Senior Member, IEEE Francois-Xavier Socheleau

A Multifamily GLRT for CFAR Detection of Signals in a Union of Subspaces

Keywords: Multifamily likelihood ratio test, nested models, subspace detection

We consider the problem of detecting an unknown signal that lies in a union of subspaces (UoS) and that is observed in additive white Gaussian noise with unknown variance. The main contribution of this paper is the derivation of a detector that can accommodate a union made of nested subspaces. This detector includes the generalized likelihood ratio test (GLRT) as a special case when the subspace dimensions are all identical. It relies on the framework of multifamily likelihood ratio tests (MFLRT) and is shown by numerical examples to achieve better performance than existing detectors.

I. INTRODUCTION

Testing whether a signal lies within a known subspace is a well-studied problem in the framework of matched-subspace detectors [START_REF] Scharf | Statistical Signal Processing: Detection, Estimation, and Time Series Analysis[END_REF]- [START_REF] Pastor | Random distortion testing with linear measurements[END_REF]. However, in practice, signals are often generated by multimodal processes so that there is not one but several possible subspace models. Among the set of possible models, the active subspace that generated the observed signal may be unknown a priori. This kind of signals complies with what is referred to as the UoS model. More precisely, x ∈ R N is an unknown signal belonging to some union of M known subspaces, i.e., x ∈ M i=1 S i , if and only if there exists i 0 such that x ∈ S i0 [START_REF] Lu | A theory for sampling signals from a union of subspaces[END_REF]. In other words, x belongs to one of the subspace S i , but we do not know a priori to which one. UoS examples include signals with unknown spectral support [START_REF] Lu | A theory for sampling signals from a union of subspaces[END_REF], spectral signatures of radar targets [START_REF] Gini | Radar detection and preclassification based on multiple hypothesis[END_REF] or sparse representations [START_REF] Wimalajeewa | Subspace recovery from structured union of subspaces[END_REF].

Signal detection under the UoS model has been partly addressed in the literature in the specific case where all subspaces are distinct, i. e., there are no subspaces in the union such that S i ⊂ S , for i = [8]- [START_REF] Lodhi | Detection theory for union of subspaces[END_REF]. All these works resort to the GLRT and estimate the active subspace index i 0 as the one that maximizes the likelihood functions. By assuming distinct subspaces, these methods fail to handle nested models properly. In our UoS context, nesting occurs when the signal can be expressed as the sum of an unknown number of basis vectors so that S i ⊂ S , for 1 ≤ i < ≤ M . Examples include signals of unknown duration, periodic signal with an unknown number of harmonics, signals transmitted through an unknown multipath channel, etc. In that case, the GLRT always chooses the subspace model with the largest dimension leading to possible detection losses. More generally, problems arise with the GLRT when subspaces are of different dimensions and are not pairwise disjoint.

F.-X. Socheleau is with IMT-Atlantique, Lab-STICC, France (e-mail: {fx.socheleau}@imt-atlantique.fr).

In [START_REF] Kay | The multifamily likelihood ratio test for multiple signal model detection[END_REF], the MFLRT has been introduced as a general solution to accommodate nested signal models. The idea is to add a penalty term to the GLR statistic to counteract its tendency to increase with the model order. More recently, a new normalizing transformation has been introduced in [START_REF] Kay | A new random variable normalizing transformation with application to the GLRT[END_REF]. It generalizes the results of [START_REF] Kay | The multifamily likelihood ratio test for multiple signal model detection[END_REF] and relies on the Legendre transform of the cumulant generating function (LT-CGF) of the test statistic. Although effective for various applications [START_REF] De Maio | GLRT versus MFLRT for adaptive CFAR radar detection with conic uncertainty[END_REF], [START_REF] Harel | Extension of the MFLRT to detect an unknown deterministic signal using multiple sensors, applied for precipitation detection[END_REF], these results are not directly applicable to our context since the GLR statistic involved in our problem does not satisfy the required properties.

The main contribution of this letter is to adapt the LT-CGF framework to the detection of an unknown signal in a UoS, with possibly overlapping or nested subspaces. The additive noise is assumed to be white Gaussian with unknown variance. The proposed detector is obtained by establishing a convergence result on the LT-CGF of a Fisher-z random variable. This approach generalizes the standard GLRT under the UoS model and includes it as a special case when the subspace dimensions are all identical. The paper is organized as follows. Sec. II is devoted to the problem formulation. The derivation of the Multifamily CFAR GLRT is presented in Sec. III. Numerical results are provided in Sec. IV, followed by conclusions in Sec. V.

Notation: We denote by X n d → X the convergence in distribution as n → +∞. O(•), O p (•) and o p (•) designate the usual asymptotic notations for non-random and random variables, respectively [START_REF] Bishop | Discrete multivariate analysis: theory and practice[END_REF]Ch. 14]. That is,

a n = O(b n ) if the ratio a n /b n is bounded for large n, X n = O p (b n ) if the random sequence X n /b n is stochastically bounded and X n = o p (1) if X n converges to zero in probability. Finally, f (m) (x) denotes the m-th derivative of f with respect to x. II. PROBLEM FORMULATION Let x ∈ M i=1 S i
, where S i are subspaces of R N . No particular assumption is made with respect to the intersection of the subspaces and we do not assume all subspaces to have the same dimension but simply that n i = dim(S i ) < N , ∀ i. The detection problem addressed in this paper can be stated as

H 0 : y = w H 1 : y = x + w, (1)
where y is the observation vector and w is a zero-mean white Gaussian vector with unknown variance. Because of the unknown parameters involved in the problem, no uniformly powerful test exists and the likelihood ratio test cannot be applied. A standard approach is then to resort to GLRT techniques that replace the unknown parameters with their maximum likelihood estimates.

Let H i ∈ R N ×ni be a basis for the n i -dimensional subspace S i and let P Si = H i H T i H i -1 H T i be its projection matrix. The GLRT for Problem (1) is a straightforward extension of the well-studied CFAR matched-subspace detector [START_REF] Scharf | Statistical Signal Processing: Detection, Estimation, and Time Series Analysis[END_REF]Ch. 4] and can be written as [START_REF] Lodhi | Detection theory for union of subspaces[END_REF] T GLRT (y) = max

1≤i≤M L i (y) H1 ≷ H0 η GLRT , (2)
where η GLRT is the threshold chosen so as to satisfy some fixed test size. L i (y) is defined as twice the generalized loglikelihood ratio of y when x is assumed to belong to subspace S i . This ratio can be expressed as [16, pp. 372]

L i (y) = N log y T y y T (I N -P Si ) y , (3)
where I N is the N ×N identity matrix. As expressed in (2), the GLRT will always select the subspace containing the greatest amount of signal energy. As long as the subspaces have similar dimensions and/or are pairwise disjoint, this may be fine but in any other cases this is problematic. The worst case happens when the subspaces are nested, i.e., when S i ⊂ S , for 1 ≤ i < ≤ M . In this scenario, the GLRT will always implement the test statistic with the matrix P S M that projects the data in the larger subspace. If the actual signal subspace is smaller than S M , the GLRT will include noise-only samples under either hypothesis, degrading the performance of the detector. For instance, this problem is illustrated in [START_REF] Kay | The multifamily likelihood ratio test for multiple signal model detection[END_REF] for a signal of unknown samples and unknown length corrupted by an additive white Gaussian noise with known variance. For this signal model, but now with an unknown noise variance, the GLRT (2) becomes a CFAR energy detector. More precisely, each basis satisfies

H i = I ni O (N -ni)×ni (4)
where O n×m is the n × m null matrix and n i < n for i < . The test statistic in ( 2) is then a monotonically increasing function of the ratio

N -1 k=0 |y(k)| 2 / N -1 k=n M |y(k)| 2
, where y(k) is the k-th entry of y. If the actual signal length is smaller than n M , the GLRT will not be effective since this ratio will be overestimated. More generally, this problem arises for any signal expressed as the sum of an unknown number of basis vectors.

Improvement of the GLRT can be achieved by using a model order selection criterion in the decision procedure. A well known criterion is the minimum description length (MDL) criterion [START_REF] Rissanen | Modeling by shortest data description[END_REF], which leads to the following test 1

T MDL (y) = max 1≤i≤M {L i (y) -n i log N } H1 ≷ H0 η MDL . (5)
In [START_REF] Kay | The multifamily likelihood ratio test for multiple signal model detection[END_REF], a multifamily likelihood ratio test (MFLRT) has also been proposed to accommodate nested models. It implements 1 Other criteria such as AIC or GIC could also be considered [START_REF] Stoica | Model-order selection: a review of information criterion rules[END_REF].

the following decision rule:

T MFLRT (y) = max 1≤i≤M L i (y) -n i log L i (y) n i + 1 × u L i (y) n i -1 H1 ≷ H0 η MFLRT , (6)
where u (•) is the unit step function. As illustrated in Sec. IV, by adding a penalty term, both tests ( 5) and ( 6) can provide a better performance than the GLRT. However, by expanding the scope of a new normalizing transformation introduced in [START_REF] Kay | A new random variable normalizing transformation with application to the GLRT[END_REF], we will show that the performance can be further improved.

III. MULTIFAMILY CFAR GLRT A. Preliminary results

To begin, we give some preliminary definitions and recall the main result of [START_REF] Kay | A new random variable normalizing transformation with application to the GLRT[END_REF]. Let K X (t) be the cumulant generating function (CGF) of some random variable X, that is,

K X (t) = log E e tX , (7)
where E denotes the mathematical expectation. The Legendre transform (LT) of K(•) is then defined as

K * X (x) = sup t∈T (tx -K X (t)) , (8)
where

T = {t : |K X (t)| < +∞}. Next, define g X (x) = 2K * X (x)u (X -E (X)) . (9)
Proposition 1.

This proposition states that, as n tends to +∞, Y n equals zero with probability 1/2 and has a χ 2 1 PDF with probability 1/2. By applying a quadratic transformation to the standardized random variable X n , a similar result could have been obtained by invoking the central limit-theorem (CLT). However, as illustrated in [START_REF] Kay | A new random variable normalizing transformation with application to the GLRT[END_REF], the asymptotic regime can be reached faster than the CLT in some cases. Prop. 1 is relevant for nested detection problems as it can be used to equalize the modelorder dependent statistics involved in a GLRT solution. The function g, defined in [START_REF] Wimalajeewa | Subspace recovery from structured union of subspaces[END_REF], actually transforms the (asymptotic) GLR statistic to a χ 2 1 random variable independent of the model order n. As shown in [START_REF] Kay | A new random variable normalizing transformation with application to the GLRT[END_REF], if L i (y) can be expressed as the sum of n i χ 2 1 random variables under H 0 , then g takes the form of the expression enclosed in braces in [START_REF] Pastor | Random distortion testing with linear measurements[END_REF]. The MLFRT ( 6) is thus a specific case of a more general theory resulting from the application of Prop. 1. Unfortunately, this proposition is not directly applicable to Problem (1) because the GLR statistic (3) is not expressed as a sum of independent random variables. However, by transforming (3), we will show that a similar result to that of Prop. 1 can be obtained for our specific problem.

B. Proposed detector

Let Li (y) be the monotonically increasing function of L i (y) defined as

Li (y) = 1 2 log y T P Si y y T (I N -P Si ) y × N -n i n i . (11)
The proposed detector is based on the derivation of a modelorder dependent penalty factor that results from the analysis of the LT-CGF of Li (y) under H 0 . Under H 0 , Li (y) is half the logarithm of an F-distribution variate and therefore follows Fisher's z-distribution with n i and N -n i degrees of freedom. Let ρ i = N/n i > 1, for t < n i (ρ i -1), the CGF of Li (y) under H 0 satisfies [START_REF] Aroian | A study of RA Fisher's z distribution and the related F distribution[END_REF]

K Li (t) = t 2 log (ρ i -1) -log Γ n i (ρ i -1) 2 Γ n i 2 + log Γ n i + t 2 Γ n i (ρ i -1) -t 2 , (12)
where

Γ(z) = +∞ 0 t z-1 e -t dt. The LT of K Li is then defined as K * Li (x) = sup t<ni(ρi-1) tx -K Li (t) and g Li (x) = 2K * Li (x)u Li (y) -µ i , (13)
where

µ i = E Li (y) = 1 2ni 1 ρi-1 -1 . Proposition 2.
Let Y ni be the random variable defined as Y ni = g Li ( Li (y)) and assume that the ratio

ρ i = N/n i > 1 is kept constant as n i → +∞. Under H 0 , Y ni d → Y
, where the PDF of Y is defined in Eq. [START_REF] Lodhi | Detection theory for union of subspaces[END_REF].

Proof: See Appendix A. Applying the transformation g Li to the statistic Li (y) is therefore a way to equalize the model-order dependent PDFs involved in our problem. Unfortunately, g Li ( Li (y)) has no closed-form expression. However, based on the asymptotic analysis provided in App. B, we are in position to formulate an explicit approximation of g Li ( Li (y)), that is

g Li Li (y) ≈ 2 Li (y)h i ( Li (y)) -K Li (h i ( Li (y))) × u Li (y) -µ i , (14)
where

h i (x) = n i ρ i (ρ i -1)e -2x + 1 -1 . ( 15
)
Finally, to solve Problem (1), we propose the following test, referred to as the MFCLRT,

T MFCLRT (y) = max 1≤i≤M g Li Li (y) H1 ≷ H0 η MFCLRT , ( 16
)
where the approximation ( 14) is used for g Li . This test does not depend on unknown parameters and therefore ensures the CFAR property. Moreover, it can be shown that g Li Li (y) is a decreasing function of n i so that it counteracts the tendency of the GLR statistic to increase with n i . Also, note that if the subspace dimensions are all identical (n i = n, ∀ i), then the tests ( 2), ( 5), ( 6) and ( 16) are all equivalent since their statistics are all monotonically increasing with L i (y).

IV. NUMERICAL RESULTS

To illustrate the performance of the MFCLRT ( 16), we consider the detection of a known signal s in an unknown deterministic multipath channel h. More precisely, x(k) = n M -1 l=0 h(l)s(k-l), where n M denotes the number of channel taps. In the simulation, the channel has an exponential decay power-delay profile, i.e., h(k) = e -k log [START_REF] Lodhi | Detection theory for union of subspaces[END_REF] 2k 0 , where k 0 > 0 denotes the 10 dB-maximum delay spread. 2 Without loss of generality, we assume that s(k) = d(k), where d(k) is the unit impulse. 3 Since the channel is unknown, the detector is unaware of the number of significant echoes that must be taken into account in the test statistic. The set of all possible echoes therefore constitutes a union of nested subspaces, each of which corresponds to a set of delays. More precisely, since s(k) = d(k), each subspace basis is defined as in Eq. ( 4) for any 1

≤ n i ≤ n M .
We investigate the performance of the proposed detector in terms of probability of detection P d . The probability of false alarm P f a is set to 10 -3 . The detection thresholds and the probabilities of detection are obtained with 100/P f a and 10 4 independent trials, respectively. The signal length N is set to 100 and n M = 50. The energy-to-noise ratio (ENR) is defined as ENR = 10 log 10 x T x/σ 2 , where σ 2 is the noise variance.

Fig. 1 shows the probability of detection as function of the ENR for the four tests (2), ( 5), ( 6) and ( 16). k 0 is set to 15. The MFCLRT clearly outperforms the other tests as it better estimates the order of the channel. The GLRT includes all the echoes in the statistic, even those with insignificant power, and therefore performs the worst. A detailed analysis (not shown here) reveals that the MDL test tends to only consider the first few powerful echoes, whereas the MFLRT often underpenalizes insignificant echoes.

The effect of the model order is analyzed in Fig. 2 by varying the value of the maximum delay spread k 0 . Once again, the MFCLRT outperforms the other tests for any value k 0 . Both the MFCLRT and the MFLRT converge to the performance of the GLRT as k 0 increases. This results from the increase in the number of significant echoes as k 0 increases, leading to a model order close to the maximum n M . Surprisingly, a collapse of performance rapidly occurs for the MDL test. This can be explained by the penalty term in Eq. ( 5) that linearly increases with n i so that the channel tail is too often ignored.

V. CONCLUSION

We have shown that existing methods, such as the GLRT, are not adapted to the detection of signals in a union of possibly nested subspaces. By adapting to our context the CGF-based transformation presented in [START_REF] Kay | A new random variable normalizing transformation with application to the GLRT[END_REF], we have derived a new detector capable of equalizing the model-order dependent statistics. Although the equalization is of an asymptotic nature (as the subspace dimension becomes large), the resulting test turns out to be very efficient, even for low-dimensional subspaces.

g Li ( Li (y)) = u Li (y) -µ i ×    Li (y) -µ i 2 σ 2 i + 1 3 K * (3) Li (ξ i ) σ 3 i Li (y) -µ i 3 σ 3 i    , (17)
where ξ i = µ i +θ( Li (y)-µ i ) and 0 < θ < 1. It is known that the distribution of a Fisher's z variable approaches normality when its degrees of freedom tends to infinity [START_REF] Aroian | A study of RA Fisher's z distribution and the related F distribution[END_REF]Sec. 15]. More precisely, let V ni = Li(y)-µi σi . For any

ρ i = N/n i > 1, V ni d → N (0, 1). Therefore, by the continuous mapping theo- rem, V 2 ni d → χ 2 1 .
The main step of the proof is then to show that the second term on the r.h.s of ( 17) is bounded in probability by a decreasing function of n i .

K * (3) Li (ξ i ) = - K (3) Li (t * i (ξ i )) K (2) Li (t * i (ξ i )) 3 , (18)
where t * i (x) = argsup t<ni(ρi-1) tx -K Li (t) . Based on the expression of the CGF defined in [START_REF] Kay | A new random variable normalizing transformation with application to the GLRT[END_REF], we observe that the ratio (18) will depend on the 3 rd and 2 nd -order derivative of the logarithm of the gamma function. These derivatives are known as the polygamma functions of order 2 and 1 [20, Ch. 6.4], respectively. Based on the asymptotic expansion given [20, Eq. (6.4.11)], we deduce that the m-th order polygamma function

Ψ (m) (z) satisfies Ψ (m) (z) = O(z -m ) (z → ∞).
Moreover, as shown in App. B, ∀ x > 0, t * i (x) is strictly increasing in n i . Consequently, ∀ x > 0,

K (3) Li (t * i (x)) K (2) Li (t * i (x)) 3 = O Ψ (2) (n i )
Ψ (1) (n i )

3 = O (n i ) . ( 19
)
In addition, provided that Li (y) > µ i , ξ i is strictly positive. Therefore, based on Eqs ( 18) and ( 19), we get K * (3) Li (ξ i ) = O p (n i ), so that APPENDIX B ASYMPTOTICS OF t * i (x) Let t * i (x) be defined as t * i (x) = argsup t<ni(ρi-1) G i (t, x) where G i (t, x) = tx -K Li (t) . From (12), we deduce that

∂G i (t, x) ∂t = 1 2 2x -log(ρ i -1) -Ψ (0) 1 2 (n i + t) + Ψ (0) 1 2 (n i (ρ i -1) -t) , (21)
where Ψ (0) (•) is the digamma function. For any real variable z, this function satisfies [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF]Eq. (6.3.18)]: Ψ (0) (z) = log(z)+ o(z -1 ). Therefore, as n i → ∞, we have | t=t * i (x) < 0, ∀ x, so that t * i (x) is an (asymptotic) supremum. Also, observe that ∀ x > 0, t * i (x) is strictly increasing in n i .

∂G i (t,

[ 12 ]

 12 Let X n = n k=1 U k where the U k 's are IID random variables with finite mean and variance. Also, assume that E |U k -E(U k )| 3 < ∞. Define the random variable Y n as Y n = g Xn (X n ). Then, Y n d → Y , where the (generalized) probability density function (PDF) of Y satisfies f Y (y) =

Fig. 1 .

 1 Fig. 1. P d as a function of ENR for the GLRT, the MDL test, the MFLRT and the MFCLRT. P f a = 10 -3 , N = 100, n M = 50, k 0 =15.

Fig. 2 . 1 ρi- 1 -1 and σ 2 i = 1 2ni 1 ρi- 1 + 1 .

 211111 Fig. 2. P d as a function of k 0 for the GLRT, the MDL test, the MFLRT and the MFCLRT. P f a = 10 -3 , N = 100, n M = 50, ENR=18 dB.

First, since σ 2 i 3 /σ 3 i=

 233 is finite, then[START_REF] Bishop | Discrete multivariate analysis: theory and practice[END_REF] Li (y) -µ i O p (1). In addition, using [12, App. A], we note that the third derivative of the LT-CGF satisfies

( 20 )

 20 In summary, g Li ( Li (y)) = u (V ni ) V 2 ni + o p (1) with V ni d → N (0, 1). Therefore, according to the continuous mapping theorem and to [15, Th. 4.4-3], g Li ( Li (y)) converges in distribution to the random variable Y whose PDF is expressed in Eq. (10).

We recall that this is the time delay (in number of samples) after which the multipath power falls 10 dB below the maximum.

s(k) can be considered as the output of a correlation-based channel sounder for instance.


Pascal E Reiller
email: pascal.reiller@cea.fr

Julien Brevet

Antonio Nebbioso

Alessandro Piccolo

Europium (III) Complexes with Size-Fractions Separated from a Humic Acid by HPSEC only Evidence Few Differences in their Chemical Environment by Time-Resolved Luminescence Spectroscopy

Keywords: Humic acid, Europium, fractionation, complexation, luminescence, fluorescence

The size fractionation of a humic acid (HA) by high performance size exclusion chromatography (HPSEC) was used as a proxy for the filtration effect during HA transport through a porous medium with minimum specific chemical interactions. The modification of the Eu(III)-HA complexes' formation with the different size-fractions, as compared to the bulk HA, were studied in time-resolved luminescence spectroscopy (TRLS). Clear modifications in Eu(III)-HA complexes' structures were shown and related to the molecular characteristics of the separated size-fractions. The properties of most of size-fractions did not induce a major alteration of the affinity towards Eu(III). Only the most hydrophilic fractions eluted in the tail of the chromatographic peak, representing about 11% of total fractions-weight, gave some significantly different parameters. Using a simplistic complexation model, it was found that the available complexation sites decreased with the size reduction of humic fractions.

Introduction

Humic substances (HS), mainly composed of humic acids (HA) and fulvic acids (FA), are the main surrogate of natural organic matter. They play an important role in the transport of lanthanides (Ln) and actinides (An) in the environment [START_REF] Mccarthy | Lanthanide field tracers demonstrate enhanced transport of transuranic radionuclides by natural organic matter[END_REF][START_REF] Fujikawa | Strong association of fallout plutonium with humic and fulvic acid as compared to uranium and 137 Cs in Nishiyama soils from Nagasaki, Japan[END_REF][START_REF] Santschi | Organic nature of colloidal actinides transported in surface water environments[END_REF][START_REF] Pourret | Colloidal control on the distribution of rare earth elements in shallow groundwaters[END_REF]. The exact nature of the Ln/An-HS interactions during transport and especially during sorption mechanisms is still unclear.

Particularly, the fractionation of natural organic matter on mineral surfaces introduces some difficulties in modelling sorption of metals in ternary system, i.e., metal/HA/surface. In the case of FA, the modelling approach was partly successful, although the FA's sorptive behaviour may be explained with electrostatic considerations [START_REF] Heidmann | Sorption of Cu and Pb to kaolinite-fulvic acid colloids: Assessment of sorbent interactions[END_REF][START_REF] Weng | Cu 2+ and Ca 2+ adsorption to goethite in the presence of fulvic acids[END_REF], and assumed to be more or less similar to that of simple organic molecules [START_REF] Schulthess | Competitive adsorption of aqueous carbonic and acetic acids by an aluminium oxide[END_REF]. The FA sorption is stable or tends to decrease with ionic strength [START_REF] Schlautman | Adsorption of aquatic humic substances on colloidal-size aluminum oxide particles: Influence of solution chemistry[END_REF][START_REF] Filius | Adsorption of fulvic acid on goethite[END_REF][START_REF] Weng | Adsorption of humic substances on goethite: Comparison between humic acids and fulvic acids[END_REF]. Conversely, HA have a different behaviour since a sorption increase on oxides is occurring [START_REF] Murphy | Interaction of hydrophobic organic compounds with mineral bound humic substances[END_REF][START_REF] Schlautman | Adsorption of aquatic humic substances on colloidal-size aluminum oxide particles: Influence of solution chemistry[END_REF][START_REF] Weng | Adsorption of humic substances on goethite: Comparison between humic acids and fulvic acids[END_REF]. Hence, it is still difficult to propose sound interpretations of HA effects on the modelling of ternary systems in a wide parametric space [START_REF] Vermeer | Metal ion adsorption to complexes of humic acid and metal oxides: Deviation from the additivity rule[END_REF][START_REF] Lippold | Effect of humic matter on metal adsorption onto clay materials: Testing the linear additive model[END_REF].

The intimate structure of HS is still a matter of debate. Nevertheless, a consensus is emerging from small-angle scattering methods [START_REF] Tombácz | Fractal structure of polydisperse humic acid particles in solution studied by scattering methods[END_REF], atomic force microscopy [START_REF] Österberg | Direct observation of humic-acid clusters, a nonequilibrium system with a fractal structure[END_REF][START_REF] Balnois | Atomic force microscopy of humic substances: Effect of pH and ionic strength[END_REF], high-performance size-exclusion chromatography (HPSEC) [START_REF] Piccolo | Effects of mineral and monocarboxylic acids on the molecular association of dissolved humic substances[END_REF][START_REF] Piccolo | Chromatographic and spectrophotometric properties of dissolved humic substances compared with macromolecular polymers[END_REF], electrospray ionization mass spectrometry [START_REF] Plancque | Molecular structure of fulvic acids by electrospray with quadrupole time-of-flight mass spectrometry[END_REF][START_REF] These | Determination of molecular formulas and structural regularities of low molecular weight fulvic acids by size-exclusion chromatography with electrospray ionization quadrupole time-of-flight mass spectrometry[END_REF], electrophoresis [START_REF] Duval | Humic substances are soft and permeable: evidence from their electrophoretic mobilities[END_REF], or nuclear magnetic resonance [START_REF] Conte | Advanced CPMAS-13 C NMR techniques for molecular characterization of size-separated fractions from a soil humic acid[END_REF][START_REF] Conte | Spectroscopic and conformational properties of size-fractions acid separated from a lignite humic[END_REF], on the fact that HS are formed of nanometric sub-entities that associate in larger bodies in the case of HA. The cohesion of these associations is assured through short range interactions, which can be largely modified in contact with strong sorbing mineral surfaces like Fe or Al oxides [START_REF] Gu | Adsorption and desorption of natural organic matter on iron oxide: Mechanisms and models[END_REF][START_REF] Hur | Molecular weight fractionation of humic substances by adsorption onto minerals[END_REF][START_REF] Reiller | Sorption of Aldrich humic acid onto hematite: Insights into fractionation phenomena by electrospray ionization with quadrupole time-offlight mass spectrometry[END_REF][START_REF] Claret | Fractionation of Suwannee River fulvic acid and Aldrich humic acids on α-Al2O3: spectroscopic evidence[END_REF].

During transport in a particular environment, HA associations can be submitted to different interactions. First, a 'simple' filtration effect, which can alter the HA structure as by performing a size-exclusion 'sorting out' different constituents making up the associations.

Second, a sorptive fractionation induced by chemical interactions with minerals in the environment, which induces a separation of constituents based on their affinity to the mineral phase [START_REF] Gu | Adsorption and desorption of natural organic matter on iron oxide: Mechanisms and models[END_REF][START_REF] Hur | Molecular weight fractionation of humic substances by adsorption onto minerals[END_REF][START_REF] Reiller | Sorption of Aldrich humic acid onto hematite: Insights into fractionation phenomena by electrospray ionization with quadrupole time-offlight mass spectrometry[END_REF][START_REF] Claret | Fractionation of Suwannee River fulvic acid and Aldrich humic acids on α-Al2O3: spectroscopic evidence[END_REF]. In fact, it is difficult to obtain these parameters independently. Christl et al. showed that the functionality (Christl and Kretzschmar, 2001) and metal binding (Christl, et al., 2001) of different size-fractions of a HA from ultrafiltration were comparable within a narrow range. [START_REF] Claret | Fractionation of Suwannee River fulvic acid and Aldrich humic acids on α-Al2O3: spectroscopic evidence[END_REF] noted that the influence of sorptive fractionation of HS on α-Al2O3 was important from the evolution of the chemical environment of europium(III) by time-resolved luminescence spectroscopy (TRLS).

The aim of this work was to study the differences in chemical environment of Eu(III), if any, when complexed by HPSEC-separated humic size-fractions, as a proxy for the filtration effect during HA transport through a porous medium with minimum specific chemical interactions.

Material and Methods

Humic acid.

The humic acid from Ginchi (GHA) was extracted from a recently cultivated forested vertisol of Ethiopian highlands (09°01'N, 38°20'E, 2300 m asl, see [START_REF] Spaccini | Carbohydrates in waterstable aggregates and particle size fractions of forested and cultivated soils in two contrasting tropical ecosystems[END_REF], for further details). Soil properties, isolation and purification procedures, and GHA characteristics are reported elsewhere [START_REF] Spaccini | Carbohydrates in waterstable aggregates and particle size fractions of forested and cultivated soils in two contrasting tropical ecosystems[END_REF][START_REF] Spaccini | Changes of humic substances characteristics from forested to cultivated soils in Ethiopia[END_REF]. The previously freezedried GHA was suspended in distilled water and titrated to pH 7 with a CO2-free solution of 0.5 mol L -1 KOH by an automatic titrator (VIT 90 Videotitrator, Radiometer, Copenhagen, Denmark) under N2 atmosphere and stirring. After having reached the constant pH 7, the solution containing potassium-humates was left under titration for 2 more hours, filtered through a Millipore 0.45 µm, and freeze-dried. Potassium humates were pretitrated at pH 7 to limit HPSEC interferences.

Preparative HPSEC separation.

The mobile phase for HPSEC consisted of a 242 mg L -1 Na2HPO4, 248 mg L -1 NaH2PO4•2H2O, 820 mg L -1 CH3CO2Na, milli-Q water solution at pH 7 and added with 300 mg L -1 NaN3 as bacteriostatic agent. The same solution was used to dissolve the potassium humates to a concentration of 600 mg L -1 . The humic solution was filtered through glass microfibre filters (Whatman GF/C) and loaded into a rheodyne rotatory injector, equipped with a 5 mL sample UV detector set at 280 nm, and a Gilson FC205 fraction collector, to automatically collect humic fractions in continuous. The elution flow-rate was set at 1.5 mL min -1 and all chromatographic runs were automatically recorded by a Unipoint Gilson Software. Ten fractions were separated with HPSEC elution: within a 37.5-135.0 mL interval, nine fractions were collected changing vial every 7.5 mL, whereas 30.0 mL were collected for the tenth fraction. The ten isolated size-fractions were first freeze-dried to reduce their volume, resuspended in 5 mL of deionised water, dialyzed (Spectra/Por 6 dialysis tube, 1kDa MW cut-off) against deionised water, and freeze-dried again. Out of 110 injections of the HA solution (330.0 mg), the total weight recovered in the ten isolated fractions was about 90% (297.5 mg) of the initial injected weight. The preparative HPSEC profile of the Ginchi HA showed two main peaks (Figure 1a). The first one (Fraction G1-G4) encompassed the majority of the mass distribution (55%, see Figure 1b). Fraction G5 shows an increase in absorbance but it does not result in an increase in mass percentage relative to Fraction G4.

Typically, the HPSEC chromatograms of humics reported in literature do not show the resolution that is proper of homogeneous materials, since the large number of heterogeneous molecules, which is accounted to be more than 10000 in humic acids [START_REF] Moulin | Direct characterization of iodine covalently bound to fulvic acids by electrospray mass spectrometry[END_REF][START_REF] Piccolo | Electrospray ionization mass spectrometry of terrestrial humic substances and their size fractions[END_REF][START_REF] Reiller | Sorption of Aldrich humic acid onto hematite: Insights into fractionation phenomena by electrospray ionization with quadrupole time-offlight mass spectrometry[END_REF], and their absorption spectra overlapping is the very cause of the signal broadening in the chromatogram. Moreover, one must not forget that our separation of humic fractions was done in the preparative mode (5 mL of loading volume), that has an even lower resolution than analytical HPSEC, as reported elsewhere [START_REF] Piccolo | Reduced heterogeneity of a lignite humic acid by preparative HPSEC following interaction with an organic acid. Characterization of size-separates by Pyr-GC-MS and H-1-NMR spectroscopy[END_REF][START_REF] Conte | Advanced CPMAS-13 C NMR techniques for molecular characterization of size-separated fractions from a soil humic acid[END_REF][START_REF] Conte | Spectroscopic and conformational properties of size-fractions acid separated from a lignite humic[END_REF]. The preparative HPSEC has the goal to obtain quantitative amounts of humic fractions which can be subjected to further analysis more than determining the humic molecular size distribution. In the preparative mode also the notations of void and total volumes loose the importance that is commonly attributed to analytical HPSEC. The important issue is to obtain highly reproducible chromatograms in order to isolate meaningful and quantitative fractions during the repetitive

G H A 1 G H A 2 G H A 3 G H A 4 G H A 5 G H A 6 G H A 7 G H A 8 G H A 9 G H A

Solution

Europium (III) stock solution was obtained from the dissolution of Eu2O3 (Johnson Matthey, 99.99%) in HClO4. It is assumed that the humic complexation is complete with C(Eu)

= 10 µmol L -1 and C(HS) = 200 mg L -1 [START_REF] Marang | Combining spectroscopic and potentiometric approaches to characterize competitive binding to humic substances[END_REF] at pH 5, even if the ratios between Eu(III) and the number of available sites in the different fractions can be slightly different [START_REF] Christl | Chemical heterogeneity of humic substances: characterization of size fractions obtained by hollow-fibre ultrafiltration[END_REF]Christl and Kretzschmar, 2001;Christl, et al., 2001). The ionic strength was fixed with 0.1 mol L -1 (NaClO4), and pH 5 was adjusted using freshly prepared NaOH and HClO4. The dissolution of GHA, bulk and fractions, were done at pH 10 overnight and then adjusted to pH 5. The pH measurements were done using a combined-glass electrode (Radiometer Analytical XC111) calibrated for its linear response with a 0.01 mol L -1 HClO4 solution, an equimolar 0.02 mol L -1 NaH2PO4/Na2HPO4 solution and an equimolar 0.02 mol L -1

Na2CO3/NaHCO3 solution, all containing NaClO4 to keep [Na + ] constant at 0.1 mol L -1 (pH = 2.0, 6.8, and 9.9, respectively). The electrode filling solution was modified with NaClO4 0.1 mol L -1 , NaCl 10 -2 mol L -1 to prevent KClO4 precipitation in the frit of the electrode. The pH 5 value was chosen to limit the formation of hydroxo and carbonato complexes of Eu(III).

Time-resolved luminescence spectroscopy

Europium (III) has been used to probe its laser-induced luminescence properties in contact with humic substances. This technique has been used to study the lanthanide (III)-HS interactions, and the rationale was to either determine interaction constants [START_REF] Dobbs | Characterization of metal binding sites in fulvic acids by lanthanide ion probe spectroscopy[END_REF][START_REF] Chung | Comparison of synchronous and laserinduced fluorescence spectroscopy applied to the Eu(III)-fulvate complexation[END_REF][START_REF] Marang | Combining spectroscopic and potentiometric approaches to characterize competitive binding to humic substances[END_REF], or gather information on the lanthanide (III) chemical environment [START_REF] Thomason | Fluorescence studies of metal humic complexes with the use of lanthanide ion probe spectroscopy[END_REF][START_REF] Shin | A study of Eu(III)-humate complexation using Eu(III) luminescence spectroscopy[END_REF][START_REF] Li | Studies of metal-binding sites in natural organic matter and their role in the generation of disinfection by-products using lanthanide ion probes[END_REF][START_REF] Kumke | Fluorescence quenching and luminescence sensitization in complexes of Tb 3+ and Eu 3+ with humic substances[END_REF][START_REF] Claret | Fractionation of Suwannee River fulvic acid and Aldrich humic acids on α-Al2O3: spectroscopic evidence[END_REF][START_REF] Brevet | Spectral and temporal luminescent properties of Eu(III) in humic substances solutions from different origins[END_REF][START_REF] Marang | Spectroscopic characterization of the competitive binding of Eu(III), Ca(II), and Cu(II) to a sedimentary originated humic acid[END_REF][START_REF] Reiller | Bi-exponential decay of Eu(III) complexed by Suwannee River humic substances: Spectroscopic evidence of different excited species[END_REF]. The observed luminescence corresponds to the 5 D0→ 7 F0 transition (electric and magnetic dipole forbidden, maximum around 580 nm), the 5 D0→ 7 F1 transition (magnetic dipole, maximum around 593 nm), and the 5 D0→ 7 F2 'hypersensitive' transition (electric dipole, maximum around 615 nm). These emission lines come from transitions of the 5 D0 excited state to the ground 7 Fj manifold [START_REF] Bünzli | Luminescent probes[END_REF].

The excitation laser beam was generated by a 355 nm tripled output of a Continuum Nd-YAG laser, coupled to an optical parametric oscillator system (Panther II, Continuum, USA).

The wavelength was tuned to 394 nm, which corresponds to the 7 F0→ 5 L6 transition of Eu(III),

providing about 1 mJ of energy in a 5 ns pulse with a repetition rate of 10 Hz. After inner conversion, the 5 D1 level is transferring energy to the 5 D0 level and the transitions to the 7 Fj manifold can be observed. Additionally, HS are able to absorb the laser emission at 394 nm and part of the absorbed energy is transferred from the 3 ππ* triplet level to the central europium ion. The time-resolved luminescence signal is collected at 90° and focused into a Acton spectrometer (slit 1 mm) using either a 600 lines mm -1 or a 1800 lines mm -1 grating.

The signal is collected during a gate width W = 300 µs, after a gate delay D = 10 µs after excitation by the laser flash. Emission spectra were recorded using a CCD camera cooled at -15°C.

Results and Discussions

3.1. 5 D0→ 7 F0 and 5 D0→ 7 F1 transitions at 1800 lines/mm.

The 5 D0→ 7 F0 and 5 D0→ 7 F1 transitions of the Eu(III)-GHA complex is presented in Figure 2, together with Eu(III) complexed by Suwannee River FA (SRFA), and HA (SRHA) spectra previously obtained [START_REF] Reiller | Bi-exponential decay of Eu(III) complexed by Suwannee River humic substances: Spectroscopic evidence of different excited species[END_REF]. It can be noted that the 5 D0→ 7 F0 transition, which is not apparent in the Eu 3+ spectrum for magnetic and electric reasons, becomes important in Eu(III)-HS complexes due to loss of symmetry [START_REF] Judd | Optical absorption intensities of rare-earth ions[END_REF][START_REF] Ofelt | Intensities of crystal spectra of rare-earth ions[END_REF]. The transition is slightly red-shifted in the Eu(III)-GHA complex as compared to Eu(III)-SRFA and -SRHA complexes. The maximum and full-width at mid-height (w) of this transition are obtained from a fitting with a Lorentzian-Gaussian peak [START_REF] Mcnemar | The resolution of laser-induced europium(III) ion excitation-spectra through the use of the Marquardt nonlinear-regression method[END_REF],

Ii = Imax,i exp       -0.5       λi -λ0, i σLG, i ²       λi -λ0, i σLG, i ² + 1 (1)
where Imax,I is the maximum intensity of the i th peak, λ0,i is the wavelength of this maximum, and σLG,i is the 'standard deviation', which can be related to wi = 1.46 σLG,i. As shown by [START_REF] Reiller | Bi-exponential decay of Eu(III) complexed by Suwannee River humic substances: Spectroscopic evidence of different excited species[END_REF], the mono-component fit of the transition is satisfactory in consideration of the existing signal-to-noise ratio (Figure S1 of SI). The peak maximum gave the value of λEu-GHA = 579.26 nm with σLG = 0.75 nm. The red-shift of the peak maximum was significant in comparison to λEu-SRHA = 579.03 nm and λEu-SRHA = 578.98 nm [START_REF] Reiller | Bi-exponential decay of Eu(III) complexed by Suwannee River humic substances: Spectroscopic evidence of different excited species[END_REF]. Application of the relation between the number of coordinated ligands and wave number of the 7 F0→ 5 D0 transition υ0 (17263.5 cm -1 ), i.e., CN = 0.237 Δυ + 0.638 [START_REF] Choppin | Correlation between ligand coordination number and the shift of the 7 F0→ 5 D0 transition frequency in europium(III) complexes[END_REF], yielded a CN = 3.6 ± 0.7. This was slightly larger than for both Eu-SRFA and -SRHA complexes, i.e. CN = 2.0 and 1.6 (± 0.7), respectively [START_REF] Reiller | Bi-exponential decay of Eu(III) complexed by Suwannee River humic substances: Spectroscopic evidence of different excited species[END_REF]. This suggests that the chemical environment for Eu(III) in complexes with GHA may be somewhat different from those with Suwannee River extracts.

The σLG value, resulting in wGHA = 1.1 nm, compared well with those obtained with Suwannee River extracts, i.e., σLG,SRFA = 0.71 nm (wSRFA = 1.04 nm) and σLG,SRHA = 0.72 nm (wSRHA = 1.05 nm) [START_REF] Reiller | Bi-exponential decay of Eu(III) complexed by Suwannee River humic substances: Spectroscopic evidence of different excited species[END_REF] or other humic extracts [START_REF] Shin | A study of Eu(III)-humate complexation using Eu(III) luminescence spectroscopy[END_REF]. This further confirmed the large distribution of this transition in humic solution, as compared to simple molecules [START_REF] Horrocks | Lanthanide ion luminescent probes of the structure of biological macromolecules[END_REF][START_REF] Yeh | A spectroscopic investigation of temperature effects on solution complexation in the Eu 3+ -acetate system[END_REF] or virus binding sites [START_REF] Burroughs | Characterization of the 5-fold Ca 2+ binding-site of satellite tobacco necrosis virus using Eu 3+ luminescence spectroscopy -A marked size-selectivity among rare-earth ions[END_REF], and also revealed the general large distribution of chemical environments in HS.

The peak ratios 5 D0→ 7 F0/ 5 D0→ 7 F1 are also noticeably different between Eu(III)-SRFA and -SRHA complexes and Eu(III)-GHA complex. The value of the maximum intensity ratio of the peaks, I( 7 F0/ 7 F1) = 0.77, is lower than the one obtained for Eu(III)-SRFA and -SRHA, i.e., 0.87 and 0.99, respectively [START_REF] Reiller | Bi-exponential decay of Eu(III) complexed by Suwannee River humic substances: Spectroscopic evidence of different excited species[END_REF]. This suggests that the loss of symmetry in the case of Eu(III)-GHA is lesser than for Eu(III)-SRFA and -SRHA. However, the area ratio (trapezoid method) of Eu(III)-GHA, i.e. A( 7 F0/ 7 F1) = 0.134, is comparable to those of Eu(III)-SRFA and -SRHA that showed values of 0.139 and 0.150, respectively [START_REF] Reiller | Bi-exponential decay of Eu(III) complexed by Suwannee River humic substances: Spectroscopic evidence of different excited species[END_REF]. This illustrates that 5 D0→ 7 F0 FWMH is slightly greater than Eu(III)-SRHA and -SRHA, and that 5 D0→ 7 F1 transition of Eu(III)-GHA is visually narrower than the Eu(III)-SRHA transition. The magnetic dipole transition 5 D0→ 7 F1 is not supposed to be largely influenced by the subtle changes in symmetry [START_REF] Richardson | Terbium(III) and europium(III) ions as luminescent probes and stains for biomolecular systems[END_REF] that seem to exist among the different chemical environments provided by the different humic fractions. Only a close inspection at 1800 lines mm -1 permits to reveal these modifications directly in aqueous solution at ambient temperature.

3.2. 5 D0→ 7 F2 at 1800 lines/mm.

The 5 D0→ 7 F2 hypersensitive transition of the Eu(III)-GHA complex is shown in Figure 3, as compared to Eu(III)-HS complexes previously obtained with Suwannee River HA and FA [START_REF] Brevet | Spectral and temporal luminescent properties of Eu(III) in humic substances solutions from different origins[END_REF][START_REF] Reiller | Bi-exponential decay of Eu(III) complexed by Suwannee River humic substances: Spectroscopic evidence of different excited species[END_REF], Gorleben AH, and Leonardite HA [START_REF] Brevet | Spectral and temporal luminescent properties of Eu(III) in humic substances solutions from different origins[END_REF]. The modification of the transition's structure is even more spectacular, as compared to 5 D0→ 7 F1. The most outstanding difference among the different complexes was the shoulder ca.

612 nm that was more prominent for Eu(III)-SRFA than for any other extract. The HS from purely aquatic medium, i.e. SRHA (Figure 3), and Kleiner Kranichsee bog HA and FA (Germany, see Fig. 4 in [START_REF] Brevet | Spectral and temporal luminescent properties of Eu(III) in humic substances solutions from different origins[END_REF], exhibited mostly the same transition shape, with a lower 612 nm shoulder. Finally, Leonardite HA and Gorleben HA, together with commercial Aldrich HA, showed a narrow-shaped transition with the lowest intensity for 612 nm shoulders. The origin of these different 612 nm shoulders is not easily settled. One can be tempted to attribute this shoulder to the result of a Stark level. A Stark level can be identified in solid state or in solution at low temperature (4K). In aqueous solution we do not have the possibility to decompose the signal in overlapping Stark levels which are broadened with temperature. In addition, HS are composed of aggregates of molecules which offer distributions of functionality that lead to overlapping. Hence, the origin of the 612 nm shoulder is not easily settled and would need further developments that are out of the scope of this study.

Our results show that the 612 nm shoulder of GHA is comparable to other 'terrestrial' HA extracts. Nevertheless, the wavelength span greater than 615 nm is not comparable to Leonardite HA nor to Gorleben HA (see Brevet, et al., 2009, and reference therein). This part of the spectrum is positioned in between the two families of Eu(III)-HS complexes. One can interpret these differences in peak's shapes with the degree of humification. Since GHA is originated from a forest soil, its diagenesis is presumably more advanced than in Suwannee River, although it should be less advanced than in a lignite like Leonardite HA, or in an oxidized extract from a sedimentary deposit like Gorleben [START_REF] Buckau | Origin and mobility of humic colloids in the Gorleben aquifer system[END_REF].

Comparisons of the complexometric titrations.

The complexometric titration curves for the different extracts are reported in Figure 4.

These curves were obtained from the evolution of the 5 D0→ 7 F2/ 5 D0→ 7 F1 peak area ratios [START_REF] Dobbs | Characterization of metal binding sites in fulvic acids by lanthanide ion probe spectroscopy[END_REF][START_REF] Claret | Fractionation of Suwannee River fulvic acid and Aldrich humic acids on α-Al2O3: spectroscopic evidence[END_REF]. Three groups of fractions can be proposed. First, the fractions for which complexation properties were mostly identical to the original bulk GHA,

i.e. G2, G3, G4, G7, and G8. Second, G1, G5, and G7 showed only slightly lower complexation strengths, as compared to bulk GHA. Third, the most retarded G9 and G10 fractions appeared increasingly different from bulk GHA. It can then be seen that most of fractions could be fairly well compared with the bulk GHA, except for the 'smallest' or 'lightest' fractions, which were significantly different in terms of complexation strength.

The interactions can be quantified using a simplistic model [START_REF] Moulin | Complexation behaviour of humic substances towards actinides and lanthanides studied by time-resolved laser-induced spectrofluorimetry[END_REF], where humic substances are considered as mixture of discrete sites omitting the charges,

Eu + HA ⇄ EuHA β (L mg -1 ) = C(EuHA) C(Eu) C(HA) (2)
where c(HA) = Cc C(HA) -C(EuHA). The obtained parameters, namely the complexation constant β, and complexation capacity of the extracts Cc (mmol g -1 ), are reported in Figure 5. This model may appear too simplistic, but (i) the amount of the different fractions were not sufficient to perform reliable acid-base titrations [START_REF] Janot | Using spectrophotometric titrations to characterize humic acid reactivity at environmental concentration[END_REF], and (ii) the application of generic parameters, as those selected otherwise (Christl and Kretzschmar, 2001;Christl, et al., 2001;[START_REF] Milne | Generic NICA-Donnan model parameters for proton binding by humic substances[END_REF], may not be directly applied to our case. As by our approach, the magnitude of log β (complexation strength) was not the most influencing parameter, while it was instead the number of available complexation sites that explained most of modifications in GHA separated fractions.

A complexation constant with a mean value of log β = 6.7 ± 0.7 (2σ) can be proposed for the bulk GHA and the ten different size-fractions. This value is in agreement with other determination under comparable total europium concentration and pH [START_REF] Caceci | The interaction of humic acid with europium (III). Complexation strength as a function of load and pH[END_REF][START_REF] Bidoglio | Complexation of Eu and Tb with fulvic acids as studied by time-resolved laser-induced fluorescence[END_REF][START_REF] Yamamoto | Systematics of stability constants of fulvate complexes with rare earth ions[END_REF][START_REF] Kautenburger | Complexation studies with lanthanides and humic acid analyzed by ultrafiltration and capillary electrophoresis-inductively coupled plasma mass spectrometry[END_REF]. A mean value of 0.18 ± 0.05 mmol g -1 was calculated for the Cc complexing capacity of bulk GHA, and the G1 to G8 fractions. The Cc values of 0.11 ± 0.01 and 0.09 ± 0.02 mmol g -1 were obtained for the G9 and G10 fractions, respectively.

These findings may be related to the modification of HS during the size fractionation. The low molecular-weight molecules are assumed to be more retarded in the gel structure. When the low molecular weight molecules finally reaches the outlet, their complexation strength seems to be weaker than that of materials that has move more freely through the gel. This effect is nevertheless less intense in this case than observed in a previous work after the sorptive HS fractionation on α-alumina [START_REF] Claret | Fractionation of Suwannee River fulvic acid and Aldrich humic acids on α-Al2O3: spectroscopic evidence[END_REF].

0 1 2 3 4 5 6 7 8 9 G i n c h i G i n c h i 1 G i n c h i 2 G i n c h i 3 G i n c h i 4 G i n c h i 5 G i n c h i 6 G i n c h i 7 G i n c h i 8 G i n c h i 9 G i n c h i

Evolutions of spectra for the different fractions.

The chemical environments provided by the different size-fractions were also estimated from changes in luminescence spectra of different complexes. Spectra of solutions containing 10 µmolEu L -1 and 200 mgHA L -1 were acquired using the 600 lines mm -1 grating. The different spectra are shown in Figure S2 of SI, and the different A( 7 F2/ 7 F1) and A( 7 F0/ 7 F1) values are reported in Figure 6. The shape of both 5 D0→ 7 F1 and 5 D0→ 7 F0 transitions did not evolve. This was expected for the former magnetic transition, and suggests a small if not absent modification in the centro-symmetry of complexes from the latter transition. While the evolution of A( 7 F0/ 7 F1) was absent, A( 7 F2/ 7 F1) decreased from 3.6-4.0 to 3.2 with decreasing size of separated fractions (Figure 6). This effect cannot be attributed to a difference in the saturation of humic sites since it appeared to be already attained (Figure 4). Although log β and Cc values were almost identical for bulk GHA and the G1 to G8 fractions, the symmetry around Eu(III) was clearly and progressively evolving with changes in fractions' sizes. This could be interpreted as a gradual modification in the composition of the different fractions.

Smaller-sized HPLC-fractions are known to be increasingly enriched in carbohydrate-like molecules, whereas a progressive decrease in the amount of aromatic structure was found [START_REF] Piccolo | Reduced heterogeneity of a lignite humic acid by preparative HPSEC following interaction with an organic acid. Characterization of size-separates by Pyr-GC-MS and H-1-NMR spectroscopy[END_REF][START_REF] Conte | Spectroscopic and conformational properties of size-fractions acid separated from a lignite humic[END_REF]. Thus, the observed decrease of A( 7 F2/ 7 F1) may be due to both an increasing number of carbohydrate molecules, exhibiting much lower A( 7 F2/ 7 F1) value [START_REF] Plancque | On the use of spectroscopic techniques for interaction studies, part I: Complexation between europium and small organic ligands[END_REF], and a decreasing number of aromatic molecules [START_REF] Conte | Spectroscopic and conformational properties of size-fractions acid separated from a lignite humic[END_REF]. This varying molecular distribution during GHA fractionation modifies the indirect excitation of the central europium (III) atom via the 3 ππ * triplet level. We must also precise here that no assumption is made hitherto on the functionality of these aromatic molecules. The influence of aromatic moieties on Eu(III) luminescence do not mean that europium(III) is linked to phenolic moieties at pH 5, which is unlikely (ca. 1% of the total Eu-HS complex following [START_REF] Milne | Generic NICA-Donnan model parameters for metal-ion binding by humic substances[END_REF] for a generic HA), but that aromatic moieties contribute to the excitation. The energy transfer could be due to a resonance phenomenon or due to an exchange mechanism, i.e., Förster resonance energy transfer (FRET) or Dexter energy transfer, respectively [START_REF] Turro | Modern molecular photochemistry[END_REF]. FRET can be operative over distances as long as 100 nm depending on the particular donor-acceptor pair. The energy transfer according to

Dexter requires an overlap of the electron orbital of the donor and the acceptor. As the distance between the ligands and Eu(III) is relatively short, the Dexter energy transfer could be favoured. So far, in the case of HA, the link between the binding sites and the chromophores engaged in the energy transfer has not been demonstrated unequivocally.

Thus, the resonance phenomenon cannot be ruled out.

The evolution of the 5 D0→ 7 F2 transition was further checked using a grating of 1800 lines mm -1 . The main difference among fractions was the 612 nm shoulder, as already discussed for G5 and G9, although no systematic evolution could be revealed (data not shown). 3.5. Luminescence decay-times analyses of GHA.

The luminescence decays of the Eu(III)-HS complexes are generally described by a bi-exponential function, which is described for our fully integrative system by

F =  i=1 n    D D+W F o i exp       - t τi dt =  i=1 n F o i i exp       - D i       1 -exp       - W i (3) F = F°           x1 τ1 exp       - D τ1       1 -exp       - W τ1 + (1 -x1) τ2 exp       - D τ2       1 -exp       - W τ2
where F is the luminescence signal, F°I, and τi are the initial luminescence and the decay time of component i, respectively, x1 the proportion of the first decay in the global signal, D the delay after the laser excitation, and W the gate width. The possible influence of photochemical reactions of humic substances on the complexation strength can be neglected out under our experimental conditions as seen previously [START_REF] Monsallier | Kinetic investigation of Eu(III)-humate interactions by ion exchange resins[END_REF][START_REF] Reiller | Bi-exponential decay of Eu(III) complexed by Suwannee River humic substances: Spectroscopic evidence of different excited species[END_REF].

The decay time evolution was obtained from the 5 D0→ 7 F2 peak area for different Eu(III)-GHA complexes at delays between 10 and 505 µs (Figure S3 and Table S1 of SI). The inevitable correlation between the parameters is shown in Table S1 of the SI. This showed, as for other extracts [START_REF] Chung | Comparison of synchronous and laserinduced fluorescence spectroscopy applied to the Eu(III)-fulvate complexation[END_REF][START_REF] Brevet | Spectral and temporal luminescent properties of Eu(III) in humic substances solutions from different origins[END_REF][START_REF] Marang | Spectroscopic characterization of the competitive binding of Eu(III), Ca(II), and Cu(II) to a sedimentary originated humic acid[END_REF][START_REF] Reiller | Bi-exponential decay of Eu(III) complexed by Suwannee River humic substances: Spectroscopic evidence of different excited species[END_REF], a bi-exponential decay with τ1 = 44 ± 2 µs and τ2 = 184 ± 3 µs. The biexponential decay can only happens if two deexcitation processes from two different excited states are occurring. In our case the deexcitation states seem to be very close in structure [START_REF] Reiller | Bi-exponential decay of Eu(III) complexed by Suwannee River humic substances: Spectroscopic evidence of different excited species[END_REF], and τ1, seemed to be characteristic of a fast reaction in the excited state for Ln(III) and An(III) [START_REF] Billard | Equilibrium constants in aqueous lanthanide and actinide chemistry from time-resolved fluorescence spectroscopy: The role of ground and excited state reactions[END_REF][START_REF] Freyer | Formation of Cm humate complexes in aqueous solution at pHc 3 to 5.5: The role of fast interchange[END_REF][START_REF] Reiller | Bi-exponential decay of Eu(III) complexed by Suwannee River humic substances: Spectroscopic evidence of different excited species[END_REF]. Hence this 'species' only occurs during the excitation and is not involved in the environmental reaction without direct excitation. S1 of SI.

3.6. Link with the structure of HPSEC-fractions. [START_REF] Conte | Advanced CPMAS-13 C NMR techniques for molecular characterization of size-separated fractions from a soil humic acid[END_REF]2007) have shown the structural modification of size-fractions during HPSEC separation for two different humic acids. Enrichment in aliphatic carbon and a decrease in aromatic carbon were assessed by 13 C-CPMAS-NMR spectra as humic sizefractions were progressively eluted in HPSEC. In our study, we noted that the evolution of A( 7 F2/ 7 F1) followed closely the expected change in aromatic content during the HPSEC separation of size-fractions [START_REF] Conte | Advanced CPMAS-13 C NMR techniques for molecular characterization of size-separated fractions from a soil humic acid[END_REF]. In fact, we found an initial decrease for the first size-fraction relative to the bulk sample, and then a peak followed by a monotonous decrease. Comparison with HPSEC separation from a lignite humic acid is more difficult since the smallest-sized fractions were not collected and analyzed [START_REF] Conte | Spectroscopic and conformational properties of size-fractions acid separated from a lignite humic[END_REF]. [START_REF] Conte | Advanced CPMAS-13 C NMR techniques for molecular characterization of size-separated fractions from a soil humic acid[END_REF]2007) also found that the smaller molecular-sized molecules were chemically alike and different from previously eluted fractions [START_REF] Conte | Advanced CPMAS-13 C NMR techniques for molecular characterization of size-separated fractions from a soil humic acid[END_REF]. Our complexation results indicate that fractions G9 and G10 behaved differently from the rest of size-fractions. Our findings also relate the decrease in number of complexation sites to the decrease of carboxylic carbons as shown by [START_REF] Conte | Advanced CPMAS-13 C NMR techniques for molecular characterization of size-separated fractions from a soil humic acid[END_REF], even though the evolution is not as spectacular. We noted that the lower-size fractions used by [START_REF] Conte | Advanced CPMAS-13 C NMR techniques for molecular characterization of size-separated fractions from a soil humic acid[END_REF] were also poorer in aromatic molecules than other fractions.

Finally, the G9 and G10 fractions were the least abundant among the GHA fractions, i.e., 4% and 7% in weight, respectively. Hence, from a complexation point of view, 89% of fractions behaved similarly to the bulk GHA, despite the evident modifications found for the Eu(III) complexation environment.

Conclusion

Structures of the Eu(III) complexes of Ginchi HA and of its HPSEC-fractions, probed using TRLS, are slightly but significantly different. Nevertheless, these differences are not sufficient to changes the decay time of the Eu(III) luminescence within the complexes. Hence, even if the HPSEC processes induce changes within the structure of the fraction as compared to the bulk HA, ca. 90% of the Eu(III) complexes with the HA fractions are very much similar and could be considered as equivalent to bulk HA in a first approximation. Only ca. 10% of HA fractions can be considered to present a somewhat stronger affinity for Eu(III). They represent the lighter fractions which are retarded in the structure of the immobile phase of HPSEC.

HPSEC can be considered as a forced migration process which minimizes chemical interactions between the mobile (HS) and the immobile phase, i.e., mostly physical interactions are in stakes in a dynamic system. Hence, one can also consider that even if the HS fractions submitted to this physical 'sorting out' are different, they share mostly the same type of interaction strength with the bulk HA. These subtle structural modifications are less important than those evidenced on oxides [START_REF] Hur | Molecular weight fractionation of humic substances by adsorption onto minerals[END_REF][START_REF] Reiller | Sorption of Aldrich humic acid onto hematite: Insights into fractionation phenomena by electrospray ionization with quadrupole time-offlight mass spectrometry[END_REF][START_REF] Claret | Fractionation of Suwannee River fulvic acid and Aldrich humic acids on α-Al2O3: spectroscopic evidence[END_REF]. Particularly, the modification of the Eu(III) environment is clearly less important after a 'physical' sorptive fractionation as compared to a 'real' or 'chemical' sorptive fractionation evidenced by [START_REF] Claret | Fractionation of Suwannee River fulvic acid and Aldrich humic acids on α-Al2O3: spectroscopic evidence[END_REF].

  loop. The HPSEC system consisted of a Gilson autosampler model 231, a Gilson 305 pump, a preparative Biosep SEC-S-2000 (600 mm, 21.2 mm id) column, preceded by a Biosep SEC-S-2000 guard column (78.0 mm, 21.2 mm id) both from Phenomenex (USA), a Gilson 116

  chromatographic runs. In preparative HPSEC mode, adsorption is concomitant with sizeexclusion separation to a larger extent (mL vs. µL of loading volume) than in analytical HPSEC, thereby showing a longer tailing chromatogram.

Figure 1 .

 1 Figure 1. HPSEC profile of the Ginchi humic acid. The vertical bars mark the time intervals

Figure 2 .

 2 Figure 2. Eu(II) 5 D0 → 7 F0 and 5 D0 → 7 F0 transitions at C(Eu) = 10 -5 mol L -1 , C(HS) = 200 mg L -1 ,

Figure 3 .

 3 Figure 3. Eu(III) 5 D0 → 7 F2 transitions at C(Eu) = 10 -5 mol L -1 , C(HS) = 200 mg L -1 , pH 5, I = 0.1

Figure 4 .

 4 Figure 4. Comparison of the complexometric titration curves of C(Eu) = 10 -5 mol L -1 with

Figure 5 .

 5 Figure 5. Comparison of complexation parameters for the different HPSEC fraction of GHA

Figure 6 .

 6 Figure 6. Evolution of A( 7 F2/ 7 F1) (up) and A( 7 F0/ 7 F1) (down) of Eu(III) complexed by the

Figure 7 .

 7 Figure 7. Evolution of fluorescence decay times of Eu(III) complexed by the different size-

Acknowledgment.

This work was financed through the MRISQ (DEN/DISN/AvC) and RBPCH (DEN/DISN/RSTB) projects of CEA.

Conversely, τ2 could be interpreted as a function of Eu(III) chemical environment in humiccomplexes. The differences in spectra at low and high delay were evidenced in [START_REF] Reiller | Bi-exponential decay of Eu(III) complexed by Suwannee River humic substances: Spectroscopic evidence of different excited species[END_REF]. The application of the relation between τ2 and the number of water molecules in the first hydration sphere proposed by [START_REF] Kimura | Comparative study on the hydration states of Cm(III) and Eu(III) in solution and in cation exchange resin[END_REF], yielded a number of 5 ± 0.5 remaining water molecules out of 9. This is in agreement with the coordination number of 3.6, as determined from λmax( 7 F0). It is to be pointed out that the particular dependence of Eu(III)-HS complexes on the number of remaining water molecules in the first coordination sphere, has not been verified up to now. Moreover, the possibility of internal quenching leading to a different expression is possible [START_REF] Horrocks | Lanthanide ion probes of structure in biology. Laserinduced luminescence decay constants provide a direct measure of the number of metalcoordinated water-molecules[END_REF][START_REF] Supkowski | On the determination of the number of water molecules, q, coordinated to europium(III) ions in solution from luminescence decay lifetimes[END_REF][START_REF] Kuke | Intramolecular deactivation processes in complexes of salicylic acid or glycolic acid with Eu(III)[END_REF].

The fitting of the luminescence decays obtained for the Eu(III) complexes are reported in Table S1 of the SI. The evolution of decay times with different GHA size-fractions are reported on Figure 7. In line with previous studies in aqueous solution [START_REF] Brevet | Spectral and temporal luminescent properties of Eu(III) in humic substances solutions from different origins[END_REF][START_REF] Marang | Spectroscopic characterization of the competitive binding of Eu(III), Ca(II), and Cu(II) to a sedimentary originated humic acid[END_REF][START_REF] Reiller | Bi-exponential decay of Eu(III) complexed by Suwannee River humic substances: Spectroscopic evidence of different excited species[END_REF], τ1 did not show a major trend. We also verified this lack of dependence in Eu-HA sorbed on α-alumina (Janot, et al., submitted). If this fast decay time was due to a fast exchange, then its lack of modification with origin or mode of preparation of humic size-fraction would be explainable. The very slight decrease observed for τ2 could eventually be reconciled with a difference of one H2O molecule in the first hydration sphere.

However, the uncertainty of this determination (σ = 0.5) prevents the reliability of the trend. Hence, there is not a major difference between all these samples.

Supporting Information Available.

The HPSEC profile of the Ginchi HA; the proportion in weight and cumulated percentage of the ten HPSEC size-fractions; the 5 D0→ 7 F0 transition, fit and residuals, with one component for Eu(III)-Ginchi AH complexes; the normalized spectra of Eu(III) complexed by the different size-fractions; The time-resolved luminescence spectra as a function of delay time of Eu-GHA complex; the fitting and correlation matrices of the luminescence decays of Eu(III) complexed by the GHA bulk and size-fractions.


Eric Aristidi
email: eric.aristidi@oca.eu

Eric Cottalorda

Marcel Carbillet

Lyu Abe

Karim Makki

Jean-Pierre Rivet

David Vernet

Philippe Bendjoya

The power spectrum extended technique applied to images of binary stars in the infrared

Keywords: Lucky imaging, speckle, Post adaptive-optics imaging

We recently proposed a new lucky imaging technique, the Power Spectrum Extended (PSE), adapted for image reconstruction of short-exposure astronomical images in case of weak turbulence or partial adaptive optics correction. In this communication we show applications of this technique to observations of about 30 binary stars in H band with the 1m telescope of the Calern C2PU observatory. We show some images reconstructed at the diffraction limit of the telescope and provide measurements of relative astrometry and photometry of observed couples.

INTRODUCTION

The speckle interferometry technique introduced by Labeyrie 1 is well adapted to the measurement of binary stars relative astrometry (and sometimes differential photometry), and active groups are still using it routinely to monitor double stars motions and refine orbits. 2,3 Its major drawback is that it cannot provide true images, and several improvements were proposed, generally based on the computation of higher-order statistical quantities of the speckle patterns. The bispectral analysis [START_REF] Weigelt | Triple-Correlation Imaging in Optical Astronomy[END_REF] is generally recognized as state-of-the art of the speckle imaging, but it is a somewhat heavy process that requires a lot of CPU time and memory.

The Lucky imaging (LI) 5 is a simple alternative for modest telescopes or in case of weak turbulence: Huffnagel 6 calculated that LI techniques give good performances when D/r 0 7, D being the telescope diameter and r 0 the Fried parameter. LI processing have be combined successively with Adaptive Optics, [START_REF] Velasco | SPIE Conference Series[END_REF] making is efficient with larger telescopes or poorer seeing conditions. The LI technique relies on two essential points: the image selection and the alignment process. A review was made by Garrel et al. [START_REF] Garrel | [END_REF] who proposed a selection criterion based on the Fourier transform of images; their technique, referred to as Fourier-Lucky (FL) imaging hereafter, is today one of the most efficient LI algorithms.

We recently began to work on a new technique, 9 the "Power Spectrum-Extented", which is a combination of a LI and speckle interferometry. The algorithm is described in details in this conference by Cottalorda et al. [START_REF] Cottalorda | SPIE proc[END_REF] It was developed within the framework of the Calern Imaging Adaptive Optics (CIAO) project. [START_REF] Carbillet | SPIE proc[END_REF] This AO bench is developed at the Epsilon telescope (diameter 1.04m) of the Centre Pédagogique Planètes et Univers (C2PU) facility 12 located at the Calern observing site (South of France), and is designed to operate in the visible and the near-infrared (H-band). Several test campaigns were made during the conception of this instrument; one of them was dedicated to binary stars observation in the infrared: it is the purpose of this presentation.

OBSERVATIONS AND DATA PROCESSING

Observations were carried out with the Epsilon telescope. It was used in Cassegrain configuration (focal ratio of 12.5), equipped with a simple optical bench hosting an infrared H filter (central wavelength λ = 1650 nm, bandpass δλ = 350 nm), a magnification lens and a near-infrared camera Ninox SWIR 640 from Raptor Photonics. 13 A photo of the bench is shown in Fig. 1 The main characteristics of the camera are displayed in Table 1. It proposes two gain modes (low-gain and high-gain), all our observations were made with the high-gain mode. The camera allows short-exposure time down to 10ms in high gain mode, and is well adapted to speckle or lucky imaging applications. It is cooled down to -20 • C by a Peltier module, even more with the circulation of a cooling liquid (that we did not use for the present observations).

However it has a somewhat strong noise which limit performances. Hence the limiting magnitude of our instrumentation is about H = 6 with an exposure time of 10 ms. The angular resolution of the telescope at λ = 1.6µm is 0.3 arcsec. A few hundreds of bright double stars with an angular separation ρ > 0.3 and a magnitude H < 6 were selected for observations from the Washington Double Star (WDS) catalog. [START_REF] Mason | Washington double star catalog[END_REF] Observations were carried out in June 2016. About 60 binary stars were observed during 8 nights, leading to 27 positive detections of companions which are presented in Table 3. Negative detections can be due to a too large magnitude difference, or bad seeing conditions.

The median seeing at Calern in the visible (λ = 500nm) is 1.1 , 15 corresponding to 0.9 in the H band (λ = 1650nm). Indeed during our observing run, the seeing monitor recorded values between 0.6 and 3 at λ = 500nm (0.5 and 2.4 at λ = 1650nm). This corresponds to weak D/r 0 ratios (between 1.5 and 7) which are totally adapted to PSE applications. In particular the night of June 23rd to 24th, conditions were exceptional with a seeing (at λ = 500nm) between 0.6 and 0.8 arcsec (D/r 0 between 1.7 and 1.9 in H band) during the full night, and we obtained almost diffraction-limited images.

The coherence time was very low during the first nights (around 2ms at λ = 1650nm), and increased up to 80ms (at λ = 1650nm) during the night of June 23rd to 24th (indeed the coherence time is proportional to the Fried parameter [START_REF] Aristidi | [END_REF] and good seeing conditions correspond generally to slow turbulence). That night, we could increase the exposure time to 100ms to gain sensitivity.

Data processing

For each object a cube of several thousands of short-exposure images was recorded. After subtraction of the mean dark current (mandatory for this camera), image cubes were processed using the PSE algorithm. [START_REF] Cottalorda | SPIE proc[END_REF] We recall here the main steps: Classify image quality according to a criterion based on the estimation of an instantaneous Fried parameter r 0 , and select a percentage of images for processing, Align all selected images to the first one (or to the best one). This alignment is performed in the Fourier plane, making use of the phase of the cross-spectrum between images, Sum centered images, and extract the phase of the Fourier transform of this sum, Calculate the average power spectrum of images, and extract the modulus of the Fourier transform of the object (following the well known speckle interferometry technique introduced by Labeyrie 1 ), From the modulus and the phase of the Fourier transform calculated above, derive the final image.

This technique appeared to be fast and efficient on our images. Some comparisons with other data processing methods (shift-and-add (SA), FL, speckle) are presented in Sect. 3. Accurate relative astrometry and photometry could be extracted from reconstructed images of the binary stars, even in the worst seeing conditions. .

Scale calibration

The scale and position angle calibration was done by taking a sequence of short-exposure images of the bright and large double star ζUma (STF 1744AB). This slow motion couple does not have an orbit yet: its position angle has moved by only 10 • since its first measurement in 1755, [START_REF] Mason | Washington double star catalog[END_REF] and it is therefore a good object for calibration. A recent measurement 16 (epoch 2015.351) gave a separation of ρ = 14.45 ± 0.05 and a position angle θ = 153.0 ± 0.3 • .

To calculate the pixel scale, we computed the average autocorrelation of images of the binary (this gives a better accuracy than a classical shift-and-add algorithm). This kind of processing is well known in speckle interferometry to measure double-star separation. This function exhibits 3 peaks whose distance is the separation of the binary stars in pixels. This gave a pixel scale of ξ = 0.0782 ± 0.0003 arcsec. Uncertainties on the pixel scale and the position angle of this calibration star are taken in account in the error bars on ρ and θ presented in the table 3.

RESULTS

Fig. 2 shows examples of reconstructed images by the PSE algorithm, for 3 binary stars of different separations. Images appear to be diffraction-limited and exhibit Airy rings around stars. In particular, the pair αCom is well resolved, while it is close to the diffraction limit of the telescope (ρ = 0.36 ). A comparison between different data processing techniques is shown in Fig. 3. We took the example of the double star ADS 11871 which appear to be a good test case since it has a large magnitude difference (∆m = 1.8) and it was observed under bad seeing conditions ( = 2.5 at λ = 500nm corresponding to 2 at λ = 1650nm). The long-exposure image (b) was obtained by adding the whole set of 13 000 individual frames. The SA image (c) was computed by centering all frames on the brightest pixel, them adding the whole image set. It is the fastest and the easyest of the algorithms that we tested, and it clearly shows the binary. However the image is embedded is a diffuse halo of the size of the seeing disc. The FL image (d) was made using Fourier-lucky algorithm. [START_REF] Garrel | [END_REF] The centering is made, as for the SA, on the brightest pixel. The image shown was calculated from a selection of the 50% best frames (according to the criterion described in Sect.2). As expected, there is a substantial improvement over the SA image. The PSE image (e) was also computed on a selection of the best 50%, and its quality is close to the FL image, at one can see on profiles displayed on the plot (f). We could estimate the Strehl ratio S t of reconstructed images, using the formula by Tokovinin 17 after having removed the companion star from images. We found Image (c), SA : S t = 8% Image (d), FL : S t = 20% Image (e), PSE :

S t = 24%
showing a slight advantage of the PSE method on the FL one, for this particular example. Table 2 gives the average error bars on mesurements of the separation ρ and the position angle θ for the four processing methods (SA, FL, speckle and PSE). Speckle remains the most accurate technique, but we can see that PSE and FL give comparable uncertainties and are far better than SA. Same for the average Strehl ratios dispayed for the 3 techniques SA, FL, PSE in Table 2 The "ghost" effect

The "ghost" effect is illustrated by the Fig. 4. It happens on images of binary stars having a small magnitude difference. In that case, the classical centroid calculations used in the methods of SA and LI are not efficient.

The brightest pixel or speckle can correspond to one star or the other, depending on atmospheric fluctuations. Reconstructed images using this alignment show a ghost image at a position symmetric to the actual companion (Fig. 4, right), resulting into a quadrant ambiguity. To solve this problem, we align images by computing crossspectra between consecutive images, as explained in Sect. 2. The cross-spectrum is a function which takes into account the complete structure of the images (not only the brightest speckle), and gives the global shift between them, providing that they are similar, i.e. that tip-tilt is the dominant effect. It works well if the time lag between images is short. The example on the double star i Boo (Fig. 4, left) shows no ghost effect, and allows, in particular, to compute the magnitude difference between the two stars. In our target list, about 30% of objects presented a ghost effect on the lucky-imaging image: all could be successfully reconstructed by the PSE technique.

Measurements on binary stars

The pair separations ρ and their position angle θ were measured on reconstructed images using the software GdPisco by J.L. Prieur (Observatoire Midi-Pyrénées). 18 This software is used routinely to process speckle images from the PISCO instrument 3 at the C2PU telescope. The magnitude difference was estimated by performing a fit of two Airy discs on the reconstructed images. Alternate values for the separation and position angle could be derived from this fit, and compared to GdPisco estimates. The agreement was found within the error bars.

Observations are presented in Table 3. For each couple, we report the WDS number, [START_REF] Mason | Washington double star catalog[END_REF] the discoverer designation and components involved (Col. 2), the star name (Col. 3), the epoch of observation in Besselian years (Col. 4), the seeing measured by the CATS station [START_REF] Aristidi | [END_REF] given at λ = 500nm (Col. 5), the exposure time in ms (Col. 6), the total number of frames (Col. 7), the percentage of frames selected by the algorithm (Col. 8), the angular separation in arcsec with its uncertainty (Col. 9), the position angle (Col. 10) and the H magnitude difference (Col. 11). If an orbit is available, the orbit reference (same designation as in the WDS orbit catalog [START_REF] Matson | Sixth catalog of orbits of visual binary stars[END_REF] ) is given in Col. 12 as well as the computed separation and position angle (Cols. 13 and 14).

The magnitude difference could not be determined in a reliable way for the pairs WDS 16326+4007 and WDS 16581+1509, due to a strong astigmatism on images. For the case of WDS 17335+5734 the pair is too 3 with respect to available orbits. Error bars correspond to uncertainties given in cols 9-10 of Table 3. The O-C for WDS 17335+5734, showing a large discrepancy for the position angle (∆θ = -30.5 • ) is not drawn on the graph.

close to be seen on the reconstructed image. The separation and the position angle were derived from the modulus of the Fourier transform of the image (by adjusting a cosine function), and the absolute quadrant could not be determined. They are flagged with a asterisk in the Table 3 When an orbit is available (cols 12-14 of Table 3), the Observed minus Computed (O-C) residuals ∆ρ and ∆θ could be calculated. They are plotted in Fig. 5. The median values computed with the residuals are ∆ρ median = -0.02 and ∆θ median = 0.36 • . The small values obtained for these offsets provide a validation of our calibration procedure (see Sect. 2).

Large residuals have been found for the star WDS 17335+5734: ∆ρ = 0.14 , ∆θ = -30.5 • . The separation and position angle of this star were measured in the Fourier plane, as its separation is lower than the diffraction limit of the telescope. Looking at our data, we think that the orbit is not reliable, since it predicts a separation of 0.089 which is clearly too small (it would have been undetectable with our instrumentation).

CONCLUSION

In this paper, we have presented measurements of close binary stars in the infrared (H-band) obtained with a 1m telescope with a commercial near-IR camera. This was a first test run which allowed to check the limits of our experiment. The limiting magnitude will definitely increase when the CIAO bench will be fully functional, allowing larger exposure times. Observations of close binaries in the H-Band are difficult to find in the literature; in our target list of Table 3, we found only two couples (HU 644AB and AC 15AB) whose relative photometry has been measured at this wavelength. [START_REF] Henry | [END_REF] Though such observations are of importance for cool stars, especially close red dwarves with short orbital periods. This presentation was also the first astronomical applications of our new lucky imaging technique, the PSE, which seem promising as it gives results comparable (and sometimes slightly better) than the Garrel FL algorithm, considered as one of the most efficient LI technique. Indeed the PSE has the advantage to be very fast: on a set of 6000 frames of size 128×128, our Python-based PSE program takes ∼50s (with a selection rate of 20%) while 470s are needed for the FL algorithm. This is a gain of CPU time of a factor ∼9, interesting for real-time processing during observations.

Figure 1 .

 1 Figure 1. Left: Optical bench at the Cassegrain focus. Right: Quantum efficiency curve of the Ninox 640 sensor (from Raptor website).

Figure 2 .

 2 Figure 2. Exemple of images reconstructed by the PSE algorithm, for the 3 stars 16Vul (ρ = 0.75 ), ζHer (ρ = 1.27 ) and αCom (ρ = 0.36 ) The yellow circle at the bottom right corner represents the Airy disc size (radius = 0.3 ).

Figure 3 .

 3 Figure 3. Data processing on the star ADS 11871. (a) example of a raw image. (b) long-exposure image. (c) Reconstructed image by SA. (d) Reconstructed image by FL with 50% selection. (e) Reconstructed image by PSE processing (50% selection). (f) Intensity plot of reconstructed images along the line joining the stars.

Figure 4 .

 4 Figure 4. Reconstruction of the binary star i Boo, having a small magnitude difference (∆m = 0.09). Left: PSE reconstruction. Right: FL reconstruction. The latter shows a ghost companion due to misalignment of images in the shift-and-add process.

Figure 5 .

 5 Figure 5. Observed minus Computed (O-C) residuals of measurements of Table3with respect to available orbits. Error bars correspond to uncertainties given in cols 9-10 of Table3. The O-C for WDS 17335+5734, showing a large discrepancy for the position angle (∆θ = -30.5 • ) is not drawn on the graph.

Table 1 .

 1 Characteristics of the Ninox 640 camera (High-Gain mode)

	Nb of	Pixel	Dynamic	Expos.	Frame	Bandwidth RON	Dark	Cooling
	pixels	size	range	time	rate	(µm)		current	temp.
	640×512 15×15 µm	14 bits	10ms-27s ≤120 Hz	0.4-1.7	37e

-1500 e -/s -20 • C

Table 2 .

 2 . Average uncertainty on the separation (δρ), position angle (δθ) measurements ans Strehl ratios for the different processing techniques. No Strehl ratio is given for speckle since it is a measurement on autocorrelations.

		δρ (arcsec)			δθ ( • )	Strehl ratio (%)
	SA	FL Speckle PSE SA	FL Speckle PSE SA FL PSE
	0.16 0.13	0.11	0.14 0.79 0.54	0.50	0.52 15 41	36

Table 3 .

 3 Table of measurements.

Acknowledgments

We wish to thank the CIAO group, in particular F.X. Schmider, F. Vakili and F. Martinache for the time and energy they spent to write funding applications to buy the camera. This research has made use of the Washington Double Star Catalog maintained at the U.S. Naval Observatory, and of the SIMBAD data-base, operated at CDS,