the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 24 Feb 2026
A Continuous Implicit Neural Representation Framework with Gradient Regularization for Sea Surface Height Reconstruction From Satellite Altimetry
Abstract. Satellite altimetry provides valuable measurements of sea surface height (SSH) but is characterized by irregular spatiotemporal sampling and substantial data gaps arising from orbital configurations, sensor limitations, and environmental conditions. These sampling properties pose challenges for constructing continuous and dynamically consistent SSH fields. In this study, we develop an interpolation framework based on implicit neural representations (INRs), in which SSH is represented as a continuous function of space and time. The framework employs sinusoidal representation networks (SIREN) to enable smooth gradients and efficient spectral representation. To improve reconstruction in regions with sharp spatial transitions, such as fronts and eddy boundaries, we incorporate a total variation (TV) regularization term, allowing the model to preserve abrupt features while maintaining global smoothness. The combination of a continuous, differentiable INR formulation with gradient-based regularization provides a compact and flexible approach for SSH reconstruction. We evaluate the proposed framework using both multi-mission satellite altimetry observations and high-resolution numerical simulations. Experiments conducted indicate that the proposed SIREN–TV framework can recover fine-scale and locally sharp structures while preserving the large-scale variability of the SSH field. The method maintains a level of global accuracy comparable to existing interpolation and data-assimilation approaches, but provides enhanced spatial detail in regions affected by strong gradients, fronts, or mesoscale activity. In addition, the continuous and fully differentiable representation enables direct computation of spatial derivatives, facilitating higher-order oceanographic diagnostics. These results suggest that INR-based formulations offer a promising complementary avenue for SSH interpolation under sparse and irregular sampling configurations.
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Status: open (until 11 Jun 2026)
- RC1: 'Comment on egusphere-2025-6389', Anonymous Referee #1, 10 Apr 2026 reply
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RC2: 'Comment on egusphere-2025-6389', Anonymous Referee #2, 08 Jun 2026
reply
Reviewer 1 gave a good summary of the paper and has excellent comments. I am just going to supplement the discussion, starting from that review.
Strengths:
The smoothness and differentiability of the method are pluses for ocean representation.
Their examples show improved performance over most of the tested methods, and they evaluate an impressive array of methods.
Weaknesses:
Some people may find it hard to learn about the method from just these few examples.
These methods are changing rapidly, and one wonders how long it will be until this method is itself superseded.
The black box nature of the method makes it difficult to build understanding. Anything the authors can do to counteract this point of view would increase the impact of their work.
The overarching questions after reading this paper are:
What are the lessons a reader will take away from it for other applications?
When do we expect this method to work well, and what are the limitations? E.g., in the first experiment, how variable could the results be if it were tried on other time ranges, and how much time coverage is needed to have a good training data set? Is their technique of training on the first 15 days and the last 15 days something that they learned was good, or is it the first thing they tried?
They should also make the language clearer about whether the training and test datasets overlap on Jan 15 and Mar 15, which is currently somewhat ambiguous, although I assume they do not overlap.
A classic reviewer question is to ask about the sensitivities of the method, and one lesson we might take from these results is how long it takes for this method to accurately learn the dynamics of each region.The presentation should be made more accessible to earth scientists, who may be less familiar with formal mathematical descriptions. In particular, the formal math exposition, it leaves out small details that would have helped me to understand.
As an example, in line 104, they describe a general mapping from spatial coordinates to physical quantities. I wish they could have noted that there were $m$ spatial coordinates and $n$ corresponding physical quantities. This only adds two characters to the exposition, but it would have made it much clearer.
Along the same lines, when they introduce the constraints on line 109, they could have given an example, as they do later with the altimeter, which would have helped.
OI can analytically produce gradient estimates and has the additional benefit of being easily understandable, transparent, and generating uncertainty maps. For example: An Improved Mapping Method of Multisatellite Altimeter Data P. Y. Le Traon, F. Nadal, and N. Ducet DOI: https://doi.org/10.1175/1520-0426(1998)015<0522:AIMMOM>2.0.CO;2
Maps from the Mid-Ocean Dynamics Experiment: Part I. Geostrophic Streamfunction, James C. McWilliams, 1976 DOI: https://doi.org/10.1175/1520-0485(1976)006<0810:MFTMOD>2.0.CO;2 and: An Objective Analysis of the POLYMODE Local Dynamics Experiment. Part II: Streamfunction and Potential Vorticity Fields during the Intensive Period Bach Lien Hua, James C. McWilliams, and W. Brechner Owens,1986, DOI: https://doi.org/10.1175/1520-0485(1986)016<0506:AOAOTP>2.0.CO;2
This extends to the other analytical methods discussed, which generally bring in trusted dynamical frameworks to do the time evolution. It is unfortunate that we currently have to give up explainability and uncertainty quantification to get improved performance.
When they introduce the SIREN framework, again they use a very formal presentation without a translation into more geophysical language, which doesn't appear until equation 4, 2 pages later. Perhaps an expanded exposition could be in an appendix if they don't want to expand the main body.
On line 164, they could have noted that $k$ is the layer number. It's obvious eventually, but please say it here and help the reader.
The normalization of the coordinates is important in regression. This is a good sign of the care they took in their analysis.
Equations 4 and 7 show no weighting in the norm. Normally, there would be a metric tensor as part of the quadratic product, or at least a diagonal matrix that scales by local uncertainty. Their regions may be small enough that they can get away with this, but it should be acknowledged.
Reviewer one may hint at this by asking whether variable a is changed regionally and how large the total variation constraint is.
Also, for comparison to Bayesian methods, they can characterize this as maximizing a likelihood with Gaussian distributions for the observation uncertainty and the total variation metric. This would also give guidance for how to weight the trade-off between data fit and structure, which is currently left to the parameter a.
Regarding the parameter a: on line 203, it seems to be renamed as \lambda, as flagged by reviewer 1
Reviewer one also asked about the correlation between training and test data, which is an important point. The autocorrelation decay times of the mapped SSH fields and the model SSH fields could be used as a time scale to see how independent the values are.
My reading of the text is that, in addition to withholding the Cryosat-2 altimeter, they train on the first 15 days of January and the last 15 days of March and then test on the withheld data in between. In this case, why withhold any altimeter data? Should we expect that Cryosat would be that different from the other altimeters, so why not compare to all of them?
This would also allow us to look at the presumably increasing and decreasing area-averaged RMS error day by day during the period between the two training data sets.
In their exposition of the spatial gradients, they don't mention the metric terms from lat and lon to distance, the scaling from SSH to geostrophic velocity, or the concerns about cyclogeostrophy and other ageostrophic features at small scales. If they mean gradients instead of velocities, then why call them $v_lat$ and $v_lon$? However, later they describe the ability to estimate velocity as a benefit of the method, so these concerns are valid
In section 5.1, line 231, they introduce the "RMSE-based score". It would be easier to understand as an RMSE skill score.
The notation in equation 12 is a little hard to follow. I could be helped by writing PSD as an explicit function of w and f, and perhaps changing to more traditional notation for frequency and wavenumber.
On line 239, they give minimum resolvable wavelengths in x and time, but don't include why. Their total variation metric is isotropic, but that doesn't mean that their learned weights will provide isotropic estimates. I suggest that they discuss this.
Figures 3 and 4 seem to be somewhat redundant, since 3 is the absolute error at points and 4 is a pixel-averaged RMS error. They should address this question for other readers, and, if it is a repeat, what is the reason for plotting such similar measures? It might instead be useful to show a cumulative distribution function for the signed differences between the observations and the reconstruction methods.
The caption for Figure 5 seems confused. The first panel is not the along-track SSH time series, as far as I can tell. It seems to be a count of daily observations in that time range.
In Figure 5, the performance of the INR is always above the OI, but this figure prompts the desire for error bars on the differences. How variable are the performances at different times (for the same training and testing rubric) and/or with different training and testing ranges?
For Table 1 and its discussion, they use the notation $u(RMSE_S)$ without comment, but the original discussion in section 5.1 just uses $RMSE_S$. I don't see the point of the extra $u()$.
For Figure 7, they should tell the dates they chose to plot, so the reader can have some idea of the evolution, and say why they chose these times. I can understand choosing arbitrary times, but random is a very specific term and could mean irregularly spaced, which triggers the question, why not pick four regularly spaced dates during the period?
Summary:The method performs well in their two examples, but the lack of UQ and the lack of transparency, e.g., the meaning of the weights learned in the neural net, are barriers to learning about the dynamics, such as the time scales used in the map.
Citation: https://doi.org/10.5194/egusphere-2025-6389-RC2
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- 1
Review of “A Continuous Implicit Neural Representation Framework with Gradient Regularization for Sea Surface Height Reconstruction From Satellite Altimetry” by Li et al.
The authors investigate the potential of an Implicit Neural Representation (INR) framework for reconstructing Sea Surface Height (SSH) fields from sparse altimetry observations. The approach is evaluated in two regional configurations: a realistic setting over the western Mediterranean basin and a simulation-based experiment in the Gulf Stream region. The study uses existing data challenge frameworks to benchmark the proposed method against established approaches (e.g. DUACS OI, BFN, 4DVARNET etc… ). The results suggest improved reconstruction performance with INR in both Observing System Experiments (OSE) and Observing System Simulation Experiments (OSSE), highlighting the potential of the method for recovering SSH fields from incomplete satellite measurements.
Overall, the manuscript is relatively well written, and the results are clearly presented. The study is relevant and the proposed approach appears promising. That said, several aspects could benefit from additional clarification and further discussion to strengthen the manuscript prior to publication in "Geoscientific Model Development". For this reason, I recommend the manuscript should be reconsidered after major revision. I outline below several points that the authors may wish to consider.
Major Points
Minor Points