Shen Qian
email: shenqian@kanagawa-u.ac.jp

Jiguang He

Xiaobo Zhou

Takamasa Imai

Tad Matsumoto

Outage Analysis for Correlated Sources Coding over NOMA in Shadowed κ-µ Fading

Keywords: NOMA, shadowed κ-µ fading, Slepian-Wolf theorem, outage probability, -outage achievable rate

We consider correlated sources coding over a up-link non-orthogonal multiple access shadowed κ-µ fading channel. The sufficient condition for lossless coding is determined by the intersection of the Slepian-Wolf region and multiple access channel region, assuming source-channel separation holds. The exact expression for the outage probability upper bound is derived by dividing the sufficient conditions into three cases. The accuracy of the analytical results is verified by the Monte-Carlo simulations. The analytical results indicate that more than 2 nd order diversity gain can be achieved with a larger ratio of line-of-sight dominant component in single cluster or multiple clusters with non-line-of-sight component. It is also found that the shadowed κ-µ fading well represents one-sided Gaussian, Rayleigh, Rician, and Nakagami-m fading in calculating the outage probability. Furthermore, the -outage achievable rate is analyzed, which is found to be larger with higher source correlation and/or average signal-to-noise ratio.

I. INTRODUCTION

Non-orthogonal multiple access (NOMA) has been considered as a promising multiple access technique for the fifth generation (5G) and beyond 5G mobile networks. The basic concept of NOMA is that multiple users are served with the same wireless resource, such as a time slot, frequency, or an subcarrier [START_REF] Wan | Non-orthogonal multiple access for cooperative communications: Challenges, opportunities, and trends[END_REF]. Compared with the conventional orthogonal multiple access schemes which serves each user with orthogonal resource, NOMA can achieve higher spectral efficiency, larger system throughput, and lower transmission latency [START_REF] Saito | System-level performance evaluation of downlink non-orthogonal multiple access (NOMA)[END_REF]. The connection capacity of Internet of Things (IoT) systems can be increased by NOMA compared to its orthogonal counterpart [START_REF] Mostafa | Connectivity maximization for narrowband iot systems with NOMA[END_REF].

Most literature on NOMA assume the information sources to be transmitted are independent. However, in the emerging IoT applications, such as surveillance networks, smart agriculture or vehicle-to-everything (V2X) communications, the data sequences, obtained by the sensor nodes, are usually correlated with each other. For example, in smart agriculture scenario, the densely deployed sensors usually obtain the similar information since the sensor nodes are close to each other geographically. Recently, work about correlated source transmission with NOMA has attracted considerable attentions. The uplink NOMA with correlated sources can be regarded as many-to-one or reach back channel problem [START_REF] Murugan | Correlated sources over wireless channels: cooperative source-channel coding[END_REF] since the correlated observations are available at the transmit nodes. The achievable rate region was analyzed in a cellular downlink NOMA transmission system with correlated information sources [START_REF] Chung | Noma for correlated information sources in 5g systems[END_REF].

In particular, the coding schemes were proposed based on Slepian-Wolf or Wyner-Ziv coding, depending on whether a lossy or lossless recovery of these information sources is needed at the destination [START_REF] Yassaee | Slepian-wolf coding over cooperative relay networks[END_REF]. Ref. [START_REF] Khan Beigi | Ultra-reliable energyefficient cooperative scheme in asynchronous NOMA with correlated sources[END_REF] employed a cooperative uplink NOMA to take advantage of the source/channel correlations by using an iterative joint detecting and decoding strategy at the receiver. The correlation among sources can be considered as a metric for user pairing and the spectraland power-efficiency can be improved. A minimum mean square error (MMSE) detector was proposed in [START_REF] Khan Beigi | NOMA-based network coding in IoT networks with correlated sources[END_REF] for iterative joint detecting and decoding the correlated sources transmitted over asynchronous NOMA channels. Cheng et al. derived the outage probability of a lossy transmission of binary symmetric source with downlink NOMA and successive refinement [START_REF] Cheng | Down-link NOMA with successive refinement for binary symmetric source transmission[END_REF]. Zhou et al. [START_REF] Zhou | Outage probability of correlated binary source transmission over fading multiple access channels[END_REF] and He et al. [START_REF] He | Performance analysis of lossy decode-and-forward for non-orthogonal MARCs[END_REF] calculated the outage probability upper bound for a two-correlated-source-coding system and for the two hop non-orthogonal multiple access relay channel, respectively, based on the derived sufficient condition. Furthermore, Song et al. [START_REF] Song | Outage probability of one-source-with-one-helper sensor systems in block rayleigh fading multiple access channels[END_REF] discussed the case where only one of the sources needs to be recovered at the destination, whereas the other source serves as a helper.

The previous work only assumed that the channels experienced non-line-of-sight fading and ignored the shadowing. However, in the IoT applications, wireless devices may be blocked by obstacles such as buildings or vegetation. Therefore, line-of-sight (LOS) and non-LOS components co-exist, in general, and the links will be heavily susceptible to possible deep fading caused by shadowing [START_REF] Zeng | Enabling ultrareliable and low-latency communications under shadow fading by massive MU-MIMO[END_REF]. To the best of the authors' knowledge, the outage analysis of correlated source transmission in shadowed fading with the coexistence of both LOS and non-LOS components has not yet been reported in the literature.

In this work, we derived the outage probability upper bound of correlated source coding in uplink NOMA with shadowed κ-µ fading. The shadowed κ-µ fading [START_REF] Paris | Statistical characterization of κ-µ shadowed fading[END_REF] is an universal distribution model which contains other well-know fading

I TYPICAL FADING MODELS RELATED TO THE κ-µ SHADOWED FADING Fading Model m κ µ Rayleigh - 0 1 Rician ∞ K 1 Nakagami-m - 0 m N κ-µ ∞ κ µ
One-sided Gaussian ∞ 0 0.5 sources. A fixed bit flipping probability p is set as a parameter in this work.

B. Channel Model

We assume that the sources-to-destination links suffer from independent and identically distributed (i.i.d) shadowed κµ fading which considers the dominant component of the LOS path affected by the shadowing and subject to random fluctuations.

The probability density function of instantaneous signal-tonoise ratio (SNR) γ of the shadowed κ-µ fading is given by [START_REF] Paris | Statistical characterization of κ-µ shadowed fading[END_REF]

f κµ (γ) = µ µ m m (1 + κ) µ Γ(µ)γ(µκ + m) m γ γ µ-1 exp - µ(1 + κ)γ γ 1 F 1 m, µ; µ 2 κ(1 + κ)γ (µκ + µ)γ , (1)
where 1 F 1 (•) and Γ(•) are the confluent hypergeometric function and Gamma function, respectively. γ denotes the average SNR. Here, the parameter m (m ≥ 0) in ( 1) and ( 2) denotes the shadowing severity in the dominant components, where m = 0 indicates complete shadowing and m → ∞ corresponds to no shadowing in the resultant dominant component. κ (κ ≥ 0) denotes the ratio between the power of the dominant component to the total power of the scattered components and µ is the number of multipath clusters. The cumulative density function of the shadowed κ-µ fading is given by [START_REF] Paris | Statistical characterization of κ-µ shadowed fading[END_REF]

F κµ (γ) = µ µ-1 m m (1 + κ) µ ) Γ(µ)(µκ + m) m 1 γ µ Φ 2 µ -m, m; µ + 1; - µ(1 + κ)γ γ , - µ(1 + κ)mγ γ(µκ + m) , (2)
where Φ 2 (•) is the bivariate confluent hypergeometric function. By specializing the parameters m, κ, and µ in (1) and (2), some typical fading models can be obtained from the shadowed κ-µ fading. We show the parameter specializations for the shadowed κ-µ representing the one-side Gaussian, Rayleigh, Nakagami-m (with shape factor m N ) , κ-µ, and Rician (with Rician parameter K) in Table I

III. OUTAGE PROBABILITY ANALYSIS

In this section, we define the admissible rate region for the information of b 1 and b 2 and derive the outage probability upper bound of the Slepian-Wolf coding.

A. Admissible Rate Region for Source Coding

According to Shannon's source-channel separation theorem, the joint source-channel coding for b 1 and b 2 can be separated into a two-stage scheme: distributed source coding and MAC transmission. For the distributed lossless source coding, according to the Slepian-Wolf theorem, the admissible rate region is the set of source coding rate pair (R s 1 and R s 1 ), satisfying the following inequalities:

     R s 1 ≥ H(b 1 |b 2 ), R s 2 ≥ H(b 2 |b 1 ), R s 1 + R s 2 ≥ H(b 1 , b 2 ), (3)
which is referred to as the Slepian-Wolf region R SW , as shown in Fig. 2. H(•|•) and H(•, •) denote the conditional and the joint entropy, respectively.

B. Achievable Capacity Region of MAC

The achievable capacity region of the MAC, R MAC , is defined as a closure of the convex hull of the rate pair (R c

1 and R c 2 ) satisfying      R c 1 ≤ C(γ 1 ), R c 2 ≤ C(γ 2 ), R c 1 + R c 2 ≤ C(γ 1 + γ 2 ), (4)
where C(γ) = log 2 (1 + γ) is the channel capacity with the instantaneous SNR γ by assuming the dimension of the channel input is 1.

R c i (i ∈ (1, 2
)) is the normalized spectrum efficiency including the channel coding rate and the modulation multiplicity of the corresponding channel. The region is illustrated as a bounded pentagonal, as shown in Fig. 3.

A c h i e v a b l e R e g i o n ℝ M A C 1 Ŕ R c 1 1 Ŕ R c 2 1 Ŕ C(γ 1 ) 1 Ŕ C(γ 2 ) 1 Ŕ C ( γ 1 1 + γ 2 ) 1 Ŕ C ( γ 2 1 + γ 1 ) R s 1 ℝ u-SW H(b1) Fig. 3. Achievable region R MAC of MAC transmission.
The sufficient condition for lossless coding of b 1 and b 2 is obtained by combining (3) with (4) as

     H(b 1 |b 2 ) Ŕ ≤ R s 1 Ŕ ≤ C(γ 1 ), H(b 2 |b 1 ) Ŕ ≤ R s 2 Ŕ ≤ C(γ 2 ), H(b 1 , b 2 ) Ŕ ≤ (R s 1 + R s 2 ) Ŕ ≤ C(γ 1 + γ 2 ), (5)
where Ŕ defines the total rate of the joint source-channel coding. For the sake of brevity, we assume that Ŕ = Ŕ1 = Ŕ2 , where

Ŕ1 = R c 1 R s 1 and Ŕ2 = R c 2 R s 2
. Eq. ( 5) indicates that if R MAC and R SW have an intersection as shown in Fig. 4, the success of the Slepian-Wolf coding over MAC can be guaranteed. On the contrary, the success of the Slepian-Wolf coding over a MAC can not be guaranteed when R MAC and R SW have no intersection with each other.

Since the source-channel separation theorem holds for point-to-point transmission in general, ( 5) is only the sufficient condition for transmitting one source with one helper over a MAC losslessly. This also leads to that the derived outage expression is the upper bound of practical performance.

C. Outage Probability Calculation

The destination can not guarantee the reconstruction of b 1 with an arbitrarily small error probability when R MAC and R SW do not have an intersection, that is, R MAC ∩ R SW = Ø. The outage event can be divided into three cases as shown in Fig. 5. The sufficient conditions of the three cases are expressed as

     case 1 : C(γ1) Ŕ < H(b 1 |b 2 ), case 2 : H(b 1 |b 2 ) < C(γ1) Ŕ < H(b 1 ), C(γ1+γ2) Ŕ < H(b 1 , b 2 ), case 3 : H(b 1 ) < C(γ1) Ŕ , C(γ2) Ŕ < H(b 1 |b 2 ), (6)
according to [START_REF] Chung | Noma for correlated information sources in 5g systems[END_REF]. In case 1, since C(γ1) Ŕ is smaller than H(b 1 |b 2 ), the outage always happens regardless of the value of C(γ2) Ŕ . In case 2 and case 3, the outage events related to both the value of C(γ1)

Ŕ and C(γ2) Ŕ . Since C(γ1) Ŕ < H(b 1 |b 2 ), A c h i e v a b l e R e g i o n ℝ M A C 1 Ŕ R c 1 1 Ŕ R c 2 1 Ŕ C(γ 1 ) 1 Ŕ C(γ 2 ) 1 Ŕ C ( γ 1 1 + γ 2 ) 1 Ŕ C ( γ 2 1 + γ 1 ) R s 2 R s 1 ℝ MAC H(b2|b1) H(b2) 1 Ŕ C(γ 1 ) 1 Ŕ C(γ 2 ) H(b1|b2) H(b1) ℝ SW Fig. 4. No outage when R MAC ∩ R ! = Ø. H(b 1 |b 2 ) < C(γ1) Ŕ < H(b 1 ) and H(b 1 ) < C(γ1)
Ŕ are distinctive, the outage probability P shadowed κ-µ out of Slepian-Wolf coding in NOMA shadowed κ-µ fading is expressed as

P shadowed κ-µ out = P out case 1 + P out case 2 + P out case 3 = Pr C(γ 1 ) Ŕ < H(b 1 |b 2 ) + Pr H(b 1 |b 2 ) < C(γ 1 ) Ŕ < H(b 1 ), C(γ 1 + γ 2 ) Ŕ < 1 + H(b 1 |b 2 ) + Pr H(b 1 ) < C(γ 1 ) Ŕ , C(γ 2 ) Ŕ < H(b 1 |b 2 ) . ( 7
)
With the assumption that Gaussian code book is used, the relationship between the instantaneous channel SNR γ i (i ∈ (1, 2)) and its corresponding rates Ŕ are written as Ŕ•H(b 1 ) = log 2 (1 + γ 1 ) and Ŕ • H(b 2 ) = log 2 (1 + γ 2 ). Then, with the assumption that each link experiences statistically independent block fading, the outage probability is written as

P out case 1 = 2 ŔH(p) -1 0 fκµ(γ 1 )dγ 1 = µ µ 1 -1 1 m m 1 1 (1 + κ) µ 1 ) Γ(µ 1 )(µ 1 κ 1 + m 1 ) m 1 1 γ1 µ 1 •Φ 2   µ 1 -m 1 , m 1 ; µ 1 + 1; - µ 1 (1 + κ 1 ) 2 ŔH(p) -1 γ1 , - µ 1 (1 + κ 1 )m 1 2 ŔH(p) -1 γ1 (µ 1 κ + m 1 )   , (8)
P out case 2 = 2 ŔH(p) -1 0 fκµ(γ 1 ) 2 Ŕ(1+H(p)) -1-γ 1 0 fκµ(γ 2 )dγ 2 dγ 1 = 2 ŔH(p) -1 0 µ µ 1 1 m m 1 1 (1 + κ 1 ) µ 1 Γ(µ 1 ) γ1 (µ 1 κ 1 + m 1 ) m 1 γ 1 γ1 µ 1 -1 • exp - µ 1 (1 + κ 1 )γ 1 γ1 1 F 1 m 1 , µ 1 ; µ 2 1 κ 1 (1 + κ 1 )γ 1 (µ 1 κ 1 + µ 1 ) γ1 • µ µ 2 -1 2 m m 2 2 (1 + κ 2 ) µ 2 ) Γ(µ 2 )(µ 2 κ 2 + m 2 ) m 2 1 γ2 µ 2 •Φ 2   µ 2 -m 2 , m 2 ; µ 2 + 1; - µ 2 (1 + κ 2 ) 2 Ŕ(1+H(p)) -1 -γ 1 γ2 , - µ 2 (1 + κ 2 )m 2 2 Ŕ(1+H(p)) -1 -γ 1 γ2 (µ 2 κ 2 + m 2 )   dγ 1 (9)
and

P out case 3 = ∞ 2 Ŕ -1 fκµ(γ 1 )dγ 1 2 ŔH(p) -1 0 fκµ(γ 2 )dγ 2 = 1 - µ µ 1 -1 1 m m 1 1 (1 + κ) µ 1 ) Γ(µ 1 )(µ 1 κ 1 + m 1 ) m 1 1 γ1 µ 1 •Φ 2   µ 1 -m 1 , m 1 ; µ 1 + 1; - µ 1 (1 + κ 1 ) 2 Ŕ -1 γ1 , - µ 1 (1 + κ 1 )m 1 2 Ŕ -1 γ1 (µ 1 κ + m 1 )     • µ µ 2 -1 2 m m 2 2 (1 + κ 2 ) µ 2 ) Γ(µ 2 )(µ 2 κ 2 + m 2 ) m 2 1 γ2 µ 2 •Φ 2   µ 2 -m 2 , m 2 ; µ 2 + 1; - µ 2 (1 + κ 2 ) 2 ŔH(p) -1 γ2 , - µ 2 (1 + κ 2 )m 2 2 ŔH(p) -1 γ2 (µ 2 κ 2 + m 2 )   . (10)
Since the derivation of the explicit expression of P out case 2 and P out case 3 is excessively complex, we use the recursive adaptive Simpson quadrature algorithm to numerically calculate the integrals in ( 9) and [START_REF] Zhou | Outage probability of correlated binary source transmission over fading multiple access channels[END_REF]. Note that infinite frame length is assumed, which results in the derived outage probability are the upper bounds for the performance of the practical signaling schemes. Therefore, the analytical results in this work serve as references in practical coding/decoding design for transmitting correlated sources over fading MAC.

IV. NUMERICAL RESULTS AND DISCUSSIONS

Fig. 6 shows the outage performance in shadowed κ-µ fading comparing with those in one-sided Gaussian, Rician, and Nakagami-m fading. In the shadowed κ-µ fading, the parameters are set as κ 1 = κ 2 = 0, µ 1 = µ 2 = 0.5, and m 1 = m 2 = ∞ for comparing with one-sided Gaussian channel; the parameters are set as κ 1 = κ 2 = 3, µ 1 = µ 2 = 1, and m 1 = m 2 = ∞ for comparing with Rician channel; the parameters are set as κ 1 = κ 2 = 0, µ 1 = µ 2 = 3 for comparing with Nakagami-m channel. Correspondingly, the factor in Rician channel is set as K = 3 and the shape factor in Nakagami-m channel is set as m N = 5. We observe that with the parameter specifications indicated in Table I shadowed κ-µ well represents the one-sided Gaussian, Rician, and Nakagami-m, both in the fully correlated sources case (p = 0) and in the independent sources case (p = 0.5), with both non-LOS and LOS components existing in the channels. It is also found that the more than 2 nd diversity order can be achieved with larger ratio of LOS dominant component in single cluster (κ = 3, µ = 1) or with multiple clusters with non-LOS component (κ = 0, µ = 3). Fig. 7 plots the theoretical outage probability against the average SNR with the parameter p varying from 0 to 0.5. The outage probabilities of correlated source transmission over orthogonal channels are offered as benchmarks. From the outage curves, we could observe that the orthogonal transmission outperforms its non-orthogonal counterpart when the two sources are highly correlated (small p). It indicates that the correlation between the two sources is exploited at the destination. In contrast, when the correlation decreases (larger p), the advantage of orthogonal transmission over non-orthogonal transmission disappears in terms of the outage performance. However, with the same p, the same diversity gain can be observed in the outage curves with either orthogonal or nonorthogonal transmission. The -outage achievable rates with respect to the average SNR are shown in Fig. 9, where is set at 0.01. It is observed that for the same p, the -outage achievable rate increases as the average SNR increases. It is also found that the - Average SNR (dB) outage achievable rate decreases as the values of p increase (the sources correlation becomes smaller). This indicates that with higher average SNR, the source correlation can be more utilized to eliminate redundancy after source coding, which leads to higher -outage achievable rate.

, the R s 1 ℝ SW ℝ MAC 1 Ŕ C(γ 1 ) 1 Ŕ C(γ 2 ) H(b2|b1) H(b1|b2) H(b1) H(b2) R s 2 R s 1 ℝ SW ℝ MAC 1 Ŕ C(γ 1 ) 1 Ŕ C(γ 2 ) H(b2|b1) H(b1|b2) H(b1) H(b2) R s 2 R s 1 ℝ SW ℝ MAC 1 Ŕ C(γ 1 ) 1 Ŕ C(γ 2 ) H(b2|b1) H(b1|b2) H(b1) H(b2) (a) (b) (c)

V. CONCLUSION

In this work, we have analyzed the outage probability for the Slepian-Wolf coding over up-link NOMA shadowed κ-µ fading channel. The outage probability upper bound expression has been derived based on the sufficient condition for lossless coding determined by the intersection of the Slepian-Wolf region and the multiple access channel region. It has been found that the outage performance of the orthogonal trans-mission is superior compared to its non-orthogonal counterpart with high correlation between the two sources. The -outage achievable rate increases with increasing the average SNR and/or with increasing the source correlation. The accuracy of the analytical results has been verified by a series of Monte Carlo simulations.

Only two correlated sources case is considered in this work, a promising future direction is to extend to the multiple correlated sources scenario by utilizing multiple-user MAC and multiple-source Slepian-Wolf theorem.

Fig. 1 .

 1 Fig. 1. An illustration of encoder and joint decoder for correlated sources coding over a MAC.

Fig. 2 .

 2 Fig. 2. Admissible region R SW specified with Slepian-Wolf theorem.

Fig. 5 .< 1 +

 51 Fig. 5. Outage occurs when R MAC ∩ R SW = Ø. (a) case 1:H(b 1 |b 2 ) Ŕ ≤ R s 1 Ŕ ≤ C(γ 1 ); (b) case 2: H(b 1 |b 2 ) < C(γ 1 ) Ŕ < H(b 1 ), C(γ 1 +γ 2 ) Ŕ <

Fig. 6 .

 6 Fig. 6. Outage probability versus average SNR (dB) in different fading, where γ 1 = γ 2 . κ = κ 1 = κ 2 = 0, µ = µ 1 = µ 2 = 0.5, and m = m 1 = m 2 = ∞ for one-sided Gaussian; κ = κ 1 = κ 2 = 3, µ = µ 1 = µ 2 = 1, and m = m 1 = m 2 = ∞ for Rician faidng with K = 3; κ = κ 1 = κ 2 = 0 µ = µ 1 = µ 2 = 3 for Nakagami-m fading with shape factor m N = 5.

Fig. 7 .

 7 Fig. 7. Comparison between the outage probabilities of orthogonal and nonorthogonal transmission, where γ 1 = γ 2 . κ = κ 1 = κ 2 = 3, µ = µ 1 = µ 2 = 1, and m = m 1 = m 2 = ∞.

Fig. 8

 8 plots the outage probabilities with different parameters (κ, µ, and m) in shadowed κ-µ fading. It is shown that with κ = κ 1 = κ 2 = 0 and µ = µ 1 = µ 2 = 1 as Rayleigh fading, κ = κ 1 = κ 2 = 2, µ = µ 1 = µ 2 = 1, and m 1 = m 2 = ∞ as Rician faidng, and κ 1 = κ 2 = 0 and µ = µ 1 = µ 2 = 2 as Nakagami-m fading, the Monte Carlo simulation results are in excellent agreement with the analytical results.

Fig. 8 .

 8 Fig. 8. Outage probability and versus average SNR (dB) with different parameters in shadowed κ-µ fading, whereγ 1 = γ 2 . κ = κ 1 = κ 2 = 0 and µ = µ 1 = µ 2 = 1 for Rayleigh; κ = κ 1 = κ 2 = 2, µ = µ 1 = µ 2 = 1, and m 1 = m 2 = ∞ for Rician; κ = κ 1 = κ 2 = 0 and µ = µ 1 = µ 2 = 2for Nakagami-m.

Fig. 9 .

 9 Fig. 9. -outage achievable rate versus average SNR (dB) as a function of p, where γ 1 = γ 2 and = 0.01. κ 1 = κ 2 = 1, and µ 1 = µ 2 = 1, and m 1 = m 2 = 1.

TABLE

  and will demonstrate the validity of the value in Table I for outage calculations in the following sections.

ACKNOWLEDGMENT

This research was supported in part by Japan Society for the Promotion of Science, KAKENHI 21K17738, and also in part by the National Natural Science Foundation of China under Grant 62072330.


Nadia Bouzid

Christelle Anquetil

Rachid Dris

Johnny Gasperi

Bruno Tassin

Sylvie Derenne

Quantification of Microplastics by Pyr-GC-MS in Sediments: Challenges and Implications

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The conventionally used protocols of density separation followed by digestion of organic matter do not allow for complete isolation of MP from the associated organic and mineral matter. This effect is important to consider when using Pyr-GC/MS for the quantification of MP in sediment samples because it can generate interferences.

Nadia Bouzid , Christelle Anquetil , Rachid Dris

Conclusion

The establishment of calibration curves in the presence of a sediment matrix has highlighted an attenuation of the signal for PP, PS, and PVC. Because of potential interferences with residual organic matter and mineral matrix effects, the use of an internal standard is emerging as a prerequisite for the quantification of plastic polymers in complex environmental samples such as sediments.

Results

② Matrix effect in Pyr-GC-MS

Pyrolysis cups for calibration curves prepared by weighing polymer particles from 1 to 30 µg and adding inert matrix or sediment matrix

The attenuation reflects the adsorption of the pyrolysis product onto or within clay sheets, as reported in the literature [START_REF] Fabbri | Analysis of Polystyrene in Polluted Sediments by Pyrolysis-Gas Chromatography-Mass Spectrometry[END_REF][START_REF] Espitalie | Role of Mineral Matrix in Kerogen Pyrolysis: Influence on Petroleum Generation and Migration[END_REF] .

Treatment steps efficiency on a sediment sample? Density separation + organic matter digestion

Residue analyzed by FT-IR imaging 2% of particles identified as a plastic polymer

①

  Additional pyrolysis cups were prepared with fixed amount of polymer particles and increasing amount of matrix sediment Indicator compounds for PP, PS, PET and PVC reviewed and selected 1 taking into account potential interactions with substances present in the residue after treatment

1 , Johnny Gasperi 3 , Bruno Tassin 1 , Sylvie Derenne 2 1 Leesu, Ecole des Ponts, Université Paris Est Créteil, 2 UMR METIS, Sorbonne Université, 3 LEE, Université Gustave Eiffel Literature cited

Acknowledgments

This study was carried out in the framework of the Sedi-Plast research program funded by the French Research Agency (ANR-19-CE34-0012).


Alex Coiret

Pierre-Olivier Vandanjon

Road safety assessment in curves based on a road embedded tire to road friction sensor

Keywords: road infrastructure, energy moderation, ecodriving, speed policy

Road safety relies mostly on available tire to road grip regarding to the forces required by a vehicle in a given dynamic situation. For example safety issues can occur when deceleration required forces exceeding the tire/road grip potential. Issues are much critical while traveling in a curve with a coupled longitudinal/transverse grip demand.

Instrumented vehicles can be used in order to evaluate the available tire to road grip in rolling situations, but it remains linked to a dedicated vehicle and models have to be use to extrapolate these evaluations to various vehicles.

The aim of this work is to propose a sensor embedded in the road infrastructure to evaluate the longitudinal and transverse grip demand of any traveling vehicle.

A 6 axis sensor is embedded in a road test track, under a piece of the real road upper layer. Measurement are made at a acquisition rate of 2500 Hz and forces measuring ranges are of 2500 daN (vertical axis) and 750 daN (tangential). Measurements are compared to on-board measurement with a vehicle instrumented with dynamometric wheels. Three situations are experienced in a 100 meters curve: traveling at a given speed and traveling while accelerating or braking moderately, and for three vehicle speeds of 40, 60 and 80 km/h. Force evaluation from the two systems are differing of only 5% for the vertical force and 10% for the tangential forces. The differences are lower for the higher grip demands. For example moderate braking from 70 km/h in curve lead to 0.23 and 0.25 longitudinal friction coefficients (LFC) for road sensor and dynamometric wheel, and to 0.17 and 0.19 transversal friction coefficients (TFC).

In perspectives, evaluations of an excessive grip demand could result in solutions as lowering the designed speed or improving the road surface layer.

Introduction

Road safety relies on passive and active techniques, with respective relevance before and after an accident occurrence. Passive safety devices as airbags are important but the active safety is in the first line to avoid an accident and its consequences. Tire to road grip level is one of the most important factor of the road active safety factors, since this grip represents the only mean to develop decelerating or steering forces.

For example safety issues can occur when deceleration required forces exceeding the tire/road grip potential. Issues are much critical while traveling in a curve with a coupled longitudinal/transverse grip demand. In [START_REF] Apuzzo | An exploratory step for a general unified approach to labelling of road surface and tyre wet friction[END_REF] a large review is proposed to qualify the tire to road grip depending on surfaces conditions such as wetting state, pollutants and temperature, and depending on vehicle and driving conditions. [START_REF] Apuzzo | An exploratory step for a general unified approach to labelling of road surface and tyre wet friction[END_REF] points out the combined importance of tires, which can be performance-labeled, and pavement classification.

In [START_REF] Mussot | Model learning of the tire-road friction slip dependency under standard driving conditions[END_REF] it is again stated that grip level is difficult to estimate under standard driving conditions. Therefore a lot of estimation methods can be distinguished such as sensor-based methods, non-parametric methods using generally black-box models and parametric methods including physical models. One of the most popular learning model is also presented, the Pacejka's model, combining likelihood and Monte Carlo Markov Chain.

From a driving task point of view, the knowledge of this grip level is of relevance to adapt the driving behavior, which could involve an experienced or inexperienced driver or even no driver at all in the case of the autonomous vehicles. It is an important information for the road manager in the process of road maintenance.

As a technical solution [START_REF] Andrieux | A Novel Approach to Real Time Tire-Road Grip and Slip Monitoring[END_REF] developed an approach to estimate the grip level in real time with on-board sensors, with the aim to detect slip variation of 1 to 1000. This work shown the possibility to distinguish a wet roadway from a dry one, but without achieving a fine grip estimation.

In the research field instrumented vehicles can be used in order to evaluate the available tire to road grip in rolling situations. In [START_REF] Brüggemann | Structural optimization of a wheel force transducer component for more realistic acquisition of vehicle load data and fracture mechanical evaluation[END_REF] it is reminded that for the forces acting of the wheels should be estimated for the real operating conditions and not only for only the vehicle weight and load. In this study the wheel weight is considered as an important factor thus being unsprung masses and dynamometric wheels of the same weight as standard wheels are proposed in order to estimated the wheel load in given driving situations. Such wheels remains too expansive to be largely deployed on vehicles (Fig. 1).

In [START_REF] Mendoza-Petit | Characterization of the loss of grip condition in the Strain-Based Intelligent Tire at severe maneuvers[END_REF] a strain-based intelligent tire is developed to estimate the situations of loss of control, by calculating the grip level. Authors are underlining that the limit of friction is not known and that it depends on road properties, wetting states, tire wear and so on. In this work an intelligent algorithm is proposed to estimate this limit of adherence, by the integration of the LuGre model. For that the transition between stick/slip and full slip is evaluated by the means of the force sensor. According to [START_REF] Mendoza-Petit | Characterization of the loss of grip condition in the Strain-Based Intelligent Tire at severe maneuvers[END_REF] the early detection of grip level by this type of intelligent tire could help the control systems to react against vehicle's loss of control. This approach is promising but the road conditions remains difficult to estimate by the on-board systems. Force sensing bearings have been experimented to evaluated the grip in order to enhance the vehicle control systems [START_REF] Zuurbier | Vehicle dynamics control based on force-sensing wheel bearings, vehicle dynamics[END_REF] or to investigate the potential of estimation of vehicle sideslip [START_REF] Madhusudhanan | Vehicle sideslip estimator using load sensing bearings[END_REF] but these systems still lack of road conditions estimation in order to detect the maximum limit of the grip potential.

Road sensors approaches exists to estimate grip level, such as Stress-In-Motion system (SIM) developed since 1994 [START_REF] Beer | Evaluation of non-uniform tyre contact stresses on thin asphalt pavements[END_REF][START_REF] Beer | Stress-In-Motion (SIM) system for capturing tri-axial tyre-road interaction in the contact patch[END_REF] for capturing tri-axial tire to road interaction. It consist in a measuring pad with a transverse array of 21 sensing elements (Fig. 2), but road texture and conditions are not taken into account. The aim of this work is to propose a sensor embedded in the road infrastructure to evaluate the longitudinal and transverse grip demand of any traveling vehicle. This sensor will not be more able to determine the grip limit of a given vehicle compared to on-board systems, but while measuring applied forces for all traveling vehicles it should allow the determination of a statistical grip limit. This sensor consist on a 3D load cell place underneath a portion of the road surface. The knowledge of the actual grip demand for a whole traffic, and its evolution in time, and facing environmental conditions as wetting state and temperature will allow a statistical and dynamical determination of grip level. Thus a road benefiting of such a sensor could propose a dynamical speed enforcement, adapted to road wear, wetting state and seasonal temperature. Moreover it could be a decision making tool for road surface management and infrastructure works.

Experimental setup

A 6 axis sensor, forces and moments, is embedded in a road test track, under a piece of the real road upper layer (Fig. 1). Measurement are made at a acquisition rate of 2500 Hz and forces measuring ranges are of 2500 daN (vertical axis) and 750 daN (tangential). Measurements are compared to on-board measurement with a vehicle instrumented with a dynamometric wheel. Three situations are experienced in a 100 meters curve (Fig. 2): traveling at a given speed and traveling while accelerating or braking moderately, and for several vehicle speeds of about 40, 60 and 80 km/h. Forces and moments are recorded simultaneously by the road sensor and the dynamometric wheel. Several speeds are experienced, while driving at constant speed, or accelerating or braking moderately. The speed is determined by the means of an optical on-board system (correvit). In this section measurements from the two systems are compared in the perspective to use the road sensor alone as a long-term grip monitoring system on open roads.

Vehicle traveling at constant speed

The Fig. 3 presents the forces and moments evaluated by the road sensor for the 40km/h speed case. The left rise and right lowering parts of each signal correspond to the entrance and leaving of the tire inside and outside the measuring area. Signals are selected within this area to establish mean values that are given in Table 1. 2.), which is quite satisfying if considering that each system errors are added together.

Vehicle traveling while accelerating moderately

With a slight acceleration, for two speed cases, the differential error is only about 3 and 6% for Fz, and about 12 and 13% for Fx and Fy at the 50km/h speed case. The low speed case of 30km/h is showing out large differential errors of more than 30%. It should be remembered that the road system is dimensioned for 750kN and so this speed case is only using one tenth of the sensor full scale.

Vehicle traveling while braking moderately

With a moderate braking maneuver, for two speed cases, the differential error is only about 4% / 5% for Fz, and between 10 / 14% for Fx and Fy at the 47 and 56km/h speed cases (Table 5. and 6.). Intended test speed were of 60 and 70 km/h but by braking the vehicle reached the road sensor at sensible lower speeds. Road sensor signals are given in Fig. 4 to show the negative braking Fx force and positive Fy transverse force. Road safety is very sensitive to potential tire to road grip. The maximum grip level can be estimated by on-board systems or road systems but these systems are often very complex and are dedicated to research purposes. This work is about the use of a quite simple and robust system embedded in a road and able to estimate the grip demand of any vehicle. This road sensor is compared to a highend dynamometric wheel and results are confirming a quite good correlation between the measurement from the two systems.

Indeed tangential forces differences are lower than 10% to 15% for moderate speeds and for rolling, accelerating or braking situations. Moreover measurement differences are lowering when speeds are increasing, so that differences should be very low for critical high speeds, since large forces are more in agreement with the sensors full scale. As a result longitudinal and transverse friction coefficient estimated from the two systems are rather close, in a 10% confidence interval. In perspective a road embedded sensor could be use in a given road to record all vehicle grip demand and to statistically verify is the road is still able to offer a satisfying grip level. Environmental grip variations could be monitored too in order to take temporary management decisions as speed limitations.

Figure 1 .

 1 Figure 1. Dynamometrical wheel (source: kistler.com)

Figure 1 .

 1 Figure 1. Stress-In-Motion (SIM) system (source: [9], fair use)

Figure 1 .Figure 2 .

 12 Figure 1. Road sensor: sensor detailed view (left), and final installation in the road pavement (right)

Figure 3 .

 3 Figure 3. Road sensor measurements (Dan and N.m) in function of time (s); V40 speed caseMeans values are indicating that the vertical force Fz is given similarly by the two systems, with a difference of about 5%. Fx values are low for this constant speed case, especially for the 40km/h speed for which systems are underestimating the probably higher value. Nevertheless Fx and Fy values are differing from 5, 7 and 15% (Table2.), which is quite satisfying if considering that each system errors are added together. Table 1. Measuring results; constant speed cases

Figure 4 .

 4 Figure 4. Road sensor measurements (Dan and N.m) in function of time (s); B70 speed case3.4 Tire to road grip estimation from the two measurement systemsUsually for road grip estimation the longitudinal and transverse friction coefficient LFC and TFC are considered. For the two braking cases these coefficient are computed and are given in Table7. Road and wheel sensors are differing only about 10%, for nonetheless low grip solicitations, since grip values are usually of 1 for dry conditions and 0.5 for wet conditions.

Table 1 .

 1 Measuring results; constant speed cases

	Speed label	sensor	Vxopt	Vyopt	Fx (daN)	Fy (daN)	Fz (daN)
	V40	road			4	42	435
	V40	wheel	41,6	-0,4	-1	35	460
	V60	road			55	86	420
	V60	wheel	59,7	-0,8	44	69	470
	V80	road			93	194	490
	V80	wheel	81,9	-2	98	227

Table 2 .

 2 Force ratio between the wheel and road systems

	Speed label	DFx	DFy	DFz
	V40	125,00	16,67	-5,43
	V60	20,00	19,77	-10,64
	V80	-5,10	-14,54	-6,84

Table 3 .

 3 Measuring results; acceleration cases

	Speed label	sensor	Vxopt	Vyopt	Fx (daN) Fy (daN) Fz (daN)
	A30	road			47	40	423
	A30	wheel	36	-0,2	30	25	437
	A50	road			54	84	438
	A50	wheel	53,8	-0,7	47	74	467

Table 4 .

 4 Force ratio between the wheel and road systems

	speed label	DFx	DFy	DFz
	A30	36,17	37,50	-3,20
	A50	12,96	11,90	-6,21

Table 5 .

 5 [Table caption] Measuring results; braking cases

	Speed label	sensor	Vx_opt	Vy_opt Fx (daN) Fy (daN) Fz (daN)
	B60	road			-120	50	475
	B60	wheel	46,8	-0,7	-135	57	494
	B70	road			-113	82	486
	B70	wheel	55,6	-0,6	-126	95	513

Table 6 .

 6 absolute and relative differences; braking cases

	Speed label	D Fx	D Fy	D Fz
	B60	11,11	-12,28	-3,85
	B70	10,32	-13,68	-5,26

Table 7 .

 7 Road and wheel sensors are differing only about 10%, for nonetheless low grip solicitations, since grip values are usually of 1 for dry conditions and 0.5 for wet conditions.Such results are confirming that the road sensor is able to continuously measure the LFC and TFC mobilized by any vehicle traveling on it, on a given road. If the manager has the aim to rebuilt the road surface at a 0.3 LFC value for example, the road can be considered in good condition as long as LFC values over 0.5 are regularly recorded. Mean values of mobilized grip could be useful to estimate environmental impact as rain or low temperatures.

	Speed label	Sensor	LFC	TFC
	B60	road	-0,25	0,11
		wheel	-0,27	0,12
	B70	road	-0,23	0,17
		wheel	-0,25	0,19

Table 7 .

 7 longitudinal and transversal coefficients


Pauline Bredy

David Farrusseng

Yves Schuurman

Frédéric Meunier

Frederic C Meunier

On the link between CO surface coverage and selectivity to CH4 during CO2 hydrogenation over supported cobalt catalysts

Keywords: cobalt, carbon dioxide, Fisher-Tropsch

An operando IR study was carried out to investigate the loss of selectivity to higher hydrocarbons observed over cobalt-based Fischer-Tropsch catalysts when CO is replaced with CO 2 . Our data unambiguously show that the structure and oxidation state of the surface of metallic cobalt remains the same whether CO 2 or CO are cofed with H 2 . The selectivity to CH 4 appears to increase monotonously with decreasing surface coverage of adsorbed CO.

The switch from Fischer-Tropsch synthesis to methanation using CO 2 -rich feeds over metallic cobalt can therefore be assigned to surface coverage effects only.

Introduction.

The need to curb CO 2 emissions and the prospects of a future greater availability of dihydrogen produced from nuclear and renewable energies have raised interest in carbon capture and utilisation (CCU) 1 . The hydrogenation of CO 2 to methanol [START_REF] Luk | Status and prospects in higher alcohols synthesis from syngas[END_REF] , methane (Sabatier reaction) [START_REF] Messou | Origin of the synergistic effect between TiO 2 crystalline phases in the Ni/TiO 2 -catalyzed CO 2 methanation reaction[END_REF] and higher hydrocarbons (Fischer-Tropsch synthesis, FTS) [START_REF] Riedel | Comparative study of Fischer-Tropsch synthesis with H 2 /CO and H 2 /CO 2 syngas using Fe-and Co-based catalysts[END_REF][START_REF] Hazemann | Selectivity loss in Fischer-Tropsch synthesis: The effect of carbon deposition[END_REF]6 are all receiving renewed interests.

The production of higher hydrocarbons directly from CO 2 and H 2 is yet more difficult than that from syngas CO and H 2 , especially over cobalt-based FTS catalysts that are poorly active for the reverse water gas-shift reaction (RWGS, Eq. 1) and do not readily form CO in situ:

CO 2 + H 2  CO + H 2 O (Eq. 1)
Fe-based materials are highly active for RWGS, but are intrinsically less active and selective than cobalt for the synthesis of higher hydrocarbons from syngas [START_REF] Khodakov | Advances in the Development of Novel Cobalt Fischer-Tropsch Catalysts for Synthesis of Long-Chain Hydrocarbons and Clean Fuels[END_REF] . Thus, there is an interest in understanding better the reactivity of CO-CO 2 -H 2 mixtures over cobalt surfaces.

Various groups have reported investigations of FTS using CO-CO 2 -H 2 mixtures over cobalt-based materials. High selectivity to CH 4 observed with CO 2 -rich feed was hypothesised to arise from a lower surface coverage of CO, which in turn would lead to a higher surface coverage with hydrogen, favouring methane formation [START_REF] Riedel | Comparative study of Fischer-Tropsch synthesis with H 2 /CO and H 2 /CO 2 syngas using Fe-and Co-based catalysts[END_REF][START_REF] Visconti | CO 2 hydrogenation to hydrocarbons over Co and Fe-based Fischer-Tropsch catalysts[END_REF] . Others have proposed that CO and CO 2 hydrogenation proceeded through different reaction pathways [START_REF] Zhang | CO and CO 2 hydrogenation study on supported cobalt Fischer-Tropsch synthesis catalysts[END_REF] .

Metallic cobalt is often considered as the most active phase during CO and CO 2 hydrogenations over Co-based materials [START_REF] Khodakov | Advances in the Development of Novel Cobalt Fischer-Tropsch Catalysts for Synthesis of Long-Chain Hydrocarbons and Clean Fuels[END_REF][START_REF] Visconti | CO 2 hydrogenation to hydrocarbons over Co and Fe-based Fischer-Tropsch catalysts[END_REF][START_REF] Visconti | Fischer-Tropsch synthesis on a Co/Al2O3 catalyst with CO2 containing syngas[END_REF][START_REF] Paredes-Nunez | CO Hydrogenation on Cobalt-Based Catalysts: Tin Poisoning Unravels CO in Hollow Sites as a Main Surface Intermediate[END_REF] . However, it was recently shown that cobalt oxides could be the main phases responsible for the formation of hydrocarbons (mostly CH 4 ) over Co-based catalysts used for CO 2 hydrogenation [START_REF] Have | Uncovering the reaction mechanism behind CoO as active phase for CO 2 hydrogenation[END_REF] . The oxidation of a fraction of metallic cobalt could thus potentially be a major cause for the increased selectivity to methane when increasing the CO 2 proportions in CO-CO 2 -H 2 mixtures.

3/13

Cobalt oxidation by CO 2 is a realistic hypothesis since using CO 2 in place of CO increases the oxygen chemical potential of the feed and the concentration of water, formed through the RWGS (Eq. 1). High partial pressures of water, the main product of the FTS reaction, is known to lead to cobalt oxidation [START_REF] Wolf | Role of CO in the Water-Induced Formation of Cobalt Oxide in a High Conversion Fischer-Tropsch Environment[END_REF] . As a matter of fact, the use of CO 2 in place of CO was shown to lead to the oxidation of a part of iron carbide into iron oxide [START_REF] Gnanamani | Fischer-Tropsch synthesis: Mössbauer investigation of iron containing catalysts for hydrogenation of carbon dioxide[END_REF] .

The objective of this work was to investigate the surface of metallic cobalt under CO-CO 2 -H 2 mixtures by operando diffuse reflectance spectroscopy (DRIFTS) and rationalise CO/CO 2 ratio modified the CO surface coverage and that the cobalt metallic surface was not modified, neither by reconstruction nor by oxidation.

Experimental

The

4/13

Operando DRIFTS experiments were performed using a modified high temperature DRIFT cell from Spectra-Tech fitted with KBr windows placed in a

Collector II assembly described elsewhere [START_REF] Li | Effective bulk and surface temperatures of the catalyst bed of FT-IR cells used for in situ and operando studies[END_REF] . The spectrophotometer used was a Nicolet 8700 (ThermoFischer Scientific) fitted with a liquid-N 2 cooled MCT detector.

The DRIFT spectra were recorded at a resolution of 4 cm -1 and 16 scans were averaged. The DRIFTS spectra are reported as log (1/R), where R is the sample reflectance. This pseudo-absorbance gives a better linear representation of the band intensity against surface coverage than that given by the Kubelka-Munk function for strongly absorbing media such as those based on metals supported on oxides [START_REF] Sirita | Quantitative Analysis of Adsorbate Concentrations by Diffuse Reflectance FT-IR[END_REF] . The contribution of gas-phase CO was subtracted using a CO(g) spectrum collected under the same experimental conditions over powdered KBr.

The reaction products were measured using a 2-m path length IR transmission gas cell fitted in a Nicolet IS10 spectrophotometer from ThermoFischer Scientific.

Methane, CO 2 , CO and water concentrations were determined through calibration curves. All hydrocarbons other than methane were quantified altogether by integrating the IR signal between 3250-2800 cm -1 after removing methane contribution, which was mostly related to methylene -CH

Results and discussion

The metal loadings measured by ICP are given were similar to the expected nominal values (Table 2). (Bottom) under various CO and CO 2 -containing feed described in Table 1 The selectivity to methane increased when CO 2 was also introduced in the feed alongside CO (Red symbols in Figure 1). The increase with respect to the relative CO surface coverage was sharp in the case of Co/Siralox, reaching more than 95 % The variation of selectivity to CH 4 with respect to relative CO surface coverage was more gradual in the case of the Co/TiO 2 : methane selectivity to CH 4 was higher than 90 % only if the CO relative coverage had decreased by more than half.

These data show that a monotonous relationship existed between the relative CO surface coverage and the selectivity to CH 4 over all four catalysts. It is likely that the decrease of CO coverage led to a similar increase of hydrogen coverage, as suggested by other authors [START_REF] Riedel | Comparative study of Fischer-Tropsch synthesis with H 2 /CO and H 2 /CO 2 syngas using Fe-and Co-based catalysts[END_REF][START_REF] Visconti | CO 2 hydrogenation to hydrocarbons over Co and Fe-based Fischer-Tropsch catalysts[END_REF] . The nature of the support appeared to lead to a different curve (Figure 1), possibly due to electronic effects.

For each catalyst and at each temperature the decomposition of the carbonyl band obtained under the CO-H 2 mixture was monitored following the removal of CO from the feed. This was done to determine the extent of the carbonyl band shift versus coverage for the metallic surface.

The carbonyl surface species over cobalt are typically converted into CH 4 rather than desorbed as CO, as signal decay is much faster under H 2 than it is under He [START_REF] Paredes-Nunez | CO Hydrogenation on Cobalt-Based Catalysts: Tin Poisoning Unravels CO in Hollow Sites as a Main Surface Intermediate[END_REF] .

The evolution of the carbonyl DRIFTS band is shown for the Pt-free Co/Siralox in Figure 2, similar profiles being observed for the other samples (not shown). The linear CO were shown to be spectator species, whereas multi-bonded CO were reaction intermediates leading to hydrocarbon formation [START_REF] Paredes-Nunez | CO Hydrogenation on Cobalt-Based Catalysts: Tin Poisoning Unravels CO in Hollow Sites as a Main Surface Intermediate[END_REF] . The interconversion of the linear and multi-bonded CO species through surface diffusion was proposed to be fast.

The observed band evolution reported in Figure 2 can be explained as follows. The carbonyls at ca. 1940 cm -1 corresponded to multi-bonded species and were readily converted into hydrocarbons, mostly methane in the absence of gas-phase CO.

These sites were then readily replenished by linear carbonyl surface-diffusing from the terrace sites. Globally, this resulted in a decay and shift (because of decreased dipole-dipole interaction on the terraces) of the linear CO signal and a slower decay and absence of shift of the signal of multi-bonded CO, which are isolated from each other and not lead to dipole-dipole coupling interactions [START_REF] Paredes-Nunez | CO Hydrogenation on Cobalt-Based Catalysts: Tin Poisoning Unravels CO in Hollow Sites as a Main Surface Intermediate[END_REF] .

The plots of the main carbonyl DRIFTS band position versus the total band area are given in Figure 3 for the Pt-free samples. The data points represented by coloured crosses were measured under various CO-CO 2 -H 2 mixtures, while the blue thick line is associated with the data collected following CO removal described in Figure 2. All

9/13

the data points collected under the various feeds fell exactly of the CO removal line, which was related to the metallic cobalt surface obtained under the CO+H 2 mixture.

These data indicate that feed composition only affected the relative CO surface coverage and that the metal surface structures and oxidation states were identical whatever the CO-CO 2 -H 2 mixture used.

10/13

It should be stressed that the plots giving carbonyl band position versus band areas are highly sensitive to the structure of the surface. A chlorine-modification induced by the presence of 50 ppm of trichloroethylene over a similar cobalt catalyst led to a plot shift of more than 60 cm -1 18 . In the present case, all data points are located on the same CO removal line pertaining to metallic cobalt within 5 cm -1 , which is the estimated precision of the measure.

While our data show that the state of the surface of metallic cobalt is not modified by varying feed composition in terms of CO/CO 2 proportions, we need to stress that some CoOx phases may still have been present under reaction conditions. Those were shown to lead mostly to methane formation [START_REF] Have | Uncovering the reaction mechanism behind CoO as active phase for CO 2 hydrogenation[END_REF] . These authors proposed that two reaction pathways occurred, one through CO(ads) on metallic cobalt and one through carbonate/formate on CoOx. Formates were in fact also observed on our catalysts and their reactivity will be discussed in a subsequent paper.

Our data unambiguously show that the structure and state of the surface of metallic cobalt remains the same whether CO 2 or CO are cofed with H 2 . The selectivity to CH 4 appears to increase monotonously with decreasing surface coverage of CO. The switch from Fischer-Tropsch synthesis to methanation using CO 2 -rich feeds over metallic cobalt can therefore be assigned solely to surface coverage effects (Figure 4). Our results clearly indicate that high CO coverage is a key requirement to obtain high selectivity for FTS synthesis in the presence of CO 2 , which acts as a diluent.

Metallic cobalt

Notes and references

There are no conflicts to declare. PB collected and analysed all the data. FCM validated and supervised data analysis. YS and DF acquired funds and conceptualized the project. FCM drafted the manuscript that was reviewed and edited by all authors.

1 M.A.A. Aziz, A. A., Jalil, S. Triwahyono and A. Ahmad, "CO 2 methanation over heterogeneous catalysts: recent progress and future prospects", Green Chem., 2015,

Figure 1 :

 1 Figure 1: Selectivity to CH 4 versus the relative CO surface coverage over supported non-

Figure 2 :

 2 Figure 2: Evolution of the carbonyl DRIFTS band under H 2 at 210 °C following the removal of CO after steady-state under the CO-H 2 feed. The red spectrum was collected under CO+H 2 just before removing CO

Figure 3 :

 3 Figure 3: Positions of the main carbonyl DRIFTS band versus the band total area measured at 210 °C over (Top) Co/Siralox and (Bottom) Co/TiO 2 under various CO and CO 2 -containing feed. The blue thick lines are the data corresponding the band decomposition following CO

Table 1 .

 1 steady-state under the CO-H 2 mixture, to determine carbonyl band shift as a function of coverage. The CO surface coverage obtained under the CO-H 2 mixture was used as reference to normalise that obtained under different feeds. CO DRIFTS signals are proportional to CO surface coverages in the range of low pseudoabsorbance values observed here, as in the case of transmission IR data[START_REF] Couble | Quantitative analysis of infrared spectra of adsorbed species using transmission and diffuse reflectance modes Case study: Heats of adsorption of CO on TiO2 and CuO/Al2O3[END_REF] . The sequence was then repeated at 210 and 180 °C. The different feed compositions used are listed in Table1. The COx conversions were always lower than 5 %, ensuring differential conditions. Gas flow rates (in mL min -1 ) used during this study

	High purity gases He (99.999%), H 2 (99.999%), CO 2 (99.995%) and 5% CO/He
	(99.9%) from Air Liquide or Messer were used without further purification. The
	samples were pre-reduced in situ at 350 °C for 1 h, which led to the reduction of
	essentially all cobalt oxide phases, as indicated by other experiments realised with
	different reduction temperatures and contact times (not shown).

2 -stretching vibrations. An average molar absorption coefficient equal to half that of CH 4 (which contains four C-H bonds) was used. This approximate method was sufficient to estimate trends for the selectivity to methane.

Operando DRIFTS data were then collected at 250 °C under the CO 2 -H 2 mixture until steady-state was reached. Various CO-CO 2 -H 2 mixtures were then used.

Carbonyl desorption/reaction was also monitored after CO was removed from the feed 5/13 following

Table 2 .

 2 Metal loadings measured by ICP was obtained under the CO-H 2 feed for all catalysts (Figure1, black symbols). The TiO 2 -based materials were the most selective to higher hydrocarbons, with selectivity to methane as low as 60 %, while the Siralox-based samples lowest selectivity to methane was about 80 %. The presence of Pt had little influence when using the CO-H 2 mixture.

		Co (wt.%)	Pt (wt. ppm)	Co particle size a
	Co/Siralox	14. 8  0.2	-	12 ± 3 nm
	Pt-Co/Siralox	14. 5  0.2	455  20	18 ± 3 nm
	Co/Titania	13. 8  0.2	-	15 ± 3 nm
	Pt-Co/Titiania	14. 0  0.2	560  20	15 ± 3 nm

a determined by XRD analysis

The operando DRIFTS data are only discussed for the data at 210 °C, since similar results were obtained at 250 and 180 °C (not shown). The lowest selectivity to 6/13 methane

  Figure 4: Schematic representation of the state of the surface under reaction conditions. The CO surface coverage under CO-H 2 is very high, limiting methanation. In contrast, CO surface coverage is much lower under CO 2 -H 2 , leading to increased selectivity to methane. The structure of the surface remains intact for all CO-CO 2 -H 2 mixtures.

	H 2 + CO 2				CH 4 C 2+ H y						H 2 + CO
	O C	H	H	O C	H	H	O C	O C	O C	H	O C	O C	O C	O C
									Metallic cobalt
										11/13


Guilhem Coq

Christian Olivier
email: olivier@sic.sp2mi.univ-poitiers.fr

Olivier Alata
email: alata@sic.sp2mi.univ-poitiers.fr

Marc Arnaudon
email: arnaudon@math.univ-poitiers.fr

In this paper we give a short theoretical description of the general predictive adaptive arithmetic coding technique. The links between this technique and the works of J. Rissanen in the 80's, in particular the BIC information criterion used in parametrical model selection problems, are established. We also design lossless and lossy coding techniques of images. The lossless technique uses a mix between fixed-length coding and arithmetic coding and provides better compression results than those separate methods. That technique is also seen to have an interesting application in the domain of statistics since it gives a data-driven procedure for the non-parametrical histogram selection problem. The lossy technique uses only predictive adaptive arithmetic codes and shows how a good choice of the order of prediction might lead to better results in terms of compression. We illustrate those coding techniques on a raw grayscale image.

INTRODUCTION

Arithmetic Coding (AC) is an efficient binary coding technique. We use it here in one of its most general form : the predictive and adaptive one. Even though those aspects of AC are known, it is quite hard to find literature dealing with both of them ; as well as to determine which aspects are actually used in image coding norms such as JPEG and JPEG2000. We try here to answer the first issue but could not collect useful informations about the second. This paper does not seek compression efficiency but wants to show how different AC processes may be used in both parametrical ( §3) and non-parametrical ( §4) model selection problems. This explains why we choose to work on raw images.

After a description of AC algorithm in §2, we take a closer look at the resulting codelength. To this end, we use works of J. Rissanen in [START_REF] Rissanen | Generalized Kraft inequality and arithmetic coding[END_REF][START_REF] Rissanen | Complexity of strings in the class of Markov sources[END_REF] and especially [START_REF] Rissanen | Stochastic complexity and modeling[END_REF]. The main conclusion of §3 is that the codelength enters the family of information criteria, a widely used tool in the vast problem of model selection. We aim at showing that the adaptive aspect of the AC used here is an essential feature.

Next, we design in §4 a new lossless coding technique. It uses a mix between AC, which is compression efficient, and fixed-length coding, which is not. It is shown in §4.2 that correctly mixing those two methods gives better compression efficiency than using only AC. The most important parameter to be adjusted in order to get that "correct" mix is the order of prediction. Moreover, that method is shown in §4.3 to have a direct application in the histogram selection problem.

Finally we design in §5 a lossy coding technique which, once again, shows the importance of the order of prediction.

GENERALITIES ON ARITHMETIC CODING

Multiple Markov Chain

The notion of Multiple Markov Chain (MMC) leads to arithmetic coding. Let E = {a 1 , . . . , a m } be a finite set with m elements. An E-valued process (X n ) n∈N * is an order k MMC if k ∈ N is the smallest integer satisfying the law equality (X n |X n-1 , . . . , X 1 ) = (X n |X n-1 , . . . , X n-k ) for all n. We will always work in the case where that law does not depend on n ; the chain is said homogeneous. An order 0 MMC is a sequence of independent random variables.

If X is an order k MMC, we will suppose that X 1 , . . . , X k are independent and uniformly distributed on E. For i ∈ E a state and j ∈ E k a multiple state, we denote by θ(i| j) the probability to see i after j. Consequently, choosing (m -1)m k real numbers θ(i| j) for j ∈ E k and i ∈ {a 1 , . . . , a m-1 } is enough for describing the evolution of X. Let θ denote such a parameter and x n = x 1 , . . . , x n be a sequence of elements of E, the likelihood of x n relatively to θ writes as :

(x n |θ) = 1 m k ∏ j∈E k ∏ i∈E θ(i| j) n(i| j) (1)
where n(i| j) is the number of occurences of i after j in x n .

Predictive adaptive arithmetic coding : PAAC

We deal here with a general AC which is both kpredictive and adaptive ; we shorten it to k-PAAC. Predictive means we code using orders k that may be greater than 1, hence a prediction of the future state of the chain from the current state. Adaptive means we do not need any prior knowledge on the chain, except its order ; we learn how to predict the future step by step. Both notions have been formally introduced and studied by Rissanen [START_REF] Rissanen | Generalized Kraft inequality and arithmetic coding[END_REF][START_REF] Rissanen | Complexity of strings in the class of Markov sources[END_REF][START_REF] Rissanen | Stochastic complexity and modeling[END_REF]. For a more concrete description of arithmetic coding, we refer to [START_REF] Witten | Arithmetic coding for data compression[END_REF] ; note that this paper does not mention the predictive aspect. Let us now give a theoretical description of the general k-PAAC algorithm.

Let x n = x 1 , . . . , x n be a chain of elements of E to be encoded and I c be the current interval firstly set to I c = [0, 1). For n ≥ t ≥ 1 we note x t = x 1 , . . . , x t . The only prior we need is an order of coding k ≥ 0, then the algorithm works as follows.

Suppose that the t ≥ 0 first symbols are dealt with ; t = 0 means we have not started the coding yet. To deal with the (t + 1)-th symbol we actualize transition probabilities as follows :

θ(t) (i| j) = n (t) (i| j) + 1 n (t) ( j) + m
where i ∈ E, j ∈ E k , n (t) (i| j) and n (t) ( j) denote the respective number of occurences of i after j and of j in the chain x t ; n (t) ( j) must not count an occurence of j at the very end of that chain. If k = 0, the multiple states j vanish and we set n (t) ( j) = t. Those probabilities reflect what we know of the chain at the time t of the coding process ; they are the adaptive aspect. We then set j = x t-k+1 , . . . , x t the current state and split the current interval I c in m smaller intervals according to the probabilities θ(t) (i| j), i ∈ E. This way, we associate to each possible future state i ∈ E an interval whose length is proportional to the probability with which we expect it. The (t + 1)-th symbol is dealt with by choosing for new I c the interval corresponding to i = x t+1 .

Once the last symbol x n has been dealt with, we are left with an interval I c = [low,high). Let . denote the superior integer part, there exists two consecutive dyadic numbers with lengthlog(high-low) in I c . We take as the arithmetic code of x n the sequence of bits given by the fractionnal part of the biggest one. If encoder and decoder agree on the order k of coding, that sequence of bits is decodable, we refer again to [START_REF] Witten | Arithmetic coding for data compression[END_REF].

For illustration in table 1, we take m = 2, E = {a, b} and encode x 4 = abaa at order k = 1. In the splits, we allow the left interval to a.

TAB. 1 -Order 1 PAAC of the chain abaa.

t x t I c θ(t) (.|.) Split 0 / 0 [0, 1) (a|a) = 1/2 (a|b) = 1/2 [0, 1 2 , 1) 1 a [0, 1 2 ) (a|a) = 1/2 (a|b) = 1/2 [0, 1 4 , 1 2 ) 2 ab [ 1 4 , 1 2 ) (a|a) = 1/3 (a|b) = 1/2 [ 1 4 , 3 8 , 1 2 ) 3 aba [ 1 4 , 3 8 ) (a|a) = 1/3 (a|b) = 2/3 [ 1 4 , 7 24 , 3 8 ) 4 abaa [ 1 4 , 7 24 )
(a|a) = not used (a|b) = not used not used log(1/4 -7/24) = 5 Code : 01001 ; predecessor : 01000 Both 1/4+1/32 and 1/4 belong to I c

This example shows the following general fact about k-PAAC : the more unexpected behaviours occur in the chain, the smaller is the last I c , the longer is the code. For instance at step t = 4 we expected b with probability 2/3, and observed a. This caused us to choose the small interval I c = [1/4, 7/24). For comparison, if b had occured the code would have been 0110 which is 1 bit shorter. This leads us to the notion of information criteria (IC).

INFORMATION CRITERIA

Let us show how the PAAC may be used to solve a model selection problem being : if x n is a realisation of an unknown MMC ( §2.1), which is its order ? More precisely, we will see how the adaptive aspect of the PAAC is involved.

Coding approach of the model selection problem

As mentionned earlier the k-PAAC length of x n , say L(x n |k), is ruled by the unexpected events in x n : the more unexpected events, the longer the code. Consequently, if x n is ruled by an unknown order k MMC and we try to k-PAAC it at an order k = k , many unexpected events might occur : either because k < k and we do not look far enough in the past, or because k > k and we take into account informations relative to a too far away past which has actually no influence on the future. Thus the minimization of L(x n |k) is an appropriate tool for seeking k .

The works of Rissanen will confirm that idea and establish a link with Information Criteria (IC).

Rissanen's result

In [START_REF] Rissanen | Stochastic complexity and modeling[END_REF] it is shown that L(x n |k) asymptotically behaves as :

BIC(x n |k) = -log (x n | θk ) + (m -1)m k 2 log n ( 2
)
where θk is the maximum likelihood (ML) estimator of order k for x n , i.e. the parameter that maximizes [START_REF] Akaike | A New Look at the Statistical Model Identification[END_REF]. BIC stands for Bayesian Information Criterion and enters the formalism of IC first introduced by Akaike [START_REF] Akaike | A New Look at the Statistical Model Identification[END_REF] ; let us mention [START_REF] Schwarz | Estimating the dimension of a model[END_REF][START_REF] Nishii | Maximum likelihood principle and model selection when the true model is unspecified[END_REF][START_REF] El | Order selection, stochastic complexity and Kullback-Leibler information[END_REF] in addition to [START_REF] Akaike | A New Look at the Statistical Model Identification[END_REF][START_REF] Rissanen | Stochastic complexity and modeling[END_REF] as important steps in the theory of IC.

Here is the idea behind IC : the first term of the criterion (2), referred to as the ML term, decreases as k grows. This is mainly because the ML estimator θk fits the datas more accurately if we let him look far away in the past. This phenomena is known as overparametrization and is the major problem to be solved in model selection, it appears on figure 1. On the other hand, the second term, the penalty, increases as k grows due to (m -1)m k which is the number of free parameters in the MMCs model of order k. Therefore, the minimization of IC over k realizes a balance between the data fitting, measured by the ML term, and the complexity of the model needed to obtain such a fitting, measured by the penalty.

The quantity BIC(x n |k) is much faster to compute than L(x n |k) the encoder should use BIC before encoding to find which order will achieve the minimum codelength.

One can design a non-adaptive order k-predictive arithmetic coding process whose codelength would be exactly log (x n | θk ) = ML . However, this process requires to send the parameter θk for decodability and, especially, it no longer answers the problem of order selection since ML suffers the overparametrization issue. In terms of IC, the adaptive aspect of the process creates the penalty term which avoids overparametrization, see again figure 1.

Comparison of actual codings with criterion

We generate a realization x n of an order k = 5 MMC with m = 2 and n = 25000. For k = 0, . . . , 10 we encode it with k-PAAC process. We also compute the criterion BIC(x n |k) and the quantity ML =log (x n | θk ). Results are presented on figure 1 divided by n to express them as a bitrate.

As expected, BIC and k-PAAC curves present a minimum at k = k while the ML method overparametrizes at k = 9.

Note that, when computing BIC, it is desirable to have enough observations compared to the number of free parameters, empirically :

n ≈ α(m -1)m k with α ≥ 20 (3)
FIG. 1 -Superposition of codelengths and criteria.

would be good. If n is too small behind the number of transition probabilities to be estimated, those transitions do not occur often in the chain and their estimation is weak, resulting in the penalty to dominate the ML term. An alternative would be to compute the number of transitions actually observed in the chain and plug them in (2) instead of (m-1)m k .

LOSSLESS CODING OF RAW IMAGES

Let [[p, q]] be the set of integers from p to q. Let us choose an r × c greyscale image and set n = rc. Firstly, the image has to be turned into a vector x n ∈ I n . For order k ≥ 1 codings, the way this linearization is done does matter since one does not want to lose proximity information on the pixels. We have chosen the "zigzag" linearization used in 8×8 blocks of DCT transform in JPEG norm [START_REF] Zaid | Transform image coding with global thresholding : application to baseline jpeg[END_REF]. Other transformations have been tested and results are quite similar. Let us now describe our lossless coding method.

Lossless coding method

It is a two-part coding technique. In first, choose a partition P of I = [0, 255] ; that is a set of m disjoined intervals (I j ) j∈[ [1,m]] whose union is I. Then, from x n , form a new chain y n as follows :

∀i ∈ [[1, n]], y i = m ∑ j=1 j ¡ I j (x i ).
(4) That is, each y i denotes the number of the interval of P in which x i falls. The chain y n has values in E = [ [1, m]]. For k an order, we denote by L(y n |k, P) its k-PAAC codelength. If m = 1, we set L(y n |k, P) to 0.

Secondly, we denote by A j the number of integers in I j . Once y i = j is known one needs, in order to recover x i ∈ I j , to specify which one of those integers x i actually is. This is done for each x i ∈ I j by a simple code with fixed length log A j . Therefore, the number of bits required to recover x n from y n is L(x n |y n ) = ∑ m j=1 n j log A j . For decodability, one should also send the partition chosen to encode. We do not take this into account here since the FIG. 2 -Lossless estimated bit-rates of Lena at order 0,1,2. codelength required to this end is very small compared to the quantities L(y n |k, P) and L(x n |y n ) we work on.

Let us note L(x n |k, P) := L(y n |k, P) + L(x n |y n ) the total lossless codelength of x n with help of the partition P.

Choice of partition and order of prediction

As m grows L(y n |k, P) also grows because y n has values in [ [1, m]]. By opposition L(x n |y n ) decreases since the intervals I j get smaller. Consequently, there should exist a partition P which balances those two phenomena by minimizing the codelentgh L(x n |k, P). This argument takes place in the theory of Minimum Description Length (MDL) introduced by Rissanen and for which we refer to Grunwald and al. [START_REF] Peter | Advances in Minimum Description Length : Theory and Applications (Neural Information Processing)[END_REF].

We estimate L(y n |k, P) by BIC(y n |k), see §3. We then define the following criterion as an estimation of the lossless order k coding of x n with the partition P : CRIT(x n |k, P) = BIC(y n |k) + L(x n |y n ).

(

We restrict ourselves to regular partitions ; i.e. partitions P(m) whose intervals all have length 256/m. We work with the 512 × 512 greyscale Lena image.

Figure 2 presents, for m ranging from 1 to 256 the estimated bit-rate CRIT(x n |k, P(m))/n for k = 0, 1, 2. For k = 1, the condition (3) is satisfied for m up to 115 but we still give the k = 1 curve up to m = 256 for completeness. The algocomplexity increases considerably with the order k and computations for k ≥ 2 shows no significant improvements ; in the case k = 2 we went up to m = 30 which makes α about 10.

Note that our coding technique with P(1) is equivalent to the pgm format1 . In the other extreme case, with P(256) we get y n = x n and L(x n |y n ) = 0 ; this means we directly encode the chain x n with the k-PAAC process. Considering this, figure 2 shows how a mix of those two methods leads to better bit-rates. The minimization of the criterion (5) tells us which partition is to be chosen in order to get the correct mix.

More important, 1-PAAC is clearly seen to reaches better bit-rates than 0-PAAC : roughly 7 bpp with huge P(200) partition for 0-PAAC against 5.4 bpp with P(50) for 1-PAAC. Note that the order k chosen for the coding process only affects the first term BIC(y n |k) of the criterion (5), hence we FIG. 3 -Laplace distribution and histogram chosen by [START_REF] Rissanen | Generalized Kraft inequality and arithmetic coding[END_REF]. may also give the following interpretation of the curves in figure 2 : no matter how we quantize them via a partition, the grey scales in our image should not be considered independent but rather of order 1. Unsurprisingsly, that dependance of a pixel greyscale on its neighboors may be shown this way on most of common images which content is comprehensible by the human brain.

Histogram selection statistical problem

It is interesting to note that the criterion (5) may be directly extended to the histogram selection statistical problem : if f is an unknown density on an interval I and x n is a sample from this density, which partition of I is to be chosen for building an histogram estimator of f ?

For such a partition P, by independence of x n and formula (4), it is readily seen that the y i 's are independent so that the 0-PAAC of y n will be the best. Let us denote by L j the length of I j and suppose that each I j contains a number of real numbers proportional to L j . Then, up to terms which do not depend on P and after little calculations, the estimated lossless order 0 codelength of x n using P is :

CRIT(x n |0, P) = BIC(y n , 0) + L(x n |y n ). CRIT(x n |0, P) = - m ∑ j=1 n j log n j nL j + m -1 2 log n. ( 6
)
This criterion is in shape really similar to the one used by Birgé and al. in [START_REF] Birgé | How many bins should be put in a regular histogram[END_REF] except it has a coding background which justifies its use. Moreover it is not restricted to regular partitions of I. If I is supposed to contain R real numbers, there could be 2 R-1 partitions to be tested, which is huge. Rissanen and al. presented in [START_REF] Rissanen | Density estimation by stochastic complexity[END_REF] a dynamic programing method which shrinks to O(R 2 ) the number of computations required to find which one of the 2 R-1 partitions achieves the minimum of [START_REF] Rissanen | Generalized Kraft inequality and arithmetic coding[END_REF]. For illustration, we present in figure 3 the partition chosen on a 2000-sample from the Laplace distribution used to represent DCT coefficients in the JPEG norm. We assume that I = [-5, 5] and R = 200.

LOSSY CODING OF RAW IMAGES

We keep the same linearization as in §4 to turn an image into a vector x n and now describe our lossy coding method. FIG. 4 -Estimated Lena's bit-rates for 0-PAAC and 1-PAAC.

Lossy coding method

For P a partition of [0, 255] in m intervals, we define the [[1, m]]-valued chain y n as in (4). Next, we quantize the datas x n on P at their barycenter. That is, for each j ∈ [[1, m]], we consider all x i 's falling into I j , compute their barycenter, round it to the closest integer B j and finally set all those x i 's to B j . This gives a new image with only m grey levels, this is where the loss occurs. Moreover, that quantization creates an injective map :

B : [[1, m]] -→ [[0, 255]] j -→ B j
With the help that map, the decoder is able to reconstruct the quantized image from only the chain y n ; therefore B is to be sent. However, the coding of such a map is very short compared to the codelength of the chain y n , so we drop it. Now we are left to encode y n with the k-PAAC process, hence the estimation of the lossy codelength of our image by the BIC criterion (2) :

BIC(y n |k) = -log (y n | θk ) + (m -1)m k 2 log n.

Influence of the order on bit-rates

still restrict ourselves to regular partition P(m) and work with Lena. Figure 4 presents the estimated bit-rates BIC(y n |k)/n for m ranging from 1 to 256 and orders k = 0, 1. For any m, the fact that the k = 1 curve is under the k = 0 curve means, as in §4 and via IC interpretation, that the chain y n is of order 1 rather than order 0.

Comparison involving distortion

Each value of m a certain quantization, thus a certain distortion. We measure this distortion by the Peak Signal to Noise Ratio (PSNR) and plot it against the corresponding bit-rate of 0-PAAC and 1-PAAC in figure 5. For illustration, we present in figure 6 the two quantized Lena images obtained for m = 3 and m = 13 with their respective PSNR. We also give bit-rates achieved by 0-PAAC and 1-PAAC on each of those image. For instance, this shows that at an imposed rate of about 1.4 bpp, the 1-PAAC allows to encode Lena with a PSNR of 33.15 dB while the 0-PAAC only gives 22.11 dB.

PERSPECTIVES

As mentionned in the introduction we did not provide efficient compression results by intentionally working on raw images. Therefore it would be interesting to insert the discussed binary coding methods after, for instance, the wavelet transform block of the JPEG2000 norm. In order to compress, one should in first determine with the BIC criterion (2) the order of the sequence of wavelet coefficients and then use the criterion [START_REF] Nishii | Maximum likelihood principle and model selection when the true model is unspecified[END_REF] to determine the partition which allows to encode those coefficients efficiently.

FIG. 5 -

 5 FIG.5-Estimated Lena's bit-rates/PSNR for 0 and 1-PAAC.

http ://www.imagemagick.org/script/formats.php

ACKNOWLEDGMENTS

The authors would like to thank the PIMHAI, INTERREG IIIB "Arc Atlantique" project for its support in the writing of this paper.

INFORMATION CRITERIA AND ARITHMETIC CODINGS : AN ILLUSTRATION ON RAW IMAGES