Maitre Myriam Guedj

Mathilde Lochmann

Maitre De Conférences María

Teresa Muñoz Sastre

Etienne Mullet

Full Felix Neto
email: fneto@fpce.up.pt

Patricia Rulence Paques
email: patricia.rulencepaques@gmail.com

Vera Germano
email: germano.vera.cruz@u-picardie.fr

Cruz

Marie Préau
email: marie.preau@univ-lyon2.fr

Keywords: Mammography, Breast cancer, Motive, Reversal theory Secondary Mammographie, Cancer du sein, Motif, Théorie du renversement Mammography, Breast cancer, Motive, Reversal theory Mammographie, Cancer du sein, Motif, Théorie du renversement

Introduction: This study examines the motives of women for agreeing or refusing to have a mammogram. Method: Three hundred and ten participants were recruited. Two questionnaires listing the motives for undergoing or refusing to undergo a mammogram were developed. Intolerance of uncertainty, worry, anxiety, personality were respectively rated by: EII, QIPS, GAD-7, BHI-24. Result: Seven facilitators were interpreted within the framework of Reversal Theory: Interest of early diagnosis, Quality of care, Habit, Collectivist perspective, Following the doctor's advice, Cancer(s) in entourage, Hypochondriac beliefs. Six barriers were revealed: Absence of cancer in entourage, Anxiety, Physical and moral pain, Lack of information, Spatiotemporal difficulties, Aspiration for freedom. The frequency of mammography was positively linked to the age, number of children, Habit, and negatively to Physical and moral pain, Lack of information and Spatiotemporal difficulties. Age and Habit had a positive effect on the frequency of screening while the Spatiotemporal difficulties had a negative effect. Conclusion: This study provides a better understanding of the motives that lead women to accept or refuse performing this examination. It could be useful to the medical profession and more broadly to the field of public health. Secondary Abstract: Introduction : Cette étude examine les motifs des femmes à accepter ou refuser la réalisation d'une mammographie. Elle investigue également les liens entre les facteurs motivationnels, l'âge, le nombre d'enfant(s), l'intolérance à l'incertitude, l'inquiétude, l'anxiété, la personnalité et la fréquence de réalisation des mammographies. Méthode : Trois-cent-dix participantes ont été recrutées. Deux questionnaires répertoriant les motifs à accepter et à refuser la réalisation d'une mammographie ont été élaborés grâce aux données de la littérature et à des entretiens semi-directifs. L'intolérance à l'incertitude, l'inquiétude, l'anxiété et la personnalité ont été respectivement évaluées par l'EII, le QIPS, le GAD-7 et le BHI-24. L'âge, le nombre d'enfant(s) ainsi que la fréquence de réalisation des mammographies ont également été renseignés. Résultat : Sept facteurs motivationnels facilitateurs ont été mis en lumière et interprétés au regard de la Théorie du Renversement : « Intérêt du diagnostic précoce », « Qualité de la prise en charge », « Habituation », « Perspective collectiviste », « Suivre l'avis du médecin », « Cancer(s) dans l'entourage », « Croyances hypocondriaques ». Six facteurs motivationnels barrières ont également été révélés : « Absence de cancer dans l'entourage », « Angoisses », « Douleur physique et morale », « Difficultés spatio-temporelles », « Manque d'informations », « Aspiration à la liberté ». La fréquence de réalisation des mammographies est associée positivement à l'âge, au nombre d'enfant(s), au facteur motivationnel « Habituation » et négativement aux facteurs motivationnels « Difficultés spatio-temporelles », « Douleur physique et morale » et « Manque d'informations ». L'âge et le facteur « Habituation » avaient un effet prédicteur positif sur la fréquence mammographique tandis que le facteur « Difficultés spatio-temporelles » avait un effet prédicteur négatif.

ORCID : https://orcid.org/0000-0002-1707-3914 etienne.mullet@wanadoo.fr E-mail et phone of correspondant author* : myriam.guedj@univ-tlse2.fr, 05.61.50.41.64 Résumé :

Introduction : Cette étude examine les motifs des femmes à accepter ou refuser la réalisation d'une mammographie. Elle investigue également les liens entre les facteurs motivationnels, l'âge, le nombre d'enfant(s), l'intolérance à l'incertitude, l'inquiétude, l'anxiété, la personnalité et la fréquence de réalisation des mammographies.

Méthode : Trois-cent-dix participantes ont été recrutées. Deux questionnaires répertoriant les motifs à accepter et à refuser la réalisation d'une mammographie ont été élaborés grâce aux données de la littérature et à des entretiens semi-directifs. L'intolérance à l'incertitude, l'inquiétude, l'anxiété et la personnalité ont été respectivement évaluées par l'EII, le QIPS, le GAD-7 et le BHI-24. L'âge, le nombre d'enfant(s) ainsi que la fréquence de réalisation des mammographies ont également été renseignés.

Résultat : Sept facteurs motivationnels facilitateurs ont été mis en lumière et interprétés au regard de la Théorie du Renversement : « Intérêt du diagnostic précoce », « Qualité de la prise en charge », « Habituation », « Perspective collectiviste », « Suivre l'avis du médecin », «

Introduction

In France, Europe and the United States, breast cancer is the most common cancer in women.

It is also the one with the highest mortality rate (17.9% of female cancer deaths in France) [START_REF] Ligier | Projection de l'incidence et de la mortalité par cancer en France métropolinaine en 2017[END_REF]. The Organized Screening (OS) program for breast cancer became widespread in France in 2004 with the following objectives: (a) lowering the specific mortality rate for breast cancer, (b) reducing the recourse to mastectomy, and (c) decreasing the incidence of late-stage breast cancer [START_REF] Lousdal | Effect of organised mammography screening on stage-specific incidence in Norway: population study[END_REF]. Mammography screening is secondary prevention: it aims to intervene early in order to improve the progression of breast cancer. Since the work of Halsted, it has often been accepted that the smaller and the earlier this cancer is treated, the more curable it is. According to the National Cancer Institute (INCa, 2013), OS has made it possible to reduce specific mortality from breast cancer by 15 to 21% worldwide.

The question of breast cancer screening by mammography is nevertheless controversial. The main criticisms targeting OS are the following: (a) the incidence of breast cancer has increased since its introduction, (b) the phenomenon of overdiagnosis is considered to be underestimated by some, (c) stabilization (not a decrease) of the advanced stages has been observed and finally, (d) the regions where screening is the most intense are not associated with a low mortality rate specific to breast cancer [START_REF] Autier | Mammography screening: A major issue in medicine[END_REF]. Other negative repercussions linked to the examination are reported such as (e) irradiation [START_REF] Yaffe | Risk of Radiation-induced Breast Cancer from Mammographic Screening[END_REF], (f) the painful aspect of the mammogram, (g) the psychological impact (anxiety, depression), and (h) the occurrence of false positives and therefore the implementation of unnecessary treatments and/or invasive procedures [START_REF] Gøtzsche | Screening for breast cancer with mammography[END_REF].

In France, the OS program recommends that women aged 50 to 74 should have a mammogram every two years. However, the level of participation has never managed to reach the 70% coverage rate targeted by the cancer plan (INCa, 2014) and has even been decreasing in recent years. The participation rate fell from 52.7% (peak in 2012) to 50.3% in 2018. Note that mammography screening is not limited to this age group. Thus, Individual Screening (IS) appears usual from the age of 40 in France (Haute Autorité de la Santé [HAS], 2013). Women under 40 who are worried about having breast cancer or have a family history may also decide to have a mammogram [START_REF] Aquaviva | Baromètre santé 2005[END_REF].

Women's attitude to this test appears to depend on a multitude of factors. Some are seen as facilitators, others as barriers [START_REF] Püschel | If I feel something wrong, then I will get a mammogram": Understanding barriers and facilitators for mammography screening among Chilean women[END_REF]. Among the facilitating factors, one can mention: (a) the place of perceived control in health (i.e., individuals attribute the health events they experience to an internal or external causality) (Epstein, 2014), (b) personality traits such as conscientiousness [START_REF] Siegler | Personality and breast cancer screening behaviors[END_REF] or type A personality traits (i.e., personality traits defined by competitiveness, sense of urgency and hostility, but also associated with Consciousness and problem-centered coping strategies) [START_REF] Ramanaiah | Type A Behavior and the Five-Factor Model of Personality[END_REF], (c) worry about cancer (Consedine et al., 2004) , and (d) the perceived risk of having cancer [START_REF] Ferrat | Understanding barriers to organized breast cancer screening in France: women's perceptions, attitudes, and knowledge[END_REF].

Among the barrier factors, one can mention: (a) fear of disease in general [START_REF] Lostao | Health beliefs and illness attitudes as predictors of breast cancer screening attendance[END_REF], (b) fatalism [START_REF] Zardanjani | Fatalism in breast cancer and performing mammography on women with or without a family history of breast cancer[END_REF], (c) shame (Püschel et al., 2009), (d) personality traits such as emotivity [START_REF] Esfahani | A qualitative exploration of personality factors in breast cancer screening behavior[END_REF] or lack of openness [START_REF] Siegler | Personality and breast cancer screening behaviors[END_REF], (e) fear of pain [START_REF] Kalecinski | Dépistage organisé ou individuel du cancer du sein ? Attitudes et représentations des femmes[END_REF][START_REF] Lecompte | Dépister les cancers du sein… Représentations des médecins, informations délivrées aux femmes et enjeux du dépistage organisé[END_REF], (f) the absence of a family history of breast cancer [START_REF] Duport | Facteurs d'adhésion au dépistage organisé du cancer du sein : étude Fado-sein[END_REF], and (g) the fact of having children [START_REF] Esfahani | A qualitative exploration of personality factors in breast cancer screening behavior[END_REF].

Motives for agreeing or refusing to have a mammogram

Few studies have focused on the motives for agreeing or refusing to undergo mammography screening. Qualitative studies have shown that the motives given for undergoing mammography screening were related to fear of disease, a desire to control one's health, the influence of breast cancer experiences in others and recommendations from healthcare professionals [START_REF] Ferrat | Understanding barriers to organized breast cancer screening in France: women's perceptions, attitudes, and knowledge[END_REF][START_REF] Nekhlyudov | Beliefs and expectations of women under 50 years old regarding screening mammography: a qualitative study[END_REF]. The motives given for not having the exam related to fear, low priority given to health issues, traumatic experience of mammograms, worry about ionizing radiation, and the insensitive attitude of radiologists [START_REF] Kalecinski | Dépistage organisé ou individuel du cancer du sein ? Attitudes et représentations des femmes[END_REF].

Quantitative studies have shown that the reasons given for engaging in mammography screening were related to fear of breast cancer, an overestimation of the risk of suffering from it, doctor's recommendations and wanting to take care of one's health. The reasons given for not undergoing the examination were related to the lack of time, a fatalistic attitude in the event of a positive diagnosis, or increased risk of invasive procedures linked to this screening [START_REF] Duport | Facteurs d'adhésion au dépistage organisé du cancer du sein : étude Fado-sein[END_REF][START_REF] Rakowski | Assessing elements of women's decisions about mammography[END_REF].

The present study

The specific determinants of having or refusing a mammogram have been extensively studied, but only a few studies have looked at the motives reported by women. Apter's Reversal Theory (RT) [START_REF] Apter | Motivational styles in everyday life: A guide to reversal theory[END_REF]) is a motivational theory that offers the possibility of bringing together and synthesizing the available scientific data and of structuring them around large intelligible factors [START_REF] Kpanake | Skin Bleaching Among Togolese: A Preliminary Inventory of Motives[END_REF][START_REF] Makris | A systematic inventory of motives for becoming an orchestra conductor: a preliminary study[END_REF]. In addition, exploring a research theme through the lens of this theory offers the resources and the perspective necessary to take into account reasons that have sometimes never been examined before [START_REF] Mullet | Putting Reversal Theory's Model of Four Domains of Experience in the Hot Seat[END_REF].

RT considers that motives are dynamic and fluctuating entities influencing the way that humans engage with their environment. [START_REF] Apter | Motivational styles in everyday life: A guide to reversal theory[END_REF] identified ten motivational states divided into four domains: 1) Telic and Paratelic in the domain of Way-Goals; 2) Conformist and Negativist in the field of Rules; 3) Mastery and Sympathy in the field of Transactions; 4) Autocentric, Intra-autic, Allocentric and Pro-autic in the field of Relations.

The Telic state is goal-oriented. For example, a possible Telic motive in connection with the refusal to have a mammogram could be: "X-rays are harmful to health". The Paratelic state pays attention to pleasant behavior in the present moment. In the Conformist state, rules determine the way in which behaviors are ordered, while the Negativist state sees the rules as restrictions and seeks to oppose them. The Mastery state sees the transaction as a way to take or, conversely, to leave, while the Sympathy state considers the transaction as a way to give or receive. The Autocentric state aspires to be the center of the interests and concerns of others. The Intra-autic state is attentive to its own interests and concerns. The Allocentric state is attentive to the needs and interests of others. Finally, the Pro-autic state aspires to live the experiences of others by proxy.

Thus, in each domain we systematically find two or four opposing motivational states of which only one can be active at the same time. On the other hand, several domains can be activated simultaneously: combinations between a state of the domain of Transactions and a state of the domain of Relations are particularly frequent (e.g., Allocentric / Mastery or Autocentric / Sympathy) [START_REF] Apter | Reversal theory: The Dynamic of Motivation, Emotion and Personality[END_REF]. In everyday life, under the influence of various factors, constant shifts called "reversals" can occur from one motivational state to another within the same domain.

The objectives of this research consisted in (a) interpreting the motivational factors inventoried with regard to the dimensions of RT and identifying two factor structures specific to the motives for agreeing and refusing to have a mammogram, (b) exploring whether the motivational factors are linked to the frequency of having mammograms, and beyond, the demographic and personality characteristics such as: age, number of children, intolerance of uncertainty, anxiety, anxiety, and personality traits. Anxiety related to mammography has often been invoked in the context of the disease, but it has never been measured as an individual variable in its own right.

Since, to our knowledge, no study has investigated the motives for consenting to or refusing mammography through the lens of RT, we investigated how the reasons for agreeing or refusing to have a mammogram would organise themselves around the ten motivational states of RT. We also wondered whether there were relationships between intolerance of uncertainty, worry, anxiety and the frequency of screening.

Method

Participants

Four hundred and seventy women were recruited in France by word of mouth, associations, social networks and a call for volunteers within Toulouse Jean-Jaurès University. Three hundred and ten of them agreed to take part (refusal rate of 34%). The participants were between 18 to 92 years old (M = 48.92; SD = 13.95), with the following age-group distribution: 28 participants were 18 to 29 years old (9%), 49 were 30 to 40 years old (16%), 74 were 41 to 50 years old (24%), 112 were 51 to 63 years old (36%), 36 were 64 to 74 years old (12%), 7 were 75 to 90 years old (2%), and 4 did not specify their age (1%).

In our sample, 66% of the participants (n = 205) had a family history of breast cancer.

Sixty-nine percent of participants reported having children (n = 212): 23% had 1 (n = 69), 31% had 2 (n = 96), 15% had 3 or more (n = 47), 18% had not (n = 56) and 13% did not provide this information (n = 42). Out of the entire sample, 80% of participants had already had at least one mammogram (n = 248), 18% had never had one (n = 57) and 2% did not wish to disclose this information (n = 5). In addition, 20% of the participants had already refused to have this examination (n = 63), 68% had never refused to have it (n = 209) and 12% did not provide this information (n = 38).

Material

Two questionnaires were constructed based on the motives identified in the literature and through the completion of a pre-survey.

Pre-survey

The objective of the pre-survey was to identify, through semi-structured interviews, the motives that led women to agree or refuse to have a mammogram. This procedure made it possible to inventory a large number of motivational factors, some of which have been little studied in the literature. Twelve participants, more or less favorable to mammographic screening, aged 25 to 61, freely agreed to participate. Four healthcare professionals also took part in this pre-survey, with a view to varying the data sources. Semi-structured interviews lasting an average of 30 minutes were carried out. They were all recorded, transcribed and then analysed. Content analyses were carried out on the entire corpus and several dimensions emerged such as lack of information, need to be reassured, the phenomenon of habit, social influences, mistrust, opposition to medicine or to injunctions, desire to take preventive action or verify, desire to take care of oneself or to reassure one's entourage, dehumanization of care, fear of cancer, painful aspect of the examination, collectivist perspective or even identification with sick or healthy relatives. All of the motives identified during this presurvey were interpreted with regard to the dimensions of RT and transposed into items in order to develop two questionnaires.

Survey

The first questionnaire listed 105 facilitating motives and the second 115 barrier motives.

Each item started out the same way: "One of the reasons that would drive me to have a mammogram is that…" or "One of the reasons that would drive me to refuse to have a mammogram is that…". These repetitions allow the respondent to adopt an open attitude towards these motives. Too often, we tend to consider that to one action corresponds one motive, and only one. All theoretically possible motivational states of RT were represented in each of the two questionnaires. To answer each item, the participants positioned themselves on an 8-point response scale (1-8) ranging from "strongly disagree" on the left to "strongly agree" on the right. Four additional tools were also used: the Uncertainty Intolerance Scale (EII) [START_REF] Freeston | Why do people worry?[END_REF], the Penn State Worry Questionnaire (PSWQ) [START_REF] Meyer | Development and validation of the Penn State Worry Questionnaire[END_REF], the General Anxiety Disorder 7 (GAD-7) [START_REF] Spitzer | A brief measure for assessing generalized anxiety disorder: the GAD-7[END_REF], and the 24-item Brief HEXACO Inventory (BHI) [START_REF] De Vries | The 24-item Brief HEXACO Inventory (BHI)[END_REF]. Lastly, a questionnaire to collect sociodemographic data as well as the frequency of mammography screening was included in the research protocol. To assess the latter variable, participants had to answer the following question "How often do you have a mammogram?" by positioning themselves on a 7-point Likert scale ("never", "every 8 to 10 years and more", "every 5 to 7 years", "every 4 years", "every 3 years", "Every 2 years", "once a year").

Procedure

The participants were asked to read an information sheet and then to answer the questionnaires in a quiet public place or at their home. Both paper and online versions were made available to them. The total test lasted on average 45 minutes. In order to limit the order effect, half of the participants answered first the questionnaire listing the facilitating motives and then the questionnaire listing the barrier motives. The second half of the sample responded in reverse order. This research received a favorable notification from a Research Ethics Committee (CER).

Results

An exploratory factor analysis with principal component extraction was carried out on the 105 facilitating patterns using Statistica 12 software, on a first part of the sample (n = 168). In the scree test, seven factors with eigenvalues between 5.75 and 1.19 and explaining 65% of the total variance were observed. This solution was chosen and was subjected to a standardized VARIMAX rotation. Items that were complex or had too low saturations were gradually removed until the final model was obtained.

The first factor called "Interest of early diagnosis" explained 10% of the variance and referred to the Telic motivational state of RT. It was named thus because its items expressed the positive consequences linked to early diagnosis. The mean score of the items composing it was M = 7.18 (SD = 1.25). The second factor explained 10% of the variance and referred to the combination of Sympathy/Autocentric RT motivational states. It was called "Quality of care" because its items expressed the importance for women that radiologists should have technical but also interpersonal skills. The mean score of the items composing it was M = 5.38 (SD = 1.92).

The third factor explained 11% of the variance and referred to the Conformism state of RT. It was called "Habit" because its items expressed how mammography could become, over time, a full-fledged health habit. The mean score of the items composing it was M = 4.41 (SD = 2.12). The fourth factor explained 9% of the variance and referred to the Conformism state of RT. It was called the "Collectivist Perspective" because its items expressed an aspiration to undergo a mammogram linked to collective interest. The mean score of the items composing it was M = 2.50 (SD = 1.63).

The fifth factor explained 9% of the variance and referred to the combination of Mastery/Allocentric RT motivational states. It was called "Follow the doctor's advice" because its items expressed the influence of the doctor's advice on women's attitudes towards mammography. The mean score of the items composing it was M = 6.20 (SD = 1.52). The sixth factor explained 7% of the variance and referred to the Pro-autic state of RT. It was called "Cancer(s) in the entourage" because its items expressed the fact that cancer experiences in the near environment could be lived by proxy. The mean score of the items composing it was M = 5.13 (SD = 2.65). Finally, the seventh factor explained 9% of the variance and referred to the Intra-autic state of RT. It was called "Hypochondriac Beliefs" because its items expressed hypochondriac fears. The mean score of the items composing it was M = 4.13 (SD = 1.83).

The same procedure was applied to the 115 barrier motives. A six-factor solution presenting eigenvalues between 6.87 and 1.12 and explaining 70% of the variance was retained. The first factor explained 13% of the variance and referred to the Pro-autic state of RT. It was called "Absence of cancer in the entourage" because its items expressed immersion in an environment where this pathology does not exist (identification with a healthy entourage).

The mean score of the items composing it was M = 1.91 (SD = 1.45).

The second factor explained 14% of the variance and referred to the combination of Intra-autic/Mastery RT motivational states. It was called "Anxiety" because its items expressed apprehension specific to the discovery of breast cancer and the lack of control over its consequences. The mean score of the items composing it was M = 2.99 (SD = 2.03). The third factor explained 13% of the variance and referred to the combination of Allocentric/Mastery RT motivational states. It was called "Physical and Moral Pain" because its items expressed the physiological and psychological impact that the exam could have on women. The mean score of the items composing it was M = 3.17 (SD = 1.94).

The fourth factor explained 10% of the variance and referred to the intra-autic motivational state of RT. It was called "Spatio-temporal difficulties" because its items expressed practical constraints such as lack of time or accessibility. The mean score of the items composing it was M = 2.46 (SD = 1.76). The fifth factor explained 11% of the variance and referred to the Paratelic state of RT. It was called "Lack of information" because its items expressed the problem of the insufficiency of the information provided, questioning the practice of mammography itself. The mean score of the items composing it was M = 3.46 (SD = 1.90). Finally, the sixth and final factor explained 9% of the variance and referred to the negativist state of the RT. It was called "Aspiration for Freedom" because its items expressed opposition to being pressured into doing a health exam. The average score of the items composing it was M = 1.74 (SD = 1.09).

Confirmatory factor analyses were performed on a second part of the sample (n = 126). A first confirmatory factor analysis was carried out on the motives for undergoing a mammogram using Statistica 12 software (see Table 1). Of the 26 items that emerged from the exploratory factor analysis, 22 were retained for confirmatory factor analysis. Four items were removed because they saturated less than the other items on their dimension. The GFI value was .84; the CFI value was .91; the RMSAE value was .05 [.00-.08]; the Chi²/Df ratio (265.55/188) was 1.41, the p-value was .001. The fit index of the data model confirmed that this solution of the motives for having a mammogram in seven factors was satisfactory. The internal consistency of the seven patterns retained was generally good (respectively from I to VII,



A second confirmatory factor analysis was carried out on the motives for refusing to have a mammogram (see Table 2). To conduct this analysis, all the items that emerged from the exploratory analysis were kept. The GFI value was .80; the CFI value was .87; the RMSAE value was .08 [.00-.08]; the Chi²/Df ratio (406.81/194) was 2.10, the p-value was .001. Although this model was slightly less satisfactory, it was still acceptable since the values of the indices approached the significance thresholds and the internal consistency of the factors composing it was excellent (respectively from I to VI, .

[Insert here Table 1] [Insert here Table 2] On this same sample (n = 126), bivariate correlation analyses were then carried out between motivational factors, variables such as age, number of children, intolerance to uncertainty, worry, anxiety, personality traits and mammogram frequency (see Table 3). The frequency of mammography screening was strongly associated with age r = .64, p ˂ .01, with the "Habit" factor, r = .55, p ˂ .01, and more moderately with the number of children(s), r = .25, p ˂ .01.

It was negatively associated with the factors "Spatio-temporal difficulties", r = -.40, p ˂ .01, "Physical and moral pain", r = -.31, p ˂ .01, and "Lack of information", r = -.25, p ˂ .01. In other words, the older a woman was, had a high number of children, was accustomed to mammography, had little regard for spatio-temporal constraints, perceived this examination less as physically and psychologically painful, had less the feeling of lacking information, the more frequently she underwent mammograms. No personality trait was significantly associated with the frequency of mammogram screening. No association was found between intolerance of uncertainty, worry, anxiety and frequency of the exam.

A series of regression analyses was carried out on the variable "Frequency of mammogram screening". Intolerance to uncertainty, worry and anxiety were tested first and explained 4% of the variance (F(1,114) = 4.24, p = 0.04): only worry was significant (β = -0.19).

Personality traits were tested second and explained 7% of the variance (F( 2 and "Spatio-temporal difficulties" (β = -0.31) were the only significant ones.

A final step-by-step regression analysis was performed on the predictors integrated simultaneously. Age (β = 0.46) as well as the factors "Habit" (β = 0.30) and "Spatio-temporal difficulties" (β = -0.21) were related to the frequency of mammogram screening. These variables explained 54% of the variance (F(3, 112) = 43.66, p ˂ 0.01). Their high intensities neutralized the low weight of the "Worries" and "Consciousness" predictors that emerged during the first regression analyses. Thus, being older, being more sensitive to the argument that mammography is common practice and being less sensitive to the motive inherent in space-time constraints, predicted a regular frequency of undertaking this examination.

Discussion

The first objective of this study was to identify the motivational barriers and facilitators to undergoing a mammogram and to interpret them with respect to RT.

The motives in favour of screening appeal, in the order of importance given by women on the response scale, to the Telic state ("Interest of early diagnosis"), the combination of the Allocentric/Mastery states ("Following the doctor's advice"), the combination of the Sympathy/Autocentric states ("Quality of care"), the Pro-autic state ("Cancer in the entourage"), the Conformism state ("Habit"), the Intra-autic state ("Hypochondriac Beliefs") and the Conformism state again ("Collectivist Perspective"). The motives for refusal appeal, in the order of importance given by the women on the response scale, to the Paratelic state ("Lack of information"), the combination of the Allocentric/Mastery states ("Physical and moral pain"), the combination of the Intra-autic/Mastery states ("Anxiety"), the Intra-autic state ("Spatio-temporal difficulties"), the Pro-autic state ("Absence of cancer in the entourage") and the Negativist state ("Aspiration for freedom"). Most of the patterns identified in this research are consistent with the evidence in the literature, but RT has allowed us to deepen our understanding.

The factors "Following the doctor's advice" and "Quality of care" already highlighted elsewhere [START_REF] Domenighetti | Women's perception of the benefits of mammography screening: Populationbased survey in four countries[END_REF][START_REF] Lecompte | Dépister les cancers du sein… Représentations des médecins, informations délivrées aux femmes et enjeux du dépistage organisé[END_REF] indicate that while women generally prefer the doctor to have control of their health, they aspire to control the behavior of the technician performing the examination and in particular wish it to be sympathetic and human.

The factor "Cancer(s) in the entourage" has also been mentioned in the literature [START_REF] Duport | Facteurs d'adhésion au dépistage organisé du cancer du sein : étude Fado-sein[END_REF][START_REF] Ferrat | Understanding barriers to organized breast cancer screening in France: women's perceptions, attitudes, and knowledge[END_REF][START_REF] Esfahani | A qualitative exploration of personality factors in breast cancer screening behavior[END_REF]. It points to the fact that the experience of breast cancer in the entourage can be lived by proxyor at least that an identification process can take placewhich makes it easier to decide to have a mammogram.

The factor "Hypochondriac beliefs" emerged in our study as a facilitating motive whereas the work of [START_REF] Lostao | Health beliefs and illness attitudes as predictors of breast cancer screening attendance[END_REF] suggested that having hypochondriac beliefs was associated with a decrease in mammography practice. Our understanding of the motives for doing this exam, which revolve around fear of disease and the anxieties inherent in cancer, seems tenuous. It is true that in individuals with a form of hypochondria, there is complacency with regard to medical exploration [START_REF] Lionet | Les fonctions du masochisme dans l'aménagement somatopsychique du patient douloureux[END_REF].

The factor "Interest of early diagnosis" is one of the main arguments used by health professionals [START_REF] Perry | European guidelines for quality assurance in breast cancer screening and diagnosis. Fourth edition-summary document[END_REF]. The goal to be achieved by performing a mammogram is therefore to detect cancer as early as possible in order to benefit from the best care. Finally, the "Habit" and "Collectivist Perspective" factors are motives which have been mentioned very little in the literature dealing with mammography. Regarding the "Habit" factor, it appears that over time mammography becomes a mechanical behavior until it evolves into a health habit. Regarding the factor "Collectivist perspective", the 100% coverage of mammography by Health Insurance certainly reinforces the idea that this examination is a gift offered by national solidarity in order to serve the common interestwhich undoubtedly directs women towards a more collectivist than individual positioning. This is consistent with the results of the study by [START_REF] Domenighetti | Women's perception of the benefits of mammography screening: Populationbased survey in four countries[END_REF] reporting that women residing in England and Italy (countries where the health service is public) overestimate the benefits of screening compared to women residing in Switzerland and United-States (countries where the health service is private).

Among the six motivational barriers identified, all have been cited in the literature. Note that few of them report a total rejection of mammography screening and that, judging from the average responses of the participants, the reasons for refusal are much weaker than the reasons for acceptance. However, these motivational factors are still good indicators for the medical profession and the preventive health field. The factor "Absence of cancer(s) in the entourage" shows that not being around women with breast cancer probably induces a feeling of less vulnerability [START_REF] Duport | Facteurs d'adhésion au dépistage organisé du cancer du sein : étude Fado-sein[END_REF][START_REF] Ferrat | Understanding barriers to organized breast cancer screening in France: women's perceptions, attitudes, and knowledge[END_REF][START_REF] Esfahani | A qualitative exploration of personality factors in breast cancer screening behavior[END_REF].

The "Anxiety" factor has also been addressed in the literature [START_REF] Consedine | The contribution of emotional characteristics to breast cancer screening among women from six ethnic groups[END_REF][START_REF] Ferrat | Understanding barriers to organized breast cancer screening in France: women's perceptions, attitudes, and knowledge[END_REF]). It appears here to be a barrier motif which contradicts the results of [START_REF] Consedine | The contribution of emotional characteristics to breast cancer screening among women from six ethnic groups[END_REF] stating that concerns about cancer are positively associated with mammography practice. The terminologies used to describe the anxiety associated with mammography vary from study to study, which no doubt contributes to maintaining these discrepancies in results. In our research, RT explains that barrier anxiety refers to mental insecurity related to the lack of control over breast cancer and its consequences. Thus, the subject's concerns center on the inability to control future unpleasant events in the event of a positive diagnosis. The "Physical and moral pain" factor has also been highlighted in previous studies [START_REF] Kalecinski | Dépistage organisé ou individuel du cancer du sein ? Attitudes et représentations des femmes[END_REF][START_REF] Lecompte | Dépister les cancers du sein… Représentations des médecins, informations délivrées aux femmes et enjeux du dépistage organisé[END_REF]. However, RT enriches this knowledge by specifying that the difficulty inherent in physical and moral pain is, in part, due to the fact of not being able to control the degree of interest shown by health professionals in the patient during the examination.

The "Spatio-temporal difficulties" factor has already been noted in several studies [START_REF] Duport | Facteurs d'adhésion au dépistage organisé du cancer du sein : étude Fado-sein[END_REF]ORS, 2001). It essentially refers to the issue of time management or the difficulties inherent in making an appointment. The "Lack of information" factor has often been deplored in the literature [START_REF] Cases | Ensemble améliorons le dépistage du cancer du sein[END_REF][START_REF] Hersch | The importance of enabling informed decision making for women considering breast cancer screening[END_REF]. It is suggested that the goal that women seek to achieve by refusing to have a mammogram is to protect themselves from an examination they feel suspicious of due to the vagueness of the information provided.

The "Aspiration for freedom" factor has also been mentioned in the literature, but succinctly and only in a few studies [START_REF] Kalecinski | Dépistage organisé ou individuel du cancer du sein ? Attitudes et représentations des femmes[END_REF][START_REF] Lecompte | Dépister les cancers du sein… Représentations des médecins, informations délivrées aux femmes et enjeux du dépistage organisé[END_REF]. In this factor, the rule is experienced as a restriction which women seek to oppose by adopting an unconventional attitude in order to feel free.

The second objective of this study was to examine whether motivational factors, and variables such as age, number of children, intolerance of uncertainty, worry, anxiety, and personality were linked to the frequency with which mammograms were performed. First, intolerance of uncertainty, worry and anxiety were not related in our study to the frequency of having mammograms (neither were the motivational factors "Anxiety" and "Hypochondriac Beliefs"). Thus, although the literature has been very interested in this type of variable in this particular context, their links with mammographic frequency remain insignificant in our study. Personality traits are also unrelated to how often mammograms are performed. The observations of [START_REF] Siegler | Personality and breast cancer screening behaviors[END_REF] about a positive effect of the Awareness factor and a negative effect of the Openness factor on the decision to undergo mammography are therefore not verified here. Regarding socio-demographic characteristics, it was observed that the more children the women have, the higher the frequency of having mammograms. In addition, it was also observed that the older women get, the more frequently they tend to have this exam which appears consistent with the age groups targeted by breast cancer screening. It therefore appears that women's attitudes towards mammography are not linear and are likely to vary over time. RT claims that human motivations are subject to fluctuations. In addition, four motivational factors -"Habit", "Physical and moral pain", "Spatio-temporal difficulties" and "Lack of information"maintain a relationship with the frequency of screening. The factors "Spatio-temporal difficulties" and "Physical and moral pain" involve transactional emotions [START_REF] Apter | Motivational styles in everyday life: A guide to reversal theory[END_REF]. Thus, the frequency of having a mammogram is negatively linked to individual concerns which nevertheless form part of the social universe of women. They focus their attention on their personal interests and on their desire to be recognized for who they are that is, full individuals who must be distinguished from others. In addition, the factors "Habit" and "Lack of information" involve somatic emotions [START_REF] Apter | Motivational styles in everyday life: A guide to reversal theory[END_REF]. Thus, mammographic frequency is positively linked to internal feelings such as the desire to be integrated into the community by adopting the majority behavior. It is also negatively linked to internal feelings such as trying to protect oneself from an exam for which there is insufficient information.

The question then arises as to how these elements of knowledge can be discussed in the medical consultation. And how can they help initiate new thinking in terms of breast cancer screening policy? It seems necessary for health actors (ministry and doctors included) to provide women with clear and fair information about the benefits and risks of mammography.

Screening policies should certainly value informed choice rather than maximizing the use of mammography [START_REF] Hersch | Use of a decision aid including information on overdetection to support informed choice about breast cancer screening: a randomised controlled trial[END_REF]. In addition, the physician can listen to and understand women's need for control and freedom, while being careful not to clumsily adopt a defensive attitude that may be counterproductive. It could be interesting in future research to question how physicians are emotionally affected by the attitude and arguments of reluctant women.

Better understanding the emotions at play during this type of consultation can help to reorganize the framework and the content of the doctor/patient exchanges. In addition, detecting the health habits of women resistant to mammography is also crucial.

Limits

This study has several limitations. First, the research protocol was relatively substantial: cognitive fatigue may have appeared at the end of the test. Second, this study had a virtual nature: it is possible that in a real situation, the motives given are not the same as in a hypothetical situation (especially for the women who have never had a mammogram) (Coniasse [START_REF] Brioude | Déterminants psychologiques de l'acceptation et du refus de participer à un essai clinique destiné à prevenir la maladie d'Alzheimer en population âgée fragilisée[END_REF]. Finally, third, the sample did not appear to be fully representative.

It was mainly made up of women who had already agreed to have at least one mammogram (those refusing to have one mostly refused to participate in the study). According to the work of [START_REF] Mulot | Comment les représentations des rapports de sexe influencent-elles la prévention du sida ?[END_REF] on the wearing of condoms "the answers given in the context of quantitative surveys correspond to the official public register, that of the norm, and do not always reflect the reality of effective behaviour". We are then faced with a paradox: how to readjust largescale public health policies such as organized screening if one is unable to consider the opinion of those who are the most resistant? Doesn't that make the turnout problem insoluble?

It might be interesting to understand why women opposed to mammography also oppose research into it. Is this an expression of a broader mistrust that needs to be further explored?

The lack of access to certain populations is in any case a problem that affects all research in general.

Statements and Declarations

Conflict of interest : The authors declare no conflict of interest.

  PhD in health psychology. Laboratoire CERPPS (Centre d'Études et de Recherches en Psychopathologie et Psychologie de la Santé) Université de Toulouse, UT2J, France 5, Allées Antonio-Machado, F-31058 Toulouse Cedex 9, France ORCID : https://orcid.org/0000-0002-5722-1555 mathilde.lochmann@univ-tlse2.fr GUEDJ Myriam*, lecturer in health psychology Laboratoire CERPPS (Centre d'Études et de Recherches en Psychopathologie et Psychologie de la Santé) Université de Toulouse, UT2J, France 5, Allées Antonio-Machado, F-31058 Toulouse Cedex 9, France ORCID : http://orcid.org/0000-0002-8855-1266 myriam.guedj@univ-tlse2.fr MUÑOZ SASTRE María Teresa, professor in health psychology Laboratoire CERPPS (Centre d'Études et de Recherches en Psychopathologie et Psychologie de la Santé) Université de Toulouse, UT2J, France . Title / Author / Data 5, Allées Antonio-Machado, F-31058 Toulouse Cedex 9, France mtmunoz@univ-tlse2.fr MULLET Étienne, director of study École Pratique des Hautes Études (EPHE) Les Patios Saint-Jacques, 4-14 rue Ferrus, 75014 Paris, France

  ,113) = 4.21, p = 0.01): only the Consciousness trait was significant (β = -0.23). Sociodemographic characteristics were tested third and explained 42% of the variance (F(2,113) = 40.34, p ˂ 0.01): only age was significant (β = 0.61). Motivational factors were tested fourth and explained 38% of the variance (F(3, 112) = 23.01, p ˂ 0.01): the factors "Habit" (β = 0.48)

Table 1

 1 Results of the confirmatory factor analysis concerning the motives for agreeing to have aNote. I, Interest of early diagnosis; II, Quality of care; III, Habit; IV, Collectivist perspective; V, Following the doctor's advice; VI, Cancer(s) in entourage; VII, Hypochondriac beliefs

	mammogram
					Factors
	One of the reasons that would drive me to have a mammogram is that…	I	II	III	IV	V	VI	VII	M	SD
	… early diagnosis could increase the chances of recovery.	.83							7.36 1.52
	… early diagnosis could allow better care	.78							7.49 1.32
	… I want us to be able to detect a tumor as soon as possible.	.73							7.35 1.62
	… early diagnosis could allow less burdensome treatment.	.69							7.22 1.42
	… the radiology center team is friendly.		.87						4.75 2.51
	… the radiology center team is human.		.79						6.13 2.23
	… I've got into the habit of doing this exam.			.83					5.33 2.81
	… as a woman, I am used to breast examinations.			.76					5.58 2.38
	… It is a common exam for me.			.73					4.57 2.58
	… I tolerate the pain of the exam well.			.64					4.23 2.66
	… I would feel guilty about the public money that would be spent is I was diagnosed with cancer late.				.76				2.31 2.01
	… I would feel guilty for refusing what social security offers me.				.75				2.82 2.31
	… I want to get the sympathy of my relatives who tell me to do it.				.71				2.08 1.72
	… I perform this exam in a collective interest, out of civic duty.				.60 .32			3.06 2.23
	… my gynecologist encouraged me to do so.	.30				.79			6.74 1.82
	… my doctor convinced me.					.77			5.99 2.19
	… I have a family history of breast cancer.						.85		5.91 2.88
	… relatives have had breast cancer.						.81		5.93 2.65
	… the possibility of having breast cancer regularly scares me.							.77 4.99 2.45
	… I often feel like I'm going to get sick.		.35					.67 2.67 1.88
	… even though my doctor thinks I have nothing, I still have some doubts.					.41		.63 5.15 2.67
	… I fear dying and leaving my children, my family.				.36			.56 4.76 2.44
	M	7.34 5.44 4.92 2.56 6.36 5.92 4.38
	SD	1.23 2.14 2.01 1.57 1.76 2.42 1.73
	Cronbach alpha	.80	.76	.78	.74	.70	.69	.70

Table

  Click here to access/download;Table;Tables.docx

Table 2

 2 Results of the confirmatory factor analysis concerning the motives for refusing to have aNote. I, Absence of cancer in entourage ; II, Anxiety ; III, Moral and physical pain ; IV, Spatio-temporal difficulties ; V, Lack of information ; VI, Aspiration to freedom

	mammogram
				Factors
	One of the reasons that would drive me to refuse to have a mammogram is that…	I	II	III	IV	V	VI	M	SD
	… we never talk about cancer in my closedenvironment.	.79						1.68 1.57
	… none of my relatives have had breast cancer.	.79						1.43 1.39
	… no one does this exam in my family.	.78						1.41 1.33
	… I have no family history of breast cancer.	.78					.31 1.71 1.72
	… I am worried about waiting for the exam results.		.79					2.96 2.43
	… this exam makes me think of my own death.		.78					2.46 2.25
	…I am afraid I won't know how to deal with it if I find out that I have cancer.		.77					3.04 2.44
	… I don't want cancer to be found because I won't be able to live normally.		.59	.33				2.16 2.08
	… I find the exam violent.			.88				3.28 2.45
	… I am afraid of suffering during the exam, afraid of pain.			.84				2.88 2.46
	… I find the examination dehumanizing (loss of dignity).			.74				2.52 2.22
	… radiologists have no regard for the pain experienced during the exam.			.71	.31			3.48 2.22
	… I have a lot of difficulty making an appointment (medical or otherwise).				.82			2.11 1.94
	… I don't have the time.				.75			2.46 2.22
	… I can't put up with having to schedule an appointment.				.70		.39 2.06 1.90
	… we are only given information on why to do it, and not on why not to do it.					.78		3.40 2.54
	… there is not enough information to make an informed decision.			.35		.76		3.14 2.50
	… some scientific researchers question this examination.					.75	.35 3.38 2.35
	… there is a lack of quantitative information on the benefits of screening.		.38			.68		3.43 2.40
	… I am satisfied with myself when I resist the "injunction to do".						.79 1.71 1.63
	… I don't want to encourage other people to perform this exam by doing it myself.	.38					.74 1.42 1.23
	…I can't put up with others telling me what to do.						.70 1.90 1.74
	M	1.57 2.65 3.04 2.21 3.34 1.69
	SD	1.34 1.93 2.12 1.65 2.07 1.38
	Cronbach alpha	.87	.86	.90	.75	.87	.81

Acknowledgment : No acknowledgment. Ethic approval and consent to participate : This research received a favorable notification (N ° 2018-119) from the Research Ethics Committee (CER) of Toulouse, France. Research Data Policy and Data Availability Statements: The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.


Blaise Raybaud

Etienne Vergnault

Angela Disdier

Philippe Thony

Jean-Jacques Roux

Identification of BRDF parameters with spectral measurements in the visible light spectrum towards solar irradiation evaluation in urban environment for photovoltaïc technologies

Keywords: Formal analysis, Data Curation, Writing), Etienne Vergnault (Methodology, BRDF, Ray tracing simulation, Building integrated photovoltaic facade, Urban energy simulation, Urban facade, Optical

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

• cos + (a) = cos(a)+| cos(a)| 2 with a ∈ [0 : 2π]

Introduction

Studies of solar potential in urban environments are today mostly useful for numerous applications in various domain, as local production of solar energy in urban areas [START_REF] Lobaccaro | Solar energy in urban environment: How urban densification affects existing buildings[END_REF], access to natural light towards better life experience of citizen [START_REF] Balocco | Natural light design for an ancient building: A case study[END_REF] or simulation of radiative transfer in visible and thermal spectral ranges to avoid overheating in districts (radiative trapping at district scale, redistribution of energy). These numerical studies have different objectives: to assess the solar potential for photovoltaic energy conversion [START_REF] Brito | The importance of facades for the solar PV potential of a mediterranean city using LiDAR data[END_REF][START_REF] Saretta | A calculation method for the bipv potential of swiss façades at lod2.5 in urban areas: A case from ticino region[END_REF], to validate access to natural light in buildings for human comfort and health [START_REF] Lobaccaro | Boosting solar accessibility and potential of urban districts in the nordic climate: A case study in trondheim[END_REF], but also to evaluate the exterior facade temperatures of the various buildings of a dense district [START_REF] Wang | Monte carlo simulations of radiative heat exchange in a street canyon with trees[END_REF][START_REF] Bijl | Modelling of sample surface temperature in an outdoor weathering test[END_REF][START_REF] Manni | Exploiting selective angular properties of retro-reflective coatings to mitigate solar irradiation within the urban canyon[END_REF]. Current simulations of solar light propagation in urban environment are generally based on visual rendering strategies used in graphics rendering. Generally, back reflected light on walls [START_REF] Vallati | Effects of radiative exchange in an urban canyon on building surfaces' loads and temperatures[END_REF] is considered as scattered light without considering the spectral or polarization properties.

We also have to distinguish light behaviour and surface properties in at least two spectral ranges: visible light (visual comfort, aesthetics, PV conversion) and thermal radiation (thermal comfort and emissivity, urban heat islands and greenhouse effects). The objective of this study is, firstly, to evaluate the impact of these properties of light for precise energy estimation and, secondly, to take into account the exact nature of light reflections in the visible spectral range (as opposed to thermal spectral range) in order to verify the hypothesis of diffuse light reflection currently assumed in most studies [START_REF] Langovoy | Empirical BRDF models for standard reference materials[END_REF]. Raybaud's thesis [START_REF] Raybaud | Evaluation de l'impact des propriétés optiques largebande de l'environnement sur le productible (énergie incidente) en milieu urbain, phdthesis[END_REF] shows the impact that such patterns can have on the energy distribution within a street. Figure 1 The study of optical properties leads to select a set of materials and optical behaviour as reflection, transmission, absorption and diffusion. These properties are described through mathematical functions, whose name often appears in BxDF format for Bidirectional Distribution "x" Function, x standing for a specific optical behaviour. BxDF distribution functions are mainly used in computer graphics rendering [START_REF] Yafei | Accurate graphics BRDF material properties and illumination for dirt and mudbrick structures[END_REF]. However, these methods are being studied for a better assessment of the impact of the sun on heat transfer in urban areas, in particular for "glass" materials [START_REF] Wang | Simulation and validation of solar heat gain in real urban environments[END_REF] or Hight Reflective Materials (HMRs) used to reduce the Urban Heat Island (UHI) effect [START_REF] Yuan | Geometrical-optics analysis of reflective glass beads applied to building coatings[END_REF]. The visual rendering of virtual scenes is similar to solar light and natural lighting studies in urban environments, especially when surface properties are considered. It is then interesting to look at the functions used in the simulation of visual rendering. These functions include in particular: BTDF (transmittance), BRDF (reflectance), BSSRDF (surface scattering in reflection), BSSTDF (surface scattering in transmission) or BSDF (BSS-RDF and BSSTDF), as shown on Figure 2. These surface properties depend on incidence and viewing angles, wavelength, polarization, and also local space coordinates. When considering the simulation of light propagation in urban scenes, functions related to the BRDF family are commonly used [START_REF] Wu | Performance assessment of the BTDF data compression based on wavelet transforms in daylighting simulation[END_REF][START_REF] Wang | Simulation and validation of solar heat gain in real urban environments[END_REF]. Radiance software is a priori the most adapted to take into account such models but the handling is complex [START_REF] Lee | Modeling the direct sun component in buildings using matrix algebraic approaches: Methods and validation[END_REF]. Measuring reflectance data to obtain a realistic BRDF of outdoor environments is also a challenge today [START_REF] Jurado | An efficient method for acquisition of spec-tral BRDFs in real-world scenarios[END_REF]. The light reflected on the opaque materials is generally divided into two components: Lambertian retrodiffusion and specular reflection [START_REF] Yang | A modified version of BRDF model based on kubelka-munk theory for coating materials[END_REF]. These distribution functions are also used in yield calculations of bifacial photovoltaic modules [START_REF] Guerrero-Lemus | Bifacial solar photovoltaics -a technology review[END_REF], with the aim of studying the impact of local environment on the electricity production of the PV modules. Transmission functions (BTDF) are often used to evaluate the effects of glazing on interior lighting [START_REF] Ayoub | A review on light transport algorithms and simulation tools to model daylighting inside buildings[END_REF][START_REF] Lee | Modeling the direct sun component in buildings using matrix algebraic approaches: Methods and validation[END_REF]. [START_REF] Collin | Discrete ordinate method for polarized light transport solution and subsurface BRDF computation[END_REF] for such application. This is particularly true if incident light is described with polarisation vector decomposition in the simulation process, even if the human eye cannot distinguish light polarization.

Initial work from Phong results in a widely used method for computer generated pictures and visual rendering modelling [START_REF] Phong | Illumination for computer generated pictures[END_REF]. This approach is based on the reconstruction of the scene in the direction of the viewer, with contributions from specular and diffused light, as is the Lafortune model [START_REF] Lafortune | Non-linear approximation of reflectance functions[END_REF].

The Blinn-Phong model [START_REF] Blinn | Models of light reflection for computer synthesized pictures[END_REF] proposes a different formulation by including the consideration of the local nature of the surface with micro-facets. This model therefore corresponds well to a BRDF and thus ensures energy conservation. However, the specular term can become difficult to standardize.

This depends in particular on the function of the chosen specular term [START_REF] Blinn | Models of light reflection for computer synthesized pictures[END_REF].

Other models have thus been progressively developed in order to better take into account the specular reflection. The surface roughness of the materials and then the polarization of light were also integrated into the models. The model from Cook-Torrance-Sparrow [START_REF] Cook | A reflectance model for computer graphics[END_REF][START_REF] Torrance | Theory for off-specular reflection from roughened surfaces*[END_REF] is the first physical rendering BRDF model. The surface is assimilated to a set of elementary mirror surfaces respecting Fresnel's laws [START_REF] Born | Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light[END_REF]. Under these assumptions, the model corresponds to a statistical distribution of the orientation of the set of surfaces. Several other models are based on a theory of micro facets, like [START_REF] Kelemen | A microfacet based coupled specularmatte brdf model with importance sampling[END_REF].

Studies of solar potential in urban areas often use strong assumptions for reflection models while BRDF can encompass much more details but are sometimes impractical for ray tracing algorithms. We aim at designing a BRDF for solar potential evaluation that is suitable for ray tracing and at identifying its coefficients from spectrophotometry measurements.

The first part of this article presents the global methodology used for solar energy evaluation and more precisely, for the integration of measured optical surface properties. Then, Section 2 presents some classical BRDF models, their limitations regarding our objectives and the hypotheses, parameters and model used in this article, as long as the measurement methodology and model fitting strategy. The third section displays reflection's results obtained with the spectrophotometer on a selected cladding material sample (a white manufactured High Pressure Laminate (HPL)) panel with color coating: the identified BRDF is compared to reference reflection coefficients.

Then, Section 4 discusses the use of the proposed BRDF in a Ray-Tracing solar irradiation evaluation code and compares simulation results with experimental data on the same setup. Lastly, this study is concluded and some directions for future research are proposed.

Methodology

Overall methodology for solar irradiation evaluation. The aim of this work is to implement real properties of light reflection in solar energy evaluations but without complex light electromagnetic description. This approach was motivated by the optical reflection characterisation of several real construction materials. These measurements are analysed through models as simple and realistic as possible, and then used to feed a radiative transfer simulation tool. Analogous to existing irradiance models, a ray tracing method based on Monte Carlo statistical sampling allows us to calculate the incident flux in the visible wavelength range (380nm to 740nm). To get an efficient implementation of reflection properties in the ray tracing calculations, an integration of the reflections using Bidirectional Reflectance Distribution Functions (BRDF) is desired. However, for easy integration, it is necessary to reduce the number of parameters describing BRDF model. This will ensure rapid calculations when we will consider wide and dense urban areas for precise solar energy evaluation. Mainly used for computer graphic rendering, usual BRDF models are sometimes not strictly compliant with optical laws or are not physically sound for calculating heat transfer in the mid and far infrared spectral range.We choose here to build the BRDF function in the visible range combining specular and diffuse optical behaviors, making sure that light energy is conserved during reflection and propagation.

The BRDF model proposed in this article is intended to be integrated into the chain of calculation of solar radiation in urban areas (Figure 3). When simulating solar radiation, the optical properties of the materials are often neglected and only one coefficient like the albedo is taken into account.

The BRDF integration should allow a better evaluation of the geometric distribution of reflections. 22. We integrate as many characteristics of the sensors as possible in our calculations when identifying the parameters. However, some assumptions remain: the incident beam in the spectrophotometer is considered as perfectly parallel and the spectral response of the sensors is assumed perfect, which may induce a bias in spectral measurements, as will be discussed in 5.

The following workflow (Figure 4) summarises the approach proposed in this article:

Definition of a BRDF

In the following, a BRDF is defined as a ratio of reflected radiance dL o (Wm -2 sr -1 [START_REF] Langovoy | Empirical BRDF models for standard reference materials[END_REF][START_REF] Montes | An Overview of BRDF Models[END_REF][START_REF] Shi | Measurement and modeling of bidirectional reflectance distribution function (BRDF) on cutting surface based on the coaxial optical microscopic imaging[END_REF], power per unit area of reflective material per unit solid angle of observer) observed over an infinitesimal solid angle to incident irradiance dE i on the sample (Wm -2 , power per unit area of reflective material) :

f r = dL o dE i . (1)
Other sources use an equivalent different definition, but starting from the emitted radiance dL i by a light source:

f r = dL o dL i cos θ i dω i , (2)
as the received irradiance is linked to the light source radiance by:

dE i = dL i cos θ i dω i (3)
However, since our goal is a reconstruction of a BRDF from measurements made with a goniospectrophotometer, the incident irradiance dE i 0 (along the direction of the incident beam) is also constant and should not be confused with dE i , see Fig. 5:

dE i 0 = dL i dω i (4)
A BRDF function is expressed in sr -1 and describes the angular distribution resulting from the reflection of light reflected on a surface. As detailed above, the optical properties depend on the position of the reflection point and the angular position of the source and the observer. Only spatially uniform functions (uniform surface) are considered here and the dependence is limited to the zenithal incidence angle, θ i , the azimuthal incidence angle ϕ i , the reflected zenithal angle θ o and the reflected azimuthal angle ϕ o . The light polarization x can also be considered:

f r (θ i , ϕ i , θ o , ϕ o , x) = dL o (θ o , ϕ o , x) dE i (θ i , ϕ i , x) (5)
The goniospectrophotometer only performs measurements in the plane of incidence (containing both the normal to the plane of the sample and the direction of the incident light, see Figure 5). Using a spherical coordinate system with vertical direction normal to the sample, the dependency of the BRDF on ϕ i is not relevant. The dependency on ϕ o can be measured partially, as ϕ o is either 0 or π. The BRDF can then be written (omitting the polarization): According to the definition, a BRDF has to check the two different properties:

f r (θ i , θ o , ϕ o ) = dL o (θ o , ϕ o ) dE i (θ i ) (6)
• Helmholtz reciprocity:

f r (θ i , ϕ i , θ o , ϕ o ) = f r (θ o , ϕ o , θ i , ϕ i ) (7)
• Compatibility with energy conservation:

2π f r (θ i , ϕ i , θ o , ϕ o ) cos θ o dω o ≤ 1. (8)
3.2. BRDF measurements

Measurement protocol

The study presented in this article focuses on a sample of a white exterior cladding material from a recognized brand. In the following, the sample will be called "the sample".

Four different angles of incidence were used: θ inc = 20

Raw measures

In this article, only results obtained with one white cladding material are presented, see Figure 7. A behavior similar to what is discussed below has been observed for other samples, with varying proportions of the diffuse and specular components. On the measurements carried out, the energy distribution measured in reflection appears to be dependent on the S or P polarization of the incident light.

For both polarizations, a diffuse behaviour (the circular part of the curve), mixed with a specular-like one is observed. The intensity of the peaks decreases with the angle of incidence for the P-polarization and increases for the S-polarization. The position of the peak appears to match the theoretical reflection angle given by the Snell-Descartes laws, revealing a geometric effect in the energy distribution.

The amplitude of this reflection presents different attenuation phenomena depending on the polarization. For an increasing angle of incidence, the reflection of the P-polarized light decreases while that of the S-polarized light increases. This observation recalls the laws of geometric optics described by Fresnel. However, this specular effect is not geometrically perfect but appears to be smeared around the theoretical angle of reflection.

Measurements reading

The ARTA measures the fraction r of the energy reflected by the sample that is intercepted by the sensor. This is found (at least for the cladding sample) to depend on the angle on incidence θ i , the angle of observation

r(θ i , θ o , x) = dE o dE i = dL o dΩ dE i = f r (θ i , θ o , x)dω c . (9)
In the following, the link between the BRDF and ARTA measurements is established, then the a model BRDF is built, adapted from the Fresnel equations for the specular part.

BRDF model construction

In computer graphics, as Kumar [START_REF] Kumar | Surface texture evaluation using 3d reconstruction from images by parametric anisotropic BRDF[END_REF] points out in his article, it is usual to consider that the light reflection on a given surface is broken down into three From the observed behavior, our BRDF model will have a diffuse part and a smeard-specular term, similar to the Blinn-Phong model [START_REF] Blinn | Models of light reflection for computer synthesized pictures[END_REF], with the reflected radiant intensity written:

I o = I s o + I d o . (10)

Fresnel equations

Fresnel equations are used to describe the energy distribution of an electromatrical radiation at the interface of two optical materials. These equations detail the difference in polarization for reflected and transmitted energy fluxes. These equations are verified under the following assumptions: the interface between the two media is flat and both media are homogeneous and isotropic. Here, as both media are non-magnetic, their impedances only depend on the refractive indices of the two media (Z i = Z 0 n i ) and the refractive index of air is n 1 = 1. Eventually, only the reflectance behaviour is of interest here and writes:

R s (n 2 , θ i , θ o ) = cos θ i -n 2 1 -1 n 2 sin θ i 2 cos θ i + n 2 1 -1 n 2 sin θ i 2 2 (11)
R p (n 2 , θ i , θ o ) = 1 -1 n 2 sin θ i 2 -n 2 cos θ i 1 -1 n 2 sin θ i 2 + n 2 cos θ i 2 ( 12
)
Although it is not transparent, the sample has an extinction angle : for a given angle of incidence, the P-polarized light is not reflected. This angle, noted θ B is called the Brewster angle: θ B = atan(n 2 ). However, in the absence of fine increments in the angle of incidence around the Brewster angle, this relationship is impractical for determining the refractive index n 2 of the sample and in the following, n 2 will be kept as a parameter of our BRDF model.

Model for the smeared-specular term

The specular part is supposed to follow Fresnel's law hence: Φ s o = k s R x Φ i , with R x the Fresnel reflection coefficient for polarization x. Furthermore, the radiant intensity (W/sr) is supposed be a lobe centered on the specular direction:

dI s o (θ o , θ i , x) = ρ s R x α + 1 2π cos + (θ s ) α E i S i , (13)
with θ s the azimutal angle defined from the specular direction. For values of θ s larger than π/2, there is no contribution of specular reflection, as noted by the use of the positive part of the cosine. As in the Lafortune model, the specular term is described by two coefficients ρ s and α. The coefficient ρ s corresponds to the magnitude of the specular reflection whereas the coefficient α corresponds to the glossy effect of the specular reflectance and governs the width of the glossy lobe while α+1 2π is a ponderation factor. This specular model results in the following BRDF:

f s (θ i , θ o , x) = ρ s R x α + 1 2π cos + θ s (θ i , θ o ) α ( 14
)

Model for the diffuse term

The diffuse term corresponds to the incident light being scattered in all directions, with the same probability and independently of the polarization.

However, the radiant intensity depends on the observation angle. This is translated into Lambert's cosine law. A part ρ d of the incident flux is reemitted uniformly in all directions:

dI d o (θ o ) = ρ d 1 π cos(θ o )E i S i , (15)
associated with a diffuse part for the BRDF:

f d (θ i , θ o ) = ρ d 1 π cos(θ o ). ( 16
)

BRDF model and interpretation of the ARTA outputs

Both modes of reflection a brought together in the following BRDF model:

f r (θ i , θ o , x) = ρ d 1 π cos(θ o ) + ρ s R x α + 1 2π cos + θ s (θ i , θ o ) α . ( 17
)
The ARTA output is a ratio of energy, that is r = Φ o /Φ i . Assuming an infinitesimal angle of aperture dω c , Φ o = I o dω c . The ARTA measurements are performed with the receiver in the plane of incidence, that implies θ s = -θ o cos(φ o ) -θ i when φ o = π and from there:

r(θ i , θ o , φ o , x) = ρ d 1 π cos(θ o ) + ρ s R x α + 1 2π cos + (θ o -θ i ) α dω c . ( 18
)

reflection coefficient calculation

For the two BRDF components and for each polarization, the reflection coefficients are obtained by integrating the BRDF on the upper half-space of the sample. They depend on the wavelength. Their dependence in polarization and angle of incidence are taken into account through the Fresnel coefficient (Equation 19) :

k d (x) = π/2 θo=0 2π ϕ i =0 f d (θ i , θ o , φ o , x) sin θ o dθ o dφ o = ρ d (x) k s (θ i , x) = π/2 θo=0 2π ϕ i =0 f s (θ i , θ o , φ o , x) sin θ o dθ o dφ o = ρ s R x (θ i ) (19)
The global reflection coefficient is obtained by adding the diffuse and specular reflection coefficients (Equation 20) :

k g (θ i , x) = k d (x) + k s (θ i , x). ( 20
)
in order to take both polarizations into account in the global reflection coefficient, the contribution of each polarization is averaged (Equation 21):

k g (θ i ) = k g (θ i , S) + k g (θ i , P ) 2 . (21)
Those reflection coefficients still depend on the wavelength. When used in Section 5, they are averaged over the visible spectral range.

Model fitting

Model fitting for a small aperture sensor

To fit the parameters of the BRDF model, a least squares method is used between Eq (18) and the ARTA measures. The model is fitted over the entire set of measurements, i.e. all reflection and incidence angles. Concerning the polarization, the model is approximated either on both polarizations at the same time, or one at a time. The approximation of the model is performed using the Matlab function lsqcurvefit. This procedure lead to the identification of the parameters in our BRDF model. However, the smeared-specular term being quite narrow in width and the ARTA sensor having a rather large aperture are in contradiction with Eq (18) therefore it is substituted with a model taking into account the variability of the BRDF over the width of the ARTA sensor.

Correction of the model fitting

In this study, the sensor used by the ARTA has an aperture of 26mm in the plane of incidence and the height of this aperture is 18mm. In this configuration, this corresponds to an aperture ranging from -13 • to +13 • with respect to the defined measurement angle. Along to the axis orthogonal to the measurement plane, the opening angle is from -9 • to +9 • with respect to the fixed measurement angle. This gives an opening angle of the solid-state sensor dω c = 0.0516 sr. The variation of the modelled BRDF over the aperture angle can be taken into account. By minimizing the difference between the theoretical and actual measurements, the calibration of the BRDF model coefficients can be refined. Taking into account the aperture according to the azimuth angle only, the following relation is obtained (Equation 22) :

r(θ i , θ o , φ o , x) = dω c 2∆θ θo+∆θ θo-∆θ f r (θ i , θ, φ o , x)dθ, (22)
This new relationship [START_REF] Phong | Illumination for computer generated pictures[END_REF] is used in the model fitting procedure, instead of (18).

Estimating the uncertainties

By performing a non-linear regression, it is possible to evaluate the uncertainties of the parameters of the fitted model, i.e. ρ s , ρ p and α, as well as possibly n 2 . The use of the function nlparcy proposed by Matlab ® provides the confidence intervals 95% for each coefficient. By injecting these uncertainties into the coefficient calculations, we can evaluate the uncertainties for the coefficients of the final Lambertian and specular reflections.

However, the function nlparcy Matlab ® requires the covariance matrix or the Jacobian matrix from the BRDF data approximation. A use of the function nlinfit (also from Matlab ® ) is then necessary. For each wavelength, the uncertainties obtained on the different coefficients are propagated in the calculation of the final reflection coefficients.

4. Results: BRDF of a white cladding HPL pannel

Reference reflection coefficients

In order to verify the results obtained thanks to the BRDF model fitting, the samples are also tested using an integrating sphere mounted in a Lambda 950 Perking Elmer ® . This device measures the overall reflection coefficient (diffused plus specular) depending on the wavelength, under an incidence

θ i = 8 • .
The integrating sphere has an uncertainty of 1%. These measurements are then compared to the global reflection coefficient obtained from the BRDF fitting, as reconstituted using the ARTA measures, for different wavelengths, under the same incidence.

Results of model fitting

Results by polarization. The obtained BRDF, after model fitting, is plotted for each polarization (Figure 9). However, for the cladding sample, the differences between corrected and uncorrected models, remain tiny and imperceptible. Concerning the P polarization, and for an angle of incidence of 45 • , the model presents difficulties in correctly fitting the measurements.

The specular reflection seems to be too important in this configuration only. These fluctuations are essentially driven by the diffuse coefficient, the specular coefficient being constant over the wavelengths (Figure 10). Regarding the refractive index of the sample surface material, the trends are varied between the two polarizations. If the refractive index is almost constant for the S polarization, the refractive index is decreasing with longer wavelenght for the P polarization (Figure 12). This behavior (having a different refraction index for both polarizations) is observed for birefringing materials. However, here, the two different values come form identifying a parameter inside a model and are too far apart for the sample material to be considered as birefinging. The refractive index in the model can be forced to the same value for both polarizations and this yields a value close to 1.24 (Figure 12). Again, sharp changes around the 860nm wavelength due to sensor change are observed. The interest in making reflection models more complex in order to take into account different reflection modes seems limited, especially for large-scale simulations. Windows can generate significant reflections at certain angles of incidence and a better understanding of the reflection patterns of materials in the urban environment could provide more information on this point. On this point, these models can also be of interest for glare and visual comfort issues. They can also help to better simulate the spectrum incident on photovoltaic modules in order to better assess their integration and camouflage into the urban landscape.

Conclusion

This study considers the influence of light wavelength and polarization, angle of incidence and angle of observation on the reflectance of a cladding material used for building construction. This material appears highly reflective (to the eye) at grazing angles of incidence, rising the need for a BRDF model that is able to reproduce such a behavior. Inspired by the Fresnel equations, the specular term combined with a usual diffuse term in our BRDF model are able to match the observations carried out on the ARTA device. The largest discrepancy between the reconstructed BRDF and the measurements with the integrating sphere lies in the 400-860nm range of wavelength. Our best bet is that our calibration for this particular sensor is not accurate. With only two parameters (governing importance sampling in RT algorithms), our BRDF model can be used to verify the influence of the material properties on the energy distribution in an urban district, when the effect of such materials (sharp angular reflectance variations) is investigated.

One obvious question for further research is the wavelength and polarisation dependency of the BRDF. While it might not be of importance for energy simulations, visual integration of PV panels in the buildings' facades should strongly rely on color aspects. Then, our material sample has been tested as new. Exposure to the outdoor weather and solar rays will induce aging and maybe dirtiness, modifying the optical properties. Some further research is necessary to confirm that the specular part of the BRDF does not fade out with time.

  shows the extreme cases, where the reflections are either only diffuse (left figure) or only specular (right figure).

Figure 1 :

 1 Figure 1: Comparison of energy distributions as a function of reflection mode (diffuse in the left figure; specular in the right figure) (from Raybaud [11])

Figure 2 :

 2 Figure 2: Illustration of a BRDF and a BSSDF

Figure 3 :

 3 Figure 3: Evaluation strategy for simulating improved urban solar radiation taking into account the BRDF.

Figure 4 :

 4 Figure 4: Workflow of the approach proposed in this article.

Figure 5 :

 5 Figure 5: Illustration of the spatial variables used in the article.

  θ o and the polarization of light x. With dE i the incident energy (W/m 2 ) and dE o the energy intercepted by the sensor that has an angle of aperture dω c

Figure 7 :

 7 Figure 7: Measurements from ARTA: percentage of incident energy reflected by the sample

  modes (Figure (8)). These three terms are a specular term, a uniform term and a bright term. If the reflecting surface is itself a light source, a fourth term is considered. It represents the energy emitted by the surface itself. The uniform component describes the set of non-directional reflections, while the specular and bright components describe the set of reflections with a direction-dependent probability. The term specular describes the reflection resulting from Snell-Descartes laws. This mode of reflection is similar to a Dirac function in the theoretical direction of reflection. The term glossy represents a modulation around this theoretical direction.

Figure 8 :

 8 Figure 8: Schematic representation of the light reflection mode.

Figure 9 :

 9 Figure 9: Fitted model from the ARTA measurments

Figure 10 :Figure 11 :

 1011 Figure 10: Global reflection coefficients for both polarization depending of the wavelength obtained with the BRDF model, for non corrected and corrected parameters, compared to the Integrating Sphere measures.Final coefficients. For each wavelength, the model coefficients are approximated. Then, for both non corrected and corrected parameters, the model allows us to obtain the reflection coefficients as a function of the wavelength. reflection coefficients can be calculated as a function of polarization (Figure10), or averaged over the polarizations as shown Figure11. The reflection coefficient appears to be largely underestimated before the 860nm gap, with 10% of reflection less than the reference measurement. But the estimation is inside the uncertainty margin after this jump. The jump is due to the impact of the diffused reflection coefficient which is the principal influencer for the global reflection coefficient, whereas the specular reflection coefficient is relatively constant (around 4) over all wave lengths, except at 870nm and 880nm, near to the sensor change.

Figure 12 :

 12 Figure 12: Diffraction index fitted from the ARTA measurments.

Figure 13 :

 13 Figure 13: Energy contribution of specular reflections evaluated with a reflection coefficient of 4% corresponding to the identified value in the case of white cladding (from Raybaud [11])

  The results confirm the need to consider polarization and angular dependency when considering physical modeling of reflections, especially for angular energy distribution. This is usually done through ray-tracing algorithms, where a BRDF model is required. A new BRDF model, integrating the polarization dependence based on the Fresnel coefficients, is then proposed and validated on different samples. This model has only four parameters for each wavelength and polarization. Considering the usual split between diffuse and specular reflectance in light simulation, these four parameters can be reduced to two (diffuse and specular global reflection coefficients) keeping the same energy ratios.

  Requirements for a BRDF to be used in urban solar irradiation simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.1.1 Definition of a BRDF . . . . . . . . . . . . . . . . . . 10 3.2 BRDF measurements . . . . . . . . . . . . . . . . . . . . . . . 11 : reflection coefficients for the x th polarization from the Fresnel's law. • f r (θ i , θ o , x): BRDF function [sr -1 ]. • Z 0 : Impedance in vacuum. • Z x : Impedance of the x th medium. • n x : Refractive index of the x th medium.

	Contents 1. Nomenclature
	1 Nomenclature • θ o : Angle of measurement. • θ i : Angle of incidence.	• I s o : Specular component of 4 the reflected radiant intensity
	2 Introduction • θ mes,ini : First angle for which 3 Methodology a measurement in reflection is carried out (at fixed angle of in-cidence). • θ mes,f in : Last angle for which a measurement in reflection is carried out (at fixed angle of in-cidence). • θ step : Angular measurement pitch with the ARTA. • N : number of measurements performed (at fixed angle of in-cidence). • ϕ • L o : Radiance (power by unit area of reflecting material per 3.1 5 Discussion: using the BRDF in a solar irradiation simula-[W/sr]. • I s Diffuse component of 5 o : the reflected radiant intensity 7 [W/sr]. • M o : Outgoing power of reflect-ing material [W]. • M i : Incident power on reflect-ing material [W]. • R • ρ x : reflection index of the BRDF model, with x the corre-sponding component (specular tion 21 6 Conclusion 22 unit solid angle of observer) or diffused). [Wsr -1 m -2 ]. • α: Power coefficient of the • L i : Radiance (power by unit specular component of the area of reflecting material per BRDF function. unit solid angle of light source) • k x : Final reflection indexes ob-[Wsr -1 m -2 ]. tained from the BRDF model, • E i : Irradiance (power per unit with x the corresponding com-area at reflecting material sur-ponent. face) [Wm -2 ]. • k g : Global final reflection. • E i 0 : Irradiance (power per unit • S i : Illuminated surface of the area of incident beam cross-section) [Wm -2 ]. sample [m 2 ].
	7 Acknowledgements • I o : Reflected radiant intensity	23
	[W/sr].

3.2.1 Measurement protocol . . . . . . . . . . . . . . . . . . 11 3.2.2 Raw measures . . . . . . . . . . . . . . . . . . . . . . 12 3.2.3 Measurements reading . . . . . . . . . . . . . . . . . . 12 3.3 BRDF model construction . . . . . . . . . . . . . . . . . . . . 13 3.3.1 Fresnel equations . . . . . . . . . . . . . . . . . . . . . 14 3.3.2 Model for the smeared-specular term . . . . . . . . . . 14 3.3.3 Model for the diffuse term . . . . . . . . . . . . . . . . 15 3.3.4 BRDF model and interpretation of the ARTA outputs 15 3.3.5 reflection coefficient calculation . . . . . . . . . . . . . 15 3.4 Model fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.4.1 Model fitting for a small aperture sensor . . . . . . . . 16 3.4.2 Correction of the model fitting . . . . . . . . . . . . . 16 3.4.3 Estimating the uncertainties . . . . . . . . . . . . . . 17

4 Results: BRDF of a white cladding HPL pannel 17 4.1 Reference reflection coefficients . . . . . . . . . . . . . . . . . 17

4.2 Results of model fitting . . . . . . . . . . . . . . . . . . . . . 17 o : Angle in which the observer is located in the plane of the sample, in relation to the axis #» y . • ϕ i : Angle of the incidence ray in the plane of the sample, in relation to the axis #» y . x

  • , 30 • , 45 • , 55 • . The angle of incidence is noted θ inc .For each angle of incidence, different angles of measure (θ o ) are used, staggered in steps of θ step = 3 • . Because of the physical dimensions of the apparatus, the measurement angle cannot be closer than 15 • to the angle of incidence. Table6indicates the number of measurements made for each configuration. The first column titled θ inc corresponds to the angle of incidence of the sample. The columns θ mes,ini and θ mes,f in correspond respectively to the minimum and maximum angles (with respect to the direction of incidence) for which a reflectance measurement was made. θ step indicates the measurement angle increment. θ inc θ mes,ini θ mes,f in θ step Table of different measurement configurations.

	20	15	105	3
	30	15	114	3
	45	15	126	3
	55	15	141	3
	Figure 6:

Acknowledgements

The authors would like to thank the French Alternative Energies and Atomic Energy Commission (CEA) for funding this research. The study presented in this manuscript is based on the PhD thesis of Raybaud Blaise [11].