Microwave radiometry is a promising all-weather method for the remote sensing of seasonal soil thawing and freezing cycles in the Arctic region. The microwave emission changes significantly at the phase transitions of water in wet soil, thereby making it possible to identify the frozen/thawed (FT) state of the soil. Recently, both single- and multifrequency radiometric methods for identifying the FT soil state have been proposed. In single-frequency methods, as implemented in the algorithms of the Soil Moisture and Ocean Salinity (SMOS) (Rautiainen et al., 2016, 2014; Roy et al., 2015) and Soil Moisture Active Passive (SMAP) (Derksen et al., 2017; Dunbar et al., 2016) satellites, the polarization index (PR) – PR=12TbV(1.4)-TbH(1.4)TbV(1.4)+TbH(1.4) – is used as an indicator of the FT soil state. Here, TbH(1.4) and TbV(1.4) are the brightness temperatures measured at horizontal (H) and vertical (V) polarizations, respectively, at a frequency of 1.4 GHz (and a viewing angle of ∼ 40∘). The time series of the PR are normalized to average seasonal maximum (winter) and minimum (summer) PR values, which are determined for specific landscape types (Rautiainen et al., 2014). The decision on the FT state of the soil is made when the normalized PR passes through zero. To date, a satellite-based product for identifying the FT soil state based on multifrequency radiometric measurements has not been created.

However, the possibility of identifying the FT soil state using the polarimetric measurements by multifrequency radiometers, such as the Scanning Multichannel Microwave Radiometer (SMMR), the Special Sensor Microwave/Imager (SSM/I), the Advanced Microwave Scanning Radiometer for EOS (AMSR-E) and AMSR-2 (at a viewing angle of ∼ 50–55∘), has been studied. Multifrequency methods for the identification of the FT soil state are based on the assessment of effective temperature, emissivity (Zhao et al., 2011, 2017; Hu et al., 2017, 2019) or reflectivity (Muzalevskiy and Ruzicka, 2020; Mizalevskiy et al., 2021) as well as on the spectral gradient of brightness temperature (ΔTb / Δf) in a wide frequency range from f=10.7 to f=37 GHz (Zuerndorfer et al., 1989, 1990; Zuerndorfer and England, 1992). Research (Zuerndorfer et al., 1989, 1990; Zuerndorfer and England, 1992) has shown that the correlation diagram between ΔTb / Δf in the frequency range from 10.7 to 37 GHz and the brightness temperature Tb(37) measured at a frequency of 37 GHz is a good indicator of the FT soil state (testing was mainly carried out for the Great Plains area, USA). Using this method, the brightness temperatures, measured by SSM/I (SMMR), were averaged between the horizontal and vertical polarizations. Results showed that low values of Tb(37) and negative values of ΔTb / Δf are effective criteria for assessing the frozen state of topsoil. In a study by Zhao et al. (2011) of agricultural areas in the Haihe River valley in China, ground radiometric measurements showed that the identification of the FT soil state using the effective emissivity had higher confidence than that using the spectral gradient of radio brightness temperature. In this case, the effective emissivity was calculated as the ratio of brightness temperatures TbH(18.7) / TbV(36.5), measured at a frequency of 18.7 GHz with horizontal polarization (H) and at a frequency of 36.5 GHz with vertical (V) polarization. The algorithm created for identifying the FT soil state (Zhao et al., 2011, 2017; Hu et al., 2017, 2019) was implemented on the basis of Fisher's discriminant analysis (Fisher, 1936), in which the discriminant function was a linear combination of two attributes: effective emissivity, TbH(18.7) / TbV(36.5), and brightness temperature, TbV(36.5). In the article by Muzalevskiy and Ruzicka (2020), modified polarization index (MPR) =12ΓH(1.4)+ΓV(1.4)ΓH(1.4)-ΓV(1.4) was proposed as an indicator of the topsoil FT state, where ΓH(1.4) and ΓV(1.4) are the effective reflectivity of topsoil with respective horizontal and vertical polarizations, estimated at a frequency of 1.4 GHz. In contrast to the effective emissivity (Zhao et al., 2011, 2017; Hu et al., 2017, 2019), the ratio of reflectivities in the MPR index minimizes the effect of soil roughness and canopy optical thickness. Effective reflectivities ΓH,V(1.4)≡1-TbH,V(1.4)TbV(6.9) were estimated based on brightness temperatures, measured by SMAP at a frequency of 1.4 GHz and by the Global Change Observation Mission – Water Satellite 1 (GCOM-W1) with vertical polarization at frequency of 6.9 GHz (as an estimation of the effective temperature of topsoil; Muzalevskiy et al., 2016). A threshold level of ∼ 1.0–1.2 in the proposed MPR index showed a significant correlation with the transitions of the topsoil temperature through 0 ∘C at 12 test sites located on the North Slope of Alaska in the USA, in Northern Canada, in Finland and in Russia (Muzalevskiy and Ruzicka, 2020; Muzalevskiy et al., 2021). The proposed method is about 10 d more accurate (Muzalevskiy et al., 2021) than the standard SMAP-SPL3FTP_E product with respect to determining the FT state (Dunbar et al., 2016), when compared with weather station data. To date, the possibility of using reflectivity spectral gradients as an indicator of the FT soil state has not been studied. In addition, the advantages of a wide-frequency-range brightness temperature spectral gradient in the L-band as an indicator of the topsoil FT state have not been studied. The development of multifrequency algorithms for classifying the topsoil FT state is of interest, especially considering that the Copernicus Imaging Microwave Radiometer (Kilic et al., 2021), equipped with a high-spatial-resolution multispectral polarimetric radiometer (1.4–36.5 GHz, 55–5 km), is expected to be launched in 2028/2029.

Ten test sites equipped with soil–climatic weather stations and located in the northern regions of the USA, Canada, Finland and Russia were selected to investigate spectral gradient methods for identifying the FT state of tundra and boreal forest topsoil. The coordinates of the test sites and their landscape classifications are summarized in Table 1. The landscape classification of the test sites is based on a database from ESA (2017). The statistics are given for the averaged pixel footprint area (44 km × 44 km), the centers of which coincided with the coordinates of the weather stations at the test sites.

At the North Slope of Alaska test sites, soil temperature was measured at the surface (depth of 1 cm). The FB test site is located on a coastal plain on a flat alluvial terrace with a moss tundra landscape comprising moist nonacidic sedges and prostrate shrubs. The SG, HV and IM sites are located on hilly terrain with dominantly moist acidic and nonacidic tussock tundra to the north and considerable shrub growth to the south. The BI test site is characterized by gentle hills with sparse vegetation in the form of mosses and grassy meadows. Boreal forest test sites are represented by SO and SA in Finland and by KJ in Canada. The SO and SA sites are comprised of forests of differing densities, and non-forest area is covered with juniper, heather, and a thin layer of lichen and moss. KJ is located in typical Canadian taiga with a low-density black spruce–lichen woodland (sandy soils) landscape. The soil temperature at the SO, SA and KJ test sites was measured at a depth of 2–5 cm. The MS test site is located on the Yamal Peninsula in moist and dry dwarf shrub–moss–lichen tundra in combination with sedge–moss mires. The soil temperature at MS was measured at a depth of 2–5 cm. The CH area is dominated by wetlands covered with larch forests of varying canopy cover percentages (ranging from 13 % to 75 %). At the CH test site, the soil temperature was measured at five test plots (which were averaged for further analysis) at the interface between the organic and mineral soil layers, at depths of 6–10 cm below the surface depending on the organic layer thickness (Loranty and Alexander, 2021). At the KJ, MS and CH test sites, water objects occupied more than 10 % of the pixel area.

The brightness temperatures of the test sites were measured at vertical and horizontal polarizations by the SMAP and GCOM-W1 satellites at frequencies of 1.4 and of 6.9, 10.7, 18.7 and 36.5 GHz, respectively. Ascending SMAP and GCOM-W1 orbits were chosen. SMAP polarimetric brightness temperature data (SPL3FTP_E) for Northern Hemisphere azimuthal projections on a 9 km Equal-Area Scalable Earth Grid (EASE-Grid 2.0) were used over the test sites for the period of soil temperature observations by the weather stations (see Table 1). GCOM-W1 brightness temperature data were acquired from the L1R product, where the brightness temperatures of high-resolution channels are resampled to the effective pixel area of the lower-resolution channel at a frequency of 6.9 GHz, gridded with a 12.5 km cell size. Pixels closest to the coordinates of weather stations, with the exception of MS, were used in the analysis. MS is located at the coast of the Kara Sea; thus, a pixel whose center is far more than 50 km from the sea was chosen. The daily 9 km EASE-Grid freeze/thaw SMAP product (SPL3FTP_E, both ascending and descending orbits) based on the polarization index (Dunbar et al., 2016) was used for the identification of the FT state of land at the test sites.

The spectral components of brightness temperature are formed by the emitting layers of a ground half-space with different thicknesses. For this reason, the difference between brightness temperatures measured in the different parts of the frequency spectrum is related to the vertical (in the direction of the 0z axis) gradient ΔTgΔzz=0 of ground temperature Tg and, hence, to the direction of the heat flux J0=KΔTgΔzz=0, where K is the thermal conductivity, through the soil surface. Indeed, on the basis of the phenomenological theory of emission (Zuerndorfer and England, 1992), the brightness temperature Tbp(f) of a dielectric half-space with a linear temperature profile can be represented as follows: 1Tbp(f)=1-Γp(f)Tg0+1-Γp(f)ΔTgΔzz=0zeff(f), where f is the frequency electromagnetic field, Γp(f) is the ground reflectivity for vertical (p= V) or horizontal (p= H) polarization, Tg0 is the temperature of the ground surface, and zeff(f) is the thickness of emitting layer. From Eq. (1), it follows that the density of the spectral gradient of the brightness temperature is directly proportional to the heat flux through the boundary of the ground surface: 2ΔTbp(f)Δf=-Tg0ΔΓp(f)Δf+1KJ0Δ(1-Γp(f))zeff(f)Δf. By analogy with the density of the spectral gradient of brightness temperature (Eq. 2), the density of the spectral gradient of reflectivity can be introduced: ΔΓp(f)/Δf.

For a dielectric inhomogeneous and non-isothermal half-space, the reflectivity at frequency f, in accordance with the method of Muzalevskiy et al. (2021) and Muzalevskiy and Ruzicka (2020), can be estimated based on the following equation: 3Γp(f)=1-Tbp(f)/Teff. In Eq. (3), Teff and Γp(f) can be interpreted as the respective effective temperature and reflectivity of the layered structure of canopy and ground, which takes soil surface roughness and canopy optical thickness into account using one combined parameter (Fernandez-Moran et al., 2015, their Eq. 10). The effective temperature, Teff, can be estimated based on the measurements of brightness temperature, TbV(6.9), by the GCOM-W1 satellite at a frequency of 6.9 GHz for vertical polarization, the values of which correlate with the surface temperature of tundra soil (Muzalevskiy et al., 2016). The physical basis for this estimate is the observation angle of AMSR-2/GCOM-W1, which is close to Brewster's angle (55∘), and this leads to a decreased impact of reflectivity, ΓV, on the measured brightness temperature, TbV(6.9). Indeed, reflection coefficient measurements show that Brewster's angle decreases from 60 to 57∘ as root-mean-square (RMS) heights of soil surface roughness increase from 0.25 to 0.93 cm (De Roo and Ulaby, 1994; see also Wang et al., 1983, their Fig. 2). The roughness of natural tundra soils has much higher values, which vary over a wide range, from 1.06 to 4.28 cm (Watanabe et al., 2012). The presence of vegetation (snow) cover on a rough soil surface leads to blurring and flattening of the V polarization angular dependence of reflectivity (Rodriguez-Alvarez et al., 2011, their Figs. 7, 8) and brightness temperature (Lemmetyinen et al., 2016, their Figs. 5–7; Chang and Shiue, 1980, their Figs. 3–5) in the region of Brewster's angles, due to the interference phenomenon. Within the error measurement of brightness temperature of ±1.3–1.5 K (Piepmeier et al., 2017; Gao et al., 2019) as well as the accuracy of emission models of ±4–5 K (Wigneron et al., 2011), Brewster's angle can be determined within a wide range from about 45 to 65∘ (Lemmetyinen et al., 2016, their Figs. 5–7; Chang and Shiue, 1980, their Figs. 3–5). The error measurement of brightness temperature and the accuracy of emission models also make it possible to neglect the variations in H polarization brightness temperatures of snow- or vegetation-covered soil over a range of observation angles from 40 to 55∘ (Roy et al., 2018, their Fig. 3; Lemmetyinen et al., 2016, their Fig. 7; Chang and Shiue, 1980, their Figs. 3–5). In this regard, the use of brightness temperatures measured at different angles in Eq. (3) is approximate. As a result, and as expected from `Eq. (3), TbV(6.9) becomes mainly directly proportional to Teff. For this reason, the reflectivity was further estimated only for horizontal polarization. Further, estimates of ΓH(f) will be considered to be the apparent values of reflectivity, as the absolute value of TbV(6.9) does not coincide with the actual values of Teff or Tg0 but is rather only proportional to them.

The time series of brightness temperatures, TbH, measured by the SMAP and GCOM-W1 satellites, and the reflectivity, ΓH, calculated based on Eq. (3), are shown for the HV test site in Fig. 1a and b, respectively. For the other test sites, such dependencies are similar. The time series of TbH and ΓH (see Fig. 1a and b, respectively) have a pronounced seasonal variation, with a periodic change in maximum and minimum values. On the dates corresponding to the moment of soil thawing or freezing, the order of brightness temperature and reflectivity values is inverted, along with an increase or decrease in frequency. Several such time periods are marked by dashed rectangles in Fig. 1a and b (see the period from 2015 to 2016).

In Fig. 1, these areas are shown on a larger scale. Indeed, as can be seen from the aforementioned figure, the brightness temperature values decrease with increasing frequency in winter (before 7 May 2016 and after 1 November 2015), i.e., a negative gradient is observed, but a positive gradient is observed in summer (see Fig. 1c, d). The seasonal variation in reflectivity has an opposite spectral gradient (see Fig. 1e, f) with respect to the spectral gradient of brightness temperature. In the transition period (7–19 May 2016 and October–November 2015; see Fig. 1), the spectral gradient of brightness temperature and reflectivity is minimal. In these time intervals, for any pairs of TbH(f1) and TbH(f2) or ΓH(f1) and ΓH(f2) at different frequencies (f1 and f2), there is a point of zero spectral gradient (the point of sign change in the spectral gradient), at which the direction of the heat flux must change, in accordance with Eq. (2). Unlike the methods in which the FT index threshold level is set on the basis of calibration or normalization (Rautiainen et al., 2016; Derksen et al., 2017; Muzalevskiy and Ruzicka, 2020; Muzalevskiy et al., 2021), the zero-threshold level of spectral gradients of brightness temperatures or reflectivity does not require calibration for the identification of the FT topsoil state. FT classification based on the spectral gradient method was applied for the first time by Zuerndorfer et al. (1989).

Detailed depictions of the spectral gradient densities of brightness temperature and reflectivity are given in Fig. 2a and b, respectively, for the HV test site. To this end, the spectral gradient densities of brightness temperature and reflectivity were calculated for the following frequency ranges: 1.4–6.9, 1.4–10.7, 1.4–18.7 and 1.4–36.5 GHz. The largest variations in brightness temperatures ΔTbH(f) and reflectivites ΔΓH(f) with frequency (see Fig. 1) are observed in winter, whereas the largest variations in the spectral gradient densities of brightness temperature ΔTbH(f)/Δf (see Fig. 2a) and reflectivity ΔΓH(f)/Δf (see Fig. 2b) are observed in summer. The spectral gradient of brightness temperature and reflectivity is highest in the frequency range of 1.4–36.5 GHz and lowest in the frequency range of 1.4–6.9 GHz. This is due to a significant contrast in temperatures and permittivities between the shallow and deeper emitting layers of the ground at frequencies of 36.5 and 1.4 GHz, respectively, compared with radiation layers that are close in thickness at frequencies of 1.4 and 6.9 GHz. At the same time, per unit interval of the frequency spectrum Δf, the spectral gradient densities of brightness temperature ΔTbH(f)/Δf and reflectivity ΔΓH(f)/Δf seem to be larger for the narrower (1.4–6.98 GHz) than for the wider (1.4–36.5 GHz) frequency band. The amplitudes of seasonal variations in ΔTbH(f)/Δf are synchronous with variations in the surface temperature of soil. The time series of ΔΓH/Δf varies in antiphase with the soil surface temperature (see Fig. 2b). Figure 2 shows that the time points corresponding to the transitions in ΔTbH(f)/Δf and ΔΓH/Δf through zero are well correlated with the time of the soil surface temperature Ts0 transition through 0 ∘C.

Similar patterns in the behavior of ΔTbH(f)/Δf and ΔΓH(f)/Δ are also observed for other test sites, except for SO and SA. As an example, ΔΓH(f)/Δf for the KJ, CH, SO and SA test sites are shown in Fig. 2c, d, e and f, respectively. In some years, multiple passing of ΔΓH(f)/Δf through zero during winter can be observed for the SO and SA test sites (see Fig. 2e and f, respectively), which does not allow for unambiguous identification of the FT states. These processes will be explained below. Moreover, it should be noted that the presence of significant wetland or open water at the CH test site is not detected in the behavior of ΔΓH(f)/Δf (ΔTbH(f)/Δf), in comparison with other test sites.

For further analysis, seasonal variations in ΔTbH(f)/Δf and ΔΓH/Δf will be used, the amplitudes of which take maximum (see Fig. 2, curve 1) and minimum (see Fig. 2, curve 4) values in the frequency ranges of 1.4–36.5 and 1.4–6.9 GHz, respectively. The soil will be considered frozen or thawed when the soil surface temperature, Ts0, transits through 0 ∘C, according to weather stations installed at the test sites.

The synchronicity of the transition through the zero spectral gradient density threshold of brightness temperature ΔTbH/Δf and reflectivity ΔΓH/Δf with the soil surface temperature crossing 0 ∘C was estimated (see Fig. 3).

From the correlation analysis of the data presented in Fig. 3, it follows that the ΔTbH(f)/Δf (ΔΓH/Δf) in the frequency range of 6.9–1.4 GHz (36.5–1.4 GHz) determines the thawed state of the topsoil 8.4–8.9 (1.2–3.3) d earlier, relative to soil surface temperature (see the bias estimations in Table 2).

In the frequency range of 6.9–1.4 GHz, the frozen topsoil state is determined 5.6–5.7 d later on average (see both indexes in Table 2). In the frequency range of 36.5–1.4 GHz, ΔTbH/Δf makes it possible to determine the frozen topsoil state 3.4 d earlier (see bias in Table 2); in the frequency range of 6.9–1.4 GHz, ΔΓH/Δf makes it possible to determine the frozen topsoil state 6.5 d later (see bias in Table 2). In general, both indexes, ΔTbH/Δf and ΔΓH/Δf, have similar RMSE values with respect to predicting the FT topsoil state. At the same time, for the identification of a thawed topsoil state, ΔΓH/Δf has a greater accuracy (bias and RMSE are lower and R2 is higher compared with ΔTbH/Δf; see Table 2) in the spectral range of 36.5–1.4 GHz. ΔTbH/Δf is more suitable for identifying a frozen topsoil state in the frequency range of 36.5–1.4 GHz (see Table 2), as it has a smaller bias (at comparable RMSE and R2 values) compared with ΔΓH/Δf. Spectral gradient methods have improved accuracy with respect to identifying the FT topsoil state in relation to both the MPR index (see Fig. 3c and the bold information in Table 2) and the standard SMAP product SPL3FTP_E (see Fig. 3d and the bold information in Table 2).

It should be noted that the correlation analysis (in the case of the frozen state) did not take the SO and SA test sites into account, as the ΔTbH/Δf and ΔΓH/Δf indexes led to a systematic error of about 1.5 months, due to unstable soil freezing (soil surface temperature for most of the winter ranged from from 0 to between -2 and -5 ∘C according to weather stations; see Fig. 2f and e). This systematic error may be because, in contrast to the other test sites, SO and SA stand out with the largest share of forest, from 70 % to 85 %, in the footprint (see Table 1). Apparently, such a significant share of forest contributes to the formation of a thicker forest litter layer and increased accumulation of snow cover compared with the rest of the test sites (e.g. sites KJ and CH, in which the share of forest is less than 60 % and 53 %, respectively). Thicker forest litter and snow cover create additional inertia in the thermal exchange between air and soil. Indeed, according to data from the SO and SA stations, these test sites have low negative soil surface temperatures in winter as well as an extended period of zero-curtain effect (see Fig. 2e and f for SO and SA, respectively). As a result, in some years, the unstable FT transition state of topsoil is reflected in the unstable transition of ΔΓH(f)/Δf (ΔTbH/Δf) through zero during winter, which explains the observed systematic error for the SO and SA test sites. Similar unstable transitions of brightness temperature have been observed in ground-based radiometric experiments at the SO test site (Lemmetyinen et al., 2016, their Fig. 2 “wetland 2013–2014”); thawed soil under a dry snow layer is a common phenomenon (Kumawat et al., 2022).

In this article, spectral gradient methods were used to identify the FT topsoil state of northern regions based on brightness temperature measurements from the SMAP and GCOM-W1 satellites across a wide frequency range. As criteria for determining the FT topsoil state, the spectral gradient densities of brightness temperature and reflectivity were used in the frequency range from 1.4 to 6.9 GHz and from 1.4 to 36.5 GHz. Both criteria give a comparable accuracy with respect to forecasting the FT topsoil state for tundra and boreal forest regions. However, the spectral gradient density of reflectivity should be used to improve the accuracy of thawed topsoil state predictions, whereas the spectral gradient density of brightness temperature (the frequency range of 1.4–36.5 GHz) should be used for identification of frozen topsoil state. The proposed method makes it possible to identify forest soils that are in a transitional state (the soil surface temperature is about 0 ∘C or has small negative values), which is revealed in the multiple transitions of spectral gradient densities of brightness temperature and reflectivity through zero (more pronounced in the frequency range of 1.4–6.9 GHz). The methods considered in this work do not fully separate the contributions of the FT topsoil state from those of waterbodies. The freezing processes of open-water areas with the formation of ice, the description of which requires a large number of additional data on snow cover thickness, air and water temperature, and wind speed are particularly difficult to account for. We did not find any significant differences in the behavior of the spectral gradient densities of brightness temperature and reflectivity measured for the test site with a high share of wetland (20 %) and open water (19 %) compared with the other test sites. Despite all assumptions made in the proposed method, the identification of FT soil surface states is possible with a relatively high determination coefficient of 0.775–0.834, a small root-mean-square error of 6.6–10.7 d, and a bias from -3.4 to +6.5 d. Further validation of this methodology, requires expanding the number of test sites, as the current analysis is limited by the small number of soil–climatic weather stations with available and up-to-date soil active-layer temperature data.