Rutger De Wit
email: rutger.de-wit@umontpellier.fr

Pénélope Chaubron-Couturier

Florence Galletti

Diversity of property regimes of Mediterranean coastal lagoons in S. France; implications for coastal zone management

Keywords: property regimes, coastal lagoons, Maritime Public Domain, commons, nature conservation, governance, Coastal Law, Marine Protected Areas

Highlights 1-Today 87 % of the surface of the Mediterranean lagoons in France is public domain 2-It comprises DPM (65 %) and domain of the Conservatoire du littoral (22 %) 3-Variability and changes of property regimes reflect a historic process.

Introduction

In many countries, e.g. France, Spain, Portugal, Greece, Turkey and several states in the USA, the national coastal seas and the coastlines are public properties often legally owned by these national States. This can be considered as a legacy of the public domain defined by Roman Law [START_REF] Fenn | Justinian and the Freedom of the Sea[END_REF]. Such property regimes are important both for defining who will have access to these ecosystems, rights for natural resource extraction (Schlager and Orstrom, 1992) and other human uses. The public domain is open for use by the citizens of these states. It is, however, clear that in many cases the second principle of Hugo Grotius, i.e., anything can be used without loss to anyone else is often not applicable in the coastal lagoons, which are therefore not fully a space res omnium communis (Thompson, 2005). Rather the uses need to be regulated by Government or by the local communities as in Community-based management [START_REF] Berkes | From community-based resource management to complex systems[END_REF]. Coastal lagoons have been used by coastal populations since prehistoric times. Traditional uses include fishing, hunting, navigation and salt-extraction [START_REF] Anthony | Coastal lagoons and climate change: ecological and social ramifications in U.S. Atlantic and Gulf coast ecosystems[END_REF][START_REF] De Wit | Seventy-year chronology of Salinas in southern France: Coastal surfaces managed for salt production and conservation issues for abandoned sites[END_REF]. More recently, particularly since the 19 th century, aquaculture, tourism and recreational uses have become increasingly common in coastal lagoons. Finally, there is increasing awareness of the need for protecting the coastal biodiversity, particularly by creating protected areas for the conservation of habitats and species. Coastal lagoons represent a so-called priority habitat (N° 1150 *) according the EU Habitats Directive (1992), meaning that member states should protect the coastal lagoons and assure their conservation status (De Wit et al., 2020). The Convention on Biological Diversity (CBD) recognizes sovereign rights over biodiversity within national territories (Thompson, 2005), which include the coastal zones. Hence, property regimes in coastal lagoons have to be considered for implementing biodiversity conservation in these ecosystems.

Coastlines are transient both on short-term and long-term timescales, as sea level elevation varies along astronomic and meteorological cycles and land may accrete or erode. This creates difficulties in delineating the public property along the coastline [START_REF] Yavuz Özalp | Development and management of private property rights on coastal areas[END_REF] with sometimes subtly different approaches within the same State [START_REF] Mcglashan | Defining the foreshore: coastal geomorphology and British laws[END_REF]. In addition, the delineation can be further confused by the presence of transient environments. In this paper, we study the property regimes of the Mediterranean coastal lagoons in S. France. In this country, many of the coastal lagoons, but not all, belong to the maritime public domain (In French: Domaine Publique Maritime, abbreviated as DPM). This is based on the French law on the Public Maritime Domain (28 November 1963) and article 2111-4 of the French General Code of the property of public persons (Code général de la propriété des personnes publiques, see website), which states that the DPM includes (ii) the floor and below-floor earth layers of those "saline ponds" that communicate, in a natural way, permanently with the sea. In the official French text, the article uses the term "étangs salés"; étang translates as pond or pool (Harrap's dictionary), while it is used in vernacular language for the coastal lagoons in South France and used for the official geographic names, as e.g. Etang de Vic.

The wording of the French legal text may thus create a conflict with the scientific definition of a coastal lagoon. according the most commonly accepted definition in coastal sciences, coastal lagoons are described as "inland water bodies, found on all continents, usually oriented parallel to the coast, separated from the ocean by a barrier, connected to the ocean by one or more restricted inlets which remain open at least intermittently, and have water depths which seldom exceed a few meters" (Kjerve, 1994).

The aim of this paper is to document the current property regimes in French Mediterranean coastal lagoons and discuss their implications for the management of these coastal ecosystems, with a particular focus on nature conservation, natural resource extraction and other human uses. In addition, the aim is to understand the genesis of the current situation, particularly the existence of the private ownership and, therefore, two hypotheses can be forwarded for explaining private properties of coastal lagoons, i.e., H1: The private property existed in the past before the DPM was implemented and has been passed through inheritance, H2: The legal text is not sufficiently precise to include all types of coastal lagoons and this has favoured the takeover of properties by private persons or institutions.

Finally, we will discuss the actions to be taken to improve the current situation, particularly with respect to conservation of coastal lagoons.

Study site and methods

Study sites

Study sites include the permanent coastal lagoons along the Mediterranean coastlines of continental France and the island of Corsica. These coastal lagoons occur in three administrative Regions, i.e., i) Occitanie (littoral zone of the Gulf of Lion, where they occur along 50 % of the coastline), ii) Province-Alpes-Côte d'Azur and iii) Corsica (littoral of the Tyrrhenian Sea). In French étangs littoraux is the generic name for coastal lagoons, and the word Étang, used in their official geographic names, is abbreviated hereafter as Et. (e.g., Etang de Vic written as Et. de Vic). Coastal lagoons are separated from the sea by a coastal barrier and have permanent or temporary connections with the adjacent sea through one or several inlets (Kjerve, 1994). Nowadays in most of the French Mediterranean lagoons the inlets have been modified by humans to assure a permanent connection with the sea. Naturally-moving inlets still occur for the Etangs de La Palme and Ayrolle in Occitanie, and several lagoons in Corsica. Several of the coastal lagoons in deltaic settings have been separated by several km's from the sea by progression of the delta, while still connecting with the sea through wetlands or artificial canals.

The lagoons close to Montpellier have been heavily modified. Originally, the Etang de Mergueil was a large lagoon that ranged from Frontignan (close to Etang de Thau) in the SW up to the Camargue in the NE. This lagoon has been compartmentalized by canals, dikes and roads into 10 different 'étangs', i.e. the larger Etang de l'Or and a complex of nine water bodies designated as the Palavas Lagoon Complex (in French Complexe lagunaire palavasien). Only two of these water bodies (Et. d'Ingril Sud, Et. du Prévost) have an inlet connecting directly to the Mediterranean Sea. The building of the Rhôneto-Sète canal, which started in the 18 th century, has separated four permanent lagoons (Etangs de l'Ingil Nord, de Vic, de l'Arnel, and du Méjean) from the coastline and exchanges of these lagoons with the sea occur through this canal. Etangs de Pierre Blanche and le Grec are located on the coastline and separated from the sea by a barrier without an inlet; these two lagoons also exchange with the sea through the Rhône-to-Sète canal.

Methods

French legal text and historical documents were studied to describe the specificities of current French law and the chronology of its implementation. Most of the French text is now available on-line on internet at the website Légifrance (https://www.legifrance.gouv.fr) and the pertinent sections are mentioned in the reference list.

The cartography was based on using Q-GIS (version La Palma 2.18.15). Shapefiles for coastal lagoons were obtained from the Data bases Ocsol in Occitanie and PACA and CORINE land cover for Corsica (CLC 521). Shapefiles for cadastral parcels and municipalities for coastal lagoons were obtained from the French national geographic institution (IGN). Openstreet maps were used as a plug-in. Information of properties regimes were obtained from documents, the website for the Conservatoire du littoral, the website of the Direction départementale des territoires et de la mer (DDTM), and interviews with coastal zone managers.

.

Results

Analyses of the chronology of laws and jurisprudence in France related to property rights of coastal lagoons

Figure 1 lists the most important legislation and historic events that have important implications for property regimes in coastal lagoons. The Edict of Moulins was signed in 1566 by King Charles IX, with the aim to clarify a distinction in the Kingdom of France for the Royal property of the King (Domaine Royal) between the properties at his disposal for private issues (Domaine casuel) and the property that was eternally associated to the kingdom (Domaine fixe), i.e. the King should legate the Domaine fixe to his successor. For the latter property, the Edict introduced the characteristics of inalienability and imprescriptibly meaning, respectively, that its property rights cannot be changed in the future and neither transferred nor sold to somebody else, and that the inalienability is exempt from subscription. Today, the current State Domain Code and the French General Code of the property of public persons (Code général de la propriété des persones publiques, CGPPP) inherited this difference, as for public property it differentiates between "Domaine Public" and "Domaine Privé" (L.3111-1 of the CGPPP) in analogy to the former "Domaine fixe" and "Domaine casuel", respectively. Hence, the Domaine Privé de l'Etat (DPE) translates as "private property of the State", which in English is an oxymora as private property is opposed to public property. The different forms of public domain are subjected to inalienability and imprescriptibly, while the "private property of the State" (DPE) is not and can be sold to third parties.

More than a century later (1681), the Colbert ordinance of the Navy introduced the Natural Maritime Domain (Domaine Public Maritime, abbreviated DPM) stating that DPM "shall be considered as the edge and shore of the sea including all that the tide covers and uncovers during the new and full moons and how far the biggest wave of March can reach on the beach". Because the original ordinance only mentioned the shores of the sea it left the issue of the coastal lagoons unsettled. The issue of how to accommodate the DPM for coastal lagoons was settled in the following centuries. Today, the DPM comprises the artificial DPM (harbours, sea-defense structures, navigation channels, etc.) and the natural DPM (Domain Public Maritime naturel). Table 1 gives the legal text in French for the natural DPM and its translation in English. The second point, i.e. des étangs salés, clearly refers to coastal lagoons. The DPM definition of the étangs salés (Table 1) raises many problems for considering the entire range of environmental conditions for coastal lagoons and creates conflicts with the aforementioned scientific definition of coastal lagoons by Kjerve (1994).

During, the 19 th century, many ambiguities were solved by the jurisprudence of the Criminal Chamber of the Cassation Court. Hence, in 1848 the judgement stipulated that coastal lagoons that do not have an inlet, but communicate with the sea only through a man-made canal do not belong to the natural DPM [START_REF] Cuënot | Journal du Palais. Recueil le plus ancien et le plus complet de la jurisprudence francaise[END_REF]. Etang salé suggest that salinity should be similar to the adjacent sea (i.e. a salinity of about 35 in the adjacent Gulf of Lion) and raises the question of whether mesohaline (salinity between 5 and 18) and oligohaline coastal lagoons (salinity lower than 5) could be included in the natural DPM? However, salinity measurements were not yet commonly used in the 19 th century. While a first judgment in 1842 first specified that the lagoon Le DPM naturel est inaliénable et constitué :

1-du sol et du sous-sol de la mer, compris entre la limite haute du rivage (c'est-à-dire celle des plus hautes mers), et la limite, côté large, de la mer territoriale ;

2-des étangs salés en communication directe, naturelle et permanente avec la mer ;

3-des lais et relais de la mer.

In France the natural Maritime Public Domain is inalienable and encompasses:

1-the seafloor and its subsurface strata between the upper level of the shoreline (i.e. those of the highest tides), and the limit of the territorial sea);

2-the "salty ponds" (étangs salés) = coastal lagoons with a direct, natural and permanent communication with the sea;

3-sediment deposits of the sea along the coastline.

should be home to the same fish species as in the adjacent sea, later this was rectified by a judgment in 1861 stipulating that the lagoon may also hold a mixture of freshwater and seawater species. This allowed to include many of the meso-and oligohaline coastal lagoons in the DPM. Finally, a judgment in 1860 stipulated that the inlet maybe temporarily closed. An example is the Etang de la Palme, where the inlet completely closes off every year (Larue and Roquet, 2016). Finally, while astronomic and meteorological tides have been taken into account since the Navy ordonnance of Colbert, more precise procedures for delimitation of the DPM were introduced in 1973 by a judgment (arrêt) of the Conseil d'Etat (the Kreitmann case-law).

While, these abovementioned Court trials focused on challenging the definition of the DPM itself, claims of private ownership of coastal lagoons have also been based on invoking pre-existing private property rights. Hence, jurisprudence followed by its inclusion in law, fixed the date of retroactivity of the DPM at 1566, the year of the Edict of Moulins. This means that public domains cannot be retroactively enforced prior to the Edict of Moulins [START_REF] Torquebiau | L'aménagement du littoral languedocien et la situation juridique des étangs côtiers[END_REF] implying that all private and Church properties existing before 1566 were to be respected and could thus not be included in the DPM.

Therefore, the Royal and the Navy's administrations, under the Old Regime period and the Second Empire, respectively, carried out an inventory and put into place restitution procedures for rights prior to the Edict of Moulins (Feral, 2000). Numerous documents were produced by families from Languedoc and Camargue, attesting to ownership by their ancestors before 1566 of coastal lagoons (étangs salés) and other water bodies, as canals etc., as well as fisheries and maniguières. The latter are fishing rights linked to the land ownership of the coastal lagoons (Feral, 2020). The Minister of the Navy, in a decision of 30 July 1964, drew up a list recognising a large number of private owners of coastal lagoons and canals in the French Mediterranean (about 200 private owners of coastal lagoons, canals, fisheries, islets, rivers, streams etc.). In Roussillon (Pyrénées orientales Department), Et. De Canet and a small part of the Salses-Leucate Lagoon (Anse de la Roquette) were recognized as private property on the basis of Spanish titles after 1566 but before the Treaty of the Pyrenees of 1659 (Feral, 2020). Remarkably, in Corsica, annexed by France in 1768 and part of the Republic of Genova before, the seven coastal lagoons remained private property until quite recently.

Private ownership was not always accepted by all sectors in society, as there have been many conflictual cases between the administration and private owners of coastal lagoons, first in front the civil courts [START_REF] Torquebiau | L'aménagement du littoral languedocien et la situation juridique des étangs côtiers[END_REF] and later in front of the administrative courts. The Navy administrations and coastal lagoon fishers organized in corporatist and professional organizations, the Prud'homies de Pêche (Féral, 1978;Féral, 1987), defended the idea of the coastal lagoon as a natural public domain for guaranteeing freedom of fishing. This resulted in an arm wrestling's with landlords [START_REF] Hostiou | Le domaine public naturel : consistance et délimitation[END_REF]Féral, 2020a) In contrast, the natural DPM is nowadays managed by the Direction Départementale des Territoires et de la Mer (DDTM), which is a service of the French State that has a branch in all coastal Departments. Human uses, e.g., shellfish farming, salt extraction and touristic infrastructure are allowed on a temporal basis only through concession contracts. Fisheries has been traditionally regulated by a type of community-based ecosystem management through the Prud'homies de Pêche [START_REF] Feral | La prud'homie des pêcheurs de Palavas, étude de la mort d'une institution[END_REF], Féral, 1987).

Cartography of property regimes in French Mediterranean coastal lagoons

Starting at the Spanish-French border in Occitanie, following the coastline clockwise and ending in Corsica, Figures 2,3,4, and 5 successively depict the cartography of property regimes and Table 2 provides quantitative data for the French Mediterranean coastal lagoons. The Etang de Canet, in the Roussillon region (Pyrénées orientales Department), which in principle qualifies well for the DPM, was private property, of which most has now been acquired by the Conservatoire. Thau lagoon, and almost all lagoons in the Aude Department are DPM (Fig. 2). However, as described before one smaller cadastral parcels, the Anse de la Roquette (44.9 ha) in the SW part of Etang Salses-Leucate, excluded from the DPM is private property. Moreover, in Etang Bages-Sigean there is another anomaly as one cadastral parcel in the SW is owned by the Aude Department. None of the lagoons in the Camargue are DPM (Fig. 4A). In the Camargue of the Gard Department, the Etangs Charnier, Grey and Scamandre are owned by the local municipalities (Vauvert and Saint-Gilles). In the Bouche du Rhône Department, the Conservatoire has acquired more than 94 % of the surface of Et. de Vaccares, while some smaller private properties still remain along the W., N., and E. shorelines. The four lagoons in the SE corner of the map in Fig. 4A all belong nowadays to the Conservatoire.

Some smaller lagoons in

Before 2009, these lagoons were privately owned by the Groupe Salins company and managed as salt preconcentration ponds for a major salt work in Salin de Giraud [START_REF] De Wit | Seventy-year chronology of Salinas in southern France: Coastal surfaces managed for salt production and conservation issues for abandoned sites[END_REF]. Since their acquisition by the Conservatoire, the management has changed and the lagoons are nowadays managed for nature conservation purposes. Close to Marseille (Fig. 4B), the Etang de Berre, the largest French Mediterranean lagoon (156 km 2 ), is DPM with the particularity that its surface is not attributed to any municipality.

The adjacent Et. de Bolmon has been acquired by the Conservatoire. Finally, in the Var Department, the Et. de Villepey (Fig. 5C) is also owned by the Conservatoire.

In Corsica, none of the seven coastal lagoons are DPM (Fig. 5). All these coastal lagoons 1) Negotiations for the acquisition of cadastral parcels in Et. Balistra by the Conservatoire have started since 2015. 2) Note that the sum of cadastral parcels is slightly above 100 % of total lagoon surface because the shorelines for the coastal lagoons often do not fit to the cadastral boundaries.

Discussion

The property regimes of Mediterranean coastal lagoons are remarkably variable in South France (Figs. 2,3,4,and 5), despite the fact that French legislation (GCPPP) exists which accommodates the status of public property as Domaine Public Maritime (DPM) for these ecosystems. To the best of our knowledge this is a first extensive study of property regimes of coastal lagoons in a Mediterranean country. We performed a search in web of science with the search words "Law" and "Coast" and "Property" that yielded 123 documents. A screening of these documents showed that none of them corresponded to a comparable study on property regimes of coastal lagoons. In this respect, we could only compare our study with an FAO report (Cataudella et al., 2015) that succinctly states that many of the Venetian valli in Italy have private owners; while in Albania, Greece and Tunisia, all coastal lagoons are public domain and in Egypt, four lagoons belong to the public domain and one is owned by a public company. This shows that the public property of the sea and the coastal aquatic ecosystems not only traces back to Roman law (see Introduction), but rather also presents a tradition in Islamic law [START_REF] Facchini | Droit de propriété et performance environnementale sur les deux rives de la Méditerranée : essai interpretation[END_REF][START_REF] Khalilieh | Islamic Law of the Sea[END_REF]. We believe that it is very important to study the property regimes of coastal lagoons as these have important implications for their management, particularly in view of increasing needs for managing these ecosystems for the conservation of habitats and species (c.f., E.U. Habitats Directive, 1992). While, some scholars from neo-liberal schools of thought argue in favour of privatization of ecosystems for biodiversity conservation, it has been shown that in most cases in the coastal zone privatization has a very negative impact both for biodiversity conservation as well for local populations [START_REF] Cabral | Transition from common to private coasts: consequences of privatization of the coastal commons[END_REF]. However, dedicated NGO's may act as private owners with a prime objective for nature conservation. For properties was in agreement with the jurisprudence stating that retroactivity of DPM implementation is not applicable before 1556 (Edict of Moulins). Nevertheless, these properties were confiscated during the French revolution and sold ('vente des biens nationaux' see Fig. 1) to private persons. In 1949 these private properties were bequeathed to the diocese of Montpellier and thus returned to the Church [START_REF] Barruol | Maguelone de l'antiquité à nos jours[END_REF]. Roussillon (with Et. de Canet and part of Et. Salses-Leucate) and Corsica (7 coastal lagoons) were integrated in the Kingdom of France after 1566 and Corsica even after implementation of the DPM. Until recently, full private ownership of coastal lagoons was complete in Corsica and prevailing in Roussillon. The private ownership existing in these territories before their annexation by France was originally respected.

The Conservatoire has been particularly active in acquiring property in these two regions (Fig. 5).

The vagueness of the juridical text (Table 1), which does not cope with the full range of environmental conditions and thus conflicts with the scientific definition (Kjerve, 1994) of coastal lagoons, has given rise to many trials at the Cassation Court during the 19 th century (Fig. 1). The Navy ordonnance of Colbert defined the coastline and the sea as a public domain and included already some first descriptions on how to delimit this domain taking the astronomic and meteorological tides into account, although it did not mention the lagoons specifically. More precise procedures were introduced in 1973 by a judgment (arrêt) of the Conseil d'Etat (the Kreitmann case-law). The jurisprudence of 1848 stated that lagoons without a direct inlet and communicating with the sea through a man-made canal, do not qualify as DPM. This is in support for the second hypothesis, i.e., the legal text is not sufficiently precise to include all types of coastal lagoons and this has favoured the takeover of properties by private persons or institutions. Indeed, the immediate result of the 1848 judgement was that Et. du Grec became private property.

In general, the jurisprudence implied that 6 of the 8 water bodies existing in the Palavas lagoon complex in the 19 th century (Etangs du Méjean, de l'Arnel, de Vic, d'Ingril Nord, de Pierre Blanches et du Grec, see Methods) do not qualify for DPM. This explains the virtual absence of DPM in these lagoons (only a small section of Et. du Méjean is DPM)

and the widespread occurrence of private state property (DPE), municipal properties and private properties (see Fig. 3). The Et. les Mouettes was created after 1950 as a result of road construction within Et. d'Ingril Sud, and currently is property of the Conservatoire. Natural processes, particularly accretion in deltas, may also contribute to separate the coastal lagoon from the sea. In South France, these deltas have been heavily modified by human intervention, meaning that the connection of these lagoons is now often through man-made channels. This may explain the previous widespread occurrence of private properties in the Rhône delta (Fig. 4) and in the Aude delta (Et. de Vendres, Fig. 2). In contrast to the aforementioned, other judgments by the Cassation Court during the 19 th century provided the necessary support for maintaining other coastal lagoons as DPM. Hence, coastal lagoons with temporary inlets (e.g. Et. de la Palme) and with lower salinities than seawater (mesohaline and oligohaline conditions) may qualify as DPM. On the other hand, it is surprising that the condition of naturalness of the inlet ( These Public Domain status represent an institutional solution for the commons, and coastal nature should indeed be considered as valuable commons. Nevertheless, while the Public Domain of the Conservatoire has really been created for conservation of the littoral nature commons, the DPM was not designed for this specific purpose. In addition, the DPM is not necessarily immutable as it may gain or lose surface with the shifting shoreline and infilling of coastal water bodies. Hence, the surfaces that are no longer submerged during astronomic and meteorological high tides leave the DPM, thus becoming private State property (DPE) and can then be sold to others [START_REF] Hostiou | Le domaine public naturel : consistance et délimitation[END_REF].

In addition, despite the inalienability, general interest can be invoked by the French Beside the fragmented property regimes of Mediterranean coastal lagoons, these ecosystems are also often located on the territories of different municipalities. Etang de Thau, which is entirely DPM (Fig. 2), is spread out over 9 municipalities. The Palavas lagoon complex with its highly fragmented property regimes (Fig. 3) covers 7

municipalities. This means that intercommunal (multi-municipal) collaboration has often been used to act as a coordinator for the management of these coastal lagoons. In practice this means that the intercommunal collaboration structures can act as the management partners of the Conservatoire and that they often hold the role of coordinator for the Natura 2000 lagoon sites. The GEMAPI law (2017), based on earlier decentralization laws, regulate the management of aquatic systems and prevention of inundations. The GEMAPI law (2017) has re-enforced the competencies of the intercommunal collaboration structures. Nevertheless, the GEMAPI law (2017) has also created institutional complications as the competences for environment have been transferred from the municipalities to larger administrative units as metropoles and agglomerations for those municipalities that are members of such units. In practice this means that for managing the coastal lagoons the metropoles and agglomerations have to collaborate with rural municipalities for the management of the lagoon ecosystems. In S.

France, coastal lagoons included in the Natura 2000 network are not recognized as Marine Protected Areas [START_REF] Chaboud | Les aires marines protégées, une catégorie particulière de territoires pour le droit et l'économie[END_REF], but rather treated as Protected

Areas on the continent. The Natural Regional Parks of the PNR Narbonnais en and colleagues [START_REF] Schlager | Property-Rights Regimes and Natural Resources: A Conceptual Analysis[END_REF][START_REF] Ostrom | Private and Common Property Rights[END_REF] for the actions allowed for the different types of actors is very useful in this respect. The Law for the creation of the Conservatoire du Littoral (1975) requests that citizens can visit the sites of the Public Domain of the Conservatoire. Hence, these areas may not be completely closed as fortresses for nature conservation (In French: "La Nature sous cloche" approach). The modalities of the visits should of course be regulated by the managers and agreed with the Conservatoire in the management master plan. Hence, the citizens typically have Access sensu [START_REF] Ostrom | Private and Common Property Rights[END_REF], i.e., the right to enter a defined physical area and enjoy nonsubtractive benefits (for example, hike, canoe, sit in the sun).

In the DPM, human uses, e.g., shellfish farming, salt extraction and touristic infrastructure are allowed on a temporal basis only through concessions. The concession holders thus hold temporary Withdrawal rights [START_REF] Ostrom | Private and Common Property Rights[END_REF].

Fisheries has been traditionally regulated by a type of community-based ecosystem management through the Prud'homies de Pêche [START_REF] Feral | La prud'homie des pêcheurs de Palavas, étude de la mort d'une institution[END_REF], Féral 1987). For this resource the fishers affiliated to the Prud'homies de Pêche, can be considered as Proprietors sensu [START_REF] Schlager | Property-Rights Regimes and Natural Resources: A Conceptual Analysis[END_REF] who have rights of access, extraction, management of the resource and exclusion, i.e. by excluding non-members from fishing in the lagoons. In conclusion, the Public domains on the coast should be managed as commons for the French citizens and for the human population in general with a major emphasis on the conservation of habitats and species. Any exploitation should be sustainable and not impair the natural heritage. This target is supported by the EU Habitats Directive (1992), which designated coastal lagoons as a priority habitat, thus recognizing that the EU member states that collectively have a specific responsibility for the conservation of this habitat.

Coastal lagoons thus represent a valuable commons for humanity because of their biodiversity, natural and cultural heritage, and natural resources. For their efficient protection, it appears most efficient that the surfaces of these water bodies are withdrawn from the over-heated economic markets on the coast worldwide to guarantee their long-term protection. Other methods for protection include legal regulation and contract-based solutions. As shown above, the latter may create complications for the managers when the property regimes are fragmented. Moreover, in the European Union, the Water Framework Directive recognizes water as a common resource. This creates a duality for the owner of a given water body, who owns the bottom and the sediment, but has to respect the water that exchanges along an aquatic continuum as a common good. and the environmental Code, and the link with their management as protected areas.

Furthermore, using the socio-ecosystems frameworks of Orstrom and Berkes it will be most interesting to study how the different rights for resource exploitation and access of these commons are regulated in national laws and customs.

Interactions" within the framework of the Human-environment observatory "Mediterranean coastline".

Legends and Footnotes to Tables:

Table 1 legend: 1) Negotiations for the acquisition of cadastral parcels in Et. Balistra by the Conservatoire have started since 2015. 2) Note that the sum of cadastral parcels is slightly above 100 % of total lagoon surface because the shorelines for the coastal lagoons often do not fit to the cadastral boundaries.

Legends to Figures:

Figure 1: Timeline listing the most important legislation and historic events that have important implications for property regimes in coastal lagoons.

Figure 1 :

 1 Figure1: Timeline listing the most important legislation and historic events that have important implications for property regimes in coastal lagoons.

  , who often claimed the exclusive private ownership of coastal lagoons with their rights to prohibit access and fishing by others. As shown before several procedures rehabilitated private property of coastal lagoons on the shores of the Mediterranean, even though their objective geographical location and hydrological features qualifies them to be part of the DPM. In this respect, a last notable case concerned the Et. de Vic opposing the fishers of Palavas and the Société des Salins de Villeneuve. The Trial (T.G.I. Montpellier, 26th June of 1964) decided to recognise private ownership and fishing rights based on an arbitration award by the Prévost de Maguelonne dating to 8 September 1190, which was taken up by a Judgment of the Parliament of Toulouse in 1617 and, therefore, prior to the Edict of Moulin of 1566 (Feral, in press). In addition, on the particular occasion of the French revolution in 1789, some Church and state property has been sold to private owners (vente des biens nationaux). Altogether, this explains why a large part of the coastal lagoon surface was private property before 1980. The Conservatoire de l'espace littoral et des rivages lacustres, known as the Conservatoire du littoral was created to prevent the destruction of natural habitats by the galloping urban and touristic developments. Enforced by Law (Law for the creation of the Conservatoire du Littoral, 1975) the Conservatoire du Littoral, hereafter referred to as the Conservatoire, is a public institution that has the prime mandate to safeguard natural areas along the coast. Therefore, it has acquired properties along the coast mainly by buying it from private owners. Occasionally, it hasq also used expropriation procedures and benefited from bequest. Upon acquisition, the property is formally private property of the Conservatoire, but normally within a year the property is shifted to the Public Domain as 'Domain Public du Conservatoire', which means that it has become imprescriptible and inalienable. Hence, its prime objective as a protected natural area cannot be changed and this is exempt from subscription. Later appeared the Nature Conservation Law in 1976, the Coastal Law (1986) and the EU Habitats directive in 1992, transposed in French law since 1995 by Loi n° 95-101, EU Habitats Directive, transposition into French Law (1995) and the Biodiversity Law (2016). Coastal lagoons are a priority habitat (i.e., N° 1150 *) in the EU Habitats directive (1992). On 1 December 1986, France also ratified the Convention on Wetlands of International Importance especially as Waterfowl Habitat (1971), known as the Ramsar Convention. By law (Law for the creation of the Conservatoire du Littoral,1975), the management of the Public Domain of the Conservatoire is delegated to local partners. When seeking partnerships for this management, the Conservatoire invites local municipalities in the first place. Other decentralized public institutions, as Departments and Régions, as well as NGO's can also act as managers. These local partners establish a master plan for the management in collaboration with the Conservatoire, who has to give his final approval.

  Figure 2, are either owned entirely by the Conservatoire (Bagnas) or shared among private property and the Conservatoire (Vendres, Pissevaches). The lagoons close to Montpellier (Fig. 3) show a remarkable patchwork of property regimes. Only Etang D'Ingril Sud, Etang de l'Or and a small section of Etang de Méjean are DPM. Three of the four lagoons that have been separated from the coastline by the building of the Rhône-to-Sète canal (see 2.1 Study sites) are either largely (Et. d'Ingril Nord, Et. d'Arnel) or for a small part (Et. Méjean) private property of the State (DPE). The abandoned Salinas in Et. d'Ingril have been acquired by the Conservatoire. The municipality of Lattes owns about half of the Et. de Méjean, a small part of this lagoon along the NE shoreline is also private property. Etang de Vic and most of Etang Pierre Blanche are Public Domain of the Conservatoire, who acquired most of this by acquisition from private owners, while some cadastral parcels were transferred from the private State domain (DPE) to the Conservatoire. The municipality of Villeneuve-lès-Maguelone owns cadastral sectors in the Et. Pierre blanche and Et. de Prévost. The rest of the latter lagoon is private property (propriety of the diocese of Montpellier and managed by the 'Compagnons de Maguelone', an NGO in charge of maintaining the heritage of the Romanesque cathedral de Maguelone and engaged in social programmes). The Et. le Grec is entirely private property, which is based on the abovementioned 1848 judgment by the Cassation Court (Cuënot et al., 1851). The canal Rhône-to-Sète is Domaine Public Fluvial, which is managed by Voies Navigables de France (VNF).

  were private properties until quite recently. The Department 'Haute Corse' acquired in 1994 the property of the largest lagoon in Corsica, i.e. Et. de Biguglia. Concerning the other six lagoons, two are still entirely privately owned (Et. de la Diana, Et. de la Ballistra), while others have been fully (Et. d'Urbino, Et. Santa Gulia, Et de Palu) or partially (Et. de Terrenzana) acquired by the Conservatoire and became Domaine Public du Conservatoire.Table 2: Forty coastal lagoons in S. France and Corsica (Regions Occitanie, Province-Alpes-Côte d'Azure, Corse) listing their surfaces (ha.) from land cover data bases (equivalent to Corine Land Cover class 521, see Methods), mean depths and surfaces (in ha.) for the different classes of property regimes, and number of municipalities and Departments involved. Most water depths obtained from Le Fur et al., 2018).

  example, the National Trust and Natuurmonumenten have acquired large natural areas along the coasts in the UK and Netherlands, respectively. In France, no large dedicated NGOs exist that acquire large areas for nature conservation purposes, while the smaller NGO ASPAS (Association pour la protection des animaux sauvages) actively acquiring properties for nature conservation purposes has no properties on the coast. Therefore, nature conservation on the French coasts has mainly relied on public policies. Hence, the Conservatoire du Littoral was created in 1975 to acquire cadastral parcels along the coast and safeguard them as public domain protected areas. Therefore, it is entitled to use land acquisition, bequest and expropriation procedures. Before 1980, private properties existed in 23 of the 40 coastal lagoons, representing about a quarter of the total lagoon surface. Since 1980, the Conservatoire also became active to acquire these private coastal lagoon properties.How can the importance of previous private property of coastal lagoons be explained and why are property regimes still highly variable in some areas? In the introduction, we forwarded two hypotheses for explaining private property of coastal lagoons, which apparently is in conflict with the definition of the Maritime Public Domain (DPM) in the French law CGPPP. The first hypothesis, i.e. the private property existed in the past before the DPM was implemented and has been passed through inheritance, certainly applies for the Et. de Prévost and Et. de Pierre Blanche, where Church property of the diocese of Maguelone existed since the 12 th century. The diocese was transferred in 1536 to Montpellier, while keeping its properties in the lagoons and surroundings. The lagoons were exploited for fisheries and salt extraction. Hence, the respect of these

  State to declassify DPM into DPE (e.g., road construction through Et. d'Ingril sud, see above). The co-occurrence of the DPM and the Public Domain of the Conservatory in a coastal lagoon territory also create some difficulties for a coherent management. While the DPM is managed by the DDTM, for the management of the Public Domain of the Conservatory, by law (Law of the Conservatoire, 1975), this institution has to establish a partnership with a local entity to whom it should delegate the management. Hence, because of these legal constraints, the DPM and the Public Domain of the Conservatoire will have different managers, which does not always facilitate a coherent approach.

  Méditerranée and PNR Camargue comprise many lagoons. Four major coastal lagoon areas in continental S. France (Et. Salses-Leucate, Narbonnais lagoon complex, Palavas lagoon complex and Camargue) and three lagoons in Corsica (Ets. Biguglia, Urbino and Palu) have been designated as wetlands of international importance according the Ramsar convention. Other Protected Area status also occur. Recent developments of the French environmental Code request to take ecological connectivity into account in marine and coastal spatial planning. The difficulty is now to integrate these ecological objectives into public domain law, which includes administrative legislation.The DPM and the Public Domain of the Conservatoire in a coastal lagoon do not exclude human access and natural resource exploitation. The terminology developed by Ostrom

Figure 2 :

 2 Figure 2: Geographic distribution, property regimes and municipalities of coastal lagoons in the Occitanie region (from the Spanish-French border to Frontignan).

Figure 3 :

 3 Figure 3: Geographic distribution, property regimes and municipalities of coastal lagoons in the Occitanie region close to Montpellier, the Palavas lagoon complex and Etang de l'Or. Upper left panel note Canal Rhône-to-Sète which is Fluvial Public Domain (Domaine Public Fluvial, DPF)

Figure 4 :

 4 Figure 4: Geographic distribution, property regimes and municipalities of coastal lagoons in the Camargue and Province-Alpes-Côte d'azur region.

Figure 5 :

 5 Figure 5: Geographic distribution, property regimes and municipalities of coastal lagoons in Corsica.

Table 1 :

 1 Definition of the natural Maritime Public Domain according French General Code of the property of public persons (Code général de la propriété des persones publiques, CGPPP) in French (left column), and its translation by the authors in English (right column).

Table 2

 2 Ingril sud. Table 2 also list the year of the first acquisitions by the Conservatoire in the different lagoons. This public body currently has property in twenty of 40 lagoons (22% of the total lagoon surface), which has been designated as Public Domain of the Conservatoire and safeguarded for nature conservation purposes, as such being also inalienable and imprescriptible. In addition, the privately owned Et. de Biguglia was acquired by the 'Haute Corse' Department in 1994. Another aspect shown by theFigures 2, 3, 4 and 5 and Table 2is that the area of many of the larger lagoons belong to different municipalities, up to 9 different municipalities as for Et. de Thau. The Et. de Salses-Leucate has even parts in two different Departments (Pyrenées orientales, Aude).

	Berre). Despite the fact that this is a minority of lagoons, DPM represents almost 65 % of
	the total lagoon surface as it includes all the larger lagoons. Private state property of the
	(DPE) only covers 1.3 %. Recently, 137.7 ha of private state property (DPE) in the
	Palavas lagoon complex (cf. Fig. 3) has been retroceded to the Conservatoire. Properties
	by Departments and municipalities represents 5.7 and 3.1 %, respectively. Currently in
	2020, private properties by third persons still existed in 12 of the 40 lagoons, although
	its surface only represents a very small 3.3 % of the total surface. Before 1975, about 25

resumes the information in a quantitative way. The total lagoon surface is about 577 km 2 . DPM occurs in twelve of the 40 lagoons, either the lagoon is partially DPM (e.g., Et. Salses-Leucate, Et. Bages-Sigean, Et. de Méjean) or entirely DPM (Et. de Thau, Et. de % of the lagoon surface was privately owned, but most of that has now been acquired by the Conservatoire. It also recovered surface from the DPM (Et. les Mouettes) following

  Table 1) has only been challenged in Court a couple of times without success (e.g., C.A. Montpellier of 29 June 1849 and 6 February 1849), as nowadays most coastal lagoons in Occitanie have human-modified inlets.Fragmented ownership of coastal lagoons is considered as highly problematic by coastal lagoon managers, particularly for those involved in nature conservation. Private ownership creates a work load for the managers, as they have to check that the different owners all respect the legislation and have to negotiate with them for other issues (e.g. Agri-environmental contracts, concertation for Natura 2000 objectives etc.). In this respect, it is very positive that the Conservatoire has acquired properties in 20 of 40 lagoons, which has resulted in reducing private lagoon property from around 25 % to the still remaining 3.3 %. The private domain of the State can also be problematic as these areas depend directly on the fiscal services of the State often without clear targets for the management of these sites. It further creates uncertainties as the private domain of the state can be sold. The public domain of the State on the coast, either the Public Maritime Domain (DPM) and the Public Domain of the Conservatoire, confer large advantages for nature protection as these domains are imprescriptible and inalienable.

  Maritime public domain can thus present a good opportunity for environmental protection of coastal lagoons in many countries where this National State property regime is rooted in Roman and Islamic laws. However, in this paper we show that this principle was not fully applied in France and that this incongruence can be explained by a historical process. Furthermore, the example in France shows that its Maritime public domain (DPM) was historically not designed for nature protection, as this represent a demand that has emerged only since the 1970's for coastal lagoons. Hence, we call for a comparative study on legislation and property regimes of coastal lagoons and study how different countries have managed resource exploitation, conservation and adopted solutions to emerging problems and needs. As mentioned dedicated NGO's can perform a role to withdraw surfaces from economic markets for long-term nature protection, while the solution in France has been acquisition by a public body, the Conservatoire, with a specific mandate for nature protection. Nevertheless, on a European and global level there is very little quantitative data available concerning property regimes of coastal lagoons, implication of public law

Table 1 :

 1 Definition of the natural Maritime Public Domain according French General Code of the property of public persons (Code général de la propriété des persones publiques, CGPPP) in French (left column), and its translation by the authors in English (right column).

	Table 2 legend:

Table 2 :

 2 Forty coastal lagoons in S. France and Corsica (Regions Occitanie, Province-Alpes-Côte d'Azur, Corsica) listing their surfaces (ha.) from land cover data bases (equivalent to Corine Land Cover class 521, see Methods), mean depths and surfaces (in ha.) for the different classes of property regimes, and number of municipalities and Departments involved. Most water depths obtained from Le Fur et al., 2018).

Table 2 footnotes :

 2:

Acknowledgements.

We thank the coastal lagoon managers Laurianne Versluys (Perpignan Méditerranée Métropole), Laurence Fonbonne (Syndicat RIVAGE Salses-Leucate), Julie Bertrand (ADENA), Hélène Fabrega (SIEL), Yves Cherainertrand (SNPN), Gaël Hemery (PNR de la Camargue), Elisabeth Le Corr (GIPREB), and François Galeazzi ('Haute Corse' Department) for providing information and sharing their opinions.

Funding

This study has been financially supported by the DRIIHM LabEx (ANR-11-LABX-0010_DRIIHM), "Device for Interdisciplinary Research on human-environments

Conflict of Interest

The authors declare no conflict of interest.


Mariia Nesterova

Franziska Schmidt

Eugen Brühwiler

C Soize

Generalized Pareto Distribution

published or not. The documents may come

INTRODUCTION

The complexity of predicting the behavior of such structures as bridges at the end of their design life is an important topic for modern research. Due to the fact that many European bridges, as well as bridges all over the world, are coming to the end of their design life, the question of the extension of their operational life is vital. Moreover, the study of such objects allows improving design rules for future structures and possibly some profit from an economic point of view.

Usually, in bridge engineering, for the forecasting of the return levels of actions within the time period of the interest, the Extreme Value Theory (EVT) is used [START_REF] Coles | An introduction to statistical modeling of extreme values[END_REF]) such as for background works on Eurocodes [START_REF] Sedlacek | Background document to EN 1991-Part 2 -Traffic loads for road bridges -and consequences for the design[END_REF]. One of the most efficient approaches to be used is the Peaks Over Threshold (POT), which proved to work well in diverse fields: wind engineering [START_REF] Agarwal | Simulation of offshore wind turbine response for long-term extreme load prediction[END_REF], precipitation predictions with nonstationary data [START_REF] Roth | Projections of precipitation extremes based on a regional, non-stationary peaks-over-threshold approach: A case study for the Netherlands and northwestern Germany[END_REF], electricity demand estimation with a time-varying threshold [START_REF] Sigauke | Modelling non-stationary time series using a peaks over threshold distribution with time varying covariates and threshold: An application to peak electricity demand[END_REF] and others.

Concerning bridges, the mentioned method is successfully used to envision the forthcoming situation with an object [START_REF] Zhou | Statistical analysis of traffic loads and their effects on bridges[END_REF][START_REF] Treacy | The use of monitored data in the verification of structural and fatigue safety of existing posttensioned concrete highway bridges[END_REF], based on the long-term monitoring of traffic actions. For the proper use of the EVT, usually only extreme, meaning the highest, actions are considered. For instance, heavy (even overweighted) trucks are taken into account without paying attention to light cars.

Since the number of light duty vehicles in most cases is quite large, they may cause fatigue of certain elements of a bridge. An interesting question is if this case can significantly affect the structure and how to anticipate it. Therefore, the focus of the current paper is mainly on the fatigue loading.

The interest of this study lays in fitting the generalized Pareto distribution (GPD), as a principal distribution of the POT approach, to the distribution of the number of fatigue cycles caused by different vehicles and axles of trucks. There were several similar studies carried out in various areas. For instance, in the area of composites, the stiffness-based model is introduced by [START_REF] Shiri | Prediction of Remaining Fatigue Cycles in Composite Materials Under Uncertainty[END_REF] for fatigue damage and life prediction or (Adasooriya, 2016) compares a new probabilistic fatigue assessment approach with a deterministic approach for railway bridges.

As it was recently proposed (Cetin, 2013), the different aspects of material fatigue may be considered as an extreme value dependent process. For a certain observed object, it is important to understand which case is the most unfavorable for a bridge: high amplitude stresses caused by rare extreme events or small stresses by a high number of cycles.

METHODOLOGY

General procedure

In general, to estimate the probability of occurrence of certain load effects (return level) within specified periods of time (return period), the EVT might be applied to structures or structural elements.

First of all, it is necessary to choose an element in the studied structure that is exposed to the most unfavorable action or combination of actions. For example, in a case of bridges, it can be an element in the middle of a long span, where the bending moment is high. For reinforced concrete bridges, the main interest would be in those rebars with higher tensile stresses; for steel bridges, it might be the bottom flange of the girder, in details, welded parts.

Then, recorded data are needed to make predictions of the future behavior of a structure. One of the possibilities is measurements by strain gauges installed on the surfaces of the observed elements. Several gauges installed at different levels can help to get the stress redistribution in a cross-section. The frequency of measurements is important to obtain correct values of stresses from the measured strain. For instance, in bridges, to detect passing cars, the frequency of 50 Hz can be sufficient. The length of monitoring is not recommended to be less than 3 months in a case of traffic loads [START_REF] Treacy | The use of monitored data in the verification of structural and fatigue safety of existing posttensioned concrete highway bridges[END_REF] due to the instability of predictions.

Usually, such measurements are used to apply EVT based on block maxima or threshold models [START_REF] Coles | An introduction to statistical modeling of extreme values[END_REF], where only extreme effects are considered. Data are fitted, respectively, to generalized extreme value distribution (GEVD) or GPD with certain parameters. These parameters are used to estimate return levels. To identify extreme values of load effects (LEs) represented, in this case, by fatigue cycles, the POT method is used as it is briefly shown in Figure 1.

Common use of GPD

It has been recently adopted for predictions of extreme traffic actions [START_REF] Zhou | Statistical analysis of traffic loads and their effects on bridges[END_REF][START_REF] Zhou | A mixture peaks over threshold approach for predicting extreme bridge traffic load effects[END_REF]. Peak values of LEs 𝑋 𝑖 that lay above a certain threshold 𝑢 are fitted to the GPD. The conditional distribution function can be expressed as: where 𝑌 = 𝑋 -𝑢 is a vector of (positive) threshold excesses 𝑌 𝑖 , so that 𝑌 𝑖 = 𝑋 𝑖 -𝑢 and 𝐹(𝑢) is the cumulative distribution function of random variable 𝑋.

𝐹 𝑢 (𝑦) = 𝑃[𝑌 ≤ 𝑦|𝑋 > 𝑢] = 𝐹(𝑦 + 𝑢) -𝐹(𝑢) 1 -𝐹(𝑢)
The main principe of the POT approach is based on the method described by [START_REF] Pickands | Statistical inference using extreme order statistics[END_REF] and applied later for traffic loads: 𝐹 𝑢 (𝑦), under mentioned assumptions and conditions: 𝑦 = 𝑥 -𝑢, 𝑦 ≥ 0, 𝜎 > 0 and 𝑥 ≥ 𝑢 for 𝜉 ≥ 0, 𝑢 ≤ 𝑥 ≤ 𝑢 -𝜎/𝜉 for 𝜉 < 0; tends to the upper tail of a GPD:

𝐺(𝑥; 𝜉; 𝜎; 𝑢) = { 1 -[1 + 𝜉 ( 𝑥 -𝑢 𝜎 )] - 1 𝜉 , 𝜉 ≠ 0 1 -exp (- 𝑥 -𝑢 𝜎 ) , 𝜉 = 0
Conditions for the application of the EVT are the following:  identical distribution of random variables 𝑋 𝑖 ,  random variables 𝑋 𝑖 are independent,  threshold 𝑢 is necessarily high.

For a long period, observations can be based on the cumulative distribution function of extreme values over a shorter period (Crespo-Minguillon & Casas, 1997).

Provided by [START_REF] Coles | An introduction to statistical modeling of extreme values[END_REF], for the probability 𝑃[𝑋 ≤ 𝑥|𝑋 > 𝑢] with 𝜁 𝑢 = 𝑃{𝑋 > 𝑢}, the solution to the function is:

𝑅 𝐿 (p) = { 𝑢 + 𝜎 𝜉 [(𝑝𝜁 𝑢 ) 𝜉 -1], 𝜉 ≠ 0 𝑢 + 𝜎log (𝑝𝜁 𝑢 ), 𝜉 = 0
where 𝑅 𝐿 (p) is p-observation return levela quantile that exceeds once every p observations with large enough p to provide 𝑅 𝐿 (p) > 𝑢.

The POT approach has also its difficulties such as selecting of an optimized threshold. On one side, it should be reasonably high, so, that extreme event types are not mixed, in order to avoid their convergence. On the other side, the threshold must be low enough to provide a necessary number of peaks for obtaining reliable results. The correct choice of parameter estimators 𝜎, 𝜉 for the GPD is also a drawback of the approach.

Case of fatigue cycles

In order to prove that there is no fatigue in an existing structure, European norms (EN 1993-1-1) provide S-N curves for diverse steel details or elements, and it is stated that there is no any danger under the value of five millions cycles of constant fatigue loading. Recent simulations bring the endurance limit in certain cases up to ten millions cycles (Baptisa, 2017).

Traffic actions on bridges are not constant as small frequent cycles are followed by high amplitude cycles rather randomly. There were several recent studies focused on the fatigue simulation on bridges based on measurements. (Donghuang, 2017[START_REF] Echard | A reliability analysis method for fatigue design[END_REF]. In the current study, the following suggestion is made: fitting of the GPD to a distribution of fatigue cycles, caused by vehicles or axles of different weights (Fig. 3). In certain cases, the proposed method takes into account all traffic actions on a bridge, though, the threshold 𝑢 from the described previously approach is simply equal to zero.

The main idea is to verify the possibility and the efficiency of applying such method to predict the level of the fatigue of bridges (or their structural elements) due to high cycle fatigue load. Here, the term 'high cycle fatigue' is used to determine the cycles from frequent light vehicles.

The use of the GPD with its shape and scale parameters makes it be possible to assess values of return levels of fatigue cycles for different levels of loading. Taking into account that 𝑢 = 0, the formula of the return level takes the following format:

𝑅 𝐿 (p) = { 𝜎 𝜉 [(𝑝𝜁) 𝜉 -1], 𝜉 ≠ 0 𝜎log (𝑝𝜁), 𝜉 = 0
where 𝜁 = 𝑃{𝑋 > 0} is the probability of a positive 𝑋 which is valid for all ELs in this case as number of cycles is always a positive value.

Normally, the value of ζ u can be found as a ratio between all 'extreme' LEsvalues that lay over a threshold 𝑢and all LEs. Considering that all cycles are taken into account, this probability should be 𝜁 = 1 which would give the return level of a maximum number of cycles 𝑁 𝑡𝑜𝑡 . On the other hand, knowing the probability of a number of cycles 𝑁 𝑖 for each range ∆ 𝑖 of LEs 𝜁 𝑖 = 𝑁 𝑖 /𝑁 𝑡𝑜𝑡 , ∑ 𝜁 𝑖 = 1, return levels are estimated for each ∆ 𝑖 that gives the prediction of the distribution of cycles at the return period of the interest. So, the procedure consists of a few steps shown in Figure 2. The proposed algorithm is applied to monitoring data from two European bridges of totally different geometry and usage. Each bridge has the specific equipment, as well, a particular geometry, materials and loading levels. In addition, a different type of monitoring was used in each country: Weight-In-Motion (WIM) system in France, strain gaugesin Switzerland.

Millau Viaduct

The viaduct itself is a large structure of 2460 m long with pylons of 77 to 245 m high with the height of towers of almost 90 m (Fig. 2). Its orthotropic steel deck suspended by cables is stiff and provided with joints that permit openings up to 80 cm in horizontal directions. Each span of 245 m is composed of six meters long elements, each of them works almost as a simply supported beam. That allows assessing local effects in the complex structure of the bridge.

The final period of 180 days of data was obtained from the WIM system. Recordings were made between October 2016 and July 2017 with a few interruptions. The system itself was located in the middle of the first span of the viaduct and recorded all trucks passing the bridge: their weights, speeds, dimensions. The deck of the bridge has two lanes in each directionnormal and high-speed. Monitoring was done only in one, more heavily loaded direction, which does not allow precise evaluating of effects on the viaduct, however, permits to use recorded data for the mathematical model and to make theoretical predictions in terms of LEs given by bending moments. Figure 4 shows the scheme of the viaduct with the place of the installed WIM system. Tables 1 &2 give categories of vehicles recorded on the bridge with an example of recordings.

Anneau bridge

The viaduct consists of seven composite steelconcrete spans of 25.6 m (two steel box girders and a reinforced concrete deck slab). There are only two lanesone in each direction. Strain gauges were installed directly on the reinforcement bars in transversal and longitudinal directions of the deck slab in the middle of the second and the forth spans (Fig. 5), as well as at the top and the bottom of a steel girder.

For each case, a pair of strain gauges was installed to record signals under the central line of the bridge and close to the middle of the lane. The highest values of stresses were obtained in the transversal reinforcement bars of the span 2, which makes this exact detail an object for the study.

Measurements of the strain have been done for one year (August 2016 to July 2017) with only short interruptions which provides almost one full year of monitoring. Moreover, strain gauges allow assessing load effects (stresses) directly, see Section 4.2. MONITORING DATA

Millau Viaduct

First, speaking about fatigue of a bridge, stresses and number of cycles for each stress range are needed. Therefore, critical elements subjected to fatigue loading have to be chosen. In the case of the Millau bridge, values of stresses in the deck are not measured directly. Therefore, it is impossible to observe the presence of dynamic effects; however, the estimation of fatigue is still achievable by using the results of WIM measurements. The system provides not only total weights of each vehicle but also the weight of each axle and distances between them.

Using long-term WIM data for evaluation of fatigue is possible even though it does not give stress cycles directly. Since the system provides forces from each axle of every passing truck that gives stress cycles in elements of the deck. Knowing the exact action applied to the bridge stress in the whole cross-section might be evaluated, in this study, cycles are defined in terms of actions.

There are different combinations of axles possible: some of them are grouped by two or three, as it was demonstrated in Table 1. Assuming that the average distance between axles in a group is 𝑑𝑥 < 3𝑚, and using the recorded distances 𝑑𝑥 between each pair of axles, all recorded types of vehicles are counted and shown on the pie chart (Fig. 6). As it can be observed from the chart, the most frequent vehicle type is the one with five axles where three last are grouped together.

The value of the stress given by a group of axles is usually higher than the one caused by each axle separately and lower than their sum. So, in this case, to calculate cycles, the common action of three axles have to be considered. Also for the types 112, 12, 122 and 1211, the group of two axles should be taken into account.

As it can be found on Figure 7, P1, P2 and P3 are point loads from each axle of the group of three axles, and P1*, P2* and P3* are moments that can be obtained theoretically if each axle would pass the span individually. So, it is necessary to take into account that for the response of the structure, there is an influence between axles in groups of two or three. It could be performed theoretically as two or three point loads, presenting on the 6-meters span at the same time.

As it was mentioned in the Section 3.1, the current study is focused on local effects, therefore, 6meters elements of the first span of the deck are considered. For simplicity, influence lines are obtained as the ones for the simply supported beam (Fig. 7). This gives the absence of the bending moment (or the values are neglectable) at "supports"connections between each span. So, for large distances between single axles of vehicles, a weight of each axle can be taken separately as there is almost no influence in terms of stresses. However, for the groups of axles, different values of bending moments are considered as for two or three point loads.

Due to the absence of direct measurements of stresses in any part of the deck, cycles for fatigue are proposed by values of bending moments in the middle of the 6-meters-span, where WIM system was installed. Influence lines for a whole structure can be applied in order to assess bending moments in other parts of the bridge if needed. As well, stresses can be evaluated in the whole cross-section. Here, for simplicity, only measured actions (bending moments) are studied. As the most frequent vehicles are 113, 11 and 112, bending moments in the midspan are calculated only for single, double and triple axles, as following:  Single point load 𝑃 𝐼 is considered if the distance between axle I and other axles is larger than 𝐿 𝑠𝑝𝑎𝑛 /2 = 3𝑚,  Double axles are presented by two point loads 𝑃 𝐼𝐼 , when the distance between them is smaller than 3𝑚; the maximum moment is obtained from both positions: either 𝑃 𝐼𝐼-1 or 𝑃 𝐼𝐼-2 above the center,  Triple axle load is given by three point loads 𝑃 𝐼𝐼𝐼 with three possible positionseach of them at the midspan one by one.

The exceptional combinations of axles removed from the calculations as not reliable. The main point here is to assess fatigue from frequent vehicles, therefore, rare trucks with more than 5 axles are removed from the initial data. The distribution of all LEs is given in Figure 8. There are high amplitudes of LEs (in grey) that are treated by EVT and the approach explained in Section 2.2, the distribution of ELEs with the 𝑇𝐻 = 215 𝑘𝑁𝑚 and the model check is shown on the figure 10, the estimated return periodin Figure 11. Compared with the fitting the GPD to N, it can be concluded that the approach is working the same way for different types of LEs: bending moments or number of cycles.

Anneau bridge

Overall, 303 days of monitoring were completed between August 2016 and July 2017. After the detailed analysis of received data, it was detected that the highest values of stresses appeared in the transversal rebar of the span 2. Therefore, this exact rebar is studied for the current paper. Here, the monitored data are represented by measured strain (𝜀, 𝜇𝑚/𝑚) directly in rebars, which allows assessing stresses easily knowing Young's modulus of steel 𝐸 = 210000 𝑁/𝑚𝑚 2 = 0,21 𝑁/𝜇𝑚 2 (EN 1993(EN -1-1, 2005)):

𝜎 = 𝐸 • 𝜀, [𝑀𝑃𝑎]
Fatigue cycles are counted using the "rainflow" method (ASTM E1049-85) that provides numbers of cycles for each stress amplitude (Fig. 12). To avoid the "noise" of the signal, all amplitudes under 1 MPa were removed from calculations and the histogram of stress amplitudes of different ranges is shown in Figure 15. The case of high stresses and their extrapolation in time is a question of the EVT and its classical application. As it was shown in the previous section, for the case of Millau. The GPD is fitted to all LEs (given here by stress cycles) as it was described in Section 2.4, fitting with the model check is shown in Figure 13.

Stress amplitudes that are smaller than 1 MPa are removed from the distribution, assuming that such stress as the noise of the system. However, stresses over 1 MPa represent cars passing or light trucks passing not exactly above the installed strain gauge. The high amplitude cycles, over 50 MPa, should be fitted to a different GPD since a number of cycles produced by them is too small (Fig. 14)

Assuming that the fitting is successful (according to the model check, Figure 13), parameters of the GPD can be normally used further for the estimation of return levels. However, in this case, the ratio between 'exceedances' and all effects is equal to one due to the absence of any threshold. What is interesting in the case of fatigue, the estimation of cycles at each range of stress amplitudes -Figure 15.

Though, the probability of occurrence of each range of 5 MPa is found for the amount of data measured for 303 days (Fig. 16). Inserting these values into the formula, return levels are calculated for several values of the return period. The method proposed in Section 2.4 is applied to the distribution of stress cycles with ranges each 5 MPa. The results for return periods of 100, 500 and 1000 years are shown in the Figure 17.

Knowing that in this particular case, all observed stresses lay under the cut-off limit (EN 1993(EN -1-9, 2005) ) of the S-N curve for steel reinforcement bars, the applied algorithm only provides the estimated numbers of cycles over certain return periods (Fig. 17). Studied stresses caused by traffic actions are not high enough to cause any fatigue danger even in 1000 years.

CONCLUSIONS

The algorithm of application of the Generalized Pareto Distribution to fatigue cycles is proposed in the current paper. Results of two studied examples of monitored objects (the Anneau bridge, Switzerland, and the Millau viaduct, France) permit the following summary:  Monitoring data are required for long-term predictions of fatigue life. Moreover, demonstrated in the section 4.2 results for the Anneau bridge allow the following conclusion: measurements by strain gauges significantly simplify the study due to the direct access to stress ranges distribution.  Estimation of return levels of N (number of cycles) at given period of time is possible using EVT and in particular, GPD with its parameters, however, more study of various objects is necessary to make more solid conclusions.  The proposed algorithm works in described specific cases (one year of monitored stresses in the deck of the Anneau bridge and a half a year of recorded vehicles weights on the Millau viaduct), however, it does not show fatigue danger in perspective. Although, it might be applied to other structures with larger traffic volume.

Predicting the number of fatigue cycles in bridges is an actual topic of the modern research. Extending the service duration of existing bridges gives the motivation for novice theoretical approaches that are being proposed nowadays. The experimental part such as monitoring of real objects is undoubtedly useful for the developing of new mathematical models. It leads to the possibility of the application of the EVT for fatigue predictions. Since the traffic growth was not taken into account in the described method, as well, it requires further model uncertainties verification, the current study is being continued by authors.

ACKNOWLEDGEMENT

This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 676139. The grant is gratefully acknowledged.

The WIM measurements for the Millau viaduct were provided by EIFFAGE, France, and the data from monitoring of the Anneau bridgeby EPFL, Switzerland, which is highly appreciated.

Figure 1 .

 1 Figure 1. Representation of the POT approach

Figure 2 .

 2 Figure 2. The algorithm of the proposed method

Figure 3 .

 3 Figure 3. Fitting the GPD to N, number of cycles

Figure 6 .Figure 7 .

 67 Figure 6. Main types of axles groups

Figure 8 .

 8 Figure 8. Bending moments, M, as a function of N, log-scale

Figure 12

 12 Figure 12. 'rainflow' counting

Figure 13 .

 13 Figure 13. Fitting the GPD to N for stress range 1…50 MPa

Table 1

 1

		. Example of vehicles types
	N	Vehi-	Num-	Distance between axles, m
		cle	ber of	1 and	2 and	3 and	4 and
		type	axles	2	3	4	5
	1	113	5	3.64	5.50	1.27	1.23
	2	40	2	5.27	-	-	-
	3	61	4	3.69	6.73	1.31	-
	4	100	3	8.7	3.61	5.1	-
	5	56	3	6.8	1.46	-	-
	Table 2. Axle weights
	N	Group of	Weight of each axles, kN
		axles	1	2	3	4	5
	1	113	15	22	17	17	17
	2	11	41	119
	3	112	32	139	34	34	-
	4	111	46	65	5	-	-
	5	12	78	142	142	-	-