Thérèse Encrenaz

Bruno Bézard

Thomas K Greathouse

S Holmes

Matthew J Richter

D Nuccitelli

John H Lacy

Sebastien Lebonnois

Franck Lefèvre

Sushil K Atreya

F Lefevre

A S Wong

T Owen

F Forget

Ground-based high-resolution IR spectroscopy of Mars: H2O and H2O2 mapping, search for CH4, and determination of CO2 isotopic ratios

Ground-based high-resolution IR spectroscopy of Mars : H 2 O and H 2 O 2 mapping, search for CH 4 , and determination of CO 2 isotopic ratios Introduction: H2O2 has been proposed as an oxidizer of the martian regolith, which may have sterilized the soil down to a few meters below the surface [START_REF] Bullock | A coupled soilatmosphere model of H2O2 on Mars[END_REF]. In fact, photochemical models have predicted the formation of H2O2 in the martian atmosphere, with column densities in the range 10 15 -10 16 cm -1 . H2O2 is formed by the combination of 2 HO2 radicals resulting from a reaction involving H, O2 and CO2 [START_REF] Krasnopolsky | Photochemistry of the martian atmosphere (mean conditions)[END_REF]. Since H is produced from the photolysis of water vapor, to first order, photochemical models expect H2O and H2O2 abundances to be correlated, if all else were equal. Any H2O-H2O2 correlation, however, is expected at a given altitude, and not necessarily in the column integrated sense.

The detection of H2O2 requires high-resolution spectroscopy, either in the submillimeter range or in the infrared (Encrenaz et al., 2004a). A good candidate in the IR is the ν6 fundamental band around 8 µm. For many years, attempts to detect H2O2 have been unsuccessful [START_REF] Krasnopolsky | High-resolution spectroscopy of Mars at 3.7 and 8 µm : a sensitive search for H2O2, H2CO, HCl and CH4 and detection of HDO[END_REF]. In particular, a stringent upper limit of 9 10 14 cm -2 was obtained by [START_REF] Encrenaz | A stringent upper limit of the H2O2 abundance in the martian atmosphere[END_REF], for a Ls value of 112°, when the mean H2O column density was relatively high (30 pr-µm). During the 2003 opposition, new attempts were performed. They led to the first detection of H2O2 by [START_REF] Clancy | A measurement of the 362 GHz absorption line of Mars atmospheric H2O2[END_REF] in the submillimeter range, and the first mapping of H2O2 at 8 µm (Encrenaz et al., 2004b). These latter observations are described below, together with a mapping of H2O, a search for CH4 and a determination of the 18 O/ 17 O and 13 C/ 12 C in martian CO2 [START_REF] Encrenaz | Infrared imaging spectroscopy of Mars : H2O mapping and determination of CO2 isotopic ratios[END_REF].

The 2003 observations:

Observations of Mars were carried out on June 19-20, 2003 (Ls = 209°) using the Texas Echelon Cross Echelle Spectrograph (TEXES) [START_REF] Lacy | A sensitive high-resolution grating spectrograph for the mid-infrared[END_REF] mounted at the 3-m NASA/Infrared Telescope Facility (IRTF). TEXES is a mid-infrared spectrograph covering the 5-25 µm range with high sptial and spectral resolution capabilities (R= 77000, pixel size = 0.3 arcsec). We scanned the northern and southern hemispheres of the planet with a 1.4 x 8 arcsec slit, oriented N-S and shifted from W to E with 0.5 arcsec steps. After binning, our spatial resolution was 1.5 arcsec. Th doppler shift was +0.045 cm -1 , corresponding to a martian radial velocity of -11 km /s.

H2O2 mapping

Two spectral ranges were covered : 1230-1236 cm -1 (8.09-8.13 µm) and 1237-1244 cm -1 (8.04-8.08 µm). About 6 transitions of H2O2 were identified in each spectral range. Fig. 1 shows the spectrum of Mars in the 1237-1243 cm -1 range, averaged over an area where the H2O2 line depths (in radiance units) were found to be maximum, in order to maximize the S/N ratio. This area covers the latitude range 10N-35S and the longitude range 20E-30W relative to the subsolar point. Figure 2 shows an enlargement of the 1241.5-1241.7 cm -1 range, which shows a doublet of H2O2 lines bracketing a central CO2 martian line. It can be seen that the best fit is obtained for a H2O2 mixing ratio of 4 10 -8 , corresponding to a column density of 7.5 10 15 cm -2 . By dividing the averaged depth of the 2 H2O2 lines by the depth of the CO2 line, we have mapped the H2O2/CO2 mixing ratio over the martian disk. The result is shown in Fig. 3, and a comparison with the GCM predictions [START_REF] Forget | Improved general circulation models of the martian atmosphere from the surface and above 80 km[END_REF] is shown in Fig. 4. It can be seen that the overall agreement is satisfactory in terms of absolute abundances and global distribution ; both observed and computed H2O2 mixing ratios behave similarly in the southern hemisphere, with a maximum at the morning limb and around 20S. However, in the northern hemisphere, the model predicts a maximum of the H2O2 mixing ratio which does not appear in the observations ; this discrepancy remains to be understood. martian HDO transition appears in the wing of a stronger H2O terrestrial absorption, after removal of this continuum with a 2 nd -order polynomial. Assuming a D/H martian ratio equal to 5 times the terrestrial value [START_REF] Krasnopolsky | High-resolution spectroscopy of Mars at 3.7 and 8 µm : a sensitive search for H2O2, H2CO, HCl and CH4 and detection of HDO[END_REF], we find that this line is best fitted with a H2O mixing ratio of 3 10 -4 . Dividing again the HDO line depth by the CO2 line at 1241.6 cm -1 , we obtained the H2O mixing ratio distribution shown in Fig. 6. Comparison with the GCM prediction (Fig. 7) shows a good agreement for both the absolute abundance and the spatial distribution.

Search for CH4

Several strong terrestrial lines of CH4 appear in the 1237-1244 cm -1 range. We have searched for a possible martian line in the blue wing of 2 of these CH4 transitions (1242.7 cm -1 and 1238.7 cm -1 ), at the expected doppler-shifted positions [START_REF] Encrenaz | Infrared imaging spectroscopy of Mars : H2O mapping and determination of CO2 isotopic ratios[END_REF]. No absorption was detected over the disk. We derived, for the CH4 mixing ratio, upper limits of 20 ppb in the morning side and 70 ppb in the evening side (the limit is less stringent in the latter case because the CO2 lines are less contrasted). This result does not contradict previous results about a possible detection of methane [START_REF] Krasnopolsky | Detection of methane in the martian atmosphere : evidence for life ?[END_REF][START_REF] Mumma | Detection and mapping of methane and water on Mars[END_REF][START_REF] Formisano | Detection of methane in the martian atmosphere[END_REF].

In particular the region north of Hellas where a high CH4 abundance (250 ppb) was reported by [START_REF] Mumma | Detection and mapping of methane and water on Mars[END_REF] was not in our field of view at the time of our observations.

Oxygen and carbon isotopes in martian CO2

Several weak transtions of CO2 isotopic bands appear in the spectral interval recorded by TEXES : (627) and ( 628) bands are present at 1237-1244 cm -1 , while ( 638) and ( 628) appear at 1229-1236 cm -1 . Using these transitions we have inferred the following results [START_REF] Encrenaz | Infrared imaging spectroscopy of Mars : H2O mapping and determination of CO2 isotopic ratios[END_REF]: These determinations are consistent with earlier in-situ measurements by Viking [START_REF] Nier | Composition and structure of the martian atmosphere : preliminary results from Viking 1[END_REF][START_REF] Owen | The composition and early history of the atmosphere of Mars[END_REF] and show no evidence for a departure from a terrestrial value. Our 18 O/ 17 O ratio is higher than the value inferred by [START_REF] Krasnopolsky | Oxygen and carbon isotope ratios in martian carbon dioxide : measurements and implications for atmospheric evolution[END_REF] which implied a depletion of the heavier isotope with respect to terrestrial values. Our results are consistent with current models of the martian atmospheric evolution, and do not require any escape nor fractionation mechanism.

The 2005 observations

New measurements of Mars with TEXES have been obtained on Nov. 30 -Dec. 4, 2005 (Ls = 332°). The following spectral intervals have been recorded : 1235-1244 cm -1 (HDO, H2O2, CH4), 1348-1354 cm -1 (HDO, CH4) and 937-944 cm -1 ( 18 O/ 16 O). A preliminary reduction seems to indicate that both H2O2 and H2O are globally les abundant than in 2003 by about 30-50%.

In all spectra, the CO2 martian lines also show an interesting pecularity : in the dark side of the planet, close to the East limb (morning side), the lines appear in emission. This probably indicates a temperature inversion between the surface and the atmosphere, the surface being colder before dawn. These data wil be used to constrain the diurnal variations of the thermal profile over the martian disk.

In summary, these data illustrate the capability of ground-based high-resolution spectroscopy to detect and monitor minor atmospheric species on Mars. In the case of H2O, these instantaneous maps provide an important complement to the long-term mapping achieved by the Mars Express instruments, and earlier by MAWD aboard Viking and TES aboard MGS.

Fig. 1 .

 1 Fig. 1. The TEXES spectrum of Mars in the region of maximum S/N (surrounding the subsolar point) between 1237 and 1244 cm -1 . The red curve

Fig. 3 .

 3 Fig. 3. Map of the H2O2/CO2 mixing ratio, in multiples of 10 -8 . The subsolar point is indicated with a white dot.

Fig. 4 .

 4 Fig.4. Map of the H2O2/CO2 ratio, as modelled by the GCM for the conditions of our observations.

Fig. 5 .

 5 Fig. 5. The HDO martian line observed in the subsolar spectrum of Mars. Models correspond to H2O mixing ratios of 200 ppm (green), 300 ppm (red) and 400 ppm (blue).

Fig. 6 .Fig. 7 .

 67 Fig. 6. Map of the H2O/CO2 ratio. A line depth ratio of 0.5 corresponds to a H2O mixing ratio of 30 ppm. The subsolar point is indicated with a white dot.

[

  18 O/ 17 O(Mars)]/[ 18 O/ 17 O(Earth)]=1.03+/-0.09 [ 13 C/ 12 C(Mars)]/[ 13 C/ 12 C(Earth)]=1.00+/-0.11


R Fablet
email: rfablet@irisa.fr

P Bouthemy
email: bouthemy@irisa.fr

Statistical motion-based object indexing using optic flow field

In this paper, we propose an original approach for content-based video indexing and retrieval. It relies on the tracking of entities of interest and the analysis of their ap parent motion. To characterize the dynamic information at tached to these objects, we consider a probabilistic mod eling of the spatio-temporal distribution of the optic flow field computed within the tracked area after canceling the estimated dominant motion due to camera movement. This leads to a general statistical framework for motion-based video classification and retrieval. We have obtained promis ing results on a set of various real image sequences.

Introduction and problem statement

In order to cope with the increasing development of dig ital video libraries, new methods are to be defined to ac cess and manipulate this tre mendous amount of informa tion, which implies a content-based analysis of these visual documents [START_REF] Aigrain | Content-based representa tion and retrieval of visual media : A state-of-the-art review[END_REF]. The first step consists in extracting the ele mentary temporal units (shots) which compose the video.

Then, the content of each extracted shot is characterized based on key-frame selection [START_REF] Muftit Ferman | Efficient filtering and clus tering methods for temporal video segmentation and visual summa rization[END_REF], mosaic image construc tion [START_REF] Irani | Video indexing based on mosaic represen tation[END_REF], extraction and tracking of entities of interest, [START_REF] Courtney | Automatic video indexing via object motion analysis[END_REF][START_REF] Gelgon | Determining a structured spatio temporal representation of video content for efficient visualization and indexing[END_REF].

The description of the motion content attached to entities of interest usually consists in determining the trajectory of these objects exploiting 2D parametric motion models and in extracting qualitative pertinent features such as the direc tion of the displacement [START_REF] Gelgon | Determining a structured spatio temporal representation of video content for efficient visualization and indexing[END_REF]. However, in case of complex motion (fluid motion, crowds, sport events), motion infor mation cannot be easily handled in such a way. Therefore,

Tracking entities of interest

The first step of our indexing scheme consists in extract ing entities of interest in the video shots. To this end, we consider the semi-automatic tracking technique presented in [START_REF] Gelgon | A region tracking technique with failure detection for an interactive video indexing environment[END_REF]. This tracking technique can cope with a variety of chal lenging situations such as complex motions, changes in illu mination or partial occlusions [START_REF] Gelgon | A region tracking technique with failure detection for an interactive video indexing environment[END_REF]. In Fig. 1, we display two examples of tracking. The first one involves a hockey player and the second one a specific area of a rugby playing fi eld in a video shot acquired with a mobile camera. Whereas the first case could also be processed using a motion de-

1
tection module even if it represents a not so easy situation, the second one cannot be addressed with usual techniques developed for motion segmentation. This tracking module offers new functionalities in the context of video indexing, especially considering the formulation of video queries. In many situations such as sport events, the information of in terest can be not only a single player but also a group of players or more generally a given zone of the scene.

Motion distribution characterization

We aim at adapting statistical tools presented in [START_REF] Fablet | Statistical motion-based video indexing and retrieval[END_REF] to cope with motion characterization within a tracked area in the video shot. In [START_REF] Fablet | Statistical motion-based video indexing and retrieval[END_REF], our approach relies on the analysis of the spatio-temporal distribution of local motion-related measurements directly computed from the spatio-temporal derivatives of the image intensity function. When focusing on an area of interest, the use of dense optic flow field be comes reasonable w.r.t. required computational load, while providing complete motion information in terms of direc tion and magnitude. Besides, we want to evaluate the actual motion of the tracked object. To this end, we compute the residual optic flow field after canceling the estimated domi nant image motion assumed to be due to camera movement.

Dominant motion estimation

We model the dominant motion between two successive images as a 2D affine motion model. The estimation of the six affine motion parameters is achieved with the robust gradient-based estimation method described in [START_REF] Odobez | Robust multiresolution estimation of parametric motion models[END_REF]. The use of a robust estimator ensures the motion estimation not to be sensitive to secondary motions due to mobile objects in the scene. The minimization is performed by means of an iterative reweighted least-square technique embedded in a multiresolution framework.

Residual optic flow estimation

To estimate the residual optic flow field, we exploit the technique described in [START_REF] Mernin | A multigrid approach for hierarchical mo tion estimation[END_REF]. The problem is stated as the global minimization of an energy function which involves robust estimators both in the regularization energy term to preserve motion discontinuities and in the data-driven en ergy term to discard the optic flow constraint when not valid. This minimization is efficiently performed through a multigrid algorithm which exploits different levels of lo cal parameterization of the flow field.

In practice, when considering two successive images of a video, we first compute the dominant image motion. Then, focusing on the area of interest, the flow field resulting from the estimated 2D affine motion model is used as an initial ization for the dense optic flow field estimation in this area.

Finally, the residual optic flow field to be used in the motion characterization stage is the difference between the com puted dense optic flow field and the dominant motion field.

Statistical motion distribution modeling

The characterization of motion information within the area of interest relies on a statistical modeling of the dis tribution of the occurrences of the computed residual optic flow field. Such a statistical representation will be denoted as model V. The conditional likelihood Pv ( o) of sequence of residual optic flow fields o computed within the tracked area over the shot is given by: [-8, 8)). Model estimate V is then defined by the set of model components { Pv ( .X)} given by: >-EA

k =K Pv(o) = IT IT Pv (ok(r)) (1
V.X EA, Pv(.X) = H0(.X)/ [L H0(.X')] ( 2 )

A'EA

In addition, given a model V and a sequence of resid ual optic flow fi elds o, the conditional log-likeliliood, LFv(o) = ln Pv(o) issued from relation [START_REF] Aigrain | Content-based representa tion and retrieval of visual media : A state-of-the-art review[END_REF], can be simply expressed as a dot product between histogram H0 and the set of model potentials V { ln Pv ( .X)} >-EA :

LFv(o) = L H0(.X) • lnPv(.X) ( 3 )
.AEA

Statistical motion-based object classifica tion and retrieval

The statistical representation of the spatio-temporal mo tion distribution within the tracked region can be exploited for motion-based video indexing and retrieval using partial query. Considering a set of video sequences, the associated stored set of extracted regions of interest and their associ ated motion distributions, we aim at retrieving, from this

Bayesian retrieval

Similarly to [START_REF] Fablet | Statistical motion-based video indexing and retrieval[END_REF], we formulate the retrieval process as a Bayesian inference issue. In fact, considering a query q and a stored set of videos (n)nEN, we determine the best matches n * according to a MAP criterion expressed using the Bayes rule as follows: n* = argmaxP(qln)P(n) n

The distribution P ( n) represents the a priori knowledge on the processed database. In our case, we will introduce no a priori, which implies an uniform distribution P(n).

Knowing the statistical model v n attached to an element n of the video base and computed as described in Section 3, P(qln) is expressed as the conditional likelihood Pvn (oq) of the quantized residual optic flow fields oq estimated for query q, w.r.t. model v n :

n* = argmaxPvn (oq) n

Hierarchical classification and retrieval (4)

To handle large video databases, it is generally required to build a hierarchical structure of the database related to content similarity. To this end, we consider an ascendant hierarchical classification scheme [START_REF] Fablet | Statistical motion-based video indexing and retrieval[END_REF]. It consists in iter atively forming new clusters in the hierarchy of entities of interest within video shots of the processed base by merging in tum pairs of elements which minimize a given similarity measure. This process results in a binary tree.

Considering two elements of the database n1 and n2, their associated models Vn, and Vn2 and the quantized residual optic flow fields on, and On2, the similarity mea sure D ( ni, n2) is a symmetric version of the Kullback Leibler divergence K L(n1 lln2):

K L(n1 l l n2) is approximated using an empirical average of the ratios of log-likelihood for the distributions attached to models Vn, and Vn2 (see [START_REF] Fablet | Statistical motion-based video indexing and retrieval[END_REF] When performing retrieval operations over a given video base, its hierarchical representation is efficiently used to sat isfy a video query. First, the node of the highest level of the tree which verifies the MAP criterion is chosen. Then, at each step, we select the child node of the current selected node, still using the MAP criterion, until a given number of answers or a given precision is reached.

S Results and concluding remarks

We we will evaluate this approach on a larger database. We are also investigating other means of designating the region of interest involving automatic motion detection module.

  we aim at deteriminning a new characterization of the mo tion distribution attached to the considered entities of in terest. We compute the residual optic flow field within the tracked area after canceling the estimated dominant motion due to camera movement. Exploiting a statistical frame work, we consider a hierarchical motion-based classifica tion stage, and we define a statistical retrieval scheme with query by example. This paper is organized as follows. Section 2 describes how we interactively extract entities of interest and automat ically track them in a video shot. In Section 3, the statistical modeling of the motion distribution based on the estimation of residual optic flow fields is presented. We introduce the statistical framework for motion-based classification and re trieval in Section 4. Finally, Section 5 contains experimen tal results and concluding remarks.

Figure 1 .

 1 Figure 1. Results of tracking of a hockey player ( a-b e) and of a specified area of a rugby playing field ( d e-/). The bounding polygon encompassing the tracked area is displayed in white.

  ) k =l rERk where K is the length of the shot, Rk the tracked region in image k of the shot, Ok ( r) the residual velocity at point r in image k and Pv (Ok ( r)) the conditional likelihood of the occurrence of velocity Ok ( r) w.r.t model V. To estimate model V for a given tracked area over a shot, we exploit a Parzen window density estimator. Since we consider a simple kernel estimate, it comes to compute the histogram H0 of the occurrences of a quantized version of residual optic flow fields o within the tracked area. Let us note A the range of quantized residual velocities (in prac tice, the horizontal and vertical components of the residual optic flow vectors are quantized over sixteen levels within

  database, examples similar to a video query in terms of mo tion content. The general idea is to define an appropriate similarity measure between shots and to determine the clos est matches according to this measure.As far as query formulation is concerned, we can also supply the user with an interactive interface to express the video query. Using the tracking module presented in Sec tion 2, the user can specify an area of interest in the first image of the query shot and this region is automatically tracked along the shot. Then, the residual optic flow fields within the tracked region are computed at the successive instants of the shot. We will consider this tracked area as the video query in the sequel. This scheme allows the user to formulate a wide range of partial video queries such as tracking specific entities (objects, characters) or focusing on a particular area of the scene involving different entities.

  for details). It indeed comes to approximate K L(n1 lln2) by the following ratio: When evaluating the similarity measure between two clus ters C 1 and C 2 , we exploit the following definition of D: In the ascendant classification procedure, the creation of a new cluster in the hierarchy requires to attach to the corre sponding node a statistical model in order to perform later the retrieval process through this indexing structure. Since the occurrence histogram of the quantized residual optic flow vectors for the regions of interest of a group of video shots is the merge of individual occurrence histograms, the associated model can be easily estimated using relation (2).

  have carried out experiments on a set of real image sequences. We have paid a particular attention to choose a video set representative of various motion situations : sport videos (basket, rugby, hockey, ... ), rigid motion situations (cars, train, ... ), and low motion activity examples. Finally, we consider a database of 50 video sequences. For each element of this database, we have selected en tities of interest over which we have performed the estima tion of their residual optic flow fields. Then, we have com puted the associated occurrence histograms and ML statis tical models. Afterwards, we have applied the hierarchical motion-based classification scheme. In Fig. 2, we report three examples of retrieval operations. For each query, four answers are sought. The first example involves a news pro gram where the entity of interest is the anchor. It presents a very weak motion activity. The four retrieved answers be long also to this class of video. The second example is a shot of a hockey game with a close-up on a player. The proposed answers involve also video with important motion activity and a focus on a particular player. The last query is concerned with a specific part of the playing field in a hockey game. The system again delivers correct examples similar to the query in terms of motion content, since they also correspond to a wide-angle shot of the playing field.

4 L

 4 = -0.210 L = -0.260 L = -0.266 L = -0.286

Figure 2 .

 2 Figure 2. Examples of motion-based retrieval operations. For each query shot and for each retrieved shot, we display the first image, and the entity of interest is delimited by a white polygon. Besides.for each reply, we report the value of log-likelihood L of the query w. r.t. statistical motion model V attached to this reply.

Acknowledgments: The authors would like to thank E. Memin for providing the code for optic flow estimation, and INA (lnstitut National de I' Audiovisuel) for supplying the MPEG-I news sequences.


Jean Robert

Kala Kamdjoug

Philippe Lenca
email: philippe.lenca@enst-bretagne.fr

Jean-Pierre Barthélemy
email: jp.barthelemy@enst-bretagne.fr

Jean-Pierre Barthélemy

J R Kala Kamdjoug

J.-P Barthélemy

Assessment of actions in a

Keywords: Refunding, Microfinance, Multi-actor, Multicriteria, Decision rules, Judgment

come

Introduction

Actions that occur within human organizations usually entail many decisionmaking processes. Whenever analysing these processes it is essential to take the multitude of actors involved into account. It is rare to find concrete applications where a single point of view is sufficient to uncover all the necessary information and several points of view must be taken into consideration and any accurate analysis of a decision-making process should be made from a multi-criteria point of view [START_REF] Roy | Multicriteria analysis: survey and new directions[END_REF].

The decision-making methodology defined and discussed in this article is based on work described in [START_REF] Jessi | An integrated framework for group decision support system design[END_REF], [START_REF] Islei | Group decision making: suppositions and practice[END_REF], and [START_REF] Marchant | Les systèmes d'aide à la décision en groupe[END_REF], where decision-makers come to a consensus on a single working model while keeping separate preferences for the evaluation of actions in a multi-criteria framework. It has been created with the dual purpose of extracting human decision-making strategies from real-life situations and providing decision-makers with a tool for guiding them in efficient decision making.

This decision-making methodology can be conceptualized according to two approaches: the process approach and the recommendation approach. The process approach involves reference actions which are those actions that decision-makers can use as if then rules for carrying out their decisions (e.g. acceptance/rejection). In a multi-actor context, these reference actions are computed by the aggregation of individual decision rules. Sub-sequently, the recommendation approach involves a new Multi-Criteria Decision Aid method (MCDA), Analysis by Trichotomy and Risk Evaluation Evidence (ATREE). ATREE is designed to help decision-makers in the final deliberation process.

This article is organized as follows: Sect. 2 outlines the working hypotheses of our methodology. Section 3 describes the decision space and the procedure used to extract expert decision strategies. Section 4 describes the conceptual framework and the process approach used to determine reference actions. Section 5 describes the MCDA method ATREE, and finally, Sect. 6 is devoted to a practical application of this methodological tool in the context of the refinancing of Microfinance Institutions (MFIs) organized as a network.

The working hypotheses

At the individual level, we consider expert decision-makers as individuals having a great experience in their domain of expertise. An expert is someone who can make sense out of chaos [START_REF] Shanteau | Psychological characteristics of expert decision makers[END_REF]. We assume that each decisionmaker uses stable strategies to evaluate actions (i.e. situations, alternatives, etc.) described by a set of criteria for a specific task (categorization, selection/elimination, etc.) [START_REF] Barthélemy | Choice basis: A model for multi-attribute preference[END_REF]. The decision-maker uses thresholds on criteria values in order to take a decision. This threshold hypothesis is a monotonicity property and implies that the criteria and the decision categories are linearly ordered. Decision rules can then expressed in terms of conjunction/disjunction on criteria value thresholds.

At the group level, a decision-maker is expected to adhere to a principle of goodwill that is composed of three requirements. According to the first requirement, he must not be motivated by personal interest. In the second, he must agree with group decisions that are based on a consensus arrived at during a committee meeting. Finally, he must cooperate with the group even if his point of view conflicts with those of others.

In this way, the proposed methodology is intended to support individual as well as group decision making.

Finally, our working hypotheses can be summarized in the four following base hypotheses:

H 1 : The decision-makers describe actions using m criteria in order to express their preferences. These criteria are ordinal. H 2 : The decision-maker accepts or rejects a given action. Therefore, it is not a choice task (to choose the best action in a given set), but a judgment task (to decide if an action is acceptable or not). H 3 : The decision-maker is assumed to be a rational being. This means that he will automatically accept an action that, according to all the criteria, is better than the one already adopted. Inversely, he will reject any action that, in the same context, is less preferable than one that has already been rejected. H 4 : Decisions must be made in a group and the decision-makers involved are required to cooperate and bear in mind that they are working towards the success of a common goal, even though they have conflicting points of view.

One objective of this study is to imitate as closely as possible expert decision-maker behaviour.

First, our approach should take into account cognitive constraints. Amongst these, the main cognitive hypotheses for the process of decision-making are: decision making viewed as the articulation of elementary strategies [START_REF] Montgomery | On decision rules and information processing strategies for choices among multi-attribute alternatives[END_REF][START_REF] Svenson | Process description of decision making[END_REF]; decision making as a search for dominance structure [START_REF] Montgomery | Aiding Decision Process, chapter Decision rules and the search for a dominance structure: toward a process model of decision-making[END_REF].

Second, our approach should be based on decision-maker expertise. Generally, to assist expert decision-makers, it is essential to understand their strategies, but, if this is not the case, it will be be necessary to obtain this vital information from them. Indeed, expert decision-makers use stable and efficient dominance structures, therefore it should be possible to acquire them [START_REF] Barthélemy | Choice basis: A model for multi-attribute preference[END_REF][START_REF] Barthélemy | A model of selection by aspects[END_REF].

On the basis of these four hypothesis, we propose a three-step methodology:

(1) extraction of each individual's decision strategies in a multi-criteria framework taking into account cognitive modelling; (2) constitution of a common consensus base of decision strategies (construction of an epistemic decision-maker representing the group of decisionmakers); (3) recommendation and decision support for the group of decision-makers based on the consensus base.

Acquisition of decision strategies

The purpose of this section is to briefly describe the process by which experts' decision strategies are obtained from real-life situations. It is divided into the two following parts: the definition of the decision space as it is used in this study and the extraction of decision strategies.

Decision space

Let us denote by A the set of actions. As mentioned in our working hypothesis we consider actions a ∈ A described by a set of m criteria. Each criterion g j , j ∈{ 1, 2, ..., m} is defined on an ordinal scale j ={ 0 < 1 < ••• < c j-1 } (e.g. g j takes c j values). At this point we will not present or develop fully the way in which these criteria can be obtained (e.g. by interviewing experts, interactive processes, etc.) because it is a long, drawn-out complex process. In addition, the family of criteria should be built in such a way that decision-makers' opinions are as well represented as possible (i.e. the family of criteria should be coherent; [START_REF] Roy | Aide Multicritère à la Décision : méthodes et cas[END_REF].

Set A has a natural representation in the direct product of the m total orders = m j=1 j . An action a∈A is described by an element x = (g 1 (x), g 2 (x), ..., g m (x)) ∈ where g k (x) indicates the value of x on the criterion g k . is ordered by the classical order : ∀x, y ∈ , x y ⇔∀j, x j j y j (literally, x ∈ is better than y ∈ or x dominates y).

Extraction of decision strategies

Each decision-maker has his own decision rules that we one can express in the classical way of production rules if <condition> then <conclusion>, where:

• <condition> expresses a sequence of tests on the criteria. As the criteria are ordinal (hypothesis H 1 ) and the experts rational (hypothesis H 3 ), all tests have the following structure <g 1 (.) σ 1 ... and g j (.) σ j ... and g m (.) σ m >, where σ j is the minimal value to be respected on g j (.), ∀j ∈{1, ..., m}. Of course, the thresholds σ i (for criterion g j ) can change from one conjunction to another and from one decision-maker to another.

• <conclusion> is acceptance or rejection (hypothesis H 2 ). Due to the nature of the preceding tests (<condition>), only the acceptance cases are considered. However, if none of the tests are valid, the conclusion is the rejection of the action. This indicates that an action is rejected when it does not fulfill any of the acceptance requirements.

An expert decision-maker's strategies, i.e. a set of production rules, is then a disjunction of conjunctions (as defined above).

The decision rules of a decision-maker divide into two subsets: the set a of accepted elements and the set r of rejected elements. The H 3 hypothesis implies that no element of r dominates an element of a . In addition, the minimal elements of a constitute the acceptance rules for decision making. These decision rules constitute an antichain which is a subset of incomparable elements in . In this case, an antichain is a disjunction of decision rules as well as a set of minimal conditions that, according to the dominance principle, must be met before an action can be accepted. Thus, learning decision-makers' strategies is equivalent to seeking an antichain in the decision space [START_REF] Pichon | Un algorithme de partition d'un produit direct d'ordres totaux en un nombre minimum de chaînes[END_REF]. Note that we can use the same representation space for coding objects of A or decision strategies. We will then use and A equally.

In order to imitate decision-makers' behaviour as closely as possible, particularly so as to take into account cognitive constraints, we used the Moving Basis Heuristics (MBH) model [START_REF] Barthélemy | Choice basis: A model for multi-attribute preference[END_REF], 1987). This model is based upon a multi-attribute representation space that uses several attributes to describe the set of actions. The model assumes that the decision-maker behaves rationally in the sense that he operates optimally when solving problems within his scope of expertise. Nevertheless, this rationality is bounded [START_REF] Simon | Model of Thought[END_REF] by the decision-maker's cognitive abilities (short-term memory and computing capacity) and job satisfaction (pleasure, risk, etc.).

In this framework, MBH pre-supposes the coordinated use of four types of rules: lexicography, threshold, conjunction and disjunction. However, during the decision-making process, the main rule used by the decision-maker is the dominance principle, all the other rules being only used to provide a dominance structure as quickly as possible [START_REF] Montgomery | Aiding Decision Process, chapter Decision rules and the search for a dominance structure: toward a process model of decision-making[END_REF]. The extraction of decision strategies is achieved using the Acquisition Par Apprentissage de Connaissances Humaines Expertes: Acquisition by Learning of Expert Knowledge (APACHE) tool which is an implementation of MBH [START_REF] Lenca | Application of expertise acquisition in savings plans[END_REF][START_REF] Lenca | Acquisition automatique et analyse de processus de décision[END_REF].

In the present methodology, there is a set of rules that describes the precise moment at which a decision-maker endorses an action. A decision-maker i endorses action o when o dominates at least one element of his set of decision rules denoted R i (i.e. the set of minimal elements of a computed by the APACHE tool; [START_REF] Lenca | Acquisition automatique et analyse de processus de décision[END_REF]. Thus, it must compile all the decision strategies of the group and reach a consensus, a sort of cumulative decision strategy [START_REF] Kamdjoug | A multicriteria aid and multiactor method: determining a corpus of reference actions-application to the resolution of micro-credit problems[END_REF].

The rule which we call the cumulative decision strategy rule is stated thus: for every action o ∈ , we count the number t(o) of decision-makers who endorse it. This value t(o) is the total of o. The reference actions can then be guaranteed by the following cohesion principle: ∀o, o ′ ∈ ,i ft (o) ≥ t(o ′ ), then we cannot have o ′ dominating o. We thus use two thresholds τ 1 τ 2 0(τ 1 ≤ d, where d is the number of decision-makers), and o∈ is accepted (resp. rejected

) if t(o) ≥ τ 1 (resp. if t(o)<τ 2 ).
In the above process, we obtain two subsets of : R a composed of the minimal elements of the subset a of accepted elements and R r which is made up of the maximal elements of the subset r of rejected elements. Therefore, is divided into three subsets a , d and r bounded by the borders R a and R r .T h es e t d contains elements o for which t 2 ≤ t(o) ≤ t 1 . Therefore, R a and R r are the decision making rules of the consensual decision-maker: R a for the acceptance subset a and R r for the refusal subset r . Unfortunately, for the elements of d , we cannot decide whether to accept or reject them. Thus, in order to make recommendations for these elements, it is necessary to formulate an additional process to deal with them. This is the third step of our methodology presented in the next section.

A description of ATREE

Once the consensus procedure described above has been carried out, then the recommendations for actions in a and r are obvious. Nevertheless, in order to come up with recommendations for actions in d , it is still necessary for these actions to undergo additional treatment. This treatment is done using an ad hoc multicriteria decision making methodology called Analysis by Trichotomic and Risk Evaluation Evidence (ATREE) which is based on two working hypotheses:

• Each expert evaluates and determines the contribution level of the criteria involved in a recommendation. This evaluation is based on the decisionmaker's decision rules. • Each criterion contributes to the acceptance or rejection of an action.

Each criterion has a level at which it contributes to the recommendation for a given action in d . This level of contribution is not necessarily numerical (for example it can be low, medium or high) but we will estimate it with numerical values that we call likelihoods.

Definition of criterion likelihoods

The acceptance or rejection of an element o ∈ has consequences that impact the contribution level of each criterion g j (j ∈{ 1, ..., m}). The acceptance of o tends to increase the contribution level of those criteria on whose basis o is positively evaluated. Similarly, the rejection of o tends to decrease the contribution level of those criteria on whose basis o is negatively evaluated.

Therefore, four likelihoods for each criterion g j can be defined:

• p a j,+ (o) is the likelihood that the contribution of the criterion g j is high when o is accepted, which can reinforce its acceptance;

• p a j,-(o) is the likelihood that the contribution of the criterion g j is low when o is accepted, which can weaken its acceptance; • p r j,+ (o) is the likelihood that the contribution of the criterion g j is high when o is rejected, which can reinforce its rejection; • p r j,-(o) is the likelihood that the contribution of the criterion g j is low when o is rejected, which can weaken its rejection.

Before proceeding to the evaluation of these likelihoods, it is essential to define a sort of numerical evaluation for each criterion g j that can be used to carry out arithmetic operations. This numerical evaluation, denoted γ j (o) ( j ∈{1, ..., m}), measures the magnitude of a criterion g j that defines action o and is defined as follows:

when o ∈ d , γ j (o) ∈[-1, 1],
where -1 corresponds to the lowest value on the scale of g j and 1 to the highest value. Depending on the magnitude of the difference in preference that exists between any two consecutive values on the evaluation scale for g j , the measurement γ j can be linear, by step, concave, convex, etc.

The evaluation of likelihoods must of course follow our working hypothesis especially the dominance one (hypothesis H 3 ) that implies that decision categories are linearly ordered. In [START_REF] Kamdjoug | Aide à la décision pour la gestion de micro-crédits en Afrique : application à la sélection des CVECA au refinancement[END_REF] we also add a second constraint, domination. Domination requires that the values of the likelihoods of an action o should be in conformity with the number of elements in d that o dominates (this property must be verified of course for the elements which dominate o). Therefore, we propose the following functions for calculating the likelihoods for criterion g j for action o ∈ d . These functions respect the two requirements.

p a j,+ (o) = γ j (x)>ρ j ,x∈[o] - γ j (x) a j
,( 1 )

p a j,-(o) = γ j (x)<-ρ j ,x∈[o] + -γ j (x) r j , ( 2 )
p r j,+ (o) = γ j (x)<-ρ j ,x∈[o] - -γ j (x) a i ,( 3 ) p r j,-(o) = γ j (x)>ρ j ,x∈[o] + γ j (x) r j , ( 4 )
where the threshold ρ j ≥ 0 is used to define the notions of strongly evaluated action o (γ j (o)>ρ j ) and weakly evaluated action (γ j (o)<-ρ i ) for each criterion g j ;t h eγ j (x) are linear functions in this study: γ j (x) =-1f o rx = 0, γ j (x) = 1 for x = c j -1 and γ j (x) = 2x c j -1 -1for0< x < c j -1; the quantities a j and r j guarantee that p a j,+ (o), p a j,-(o), p r j,+ (o) and p r j,-(o) remain limited to the interval [0, 1] and are defined by: a j = max

The degrees of confidence

The likelihoods defined above explain in a measurable way how each criterion contributes to the rejection or the acceptance of an action. The objective of this section is the calculation of global measurements called degrees of confidence for acceptance and rejection.

For a given action o ∈ A d ,thedegree of confidence for acceptance is defined as follows:

a (o) = 1≤j≤m φ a j (o)(1 + p a j,+ (o) -p a j,-(o)),( 5
)
where φ a j is a positive function so that, for o, o ′ ∈ ,i fg j (o) ≤ g j (o ′ ), then φ a j (o) ≤ φ a j (o ′ ). This function is intended to regulate the value of a (o). Likewise, for a given action o ∈ the degree of confidence for rejection is defined as follows:

r (o) = 1≤j≤m φ r j (o)(1 + p r j,-(o) -p r j,+ (o)),( 6
)
where φ r j is a positive function so that, for o, o ′ ∈ ,i fg j (o) ≤ g j (o ′ ), then φ r j (o) ≥ φ r j (o ′ ). It is intended to regulate the value of a (o). a (o) is an additive form which measures the proximity of o to a , and r (o) is an additive form which measures the proximity of o to r . The principles of dominance and domination, presented in Sect. 5.1, guarantee that a is a monotonous-increasing function and r is monotonous-decreasing function, just as is demonstrated in [START_REF] Kamdjoug | Aide à la décision pour la gestion de micro-crédits en Afrique : application à la sélection des CVECA au refinancement[END_REF].

Several regulating functions might be considered, but the majority introduce a significant scale-based bias. Exponential functions make an ideal compro-mise in a way that is suitable to the present methodological context. These exponential functions are defined as follows: φ a j (o) = e ω j γ j (o)+α j , φ r j (o) = e -ω j γ j (o)-α j , where ω j 0, α j 0. (7)

Finally, for a given action o ∈ d , the acceptance of o is recommended when the value of a (o) is sufficiently greater than the value of r (o), otherwise rejection is recommended:

if a (o) a (o) + r (o) ≥ η(where η ∈[0.5, 1]) then o is accepted, else o is refused.
The threshold η ∈[0.5, 1] defines exactly at what point an action is accepted or rejected. The precise limits of this threshold are established empirically, during discussions with the decision-makers.

A practical application of the proposed methodology to the refinancing of MFIs

Very few projects in the field of MCDA studies deal with microfinance. Nevertheless, a study by [START_REF] Kamdjoug | Human and social criteria for multicriteria decision aid in microcredit: how to choose a cveca installation site[END_REF] applies the ELECTRE III method [START_REF] Roy | Aide Multicritère à la Décision : méthodes et cas[END_REF] to the problem of new MFI site selection. However, existing multi-criteria methods were not adapted to the selection process for MFI seeking refinancing. Therefore, to meet this theoretical and methodological need, the present methodology is proposed. This methodology was tested with success on the selection process for candidates in the second year ENST Bretagne engineering programme [START_REF] Kamdjoug | Aide à la décision pour la gestion de micro-crédits en Afrique : application à la sélection des CVECA au refinancement[END_REF]. Following the positive results of this test, the methodology was applied with confidence in the field of microfinance. In this section, we describe how the methodology was applied to the microfinance decision-making process of an organized network of rural financial institutions in Cameroon. Five Microfinance and Development (MIFED) experts were involved. Their task was to decide whether a Microfinance Institution (MFI) is financially viable or not. The group of institutions used to test the theoretical model was composed of 18 real MFIs.

Evaluation criteria

Five criteria were designed to identify financially viable MFI on the basis of their sustainability and ability to provide quality services to their clients:

A: The real funding needs of the MFI. This criterion gauges if the income generating activities of the MFI clients can justify the quantity of funds requested. This is because loan repayment feasibility and timing depend very much on MFI lending activity as well as on the business activity of the MFI clients. B: MFI credit portfolio management. This criterion assesses the quality of credit portfolio management using instruments such as financial ratios, credit recovery and loan repayment periods. C: MFI -active member relations. This criterion evaluates MFI managerial staff-support team relations, member dedication to operational efficiency and the level of involvement of women in managerial positions. D: MFI information disclosure levels. This criterion evaluates the MFI on the basis of the frequency of work-oriented meetings between different levels of MFI management and the real impact these sessions have on MFI members. E: MFI outreach and growth potential. This criterion evaluates the MFI on the basis of the relative increase in the number of MFI clients and the size of the average savings and loan transaction.

These criteria allow us to gauge MFI administration levels, MFI sustainability levels and MFI efficiency in administering financial services that support productive client micro-projects. Even though the criteria are based on a mixture of quantitative and qualitative data, we use ordinal scales to evaluate them. Each criterion is evaluated using the following evaluation scale and codes: 0 for "totally unacceptable", 1 for "unacceptable", 2 for "acceptable" and 3 for "very acceptable". Therefore, the decision space contains 1,024 objects.

Decision-makers rules

Table 1 presents, the set of decision rules, for each decision-maker. For example, decision rule (2 3 1 0 0) of Expert 3 should be interpreted as Expert 3 recommends accepting the MFI whenever it is judged "acceptable" on the basis of criterion A, "very acceptable" for criterion B and "unacceptable" for criterion C (no matter the outcome of the evaluation of criteria D and E).

Table 2 gives the evaluation scale proportions and standard deviations for the five decision rule criteria. 1 These proportions and standard deviations give us information regarding the rules applied by each expert. When an expert attaches great importance to a criterion, this criterion is considered to be proportionally high with minimal standard deviation. For example, in Expert 3's decision rules, criterion B is said to be proportionally high (1) with min-1 In order to compute the proportion and standard deviation for each criterion, we proceed in the following way: for each expert, we use the value assigned to each criterion by each decision rule (e.g. x i ). From this value, we construct a Bernoulli variable using the number 1, when x i > 1-otherwise 0 is used. Then we calculate the sum of the Bernoulli variables obtained from the decision rules of the corresponding expert. This constitutes a Binomial variable. The ratio of the sum of the Bernoulli variable over the number of decision rules corresponds to the proportion of the criterion for that expert. The standard deviation of the criterion is the standard deviation of the Binomial variable obtained by summing the Bernoulli variables.

Table 1 The experts' decision rules

Expert1

{( 22000),( 13310),( 13021),( 12231),( 13102),( 13012),( 02222),(123 13),( 03313),( 31323),(12033)} Expert2 {(33100),( 22200),( 33010),( 32110),( 22020),( 13120),( 31220),(212 30),( 12230),( 32001),( 23101),( 23011),( 12221),( 21321),( 12031),(2 1131),( 23002),( 22102),( 22012),( 13022),( 12122),( 03332),( 22003),( 13303),( 21213),( 21023),(03233)} Expert3 {( 23100),( 22200),( 03200),( 02300),( 22110),( 02210),( 22020),(120 30),( 03030),( 12201),( 23011),( 03021),( 23002),( 22102),( 02202),(2 2012),( 13012),( 12022),( 02032),( 22003),( 13103)} Expert4 {(32200),( 22300),( 13300),( 33110),( 22210),( 03320),( 33030),(122 30),( 30330),( 02330),( 13211),( 31311),( 03311),( 21221),( 12221),(0 3221),( 32031),( 20331),( 33102),( 12302),( 22022),( 02222),(3032 2),( 20232),( 22203),( 21213),( 03213),( 12133)} Expert5 {( 22200),( 31300),( 23110),( 30210),( 32020),( 22120),( 20220),(2203 0),( 33001),( 32101),( 22021),( 30131),( 32002),( 23002),( 30202),(21 202),( 20302),( 31022),( 20122),( 20032),( 22113),( 20213),( 20023)} imal standard deviation (0). Since criterion B measures the credit portfolio management quality, it implies that Expert 3 is more likely to reject an MFI with an inclination towards risk-incurring credit portfolio management. And so, according to Expert 3's decision rules, if the MFI is to become sustainable, it must maintain an average level of acceptability with respect to criterion B.

Consensus base

A closer look at the decision rules in Table 1 reveals that, in some cases, there are differences in the way that the experts formulate their decision rules. This indicates that, in the beginning, the experts do not assign criteria the same level of contribution to the decision even though they must eventually reach a consensual final decision. In practice, however, it is difficult to capture and conceptualise the negotiating process that the experts carry out when working to reach a cumulative consensus on their final recommendations for the MFI. 03313),( 12313),( 31330),( 12033),( 13212),( 13302),( 21213), ( 21231),( 21321),( 31221),( 02222),( 12221),( 12230),( 13022), ( 13031),( 13121),( 13310),( 22003),( 22012),( 22102),( 23002), ( 23011),( 23101),( 23110),( 32002),( 32101),( 32110),( 33001), ( 33100),( 22020),( 22200)} [Majority vote] Therefore, if one is to determine how common decision rules are made, it is necessary to explore different ways in which experts might reach a cumulative consensus. We propose and test three alternative voting rules (or cumulative decision rules): for the unanimity vote, the suggested values are τ 1 = 5 and τ 2 = 1, the majority values are τ 1 = 3 and τ 2 = 1, and the intermediate values are τ 1 = 4 and τ 2 = 2.

τ 1 = 3 τ 2 = 1 R a = {(
Table 3 presents the cumulative decision rules of the consensus. There are 31 decision rules for representing the minimum conditions for obtaining an "acceptance" recommendation for MFI refinancing when the majority vote option is applied. The number is reduced to 9 decision rules when the unanimity vote is applied. These results illustrate the diversity of the minimal profile for representing when an MFI is approved for refinancing under a majority vote.

Note that, from a practical point of view, the theoretical model applies the decision strategies to sample data taken from a village bank. Then model recommendations are compared to real decisions that have been taken in the past.

τ 1 = 3, τ 2 = 1 τ 1 = 4, τ 2 = 2 τ 1 = 5, τ 2 = 1 Decisions (3 2 2 2 2) Accepted Accepted Accepted Accepted (3 3 3 3 2) Accepted Accepted Accepted Accepted (2 2 3 3 2) Accepted Accepted Accepted Accepted (2 2 2 2 1) Accepted Accepted Accepted Accepted (2 0 2 1 1) Rejected Rejected Rejected Rejected (2 1 1 2 1) Rejected Rejected Rejected Rejected (2 3 3 3 2) Accepted Accepted Accepted Accepted (2 0 1 2 1) Rejected Rejected Rejected Rejected (1 1 2 1 2) Rejected Rejected Rejected Rejected (2 2 2 3 2) Accepted Accepted Accepted Accepted (3 3 2 3 2) Accepted Accepted Accepted Accepted (2 0 1 2 1) Rejected Rejected Rejected Rejected (3 3 3 3 2) Accepted Accepted Accepted Accepted (2 1 2 3 2) Rejected Rejected Rejected Accepted (1 0 0 2 1) Rejected Rejected Rejected Rejected (2 1 2 1 1) Rejected Rejected Rejected Accepted (2 2 3 3 3) Accepted Accepted Accepted Accepted (2 1 1 1 1) Rejected Rejected Rejected Rejected
Finally, parameters τ 1 and τ 2 are adjusted to best fit recommendations and past decisions. These two thresholds can also be used as control parameters that will allow "easy" or "difficult" conditions for refinancing.

Analysis and recommendations

The acceptance and rejection subsets lead to fairly well defined scenarios that leave limited space for ambiguity. The subset for dubious MFI, however, is an entirely different matter and needs to be adjusted before it can be useful for formulating recommendations from the present model. For this reason, before processing the MFI in the dubious subset, we must first adjust our model's ATREE parameters so that it will produce results that are closer to the experts' recommendations (see Table 4). Furthermore, there is the matter of dubious subset voting rules. In the case of the unanimity vote, no MFI is in the dubious subset, therefore it is not possible to change the recommendations by modifying the parameters. But in the case of intermediate and majority votes, only one MFI,(21232),isinthedubious subset's.

A closer analysis of Table 4 demonstrates that all MFI shown to have poor credit portfolio management were rejected by our model (which is because they were assigned a value of 0 or 1 for criterion B). This recommendation confirms the importance of meticulous MFI credit management and is consistent with the decision-makers' concerns. In the case of two MFI however, we observe two contradictions between the model's recommendations and those of the experts: (21211)and(21232).T hesetwoMFIwereaccepted by the decision-makers even though their credit portfolio management is considered to be risk-incurring. It is for this reason that our model rejects them. When this discrepancy was raised with the experts, they justified their acceptance of these two MFI by pointing out that, at the time of the evaluation, the credit management team for those MFI had just been changed and the new team was seen to be practising sound management. Unfortunately, the experts' recommendations on these two MFI (values (21211 )a n d( 21232) )w e r eb a s e d on additional information that the model could not take into account. In such situations, the decision-makers can disregard the model's recommendation and base their decision on background knowledge that their professional experience has shown to be significant.

Conclusion

This paper proposes a three-step multicriteria decision aid methodology to assist a group of decision-makers in an acceptance/rejection recommendation task. The first phase is based on decision-makers' strategies, the second one on the construction of a consensus base and the last on an original multi-criteria decision aid tool.

We also present a case study in microfinance decision making for 18 concrete decisions taken by a group of five experts. The recommendations provided by our methodology are very similar to those of the experts.

Moreover our approach allows us to build a system that accumulates and shares expert knowledge. This system would allow MIFED experts to process a greater number of refinancing applications with more efficiency than is currently possible. It would also enable foreign and local experts to exchange knowledge and resources. The usefulness of this methodological tool can be justified for two reasons. First, there are not enough microfinance experts to meet current demands in Cameroon and second, those foreign consultants who have made themselves available to supervise the project only do so temporarily.

  y∈ d Ŵ j ([y] -) and r j = max y∈ d Ŵ j ([y] + ), where for ⊆ d , Ŵ j ( )= x∈ |γ i (x)|; [o] + ⊂ d contains the elements which dominate o and [o] -⊂ d contains the elements which are dominated by o.

Table 2

 2 Statistics on the criteria in expert decision rules

		Statistics	ABCDE
	Expert 1	Proportion	2/11	10/11	6/11	5/11	7/11
		Standard deviation	1.28	0.95	1.65	1.65	1.59
	Expert 2	Proportion	18/27	21/27	10/27	14/27	11/27
		Standard deviation	2.45	2.16	2.51	2.60	2.55
	Expert 3	Proportion	9/21	21/21	6/21	6/21	9/21
		Standard deviation	2.27	0.00	2.07	2.07	2.27
	Expert 4	Proportion	16/28	21/28	22/28	15/28	10/28
		Standard deviation	2.62	2.29	2.17	2.64	2.53
	Expert 5	Proportion	23/23	11/23	8/23	10/23	11/23
		Standard deviation	0.00	2.40	2.28	2.38	2.40

Table 3

 3 Consensus decision rulesThresholdsDecision rules R a and R r

Table 4

 4 The theoretical model versus the experts' recommendations cveca

The construction of reference actionsThe focus of this section is to construct a kind of conceptual decision-maker that combines an individual decision-maker's decision strategies in a single set of rules, i.e. a set of reference actions. Typically, when faced with a specific kind of problem, a group of decision-makers will not have a single set of decision making rules, but rather they will have variety of decision strategies. A large review of this theoretical context has been done recently in[START_REF] Jabeur | Une démarche générale d'aide aux membres d'un groupe à la recherche d'un résultat de consensus[END_REF].Thus, when constructing a decision making model, it is essential that, as stated in hypothesis H 4 , decision-makers cooperate, even though they have conflicting points of view. This enables a consensus to be reached: at the end of the process they agree to adopt one single decision. This final decision is the cumulative of their individual decision processes. The set of reference actions corresponds to a consensus of decision strategies. Of course it may contain some rules that do not correspond to any individual strategies (or real actions) but it represents the group of decision-makers' strategies. This set is a construction obtained with the decision-makers' agreement.

R r = {(33000),( 32100),( 32010),( 23010),( 23001),( 22101),( 22011), ( 22002),( 13210),( 13201),( 13120),( 13111),( 13102),( 13030), ( 13003),( 12130),( 12121),( 03310),( 31310),( 31130),( 21320), ( 21131),( 20330),( 12320),( 12311),( 12023),( 03230),( 03212), ( 02321),( 02231),( 31212),( 31122),( 31113),( 31032),( 30321), ( 30231),( 21312),( 12213),( 03303),( 02313),( 31303),( 30313), ( 30223),( 30133),( 20323),( 03133),(11333 31323),( 03233),( 03332),( 12133),( 13303),( 12222),( 12231), (13221),( 13320),( 22113),( 23111),( 32003),( 32012),( 32102), ( 32111),( 33011),( 33101),( 33110),( 22021),( 22030),( 22120), ( 23002),( 32020),( 22200)} [Intermediate vote] R r ={(32100),( 23100),( 23001),( 22002),( 33010),( 32011),( 22111), (13021),( 13301),( 13220),( 13211),( 13130),( 12320),( 31320), ( 31230),( 21330),( 21222),( 13203),( 13113),( 12312),( 12303), ( 12213),( 12132),( 12123),( 03312),( 03303),( 03213),( 02313), ( 31312),( 31303),( 03331),( 03133),( 31133),( 11333),(30333 22203),( 32031),( 33030),( 33102),( 22022),( 33110),( 22210), 33000),( 32100),( 32010),( 23010),( 23001),( 22101),( 22011), ( 22002),( 13210),( 13201),( 13120),( 13111),( 13102),( 13030), ( 13003),( 12130),( 12121),( 03310),( 31310),( 31130),( 21320), ( 21131),( 20330),( 12320),( 12311),( 12023),( 03230),( 03212), ( 02321),( 02231),( 31212),( 31122),( 31113),( 31032),( 30321), ( 30231),( 21312),( 12213),( 03303),( 02313),( 31303),( 30313), ( 30223),( 30133),( 20323),( 03133),(11333)}


LATMOS F Montmessin
email: franck.montmessin@latmos.ipsl.fr

T Mcconnochie

D M Venhaus

R C Wiens

First Results from Atmospheric Observations of CO2, H2O, and CO from Supercam on Mars2020-Pereverance Rover

Introduction:

The SuperCam instrument onboard the Mars2020-Perceverance rover [1,2] has the capability of performing several active and passive techniques. Passive spectroscopic measurements of the atmosphere are possible in the 0.4-0.85 (VIS) and the 1.3-2.6 (IR) micron ranges [3,4]. Since landing in Jezero crater in February 2021, SuperCam has performed numerous atmospheric observations. The technique used is called "passive sky", and has already been successfully conducted by Chem-Cam on the Mars Science Laboratory rover [5]. Su-perCam extends on the Capabilities of ChemCam, by also being able to probe the atmosphere in the critical IR window which includes absorption and scattering characteristics of gases and aerosols of particular interest.

Passive sky measurements have typically been carried out every two weeks, providing a consistent monitoring of key quantities such as CO2, O2, H2O and CO abundances along with cloud and aerosol properties. Particular attention was given to joint measurements of O2 and CO, as they provide unique insights into the Martian chemical cycle and have never before been measured at the same time from the surface. The results presented here will focus on the retrieval of CO2, H2O and CO volume mixing ratios.

Primary science objectives. CO has been observed [6] to follow the expected seasonal cycle of a chemically-inert non-condensable trace gas. To resolve the expected seasonal cycle of CO, our target measurement precision is +/-100 ppm. Near-surface abundances of water vapor are notoriously difficult to obtain, but are particularly valuable for assessing possible surface-atmosphere exchange processes. Our primary objective for water vapor is therefore to routinely sample the daytime H2O column, for direct comparison with nighttime mixing ratio measurements made by the Mars 2020 Mars Environmental Dynamics Analyzer (MEDA) humidity sensor [7], but also to help connect in-situ measurement by MEDA with measurements from orbit. We target a precision for precipitable water column of +/-1 precipitable microns, comparable to ChemCam [8].

Method:

The SuperCam instrument suite consists of a ChemCam-heritage reflection spectrometer, 385-465 nm ("violet"), < 0.2 nm res. [3], an intensified transmission spectrometer, 536-853 nm, 0.3 -0.7 nm res. [3], and an acousto-optic-tunable-filter (AOTF)based IR spectrometer, 1300 -2600 nm, 20 -30 cm - 1 res. [2,4].

The passive sky observing strategy is the same as the ChemCam MSL approach, where SuperCam is pointed to the sky at two different elevation angles thus yielding two different path lengths through the absorbing gas of interest. By ratioing the two obtained spectra, one removes most instrument response and solar spectrum uncertainties that are ~10x and ~100x larger than the signals of interest for the IR AOTF and transmission spectrometers, respectively. Due to sunsafety constraints, all sky spectra are effectively out of focus yielding an effective field of view of ~3º diameter. To obtain the desired sensitivities for O2 and H2O vapor, we use ~20 minutes of total integration time to provide multiple visits to each pointing position.

First results:

Initial results include several retrievals of water vapor mixing ratios along with a tentative determination of CO abundances in a limited number of measurements. H2O was found to vary between 24 and 147 ppmv with a 5 ppmv uncertainty from H2O and CO2 infrared signatures, in rough agreement with previous MEDA RH measurements [START_REF] Tamppari | AGU fall meeting[END_REF].

CO mixing ratios range between 900 and 1700 ppmv, values which correspond well with concentrations observed from orbit, such as with the ACS instrument onboard the ExoMars Trace Gas Orbiter [START_REF] Fedorova | [END_REF].

Figure 1 :

 1 Figure 1: Fitting attempt of CO, CO2 and H2O in the IR. Data collected on Sol 111.


LESIA T Bertrand

M Wolff

RETRIEVAL OF AEROSOL PROPERTIES AT JEZERO CRATER USING THE SUPERCAM INSTRUMENT ON-BOARD THE NASA MARS 2020 PERSEVERANCE ROVER

Introduction:

SuperCam on-board the Mars2020-Perseverance rover is a suite of remote sensing instruments [1,2] that includes passive visible and near-infrared (VISIR) spectroscopy covering the following channels: 385-465 nm, 536-853 nm, and 1300-2600 nm. At these wavelengths, the scattering of light by aerosols is strongly sensitive to the particle size. By observing the Martian sky light scattered by aerosols (dust and water ice clouds), at two different elevation angles with the VISIR technique, and by comparing the measurement with the results of a multiple scattering radiative transfer model, we are able to retrieve and monitor aerosols properties such as the particle sizes and phase functions. This "passive sky" technique has been demonstrated in the visible range with ChemCam on the Mars Science Laboratory (MSL) rover [3]. With SuperCam, the atmosphere is probed for the first time from the ground in both the visible and near-infrared domains [4], which provides further information on the aerosol properties.

The monitoring of these properties is important because they are considered significant factors in determining the radiative effect of aerosols and therefore in controlling the climate of Mars (e.g. [5]). In particular, the particle sizes have been observed to change depending on location and season. Typical values for dust sizes during non-storm events have been reported with an effective radius (of equivalent volume sphere) around ~1-1.5 μm (+/-50%), and variances around 0.2-0.5 from a broad set of spacecraft and instruments at the surface and in orbit. During major dust events, an increase in dust particle size up to 3-4 µm is a general trend [6][START_REF] Kahre | The Atmosphere and Climate of Mars[END_REF][START_REF] Chen-Chen | [END_REF].

Our objectives are to retrieve aerosol properties and monitor their seasonal evolution with the VISIR passive sky technique. Here we will focus on the characterization of dust particles using the available dataset and the multiple scattering model DISORT.

The effect of dust particle sizes on the climate, which is still not well understood, will also be discussed in light of recent modeling results, exploring multimodal dust particle size distributions [START_REF] Urata | [END_REF].

Method:

SuperCam passive sky observations: Our primary observing strategy is the same approach used for MSL ChemCam "passive sky" observations [3]: ratioing instrument signals from two pointing positions with different elevation angles, which eliminates solar spectrum and instrument response uncertainties (Fig. 1). Fig. 1. Geometry of the passive sky observation.

DISORT: Multiple scattering radiative transfer model:

We use the multiple-scattering program DISORT (Discrete Ordinates Radiative Transfer Program for a Multi-Layered Plane-Parallel Medium, [START_REF] Thomas | [END_REF]11]) to model the Martian sky, replicate the passive sky observation by SuperCam, and fit the observed sky radiances, taking into account the diffusion of solar radiation by aerosols (dust and water ice clouds). We use cylindrical dust particles and spherical ice particles. Dust refractive indices are derived from MRO/CRISM data [12], and the library for scattering properties are calculated for a grid of effective radius r eff =0.01,0.05,0.1,0.2,0.3,…3.0 µm assuming a variance of the distribution of v eff =0.3. The input parameters of the model are gathered by a python preprocessing tool (https: //github.com/ kconnour/pyRT_DISORT) and include the column optical depth seen by the rover, the elevation and azimuth angles of the Sun and of the instrument at the time of observation, the surface albedo, the effective radius of the aerosol particles and their vertical distribution.

Preliminary results:

We analyzed the passive sky observations obtained during the first 200 sols of the mission, corresponding to the northern spring and summer on Mars (no major dust storm). Preliminary results indicate relatively small particle sizes for dust of around 1 µm (Fig. 2), which is broadly consistent with ChemCam passive sky dust particle size results for the comparable season [3].

We are currently improving the model to take into account water ice clouds, spectral surface albedo, CO 2 absorption, as well as an optimization tool to allow for better fits of the data. These improvements will allow us to extract the maximum amount of information from the SuperCam passive sky spectra. The model matches the near-infrared data relatively well using 0.8-1 µm dust particles. However, the UV-VIS data are not well reproduced. We are currently investigating this issue.

Fig. 2 .

 2 Fig. 2. Mean ratio of the low elevation spectra to the high elevation spectra. Black points are observations for sol 133 (solar longitude Ls=68°). DISORT model results are shown for different choices of dust particle sizes.The model matches the near-infrared data relatively well using 0.8-1 µm dust particles. However, the UV-VIS data are not well reproduced. We are currently investigating this issue.