Our objective is predicting the temporal evolution of sea level in the Adriatic basin on a two-dimensional grid (i.e., dense prediction) with a 3 d horizon at hourly resolution. The horizontal and vertical grid sizes are H=94 and W=115 respectively, spanning a geographical region bounded by latitudes 40.00 to 45.87° N and longitudes 12.20 to 18.85° E. The input to the prediction model consists of past 72 h of available SSH observations from N=11 Adriatic tide gauges together with their respective astronomic tides, as well as the past and future 72 h of gridded geophysical variables obtained by the atmospheric and ocean numerical models.

The training time window spans two intervals: from 2000 to May 2019 and from 2021 to 2022. The testing time window covers the period from June 2019 to the end of 2020. This specific testing interval was selected due to the occurrence of numerous high sea level and coastal flood events in the northern Adriatic and matches the period chosen in our previous work .

The following tide gauges along the Adriatic coast are considered in this study for SSH measurements: Koper, Venice, Ancona, Ortona, Vieste, Neretva, Ravenna, Sobra, Stari Grad, Tremiti and Vela Luka (see Fig. ). To characterize the inter-dependencies within this network, we analyzed the correlation between station records, observing strong clustering between neighboring stations (see Appendix ). Their SSH availability ranges from 15 % to 90 % during years 2000–2022 , which has to be accounted for during training and testing, as the model is required to cast predictions also when data from some tide gauges is missing. The raw SSH data, originally recorded at 1 min or 10 min intervals (depending on the station), was filtered using the methods from to eliminate three types of sensor errors: (i) sensor freeze, which results in a constant output value for an extended period of time, (ii) extreme outliers, and (iii) extreme jumps in the signals. Subsequently, the measurements were downsampled to hourly resolution. To prevent aliasing and reduce high-frequency noise, we applied a Gaussian smoothing kernel (σ=25 min) using a weighted moving average prior to subsampling. The Gaussian weighted averaging was implemented with dynamic weight normalization to robustly handle missing time-steps in the raw high-frequency series. For each location, the astronomical tides in 1 year intervals were computed using the UTIDE Tidal Analysis package for Python .

Following , the sea level readings are classified as low if they fall below the 1st percentile and as high if they exceed the 99th percentile of the observed values at the given location (see , for exact thresholds). During evaluation (Sect. ), several metrics are computed separately for all sea level values and for the aforementioned extreme classes. This enables assessing the model's ability to predict both tails of the sea level distribution: (i) high values, which are crucial for coastal flood warnings, as well as (ii) low values, which may restrict marine traffic in the shallow northern region of the Adriatic basin.

For geophysical variables, we employed ERA5 reanalysis data for training and ECMWF Ensemble Prediction System (EPS) data for evaluation. The reason for using the ERA5 data in training is that our prior research indicated improved performance in the multi-point (i.e., spatially sparse) prediction setup. However, the evaluation is carried out on the ECMWF EPS dataset to faithfully reflect the operational forecasting setup in which reanalysis is unavailable and ensemble forecasts are typically used. The following parameters from ERA5 reanalysis were used: (i) 10 m winds, (ii) mean sea level air pressure, (iii) sea surface temperature (SST), (iv) mean wave direction, (v) mean wave period and significant height of combined wind waves and (vi) the swell data. Following our previous work , all input fields were spatially cropped to the Adriatic basin and subsampled to a 9×12 spatial grid.

For operational forecasting and evaluation, an ensemble of nens=50 atmospheric members is used. HIDRA-D processes each member separately, generating 50 dense sea level forecasts, which are then averaged to produce the final prediction.

To enable supervised training of the network on the entire basin, the along-track sea level anomalies (SLA) from altimeter satellites were acquired from the Copernicus marine service product SEALEVEL_EUR_PHY_L3_MY_008_061. The SLA variables are provided relative to a 20 year mean (1993–2012) with a 1 Hz (∼7 km) sampling resolution. This dataset incorporates data from all available altimeter missions, including Sentinel-6A, Jason-3, Sentinel-3A, Sentinel-3B, Saral/AltiKa, Cryosat-2, Jason-1, Jason-2, Topex/Poseidon, ERS-1, ERS-2, Envisat, Geosat Follow-On, HY-2A, HY-2B, and others. The ADT values used in this study are computed as the sum of the provided variables: SLA_filtered, ocean_tide, mean dynamic topography (MDT) and dynamic atmospheric correction (DAC). Note that ocean_tide (ocean tide correction, or OTC) and DAC are explicitly added back to the anomaly. This effectively reverses the standard altimetry corrections, restoring the high-frequency tidal and atmospheric surge signals that were filtered out, thereby reconstructing the instantaneous total water level observed by tide gauges. This is consistent with how ADT is treated in the CMEMS NEMO model for the purposes of data assimilation (Ali Aydogdu, CMCC, personal communication, 2023) and thus enables comparisons to the NEMO model .

For comparisons with hourly measurements from our model or NEMO, the time of a satellite ADT measurement was not rounded to the nearest hour, recognizing that sea level can change rapidly. Instead, hourly time series from our model or NEMO were linearly interpolated to the exact time of each satellite ADT observation. The spatial locations of the satellite ADT measurements were binned into a grid with size 94×115 equal to the spatial output of our model. In cases where multiple satellite ADT measurements fall within the same grid cell, the average of those measurements is taken. For visualization purposes, ADT values are adjusted to have the mean equal to zero. This adjustment is consistent with the model training and evaluation setup, where mean removal is applied as a pre-processing step (see Sect. ).

To validate the satellite altimetry data and assess its accuracy in coastal regions, it is crucial to compare it with independent, in situ measurements. Tide gauges provide reliable SSH observations at specific coastal locations. Therefore, to verify the correlation between satellite ADT and tide gauge SSH measurements, in Fig.  we present pairs of ADT and SSH measurements. A satellite ADT measurement is assigned to a tide gauge if it falls within 20 km of its location. If multiple ADT measurements from the same satellite track satisfy this condition, we retain only the closest one. To determine SSH at the time of satellite ADT measurement, we perform linear interpolation between the two nearest SSH observations. To remove bias and focus on the dynamic components, we subtract the mean values of SSH and ADT at each location. Although the measurements come from completely different sources, the visualizations indicate a strong correlation between ADT and SSH, with an average absolute difference of 4.0 cm.

Despite the availability of ADT data from multiple satellites, its spatial and temporal coverage remains highly uneven. Due to the repetitive nature of satellite orbits, satellite ADT observations are concentrated along specific ground tracks rather than being uniformly distributed. Figure  visualizes the spatial distribution of satellite ADT measurements, showing that the majority of observations originate from a limited number of tracks, leaving vast regions with very few satellite ADT measurements.

Figure  illustrates the frequency of tracks between the years 2000 and 2020. Given the high variability of sea level dynamics, satellite ADT data presents an extremely sparse source of ground truth. On average, 2.4 tracks per day are recorded, meaning that in 90 % of hourly intervals, no satellite ADT measurements are available beyond tide gauge measurements. In the remaining 10 % of cases, a single track is present, covering an average distance of 156 km, corresponding to approximately 25 grid points. In our setup, the Adriatic basin covers 3413 grid points, meaning that a single track typically covers less than 1 % of the total number of grid points.

A significant challenge in combining satellite altimetry (ADT) and tide gauge data is the lack of calibration between the two sensor types. As defined in Sect. , ADT represents the level relative to a reference geoid. In contrast, tide gauges measure SSH relative to a local vertical datum, often tied to a local reference ellipsoid. These local reference points are distinct for each tide gauge and their vertical offset from the geoid is generally not readily available. Consequently, without knowing the transformation between tide gauge SSH measurements and satellite ADT measurements, tide gauge data cannot be directly used for supervising the training of a model designed to predict ADT.

To address this discrepancy, we perform a bias correction for each tide gauge i, to convert it into ADT. The following model is used to transform the tide gauge measurements yigauge into ADT (yiADT): 1yiADT=yigauge+bi, where yigauge is the raw measurement from tide gauge i, and bi represents the unknown vertical bias for that specific tide gauge relative to the geoid. This displacement primarily represents the offset between the vertical datums, which we assume to be physically constant over the timescales of this study. While vertical land movements can induce slow temporal variations in this offset, these changes are not taken into account here; therefore, we approximate bi as a constant.

Within our model framework, these unknown bias terms, bi, are treated as learnable parameters and are estimated concurrently with the main model parameters during the training process. As a pre-processing step, both the satellite ADT dataset and each individual tide gauge time series are independently centered by removing their respective means. Consequently, the training process optimizes the bias parameters bi to align the centered tide gauge observations with the satellite ADT measurements in their vicinity, effectively connecting the two data sources.

The numerical ocean model used in this paper is the Copernicus Marine Environment Monitoring Service (CMEMS) product MEDSEA_ANALYSISFORECAST_PHY_006_013 , based on the Nucleus for European Modelling of the Ocean (NEMO) v4.2 . The Mediterranean Sea Physical Analysis and Forecasting model (MEDSEA_ANALYSISFORECAST_PHY_006_013) spans a regular grid with a 1/24° (approximately 4 km) horizontal resolution and 141 vertical z*-levels with partial cells to accurately represent the model topography. It employs a staggered Arakawa C-grid with land area masking. Tidal forcing is represented by eight tidal constituents (M2, S2, N2, K2, K1, O1, P1, Q1). The model is forced at its Atlantic lateral boundary by the Global analysis and forecast product (GLOBAL_ANALYSISFORECAST_PHY_001_024) and by a combination of daily climatological fields from a Marmara Sea model and the global analysis product in the Dardanelles Strait. Atmospheric surface forcing is provided by the ECMWF deterministic model. The model was initialized from the World Ocean Atlas (WOA) 2013 V2 winter climatology on 1 January 2015. In situ vertical profiles of temperature and salinity from ARGO, Glider, and XBT, as well as SLA data from multiple satellites (including Jason 2 & 3, Saral-Altika, Cryosat, Sentinel-3A/3B, Sentinel6A, and HY-2A/B) are assimilated via the OceanVar (3DVAR) scheme. The hydrodynamic part of the model is coupled to the wave model WaveWatch-III. We refer the reader to for further details.

NEMO computes sea level as a local deviation from the geoid, in theory making it directly comparable to satellite ADT measurements in our setup. However, when analyzing the mean difference between satellite ADT observations and NEMO forecasts (extracted at the satellite ADT measurement locations and linearly interpolated to the corresponding timestamps), we find that, on average, ADT observations are 34.61 cm higher than NEMO forecasts in the training set period. To correct this systematic bias, we apply an offset of 34.61 cm to the NEMO forecasts.

In the analysis of sea level forecasts produced by NEMO, we use the same region defined by HIDRA-D. The region is subsampled from a 141×185 grid to a 94×115 grid to match the resolution of HIDRA-D. This subsampled version is also used when comparing the model to satellite ADT data, ensuring that the metrics are computed on the same set of ADT measurements in the same spatial locations.

The goal of this study is to predict the temporal evolution of the ADT field, denoted as Y, on a dense two-dimensional grid over the entire Adriatic basin. The target output is a sequence of hourly grid maps over a forecast horizon of T=72 h. The spatial domain is defined by a grid of size H×W (94×115), covering the latitudes 40.00 to 45.87° N and longitudes 12.20 to 18.85° E.

The model approximates the mapping function F such that Y=F(XSSH,Xgeo). The inputs consist of sparse SSH (XSSH), which encompasses the past 72 h of hourly SSH observations from N=11 tide gauges along with their astronomical tide components, and geophysical forcing (Xgeo), a tensor containing the past 72 h and forecasted future 72 h of atmospheric and oceanic variables (e.g., wind, pressure, SST) derived from numerical models.

A fundamental challenge in this setting is that the ground truth for the dense output Y (satellite altimetry) is spatially sparse and temporally intermittent. Furthermore, the input tide gauge measurements (ygauge) and the target satellite data (yADT) utilize different vertical datums. Therefore, the problem formulation includes the simultaneous estimation of a set of station-specific bias parameters, {bi}i=1N, to align the inputs with the prediction target.

The primary objective of the HIDRA-D model is to forecast dense sea levels across the basin. A key challenge here is that available inputs are limited to past SSH measurements, which are sparsely located at tide gauge positions, and can be supplemented with geophysical variables such as wind, air pressure, sea surface temperature (SST), and waves. To address the need for a comprehensive understanding of the sea basin's state from these sparse inputs, we leverage our previously developed multipoint SSH prediction model, HIDRA3 , described in Sect. . HIDRA3 has proven to be effective in generating a rich latent representation of the sea basin's state, which can subsequently be decoded into accurate SSH predictions at multiple tide gauge locations, including those that were absent from the input. In HIDRA-D, we take this latent representation derived from HIDRA3, along with the aforementioned geophysical features, as the foundation to forecast dense sea level values (see Fig.  for architecture). For this purpose, we introduce the Dense decoder module (detailed in Sect. ). However, a general challenge in directly forecasting dense sea level fields is the very large output dimensionality, which would typically necessitate a model with a large number of parameters. Furthermore, observed sea level variations are not characterized by an equal distribution of frequencies; rather, low-frequency components account for the majority of sea level fluctuations. Consequently, to make the prediction task more tractable and efficient, we design the Dense decoder module to forecast only these dominant low-frequency components along with their corresponding phases. This forecasting is performed in the 2D Fourier domain for each of the 72-hourly prediction points. The final dense predictions are then efficiently reconstructed by applying the inverse 2D discrete Fourier transform (2D IDFT) to these predicted Fourier coefficients.

HIDRA-D is trained end-to-end using a multi-component loss function designed to effectively leverage diverse data sources and provide targeted supervision. While the primary goal is to predict dense sea level maps, relying solely on satellite ADT measurements (denoted as yADT) can be insufficient for robustly training the entire network, especially its backbone components. To address this, we incorporate tide gauge SSH measurements (denoted as ygauge), which, despite their spatial sparsity and missing values, provide valuable supervisory signals.

To foster robust learning of the HIDRA3 module's internal representations, which form the foundation for subsequent dense predictions, a dedicated loss term, L1, is applied. This loss directly supervises the HIDRA3 module by comparing its SSH predictions (ypgauge) to local SSH measurements from tide gauges (ygauge): 4L1=(ypgauge-ygauge)2‾. Here, the overline symbol (…)‾ denotes the mean taken over all timesteps and all available tide gauge stations. This targeted supervision provides an additional training signal specifically for the HIDRA3 backbone.

The dense 2D sea level predictions ypgrid are primarily trained using satellite ADT observations. The loss term L2 measures the mean squared error between the dense ADT predictions (ypgrid) and satellite ADT observations (yADT): 5L2=(ypgrid-yADT)2‾. As the dense predictions (ypgrid) have an hourly resolution, we linearly interpolate between two neighboring predictions to match the precise timestamps of the asynchronous ADT measurements from satellites. L2 extends the learning beyond tide gauge locations, leveraging the broader spatial coverage of satellite data.

To further enhance the training of the Dense decoder and utilize all available data, tide gauge data are also incorporated into its supervision through the loss term L3. This is particularly beneficial for improving predictions at coastal locations where tide gauges are situated. L3 measures the mean difference between the dense sea level predictions at tide gauge locations (yp,igrid) and the corresponding tide gauge information. Since tide gauge observations (ygauge) typically measure local SSH relative to a local datum, they must be transformed to be comparable with the network's predictions. Recall from Sect. , that this transformation is achieved by adding learned station-specific displacements bi to the tide gauge SSH observations yigauge, effectively converting them into local ADT equivalents.

A challenge with tide gauge data is the frequent unavailability of measurements. To ensure that L3 is defined over a comprehensive set of points and to provide continuous guidance, missing tide gauge observations yigauge are replaced by the corresponding point SSH predictions (yp,igauge) from the HIDRA3 module. This is possible because the HIDRA3 module outputs predictions for all tide gauges, even when input measurements for some tide gauges are missing. Consistent with the treatment of yigauge, these HIDRA3 predictions are also transformed into the space of ADT by adding the same station-specific displacement bi. Thus, L3 is defined as: 6L3=(yp,igrid-(yigauge+bi))2‾where observation yigauge exists,(yp,igrid-(yp,igauge+bi))2‾where observation yigauge does not exist. This approach ensures that the Dense decoder module is consistently supervised at all tide gauge locations, leveraging either direct (transformed) observations or transformed HIDRA3 predictions as targets.

The final composite loss function, used for gradient backpropagation during the end-to-end training of the HIDRA-D model, is a weighted sum of these individual loss terms: 7L=αL1+βL2+γL3. The weights were selected to enforce a hierarchical training structure: α=100, β=1, and γ=1. The significantly higher magnitude of α is a design choice intended to act as a soft constraint, ensuring that the HIDRA3 backbone prioritizes the accuracy of point-based SSH predictions (L1) above all else. This prevents the optimization of the dense reconstruction losses (L2 and L3) from degrading the quality of the underlying backbone representations. Consequently, the backbone learns primarily from the high-fidelity tide gauge data, while the Dense decoder adapts to these representations to satisfy the basin-wide constraints. The weights β and γ are set equally as L2 and L3 operate on largely spatially disjoint sets of points (sparse satellite tracks versus fixed tide gauge locations) and therefore do not require competitive weighting.

We use the AdamW optimizer with initial learning rate of 10-5 and a weight decay of 0.001. A higher initial learning rate of 10-3 is used for training the displacements bi. We utilize a cosine annealing schedule to progressively decay the learning rate from its initial value η to a minimum value of η/100 over the course of training. Parameters are initialized using a standard Xavier initialization , and parameters of the HIDRA3 module are initialized as described in . During training, tide gauge failures are simulated by randomly deactivating a subset of tide gauges with a probability of 0.5. The model is trained for 50 epochs with a batch size of 128 data samples. Prior to training, all input data is standardized by subtracting the mean and dividing by the standard deviation. The mean is computed independently for each tide gauge location, while a single standard deviation is determined across all locations. Hyperparameters were tuned using a validation set separate from the test set. To maximize the data available for learning, the final models were trained on the full training dataset (see Sect. ). Each geophysical variable and satellite ADT data undergoes independent standardization. Training requires approximately 12 h on a system equipped with an NVIDIA A100 Tensor Core GPU.

To assess the accuracy of dense sea level predictions, we compare them against ADT values (i.e., satellite along-track measurements) on the test set, spanning the period from June 2019 to the end of 2020. Table  presents the mean absolute error (MAE) and root mean squared error (RMSE) for HIDRA-D and NEMO based on satellite ADT data. The performance of the model against in situ tide gauge measurements requires a specific cross-validation setup to avoid data leakage and is detailed separately in Sect.  and Table . Results indicate that overall HIDRA-D outperforms NEMO, with both MAE and RMSE being significantly lower. Furthermore, HIDRA-D exhibits a very low bias of -0.17 cm. Since we performed a bias correction of NEMO to align it with satellite ADT data (see Sect. ), its bias is effectively zero.

To further analyze the spatial and temporal structure of the discrepancies between the models, Fig.  illustrates the root mean square difference (RMSD) between HIDRA-D and NEMO across the basin for different forecast lead times. The metric is computed over the entire test period. Visually, the highest RMSD values (reaching approx. 8 cm) are concentrated in the northern Adriatic, likely due to the complex shallow-water dynamics in that region. Comparisons with available satellite ADT measurements (latitude >43.5°) confirm that while both models exhibit higher errors in this area, HIDRA-D performs better with an RMSE of 5.37 cm compared to 6.79 cm for NEMO. In contrast, the central and southern parts of the basin generally exhibit lower differences, mostly ranging between 4 and 6 cm. Notably, both the spatial pattern and the magnitude of the RMSD remain remarkably stable across all lead times, indicating that the divergence between the two models does not grow significantly as the forecast horizon extends from T+1 to T+72 h.

To visually compare the forecasts, we present two examples of dense forecasts generated by HIDRA-D and NEMO, along with the difference between these forecasts. Forecasts under calm atmospheric conditions are presented in Appendix  (Fig. ), while Fig.  depicts the forecasts of dense sea level during a storm surge event. Note that the HIDRA-D predictions are smoother, containing lower spatial frequencies than predictions from NEMO. This result is expected because HIDRA-D does not model processes with spatial scales below the barotropic Rossby radius, λR. These smaller-scale processes can be caused by other ocean processes, such as ocean currents, which are not explicitly included in the HIDRA-D model. Despite this difference, both models generally produce forecasts of a comparable magnitude. Due to space limitations only selected parts of the forecasts are presented here, for visualizations of the entire forecast and additional forecasts, we direct the reader to the video supplement of the paper.

Figure  compares HIDRA-D and NEMO predictions with along-track satellite ADT observations. Selected dates correspond to the first day of each month from June 2019 to May 2020, with the longest track in the basin chosen for each day. The forecasts were obtained for the same day as the satellite altimetry observations, with model predictions interpolated to match the satellite observation times. The results indicate that while both HIDRA-D and NEMO predictions are both smoother than altimetry observations, HIDRA-D predictions better capture the overall trend reflected in the measurements.

The methodology described in Sect.  involves learning the vertical displacement, bi, for each tide gauge i. A positive bi indicates that, on average, measurements from tide gauge i are lower than satellite ADT measurements in its vicinity. The resulting displacements for the different tide gauge locations are shown in Table . The values vary in both sign and magnitude, ranging from -0.8 cm at Koper to +5.6 cm at several Italian locations. This variation underscores the necessity of estimating these displacements individually for each tide gauge, as a single, uniform offset would not suffice to accurately align all tide gauge records to the common ADT reference.

This section analyzes the HIDRA-D prediction accuracy at coastal locations to estimate the potential for its practical application in sea level prediction at locations without tide gauges. The evaluation setup involves removing, one at a time, one tide gauge from the total set of N=11 tide gauges, and training a separate model for each excluded tide gauge. Each separate model for each of the eleven tide gauges is therefore trained on 10 remaining tide gauges. When calculating scores or visualizing, we always use the model that was not trained on data from the corresponding tide gauge. When presenting results from such setup, we use the notation HIDRA-DN, where N=11 refers to the number of models trained and used in the evaluation. Note that since we do not have any information about tide gauges' local vertical datums, bias correction over the test set is applied for each excluded tide gauge and model separately.

We evaluate the models using standard performance measures from : mean absolute error (MAE), root mean squared error (RMSE), accuracy (ACC), bias, recall (Re), precision (Pr), and the F1 score. All metrics presented in Table  are averaged over all forecast lead times (from T+1 to T+72 h) and across all tide gauge locations. The table categorizes performance for all sea level values (overall) and separately for low and high sea level values (see Sect.  for definitions). Comparing HIDRA-DN and NEMO, the results indicate that HIDRA-DN achieves lower errors than NEMO overall, with a particularly significant reduction in error for low SSH values.

In contrast, for high sea level values, NEMO outperforms HIDRA-DN. Figure  shows RMSE scores for each tide gauge location for further insights. While HIDRA-DN achieves comparable error levels to NEMO at many locations, it exhibits significantly higher RMSE at Koper, Venice, and Neretva for high sea level values. This discrepancy likely arises from three connected reasons. First, the primary supervision for the dense field comes from satellite ADT data, which is spatiotemporally sparse (Fig. ); the probability of a satellite track capturing the peak of a transient short-duration storm surge is low, leading to a training set imbalanced against extreme dense events. Second, deep learning models trained with mean squared error objectives tend to produce smoothed outputs to minimize global error, occasionally underestimating sharp peaks (regressing to the mean). Third, extreme values at specific locations are often driven by unresolved local topographic effects in bays and harbors. In the leave-one-out HIDRA-DN setup, the model must infer these local dynamics without ever seeing training data from that specific coastline geometry, whereas NEMO explicitly solves physical equations using high-resolution bathymetric grids. These findings suggest that HIDRA-DN is highly effective for open waters and general basin dynamics but faces limitations in resolving localized coastal extremes at locations where it has not been explicitly trained.

It is important to note, however, that these conclusions apply to an arbitrary point on the coastline where no training data is provided. At training tide gauge locations, HIDRA-D performs marginally worse than HIDRA3. The average RMSE across all stations increases from 3.28 to 3.61 cm. For high sea level values, the RMSE increases from 5.61 to 6.72 cm. For low sea levels, the performance is more similar, with an RMSE of 4.24 cm for HIDRA3 and 4.41 cm for HIDRA-D.

To visually observe the model outputs, we compare the forecasted SSH time series generated by HIDRA-DN and NEMO against tide gauge measurements. Comparison during a storm surge event is illustrated in Fig. , while performance during calm atmospheric conditions is provided in Appendix  (Fig. ). We can see that both models capture the sea level dynamics well. For additional visualizations, including further forecasts and failure cases, we refer the reader to the video supplement.

HIDRA-D is further evaluated by removing subsets of nearby tide gauges within the Adriatic basin. Specifically, we conduct two separate training experiments. As demonstrated in the correlation analysis (Fig. ), the Adriatic tide gauges form two distinct clusters with high internal correlation: the northern group (Koper, Venice, Ravenna, Ancona) and the central/southern group. To rigorously test the model's ability to infer dynamics without relying on highly correlated neighbors, we first exclude tide gauge data from Koper, Venice, Ravenna, and Ancona. We refer to this model as HIDRA-DS, as it relies solely on measurements from tide gauges in the southern Adriatic. In the second setup, we use only the northern tide gauges, and denote the resulting model by HIDRA-DN. Each model is then tested on the locations that were excluded from its training.

Figure  presents the RMSE scores, which are compared against those of HIDRA-DN. The results indicate that the performance degradation remains minimal, despite the fact that the input SSH measurements originate from locations hundreds of kilometers away. This suggests that HIDRA-D effectively captures the overall dynamics of the Adriatic basin, even in scenarios with remote and spatially-limited tide gauge coverage.

In Sect. , the northern Adriatic barotropic Rossby radius was used to define the spatial scale threshold as λR=150 km. This threshold represents the lower limit for the forecasted wavelength in the dense output of HIDRA-D and determines the size of the non-zero element regions in the Fourier matrix F, which for λR=150 km corresponds to 5×5 and 4×5 submatrices. To investigate the influence of this hyperparameter, we conducted an ablation study by training two model variants where the dimensions of the predicted Fourier submatrices were increased or decreased by one element in each dimension. This modification required changing the output dimension of the final dense layer to match the number of coefficients in these resized submatrices. Note that the final spatial grid size (94×115) remains unchanged. These variants are hereafter referred to as HIDRA-D4×4 (utilizing 4×4 and 3×4 submatrices) and HIDRA-D6×6 (utilizing 6×6 and 5×6 submatrices). For the cross-validation setup, they are denoted as HIDRA-D4×4N and HIDRA-D6×6N.

The results of this ablation study are presented in Table . The evaluation on the satellite ADT measurements shows that the MAE is largely unaffected by changes in the submatrix size. However, a different trend emerges in the cross-validation scenario, where models were evaluated on tide gauges that were excluded from training. In this case, our proposed model HIDRA-DN, achieved the lowest MAE. This supports our selection of λR.