The long wavelength part of the MSS is derived along the highly accurate nearly uninterrupted mean profiles of the Exact Repeat Missions (ERM) derived using TOPEX/Jason-1/Jason-2 observations (T/P reference). This reference mission is indicated in Table , along with the rest of the satellites and orbital configurations included in the MSS. This is done in a similar fashion as the DTU21MSS . where ERM (other than the T/P orbit) and GM missions have been fitted to the 20 year T/P reference period by using the T/P reference as the constraint. This is similar to other MSS methods, utilizing a reference orbit and then fitting other ERM and GM missions to fill the gaps At each ground track location we determine a 4 parameter solution to account for the mean and largest time-variable signals using the following equation: 1SSH=MSS+a(t-t0)+bcos⁡ωa(t-t0)+csin⁡ωa(t-t0), where a is the sea level trend and b and c are the coefficients of the in-phase and in-quadrature of the annual signal. In order to ensure that the MSS is determined wrt. to certain mean period the t0 is fixed to 1 January 2003, the center of the T/P reference orbit . To ensure later stages benefits most from the novel SWOT data at short wavelengths, a low-pass gaussian filter with 0.5-gain at 10 km full wavelength has been applied to low and mid latitudes (<70°).

Over steep geoid gradients it is important to account for the fact that the point of closest approach affecting the sea level observations are not at nadir but up to a few km off-nadir. The corrections was developed by and ranges up to nearly 40 mm across the Aleutian trench in the northern Pacific Ocean. The correction is dependent on the altitude of the satellite and and is applied to the LRM satellite data as suggested in .

The SWOT satellite is a wide-swath altimeter mission, launched December 2022 into a 1-d repeat Calibration/Validation (Cal/Val) orbit . After three months of observations in the Cal/Val orbit, SWOT switched to the science orbit with a 21-d repeat and almost global coverage. With an inclination of 77.6° the northernmost Arctic Ocean is not yet covered.

SWOT carries the KaRIn altimeter, observing the Sea Surface Height (SSH) in two 50 km swaths on each side of nadir, with a 20 km nadir gap. To get the high range precision needed, corrections for the long-wavelength roll error is needed . This is done at crossover-locations and interpolated in between . In the science orbit, the distance between the crossover locations are much shorter than those in the Cal/Val orbits, and the quality of the operational SSH observations are therefore expected to be of a higher quality .

For the DTU25MSS model we use the SWOT Level 3 ocean data product, that is produced by and is based on the Level 2 ocean data products. Specifically we use the L3 Expert (2 km) version 2.0.1 and L3 Unsmoothed (250 m) version 1.0.2 . The 2 km data used covers the period from the 26 July 2023 to 22 April 2025, a period of 1.75 years (cycle 1–31). The 250 m data has the same start date (from cycle 1) but at the moment of writing only the first 16 cycles was processed to L3, stretching to the 17 June 2024, thereby covering 0.9 years. We use the Level 3 data products to utilize the state of the art Level 2 corrections as well as further editing that has been used in the process of creating the L3 product. This is important, as the MSS is a global product and a robust global framework provides a more stable environment for the development of a global model. Importantly, another factor is the implementation of additional altimeters in the cross-track roll correction for SWOT. The implementation of independent Sentinel-6MF observations further improves the roll-correction and improves the subsequent observations .

The preprocessing of the SWOT data mainly revolves around the long-wavelength correction, and the stacking of the data is performed to reduce the influence of temporal signals. For each cycle and pass, SWOT measures the full SSH, and is corrected for all geophysical and atmospheric effects , see Fig. a1. Subtracting a given MSS model yields the sea level anomaly (Fig. a2).

In order to fix the SWOT observations to the same reference period as the DTU25_LWMSS, as well as removing long-wavelength mesoscale oceanographic features, each cycle and pass is corrected. An example of the correction surface is seen in Fig. a3. The parameters β that is estimated is a bias and tilt in along-track and cross-track directions. This will also take into account any residual roll errors that might still be present in the data. This acts as a high-pass filter, removing the long wavelengths corresponding to the order of the fitted plane, and the short wavelength features emerges (Fig. a4).

In Fig. b1–b4 the same steps have been done for all passes for SWOT cycle 2 to illustrate the large variability of the SLA in the region off South Africa, and the subsequent small scale details revealed when fixing the long wavelengths to the reference surface. In Fig. c1–c4 a stack of 10 cycles is shown to illustrate the consequences of stacking along with this correction. In Fig. c2 mesoscale features are visible as this is a mean of approximately 7 months of data, and will have covered a specific oceanographic seasonality. By applying the correction to the data, the mesoscale features are removed and we are left with the short wavelengths (Fig. c4) which is gridded and used to construct the updated DTU25_2 kmMSS.

The correction surface is determined one for each swath in 50km×50km tiles, where the estimation matrix A is created 2A=1c0a0c02…a0o⋮⋮⋮⋮⋮⋮1cNaNcN2…aNo∈R(NcNa)×(1+o), and where c(⋅) and a(⋅) corresponds to the cross-track and along-track distances respectively, o determines the order of the plane used for the correction and Nc and Na is the number of points in the cross-track and along-track direction. An order o=0 or o=1 is used depending on the oceanographic variability of the region (more details in later section). For the given area we then remove the rows from the matrix that contains either invalid points sorted out from the outlier sorting or points not wanted for the correction, such as being too close to the coast. The parameter is then determined from least-squares fitting on the SLA 3SLA=SSH-DTU25LWMSS, as seen in Fig. a2. The correction surface is then determined from 4β=(A⊤A)-1A⊤SLA where the along-track overlap of 80 % with a Hann window in order to ensure a smooth solution. The subsequent corrected SLA is determined from: 5SLAc=SLA-Aβ. These corrected SLAc observations are then used in the further processing of SWOT. To further reduce the temporal signal and reduce the noise we take the temporal average for each observation-location over all cycles available (N); 6〈SLAc〉=1N∑nNSLAc(n). The 〈SLAc〉 is used in the further gridding and subsequent update of the reference MSS. The influence of the parametric correction was largest in the beginning of the SWOT science orbit, where a low number of cycles was available (<10 cycles). However, with the large number of cycles available at the time of writing (31 cycles) the importance of the correction has decreased significantly compared to at the beginning of the SWOT mission.

We use the L3 Expert (2 km) data product in the open ocean (>5 km from the coast), as the upwards continuation of the gravity signals will result in signals at shorter wavelengths to be too attenuated to be captured. The data is provided on a fixed grid at 2 km spacing and therefore is a significantly reduced dataset as well as computationally easy to compute the temporal average. This reduces the time and memory complexity of computing the MSS solution in the open ocean.

The gridding of the observations is done by least-squares collocation, similar to the construction of the older DTU models . Essentially the updated MSS is constructed by determining the data covariance C0 and defining the covariance function C(s). The covariance function used is 7Cr(s)=1+srexp⁡-sr8C(s)=C0αCr1(s)+(1-α)Cr2(s) which consists of two second-order Gauss–Markov covariance functions C|r(s) with their corresponding correlation lengths r1 and r2, each part weighted by the factor α, and the distance s is determined by the geodetic distance between points (ϕ1,λ1) and (ϕ2,λ2), and the correlation lengths is defined as C(r)=12C0. The two-part covariance function was determined by inspection of empirical covariances from SWOT 〈SLAc〉, which on average matched two length scales λ1=2.5km and λ2=10km, resembling the SWOT data sampling spacing and the scale of the residual MSS submesoscale features. The weighing is determined as α=0.75. Additionally, an associated observation uncertainty σϵ2 is determined and used to provide higher importance to observations with low noise. Uncertainty estimation of the SWOT data is further explored in Sect. .

The prediction grid SLA^ is then determined from the 〈SLAc〉 at locations x=(ϕ,λ) at each gridpoint p=(ϕp,λp) 9SLA^(p)=C(x,p)⊤(C(x,x′)+σϵ2I)-1〈SLAc〉, where I is the identity matrix and the input to the covariance function is the pointwise geodetic distances C(⋅,⋅)=C(dgeo(⋅,⋅)). To get the full model, we add back the long wavelength signals that were previously removed: 10DTU252 kmMSS=SLA^+DTU25LWMSS. Additionally, the associated grid uncertainty σSLA^ can be determined from 11σSLA^(p)=diag(C(p,p′)-(C(x,x′)+σϵ2I)-1C(x,x′)). This framework involves repeated inversion of large matrices. However the mathematical equivalence to Gaussian Process Regression enables usage of software frameworks, such as GPyTorch which is utilized in this project to handle large datasets , necessary for utilizing SWOT data for global grids.

This is computed globally in tiles of 6°N×10°E size at 0.01° resolution (∼1.1km at the equator), with a 50 % overlap between tiles. The overlapping tiles are then combined with a cosine weighting function to ensure no edge effects. Tiles with either no SWOT data (the entire tile is above/below ±77.6°N) or completely over land, we return the DTU25_LWMSS solution, but without the lowpass filter normally applied on the reference field.

In order to handle the increased amount of data when inverting the matrix in collocation, the gridding process is based on an iterative nearest-neighbor scheme. In each tile 10×10 grid points are selected, and a common nearest-neighbor search in the SWOT datapoints is done, starting with the 100 nearest points. A check is done to determine if at least 5 % of the points are further away than 50 km, to ensure no clipping of the covariance function. If not, we increase with 50 points and check again iteratively. The nearest neighbor search is done with the FAISS library, which significantly increases the lookup speed . A common number of points is the nearest 300 points.

The effect of the correction for MSS determination is shown in Fig. . This is the same area as in Fig. . The rows (a-d) show the MSS (determined from SLA^+DTU25LWMSS), associated uncertainty σSLA^2, the SLA^ and highpass filtered MSS at 20 km wavelength. The columns 1–4 are the different correction levels, with the first column only removing one overall bias from the SWOT data, and the next three columns using correction order o=0 to o=2. The region just off South Africa has high oceanographic variability, resulting in a high need for correction. Quiet regions show almost no need for correction. However, we see that a simple bias results in mesoscale signal still present in the data (resembling that of Fig. ) as well as edge effects due to small differences in height in the different passes.

The effect of higher order correction reduces uncertainty due to more oceanographic variability being removed, the mean SLA becomes smaller and the edge effect decreases. The effect on the spectral properties of the MSS is seen in Fig. e1 and e2, where the north-south and east-west direction power spectral density is computed. It can be seen that higher order corrections have smaller differences at long wavelengths. The effect at short wavelength is most pronounced in the east-west direction, due to the uncorrected edge effects being primarily in the east-west direction, and aliased to higher wavelengths in the north-south.

While higher order correction might seem better, higher orders correction might remove too much of the signal from SWOT. Comparing the area with Sentinel-3A&B data in Fig. , the second order correction showed a worse correspondence, compared to the first order. Globally in quiet regions, the first order was worse compared to the zeroth order, indicating too much of the signal was removed. In each tile when computing the DTU25_2 kmMSS, the initial correction is the zeroth order, but if the 90th-percentile of absolute SLA exceeds 10 cm, a first order correction is used instead.

Inspecting the power spectra in Fig. e1 and e2, the main difference between the reference surface and the corrected MSS is the lower power level at short wavelengths, cased by the lower noise in the SWOT data. The effect is predominantly located at wavelengths shorter than O(20km). This corresponds to the band in which SWOT has shown to have excellent performance, and from where we can utilize the observations best . At the shortest wavelengths (O(2km), outside of scope of figure) there is a small increase in power due to small uncorrected height differences between gridded swaths at wavelengths matching the correlation length. This could be compensated by increasing the correlation length, however this will degrade the quality of the SWOT data for each swath. Due to the very small wavelength, this effect is ultimately removed when interpolating the MSS onto observations in practical use, and has thus not been corrected in this version, however will be a subject for further versions.

Atolls are ring-shaped island that encircle a central lagoon of open water and is a good way to visually demonstrate the benefit of SWOT for small scale signals. French Polynesia is located in the Pacific Ocean and should be well resolved by nadir altimetry as well as by the new 2 km SWOT ocean data. This is true, up to a certain extent, as the enclosed waters of the atolls are either poorly resolved or not captured at all with the coarse resolution used in the open ocean.

In Fig.  we see the MSS variation for the island chains in French Polynesia with either DTU21MSS (a) or DTU25MSS (b), highpass filtered to showcase the spectral band where we gain new information from SWOT. In the open ocean we generally see a lower noise level in DTU25MSS compared to DTU21MSS, and some small scale seamounts are discernible. However looking specifically at the atolls, we see how the 250 m data enables us to obtain a much sharper delineation between open ocean and the lagoons (Fig. a1, b1, and d), matching the enclosed areas as observed with Sentinel-2 (Fig. c).

While the atolls in region a1/b1 was captured by nadir altimetry due to their size (wider than 20 km), in region a2/b2 we see small scale atolls that were not captured at all in the older MSS. With the 250 m data, we capture the circular shape of the small atolls, down to the small Tepoto atoll with a lagoon width at 1 km. The higher mean water level compared with the open ocean in the surrounding area is in line with the known behavior of atolls , and the introduction of the smaller scale atolls should improve the MSS reference for regional studies in the global oceans.

Tidal flats are complex coastal features that are both small scale and dynamic in time. They encompass river outlets as well as non-permanent islands, and have been challenging to map from satellite altimetry. With swath altimetry from SWOT, it has been show that it's possible to map the tidal flats, as well as constructing an elevation model from the data .

Looking at the Wadden Sea, a coastal region in the Netherlands, Germany and Denmark as observed from Sentinel-2 in Fig. a, the complex coastal structure is seen. Shown in Fig. c, is DTU21MSS with black dots illustrating all available altimetry data used to construct the model. What is clear is the absence of valid data in the coastal zone, which constrains older models in their validity in this region. With SWOT we are able to observe all the way to the coasts which encompasses several smaller islands and peninsulas.

However these features are dynamic in time, at different timescales. While they might be physical present during the sampling time of SWOT, this might not be valid during the entire 30 years where satellite altimetry has been available. This might lead to a problematic reference surface, however as we saw before the conventional altimetry already exhibits problems in this area. Further discussion and studies into incorporating new complex features into a reference surface would be needed, as well as their impact on the resulting observations, depending on the use of the product. We have elected to keep it as a real physical signal in the product, to use as a reference for the SWOT mission.

Another challenging region which historically has caused problems are fjords, due to their long and shallow reaches , as well as archipelagos as the many small islands causing contamination in the return waveforms in the conventional altimetry .

Encompassing both, the west coast of Norway has a lot of islands as well as very shallow and long fjords. This has caused earlier MSS solutions to rely on either extrapolation or data filling from alternative data sources, due to having no reliable satellite altimetry data in the coastal region, as seen in Fig. b, with the point illustrating all available altimetry used for DTU21MSS between 59.7–62.3°N for reference.

We see that the MSS has been updated in the fjords, with Hardangerfjord becoming lower and Sognefjord increasing in height, indicating new height data in these fjords. Validation of these regions is challenging, as no other radar altimeter can get reliable data in the shallow fjords, however future studies of the ability of SWOT to map the fjords might be possible with the small footprint of ICESat-2 .

What we see at Sognefjord is a sudden stop in the update, which is caused by the lack of new data east from here, most likely caused by the distinction between open-ocean and inland-water classification. As new versions of the SWOT data processing is released, improving its quality, future versions of the MSS would be improved in turn.

In order to determine the effect of using SWOT in the coastal zone, we subdivide the Sentinel-3A&B observations mentioned before, to encompass the area around the coastal zone (1–20 km). Here only the full heights are used, as it is not possible to high-pass filter such short profiles. The resulting comparison globally as well as in the three cases shown in Sect.  are seen in Table . Here both the DTU25_2 kmMSS and the DTU25MSS are used, in order to see the effect of including the coastal features with the 250 m data. In Table  the HybridMSS and CLS22MSS are almost identical, due to the HybridMSS primarily consisting of the CLS22MSS in the coastal zone .

As Sentinel-3A&B operates in SAR mode, the footprint along the track is only O(300m), but the footprint size in the along-track direction is O(7km). By including features of size O(2.5km) in the MSS, even though the feature might be physical, it might not necessarily reflect the observation as seen from Sentinel-3A&B.

In order to see the effect of this, the standard deviation (SD) of the Sentinel-3A&B observations with reference to the MSS models, as a function of distance to the coast, is seen in Fig. . This is three very different areas, and we see the benefit as well as the problems in some of the cases.

In Fig. a the Weddel sea from Fig.  is seen, with overlay of Sentinel-3A&B measurement locations. The color of each locations is the difference between the agreement between Sentinel-3A&B mean profiles and the MSS models DTU25MSS and HybridMSS, determined as 17diff=|SSHS3-DTU25|μ=0-|SSHS3-Hybrid|μ=0, where SSHS3 is the mean profile of Sentinel-3A&B and μ=0 means we subtract the mean before computing the absolute differences. This results in white being equal agreement between Sentinel-3A&B and the MSS models, red is worse agreement for DTU25MSS compared with the HybridMSS, and blue is better agreement for DTU25MSS. This is computed for Fig. b and c as well.

In Fig.  there are corresponding plots showing the standard deviation of the areas a-c, as a function of distance to the coast. We see how all models generally have the same uncertainty far from the coast, as we also saw in Table  that the overall difference between the models are small. However close to the coast (<10 km) the differences increase, which also matches where we start seeing colors in the points on the map.

For the Weddel Sea in Fig. a, almost all models have the same performance, except the DTU25MSS which includes the 250 m data which we saw became much more complex in the tidal flats. Inspecting the location of the differences we see that some locations have a much better agreement while some are much worse, which is underlined if we computed the median absolute error instead of the standard deviation. However the cause of this could be due to the time-varying aspect of tidal flats or the large and anisotropic footprint and reflection of Sentinel-3A&B. Further studies into incorporating these complex regions into MSS reference surfaces might provide better agreement with nadir altimeters.

For atolls in the open ocean, the case is the opposite, with a large increase for the current models close to the coast, and a very small standard deviation when including the 250 m data. Looking closer to some examples in Fig. b1, the few points that lie in the central lagoons seem to have benefited overall quite well from the increased height.

To determine to effect of switching to a new MSS reference when using SWOT, we compare oceanographic features-of-interest based on different reference surfaces, including the experimental reference frame at the 2023 epoch. The three features are the zeroth, first and second order derivatives of the sea surface, reflected as the SLA, sea surface currents (SSC) and the relative vorticity (ξ). These are determined in two cases; (a) for a single pass thereby reflecting the effect on an instantaneous observation, and (b) for the full stack of observations. The stacking of SWOT data resembles an MSS which, compared with the MSS reference models, gives an indication of the residual effect of unmodeled geodetic features leaking into the ocean features as observed from SWOT.

Firstly for the single pass, the SLA is determined as 19SLA=SSH-MSS(⋅), with the different MSS models as reference. The SSC are then determined from the derivative of the Absolute Dynamic Topograhy (ADT) in x- and y-directions to use the geostrophic equations: 20u=-gf∂ADT∂y,v=gf∂ADT∂x,SSC=u2+v2 where f=2Ωsin⁡ϕ, g is the gravitational acceleration, x and y resembles the two axis of the SWOT sampling grid, and 21ADT=SSH-N=SSH-(MSS(⋅)-MDT), where N is the geoid and MDT is the Mean Dynamic Topography . The derivative is computed in the SWOT native grid of 2km×2km, however to reduce the smallest scale effects the SLA (before computing the derivative) is smoothed with a 6km×6km lowpass boxcar filter. Computing the derivative minimizes the effect of the long wavelength MDT, and the same MDT is used for all situations, but is still included in order to reflect the numeric value of the features. The assumptions needed for geostrophic balance does not hold at these scales, however this can be used as a proxy to determine the energy level of submesoscale features or resulting omitted geodetic signals present in the data, and how they would show up as sea surface currents when using the geostrophic equations at this scale.

Lastly the vorticity is determined as 22ξ=∂v∂x-∂u∂y. This is again determined from the native 2 km grid of SWOT, with the 6km×6km lowpass boxcar filter applied on the SLA. The derivative has been applied a second time, which results in a smaller overall swath. The small spatial scale of the derivative results in noisy estimates of the vorticity, but can be used to evaluate the relative difference between the MSS fields at these small scales.

The same is computed for the stack, however in the equations shown before the SSH is replaced by 1N∑iNSSHi for N passes in the stack. As this is the definition of the MSS (when N→∞), this indicates the effect of unmodeled geodetic features leaking into the oceanographic variables.

In Fig. a and d we see the effect of changing the MSS on SWOT data for a single pass (left) and for the full stack (right). On the left we see that there is not a large difference between the different MSS models, indicating that the signal observed by a SWOT pass is dominant, compared to the MSS errors. On the right the difference with DTU25MSS is much smoother, compared with the other reference surfaces. As we only updated the MSS with the short wavelengths, a long-wavelength mean signal is expected to be present in the data, which is what we see. This is clear when using the DTU25MSS_2023X, as the main difference between SWOT and the current MSS models is the larger scale changes, some of which can be attributed to sea level rise. In the following derivatives it can be seen how the DTU25MSS_2023X is only different from the DTU25MSS at long wavelengths.

For the SSC in Fig. b we see a slight decrease in the noise but the majority of the signal is still left in the data, however for the stack in figure Fig. e the strength of the residual signal is significantly decreased. This is even more discernible with the vorticity, as the signal for a single pass looks almost identical with only a small decrease, however there is a clear effect for the full stack.

As the majority of the improvement is at very short wavelengths by SWOT, and the derivative acts as a high-pass filter, it is expected that the improvement will become more clear at higher derivatives. The second order derivative resembles the process with which gravity is determined from sea surface observations, which illustrates why SWOT has been able to capture marine gravity at an unprecedented level .

Repeating the same analysis, this is done for the independent SWOT data from the Cal/Val orbit in Fig. . We see the same as in Fig. , indicating this effect is not only caused by including the same SWOT data into the reference. However we also see the effect of hybridization of the HybridMSS, which causes small scale effects at certain locations , primarily noticeable when stacking the highly accurate SWOT data.