Justafort Jules

Bénédique Paul

Myriam Adam

Nadine Andrieu
email: nadine.andrieu@cirad.fr

Nadine Andrieu Co

producteurs de stratégies d'adaptation au changement climatique : le cas des exploitations agricoles en Haïti

Keywords: exploitations agricoles, innovation, modélisation, zones insulaires farming systems, innovation, modeling, insular areas

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

Introduction

Le climat est en train de changer partout dans le monde et plus rapidement que prévu (IPCC, 2021a(IPCC, , 2021b)).

Les projections de changement climatique pour les zones insulaires de la Caraïbe prévoient une augmentation des températures, de l'intensité des cyclones et de l'occurrence des sécheresses prolongées (IPCC, 2021a(IPCC, , 2021b)), avec des conséquences négatives pour l'agriculture haïtienne qui compte environ un million de petites exploitations agricoles (MARNDR, 2012). À l'horizon 2021-2040, une augmentation de 1,5 degré de la température globale va impacter les systèmes naturels et anthropisés, en particulier dans les zones hautement vulnérables telles que Haïti (IPCC, 2022). Haïti fait partie des trois pays déjà les plus affectés par les risques climatiques sur la période allant de 2000 à 2019 [START_REF] Eckstein | Global Climate Risk Index 2021[END_REF]. Un des défis pour la recherche et le développement en Haïti est donc d'aider les producteurs à concevoir des stratégies innovantes à court terme pour faire face au changement climatique.

La « co-conception » de systèmes agricoles innovants désigne un processus visant à concevoir de nouvelles stratégies, pratiques, systèmes de cultures ou d'élevage, permettant à l'agriculteur et sa famille de faire face à des défis multiples [START_REF] Meynard | Re-design and co-design of farming systems. An overview of methods and practices[END_REF]. Dans ce processus, agriculteurs, chercheurs et d'autres parties prenantes utilisent des démarches méthodologiques variées telles que des ateliers multi-acteurs, le prototypage participatif, la simulation via des modèles numériques (Le [START_REF] Gal | How does research address the design of innovative agricultural production systems at the farm level? A review[END_REF][START_REF] Périnelle | Combining on-farm innovation tracking and participatory prototyping trials to develop legume-based cropping systems in West Africa[END_REF][START_REF] Jeuffroy | Design workshops for innovative cropping systems and decision-support tools: Learning from 12 case studies[END_REF]. Dans ce dernier cas, la simulation est utilisée pour comparer ex-ante les performances de différentes alternatives par rapport à une situation initiale. Ces alternatives sont celles souhaitées par les agriculteurs, le rôle du chercheur étant alors d'outiller le processus en adaptant ou développant les modèles utilisés. Il existe relativement peu de démarches de co-conception de systèmes de production innovants appliquées aux enjeux du changement climatique [START_REF] Schaap | Participatory design of farm level adaptation to climate risks in an arable region in The Netherlands[END_REF][START_REF] Andrieu | Co-designing climate-smart farming systems with local stakeholders: A methodological framework for achieving large-scale change[END_REF][START_REF] Naulleau | How can winegrowers adapt to climate change? A participatory modeling approach in southern France[END_REF]. Dans les approches existantes, la simulation est utilisée pour explorer avec les producteurs différentes stratégies sous différents scénarios de changement climatique ou pour évaluer si ces stratégies améliorent les capacités d'adaptation des exploitations. Dans ces démarches, les solutions explorées sont situées et correspondent aux attentes et contraintes spécifiques des producteurs. Dans des contextes caractérisés par une forte vulnérabilité socio-économique des exploitations, un des défis est de combiner les attentes des producteurs en termes de couverture des besoins immédiats de leur famille avec les enjeux du changement climatique.

L'objectif de cette étude était d'analyser la pertinence d'une démarche de co-conception de stratégies d'adaptation des exploitations agricoles au changement climatique dans une zone hautement vulnérable telle que Haïti.

L'étude a été menée à Saint-Michel-de-l'Attalaye. La démarche passe par un diagnostic du fonctionnement et des performances technico-économiques des exploitations, puis par l'exploration de stratégies d'adaptation pensées par les producteurs eux-mêmes. Nous discutons ensuite de la contribution de cette étude aux démarches de co-conception de systèmes innovants pour faire face au changement climatique dans des zones hautement vulnérables. exploitation. Les agriculteurs adaptent la densité de semis en diminuant le nombre de grains par poquet. Ils produisent du compost, épandent de la cendre de bois sur les parcelles et ajoutent d'autres cultures dans les associations culturales. L'entraide est très forte dans ce groupe.

Démarche méthodologique

La démarche méthodologique s'est déroulée en deux étapes :

le diagnostic du fonctionnement et des performances initiales d'exploitations représentatives des différents types rencontrés dans la zone ; la co-conception de stratégies d'adaptation avec les producteurs.

Diagnostic des performances technico-économiques des exploitations de la zone

Nous avons réalisé des enquêtes directives auprès d'un échantillon de 24 agriculteurs (8 agriculteurs par type). Les 8 exploitations par type ont été sélectionnées au hasard dans la base de données de 393 exploitations ayant permis de construire la typologie (Sect. 2.1). Puis, nous avons confirmé leur disponibilité et consentement pour l'étude. Le guide d'enquête comprenait plusieurs modules (Tab. 1).

Pour traiter les données d'enquête, nous avons utilisé un outil de simulation de fonctionnement de l'exploitation agricole développé en Guadeloupe [START_REF] Rasse | Construction d'un outil de modélisation de la Petite agriculture familiale agroécologique de Guadeloupe : tensions et synergies entre productivité et services écosystémiques[END_REF], que nous avons adapté au contexte haïtien. Cet outil développé sous Excel vise à calculer différents indicateurs technico-économiques (production de biomasse, couverture des besoins caloriques du ménage, valeur ajoutée brute-VAB, revenu) et environnementaux (indice de fréquence de traitements phytosanitaires-IFT, efficience azotée, émissions de CO 2équivalent), en vue d'évaluer les performances multiples associées à différents changements possibles (introduction d'un nouveau système de culture ou d'élevage, d'une nouvelle pratique, d'un changement de surface ou d'effectifs des animaux) au sein d'une exploitation agricole [START_REF] Rasse | Construction d'un outil de modélisation de la Petite agriculture familiale agroécologique de Guadeloupe : tensions et synergies entre productivité et services écosystémiques[END_REF] Pour adapter l'outil au cas d'étude haïtien, nous avons considéré de nouvelles cultures telles que le pois Congo (Cajanus cajan), le maïs (Zea mays), le riz (Oriza sativa), etc. Pour chacune de ces cultures, nous avons dû renseigner les paramètres suivants : rendement, valeur calorique, matière azotée totale et matière sèche en nous basant sur la bibliographie (USDA-Food, sd). Nous avons converti les unités de mesure utilisées par les agriculteurs (sacs, marmites) à celles utilisées dans l'outil grâce à une base de données de la sous-structure agricole et informatique du Ministère de l'agriculture (2022).

Les données de chaque exploitation ont été entrées dans l'outil. Ainsi, 24 fichiers de simulation ont été construits. Des tests de normalité des données (Shapiro-Wilk normality test), d'homogénéité des variances (Levene test of homogeneity of variances en cas de test de normalité inférieur à 0,05 ou Barlett test dans le cas contraire), puis des analyses de variance (ANOVA) ont été réalisés sous R afin d'identifier si les différences observées entre types sur les indicateurs obtenus pour les 24 exploitations étaient significatives.

Co-conception des stratégies

La co-conception des stratégies d'adaptation a été réalisée avec un échantillon de 9 agriculteurs (trois par type) sélectionnés parmi les 24 ayant été interrogés lors de la phase de diagnostic. Le choix de ces agriculteurs s'est basé sur leur disponibilité pour une deuxième phase d'entretien et leur souhait d'explorer des stratégies d'adaptation face au changement climatique.

Les entretiens individuels ont débuté par la restitution des résultats du diagnostic : les performances technico-économiques des stratégies initiales (SC0). Ensuite, la parole a été accordée à l'agriculteur pour qu'il puisse exprimer les éventuels changements qu'il souhaitait apporter à SC0.

Les grandes questions ayant guidé cette phase de coconception sont résumées ainsi : -Au cours de ces dernières années, quels sont les changements lie ́s aux contextes e ́conomique et climatique apporte ́s dans les exploitations ? -Considérant le fonctionnement de l'exploitation et ses performances technico-e ́conomiques actuelles, quels sont les changements envisagés pour les améliorer ? -Compte tenu des effets du changement climatique perçus, quelles seront les nouvelles pratiques et techniques à mobiliser pour s'adapter au changement climatique ?

Les nouvelles stratégies (SC1) décrites par chacun des 9 producteurs ont été entrées dans l'outil de simulation. Les résultats obtenus pour les nouvelles stratégies (SC1) ont été ensuite comparés à ceux des stratégies du diagnostic (SC0).

Après les entretiens individuels, un atelier a été organisé avec les 24 agriculteurs de l'étape de diagnostic, dans l'objectif de contextualiser les changements individuels décrits au regard des grands changements de l'agriculture de Saint Michel de l'Attalaye (saisons de culture, apparition ou disparition de nouvelles cultures dans la zone) et du changement climatique perçu.

Résultats

Fonctionnement et performances technicoéconomiques des exploitations

À Saint Michel de l'Attalaye, les activités de l'exploitation sont gérées par le chef du ménage agricole accompagné de son conjoint et/ou de quelques membres de la famille. En période de pointe, certains agriculteurs ont recours à la coumbite, une pratique qui vise, dans un collectif d'agriculteurs, à s'entraider pour l'exécution de tâches agricoles. Il existe également des prêts et/ou des locations d'outils, équipements de travail du sol ou de transformation : par exemple location de charrue, de moulins à canne à sucre fixes ou mobiles, ou de guildive, petite distillerie traditionnelle utilisée pour produire un alcool à base de jus de canne appelé clairin.

L'élevage (bovins, caprins, équins, porcins, etc.) est largement pratiqué. Les animaux sont gardés à la corde aux alentours de la maison pendant la saison de culture (mi-mars à décembre) et sont en vaine pâture sur les parcelles après les récoltes de janvier à début mars. Ils sont néanmoins ramenés le soir et gardés à la corde aux alentours de la maison pour réduire les risques de vol.

Dix-sept systèmes de culture différents, tous types d'exploitation confondus, ont été identifiés, traduisant le niveau important de diversification de l'agriculture de la région. Les systèmes de culture les plus représentatifs sont : la culture semi-pérenne de canne à sucre (4 à 8 ans), l'association pois Congo x maïs en culture pure ou intercalaire sur les parcelles de canne à sucre en début de plantation, l'association riz pluvial x pois Congo sur les sols argileux et l'association maïs x papaye x pois Congo x banane, le maïs et le pois congo étant cultivés en début de plantation de la banane.

Pour un même système de culture, l'organisation du travail sur l'année n'est pas très différente d'un type d'exploitation à l'autre. Mais, le retard des pluies, le manque de disponibilité de semences et/ou de moyens pour payer les prestations de préparation du sol peuvent faire varier le calendrier cultural.

Habituellement, la préparation du sol se fait par traction animale. Mais sur les collines, les agriculteurs utilisent pioche, houe et louchette pour labourer le sol, car ces outils sont mieux adaptés à ces types de relief. C'est particulièrement le cas des agriculteurs du type 2, qui possèdent les parcelles les plus pentues.

Les interactions entre culture et élevage sont fortes dans les exploitations. Par exemple, sur celles où sont présents des équins et bovins, la force de travail de ces derniers est employée pour le transport de matériaux et de récoltes, le labour et le hersage du sol, et le broyage de la canne à sucre. En retour, les résidus de récolte servent à alimenter le troupeau : c'est le cas des parcelles de canne à sucre, qui jouent un rôle particulier en raison de leur forte capacité de production de biomasse et de la palatabilité de cette culture. Les agriculteurs valorisent aussi les fumures animales pour fertiliser les parcelles via la vaine pâture en hiver. Des différences de performances technico-économiques ont été observées entre types d'exploitations.

Le type 1 présente un taux moyen de couverture des besoins caloriques du ménage de 49 %, suivi du type 3 (39 %) et du type 2 (seulement 9 %) (Fig. 1). Le type 2 produit essentiellement de la canne à sucre, tandis que les types 1 et 3 produisent principalement des cultures vivrières (Sect. 2.2). Il existe une différence significative entre les trois types d'exploitations pour cet indicateur (Tab. 2).

Les exploitations du type 1 présentent les revenus moyens les plus élevés et la meilleure valorisation de la journée de travail. Ensuite viennent les types 2 et 3 (Fig. 2). Les différences sont significatives entre les types pour cet indicateur (Tab. 2).

Les émissions moyennes de gaz à effet de serre sont plus élevées pour les exploitations du type 3 (1,4 tonne CO 2 -éq/an) et plus faibles pour celles du type 1 (0,8 tonne CO 2 -éq/an) (Fig. 3). La quantité de CO 2 -éq émise est liée au nombre de bovins, le type 3 possédant le plus grand nombre de bovins, ensuite viennent les types 2 et 1. Les différences ne sont toutefois pas significatives entre les trois types pour cet indicateur (Tab. 2).

Stratégies (SC1) conçues par les agriculteurs

Le tableau 3 présente les nouvelles stratégies (SC1) conçues par chacun des 9 agriculteurs. Tous types confondus, les producteurs cherchent à agrandir les surfaces cultivées pour diversifier davantage les systèmes de cultures. Les producteurs des types 1 et 3 souhaitent en outre agrandir les surfaces en Tableau 3. Nouvelles stratégies (SC1) conçues par les 9 agriculteurs. Table 3. New strategies (SC1) designed by the 9 farmers.

Type Stratégies d'adaptation conçues avec les agriculteurs Type 1 Diversification et extension des surfaces cultivées

Exploitation 1 : Une nouvelle parcelle en canne (þ 1 ha). Augmentation de la taille du troupeau de 2 bovins à traction.

Exploitation 2 : Augmentation de la production de banane et papaye (þ 0,6 ha). Une nouvelle parcelle en canne (þ 2,6 ha). Une nouvelle parcelle maïs et sorgho (Var. pa pè lesèk) et pois inconnu et pois Congo (þ 1,3 ha).

Exploitation 3 : Augmentation de la production de canne (þ 1,9 ha). Une nouvelle parcelle en banane et papaye (þ 1,3 ha). Augmentation de la production de compost et usage de bagasse de canne pour chauffage du jus de canne.

Type 2 Augmentation du troupeau et banane

Exploitation 4 : Augmentation de la production de canne (þ 1,3 ha). Une nouvelle parcelle en banane et haricot (þ 0,6 ha). Augmentation de la taille du troupeau de 2 vaches.

Exploitation 5 : Augmentation de la taille du troupeau de 3 bovins à traction (þ 0,6 ha). Une nouvelle parcelle en banane et igname (þ 0,6) et en manioc (þ 0,6 ha).

Exploitation 6 : Une nouvelle parcelle de 1,3 ha proche d'un cours d'eau en banane et papaye. Augmentation de la taille du troupeau de 2 vaches.

Type 3 Augmentation des cultures de « subsistance » (maïs, pois Congo, riz, sorgho, patate douce)

Exploitation 7 : Reprise de la culture de patate douce (1,3 ha dans les 1,9 ha réservés au pâturage). Une nouvelle parcelle en canne (þ 3,2 ha). Augmentation de la taille du troupeau de 1 vache et de 2 taureaux à traction.

Exploitation 8 : Une nouvelle parcelle en arachide et maïs et sorgho (Var. pa pè lesèk) et pois Congo (þ 0,6 ha). Augmentation de la production de canne (þ 2,6 ha).

Exploitation 9 : Augmentation de la production de canne (þ 0,6 ha). Une nouvelle parcelle en riz et pois Congo (þ 1,3 ha). Augmentation de la taille du troupeau de 1 vache. canne. Les producteurs des types 2 et 3 souhaitent augmenter les effectifs animaux. Parmi les raisons qui incitent les producteurs à proposer ces changements, on note :

l'irrégularité des pluies et les périodes de sécheresse prolongée. Afin de limiter les risques de mauvaises récoltes liés à des inondations ou des sécheresses prolongées, certains agriculteurs souhaitent multiplier le nombre de parcelles cultivées dans des localisations différentes. Selon eux, certaines zones sont plus résilientes que d'autres aux risques climatiques, du fait de la pente et de l'inclinaison du terrain, de la distance aux cours d'eau pour accéder à l'irrigation, ou des conditions pédoclimatiques ; la présence d'infrastructures productives. Certains agriculteurs, tous types d'EA confondus, veulent mettre en place de nouvelles parcelles en canne à sucre en raison de l'installation d'une guildive à proximité ; la nécessité de répondre aux besoins du ménage. En raison de l'inflation (48,3 % en décembre 2022 selon les estimations de l'Institut haïtien de statistique et d'informatique -IHSI) et des prix des produits alimentaires qui tendent à augmenter, certains agriculteurs, du type 2 notamment, veulent valoriser des lopins de terre en cultures vivrières ; le prix et la disponibilité des prestations de labour. La plupart des agriculteurs des types 2 et 3 veulent renforcer leur système d'élevage, notamment en bovins pour la traction animale, et diminuer le coût des services de labour. C'est aussi un moyen de diversifier leurs sources de revenu par la vente de services de labour.

Évaluation comparée des stratégies

Les figures 4, 5 et 6 comparent les stratégies initiales (SC0) et les nouvelles stratégies proposées par les 9 agriculteurs (SC1).

Les agriculteurs prévoient de diversifier davantage leur système de production agricole en vue de mieux couvrir les besoins alimentaires de leur famille. Dans l'ensemble, la couverture des besoins caloriques du ménage (RecBesM) augmente de SC0 à SC1 (Fig. 4). Ce taux est en moyenne plus élevé dans les exploitations de type 1, suivi du type 2, puis du type 3, les différences étant significatives (Tab. 4). La plus forte augmentation entre SC0 et SC1 est observée pour les agriculteurs du type 2, qui misent sur le revenu de la canne à sucre pour mettre en place des parcelles vivrières afin d'améliorer la couverture des besoins du ménage.

Le revenu agricole est nettement amélioré dans SC1 par rapport à SC0 (Fig. 5). Dans ces nouvelles stratégies (SC1), les agriculteurs augmentent la taille de la superficie des parcelles cultivées et de leur troupeau. Il y a certes une augmentation des charges intermédiaires, mais l'amortissement reste le même car les agriculteurs n'ont pas prévu d'acquérir de nouveaux outils agricoles dans SC1.

En comparant les deux stratégies (Fig. 6), les émissions de CO 2 augmentent de SC0 à SC1 pour tous les types, car les agriculteurs augmentent la taille de leur troupeau, notamment ceux du type 2.

Discussion

4.1 Quelles stratégies d'adaptation pour les agriculteurs de Saint Michel de l'Attalaye ?

Dans tous les types d'exploitations, les exploitants, ne voulant pas rompre brutalement avec leurs pratiques initiales, ont proposé des stratégies d'adaptation incrémentales visant à maintenir la nature et l'intégrité de leur système agricole sans modification de ses attributs fondamentaux (GIEC, 2018).

Néanmoins, les stratégies envisagées varient d'un type d'exploitation à l'autre, avec en particulier l'accent mis sur la banane et l'élevage dans le type 2, et sur les cultures de subsistance dans le type 3. Pour les trois types d'exploitations, la canne prend davantage d'importance dans les stratégies. Cela peut s'expliquer par la résilience de cette culture face au changement climatique [START_REF] Garcia | Rapport spe ́cial du GIEC sur les conse ́quences d'un re ́chauffement plane ́taire de 1,5 °C par rapport aux niveaux pre ́industriels et les trajectoires associe ́es d'e ́missions mondiales de gaz àeffet de serre, dans le contexte du renforcement de la parade mondiale au changement climatique, du de ́veloppement durable et de la lutte contre la pauvrete[END_REF], mais aussi par les possibilités de diversification des revenus qu'elle permet via la production de sirop et de clairin, qui contribuent à la résilience économique des exploitations. Les stratégies proposées par les agriculteurs visent également toutes à augmenter les capacités de production pour améliorer les performances des exploitations en termes de couverture des besoins en calories et de revenu agricole. L'augmentation envisagée des surfaces dans la plupart des stratégies passe par la vente de sirop, de clairin ou d'animaux. Néanmoins, l'acquisition de surfaces sera plus difficile pour les plus petits producteurs (en particulier du type 2), souvent en insécurité foncière. La quasi-totalité des exploitants veulent aussi augmenter la taille de leur cheptel, notamment en bovins, mais en gardant les mêmes règles de conduite. Par conséquent, les émissions de CO 2 -éq simulées augmentent. Lorsque les stratégies d'adaptation des agriculteurs induisent une augmentation à moyen terme des émissions de gaz à effet de serre, il s'agit d'une mal-adaptation environnementale [START_REF] Barnett | Maladaptation[END_REF]Magnant, 2013). On peut questionner la pertinence de cet indicateur pour de petites exploitations pratiquant une agriculture pluviale non irriguée, dotées d'outils traditionnels et de faibles moyens financiers, n'utilisant pas d'engins motorisés mais voulant augmenter la taille de leur cheptel de quelques bovins afin d'assurer leur survie. Il invite cependant à réfléchir aux pratiques d'alimentation des animaux ou de valorisation des déjections permettant de limiter les émissions liées au développement de l'élevage.

Quels enseignements pour la co-conception de systèmes faisant face au changement climatique en zone hautement vulnérable ?

De nombreux travaux ont démontré que les producteurs, dans des démarches de co-conception, préfèrent explorer des changements incrémentaux qu'ils pourront intégrer de manière progressive dans leur stratégie de production [START_REF] Périnelle | Combining on-farm innovation tracking and participatory prototyping trials to develop legume-based cropping systems in West Africa[END_REF][START_REF] Andrieu | Scenarios for an agroecological transition of smallholder family farmers: A case study in Guadeloupe[END_REF]. Face à l'urgence du changement climatique (IPCC, 2022), un défi pour le chercheur qui accompagne le processus de conception est d'aider les agriculteurs à explorer des changements de rupture. [START_REF] Jeuffroy | Design workshops for innovative cropping systems and decision-support tools: Learning from 12 case studies[END_REF] montrent la nécessité d'éviter les effets de fixation qui maintiennent les propositions des acteurs dans ce qui est connu. Combiner ces simulations avec l'utilisation de démarches d'anticipation pourrait permettre d'explorer des changements de rupture [START_REF] Vervoort | Co-conception avec les producteurs de stratégies d'adaptation au changement climatique : le cas des exploitations agricoles en Haïti[END_REF].

Nous avons fait le choix de laisser les producteurs explorer les stratégies qu'ils souhaitaient et de plutôt les outiller dans cette réflexion. Ces stratégies étaient différentes entre types, ce qui montre la nécessité de prendre en compte cette diversité pour accompagner l'adaptation des producteurs au changement climatique, et en particulier accompagner les changements structurels souhaités par les producteurs les plus pauvres.

Les solutions proposées par les agriculteurs pour s'adapter au changement climatique visent, sans surprise, à répondre en premier lieu à des enjeux socio-économiques et alimentaires. Dans un contexte où les agriculteurs sont plus qu'ailleurs vulnérables au changement climatique, favoriser leur adaptation à ce changement ne peut donc se faire sans mesurer les cobénéfices en termes de sécurité alimentaire ou de revenus [START_REF] Osorio-García | Can an innovation platform support a local process of climate-smart agriculture implementation? A case study in Cauca, Colombia[END_REF]. Ces stratégies reposent sur une diversification accrue des systèmes de production, un des principes clés de l'agroécologie définie comme effective pour s'adapter au changement climatique (IPCC, 2022).

Tout comme dans la démarche DEED À Describe, Explain, Explore and Design À [START_REF] Descheemaeker | Which options fit best? Operationalizing the socio-ecological niche concept[END_REF], cette étude vise à être combinée à des expérimentations afin d'explorer les systèmes de cultures ou d'élevage les plus prometteurs et leurs contraintes de mise en oeuvre.

Conclusion

Cette étude, menée Saint Michel de l'Attalaye en Haïti, a permis de diagnostiquer les performances technico-économiques initiales des exploitations agricoles, puis d'aider les agriculteurs à concevoir des stratégies d'adaptation au changement climatique. Le diagnostic montre que l'agriculture de la zone est caractérisée par de petites exploitations familiales de subsistance, dotées de faibles moyens de production et faiblement émettrices de gaz à effet de serre par hectare. En dépit de leurs faibles ressources, elles disposent d'une panoplie de techniques et de pratiques agricoles pour s'adapter au changement climatique. À travers la démarche de co-conception, l'étude montre que les changements de stratégies envisagés par les agriculteurs varient d'un type d'exploitation à l'autre, mais reposent tous sur l'augmentation de la diversification des cultures et de l'élevage. Ces changements visent surtout à augmenter leur niveau de production pour améliorer les revenus et la sécurité alimentaire des familles sous contrainte de changement climatique. Le défi est d'accompagner la mise en oeuvre de ces changements sur le terrain, en particulier les changements structurels souhaités par les agriculteurs les plus pauvres. Une perspective est de combiner les simulations avec de l'expérimentation de terrain. D'autres boucles de co-conception intégrant des outils d'anticipation pourraient également être mobilisées pour identifier des changements de rupture avec les producteurs.

Remerciements. Nous remercions les agriculteurs ayant participé à cette recherche. Cette recherche a bénéficié du support financier du MARNDR et de la BID, dans le cadre du Programme de mitigation des désastres naturels (PMDN2), accord de subvention n°LOT2-MARNDR/PMDN/SCBD-2/18.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Couverture moyenne des besoins caloriques du ménage pour les trois types d'exploitations. Fig. 1. Average coverage of the household caloric needs for the three types of farms.

Fig. 3 .

 3 Fig. 3. Emissions moyennes de gaz à effet de serre pour les trois types d'exploitations. Fig. 3. Average greenhouse gas emissions for the three farm types.

Fig. 4 .Fig. 5 .

 45 Fig. 4. Couverture des besoins caloriques du ménage pour la stratégie initiale (SC0) et pour la nouvelle stratégie (SC1) pour les trois types d'exploitations, chaque box-plot représente la dispersion des données simulées pour les trois exploitations de chaque type. Fig. 4. Coverage of the household caloric needs for the initial strategy (SC0) and for the new one (SC1) for the three types of farms, each boxplot represents the dispersion of the simulated data for the three farms of each type.

Fig. 6 .

 6 Fig. 6. Emissions de gaz à effet de serre (CO 2 -éq) pour la stratégie initiale (SC0) et la nouvelle stratégie (SC1) pour les trois types d'exploitations, chaque box-plot représente la dispersion des données simulées pour les trois exploitations de chaque type. Fig. 6. Greenhousegas emissions (CO 2 -eq) for the initial strategy (SC0) and for the new one (SC1) for the three types of farms, each box-plot represents the dispersion of the simulated data for the three farms of each type.

Table 1 .

 1 Tableau 1. Liste des variables clés collectées lors des enquêtes. List of key variables collected during surveys. , car les producteurs n'étaient pas en mesure de fournir des données précises pour les produits animaux autoconsommés.Le revenu agricole concerne le revenu total généré par l'exploitation après avoir enlevé toutes les dépenses du produit brut total.Les émissions liées à l'épandage d'engrais minéraux ou organiques, l'utilisation de carburant, les émissions entériques des animaux, ainsi que celles liées à la valorisation de leurs déjections sont calculées. Les calculs sont basés sur les équations suggérées par le GIEC pour le « tier 1 » (les équations détaillées sont dansIPCC, 2006IPCC, , 2021aIPCC, et 2021b)).À Saint-Michel-de-l'Attalaye, les exploitations ne sont pas motorisées ; il n'y a pas eu d'épandage d'engrais minéral sur les parcelles au cours de l'année des enquêtes. Par conséquent, l'évaluation des émissions (en CO 2 équivalent) s'est basée sur le nombre d'animaux présents sur les exploitations, notamment le nombre de bovins.

	. La

consommées

Matériels et méthodes 2.1 Zone d'étudeL'étude a été réalisée à Saint-Michel-de-l'Attalaye, commune du département de l'Artibonite d'Haïti. Avec une superficie de 613,74 km 2 , c'est l'une des communes les plus vastes de la République d'Haïti(IHSI, 2009). Elle est majoritairement rurale et présente des hétérogénéités physiques remarquables[START_REF] Paul | Diagnostic agro socio-économique de la commune de Saint Michel de l'Attalaye[END_REF].Le climat dominant, de type tropical, est, selon la classification de Köppen, un climat de savane avec hiver sec. La température et la pluviométrie annuelles moyennes sont respectivement de l'ordre de 23 °C et de 1298 mm (Climate Data, 2021). Il y existe trois cours d'eau permanents qui s'écoulent du nord au sud à Saint-Michel-de-l'Attalaye[START_REF] Paul | Diagnostic agro socio-économique de la commune de Saint Michel de l'Attalaye[END_REF]. Mais, malgré leur présence, la commune est dépourvue de système d'irrigation. La population totale de Saint Michel de l'Attalaye est estimée à 136 876 habitants, avec 41,5 % de la population âgée de moins de 15 ans, 53 % âgée de 15 à 64 ans et 5,5 % âgée de 65 ans et plus(IHSI, 2009). La densité démographique est de 245 habitants/km 2 .L'économie repose sur l'agriculture et l'élevage(MARNDR, 2012). Des enquêtes menées auprès de

393 exploitations, suivies d'une analyse en composantes principales et d'une analyse des clusters, ont permis d'identifier trois grands types de producteurs en fonction de l'adoption de pratiques innovantes en lien ou non avec le changement climatique[START_REF] Paul | Diagnostic agro socio-économique de la commune de Saint Michel de l'Attalaye[END_REF], une pratique pouvant être innovante dans un contexte donné alors qu'elle est couramment mise en oeuvre dans un autre.Type 1 -Des exploitants proactifs qui innovent pour faire face à des irrégularités des pluies et des sécheresses prolongées. Les innovations observées sont : l'adaptation de la densité de semis, la mise en place de structures antiérosives (rampes), le ramassage de fumier animal provenant des lieux de parcage des animaux et la production de compost à partir des résidus de culture et des déjections animales. Il y a, en outre, l'abandon de certaines cultures (haricot) au profit de cultures telles que le pois Congo, la canne à sucre, ou le maïs à cycle court. La superficie moyenne de ces exploitations est de 3 ha. Certains des exploitants de ce type ont l'habitude de participer à des séminaires de formation sur l'agriculture. Ce sont pour la plupart des délégués d'associations d'agriculteurs et des bénéficiaires directs de projet. Ils commencent progressivement à planter des arbres sur leurs parcelles.Type 2 -Des exploitants confrontés à des irrégularités des pluies et des sécheresses prolongées, mais qui innovent plutôt pour améliorer des sols dégradés. Beaucoup de ces exploitations sont en zone de pente et les sols se dégradent facilement. La canne à sucre est largement cultivée, notamment sur les pentes. Les agriculteurs aménagent leurs parcelles avec des structures antiérosives : rampes, murettes en pierre et canaux de contour. La superficie moyenne de ces exploitations est de 1,5 ha.Type 3 -Des exploitants agricoles dont les systèmes de production reposent traditionnellement sur la combinaison de systèmes de culture et d'élevage, et qui tendent à se diversifier davantage avec le changement climatique (principalement la sécheresse). Ces exploitations ont des superficies supérieures ou égales à 2 ha, avec le plus grand nombre de parcelles par

J.Jules et al. : Cah. Agric. 2023, 32, 27


Aurélie Antunes

Philippe Robin

Gilles Mourier

Michel De Waard

Rémy Béroud

Denis Servent

Evelyne Benoit

Evidence of rattlesnake crotalphin effects on tetrodotoxin-sensitive Na channels of mouse dorsal root ganglion neurons

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

Evidence of rattlesnake crotalphine effects on tetrodotoxin-sensitive Na channels of mouse dorsal root ganglion neurons

Aurélie ANTUNES 1,2 , Philippe ROBIN 1 , Gilles MOURIER 1 , Michel DE WAARD 2,3,4 , Rémy BEROUD 2 , Denis SERVENT 1 and Evelyne BENOIT 1

Crotalphine induces decreased DRG neuron viability

Linear relationship (r 2 = 0.988) between the peak Na current blockade and the leakage current amplitude in the absence and in the presence of 5, 10 and 50 μM of toxin. Means ± S.E.M. of data obtained from 4-9 neurons. *: P ≤ 0.049.

A decrease of DRG neuron, but not of CHO cell, viability is detected in the presence of crotalphine (50 μM), using the propidium iodide fluorescence assay (a,b,c). Similarly, a decrease of DRG neuron and UACC-257 cell viability is detected in the presence of crotalphine (50 μM), using the trypan blue assay (d). Note that the toxin induces a maximal decrease of 28% of cell viability. Means ± S.E.M. of data from 3-12 cell cultures. *: P ≤ 0.05, **: P ≤ 0.005 and ***: P ≤ 0.0005 (b,c,d).

CONCLUSION.

Taken together, these results point out, for the first time, the effectiveness of crotalphine to act on the Na V 1.7 channel subtype, directly and/or indirectly by affecting cell membrane integrity. This mechanism of action could contribute to the peptide analgesic properties.

RESULTS.

Crotalphine inhibited the TTX-sensitive Na current of mouse DGR neurons with a micromolar range affinity (IC 50 = 18.70 μM) (1). In addition, the toxin had no effect on activation or inactivation kinetics (2) and voltage-dependence of the TTX-sensitive Na current (3). Thus, crotalphine appears to block the channel pore without affecting its biophysical properties. The leakage current amplitude was significantly enhanced with increasing blockade of TTX-sensitive Na current (4), likely due to a decreased DRG neuron viability (5).

INTRODUCTION.

Crotalphine is an analgesic peptide identified in the venom of the South American rattlesnake Crotalus durissus terrificus. Although its antinociceptive effects are well documented, its direct mechanisms of action are still under investigation. The aim of the present work was to study the action of the crotalid peptide on the sodium channel subtype Na V 1.7, a genetically validated pain target. To this purpose, the effects of crotalphine were evaluated on the tetrodotoxin (TTX)-sensitive Na current of dorsal root ganglion (DRG) neurons of adult mice, using the whole-cell patch-clamp configuration, and on the cell viability, using the propidium iodide fluorescence and trypan blue assays.

Crotalphine induces increased leakage current amplitude

  Traces of TTX-sensitive Na currents recorded before (in black) and after (in pink) application of 50 μM of crotalphine. On the right, the toxin peak current is normalized to that of control.Representative experiment of the time courses of crotalphine (50 μM) effects on the leakage (light circles) and peak Na (dark circles) current amplitudes, before and after toxin addition (arrow). Note the crotalphine-

1

  Université Paris-Saclay, CEA, Institut des sciences du vivant Frédéric Joliot, DMTS, SIMoS, EMR CNRS/CEA 9004, Gif-sur-Yvette, France 2 Smartox Biotechnology, Saint-Egrève, France 3 L'Institut du thorax, INSERM, CNRS, Université de Nantes, Nantes, France 4 LabEx "Ion Channels, Science and Therapeutics", Valbonne, France curve of crotalphine steady-state effects on TTX-sensitive Na current. Each value is expressed as percentage of that determined before toxin application. Kinetic parameters of activation (Time A and Slope A ) and inactivation (Time I and Slope I ) of TTX-sensitive Na current recorded without and with 50 μM of crotalphine (means ± S.E.M. of 4 neurons). Z = pyroglutamic acid (non-standard amino acid) Current-voltage relationships (left panel), and steady-state inactivation-(open circles) and conductance-(filled circles) voltage relationships (right panel), before (black circles) and after (green circles) exposure of neurons to 50 μM of crotalphine. Each value is expressed as percentage of maximal peak amplitude of current at strongly negative pre-pulse voltages or maximal conductance calculated at strongly positive test-pulse voltages.


On the False Alarm Probability of the Normalized Matched Filter for off-grid targets: A geometrical approach and its validity conditions Pierre Develter1,2 , Jonathan Bosse 1 , Olivier Rabaste 1 , Philippe Forster3 , Jean-Philippe Ovarlez 1,2 Abstract-Off-grid targets are known to induce a mismatch that dramatically impacts the detection probability of the popular Normalized Matched Filter. To overcome this problem, the unknown target parameter is usually estimated through a Maximum Likelihood strategy resulting in a GLRT (Generalized Likelihood Ratio Test) detection scheme. While the test statistic for the null hypothesis is well known in the on-grid case, the off-grid scenario is more involved, and to the best of our knowledge, no such theoretical result is available. This paper fills this gap by proposing such an expression under circular compound Gaussian noise with a known covariance matrix thanks to a geometrical approach. It is exact, provided that the probability of false alarm is low enough: we derive the conditions guaranteeing the exactness of the relationship and we show numerically that the formula still yields a good approximation of the probability of false alarm when this condition is not met.

Index Terms-Radar detection, Off-Grid, GLRT, PF A- threshold relationship, Theory of Tubes

I. INTRODUCTION

In domains as varied as telecommunications, psychology, or radar, detection theory is a crucial tool to separate ambient noise from signals of interest. In detection theory, received signals are submitted to a hypothesis test to discriminate useful signals from noise. Detection of signals with unknown deterministic parameters is classically addressed with a Generalized Likelihood Ratio Test (GLRT) that replaces the unknown parameters with their Maximum Likelihood Estimators (MLE) in the Likelihood Ratio detection test [START_REF] Scharf | Statistical signal processing: detection, estimation, and time series analysis[END_REF]. When analytical MLE solutions are not available for signal parameters of interest, most detection strategies assume for ease of implementation that those parameters lie over a discrete set, called the grid, usually chosen so that all tests on the grid are statistically independent when the tested signal consists of noise. However, parameters have no reason to fall precisely on the grid since they are often distributed over a continuous range. This induces a mismatch between the tested and true target parameters that deteriorates the detection performance of most state-of-theart tests made under the on-grid assumption. This paper will consider this very general problem in the radar context where unknown parameters can include Doppler shift, distance, or direction without loss of generality. The Normalized Matched Filter is the GLRT associated with the problem of detecting a known signal of unknown amplitude in Gaussian ambient noise of unknown power. It is also widely used for adaptive radar in non-Gaussian contexts [START_REF] Pascal | On a SIRV-CFAR detector with radar experimentations in impulsive noise[END_REF] for example when the noise is distributed according to a Complex Elliptically Symmetric (CES) distribution [START_REF] Ollila | Complex Elliptically Symmetric distributions: Survey, new results and applications[END_REF]. The off-grid mismatch impact is particularly dramatic for the Normalized Matched Filter (NMF) test [START_REF] Rabaste | Off-grid target detection with Normalized Matched Subspace Filter[END_REF]. In some cases, the detection probability may even vanish to 0 when the Signal to Noise Ratio (SNR) tends to infinity [START_REF] Rabaste | Off-grid target detection with Normalized Matched Subspace Filter[END_REF], especially for low Probabilities of False Alarm (P F A ), familiar in radar context. To overcome this problem, the most apparent solution consists of testing over the whole continuous support of the signal parameter, not just the grid: this is the true off-grid GLRT. However, to the best of our knowledge, the analytical expression of the null hypothesis statistic and the related P F A is unknown in the literature for this GLRT.

The robustness of detectors to mismatched signals is a wellexplored topic in the literature. Several types of mismatches are addressed thanks to the derivation of suited GLRTs, such as mismatch lying in a cone [START_REF] Maio | Robust adaptive radar detection in the presence of steering vector mismatches[END_REF]- [START_REF] Besson | Adaptive detection with bounded steering vectors mismatch angle[END_REF], quadratically constrained [START_REF] De Maio | Fractional QCQP with applications in ML steering direction estimation for radar detection[END_REF], among others [START_REF] Bandiera | Advanced Radar Detection Schemes Under Mismatched Signal Models[END_REF]. Sometimes, another approach is privileged, and other signal models are used to reduce interference from orthogonal signals [START_REF] Bandiera | An ABORT-like detector with improved mismatched signals rejection capabilities[END_REF]. Those works do not deal specifically with the very specific non-linear off-grid mismatch and are sub-optimal for this problem. Furthermore, the low P F A used in Radar context makes things even harder: for example, the cone approach can be used to deal sub-optimally with offgrid targets but is limited to high P F A in this case, which is not always realistic in practice [START_REF] Bosse | Adaptive subspace detectors for off-grid mismatched targets[END_REF].

While the off-grid issue has been studied extensively in inverse reconstruction problems [START_REF] Tang | Compressed sensing off the grid[END_REF], [START_REF] Lasserre | Bayesian sparse Fourier representation of off-grid targets with application to experimental radar data[END_REF] (in those contexts, the decrease in performance due to off-grid targets is particularly dramatic) and in sparse estimation problems (especially in offgrid Direction Of Arrival estimation in Radar contexts, see for example [START_REF] Dai | Root sparse Bayesian learning for off-grid DOA estimation[END_REF], [START_REF] Chen | Off-grid DOA estimation using sparse Bayesian learning in MIMO radar with unknown mutual coupling[END_REF]), off-grid target detection has received less interest as far as we are aware of.

In the Bayesian context, when all target parameters are distributed according to a uniform probability distribution, it can be shown that the best detector in the sense of Neyman-Pearson criterion is the Average Likelihood Ratio Test (ALRT) derived in [START_REF] Selin | Detection of coherent radar returns of unknown Doppler shift[END_REF] assuming the signal amplitude as known and in [START_REF] Brennan | A comparison of averagelikelihood and maximum-likelihood ratio tests for detecting radar targets of unknown Doppler frequency[END_REF] assuming a random Gaussian signal amplitude.

In this paper, we focus on the deterministic case where the target amplitude and unknown parameters are supposed deterministic. In this context, the off-grid GLRT can be costly to implement but it offers very good detection performance.The most common way to perform GLRT detection is to over-sample the grid: other approaches include approximating the GLRT either with a subspace approach, by approximating the cell with Discrete Spheroidal Sequences subspace [START_REF] Rabaste | Off-grid target detection with Normalized Matched Subspace Filter[END_REF] or the Taylor expansion of the signal [START_REF] Besson | Matched direction detectors and estimators for array processing with subspace steering vector uncertainties[END_REF] in order to use subspace detectors [START_REF] Scharf | Matched subspace detectors[END_REF] or using a Monopulse-inspired method [START_REF] Develter | Off-grid radar target detection with the Normalized Matched Filter: A monopulsebased detection scheme[END_REF].

The analysis of the off-grid GLRT is often delicate. In particular, the theoretical P F A is difficult to investigate since the test quantity is the maximum of non-independent random variables, yet it is highly important as it is needed in order to implement the detector.

The search for the probability of a stochastic process or random field exceeding a threshold has been the subject of numerous works in the applied statistics literature (see for example [START_REF] Rice | Mathematical analysis of random noise[END_REF]- [START_REF] Worsley | Estimating the number of peaks in a random field using the Hadwiger characteristic of excursion sets, with applications to medical images[END_REF]). However, those works mainly focus on real processes and random fields and to our knowledge, few articles [START_REF] Hayward | CFAR detection of targets with unknown Doppler shifts[END_REF] link them with the detectors used by the signal processing community.

Based on earlier results on the volume of tubes on sphere [START_REF] Hotelling | Tubes and spheres in n-spaces, and a class of statistical problems[END_REF], [START_REF] Johnstone | On Hotelling's formula for the volume of tubes and Naiman's inequality[END_REF], we showed in [START_REF] Develter | On the false alarm probability of the Normalized Matched Filter for off-grid target detection[END_REF] how to obtain an upper bound for the P F A -threshold relationship. In this paper, we dive once again into tube theory to show how it also enables an original and simple derivation of the on-grid P F A -threshold relationship. Moreover, starting from studies on overlap [START_REF] Johansen | Hotelling's theorem on the volume of tubes: some illustrations in simultaneous inference and data analysis[END_REF], [START_REF] Kuriki | Tail probabilities of the maxima of multilinear forms and their applications[END_REF] we derive new conditions of equality of the proposed upper bound. We show that above a certain limit threshold, equality is met. Closed-form expressions for this limit threshold are derived under white noise. As we will show, the proposed formula is proved to be exact under P F A that are not too high, which is the standard regime of application of the NMF.

Section II presents the signal model, the off-grid problem, and the true GLRT formulation. Section III introduces the tube formalism and gives an original derivation of the P F Athreshold relationship using a formula related to the surface of tubes embedded on hyper-spheres. Section IV showcases the domain of validity of the relationship. It features the derivation of local overlap criteria for tubes around general multi-dimensional manifolds embedded on hyper-spheres. In Section V we check the validity of our derivations for our application by comparing the theoretical thresholds to Monte-Carlo simulations. The relationship between the correlation of the noise and the size of the domain of validity is also examined.

Notations: Matrices are in bold and capital, vectors in bold. For any matrix A or vector, A T is the transpose of A and A H is the Hermitian transpose of A. I is the N ×N identity matrix and CN (µ, Γ) is the circular complex Normal distribution of mean µ and covariance matrix Γ. S n-1 is the unit sphere in R n . The real and imaginary part operators of a complex number are denoted by Re(.) and Im(.). The operator u is the angle of a complex number u.

denotes the Hadamard product. Γ(.) is the gamma function. . denotes the classical l 2 norm for vectors.

II. PROBLEM FORMULATION

A. Signal Model

A very common detection problem in radar as well as in other domains consists in detecting a complex signal d ∈ C N corrupted by an additive noise n (clutter, thermal noise, etc.). This problem can be stated as the following binary hypothesis test, where the goal consists in deciding between two hypotheses H 0 and H 1 :

H 0 : r = n, (noise only) H 1 : r = α d(θ) + n,
(signal plus noise)

where r is the complex vector of size N of the sampled received signal, α is an unknown complex target amplitude and d(θ) stands for a generally known steering vector characterized by unknown target parameters θ (time-delay, Doppler or angle in radar). In the sequel, we will assume that n is a zeromean complex circular Gaussian noise vector with unknown variance σ 2 i.e. n ∼ CN (0, σ 2 Γ) and known shape matrix Γ (covariance matrix up to a scale factor). This context is known as a partially homogeneous Gaussian environment. All the results in this paper still apply to any other spherically invariant distribution. In this work, we will assume θ to be a deterministic unknown scalar. Although all general results of this paper apply to any signal model d(θ) (including chirp signals, in which case the unknown parameter θ models the unknown range of the target), in order to derive closed-form solutions, we choose the following common signal model:

d(θ) = 1 √ N 1, e 2iπθ , . . . , e 2iπ(N -1)θ T . (1)
This model of steering vector is very common in spectral analysis and often encountered in radar Range-Doppler detection schemes (in which case θ represents the Doppler shift of the target) where the problem consists in estimating a complex sinusoid embedded in noise after range Matched Filter processing.

B. Normalized Matched Filter (NMF)

In classical detection theory, for unknown parameters {λ i } i∈[0,1] depending on each hypothesis {H i } i∈[0,1] (either parameters of interest and/or nuisance parameters), the usual procedure relies on the Generalized Likelihood Ratio (GLR) statistic, namely the ratio Λ(r) between the Probability Density Function (PDF) f H1 (.) of the data under H 1 and the PDF f H0 (.) under H 0 where the unknown parameters are replaced by their ML estimate:

Λ(r) = max λ1 f H1 (r) max λ0 f H0 (r) H1 ≷ H0 w 2 ,
where w 2 is the detection threshold set according to a desired P F A .

When λ 1 = {α, σ} and λ 0 = {σ} with θ known, the corresponding GLRT is known as the NMF (Normalized Matched Filter) [START_REF] Scharf | Statistical signal processing: detection, estimation, and time series analysis[END_REF]:

d(θ) H Γ -1 r 2 d(θ) H Γ -1 d(θ) r H Γ -1 r H1 ≷ H0 w 2 .
(

) 2
This test is also widely used for adaptive radar in non-Gaussian contexts [START_REF] Pascal | On a SIRV-CFAR detector with radar experimentations in impulsive noise[END_REF], [START_REF] Conte | Asymptotically optimum radar detection in compound-Gaussian clutter[END_REF] for example, when the noise is distributed according to a Complex Elliptically Symmetric (CES) distribution [START_REF] Ollila | Complex Elliptically Symmetric distributions: Survey, new results and applications[END_REF]. Its statistic under H 0 , in this case, is the same as in the Gaussian case. Equivalently, Eq. ( 2) can be rewritten with normalized whitened vectors:

s(θ) H u 2 H1 ≷ H0 w 2 , (3)
where

s(θ) = Γ -1/2 d(θ) Γ -1/2 d(θ) and u = Γ -1/2 r Γ -1/2 r .
The corresponding P F A -threshold relationship is well known and is found using statistical tools in [START_REF] Scharf | Statistical signal processing: detection, estimation, and time series analysis[END_REF]:

P F A = 1 -w 2 N -1 . (4)
Note that the NMF expression (3) has a simple geometric interpretation. It is indeed the squared cosine of the angle between s (θ) and u. The threshold characterizes the squared cosine of the angle cos -1 w. A target is detected when the vector angle is below this limit angle.

C. Detecting on a grid and related issues

The test (3) was derived with the parameter θ supposed to be known. In practice, this is not the case, and this is why tests are made for several fixed parameter values. The collection of parameters θ where detection tests are run is called the grid. For our model, the usual grid is generally

G = k N , k ∈ [0, 1, . . . , N -1]
. Fourier resolution cells for this grid are then the following

D k = k N - 0.5 N , k N + 0.5 N , (5)
where k ∈ [0, 1, . . . , N -1].

When the point θ where the NMF is tested is different from the true parameter θ 0 of the target, the target is said to be off-grid. This induces a mismatch δ θ = θ -θ 0 between the real target steering vector s(θ 0 ) and the steering vector s(θ) under test. Unfortunately, it was shown in [START_REF] Rabaste | Geometrical design of radar detectors in moderately impulsive noise[END_REF] that the NMF detector is very sensitive to steering vector mismatch, potentially leading to a dramatic deterioration of the detection performance: the detection probability can even tend to 0 when the SNR tends to infinity. This phenomenon occurs [START_REF] Rabaste | Off-grid target detection with Normalized Matched Subspace Filter[END_REF] for P F A as high as 10 -3 in the chosen resolution cell of width 1/N . Figure 1 represents the average probability of detection of the NMF as a function of the SNR for δ θ uniformly distributed in D 0 : the asymptotic probability is well below 1 for a threshold corresponding to a P F A of 10 -6 . Note that this simulation was done under white noise. The NMF behavior can be even worse when Γ = I. In this case, the detection probability depends on the considered cell D k .

D. The off-grid GLRT

Instead of assuming that the target parameter lies on a grid, it is more realistic to assume that it is unknown. This leads to the general off-grid GLRT procedure, which gives in our case: where D is the search domain relative to the unknown parameter θ. Usually, for the steering vector defined in Eq. ( 1), D is defined as one of the Fourier resolution cells D k . This detector corrects the off-grid issue of test (3) as can be seen in Figure 1: its probability of detection is close to that of the oracle detector that knows the positions of the targets. In the next section, we derive its P F A -threshold relationship, the difficulty consisting here to evaluate the statistic of the maximum of a continuum of non-independent random variables.

GLRT (u, D) = max θ∈D s(θ) H u 2 H1 ≷ H0 w 2 , (6)

III. AN ANALYTICAL P F A -THRESHOLD RELATIONSHIP WITH A GEOMETRICAL INTERPRETATION

Through geometrical considerations, Hotelling [START_REF] Hotelling | Tubes and spheres in n-spaces, and a class of statistical problems[END_REF] derived a methodology to study statistical tests over the real sphere thanks to the computation of the surfaces of tubes around a curve. Indeed, for a spherically invariant noise on the sphere, evaluating the P F A of a test reduces to computing the surface of the acceptance zones, which are tubes. His formula for the surface of tubes is presented in Section III-A. It enables us to provide an alternative derivation of the well-known P F Athreshold relationship (4) for the NMF in Section III-C thanks to a simple rewriting of the NMF test quantity in Section III-B. Unfortunately, as will be explained in Section III-D, Hotelling's formula cannot be directly applied to the case of the off-grid NMF expressed in [START_REF] Besson | Detection of a signal in linear subspace with bounded mismatch[END_REF]. Indeed, as we will show, finding the P F A in the radar case with one unknown parameter requires the computation of the volume of a tube around a manifold of dimension M = 2: this is addressed in Section III-D thanks to [START_REF] Johnstone | On Hotelling's formula for the volume of tubes and Naiman's inequality[END_REF].

A. Hotelling's original geometrical approach

In this section, we present Hotelling's original theorem for computing the surface of tubes on a sphere. Consider a curve γ(ξ) on the sphere radius φ is defined as the set of points with geodesic distance to the curve inferior to φ. Formally:

S n-1 , with ξ ∈ [0, b]. A tube T of geodesic
T = u ∈ S n-1 , ∃ ξ ∈ [0, b], u T γ(ξ) > cos(φ) .
T can be seen as the union of the spherical caps SC ξ = u ∈ S n-1 , u T γ(ξ) > cos(φ) . A graphical example of a tube is provided in Figure 2. In [START_REF] Hotelling | Tubes and spheres in n-spaces, and a class of statistical problems[END_REF], Hotelling gives a formula for computing the surface of T for a closed curve (γ(0) = γ(b)).

Theorem 3.1: [START_REF] Hotelling | Tubes and spheres in n-spaces, and a class of statistical problems[END_REF] The surface enclosed by a tube of geodesic radius φ around a closed curve on the real unit sphere S n-1 is the product of the length of the axial curve by the volume of the n -2 ball of radius sin φ:

π (n-2)/2 Γ n 2 sin n-2 (φ) . (7)
When dealing with a non-closed curve, one has to add the surface of the two end semi-spherical caps to Hotelling's formula in order to find the surface of T .

Note that, in general, for Hotelling's formula to hold, it is necessary that each point in the tube belongs to a unique crosssection. Following Hotelling, this restriction will be called the non-overlap condition. Overlap phenomenons can happen when a tube draws back into itself (non-local overlap) and when its curvature becomes too high (local overlap). Nonoverlap is locally guaranteed when φ is low enough. More specifically, for a curve of constant radius of curvature ρ, Hotelling shows in [START_REF] Hotelling | Tubes and spheres in n-spaces, and a class of statistical problems[END_REF] that the condition for having no local overlap is the following:

sin φ ≤ ρ . (8)
In case of overlap, the surface given by Hotelling's theorem becomes an upper bound.

A study of overlap phenomenon is given in Section IV.

B. The complex manifold

The goal of this section is to rewrite the NMF test quantity (3) using real vectors in order to apply Hotelling's formula [START_REF] Besson | Adaptive detection with bounded steering vectors mismatch angle[END_REF]. Indeed, for any α ∈ [0, 2π], let us remark that Re s (θ)

H u exp (-iα) ≤ s (θ) H u , those two quantities 𝛾(𝜃, 𝛼) cos -1 𝓌 𝒮𝒞 𝛼 Fig. 3: T (in violet) embedded on the unit sphere S 2 in R 3 . SC α is drawn in blue. [START_REF] Hohenwarter | GeoGebra -ein Softwaresystem für dynamische Geometrie und Algebra der Ebene[END_REF] being equal for α = s (θ)

H u. We then have, decomposing s (θ) and u into real and imaginary parts:

Re s (θ) H ue -iα = γ 1 (θ) T u cos α + γ 2 (θ) T u sin α ,
where

γ 1 (θ) = Re(s (θ)) Im(s (θ)) , γ 2 (θ) = -Im(s (θ)) Re(s (θ)) and u = Re(u) Im(u) is a 2N -real valued noise vector drawn uniformly on S 2N -1 under H 0 . Denoting γ(θ, α) = γ 1 (θ) cos α + γ 2 (θ) sin α , (9)
we have, therefore:

max α∈[0,2π] γ(θ, α) T u = s (θ) H u . ( 10
)
We see that the complex case leads to consider a 2D real manifold γ(θ, α). The GLRT (6) reads:

max θ,α γ(θ, α) T u H1 ≷ H0 w . (11)
C. An original alternative derivation of the on-grid PFAthreshold relationship

In this section, we provide a simple alternative derivation to [START_REF] Scharf | Statistical signal processing: detection, estimation, and time series analysis[END_REF] for the on-grid P F A of (3). Indeed, finding the P F A reduces to a simple geometrical problem, which enables us to apply Hotelling's theorem. Readers interested in the derivation of the P F A for the GLRT (6) can skip to Section III-D. In this section, θ is considered fixed in D: we deal with the on-grid case.

A false alarm occurs when u ∈ T , where

T = u ∈ S 2N -1 , max α∈[0,2π] γ(θ, α) T u > w .
The tube T we deal with here is represented in Figure 3 in R 3 . Note that T is drawn around a closed curve: indeed γ(θ, 0) = γ(θ, 2π). Since u has been whitened, u is uniformly distributed over the unit 2N -sphere under the null hypothesis, the P F A is the ratio of the surface of T and the surface of the unit sphere.

We can apply Theorem 3.1 to the tube T previously defined around the curve γ(θ, α) with fixed θ and parameterized by α to find the P F A of the NMF (4). The length of the axial curve is equal to 2π. In this case, the tube is closed and one does not need to add end spherical caps to Hotelling's formula. Furthermore, we prove that T does not overlap in Section A of the appendix so that the surface given by Hotelling's formula is thus exact in this case. Applying Theorem 3.1 with n = 2N , φ = cos -1 w gives:

Surface(T ) = 2π π N -1 Γ(N ) sin 2(N -1) (φ) , = 2 π N Γ(N ) 1 -w 2 N -1 . (12)
Dividing ( 12) by the surface 2 π N Γ(N ) of S 2N -1 leads to the expected result (4). This geometrical approach provides an alternative to the traditional one based on statistical tools [START_REF] Scharf | Statistical signal processing: detection, estimation, and time series analysis[END_REF].

D. Extending Hotelling's approach to the GLRT Unfortunately, Hotelling's result is not immediately applicable to the considered GLRT [START_REF] Bosse | Adaptive subspace detectors for off-grid mismatched targets[END_REF]: since a maximization on the parameter θ is introduced, the surface of interest is spread around a 2D manifold as will be seen now.

The acceptance region in this case is a new tube T around the two-dimensional manifold γ(θ, α):

T = u ∈ S 2N -1 , max θ∈D,α γ(θ, α) T u > w .
Note that in this case, T follows a manifold that is often not closed since, writing D = [θ 1 , θ 2 ], γ(θ 1 , α) = γ(θ 2 , α) in general. Unlike previously, when computing the surface of the tube, a term accounting for its boundaries will appear.

Hotelling's result does not cover this multi-dimensional manifold case as it gives the surface of a tube around a curve. However, in [START_REF] Johnstone | On Hotelling's formula for the volume of tubes and Naiman's inequality[END_REF] and [START_REF] Knowles | On Hotelling's approach to testing for a nonlinear parameter in regression[END_REF], this result is extended to a special case of two-dimensional manifolds embedded on S n-1 which is of interest to us:

Theorem 3.2: [26] For i ∈ [1, 2], let γ i : [θ 1 , θ 2 ] → S n-1 be regular curves. Assume γ 1 (θ) T γ 2 (θ) = 0 for all θ. Let Z(θ) = γ 1 (θ) T u 2 + γ 2 (θ) T u 2 1/2
where u is uniformly distributed on S n-1 . Then for 0 < w < 1, we have:

P max θ1≤θ≤θ2 Z (θ) > w ≤ (1 -w 2 ) (n-2)/2 + Γ n 2 w (1 -w 2 ) (n-3)/2 2π 3/2 Γ n -1 2 × θ2 θ1 2π 0 γ1 (θ) cos Ω + γ2 (θ) sin Ω 2 - γ1 (θ) T γ 2 (θ) 2 1/2 dΩ dθ , (13)
where γi (θ) is the derivative of γ i (θ) with respect to θ.

When there is no overlap (see Section IV for a detailed study), i.e. low P F A regimes, this inequality becomes equality.

It turns out that we can reformulate our problem in order to fulfill the assumptions of the above theorem with n = 2N . Indeed, using ( 9) and [START_REF] Bandiera | An ABORT-like detector with improved mismatched signals rejection capabilities[END_REF], we can check that:

s (θ) H u 2 = γ 1 (θ) T u 2 + γ 2 (θ) T u 2 ,
so that Theorem 3.2 gives us the desired P F A (when equality holds in ( 13)). Follows our result:

Corollary 3.2.1: In the absence of overlap (low P F A regimes), the P F A for the GLRT (6) for a search interval D = [θ 1 , θ 2 ] with the steering vector d (θ) defined in (1) is given by:

P F A = (1 -w 2 ) N -1 + (14) Γ(N ) w (1 -w 2 ) N -3 2 π 1/2 Γ N -1 2 θ2 θ1 γ1 (θ) T P ⊥ γ2(θ) dθ.
where

P ⊥ γ2(θ) = I -γ 2 (θ)γ 2 (θ)
T is the orthogonal projector on γ 2 (θ). Under white noise (Γ = σ 2 I), this result simplifies to:

P F A = (1 -w 2 ) N -1 + (15) π 3 Γ(N ) w (1 -w 2 ) N -3 2 Γ N -1 2 N 2 -1 1 2 (θ 2 -θ 1 ) .
When D = [0, 1], the first term has to be removed from the equations. The integral in ( 14) can be easily evaluated numerically.

Proof: First, notice that the derivatives γ1 (θ) and γ2 (θ) are orthogonal, and that γ1 (θ) = γ2 (θ) . Thus, for all Ω ∈ [0, 2π], we have that

γ1 (θ) cos Ω + γ2 (θ) sin Ω 2 = γ1 (θ) 2 ,
which does not depend on Ω. The double integral simplifies:

θ2 θ1 2π 0 γ1 (θ) cos Ω + γ2 (θ) sin Ω 2 - γ1 (θ) T γ 2 (θ) 2 1/2 dΩ dθ , = 2π θ1 θ2 γ1 (θ) 2 -γ1 (θ) T γ 2 (θ) 2 1/2 dθ, = 2π θ2 θ1 γ1 (θ) T P ⊥ γ2(θ) dθ, (16)
In the case of white noise, this integral can be computed analytically. Let x be the following vector:

x = 2π [0, 1, . . . , N -1] T , so that γ1 (θ) = x x γ 2 (θ) , (17)
γ2 (θ) = - x x γ 1 (θ) . (18)
Then:

γ1 (θ) = γ2 (θ) = 2π (N -1)(2N -1) 6 . ( 19
)
and

γ1 (θ) T γ 2 (θ) = x x γ 2 (θ) T γ 2 (θ) , = 2π N N -1 k=0 k = π (N -1) . (20)
Then, injecting [START_REF] Scharf | Matched subspace detectors[END_REF] and ( 20) into ( 16):

2π θ2 θ1 γ1 (θ) T P ⊥ γ2(θ) dθ = 2(θ 2 -θ 1 )π 2 (N 2 -1) 3 .
(21) Replacing the double integral in [START_REF] Lasserre | Bayesian sparse Fourier representation of off-grid targets with application to experimental radar data[END_REF] with [START_REF] Rice | Mathematical analysis of random noise[END_REF] gives the expected result.

Interestingly, note that the first term in [START_REF] Chen | Off-grid DOA estimation using sparse Bayesian learning in MIMO radar with unknown mutual coupling[END_REF] represents the surface of the two semi-spherical caps at the extremities of the tube. As such, it is equal to the P F A of the NMF expressed in [START_REF] Rabaste | Off-grid target detection with Normalized Matched Subspace Filter[END_REF]. The second term shows the influence of the manifold induced by the off-grid nature of the problem. It is analogous to the one-dimensional case of Theorem 3.1, divided by the surface of S 2N -1 . Here, θ 2 -θ 1 plays the role of the manifold length, and the rest of the rightmost term is the surface of the cross-section divided by the area of S 2N -1 .

The relationships given in Corollary 3.2.1 are upper bounds in the presence of overlap. In this case, they still hold interest in the radar context where controlling the P F A is fundamental. The next section investigates the conditions under which no overlap happens.

IV. ON THE APPEARANCE OF OVERLAP

The goal of this section is to determine for which thresholds formula (13) (formula [START_REF] Chen | Off-grid DOA estimation using sparse Bayesian learning in MIMO radar with unknown mutual coupling[END_REF] in the case of white noise) holds equality. This requires us to investigate the conditions under which overlap occurs. We will start by exhibiting general results from [START_REF] Johansen | Hotelling's theorem on the volume of tubes: some illustrations in simultaneous inference and data analysis[END_REF], [START_REF] Kuriki | Tail probabilities of the maxima of multilinear forms and their applications[END_REF] as well as some original results on shiftinvariant manifolds before delving into our specific problem.

Let us consider an M -dimensional manifold M = {γ(ξ), ξ = (ξ 1 , . . . , ξ M ) ∈ D} defined on the search domain D over S n-1 and let us consider the tube T over S n-1 around M consisting of the points u satisfying u T γ(ξ) > w for some ξ in D. M can be seen, loosely speaking, as the axis of T . The cross-section CS ξ defined in [START_REF] Johansen | Hotelling's theorem on the volume of tubes: some illustrations in simultaneous inference and data analysis[END_REF] at a point ξ belonging to M is the set of points of T orthogonal to the derivatives of γ in ξ. Formally:

CS ξ = u ∈ T , u T ∂γ ∂ξ T = 0, u T γ(ξ) > w . ( 22
)
Excluding the edge effects defined later in this section, the tube defined as the union of cross-sections overlaps if and only if a point u of T belongs to more than one cross-section: in this case, Hotelling's geometrical approach and its extensions lead to an overestimation of the Probability of False Alarm. Otherwise, the equality holds true in [START_REF] Lasserre | Bayesian sparse Fourier representation of off-grid targets with application to experimental radar data[END_REF]. There are two types of overlap defined in [START_REF] Hotelling | Tubes and spheres in n-spaces, and a class of statistical problems[END_REF]: local overlap, which derives from local differential properties of the manifold generating the tube, and non-local overlap, which depends on the overall shape of the tube. Global overlap encompasses both types of Fig. 4: Illustration of all the phenomena leading to an overestimation of the P F A when using [START_REF] Chen | Off-grid DOA estimation using sparse Bayesian learning in MIMO radar with unknown mutual coupling[END_REF].

overlap. It is linked to a limit overlap threshold w lim and there is no overlap if: w ≥ w lim .

The limit threshold can be equivalently seen as an angle φ lim such that cos φ lim = w lim , and there is no overlap if

φ = cos -1 w ≤ φ lim .
In the sequel, the conditions are expressed in terms of φ lim .

In addition to those phenomena, edge effects also have to be considered. They can appear when dealing with nonclosed manifold and cause the same problems as overlap i.e. an overestimation of the P F A . In the 1D case illustrated in Figure 2, edge effects would occur when the green semi-spherical caps at the end overlap each other. A manifold is said to be closed along dimension k if its k-th variable ξ k belongs to an interval [ξ k1 , ξ k2 ] such that γ(ξ 1 , . . . , ξ k1 , . . . , ξ M ) = γ(ξ 1 , . . . , ξ k2 , . . . , ξ M ) for all ξ i , i = k. A manifold is said to be closed if it is closed along all its dimensions.

Figure 4 summarizes all the phenomena that can arise.

A. The general case

In this section, we give the conditions under which no overlap happens under general conditions for any M -dimensional manifold γ(ξ 1 , . . . , ξ M ) embedded in S n-1 .

In [START_REF] Johansen | Hotelling's theorem on the volume of tubes: some illustrations in simultaneous inference and data analysis[END_REF], a criterion for characterizing the overlap of a tube embedded on a sphere around a curve is introduced that is a direct consequence of the fact that the union of the cross-section needs to be disjoint. It turns out that the arguments used by the authors can be generalized in order to find overlap criteria for tubes around any M -dimensional manifolds, as suggested in [START_REF] Kuriki | Tail probabilities of the maxima of multilinear forms and their applications[END_REF].

Theorem 4.1: [START_REF] Johansen | Hotelling's theorem on the volume of tubes: some illustrations in simultaneous inference and data analysis[END_REF], [START_REF] Kuriki | Tail probabilities of the maxima of multilinear forms and their applications[END_REF] Let γ(ξ) be a C 2 M -dimensional manifold parameterized by ξ = (ξ 1 , . . . , ξ M ) ∈ D. Let φ lim be the limit angle for which no overlap occurs, related to w lim [START_REF] Worsley | Estimating the number of peaks in a random field using the Hadwiger characteristic of excursion sets, with applications to medical images[END_REF] by cos(φ lim ) = w lim . Let P ξ be the projection onto the subspace spanned by γ(ξ ) and its derivatives ∂γ ∂ξ . φ lim is given by:

cot 2 φ lim = sup ξ,ξ ∈D 2 1 -γ(ξ) T P ξ γ(ξ) 1 -γ(ξ) T γ(ξ ) 2 , sup ξ,ξ ∈D 2 h(ξ, ξ ) . ( 24
)
The criterion [START_REF] Hayward | CFAR detection of targets with unknown Doppler shifts[END_REF] encompasses both local and non-local overlap:

φ lim = min{φ local , φ non-local } , (25)
where φ local and φ non-local are the limit angles such that local and non-local overlaps occur. Local overlap occurs when ξ tends to ξ, and non-local overlap arises when the sup of h in ( 24) is attained for ξ = ξ . Note that [START_REF] Hayward | CFAR detection of targets with unknown Doppler shifts[END_REF] does not take into account edge effects.

It can be simplified in the case of a shift-invariant manifold:

Definition 4.1: A real manifold γ(ξ) is said to be shiftinvariant when, for any ξ, ξ , the scalar product γ(ξ) T γ(ξ ) depends only on ξξ :

γ(ξ) T γ(ξ ) = f (ξ -ξ ) ,
where f is an even function. Then, similarly to the case of a single parameter, we have the following property: Proposition 4.1: For a shift-invariant manifold γ(ξ), h as defined in ( 24) is a function of ξξ : h(ξ, ξ ) = g(ξ -ξ ). Consequently,

cot 2 φ lim = sup x∈E g(x) , ( 26
)
where E is the image of D×D by the function (ξ, ξ ) → ξ-ξ . Proof: The proof is given in Appendix B. The implications of this result are detailed in the following sections.

1) On local overlap: In this section, we discuss the occurrence of local overlap around a M -dimensional manifold. The results of this section are particularized to our complex signal model (1) in Section IV-B1.

Local overlap is linked to the curvature of the manifold. To illustrate this, consider the case of a tube around a curve in Euclidean space drawn in Figure 5: there is local overlap whenever the tube radius is greater than the radius of curvature of the curve.

In [29, Annex A.2], a local overlap criterion is developed. Recall that it corresponds to the case where ξ tends to ξ.

We define the local overlap angle φ local in ξ similarly in the multi-dimensional case as:

Corollary 4.1.1: [29, Annex A.2] In the case of a tube around an M -dimensional manifold γ, using the same nota-Fig. 5: Illustration of local overlap in 1D in the Euclidean case. Here, the radius of the tube (in red) is greater than the radius of curvature. This causes overlap: see, for example, that the point u belongs to both cross-sections CS ξ and CS ξ . tions as before with β representing the directions of convergence from ξ to ξ, the limit local overlap angle is given by:

cot 2 φ local = sup ξ∈D sup β∈S M -1 lim →0 h(ξ, ξ + β) , = sup ξ∈D sup β∈S M -1 i,j β i β j (I -P ξ ) ∂ 2 γ ∂ξ i ∂ξ j 2   i,j β i β j ∂γ T ∂ξ i ∂γ ∂ξ j   2 .( 27
)
Readers can refer to [START_REF] Kuriki | Tail probabilities of the maxima of multilinear forms and their applications[END_REF]Annex A.2] for an interpretation in terms of principal curvatures.

Proof: The proof is given in [START_REF] Kuriki | Tail probabilities of the maxima of multilinear forms and their applications[END_REF]Annex A.2]. Formula ( 27) can be simplified in the case of a shiftinvariant manifold: [START_REF] Johnstone | On Hotelling's formula for the volume of tubes and Naiman's inequality[END_REF] shows that the maximization on ξ can be dropped in [START_REF] Develter | On the false alarm probability of the Normalized Matched Filter for off-grid target detection[END_REF]. Indeed, h(ξ, ξ ) only depends on the difference ξξ . The following developments allow us to go further: Corollary 4.1.2: Let γ(ξ) be a M -dimensional shiftinvariant manifold, and let G be its first fundamental form:

G = ∂γ ∂ξ T ∂γ ∂ξ . ( 28
)
Then G does not depend on ξ. Besides, let

G = G T /2 G 1/2 , ( 29
)
and µ be a reparametrization of γ defined by

µ = G 1/2 ξ , (30)
Then ∂γ T ∂µ i ∂γ ∂µ j = δ ij , (31)
and ( 27) reduces to:

cot φ 2 local = sup β∈S M -1 i,j β i β j ∂ 2 γ ∂µ i ∂µ j 2 -1 . ( 32
)
Proof: The proof is given in the appendix C.

2) On non-local overlap: Non-local overlap arises when the tube draws back into itself, as shown in blue in Figure 4 for a tube spanned by a curve. In [START_REF] Johansen | Hotelling's theorem on the volume of tubes: some illustrations in simultaneous inference and data analysis[END_REF], it is shown in the case M = 1 that the limit angle around a closed manifold γ(ξ) linked to this type of overlap can be characterized entirely by looking at the pairs of points (ξ, ξ ) that minimize locally the distance γ(ξ) -γ(ξ ) , with ξ = ξ . In such case, [START_REF] Hayward | CFAR detection of targets with unknown Doppler shifts[END_REF] reduces to an intuitive geodesic distance criteria when ξ = ξ . In the general case, M is arbitrary, and the following holds: Proposition 4.2: Consider a tube around the M -dimensional manifold γ lying on the sphere. The set of pairs ξ, ξ that characterizes non-local overlap is:

Ξ = ξ, ξ , ξ = ξ , γ(ξ) -γ(ξ ) T ∂γ ∂ξ k = 0 and γ(ξ) -γ(ξ ) T ∂γ ∂ξ k = 0, ∀k ∈ [1, M ] . ( 33
)
Then, if the manifold is closed:

φ non-local = min (ξ,ξ )∈Ξ 1 2 cos -1 γ(ξ) T γ(ξ ) . ( 34
)
For a non-closed manifold, a term accounting for the boundaries must be taken into account:

φ non-local = min min (ξ,ξ )∈Ξ 1 2 cos -1 γ(ξ) T γ(ξ ) , E , (35)
where

E = inf (ξ,ξ )∈B×D cot -1 h (ξ, ξ ) . ( 36
)
Proof: It is possible to adapt the proof of proposition 4.2 of [START_REF] Johansen | Hotelling's theorem on the volume of tubes: some illustrations in simultaneous inference and data analysis[END_REF] for an M-dimensional manifold using the principal curvature interpretation of [START_REF] Develter | On the false alarm probability of the Normalized Matched Filter for off-grid target detection[END_REF] and by considering frontiers around the local minima. The extension to the case of a nonclosed manifold is then straightforward. In the non-closed case, the limit angle φ non-local also encompasses instances of local overlap when ξ tends to ξ ∈ B. Plugging it in (25) still yields the right exact limit angle φ lim .

As noted in [START_REF] Johansen | Hotelling's theorem on the volume of tubes: some illustrations in simultaneous inference and data analysis[END_REF], this formulation is not necessarily simpler to use than [START_REF] Hayward | CFAR detection of targets with unknown Doppler shifts[END_REF] since it can be more involved to find the set of pairs Ξ than to compute [START_REF] Hayward | CFAR detection of targets with unknown Doppler shifts[END_REF]. However, in Section IV-B2, we show that in our specific case, it enables us to dramatically reduce the computational complexity of the search.

3) On edge effects: The formula [START_REF] Hayward | CFAR detection of targets with unknown Doppler shifts[END_REF] does not take into account edge effects that can arise when dealing with nonclosed manifolds. Indeed, noting B the boundaries of D, [START_REF] Hayward | CFAR detection of targets with unknown Doppler shifts[END_REF] is defined for ξ ∈ D, ξ ∈ D \ B. Edge effects appear when a point u is such that there exist two distinct points ξ 1 , ξ 2 in B such that u T γ(ξ 1 ) > w and u T γ(ξ 2 ) > w. It is illustrated in Figure 4 in the case M = 1. One has to check that the limit angle φ lim is big enough so that edge effects do not appear.

B. Application to the complex signal model (1)

We apply here the general results of the previous section to our case of interest where γ is given by ( 9), defined on D × [0, 2π]. This first result gives the limit global overlap threshold for our application: Corollary 4.1.3: With γ defined as in ( 9), the limit angle for no global overlap is:

cot 2 φ lim = sup θ, α ∈ D × [0, 2π] θ , α ∈ D × [0, 2π] 1 -γ(θ, α) T P θ ,α γ(θ, α) (1 -γ(θ, α) T γ(θ , α )) 2 ,
(37) where P θ ,α = I -P ⊥ θ ,α with P ⊥ θ ,α defined later in (40). Proof: We simply inject (9) in [START_REF] Hayward | CFAR detection of targets with unknown Doppler shifts[END_REF]. Under white noise, for one cell D k , one can check numerically that the corresponding limit P F A is equal to:

P F Alim ≈ 10 -2.52 , (38)
Fortunately, this P F A is well above the common P F A encountered in radar applications.

In the following sections, we discuss how to find the limit angles φ local and φ non-local . Indeed, the search domain in (37) is 4-dimensional and so the criterion can be heavy to evaluate. We can accelerate the search of the global limit angle by first finding the local and non-local overlap angle φ local and φ non-local through ( 27) and (34) then combining them using [START_REF] Hotelling | Tubes and spheres in n-spaces, and a class of statistical problems[END_REF].

1) On local overlap: The following corollary gives φ local in our case: Corollary 4.1.4: For our manifold γ defined in (9), we have:

cot 2 φ local = sup θ,α∈D×[0,2π] sup ϕ∈[0,2π] J(θ, α, ϕ) , (39)
where and

J(θ, α, ϕ) = P ⊥ θ,α cos 2 ϕ ∂ 2 γ ∂θ 2 + sin 2ϕ ∂ 2 γ ∂θ∂α + sin 2 ϕ ∂ 2 γ
P ⊥ θ,α = I -M θ,α M T θ,α M θ,α -1 M T θ,α , (40)
with M θ,α = γ(θ, α),

. This expression can be easily evaluated setting β = [cos(ϕ), sin(ϕ)] and then maximizing on ϕ.

Proof: We simply inject ( 9) in ( 27), with M = 2. The derivatives of γ are found from a straightforward derivation using ( 17) and [START_REF] Besson | Matched direction detectors and estimators for array processing with subspace steering vector uncertainties[END_REF]:

∂γ ∂θ = I -γ(θ, α)γ(θ, α) H (cos α γ 2 (θ) -sin α γ 1 (θ)) x x , ∂γ ∂α = -sin α γ 1 (θ) + cos α γ 2 (θ) .
It is possible to find the analytical limit threshold for local overlap under white noise.

Corollary 4.1.5: Under white noise with the signal model as in (1), the limit local angle (39) is:

φ local = tan -1 √ 5 -C 2 ,
and the limit local threshold is

w 2 local = cos 2 tan -1 √ 5 -C 2 , (41)
where

C = 3 5 3N 2 -7 (N -1) 2 . ( 42
)
Proof: The proof is given in Appendix D.

Note that lim

N →∞ C = 9 5
: for N large enough,

w 2 local ≈ cos 2 tan -1 2 √ 5 . (43)
For N = 10, according to (15) this corresponds to the limit:

P F Alocal ≈ 10 -2.52 = P F Alim .
thus local overlap is the limiting factor for this noise setting.

2) On non-local overlap: First, consider the case where the target is searched over the whole spectral domain i.e. D = [0, 1]. This corresponds to the operational context where a single target is searched in the scene. In this case, the manifold γ in ( 9) is closed. In order to compute the non-local limit angle φ non-local , one should evaluate criterion (34). The following corollary simplifies the criteria.

Corollary 4.1.6: Consider a tube lying on the sphere around the manifold γ defined in [START_REF] Bandiera | Advanced Radar Detection Schemes Under Mismatched Signal Models[END_REF]. Define Ξ as:

Ξ = (θ, θ ) , θ = θ , ∂ s(θ) H s(θ ) ∂θ = 0 . (44)
Then when D = [0, 1], φ non-local in (34) reduces to:

φ non-local = min (θ,θ )∈Ξ 1 2 cos -1 s(θ) H s(θ ) . (45)
and otherwise, when

D = [θ 1 , θ 2 ] [0, 1], φ non-local in (35) reduces to φ non-local = min min (θ,θ )∈Ξ 1 2 cos -1 s(θ) H s(θ ) , E . (46)
where E has been defined in (36), with B = {θ 1 , θ 2 } × [0, 2π] and ξ = (θ, α), ξ = (θ , α ). Proof: The proof is provided in the appendix. This simplification allows us to simply investigate the critical points with (θ = θ ) of ambiguity maps s(θ) H s(θ ) such as the one drawn on Figure 6. In the case of a closed manifold i.e. D = [0, 1], the search (45) becomes two-dimensional. When D = [θ 1 , θ 2 ] [0, 1], one has to evaluate (36) in order to compute (46). The search (46) is thus three-dimensional.

Under white noise, γ is shift-invariant. The search space can be further simplified and is of dimension 1: indeed, in this case, the product s(θ) H s(θ ) depends only on the difference δ = θ -θ. Finding the local maxima of s(θ) H s(θ ) simply reduces in finding the local maxima of s(θ) H s(θ + δ) for any fixed θ. s(θ) H s(θ + δ) represents the autocorrelation of s, and it is well known that:

s(θ) H s(θ + δ) = 1 N sin(πδN ) sin(πδ) . (47)
The autocorrelation of s is represented in Figure 7. In this case, in a single cell D k the set Ξ is empty since the derivative of s(θ) H s(θ + δ) only vanishes for δ = 0. If D = [0, 1], φ non-local can be readily obtained from the first secondary lobe. C. On edge effects

If D = [θ 1 , θ 2 ] [0, 1], then B = {θ 1 , θ 2 } × [0, 2π
] and one has to take into account edge effects. The result in appendix A shows that the tubes around the sub-manifolds {γ(θ 1 , α), α ∈ [0, 2π]} and {γ(θ 2 , α), α ∈ [0, 2π]} do not self-overlap. We simply check that those tubes do not overlap with each other: Proposition 4.3: Consider the tube around the manifold γ defined in [START_REF] Bandiera | Advanced Radar Detection Schemes Under Mismatched Signal Models[END_REF]

on a cell D = [θ 1 , θ 2 ] [0, 1].
Barring the unlikely case where the length of the manifold for fixed α is smaller than φ, no edge effects appear if

φ < φ edge 1 2 cos -1 s(θ 1 ) H s(θ 2 ) . (48)
In particular, under white noise, if [θ 1 , θ 2 ] is a cell D k as defined in [START_REF] Maio | Robust adaptive radar detection in the presence of steering vector mismatches[END_REF], s(θ 1 ) H s(θ 2 ) = 0, so that φ edge = π/4. Proof: No edge effect occurs if:

φ < min α1,α2 1 2 cos -1 γ(θ 1 , α 1 ) T γ(θ 2 , α 2 ) . (49)
With (58), the minimum (49) is reached for α 1α 2 = s(θ 1 )s(θ 2 ). In this case, γ(θ 1 , α 1 ) T γ(θ 2 , α 2 ) = s(θ 1 ) H s(θ 2 ) , and the rightmost term of (49) reduces to φ edge in (48).

V. NUMERICAL RESULTS

Let us check the validity of Eq. ( 15). Figure 8-(a) presents the P F A -threshold relationship given by Eq. ( 15) and empirically computed thresholds using 10 8 complex circular white Gaussian noise samples for a steering vector size of N = 10. The continuous research over the domain D is replaced by a discrete search using 30 tests in the cell, where D = D 0 .

The formula seems to fit very well when the P F A is low enough (or, equivalently, if the threshold w is high enough). It is not valid for P F A close to 1 because of overlap (it even exceeds 1). However, such high P F A have no practical interest for standard applications.

It is not trivial to verify the limit overlap value (43) simply by looking at Figure 8-(a): overlap stops having a significant impact on the relationship well before attaining w lim . Let us check our value of w lim by exhibiting a well-chosen point u of the tube belonging to more than one cross-section for a threshold w very close to w lim , with w < w lim , under white noise. Indeed, consider (for any θ, α ∈ D × [0, 2π] since γ is shift invariant), the point of the tube

u = cos(φ) γ(θ, α) + sin(φ) n , (50)
with cos φ = w and n is the unit norm vector such that:

n ∝ γ(θ, α) + i,j∈[1,2] β i β j ∂ 2 γ ∂µ i ∂µ j ,
where µ is the parametrization defined in [START_REF] Conte | Asymptotically optimum radar detection in compound-Gaussian clutter[END_REF], and

(β 1 , β 2 ) = arg max (β1,β2)∈S 1 i,j β i β j ∂ 2 γ ∂µ i ∂µ j 2 ,
where the right quantity is maximized numerically with β 1 = sin(ϕ), β 2 = cos(ϕ). Note that, using (53), n is orthogonal to γ(θ, α) so that u is indeed a point of the tube since u T γ(θ, α) = cos φ = w and u = 1. Figure 9 shows that u belongs to only one cross-section when φ < φ lim and to three cross-sections when φ > φ lim . Indeed, defining the complex vector u by u = Re(u) Im(u) , see that the derivative of the product u H s(θ + δ) 2 vanishes to 0 above the threshold once in the first case, and three times in the second. Using [START_REF] Bandiera | An ABORT-like detector with improved mismatched signals rejection capabilities[END_REF], this means that u T ∂γ ∂θ = u T ∂γ ∂α = 0 for 3 values

ξ i = θ + δ i , ∠u H s(θ + δ i
) so that u belongs to 3 crosssections CS ξi according to the definition [START_REF] Adler | On excursion sets, tube formulas and maxima of random fields[END_REF]. Even though it is hard to detect visually in Figure 8-(a), overlap occurs right before the limit threshold value w lim found in (41): the limit threshold is very conservative and formula [START_REF] Chen | Off-grid DOA estimation using sparse Bayesian learning in MIMO radar with unknown mutual coupling[END_REF] can be used as a good approximate of the true P F A threshold relationship for thresholds well below that.

We have also compared the formula with empirical thresholds for colored noise (Γ = I). For this purpose, we used the following well-known model of covariance matrices:

Γ(ρ) = T o 1 ρ . . . ρ N -1 , (51)
where T o(.) is the Toeplitz matrix operator, and ρ is a scalar that defines the level of correlation of the noise. Results can be observed in Figures 8-(b) for the edge cell D 0 where detection performance is lower on average. Zooming on the leftmost part of the curves, it can be seen that the overlapping phenomenon for low P F A values tends to increase slightly with ρ: the gap between the curves widens slightly and lasts a bit longer as noise becomes more correlated. This is not surprising, as correlated noise bends the manifold, increasing the likeliness of both local and non-local global overlap. The formula is still a good approximation of the P F A -threshold relationship well before the limit threshold for no overlap w lim .

To verify this behavior, we plot the relationship between ρ and the minimum threshold for which there is no overlap w lim computed thanks to (25) on Figure 10. The components w local and w non-local are obtained thanks to (39), (35). As we suspected, w lim tends to increase with ρ. The arising of nonlocal overlap can be explained by looking at Figure 6: when ρ increases, a side-lobe enters cell D 0 as Γ then modelizes noise centered around θ = 0.

VI. CONCLUSION

This article addresses the off-grid detection problem using the NMF-GLRT by finding an analytical P F A -threshold relationship. We analyzed its domain of validity thanks to the application of results on the overlap phenomena for tubes around multi-dimensional manifolds to the off-grid signal model. Several new closed-form expressions under white noise have been expressed. This shows that our relationship is valid for most common radar applications for which the P F A is low enough.

APPENDIX

In the first section of the appendix, we prove that the tube T , defined in [START_REF] Tang | Compressed sensing off the grid[END_REF] for fixed θ, does not overlap. Then, we provide proofs for the corollaries of Section IV.

A. On the absence of overlap of the tube T (12) for fixed θ Since ∂γ(θ, α) ∂α = 1, γ(θ, .) is parameterized by arc length. The radius of first curvature is then defined as

ρ = ∂ 2 γ(θ, α) ∂α 2 -1 = 1.
Then, since sin(cos -1 w) < 1 for all w, there is no local overlap according to [START_REF] De Maio | Fractional QCQP with applications in ML steering direction estimation for radar detection[END_REF].

Let us prove there is no non-local overlap either by searching the pairs of points of interest (α, α ) in Ξ, that verify:

γ(θ, α ) T γ(θ, α) = γ(θ, α) T γ(θ, α ) = 0 .
Those conditions imply that:

(cos α γ 1 (θ)+sin α γ 2 (θ)) T (cos αγ 2 (θ)-sin αγ 1 (θ)) = 0 ,
that leads to the condition:

-cos α sin α γ 1 (θ) + sin α cos α γ 2 (θ) = 0 , and equivalently: sin(α -α ) = 0. (b) Highly correlated noise : ρ = 0.9.

Fig. 8: Comparison between the theoretical P F A -threshold given in ( 15) for (a) and ( 13) for (b) and the empirical Monte Carlo P F A -threshold relationships for N = 10 and for several values of ρ (51). The relationship is drawn for the search domain D 0 . The on-grid relation ( 4) is also drawn for comparison purposes. The limit overlap threshold w lim proposed in ( 23) is in purple.

Fig. 9: Illustration of the overlap phenomenon: squared projection of u defined in (50) on s(θ + δ) for θ + δ ∈ D 0 for two values of φ: φ = 0.95φ lim and φ = 1.05φ lim .

Thus, the set Ξ is defined as:

Ξ = {(α, α ) : α = α , sin(α -α ) = 0} , = {(α, α + π), α ∈ [0, π]} . For any α ∈ [0, π], we have γ(α, θ) T γ(α + π, θ) = -1, so that φ non-local = 1 2 arccos(-1) = π 2
according to (35): there is no non-local overlap for T . Thus, w lim = cos φ lim = 0 and the surface given by ( 12) is exact for any threshold w.

B. Proof of Proposition 4.1

Proof: Let γ(ξ) T γ(ξ ) = f (ξ-ξ ). The quantity h(ξ, ξ ) depends on ξ and ξ through γ(ξ) T γ(ξ ) = f (ξ -ξ ) in the denominator, and, by the way of P ξ in the numerator,

γ(ξ) T ∂γ(ξ ) ∂ξ k and ∂γ(ξ) T ∂ξ i ∂γ(ξ ) ∂ξ j for k ∈ [1, M ] and (i, j) ∈ [1, M ] 2
. By using the derivatives of f , it can be shown that those two terms and h depend only on the difference ξξ . In order to prove Corollary 4.1.2, we need to introduce the following lemma on the existence of the parametrization µ:

Lemma A.1: Let γ(ξ) be a M -dimensional shift-invariant manifold. The parametrization µ chosen in ( 30) is well defined, and its first fundamental form is equal to I:

∂γ T ∂µ i ∂γ ∂µ j = δ i,j .
Proof: Consider G = (g ij ) 1≤i,j≤N defined as in [START_REF] Johansen | Hotelling's theorem on the volume of tubes: some illustrations in simultaneous inference and data analysis[END_REF]. Note that by differentiating γ T γ = 1 twice, we have:

γ T ∂ 2 γ ∂ξ i ∂ξ j = - ∂γ ∂ξ i T ∂γ ∂ξ j . ( 52
)
Let f (ξ -ξ ) = γ(ξ) T γ(ξ ). It can be rewritten equivalently f (y) = γ(ξ) T γ(ξ + y), which gives, using (52):

g ij = ∂γ T ∂ξ i ∂γ ∂ξ j = - ∂ 2 f ∂y i ∂y j y=0 .
Thus, the coefficients g ij do not depend on ξ and so neither does G. We can define G 1/2 as in [START_REF] Kuriki | Tail probabilities of the maxima of multilinear forms and their applications[END_REF] since G is positive definite. Then, the parametrization (30) in the corollary is well defined, so that ξ = G -1/2 µ. Then: The following proves Corollary 4.1.2: Proof: Equation ( 31) has been proved in the above lemma. Let us prove [START_REF] Hohenwarter | GeoGebra -ein Softwaresystem für dynamische Geometrie und Algebra der Ebene[END_REF]. First, note that differentiating γ T γ = 1 twice yields:

γ T ∂ 2 γ ∂µ i ∂µ j = - ∂γ ∂µ i ∂γ ∂µ j = -δ ij . (53)
Next, let us introduce Christoffel symbols of the first kind:

Γ ijk = ∂ 2 γ T ∂µ i ∂µ j ∂γ ∂µ k . (54)
It can be easily verified that those symbols can be expressed as a function of the derivatives of the first fundamental form G = g ij 1≤i,j≤N :

Γ ijk = 1 2
∂g ik ∂µ j + ∂g jk ∂µ i -∂g ij ∂µ k , which gives, since according to (31) g ij = δ i,j :

Γ ijk = 0 . (55)
Let us now consider Equation ( 27) with parametrization µ.

The denominator reduces to 1:

i,j

β i β j ∂γ T ∂µ i ∂γ ∂µ j = i,j β i β j δ i,j = i β 2 i = 1 .
We thus have:

cot 2 φ local = sup β∈S M -1 i,j β i β j (I -P µ ) ∂ 2 γ ∂µ i ∂µ j 2 , = sup β∈S M -1    i,j β i β j ∂ 2 γ ∂µ i ∂µ j 2 - P µ i,j β i β j ∂ 2 γ ∂µ i ∂µ j 2    , (56)
where the maximization on µ has been omitted since the manifold γ is shift-invariant. γ, ∂γ ∂µ 1 , . . . , ∂γ ∂µ M forms an orthonormal family so that the second term of (56) is, with Pythagoras's theorem: Lemma A.2: Under white noise (Γ = I), the manifold γ as defined in ( 9) is shift invariant.

P µ i,j β i β j ∂ 2 γ ∂µ i ∂µ j 2 = i,j β i β j γ T ∂ 2 γ ∂µ i ∂µ j 2 + k i,j β i β j ∂γ T ∂µ k ∂ 2 γ ∂µ i ∂µ j
Proof: The complex manifold s(θ) can be expressed as: where f (θ -θ ) = e iπ(N -1)(θ -θ) sin(πN (θ -θ)) sin(π(θ -θ))

. Thus one finds:

γ(θ, α) T γ(θ , α ) = cos(α -α ) Re (f (θ -θ ))

+ sin(α -α ) Im (f (θ -θ )) , which shows the manifold of interest γ is shift-invariant.

We are now able to prove Corollary 4.1.5: Proof: In order to find a parametrization µ of γ satisfying the condition [START_REF] Rabaste | Geometrical design of radar detectors in moderately impulsive noise[END_REF], one can use the following vector d (v) instead of d(θ) in the derivations, changing the origin of time (d and d model the same problem):

(d (v)) n = 1 √ N exp 2iπ n - N -1 2 l -1 v ,
for 0 ≤ n ≤ N -1 and where

l = π N 2 -1 3 , v = l θ .
The parametrization µ = (v, α) for γ written with d verifies condition [START_REF] Rabaste | Geometrical design of radar detectors in moderately impulsive noise[END_REF]. Let us compute (32):

∂ 2 γ T ∂ 2 v ∂ 2 γ ∂ 2 v = C , ∂ 2 γ T ∂v ∂α ∂ 2 γ ∂ 2 v = ∂ 2 γ T ∂v ∂α ∂ 2 γ ∂ 2 α = 0 , ∂ 2 γ T ∂ 2 v ∂ 2 γ ∂ 2 α = ∂ 2 γ T ∂v ∂α ∂ 2 γ ∂v ∂α = ∂ 2 γ T ∂ 2 α ∂ 2 γ ∂ 2 α = 1 ,
where C has been defined in (42). Injecting in [START_REF] Hohenwarter | GeoGebra -ein Softwaresystem für dynamische Geometrie und Algebra der Ebene[END_REF], with β 1 = cos ϕ, β 2 = sin ϕ and maximizing on ϕ, we get: = Re e -i (α-α ) s(θ) H s(θ ) ,

≤ s(θ) H s(θ ) , (58)
with the equality attained for α -α = s(θ) H s(θ ). This implies that the only points of Ξ worth investigating are the pairs of points (θ, α), (θ , α ) with (α, α ) chosen arbitrarily so that α -α = s(θ) H s(θ ) and (θ, θ ) being a critical point of the quantity s(θ) H s(θ ) which define the set Ξ in (44).

Fig. 1 :

 1 Fig.1: Mean P D of the NMF (3) (red), its oracle counterpart that knows the positions of the targets (yellow) and the GLRT (6) (blue) as a function of the SNR in the presence of off-grid targets under white noise. P F A = 10 -6 and N = 10.

Fig. 2 :

 2 Fig. 2: Example of a tube T on S 2 around a curve γ(ξ). Since the curve is non-closed, semi-spherical caps (in green) are present at the ends of T .

∂α 2 cos 2 ϕ ∂γ ∂θ 2 + sin 2ϕ ∂γ T ∂θ ∂γ ∂α + sin 2 ϕ ∂γ ∂α 2 ,

 222

Fig. 6 :Fig. 7 :

 67 Fig. 6: Ambiguity map s(θ) H s(θ ) drawn for θ, θ ∈ D 0 for highly correlated noise (ρ = 0.95, see Eq. (51)), with N = 10. Crosses represent local maxima.

  (a) Uncorrelated noise : ρ = 0.

Fig. 10 :

 10 Fig. 10: Comparison of the limit global overlap thresholds w lim (purple), w local (blue) and w non-local (red) versus ρ obtained with (25), (39) and (35) with N = 10, in the cell D 0 .

β i β j Γ ijk 2 , = 1 ,

 21 using (53), (54) and (55). D. Proof of Corollary 4.1.5 In order to prove Corollary 4.1.5 we need to introduce the following Lemma to use Corollary 4.1.2.

e

  2iπk(θ -θ) , = f (θ -θ ) , = Re (f (θ -θ )) + i Im (f (θ -θ )) .

cot 2 φ 1 cot 2

 212 local = max ϕ C cos 4 ϕ + 6 cos 2 ϕ sin 2 ϕ + sin 4 ϕ -1 , = max ϕ (C -5) cos 4 ϕ + 4 cos 2 ϕ . (57)The maximum is obtained for cos 2 ϕ = 2 5 -C. Then, injecting this value in (57) and simplifying, we obtain:cot 2 φ local = 4 5 -C . Since w 2 local = cos 2 tan -1 φ local , result(41) holds.E. Proof of Corollary 4.1.6Proof: First, see that:γ(θ, α) T γ(θ , α ) = cos(α -α ) Re s(θ) H s(θ )+ sin(α -α ) Im s(θ) H s(θ ) ,

DEMR, ONERA, Université Paris-Saclay, F-91120 Palaiseau, France

SONDRA, CentraleSupélec, Université Paris-Saclay, F-91192 Gif-sur-Yvette, France

[START_REF] Ollila | Complex Elliptically Symmetric distributions: Survey, new results and applications[END_REF] Université Paris-Saclay, ENS Paris-Saclay, CNRS, SATIE, F-91190, Gifsur-Yvette, France.