The input variables in SSH reconstruction have different physical units and numerical ranges. Longitude and latitude are measured in degrees, whereas time is measured in days. If these raw coordinates are directly used as network inputs, the difference in numerical scale may make the training less stable. Therefore, each coordinate is shifted and rescaled before being passed to the network: 3loñ=factorslon,lat̃=factorslat,timẽ=factorttime. The spatial scaling factor factors and temporal scaling factor factort are used to bring the spatial and temporal inputs to comparable numerical ranges. The normalized coordinate is denoted by s̃=(loñ,lat̃,timẽ).

This normalization is only a preprocessing step for the network input and does not change the physical meaning of longitude, latitude, or time. This preprocessing improves numerical conditioning and makes SIREN training more stable for the sparse multi-mission altimetry data used here.

After coordinate normalization, the normalized coordinate s̃ is used as the input to the SIREN network. Starting from the general INR layer in Eq. (), we use a sinusoidal activation function to construct the hidden layers. Let z0=s̃ denote the input to the network. For the kth hidden layer, the layer output is defined as 4zk+1=sin⁡(Wkzk+bk),k=0,…,L-1, where k denotes the hidden-layer index, L is the number of hidden layers, Wk and bk are the trainable weights and biases of the kth hidden layer, and the sine function is applied element-wise. Here, k=0 corresponds to the first hidden layer, and k=L-1 corresponds to the last hidden layer. Compared with the general form in Eq. (), Eq. () specifies the activation function as sin⁡(⋅).

We employ a fully connected SIREN network consisting of an input layer, several sinusoidal hidden layers, and a final linear output layer. After the last hidden layer, the reconstructed SSH value at the queried coordinate is obtained as 5Φθ(s̃)=WoutzL+bout, where zL is the last hidden representation, and Wout and bout are the parameters of the output layer. The parameter set θ includes all hidden-layer weights and biases as well as the output-layer parameters. The differentiability of this coordinate-based representation is used in the spatial-gradient regularization introduced below.

The data-fidelity loss measures the mismatch between the observed SSH value and the reconstructed SSH value at each satellite-track location. For an observed coordinate si, let SSHtrue,i denote the satellite-observed SSH value and let SSHrec,i=Φθ(s̃i) denote the reconstructed SSH value output by the network. The data-fidelity loss adopts the mean squared error: 6Ldata(θ)=1N∑i=1NSSHrec,i-SSHtrue,i2. This term constrains the network output SSHrec,i to match the observed value SSHtrue,i at the available satellite-track locations. The unweighted mean squared error can also be related to a standard least-squares interpretation, in which the available along-track observations are treated with a common error variance. Thus, the present experiments use an ordinary least-squares data term in which the observed along-track samples contribute equally.

However, the summation in Eq. () only includes observed coordinates where satellite measurements are available. For unobserved locations, there is no corresponding SSHtrue value to directly guide the reconstruction. Under sparse and irregular sampling, the learned SSH function may therefore produce unsupported local variations. In the reconstructed field, such variations can appear as rapid SSH changes over short spatial distances and are reflected in large local spatial gradients.

Let Φθ(s̃) denote the reconstructed SSH field. The normalized spatio-temporal coordinate is s̃=(loñ,lat̃,timẽ). The spatial gradient is computed with respect to the normalized longitude and latitude coordinates: 7∇spΦθ(s̃)=∂Φθ∂loñ,∂Φθ∂lat̃=glon,glat, where glon and glat denote the two normalized-coordinate gradient components. These derivatives describe how rapidly the reconstructed SSH changes with respect to the normalized horizontal input coordinates. In the TV term, they provide a dimensionless regularization measure; no conversion to physical distance is applied.

To penalize excessive local spatial changes, we adopt a first-order isotropic TV penalty: 8LTV(θ)=1M∑j=1Mglon,j2+glat,j2, where the summation is evaluated over M sampled points in the reconstruction domain. Here, j indexes the points used to evaluate the TV regularization, while i in Eq. () indexes the observed satellite-track samples. This term measures the average magnitude of the local spatial gradient. A larger value indicates stronger short-distance SSH variations, while a smaller value corresponds to a smoother local spatial structure.

The final training objective combines the data-fidelity loss and the TV regularization term: 9Ltotal(θ)=Ldata(θ)+aLTV(θ), where a>0 controls the relative contribution of the TV regularization term compared with the data-fidelity loss. In this study, a is treated as a global hyperparameter for each experimental configuration and is kept fixed over the reconstruction domain. The coefficient a, together with the network and optimization settings, was determined using the validation datasets described in Sect. 5. A very small a may provide insufficient regularization and lead to noisy local variations, whereas an overly large a may over-smooth the SSH field and weaken dynamically meaningful sharp structures.

Figure  summarizes the processing pipeline, including coordinate normalization, the SIREN-based implicit representation, and TV regularization.

The RMSE between reconstructed and true SSH is first computed as 10RMSE=1N∑i=1NSSHrec(loni,lati,timei)-SSHtrue(loni,lati,timei)2, where N denotes the number of evaluation points. SSHrec(loni,lati,timei) is the reconstructed SSH at the ith spatio-temporal location, and SSHtrue(loni,lati,timei) is the corresponding reference SSH value.

Following the common definition of a skill score, this RMSE is normalized by the root-mean-square amplitude of the reference SSH field: 11RMSESS=1-RMSERMS(SSHtrue), where RMS(⋅) denotes the root-mean-square of the reference SSH field. A score of 1 corresponds to a perfect reconstruction, while larger values indicate better reconstruction skill.

This score is used with different averaging domains in the two experiments. In the OSE case, where the final test uses withheld along-track observations, we compute daily values RMSESS(tk) and report their temporal mean over the valid evaluation days. In the OSSE case, where the complete NATL60 reference field is available, the same RMSESS definition is applied once to all valid grid points and evaluation times. Thus, in the OSSE comparison table, one RMSESS value is computed over these valid space–time evaluation points, rather than as a temporal mean of daily scores, a raw RMSE in metres, or a separate metric; it is rounded to two decimal places.

For the OSSE, where the complete NATL60 reference field is available, we assess the scale-dependent consistency of the reconstructed SSH fields using a power spectral density skill score in the wavenumber–frequency domain. To make the notation explicit, we write the PSD as a function of spatial wavenumber k and temporal frequency f. The wavenumber–frequency PSD skill score is defined as 12PSDSS(k,f)=1-PSDSSHrec-SSHtrue(k,f)PSDSSHtrue(k,f). Here, PSD⋅(k,f) denotes the wavenumber–frequency power spectral density of a given field evaluated at spatial wavenumber k and temporal frequency f. The numerator represents the spectral energy of the reconstruction error, and the denominator represents the spectral energy of the reference SSH field. This score therefore measures the relative reconstruction skill at each wavenumber–frequency bin. A value close to 1 indicates that the reconstruction error is small relative to the reference signal at the corresponding spatio-temporal scale, whereas a value close to 0 indicates that the error energy is comparable to the signal energy.

Following the Ocean Data Challenges protocol, two PSD-score-based effective-scale diagnostics, λx and λt, are derived from the 0.5 contour of the normalized wavenumber–frequency PSD skill score PSDSS(k,f). This threshold is used to summarize the shortest spatial and temporal scales that satisfy the adopted spectral-skill criterion. Specifically, λx is defined as the shortest spatial wavelength along this contour, while λt is defined as the shortest temporal period along the same contour. Smaller values of λx and λt indicate shorter effective scales under this diagnostic.

These quantities should be interpreted as threshold-based effective-scale diagnostics, rather than as nominal grid spacings or direct observational resolutions. In particular, when λx approaches the smallest wavelength represented by the evaluation grid and spectral discretization, it indicates that the adopted spectral-score criterion remains satisfied down to the smallest scales accessible in the diagnostic, rather than guaranteeing that all physical SSH structures at that scale are fully resolved.

The OSE configuration is conducted over the Western Mediterranean Sea within the domain [1°E,20°E]×[30°N,45°N]. It assesses the reconstruction method under realistic satellite sampling and measurement conditions. This domain defines a common reconstruction target. Following the SSH MapMed OSE benchmark , the reconstruction model is fitted using real along-track SSH observations from SARAL/Altika (alg), Haiyang-2B (h2b), Jason-3 (j3), Sentinel-3A (s3a), and Sentinel-3B (s3b) from 1 January–15 March 2021. The benchmark evaluation uses CryoSat-2 new-orbit observations (c2n) from 15 January–15 March 2021. The same non-CryoSat-2 input missions from 16–31 March 2021 and the corresponding CryoSat-2 observations form the validation dataset for hyperparameter selection.

Two classes of comparative experiments are conducted to assess reconstruction performance.

We first compare our approach with the optimal interpolation (OI), which serves as the present-day standard for DUACS products provided by AVISO. This detailed comparison is conducted from multiple perspectives, including the orbital distribution of absolute errors, the spatial distribution of residuals on the grid, the temporal evolution of the RMSE skill score, and the spectral performance assessed via PSD.

Second, to further validate the robustness and general applicability of our approach, we compare it with several representative state-of-the-art methods, namely the Covariance-based optimal interpolation (BASELINE OI), the back-and-forth nudging algorithm combined with a quasigeostrophic model (BFN_QG) , BFN_QG with Dirichlet boundary conditions (BFN_QG with coasts), and Wavevar .

Figure  provides a combined error diagnostic for the INR-based reconstruction and the OI baseline. Figure a shows the cumulative distribution function (CDF) of signed SSH residuals at the withheld along-track observation locations. The residual is defined as SSHrec-SSHobs, so positive values indicate overestimation and negative values indicate underestimation relative to the withheld observations. Compared with OI, the INR residuals are more tightly concentrated around zero, indicating a smaller residual spread and little systematic offset. The mean residual, standard deviation, and RMSE are 0.000, 0.026, and 0.026 m for INR, respectively, compared with 0.004, 0.039, and 0.039 m for OI. This result suggests that INR is more consistent with the withheld along-track observations than OI.

Figure b and c show gridded RMSE maps for the INR-based reconstruction and OI, respectively. The maps are obtained by assigning the withheld along-track residuals to spatial grid cells and computing the RMSE within each cell using Eq. (), with the withheld observations used as the reference values. These RMSE maps complement the residual CDF by showing where the remaining errors are located. Overall, the INR-based reconstruction exhibits lower RMSE values than OI over much of the domain. Some localized high-RMSE regions remain for both methods, especially near parts of the domain boundary and coastal areas, but they are weaker and less extensive in the INR map.

The temporal behaviour of the reconstruction is assessed using the RMSE skill score, denoted as RMSESS and computed according to Eq. (). Figure  summarizes the temporal variability of the evaluation data and the corresponding reconstruction skill. Figure a shows the daily number of available along-track observations used for the evaluation. The observation count varies noticeably over time, reflecting the non-uniform temporal sampling of the along-track data.

Figure b shows the daily RMSE skill scores for the INR-based reconstruction and OI. The INR method yields higher RMSESS values than OI over most valid paired evaluation days. Across 49 paired evaluation days, the mean daily RMSESS is 0.748 for INR and 0.631 for OI, with standard deviations of 0.120 and 0.151, respectively. This indicates that INR improves the average daily test skill and also shows a slightly smaller day-to-day spread than OI.

Figure c further reports the paired daily difference between INR and OI. We define this difference as Δ=RMSESSINR-RMSESSOI. Positive values indicate days when INR outperforms OI. The paired daily difference is 0.117±0.095, with an approximate 95 % confidence interval of [0.090,0.143], and INR outperforms OI on 95.9 % of the evaluated days. The positive confidence interval and the large fraction of positive daily differences indicate that the INR improvement is not driven by a few isolated dates, but is maintained over most of the OSE evaluation period. The remaining day-to-day variability indicates that the improvement is not identical on every date. Some dates are more difficult because the withheld tracks provide fewer samples or sample more challenging parts of the domain.

Spectral diagnostics for the OSE are presented in Fig. . Because the withheld CryoSat-2 reference is available only on sparse along-track segments, these diagnostics are one-dimensional along-track spatial spectra rather than the wavenumber–frequency diagnostic of Eq. (). Figure a shows the one-dimensional along-track spatial PSDs of the INR reconstruction, the withheld CryoSat-2 reference SSH, and the OI baseline. Following the publicly available SSH MapMed OSE benchmark evaluation procedure , the reconstruction is sampled at the CryoSat-2 locations and times. Continuous CryoSat-2 tracks are separated at time gaps greater than 4 s and partitioned into nominal 300 km segments with 50 % overlap; Welch spectra are computed from the pooled valid segments. Figure b shows the corresponding along-track PSD skill score, where PSDerr is the one-dimensional spatial PSD of the reconstruction error and PSDref is the one-dimensional spatial PSD of the CryoSat-2 reference SSH on these same segments. The 0.5 crossing is used to derive the PSD-score-based effective spatial scale. Under this diagnostic, INR reaches an effective spatial scale of 106 km, whereas OI reaches 154 km. This indicates that, along the withheld CryoSat-2 tracks, the INR reconstruction remains spectrally closer to the reference than OI over a broader range of spatial wavelengths, including shorter wavelengths under the adopted PSD-score criterion.

This section provides a broader evaluation of the proposed approach by comparing it against several representative reconstruction methods. The assessment relies on the two quantitative metrics introduced in Sect. , namely the temporal mean of the RMSE skill score, denoted as RMSESS‾, and the PSD-score-based effective spatial scale λx. Here, RMSESS‾ is defined as 13RMSESS‾=1Nday∑k=1NdayRMSESS(tk), where Nday denotes the number of valid evaluation days, and RMSESS(tk) denotes the RMSE skill score computed from Eq. () on the kth valid evaluation day. The results are summarized in Table .

As shown in Table , INR achieves a slightly lower RMSESS‾ than WaveVar, but it gives the smallest λx among all compared methods. This indicates that INR does not provide the best temporal-mean global error score in this comparison, but shows a more favorable PSD-score-based spatial effective-scale diagnostic under the adopted PSD-skill criterion. This behavior is consistent with the characteristics of the proposed framework. The SIREN-based representation describes SSH as a continuous function of spatial and temporal coordinates, while the gradient-based regularization constrains unstable or noisy variations in the learned field. Together, these two components help maintain competitive overall reconstruction accuracy while improving the spectral consistency and spatial coherence of the reconstructed SSH field.

The OSSE configuration is conducted over a dynamically active portion of the Gulf Stream region, spanning 65–55° W and 33–43° N. It uses the NATL60 SSH simulation, based on the Nucleus for European Modelling of the Ocean (NEMO) , as the known reference field. This domain defines a common reconstruction target. Following the NATL60 OSSE benchmark , the reconstruction model is fitted using pseudo-altimetric observations from the four-nadir-plus-SWOT configuration during the assessment period from 22 October–2 December 2012. The complete NATL60 SSH simulation over the same domain and period provides the reference for full-domain evaluation. Pseudo-altimetric observations from the four-nadir-plus-SWOT configuration and the corresponding NATL60 SSH fields from 2 January–30 September 2013 are used as the validation dataset for hyperparameter selection.

The experimental setup aims to comprehensively evaluate the proposed approach's performance and robustness. First, we compare the reconstructed SSH with the model-simulated SSH reference using both direct SSH errors and spatial-gradient diagnostics. The gradient diagnostics are used to examine whether the reconstructed field captures major spatial transitions, such as fronts and eddy boundaries. Second, to further validate the general applicability of our method, we compare it with several existing state-of-the-art interpolation and reconstruction methods, including DUACS a traditional covariance-based optimal interpolation method,, three model-based data assimilation schemes: BFN a data assimilation method that follows quasi-geostrophic dynamics,, DYMOST dynamic OI accounting for the SSH nonlinear temporal propagation,, and MIOST multiscale OI,. Finally, we compare with the supervised 4DVarNet method based on a neural network architecture backed by a variational formulation,. Performance is assessed using the statistical metrics defined in Sect. , including the mean RMSE skill score RMSESS‾ and the spectral diagnostics λx and λt. These metrics summarize reconstruction accuracy and the PSD-score-based effective spatial and temporal scales under the Ocean Data Challenges benchmark protocol.

Figure  shows qualitative examples at four selected times covering the evaluation period. Specifically, we divide the evaluation period into four approximately equal sub-periods and use the middle valid snapshot from each sub-period. The four columns compare the along-track observations, the reconstructed SSH fields, and the corresponding physical SSH-gradient magnitudes.

The physical SSH-gradient magnitude shown in the bottom row is a visualization diagnostic distinct from the normalized-coordinate gradients used in the TV regularization. It is derived from automatic differentiation of the SIREN representation and explicitly rescaled to physical distance. This diagnostic uses one of the key properties of the INR framework, namely that the reconstructed SSH field is represented as a continuous and differentiable function of the spatio-temporal coordinates. Let η(lon,lat,t)=Φθ(s̃) denote the reconstructed SSH field, where s̃=(loñ,lat̃,timẽ) is the normalized spatio-temporal coordinate, and lon and lat denote longitude and latitude in radians. Automatic differentiation first gives the derivatives of Φθ with respect to the normalized input coordinates. These normalized-coordinate derivatives are then rescaled to derivatives with respect to longitude and latitude: 14∂η∂lon=∂Φθ∂loñ∂loñ∂lon,∂η∂lat=∂Φθ∂lat̃∂lat̃∂lat. These longitude and latitude derivatives are then converted into physical-distance derivatives using the Earth-radius scale factors: 15∂η∂x=1Rcos⁡(lat)∂η∂lon,∂η∂y=1R∂η∂lat, where R is the Earth radius. The physical SSH-gradient magnitude is defined as 16|∇η|=∂η∂x2+∂η∂y21/2. For visualization, we plot 105|∇η|, corresponding to units of m per 100 km. Therefore, the bottom-row panels should be interpreted as physical SSH-gradient magnitudes derived from the differentiable SSH representation, rather than as direct velocity or vorticity fields. Their further conversion into geostrophic velocity would require additional dynamical assumptions.

The reconstructed SSH fields in the middle row reproduce the main mesoscale SSH patterns across the four snapshots, including eddy-like structures and their surrounding transition regions. The corresponding gradient-magnitude maps in the bottom row emphasize regions where the reconstructed SSH field changes rapidly, such as eddy peripheries and frontal-like transition zones. These high-gradient regions are spatially consistent with the structures visible in the reconstructed SSH fields, indicating that the learned field preserves coherent spatial variations rather than producing isolated noisy gradients.

The gradient diagnostic provides a complementary view of the reconstructed SSH field by showing whether rapid spatial transitions, such as fronts and eddy boundaries, are represented as coherent structures.

While Fig.  shows four selected snapshots, the full temporal evolution of the along-track observations, reconstructed SSH fields, and corresponding physical SSH-gradient magnitudes is provided in Video Supplement S1. Together with the snapshot examples, the video provides a more complete visual assessment of the reconstruction behavior over the evaluation period.

Since the OSSE configuration provides the complete NATL60 SSH field as the reference, the full spatio-temporal reconstruction error can be directly evaluated. The error field and the two RMSE summaries used below are defined together as 17e(xm,yn,tj)=SSHrec(xm,yn,tj)-SSHtrue(xm,yn,tj),RMSEspace(tj)=1NxNy∑m=1Nx∑n=1Nye(xm,yn,tj)21/2,RMSEtime(xm,yn)=1Nt∑j=1Nte(xm,yn,tj)21/2. Here, SSHrec denotes the reconstructed SSH field and SSHtrue denotes the NATL60 reference field. All errors in this analysis are computed in the original SSH physical space, with units of metres.

Figure  presents the spatio-temporal reconstruction error of the proposed method in the OSSE experiment. Figure a–d show the instantaneous SSH reconstruction error at the same four selected dates as those used in the preceding INR reconstruction figure. These dates were obtained by dividing the evaluation period into four approximately equal sub-periods and selecting the middle valid snapshot from each sub-period. The instantaneous error maps show that errors are relatively small over much of the domain. Larger errors are mainly localized in dynamically active regions with stronger SSH variability and sharper gradients. This indicates that energetic mesoscale structures remain more challenging to reconstruct accurately under sparse satellite-like sampling.

Figure e shows the spatially averaged RMSE over time. This time series summarizes how the domain-mean reconstruction error varies among the evaluated time slices. The RMSE is relatively larger near the beginning and then fluctuates during the evaluation period. Larger RMSE values indicate time slices in which the SSH structures and sampling constraints are more challenging for the reconstruction.

To summarize the spatial distribution of the reconstruction error over the full evaluation period, Fig. f shows the time-mean RMSE map. The time-mean RMSE remains low over most of the domain, while several localized regions exhibit relatively larger values. These localized high-error regions are mainly associated with areas where the SSH field has stronger spatial variability and sharper gradients. This result further shows that the proposed method can reconstruct the broad SSH pattern and maintain spatial coherence over most of the domain, while dynamically active mesoscale structures remain more difficult to reproduce accurately.

Table  reports the performance of the proposed INR method in comparison with several representative state-of-the-art approaches. Unless otherwise specified, INR denotes the full proposed model with TV regularization, while “INR without TV” is used only as an ablation variant. In terms of the OSSE RMSESS computed over all valid grid points and evaluation times, INR performs on par with the top-performing schemes. Although its RMSESS value is slightly lower than that of 4DVarNet v2022, INR attains the smallest diagnosed spatial scale λx under the adopted Ocean Data Challenges PSD-score protocol. Both INR configurations also reach the 2.05 d lower boundary of the common daily-sampled temporal diagnostic, whereas the non-INR methods cross the 0.5 threshold at longer periods. This boundary-limited result establishes the relative advantage of INR within the common benchmark, but it does not rank performance at periods shorter than 2.05 d.

The effect of the TV regularization term is isolated by comparing the full INR model with an ablation variant in which the TV term is removed while all other settings are kept unchanged. The ablation result shows that removing the TV term has little influence on the normalized RMSE score, which remains approximately 0.94. Both configurations reach the 2.05 d lower boundary of the temporal diagnostic, so this boundary-limited value cannot be used to distinguish their sub-2.05 d performance. In contrast, the diagnosed spatial scale λx, derived from the 0.5 contour of the PSD-score field, changes from 0.12° without TV regularization to 0.03° with TV regularization. This indicates that the TV term mainly affects the spatial PSD-score diagnostic, by reducing isolated large gradients and part of the high-wavenumber error energy, while the domain-averaged pointwise error changes little.

As reported in Table , INR gives the smallest PSD-score-based effective spatial scale and reaches the lowest temporal value accessible under the common evaluation criterion. To clarify how these diagnostics are obtained and how they should be interpreted, Fig.  shows the PSD-based score in the wavelength–period domain. The score is generally high for long temporal periods, especially above approximately 15–20 d, over a broad range of spatial wavelengths, indicating that slowly varying SSH components are well reconstructed. Relatively high scores are also observed at broad spatial wavelengths, roughly larger than 3°, suggesting that large-scale SSH variability is captured with small relative spectral error. In contrast, the main low-score region is concentrated at shorter temporal periods and intermediate spatial wavelengths, approximately within 3–10 d and 0.5–2°. This indicates that rapidly evolving and spatially complex SSH variability remains more challenging to reconstruct.

The temporal axis terminates at 2.05 d because the PSD calculation uses 41 daily SSH fields. Its largest positive temporal frequency is therefore 20/41d-1, corresponding to a shortest represented temporal period of 41/20=2.05d. The black 0.5 contour reaches this lower boundary. Thus, within the common benchmark, INR maintains the PSD-score criterion down to the highest temporal frequency accessible to the daily-sampled diagnostic, whereas the non-INR methods in Table  cross the criterion at longer periods. This is a valid relative comparison under the shared protocol. However, the boundary-limited 2.05 d value neither establishes skill at periods shorter than 2.05 d nor provides an exact temporal resolution. Some high-score colors adjacent to this boundary should likewise not be interpreted as sub-2.05 d skill. In addition, the PSD-based score is a relative spectral-error measure and can be sensitive where the reference spectral energy is weak. Following the Ocean Data Challenges protocol, the spatial scale λx=0.03° remains a contour-derived comparative diagnostic, rather than a nominal grid spacing or a guarantee that all SSH structures at this scale are fully resolved.

In addition to the quantitative comparison above, we provide a qualitative comparison with the benchmark OI baseline. The comparison is shown for the four-nadir setting, for which reproducible gridded OI outputs are provided by the benchmark. This allows the spatial reconstruction patterns, reconstruction errors, and derivative-based diagnostics of OI and INR to be compared on the same input observations and evaluation grid.

Figure  compares the NATL60 reference SSH field, the OI baseline reconstruction, and the INR reconstruction at three representative evaluation times. The rows correspond to cases selected according to the 0.1, 0.5, and 0.9 quantiles of the OI SSH reconstruction error, respectively. Both methods reproduce the broad SSH distribution from sparse along-track observations, while noticeable differences appear in dynamically active regions with strong spatial variability. The error maps show that the largest residuals tend to occur near strong-gradient frontal and eddy-like features visible in the reference, indicating that these regions remain challenging under sparse observational constraints. Compared with the OI baseline, the INR error maps in these selected cases contain fewer broad high-amplitude residual patches and show a more spatially continuous SSH pattern. This qualitative comparison complements the quantitative metrics by showing where the main residual patterns occur in the selected OSSE examples.

To further examine the behavior of the reconstructed derivative fields, we compare the SSH gradient magnitude derived from the NATL60 reference field, the OI baseline reconstruction, and the INR reconstruction. This derivative-based diagnostic complements the SSH snapshot comparison by focusing on fronts, eddy boundaries, and other high-gradient regions. The same three evaluation times as in Fig.  are used for consistency.

Figure  shows the corresponding gradient-magnitude fields and gradient diagnostic errors. The OI-derived gradients are comparatively diffuse and show some small-scale artifacts in these examples, whereas the INR-derived gradients follow the main frontal and eddy-like bands more continuously. The diagnostic error maps also show that sharp fronts and eddy boundaries remain the main sources of gradient discrepancies for both methods, partly because small spatial displacements can lead to pronounced derivative errors. Some very narrow gradient filaments in the NATL60 reference are absent from both reconstructions, as expected under sparse satellite-like sampling. We therefore use this comparison mainly to assess the dominant gradient structures and the location of large gradient errors. Overall, the derivative-based comparison indicates that the gradient-regularized INR framework reduces some spurious derivative artifacts relative to the OI baseline while retaining the dominant gradient structures visible in the reference.