Due to its insulating and draining role,
assessing ground vegetation cover properties is important for high-resolution hydrological modeling of permafrost regions. In this study,
morphological and effective hydraulic properties of Western Siberian Lowland
ground vegetation samples (lichens,

Porosity values are in line with previous values found in the literature, showing that arctic cryptogamic cover can be considered an open and well-connected porous medium (over 99 % of overall porosity is open porosity). Meanwhile, digitally estimated hydraulic conductivity is higher compared to previously obtained results based on field and laboratory experiments. However, the uncertainty is less than in experimental studies available in the literature. Therefore, biological and sampling artifacts are predominant over numerical biases. This could be related to compressibility effects occurring during field or laboratory measurements. These numerical methods lay a solid foundation for interpreting the homogeneity of any type of sample and processing some quantitative properties' assessment, either with image processing or with a pore network model. The main observed limitation is the input data quality (e.g., the tomographic scans' resolution) and its pre-processing scheme. Thus, some supplementary studies are compulsory for assessing syn-sampling and syn-measurement perturbations in experimentally estimated, effective hydraulic properties of such a biological porous medium.

Covering a quarter of the Northern Hemisphere's land surface (Brown et al.,
1997), permafrost soils are the most representative soil types in arctic and
subarctic regions. Permafrost is a soil layer in which temperature remains
below 0

Previous studies have addressed this quantification through field observations (Olefeldt and Roulet, 2014; Streletskiy et al., 2015; Throckmorton et al., 2016; O'Connor et al., 2020) or field and laboratory experiments (Vedie et al., 2011; Roux et al., 2017; Wagner et al., 2018). Some recent studies have also dealt with this question using a modeling approach (Bense et al., 2012; Genxu et al., 2017; Burke et al., 2020; Du et al., 2020; Fabre et al., 2017).

Bryophytes (mosses) and lichens are widely present in tundra and taiga
environments. The dominant ground cover consists of

Ground vegetation transfer properties are key information for high-resolution hydrological modeling of permafrost-related catchments. Thus,
reliable estimates of them are necessary for water flux studies for boreal
soils and for climate change impact assessment of the hydrology of high-latitude continental surfaces. Therefore, some recent efforts have been put
into emphasizing the role of the cryptogamic layer in Earth system models (Stepanenko et al., 2020; Shi et al., 2021). Devoted modeling tools have also been created to predict

Many studies are available for the decayed

Only a few studies were conducted on the living part of this upper permafrost layer. Hence, quantitative assessments of some key hydrological properties of ground vegetation layers are needed, such as total, open and enclosed porosity, hydraulic conductivity and specific surface area. In terms of hydraulic properties, hydraulic conductivity has been assessed in the laboratory using constant or falling-head permeameters (Quinton et al., 2000; Price et al., 2008; Hamamoto et al., 2016; Weber et al., 2017) or via field measurements (Päivänen, 1973; Crockett et al., 2016; this study). The results are presented in Table 2, with some peat results for comparison. Otherwise, arctic lichens have received little attention to date. To our knowledge, only one study has estimated lichens' hydraulic properties, considering unsaturated hydraulic conductivity without taking into account macropores (Voortman et al., 2014). However, the specific surface areas of some other lichen species are documented in the literature (Adamo et al., 2007). Some studies quantified arctic lichen properties in response to acid rain (Tarhanen et al., 1999) to clarify their interaction with the rhizosphere (Banfield et al., 1999) or in relation to their albedo properties (Bernier et al., 2011). Some transmembrane transfer properties are also available in the literature (Potkay et al., 2020)

Notation glossary.

Synthesis of saturated hydraulic conductivity (m s

Thus, many field and experimental studies are available throughout the literature. However, field work and experimental studies are known to bring their own difficulties, due to the local conditions' variabilities, sampling biases, disturbances and measurement uncertainties. The aim of this study is to assess some of the transfer properties that are well documented with field and experimental studies with an innovative numerical scheme. Indeed, the use of numerical workflows enhances reproducibility and intercomparison capability between samples. Numerical workflows are often used when experimental studies are complicated to implement (reservoir engineering, aerodynamics, micro-fluidics).

In this work, such numerical workflow is intended to be used for evaluating
the hydrological transfer properties of representative vegetation types of
the Western Siberian Lowlands. To this end, natural samples collected from
the Western Siberian Lowlands are digitally analyzed to characterize some
morphological and hydraulic transfer properties. Thus, in contrast to previous works compiled in Table 2, this study aims to assess the hydraulic
properties of lichens and

To validate this hypothesis, a thorough analysis of sample homogeneity is carried out, based on porosity, as it is the main driver of flow dynamics in porous media (Koponen et al., 1996, 1997). This enables the classification of samples according to their homogeneity. Indeed, for homogeneous samples, a smaller sample region can be considered an effective medium sharing the same properties as the whole sample. Multidimensional porosity description leads to a statistical study of the existence of a representative elementary volume (REV). Two standard porous-medium modeling methodologies are used throughout this study: direct numerical simulations on computed representative elementary volumes (DNS-REVs) and pore network modeling with a built-in solver (PNM). The impossibility of collecting a substantial number of samples is compensated for by a statistical quantification of a REV for each sample. This implies that the REV is smaller than the sample, and hence sampling size is chosen to match sizes that were used in the previous literature, such as Weber et al. (2017).

Samples were collected at Khanymei Research Station (N63

Sampling is thoroughly conducted to minimize structural perturbations. In order to achieve this, each sample's surroundings are cleared with special care prior to extraction. Then, the sample is extracted using a ceramic
knife directly at the right dimensions to fit in a high-density polyethylene box, where it remains from the moment of sampling and drying to
the tomographic examination. Additionally, four in situ hydraulic conductivity measurements are performed on various

Sampling collection method overview.

The samples are then dried at 40

X-ray computed tomography (X-CT) has been widely studied and is extensively used for medical purposes and geoscientific applications (Christe et al.,
2011). Tomography is a non-destructive technique which enables the
observation of pore structure data at micron scale, especially for pore
space assessment in sedimentary rocks. X-CT scanning has been acknowledged
as being an efficient method for accessing morphological information, such
as the pore structure of peat soils (Turberg et al., 2014). Cnudde and Boone
(2013) published an exhaustive review of X-ray tomography applications for the Earth sciences. Rezanezhad et al. (2016) demonstrated that X-CT peat scanning showed a satisfactory spatial resolution for the study of peat's
pore morphology. Since bryophytes can be assumed to represent a cluster of
individuals, X-CT permits the segmenting of each plant structure, which
cannot otherwise be achieved without destructive techniques. Tomographical
scans of studied samples are produced using EasyTom^{®}XL (RX Solutions, France) with a maximal X-ray emission source set to 90 kV. The
obtained resolution, after tridimensional reconstruction, is 94

The sampling locations and processing facilities were far away from each
other. To ensure structural preservation, special care is taken throughout
the sampling, transportation and scanning operations. The samples are
oven-dried for 48 h at 40

To ensure the dry samples' representativity, we used an analogous drying
experimental protocol to the one carried out by Kämäräinen et al. (2018). This experiment is conducted on similar

A comparative study between each of the two sample lots and the lone
individual show that drying does not affect structural preservation. Our
validation experiment converges with the results found by
Kämäräinen et al. (2018). This also confirms hyaline cells'
structural durability: the early work of Puustjärvi (1977) showed that hyaline cells were well preserved during biological decay. Drying impacts aside,

Global porosity (

Porosity is computed on bidimensional horizontal slices along the

The samples could then be classified into three types according to the porosity profile along the vertical axis (Table 3 and Supplement S2). As
porosity appears to be almost constant over the

Type I: constant high porosity along the

Type II: low basal porosity, linearly increasing to the top of the sample

Type III: no specific trend observed in vertical porosity

Computed global porosity (

Open and connected porosities (

The specific surface area is deduced using the same shape analysis and
labeling tools included in IPSDK™. Integrating the surface
between both phases (void and solid) yields the total surface

Specific surface area is conventionally expressed in relation to density (m

Pore size distribution is calculated using ImageJ-Fiji's implemented image segmentation tools on the binarized image stacks. In each stack's image, a Euclidean distance transformation of the matrix phase from the void phase is first applied. Then, for each isolated void patch, the Feret diameter is computed.

In this study, the collected samples are assumed to form a complex fibrous porous medium. Resolving mechanistic equations in such large domains is not straightforward due to the extensive computational resources required. Conversely, resolving such equations on an arbitrary cropped sample would not aid the hydraulic property assessment. To make the link between microscale and macroscale phenomena, a reproducible pattern is required to avoid microscale heterogeneities and lack of information due to a diminutive sample size. To do this, finding a representative region that validates scale separation assumptions with both microscale and macroscale heterogeneities is compulsory, thus defining the volumetric average of a microscale property that is continuous and informative at a macroscale. One of the first volume-averaging methods consists of finding a statistical REV for the given studied property.

Indeed, REV is a theoretical concept clarifying the definition of the
macroscopic scale (Darcy scale) and the microscopic scale (pore scale) and
characterizing a given porous medium. This REV can be assumed to be a specific sample volume in which transfer-governing equations (single-phase flow, for example) may be defined along with the associated effective properties. A proper mathematical definition of a REV is given in Bachmat and Bear (1987),
Quintard and Whitaker (1989) and Whitaker (1999) along with a thorough definition of volume-averaging methods. A generic profile for a given property

Schematic representation of fluctuations of a generic property

The fluctuation profile shows three main domains. Here, the REV is defined as the smallest volume for which statistical fluctuations of a given property in a given space are sufficiently low to consider its average value an effective property. Finding the representative elementary volumes of some key properties (e.g., porosity and intrinsic permeability) is a routine workflow in porous-medium sciences. It is often used for fractured oil reservoirs (Durlofsky, 1991) or artificially packed glass bead media (Leroy et al., 2008). A REV is, by definition, large when compared to characteristic lengths of heterogeneities at a microscopic scale but small when compared to characteristic lengths of heterogeneities at the macroscopic scale. Thus, the properties computed for a REV of a porous medium may be defined and computed as continuous functions of space and even constant, in the case of a homogeneous porous medium, as defined by Bear (1972). In general, REVs are described on the basis of morphological characteristics such as porosity, although a distinct REV can be found for any given porous-medium property. Porosity and hydraulic-conductivity-related REVs are characterized throughout this study, leading to two different sizes, one for each property.

From previously binarized image stacks, a statistical REV analysis is conducted using dedicated high-performance image processing Python libraries (IPSDK™), encapsulated in a specifically designed batch process for which the flowchart is shown in Fig. 3.

Flowchart of the representative elementary volume of porosity (REV

First, porosity (Eq. 1) is computed for a given sub-sampling volume within the whole sample. Then, the sub-sampling volume location is incrementally reduced and moved in every spatial direction. For each sub-volume, intermediate porosities are computed. The average and standard deviations are stored for each chosen sub-sampling volume. Then, an algorithmic routine is used to find the maximal size that satisfies a given threshold (1 %, 3 % or 5 % of porosity fluctuation). These thresholds define the statistical representativity of these REVs. Thus, a REV satisfying a 1 % threshold can be assumed to be a high-grade REV, whereas the 5 % threshold corresponds to lower-grade REVs. For 10 of the 12 studied samples, a REV of porosity is found. The two remaining samples, Hollow1.2 and Peat2.2, do not exhibit a REV for the chosen thresholds. A collection of tridimensional reconstructions of the samples as well as some examples of REVs for each sample are shown in Supplement S1. The graphical plots for each studied sample are available in Supplement S3. The black dot represents the smallest reached representative elementary volume for each sample.

Hydraulic conductivity is estimated through single-phase flow computations performed by solving a Navier–Stokes equation in the pore space of the considered sample. The concept is to carry out the numerical simulation of fluid flow, reproducing the conditions occurring in a constant-head (CHP) permeameter. Then, a sample's hydraulic conductivity is computed from the obtained velocity field. A virtual CHP is created by imposing a constant pressure on two opposite faces to one direction (inlet and outlet). Water-tight wall boundary conditions are applied to other faces, as shown in a conceptual representation of the initial and boundary conditions available in Supplement S2.

Due to computation time limitations, the biggest studied sub-volume with
this approach corresponds to a quarter of the total sample. In Sect. 2.4,
we stated that the REVs of effective physical properties were valid for that
particular physical property. Thus, a hydraulic conductivity REV is
required to statistically assess hydraulic conductivity. For that purpose, instead of counting voxel value algorithms (as made for porosity),
retrieving a representative elementary volume for hydraulic conductivity requires extensive fluid mechanics simulations. Here, a laminar single-phase
flow induced by a pressure gradient is computed for each sub-sample, being
consistent with the idea of reducing and moving a defined sub-volume inside
the overall sample. From open-porosity data in Table 3, we can show that most porosity is connected to each one. This leads to the assumption that
considering all pores can be considered effective in permeability-driven phenomena. As these simulations are resource-costly,
Type-I samples (constant porosity) are selected as they are sufficiently homogeneous for the establishment of REVs. Other types are treated by
another method presented in Sect. 2.4.3. The implemented method relies on
Mohammadmoradi and Kantzas (2016) in conjunction with automatic mesh manipulation tools (

For each sample, four potential REV sizes are computed (23.5, 15.7,
11.8 and 9.4 mm), consisting of 8, 27, 64 and 125 simulations on the

A careful convergence study is also conducted so that numerical errors,
associated with discretization resolutions and iterative procedures for the
approximated inversions of the linear systems involved, are low enough to be neglected in the analysis of the results. Inlet pressures are chosen to
avoid turbulent flows (Re

To avoid artifacts related to the physics of a specific fluid, the

In continental surface hydrology, liquid water's physical property
variations (e.g., volumetric mass

This method is suitable for samples meeting porosity homogeneity requirements, classified into Type-I samples. However, another method is needed to compensate for Type-II and Type-III sample heterogeneity, as using direct numerical simulations on a complete usable volume is prohibitive in terms of computational resources. The results obtained for hydraulic conductivity REV computations are available in Supplement S4.

A double-ring infiltrometry test was also conducted during the sampling campaign. For the sake of comparison, hydraulic conductivity values obtained using this method are also shown in Table 2.

Obtained representative elementary volume based on porosity (REV

For the samples that do not exhibit a REV for hydraulic conductivity (Type-II and Type-III samples), the hydraulic conductivity is then studied using a
pore network model, generated from the binarized image stacks. Pore network models are based on the structural simplification of a complex pore
structure (rocks or reactive porous industrial media, for example) into a
two-state model: spheres and throats. This method often uses various image
processing and segmentation tools to generate a network of spheres and
linking throats, based on an initial tridimensional volume. Introduced by
Fatt (1956), pore network modeling was first studied in conjunction with predefined network properties. Then, pore network generation was adapted to
model some porous media, scanned with X-ray tomography using image-processing algorithms as accurately as possible (Dong and Blunt, 2009). Various algorithms are used to create the internal pore
network structure, such as the

For each binarized Type-II and Type-III image stack, a direct pore network extraction is conducted using the SNOW algorithm implemented in the

In Supplement S5, a comparative study is described based on Type-I samples between both developed workflows. Then, some clues are given as to whether DNS or PNM is suitable for a given sample.

The global-porosity and open-porosity proportion (p

On average, lichens are the most porous of the collection, and peat is the least porous. Porosity values are in line with previously obtained data from
the literature for the highest porous media of the collection (Yi et al., 2009). However, important variability can be observed for

Planar porosity plot along the

No specific trend can be accessed from the

Type I: stable high-porosity profile samples, excluding border effects (

Type II: medium- to high-porosity profile samples associated with a progressive increase from the bottom to the top: Hollow1.2, Hollow1.4, Peat2.2, Peat2.3

Type III: medium to low porosity associated with no specific trend porosity profiles: Mound1.1, Mound2.4, Mound2.5.

Open and connected porosity (

Pore size distribution (Fig. 5) is heterogeneous in each sample, and the sizes are concentrated between 0.01 and 1.00 mm of pore radii. The median pore size varies from 0.23 mm (for peat samples) up to 0.88 mm (for lichen samples).

Inscribed pore size distribution by classified type using particles' Feret diameter measurement. An averaged value is computed for each sample type, each color nuance representing each type.

Intermediate median pore size values can be found for mound

Specific surface area (m

Specific surface area values seem to be uneven between each sample type. For instance, low specific surface areas can be observed for some hollow

Representative elementary volumes for porosity have been computed when possible. For samples exhibiting a REV, porosity has been computed using Eq. (1) applied to the REV. For samples admitting no REV, porosity has still been computed using Eq. (1) but applied to the whole usable volume of the sample. A REV retrieval algorithm was applied to all 12 studied samples, although 2 of them (Hollow1.2 and Peat2.2) did not admit a REV. Obtained REV sizes are shown in Table 4. Some examples of tridimensional visualizations of REVs of porosity are shown in Supplement S1. Due to the numerous graphs obtained during REV computation, tridimensional porosity plots are available in Supplement S3.

REV^{®} Xeon^{®} E5-2680 v2 (2.80 GHz) processors and 128 GB of RAM, using high-performance Python image-processing libraries (IPSDK™). Graphical synthesis of the digital porosity assessment is presented in Fig. 7.

Numerical porosity estimations (%) for each sample. An averaged value is computed for each identified sample type (I, II, III) with corresponding color nuances. Peat 2.2 and Hollow1.2 did not admit any REV.

Due to the time and computational resources needed to achieve a careful study of a representative elementary volume of hydraulic conductivity, only Type-I samples were studied by DNS, as they represent the most homogeneous samples of the collection. Computed REVs of the hydraulic conductivity sizes are given in Table 5. Diagonal hydraulic conductivity tensor components are shown in Fig. 8, and box plots are available in Supplement S4. Computations for the largest sub-sample size (on a 23.5 mm edge) showed higher component hydraulic conductivity values than for the three smaller sizes. This discrepancy can be related to an insufficient computation number for obtaining a good average value, hence the wider statistical spread around the mean value. Moreover, the higher values for the largest studied sizes can also be correlated with heterogeneous hydraulic conductivity behavior, as theoretically shown in Fig. 2, such as effects related to the existence of macropores. An example of a pressure field obtained on a sub-sample of Hollow2.8 through DNS is shown in Fig. 9b.

Diagonal components of the hydraulic conductivity tensor (m s

Diagonal components of the hydraulic conductivity tensor (m s

For three of the Type-I samples, REV

Numerical estimations of hydraulic conductivity are presented in Fig. 8. For each sample of Type I, the axial components of the hydraulic conductivity tensor are given, based on the representative elementary volume of hydraulic conductivity. For Type-II and Type-III samples, hydraulic conductivity estimates are given based on pore network modeling. An example of pressure field computation is shown on sample Mound2.5 in Fig. 9a. Using a pore network allows the estimation of properties in a model based on the whole sample. The use of a pore network is an affordable alternative to direct numerical simulations at the cost of accuracy.

The values obtained vary from

Digital assessments of the morphological and hydraulic properties of

Due to technical limitations, scanning devices have a minimal resolution
that causes a loss of information, acting as a threshold. In this study,
minimal resolution fluctuated between 88 and 94

As described in previous sections of this study, the samples collected are considerably porous. Porosity values are in line with past results found in the literature (Yi et al., 2009; Kämäräinen et al., 2018), with porosities above 90 % for some of the samples. Interestingly, a volumetric digital-specific surface can be well linked with the porosity of complete samples as well as the average porosities found for representative elementary volumes.

A clustering can be seen for the three studied sample types (Fig. 10),
although mathematical relations between specific surface and porosity are
not well defined for such porous media. The specific surface values obtained are of the same magnitude as previous values obtained for other natural moss
and lichen species using geometrical calculations (

Specific surface (m

In Supplement S5, a comparison between direct numerical simulations and pore network modeling is made showing that pore network modeling is suitable for
bypassing the heterogeneity issues observed in our samples. Indeed, the obtained porosity values with PNM are in a 5 % threshold compared to voxel-counting results (Eq. 1). The hydraulic conductivities computed by PNM and DNS are more contrasted, with 1 to 2 orders of magnitude of difference. One should bear in mind that the range of hydraulic conductivity
of natural porous media is huge, with up to 15 orders of magnitude between coarse gravel (

The obtained numerical hydraulic conductivities tend to show high and relatively isotropic hydraulic conductivity tensor values. Hydraulic conductivities found using DNS are sizably higher than previous values found in the literature using field percolation (Table 2), often by up to 1 to 3 orders of magnitude. The hydraulic conductivities found using pore network modeling seem to be more in line with the values in Table 2 as well as the field experiment results shown in Table 6. Nevertheless, it should be kept in mind that the results obtained by this method are less structurally accurate that those obtained from DNS, since they rely on a simplified description of the pore structure. Some clues can be advanced to explain this discrepancy, the first being the impact of numerical reconstruction routines and mesh generation procedures (discussed in Sect. 4.1), the latter being moss compression during field experiments.

Hydraulic conductivity values (m s

Our digital, constant-head permeameter experiments were conducted in a fully saturated medium. Technically unreachable porosity (porosity that is smaller than the minimal scanning resolution) is assumed to play a negligible role in transfers through a saturated medium, reacting as enclosed porosity. In the case of low-permeability porous media, such sub-resolution porosity may affect flow (Soulaine et al., 2016). However, in the case of highly porous and connected media like mosses and lichens, the effects related to sub-resolution porosity are assumed to be low when compared to the effects of the large macropores, which has been shown by Baird (1997). It should also be noted that most of the porosity is opened and connected in our case.

However, moss and lichen samples are compressible (Golubev and Whittington, 2018; Howie and Hebda, 2018; Price and Whittington, 2010). Field percolation experiments induce a sizeable and rapid mass imbalance on this bryophytic cover, compacting the pore space more than would occur under natural rainfall conditions. This might notably affect flow patterns in macropores and explain the lower hydraulic conductivities found in field experiments. Indeed, some clues are given with the results of Weber et al. (2017) on hydraulic conductivity variations according to water saturation. Therefore, the numerical hydraulic conductivity assessments carried out in this study enable property quantification of the medium without perturbation, such as compression of the biological pore structure, which is not possible in field experiments.

A numerical assessment of morphological and hydraulic properties was carried
out on digital X-CT reconstructions of samples of

The methods developed for this application show that a numerical work scheme based on image processing allows retrieval of the morphological properties of any variety of sample. Using such a method permits a nearly unlimited number of property assessments of the same sample, whereas an experimental work scheme requires many samples. Numerical methods enable a qualitative classification of the overall homogeneity of a sample, which is not easily doable using solely experimental methods. Image processing seems to be a satisfactory method, provided that the studied sample is sufficiently homogeneous for the studied property. For heterogeneous samples, image processing is not optimal. However, in the absence of another method, pore network modeling allows us to obtain some information on the studied property which is close to the one found for the homogeneous samples using image processing.

These results provide firm ground for quantitative hydrological modeling of the bryophytic cover in permafrost-dominated peatland catchments, which is crucially important for a better understanding of the global climate change impacts on arctic areas. Using numerical methods potentially enables the assessment of moss and lichen's structural hydraulic conductivity without disturbance by any biological or physical phenomena. Therefore, the porous-medium approaches developed throughout this study led to unprecedented qualitative and quantitative descriptions of such peculiar, highly porous biological media.

These physical properties can then be used as input parameters to describe ground vegetation layers in high-resolution hydrological models of arctic hydrosystems and extensively refine simulations of this critical compartment of boreal continental surfaces. For example, they will be used in further modeling studies of permafrost under climate change at the Khanymei Research Station in the framework of the HiPerBorea project (hiperborea.omp.eu). Further studies are needed to assess variable water content consequences for peat and vegetation pore structure. Indeed, water content is one of the main drivers controlling effective transport properties, such as unsaturated flow, volume change and thermal conductivity.

Material is available on the project's
website (

The supplement related to this article is available online at:

Founding acquisition and project administration was handled by Laurent Orgogozo. The conceptualization, supervision and validation of this paper were conducted by Manuel Marcoux and Laurent Orgogozo. The original investigation, analysis, data curation, software engineering and writing of the original draft were done by Simon Cazaurang. Resources and field methodology were brought by Sergey Loiko, Artem Lim, Stéphane Audry, Liudmila Shirokova, Oleg Pokrovsky and Georgiy Istigechev. The original internal review upon submission was conducted by Manuel Marcoux, Laurent Orgogozo, Sergey Loiko, Artem Lim, Stéphane Audry, Liudmila Shirokova and Oleg Pokrovsky.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was granted access to the HPC resources of
the CALMIP supercomputing center under the allocation 2020-(p12166). Partial support from the Tomsk State University Development Program
(“Priority-2030”) is also acknowledged. Sergey Loiko and Artem Lim thank the Russian Science Foundation (project no. 18-77-10045) for supporting the field work and Georgiy Istigechev for his help with the double-ring infiltrometry.
The authors thank Laurent Bernard and Romain Abbal of Reactiv'IP for their support regardin the developments of the digital characterizations with IPSDK (tm) The authors also want to thank the Clarens Mire steering committee (

This research has been supported by the Agence Nationale de la Recherche (grant no. ANR-19-CE46-0003-01), the Centre National de la Recherche Scientifique (grant BryophyGel “Défi InFinitTI 2018 – 237579”), the Ambassade de France à Moscou (PHC Kolmogorov project 2017 N 38144TB), and the Russian Science Foundation (project no. 18-77-10045).

This paper was edited by Philippe Ackerer and reviewed by Xiaoying Zhang and one anonymous referee.