resolution SSS fields. Thus, X and X^ are always defined on the high-resolution grid, whereas Y is defined on the coarse observation grid. In the next subsection, we describe the diffusion-based generative model used to implement this mapping.

The backbone of our diffusion model is a U-Net architecture, widely used in image generation and reconstruction tasks . It follows the usual encoder-decoder structure: a sequence of convolutional and attention blocks gradually reduces the spatial resolution while increasing the number of feature channels, then a symmetric decoding path progressively upsamples the representation back to the original 1/12° grid. In practice, we use a multi-channel U-Net that takes as input the stacked SSS, SST and SSH fields (when available) together with a sinusoidal time embedding of the diffusion step, as in standard DDIM implementations . Self-attention layers are included at the coarsest resolutions to capture long-range spatial dependencies, while convolutional residual blocks handle local interactions. Skip connections link encoder and decoder blocks at matching scales, allowing the network to combine local fine-scale information with broader contextual features and to better preserve sharp fronts and filaments (details about the architecture in Appendix A). This design provides a good balance between expressiveness and computational cost for high-resolution oceanographic images.

We present hereinafter the two phases of the classical diffusion model and we propose a new algorithm, the Gradient-Enhancing (GE) algorithm that allows to increase the contrast in the generated SSS 1/12° samples.

Diffusion models are a class of probabilistic generative models that learn the probability distribution of a set of training samples in order to generate new, statistically consistent realizations . In our case, a training sample x0 corresponds to a high-resolution image on the 1/12° grid (e.g. including SSS, and in some configurations also SST and SSH channels). Training consists in two phases: a “forward” noising process where gaussian noise is added to the target sample, and a “backward” process where the network learns to gradually remove the noise to retrieve the original sample.

In the forward diffusion process, Gaussian noise is therefore progressively added to an image x0 over a fixed number T of steps (typically T=1000). At each step t, a coefficient αt∈(0,1) controls the variance of the added noise. After sufficiently many steps, the original information is effectively lost and the sample becomes close to pure isotropic Gaussian noise. The forward process can be written in closed form as 3xt=α¯tx0+1-α¯tϵ,ϵ∼N(0,I), where α¯t=∏s=1tαs is the cumulative product of the noise schedule. This formulation allows xt to be sampled directly from x0 without explicitly simulating all intermediate steps.

The model then learns to reverse this process during the denoising phase, recovering x0 step by step by predicting (and removing) the noise ϵ added at each step. In the ϵ-prediction formulation , the reverse dynamics can be written as 4xt-1=1αtxt-1-αt1-α¯tϵθ(xt,t)+σtϵ, where ϵ∼N(0,I) is fresh Gaussian noise injected to maintain stochasticity in the reverse process, σt controls its variance, and ϵθ(xt,t) is the neural network prediction of the noise at step t.

The network parameters θ are trained by minimizing the mean-square error between the true noise and its prediction: 5Lθ=Ex0,ϵ,tϵ-ϵθ(xt,t)2, where t is typically sampled uniformly from {1,…,T} and xt is obtained from x0 using the forward process above. At this stage, the training is unconditional with respect to the coarse observation: the model learns a prior distribution over high-resolution multichannel images drawn from GLORYS, without yet involving the coarse SSS observations Y or the observation operator H. This prior represents the joint GLORYS distribution of SSS with SST and/or SSH, including their spatial organization and cross-variable relationships. Time is not explicitly modeled in the present architecture but relies on the joint information between SSS and auxiliary variables at the daily situation scale. Extending the framework to a spatio-temporal diffusion model may allow temporal coherence to be learned more directly. Conditioning on (Y,Z) to perform downscaling is introduced at inference time and described in the next subsection.

Once training is complete, the model is able to generate high-resolution samples that are consistent with the learned joint distribution of SSS, SST and/or SSH as represented in GLORYS. The conditioning on a given observation is then performed during inference. Conceptually, the learned diffusion prior plays a role similar to a statistical background, while the coarse SSS observation provides the constraint imposed through the observation operator.

Several conditioning strategies have been proposed in the literature ; here we follow the pseudo-inverse guidance method of , which offers a good compromise between performance, simplicity and computational cost and can be applied to a wide range of inverse problems beyond super-resolution (see Sect. ).

We rely on the observation model given in Eq. (), where X denotes the high-resolution SSS field on the 1/12° grid, Y the coarse SSS field on the 1/3° grid, and H the observation operator (spatial averaging and subsampling). We denote by H† a pseudo-inverse of H that lifts residuals from observation space back to state space (in practice, an upsampling operator from 1/3 to 1/12°).

During inference, let xt be the current latent state in diffusion step t, and ϵθ(xt,t) the predicted noise. As in the standard ϵ-prediction formulation, we first form a one-step estimate of the clean sample, 6X^t=xt-1-α¯tϵθ(xt,t)α¯t. We then compute the mismatch between this estimate and the observation in the coarse space, 7rt=Y-HX^t, and project this residual back onto the high-resolution grid using the pseudo-inverse H†. This defines a guidance term 8gt∝H†rt=H†(Y-HX^t), which nudges the reverse diffusion towards the subset of the learned distribution that is compatible with the observations. In practice, gt is added to the standard reverse update for xt-1 (see Algorithm A1 in Appendix A for a complete description of the guided inference procedure).

A key advantage of this strategy is that the conditioning is entirely decoupled from the training: the diffusion model is first trained as an unconditional generator on high-resolution GLORYS fields, and different inverse problems (e.g. super-resolution, noisy or masked observations) are handled at inference time simply by changing the operator H and the corresponding pseudo-inverse H†. Because the reverse process remains stochastic, starting from different realizations of the initial Gaussian noise yields different high-resolution samples that all satisfy the same coarse SSS observation Y, allowing us to explore a range of plausible fine-scale fields instead of a single averaged solution. A schematic representation of the conditional inference phase is shown in Fig. .

Because the reverse diffusion process remains stochastic, different random initializations give rise to different plausible denoised patches for the same coarse SSS constraint. This raises two related questions: how to select a single displayed reconstruction, and how to interpret the spread across generated samples. In this study, we compute the ensemble mean image and select the generated member that is closest to this mean in ℓ2 norm. This closest-to-mean member acts as a medoid-like representative of the conditional ensemble: it remains close to the ensemble central tendency while preserving the spatial texture of an individual realization. By contrast, the ensemble mean may reduce pixel-wise error but can smooth fronts and filaments when plausible structures are slightly displaced across members. The choice between an ensemble mean, a representative member, or another criterion is therefore application dependent. The ensemble variance can also be interpreted as a first diagnostic of reconstruction ambiguity.

In the following, we propose the Gradient-Enhancing (GE) algorithm to select the SSS 1/12° field with the highest contrast, enhancing small structures.

The image gradient refers to the directional change in the pixel intensity in both the x-axis and the y-axis of an image. It is a key ingredient in image processing and computer vision, particularly in detecting edges or in processing images with edges. The problem of too-low image gradients is common in image reconstruction tasks. Deterministic models optimized with mean-square-error objectives tend to approximate the conditional mean of possible high-resolution solutions, which can smooth fronts and filaments when several plausible fine-scale configurations are compatible with the same coarse observation. During inference, the GE procedure guides the generative process towards higher-contrast samples in the learned distribution. The idea is to favor, at each denoising step, realizations that exhibit stronger local gradients while still remaining consistent with the learned distribution and the coarse SSS constraint. The method is built on top of the pseudo-inverse guidance described above, by adding an additional contrast-based guidance term.

The complete GE algorithm is detailed in Appendix A1 and is applied each time step during the inference phase. In the diffusion timestep t, we consider the one-shot prediction of the network, that is, the current estimate of the final fully denoised image available in DDIM. This estimate gets better and better as the denoising process goes on. On kernels of 2×2 pixels of the image, we enhance the local contrast by subtracting the kernel's mean to the 4 pixels, multiplying by a scaling factor, then adding the mean back. In regions where pixel values deviate strongly from the local kernel mean, indicating the presence of a front or other fine-scale structure, the contrast is amplified. In smoother regions, the effect remains weaker so that no spurious small-scale features are created. The resulting contrast-enhanced image is then fed back into the update for t-1, thereby nudging the network towards higher-contrast samples, which still lie within the learned distribution. The process is sensitive to instabilities, especially for high gradients that can explode over numerous time-steps of multiplicative scaling. The following scaling term provides a damping effect on the highest gradients of the image to prevent those instabilities:

s=γ-δ|X^t|max⁡|X^t| where max is computed over all kernels.

The scaling factor s is equivalent to γ-δ for high gradients and to γ for small gradients; both parameters are kept close to one and tuned so that the final gradient distribution of the generated images matches as closely as possible that of the test set. In summary, we want to reduce the overall proportion of low gradients, enhance the overall proportion of high gradients, and do so at the right places in the generated samples. In the following, we refer to the diffusion model combined with this gradient-enhancing (GE) procedure as DIFF-SST-SSH-GE.

For the first three experiments (DIFF-SST, DIFF-SSH, DIFF-SST-SSH), we generate K=15 samples with independent noise seeds, compute the ensemble mean, and select the single realization that is closest to that mean in ℓ2 norm. This closest-to-mean member provides a pragmatic proxy for the center of the conditional distribution while preserving the texture of an individual realization. An example of different plausible SSS outputs of DIFF-SST-SSH is illustrated in Fig. . Inference also generates the associated 1/12° SST and/or SSH channels, depending on the configuration, because the diffusion prior represents a multichannel GLORYS-consistent state. The evaluated downscaling target, however, remains SSS. The variance map displays the pixel-wise ensemble variance across the generated high-resolution SSS patches, considered as a first diagnostic of reconstruction ambiguity. As expected, the spread is higher in dynamically active regions where fine-scale structures are more contrasted or slightly displaced across samples.

For DIFF-SST-SSH-BEST: from K=40 samples, we pick the realization that minimizes RMSE to the reference; this is not operational (it requires ground truth) but indicates the headroom available from the same trained prior.

For DIFF-SST-SSH-GE: we apply the gradient-enhancing guidance during inference and use the same central selection (closest-to-mean over K=15 guided samples), yielding higher-contrast outputs while remaining distribution-consistent.

In Table  we give for the six experiments the mean RMSE (psu) and SSIM during the test year 2020 over the test domain. These performances are computed from a daily basis. Conditioning on high-resolution auxiliaries at 1/12° improves SSS reconstructions from coarse SSS at 1/3°. DIFF-SST outperforms DIFF-SSH and adding together SST and SSH improves the global RMSE. Indeed, as mentioned in Introduction, SST adds more information than SSH; SST provides the most precise guidance for the reconstruction of SSS. SST exhibits correlated structures at a very fine scale, unlike SSH, which is much smoother in the current GLORYS-based setup. A yearly average quantify these gains: DIFF-SST improves RMSE by ∼ 21 % relative to DIFF-SSH (0.045 vs. 0.057 psu); adding SSH on top of SST yields a further ∼ 11 % improvement (0.040 vs. 0.045 psu for DIFF-SST-SSH vs. DIFF-SST), indicating complementary mesoscale constraints from SSH.

To better understand the behavior of the six models, we present daily time series of RMSE over the year 2020. The y-axis is the mean RMSE computed over the Gulf Stream domain. As the order of magnitude is different for the six experiments, we compare DIFF-SST and DIFF-SSH in Fig.  and the 4 other models in Figs. . For the six experiments, across the seasonal cycle, errors peak in late spring–summer (day ∼ 170–200) and decrease towards winter. This timing coincides with seasonal maxima of eddy kinetic energy and frontal activity in the Gulf Stream region .

As expected, in Fig.  DIFF-SST (magenta line) systematically outperforms DIFF-SSH (green line). In Fig. , DIFF-SST-SSH (magenta line) and RESAC (yellow line) display very similar daily RMSE time series, while DIFF-SST-SSH-BEST (blue line) provides a baseline/lower envelope because it uses the known reference for sample selection. DIFF-SST-SSH-GE (green line) has a larger RMSE than the standard DIFF-SST-SSH configuration. Thus, GE does not improve pixel-wise scores; rather, it is designed to test whether contrast-oriented inference can recover sharper structures at the cost of a moderate degradation in RMSE/SSIM. This distinction is important because RMSE and SSIM alone do not fully characterize frontal contrast and high-frequency variability.

For that purpose we compare different snapshots and then quantify the structural differences using gradient and spectral diagnostics. Visual differences in Fig.  are subtle at the scale of the full patch, and the figure should be interpreted as a qualitative illustration rather than as direct proof of sharpening. The effect of GE on small-scale structure is therefore assessed more directly in the next subsection using the gradient distribution and temporal Fourier spectrum. For all indices (RMSE, SSIM) computed yearly or daily the ranking of the models remains stable (Fig. ). Thus, as using both auxiliary variables gives the best performances, DIFF-SST-SSH only will be used for the last comparisons.

To assess fine-scale structure beyond pixelwise error, we analyse spatial gradient-magnitude distributions and temporal Fourier spectra over the 2020 test year. The reference gradient distribution (Fig. , left) is strongly peaked at low gradient values, corresponding to smooth regions, with a long tail representing fronts and energetic structures. The right panel shows the difference between model and reference gradient distributions for DIFF-SST-SSH, DIFF-SST-SSH-GE, and RESAC. All methods overestimate the proportion of weak gradients and underestimate intermediate-to-strong gradients, consistent with residual smoothing. DIFF-SST-SSH-GE reduces this bias over the low-to-intermediate range, indicating improved contrast and sharper reconstruction of fine-scale structure.

Temporal spectra provide a complementary view (Fig. ). DIFF-SST-SSH and RESAC under-represent high-frequency (weekly-daily) variance, while DIFF-SST-SSH-GE remains closer to the reference across most frequencies and becomes slightly positive at the highest frequencies, consistent with recovered small-scale temporal energy. DIFF-SST-SSH-BEST does not uniformly improve all spectral ranges, illustrating that selecting a sample solely by minimizing RMSE does not guarantee optimal spatio-temporal consistency. Since BEST selection is not applicable operationally, it is interpreted only as a performance headroom indicator.

Overall, when pixelwise accuracy is the primary objective, DIFF-SST-SSH provides a strong operational choice with performance comparable to RESAC. When the fidelity of fronts and filaments is critical (e.g. frontal detection, feature tracking, or Lagrangian applications), DIFF-SST-SSH-GE offers a sharper representation of fine scales at a small cost in RMSE/SSIM. Across experiments, diffusion-based reconstructions are visually sharper than RESAC.

High-resolution SST is the dominant auxiliary constraint for SSS downscaling in the Gulf Stream. Across the full 2020 test year, conditioning on SST yields substantially lower error than conditioning on SSH alone (Table ; Fig. ). This ranking is robust throughout the seasonal cycle, including the most energetic period (day ∼ 170–200) when reconstruction is most challenging (Figs.  and ). This is consistent with the visual comparison of the three variables (Fig. ): SST shares filamentary and frontal structures with SSS at finer scales, whereas SSH fields used here are smoother and mostly express mesoscale organization. In other words, SST appears to supply the local small-scale information that the model can “lock onto”, while SSH provides a weaker but complementary contextual constraint on the position and intensity of mesoscale features.

Adding SSH on top of SST provides a smaller but consistent gain. DIFF-SST-SSH improves on DIFF-SST in yearly-mean RMSE (Table ) and remains competitive day-by-day (Fig. ). The improvement is modest compared to the DIFF-SSH/DIFF-SST gain, but it is physically plausible. Yet this result should be interpreted with caution. It does not necessarily imply that SSH is intrinsically less relevant for constraining salinity fine scales; rather, it may reflect the comparatively smooth character of the SSH fields used in the present GLORYS-based setup. This point is particularly important in light of recent SWOT observations, which reveal a much richer fine-scale sea-surface height variability than previously accessible from conventional altimetry . Recent studies also suggest that part of this small-scale and high-frequency SSH variability is still underestimated or only partially represented in present-generation 1/12° model-based products . Therefore, the weaker performance of SSH in our framework may be tied to the effective resolution and information content of the SSH product used here, rather than to the physical usefulness of SSH itself. Revisiting this question with SWOT-based or SWOT-constrained SSH products is a natural and important next step.

Diffusion with pseudo-inverse guidance is comparable to a strong CNN baseline under pixelwise metrics, while providing a conditional ensemble. With the closest-to-mean selection strategy (Sect. ), DIFF-SST-SSH and RESAC yield very similar yearly-mean RMSE/SSIM (Table ) and similar temporal evolution (Fig. ). The diffusion approach should therefore not be interpreted as providing a large RMSE improvement over RESAC in this controlled experiment. RESAC remains a strong deterministic baseline, specifically designed for scale-adaptive downscaling. The added value of the diffusion framework is different: inference naturally produces an ensemble of plausible high-resolution fields conditioned on the same coarse observation (Fig. ), and provides a diagnostic of reconstruction ambiguity.

Contrast-oriented guidance shifts the solution toward stronger small-scale contrast, but trades off against RMSE/SSIM and can over-amplify variability. The GE procedure increases gradient contrast relative to the standard DIFF-SST-SSH configuration (Figs.  and ). At the same time, DIFF-SST-SSH-GE degrades RMSE/SSIM relative to DIFF-SST-SSH (Table ), which is expected because RMSE favors smoother reconstructions and penalizes phase/position errors strongly. The gradient histogram also shows that GE reduces the proportion of weak gradients relative to the reference, and the temporal spectrum can exceed the reference over part of the frequency range. These diagnostics indicate that GE should be interpreted as a contrast-enhancing variant with a structural trade-off, not as a uniformly superior reconstruction. Model choice should therefore depend on the scientific objective: for applications prioritizing pixelwise accuracy, DIFF-SST-SSH is preferable; for applications where front and filament fidelity matters, GE-like guidance may be useful but should be tuned carefully.

The additional diagnostics clarify how errors differ between approaches. Gradient histograms show that all methods overpopulate weak gradients and under-represent part of the mid-high tail (Fig. ), indicating residual over-smoothing relative to GLORYS. The GE variant partially corrects this by enhancing low/intermediate gradients, but the negative difference at the weakest gradients suggests that it may also over-sharpen the field by converting too many weak gradients into stronger ones. The temporal FFT analysis provides an independent view: DIFF-SST-SSH and RESAC under-represent high-frequency variance, while DIFF-SST-SSH-GE restores more energy but can exceed the reference over part of the spectrum (Fig. ). Taken together, these results support a nuanced conclusion: the diffusion prior with standard selection is competitive under pixelwise scores, but its default samples remain slightly too smooth; targeted inference-time guidance can recover part of the missing variability, at the risk of over-amplification if the contrast guidance is too strong.

A related implication concerns sample selection. The “BEST” choosing strategy indicates headroom from the same trained prior (Table ), but it is not operational and should not be over-interpreted as achievable performance. More importantly, the fact that “BEST” does not uniformly improve all diagnostics (e.g. FFT spectral behavior can differ) highlights that selecting samples solely to minimize RMSE does not guarantee spatio-temporal realism. This motivates future work on task-aware or physics-aware selection criteria that can be computed without access to truth.

The present study was designed as a controlled methodological experiment, using GLORYS as a dynamically constrained high-resolution reference and synthetically degraded SSS as coarse observations. This framework is well suited to isolate the behavior of the diffusion-based downscaling method, but it also defines the main limitations of the present work and the conditions required for transfer to real satellite SSS products.

First, GLORYS is not assumed to be a perfect truth. It combines an ocean model with heterogeneous assimilated observations, and local inconsistencies between SSS, SST and SSH may remain. In this paper, GLORYS provides a complete multivariate gridded reference from which the diffusion model learns a GLORYS-consistent joint prior over SSS, SST and SSH. The generated reconstructions should therefore be interpreted as physically plausible within the GLORYS framework, not as guaranteed representations of the real ocean at all scales.

Second, the use of GLORYS avoids many of the complexities of L-band satellite retrievals, including sensor-specific biases, correlated retrieval errors, and irregular observation patterns . While this is appropriate for method development, it means that the reported metrics should not be interpreted as direct estimates of performance expected for SMOS, SMAP, or multi-satellite products such as CCI products . Real Level-4 SSS products are not clean low-pass versions of the true SSS field. For example, ESA CCI SSS is a multi-sensor Level-4 product derived from L-band radiometry; although it is sampled on a 0.25° grid, its effective spatial resolution is about 50 km and it provides uncertainty and quality information such as random error, unexplained variability, number of observations, outliers, and land/ice/SSS quality flags . Such information should be used in future work to weight the observation guidance spatially and to mask unreliable pixels.

Third, the observation operator H used in this study mainly reflects spatial averaging and sub-sampling (Eq. ). For real applications, H should mimic the targeted satellite product much more faithfully, including its effective footprint, grid, masks, and spatially heterogeneous uncertainty. Because pseudo-inverse guidance injects observational information at each denoising step , the realism of H is a central requirement for domain transfer. In practical applications, the guidance term should therefore be uncertainty aware, matching H(X) to the satellite observation Y on the satellite grid while down-weighting uncertain, missing, or contaminated pixels.

Fourth, the auxiliary variables also have product-dependent limitations. High-resolution SST is an effective constraint in the present experiment, but observed infrared SST is unavailable under clouds. Daily gap-free Level-4 SST products may be distributed on fine grids, yet their local information content can degrade under persistent cloud cover because they rely on interpolation, microwave SST, previous clear-sky observations, in-situ data, and the analysis method . In such regions, SST should be treated as an uncertain auxiliary constraint rather than as a uniformly reliable high-resolution observation. Similarly, the relatively weaker contribution of SSH in our experiments should not be regarded as definitive, as it may partly reflect the smooth character of the SSH fields in the present GLORYS-based setup. This balance may evolve as higher-resolution SSH observations become available, particularly through SWOT, which may provide a richer fine-scale dynamical constraint than that represented in current-generation reanalyses.

Fifth, the role of in-situ observations should be stated, such as Argo profiles within the proposed transfer framework. The high-resolution prior would still be learned from gridded reanalysis or simulation fields, while the observational constraint would be provided by a satellite SSS product through a product-aware observation operator. Argo, thermosalinograph data, moorings, drifters, and regional campaigns would then provide independent validation, calibration, and bias assessment.

Finally, operational deployment requires sample selection and uncertainty-calibration strategies that do not rely on access to the true high-resolution field. The closest-to-mean member used here provides one representative realization, while the ensemble spread shown in Fig.  provides a first diagnostic of reconstruction ambiguity. A calibrated uncertainty product would require testing the relationship between ensemble spread and actual reconstruction error using independent validation data. In that context, the diffusion framework is not only a source of multiple plausible reconstructions, but also a way to diagnose when the solution is weakly constrained, for example near coasts, under cloud-degraded auxiliary SST, or under strong satellite SSS retrieval uncertainty.

In a controlled Gulf Stream experiment, diffusion models guided by pseudo-inverse conditioning can downscale coarse SSS to 1/12° with accuracy comparable to a strong convolutional baseline, while naturally providing a conditional ensemble of plausible high-resolution fields. In the current GLORYS-based context, high-resolution SST is the primary source of fine-scale guidance; SSH adds complementary mesoscale context. A contrast-oriented guidance term can partially recover missing gradients and high-frequency temporal variance, at a small cost in RMSE/SSIM, highlighting an application-dependent trade-off between distortion metrics and structural realism .

The next methodological frontier is transferability to satellite SSS. This requires (i) realistic product-aware observation operators and noise/mask models in H, (ii) uncertainty-aware weighting of satellite SSS and auxiliary SST/SSH constraints, (iii) strategies to mitigate reanalysis-to-satellite distribution shift, and (iv) operational sample selection and uncertainty calibration using independent in-situ validation. With these steps, diffusion-based downscaling has the potential to enrich present SMOS/SMAP/CCI products with GLORYS-consistent fine-scale structure, and to provide ensemble-based reconstructions that are directly usable for process studies and, ultimately, data-assimilation-oriented applications.