Prior to the space-time weighted fusion, a simple global one-dimensional look-up-table correction is applied to reduce the systematic SWH-dependent bias of the WW3-ST6 background relative to the calibrated CCI Sea State altimeter observations. This correction is analogous to a nonlinear calibration. Collocated WW3-ST6 and CCI Sea State SWH pairs are grouped according to the WW3-ST6 SWH value, and the mean altimeter-minus-WW3 difference is calculated within each 0.1 m SWH bin. The resulting bin-mean correction is then applied to the WW3-ST6 SWH field and to the model-equivalent SWH values collocated with each altimeter record. For SWH values above 12 m, where collocated samples are relatively sparse, a simple linear-regression correction is used instead of fitting individual bins, in order to avoid overfitting in the high-wave range. All subsequent data-fusion procedures are therefore based on the look-up-table-corrected hindcast background rather than on the original hindcast.

The data-fusion method employed in this dataset is a space-time weighted fusion approach inspired by optimal interpolation and objective analysis, originally developed for meteorological and oceanographic data analysis (e.g., Bretherton et al., 1976). In this framework, sparse observations are used to correct a background field through distance- or covariance-dependent weights. The present study modifies this framework for SWH reconstruction to give strong local influence to calibrated altimeter SWH observations within the fused product with the corresponding formula expressed as follows: 1A(i,j,t)=M(i,j,t)+∑k=1Nw(i,j,t)k⋅Ok-Mk2w(i,j,t)k=exp⁡-dk2(i,j,t)/2R(i,j,t)2/∑k=1Nexp⁡-dk2(i,j,t)/2R(i,j,t)23R(i,j,t)=mindk(i,j,t)(k=1,2,3,…,N)4dk(i,j,t)=C+Sk/S12+Tk/T12+QMk-M(i,j,t)/Mk2Sk<Sthr&Tk<Tthr&ri,j,k>rthr5dk(i,j,t)=+∞Sk>SthrorTk>Tthrorri,j,k<rthr where i, j, and t represent longitude, latitude, and time, respectively; M (model) and A (analysis) represent the NWM hindcast data before and after data fusion, respectively; k∈{1,2,…,N} represents the index of altimeter observations; Ok and Mk represent the values of altimeter-observed and corresponding collocated modelled SWH at the kth altimeter record; wk represents the weight factor at spatio-temporal location (i, j, t) for the kth altimeter record; dk represents the distance from the model location (i, j, t) to the kth altimeter record in a super-space spanned by space, time, and SWH, which is defined in Eq. (4); R represents the distance to the nearest altimeter record; C is a constant to mitigate the impact of cases with very small distances; S and T represent the spatial distance and the time difference between (i, j, t) and the kth altimeter record, respectively, and S1 and T1 represent coefficients that make S and T dimensionless; Q is also a scaling coefficient that ensures the quantities of SWH, time, and space are comparable; ri,j,k is the correlation coefficient (CC) between the NWM-hindcast SWHs time series at grid location (i, j) and the time series at the location of the kth altimeter record within the corresponding month; Sthr, Tthr, and rthr are threshold values for Sk, Tk, and ri,j,k, respectively: An observation is excluded from consideration if the space-time separation or the correlation between the model point and the observation point exceeds these thresholds.

This method implicitly gives strong confidence to calibrated altimeter SWH observations relative to the non-assimilative WW3 background. The uncertainty of the resulting product is therefore assessed empirically through independent buoy validation, leave-one-satellite-out altimeter validation, parameter-sensitivity tests, and background-sensitivity experiments, rather than through a formal analytical error-covariance model.

To reduce potential space-time representativeness errors, all altimeter observations located within 75 km of coastlines were excluded, as SWH can vary substantially over short distances in nearshore regions due to bottom-wave interactions and coastal sheltering.

Here, C, S1, T1, Q, Sthr, Tthr, and rthr are set to 8, 50 km, 90 min, 8000, 2000 km, 24 h, and 0.7, respectively, based on a straightforward empirical tuning procedure which was conducted using the 2011 CCI-Sea State dataset. Specifically, Jason-2 data were withheld, and the remaining three altimeter missions (CryoSat-2, ENVISAT, and Jason-1) were fused with the WW3-ST6 hindcast data. The fused results were then validated against the independent Jason-2 observations, and the combination of tuning parameters (C, S1, T1, Q, Sthr, Tthr, rthr) that yielded relatively small overall validation errors was selected. The error metrics considered in this evaluation include bias, RMSE, and correlation coefficient (CC): 6Bias=1n∑i=1nyi-xi7RMSE=1n∑i=1nyi-xi2 8CC=∑i=1nyi-y‾xi-x‾/∑i=1nyi-y‾2∑i=1nxi-x‾2 where x and y denote the SWH from the dataset to be evaluated and the reference data (in this case, fused dataset without Jason-2 and Jason-2 from CCI-Sea State), respectively; n is the sample size, and the bars over them denote their mean values. It is worth noting that the fused results are not sensitive to these tuning parameters near their optimal values. Moderate changes in these tuning parameters only weakly affect the fused results. This is because most target grid points are simultaneously constrained by spatial separation, temporal separation, modelled-SWH similarity, background-field correlation, threshold-based observation selection, and the base-distance term. The fusion therefore mainly represents an interpolation of altimeter-minus-background increments among dynamically related observations, rather than an extrapolation controlled by a single parameter.

With respect to the construction of the weights, while conventional objective analysis and optimal interpolation methods often define weights primarily according to spatial separation, our approach introduces several additional terms to better reflect the characteristics of wave fields:

Introduction of the temporal distance. The distance definition incorporates the temporal separation between the model grid point to be corrected and the observation point. This accounts for the fact that the sea state at a given location is influenced not only by nearby observations in space but also by those in time. Consequently, both past and future observations are allowed to contribute to the correction at a given time, unlike most operational wave assimilation schemes, which generally exclude future data when updating past states.

During the data fusion process, the number of altimeter satellites used plays a critical role. In principle, fusing more satellites increases the spatiotemporal coverage of the observations, which should, in turn, improve the accuracy of the fused dataset. However, as pointed out by Jiang (2020), the growing number of altimeters over time also increases the likelihood of sampling extreme events. This may introduce undersampling biases in long-term climate analyses. Although the fusion process can mitigate this issue to some extent, it may still remain a potential source of error in climate-related applications.

To address this concern, we adopt a strategy similar to that used in the Maps of Sea Level Anomalies product of AVISO, generating two complementary datasets: a “multi-sat” product, which integrates observations from all available altimeter satellites in CCI-Sea State, and a “two-sat” product, which consistently fuses data from only two altimeters. The “multi-sat” product prioritizes accuracy by maximizing observational coverage, whereas the “two-sat” product is designed to reduce sampling-related temporal inhomogeneity by maintaining a relatively stable number of observations and broadly consistent orbital sampling characteristics. Here, sampling consistency refers to reducing temporal changes in observation density and orbital sampling characteristics and is distinct from the cross-mission calibration consistency already provided by the CCI Sea State dataset. From an application perspective, the “multi-sat” product is better suited for tasks that demand higher accuracy, such as model evaluation and the training of deep learning algorithms. In contrast, the “two-sat” product is more appropriate for long-term climate analyses, where temporal consistency is of greater importance.

For the “two-sat” product, the selection of satellites is guided by sampling characteristics: one satellite with an orbital inclination of approximately 66° is paired with another in a near-polar orbit to maintain broadly consistent orbital sampling characteristics over time (Table 1). In cases where a given satellite experiences temporary data gaps, observations from another satellite with a similar orbital configuration are preferentially used as substitutes. For example, in the CCI-Sea State v3 dataset, Jason-2 data were missing during several days in June 2017 and March 2019. In such instances, Jason-3 data were used as replacements for Jason-2. Although the CCI Sea State v4 record starts in August 1991, the fused product starts in October 1992 because, before this time, only one altimeter mission is available in the CCI record and the prescribed two-satellite configuration cannot be constructed. For consistency, the multi-sat product is released over the same temporal period as the two-sat product.

The CMEMS gridded dataset, the WW3-ST6 dataset, and the fused dataset (“two-sat” version) were all bilinearly interpolated to each buoy location, and subsequently compared against the corresponding SWH of buoy observations. Figure 2a–c shows scatter plots comparing buoy observations with the three datasets for instantaneous values at 12:00 UTC.

The comparison of instantaneous values at 12:00 UTC between SWHs from the CMEMS gridded dataset and NDBC buoys (Fig. 2a) shows a bias of nearly zero, an RMSE of 0.37 m, and a CC of 0.93. Although these error metrics may not appear poor at first glance, they are substantially worse than the error metrics obtained from comparisons between along-track altimeter SWHs and buoys over the same period (e.g., Jiang, 2023). Moreover, they are significantly inferior to those from the NWM hindcast (WW3-ST6), which does not assimilate any observations (Fig. 2b, bias = 0.002 m, RMSE = 0.31 m, CC = 0.95). This finding supports our point raised in the Introduction Section that the accuracy of “independent” Level-4 gridded altimeter SWH products is often much lower than that of raw NWM outputs. However, after fusing WW3-ST6 with along-track altimeter observations, the RMSE further decreased to 0.27 m and the CC increased to 0.96, demonstrating the effectiveness of the data fusion approach.

We further evaluated the global performance of the fused dataset using altimeter observations from 2011. This year was selected because four altimeter missions were available simultaneously, enabling multiple leave-one-satellite-out and two-satellite-combination experiments. In each experiment, one or more satellite missions were excluded from the fusion and then used as withheld references for evaluation. In this validation, one satellite was excluded from the fusion, while the remaining altimeter missions were fused with WW3-ST6. The excluded satellite was then used as an independent reference for evaluation. For example, in Fig. 3, Jason-2 was withheld and subsequently compared against the fused and unfused datasets. It should be noted that, when Jason-2 is used as the withheld reference, Jason-2 observations are excluded from the space-time fusion but were included in the global one-dimensional look-up-table correction. The comparison is therefore independent with respect to the fusion step, while the possible influence of the look-up-table step is expected to be minor because this global correction has a very small effect on RMSE and CC, as shown in Fig. 3b.

Figure 3a shows the comparison between the original WW3-ST6 hindcast and Jason-2. The results indicate that even without assimilation, WW3-ST6 already achieves a relatively high accuracy, with a bias < 0.1 m, an RMSE of 0.34 m, and a CC of 0.97. Figure 3b presents the outcome after applying a one-dimensional look-up table correction to WW3-ST6 SWHs using CCI-Sea State data prior to fusion. Although this calibration reduces the bias to a smaller value, its effect on RMSE and CC is negligible, with only a 0.01 m decrease in RMSE and a 0.001 increase in CC. This suggests that the pre-fusion correction was not the main contributor to accuracy improvement. In contrast, the fused dataset (Fig. 3c) shows substantial enhancement relative to Jason-2, with the bias remaining negligible, the RMSE reduced by nearly one third to 0.23 m, and the CC increased by more than 0.015 to 0.985, highlighting the high accuracy of the fused product.

Tables 2–4 summarize the evaluation results of bias, RMSE, and CC, respectively. In each experiment, two of the four available satellites were fused with WW3-ST6, while the remaining satellites were used as independent references. It is noted that these experiments are intended to evaluate the stability and achievable accuracy of the fusion method under different observational configurations, rather than to provide a complete year-by-year assessment of the full record, thus, only one year of data is used.

As shown in the tables, when any pair consisting of one satellite with a 66° inclination and one with a near-polar inclination was selected for fusion, all error metrics improved markedly compared with the original WW3-ST6 results. Specifically, the bias, which originally ranged from 0.03 to 0.06 m, was reduced to within 0.01 m. Such a small bias can be regarded as small relative to the mean SWH, but can be meaningful for long-term climate analyses. Meanwhile, the RMSE decreased from 0.36–0.39 m to 0.23–0.25 m, representing a reduction of more than 30 %, and the CC increased from about 0.95 to 0.98, also reflecting a substantial improvement. These results demonstrate the robustness and accuracy of the fusion method under different satellite-combination configurations. In particular, the method consistently reduces bias and RMSE and increases CC relative to the original WW3-ST6 background, regardless of which pair of satellites is used for fusion and which mission is used as the withheld reference. Since the final fused dataset is generated using the same fusion framework, the same calibrated CCI Sea State observations, and the same WW3-ST6 background, these results provide strong evidence supporting the robustness and accuracy of the resulting dataset.

Comparison of the results in Tables 2–4 with those in Fig. 3 indicates that, when Jason-2 is used as the independent reference, the improvement in accuracy from fusing three satellites instead of two is marginal, with the RMSE reduced by only about 0.01 m and the CC increased by just 0.001. Similar results are obtained when other satellites are used as the reference, leading to the consistent conclusion that the additional reduction in errors is insignificant. This can be attributed to the fact that, after fusing two satellites, the overall accuracy of the fused dataset is already very close to the intrinsic accuracy of radar altimeter SWH measurements, which corresponds to a global RMSE of about 0.2 m according to previous studies (e.g., Ribal and Young, 2019; Jiang, 2020, 2023). This further demonstrates the robustness of the fusion method.

Although these results suggest that adding more satellites beyond two does not substantially improve accuracy, we still generated a “multi-sat” version of the dataset. This is because, despite the diminishing marginal benefits in terms of accuracy, additional satellites can still provide slight improvements. Moreover, the inclusion of more satellites may help to homogenize the spatial distribution of errors – reducing potential regional biases, especially in gaps between the ground tracks of the fused satellites. Importantly, as long as the altimeter observations themselves are more accurate than the background field (a condition met by all current missions), increasing the number of fused satellites will not degrade the overall accuracy.

Figure 4 shows the spatial distribution of errors corresponding to the data presented in Fig. 3. The upper and lower panels respectively display the comparisons between the WW3-ST6 and Jason-2 SWH data before and after data fusion. The three columns represent the spatial distributions of bias, RMSE, and CC. Before fusion, the data exhibit opposite bias patterns between the high latitudes of the Northern and Southern Hemispheres, with the maximum bias reaching nearly 0.5 m. Such large biases also lead to RMSE values exceeding 0.5 m in high-latitude regions. Areas with low CC values are mainly concentrated in low-latitude regions, where the mean SWH is smaller and its annual variability is limited. In high-latitude regions, although the absolute errors are larger, the mean SWH and its variability are also higher, allowing the CC to remain above 0.9. After fusion, except for a few areas near sea ice, the bias is largely reduced to within ±0.1 m, the RMSE decreases to below 0.5 m, and the CC shows a clear improvement across both high- and low-latitude regions.

The purpose of the “two-sat” version is to provide a dataset with improved temporal homogeneity for wave climate studies. However, maintaining a constant number of fused altimeters does not by itself guarantee the homogeneity of the time series. The replacement of satellite missions may still introduce artificial discontinuities through differences in orbit, sampling geometry, instrument characteristics, or residual inter-mission calibration uncertainties.

Therefore, we analyzed the possible discontinuities associated with satellite mission transitions in the “two-sat” product. Figure 5 shows the time series of monthly mean SWH from the WW3-ST6 hindcast background, the “two-sat” fused dataset, and the fused-minus-background difference during 1992–2023 for the global ocean and different latitude bands. The fused SWH generally follows the temporal variability of the WW3-ST6 background, including the pronounced seasonal cycle in the mid- and high-latitude bands, while applying a relatively stable positive or negative correction depending on region. The fused-minus-background difference also exhibits clear regional and seasonal characteristics, reflecting systematic differences between the altimeter-constrained analysis and the numerical background.

Importantly, these differences evolve smoothly across the major satellite-transition times marked by the vertical dashed lines. Although the magnitude and seasonality of fused-minus-background difference vary among latitude bands, particularly in the mid- and high-latitude oceans where wave variability is stronger, no abrupt shift appears to be synchronized with the replacement of satellite missions. This indicates that the two-satellite configuration does not introduce visually detectable step-like discontinuities in the monthly mean SWH record. Therefore, no obvious discontinuity synchronized with the mission transitions is observed in either the fused SWH or the difference time series, supporting the temporal homogeneity of the two-satellite product across mission transitions.

One of the primary applications of the fused dataset is in SWH-based wave-climate studies. The fused product substantially reduces the undersampling errors that have long limited the use of altimeter data in SWH-based wave-climate studies. Jiang (2020) showed that, when only two altimeter missions are available, undersampling can introduce an RMSE of approximately 0.17 m in the monthly mean SWH on a coarse 2° grid (assuming the altimeter SWH is error-free). While such errors may not strongly impact climatological means, they can still introduce additional uncertainty in trend estimates. More importantly, even on such a coarse 2° grid, the undersampling effect can still cause systematic underestimations exceeding 0.1 m in the monthly 90th percentile, with RMSEs larger than 0.4 m, and even greater biases at higher percentiles: the monthly 99th percentile may suffer from systematic underestimations over 0.7 m and RMSEs over 0.8 m. These biases decrease as the number of altimeter missions increases, but if not explicitly corrected (which may be a difficult task), they can lead to artificial trend errors far larger than the true climate signal. This suggests that previous studies on extreme SWH climate based solely on along-track altimeter data (e.g., Izaguirre et al., 2011; Patra et al., 2020; Takbash et al., 2019; Young and Ribal, 2020) may be affected by such limitations, although the random error of altimeters may partially compensate for undersampling errors in extreme value analysis, which represents a topic worthy of further investigation in future work.

The fused dataset effectively mitigates these undersampling issues. In particular, it provides spatially complete SWH fields on a regular 0.5°×0.5° grid, thereby reducing the need to account for the strong grid-size dependence that arises when sparse along-track observations are directly gridded. Compared with the gridding of along-track data, the fused dataset also provides a more convenient and reliable basis for SWH-based wave-climate studies.

Figure 7a–d shows the global mean SWH distribution for January 2011 derived from the fused dataset and the original CCI-Sea State L4 data, together with their absolute and relative differences (with the fused dataset re-averaged to 1° resolution for the difference calculation). In Fig. 7a, the distribution exhibits typical features of global wave fields in boreal winter, with high SWH in the Northern Hemisphere westerlies driven by extratropical storms. In comparison, the waves generated by the Southern Hemisphere westerlies are relatively smaller. During this month, the lowest mean SWH occurs in the region near the Arabian Sea in the Indian Ocean. The monsoon is inactive in this region and swells generated in the Southern Ocean cannot propagate into this area due to the blocking effects of the African continent and Madagascar. The original CCI–Sea State L4 data exhibit qualitatively consistent spatial patterns with the fused product (Fig. 7b). However, because of undersampling, the altimeter ground tracks remain clearly visible in the monthly mean fields of the CCI–Sea State L4 product. Figure 7c and d shows the absolute and relative differences between Fig. 7a and b, respectively. In regions with large mean SWH, undersampling associated with satellite orbits can lead to biases approaching ±1 m. In regions with smaller mean SWH, although the absolute sampling error is typically within ±0.3 m, this seemingly small absolute error may still correspond to a relative error of up to 20 %. Such undersampling errors can exert a significant influence on certain types of SWH-based wave-climate studies.

While the differences between the monthly mean SWH distributions in Fig. 7a and b are not particularly pronounced, Fig. 7e–h respectively present the global distribution of the 95th-percentile SWH in January 2011 derived from the fused dataset, the original CCI-Sea State data at 1° resolution, and their difference (with the fused dataset also re-averaged to 1° for comparison). The differences between Fig. 7e and f are even more pronounced. When a storm is successfully captured by the altimeter, the 95th percentile SWH derived from Fig. 7f is significantly higher than that in Fig. 7e. Conversely, if the altimeter fails to sample the waves generated by a storm (particularly the high SWH values during the storm), the 95th-percentile SWH in Fig. 7f becomes notably lower than that in Fig. 7e. This issue makes the orbital patterns and data discontinuities in Fig. 7f more apparent. Such undersampling can easily lead to errors exceeding 2 m (Fig. 7g) in the monthly 95th-percentile SWH and relative errors of over 40 % (Fig. 7h).

It is worth noting that in January 2011, four altimeter satellites were operating simultaneously, which considerably reduced undersampling errors compared with periods when only two satellites were in orbit. Nevertheless, even under these favourable conditions, such large errors still occur in the estimation of extreme wave heights, underscoring the necessity of data fusion.

A closer inspection further reveals that underestimation occurs more frequently than overestimation in the estimation of extreme SWH in Fig. 7f. This is because extreme wave events are generally less likely to be captured by the altimeter sampling, which has been pointed out by previous studies (Jiang, 2020). Jiang (2020) has further demonstrated that as the number of satellites increases, the likelihood of such overestimation gradually diminishes, which may, in turn, affect the analysis of long-term climate trends.

Previous studies have also pointed out that the use of a 1° grid tends to underestimate altimeter-based SWH percentiles, and that coarser grids of 2 or 3° may be more appropriate (Young and Ribal, 2020; Takbash et al., 2019; Jiang, 2020). However, coarser grids inevitably lead to the loss of spatial detail. In addition, as the computation area increases, the physical meaning of percentile estimates derived from altimeter observations becomes less clear: They will shift from the temporal percentiles of spatially averaged SWH within a given grid cell to the spatio-temporal percentiles of the broader region corresponding to the grid size. In contrast, percentiles computed from the fused dataset not only retain higher spatial resolution but also have a well-defined physical interpretation. For instance, when the 95th or 99th percentile is calculated at the native 0.5° resolution of the fused dataset, the resulting value corresponds directly to the extreme percentile of the time series of spatially averaged SWH within each 0.5° grid cell.

In recent years, with the rapid development of artificial intelligence (AI), data-driven modeling approaches have re-emerged as a promising alternative to traditional numerical simulations. Compared with numerical models, AI-based methods offer lower computational cost and higher efficiency, and they have already achieved remarkable success in weather forecasting. Large-scale AI models such as Pangu, GraphCast, GenCast, and Aurora have, in some cases, delivered forecasting skill surpassing that of numerical weather prediction (NWP) systems at a fraction of the computational expense. Similarly, we believe that AI-based wave models represent a promising future research direction.

A key requirement for training high-performing AI models is the availability of large, accurate, and well-structured datasets. Although ERA5 SWH data already provide a reasonable source of training samples, as noted in Sect. 1, ERA5 does not incorporate future observations to constrain past states, leaving room for improvement. In one of our recent studies, we used a five-year subset of the fused dataset generated using the present framework to train and fine-tune an AI wave model for global SWH (Wang and Jiang, 2024). Hindcast experiments demonstrated that the AI model fine-tuned with the fused dataset not only outperformed the counterpart trained directly on ERA5 but also significantly exceeded the hindcast skill of WW3-ST6. This indicates that the fused dataset enables AI wave models to achieve, in terms of SWH hindcast accuracy, performance that rivals or even surpasses state-of-the-art NWMs (Fig. 8). Figure 8 is reproduced from an independent-test-year results of Wang and Jiang (2024), in which 2020 was withheld from model training and used exclusively for evaluation. It is noted that Wang and Jiang (2024) focused on AI model development and used the fused SWH fields only as training data. It did not provide the full long-term dataset, the two product configurations, the systematic validation, or the public data documentation presented here.

More details of this AI wave model and how this fused dataset is used can be found in Wang and Jiang (2024). Meanwhile, it is noted that we additionally provide the R value in Eq. (3), which represents the spatiotemporal distance to the nearest altimeter observation, to further adapt the fused dataset for AI training applications. Intuitively, a smaller R value indicates stronger influence from altimeter measurements, which allows it to be incorporated into the training loss function as a weight, so that data points closer to direct observations can be assigned higher importance. This R value and corresponding weighting strategy were also employed in Wang and Jiang (2024).