B Caron Ifra-Llacan
email: caron.bernard@yahoo.fr

Documenting linguistic varieties of the South-Bauchi group

Introduction

In this paper, we want to address the question "What grammar for what language(s)?". The answer to this question will involve personal reflexions, drawn from personal experience, with all the value and limitations of a personal experience as a specialist of a group of 15 languages or so Chadic languages, spoken in the south of Bauchi State, Nigeria, all of them dominated by Hausa, and the majority being in serious danger of extinction.

Size matters

The size of languages entails different problems, and different choices.

Languages with a large number of speakers (over 1 million) imply the need for a reference grammar; with a view to standardization, teaching and the development of the language. The problems that arise are essentially theoretical, methodological and political: which type of orthography, using which system should be implemented? What should it represent? Full or partial tone marking? Which script: Roman or Ajami? With or without diacritics? Which lect should be chosen for prescriptive purposes? Both in terms of geography (which dialect?) or sociology (which sociolect?).

The linguist dealing with languages spoken by a smaller number of speakers faces a different situation. They are often minority, sometimes endangered languages. This entails problems of bi-or multilingualism, even language attrition. In a country such as Nigeria, the number of little described languages, related to the scarcity of financial and human resources means that a few, not many languages will get a grammatical description. How should these 'lucky' languages be selected? The decision as to which language should be described is often taken on practical, non-linguistic reasons such as accessibility to the linguist of the villages and/or speakers. That is how we decided to study Geji, rather than Pelu, and Zaar rather than Guus (aka Sigidi). Let us have a look at South-Bauch West languages whose situation as minority or severely endangered languages entails specific problematics.

South-Bauchi West (SBW) languages

The situation of SBW language varies from prosperous to dying and even dead:

A few languages are sound (Zoɗi, Polci, Zakshi), even prosperous (Zaar). The basis of their economy is agriculture in the plains, where their speakers have access to primary and secondary education, in Hausa and English. They have been and continue to be in contact with speakers of other languages, whether Chadic (SBW and non-SBW languages such as Angas) or non-Chadic (Afizere, Pyem, Ɓoi, Shall, etc.).

The rest are isolated, endangered languages spoken on hills (whereas the less-endangered ones are spoken in the plains where their population could develop numerically and otherwise) by communities of aroung 600 speakers. Two of those language (ex: Zaranda ; Luri) are spoken in communities that have split, creating a situation with an "old" village up in the hills with a few older people, living in terrible conditions, using the "old" language and a "new" village where the majority of the population has moved down the hill, by the road, and switched to Hausa, Jaar, etc.

What type of grammar for SBW languages?

Given their socio-linguistic situation, what type of grammar is suitable, or can be reasonably envisaged for SBW languages? This in turn is determined by the usage these grammars are intended for. These fall under various categories:

 Documentation: referring to the title of the conference, this can be done with a view to archiving the languages and cultures before they disappear, building an archive with a primary concern for museology. This type of work should be as exhaustive as possible. This type of general, all-encompassing grammar can, in the end, fulfill the aims of the other three types by providing the relevant data. However, they can only be achieved by teams of researchers with specialist competence in different fields such as linguistics, anthropology, sociology, geography, . This type of conjoined effort, and the finance required, is rarely available in Nigeria, and certainly not for SBW languages.

 Typology of languages and cultures: this type of description relies heavily on the former one, and is therefore event more time-and human resources-consuming.

 Language change, studied from the point of view of socio-linguistics and typology generally requires purely linguistic and socio-linguistic material, and little cultural data. However, socio-linguistic studies involve data covering large populations and/or a minute study of language use with several variables (age, sex, situations, linguistic genres, etc.) wich consumes time and human resources.

 History of population: with the help of reconstruction, the data collected should tell us more about the population and its movements in the Chad area over the past four hundred years. The data needed for this type of work is generally limited to lexicon (both basic and cultural) and morphology. Given that we are dealing with a group of genetically related languages, let us not forget another type of grammar, i.e. the comparative grammar, which is both heuristic and hermeneutic.

Now, what are the tools available to reach that kind of description?

Theory & methodology

The structuralist foundation of our activity as linguists has come under heavy criticism from various sides. We will try to briefly retrace those tendencies, and evaluate what this leaves us with. Let us begin with how the scientific object of linguistics is generally defined, starting with Saussure, who has remained the driving force behind our descriptive activity as linguists, despite those criticism. Saussure's aim in laying the foundation of linguistics as a science is to find stable units in the continuous flow of speech, and in the geographical and individual variations. For Saussure, the object of linguistics is language, defined as an instrument of communication 2 between speakers in a given society. This definition finds the stability Saussure was looking for in the abstraction of a population of speakers sharing the same language. Now, the concept of people is ductile, in Africa as well as anywhere (cf. Central Europe), and is politically loaded. The stable association of a language with a population, a culture and a territory is not the rule in the areas where we (as Africanists) work. SBW languages are a good example. SBW languages, like Plateau languages, are generally considered to be spoken by small population units living in quasi-autarcy on top of hills which enjoy a good pluviometry which in turn allowed these populations to be self-sufficient. This seclusion The fear of slave-hunters reinforced. This situation was used as an explanation for the remarquable language diversity in this area, where languages changed from one village to another within a distance of about 10 to 20 kilometers, each village counting an average of 600 speakers. However, far from being isolated, these populations developed very close ties:

Though there were inter-group conflicts, there were also social intercourse and intermarriage between the different groups. This is perhaps one explanation for the similarity of the various langages. Thus, with continued close contacts 2 Chomsky changed it to the concept of "competence", i.e. the grammatical competence of the speaker. The way his theory pushed the whole process of data collection into the background under what he called the "black box" prevented his theory from becoming popular with those who were concerned with documenting languages. This is generally the case with more theoretically oriented linguistics. In France, Antoine Culioli's "Theory of Predicative and Enunciative Operations", although it has given birth to a few descriptive grammars (Béarth 1971, Caron 1990) has had as little success with descriptivists.

and intermarriage, the various groups came to share socio-political institutions. [...] Indeed, the plateau has been an important area of culture fusion. (Abubakar 1980 : 183)

Moreover, the situation of the present SBW languages has not always been stable. Their migration has been retraced by E. [START_REF] Mohammadou | From Lake Chad to Bauchi along the Northern Migratory Corridor: The Zaar Case. Man and the Lake[END_REF] from the Kanem empire North of Lake Chad to their present situation over the past 400 years. The splinter groups detached from the migration movement settled on various hills on the way, and gave birth to various Chadic languages. The map below summarizes this vast movement, with SBW languages represented under the cover name "Zaar" (6): Following the European colonization, the SBW populations have come down from the hills and settled in the plains. A second movement is taking place now with the Zaars moving north of their situation at the beginning of the 20 th Century (roughly around Lusa and present-day Bogoro), up to the north of Tafawa Balewa where they are fighting for political power with the Hausa-Fulani. This mobility accounts for numerous language contacts with Chadic and non-Chadic languages, resulting in a certain amount of heterogeneity in the languages. Traces of these contacts are found in SBW in the coexistence of 2 distinct phonological systems: one for the core lexicon, and a second one for more recent loanwords. This heterogeneity within languages has called for new concepts. This is Nettle about Fyem:

So significant are the relationships between groups on the plateau that I will suggest that language evolution there should be conceptualised not as the splitting from one or two proto-languages of daughters which subsequently changed by their own intenal dynamic, but rather as a continual flux of intergroup diffusion, both of words and of grammatical structures. (Nettle 1998 : 3)

The rhizome model, developed by Deleuze and Guattari, as opposed to the tree-like model used for classical reconstructions is better adapted to this type of situation:

Plus généralement, il se peut que les schémas d'évolution soient amenés de plus en plus à abandonner le vieux modèle de l'arbre et de la descendance. [...] Les schémas d'évolution ne se feraient plus seulement d'après des modèles de descendance arborescente, allant du moins différencié au plus différencié, mais suivant un rhizome opérant immédiatement dans l'hétérogène et sautant d'une ligne déjà différenciée à une autre [...] Des communications transversales entre lignes différenciées brouillent les arbres généalogiques. [...] Le rhizome est une antigénéalogie. (Deleuze & Guattari 1976 , p. 30)

From a sociolinguistic point of view , new concepts are emerging, reviving old debates concerning mixed languages, language continuum, sprachbund, etc. SBW language are a good example of language continuum. It has been argued in (Caron 2003) that Guus (aka Sigidi) and Zaar which are considered by their speakers to be distinct languages, is closer to the Kal language than Kal is to the other "dialects" of Zaar such as Bogoro or Gambar Lere (GL). See the tables below concerning percentage of common roots and instances of these roots compared to Proto-Chadic (P.C.). In order to account for this type of sociolinguistic situation, Robert Nicolaï proposes to replace the study of language as a tool for communication within a society with the study of individual repertories working as a mille-feuilles.

J'entends par « feuilletage » le

Temporary conclusion : What now?

Should we indulge into this type of post-modern linguistics based on the deconstruction of classical structuralist foundations of Saussure's concepts? If the situation is complex indeed, and calls for new concepts in the interpretation of facts, what should we do when facing the situation of SBW languages as described above? We have decided to take pragmatic steps, using tried and tested concepts and techniques based on Saussure and the Prague School (sign, information structure, etc.).

The new technologies (digital recording of audio and video material) applied to new questionnaires aimed at capturing the language used by groups and individuals in various locations and pragmatic situations will elicit all the material needed for any theoretical analysis. Well-documented corpora are the first and necessary step to any description of languages, whether endangered or not.

In the case of SBW languages, our aim is to produce a comparative grammar, with a core grammar and vocabulary working as the yardstick to which individual SBW languages should be compared.

SBW Comparative grammar: a synopsis

Core grammar

Although this core grammar could be regarded as a proto-language, it does not pretend to have any historical reality. Let us see a sample taken from Caron 2006:

TAM systems All SBW's TAM systems conform to the same structure: conjugation is expressed in a pre-verbal morpheme, which includes a person mark. The verb itself is not touched by TAM, with two exceptions: floating tones (or tone progagation) and the occasional use of Verbal Nouns for Continuous and Future. When the subject is nominal, the personal mark is dropped and a variant of the TAM marker appears alone.

What can be noted though is the common existence of a 'zero' TAM (H. Jungraithmayr's 'Grundaspekt'), where only the bare Pro is used to conjugate the verb. We have chosen to name this TAM Aorist when it is opposed to the couple Completive + Incompletive, and Completive when it is opposed to an Incompletive only. We have named Continuous the TAM with a locative structure involving Pro's having the form of subjects of non-verbal predicates, and a Verbal Noun. See below an example in Zoɗi : Incompletive. Except for 1S, the inflexion follows the pattern « Pro + àà ». It is followed by a VN. With a noun subject, the Incompletive mark is à.

Future. The inflexion follows the pattern « Pro + VV » in the singular and « Pro + n » in the plural. With a noun subject, the Future mark is níí.

Sub-systems of various nature and extension

These sub-systems are aimed at illustrating variation. Each sub-system could be a language, or a structure common to a group of idioms.

Transversal modules

These modules are aimed at analyzing common features that do not correspond to the core grammar. They could be accounted for by studying external influences on the core SBW system.

What languages ?

The languages that are sufficiently documented to be introduced in the comparative grammar are:

  The inflexion follows the pattern « Pro + Ø ». The Perfect is a combination of the Completive + CaK at the end of the Verb Phrase. The habitual is marked by a inserted between the Completive subject pronouns and the verb.

caractère propre à tout répertoire linguistique de pouvoir fonctionner comme ressource dans la re-élaboration de variétés linguistiques et d'usages langagiers constitués à travers la refonctionnalisation de traits, de formes et de fragments discursifs matériellement disponibles. Parler de « feuilletage » permet ainsi d'appréhender la notion de superposition et la multiplicité des usages et des variétés disponibles dans le répertoire, sans leur attribuer a priori une homogénéité structurelle, au sens où une telle qualité est présupposée pour une description structurale ou une identification collective. (Nicolaï 2000: 10)

For proto-Chadic : Jungraithmayr & Ibriszimow 1994.

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MD, PhD Mathie Lorrot
email: mathie.lorrot@rdb.aphp.fr

PhD Fabienne Bon

M J El Hajje

MD Serge Aho

MD Marjorie Wolfer

BS Hélène Giraudon

BS Jérôme Kaplon

MD Elisabeth Marc

Jean Raymond

MD, PhD Pierre Lebon

MD Marie-Joëlle El-Hajje

MD Josette Raymond

MD, PhD Pierre Pothier

MD Dominique Gendrel

Epidemiology and clinical features of gastroenteritis in hospitalised children: prospective survey

Keywords: gastroenteritis, rotavirus, norovirus, enteric bacteria, clinical severity, gastroenteritis severity in children

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

Acute gastroenteritis (AGE) is one of the most common diseases in humans, and continues to be a cause of high morbidity and mortality in children worldwide. Children under 5 years of age are particularly prone, and it is calculated that, in this group, there are more than 700 million cases of acute diarrhoea every year [START_REF] Snyder | The magnitude of the global problem of acute diarrhoeal disease: a review of active surveillance data[END_REF]. Worldwide estimates indicate a mean of between 3.5 and 7 episodes of diarrhoea during the first 2 years of life, and over 11 000 deaths per day throughout the world, particularly in developing countries [START_REF] Kapikian | Overview of viral gastroenteritis[END_REF]. In developed countries, deaths from diarrhoea are less common, but AGE is a major cause of morbidity in childhood, leading to many hospitalizations and doctor's visits [START_REF] Parashar | Global illness and deaths caused by rotavirus disease in children[END_REF][START_REF] Glass | The epidemiology of rotavirus diarrhea in the United States: surveillance and estimates of disease burden[END_REF][START_REF] De Wit | Etiology of gastroenteritis in sentinel general practices in the netherlands[END_REF].

Much of the gastroenteritis in children is caused by viruses. Group A rotaviruses are the leading cause of severe acute gastroenteritis in infants and children throughout the world, causing an estimated 500,000 -600,000 deaths each year. All children are infected by the age of 2-3 years and the first infection can usually be associated with acute diarrhoea, which can be severe. Other viral agents of AGE are enteric adenoviruses, caliciviruses and astroviruses.

With the improvements in molecular methods of viral diagnosis over the past decade, the role of noroviruses in the etiology of childhood diarrhoea is increasingly recognized. In contrast, the role of Aichi viruses, which belong to the picornavirus family, is not well known [START_REF] Ambert-Balay | Prevalence and genetic diversity of Aichi virus strains in stool samples from community and hospitalized patients[END_REF][START_REF] Pang | Human caliciviruses in acute gastroenteritis of young children in the community[END_REF].

To date, there are few data available about the enteric viruses and bacteria in circulation in France. The prevalence and diversity of the four major viruses (rotaviruses, caliciviruses, astroviruses and adenoviruses) are documented for gastroenteritis in children consulting a physician or a hospital emergency service [START_REF] Bon | Prevalence of group A rotavirus, human calicivirus, astrovirus, and adenovirus type 40 and 41 infections among children with acute gastroenteritis in Dijon, France[END_REF][START_REF] Buesa | Molecular epidemiology of caliciviruses causing outbreaks and sporadic cases of acute gastroenteritis in Spain[END_REF][START_REF] Chikhi-Brachet | Virus diversity in a winter epidemic of acute diarrhea in France[END_REF]. In contrast, their role in communityacquired severe AGE requiring hospitalization have rarely been investigated [START_REF] Kapikian | A rotavirus vaccine for prevention of severe diarrhoea of infants and young children: development, utilization and withdrawal[END_REF]. The first study of hospitalized gastroenteritis conducted in Saint Vincent de Paul children's hospital (Paris) from January 1997 to December 2000 reported that 50.89% of 725 children hospitalized for AGE were infected with rotavirus. The role of other viral pathogens was not investigated [START_REF] Moulin | Hospitalization for acute community-acquired rotavirus gastroenteritis: a 4-year survey[END_REF].

We have conducted a prospective study to determine the viral and bacterial etiology of AGE in children hospitalized for AGE. In collaboration with the National Reference Center (NRC) of enteric viruses based in Dijon, we systematically searched for viral pathogens (rotavirus, norovirus, sapovirus, astrovirus, enteric adenovirus and Aichi virus) using sensitive molecular methods. Bacterial pathogens (Salmonella, Shigella and Camplylobacter spp.) were also systematically screened for. We investigated the clinical pattern of the disease in these children and the severity of the episodes of AGE. This study provides precise data on the frequency and the severity of bacterial and viral pathogens among children hospitalized for AGE in a Parisian paediatric hospital during a 2-year period.

Patients and methods.

Population, clinical definitions and specimen collection.

The study was carried out in the Parisian teaching hospital Cochin-Saint Vincent de Paul, which is situated at the south of Paris and provides easy access to appropriate medical care. At this hospital, there are a mean of 23,000 consultations for all causes in children every year; 15 to 20 % of the consultations for diarrhoea lead to hospitalization [START_REF] Moulin | Hospitalization for acute community-acquired rotavirus gastroenteritis: a 4-year survey[END_REF].

After its approval by the Ethics Committee of the Cochin-Paris V Faculty of medicine, the study was prospectively performed between November 2001 and May 2004 and included all children between 0 months and 15 years of age hospitalized with acute diarrhoea. Acute diarrhoea was defined as at least 3 looser-than normal stools within a 24-h period. All admissions to the hospital were based on clinical examinations at the request of the hospital emergency team. Patients with chronic diarrhoea (> 10 days) were excluded. For each child, clinical data including, age, sex and admission date were prospectively collected at inclusion. Each episode was graded using the 20-point severity score scale [START_REF] Ruuska | Rotavirus disease in Finnish children: use of numerical scores for clinical severity of diarrhoeal episodes[END_REF][START_REF] Joensuu | Randomised placebocontrolled trial of rhesus-human reassortant rotavirus vaccine for prevention of severe rotavirus gastroenteritis[END_REF]. Furthermore, the occurrence of bloody diarrhoea, the use of intravenous rehydration and the length of hospitalization were documented. Stool samples were systematically collected to screen for etiologic agents of diarrhoea. Bacteria (Salmonella, Shigella and Campylobacter spp.) were detected by routine cultivation in the Bacteriology Laboratory of the Saint Vincent de Paul Hospital. After collection, the samples were stored frozen (-20°C) at the Virology Laboratory. The stools were sent together to the enteric viruses NRC (Dijon), for viral investigation.

Laboratory methods.

The presence of pathogenic bacteria was determined by routine culture assays on selective media. Stool samples were screened for the presence of group A rotaviruses, adenovirus types 40 and 41 and astroviruses by commercial enzyme immunoassay (EIA) kits (respectively: Biomérieux, France; Oxoïd Ltd, UK; Meridian Diagnostics Inc, USA). All positive samples were confirmed and characterized by molecular biology methods (PCR or RT-PCR).

Noroviruses, sapoviruses and Aichi virus were detected by RT-PCR.

Nucleic acids were extracted from 20% stool suspensions in phosphate-buffered saline with a QIA Amp viral RNA kit (Qiagen, Hilden, Germany). The Rotavirus RT-PCR used the primer sets Beg9/End9 [START_REF] Gouvea | Polymerase chain reaction amplification and typing of rotavirus nucleic acid from stool specimens[END_REF] and Con2/Con3 [START_REF] Gentsch | Identification of group A rotavirus gene 4 types by polymerase chain reaction[END_REF], The astrovirus RT-PCR used the primers Mon 244 and Mon 245 [START_REF] Noel | Typing of human astroviruses from clinical isolates by enzyme immunoassay and nucleotide sequencing[END_REF] and the adenovirus PCR used the primers Hex1DEG/Hex2DEG [START_REF] Allard | Rapid typing of human adenoviruses by a general PCR combined with restriction endonuclease analysis[END_REF]. The norovirus and sapovirus RT-PCR used primer sets in separate reactions. The primer sets SR80/NVP110 [START_REF] Noel | Parkville virus: a novel genetic variant of human calicivirus in the Sapporo virus clade, associated with an outbreak of gastroenteritis in adults[END_REF] and JV12/JV13 [START_REF] Vinje | Molecular detection and epidemiology of small roundstructured viruses in outbreaks of gastroenteritis in the Netherlands[END_REF] were used to amplify a fragment of the RNA polymerase gene of sapoviruses and noroviruses, respectively. The primer sets G1SKF/G1SKR and G2SKF/G2SKR [START_REF] Kojima | Genogroup-specific PCR primers for detection of Norwalklike viruses[END_REF] were used to amplify a fragment of the capsid gene of genogroup I and II noroviruses, respectively. The Aichi virus RT-PCR used the primers Ai6261/6779 targeting the RNA polymerase gene [START_REF] Yamashita | Application of a reverse transcription-PCR for identification and differentiation of Aichi virus, a new member of the Picornavirus family associated with gastroenteritis in humans[END_REF].

Genotyping of norovirus, sapovirus, astrovirus and Aichi virus was performed by direct sequencing of the PCR products using the ABI Prism Big Dye Terminator Cycle Sequencing Ready Reaction Kit and an automated sequencer (model 373A DNA sequencing system, Applied Biosystems).

Statistical methods.

Statistical analysis was performed with the Mann-Whitney U test, which was used to compare the medians of clinical symptoms scores in associated rotavirus and norovirus cases. For comparison of proportions, chi-square or Fisher's exact test was used. All tests were twotailed and considered significant when the P value was ≤ 0.05.

Results

Study population and general screening.

A total of 552 children were hospitalized for gastroenteritis between November 2001 and May 2004. Clinical data and stool samples were obtained from 457 (82.8%) children. The mean age of these children was 15.9 months (standard deviation, 22.13; range, 0.2 -156 months) and the median was 9.25 months. The sex ratio was 1.26. The mean Vesikari score was 11.7 (standard deviation, 3.16; range, 4-19). Among the 95 sets of data rejected in this study, 13 were rejected for the lack of clinical data and 82 for the absence of a faecal sample.

Comparison of the data of patients at the time of enrolment showed that there was no significant difference between the data (mean age, sex ratio and Vesikari score) of included and excluded cases (p> 0.05, Mann-Whitney U-test), (data not shown).

Laboratory investigations detected at least one pathogen (virus and/or bacteria) in 330 stool samples (72.2%) of the 457. Viral pathogens were detected in 305 (66.7%) samples, and bacterial pathogens in 31 (6.8%) samples.

Among the viruses, group A rotaviruses were detected in 240 samples (223 in single infections, 17 in mixed infection) and the most prevalent G type were G1 (114 isolates, 47.5%), G4 (38 isolates, 15.8%), G3 (38 isolates, 15.8%), G9 (6 isolates, 2.5%) and G2 (6 isolates, 2.5%). The P [START_REF] Bon | Prevalence of group A rotavirus, human calicivirus, astrovirus, and adenovirus type 40 and 41 infections among children with acute gastroenteritis in Dijon, France[END_REF] type was by far the most frequent accounting for 233 rotaviruses strains (97 % of rotaviruses strains). The second most frequent group of viruses was norovirus with 55 isolates (38 in single infections and 17 in mixed infections). Molecular characterization showed that genogroup II (GGII) strains were clearly predominant with 50 isolates (90.9%) including 28 GGII.4 (50.9%), 18 GGIIb (32.7%), 3 GGII.2 (5.5%) and 1 GGII.7 (1.8%). The remaining 5 noroviruses belonged to the GGI (genotypes 1, 2, 4 and 6).

The other viruses were less frequency detected: enteric adenoviruses were detected in 16 samples, (3.5%), astrovirus in 7 (1.5%), Aichi in 4 samples (0.9%) and sapovirus in 2 samples (0.4%). Bacterial pathogens were 13 Campylobacter, 14 non typhi Salmonella, 5 Shigella.

Single pathogen infections were responsible for 307 cases of gastroenteritis, most of which were viral agents (283 samples versus 25 samples (24: 1 bacterium; 1: 2 bacteria) for bacteria). Among these, rotaviruses were responsible for 223 cases (48.8%) and noroviruses for 38 cases (8.3%). The other enteric viruses were adenovirus type 40/41, astrovirus, Aïchi virus and sapovirus respectively in 16, 3, 2 and 1 cases. Single bacterial pathogens were found in 24 cases (12 Campylobacter, 9 non typhi Salmonella sp., 3 Shigella).

Mixed infections were detected in 23 samples and mainly involved combinations of viruses (16 samples) especially rotavirus and norovirus. Dual infections with rotavirus and norovirus were detected in 10 samples, with norovirus and astrovirus or sapovirus in 1 sample, with rotavirus and astrovirus or Aïchi virus in 1 sample. Triple infections with norovirus, rotavirus and astrovirus were detected in 2 samples. Mixed infections involving viruses and bacteria were detected in 6 samples: non typhi Salmonella was associated with rotavirus in 2 samples, with norovirus in 1 sample and with norovirus-rotavirus in 1 sample; Campylobacter was associated with norovirus and Shigella with Aïchi virus in 1 sample. There was only one dual bacterial infection, which associated Salmonella and Shigella (Table 1).

Monthly distribution.

The peak of the gastroenteritis epidemic was due to rotavirus and was observed during the winter months, the highest peak occurred in January. Interestingly, a second lower peak of rotavirus incidence was observed in September. Rotaviruses continued to be detected in the interepidemic season but, during these 2 years of surveillance, rotaviruses were not detected in June Norovirus AGE have a seasonal pattern similar to that of rotavirus AGE with a main peak observed in January and a second peak in September (Figure 1). No consistent seasonal pattern was observed for enteric adenovirus, astrovirus and bacterial AGE.

Etiology of hospitalized AGE in relation to age of the children.

The proportion of the pathogens detected among children in different age groups is presented in Table 1.

Most hospitalizations for AGE were reported in children less than 2 years of age (385 children, 84.2%). In these children, 267 (69.3%) had a viral etiology: rotavirus and norovirus were detected in 210 (54.5%) and 47 (12.2%) stool samples, respectively; a single bacterial etiology was identified in 14 stool samples (3.6%) and mixed bacterial and viral infections in 5 stool samples (1.3%). In 99 cases of AGE (25.7%), no acknowledged pathogen was identified.

One hundred and seventy-seven hospitalized children (38.7%) were less than 6 months of age.

In these children, 97 (54.8%) had a viral etiology: rotavirus and norovirus were detected in 77 (42.4%) and 15 (8.5%) stool samples respectively. Single bacterial etiology was identified in 8 stool samples (4.5%) and mixed bacterial and viral infections in 2 stool samples (1.1%). In 67 cases of AGE (37.8%), no acknowledged pathogen was identified.

Clinical severity criteria in children hospitalized for rotavirus or norovirus AGE.

The clinical severity of the disease was analyzed in children infected by a single virus (rotavirus or norovirus). The 223 children infected with rotavirus and the 38 children infected with norovirus were similar in terms of age and sex ratio (p> 0.05, Mann-Whitney U-test).

The mean age was 13.65 months (median: 9 months) for rotavirus infections versus 14.9 months (median: 9.5 months) for norovirus infections. The sex ratio was 1.27 for rotavirus infections versus 1.23 for norovirus infections.

The correlation between the severity of the AGE (severity score, intravenous rehydration, length of hospitalization) and the virus (rotavirus and norovirus) is reported in table 2.

Rotavirus AGE were more severe in infants above 6 months than in infants from 0 to 6 months (p<0.0001). Rotavirus AGE were significantly more severe than norovirus AGE with respect to the Vesikari score (12.6 versus 10.47, p<0.001), the duration of the hospitalizations (3.2 days versus 1.85, p<0.001) and the requirement for intravenous rehydration (77.13% versus 55.26%, p= 0.005). This contrast was more marked in children above 6 months (p<0.007 for severity score and p<0.001 for the length of hospitalization and p=0.03 for the need for intravenous rehydration).

These differences in severity according to age were not observed in norovirus gastroenteritis.

DISCUSSION

This is the first prospective study to provide data for cause-specific hospitalizations due to community acquired gastroenteritis in children in France. This study was conducted from November 2001 to May 2004 in a Parisian paediatric hospital, but these results are still relevant today. The management of acute gastroenteritis is currently the same with use of an oral rehydration solution and/or intravenous fluid replacement; the trends of hospitalization should continue to show the same pattern. The two rotavirus vaccines (RotaTeq and Rotarix) were licensed in France in 2006, but rotavirus vaccination is not included in the French official recommendations for childhood immunizations, and vaccine coverage is still very low, less than 10 percent.

Our results show that group A rotavirus caused more than 50 % of the total hospitalizations for gastroenteritis per year in a paediatric unit in Paris (France) and was the leading cause of severe diarrhoea in young children. In addition, they highlight the role of norovirus in France as the second viral etiologic agent causing 12% of hospitalizations for AGE in children. In contrast, the other enteric viruses including adenovirus type 40 and 41, astrovirus, sapovirus and Aichi virus were a minor cause of hospitalization for childhood gastroenteritis.

Altogether, our results show that viruses are by far the most frequent etiology in children hospitalized for gastroenteritis (66.7%), whereas a bacterial etiology was found in 6.8% of cases, and the etiology of 27.8% of the AGE was not elucidated.

Previous studies underlined the importance of viruses in children hospitalized for gastroenteritis especially in developed countries [START_REF] Clark | A review of viral gastroenteritis[END_REF][START_REF] Parashar | Rotavirus and severe childhood diarrhea[END_REF]. However, few studies have investigated prospectively the relative contributions of the different enteric viruses and bacteria in children admitted to hospital for gastroenteritis in developed countries. In accordance with our results, studies conducted in other developed countries in hospitalized children found a large predominance of viruses (56 to 70%) versus bacteria (4 to 17%) [START_REF] Kirkwood | A 5-year study of the prevalence and genetic diversity of human caliciviruses associated with sporadic cases of acute gastroenteritis in young children admitted to hospital in Melbourne, Australia (1998-2002)[END_REF][START_REF] Ambrozova | Viral gastroenteritis in children][END_REF]. A large retrospective study conducted in Australia found the same ratio between viral and bacterial etiologies in children [START_REF] Barnes | Etiology of acute gastroenteritis in hospitalized children in Melbourne, Australia, from April 1980 to March 1993[END_REF]. In contradiction with these studies, Boga et al [START_REF] Boga | Etiology of sporadic cases of pediatric acute gastroenteritis in asturias, Spain, and genotyping and characterization of norovirus strains involved[END_REF] in a study conducted in Asturias found bacteria and viruses with a similar frequency, Campylobacter jejuni and Salmonella spp were respectively the second and third main cause of gastroenteritis in children. Among the bacterial etiologies, we also found a large predominance of C. jejuni and Salmonella spp, but, on the whole, bacteria were a minor cause of hospitalization for diarrhoea in children.

Among the viruses, rotaviruses were by far the main cause of gastroenteritis in hospitalized children, far ahead of noroviruses, which were found in 8-15% of children hospitalized for gastroenteritis [START_REF] Bon | Prevalence of group A rotavirus, human calicivirus, astrovirus, and adenovirus type 40 and 41 infections among children with acute gastroenteritis in Dijon, France[END_REF][START_REF] Kirkwood | A 5-year study of the prevalence and genetic diversity of human caliciviruses associated with sporadic cases of acute gastroenteritis in young children admitted to hospital in Melbourne, Australia (1998-2002)[END_REF][START_REF] Oh | Viral agents of acute gastroenteritis in German children: prevalence and molecular diversity[END_REF][START_REF] O'ryan | Human caliciviruses are a significant pathogen of acute sporadic diarrhea in children of Santiago, Chile[END_REF][START_REF] Simpson | Infantile viral gastroenteritis: on the way to closing the diagnostic gap[END_REF]. These studies and our results contrast with other sentinel studies conducted by physicians in the general population [START_REF] De Wit | Etiology of gastroenteritis in sentinel general practices in the netherlands[END_REF][START_REF] Pang | Human caliciviruses in acute gastroenteritis of young children in the community[END_REF][START_REF] Chikhi-Brachet | Virus diversity in a winter epidemic of acute diarrhea in France[END_REF], in which the incidence of norovirus ranged from 10 to 20%. This is close to and in some cases even higher than the incidence of rotavirus infections [START_REF] Chikhi-Brachet | Virus diversity in a winter epidemic of acute diarrhea in France[END_REF]. One possible explanation of the relatively higher incidence of norovirus is that these sentinel studies highlight the diagnosis of mild infections, whereas more severe cases of gastroenteritis are admitted to the emergency department of the hospital.

Sequence typing of the rotavirus strains showed that the main type was G1;P [START_REF] Bon | Prevalence of group A rotavirus, human calicivirus, astrovirus, and adenovirus type 40 and 41 infections among children with acute gastroenteritis in Dijon, France[END_REF] (47.5%) followed by the two other conventional types: G3;P [START_REF] Bon | Prevalence of group A rotavirus, human calicivirus, astrovirus, and adenovirus type 40 and 41 infections among children with acute gastroenteritis in Dijon, France[END_REF] (15.8%) and G4;P [START_REF] Bon | Prevalence of group A rotavirus, human calicivirus, astrovirus, and adenovirus type 40 and 41 infections among children with acute gastroenteritis in Dijon, France[END_REF] (15.8%) whereas the G9;P [START_REF] Bon | Prevalence of group A rotavirus, human calicivirus, astrovirus, and adenovirus type 40 and 41 infections among children with acute gastroenteritis in Dijon, France[END_REF] type was seldom detected (2.5%). This pattern of genotype distribution with the low prevalence of the G9 type was close to that reported in our previous works since 1995 [START_REF] Bon | Prevalence of group A rotavirus, human calicivirus, astrovirus, and adenovirus type 40 and 41 infections among children with acute gastroenteritis in Dijon, France[END_REF][START_REF] Chikhi-Brachet | Virus diversity in a winter epidemic of acute diarrhea in France[END_REF], but considerably different from that we reported during the 2004/2005 season following the present study. In this latter season, G9;P [START_REF] Bon | Prevalence of group A rotavirus, human calicivirus, astrovirus, and adenovirus type 40 and 41 infections among children with acute gastroenteritis in Dijon, France[END_REF] emerged as by far the major genotype (54.7%) whereas G1;P [START_REF] Bon | Prevalence of group A rotavirus, human calicivirus, astrovirus, and adenovirus type 40 and 41 infections among children with acute gastroenteritis in Dijon, France[END_REF] was very rarely detected [START_REF] De Rougemont | Unexpected substitution of dominant rotavirus G genotypes in French hospitalized children over five consecutive seasons[END_REF]. Among noroviruses, there was a clear predominance of genogroup II, and especially the GGII.4 genotype, which is currently reported in adult gastroenteritis outbreaks [START_REF] Kroneman | Data quality of 5 years of central norovirus outbreak reporting in the European Network for food-borne viruses[END_REF].

Norovirus genotype GII.4 has also been reported as the main norovirus genotype in sporadic cases of gastroenteritis in children [START_REF] Buesa | Molecular epidemiology of caliciviruses causing outbreaks and sporadic cases of acute gastroenteritis in Spain[END_REF][START_REF] Kirkwood | A 5-year study of the prevalence and genetic diversity of human caliciviruses associated with sporadic cases of acute gastroenteritis in young children admitted to hospital in Melbourne, Australia (1998-2002)[END_REF][START_REF] Boga | Etiology of sporadic cases of pediatric acute gastroenteritis in asturias, Spain, and genotyping and characterization of norovirus strains involved[END_REF]. In accordance with these previous reports, our results show that norovirus GGII.4 was predominantly detected in the faeces of children. GGII.4 norovirus was observed throughout the study with a peak during the winter of 2002/2003. It is important to note that this peak corresponded to an increase in gastroenteritis outbreaks throughout Europe caused by a new variant (HU/NoV/Farmington Hill/2002/USA, accession number AY502023) [START_REF] Lopman | Increase in viral gastroenteritis outbreaks in Europe and epidemic spread of new norovirus variant[END_REF][START_REF] Zheng | Norovirus classification and proposed strain nomenclature[END_REF]. The second most frequent norovirus genotype found in this study was GGII.b. Several studies have reported this genotype as a cause of sporadic and outbreak cases of gastroenteritits in children, and some of these studies suggested that GGIIb was closely associated with infections in children [START_REF] Liu | Identification of viral agents associated with diarrhea in young children during a winter season in Beijing, China[END_REF][START_REF] Rachakonda | Genetic diversity of noroviruses and sapoviruses in children with acute sporadic gastroenteritis in New Delhi, India[END_REF][START_REF] Arista | Heterogeneity and temporal dynamics of evolution of G1 human rotaviruses in a settled population[END_REF][START_REF] Waters | Etiology of community-acquired pediatric viral diarrhea: a prospective longitudinal study in hospitals, emergency departments, pediatric practices and child care centers during the winter rotavirus outbreak, 1997 to 1998. The Pediatric Rotavirus Epidemiology Study for Immunization Study Group[END_REF][START_REF] Beersma | Norovirus in a Dutch tertiary care hospital (2002-2007): frequent nosocomial transmission and dominance of GIIb strains in young children[END_REF]. In this study, the highest rates of GGIIb detection were observed in January-February 2002 and the fall and winter of 2003-2004. These periods coincide with GGIIb adult outbreaks in France [START_REF] Bon | Molecular epidemiology of caliciviruses detected in sporadic and outbreak cases of gastroenteritis in France from December 1998 to February 2004[END_REF]. So, the rate of detection of GGIIb and GGII.4 norovirus strains identified in children are similar to those circulating in the adult population in the same period. These results and more recent data (data not shown) do not support the hypothesis that one of these strains could play a predominant role in children.

The severity of norovirus and rotavirus gastroenteritis excluded mixed infections and was evaluated on three criteria, the severity score defined by Vesikari [START_REF] Ruuska | Rotavirus disease in Finnish children: use of numerical scores for clinical severity of diarrhoeal episodes[END_REF], the need for intravenous rehydration and the length of the hospitalization. In the population of children studied, norovirus infections were significantly less severe than rotavirus infections. The literature concerning symptom severity in norovirus and rotavirus gastroenteritis is heterogeneous [START_REF] Patel | Systematic literature review of role of noroviruses in sporadic gastroenteritis[END_REF]. A number of previous studies found lower levels of disease severity in norovirus than in rotavirus gastroenteritis, and most of these were carried out in developed countries [START_REF] Pang | Human caliciviruses in acute gastroenteritis of young children in the community[END_REF][START_REF] Gomara | Structured surveillance of infantile gastroenteritis in East Anglia, UK: incidence of infection with common viral gastroenteric pathogens[END_REF][START_REF] Chen | Molecular epidemiology and clinical manifestations of viral gastroenteritis in hospitalized pediatric patients in Northern Taiwan[END_REF][START_REF] Colomba | Viral gastroenteritis in children hospitalised in Sicily, Italy[END_REF][START_REF] Gourlain | Epidemiology of acute viral gastroenteritis in children hospitalized in Rouen, France[END_REF][START_REF] Medici | Molecular epidemiology of norovirus infections in sporadic cases of viral gastroenteritis among children in Northern Italy[END_REF]. In contrast, other studies found no significant differences between the severity of rotavirus and norovirus infections [START_REF] Nakagomi | Norovirus infection among children with acute gastroenteritis in Recife, Brazil: disease severity is comparable to rotavirus gastroenteritis[END_REF][START_REF] Sakai | Clinical severity of Norwalk virus and Sapporo virus gastroenteritis in children in Hokkaido, Japan[END_REF][START_REF] Nguyen | Diversity of viruses associated with acute gastroenteritis in children hospitalized with diarrhea in Ho Chi Minh City, Vietnam[END_REF][START_REF] Sdiri-Loulizi | Acute infantile gastroenteritis associated with human enteric viruses in Tunisia[END_REF]. One such study was conducted by us in Tunisia, and we attributed the severity of the norovirus infection to a lack of medical care at home and the delay to hospitalization for children infected with norovirus. This could also be the case for the studies conducted in countries with poor sanitation and hygiene.

In conclusion, these results highlight the main role of rotavirus in hospitalized gastroenteritis among French children less than 2 years of age. During the study until the end of the 2003-2004 winter season the G1;P [START_REF] Bon | Prevalence of group A rotavirus, human calicivirus, astrovirus, and adenovirus type 40 and 41 infections among children with acute gastroenteritis in Dijon, France[END_REF] genotype was widespread while G9:P[8] remained rare. A majority of severe rotavirus gastroenteritis requiring hospitalization may be prevented by the widespread use of rotavirus vaccines [START_REF] Vesikari | Safety and efficacy of a pentavalent human-bovine (WC3) reassortant rotavirus vaccine[END_REF][START_REF] Boom | Parashar UD Effectiveness of pentavalent rotavirus vaccine in a large urban population in the United States[END_REF][START_REF] Ruiz-Palacios | Safety and efficacy of an attenuated vaccine against severe rotavirus gastroenteritis[END_REF]. Noroviruses are the second leading causative agent of gastroenteritis in hospitalized young children and such infections are less severe than those caused by rotavirus. The norovirus genotype pattern in children shows a clear predominance of genotype GII.4 followed by genotype GGIIb. This pattern reflects the norovirus genotypes in circulation within the population rather than a predominance of particular genotypes in children. NS= Not significant a) Rotavirus severity score: 0-6 months versus 6-24 months (p<0.0001) or >24 months (p=0.008). b) Norovirus severity score: p not significant for all age groups.

Figure 1 :

 1 Figure 1: Monthly distribution of viruses in children hospitalized for gastroenteritis (January 2002-May 2004)

Table 1 :

 1 Etiology of gastroenteritis in 457 hospitalized children in relation to age.

				Age (months)			Total
		0-6	>6-12	>12-24	>24-60	>60-180
	Rotavirus	77	63	59 (57.8%) 22 (44.9%) 4 (17.4%) 223
		(43.5%)	(59.4%)				(48.8%)
	Norovirus	15 (8.5%) 10 (9.4%) 8 (7.8%)	3 (6.1%)	2 (8.7%)	38 (8.3%)
	Adenovirus	6	5	5	0	0	16 (3.50)
	Astrovirus	1	2	0	0	0	3
	Aïchivirus	0	0	1	0	1	2
	Sapovirus	0	0	1	0	0	1
	Bacteria	8	2 a	4	6	5	25 a
							(5.4%)
	Virus +	2 b ; c	0	3 d ; e	1 f	0	6
	Bacteria
	Virus + virus 3 g ; h ; i	9 j ; k ; l ; m	4	0	0	16 (3.5%)
	NPI*	67 (37.8	15 (14.1	17 (16.7	17 (34.7	11 (47.8	127 (27.8
		%)	%)	%)	%)	%)	%)
	Total	177	106	102	49	23	457
		(38.7%)	(23.2%)	(22.32%)	(10.72%)	(5.03%)
	Values are numbers of cases, with percentages in parentheses.
	* NPI: No acknowledged enteric pathogens.

Table 2 :

 2 Clinical features according the etiologic agent (rotavirus or norovirus, mixed infections excluded) related to the age of the children with gastroenteritis.

	Age	0-6 months (177)	6-24 months (208)	> 24 months (72)
		Rotavirus (75)	Norovirus (15)	p value	Rotavirus (122)	Norovirus (18)	p value	Rotavirus (26)	Norovirus (5)	p valu
	Severity score, 1-20 points: mean value (+/-sd)	11.1 a (+/-2.89)	9.93 b (+/-2.46)	NS	13.48 a (+/-2.54)	11.39 b (+/-3.05)	p<0.002	12.88 a (+/-2.94)	8,8 b (+/-2.28)	p=0.0
	Length of hospitalization in days (+/-sd)	3.26 (+/-1.84)	2.17 (+/-1.06)	p=0.0293	2.90 (+/-1.39)	1.78 (+/-1.06)	p=0.0013	2,85 (+/-1.26)	1,2 (+/-0.45)	p=0.00
	Intravenous rehydration (%)	40 (53.3%)	5 (33.33%)	NS	107 (87.7%)	12 (66.67%)	p=0.031	25 (96.1%)	4 (80%)	NS
	Bloody stools (%)	0	0		3 (2.46%)	2 (11.11%)		1 (3.8%)	0 (0%)

Acknowledgments

We thank Philip Bastable for editorial assistance. This work was supported by the national PHRC 2001 (Programme Hospitalier de Recherche Clinique) from AP-HP (Assistance Publique -Hôpitaux de Paris).

We thank the physicians and their patients for participation in this study.


Xavier Bonnefond

Sébastien Marinesque

Application of a nudging technique to thermoacoustic tomography

ThermoAcoustic Tomography (TAT) is a promising, non invasive, medical imaging technique whose inverse problem can be formulated as an initial condition reconstruction. In this paper, we introduce a new algorithm originally designed to correct the state of an evolution model, the back and forth nudging (BFN), for the TAT inverse problem. We show that the flexibility of this algorithm enables to consider a quite general framework for TAT. The backward nudging algorithm is studied and a proof of the geometrical convergence rate of the BFN is given. A method based on Conjugate Gradient (CG) is also introduced. Finally, numerical experiments validate the theoretical results with a better BFN convergence rate for more realistic setups and a comparison is established between BFN, CG and a usual inversion method.

Introduction

ThermoAcoustic Tomography (TAT) is a hybrid imaging technique that uses ultrasound waves produced by a body submitted to a radiofrequency pulse, uniformly deposited throughout the body. The absorption of this initial energy causes a non-uniform thermal expansion, leading to the propagation of a pressure wave outside the body to investigate. This wave is then measured all around the body with piezoelectric transducers or, more recently, thanks to interferometry techniques.

It appears that the absorption of the initial pulse is highly related to the physiological properties of the tissue [START_REF] Joines | The measured electrical properties of normal and malignant human tissues from 50 to 900 mhz[END_REF]. As a result, the magnitude of the ultrasonic emission (i.e. thermoacoustic signal), which is proportional to the local energy deposition, reveals physiologically specific absorption contrast. See [START_REF] Stantz | Thermoacoustic CT of the breast[END_REF][START_REF] Kruger | Thermoacoustic CT of the breast[END_REF][START_REF] Paltauf | Photoacoustic tomography using a Mach-Zehnder interferometer as acoustic line detector[END_REF][START_REF] Xu | Photoacoustic imaging in biomedicine[END_REF] for an introduction to the experimental setup. Considering that the initial illumination is a Dirac distribution in time, the problem of recovering the absorptivity of the investigated body from the thermoacoustic signal is equivalent to recovering the initial condition of a Cauchy problem involving the wave equation from the knowledge of the solution on a surface surrounding the imaging object [START_REF] Scherzer | Variational methods in imaging[END_REF]. Note that the experimental constraints do not allow the acquisition surface to completely surround the tissues to investigate, so that one cannot expect a better situation than measurements on a half-sphere (as in breast cancer detection for example). Moreover, several experimentation constraints limit the number of sensors, so that the methods of reconstruction considered must be robust to the space sampling.

Many works dealing with TAT have been achieved during the last decade (see [START_REF] Kuchment | Mathematics of Photoacoustic and Thermoacoustic Tomography[END_REF] for an overview, or [START_REF] Agranovsky | On reconstruction formulas and algorithms for the thermoacoustic and photoacoustic tomography[END_REF][START_REF] Wang | Photoacoustic imaging and spectroscopy[END_REF]), and many of the methods used in these papers strongly depend on a set of additional assumptions, among which:

• the homogeneity of the tissues, leading to a constant speed of sound for the pressure wave;

• the lack of frequency-dependent attenuation, so that the pressure wave obeys the classical, undamped, wave equation;

• the complete data situation, where the acquisition surface is considered to enclose the whole body.

The study of TAT in the case of an homogeneous medium without attenuation gave rise to explicit inversion formulas. These latter usually require that the acquisition set is a closed surface, even if they can be approximated in the limited view situation (where the surface is a half-sphere for example). Among these, filtered backprojection formulas, even though they have been very successfully implemented in several situations, seem to face some issues when the source has support partly outside the observation surface, or when the data sampling is not sufficient (see [START_REF] Finch | Determining a function from its mean values aver a family of spheres[END_REF][START_REF] Haltmeier | Filtered backprojection for thermoacoustic computed tomography in spherical geometry[END_REF]). Fourier's type formulas, which offer a much better numerical efficiency, apply on very specific geometries (see [START_REF] Kunyansky | A series solution and a fast algorithm for the inversion of the spherical mean Radon transform[END_REF][START_REF] Norton | Ultrasonic reflectivity imaging in three dimensions: Exact inverse scattering solutions for plane, cylindrical and spherical apertures[END_REF][START_REF] Xu | Exact frequency-domain reconstruction for thermoacoustic tomography: II. Cylindrical geometry[END_REF]) or imply some additional approximations, like an interpolation from spherical to cartesian coordinates [START_REF] Anastasio | Application of inverse source concepts to photoacoustic tomography[END_REF][START_REF] Cox | Photoacoustic tomography with a limited-aperture planar sensor and a reverberant cavity[END_REF][START_REF] Haltmeier | Thermoacoustic tomography and the circular Radon transform : exact inversion formula[END_REF][START_REF] Haltmeier | A reconstruction algorithm for photoacoustic imaging based on the nonuniform FFT[END_REF][START_REF] Xu | Exact frequency-domain reconstruction for thermoacoustic tomography: I. Planar geometry[END_REF][START_REF] Xu | Reconstructions in limited-view thermoacoustic tomography[END_REF].

Even though some of these techniques give rise to very efficient reconstruction schemes (especially in [START_REF] Kunyansky | A series solution and a fast algorithm for the inversion of the spherical mean Radon transform[END_REF]), so far it is not clear if they can be extended to less restrictive acquisition geometries or to an attenuated wave with non constant speed (see [START_REF] Kunyansky | Thermoacoustic tomography with detectors on an open curve: an efficient reconstruction algorithm[END_REF] for an answer about the acquisition surface).

Moreover, since the work of Burgholzer et al (see [START_REF] Burgholzer | Exact and approximative imaging methods for photoacoustic tomography using an arbitrary detection surface[END_REF]), the time reversal method (first suggested in [START_REF] Finch | Determining a function from its mean values aver a family of spheres[END_REF]) has been applied with success to the TAT (see e.g. [START_REF] Hristova | Time reversal in thermoacoustic tomography -an error estimate[END_REF][START_REF] Hristova | Reconstruction and time reversal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous media[END_REF]). Even though the first known results on this method required a complete data framework, recent works by Stefanov et al extended it to a Neumann Series method in a fairly general situation (incomplete data, known variable sound speed and external source) with very good results in [START_REF] Qian | An efficient Neumann-series based algorithm for Thermoacoustic and Photoacoustic Tomography with variable sound speed[END_REF][START_REF] Stefanov | Thermoacoustic tomography with variable sound speed[END_REF].

In this paper, we show that the framework of TAT lends itself to the application of some data assimilation techniques. These latter, mainly used in geophysics, aim at correcting the state of an evolutionary model by means of data, in order to obtain a good approximation of the real state (see [START_REF] Talagrand | Assimilation of Observations, an Introduction[END_REF] for an introduction). The particular data assimilation method proposed here is based on a nudging technique: given an evolution model of the state and direct observation (our data), it consists in adding, inside the model equation, a newtonian recall of the state solution to the observations (or data), which is usually called the feedback or nudging term [START_REF] Auroux | The Back and Forth Nudging algorithm for oceanographic data assimilation[END_REF]. As we shall see, from a practical point of view, this method can be successfully used to manage the usual issues of the TAT inverse problem as incomplete data, external source and variable sound speed (when given, however). So far, however, the theoretical convergence result for the nudging technique is based on a classical result about stabilization of the wave equation [START_REF] Lions | Exact controllability, stabilization and perturbations for distributed systems[END_REF][START_REF] Russel | Controllability and stabilizability theory for linear PDE's:recent progress and open questions[END_REF], which requires somehow a geometric optics condition (see [START_REF] Bardos | Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary[END_REF] or [START_REF] Liu | Locally distributed control and damping for the conservative systems[END_REF] for an overview). This geometric condition, whose verification is not a simple matter, is not automatically satisfied in an incomplete data framework or with a variable sound speed. Consequently the proof provided in this article only stands in a very favorable situation (3d case, constant speed and complete data), but does not depend on usual stabilization techniques and yields explicit convergence rates.

Different nudging terms can be found in the literature, and most of them have been used to assimilate data in physical oceanography as in [START_REF] Krishnamurti | Physical initialisation for numerical weather prediction over the tropics[END_REF][START_REF] Lorenc | The meteorological office analysis correction data assimilation scheme[END_REF][START_REF] Lyne | A data assimilation experiment and the global circulation during the FGGE special observing periods[END_REF] and more recently in [START_REF] Auroux | Symmetry-based observers for some water-tank problems[END_REF]. Fundamental articles offer the basis of most popular techniques as Kalman filter [START_REF] Kalman | A new approach to linear filtering and prediction problems[END_REF], Kalman-Bucy filter [START_REF] Bucy | Filtering for Stochastic Processes with Applications to Guidance[END_REF] and Luenberger observer [START_REF] Luenberger | Observers for Multivariable Systems[END_REF].

Since the basic nudging method showed its limits in real conditions by using the data once, Auroux and Blum proposed to extend the method by adding a resolution backward in time and iterating the process, thus defining the back and forth nudging (BFN) algorithm. They presented this technique in [START_REF] Auroux | A nudging-based data assimilation method for oceanographic problems: the Back and Forth Nudging (BFN) algorithm[END_REF][START_REF] Auroux | The Back and Forth Nudging algorithm for oceanographic data assimilation[END_REF] and implemented it with computationally efficient results in comparison with traditional data assimilation methods, like optimal filters (from extended Kalman's to particle filters) or other optimal minimization techniques (variational methods), which appear costfull respectively in memory needs and computing time -see the brief introductions of Talagrand in [START_REF] Talagrand | Assimilation of Observations, an Introduction[END_REF][START_REF] Talagrand | Variational assimilation of meteorological observations with the adjoint vorticity equation. I : Theory[END_REF][START_REF] Talagrand | Variational assimilation of meteorological observations with the adjoint vorticity equation. II : Numerical Results[END_REF] for further details about data assimilation and a relevant bibliography. Recently, a more general formulation of the BFN using observers has been presented by Ramdani, Tucsnak and Weiss in [START_REF] Ramdani | Recovering the initial state of an infinite-dimensional system using observers[END_REF].

As inverse problems are often solved by means of variational techniques, we also introduce a least squares method based on the Conjugate Gradient (CG).

The two methods introduced are compared to the Neumann Series presented in [START_REF] Qian | An efficient Neumann-series based algorithm for Thermoacoustic and Photoacoustic Tomography with variable sound speed[END_REF], which is one of the best existing method which fits the variable speed case and partial data settings.

This paper is organized as follows. In Section 2, we introduce the TAT inverse problem in a quite general form, i.e. without the use of the usual assumptions, then we describe the BFN algorithm and we state the main result of this article: the convergence Theorem 3. Then Section 3 is devoted to the proof of this result. In Section 4, we describe the variational method that uses CG algorithm. Finally, we give numerical results in Section 5.

Notation

In the following, we shall use this notation:

• The open ball with center x ∈ R 3 and radius r > 0 is denoted by B(x, r);

• Its boundary, the sphere with center x and radius r, is noted S(x, r);

• For every x ∈ R 3 and 0 < r 1 < r 2 , we define the spherical shell of center x and radii r 1 , r 2 :

A(x, r 1 , r 2 ) = y ∈ R 3 |r 1 < x -y < r 2 ,
and if x = 0: A(r 1 , r 2 ) = A(0, r 1 , r 2 );

• For any set S of R 3 and r 1 > 0, we denote by T (S, r 1 ) the set S extended to the points at a distance lower than r 1 from S:

T (S, r 1 ) = y ∈ R 3 |∃x ∈ S, y -x < r 1 .
2 Presentation of the method

The general TAT problem

In the following, we will denote by p exact (x, t) the pressure wave resulting from the thermal expansion of the body. We will make the assumption that the measurement process of this pressure is subject to some perturbation p noise , such that the actual data can be written p data := p exact +p noise . In thermoacoustic tomography, the data are nothing but {p data (x, t)|x ∈ S }, where S is a surface surrounding the body to investigate. The pressure p exact satisfies the Cauchy problem (see [START_REF] Scherzer | Variational methods in imaging[END_REF] for a detailed calculation):

Lp exact (x, t) = E dep (x, t), (x, t) ∈ R 3 × [-1, ∞), p exact (x, -1) = 0, x ∈ R 3 , ∂ t p exact (x, -1) = 0, x ∈ R 3 ,
where L is the differential operator governing the wave, most likely the wave operator, or a damped wave operator, and E dep is the energy deposited in the body around the time t = 0 (so we have to consider a negative initial time, e.g. t = -1). This energy can be written:

E dep = e(x) dj dt (t), e(x) := βc c p I em (x)Ψ(x),
where e is called the normalized energy deposition function. Here β is the thermal expansion coefficient, c p is the specific heat capacity, c the sound speed and I em is the radiation intensity of the initial energy pulse. All these parameters, including the time shape j of the pulse, are known and we see that the knowledge of E dep is sufficient to compute the absorption density Ψ inside the body, which is our purpose.

So far, the general thermoacoustic problem can be formulated in the following way:

Let p exact (x, t) be solution of:

Lp exact (x, t) = f 0 (x) dj dt (t), (x, t) ∈ R 3 × [-1, ∞), p exact (x, -1) = 0, x ∈ R 3 , ∂ t p exact (x, -1) = 0, x ∈ R 3 , (1)
where j is known and f 0 is the object to reconstruct, with compact support in the unit ball B(0, 1). Can we compute f 0 , or a good approximation of f 0 , from the knowledge of p data = p exact + p noise on a surface S surrounding B(0, 1) ?

Reduction to the homogeneous case

As we have seen, even though it is not possible in practice, the initial energy deposition is meant to be a Dirac pulse in time at t = 0, that is dj dt = δ ′ 0 . Let q 0 be solution, which is no longer to be considered for negative time values, where it is the null solution, of the Cauchy problem:

Lq 0 (x, t) = f 0 (x)δ ′ 0 (t), (x, t) ∈ R 3 × R + , q 0 (x, 0) = 0, x ∈ R 3 , ∂ t q 0 (x, 0) = 0, x ∈ R 3 ,
which is equivalent to:

Lq 0 (x, t) = 0, (x, t) ∈ R 3 × R + , q 0 (x, 0) = f 0 (x), x ∈ R 3 , ∂ t q 0 (x, 0) = 0, x ∈ R 3 , (2)
according to Duhamel's principle (see [10]), provided that L is a (damped) wave operator. Then p data is solution of (1) if and only if p data = j * t q 0 , which means that a deconvolution operation on the thermoacoustic signal p exact leads to the knowledge of the solution q 0 of (2) on the same surface S surrounding the object.

In the following, we will still denote by p exact the solution of ( 2), assuming that we know this latter for all x in S , and all t in R + .

Some useful facts about the standard wave equation

In this section we give some basic properties of the following wave equation:

L 0 u(x, t) = 0, (x, t) ∈ R 3 × R + , u(x, 0) = l(x), x ∈ R 3 , ∂ t u(x, 0) = h(x), x ∈ R 3 , (3)
where l and h are two C ∞ functions with compact support in B(0, 1), and L 0 := ∂ tt -∆. All this material can be found in [START_REF] John | Plane Waves and Spherical Means[END_REF] for example. According to Huyghens' principle in the strong sense, for every couple (x 0 , t 0 ) ∈ R 3 × R + , the classical solution u(x 0 , t 0 ) of ( 3) only depends on the values of l and h on the backward characteristic cone:

x -x 0 = |t -t 0 |.
As a matter of fact, when the initial conditions l and h have their support in B(0, 1), the solution at any time t has its support in B(0, 1 + t).

Huyghens' principle leads to the energy conservation for the solution u of (3). Definition 1. For every time T , we call energy of u the quantity:

E u (T ) := 1 2 u(•, T ) 2 H 1 0 (R 3 ) + ∂ t u(•, T ) 2 L 2 (R 3 ) .
The variation of the energy of u is estimated from the following identity:

∂ t u (∂ tt u -∆u) = 0 = ∂ t ∇u 2 2 + (∂ t u) 2 2 - 3 i=1 ∂ x i (∂ t u ∂ x i u).
Since u has a compact support, integrating for 0 < t < T and over the whole space yields:

E u (T ) = E u (0).
Note that if h is zero, we have:

E u (0) = 1 2 l 2 H 1 0 (R 3 ) .

Nudging

Now, let us introduce the application of the BFN algorithm to the general TAT problem.

Assume that the observation surface S is included in the sphere S(0, 1). As exposed in Section 2.1, our purpose is to compute an approximation of the original object f 0 from an incomplete set of data φp data , where p data is a the solution (possibly contaminated by noise) of the wave equation ( 2) and where φ allows the knowledge of p data only on the observation surface S . Theoretically speaking, φ should be 1 S , but in practice the discretization process allows us to choose a C ∞ function with compact support. Indeed, given a resolution, there exists ε > 0 such that any function supported in T (S , ε) and equal to 1 on S has the same discrete counterpart as 1 S (see Figure 1). This is the reason why we will make the following assumptions on φ:

i ∀x ∈ R 3 , 0 ≤ φ(x) ≤ 1;
ii φ has a compact support in T (S , ε);

iii ∀x ∈ T (S , ε/2), φ(x) = 1.
Finally, in order to avoid some technical difficulties, we will assume that the initial object f 0 has his support in B(0, 1 -ε) and is such that every Cauchy problem we shall consider has a classical solution, say C ∞ .

We may proceed as follows:

Definition 2 (Iterate of the BFN algorithm for TAT).

[Forward evolution]

From an rough estimate f 0,1 of the object to reconstruct f 0 , defined as an approximate of the initial condition supported in B(0, 1-ε), we first consider a solution p 1 of Problem (3) with initial conditions l = f 0,1 and h = 0 (as usual in TAT).

We compute this solution until time T = 2, chosen so that both p 1 (•, t) and p exact (•, t) (just as every function with compact support in the ball B(0, 1 -ε)) vanish on B(0, 1 + ε) for all t ≥ T (cf Huyghens' principle in Section 2.3).

Body S T (S , ε)

Figure 1: The body to investigate is surrounded by the acquisition surface S , and generates a pressure wave.

Thus at this final time, in the noise-free situation, the innovation term p 1 -p data , defined as the difference between the solution p 1 of Problem (3) and the data, is still a solution of (3) with l = f 0 -f 0,1 and h = 0 and has an energy matching up to:

E p 1 -p data (T ) = E p 1 -p data (0) = 1 2 ∇(f 0 -f 0,1 ) 2 H 1 0 (R 3 ) ,
since the conservation of energy for the wave equation holds in this situation.

[Backward evolution]

Then we apply the backward nudging method to make the solution evolve back in time, with, as announced, the addition in the backward equation of a newtonian feedback, which adjusts the solution along its evolution by recalling it to the observed data.

Namely, we add to L the feedback correction term kφ∂ t (• -p data ), for some nudging parameter k ∈ R + . And we compute the backward solution starting from the final state of the forward implementation, i.e. p 1 (x, T ) and ∂ t p 1 (x, T ), untill the initial time t = 0.

We obtain a corrected solution p1 of the Cauchy problem, called back-ward nudging problem:

Lp 1 (x, t) = kφ(x)∂ t (p 1 -p data )(x, t), (x, t) ∈ R 3 × [0, T ], p1 (x, T ) = p 1 (x, T ), x ∈ R 3 , ∂ t p1 (x, T ) = ∂ t p 1 (x, T ), x ∈ R 3 . (4)
In order to obtain an equation in forward time, let us consider the map t → p1 (T -t), still denoted by p1 . The Cauchy problem becomes:

Lp 1 = -kφ(x)∂ t (p 1 + p data )(x, T -t), (x, t) ∈ R 3 × [0, T ], p1 (x, 0) = p 1 (x, T ), x ∈ R 3 , ∂ t p1 (x, 0) = -∂ t p 1 (x, T ), x ∈ R 3 , ( 5
)
where L is the operator L backwards in time.

Obviously, in practice, when L is not the wave operator, there is no reason for this operator to be reversible. Nevertheless the newtonian feedback may act as a regularization term and keep the computation stable.

[Update of the estimate]

Finally, after this back and forth evolution, a new estimate p1 (T ) is obtained, but in order to iterate the process we need the new estimate to be supported in B(0, 1 -ε), just like f 0 and f 0,1 . This is simply done by defining:

f 0,2 (x) := 1 B(0,1-ε) p1 (x, T ), x ∈ R 3 .
The last step can be easily justified by the fact that, since f 0 is supported in B(0, 1 -ε):

f 0 -p1 (•, T ) H 1 0 (R 3 ) ≥ f 0 -1 B(0,1-ε) p1 (•, T ) H 1 0 (R 3
) . This scheme is then iterated, constructing a sequence of estimates (f 0,n ) n∈N * , which may converge to f 0 . The following theorem, proved in Section 3, ensures that the convergence is geometrical in H 1 0 (R 3 ) under the assumptions:

(i) L is the standard wave operator ∂ tt -∆;

(ii) The data are noise-free, i.e. p data = p exact (see Section 2.1), and the observation surface S is the whole sphere S(0, 1);

(iii) The object to reconstruct f 0 is a C ∞ function with compact support in B(0, 1);

(iv) f 0,1 and f 0,2 are two consecutive estimates of f 0 obtained as described in Definition 2.

Theorem 3. Under the assumptions (i)-(iv), there exists 1 > s > 0 only depending on f 0 , φ and the nudging parameter k > 0 such that:

f 0 -f 0,2 H 1 0 (R 3 ) ≤ s f 0 -f 0,1 H 1 0 (R 3 ) . (6)
We only consider this ideal framework for the sake of the proof, but numerical results given in Section 5 apply with more realistic assumptions and show a better convergence rate.

Proof of Theorem 3

This section is devoted to the proof of Theorem 3. We study the case where L = L 0 := ∂ tt -∆ is the standard wave operator. Let φ, f 0 and p data be as in Theorem 3. In particular, for the sake of the proof, we make the assumptions that the data are noise-free and that, since S = S(0, 1), the set T (S , ε/2) is equal to A(1 -ε/2, 1 + ε/2). We shall use the notation:

L k := ∂ tt -∆ + kφ∂ t , k ∈ R + .
We first study the existence of the objects used in the BFN approximation scheme.

Solution of the backward nudging equation

The proof of the following theorem can be found in [START_REF] Bers | Partial differential equations[END_REF], Chapter 2.8.

Theorem 4. Given three C ∞ -functions φ, l and h with compact support and k ∈ R + , T > 0 there exists a unique C ∞ solution p of the Cauchy problem:

L k p(x, t) = -kφ(x)∂ t p data (x, T -t), (x, t) ∈ R 3 × [0, T ], p(x, 0) = l(x), x ∈ R 3 , ∂ t p(x, 0) = h(x), x ∈ R 3 . (7)
Moreover, for every (x 0 , t 0 ) ∈ R 3 × R + , p(x 0 , t 0 ) only depends on the values of l, h, φ and p data (•, T -•) in the backward characteristic cone:

x -x 0 ≤ |t -t 0 |.
In particular, this means that for every sketch f 0,1 with compact support and of class C ∞ , the solution p1 of the backward nudging equation ( 7) exists, and for all t ∈ [0, T ] the function p1 (•, t) has a compact support. This property will be helpful to apply an energy method. Now, it is easy to show that if p1 is solution of ( 7) with initial conditions l(x) = p1 (x, 0) = p 1 (x, T ) and h(x) = ∂ t p1 (x, 0) = -∂ t p 1 (x, T ), then the function u := p1 -p data (•, T -•) is solution of the Cauchy problem:

L k u(x, t) = 0, (x, t) ∈ R 3 × [0, T ], u(x, 0) = p 1 (x, T ) -p data (x, T ), x ∈ R 3 , ∂ t u(x, 0) = -∂ t (p 1 -p data )(x, T ), x ∈ R 3 , (8)
and satisfies:

E u (0) = E p 1 -p data (T ) = 1 2 f 0,1 -f 0 2 H 1 (R 3 ) .
Since, from the definition of f 0,2 :

E u (T ) ≥ 1 2 p1 (•, T ) -f 0 2 H 1 (R 3 ) ≥ 1 2 f 0,2 -f 0 2 H 1 (R 3 ) ,
we only need to prove that:

E u (T ) ≤ sE u (0), for some 0 < s < 1,
to complete the proof of Theorem 3. This is the purpose of the next section.

Energy inequality

We have the following identity for every solution u of (7):

∂ t uL k u = 0 = ∂ t ∇u 2 2 + (∂ t u) 2 2 + kφ(∂ t u) 2 - 3 i=1 ∂ x i (∂ t u ∂ x i u).
Recall that u has a compact support for every t, thus, integrating the latter equality over the whole space and for 0 < t < T , we get:

E u (T ) -E u (0) = -k T 0 R 3 φ(∂ t u) 2 . (9)
Consequently, the loss of energy of u is exactly the amount of its 'energy' passing through the support of φ during time T .

In [START_REF] Finch | Determining a function from its mean values aver a family of spheres[END_REF], in order to invert the spherical Radon transform, Finch et al proved the following trace identity, which will be usefull to estimate (9):

Theorem 5. Suppose h i ∈ C ∞ 0 (B(0, ρ)
) and u i is the solution of (3) for l = 0 and h = h i , i = 1, 2. Then we have the identity

1 2 R 3 h 1 h 2 = -1 ρ ∞ 0 S(0,ρ) tu 1 ∂ tt u 2 .
From this we will obtain a key estimate of the 'energy' passing through the sphere S(0, ρ). Indeed, if u is solution of (3) with h = 0, taking h 1 = f and h 2 = ∆f in the theorem yields:

1 2 R 3 f ∆f = -1 ρ ∞ 0 S(0,ρ) t(∂ t u) 2 ,
which gives:

1 2 f 2 H 1 0 (R 3 ) = 1 ρ ∞ 0 S(0,ρ) t(∂ t u) 2 . ( 10
)
Let us introduce the solution u 0 of the Cauchy problem:

L 0 u 0 (x, t) = 0, (x, t) ∈ R 3 × [0, T ], u 0 (x, 0) = p 1 (x, T ) -p data (x, T ), x ∈ R 3 , u 0t (x, 0) = -∂ t p 1 (x, T ) + ∂ t p data (x, 0), x ∈ R 3 , which is nothing but (p 1 -p data )(•, T -•).
Then u = u 0 + v where v is solution of:

L 0 v(x, t) = -kφ∂ t (u 0 + v)(x, t), (x, t) ∈ R 3 × [0, T ], v(x, 0) = ∂ t v(x, 0) = 0, x ∈ R 3 .
The calculation of the energy of v yields:

E v (T ) = -k T 0 R 3 φ∂ t u 0 ∂ t v -k T 0 R 3 φ(∂ t v) 2 .
Thanks to the Cauchy-Schwarz inequality, one has:

E v (T ) ≤ k φ∂ t u 0 L 2 ([0,T ]×R 3 ) H(T ) -kH(T ) 2 , (11)
where:

H(T ) := φ∂ t v L 2 ([0,T ]×R 3 )
.

Moreover, we have:

E v (T ) ≥ R 3 (∂ t v) 2 (•, T ) ≥ R 3 φ(∂ t v) 2 (•, T ) = d dT H 2 (T ).
So that, dividing by H(T ) in ( 11) we find:

d dT H(T ) + kH(T ) ≤ k φ∂ t u 0 L 2 ([0,T ]×R 3 ) .
A classical calculation shows that this differential inequation yields:

H(T ) ≤ T 0 k φ∂ t u 0 L 2 ([0,s]×R 3 ) e k(s-T ) ds,
and since the map s → √ φ∂ t u 0 L 2 ([0,s]×R 3 ) is increasing with s, we finally get:

H(T ) ≤ φ∂ t u 0 L 2 ([0,T ]×R 3 ) (1 -e -kT ).
This inequality expresses the fact that despite the feedback part of Equation [START_REF] Bers | Partial differential equations[END_REF], its solution u keeps a certain amount of energy on the support of φ, which is proportional to the same energy for u 0 . Indeed:

φ∂ t u L 2 ([0,T ]×R 3 ) ≥ φ∂ t u 0 L 2 ([0,T ]×R 3 ) -H(T ) ≥ φ∂ t u 0 L 2 ([0,T ]×R 3 )
e -kT , thus:

T 0 R 3 φ(∂ t u) 2 ≥ T 0 R 3 φ(∂ t u 0 ) 2 A e -2kT . (12)
The last step of this proof is the estimation of A. We have:

A ≥ A(1-ε/2,1+ε/2) T 0 (∂ t u 0 ) 2 = A(1-ε/2,1+ε/2) T 0 (∂ t p 1 -∂ t p data ) 2 (T -t),
so that:

A ≥ A(1-ε/2,1+ε/2) T 0 (∂ t p 1 -∂ t p data ) 2 = 1+ε/2 1-ε/2 T 0 S(0,ρ) (∂ t p 1 -∂ t p data ) 2 .
Since p 1 -p data vanishes on A(1 -ε/2, 1 + ε/2) for every t > T :

A ≥ 1 T 1+ε/2 1-ε/2 ∞ 0 S(0,ρ) t(∂ t p 1 -∂ t p data ) 2 .
Finally, applying (10) to p 1 -p data , one has:

A ≥ 1 T 1+ε/2 1-ε/2 ρ 1 2 f -f 0,1 2 H 1 0 (R 3 ) dρ, which yields: A ≥ ε T E u (0). ( 13
)
Combining the above estimate with ( 9) and ( 12), we get:

E u (T ) ≤ (1 - kε T e -2kT )E u (0),
which was the inequality to prove, with s := 1 -kε T e -2kT < 1. Remark 6. Since we know that T = 2, there exists an optimal choice for the nudging parameter k which leads to s = 1 -ε 8e . Even though this theoretical convergence rate does not seem satisfactory, we shall see in Section 5 that the actual convergence is much faster in practice.

Variational approach

We deal now with a variational formulation of the inverse TAT problem, that leads to some new reconstruction techniques.

In this section, we consider the reconstruction of an object f 0 ∈ H 1 0 (B(0, 1)) from a set of observations p data = p exact + p noise . Here, p exact := W f 0 where the linear operator W maps an initial condition f to its related solution of the Cauchy problem (3) with a null derivative initial condition h = 0, restricted to a non-empty, open observation set A ε ⊂ A(1 -ε, 1 + ε). Namely, W can be written W = ΦW , with:

W : H 1 0 (B(0, 1)) -→ C 0 ([0, T ]; H 1 0 (B(0, R))) f -→ W f,
where W f is the weak solution of (3) with the initial conditions l = f and h = 0. Even though W f is defined on the whole space R 3 , we can choose R > 0 large enough for any solution of (3) to keep a null trace on S(0, R), such that the restriction of W f to B(0, R) is in H 1 0 (B(0, R)) for every t ∈ [0, T ]. The operator Φ is the restriction operator:

Φ : L 2 ([0, T ] × R 3 ) -→ L 2 ([0, T ] × A ε ) g -→ g | Aε .
Our purpose is to solve the inverse problem:

W f = p data ,
by means of the following minimization problem:

(P α ) Minimize J α (f ) s.t. f ∈ H 1 0 (B(0, 1)).
Here, J α is the Tikhonov functional:

J α : H 1 0 (B(0, 1)) -→ R + f -→ 1 2 T 0 W f -p data 2 L 2 (Aε) dt + α 2 f 2 L 2 (B(0,1)) ,
with α > 0. Since W is a bounded linear operator, J α is a strictly convex (thanks to the regularization parameter α), differentiable and coercive functional, and any solution of (P α ) is characterized by:

∇J α (f ) = W * (W f -p data ) + αf = 0.
For any α > 0, the above equation has a unique solution:

f α = (W * W + αI) -1 W * p data ,
so that Problem (P α ) is well-posed. We have chosen to solve this latter by means of the conjugate gradient method, which requires, at each iteration, the computation of ∇J α . Now, one has, for every ψ ∈ C ∞ 0 (B(0, 1)) and f ∈ H 0 1 (B(0, 1)):

∇J α (f ), ψ L 2 (B(0,1)) = W * (W f -p data ), ψ + α f, ψ = W f -p data , W ψ G f,ψ +α f, ψ ,
with:

G f,ψ = T 0 Aε (W f -p data )W ψ = T 0 B(0,R) 1 Aε (W f -p data )W ψ.
Considering the weak solution u * of the adjoint equation:

∂ tt u * (x, t) -∆u * (x, t) = 1 Aε (W f -p data ), u * (x, T ) = 0, ∂ t u * (x, T ) = 0, (14)
and since W ψ ∈ C ∞ 0 (B(0, R)), we have:

G f,ψ = T 0 B(0,R) ∂ tt u * W ψ + T 0 B(0,R) ∇u * , ∇W ψ .
Since B(0, R) has been chosen large enough, applying Green's formula and integrating by parts yield:

G f,ψ = T 0 B(0,R) u * LW ψ - B(0,R) ∂ t u * (x, 0)W ψ(0) dx = -∂ t u * (•, 0), ψ .
Finally:

∇J α (f ) = -∂ t u * (•, 0) + αf.
This calculation ensures that each iteration of the conjugate gradient algorithm necessitates successively the computations of W f and of its adjoint state u * , whose algorithmic complexity is comparable to one iteration of the BFN algorithm. This should be noted that the computation of the adjoint state requires the storage of W f -p data on A ε , which is not required in the BFN algorithm.

Note that even if we have formulated the calculation of ∇J α in a continuous framework, its discrete counterpart leads to similar considerations. Obviously, the behaviour of the method strongly depends on the scheme chosen to compute solutions of (3).

Numerical implementation

In order to test the numerical behavior of the BFN method, we have implemented it in several situations. In particular, we have considered the case where the data are available only on a half-sphere, so that the illustration fits the practical experimental set-up.

Implementation framework

We now describe the numerical context chosen to compare Back and Forth Nudging (BFN) and the Conjugate Gradient (CG) techniques to a reasonable algorithm, namely the Neumann Series (NS) (since it is shown in [START_REF] Qian | An efficient Neumann-series based algorithm for Thermoacoustic and Photoacoustic Tomography with variable sound speed[END_REF] that the use of NS significantly improves the Time Reversal (TR) algorithm, we mainly chose to consider NS results).

We work now in a 2D framework for numerical costs reasons. The object to reconstruct is in [-0.5, 0.5] 2 gridded with 256 2 pixels, such that the space step is δx = 1/256.

Sensors are located on the circle S(0, √ 2/2) surrounding the object, on which various distributions are considered. If no additional information is given, 800 detectors are used when the data are complete. Nevertheless some reconstructions were computed with less detectors in order to bring to light the robustness of the techniques. When a multiplicative white Gaussian noise is added to the data, their level is set to 15%.

The final time T = √ 2 is then taken minimal in order to leave waves traveling to the sensors when sound speed is constant with value 1. Note that the four studied methods can significantly be improved by increasing T (cf. [START_REF] Qian | An efficient Neumann-series based algorithm for Thermoacoustic and Photoacoustic Tomography with variable sound speed[END_REF]). We use the domain [-0.5 -√ 2 -ε; 0.5 + √ 2 + ε] 2 for the computation, where ε represents a few pixels, which is large enough to avoid reflection effects on its boundary.

The model is computed by means of the classical finite difference time domain (FDTD) method (explicit Euler scheme and classical five points discretization of the Laplace operator ∆ δx ).

Figure 2 represents the 2-dimensional objects to reconstruct, latter denoted by f 0 . Squares able to give rise to typical phenomenons of observation of wave-like systems while the second object, known as the Shepp-Logan phan-tom, allows good analysis and comparisons of the ability of the methods. The third object has been considered for combining sharp contours with smoother areas. Each is implemented as a 256 by 256 matrix. Four sound speeds are used. First a constant one c = 1, then three other one, variable, which are chosen as in [START_REF] Qian | An efficient Neumann-series based algorithm for Thermoacoustic and Photoacoustic Tomography with variable sound speed[END_REF], respectively defined as: Concerning the parameters of the algorithms, the regularization parameter α in CG method equals 0. Other values were tested that stabilized the algorithm but always blur the reconstructions so that we get higher errors than without regularization. The observed convergence rates of CG and NS methods are fairly high, so that only 10 iterations yield a good approximation of the best achievable solutions. With regard to the two nudging coefficients, while we can theoretically affirm that the larger it is, the faster the solution tends to the observations (thanks to the energy equality ( 9)), the discretization of the time-derivative of the solution imposes a coefficient k not larger than 1/δt. In order to obtain the best convergence and to make the choice of k less arbitrary (and fitting with the limits of the numerical method), it is defined as 0.9/δt.

c(x, y) = 1 + 0.2 sin(2πx) + 0.1 cos(2πy), c(x, y) = 9(x 2 + y 2 ) 1 + 9(x 2 + y 2 ) + exp(-90(x 2 + y 2 )) (15)
The following tables give the relative mean square error obtained with the best estimation obtained after at most 10 iterations and, in brackets, the corresponding number of iterations required. Note that BFN, CG and NS furnishes convex error plots with a minimum.

Finally, as it has been suggested in [START_REF] Ramdani | Uniformly exponentially stable approximations for a class of second order evolution equations -Application to LQR problems[END_REF], we also have used an artificial numerical attenuation to offset spurious high-frequency effects and noise, with interesting consequences. Then for BFN, NS and CG techniques, the numerical model becomes:

u n+1 -2u n + u n-1 δt 2 = ∆ δx u n + δx ε ∆ δx u n -u n-1 δt ,
where the value for ε is empirically set and an optimal value appears for each method, which are approximately 2 for BFN and 1.7 for CG and NS in a complete data situation.

The attenuation being only introduced in a numerical improvement view, it is set to attenuate both back and forth solutions, should the case arise, and when the data are corrupted by noise.

Implementation results

The first results are given in About the propagation of the singularities and their observation, we put the light on the effect of getting incomplete data (in a noiseless situation) in Fig. 4. The sensors are located on the upper half circle, that illustrate the practical case where the observed body cannot be surrounded by the observation surface.

In this situation (constant speed and noiseless data on the half circle), the three methods are compared in Fig. 5 and6. NS is excellent, while BFN equivalent to CG convergence in about 5 iterations, but keeps on improving the reconstruction along next iterations. As always, even if its reconstruction is the worst one, CG converges in less iterations and yields a good rough estimate faster than the others. which are still homogeneously located on the observation circle, but only one out of σ is considered in the data. The results shown in Table 4 highlight some redundancy aspect of the data since the reconstructions have a similar quality in all situations, particularly with CG which is very robust here.

That is obvious at convergence of the algorithms, but one must consider more than 100 iterations for BFN and NS when σ = 8. Finally, if the intensity is not well recovered, forms are reconstructed with few artifacts. Then we sum up the different unfavorable conditions from above and show the results on Fig. 7 and8. Here data are acquired on the half circle, only one out of two sensors is considered, a 15% level noise is added and the speed is constant. The last object, a skull, has been chosen in this last experiment in order to consider a more complex structured object. Finally, even if the reconstructions obtained in this last situation are not BFN CG TR NS 30.7(4) 37.8(10) 59.9 37.9(4) satisfying, it may fit with very bad real conditions (where the noise level can be far greater than 15% for example) and allows us to test the robustness of the methods. In this view, Table 5 shows the errors if we consider variable speeds in a similar case but with σ = 1.

Speed

Implementation conclusions

CG reacts much badly to noise addition above all, but it provides good reconstructions after a couple of iterations in comparison with BFN and NS, so that it can be used to get a rough estimate for another method. Moreover CG keeps interesting when data are incomplete, not overreacting. On the whole, BFN and NS provide good and quite equivalent reconstructions with a similar cost. Finally, NS furnishes better reconstruction, but appears less robust than BFN to noise and to restriction of the number of sensors.

Numerical attenuation provides a good regularization that offsets noise for NS and BFN. Notice that this artificial numerical attenuation may allow us to consider lossy medium in back and forth evolutions as the physical loss does not exceed the artificial one.

Depending on whether data are complete or not, one may choose CG to get a first rough estimate, then NS or BFN to reconstruct the object. It has to be said that not stopping iterations to 10 often improves results in noiseless and/or complete data situations, but that this number of iterations provides the best estimate in less favorable settings.

Conclusions

We have established the theoretical geometrical convergence of the TAT-BFN algorithm, with an explicit convergence rate in an ideal situation. In principle, this method enables to make weak assumptions in the TAT modeling (variable sound speed, damping, incomplete data, etc.), since the model is considered as a weak constraint. Even though the proof of the convergence has only been stated in an ideal framework, the numerical implementation of the algorithm showed that the BFN method remains efficient in more general situations. Indeed this technique still yields robust reconstructions in the noisy and incomplete data situation. Even though the convergence rates of the Conjugate Gradient and of the Neumann Series methods appear to be better, the BFN provides more satisfactory asymptotic behavior and minimal relative error, along with a low numerical complexity, in quite unfavorable observation situations.

From these preliminary results, one should now focus on their generalization to more realistic wave equations. For example, one may introduce some damping in the model and study the theoretical validity and convergence of the methods. Our first attempts in this direction (mainly numerical implementations) suggest future interesting developments.

Figure 2 :

 2 Figure 2: The initial objects f 0 : squares, Shepp-Logan and skull.

-0. 4

 4 exp(-10(3 x 2 + y 2 -2) 2 ), c(x, y) = 1.25 + sin(2πx) cos(2πy).They are plotted on figure3and abbreviated respectively NTS, TS1 and TS2, as the first is a Non-Trapping Speed while the two other are Trapping Speeds. No cutoff is used to smooth the transition between variable internal speed and constant external speed (always being 1).

Figure 3 :

 3 Figure 3: The different variable speeds: NTS, TS1 and TS2 (from equations (15)).

Figure 4 :

 4 Figure 4: Reconstructions by 100 BFN iterations. Squares observed singularities. Complete (1.1% error) and half-circle (10.4%) noiseless data with constant speed.

Figure 5 :Figure 6 :

 56 Figure 5: Incomplete noiseless data on the half circle from Shepp-Logan phantom with constant speed: relative mean square error plots (BFN in solid line, CG in dashed-dotted line and NS in dashed line).

Table 4 :

 4 Partial noiseless data (one out of σ sensor records data) from Shepp-Logan phantom with constant speed: relative mean square errors (number of iterations, limited to 10).

Figure 7 :

 7 Figure 7: Incomplete partial data (σ = 2) on the half circle with 15% noise from skull phantom and with constant speed: relative mean square error plots (BFN in solid line, CG in dashed-dotted line and NS in dashed line).

Figure 8 :

 8 Figure 8: Incomplete 15% noise data on the half circle from skull phantom with constant speed and partial data (σ = 2): reconstructions with BFN (top left), TR (top right), CG (bottom left) and NS (bottom right) and relative mean square errors table (error, number of iterations, limited to 10).

Table 1

 1

	in a noiseless complete data situation.

Table 1 :

 1 Noiseless complete data for Shepp-Logan phantom: relative mean square error (number of iterations, limited to 10). iterations as we can see in Table2for the BFN, where 100 iterations instead of 10 are computed.

	c ≡ 1 1	NTS 1.5(100) 1.2(100) 4.1(100) TS1 TS2

Table 2 :

 2 Noiseless complete data from Shepp-Logan phantom: relative mean square error (BFN, 100 iterations).

	When adding a 15% level noise in the constant speed situation, one sees on
	Table 3 that any method offers an interesting reconstruction: one gets less
	than 20% error for 15% noise thanks to the attenuating scheme.
	BFN	CG	NS
	17.6(10) 19.8(2) 17.1(2)

Table 3 :

 3 Noisy complete data (15% noise) from Shepp-Logan phantom with the constant speed. Reconstruction with attenuating sheme: relative mean square error (number of iterations, limited to 10).

Table 5 :

 5 Incomplete 15% noise data on the half circle from Shepp-Logan phantom: relative mean square errors (number of iterations, limited to 10).

		map BFN	CG	NS
	c ≡ 1 NTS	36.4(8) 40(10) 38.7(3) 36.3(8) 40.6(10) 38.6(3)
	TS1	46.3(4) 50.9(8) 51.9(3)
	TS2	39.1(8) 42.3(9) 41.5(4)