Since the launch of SMOS in 2009, ESA has been generating brightness temperature full-polarization data products from it. In this study, we focus on the official SMOS Level 1B (L1B) product version 504 data north of 60∘ N from 2014 to estimate SIC. The L1B data contain the Fourier components of TB at the antenna reference frame , from which one can obtain temporal snapshots of the spatial distribution of TB (i.e. an interferometric TB image) by performing an inverse Fourier transform. The TB data are geo-referenced at an Equal-Area Scalable Earth (EASE) Northern Hemisphere grid of 25 km on the side. The radiometric accuracy of individual TB observations from SMOS is ∼ 2 K at boresight, and it increases on the extended alias-free field of view . Proceeding from L1B data, though computationally more demanding than the more traditional L1C data products, has several benefits. For example, it allows one to change the antenna grid from the operational size of 128×128 pixels to 64×64 pixels. As shown by , the smaller grid is optimal in that it helps mitigate some of the spatial correlations between measurements that are present in the larger grid.

We correct TB for a number of standard contributions such as geomagnetic and ionospheric rotation and atmospheric attenuation . The galactic reflection is not significant at high latitudes, and no correction was applied. We then filter out outliers (defined as those estimates that deviate by more than 3σ from the mean value, where σ is the radiometric accuracy at the given point in the antenna plane). We also filter out TB observations in regions of the field of view that are known to have low accuracy due to aliasing , Sun reflections, and Sun tails.

To lower the noise level, we averaged TB measurements from both ascending and descending orbits over periods of 3 days, which thus define the time resolution of our SIC maps. We also averaged acquisitions in incidence angle binnings of 2∘. Since some incidence angles could be missing due to the SMOS acquisition feature and interferences, we use a cubic polynomial fit to interpolate TB measurements to have the full range of incidence angles in each grid position.

We use SIC maps from the database of the Ocean and Sea Ice Satellite Application Facility (OSI SAF product version OSI-401a) of the European Organisation for the Exploitation of Meteorological Satellites (EUMETSAT) for comparison with the products we are obtaining.

These are computed from brightness temperature observations from SSMIS at 19 and 37 GHz, are corrected for atmospheric effects using forecasts from the European Centre for Medium-Range Weather Forecasts (ECMWF), use monthly dynamic tie points, are available on a polar stereographic 10 km grid for both polar hemispheres, and include SIC uncertainty estimates . In this study, we used daily SIC maps in the Arctic Ocean from the OSI SAF Northern Hemisphere products of the year 2014.

We also used SIC estimates from ice charts generated from various sensors by the National Ice Center to identify regions of interests to compute the 100 % ice tie points.

It is rarely possible to obtain the ancillary geophysical data such as sea ice temperature, salinity, and ice thickness that are required to estimate brightness temperature from a microwave-emission model. Therefore, making assumptions and approximations becomes critically important. It is possible, however, to define a number of indices resulting from a combination of brightness temperature observations that are less sensitive to the unknown physical parameters. For example, estimates of soil moisture or sea ice concentration from radiometric measurements are often derived by combining TB measurements obtained from different polarizations, frequencies, and angles of incidence .

Hereafter, we use two indices, the polarization difference (PD) index and the angular difference (AD) index. The PD index is defined as the difference between TB measurements obtained at vertical TBV and horizontal TBH polarizations as PD(θ)=TBV(θ)-TBH(θ).

The AD index is defined as the difference between two vertical polarization TB measurements obtained at two different angles of incidence as AD(θ)=TBVθ+Δθ-TBV(θ).

Figures  and show the variation of PD and AD for the thick-ice model with angle of incidence, respectively. In defining AD, we use vertical rather than horizontal polarization because identification of the three media is facilitated by the larger dynamic range and non-crossing signatures of vertical polarization (Fig. ). We choose Δθ= 35∘ angle difference because this value represents a good compromise between sensitivity of the index and radiometric accuracy in the case of SMOS and, importantly, is also well supported by the wide range of satellite viewing angles that characterizes SMOS.

Although the polarization ratio (PR) is also a commonly used index, we have chosen PD after verifying that its dynamic range is larger than that of PR and suspecting that PD would yield higher-accuracy estimates given the SMOS error budget.

Tie points are widely used for retrieving SIC with higher-frequency radiometers, as well as in other fields such as photogrammetry e.g.. In this study, we use tie points as the typical TB values for 100 and 0 % concentrations, which permit us to compute the sea ice concentration. Tie points can therefore be viewed as SIC calibration points because their expected radiation can be unambiguously determined.

Figure  shows theoretical PD tie point values for open water and sea ice, as well as ice with a snow layer. The values for an angle of incidence of 50∘ are marked by solid red circles. This angle represents a good compromise in PD contrast between the two media and SMOS accuracy . The two bounding values are 62.9 K for seawater and 26.8 K for ice with snow cover (Table ). The large difference between tie point values suggests that it is possible to estimate SIC at the L band.

Figure  shows theoretical AD tie point values for difference in incidence angle Δθ= 35∘ and angles of incidence up to θ= 30∘, which, per Eq. (), represents the TBV difference between θ= 60∘ and θ=25∘. The values for an angle of incidence of 25∘ are marked by solid red circles, for which the tie points are 51.8 K for seawater and 8.6 K for ice with snow cover (Table ). Hereafter, PD and AD are evaluated at the incidence angles of θ= 50∘ and θ= 25∘, respectively.

Figure  shows that TB at nadir increases non-linearly as a function of ice thickness up to the saturation value of ∼ 250 K, which is reached when ice becomes about 70 cm thick. Notice that TB estimates start at an ice thickness of 5 cm because there is a discontinuity in the Burke model as the thickness of ice tends to zero e.g.. Compared with TB, the total variation of both AD and PD with ice thickness is significantly smaller, and they are therefore better suited to estimate sea ice concentration.

In this section, we calculate the sensitivity of SIC estimates to changes in surface emissivity due to variations in the physical properties of sea ice (i.e. salinity, temperature, and thickness). We work with estimated SIC derived from the three indices TB, PD, and AD. This is done following a standard error propagation method (as also used in ). It is important to determine how changes in ice conditions affect SIC estimates through those three indices to try to minimize SIC errors obtained using SMOS.

Table  lists the sensitivities, according to our theoretical model, of the indices I (I=TB, PD, and AD) to the geophysical variables of ice and seawater – physical temperature (i.e. δI/δT), salinity (δI/δS), and thickness (δI/δd) – evaluated within the ranges of Twater=[2,15], Swater=[10,38], Tice= [-20, -5], and Sice=[2,12]. It should be noted that those sensitivities are calculated using the model and the nominal Arctic temperature and salinity values defined in Sect. . In order to assess which index is least sensitive to changes in a given geophysical variable, we calculate absolute sensitivities, defined as the sensitivities multiplied by the dynamic range of the measurements.

Knowing the value of the tie points of sea ice (SIC = 100 %) and seawater (SIC = 0 %), one can compute the average slopes of the SIC estimates to their corresponding parameters TB, PD, and AD (i.e. δSIC/δTB, δSIC/δPD, and δSIC/δAD). From data in Table , we obtain the average slopes as δSIC/δTB=0.65, δSIC/δPD=2.32, and δSIC/δAD=2.77. These slopes can be used to propagate TB, AD, and PD errors to errors in the SIC estimates.

We assume reasonable values for the variability of the physical parameters on which our emissivity model depends – namely T, S, and d of ice (generically denoted by g) – as follows: ΔT=5  K, ΔS=4 gkg-1, and Δd=30 cm. Using the values in Table  and the calculated average slopes, one can finally compute the errors in SIC estimates associated with the geophysical variability of g when the index I is used to evaluate SIC as ΔSICg=δSICδI⋅δIδg⋅Δg.

To evaluate the final impact of geophysical variability on the SIC evaluation using the index I, we compute the root sum square (RSS) of the SIC uncertainties due to the geophysical parameters (Table ). The table shows that AD is the most robust index to retrieve SIC, slightly better than PD, and significantly better than TB. Because TB is theoretically more sensitive to thin ice than the other two indices, one can expect that the use of TB to retrieve SIC would result in larger SIC errors. Moreover, the uncertainty distribution of TB is too broad, especially due to thickness, and thus less adequate to fulfil the statistical hypotheses used to derive SIC. Despite the uncertainties in the theoretical physical model of ice, we consider the differences significant enough to focus on inversion algorithms using the PD and AD indices, and not on TB.

Following the theoretical analysis, we now turn to evaluating its performance empirically. We therefore select several regions of interest in the Arctic Ocean where SIC has been determined to be either 0 or 100 % by other sensors and methods. To identify such regions, we use SIC maps from OSI SAF and from the National Ice Center. In particular, we selected the open-seawater region within latitudes 55–70∘ N and longitudes 20∘ W and 25∘ E, which comprises more than 2000 pixels in a typical SMOS image. For sea ice, we selected the MY ice region within latitudes 78–83∘ N (the northernmost latitude observable by SMOS) and longitudes 75–150∘ W, which comprises about 1000 pixels per SMOS image. We expect some level of uncertainty associated with the selection of the region to compute the 100 % tie points for summer periods stemming from known errors in the summer SIC products by OSI SAF .

We calculated SMOS brightness temperatures of these target regions to evaluate their potential as empirical tie points for seawater and sea ice. The temporal variation, in 2014, of the spatially averaged (median) TB at nadir of the two geographic regions is shown in Fig. . The values are consistent with the modelled values in Table . For the seawater region, the figure shows that the brightness temperature is constant, at about 99 K, to within ∼ 2.5 K (1σ standard deviation) throughout the year. For the ice region, TB is also stable during the non-summer months, but it drops by about 20 K during the summer season due to changes in surface emissivity associated with snow and ice melt and concurrent formation of meltwater ponds. The factor-of-2 increase in formal error in summer relative to winter is also an indication of increased radiometric variability in surface conditions (as shown in Table ).

Figure  shows that the temporal radiometric stability of the seawater region during 2014, and that of sea ice during the non-summer months, is also reflected in the AD and PD indices, as one would expect. This suggests that a different set of tie points during winter and summer periods could be beneficial for the quality of the SIC retrievals. On the other hand, the AD and PD tie point values are very stable during winter and spring (November to June), indicating that values are robust to variations in physical temperature and that it may not be necessary to compute tie point values often (daily or monthly), as done with the OSI SAF product.

Figure  shows a 2-D scatter plot of AD and PD indices for the two regions defined above during March (winter tie point) and July (summer tie point) 2014. The index values associated with seawater and with ice group form two well-differentiated clusters, which implies that the two types of regions can be clearly segregated using these indices. This is also true for the summer tie points even though in this case the dispersion is larger and values are closer to sea tie points, as expected following Fig. .

The modelled (with snow and without) and observed TB, AD, and PD tie point values for winter and summer 2014, as well as the standard deviation (σ) of the measurements, are listed in Table . It is encouraging that most of the values are in agreement at about 2σ despite underlying model assumptions such as uniform sea ice temperature and specular ocean surface. Another important result is that the observed SMOS data are closer to the model when snow is considered.

The brightness temperature of mixed pixels – that is, ocean pixels partially covered by sea ice – can be expressed as a linear combination of the brightness temperature of ice and seawater weighted by the percentage of each surface type e.g.: TBmixed=CTBice+(1-C)TBwater, where C is the fraction of ice present in a pixel, with C=1 corresponding to 100 % of ice and C=0 to 0 % of ice, or equivalently 100 % of seawater. Since AD and PD (Eqs. –) depend linearly on brightness temperature, Eq. () can be used to express both AD and PD.

There are several possible strategies to estimate sea ice concentration at a given pixel from the AD and PD values measured at that pixel. The simplest approach is to consider that the values of the tie points are good representatives of the values of AD and PD at the respective medium, i.e. seawater and sea ice, such that AD≈CADice+(1-C)ADwater,PD≈CPDice+(1-C)PDwater.

Concentration C can thus be retrieved from the value of either AD or PD by inverting the associated linear equation. In general, C can also be evaluated simultaneously with the AD and the PD observations by averaging the values obtained from both indices as C=12AD-ADwaterADice-ADwater+PD-PDwaterPDice-PDwater.

This is known as the linear estimation of SIC. However, this approach might be too simple, as the values of AD and PD on ice and seawater can have some non-negligible dispersion due to geophysical conditions and to radiometric noise.

In this paper, a new inversion algorithm to estimate C is presented, which considers that AD and PD have known distributions, and by combining the observations it is possible to infer the value of C that is statistically more probable.

The distributions of the SMOS AD and PD are unimodal and symmetric (not shown), thus allowing us to approximate them by Gaussians and considering the pure ice and pure sea measurements as independent. Therefore we can easily use a MLE approach. The MLE has many optimal properties in statistical inference such as e.g. sufficiency (the complete information about the parameter of interest is contained in the MLE estimator), consistency (the true value of the parameter that generated the data is recovered asymptotically, i.e. for sufficiently large samples), efficiency (asymptotically, it has the lowest-possible variance among all possible parameter estimates), and parameterization invariance (same MLE solution obtained independent of the parametrization used).

Assuming the linearity superposition of indices (Eq. ), it follows that the distributions of AD and PD (fAD, fPD) in a general ocean pixel can be expressed as fAD∼NCAD‾ice+(1-C)AD‾water,C2σADice2+(1-C)2σADwater2,fPD∼NCPD‾ice+(1-C)PD‾water,C2σPDice2+(1-C)2σPDwater2, where the bar over the AD and PD indices refers to their mean values, the subindex identifies the medium, and σ is the associated standard deviation for each index and media. To obtain the mean and standard deviation values, we used the SMOS measurements at the regions for generating tie points and periods discussed in Sect. . The symbol N means normal probability density function.

As a first approximation, we have considered AD and PD two independent variables. It thus follows that the likelihood function L is equal to the product of their distributions or, equivalently and conveniently, to their sum (recall that the likelihood is the logarithm of the probability density function): l^=ln⁡(L)=ln⁡(fAD)+ln⁡(fPD). The MLE of SIC is the value of C that maximizes the likelihood function l^.

We have calculated AD and PD values from SMOS brightness temperature and used the MLE approach to obtain SIC estimates over the Arctic Ocean for the year 2014. We have estimated SIC using different tie points, characterized by their central value and dispersion. For seawater, we have used a single year-round median value and the associated standard deviation for each index. For ice tie points, we have used two sets of values, as suggested by the results in Fig. . For the first set, we have computed for all years the median of the tie points between December and May (Table ), i.e. the winter–spring months, when Arctic sea ice extent is close to its annual maximum. For the second set, we have used those same winter–spring values for the months of October through May but the average of the summer values for the months between June and September (Table ). We have used neither the October nor the November data to compute ice tie point values because these are months of maximum extension of thin ice, and underlying emission through thin ice could cause some errors on the SIC estimates (Fig.  and Table ).

The root-mean-square (rms) error of SIC retrievals relative to OSI SAF over the Arctic Ocean is shown in Fig. . Four types of retrievals and two sets of tie points are compared. Introducing a specific set of summer tie points (black solid line) reduces the rms error with respect to using only one unique tie point for the whole year (black dotted line). The rms reduction is about 24 and 12 % in July and August, respectively, and to smaller degree in June and September. Therefore, we will hereafter use a different set of tie point values in summer and winter.

Furthermore, using the set of summer–winter tie points, results from four types of inversions that stem from combinations of linear and MLE method and indices are compared in Fig. . The lowest rms values through all months in 2014 but January are obtained with the MLE inversion algorithm and the AD index alone. The evolution throughout the year of the rms obtained with the linear retrieval method is similar in the case of the MLE method, albeit with a ∼ 5–10 % increased noise level. Larger rms values and increased temporal variability are observed when the PD index is also used. The rms error of all retrievals is largest in fall, in particular if the PD index is used. Those are months of ice formation; therefore vast regions become covered with frazil ice, nilas, and thin young ice, following the minimum ice extension of September. All methods converge to similar results in September, since this period is the one with minimum ice extension and minimum thin ice is expected (thus resulting in very small difference between using AD or AD and PD methods).

The spatial variation of the difference in MLE SIC retrievals when using only the AD index and when using the AD and PD indices for the period 2–5 November 2014 is shown in Fig. . As expected, the largest differences are associated with regions of thin ice formation, in particular in the Laptev Sea, Kara Sea, and along the edge of the ice pack both in the western Arctic and in the Atlantic sector. Together, the spatio-temporal snapshots in Figs.  and highlight the sensitivity of PD to the presence of thin ice, which naturally leads to an increase of the retrieval error when PD is used. This conclusion is not fully consistent with the analysis done using the models in Sect.  on the dependence of the indices (TB, PD, AD) on ice thickness. Table  shows that, theoretically, PD is slightly less sensitive to thin ice than AD. However, the AD index is the least sensitive (lowest RSS) to variations of all the analysed variables. Therefore, we will hereafter use the AD index, summer–winter tie point values, and an MLE-based estimator for SIC retrievals.

We have evaluated the mutual consistency of the SMOS SIC retrievals, and in the process we have determined which is the approach that leads to the minimum error in the retrieval of SIC. We now evaluate the accuracy of those retrievals. Although a representative (in the space–time domain) ground-truth dataset that allows us to assess the accuracy of SMOS retrievals does not exist, the SIC estimates from OSI SAF are a good option for cross-check. They are independent from SMOS, the spatio-temporal sampling and resolution of their products are commensurate with those of SMOS, and their error budget is available.

The spatial distribution of SMOS SIC in the Arctic Ocean has been estimated from SMOS data for the 3-day period 2–5 March 2014, and it has been compared with the OSI SAF SIC product on 4 March 2014. The largest differences between both algorithms are located at the margins of the sea ice cover, where thinner ice can be expected (see Fig. ). March is the month of maximum sea ice extent, but the results for other winter months are similar.

On the other hand, November is the month of maximum extension of thin young ice, especially through the Beaufort Sea since ice in the Laptev and Kara seas remains thin during winter . Significantly larger differences between SMOS and OSI SAF products are now observed over a much wider area of the Arctic Ocean including the Barents, Kara, Laptev, East Siberian, and Beaufort seas (Fig. ).

The brightness temperature measured by a passive microwave radiometer increases with sea ice thickness up to a saturation value. Such an increase is more gradual for low frequencies and horizontal polarization e.g.. At the SMOS L band, the increase of emissivity with ice thickness reaches saturation for an ice thickness that is about 60 cm, depending on ice salinity and temperature , whereas at the OSI SAF frequencies (19 and 37 GHz) it is only a few centimetres . For example, for pixels that are 100 % covered by thin ice of, say, 25 cm thickness, the AD and PD values for those pixels will be slightly different than the tie point value of ice because the value of ice tie points was computed from thick MY ice (see Fig. ) for model analysis. This contrast leads to a difference in classification of such pixels, which will be considered mixtures of water and ice in the case of SMOS and as 100 % ice with OSI SAF. In other words, the estimation of SIC of a sea covered by frazil ice and nilas will be higher for OSI SAF than for SMOS.

To further analyse this classification difference, we have calculated the probabilities of SMOS SIC conditioned by values of OSI SAF SIC using a full year, 2014, of Arctic-wide estimates. The probability of estimating a SIC value with SMOS that is less than or equal to 5 % when the estimated OSI SAF SIC is 0 % is shown in Fig.  (red line). As expected, the conditioned probability is very high throughout the year. This implies that both products have a similar ability to detect (close to) 100 % ocean pixels. This implies that the probability of having high SMOS SIC values when OSI SAF is low is almost zero, which also means that the rate of triggering false alarms on ice detection with SMOS is low.

However, the probability of estimating a SMOS SIC equal to or higher than 90 % while the OSI SAF SIC is 100 % is not constant during the year and decrease with respect to the previous case. During the winter period (between January and April), the conditioned probability is notably high (near 0.9) (see Fig.  blue line). Then it decreases sharply during spring and most notably in summer. This change in the conditioned probability starting in the spring could stem from a change in ice properties. As the snow becomes wetter with the onset of the melt season, the observed emissivity starts to change, and this varies with the observating frequency (different scattering response). The observed increase of the conditioned probability in June could be due to the use of a summer tie point (applied from June to September), which improved the rms with respect to OSI SAF as shown in Fig. . The low conditioned probability in fall can be explained by the presence of thin ice.

We have analysed the spatial distribution of the conditioned probability of SIC estimates for the months of March and November. Those regions where OSI SAF SIC is more than 0.9 while SMOS SIC is less than 0.9 (light blue colour in Fig. ) outline the edge of the ice cover. This is in good correspondence with the expected areas of thin ice. Additionally, this condition is extended when analysing November data (Fig. b), when thin ice is more frequent in the Arctic.

During the winter months, the spatial coefficients of determination (r2) between SMOS and OSI SAF SIC are high (more than 0.65), which again is consistent with our interpretation about the role of thin ice in SMOS SIC (see Fig. ). As melt starts, the correlation between SIC estimates continues to be high, thanks to the use of the summer tie point in the retrieval. In September, ice cover extent is at minimum because ice growth has not started yet, there is almost no thin ice, and the correlation remains high. The correlation drops in the fall (between October and December) because ice growth starts by freezing of the sea surface, producing large amounts of new thin ice. To compute these values, we have only included SIC values between 0.05 (5 %) and 0.95 (95 %) when computing correlations to avoid the two extreme values leading to too-high, non-significant values of correlation.