Sylvie Monjean-Decaudin

Joëlle Popineau-Lauvray

How to apply comparative law to legal

Since the 1950's, translating legal texts has gained momentum in an increasingly globalized world. International organisations, the European Union, corporations, international and national courts, companies and private individuals have increasingly been relying on all types of legal translations. They include rules of law with most of them leading to legal consequences. How should the translator deal with legal texts? How to assess whether s/he is experienced enough to face the complexity of a semantic transfer when s/he translates texts from a source law to another target law? This article explains new advances in juritraductology (the science of translating legal texts) and gives examples of how they might be used in legal translation. The first step is to highlight the specificities of a legal translation by suggesting a method for assessing its level of legal complexity. The second step describes the 3-stage methodology of a legal translationa first semasiological stage, a second stage based on comparative law and a last onomasiological stage -and is exemplified by examples taken from Common Law and the French law. Our demonstration combines legal theory and legal translation practice and aims at providing researchers, translators and teachers with the know-how needed to complete the entire translation process of a legal text.

Since the 1950's, translation has gained momentum in a newly rebuilt and increasingly globalized world, either because of international reasons linked to WWII or national ones. Countries exchange political views, have trading relationships, share economic, financial and legal ideas, leading to a higher demand for translation. Legal texts are thus translated in different legal contexts and meet the requests of international or national institutions, companies or individuals for specific purposes. The legal translator is therefore entrusted with the task of transferring a wide range of legal texts from one language to another, with each of them having specific legal effects based on a specific context. How should the translator deal with a legal text? Should s/he evaluate its level of legal complexity to to be able to estimate the issues and stakes in his/her translation? How can s/he assess s/he has gained enough experience to face the challenges posed by transferring the meaning between the source law and the target law systems? How should s/he deal with it? Juritraductology (or the science of legal translation) offers a new conceptual framework to translation studies applied to law: this article presents it as the basis of a new theoretical and methodological tool for analysing translation and its process (part 1). Prior to this, the translator must analyse the legal text to translate, its legal context and assess its level of legal complexity (part 2). Juritraductology offers a three-stage process in which comparative law plays an important role (part 3).

Juritraductology: a new conceptual framework in law and translation sciences

Juritraductology is a cross-interdisciplinary field of study that emerged in France in the early 21st century (Monjean-Decaudin, 2012). This novel approach differs from Legal Translation Studies, or traductologie juridique in French, which mainly focus on the complexity of a legal translation and/or the translation of legal concepts. Legal translation studies are a subdivision combining (juri)linguistics and translation studies and offer various interesting and worthy approaches and solutions. Besides Legal Translation Studies, comparative law specialists have developed their own approach to find solutions when translating concepts of law in a specific legal system. However, both theories have developed in a distinct way and to a certain extent remain independent: translation theories are disconnected from theories of law.

Juritraductology, a field of study developed by the Cerije in 2012, offers a new crossdisciplinarity, building bridges between translation studies and law. Combining theories makes it thus possible to establish the required connection between the legal right to translate and legal translation, which form the bedrocks of juritraductology. In other words, the juritraductological thinking is conceived as a science enabling ideas to interact with each other without any disciplinary frontiers. The approach can encompass all laws and all languages, because only a broader approach can help nurture the epistemological debate in this field of study and thus enrich the targeted applications.

Juritraductology has several epistemological founding principles, as shown in figure 1 (see below). Law and linguistics, which are the two "nourishing" sciences, merge to form legal linguistics. Translation studies mainly originated from linguistics and became an independent field of study in the 1970's. The legal right to translate has its roots in Law. And legal translation has it roots in Legal linguistics and translation studies. Both combine and create the two main fields of study in juritraductology. The field is undoubtedly a cross-disciplinary area. Two remarks can be made: firstly, juritraductology does not place disciplines side by side but instead combines them; secondly, law has a determining influence on translation, thus explaining the consequences on legal translation, which have remained unexplored up to now.

Figure 1: Juritraductology and its epistemological founding principles Juritraductoloy as a cross-disciplinary field of study shares common knowledge with the three other fields of study already mentioned at different levels. Firstly, juritraductology shares focus and knowledge with linguistics and translation studies. Both build the general framework for the conceptualization and application phases leading to juritraductology. A juritraductological approach does not need to fully put into practice all approaches and concepts used in linguistics and translation studies. Juritraductology selects particular aspects dealing with in TS are especially useful for a juritraductological approach to legal translation. For example, the founding principles and the methods used in translation studies seem particularly well-adapted to legal translation. The theories describing general translation problems and the approaches of all the people involved in translation help define the field of study of juritraductology.

Secondly, juritraductology projects itself as an approach strongly connected to law in general and comparative law in particular. Juritraductology must meet the demanding requirements set by law (law is both a "consumer" and "a producer" of legal translations). Juritraductology embraces all the linguistic and translational demands contained in law. It aims at describing all legally relevant translation issues and thus expands the scope of legal translation. The subdivisions of law it considers are various, namely international law, European law, national law (either civil, commercial, customs, electoral, tax, criminal, rural, welfare, and environmental law, arts law, communication law and policy, consumer protection laws, intellectual property law, public health law, traffic law, home security, insurance law, cultural property law, labour law, tourism law, insurance law, local government law, law concerning foreign nationals, forestry law, financial law, cyber-security law, transportation law, and administrative, civil, tax, military and criminal law procedures. Moreover, the fields of law juritraductology crosses are closely associated with legal language.

Thirdly, the combined interactions of fields which define the scope of study of juritraductology is legal linguistics. Legal linguistics assists juritraductology in the role it may play as an auxiliary branch of law. As it is the case for translation studies, it forms the discursive environment in which juritraductology is designed. Gérard Cornu uses a metaphor and says that a legal translator picks a word studied by general linguistics, as a farmer would pick a tool at a large department store (Cornu 2005, p. 26). Likewise, juritraductology can choose the most fertile and appropriate fields among the branches of general linguistics and translation studies. Therefore, juritraductology deals with theoretical and practical questions in legal translations whenever they are of legal interest, among which questions arisen before translating.

II. Before translating

It is commonly accepted that we translate legal texts for a purpose, which is mainly pragmatic and sometimes mandatory due to a legal context, and for legal specific needs? In most cases, law is translated because a legal situation arises, an act or a deed requires it (Monjean-Decaudin 2010). Juritraductology aims at exploring the translation carried out in various legal contexts. In addition, it shows the varying levels of legal effect. This approach is a novel approach as it has never been explored before. It highlights the impact of law and its consequences on translation. In practical terms, this approach presents itself as a targeted or functionalist activity (Monjean-Decaudin 2016) and offers an introductory framework for any translation process.

Before translating, the translator must understand the legal context in which the source text has been written to fully understand the issues involved in the text and its translation. In other words, s/he must understand why, for whom and for what purpose law is being translated in a context (1). Then, depending on the context, s/he must be able to measure its level of legal complexity (2).

What are the different contexts for translating law?

There are mainly four legal contexts for which a translation is required. The list below is not exhaustive.

a. In international public law

Translation is carried out within the framework of institutions and organisations, both at international and regional level. Translating helps create a supranational regime in several languages, for example when drafting an international treaty. Each international or regional organization adopts its own language system to determine the official languages it has. The number of official languages determines both the language pairs and the volume of translation. The language system used in the International Monetary Fund is different from that of the European Union. By declaring English the only official language, the world of finance has deliberately become unilingual. Translations carried out by the IMF's language services mainly have an informative role for inter-state communication. As a result, these translations do not involve any legal consequences, as evidenced by the introductory IMF disclaimer stating: "While every effort is made to ensure the accuracy of translations, the version of any document used by the IMF is the English version, as American English is the working language of the IMF." On the contrary, the European Union is a paradigmatic example of institutional multilingualism, soundly described by Umberto Eco in his famous motto: "Translation is the language of Europe". Today, with 24 official languages and 552 language combinations, the European Commission's Directorate-General for Translation is one of the world's largest translation services. The texts written in all the official languages of the Member States are published in the Official Journal of the European Union to produce legal effects in the legal orders of the Member States. The decisions made by the Court of Justice of the European Union are published in all the official languages in order to ensure a uniform law enforcement across the EU (Monjean-Decaudin 2015, p. 95).

b. In international private law

Cross-national exchanges between physical and/or legal persons with different nationalities are governed by private international law. Increased exchanges have led to a dramatic increase in legal translation. Key economic players expand their trade across borders, people travel and work in a foreign country, they marry foreigners or even commit crimes and offenses abroad: the daily life brings about many acts and deeds that need a translation. When you adopt a child abroad you need a translation; when you transfer your patent to a foreign company, you need a translation; when you hire a foreign worker, you need a translation. There are many cases in which translations are needed, be it administrative documents, private deeds or authentic instruments. Under private international law, people need a translation to assert their rights or to acknowledge their status at a local government office in a foreign country (for example, when foreign nationals get married or divorce, when a property located in a foreign country is left to a person, etc.). All these translations have legal effects. However, they are limited to the legal situation of each person involved.

c. In a judicial context

In a globalized world, people subject to trial can travel abroad, where again international justice and legal cooperation is based on translation. In this context, legal translation is most frequently carried out by sworn translators for justice services.

Judicial translation can be used in all types of proceedings, either in civil, criminal or administrative matters. Both written and oral translations fulfil two distinctive functions (Monjean-Decaudin, 2012).

Firstly, judicial translation is a communication tool for the judicial system. In this case, the translation may circulate or not. When there is a cross-border dispute, the translation travels to the foreign country. It is carried out to help judicial cooperation when exchanging acts and information between judicial authorities. The translation of a European arrest warrant or an international letter rogatory is, by its nature, a "circulating" translation. Where the translation allows the national court to understand the content of a document written in a foreign language, the translation is, in this case, a "noncirculating" one. It is not carried out to be sent to another State but to inform the national judge about the content of a document provided to the court. Documents may be testimonies, contracts, code articles, etc.

On the other hand, translation is a procedural safeguard for suspects/defendants speaking a foreign language. It is more obvious in criminal than in civil matters, and in oral than written translations. But in all cases, translating a document aims at safeguarding the right to a fair trial to individuals who do not understand the language of the proceedings. The Directive 2010/64/EU of the European Parliament and of the Council of 20 October 2010 on the right to interpretation and translation in criminal proceedings reasserts the importance of the right to interpretation and translation in criminal proceedings (Monjean-Decaudin, 2011).

d. In a scientific context

We all live in a multilingual and translated world but we also work in a context of multilingual and translated law.

Translation is used to expand legal knowledge on legal doctrine or normative texts (constitution, code, laws, etc.) and court decisions. Legal translation is carried out when law is imported and exported. We can either know a foreign law or make our own law known to the rest of the world through translation. Translation is also an integral part of comparative law. Examples which prove this include, for example, the translation of the description of major contemporary legal systems into different languages, the drafting of bilingual dictionaries or the publication of the Henri Capitant Law Review (Goré 2011 : 114).

In this context, the translator is generally both a lawyer and a comparative law specialist (Sacco 2011 : 23) and the effects of translation are important for law.

Once the context has been clearly defined, let us consider the degré de juridicité.

Le degré de juridicité -the level of legal complexity in a text

All translations do not systematically lead to legal consequences, but all legal translations, per se, have some legal complexity. Depending on the context, these consequences are more or less significant. From a methodological point of view, before translating a legal text, the translator must assess the level of legal complexity of a text (Decaudin 2007 : 94-95).

To assess this level, s/he must study two distinct indicators. The first indicator takes into account the scope of legal knowledge which is required to understand and translate the text and/or legal concepts. The second indicator identifies the resulting legal consequences when the translation is done. Thus, the more the text contains law and legal concepts, -i.e. the more it requires in-depth legal knowledge to be understood-, the higher its level of legal complexity. Moreover, the more legal consequences the translation involves -i.e. if the rule contained in the text or concept is binding and has legal consequences-, the higher its complexity.

When translating, two cases can arise: the legal text to translate contains only one of the two abovementioned criteria or contains both of them. Let us consider examples containing only one criterion.

In a first example, the number of highly specialized terms in the source text requires a sharp law knowledge to understand and translate them. For instance, a book on procedural law written by a law professor contains many legal concepts requiring an accurate knowledge when translating. On the other hand, a university textbook for lawyers (students, teachers, etc.) has no binding force or legal effect and, therefore, the resulting translation has no legal effect. Another example is about translating a holographic will and testament written in French and containing only some words: " Je donne toute ma fortune après ma mort à..." ["I leave all my property to … when I am gone" (my translation)]. This is an example of a will and testament in France. The French text does not contain legal concepts, and is therefore quite easy to translate at first sight. While a rather simple vocabulary does not require a sharp legal knowledge or a highly specialized terminology, nonetheless, legal consequences will be effective after translating this will and testament.

Let us now consider texts containing both criteria. In that case, legal effects add up to a highly specialized terminology, resulting in a sharper complexity and the utmost legal value a text may have. Translating laws, court decisions, some agreements and contracts are examples of this. The translator must make sure to transfer the correct meaning of texts while taking into account the challenges and stakes of his/her translation. The legal complexity of a translation will therefore depend on its clients and function. Translating a final judgment of a court has a lower legal complexity than translating a European regulation: the effects of a final judgment concern the parties involved in the trial whereas European regulations will apply to all EU Member-states.

From the very beginning of the legal translation process, the series of steps depends on the legal complexity of the text. A methodology specifically designed for legal translation makes it possible to overcome the difficulties shown in legal terms and concepts.

Part 2. How to translate a legal text?

Prior to any other task, the translator must carefully read the text where terms and legal concepts are spotted. S/he will put the text in a specific legal culture (for example, Common Law culture, Romano-Germanic culture) and in the subdivision of law covered by the source text (e.g. public law, private law, criminal law, civil law). When reading is completed, the comparative-law step takes place, in which the translator simultaneously takes into account the source language and law system, with the aim of translating a text from a source law system to a target law system. As a result, a text may be more or less difficult to translate, depending on whether the transfer between language and legal systems is compatible or not: Common Law and the French Law System share common roots. Translating from English into French is more compatible than translating from English into Chinese for example. The lack of common references between two distant law systems may be a further difficulty for the translator (Monjean-Decaudin 2013 : 6).

Whatever the legal systems concerned, comparative law consists in comparing two systems with a view of performing the translation process of a legal text. Among the different approaches described by authors, comparative law is part of the translation process [START_REF] Gémar | La traduction juridique[END_REF](Gémar , Šarčević 1997)).

More precisely, the methodological approach chosen and developed in legal translation studies is a three-step approach comprising a semasiological step, a comparative-law step and an onomasiological step. Such an approach has been implemented in translation classes (Popineau à paraître).

We will exemplify each step by giving pairs of English and French terms: -Crime (in English) and crime (in French); -Copyright and droit d'auteur; -Act of God and catastrophe naturelle.

The three-step approach will answer the following questions: is the English false friend crime an equivalent of the French term crime? Can the French term droit d'auteur be translated by the English term copyright? Are Act of God and catastrophe naturelle comparable, and therefore translatable?

The 3-step approach requires language in both source and target languages, each step being explained in the corresponding language. Quotations in French and English are part of the demonstration and may not always be translated.

Step 1: the semasiological step In the semasiological step, the goal is to understand the source text and its building blocks both on semantic and conceptual levels. As these blocks contain legal knowledge, the semasiological step requires an in-depth documentary research into the legal system of the source language. Understanding is a dual concomitant process: it implies understanding both source and target languages and systems, conveying a meaning in words. "Les mots sont un passage obligé pour les concepts juridiques [words are a necessary step for conveying legal concepts] (Legeais 2008 : 267). The abstract approach that is part of legal translation consists in legally defining or not a concept. To this end, this step consists in carrying out a documentary research to define and put a legal concept into a context.

Let us give a first example: crime (in French) In French law, crime is part of a trilogy: contravention, délit et crime. In the first semiological step, a documentary research leads to finding definitions. In French criminal law, contravention is "une espèce d'infraction appartenant à une catégorie située en bas de l'échelle de la gravité (Cornu 2016, p. 263)" [a kind of wrongdoing with a less severe punishment] ; a délit is a "espèce d'infraction moins grave que le crime et plus grave que la contravention (Cornu 2016 : 320) [a wrongdoing less severe than crime but more severe than contravention]; and finally, crime is a "espèce d'infraction pénale appartenant à la catégorie des plus graves d'entre elles (Cornu 2016 : 288) [the most severe wrongdoing of the group]. The definitions show that the French term infraction is a general hypernym into which contravention, délit and crime fall, with severity being the classifying criterion.

Crime in French is often translated by and compared to crime in English. Are they true synonyms? Comparing French crime in source law to English crime in target language will be carried out in Step 2.

We can also show how significant this step is by giving a second example: droit d'auteur. The same documentary approach is followed. In French law, droit d'auteur is defined as "droit de propriété incorporelle exclusif et opposable à tous, qui comprend l'ensemble de prérogatives morales (droit de divulgation, droit à la paternité, droit à l'intégrité de l'oeuvre, droit de repentir ou de retrait) et patrimoniales (droit de reproduction, droit de représentation, droit de suite) dont jouit l'auteur sur son oeuvre du seul fait de sa création (Cornu 2003 : 66)." [Summary: Droit d'auteur comprises two types of rights: a moral right and a patrimonial right]. Droit d'auteur is usually translated by copyright into French. Are they true cognates or false friends?

Finally, yet importantly, a documentary research on Act of God shows that: -Act of God is "a natural and unavoidable catastrophe that interrupts the expected course of events2 ". When a distributor contract between an English and a French firm needs to be translated, what is the best equivalent of Act of God in a French distributor contract? Once the semasiological phase is completed, once the meaning of a concept in the source language and law is found, the second step is to find a possible equivalent concept in the target law and language: the comparative-law step begins.

Step 2: the comparative-law step "Comparer consiste à établir des rapports de ressemblances et de différences entre les termes d'un savoir, puis à en mesurer l'ampleur, à en chercher les raisons et à en apprécier la valeur » [Comparing is finding how similar and different terms may be in a field of study, evaluating how significant their differences are, explaining the underlying reasons for such differences and finally evaluating them] (Laithier 2009 : introduction) In this respect, and prior to a correct translation, legal concepts usually require an in-depth comparative study. The legal translator becomes " le savant exégète qui s'abîme dans les profondeurs du texte écrit" [a scholarly expert who delves into the depths of the written text] (Ost 2009 : 111) and into the legal system and meaning conveyed by the source language. Before any translation, the translator should look for a possible equivalent in the target language. This possible equivalent should fulfill the first and foremost role of translation: conveying the source text meaning to the audience reading the target translated text.

Let us turn back to our first example: crime (in French) The documentary research carried out in Step 1 shows that crime in French belongs to a triplet, with infraction being the hypernym in French (Figure 2); a severity criterion ranks them from less to more severe punishment.

Let us now turn to the English false friend crime. Is crime in English a possible equivalent to crime in French?

Step 2 aims at comparing legal concepts in both source and target language.

Crime in English is "an offense against public law usually excluding a petty violation. Furthermore, felony and misdemeanor3 are words accompanying the definition: a crime can be a felony or a misdemeanor according to the dictionary. Our documentary research continues with definitions of the above-mentioned words: a felony is "a crime that has a greater punishment imposed by statute than that imposed on a misdemeanor" and misdemeanor is "a crime that carries a less severe punishment than a felony"; again, severity is a ranking criterion for each word. Crime seems to be an hypernyms for both words.

Both are hypernyms, into which two terms can be found in English (misdemeanor and felony) and three terms in French (contravention, délit and crime). If we compare definitions, similarities do appear: severity is a significant criterion for both words; moreover syntax shows comparatives in both English and French (more.. than.., less.. than…,; moins.. que, plus.. que):

"Espèce d'infraction moins grave que le crime et plus grave que la contravention" "Espèce d'infraction pénale appartenant à la catégorie des plus graves d'entre elles" "a crime that has a greater punishment imposed by statute than that imposed on a misdemeanor" "a crime that carries a less severe punishment than a felony"

Step 2 shows that crime in French and crime in English share similarities.

Let us take the second example. Droit d'auteur in French is commonly translated by copyright into English. Are they true equivalent concepts? Firstly, they have deeply different founding principles: « Tandis que le droit d'auteur se consolide en Europe, les Etats-Unis adoptent un corpus législatif d'une autre nature autour de la notion de copyright. Leurs fondements respectifs sont profondément distincts. »[My translation :While the French droit d'auteur is being strengthened in Europe, the United States is adopting different laws and legal texts dealing with copyright. They have deeply different founding principles.] (Benhamou et Farcy, 2009 : 22).

Although both are proprietary rights, copyright is "utilitariste et économique [a use-oriented and economic right]" whereas « le juridique prime sur l'économique dans le droit d'auteur" [the legal side is heavier than the economic side in French droit d'auteur] (Ibid. 23).

Secondly, copyright is more restrictive and is "the legal right to have control over the work of a writer, artist, musician, etc4 ", or in other words, « droits exclusifs du titulaire du copyright sur son oeuvre » (Cornu 2003, p. 66) [the exclusive rights of the copyright owner on his/her work]. "If you own the copyright on something, it is your intellectual property, and other people must pay you to broadcast, publish, or perform it5 ", the Merriam-Webster dictionary says, without mentioning any moral rights.

The French droit d'auteur is indeed a dualistic right as it comprises two types of rights: « le droit d'auteur français est qualifié de droit dualiste en raison de la coexistence de droits de différentes nature : d'une part le droit moral appartenant à la famille des droits de la personnalité qui conserve à l'auteur un pouvoir de contrôle sur sa création même s'il en a cédé les droits, et d'autre part les droits patrimoniaux qui lui permettent de tirer profit de l'exploitation de son oeuvre » [The French droit d'auteur comprises two types of rights ; on the one side, a moral right belonging to the rights of the personality by which the creator keeps a right of controlling his/her work even if s/he transferred proprietary rights ; on the other end, proprietary rights by which s/he can financially profit from his/her work] (Cornu 2003 : 67).

Thirdly, The French definition lists many neighboring rights: " le droit de divulgation, le droit à la paternité, le droit à l'intégrité de l'oeuvre, le droit de repentir ou de retrait sont les droits moraux entrant dans la sphère du droit d'auteur ; s'y ajoutent le droit de reproduction, droit de représentation, droit de suite" Translating neighboring rights into English is difficult because the French neighboring rights do not exit in Common Law and no equivalents are given in comparativelaw dictionaries (Ibid.,p. 66 and p. 81). In this law comparative step, copyright appears to be a shortened truncated French droit d'auteur with missing moral rights.

Finally, Act of God is commonly translated by catastrophe naturelle into French.

In French law6 catastrophe naturelle is "un agent naturel ayant une intensité anormale », in a broader definition, [qui sont] « les dommages matériels directs ayant eu pour cause déterminante l'intensité anormale d'un agent naturel, lorsque les mesures habituelles à prendre pour prévenir ces dommages n'ont pu empêcher leur survenance ou n'ont pu être prises ». Both definitions mention a natural and unavoidable factor and seem to be exact equivalents.

Step 3: the onomasiological step A translation is given in this last -but not least-step. It is about deciding which criteria carry the equivalence. The choice consists in deciding on an acceptable term both on the linguistic and legal levels; a large variety of translations can be given.

Let us look at the French trilogy formed by contravention, crime, délit. We are facing a translation imbalance. As shown in step 1 and 2, three French terms refer to two English terms. The hypernym crime can be translated by its French counterpart infraction; but the number of hyponyms is different (2 against 3). The translator is facing a choice: s/he must decide which term s/he would choose or discard to keep a balance when translating. Two translation strategies are possible: -Crime is translated by the legally correct French term infraction; -The translator decides to introduce an adjective to best describe the severity (mineure and majeure, minor and major) of the wrongdoing. The latter seems a satisfactory solution: infraction mineure does legally correspond to misdemeanor, which is "a crime punishable by a fine and by a term of imprisonment not to be served in a penitentiary and not to exceed one year"; and infraction majeure does legally correspond to felony.

Comparing the French droit d'auteur to Common Law copyright shows fundamental conceptual differences between the two words: when comparing both terms, droit d'auteur does not sound a legally acceptable translation for copyright into English and vice-versa: « Les mêmes termes de chacun des droits peuvent recouvrir des sens différents. L'exemple classique est le terme « droit d'auteur » qui, pour les Canadiens est la traduction de « Copyright » et qui n'est pas un équivalent du droit d'auteur belge et français. » [Similar terms may have different meanings in different law systems. A standard example is the word « Droit d'auteur » which is translated by copyright in Canada and which is not an equivalent of the droit d'auteur in Belgium and in France"] (Cornu 2003 : 20).

However, the translator must translate, as it is his/her duty and what s/he is being paid for. Borrowing a foreign word is a possible strategy when terms have no equivalent counterparts in target legal languages. Borrowing the French term droit d'auteur into English and the English copyright into French contracts is a way of both transferring an odd meaning into a text and keeping the legal differences when no equivalent legal concept exists in the target language.

We can finally complete our comparative law step with our third example: Act of God and catastrophe naturelle. Is Act of God the best possible and legally correct translation for the French catastrophe naturelle? It seems both are comparable, and thus translatable. Moreover, they both appear in contracts where long lists of "unexpected, disruptive event[s] that may excuse a party from performing duties under a contract", for instance in:

[…] Company shall not be in default by reason of any failure in its performance under this Agreement if such failure results from, whether directly or indirectly, fire, explosion, strike, freight embargo, Act of God or of the public enemy, war, civil disturbance, act of any government, de jure or de facto, or agency or official thereof, material or labor shortage, transportation contingencies, unusually severe weather, default of any other manufacturer or a supplier or subcontractor, quarantine, restriction, epidemic, or catastrophe, lack of timely instructions or essential information from Distributor, or otherwise arisen out of causes beyond the control of the Company, with the French translation being:

[…] La Société ne saurait être tenue pour responsable de l'inexécution du présent contrat, dans le cas où ladite inexécution résulte de façon directe ou indirecte de : incendie, explosion, grève, embargo de marchandises, catastrophe naturelle, attentat terroriste, guerre, troubles, fait du prince de jure ou de facto, pénurie de matériaux ou de main d'oeuvre, aléas liés au transport, conditions météorologiques extrêmes, inexécution émanant d'un fabricant, fournisseur et sous-traitant, mise en quarantaine, contingentement, épidémie ou désastre, défaut d'information en temps et en heure de la part du Distributeur, ou de toute cause indépendante de la volonté de la Société.

"Unforeseeable circumstances that prevent someone from fulfilling a contract", such as fire, explosion, labor shortage, etc, are the exact definition of the French legal concept force majeure, which is «un événement imprévisible et irrésistible qui, provenant d'une cause extérieure au débiteur d'une obligation ou à l'auteur d'un dommage (force de la nature, fait d'un tiers, fait du prince) le libère de son obligation ou l'exonère de sa responsabilité (my translation : in French, force majeure is an unforeseen and compelling event that frees the lienee from his/her obligation or exonerates the person who caused the damage from his/her liability (natural event, act of a third party, or government fiat)" (Cornu 2016 : 471). Force majeure is a hypernym into which both catastrophe naturelle in French and Act of God in English fit. Therefore, the above contract can be translated in a shorter way into both French and English:

[…] La Société ne saurait être tenue pour responsable de l'inexécution du présent contrat, dans le cas où ladite inexécution résulte d'un cas de force majeure.

[…] Company shall not be in default by reason of any failure in its performance under this Agreement if such failure results from force majeure.

Force majeure is a frequent legal borrowing in English contracts, as force majeure clauses are acceptable in Common Law systems such as in the US and in the UK.

Conclusion

Proof has been given that translators may avoid usual translation pitfalls when applying the 3-step legal approach. Although the translation strategies suggested in the different cases may differ, the resulting translations are legally correct and consider both source and target legal contexts and systems after comparing both source and target legal contexts and systems. Translators avoid false friends (crime and crime) as well, thanks to a complete and well-organized documentary research. Comparative law highlights major conceptual differences and eliminates common mistakes or mistranslations (copyright and droit d'auteur). Comparable and translatable concepts do exist (Act of God and catastrophe naturelle) and may sometimes fall into hypernyms to give shorter but still legally correct translations (force majeure). The three examples given in this paper have shown that juritraductology is a legitimate academic discipline in a broader translation approach which is both well-thought of and well suited for legal texts.

Thanks to Carol Jagot-Lachaume and Solweig Franzinetti for proofreading this article.

http://www.memidex.com/

https://www.merriam-webster.com/dictionary/crime#legalDictionary

http://www.memidex.com/copyrighting

https://www.merriam-webster.com

https://www.legifrance.gouv.fr/ -Loi n° 82-600 du 13 juillet 1982.


Mohamed Anis Fekih
email: mohamed-anis.fekih@insa-lyon.fr

Ichrak Mokhtari
email: ichrak.mokhtari@insa-lyon.fr

Walid Bechkit
email: walid.bechkit@insa-lyon.fr

Yasmine Belbaki
email: dy_belbaki@esi.dz

Hervé Rivano
email: herve.rivano@insa-lyon.fr

On the regression and assimilation for air quality mapping using dense low-cost WSN

The use of low-cost Wireless Sensor Networks (WSNs) for air quality monitoring has recently attracted a great deal of interest. Indeed, the cost-effectiveness of emerging sensors and their small size allow for dense deployments and hence improve the spatial granularity. However, these sensors offer a low accuracy and their measurement errors may be significant due to the underlying sensing technologies. The main aim of this work is to reconsider and compare some regression approaches to assimilation ones while taking into account the intrinsic characteristics of dense deployment of low cost WSN for air quality monitoring (high density, numerical model errors and sensing errors). For that, we propose a general framework that allows the comparison of different strategies based on numerical simulations and an adequate estimation of the simulation error covariances as well as the sensing errors covariances. While considering the case of Lyon city and a widely used numerical model, we characterize the simulation errors, conduct extensive simulations and compare several regression and assimilation approaches. The results show that from a given sensing error threshold, regression methods present an optimal sensor density from which the mapping quality decreases. Results also show that the Random Forest method is often the best regression approach but still less efficient than the BLUE assimilation approach when using adequate correction parameters.

Introduction

Air pollution is one of the major concerns in many cities worldwide. Indeed, millions of people live in cities where pollutant concentrations exceed standard health limits many times each year which poses serious health concerns. According to the World Health Organization (WHO), seven million deaths were attributable to the indoor and outdoor air pollution in 2016 [START_REF]Burden of disease from the joint effects of household and ambient air pollution for 2016[END_REF]. Indeed, long-term exposure to air pollution can cause cancer and damage to the immune, neurological, and respiratory systems. In extreme cases, it can even cause death. In order to reduce the impact of air pollution, an efficient monitoring system is needed where the main aim is to generate accurate pollution maps in real time. Traditional monitoring stations are equipped with multiple sensors measuring a number of pollutants, such as carbon monoxide (CO), nitrogen oxides (NOx), ozone (O3) and particulate matter (PM) [START_REF] Kumar | The rise of low-cost sensing for managing air pollution in cities[END_REF]. These monitoring stations give accurate concentrations but they are too big and expensive to be deployed everywhere [START_REF] Schneider | A network of low-cost air quality sensors and its use for mapping urban air quality[END_REF]. In addition to measurement-based traditional monitoring, air pollution maps can be also obtained using numerical simulations of physical models. These models simulate the air pollution dispersion based on the locations of pollution sources, emission rates and meteorological data to estimate the pollutant concentrations.

The limits of traditional monitoring stations in terms of cost, size, flexibility and spatial granularity led to the emergence of small and low-cost air quality sensors. The wireless connection these sensors, forming a low-cost Wireless Sensor Network (WSN), presents many advantages compared to traditional air pollution monitoring solutions. Indeed, the cost-effectiveness allows for dense and large deployments and hence improves the spatial granularity [START_REF] Arroyo | Wireless sensor network combined with cloud computing for air quality monitoring[END_REF][START_REF] Hasenfratz | Deriving high-resolution urban air pollution maps using mobile sensor nodes[END_REF][START_REF] Anjomshoaa | City scanner: Building and scheduling a mobile sensing platform for smart city services[END_REF]. Moreover, the size and cost considerations offer more deployment flexibility. However, low cost WSN present some problems and challenges to be addressed. The most significant of these is the low accuracy of the sensing probes. For instance, most low-cost gas probes are electrochemical which means they are very sensitive to temperature and humidity depending on the electrolyte. Moreover, this kind of probes present cross-reactivity with similar molecule types [START_REF] Gerboles | Measuring air pollution with low-cost sensors[END_REF]. Optical particle counter probes usually used as low-cost PM sensors are also less accurate than ground stations. Indeed, the given particle counts is sensitive to many parameters such as particle shape, color, density, humidity and refractive index. In addition, the conversion from particle counts to PM mass is based on theoretical models [START_REF] Gerboles | Measuring air pollution with low-cost sensors[END_REF]. Furthermore, the high density of these sensors pushes us to reconsider the deployment approaches [START_REF] Boubrima | Optimal WSN deployment models for air pollution monitoring[END_REF][START_REF] Boubrima | On the deployment of wireless sensor networks for air quality mapping: Optimization models and algorithms[END_REF] as well as the collected data analysis for air quality mapping.

In this paper we tackle these challenges while comparing regression and assimilation approaches, taking into account numerical model errors and sensing errors. To the best our knowledge this is the first work to compare regression and assimilation methods for air quality mapping using dense low-cost WSN data. The main contributions of this work can be summarized as follows:

1. We derive a generic framework that allows us to compare different regression and assimilation approaches. 2. We characterize the errors of the SIRANE numerical model [START_REF] Soulhac | The model sirane for atmospheric urban pollutant dispersion; part i, presentation of the model[END_REF][START_REF] Soulhac | The model sirane for atmospheric urban pollutant dispersion; part ii, validation of the model on a real case study[END_REF][START_REF] Soulhac | The model sirane for atmospheric urban pollutant dispersion. part iii: Validation against no2 yearly concentration measurements in a large urban agglomeration[END_REF] based on a realistic data set. 3. We conduct extensive simulations and compare four regression approaches and one assimilation method taking into account multiple errors of both measurements and the numerical model. 4. We derive some guidelines and insights for the choice of adequate approach depending on different parameters such as the deployment density, the simulation error, the sensing error, etc.

Air quality monitoring has attracted great attention in recent years. In order to tackle the pollution problem, we first need to build fine grained estimation maps. These maps allow us to assess the exposure of the population to pollution, detect highly polluted areas and warn the population if the recommended thresholds are exceeded. Generally, two types of approaches are used for air pollution mapping: spatial interpolation and data assimilation. Spatial interpolation methods have been largely used in the literature and several works applied Land-Use Regression (LUR) techniques. Kerckhoffs et al. [START_REF] Kerckhoffs | A national fine spatial scale land-use regression model for ozone[END_REF] propose a multiple regression model that generates average annual concentrations of Ozone (O 3 ) at a fine spatial scale (50m × 50m). Ozone concentrations were measured at 90 sites covering the whole Netherlands. All sites were measured simultaneously during four campaigns of two weeks each spread over the seasons. Several explanatory variables were used, characterising mainly roads, buildings, traffic and population density. This model allowed to reach a determination coefficient of 0.77.

Hasenfratz et al. [START_REF] Hasenfratz | Deriving high-resolution urban air pollution maps using mobile sensor nodes[END_REF] used mobile sensor nodes to collect ultra fine particle (UFP) data. Sensors were installed on top of public transport vehicles in the city of Zurich. Based on the collected measurements and land-use data, a LUR model was developed to create pollution maps with a spatial resolution of 100m × 100m. Twelve explanatory variables representing the traffic, the population and the city's characteristics were examined to build the air quality models for UFP. The accuracy of models across various time scales were compared. Results show that low temporal resolution mapping achieves good results while high temporal resolution estimation presents high errors. To increase the accuracy of these models, past measurements were used in the modeling process which permitted to reduce the root mean square error by 26%.

Marjovi et al. [START_REF] Marjovi | Extending urban air quality maps beyond the coverage of a mobile sensor network: data sources, methods, and performance evaluation[END_REF], proposed two approaches for estimating the pollution level of UFP at desired timelocation pairs in Lausanne in Switzerland. The first is a log-linear regression built over a virtual dependency graph based on land-use data. The second is a deep learning framework which can capture automatically the relationships between data based on autoecoders. Different land use, meteorological and traffic data were used to build these models, among them altitude, scope, density of population, buildings, industries, etc. The two approaches were evaluated against three canonical modeling techniques namely Basic Log-Linear regression model (BLL), Network-based Log-Linear regression (NLL) and Basic Log-Linear regression with Land-Use (BLL-LU). The results demonstrated the superiority of the proposed approaches and specifically the deep learning model over the canonical techniques.

Adam-Poupart et al. developed in [START_REF] Adam-Poupart | Spatiotemporal modeling of ozone levels in quebec (canada): a comparison of kriging, land-use regression (lur), and combined bayesian maximum entropy-lur approaches[END_REF] three interpolation methods: i) a land-use mixed-effects regression, ii) a Bayesian maximum entropy using both O 3 monitoring station data and the land-use-mixed outputs (BME-LUR), and iii) a kriging model for estimating the daily daytime average ozone concentrations from routine monitors in Quebec (Canada). Results showed the superiority of the BME-LUR as the best predictor followed by the LUR method and the kriging.

Data assimilation aims to efficiently combine numerical models with observations. It has been initially developed and used for meteorology until the 90's [START_REF] Bocquet | Data assimilation in atmospheric chemistry models: current status and future prospects for coupled chemistry meteorology models[END_REF]. After that, it has been applied in many other fields such as atmospheric chemistry and agronomy. Today it is used in many applications like the optimization of observation network and the correction of several numerical models' errors in different fields.

In the air quality field, several assimilation-based approaches were proposed in the literature. Tilloy et al. [START_REF] Tilloy | Blue-based no 2 data assimilation at urban scale[END_REF] used the Best Linear Unbiased Estimator (BLUE) method to combine NO 2 ground observations provided by 9 fixed monitoring stations across Clermont-Ferrand city in France and a simulation at urban scale. The simulation has been carried out over the city every three hours for the whole year 2008 using a model called ADMS Urban. BLUE was computed every three hours when new simulated concentrations were available and results showed that data assimilation corrected the problem of under estimation of NO 2 concentrations when using only the model.

Kumar et al. [START_REF] Kumar | Data assimilation of surface air pollutants (o3 and no2) in the regional-scale air quality model aurora[END_REF] used a bias-aware optimal interpolation combined with Hollingsworth-Lönnberg method to estimate error covariance matrices to perform corrections on hourly simulated O 3 and NO 2 concentrations. The simulation has been performed using the regional-scale air quality model AURORA over Belgium for a summer and a winter month (June and December respectively). The monitoring network was composed of 73 monitoring stations. Results showed that data assimilation improved the simulation results with a root mean square error that went down from 27.9 to 12.6 for O 3 and from 17.4 to 11 for NO 2 .

In another work [START_REF] Wang | Assimilation of omi no2 retrievals into a regional chemistry-transport model for improving air quality forecasts over europe[END_REF], NO 2 observations from Ozone Monitoring Instruments (OMI) aboard NASA Aura satellite during November-December were combined with results of an air quality Model to improve air quality forecasts resulting in a spatial resolution of around 40 km x 55 km. The simulations showed that the RMSE was reduced by 15%. However, the resolution offered by satellite data is very low and does not allow air quality characterisation at a human scale.

The majority of data assimilation solutions are based in their validation on reference stations whose data are accurate. However it is not feasible to equip a large area with high-accuracy sensors due to their cost and complexity. Therefore, our first objective in this work is to propose a framework that allows comparing assimilation approaches to regression ones when using low-cost sensors. Our framework is based on numerical models, the characterization of their errors and the characterization of sensing errors. Another objective is to use this framework to compare the performances of different solutions and to derive some guidelines and insights.

Brief Overview on Regression and Data Assimilation for spatial mapping

We present in the section a brief background of some regression and assimilation approaches that we compare later. The main used notations are presented in Table 1. The observational vector (represents pollution measurements) σ m Sensing (measurement) standard deviation error x a

The analyzed vector (represents the best estimation given by a mapping approach) R

The observational error covariance matrix B

The background error covariance matrix H

The observation operator K

The Kalman gain matrix

Spatial interpolation

Spatial interpolation bring together a set of techniques that estimate the concentration values at a given point as a weighted average of the measurements at surrounding observation points [START_REF] Xie | A review of urban air pollution monitoring and exposure assessment methods[END_REF]. Land-Use-Regression (LUR) assume that pollutant concentrations at a given location depend on the surrounding environment, land-use and traffic characteristics [START_REF] Jerrett | Modeling the intraurban variability of ambient traffic pollution in toronto, canada[END_REF].

Linear regression seeks to establish a linear relationship between the studied variable called the dependent variable and the variables likely to explain the phenomenon called independent or explanatory variables. The resulting model is then used to predict the concentration of a pollutant in a given location where no sensor is deployed. The most used explanatory variables in the air quality field are: humidity, temperature, wind speed, population, traffic intensity and road length [START_REF] Xie | A review of urban air pollution monitoring and exposure assessment methods[END_REF]. K nearest neighbors (KNN) regression uses 'feature similarity' to predict values of any data points where no sensor is deployed. Similarity between the point to be predicted and the observation points is defined according to a distance metric applied to independent variables. The k nearest neighbors are the k closest points to the predicted point based on the defined metric. The concentrations at these neighbors are then used to estimate the concentration at the predicted point. Today other recent machine learning techniques such as random forest and Xgboost have emerged as serious competitors to state-of-the-art methods. Random forest (RF) also called decision forest [START_REF] Chen | Xgboost: extreme gradient boosting[END_REF] performs, in its most classical form, parallel learning on multiple decision trees built randomly and driven on different subsets of data using a bagging approach [START_REF] Biau | Analysis of a random forests model[END_REF]. The main idea behind this technique is that the aggregate of the results of multiple predictors gives a better prediction than the best individual predictor. Indeed this allows to average noisy and unbiased models and thus create a model with low variance. Extreme gradient boosting abbreviated as Xgboost is also an ensemble learning method that combines the outputs from individual trees to make a prediction exactly like random forest. Xgboost and RF differ in the way the trees are built and the order and the manner the results are combined. In Xgboost, trees are built one at time, where each new tree helps to correct errors made by the previously-trained tree.

Data assimilation

One of the best assimilation methods used for air quality mapping is the Best Linear Unbiased Estimator (BLUE) [START_REF] Tilloy | Blue-based no 2 data assimilation at urban scale[END_REF]. BLUE uses the errors of both the background and the observations to estimate the state vector as follows:

x a = x b + K(y m -Hx b ) (1)
Where x a is the analyzed vector, x b the background vector (simulated values using numerical models in our case), K the Kalman gain, y m the observations vector (measurements given by the deployed sensors in our case), H the observation operator which is a matrix where h i j = 1 if the sensor i is deployed at position j and h i j = 0 otherwise. The equation 1 means that the analysed vector is a correction of the background vector with K(y m -Hx b ).

While minimizing the sum of the squares of the estimation errors, BLUE gives the optimal Kalman gain as follows :

K = BH T (HBH T + R) -1 (2)
Where B and R are the covariance matrices of the background estimate and the observations respectively.

Methods and models

We present in this section the general methodology that we propose to compare different mapping approaches. We present then our work to characterize the simulation errors and to generate the ground truth and measurements data sets. We present at the end of the section, the developed regression mapping models.

General Methodology

In order to compare different regression and assimilation approaches, we propose first to generate a large number of ground truth data sets based on physical models (numerical simulations) and an adequate estimation of the simulation error variance-covariance matrix. We explain in the following sub-sections the approach that we use in order to estimate the simulation error variance-covariances.

Once the ground truth data sets generated, we generate for each one a large number of observations (pollutant concentrations) in positions where low-cost sensors are deployed. This generation will take into account the ground truth realizations as well as the measurements' errors variance-covariance matrix. Based on the measurement realizations, simulation values and ground truth realizations, different regression and assimilation methods can be developed and compared.

Area of interest and simulation data set

Without loss of generality, we consider in this work, simulations generated by the model SIRANE [START_REF] Soulhac | The model sirane for atmospheric urban pollutant dispersion; part i, presentation of the model[END_REF][START_REF] Soulhac | The model sirane for atmospheric urban pollutant dispersion; part ii, validation of the model on a real case study[END_REF][START_REF] Soulhac | The model sirane for atmospheric urban pollutant dispersion. part iii: Validation against no2 yearly concentration measurements in a large urban agglomeration[END_REF]. SIRANE is a stationary model designed for urban areas and is widely used by the certified associations of air quality monitoring in France. The used simulations give the Nitrogen-Dioxide concentrations in Lyon city (France) in 2008. We consider in this work a 25km 2 area of interest with a spatial resolution of 20m x 20m. This area corresponds to the center of Lyon and its immediate vicinity (see Figure 1 (a)).

Characterization of the variance of simulation errors

In order to characterize the variance of the simulation errors, we used NO 2 concentration values provided by 16 reference stations in Lyon and compared them to the simulated concentrations of SIRANE . We considered monthly values in both cases. For each simulated value z, we computed the standard deviation of the errors associated to simulated values in z ± 5ug/m 3 . Figure 2 shows that the model's error standard deviation, σ b , depends linearly on the pollutant concentration starting from a given threshold z 0 :

σ b (z) = α(z -z 0 ) + β , z ∈ [z 0 , +∞[ ( 3
)
Were z is the simulated value. As shown in Figure 2, a linear regression with z 0 = 24 ug/m 3 results in α = 0.344 and β = 2.27 ug/m 3 with a high R 2 value ( R 2 = 0.94).

Ground truth and measurements generation

Based on the variances of simulation errors and correlation coefficients w pq , we generate the variancecovariance matrix. In this work, we consider the correlation coefficient as a function of the distance [START_REF] Boubrima | On the deployment of wireless sensor networks for air quality mapping: Optimization models and algorithms[END_REF] given as: w pq = e -δ ḋ pq , were δ is the attenuation coefficient of the correlation function and d pq is the euclidean distance between points p and q. Assuming that the simulation error follows a multivariate normal distribution, we generate a large number of ground truth data sets based on the simulated values and the computed variance-covariance matrix.

Once the ground truth data sets generated, we generate for each ground truth set, a large number of observations (measurements) in positions where low-cost sensors are deployed. Sensors' measurements on their side are generated using a normal distribution since the sensing errors are not correlated with each other. Hence, the measurement error variance-covariance matrix that we note R is given by: R = v 0 I, where v 0 is the vector of measurements' error variance and I is the identity matrix.

Dependant variables for regression methods

The performance of regression methods rely mainly on the choice of the variables likely to explain the best the phenomenon under study. For our study we collected more than 120 relevant variables to characterise the streets, traffic load, land-use, population density and environment of the area of interest. Land-use data and traffic load were taken from Data Grand Lyon1 and Open street map. Meteorological variables were obtained from "Météo-France"2 . Satellite data were provided to us by USGS Earth Explorer 3 .

We perform a pretreatment on these data to eliminate redundant and non-significant variables. For this purpose, we first perform a Student test to eliminate the non-significant variables. We calculate then the variance inflation factor (VIF) in order to check whether the information provided by a variable is present in one or more other variables of the model. We eliminate variables with a high VIF in order to reduce multicollinearity. Finally, we calculate the adjusted R2, which evaluates the contribution of each variable to the model, before and after adding each variable and then we eliminate the non-significant variables to the model. Following these steps, we validated around 35 % of the initial variables. We divided the resulting variables into four categories listed in table 2:

Performance evaluation

We compare in this section different mapping strategies while taking into consideration the intrinsic characteristics of dense deployment of low cost WSN for air quality monitoring. We first evaluate and com-pare some regression approaches regarding different metrics and conduct after that comparisons against the BLUE assimilation method.

Choice of the parameter k in KNN regression

The k parameter in KNN regression gives the number of the closest points to the predicted one that will be used to estimate the predicted value. On one hand, using a small k means restraining the set of a given prediction and forcing the regression to be more blind to the overall distribution. On the other hand, a higher k averages more points values and hence is more resilient to outliers. However, larger values of k will have smoother decision boundaries. To choose a good value of k, we evaluated the mean absolute error (MAE) for different values of k. The results are presented in table 3. We notice the parameter k has no significant impact on the MAE for k ∈ [START_REF] Schneider | A network of low-cost air quality sensors and its use for mapping urban air quality[END_REF][START_REF] Adam-Poupart | Spatiotemporal modeling of ozone levels in quebec (canada): a comparison of kriging, land-use regression (lur), and combined bayesian maximum entropy-lur approaches[END_REF]. However, the minimum MAE is reached when k=5. k is then set to 5 for the following comparisons.

Comparing regression-based approaches in function of the fraction of deployed sensors

We consider in this work four regression approaches: linear regression, KNN, Random forest and Xgboost. We compared these approaches on 30 realizations of ground truth obtained from a simulation with α= 0.05. First, we evaluate the behaviour of these methods regarding the MAE metric when increasing the fraction of deployed sensors. For each ground truth data set, all approaches are executed ten times with different measurement realizations (with σ m =1 ug/m 3 ). The results, depicted in Figure 3 (a), show that all methods present a better performance when the percentage of sensors increases. Indeed the more sensors are deployed, the lowest the error is and the better the estimate is. Random forest outperforms all the methods reaching an MAE of 1.48 ug 3 when 40% of sensors are deployed, it is followed nearly by xgboost and then KNN and linear regression.

Impact of sensor errors on regression methods

In order to assess the impact of sensing errors and the high density on the behaviour of the regression methods, we evaluate these methods regarding the fraction of deployed nodes with different sensing errors. For that, we plot in Figure 3 that when the error of sensing is relatively high (>=3), increasing the number of deployed sensors does not necessarily improve the results, but instead decreases the performances when the density exceeds an optimal sensor density. In this case, when the sensors have a significant measurement error it is not interesting to deploy more sensors to improve the performance. The results of the four regression approaches are similar with different sensing error threshold and different optimal sensor density.

In order to compare the regression methods regarding the sensing errors σ m , we fixed the percentage of deployed sensors to 30% and varied σ m . Results, plotted in Figure 3 (c), show that all regression methods are sensitive to sensor error. Indeed the MAE is higher as the sensing error increases. From the different curve slaves, we can also notice that Xgboost is less sensitive to sensing errors than random forest. Starting from a given threshold, Xgboost outperforms random forest. We also notice that linear regression is less sensitive to sensing errors compared to KNN. These results give guidelines regarding the method choice in function of the sensing errors.

Assimilation results depending on simulation errors

The data assimilation results are obtained over 30 realizations of ground truth and with different values of α increasing from 0.05 to 0.4. We recall that the higher α is, the lower is the numerical model quality. Figure 4 (a) depicts the data assimilation MAE results with two different values of α in function of the fraction of deployed sensors used to correct the model. As expected, an assimilation with an accurate simulation output (i.e. small α) performs better than an assimilation with a less accurate physical model. Moreover, the higher the fraction of deployed sensors is, the lower is the impact of the numerical model errors.

Assimilation results depending on sensing errors

To evaluate the effect of the sensing error, we perform BLUE assimilation with a fixed α (0.05) and different sensing errors σ m . As illustrated in Figure 4 (b), the MAE decreases as we increase the fraction of deployed sensors. We can observe that even with a σ m =7, the assimilation still can correct the model thanks to the consideration of both the simulation error covariance matrix and sensing error covariance matrix.

Regression Vs Assimilation

In this last simulation, we compare the different regression methods to the BLUE assimilation one with different values of α. Figure 4 (c) shows that the implemented assimilation method (BLUE) has 40% better air quality estimation compared to Random Forest which offers the best performances of regression methods. This is due to the consideration of the model's error as well as the sensing errors in the assimilation process. BLUE performs very well when the simulation error covariance matrix and sensing error covariance matrix are well estimated.

Conclusions

One of the main challenges of low-cost WSN is their low accuracy. In this work, we proposed a general framework that allows the comparison of different regression and assimilation strategies based on numerical simulations and an adequate estimation of the simulation error covariances as well as the sensing errors covariances. We also studied four regression approaches and one assimilation method and compared them in terms of pollution estimation quality while considering the errors of both measurements and the numerical model. We have shown that data assimilation is less sensitive to the variations of measurement errors. Moreover we have seen that a big number of sensors is not always good for regression methods when sensors present an important sensing error. However this is not the case for assimilation when numerical model errors and sensing errors are well estimated. As future research work, we intend using more realistic errors characterization by introducing some noise to R and B matrices. We also plan to extend our framework by adding new approaches and test it with real data sets.

Fig. 1

 1 Fig. 1 NO2 concentrations of the area of interest (a) a realization of the ground truth; (b) Random forest regression based estimation; (c) BLUE assimilation based estimation

Fig. 2

 2 Fig. 2 Error standard deviation Vs Simulation Values

  (b) the MAE of the Xgboost method, considering three different sensing standard deviation error (σ m = 1, 3 and 7 ug/m 3 ) while varying the percentage of deployed sensors. The results show

Fig. 3

 3 Fig. 3 (a) MAE Vs percentage of deployed sensors, σ m = 1 ug/m 3 ; (b) MAE of Xgboost method Vs percentage of deployed sensors,s ∈ [1, 3, 7]; (c) MAE Vs Standard error deviation, Fraction of deployed sensors = 0.3

Fig. 4

 4 Fig. 4 (a) Assimilation's MAE Vs percentage of deployed sensors σ m = 3 ug/m 3 ; (b) Assimilation's MAE Vs percentage of deployed sensors, α = 0.05; (c) MAE Vs values of α, σ m = 1 ug/m 3 , fraction of deployed sensors = 0.3

Table 1

 1 Main notations used in this work

	Symbol	Description
	x t	Ground truth vector of pollution concentrations
	x b	Background vector (simulated values)
	σ b	Simulation (numerical model) standard deviation error
	y m

Table 2

 2 Main categories of dependant variables

	Categories	Example of Variables
	Land use	Number of buildings in a given buffer
	Traffic and road networks	Number and length of roads in a given
		buffer
	Population	Density of population in a given buffer
	Meteorology	Temperature, humidity

Table 3

 3 MAE vs k value of KNN regression (σ m = 2 ug/m 3 , Fraction of deployed sensors = 0.3)

	k	3	5	7	9	11	13	15
	MAE 3.246 3.230 3.234 3.240 3.245 3.251 3.256

https://data.grandlyon.com/

https://donneespubliques.meteofrance.fr/

https://earthexplorer.usgs.gov/

Acknowledgements This work has been supported by the "LABEX IMU" (ANR-10-LABX-0088) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).


Olivier Gargominy1

Jean-François Butaud2

Benoît Fontaine3

Vincent Prié4

Dario Zuccon4

Sandrine Tercerie5

Joana Hauata6

Patrinat

RESEARCH NOTE Discovery of the only living population of Pupoidopsis hawaiensis (Gastropoda: Pupillidae) in the last 50 years

Keywords: upraised coral reefs, Tuamotu, Anaa, conservation

Pupoidopsis hawaiensis Pilsbry & C.M. Cooke, 1921, and its monospecific genus, is the only representative of Pupillidae in tropical Pacific Islands. Although most of native species from that region are single island endemics, P. hawaiensis has been recorded from Hawaii to the Tuamotus. However, it has not been recorded alive since 1965. Here we report the only modern living population of that species from a single site in Anaa (Tuamotu archipelago). A single COI haplotype and a single 28S allele support the hypothesis of self-fertilization. The species should be listed as Critically Endangered, and included in the list of protected species of French Polynesia. An Action Plan for its conservation should be undertaken, with protection status given to its last known stronghold in Anaa.

Pupoidopsis hawaiensis Pilsbry & C.M. Cooke, 1921 (Pupillidae, Pupoidinae) is noteworthy because this species, the sole representative of the genus, is endemic to the Pacifi c Islands and has historically been reported with a wide distribution in this area, while most other native species are single island endemics. It is also the unique indigenous representative of the Pupillidae in tropical Pacifi c islands. First described as an extinct genus and species from Oahu, Molokai, and Maui in the Hawaiian islands (Pilsbry 1920(Pilsbry -1921)), it is now known from Hawaii, Line islands (Kiribati) and Tuamotu archipelago (French Polynesia). It has only been found alive four times: in Kiritimati -Christmas (Line islands, Kiribati) in 1924 [START_REF] Anonymous | Pupoidopsis hawaiensis on Christmas Island[END_REF][START_REF] Cooke | Distribution and anatomy of Pupoidopsis hawaiiensis[END_REF] and 1965 [START_REF] Christensen | Pupoidopsis hawaiensis Pilsbry & Cooke, 1921 (Gastropoda: Pupillidae): extirpated in Hawai'i but a possible survivor in Kiribati and French Polynesia[END_REF] and in Tabuaeran -Fanning (Line islands, Kiribati) and Hao (Tuamotu) in 1934 [START_REF] Christensen | Pupoidopsis hawaiensis Pilsbry & Cooke, 1921 (Gastropoda: Pupillidae): extirpated in Hawai'i but a possible survivor in Kiribati and French Polynesia[END_REF]. All collectors report the species as belonging to the lowland fauna, either on high islands or atolls. The heavy human impact on lowland habitats is thought to have been the cause of extirpation of P. hawaiensis from its former distribution [START_REF] Gargominy | Disjunct distribution in the Pacifi c Ocean: Pupoidopsis haiwaiensis in French Polynesia, Kiribati and Hawaii, South Pacifi c Islands[END_REF].

Here we report the presence of P. hawaiensis from Anaa (Tuamotu) for the fi rst time, and document the only currently known living population of this species.

MATERIALS AND METHODS

A malacological survey of Anaa (Tuamotu Islands, French Polynesia) was conducted January 4 -18, 2018, on behalf of the NGO Pu Tahi Haga no Ganaa and funded by BEST 2.0 [START_REF] Gargominy | Inventaire des Mollusques continentaux d'Anaa[END_REF]. Data were collected from 40 localities, representing a total of 676 lots, 22,000 specimens and 38 taxa. Specimens are deposited in the collection of the Museum national d' Histoire naturelle (Paris, France), under numbers MNHN-IM-2018-46, -71, -80, -129, -141, -302, -333, -491, -527, -562, -584, -624 andMNHN-IM-2013-76001 to 76005. Five specimens of P. hawaiensis were prepared for molecular analyses (MNHN-IM-2013-76001 to 76005). Total genomic DNA was extracted using the Qiagen DNA Mini Kit and following the manufacturer's protocol. Two standard genes fragments were selected for PCR amplifi cation: the mitochondrial cytochrome oxidase I (COI, amplified with primers LCO1490 and HCO2198, [START_REF] Folmer | DNA primers fo r amplifi cation of mitochondrial cytochrome c oxidase subunit I from diverse metazoan invertebrates[END_REF]) and the nuclear large subunit rRNA gene (28S, amplifi ed with primers C1 and D2, [START_REF] Jovelin | Phylogenetic relationships within the polyopisthocotylean monogeneans (Platyhelminthes) inferred from partial 28S rDNA sequences[END_REF]. Sequences were compared to other Pupillidae sequences available in Genbank. We assembled two datasets: the COI (655bp) and the 28S (784bp) datasets, including 10 and nine other Genbank Pupillidae sequences, respectively. We used Valloniidae sequences as outgroup. Both datasets were analysed under the maximum likelihood criterion, using RAxML v. 7.0.3 [START_REF] Stamatakis | RAxML-VI-HPC: maximum likelihood-based phylogene tic analyses with thousands of taxa and mixed models[END_REF], selecting a GTR+Γ+I model and random starting tree, with empirical base frequencies and estimated α-shape parameters and GTR-rates. Nodal support was estimated using 100 bootstrap replicates.

RESULTS

Pupoidopsis hawaiensis shells were collected in one fourth of the collecting sites (11 stations out of 40). These stations are distributed all over the island (Fig. 1), but the species was recorded in greater abundance on the south-western part of the island (Otepipi, Okuku, Hapuku-Putuahara and Tekahora islets). Pupoidopsis hawaiensis was found alive at a single station on the motu Otepipi (Figs. 1, 2, 3). The area was no more than 100 square meters. It was quite abundant, with up to ten individuals fixed under a single limestone pebble a dozen centimeters in diameter. It was only found on the ground under rocks. At this site, the vegetation was dominated by the native trees and shrubs Allophylus rhomboidalis (Nadeaud) Rdlk., Cyclophyllum barbatum (G.Forst.) N.Hallé & J.Florence, Glochidion wilderi J.Florence, Guettarda speciosa L., Pisonia grandis R .Br., Planchonella tahitensis (Nadeaud) Pierre ex Dubard, Premna serratifoli a L., Sophora tomentosa L., Timonius unifl orus (Banks ex C.F.Gaertn.) Govaerts, Xylosma suaveolens (J.R.Forst. & G.Forst.) G.Forst. The understory was very scarce on this rocky station, with native ferns Asplenium polyodon G.Forst. and Microsorum grossum (Langsd. & Fisch.) S.B.Andrews. Among the fi ve sequenced specimens, we recovered a single COI haplotype and a single 28S allele (GenBank accession numbers MT457074-MT457078 and MT457405-MT457409, respectively). The phylogenetic reconstruction places Pupoidopsis hawaiensis in the Pupillidae (Fig. 4). In the COI tree Pupoidopsis is sister to Pupoides albilabris, while in the 28S tree it is basal to a clade containing Pupoides albilabris, Pupilla and Gastrocopta, albeit with no bootstrap support.

DISCUSSION

Although the phylogenetic reconstruction confi rms that Pupoidopsis hawaiensis belong to the Pupilloidea, the limited number of published sequences available in Genbank prevent us to clarify its place within the family. A better resolution requires a larger, multigene dataset and the inclusion of the type species of Pupoides (Bulimus nitidulus L. Pfeiffer, 1839, type locality in Cuba). The fact that among the five sequenced specimens, we recovered a single COI haplotype and a single 28S allele is congruent with [START_REF] Cooke | Distribution and anatomy of Pupoidopsis hawaiiensis[END_REF] who observed that most specimens are female by lacking penis, and we suggest that the species could reproduce by self-fecundation. This probable self-fertilization, allowing colonization of a new island with a single individual, is congruent with the exceptionally wide distribution of the species within the Pacific. Anaa is an uplifted atoll, i.e. typically composed of lowland habitats either on sand or on rocky calcareous substrate. Preferred habitat for P. hawaiensis in this island is feo, which corresponds to areas of upraised limestone (coral reef) up to few meters high, or areas neighbouring feo with great density of limestone pebbles on the ground. Feo represent areas where vegetation is in the best state of conservation on this atoll, as these areas were not converted to coconut plantations due to the rocky substrate. This relatively good state of conservation of feo probably accounts for the fact that P. hawaiensis still survives there. However, we are unable to explain why the species has been extirpated everywhere in Anaa except in a single place on Otepipi. No rat predation on this species is recorded, but it is worth noting that just as elsewhere in Anaa, Otepipi harbours two rat species (Rattus exulans and R. rattus). As a comparison, the land snail Mautodontha daedalea (A.A. Gould, 1846), endemic to Makatea and Anaa atolls in the Tuamotu, was found alive on fi ve stations in Anaa (with 12 additional stations where shells only were found). Despite the fact it belongs to the Endodontidae, a family which was much impacted by habitat perturbations [START_REF] Solem | Endodont oid Land Snails from Pacific Islands (Mollusca: Pulmonata: Sigmurethra). Part 1. Family Endodontidae[END_REF][START_REF] Abdou | Nouveaux gastéropodes Endodontidae et Punctidae (Mollusca, Pulmonata) récemment éteints de l'archipel des Gambier (Polynésie)[END_REF][START_REF] Zimmermann | Quatre espèces nouvelles d'Endodontidae (Mollusca, Pulmonata) éteints de Rurutu (I les Australes, Polynésie française)[END_REF][START_REF] Sartori | Anthropogenic extinction of Pacifi c land snails: A case study of Rurutu, French Polynesia, with description of eight new species of endodontids (Pulmonata)[END_REF][START_REF] Sartori | Radiation and decline of endodontid land snails in Makatea, French Polynesia[END_REF], it is still more represented than P. hawaiensis. Pupoidopsis hawaiensis may still survive in Anaa in other stations, but the contrast between the station on Otepipi, where live individuals were abundant, and other stations were only empty shells were found suggests that if it is still extant somewhere else, its abundance is much lower than it used to be. We hypothesize that it has disappeared from 1 0 out of 11 stations, i.e. a reduction of more than 90% of its former range on this single atoll. As stated by [START_REF] Christensen | Pupoidopsis hawaiensis Pilsbry & Cooke, 1921 (Gastropoda: Pupillidae): extirpated in Hawai'i but a possible survivor in Kiribati and French Polynesia[END_REF], some islands in Kiribati and Tuamotu may still support living populations of P. hawaiensis, as it is now proved for Anaa. However, the species must be highly threatened, since it is extinct in most of its known former range (Kiribati, Hawaiian archipelago -Christensen ( 2013)) and close to extinction in Anaa. As for most invertebrates which are under-studied, we cannot rule out the fact that it is still extant somewhere else. However, it has never been found in French Polynesia, where extensive collections have been done (for instance, Society islands, Austral islands, Gambier islands, Makatea -e.g. [START_REF] Gargominy | Beyond the alien invasion: A recently discovered radiation of Nesopupinae (Gastropoda: Pulmonata: Vertiginidae) from the summits of Tahiti (Society Islands, French Polynesia)[END_REF][START_REF] Christensen | First records of the invasive predatory land snail Gulella (Huttonella ) bicolor (Hutton, 1834) (Gastropoda: Streptaxidae) from the Society Islands, French Polynesia[END_REF][START_REF] Sartori | Anthropogenic extinction of Pacifi c land snails: A case study of Rurutu, French Polynesia, with description of eight new species of endodontids (Pulmonata)[END_REF][START_REF] Sartori | Radiation and decline of endodontid land snails in Makatea, French Polynesia[END_REF][START_REF] Abdou | Nouveaux gastéropodes Endodontidae et Punctidae (Mollusca, Pulmonata) récemment éteints de l'archipel des Gambier (Polynésie)[END_REF]. Additional surveys should be undertaken in neighboring islands (especially Niau, another uplifted atoll) to fi nd the species. Pupoidospis hawaiensis is currently classifi ed as data defi -cient DD in the IUCN Red List [START_REF] Cowie | Pup oidopsis hawaiensis. The IUCN Red List of Threatened Species[END_REF]. We propose that it should be listed as Critically Endangered B1,B2ab (i,ii,iii,iv,v) (only one living popula tion on a single atoll and general decline of extent of occurrence, area of occupancy, quality of habitat, number of locations, number of mature individuals). We recommend that it should be included in the list of protected species of French Polynesia [START_REF] Anonymous | Pupoidopsis hawaiensis on Christmas Island[END_REF], and an Action Plan for its conservation should be undertaken. The feo of Otepipi could also be given a protection status, for instance through the general management plan (PGA) of the Anaa municipality. Feo also provide strong protection against natural disasters as barriers to the ocean.

ACKNOWLEDGMENTS

We thank the European Commission which founded part of this work through BEST 2.0 project 1264 "Sauvegarde de la fl ore menacée de l'atoll de Anaa et défi nition de zones prioritaires de conservation basées sur la flore et la malacofaune sur les atolls soulevés de Anaa et Niau aux Tuamotu". We thank also Maima Matai and Mare Paeahi who helped for malacological collections in the field. Frédéric Alain Jacq prepared the map of Figure 1. This work was supported by 'Service de Systématique Moléculaire' (UMS2700 Acquisition et Analyse de Données pour l'Histoire naturelle, MNHN). We also thank two anonymous reviewers for helpful comments on the manuscript.