J Rochon

D Thiery

Physico-chemical exchanges and transfer of pollution in aquifers

This paper is divided into two parts. The first one is a mathematical analysis of different laws for physic and chemical exchanges De curing during a natural flow in an aquifer. The second part shows, with field experireents at Bonnaud (Jura), how it is possible to determine the parameters of an exchange law in order to forecast the long term evolution of the concentration of a solute which has reached the aquifer.

1 INTRODUCTION , ' ;.) t = time x = distance
Hany exchanges la'ds may be studied : an exchange law may be reversible or not, its equilibrium equation (01.' "isotherm") may be linear or not (e.g. Langmuir scheme), see J. [START_REF] Rochon | Propagation de substances miscibles en interaction physico-chimique avec Ie substrat, Approche simplifiee pour I'utilisation en hydrogeologie[END_REF] ~'le choose to study some of these la',/s which seem to be the most useful. homogeneous and isotropic aquifer, -uniform and perm~nent flow.

The evolution of the concentration will result of three phenomenous :

-advection : it is displacement of the tracer with the flow, -dispersion, -physic and chemical exchanqes between phase.

~~is assuoc€d that there are no exchanges with the lower and upper boundaries of the .~quifer. The general equation is :

)':1) J'e _IJ.

~-

'"

if Sa is equal to infinity which means no saturation effect, the system of equations is similar to the system describing the linear exchanges.

As the solution depends on four dimensionless numbers, it is not possible to draw a set of type curves.

We choose to draw type curves for instantaneous exchanges corresponding to N e equal to lnfini ty.

If the Peclet number is determined, only two dimensionless number are left : Ca and Sa. Type curves have been drmm to study the influence of each of these numbers. See graphs 3 and 4.

Ionian exchanges

The hypothesis are the following : -the sum of the cations in the liquid phase is T, -the sum of the cations in the solid phase is Qa, ~the anions are not adsorbed by the solid phase, -the total quantity of injected cations is m.

The relation between phases is the follow ing :

S _ ~Q,C .-T<{e..1)C when the equilibrium is reached or at any time, if the exchange kinetic is instantañ eaus. k is the constant of the mass action law.

The sum of all the •cations in solution was equal to To. At the time of injection, it is increased at the injection point and then is described by the law Qf dispersion without exchanges :

JT .,1> )'T _At 1T ')t -),,' 'h:
The reason is that the anions are not adsorbed which is an hypothesis, so the sum of the anions in solution is not involved in exchanges. The electrical neutrality tells that the sum T of the cation too will not be involved in exchanges (see J. [START_REF] Rochon | Echange cationique au cours d'un ecoulement unidirectionnel en milieu poreux sature[END_REF] .

Assuming that the exchange kinetic is instantaneous, it is possible to derive the following equilibrium relation which holds at any time The equilibrium relation has the same expression than Langmuir one, when T R is equal to unity, i.e. for low concentrations of injection.

The dimensionless system of differential equations is :

'JCR • .i.~_ ~_ Co. .(T~-:& ~'Jr,) ?t.-f'e ''1 ?x. (T•• c.fS.)' J~. -:n.
\"lhen Co .« To, T "" 1 and is constant, and these equations ~re similar to Langmuir scheme. However the second equation which has no exchanges term shows that the variation of T is much faster than the variation of C. After a time equal to about twice the advection time the evolution of concentrations is identical to Langmuir scheme.

3.2

The field experiment to be analysed Among all the experiments three of them have been selected and are analysed in the following lines. Each injection took place in well C and measurements were in well D, 13 meters further. The main characteristics of these experiments are gathered in table 1. The aquifer has the following characteristics :

-the water table is of limited thickness (3 t~4 m) which helps for the measurements, -the aquifer is confined which implies no influences of the fluctuation of the water table and no exchanges with the non saturated zone, -.the aquifer is not thick and is well defined, -the naturel variations of level are small, -the hydraulic conductivity is high which helps the advection of the injected tracers (transmissivity : 10-3 m 2 /s) In order to get a uniform gradient and increase'the velocities to keep the experiment duration reasonable, a uniform flow has been imposed and controlled by a set of injection wells (M) and pumping wells (N) with constant rates (see graph 6) .

The three experiments have been analysed separately by ajustment of the parameter with a mathematical model. The number of'parameteã nd also the crosspart of the aquifer in takes place.

Other experiments \•lith "perfect" tracers like tritium for which there are no exchanges have been performed on the same experiment field. The analysis of these experiments (P. Peaudecerf, J.P. [START_REF] Peaudecerf | Application of a mathematical model to the characterisation of dispersion effects on groundwater quality[END_REF] These parameters lead to a dispersivity a = Diu = 0.66 m and a Peclet number:

Pe = uLID = 19.8 (L = distance = 13 m) .
This high value of Peclet number shows that the ad~ection term is much larger than the diffusion one.

The experiments with the other matters (I~Na, Rhodamine vIT, Uranine) have been analysed with these same values for D, u and n. The analysip have been done only;byJajusting the parameters of exchanges between phases.

Choice of the model of exchange

Var.ious injections have been realised on the same location with the same tracers but with different injected mass (ratio. of 10). None of these test show any mass effect i in other words, for a given tracer, and after a given duration the concentration is proportionnal to the injected mass of tracer. l'loreover, as the capacity of cationic exc hange Q o of the aqUifer and the chemical nature of the water To are not known, it has not been possible to try any interpretation with ionian exchange scheme or Langmuir scheme. The model ",hich has been chosen is then the reversible and linear exchanges. It must be noted that, as the duration of the measurements is sometimes short and as some incidents occur during the experiments, it should have been illusory to use a more complicated model ",ith more parameters.

Determination of the parameter

For each experiment two parameters have been determined : -the kinetic constant k (corresponding to the half exchange time :C td = Ln2/k ), _ the equilibrium constant k . c

The corresponding dimensio~less parameters are :

-number of e~changes : Ne = L.kc/u -retention capacity: Ca ~k e (l-n)/n The interpretation scheme is one dimensional so the transverse difusion is neglected The flow is assumed to take place in a prism of aquifer of section A (thickness hand ",idth 1) in which the concentration is uniform. The thickness of the aquifer ranges between 2.9 m at injection point to 2.5 m 217 at meas,urement point. The Vlidth 1 is not Kno;'ln. It is possible to compute it by the ratio ot the injected mass to the initial injected concentration ",hich is a parameter of the model.

The parameters obtained after calibration are gathered in table 2. The comparison of the measured and computed concentration appear on graph 7 to 9. These graphs show that the calibration is reasonable, keeping in mind the incidents "'hich occur during the experiments and also the fact that the concentrationimeasurements seems sometimes affected by erratic variations.

Interpretation of the exchange parameters

The experiment ",ith Sodium Iodide (I-Na) shows little exchange because the Retention Capacity is small (Ca = 0,45) and the kinetic 9f exchange is slow (Ne = 1.0).

The experiments Vlith Rhodamine and Uramine

give results ",hich are quite similar (see graph graph 8 and 9) ",ith much more exchanges : the Retention capacity ranges from 2 to 3, the Number of Exchanges is equal to about 2.5. The computed width of the equivalent prism of aquifer is equal to 7 or R meters for the 3 experiments. This width does not describe the lateral extension of the plume of tracer but rather the width of aquifer influenced by the pumping at the measurement point D.

CONCLUSION

The use of models makes it possible to predict the migration of a pollution in the water of an aquifer, taking into account the physic and chemical interactions of the aquifer. To determine the exchange parameters it is necessary to do the following :

-perform a first experirr.ent on the field uranine Graph 8.

~~~.

OC"f,t,J. Vlith a "good" tracer (Le. vlith no or little exchanges) in order to determine the parameters describing advection and dispersion, perform a second test ~preferably a field experiment-Vlith a solute similar to the product to study but Vlhich is not a pollutant. ~fuen such an experiment may not be performed (for economical reaso~s or because there is no solute Vlith properties similar to the polluant to study) a dynamic -test may be realised in laboratory with a sample of the ground and Vlith the pollutan~. From this second experiment, the eXChange param eters will be deducted. However a laboratory test will only give a rough estimation of the parameters by not taking into account the heterogeneity of the real field.

2. 1 .

 1 Linear exchanges 'dith linear exchange kinetic 2 INTRODUCTION TO THE FIRST PART t,= ~/;" 5, = S . .f,.,L•"/", which may be 'driten with dimensionless variables : P e =j,L it is the Peclet number tJo.=b1k it is the nwnber• of Echanges du-~~ing the advection along the distance L C~=~\it is the Exchange Capacity of n the aquifer This system of two dimensionless differential equations has been solved by finite differences with B.R.G.M. numerical code M.O.D.E.L. (J. Ausseur, B. Kanehiro, J.P, Sauty, 1979) and made it possible to design dimensionless type curves describing the Three dimensionless numbers appears in the liquid phase (wa-This part will deal with the study of the concentrations in the water of an aquifer after an injection of tracer. The hypothesis relative to the aquifer are :

  following a slug injection of tracer. The calculations have boon performed for a Peclet number equal to 10, a number of exchange ranging from 0.1 to 10 and a Retention Capacity from 0.1 to 5. Graphs 1 and 2 shows the type curves for Ca = 1 and Ca = 2. 2.2. Exchanges according to Langmuir isotherm The eqUilibrium equation is = ~.C 1. C/CJ half saturation concentration in the liquid phase (water) equilibrium concentration in the solid phase (aquifer) The exchange kinetic is of second order : ;t= ~c(~eC. S•• 5) S. The tnansformation of the variable is the same as in the previous chapter. The dimensionless equation may be written: 2£llt=*.~_'2fB_c~.~J.(C._C _c <: /5) 'Jtl\, re "'~1 "'''It "'1.\ R-"ft/~C\ 't= N. (C.-S._C.\/!>",) 214 with a fourth dimensionless number s _CJ .L." it is the Saturation Constant.

  t15 N 6 //7 //6 //0 I P Graph 6. Situation of wells All the tracer experiments in uniform flow which are analysed in this part took place in 1974 at Bonnaud, near Lons~le-Saunier (Jura -France). The experiments were designed together by two teams : the Bureau de Recherches Geologiques et Ninie-res (ERG!.I) and the Section d'Application des Radioelements du Commissariat a l'Energie Atomique (SAR-CEA). The aim of the research was to study extensively with field experiments the underground migration of various matters soluble in water. All the results are described in B. Gaillard, J. Guizerix, J. J.1argat, J. Nolinari, P. Peaudecerf, 1976. 3 INTRODUCTION TO THE SECOND PART 3.1 Description of the experiment field

  made it possible to determine the parameters describing adve~tion and diffusion :~effective velocity u = 3.8 10-5 mls {3,

Table 2 .

 2 Parameters of exchange

		I-Na	Rhodanine uranine ,>-r
	kinetic constant kc (10~6 s -1)	3.0	B.O	7.5
	half exchange time td (days)	2.7	1.0	1.1
	equilibrium constant ke	0.22	1.12	1.49
	number of exchan ges Ne	1.03	2.74	2.57
	Retention capa~. city Ca	0.45	2.27	3.03
	equivalent width of prism 1 (m)	6.7	B.4	6.7

  BIBLIoGRAPHY Ausseur, J.Y., Kanehiro, B., Sauty, J.P. Mass transfer through fractured rocks "M.O.D.E.L. tI Code. Convection Dispersion Exchange Decay, Bureau de Recherches G~o logiques et Hinieres, rapport n°79 SGN 748 HYD (to be pr~nted) Gaillard, B., Guizerix, J., Hargat, J., l.lolinari, J., Peaudecerf, P., 1976, Action thematique programmee "hydrogeologie", Etude des caracteristiques de transfert de substances chimiques dans les nappes (site de Bonnaud -Jura), Bureau de Recherches Geologiques et Minieres, rapports 75 SGN 056 m4E, 76 SGN 235 ill4E, 76 SGN 236 M'ffi, 76 SGN 240 AME, 76 SGN 226 m,ffi.

	NOTATIONS
		~',,;
	C	Concentration in the liquid phase
	C a	Retention capacity
	k	Equilibrium Constant


Mathieu Fauvel

Charles Bouveyron

Stéphane Girard

Parsimonious Gaussian Process Models for the Classification of Multivariate Remote Sensing Images

Keywords: Kernel, remote sensing images, Gaussian process, parsimony, hyperspectral

A family of parsimonious Gaussian process models is presented. They allow to construct a Gaussian mixture model in a kernel feature space by assuming that the data of each class live in a specific subspace. The proposed models are used to build a kernel Markov random field (pGPMRF), which is applied to classify the pixels of a real multivariate remotely sensed image. In terms of classification accuracy, some of the proposed models perform equivalently to a SVM but they perform better than another kernel Gaussian mixture model previously defined in the literature. The pGPMRF provides the best classification accuracy thanks to the spatial regularization.

INTRODUCTION

In a multivariate remote sensing images, a pixel is represented by a vector x ∈ R d for which each component is a measurement corresponding to specific wavelengths [START_REF] Chang | Hyperspectral imaging. Techniques for spectral detection and classification[END_REF]. The classification of such images requires algorithms that are robust to the number d of spectral wavelengths and that are able to include additional spatial information in the classification process [START_REF] Fauvel | Advances in Spectral-Spatial Classification of Hyperspectral Images[END_REF].

Kernel methods, such as SVM, have shown good abilities in classifying images with a large number of spectral bands [START_REF]Kernel Methods for Remote Sensing Data Analysis[END_REF]. The use of a kernel function that defines a measure of similarity between two samples, here two pixel-vectors, make them robust to the spectral dimension. However, including spatial information in the classification process is not easy, and the resulting algorithms usually involve a separate step for the extraction/inclusion of the spatial information [START_REF] Fauvel | Advances in Spectral-Spatial Classification of Hyperspectral Images[END_REF].

On the contrary, conventional statistical method such as Markov random fields (MRF) model the spatial relationship between adjacent pixels in a proper way by using a local energy function

U (yi|xi, Ni) = -ln p(xi|yi) + ρE(Ni) ( 1
)
where yi is the label, p(xi|yi) the conditional probability of having xi given yi, E is an energy term that characterizes the local context of the pixel, Ni represents the neighborhood of xi in the spatial domain and ρ is a positive parameter. The statistical modeling in the spectral domain usually suffers from the increase of the dimension. With the conventional Gaussian assumption, p(xi|yi = c) ∼ N (µc, Σc), the number of parameters of the model scales with the square of the number of spectral variables, which makes difficult reliable estimations.

Several approaches have been proposed to combine MRF and kernel methods. The main problem is to compute properly the conditional probability in the MRF energy function. Indeed with kernel functions, the samples x are implicitly mapped to φ(x) that live on a feature space. For the commonly used Gaussian kernel function, the dimension of the feature space is infinite, and probability functions cannot be defined. Several strategies were proposed to overcome this difficulty. In [START_REF] Tarabalka | SVM and MRF-Based Method for Accurate Classification of Hyperspectral Images[END_REF], it was proposed to use SVM to estimate the conditional probability p φ(xi)|yi in the MRF energy function. However, the estimated probability is not a true probability, but a scaled version of the SVM output [START_REF] Platt | Probabilistic outputs for support vector machines and comparisons to regularized likelihood methods[END_REF]. In [START_REF] Moser | Combining support vector machines and markov random fields in an integrated framework for contextual image classification[END_REF], the authors defined theoretically the conditional probability in the kernel feature space, but they assumed that the covariance matrices were common to the different classes in the feature space. A similar assumption regarding the covariance matrix was done in [START_REF] Dundar | Toward an optimal supervised classifier for the analysis of hyperspectral data[END_REF]. Although the performances of the above mentioned method were good, such assumption about the covariance matrix could limit the effectiveness of the methods.

In this paper, a family of parsimonious Gaussian process models is proposed to compute p φ(xi)|yi in eq. ( 1). These models allow to build from a finite set of training samples, a Gaussian mixture model in the kernel feature space, even in the infinite dimensional case. They assume that the data of each class live in a specific subspace of the kernel feature space [START_REF] Bouveyron | Kernel discriminant analysis and clustering with parsimonious Gaussian process models[END_REF].

The remainder of the paper is organized as follows. Section 2 presents the family of parsimonious Gaussian process models. Section 3 focuses on the experimental results on one real hyperspectral images. Finally, conclusions and perspectives are discussed in Section 4.

CLASSIFICATION WITH PARSIMONIOUS GAUSSIAN PROCESS MODELS

2.1. Gaussian process in the kernel feature space

Let S = (xi, yi) n i=1 be a set of training samples, where xi ∈ J, J ⊂ R d , is a pixel and yi ∈ {1, . . . , C} its class, and C the number of classes. For short, in the followingln p φ(xi)|yi will be referred to Ω(φ(xi), yi).

In this work, the conventional Gaussian kernel function

k(xi, xj) = exp - xi -xj 2 R d 2σ 2 , σ > 0, (2)
is used. Its associated feature space is F and the mapping function is φ : R d → F . From the Mercer theorem, we have k(xi, xj) = φ(xi), φ(xj) F and the kernel evaluation can be written as (the series converges absolutely and uniformly for almost all (xi,xj)) [START_REF] Schölkopf | Input space versus feature space in kernel-based methods[END_REF]:

k(xi, xj) = d F m=1 emvm(xi)vm(xj) (3)
where dF = dim(F ), {em, m = 1, 2, . . .} is the sequence of positive eigenvalues in decreasing order of the integral operator T k ,

(T k f )(z) = J k(z, x)f (x)dx, associated to k and {vm : R d → R, m = 1, 2, .
. .} is the sequence of corresponding normalized eigenfunctions. From eq.( 3), φ can be defined as

φ : x → √ emvm(x), m = 1, 2, . . . (4)
For the Gaussian kernel dF = +∞ [START_REF] Schölkopf | Input space versus feature space in kernel-based methods[END_REF]. Therefore the conventional multivariate normal distribution cannot be defined.

To overcome this, let us assume that φ(x), conditionally on y = c, is a Gaussian process with mean µ c and covariance function Σc. Hence, for all r ≥ 1, random vectors on R r defined by [φ(x)1, . . . , φ(x)r] are, conditionally on y = c, a multivariate normal vectors. Therefore, it is possible to write for yi = c

Ω(φ(xi), yi) = r j=1 φ(xi) -µ c , qcj 2 2λcj + ln(λcj) 2 + γ ( 5
)
where λcj is the j th eigenvalue of Σc in decreasing order, qcj its associated eigenvector and γ a constant term that does not depend on c. If the Gaussian process is not degenerated (i.e., λcj = 0, ∀j), r has to be large to get a good approximation of the Gaussian process.

Unfortunately, only a part of the above equation can be computed from a finite training sample set:

Ω(φ(xi), yi) = rc j=1 φ(xi) -µ c , qcj 2 2λcj + ln(λcj) 2 computable quantity + r j=rc+1 φ(xi) -µ c , qcj 2 2λcj + ln(λcj) 2
non computable quantity [START_REF] Moser | Combining support vector machines and markov random fields in an integrated framework for contextual image classification[END_REF] where rc = min(nc, r) and nc is the number of training samples of class c.

Parsimonious Gaussian process

To make the above computational problem tractable, it is proposed to use a parsimonious Gaussian process model in the feature space for each class.

Definition 1 (Parsimonious Gaussian process) A parsimonious Gaussian process is a Gaussian process φ(x) for which, conditionally to y=c, the eigen-decomposition of its covariance operator Σc is such that A1. It exists a dimension r < +∞ such that λcj = 0 for j ≥ r and for all c = 1, . . . , C.

A2. It exists a dimension pc < min(r, nc) such that λcj = λ for pc < j < r and for all c = 1, . . . , C.

The assumption A1 is motivated by the quick decay of the eigenvalues for a Gaussian kernel [START_REF] Braun | On relevant dimensions in kernel feature spaces[END_REF]. Hence, it is possible to find r < +∞ such as λcr ≈ 0. The assumption A2 expresses that the data of each class live in a specific subspace of size pc, the signal subspace, of the feature space. The variance in the signal subspace for the class c is modeled by the parameters λc1, . . . , λcp c and the variance in the noise subspace, common to all the classes, is modeled by λ. This model is referred to by pGP 0 . Common between groups Free Common pGP 5

Common within and between groups Free Free pGP 6

Common within and between groups Free Common

From this model, it is possible to derive several sub-models. Table 1 lists the different models that can be built from pGP 0 . For the model pGP 1 , it is additionally assumed that the data of each class share the same intrinsic dimension, i.e., pc = p, ∀c ∈ {1, . . . , C}.

In the model pGP 2 , the variance of Fc is assumed to be equal for all eigenvectors, i.e., λcj = λc, ∀j ∈ {1, . . . , pc}. For the model pGP 4 , it is assumed that the intrinsic dimension is common to every class and the variance is common between them, i.e., λcj = λ c ′ j , ∀j ∈ {1, . . . , p} and c, c ′ ∈ {1, . . . , C}. In term of parsimony, pGP 0 is the least parsimonious model while pGP 8 is the most parsimonious of the proposed models.

In the following, only the model pGP 0 is discussed. Similar results can be obtained for the other models.

Proposition 1 Letting pM = max(p1, . . . , pC ), eq. ( 5) can be written for pGP 0 as

Ω(φ(xi), yi) = pc j=1 1 λcj - 1 λ φ(xi) -µ c , qcj 2 2 + 1 2λ φ(x) -µ c 2 + pc j=1 ln(λcj) 2 + (pM -pc) ln(λ) 2 + γ ′ (7)
where γ ′ is a constant term that does not depend on the index c of the class.

The computation of eq. ( 7) is now possible since pc < nc, ∀c ∈ {1, . . . , C}. In the following, it is shown that the estimation of the parameters and the computation of eq. ( 7) can be done using only the kernel evaluation, as in standard kernel methods.

Model inference

Let us define the centered Gaussian kernel function according to class c as:

kc(xi, xj) = k(xi, xj) + 1 n 2 c nc l,l ′ =1 y l ,y ′ l =c k(x l , x l ′ ) - 1 nc nc l=1 y l =c k(xi, x l ) + k(xj, x l ) . (8)
The associated normalized kernel matrix Kc of size nc × nc is defined by

(Kc) l,l ′ = kc(xl, x l ′ ) nc . (9)
With these notations, the following result holds for pGP 0 .

Ω(φ(xi), yi) = 1 2nc pc j=1 1 λcj 1 λcj - 1 λ nc l=1 y l =c β cjl kc(xi, x l ) 2 + 1 2 λ kc(xi, xi) + pc j=1 ln( λcj) 2 + (pM -pc) ln( λ) 2 (10)
Table 2. Information classes for the University Area data set and classification accuracy. SVM refers to the support vectors machine, GMM refers to the Gaussian Mixture Model, KGMM refers to the kernel GMM proposed in [START_REF] Dundar | Toward an optimal supervised classifier for the analysis of hyperspectral data[END_REF]and OA refers to overall accuracy. Proposition 2 For c = 1, . . . , C and the model pGP 0 , eq. ( 7) can be computed with eq. ( 10), where β cjl is the l th component of the normalized eigenvector β cj associated to j th largest eigenvalue λcj of Kc and

λ = 1 C c=1 πc(rc -pc) C c=1 π trace(Kc) - pc j=1 λcj ( 11
)
and πc = nc/n. See [START_REF] Bouveyron | Kernel discriminant analysis and clustering with parsimonious Gaussian process models[END_REF] for the proof.

The estimation of pc is done by looking at the cumulative variance for the sub-models pGP 0,2,5 . In practice, pc is estimated such as the percentage of the cumulative variance is higher than a given threshold t h :

pc j=1 λcj nc j=1 λcj > t h . ( 12
)
For the other sub-models, p is a fixed parameter given by the user.

EXPERIMENTAL RESULTS

In this section, results obtained on one real data set are presented. The data set is the University Area of Pavia, Italy, acquired with the ROSIS-03 sensor. The image has 103 spectral bands (d = 103) and is 610×340 pixels, see Figure 1.(a). Nine classes have been defined by a photo-interpret for a total of 42776 referenced pixels, see Table 2.(a). 50 pixels for each class have been randomly selected from the samples for the training set, and the remaining set of pixels has been used for validation. The process has been repeated 50 times, each time a new training set has been generated and the variables have been scaled between -1 and 1. The mean result in terms of overall accuracy (percentage of correctly classified pixels) are reported. To assess the statistical significance of the observed differences in terms of classification accuracy, a Wilcoxon rank-sum test has been applied over the 50 repetitions. It tests if the data from two populations are samples from distribution with equal medians. In the experiments, it is used to test whether the 50 classification accuracy are significantly different (not equal medians) or not (equal medians).

Two sets of experiments have been conducted. First, the proposed models have been compared to three others models: the support vectors machines (SVM), a conventional Gaussian mixture model (GMM) and a Kernel GMM defined in [START_REF] Dundar | Toward an optimal supervised classifier for the analysis of hyperspectral data[END_REF]. These models use only the spectral information from the remote sensing image. Second, the model pGP 1 has been used to build a pGPMRF classifier that uses both the spatial and the spectral information.

Comparison with others spectral classifier

In this section, the proposed models are compared to other classifiers in terms of classification accuracy. The Gaussian kernel was used for each kernel method. All the parameters have been selected by a five fold cross-validation. The ranges of the tested values are reported in Table 3. A small regularization term (τ = 10 -5 ) has been added in the GMM for the inversion of the covariance matrix.

Results are reported in Table 2.(b). The three best results in terms of classification accuracy are obtained for pGP 0 , pGP 1 and SVM, in boldface in the table. The differences between them are not significant according to the Wilcoxon test. All the other methods provide results that are significantly worst in terms of classification accuracy.

From the results, and for that experimental protocol (small training set and University area data set) the proposed models pGP 0 and pGP 1 provides classification accuracies similar to those obtained with SVM. They outperformed in terms of classification accuracy the KGMM proposed in [START_REF] Dundar | Toward an optimal supervised classifier for the analysis of hyperspectral data[END_REF] and the conventional GMM.

Classification with the pGPMRF

In this section, the model pGP 1 is used to compute the conditional probability in eq.( 1). For the spatial energy term in eq.( 1), a conven- tional Potts model is used [START_REF] Li | Markov Random Field Modeling in Image Analysis[END_REF]:

E(Ni) = x j ∈N i δ(yi, yj) (13)
where δ is the delta function. A second order neighborhood is considered, i.e., Ni is the set of 8 surrounding pixels of pixel xi. For the optimization, a Metropolis algorithm is used. It minimizes the global energy, UG = i U (yi|xi, Ni), by an iterative minimization of the local energy (1). For details, see [START_REF] Li | Markov Random Field Modeling in Image Analysis[END_REF]. ρ was set to 16. The thematic maps obtained for one repetition is reported in the Figure 1 and the mean overall accuracy is reported in the Table 2. Using the MRF modeling leads to an mean improvement of 8.3% in terms of overall accuracy in comparison with the use of pGP 1 alone. The Wilcoxon test shows that the improvement is significant. The Figure 1.(c) is much more homogeneous than Figure 1.(b), thanks to the spatial regularization operated by the pGPMRF.

CONCLUSIONS AND PERSPECTIVES

A family of parsimonious Gaussian process models was presented in this article. They make possible the computation of the Gaussian mixture model when the original samples are mapped into an infinite dimensional space, such as ones associated to the Gaussian kernel function. By assuming that the data of each class are located in a specific subspace of the kernel feature space, it was shown that all the computations can be expressed with kernel evaluation.

Experimental results exhibit, for a small size of training set and for the University area data set, that two models pGP 0,1 provide similar performances with SVM in terms of classification accuracy, and outperformed another kernel GMM that were proposed in [START_REF] Dundar | Toward an optimal supervised classifier for the analysis of hyperspectral data[END_REF].

In order to take into account the spatial correlation in the image, a pGPMRF was build using the conditional probability provided by the model pGP 1 . The classification accuracy was increased by 8%. Hence the proposed models associated to the MRF model are appropriated for the classification of multivariate remote sensing image, in particular when the number of spectral bands is large.

Further analysis are required to better assess the performance of the proposed models. The effect of the size of the training set should be investigated. When nc grows, the size of the associated feature spanned by each class is possibly larger. Therefore, models that are more parsimonious, e.g. pGP 4 , may provide better results than pGP 0,1 .

Regarding the pGPMRF, the Potts model is a very simple model and more sophisticated models can be used [START_REF] Li | Markov Random Field Modeling in Image Analysis[END_REF]. Furthermore, the selection of the parameter ρ in eq. ( 1) value can benefit of a dedicated optimization algorithm.

Fig. 1 .

 1 Fig. 1. (a) RGB color composition for the University area, thematic map obtained with (b) pGP 1 and (c) the pGPMRF.

Table 1 .

 1 List of the sub-models of the parsimonious Gaussian process model. Fc refers to the signal subspace of the considered class.

	Model	Variance inside Fc	q cj	pc
	pGP 0	Free	Free	Free
	pGP 1	Free	Free Common
	pGP 2	Common within groups	Free	Free
	pGP 3	Common within groups	Free Common
	pGP 4

Table 3 .

 3 Grid search setting for the cross validation. t h corresponds to the threshold value on the cumulative variance, Cp is the regularization term for the SVM and λr refers to the ridge regularization term in KGMM.

	Method	Parameter	Range
		σ 2	{2 -3 , 2 -2 , . . . , 2 6 }
	pGP	t h	{0.900, 0.911, . . . , 0.999}
		p	{5, 10, . . . , 45}
	SVM	σ 2 Cp	{2 -3 , 2 -2 , . . . , 2 4 } {10 -2 , . . . , 10 4 }
	KGMM	σ 2	{2

0 , 2 1 , . . . , 2 8 } τ {10 -13 , . . . , 10 -6 }


Mickaël Causse
email: mickael.causse@isae.fr

Frédéric Dehais
email: frederic.dehais@isae.fr

Mahé Arexis
email: mahé.arexis@edu.univ-fcomte.fr

Josette Pastor
email: josette.pastor@inserm.fr

Cognitive aging and flight performances in general aviation pilots

Unlike professional pilots who are limited by the FAA's age rule, no age limit is defined in general aviation. Our overall goal was to examine how age-related cognitive decline impacts piloting performance and weather-related decision-making. This study relied on three components: cognitive assessment (in particular executive functioning), pilot characteristics (age and flight experience) and flight performance. The results suggest that in comparison to chronological age, cognitive assessment is a better criterion to predict the flight performance, in particular because of the inter-individual variability of aging impact on cognitive abilities and the beneficial effect of flight experience.

INTRODUCTION

The population of general aviation (GA) pilots is getting older in the USA (D. J. [START_REF] Hardy | Cognition and flight performance in older pilots[END_REF] and in European countries like France where 41% of private pilots are more than 50 years old (BEA 1 , 2008). Unlike professional pilots who are limited by the FAA's "age 65" rule, no such restriction exists for GA pilots. It is worth examining how agerelated cognitive decline may impact piloting performance in GA.

Casualties in GA

Contrary to commercial aviation (CA) pilots, GA pilots have not necessarily experienced a professional training, fly mostly on their own, without any co-pilot and with very few assistance systems. They have less support from the air traffic control and are more affected by weather conditions. Not surprisingly, in GA, the accident rate is considerably higher than in CA [START_REF] Li | Crash risk in general aviation[END_REF]. Li and coworkers (2001) analyzed NTSB2 data files and showed that pilot errors were a probable crash cause in 38% of the airline crashes and of 85% in the GA. A main cause of accidents in GA is related to the loss of situation awareness of the aircraft position: since the light aircrafts' altitude is often relatively low, the loss of the aircraft's position awareness can provoke hazardous heading deviation [START_REF] Gibson | Loss of situation awareness-Causes and consequences[END_REF]. Indeed it is interesting to denote that the greatest part of the GA fatalities occurs on route, away from the airports, but 46% of the crashes occur at airports [START_REF] Li | Correlates of pilot fatality in general aviation crashes[END_REF] and are related to poor decision making by pilots who do not sufficiently take into account external cues (landing distance available, crosswind speed, traffic…).

Age and/or cognitive performance: casualty factors in GA?

Determining which factors are predictive of such errors is a great challenge to improve safety in GA. Several studies have revealed significant aging issues on accident rates in GA [START_REF] Harkey | Age-related changes in selected status variables in general aviation pilots[END_REF][START_REF] Kay | Effects of aging on aviation-related cognitive function in commercial airline pilots. Paper presented at the In Proceedings of the 72nd Meeting of the Aerospace Medical Association[END_REF][START_REF] Taylor | Pilot age and expertise predict flight simulator performance: A 3-year longitudinal study[END_REF], though these results are criticized [START_REF] Li | Human factors in aviation crashes involving older pilots[END_REF][START_REF] Rebok | Pilot Age and Error in Air-Taxi Crashes[END_REF]. Hardy and colleagues (2007) examined the effect of age on pilot cognition in a 28-62 years old sample and showed that cognitive performance began to decline very early, from 40 years old. Although this early decline could be partly related to a variety of medical issues, and in spite of the absence of a definitive consensus, the fall of cognitive performance is strongly suspected to be partly responsible for the increased accident rate with age. This raises the importance of monitoring the pilot cognitive functioning as long as the decline of these abilities represents a much higher risk of accident than a sudden physical incapacitation [START_REF] Schroeder | Pilot age and performance: an annotated bibliography[END_REF]. Substantial literature focuses on the evaluation of the cognitive state to predict flight performance [START_REF] Taylor | Relationship of CogScreen-AE to flight simulator performance and pilot age[END_REF] or aeronautical decisionmaking [START_REF] Wiggins | Expertise in aeronautical weather-related decision making: A cross-sectional analysis of general aviation pilots[END_REF], but its conclusions remain contradictory. Several reasons may explain the difficulty to draw a definitive conclusion on the effects of aging on flight performance in GA pilots. There is a great inter-individual variability in the deleterious effects of aging on cognition [START_REF] Buckner | Memory and Executive Function in Aging and AD Multiple Factors that Cause Decline and Reserve Factors that Compensate[END_REF]; studies do not necessarily focus on the cognitive functions that are the most impacted by aging; few researches attempt to link, in the same sample, cognitive performance to flight abilities; a large part of the studies is interested on safety aspects like communications (Morrow, Menard, et al., 2003); few researches are exclusively related to the GA population; finally, another source of complexity arises from the suspected compensative role of flight experience on aging effects [START_REF] Morrow | The influence of expertise and task factors on age differences in pilot communication[END_REF]. Different measurements of cognitive efficiency have been identified as crucial to the piloting ability, for instance: time-sharing [START_REF] Tsang | Age, attention, expertise, and time-sharing performance[END_REF], speed of processing [START_REF] Taylor | The effects of information load and speech rate on younger and older aircraft pilots' ability to execute simulated air-traffic controller instructions[END_REF], attention [START_REF] Knapp | F-16 Class A mishaps in the US Air Force, 1975-93[END_REF] or problem solving [START_REF] Wiggins | Expertise in aeronautical weather-related decision making: A cross-sectional analysis of general aviation pilots[END_REF]. One of the most promising approaches consists in administering a tests battery to pilots, such as Cogscreen-AE [START_REF] Horst | COGSCREEN-Personal computer-based tests of cognitive function for occupational medical certification[END_REF], and to correlate their score with their performance during flight simulator sessions or in real flight conditions [START_REF] Yakimovitch | Flight performance and CogScreen test battery in Russian pilots[END_REF]. Taylor and colleagues (2000) were able to predict 45% of the variance of the flight simulator performance with four Cogscreen-AE predictors (speed/working memory, visual associative memory, motor coordination and tracking) in a cohort of 100 aviators (aged 50-69 years). However, the identification of the most relevant cognitive functions to predict flight performance remains a key issue.

Executive functions: early markers of age-related cognitive decline?

Contrary to studies that involved Cogscreen-AE, a large battery in terms of explored cognitive functions, we propose to focus specifically on executive functions (EFs). Indeed, these cognitive functions are among the earliest ones to be impacted by aging [START_REF] Raz | Aging of the Brain and Its Impact on Cognitive Performance: Integration of Structural and Functional Findings. The handbook of aging and cognition[END_REF] and represent excellent clues of aging effects on cognitive performance.

Functional neuroimaging brings evidence that the brain is subject to anatomical and physiological modifications in normal aging [START_REF] Cabeza | Aging gracefully: compensatory brain activity in high-performing older adults[END_REF]. The prefrontal cortex appears to be the earliest affected region [START_REF] Tisserand | Special issue on the involvement of prefrontal networks in cognitive ageing[END_REF] and may prominently account for age-related cognitive changes [START_REF] West | Effects of increased response dominance and contextual disintegration on the Stroop interference effect in older adults[END_REF].

Because the prefrontal cortex plays a dominant role in the implementation of EFs, it is not surprising that aging provokes a selective alteration of reasoning [START_REF] De Neys | Logic and belief across the lifespan: the rise and fall of belief inhibition during syllogistic reasoning[END_REF] or inhibition, updating and set-shifting [START_REF] Fisk | Age-related impairment in executive functioning: Updating, inhibition, shifting, and access[END_REF]. However, the executive changes vary considerably across people. The complex interactions between the cerebral structures underlying EFs [START_REF] Buckner | Memory and Executive Function in Aging and AD Multiple Factors that Cause Decline and Reserve Factors that Compensate[END_REF], sociocultural background and genetic factors [START_REF] Nagel | Human aging magnifies genetic effects on executive functioning and working memory[END_REF] may explain the heterogeneity of this decline. The study of EFs has appeared recently in aeronautics, for instance, [START_REF] Hardy | Age-Related Group and Individual Differences in Aircraft Pilot Cognition[END_REF] found very significant age-related differences in pilots' executive functioning (e.g. inhibition, setshifting) and [START_REF] Taylor | Cognitive ability, expertise, and age differences in following air-traffic control instructions[END_REF] established a relationship between interference control and the ability to follow air traffic instructions. These cognitive functions underlie goal-directed behavior and adaptation to novel and complex situations [START_REF] Royall | Executive Control Function: A Review of Its Promise and Challenges for Clinical Research. A Report From the Committee on Research of the American Neuropsychiatric Association[END_REF]. They allow the inhibition of automatic responses in favor of controlled and regulated behavior, in particular when automatic responses are no longer adequate to the new environmental contingences.

They also encompass decision-making [START_REF] Sanfey | Phineas gauged: decision-making and the human prefrontal cortex[END_REF] or reasoning abilities [START_REF] Decker | Evidence of construct similarity in executive functions and fluid reasoning abilities[END_REF]. According to Miyake [START_REF] Miyake | The unity and diversity of executive functions and their contributions to complex "frontal lobe" tasks: A latent variable analysis[END_REF], three major low-level EFs are moderately correlated, but clearly separable: set-shifting between tasks or mental sets ("shifting"), inhibition of dominant or prepotent responses ("inhibition"), and updating and monitoring of working memory (WM) representations ("updating"). According to our hypotheses, EFs are crucial for piloting. Indeed, contrary to most jobs that are characterized by maintenance functions and very little integration of new information, which can hide effects of cognitive decline [START_REF] Park | Aging, cognition, and work[END_REF], piloting activity takes place in a dynamical, uncertain and rapidly changing environment where new information must be integrated and updated continuously. EFs appear critical for handling the flight, monitoring engine parameters, planning navigation, maintaining an up-to-date situation awareness -SA - [START_REF] Endsley | Situation awareness in aviation systems[END_REF], correctly adapting to traffic and environmental changes and

Aims and Hypotheses

In this experiment, we proposed to specifically evaluate three low-level EFs (shifting, inhibition and updating) and to link, in the same sample, their efficiency to flight navigation performance and decision making. Although it is little explored, decision making is a very important issue regarding pre-cited flight safety statistics and especially during the critical landing phase. This phase requires following an arrival procedure through several waypoints and implies formalized sequences of actions (e.g. to adjust engine parameters, to extend the flaps…). It also requires decision-making processes based upon rational elements like the maximum crosswind speed for a given aircraft. In spite of the presence of such formalized rules and procedures, numerous pilots make erroneous decision.

We have also taken into account a well established general ability: reasoning. Reasoning reflects fluid intelligence and supports processes relevant for many kinds of abilities (verbal, spatial, mathematical, problem solving etc.) and adaptation to novelty. It is a concept strongly related to executive functioning [START_REF] Decker | Evidence of construct similarity in executive functions and fluid reasoning abilities[END_REF][START_REF] Roca | Executive function and fluid intelligence after frontal lobe lesions[END_REF]. We also assessed the speed of processing because it represents a reliable measure of general cognitive decline and it modulates the overall cognitive efficiency [START_REF] Salthouse | Working-memory mediation of adult age differences in integrative reasoning[END_REF]. Finally, we have also taken into account age to assess its participation to the flight performance variation and we have controlled total flight experience in the analysis. Indeed, experience is well known to improve flight performance and to protect against aging effects [START_REF] Harkey | Age-related changes in selected status variables in general aviation pilots[END_REF][START_REF] Li | Driving-while-intoxicated history as a risk marker for general aviation pilots[END_REF][START_REF] Morrow | Expertise and age differences in pilot decision making[END_REF][START_REF] Taylor | Pilot age and expertise predict flight simulator performance: A 3-year longitudinal study[END_REF]. Our hypothesis was that the chronological age is not a sufficient criterion to predict piloting performance and decision-making relevance and that cognitive performance is a much more relevant criterion.

METHODS

Participants

The participants were 32 private licensed pilots rated for visual flight conditions. The pilots that had no longer flown during the past two years were excluded because of the potential impact on flight simulator performances. All participants had past experience with a computer-based flight simulator. Inclusion criteria were male, right handed, as evaluated by the Edinburgh handedness inventory [START_REF] Oldfield | The assessment and analysis of handedness: the Edinburgh inventory[END_REF], native French speakers, under-or postgraduate. A professionally trained clinical psychologist neuropsychologically examined all participants. Non-inclusion criteria were expertise in logics, airline pilots and sensorial deficits, neurological, psychiatric or emotional disorders and/or being under the influence of any substance capable of affecting the central nervous system. Due to their influence on decision-making processes, emotional disorders identification was based on impulsivity and anxiety assessment with the Barratt Impulsiveness Scale (BIS-10, [START_REF] Bayle | Factor analysis of french translation of the Barratt impulsivity scale (BIS-10)[END_REF] received the instructions, a flight plan and various technical information related to the aircraft (e.g. aircraft's crosswind limit). Basically, the scenario was to take off, reach a waypoint with the help of the aircraft radio navigation system and finally, land on a specified airport. The pilots were instructed that they were in charge of all the decisions and that they could only receive an informative weather report before landing. In order to increase the subject's workload, on route, the pilots had to perform a mental arithmetic calculation of the ground speed (thanks to the embedded chronometer). Moreover, a failure of the compass was scheduled. After this failure, the pilots had to navigate thanks to the magnetic compass, which presents the particularity to be difficult to use as it is anti-directional. The flight scenario lasted for approximately 45 min.

Flight performance

The assessment of flight performance was founded on flight path deviations (FPD), expressed in terms of the amount of angular deviation in the horizontal axis from the ideal flight path. This measure is widely used as an indicator of the primary flight performance [START_REF] Hyland | Experimental evaluation of aging and pilot performance[END_REF][START_REF] Leirer | Marijuana, aging, and task difficulty effects on pilot performance[END_REF][START_REF] Yakimovitch | Flight performance and CogScreen test battery in Russian pilots[END_REF]. The deviation was measured from take-off to the waypoint before the landing decision, in order to assess the same flight segment for all the pilots. Indeed, after the waypoint, some pilots quitted the flight before others (because of the no-landing decision).

Crosswind landing decision

After the waypoint and before reaching the runway threshold, the pilots must state whether the meteorological conditions, as provided by the automatic information system of the arrival airport, were compatible with a landing or necessitated a go-around and a diversion. For this purpose, the pilots had to assess the crosswind component using a commonly utilized formula3. This formula is part of the basic knowledge of pilots and a rather important wind (i.e. 10-15 knots) systematically leads the pilot to consider it. The calculation result exceeded of 6 knots the aircraft's maximum crosswind limit specified in the documentation provided to the pilots at the time of flight preparation. The measured dependent variable was binary: correct when the pilot decided to divert before the runway threshold, incorrect when the pilots continued the landing beyond the runway threshold.

Neuropsychological test battery

Target hitting test

It provides a basic psychomotor reaction time [START_REF] Loubinoux | Modulation of behavior and cortical motor activity in healthy subjects by a chronic administration of a serotonin enhancer[END_REF]. The instruction was to click as fast as possible on each target. The performance was measured by a velocity index inspired by the Fitts' law (1954). The index is the average ratio of the base 10 logarithm of the distance in pixels between two targets, divided by the time in seconds to go from the first target to the second. The index is considered as a measure of the speed of processing. The 2-back test It aims at assessing WM, in particular maintenance and updating abilities [START_REF] Chen | Sub-processes of working memory in the N-back task: An investigation using ERPs[END_REF]. Participants viewed a continuous stream of stimuli and had to determine whether the current stimulus matched on a specific dimension (shape, for our test) the stimulus 2-back in the stimuli sequence. The percentage of correct responses was collected. It is considered as a measure of working-memory updating ability.

The reasoning test

It has been used in a previous study to assess executive functioning [START_REF] Causse | Monitoring Cognitive and Emotional Processes Through Pupil and Cardiac Response During Dynamic Versus Logical Task[END_REF]. The goal of the task is to solve syllogisms by choosing, among three suggested solutions, the one that is a logical conclusion. Syllogisms are based on a logical argument in which one proposition (the conclusion) is inferred from a rule and from another proposition (the premise). We used four existing forms of syllogisms: modus ponendo ponens, modus tollendo tollens, setting the consequent to true and denying the antecedent.

Each participant had to solve 24 randomly displayed syllogisms. The measurement was the percentage of correct responses.

The Wisconsin Card Sorting test -WCST -(Berg, 1948)

It gives information on the subject's abstract reasoning, discrimination learning and shifting abilities [START_REF] Eling | On the historical and conceptual background of the Wisconsin Card Sorting Test[END_REF]. The test version here is a computer implementation that is very similar to the clinical version of the WCST [START_REF] Heaton | A manual for the Wisconsin card sorting test: Odessa, FL: Psychological Assessment Resources[END_REF].

Participants had to sort cards according to three different unknown categories (color, shape, number); an audio feedback indicated whether the response was correct or not (yes/no). When the participant categorized successfully ten cards, the target category was automatically changed. The task ended when six categories were achieved (color, shape, number, color,

The Spatial Stroop test

It assesses the conflict between the meaning of a location word (e.g. "left") and the location where the word is displayed. The ability to restrain a response according to the localization of the word gives information on inhibition efficiency. This conflict appears to be provoked by the simultaneous activation of both motor cortices [START_REF] Desoto | When in doubt, do it both ways: Brain evidence of the simultaneous activation of conflicting motor responses in a spatial Stroop task[END_REF]. Our test encompasses four control conditions. "Stroop neutral meaning" (SNM): a motor answer is given with the appropriate hand according to the word meaning displayed centrally on the screen; "Stroop neutral position" (SNP): the response is given according to the location of a string of XXXXX, displayed at the left or the right of the screen; "Stroop meaning incompatible/compatible" (SMI/SMC): the response is given according to the meaning of the word, compatible or incompatible with its location at the screen. In order to get the pure effects of inhibition, the interference score was calculated to control reading and localization effects by: SMI -(SNP*SNM) / (SNP+SNM).

RESULTS

Statistical analysis

The relationship between age and neuropsychological variables and the ability of the neuropsychological variables to predict piloting performance was tested using regression analyses. The Bonferroni-Holm [START_REF] Holm | A simple sequentially rejective multiple test procedure[END_REF] familywise error rate. The landing decision being a Boolean variable, we performed one-way ANOVAs using decision as a categorical variable to examine whether the cognitive efficiency differed according to the pilot's decision relevance.

Pilot characteristics

The mean age of our sample was 47.28 years (SD = 15.87). The mean level of education was very high (15.4 years, SD = 2.25) and did not significantly correlate with age (p = .462, r = -.14). In all pilots, impulsivity and anxiety trait level were within a normal range (respectively, mean = 40.81, SD = 9.29; mean = 36.42, SD = 7.60). The mean total experience was of 1545 hours of flight (Range = 57-13000). As expected the Bravais-Pearson correlation revealed that there was no significant correlation between age and total flight experience (p = .117, r = .28). This is an important outcome as it helps to disentangle between aging and experience impacts on piloting performance.

Aging effects on cognitive variables

Table 1 shows aging effects on all assessed cognitive variables. With the exception of reasoning performances, Bravais-Pearson correlation demonstrated that all the neuropsychological abilities declined with age. Reasoning performances solely showed a trend to diminish. The rather steep slope of the decrease of velocity with age (see Figure 1) existed also in the other cognitive abilities that showed a marked decline around 55 years of age. For instance, regarding the update in WM, ten participants out of eleven of more than 55 years of age had a correct response rate below 70%, which was never the case for any

Figure 1 about here Table 1 about here Explanatory variables of piloting performance

The mean FPD amplitude was 29.36 (SD = 11.92). We first examine the predictability of age and neuropsychological performances on the piloting ability. We conduct flight experience-partialled correlation to control for flight experience effects, see Table 2.

Table 2 about here

It is interesting to note that flight experience-partialled correlation between reasoning and FPD was the most significant (see Figure 2), suggesting that this general ability was among the most critical to reach a good piloting performance. In addition, speed of processing and updates in WM were also significantly predictive of FPD. The higher the reasoning, the speed of processing and updates in WM abilities were, the smaller the FPD was. In contradiction with our expectations, analysis showed that age was correlated with piloting performance. We therefore conducted 5 others partial correlations between age and piloting performance and we controlled flight experience plus cognitive effect. None of these 5 correlations remained

Figure 2 about here

Cognitive variables related to the crosswind landing decision

Results showed that 50% of the pilots erroneously kept on landing in spite of adverse wind conditions. ANOVA revealed that pilots who were more akin to make the good decision to go-around demonstrated better updating in WM and set-shifting score (respectively, F(1,30) = 9.76, p = .003, ² p = .25; F(1,30) =5.33, p = .027, ² p = .15), see Figure 3 and A first objective of this study was to assess the effects of normal aging on cognitive functioning. Our sample of pilots was submitted to a neuropsychological battery that taped three crucial low-level EFs [START_REF] Miyake | The unity and diversity of executive functions and their contributions to complex "frontal lobe" tasks: A latent variable analysis[END_REF]. EFs analysis on GA pilots is rather uncommon and is of interest as this specific population usually presents a high level of education, which is supposed to be protective from aging effects. However, in spite of this high level of education, analysis revealed that pilots demonstrated a significant decrease in every considered EF. Consistently with literature [START_REF] Gregoire | Effect of age on forward and backward digit spans[END_REF][START_REF] Taylor | Pilot age and expertise predict flight simulator performance: A 3-year longitudinal study[END_REF][START_REF] Verhaeghen | Facts an& fiction about memory aging: A quantitative integration of research findings[END_REF]Yesavage, Dolhert, & Taylor, 1994), updating in WM was sensitive to age effects. The decline was particularly obvious from 55, where ten out of eleven pilots had a lower performance than all the younger ones. In the same way, set-shifting and inhibition were also significantly impacted by age. Various studies have highlighted that older adults demonstrate lower mental flexibility [START_REF] Ashendorf | Exploring Age-Related Decline on the Wisconsin Card Sorting Test[END_REF][START_REF] Fristoe | Examination of age-related deficits on the Wisconsin Card Sorting Test[END_REF] and a greater sensitivity to interference compared to younger adults [START_REF] Bruyer | Effects of age and education on the Stroop interference[END_REF][START_REF] West | Effects of increased response dominance and contextual disintegration on the Stroop interference effect in older adults[END_REF]. Taken together, these results confirmed that the three taped executive functions are vulnerable to aging effects [START_REF] Fisk | Age-related impairment in executive functioning: Updating, inhibition, shifting, and access[END_REF].

We also assessed two well-established general abilities that modulate many kinds of cognitive functions: reasoning and speed of processing. Reasoning did not appear to be consistently impacted by aging, the decline of correct responses in the syllogism resolution task was only near significance threshold. De Neys (2009) showed that reasoning performances begin to drastically decline only after 65. It makes sense that the linear regression was not significant given the large age range of our sample (22-78). Finally, the observed reduced speed of processing was also a classical effect of normal aging [START_REF] Birren | Speed of response as a function of perceptual difficulty and age[END_REF][START_REF] Houx | Stroop interference: aging effects assessed with the Stroop Color-Word Test[END_REF][START_REF] Salthouse | The processing-speed theory of adult age differences in cognition[END_REF]. According to [START_REF] Salthouse | A theory of cognitive aging[END_REF], the speed at which the neural connections are performed determines the number of nervous centers that cooperate at the same time, and therefore, modulates intellectual performance. Thus, speed of processing is considered as a factor of cognitive deterioration, and not as a symptom of such degradation. This overall cognitive decline, despite a high mean level of education (15.4 years) may appear counterintuitive. However, [START_REF] Tucker-Drob | The cognitive reserve hypothesis: A longitudinal examination of age-associated declines in reasoning and processing speed[END_REF] showed that a high level of education has no consistent neuroprotective effect. According to him, education increases cognitive skills in advanced age, but does not influences the rate of cognitive decline. In other words, higher cognitive performances in advanced age associated with a high level of education are allowed by preexistent superior cognitive abilities. The rate of cognitive decline remains the same as for less educated people. In this perspective, it was coherent that, in spite of inter-individual variability, linear regressions could reveal age related cognitive decline in the tested pilots.

Our results suggest that EFs are crucial for piloting. Indeed, the pilot's activity takes place in a dynamical and changing context where new information must be integrated and updated continuously. Update in WM appeared to be essential in this context, most likely to maintain up-to-date SA as suggested by a previous work that showed that losing SA provokes hazardous heading deviation [START_REF] Gibson | Loss of situation awareness-Causes and consequences[END_REF]. Besides, a study by Taylor and coworkers (2000) found that WM and speed of processing are predictive of the piloting performance. We are in line with these results as the index of speed of processing, which is a reliable measure of general intellectual performance [START_REF] Salthouse | Working-memory mediation of adult age differences in integrative reasoning[END_REF], was also related to piloting performance in our study. In addition, our results suggest that reasoning ability appeared to be one of the most predictive functions of piloting ability. This latter outcome confirmed previous results of Causse and coworkers (in press) and [START_REF] Wiggins | Expertise in aeronautical weather-related decision making: A cross-sectional analysis of general aviation pilots[END_REF] that showed strong links between reasoning and piloting performance. This is particularly interesting as this function was not clearly correlated with age in our study. In this perspective, the assessment of reasoning performance appears to be critical for safety and not According to our hypotheses and other authors [START_REF] Li | Human factors in aviation crashes involving older pilots[END_REF][START_REF] Rebok | Pilot Age and Error in Air-Taxi Crashes[END_REF], analysis that controlled for cognitive effects showed that chronological age per se was not a relevant variable to predict piloting performance. [START_REF] Schroeder | Pilot age and performance: an annotated bibliography[END_REF] has pointed out the necessity to use neuropsychological tests rather than relying on age. We must admit that a more extreme sample size including a great proportion of pilots aged of more than 60 would have increased the strength of our conclusions. Unfortunately we had difficulties to recruit several pilots of this category as they were reluctant to take part to the experiment. We plan to perform a future study focused on such a population (60+) in collaboration with airfield medical staff to facilitate recruitment. Nevertheless, our results tend to confirm the limitation of using the chronological age as a single criterion to decide whether a given pilot is able to fly or not.

We also addressed weather-related decision-making. In agreement with several authors (Causse, et al., in press;[START_REF] Dehais | GHOST: experimenting conflicts coutermeasures in the pilot's activity[END_REF][START_REF] Goh | Human factors analysis of accidents involving visual flight rules flight into adverse weather[END_REF] [START_REF] Muthard | Factors that mediate flight plan monitoring and errors in plan revision: Planning under automated and high workload conditions[END_REF][START_REF] Orasanu | Cognitive and contextual factors in aviation accidents: Decision errors[END_REF], our results confirmed the difficulty of pilots to revise their flight plan, especially during the final approach where a great number of them keep on landing in adverse meteorological conditions [START_REF] Rhoda | An assessment of thunderstorm penetrations and deviations by commercial aircraft in the terminal area[END_REF]. Indeed, this was the case for 50% of our volunteers. ANOVAs showed that updating and set shifting may have been particularly related to accurate decision-making. These abilities were probably particularly essential to integrating the degradation of meteorology during the course of flight scenario and to recollect the aircraft's maximum crosswind limit at the time of the approach.

In our experiment, this inability to call to mind critical information, to maintain an up-to-date SA and to inhibit the no more appropriate flight plan seems to lead pilots to erroneously persist on landing. In this sense, [START_REF] Muthard | Factors that mediate flight plan monitoring and errors in plan revision: Planning under automated and high workload conditions[END_REF] has shown the great difficulties encountered by some pilots to integrate critical contextual changes such as deteriorating weather, and a survey of [START_REF] Ebbatson | Crosswind Landings in General Aviation: A Modified Method of Reporting Wing Information to the Pilot[END_REF] has demonstrated that 30% of the pilots are unable to recall the crosswind limit of their aircraft. Given that the 2-Back task also assesses the maintenance in WM, another additional explanation is that the participants did not keep in mind the whole radio-communicated message, in particular the critical wind speed data. This is consistent with the prior findings of Morrow (2003) and [START_REF] Taylor | Cognitive ability, expertise, and age differences in following air-traffic control instructions[END_REF] which showed that poor WM performances degraded the ability to follow ATC radio communication. These outcomes provide new insight for a particular category of accidents called Plan Continuation Errors -PCE -when pilots fail to perceive the changing context of the airspace and subsequently to inhibit the flight plan and consider alternative ones. This phenomenon has been demonstrated both in CA [START_REF] Rhoda | An assessment of thunderstorm penetrations and deviations by commercial aircraft in the terminal area[END_REF][START_REF] Orasanu | Errors in Aviation Decision Making: Bad Decisions or Bad Luck[END_REF]GA (BEA, 2000). The BEA revealed that these pilots' trend to land (called the get-home-itis syndrome in the study) has been responsible for more than 41.5% of casualties in light aircrafts (BEA, 2000). The identification of the brain mechanisms related to pilot errors requires much progress.

However, the results of this study confirm that neuropsychological evaluation is a reliable means for predicting piloting and decision-making performances. This is an important issue as long as the cognitive decline is subtle and may impact flight safety. These types of experiments pave the way to the development of dedicated software including neuropsychological tests that could be administered for pilot certification and during their medical examination. This could help preventing dangerous behaviors, in particular by detecting subtle, though crucial, cognitive impairments. In addition, cognitive decline can reflect the onset of a certain neuropathology or be transient, and reflect the adverse effects of substance consumption (medication, alcohol…), chronic stress, mental fatigue, depression etc.

In such cases, its early detection could help pilots, by advising them and directing them to a medical staff.

  performing accurate decision making by inhibiting wrong behavioral responses. In a light aircraft, the basic analogical and separated instrumentation requires mental effort and reasoning capabilities to maintain SA. Eventually, since EFs modulate mental flexibility, responses or the capacity to maintain up-to-date SA, they are critical to relevant aeronautical decision-making.

  and the Spielberger state anxiety inventory (STAI Y-A, Bruchon-Schweitzer & Paulhan, 1993). All participants received complete information on the study's goal and experimental conditions and gave their informed consent.

  The flight scenario has been setup in collaboration with flight instructors to reach a satisfying level of difficulty and realism. To familiarize the participants with the PC-based flight simulator and to minimize learning effects in order to obtain reliable flight simulator performances, each volunteer underwent a training session. Before the navigation, they

  ) or when the deck of 128 cards was used. The total numbers of errors was derived from the individual cards' records. This number is considered as a measure of setshifting ability.

  Similarly, set-shifting strongly declined after 55, where five pilots had a higher interference index than any other pilots.

  outcome showed that age effect on piloting performance was mediated by the cognitive performance decline.

Figure 4 .

 4 Figure4. These results suggest that these two abilities were particularly critical for decision-

Figure

  Figure 3 about here

  pilots. Reasoning processes are very close to EFs and reflect fluid intelligence, a central cognitive ability linked to various types of mental activity (calculation, problem solving etc.) and is essential to adaptation to novel problems. Complex and novel problems cannot be solved directly by referring to a store of long-term knowledge but require analytic or fluid reasoning. The complexity of our flight scenario with unexpected events such as compass failure may have contributed to a strong involvement of reasoning abilities. The pilots had to perform numerous operations during the navigation to estimate their position and they must use radio navigation systems to reach waypoints. In addition, the scheduled compass failure required pilots to use the anti-directional magnetic compass as backup. The use of this instrument is counterintuitive and may have caused additional difficulties.

Figure 1 .

 1 Figure 1. Speed of processing index regressed on age (n = 32).

Figure 2 .

 2 Figure 2. FPD regressed on reasoning performances (n = 32).

Figure 3 .

 3 Figure 3. Update in WM performances according to the landing decision (n = 32).

Figure 4 .

 4 Figure 4. Set shifting performances according to the landing decision (n = 32).

  correction was applied to control the

	F
	o r
	P e
	e
	r
	R e
	v
	i e
	w
	O n
	l y
	13

  ;

	F
	o r
	P e
	e
	r
	R e
	v
	i e
	w
	O n
	l y
	19

Table 1 .

 1 Neuropsychological performances regressed on age (n = 32) (* p .05; ** p .01;*** p .001).

	Variables	r	R²	Corrected p-value
	Update in WM	-.65	.42	<.001***
	Set-shifting	+.40	.16	.042*
	Inhibition	+.57	.32	.002**
	Reasoning	-.31	.09	.0829
	Speed of processing	-.72	.52	<.001***

Table 2 .

 2 Flight experience-partialled correlation between age and neuropsychological performances with piloting performance (n = 32) (* p .05; ** p .01).

	Variables	r	R²	Corrected p-value
	Age	50	.25	.020*
	Update in WM	-.41	.17	.022*
	Set-shifting	-.23	.5	.452
	Inhibition	+.19	.3	.322
	Reasoning	-.54	.30	.006**
	Speed of processing	-.45	.20	.050*

French Accident Investigation Bureau.

National Transportation Safety Board: independent U.S. federal government agency responsible for civil transportation accident investigation.

Crosswind (in knots) = effective wind (in knots) * sin (angle between runway and wind direction). Moreover, pilots have mnemonic methods to simplify this calculation.

Acknowledgment

The authors would like to thank Dr Jonathan Levy for his comments and guidance in the write-up of this manuscript. We would also like to express our sincere gratitude to the Lasbordes Airfield ISAE pilots who volunteered their time to complete this research. The study was supported by a "Gis Longévité" grant, and the Midi-Pyrenees Regional Council grants 03012000 and 05006110.


Tridimensional and multi-layer modelling of transfers in unsaturated porous media Modelisation multi-couche et en 3 dimensions des transfelts atravers la zone non saturee

Waste disposal sites are usually protected from rainfalt by sloping capiltary barriers which cannot be modelled by classkal unsaturated zone monodimensional models. A 3-dimensional and multilayer fil\ite differences model has been developed. This model-MARTHE -has a system of nested grids with irregular rectangular meshes which guarantees a higher accuracy in given areas. The time discretization follows a special implicit scheme which allows large time steps with perfect flow balance. The flow equations are solved by a preconditioned conjugale gradient solver which is fast and accurate. An example of its application: the simulation of flows through a capillary barrier receiving rainfall is described. RESUME: Dans les sites de stockage de dechets en surface, l'isolation vis-a.-vis des pluies est generalement realisee par des barrieres capillaires inclinees dont Ie fonctionnement ne peut etre simu16 par les modeIes monodimensionnels classiques en zone non saturee. Un modele adifferences finies, it la fois multicouches et tridimensionnel a donc ete realise. Ce modele -Ie modele MARTHEutilise un mailtage forme de paralteIepipMes irreguliers avec possibilite de mailtages emboltes (gigognes), pour assurer une precision maximale en certaines zones. La discretisation du temps se fait selon un schema particulier, totalement implicite, permettant de grands pas de temps de calculs tout en conservant des bilans parfaitement equilibres. La resolution des equations est assuree par la .methode des gradients conjugues avec preconditionnement qui est tres performante. Un exemple d'application du modele est presente: la simulation des econlements atravers nne barriece capillaire soumise ala pluie.

INTRODUCTION

Wastes stored in superficial formations must be protected from the infiltration of rainfall which could generate and transport leachate to the underlying aquifers. A capillary barrier made by the superposition of two graded sloping layers of granular material is a very efficient way of preventing rainfall infiltration. As the layers are sloping, the design and management of snch a capillary barrier can only be managed in transient state through 2 or 3~dimensional numerical models of transport in unsaturated porous media.

The main difficulty is dne to the non-linear character of the f1qw eqnation from which the velocities are detetmined. From the velocities, and the variations of water content, the transport can be calculated llsing the random walk method with particles (Kinzel bach, 1988) in a scheme adapted to transient state (Thiery andlung, 1990,1991;[START_REF] Thiery | Modelling contaminant transport through the unsaturated zone in transient state with a random walk particles scheme[END_REF].

FLOW MODELLING

The following assumptions are made:

• isothermal flow;

• non-swelling soils (the porosity is invariant);

• incompressible fluid (water);

• infinite penneability for air.

With these assnmptions the general flow equation is:

dB div[K GradB] = Ss ift + q (I)
where: H: hydraulic head at location x, y, z [L] q : volumetric density of inflow (injection) or outflow (abstraction) [T-l] Ss: specific storage [L-l] . K: hydraulic conductivity for water [LT-l].

Spatial discretization

Discretization is by finite differences with a pure implicit method. Writing the balance equation between the central mesh and its 6 neighbours (North, South, East, West, Up and Down) yields:

K=f(h) =f(H-Z)

Storage

The storage term is divided into two components:a term proportional to the variation of water contentD8 -a compressibility t~rm. according to the relative saturation S/es STO = VOL. [D8 + (8/8s) . Ss . DH] I Dt (3)

Linearization

The balance equation in each mesh is non-linear;

there are several methods to solve it, as described by [START_REF] Cooley | Some new procedures for mineral solution of variability saturated flow problems[END_REF] or more recently by [START_REF] Thiery | Calculation of natural aquifer recharge from rainfall with an unsaturated zone model solving Richard's equation[END_REF]Thiery ( , 1990a)), [START_REF] Celia | A general mass-conservative numerical solution for the unsaturated flow equation[END_REF], [START_REF] Ross | Efficient numerical methods for infiltration using Richard's equation[END_REF] and [START_REF] Kirkland | Algorithms for solving Richard's equation for variability saturated soils[END_REF]. The method that we chose is described by [START_REF] Thiery | Calculation of natural aquifer recharge from rainfall with an unsaturated zone model solving Richard's equation[END_REF]. This is an iterative method (with underrelaxation) which always yields perfect balance and, in contrast to the approach of [START_REF] Cooley | Some new procedures for mineral solution of variability saturated flow problems[END_REF], is independent of soil functional relations between pressure, water content and hydraulic conductivities. The values of K and ,... STO are the most recent estimations in the ,.) iterative process corresponding to the end of the time step. At iteration k+1:

D8 k + 1 = (He -Hb)• [(8 L 8b)/(

Resolution

In a grid of N meshes it is necessary. at each iteration, to solve a linear system of N equations identical to equation (8). This linear system is synmletrical and generates. a symmetrical square matrix of dimension N x N. This matrix is very sparse and has non-zero terms only along 7 diagonals; the centre and the 6 neighbours in the 6 directions. This system is solved very efficiently by a preconditioned conjugate gradient -as described by [START_REF] Hill | Solving groundwater flow problems by conjugate-gradient methods and the strongly implicit procedure[END_REF]specially adapted to this type of matrix, which is very fast and needs very little memory. As the system is highly non-linear, an under-relaxation coefficient is used to avoid oscillations during the iterative process.

Discretization oftime

Time steps are automatically determined to ensure optimal convergence and are indexed on the variations of saturation in the meshes.

Boundary conditions ~~~c'

Four types of boundary condition are possible: a) prescribed potential (hydraulic head), pressure or water content b) no-flow boundary c) seepage face d) unit gradient of hydraulic head (natural drainage). Boundary conditions a) and b) are classical; the implementation of the third boundary condition (seepage face) is as a "tentative prescribed hydranlic head". If the computed hydraulic head at a given iteration becomes greater then the elevation of the node, the head is prescribed to the value of the elevation and there will be seepage. Then, for future iterations, if the calculated flow is negative (seepage), the hydraulic head stays prescribed, otherwise if it is positive (inflow), it is liberated and the hydraulic head will be lower than the elevation (no seepage). This scheme is efficient, as it is shown below. The fourth boundary condition (unit gradient) is implemented by simulating, in the meshes concerned, a density of outflow equivalent to the hydraulic conductivity corresponding to the local hydraulic head.

Internode weighting

The exchange coefficient between adjacent meshes may be Harmonic mean, Geometric mean (as advised by Vauelin et al., 1979) or Upstream.

VERIFICATION OF MODEL

The method described in the previous section has been implemented in the 3-dimensional, Finite Difference Model MARTHE described by Thiery (1990b). The model has been verified in the unsaturated zone by several severe tests published in the literature, among which are [START_REF] Cooley | Some new procedures for mineral solution of variability saturated flow problems[END_REF], [START_REF] Celia | A general mass-conservative numerical solution for the unsaturated flow equation[END_REF] and many others.

Flow through a square embankment

Tins is test No I as described by [START_REF] Cooley | Some new procedures for mineral solution of variability saturated flow problems[END_REF] -flow through a square bank of porous medium of size 10. Hydraulic head is prescribed at 10 (nnits) over the left boundary and 2 over the right bonndary. The right boundary is a seepage face. The functional relations of the soil are:

hth, = [(Bs -B) f (B -Br)]bt K/K s = [(B -Br)/(fh -Br)]bk K/Ks = [1 + (hthtl1/b']-bk
where:

h t = 1.778; b t = 0.25; bk = 4; K s = 10-2

Figure 1 displays the calculated hydraulic head and the free snrface; there is a seepage face at the left boundary from elevation 2 to elevation 4.8. The results are similar to those obtained by [START_REF] Cooley | Some new procedures for mineral solution of variability saturated flow problems[END_REF].

Test 2. Transient seepagejrom a stream

This is test No 6 as described by [START_REF] Cooley | Some new procedures for mineral solution of variability saturated flow problems[END_REF] -the evolution of an infiltration profile and the response of a ,vater table to infiltration. The soil parameters are identical to those from test No 1 with the additional values: B r =5%; B s =25%; Ss = 10-4 . The aquifer is initially at elevation 2 and the river is suddenly filled with.water to elevation 18. Fignre 2 displays the free snrface (h = 0) at dates 35, 90, 194,226,302 and infinite (thick line). The water table has risen at time 194 and a short time later (t= 226) the whole region under the river is saturated. The results were obtained with a relaxation coefficient equal to 0.7 and only weighting by geometric mean was efficient. The results are exactly similar to those described by [START_REF] Cooley | Some new procedures for mineral solution of variability saturated flow problems[END_REF]. The calculation in steady state. necessitated a relaxation coefficient not greater than 0.2. Fignre 3 displays the calculated hydraulic head in steady state and the free surface. There is a small seepage face at the bottom of the lower aquifer (over a height of less than one meter). Within the confining bed there is an "inverted" free surface because of the drainage into the lower aquifer. These results are similar to those of Cooley. A relaxation coefficient not greater than 0.5 was necessary to obtain a smooth convergence. ------------------1----------------Freetsurfoce '"

1-

• GJ' ~~~, ::~" "~~"' c-" -.:.-=::-,. '~" \

!----IO _ _ hl 12

f=i~=-----=::::::::::--" " " "0, "<' The model MARTHE has been used to simulate a capillary barrier formed by a sandwich of 2 sandy layers separated by a semi-pervious layer of clay. The dimensions of the barrier are 50 x 6.6 m and the slope is 7.5%. The hydraulic conductivity is described by the relation:

KfK s = 1/[1 + (h/hk)bk]
and the retention curve by: h/h t = [I/Se -I]bt hllt t = [(I/Se)Il(I-bt) -I]bt where: Se = (S -Sr)/(Ss -Sr)

The parameters are: sand: h, = 0.15 m; b, = 0.4; ; hk = 0.20 m; bk = 3; K, = 10• 6 m/s; Sr=2%; Ss=25% clay: h, =20m; b,=0.5;; hk=O.1 m; 1Jk= 1; K, = 10-10 m/s; Sr = 10%; Ss = 32% (clay parameters were adapted from [START_REF] Brun | Cin6tique d'infiltration,au sein d'une couche d'argiIe compactee -Etude experimentale et numerique[END_REF] The barrier receives a rainfall equal to 150 mm/year.

Figure 4 displays the calculated hydraulic head and the free surface in steady state. With these parameters, most of the rainfall•is drained in the upper sandy layer, the rest (4.3 nIDl, i.e., 2.9%) flows quasi-vertically through the clay, which is saturated except for the lower half meter, and reaches the lower sandy layer

  Hk -Hb)] (4) where: k = iteration number Hb =H(t-Dt) 8b = 8 (t-Dt) D8 may be written as D8 = C . DH where C has the dimension of a specific storage or a specific moisture capacity; C = (8 k -8b)/(Hk -Hb) Equation (3) then writes as: STO = VOL. [C• DH + (8/8 s ) . Ss• DH] IDt the following algebraic equation for mesh c: ~T" oB, -(~Tv +EM)aBo =-Qi-EMoHb (8)

Fig. 1

 1 Fig. 1 Flow through a square embankment.

  Fig. 2 Transient seepage from a stream.

Fig

  Fig. 3 Drainage from a canyon wall.

Fig. 5

 5 Fig. 5 Nested grid for the capillary barrier.

Acknowledgement: This is BRGM Contribution No 93005; this work was financially supported by a BRGM research project.