Quentin Nogues
email: quentin.nogues33@gmail.com

Aurore Raoux

Emma Araignous

Aurélie Chaalali

Tarek Hattab

Boris Leroy

Frida Ben

Rais Lasram

Valérie David

François Le

François Le Loc'h

Jean-Claude Dauvin

Nathalie Niquil

Cumulative effects of marine renewable energy and climate change on ecosystem properties: sensitivity of ecological network analysis

Keywords: Cumulative impact, Linear inverse modeling, Marine Renewable Energies, Ecological Network Analysis, ENA food web. Graphical

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The world is in constant evolution, and human activities have deeply changed the rate of this evolution (Halpern et al., 2008b). Due to constant human activity, ecosystem functioning can no longer be dissociated from the dynamics of anthropogenic activities [START_REF] Vitousek | Human Domination of Earth's Ecosystems[END_REF]. Although coastal ecosystems are already heavily stressed, they are going to be increasingly exploited owing to the ever-increasing human population and demand for resources [START_REF] Halpern | Spatial and temporal changes in cumulative human impacts on the world's ocean[END_REF]. The English Channel -an epicontinental sea -is considered as one of the most anthropized marine ecosystems in the world (Halpern et al., 2008b;[START_REF] Dauvin | Chapter 6: The English Channel: La Manche[END_REF]. It is subjected to multiple anthropogenic drivers such as marine transport, fishing, sediment dredging and aggregate extraction (Dauvin et al., 2004;[START_REF] Dauvin | Chapter 6: The English Channel: La Manche[END_REF]. It is also a hotspot for the future development of an offshore wind farm (OWF) in

France [START_REF] Pezy | The environmental impact from an offshore windfarm: Challenge and evaluation methodology based on an ecosystem approach[END_REF][START_REF] Raoux | Benthic and fish aggregation inside an offshore wind farm: Which effects on the trophic web functioning?[END_REF]. The effects of OWFs on ecosystems are divided in two phases: (i) the construction phase and (ii) the operational phase. The construction phase is characterized by short-lasting heavy disturbance (extreme noise from pile driving and vessels, cabletrenching disturbance), while the operational phase is associated to long-lasting effects on the environment [START_REF] Hammar | Assessing ecological risks of offshore wind power on Kattegat cod[END_REF]. One of the long-lasting effects of OWFs is the creation of new habitats and shelters for benthic and demersal species through the introduction of hard substrates in the surrounding soft substrate habitats [START_REF] Wilhelmsson | Fouling assemblages on offshore wind power plants and adjacent substrata[END_REF]. The increased habitat heterogeneity can lead to changes in the abundance, biomass, and species richness of benthos and fish [START_REF] Coates | Enrichment and shifts in macrobenthic assemblages in an offshore wind farm area in the Belgian part of the North Sea[END_REF][START_REF] Wilhelmsson | The influence of offshore windpower on demersal fish[END_REF], known as the reef effect. This reef effect is likely to be the main ecosystem effect of OWFs during the exploitation phase and can potentially affect the whole food web [START_REF] Bergström | Effects of an offshore wind farm on temporal and spatial patterns in the demersal fish community[END_REF]. Only the exploitation phase is considered in this study, and the construction phase is ignored [START_REF] De Mesel | Succession and seasonal dynamics of the epifauna community on offshore wind farm foundations and their role as stepping stones for non-indigenous species[END_REF][START_REF] Langhamer | Artificial reef effect in relation to offshore renewable energy conversion: State of the art[END_REF][START_REF] Petersen | Offshore Windmill Farms: Threats to or Possibilities for the Marine Environment[END_REF].

To help managers and decision-makers in their sustainable development mission, it is a priority to understand how OWFs act on ecosystem dynamics, which is mainly driven by trophic interactions between species. Trophic network modeling can help to study those interactions. In a network, a stimulus on one part of the model can cascade throughout the network. Trophic network modeling is therefore an interesting tool to understand how drivers act on the whole ecosystem.

To analyze how food webs are affected by drivers, [START_REF] Ulanowicz | Ecology, the ascendent perspective[END_REF][START_REF] Ulanowicz | Quantitative methods for ecological network analysis[END_REF] used ecological network analysis (ENA) indices. ENA indices can assess properties like the size, function or organization of a network through multiple metrics related to ecological processes, such as omnivory, recycling, overhead of flows [START_REF] Borrett | Walk partitions of flow in Ecological Network Analysis: Review and synthesis of methods and indicators[END_REF][START_REF] Heymans | Regime shifts in the Northern Benguela ecosystem: Challenges for management[END_REF]Niquil et al., 2012a;[START_REF] Ulanowicz | Growth and Development : Ecosystems Phenomenology[END_REF]. These indices have been considered to be good potential indicators of the ecosystem health status and were thus cited by the OSPAR convention as promising indicators [START_REF] Niquil | Ongoing research on ecosystem health indicators for food webs in the MSFD context[END_REF][START_REF] De La Vega | How to include ecological network analysis results in management? A case study of three tidal basins of the Wadden Sea, south-eastern North Sea[END_REF][START_REF] Safi | Vitamine ENA: A framework for the development of ecosystembased indicators for decision makers[END_REF]. They were subsequently employed to understand the effects of human activities (e.g., the extension of a harbor) on ecosystems (Tecchio et al., 2016), or even to develop different management scenarios [START_REF] Heymans | Regime shifts in the Northern Benguela ecosystem: Challenges for management[END_REF][START_REF] Tomczak | Ecological Network Indicators of Ecosystem Status and Change in the Baltic Sea[END_REF]. [START_REF] Raoux | Benthic and fish aggregation inside an offshore wind farm: Which effects on the trophic web functioning?[END_REF] explored a new way to look at the potential effects of OWFs through food web models and flow analysis. These authors implemented the Ecopath with Ecosim (EwE) approach [START_REF] Christensen | Ecopath with Ecosim: Methods, capabilities and limitations[END_REF][START_REF] Heymans | Ecopath Theory, Modeling, and Application to Coastal Ecosystems[END_REF][START_REF] Polovina | Model of a coral reef ecosystem -I. The ECOPATH model and its application to French Frigate Shoals[END_REF] to model the food web at a planned OWF (Courseulles-sur-mer, Bay of Seine, English Channel, France). One of their main results

was that the total ecosystem activity, recycling, and the ecosystem maturity increased after OWF construction, due to the reef effect [START_REF] Raoux | Benthic and fish aggregation inside an offshore wind farm: Which effects on the trophic web functioning?[END_REF][START_REF] Raoux | Measuring sensitivity of two OSPAR indicators for a coastal food web model under offshore wind farm construction[END_REF]. Results also indicated an anticipated increase in detritivory flows and an increased benthic fish biomass after OWF construction.

However, one of the main weaknesses of impact studies is that they focus on a single ecosystemic driver (Raoux et al., 2018), and ignore the other drivers of disturbance with long-lasting effects such as climate change [START_REF] Hoegh-Guldberg | The impact of climate change on the world's marine ecosystems[END_REF]. The main issue when ignoring combined drivers is to miss cumulative effects that may alter the magnitude or direction of predicted changes [START_REF] Breitburg | Understanding Effects of Multiple Stressors: Ideas and Challenges[END_REF][START_REF] Vinebrooke | Impacts of multiple stressors on biodiversity and ecosystem functioning: the role of species co-tolerance[END_REF][START_REF] Fu | Risky business: The combined effects of fishing and changes in primary productivity on fish communities[END_REF][START_REF] Planque | How does fishing alter marine populations and ecosystems sensitivity to climate?[END_REF]. Hence, irrelevant or misleading recommendations can be issued when cumulative effects are not included (Halpern et al., 2008).

We propose to investigate the cumulative effects of climate change and one OWF using ecological network analysis on a food web model. To do so, we transformed the We used those scenarios combining the effects of climate change and the OWF to compute a series of ENA indices to study the effects of each driver separately and then of the combined drivers on the food web of Courseulles-sur-Mer. We explored the capacities of ENA indices to study cumulative effects. Finally, we discussed the advantages and weaknesses of a framework based on ENA indices to investigate cumulative impacts and provide management recommendations.

Materials and methods

Study area

This study is focused on the area of the future OWF of Courseulles-sur-Mer (Normandy, France), that will be located in the lower middle part of the Bay of Seine (Figure 1). The bay opens onto the English Channel in its northern part, its depth is relatively shallow with a mean of 30 m at the future OWF location. The OWF area will cover 50 km², and 64 offshore wind turbines are planned. The OWF will be built on coarse sand and sandy gravels harboring benthic communities of the Bay of Seine [START_REF] Baffreau | Mapping benthic communities: An indispensable tool for the preservation and management of the eco-socio-system in the Bay of Seine[END_REF].

Cumulative impact models

In order to test ENA indices in cumulative impact scenarios, we created trophic models based on two studies: (i) a trophic modeling of the expected reef effect caused by the future Courseulles-sur-Mer OWF [START_REF] Raoux | Benthic and fish aggregation inside an offshore wind farm: Which effects on the trophic web functioning?[END_REF], and (ii) results from niche models simulating climate change [START_REF] Rais Lasram | An open-source framework to model present and future marine species distributions at local scale[END_REF].

Wind farm effect on the food web

The construction of turbines and their associated scour protection creates new hard-substrate habitats and shelters such as artificial reefs. These new hard-substrate habitats increase habitat heterogeneity, leading to increased abundance, biomass, and species richness of the benthic macrofauna (e.g., the mussel (Mytilus edulis)) and of demersal fish (e.g., the pouting (Trisopterus luscus)) [START_REF] Coates | Enrichment and shifts in macrobenthic assemblages in an offshore wind farm area in the Belgian part of the North Sea[END_REF][START_REF] Raoux | Benthic and fish aggregation inside an offshore wind farm: Which effects on the trophic web functioning?[END_REF][START_REF] Wilhelmsson | The influence of offshore windpower on demersal fish[END_REF].

The reef effect of the future Courseulles-sur-Mer OWF was modeled using trophic network models quantifying the flows of energy between organisms, as described in [START_REF] Raoux | Benthic and fish aggregation inside an offshore wind farm: Which effects on the trophic web functioning?[END_REF]. This trophic model was based on the Ecopath with Ecosim approach [START_REF] Christensen | Ecopath with Ecosim: Methods, capabilities and limitations[END_REF][START_REF] Polovina | Model of a coral reef ecosystem -I. The ECOPATH model and its application to French Frigate Shoals[END_REF]. A first Ecopath model was built to model the ecosystem before the establishment of the OWF.

Ecopath is a mass balance model, meaning that the sums of the inflows and outflows of the system are equal. It was used as a basis for an Ecosim model. Ecosim is a time dynamic model based on the mass balance Ecopath model [START_REF] Pauly | Ecopath, Ecosim, and Ecospace as tools for evaluating ecosystem impact of fisheries[END_REF]. It was used to model the reef effect of the future Courseulles-sur-Mer OWF and to create an Ecopath model 30 years from the OWF construction. Two Ecopath models were thus created: a baseline Ecopath model called BOWF (before offshore wind farm), and a model called REEF, based on the reef effect demonstrated to be the main long-term effect of the farm [START_REF] Raoux | Benthic and fish aggregation inside an offshore wind farm: Which effects on the trophic web functioning?[END_REF]. Both models were composed of the same 37 compartments of species, with similar trophic interactions [START_REF] Leguerrier | Numerical analysis of the food web of an intertidal mudflat ecosystem on the Atlantic coast of France[END_REF].

While the Ecopath approach is a well acknowledged mass balance approach, it can be criticized for its limitations in the modeling of low-trophic-level compartments -which are often subjected to high inter and intra year biomass variability -and because the method is based on a manual balancing procedure, which is not based on uncertainty evaluation of the input parameters. In order to take into account the uncertainty in the input parameters but also to define uncertainty in the outputs (flow values or ENA indices), another mass balance modeling approach is more suitable, i.e., the LIM linear inverse models, Monte Carlo Markov Chain or LIM-MCMC (N. Niquil et al., 2012;van Oevelen et al., 2010a). Linear inverse modeling relies on a set of linear equalities and inequalities to estimate unknown flows in food webs. The equalities set the mass balance of the model, while the inequalities correspond to the confidence intervals of the measured flows. The Monte Carlo method is a sampling procedure used to determine the complete coverage of the range of possible solutions. Which means that while Ecopath only considers one value for each parameter/flow, LIM-MCMC integrates a range of values to represent the variability of living organisms [START_REF] Meersche | xsample() : An R Function for Sampling Linear Inverse Problems[END_REF]Niquil et al., 2012;van Oevelen et al., 2010). This variability is also very useful to compare models statistically.

We thus used the BOWF and REEF Ecopath models of [START_REF] Raoux | Benthic and fish aggregation inside an offshore wind farm: Which effects on the trophic web functioning?[END_REF] and transformed them into linear inverse models (LIMs). Because the two Ecopath models used the same topology, we based our topology on them. However, we had to reduce the number of biological compartments to shorten the computation time. The number of compartments decreased from 37 to 19 (Figure 2, Table A.1), and aggregation was based on the diet of each Ecopath compartment (Table A.1, A.3 & A.4;[START_REF] Leguerrier | Numerical analysis of the food web of an intertidal mudflat ecosystem on the Atlantic coast of France[END_REF].

Keeping the same topology between the LIM BOWF model and the LIM REEF model allowed us to compare the results of the two models free of any structural bias (Table A.3 & A.4). We defined the mass balance equation for each compartment. In a mass balance equation, the input of a compartment is equal to its output. The input is the consumption of the compartment, while the output is the excretion, the respiration and the production of the compartment available for the system (predation).

Production (P) = consumption (Q) -excretion (U) -respiration (R)
Regarding the constraints of the model, we set minimum and maximum values for parameters such as the production to biomass ratio (P/B), the production to consumption ratio (P/Q), the respiration to consumption ratio (R/Q), and the excretion to consumption ratio (U/Q) (Table A .2). We used constraint values from the literature for some compartments, while others were computed with the Ecopath values and pedigrees. Pedigrees were used to define the uncertainty around the Ecopath values to determine minimum and maximum values (Table A .2). When the Ecopath values were considered very reliable ('Fish, planktivorous', 'Fish, benthos feeders', 'Fish, piscivorous' and 'Flat fish'), the uncertainty around the P/B, P/Q and U/Q values was set to 10 %. The diet constraints were calculated with the same method for all compartments of the models (BOWF and REEF). Finally, we added primary production values to drive the two LIM models: while the Ecopath model does not consider primary production values, the LIM model needs minimum and maximum values of potential primary production [START_REF] Behrenfeld | Photosynthetic rates derived from satellite-based chlorophyll concentration[END_REF][START_REF] Napoléon | Multi-parametric relationships between PAM measurements and carbon incorporation, an in situ approach[END_REF].

The xsample function from the LIM package was used to compute the LIM models (van den [START_REF] Meersche | xsample() : An R Function for Sampling Linear Inverse Problems[END_REF]. The LIM-MCMC model depends on two parameters for its computing: the jump and the number of iterations. With the equations and the constraints, the LIM MCMC model creates a polytope of solution that includes all the possible values for each flow of the system. The number of iterations indicates the number of solutions sampled in the space of solution. The mean distance between iterations is the jump. If the sampling goes outside the space of solution, it will bounce back (mirror technique) inside the space. A jump of 0.05 gC.m -2 and 500,000 iterations was chosen after testing the mean shifting value/variability of the biggest flows in order to properly sample solutions.

Climate change simulation

In order to assess cumulative impacts through a trophic modeling approach, we had to integrate the climate change effect in both LIM trophic models. To evaluate the effect of climate change, we used results from the niche models published in Ben Rais [START_REF] Rais Lasram | An open-source framework to model present and future marine species distributions at local scale[END_REF]. Niche model algorithms are correlative approaches aimed at identifying the potential species niches by correlating species occurrences to environmental variables such as temperature, salinity or other parameters. Niche models were built with an ensemble modeling procedure combining different modeling techniques.

Future climate change projections for 72 species of the Bay of Seine were based on the "business as usual" representative concentration pathway scenarios (Intergovernmental Panel on Climate Change, RCP 8.5) at the 2091-2100 time steps [START_REF] Rais Lasram | An open-source framework to model present and future marine species distributions at local scale[END_REF].

The compartments of the LIM trophic model were not all simulated in the niche model approach.

Consequently, we focused on integrating the niche model results of the commercially exploited compartments, which included 4 fish compartments -Fish, benthos feeders (FBF); Flat Fish (FFI); Fish, Piscivorous (FPI); Fish, Planktivorous (FPL) -and 2 mollusk compartments -Cephalopods (CEP); King Scallop (KSC). Those compartments represented the compartments impacted by climate change.

To integrate the results of the niche models into the trophic models, we linked the evolution of the suitability index of the niche models to biomass according to [START_REF] Chaalali | From species distributions to ecosystem structure and function: A methodological perspective[END_REF]. We used the evolution of the suitability index C to multiply the biomass of the species by (1-C) to reflect the biomass changes:

where P ref is the suitability index of the species in the current niche model and P rcp the suitability index of the species in RCP 8.5.

As the production to biomass ratio (P/B) of fish is based on temperature [START_REF] Lassalle | Lower trophic levels and detrital biomass control the Bay of Biscay continental shelf food web: Implications for ecosystem management[END_REF], the P/B ratios of all the fish compartments impacted by climate change were modified following the temperature values of the RCP 8.5 scenarios (Table A.6). We used the temperature values corresponding to the habitats of each compartment (surface temperature for the pelagic compartments, sub-surface temperature for the bentho-pelagic compartments, and bottom temperature for the benthic and demersal compartments). In a mass balanced condition, the P/B ratio of fish species is considered to be equal to the instantaneous coefficient of total mortality (Z) (Figure 3):

where M is natural mortality and F is fishing mortality, and M is calculated from the empirical equation:

where K is the curvature parameter of the von Bertalanffy growth function (VBGF), while L∞ is the asymptotic length and T the mean environmental temperature in °C depending of the depth.

We computed new P/B ratios by changing the temperature in the natural mortality equation (M) without changing fishing mortality (F) so as to be coherent. The evolution of natural mortality was then applied to the P/B ratio of each fish compartment. If natural mortality increased by 10 % for a fish species owing to the new temperature, we increased the P/B ratio of the species by 10 % in the climate change scenario.

We had to use different methods for the Mollusk compartments. There was no equation connecting the P/B ratio with temperature for cephalopods. We therefore used the cephalopod feeding rate equation because of its link to temperature [START_REF] O'dor | Energy and nutrient flow[END_REF]

:
where W is the body mass in g and T is temperature in degrees Celsius.

We computed a new feeding rate for cephalopods with the niche model. The evolution of the feeding rate between the current and the future RCP 8.5 temperatures allowed us to estimate the P/B ratio of the cephalopod compartment. We used a food conversion rate of 30 % for cephalopods [START_REF] Boyle | Cephalopods: Ecology and Fisheries[END_REF].

For king scallops, we used the growth rate equation (Laing, 1999). The equation was verified by comparing the temperature estimated from the RCP 8.5 scenarios with the temperature ranges applied in growth experiments [START_REF] Chauvaud | Variation in size and growth of the Great Scallop Pecten maximus along a latitudinal gradient[END_REF][START_REF] Laing | Effect of temperature and ration on growth and condition of king scallop (Pecten maximus) spat[END_REF]. The temperature estimated with the RCP model was sustainable by Pecten maximus in situ and ex situ (15.3°C), and resulted in increased production:

We thus changed the P/B ratio of the king scallop compartment according to the evolution of its growth rate between the current temperature and the future temperature under the RCP 8.5 scenario.

Ecological network analysis

With their 19 compartments each, the LIM models contained a flow matrix of 144 flows. We computed multiple metrics to analyze the ecological properties of each model. We first computed the throughflow/activity of (i) each compartment (sum of all the inflows/outflows) (Table A.1), (ii) the import (IMP), and (iii) respiration (RES) to understand the effects of the climate change and of the OWF at the flow matrix level. We also computed ENA indices (Table 1). ENA indices summarize the emergent properties of the ecosystem. A range of indices was chosen according to previous recommendations about their use by ecosystem managers [START_REF] Fath | Ecological network analysis metrics: The need for an entire ecosystem approach in management and policy[END_REF][START_REF] Safi | Vitamine ENA: A framework for the development of ecosystembased indicators for decision makers[END_REF]. where is the flow between compartments j and i. [START_REF] Finn | Measures of ecosystem structure and function derived from analysis of flows[END_REF][START_REF] Ulanowlcz | Symmetrical overhead in flow networks[END_REF].

Average Mutual

Information (AMI)

Average mutual information measures the organization of a network, the more web-like the system is, the lower the AMI should be. [START_REF] Gallager | Information Theory and Reliable Communication[END_REF][START_REF] Latham | Network flow analysis algorithms[END_REF])

Ascendency (A)
Ascendency is a measurement of the growth of the system, integrating its size (T..) and its organization (AMI). [START_REF] Ulanowicz | An hypothesis on the development of natural communities[END_REF][START_REF] Ulanowicz | Growth and Development : Ecosystems Phenomenology[END_REF][START_REF] Latham | Network flow analysis algorithms[END_REF] Development capacity (DC)

The total development capacity is the upper limit of ascendency, it represents the maximum development of the system. [START_REF] Latham | Network flow analysis algorithms[END_REF])

Overhead (R)
The overhead is the "reserve" where TL is the trophic level of i or j. [START_REF] Libralato | System Omnivory Index[END_REF] Recycling index or Finn Cycling Index (FCI)

The recycling index is the fraction of energy recycled in the system.

where TST is the total system throughflow and TSTc the cycled total system throughflow. [START_REF] Odum | Trends Expected in Stressed Ecosystems[END_REF] Mean Trophic

Level (MTL)

The mean trophic level is the mean trophic level of the where B is the biomass of i or j. We followed this framework (Figure 4) to analyze the combined effects of climate change and of the reef effect depending on the impacted compartment. We used the same methodology as [START_REF] Fu | Risky business: The combined effects of fishing and changes in primary productivity on fish communities[END_REF] to characterize the combined effect on each ENA index and for each scenario. Then, we summed the variation ( ) of each single effect in a same scenario to get the reef + climate change additive effect:

We also computed the cumulative effect as the variation between the initial BOWF model and the cumulative impact scenario (reef effect and climate change effect) for each ENA ( ):

To determine the combined effect, we compared the additive effect variation with the cumulative effect variation. If the two variations were equal, the effect was considered additive:

If the additive effect variation was lower than the cumulative effect variation and both variations were positive or negative, the response was synergistic positive or negative:

Synergistic positive:

Synergistic negative:

If the additive effect variation was higher than the cumulative effect variation while both variations were positive or negative, the response was dampened positive or negative.

If the additive effect variation was opposite to the cumulative effect variation, the response was antagonistic. When the combined effect variation was positive, it was a positive antagonism, and vice versa for negative antagonism (Figure 5). differences even for tiny effects [START_REF] Tecchio | Evaluating ecosystem-level anthropogenic impacts in a stressed transitional environment: The case of the Seine estuary[END_REF]. The non-parametric Cliff delta [START_REF] Cliff | Dominance statistics: Ordinal analyses to answer ordinal questions[END_REF] was chosen for its better suitability (Valérie Girardin & Justine Lequesne, pers. com., Laboratoire de Mathématiques Nicolas Oresme, [START_REF] Tecchio | Evaluating ecosystem-level anthropogenic impacts in a stressed transitional environment: The case of the Seine estuary[END_REF]. The same threshold as [START_REF] Romano | Exploring methods for evaluating group differences on the NSSE and other surveys: Are the t-test and Cohen's d indices the most appropriate choices?[END_REF] and [START_REF] Macbeth | Cliff 's Delta Calculator: A non-parametric effect size program for two groups of observations[END_REF] was used considering the differences between the datasets: negligible if the Cliff delta (| ∂Cliff |) was < 0.147, low if < 0.33, medium if < 0.474, or strong if > 0.474.

The Cliff delta was used to compare single effect models ("reef only" and "climate change only" scenarios) to the initial reference BOWF model and to compare the REEF model with the cumulative impact scenarios (reef effect + climate change effect). This was applied to the throughflow/activity (input/output) of each compartment as well as on the ENA. The delta also allowed us to compare the additive effect variation ( ) with the cumulative effect variation ( . If the difference was considered medium (| ∂Cliff | < 0.33), the cumulative effect was considered different than the additive effect. We determined the cumulative effect depending on the difference between the additive effect variation and the cumulative effect variation (Figure 5).

To reduce the number of models when describing the ENA indices, we used an average unweighted pair group method with arithmetic mean (UPGMA) based on Euclidean distance. The UPGMA compared all the different models according to their mean ENA values.

Results

Reef effect on the food web

When the reef effect of the offshore wind farm was modeled in the LIM model, it resulted in an increased activity of the benthic compartment and in a decreased activity of the pelagic compartment. The reef effect also resulted in increased detritus consumption, increased phytoplankton activity and increased import (Figure 6).

Effects of climate change on the food webs before offshore wind farm construction

When the BOWF models were impacted by the climate change scenarios, three compartments displayed a strong increase in activity and 11 displayed a strong decrease in activity. Throughout the compartments and scenarios, climate change had mainly a negative effect on compartment activity.

Negative effects on activity can be divided into two types: (i) the direct effect of climate change on the targeted compartment (for instance, lower biomass of the cephalopod compartment in the cephalopod scenario leading to lower cephalopod activity), and (ii) the indirect / cascading effect of climate change on activity (lower biomass of the cephalopod compartment in the cephalopod scenario leading to lower cetaceous activity) (Figure 6).

Combined effects of climate change and of an offshore wind farm on the food web

When the offshore wind farm (REEF) models were impacted by the climate change scenarios, 8 compartments showed a strong increase in activity and 13 a strong decrease in activity. The REEF scenarios displayed greater changes due to climate change than the BOWF scenarios (38 strong variations in total for the REEF scenarios versus 30 for the BOWF scenarios), with a relatively more positive effect on activity. Similarly to the BOWF scenarios, the negative effects on activity were of two types: (i) the direct effects, and (ii) the indirect / cascading effects. While the direct effects of climate change appeared to be similar in the BOWF and REEF scenarios, the indirect / cascading effects were mostly different. There were four similarities between the BOWF and REEF models among the twenty indirect effects of climate change: (i) the indirect increase in invertebrate predator activity in the total scenario and the "fish, benthos feeders" scenario, (ii) the indirect decrease in phocidae activity in the total scenario and in the "fish, piscivorous" scenario, (iii) the indirect decrease in cetaceous activity in the total scenario and in the "fish, piscivorous" scenario, and (iv) the indirect decrease in import in the total scenario and the "fish, benthos feeders" scenario (Figure 6).

Changes in the structure and functioning of the food web

The flow matrices were used to compute the ENA indices. The UPGMA discriminated five groups of models. The models including the sole climate change effect and the BOWF model formed three groups (with two groups composed of only one model). The models including the reef + climate change effect formed two groups (Figure 7,A).

In both the BOWF and REEF scenarios, the "fish, benthos feeders" (FBF) scenario and the total (TOT) scenario showed differences with the rest of the climate change scenarios (Figure 7 The Cliff delta was used to compare the ENA results of: the BOWF model and the single effect models (reef effect or climate change effect), and between the REEF model and the cumulative models (reef effect + climate change effect).

Cumulative effects on ENA indices

The cumulative impact assessment showed a wide variety of cumulative effects -additional, synergetic, dampened, and antagonistic. Six of the seven scenarios showed at least one cumulative effect on their ENA (0.33 < | ∂Cliff |). As for ENA indices, seven out of eight indices in total showed at least one cumulative effect (ΔIksep ≠ ΔIkcum) (Figure 8).

Discussion

This study assesses the cumulative impacts of climate change and an OWF on the Bay of the Seine food web, based on LIM models. Because of the complexity of modeling climate change in its full extent [START_REF] Larocque | Common challenges for ecological modelling: Synthesis of facilitated discussions held at the symposia organized for the 2009 conference of the International Society for Ecological Modelling in Quebec City, Canada[END_REF], this work was focused on the effect of climate change on the indigenous species of the bay, more particularly on commercially exploited species. That is why it is a sensitivity analysis of these cumulative impacts on ENA indices. First, our sensitivity analysis revealed that each effect taken separately had an impact on the functioning and organization of the ecosystem. The "fish, benthos feeders" compartment appeared particularly sensitive to the two single effects. When the effects were combined, we observed significant cumulative effects on multiple ENA indices for multiple scenarios. Those significant cumulative effects led to unexpected effects on the ecosystem. Overall, ENA indices appeared sensitive to the cumulative impact.

The structuring role of the reef effect

The results of the REEF model seem to suggest a regime shifting towards a new stable state. An ecosystem shift is documented as a change at several trophic levels leading to ecosystem restructuring [START_REF] Andersen | Ecological thresholds and regime shifts: approaches to identification[END_REF][START_REF] Lees | Characterizing regime shifts in the marine environment[END_REF]. The restructuring of the system should be visible on the ecosystem structure but also on its functioning. The restructuring due to the reef effect was already predicted by [START_REF] Raoux | Benthic and fish aggregation inside an offshore wind farm: Which effects on the trophic web functioning?[END_REF]2019) based on the evolution of the mean trophic level (MTL 2). We modeled a similar decrease in MTL 2 between the BOWF and REEF models. A decline in MTL 2 would explain a transition in the food web from long-lived high-trophic-level organisms like piscivorous fish towards short-lived low-trophic-level organisms such as invertebrates and planktivorous fish [START_REF] Pauly | Fishing down marine food webs[END_REF]. In the reef effect case, the biomass of the higher trophic level was not reduced, whereas the biomass of low-trophic-level organisms increased. Like [START_REF] Raoux | Benthic and fish aggregation inside an offshore wind farm: Which effects on the trophic web functioning?[END_REF]2019), we consider this change as a restructuring of the community towards a new stable state rather than an unsustainable alteration of the ecosystem [START_REF] Holling | Engineering resilience versus ecological resilience[END_REF]. We improved over [START_REF] Raoux | Benthic and fish aggregation inside an offshore wind farm: Which effects on the trophic web functioning?[END_REF]2019) by also observing a change in the ecosystem's functioning. This change was noticeable through the different responses of the BOWF and REEF models to climate change. When we impacted the two models with the climate change scenarios, very few similarities between the BOWF response and the REEF response emerged. The differences in the cascading responses to climate change between the BOWF and REEF models may be explained by a shift in the system [START_REF] Andersen | Ecological thresholds and regime shifts: approaches to identification[END_REF][START_REF] Lees | Characterizing regime shifts in the marine environment[END_REF]. Our ENA results reflect a shift from a bentho-pelagic system towards a more benthic system. The system appeared more active with lower-trophic-level species relying less on herbivory and more on detritivory and recycling. Alongside the increase in redundant pathways, the system appeared more resistant to changes.

The ENA variations that we predicted between the BOWF model and the REEF model described a more resistant OWF system. Resistance is defined as the ability of a system to maintain its original state in a disturbed context as described by [START_REF] Holling | Engineering resilience versus ecological resilience[END_REF], and in accordance with 'ecological resilience' as described by [START_REF] Harrison | Stability under Environmental Stress: Resistance, Resilience, Persistence, and Variability[END_REF]. Many ENA indices have been related to the notion of resistance. [START_REF] Lassalle | Lower trophic levels and detrital biomass control the Bay of Biscay continental shelf food web: Implications for ecosystem management[END_REF] hypothesized that an increase in the detritivory to herbivory (D/H) rate was associated with a system more resistant to primary production variation. Recycling (FCI) may act as a buffer during disturbances and increase the resistance of the system [START_REF] Saint-Béat | Trophic networks: How do theories link ecosystem structure and functioning to stability properties? A review[END_REF]. The overhead/redundancy of the flow (RoC and RoCi) may also act as a buffer during disturbances to maintain the system and thus increase its resistance [START_REF] Ulanowlcz | Symmetrical overhead in flow networks[END_REF].

We predicted an increase of each of those indices (D/H, FCI, RoCi). Our results are less conflicting than the ones observed by [START_REF] Raoux | Measuring sensitivity of two OSPAR indicators for a coastal food web model under offshore wind farm construction[END_REF] and describe a more coherent picture of the system, with benthic compartments playing a key role in the resistance of ecosystems [START_REF] Norling | Structural and functional effects of Mytilus edulis on diversity of associated species and ecosystem functioning[END_REF][START_REF] Dame | A Statistical Test of Network Analysis: Can it Detect Differences in Food Web Properties?[END_REF][START_REF] Raoux | Measuring sensitivity of two OSPAR indicators for a coastal food web model under offshore wind farm construction[END_REF][START_REF] Wang | Effects of established offshore wind farms on energy flow of coastal ecosystems: A case study of the Rudong offshore wind farms in China[END_REF].

The effect of climate change on the system also appeared to validate the more resistant state of the REEF model, especially on the "fish, benthos feeders" scenario (FBF). The BOWF model led to more changes in ENA indices than the REEF model: Its lower omnivory level (SOI) revealed a less complex system evolving from a web-like system towards a more linear system [START_REF] Libralato | System Omnivory Index[END_REF]. As the complexity of a system is an indicator of flexibility [START_REF] Fagan | Omnivory as a Stabilizing Feature of Natural Communities[END_REF][START_REF] Lobry | Diversity and stability of an estuarine trophic network[END_REF], we can expect a less stable and more vulnerable system. We also predicted that the system would lose part of its recycling capacity (FCI) and be less mature due to the lower redundancy of internal flows (RoCi) [START_REF] Christensen | Ecopath with Ecosim: Methods, capabilities and limitations[END_REF][START_REF] Odum | The Strategy of Ecosystem Development[END_REF][START_REF] Ulanowicz | The dual nature of ecosystem dynamics[END_REF][START_REF] Ulanowlcz | Symmetrical overhead in flow networks[END_REF]. On the other hand, the only modeled effect of climate change on the "fish, benthos feeders" scenario of the REEF model was a higher detritivory to herbivory rate (D/H) of the system. As such, the reef effect seemed to increase the resistance of the system to climate change. However, one must not forget that these results depend on the way the climate change effect was modeled. The reef effect could also promote other effects of climate change, like the establishment of invasive species [START_REF] Langhamer | Artificial reef effect in relation to offshore renewable energy conversion: State of the art[END_REF], but their study would need specific field experiments.

Role of the compartment impacted by climate change in the response of the ecosystem

Our analysis of the different ENA responses showed a pattern segregating the responses of the scenarios depending on the compartment impacted by climate change. The "fish, benthos feeders" (FBF) and the total (TOT) climate change scenarios showed important estimated changes in their structure and functioning, while the other climate change scenarios seemed close to their reference model. This happened in both the BOWF and REEF models. In both cases, the "fish, benthos feeders" (FBF) and the total (TOT) climate change scenarios impacted the "fish, benthos feeders" compartment (FBF). This leads us to think that the importance of the "fish, benthos feeders" compartment isn't defined by the aggregating role of the reef effect [START_REF] Reubens | Aggregation and feeding behaviour of pouting (Trisopterus luscus) at wind turbines in the Belgian part of the North Sea[END_REF], but rather by its position in the food web and thus its keystoneness. The keystoneness of a species defines its structuring role [START_REF] Power | Challenges in the Quest for Keystones[END_REF][START_REF] Libralato | Keystone Species and Keystoneness[END_REF] and is not an intrinsic property of any given species [START_REF] Fauth | Identifying potential keystone species from field data -An example from temporary ponds[END_REF]. The fact that the "fish, benthos feeders" compartment maintained its role despite the shift in the regime of the system might be due to the central position of its trophic niche in both systems.

The "fish, benthos feeders" (FBF) compartment seemed to play two roles in both models: first, it regulated the "invertebrates, benthic predators" compartment (IPR), and second, it ramified the system through its high omnivory. This was predicted when the "fish, benthos feeders" were impacted by the climate change effect (Figure 7). The trophic niche of the "fish, benthos feeders" was thus claimed by the "invertebrate, benthic predators", and consequently changed the functioning of the food web. This result was different when considering the ecosystem shift caused by the reef effect: the "invertebrate, benthic predators" seemed more capable to replace the "fish, benthos feeders" in their niche. This replacement seemed to be facilitated by the shift of the ecosystem towards a more benthic system, in which the invertebrate benthic predators might be more successful in filling the same trophic role as the fish benthos feeders. As such, the system seemed less altered by the effect of climate change on the "fish, benthos feeders" scenario in the offshore wind farm system. However, we should remain careful because changing communities can have wider effects on the ecosystem likely to go undetected by trophic modeling approaches [START_REF] Fontaine | The ecological and evolutionary implications of merging different types of networks[END_REF][START_REF] Kéfi | Network structure beyond food webs: Mapping non-trophic and trophic interactions on Chilean rocky shores[END_REF][START_REF] Kéfi | More than a meal... integrating non-feeding interactions into food webs[END_REF]. As such, it is necessary to monitor such keystone species to maintain the ecosystem properties before, during and after the exploitation of the offshore wind farm. Previous studies have indeed already highlighted the advantages of monitoring keystone species (Payton et al., 2002;[START_REF] Libralato | A method for identifying keystone species in food web models[END_REF][START_REF] Gelcich | Navigating transformations in governance of Chilean marine coastal resources[END_REF].

Understanding the mechanism behind the unexpected effect of the cumulative impact

Our scenarios show a wide variety of cumulative impacts, in line with previous studies that consider the additive effect to be the main type of combined effect in marine ecosystems, with a significant role of the cumulative impact on ecosystems (Crain et al., 2008). We evidenced synergetic and antagonistic effects, which proved the ability of ENA indices to reflect unexpected effects visible only when studying cumulative impacts [START_REF] Jackson | Historical overfishing and the recent collapse of coastal ecosystems[END_REF][START_REF] Przeslawski | Synergistic effects associated with climate change and the development of rocky shore molluscs[END_REF]. Cumulative impacts are often studied using community or population level metrics (Crain et al., 2008). By observing the same type of response with ENA, we demonstrate that cumulative effects are also significant at the functional level of the ecosystem. This is useful as it allows us to follow the properties of the ecosystem and its state in relation to the cumulated impact. Using functional metrics like ENA indicators might also help us better understand cumulative effects. This strengthens the importance of incorporating multiple scenarios of events in future ecological network analysis.

The strongest synergetic effect was predicted on the detritivory to herbivory (D/H) index of the "fish, benthos feeders" scenario (FBF). This synergistic effect seems to result from the shifting regime of the Courseulles-sur-Mer ecosystem. As the system shifted from a benthopelagic system to a more benthic system, the effect of climate change on the ecosystem functioning seemed to change (Figure 7). In the BOWF model, the climate change effect on the fish benthos feeders resulted in an increased activity of the invertebrate benthic predators. This higher invertebrate benthic predator activity did not lead to an increase in detritivory because the system was not producing enough benthic detritus. Yet, this changed with the reef effect. As a system becomes more benthic, it produces more benthic detritus (Norling andKautsky, 2008, 2007;[START_REF] Raoux | Benthic and fish aggregation inside an offshore wind farm: Which effects on the trophic web functioning?[END_REF]. This increase in benthic detritus became available to the invertebrate benthic predators, and increased the detritivory to herbivory rate of the system. Many invertebrate benthic predators are indeed also scavengers. This changing response of the ecosystem to climate change due to the regime shift induced this unexpected effect on the detritivory to herbivory rate. This higher detritus consumption with the higher detritivory to herbivory rate of the OWF ecosystem resulted in a synergistic effect, as predicted by the "fish, benthos feeders" scenario (Table 2). An interaction between effects has already been shown to modify the effect of one or multiple events, e.g., in chemical reactions [START_REF] Pelletier | Ecotoxicological Effects of Combined UVB[END_REF] and physiological/ecological reactions [START_REF] Przeslawski | Synergistic effects associated with climate change and the development of rocky shore molluscs[END_REF]Christensen et al., 2006). We observed it at the functional level of the ecosystem. Integrating the evolving interactions between effects in a shifting ecosystem could help us better anticipate these effects. For us to detect those shifts, monitoring the entire ecosystem using ecological network analysis is needed, using local and relevant data on all trophic levels [START_REF] Araignous | Recommandations pour une approche écosystémique des aires d'implantationd'énergies marines renouvelables : Cas d'étude du parc éolien offshore de Courseulles-sur-mer[END_REF].

While two effects can interact with each other toward a synergistic effect, they may also interact negatively, with a resulting antagonistic or dampened effect. This was mainly modelled for the detritivory to herbivory rate of the "flat fish" scenario (Table 2). When considered independently, the climate change effect and the reef effect increased the detritivory to herbivory rate. However, when the effects were combined, the cumulative effect on the detritivory to herbivory rate was dampened.

This could be explained by the fact that the regime shift made the ecosystem impervious to the climate change effect. This mechanism of resistance has already been acknowledged at the physiological level or at the population level [START_REF] Vinebrooke | Impacts of multiple stressors on biodiversity and ecosystem functioning: the role of species co-tolerance[END_REF]. ENA proved able to do the same at the functional level of the ecosystem.

The regime shift due to the reef effect seemed to explain multiple cumulative effects. However, sometimes it was uncertain whether the cumulated effect resulted from the model structure or not.

This was visible on the total overhead of the "fish, benthos feeders" scenarios. In this case, the dominating effect of climate change overcame the cumulative effect (Table 2). With the "fish, benthos feeders" compartment relying significantly on imports (like cetaceous, seals and birds), a reduction of its biomass will lower the import overhead. As the import overhead is a part of the total overhead (Ulanowicz and Norden, 1990), the total overhead will decrease even though the internal overhead remains unchanged. This could be the result of the model structure itself, as the dependency of the "fish, benthos feeders" compartment to import is mainly due to the reduced scale of the trophic model. Moreover, ENA indices are known to be potentially highly influenced by the model structure [START_REF] Baird | On the consequences of aggregation and balancing of networks on system properties derived from ecological network analysis[END_REF][START_REF] Johnson | The effects of aggregation on the performance of the inverse method and indicators of network analysis[END_REF]. The lower import overhead could also be the sign of a more isolated system compared to the rest of the English Channel ecosystem [START_REF] Baird | Comparative study on the trophic structure, cycling and ecosystem properties of four tidal estuaries[END_REF]. Using a single ENA index to describe a changing system is not suitable to describe a changing ecosystem; a larger pool of indices is more adapted to reach an overall picture of the ecosystem organization and functioning (de la [START_REF] De La Vega | How to include ecological network analysis results in management? A case study of three tidal basins of the Wadden Sea, south-eastern North Sea[END_REF]. The same is true for combined impacts, which make it possible to predict different types of cumulative effects for different parameters of the ecosystem. While an index can describe a synergetic effect of the combined reef and climate change effects, another index may describe an antagonistic effect. This point should be emphasized when working on a limited number of indices, as multiple indices can be impacted differently. Interactions between events thus appear to be radiative -not unidirectional -, i.e., different properties of an ecosystem can evolve in different ways. This emphasizes the need for complementary indices to provide a holistic view of the ecosystem [START_REF] Fath | Ecological network analysis metrics: The need for an entire ecosystem approach in management and policy[END_REF][START_REF] Safi | Vitamine ENA: A framework for the development of ecosystembased indicators for decision makers[END_REF], especially while addressing the effect of cumulative impacts.

Ecological network analysis indices seem to be suitable for addressing a wide range of cumulative effects. Along with an integrative approach, this ability to study cumulative impacts in their overall diversity makes ENA indices the ideal ecological indicators necessary for the future of ecological science and management (de la [START_REF] De La Vega | How to include ecological network analysis results in management? A case study of three tidal basins of the Wadden Sea, south-eastern North Sea[END_REF][START_REF] Fath | Ecological network analysis metrics: The need for an entire ecosystem approach in management and policy[END_REF][START_REF] Safi | Vitamine ENA: A framework for the development of ecosystembased indicators for decision makers[END_REF]. It is now time to implement those indices in more complex cumulative impact scenarios to describe the mechanisms behind cumulative impacts. zone III, positive dampened effect; zone IV, negative synergistic effect; zone V, negative antagonistic effect; and zone VI, negative dampened effect.

Figure 1 :

 1 Figure 1: Localization of the future offshore wind farm of Courseulles-sur-Mer in the Bay of Seine.

  Four steps are necessary to create a LIM model, in which we determined: -The topology of the network based on the possible flows -The mass balance equations -The constraints of the model -The computing parameters (jump and iterations).

Figure 2 :

 2 Figure 2: Functional compartments of the Courseulles-sur-Mer ecosystem model organized according to their trophic level, based on the BOWF trophic levels.

Figure 3 :

 3 Figure 3: Schematic representation of the climate change modeling approach.

  (Herbivory) in the ecosystem.where DET is the flow of detritus consumption and PP the flow of consumption by

  network's compartments. It is based on the compartments' trophic levels. Its values can be 1 (primary producers and detritus), 2 (all consumers), and can reach > 4 (top predators).

  different types of cumulative effects Sixteen models were computed to test ENA indices in multiple scenarios. The models included: -The reference BOWF model -The REEF model -The climate-change-only scenarios, based on the LIM BOWF model, with each of the selected compartments impacted by the climate change effect and an extra model (BOWFTOT) combining all the compartments impacted at once -The cumulative impact scenarios, based on the LIM REEF model, with each of the selected compartments impacted by the climate change effect and an extra model (REEFTOT) combining all the compartments impacted at once. Each model was named according to its initial model (BOWF or REEF) and to the name of the impacted compartment if impacted by climate change (Figure 4).

Figure 4 :

 4 Figure 4: Modeling framework of the sixteen models all represented by gray boxes, with the initial BOWF model, single effect models including seven climate change scenarios and the REEF model, and the combined models including seven cumulative scenarios.

First

  , we computed the relative index variation between the reference BOWF model and a single effect model (reef effect only or climate change only) for each ENA: where I c is the index value of the reference model and I k the index value of a single-effect model (reef only or CC only).

Figure 5 :

 5 Figure 5: Different types of cumulative effects on the different index values (e.g., ENA index).

Figure 6 :

 6 Figure 6: Variation of all the throughflow/activity (input/output) in gC.m² between: the BOWF model and the single effect models (reef effect or climate change effect, a group on the left), and between the REEF model and the cumulative models (reef effect + climate change effect, b group on the right). Red, increased activity; blue, decreased activity; bold-bordered square, strong variation according to the cliff delta (| ∂Cliff | < 0.474).

  , B). Those models showed a greater number of strong differences (| ∂Cliff | > 0,474) with their respective reference model compared to the other scenarios. Four strong differences were observed under the BOWFTOT model compared to BOWF, six under BOWFFBF compared to BOWF, and three under REEFTOT and REEFFBF compared to REEF.The climate change effects on the "fish, benthos feeders" (FBF) scenario and the total (TOT) scenario appeared as structuring as the REEF effect itself (Figure7, A& B). The climate change effect strongly impacted (| ∂Cliff | > 0.474) four indices of the total scenario and six of the "fish, benthos feeders" scenario, while the reef effect strongly impacted six indices. The remaining climate change scenarios had little effect on the system (Figure 7, B).

Figure 7 :

 7 Figure 7: A) Following UPGMA to determine different groups of models, 5 groups were selected. B)

Figure 8 :

 8 Figure 8: Cumulative effects of the different scenarios on multiple ENA indices (+/-SD). Each cumulative effect is represented in a zone. Zone I, positive synergistic effect; zone II, positive

  Ecological network analysis indices: sensitive tools to manage the cumulative impact of the REEF and climate change effects.

Figure 1 :

 1 Figure 1: Localization of the future offshore wind farm of Courseulles-sur-Mer in the Bay of Seine.

Figure 2 :

 2 Figure 2: Functional compartments of the Courseulles-sur-Mer ecosystem model organized according to their trophic level, based on the BOWF trophic levels.

Figure 3 :

 3 Figure 3: Schematic representation of the climate change modeling approach.

Figure 4 :

 4 Figure 4: Modeling framework of the sixteen models all represented by gray boxes, with the initial BOWF model, single effect models including seven climate change scenarios and the REEF model, and the combined models including seven cumulative scenarios.

Figure 5 :

 5 Figure 5: Different types of cumulative effects on the different index values (e.g., ENA index).

Figure 6 :

 6 Figure 6: Variation of all the throughflow/activity (input/output) in gC.m² between: the BOWF model and the single effect models (reef effect or climate change effect, a group on the left), and between the REEF model and the cumulative models (reef effect + climate change effect, b group on the right). Red, increased activity; blue, decreased activity; bold-bordered square, strong variation according to the cliff delta (| ∂Cliff | < 0.474).

Figure 7 :

 7 Figure 7: A) Following UPGMA to determine different groups of models, 5 groups were selected. B) The Cliff delta was used to compare the ENA results of: the BOWF model and the single effect models (reef effect or climate change effect), and between the REEF model and the cumulative models (reef effect + climate change effect).

Figure 8 :

 8 Figure 8: Cumulative effects of the different scenarios on multiple ENA indices. Each cumulative effect is represented in a zone. Zone I, positive synergistic effect; zone II, positive antagonistic effect;

Table 1 :

 1 ENA indices computed with the LIM models results. 264

	Name	Objective	Calculation	References
	Total activity of the system or Total System Throughput (T..)	The total system throughput is equal to the sum of all the flows in the system. It is a network level indicator of the size and activity of the system.

Table 2 :

 2 Representation of the different mechanisms of the cumulative effects based on the LIM model results.

	Scenario ENA index	Climate change effect alone	Reef effect alone	Combined effects	Type of effect cumulative	Mechanism
	Fish,	Detritivory					Interaction between the two
	benthos feeders	to herbivory	+	+	+++ Positive synergistic	effects due to the regime shift caused by the reef
	(FBF)	(D/H)					effect.
	Fish, piscivorous (FPI)	Internal relative overhead (RoCi)	+	+	-	Negative antagonistic	Interaction between the two effects due to the regime shift caused by the reef effect.
		Detritivory					Resilience of the ecosystem
	Flat fish (FFI)	to herbivory	+	+	+	Positive dampened	to the climate change effect due to the regime shift
		(D/H)					caused by the reef effect.

Table Captions Table 1 :

 Captions1 ENA indices computed with the LIM models results.

Table 2 :

 2 Representation of the different mechanisms of the cumulative effects based on the LIM model results.

	898	Annex:

Table A .1: Ecopath groups from Raoux et al. (2017) compared to LIM groups. 899 Ecopath compartments from Raoux et al. (2017)

 A

			LIM trophic compartments	LIM Symbol
	1 Bottlenose dolphins 2 Harbour porpoises	1 Cetaceans	CET
	3 Harbour seals 4 Grey seals	2 Phocidae	PHO
	5 Plunge and pursuit, diverse seabirds 6 Surface feeders, seabirds	3 Birds	BIR
	7 Benthopelagic cephalopods 8 Benthic cephalopods	4 Cephalopods	CEP
	9 Fish, mackerel
	Fish, European pilchard
	Fish, European sprat	5 Fish, planctivorous	FPL
	Fish, planctivorous
	Fish, atlantic horse mackerel
	Fish, European seabass
	Fish, sharks and rays
	Fish, gurnard
	Fish, pouting	6 Fish, benthos feeders	FBF
	Fish, poor cod
	Fish, benthos feeders
	Fish, sea bream
	Fish, Atlantic cod
	Fish, whiting	7 Fish, piscivorous	FPI
	Fish, piscivorous
	Fish, sole
	Fish, European plaice	8 Fish, flat fish	FFI
	Fish, other flatfish
	Benthic inv., predators	9 Invertebrates, predators	IPR
	Benthic inv., filter feeders	10 Invertebrates, filter feeders	IFF
	Benthic inv., bivalves	11 Bivalvia	BIV
	King scallop	12 King scallops	KSC
	Benthic inv., deposit feeders	13	Invertebrates, deposit feeders	IDF
	Suprabenthos	14 Suprabenthos	SUP
	Meiofauna	15 Meiofauna (Nematodes)	NEM
	Zooplankton	16 Zooplankton	ZOO
	Bacteria	17 Bacteria	BAC
	Phytoplankton	18 Phytoplankton	PHY
	Detritus	19 Detritus	DET
	900
	901

Table A .5: Biomass modifications caused by the climate change effect (RCP 8.5).

 A

	Compartment \ Biomass	BOWF	REEF	BOWF with RCP 8.5	REEF with RCP 8.5
	CEP	1.98E-02	2.46E-02	6.60E-03	8.18E-03
	FPL	5.92E+00	4.76E+00	2.53E+00	2.01E+00
	FBF	3.20E+00	6.69E+00	6.25E-01	9.29E-01
	FPI	2.68E-01	1.02E-01	1.81E-02	4.40E-03
	FFI	7.85E-02	1.78E-01	3.07E-02	6.06E-02
	KSC	7.70E-01	7.43E-01	3.31E-03	3.20E-03

Table A .6: P/B ratio modifications as a result of climate change effect modeled over the 2091-2100 912 period (RCP 8.5). 913

 A

	Group	Scientific name	P/B current	ecology	T° current	Linf	M	F	T° RCP 8,5	M RCP 8,5	P/B RCP 8,5
		Mullus surmuletus	1.42	Demersal	13.01	46.83	0.20	1.22	15.30	0.21	1.43
		Mustelus mustelus	0.37	Demersal	13.01 175.00	0.12	0.25	15.30	0.13	0.38
		Labrus bergylta	1.42	Demersal	13.01	59.45	0.19	1.23	15.30	0.20	1.43
		Zeus faber	0.86 Benthopelagic 13.01	69.30	0.25	0.61	15.40	0.26	0.87
		Scyliorhinus canicula	0.44	Demersal	13.01	81.25	0.22	0.22	15.30	0.23	0.45
		Trigloporus lastoviza	0.55	Demersal	13.01	39.50	0.77	-0.22 15.30	0.82	0.60
	Fish, Benthos	Scyliorhinus stellaris	0.44	Demersal	13.01	-	-	-	15.30	-	0.45
	Feeders	Raja montagui	0.44	Demersal	13.01	78.02	0.33	0.11	15.30	0.35	0.46
		Callionymus lyra	0.82	Demersal	13.01	21.25	0.92	-0.10 15.30	0.97	0.87
		Dicentrarchus labrax	0.54	Demersal	13.01	82.70	0.20	0.34	15.30	0.21	0.55
		Trisopterus luscus	1.32 Benthopelagic 13.01	41.00	0.79	0.53	15.40	0.83	1.36
		Spondyliosoma cantharus	0.58	Demersal	13.01	42.20	0.36	0.22	15.30	0.38	0.60
		Trisopterus minutus	1.50 Benthopelagic 13.01	22.00	0.71	0.79	15.40	0.75	1.54
		Pollachius pollachius	0.62 Benthopelagic 13.01	85.60	0.27	0.35	15.40	0.29	0.64
	Fish,	Gadus morhua	1.20 Benthopelagic 13.01 123.67	0.18	1.02	15.40	0.19	1.21
	Piscivorous	Merlangius merlangus	1.07 Benthopelagic 13.01	42.70	0.52	0.55	15.40	0.55	1.10
		Limanda limanda	1.14	Demersal	13.01	25.60	0.79	0.35	15.30	0.84	1.19
	Flat Fish	Platichthys flesus Solea solea	0.56 0.70	Demersal Demersal	13.01 13.01	40.80 42.40	0.49 0.44	0.07 0.26	15.30 15.30	0.51 0.46	0.58 0.72
		Pleuronectes platessa 0.85	Demersal	13.01	59.40	0.22	0.63	15.30	0.23	0.86
		Clupea harengus	0.75 Benthopelagic 13.01	33.10	0.50	0.25	15.40	0.53	0.78
		Engraulis encrasicolus 0.58	Pelagic	13.02	18.60	0.79	-0.21 15.40	0.84	0.63
		Hyperoplus lanceolatus	1.12 Benthopelagic 13.01	29.40	0.64	0.48	15.40	0.67	1.15
	Fish, Planctivorous	Ammodytes tobianus Scomber scombrus Trachurus trachurus	1.12 Benthopelagic 13.01 0.83 Pelagic 13.02 0.55 Pelagic 13.02	19.70 39.40 40.20	1.08 0.57 0.37	0.04 0.26 0.18	15.40 15.40 15.40	1.14 0.61 0.39	1.18 0.87 0.57
		Chelidonichthys lucernus	0.55	Demersal	13.01	48.40	0.34	0.21	15.30	0.36	0.57
		Sardina pilchardus	0.99	Pelagic	13.02	19.50	0.64	0.35	15.40	0.68	1.03
		Sprattus sprattus	1.34	Pelagic	13.02	17.50	0.59	0.75	15.40	0.62	1.37
	914
	915

Table A .7: Keystoneness values for each compartment of the model as computed in Libralato et al. (2006).

 A

	Compartments	Mean BOWF keystoneness	Mean BOWF biomass	Mean REEF keystoneness	Mean REEF biomass
	FBF	1.75E-01	3.19E+00	1.87E-01	6.69E+00
	BAC	4.48E-02	7.50E-01	5.26E-02	7.70E-01
	IPR	1.20E-02	2.94E+00	2.47E-02	3.01E+00
	PHY	-1.82E-03	3.24E+00	4.16E-03	3.24E+00
	FPI	-2.28E-02	2.68E-01	-8.60E-03	1.02E-01
	ZOO	-4.53E-02	1.72E+00	-3.80E-02	1.79E+00
	PHO	-1.77E-01	9.41E-04	-1.62E-01	2.76E-03
	NEM	-1.97E-01	9.70E-01	-1.89E-01	1.06E+00
	CEP	-3.06E-01	1.98E-02	-2.94E-01	2.47E-02
	DET	-3.62E-01	1.90E+01	-3.57E-01	1.90E+01
	FPL	-3.64E-01	5.92E+00	-3.57E-01	4.76E+00
	KSC	-3.70E-01	7.70E-01	-3.61E-01	7.43E-01
	FFI	-3.94E-01	7.86E-02	-3.85E-01	1.78E-01
	BIV	-4.11E-01	1.95E+01	-4.53E-01	4.29E+01
	SUP	-4.14E-01	2.00E+00	-4.06E-01	1.71E+00
	BIR	-4.89E-01	1.70E-02	-4.75E-01	2.25E-02
	IFF	-6.21E-01	3.12E+00	-6.18E-01	4.78E+00
	IDF	-6.21E-01	3.57E+00	-6.16E-01	2.98E+00
	CET	-6.61E-01	4.29E-04	-6.47E-01	1.51E-04

Acknowledgements

This work was funded by the Normandy Region (RIN Trophi-Service project), and was part of the TROPHIK project, which benefited from France Energies Marines and State financing managed by the National Research Agency under the Investments for the Future program (reference ANR/FEM EMR-ITE ANR-10-IED-0006-12). NN, EA, FBRL and FFL are also supported by the APPEAL project (ANR-10-IED-0006-25). We also thank for their help in compiling the datasets and for giving expert advice Jeremy Lobry, Géraldine Lassalle, Mathilda Haraldsson and Maud Thermes and all the partners and collaborators of the TROPHIK project for their help in compiling the datasets and for giving expert advice. We thank Annie Buchwalter for English correction.

Invertebrates

ZOOplankton

-0.00 0.00 -0.00 0.50 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.10 0.00 0.00 --0.12 0.31 -0.

Invertebrates, PRedators


Milan Courcoux-Caro

Charles Vanwynsberghe

Cédric Herzet

Alexandre Baussard

Sequential Sensor Placement using Bayesian Compressed Sensing for Source Localization

Keywords: Acoustic, source localization, sequential sensor placement, compressed sensing

This paper deals with the sensor placement problem for an array designed for source localization. When it involves the identification of a few sources, the compressed sensing framework is known to find directions effectively thanks to sparse approximation. The present contribution intends to provide an answer to the following question: given a set of observations, how should we make the next measurement to minimize (some form of) uncertainty on the localization of the sources? More specifically, we propose a methodology for sequential sensor placement inspired from the "Bayesian compressive sensing" framework introduced by Ji et al. Our method alternates between a step of sparse source localization estimation, and a step to choose the sensor position that minimizes the covariance of the estimation error. Numerical results show that an array designed by the proposed procedure leads to better performance than sensors positioned at random.

I. INTRODUCTION

Sensor arrays are commonly used to achieve source localization (SL) in acoustic or radar applications. The last decade has shown promising results in the use of sparse approximation to solve this problem, when the measured field (acoustic or electromagnetic) results from a few number of sources. Moreover, when performed with an array of random sensor positions, it shows close connections with compressed sensing (CS) [START_REF] Fortunati | Single-snapshot doa estimation by using compressed sensing[END_REF]- [START_REF] Edelmann | Beamforming using compressive sensing[END_REF].

This paper investigates the design of the array geometry to obtain efficient compressed measurements in SL. An extensive literature exists when it is based on optimizing the directivity pattern (typically by reducing the side lobe level peak, see e.g., [START_REF] Boeringer | Particle swarm optimization versus genetic algorithms for phased array synthesis[END_REF], [START_REF] Yan | Sidelobe reduction in array-pattern synthesis using genetic algorithm[END_REF]) but it is rather dedicated to beamforming techniques. In sound-field control applications, a review of joint source and sensor placement methods show how to enhance the stability and accuracy of the synthesized field [START_REF] Koyama | Optimizing source and sensor placement for sound field control: An overview[END_REF]. However, to our knowledge, only few studies are connected to the CS framework; we notice the work by Gaumond et al for the array geometry design by Statistical Restricted Isometry Property (StRIP), applied to underwater acoustics [START_REF] Gaumond | Sparse array design using statistical restricted isometry property[END_REF].

In this contribution we propose a sequential sensor placement to design the array geometry. The key idea stems from the following question: given a set of measurements obtained from an array of sensors, how to place a new sensor so that This work was funded by the the French defense procurement agency DGA.

the new measure minimizes the uncertainty on the angles of arrival? This idea can be identified with the so-called "Bayesian experimental design" and has recently been studied to iteratively position sensors in a stationary acoustic scene [START_REF] Gilquin | Iterative positioning of microphone arrays for acoustic imaging[END_REF]. In the latter contribution, the sparse nature of the sought signal is not exploited. In the present work, we propose an algorithm for sensor placement, inspired from the "Bayesian compressive technique" introduced in [START_REF] Ji | Bayesian compressive sensing[END_REF], [START_REF] Seeger | Compressed sensing and bayesian experimental design[END_REF], and exploiting a sparse prior on the source locations. Similarly to [START_REF] Gilquin | Iterative positioning of microphone arrays for acoustic imaging[END_REF], we assume that the measured acoustic field is stationary all along the sensing process. The proposed methodology therefore targets applications where processing time is not crucial but where the focus is on reducing the number of sensors for, e.g., economic constraints.

The paper is organized as follows. In section II, we formally state the "source localization" and "sensor placement" problems. In section III, we show that a closed-form expression for the error-covariance matrix appearing in our sensor-placement problem can be obtained by using a proper Bayesian hierarchical model. In section IV, we present a low-complexity greedy procedure to find an approximate solution of the sensorplacement problem. Finally, in section V we illustrate the performance gain obtained by the proposed adaptive placement of the sensors for source localization.

II. MODEL AND GENERAL STRATEGY

Consider a linear array of aperture D which consists of M sensors. Their positions are defined in the vector p ∈ [-D 2 , D 2 ] M . We suppose that the measured signal y(p) ∈ C M results from S uncorrelated narrowband sources in the far field. In the SL literature, one classic approach is to model the forward problem on grid as:

y(p) = A(p)x + w, (1)
where w is an uncorrelated noise, supposed to be a zeromean complex Gaussian with variance α -1 0 . The matrix A(p) ∈ C M ×N contains N steering vectors associated to plane waves propagating from angles

θ 1 , • • • , θ N ∈ [-π 2 , π 2
]. For convenience we define A(p) by its rows, such that A(p) = [r(p 1 ) T , . . . , r(p M ) T ] T and:

r(p m ) = [e -j 2π
λ pmsin(θ1) , . . . , e -j 2π λ pmsin(θ N ) ],

with λ the source wavelength. Thus, finding the S sources consists in identifying an S-sparse vector x ∈ C N . The 241 978-9-0827-9705-3 EUSIPCO 2020 magnitude of the non-zero values gives the source amplitudes, and their indices relate to their angular positions in the far field. The goal of this paper is to design the geometry of the sensor array (that is the choice of p) so that the measurements minimize some form of uncertainty of the ground-truth value of x. The main idea pursued hereafter is as follows: given some "posterior" covariance error matrix on x, say Σ(p) ∈ C N ×N , our goal is to identify the sensor position minimizing the determinant of Σ(p):

p = argmin p∈[-D 2 , D 2 ] M log |Σ(p)|. (3)
In the literature, this approach is known as "D-optimal" design [START_REF] Fedorov | Theory of optimal experiments[END_REF]. Solving (3) entails nevertheless several problems. First of all, evaluating the covariance matrix Σ(p) may be a difficult task depending on the prior model used for x. We address this problem in section III by relying on a Gaussian hierarchical model as in [START_REF] Yang | Off-grid direction of arrival estimation using sparse bayesian inference[END_REF]. Second, solving numerically optimization problem (3) may be a difficult task. In this paper, we consider a greedy approach to find an approximate solution of this problem, see section IV.

III. SOURCE LOCALIZATION BY MAXIMUM A POSTERIORI

In order to solve the optimization problem (3), we first introduce the SL estimator to infer both an empirical estimation of the covariance matrix Σ(p), and the unknown source vector x. It relies on the Sparse Bayesian Inference (SBI) [START_REF] Yang | Off-grid direction of arrival estimation using sparse bayesian inference[END_REF], whose MAP resolution is described below.

A. Bayesian hierarchical model

First, the noise w follows the uncorrelated complex circular distribution with variance α -1 0 , that is CN (0, α -1 0 I M ). The MAP infers α -1 0 since it is unknown, and uses the following non-informative Gamma prior of shape c and rate d:

P(α 0 |c, d) = Γ(α 0 |c, d), (4)
with c, d → 0 + , as done in [START_REF] Yang | Off-grid direction of arrival estimation using sparse bayesian inference[END_REF], [START_REF] Wu | Complex multitask bayesian compressive sensing[END_REF].

Second, the present problem is undetermined, thus N M > S. An intuitive choice for the prior of x could be the Laplace distribution to favor sparsity, but it does not lead to a tractable derivation of the MAP. Rather, we substitute this choice by a normal distribution prior on x:

P(x|α) = CN (0, diag(α) -1 ), (5)
combined with a Gamma hyper-prior for each element of α in the diagonal covariance matrix, that is:

P(α|ρ) = N n=1 Γ(α n |1, ρ). (6)
The shape parameter equals 1, and the rate ρ is a small positive constant as suggested in [START_REF] Yang | Off-grid direction of arrival estimation using sparse bayesian inference[END_REF]. Such a hierarchical model is the basis of the well-known Relevance Vector Machine procedure [START_REF] Tipping | Sparse bayesian learning and the relevance vector machine[END_REF], and appears in various works to express a simple surrogate of a sparse prior on x [START_REF] Yang | Off-grid direction of arrival estimation using sparse bayesian inference[END_REF], [START_REF] Seeger | Bayesian inference and optimal design for the sparse linear model[END_REF].

B. MAP estimator by Expectation-Maximization

Consider the estimation of x in a MAP sense:

x = argmax x P(x|y(p), α, α 0 ). (7)
Taking model ( 4)-( 6) into account, a closed-form solution of x can be found since all hyper-priors are conjugates of the Gaussian priors on x and w. In fact, the posterior distribution is also Gaussian, and writes P(x|y(p), α, α 0 ) = CN ( x p , Σ p ). Its mean and covariance read:

x p = α 0 Σ p A(p) H y (8)
Σ p = (α 0 A(p) H A(p) + diag(α) -1 ) -1 . ( 9
)
The variables α and α 0 are latent and must therefore be estimated as well. This can be done via an Expectation-Maximization (EM) procedure as described in [START_REF] Yang | Off-grid direction of arrival estimation using sparse bayesian inference[END_REF]. Note that Σ p is obtained posterior to the measurements with sensor positions fixed, and thus is not parametric. Unlike the expression of Σ(p) in (3), it is not a function of the vector p.

The SBI algorithm is implemented as follows: after initializing α and α 0 , one computes x p and Σ p from ( 8)-( 9). Then, α and α 0 are re-evaluated via an EM update. These steps are repeated until convergence of the procedure.

IV. GREEDY SENSOR PLACEMENT

In this section, we propose a greedy procedure to find an approximate solution of (3). It starts with a small number of sensors M 0 , from which a first estimation of Σ p is computed. Then, the most informative sensor position, named p * , is chosen to reduce the estimation covariance error. Again, Σ p is derived with the new measurement, named y * , to continue and repeat the placement until the array consists of M sensors. Because of the incremental nature of the approach, the size of the vector y (resp. p) increases with the addition of the new element y * (resp. p * ) at each iteration.

The greedy procedure consists in finding the sensor position p * which maximizes the reduction of the ellipsoid error volume. According to eq. ( 3), it writes:

p * = argmin p * ∈[-D 2 , D 2 ] log Σ p p * , (10)
by keeping the previous positions p constant so that the covariance is parametric to p * only. By relying on eq. ( 9), one can write the hypothetical expression of Σ [p T , p * ] T if y * was known. If (a) the hyper-parameters α 0 and α are supposed to be approximately the same with one new measurement, and (b) by using the Matrix determinant lemma, one can evidence that:

|Σ [p T , p * ] T | ≈ | Σ p | 1 + α 0 r(p * ) Σ p r(p * ) H , ( 11
)
with r(p * ) the candidate row vector computed at the position p * . Thus, by defining δh(p * ) = log(1 + α 0 r(p * ) Σ p r(p * ) H ), problem (10) can be rewritten as:

p * = argmax p * ∈[-D 2 , D 2 ]
δh(p * ).

Algorithm 1 Sequential sensor placement for sparsitypromoting Source Localization by a greedy approach.

Require: M 0 , M , y ∈ C M0 , p = [p 1 , . . . , p M0 ] T for m = M 0 + 1, M 0 + 2 . . . , M do -Localize sources by SBI to compute Σ p -Find the next optimal sensor position p * -eq. ( 12) -Acquire the new measurement y * -Update variables y ← [y T , y * ] T and p ← [p T , p * ] T end for return p

The quantity maximized in the log function corresponds, in ratio, to the volume reduction of the ellipsoid error by the current sequence. Note that, in previous works, a similar result is obtained via the definition of the Shannon entropy [START_REF] Ji | Bayesian compressive sensing[END_REF], [START_REF] Seeger | Bayesian inference and optimal design for the sparse linear model[END_REF]. Indeed, the entropy of a multivariate normal distribution of covariance Σ equals |Σ| up to a constant. Consequently, given the assumptions, reducing the entropy of the unknown x amounts to accomplishing the so-called D-optimal design as described in equation ( 3).

Finally, we remark that finding p * is doable by an extensive search on the line [-D 2 , D 2 ] regularly sampled on H candidate positions. In this way, the computation remains very fast, and the longest step of the strategy remains the SBI step, necessary to update the empirical covariance matrix. The proposed method is summarized in the Algorithm 1.

V. NUMERICAL EXPERIMENTS

This section presents results from numerical simulations of the SL problem in the far field. It compares the performance of the SL, when the sequential sensor placement is either from the proposed method, or done randomly as in the classical CS framework [START_REF] Xenaki | Compressive beamforming[END_REF].

A. Simulation setup

In the considered scenario, we configure the array with an aperture D = 10 m and an SNR = 12 dB. The sensor noise variance α -1 0 is calculated according to this SNR. The forward model regularly samples the interval [-π 2 , π 2 ] on N = 181 angles. S = 10 far-field uncorrelated sources emit at the wavelength λ = 1.0 m. Their angular positions are sufficiently separated to respect the necessary condition for the sparse source recovery -see [START_REF] Xenaki | Grid-free compressive beamforming[END_REF] for more details. But to our knowledge, in the SL framework, no condition on the minimal number of measurements guaranteeing the localization of S sources exists.

The sequential array design starts from M 0 = 2 sensors, located at ±D/2 to respect the aperture. Note that there is an ambiguity in the SL with a low number of sensors. Nevertheless, we arbitrarily limit M 0 to 2 in order to reduce the effect of the initial condition imposed by the user. 43 sensors are added one by one to reach M = 45 in total. At each of the 43 iterations, the optimum criterion seeks the next sensor into the interval [-D 2 , D 2 ] regularly sampled on H = 10001 points.

B. Performance score

In this paper, the SL is assessed as a source detection tool. To do so, the chosen performance score is the Jaccard index J, a measure of similarity by calculating the ratio between the intersection and the union of 2 sample sets. Here, the sample sets are the detected sources and the ground truth. The Jaccard index is a function of its true positive (TP), false positive (FP) and false negative (FN) rates. As in [START_REF] Denoyelle | The sliding frank-wolfe algorithm and its application to superresolution microscopy[END_REF], it writes:

J = TP TP + FP + FN . ( 13
)
Note that 0 ≤ J ≤ 1, and J = 1 reveals an exact detection, whereas J = 0 if none of the sources is correctly detected. To compute the TP, FP and FN rates, we pair the spikes from the estimation x with the ground truth positions if the distance between them is lower than a tolerance radius r. The number of reconstructed pairs is TP, the unpaired spikes from x give the FP, and the unpaired ground truth positions give the FN. Knowing that the angular grid equally samples [-π 2 , π 2 ] , the tolerance radius r arbitrarily equals the angular step |θ n+1θ n |. In all the results below, the score J is averaged over several realizations to obtain statistical trends.

C. Relevance of the sequential design for SL

In this section, we analyze how much the score δh is relevant to sequentially improve the source detection score J. To do this, we test sensor positioning as a function of δh values. First, δh(p * ) is linearly scaled between 0 and 1, to obtain δh(p * ). Accordingly, δh(p * ) = 1 (resp. 0) is the most (resp. least) informative position candidate. Then, we run the sequential placement and choose randomly the M sensor positions p * such that δh( p * ) ∈ [0, 0.25]. The same process is repeated 4 times, from the least to the most informative intervals [0.25, 0.5], [0.5, 0.75], and [0.75, 1]. Note that a fully random placement corresponds to δh( p * ) ∈ [0, 1], and the optimal choice is obtained for δh( p * ) = 1. the performance increases quicker if the interval is closer to δh = 1. The best average Jaccard index is given with the choice "δh = 1", and a fully random choice improves the performance at a rate between δh ∈ [0.25, 0.5] and δh ∈ [0.5, 0.75]. Note that the choice "δh = 1" only needs 25 sensors to reach the best detection performance J ≈ 0.95, against 40 when fully-random positioning is considered. Note that J does not reach 1 in average, since a small portion of the realizations never detect all sources by SBI.

In order to evaluate the performances (or robustness) of the method according to the noise level, it is proposed in Fig. 2 to plot the optimal (that is "δh = 1") sequential placement results when the SNR range from 0 to 12 dB. The curves follow a similar trend and the best detection performance, corresponding to J ≈ 0.9, is only reached for SNR = 12 dB (for the considered range of sensors). It appears that the method is quite affected by the noise level. One way to overcome this problem could probably be to deal with multisnapshot measurements.

D. Comparison with other methods

In this section, we propose to compare the proposed method with 3 state of the art approaches: random, uniform linear and the Golomb ruler [START_REF] Gaumond | Sparse array design using statistical restricted isometry property[END_REF]. The Golomb geometry is the only method found by the authors, that use the CS framework in source localization. Note that the Golomb array size does not exist for M > 27. Also note that the optimal and random geometries are designed by an iterative addition of sensors, however linear and Golomb arrays can not follow this iterative procedure while keeping a fixed aperture D. It means that these two geometries redefine all sensor positions at each iteration.

Fig. 3 shows the results obtained by averaging 100 realizations for a fixed SNR = 12 dB. Note that, this time, for each realization, the position and the initial phase of the sources are randomly fixed. The proposed method and the linear design have similar results. To reach, for example, J = 0.9, the random placement needs in our simulations 25 additional sensors compare to our method. On can note that 21 sensors are needed to reach the best detection performance. This is related with the Shannon-Nyquist conditions, saying that M 2D λ -1 = 19 to avoid spatial aliasing. Finally, despite having a better Statistical Restricted Isometry Property (StRIP) than random geometries [START_REF] Gaumond | Sparse array design using statistical restricted isometry property[END_REF], the Golomb array poorly succeeds in detecting the sources in the presented simulation. Consequently, with enough sensors (e.g. M = 40 to 45 in Fig. 3) all methods (but Golomb) can finally detect the S sources.

Thus, the proposed approach is of interest when the number of sensors is limited. Note that the uniform linear array is highly dependent of the highest frequency of the signal contrary to our method.

VI. CONCLUSIONS

This paper proposes a sequential sensor placement using Bayesian experimental design to localize sources. It is based on a greedy optimization procedure, and relies on the inference of the a posteriori error covariance matrix. To our knowledge, the proposed placement method is the first in SL to be driven by both the model and the measured data. Since it relies on a sparsity-promoting prior, it is strongly connected to Bayesian Compressive Sensing introduced in [START_REF] Ji | Bayesian compressive sensing[END_REF], [START_REF] Seeger | Compressed sensing and bayesian experimental design[END_REF] although, unlike these contributions, the present paper deals with a parametric sensing matrix of steering vectors.

The numerical results show that the proposed approach is relevant and gives better results than the state of the art (uniform linear array and CS geometries), in the context of source detection and localization. In practice, it is useful if one needs to minimize the number of sensors (e.g. for economic purposes) to find all the sources. For applications, future research directions could extend the present study to arbitrary array dimensions (i.e., planar or volumetric), or to the use of "off-the-grid" localization methods.

Fig. 1 .

 1 Fig. 1. SL performance for 10 sources, in a sequential placement of 43 sensors (with M 0 = 2 initial sensors). Jaccard index averaged on 100 realizations, with fixed sources and different noise w.

Fig. 1 Fig. 2 .

 12 Fig.1corresponds to the averaged score J obtained from 100 realizations when the SNR and the source positions are fixed. The results show that the considered criterion is consistent for choosing informative sensor positions. Indeed,

Fig. 3 .

 3 Fig. 3. SL performance for 10 sources. Sequential selection of 45 sensors (with K 0 = 2 initial sensors). Jaccard index averaged on 100 realizations, with different sources (position and phase).


Nathalie Lamoureux

Sebastian Hesse

Nancy Faßheber

Gernot Friedrichs

Pascale Desgroux

NCN absorption cross section determination; Re-assessment of the zero-pressure radiative lifetime measurements

Results

Motivations and Objectives

Radiative lifetime measurements of3 NCN using a picosecond laser □ Constant gas flow rate of NCN 3 (1 ppm diluted in Ar). □ Total pressure was increased by increasing the gas flow rate of Ar. □ Gas flow rate was adjusted at each pressure in order to have a complete renewal of the gas mixture every second.

1-

  University of Lille, CNRS, URM8522 -PC2A Physicochimie des Processus de combustion et l'atmosphère, F-59000 Lille, France 2 -Christian Albrechts University, Institute of Physical Chemistry, Kiel, Germany Experimental Approach

  □ Cyanogen azide (NCN 3 ) served as NCN source □ NCN 3 (extremely explosive) synthesis from NaN 3 and BrCN □ Product characterisation by FTIR NCN 3 decomposition □ 1 NCN is generated from the 193 nm excimer laser photolysis of NCN 3 , followed by collision induced intersystem crossing (CIISC) forming 3 NCN NCN 3 + h  NCN( 1 ) + N 2 NCN( 1 ) + M  NCN( 3 ) + M □ [ 3 NCN] = [ 1 NCN] 0 was assumed after a delay time of 400 -1000 µs. (Dammeier et al., Int. J. Chem. Kin. 45 (2013)) NCN 3 diluted in Ar was introduced into the reactor at 298 K and low pressures (p < 10 mbar). • 3 NCN fluorescence lifetimes could be measured down to pressures of p = 1 kPa. • Measurements confirmed the value of  = 183 ns measured by Smith et al., hence the previously used lifetime is not the reason for the disagreement in the NCN absorption cross section. • We also could not identify potential collisionless intersystem crossing processes from 3 NCN to 1 NCN that may have interfered with the lifetime measurements. • NCN yield from the NCN 3 decomposition needs to be reassessed. • Analysis of new shock tube measurements for NCN absorption cross section are in progress. Aim  Measurement of absolute NCN concentrations in flames by CRDS.  Species concentration of NCN can be determined from the absorbance A and the absorption cross section    NCN  Determination of the absorption cross-section  From spectral simulations and the experimental radiative lifetime  Temperature dependent simulation of the absorption spectrum using PGOPHER (Western, JQSRT 186 (2017))  Excitation LIF spectra recorded at room temperature (Smith et al. , J. Chem. Phys. 91 (1989)) and 1660 K (Lamoureux et al., Combust. Flame 160 (2013))  Absorption spectrum recorded at room temperature (Dammeier and Friedrichs, J. Phys. Chem. A 114 (2010))  Excellent ability of the simulated spectrum to fit the experimental ones over a wide temperature range  The hot band (010) -(010) peaks in the vicinity of the cold band (000) -(000) (separation by only 0.03 nm) Watson, J. Mol. Spectrosc. 233 (2005))  Using the radiative lifetime, which was reported only once for NCN (Smith et al., 1989), one obtains:  cm molecule 3.3 10 exp 1.80 10 K (±22%)  From experimental shock tube measurements (thermal decomposition of NCN 3 in Ar bath)  Has been measured directly only once (Dammeier and Friedrichs, 2010), yielding  cm molecule 1.43 10 exp 1.91 10 K (±25%) Issues  Disagreement between the experimental and the calculated NCN absorption cross-section by a factor of 2.6.  What are the reasons? 1.) Radiative lifetime or 2.) Expected NCN concentrations in shock tube experiments.

NCN fluorescence lifetime □ Excitation of 3 NCN by using the ps-laser at  = 329.1 nm. □ 3 NCN fluorescence lifetime collected at 329 nm by fast PMT.

Acknowledgements

This work is a contribution to the CPER research project CLIMIBIO. The authors thank the French Ministère de l'Enseignement Supérieur et de la Recherche, the Hauts de France Region and the European Funds for Regional Economical Development for their financial support to this project.