Angèle Laurent

Claudio Satriano

Pascal Bernard

Pascal Bernard Detection

Detection, location and characterization of VLF events during the 2018-2019 seismovolcanic crisis in Mayotte

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

Detection, location and characterization of VLF events during the 2018-2019 seismovolcanic crisis in Mayotte

Angèle Laurent (1) , Claudio Satriano (1) , Pascal Bernard (1) (1) Institut de Physique du globe de Paris, Université Paris Diderot-PRES Paris Université -Paris, France (alaurent@ipgp.fr)

Context :

Mayotte volcano-seismic crisis started on 10 may 2018, 3 000 events manually localised in two swarms at 10 and 30 km, on sea, on the East side of Mayotte island. OBS with hydrophone channel have been deployed during severals marine campaigns. During Mayobs1, a new submarine volcano is discover around 50 km on East of Mayotte island [Feuillet, et al, 2019]. GNSS data have showed that the island is moving of a dozen of centimetres downward and toward East and vertically. These movements are interpreted as the deflation of a source at 30 km deep on the East of Mayotte. The total estimate volume on November was around 2 km 3 [Lemoine et al, 2019].

Overview:

Results : VLFs catalog :

All along the seismic crisis of Mayotte, we detect more than 300 events. We observe that the central period is increasing until almost 16s (oct 2018) and since is in constant decay. This change of central period of VLFs might be interpreted like the source degradation.

Classification :

We performed an amplitude ratio between three band of frequencies : Hight Frequencies (HF) : 10 -20 Hz Med Frequencies (MF) : 1 -5 Hz Low Frequencies (LF) : 0.05 -0.1 Hz → 10 -20 s The high frequency band highlights the volcanotectonic earthquake (VT) occurring all along the Mayotte crisis. The medium frequency band enhance the long period earthquake (LP) and the low frequency band reclaim the very low frequency signals (VLF). The aim of this classification is to determine the possible link between VT, LP and VLF, all three present in the volcanic crisis of Mayotte. Vt or LP events that occur before a VLF can be thought as precursors or triggers, in this case they can inform us about the current phenomenon. We show three cases : VLF in the coda of a VT event (A), LP at the beginning of a VLF (B) and a VLF that seems to be "alone" (C).

Localisation :

The retrieved epicentral locations are 20 km east of Petite Terre. VLFs are closer to the seismic swarm than to the volcano.

Resonator models :

:

☞ Confirmation of the VLFs localisation by amplitude ratios at station pairs.

☞ A automatic classification of VLFs waveforms with or without VT or LP "precursors". Analyse the frequency content and the localisation of the VT and LP signals before a VLF.

☞ To better understand the source phenomenon we should use a numerical modelling of the VLF waveform signal as a dumping oscillator through a medium (soil and/or water). Event depth (Z) depends on chosen velocity: Z>20 km @ 2.5 km/s; Z<5 km @ 4.5 km/ s. A velocity of 3.5 km/s gives depths between 5 and 20 km, in better accordance with the chosen velocity value.

Methodology :

STA/LTA detector :

We compute STA/LTA ratio on the filtered (10-20 s) signal and use a triggering threshold of 4 (red dashed line). We only keep detected events with central period between 16 and 14.5 s, and Ra ≥ 3.

Spectrogram detector :

We compute the spectrogram of signals filtered between 10 and 20 s and 1-bit normalised. We then search for monochromatic transients (horizontal lines) on the spectrogram by looking at the derivative of f_max(t).

Back-projection localisation :

We cross-correlate for each pair of stations (broadband and hydrophones) VLF signals filtered between 10 and 20 s and 1-bit normalised. Station-pair cross-correlations are then mapped in 3D space using a constant velocity. Cross-correlation maps are finally stacked to find the maximum likelihood source location.

  VLF major event is recorded 2019-11-11, with a duration of 25 minutes, a central period of 15s and Rayleigh waves polarisation. The wave polarisation is retrograde and longitudinal compared to the direction of wave propagation. The hypothesis of [Lemoine et al, 2019] is oscillating fluids into a magmatic reservoir. A similar case is seen at Le Rocard submarine volcano, interpreted by [Talandier et al., 2016] as a migration of resonance fluids in an open superficial and pressurised pipe. A max /A mean Mayobs 1 : 2019-02-24 -2019-05-10 Mayobs 2 : 2019-05-27 -2019-06-14 Mayobs 6 : 2019-07-11 -2019-09-28

  Fazio et al, 2018 proposed severals Tornillos sources models from laboratory experiences, we choose an open-closed pipe (a) and a rectangular cavity (b) to modelled our resonating source. It helps us to better constrain the physical dimensions of our system. According to the central period of 15s, the source of the VLFs may have a dimension of several kilometers.

  Fazio, M., Alparone, S., Benson, P. M., Cannata, A., & Vinciguerra, S. (2019). Genesis and mechanisms controlling tornillo seismo-volcanic events in volcanic area. Sci Rep 9, 7338. https://doi:10.1038/s41598-019-43842-y Feuillet, N., et al (in prep). Birth of a large volcanic edifice offshore Mayotte through lithosphere-scale rifting. Lemoine, A., Bertil, D., Roullé, A., & Briole, P. (2019, February 28). The volcano-tectonic crisis of 2018 east of Mayotte, Comoros islands. https://doi.org/10.31223/ osf.io/d46xj Poiata, N., C. Satriano, J.-P. Vilotte, P. Bernard, & K. Obara. (2016). Multiband array detection and location of seismic sources recorded by dense seismic networks, Geophys J Int, 205(3), 1548-1573, doi:10.1093/gji/ ggw071 Talandier, J., Hyvernaud, O., & Maury, R. C. (May 2016).Unusual seismic activity in 2011 and 2013 at the submarine volcano rocard, society hot spot (french polynesia). Geophysical Research Letters, 43(9) :4247-4254.https://doi.org/10.1002/ 2016gl068342.


domain: math info info.info-ro

text:
Marinela Shehaj
email: marinela.shehaj@hds.utc.fr

Ilya Kalesnikau
email: i.kalesnikau@tele.pw.edu.pl

Dritan Nace
email: dritan.nace@hds.utc.fr

Michał Pióro
email: m.pioro@tele.pw.edu.pl

Ermioni Qafzezi
email: ermioni.qafzezi@etu.utc.fr

Modeling Transmission Degradation on FSO Links Caused by Weather Phenomena for WMN Optimization

Keywords: free space optics, wireless mesh networks, robust optimization, uncertainty sets, hitting sets

This work deals with wireless mesh networks (WMN) composed of FSO (free space optics) links. Although FSO links realize broadband transmission at low cost, their drawback is sensitivity to adverse weather conditions causing transmission degradation on multiple links. Hence, designing such FSO networks requires an optimization model to find the cheapest configuration of link capacities that will be able to carry an acceptable level of the demanded traffic in all weather states that can be foreseen in network operation. Such a model can be achieved using robust optimization techniques, and for that it is important to find a tractable way of characterizing possible link (capacity) degradation states corresponding to weather conditions not known in advance. In the paper we show how weather records can be translated to vectors of link capacity available (after degradation) in particular weather states, and how the resulting link degradation states may be represented mathematically in a compact and tractable way to be exploited in optimization. To solve the second task we will make use of a generalization of a combinatorial problem of finding a minimum hitting set to deduce a compact set approximating a given set of link degradation states. Finally, we illustrate the effectiveness of our model by means of a numerical study.

I. INTRODUCTION

Context and motivation. The FSO networks considered in this paper are characterized by fluctuations of capacity currently available on the links caused by adverse weather conditions. For this reason, optimization of FSO becomes complex as it has to cope with degradation of data transmission on the FSO links at the network design stage. As a continuation of our previous studies, this issue is undertaken below by introducing a new way of characterizing link degradation states for optimization purposes. Here, an important element is the use of proper, channel condition-dependent, signal modulation and coding schemes (MCS) at the transmitters. This allows affected link transmissions to remain operational, assuring proper data decoding at the receivers at the expense of decreased data transmission rate, i.e., link capacity. Thanks to that, the influence of degraded weather condition on link capacity (bit-rate) is kept under control but at some price that one needs to evaluate and optimize. In this study, the fraction of the maximum attainable (i.e., nominal) link capacity lost due to a change of its MCS is referred to as link degradation ratio. Typical values of such ratios are 0.50, 0.75 and 1, and each particular weather condition that can occur defines a degradation state referred to as the link degradation state.

Problem studied. In our previous papers [START_REF] Nace | An optimization model for robust FSO network dimensioning[END_REF], [START_REF] Shehaj | Dimensioning of hybrid FSO/fiber networks[END_REF] we presented optimization techniques addressing the problem of link capacity fluctuations in FSO networks caused by adverse circumstances, such as bad weather affecting the received signal strength in wireless networks. This problem is complex since it requires consideration of the state-dependent capacities for all network traffic demands in all possible network states caused by weather degradations. Closely related to that is the issue of preparing the data required to characterize the set of link degradation states for the dimensioning problem. Such data are obtained by translating weather states into network states characterized by link degradations ratios. We take into account how different weather factors as rain, snow, visibility, etc. impact the transmission quality measured by means of link margin (see [START_REF] Bouchet | Free-Space Optics: Propagation and Communication[END_REF]). Typically, a given weather condition affects a subset of network links, and each affected link loses a portion (fraction) of its nominal capacity -the quantity called link degradation ratio above. Then, each particular weather condition that may occur defines a network state referred to as link degradation state. We will describe a full procedure to convert the weather data to the link degradation states set used in our optimization approach. This is the first question studied in this paper. On the other hand, this gives rise to a large number of such states to deal with, while their prediction is based solely on the weather conditions observed in the past. Thus, the set of the considered link degradation states, called the reference set of states, typically contains a large number of states of order of number of hours included in a given reference time horizon (i.e. 24 × 365 = 8760 for a period of one year). Furthermore, we need to dimension the network to deal with future weather states as well, which motivates the need for a compact characterization of the reference sets. A certain type of such compact sets (called uncertainty sets) was studied in [START_REF] Nace | An optimization model for robust FSO network dimensioning[END_REF], [START_REF] Shehaj | Dimensioning of hybrid FSO/fiber networks[END_REF], [START_REF] Pióro | Optimizing flow thinning protection in multicommodity networks with variable link capacity[END_REF]. In this paper we propose a version of uncertainty sets that estimate the original sets of degradation states more accurately. Our alternative characterization of the link degradation states captures the incidence of a group of links in several degradation states; more precisely, we find the most frequent subsets of links simultaneously affected by the weather phenomenona.

Contribution. The contribution of this paper is twofold. First, we first present a way how weather conditions can be converted to corresponding link capacity degradation vectors. We enumerate weather factors affecting the link margin, give an integrated formula for link margin, and explain how the value of link margin is related to the proper MCS choice. Second, we propose an enhanced characterization of link degradation states that captures everyday fluctuation of weather conditions. For that we define a special kind of uncertainty set, called a covering K-set, using a generalization of the minimum hitting set problem. This uncertainty set, which is new, comes as an alternative of two former uncertainty sets already proposed and investigated before in [START_REF] Nace | An optimization model for robust FSO network dimensioning[END_REF], [START_REF] Shehaj | Dimensioning of hybrid FSO/fiber networks[END_REF], [START_REF] Pióro | Optimizing flow thinning protection in multicommodity networks with variable link capacity[END_REF]. The presented considerations are illustrated by a numerical study. In our view the p-covering model is more natural as it better captures the specificities of the set S because it represents the frequency of subset of links used in it, contrary to previously proposed sets which capture only the topological aspects of the network. Paper organization. In Section II we present a mathematical formulation of the problem of converting the data weather to an explicit list of link availability states (degradation states may be immediately obtained). In Section III we recall first our previous works in approximating the link degradation states set through the so called link K-set and node K-set, which are known as the uncertainty set. Then we introduce a new type of uncertainty set based on a generalization of the minimum hitting set problem. Next, in Section IV, we present a numerical study that illustrates effectiveness of the proposed uncertainty set representation. Finally, in Section V, we give concluding remarks.

II. IMPACT OF WEATHER CONDITIONS ON LINK CAPACITY

This section is dedicated to a general presentation of the impact of weather conditions in FSO data transmission performance and the concept of link margin. Further, we devote special attention to the problems associated with converting the link margin into availability ratio and the modulation schemes that better fit our system. Finally, we consider an example of FSO system in different weather condition and the modulation scheme that we apply.

A. Conversion formulae

The link margin of a particular FSO link is a function of the atmospheric conditions, as well as of the distance between the terminals of the link, the power of the transmitter, and the sensitivity of the receiver. Link margin can be considered as the extra-power available at the receiver (measured in milliwatts [mW]), compared to the minimum requirement of keeping the link active. Considering the technical data provided by the manufacturer and the weather conditions for a specific geographical zone, we can compute the link margin in order to obtain the availability ratio of the links using (after [START_REF] Bouchet | Free-Space Optics: Propagation and Communication[END_REF]) the following formula:

(1)

M Link = P e + |S r |-Att Geo -P Syst -Att F og -Att Rain -Att Snow
where M Link is the link margin (in dB), P e is the emitted power, expressed in decibels per mW, i.e., in dBm, S r is the receiver sensitivity in decibels per milliwatt (in dBm), Att Geo is the geometric attenuation (in dB), P Syst is the equipment loss (in dB) given by the manufacturer (possibly multiplied by 2), Att F og is the fog attenuation (in dB), Att Rain is the rain attenuation (in dB), Att Snow corresponds to the snow attenuation (in dB). Note that in formula (1) one group of the factors are equipment dependent, while the others are weather dependent. We will not discuss the particular factors here (for details see [START_REF] Bouchet | Free-Space Optics: Propagation and Communication[END_REF]) but focus on the modulation and coding schemes used to evaluate link degradation ratios for different weather scenarios, see Tables I andII.

B. Adaptation of modulation and coding schemes

Bad weather conditions deteriorate the FSO link, and in severe cases the transmission capacity of a link can be lost even entirely. In order to keep the link active, the data rates of the affected links can be increased by changing the signal modulation scheme, depending on the receiver sensitivity.

The advantage of using QAM is that it is a higher order form of modulation and as a result the data rate of a link can be increased. While higher order modulation rates are able to offer much faster data rates and higher levels of spectral efficiency for the radio communications system, this comes at a price. As a result, many radio communications systems now use dynamic adaptive modulation techniques. They sense the channel conditions and adapt the modulation scheme to obtain the highest data rate for the given conditions. Selecting the right order of QAM modulation for any given situation, and having the ability to dynamically adapt it can enable the optimum throughput to be obtained for the link conditions for that moment. Reducing the order of the QAM modulation enables lower bit error rates to be achieved and this reduces the amount of error correction required. In this way the throughput can be maximised for the prevailing link quality [START_REF] Hanzo | Quadrature Amplitude Modulation: From basics to Adaptive Trellis-Coded, Turbo-Equalised and Space-Time Coded OFDM, CDMA and MC-CDMA Systems[END_REF], [START_REF] Kahn | Modulation and detection techniques for optical communication systems[END_REF].

If we note M link (the link margin of link i in dB) and M * linkmax ( the maximum link margin of our system in the best conditions in dB), in general we can do the correspondence shown in Table I between link margin of a FSO system and the modulation scheme that we apply. ; link margin modulation scheme degradation ratio

M link > 0.5M * linkmax 16-QAM 0% 0.25M * linkmax < M link ≤ 0.5M * linkmax 4-QAM 50% 1 < M link ≤ 0.25M * linkmax BPSK 75% otherwise
loss of communication 100%

C. An example

In order to explain the elements that determine the link margin we consider the following example. We assume an FSO transmission system with the distance between the transmitter and receiver equal to 7 km, which uses a wavelength of 1550 nm for data transmission. Four different weather conditions are taken into account: sunny, heavy rainfall, light snowfall, and heavy snowfall. Table II illustrates the link margin and the modulation and coding scheme applied to this system. For this example we have assumed that the M * linkmax is the maximum links margin reached for a visibility of 15 km and it corresponds to 49.39 dB.

A. Optimization problem

In our considerations we assume an undirected network graph G = (V, E) composed of the set of nodes V and the set of undirected links E ⊆ V |2| , where V |2| is the set of all 2-element subsets of V, i.e., each link e ∈ E represents and unordered pair {v, w} of nodes for some v, w ∈ V (v = w). The unit capacity cost on a given link e is denoted by ξ(e). As the network links are exposed to adverse weather conditions, we consider an explicit list of link degradation states S induced by considered set of weather conditions. For each state s ∈ S the link degradation ratios β(e, s) are computed for all links using the formulae described in Section II (and summarized in Table I) applied to the weather condition corresponding to state s.

The considered optimization (network dimensioning) problem consists of finding the values of nominal link capacities y e , e ∈ E, such that their total cost C(y) := e∈E ξ(e)y e is minimized, and in all states the same (given) traffic demand matrix is realized -this implies 100% traffic protection. We assume the global rerouting mechanism [START_REF] Pióro | Routing, Flow, and Capacity Design in Communication and Computer Networks[END_REF] to protect the traffic flows -this in turn means that in each state s ∈ S the traffic matrix can be realized from scratch in available link capacities which are given by β(e, s) • y e , e ∈ E. A mathematical formulation of this problem, not given here for the space reason, can be found in [START_REF] Nace | An optimization model for robust FSO network dimensioning[END_REF].

In fact, for given link degradation state list S, the considered problem is not too difficult to solve by the commercial solvers (like CPLEX). However, an issue appears here since when the network is to be optimized, the list of states is not known and can only be estimated based on historical weather records for the network area. Such an estimation can be done for, say, a coming year by using the data for the past year but this does not guarantee a reliable estimations due to weather variations and enormous number of possible link degradation states implied by future weather conditions.

To deal with this problem the so called uncertainty sets can be used [START_REF] Nace | An optimization model for robust FSO network dimensioning[END_REF]. Such sets, described in a compact way, can be used instead of the explicit list of link degradation states S, and when properly constructed they can provide solutions that cover most of the real weather conditions with a reasonable total capacity cost C(y). To apply this approach the problem formulation must be adjusted in order to use an uncertainty set instead of the explicit list S to characterize link degradation states; such formulations for certain types of uncertainty sets (the so called K-sets), together with a specialized solution algorithm, are given in [START_REF] Nace | An optimization model for robust FSO network dimensioning[END_REF] and [START_REF] Shehaj | Dimensioning of hybrid FSO/fiber networks[END_REF]. As compared to S, these K-sets contain a limited number of link degradation sets, and are compactly described through a system of inequalities. They approximate S through an uncertainty set including all states with up to K links or K nodes degraded with the same link degradation ratio β(e), e ∈ E; in the latter case, all links incident to the degraded nodes are assumed to be affected. Hence, we may say that parameter K specifies the maximum number of areas (each of such area is composed of one or several links), that can be simultaneously affected.

In our approach, we have assumed global rerouting, that is we are allowed to reroute all traffic, affected or not, for each network state. Then, we solve the link dimensionning problem for various weather scenarios. More precisely, we first optimize link capacity (i.e., we solve the network dimensionning problem) taking into account all weather states for the time horizon, for which we produce the reference set of states. Then we dimension the network assuming a special (uncertainty) set of link degradation states. To solve the optimization problem we have used a cut generation algorithm. It is an iterative algorithm which results from application of the classical Bender's decomposition method to our optimization problem. In each iteration the master problem involving only the capacity variables y is solved and then its optimal solution y * is tested for feasibility with respect to the chosen uncertainty set. If the test is positive, the algorithm is stopped and y * is optimal for the main problem. If not, new inequalities deduced from the feasibility tests are added to the master problem and the algorithm is reiterated.

B. Uncertainty sets

Let F ⊆ 2 E be a given family of subsets of the set of links E and let ∆(e, f ) denote the binary incidence coefficient equal to 1 if, and only if, e ∈ f for f ∈ F. Now, for a given integer parameter K ≥ 1, consider a set U(K, F) ⊂ {0, 1} |E|+|F | described by the following system of inequalities in binary variables u = (u e : e ∈ E) and y = (y f : f ∈ F):

f ∈F y f ≤ K (2a) u e ≤ f ∈F ∆(e, f )y f , e ∈ E (2b)
u e ∈ {0, 1}, e ∈ E; y f ∈ {0, 1}, f ∈ F. (2c)
The purpose of introducing U(K, F) is to define the so called uncertainty set, which is a set of u vectors U (K, F) that describe a list of degradation states. The definition is as follows:

U (K, F) := {u : ∃y ∈ {0, 1} |F | , (u, y) ∈ U(K, F)}. ( 3
)
The idea is that each vector u ∈ U (K, F) defines a set s(u) := {e ∈ E : u e = 1} of degraded links. Assuming that each link is in all states degraded with the same link degradation ratio β(e), we get a full description of the link degradation states determined by a given uncertainty set.

In fact, the so introduced notion of uncertainty set is a generalization of some previously used notions. To illustrate this point let us first consider the family F = {{e} : e ∈ E}. Since in this case we can identify the singletons {e} composing family F with the corresponding links, the set U(K, F) ⊂ {0, 1} 2|E| is defined by:

e∈E y {e} ≤ K (4a) u e ≤ e ∈E ∆(e, {e })y {e } , e ∈ E (4b) u e ∈ {0, 1}, e ∈ E; y {e} ∈ {0, 1}, e ∈ E. (4c)
Clearly, the resulting uncertainty set U (K, F) is defined simply like this:

e∈E u e ≤ K (5a) u e ∈ {0, 1}, e ∈ E. (5b)
The above set describes in a compact way the list of states s(u), u ∈ U (K, F), containing all the states corresponding to simultaneous degradation of a subset of k links (there are |E| k of such states) for all k = 0, 1, . . . , K. The so defined uncertainty set corresponds to the notion of link K-set considered in [START_REF] Nace | An optimization model for robust FSO network dimensioning[END_REF].

Another kind of a known uncertainty set is obtained for F = {δ(v) : v ∈ V }, where δ(v) denotes the set all links in E incident with node v ∈ V. Then the set is as follows:

v∈V y δ(v) ≤ K (6a) u e ≤ v∈V ∆(e, δ(v))y δ(v) , e ∈ E (6b) u e ∈ {0, 1}, e ∈ E; y δ(v) ∈ {0, 1}, v ∈ V. (6c)
Simplifying the notation by using variables y v instead of y δ(v) , we can rewrite (6) like this:

v∈V y v ≤ K (7a) u {v,w} ≤ y v + y w , {v, w} ∈ E (7b) u e ∈ {0, 1}, e ∈ E; y v ∈ {0, 1}, v ∈ V. (7c)
The set U (K, F) resulting from ( 7) corresponds to the notion of node K-set considered in [START_REF] Shehaj | Dimensioning of hybrid FSO/fiber networks[END_REF].

C. Covering K-sets

Below we will show a way to find families F for constructing uncertainty sets for a given (possibly very large) state list S.

Let us assume that all states s in S (each representing a set of simultaneously degraded links) have cardinality greater or equal to a given integer parameter p. Now, let X |p| denote the family of all p-element subsets of a given set X, and consider the following binary programming problem:

min f ∈E |p| x f (8a) f ∈s |p| x f ≥ K, s ∈ S (8b) x f ∈ {0, 1}, f ∈ E |p| . (8c)
Finally we define the family F := {f ∈ E |p| : x * f = 1} (where x * is an optimal solution of ( 8)) and call the so obtained uncertainty set U (K, F) the p-covering K-set of the list of states S. Note that formulation ( 8) is actually a generalization of minimum hitting set problem -a well known N P -hard combinatorial problem [START_REF] Garey | Computers and Intractability[END_REF].

The so obtained family F may be seen as containing those subsets (of reasonable size p) of links that are frequently simultaneously affected, and we assure protection against link degradation states that contain all links included in up to K of such subsets.

To illustrate the concept of p-covering K-set, consider a simply network instance with 11 links (E = {e 1 , e 2 , . . . , e 11 }) and the set S = {s 1 , s 2 , . . . , s 7 } composed of 7 states given in Table III.

Solving the binary problem (8) for p = 2, K = 1, we obtain the following family F (composed of three 2-link subsets of E) defining the corresponding 2-covering 1-set: {{e 2 , e 4 }, {e 6 , e 8 }, {e 7 , e 9 }}. For p = 2 and K = 2, the Note that an advantage of using covering K-sets is that they are defined using the information of the actual list of states S that is to be approximated; this feature is not present when specifying link or node K-sets.

IV. NUMERICAL RESULTS

Below we present results of a numerical study illustrating the performance of our optimization approach for various weather scenarios. All numerical calculations were performed on an Intel (R) Core (TM) i7-4610M, 3.0 GHz computer with 16 GB of RAM. The algorithms were coded in Java and the optimization problems were solved with IBM ILOG CPLEX 12.6 (using Concert Technology) running with the default setting.

In our numerical tests, we have dimensionned the network assuming a special (uncertainty) set of link degradation states represented by the so-called link, node K-set or p-covering Kset where K specifies the number of areas (represented by network links, nodes or group of links) that can be simultaneously affected by weather conditions. For all the considered K-sets the degradation ratio β(e) = 0.50 was assumed uniformly for all affected links in the network. This value represents the degradation ratio of more than 80% of affected links all over the degradation states (not considering the normal states, i.e. corresponding to sunny days with no link affected), while two-third of such states contain only links with the same degradation ratio β(e) = 0.50. We run the optimization approach for a selected set of values of parameter K, checking to what extent the demand traffic matrix is satisfied by the Kset based on solution in all states in the reference degradation set. Finally, we have examined the percentage of carried traffic achieved for different K-sets as compared to carried traffic observed when all states in the reference set are tested for the obtained dimensionned network.

A. Network nobel-germany

We have tested our optimization approach on a moderatesize network example derived from the instance called nobelgermany that can be found in SNDlib [START_REF] Orlowski | SNDlib 1.0 -Survivable Netwrok Design Library[END_REF]. The instance is composed of 17 nodes and 26 undirected links, and there are 120 undirected traffic demands. We have used three different weather scenarios constructed for the purpose of this study.

Indeed, using the weather records from [START_REF]World Weather Online[END_REF], [START_REF]Köppen-Geiger climate classification[END_REF], we have built up three different weather scenarios: the first scenario is typical for southern Europe (called "Mediterranean climate" scenario), the second one is typical for middle Europe (called "moderate continental climate" scenario), and the third is typical for northern Europe (called "temperate continental climate" scenario). These scenarios were then used to estimate link degradation state for each hour of a typical year for nobelgermany. A detailed description of such an estimation process is given in [12], while other details on the obtained network instances are given in [START_REF] Shehaj | Link dimensioning of hybrid FSO/fiber networks resilient to adverse weather conditions[END_REF].

Tables IV, V and VI show the results for different types of uncertainly sets (link K-sets, node K-sets) compared to p-covering K-sets approach described above (for p = 2). Such sets have been computed in advance with respect to the reference set of states. Typically, we have used formulation [START_REF] Garey | Computers and Intractability[END_REF] with p = 2 and K ∈ {2, 3} to compute such subsets. Although the problem is N P -hard, the computation time for the considered instances does not exceed 3 minutes. The number of subsets are for K = 2 and K = 3 respectively 18 and 20 for "cold continental climate" scenario, 45 and 60 for "temperate continental climate" scenario, and finally 67 and 113 for "Mediterranean climate" scenario. In each table, the uncertainty set used is pointed in the first column where link(K) represents a link K-set, node(K) a node Kset, and set(K) a covering K-set. The second column gives the optimal capacity cost C * (K). The third column represents the average carried traffic AT (K). The last column shows the computation time in [s]. For our results we have considered only the states where the network is connected. (It happens that in 10-15% of the time the weather conditions are extremely bad so that can cause the disconnection of the network.) When comparing the total capacity cost and the average carried traffic percentage for different types of uncertainty sets, namely for link K-sets, node K-sets, and 2-covering K-

B. Paris Metropolitan Area network

In this section we will study of a realistic FSO network instance with 12 nodes and 21 links. We call this instance PMAN (Paris Metropolitan Area network) since it was created using the real data for the Paris Metropolitan Area. It should be noted that although all the data used to construct the PMAN instance are real, the instance itself does not exist in reality and was elaborated only for our own purpose. A full characterization of PMAN is available in [START_REF] Nace | An optimization model for robust FSO network dimensioning[END_REF].

We examined the weather conditions in Paris Metropolitan Area observed during one year period from January 1, 2016 until December 31, 2016 available in [START_REF]World Weather Online[END_REF] and extracted a set W of all 366 × 24 = 8784 hourly weather states. Then we translated each such obtained weather state to the corresponding link degradation states. We observed that among all 8784 states there are 7531 connected states (85.74%) and 1253 (14.26%) disconnected states. Among the 7531 connected states, 68.46% are nominal states with no degraded links.

The individual traffic demands in PMAN (the demands are directed and there are 132 of them) are of the order of several Gbps, except for the traffic between Paris 1 and Paris 2 with much higher demand of 74.38 Gbps (in each direction). The full traffic matrix for PMAN can be found in [START_REF] Nace | An optimization model for robust FSO network dimensioning[END_REF]. An M = 10 Gbps FSO module is assumed, as for example the module of the LightPointe's AireLink 80 10Gbps system [START_REF]GHz Radios[END_REF]; the cost of one such module on each link is equal to 1. In our example we have considered all states (8784 states) including 1252 cases when the network is disconnected. As before we have tested our optimization approach for the three uncertainty sets on PMAN as shown in Table VII. Comparing to nobel-germany instances, the results for PMAN pleads clearly in favor of the new uncertainty set proposed in this study. One may note that set(2) has a lower cost compared to link(3) or node(1) while giving one of the highest traffic satisfaction for all uncertainty scenarios. We can observe that in some cases the obtained results don't exhibit the expected relation between the cost and the traffic satisfaction, that is, for the same uncertainty set type, higher is the cost higher the traffic satisfaction should be. The reason for that stands in the fact that these criteria are optimized separately. In our optimization approach we build a link cost-efficient network for some given uncertainty set and we measure what is the highest traffic satisfaction this network can provide for each state of the reference set. Hence, it may happen that one network instance with a given dimensioning distribution performs better than another instance with a higher dimensioning cost but with a different distribution of capacities. This may be observed in Table VII for cases link(3) and link [START_REF] Hanzo | Quadrature Amplitude Modulation: From basics to Adaptive Trellis-Coded, Turbo-Equalised and Space-Time Coded OFDM, CDMA and MC-CDMA Systems[END_REF] or set [START_REF] Shehaj | Dimensioning of hybrid FSO/fiber networks[END_REF] and set(3).

V. CONCLUDING REMARKS

In the paper we have presented some new results on a robust optimization problem related to FSO network dimensioning resilient to link capacities degradation caused by weather conditions. Our solution approach optimizes link capacity taking into account all weather states for a given time period, for which we determine a reference set of (link capacity degradation) states. One of the main issues here is potential (large) size of the reference set, which may make the optimization problem computationally untractable for large networks. Furthermore, we need to construct such a reference set of states using historical weather data records, which may not fit the future states for which the network is being optimized.

To handle this, in our previous work we used the so called link K-sets (where up to K arbitrary links can be degraded simultaneously) and node K-sets (where up to K arbitrary nodes can be degraded simultaneously). In this paper we have investigated a more general notion of K-set, called pcovering K-set, where the corresponding link groups are more appropriate for the reference set approximation, and show how to build such groups of links.

In contrast to previous uncertainty sets, the new set is composed of subsets of links being the most frequently met in the states of the reference set. This is a qualitative step forward in achieving a more natural compact approximation of the reference set. Unlikely to the link and node K-sets, the new set is built by examining explicitly all subsets of the reference set. Next, we test these new uncertainty sets for a selected set of values of parameters p and K (p = 2 and K = 2, 3) and check to what extent the demand traffic matrix is satisfied by the obtained solution in all states in the reference set of states S, and at what cost.

The presented approach has been illustrated by means of a moderate-size network instance (nobel-germany) for three different weather scenarios, as well as using a realistic instance representing the Paris Metropolitan Area Network. The numerical results show that using p-covering K-sets for state modeling gives encouraging results when compared to the previously considered link and node K-set uncertainty sets. This is in particular true for PMAN, a realistic network instance.

As far as the future work is concerned, we will investigate a natural extension of the above-described p-covering set model admitting multiple link degradation ratios.

TABLE I :

 I Link margin and corresponding modulation schemes.

TABLE II :

 II Link margin for 4 typical weather conditions.

		sunny	heavy	moderate	heavy
		day	rain	snow	snow
			[10 mm/h] [2.2 mm/h]	[10 mm/h]
	emitted power [dBm]	51	51	51	51
	visibility [km]	15	5.5	3.0	0.2
	receiver sensitivity [dBm] -38.23	-38.23	-38.23	-38.23
	beam divergence [mrad]	2.5	2.5	2.5	2.5
	aperture surface [m 2 ]	0.025	0.025	0.025	0.025
	Att Geo [dB]	39.83	39.83	39.83	39.83
	Att F og [dB]	0.00026 0.0013	0.059	594.33
	Att Rain [dB]	0	35.23	0	0
	Att Snow [dB]	0	0	47.10	144.89
	system losses [dB]	0	0	0	0
	link margin [dB]	49.39	14.17	0.63	-689.83
	modulation scheme	16-QAM 4-QAM	BPSK communication lost
	III. COMPACT APPROXIMATION OF THE SET OF
	DEGRADATION STATES

TABLE III :

 III An example of 2-covering K-sets. {e 2 , e 3 }, {e 5 , e 10 }. In the above table, columns 3 and 4 illustrate how the sets in S are actually covered by the sets from the corresponding families F.

	state set	composition	p = 2, K = 1 p = 2, K = 2
	s1	{e2, e3, e5, e7, e9, e11}	{e7, e9}	{e7, e9}, {e2, e3}
	s2	{e2, e4, e6, e7, e8}	{e2, e4}	{e2, e4}, {e6, e8}
	s3	{e3, e5, e6, e8, e10, e11}	{e6, e8}	{e6, e8}, {e5, e10}
	s4	{e1, e2, e4, e7, e8, e9, e10}	{e7, e9}	{e7, e9}, {e2, e4}
	s5	{e2, e3, e6, e7, e9, e10}	{e7, e9}	{e7, e9}, {e2, e3}
	s6	{e1, e2, e4, e6, e8, e11}	{e2, e4}	{e2, e4}, {e6, e8}
	s7	{e2, e4, e5, e6, e9, e10}	{e2, e4}	{e2, e4}, {e5, e10}
	resulting family defining the 2-covering 2-set has 2 additional
	subsets:

TABLE IV :

 IV Mediterranean climate.

	uncertainty cost carried traffic computational
	set	C * (K) AT (K)[%]	time [s]
	link(1)	149	99.07	75.41
	node(1)	179	99.57	66.21
	link(2)	188	99.64	73.61
	set(2)	207	99.72	152.31
	link(3)	208	99.76	102.38
	link(4)	221	99.79	95.45
	node(2)	221	99.81	692.24
	set(3)	223	99.79	171.06
	link(5)	228	99.76	112.56
	link(6)	229	99.82	114.34
	link(7)	230	99.82	134.05
	link(8)	230	99.82	163.04
	link(9)	231	99.82	137.26
	link(10)	231	99.82	108.76

TABLE V :

 V Temperate continental climate.

	uncertainty. cost carried traffic computational
	set	C * (K) AT (K) [%]	time [s]
	link(1)	149	92.11	65.03
	node(1)	179	94.50	70.11
	link(2)	188	94.51	82.56
	set(2)	203	95.41	140.02
	link(3)	208	95.87	109.06
	set(3)	218	96.71	123.11
	node(2)	221	96.90	729.97
	link(4)	221	96.67	103.95
	link(5)	228	96.47	122.43
	link(6)	229	96.94	129.25
	link(7)	230	97.03	155.91
	link(8)	230	97.02	166.05
	link(9)	231	97.02	136.18
	link(10)	231	97.15	111.53

TABLE VI :

 VI Cold continental climate.

	uncertainty. cost carried traffic computational
	set	C * (K) AT (K)[%]	time [s]
	link(1)	149	81.06	68.58
	node(1)	179	84.82	72.38
	link(2)	188	84.83	80.61
	set(2)	190	85.22	96.95
	set(3)	205	87.35	120.74
	link(3)	208	87.19	110.31
	link(4)	221	88.85	110.68
	node(2)	221	89.46	742.51
	link(5)	228	88.79	118.61
	link(6)	229	89.73	122.52
	link(7)	230	89.97	150.59
	link(8)	230	89.92	168.46
	link(9)	231	90.07	135.62
	link(10)	231	90.08	103.61
	sets, we observe that in all tables, using 2-covering K-sets is
	competitive as compared to using link K-sets. In particular,
	consider the results for 2-covering 3-set given in Tables V and
	VI. (Recall that using this set, i.e., set(3), protects the traffic
	against all combinations of simultaneous link degradation in
	three sets of two links each; these sets compose the family
	obtained in the construction of this set, see Section III-C.) For
	this set we achieve costs (218, 205) carried traffic percentages
	(96.71%, 87.35%). Considering results for link(4) in Table V
	and for link(3) in Table VI, we observe higher costs (221,
	208) and lower carried traffic percentage (96.67%, 87.19%).
	Yet, although not always comparable, in some cases (as in
	Table IV), node K-sets perform better than the other two K-
	set types approaches; for instance see the node(2) result and
	compare it with set(3) and link(4)). Note that as compared to
	node K-sets, 2-covering 3-sets (set(3)) solutions are cheaper
	for temperate and cold continental climate at the same time
	achieving competitive percentages of carried traffic.

TABLE VII :

 VII PMAN network.

	uncertainty cost carried traffic computational
	set	C * (K) AT (K)[%]	time [s]
	link(1)	45	93.39	31.76
	link(2)	48	93.63	39.76
	set(2)	50	93.70	93.47
	link(3)	51	93.67	47.24
	set(3)	51	93.64	86.51
	node(1)	52	93.51	87.23
	link(4)	54	93.71	50.06
	node(2)	54	93.48	435.24
	link(5)	54	93.45	49.01
	link(6)	54	93.54	52.74
	link(7)	54	93.45	51.41
	link(8)	54	93.47	72.86
	link(9)	54	93.47	77.84
	link(10)	54	93.47	72.50

Acknowledgements: This paper is based on work from COST Action CA15127 (Resilient communication services protecting enduser applications from disaster-based failures RECODIS) supported by COST (European Cooperation in Science and Technology). The work of the French authors was carried out in the framework of the Labex MS2T funded by the French Government through the program "Investments for the future" managed by the National Agency for Research [Reference ANR-11-IDEX-0004-02] and was partially financed by the PGMO Project -Programme Gaspard Monge. The Polish authors were supported by the National Science Centre, Poland, grant number 2015/17/B/ST7/03910 "Logical tunnel capacity control -a traffic routing and protection strategy for communication networks with variable link capacity".


Romain Guyard

Véronique Cherfaoui

VANET distributed data fusion for traffic management

In this article, we propose a distributed fusion algorithm to detect traffic congestion through the exchange of messages in vehicle network. This algorithm is based on the Dempster-Shafer theory that manages the uncertainties on data and sources of information. Each vehicle updates its database with local measurements (speed and interdistance) and information received from other vehicles and can calculate its route. Thanks to the collaboration, smart cars can avoid congested roads and take a better path to their destination. Several variants of the algorithm are studied and compared to a centralized approach through experiments carried out on the SUMO simulator using real urban road networks.

I. INTRODUCTION

In the domain of traffic management, most algorithms use a centralized server (eg. Waze, Google map...) that collects data (position, speed...) sent by connected vehicles. The server computes a map of congested roads. This map is then sent back to the vehicles that can use it to find the fastest path. In some cases the path is directly computed by the central server.

In this paper we study an other method of collaboration based on direct connection between cars in the Vehicular Ad-Hoc Network (VANET) context. The VANET has some characteristics radically different from regular computer networks. Nodes are dynamical and connections between nodes are only temporary since their relative positions change all the time. A WiFi norm called 802.11p has been proposed for VANET. Messages can only be sent in a broadcast mode since there is no possible durable connection between nodes.

Traffic management has been heavily studied and different methods has been used to detect traffic jams in VANET.

A common solution is to analyze a local map of cars like proposed in [START_REF] Nadeem | Trafficview: traffic data dissemination using car-to-car communication[END_REF] with the application Trafficview. Trafficview generates an instantaneous snapshot of the traffic and not an average over time. Moreover it is sensitive to Sybil attacks2 and needs trust between nodes which can not be accepted in real conditions. [START_REF] Younes | A performance evaluation of an efficient traffic congestion detection protocol (ecode) for intelligent transportation systems[END_REF] proposes an efficient traffic congestion detection protocol (ECODE) for intelligent transportation systems. This method, as in [START_REF] Lin | Road congestion detection by distributed vehicle-to-vehicle communication systems[END_REF], sends the position of cars in the network which can be a privacy concern. [START_REF] Fukumoto | Analytic method for real-time traffic problems by using contents oriented communications in vanet[END_REF] proposed to count the number of messages exchanged to determine the traffic level. This mechanism can be easily implemented in complement of other VANET applications but it can be inaccurate and can be modified artificially by a dysfunction or by an attacker. In [START_REF] Dornbush | Streetsmart traffic: Discovering and disseminating automobile congestion using vanet's[END_REF], an adaptation of K-mean to V2V is used as well as a distributed hierarchical agglomerative clustering system. The method can detect the status of roads by fusing position of cars in the network. Traffic patterns recognition can be used to detect traffic congestion as shown in [START_REF] Vaqar | Traffic pattern detection in a partially deployed vehicular ad hoc network of vehicles[END_REF]. The author of [START_REF] Miller | Vehicle-to-vehicle-to-infrastructure (v2v2i) intelligent transportation system architecture[END_REF] introduces the concept of vehicle to vehicle to infrastructure (V2V2I). With this method average speed are computed using V2V and then the results are centralized to an infrastructure.

In this paper, we propose an adaptation of a distributed algorithm originally proposed in [START_REF] Ducourthial | Experiments with Self-stabilizing Distributed Data Fusion[END_REF] to perform traffic management using V2V communications. This auto-stabilizing algorithm use distributed data fusion to improve the knowledge of each vehicle. Since this method doesn't send data to a third part organization, it keeps personal data private. A centralized algorithm becomes unusable when the central server fail; whereas a decentralized algorithm is more robust in this case. Finally, our method doesn't require internet connection. In a crowdy road, mobile network can be saturated and thus unusable. This algorithm is auto-stabilizing and is resistant to transient errors due to broken or dirty sensors as proven in [START_REF] Ducourthial | Self-stabilizing Distributed Data Fusion[END_REF]. The algorithm is also resilient to deliberately modification of local data.

In the first section, we present the distributed data fusion algorithm and its variants based on Dempster Shafer theory (or belief functions theory) to model uncertainties in traffic management scenarii. The next section describes the implementation of the algorithm in the SUMO simulator. Finally, results of the simulations are presented and discussed.

II. DISTRIBUTED DATA FUSION ALGORITHM

The method described in this paper is based on the algorithm presented in [START_REF] Ducourthial | Experiments with Self-stabilizing Distributed Data Fusion[END_REF]. The objective is to fuse data coming from other vehicles with local data to determine the level of congestion of roads. Using this information, each car can find the fastest path to its destination.This algorithm has multiple steps:

1) Modeling of local data: each vehicle observes its environment, generates local data and models a representation using belief functions. Local values are independent of each others.

1) Data reception: each vehicle receives data coming from all other vehicles in neighborhood. 2) Data fusion: information from other vehicles and the local representation are fused. 3) Data sending: each vehicle sends the fused data to other vehicles. The algorithm updates estimations of lane average speed on a road map. Each lane of each road has a unique ID that every cars knows. The average speed can be used to determine the shortest path using for example a Disjktra algorithm.

A. Representation of road congestion status

Car sensors are imperfect. Data generated with imperfect sensors give imperfect results. Moreover in VANET, it is not possible to fully trust other vehicles. Cars can have broken or dirty sensors or an attacker can try to benefit from sending false information in order to have a better driving experience. Thus, we propose to use Dempster Shafer theory that is able to deal with uncertainties.

1) Belief functions: Dempster-Shafer theory introduced in [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF] is often used to model both imprecision and uncertainties of an agent. The Dempster-Shafer framework models confidence in events using belief functions. Belief functions can be expressed by different representation. The most common representation is the mass function. If we consider a frame of discernment Ω the finite set of all possible events, then the mass function are values attached to the power set of Ω following the equation 1. Let be a variable ω ∈ Ω and A ⊆ Ω a element of 2 Ω . The quantity m(A) is interpreted as the part of the belief allocated to the hypothesis "the answer ω is in the subset A of Ω".

0 ≤ m(A) ≤ 1, A⊆Ω m(A) = 1 (1)
The mass m(Ω) represents the ignorance and m(∅) the inner conflict. A "focal element" of m is an element A of Ω such that m(A) > 0. A "simple mass function" has one or two focal element including Ω. If Ω is the only focal element then the mass function is called "vacuous" and represents total ignorance. If Ω is not a focal element, then the mass function is called "dogmatic". With imperfect sensors, most of mass functions are non-dogmatic and we will thus assume it is the case in this paper. Mass functions can be combined to improve the knowledge and reduce uncertainties. The wellknown Dempster conjunctive rule is shown in equation 2.

     (m 1 ⊕ m 2 )(∅) = 0 A = ∅ (m 1 ⊕ m 2 )(A) = 1 1-K B C=A m 1 (B)m 2 (C) where K = B C=∅ m 1 (B)m 2 (C)
(2) This rule requires the independence of sources. When a source uses multiple times the same information to generate data, data incest can occur. To avoid data incest, the fusion operator has to be idempotent. An idempotent fusion operator called cautious has been proposed in [START_REF] Denoeux | Conjunctive and disjunctive combination of belief functions induced by nondistinct bodies of evidence[END_REF]. This operator is applied on the conjunctive decomposition presented by [START_REF] Smets | The canonical decomposition of a weighted belief[END_REF] using the operator minimum. This operator is idempotent, associative and commutative and noted .

2) Modeling road congestion status: The congestion of a road is represented by the following frame of discernment: Ω = {F luidF ree, F luidSparse, SlowSparse, CongestedDense}. A F luidF ree state represents a road without any cars or with cars that can drive at the maximal speed without any cars near. The F luidSparse state is also a state where cars can drive at maximal speed but the road contains cars close to the vehicle, which is a risk of traffic jam. Adding more cars to the road will make all the cars to slow down. In the SlowSparse state cars are to many to keep maximum speed. Finally, the CongestedDense state is the state where cars are too many and thus a traffic jam is formed. The states of Ω is summarized on the figure 1.

Two sensors are used to evaluate the congestion of the road : the speed sensor LIDARs that detect the distance between cars. A basic belief assignment function is used to transform the measurement to a belief function m local . Each sensors measurement is converted to belief function separately and are then fused together.

3) Representation of the average speed on a road: The car average speed is represented by the frame of discernment Ω s = {F luid, Slow, Congested}The speed value has a range of [0, V max ], with V max the speed limit of the road. Let

m local = (m local s ) Ωs↑Ω ⊕ (m local d ) Ω d ↑Ω (3)
The speed sensor gives information about the states {DenseCongested} and {SlowSparse} and also the union of the states {F luidF ree, F luidSparse} since the speed data can not distinguish the difference between a road where cars drive at maximum speed with a "empty" road.

The inter-distance can give information about the states {F luidF ree} and {CongestedDense} and the union of the state {F luidSparse, SlowSparse} since cars inter-distance can not distinguish the difference between a road with some cars but driving at normal speed and a road with some cars but slowed. To have an information about the full frame of discernment, the two belief functions generated from speed sensor and the inter-distance are then combined using the Dempster's rule.

B. Data reception

The transmission of data is done using a WiFi norm called WiFi 802.11p for VANET. Data are broadcasted to other cars. Every car sends the fused data m public to other cars in the neighborhood. A message is constituted of a collection of belief function representing an estimation of the status of roads. This collection, noted [m public ], contains only non vacuous belief functions. The car collects all the messages received and saves them until the next step. When a new step starts, the saved data is fused and the database is cleared to be able to receive new messages for the following step.

We consider in the following that the messages are always received by every cars in range.

C. Data fusion

Fusion of data is done by an idempotent operator since every car uses information that have been already fused. Therefore, the algorithm uses the cautious operator.

Belief functions coming from other vehicles are representation of a distant measurement. in space and in time. Since some time has passed since the data has been generated, data received are less representative of the current situation and thus, they should be discounted before the fusion. The discounting operation reduces the importance of sources that are not precise or can not be trusted. The discounting proposed by Shafer in [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF] is given by the equation 4. m α (A) is the mass discounted with a factor of α ∈ [0, 1]. The value of α depends on the reliability of the source.

A = Ω α m(A) = αm(A) α m(Ω) = (1 -α) + αm(Ω) (4)
The discounting in the proposed algorithm has been studied in a previous paper [START_REF] Guyard | Study of discounting methods applied to canonical decomposition of belief functions[END_REF]. The time between each step is considered as the same and thus we use a fixed discounting. The equation 5 shows the general fusion formula.

m public = f use(m local , α [m public ][0] α [m public ][1] ...) (5)
In [START_REF] Guyard | Study of distributed data fusion using dempster's rule and cautious operator[END_REF] we proposed 4 different ways of data fusion, shown in figures 3, 4, 5 and 6 that are going to be tested in this scenario. The idea of these scenarios is to use Dempster's rule that increases the knowledge when data comes from two independent sources or the cautious operator when sources are dependent. Local value is always independent of values coming from the network. It is then possible to combine it with Dempster's rule the local value and the result of the cautious operator applied to all values sent by other cars (distributed values). Figure 4 (where ⊕ represents Dempster's rule and the cautious rule) shows the fusion diagram of this proposition. As proposed in [START_REF] Zoghby | Distributed data fusion in VANETs[END_REF], since local data is timely independent, we can add a temporal fusion of the local value with Dempster's rule. This operation is done before the network data fusion as shown in Figure 5. Finally, we propose a fourth fusion diagram in Figure 6 that assumes that nodes send both distributed and local values. Dempster's rule can be used to combine neighbor local values since independent local data is fused only once.

D. Data sending

Fused data m public are broadcasted to other vehicles. Since we fuse data for all roads of the map we have to send a belief function for each lanes of each roads. The WiFi 802.11p doesn't allow packaging and thus data has to be sent at once. Therefore, we must limit the size of message at 1500 bytes WiFi 802.11p packet. A mass can be encoded using only one byte since a precision of 1 256 is enough. If we consider the conflict normalized and the sum of all masses equal to 1, a belief function can be represented using 14 bytes. We also need to add a lane identifier which can be represented using 4 bytes, assuming each car having the same map. Thus, we can send 107 belief functions. In order to choose which road information should be sent, the algorithm sorts the belief function based on the mass on the ignorance. The more commit the belief function are the more likely they are going to be sent.

III. SIMULATION

We have developed a simulated scenario with SUMO simulator [START_REF] Behrisch | Sumosimulation of urban mobility: an overview[END_REF], [17] to test each variation of the algorithm.

A. Scenario

The scenario must test the different characteristics of the variants of the algorithm. The traffic is simulated in multiple real cities in order to have realistic results. A constant heavy traffic is kept in order to cause traffic congestion. To do this, we set a fixed number of cars that drive in the city. Each car starts at a random location and travel to another random location in the city. Each time a car arrives at destination, it is removed from the simulation and a new car with a new path is generated.

The algorithms are compared by counting the number of cars that have reached destination after a fixed time.

To obtain realistic results, we have tested on two scenarios: one in a district of Paris and in the city of Compiègne (average french city) The two cities have a very different topology.

B. Pathfinding algorithm

The generated distributed belief function should be used to found the fastest path to a destination. The decision of the shortest path can be done using Disjktra's algorithm. The weight on each edge is the travel time computed using the maximum speed and the length of a lane. The result of the data fusion algorithm should be used instead of using the maximum speed. In order to use a belief function in Disjktra's algorithm, the average speed s e should be estimated. We use the plausibility functions as proposed by [START_REF] Lefevre | Belief function combination and conflict management[END_REF].

The plausibility of a set of a mass function is an alternative representations of mass functions. The plausibility is noted P l and it is described by the equation 6. The set of plausibility on the bayesian elements is called the contour function and it is noted pl.

P l(A) = B∩A =∅ m(B) (6)
A transformation function is used to change the plausibilities back to values and thus estimate the original speed modelized by the belief function. To determine our transformation function we have used the special case where the local speed values s are transformed into belief function m local s using the sigmoid functions. Our goal is to inverse the sigmoid function to find an estimation of the speed. The transformation function should return the original value s when applied on the particular case m local s = S s (s). The belief functions has to be transformed from Ω to the original speed set set of discernment Ω s using a method called minimization and noted ↓ as shown in equation 7.

m s = m Ω↓Ωs (7)
The plausibilities of the belief function generated with the speed sensor and discounted is used to make the estimation. In order to have a bijective function, the function P l Sum (m) = P l(m({Slow, F luid})) -P l(m({Congested})) + P l(m({F luid})) -P l(m({Congested, Slow})) is considered. The function P lD is then defined by P lD(s) = P l Sum (S s (s)) with S s (s) = m local s the speed sigmoid function. To find the speed using the plausibilities functions, we must invert the function P lD. This inverse is plotted in figure 7. It is then possible to find the estimate speed s e with the equation 8.

s e = P lD -1 (P l Sum (m public ) Ω↓Ωs )) (8)

C. Implementation

A C++ program has been implemented using SUMO simulator to test the scenario. To simulate the WiFi-P connection we have made our own implementation of a communication system. The theoretical range of the WiFi 802.11p is 2 km which is unrealistic in urban driving. The study in [START_REF] Demmel | Ieee 802.11 p empirical performance model from evaluations on test tracks[END_REF] showed that in ideal conditions the range can be up to 500m thus we have chosen to limit the range to 200m. The message length is limited to 1500 bytes, headers included. We suppose that our application will not be the only one that need car communications capabilities thus the data size has been limited to 20% of the maximum defined by the norm. The cities road networks are imported to SUMO simulator from OpenStreetMap.

IV. RESULTS

In the figure 8 we can see a part of the city of Paris simulated in SUMO and the figure 9 is Compiegne. Roads are displayed in red when its occupancy is unknown, green is the road is free and yellow when it's congested. Then, the performance of each algorithm grows at their own rates. In Compiègne, there is a significant difference between scenarios when about 500 cars are simulated at a given time. In Paris, only 50 cars are enough but the deviation is smaller. After an hour, the centralized average out performs the other algorithms. The second best is the Dempster at one hop, followed by the two other algorithms that mix cautious rule and Dempster's rule. Finally the "cautious only" algorithm is the worse, but still better than no fusion. We can see in the Compiègne scenario, with this particular random seed, that the "Dempster at one hop" has the same performance than the "no fusion" scenario. The "Dempster at one hop" algorithm have a bigger standard deviation than the other algorithms. It stays the better distributed algorithm in average but in some cases it can be the worst.

B. Summary

The SUMO simulator also generates pollution emission information. The car average CO 2 pollution emission is computed for each algorithm. We can observe the link between pollution and time time spent in traffic jams. The performance of each algorithm are summarized in the table I. We added a "no fusion" algorithm where every cars choose the shortest path without taking in account the occupancy of the roads. The performance metric M is the percentage of cars that have arrived to their destinations (nbArrived) in comparison with method without fusion (nbN oF usion): M = nbArrived nbN oF usion ×100. In this table, the result is an average value of 10 iterations of the scenario.

The centralized method has the best performance but there are drawbacks. If a central server fails, the whole system stops working. It also needs internet connectivity to operate since the private data of the users are sent to a third party. The "cautious only" algorithm gives better results than doing no fusion but both "Dempster local value" and "Dempster after cautious" are better. Finally, the best distributed algorithm is the "Dempster at one hop" but it needs to send car's local values to neighbors. The privacy of this last algorithm is better than the centralized version since only neighbors have access to private data and not a central server that aggregate everyone's positions in the network.

V. CONCLUSION

In the paper we have shown an new application for a distributed data fusion algorithm based on the Dempster Shafer theory. We have demonstrated the utility of distributed data fusion to regulate traffic. We have shown the difference between the different variants in different realistic scenario. Despite slightly worse results than with centralized fusion, distributed data fusion shows promising results since it preserves users' private data by not sending personal information to a third party and are more resilient to malfunctions. in the absence of a single point of failure. We have also shown that data fusion reduces the amount of emissions. Future work concerns the improvement of the mass building model and the validation from real data acquired on the platforms of the Heudiasyc laboratory.

Figure 1 .

 1 Figure 1. Road states in Ω

  V the proportion of V max . Sigmoids are used to represent the average speed of a road. These functions change the speed value to a belief function m local s . The sigmoid function S s () is shown on the figure 2. The sensors are imperfect thus the sigmoid function does not generate a dogmatic belief function. The uncertainty is represented by mass m(Ω s ) = 0.3. The parameters of the sigmoids are chosen to make the curves smooth with empiric values. In a real scenario, they should be chosen by experts of the domain or by learning methods approach. 4) Representation of the congestion of the road: The road congestion is represented by an other frame of discernment Ω d = {F ree, Sparse, Dense}. The congestion

Figure 2 .

 2 Figure 2. Speed basic belief assignment

Figure 3 .Figure 4 .Figure 5 .

 345 Figure 3. Cautious operator onlyFigure 4. Dempster's rule before sending

Figure 7 .

 7 Figure 7. Transformation function P lD -1

Figure 8 .

 8 Figure 8. Map of a portion of Paris (Montmartre) in SUMO simulator

Figure 9 .

 9 Figure 9. Crop of Compiègne's map in SUMO simulator

Figure 10 .Figure 11 .

 1011 Figure 10. Number of cars that have reach their destination after a given simulation time with 50 vehicles in a part of Paris

  VI. ACKNOWLEDGMENTSThis work was carried out in the framework of the challenge DAPAD (Distributed and Augmented vehicle Perception to support Autonomous Driving) funded by the Labex MS2T, supported by the French Government, through the program "Investments for the future" managed by the National Agency for Research (Reference ANR-11-IDEX-0004-02)

The authors are with Sorbonne Universités, Université de Technologie de Compiègne, CNRS Heudiasyc UMR 7253, France

The Sybil attack is an attack wherein a reputation system is subverted by forging identities in peer-to-peer networks.