Laure Thierry De Ville

Romain Chenuil

David

L Thierry D E Ville D 'avray

D Ami

A Chenuil

R David

J.-P Féral

Application of the ecosystem service concept at a small-scale: The cases of coralligenous habitats in the North-western Mediterranean Sea

Keywords: Experts' knowledge, Marseille, Port-Cros, Appraisals, Interviews, Workshops

come

Introduction

Coralligenous habitats are consid ered to be some of the more complex marine habitats [Paoli et al., 2016b] and their 1666 species (at least) reported by [START_REF] Ballesteros | Med iterranean coralligenous assemblages: a synthesis of present knowled ge[END_REF] contribute actively to the biodiversity of the Med iterranean Sea. Such great biod iversity and coastal location suggest that these habitats provid e many ecosystem services. However, services from marine ecosystems are less stud ied than those of terrestrial ecosystems [START_REF] Beaumont | Id entification, definition and quantification of good s and services provid ed by marine biod iversity: implications for the ecosystem approach[END_REF] and there is almost no literature d ealing with ecosystem services provid ed by coralligenous habitats [Paoli et al., 2016a], while there is a relative abund ance of literature on ecosystem services provid ed by other marine habitats: seagrasses [START_REF] Vassallo | The value of the seagrass Posidonia oceanica: a natural capital assessment[END_REF]Nord lund et al., 2016], coral reefs [START_REF] Moberg | Ecological goods and services of coral reef ecosystems[END_REF], d eep marine ecosystems [START_REF] Armstrong | Services from the deep: steps toward s valuation of d eep sea good s and services[END_REF], coastal ecosystems [START_REF] Liquete | Current status and future prospects for the assessment of marine and coastal ecosystem services: a systematic review[END_REF].

Since [START_REF] Marion | Esquisse d'une topographie zoologique du Golfe de Marseille[END_REF] first identified them in Marseille, coralligenous habitats have been the subject of a relatively small number of studies, which has been increasing in the last d ecad e. Since the review of [START_REF] Ballesteros | Med iterranean coralligenous assemblages: a synthesis of present knowled ge[END_REF], various indicators have been created during the last five years specifically to measure the health status of coralligenous habitats [START_REF] Cahill | A multispecies approach reveals hot spots and cold spots of diversity and connectivity in invertebrate species with contrasting d ispersal mod es[END_REF][START_REF] Deter | A preliminary study toward an ind ex based on coralligenous assemblages for the ecological status assessment of Med iterranean French coastal waters[END_REF][START_REF] Gatti | Coralligenous reefs state along anthropized coasts: application and validation of the COARSE index, based on a rapid visual assessment (RVA) approach[END_REF][START_REF] Doxa | Mapping biod iversity in three-d imensions challenges marine conservation strategies: the example of coralligenous assemblages in North-Western Mediterranean Sea[END_REF][START_REF] Sartoretto | An integrated method to evaluate and monitor the conservation state of coralligenous habitats: the INDEX-COR approach[END_REF]. These ind icators confirm the interest of ecologists in coralligenous habitats, while literature dealing with the economics aspect of coralligenous habitats is in its infancy [Paoli et al., 2016a].

Coralligenous habitats are encountered along most of the Med iterranean coasts but they are mainly stud ied in the Northwestern region. Fig. 1 shows typical coralligenous habitats of Marseille. Coralligenous habitats have been observed from 20 m to 120 m und er the surface of the sea [START_REF] Ballesteros | Med iterranean coralligenous assemblages: a synthesis of present knowled ge[END_REF]. These habitats are very complex and heterogeneous assemblages, which d iffer from one Mediterranean region to another. Thus ecologists hardly agree on a common definition. In this stud y we used the most consensual d efinition of coralligenous habitats: hard substrates of biogenic origin that are mainly produced by the accumulation of calcareous encrusting algae growing in dim light conditions, and unique calcareous formations of biogenic origin in Mediterranean benthic environments [START_REF] Ballesteros | Med iterranean coralligenous assemblages: a synthesis of present knowled ge[END_REF]. Coralligenous habitats are threatened by large-scale events, waste waters [START_REF] Hong | Étud e faunistique d 'un fond d e concrétionnement d e type coralligène soumis à un grad ient d e pollution en Méd iterranée nord -occid entale (Golfe d e Fos)[END_REF], physical impacts from d iving [START_REF] Garrabou | The impact of d iving on rocky sublittoral communities: a case stud y of a bryozoan population[END_REF][START_REF] Harmelin | Population structure and partial mortality of the gorgonian Paramuricea clavata (Risso) in the north-western Med iterranean (France, Port-Cros Island )[END_REF] or fishery activities [START_REF] Bell | Effects of depth and marine reserve fishing restrictions on the structure of a rocky fish assemblage in the north-western Med iterranean Sea[END_REF][START_REF] Garcia-Rubies | Effects of total fishing prohibition on the rocky fish assemblages of Med es island s marine reserve[END_REF]] and invasive species [START_REF] Piazzi | Invasion of Caulerpa racemosa (Caulerpales, Chlorophyta) in the Med iterranean Sea: the balance of thirteen years of spread[END_REF]. Recent stud ies showed that sea acid ification is also a threat to coralligenous habitats [START_REF] Martin | Response of Med iterranean coralline algae to ocean acid ification and elevated temperature[END_REF][START_REF] Martin | One-year experiment on the physiological response of the Med iterranean crustose coralline alga, Lithophyllum cabiochae, to elevated pCO 2 and temperature[END_REF][START_REF] Nash | Mineralogical response of the Mediterranean crustose coralline alga Lithophyllum cabiochae to nearfuture ocean acidification and warming[END_REF]. Their extremely slow d evelopment makes their recovery from these impacts very limited . In this stud y we focused on sites in the Northwestern Med iterranean Sea where coralligenous habitats are T closest to the surface, which favours their observation and interaction with human beings.

The concept of ecosystem services has been used since the 1960's in order to raise awareness of human dependency on nature and the need to preserve nature for the sake of human well-being. The Millennium Ecosystem Assessment (MEA) [2005], the proceeding of The Economics of Ecosystems and Biod iversity (TEEB, 2010) and the Common International Classification of Ecosystem Services (CICES) [START_REF] Haines-Young | The links between biod iversity, ecosystem services and human well-being[END_REF] show a common international and cross-disciplinary acad emic effort to d efine and classify ecosystem services in order to facilitate their assessment. However, there is a break between the MEA-TEEB and the CICES concerning both the d efinition and the classification of ecosystem services. The MEA and TEEB ad opted wide und erstand ings of ecosystem services and both includ ed supporting services as a category of ecosystem services. However, these classifications can lead to some confusion when the economic contributions of ecosystem services are assessed . That is why the CICES followed the more restrictive d efinition of ecosystem service proposed by [START_REF] Boyd | What are ecosystem services? The need for standardized environmental accounting units[END_REF] with the aim to avoid d ouble counting of some categories. In this stud y we opted to includ e the supporting services, since our aim at this step was not to make an economic valuation but to gather the maximum available knowled ge regard ing the services provid ed by coralligenous habitats and to test the application of the concept of ecosystem service with experts at a local scale.

As underlined by [START_REF] Levrel | How to overcome the implementation gap in ecosystem services? A user-friend ly and inclusive tool for improved urban management[END_REF], "one of the main issues with the concept of ecosystem services is its absence on the ground in concrete operational d ecision-making". The goal of the present study is to apply the ecosystem service concept to a complex and relatively unknown ecosystem at an operational management scale. We broke this down into two objectives. First we aimed to identify ecosystem services provid ed by coralligenous habitats at small-scale. Second we aimed to identify criteria of existence and importance of ecosystem services and used them to rank the services. The study area consisted of two French marine sites contrasted in anthropogenic pressure in order to cover the two situations d espite the small-scale. The method ology was based on the opinions of 43 experts collected through three complementary survey methods: semi-directive interviews, an online questionnaire and workshops.

Materials and methods

Study area

The stud y area consisted of two French marine sites that we called "Marseille site" and "Port-Cros site". The d istance between these two sites is about 100 km (about 62 miles). The so-called Marseille site encompassed the coastal and marine zone from Martigues to La Ciotat, includ ing the Côte Bleue marine park (existing since 1983) and the Calanques national park (existing since 2012). The Port-Cros site includ ed the marine core and ad jacent area of Port-Cros national park (existing since 1963). Fig. 2 d isplays the two sites, and ind icates the location of coralligenous habitats and the marine protected areas. Marseille and Port-Cros present similarities in terms of coralligenous habitats but d ifferences in terms of anthropogenic pressure, economic and social contexts. Both sites harbor abundant coralligenous habitats, consid ered as some of the most beautiful across the French Mediterranean coast [START_REF] Tribot | Taxonomic and functional d iversity increase the aesthetic value of coralligenous reefs[END_REF]. Those coralligenous habitats are mostly on vertical cliffs, between d epths of 20 m and 50 m [START_REF] Hong | Étud e faunistique d 'un fond d e concrétionnement d e type coralligène soumis à un grad ient d e pollution en Méd iterranée nord -occid entale (Golfe d e Fos)[END_REF][START_REF] Laborel | Le concretionnement algal "coralligène" et son importance géomorphologique en Méd iterranée[END_REF], and their communities are often d ominated by gorgonians. These particularities make them d ifferent from coralligenous habitats of other locations in the Med iterranean Sea. There is an abund ance of historical ecological d ata in those areas [START_REF] Marion | Esquisse d'une topographie zoologique du Golfe de Marseille[END_REF][START_REF] Pérès | Notes sur les fonds coralligènes de la région de Marseille[END_REF][START_REF] Hong | Étud e faunistique d 'un fond d e concrétionnement d e type coralligène soumis à un grad ient d e pollution en Méd iterranée nord -occid entale (Golfe d e Fos)[END_REF][START_REF] Hong | Contribution à l'étud e des peuplements d 'un fond coralligène d ans la région marseillaise en Méd iterranée Nord -Occid entale[END_REF][START_REF] Witkowski | L'ichtyofaune associée aux fonds coralligènes d e l'île d e Port-Cros (Var, Provence, France)assemblages et rythmes d iurnes[END_REF]. Marseille is the second largest city in France with about 869,800 inhabitants (INSEE, 2015). This city is located in a "département"1 of almost 2 million inhabitants and is the first French Mediterranean harbor. Next to Port-Cros site, the main city, Hyères, counts about 57,500 inhabitants in a "département" of about 1 million inhabitants. Thus, human impacts on natural environment are likely to be much higher in the Marseille site than in the Port-Cros site. Both sites includ e islands where most of the coralligenous habitats are found . Marseille islands are closer to the mainland than Port-Cros island s, thus the former are more accessible than the latter. In both sites diving activities are intense (but more regulated in Port-Cros), while professional artisanal fishing activities are d eclining. A recent stud y id entified d ifferences between the bay of Marseille and the Port Cros national park area (including coralligenous habitats [START_REF] Cahill | A multispecies approach reveals hot spots and cold spots of diversity and connectivity in invertebrate species with contrasting d ispersal mod es[END_REF]): Port-Cros coralligenous d isplayed particularly high intraspecific genetic d iversity (shared with Corsica, another insular and preserved location), which was suggested to be a consequence of a lower human pressure (e.g. much lower population d ensity) in these areas. Various ind icators of impacts also confirm a higher pristineness of the Port-Cros site, although there is consid erable variation in environment quality within the Bay of Marseille (and between it and the Calanques) [START_REF] Deter | A preliminary study toward an ind ex based on coralligenous assemblages for the ecological status assessment of Med iterranean French coastal waters[END_REF][START_REF] Sartoretto | An integrated method to evaluate and monitor the conservation state of coralligenous habitats: the INDEX-COR approach[END_REF].

Preliminary list of ecosystem services based on literature

The literature search was performed before the survey in ord er to identify potential ecosystem services provided by coralligenous habitats as a list to be completed and valid ated thanks to expert's unpublished knowledge. The literature was investigated in 2015 to identify services to submit to experts d uring the survey.

The literature search includ ed peer-reviewed articles available on Google Scholar, reports available in the studied protected marine areas and scientific books. The only d ocument that explicitly mentioned the ecosystem services provid ed by coralligenous habitats was [START_REF] Mangos | Valeur économique d es bénéfices soutenables provenant des écosystèmes marins méd iterranéens[END_REF]. This was a large-scale project that estimated the economic benefits of five marine ecosystems at the Med iterranean Basin scale. This work contrasted with our objective, which was to valid ate the existence of ecosystem services provid ed by a specific ecosystem at a small-scale. Some ecosystem services provid ed by coralligenous habitats can be id entified thanks to other stud ies which d id not mention explicitly ecosystem services, or d id mention ecosystem services that are provid ed by other ecosystems sharing with coralligenous habitats the relevant characteristics implied in the supply of the service mentioned. A preliminary list of ecosystem services potentially provided by coralligenous habitats was assembled from the literature independently from location. The preliminary list was completed with the input from experts' interviews (the first step of our survey). Then, a list including 15 ecosystem services was submitted to the experts via the online questionnaire (the second step of our survey) and/or during workshops (the third step of our survey).

Expert survey

Since the literature search revealed few stud ies clearly describing the ecosystem services provided by coralligenous habitats, a survey was implemented based on an expert panel of 43 ind ivid uals. The survey was carried out between October 2015 and June 2016 and included three sequential steps: at first, ind ivid ual face-to-face interviews to id entify and d escribe the ecosystem services, then, an online questionnaire to start the validation process and at last, workshops to finally validate a list of services and their importance. This three-step approach enabled us to gather ind ivid ual and shared points of view, and to hand le open and closed answers in a funnel-shaped way. Our survey protocol was inspired by the method s Delphi and focus groups. [START_REF] Skulmoski | The Delphi method for grad uate research[END_REF] described Delphi as a method that uses several rounds of survey among a panel of experts to reach a consensus, and is "a flexible research technique well suited when there is incomplete knowled ge about a phenomenon". All the 43 experts answered the online questionnaire, but only 8 of them were interviewed face-to-face and 16 participated to the workshops. Among the 8 experts interviewed , 2 (researchers) participated also to the workshops; the others (mainly fishermen) were not available for the workshops. The d etailed process of the expert survey is d isplayed in Fig. 3.

Experts selection

Following [START_REF] Krueger | The role of expert opinion in environmental mod elling[END_REF] we consid ered as an expert any person having "relevant and extensive knowled ge or in-d epth experience in relation to" coralligenous habitats and /or their use or management in one or both of the sites stud ied . 43 ind ivid uals from the stud y sites formed the expert pool: Marseille (25) and Port-Cros (18). They were selected for their profiles: researchers (ecologist or economist), managers of marine protected areas, professional artisanal fishermen, representatives of d iving activities (fed eration or d iving structures), professional d ivers, and people monitoring coralligenous habitats. Their d istribution among profiles and sites is presented in Table 1. Our panel represented most of the available expertise since the

Interviews

8 out of 43 experts were interviewed . Only few interviews were carried out because the total number of experts in the two localities is small. The experts selected for the interviews had very specific knowled ge on at least one of the following subjects: artisanal fishery (including red coral), diving, coralligenous habitats general ecology or characteristics of a taxonomic group (algae, sponges, fishes, red coral). The fishermen2 selected represented the d iversity of fishing practices, targeted species and sites. Interviews were carried out to obtain some d eep knowled ge that could not be gathered through online questionnaire or workshop. The interviews helped to complete the preliminary list of ecosystem services and to confirm our findings from the literature. Thus it was a preliminary step to prepare the online questionnaire and workshops. Interviews were conducted individually faceto-face in a semi-d irective way, lasting between 40 and 90 min, and were record ed and transcribed . The interviews were cond ucted as follow. In the first part the expert had to presence his own experience about coralligenous habitats and his activity (fishing, diving, research). In the second part of the interview the expert had to present his definitions of coralligenous habitats and ecosystem services and compare them with definitions given by the interviewer. Thereafter each expert was asked about the list of services provid ed by coralligenous habitat and to d escribe the most relevant services accord ing to its experience.

Online questionnaire

The questionnaire was performed online, using the software © LimeSurvey (version 2.06+ Build 150731). It contained 25 questions grouped in 6 parts: vocabulary issues, list of services, importance of services, pressures, management, and respond ents' profiles. The entire questionnaire is available as supplementary material. The online questionnaire was fully completed by the 43 experts who each answered about one of the sites. Respond ents took 30 min on average to answer the questions. In this paper we focus on the list of services and their importance.3 Thus we were interested in two main issues: first the valid ation of ecosystem services and second the importance and ranking of ecosystem services.

Rankings in order to validate the existence of ecosystem services.

The first issue was treated as follows: the list of 15 ecosystem services was provid ed to the expert. After each item, the expert is given the following question "Is this an ecosystem service provid ed by coralligenous habitats?" A Likert-type scale with a set of five answers was offered to each respondent: Yes I'm sure/Yes I think/I d on't know/No I d on't think so/No I'm sure it's not. To aggregate individual answers, the following process was implemented: we ranked

Table 1

Profiles of the 18 experts of Port-Cros site (PC) and 25 experts of Marseille site (MRS).

An expert can correspond to several profiles. the services accord ing to the frequency of answers across the 43 experts. Three d ifferent rankings4 among mod alities were considered: A) a ranking that gave priority to positive answers, B) a ranking that gave priority to negative answers, C) a ranking that gave priority to the d egree of certainty of answer. We ranked the 15 services accord ing to each previous ranking of mod alities. This allowed us to consider ranking consistency and to d etermine ex post frequency threshold s in ord er to classify the 15 ecosystem services. Threshold s were chosen in such a way that consistency in each category of services be maximized.

Rankings of the importance of ecosystem services.

To explore this issue, we used a set of four questions in the online questionnaire. First, experts were asked to state a level of importance of services provided in the studied site through the following question "According to you, how important is this service [in the studied site]?" A Likert-type scale with a set of four items ("strong importance", "med ium importance", "low importance" and "Do not know") was used . To confirm this judgment, thereafter experts had to select at least six services that they considered as main services provided by coralligenous habitats in the studied site. Then, experts had to rank those six services from 1 to 6: "1" for the most important, "2" for the subsequent service, and so on. When experts didn't select a service in the first part of the questionnaire, this service was not proposed to them for the following questions and thus the item "d o not know" was automatically generated and includ ed in the analysis. We got three d ifferent rankings about importance that allowed us to explore the consistency of ranking. Lastly, a question enabled exploration of the specific meaning of "importance" to each expert. Each of them was asked to d efine his/her own criterion of importance, the one s/he used to rank services. We proposed four definitions of importance as follows: a service is more important if… "it is at the basis of other services"/"it is perceived by more people"/"it impacts more people"/"it is more threatened".

Further analyses.

In ord er to cross-reference the results from questions about existence and importance we transformed the mod alities of acceptance into scores from 1 to 5, and the modalities of importance from 1 to 4. For each ecosystem services, the mean and med ian scores of the 43 answers were then calculated to analyse correlation between both items "existence" and "importance". A comparison of the results of d ifferent groups of experts representing the two geographical groups (Marseille and Port-Cros) was undertaken using the Fisher exact test.

Workshops

We conducted a one-day workshop in each study site: one took place on the 3rd of December 2015 includ ing 11 experts from the Marseille site; a second took place on the 16th of June 2016 with 6 experts from the Port-Cros site. The main objective of the workshops was to help the interpretation of the answers obtained through the online questionnaire, and to investigate disagreements. Experts were encouraged to d ebate, especially about the controversial points, and to seek consensual answers. The workshop focused on the valid ation of the ecosystem services and their importance: we presented a synthesis of the answers to the online questionnaire; we then animate a debate on nonconsensual answers between experts and end ed by a vote by show of hand.

Results

Preliminary list of 15 ecosystem services selected from literature search and experts interviews

The 15 ecosystem services id entified through the literature review and the interviews of experts are presented in the Table 2.

Validation and importance of ecosystem services according to experts'

answers to the online questionnaire

Validation of the existence of ecosystem services

The first result of the online questionnaire was the ranking of the 15 ecosystem services accord ing to the valid ation status given by the 43 experts. Analysis of the answers to the online questionnaire indicated four status categories of services with respect to the experts' opinion as follows. As observed in Fig. 4, the merging of services in categories is consistent regardless of the rankings of modalities chosen (see cases A, B, and C). For instance, in the category 1, labelled "unanimously valid ated services", were includ ed the services that gather only positive answers with more than 90% of experts being sure of their answer. In the category 2, labelled "services accepted by the majority", were includ ed services gathering more than 80% of positive answers, and not falling in the category 1. In the category 3, labelled "uncertain services", were included services not falling in the previous categories and gathering more than 20% of "I d on't know" answers and fewer than 10% of negative answers. In the category 4, labelled "services refuted by the majority", we put services gathering more negative and uncertain than positive answers.

Rankings of the importance of the ecosystem services

A second result from the online questionnaire was that the three method s of rankings converged to a unique ranking of importance of the 15 ecosystem services. It must be underlined that the 15 services of the preliminary list were selected at least by one expert to be a part of the most important service (see Fig. 6) and also to be of strong importance (see Fig. 5). The ranking of services mad e through the answers to different questions about importance of ecosystem services are presented in Figs. 5, 6 and Table 3. The services classified as most important were "biod iversity" and "habitat and refuge". The services "d iving", "research", "nursery", "inspiration" and "food" were classified as "services of strong importance". The services "red coral" and "carbon sequestration" were classified as services of med ium importance. Finally, the services "water filtration", "angling", "spearfishing", "aquarium" and "coast stabilization" were classified as services of low importance. Only the service "bio-ind icator" was subjected to non-concord ance between rankings method s: it was selected by 65% of the experts as part of the most important services, was stated of "strong importance" by 68% but then ranked in the last position when experts had to directly compare it to other services. Thus we d id n't attribute to it a consensual level of importance. As a general remark we highlight that experts considered supporting services as the most important, then provisioning and cultural services. Regulating services were placed after, but experts recognized a lack of knowledge concerning the input-output related to the ecological processes und erlying these services.

Correlation analyse

A third result is a positive relationship between valid ation and importance of ecosystem service, d isplayed in Fig. 7. This figure distinguishes services validated and very important ("biod iversity", "habitat and refuge", "bio-ind icator", "research", "d iving", "inspiration" and "nursery"), services validated and of medium importance ("food", "red coral", "angling" and "spearfishing"), services controversially validated and ranked ("carbon sequestration" and "aquarium"), and services not valid ated ("water filtration" and "coast stabilization").

Table 2

Preliminary list of 15 potential ecosystem services provid ed by coralligenous habitats, submitted to experts via the survey.

Definitions

Illustrating references Comments

Provisioning ecosystem services Food Wild stocks of species that professional fishermen fish, restaurants serve, people eat and savour. Species examples: Scorpaena scrofa, Palinurus elephas, Homarus gammarus. [START_REF] Harmelin | Ichthyofaune d es fond s rocheux d e Méd iterranée: structure d u peuplement d u coralligène d e l'île d e Port-Cros (Parc National)[END_REF][START_REF] Mangos | Valeur économique d es bénéfices soutenables provenant des écosystèmes marins méd iterranéens[END_REF][START_REF] Witkowski | L'ichtyofaune associée aux fonds coralligènes d e l'île d e Port-Cros (Var, Provence, France)assemblages et rythmes d iurnes[END_REF], INPN-MNHN 1170-14.

Red coral

Wild stocks of red corals that professional coral fishermen harvest, jewelers shape and sell, people enjoy as ornamental or jewelry. [START_REF] Liverino | Red Coral. Jewel of the Sea[END_REF][START_REF] Ascione | The art of coral: myth, history and manufacture from ancient times to the present[END_REF][START_REF] Santangelo | Red coral fishing trends in the western Med iterranean Sea d uring the period 1981-1991[END_REF][START_REF] Santangelo | Red coral: conservation and management of an over exploited Med iterranean species[END_REF][START_REF] Paolini | La fascination du corail[END_REF][START_REF] Ballesteros | Med iterranean coralligenous assemblages: a synthesis of present knowled ge[END_REF][START_REF] Tsounis | Red coral fishery at the Costa Brava (NW Mediterranean): case stud y of an overharvested precious coral[END_REF][START_REF] Mangos | Valeur économique d es bénéfices soutenables provenant des écosystèmes marins méd iterranéens[END_REF][START_REF] Allemand | Les coraux précieux. Fiche scientifique[END_REF] Targeted species are not only found in coralligenous habitats even it might be their preferable habitats. Fishing is usually done not directly on coralligenous habitats but at the ed ge. status "no answer" was generated automatically when an expert did not select the service in the question 3.

Fig. 6. Number of experts who selected each service at question 9 ("Select the most important services") of the questionnaire. Experts must select at least 6 services and could select only the services they previously selected as "existing" services in the previous question 3.

0 0 0 1 1 2 2 Aquarium 0 0 1 1 1 1 1 Bio-indicator 0 0 0 0 0 0 0

Criteria of importance of the ecosystem services

Finally the online questionnaire revealed that a criterion was widely shared by expert to rank the services: "a service is more important if it is at the basis of other services". It was selected by 70% of the experts. 40% of the experts also used the criteria "impacts more people" and "is more threatened ". Only 20% of them d eclared using the criteria "is perceived by more people" (see Fig. 8). These results confirm that experts placed the supporting services at the head of the ranking.

Little divergences observed between Marseille and Port-Cros

Among the fifteen ecosystem services, there are only little divergences between the experts' opinions of Marseille (25 experts) and Port-Cros (18 experts) concerning main issues "valid ation of services" and "importance of services" from the online questionnaire: "red coral", "water filtration", "nursery", "coast stabilization". The service "coast stabilization" was the one triggering the most divergent opinions: it was consid ered as not existing by 72% of the experts of Marseille but only by 16% of the experts of Port-Cros (Fischer exact p-value = 0.0051). The service "red coral" was considered existing and important by 96% experts of Marseille while only by 61% experts of Port-Cros (Fischer exact p-value = 0.0062). On the contrary the ecosystem service "water filtration" was considered important by 44% of the experts of Port-Cros but not by the experts of Marseille (Fischer exact p-value = 0.0007). Likewise, the service "nursery" was considered important by 83% of the experts of Port-Cros but by only 36% of the experts of Marseille (Fischer exact p-value = 0.0059).

Second step of validation and ranking of services: through the workshops

The workshops were the last step of our survey protocol and allowed us to get precise information about the ecosystem services available at each of the study sites according to the experts. During the workshops, the participants collectively revised some of their initial assessments and converged to a consensus. They finally stated that they could only refute the ecosystem service "coast stabilization". Their argument was that coralligenous habitats able to play this role are the flat types, which were not represented in our study sites, and usually are at depths too d eep to impact the coastline. They weren't confid ent enough to categorically refute the other controversial services "water filtration" and "carbon sequestration". The pros argument given by experts for the "carbon sequestration" service is the presence of many calcareous species (especially red algae) in coralligenous habitats, which used carbon to grow and then fossilize. The pros argument given by experts for the service "water filtration" is the presence of many filterers such as sponges. The cons argument for both services is the effective regulating impact of coralligenous habitats at the global Med iterranean scale and the capacity of calcareous species to capture carbon and the time of sequestration is unknown. Thus, these two services remain controversial until the knowled ge gaps are filled by focused studies. Experts stated that the services "d iving", "food ", "inspiration", "research" and "bio-ind icator" were actually provid ed by coralligenous habitats in the stud y sites, and surely provid ed by most of the coralligenous habitats all over the Mediterranean Sea. All experts validated "Biod iversity" and "habitat and refuge" as services provid ed by coralligenous habitats of any type and any site. The services "red coral", "aquarium", "spearfishing", and "angling" were valid ated with less confidence due to their non-estimated but probably low benefits or low number of beneficiaries. The service "red coral" was provided in one of the stud y sites. It should be provid ed in all areas where resources is available and where harvesting is possible. However it may be anecdotal in term of population impacted by the benefits. The same may be true for the service "aquarium" which is certainly even more anecdotal. The services "spearfishing" and "angling" are recreational activities practiced in the studied sites, but the proportion of activity practiced on coralligenous habitats was unknown, it could be either low or high, experts were not able to estimate any tendency. The validation statuses of the 15 ecosystem services presented in the preliminary list are sum up in Table 4. Fig. 7. Mean of the scores given by the 43 experts concerning the valid ation of ecosystem services (answers to the question 3:"Is it an ecosystem service provid ed by coralligenous habitats?") and the importance of the ecosystem services (answers to the question 5 "How important is this ecosystem service?") obtained through the online questionnaire.

Discussion

We found that even if less known and quite different, coralligenous habitats provide services quite similar to those provided by coral reefs: food , ornamental and aquarium resources, habitat maintenance, recreation and tourism opportunities [START_REF] Elliff | Ecosystem services provid ed by coral reefs in a Southwestern Atlantic Archipelago[END_REF][START_REF] Moberg | Ecological goods and services of coral reef ecosystems[END_REF]. But coralligenous services are quite d ifferent from those provided by Posidonia oceanica seagrass (considered as one of the most important Med iterranean marine ecosystem together with coralligenous habitats), that are primarily "nursery", "food " and "carbon sequestration" [START_REF] Campagne | The seagrass Posidonia oceanica: ecosystem services id entification and economic evaluation of good s and benefits[END_REF]. The stud y from Paoli et al. [2016a] identified some services provided by coralligenous habitats of Italian sites that are similar to this stud y: such as "natural medicine availability" (research), "ornaments" (red coral), "landscapes" (diving), "habitat", "storage of energy and matter" (food ), and they ad d ed the role of sed iment and nutrients retention by the planar structure of gorgonians. Paoli et al. [2016a] reported also the controversial status of "waste treatment" (water filtration), "breathable air" (carbon sequestration) and "nursery".

This stud y focused on a small-scale approach to id entify and rank ecosystem services. We carefully treated the issues of existence and importance separately. However, the results showed an evident positive correlation of the answers to the two questions. This would have been an unexpected result for a large-scale study, but for a small-scale study it confirms that a minimum level of "importance" (in term of global benefits on people) confirms the existence of a service. In most largescale stud ies this minimum "level" of importance is not investigated. For the application purpose, our case stud y highlights the need for a frame of reference to id entify ecosystem services. In this stud y we constrained the geographical scale (the two stud ied sites), but experts triggered on a minimum threshold of population impacted and the perceived value of the benefits to id entify ecosystem services. For example the services "red coral" or "aquarium" impact very few local people but possibly have a high value (specially a high economic or/ and cultural value) for these people and thus reach a minimum threshold to be consid ered as a service. The existence of a service "carbon sequestration" might depend on the temporal and geographical frame: the impact might be significant only at large scale and long term. These reflections lead us to think that, for the id entification of an ecosystem service, a frame of reference should be calibrated in term of geographical scale, size of population impacted , benefits value and temporal scale. For each of these criteria, a minimum threshold should d etermine the existence of the service and then its importance can be estimated for each criterion based on the d istance from this minimum threshold. The category of well-being impacted should also be specified in ord er to balance importance: d oes the service satisfy essential basic needs for survival or those needs related to supplementary well-being? For example, the existence of favourable d iving spots represents immediate benefits, is essential to the local professionals of diving, and is profitable to d ivers from local to d istant d ivers. Depend ing on the population framed , the service would be consid ered and valuated differently. To this perspective, only stud ies at small-scales such as the one we implemented can provid e meaningful input to policies aimed at managing the local environmental. Apart from the evid ent positive correlation between existence and importance, we observed slight variations in this correlation. For example the experts mostly accepted the services "spearfishing", "angling" and "red coral" but considered them of med ium-low importance. The three activity-related services were consid ered not being practiced by a large number of people and were not supportive of other services. These results are consistent with the criterion of importance highlighted by experts. We are aware that the criteria of importance of the services must be considered cautiously, and that it is incorrect to balance the importance of a service that is essential for the living of a small part of the population (such as the artisanal fishermen or professional of d iving structure), with the importance of services which affect the recreational activities, and with the importance of a service which helps for the maintenance of a healthy environment at an unknown level.

The criterion of importance mostly used by experts was "a service is more important if it is at the basis of other services". This suggests that the concerned services are ind irect. This criterion is not compatible with the definition of ecosystem services refers to the Common International Classification of Ecosystem Services (CICES) [START_REF] Haines-Young | Common International Classification of Ecosystem Services (CICES) V5.1 and Guid ance on the Application of the Revised Structure[END_REF] program and with an economic valuation perspective. The large number of environmental experts compared to economists may explain this result, since the questionnaire showed that environmental experts mostly didn't take in account the double-counting issue while economists were really aware of it. The two other criteria to decide of the "importance" of an ecosystem service that are mostly cited by experts were "is perceived by more people" and "impacts more people". These criteria justify the elimination of very anecdotal services. Indeed, the experts confirmed that the service "aquarium" is anecd otal if we consid er d irect beneficiaries only, that is one fisherman of living organisms in the stud y sites, but may impact many visitors in public aquariums. Experts did not consider the threatened status as a criterion of importance of a service. The most threatened services, certainly "red coral" and "food ", were not ranked based on this criterion, but on the number of beneficiaries.

Since services provided by an ecosystem to human beings crucially d epend on the natural and socio-economical environments in which they are provided, the objective of having two locations was to validate ecosystem services provid ed by coralligenous habitats in d ifferent natural and socio-economical environments. However we consid ered the few significant d ifferences in the services bund le of two nearby sites d ifferently exposed to anthropogenic pressures and for further studies we expect that the comparison between very distant and heterogeneous sites would highlight greater d ifferences in the service bund les. The origin of the d ifferences observed between the two stud ied sites may come from the "supply sid e" (morphotypes, ecological communities, accessibility of the coralligenous habitats) or from the "d emand side" (peoples' perceptions, cultural habits, size of the local human popula-tion), or may arise from contextual constraints as local specific reg-ulations. It can be note that an ecosystem service exists if it is both supplied and d emand ed . Our results showed that opinions of experts from both sites were d ivergent for the service "red coral", "nursery", "water filtration" and "coast stabilization". The service "red coral" was more accepted and consid ered more important by the experts of Marseille than by those of Port-Cros. This result is consistent with the fact that there are four active harvesters of red coral in Marseille area, ences should be und ertaken to get a robust answer at this small-scale where dataset usually does not exist. Concerning the services "nursery", "water filtration" and "coast stabilization", there are classified as regulating services with large-scale impact. Thus, we consid er that the average answer (average of the two sites) is more reliable than sitespecific answers because, at the current scientific knowled ge, there is no stud y to support the d ifference in regulating services provid ed by two nearby sites. Finally, this stud y was original by the method ology used in the sense that it simplified the process of collecting responses inspired by the Delphi method ology, but provid ed information that was relevant and accurate. In the absence of scientific evidence, experts' experience was the best knowled ge proxi available even if the process could not avoid a part of subjectivity and if consensus d oes not guarantee scientific veracity. This stud y relied on 43 experts for two North-western Mediterranean sites. This sample is at least as robust than those in other stud ies treating similar issues at a large scale: for example [START_REF] Beaumont | Id entification, definition and quantification of good s and services provid ed by marine biod iversity: implications for the ecosystem approach[END_REF] consulted 21 experts about ecosystem services provided by marine ecosystems globally, Nord lund et al. [2016] consulted 91 experts to deal with the different types of seagrasses in the whole world.

C onclusions and perspectives

This study presented a first attempt to list the services provided by coralligenous habitats, valid ated by a panel of experts, plus a complementary list of potential services that should not be eliminated before further researches. Carrying out surveys at local scales allowed us to actually d ecid e on the existence and importance of services in the stud y area. Accord ing to the expert panel, we can consid er that the major services id entified in this stud y (food , red coral, d iving site, d iscovery potential) are very likely to be found in other coralligenous sites. Moreover, our survey protocol could be easily extend ed to other Med iterranean sites to confirm it.

Provisioning and cultural services are quite evid ent and mostly accepted. But there is a significant lack of knowledge about the regulation of some ecosystem services, the importance of which is very uncertain. To start filling the knowledge gap regarding these services, which may be essential if effective, scientific stud ies in two areas are clearly needed: 1) the capacity of calcareous species to sequester carbon and 2) to quantify the provisioning and cultural services, specific d ata should be collected about the recreational activities and the resources caught and harvested in coralligenous habitats. Moreover, apart from its status of function or service, the "nursery" role, as d efined by [START_REF] Beck | The role of nearshore ecosystems as fish and shellfish nurseries[END_REF], of coralligenous habitats could be verified only after the observations of juveniles living exclusively in coralligenous habitats and moving to another habitat for their adult phase. Juveniles are not easy to observe and id entify, thus their observation would need a specific experiment that have not been performed on coralligenous habitats to date.

The application of the ecosystem service concept to coralligenous habitats at a very local scale showed that the current wid espread definition of ecosystem services used for the CICES must be adapted to be applied at small-scale and that the id entification of service should always be based on a referential frame (geography, time, population, benefits scales) to allow concrete operational decision-making. Indeed, our stud y highlighted few d ifferences in the supplying and perception of services between two close-by sites with relatively similar coralligenous habitats but d ifferent socio-economic context. Thus we expect that further stud ies mad e in more d istant sites involving differences in supply and d emand would highlight further site-related d ifferences in the bund le of services.

However, even if current knowledge on coralligenous habitats does not allow quantifying precisely the benefits precisely, it is sufficient to use non-market valuation method s for the valuation of most of the services provid ed . Thus, an attempt of valuation was mad e after this stud y in the same stud y area Multiple choices (4) : "Irreplaceable : only provided by coralligenous habitats", "Partly replaceable : other habitats can provide the same service but at lower quality or quantity", "Totally replaceable : other habitats provide this service at the same level at least", "I don't know" (Not compulsory) Multiple choices (5) : "A service is more important especially as it forms the basis of others", "A service is more important especially as it is perceived by a large population", "A service is more important especially as it impacts a large population", "A service is more important especially as it is threatened", "other reason". (Not compulsory)

25

Fig. 1 .

 1 Fig. 1. Typical coralligenous habitats of Marseille. Photocred its: Fréd éric Zuberer (OSU Pythéas)

Fig. 2 .

 2 Fig. 2. Map of the two stud ied areas. Top: the Marseille site (includ ing the bay of Marseille, the marine park of the Côte Bleue and the national park of the Calanques). Down: the Port-Cros site (including the national park of Port-Cros and its marine adjacent area in the bay of Hyères). The park core is the most protected area and regulated by specific regulations.

Fig. 3 .

 3 Fig. 3. Protocol applied for the survey.

Fig. 4 .

 4 Fig. 4. Answers to the question 3 "Is this an ecosystem service provided by coralligenous habitats?" of the online questionnaire. The three cases (A, B, C) corresponds to three possibilities of ranking the modalities.

Fig. 5 .

 5 Fig.5. Experts' answers to the question 5 of the online questionnaire "Accord ing to you, how important is this service [in the stud ied site]?". The status "no answer" was generated automatically when an expert did not select the service in the question 3.

Fig. 8 .

 8 Fig. 8. Percentage of experts who declared that they used the criterion in their ranking of the importance of services. Legend : green = yes, red = no. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

  [Thierry d e Ville d 'Avray, 2018]. The online questionnaire : parts 1, 2, 3 and 6 n1 : Number of participants answering the particular question about Marseille area n2 : Number of participants answering the particular question about Port-Cros area N : Sum of participants answering the particular question to you, are the following ecosystem services provided by coralligenous habitats ? -> ES-list Multiple choices (5):"Yes I'm sure", "Yes I think", "I don't know", "No, I don't think so", "No, I'm sure"

  the services that you selected in order of importance ("1" for the most important) Multiple numerical input (controlled) (Not compulsory) 11) On which criterion did you base your ranking ?

Table 3

 3 Answers

	Biodiversity	35	37	41	42	42	42	42
	Habitat and refuge	5	26	32	35	35	35	35
	Nursery	2	7	16	19	22	23	23
	Food	0	7	11	19	24	28	28
	Diving site	1	4	10	13	16	22	22
	Research	1	3	6	9	13	18	20
	InspiraƟon	2	5	7	9	14	17	19
	Red coral	3	3	5	10	13	16	18
	Carbone sequestraƟon	2	6	7	8	11	12	12
	Water filtraƟon	2	3	0	7	8	8	8
	Coastline stabilizaƟon	1	2	2	2	4	4	4
	Angling site	0	1	2	2	3	4	4
	Spearfishing site

obtained from the question 10 of the online questionnaire: "Rank the services you have selected as most important". The table shows the cumulative numbers of positions 1 to 7 in the ind ivid ual ranking of experts. 1

1 + 2 1 to 3 1 to 4 1 to 5 1 to 6 1 to 7

Table 4

 4 Validation status of the 15 ecosystem services presented in the preliminary list accord ing to experts. Cros area. Then our survey protocol allows us to id entify this d ifference. Unfortunately, it d oes not allow us to explain this d ifference that can be d ue to a higher presence of exploitable red coral in Marseille, or d ue to local regulations or to the local socioeconomic context. Specific stud ies combining natural and social sci-

	Ecosystem service	Status
	Food	unanimously validated
	Diving spot	unanimously validated
	Research	unanimously validated
	InspiraƟon	unanimously validated
	Red coral	majoritarly validated
	Aquarium	majoritarly validated
	Angling spot	majoritarly validated
	Spearfishing spot	majoritarly validated
	Bio-indicator	majoritarly validated
	Biodiversity bequest	majoritarly validated
	Carbon sequestraƟon	controversial
	Water filtraƟon	controversial
	Coast stabilizaƟon	refuted
	Ecosystem funcƟon	Status
	Habitat and refuge	unanimously validated
	Nursery	controversial

Département: a French ad ministrative division.

In many cases their professional constraints prevented them from attending the workshops, but they were able to grant us in-situ interviews earlier in the survey procedure.

These are Parts 2 and 3 of the questionnaire.

This approach is in the spirit of a lexicographical ord er where consumer ranks bud d le of good s first accord ing to the quantity of good 1 whatever the quantities of the other goods; if two bundles have the same quantity of good 1, then the consumer compares quantities of good 2 and so on.

Acknowledgements

The work of Laure Thierry de Ville d'Avray was supported by a grant from Aix-Marseille University. This work was also supported by the ERA-NET (SeasEra) through CIGESMED Program (contract ANR-12-SEAS-0001-01) and by the Observatoire Hommes-Milieu (OHM) littoral Méditerranée through EVACOR and EVACOR2 programs. We thank the three marine parks (parc marin d e la Côte Bleue, parc national des Calanques, parc national de Port-Cros) for their logistic support and/or their help to identify the relevant local experts. We are very grateful to all the experts who shared their experience, and took the time to participate to the survey. We thank our colleagues who helped testing the questionnaire and in particular Thomas Schohn for his support to transcribe experts' speech.


Paul -Quentin

Elias Dphy

Nicolas Severac

Jean-Marc Luyssen

Jean-Pierre Tobeli

François Lambert

Reynald Bur

Aurélien Houard

Yves-Bernard André

Sylvain Albert

André Mysyrowicz

Ivan Doudet

Experimental Investigation of Linear Energy Deposition Using Femtosecond Laser Filamentation in a M=3 Supersonic Flow

The goal of this study is to demonstrate experimentally the "Laser Spike" concept in a supersonic flow. The "Laser Spike" uses a thin filament of plasma created by a femtosecond laser. This plasma filament formed by the TeraWatt laser pulse acts as a linear energy deposition upstream of the shock wave created by a blunt body. It has been found that the laser energy deposition induces a significant transient reduction of the drag. This reduction has been quantified using a drag balance. These experiments have been simulated numerically by modeling the plasma filament as a source term in the energy equation. It is shown that the plasma filament created by the ultra-short laser pulse forms a low-density heated core that interacts with the detached bow shock. This interaction produces an inflating recirculating bubble and leads to a transient reduction of the drag. These findings demonstrate the interest of ultrashort laser for flow control. Drag reduction, flow control and trajectory control are among the possible applications of the "laser spike" concept. Sonic boom alleviation could also be improved with this device.

I. Introduction

FF-body energy deposition in supersonic flows has been considered for many years, both experimentally and numerically, as a way to reduce the drag, alleviate the sonic boom or provide a steering mechanism. These considerations are particularly interesting for the development of more efficient supersonic and hypersonic vehicles system, since they may translate in longer cruising range, quieter operation and agile steering capabilities, respectively. Among the different methods to deposit the energy, microwave-induced breakdown [START_REF] Kolesnichenko | Basics in Beamed MW Energy Deposition for Flow / Flight Control[END_REF]- [START_REF] Kourtzanidis | Gas heating effects on the formation and propagation of a microwave streamer in air[END_REF] and laser-induced breakdown [START_REF] Yan | Laser energy deposition in quiescent air[END_REF]- [START_REF] Ghosh | Numerical simulation of the fluid dynamic effects of laser energy deposition in air[END_REF] have been extensively investigated as these processes enable a volume energy deposition through the focusing of electromagnetic energy. This latter method has been investigated in wind tunnel experiments, both in single pulse mode [START_REF] Adelgren | Control of Edney IV Interaction by Pulsed Laser Energy Deposition[END_REF], [START_REF] Yang | Single Pulse Laser Energy Deposition in Quiescent Air and Hypersonic Flows[END_REF] or in repetitive mode [START_REF] Sasoh | Supersonic Drag Reduction with Repetitive Laser Pulses Through a Blunt Body[END_REF], [START_REF] Pham | Impacts of Laser Energy Deposition on Flow Instability over Double-Cone Model[END_REF], on blunt body at supersonic speed. In both cases, significant modifications of the flowfield near detached shock wave were observed, coupled to a decrease of the drag. In these experiments, the energy deposition is localized in a volume close to the focal point of the laser. The deposition produces a spot of heated gas that subsequently interacts with the flowfield. Because of this localized nature, the interaction is transient and quasi-steady effects are sought using high repetition rate. Recently, as a way to improve the interaction time and reduce the repetition frequency, it has been suggested to use elongated and linear energy deposition [START_REF] Kremeyer | Computational Study of Shock Mitigation and Drag Reduction by Pulsed Energy Lines[END_REF]- [START_REF] Elias | Numerical Simulations on the Effect and Efficiency of Long Linear Energy Deposition Ahead of a Supersonic Blunt Body : Toward a Laser Spike[END_REF]. This idea is guided by the recent progress of femtosecond lasers, which are known to create thin ionized channels [START_REF] Adelgren | Control of Edney IV Interaction by Pulsed Laser Energy Deposition[END_REF][13] [START_REF] Couairon | Femtosecond filamentation in air[END_REF]. These channels can be used to guide an electric discharge and there is currently an ongoing research effort to couple more energy to this ionized channel using various mechanisms (more powerful nanosecond lasers [START_REF] Zhou | Revival of femtosecond laser plasma filaments in air by a nanosecond laser[END_REF], pulsed discharges or microwave excitation [START_REF] Michael | Subcritical microwave coupling to femtosecond and picosecond laser ionization for localized, multipoint ignition of methane/air mixtures[END_REF]). This O 2 concept has been dubbed the "Laser Spike", as it aims at perturbing and deflecting the upstream flow in a fashion that recalls aerospikes. Assuming an elongated or filamentary energy deposition in front of a supersonic body, numerical simulations have consistently shown a significant drag reduction. However, the actual method to obtain this elongated energy deposition has not been demonstrated experimentally, nor tested in a wind tunnel. A recent experimental work [START_REF] Point | Energy deposition from focused terawatt laser pulses in air undergoing multifilamentation[END_REF] has shown that significant gas heating can be obtained with femtosecond pulses only, provided that the pulse energy is large enough (energy range between 5 mJ and 250 mJ, pulse duration 50-200 fs). Temperature rise of several hundred Kelvin in air have been measured in a filamentary volume of length between 5 and 10 cm; in addition, the conversion efficiency of the pulse energy to heat can reach up to 60%. The linear energy deposition is in the range 0.1-4 mJ/cm, which is consistent with the requirements derived through parametric numerical simulations [START_REF] Elias | Numerical Simulations on the Effect and Efficiency of Long Linear Energy Deposition Ahead of a Supersonic Blunt Body : Toward a Laser Spike[END_REF]. In addition, this avoids the use of a secondary power source. This key result has opened the way to an experimental demonstration of linear energy deposition in a supersonic flow. The goal of this paper is to report on recent experiments aimed at demonstrating the « Laser Spike » concept. These experiments have been performed in Onera's R1Ch wind-tunnel, using the ENSTAMobile laser chain to produce ultrashort laser pulses in the TeraWatt range. The first part of the paper will describe the experimental setup used for the experiments. The second part of the paper will present the results of this experimental investigations. First, the energy deposition in quiescent air will be described. Second, the effect of the femtosecond laser energy deposition in the M=3 airflow will be presented. Schlieren characterization will show the details of the interaction and flow modifications. The drag and pressure measurements will also be presented and discussed. Finally, the experimental results will be compared to numerical simulations of the laser energy deposition.

II. Experimental setup A. Wind-tunnel

The tests are performed in Onera's R1Ch blow-down wind tunnel. This facility is an open-jet wind-tunnel and uses a M=3 axisymmetric nozzle with an exit section 310 mm in diameter. The maximum stagnation conditions are 15 bar and 380 K for the pressure and temperature, respectively, yielding a maximum mass flow rate of 60 kg/s. When operating at low stagnation pressure (~3-5 bar) the test chamber is pumped down to facilitate the priming of the nozzle. The supersonic jet is collected and slowed down in a supersonic diffuser and collected in a 500 m 3 vacuum sphere. For this project, the useful testing time varies between 15 s to 25 s, depending on the flow conditions. The range of stagnation pressure used in the experiments is: 2.5<Pi (bar)<8.5, for a stagnation temperature of ~305 K. The terawatt laser is placed next to the wind-tunnel test section. The laser beam is guided using 3 inches mirror in the test chamber. After the focusing lens, the beam is guided inside the test model holding sting and emerges from a diaphragm at the tip of the model.

B. Test model

The test model is a generic blunted body (HB2 shape [START_REF] Cérésuela | Mesure d'efforts et de pression sur la maquette balistique étalon HB2[END_REF]) which is a blunted cone (50° apex angle) with a base section of 60-mm in diameter (Figure 2). The nose of the model is hemispheric (radius 18 mm). As discussed above, the model is hollowed to provide an optical path for the laser beam. The tip of the model has a 3-mm diaphragm that provides the exit path for the focusing beam, as shown in Figure 3.

C. Laser system

The EnstaMobile system is a research TW laser which can deliver laser pulses with a duration as low as 50 fs. Its center wavelength is 800 nm and its maximum pulse energy measured at the tip of the test model is 150 mJ. The laser operates with a repetition frequency of 10 Hz. The laser master clock at 10 Hz defines the time reference for the synchronization of all the diagnostics used in the study. A fast photodiode is used to time the laser pulse.

The ultra-short laser pulse is focused with a lens and forms a plasma filament near the focal point of the lens [START_REF] Couairon | Femtosecond filamentation in air[END_REF], [START_REF] Durand | Kilometer range filamentation[END_REF]. The location of the filament core in front of the model can be displaced by moving the focusing lens L

D. Drag balance

The drag balance is composed of two parts :

 a weighted plate that holds the test model  a fixed plate mounted on the holding sting

The resulting drag force acting on the weighted plate is measured using three piezoelectric Kistler force sensors (type 9134B) mounted at 120° around the model axis. These sensors measure the axial component of the drag. The overall sensitivity of the balance is 50 N/V.

E. Diagnostics

In addition to the internal drag balance, the model is fitted with two high-bandwidth pressures gauges (PCB 132A31). Their quoted bandwith is 500 kHz. The flow topology is characterized using a pulsed Schlieren system in a Z-configuration [START_REF] Settles | Schlieren and Shadowgraph Techniques: Visualizing Phenomena in Transparent Media[END_REF]. A pulsed laser diode (Cavitar Smart, central wavelength 688 nm) is used as the point light source (using an emission slit). A narrow

Pressure sensors Z

bandpass optical filter centered on the laser wavelength is used on the detection side to filter out any stray light emitted by the ultra-short laser pulse, consisting mainly in a broadband emission. The nanosecond laser emits variable-length laser pulses. For this study, 100-ns long pulses are used. This duration is short enough to collect nearly instantaneous snapshots of the flow. The laser pulses are synchronized on the Enstamobile master clock, but with a controlled time delay. Owing the very good repeatability of the laser-flow interaction (see below), it is possible to probe different instants of the phenomena by gradually changing the delay.

III. Results

A. Energy deposition in quiescent air

First, the effect of the laser pulses in quiescent air is investigated. This has already been investigated in a laboratory setup [START_REF] Point | Energy deposition from focused terawatt laser pulses in air undergoing multifilamentation[END_REF] using a Mach-Zender interferometer, in particular to measure the number density variations in the gas.

Here, the goal is to check the location of the laser-induced quasi-linear energy deposition in front of the test model. The cumulated emission of several tens of laser pulses is shown in Figure 4. The Schlieren images in Figure 5 show that the laser-induced plasma filament initially forms a central heated core (length ~60 mm). The plasma filament formed by the laser heats a column of gas. Compared to the characteristic timescale of the flow, this heating is nearly instantaneous. This causes a significant pressure rise in the heated core. The high-pressure relaxes in a quasi-cylindrical blast wave that expands radially (T= 8.8 µs and later). The heated channel relaxation is then driven by thermal conduction and natural convection, but with a much longer timescale (~ a few ms). The maximum diameter of the heated core is 5 mm, which is consistent with previous findings in a laboratory experiment [START_REF] Point | Energy deposition from focused terawatt laser pulses in air undergoing multifilamentation[END_REF].

B. Energy deposition in a supersonic flow

When the test model is in the supersonic airflow, the activation of the ultra-short laser pulses induces a significant perturbation of the flow. First, the drag balance signal shows that the laser-induced energy deposition cause significant fluctuations in the drag, as shown in Figure 6. The time evolution of the balance signal is typical of the results obtained for a variety of experimental conditions. First, one notices a slight increase of the drag, between 100 µs and 170 µs after the laser pulse. Then, the drag signal decreases sharply and then evolves roughly as a damped oscillator for 1-2 milliseconds. This behavior is entirely correlated to the laser activation, as shown in Figure 7.

mm

The interaction of the heated channel created by the ultra-short laser pulse with the supersonic airflow is investigated for the test run 1632, typical of the results obtained in this study. The flow conditions for this run are given in Table 1. The laser operating conditions are given in Table 2. The laser pulse is generated every 100 ms (10 Hz repetition rate). During the whole test run, the signals from the drag balance and pressure sensors are recorded using a fast data acquisition system (2.5 Msamples / s). The Schlieren setup is used to recover the time evolution of the interaction. By changing progressively the time delay between the femtosecond pulse and the nanosecond laser pulse used to probe the flow, it is possible to recover the time evolution of the interaction with a good temporal resolution. This is possible because of the very good shot-toshop repeatability of the phenomenon, as shown in Figure 8. The drag signal is very repeatable. The dispersion is greater for the unsteady pressure probes. This is caused by the turbulent pressure fluctuation during the interaction. The repeatability is further checked by set a fixed delay for the Schlieren laser pulse and checking that the flow features remains nearly unchanged.

Figure 9 and Figure 10 present the different phases of the interaction. First the formation of the heated channel and the cylindrical blast wave is observed (T0+0.3 µs to T0+8.9µs), as in the tests in quiescent air (Figure 5). These features are also revealed in the zoomed viewed in Figure 11. The heated core is convected downstream and interacts with the detached bow shock. Because of the high temperature in the heated core, the flow becomes locally subsonic and enables the formation of a recirculating bubble expanding upstream from T0+4.6 µs to T0+35.7 µs in Figure 9 and Figure 11. Then the recirculating bubble expands radially between T0+60.5 µs and T0+112.1 µs (Figure 9 et Figure 10), and is convected downstream by the flow. One should note that, during this phase, the recirculating flow is probably turbulent. In addition, during the convection downstream of the bubble, the flow becomes unstable and no longer axisymmetric. The flow recovers its initial configuration approximately 250 µs after the laser energy deposition.

IV. Simulation

The previous works on line energy deposition were only theoretical due to the lack of experimental data. The observations reported above a first set of data points needed to perform a representative simulation.

A. Simulation details

The numerical simulation of the laser energy deposition is done for the conditions of the test run 1632 whose conditions are given in Table 1. The ONERA compressible flow solver CEDRE [START_REF] Refloch | CEDRE software[END_REF]is used for this purpose. Since most of the interaction shows a cylindrical symmetry, 2D axisymmetric simulation are considered. Despite the high Reynolds number of the flow, the flow is supposed to be laminar. The mesh is refined near the wall and at the location of the energy deposition such as the first cell thickness corresponds to a dimensionless wall distance 𝑦 + ∼ 2 -4.

Figure 12 shows the computational domain used for this simulation. The mesh near the wall and at the location of the energy deposition has refined structured patches to improve the spatial resolution. The flow solver is second order accurate in space and time, using an implicit Runge-Kutta time integration. The eulerian fluxes are computed using a hybrid HLLC scheme [28].

Here, 𝜎 0 is the source term amplitude in W.m -3 , 𝑓(𝑟) is the radial modulation, 𝑔(𝑥) is the axial modulation and 𝐻 𝑇 (𝑡) is a gate function of duration T. Note that the actual duration of the laser energy deposition is around T~ 100-200 fs, while for the simulation the energy deposition duration it T=50 ns. This difference is needed to avoid unpractical timesteps for the simulation. For the given level of energy, we check that the simulated temperature rise The amount of energy 𝐸 0 deposited in the flow is given by a spatial and temporal integration of the source term:

𝐸 0 = 𝜎 0 𝑇 ∫ 2𝜋𝑟𝑓(𝑟)𝑑𝑟 ∞ 0 ∫ 𝑔(𝑥)𝑑𝑥 ∞ -∞ (2)
For this study, the spatial modulations are Gaussian functions, parametrized by their full width at half maximum (FWHM) :

𝑓(𝑟) = exp (- 𝑟 2 (𝐹𝑊𝐻𝑀 𝑟 ) 2 4 𝑙𝑛(2)) 𝑔(𝑥) = exp (- 𝑥 2 (𝐹𝑊𝐻𝑀 𝑥 ) 2 4 𝑙𝑛(2)) ∫ 2𝜋𝑟𝑓(𝑟)𝑑𝑟 ∞ 0 = 𝜋 4 ln(2) 𝐹𝑊𝐻𝑀 𝑟 2 ∫ 𝑔(𝑥)𝑑𝑥 ∞ -∞ = √𝜋 𝐹𝑊𝐻𝑀 𝑥 2√ln(2)
With these assumptions the source term amplitude is :

𝜎 0 = 𝐸 0 𝑇𝑉 (3)
In the above expression, 𝑉 = ( 𝜋 4ln

) 3 2 𝐹𝑊𝐻𝑀 𝑟 2 𝐹𝑊𝐻𝑀 𝑥 is the characteristic volume of the energy deposition.

B. Parametric analysis

A parametric analysis is performed by changing the energy 𝐸 0 , the width and the length of the energy deposition.

For each case a comparison is made with the experimental results, as shown in Figure 13. The computed drag is also compared with the drag signal measured experimentally (not shown here). The best agreement is obtained for two sets of parameters, giving a range of deposited energy between 10 mJ and 15 mJ and a full width at half maximum of 20 mm along the axis, and 0.6 mm along the radius.

C. Analysis of the interaction

The good agreement obtained between the simulations and experiments is used to obtain a deeper understanding of the laser-flow interaction. The following figures show the evolution several field quantities during the interaction. The low-density heated core formed by the laser energy deposition appears clearly in Figure 14. The heated core interacts with the bow shock and form a detached recirculating bubble that extends upstream along the path of the convected heated core, as shown in Figure 15. This upstream propagation ceases when there is no more heated gas to feed the bubble. At this instant the drag force reach its maximum (approximately 10% increase). The recirculating bubble extends radially (Figure 17) and is subsequently convected downstream by the incoming flow. The pressure on the model nose decreases significantly (Figure 18), which leads to a 50% drop in the total drag on the body. Finally, the flow recovers its initial configuration ((Figure 19). The pressure on the model stagnation point recovers its initial value, and so does the drag.

V. Conclusion

In this study, the significant effect of femtosecond filaments on a supersonic flow has been experimentally demonstrated. It has been found that the low-energy laser spike (~10 mJ deposited) induces a significant transient reduction of the drag. This reduction has been quantified. Finally, these experimental findings have been simulated numerically. It is shown that the plasma filament created by the ultra-short laser pulse forms a low-density heated core that interacts with the detach bow shock. This interaction produces an inflating recirculating bubble and leads to a transient reduction of the drag.

The observations reported here demonstrate the potential interest of ultra-short laser for flow control applications. Further work will have to address the effect of repetitive laser pulses to obtain a quasi-steady effect on the flow.

[28] E. F. Toro, M. Spruce, and W. Speares, "Restoration of the contact surface in the HLL-Riemann solver," Shock Waves, vol. 4, no. 1, pp. [START_REF] Durand | Kilometer range filamentation[END_REF][START_REF] Settles | Schlieren and Shadowgraph Techniques: Visualizing Phenomena in Transparent Media[END_REF][START_REF] Refloch | CEDRE software[END_REF][28][29][30][31][32][33][34]1994.

Figure 1 -Figure 2 -

 12 Figure 1 -Detail of the experimental setup. The top picture shows the R1Ch facility test section and the exit port of the Enstamobile laser. The laser beam path is shown in red. The detail of the injection in the test model is shown in the two detailed CAD views

Figure 3 -

 3 Figure 3 -Detail of the 3-mm diaphragm at the model nose tip. The dimensions of the diaphragm are shown on the left. The picture on the right shows the nose of the model.

Figure 4 -

 4 Figure 4 -Long-exposure (4 s) image showing the cumulated emission of 40 laser pulses (50 fs, ~ 150 mJ/ pulse) in quiescent air.

Figure 5 -

 5 Figure 5 -Schlieren picture of the filamentary energy deposition upstream of the blunt body in quiescent air (ambient temperature, atmospheric pressure)

  Freestream flow velocity (m.s -1 ) 631

TimeFigure 7 -Figure 6 -

 76 Figure 7 -Sample drag signal (test run 1632). The time axis is referenced to the laser pulse activation (dashed red line).

Figure 8 -

 8 Figure 8-Drag (top), lower pressure (middle) and upper pressure (bottom) signals versus time for 30 laser pulses. The time scale reference is the given by the fast photodiode measuring the ultra-short laser pulse. For the pressure signals, the blue curve is the mean of the 30 sampled signals.

Figure 9 -

 9 Figure 9 -Schlieren images of the flow during the interaction.

Figure 10 -

 10 Figure 10-Schlieren images of the flow during the interaction.

Figure 11 -

 11 Figure 11-Detailed view of the interaction zone between the heated channel and the bow shock.

Figure 12 -

 12 Figure 12 -Domaine de calcul. Les zones de maillage quadrangulaires sont discrétisées à l'aide d'un maillage structuré.

  walls and pressure rise in the heated channel is independant of T as long as 𝑇 ≪ 𝜏 𝐹𝑙𝑜𝑤 , where 𝜏 𝐹𝑙𝑜𝑤 is a characteristic time of the flow (~ 1 µs).

Figure 13 -Figure 14 -

 1314 Figure 13-Comparaison between the experiments and. The blue lines mark the positions of the flow structures on the experiments. The columns correspond to three different times.

Figure 15 -

 15 Figure 15 -Density (upper left), temperature (upper right), Mach (lower left) and pressure fields at t=36 µs.

Figure 16 -

 16 Figure 16-Density (upper left), temperature (upper right), Mach (lower left) and pressure fields at t=65 µs.

Figure 17 -

 17 Figure 17 -Density (upper left), temperature (upper right), Mach (lower left) and pressure fields at t=100 µs.

Figure 18 -

 18 Figure 18 -Density (upper left), temperature (upper right), Mach (lower left) and pressure fields at t=149 µs.

Figure 19 -

 19 Figure 19 -Density (upper left), temperature (upper right), Mach (lower left) and pressure fields at t=200 µs.

Table 1 -Test run 1632 flow conditions.

 1

Table 2 Laser operation parameters, test run 1632.

 2

	Pulse energy ( laser output port) (mJ)	170 ± 10
	Focal length of focusing lens (mm)	1500
	Approximate length of the energy depostion (mm)	55 ± 2
	Distance of the heated channel center to the nosetip (mm) 25 ± 2

Acknowledgments

This study has been partly funded by the DGA ("Délégation Générale à l'Armement") through the ANR ASTRID project ANR-00008-015-001. PQ Elias acknowledges the support of Y. Doutreleau during the course of this project.


W Krouka
email: wafa.krouka@u-pem.fr

F Sarrazin

E Richalot

Influence of the reverberation chamber on antenna characterization performances

Keywords: Antenna characterization, reverberation chamber, mode stirring, chaoticity

Antenna characterization using reverberation chamber is being widely used and therefore several measurement techniques have been proposed. In this paper, the characterization technique based on two antennas is used in order to determine the radiation efficiency of a patch antenna. This work is conducted in two different configurations of a reverberation chamber (classical configuration and chaotic configuration). We evaluate the stirring techniques used (mechanical and frequential) in both configurations by calculating the number of independent samples. Statistical behavior of both cavity configurations (field homogeneity and isotropy) are also evaluated through the calculation of the enhanced backscatter parameter and the standard deviation of the measured parameters. Different measurements and calculation results are compared.

I. INTRODUCTION

Reverberation Chamber (RC) measurements have been extended to antenna characterization after being mainly limited to electromagnetic compatibility measurements. Indeed, RC is an interesting alternative to Anechoic Chamber (AC) to perform antenna radiation efficiency measurement. The original approach consists in measuring the power received by an antenna under test and comparing it to the power received by a reference antenna of known efficiency [START_REF] Piette | Antenna radiation efficiency measurements in a reverberation chamber[END_REF]. Other techniques, based on quality factor measurement, avoid the use of a reference antenna [START_REF] Christopher | Reverberation chamber techniques for determining the radiation and total efficiency of antennas[END_REF] [START_REF] Besnier | Antenna efficiency measurement from quality factor estimation in reverberation chamber[END_REF]. RC measurement accuracy lays on the uniform and isotropic incident field impinging on the antenna under test within the usable volume. These ideal properties are only obtained statistically by averaging the field over a number of independent realizations.

There are two main mode stirring techniques that allow obtaining different field configurations inside the RC. Mechanical stirring permits to change the boundary conditions of the electromagnetic field inside the RC using a rotating metallic stirrer. Besides, frequency stirring is also used by taking measurement samples over a frequency range. Stirring techniques and even their combination have been studied in [START_REF] Madsén | Models for the number of independent samples in reverberation chamber measurements with mechanical, frequency, and combined stirring[END_REF] [START_REF] Lemoine | Evaluation of frequency and mechanical stirring efficiency in a reverberation chamber[END_REF]. The random behavior of the electrical field inside the RC in addition to the number of measurements obtained through stirring allows us to apply statistical treatment on the electrical field. Recently, chaotic cavities have aroused great interest in the RC community [START_REF] Gros | Statistics of the electromagnetic response of a chaotic reverberation chamber[END_REF] as they present intrinsically uniform and isotropic field (without the need of any stirring process) compared to classical parallelepiped regular cavities used as RC. The objective of this paper is to evaluate the accuracy improvement of antenna efficiency measurement performed inside a chaotic RC compared to a classical one.

Measurements presented in this paper are conducted in two different configurations of an RC (2.95 m x 2.95 m x 2.35m) : the classical configuration (Fig. 1.a) and the chaotic configuration (Fig. 1.b) in which the geometry of the walls inside the RC has been modified by adding six metallic hemispheres (of radius 40 cm) in order to make its behavior more chaotic. Previous studies have already been conducted in this context [START_REF] Gros | Statistics of the electromagnetic response of a chaotic reverberation chamber[END_REF].

II. ANTENNA EFFICIENCY MEASUREMENT TECHNIQUE

In [START_REF] Christopher | Reverberation chamber techniques for determining the radiation and total efficiency of antennas[END_REF] different RC measurement techniques for determining antenna efficiency have been exposed. The approach based on two antennas is applied throughout this paper as it presents a good tradeoff between measurement accuracy and complexity. Two identical patch antennas are considered. They are made of a half-disc slot excited by a monopole and printed on FR4 substrate (Fig. 2). They are broadband antennas matched between 1.75 and 2.7 GHz (S 11 minor than -6 dB), so their operating frequencies are much higher than the Lowest Usable Frequency (LUF) of the RC (around 400 MHz). In order to determine the antenna efficiency, we need to calculate the chamber's time constant τ RC and the backscatter coefficient e b . The parameter τ RC represents the delay time that takes the energy inside the chamber to reverberate. It is determined from the power delay profile (PDP) as the inverse of its slope. The latter is calculated using the chamber's time domain response generally obtained using scattering parameter S 21 [START_REF] Piette | Antenna radiation efficiency measurements in a reverberation chamber[END_REF].

 𝑃𝐷𝑃(𝑡) = 〈|ℎ(𝑡, 𝑛)|²〉   ℎ(𝑡, 𝑛) = 𝐼𝐹𝑇[𝑆 21 𝑛 (𝑓)] 
with 'n' the number of independent samples obtained using stirring techniques.

RC quality factor is mainly related to different losses in the chamber. It can be calculated from the chamber's time constant τ RC [START_REF] Besnier | Antenna efficiency measurement from quality factor estimation in reverberation chamber[END_REF].

 𝑄 = 𝜔𝜏 𝑅𝐶  

with ω = 2πƒ (ƒ is the frequency). As electric field in RC, measured scattering parameters within an RC can be decomposed into stirred and unstirred components.

 𝑆 𝑖𝑗 = 𝑆 𝑖𝑗,𝑠 + 𝑆 𝑖𝑗,𝑢𝑠   with i,j={1,2}, the indexes relative to the first and the second antenna. The unstirred component represents the part that is not influenced by the stirrer position and is deterministic (ideally equal to the measured parameter in AC). On the other hand, stirred component is the statistical random part. It is calculated using [START_REF] Lemoine | Evaluation of frequency and mechanical stirring efficiency in a reverberation chamber[END_REF]. In our case, uncorrelated component is obtained by calculating the average of the measured parameter in RC over all stirrer positions.

 𝑆 𝑖𝑗,𝑠 = 𝑆 𝑖𝑗 -〈𝑆 𝑖𝑗 〉 

Using stirred components of the measured scattering parameters, we can calculate the enhanced backscatter coefficient [START_REF] Gros | Statistics of the electromagnetic response of a chaotic reverberation chamber[END_REF]. This coefficient evaluates the homogeneity of the electrical field inside the RC and is theoretically equal to 2 if the used reverberation chamber is well stirred [START_REF] Zhihao Tian | Enhanced backscatter coefficient measurement at high frequencies in reverberation chamber[END_REF]. In our case, the used antennas are considered identical. Thus, we consider 〈|𝑆 22,𝑠 |²〉 and 〈|𝑆 11,𝑠 |²〉 as equivalent, and the enhanced backscattering coefficient is then calculated as [START_REF] Gros | Statistics of the electromagnetic response of a chaotic reverberation chamber[END_REF].

 𝑒 𝑏 = 〈|𝑆 11,𝑠 |²〉 〈|𝑆 21,𝑠 |²〉  
After calculating the previous quantities, antenna efficiency can be determined using [START_REF] Zhihao Tian | Enhanced backscatter coefficient measurement at high frequencies in reverberation chamber[END_REF] for the total antenna efficiency.

 𝜂 𝑡𝑜𝑡 =√ 𝐶 𝑅𝐶 𝜔 𝑒 𝑏 〈|𝑆 11,𝑠 |²〉 𝜏 𝑅𝐶  
where 𝐶 𝑅𝐶 is a frequency dependent chamber's constant and is equal to

16𝜋 2 𝑉 𝜆 3
, 'V' being the RC volume and 'λ' the wavelength. In order to obtain antenna radiation efficiency, we apply a correction to the scattering parameter dependent quantity 〈|𝑆 11,𝑠 |²〉 . This correction aims to overcome the mismatch of the antenna.

III. STIRRING PERFORMANCES

In order to evaluate the measurement accuracy in an RC, the patch antenna under test is first of all characterized in an AC; the obtained reflection coefficient will serve as a reference in the following. Measurements are then performed in the RC using a Rohde&Schwarz ZNB8 VNA over the 1 GHz-3.5 GHz frequency band.

Mechanical and frequency stirring techniques are applied to the measured parameters in RC in order to enhance measurements precision. To better understand the impact of both stirring techniques, they are compared in both configurations of the RC in the next two parts.

A. Mechanical Stirring

Mechanical stirring consists in rotating the stirrer and performing measurements for each position of the stirrer. In our case, we use a rotation step of 5 degrees; thus, a complete rotation of the stirrer provides us with 72 measurements. In order to calculate the number of uncorrelated stirrer positions we evaluated the correlation coefficient between the received power at a fixed antenna location versus the stirrer rotation angle. Adopting the same criterion as in [START_REF] Besnier | Antenna efficiency measurement from quality factor estimation in reverberation chamber[END_REF] and [START_REF] Madsén | Models for the number of independent samples in reverberation chamber measurements with mechanical, frequency, and combined stirring[END_REF], we consider that data are uncorrelated once the correlation coefficient is lower than 1/e. The Fig. 3 presents the correlation coefficient calculated over a 600 MHz window centered on 2 GHz, the number of independent stirrer positions is estimated to be 72 in both cavity configurations, i.e. all positions that have been actually measured can be considered as uncorrelated. In order to study the impact of mechanical stirring on measurements in RC we calculate the STandard Deviation (STD) of the mechanical stirred S 11 relatively to the measurements of the same parameter in the AC (Tab. 1). We can notice the lower fluctuation level in the chaotic RC configuration compared to the classical RC. Calculation results of STD present lower values for both chamber configurations when mechanical stirring is applied relatively to results calculated for one fixed stirrer position (without stirring). Also, we clearly notice that the chaotic configuration presents better performances even when mechanical stirring is not applied, which is consistent with the chaotic cavity theory. We note that the electrical field homogeneity can be enhanced simply by making the RC structure more chaotic.

B. Frequency stirring

To emulate the effect of a frequency stirring process, the measured complex reflection coefficient is averaged over a sliding window of a chosen width and the result is associated to the central frequency of the averaging bandwidth. As the frequency bandwidth is usually arbitrary chosen, we suggest to study its impact on the results in terms of STD compared to the AC. It is equivalent to vary the number of uncorrelated data considered in the average calculation. Results are presented in Fig. 4. We observe that lower STD values are obtained in the chaotic RC configuration compared to the classical RC configuration, no matter the frequency stirrer window width. As expected, the STD decreases as we perform the frequency stirring for reasonable frequency widths. Then, it starts increasing because the considered window becomes very large compared to S 11 fluctuations. A minimal STD is obtained with a window of 106 MHz in the classical configuration and 98 MHz in the chaotic configuration (Fig. 4). Therefore an averaging window of 100 MHz will be used in the following. The Fig. 5 presents the frequency stirred antenna reflection coefficient obtained with a 100 MHz bandwidth averaging window in both RC configurations and compares them to the reflection coefficient measured in an AC. We notice that the curve obtained in the chaotic configuration is smoother than the one associated to the classical RC, as expected from previous results. In order to evaluate the number of uncorrelated data over the averaging window we calculate the coherent bandwidth using the formula Δf = f/Q [START_REF] Christopher | Reverberation chamber techniques for determining the radiation and total efficiency of antennas[END_REF] where 'Q' is the quality factor whose calculation will be detailed in Section (IV.A). Concerning the number of independent frequency samples, for a frequency of 2 GHz, the calculated number of uncorrelated samples over a 100MHz window is around 400. This large value compared to the number of independent stirrer positions explains the better efficiency of the frequency stirring technique in both configurations. The enhancement of independent data samples number results in a better measurement precision.

This study on the frequency stirring shows the necessity to have a large number of uncorrelated samples to obtain a precise determination of the reflection coefficient. Therefore, in the following, to overcome the limitation of the low number of independent stirrer positions, the obtained parameters when considering the mechanical stirring are smoothed on a sliding frequency window of 100 MHz. Table 2 shows the good accuracy of the antenna reflection coefficient characterization using this approach. Large fluctuations are observed with measurements performed at a fixed stirrer position in the chaotic RC without considering any stirring technique (green curve). Mechanical stirring leads to a decrease of the response fluctuations (red curve). Moreover, a better result is obtained when applying, after the mechanical stirring step, a smoothing over a sliding window of 100 MHz over all the frequency range (blue curve).

RC configuration

IV. ANTENNA EFFICIENCY CALCULATION

A. Quality factor

Quality factor Q has already been presented in Section II. Using (3), we calculate Q in both RC configurations. Results are presented in Fig. 7. We notice a decrease of the quality factor in the chaotic RC configuration compared to the classical one. To understand this difference, we consider the quality factor associated to losses localized on the cavity walls in a well-operating RC (where the field is homogeneous). In this case, the quality factor can be approximated as

𝑄 = 3 2.

𝑉 𝑆𝛿



where δ is the wall penetration depth. Thus, the quality factor is proportional to V 𝑆 ⁄ ratio.

Indeed, the insertion of the six hemispheres impacts the volume and the surface of the cavity. If V 0 , S 0 stand for the volume and the surface of the conventional RC, and V 1 , S 1 stand for the same quantities in the chaotic RC respectively, then, the quality factors Q 0 and Q 1 in these two cavities verify

Q 0 /Q 1 = V 0 .S 1 / (V 1 .S 0 ) = 1.12.
Besides, the average ratio over the measure frequency band of the quality factors extracted from measurements in both cavities is of 1.15. The agreement of this value with the theoretical prediction confirms the fact that the quality factor variation is due to the cavity geometries.

B. Antenna efficiency

In this section, we present calculation results of the enhanced backscatter coefficient (Fig. 8) and both total (Fig. 9) and radiation (Fig. 10) antenna efficiencies. Results have been smoothed using a 100 MHz sliding window in order to blur the frequency fluctuation of the parameters.

When considering the chamber's classical configuration, e b coefficient presents higher values in low frequencies. However, for frequencies above 1.5 GHz, and for both chamber configurations, the enhanced backscatter coefficient oscillates around two, which represents the theoretically ideal value of the backscatter coefficient. This result shows a good homogeneity of the electrical field inside the reverberation chamber and so a good stirring performance. Once the enhanced backscattering coefficient as well as the quality factor are known, it is possible to compute the antenna total efficiency from [START_REF] Zhihao Tian | Enhanced backscatter coefficient measurement at high frequencies in reverberation chamber[END_REF]. Results are presented in Fig. 9. We can see that both results (classical RC and chaotic RC) obtained when considering mechanical stirring associated to frequency smoothing are very close (with a maximal difference

RC configuration

Classical Chaotic SDT without smoothing 0.0853 0.0560 STD with smoothing 0.0279 0.0258 of 4 %). Indeed, as we work at a very high frequency compared to the RC LUF, even the classical cavity behaves close to an ideal RC. Also, we note that the only mechanical stirring is not sufficient to retrieve a smooth efficiency and the fluctuations in this case reach up to 30 % of the efficiency. The antenna radiation efficiency can then be obtained from the total one by taking into account mismatches. Results are presented in Fig. 10. As the patch antenna is well matched in the 1.7-2.7 GHz frequency band, the radiation efficiency is very close to the total one.

V. CONCLUSION

The radiation and total efficiencies of a slotted-based patch antenna have been evaluated within two configurations of a unique RC: the classical configuration and the chaotic one. Both mechanical and frequency stirring processes have been considered to average the measurement over several uncorrelated data. The performance study of these two stirring approaches shows that the chaotic cavity allows obtaining lower fluctuations of the reflection coefficient around its average value compared to the classical cavity. This enhancement of the RC properties thanks to the chaotic behavior is also confirmed by the enhancement backscattering coefficient in the lowest frequency range but the impact of the cavity configuration is not highlighted in the computed efficiency. Indeed, the considered frequency range is much higher than the RC LUF and thus, the RC is close to an ideal behavior, even in its classical configuration. Further work will include antenna efficiency measurement in a frequency range close to the LUF where the impact of the cavity chaoticity is expected to be enhanced.

Fig. 1 .

 1 Fig. 1. Reverberation chamber configurations: (a) classical configuration (b) chaotic configuration.

Fig. 2 .

 2 Fig. 2. Characterized patch antenna.

Fig. 3 .

 3 Fig. 3. Correlation coefficient versus stirrer step between received power for a fixed antenna location in chaotic RC over a 600 MHz window centered on 2 GHz.

Fig. 4 .

 4 Fig. 4. Standard deviation of the complex S11 parameter obtained in the RC and averaged over variable frequency windows relatively to S11 coefficient measured in the anechoic chamber.

Fig. 5 .

 5 Fig. 5. Frequency stirred S11 parameter using a 100 MHz averaging bandwidth.

Fig. 6 .

 6 Fig. 6. Scattering parameter of the patch antenna in the chaotic RC without and after applying mechanical stirring techniques and frequency smoothing.

Fig. 7 .

 7 Fig. 7. Quality factors of the classical and the chaotic reverberation chambers.

Fig. 8 .

 8 Fig. 8. Enhanced backscattering coefficient versus frequency in both RC configurations: classical and chaotic.

Fig. 9 .

 9 Fig. 9. Total efficiency of the patch antenna.

Fig. 10 .

 10 Fig. 10. Radiation efficiency of the patch antenna.

TABLE I .

 I STANDARD DEVIATION OF COMPLEX MECHANICAL STIRRED SCATTERING PARAMETER S11 IN REGARD TO THE COEFFICIENT MEASURED IN AC.