*5.1. L-band Response to SIT*

Using the ground truth as described in the previous section allows establishing the relation between SMOS and SMAP L-band First Stokes parameter, the PD and the SIT parameter. Figures 3 and 4 show some L-band parameters as functions of the ground truth, which is filtered with a 95% SIC threshold in all cases. It should be noticed that recent efforts for characterizing the L-band response to ice thickness were hindered by the distribution and characteristics of the ground truth used [2,14,22]. On the one hand, the spatial distribution of the ground truth is limited to those regions sampled by ships. On the other hand, campaigns are normally carried out at the end of winter or in spring, therefore a thickness seasonal bias can be expected [14,22]. The author in [2] uses the Ocean ReAnalysis System 5 (ORAS5) modeled ice thickness for their comparison against SMOS SIT. They notice an overestimation of the ORAS5 with respect to SMOS SIT. This is attributed to the simplified representation of thin ice by the model (see [2] for further details).

Figure 3 displays the relationship between the First Stokes and PD parameters and the daily ULS ground truth. The First Stokes parameter shows a large dispersion for thin ice which decreases with increasing ice thickness. Such a decrease in dispersion reveals a saturation of the response of TB to large SIT values (i.e., above 0.6 m). The signal saturates at a value of ∼240 Kelvins. The standard deviation of the TB at large SIT values is in fact an estimation of the radiometric error of the observation system, because at such SIT values, the signal no longer responds to the geophysical parameters and the variability can be mainly attributed to the radiometric noise.

A theoretical relationship between TB and ice thickness based in the incoherent Burke model can be found in [10] and [12]. We note here that larger dispersion is seen for very thin ice while other comparisons between these parameters show that the largest dispersion is found at around 20 cm ice thickness [12,13]. Probably, several physical mechanisms not considered in the theoretical model can be affecting the TB response to thin ice, such as the impact of ice surface temperature and ice surface roughness variations, among others.

The relationship between SMOS PD and the ground truth can be found in Figure 3b. The PD presents a constant but larger standard deviation (with respect to the case of the First Stokes parameter) for ice thinner than 0.6 m, and for thicker ice the PD saturates at a value slightly below ∼30 Kelvins. The PD intersects with the ordinate axis at ∼70–80 Kelvins. In contrast, [13] indicates an intersection at around 50 Kelvins. We attribute this difference either to a wrongly classified thickness with the CFDD model by [13] or to the inclusion of pixels with different SICs in their analysis.

**Figure 3.** (**a**) Soil Moisture and Ocean Salinity (SMOS) First Stokes parameter calculated for 0–40◦ incidence angles against ULS derived SIT. (**b**) SMOS Polarization Difference (PD) at 50◦ incidence angle. The blue line represents the average and the red lines plus and minus one standard deviation.

In order to confirm the L-band response to SIT, as shown by SMOS data (see Figure 3), we perform the same analysis but with SMAP TB. The comparison between SMOS and SMAP TB responses to ice thickness is shown in Figure 4a. The L-band signal response from both sensors presents very similar patterns, including larger dispersion for thin ice and saturation for thick ice. Nonetheless, SMAP TBs are slightly lower for thick ice than those of SMAP, something that can be a consequence of SMOS residual biases [41].

We estimated the standard deviation of the ice thickness retrieval by propagating the TB error to the SIT according to Equation (2). Surprisingly, the standard deviation of ice thickness increases with increasing thickness, as opposed to what happens with the TB. This is due to the formulae used in the thickness retrieval and the larger sensitivity of TB to SIT for thin ice.

The limit of validity of the empirical retrieval is located around the 0.5 m SIT [11,13]. In order to further constrain this limit, we analyzed the factor within the exponential term in Equation (2), *γd*, which corresponds to a normalized thickness. The scatter plot of Figure 4b evidences the relation between ground truth and the normalized thickness. As expected, we see an almost linear relation for ice thinner than 0.6 m. Besides, the dispersion of the parameter increases with ice thickness and finally, for thicker ice, the sensitivity of TB to SIT is completely lost. The scatter points in that region are distributed around a horizontal line (hence, the attenuation factor *γ* is meaningless in that region). Obviously, this limitation applies only to the pure empirical retrieval. The thickness retrieval can be extended to thicker ice using a radiative transfer model and assuming a lognormal distribution [12]. However, this approach can be sensitive to the auxiliary fields used, such as SIC, temperature and salinity, whose uncertainties can be difficult to quantify [2].

**Figure 4.** (**a**) SMOS and Soil Moisture Active Passive (SMAP) Brightness Temperatures (TB) First Stokes response to ULS derived ice thickness at 40◦ incidence angle. (**b**) Normalized thickness from the empirical retrieval against ground truth (see Equation (2)).

#### *5.2. Assessment of UH SIT*

The direct comparison between the UH ice thickness and the ground truth reveals that the product underestimates thin sea ice and overestimates thickness larger than 0.3 m (see Figure 5a). This underestimation for thin ice and its relationship with SIC has been widely reported in the literature [2,12]. Considering that CFDD is already overestimating the ULS ground truth for thick ice (see Figure 2), a larger thickness overestimation of the UH product with respect to ULS is expected for thick ice (see Figure 5b and Section 4). The ice thickness standard deviation increases with thickness (see Figure 5b). The observed decrease in the standard deviation for large thickness is due to the decrease of the scatter region analyzed in the calculation.

The CFDD thickness and the ULS scatter plots against the UH SIT reveal with their similarity that the proposed homogeneity hypothesis works fine during the freeze-up periods. This situation provides a solid ground for extending the ULS ground truth with the CFDD ice thickness.

The Probability Density Functions (PDFs) for the freeze-up period for SIC over 90% reveals that the modes values are of similar value for UH and CFDD SITs (see Figure 5). Moreover, the shape of both PDFs are similar. However, UH PDF shows a smoother shape, with gentle slopes and a larger amount of occurrences for thickness above 0.5 m (see Figure 5d). Interestingly, a different PDF is observed for UH SIT during October, at the beginning of the freeze-up period (see Figure 5c). In this case, the modes of both distributions are coincidental as was the case for the whole-winter period. However, the shapes of the distributions are markedly different, even when considering that the CFDD model SIT is around observable values for the SMOS-based L-band SIT retrieval valid range (i.e., from 0.1 to 0.6 m), indicating poor-quality SMOS SIT retrievals, even at these favorable conditions (i.e., for high SIC conditions). A possible explanation for this change on the distribution shape could be the application of the lognormal distribution to the largest thickness values [12]. This change in the shape of the distribution is no longer visible for lower SICs.

The analysis of the distributions with lower SIC values (see Figure 6) indicates that the UH product bias increases with decreasing SIC. Furthermore, the UH SIT distribution is more peaky with decreasing SIC. The increasing negative bias of the product with decreasing SIC has already been reported by [12]. The peaky behavior of the distribution might reveal that the retrieval algorithm is saturated for low SIC values. On the other hand, the thickness distribution for SICs between 70% and 90%, while having a bias, presents a very similar shape to that of the CFDD SIT (see Figure 6c). The latter facilitates a possible readjustment of the product using a simple shift. The distribution of SIT in cases with lower SICs has both a large bias and peaks at lower SIT values (see Figure 6d). In this case, a readjustment would require both a shift and a rescaling provided that the ice thickness signal is still present in the measured TB and is not contaminated by the ocean signal.

**Figure 5.** (**a**) Comparison between the ULS ground truth and the University of Hamburg (UH) ice thickness. Soil Moisture and Ocean Salinity (SMOS) thickness uncertainty and saturation ratio are always below 1 m and 90% respectively. The red line represents the identity line. Ground truth was acquired for the freeze-up periods between 2010 and 2017. (**b**) Comparison between the modeled CFDD SIT and UH ice thickness for the freeze-up periods between 2010 and 2017. SMOS thickness uncertainty and saturation ratio are always below 1 m and 90% respectively. The mean and standard deviations are depicted with a blue and red lines respectively. (**c**) Probability Density Functions (PDFs) of CFDD and UH SIT during October between the years 2010 and 2017. (**d**) PDFs of CFDD and UH SIT for case during the freeze-up periods between 2010 and 2017 with SIC over 90% in all cases.

The UH SIT product well reproduces the distribution for the whole freeze-up period for high SIC values (see Figure 5d). However, we recognize the inability of the UH algorithm to represent the real distribution of SITs for high SIC (i.e., above 90%) during specific periods of the year (e.g., October). However, this distribution facilitate that UH product better represents the end of winter fully developed ice for high SICs. During the same period of the year, when lower SICs are considered, UH product departs from CFDD ice thickness. Moreover, we see both the increasing underestimation and kurtosis (peakiness) of the distributions with decreasing SICs (see Figure 6). The satellite sampling which permits a daily coverage for the whole Arctic region [15] contrasts with the limitations of the thickness retrieval for specific periods of the year (see Figure 5c). The inability of the UH product to capture the temporal trend of the ice thickness at high SICs is a significant concern since the satellite temporal resolution is left unexploited (i.e., the retrievals do not properly reflect the real ice thickness distribution for a given time). The latter leaves the door open for future improvements with a temporally enhanced thickness retrieval.

**Figure 6.** Comparison between the modeled CFDD SIT and the UH ice thickness. (**a**) SIC is between 70% and 90%. (**b**) SIC is between 30% and 50%. SMOS thickness uncertainty and saturation ratio are always below 1 m and 90% respectively. The mean and standard deviations are depicted with a blue and red lines respectively. (**c**) PDFs of CFDD and UH SIT for case a. (**d**) PDFs of CFDD and UH SIT for case b.

#### *5.3. Assessment of UB SIT*

The following comparisons were carried out filtering the UB SIT based on the UH restrictions of saturation and uncertainty. The methodology makes the comparison between products and ground truth more consistent and reliable, because we are comparing the same datasets. The analyzed UB SIT cells are sometimes saturated and sometimes fall within the observable thickness range, when saturated they are also filtered (i.e., above 0.5 m). The scatter plot between the buoy ULS thickness and the UB SIT indicates that the UB product underestimates SIT for the whole freeze-up period (see Figure 7b). The UB thickness shows a large amount of occurrences near the saturation limit. We interpret this as an indication that the algorithm is trying to represent in the range of half a meter CFDD thickness values that are clearly over their own retrieval limit (see Figure 7d). We also observe that ice thickness standard deviation increases with thickness (see Figure 7b). In contrast with the UH product, only a slight SIT underestimation is present in the UB product for thickness below 0.4 m, indicating a relatively higher accuracy of the latter within the mentioned thickness range.

Here again, as what happened with the UH SIT, the CFDD thickness and the ULS scatter plots' similarity supports the proposed homogeneity hypothesis. Therefore, we have further support for using the CFDD SIT as an Arctic-wide proxy of SIT ground truth.

Indeed, the UB SIT marginal distribution resembles that of CFDD, for the month of October and SIC above 90% (see Figure 7c). The CFDD thickness values are within the observable L-band thickness range (0–0.5 m), and therefore they are expected to be resolved as they do (as opposed to UH product, see Section 5.2). The good fit comes as no surprise since the UB product is based on the CFDD model

and the NCEP/NCAR surface reanalysis auxiliary data [13]. The observed good fit during October is no longer seen for later periods (i.e., from November to January) when UB is not able to resolve CFDD thickness values lying over the retrieval limit (see Figure 7d).

The UB SIT behavior when considering decreasing SIC values is similar to that of the UH product, i.e., an increasing bias and a more peaky distribution on the thinner thickness range (see Figure 8). The distributions reveal that the UB product is wrongly classifying thickness above 0.5 m, by setting them to a fixed value of 0.5 m. The wrongly classified CFDD thickness, calculated after filtering the saturated UB SIT, amount to 50% of the total (this percentage is calculated for the whole freeze-up period, considering values of SIC above 90%). The UB thickness products caps the retrievals at 0.5 m [13] taking into account the limits of the empirical retrieval. By doing so, they are indirectly forcing an artificial saturation visible in all comparisons. On the positive side, we see the related bias of the acquisition for lower SICs, a similar behavior seen for both UB (see Figure 8c,d) and UH (Figure 6a,d) SIT, although the former shows a slightly smaller bias (with respect to ground truth) than the latter.

**Figure 7.** (**a**) Comparison between the ground truth measured by the upward looking sonar and the University of Bremen (UB) ice thickness. SMOS thickness uncertainty and saturation ratio are always below 1 m and 90% respectively. The red line represents the identity line. Ground truth was acquired for the freeze-up periods between 2010 and 2017. (**b**) Comparison between the modeled CFDD SIT and the UB ice thickness for the freeze-up periods between 2010 and 2017. SMOS thickness uncertainty and saturation ratio are always below 1 m and 90% respectively. The mean and standard deviations are depicted with a blue and red lines respectively. (**c**) PDFs of CFDD and UB SIT during October between the years 2010 and 2017. (**d**) PDFs of CFDD and UB SIT for the freeze-up periods between 2010 and 2017 with SIC over 90% in all cases.
