Maxime Bertoux
email: maxime.bertoux@cantab.netorcid.org

Elisa De Paula

França Resende

Leonardo Cruz De Souza

Klüver & Bucy Syndrome: An investigation of social and affective cognition

Keywords: Klüver-Bucy syndrome, social cognition, decision-making, reversal learning

Introduction

Klüver-Bucy syndrome (KBS) is a neurobehavioural syndrome that has firstly been described in rhesus monkeys after bilateral resection of anterior and medial temporal regions, including the amygdala [START_REF] Hayman | Klüver-Bucy syndrome after bilateral selective damage of amygdala and its cortical connections[END_REF][START_REF] Marlowe | Complete Klüver-Bucy syndrome in man[END_REF]. It has rarely been observed in humans and involves visual agnosia, hypermetamorphosis (an excessive attention to visual stimuli with a tendency to touch every perceived stimuli), oral tendency (a strong urge to bring objects to the mouth), placidity (a diminution or loss of anger and fear responses, blunted affect and obedience), hyper-sexuality and changes in dietary habits [START_REF] Kluver | Psychic blindness and other symptoms following bilateral temporal lobectomy in rhesus monkeys[END_REF]. The syndrome can be either complete or incomplete and can be a chronic or a transient condition. In human, fragments of this syndrome have been observed due to a variety of different conditions such as following temporal lobes resection [START_REF] Anson | Post-ictal Klüver-Bucy syndrome after temporal lobectomy[END_REF][START_REF] Ghika-Schmid | Klüver-Bucy syndrome after left anterior temporal resection[END_REF][START_REF] Obrador | Temporal lobotomy[END_REF][START_REF] Terzian | Syndrome of Klüver and Bucy Reproduced in Man by Bilateral Removal of Temporal Lobes[END_REF], systemic lupus erythematosus [START_REF] Lin | Kluver-Bucy syndrome in one case with systemic lupus erythematosus[END_REF][START_REF] Oliveira | Kluver-Bucy syndrome in systemic lupus erythematosus[END_REF], intoxication [START_REF] Sandson | Kluver-Bucy syndrome associated with delayed post-anoxic leucoencephalopathy following carbon monoxide poisoning[END_REF], stroke [START_REF] Pitt | Kluver-Bucy syndrome following heat stroke in a 12-year-old girl[END_REF] or as a consequence of a traumatic brain injury [START_REF] Asensio | Klüver-Bucy syndrome as a result of minor head trauma[END_REF][START_REF] Kwiatkowski | Postraumatic klüver-Bucy syndrome--description and comparison of two clinical cases[END_REF][START_REF] Morcos | A case of acute-onset partial Kluver-Bucy syndrome in a patient with a history of traumatic brain injury[END_REF][START_REF] Olson | Klüver-Bucy syndrome as a result of minor head trauma[END_REF][START_REF] Pradhan | Kluver Bucy syndrome in young children[END_REF], but herpes encephalitis is considered to be the most frequently reported aetiology [START_REF] Jha | Kluver-Bucy syndrome -An experience with six cases[END_REF].

Although most studies emphasize the cause of the syndrome and not its cognitive profile, a few previous neuropsychological observations have been made previously in adults or children [START_REF] Cohen | Long-term neuropsychological follow-up of a child with Klüver-Bucy syndrome[END_REF][START_REF] De Tiège | Postinfectious immune-mediated encephalitis after pediatric herpes simplex encephalitis[END_REF], including rare cases presenting a complete syndrome (e.g. [START_REF] Marlowe | Complete Klüver-Bucy syndrome in man[END_REF]. These past neuropsychological observations have been important to document the general cognitive profile of KBS as well as the neural correlates of the rare symptoms composing this syndrome. However, complete neuropsychological investigations of KBS have been quite rare in the past, mainly because the syndrome is frequently associated with language impairments and could involve a lack of compliance for cognitive testing, which might stand in the way of an exhaustive cognitive assessment. Furthermore, another limit to previous investigations of cognition in KBS is the total absence of findings related to social and affective cognition. Social neurosciences have grown in interest and importance after the 90's and social cognition as a cognitive dimension has been quite neglected during the past decades in clinical medicine. As a growing number of psychiatric and neurologic conditions have since been found to be characterized by social cognitive dysfunctions [START_REF] Kennedy | The social brain in psychiatric and neurological disorders[END_REF], social and affective cognition is now increasingly regarded as a main cognitive domain that should be evaluated in clinical setting, as suggested by the last edition of the Diagnostic and Statistical Manual of Mental Disorders [START_REF] Apa | American Psychiatric Association . Diagnostic and statistical manual of mental disorders[END_REF]. Nowadays, critical functions such as theory of mind (the ability to infer what others believe, think or feel) and emotion recognition, two welldescribed processes known to carry normal social interactions, can be assessed in clinical practice with different validated tools.

The link between social behaviour, social cognition and temporal or limbic structures is also well established [START_REF] Kennedy | The social brain in psychiatric and neurological disorders[END_REF]. However, to our knowledge, no previous report of KBS has included social and affective cognitive assessments. The present observation intends to fill this gap in describing the nature and extend of the social cognitive and affective dysfunctions caused by polar frontotemporal damages in a patient with KBS.

Case history

We describe the case of ASP, a 35 years-old woman with a history of severe impairments in her social and affective abilities that started after an episode of herpes encephalitis in 1981, at the age of 9 years old. These difficulties were so important that she was never able to live an ordinary life, despite no major general cognitive deficits.

At 9 years old, she presented with a sudden headache, fever, vomiting, somnolence, seizures and cutaneous rash. Computed tomography head-scan showed bilateral temporal hippo-intensities. Lumbar puncture showed pleocitosis and elevated proteins in the cerebrospinal fluid (CSF). Electroencephalogram (EEG) showed bilateral temporal pseudo periodic complexes. Herpes viruses 7 and 16 were isolated from her throat, and she was treated with vidarabine for herpes encephalitis. She recovered gradually and awakened from the coma with akinetic mutism. At the discharge, her language was almost normal but she presented a behavioural symptomatology characterized by grasping, hyperphagia and excessive chewing.

One year after, at 10 years old, her behaviour has worsened, with hyperkinetic movements, logorrhoea, aggressiveness, agitation and childish behaviour retrieved in addition to the initial symptomatology. Her CSF and EEG were still abnormal and ASP received a second treatment with acyclovir. Afterwards, she developed hyperbulimia (she ate potatoes peelings, raw pasta, sugar cubes), hyper-sexuality (with masturbation in public) and loss of social rules (she spit on any ground frequently and used excessively obscene words frequently).

When she was 12 years old, ASP started to present behavioural stereotypies during social interactions. She was described as aggressive and also had important difficulties to maintain her attention to a particular task and a strong tendency to react to every stimulus. She had a perseverative, impulsive and exploratory behaviour. As a consequence, she ran away from her home and from the hospital many times. During these runaways, she suffered from sexual abuses. Although her behaviour deficits critically impacted her schooling and social functioning with others students, she was however able to go to an adapted school. There, she was described as being emotive but paradoxically, she couldn't express her feelings well.

During her teenage years, until 16 years old, ASP's behaviour stabilized, although it remained some aspects of what was called a "frontal syndrome", as well as occasional seizures. At this time, her cognitive profile was characterized by the presence of perseverations, stereotypies and a poor (reduced and non-spontaneous) language with maintenance difficulties in attention tests. However, her autonomy was described as being increased at this time.

At 27 years old, a standard neuropsychological assessment was described as being subnormal, with a good global efficiency as well as normal attention, praxis and memory abilities (except for spontaneous recall). Executive functioning (planning, visual conceptualisation, Luria's graphic sequence) and working memory were described as being normal as well, with the exception of verbal fluencies that were reduced, especially in its lexical (i.e. non-semantic) dimension. Similarly to all the neuropsychological examinations performed in the past, there was a clear dissociation between the observed pathological behaviour and the normal/subnormal neuropsychological performance.

At 33 years old, ASP moved to a residency for disabled people where she shared a room with another woman. Since then, she started to have some responsibilities in the residency, such as preparing some dinners, performing grocery shopping and housework for the community. She succeeded to manage most of the aspects of her daily life but still had some restriction, such as with sugar consummation (she has a strong tendency to eat it compulsively in any form or to store it in her pockets). She took silk painting and cookery courses but failed all attempts of professional reinsertions, partially due to repeated runaway during stress situations, including from the residency.

From the acute episode of herpes encephalitis at the age of 9 and her medical and psychological state at 35 years old, the symptoms of ASP evolved. Although a "complete KBS" was retrieved during her child years, amnesia and hypersexuality disappeared over time. Placidity, dietary changes, hyperorality, visual agnosia and lack of emotional response remained the observable symptoms of what is called an "incomplete KBS".

Clinical observations

At 35 years old, ASP consulted for a neurological follow-up at the Neuropsychology Centre within the neurology department of the Pitié-Salpêtrière Hospital in Paris, France. An extensive neuropsychological assessment was performed at this time. Beside a preserved general cognitive efficiency (Mini-Mental State Exam = 29/30; Mattis Dementia Rating Scale (MDRS) = 121/144), cognitive functions were normal, including executive functioning and verbal episodic-like memory. In particular, programming, planning, inhibition and flexibility abilities, visual conceptual reasoning and attentional control (including sustained attention) were normal. On this domain, her cognitive performance was stable compared to the previous one performed at 27 years old (Table 1). Similarly to previous cognitive assessments, ASP still presented naming and identification difficulties for objects and famous faces (visual agnosia), as well as strongly decreased verbal fluency abilities (which explain the decreased total score of the MDRS).

(Please insert Table 1 around here)

A social cognitive and affective assessment was also performed (Table 2). For the purpose of this study, we administered the same tests to 10 healthy controls matched on age (40 years old) and education level, except the Bubbles test and the Creative Thinking Test, for which controls data from previously published studies [START_REF] Adolphs | A mechanism for impaired fear recognition after amygdala damage[END_REF][START_REF] De Souza | Poor creativity in frontotemporal dementia: A window into the neural bases of the creative mind[END_REF] were considered.

The Iowa Gambling Task [START_REF] Bechara | Deciding advantageously before knowing the advantageous strategy[END_REF] was used to assess problem solving in the format of a decision-making task. This test is based on cards selection among 4 decks. Every deck of cards offers rewarding or punishing cards as well as, sometimes, rewarding and punishing cards (at the same time). Rewards and punishments were virtual money. As the principle of the test is not explicit and it is only told to participants that they have to maximize their gains, The reversal-learning paradigm involves the adaptation of behaviour to changes of stimulus-reward contingencies. Adapted from [START_REF] Rolls | Emotion-related learning in patients with social and emotional changes associated with frontal lobe damage[END_REF] paradigm, participants have to learn through successive trials to choose one stimulus appearing alone in the screen, which resulted in a gain, and to withhold the choice of a second stimulus to avoid a punishment. The two stimuli appear one by one in a random order on the screen and rewards/punishment are presented through visual and auditory feedback. A criterion of 9 correct responses within a sequence of 10 consecutive trials is needed to consider the rule as learned. Contingencies are then reversed, with the previously rewarding stimulus becoming punishing and the second one becoming rewarding. In this test, ASP failed to reverse the first learned rule, although she tried for 144 trials.

Theory of mind (ToM) was assessed through the reduced and modified faux pas test, a part of the mini Social cognition and Emotional Assessment -mini-SEA, [START_REF] Bertoux | Sensitivity and specificity of ventromedial prefrontal cortex tests in behavioral variant frontotemporal dementia[END_REF]. In this test, participants have to detect the presence of social faux pas in some stories and to provide answers to five questions assessing the understanding of the faux pas and its social consequences [START_REF] Stone | Frontal lobe contributions to theory of mind[END_REF]). ASP's performance was characterized by subnormal abilities to detect social faux pas. Although she did not detect the faux pas present in some stories, she was able to explain the problematic situation and its consequences in the stories in which she identified a faux pas.

Facial emotion comparison, which involved to decide whether two emotions presented visually were similar or not, was subnormal in ASP. Facial emotion recognition, which is the second test composing the mini-SEA [START_REF] Bertoux | Sensitivity and specificity of ventromedial prefrontal cortex tests in behavioral variant frontotemporal dementia[END_REF] involved the recognition of 35 facial emotions. Recognition abilities were impaired in ASP, particularly for fear, disgust, anger and sadness.

We decided to deepen our investigation of this emotion recognition deficit by using the "Bubbles" paradigm [START_REF] Gosselin | Bubbles: a technique to reveal the use of information in recognition tasks[END_REF] to measure how ASP makes use of information from specific facial features in order to identify emotion expressions. This paradigm involves the presentation of only small regions of a randomly sampled face space expressing an emotion. Participants have then to identify the emotion presented. As the other regions of the face are scrambled, the method allows to statistically identify the extent to which specific facial features (eyes, mouth, nose regions) contribute to the recognition of the emotion. This approach was previously used in autism [START_REF] Adolphs | Distinct face-processing strategies in parents of autistic children[END_REF] and following amygdala damage [START_REF] Adolphs | A mechanism for impaired fear recognition after amygdala damage[END_REF]. In ASP, this approach allowed us to identify a massive impairment of facial fear recognition and a strong reduction of the visual exploration of the eye-region of the face.

Finally, we used the Torrance Test of Creative Thinking [START_REF] Torrance | Torrance Tests of Creative Thinking[END_REF]

Discussion

This case report is the first attempt to document and characterize the social and affective cognition in KBS. The clinical observation of ASP, a 35 years old woman with an incomplete KBS, reveals that following amygdala, temporal pole and orbitofrontal damages, the patient presented specific social and affective cognition deficits. In more details, she was unable to reverse a previously reinforced rule, suggesting adaptation difficulties in relation to stimulus-reward association changes.

She was also strongly impaired in the recognition of facial emotion expressions, particularly for the recognition of fear. ASP also had difficulties in detecting social faux pas in stories depicting social interactions. Finally, she had reduced originality, fluency and flexibility in a context involving creative thinking.

Although the link between temporal or limbic cerebral lesions (causing the symptomatology of KBS) and dysfunctions of social dimension of behaviour and cognition are now well documented [START_REF] Rankin | Structural anatomy of empathy in neurodegenerative disease[END_REF][START_REF] Rankin | Patterns of cognitive and emotional empathy in frontotemporal lobar degeneration[END_REF][START_REF] Davis | The amygdala: vigilence and emotion[END_REF], social and affective cognition has never been explored in KBS before. Considering ASP's cognitive profile, we believe that social cognition dysfunctions in KBS have been underestimated in the past and could be a core feature of KBS. The specific impairments described through the social and affective cognitive examination that we performed echoed to ASP history and daily life difficulties across her lifespan. In particular, the impossibility to reverse a previously rewarding behaviour that becomes punishing despite good conceptual, inhibition and flexibility abilities could explain real life difficulties to cope with contexts becoming problematic or negative after having been positive previously.

In addition, ASP's severe difficulty to recognize negative facial emotions was striking. Positive emotions, such as surprise and happiness, as well as neutral expressions, were recognized within the normal range, but the patient struggled to recognize anger, disgust, sadness and fear. Therefore, we deepened the investigation of these impairments and observed a marked deficit in processing the eye regions of the face at the Bubbles test. The facial eye-region carries important information helping to identify emotion expressions such as anger or fear [START_REF] Kestenbaum | Neural and behavioral correlates of emotion recognition in children and adults[END_REF]. This deficit, leading to emotion recognition difficulties, could have played a critical role in ASP social difficulties. Significant evidences of a link between amygdala and negative emotion recognition have been shown before [START_REF] Davis | The amygdala: vigilence and emotion[END_REF], however, for ASP, whether lesions of the amygdala caused by the encephalitis are responsible for such deficits in processing the eye-region of the face remains an open question. According to [START_REF] Adolphs | A mechanism for impaired fear recognition after amygdala damage[END_REF], the amygdala is a critical region allowing the visual system to look for and fixate the attention on other people's eyes in order to make use of such information to identify fear expression.

Our results echoed those from their patient, SM, who presented this exact pattern of fear recognition deficit due to an inability to use information carried by the eye regions of the face. However, in contrary to SM, ASP was also impaired in the recognition of anger, sadness and disgust. Although amygdala lesion could be responsible for these impairments [START_REF] Davis | The amygdala: vigilence and emotion[END_REF], difficulties in recognizing anger were also previously associated with the integrity of orbitofrontal and anterior cingulate areas and impairment in sadness recognition was related to inferior and middle temporal regions [START_REF] Blair | Dissociable neural responses to facial expressions of sadness and anger[END_REF]. Furthermore, disgust identification is classically associated with the anterior insula [START_REF] Woolley | Impaired Recognition and Regulation of Disgust Is Associated with Distinct but Partially Overlapping Patterns of Decreased Gray Matter Volume in the Ventroanterior Insula[END_REF]. All these mentioned regions, i.e., the amygdala, orbitofrontal, inferior temporal and insula were profoundly damaged in ASP by herpes encephalitis at the age of 9 years old, which prevent any attempt to map these cognitive deficits to more specific brain lesions. In addition, although it can be assumed that she acquired the skills allowing emotion recognition before the encephalitis, given the relative consensus regarding the acquisition of such abilities at 7 years old [START_REF] Durand | The development of facial emotion recognition: the role of configural information[END_REF][START_REF] Vicari | Recognition of facial expressions of emotions in school-age children: the intersection of perceptual and semantic categories[END_REF], we cannot definitely rule out the possible developmental difficulties that ASP may have had in addition to the acquired difficulties following her brain lesions. Indeed, past evidences also showed that facial emotion recognition skills continue to improve until the adulthood [START_REF] Herba | Annotation: Development of facial expression recognition from childhood to adolescence: behavioural and neurological perspectives[END_REF].

Similarly, developmental difficulties may also explain ASP's subnormal performance in the faux pas test, an assessment of theory of mind abilities that involves the understanding of social context or rules. In contrary to patients with marked difficulties to infer what other peoples think, believe or feel, ASP showed difficulties in detecting the faux pas but had no difficulties to explain them through the test when they were detected. Therefore, we concluded that her inference skills were normal, raising the possibility that other variables could have impacted her performance at this test. In particular, we believe that social norms processing may be disturbed in ASP. Amygdala, temporal lobe, insula and medial prefrontal cortex are assumed to be critical for social norms acquisition and enforcement [START_REF] Buckholtz | The roots of modern justice: cognitive and neural foundations of social norms and their enforcement[END_REF][START_REF] Gu | Necessary, yet dissociable contributions of the insular and ventromedial prefrontal cortices to norm adaptation: computational and lesion evidence in humans[END_REF]. These neural dysfunctions could explain ASP difficulties to detect the faux pas, which could be considered as social norms violation. In addition, considering that the herpes encephalitis occurred at an early age in ASP, where a child is able to respond to third-party norm violations [START_REF] Vaish | Three-year-old children intervene in third-party moral transgressions[END_REF] but is still in the process of integrating social norms and automatisms [START_REF] Blackmore | The social brain in adolescence[END_REF], we think this developmental disruption could have impacted ASP ability to analyse and explain social faux pas and to adapt herself to complex social contexts, despite good mentalizing abilities.

We also found that ASP had impaired creative thinking, in both visual and verbal domains. This impairment was evidenced by the TTCT, which was designed to assess the key dimensions of creativity. This test has been employed in both clinical and research contexts and provides scores of fluency (the number of productions), flexibility (the ability to provide different categories of productions), originality (the ability to create rare productions) and elaboration (the ability to enrich productions with details); all these creative domains being affected in ASP. It has been demonstrated that brain diseases can impair creative production [START_REF] Schott | Pictures as a neurological tool: lessons from enhanced and emergent artistry in brain disease[END_REF][START_REF] De Souza | Frontal lobe neurology and the creative mind[END_REF]. More specifically, reduced creative performance has been related to prefrontal damages [START_REF] De Souza | Poor creativity in frontotemporal dementia: A window into the neural bases of the creative mind[END_REF]. From a cognitive point of view, creativity may be defined as "the ability to produce work that is both original (new, unusual, unexpected) and valuable (useful, adaptive, appropriate)" [START_REF] Dietrich | The cognitive neuroscience of creativity[END_REF][START_REF] Sternberg | The concept of creativity: Prospects and paradigms[END_REF]. In line with this definition, it is considered that creative thinking relies on different cognitive abilities, such as flexibility, abstract thinking, planning and attention [START_REF] De Souza | Poor creativity in frontotemporal dementia: A window into the neural bases of the creative mind[END_REF][START_REF] Gonen-Yaacovi | Rostral and caudal prefrontal contribution to creativity: a meta-analysis of functional imaging data[END_REF]. Besides these "classical" functions, it is highly probable that affective cognition plays an important role in creative thinking and production. Amygdala and orbitofrontal cortex in particular have been pointed out as key structures carrying the emotional and motivational dimension of creativity. In addition, the medial prefrontal cortex and temporal pole, key regions in theory of mind processing, have also been shown to be involved in the creative-ideation processes of creativity (Bashwiner et al., 2016). The present report reinforces the concept of creativity as a complex and sophisticated set of multidomains neurocognitive abilities, depending on the integrity of prefrontal and temporal regions and requiring the efficient integration of different cognitive representations, including affective ones.

Finally, this case report also highlights two important dissociations that are relevant for clinicians and the field of neuropsychology. First is the dissociation between ASP's daily life difficulties and her normal/subnormal "classic" neuropsychological profile (with the exception of visual agnosia). Indeed, despite the close support of her family, ASP's life was and is still marked by difficulties to adapt herself to her social environment. However, these important social adaptation difficultiesstrongly impacting her daily activitieswere not reflected at all in her standard neuropsychological assessment. For example, given ASP history, one could have expected inhibition impairment and major executive dysfunctions, but her profile to all executive measures was normal. The second dissociation that should be highlighted is between the classic neuropsychological performance and the social and affective cognitive scores. ASP indeed presented a clear dissociation between preserved executive, memory or attention functioning and impaired social and affective cognition. Taken together, these dissociations point out to an independence between the social dimensions of behaviour/cognition and the more "classic" neuropsychological assessment.

Social interactions being critical in almost every aspects of Human life, these two dissociations cast doubt about the ecological validity of the classic neuropsychological assessment. Indeed, most of our daily life activities are motivated and/or modulated by social goals and contexts, however, our current conception of mental functioning assessment historically exclude its social and affective dimensions, although it could be considered as the most important one. On the one hand, the current neuropsychological assessment does not allow evaluating to what extend a cognitive dysfunction impacts the social and affective life; and on the other hand, it generally does now allow to assess whether the more specifically social processes are preserved or not.

Based on this observation, together with others from different conditions (e.g. [START_REF] Bertoux | In two minds: executive functioning versus theory of mind in behavioural variant frontotemporal dementia[END_REF], we believe there is a need for new, ecologically relevant neuropsychological tools that could integrate the missing social dimension in the assessment of cognitive functions such as memory, decision-making, etc., in order to improve the ecological validity of the available clinical tools. These new tools should overcome the limitations of currently available social-cognitive tests. They should less rely on language or executive functions, provide purest measure of specific processes and be highly sensitive but also specific to their dysfunctions. In addition, they should present good psychometric values of validity and reliability, which are not always available for tests issued from experimental studies. This case also highlights the need to cast light on social norms acquisition and identification processes as currently, no test exists to assess this dimension of social cognition.

In addition, this dissociation between social cognition and more classical cognitive functions suggest that KBS could represent an efficient prototype to study the processes allowing social adaptation, similarly to behavioural variant frontotemporal dementia (bvFTD). However, a limit to such approach is that group studies on KBS would be difficult to conduct due to the rarity of the syndrome. This is why we consider that the case of ASP has both clinical and theoretical relevancy to the field, although we acknowledge that this single case approach implies some methodological limitations that cannot be overcomed. In this report, limitations mostly concern the variation of some neuropsychological scores between the two last assessments, for which we cannot provide any definitive interpretation, as well as the inclusion of a small (<15) sample of control participants, which could have hindered our interpretations.

Taken together, this first observation of social cognition in KBS shows that this syndrome can manifest as a social and affective cognitive disorder, which critically impact behaviour in turn. Whether complete or incomplete KBS could similarly impact social cognition remains a question that need to be answered by future studies. This report also provides critical input on the role of polar temporal, orbitofrontal and amygdala in social behaviour and adaptation. Finally, it does show that currently, the neuropsychological tools that are classically used for clinical examinations in neurology and psychiatry does not capture the social and affective dimensions of cognition and behaviour. New tests are then strongly needed to affine our measure of social cognition, as well as to transform the current neuropsychological assessment into a more ecological evaluation, closer to daily life activities and difficulties.

  participants have to understand that specific decks offer a positive ratio of rewards/punishments (C & D) although others offer the inverse ratio (A & B). They thus have to choose adequately in order to win money. In this test, although ASP's overall ratio of choices ((C+D)-(A+B)=-6) indicates that she picked more cards in disadvantageous decks (i.e. C & D) than in advantageous decks of cards, ASP showed a general tendency to switch from disadvantageous decks to advantageous decks over the course of the test, which is the expected normal performance. Alternative measures show that under condition of ambiguity (i.e. in the first 40 trials), ASP has a negative performance ((C+D)-(A+B)=-8) but in later trials, under known risk, she has a positive performance ((C+D)-(A+B)=4). This negative performance in the earlier trials should also be put in perspective with ASP's reversal difficulties, which was observed in the following test.

  to assess originality, verbal/visual fluency and flexibility in a creative context. This test showed decreased originality, fluency and flexibility for both verbal and visual modalities. (Please insert Table 2 around here) Brain scintigraphy showed an absence of cerebral perfusion in the right temporal pole and bilateral orbitofrontal cortices as well as a severe hypoperfusion in medial, ventral and cingulate prefrontal cortices (Figure 1). Brain MRI showed right greater than left grey matter atrophy in the temporal lobes, especially in the right anterior temporal pole, associated with an atrophy of frontal ventral anterior regions and a sequelae cavity in the right hippocampus. (Please insert Figure 1 around here)

Figure 1

 1 Figure 1 Title and legend Figure 1 -Brain scintigraphy (single-photon emission computed tomography) of ASP showing remarkable hypoperfusion in the temporal regions, with no flux detected in the right anterior temporal lobe and orbitrofrontal cortex. From top to bottom: coronal, sagittal and transverse planes.

Table 1 -

 1 ASP's profile of performance on standard cognitive assessment at 27 and 35 years old. * indicates a decrease compared to normative data for her age and education level. N.A. : Non

	Standard Cognitive Tests	27y old	35y old
	Mini Mental State Examination	30/30	29/30
	Frontal score	54/60*	52/60*
	Wisconsin Card Sorting Task Revised	20/20	20/20
	Criteria	6	6
	Errors	2	0
	Perseverations	1	0
	Verbal Fluency	4/10*	3/10*
	Lexical fluency (1min)	3	2
	Semantic fluency (1min)	10	5
	Graphic Sequences	10/10	10/10
	Environmental dependency assessment	20/20	19/20
	Verbal Similarities (WAIS)	9/19	9/19
	Tower of London
	Time score	32	32
	Score	10	10
	Free and Cued Selective Reminding Test
	Encoding	16/16	16/16
	Free recall	20/48	32/48
	Total recall	48/48	48/48
	Delayed total recall	16/16	16/16
	Rey complex figure
	Rey complex figure copy	36/36	36/36
	Rey complex figure 3m recall	22/36	15/36
	Picture identification	N.A.	28/35*
	Picture naming	N.A.	24/35*
	Famous faces
	Identification	N.A.	8/35*
	Naming	N.A.	7/35*
	Knowledge	N.A.	32/35
	available

Table 2 -

 2 ASP's profile of performance on social and affective cognitive assessment and controls' mean and (SD). * Norms of control women of comparable age (37.5 years old) from[START_REF] Adolphs | A mechanism for impaired fear recognition after amygdala damage[END_REF]. ** Norms from de Souza et al., 2010. N.A. : Non available, see[START_REF] Adolphs | A mechanism for impaired fear recognition after amygdala damage[END_REF]

Acknowledgments

We are indebted to Dr. Nathalie Guyader for the programming and interpretation of the Bubbles test and to ASP's mother. We also thank Pr. Jean Luc Truelle. MB is funded by DistALZ.

Disclosure statement

The authors report no conflicts of interest.


E Simon

M Piot

Erick Fellah

C Ogam

Depollier

R Roncen
email: remi.roncen@onera.fr

Z E A Fellah

F Simon

E Piot

M Fellah

E Ogam

C Depollier

Bayesian Inference for the Ultrasonic Characterization of Rigid Porous Materials Using Reflected Waves by the First Interface

HAL is

Introduction

Porous materials find widespread applications, ranging from hydrology [START_REF] Vrugt | Treatment of input uncertainty in hydrologic modeling: Doing hydrology backward with markov chain monte carlo simulation[END_REF] to medicine (cancellous bone [START_REF] Haire | Biot theory: a review of its application to ultrasound propagation through cancellous bone[END_REF]) and sound absorption [START_REF] Allard | Propagation of Sound in Porous Media: Modelling Sound Absorbing Materials 2e[END_REF]. The estimation of their micro-structural properties, such as porosity, tortuosity and characteristic lengths, plays a crucial role in the characterization of their physical behavior. One common goal of these areas of physics is the non intrusive, robust and cheap inference of all the required intrinsic properties of porous media that are necessary to describe their physical behavior. Due to the inherent measurement system noise, the quantitative assessment of the uncertainty of the identified parameters is equally important.

The most widespread model to characterize wave propagation inside poroelastic media is Biot's model [START_REF] Biot | Theory of propagation of elastic waves in a fluid-saturated porous solid. i. low-frequency range[END_REF][START_REF] Biot | Theory of propagation of elastic waves in a fluid-saturated porous solid. ii. higher frequency range[END_REF][START_REF] Biot | Mechanics of deformation and acoustic propagation in porous media[END_REF][START_REF] Biot | Generalized theory of acoustic propagation in porous dissipative media[END_REF], which accounts for the presence of three different waves inside the poroelastic material: two compressional waves and one shear wave. In some applications, one is mostly interested in acoustic propagation in rigid porous media where the solid phase is immobile and only the airborne compressional wave propagates; this configuration is the focus of the present study. The equivalent fluid model of Johnson-Champoux-Allard (JCA [START_REF] Johnson | Theory of dynamic permeability and tortuosity in fluid-saturated porous media[END_REF][START_REF] Champoux | Dynamic tortuosity and bulk modulus in air-saturated porous media[END_REF]) is widely used for sound propagation in rigid porous media and requires the knowledge of 5 intrinsic material properties (porosity, tortuosity, static viscous permeability, viscous and thermal characteristic lengths). The JCA model is well adapted to describe the acoustical behavior of porous materials with rigid skeletons having arbitrary pore shapes. The model is obtained by the connection between asymptotic developments in the low and high frequency regions. In the high frequency regime, where the viscous boundary layer is much smaller than the pore radius, the porosity, tortuosity and characteristic lengths are sufficient to define the porous wave behavior. These are the parameters inferred in the present study.

Different inverse problems for parameter identifications have been attempted in the past regarding air-saturated porous media, mostly in the audio frequency regime [START_REF] Zieliński | Normalized inverse characterization of sound absorbing rigid porous media[END_REF][START_REF] Atalla | Inverse acoustical characterization of open cell porous media using impedance tube measurements[END_REF][START_REF] Cobo | A comparison of impedance models for the inverse estimation of the non-acoustical parameters of granular absorbers[END_REF][START_REF] Hentati | Multi-levels inverse identification of physical parameters of porous materials[END_REF][START_REF] Vanhuyse | Global optimisation methods for poroelastic material characterisation using a clamped sample in a kundt tube setup[END_REF]. The largest unaddressed issues of these studies are the modeling deficiency at these frequencies and the scarcity of uncertainty considerations. Being in the high frequency domain allows the use of models that are exactly defined in the asymptotic limit, overcoming the modeling limitations of audiofrequency based inverse methods. In earlier studies, Fellah et al. [START_REF] Fellah | Measuring the porosity and the tortuosity of porous materials via reflected waves at oblique incidence[END_REF][START_REF] Fellah | Ultrasonic measurement of the porosity and tortuosity of air-saturated random packings of beads[END_REF] formulated the reflection coefficient at the first interface of a rigid porous material in the high frequency limit. Multiple obliquely incident waves in the high frequency domain were used to obtain a direct relationship between the reflected waves and two parameters of interest, namely, the porosity and tortuosity. To extend the identification to other parameters, Fellah et al. [START_REF] Fellah | Ultrasonic characterization of porous absorbing materials: Inverse problem[END_REF] then introduced an ultrasonic characterization technique to identify the porosity, tortuosity and the two characteristic lengths, by using both reflected and transmitted waves and an inverse method based on time-domain signals, and by assuming a given ratio between the viscous and thermal characteristic lengths. Groby et al. [START_REF] Groby | Analytical method for the ultrasonic characterization of homogeneous rigid porous materials from transmitted and reflected coefficients[END_REF] then developed an analytical method using both the reflected and transmitted signals in the ultrasonic regime to extract the values of the porosity, tortuosity and the viscous and thermal lengths, in the frequency domain; it was argued that for highly resistive materials, the method might yield wrong results. Later, Fellah et al. [START_REF] Fellah | Simultaneous determination of porosity, tortuosity, viscous and thermal characteristic lengths of rigid porous materials[END_REF] focused on low resistivity materials and solved the inverse scattering problem in the ultrasonic regime for the transmission mode only, to identify the porosity, tortuosity and characteristic lengths using time domain signals. Using reflected waves from the first interface is particularly suited to characterizing highly resistive materials, because they have large reflection coefficients. However, the previous studies have not been able to yield the identification of the characteristic lengths using only reflected waves from the first interface. Only the porosity and tortuosity have been measured in this configuration [START_REF] Fellah | Measuring the porosity and the tortuosity of porous materials via reflected waves at oblique incidence[END_REF][START_REF] Fellah | Ultrasonic measurement of the porosity and tortuosity of air-saturated random packings of beads[END_REF].

The primary aim of the present study is to extend the previous results of [START_REF] Fellah | Measuring the porosity and the tortuosity of porous materials via reflected waves at oblique incidence[END_REF][START_REF] Fellah | Ultrasonic measurement of the porosity and tortuosity of air-saturated random packings of beads[END_REF] to identify the characteristic lengths of porous materials, using only the knowledge of obliquely reflected waves at the first material interface, in the high-frequency domain. In previous work [START_REF] Fellah | Measuring the porosity and the tortuosity of porous materials via reflected waves at oblique incidence[END_REF][START_REF] Fellah | Ultrasonic measurement of the porosity and tortuosity of air-saturated random packings of beads[END_REF], the dispersive contribution to the reflection operator has been neglected. In this article, we take into account the dispersive part of the reflection coefficient (fractional derivative of order -1/2 in the time domain and 1/ √ jω term in the frequency domain). The influence of the characteristic lengths on the reflection operator is taken into account, which in turn makes their identification possible when using only reflected waves.

The secondary objective is to apply to the ultrasonic time-domain identification process a statistical Bayesian inference method. This enables the systematical encompassing of measurement uncertainties and prior information into the identification of porous media properties, as was done previously in [START_REF] Niskanen | Deterministic and statistical characterization of rigid frame porous materials from impedance tube measurements[END_REF][START_REF] Chazot | Acoustical and mechanical characterization of poroelastic materials using a bayesian approach[END_REF] in the frequency domain at audio-frequencies when using impedance tube measurements as observational inputs.

The present article is organized as follow: the acoustical model is presented in Section II, where the high frequency limit of the JCA model is detailed and where the reflection operator formulation is given. The reflected wave is then expressed in the time domain. A critical incident angle is presented, relating the tortuosity to the characteristic length ratio. The inference methodology is described in Section III, where Bayes' theorem is recalled and the numerical exploration of the parameter space is detailed. Results for the inference of three widely different porous materials are presented in Section IV. Concluding remarks are presented in Section V.

Acoustical Model

In the study of porous media, two different cases are typically considered. When the fluid and solid phases have coupled contribution to the acoustic attenuation in the material, the Biot model is used to describe the wave propagation in both the solid and fluid phases, resulting in the occurrence of two compressional and one rotational waves. When the solid phase can be assumed rigid, which is often the case in air-saturated porous media, and the frame motion can be neglected, one usually resorts to using an equivalent fluid model where only the compressional wave propagation in the fluid phase of the porous material is considered. The classical properties of density and compressibility are replaced by equivalent properties in order to take into account viscous and thermal dissipation in the pores. This is the approach we adopt in this work. Let a macroscopically homogeneous rigid porous material of porosity φ be filled with a fluid of density ρ f and dynamic viscosity η. Assuming weak fluid-frame coupling and a wavelength much larger than the pore size, one can apply continuum mechanics to the fluid phase in the material and identify the whole porous medium as an equivalent fluid material of effective properties to account for viscous and thermal dissipation arising from the fluid-frame interaction. As initially proposed by Zwikker and Kosten [START_REF] Zwikker | Sound absorbing materials[END_REF], one can decouple the contribution of viscous and thermal dissipation in the pores. To do so, one defines the dynamic tortuosity α(t) that encompasses the viscous-inertial fluid/structure interactions, and the dynamic compressibility β(t) that takes into account all the thermal fluid/structure interactions. In the time domain, these factors are operators and the equations controlling the wave behavior are the Euler equation and a constitutive equation, given in the time domain by

ρ f t 0 α(t -τ ) ∂v(τ ) ∂τ dτ = -∇p, 1 K f t 0 β(t -τ ) ∂p(τ ) ∂τ dτ = -∇ • v. (1)
In the previous equations, v is the macroscopic velocity and p the macroscopic pressure, obtained by averaging the microscopic velocity and pressure fields over a representative elementary volume; ρ f is the fluid density and K f is the fluid bulk modulus.

Of primary interest to this study is to determine parameters defined in the high frequency limit, namely the porosity φ, the tortuosity α ∞ , the viscous characteristic length Λ (unit m, [START_REF] Johnson | Theory of dynamic permeability and tortuosity in fluid-saturated porous media[END_REF]) and the thermal characteristic length Λ (unit m, [START_REF] Champoux | Dynamic tortuosity and bulk modulus in air-saturated porous media[END_REF]). The theoretical expressions of α(t) and β(t) in the time domain, in the high frequency limit, are given by Johnson et al. [START_REF] Johnson | Theory of dynamic permeability and tortuosity in fluid-saturated porous media[END_REF] and Champoux & Allard [START_REF] Champoux | Dynamic tortuosity and bulk modulus in air-saturated porous media[END_REF], and are written in time domain [START_REF] Fellah | Transient acoustic wave propagation in rigid porous media: A time-domain approach[END_REF] as

α(t) = α ∞ δ(t) + 2 Λ η πρ f H(t) √ t , β(t) = δ(t) + 2(γ -1) Λ η π Pr ρ f H(t) √ t , (2)
where Pr is the Prandtl number, η is the fluid dynamic viscosity, γ = 1.4 is the fluid heat capacity ratio, δ(t) is the Dirac delta function (which represents the instantaneous, non-dispersive response of the material), and H(t) is the Heaviside function (H(t < 0) = 0, H(t ≥ 0) = 1), used for causality reasons. The time convolution of t -1/2 is interpreted as a fractional derivative operator of order ν = 1/2, following the definition in Samko [START_REF] Samko | Fractional integrals and derivatives: theory and applications[END_REF]:

D ν [g(t)] = 1 Γ(-ν) t 0 (t -u) -ν-1 g(u)du, (3)
where Γ(x) is the gamma function. These terms in α(t) and β(t) represent the dispersive kernel, expressing the dispersive nature of the viscous and thermal effects in the pores at high frequencies. For a more in-depth analysis of fractional derivatives in acoustical modeling, the reader is referred to [START_REF] Monteghetti | Design of broadband time-domain impedance boundary conditions using the oscillatory-diffusive representation of acoustical models[END_REF] and the references therein. Combining Eqs. 1 with Eqs. 2 yields

ρ f α ∞ ∂v ∂t + 2 ρ f α ∞ Λ η πρ f t 0 ∂v/∂τ √ t -τ dτ = -∇p, (4)
1 K f ∂p ∂t + 2 γ -1 K f Λ η π Pr ρ f t 0 ∂p/∂τ √ t -τ dτ = -∇ • v. (5)
The dispersive nature of porous media is taken into account via a memory effect, meaning that part of the material's response is not instantaneous and is associated with a time delay.

Reflection scattering operator

The direct scattering problem requires finding the pressure field reflected by the first interface of the porous material. Let us consider here a rigid-frame, air-filled isotropic porous medium inside a semi-infinite region 0 ≤ x < +∞, as depicted in Fig. 1. A sound pulse impinges on the medium from the left with an incident angle θ i . We are mostly interested here in determining the reflected wave, or, equivalently, the reflection scattering operator R(t) that describes the relationship between incident (p i (x, t)) and reflected (p r (x, t)) plane waves, as

p r (x, t) = t 0 R(τ )p i (x, t -τ ) dτ. ( 6
)
To derive the required operator, continuity of pressure and velocity on the first interface of the material is assumed, resulting in the interface conditions [START_REF] Gurevich | Interface conditions for biot's equations of poroelasticity[END_REF] p(0 -, t) = p(0

+ , t), v(0 -, t) = φv(0 + , t), (7)
where a porosity of 1 has been taken for the ambient fluid (x < 0), and where thesuperscript stands for the left domain, and the + superscript stands for the right domain. In order to manipulate the following equations more easily , we make use of the Laplace transform method, assuming an idle medium as initial condition. We note P (x, s) (resp. V (x, s)) the Laplace transform of p(x, t) (resp. v(x, t)) defined by

P (x, s) = L [p(x, t)] = ∞ 0 exp (-st) p(x, t)dt, (8)
and the inverse Laplace transform by p(x, t) = L -1 [P (x, s)]. For the Laplace coefficients, we use the same symbols as the time operators, but remove the tilda. Using the relations L [δ(t)] = 1 and L H(t) √ πt = 1 s , one can transform the expressions of α(t) and β(t) in the Laplace domain into

α(s) = α ∞ 1 + 2 Λ η sρ f 1/2 β(s) = 1 + 2(γ -1) Λ η s Pr ρ f 1/2 . ( 9
)
The expression of the pressure field in the region at the left of the material is the sum of the incident and reflected fields

p 1 (x, t) =p i t - x c f cos θ i - y c f sin θ i + p r t + x c f cos θ i - y c f sin θ i , (10)
where

c f = K f /ρ f ≈ 340 m.s -1
is the sound speed in air. In the Laplace domain, the fields on the left side of the material become, for

x < 0 [27]                        P 1 (x, s) = e -s x c f cos θ i +R(s, θ i )e s x c f cos θ i e -s y c f sin θ i V 1 (x, s) = cos θ i Z f e -s x c f cos θ i -R(s, θ i )e s x c f cos θ i e -s y c f sin θ i (11)
where R(s, θ i ) is the Laplace transform of the reflection kernel of interest,

Z f = ρ f c f ≈ 418 kg.m -2 .s -1
is the characteristic impedance of the material, and V 1 (x, s) is the x component of the velocity. Inside the material, the fields are expressed, for 0 ≤ x < +∞, as

P 2 (x, s) = A(s)e -s x cm cos θ m e -s y cm sin θ m V 2 (x, s) = cos θ 1 Zm A(s)e -s x cm cos θ m e -s y cm sin θ m ( 12
)
where A is an unknown parameters, θ m is the refracted wave angle in the porous material, relative to its normal, and V 2 (x, s) is the x component of the velocity. The phase velocity in the equivalent fluid material is defined as c m = c f / α(s)β(s), while the characteristic impedance of the material is given by

Z m = Z f α(s)
β(s) . Using the boundary conditions of Eq. 7, one obtains the equation system

1 + R(s, θ i ) = A(s), 1 -R(s, θ i ) = φ Z f cos θ m Z m cos θ i A(s). (13)
Solving the system 13 yields the reflection coefficient R(s, θ i ) as

R(s, θ i ) = 1 -E(s) 1 + E(s) , E(s) = cos θ m cos θ i φ β(s) α(s) , cos θ m = 1 - sin 2 θ i α(s)β(s) , (14)
and where the Snell-Descartes law has been used. Using the inverse Laplace transform to go back into the time domain, the reflection coefficient at the first interface described in Eq. 14 acts as an operator whose expression is:

R(t, θ i ) = 1 -E 0 1 + E 0 δ(t) + 2E 0 η/ρ f α ∞ -sin 2 θ i (1 + E 0 ) 2 × α ∞ -2 sin 2 θ i Λ - α ∞ (γ -1) Λ √ Pr H(t) √ πt , (15)
with

E 0 = φ α ∞ -sin 2 θ i α ∞ cos θ i . ( 16
)
Once the reflection operator is known, one can convolve the incident signal with it in order to yield the numerical reflected signal in the time domain, for a given incident angle θ i , which can in turn be compared to the experimental reflected signal, allowing one to carry on with the inverse method, as presented in Section III. Anticipating Section II.C, let us consider a porous material (P1) with certain properties (given in Table 1). An incident transient pressure wave p i impinges on the material and is partly reflected. In Figure 2, we plot the experimental incident signal, the numerically calculated reflected signal and the reflection operator used in the convolution.

Critical angle at oblique incidence

An important property of the reflection coefficient can be identified by first simplifying Eq. 15 as

R(t, θ i ) = R0 (t, θ i ) + R1 (t, θ i ), (17)
with

R0 (t, θ i ) = 1 -E 0 1 + E 0 δ(t), (18)
and

R1 (t, θ i ) = 2E 0 η/ρ f α ∞ -sin 2 θ i (1 + E 0 ) 2 × α ∞ -2 sin 2 θ i Λ - α ∞ (γ -1) Λ √ Pr H(t) √ πt , (19)
where R1 corresponds to the dispersive part of the reflection coefficient. One observes that R1 can be canceled if there exists a critical angle θ c,1 verifying

sin 2 θ c,1 = 1 2 α ∞ 1 - Λ Λ γ -1 √ Pr . (20)
Measuring such an angle can be done by detecting, for instance, a change in the sign of the imaginary part of the total reflection coefficient, which can be readily measured by first transforming the incident and reflected signals in the frequency domain via an fast Fourier transform, and calculating the ratio between the two signals. The higher the tortuosity and the lower the ratio Λ/Λ , the higher the critical angle becomes. This is the first time that this relationship has been highlighted, and its knowledge used in inverse identification methods. There also exists an angle θ c for which R(t, θ c ) = 0, as was shown initially in [START_REF] Fellah | Characterization of porous materials with a rigid frame via reflected waves[END_REF]. While its expression is time dependent, estimating its value numerically can be done quickly. This angle will not be used in the inverse problem, as it was not detected experimentally. In the same fashion, an angle θ c,0 for which R0 (t, θ c,0 ) = 0 exists and is relatively close to θ c . Again, its value was not measured experimentally for the cases at hand and thus its expression was not used further during the inference.

Illustrations of the direct problem

For completeness, let us consider 3 different numerically generated porous materials (P1, P2 and P3) and display their reflection coefficient and critical angles (properties were chosen close to the ones inferred later on). The material properties are summarized in Table 1, along with the critical angle θ c,1 for which R1 cancels out, and the critical angle θ c for which the full reflection cancels out, at 190 kHz. The spread of these two values is increased for low porosity materials that typically display a higher reflection coefficient. We now display the full reflection coefficient against the 1: Synthetic porous media: summary of their properties angle of incidence at 190 kHz for all three materials studied, in Figure 3 (left). The dispersive part of the reflection coefficient, R 1 (s, θ i ), is also displayed in Figure 3 (right). When looking at low values of incidence angles, the dispersive contribution is more important for materials of type P1 and P2 (associated with high flow resistivities). This result is in agreement with our initial comments that the dispersive correction of the reflection is greater for resistive, low characteristic length materials. However, this is surprisingly not the case anymore for high values of the incidence angle, for which R 1 (s, θ i ) takes more important values in the case of material P3 (associated with low flow resistivities). This raises the question of where (in terms of the incident angle) one should observe a given material in order to optimally determine its parameters. Our examples suggest that highly resistive materials should be observed at low angles of incidence, while low resistivity materials could yield better identification results when observed at higher incidence angles, provided that the added uncertainty stemming from grazing angles of incidence remains low. In Figure 4, we show a decomposition of the reflected waves for the three materials considered, at an angle where the dispersive nature of the material is the most marked: using Figure 3(right) yields an angle of 0°for materials P1 and P2, and an angle of 80°for material P3. This decomposition consists in the reflected wave associated with the instantaneous response of the material (convolution of the incident wave with R0 (t, θ i )) and the dispersive part of the reflection (convolution of the incident wave with R1 (t, θ i )), the sum of which yields, by linearity of the convolution operator, the full reflected wave (associated with the convolution of the incident wave with R(t, θ i )). As expected, the contribution of the dispersive nature of porous media is significant and has to be taken into account, even for low resistivity materials, when high angles of incidence are concerned. This result has not been mentioned in the literature before, to the best of the authors' knowledge, and opens the way for new characterization methods, focusing the observation on regions of interest depending on prior knowledge on the porous material nature and pore scales (which is often a readily available piece of information). Since R1 (t, θ i ) is a significant contribution to the reflection and depends on the parameters φ, α ∞ , Λ, Λ to be identified, we can now proceed with the inference method.

Name φ α ∞ Λ(µm) Λ (µm) θ c,1 (°) θ c (°) P1 0.

Statistical inference

In the context of inverse problems for parameter estimation, statistical inference in the Bayesian framework consists in recasting parameters of interest as random variables associated with probability densities encompassing established information, or new information collected through measurements. The solution of such an inverse problem is no longer a single value for the estimated parameters of interest, but their posterior probability density [START_REF] Smith | Uncertainty quantification: theory, implementation, and applications[END_REF]Chap. 8]. A good review on inverse problem theory can be found in [START_REF] Tarantola | Inverse problem theory and methods for model parameter estimation[END_REF].

Bayes's theorem

Statistical inverse problems rely on information from both the observed experimental data d exp and the prior knowledge on the parameters m. In the present article, d exp encompasses all the measured reflected signals in the time domain, and is thus represented by a matrix of size N d × K, where N d is the number of observed discrete time steps and K is the number of different incident angles considered. The parameters of interest are m = (φ, α ∞ , Λ, Λ ). These two states of information are combined to express our degree of knowledge about the true values of m, improved by the information contained in the observation d exp . We define a marginal posterior probability density π(m|d exp ) for the model parameters m, given a set of observable parameters d exp . Using Bayes' theorem, this becomes

π(m|d exp ) = π(d exp |m) π(m) π(d exp ) , (21)
where L(m) ≡ π(d exp |m) is the likelihood function, π(m) is the prior density, and π(d exp ) is the Bayesian evidence. Prior and posterior densities represent the state of knowledge, or degree of belief we have on the parameters, before and after having observed a certain piece of data d exp , respectively. The posterior probability π(m|d exp ) reflects all the information inferred on m, conditional on the observation d exp , and is the quantity to be found. The evidence π(d exp ) represents the probability of the experimental data being generated by the given parameters and models; the evidence is generally a term difficult to evaluate numerically: its calculation can be conveniently circumvented by the MCMC method detailed in Subsection III.D. The aforementioned terms are detailed in the following.

Likelihood

The information on the physical correlations between m and d exp as predicted by a physical model, here a convolution between incident signal and reflection operator, is represented by a deterministic forward operator G, and a noise accounting for our uncertainty relative to the experiment, at each observed time step, and for all angles. Thus, is of size the size of d exp . For each incidence angle θ k ,

d exp k = G(m * , θ k ) + k , (22)
with m * unknown and where the components of k are assumed independent and identically distributed random variables of zero mean Gaussian density π . The forward operator is represented in the discrete-time way by noting G(m,

θ k ) = (G 1 (m, θ k ), . . . G j (m, θ k ), . . . , G N d (m, θ k ))
, where N d is the number of time-steps observed in the experimental reflected signal, and with

G j (m, θ k ) = t j 0 R(m, τ, θ k )p i (x meas , t j -τ )dτ, (23)
where the dependency on the parameters m has been made explicit in the writing of the reflection kernel. The measurement location is noted by x meas , and t j is the j th observed time step. The j th component of d exp becomes, for the k th incident angle:

d exp j,k = G j (m, θ k ) + j,k , (24)
The likelihood of an observation conducted with a given incident angle θ k takes the form

L(m, θ k ) = π (d exp k -G(m, θ k )), (25)
which reads, assuming π follows a zero mean Gaussian distribution,

L(m, θ k ) = n d j=1 1 2πσ 2 j,k exp   - d exp j,k -G j (m, θ k ) 2 2σ 2 j,k    , (26)
Parameter φ α ∞ Λ(µm) Λ /Λ Prior Min 0.2 1.0 1.0 1.0 Prior Max 0.99 3.0 1000 10

Table 2: Prior boundaries where σ 2 j,k denotes the noise variance associated to a time step and an incident angle, and is either determined experimentally or set a priori, for each signal. The larger the value of σ 2 j,k , the larger the estimated uncertainty on the observation. The value of the standard deviation σ j,k is taken in this work to 2% of the absolute value of the considered experimental reflected signal, to encompass our uncertainty in the acquisition chain and the experimental conditions (temperature, pressure, etc...). For numerical reasons, the logarithm of the likelihood is used instead of Eq. 26. The total log-likelihood is then obtained as

log (L(m)) = K k=1 log (L(m, θ k )) , ( 27
)
where K is the number of incident angles used. Similar to classical inverse problems where a cost function has to be minimized in order to retrieve the parameters of interest, one can view the likelihood as the probabilistic equivalent of such a quantity, effectively measuring the probability of the experience being observed, for a given set of parameters.

Prior model

The prior probability π(m) contains the information obtained on the model parameters independently of the experimental data used during the inference. It reflects the user's prior insight into the problem, generalizing the classical constraints used during deterministic inversion, assigning weights to the more probable values. When no prior information is available, a wide-range, non-informative improper homogeneous density represents well one's ignorance and merely sets boundaries for each parameter, as in classical deterministic inverse problems. In the present article, we first assume prior independence among the unknown parameters, setting

π(m) = nm i=1 π i (m i ). (28)
In the absence of observation, there is no known correlation between the parameters. As no statistical information on the parameters was accessible prior to the inference, a uniform probability π i (m i ) ∼ U(m i,min , m i,max ) is assumed for the priors, with m i,min and m i,max the boundaries, given in Table 2. To further constrain the problem, when possible, one can include a known correlation between the parameters, using the approximate knowledge of the critical angle defined in Eq. 20.

As this information is gained through the observation of the signals in the frequency domain, it can still be seen as a prior (independent) knowledge, since the inference is performed on the reflected signals in the time domain. This method is applied for the inference of material M2 in Section IV.C.

MCMC: posterior sampling

As the posterior probability density defined in Eq. 21 is only known up to a normalizing constant, one still needs to devise a technique to draw samples directly from the posterior distribution, without having to calculate the evidence, which is especially arduous in high-dimensional parameter spaces. To address this issue, a technique called Markov Chain Monte Carlo (MCMC) was exploited.

The central idea behind MCMC is to draw samples directly from the posterior distribution without having to calculate the normalizing factor, by designing a random-walk algorithm that explores the posterior density and moves preferentially towards regions of high probability densities, efficiently exploring the area of interest and creating a Markov chain whose stationary distribution is the distribution of interest [START_REF] Gilks | Markov chain Monte Carlo in practice[END_REF]. This is done with a Metropolis-Hasting algorithm [START_REF] Metropolis | Equation of state calculations by fast computing machines[END_REF][START_REF] Hastings | Monte carlo sampling methods using markov chains and their applications[END_REF], consisting in an acceptance/rejection step that creates the transition kernel of a Markov chain converging to the posterior distribution. The MCMC strategy is coupled with a Differential Evolution algorithm [START_REF] Storn | Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces[END_REF]: multiple chains are run in parallel and learn from each other through a mutation operator, thus solving the problems of selecting a direction and a scale for the jump from the current point to the proposal point [START_REF] Ter | Genetic algorithms and markov chain monte carlo: Differential evolution markov chain makes bayesian computing easy[END_REF][START_REF] Ter | A markov chain monte carlo version of the genetic algorithm differential evolution: easy bayesian computing for real parameter spaces[END_REF]. In the present study, the latest improvements on MCMC techniques were integrated, making use of multi-try sampling from past states, snooker updates and self-adaptive randomized subspace sampling. The full details on this method, coined MT_DREAM_ZS, are given in [START_REF] Laloy | High-dimensional posterior exploration of hydrologic models using multipletry dream(zs) and high-performance computing[END_REF] and the references therein.

In the present work, 3 chains were used on which 10 5 iterations were performed, with the first 10% of each chain being discarded as burn-in to allow decorrelation with the initial samples. The chains were initialized by sampling from the prior's support. Convergence was checked a posteriori, through a Gelman-Rubin diagnostic [START_REF] Gelman | Inference from iterative simulation using multiple sequences[END_REF] and a visual check of the chain samples.

Results

In this section, three widely different materials are tested and their properties inferred. The first material, M1, consists in a random packing of spheres and is highly resistive; material M2 is a resistive foam and material M3 is a classical foam used as a sound absorbing material, of low estimated resistivity. Due to its low resistivity and the low values of incident angles, the reflected wave amplitudes of M3 are lower than for the two other materials, and it is interesting to test the limits of the present method in such a case. For the first and third materials, no critical angle θ c,1 was observed; for the second material, it was estimated. For all materials, the inference is summarized through the display of posterior probability density functions (pdfs) of all parameters, along with joint-pdfs to reveal the correlations between parameters. These were evaluated from the chain samples through a kernel density estimation. Other experimental results, obtained with reference methods developed in [START_REF] Champoux | Air-based system for the measurement of porosity[END_REF][START_REF] Brown | Evaluation of the viscous characteristic length of air-saturated porous materials from the ultrasonic dispersion curve[END_REF][START_REF] Fellah | Ultrasonic measurement of the porosity and tortuosity of air-saturated random packings of beads[END_REF][START_REF] Leclaire | Determination of the viscous and thermal characteristic lengths of plastic foams by ultrasonic measurements in helium and air[END_REF], are displayed as well. The Maximum A Posteriori (MAP) estimator was chosen as the reference probabilistic estimator in this study, since it is closest in meaning to the estimates obtained through usual deterministic methods, related to a maximum likelihood estimation. The MAP m MAP estimate is defined as

m MAP = arg max m π(m|d exp ). (29)

Experimental procedure

Experiments were performed in air with two broadband Ultran NCT202 transducers with a 190kHz center frequency and 6dB bandwidth extending from 150 to 230kHz. The associated wavelength is λ = c f /f ≈ 1.8 mm, which remains larger than the pore sizes of all the materials considered in this study. An optical goniometer was used to position the transducers. Uncertainties on the angle measurements are not considered in the present work. Pulses of 400V amplitude were provided by a 5052PR Panametrics pulser/receiver. The signals received were amplified to 90dB and filtered above 1MHz to avoid high frequency noise. Electronic interference was removed by 1000 acquisition averages. Measuring the reference incident signal was done with the help of a rigid metal plate acting as an acoustic mirror. The experimental setup is shown in Figure 5.

M1 identification: sphere packing

The waves reflected from material M1 were measured at 3 different angles of incidence (13.5°, 17°, 66.6°). Such an interval is interpreted as a percentage probability that the true value of a given set of 2 parameters falls within the credible region. These results are summarized in Table 3 where the MAP and standard deviation of each parameter are reported. Results obtained in [START_REF] Fellah | Ultrasonic measurement of the porosity and tortuosity of air-saturated random packings of beads[END_REF] using classical identification techniques on the same material are also recalled for reference. These reference measurements did not account for the dispersive contribution to the reflection operator, thus explaining the difference between obtained values. Interestingly, the porosity remains quite close in both cases. While φ, α ∞ and Λ densities have a Gaussian shape, the density of Λ /Λ is less informative, indicating the asymmetric loss of sensitivity relative to the likelihood function when Λ increases (seen by the flattening of the pdf). Note that the extreme limit imposed by the prior Λ /Λ = 10 is reached for this material, and that a larger boundary would have extended the density support. A joint-pdf of approximate elliptic shapes such as the ones between φ, α ∞ and Λ in Fig. 6 denotes a positive correlation between parameters. This indicates that an increase in the estimation of one parameter results in an increase in the correlated parameter. These correlations need to be taken with caution, as they do not necessarily denote a physical correlation between the parameters. For instance, a material with very high porosity tends to have lower tortuosity, which is opposite to what could be understood from the joint-pdf. These correlations stem from the modeling structure and the experimental observations used in the inference. Correlations between parameters are expected to be strong, due to their association in the reflection coefficient expression, in Eq. 15. For example, if one looks at the expression of E 0 (Eq. 16), which intervenes in the definition of R0 , one sees that at θ i = 0, E 0 = φ/ √ α ∞ . A simultaneous increase in both φ and α ∞ could then leave E 0 unchanged, which in turn would result is a sensibly similar likelihood function, thus explaining the positive correlations between those two parameters during Bayesian inference process. Taking into account R1 would change these correlations, but the idea remains the same. Should a more informative prior be stipulated for the porosity, the posterior marginal probability density of φ would be shifted towards the prior, and α ∞ and Λ would be shifted in the same direction (increase or decrease). In the case where no prior is used, the Bayesian inference is called objective [START_REF] Xiang | Objective bayesian analysis in acoustics[END_REF]. An objective inference allows to learn about correlations between parameters, stemming from the modeling structure or the experimental observation. Knowing these correlations, one can then anticipate the influence of setting a more informative prior on a certain parameter.

In second level Bayesian inference [START_REF] Xiang | Objective bayesian analysis in acoustics[END_REF], the prior modeling itself can enter the inference process, resulting in extra parameters in the identification, and model selection. In the present study, only the first level Bayesian inference (model calibration) was considered. [START_REF] Fellah | Ultrasonic measurement of the porosity and tortuosity of air-saturated random packings of beads[END_REF].

The signals calculated using the Markov chain samples are compared with the experimental reflected signals in Figure 7 for a particular incident angle used in the inference. The Fourier transform is also used to compare the spectral density of calculated and experimental signals for the same two angles. Signals are normalized using the maximum value of the experimental signal. The remaining small discrepancies between signals can be explained in the following two ways: the rigid skeleton hypothesis may not be adequate in the case of high amplitude waves for sphere packing materials such as this one, due to a movement between the spheres. Furthermore, the geometry of the representative pore itself is somewhat out of the scope of the JCA model, where it is assumed that no constriction is present between pores. Due to the sphere curvatures, it is possible that the model corrections of Pride [START_REF] Pride | Drag forces of porous-medium acoustics[END_REF] are necessary to represent more accurately the wave propagation in such a material. This is left for a future study.

M2 identification: high resistivity foam

The waves reflected from material M2 were measured at 4 different angles of incidence (0°, 23°, 39.7°, 49.8°). Figure 8 shows the pdfs (diagonal elements) and joint-pdfs (extra-diagonal elements) inferred on the material M2 properties. These results are summarized in Table 4 where the MAP and standard deviation of each parameter are reported. Results obtained using classical identification techniques are also recalled for reference. The small discrepancies between reference and MAP values can be attributed to a lower sensitivity of the porosity in the transmission mode at high frequencies [START_REF] Fellah | Direct and inverse scattering of transient acoustic waves by a slab of rigid porous material[END_REF]. All the parameters display a pdf of Gaussian shape. Once again, elliptic joint-pdfs denote the correlations between parameters, stemming from the modeling used. [START_REF] Champoux | Air-based system for the measurement of porosity[END_REF][START_REF] Brown | Evaluation of the viscous characteristic length of air-saturated porous materials from the ultrasonic dispersion curve[END_REF][START_REF] Leclaire | Determination of the viscous and thermal characteristic lengths of plastic foams by ultrasonic measurements in helium and air[END_REF].

Parameter

The results presented here take into account prior information obtained on the critical angle defined in Eq. 20, estimated between θ i = 39.7°and θ i = 49.8°. Indeed, as can be seen in Figure 9, the phase of the experimental coefficient is seen to become negative between these two values. The prior density function is updated as

π(m) × π θc (a θc ) ← π(m), (30)
where

a θc = 1/2α ∞ 1 -(Λ/Λ )(γ -1)/ √
Pr and π θc is a normal distribution centered on (sin 2 θ i + sin 2 θ i )/2 with standard deviation (sin 2 θ i -sin 2 θ i )/6. The standard deviation is chosen so as to get 99% of the probabilistic weight between the observed boundaries. This choice is made in light of our lack of certainty regarding the exactness of relation 20, which, in particular, does not hold any longer when the Pride correction [START_REF] Pride | Drag forces of porous-medium acoustics[END_REF] is taken into account.

To further validate the inference, the signals calculated using the Markov chain samples are compared with the experimental reflected signals in Figure 10 for a particular incident angle that was used for the inference. The Fourier transform is also used to compare the spectral density of calculated and experimental signals at the same two angles. Signals are normalized using the maximum value of the experimental signal.

It should be noted that without the extra prior knowledge on the critical angle, the inference outcome is less conclusive, as the ratio Λ /Λ is clearly not a well identified parameter. As was noted for the identification of material M1, the ratio Λ /Λ tends to lose sensitivity to the likelihood function as its value increases (asymmetric sensitivity function): once a certain value has been reached, it is virtually impossible to distinguish the likelihood, which makes the inverse problem challenging to solve. Regularizing the inference through prior information is one of the strengths of such a statistical approach, which would have otherwise failed here.

M3 identification: low resistivity foam

The waves reflected from material M3 were measured at 4 different angles of incidence (0°, 17°, 23°, 35°). Figure 11 shows the pdfs (diagonal elements) and joint-pdfs (extra-diagonal elements) inferred on the material M3 properties. These results are summarized in Table 5 where the MAP and standard deviation of each parameter are reported. Results obtained using classical identification techniques on the same material are also recalled for reference. For highly porous materials such as the one considered here, it is common to have ratios of Λ /Λ between 2 and 3. To take this prior

π(m) × π Λ /Λ (Λ /Λ) ← π(m), (31)
where π Λ /Λ ∼ N (2.5, 1). The standard deviation was taken as 1.0 to explain our uncertainty in the guidelines for the prescription in Λ /Λ, and to avoid shifting too strongly the parameter towards the prior: it is estimated that few foam materials such has the one considered here have a ratio of characteristic lengths higher than 4. The influence of a priori information on the posterior has not been studied further here. The results in Table 4 indicate a posterior probability close to the prior mean, with a standard deviation about half as large. This translates the fact that information from the observed quantities contained identifiable knowledge on Λ /Λ for us to extract. Despite this prior, the standard deviation of Λ remains large, compared with the other inferences performed in this study. Values for the porosity and tortuosity, however, remain well defined and close to the expected values for this type of foam. The observed discrepancies between values can be attributed to the method reaching its limits for this kind of low resistivity material, where the reflected wave amplitude is much smaller than the incident wave amplitude 12 for a particular incident angle that is used for the inference. The Fourier transform is also used to compare the spectral density of calculated and experimental signals. Signals are normalized using the maximum value of the experimental signal.

In future studies, higher values of incident angles will be used for the identification of low resistivity materials such as this one, as this appears to be the observable region where the dispersive part of the reflection is the greatest.

Discussion

One of the key strengths of Bayesian inference is updating one's belief on a certain hypothesis or parameter through the observation of new data via a robust and coherent approach. In the present work, only a modest use of this strength was tested: the user would certainly benefit from including previous experiments on given materials in terms of informative priors, which would further regularize the inverse problem at hand. While reflected waves were the primary concern of the present study, one is prone to wonder why the transmitted waves were not measured and used as well, in order to gain more information through this observational quantity. This question is especially valid for material M3, a low resistivity foam that is more often tested in transmission measurement mode, since its reflection coefficient is low. Again, the goal set in the present study was purely to introduce a new measuring experiment (the first interface reflected wave at oblique incidence), and a coherent way to aggregate different observations on a given material through first level Bayesian inference. It is of course strongly advised to make use of transmission results whenever readily available. Here, the only additional measurement that was used is the critical angle for which the scattering operator cancels out, for material M2.

Conclusion

This article proposes a method for the statistical Bayesian inference of homogeneous porous media with a rigid frame, using reflected ultrasonic waves from the first interface in the time domain and the Johnson-Allard model. The reflection operator of the material at the first interface has been worked out in the asymptotic regime in the high frequency limit, for obliquely incident waves. This result has been further analyzed to yield an interesting relationship between a particular angle of incidence, directly relating the tortuosity to the characteristic length ratio of the material. This new knowledge has been used as prior information to improve the regularization of the inverse problem. Furthermore, a detailed analysis of the diffusive kernel of the reflection operator has given new insights into the identification strategy: for highly resistive materials, observing the reflected pressures at low oblique incidence angles is the best strategy, since this is where the diffusive kernel is the most influential; for low resistivity materials, by contrast, it appeared that the diffusive kernel's influence is truly significant mainly for high values of incidence angles, where the full reflection coefficient is greater than for low incidence angles.

In the context of first level Bayesian inference, the information contained in the experimental observation and the user knowledge obtained independently of the experiments has been combined to express the posterior probability density of the parameters of interest: the porosity, tortuosity and characteristic lengths. This method offers the possibility to coherently take into consideration prior knowledge of the material properties and to allow uncertainties in the experiment. It is a robust way to take into account new experiments such as the one presented in this study, and combine the extracted knowledge with previous experiments. Three different materials have been tested and their properties inferred: one granular resistive material of low porosity, one resistive foam of low pore size and medium porosity, and a classical sound absorbing, low resistivity and high porosity foam. It has been shown that using the knowledge of the reflected waves by the first interface only, in the high frequency limit, the extraction of information on the target parameters is possible, even in the case of the low resistivity material where the reflection coefficient is low.

Figure 1 :

 1 Figure 1: Schematic of the scattering problem

Figure 2 :

 2 Figure 2: (Color online) Time domain representation of reflected pressure signals (left axis) and reflection operator (right axis)

Figure 3 :

 3 Figure 3: (Color online)For the three synthetic materials P1, P2 and P3, at 190 kHz. Left: Absolute value of the total reflection coefficient |R(s, θ i )| as a function of the incident angle. Right: Absolute value of the scattering part of the reflection coefficient |R 1 (θ i )|

Figure 4 :

 4 Figure 4: (Color online) Reflected waves in the time domain associated with three different reflection operators: R, R0 , R1 .For material P1 at angle of incidence θ i = 0°(left), material P2 at angle of incidence θ i = 0°(middle), and material P3 at angle of incidence θ i = 80°(right). The angles are chosen according to Figure3(right), corresponding to the maximum of R1 .

Figure 6 Figure 5 :

 65 Figure 5: Simplified schematic of the experimental apparatus

igure 9 :

 9 (Color online) Imaginary part of the reflection coefficient for material M2, at four different incident angles.

Figure 10 :

 10 Figure 10: (Color online) Normalized experimental signals : and normalized MAP estimate: at θ i = 23°. Left: normalized time signals. Right: normalized frequency spectrum.

Figure 11 :

 11 Figure 11: (Color online) M3 inference pdfs (the graphs on the diagonal) and joint-pdfs (the offdiagonal scatter plots)

  Table

		5	1.7	30	90	57.8	74.5
	P2	0.8	1.5	50	150	52.6	62.7
	P3	0.99 1.04	200	400	39.0	46.6

Table 3 :

 3 M1 inference summary (N/A: non-available). Reference results are obtained with methods developed in

		1.77
	1.68 , 1
		1.59
		1.51
	112.76
		83.93
	$
		55.11
		26.28
		10.29
	$	7.06
	$' /	3.82
		0.58
		0 . 4 2	0 . 4 3	0 . 4 3 ?	1 . 5 1 0 . 4 4	1 . 5 9	1 . 6 8 , 1	2 6 . 2 8 1 . 7 7	5 5 . 1 1	8 3 . 9 3 $	0 . 5 8 1 1 2 . 7 6	3 . 8 2 $' / $ 7 . 0 6	1 0 . 2 9
	Figure 6: (Color online) M1 inference pdfs (the graphs on the diagonal) and joint-pdfs (the
	off-diagonal scatter plots)

Table 4 :

 4 M2 inference summary. Reference results are obtained with methods developed in

		φ	α ∞ Λ(µm) Λ /Λ
	Reference values	0.75 1.45	50	2 -3
	Inverted MAP value 0.80 1.44	45.4	2.3
	Standard deviation 0.003 0.02	3.2	0.22

Table 5 :

 5 M3 inference summary. Reference results are obtained with methods developed in[START_REF] Champoux | Air-based system for the measurement of porosity[END_REF][START_REF] Brown | Evaluation of the viscous characteristic length of air-saturated porous materials from the ultrasonic dispersion curve[END_REF][START_REF] Leclaire | Determination of the viscous and thermal characteristic lengths of plastic foams by ultrasonic measurements in helium and air[END_REF]

	Parameter	φ	α ∞	Λ(µm) Λ /Λ
	Reference values	0.97	1.06	263	1.63
	Inverted MAP value 0.99	1.11	283	2.92
	Standard deviation 0.004 0.011	54.2	0.47

Acknowledgment

The first author would like to thank the Occitanie region for its financial support during his PhD. C. Depollier is supported by Russian Science Foundation grant number 14-49-00079. The authors appreciate the critical reviews and useful comments made by the anonymous reviewers.