The bottom-most infinite half-space representing the ground was described using the following parameters: Tg, εfrozen, Hr,gs, Qr,gs, Nr,gsp (see Fig. ). The ground-snow interface reflectivity sgsp was obtained from Eqs. () and (). This study aimed to retrieve the ground surface temperature Tg by considering a fixed and constant ground permittivity in frozen conditions. Various models describe the ground permittivity at 1.4 GHz , but very few in the case of frozen ground . The permittivity of a frozen ground was set to εfrozen = 5.0 + 0.5 i, similar to past studies and SMOS algorithm . We considered the ground surface reflectivity as in Eq. () accounting for various effects including roughness using four parameters (Hr,gs, Qr,gs, Nr,gsH and Nr,gsV). The polarization mixing ratio Qr,gs as well as the angular effects parameters Nr,gsH and Nr,gsV) were set to zero, as suggested by several studies . The value of Hr,gs was optimized for all the sites using a range of 0 to 1 with 0.1 increments.

The layer accounting for the snow was defined by its effective temperature Ts, permittivity εs, layer internal transmissivity ts and reflectivity rs (Fig. ). According to and , dry snow is considered transparent at L-band, i.e., its internal transmissivity and reflectivity are ts=1 (no absorption) and rs=0 (no volume scattering), respectively. Consequently, our model became independent of Ts. However, showed that air–snow interface impacts on impedance matching can not be ignored, i.e., the snow surface reflectivity ssp∗≠0. We considered refraction (Eq. ) and reflection for a smooth air–snow interface (Eq. ). The dry snow permittivity was set to εs=1.53 according to Eq. (4) of for a mean snow density ρs = 300 kgm3, which corresponds to the high Arctic snowpack average density observed by and . We assume a snowpack with the same parameters above the ground and the ice-covered water bodies.

During winter, water bodies are fully covered by an ice layer with liquid water remaining below the ice layer . The ice layer was defined by its permittivity εi = 3.18 and considered transparent (internal transmissivity ti=1 and internal reflectivity ri=0). However, smooth surface refraction (Eq. ) and reflection sisp∗ (Eq. ) were taken into account at the ice–snow interface. Similarly to the ground layer, the liquid water layer was defined with Tw, εw, Hr,wi, Qr,wi and Nr,wip (Fig. ). The water temperature Tw was considered constant throughout winter and equal to 2 °C . We consider fresh water whose L-band permittivity εw was fixed to 86 + 13 i . The water–ice interface reflectivity swip was obtained from Eq. (), accounting for the water–ice interface heterogeneity: Qr,wi, Nr,wiH and Nr,wiV were set to 0 ; the value of Hr,wi was optimized for all the sites in a range of 0 to 2 with an iteration step of 0.1. The water body νwi accounted for the area percentage of the considered SMOS node covered by water bodies based on the water class from ESA CCI land cover (Table ).

Figure  depicts a schematic of the MEMs and Table  lists a summary of the input parameters. This study tested two configurations: one considering a homogeneous scene with only ground (hereafter named MEM_G) and one with a heterogeneous scene composed of ground and snow and ice covered water bodies (hereafter named MEM_G+WI).

In the MEM_G configuration, we retrieved Tg by testing Hr,gs values from 0 to 1 with 0.1 increments. Figure  shows the biases obtained with all tested Hr,gs and biases obtained with Tg-ERA5 for each site, with respect to Tg-insitu. For all sites, the bias changed in the negative direction with increasing Hr,gs. For sites with νwi≤0.04, the biases went from positive down to negative values with increasing Hr,gs, except for Awuna2 and Umiat whose biases remained positive. For sites with νwi≥0.20, the biases of numerous sites remained negative and went down close to -30 °C. This suggests that the water bodies strongly affect the Tg retrieval bias. That is why we optimized the value of Hr,gs only on sites less affected by water bodies. For sites with νwi≤0.04, the bias was minimized with Hr,gs = 0.8 (average = 0.2 °C, median = -0.2 °C, Q1 = -1.6 °C, Q3 = 0.8 °C, range = 2.4 °C). Surprisingly, the sites with the highest νwi (between 0.44 and 0.59) showed positive biases for some Hr,g.

The results in Sect.  showed that the Tg retrieval bias strongly depends on water fraction. The MEM_G+WI model accounted for the presence of frozen water bodies (i.e., νwi≠0) in the TB calculation (Fig. ). In this configuration, Tg was retrieved with different tested Hr,wi values from 0 to 1 with 0.1 increments; Hr,gs was set to 0.8 as shown in Sect. . For each site, Fig.  shows the biases obtained with various Hr,wi and compared with Tg-ERA5 bias with respect to Tg-insitu. The higher Hr,wi the more negative the bias, while the slope of the variations is linked to νwi. As expected, for sites with νwi≤0.04, the biases showed little variations for all Hr,wi. At Piksiksak (νwi=0.04) bias went from 5.2 °C (Hr,wi=0) down to 2.0 °C Hr,wi=1. For sites with νwi≥0.20, the biases varied considerably with increasing Hr,wi. For instance at Atqasuk (νwi=0.24), the bias decreased from 16.5 to -7.8 °C with Hr,wi=0 and Hr,wi=1. At East Teshekpuk (νwi=0.41), the bias for the Hr,wi extrema decreased from 37.0 to -16.7 °C. For the sites with the highest νwi (between 0.44 and 0.59), all the biases remained greater than 15 °C for the tested Hr,wi range. Consequently, we do not consider them in the following analysis of the water body correction method. For the sites with 0.20≤νwi≤0.41, the bias was minimized with Hr,wi = 0.7 (average = 0.7 °C, median = 0.2 °C, Q1 = -2.9 °C, Q3 = 2.8 °C, range = 5.7 °C).

The R, bias, and ubRMSD using MEM_G with Hr,gs = 0.8 and MEM_G+WI with Hr,wi = 1 were compared to Tg-ERA5 metrics in Fig. . For the sites with νwi≤0.04, when accounting for the water bodies with MEM_G+WI, we selected Hr,wi = 1 for the ice–water interface as it minimized the bias average of these sites (average = 0.6 °C). Each metric (in gray) is given with its confidence limits at 5 % (orange) and 95 % (blue). This representation enables us to show the dispersion of the metrics for all the considered sites. The R values of the retrieved Tg (median = 0.60 for both MEM_G and MEM_G+WI) were better than ERA5 (median = 0.51). Moreover, in the case of ERA5, the interquartile range was larger (Q1 = 0.33, Q3 = 0.55, range = 0.22) and the 5 % confidence limit dropped to negative values. All the biases are centered around zero (mean = 0.2 °C for MEM_G, 0.6 °C for MEM_G+WI and -0.8 °C for ERA5), and all the absolute biases were lower than 5 °C, except an outlier for ERA5 with a strong negative bias = -13.1 °C (Kelly Station, according to Fig. ). The ubRMSD from both inversions (median = 2.1 °C for both MEM_G and MEM_G+WI) were significantly smaller than those from ERA5 (median = 3.9 °C).

The overall R, bias and ubRMSD for MEM_G+WI with different Hr,wi are summarized in Fig.  with the corresponding MEM_G (with Hr,gs = 0.8) and ERA5 metrics. Similarly to Fig. , the boxes show the metric dispersion. The R values remained the same for all Hr,wi and equal to the R reached with MEM_G (median R = 0.21), but lower than ERA5 (median R = 0.62). The biases went more negative with increasing Hr,wi values. The bias was minimized for Hr,wi = 0.7 (Sect. ), with a median value (0.2 °C) which was closer to zero than the bias with MEM_G (median = -13.0 °C) and with ERA5 (median = 2.3 °C). Yet, for bias, the interquartile range for Hr,wi = 0.7 (Q1 = -2.9 °C, Q3 = 2.8 °C, range = 5.7 °C) remained much larger than ERA5 (Q1 = 0.8 °C, Q3 = 3.2 °C, range = 2.4 °C), which meant that the bias remained higher for some of the sites. A wider range (Q1 = 4.4 °C, Q3 = 6.6 °C, range = 2.2 °C) was also observed for the ubRMSD for all the Hr,wi and MEM_G (Q1 = 3.7 °C, Q3 = 5.3 °C, range = 1.6 °C) with respect to ERA5 (Q1 = 3.2 °C, Q3 = 3.5 °C, range = 0.2 °C).

We used a frozen ground permittivity of εfrozen = 5 + 0.5 i, as defined by and which was commonly used in various studies . The emission depth of L-band observations is usually associated with the first 5 cm of the ground . However, the emission depth varies with the ground state and texture, based on the ground attenuation constant α (δe=1/2α ), and consequently the ground complex dielectric constant εg. For εfrozen = 5.0 + 0.5 i, the calculation based on shows that the associated emission depth δe≃ 15 cm. When it comes to frozen ground, the effective depth is still not well defined and it becomes even more complex with a snow layer on top of the ground. estimated the emission depth of frozen ground at a maximum of 50 cm, but observed a TB saturation only when reaching a 30 cm frost depth. By computing metrics for Tg-insitu at all the available depths for the sites with νwi≤0.04, we found that R was better than ERA5 (median = 0.51) for depth down to 30 cm (median range from 0.57 to 0.74) (Fig. ). For in situ measurements down to 45 cm, the median absolute biases were smaller than 1.5 °C and the median ubRMSD were smaller than 2.5 °C. These results suggest that the sensitivity depth is in fact down to 50 cm or less. For deeper Tg-insitu, the correlation decreased to negative values (median R = -0.18 for depth = 120 cm). Note that for the period of this study (focused on Tg-insitu < -5 °C at 5 cm depth) the ground was fully frozen down to 50 cm for the 11 USGS sites that provide ground temperatures down to 120 cm. Due to potential shallow frozen soil, emissions from the underlying unfrozen soil should be taken into account in the early winter . As observed by , the observed signal can encompass a contribution of the boundary between frozen and unfrozen soil, which was not taken into account in our modeling.

Concerning the ground surface parameters, the commonly used H–Q–N empirical model has been tuned for soil moisture (SM) and vegetation optical depth (VOD) retrievals in many studies . Hence, its parametrization should be optimized for Tg retrievals in the Arctic environment. We found the optimized set of values Hr,gs=0.8, Qr,gs=0, Nr,gsH=0, =Nr,gsV=0 for the snow-ground interface, which is consistent with . This parametrization depends on the chosen ground permittivity value. According to the Fresnel reflection coefficients (Eq. ), increasing ground permittivity leads to a decrease of the emissivity. Using the H–Q–N model (Eq. ), increasing Hr,gs means an increase of the emissivity. Thus, the soil parametrization requires a joint optimization of εg and Hr,gs.

We optimized Hr,gs based on a permittivity of a frozen ground value of εfrozen = 5 + 0.5 i, but this value could be re-evaluated. The soil permittivity depends on the soil liquid water content and other characteristics (e.g., texture and bulk density). Based on a review of ground permittivity models (see Sect. ), we investigated other potential values for frozen soil permittivity. For a frozen ground (Tg < -5 °C), we assumed the water to be completely frozen and thus SM negligible, i.e., SM≃ 0 m3m-3 . Soil property information (clay fraction, sand fraction, soil organic content and bulk density) was extracted at each site location from the SoilGrids 250 m v2.0 database for the 0–5 cm soil layer (Table  in the appendices). The Soil Organic Carbon (SOC) content was very high at all the sites, as expected in the Arctic region, i.e 5 to 10 times higher than the global mean 40 gkg-1 (according to SoilGrid v2.0), and so was the bulk density. Dielectric constant models like the commonly used Mironov model do not use the SOC information to compute the permittivity. It was first designed considering SM and clay content . It was then further developed to use SM, Tsg (here set as -20 °C), and bulk density . was based on silt, clay, sand content and bulk density. defined a soil permittivity model tailored for high organic content soils, whereas Park's model was updated to consider soil organic content . The permittivities computed with these models for our sites are listed in Table  in the appendices. The obtained εfrozen real parts went from 1 to 4, while the imaginary parts ranged from 0 to 0.1. This comparison of various permittivity models that depend on soil texture showed that the permittivity variability for frozen arctic soils was low and made legitimate the use of a fixed value for the ground permittivity. However, the obtained permittivities were significantly lower than εfrozen = 5.0 + 0.5 i. This could be an evidence that SM > 0 m3m-3, even in frozen ground conditions (Tg < -5 °C). In addition, a permittivity equal to εfrozen = 5.0 + 0.5 i may result from a soil surface which was saturated with water at freezing time. But, as the Arctic soil shows high SOC and high bulk density (Table ), it may not satisfy this water saturation condition. For the imaginary part of the permittivity, showed a decrease with decreasing temperatures. In situ measurements of frozen ground permittivity could be valuable, simultaneously with tower-based radiometer observations in the Arctic tundra environment. Some probes seem efficient for this task, such as the one described in . Using a constant permittivity, calculated under the assumption of a homogeneous ground, is a practical solution for our model, as it reduces the number of free parameters and auxiliary data. However, dielectric mixing models enable to characterize heterogeneous materials and could better fit the Arctic soils local behavior.

Snow cover was present for all ground temperature observations used in Tg retrievals (i.e., the observed snow depth was above 10 cm), motivating the use of a snow layer in the MEM model. and suggested that snow emissions at L-band are related to the bottom 10 cm of the snow layer. The typical Arctic snow profile consists of a dense wind slab of high density (ρ≃ 300–400 kgm-3) but with a depth hoar underneath with lower density (ρ≃ 250 kgm-3) . However, the impact in terms of εs is low in the model of and hence, we used (εs(ρ = 300 kgm-3) ≃ 1.5 and εs(ρ = 250 kgm-3) ≃ 1.4) in the present study. In addition, our model does not account for the inclusion of ice crusts in the snowpack (e.g., after rain-on-snow (ROS) events) , nor low vegetation (e.g., shrubs or mosses) that could be observed in the tundra environment and might add complexity to the snowpack microwave emission. In fact, observed a decrease in horizontal polarization as the impact of ice crust formation, but underlined the difficulty of modeling and quantifying such event at L-band. As for the vegetation, multiple effects may mitigate the Tg. The presence of shrubs leads to a snow accumulation with a lower density than on herbaceous areas, which means more thermal insulation from the snowpack . However, also observed thermal exchanges between air and soil through the branches. As these effects are observed at local scale, it is difficult to model it at the SMOS scale (≃ 40 km). Various temporal matching between in situ measurements Tg-insitu and the retrieved Tg were tested (not shown): closest measurement to the satellite overpass time or daily maximum, minimum or mean. The metrics remained similar because we observed very few daily variations of Tg due to the snow insulation effect.

We used the water fraction for a 40 km resolution, but showed that a working area of approximately 123 km × 123 km is required to capture all the microwave signal that contributes to the SMOS observed BT. In fact, due to the multiple observation angles, the size and shape of the elliptical footprint vary. Using an average single round buffer for all the angles is a potential error source. For sites located near the coast, the nearby presence of the ocean is non-negligible. The considered water body areas may also vary over time. Dynamic water maps could improve the TB correction, even more if they provide us with information on the water state (e.g., frozen, snow and ice covered, etc.). The water bodies greatly affect the passive microwaves observations in summer in the Arctic area . Including water bodies in the MEM in winter is even more difficult because even if their surface is fully covered with ice, they may not be completely frozen in depth . We tested various modeling configurations for the water bodies (ice only, liquid water only, ice on top of liquid water with a smooth interface, not shown). None were fully successful, but introducing the Hr,wi parameter worked better. Indeed, it represents the surface roughness at the ice–water interface, which is not flat and significantly affects microwave observations. Yet, different models should be applied depending on water body characteristics (e.g., depth) as shallower lakes could freeze down to the bottom or sea ice may be formed on the coastal areas.

Figures  and  show that using a unique Hr,wi for all the sites does not ensure fully optimized Tg. To better understand the possible impact of snow and ice covered water bodies and model configuration, we present the Inigok site with a high water fraction of νwi=0.23. Figure  shows varying performance of the time series of Tg,MEMG, Tg,MEMG+WI and Tg-ERA5 compared to Tg-insitu. The Tg,MEMG time series showed a negative bias that was well corrected in the Tg,MEMG+WI time series. The Tg-ERA5 time series did not show a systematic bias with the Tg-insitu time series. However, the ERA5 dynamic was quite different from the in situ measurements. While Tg-insitu and Tg seemed linked to air temperature when it rises above -10 °C (e.g., in early 2014), but with a lag. This was not observed for Tg-ERA5, while it appeared in the retrieved Tg,MEMG and Tg,MEMG+WI. This could be linked to wet snow events, that increase the snowpack thermal conductivity and consequently the link between air temperatures and Tg. They also challenge the snowpack transparency hypothesis , that could no longer be valid, and could lead to an increase in the retrieved Tg values. Using MEM_G or MEM_G+WI did not affect the time series dynamic, as shown by the similar R and ubRMSD in Fig. . However, a strong interannual difference is observed. In winter 2014, we found R = 0.46 for Tg-ERA5, while we obtained R = 0.29 for both Tg,MEMG and Tg,MEMG+WI (see Fig.  in the appendices). By contrast, in winter 2019, a correlation of R = -0.03 is obtained with ERA5, while R = 0.61 using MEM_G or MEM_G+WI. These discrepancies between years suggest that ice conditions change throughout the years and further ice parametrization would be needed to obtain satisfactory Tg retrievals for scenes with high water body fractions.

The similarities of behaviors of in situ and retrieved time series also varied during a single season. Figure  focuses on the retrievals using MEM_G+WI with different Hr,wi at Inigok from December 2016 to May 2017. This period corresponds to the winter season with best correlation according to Fig.  (R = 0.74). The retrieved Tg and Tg-insitu were averaged per month and plotted with their standard deviation. Each graph of Fig.  corresponds to a different Hr,wi used in the modeling. The difference between the monthly averaged Tg and the monthly averaged Tg-insitu is denoted ΔT‾. December (ΔT‾ = -0.5 °C), January (ΔT‾ = -1.8 °C) and February (ΔT‾ = 0 °C) Tg are in good agreement with Tg-insitu for Hr,wi = 0.7. However, in March, Hr,wi = 0.8 provides better results (ΔT‾ = -0.2 °C). The best Hr,wi is 0.9 for April (ΔT‾ = -0.5 °C) and May (ΔT‾ = -0.5 °C). This suggests a possible evolution of the ice conditions throughout the winter that affects the ice–water surface rugosity and Tg inversion. This is in agreement with synthetic aperture radar (SAR) studies which take into account roughness parameters over lakes to represent the impact of the roughness at the water–ice interface on microwave signal. linked the water–ice interface roughness with the growth of tubular bubbles during ice formation, leading to higher roughness. Slushing water in ice cracks at the end of the freezing season induces more complexity than our three horizontal layer modeling for water bodies . Ground-based radiometric observations would be highly beneficial to better understand the seasonal effect of water–ice interface roughness on TB in Arctic regions. Such observations may also help the development of a more complex model to better describe the L-band emissions of the circumarctic lakes and their variations through the seasons.