Empirical observations reveal a wide range of different microwave signatures in active or passive remote sensing over snow covered areas as shown e.g., by . The lack of realistic models to understand these signatures was the motivation for efforts leading to the Microwave Emission Model of Layered Snowpacks (MEMLS) in the 1990s . Initially the microwave emission behavior of single snow layers was investigated by and later by . The measurements led to an empirical approach for the scattering coefficient of snow in the frequency range 5–100 GHz and correlation-length range 0.05–0.3 mm as well as to a first version of MEMLS . Empirical relations for the scattering coefficient have also been implemented in the Helsinki University of Technology (HUT) model developed by and later adapted by . MEMLS was extended to coarse-grained snow for correlation lengths up to 0.6 mm . The snow microstructure was characterized by an exponential correlation function which allows computing the scattering coefficient analytically using the improved Born approximation (IBA) .

As an advantage of IBA and the characterization of snow in terms of correlation functions, the most relevant snow input parameters of MEMLS, correlation length and density, can be measured directly and objectively by various methods. Other models may require e.g., a conversion of measured parameters to model-effective ones . The exponential correlation length could be e.g., obtained by micro-computed tomography (μCT) from a fit to the reconstructed three-dimensional microstructure . Snow density and correlation length can be also obtained efficiently from field measurements using high-resolution penetrometry (SnowMicroPen – SMP) . Alternatively, optical methods can be used, e.g., , to measure the specific surface area (SSA) and use an empirical relation to compute the exponential correlation length . The latter method is appealing since SSA is commonly available. Accordingly, MEMLS was widely used for various questions related to passive microwave remote sensing .

In recent years, there was an increasing interest of the snow remote sensing community in active microwave measurements, which was mainly driven by the Cold Regions Hydrology High-Resolution Observatory CoReH2O and related activities. However, single-layer models for the radar signal as presented in or are mainly used for efficient operation in retrieval schemes. For the sake of low complexity, these models are naturally based on strongly simplifying assumptions, e.g., treating snow as a collection of independent scatterers. However, scatterers are densely packed in snow and strongly interact with each other. More realistic models based on dense media radiative transfer (DMRT) have been developed , including the possibility of using the numerical solution of Maxwell's equations for the single-layer scattering coefficients . The DMRT-based models however require at least two microstructural input parameters, which can be presently obtained only by μCT and often require time consuming casting procedures in the field.

To cope with recent requirements in active microwave remote sensing, while relying on an established, physical model of intermediate complexity, it is the aim of the present paper to extend MEMLS and develop a first version of MEMLS3&a. Thereby, we can build on the description of the microstructure in terms of the exponential correlation length as a single, objective parameter which can be derived from in situ field measurements. For the backscattering model, we shall extend the description of the snowpack in MEMLS to account for a slightly undulated snow surface as shown in Fig. . The slightly undulated patches should be small enough to leave the emission largely unaffected but large enough to allow for specular backscattering at near-vertical incidence.

The paper is organized as follows: in Sect.  we present the development of the model and the calculation of the total backscatter with its specular and diffuse components. In Sect.  the validation data consisting of active and passive microwave measurements from Sodankylä, Finland, are described. Section  presents the validation of both MEMLS and MEMLS3&a using the Sodankylä data, followed by a discussion (Sect. ) and the conclusions (Sect. ). Details about the calculation of the specular reflectivity are given in the Appendix.

In MEMLS the snow cover is considered as a stack of n horizontal layers with planar boundaries at the snow surface and between snow layers. Each layer is characterized by snow parameters (layer thickness, correlation length, density, liquid water content and temperature) that determine the layer-radiative properties. Also the salinity can be taken into account layerwise. The snow–ground interface is characterized by a reflectivity s0. A sandwich model is used to combine internal scattering and reflections at the interfaces. Internal volume scattering is accounted by a two-flux model (up- and downwelling streams) derived from a six-flux approach (fluxes in all space directions). The absorption and scattering coefficients are functions of the six-flux parameters. The absorption coefficient can be obtained from density, frequency, temperature and salinity; the scattering coefficient depends on the correlation length, density and frequency. For a detailed description of MEMLS we refer to the technical documentation . In the following, we focus on the backscatter model by considering the total backscatter as a sum of specular and diffuse components. Since the total reflectivity of a snowpack is related to its emissivity, it can be derived from passive observations alone. Thereby, active and passive observables can be appropriately combined to obtain a prediction for the radar backscatter.

We used snow input data generated from three different snow measurement methods to run model simulations which are compared to backscatter measurements from ESA's SnowScat scatterometer for validation . All measurements were made on 1 March 2012 at the test site of the Finnish Meteorological Institute (FMI), in Sodankylä, Finland, during ESA's Nordic Snow and Radar Experiment (NoSREx) III . The snow measurements were conducted directly in the field of view of the scatterometer and the radiometer in order to minimize the influence of the spatial variability of the snowpack.

As shown in Sect. , MEMLS3&a simulations were in reasonable agreement with SnowScat observations. To achieve this agreement, however, several parameters were chosen to match model and observation. This was necessary, since the active part contains, in contrast to the passive part, empirical parameters (ss0,q and m) which could not be measured. Likewise, the ground parameters s0 and rmsg are subject to uncertainties.

The specular part of the snow–ground reflectivity ss0 was chosen to be proportional to s0 and the same factor of 0.75 could be applied for all frequencies and polarizations to convert s0 into ss0. With ss0=0.75s0, the main part of the snow–ground reflectivity is specular. This requires the ground to be smooth and the overlaying snow layer to be transparent. The vegetation is subject to very low temperatures and a steady temperature gradient, which forces the water of the soft vegetation (lichen, mosses, shrubs (myrtillus species); Fig. ) to move upwards into the snow. Given the height of the vegetation of less than 10 cm, it seems reasonable to assume that the vegetation dries out during winter and can be treated as fully transparent for the present microwave frequencies. This allows the soil interface to act as specular reflector, which is then accounted for by ss0 in the model. Though being reasonable, a sound justification of this line of argumentation requires further investigations.

The cross-polarization in MEMLS3&a is solely determined empirically via the parameter q. This pragmatic approach was chosen since the physical origin of cross-polarization in snow is still the subject of ongoing research. In the DMRT based approach , cross-polarization emerges from non-spherical shapes of aggregated sphere clusters. A different route to cross-polarization can be taken via the discrete dipole approximation (DDA), e.g., from or , which principally accounts for multiple reflections and polarizations inside a given snow volume. DDA requires the full three-dimensional description of the microstructure, which can be provided by μCT. A comparison to such a model could further elucidate the justification and the value of the parameter q.

Another parameter chosen empirically is the mean slope of surface undulations m. In principle, this parameter could be obtained from the analysis of the surface height, similar to what has been done in and for fresh snow. In a simple reasoning, the mean squared slope m can be expressed as the ratio between the standard deviation of the surface height and the lateral correlation length of the height correlation function. According to m would then take a value of 0.14 for fresh snow which is in the same order of magnitude as applied in our simulations (m=0.1). This small-scale roughness of the snow surface is not taken into account by the model, where only slight surface undulations are allowed.

The individual magnitudes of the specular and diffuse contributions are shown in Fig. . Towards higher frequencies, the diffuse component increases and outweighs the specular reflectivity from 12.5 GHz for v-pol and from 14.5 GHz for h-pol. Note that the magnitude of the specular component also depends on the undulation of the surface and therefore on the value of m. However, a pronounced impact of m is limited to small incidence angles (Fig.  and Sect. ) for reasonable values of m (m≈0.1).

In contrast to MEMLS3&a, MEMLS does not require free empirical parameters. In this regard, we attribute the fact that MEMLS3&a matches the SnowScat observation better than MEMLS the SodRad observations to the additional free parameters in MEMLS3&a, foremost ss0 and q. However, for the passive simulations, parameters also had to be chosen without direct experimental justification, namely s0 and rmsg, which determine the contribution of the snow–ground interface. This contribution is dominant and critical in our frequency range, as dry snowpacks thinner than ∼1 m are highly transparent. Unfortunately, the knowledge about the scattering at the ground surface is limited. Therefore, the snow–ground reflectivity s0 was modeled using the model of . This model is an empirical parametrization of the Fresnel formula depending on the standard deviation of the soil surface height rmsg and the soil permittivities. For the soil permittivities, provide experimental data and an empirical model based on experimental data, but dielectric models for the permittivities of frozen soils are still under development. For rmsg of the soil below the snowpack no measurements were available. In addition, the model of does not account for vegetation, which is in our case consistent with the argument on transparency given above. We note that estimating the snow–ground reflectivity is critical for all microwave models, which was also concluded from recent experiments . However, at 10 GHz, the frequency which is most influenced by the soil; MEMLS and SodRad were in good agreement.

In contrast, the mismatch between model and measurements was largest at 36 GHz and is most sensitive to details of the snow microstructure. MEMLS assumes an exponential fit of the density correlation function of the snow microstructure. The exponential fit is a reasonable starting point but small deviations can have a large influence on scattering. As detailed by , the correlation function of snow can take different shapes and its representation by means of a single correlation length might be inappropriate. Instead the Teubner–Strey form, a two-scale form for bicontinuous media might be more appropriate. The inclusion of other types of correlation functions into MEMLS is possible by adapting the calculation of the scattering coefficient. We thus believe that the present model provides a suitable test case to investigate the impact of more sophisticated representations of the snow microstructure.

We further tried to assess the influence of the spatial variability of the snowpack. The standard deviation obtained from the 15 MEMLS runs is 8 K at 36.5 GHz, h- and v-pol, implying a non-negligible influence of the location of the in situ snow measurements on the modeled brightness temperatures.

We also found that the higher values measured by SodRad throughout the whole frequency range at h-pol for an azimuth angle of 140∘ indicate an effect of the surrounding environment, such as trees, which were closer to the field of view at this azimuth angle. The spatial variability of the snowpack together with the influence of the environment is potentially able to bias simulated and measured brightness temperatures.

The degree of complexity of existing models simulating microwave backscattering from snow range from single-layer approaches to numerical solutions of Maxwell's equations . In this context, we propose MEMLS3&a as a model of intermediate complexity. In contrast to the HUT model , which has comparable complexity, MEMLS avoids traditional grain size as input parameter, which is prone to uncertainties in the visual estimation method . The advantage of MEMLS3&a (as well as MEMLS) is the correlation length as microstructural quantity, which can be obtained from objective measurements without conversion and, given the SMP retrieval method, with high efficiency in the field.

Presently, models differ not only in the representation of snow microstructure but also in the solution of the radiative transfer or the type of interfaces between the layers, which makes it difficult to attribute the discrepancies in model performance to a particular part of the model. A comparison by of at least the passive models showed that no model was able to reproduce all of the investigated microwave observations. For a detailed model assessment in view of future developments, various effects (spatial variability, snow microstructure, soil) must be isolated. A promising way is by using measurements of specifically prepared snow slabs, as already presented by . Together with complete 3-D microstructural information, these types of idealized experiments will allow us to minimize spatial variability, avoid the influence of the ground and compare different microstructural concepts for scattering coefficients. Together with available multi-layer models like MEMLS3&a, DMRT-ML or the DMRT-QMS package , this will clarify our understanding of the processes involved in microwave emission and scattering of snow.

We adapted the MEMLS to include backscattering and presented a detailed description of the relevant parameters and their derivation. The reflectivity was decomposed into diffuse and specular components, and the snowpack was allowed to be slightly undulated. This procedure could be applied to other passive microwave models as well. Model simulations were in reasonable agreement with scatterometer observations, if the specular snow–ground reflectivity ss0 and the cross-polarization ratio q were chosen accordingly. We found that the contribution of the snow–ground interface is a critical parameter, which needs further investigation. The empirical formulation of the cross-polarization ratio q is a limitation with respect to other existing microwave models. MEMLS3&a is a model of intermediate complexity, which avoids fitting procedures of the scattering efficiency of snow in combination with SMP or μCT measurements. This eliminates a main uncertainty of snow characterization in microwave remote sensing.

MEMLS3&a is integrated in the standard release of MEMLS as a separate sub-routine. Both versions, active and passive are built on the same set of core functions.