Emission model for SMOS retrieval

In contrast to the first approaches to retrieve Ice thickness from SMOS L-band brightness temperatures (e.g. Reference Kaleschke, Tian-Kunze, Maaß, Mäkynen and DruschKaleschke and others, 2012), the emission model presented by Reference Maaß, Kaleschke, Tian-Kunze and DruschMaaß and others (2013) accounts for a Snow layer on top of the Ice. This model is referred to as M2013 in the following. The M2013 model is based on the emission model described by Reference Burke, Schmugge and ParisBurke and others (1979), which is used to consider a semi-infinite (half-space) layer of air on top, a layer of Snow on top of a layer of Ice, and a semi-infinite layer of sea water at the bottom. The emission model describes the brightness temperature above Snow-covered sea Ice as a function of the water temperature and permittivity, and of the temperatures, permittivities and thicknesses of the Snow and the Ice layer, respectively. In the model, the permittivities are calculated from empirical relationships. Water permittivity mainly depends on water temperature and salinity (Reference Klein and SwiftKlein and Swift, 1977). Ice permittivity can be approximately described as a function of brine volume fraction (Reference Vant, Ramseier and MakiosVant and others, 1978), which depends on Ice salinity and the densities of the Ice and the brine (Reference Cox and WeeksCox and Weeks, 1983), which in turn mainly depend on Ice temperature (Reference PounderPounder, 1965; Reference Cox and WeeksCox and Weeks, 1983). The permittivity of dry Snow can be estimated from Snow density and Snow temperature (Reference Tiuri, Sihvola, Nyfors and HallikainenTiuri and others, 1984). If not given as input to the M2013 model, the bulk temperatures of the Snow and the Ice layer are estimated from the surface temperature, using a simple heat transfer equation (Reference UntersteinerUntersteiner, 1964; Reference Maykut and UntersteinerMaykut and Untersteiner, 1971) and climatological values for the thermal conductivities of Ice and Snow (Reference Yu and RothrockYu and Rothrock, 1996). As a result, the Snow-cover thickness affects the Ice temperature in the model. In Reference Maaß, Kaleschke, Tian-Kunze and DruschMaaß and others (2013), this impact is assumed to be the main reason for the (indirect) dependence between the brightness temperature and the Snow thickness and is the basis for their Snow thickness retrieval over thick Ice. Thus, the relationship between Ice surface temperature T _Surface and Snow/Ice interface temperature T _Snow-Ice (from which the bulk Ice temperature is inferred by assuming a constant temperature gradient in the Ice) is essential, and here we apply two different approaches to compare brightness temperatures derived from the M2013 model with those derived from the more sophisticated MEMLS-based model (Reference Tonboe, Dybkjær and HøyerTonboe and others, 2011), which is referred to as T2011 throughout this study and is introduced in the next subsection. We compare T2011 brightness temperatures with (1) brightness temperatures as obtained from the M2013 model when using the T2011 model’s T _Surface as input parameter and calculating the Ice temperature from the simple heat transfer equation in the M2013 model (denoted by Tb) or with (2) brightness temperatures as obtained from the M2013 model when using both T _Surface and T_Snow-Ice from the T2011 simulations as input parameters (denoted by Tb*). By using the comparisons with Tb*, we evaluate the potential of the emissivity calculations in the M2013 model independently of the embedded simple heat equation model, whereas Tb is used to identify the difference between the previously used retrieval approach and the T2011 model with regard to the potential for a Snow thickness retrieval from L-band measurements.

The input parameters for the M2013 model are sea-water temperature, sea-water salinity, surface temperature (and optionally Snow/Ice interface temperature), bulk Ice salinity, bulk Snow density, and the Ice and Snow thicknesses. For the simulations presented here, we assume that the water has a salinity of 33, is at the freezing point of sea water and has a constant water temperature of -1.8°C. In this study, the remaining input parameters of the M2013 model are provided by the T2011 model.

Combined thermodynamic and emission model

Our relatively simple M2013 model presented above is compared to a more sophisticated combined thermodynamic and emission model, which is described by Reference Tonboe, Dybkjær and HøyerTonboe and others (2011). In this T2011 model, a one-dimensional Snow and Ice thermodynamic model is driven by the European Centre for Medium-Range Weather Forecasts (ECMWF) reanalysis ERA40 data, namely the surface air pressure, the 2 m air temperature, the 10 m wind speed, the incoming shortwave solar radiation and longwave radiation, the dew-point temperature and the precipitation data. The thermodynamic model has been developed to provide realistic microphysical input from these parameters to the emission model. The emission model is a sea-Ice version (Reference Tonboe, Heygster, Pedersen, Andersen and MätzlerTonboe and others, 2006) of MEMLS (Reference Wiesmann and MätzlerWiesmann and Matzler, 1999). The T2011 model has a time step of 6 hours. The vertical resolution in the Ice is 0.05 m, while the vertical resolution in the Snowpack depends on individual precipitation events and the subsequent metamorphosis. The output parameters of the thermodynamic model are, for example, the Snow freeboard, Snow and Ice thickness, the surface density and temperature, the correlation length of the surface Snow, the Snow/Ice interface temperature, the average Snow density, the average correlation length of the Snow, and the average Snow temperature and salinity.

The combined thermodynamic and emission model T2011 was used to simulate the 1.4GHz brightness temperature at 50° incidence angle at horizontal and vertical polarization for six locations of multi-year Ice in the Arctic (Fig. 1). The simulations were performed for the period 1 September 1999 to 31 May 2000 and were initiated with an isothermal 2.5 m thick Ice floe at 270 K with a 5 cm thick layer of old Snow on top. In each simulation, the Ice thickness gradually increases from 2.5 m to almost 4.5 m at the end of the simulation period. The initial salinity profile consists of low salinities (0.5-1.0) in the upper 0.2 m of Ice and a constant salinity of 2.5 in the rest of the Ice column. The salinity of newly formed Ice at the Ice/water interface depends on growth rate. As input to the M2013 model we use bulk Ice salinity, i.e. the average salinity over the Ice column. As new Ice forms at the bottom of the multi-year Ice, bulk Ice salinity gradually increases from 2.4 at the beginning of the simulations to -3.6 at the end because the relative fraction of the more saline Ice formed during the current Ice growth season gradually increases.

Fig. 1. The six multi-year Ice profile positions of the model simulations described in Reference Tonboe, Dybkjær and HøyerTonboe and others (2011).

Comparison of brightness temperatures from the two models

Figure 2 shows the time series of the horizontally and vertically polarized brightness temperatures at 1.4 GHz for an incidence angle 6˃ = 50° as simulated with the T2011 model from 1 September 1999 to 31 May 2000. Additionally, we show the T2011 simulations of the Ice parameters that are input parameters to the M2013 model and the brightness temperatures simulated with the M2013 model using these parameters. From a first visual inspection of the time series, the following may be observed:

Fig. 2. Time series for the six T2011 model simulations every 6 hours from 1 September 1999 to 31 May 2000. Numbers 1-6 refer to the locations given in Figure 1. (a) The horizontally polarized brightness temperature (TBH) at 1.4 GHz (θ = 50°), as obtained from the T2011 (red) and the M2013 model (blue; Tb), as well as the T2011 simulations of surface temperature T _Surface (green). (b) The brightness temperatures at vertical polarization (T2011 in red, M2013 in blue; Tb) and the Snow/Ice interface temperature T _Snow-_Ice from the T2011 model (green). (c)TheT2011 simulations of Ice thickness (blue) and Ice salinity (red). (d) The T2011 Snow thickness (blue) and Snow density (red). The parameters in (c,d) are used as input parameters to the M2013 model for all simulations, while T _Surface is used as input for the M2013 simulations denoted with Tb and both, T _Surface and T _Snow- _Ice, for the M2013 simulations denoted with Tb*. The dotted lines indicate the results from the first 100 time steps of the simulations, of which the simulated brightness temperatures are excluded from the analysis (but shown in Figs 6 and 7).

1. 1. At the beginning of the simulations, the T2011 simulations for horizontal polarization exhibit a very high variability and show very high values.

2. 2. In general, the T2011 model simulations show a higher variability of brightness temperatures at both polarizations than the simulations with the M2013 model.

3. 3. In the first part of simulation No. 3, the Snow density takes very high values, up to ˃900kgm ⁻³causing the brightness temperature at horizontal polarization to decrease considerably, according to both the T2011 and the M2013 model. These high density values occur when a precipitation event coincides with air temperatures above 0°C. The resulting precipitation is assumed to be rain, which then freezes when the temperature falls below 0°C, so the considered layer has the density of pure Ice. However, as there is no mechanism for drainage in the model, it treats this layer as Snow.

In order to further investigate the brightness temperatures modeled with the T2011 and the M2013 model, and the main factors controlling their evolution, we take all brightness temperatures from all six simulations and all time steps to compare the results. However, due to findings 1 and 3 above, we exclude some simulations from the correlation analysis below: The T2011 simulations start with a prescribed initial profile and need some time to adjust to the observed situation. Thus, we exclude the brightness temperature simulations of the first 100 time steps, i.e. of the first 25 days. Additionally, because the simulations with very high Snow densities behave exceptionally, we exclude simulations with _ρSnow ˃ 600 kg m ⁻³ from the correlation analysis.

A more quantitative analysis reveals the following findings and their implications for further comparisons:

1. 1. At horizontal polarization, the T2011 brightness temperatures can be divided roughly into two regimes if we consider the Ice covered by a relatively thin and light Snow layer separately from the Ice covered by a Snow layer with a high density and/or thickness. The separation appears to be most successful (in terms of explaining the resulting two distinct clusters of brightness temperatures) when we divide the simulations into two approximately commensurate clusters by separating the simulations assigned a Snow density of ρ_Snow ˂ 400 kg m–³ and a Snow thickness of d _Snow ˂ 0.25 m from the remaining simulations (Fig. 3).

2. 2. In the M2013 model, the Snow/Ice interface temperature T_Snow-Ice is almost a linear function of the surface temperature T _Surface of the Snow-covered sea Ice, while in the T2011 model, T_Snow-Ice is influenced more by other parameters (Fig. 4). Separating the results for a relatively light and thin Snow cover from the remaining cases leads to a distinct separation of T _SNOW-Ice as a function of T _Surface in the M2013 model, but not in the T2011 model. As mentioned before, due to the different approaches used to determine T_Snow-Ice (and thus the Ice temperature) in the two models, we either use both T_Surface and T _Snow-Ice from the T2011 simulations (Tb*; see Fig. 5–7), or only T _Surface from the T2011 simulations (Tb; see Table 1; Fig. 2 and 8), as input to the M2013 model.

Fig. 3. Two-dimensional histogram of the Snow density and Snow thickness values as they occur for all six locations and times during the T2011 simulations. The box indicates the cases with a relatively light and thin Snow cover (ρ_Snow ˂ 400 kg m^{- 3} and d _Snow ˂ 0.25 m), which are considered separately in the following.

Fig. 4. Surface temperature T _Surface versus Snow/Ice interface temperature T_Snow _ _Ice as obtained from the T2011 model (reddish colors) and as obtained from the M2013 model (bluish colors), respectively. Lighter colors indicate cases where the Ice is covered by a relatively light and thin Snow cover (see Fig. 3).

Fig. 5. Horizontally (left) and vertically (right) polarized brightness temperatures as obtained from the T2011 model and as obtained from the M2013 model (Tb*), respectively. Light purple colors indicate cases where the Ice is covered by a relatively light and thin Snow layer, while dark purple colors indicate cases where the Ice is covered by a heavier or thicker Snow layer (see Fig. 3). Blue dots indicate cases with high Snow densities ρ_Snow ˃ 600 kgm ⁻³and gray dots indicate simulations from the first 100 time steps, which are shown as dotted lines in Figure 2. The simulations indicated by the blue or gray dots are excluded from the further analysis.

The comparison of all brightness temperatures at horizontal polarization as obtained from the T2011 and the M2013 emission model (Tb*) confirms that the T2011 simulations cover a broader range of brightness temperature values (Fig. 5). Separating the simulations with respect to their Snow density and Snow thickness appears to explain the two observed clusters. The coefficients of determination, i.e. the squared correlation coefficients (Fig. 5), are r² = 0.84 for simulations with a relatively light and thin Snow cover and 0.53 for the remaining simulations. At vertical polarization, Snow density and Snow thickness have a negligible impact on brightness temperatures (Fig. 5), and we determine only one coefficient of determination. The vertically polarized brightness temperatures from the T2011 and the M2013 model agree better (r²= 0.95) than the horizontally polarized brightness temperatures. Yet, at vertical polarization, brightness temperatures in the M2013 model are generally higher than in the T2011 model, and their difference increases with decreasing brightness temperature. For comparison, if we use only the surface temperature from the T2011 model as input to the M2013 model and estimate Ice temperature with the simple heat transfer equation given in the M2013 model, agreement between the brightness temperatures from the M2013 (Tb) and the T2011 model is lower, as indicated, for example, by the corresponding coefficients of determination, which are r² = 0.72 and 0.34 at horizontal polarization, and 0.65 at vertical polarization (compared to 0.84, 0.53 and 0.95 for Tb*).

The next step is to identify which Ice parameters mainly determine brightness temperature. At horizontal polarization, the brightness temperature in the M2013 model (Tb*) is mainly determined by T_Snow-Ice (Fig. 6), and thus, because we assume a linear temperature gradient within the Ice, by bulk Ice temperature. If we assume a linear relationship between T_Snow-Ice and brightness temperature, T _Snow-Ice explains 95% of brightness temperature variability for the simulations considered here (i.e. r² =0.95). For comparison, if we do not use T_Snow-Ice as an input to the M2013 model, but instead estimate T_Snow-Ice from T _Surface and the Ice conditions, using the simple heat equation in the M2013 model, the relationship between T_Snow-Ice and horizontally polarized M2013 brightness temperature (Tb) is weaker (r² = 0.61). For the T2011 model, the impact of T_Snow-Ice on the horizontally polarized brightness temperature is lower (Fig. 6) than for the M2013 model, although T_Snow-Ice can still be interpreted as accounting for more than half of the brightness temperature variability: r² =0.89 for cases with a relatively light and thin Snow cover and 0.55 for the remaining cases. Because, at vertical polarization, cases with a relatively light and thin Snow cover do not differ from the remaining cases, we consider all simulations together in Figure 7. Both models suggest that brightness temperature at vertical polarization is almost solely a function of T_Snow-Ice. the coefficients of determination for the two parameters are r ²=0.97 for the T2011 model and 0.99 for the M2013 model (Tb*). For comparison, r² is 0.73 if only T _Surface is used as input to the M2013 model (Tb). For high Snow/Ice interface temperatures, the brightness temperatures from the two models are similar, while the difference increases for lower Snow/Ice interface temperatures (Fig. 7).

Fig. 6. Horizontally polarized brightness temperatures as obtained from the T2011 model (left) and as obtained from the M2013 model (Tb*; right) versus the Snow/Ice interface temperature T _Snow-Ice. Colors are explained in the Figure 5 caption.

Fig. 7. Vertically polarized brightness temperatures as obtained from the T2011 model (red) and as obtained from the M2013 model (blue; Tb*) versus the Snow/Ice interface temperature T _Snow-_Ice.

Impact of Ice parameters on potential Snow thickness retrieval

Here we investigate the differences between the two models in terms of the potential for Snow thickness retrieval from L-band brightness temperatures and the impact of Ice conditions. In the M2013 model, the relationship between surface temperature and Ice temperature is mainly determined by Snow thickness, and this dependence was the basis for a first retrieval of Snow thickness from L-band brightness temperatures over thick sea Ice (Reference Maaß, Kaleschke, Tian-Kunze and DruschMaaß and others, 2013). Thus, we now consider the M2013 brightness temperatures as simulated for the surface temperatures from the T2011 simulations and for the Ice temperature as estimated within the M2013 model (Tb). In the first part of this section, we found that Ice temperature is the main Ice parameter influencing brightness temperature. In order to exclude the influence of the surface temperature, we keep it at a constant value by selecting only data with specific surface temperatures from the whole simulation dataset. Thus, we try to investigate which of the remaining Ice parameters significantly influence brightness temperature.

Correlation analyses (not shown here) indicate that the main factors influencing the simulated brightness temperatures and the Ice temperature are Snow thickness d _Snow, Ice thickness d _Ice and Snow density _ρSnow. In order to further investigate these parameters, we proceed as follows: In the considered cases, the total ranges of values are d _Snow = 0.05...0.50 m, d _Ice = 2.7...4.5 m, and ρSnow = 200...500kgm-³. For each parameter (d _Snow, d _Ice or ρSnow), we select from the simulation dataset only the data associated with an almost constant surface temperature (±1 K) and with roughly constant values for the two remaining parameters. This latter requirement is instantiated by selecting only the data in which these two remaining parameters are within 13-15% of their total variabilities (i.e. ±0.033 m for d _Snow, ±0.135 m for d _Ice and ±20 kgm ⁻³ for ρSnow). We then take into account all cases in which the considered parameter varies by at least 40% of its total variability and in which we find more than 50 simulations.

For these cases, we calculate the coefficients of determination between the T2011 simulations for the considered Ice parameter (d_Snow, d _Ice or _ρSnow) and the brightness temperatures from the T2011 and the M2013 model. Additionally, we calculate the coefficient of determination between the considered Ice parameter and the Snow/Ice interface temperatures as obtained from the two models. We use the average coefficients of determination as rough estimations of the considered parameter’s impact (Table 1). However, we find that the coefficients of determination are not very stable. If we change the maximum range for the parameters that are supposed to be constant, the minimum range for the considered parameter, or the minimum number of simulations found for each case, the values of the coefficients of determination change significantly. Thus, the presented values should be interpreted with caution. At horizontal polarization, the correlation analyses for the two models agree in that brightness temperature has the highest correlation with Snow thickness and the lowest with Snow density, when the respective remaining parameters are kept constant. In contrast, at vertical polarization, the M2013 model suggests a high correlation with Snow thickness (r ² =0.82), while the correlation with Snow thickness is low in the T2011 model (r ² = 0.09), and the highest value for the T2011 model is found for Ice thickness (r ²=0.37). As for brightness temperatures, the Snow/Ice interface temperature is mainly explained by Snow thickness in the M2013 model (r ² =0.83), followed by Ice thickness (r ² =0.36). In contrast, according to the T2011 model, correlations with all three considered parameters (d _SNOW, d _{Ice, ρSnow)} are relatively low. The two models agree relatively well regarding the dependency between the horizontally polarized brightness temperature and the Snow thickness, suggesting that it may be possible to extract information on the Snow thickness of thick sea Ice from SMOS measurements. Thus, in Figure 8 the relationship between brightness temperature at horizontal polarization and Snow thickness is shown for four different Ice conditions (in terms of surface temperature, Snow density and Ice thickness).

Table 1. Average coefficients of determination r² between brightness temperatures obtained from the T2011 model (denoted by superscript T) and the M2013 model (denoted by superscript M) at horizontal (TBH) and vertical (TBV) polarization with respect to Snow thickness d_Snow, Ice thickness d _Ice and Snow density ρ_Snow, respectively. For each parameter the two remaining parameters are kept almost constant. Additionally, the average r² between the Snow/Ice interface temperature T _Snow-_Ice (denoted by T _SI) and d _Snow, d _Ice and ρ_Snow are given. N is the number of simulations found for the criteria. The r² values are significant at the 99% level, unless a different value is given in parentheses

Fig. 8. Horizontally polarized brightness temperatures as obtained from the T2011 model (red) and as obtained from the M2013 model (Tb; blue) versus the Snow thickness for Ice surface temperature (T_SURFACE), Snow density (ρSnow) and Ice thickness (d_{I CE}) values selected as given in the figure. The blue line is the brightness temperature as a function of the Snow thickness as modeled with the M2013 model for the average values of T _Surface, ρSnow and d_Ice . The red line is a fitted curve for the T2011 results and has the form the given coefficients of determination are calculated with respect to the corresponding fitted curves.

Compared to the results in Table 1, the constraints on surface temperature (±2 K) and Ice thickness (±0.20 m) are somewhat less restrictive in Figure 8 (see figure caption), and each example contains between 126 and 192 single simulations. Additionally, the coefficients of determination for these examples are calculated with respect to a root function (see Fig. 8 caption). For all four considered Ice conditions, brightness temperatures in the T2011 model increase more steeply with Snow thickness than in the M2013 model. The coefficients of determination are between 0.43 and 0.84 with respect to a root function (see Fig. 8), and 0.02 lower each with respect to a linear function (not shown here).