TCThe CryosphereTCThe Cryosphere1994-0424Copernicus PublicationsGöttingen, Germany10.5194/tc-10-2113-2016Retrieving the characteristics of slab ice covering snow by remote sensingAndrieuFrançoisSchmidtFrédéricSchmittBernardhttps://orcid.org/0000-0002-1230-6627DoutéSylvainBrissaudOlivierGEOPS, Univ. Paris-Sud, CNRS, Université Paris-Saclay, Rue du Belvédère, Bât. 504-509, 91405 Orsay, FranceUniv. Grenoble Alpes, IPAG, 38000 Grenoble, FranceCNRS, IPAG, 38000 Grenoble, FranceF. Andrieu (francois.andrieu@u-psud.fr)15September20161052113212815July201529September20153July201626July2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://tc.copernicus.org/articles/10/2113/2016/tc-10-2113-2016.htmlThe full text article is available as a PDF file from https://tc.copernicus.org/articles/10/2113/2016/tc-10-2113-2016.pdf
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 mm and the noise on the signal is above 3 %, then
it is not possible to invert the grain diameter of the snow. In contrast,
the roughness and the thickness of the slab can be determined, even with high
levels of noise up to 20 %. Experimental validations have been conducted on
spectra collected from laboratory samples of water ice on snow using a
spectro-radiogoniometer. The results are in agreement with the numerical
validations, and show that a grain diameter can be correctly retrieved for
low slab thicknesses, but not for bigger ones, and that the roughness and
thickness are correctly inverted in every case.
Introduction
Various species of ice are present throughout the solar system, from water
ice and snow on Earth to nitrogen ice on Triton , not to
forget carbon dioxide ice on Mars . The physical properties of the
cover also have an impact on the energy balance; for example, the albedo
depends on the grain size of the snow , on the
roughness of the interface , on the presence or lack of impurities and
the physical properties of impurities . The study and monitoring of theses
parameters is key to constraining the energy balance of a planet.
Radiative transfer models have proven essential for retrieving such
properties and their evolution at a large scale,
and different families exist. Ray-tracing algorithms, such as those described
in for snow, for compact
polycrystalline ice or for particulate media such as
rough ice grains in an atmosphere, simulate the complex path of millions of
rays into the surface. Such modelings are booming due to the positive
comparison between models and exact calculations (e.g., ). Analytical solutions of the radiative transfer in
homogeneous granular media have been developed, for example, by
and . They are fast, but when the
surface cannot be described statistically as a mono-layer, they must be
combined with another family of techniques such as discrete ordinate methods
like DISORT . These methods have been widely studied on
Earth snow and
other planetary cryospheres , modeling
a granular surface. Compact polycrystalline ice has, however, been
recognized to exist on several objects: CO2 on Mars
, N2 on Triton and Pluto
and probably SO2 on Io
, as suggested by the very long light path lengths
measured, over several centimeters to decimeters . In particular, the Martian climate is
mostly controlled by a seasonal CO2 cycle that results in the
condensation and sublimation of deposits constituted of a layer of nearly pure CO2 ice up to 1 m
thick, possibly contaminated with H2O ice and regolith dust .
The monitoring of the evolution of these deposits' composition would bring
key pieces of information on the water ice cycle on Mars and on the dust
transport to the surface.
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 case of sea or lake ice. We developed an approximated model
that has the capability of being able to model a
layer of ice covering a surface with radically different optical properties,
for instance a different refractive index, unlike its predecessors. It was
originally designed for the study of Martian seasonal ice deposits, using
massive spectro-imaging datasets, such as the OMEGA or
CRISM datasets. For this purpose, it is semi-analytic and
implemented to optimize the computation time. However, the algorithm was
built to be adaptable to any other spectroscopic data, from terrestrial water
ice laboratory measurements, as is the case of this work, to the study of
SO2 ice on Io or N2 ice on Pluto using the
NIMS and RALPH datasets respectively.
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
. Three kind of experiments were conducted. First, the
BRDF of only one snow layer was measured, and then it was measured again after adding
a slab ice layer at the top. The objective was to test the effect of an ice
layer at the top of the snow on the directivity of the surface. Second, the
specular lobe was closely investigated, at high angular resolution, at the
wavelength of 1.5 µm, where ice behaves as a very absorbing media.
Finally, the bidirectional reflectance was sampled at various geometries on
61 wavelengths ranging from 0.8 to 2.0µm. In order to
validate the model, we did qualitative tests to demonstrate the relative
isotropization of the flux. We also conducted quantitative assessments by
using a Bayesian inversion method in order to estimate the sample thickness,
the surface roughness and the snow grain diameter from the radiative
measurements only. A comparison between the retrieved parameters and the
direct independent measurements allowed us to validate the model.
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.
Description of the model
The model by is inspired by an existing one described
in and , which simulates the bidirectional
reflectance of stratified granular media. It has been adapted to compact
slabs, contaminated with pseudo-spherical inclusions, and a rough top
interface. In the context of this work, we suppose a layer of pure slab ice,
overlaying an optically thick layer of granular ice, as described in
Fig. . The roughness of the first interface is
described using the probability density function of orientations of slopes
defined in . This distribution of orientations is fully
described by a parameter θ¯, which can be interpreted as a mean
slope parameter at the surface, in the case of small angles. The ice matrix
is described using its optical constants and its thickness.
Scheme of the surface representation in the radiative transfer model
applied to the laboratory measurements. h represents the slab thickness and
θ¯ represents the mean slope parameter used to describe the
surface roughness.
Illustration of the radiative transfer in the surface medium.
Anisotropic transits are represented in red. F is the incident
radiation flux, Rspec and RDiff are the
specular and diffuse contributions to the reflectance of the surface respectively,
rs is the Lambertian reflectance of the granular substrate and
R0 and T0 are the total reflection and transmission
factors of the slab layer respectively. A prime indicates an anisotropic transit. The
reflection and transmission factors are different in the cases of isotropic
or anisotropic conditions. The granular and slab layers are artificially
separated in this figure to help the understanding of the coupling between
the two layers. Top: illustration of the reflections and transmission at the
first interface, used in the calculations of Rspec and the
determination of the amount of energy injected into the surface. z is
the normal to the surface, Wf the local normal to
a facet, i and e are the incidence and emergence angles respectively and
ef is the local emergence angle for a facet. Each different
orientation of a facet will lead to a different transit length in the slab.
A more detailed description can be found in .
Figure illustrates the general principle
of the model. The simulated bidirectional reflectance results from two
separate contributions: specular and diffuse. The specular contribution in
the model is estimated from the roughness parameter, the optical constants of
the matrix and the apertures of the light source and the detector. In
practical applications, the optical constants of the ice matrix and the
optical apertures are known parameters. The specular contribution for a given
geometry only depends on the orientation of slopes at the surface, which is
fully determined by θ¯. The total reflection coefficient at the
first rough interface is obtained by integrating specular contributions in
every emergent direction, at a given incidence. This gives the total amount
of energy transmitted into the system constituted of the contaminated slab
and the substrate. The diffuse contribution is then estimated through solving
the radiative transfer equation inside this system under various hypotheses.
The following considerations are made. (i) The first transit through the slab
is anisotropic due to the collimated radiation from the source, and due to there being an isotropization at the second rough interface (i.e., when the
radiation reaches the semi-infinite substrate). For the refraction and the
internal reflection, every following transit is considered isotropic.
(ii) The geometrical optics is valid. The reflection and transmission factors
of the layers are obtained using an analytical estimation of the Fresnel
coefficients described in and , as
well as a simple statistical approach, detailed in . The
contribution of the semi-infinite substrate is described by its
single-scattering albedo. Finally, as the slab layer is under a collimated
radiation from the light source, and under a diffuse radiation from the
granular substrate, the resulting total bidirectional reflectance is computed
using adding–doubling formulas .
In this work, the radiative transfer model described in
is used to simulate the reflectance factor spectrum of a pure slab of water
ice covering a layer of snow, as represented in
Fig. . This spectrum may vary in terms of three parameters:
(i) the roughness θ¯ of the slab ice surface, which characterizes
the power distribution in the specular lobe; (ii) the thickness h of the
slab ice layer, which determines the absorption in the ice layer and
(iii) the grain diameter ⊘s of the snow, which determines the
absorption in the snow.
DataSpectro-radiogoniometer
The bidirectional reflectance spectra were measured using the
spectro-radiogoniometer from Institut de Planétologie et d'Astrophysique
de Grenoble, fully described in . We collected spectra in
the near-infrared spectrum at incidences ranging from 40 to 60∘,
emergence angles from 0 to 50∘ and azimuth angles from 0
to 180∘. The sample is illuminated with a 200mm
large monochromatic beam (divergence <1∘), and the
near-infrared spectrum covering the range from 0.800 to
4.800µm is measured by an InSb photovoltaic detector. This
detector has a nominal aperture of 4.2∘, which results in
a field of view on the sample that has a diameter of approximately 20mm. The minimum angular sampling of illumination and observation
directions is 0.1∘, with a reproducibility of
0.002∘. In order to avoid azimuthal anisotropy, the sample is
rotated during the acquisition. The sample rotation axis may be very slightly
misadjusted, resulting in a notable angular drift on the emergence measured
up to 1∘.
Ice BRDF measurements
The ice samples were obtained by sawing artificial columnar water ice into circular sections
20cm in diameter. These sections were put on top
of an optically thick layer of compacted snow that was collected in Arselle,
in the French Alps. The spectral measurements were conducted in a cold
chamber at 263 K. However, the ice and the snow were unstable in the
measurement's environment, due to the dryness of the chamber's atmosphere.
The grain size of the snow showed evolution, and the thickness of a given
slab showed a decrease of 0.343mmday-1. Each sample needed
an acquisition time of 10h. For each measurement, the ice slab was
sliced, and its thickness was measured in five different locations. It was
then set on top of the snow sample, and this system was put into rotation in
the spectro-radiogoniometer for the measurement. The sample completes a full
rotation (10s) during the measurement of the reflectance at one
wavelength and one geometry. As the surface is not perfectly planar, the
measured thickness is not constant. This results in an 2σ standard
deviation in the measurement of the thickness that ranges from 0.54 to
2.7mm in our study, depending on the sample.
Specular contribution
The specular reflectance was measured on a slab sample on top of Arselle snow, 12.51mm thick. This sample is described as sample 3 in the
next paragraph. The illumination was at an incidence angle of
50∘, and 63 different emergent geometries were sampled,
ranging from 45 to 55∘ in emergence and from
170 to 180∘ in azimuth. The specular lobe is
sampled every 1∘ in emergence and azimuth, and within
47 and 53∘ in emergence and
175 and 180∘ in azimuth.
Ice on snow diffuse reflectance spectra
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 h1=1.42±0.47mm, h2=7.45±0.84mm
and h3=12.51±2.7mm, respectively, for samples 1, 2 and 3, with
errors at 2σ. Sixty-one wavelengths were sampled ranging from 0.8 to
2.0µm. Spectra were collected on 39 different points of the
BRDF for the incidence, emergence and azimuth angles: 40,50,60∘, 0,10,20 and 0,45,90,140,160,180∘, respectively. This set of angles results in
only 39 different geometries because the azimuthal angle is not defined for
a nadir emergence.
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.
Method
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 270K. The 7K difference between the
actual temperature of the room and the temperature assumed for the optical
constants has a negligible effect. We combined the datasets of
and , making the junction at
1µm, the former set for the shorter wavelengths and the
latter for the wavelengths larger than 1µm.
In order to validate the model on the specular reflection from the slab, we
chose to use the reflectance at 1.5µm, where the ice is very
absorptive. Figures
and clearly demonstrate that there is a negligible
diffuse contribution in geometry outside the specular lobe from the sample
with a 12.51mm thick pure slab. Thus, the roughness parameter
θ¯ is the only one impacting the reflectance in the model. We
chose to invert this parameter first and validate the specular contribution.
We then focused on the validation in the spectral domain, for the diffuse
contribution. We used the estimation of the roughness parameter
θ¯ obtained earlier and the spectral data in order to estimate
the slab thickness and the grain size of the snow substrate. To do this, we
assumed that the roughness was not changing significantly enough to have
a notable impact on diffuse reflectance from one sample to another. This
assumption is justified by the fact that the different columnar ice samples
were made the same way, as flat as possible, and the low value of
θ¯ retrieved as discussed in the next section. It is confirmed by
the results of Sect. , which suggest a very low roughness,
as expected. Such low roughness parameters have a negligible influence on the
amount of energy injected into the surface.
Inversion strategy
The inversion consists in estimating the model parameters m (i.e., the
slab thickness, the roughness parameter, the snow grain diameter) from the
models F(m)) (the reflectance simulations) that best reproduce
the data d (the reflectance observations). Looking for these
parameters sets is called the inverse problem. showed
that this inverse problem can be mathematically solved by considering each
quantity as a probability density function (PDF). In nonlinear direct
problems, the solution may not be analytically approached. Nevertheless, it
is possible to sample the solutions' PDF with a Monte Carlo approach as shown
in , but this solution is very time-consuming.
The actual observation is considered as a priori information on the data
ρD(d) in the observation space D. It is assumed to be
an N-dimension multivariate Gaussian distribution
G(dmes,C‾‾),
centered on dmes with a covariance matrix
C‾‾. The measurements at any given
wavelength/geometry are supposed to be independent from each other, as each
measurement of one wavelength, at one geometry is done individually. The
matrix C‾‾ is thus assumed to be diagonal and
its diagonal elements Cii are σ12,…,σN2,
with σi being the standard deviations corresponding to the
uncertainties of each measurement. The a priori information on model parameters
ρM(m) in the parameters space M is independent of
the data and corresponds to the state of null information
μD(d) if no information is available on the parameters.
We consider a uniform PDF in their definition space M. The a posteriori PDF in
the model space σM(m) as defined by Bayes' theorem
is
σM(m)=kρM(m)L(m),
where k is a constant and L(m) is the likelihood function,
L(m)=∫DρD(d)θ(d∣m)μD(d)dd,
where θ(d∣m) is the theoretical relationship of the PDF for d
given m. We do not consider errors of the model itself, so θ(d∣m)=δ(F(m)) is also noted dsim for simulated data. Therefore, the
likelihood is simplified into
L(m)=G(F(m)-dmes,C‾‾),
and in the case of uniform a priori information ρM(m), the a posteriori PDF is
σM(m)=kL(m).
This expression is explicitly
σM(m)=k⋅exp-12×tF(m)-dmesC‾‾-1F(m)-dmes,
where the superscript t is the transpose operator that applies to
(F(m)-dmes). The factor k is adjusted to
normalize the PDF. The mean value of the estimated parameter can be computed
by
m=∫Mm⋅σM(m)dm,
and the standard deviation,
σm=∫Mm-m¯2⋅σM(m)dm.
In order to speed up the inversion strategy but keep the advantage of the
Bayesian approach, we choose to sample the parameter space M with regular
and reasonably fine steps, noted i. The likelihood for each element is
L(i)=exp-12×tdsim(i)-dmesC‾‾-1dsim(i)-dmes.
The derivation of the a posteriori PDF with such formalism for specular lobe
inversion and for spectral inversion is explained in the next sections.
Specular lobe
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 pictureelement), we choose to define the “angel”
(contraction of angularelement) as a single element in
a gridded angular domain. Interestingly, angel also refers to a supernatural
being represented in various forms of glowing light. A single angel
measurement could not well constrain the model, even at different
wavelengths. Instead, a full sampling around the specular lobe should be
enough, even at one single wavelength. We chose a wavelength where the
diffuse contribution was negligible in order to simplify the inversion
strategy. We chose to focus on the 1.5µm wavelength, as we
measured a penetration depth lower than 1mm and
thus much lower than the thickness of the used sample for this channel. We first generated
a synthetic database (lookup table), using the direct radiative transfer
model. We simulated spectra in the same geometrical conditions, for
a 12.5mm thick ice layer over a granular ice substrate constituted
of 1000µm wide grains. These two last parameters are not
important since the absorption is so high in ice, such that the main
contribution is from the specular reflection, and the diffuse contribution is
negligible (the penetration depth inside a water ice slab at the
1.5µm wavelength is lower than 1 mm).
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 0.1 to 5∘
with a constant step dθ¯=0.01∘. We used
a likelihood function L defined in Eq. (), where
dsim and dmes are
ngeom-element vectors, with ngeom the number of
angels (63 in this study). They respectively represent the simulated and
measured reflectance at a given wavelength in every geometry.
C‾‾ is a ngeom×ngeom matrix. It represents the uncertainties of the data. In this
case, we considered each wavelength independently, thus generating a diagonal
matrix, containing the level of errors given by the technical data of the
instrument described by . It corresponds at this
wavelength to 2% of the signal. The roughness parameter θ¯
returned by the inversion is described by its normalized PDF:
Pθ¯(i)=Lidθ¯∑jLjdθ¯=Li∑jLj.
The value θ¯(i) in the database with the highest probability
(maximum likelihood) corresponds to the sampling step that is closest to the
maximum of the a posteriori PDF. If the PDF is close to a Gaussian, then
the solution to the inverse problem can be estimated by its mean,
θ¯=∑iθ¯iLi∑iLi,
and associated standard deviations,
σθ¯=∑iθ¯i-θ¯2Li∑iLi.
We give error bars on the results that correspond to two standard deviations,
and thus a returned value for θ¯ that is
θ¯r=θ¯±2σθ¯.
Diffuse spectra
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
(4.10-2nm), and then downsampled it at the resolution of the
instrument (2nm). We sampled the 17 085 combinations of two
parameters for the 39 different geometries: p1, the thickness of the slab
from 0 to 20mm (noted i=[1,201]) every 0.1mm (noted
dp1), and p2, the grain size of the granular substrate from
2 to 25µm every 1µm and from 25 to
1500µm every 25µm (noted j=[1,85] and the
corresponding dp2(j)). The parameters' space is thus irregularly
paved with dp(i,j)=dp1.dp2(j).
For the inversion, we used the same method as previously described, with
a likelihood function L that is written as in Eq. (). Two
different strategies were adopted. First, we inverted each spectrum
independently. Thirty-nine geometries were sampled (described in
Sect. ), and thus we conducted 39 inversions
for each sample. This time dsim and dmes
are thus respectively the simulated and measured spectra. Therefore
dsim and dmes are
nb-element vectors, where nb is the number of
bands (61 in this study) and C‾‾ is
a nb×nb matrix. As previously done (see
Sect. ), we considered each wavelength independently, thus
generating a diagonal matrix, containing the level of errors given by the
technical data of the instrument given by . The error is
a percentage of the measurement, and thus C‾‾
is different for every inversion.
Secondly, we inverted the BRDF as a whole, for each sample. For this method,
dsim and dmes are the
simulated and measured BRDF respectively and are thus nb×ngeom-element vectors (2379 in this study), where
nb is the number of bands (61 in this study) and
ngeom is the number of geometries (39 in this study) sampled.
C‾‾ is a nb×ngeom×nb×ngeom diagonal matrix, containing the errors of the data.
We represent the results the same way as previously, but there are two
parameters to inverse. For the sake of readability, we plot the normalized
marginal probability density function for each parameter. We present here the
general method for the inversion of np=2 parameters: the slab thickness
and the grain size of the substrate. The PDF for the two parameters p is
described by
Pp(i,j)=Li,jdpi,j∑i∑jLi,jdpi,j.
For a given parameter p1, the marginal PDF of the solution is
Pp1(i)=L′idp1i∑i∑jLi,jdpi,j,
with L′i=∑jL(i,j)dp2(j). The value
p1(i) in the database with the highest probability (maximum likelihood)
corresponds to the sampling step that is closest to the maximum of the a
posteriori PDF. The marginal PDF can be described by the mean,
p1=∑ip1iL′idp1i∑i∑jLi,jdpi,j,
and the associated standard deviation,
σp1=∑ip1i-p12L′idp1i∑i∑jLi,jdpi,j.
As for the roughness parameter, we give error bars on the results that
correspond to two standard deviations, and thus a returned value for p1
that is
p1r=p1±2σp1.
Numerical validations of the inversion 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.
(a) Reflectance factor at a wavelength of
λ=1.4µm vs. phase angle for snow only (black crosses)
and the same snow but covered with a 1.42±0.27mm water ice slab
(red squares). (b) Same data but normalized by the value at a phase
angle α=20∘.
On Fig. each curve corresponds to a the normalized sum of 1000
a posteriori PDFs for the grain diameter of the underlying snow resulting
from 1000 random noise draws of the same 2% level. We name this
normalized sum of PDF a “stack” in the remainder of this article.
Figure a is obtained for a low slab thickness of 1mm.
In this case, the grain diameter of the snow can be correctly estimated: the
PDF are centered on the correct value and the dispersion suggests an a
posteriori uncertainty lower than the retrieved value. When the thickness of
the slab layer increases, so does the a posteriori uncertainty on the
estimation of the grain diameter. For a slab thickness of 5mm
(Fig. b), the a posteriori uncertainty is of the same order as
the estimated value, meaning that the grain diameter cannot be retrieved. The
grain diameter of the snow thus cannot be retrieved for slab thicknesses
greater than 5mm.
Figure a represents the stack of 1000 a posteriori PDFs for the
thickness of the ice layer. These PDFs do not depend on the grain diameter of
the snow, but only on the thickness itself and the level of noise. It shows
that the thickness can be estimated, in the experimental conditions (2%
noise level), with an uncertainty of 2% for lowest thicknesses to
5% for highest ones. All obtained a posteriori PDFs for the thickness
were very close to a Gaussian function. We were thus able to sum them up by their means and standard
deviations, allowing us to plot, for example, the uncertainty of the
thickness estimation as a function of the thickness that we want to estimate
(Fig. b) and of the level of noise on the data (Fig. ).
Figure b shows the uncertainty (at 2σ) on the estimation
of the thickness of the slab layer as a function of the thickness itself, in
the experimental conditions described by , which means a
2% noise level on the signal. This relative uncertainty does not depend
on the thickness in the range of values tested. The low values for
thicknesses below 1mm are an effect of the discretization in the
LUT: the thickness has been sampled every 0.1mm. Below
1mm, this sampling step is large relatively to the values itself
and ranges from 10 to 100%. The relative uncertainty that we expect to
be about 5% is then no longer measurable, and the value drops to 0.
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. (a) The ice layer is 1mm
thick. (b) The ice layer is 5mm thick.
(a) Normalized stacks of 1000 a posteriori PDFs for the
thickness of the slab ice layer, when conducting the inversion on synthetic
data, with added random noise. The legends indicate the values for the
thickness used to create the synthetic data. (b) The a posteriori
uncertainty (at 2σ) on the thickness estimation as a function of the
slab thickness.
(a) A posteriori uncertainties at 2σ on
the grain diameter as a function of the noise standard deviation, for a
2mm thick ice layer. (b) A posteriori uncertainties at
2σ on the thickness as a function of the noise standard deviation.
(a) Measured (black) and simulated (red) reflectance factor
at 1.5µm in the principal plane for an incidence angle of
50∘. The specular lobe measured is not centered at
50∘. (b) Measured (black) and simulated (red)
reflectance factor at 1.5µm in the principal plane for an
incidence angle of 50∘. We simulated a small misadjustment of
the sample, resulting in a shift of the observation of 0.5∘ in
emergence and 0.2∘ in azimuth.
Measured and simulated reflectance factor around the specular
geometry at 1.5µm for an incidence angle of
50∘. The simulation was computed assuming the determined shift
of 0.5∘ in emergence and 0.2∘ in azimuth.
Figure shows the evolution of the a posteriori uncertainties
for the estimations of thicknesses and grain diameters as a function of the
noise level. For the grain diameters, a slab thickness of 2mm has
been used. The results show that with very low noise, i.e., lower than
0.5%, the a posteriori uncertainties of the results are of the same
order of magnitude, even for the grain diameters. When the level of noise
increases, the uncertainties of the thickness estimations increase in the
same proportions (Fig. b), unlike the uncertainties of grain
diameters (Fig. a) that increase drastically with the noise level.
The uncertainties of the grain diameters seem to be saturated for high noises. This effect is
only an edge effect due to the size of the LUT: the dispersion of the a
posteriori PDFs cannot get bigger than the range of values tested.
With the level of noise at 2% as expected for the measured spectra
, a posteriori uncertainties are expected to be about
5% on the thickness, and should be lower than 50% for the grain
diameter for low thicknesses. This means that the method should be able to
retrieve thicknesses with an uncertainty that corresponds to the level of
noise, but cannot retrieve the grain diameters of the snow when the ice layer
above is thicker than 5mm.
ResultsImpact of a slab on the BRDF
Figure shows the reflectance factor (the
ratio between the bidirectional reflectance I/F of the surface and the
reflectance of a perfectly Lambertian surface) vs. phase angle (angle between
incident and emergent directions) of the snow and the snow covered with
an ice slab 1.42mm thick (sample 1). It illustrates the two most
notable effects of a thin layer of slab ice on top of an optically thick
layer of snow. The most intuitive effect is to lower the level of
reflectance: it is due to absorption during the long optical path lengths in
the compact ice matrix as the dependance of the reflectance on the phase
angle is strongly attenuated by the addition of the ice layer. The second
effect is that the radiation is more Lambertian than that of snow
only. These data give credit to the first hypothesis of isotropization of the
radiation formulated in the model (see Sect. ).
The description of the bottom granular layer as Lambertian, defined only by
its single-scattering albedo, may be considered simplistic, but this dataset
shows that a thin coverage of slab ice, even on a very directive material
such as snow, is enough to strongly flatten the BRDF.
Specular lobe: roughness retrieval
We performed the inversion taking into account 63 angel measurements, but for
the sake of readability, Fig. represents
only the reflectance in the principle plane. The shapes and the intensities
in Fig. a are compatible, but the
measurement and simulation are not centered at the same point. The simulation
is centered at the geometrical optics specular point (emergence 50∘
and azimuth 180∘), whereas the measurement seems to be
centered around an emergence of 50.5∘. This could be due to
slight misadjustment of the rotation axis of the sample in the instrument.
This kind of misadjustment is common, and can easily result in a notable
shift up to 1∘ of the recorded measurement geometries. We
simulated different possible shifts in this range, and found a maximum
likelihood for the simulation represented in
Fig. b for a shift of 0.5∘ in
emergence, as suggested by the first plot in
Fig. a, and 0.2∘ in azimuth.
The measurements and the simulations corresponding to the maximum likelihood
are represented in Fig. . The shape and the magnitude of
the specular lobe are very well reproduced. Both lobes show a small amount of forward
asymmetry. This asymmetry is not due to the sampling as it is also
present when the simulation is not shifted (see the red curve in
Fig. ). It is due to an increase in the
Fresnel reflection coefficient when the phase angle increases for this range
of geometries. Figure shows the a posteriori
PDF for the parameter θ¯. The simulation
corresponding to the maximum likelihood was obtained with
θ¯=0.43∘. The inversion method gives a result with
a close to Gaussian shape at
θ¯=0.424∘±0.046∘. Unfortunately, we
have no direct measurement of θ¯ because the experimental
determination of the roughness of a sample requires a digital terrain model
(DTM) of its surface. These DTMs are measured with a laser beam, and are thus
very difficult to obtain for ice samples, as multiple reflections create
false measures. Still, we find a low value, which is consistent with the
production of slabs of columnar ice in the laboratory that are very flat, but
still imperfect as described in the dataset. The average slope is compatible
with a long-wavelength slope at the scale of the sample, demonstrating that
the microscale was not important in our case. Indeed, for a sample that has
a length L, a 1σ standard deviation on the thickness Δh can
be attributed to a general slope ϑ=arctanΔhL due to a small error in the parallelism of the two surfaces of
the slab. In the case of sample 3, L=20cm and Δh=1.35mm result in ϑ=0.39∘, which is
compatible with the roughness given by the inversion. We thus think that what
we see is an apparent roughness due to a small general slope on the samples,
and that the roughness at the surface is much lower than this value.
The a posteriori probability density function for the roughness
parameter θ¯, noted Pθ¯.
The inverted value at 2σ is
θ¯=0.424±0.046∘. The simulation corresponding with
the highest likelihood is obtained for θ¯=0.43∘.
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.
Diffuse reflectance: thickness and grain diameter retrievalExample of individual geometries
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 represents three examples of measured
and simulated reflectance spectra at the maximum likelihood for three
different geometries. We also represented the mismatch between these
simulations and the observations. We find an agreement between the data and
the model that is acceptable. Nevertheless, there seems to be a decrease in
quality in the fits as the thickness increases.
Figure shows an example of the
marginal PDF for the three samples that are associated with the previous
fits. The thickness is well constrained as the marginal a posteriori
probability density functions are sharp and very close to a Gaussian function. However, the
grain diameter of the substrate seems to have a limited impact on the result
since it is only slightly constrained. The marginal PDFs for the grain
diameter of the substrate are broad, and thus the a posteriori relative
uncertainties in the result are very high. Unfortunately, we have no reliable
measurement of the grain diameter of the substrate, as it is evolving during
the time of the measurements. Numerical tests show that the snow grain
diameter is not accessible for slab thicknesses above 5mm. The a
posteriori PDFs for samples 2 and 3 are then not to be interpreted.
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 40∘, emergence
10∘ and azimuth 140∘ for sample 1 (thickness:
1.42mm) (a); at incidence 40∘, emergence
20∘ and azimuth 45∘ for sample 2 (thickness:
7.45mm) (b); and at incidence 60∘ and
emergence 0∘ for sample 3 (thickness:
12.51mm) (c). The thicknesses indicated were measured
before putting the sample into the spectro-radiogoniometer, and the errors are
given at 2σ. The absolute differences are shown in blue on each
graph.
Marginal a posteriori probability density functions for
(a) the thickness of the slab Pp1(i)
and (b) the grain diameter of the snow substrate Pp2(j) for the three samples, and for the geometries described in
Fig. .
Results for 39 geometries
Figure shows the measurements and the final result of the
inversion of the thickness for the three samples, and for 39 measurement
geometries independently. The data and the model are compatible. Still, the
thickness of sample 1 is slightly overestimated. This may reveal
a sensitivity limit of the model. The thickness of sample 3 seems
underestimated. This could be partly due to the duration of the measurement:
the slab sublimates as the measure is being taken. Moreover, the specular
measurements were performed on that sample, increasing the duration
of the experiment even more. The inversion points in Fig. are sorted
by increasing incidence and, for each incidence, by increasing azimuth. There
seems to be an influence of the geometry on the returned result: it is
particularly clear for sample 2. The estimated thickness tends to increase
with incidence and decrease with azimuth. This effect disappears for large
thicknesses (sample 3).
Results of the inversions and measurements with error bars at
2σ for samples 1, 2 and 3, and for the 39 different geometries of
measurement. The inversion points (in red) are sorted by incidence (three
values), and each incidence is then sorted by azimuth (13 values: one for
emergence 0∘ and six each for the 10 and 20∘
emergences).
Full BRDF inversion
Figure shows the measure and the simulation
corresponding to the maximum likelihood at the λ=1.0µm
wavelength when conducting the inversion on the whole BRDF dataset for each
sample. The relatively flat behavior of the radiation with the phase angle is
reasonably well reproduced. The quality of the geometrical match increases
with the thickness of the sample. This is consistent with the fact that
a thicker slab will permit a stronger isotropization of the radiation. It is
also consistent with the disappearance of the geometrical dependence on the
estimation of large thicknesses noted in Fig. . The values
of thicknesses returned by the inversion are displayed in
Fig. a. They are also compatible with
the data, and the results are close to the one given by independent
inversions of each geometry (see Figs.
and ). The grain diameter returned (see
Fig. ) for sample 1 is lower, but
is compatible with the one given by independent measurements. For samples 2 and
3, the PDFs are not interpreted, as the grain diameter cannot be constrained
by the method.
Measured and simulated reflectance factor at
λ=1µm (R) for (a) sample 1,
(b) sample 2 and (c) sample 3.
Marginal a posteriori probability density functions for
(a) the thickness of the slab Pp1(i)
and (b) the grain diameter of the snow substrate Pp2(j) for the three samples.
Discussion
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 ∼2.5 h for the thickness and grain
diameter study (33 186 wavelengths sampled, 666 315 spectra). The
inversions themselves were performed in less than one-tenth of a second for
specular lobe and independent spectral inversions, and 2 s for
BRDF-as-a-whole inversions. Every calculation was computed on one Intel CPU
with 4 GB RAM. It has to be noted that once the lookup table has been
created, an unlimited number of inversions can be conducted. This means that
this method satisfies the speed criterion for the study of massive and
complex datasets. For inversions over very large databases, the code has been
adapted to GPU parallelization. It is also possible to increase the speed of
the calculation of the lookup tables by means of multi-CPU computing. This
Bayesian method has been designed to deal with the particular case of a
direct model for which the computation time per simulation decreases with the size of
the database. In our case, the direct radiative transfer algorithm is slow to
simulate only one spectrum (∼1s), but becomes fast for the
calculation of large spectral databases (75 spectra per second in the case
of the grain diameter and thickness study in this work). This particularity
makes the usual Bayesian approaches such as Markov chain Monte
Carlo methods inefficient, and in contrast, makes the method presented in
this paper efficient.
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 5% uncertainty. Moreover, for the radiative
transfer model used in this work (see Sect. ), and
in the experimental conditions described in Sect. , we could
determine on synthetic cases (see Sect. ) that a 5%
uncertainty of ice thickness estimation should be expected, and that the
grain diameter of the underlying snow could not be determined for ice
thicknesses higher than 5mm. The experimental results on the
thickness were in agreement with these estimations.
A third point to be discussed is the capability of the model to reproduce
reality. Section showed that every thickness estimation was
in agreement with independent measurements. This means that the modeling of
radiative transfer in the ice layer is satisfactory, and that the thickness
can be determined only using spectral measurements. However, this is not the
case for the estimations of the grain diameter of the snow. Indeed, when the
ice layer is thicker than 5mm, our synthetic study predicts that
it cannot be retrieved. Still, the results obtained on experimental data for
slab thicknesses greater than 5mm (blue and green curves in
Figs. and
) showed a posteriori PDFs for the
grain diameters with surprisingly low standard deviations compared to what
was obtained for synthetic data. The experimental results favor situations in
which the geometrical optics hypothesis that is fundamental in the radiative
transfer model is no longer valid. This shall not be interpreted as a result
on the grain diameter, as the synthetic test showed that it was unaccessible.
These low a posteriori uncertainties shall rather be interpreted as a
compensation effect: a behavior that cannot be reproduced by the model may be
approached by the most extreme values tested. In our case, small grain
diameter results, even if they are either not realistic or not in agreement with the
model's hypothesis, will produce an effect in the simulation that reproduces
the data better than the other cases.
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
CO2 ice deposits, but also to the study of other planetary compact
ice such as nitrogen ice on Pluto or Triton, or SO2 ice on Io.
However, our results show that this method is adapted to study the granular
material underneath only in the most favorable cases, when the uncertainties
of the data are lower than 1 %, or when the absorption in the slab
layer is weak. In the general case of a slab ice layer covering a granular
material, the retrieval method used in this work is not adapted to the study
of the bottom layer.
Conclusions
The aim of this present work is to validate an approximate radiative transfer
model developed in using several assumptions. The most
debated one is that the radiation becomes Lambertian when it reaches the
substrate. We first qualitatively validated this assumption with snow and ice
data. We then quantitatively tested and validated our method using a pure
slab ice with various thicknesses and snow as a bottom condition. The
thicknesses retrieved by the inversion are compatible with the measurements
for every geometry, demonstrating the robustness of this method to retrieve
the slab thickness from spectroscopy only. The result given by the inversion
of the whole dataset is also compatible with the measurements. We also
validate the angular response of such slabs in the specular lobe.
Unfortunately, it was not possible to measure the microtopography in detail
to compare it with the retrieved data. Nevertheless, we found very good
agreement between the simulation and the data. In future work, an
experimental validation of the specular lobe and roughness should be
addressed.
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 5mm in our case). The whole
method is thus adapted to study the top slab layer of a planetary surface
using satellite hyperspectral data, for instance Martian seasonal deposits,
that are constituted of a slab CO2 ice layer resting directly on
the regolith.
Data availability
The experimental spectra used in this work are publicly available on the SSHADE
(previoulsy GhoSST) database. The following links are permanent.
Experiments
Brissaud, O., Schmitt, B., and Douté, S.: Vis-NIR spectral bidirectional
reflection distribution fonction of sintered snow (Arselle) at -10 ∘C,
https://sshade.eu/old-ghosst/experiment/EXPERIMENT_BS_20160831_000, 2009a.
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 -10 ∘C,
https://sshade.eu/old-ghosst/experiment/EXPERIMENT_BS_20160901_000, 2009b.
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 -10 ∘C,
https://sshade.eu/old-ghosst/experiment/EXPERIMENT_BS_20160902_000, 2009c.
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 -10 ∘C,
https://sshade.eu/old-ghosst/experiment/EXPERIMENT_BS_20160903_000, 2009d.
Brissaud, O., Schmitt, B., and Douté, S.: Vis-NIR spectral bidirectional
reflection distribution fonction around specular angle 50 ∘ of slab ice
12.5 mm thick on sintered snow (Arselle) at -10 ∘C,
https://sshade.eu/old-ghosst/experiment/EXPERIMENT_BS_20160904_000, 2009e.
Simulations plotted in Fig. 10
Andrieu, F., Douté, S., Schmidt, F., Schmitt, B., and Brissaud, O.: Vis-NIR
bidirectional reflection spectrum (i=40, e=10∘, a=140∘) of simulated
slab ice 2.2 mm on snow at -3 ∘C,
https://sshade.eu/old-ghosst/experiment/EXPERIMENT_BS_20160912_001, 2015a.
Andrieu, F., Douté, S., Schmidt, F., Schmitt, B., and Brissaud, O.: Vis-NIR
bidirectional reflection spectrum (i=40, e=20 ∘, a=45 ∘) of simulated
slab ice 7.5 mm on snow at -3 ∘C,
https://sshade.eu/old-ghosst/experiment/EXPERIMENT_BS_20160912_003, 2015b.
Andrieu, F., Douté, S., Schmidt, F., Schmitt, B., and Brissaud, O.: Vis-NIR
bidirectional reflection spectrum (i=60, e=0∘, a=180∘) of simulated
slab ice 11.4 mm on snow at -3 ∘C,
https://sshade.eu/old-ghosst/experiment/EXPERIMENT_BS_20160912_006, 2015c.
Acknowledgements
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
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