We present an effort to validate a previously
developed radiative transfer model, and an innovative Bayesian inversion method designed to retrieve
the properties of slab-ice-covered surfaces. This retrieval method is adapted
to satellite data, and is able to provide uncertainties on the results of the
inversions. We focused on surfaces composed of a pure slab of
water ice covering an optically thick layer of snow in this study. We sought to retrieve
the roughness of the ice–air interface, the thickness of the slab layer and
the mean grain diameter of the underlying snow. Numerical validations have
been conducted on the method, and showed that if the thickness of the slab
layer is above 5

Various species of ice are present throughout the solar system, from water
ice and snow on Earth to nitrogen ice on Triton

Radiative transfer models have proven essential for retrieving such
properties

Compact slabs have very different radiative properties from closely packed
granular media, and radiative transfer models have been developed to study
their characteristics (e.g.,

In the present article, we test the accuracy of this approximated model on laboratory spectroscopic measurements of the bidirectional reflectance distribution function (BRDF) of pure water ice on top of snow. At the same time, we present an innovative Bayesian inversion method that was developed to retrieve the properties of solar system compact ice using satellite spectro-imaging data. In this paper, the term “inversion” is used and means “solving the inverse problem”. The slabs that are studied contain no impurity, and the surface properties we seek to retrieve are the thickness of the ice, the roughness of the ice–air interface and the grain diameter of the underlying snow. The main goals of this work are (i) to test the ability of the model to reproduce reality and (ii) to propose an inversion framework able to retrieve surface ice properties, including uncertainties, in order to demonstrate the applicability of the approach to satellite data.

We present a set of spectro-goniometric measurements of different water ice
samples put on top of snow using the spectro-radiogoniometer described in

The inversion algorithm that is tested is based on lookup tables (LUTs) that minimize the computation time of the direct model. The solution is formulated as a probability density function, using Bayesian formalism. This strategy is very useful for analyzing hyperspectral images. The thickness of ice estimated from the inversion is compared to real direct measurements. In addition, the power distribution in the specular lobe, which is determined by the roughness of the surface, is adjusted to demonstrate that the model is able to reasonably fit the data with a consistent roughness value.

The model by

Scheme of the surface representation in the radiative transfer model
applied to the laboratory measurements.

Illustration of the radiative transfer in the surface medium.
Anisotropic transits are represented in red.

Figure

In this work, the radiative transfer model described in

The bidirectional reflectance spectra were measured using the
spectro-radiogoniometer from Institut de Planétologie et d'Astrophysique
de Grenoble, fully described in

The ice samples were obtained by sawing artificial columnar water ice into circular sections

The specular reflectance was measured on a slab sample on top of Arselle snow,

The diffuse contribution was measured on three samples of different slab
thickness. The three thicknesses were measured on different locations of the
samples with a caliper before doing the spectro-goniometric measurement,
resulting in

Diffuse reflectance spectra of natural snow only were also measured before putting a slab on top of the snow. The objective was to estimate the effect of a slab layer on the BRDF.

We designed an inversion method aimed at massive data analysis. This method
consists of two steps: first, the generation of a synthetic database that is
representative of the variability in the model, and then the comparison with
actual data. To generate the synthetic database, we used optical constants
for water ice at

In order to validate the model on the specular reflection from the slab, we
chose to use the reflectance at

We then focused on the validation in the spectral domain, for the diffuse
contribution. We used the estimation of the roughness parameter

The inversion consists in estimating the model parameters

The actual observation is considered as a priori information on the data

To study the specular lobe, we have to consider the whole angular sampling of
the spot as a single data measurement. Similar to the “pixel” (contraction
of

The sampling of the parameter space, i.e., the lookup table, must correctly
represent the variability of the model according to its parameters. For this
study, we sampled the roughness parameter from

When out of the specular lobe, the radiation is controlled by the complex
transfer through the media (slab ice and bottom snow). The experimental
samples were made of pure water slab ice, without impurity. We generated the
lookup table for every measurement geometry at very high spectral resolution
(

For the inversion, we used the same method as previously described, with
a likelihood function

Secondly, we inverted the BRDF as a whole, for each sample. For this method,

In order to numerically validate the inversion method described above, two kinds of tests were conducted. First, we applied Gaussian noise and inverted every spectrum in the synthetic spectral database. We show that with a negligible noise, the parameters are always correctly retrieved with negligible uncertainties, and as the level of noise on the data increases, so do the uncertainties on the results. Secondly, we generated spectra for parameters that were not sampled in the database and tried to recover their characteristics successfully.

On Fig.

Figure

Normalized stacks of 1000 a posteriori PDFs for the grain diameter
of the snow, when conducting the inversion on synthetic data, with added
random noise. The legends indicate the values for the grain diameter used to
create the synthetic data.

Measured and simulated reflectance factor around the specular
geometry at

Figure

With the level of noise at

Figure

We performed the inversion taking into account 63 angel measurements, but for
the sake of readability, Fig.

The a posteriori probability density function for the roughness
parameter

Moreover, the value retrieved by the inversion is very well constrained as the probability density function is very sharp. This means that we have an a posteriori uncertainty on the result that is very low. The quality of the reproduction of the specular lobe by the model suggests that the surface slope description is a robust description despite its apparent simplicity. In particular, one single slope parameter is enough to describe this surface.

To reproduce diffuse reflectance we used the results obtained with the
specular measurements and assumed that the roughness of the samples did not
change much between the experiments. The range of variations in roughness
should be negligible in the spectral analysis. We simulated slabs over snow,
with the grain diameter of the substrate and the thickness of the slab as
free parameters. Figure

Reflectance factor spectra for the measure and the simulation at the
maximum likelihood, and for the geometry for which maximum likelihood was
highest, for each sample: at incidence

Marginal a posteriori probability density functions for

Figure

Results of the inversions and measurements with error bars at

Figure

Measured and simulated reflectance factor at

Marginal a posteriori probability density functions for

The two main goals of this work were (i) to develop and validate an inversion method that is adapted to the treatment of massive and complex datasets such as satellite hyperspectral datasets, and (ii) to partially validate a previously developed radiative transfer model.

The first criterion is the speed of the whole method, including the direct
computation of the LUT and the inversion. The lookup tables used for this
project were computed in 150 s for the roughness study (1763 wavelengths
sampled, 30 933 spectra) and

A second aspect is the reliability of the inversion method, regardless of the
direct model. For a given level of measurement errors, the user shall know
the quality of the retrieval of any parameter. The Bayesian statistics in our
method allowed us to determine that the thicknesses estimated in this work
were reliable, with a

A third point to be discussed is the capability of the model to reproduce
reality. Section

This work show that the radiative transfer model and the inversion method
tested are adapted to retrieve the characteristics of an ice slab overlaying
a granular layer. In particular, they are adapted to the study of the Martian

The aim of this present work is to validate an approximate radiative transfer
model developed in

The large uncertainties in the grain diameter inversion demonstrate that the bottom condition is less important than the slab for the radiation field at first order, as predicted by the synthetic tests conducted. The inconsistency between the a posteriori PDFs of the grain diameters for experimental data and numerical tests stresses that synthetic tests must be performed in order to determine which quantities can be retrieved or not in the context of the study, and to precisely calculate the expected uncertainties.

The comparison of the a posteriori uncertainties on the thickness of the
slab and of the grain diameter of the snow substrate illustrates the fact
that those uncertainties depend both on the constraint brought by the model
itself and on the uncertainty introduced into the measurement, which only the
Bayesian approach can handle. The use of Bayesian formalism is thus very
powerful in comparison with traditional minimization techniques. We propose
here a fast and innovative method focusing on massive inversions, and we
demonstrated that it is adapted to remote sensing spectro-imaging data
analysis. The radiative transfer model used in this study was proven
appropriate to study the superior slab layer, but not the bottom one, unless
the top layer is thin (thinner than

The experimental spectra used in this work are publicly available on the SSHADE (previoulsy GhoSST) database. The following links are permanent.

Brissaud, O., Schmitt, B., and Douté, S.: Vis-NIR spectral bidirectional
reflection distribution fonction of sintered snow (Arselle) at

Brissaud, O., Schmitt, B., and Douté, S.: Vis-NIR spectral bidirectional
reflection distribution fonction of slab ice 1.42 mm thick on sintered
snow (Arselle) at

Brissaud, O., Schmitt, B., and Douté, S.: Vis-NIR spectral bidirectional
reflection distribution fonction of slab ice 7.45 mm thick on sintered
snow (Arselle) at

Brissaud, O., Schmitt, B., and Douté, S.: Vis-NIR spectral bidirectional
reflection distribution fonction of slab ice 12.5 mm thick on sintered
snow (Arselle) at

Brissaud, O., Schmitt, B., and Douté, S.: Vis-NIR spectral bidirectional
reflection distribution fonction around specular angle 50

Andrieu, F., Douté, S., Schmidt, F., Schmitt, B., and Brissaud, O.: Vis-NIR
bidirectional reflection spectrum (i=40, e=10

Andrieu, F., Douté, S., Schmidt, F., Schmitt, B., and Brissaud, O.: Vis-NIR
bidirectional reflection spectrum (i=40, e=20

Andrieu, F., Douté, S., Schmidt, F., Schmitt, B., and Brissaud, O.: Vis-NIR
bidirectional reflection spectrum (i=60, e=0

The authors would like to thank the four anonymous reviewers for their comments that helped to greatly improve this paper. This work was supported by “Institut National des Sciences de l'Univers” (INSU), the “Centre National de la Recherche Scientifique” (CNRS) and “Centre National d'Etude Spatiale” (CNES), through the “Programme National de Planétologie” and MEX/OMEGA Program. Edited by: E. Hanna Reviewed by: four anonymous referees