domain: sde sdv

text:
Jérôme Baudrier
email: jerome.baudrier@ifremer.fr

Alain Lefebvre
email: alain.lefebvre@ifremer.fr

François Galgani
email: francois.galgani@ifremer.fr

Claire Saraux
email: claire.saraux@ifremer.fr

Mathieu Doray
email: mathieu.doray@ifremer.fr

Optimising French fisheries surveys for marine strategy framework directive integrated ecosystem monitoring

The French initial assessment of the Marine Strategy Framework Directive (MSFD) highlighted the lack of reliable data concerning offshore areas. During the planning of the monitoring programmes, the scientists therefore proposed to partially cover this gap by using existing fisheries research vessel surveys deployed for the purposes of the Common Fisheries Policy (CFP). This paper describes ways of improving the effectiveness of these surveys and making them better suited to delivering the information needed for the MSFD. The process took two years and became operational at the beginning of the year 2016. Testing phases from October 2013 to August 2015 had to be organized to fit within the ongoing fisheries tasks without significantly increasing the workload in terms of both time and human resources. Six fisheries research surveys henceforth collect new data, with or without additional sampling techniques. Specific examples are given with litter and hydrological data which will be used to assess the environmental status of French marine waters. The paper also identifies certain limitations regarding this approach. This French experiment enabled more efficient and effective use of current data collection efforts, while optimising vessel time and implementing an ecosystem approach in collecting data for fisheries management.

Introduction

The EU Marine Strategy Framework Directive (MSFD, 2008/56/EC) establishes a framework within which EU Member States shall take action to achieve or maintain good environmental status (GES) of their marine waters by 2020 [START_REF] Doray | Directive 2008/56/EC of the European Parliament and of the Council of 17 June 2008 establishing a framework for community action in the field of marine environmental policy (Marine strategy Framework Directive)[END_REF]. GES is based on 11 qualitative descriptors, as listed in Annex I of the MSFD. The first stage of this marine strategy was for all Member States to carry out an initial assessment of the current status of their seas. The 2012 French initial assessment highlighted that there was a significant lack of data concerning offshore areas (beyond the 20 m isobath) in comparison with coastal and transitional waters monitored for the Water Framework Directive (WFD; [START_REF]Directive 2000/60/EC of the European Parliament and of the Council of 23 October 2000 establishing a framework for community action in the[END_REF]) [START_REF] Medde | Ministère de l'écologie, du développement durable et de l'énergie), Plan d'action pour le milieu marin, Evaluation initiale des eaux marines, sous-région marine Manche -mer du Nord. Coordination technique[END_REF][START_REF] Medde | Ministère de l'écologie, du développement durable et de l'énergie), Plan d'action pour le milieu marin, Evaluation initiale des eaux marines, sous-région marine mers celtiques[END_REF][START_REF] Medde | Ministère de l'écologie, du développement durable et de l'énergie), Plan d'action pour le milieu marin, Evaluation initiale des eaux marines, sous-région marine golfe de Gascogne[END_REF][START_REF] Medde | Ministère de l'écologie, du développement durable et de l'énergie), Plan d'action pour le milieu marin, Evaluation initiale des eaux marines, sous-région marine Méditerranée occidentale[END_REF]. Crise et al. [START_REF] Crise | A MSFD complementary approach for the assessment of pressures, knowledge and data gaps in Southern European Seas: the PERSEUS experience[END_REF] came to the same conclusions for southern European seas under the Perseus experiment, where data availability was poor in open sea for most of the descriptors. Teixeira et al. [START_REF] Teixeira | A Catalogue of Marine Biodiversity Indicators[END_REF] noted that the number of indicators decreased noticeably from shallow to deep waters, due among other factors to the lack of sampling. The general conclusions of technical assessment reports highlighted for all French marine sub-regions the lack of standardized and accessible observational data for offshore areas [START_REF] Dupont | Article 12 Technical Assessment of the MSFD 2012 obligations -France[END_REF]. These knowledge gaps remain an issue for the computation of operational indicators to assess correctly the state of the marine environment that have emerged from recent legislative commitments [START_REF] Rombouts | Food web indicators under the Marine Strategy Framework Directive: from complexity to simplicity?[END_REF]. In order to collect information on anthropogenic disturbances and their impact on the environment and to develop a proper scientific and technical basis for the implementation of actions aimed at improving the quality of the marine environment, it is necessary to conduct regular surveys, collect data and analyse their results.

Creating new surveys entirely dedicated to MSFD would be very time consuming and costly, probably resulting in insufficient spatial or temporal coverage. A better strategy would be to adapt and optimize existing surveys. Monitoring programmes (MP) could be efficiently developed by using survey downtimes for additional sampling. In particular, fisheries surveys could play an important role in MSFD monitoring programmes [START_REF] Shephard | Making progress towards integration of existing sampling activities to establish Joint Monitoring Programmes in support of the MSFD[END_REF][START_REF] Doray | The PELGAS survey: ship-based integrated monitoring of the Bay of Biscay pelagic ecosystem[END_REF], as they are the only surveys that sample the EU offshore and coastal marine waters each year. The French Research Institute for the Exploitation of the Sea (Ifremer) has been carrying out standardised fisheries surveys every year for the last three decades in the seas surrounding France. These surveys use either bottom trawls or a combination of fisheries acoustics and pelagic trawl sampling. Additional sampling techniques (e.g. floating-manta trawl, WP2 plankton net, etc.) are used in some cases. The various differentbut compatible -existing monitoring methods need to be adapted and standardized to take into account regional differences. When planning monitoring schemes, care should be given to ensure suitable spatial and temporal scales. Sampling protocols in Europe are currently available for the assessment of certain indicators (see e.g. ICES Survey Protocols, http://www.ices.dk/publications/our-publications/Pages/Survey-Protocols.aspx -Accessed 05 January 2018), but should be adapted to MSFD requirements and standardized for compatibility with other regions.

This is the aim of the Joint Monitoring Programmes (JMP), that examine how much and what type of sampling is needed, at a minimum, in order to achieve quality standards and how these data could be collected most efficiently [START_REF] Shephard | Making progress towards integration of existing sampling activities to establish Joint Monitoring Programmes in support of the MSFD[END_REF][START_REF] Zampoukas | Marine monitoring in the European Union: how to fulfill the requirements for the marine strategy framework directive in an efficient and integrated way[END_REF]. Pioneering integrated surveys [START_REF] Doray | The PELGAS survey: ship-based integrated monitoring of the Bay of Biscay pelagic ecosystem[END_REF] and pilot projects (e.g. De Boois and van Hal [START_REF] De Boois | Towards a joint monitoring programme for the North Sea and the Celtic Sea: Activity C Multidisciplinary monitoring[END_REF]) have demonstrated that indicators of other ecosystem component states could be measured alongside routine biological sampling conducted for fish stock assessments during fisheries surveys in the Regional Sea Conventions (RCS) areas (e.g. International Bottom Trawl Surveys (IBTS) under the Barcelona Convention for the Mediterranean; the Bucharest Convention for the Black Sea; the Oslo-Paris Convention (OSPAR) for the NE Atlantic, and the Helsinki Convention (HELCOM) for the Baltic). However, sampling activities cannot be extended endlessly without compromising data quality [START_REF] Shephard | Making progress towards integration of existing sampling activities to establish Joint Monitoring Programmes in support of the MSFD[END_REF]. Ecosystem sampling should indeed be optimised in a coordinated and adaptive way, to ensure that the addition of new sampling protocols does not compromise the quality of standard data collection, and to optimize the utility of the sampling programme. If interesting studies have provided general principles [START_REF] Shephard | Making progress towards integration of existing sampling activities to establish Joint Monitoring Programmes in support of the MSFD[END_REF][START_REF] Kupschus | Practical implementation of ecosystem monitoring for the ecosystem approach to management[END_REF], few practical examples of the optimisation of data collection programmes for ecosystem sampling are to be found in the literature (but see e.g. [START_REF] Doray | The PELGAS survey: ship-based integrated monitoring of the Bay of Biscay pelagic ecosystem[END_REF]).

The aim of this paper is to bridge this gap by presenting a practical example of the optimisation of a data collection programme to meet MSFD ecosystem monitoring requirements. The article describes how French fisheries surveys were optimised to move on from single-species stock assessment toward MSFD compliant ecosystem monitoring.

Material and Methods

French monitoring programmes

The second stage of the marine strategy was to establish and implement monitoring programmes to measure progress towards GES. The very wide scope covered by MSFD requires additional monitoring in areas where it was not previously required by EU law [START_REF] Zampoukas | Marine monitoring in the European Union: how to fulfill the requirements for the marine strategy framework directive in an efficient and integrated way[END_REF]. In France, the definition of the MP lasted from 2012 to 2015 under the supervision of the Ministry of Environment, and coordinated by the Marine Protected Areas Agency (AAMP) and Ifremer. Scientists designated for each of the 11 descriptors undertook the design of these MP (a complete survey programme for each descriptor, including terrestrial, aerial and nautical means). In 2012, each of them suggested that vessel-based surveys should collect data for MSFD. On this occasion, they identified existing observation platforms provided by research vessels (R/V) which were mainly involved in fisheries stock assessment, as a potential way of providing additional means of monitoring. The MSFD coordination team from Ifremer therefore proposed to explore the possibilities of adding MSFD monitoring to existing vessel surveys.

A similar choice was also made by most Member States, which decided for cost-efficiency reasons to develop their MP by extending existing monitoring activities [START_REF] Shephard | Making progress towards integration of existing sampling activities to establish Joint Monitoring Programmes in support of the MSFD[END_REF]. Moreover, previous management strategies focusing on specific human impacts or individual species have been judged insufficient for the implementation of the ecosystem approach called for by new regulations such as MSFD [START_REF] Kupschus | Practical implementation of ecosystem monitoring for the ecosystem approach to management[END_REF]. In France, the PELGAS survey [16] followed an ecosystemic approach by monitoring hydrology, phytoplankton, zooplankton as well as birds and mammals, in addition to its fish stock assessment role, from 2000 in the Bay of Biscay. This survey demonstrated the potential of fisheries surveys for ecosystem monitoring [START_REF] Doray | The PELGAS survey: ship-based integrated monitoring of the Bay of Biscay pelagic ecosystem[END_REF][START_REF] Doray | Manual of fisheries survey protocols. PELGAS surveys (PELagiques GAScogne)[END_REF].

Feasibility study for optimisation of the French fisheries surveys

Following the proposals made in 2012, the optimisation of vessel surveys in France was considered a priority because it offered the means to obtain suitable offshore data at moderate cost. It was pointed out that the MP should be able to provide data for the calculation of the indicators set by the Commission Decision (e.g. [START_REF] Zampoukas | Marine monitoring in the European Union: how to fulfill the requirements for the marine strategy framework directive in an efficient and integrated way[END_REF]). The data to be collected were consequently related to the requirements of the MSFD, and particularly information on indicators of several quantitative descriptors selected by the European Commission [START_REF][END_REF]. French research scientists selected the following criteria for good environmental status relevant to the descriptors of MSFD: (i) biodiversity: distributional range, pattern and area covered by the species; population abundance, demography and genetic structure; condition of typical species and composition of species; (ii) habitat: physical, hydrological and chemical conditions; (iii) productivity: performance of key predator species using their production per unit biomass; (iv) eutrophication: nutrients, dissolved oxygen, water transparency, chlorophyll concentrations and phytoplankton community composition; (v) contaminants: concentration of contaminants measured in biota; (vi) marine litter: amount of floating litter (including micro-particles) and litter deposited on the seafloor; and (vii) energy: distribution of underwater noise. They are used for the requirements of descriptors 1 Diversity (i/ii), 4 Food webs (iii), 5 Eutrophication (iv), 8 Contaminants (v), 9 Contaminants in fish and seafood (v), 10 Litter (vi), 11 Energy and noise (vii). These descriptors were selected because they could be collected during fisheries surveys to compensate for the lack of French data in offshore areas [START_REF] Medde | Ministère de l'écologie, du développement durable et de l'énergie), Plan d'action pour le milieu marin, Evaluation initiale des eaux marines, sous-région marine Manche -mer du Nord. Coordination technique[END_REF][START_REF] Medde | Ministère de l'écologie, du développement durable et de l'énergie), Plan d'action pour le milieu marin, Evaluation initiale des eaux marines, sous-région marine mers celtiques[END_REF][START_REF] Medde | Ministère de l'écologie, du développement durable et de l'énergie), Plan d'action pour le milieu marin, Evaluation initiale des eaux marines, sous-région marine golfe de Gascogne[END_REF][START_REF] Medde | Ministère de l'écologie, du développement durable et de l'énergie), Plan d'action pour le milieu marin, Evaluation initiale des eaux marines, sous-région marine Méditerranée occidentale[END_REF].

A feasibility study was launched in October 2013 to assess the compatibility between the proposals made by the scientists, linked to the previous data, and the fisheries stock assessment operations. This preliminary stage was developed with the help of vessel-based survey coordinators. The MSFD coordination team from Ifremer sent a questionnaire concerning new protocols to implement for the requirements of the directive. The main topics in the questionnaire were: additional monitoring and parameters to be recorded, vessel survey concerned, sampling strategy, operational protocol onboard and logistic. This was applicable to all vessel surveys (coastal and off-shore) implemented by Ifremer for fisheries stock assessments. The aim was to make a first selection among the various proposals taking into account the logistical limits to the amount of work that can be done in a single survey, and the priority given to the primary survey objectives defined by the funding provider [START_REF] Shephard | Making progress towards integration of existing sampling activities to establish Joint Monitoring Programmes in support of the MSFD[END_REF]. The preliminary indications from the questionnaire lead the MSFD coordination team to exclude the vessels and surveys that were not appropriate to conduct the proposed monitoring operations (e.g. irrelevant survey areas, vessel too small…). In February 2014, an analysis of the results of this questionnaire was conducted. Six vessel surveys were chosen on the basis of four major criteria: the possibility of having more people onboard, of working on a continuous basis (day / night), of loading and deploying more sampling equipment, and the areas covered by the French surveys. These fisheries surveys are conducted with two R/V, Thalassa in the Atlantic region and L'Europe in the Mediterranean sea (see Table 1 for a presentation of their main characteristics).

As a result, the second step involved the deployment of near-operational field tests between September 2014 and August 2015 [START_REF] Baudrier | Optimisation des campagnes halieutiques pour la construction d'indicateurs DCSMM[END_REF].In addition to already known operations (e.g. seafloor litter, water nutrients, mammals and birds, etc.), two new protocols for routinely monitoring fisheries surveys were tested. The first was devoted to the collection of organisms for contaminant and food web monitoring [START_REF] Mialet | Bilan des essais et optimisation du suivi mutualisé « réseaux trophiques et contaminants » sur les campagnes halieutiques DCF 2014-2015[END_REF]. On the basis of trawling carried out by Cefas (Centre for Environment Fisheries and Aquaculture Science -UK) and by Ifremer (France), a new protocol for gelatinous zooplankton has also been tested since 2012 and 2014, respectively [START_REF] Aubert | No more reason for ignoring gelatinous zooplankton in ecosystem assessment and marine management: concrete cost-effective methodology during routine fishery trawl surveys[END_REF]. The aim of these different tests (more regular and using new protocols) was to make sure of the compatibility between suggested protocols and those from the fisheries stock assessments. If they proved to be so, MSFD protocols were adapted in order to be fully operational for the monitoring programmes. The study ended in October 2015 and a synopsis was sent to the partners to inform them of the possibility of implementing the different MSFD protocols [START_REF] Baudrier | Mise en oeuvre du programme de surveillance de la DCSMM: synthèse des essais réalisés à bord des campagnes halieutiques[END_REF]. Fig. 1 shows the main steps of the French experimentation.

The testing phase successfully demonstrated the suitability of most of them, in a cost-effective way. Gridded maps were drawn up, based on feasibility studies data, using a spatial interpolation method, the block averaging procedure (BAP; [START_REF] Petitgas | Combining acoustic and CUFES data for the quality control of fish-stock survey estimates[END_REF]). BAP enables to rapid processing of large amounts of ecosystem data, collected according to different sampling schemes, while avoiding edge effects [START_REF] Doray | The PELGAS survey: ship-based integrated monitoring of the Bay of Biscay pelagic ecosystem[END_REF]. Gridded maps were produced using a reasonably fine (0.25°x 0.25°) common grid, using the R package EchoR [START_REF] Doray | EchoR: R package for computing indices of the state of fish population and communities, based on fisheries acoustic data, R package[END_REF].

Results

The French monitoring programmes were adopted at the beginning of June 2015 by the authorities coordinating each marine sub-region. Building on the success of the feasibility study (2014-2015; Fig. 1), they all recommended the collection of the necessary data for MSFD monitoring by optimising time from vessel fisheries management surveys. Marine mammals, birds, marine litter (floating litter, sea-floor litter, micro-particles), gelatinous zooplankton, contaminants, food webs, underwater noise and hydrology were to be monitored in offshore areas [START_REF] Medde | Ministère de l'écologie, du développement durable et de l'énergie), Plan d'action pour le milieu marin, Programme de surveillance, sous-région[END_REF][START_REF] Medde | Ministère de l'écologie, du développement durable et de l'énergie), Plan d'action pour le milieu marin, Programme de surveillance, sous-région marine mers celtiques[END_REF][START_REF] Medde | Ministère de l'écologie, du développement durable et de l'énergie), Plan d'action pour le milieu marin, Programme de surveillance, sous-région marine golfe de Gascogne[END_REF][START_REF] Medde | Ministère de l'écologie, du développement durable et de l'énergie), Plan d'action pour le milieu marin, Programme de surveillance, sous-région marine Méditerranée occidentale[END_REF]. Because of the significant cost of lab analyses, organisms (cephalopods and fishes catches) are sampled for contaminants and food webs analyses only when funding is available. Following the submission of the French monitoring strategies, additional sampling for MSFD became official at the beginning of 2016 [START_REF] Baudrier | Bilan de l'optimisation des campagnes halieutiques réalisée dans le cadre du programme de surveillance de la directive cadre stratégie pour le milieu marinannée 2016[END_REF], starting with the IBTS survey. Examples of potential indicators calculated from the data collected by the PELGAS ecosystem survey can be found in Doray et al. [START_REF] Doray | The PELGAS survey: ship-based integrated monitoring of the Bay of Biscay pelagic ecosystem[END_REF][START_REF] Doray | Monitoring small pelagic fish in the Bay of Biscay ecosystem, using indicators from integrated survey[END_REF]. Today, most of the European vessel surveys try to collect data on human pressures and ecosystem state, in response to different requirements such as CFP, MSFD and other legislation or international agreements [START_REF] Shephard | Making progress towards integration of existing sampling activities to establish Joint Monitoring Programmes in support of the MSFD[END_REF]. The following sections present the vessel surveys used in France and the new information obtained from them. Subsequently, examples are given with litter and hydrological data collected for MSFD requirements and which will be used to assess the environmental status of the French marine waters. Those case studies have been chosen as (i) litter is an emerging topic, specifically covered by MSFD; (ii) hydrological data were collected using an integrated system (Pocket Ferry Box), which allowed to provide high resolution data that could complement low resolution sampling strategies generally used.

Shipborne fisheries surveys

Six shipborne surveys, related to the requirements of the European data collection framework (DCF), are used for the purposes of the MSFD French legislation: IBTS [31], PELGAS [16], MEDITS [32], PELMED [START_REF] Bourdeix | PELMED -PELagiques MEDiterranée[END_REF], CGFS [START_REF] Coppin | [END_REF] and EVHOE [START_REF] Mahé | EVHOE EValuation Halieutique de l'Ouest de l'Europe[END_REF]. Five of them are mandatory according to Table 10 of the Annex of the EU-MAP Commission implementing decision EU/2016/1251 [START_REF][END_REF]. Table 2 presents their main characteristics and Fig. 2 the areas covered by the fisheries surveys. From 2015 onwards, the CGFS survey was carried out on another scientific vessel. In order to ensure the continuity of the time series, an inter-calibration between the two vessels (R/V Thalassa and R/V Gwen Drez) was realized in October 2014 [START_REF] Auber | Intercalibration of research survey vessels: Gwen Drez and Thalassa[END_REF]. MSFD field tests began with the CAMANOC survey [START_REF] Travers-Trolet | CAMANOC cruise, RV Thalassa[END_REF] on R/V Thalassa during this intercalibration exercise in the Western English Channel (Fig. 1).

Since the surveys maintain long-term series, the DCF protocols (sampling plan, fishing gear, catch analysis protocol) are often standard (or lightly amended) and are repeated from one year to the next. Before the implementation of the feasibility study, these surveys were already multi-disciplinary. Additional activities considered for MSFD were: (i) collecting other information from the catch (marine litter, jellyfish, biota for analysis); (ii) automated and high spatio-temporal resolution sampling (Pocket Ferry Box, Automatic Identification System (AIS));

(iii) deployment of equipment (floating-manta trawl, WP2 plankton net, Niskin bottle); (iv) observations (birds, mammals, floating litter). Some of these protocols were carried out before implementation of the MSFD but most were done on a more or less regular and/or coordinated basis. MSFD MP made it possible to harmonize the protocols for all the surveys. Today, the opportunistic use of French fisheries surveys provides the only large-scale spatio-temporal sampling tool for various data, such as sea-floor litter [START_REF] Galgani | Litter on the Sea Floor Along European Coasts[END_REF] or gelatinous zooplankton [START_REF] Aubert | No more reason for ignoring gelatinous zooplankton in ecosystem assessment and marine management: concrete cost-effective methodology during routine fishery trawl surveys[END_REF], and at low cost. Table 3 shows for each survey what has been added for the requirements of the MSFD, distinguishing operational and strategic changes. Descriptors concerned by all of these data are highlighted as well.

Marine litter

Litter on the sea floor, floating litter and microplastics were assessed using the MSFD protocols as defined by GES technical group [START_REF] Galgani | Marine litter within the European Marine Strategy Framework Directive[END_REF][START_REF] Galgani | Monitoring Guidance for Marine Litter in European Seas[END_REF] and enshrined in trawl survey manuals for both MEDITS [START_REF]International bottom trawl survey in the Mediterranean. Instruction manual. MEDITS-Handbook[END_REF] and IBTS [START_REF] Ices | Manual for the International Bottom Trawl Surveys[END_REF] cruises. Experiments for floating litter and microplastics were launched more recently on pelagic surveys (PELGAS, PELMED).

Marine litter abundance at sea can be estimated either by direct observations of large debris items (e.g. ROVs), by shipbased and aerial observations for debris floating at the sea surface, by large-scale imagery applications and net trawls. Net-based assessment surveys are the most widespread and effective methods to date [START_REF] Galgani | Litter on the Sea Floor Along European Coasts[END_REF], using a similar approach to the methodology for monitoring benthic species on the sea floor or surface organisms for floating litter. Results from trawl surveys indicated efficient sampling and results for sea floor litter and floating litter, with possible time trends assessment. The approach has been extended in the past few years to the evaluation of floating microplastics, with common approaches and in a standardized manner covering all MSFD requirements [START_REF] Gago | First observation on neustonic plastics in waters off NW Spain (spring 2013 and 2014)[END_REF][START_REF] Gago | Microplastics in Seawater: recommendations from the Marine Strategy Framework Directive Implementation Process[END_REF]. Typically, results from the first Fig. 2. Main areas covered by the French standardised fisheries surveys (data from Ifremer). experiments held in 2015 and 2016 showed the potential of such an approach for assessment of annual variations in both total litter and specific items, the latter approach being critical when considering reduction measures that are dedicated to a specific type of litter. For example, the ban on non-degradable plastic bags, as decided in France in July 2016, will require a suitable tool to assess the effectiveness of the measures. Monitoring plastic sheets and bags on the sea floor is a suitable approach for this purpose, providing consistent data on quantities, distribution and time trends (Fig. 3). Gridded maps of average densities of total litter and plastic sheets and bags as measured in bottom-trawl surveys are presented. Data were obtained during MSFD optimised French fisheries surveys in 2015 and 2016, following the Guidance for monitoring marine litter in European seas [START_REF] Galgani | Monitoring Guidance for Marine Litter in European Seas[END_REF] as a standardized protocol. Highest litter densities (600-1000 litter per square meter) were found in the main river plumes (Seine, Gironde, Rhône) and around Corsica island.

Hydrology

The literature highlights that in general, the observation and monitoring of highly dynamic systems such as marine waters requires dense sampling in space and time in order to detect short-term events which might have a strong impact on the ecosystem [START_REF] Petersen | A mature system for operational monitoring[END_REF]. Focusing on the MSFD sub-regions, existing observation and monitoring programmes lack the high spatio-temporal resolution coverage required to determine the status of the marine environment and changes therein. Most of the regular and year-round observations and monitoring programmes only cover the very near coastal zone. Numbers of stations drastically decline or are non-existent offshore (beyond the 20 m isobath).

Considering the need for MSFD monitoring programmes for Eutrophication and Pelagic Habitats, the fact is that processes involved in phytoplankton bloom dynamics cover a wide range of spatio-temporal scales [START_REF] Cloern | Phytoplankton bloom dynamics in coastal ecosystems: a review with some general lessons from sustained investigation of San Francisco bay, California[END_REF][START_REF] Dickey | Emerging ocean observations for interdisciplinary data assimilation systems[END_REF], and that bloom events could last only a few days but with strong direct and indirect effects on ecosystems. The responsiveness of phytoplankton to environmental changes and the ecosystem goods and services associated with this compartment explain why phytoplankton is widely used as an indicator for the development of environmental quality assessment systems [START_REF] Doray | Directive 2008/56/EC of the European Parliament and of the Council of 17 June 2008 establishing a framework for community action in the field of marine environmental policy (Marine strategy Framework Directive)[END_REF][START_REF]Directive 2000/60/EC of the European Parliament and of the Council of 23 October 2000 establishing a framework for community action in the[END_REF][START_REF]Eutrophication status of the OSPAR maritime area. Second OSPAR integrated report[END_REF].

Consequently, since ecological studies and overall ecological assessments need to integrate physical, chemical and biological information in order to achieve a comprehensive view of the states and dynamics of the ecosystems, but also need to be in phase with its main spatial and temporal controlling scales [START_REF] Cloern | Phytoplankton bloom dynamics in coastal ecosystems: a review with some general lessons from sustained investigation of San Francisco bay, California[END_REF][START_REF] Mantua | The Pacific Decadal Oscillation[END_REF], it was relevant to test the added value of the implementation of an integrated, autonomous system called Pocket Ferry Box (PFB, 4H-JENA), the particularity of which is to be coupled with a multiple-fixed-wavelength spectral fluorometer (Algae Online Analyser, AOA, bbe Moldaenke [START_REF] Beutler | A fluorimetric method for the differentiation of algal populations in vivo and in situ[END_REF]). The aim is to gain further knowledge on high-resolution events in order to be able to explain why some parameters (or MSFD criteria) could downgrade or upgrade Good Environmental Status assessment results, mainly on the basis of low resolution sampling strategies. Moreover, high resolution data could contribute to validation and calibration steps when using satellite remote sensing and hydrodynamic-hydrological modelling derived products. All parameters were recorded at a frequency of once per minute and geographical coordinates were provided by an external GPS receiver. The system used during this trial was assembled with sensors for salinity and temperature (AANDERA conductivity sensor 3919), pH (Meinsberg MV 3010), oxygen concentration (AANDERA oxygen optode 3835), CDOM (AANDERA Cyclops 7 sensor) and spectral groups of phytoplankton using an Algae Online Analyser (bbe Moldaenke).

The implementation of high resolution in situ measurements with a Pocket Ferry Box, coupled to a spectral fluorometer and associated with optimised unsupervised spectral classifier data processing [START_REF] Rousseeuw | Hybrid Hidden Markov Model for Marine Environment Monitoring[END_REF][START_REF] Lefebvre | Poisson-Caillault Unpublished results. High resolution overview of phytoplankton spectral groups and hydrologival conditions in the eastern English Channel using unsupervised clustering[END_REF], offers the means to provide an environmental state-specific and synoptic view of the main hydrological parameters, phytoplankton biomass and spectral groups in a MSFD marine sub-region for a given period (Fig. 4). During the MSFD cruises, this system also allows a near real-time adaptive sampling strategy to capture the main hydrological and biological changes. This will contribute to the definition of eco-hydrological regions in order to focus low resolution conventional sampling strategy in these areas and consequently to optimize the costs of laboratory analysis on a lighter set of in situ samples.

Discussion

The paper presents a case-study of integrated fisheries surveys developed in France from 2013 for the needs of MSFD. The discussion focuses essentially on the strength and limitations of this approach.

Improving international partnerships

Fisheries surveys conducted within the EU Data Collection Framework have been coordinated at international level within the ICES survey planning groups since the early 2000s (e.g. Working Group on Acoustic and Egg Surveys -WGACEGG, 2004) and earlier (e.g. International Bottom Trawl Survey Working Group -IBTSWG, 1992), or at Mediterranean scale (e.g. MEDITS and MEDIAS survey groups). Despite the implementation of multi-disciplinary surveys which are sometimes international, the lack of coordination of monitoring programmes on other ecosystem components has often been highlighted (e.g. [START_REF] Heslenfeld | OSPAR Ecological Quality Objectives: the utility of health indicators for the North Sea[END_REF]). For better integration and international collaboration with regard to monitoring activities, Joint Monitoring Programmes are necessary [START_REF] Shephard | Making progress towards integration of existing sampling activities to establish Joint Monitoring Programmes in support of the MSFD[END_REF]. The pilot action "Multi-use of Infrastructure for Monitoring in the North Sea" proposed in 2012, was aimed at achieving this goal during various IBTS cruises in 2014 [START_REF] De Boois | Towards a joint monitoring programme for the North Sea and the Celtic Sea: Activity C Multidisciplinary monitoring[END_REF]. Similarly, the Working Group on Integrating Surveys for the Ecosystem Approach (WGISUR) explores the possibility of developing integrated surveys, which means joint monitoring cruises using similar protocols and instrumentation to collect data [START_REF] Zampoukas | Marine monitoring in the European Union: how to fulfill the requirements for the marine strategy framework directive in an efficient and integrated way[END_REF]. All these experiments should be developed further. For example, English and French trials on gelatinous zooplankton resulted in the successful implementation of the observation protocol in the French MSFD monitoring programme that will hopefully inspire other EU coastal Member States [START_REF] Aubert | No more reason for ignoring gelatinous zooplankton in ecosystem assessment and marine management: concrete cost-effective methodology during routine fishery trawl surveys[END_REF].

An interesting illustration is given by marine litter. Large scale trawl surveys on the seabed began long before strong support for monitoring was provided [START_REF] Galgani | Litter on the Sea Floor Along European Coasts[END_REF]. Monitoring is successful when the results can be used directly as a basis for effective management decisions [START_REF] Karydis | Marine water quality monitoring: a review[END_REF].A t present, trawl surveys provide adequate means to achieve both criteria 10DC1 (composition, amount and spatial distribution of litter on the coastline, in the surface layer of the water column, and on the seabed) and 10DC2 (same with micro-litter), with standardized protocols at RSC levels, enabling assessment of sea floor litter and both floating debris and microplastics. The use of nets in most surveys (IBTS, CGFS, EVHOE and MEDITS) gives satisfactory results in terms of sampling, but, because these gears are crafted for collecting nectobenthic rather than benthic species, they probably underestimate the quantities present. They do however provide consistent results in terms of trend assessment, location of accumulation hot spots and composition of litter (Fig. 3), and are thus in phase with MSFD requirements in terms of monitoring of both litter and the impact of reduction measures. In addition, routines are under development for the automatic calculation of indicators / criteria, prior to reporting or scientific analysis. As an integrated tool, fish stock surveys provide a powerful means to simplify, standardize and report on the basis of D10 criteria, enabling further aggregation and analysis of GES. The inclusion of standardized protocols in the manuals of the different surveys from a given marine region would be a way to provide common indicators at European level. This example illustrates the use of "operational" indicators which have wellunderstood relationships between natural state and specified anthropogenic pressures, necessary to define environmental targets and decide on management measures [START_REF] Shephard | Surveillance indicators and their use in implementation of the Marine Strategy Framework Directive[END_REF]. To gain an accurate and meaningful assessment of plastics and their influence, large-scale and longterm monitoring is needed across countries and environments. In this case, monitoring could also provide information on the efficiency of implementation of measures decided for mitigating marine pollution and deterioration [START_REF] Douvere | The importance of monitoring and evaluation in adaptive maritime spatial planning[END_REF].

Limitations

On routine fisheries surveys, the sampling plan was only designed for fisheries stock assessment, and not for other needs such as MSFD monitoring. This priority can result in the deployment of equipment and observations in areas that are of little interest for collecting data for the MSFD. Moreover, vessel space available for scientific work can be an issue. For example, in the western Mediterranean sea, MEDITS and PELMED surveys are performed on the R/V L'Europe, which is much smaller than the Thalassa used for the other surveys (EVHOE, IBTS, CGFS, PELGAS). There is not room for as many scientists and as much equipment on board as might be hoped for (e.g. 8 scientists on board L'Europe vs. 25 on board Thalassa; see Table 1). This is the main reason for the acceptance of fewer MSFD protocols for the PELMED and MEDITS surveys, and can sometimes be a problem for operating a standardized protocol. However, even on those surveys using a small vessel, protocols could be adapted so to collect important information for MSFD.

Another issue is the scheduling of the surveys: DCF dictates that they are always planned during the same season, not necessarily the best one for certain protocols (for example, phytoplankton samples during the winter period). Weather can often be a major factor disabling a number of potential opportunities for MSFD monitoring, as is the case, for example, with microplastics (floating-manta trawl) or visual observations (birds, mammals, floating litter).

A final difficulty is the fact that responsibility for CFP and MSFD is split between two different ministries in France, as it is at European level (DGMARE and DGENV). The priorities for the two policies are not the same; ways of achieving increased cooperation and coordination should be explored. The challenge is to make our monitoring surveys more efficient and better targeted to deliver all the information needed. While ambitious marine ecosystem management objectives have been adopted in Europe within the MSFD framework, funds made available to implement MSFD Monitoring Programmes have not so far been sufficient to fully achieve the MSFD objectives [START_REF] Shephard | Making progress towards integration of existing sampling activities to establish Joint Monitoring Programmes in support of the MSFD[END_REF][START_REF] Borja | Marine monitoring during an economic crisis: the cure is worse than the disease[END_REF]. The EU DCF funding so far been essential to maintain the fisheries surveys and allow for their optimisation to meet MSFD requirements. The future of MSFD monitoring in Europe will probably depend on the availability of funds for developing suitable Monitoring Programmes to meet the challenge of the MSFD objectives [START_REF] Doray | The PELGAS survey: ship-based integrated monitoring of the Bay of Biscay pelagic ecosystem[END_REF]. It will also depend on human resources with the right skills in terms of the methodologies to be developed and implemented for MSFD purposes (examples: plankton taxonomy, innovative technologies, data storage and data processing). In France, new dedicated ship-based ecosystem surveys will still be necessary, to complement existing integrated fisheries surveys mainly in terms of spatial and temporal coverage. For example, new vesselborne, MSFD hydrological surveys have been conducted in the Eastern and Western English Channel since 2017. They take place in spring and summer to complement the IBTS and CGFS fisheries surveys implemented in fall and winter, respectively. These new hydrological surveys are in the process of been optimised for collecting other MSFD data, following the same approach as for fisheries surveys. The Ministry of Environment currently works on this topic and a dedicated working group has been created (first meeting in March 2018). Likewise, offshore data on benthic communities might be collected during new MSFD hydrological surveys, as the deployment of specific equipment (grab / dredge) was deemed too time consuming to be added to fisheries surveys sampling.

Beyond DCF and MSFD monitoring strategy

The optimisation of fisheries surveys contributes towards developing a novel set of hydrological and biological observations aimed at achieving further knowledge on plankton community dynamics and structure, supporting the MSFD requirements in particular with regard to pelagic habitat (D1) and eutrophication (D5) issues. The strategy is based on a combination of conventional sampling tools providing data at low spatial and high taxonomic resolution (e.g. manual identification of zooplankton net samples up to the plankton species level) and of novel high-resolution sampling tools (Pocket Ferry Box, zooplankton imagery and optical instrumentations, single cell optical and bulk characterizations). Since, however "optimised" the fisheries surveys may be, it will never be possible to cover the full range of spatial and temporal scales characterising the different processes structuring the plankton communities, we also need to consider modelling and satellite derived products. Consequently, in situ low and high resolution data series collected during optimised surveys will help scientists to develop, to calibrate and to validate ocean colour algorithms and combine hydrodynamic-biogeochemical models [START_REF] Thyssen | High-resolution analysis of a North Sea phytoplankton community structure based on in situ flow cytometry observations and potential implication for remote sensing[END_REF][START_REF] Ford | Observing and modelling phytoplankton community structure in the North Sea[END_REF].

Conclusion and future perspectives

To conclude, the strong points of the shipborne surveys are their sustainability, their frequency (annual surveys) and the use of standardized protocols between different countries. Since they maintain long-term series, the protocols have been thoroughly validated and are repeated each year. Furthermore, they are extensively funded by the European Commission in accordance with the Common Fisheries Policy. Therefore, it was pertinent to append new monitoring protocols to meet MSFD requirements because there are real opportunities to collect additional data. Finally, this approach also allows investigation of several ecosystem compartments and MSFD indicators at the same time, rather than developing separate specific data collection for each of the indicators. This should enable a much better understanding of the health of the ecosystem as well as potential interactions between indicators. The French experiment showed that it was less costly than carrying out surveys during a separate cruise, but in either case it does involve additional costs. The implementation of MSFD MP enables the sustainable use of particular cost-effective protocols, the deployment of new ones and the standardisation of some activities (hydrological protocols, data storing, analysis standardisation, etc.). It also offers new perspectives.

For example, Doray et al. [START_REF] Doray | Monitoring small pelagic fish in the Bay of Biscay ecosystem, using indicators from integrated survey[END_REF] have shown that multiple time series (~15 years long) of potential ecosystem indicators (D1 and D3 related) collected during the PELGAS survey could be jointly analysed to extract the main large-scale trends in the indicator trajectories. This study provided a basis for hypotheses on the main changes undergone by the pelagic ecosystem of the Bay of Biscay, on the main ecological processes driving them and on the potential interactions between ecosystem components. Comparing indicator trends and proxies of external forcing also offers a means to investigate the effects of climate change and fishing on the ecosystem. The analysis of the future time series of data collected by the optimised French fisheries surveys could inform the MSFD and provide similar results over broader spatial (all EU French waters) and ecological (D1, D3, D4, D8, D9, etc.) scales, if their standardized collection is maintained over the next 15 years. coordinators and the crew of R/V Thalassa and L'Europe are thanked for their cooperation during the experiments and implementation of new monitoring protocols. We are grateful to Morgan Le Moigne who shared with us the raw data for marine litter, and Lola Turpin for her help with Fig. 1 in this paper. We are indebted to Michael Paul for improving the English of the text. We also want to thank the editor and the anonymous reviewer for their thoughtful comments and suggestions on an earlier version of the manuscript.

Fig. 1 .

 1 Fig. 1. Main steps of the French experimentation.

Fig. 3 .

 3 Fig. 3. Gridded maps of average densities of bottom (a-b) total litter and (c-d) plastic sheets and bags collected in 2015 and 2016 during the MSFD optimised French fisheries surveys (data from Ifremer).

Fig. 4 .

 4 Fig. 4. Change in the total phytoplankton biomass (dark line) and in the relative proportion of the phytoplankton spectral groups (Green-algae, Blue-green algae, Cryptophyta and Diatoms) during the CAMANOC 16th September -13th October 2014 cruise on board the R/V Thalassa (data from Ifremer).

Table 1

 1 Main characteristics of R/V Thalassa and L'Europe.

	Vessel	Thalassa	L'Europe
	Main missions	Fisheries research	Fisheries research
		Physical oceanography	Coastal environment
		Underwater systems
		deployment
	Length overall	73.65 m	29.60 m
	Overall breadth	14.90 m	10.60 m
	Draught	6.10 m	3.45 m
	Year of	1996	1993
	construction
	On-board staff	Crew: 25	Crew: 8
		Scientists: 25	Scientists: 8
	Internet link	http://flotte.ifremer.fr/fleet/	http://flotte.ifremer.fr/
		Presentation-of-the-fleet/	fleet/Presentation-of-the-
		Vessels/Deep-sea-vessels/	fleet/Vessels/Coastal-
		Thalassa	vessels/L-Europe
		Accessed 02 February 2018	Accessed 02 February 2018

Table 2

 2 Main characteristics of IBTS, CGFS, EVHOE, PELGAS, MEDITS and PELMED fisheries surveys.

	Acronym Full name	Main sampling gear	Area	Period (quarter) Time series	References
	IBTS	International bottom trawl survey in the North	Bottom trawl	South North Sea / Eastern	I	1978 -cont. / 2007 -	[31]
		Sea		Channel		cont.
	CGFS	Channel Ground Fish Surveys	Bottom trawl	Eastern Channel	II	1988 -cont.	[34]
	EVHOE	Evaluation des ressources halieutiques de l'ouest	Bottom trawl	Bay of Biscay and Celtic Sea	IV	1987 -cont.	[35]
		européen
	PELGAS	Petits pélagiques Gascogne	Acoustic & Pelagic	Bay of Biscay	II	2000 -cont.	[16]
			trawl
	MEDITS	International bottom trawl surveys in the	Bottom trawl	Gulf of Lions and East Corsica III	1994 -cont.	[32]
		Mediterranean
	PELMED Petits pélagiques Méditerranée	Acoustic & Pelagic	Gulf of Lions	III	1993 -cont.	[33]
			trawl

Table 3

 3 Additional activities considered for MSFD monitoring on the 6 French vessel surveys.

	EVHOE		Sustainable funding			New MSFD protocol			Equipment upgrade to	standardize and increase the	number of sensors	New MSFD equipment				Sustainable funding		New MSFD protocol		New MSFD protocol		New MSFD protocol			New MSFD protocol		Sustainable funding		New MSFD protocol		New MSFD protocol
	CGFS		New MSFD protocol			New MSFD protocol			Equipment upgrade to	standardize and increase the	number of sensors	New MSFD equipment				Sustainable funding		New MSFD protocol		New MSFD protocol		New MSFD protocol			Sustainable funding		Sustainable funding		Sustainable funding		New MSFD protocol
	PELMED		Sustainable funding and	protocol harmonisation		New MSFD protocol			Equipment upgrade to	standardize and increase the	number of sensors	New MSFD equipment				Not possible (pelagic trawl)		New MSFD protocol		New MSFD protocol		New MSFD protocol			New MSFD protocol		Sustainable funding		Sustainable funding		New MSFD protocol
	MEDITS		No observers (no possibility	of having one more person	onboard)	No observers (no possibility	of having one more person	onboard)	Equipment upgrade to	standardize and increase the	number of sensors	New MSFD equipment				Sustainable funding		Not possible (no time	available)	New MSFD protocol		New MSFD protocol			Not possible (no time	available)	Not possible (no time	available)	New MSFD protocol		New MSFD protocol
	PELGAS		Sustainable funding			New MSFD protocol			Equipment upgrade to	standardize and increase the	number of sensors	New MSFD equipment				Not possible (pelagic trawl)		Not possible (no time	available)	New MSFD protocol		New MSFD protocol			Sustainable funding		Sustainable funding		Sustainable funding		New MSFD protocol
	IBTS		Sustainable funding			New MSFD protocol			Equipment upgrade to	standardize and increase the	number of sensors	New MSFD equipment				Sustainable funding		New MSFD protocol		New MSFD protocol		New MSFD protocol			Sustainable funding		Sustainable funding		Sustainable funding		New MSFD protocol
	Activity	MSFD descriptor related to	Observers for mammals and birds	D1, D4		Observers for floating litter	D10		Physics and chemistry of the water	column (CTD a )	D1, D4, D5, D7	Physics and chemistry of subsurface	water (Hight Frequency Ferry Box /	Pocket Ferry Box)	D1, D4, D5, D7	Seafloor litter	D10	Microplastics	D10	Gelatinous zooplankton	D1, D2, D4, D5	Organism collection for contaminants	and food web b	D4, D8, D9	Station water nutrient samples	D5	Station phytoplankton samples	D1, D4, D5	Station zooplankton samples	D1, D4	Automatic Identification System (AIS)	D11

a Measurements: conductivity, temperature, pressure, pH, dissolved oxygen, fluorescence and turbidity. b Not every year.

Acknowledgements

This work was supported by the French Ministry for Ecology, Sustainable Development and Energy (MEDDE). Vessel survey


David Guinovart Sanjuán
email: davidgs@matcom.uh.cu

Reinaldo Rodríguez
email: reinaldo@matcom.uh.cu

Ramos Raúl

Guinovart Díaz

Julián Bravo
email: jbravo@matcom.uh.cu

Andrei Martínez-Finkelshtein
email: andrei@ual.es

Aura Conci
email: auraconci2012@gmail.com

Frederic Lebon
email: lebon@lma.cnrs-mrs.fr

Serge Dumont
email: serge.dumont@u-picardie.fr

AVERAGE OF THE ELASTIC PROPERTIES OF BIO-HETEROGENEOUS STRUCTURES

Keywords: Effective elastic properties, Asymptotic Homogenization Method, Bio-composites, Laminated structures, Numerical analysis, Finite Element Methods

Sometimes the analysis of the stress state, crack initiation and some others phenomenon in heterogeneous media is complicated. In order to facilitate the ulterior study of such problems, we apply the asymptotic homogenization method (AHM) to a general laminated shell composite and an equivalent homogeneous elastic problem with effective properties is obtained. The physical and material properties of the bio-composites are studied using the two scale of AHM. The heterogeneous elastic problem is solved. As examples of the application of the AHM the cornea is considered as a laminated shell bio-composite. The cornea is composed with three cell layers: the outer epithelial cells, stromal cells and endothelial cells. Between these layers are extracellular structures called Bowman's and Descemet's membranes. The cornea has a variable thickness. The polar coordinates system is introduced to study the geometry of this media. A function that describes the variation of the thickness is used to generate the model of the cornea. The effective elastic tensor is calculated by AHM and FEM considering two models, one where the thickness is constants along the cornea and other considering thickness variation. These two models by AHM and FEM are compared and the influence of the variation of the thickness in the effective properties of the cornea is shown. The effective stress is determinate solving the homogenized problem for constant and variable thickness.

INTRODUCTION

The cornea possesses a unique combination of mechanical stiffness, strength, and optical transparency that enables it to serve as both a protective covering and the primary refractive component of the eye. These properties depend on the composition and orientation of the cornea, and differ from the central cornea to the limbus, [START_REF] Nguyen | An inverse finite element method for determining the anisotropic properties of the cornea[END_REF]. The stress and the strain are related with the deformation and the geometry of the cornea, hence the importance of the study of the elastic properties of the cornea [START_REF] Holmes | Corneal collagen fibril structure in three dimensions: Structural insights into fibril assembly, mechanical properties, and tissue organization[END_REF], [START_REF] Garcia-Porta | Corneal biomechanical properties in different ocular conditions and new measurement techniques[END_REF], [START_REF] Voorhies | Static and dynamic stress/strain properties for human and porcine eyes[END_REF]. Otherwise, the glaucoma and myopia are the most common problems that the cornea has on these days [START_REF] Gatinel | Determination of corneal asphericity after myopia surgery with the excimer laser: A mathematical model[END_REF], [START_REF] Broman | Influence of corneal structure, corneal responsiveness, and other ocular parameters on tonometric measurement of intraocular pressure[END_REF], [START_REF] Genest | Study on the influence of corneal biomechanical properties over the short term in response to corneal refractive therapy for myopia[END_REF]. The transplants and surgery have to be very precise proceeding, then knowledge of the effective properties become an important tool because in a certain kind of eye surgery, the human eyeball is deformed sustainably by the application of an elastic band. To many author talk about the geometry of the cornea and its influence on the mechanical properties, [START_REF] Donohue | Numerical modeling of the corneaâ's lamellar structure and birefringence properties[END_REF], [START_REF] Pinsky | Computational modeling of mechanical anisotropy in the cornea and sclera[END_REF], [START_REF] Sigat | Finite element modeling of optic nerve head biomechanics[END_REF]. Using the two scale Asymptotic Homogenization Method and Finite Elements Method, we determinate the effective properties of the cornea and compare the influence of its geometry on the results.

LINEAR ELASTIC PROBLEM IN THE CORNEA

In [START_REF] Skacel | Comparison of constitutive models of arterial layers with distributed collagen fibre orientations[END_REF], [START_REF] Genest | Study on the influence of corneal biomechanical properties over the short term in response to corneal refractive therapy for myopia[END_REF], [START_REF] Li | Nonlinear analysis of static axisymmetric deformation of the human cornea[END_REF], [START_REF] Pandolfi | Three-dimensional modeling and computational analysis of the human cornea considering distributed collagen fibril orientations[END_REF], [START_REF] Cardoso | Role of structural anisotropy of biological tissues in poroelastic wave propagation[END_REF], the stress-strain relationship in the cornea has an exponential behavior and the analysis of the elastic properties is made considering a non-linear model. Otherwise, beyond a certain stress, the σ -ǫ relationship becomes almost linear [START_REF] Elsheikh | Strain-rate sensitivity of porcine and ovine corneas[END_REF].

In order to reduce the analysis of the elastic theory problem in the cornea, a linear constitutive equation for the stress and the strain is considered, following the models presented in [START_REF] Asher | Etiology of keratoconus: proposed biomechanical pathogenesis[END_REF], [START_REF] Cabrera-Fernádez | Finite element analysis applied to cornea reshaping[END_REF], [START_REF] Donohue | Numerical modeling of the corneaâ's lamellar structure and birefringence properties[END_REF], [START_REF] Fisher | The elastic constants of the human lens[END_REF], [START_REF] Anderson | Application of structural analysis to the mechanical behaviour of the cornea[END_REF].

The cornea is considered a solid periodic structure Ω bound by the surfaces Σ 1 and Σ 2 . Fixed a coordinate system x = (x 1 , x 2 , x 3 ), the geometry of the cornea is described by a function ̺ ̺ ̺(x). For small deformations, the operator relating the strain ǫ ǫ ǫ and the stress σ σ σ in the cornea is the Hooke's law,

σ σ σ = C C C(x, y) : ǫ ǫ ǫ, (1)
where C C C is the elastic tensor, regular in x and periodic in y; here y = ̺ ̺ ̺(x)/ε is the fast or local variable and ε is a very small parameter which characterizes the periodicity. Following the Einstein's convention of summing over repeated indices, the elastic problem takes the form

σ ij j + f i = 0 on Ω, (2)
with boundary conditions

u j = u 0 j on Σ 1 , σ ij • n j = S i on Σ 2 , (3)
where {•} j denotes the covariant derivative with respect to the variable x j , f i are the components of the body forces, u j are the components of the displacement, n j is the normal vector in Σ 2 and the functions u 0 i and S i are the values of the displacement and the stress in Σ 1 and Σ 2 , respectively.

The general expression of the Cauchy's formula is

ǫ mn = 1 2 (u m n -u n m ). (4)
Taking into account the symmetry of the elastic tensor C ijkl and substituting the equation (4) in equation ( 1), the contravariant components of the stress tensor are

σ ij = C ijkl (u k,l -Γ p kl u p ), (5)
where Γ p kl denotes the Christoffel's symbols of second kind. Substituting equation (5) into equation (2), the general expression of the linear elastic problem can be obtained:

̺ k,j ε C ijmn |k + C ijmn ,j + Γ i jk C kjmn + Γ j jk C ikmn (u m,n -Γ r mn u r ) +C ijmn u m,nj -Γ r mn,j u r -Γ r mn u r,j + f i = 0 on Ω, (6)
with the boundary conditions

u i = u 0 i on Σ 1 C ijmn (u m,n -Γ r mn u r ) n j = S i on Σ 2 , (7)
where { } |j = ∂ ∂yj { } denotes the differentiation with respect to the fast or local curvilinear coordinate. One of the main difficulties in solving problem ( 6)-( 7) resides in the rapidly oscillating coefficients. In this paper we reduce it to an equivalent problem with homogenized coefficients, and thus obtain the effective properties of the cornea, by using the two-scale Asymptotic Homogenization Method (AHM) described next.

ASYMPTOTIC HOMOGENIZATION METHOD

The AHM is used to determine the homogenized problem and the effective properties of the solution of ( 6)-( 7). The general expression for the two-scale asymptotic expansion is

u (ε) m = v m + ε N p (0)m v p + N lk (1)m v l,k + o(ε), (8)
where the local functions N lk m ≡ N lk m (x, y) and N p (0)m ≡ N p (0)m (x, y) are y-periodic, and N = 0, where • is the average operator, defined by

• = 1 V Y Y (•) √ gdy.
Here Y is the periodic cell, V Y is its volume, and

g = det ([g ij ])
where [g ij ] is the metric tensor.

Substituting the expansion (8) into the heterogeneous problem ( 6)-( 7) we consider the the coefficients for ε -1 , which must vanish:

̺ q,j C ijlk + ̺ p,n C ijmn N lk (1)m|p ̺ q,j |q = 0, (9)
-̺ l,j C ijmn Γ p mn + ̺ l,j C ijmn ̺ t,n N p (0)m|t |l = 0. ( 10
)
Solving the differential equation ( 9), the values of the local functions N lk (1)m|p can be determined. Equation ( 10) allows to calculate N p (0)m|t as a function of

N lk (1)m|p , ̺ l,j C ijmn ̺ t,n N p (0)m|t |l = -Γ p mn ̺ q,s C ijks N mn (1)k|q ̺ l,j |l . ( 11
)
Applying the average operator to the coefficient of ε 0 and grouping the coefficients of v k,jl , v k,l and v l , we get

C ijkl e ≡ h ijkl = C ijkl + C ijmn ̺ p,n N kl (1)m|p , (12)
as well as expressions for the homogenized coefficients,

h ilk = h ijlk ,j -Γ l mn h ikmn + Γ i jn h njlk + Γ j jn h inlk , (13)
h il = -Γ l mn h ijmn ,j -Γ i jk h kjmn + Γ j jk h ikmn Γ l mn , (14)
h ijl * = -Γ l mn h ijmn . ( 15
)
Using these expressions we write the homogenized problem, consequence of ( 6)-( 7), but now with slowly oscillating coefficients:

h il v l + h ikl v l,k + h imln v l,nm + f i = 0 on Ω, (16)
and with boundary conditions

v i = u 0 i on Σ 1 h ijl * v l + h ijlk v l,k n j = S i on Σ 2 . ( 17
)
Now, the effective linear stress has the expression

σ ij e = C ijkl e v k,l + C ijk e v k , (18)
where C ijkl e are the effective coefficients (12), v k is the solution of the homogenized problem ( 16)-( 17) and

C ijk e is C ijk e = C ijmn N k (0)m|n -Γ k mn C ijmn . ( 19
)

COMPOSITION AND GEOMETRY OF THE CORNEA

The cornea is laminate shell composite, with three cell layers: the outer epithelial cells (I), stromal cells (III) and endothelial cells (V). Between these layers are extracellular structures called Bowman's (II) and Descemet's (IV) membranes, [START_REF] Anderson | Application of structural analysis to the mechanical behaviour of the cornea[END_REF][START_REF] Dupps | Biomechanics and wound healing in the cornea[END_REF]. The layers (I)-(V) of the cornea have central thickness of 14µm, 12µm, 450µm, 10µm and 14µm, respectively. A cross-section of the cornea is shown in Fig. 1. The spherical coordinate system (r, ϕ, θ) is used to describe the three-dimensional model of the cornea [START_REF] Cardoso | Role of structural anisotropy of biological tissues in poroelastic wave propagation[END_REF], shown in Fig. 2a. In order to determine the effective properties of the cornea, we consider it rotationally In other words, its elastic properties have no variation in the θ-direction. As a consequence, we can use a two-dimensional model of the cornea, which in turn can be described by means of the polar coordinate system (r, ϕ).

In our calculations we have used the following values of the cornea parameters: its thickness is approximately T 1 = 0.5mm at its apex and T 2 = 0.67mm at the limbus, [START_REF] Pandolfi | Three-dimensional modeling and computational analysis of the human cornea considering distributed collagen fibril orientations[END_REF][START_REF] Ruberti | Corneal biomechanics and biomaterials[END_REF]; the length of the half cross-section of the cornea is D 1 = 6mm, with D 2 = 2.5mm of height. The inner radio in the center of the cornea is R 1 = 7.56mm, at the limbus the radio is R 2 = 7.51mm and the angle between the radios is α = 0.8235 (see Fig. 2b), [START_REF] Cabrera-Fernádez | Finite element analysis applied to cornea reshaping[END_REF]. 20) and ( 21), respectively.

Alternatively, we consider also the constant-thickness model, corresponding to

̺(r, ϕ) = r. ( 21
)
Using the polar coordinates system both two-dimensional models of the cornea can be compared, see Fig. 3. The positions are given by the equations

x 1 = ̺ cos(ϕ), x 2 = ̺ sin(ϕ), ( 22
)
where ̺ is as in ( 20) and ( 21).

EFFECTIVE ELASTIC PROPERTIES OF THE CORNEA

The layers which compose the cornea are isotropic materials; the cornea is oriented un such a way that the fast variation of its properties corresponds to the y-direction, where y = ̺/ε and ̺ is given either by ( 20) or ( 21). In the polar coordinate system the metric tensor is g ij = diag[1, r 2 , 1] and the general expression for the isotropic components of C ijkl is

C ijkl = λg ij g kl + µ(g ik g jl + g il g jk ), (23)
where

g ij = [g ij ] -1
. Considering the polar metric tensor, the nonzero Christoffel's symbols are

Γ 1 22 = -r Γ 2 12 = Γ 2 21 = 1 r . ( 24
)
The elastic properties for the different layers of the cornea (see Fig. 1) are given in Table 1. These values were taken from [START_REF] Cabrera-Fernádez | Finite element analysis applied to cornea reshaping[END_REF]. The elastic tensor ( 23) can be determined using the input parameters of Table 1. Considering y = ̺/ε one-dimensional, the local problems ( 9) and ( 10) take the following expressions, respectively:

∂ ∂y ̺ ,j C ijlk + ̺ ,n C ijmn ∂N lk (1)m ∂y ̺ ,j = 0, (25)
∂ ∂y -̺ ,j C ijmn Γ p mn + ̺ ,j C ijmn ̺ ,n ∂N p (0)m ∂y = 0. (26)
In order to obtain the effective coefficients, the value of the local function ∂N lk (1)m /∂y is determined by integrating the local problem (25),

∂N lk (1)m ∂y = A ilk -̺ ,p C iplk ̺ ,n C ijmn ̺ ,j -1 , (27)
where the tensor A ilk ≡ A (x). With account of the periodic condition of the local function N lk m , the average ∂N lk

(1)m /∂y = 0, and the function A ilk yields

A ilk = ̺ ,p C iplk ̺ ,n C ijmn ̺ ,j -1 ̺ ,n C ijmn ̺ ,j -1 -1 . ( 28
)
Finally, functions ∂N lk (1)m /∂y are obtained by using the values of ̺ and C ijkl , given in ( 20) and ( 23), respectively, in the equations ( 27) and ( 28).

In this paragraph, the numerical method proposed to solve problems ( 25), ( 26) is described. Since the problem is composed of two coupled PDE's with singular (discontinuous) rigidity tensor, a P 1 (piecewise linear) finite element method (FEM) is used on a weak formulation of ( 25), ( 26).

Due to the properties of the finite element discretization, after the computation of an approximation of the solution N p (0)m and N lk (1)m using the FEM method, the integrals (see formula (12) for example) for the calculus of the effective coefficients are replaced by sums over each element contribution.

In what follows we use the super-indices t v and t 0 in order to denote the computed coefficients corresponding to the thickness functions ( 20) and ( 21), respectively.

In [START_REF] Anderson | Application of structural analysis to the mechanical behaviour of the cornea[END_REF], [START_REF] Asher | Etiology of keratoconus: proposed biomechanical pathogenesis[END_REF], [START_REF] Pandolfi | Three-dimensional modeling and computational analysis of the human cornea considering distributed collagen fibril orientations[END_REF], [START_REF] Lanchares | Finite element simulation of arcuates for astigmatism correction[END_REF], [START_REF] Li | Nonlinear analysis of static axisymmetric deformation of the human cornea[END_REF] and [START_REF] Ruberti | Corneal biomechanics and biomaterials[END_REF] the variable thickness along animal or human lens is considered. Otherwise, [START_REF] Broman | Influence of corneal structure, corneal responsiveness, and other ocular parameters on tonometric measurement of intraocular pressure[END_REF], [START_REF] Last | Compliance profile of the human cornea as measured by atomic force microscopy[END_REF], [START_REF] Donohue | Numerical modeling of the corneaâ's lamellar structure and birefringence properties[END_REF] and [START_REF] Huang | Mathematical model of corneal surface smoothing after laser refractive surgery[END_REF] a constant average thickness in the cornea's structure is analyzed. In this work, we address and compare both cases in order to analyze the influence of the thickness variation on the effective properties.

We also compare the values of the effective coefficients C ijkl e obtained by AHM and the finite element method (FEM) in the both models. The effective coefficients for t 0 , found by AHM and FEM, are shown in Table 2 for the values r = 7.56 and ϕ = π/2. In the case of the varying thickness (denoted by the superscript t v ), the coefficients do depend on r and ϕ (see Tables 3 and4). Thus, in Table 2 we present also the distance between C C C t0 e and C C C tv e . We present also the values of the components C 1111 e (Table 3) and C 1112 e (Table 4) of the elastic tensor C C C tv e for the case of the varying thickness. We can appreciate a very good coincidence between the effective properties calculated by AHM and FEM. It should be taken into account that in the present situation even a small difference in the effective coefficients (for varying and constant thicknesses), as in the third column of Table 2, renders important difference in the anisotropy of the material. Namely, for the constant thickness with isotropic layers the homogenized media exhibits transversely isotropic behavior, but in the case with varying thickness the material has a monoclinic symmetry.

A important comparison between the calculation of the effective coefficients, obtained by AHM and FEM, is the time of the computational operations. The values of Table 2 have been reproduce using AHM in a average time of 0.051 seconds, however, the average time reported by FEM was 1.6 seconds.

SOLVING THE HOMOGENIZED PROBLEM OF THE CORNEA

An external uniform force is applied on a two-dimensional cornea, simulating the intra-ocular pressure (IOP). This force f occurs in the r-direction and has contravariant components (cos ϕ, 0, 0) with respect to the basis of the tangent vectors g i . This force crates an internal displacement u| Σ1 = (0, 0, 0) and the traction S| Σ2 that has the same contravariants components of the force f .

The influence of the thickness variation in the stress tensor is analyzed. We consider a particular case of the elastic problem. The variation of the properties is in the x 1 direction and the only non zero displacement is u 1 . Under these assumptions, the homogenized problem ( 16)-( 17) is

h 11 v 1 + h 111 v 1,1 + h 1111 v 1,11 + cos ϕ = 0 on r ∈ (7.56, 8.06), (29)
with the boundary conditions

v 1 = 0 on r = 7.56 h 111 * v 1 + h 1111 v 1,1 n 1 = cos ϕ on r = 8.06. (30) for ϕ ∈ [π/2, π/2 + α].
The resulting coefficients h 1111 appear in Table 2; the corresponding coefficients h 111 , h 11 and h 111 * are determined using equations ( 13)-( 15).

The problem ( 29)-( 30) is solved at different points of the cornea. The effective stress is calculated using the equations ( 18) and ( 19) for the angles ϕ = π/2 (corneal apex) ϕ = 1.9002 (a point between the apex and the limbus) and ϕ = π/2 + α (limbus of the cornea). e for variable (⋄) and constant thickness(-) coincide. However, the effective stress has different behavior for other values of ϕ, effect being more apparent at the limbus, (ϕ = 2.2778). Notice that the highest stress is located at the center of the cornea and the lowest one corresponds to the limbus.

CONSLUSIONS

The AHM is a powerful tool to determine the effective properties of the cornea quickly and with high precision. The methodology bring the possibility to aboard problems like the analysis of the bio-mechanics properties of the cornea after an instra-stromal ring segments implantation [START_REF] Martínez-Finkelshtein | Comparative analysis of some modal reconstruction methods of the shape of the cornea from the corneal elevation data[END_REF][START_REF] Martínez-Finkelshtein | Adaptive cornea modeling from keratometric data[END_REF], or the study of the variation of the corneal thickness in a healthy eye as function of the anterior face of the cornea and inner ocular pressure (IOP).

The difference of the length along the cornea is studied and its influence in the mechanical behavior of the cornea as a laminated shell bio-composite. The analytic expressions of the effective coefficients are very important result of this work. The numerical values are compared with the effective coefficients obtain by FEM, very good coincidence is reported and significant difference between the time of the computational operations.

Figure 1 :

 1 Figure 1: Cross-section through the cornea

Figure 2 :

 2 Figure 2: Geometry of the cornea: a) Three dimensional representation of the cornea given by the function (20). b) Geometry of the cross section of the cornea Taking into account the values of the parameters R 1 , R 2 , D 1 , D 2 , T 1 , T 2 and α, we can use the following function that describes the geometry of the cornea: ̺(r, ϕ) = r + (m • r + n) (ϕ -π/2) 2 , (20) with m = 0.5014, n = -3.8643, r ∈ [7.56, 8.06] and ϕ ∈ [π/2, π/2 + α].

Figure 3 :

 3 Figure 3: Comparison of the two-dimensional cornea: thickness varying (continuous line) and constant thickness along the cornea structure (discontinuous line), given by (20) and (21), respectively.

Figure 4 :

 4 Figure 4: Comparison of the behavior of the effective stress σ 11 e at different points of the cornea with constant and variable thickness along the structure.

Table 1 :

 1 Mechanical properties of the cornea.

	Layers	Young's Modulus Poisson's ratio
	Epithelial cells (I)	0.622	0.49
	Bowman's membrane (II)	0.275	0.49
	Stromal cells (III)	0.279	0.49
	Descemet's membrane (IV)	0.304	0.49
	Endothelial cells (V)	0.331	0.49

Table 2 :

 2 Effective coefficients C C C t0 e calculated by AHM and FEM. Difference between C C C t0 e and C C C tv e .

	Coefficients C C C t0 e (AHM) C C C t0 e (FEM) C 1111 e 4.8812 4.8814 C 1122 e 0.0821 0.0836 C 1133 e 4.6898 4.6936 C 2222 e 0.0015 0.0015 C 2233 e 0.0821 0.0821 C 3333 e 4.8886 4.8888 C 1112 e 0 0 C 2212 e 0 0 C 3312 e 0 0 C 2323 e 0.0017 0.0017 C 1323 e 0 0 C 1313 e 0.0957 0.0957 C 1212 e 0.0017 0.0017	C C C t0 e -C C C tv e 8 × 10 -4 (AHM) 8 × 10 -8 2 × 10 -6 2 × 10 -9 4 × 10 -8 3 × 10 -15 1 × 10 -8 3 × 10 -7 1 × 10 -5 2 × 10 -8 6 × 10 -6 1 × 10 -6 8 × 10 -8

Table 3 :

 3 Components C 1111

	e	of the elastic tensor C C C tv e for different values of r and ϕ.
	1.5708 AHM 7.56 4.88123 4.88133 4.88098 4.88132 4.88083 4.88129 1.9002 2.3943 ϕ → FEM AHM FEM AHM FEM r ↓
	7.76 4.88123 4.88133 4.88132 4.88134 4.88137 4.88133
	8.06 4.88123 4.88133 4.88182 4.88127 4.88211 4.88221
	Table 4: Components C 1112 e	of the elastic tensor C C C tv e for different values of r and ϕ.
	1.5708 r ↓ AHM FEM ϕ → 7.56 0 0 7.76 0 0 -0.0088 × 10 -4 -0.0098 × 10 -4 -0.0172 × 10 -4 -0.0193 × 10 -4 1.9002 2.3943 AHM FEM AHM FEM 0.0243 × 10 -4 0.0272 × 10 -4 0.0479 × 10 -4 0.0535 × 10 -4 8.06 0 0 -0.0584 × 10 -4 -0.0653 × 10 -4 -0.1148 × 10 -4 -0.1285 × 10 -4

ACKNOWLEDGEMENTS

The authors gratefully acknowledge to the project SHICHAN, supported by FSP (Cooperation Scientifique Franco-Cubaine) Cuba 2011-26, PROJET N o 29935XH and to the project Composite Materials from University of Havana. RRR thanks to Department of International Relations -DRI -Universidade Federal Fluminense -UFF, Edital Programa de Apoio a Projetos de Internacionalização da UFF -PIUFF -Edição 2014 for the financial suport during his stay at UFF as a visiting professor in 2015. AC gratefully thank the institutional support provided by INCT-MACC (National Institute of Science and Technology in Computer Science Medical Assistance and the Brazilian agency CNPq Project number 302298/2012-6. The fifth author (AMF) was partially supported by MICINN of Spain and by the European Regional Development Fund (ERDF) under grant MTM2011-28952-C02-01, by Junta de Andalucía (Excellence Grant P11-FQM-7276 and the research group FQM-229), and by Campus de Excelencia Internacional del Mar (CEIMAR) of the University of Almería. This work was completed during a visit of AMF to the Department of Mathematics of the Havana University, whose hospitality he wishes to acknowledge.