# Ice state evolution during spring in Richardson crater, Mars

François Andrieu1,2, Frédéric Schmidt1, Sylvain Douté3, Eric Chassefière1 1
GEOPS, Univ. Paris-Sud, CNRS, Université Paris-Saclay, Rue du Belvédère, Bât.
504-509, 91405 Orsay, France, 2LESIA, Observatoire de Paris, PSL Research
University, CNRS, Sorbonne Universités, UPMC Univ. Paris 6, Univ. Paris
Diderot, Sorbonne Paris Cité, France, 3IPAG, Univ. Grenoble Alpes, CNRS,
F-38000 Grenoble, France

###### Abstract

The Martian climate is governed by an annual cycle, that results in the
condensation of CO2 ice during winter, up to a meter thick at the pole and
thousands of kilometers in extension. Water and dust may be trapped during the
condensation and freed during the sublimation. In addition, ice may be
translucent or granular depending on the deposition process (snow vs direct
condensation), annealing efficiency, and dust sinking process. The
determination of ice translucency is of particular interest to confirm or
reject the cold jet model (also known as Kieffer model).

This work is focused on the dune field of Richardson Crater (72°S, 180°W)in
which strong interactions between the water, dust and CO2 cycles are observed.
We analyzed CRISM hyperspectral images in the near IR using radiative transfer
model inversion. We demonstrate that among the states of CO2 ice, the
translucent state is observed most frequently. The monitoring of surface
characteristics shows a decrease in the thickness of the ice during the spring
consistently with climate models simulations. We estimate a very low dust
content of a few ppmv into the CO2 ice, consistent with the formation scenario
of cold jets. The water impurities is around 0.1%v, almost stable during the
spring, suggesting a water escape from the surface of subliming CO2 ice layer.
The water ice grain size varies in a range 1 to 50 microns. From these
results, we propose the following new mechanism of small water ice grain
suspension: as a cold jet occurs, water ice grains of various sizes are lifted
from the surface. These jets happen during daytime, when the general upward
gas flux from the subliming CO2 ice layer is strong enough to carry the
smaller grains, while the bigger fall back on the CO2 ice layer. The smaller
water grains are carried away and integrated to the general atmospheric
circulation.

###### keywords:

Mars, CO2 ice, radiative transfer, inversion, spectroscopy, seasonal south
polar cap

##  1 Introduction

The Martian climate is mostly governed by a planetary scale CO2 condensation-
sublimation cycle. As the atmosphere is composed of 96% CO2 (Mahaffy et al.,
2013; Owen et al., 1977), the CO2 cycle is the dominant climatic cycle
(Leighton and Murray, 1966; Tillman, 1988; Mischna et al., 2003). The other
two cycles that play important role in the martian climate are the dust cycle
and the water cycle. General circulation models for Mars are now able to
predict at a regional scale the amount of CO2 ice present at the surface, but
the water ice and dust cycles are still under-constrained. On Mars, climatic
quantities such as CO2 ice quantity at the surface are computed into boxes
that are 3.75° in latitude and 5.25° in longitude (up to a few hundreds of
kilometers). As approximately 30% of the atmosphere condenses at the surface
during winter, the CO2 cycle has a huge radiative impact on the planet and it
is therefore of prime importance to be able to model it correctly.

One method to decipher the microphysics of the surface/atmosphere exchange
involved in this cycle is to study the details of the reflectance response of
the soil. Indeed, the full shape of the absorption bands contains informations
about the surface and atmospheric properties. Not only the chemical
composition (presence/absence of CO2 or water ice) can be derived, but also
the surface properties (grains size, abundances…). Various models have been
proposed to simulate the reflectance of the Martian surface when covered with
CO2 ice (Douté and Schmitt, 1998; Appéré et al., 2011; Singh and Flanner,
2016), but a common point of those models is that they all considered that the
ice deposits are granular, with a porosity close to 50%. Yet, defrosting
spring activity (Piqueux et al., 2003; Kieffer et al., 2006; Hansen et al.,
2013; Andrieu et al., 2014) strongly suggest that during most of the spring
(defrosting season), the CO2 ice layer is in fact in a translucent slab state.
Then it is very important to be able to model the radiative response of such a
surface, very different from a granular surface’s response.

Recently, we have built a radiative transfer model able to simulate the near
infrared reflectance of a surface constituted of a slab of CO2 ice, clean or
contaminated with water ice and mineral dust, and that lays on the Martian
regolith (Andrieu et al., 2015). We also developed an inversion method,
adapted to our study that we validated in a previous study: based on Bayesian
formalism, the method uses a reference theoretical database to estimate a
probability density function of the value of the parameters that are to be
investigated(Andrieu et al., 2016). The combination of the inversion method
with the radiative transfer model allows to retrieve ice covered surface
properties from their near infrared spectrum. The properties retrieved are the
thickness of the ice layer, and the proportions and grain sizes of impurities.
These tools allow us to follow the volatiles exchanges between the Martian
surface and atmosphere during spring, to bring better constraint to the
seasonal Martian climatic cycles.

This study focuses on the dune field inside Richardson Crater (180°W, 72°S).
This dune field was widely studied (Mohlmann and Kereszturi, 2010; Kereszturi
et al., 2011; Supulver et al., 2001; Edgett et al., 2000; Martinez et al.,
2012) because it showed intense spring activity and interactions between water
and CO2 cycle.

In this work we:

  * 1.

argue about the presence/absence of translucent CO2 ice.

  * 2.

determine the surface properties of the seasonal frost (thickness, grain-size,
dust impurities, water impurities)

  * 3.

propose a microphysical model to describe the exchanges of volatiles in
Richardson crater

##  2 Method

In this work we compare real data with synthetic data to perform an
“inversion”. To produce realistic synthetic data, we first need a realistic
direct model described in subsection 2.1. Among the initial inputs of this
model are the optical indexes of the species constitution the surface. Optical
indexes for ices are documented in the literature (Schmitt et al., 1998), but
are unknown for Martian dust. Therefore, we estimated them from spectral
measurement, as described in subsection 2.2. Then the direct model and its
inversion will be described. The final step is the inversion method used to
get information from the comparison of our synthetic spectra with real data,
detailed in the last section 2.3.

###  2.1 Direct model

##### Granular ice

The radiative transfer model used to simulate the spectra in the case of a
translucent ice is semi analytic (Douté and Schmitt, 1998), as used to
determine seasonal cap properties (Appéré et al., 2011). The surface is
supposed to be constituted of a semi infinite layer (i.e. optically thick)
that represents the CO2 deposit, contaminated with water ice and dust. The
model uses as initial input: (i) the optical constants used for the CO2 and
water ice are from (Schmitt et al., 1998), and the optical constants of the
regolith were estimated from summer measurements (see previous section) (ii)
the grain-sizes (iii) the proportions of each type of impurity present in the
ice layer.

##### Translucent ice

The radiative transfer model used to simulate the spectra in the case of a
translucent ice is semi analytic (Andrieu et al., 2015). The model uses as
initial input: (i) the optical constants of the materials constituting the
surface: CO2 and water ice from laboratory measurements (Schmitt et al.,
1998), and dust from previous consideration, (ii) the thickness of the upper
ice layer, (iii) the roughness of the upper surface, (iv) the grain-sizes and
(v) the proportions of each type of impurity present in the ice layer and (vi)
the single scattering albedo of the underlying regolith, that is supposed to
be a semi-infinite and lambertian. The number of impurity types that can be
introduce in the model is not limited in theory, but practically, it is not
possible to introduce more than two different types due to computation
limitations.

##### Linear mixture of defrosted terrain

In order to take into account the presence of defrosted area within one CRISM
pixel, we allow a linear mixture of regolith with the previous ice model (see
fig. 1). Thus, one extra parameter will be estimated: the surface proportion
of naked terrain. This possibility of sub-pixel geographical mixing appears
even at the spatial resolution of the CRISM instrument, of about 20 m per
pixel, if a mixture occurs within the same pixel. It is also possible that,
following ejections, an optically thick layer of dust covers the layer of ice
locally (see Figure 1). These two possibilities are described in exactly the
same way, considering a linear mixture between a surface described by the
radiative transfer model (Andrieu et al., 2015), and a surface consisting only
of regolith.

(a)  (b)

Figure 1: Model of the surface: a slab of CO2 ice, possibly contaminated with
water ice and regolith dust lays on the top of the lambertian regolith. A sub-
pixel linear mixing of two different surface is possible to model (a) the
possibility of completely defrosted patches within a pixel or (b) an optically
thick layer of dust covering the ice. Both situations result in the same
modeled spectrum.

###  2.2 Optical indexes of the Martian dust

The martian southern dust differ from the northern one and from regular
palagonite (Douté et al., 2007). We may estimate the optical indexes analyzing
a defrosted area, with some assumptions. Contrary to this previous work, we
did not assume a lambertian surface for the dust and correct for the
photometric effect in order to estimate the reflectance in a normal geometry.

####  2.2.1 Single scattering albedo of the regolith

Using the Hapke reflectance model (Hapke, 1984), we inverted reflectance
spectra of the ice free surface measured by the CRISM instrument to estimate
the single scattering albedo $\omega(\lambda)$, that can be easily converted
into normal reflectance. The goal is to find the parameter $\omega(\lambda)$,
assuming the other fixed parameters, taking the average martian photometric
parameters given by Vincendon, 2013, _i.e._ $\theta$ = 17, c = 0.6, b = 0.12,
$B_{0}$ = 1 and h = 0.05. The estimation is performed by a bayesian inversion,
for which we consider each wavelength independently (Fernando et al., 2013,
2015; Schmidt and Fernando, 2015). The solution is a probability density
function (PDF) of $\omega$ that can be reduced by its mean
$\left\langle\omega\right\rangle$, dependent on the wavelength.

The measured spectra data can be modeled with these photometric parameters and
with a relatively small uncertainties on $\omega$ (less than 0.05). Moreover,
it can be noted in the figure 2 that the result is very stable as a function
of the wavelength, which is a guarantee of the quality of the inversion,
because each wavelength has been inverted independently.

####  2.2.2 Optical indexes of the Martian dust

The Shkuratov et al., 1999 model is an inverse analytical model (with an
inverse analytical form) of one-dimensional geometric optics, under normal
conditions of incidence and emergence. Assuming the real part of indexes, the
size of the grains, the porosity of the surface and knowing the value of the
reflectance from the previous step, we estimate the imaginary part $k_{n}$ of
the optical index. We fix the value for porosity of the surface at 0.5, which
is the classical value for this type of surface (Hapke, 2012). For the real
part of its optical index and the size of its grains, we know the order of
magnitude, but not the precise values. We have generated optical indexes by
assuming grain sizes varying from $10\,\mbox{$\mu$m}$ to $30\mbox{$\mu$m}$ and
real parts of the optical indexes ranging from 1.3 to 1.9. Using Shkuratov’s
inverse analytic model, we can generate the PDF of $k_{n}$ from the PDF of
reflectance (see figure 2). There is an increase of $k_{n}$ with the
wavelength and the uncertainties are increasing, but remain relatively low,
around 0.001. The grain-size has a much bigger impact on the form of the
spectrum of $k_{n}$ than the real part of the optical index. If we compare the
values of the $k_{n}$ calculated from the OMEGA (Douté et al., 2007) or CRISM
data, using the same values for the other parameters (real part of the optical
index, grain size, compactness), very close results are obtained,
demonstrating the robustness of the method regarding the input data. The grain
size is not constrained. Since each grain size has coherent optical index, it
is impossible to distinguish those two parameters. We choose a grain size of
$30\,\mu$m arbitrarily. It is close to the typical martian dust (Kieffer,
2007; Vincendon et al., 2009; Forward et al., 2009). In the following the
retrieved dust content will be subject to this indetermination.

(a)  (b)

Figure 2: (a) Distribution of the values (PDF) of the imaginary part of the
optical index $k_{n}$, calculated from the values of $\omega$ of the Markov
chains obtained after the Bayesian inversion, considering a grain size of
$30\,\mu$m in the Shkuratov model. We thus propagate the distribution of the
values of $\omega$ on the values of $k_{n}$. These values being determined
from the calculation of the single diffusion albedo $\omega$, the same
discontinuities are observed, and it is not possible to calculate a value for
a wavelength situated in an absorption band of the CO2. (b) Zoom on the part
of interest located between $1\mu m$ and $2.6\mu m$ of the distribution of the
values of the imaginary part of the optical indices $k_{n}$, obtained from the
values of $\omega$ of the Markov chains. We see that just like the
distribution of $\omega$, the distribution of the values of $k_{n}$ also often
admits two maxima. The value that has been retained is the average value of
this distribution.

###  2.3 Inversion

The inversion methods has been validated in a previous study (Andrieu et al.,
2016). The principle is to build a spectral lookup table using the radiative
transfer model, and to find a solution to the inverse problem using bayesian
statistics. The solution is a probability density function (PDF) that is
defined in the parameters space. The likelihood $L$ is estimated for each
spectrum in the look up table (LUT), with:

$L=\exp\left(-\frac{1}{2}\times{}^{t}\left(d_{sim}-d_{mes}\right)\overline{\overline{C}}^{-1}\left(d_{sim}-d_{mes}\right)\right)$
(1)

where $d_{sim}$ is a simulated spectrum from the LUT, $d_{mes}$ is the data, t
is the transpose operator that applies to $\left(d_{sim}-d_{mes}\right)$, and
$\overline{\overline{C}}$ is the covariance matrix that represent the
uncertainties on $d_{mes}$. $d_{sim}$ and $d_{mes}$ are $n_{b}$ elements
vectors corresponding to respectively simulation data and measurements data,
with $n_{b}$ the number of spectral bands used ($n_{b}=247$).

Therefore, $\overline{\overline{C}}$ is a $n_{b}\times n_{b}$ matrix, where
each eigenvector represents a source of uncertainty, and the associated
eigenvalue represents the level of uncertainty. Usually this matrix is assumed
to be diagonal with the noise level variance on the diagonal element. In our
case, we introduced the uncertainties due to the atmospheric correction: the
shift and general slope that may be introduced by variations of geometry at
the surface, accentuated by the atmospherical corrections (see Fig. 5). We
represent it by two eigenvectors $\vec{t}=\frac{1}{n_{b}}\left(1,...,1\right)$
and $\vec{s}=\frac{\vec{S}}{\left\|\vec{S}\right\|}$, with
$\vec{S}=\left(-\frac{n_{b}-1}{2},\ldots,-2,-1,0,1,2,\ldots,\frac{n_{b}-1}{2}\right)$.
$\vec{s}$ characterizes the spectral slope value, and $\vec{t}$ characterizes
the ordinate at the spectral shift. As $\overline{\overline{C}}$ must be a
positive definite matrix, $\vec{s}$ and $\vec{t}$ must be orthogonal, so the
formulation of $\vec{s}$ is imposed by $\vec{t}$. The 245 other eigenvectors
represent the noise, and are build using the Gram-Schmidt process. In the
basis constituted of the eigenvectors, $\overline{\overline{C}}$ is diagonal
and its diagonal elements are the eigenvalues $\sigma_{1}^{2}$,
$\sigma_{2}^{2}$ and 245 times $\sigma_{3}^{2}$ , where $\sigma_{1}$ and
$\sigma_{2}$ respectively represent a standard deviation associated to
$\vec{t}$, $\vec{s}$, and $\sigma_{3}$ represent the standard deviation
associated to the noise. We chose $\sigma_{1}=\sigma_{2}=0.02$ and
$\sigma_{3}=0.03$, in agreement with our analysis.

The likelihood computed are then converted to _a_ _posteriori_ probability
density functions (PDF). For the parameter $m_{i}$ of the radiative transfer
model, the a posteriori PDF is:

$\mathcal{P}\left\\{m_{i}\right\\}=\frac{\mbox{L}\left(m_{i}\right)\mbox{d}p\left(m_{i}\right)}{\sum_{i}\mbox{L}\left(m_{i}\right)\mbox{d}p\left(m_{i}\right)}$
(2)

where $\mbox{d}p\left(m_{i}\right)$ is the differential element at the
sampling point $m_{i}$ of the parameters space $M$:
$\mbox{d}p\left(m_{i}\right)=\prod_{j}\partial p_{j}\left(m_{i}\right)$. To
make the solution easier to read, marginal a posteriori PDF are built: the PDF
is projected on one parameter’s axis. This allow the visualization of the
solution for a given parameter $p_{j}$ in a graph. For the parameter $p_{j}$,
there are $n_{j}$ sampling steps that are represented by the letter $k$ ($k$
in Eq. 3 ranges from $0$ to $n_{j}$). The marginal a posteriori PDF for
$p_{j}$ at the sampling step $k$ is:

$\mathcal{P}\left\\{p_{j}(k)\right\\}=\frac{\mbox{L}^{\prime}\left(k\right)}{\sum_{i}\mbox{L}^{\prime}\left(k\right)\partial
p_{j}\left(k\right)}$ (3)

with $\partial p_{j}\left(k\right)$ the differential element for $p_{j}$ at
$k$, and:

$\mbox{L}^{\prime}\left(k\right)=\sum_{i}\mbox{L}(m_{i}\mid
p_{j}\left(k\right))\prod_{l\neq j}\partial p_{l}(m_{i}\mid
p_{j}\left(k\right))$ (4)

###  2.4 Look Up Tables (LUT)

The LUT must make it possible to sample the space of the parameters in an
optimal way: representative way not to miss any information, but not too
voluminous to avoid making the calculations more complicated. A series of
tests was therefore carried out, traversing the space coarsely following a
logarithmic sampling to locate where in the parameters space there is most
variability. As we do not know precisely the optical constants of the Martian
regolith, it is useless to try to find the size of the grains. To maintain
maximum consistency, this parameter was not sampled. The set of values sampled
in the synthetic LUT is summarized in the table 1 for the slab ice model, and
in the table 2 for the granular ice model.

Parameter |  Values sampled |  Number of sampled values
---|---|---
Thickness of the CO2 slab$\left(\mbox{mm}\right)$ |  $0$ ; $0.01$ ; $0.05-1$ in steps of $0.05$ ; $1-200$ in steps of $1$ ; $200-300$ in steps of $5$ ; $300-1000$ in steps of $50$ |  $250$
Total amount of impurities $\left(\mbox{\%}\right)$ |  $0$ ; $0.005$ ; $0.01$ ; $0.05-1$ in steps of $0.05$ ; $10-100$ in steps of $10$ ; $10-100$ in steps of $10$ |  $32$
Dust proportions among impurities $\left(\mbox{\%}\right)$ |  $0$ ; $1$ ; $10$ ; $50$ ; $90$ ; $99$ ; $100$ |  $7$
Water ice proportions among impurities $\left(\mbox{\%}\right)$ |  $0$ ; $1$ ; $10$ ; $50$ ; $90$ ; $99$ ; $100$ |  $7$
Water ice grain diameter $\left(\mbox{$\mu$m}\right)$ |  1 ; $3$ ; $10$ ; $30$ ; $100$; $300$ ; $1000$; $3000$ ; $10000$ ; $30000$ |  $10$
Surface proportion of dust in linear mixing $\left(\mbox{\%}\right)$ |  $0-100$ in steps of $10$ |  $11$
Incidence |  $55\text{\textdegree}-80\text{\textdegree}$ in steps of $5\text{\textdegree}$ |  $6$
Emergence |  $0\text{\textdegree}-30\text{\textdegree}$ in steps of $10\text{\textdegree}$ |  $4$
Azimuth |  $60\text{\textdegree},150\text{\textdegree}$ |  $2$
Table 1: Values sampled in the LUT for the translucent slab model. Parameter |  Values sampled |  Number of sampled values
---|---|---
Grain diameter of CO2 ice $\left(\mbox{$\mu$m}\right)$ |  $1$ ; $3$ ; $10$ ; $30$ ; $100$ ; $300$ ; $1000$ ; $3000$ ; $10000$ ; $30000$ ; $100000$ ; $300000$ |  $12$
Total amount of impurities $\left(\mbox{\%}\right)$ |  $0$ ; $0.005$ ; $0.01$ ; $0.05-1$ par pas de $0.05$ ; $10-100$ in steps of $10$ ; $10-100$ in steps of $10$ |  $32$
Dust proportions among impurities $\left(\mbox{\%}\right)$ |  $0$ ; $1$ ; $10$ ; $50$ ; $90$ ; $99$ ; $100$ |  $7$
Water ice proportions among impurities $\left(\mbox{\%}\right)$ |  $0$ ; $1$ ; $10$ ; $50$ ; $90$ ; $99$ ; $100$ |  $7$
Water ice grain diameter $\left(\mbox{$\mu$m}\right)$ |  1 ; $3$ ; $10$ ; $30$ ; $100$; $300$ ; $1000$; $3000$ ; $10000$ ; $30000$ |  $10$
Surface proportion of dust in linear mixing $\left(\mbox{\%}\right)$ |  $0-100$ in steps of $10$ |  $11$
Incidence |  $55\text{\textdegree}-80\text{\textdegree}$ in steps of $5\text{\textdegree}$ |  $6$
Emergence |  $0\text{\textdegree}-30\text{\textdegree}$ in steps of $10\text{\textdegree}$ |  $4$
Azimuth |  $60\text{\textdegree},150\text{\textdegree}$ |  $2$
Table 2: Values sampled in the LUT for the compact granular layer model Douté
and Schmitt.

This synthetic LUT for the slab model contains $9939264$ spectra,
corresponding to $207068$ configurations of different surfaces for $48$
geometries of measurements, representing the variability of the direct model.
Each spectrum was calculated at high spectral resolution ($29037$ wavelengths
at a spectral resolution of about $0.5\text{\,cm}^{-1}$), and then resampled
at the resolution of the CRISM instrument ($247$ wavelengths used here, at a
resolution of $6.5\,\text{nm}$), using the transfer function of the sensor. It
was computed in $60\,\text{h}$, in parallel on four Intel Xeon CPU cores Xeon
E5640, $2.66\,\text{GHz}$.

To take the sub-pixel mixture into account during the inversion, the LUT is
extended by calculating the mixture with a martian dust spectrum at the
corresponding geometry and for the different desired surface ratios. In this
study, we sampled $11$ values of surface proportions of linear mixing with
dust, ranging from $0$ to $100$ in steps of $10$. The LUT compared to the data
during the inversion is thus $11$ times more voluminous than that described
previously, and therefore contains a total of $109331904$ spectra,
corresponding to $2277748$ different surface configurations.

##  3 Data

###  3.1 CRISM data

The CRISM instrument (Murchie et al., 2007) has well covered the Richardson
Crater area, and provide 12 cubes of the same area for the spring martian year
28. This time coverage is enough to follow the general time evolution of
seasonal deposits through spring: the mean time span between two acquisitions
is 9° of solar longitude, or 16.7 _sols_. Another advantage of the CRISM data
is that most of the observations are coupled with HiRISE images (McEwen et
al., 2007) of the same area, taken at the same moment, at a resolution of
25cm/px. This allows to investigate closely the relations between
spectroscopic and spatial informations. In particular, the dark dune spots,
widely active in the area (Kereszturi et al., 2011) have a spatial extension
of a few CRISM pixels, and thus their evolution can be monitored both
spectroscopically using CRISM and geomorphologically using HiRISE data.

The purpose of this work was to investigate precisely the interactions between
surface and atmosphere during the spring defrosting in areas that show
defrosting spring activity. These spring defrosting manifestations have a
typical scale of a few hundreds meters to a few kilometers. The resolution of
the instrument used to study them shall then be lower than a hundred meters.
We thus chose the instrument CRISM that provide the best possible spatial
resolution (approx. 20m per pixel) compared to others imaging spectrometers.
Moreover, CRISM acquires data always at the same local solar time, so diurnal
variations are not to be taken into account.

We searched for areas of particular interest, that presented a time coverage
by CRISM that was sufficient to follow the evolution of the seasonal deposits.
We chose the Richardson crater dune field (See Figure 3) for the following
reasons:

  1. 1.

The time sampling by CRISM was sufficient for the spring of martian year 28
(12 data cubes available).

  2. 2.

The area is fully covered by CO2 ice at the beginning of the time series and
fully defrosted at the end.

  3. 3.

This area showed an intense surface process activity.

  4. 4.

Several previous studies demonstrated the abundant presence of water ice in
this area, suggesting possible interactions between water and CO2 cycles, and
justifying the use of complex radiative transfer simulations to model the
radiation in the surface.

Cube ID |  Acquisition date |  $L_{S}\,\left(\text{\textdegree}\right)$ |  AOT |  Mean solar incidence $\left(\text{\textdegree}\right)$ |  Comments
---|---|---|---|---|---
HRS0000411F |  30/01/2007 |  $175.213$ |  n. c. |  $83.9$ |  Atmosphere saturated
HRL000043A2 |  10/02/2007 |  $181.550$ |  $0.59$ |  $80.9$ |  CO2 and water ice at the surface
HRS000049E5 |  09/03/2007 |  $197.016$ |  $1.11$ |  $75.9$ |  CO2 and water ice at the surface
FRT000052BC |  05/04/2007 |  $213.719$ |  $0.92$ |  $69.1$ |  CO2 and water ice at the surface
FRT000054E5 |  12/04/2007 |  $217.503$ |  $0.72$ |  $66.7$ |  CO2 and water ice at the surface
HRS000056C0 |  17/04/2007 |  $220.683$ |  $0.43$ |  $66.1$ |  CO2 and water ice at the surface
FRT00005AF7 |  15/05/2007 |  $238.066$ |  $1.32$ |  $60.2$ |  CO2 and water ice at the surface
FRT00005C94 |  20/05/2007 |  $241.319$ |  n. c. |  $59.8$ |  CO2 and water ice at the surface
FRT00005E38 |  26/05/2007 |  $245.220$ |  $1.17$ |  $57.9$ |  CO2 and water ice at the surface
FRT00005FF6 |  31/05/2007 |  $248.483$ |  $1.32$ |  $57.6$ |  CO2 and water ice at the surface
FRT00006102 |  04/06/2007 |  $251.104$ |  $1.51$ |  $59.0$ |  No CO2 ice at the surface
FRT00006C56 |  25/07/2007 |  $283.403$ |  $2.1$ |  $53.7$ |  No CO2 ice at the surface
Table 3: Summary of available CRISM data for Richardson crater dune field, and
for martian year 28. Dust aerosol optical thicknesses (AOT) are given by M. J.
Wolff (Wolff et al., 2009). <<n. c.>> (not calculated) means M.J. Wolff’s
algorithm could not retrieve the dust AOT. Aerosols effect were not corrected
in this case. Incidences are defined with respect to the Areoide.

The 12 available CRISM cubes for this area are summarized in table 3. The
first cube of the spring (HRS0000411F) turned out to be ineffective, because a
large number of bands were saturated in atmosphere due to the high incidence
angle and because of the generally low albedo of the surface. From Ls=251° and
after, no CO2 ice can be detected at the surface using band criterion methods
(Langevin et al., 2007), meaning the the seasonal ice deposits have completely
sublimed.

In the area, we chose to focus on a few different spots, and to investigate
deeply their evolutions. We used the combined HiRISE and CRISM data to select
a few “control points”, where no dark spot of flow activity is recorded during
the spring, and some point precisely where a dark spot appears (See Figure 3
and 4 for details). The temporal series of spectra corresponding to each study
area were then extracted from the CRISM cubes.

(a)  (b)

Figure 3: (a) localization of Richardson crater, Mars. The back is a polar
projection map centered on the south pole of Mars, from (Hansen et al., 2010),
where the limit of the seasonal CO2 ice deposits in the middle of the spring
(220°< Ls<225°) is represented in white. A MOC image of Richardson crater,
located at 72.6°S, 180.4°W, is superimposed, and then a map projected color
composition of a CRISM cube FRT000052BC (LS=213°). (b) Detail of areas of
particular interest: extract from the CRISM color composition in (a)
superimposed on an extract from HiRISE image PSP_002542_1080. Two control
points (CP1 and CP2) and two points of particular interest that show
defrosting activity (DS1 and DS2) were defined. Figure 4: HiRISE series of the
area of particular interest defined in Figure 3b, during the spring of MY28.
Red arrows indicate the positions of the points CP1, CP2, DS1 and DS2 defined
in figure 3b.

###  3.2 Noise level

The nominal CRISM instrumental noise is very low, and still was at the time
when the data were acquired (2007), with a signal to noise ratio between 400
and 500 depending on the wavelength. Nevertheless, the real data show a level
far higher, most probably due to detector electronics, atmospheric
perturbations, or geometric variations within a pixel. Computing the
covariance matrix of a set of spectra that are supposed to be close to
identical allowed us to estimate that level of noise at approximately
$3\mbox{\%}$ of the signal, consistently with previous studies (Ceamanos et
al., 2013; Fernando et al., 2013).

###  3.3 Atmospheric correction

The atmospheric gas contribution was corrected using _ab initio_ line by line
radiative transfer calculations(Douté, 2014), based on general circulation
models (GCM) predictions (Forget et al., 1999, 2006). The mineral aerosols
atmospheric contribution was corrected using aerosols optical thickness (AOT)
estimations by Wolff et al., 2009 and the method developed by Douté et al.,
2013, that take into account the radiative coupling between gas and aerosols.
No correction method has proven 100% efficient to remove the aerosols
contribution for CO2 ice covered areas (Vincendon et al., 2007, 2008; Smith,
2009; Wolff et al., 2009; Ceamanos et al., 2013; Douté et al., 2013; Douté,
2014). Therefore, we expect correction errors, that have to be taken into
account in the inversions. We chose to estimate that level of error, and to
introduce it in the inversion framework as a source of uncertainty.

(a)

(b)

Figure 5: Effect of local slopes at the surface, not taken into account in the
atmospheric corrections (Douté et al., 2013). (a) Two spectra that are
supposed to be identical before (up) and after (down) atmospheric corrections.
The black lines represent the spectra before and after correction in a flat
area, and the red ones in a slope. (b) Difference between the two supposed
identical spectra (red spectra minus black spectra from fig. a), before
(green) and after (red) atmospherical correction. There is a difference
between the two supposed identical spectra, and the atmospherical corrections
seem to enhance it, by adding a general spectral slope to the data.

As shown on figure 5, the atmospheric corrections may introduce a bias in the
data in case of a sloped facet: in this graph are represented two spectra from
the same region during summer. The whole area is homogeneously covered in dust
and thus is not expected to show any spectral variability. The variability
here is correlated with photometry, and thus attributed to a bias in
atmospherical corrections. The bias as been estimated to be a general negative
spectral slope of maximal magnitude of 0.01UR/µm (UR mean units of reflectance
factor, that is dimensionless) but also a shift in the absolute level in the
order of 0.02 UR. The covariance matrix of the uncertainties
$\overline{\overline{C}}$ has to take these effects into account.

##  4 Results

###  4.1 Best fit

Figure 6:  Best fit of translucent ice model as a comparison with the data for
temporal series of point CP1 from $L_{S}=181.5\text{\textdegree}$ to
$L_{S}=248.5\text{\textdegree}$. The best fit spectra are re-adjusted at the
level of deformation estimated by the inversion. Figure 7: Best fit of
granular ice model as a comparison with the data for temporal series for CP1
from $L_{S}=181.5\text{\textdegree}$ to $L_{S}=248.5\text{\textdegree}$. The
best fit spectra are re-adjusted at the level of deformation estimated by the
inversion.

The inversion method described above takes into account the fact that the
reflectance spectrum of the surface may be slightly deformed by a spectral
slope and level due to imperfect atmospheric correction. It is crucial to
avoid interpreting these deformations in terms of composition and surface
condition erroneously. Not only this potential bias is ignored by the
inversion algorithm thanks to the first two eigenvectors of
$\overline{\overline{C}}$ (see section 2.3), but it is estimated and can
therefore be corrected from the data. The adjustments shown in the figures 6
and 7 were done by correcting this bias according to the level estimated by
the inversion algorithm. As demonstrated by these figures, the best fits are
usually in very good agreement with the data. The worst is the first, due to
spikes at wavelengths around 1.7 $\mu$m and 2.2 $\mu$m in the observations
that are most probably artifacts.

###  4.2 Slab vs granular ice

Both slab ice or granular ice models fit reasonably well the data. Therefore,
the quality of the fit will not be determinant to decide which model is closer
to reality. Figure 8 shows an error index $EI=\left|\ln L\right|$, where L is
the likelihood. Thus high value of $EI$ means bad fit, and $EI=0$ means a
perfect fit. In terms of error index, the granular model fits better (even if
badly ) for the first two measures of the season, and then the slab model have
a lower error index. Nevertheless both models show an error index of similar
magnitude. The quality of the fit then is inappropriate to discriminate the
models.

Figure 8: Error indexes ($EI$) of the fits for point CP1 for the slab and
granular models

To test the coherence of these model, we compare them with other independent
estimates. Predictions of CO2 ice quantity (in kg.m-2) at the surface by
climate models are readily available via the Mars Climate Database
111http://www-mars.lmd.jussieu.fr/mcd_python/ (Lewis et al., 1999; Forget et
al., 1999, 2006; Millour et al., 2012, 2014). The thicknesses are then deduced
from the GCM prediction by considering an homogeneous layer, and mass density
of $1606$ kg.m-3 (Kieffer, 2007; Thomas et al., 2011). The thickness data
derived from the climate models are then plotted in the graphs with ice
thickness estimates in Fig. 9. We can see in the figure 9 that the direct
estimates realized with our algorithm and the predictions from the climate
models are consistent. Similar orders of magnitude, and often very close
estimates are observed, except for Ls = 181.5°, where we observe thicknesses
clearly greater than those predicted.

For the granular ice model, we can estimate the minimum thickness of granular
ice required for a optically thick layer. Since the ice is poorly absorbent, a
layer of 10 grains is required to fulfill this property (see fig. 9). We can
thus compute the minimum thickness in the case of a granular ice that is
inconsistent with the MCD, even with the expected behavior of sublimating. The
grain diameter in the case of a granular model shows no clear evolution during
spring (see figure 9a), and figure 9b shows that the minimum thickness
required to fulfill the granular model assumptions is highly inconsistent with
the climate models predictions. On the contrary, the thickness estimates from
the slab ice models show a decrease during spring in agreement with the values
predicted by the Mars Climate Database. From this result, the granular ice can
be ruled out and be ignored for the rest of this article.

(a) Granular (b) Slab

Figure 9:  (a) Evolution of the grain size in a granular ice layer case for
CP1. The shades of green represent the values of the marginal a posteriori of
the likelihood PDF, and the red crosses point out the maximum value of the
likelihood PDF (best fit), at each observation date. (b) Evolution of the
thickness of the ice layer as a function of Ls for CP1, in the case of a slab
layer. The GCM predicted evolution (Millour et al., 2014 ©LMD/OU/IAA/ESA/CNES)
has been added in blue diamonds, considering a density for CO2 ice of
$\rho=1606\,\mbox{kg}\mbox{m}^{-3}$ (Kieffer, 2007), and the minimal thickness
possible for a granular layer, derived from the grain-diameters given by
figure a have been added in red.

###  4.3 Thickness

As a general rule, the thickness estimated by the inversion algorithm for the
ice layer decreases during spring, as is naturally expected. Figure 10
presents the temporal ice thickness for the four points of our study from
early to late spring. They represent on the same graph the posterior marginal
probability density values (eq. 3) for each inversion.

All points are following the expected sublimation process. One can note a
local variability of the thickness, on a scale that is not taken into account
by the GCM: the ice layer is thicker on the crests of dunes (CP1) and thinner
in the inter-dunes (CP2), which is expected. Indeed, the inter-dunes show
polygonal cracks (see Fig. 3b), indicating possibly the presence of
permafrost, and most probably a water ice rich subsurface with a high thermal
inertia. Conversely, the dune is constituted of sand, with small grains with
low thermal inertia. A terrain with high thermal inertia will start to
condense CO2 much later, and the dune will therefore start the spring with a
much thicker CO2 ice layer than the inter-dunes.

(a) Point CP1 (b) Point CP2 (c) Point DS1 (d) Point DS2

Figure 10: Evolution of the thickness of the top slab ice layer as a function
of Ls for (a) CP1 (b) CP2, (c) DS1 and (d) DS2. The shades of green represent
the values of the marginal a posteriori of the likelihood PDF, and the red
crosses point out the maximum value of the likelihood PDF (best fit).

At the center of the dark spots, the thickness of the layer is lower than that
of the surrounding areas (the thicknesses at CP1 are compared with those of
DS1 and the thicknesses of CP2 with those of DS2), which is consistent with
the mechanism of cold jets proposed by Piqueux et al., 2003 and Kieffer et
al., 2006: it is expected that the ice breaks where it is most fragile, where
the thickness of the layer is the lowest. Alternatively, the points in the
dark fans DS1 and DS2 could have a lower slab thickness due to the enhanced
sublimation due to darker albedo because of the covering dust. In the dark
spots of the inter-dune, the thickness of the layer is low from the beginning
of spring, which is consistent with the activity of jets much earlier in these
zones (see fig. 3b), as seen by Pommerol et al., 2013 in the north. These are
illuminated longer during the day than the slope zones (the slopes are
oriented towards the east or the west) from the end of the polar night, and
the greater roughness of the inter-dune zones compared to the smooth zones of
the slopes favor a more important basal sublimation, and thus the activity is
early (Pommerol et al., 2013). It should be noted that the thickness remains
almost constant throughout the season, which is consistent with an effect due
to contamination by water ice proposed by (Pommerol et al., 2013), and that
will be discussed in section 4.4. In detail, DS1 and DS2 have a peculiar
behavior that thickness is decreasing from Ls=181.5 to Ls=210-220 and then
increase to finally decrease. We attribute this apparent thickening of the ice
to the dust migration within the ice, producing a optically thick layer of
dust migrating downwards.

###  4.4 Contamination

####  4.4.1 Impurities proportions

The total impurity content of the ice layer corresponds to the sum of the
volume proportions of water ice and dust in the matrix. As can be seen in
figure 11, this content seems to remain relatively stable between 0.01 % and
0.1% for the all points. The total budget of impurities has to be investigated
conjointly with the respective proportions of dust and water ice among them,
represented on figure 12. For all the points studied, dust contamination seems
very low. Indeed, the relative proportions between water ice and dust
estimated by the inversion algorithm systematically give a proportion of ice
water greater than $99\,\mbox{\%}$, or a total content of dust always less
than $5\,\mbox{ppmv}$ (see Figure 12). This result tends to reinforce the
hypothesis of a very effective cleaning of dust (Kieffer, 2007; Portyankina et
al., 2010). On the contrary, water ice is not cleaned, or at least much less
efficiently than dust because of a higher albedo. This behavior is expected,
the dust being low albedo (around 0.3) compared to water ice, which may have a
very high albedo, and the cleaning efficiency decreasing for a growing albedo
(Portyankina et al., 2010). The graphs representing the total impurity content
(fig. 11) can therefore be directly interpreted as a volumetric water content
(with an error of less than $1\,\mbox{\%}$).

(a) CP1

(b) CP2

(c) DS1

(d) DS2

Figure 11: Evolution of the total content in impurities for (a) CP1 (b) CP2,
(c) DS1 and (d) DS2. The shades of green and the red crosses are interpreted
in the same way as in the figure 10.

In details, the two points studied in the dark spots show different behaviors:
the DS1 point appears to have a level of contamination similar to that of the
control points. Conversely, the DS2 point in the inter-dune appears to be
considerably more contaminated with water ice than all other points studied,
displaying impurity contents up to 5 times higher than other locations in
early spring. The DS2 contamination then decreases in a few weeks to stabilize
at the same level as the surrounding points.

The behavior of DS2 is consistent with the jet mechanism. Indeed, the
triggering of a jet of gas is supposed to be a violent event, which tears
material from the regolith and deposits it at the top of the layer of ice. It
is therefore expected that there will be more contamination in the vicinity of
dark spots. Conversely, DS1’s behavior is unexpected: its level of
contamination is equivalent to that of the environmental control point. The
apparent inconsistency with the jet mechanism can be resolved if most of the
material ejected by the cold jet do not stay in the vicinity of the jet, but
is either carried away by the wind, roll down the slope or is efficiently
cleaned. It would also mean that the darkness of spot around the jet is a
photometric effect. In addition, the upward sublimation flux of increasing
intensity during spring favors a transport of dust and water ice grains down
from the ridge of the dune to the inter-dunes, by decreasing the friction
coefficient between the grains and the slope. This would explain the formation
of flows from dark spots, such as the one in Fig. 4 from
$L_{S}=213\text{\textdegree}$. In this case, it would be expected to find
greater contamination in the inter-dune, or at least at the foot of the
slopes. The follow-up of a control point precisely at the bottom of the slope
would make it possible to decide on this point. Another possible explanation
is that the jets in the inter-dunes contain much more water ice than the jets
in the ridges: as the inter-dunes subsurface is most probably water ice rich
(polygons) and the jet mechanism may pull water out from the subsurface to the
fan (Kereszturi et al., 2011). It will later enrich the layer in water ice.
Conversely, the dune is constituted of sand, and the jet will only pull dust,
that will then be efficiently cleaned before the next measurement (Pommerol et
al., 2011).

(a) CP1

(b) CP2

(c) DS1

(d) DS2

Figure 12: Evolution of the proportion of water ice among impurities (a) CP1
(b) CP2, (c) DS1 and (d) DS2. The shades of green and the red crosses are
interpreted in the same way as in the figure 10.

####  4.4.2 Water grain size

(a) CP1

(b) CP2

(c) DS1

(d) DS2

Figure 13: Evolution of the water ice grain diameter for (a) CP1 (b) CP2, (c)
DS1 and (d) DS2. The shades of green and the red crosses are interpreted in
the same way as in the figure 10.

The figure 13 shows that the grain size increases during the spring for all
the studied points, up to about $L_{S}=238\text{\textdegree}$ from 1 microns
to 50 microns. Then grain size suddenly decreases up to few microns at the end
of the season. Variations in water ice grain size are not expected during the
sublimation of the seasonal CO2 deposits. Indeed, water is supposed to be
thermalized by CO2 ice, as carbon dioxide has a condensation point much lower
than water. So an evolution in the grain size means that water ice may have a
significant mobility, even at the very beginning of the spring. This is
unexpected and a mechanism able to explain the variations both in grain size
and total content will be detailed in the following subsections.

####  4.4.3 Water escape

Water ice at the surface is in contact with CO2 ice throughout the spring,
which is a cold trap. Therefore H2O is not expected to sublimate as long as
the ice of CO2 has not completely disappeared. To remain the total water
budget constant in a shrinking CO2 slab, we expect an increase of volumetric
water impurity. However, the combined analysis of the previous subsections
clearly show that this is not the case. So there is an water escape process.
This escape can be explained by the spring activity of jets. Indeed, during an
ejection, part of the water ice entrained by the jet can remain suspended in
the atmosphere, carried by the ambient sublimation flux (Kieffer et al.,
2000). In this case, the fine part of the water ice grains must gradually be
eliminated over repeated ejections, and the grain size must therefore increase
during the spring, as observed (fig. 13).

Note that water ice remain throughout spring present in amounts that are far
lower than CO2 (about $0.1\%$), and that there is no clear albedo variations
such as those described by Pommerol et al., 2011. The formation of a thin
water frost layer at the top of the subliming CO2 ice layer would create an
albedo increase that is not seen here. Therefore it is supposed that water ice
is always in intimate mixing with CO2 ice. Due to removal from the bottom and
deposition at the top, there still may be a vertical variation of the water
ice content inside the CO2 ice layer, that is not taken into account in the
inversion. All quantities are vertically averaged.

####  4.4.4 Dust in linear mix

The jet mechanism proposed to explain dark spots involves the deposition of a
layer of dust and possibly water ice at the top of the layer after ejection.
If the layer is thick enough locally, one should observe, either only dust in
the center of the dark spot, or a linear mixture between two types of surface:
dust and ice. We can see in Fig. 14 that there is no significant difference in
terms of sub-pixel geographic mixing of dust and ice between points. Moreover,
it can be noted that all the points studied exhibit a similar behavior, with a
small peak of contamination around $L_{S}=220\text{\textdegree}$ that can be
explained by a possible dust storm, as also see in the atmospheric dust
content estimate (see table 3). Then, the surface proportion of dust increase
from $L_{S}=238\text{\textdegree}$ to the end. This is probably due to the
fact that from this date the CO2 has been totally sublimated on some areas of
the image. This explanation is consistent with the sudden decrease in grain
size observed in fig. 13. Indeed, CO2 ice forces the temperature of the media
in contact with it to its equilibrium temperature (about $150\,\mbox{K}$), but
when the CO2 has disappeared, dust and water ice are no longer buffered by
this CO2 condensation temperature. In this case, the temperature increases,
and the water ice finally sublimates. However, there are still ice-covered
areas of CO2 a few meters away, and water could be trapped again. Water re
condensed at the surface, forming a fine frost, explaining the sudden decrease
in the size of the observed grains. This mechanism was also observed in the
north, the sublimation of seasonal deposits leading to the formation of a ring
of water frost following the recession of the seasonal cap (Wagstaff et al.,
2008; Appéré et al., 2011; Appéré, 2012; Pommerol et al., 2013).

(a) Point CP1

(b) Point CP2

(c) Point DS1

(d) Point DS2

Figure 14: Evolution of the proportion of dust in linear mix for (a) CP1 (b)
CP2, (c) DS1 and (d) DS2. The shades of green and the red crosses are
interpreted in the same way as in the figure 10.

It is worth to remind that, on the wavelength range $1\,\mbox{$\mu$
m}-2.6\,\mbox{$\mu$ m}$, the pure dust has a spectral behavior very monotonic,
quasi-linear. In fact, it is difficult to distinguish the effects of aerosols
in the atmosphere from dust within the ice layer. It is therefore expected to
have high uncertainty on dust contamination estimates by the inversion
algorithm. Actually, this is not the case in our results: the amounts of dust
are very well constrained, even tacking into account the uncertainties due to
atmospheric correction. We attribute this behavior to the non-linear coupling
between CO2 bands and dust within the slab. The same behavior may also occur
in case of differentiation between linear surface mixing with non-linear
impurities. Again our results show that there is enough information in the
spectral data to decipher this situation, most probably due to the non-
linearities.

It would still be interesting to test the algorithm with other atmospheric
correction methods. For example, the photometric properties of aerosols can be
used to better isolate them and improve the correction (Douté et al., 2013;
Douté, 2014).

##  5 Discussion

(a)  (b)

(c)  (d)  (e)  (f)

Figure 15: Proposed model for the evolution of seasonal ice deposits during
spring. (a) At the beginning of the spring the ice layer becomes compact in
slab form ($L_{S}\sim 181\text{\textdegree}$), In the meantime, it is cleaned
of the impurities of dust and ice thanks to solar irradiation (b) Basal ice
sublimation starts, and the impurities that were trapped in the ice are
accumulated at the bottom of the layer. This is the first stage of the cold
geyser model. (c) The pressure built inside pockets under the layer reaches
the ice breaking threshold. The ice breaks and gas escapes brutally, carrying
impurities with it. (d) Small grains are lifted and put into suspension into
the atmosphere, while large one fall back to the ground. Small water ice
grains that are put into suspension will eventually sublimate and be
integrated to the global water circulation. Dust is deposited in a superficial
layer. (e) In the place of a cold jet, the deposited layer of dust and water
ice is sinking. The dust sink very rapidly, and the water ice is integrated to
the layer due to the condensation of CO2 at night at the top of the layer, and
the sublimation during the day of de CO2 ice at the bottom of the layer. (f)
If a patch of terrain is completely defrosted, then the ground is no more
thermally buffered to the equilibrium temperature of CO2 ice. The temperature
rises and the water ice sublimates. If CO2 ice is still present in the area,
it will act as a cold trap, and the just sublimated water will condense on the
closest patch of CO2 ice.

We propose a sketch (see fig. 15) to summarize the evolution of the seasonal
cap in Richardson crater in dark spots and off dark spots.

##### Mechanism for water escape

We propose a mechanism of escape of water ice, which would be incorporated
into the atmosphere, from $L_{S}=190\text{\textdegree}$, in a way consistent
with the jet mechanism proposed by Piqueux et al., 2003 and Kieffer et al.,
2006; Kieffer, 2007.

If we consider that the sublimation flux can raise grains of dust of a
supposed density of 2200 $\mbox{kg}.\mbox{m}^{-3}$ up to a radius of
$3.5\mbox{$\mu$m}$ (Kieffer et al., 2000; Kieffer, 2007), it is clear that
water ice grains with a density of about 920 $\mbox{kg.m}^{-3}$ may be
suspended to larger radii up to 10 microns. Step by step, all the grains
smaller than this radius will be eliminated (see Figure 15 d and e). Before
the abrupt decrease $L_{s}=240\text{\textdegree}$, the grain size seems to
stabilize around a value between $10\,\mbox{$\mu$m}$ and $30\,\mbox{$\mu$m}$
(see fig. 13). This value is consistent with the grain sizes that can escape.

Figure 16: Atmospheric water vapor content in the studied area, predicted by a
GCM (Millour et al., 2014 ©LMD/OU/IAA/ESA/CNES).

OMEGA observations suggest an enrichment of the atmosphere at these latitudes
from $L_{S}=250\text{\textdegree}$ (Melchiorri et al., 2009), and are not
incompatible with an enrichment earlier in the season. It would be interesting
to study these observations of atmospheric water vapor in more detail.

The water escape noted in section 4.4.1 is also consistent with an enrichment
of atmospheric vapor during the spring predicted by climatic models (Forget et
al., 2006; Millour et al., 2014), as we can see it in figure 16. This
enrichment is becoming more and more important as the season advances. This
increase in the quantity of water vapor predicted in the atmosphere by the
GCMs can be linked, on one hand, to the increase in atmospheric pressure and,
on the other hand, to forcing.

##### Rolling layer mechanism

According to MCD (Forget et al., 1999, 2006; Millour et al., 2014), at the
latitude of Richardson crater, $1.5\,\text{kg}$ of CO2 ice recondenses each
night per square meter at the beginning of the spring. We can assume that this
ice condenses mostly at the top of the layer (Pilorget et al., 2011) (because
of the opacity of the CO2 to thermal radiations, and because condensation is
exothermic). This means that from Ls=170° to Ls=250°, from the beginning tho
the end of the sublimation of the seasonal deposit, a total amount of
$112\,\text{kg}$ of CO2 ice recondenses per square meter meaning up to
$70\,\text{mm}$ of CO2 ice recondenses at the surface. To determine this
amount, we simply consider that the amount of recondensed CO2 decreases
linearly to 0 at LS=250°. This represents more than 15% of the initial
thickness of the layer. This lead us to assume that the large water ice grains
that are ejected during a cold jet event, are re-integrated to the slab layer
(see figure15e) by the condensation of CO2 above them at night.

##  6 Conclusions and perspectives

The main results provided by our data analysis are: (i) CO2 ice is most
probably in a translucent state during the whole spring, instead of a granular
one, as previously thought. (ii) Retrieved seasonal variations of translucent
ice thickness is consistent with GCM prediction. (iii) Only few ppmv of dust
are present in the CO2 ice slab during the spring. (iv) Since the water ice
content is 0.1%v throughout the spring season but the CO2 ice is shrinking, we
must conclude that a water escape occurs. The escape of water into the
atmosphere seems to be in agreement with water vapor measurements. (v) The
water grain size increases during the spring from 1 to 50 microns and suddenly
falls down again.

Using all these new constraints, we propose a new sketch to explain the
microphysics of the seasonal south polar cap. We propose a refined cryoventing
mechanism in the presence of water ice: the water ice accumulates at the
bottom of the layer as a result of the slab sublimation from the bottom. Then
water ice grains are ejected during cryoventing episodes. The larger grains
fall back on the layer, while the smaller ones are lifted by the general
sublimation flux, and integrated to the atmospheric circulation. In the last
stage, the water from already defrosted regions condenses onto CO2 ice that
acts as a cold-trap.

The small grain size of water that are incorporated into the general
circulation, should most probably sublimate. Future work, using for instance
limb measurements of MCS data, should confirm this mechanism. Also the effect
of this mechanism to the general circulation should be studied by numerical
simulations. Other study using CRISM data in other location of the south pole,
or even the north pole would be interesting in order to determine if the
proposed sketch is general on Mars or particular to the Richardson crater. In
this perspective, regional scale analysis using OMEGA and the large footprint
mode of CRISM should also be conducted.

#### Acknowledgement

We acknowledge support from the “Institut National des Sciences de l’Univers”
(INSU), the "Centre National de la Recherche Scientifique" (CNRS) and "Centre
National d’Etude Spatiale" (CNES) and through the "Programme National de
Planétologie" and MEX/OMEGA Program. The project « Multiplaneto » has been
granted in 2015 and 2016 by Défi Imag’In/CNRS.

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