Proposition for a Goldbach's conjecture demonstration Denise Vella-Chemla June 30, 2018

One tries to demonstrate Goldbach's conjecture. One defines 4 variables : X a (n) = #{p + q = n such that p and q odd, 3 p n/2, p and q primes} X b (n) = #{p + q = n such that p and q odd, 3 p n/2, p compound and q prime} X c (n) = #{p + q = n such that p and q odd, 3 p n/2, p prime and q compound} X d (n) = #{p + q = n such that p and q odd, 3 p n/2, p and q compound}

In the following, one notes E(x) the integer part of x (i.e. ⌊x⌋) and π(x) the number of prime numbers lesser than or equal to x. We have the equality above : it follows from recurrence demonstrations that can be found in [DV] a note written in octobre 2014 * .

X d (n) -X a (n) = E(n/4) -π(n) + δ(n) (1)
δ(n) takes values 0,1 or 2. 1) results from the very general fact on any subsets and intersection and union cardinalities :

Simplification of note [DV] propositions provided by Alain Connes in may 2018 : [(

#(P ∪ Q) + #(P ∩ Q) = #(P ) + #(Q) (2)
Here neglecting limit cases that contribute to δ(n)), one sees that (a)

#(P ∩ Q) corresponds to X a (n). (b) #(P ∪ Q) corresponds to E(n/4) -X d (n). (c) #(P ) + #(Q) corresponds to π(n).
Then we have a very simple proof of (1) as a consequence of ( 2).]

Let us see now a property concerning X a (n).

We decide to represent compound numbers by gray color and prime numbers by white color.

We represent odd numbers between 3 and n/2 by rectangles in the bottom of the drawing above and odd numbers between n/2 and n, complementary to n of numbers from the bottom of the drawing by rectangles in the top of the drawing. Rectangles represent contiguous columns associated to decompositions as two odds'sum, and containing x in their bottom part and n-x their complementary in their top part. Columns are contiguously positioned according to the nature of decompositions they contain (according to their type a, b, c or d).

We use those colors :

-green for #(P ∩ Q) ; -red for #(P ∪ Q) ; -blue for #(P ) + #(Q) = π(n).

(P ) + #(Q) -#(P ∪ Q) = #(P ∩ Q).
and let's replace cardinals by associated variables, we obtain

π(n) -E(n/4) + X d (n) -δ(n) = X a (n)
and we wish to have the insurance that X a (n) is always strictly positive since it counts Goldbach's n's decompositions (as sum of two primes).

Although, if we demonstrated that

X a (n) = X d (n) -E(n/4) + π(n) -δ(n) is a relation always verified,
this relation doesn't guarantee that above a certain integer range, X a (n) is always strictly positive.

We note

Credit(n) = 3 x n/2 (BooleanP rime(x) ∧ ¬BooleanP rime(n -x) ∧ BooleanP rime(n + 2 -x)) Debit(n) = 3 x n/2 (BooleanP rime(x) ∧ BooleanP rime(n -x) ∧ ¬BooleanP rime(n + 2 -x))
We find the following recurrence relation for X a (n), very accounting :

X a (n + 2) = X a (n) + Credit(n) -Debit(n) + BooleanP rime( n+2 2 )
Adding the boolean BooleanP rime( n+2 2 ) ensure X a (n)'s positivity for all 2p with p prime, 2p verifying trivially Goldbach's conjecture.

Except those trivial cases of Goldbach's conjecture verification, we wish to demonstrate that X a (n) is always greater than Debit(n). We know that X a (n) is always strictly positive below 4.10 18 (by computer calculations from Oliveira e Silva in 2014).

First we explain what ensure X a (n) positivity for numbers n = 6k + 2.

Variables values arrays in annex show that for nearly all n = 6k + 2 (notably in the second array), we have

X a (n) = Debit(n) + ǫ(n).
ǫ(n) has either value 1 (when 3 is a Goldbach's decomponent of n, 1 being compound, 3 + (n -3) decomposition is not counted by Debit(n)) or value 0.

We see studying Credit(n) and Debit(n) definitions that among prime numbers lesser than n/2, ones are counted by Credit(n) while the others are counted by Debit(n), because all prime numbers lesser than n can't be simultaneously Goldbach's decomponents of n. This argument ensure the strict positivity of Credit(n).

Let us see now why, in the case in which n is of the form 6k + 2, Debit(n) = X a (n) -ǫ(n) : in such a case, prime numbers of the form 6k ′ -1 can't be Goldbach's decomponent of n because if it were the case, n -x = (6k + 2) -(6k ′ -1) = 6(k -k ′ ) + 3 would be divisible by 3. Prime numbers x that can be Goldbach's decomponents of n are thus of the form 6k ′ + 1 ; this fact has as consequance that n + 2 -x = (6k + 4) -(6k ′ + 1) = 6(k -k ′ ) + 3 is divisible by 3 and is thus countable as a debit. We have X a (n) = Debit(n) + ǫ(n), that could implies X a (n)'s vanishing but the Credit(n) addition, Credit(n) being strictly positive permits to avoid such a vanishing.

In the case where n is of the form 6k or 6k +4, one sees that X a (n) is always strictly greater than Debit(n), what guarantees its strict positivity when one substracts Debit(n) to it. Let us try to explain why this is the case : by its definition, Debit(n) is the cardinality of a subset of the set of cardinal X a (n) (indeed, Debit(n) counts Goldbach's decompositions of n = x + (n -x) such that n + 2 -x is not prime) ; if X a (n) were equal to Debit(n), we would have, from the definition of Debit(n), for all Goldbach's decomposition of n, at the same time n -x prime and n + 2 -x prime, implying that x + (n + 2 -x) would be a Goldbach's decomposition of n + 2 (i.e. that all Goldbach's decompositions p1 + p2 of n would be inherited as Goldbach's decompositions p1 + (p2 + 2) by n + 2). But we know by congruences study † that x is a Goldbach's decomponent of n if and only if x ≡ n (mod p) for every p lesser than √ n. All those incongruences couldn't be verified all at the same time, on one side by x and n, and on the other side by x and n + 2. This has as consequence that for even numbers n of the forms 6k and 6k + 4, Debit(n) < X a (n) and it implies, by inheritance from n to n + 2, that X a (n) is strictly positive for all n 6. †. see for instance a october 2007 note, Changer l'ordre sur les entiers pour comprendre le partage des décomposants de Goldbach that can be downloaded at http://denisevellachemla.eu.

Annex 1a : variables values array for even numbers between 6 and 100

n X a (n) Credit Debit BooleanP rime( n+2 2 ) 6 1 0 0 8 1 0 0 1 10 2 0 1 12 1 1 1 1 14 2 1 1 16 2 1 1 18 2 1 1 20 2 1 1 1 22 3 1 1 24 3 1 2 1 26 3 2 3 28 2 2 1 30 3 1 2 32 2 2

Figure 1 :

 1 Figure 1 : n'decompositions contiguously positionned according to their nature


Bruno Almeida
email: bruno.r.almeida@gmail.com

Christophe Roche
email: christophe.roche@univ-savoie.fr

Rute Costa
email: rute.costa@fcsh.unl.pt

Rute Costa

Archaeological classification and ontoterminology: the case of Islamic archaeology of the al-Andalus

come

Introduction

Within archaeology, classification plays a key role in the organization of knowledge regarding the physical remains of past cultures. In this context, classification is the ordering of archaeological data into classes or groups. Typologies of archaeological data, such as artefacts, are the most relevant classification systems. According to Darvill, a typology consists of a theoretical system of types and subtypes of objects, based on "their qualitative, quantitative, morphological, formal, technological, and functional attributes" [START_REF] Darvill | The concise Oxford dictionary of archaeology[END_REF]. Classification is also used in the seriation of objects within a culture. A series, as a broad unit of classification, groups together objects according to incremental changes. According to Kipfer, a series has "a duration in time, when one culture or style develops into another, and an extent in space (the area occupied by the various cultures or styles making up the series" (Kipfer, 2000, pp. 506).

This paper describes the continuities and discontinuities between classification as a practice in archaeology and the development of an ontoterminology in the domain of Islamic archaeology of the al-Andalus. This work is carried out in the context of an ongoing PhD thesis in linguistics at the Universidade NOVA de Lisboa, Portugal, and in information and communication science at the Université de Savoie Mont-Blanc, France. An ontoterminology has been defined as "a terminology whose conceptual system is a formal ontology" [START_REF] Roche | Ontoterminology: how to unify terminology and ontology into a single paradigm[END_REF](Roche, , p. 2626)). On the other hand, an ontology consists of an "explicit specification of a conceptualization" (Gruber, 1993, p. 199), i.e. a formal and computational concept system representing shared knowledge in a domain.

The development of an ontoterminology requires an interdisciplinary approach effectively placing terminology as a subject matter between linguistics and knowledge representation and reasoning, a domain within artificial intelligence. Among other things, this justifies the theoretical distinction between terms and concepts. Terms are used and recognized by domain experts within specialized communication, i.e. in the production and reception of texts in specialized domains of activity. They are units of meaning within a given linguistic system (for instance, Portuguese or Spanish). From a linguistic point of view, terms are therefore placed within lexis and are analysis as such. Concepts, on the other hand, are analysed as units of knowledge in a given domain. They are placed in relation to one another in a concept system and represented through a formalism, following an approach in line with knowledge representation.

Classification in archaeology: from practice to theory

Archaeological classification can be thought of as an activity and as a result deriving from the practical requirements of field work. Classification schemes can be valuable resources, for example, in the relative dating of pottery sherds. In theoretical terms, Adams & Adams consider that types within typologies of artefacts have multiple dimensions, namely material, mental and representational [START_REF] Adams | Archaeological typology and practical reality: a dialectical approach to artifact classification and sorting[END_REF]. The material dimension pertains to the real-world objects whose fragments are excavated, classified, restored and later preserved in museum collections. The mental dimension refers to the mental images that archaeologists have of these objects, their attributes and the relationships established between them. Finally, the representational dimension consists of all kinds of descriptions (verbal or nonverbal), categorizations and nomenclatures proposed by archaeologists.

This approach to classification enables the distinction between real-world objects, their mental images and the representations that are made of them. According to Adams & Adams, the basic elements of a typology are concepts, descriptions, definitions, labels, categories and members [START_REF] Adams | Archaeological typology and practical reality: a dialectical approach to artifact classification and sorting[END_REF]. Concepts are the mental aspects of typologies, which make possible for archaeologists to have an understanding of types, their defining characteristics, and to communicate about them in a meaningful way.

Within the representational dimension, descriptions are the verbal or non-verbal representations of types and their characteristics. They enable communication between two or more individuals in the domain. Definitions are statements or depictions that identify the attributes that are able to distinguish one type from the remaining artefact types. [START_REF] Adams | Archaeological typology and practical reality: a dialectical approach to artifact classification and sorting[END_REF] consider that most types are never explicitly defined in archaeology, although it is theoretically possible to produce definitions for any type of artefact. Labels are identifiers of types within a typology. They consist of arbitrary words, numbers or alphanumeric codes that allow to refer to types in place of their descriptions or definitions. Names, on the other hand, are not arbitrary, since they invoke certain features of types in a given language, while also having a denotative function. According to [START_REF] Adams | Archaeological typology and practical reality: a dialectical approach to artifact classification and sorting[END_REF], most typologies employ either labels or names, but not both.

With regard to categories, [START_REF] Adams | Archaeological typology and practical reality: a dialectical approach to artifact classification and sorting[END_REF] consider that they are not purely within concepts, descriptions or definitions. Instead, categories range over the mental and representational dimensions in sorting objects for the purposes of a given typology. Lastly, members refer to the real-world objects, whose characteristics are analysed and identified as exemplars of a given type concept, i.e. they are placed within a type category.

Relationship with terminology and ontology development

This approach to classification raises the question of how the different elements of a typology can be represented in an ontoterminology, i.e. a terminology whose conceptual dimension is represented in an ontology. There is no question that a type category is represented as a class in an OWL ontology, since it consists of a set of individual objects. Conversely, type members are represented in OWL as individuals within classes.

With regard to the mental dimension of typologies described by Adams & Adams, we saw that types are defined or described in typologies through depictions or verbal statements. These can be interpreted by the terminologist or ontology developer in order to gain an understanding of the relationships between types and their defining characteristics. Characteristics of types can be represented in OWL through object properties. Type concepts, therefore, are represented by the formal description of OWL classes in an ontology through sets of axioms.

Regarding the other so-called 'representational' elements of typologies, type names consist of terms in a given language within specialized communication in archaeology. Type labels, on the other hand, are not necessarily linguistic items. Furthermore, their arbitrary nature may limit them to the purposes of a particular typology. In an ontoterminology, type names will denote OWL classes that represent type categories.

Islamic archaeology of the al-Andalus

In Portugal and Spain, Islamic archaeology is focused on the recovery and study of the material culture left behind during the eight centuries of Islamic presence in the Iberian Peninsula (from 711 to 1492). Pottery artefacts are the most studied type of material from this period due to its durability and cultural significance. The study of Islamic pottery of the al-Andalus has allowed for a deeper understanding of its culture and society, including its everyday life, eating habits, trade relations, technical development, symbolism and ideology [START_REF] Gómez Martínez | La cerámica islámica de Mértola: producción y comercio[END_REF].

Terminology and classification in Islamic archaeology

The terminology of Islamic pottery artefacts, as used by Portuguese and Spanish archaeologists, is placed in close association with the typological analysis of these objects. This approach was pioneered in Spain by [START_REF] Rosselló-Bordoy | Ensayo de sistematización de la cerámica árabe en Mallorca[END_REF]. His research was initially focused on the analysis of artefacts recovered in Mallorca, having identified 17 different series of pottery artefacts in the region, for which he proposed a terminology in Spanish and Catalan. In his later work, Rosselló-Bordoy included information from historical sources and documented finds from other terri-TOTh 2017 tories of the al-Andalus, expanding his classification to 41 different series within 10 functional classes [START_REF] Rosselló-Bordoy | El nombre de las cosas en al-Ândalus: propuesta de terminología cerámica[END_REF].

A similar approach was followed in Portugal by the CIGA (Cerâmica Islâmica do Gharb al-Ândalus) group of archaeologists. The activity of this group is focused on the study of pottery artefacts recovered in the western part of the al-Andalus, known in the Islamic period as Gharb al-Andalus. A typology and terminology of artefacts in Portuguese was proposed by the CIGA group [START_REF] Bugalhão | CIGA: projecto de sistematização para a cerâmica islâmica do Gharb al-Ândalus[END_REF]. The typology consists of 8 classes, which are based on the function of the artefact types. These classes are further specialized into a total of 48 types according to formal and functional distinctions.

The information presented for each artefact type in both typologies includes names, verbal descriptions and archaeological illustrations of typical examples, focusing on attributes such as geometrical shape and component parts.

Developing an ontology of artefact types

The knowledge shared by Portuguese and Spanish archaeologists, as expressed in the specialized texts of the domain, is currently being represented through an OWL ontology. OWL was chosen not only for its importance as a recommendation by the W3C for the semantic web, but also due to its foundations in description logics. This enables the use of reasoners within ontology editors such as Protégé [START_REF] Musen | The Protégé project: a look back and a look forward[END_REF]. Reasoners facilitate ontology development by checking for inconsistencies and drawing inferences, which allows for the automatic classification of concepts.

Within such formalism, the above-mentioned typologies of pottery artefacts can be represented and distinguished from a comprehensive conceptualization of artefact types. The categories employed in archaeological classifications may have different scopes. For example, the class of artefacts for 'other uses, not exclusively domestic' (Rosselló-Bordoy, 1991, p. 172) is used to group together artefacts that do not seem to fit within other classes (in this case, alembics). Other categories have a more precise scope, such as 'tableware' or 'lighting objects' (Bugalhão et al., 2010, pp. 460-461). Archaeological classes, therefore, may include widely different types of objects. For instance, a class such as 'ritual and recreational objects' includes drums, playing stones and basins for ablutions (Bugalhão et al., 2010, p. 462).

The example of lighting artefacts will be discussed in this section to further illustrate the similarities and differences between a classification in archaeology and an ontology in knowledge representation. The typologies are based on hierarchies in which the class of lighting objects has several subclasses. This expresses the fact that there can be several types of lighting objects made of pottery. On the other hand, the conceptualization of the domain proposed in this paper is based on the epistemological principle of genus-differentia, also referred to as 'specific difference' [START_REF] Roche | The differentia principle as a cornerstone of ontology[END_REF].

Our approach allows for a fine-grained distinction between artefact types based on their essential characteristics, either formal or functional. Furthermore, the proposed conceptualization may hold across other material types beside pottery. The use of a reasoner in Protégé allows for the automatic classification of the abovementioned typologies of lighting objects within our ontology. This makes explicit information that was conflated within the typologies of pottery artefacts.

From typologies to an ontology of artefact types: the case of lighting objects

In the typology proposed by the CIGA group, based on findings from the Gharb al-Andalus, the class of pottery lighting objects subsumes four classes denoted by the Portuguese terms candil, candeia, candeia de pé and lanterna [START_REF] Bugalhão | CIGA: projecto de sistematização para a cerâmica islâmica do Gharb al-Ândalus[END_REF]. Fig. 1 shows instances of these types of lighting objects.

On the other hand, the typology of lighting objects proposed by Rosselló-Bordoy consists of three series, denoted by the Spanish terms candil, almenara and fanal [START_REF] Rosselló-Bordoy | El nombre de las cosas en al-Ândalus: propuesta de terminología cerámica[END_REF]. These are illustrated in Fig. 2. From left to right and top to bottom: candil, candeia, candeia de pé and lanterna (Bugalhão et al., 2010, p. 471) The available information shows that candil [pt] denotes a lighting object with a closed chamber, while candeia has an open chamber.1 Candeia de pé also has an open chamber, which is supported by a tall foot. Finally, lanterna denotes a lighting object that was designed to be used in open spaces, and is characterized by a closed orifice and globular body. Rosselló-Bordoy. From top to bottom: candil, almenara, fanal (Rosselló-Bordoy, 1991, p. 174) Regarding the Spanish terms, candil [es] denotes any artefact kind for domestic illumination, open or closed, unlike in the case of the Portuguese term candil [pt], which tends to be used for denoting the closed shape [START_REF] Bugalhão | CIGA: projecto de sistematização para a cerâmica islâmica do Gharb al-Ândalus[END_REF]. On the other hand, fanal denotes a lighting object designed for open spaces. There are several variants of lighting objects for domestic use, some of which have well established terms in Spanish. The closed and open variants are denoted by the terms candil de piquera and candil de pellizco, respectively, or candil de cazoleta cerrada and candil de cazoleta abierta. The open variant with a tall foot is denoted by the term candil de pie alto [START_REF] Navarro Palazón | Siyasa: estudio arqueológico del despoblado andalusí (ss. XI-XIII)[END_REF][START_REF] Rosselló-Bordoy | El nombre de las cosas en al-Ândalus: propuesta de terminología cerámica[END_REF].

TOTh 2017

Fig. 2: Illustration of the series of lighting objects according to

An ontology of these artefact types was proposed in a previous paper following the genus-differentia principle. The differences found between the Portuguese and Spanish terminology were explained by the relationship between each lexical network and the ontology [START_REF] Almeida | Terminology and ontology development in the domain of Islamic archaeology[END_REF]. However, this ontology of lighting objects was not complete, since it excluded objects such as the one referred to by the Spanish terms almenara and policandela, whose existence in pottery is doubtful. Also absent from the ontology is a kind of portable lamp that has a disk instead of a neck, referred to in Spanish as candil de disco impreso [START_REF] Gómez Martínez | Contenedores de fuego en el Garb al-Andalus[END_REF].

Rosselló-Bordoy's seriation motivate a further problem with regard to our ontology. The candil [es] series, shown in Fig. 2, includes objects that constitute a different series of artefacts according to other authors. This series is denoted by the Spanish term lamparilla, and is characterized by its conical profile [START_REF] Vallejo Triano | Aportaciones para una tipología de la cerámica común califal de Madinat al-Zahra[END_REF]. A variant with a bifrustum-shaped body is thought to exist, and is denoted by the Spanish term lamparilla bitronconónica. A differentiating factor of these artefacts is that they lack a beak in which to hold the wick. Contrary to other domestic lighting objects, the wick would be floating in a cork within the fuel chamber [START_REF] Gómez Martínez | Contenedores de fuego en el Garb al-Andalus[END_REF].

The consideration of the artefact types mentioned above leads us to propose the concept system represented in Fig. 3 and Fig. 4. This hierarchy was formalized in OWL through the Protégé ontology editor (Fig. 5). Concepts are represented by classes in OWL. Essential characteristics are represented by object such as hasFunction or hasComponentPart, and by classes or individuals in the range of these properties, such as lighting closed spaces or tall foot. In the cases where we need to represent the absence of a characteristic, as in 'without a beak', the resulting class is defined through the negation of its sibling class. In this example, 'single lamp without beak' can be defined in Protégé as Sin-gleLamp and (not SingleLampWithBeak), since all single lamps are defined as either having a beak or not (i.e. SingleLamp ≡ SingleLampWithBeak or SingleLampWith-outBeak). This conceptualization allows to distinguish between lamps with a single chamber and lamps with multiple chambers, which includes the almenara artefact type. The specialization of single lamps, as seen in Fig. 4, allows to distinguish between lamps with a beak, such as candil [pt] or candeia, and lamps without a beak, such as lamparilla or lamparilla bitronconónica. Finally, the specialization of single lamps allows to distinguish between closed shapes, including candil de disco impreso, and open shapes, namely candeia and candeia de pé.

4.2

The archaeological typologies of artefacts within the ontology

The typologies of pottery artefacts mentioned above can be better understood by representing them within our ontology. In order to achieve this, a class hierarchy was created for each of these typologies, namely CIGALightingObject and Rossello-BordoyLightingObject (Fig. 6 and Fig. 7,respectively). This allows for a reasoner to classify each one of these classes within the conceptualization of artefacts described in the last section.

In these hierarchies, each class was described through relevant attributes gathered from the available textual and visual sources. For example, the candeia class of the CIGA group was described as having an open shape, single chamber, beak, flat base and the function of providing a light source in domestic environments. Based on this information, HermiT, a reasoner within Protégé, correctly classified candeia as a subclass of OpenSingleLampWithBeakAndFlatBase (as highlighted in the class description shown in Fig. 8).

TOTh 2017 This approach also allows us to study the differences between each classification scheme. For example, the RosselloBordoyPotteryCandil class is subsumed by Sin-gleLamp, which means that in Rosselló-Bordoy's scheme, the candil [es] class is any kind of domestic lighting object with a single chamber. On the other hand, the CIGAPotteryCandil class is subsumed by ClosedSingleLampWithBeak, showing that the candil [pt] class in the CIGA group's proposal excludes the objects referred to in Spanish as lamparillas.

Concluding remarks

This article has shown how archaeological classification relates to the development of an ontoterminology. Typologies of artefacts provide insights into the material culture of the al-Andalus. Furthermore, these typologies are directly related to the terminology of the domain in Portugal and Spain. This relationship is attested to by the proposals put forward by Portuguese and Spanish archaeologists, which are important sources for the development of an ontoterminology.

The example of lighting artefacts shows, however, that the typologies have information that is implied or conflated in their proposed ordering of artefacts. This is to be expected, since the purpose of these typologies is to express the names of artefact types in Islamic archaeology. On the other hand, the development of an ontoterminology requires a formalism in which the essential characteristics of each artefact type are made explicit.

The approach presented in this paper allows to gain a better understanding of the typologies through logical reasoning, since we can determine how each typology relates to a formal conceptualization of artefacts based on the genus-differentia principle. Furthermore, this approach enables the contrastive study of the Portuguese and Spanish terminology of the domain in reference to our ontology, which is valuable from the point of view of terminology research.

Fig. 1 :

 1 Fig. 1: Illustration of the class of pottery lighting objects according to the CIGA group.From left to right and top to bottom: candil, candeia, candeia de pé and lanterna(Bugalhão et al., 2010, p. 471)

Fig. 3 :

 3 Fig. 3: Concept system of lighting artefacts

Fig. 5 :

 5 Fig. 5: Ontology in Protégé

Fig. 6 :

 6 Fig. 6: The class of pottery lighting objects according to the CIGA group

Fig. 9

 9 Fig.9shows that HermiT was able to infer the relationship between the defined classes of our conceptualization based on genus-differentia and the artefact types identified by the CIGA group and Rosselló-Bordoy. This makes explicit information that was conflated in both classification schemes. While the types of lighting objects are placed at the same level in the typologies, the defined classes we propose express more fine-grained relationships. For example, almenara and candil [es] are both

Fig. 9 :

 9 Fig. 9: Inferred relationship between the archaeological classifications and the conceptualization of artefacts

In pottery analysis, a vessel is considered to be open if the orifice has a diameter equal to or greater than the maximum diameter of the body. On the other hand, a vessel is considered to be closed if the orifice has a diameter less than the maximum diameter of the body (Rice

, 2015).

Acknowledgements. This research has been financed by Portuguese National Funding through the FCT -Fundação para a Ciência e a Tecnologia as part of the project Centro de Linguística da Universidade Nova de Lisboa -UID/LIN/03213/2013.

Résumé

Cet article décrit les continuités et les discontinuités entre la classification en archéologie et le développement d'une ontoterminologie dans le domaine de l'archéologie islamique de l'al-Andalus. Le développement d'une ontoterminologie exige une approche interdisciplinaire qui place la terminologie comme un domaine entre la linguistique et la représentation des connaissances. L'exemple des artefacts d'éclairage présenté dans cet article démontre que les typologies archéologiques ont des informations implicites ou confondues dans leur organisation proposée d'artefacts. Alors que le but de ces typologies est d'exprimer le nom des types d'artefacts dans l'archéologie islamique, le développement d'une ontoterminologie nécessite d'un formalisme explicitant les caractéristiques essentielles de chaque type d'artefact. Cette approche permet une meilleure compréhension des typologies des artefacts par le raisonnement logique.


Claudia Aristizábal
email: claudia.aristizabal@univ-grenoble-alpes.fr

Pierre-Yves Bard
email: pierre-yves.bard@univ-grenoble-alpes.fr

Juan Camilo Gómez

Céline Beauval
email: celine.beauval@univ-grenoble-alpes.fr

INTEGRATION OF SITE EFFECTS INTO PSHA: A COMPARISON BETWEEN TWO FULLY PROBABILISTIC METHODS FOR THE EUROSEISTEST CASE

Keywords: Site Response, PSHA, Non-linear behavior, Host-to-target adjustment, stochastic simulation

Several approaches have been proposed to integrate site effects in Probabilistic Seismic Hazard Assessment (PSHA), varying from deterministic, to hybrid (probabilistic-deterministic), and finally fully probabilistic approaches. The present study compares hazard curves obtained for a soft, non-linear site with two different, fully probabilistic site specific seismic hazard methods: 1) The Full Convolution Analytical Method (AM) (Bazzurro and Cornell 2004a,b) and 2) what we call the Full Probabilistic Stochastic Method (SM). The AM computes the site-specific hazard by convolving the site-specific bedrock hazard curve, Sar(f), with a simplified representation of the probability distribution of the amplification function, AF(f) at the considered site, while the SM is built from stochastic time histories on soil corresponding to a representative, long enough catalogue of seismic events. This comparison is performed for the example case of the Euroseistest site near Thessaloniki (Greece). We generate a hazard-consistent synthetic earthquake catalogue, apply host-to-target corrections, calculate synthetic time histories with the stochastic point source approach, and scale them using an adhoc frequency dependent correction factor to fit the specific rock target hazard. We then propagate the rock stochastic time histories, from depth to surface using two different 1D site response analysis, a linear equivalent (LE) and non-linear (NL) codes, to evaluate the code-to-code variability. Lastly, we compute the probability distribution of the non-linear site amplification function, AF(f), for both site response approaches, and derive the site-specific hazard curve with both AM and SM approaches. Results are found in relatively satisfactory agreement whatever the site response code along all the studied periods. The code-to-code variability (EL and NL) is found significant, providing a much larger contribution to the hazard estimate uncertainty, than the method-to-method variability (AM and SM). However, the AM approach presents a numerical limitation, that is not encountered with the SM approach, though with a much higher computational price. The use of stochastic simulations to integrate site effects into PSHA allows to better investigate the variability of the site response physics, and a good parameterization of the input parameters, something that currently is not possible with real data due to its scarcity especially at high acceleration levels.

INTRODUCTION

The integration of site effects into Probabilistic Seismic Hazard Assessment (PSHA) is a constant subject of discussion within the seismic hazard community due to its high impact on hazard estimates. A relevant overview and enlightening examples can be found for instance in Bazzurro andCornell 2004a,b andPapaspiliou et al. 2012a,b. However, it is still treated in a rather crude way in most engineering studies, and the variability associated to the non-linear behavior of soft soils is not fully or not properly taken into account. For instance, all the various approaches discussed in [START_REF] Aristizábal | Site-specific PSHA: Combined effects of single station sigma, host-to-target adjustments and non-linear behavior[END_REF]Aristizábal et al. ( , 2018a) ) to include site effects into PSHA correspond to hybrid (deterministic -probabilistic) approaches, for which the (probabilistic) uniform hazard spectrum S ar on the specific bed rock of a given site is simply multiplied by the median site-specific amplification function AF(f; S ar ), possibly including non-linear site response for this particular ground motion level. The main drawback of those different hybrid-based approaches is that once the hazard curve on rock is convolved with the local non-linear, frequency dependent amplification function, AF(f;S ar ), the exceedance probability at site surface is no longer the one defined initially for the bedrock hazard: at a given frequency, the same surface ground motion can be reached with different reference bedrock motion (corresponding to different return periods and/or different scenario earthquakes) and different non-linear site response. This issue was investigated by Bazurro and Cornell 2004b, who concluded for their two example sites (clay and sand), that the hybrid-based method tends to be non-conservative at all frequencies and at all mean return periods with respect to their approximation of the fully probabilistic method. The present work constitutes a further step along the same direction, trying to overcome the limitation of hybridbased approaches for strongly non-linear sites by providing a fully probabilistic description of the sitespecific hazard curve. It presents a comparison exercise between hazard curves obtained with two different, fully probabilistic site-specific seismic hazard approaches: 1) The Full Convolution Analytical Method (AM) proposed by Bazurro and Cornell 2004a and 2) what we call the Full Probabilistic Stochastic Method (SM). The AM computes the site-specific hazard on soil by convolving the site-specific hazard curve at the bedrock level, S ar (f), with a simplified representation of the probability distribution of the site-specific amplification function, AF(f; S ar ), while the SM derives hazard curves directly from a large set of synthetic time histories at site surface, combining simple point-source stochastic simulations for bedrock on a hazard-consistent event catalogue, and non-linear site response for all bedrock time histories. This comparative exercise is implemented here for the Euroseistest site, a multidisciplinary European experimental site for integrated studies in earthquake engineering, engineering seismology, seismology and soil dynamics [START_REF] Pitilakis | The EUROSEISTEST Strong-Motion Database and Web Portal[END_REF].

STUDY AREA: THE EUROSEISTEST SITE

As stated in [START_REF] Pitilakis | The EUROSEISTEST Strong-Motion Database and Web Portal[END_REF], "the Euroseistest site is a multidisciplinary European experimental site for integrated studies in earthquake engineering, engineering seismology, seismology and soil dynamics. It is the longest running valley-instrumentation project worldwide, and is located in Mygdonia valley (epicenter area of the 1978, M6.4 earthquake), about 30 km to the NE of the city of Thessaloniki in northern Greece. It consists of a 3D strong-motion array and an instrumented SDOF structure (EuroProteas) to perform free and forced tests" (see Figure 1). This site was selected as an appropriate site to perform this exercise, because of the availability of extensive geological, geotechnical and seismological surveys. The velocity model of the Euroseistest basin has been investigated by several authors [START_REF] Jongmans | EURO-SEISTEST: Determination of the geological structure of the Volvi basin and validation of the basin response[END_REF][START_REF] Raptakis | Site effects at Euroseistest-I. Determination of the valley structure and confrontation of observations with 1D analysis[END_REF]Chávez-García et al. 2000), and was used to define 1D soil column for the present exercise (Figure 2a and Table 1). Epicenter and focal mechanism [START_REF] Liotier | Modelisation des ondes de volume des seismes de l'arc Ageen[END_REF]) of the mainshock (red) and epicenters of the largest events of the sequence (green, yellow) [START_REF] Carver | Aftershocks of the June 20, 1978, Greece earthquake: a multimode faulting sequence[END_REF] are indicated (from http://euroseisdb.civil.auth.gr/geotec).

(b) 3D model of the Mygdonia basin geological structure [START_REF] Manakou | 3D soil structure of the Mygdonian basin for site response analysis[END_REF].

Degradation curves are also available to characterize each soil layer, (Figure 2b,c), [START_REF] Raptakis | Data analysis of the Euroseistest strong motion array in Volvi (Greece): Standard and horizontal-to-vertical spectral ratio techniques[END_REF] and local recordings as well to calibrate the model in the linear (weak motion) domain. Several studies performed at the Euroseistest, both instrumental and numerical, have consistently shown a fundamental frequency (f 0 ) around 0.6 -0.7 Hz [START_REF] Riepl | Detailed evaluation of siteresponse estimation methods across and along the sedimentary valley of Volvi (EURO-SEISTEST)[END_REF], Raptakis et al., 2000[START_REF] Maufroy | Earthquake ground-motion in the Mygdonian basin, Greece: the E2VP verification and validation of 3D numerical simulation up to 4 Hz[END_REF], 2016and 2017) with an average shear wave velocity over the top 30 meters V S30 equal to 186 m/s. The 1D simulations performed in this study are based on the parameters listed in Table 1, which have been shown to be consistent with the observed instrumental amplification functions, AF(f), at least in the linear domain. Euroseistest shear modulus (b) and damping ratio (c) degradation curves [START_REF] Pitilakis | Geotechnical And Geophysical Description Of Euro-Seistest, Using Field, Laboratory Tests And Moderate Strong Motion Recordings[END_REF].

METHODOLOGY

The methods followed in this work correspond to two different, fully probabilistic procedures to account for highly non-linear soil response in PSHA, a Full Probabilistic Stochastic Method (SM) developed for the present work, and the Full Convolution Analytical Method (AM).

For both approaches, the first step is to derive the hazard curve for the specific bedrock properties of the considered site. As shown in Table 1 and Figure 2, this bedrock is characterized by a large Swave velocity (2600 m/s), and is therefore much harder than "standard rock" conditions corresponding to V S30 = 800 m/s. This step must thus include host-to-target corrections ("HTT", [START_REF] Cotton | Criteria for selecting and adjusting ground-motion models for specific target regions: Application to central Europe and rock sites[END_REF][START_REF] Van Houtte | Analysis of the origins of κ (kappa) to compute hard rock to rock adjustment factors for GMPEs[END_REF][START_REF] Delavaud | Testing the global applicability of ground-motion prediction equations for active shallow crustal regions[END_REF][START_REF] Biro | Importance and impact of host-to-target conversions for ground motion prediction equations in PSHA[END_REF]. For sake of simplicity in the present exercise, the hazard curve has been derived with only one GMPE [START_REF] Akkar | Empirical ground-motion models for point-and extendedsource crustal earthquake scenarios in Europe and the Middle East[END_REF], which is satisfactorily representative of the mean hazard curve obtained with seven other GMPEs deemed relevant for the European area. For sake of simplicity also, the HTT adjustments have been performed on the basis of simple velocity adjustments calibrated on KiKnet data [START_REF] Laurendeau | Derivation of consistent hard rock (1000 < VS < 3000 m/s) GMPEs from surface and down-hole recordings: analysis of KiK-net data[END_REF].

For the SM approach, the next step consists in generating a synthetic earthquake catalog sampling the magnitude-frequency distribution (better known as the Guttenberg-Richter Law) of the area source zone of the SHARE Seismotectonic model [START_REF] Woessner | The 2013 European seismic hazard model: key components and results[END_REF][START_REF] Pagani | OpenQuake engine: an open hazard (and risk) software for the global earthquake model[END_REF]. This synthetic earthquake catalog was derived using the Stochastic Event Set Calculator tool of the OpenQuake engine [START_REF] Pagani | OpenQuake engine: an open hazard (and risk) software for the global earthquake model[END_REF], and is characterized by a set of events with magnitude-distance (M,D) couples corresponding to a total duration of 50000 years. It was then used to generate compatible synthetic time histories on rock using the Boore 2003Stochastic Method (Boore 2003, 2005). By compatible we mean that the hazard curve on rock generated with the classical PSHA method, is (approximately) equal to the hazard curve built from the generated stochastic time histories. This component of the approach is not trivial however, and requires specific adjustments / corrections that are described in the results section. All such hard rock corrected and scaled time histories are then propagated from depth to surface using two different 1D non-linear site response codes, in order to obtain a large set of surface time histories corresponding to the whole catalogue of seismic events representative of the site hazard. One set of time histories on soil is based on the SHAKE91 [START_REF] Schnabel | SHAKE a computer program for earthquake response analysis of horizontally layered sites[END_REF][START_REF] Idriss | User's manual for SHAKE91: a computer program for conducting equivalent linear seismic response analyses of horizontally layered soil deposits[END_REF]Schnabel et al. 2001) linear-equivalent code (LE), while the second set was derived using the NOAH [START_REF] Bonilla | NOAH: users manual[END_REF], fully non-linear code (NL). Both codes were used with the aim to get a hint of the code-to-code variability. The next and final SM step consists in deriving the site-specific hazard curve by simply calculating the annual rate of exceedance of all surface synthetics.

The AM approach requires to describe the amplification function AF(f; S ar ) and its variability as a function of frequency and input motion level and waveform, with a piecewise linear function with appropriate lognormal distribution, for both site response analysis codes (LE and NL). The large number of computations required by the SM approach were also used to derive this simplified representation of amplification factors. The bedrock hazard curve (including HTT adjustments) could then be convolved analytically with this simplified decscription of amplification factor using the mathematical framework developed by [START_REF] Bazzurro | Nonlinear soil-site effects in probabilistic seismic-hazard analysis[END_REF], to obtain another estimate of the site surface hazard curve, to be compared with the SM estimate.

RESULTS

The site-specific soil hazard estimates at the Euroseistest using the two different fully probabilistic methods are presented below, after presentation of the intermediate step results obtained with the described methodology and with some discussion on the various required assumptions.

Rock Target Hazard

The first step consists in defining the target hazard on "standard rock" at the Euroseistest. This was done with the Akkar et al. 2014 GMPE for V S30 = 800 m/s, the Openquake Engine [START_REF] Pagani | OpenQuake engine: an open hazard (and risk) software for the global earthquake model[END_REF]) and the seismotectonic model proposed on the European SHARE project [START_REF] Woessner | The 2013 European seismic hazard model: key components and results[END_REF]. The resulting hazard curves are displayed in Figure 3 for the spectral acceleration at three different periods (0, 0.2 and 1.0 s, respectively)

Synthetic Earthquake Catalog

Once defined the target hazard on standard rock, we proceed to generate a hazard consistent stochastic seismic catalogue, comprising enough seismic events to build synthetic hazard curves for standard rock at the selected site. This was done again with the Openquake Engine [START_REF] Pagani | OpenQuake engine: an open hazard (and risk) software for the global earthquake model[END_REF]. A sensitivity analysis regarding the respective contributions to the total hazard of the various area sources surrounding the Euroseistest according to the SHARE seismotectonic model, indicated that for the considered return periods, the rock hazard is almost fully controlled by one single crustal area source, i.e., the "GRAS390" one containing the Euroseistest site. The stochastic earthquake catalog was thus generated with the Stochastic Event Set Calculator tool implemented in the Openquake Engine, after sampling the frequency-magnitude distribution (i.e., the Gutenberg-Richter law) of the GRAS390 crustal area source zone. The associated Monte Carlo simulation approach results in homogeneously distributed earthquakes inside the selected source zone. A sensitivity analysis on the catalogue length (considering 500, 5.000, 25.000, 35.500 and 50.000 years) showed that only the catalogue length of 50.000 years allows replicating the target rock hazard curve with an acceptable level of misfit (<5%) for all oscillator periods up to a return period of 5.000 years. The final, so selected catalogue consists of a total of 21806 events ranging from the minimum considered magnitude (4.5) to 7.8, in a distance range from 0 to 150 km, and a depth range from 0 to 20 km.

Host-to-target Adjustments

As discussed in more detail in [START_REF] Aristizábal | Site-specific PSHA: Combined effects of single station sigma, host-to-target adjustments and non-linear behavior[END_REF]Aristizábal et al. ( , 2018a)), the bedrock underneath the Euroseistest basin (~196 m depth) has a shear wave velocity of about 2600 m/s, much larger than the useable ranges of almost all GMPEs. It is therefore mandatory to perform "host-to-target adjustments" (HTTA) corrections [START_REF] Cotton | Criteria for selecting and adjusting ground-motion models for specific target regions: Application to central Europe and rock sites[END_REF][START_REF] Van Houtte | Analysis of the origins of κ (kappa) to compute hard rock to rock adjustment factors for GMPEs[END_REF][START_REF] Delavaud | Testing the global applicability of ground-motion prediction equations for active shallow crustal regions[END_REF][START_REF] Biro | Importance and impact of host-to-target conversions for ground motion prediction equations in PSHA[END_REF]. This type of host-to-target corrections is extensively discussed in [START_REF] Aristizábal | Site-specific PSHA: Combined effects of single station sigma, host-to-target adjustments and non-linear behavior[END_REF]Aristizábal et al. ( , 2018a)). However, the high level of associated uncertainty, and the questions recently raised about the reliability of usual V S30 -κ 0 relationships [START_REF] Ktenidou | Empirical estimation of high-frequency ground motion on hard rock[END_REF][START_REF] Kottke | VS30-k0 relationship implied by ground motion models?[END_REF][START_REF] Laurendeau | Derivation of consistent hard rock (1000 < VS < 3000 m/s) GMPEs from surface and down-hole recordings: analysis of KiK-net data[END_REF], together with the significant level of complexity and subjectivity involved in the current HTTA procedures [START_REF] Campbell | Prediction of strong ground motion using the hybrid empirical method and its use in the development of ground-motion (attenuation) relations in eastern North America[END_REF][START_REF] Atik | ost vibration theory approach[END_REF], led us to use in this paper another, much simpler and straightforward, way to account for rock to hard rock correction for probabilistic seismic hazard purposes.

This new way is based on the recent work by [START_REF] Laurendeau | Derivation of consistent hard rock (1000 < VS < 3000 m/s) GMPEs from surface and down-hole recordings: analysis of KiK-net data[END_REF], who proposed rock GMPEs valid for surface velocities ranging from 500 m/s to 3000 m/s, on the basis of a KiK-net subset. These GMPEs present the double advantage to rely only on the value of rock velocity, without needing the κ 0 values neither for the host region nor for the target site, and provide rather robust predictions of rock motion whatever the approach used for their derivation (deconvolution of surface recordings to outcropping bedrock with the known 1D profile, or correction of down-hole recordings). The "LA17" GMPE used in the following LA17 is based on a combined data set including surface recordings from stiff sites and hard rock motion estimated through deconvolution of the same recordings by the corresponding 1D linear site response. The site term in the corresponding (simple) GMPE equation is given by a simple dependence on rock S-wave velocity value V S30 in the form s 1 (T). ln(V S30 /1000), where s 1 is tuned from actual recordings for each spectral oscillator period T. The HTT adjustment factor can thus be computed as follows:

𝐻𝑇𝑇 𝐴𝑑𝑗𝑢𝑠𝑡𝑚𝑒𝑛𝑡 𝐹𝑎𝑐𝑡𝑜𝑟𝑠 = 𝐿𝐴17 𝐻𝐶 𝐻𝑎𝑟𝑑 𝑅𝑜𝑐𝑘 (𝑉𝑠 = 2600 𝑚/𝑠) 𝐿𝐴17 𝐻𝐶 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑅𝑜𝑐𝑘 (𝑉𝑠 = 800 𝑚/𝑠) = 𝑠(𝑇). 𝑙𝑛(2600/800) (1)
Since the site terms in LA17, for such hard and standard rock, do not include any non-linearity, the resulting HTT adjustment actors are ground-motion independent, and thus return period independent.

The values listed in Table 2 indicate that hard rock motion is systematically smaller than the standard rock motion at all spectral periods, with larger reduction factors at short periods. Of course these corrections are in principle valid only for Japanese rock sites, but most of HTTA methods are implicitly based on US / California type of rocks, which are often applied to many other parts of the world without special warning.

Ultimately, the hard rock hazard curve is then derived by simply scaling the AA14 hazard curve with the HTTA factors for each period (Figure 4). One might wonder why scaling AA14 curve instead of using directly LA17 in the calculations. The main reason was that AA14 (Vs=800 m/s) provide a good approximation to the mean hazard estimate at the Euroseistest on standard rock from seven explored GMPES (Aristizábal et al. 2018a), while the estimates with LA17 (Vs=800 m/s) is located outside the uncertainty range of the same selected representative GMPEs: this is probably due to the very simple functional form used in LA17, and to the fact it is elaborated only from Japanese data. Its main interest however is to provide a rock adjustment factor calibrated on actual data from a large number of rock and hard rock sites, without any assumptions regarding non-measured parameters such as κ 0 . Once the approach proposed in [START_REF] Laurendeau | Derivation of consistent hard rock (1000 < VS < 3000 m/s) GMPEs from surface and down-hole recordings: analysis of KiK-net data[END_REF] will be applied to other data sets, the corresponding results can be used to provide alternative scaling factors between hard rock and standard rock.

Synthetic Time Histories on Rock

The previously derived earthquake catalogue is then used to generate hazard consistent synthetic time histories on hard rock with the well known Ground-motion Simulation Stochastic Method (Boore 2003), as implemented in the corresponding Fortran code SMSIM (Boore 2005). The selection of this code for the generation of synthetic time histories was our particular choice, however, any other suitable method to generate synthetic time histories is also suitable. This step is undoubtedly one of the most time consuming of the entire process, since the large return periods considered here require to perform such simulations for a very large number (~21800) of events. The main input parameters for the Boore stochastic method (Boore 2003) are the moment magnitude (M W ), the distance (D hyp ), the stress drop (Δσ), the crustal amplification factor (CAF(f)), and the high frequency site attenuation factor, Kappa (κ 0 ). Considering the published, locally available data at the Euroseistest, the Boore stochastic method has been applied wit the crustal velocity structure detailed in [START_REF] Maufroy | Source-related variability of site response in the Mygdonian Basin (Greece) from accelerometric recordings and 3D numerical simulations[END_REF], a κ 0 value of 0.024s [START_REF] Ktenidou | Understanding the physics of kappa (κ): insights from a downhole array[END_REF]Perron et al. 2016) and a lognormal distribution of the stress drop with a mean value of 30 bars and an associated standard deviation of 0.68. The resulting raw hazard curves are displayed in red on Figure 4. They are found not to fit the target hard-rock hazard curve (green curves in Figure 4). Such a discrepancy is related to the lack of consistency between the theoretical, stochastic approach and the empirical AA14 GMPE; it was thus corrected by applying frequency dependent scaling factors derived from the ratios between SMSIM and AA14 hazard curves for a target hard-rock velocity of 2600 m/s. These scaling factors were applied on the Fourier spectra of the stochastic time series and a set of 21 806 modified hard-rock, synthetic time series were obtained by inverse Fourier transform. The distribution of the resulting response spectra values for the set of modified synthetics allowed deriving the scaled hazard curves displayed in black on Figure 4. This scaling allowed to obtain a very satisfactory fit of the target AA14 hazard curve for a 2600 m/s hard-rock site at the three different spectral periods, a task that is not possible to perform by common, time domain scaling techniques.

Non-Linear Site Response and resulting hazard curves

This set of ~21800 modified time histories was then considered as the outcropping hard-rock motion and propagated through the 1D soil column described in Figure 2 and Table 1 to obtain a corresponding series of ~21800 site surface motion synthetics. Two site response analysis approaches were considered, the SHAKE91 1D linear equivalent (LE) code [START_REF] Schnabel | SHAKE-91: equivalent linear seismic response analysis of horizontally layered soil deposits, the earthquake engineering online archive NISEE e-library[END_REF] and the NOAH 1D non-linear (NL) code [START_REF] Bonilla | NOAH: users manual[END_REF], respectively. The corresponding amplification factors for pga are displayed in Figure 5. These two sets of site surface synthetics were then used to derive the site-specific hazard curves in two different ways, SM and AM.

Full Probabilistic Stochastic Method (SM)

What it is called here the Full Probabilistic Stochastic Method, SM, is nothing else than the hazard curve built directly from the set of synthetic time histories at site surface. The annual rate of exceedance of a certain ground-motion level (X), is obtained by counting the number of events for which the considered ground motion parameter x is exceeding the X value, and dividing it by the catalogue length as expressed in Eq. ( 2).

𝜆 𝑥 ≥ 𝑋 = 𝑁 !"!#$% (𝑥 ≥ 𝑋) 𝐶𝑎𝑡𝑎𝑙𝑜𝑔𝑢𝑒 𝑡𝑖𝑚𝑒 𝑙𝑒𝑛𝑔𝑡ℎ ( 2 )
The resulting hazard curves at site surface are displayed in Figure 6 for three different spectral periods (PGA, 0.2s and 1.0s), for LE (SHAKE91, red solid line) and NL (NOAH, green solid line) site response computations.

Full Convolution Analytical Method (AM)

In order to obtain fully probabilistic, site-specific hazard curves, Bazurro and Cornell (2004a,b) proposed three different approaches based on convolving the bedrock specific hazard curve with more or less simple descriptions of the site amplification function, AF(f, Sar). These approaches are intended to provide more precise and reliable surface ground-motion hazard estimates than those found by means of standard attenuation laws for generic soil conditions. One of the proposed methods is called "Analytical Estimate of the soil hazard" corresponds to the Full Convolution Analytical Method (AM) or as they. It is based on a piecewise linear representation of the site amplification function AF(f, Sar) = C 0 + C 1 ln(S ar (f)) with a lognormal standard deviation σ ln(AF(f)) (see Figure 5), from which a simple, closed-form solution is obtained that appropriately modifies the hazard results at the rock level, to finally obtain the hazard curve on soil, Eq. ( 3 ).

HC !"#$ (𝑓) = HC !"#$ (𝑓) . 𝑒 ! ! ! ! ! ! ! (! ! !!) ! ( 3 )
Where C 1 and σ ln(AF(f)) are the slope and standard deviation of the piecewise-linear regression models of the AF(f) vs. S ar (f) as proposed by Bazurro and Cornell 2004b (and shown in Figure 5 for the present example), and k 1 is the slope (in log-log scale) of the straight-line tangent to the rock hazard curve at the point, thus corresponding to the exponent of the local power law representation of the bedrock hazard curve, i.e., HC [Sar(f)] = K 0 .Sar(f ) -k1 . The main advantages of this method are: (1) The amplification produced due to the soil site effects is considered as an a posteriori correction to the hazard calculations.

(2) The full meaning of the hazard curve and UHS is respected.

(3) The calculations of the hazard curve on soil are very simple and with low computational demand. (4) Fewer accelerograms are required to calculate the AF(f, S ar ,) than the SM, resulting in a much lower computational demand.

The AM hazard curves at site surface were calculated in that way again for different spectral periods (PGA, 0.2s and 1.0s), and are shown also in Figure 6 (dotted lines) The same 1D linear equivalent (LE) and 1D non-linear (NL) site response analysis as in the SM approach, using SHAKE91 and NOAH (red and green) respectively, were used to derive the piecewise linear approximations (Table 3) and the associated σ ln(AF(f)) values.

DISCUSSION

This comparison indicates that hazard curves calculated using the Full Convolution Analytical Method (AM) and the Fully Probabilistic Stochastic Method (SM), present a good agreement at all considered return periods and spectral periods (Figure 6). Nevertheless, and despite the good fit between both methods, the AM method presents an intrinsic numerical limitation that did not allow us to calculate the site surface hazard curve even at the most common, 475 years return period. This limitation is related to the values of C 1 and σ ln(AF(f)) in the modifying term of equation ( 3) : when C 1 is close to -1, and/or the site response variability σ ln(AF(f)) is large, the modifying site factor term may become very large, and so is the site hazard. As shown in Figure 5, Table 1, andFigure 7, C 1 values close to -1 may be obtained when the surface motion saturates because of strong non-linearity, and relatively large site response variability are obtained, especially at large rock motion levels and for the 1s period. This obviously raises questions about the AM effectiveness to account for non-linear site effects in PSHA. It is important though to emphasize that this limitation only affects sites with strong non-linear and/or high seismic demands such is the case in the present example. When such a limitation is faced, a possible, conservative way to cope with it could be to set a minimum value -larger than -1.0 -of the slope (C 1 ) of the amplification regression (AF(f) vs S ar (f)). This would allow the exponential term related to the soil factor in Eq. ( 3) [START_REF] Bazzurro | Nonlinear soil-site effects in probabilistic seismic-hazard analysis[END_REF] Eq. 17) to remain within "reasonable bounds", in order to continue using the AM at long return periods. However, it is important to keep in mind that setting such a minimum value for C 1 , is equivalent to setting a upper bound to the amount of non-linearity in the site response, resulting in particular in the absence of any saturation, which is not easily acceptable from a physical viewpoint.

We can also highlight that most of the variability in terms of hazard estimates are due to the code-tocode (SHAKE91, NOAH) variability and the soil column modeling, rather than the method-to-method (AM, SM) variability. As shown in Figure 7, SHAKE91 tends to predict larger values of surface PGA for high bedrock input level, while the trend is opposite for the two other spectral periods (0.2s and 10.s). One may also notice a significant non-linear behavior even at low frequencies (1.0s), and relatively low return periods, which is related to the facts that such a period is already shorter than the site fundamental period (around 1.4 s), and that the velocity contrast at the sediment-bedrock interface is quite large (see Table 1): this induces large strains at large depth, which in turn impacts the shear modulus and damping values over the whole soil column, and significantly decreases the amplification and thus the surface motion. To calibrate and validate our models, we compared the synthetic time histories on soil with current available data on site. Figure 7 displays the plot relating the spectral acceleration on hard rock at the bottom of the Euroseistest basin, TST196 Sa Rock (g), to the acceleration on soil at the surface, TST Sa Soil (g), for the three considered different spectral periods and both EL (SHAKE91) and NL (NOAH) site response codes, together with the registered acceleration values from different events reported at the Euroseistest database [START_REF] Pitilakis | The EUROSEISTEST Strong-Motion Database and Web Portal[END_REF]. The same amount of earthquakes were used at the three different spectral periods, nevertheless, only accelerations above 0.001g are displayed on this plot, this is the reason why the number of points varies from plot to plot. From visual inspection, it is possible to say that the two models (EL and NL) are consistent with the actual Euroseistest site observations, and we can expect that the real median of the site is contained between the two proposed hazard models, since most of the data is located within the scatter of the two models and follows similar shape. One could possibly argue that SHAKE91 performs slightly better for low groundmotion levels [Sa < 0.01g], while NOAH might be preferable for intermediate ground-motion levels [0.01g ≤ Sa ≤ 0.1g]; no conclusion however can be drawn as to which model performs better for high acceleration level [Sa > 0.1], since there are no strong motion recordings. Related to the real data variability, it seems that the synthetic time histories and site response computations of both models (SHAKE and NOAH) slightly underestimate the actual variability, since some of the yellow points are located outside the scatter range of both cases (red and green dots in Figure 7).

CONCLUSIONS

We found relatively satisfactory consistency between both fully probabilistic methods (AM and SM) using the two different EL-SHAKE91 and NL-NOAH site response codes along all the studied periods. On this example case study, the code-to-code variability (SHAKE91 and NOAH) is found to control the uncertainty in terms of hazard estimates, rather than the method-to-method variability (AM and SM). It must be mentioned however that the numerical limitation of the AM approach for strongly non-linear sites, already acknowledged by its authors, prevents from retrieving the site surface hazard curve for intermediate and long return periods at the Euroseistest because of the ground motion saturation. The SM does not present this numerical limitation, and the hazard curve on soil was derived even at long return periods, yet, with a much higher computational price.

By comparing the two models with real in-situ data, we observed that the two site response models (EL and NL) are consistent with the site recordings, in terms both of median values and event-to-event variability. One can thus expect that the real median hazard estimate of the site will be contained between the two proposed hazard models, since most of the in-situ data are located within their dispersions. SHAKE91 may be thought to perform slightly better for low ground-motion levels [Sa soil < 0.01g], while NOAH would be preferable for intermediate ground-motion levels [0.01g ≤ Sa ≤ 0.1g]. No conclusion can be made at high acceleration level [Sa > 0.1] since no real data is available to calibrate the model on this range. We encourage the use of synthetic simulations calibrated wit real data such as the one we proposed in this paper, since it allows a better understanding and accounting of the variability of the physical phenomena, an improved parameterization of the input values, and a more robust probabilistic analysis. In most cases, this is presently not possible with real data because of their scarcity at high acceleration levels.

Figure 1 .

 1 Figure 1. (a) Map showing the broader region of occurrence of the 20 June 1978 (M6.5) earthquake sequence.Epicenter and focal mechanism[START_REF] Liotier | Modelisation des ondes de volume des seismes de l'arc Ageen[END_REF]) of the mainshock (red) and epicenters of the largest events of the sequence (green, yellow)[START_REF] Carver | Aftershocks of the June 20, 1978, Greece earthquake: a multimode faulting sequence[END_REF] are indicated (from http://euroseisdb.civil.auth.gr/geotec).(b) 3D model of the Mygdonia basin geological structure[START_REF] Manakou | 3D soil structure of the Mygdonian basin for site response analysis[END_REF].

  (a) 1D shear wave, Vs, and compressive wave velocity, Vp, velocity profiles between TST0 and TST196 stations, located at the middle of the Euroseistest basin, at surface and 196 m depth, respectively.

Figure 3 .

 3 Figure 3. Target hazard curve on rock (Vs=800 m/s) at the Euroseistest calculated using the Akkar et al., 2014 GMPE (AA14) for three different spectral periods (PGA, 0.2s, 1.0s).

Figure 4 .

 4 Figure 4. Hazard curves for three different spectral periods (PGA, 0.2s, 1.0s): AA14 (800 m/s) HC on standard rock (blue). AA14 (2600 m/s) HC on hard rock after HTT (green). SMSIM (2600 m/s) HC built from synthetic time histories on hard rock, 2600 m/s, using Boore 2003 stochastic method (red). SMSIM (Scaled) is the HC built from synthetic time histories on hard rock, SMSIM (2600 m/s), and scaled to match AA14 (2600 m/s) HC.

Figure 5 .

 5 Figure 5. Amplification factors for pga as a function of rock pga as derived with EL-SHAKE91 (left, a) and NK-NOAH (right, b) site response computations for the set of 21806 rock synthetics. Each plot also displays a trilinear piecewise regression models of 𝐴𝐹(𝑓) on 𝑆 !" (𝑓) f.

Figure 6 .

 6 Figure 6. Median hazard curves at site surface using both equivalent-linear SHAKE91 (red) and nonlinear NOAH (green) for both AM (dotted line) and SM (continuous line) at three different spectral periods (PGA, 0.2s and 1.0s). Also shown is the hazard curve for the reference rock (2600 m/s, blue curve) 0.01 0.10 1.00 Sa (g)

Figure 7 .

 7 Figure 7. Equivalent linear (SHAKE91, red) and non-linear (NOAH, green) amplification factors compared with real data (yellow) for three different spectral periods (PGA, 0.2s and 1.0s).

Table 1 .

 1 Material properties of the Euroseistest soil profile used for the site response calculations.

Layer Depth (m) Vs (m/s) Vp (m/s) ρ (kg/m 3 )

	1	0.0	144	1524	2077	Q 14.4	φ' 47	Ko' 0.26
	2	5.5	177	1583	2083	17.7	19	0.67
	3	17.6	264	1741	2097	26.4	19	0.68
	4	54.2	388	1952	2117	38.8	27	0.54
	5	81.2	526	2200	2151	52.6	42	0.33
	6	131.1	701	2520	2215	70.1	69	0.07
	7	183.0	2600	4500	2446	-	-	-
	*Water table at 1 m depth. Vs: shear wave velocity. Vp: Compressive wave velocity. ρ: soil density. Q:
	Anelastic attenuation factor. φ: Friction angle. Ko: Coefficient of earth at rest.

Table 2 :

 2 Host-to-target adjustments factors derived using LA17 hazard curves from a standard rock (VS30=800 m/s) to the Euroseistest hard rock (VS30=2600 m/s).

	Spectral Period (s)	PGA 0.05 0.1	0.2	0.5	1.0	2.0
	HTT adjustment factor 0.47 0.45 0.38 0.51 0.70 0.78 0.86

Table 3 .

 3 Slope (C1) and standard deviation (σ) of the piecewise-linear models (PLM) of the AF( f ) Vs. Sar(f ).

	Spectral Period (s)	Piecewise-Linear Models	C 1	SHAKE91 σ[ln(AF)]	C 1	NOAH SHAKE91 σ[ln(AF)]
		PLM 1 (Blue)	-0.1089	0.15	-0.1833	0.11
	PGA	PLM 2 (Red)	-0.2906	0.19	-0.3649	0.08
		PLM 3 (Green)	-1.2140	0.19	-0.8594	0.08
		PLM 1 (Blue)	-0.1475	0.19	-0.1020	0.12
	(0.2)	PLM 2 (Red)	-0.4043	0.18	-0.4789	0.13
		PLM 3 (Green)	-1.1970	0.27	-1.1172	0.35
		PLM 1 (Blue)	-0.0561	0.27	-0.0170	0.11
	(1.0)	PLM 2 (Red)	-0.1438 -0.1438	0.14	-0.3298	0.22
		PLM 3 (Green)	-1.2900	0.08	-1.4930	0.26

ACKNOWLEDGMENTS

This work has been supported by a grant from Labex OSUG@2020 (Investissements d'avenir -ANR10 LABX56 -http://www.osug.fr/labex-osug-2020/). Special thanks are due to Fabian Bonilla for providing his non-linear code NOAH and additional technical assistance on its usage. We thank Fabrice Hollender for providing valuable geotechnical and site-specific data to build the non-linear models. All ground-motion synthetics used in this study were generated using Boore 2003 Stochastic Method and his open source code SMSIM (http://www.daveboore.com/). Hazard calculations on rock have been performed using the Openquake engine and the post-processing toolkits developed by the GEM Foundation (http://www.globalquakemodel.org/), as well as the SHARE area source model (http://www.share-eu.org/).