The equations used for the forward model are obtained under the assumption of hydrostatic equilibrium . They can be written as follows: 3Frcs=ρw-ρiρwHi+1-αcsccs-ρsρwHs4Frak=ρw-ρiρwHi+1-αakccs-ρsρwHs5Flis=ρw-ρiρwHi+1-αisccs-ρsρwHs where Frcs, Frak, and Flis are the freeboards for CryoSat-2, AltiKa and ICESat-2, and ρw, ρi and ρs are the densities for sea water, sea ice and snow respectively. Hi is the SIT and Hs is the SD. cs is the velocity of the radar pulse in the snow and c is the velocity in a vacuum. We assume that c/cs=1.28 . αcs, αak and αis are the bias factors for CS, AK and IS respectively. We use the sea ice type product from to classify each grid cell as first-year ice (FYI) or multi-year ice (MYI). We then assign sea ice bulk densities of 916.7 and 882 kgm-3 for FYI and MYI, respectively, per . For sea water, we use ρw=1023kgm-3 . ρs evolves seasonally following : 6ρs=6.50M+274.51 where M is the number of months since October (i.e. M=3 for January).

In order to formulate an inverse problem, we define a data vector d. We jointly invert a combination of CS and IS data (or CS and AK data), and define d=(dcs,dis), with dcs={(ccsk,Frcs,k), k∈[1,…,Ncs]}, and dis={(cisk,Flis,k), k∈[1,…,Nis]}, and where ccsk are the coordinates of data of index k (ccsk=(loncsk,latcsk)), Frcs,k the value of the freeboards of index k, and Ncs the number of input data for CryoSat-2 (and same for IS).

A linear inverse problem is formulated following : 7d=Gm+ϵ where m is the unknown model vector, d the data vector defined in Sect. , ϵ represents the data errors (i.e. the inability of the model to explain the data), and G is the forward model operator.

In order to parametrise the 2D maps of the variables to reconstruct (Hi, Hs and α), we use nodes and a Delaunay triangulation . Here, the nodes of the Delaunay triangulation are unknown parameters, they are independent of the data locations and will randomly move during the inversion procedure. For our inversion, the vector m is defined as m=(mHi,mHs) with (example for Hi) mHi={(cHik,Hik),k∈[1,…,nH]} where cHik=(lonHik,latHik) are the 2D coordinates of the node associated with index k (Fig. ), Hik the value for SIT associated to the node of index k, and nH the number of nodes. The model vector mHs is defined in a similar way. We also invert for the bias factor, and can thus write the vector model m=(mHi,mHs). The three nodes defining each Delaunay triangle can be linearly interpolated to any point within the triangle by computing Barycentric coordinates . This then provides a continuous field over the domain but with discontinuities in the gradient at triangle edges.

Here, the dimension of the model space (the number of nodes nHi) will be treated as an unknown variable to be inverted for. In this way, the level of spatial resolution is directly determined by the data without having to define a smoothing parameter. We summarise the model and data vectors in Table .

In order to recover the model parameters from the observed data, we use a Bayesian approach, where all quantities are defined probabilistically: 8P(m|d)=P(d|m)P(m)P(d) where P(d|m) is the likelihood function representing the probability of observing the data given a particular model, P(m) is the prior probability which represents the information we know on the model before measuring the data d and P(d) can be seen as a normalisation constant. The solution is P(m|d) and called the posterior probability distribution.

We use uniform prior distributions. For a given model parameter x, the prior distribution is: 9P(x)=1b-a,if a≤x≤b,0,else,x∈{Hi,Hs,αcs,αis}.

The intervals [a,b] for each distribution are as follows:

Hi: [0,6]m

The likelihood function is based on a mathematical model for data errors. We make the assumption that the noise on the data is a combination of several sources of uncorrelated errors, such that the final distribution should converge towards a Gaussian as per the Central Limit Theorem. Furthermore, we use diagonal covariance matrices for simplicity . Thus, the joint likelihood function is a Gaussian likelihood that can be written as 10P(d|m)=P(dcs,dis|m)=P(dcs|m)×P(dis|m)=12πσFcsrσFislexp⁡-‖g(m)-dcs‖22σFrcs2-‖g(m)-dis‖22σFlis2 with σFrcs and σFlis being the standard deviations of the input data.

To estimate the posterior solution P(m|d), we use the reversible-jump algorithm , which is an extension of the standard McMC Metropolis algorithm to models with variable levels of complexity. The algorithm produces a chain of random Delaunay models where each new proposed model is drawn as a random perturbation of the current model. We note that the range of nodes per model was between 1000 and 2000 at the end of the burn-in period (for all the parameters we inverted for). The proposed model is either accepted or rejected according to an acceptance rule based on the ratio of posterior probabilities of the current and proposed models. The solution thus obtained is a large ensemble of Delaunay models representing the posterior distribution.

At each iteration of the random walk, the algorithm makes a random choice between four perturbations of the model parameters (example for inversion of Hi and Hs only):

Value. The value of a randomly chosen node of index k in the map is perturbed as well as its associated values Hik and Hsk.

The following describes the methods for inversion at a local region, specifically around the upward-looking sonar (ULS) moorings deployed by the Beaufort Gyre Exploration Project (BGEP). The BGEP data were collected at three locations in the Beaufort Sea (BGEP-A, BGEP-B, and BGEP-D) between 2010 and 2024. Sea ice thickness was derived from draft measurements by dividing them by 0.89, as per .

In the previous Arctic basin inversion, a 360×360 EASE-2 grid was employed. The localised region of interest (ROI) was then defined by cropping this grid to encapsulate each ULS location. We find that a 55×55 grid (25 km resolution) over the Beaufort Sea provides a suitable domain: it fully encompasses each ULS location, avoids placing them near the cropped boundaries, and ensures that the Delaunay triangulation consistently covers the region. For the inversion itself, a 57×57 grid is specified. This adds a one-grid point border around the 55×55 cropped input, ensuring that no data points are located on the boundaries of the inversion domain.

Since the ROI covers a smaller area than the full Arctic basin, the initial cell count for the inversion could be proportionally reduced. By applying the ratio of the respective grid sizes, this was set to 35 cells. Following the same procedure, the maximum number of cells was derived and rounded to 100. The number of iterations was set through trial and error, and 500 000 iterations were found to be sufficient for the models to stabilise, with the first 100 000 iterations discarded as burn-in. These choices of ROI, cell limits, and iterations reduced the computation time to approximately 10 min d⁻¹. To verify the robustness of this setup, an ROI inversion was also conducted using the same parameters as the full Arctic inversion (i.e., identical iterations and cell limits). The results were effectively the same as those from the reduced setup, confirming that the lower number of iterations and cells could be used without loss of accuracy. Therefore, the reduced configuration was adopted in order to minimise computation time.

The inversion was then performed at mid-month intervals for the winter periods from 2018–2021, corresponding to the years for which comparison data are available from AWI.

Following standard practice in inversion analyses, synthetic inversions are conducted to assess the validity of the proposed method. Moreover, these experiments provide a basis for comparison with alternative interpolation schemes and serve to illustrate the specific benefits of the inversion framework.

To perform this synthetic test, it is first necessary to produce synthetic data by running the forward model for a given target model. Here we used previously computed fields of SIT and SD to calculate the freeboards, Fl and Fr, by means of Eqs. () and (). In order to simulate true input freeboards, a random Gaussian noise, but no covariances, is added to the data. Then, we sample the data following a chosen number of orbits tracks and a chosen number of freeboards per track. The data close to the North Pole are removed to simulate true satellite observations.

Finally, the inversion is carried out using these computed synthetic freeboards as input. The entire procedure is illustrated in Fig. . It is evident that the retrieved inverse SD and SIT are analogous to the true SD and SIT fields, yet they exhibit a reduced number of features and a smoother aspect in comparison to the original maps. This phenomenon can be attributed to the substantial volume of data utilised in this synthetic evaluation. The true SIT and SD fields were generated following the parametrisation described in Sect. , using 15 d of CS2 and IS2 freeboards (approximately 40 000 measurements), whereas only 9000 freeboards were used in the synthetic inversion. This reduced dataset was selected to enable faster inversions and allow for multiple synthetic experiments.

The results of this synthetic test are used as a benchmark for comparison with an alternative interpolation-based approach. Specifically, we employ the GPSat Python library , which performs Gaussian process interpolation of the freeboard data prior to SIT and SD retrieval. Since the synthetic data is already defined on a regular grid, no additional spatial binning is applied before interpolation. The interpolation is performed on a 360×360 grid at 25 km resolution, consistent with the resolution of the synthetic freeboard fields.

The procedure and the resulting SD and SIT fields are plotted in Fig. . Unlike the inversion approach, this method requires an explicit interpolation step to generate continuous freeboard fields prior to applying Eqs. () and (). In contrast, the Bayesian inversion approach directly estimates continuous SD and SIT fields through the spatial subdivision defined by the Delaunay triangles.

First, a quantitative analysis of the results depicted previously is performed on Fig. . For both the Inverse solutions and GPSat, the difference with the True SIT/SD fields are close to 0 over the Central Arctic, which might be linked with the higher data coverage over this area (Fig. a and c). We notice that the inversion solution overestimates the SIT compared to the true field on Arctic's borders (Fig. a). This overestimation is reflected on the histogram (second line on Fig. b), thus yielding a higher STD and RMSE compared to the GPSat solution (second line on Fig. d). The Inverse and GPSat solutions for SD are highly consistent, both in terms of spatial patterns (first row of Fig. a and c) and statistical performance (RMSEInverse SD=0.152, STDInverse SD=0.150; RMSEGPSat SD=0.144, STDGPSat SD=0.142).

This quantitative analysis is performed using only one model for the Inverse solutions (the mean SIT/SD map). Nevertheless, a principal benefit of the Bayesian method is that it enables the calculation of thousands of intermediate models during the inversion process. Thus, we pursue the analysis by looking at all the models computed during the inversion.

In Fig. a we can see that the distribution of models is close to the true SIT field on an example horizontal transect, that validates the visual observation made with Fig. . However, in Fig. c, the shape of the true SIT field is very different from the inverse SIT on a different horizontal transect. These discrepancies between the two studied transects can be explained by the number of input data used. As shown in Fig. , the number of freeboard data is important for the y=200 transect as it is close to the North pole. On the other side, for y=250, the amount of freeboard data is much smaller. Thus, on y=200 the solution is better constrained by the input data whereas on y=250, due to the lack of data, the probabilistic solution is wider as a lot of values of SIT are possible. This emphasises the probabilistic nature of the inverse approach introduced here: with a large amount of data, the distribution of models is restricted around the true value whereas with less data the distribution is wider.

This characteristic of the Bayesian approach can also be seen for specific locations as depicted in Fig. d and f. In Fig. d, the chosen point (red dot) is located close to the North Pole, where the amount of freeboard data is largest. Thus, the distribution of models is restricted around the true value of SIT/SD. However, in Fig. f, the chosen point is located in a region with fewer freeboard data. As expected, the distribution of models is not centred around the true value. This emphasises again the fact that, with less input data, the inversion is less constrained and thus the number of possible values for the SIT and SD fields is more important and different from the true value.

Compared to other interpolation approach, the Bayesian inversion doesn't produce a single model for the different fields (SIT, SD and any other parameter we would invert for). Instead, it produces a probabilistic solution in which all the models (removing the burn-in period) are of interest (for example to study the behaviour and covariances of the different fields inverted).

Furthermore, this Bayesian approach doesn't require interpolation and can be applied directly to the binned along-track data. These data directly determine the level of smoothing (more data implies more features). With GPSat, the level of smoothing is an arbitrary choice of the user. Instead, with our approach, the level of complexity in the solution is determined by the level of information present in the data.

Arctic maps of SIT and SD are computed for every winter from 2018–2021 using freeboards from a combination of either CryoSat-2/ICESat-2 (CS-IS) or CryoSat-2/AltiKa (CS-AK). The resulting SIT and SD maps are presented in Figs.  and , for prescribed values of αcs=1, αis=0 and αak=0. The lack of data near the pole for CS-AK is a result of AltiKa's orbit, limited to 81.49° N.

Moreover, the inverse approach set out in this study enables an inversion of the bias factor for one or both satellites. Figure  (left column) shows the results for April 2019 for an inversion with an unknown bias factor, designated as αcs, in addition to SIT and SD. The computed αcs is close or larger to 1, and even larger than 1, in most of the Arctic borders. However, over the Central Arctic, αcs is smaller and around 0.5.

Figure  (centre and right-hand columns) demonstrates the results for an inversion with both αcs and αis as unknown parameters. The findings demonstrate the efficacy of the inverse approach in generating consistent results for the bias factor (αcs close to 1, expect over the Central Arctic as for the CS-IS-3p inversion, and αis close to 0). However, the negative values obtained for αis are difficult to explain, especially because of the low uncertainty bounds for the laser freeboards. In the same time, the inverse SIT and SD computed with this inversion scheme demonstrate a poor correlation with OIB data, emphasising the necessity for further refinement of this specific inversion. Further details on the inversions for the bias factors can be found in Figs. S5–S13 in the Supplement.

This study demonstrates the capability of the Bayesian approach to simultaneously retrieve SIT and SD from along-track freeboard measurements, without requiring data interpolation or coincident satellite orbits. The methodology successfully captures well-established geographical patterns in SIT and SD, including the presence of thicker ice north of Greenland and within the Canadian Archipelago, regions dominated by MYI.

The retrieved SIT and SD estimates show promising agreement with existing datasets, exhibiting consistency with AWI SIT and AMSR2 SD products. Furthermore, the results display encouraging correlations with validation data (OIB, BGEP-ULS, MOSAiC, IceBird). The inverse method's ability to capture interannual variability is particularly promising, as this is a known weakness of commonly used SIT products . However, a systematic bias is observed, with the inversion method underestimating SIT by 0.07 m while overestimating SD by 0.12 m relative to OIB. Several potential explanations for this bias are considered. One contributing factor may be the assumption of full radar penetration in the CryoSat-2 retrievals (αcs=1). Differentiating the freeboard equations (Eqs.  and ) with respect to αcs indicates that an increase in penetration factor (αcs>1) would lead to higher SIT estimates. However, given that penetration factors are typically constrained within the range [0,1], a value exceeding unity would suggest scattering of the Ku-band within the ice pack, which is not physically expected. Such an αcs>1 could also mean that the backscatter is predominantly off-nadir. Additionally, the observed 7 cm SIT underestimation could be linked to systematic biases in the input freeboard data or the smoothing effect of the inversion approach over the OIB comparison region (left map in Fig. ). This underestimation could also be a consequence of the choice of constant values for the sea ice density. Validating the inverse SIT against the BGEP-ULS moorings suggested that the inverse approach captured the seasonal and interannual variations in ice thickness quite well. This is particularly evident in the values of r close to 1 for the three locations.

Regarding SD retrieval, the overestimation of 0.12 m compared to OIB aligns with known biases in the OIB Quick-Look SD product, which underestimates SD by approximately 0.08 m . When evaluating SD inversion performance against MOSAiC (Winter 2019–2020) and IceBird (April 2019), different trends emerge. The CS-IS-2p inversion underestimates SD by 0.19 m compared to MOSAiC. This discrepancy compared to the OIB evaluation may be attributed to temporal and regional differences, as MOSAiC measurements were conducted during Winter 2019–2020 in the central Arctic, north of Svalbard, whereas OIB data correspond to April 2019 observations with measurements made in other regions of Central Arctic. It is to note that the MOSAiC SD data contain a representation error, the main MOSAiC floe being a piece of remnant ice that was chosen for survivability and contained thicker snow than the surrounding sea ice. For IceBird, where the measurements were also performed during April 2019, the inversion yields an SD overestimation of 0.03 m, suggesting a closer agreement with OIB than with MOSAiC. The differences in measurement techniques and geographical coverage likely explain the varying correlation statistics observed across validation datasets (OIB, MOSAiC and IceBird).

The inverse method introduced in the paper can provide probabilistic inversions even for seemingly unconstrained problems: this comes from the existence of prior information and the inherent correlation scales that are introduced in the Bayesian inversion approach. Thus, given an user-defined prior probability distribution P(m), we can also invert for the bias factor α in addition of SIT and SD. The inversion framework estimates the bias factors for each satellite. We show some of these inversion results in Fig.  and more details can be found in the Supplement (Figs. S5–S13). The retrieved values align well with expected ranges for radar (Ku-band) and laser measurements, despite the use of a weakly constrained prior. Further investigation is required to refine the spatial patterns observed in the retrieved bias factor maps and their relationship with SIT and SD results. Additionally, a more in-depth analysis of these results could help to explaining some of the surprising findings, as for example the negative values obtained for αis with CS-IS-4p. Adjustments to the prior distribution or modifications to the inversion framework may be necessary to ensure more physically consistent results.

In this study, we focused on 3 satellites (CS, IS, AK) but the method could be adapted to other radar/laser altimeters and other sensors. In addition, we have only used a combination of two satellites. However, the algorithm can be easily modified to include freeboards from more than two satellites. This requires the definition of one equation per satellite (for example, we could perform the inversion with Eqs. –). The advantage of conducting the inversion using all three equations and satellites is that it leverages a larger dataset, enhancing the robustness of the analysis (3 data sources instead of 2). This means that we could perform the inversion using a smaller number of days of data (e.g. 7 d) and still have the same number of freeboards as if we used two satellites but 15 d of data. This allowance for satellites fusion and the multiplicity of altimeters in orbit needs to be further investigated and is promising with a view to future missions.

In an inverse problem, we need to define a forward model, described here by Eqs. ()–(). A change in the forward modelling would have an impact on the inverse solution. Another idea would also be to invert for a mass (ρw-ρi)/(ρw)Hi instead of inverting for SIT (cf. Eq. ). This requires a choice on the prior for this new parameter. Additionally, in this study, a constant value for the snow density (ρs) was used. Following the methods of , this value could be varied temporally, to better represent the densification of snow throughout the winter season. Similarly joint inversions of the sea ice density is also possible in principle. Given the sea ice density as a free parameter, we could define a prior distribution that includes ice age and potentially other factors as weighting. Further works on this inverse method could aim at investigating these questions.

Furthermore, while our study focuses on the Arctic, this algorithm can also be applied to estimate ice and snow thickness in the Antarctic. Additionally, the inversion can be performed over specific polar regions to enable targeted studies of particular areas with large uncertainties identified by our probabilistic inversion method.

The estimated error associated to the observations is an input parameter of the inversion that can be modified. Here, we presented our result with an error computed using calculated freeboard uncertainties. Another approach to this specific question would be to set up a hyper-parameter λ to scale data errors and to use a hierarchical Bayes approach, where this parameter is treated as an unknown in the inversion : 11σ=λσdata where σ is the standard deviation in Eq. (), λ the unknown hierarchical parameter and σdata the error vector on the input prior to the inversion.