# Forward modelling of passive microwave emissivities over snow-covered areas at continental scale

Iris de Gélis, Catherine Prigent, Carlos Jimenez, Melody Sandells

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Iris de Gélis, Catherine Prigent, Carlos Jimenez, Melody Sandells. Forward modelling of passive microwave emissivities over snow-covered areas at continental scale. Remote Sensing of Environment, 2025, 328, pp.114821. 10.1016/j.rse.2025.114821. hal-05368185

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# Forward modelling of passive microwave emissivities over snow-covered areas at continental scale

Iris de Gélis a,b,c,*, Catherine Prigent b,a,c, Carlos Jimenez a,b,c, Melody Sandells c

<figure><img src="image_3.png" /></figure>

## ARTICLE INFO

Edited by Menghua Wang

**Keywords:**

Passive microwaves

Snow

Forward model

Geophysical parameters

Continental scale

## ABSTRACT

To assimilate passive microwave data in numerical weather prediction, a comprehensive understanding of the components of the radiative transfer equation is essential. Given the significant variability of emissivity in snow-covered regions — affected by frequency, polarisation, and the macro- and microstructural properties of snow — attention must be paid to the design of a forward model. However, existing physical models are unsuitable for global-scale studies due to their reliance on numerous inputs, such as snow grain size across different layers, which are typically unavailable at larger scales. In this study, we propose a method that utilises geophysical properties accessible at the continental scale to derive accurate emissivity values for frequencies ranging from 1 GHz to 90 GHz, in both vertical and horizontal polarisations, with a focus on the incident angles of conical scanners (approximately 50°). Our approach employs neural networks to obtain a robust forward model using geophysical variables as input data. A training dataset was developed based on satellite-derived surface emissivity from the SMOS and AMSR2 instruments by subtracting atmospheric components and surface temperature modulation. The results, which accounts for the actual geophysical state of the surface and its temporal variability, outperform the emissivity climatologies. We achieved snow-covered surface emissivities at the continental scale with a correlation coefficient above 0.9 and a RMSE below 0.02 for frequencies up to 18.7 GHz, and around 0.03 for higher frequencies. Additionally, we demonstrate that, in a typical tundra snowpack where the macro- and microstructural properties of snow can be obtained, the emissivities retrieved by our neural network-based forward model are consistent with results from the physical model (SMRT). This proposed model will also support preparations for the CIMR mission.

## 1. Introduction

The need for accurate microwave surface emissivity models for all Earth surfaces has long been recognised (English, 1999; Weng et al., 2001), for the retrieval of surface and atmospheric parameters from satellites, as well as for the assimilation of surface-sensitive satellite observations in Numerical Weather Prediction (NWP). However, assimilating surface-sensitive observations still remains far more difficult over continental surfaces than over the ocean, primarily due to their typically higher microwave emissivities, as well as complex spatial and temporal variability (Bormann et al., 2017). As a consequence, in NWP, a significant portion of the observations over continental surfaces is discarded, especially over snow-covered surfaces. 'All sky' (i.e., clear as well as cloudy) passive microwave satellite observations are now successfully assimilated, delivering important benefits for the forecast (Bauer et al., 2010; Duncan et al., 2021). Advancing 'all-surface'

assimilation (Lawrence et al., 2019; Geer et al., 2022) is the next challenge in NWP developments, towards coupled land-ocean-atmosphere assimilation (de Rosnay et al., 2022), with expected improvements for atmosphere and surface characterisation, and consequently forecasts.

Snow-covered surfaces present a significant challenge due to their highly variable microwave signatures, which depend on environmental conditions and seasons, as well as on the observation characteristics (frequency, polarisation, incidence angle). Observations from both satellite and ground-based radiometry have long highlighted this variability (Kunzi et al., 1976; Mätzler, 1994; Grody, 2008). The physical properties of seasonal snowpacks exhibit substantial spatial and temporal variability, influenced by processes such as accumulation, stratification, wetness, melting and freezing cycles, and intricate metamorphism at the snow grain scale (Brun et al., 2013; Domine et al., 2019). These complexities make snow a challenging medium for radiative transfer,

<footnote>ª Estellus, Paris, 75002, France</footnote>

<footnote>ᵇ Sorbonne Université, Observatoire de Paris, Université PSL, CNRS, LIRA, Paris, 75014, France</footnote>

<footnote>ᶜ Department of Geography and Environmental Sciences, Northumbria University, Newcastle upon Tyne, UK</footnote>

<footnote>* Corresponding author at: Estellus, Paris, 75002, France.</footnote>

<footnote>E-mail addresses: iris.degelis@obspm.fr (I. de Gélis), catherine.prigent@obspm.fr (C. Prigent).</footnote>
involving ground emission, volume emission and scattering within the snowpack, and surface scattering (Foster et al., 1984; Royer et al., 2017). Below 18 GHz the snow microwave signal is little affected by volume scattering, and emanates from deeper in the snowpack. At frequencies around 18 GHz and above, a significant part of the emissivity variability is expected to come from the scattering by the snow grains, with the larger the snow grains, the stronger the scattering that tends to reduce the emissivity. The snow grains undergo metamorphisms during winter, in response to changing atmospheric conditions. From in situ measurements, Sturm and Benson (1997) evidenced that the increasing temperature gradient between the warm base of the snowpack (in contact with the ground), and the top of the snowpack (in contact with the cold atmosphere) generates grain growth, fed by the water vapour transport through the pore spaces of the snow (the temperature-gradient metamorphism, Marbouty (1980)). This process can generate very large depth hoar crystals at the bottom of the snowpack (Domine et al., 2007). Josberger and Mognard (2002) showed that these strong temperature gradients in the snowpack induce changes in the microwave emissivities, and they proposed the temperature gradient as a proxy for snow grain metamorphism. At the end of the winter season, snow melting also strongly affects the microwave signals. Especially at 18 GHz and above, liquid water in the snow leads to a large increase in microwave emissivity, compared to the emissivity of the completely frozen snowpack. The snow can undergo melting and freezing cycles, that induce strong temporal variability even at sub-daily scale (Tuttle and Jacobs, 2019). In addition, other environmental factors, such as vegetation interference and terrain relief, further complicate the microwave signatures of snow-covered areas (Cordisco et al., 2006). As a consequence, microwave remote sensing of bulk snow parameters such as snow depth or snow water equivalent is not straightforward. The variability of the snow passive microwave signatures is often dominated by snow microstructural properties (e.g., Rutter et al. (2014), Sandells et al. (2017)), making it difficult to establish a direct relationship between microwave measurements and the bulk snow properties (namely snow depth or snow water equivalent). While correlations can sometimes be established for specific geographic regions and snowpack histories, such relationships do not hold consistently at continental scales, and current microwave retrievals of snow water equivalent (e.g., Pulliainen et al. (2020)) try to account for the variability of snow micro-structures, to obtain improved performances at global scales.

Over the past decades, efforts have been made to develop radiative transfer models for simulating microwave snow emissivity, using various electromagnetic theory and semi-empirical approaches. Notable examples include the Dense Media Radiative Transfer - Quasi-Crystalline Approximation (QCA) Mie scattering of Sticky spheres (DMRT-QMS) model (Chang et al., 2014), the multi-layer Helsinki University of Technology model (HUT-nlayers) model (Lemmetyinen et al., 2010), and the Microwave Emission Model of Layered Snowpacks (MEMLS) model (Mätzler and Wiesmann, 1999; Proksch et al., 2015), which have been systematically compared by Royer et al. (2017). More recently, the Snow Microwave Radiative Transfer (SMRT) (Picard et al., 2018) was introduced, not as a new model, but as a modular platform designed to unify existing models and enable seamless experimentation by switching between them. All these models rely on detailed macro- and microstructural descriptions of the snowpack for each layer, incorporating bulk properties such as snow temperature, density, and wetness, along with fine-scale details like grain size, distribution, and arrangement. When provided with in situ snow measurements, these models have proven effective for simulating local observations (e.g., Vargel et al. (2020)). However, their application at continental scales and in NWP operations is significantly constrained by the lack of detailed snow physical description over large areas and the computational time required to run these complex models. Simplified snow emissivity models have also been developed for NWP applications (Weng et al., 2001; Hirahara et al., 2020), but they face difficulties in accurately capturing the spatial and temporal variability of snow signatures, across

the frequency range of assimilated microwave observations. For instance, Hirahara et al. (2020) reported reasonable agreement at 6 and 10 GHz where the emission signature remains relatively stable, but at higher frequencies, the model fails to replicate the highly variable scattering behaviour of snow.

Microwave emissivities have also been directly calculated from satellite observations by removing atmospheric contributions (gas and clouds) and accounting for land surface temperature modulation (Prigent et al., 1997; Karbou et al., 2008; Baordo and Geer, 2016; Munchak et al., 2020). These satellite-derived emissivities effectively capture the spatial and temporal variability of emissivities, leading to the development of the Tool to Estimate Land Surface Emissivities at Microwave and Millimeter Waves (TELSEM2). TELSEM2 provides monthly emissivity climatologies derived from 15 years of Special Sensor Microwave Imager (SSM/I) observations between 18 and 90 GHz (Aires et al., 2011; Wang et al., 2017). Compared to model-calculated emissivities, TELSEM2 emissivities exhibit an improved agreement with satellite observations in the 18 to 90 GHz frequency range, particularly over snow-covered surfaces (Prigent et al., 2015; Hirahara et al., 2020). This makes satellite-derived emissivities an attractive alternative to physically-based models for complex media like snow. NWP schemes are moving towards coupled atmospheric and surface prediction, with the simultaneous assimilation of observations that are sensitive to both atmosphere and surface properties. This requires not only precise emissivity estimates, but also to establish the link between the emissivity and the surface parameters. The existing emissivity atlases (like TELSEM2) do not reflect the real surface emission depending on the current geophysical conditions. Thereby, a critical challenge remains: relating these emissivities to geophysical parameters to provide the community with a robust forward model for emissivity estimates based on NWP-modelled snow conditions.

In the next decade, several advanced microwave conical imagers are scheduled for launch in Europe, including the Copernicus Imaging Microwave Radiometer (CIMR) (planned for 2030) (Donlon et al., 2023). CIMR is specifically designed for polar region observations and will operate at frequencies of 1.414, 6.925, 10.65, 18.7, and 36.5 GHz, with full polarimetry (V and H polarisations, as well as the 3rd and 4th Stokes parameters). It has a conical scan over a swath of 1900 km, with a 52° incidence angle at 1.4 GHz and 55° for the other channels. It will cover simultaneously this wide frequency range for the first time, with exceptional spatial resolution for channels above 1.4 GHz (up to 5 km around 18.7 and 36.5 GHz) and unprecedented measurement accuracy across all frequencies. To maximise the potential of this instrument and facilitate its assimilation into NWP models, it is essential to develop unified and robust surface emissivity parameterisations that cover their large frequency range.

Here, we propose to bridge the gap between precise physical models, which require multiple detailed inputs unavailable at large scales, and robust emissivity atlases that are not linked to current surface conditions. Thus, we develop a pragmatic approach to parameterise microwave emissivities from 1.4 to 90 GHz for both orthogonal polarisations based on geophysical variables that describe the surface. The methodology involves three main steps:

* Building a satellite-derived emissivity database: a comprehensive database of satellite-derived emissivities is constructed using existing observations across the frequency range. Observations from the SMOS mission are utilised for 1.4 GHz, while AMSR2 provides data for the higher frequencies. This database has been carefully filtered and made more robust to compensate for noisy instantaneous emissivities.

* Selecting geophysical predictors: the relationship between satellite-derived emissivities and available geophysical parameters at the continental scale is analysed to identify suitable predictors for emissivity parameterisation. NWP reanalyses, such as ERA5, offer a consistent and extensive dataset of global variables over long time periods. Using ERA5 variables as predictors
ensures both the consistency of the emissivity parameterisations and their seamless integration into comparable NWP systems.

* Developing the forward emissivity model, based on Neural Networks (NNs) techniques: the selected predictors and the corresponding satellite-derived emissivities are used to train a NN emissivity parameterisation. The NN-based emissivity parameterisation is developed to replicate the satellite-derived emissivities at the continental scale throughout the snow season. This approach can ensure robust performance across the full frequency range and changing snow conditions.

The satellite-derived microwave emissivities, along with the corresponding geophysical variables, are detailed in Section 2. Section 3 outlines the methodology for parameterising the multi-frequency microwave emissivities. It includes a joint analysis of geophysical parameters and multi-frequency emissivities to identify the most relevant geophysical variables for representing snow-covered surface emissivities. The results of the emissivity parameterisation are presented in Section 4, with further discussion provided in Section 5, including comparisons with SMRT simulations. Finally, the conclusions are summarised in Section 6.

## 2. Data

### 2.1. The satellite-derived land surface emissivities

#### 2.1.1. The satellite data

To compute the satellite-derived emissivities across the full frequency range, two current sensors are utilised: SMOS at 1.4 GHz, and AMSR2 for the frequencies above.

Since 2009, SMOS has provided multi-angular and multipolarisation observations at 1.4 GHz, using an interferometric antenna. The spatial resolution varies from 35 to 50 km, depending on the incidence angle (Kerr et al., 2010). The equator crossing time of ascending and descending orbits are at 6:00 am and 6:00 pm local time, respectively. This study relies on the L3C product from the Centre Aval de Traitement des Données SMOS (CATDS), which includes $T_B$ projected onto the ground at vertical (V) and horizontal (H) polarisations. The multi-angular data are divided into 14 fixed angle classes of 5°, spanning 2.5° to 62.5° (Al Bitar et al., 2017). For this study, the angle range centred at 52.5° is selected to match the CIMR observing incidence angle at 1.4 GHz. To minimise the Radio-Frequency Interference (RFI) effects, a threshold of 0.4 is applied to the RFI flag provided by the CATDS, filtering out affected data (Uranga et al., 2022). Note that similar tests were performed with Soil Moisture Active and Passive (SMAP) L-band data at 40° incidence angle (Entekhabi et al., 2010). For the sake of brevity, the results are not presented here but are similar to those obtained with SMOS.

The AMSR2 sensor aboard the GCOM-W1 Japan Aerospace Exploration Agency (JAXA) satellite measures at 6.925, 7.3, 10.65, 18.7, 23.8, 36.5, and 89.0 GHz, both orthogonal polarisations with a 55° incidence angle, at 1:30 am and 1:30 pm local time. The 7.3 GHz, very close in frequency to the 6.925 GHz channel, is primarily dedicated to RFI mitigation, and is excluded from the analysis. The 23.8 GHz channel is also excluded from the analysis because it is highly impacted by water vapour. Furthermore, the frequency dependence of the emissivity is expected to be smooth (Prigent et al., 2000), so removing the 7.3 and 23.8 GHz bands does not change our analysis. However, the data at 89.0 GHz are analysed: this channel will be present on the European MicroWave Imager (MWI) planned for launch in 2026 on board MetOp-SG satellites (Accadia et al., 2019). The level L1R product from JAXA is adopted. It contains $T_B$ at the swath locations with original spatial resolutions (ranging from 4 to 48 km depending on the frequency) (Maeda et al., 2015).

#### 2.1.2. Computation of the satellite-derived emissivities

The brightness temperatures ($T_B$) observed by the satellite includes the direct contribution from the surface, the downwelling atmospheric brightness temperatures at the surface ($T_{atm}^\downarrow$) reflected by the surface, and the upwelling atmospheric brightness temperatures ($T_{atm}^\uparrow$). Consequently, for each frequency and polarisation, the radiative transfer equation is expressed as:

$$T_B = T_{surf} \times \epsilon \times \tau + T_{atm}^\downarrow \times (1 - \epsilon) \times \tau + T_{atm}^\uparrow \quad (1)$$

with $T_{surf}$ the temperature at the surface, $\epsilon$ the emissivity, and $\tau$ the atmospheric transmission. Thereby, emissivity can be computed with the following formula:

$$\epsilon = \frac{T_B - T_{atm}^\uparrow - T_{atm}^\downarrow \times \tau}{(T_{surf} - T_{atm}^\downarrow) \times \tau} \quad (2)$$

The atmospheric contribution to the radiative transfer (i.e. $\tau$, $T_{atm}^\downarrow$ and $T_{atm}^\uparrow$) is computed with the gas absorption model from Rosenkranz (2017). Atmospheric meteorological data are sourced from the ERA5 reanalysis provided by Copernicus (Hersbach et al., 2020). The ERA5 hourly data are spatially and temporally linearly interpolated to match the satellite swath locations and observation times. To minimise uncertainties associated with cloud contributions, only clear-sky observations are considered: for this purpose, the ERA5 total liquid water path must be below 0.05 kg m$^{-2}$.

Eqs. (1) and (2) involve the surface temperature $T_{surf}$, which in this study is assumed to correspond to the skin temperature from ERA5 reanalysis ($T_{surf} = T_{skin}$). Over snow-covered surfaces, microwave radiation can originate from below the surface, with the depth of the signal source increasing as the frequency decreases. The penetration depth of the microwave signal also varies significantly depending on snow properties, such as wetness and grain size (Schanda et al., 1983). While microwave emission models coupled with thermodynamic models can theoretically compute the effective emitting temperature at a given frequency (Tonboe et al., 2011), these models require detailed inputs that are unavailable at continental scales and across a wide range of frequencies. Consequently, this study employs the skin temperature for the emitting temperature, regardless of frequency, to derive ‘effective’ emissivities consistently across frequencies and at a continental scale. This approach aligns with practices commonly used in NWP centres (Weng et al., 2001; Geer et al., 2022). It is important to note that this approach can result in “effective” emissivities exceeding 1, particularly at lower frequencies. This occurs because the skin temperature is often colder than the actual temperature of the emitting layer below the surface. Consequently, the calculated “effective” emissivity compensates for this difference, and can be higher than 1. The same strategy to derive satellite emissivities has been used and evaluated in earlier works, e.g., Prigent et al. (1997). For the selected frequencies and both orthogonal polarisations, satellite-derived surface emissivities are systematically computed from the launch of each instrument until 2020. Using these emissivities a climatology has also been built and will be used within this study.

### 2.2. The geophysical parameters

To ensure the emissivity parameterisation is practical and widely accessible for the community, the geophysical predictors must be globally available, publicly accessible, and cover an extended time period. The ERA5 meteorological reanalysis provided by Copernicus offers a comprehensive and consistent representation of the atmosphere and surface conditions. Available globally at a spatial resolution of 0.25° on a regular latitude-longitude grid, ERA5 provides hourly data, extending back to 1940. These reanalyses are generated using physical models combined with the assimilation of diverse observations, including in situ and satellite data (Hersbach et al., 2020). Note that our results are constrained by the accuracy of the input variables. ERA5 snow products have been compared with satellite-based datasets (Kouki et al., 2023)
and in situ measurements (Urraca and Gobron, 2021). Despite a discontinuity in 2004, ERA5 has demonstrated good accuracy, particularly in the period following that date. The main discrepancies arise in mountainous regions, where the high topographic variability and the spatial scale of our retrieval lead to reduced reliability. Indeed, given the satellite's field of view, a single observation can encompass a wide range of terrain types, from the valley to mountain top.

The link between the snow emissivities and the snow parameters has been briefly summarised in Section 1. We discuss below the parameters (marked in italics in the text) that are selected as potential predictors of emissivity and justify their use. It should be noted that the available snow related parameters are not all directly linked to microwave emissivities. Including more emissivity-related parameters would improve the parameterisation but those variables (e.g., multi-layer snow grain size) are not available at global scale.

For snow-specific parameters, ERA5 includes variables derived from a snow scheme detailed and evaluated by Dutra et al. (2010). In the model, the snowpack is represented by one layer on top of the upper soil layer, with an independent prognostic, thermal, and mass content. At each time step and location, the snow layer is characterised by a single snow temperature ($T_{snow}$), the Snow Water Equivalent (in meter of water equivalent, called snow depth in the ERA5 distribution), and Snow Density (mass of snow per cubic meter in the snow layer). Snow Depth (as defined by the snow community, i.e., the actual depth of the snow layer in m) is not directly provided in ERA5, but is calculated as follows:

$$ \text{Snow Depth} = \text{Snow Water Equivalent} \times \frac{\text{Water Density}}{\text{Snow Density}} \quad (3) $$

with the Snow Water Equivalent (in meter of water equivalent, called snow depth in the ERA5 distribution), the Snow Density provided by ERA5, and water density (1000 kg m⁻³).

In areas with little to no vegetation, the ERA5 Snow Albedo decreases linearly with snow age under non-melting conditions and exponentially during melting conditions (ECMWF, 2023). Snow albedo tends to reduce as snow grain size increases (Warren, 1982; Domine et al., 2007). As for example, Munneke et al. (2011) proposed a method for determining snow albedo based on grain size, further emphasising its relationship with snowpack properties. Consequently, Snow Albedo can serve as a valuable proxy for snow grain size close to surface snow layers. Snow Melt is defined in ERA5 as the accumulated amount of melted water from snow in the snow-covered area of a grid box (in meter of water equivalent).

The relationship between the microwave emissivities and the bulk snow properties (as provided by the ERA5 and described above) is insufficient to fully account for the spatial and temporal variability of the emissivities (e.g., Rutter et al. (2014), Sandells et al. (2017)). It has been shown that some metamorphism in the snow (and the related-changes in snow grain size) can be linked to temperature-related parameters, such as the gradient between the top of the snow and its bottom ($T_{skin}-T_{ground}$), and this parameter is added as a potential predictor of large particles such as depth hoar (Sturm and Benson, 1997; Josberger and Mognard, 2002). This parameter can be used simultaneously with snow albedo, as it provides information on snow grain size deeper within the snowpack, whereas albedo is limited to surface information. To account for the potential penetration of the signal down to the ground, the $T_{ground}$ is also added. The temperature of the air at 2 m ($T_{2m}$) is tested, as a proxy for the atmospheric environment.

To account for the effect of vegetation on snow emissivities, ERA5 provides a Leaf Area Index (LAI), based on Moderate-Resolution Imaging Spectroradiometer (MODIS) acquisitions. However, LAI is not available during nighttime, making it unreliable or unavailable for polar winter above 65°N. As a proxy for vegetation density, the European Space Agency (ESA) Climate Change Initiative (CCI) Above Ground Carbon (AGC) is selected, averaged over 2018 and 2019 (Santoro and Cartus, 2023). For a given location, this parameter is fixed in time.

To account for the effects of terrain relief, two parameters are extracted from ERA5: the Altitude of the location and the StD of Orography at that location. The StD of Orography serves as a proxy for large-scale surface roughness.

ERA5 also provides Lake Cover data, which indicates the probability of liquid or frozen lakes within a grid cell. This should be a valuable information as the presence of liquid water highly influence emitted signal.

The Sun Zenith Angle is incorporated as a predictor to capture the timing of observations, both seasonally and diurnally. This parameter distinguishes regions with minimal sunlight exposure during mid-winter from those receiving significant sunlight in late spring. It also accounts for differences in emissivities related to the time of day, such as AMSR2 observations at midday versus midnight, depending on whether the satellite is in an ascending or descending pass. This distinction is particularly crucial at the end of the snow season when melting and freezing cycles dramatically affect snow emissivity.

### 2.3. Development of the database

A full winter season is considered, in the Northern Hemisphere (continental snow-covered extent in the Southern Hemisphere is limited). Data are collected from the beginning of October 2018 to the end of May 2019. Greenland and permanent glaciers are excluded from this study, as ERA5 snow information is not available in these areas. Microwave emissivities are very sensitive to the presence of open ocean, and data within 80 km of the coast are removed to avoid any contamination. A spatial and temporal interpolation is performed to collocate the geophysical parameters with the satellite-derived emissivities at swath level. As already mentioned, the analysis is limited to clear-sky emissivities (with ERA5 total liquid water path < 0.05 kg m⁻²). Data are kept only if the Snow Depth is above 0.02 m.

Data over 10 days are aggregated onto the same Equal-Area Scalable Earth (EASE) grid 2.0 over the Northern Hemisphere at 12.5 km (Lambert's equal-area, azimuthal) (Brodzik et al., 2012). A 10-day aggregation period was chosen as a compromise between the robustness of the database and the possibility of modelling singular events. The average emissivity is kept after a removal of outliers distant from 2 StD of the mean. 10-day averages are created separately for the two overpassing times (ascending and descending orbits), as snow emissivities behave differently during the day and night, especially during the melting season when snow melting and refreezing events occur, implying large variability in the emitted signal (Tuttle and Jacobs, 2019).

Our aim is not to directly replicate instantaneous emissivities (computed with Eq. (2)), as these can be affected by noise. Therefore, it is important to ensure that the database is thoroughly cleaned to create a reliable and robust forward model.

Climatological emissivity records for both day and night at 10-day resolution for each instrument, frequency and polarisation are computed based on the emissivity database.

## 3. Methodology

### 3.1. Joint analysis of geophysical parameters and microwave emissivities

Figs. 1 and 2 present examples of satellite-derived emissivities over snow-covered regions, along with the collocated geophysical parameters, for two contrasting periods, in the middle (1–10th January 2019) and at the end (1–10th April 2019) of the winter season. The 10.7 GHz channel is not shown on maps as it is very similar to the 6.9 GHz band. The spatial and temporal variability of the snow emissivity is observed, depending on frequency, polarisation, and time of the year. As expected, the snow emissivity at a given frequency is systematically higher in the vertical polarisation than in the horizontal polarisation. For a given location and season, the snow emissivity tends to decrease
<figure><img src="image_4.png" /><figcaption>Fig. 1. Average satellite-derived emissivity over ten days for selected frequencies (at 1:30 pm for AMSR2, 6:00 am for SMOS). Only snow-covered regions are presented (Snow Depth> 0.02 m).</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 2. Examples of average geophysical parameters over ten days at 1:30 pm. Only snow-covered regions are presented (Snow Depth> 0.02 m). The two red lines on the map of Std of Orography of the beginning of April 2019 indicate the transect studied later in the manuscript. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

with increasing frequency. North of Canada and Siberia, the snow emissivity significantly drops from January to April, evidencing stronger volume scattering likely due to an increase in the snow grain sizes: the higher the frequency, the larger the emissivity decrease. The decreasing of the emissivity can also be linked to a deeper snowpack leading to more volume scattering.

Fig. 3 presents the correlation matrix between the geophysical parameters and the emissivities at different frequencies for two contrasting periods in the boreal winter (January February, and April May), over the snow-covered Northern Hemisphere (excluding Greenland, as already mentioned).

Among emissivities, during the mid-winter period, lower-frequency emissivities exhibit strong intercorrelation (Fig. 3), while the 36.5 and 89.0 GHz emissivities display distinct behaviour. Conversely, during the melting period, the correlation among lower-frequency emissivities diminishes, while the 18.7, 36.5, and 89.0 GHz frequencies exhibit stronger intercorrelation. As winter progresses, snow undergoes metamorphism, leading to an increase in grain size (Sturm and Benson, 1997). This evolution enhances volume scattering, which results in a decrease in emissivity. This behaviour is observed even at lower frequencies later in the season. For example, in Fig. 1, the 18.7 GHz channel exhibits characteristics similar to lower frequencies in January but aligns more closely with the behaviour of the 36.5 and 89.0 GHz bands in April.

The analysis of correlations between snow parameters and emissivities (Fig. 3) reveals that *Snow Depth* and *Snow Water Equivalent* are less correlated with emissivities than *Snow Density* and *Albedo*. While some correlations may be observed for specific geographic regions and snowpack history, linear relationships between microwave

signatures and *Snow Depth* / *Water Equivalent* often fail across continental scales (Cordisco et al., 2006) or throughout an entire winter season (Rosenfeld and Grody, 2000; Grippa et al., 2004). The linear correlation coefficients in Fig. 3 represent spatio-temporal correlations. Separate examinations of spatial and temporal correlations (not shown) reveal that *Snow Depth* and *Water Equivalent* have minimal spatial correlation with microwave emissivities at a given time, but locally these parameters can show higher temporal correlation with emissivities. This observation aligns with the algorithm proposed by Derksen et al. (2010), which estimates snow water equivalent based on the history of $T_B$ at 37 GHz.

For a more detailed space and time analysis, Hovmöller graphs (Fig. 4) are produced for selected geophysical parameters and emissivities, over one transect in North America, at 110°W, between 40°N and 70°N. The Hovmöller graphs clearly illustrate a progressive increase in *Snow Density* throughout the winter season. This parameter plays a critical role in determining the dielectric properties of snow in radiative transfer models (Wiesmann and Mätzler, 1999). However, rather limited correlations are calculated between the *Snow Density* and the emissivities at 36.5 GHz and above. On the contrary, an anticorrelation is observed during mid-winter between *Snow Density* and the emissivities at low frequencies. While dry snow is generally rather transparent at low band frequencies, Schwank et al. (2015) demonstrated that it still impacts emissivities, primarily due to effects at the snow-air interface. ERA5 reanalysis provides a bulk *Snow Density*. In practice snow density varies across the layers within the snowpack and a snowfall may introduce a thin layer of low density which has been shown to have an effect on high frequency emissivities above 100 GHz (Sandells et al., 2024).
As previously mentioned (Section 2.2), *Snow Albedo* can be seen as a proxy for snow grain sizes, which play a key role in the scattering properties of snow at microwave frequencies. During the mid-winter period, correlations are observed between the *Snow Albedo* and the emissivities above 18.7 GHz (Fig. 3). On the Hovmöller graph (Fig. 4), north of 50°N, the time and space patterns of the snow albedo and the 36.5 GHz emissivities (V and H) show similarities, at least up to April.

Snow melting during spring time affects the emissivities, as observed on the Hovmöller diagrams (Fig. 4). At low frequencies, the emissivities at H polarisation are mostly impacted (decreasing of emissivities caused by the presence of water). At high frequencies, both polarisation are affected, with a significant increase of the emissivities as compared to the dry snowpack (Magand et al., 2008).

Temperature-related information (e.g., snow, ground) show significant correlation with the emissivities during the full snow season. At high frequencies, correlation is highly likely linked to snow properties influenced by surrounding medium temperature (Shahroudi and Rossow, 2014; Cordisco et al., 2006), especially during spring when higher temperature implies snow melt that has a strong impact on snow emissivities. At low frequencies, the anticorrelation could be an artefact of the emissivity computation described by Eqs. (1) and (2). The actual emission temperature at low frequencies is likely not the skin temperature, but a temperature below the surface (and possibly higher than the skin temperature), at a depth corresponding to the signal penetration at these frequencies. During the melting period the anticorrelation is strengthened, probably due to higher temperatures causing snowmelt and the presence of water in the snowpack, which reduces the emissivity as seen earlier for these low frequencies.

The correlation with the gradient of temperature between the skin and the ground is also given in Fig. 3. The lower frequencies seem to be anticorrelated with this temperature gradient whereas a positive correlation appears especially at spring time for higher frequencies. We see here two distinct phenomena. First, the sensitivity of lower frequencies to ground temperature because of penetration, noticeable also when directly looking at ground temperature correlation with low frequencies (especially for the 1.4 GHz channels). Then, the temperature gradient through the snowpack controls the metamorphism of the snow grains (Josberger and Mognard, 2002), reducing emissivities at higher frequencies (scattering phenomenon).

The selected vegetation indicator, the *Above Ground Carbon (AGC)*, demonstrates a strong correlation with emissivities during both winter periods (Fig. 3), with the correlation increasing at higher frequencies. Notably, the correlations are stronger for H-polarised emissivities than for V-polarised ones. Vegetation typically exhibits high emissivity with minimal polarisation differences (Prigent and Jimenez, 2021). As a consequence, in regions where the snowpack emissivity is inherently low, the influence of the overlying high-emissivity vegetation becomes more pronounced, consistent with observations by Derksen (2008). This relationship is further supported by Fig. 4, which shows that areas with AGC values exceeding 50 Mg·ha⁻¹ north of 53°N correspond to an increase in emissivities, particularly at frequencies of 18.7 and 36.5 GHz.

The terrain relief tends to increase the emissivities, especially in H polarisation, and to depolarise the signal (Prigent et al., 1997; Mätzler and Standley, 2000). This effect is evident in Fig. 4, especially around 41°N, where the *StD of Orography* is high. In this region, emissivities increase markedly in H polarisation, accompanied by a pronounced reduction in polarisation differences. Similar patterns are observed around 45°N, highlighting the impact of terrain relief on the microwave emissivities.

The most prominent features in the emissivity Hovmöller graphs (Fig. 4) are linked to the presence of water bodies, such as small lakes near 54.5°N, Athabasca Lake around 59°N, and the gulf between the Northwest Territories and Victoria Island around 68°N. These areas are characterised by low emissivities, particularly at 6.9 and 18.7 GHz for H polarisation. During mid-winter, the lakes are likely frozen and

<figure><img src="image_6.png" /><figcaption>Fig. 3. Linear correlation between geophysical parameters and emissivities for both mid-winter (bottom left) and melting period (top right). AGC stands for Above Ground Carbon; Snow Water Eq. for Snow Water Equivalent.</figcaption></figure>

potentially snow-covered, with scattering signals becoming evident at frequencies above 18.7 GHz. Lakes are also associated with highly polarised signals (large emissivity polarisation differences), especially early in the snow season, indicating the presence of liquid water. Note that the gulf around 68°N is covered by sea ice after November, leading to a sharp emissivity increase in this region at that time. Incorporating detailed information about lakes and water bodies into the emissivity parameterisation is critical for accurately representing the high emissivity variability linked to these surfaces.

## 3.2. Statistical model for emissivity parameterisation

### 3.2.1. Description of the method

This study relies on statistical models to establish links between geophysical parameters and emissivities. As previously analysed, numerous geophysical parameters influence the microwave signal in a highly intricate manner, both spatially and temporally. Neural networks are particularly well-suited for this task, as they can effectively capture complex, non-linear relationships between variables. Neural networks have already been widely applied to regression problems in remote sensing and have recently demonstrated success in forward ocean emissivity modelling (Kilic et al., 2023). In this study, a compact Multi-Layer Perceptron (MLP) (Rosenblatt, 1961) is utilised, featuring only one hidden layer with 80 neurons. This type of neural networks can be quickly trained on standard Central Processing Units (CPUs). Various configurations were tested (different number of layers and/or neurons), and the results indicate that the architecture of the MLP does not significantly alter the outcomes. The network is trained using a Bayesian regularisation back-propagation algorithm, which ensures robust performance and reduce the probability of overfitting by constraining the weights of the NN (MacKay, 1992; Foresee and Hagan, 1997).

A separate neural network is trained for each frequency. To ensure a consistent relationship between the V and H polarisations, a single NN is employed to predict both simultaneously. This approach enforces coherence between the two polarisations while maintaining flexibility across different frequencies.
<figure><img src="image_7.png" /><figcaption>**Fig. 4.** Hovmöller diagrams across the longitude 110°W and latitudes varying from 40°N to 70°N. See the red line over Canada in Fig. 2b to locate the transect. These graphs also represent geophysical variables evolution along the season (from October 2018 to the end of May 2019). Variable without any seasonal variation (i.e., fixed variable over time) are represented by transect. Only snow-covered areas are represented here. Only some of the studied frequencies are shown here. These diagrams correspond to ascending passes. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

### 3.2.2. Training configuration

Following the joint analysis, a subset of geophysical variables was selected to train a neural network-based statistical model. Individual and various combinations of input parameters were tested to determine their effectiveness, ultimately focusing on a specific configuration that balanced model simplicity with the inclusion of essential geophysical details. Table 1 presents an overview of the 11 geophysical parameters selected for this experiment, which were chosen based on their significant relationships with passive microwave emissivities, as identified in our initial analysis.

Climatologies of emissivities, derived from satellite time series, have been generated for each 10-day aggregation as described in Section 2.3. First, these climatologies serve as a baseline method against which our results (i.e., the retrieved emissivities) are systematically compared. The reason for this is that we want to show that we can improve the current radiative transfer based on the use of climatological emissivities. Second, these emissivity climatologies are also considered as complementary inputs to the neural network parameterisation, alongside the geophysical parameters. This is a common approach in prediction problems, where a priori knowledge of the variable to be predicted can be incorporated as a predictor to further constrain the prediction problem, if this information is available. Note that our goal is not to reproduce the climatologies, but the observed emissivities (also called target emissivities) with possible interannual variations. The climatologies act only as a first guess to guide the NN towards the correct emissivities.

Finally, we refer to the three following sets of emissivity data: (1) target emissivities used to train and evaluate the NN, which are 10-day aggregated retrieved emissivities from winter 2018–2019; (2)

<table><caption>**Table 1**
List of geophysical parameters used as input of the presented experiments.</caption><thead><tr><th rowspan="2">Name</th><th colspan="3">Geophysical parameters</th></tr><tr><th>Fixed</th><th>Snow</th><th>Temperature</th></tr></thead><tbody><tr><td rowspan="5">NN 11 parameters</td><td>Lake Cover</td><td>Albedo</td><td>Snow (T<sub>snow</sub>)</td></tr><tr><td>Std of Orography</td><td>Density</td><td>Ground (T<sub>ground</sub>)</td></tr><tr><td>Sun Zenithal Angle</td><td>Depth</td><td>Gradient (T<sub>skin</sub> - T<sub>ground</sub>)</td></tr><tr><td>Above Ground</td><td>Melt</td><td></td></tr><tr><td>Carbon (AGC)</td><td></td><td></td></tr></tbody></table>

emissivity climatologies used as the first guess in the NNs and as a baseline methodology to obtain surface emissivities, they are derived from 10-day aggregation over the whole satellite period; and (3) retrieved emissivities, which are the output of the NNs.

The database of emissivity and geophysical data has been divided into two distinct subsets: one for training the neural networks and the other for statistically evaluating the method. It is important to note that the Bayesian regularisation back-propagation algorithm does not require a separate validation set to evaluate model over-fitting, as the regularisation process is inherently incorporated into the training algorithm itself (MacKay, 1992; Foresee and Hagan, 1997).

## 4. Results

### 4.1. General results

To compare our NN emissivity parameterisation with the target satellite-derived emissivity values, we evaluate different metrics. The
<figure><img src="image_8.png" /><figcaption>Fig. 5. Linear correlation coefficient *R* and the Root Mean Square Error (RMSE) computed from the comparison of retrieved emissivities and the target values. As for comparison purpose, the comparison between the emissivity climatologies and the target values is given in black. In red is given the StD of target emissivities. The solid and dashed lines correspond to the V and H polarisations, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

first metric is the linear correlation coefficient *R*. The second is the RMSE, which is calculated based on the differences between the target values (i.e., satellite-derived emissivities) and the model outputs. Since our method is built on neural networks, the bias (systematic error) is assumed to be negligible across the entire dataset, making the RMSE nearly equivalent to the standard deviation. These quantitative metrics are presented separately for each studied frequency and polarisation. Emissivity climatologies are used as a reference baseline to compare with the results produced by our method. The distributions of the satellite-derived and parameterised emissivities are also compared, as well as the distribution of their differences. Maps of the results are also evaluated.

Fig. 5 presents the statistical results for the selected NN param-
eterisations, over the studied snow-covered surfaces in the Northern
Hemisphere, from October 2018 to May 2019. The NN parameterisation
that use only geophysical parameters (yellow lines on Fig. 5) achieve a
correlation coefficient (R) above ~0.8 for all frequencies and polarisa-
tions, and an RMSE below ~0.03 for frequencies up to 18.7 GHz (both
polarisations) and below 0.045 for higher frequencies. This represents
an approximate twofold reduction in error compared to using mean
emissivities for each frequency and polarisation, as indicated by the
ratio of 2 or more between the RMSE and the standard deviation of
the initial satellite-derived emissivities (targets), shown in the top-right
panel of Fig. 5. The results consistently perform better for V than for H
polarisation, as expected from the lower emissivity variability observed
at V polarisation. Comparisons between the emissivity climatologies
and the target emissivities (black lines in Fig. 5) reveal that emissivity
climatologies outperform NN parameterisation using only geophysical
inputs. Previous studies have demonstrated the effectiveness of emis-
sivity climatologies in capturing the spatial and temporal variability of
snow signatures at frequencies of 18.7 GHz and above (Prigent et al.,
2015; Hirahara et al., 2020) on a continental scale. Incorporating emis-
sivity climatologies as inputs to the NN parameterisation (in addition
to the geophysical parameters, green lines in Fig. 5) improves the
agreement with the target emissivities. This approach treats emissivity
climatologies as a priori values for the parameterisation, while the
geophysical parameters add temporal variability to better represent
the current environmental state for the specific winter conditions.
With these additional climatological inputs, the correlation coefficients
exceed 0.9 for all frequencies and polarisations, and the RMSE falls
below 0.02 for frequencies up to 18.7 GHz and around 0.03 for higher
frequencies. Note that *R* at 18.7 GHz is consistently lower than at other
frequencies: this is attributed to the limited variability of emissivities at

this frequency compared to higher frequencies, alongside the influence
of complex scattering mechanisms.

The distributions of the retrieved emissivities (with and without
climatologies) are compared to the original emissivity distributions and
their climatologies in Fig. 6. These comparisons reveal very similar
behaviours across all frequencies and polarisations, even at high fre-
quencies where the target emissivity variability is relatively large. The
same figure also presents the distribution of emissivity errors, showing
no observable biases. This represents a significant improvement over
the results of radiative transfer models reported by Hirahara et al.
(2020), which exhibited substantial biases, particularly at frequencies
above 10 GHz.

Finally, Figs. 7 and 8 present maps of the parameterised emissivities (using 11 variables with emissivity climatologies) and the differences between the target emissivities and the outputs of our method for two distinct time periods. The parameterised emissivities capture the large-scale spatial structures for both periods, even at high frequencies where pronounced latitudinal gradients are observed, influenced by complex volume scattering mechanisms (see Fig. 1). The difference maps further confirm this overall agreement, although some regions exhibit discrepancies, particularly at higher frequencies. Two of these regions (the two transect presented by red lines in Fig. 2b) are further analysed below.

4.2. *Focus on specific regions*

The Hovmöller diagrams along 65°N in Russia span from west of
the Ural Mountains, across the Siberian lowlands, to east of the Lena
River (Fig. 9). The transects covers boreal forests and tundra snow,
following the Sturm snow type classification (Sturm et al., 1995; Sturm
and Liston, 2021). Three frequencies are shown here (6.9, 18.7 and
36.5 GHz) for brevity. Only H-polarised signals are shown, as (1) V
and H are not that different (2) H is more variable and thus more
complicated. Comparison of the target and retrieved emissivities (and
their difference) shows that the parameterised emissivities accurately
represent several key features, from west to east: relatively high snow
emissivities in forested regions, even higher emissivities over the Ural
mountain range, the sharp drop in emissivity at high frequencies (be-
ginning early in the snow season at 89.0 GHz and slightly later at
36.5 GHz), an increase in emissivity at high frequencies between 80°E
and 100°E (driven by increased vegetation density and orographic
effects), and a subsequent decrease in emissivity, reaching very low
values around 130°E (corresponding to reduced snow albedo and thus
<figure><img src="image_9.png" /><figcaption>**Fig. 6.** Top panels: Histogram of the snow-covered surfaces emissivities from the target (red), from the climatologies (black) and from our retrievals with two configurations of inputs (11 geophysical variables only (yellow) or along with emissivity climatologies (green)). Bottom panels: Histograms of errors, i.e., targets emissivities minus the emissivity climatology in black or the retrieved emissivities in the different configuration in yellow and green. The solid lines and dashed lines represent V and H polarisations respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Fig. 7.** Retrieved emissivities for ascending orbits (1.30 p.m. LST for AMSR2, 6.00 a.m. for SMOS).</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Fig. 8.** Maps of errors, i.e., differences between target and retrieved emissivities for ascending orbits (1.30 p.m. LST for AMSR2, 6.00 a.m. for SMOS).</figcaption></figure>

larger grain sizes). The parameterised emissivities also capture the low emissivities in H polarisation observed at the end of the melting season between 60°E and 90°E, which are associated with snowmelt processes. The differences between the target emissivities and those derived from emissivity climatologies are also shown. The emissivity climatologies struggle particularly at the beginning and end of the snow season, where greater inter-annual variability is expected. In contrast, the emissivity parameterisation, which incorporates geophysical parameters representative of current conditions, shows improved agreement with the target emissivities, especially during these periods

when climatologies are less effective. Note that the very low emissivities east of 130°E are significantly better reproduced by the emissivity parameterisation compared to the climatologies.

The results for a South to North transect in Canada is presented in Fig. 10. Note that this was the transect analysed in Section 3.1. This south-to-north transect traverses regions characterised by a wide variety of snow types, including prairie, montane, boreal forest, and tundra snow (Sturm et al., 1995; Sturm and Liston, 2021). The NN parameterisation reproduces emissivities at low frequencies. However, challenges arise at high frequencies south of 43°N, where the parameterisation struggles to replicate the target low emissivities. Conversely,
<figure><img src="image_12.png" /><figcaption>**Fig. 9.** Hovmöller diagrams across the latitude 65°N and longitudes varying from 40°E to 160°E and during the season from October 2018 to the end of May 2019. See the red line over Russia in Fig. 2b to locate the transect. Only some of the studied frequencies in H polarisation are represented here. These diagrams correspond to ascending passes. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

the parameterisation predicts excessively high emissivities south of 50°N. This region, located south of 50°N, has a thin snow cover, and it is not consistently snow-covered throughout the winter. In such areas with ephemeral snow, the variability in emissivity is inherently challenging to model accurately.

# 5. Discussion

## 5.1. Evaluation on specific situations

The training and testing of our method were conducted using data (emissivity and geophysical variables) from the winter of 2018–2019. As described in Section 2.3, the database consists of 10-day aggregations to reduce noise during training, whether from emissivities derived from observations or geophysical variables from ERA5 reanalysis. All results presented thus far are based on this 10-day aggregation approach. To further assess the methodology and its generalisation capabilities, additional tests are performed on two specific days: 17th December 2016, and 14th March 2017, representing different winter periods. Fig. 11 shows the statistical results for these dates, which, while slightly degraded compared to Fig. 5, remain promising. For both days, the parameterised emissivities outperform the emissivity climatologies, particularly on 17th December. Biases are minimal for both dates, and the RMSE is below ~0.04. The target and parameterised emissivity maps (Fig. 12 and Fig. 13) exhibit similar spatial patterns,

even at high frequencies. However, the difference maps reveal regions of discrepancies, which are nonetheless less pronounced compared to those observed with emissivity climatologies.

In addition to the intrinsic complexity and variability of snow emissivities, the parameterisation is subject to several sources of error. First, daily emissivities are inherently noisier than the 10-day aggregated emissivities used earlier in this study. These daily emissivity values are influenced by errors in surface temperature and atmospheric profiles, including potential contributions from clouds (Section 2.1.2). Additionally, errors may arise from the geophysical inputs used in the parameterisation. For example, the snow model in ERA5 is relatively simple, relying on a single-layer snowpack representation. Ongoing efforts focus on improving this model to enhance its accuracy and representativity (Arduini et al., 2019; Hirahara et al., 2020).

## 5.2. Comparison to physical models (SMRT): the frequency dependence of the emissivities

As discussed earlier, detailed radiative transfer models (e.g., SMRT) cannot be fed with bulk geophysical parameters such as ERA5 information for continental-scale applications for several reasons (e.g., lack of microstructure and layering information). Consequently, no attempt is made here to make a global comparison of the outputs of such physical models with the parameterised snow emissivities of this study. However, approximate consistency between the physical models and
<figure><img src="image_13.png" /><figcaption>**Fig. 10.** Hovmöller diagrams across the longitude 110°W and latitudes varying from 40°N to 70°N and during the season from October 2018 to the end of May 2019. See the red line over Canada in Fig. 2b to locate the transect. Only some of the studied frequencies are represented here. These diagrams correspond to ascending passes. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

the parameterised emissivities can be assessed locally, across frequencies and polarisations to ensure the emissivity spectra are physically realistic. We compare the frequency dependence of the parameterised emissivities with results from the SMRT model (Picard et al., 2018) for a region classified as tundra snow (Sturm et al., 1995; Sturm and Liston, 2021) in the north of Canada (65°N 105°W (±1°)), during two distinct periods of the winter. This point is chosen at a flat location without vegetation and with a low lake cover fraction. The goal is to select a site with a snow cover as homogeneous as possible across the satellite footprint to allow for meaningful comparisons. According to the classification, tundra snow is characterised by low temperatures, low precipitation, and strong winds. These conditions result in a snowpack composed of alternating layers of depth hoar and windslab, from the bottom to the top. Due to strong winds, fresh snow in these regions is rapidly transformed into windslab, which consists of small, rounded snow grains. In contrast, the basal depth hoar layer is composed of larger grains formed by the strong temperature gradient within the snowpack. This depth hoar layer typically has a lower density than the windslab layer. The snow microstructure used in SMRT is described by the sticky hard sphere formulation, with the model configured to represent typical values of snow grain size and stickiness for these types of snow layers. Specifically here, the snowpack is modelled as a two-layer structure, with a bottom layer of depth hoar and a top layer of windslab. The windslab layer is characterised by a snow grain size of 0.16 mm, as suggested by Fierz et al. (2009), and a stickiness value of 0.18. For the depth hoar layer,

the typical snow grain size is 0.32 mm (Domine et al., 2007), with a stickiness value of 0.1. Stickiness values are taken from Soriot et al. (2022) in coherence with (Löwe and Picard, 2015) study. While these values may not perfectly reflect the actual state of the snowpack (as in situ measurements would be required for precise characterisation), our goal is to determine whether the same frequency dependence can be observed in the initial satellite-derived emissivities, in the emissivity parameterisation, and in the physical model. Furthermore, given the spatial variability of the snow (Sturm and Benson, 2004; Rutter et al., 2014) and the large footprint of satellite measurements, we chose to use typical values corresponding to tundra snowpack. This allows to produce an average brightness temperature that can be compared to the model output, which operates at the scale of the satellite footprint. The thicknesses of the two layers are chosen based on the assumption that the depth hoar layer increases in thickness as the season progresses. Beneath the snowpack, the soil substrate is modelled following the formulation in Montpetit et al. (2018) for sub-arctic frozen organic soil. This model relies on two parameters: soil roughness, with an RMS value set at 0.01 m, and ground temperature. To ensure consistency between our emissivity parameterisation and the radiative transfer model, the mean *Snow Density, Snow Depth*, snow temperature (*T*<sub>snow</sub>), and ground temperature (*T*<sub>ground</sub>) are derived from the ERA5 reanalysis. Table 2 summarises the selected parameters for the two winter periods. The SMRT simulations in this study use the Improved Born Approximation (IBA) (Mätzler and Hüppi, 1989; Mätzler, 1998) for the scattering
<figure><img src="image_14.png" /><figcaption>**Fig. 11.** RMSE and Bias over two specific days. Results of our method (NN 11 parameters + Emissivity climatologies in green) can be compared to the emissivity climatologies (black). The solid lines and dashed lines represent V and H polarisations, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

formulation, combined with Discrete Ordinate and Eigenvalue Solver (DORT) solver (Picard et al., 2013).

SMRT simulates surface $T_B$, which is the product of emissivity and the emitting temperature. For comparison, our 'effective' emissivities are multiplied by the skin temperature ($T_{skin}$), following Eq. (1) in Section 2.1.2.

The results of the SMRT simulations are shown in Fig. 14 as a function of frequency, alongside the $T_B$ values derived from satellite-based emissivities and those from the parameterised emissivities. Note that although the emissivity parameterisations were performed independently for each frequency, the parameterised emissivities exhibit smooth frequency-dependent behaviour, as expected. Across all frequencies and for both dates, the SMRT simulations align well with the other $T_B$ values, showing a very similar frequency dependence. The agreement at V polarisation is particularly strong up to 36.5 GHz, although at higher frequencies, SMRT calculates slightly colder $T_B$ values. These results are encouraging, especially given the involvement of significant volume scattering in this area, as indicated by the

<caption>**Table 2**</caption>

SMRT configuration for a tundra snowpack and substrate layer at 65°N 105°W. Temperatures, density and total snow depth parameters are chosen according to ERA5 reanalysis. As in ERA5 snow consists in only one layer, temperature and density for each snow layer are assumed to be in average the ERA5 value. Grain size and stickiness are chosen as function of typical values for these type of snow layer.

<table><thead><tr><th></th><th></th><th>1–10th<br>January 2019</th><th>1–10th<br>April 2019</th></tr></thead><tbody><tr><td rowspan="5">Snow layer 1:<br>Windslab</td><td>Thickness (m)</td><td>0.3</td><td>0.2</td></tr><tr><td>T (K)</td><td>240</td><td>256</td></tr><tr><td>Grain size (mm)</td><td>0.16</td><td>0.16</td></tr><tr><td>Stickiness</td><td>0.18</td><td>0.18</td></tr><tr><td>Density (kg m<sup>-3</sup>)</td><td>170</td><td>210</td></tr><tr><td rowspan="5">Snow layer 2:<br>Depth Hoar</td><td>Thickness (m)</td><td>0.2</td><td>0.3</td></tr><tr><td>T (K)</td><td>255</td><td>256</td></tr><tr><td>Grain size (mm)</td><td>0.32</td><td>0.32</td></tr><tr><td>Stickiness</td><td>0.1</td><td>0.1</td></tr><tr><td>Density (kg m<sup>-3</sup>)</td><td>110</td><td>190</td></tr><tr><td rowspan="2">Soil layer</td><td>T (K)</td><td>265</td><td>262</td></tr><tr><td>Soil roughness RMS (m)</td><td>0.01</td><td>0.01</td></tr></tbody></table>

pronounced decrease in the signal with increasing frequency. Such scattering introduces high sensitivity to grain microstructural parameters. Yet, using a simple two-layer snowpack model, typical grain properties and soil parameters, and ERA5 data, the SMRT simulations successfully reproduce the frequency dependence observed in the satellite-derived emissivities. However, we are aware that the simulation is highly dependent on the configured snowpack. In particular, the configured snowpack may be too simplistic and not perfectly representative of the real snowpack. For example, the bias between H-polarisation SMRT simulations and our results may be due to the insufficient number of layers parameterised in the snowpack for SMRT simulations. This also highlights the difficulty of using physical models on a large scale, given the complexity of parameterising a snowpack. Nevertheless, radiative transfer models provide invaluable insights into the physical behaviour of the snowpack.

## 6. Conclusions

This study focuses on the forward modelling of snow-covered surface emissivities across various frequencies, ranging from 1 GHz to 90 GHz, considering both V and H polarisations. Our aim is to provide robust emissivities estimates on a continental scale. Existing physical models are not applicable as they mostly rely on snow properties obtained through in-situ measurements. Large scale snowpack models are also not able to produce the necessary parameters, although progress is being made (Woolley et al., 2024). While emissivity climatologies provide reliable values, they do not account for the actual surface state. As a consequence, the proposed method links globally available geophysical properties with robust satellite-derived emissivities, using neural networks. As most global geophysical variables are not directly related to snow emissivity and have inherent uncertainties, we demonstrate that emissivity climatologies remain crucial for guiding neural networks towards accurate emissivities. Adding the geophysical values as inputs achieves better retrievals of singular events compared to climatology alone, and the overall results surpass those derived from climatological data. Notably, the R correlation coefficient exceeds 0.9 for all studied frequencies and both polarisations. The RMSE is below 0.02 for frequencies up to 18.7 GHz and around 0.03 for 36.5 and 89.0 GHz, with no bias observed.

As mentioned, most of the selected geophysical variables affect snow emissivity indirectly by modifying the snowpack. This naturally limits the accuracy of simulating effective emissivity using ERA5 data. The stratigraphy of the snowpack is shaped by past winter conditions. Therefore, a future refinement of our method could involve a better consideration of the temporal evolution of geophysical parameters,
<figure><img src="image_15.png" /><figcaption>Fig. 12. Emissivities and errors for ascending orbits of 17th December 2016.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 13. Emissivities and errors for ascending orbits of 14th March 2017.</figcaption></figure>

which leads to a sequence of changes resulting in the actual stratified snowpack.

The frequency dependence of the emissivity parameterisation has been compared to SMRT results, for one location under two different winter conditions. Complex radiative transfer models such as SMRT have the ability to dissect and understand the contributions of various snowpack characteristics. These models can help quantify the sensitivity of emissivities to snow microstructure under different environmental conditions, such as temperature gradients, wind effects, or variations in snow depth. This is crucial for advancing our understanding of the factors driving emissivity variability and improve the emissivity parameterisation for large scale applications.

### CRediT authorship contribution statement

**Iris de Gélis:** Writing – original draft, Writing – review & editing, Validation, Software, Methodology, Data curation, Conceptualization.
**Catherine Prigent:** Writing – review & editing, Methodology, Funding acquisition, Conceptualization.
**Carlos Jimenez:** Writing – review & editing, Methodology, Data curation, Conceptualization.
**Melody Sandells:** Writing – review & editing.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgements

This work has been supported by the European Space Agency (ESA) as part of the Copernicus Imaging Microwave Radiometer (CIMR) preparation, specifically through the CIMR Scientific End-to-End Performance Simulator (CIMR-SCEPS) project (Contract No. 4000138129/22/NL/IA). The CERISE project (grant agreement No101082139) is funded by the European Union. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the Commission. Neither the European Union nor the granting authority can be held responsible for them. We thank Patricia de Rosnay (ECMWF) for valuable discussions. This research was also supported by the International Space Science Institute (ISSI) in Bern, through ISSI International Team project #24-623. The authors thank the three anonymous reviewers for their careful reading of our manuscript and their insightful comments and suggestions.
<figure><img src="image_17.png" /><figcaption>**Fig. 14.** Dependency in frequency of brightness temperatures ($T_B$) at 65°N 105°W (±1°) from the observation (Targets), the retrieval (NN outputs) and the SMRT physical model (SMRT outputs). Observed and retrieved $T_B$ are obtained by multiplying the skin temperature to the emissivity as explained in Eq. (1) in Section 2.1.2. The SMRT model is configured using typical snowpack values (see Table 2) from snow belonging to tundra according to Sturm classification. Observed brightness temperature are given at AMSR2 incident angle (55°) except for the 1.4 GHz band where SMOS value at 52.5° is shown. Our retrieval and SMRT output are given for the same incident angles.</figcaption></figure>

## Data availability

Data will be made available on request.

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