The Snow Microwave Radiative Transfer (SMRT) thermal emission and backscatter model was developed to determine uncertainties in forward modeling through intercomparison of different model ingredients. The model differs from established models by the high degree of flexibility in switching between different electromagnetic theories, representations of snow microstructure, and other modules involved in various calculation steps. SMRT v1.0 includes the dense media radiative transfer theory (DMRT), the improved Born approximation (IBA), and independent Rayleigh scatterers to compute the intrinsic electromagnetic properties of a snow layer. In the case of IBA, five different formulations of the autocorrelation function to describe the snow microstructure characteristics are available, including the sticky hard sphere model, for which close equivalence between the IBA and DMRT theories has been shown here. Validation is demonstrated against established theories and models. SMRT was used to identify that several former studies conducting simulations with in situ measured snow properties are now comparable and moreover appear to be quantitatively nearly equivalent. This study also proves that a third parameter is needed in addition to density and specific surface area to characterize the microstructure. The paper provides a comprehensive description of the mathematical basis of SMRT and its numerical implementation in Python. Modularity supports model extensions foreseen in future versions comprising other media (e.g., sea ice, frozen lakes), different scattering theories, rough surface models, or new microstructure models.

The number and diversity of spaceborne observations from passive and active
microwave sensors over snow-covered regions has considerably increased over
the last 3 decades. Due to the demand for global monitoring of the cryosphere
and its change, numerous algorithms have been developed to retrieve
geophysical information on snow cover extent

Several physically based models have been developed previously mainly for
passive microwave remote sensing, including HUT

As a remedy, more and more studies include predictions from different
models

The representation of snow microstructure is critical since it immediately
constrains the choice of formulation to compute the scattering coefficient.
Several empirical formulations of the scattering coefficient have been
developed as a function of traditional grain size

The IBA developed by

All examples mentioned above indicate a clear demand for a modular and
extensible approach that unifies existing knowledge and facilitates efficient
intercomparisons of model ingredients with particular focus on the
representation of microstructure. To this end we developed the Snow Microwave
Radiative Transfer (SMRT) model as a versatile tool to compute
backscattering and brightness temperature (active–passive mode) from
multilayered media, composed of bi-continuous, random microstructures
(typically snow or bubbly ice), overlying a reflective surface (typically
soil, water, or ice). The originality of this new model is the flexibility for
the user to select among various electromagnetic or microstructure
formulations at different stages of the forward modeling problem. SMRT
includes IBA, DMRT, and independent Rayleigh scattering theories to compute
the scattering and absorption coefficients and the phase function. When using
IBA, it is possible to choose between several representations of isotropic
microstructures that are prescribed by analytical forms of the ACF. This is
complemented by several soil model implementations and permittivity
formulations. Additionally, language bindings are implemented to facilitate a
direct comparison with widely used models (DMRT-QMS, MEMLS, and HUT) using
their original code. In short, SMRT is designed to enable easy and rigorous
intercomparison and exploration of electromagnetic theories, common models,
and microstructure representations. SMRT version 1.0 is written in Python
(

The paper is organized as follows. The next section gives an overview about the model architecture, the most important formulations, the code structure, and basic usage. In the third section we present an intercomparison of SMRT with other models and explore the equivalence between different microstructures. The fourth section is dedicated to the discussion of limitations and perspectives. The last section concludes the paper.

SMRT was designed to be easy to use and computationally efficient and to allow
exploration of the various approximations or formulations available for
computing snow scattering and emission in the microwave domain. Even though
the goal was to maximize flexibility and versatility, some specific choices
and compromises were nevertheless necessary: (i) SMRT is a radiative transfer
model. This implies that interlayer interferences and coherent effects are
neglected. It is not suitable for interferometric computation. (ii) SMRT
considers media composed of plane-parallel, horizontally infinite,
homogeneous layers and is therefore not suitable to compute
3-D effects. (iii) The current version is limited to isotropic
media at the microstructure scale as well as at the scale of the snowpack.
This means that microstructural anisotropy of snow is neglected

The model is centered around the radiative transfer equation to compute the propagation of radiative energy in the medium produced by thermal emission in the medium (passive mode) and received from the sky (radar beam in active mode and sky thermal emission in passive mode). In addition to the radiative transfer equation, the other main components include the electromagnetic model that describes electromagnetic behavior of snow (i.e., the effective refractive index or permittivity, absorption and scattering coefficients, and phase function) and the boundary conditions between layers (called interfaces hereinafter) and at the bottom interface (called substrate hereinafter). All these components are well isolated in the code and various formulations from the literature are available. Here, only the common elements are presented; the switchable formulations are described in the following sections and appendix.

The model solves the time-independent
radiative transfer equation assuming a horizontally homogeneous medium with
isotropic snow at the microscopic level this is

Multilayered medium modeled by SMRT. The incident radiation

Further assuming that (i) the medium is azimuthally symmetric and (ii) the
medium is composed of homogeneous layers (Fig.

The continuity conditions at layer interfaces and the boundary
condition at the bottom interface are expressed by

SMRT architecture and main components. The core components (blue) are fixed and contain no scientific code in contrast to the switchable and extensible components (orange), which define the snowpack and model configurations.

Given the main governing equations (Eqs.

The implementation of the IBA in SMRT closely
follows the original work of

Equations (

Because the IBA phase matrix in the 1–2 frame is diagonal and the fourth component of the rotation matrix is orthogonal to the three others, the fourth component of the phase matrix in the main frame is also orthogonal to the three others. Except if the full Müller matrix is required by the user, the radiative transfer equation can be solved considering only the three first components, thus reducing the computational cost. This is the way it is implemented in SMRT.

The scattering coefficient

For the absorption coefficient,

The effective permittivity is not only needed to compute the absorption
coefficient but also implicitly to compute the boundary reflection equations
(Eqs.

Different electromagnetic theories use different microstructure
representations. In the simplest setting of Rayleigh or independent Mie
scattering for a collection of spheres, the microstructure is solely
characterized by the sphere radius. The positions of the scatterers are
random and uncorrelated, meaning that interpenetration is possible. In DMRT
the microstructure is provided in terms of the Fourier transform of the
pair-correlation function

SMRT provides a unified and versatile vision of the microstructure
representation. Any microstructure model is defined by specifying the set of
required and optional parameters and by providing, at least for use with IBA,
an analytical expression of ACF, either for the real-space form or its
Fourier transform (or for both). Though IBA requires only the Fourier
transform, see Eq. (

Overall, the microstructure representation in SMRT closely follows a
library concept as commonly employed for small angle scattering
software such as in

SMRT components and different formulations available in version 1.0. Each microstructure can only be used with compatible electromagnetic models as indicated in parentheses.

Note that each microstructure model comes with its own microstructure
parameters. The exponential model (Eq.

The necessity of also including models that are defined via the real-space ACF mainly originates from the use of level-cut Gaussian random
field models in the context commonly termed bi-continuous DMRT

For running SMRT with DMRT theory, the SHS microstructure must be selected. In contrast, when using IBA, any of the above microstructure models can be selected.

The model implementation is highly modular to allow switching among several
formulations at each stage of the computation and adding new formulations
defined by users. Another feature is the extensive use of default behaviors
to facilitate an easy use by beginners but still allow experts to set
advanced formulations for specific investigations or sensitivity
studies, for example. The code is carefully encapsulated; each “science” component
(indicated by the orange color in Fig.

To illustrate the mode of operation of the model it is instructive to relate the instructions of a tiny but
fully functioning code snippet to the model operations carried out
in the background:

In the second step, the definition of the model is completed by selecting the
electromagnetic theory (that computes the scattering and absorption
coefficients, phase matrix, and effective permittivity) and the radiative
transfer solver. As mentioned before, some electromagnetic theories are only
compatible with particular microstructure models, e.g., DMRT only works with
SHS and Rayleigh works with any microstructure that defines a radius but
inherently considers independent spheres. For solving the radiative transfer
equation, only the discrete ordinate and eigenvalue (DORT) method is
currently implemented, based on

The model is implemented in Python (2.7

In addition, we provide different tools for convenience: (1) to facilitate convenient computation of time series or sensitivity study by a few, clear-cut lines of code the model can be run on lists of different snowpacks. (2) To foster comparisons between SMRT and other common existing models (MEMLS, DMRT-QMS, and HUT), we provide language bindings to seamlessly run these models within SMRT, which use the prescribed snowpack in SMRT and collect results as if they were produced by SMRT. This requires that the source code of these models is separately installed (they are not distributed with SMRT for licensing reasons). Note that this feature is currently limited to the passive mode.

As SMRT seeks to unify formulations from other models, a natural starting
point for the validation of SMRT is a model comparison (namely with DMRT-ML,
DMRT-QMS and MEMLS) to assess the validity of the implementation. This is
conducted in Sect.

For a sparse medium, i.e., when density tends to zero, many formulations must
show the same behavior as the independent spheres with Rayleigh or Mie
theory. In SMRT, it is possible to run several combinations of microstructure
and electromagnetic models as shown in Fig.

Scattering coefficient by several electromagnetic theories
(independent spheres, IND; non-sticky hard spheres, HS; sticky hard spheres,
SHS) as a function of density for sparse media described by various
microstructures. The sphere radius is
100

Comparison of brightness temperatures

We compare SMRT to results produced from original code of several DMRT
variants. Figure

Brightness temperature at V polarization (55

Comparison of brightness temperatures simulated by SMRT IBA with an
exponential autocorrelation function and MEMLS. The correlation length is
100

The previous results were obtained for small scatterers and moderate
stickiness, which is compatible with the short-range approximation. It is
therefore of interest to investigate the limits of this implementation. To
this end, Fig.

To overcome the restrictive range of validity of the DMRT QCA short range, and
considering that SMRT version 1.0 does not provide DMRT QCA in the long-range
approximation, an alternative strategy is to combine IBA with the SHS
microstructure model. Figure

Figure

Figure

Equipped with the confidence from the previous sections that SMRT is working as desired, we shall address an actual, open scientific question. Setting the correct microstructure parameters in microwave model simulations from in situ observations or snowpack simulations is notoriously difficult and nearly every study uses a different approach. To this end we demonstrate how the equivalence between different approaches can be investigated with SMRT.

The problem originates from the fact that high-level microstructural
characterization in terms of the ACF is commonly not available since complete
profiles of

These issues have been solved in different ways in literature. For the SHS
microstructure,

Grain size scaling factor

To explore if this latter approach is equivalent to choose an optimal
stickiness value, we use SMRT to find the equivalent microstructure
representations for non-sticky spheres with grain size scaling and sticky
spheres. In the following, equivalent microstructures are interpreted as
microstructures with the same density but different size parameters that produce
the same electromagnetic behavior. This is exemplified by using SMRT IBA and
matching brightness temperatures at

Though the approach of using a stickiness close to 0.1 seems more physical
compared to an empirical scaling factor, it also has weaknesses. Natural snow
is composed of grains with variable size, which more resembles a collection
of spheres with a distribution of radii (i.e., polydisperse spheres). Such
dispersion is important and generally leads to increased scattering compared
to the medium with monodisperse spheres with the mean radius of the
polydisperse spheres

The exponential autocorrelation is a different and attractive solution
because it involves only two parameters that should be fully determined by
density and SSA. However, in practice a “hidden” third parameter must be
introduced to empirically scale the correlation length in the Debye relation

Scaling factor

The numerical experiments facilitated by SMRT from this section show
how different studies, which were hitherto not amenable to a comparison
due to apparently different approaches, are now comparable and can be
shown to be nearly equivalent for particular parameter choices.
Moreover the results unambiguously show that density and SSA are not
sufficient to appropriately characterize snow microstructure for
microwave modeling purposes and that the sensitivity to a third
parameter is highly significant. Until alternative measurement
techniques or progress in modeling the microstructure evolution are
available, the initialization of microstructure models relies on

SMRT version 1.0 bears some limitations that are inherent to the architecture
as discussed in Sect.

The scope of SMRT is currently limited to a snowpack over a surface (called
substrate), which is a common approach for some applications such as soils,
but may be inappropriate for other snow-covered environments in which
volume scattering, layering within the substrate, or temperature heterogeneity
may be important. For instance snow-covered sea ice or frozen lakes need to
account for bubbly and salty ice with a nonuniform temperature profile.
While the generic plane-parallel layered structure in SMRT and the DORT
solver are readily suited for this kind of modeling, the electromagnetic
behavior of these materials needs to be additionally implemented, which is
technically easy due to the modular architecture. Bubbly ice

Considering soil as a volume scattering medium or accounting for
inhomogeneous temperatures or wetness can be treated within DMRT and
layered radiative transfer

Accurate simulations of snow on the ground in active mode would
require more advanced surface scattering models than implemented in the current
version. SMRT inherits from the soil modules implemented in DMRT-ML
and previously in HUT and MEMLS, which were tailored to the passive
mode. These modules mainly compute a specular reflection while a
faithful backscatter computation is required for the active
mode. DMRT-QMS includes an advanced rough surface treatment from
independent numerical simulations

A strong assumption in SMRT version 1.0 is the isotropy of the
microstructure. Some types of snow have been shown to be highly anisotropic,
especially due to differences between the vertical and horizontal directions

Some limitations of SMRT are inherent to the radiative transfer equation,
which does not keep track of the absolute phase. This obviously prevents
interferometric calculations and may be restrictive when the layer thickness
is smaller than the wavelength of the microwaves, that is, at low frequencies

Another limitation concerns simulations of altimetric signal or
frequency-modulated continuous-wave radar. The radiative transfer
equation solver available in SMRT version 1.0 considers the stationary
radiative transfer equation (Eq.

Finally we acknowledge that the Python implementation of SMRT bears some peculiarities. By extensively using Python dynamic capabilities, the model computation is probably less efficient than specialized code, even though numerically critical code is delegated to optimized libraries through SciPy. Because of Python, the model may be inadequate for high-performance computation. In this case SMRT may still be useful for prototyping and determining the optimal subset of formulations that could then be implemented in compiled language since a numerical reference greatly helps to achieve such an optimization step. Moreover, it is worth noting that the Python ecosystem for high-performance computing is fast improving and that SMRT code may be parallelized in the future.

A new radiative transfer model to simulate emission or radar echo from a snowpack has been presented in this paper. It is built around the radiative transfer equation and specifically tailored to model snow but in the future also other plane-parallel media in the cryosphere. SMRT differs from other models in its scope in many aspects. SMRT is not a new model with a more advanced theory, it is rather a repository of established formulations or widely used model configurations that can be easily interchanged. The novelty is thus to allow testing of different existing configurations and exploration of new ones, in particular regarding the microstructure. Using SMRT, we have highlighted the equivalence between different widely used microstructure representations (SHS and exponential autocorrelation function) and different approaches proposed in the literature to run simulations based on in situ measurements. These results show that to fully describe snow in microwave models requires at least three main metrics, the density, grain size, and another parameter characterizing larger-scale structural correlations of the ice matrix. The fact that these latter properties are presently inaccessible by other measurements or snowpack modeling contributes to the uncertainties in microwave simulations, and actually constitutes one of biggest challenges to solve.

The numerical validation of SMRT has shown the numerical equivalence with DMRT-ML for the DMRT QCA-CP electromagnetic formulation and has shown close results with DMRT-QMS under DMRT QCA under the small scatterer assumption in passive and active modes even though small differences remain unexplained. Larger differences are observed with respect to MEMLS, which we attribute to the six-flux method used by MEMLS to solve the radiative transfer equation. Regarding HUT, SMRT contains no sufficiently similar configuration to perform a validation. Nevertheless the language binding to the HUT code has been included for future comparisons with other configurations. Not all SMRT configurations and available microstructure representations have been tested in this study because of the large number of possible combinations; this is left to future work.

Several limitations of SMRT version 1.0 have been outlined that can be
readily overcome by model extensions which are supported by
modularity. The developed code is highly structured for each step of
the radiative transfer calculation. The model is designed to
facilitate future developments of existing and new formulations
without changing existing code, which should foster community-based
contributions and consolidate SMRT as a repository of the community
knowledge. Future work includes implementation of new features to
account for different media (e.g., sea ice), variants of
electromagnetic models (e.g., DMRT QCA long range) or radiative
transfer solver (e.g., six-flux solver or time-resolved radiative
transfer equation) to increase the scope of applications. In this
paper we focused on two widely used microstructure representations;
SMRT already includes other representations and new ones, such as empirical autocorrelation functions derived from

The code to generate the figures is released under the
LGPLv3 open-source library and is available at

The discrete ordinate and eigenvalue method (DORT) is a widely used method to
solve the radiative transfer equation for multilayered media. It is
particularly recommended when optical depth is thick and multiple scattering
is significant

The transformation of the radiative transfer equation into a system of linear
ordinary differential equations requires the discretization of the azimuthal
and zenith angular dependences. The

The

Introducing the matrix–vector notation for
the linear solver, we define an intensity vector

This is a nonhomogeneous system of ordinary first-order differential equations with constant coefficients.

Within each layer

At this point, the problem consists of determining the

The additional boundary condition for the bottommost layer

Finally, using Eq. (

In active mode, the downwelling beam with incidence angle

All boundary conditions provide

For the passive microwave case, only the first mode

This well-known approximation is mostly given for reference as it only
applies to the sparse medium. In this case, the effective permittivity is equal
to that of the background:

Formulations for DMRT QCA and QCA-CP are available in many studies and
briefly recalled here for completeness for the monodisperse
sphere and under the short-range approximation.
The QCA-CP version is formulated according to

The effective permittivity in the QCA approximation is given by

In the short-range approximation, the phase matrix of DMRT QCA and DMRT QCA-CP is the same as for independent Rayleigh scatterers.

The three authors have contributed to model development, validation, and writing of the paper.

The authors declare that they have no conflict of interest.

We acknowledge the European Space Agency, which supported this model development under ESTEC contract no. 4000112698/14/NL/LvH, with a contribution from the NERC National Centre for Earth Observation. We would like to thank Christian Mätzler for abundant advice. Edited by: Dan Goldberg Reviewed by: Christian Mätzler and one anonymous referee