F Labrell

H Câmara-Costa

C Dufour

J Grill

M Chevignard

Maternal Stress and Paediatric Brain Cancer: a French study

Keywords: Maternal Stress, Paediatric Brain Tumours, Assessment, Standards of Care

Background: Despite evidence that paediatric cancer induces a trajectory of parental stress, studies in case of childhood malignant brain tumours remain scarce. The present work aims to add to the scientific literature by evaluating maternal stress with a French translation of the Paediatric Inventory for Parents (PIP) in a population of mothers whose children have been diagnosed with a brain tumour. Procedure: The mothers of 35 children with malignant brain tumours completed the Paediatric Inventory for Parents (PIP), the State-Trait Anxiety Inventory (STAI) and the Family Assessment Device (FAD) at a mean time since diagnosis of 2.3 years.

Results:

The PIP showed good internal consistency and correlated significantly with the STAI and the FAD. Maternal stress scores were higher in younger mothers and in mothers of younger children. Emotional functioning was the most affected domain.

Conclusions:

Emotional stress as assessed by the PIP in mothers of children with malignant brain tumours is considerable, and should be routinely assessed, in order to implement appropriate specific psychosocial interventions when needed.

URL: http://mc.manuscriptcentral.com/wjpo Email: karen.kayser@louisville.edu Journal of Psychosocial Oncology

MATERNAL STRESS AND PEDIATRIC BRAIN CANCER Maternal Stress and Paediatric Brain Cancer: a French study

In France, an estimated one out of every 440 children is diagnosed with cancer before the age of 15, the most frequent paediatric cancers being leukaemia (29%), central nervous system tumour (23%), lymphoma (12%) and neuroblastoma (8%) [START_REF] Lacour | Incidence of childhood cancer in France: National Children Cancer Registries, 2000-2004[END_REF] . Between 2000 and 2008, the overall survival was 91.7% at 1 year, 86.9% at 2 years and 81.6% at 5 years 2 for paediatric cancers in general, and 84.3% at 1 year, 77.5% at 2 years and 71.9% for Central Nervous System (CNS) malignancies, such as brain tumours. Given the lack of studies on parental stress in case of paediatric cancer in France [START_REF] Labrell | Évaluation du stress parental en cas de cancer de l'enfant[END_REF] , the present study aimed to evaluate parental, and here maternal stress using the Paediatric Inventory for Parents (PIP) [START_REF] Streisand | Childhood illness-related parenting stress: the pediatric inventory for parents[END_REF] in a population of French mothers whose children had been diagnosed with a brain tumour. Indeed, since mothers are more frequently present than fathers during the medical visits for paediatric cancer, only mothers have been envisaged in this study.

Childhood cancer is a highly stressful experience for the family, including the child, his/her parents and the siblings [START_REF] Alderfer | Family functioning and posttraumatic stress disorder in adolescent survivors of childhood cancer[END_REF][START_REF] Cernvall | Experiential Avoidance and Rumination in Parents of Children on Cancer Treatment: Relationships with Posttraumatic Stress Symptoms and Symptoms of Depression[END_REF][START_REF] Dunn | Posttraumatic stress symptoms in parents of children with cancer within six months of diagnosis[END_REF][START_REF] Haegen | Stress, Psychosocial Mediators, and Cognitive Mediators in Parents of Child Cancer Patients and Cancer Survivors: Attention and Working Memory Pathway Perspectives[END_REF][START_REF] Long | Family Adjustment to Childhood Cancer: A Systematic Review[END_REF][START_REF] Okado | Association Between Parent and Child Distress and the Moderating Effects of Life Events in Families With and Without a History of Pediatric Cancer[END_REF][START_REF] Rodriguez | Cancer-Related Sources of Stress for Children With Cancer and Their Parents[END_REF][START_REF] Streisand | Pediatric-Specific Parenting Stress and Family Functioning in Parents of Children Treated for Cancer[END_REF][START_REF] Vrijmoet-Wiersma | Assessment of parental psychological stress in pediatric cancer: a review[END_REF] . Paediatric cancer generates a trajectory of stress from the diagnostic announcement throughout treatment, remission and possible relapses [START_REF] Eccleston | Psychological interventions for parents of children and adolescents with chronic illness[END_REF] . The psychological repercussions on parents of children diagnosed with cancer, such as anxiety or depression, are considerably greater compared to other paediatric illnesses [START_REF] Dunn | Posttraumatic stress symptoms in parents of children with cancer within six months of diagnosis[END_REF][START_REF] Vrijmoet-Wiersma | Assessment of parental psychological stress in pediatric cancer: a review[END_REF] . Because parental and child mental health are interconnected [START_REF] Wickramaratne | Children of Depressed Mothers 1 Year After Remission of Maternal Depression: Findings From the STAR*D-Child Study[END_REF] , the need to support parental mental health has long been considered essential [START_REF] Kazak | An Integrative Model of Pediatric Medical Traumatic Stress[END_REF] . Indeed, parental stress is an important dimension of parental mental health, since parents frequently experience psychological difficulties in the presence of serious health problems in their child [START_REF] Bruce | A systematic and conceptual review of posttraumatic stress in childhood cancer survivors and their parents[END_REF][START_REF] Hall | Posttraumatic Stress Symptoms in Parents of Children with Acute Burns[END_REF][START_REF] Van Oers | Anxiety and Depression in Mothers and Fathers of a Chronically Ill Child[END_REF] , even if several factors, such as family functioning, can be protective [START_REF] Yeh | Adaptational process of parents of pediatric oncology patients[END_REF] . This issue has recently led researchers to propose the Psychosocial Standards of Care for Childhood Cancer based on a systematic literature review focused particularly on parents of children with cancer [START_REF] Kearney | Standards of Psychosocial Care for Parents of Children With Cancer: Standards of Psychosocial Care for Parents of Children With Cancer[END_REF] . An important recommendation is MATERNAL STRESS AND PEDIATRIC BRAIN CANCER the assessment of parental mental health, in order to define and implement appropriate interventions according to needs.

The large majority of studies on parental stress consider paediatric cancers in general.

However, children with brain tumours have to face particular treatments (neurosurgery, chemotherapy, and/or high-dose central nervous system irradiation) and treatment-related sequelae, both physical and neurocognitive (including impaired health, endocrine deficits, cognitive deficits, learning difficulties) [START_REF] Kearney | Standards of Psychosocial Care for Parents of Children With Cancer: Standards of Psychosocial Care for Parents of Children With Cancer[END_REF][START_REF] Ne | Late complications in childhood central nervous system tumour survivors[END_REF][START_REF] Armstrong | Long-Term Outcomes Among Adult Survivors of Childhood Central Nervous System Malignancies in the Childhood Cancer Survivor Study[END_REF][START_REF] Boman | Health and persistent functional late effects in adult survivors of childhood CNS tumours: a population-based cohort study[END_REF][START_REF] Chevignard | Core deficits and quality of survival after childhood medulloblastoma: a review[END_REF][START_REF] Mulhern | Late neurocognitive sequelae in survivors of brain tumours in childhood[END_REF][START_REF] Radcliffe | Adjustment in childhood brain tumor survival: child, mother, and teacher report[END_REF] . These specific sequelae of brain injury have significant long-lasting impact on the child's academic achievement, independence, social integration and overall participation in society. In addition, research has mainly focused on treatment risk and less on the role of family concerns [START_REF] Ach | Family factors associated with academic achievement deficits in pediatric brain tumor survivors: Academic achievement in survivors[END_REF] . For instance, parents of children with brain tumours not only feel distressed about their child's later school achievement after recovery [START_REF] Hutchinson | Stress and Coping in Caregivers of Pediatric Cancer Survivors[END_REF] , they also face also long-term uncertainty and the burden of caregiving, even when their children are off treatment [START_REF] Hutchinson | Adjustment of caregivers of pediatric patients with brain tumors: a cross-sectional analysis[END_REF] . Overall, there is not enough research addressing stress among mothers of children with brain tumours, which account for 30 % of paediatric cancers in France 1 .

The assessment instruments for evaluating maternal stress generally do not take into account the particularities of paediatric cancer [START_REF] Van Der Geest | Parenting stress as a mediator of parents' negative mood state and behavior problems in children with newly diagnosed cancer[END_REF] , even less in the case of brain cancers which generate parental concern on two counts: the impact of cancer, on the one hand, and of the brain injury on the other. In the present study, we opted to use the Paediatric Inventory for Parents (PIP 4 ) as a measure of maternal stress. Although this questionnaire was not specifically developed for paediatric cancer, it was elaborated to evaluate parental stress related to severe illnesses and malignancies (e.g. leukaemia, lymphoma), and designed to explore specific categories of extreme stressors for parents (emotional functioning, medical care, communication, role function). Results from studies performed using the PIP have highlighted associations of parental stress with parental anxiety [START_REF] Streisand | Childhood illness-related parenting stress: the pediatric inventory for parents[END_REF] , and aspects of family URL: http://mc.manuscriptcentral.com/wjpo Email: karen.kayser@louisville.edu

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functioning such as communication, family roles and affective responsiveness [START_REF] Streisand | Pediatric-Specific Parenting Stress and Family Functioning in Parents of Children Treated for Cancer[END_REF] . Further, the PIP was identified as one of the few assessment instruments currently available to evaluate parental stress related to paediatric cancer in a recent literature review [START_REF] Labrell | Évaluation du stress parental en cas de cancer de l'enfant[END_REF] . For these reasons, we considered the PIP the most suitable instrument to evaluate maternal stress in the present work.

METHODS

Participants

Participants were selected from two hospitals near Paris specialized in the treatment of paediatric cancers (Gustave Roussy) and in the rehabilitation and follow-up of children with acquired brain injury, including brain tumours (Saint Maurice Hospital). Inclusion criteria were having a malignant brain tumour at least one month from diagnosis and, for the mothers, being able to understand, speak and read French. Of the 48 eligible mothers who were approached, 35 gave their informed consent and participated in the study, which provided a participation rate of 73 %. All questionnaires were rendered anonymous in order to keep data analysis confidential.

The mean age of the 35 children (19 boys, 16 girls) was 9.9 years (SD = 5.1 years, range 2.5 -22 years). The mean time since diagnosis was 2.3 years (SD = 3.05, range: 2 months-14 years). At the time of the study, 20 children were under treatment and 15 had finished their treatment.

Procedure

The study was approved by the medical ethics committees at both participating institutions.

Eligible mothers received information about the study and a consent form. They were contacted on the occasion of a medical consultation either by the treating physician or by the researchers. The mothers could complete the questionnaires directly in the waiting room of the hospital or complete them later at home and send them back by post. In both cases, the URL: http://mc.manuscriptcentral.com/wjpo Email: karen.kayser@louisville.edu

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questionnaires and aims of the study were explained to them. In addition, medical data was obtained from clinical report forms addressed to the physicians and sociodemographic data was obtained through the hospital medical reports by the researchers.

Measures

In addition to the PIP, the mothers were invited to complete two other questionnaires to assess the construct validity of the French version of the PIP, namely the State Trait Anxiety Inventory (STAI) and the Family Assessment Device (FAD). Both tools have been found to be correlated with the PIP evaluation [START_REF] Streisand | Pediatric-Specific Parenting Stress and Family Functioning in Parents of Children Treated for Cancer[END_REF][START_REF] Vrijmoet-Wiersma | Psychometric qualities of the Dutch version of the Pediatric Inventory for Parents (PIP): a multi-center study[END_REF] .

PIP

The PIP is a 42-item self-report questionnaire that measures parental stress. It comprises four domains: Communication with the child and the medical team (COM, 9 items; e.g. 'Speaking with the doctor'), Medical Care (MEDI, 8 items; e.g. 'Helping my child with medical procedures (having injections, swallowing medicine, changing dressings)'), Parental Role (ROLE, 10 items; e.g. 'Feeling uncertain about disciplining my child') and Emotional Functioning (EMO, 15 items; e.g. 'Worrying about the long-term impact of the illness').

Each of the 42 items is rated on two 5-point Likert-type Scales, one assessing the frequency of each stressor during the last week, and one assessing how difficult the issue has been for them as a parent.

The English-language PIP has recently been rated 6 on a 7-point level of evidence scale [START_REF] Guyatt | GRADE: an emerging consensus on rating quality of evidence and strength of recommendations[END_REF] and is a well-known tool present in reviews on family changes in case of childhood cancer [START_REF] Long | Family Adjustment to Childhood Cancer: A Systematic Review[END_REF][START_REF] Kearney | Standards of Psychosocial Care for Parents of Children With Cancer: Standards of Psychosocial Care for Parents of Children With Cancer[END_REF] .

Translation of the PIP into French. After contacting the first author of the PIP, this questionnaire was translated into French by a team of three people. An English native speaker was asked to perform a back translation. The translated version was pilot-tested with 20 families who were invited to point out any items that might be difficult to understand, so as to change the wordings if required. No adaptation proved necessary.

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STAI

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The French version of the State-Trait Anxiety Index 34 is a 40-item questionnaire assessing the respondent's present state of stress (State, 20 items) and his or her general inclination toward anxiety (Trait, 20 items). The STAI has high internal consistency and appropriate construct and discriminative validity across typically-developing and clinical samples [START_REF] Barnes | Reliability Generalization of Scores on the Spielberger State-Trait Anxiety Inventory[END_REF] , including families of childhood cancer survivors [START_REF] Kazak | Predicting posttraumatic stress symptoms in mothers and fathers of survivors of childhood cancers[END_REF] . The mothers were asked to rate each statement on a four-point scale ranging from 1 (not at all) to 4 (very much so) for the State scale, and from 1 (almost never) to 4 (almost always) for the Trait scale.

FAD

The short form of the French version of the Family Assessment Device (FAD) [START_REF] Speranza | The French version of the Family Assessment Device[END_REF] is a 12-item self-report measure of family functioning comprising six subscales (Problem Solving, Communication, Roles, Affective Responsiveness, Affective Involvement, and Behavioural Control) and an overall General Functioning scale. Previous research has demonstrated the adequacy of the psychometric properties of the FAD and its usefulness as a measure of family functioning in nonclinical and medical samples [START_REF] Speranza | The French version of the Family Assessment Device[END_REF][START_REF] De Haan | Reliability and validity of a short version of the general functioning subscale of the McMaster Family Assessment Device[END_REF] , such as in the case of pediatric cancer [START_REF] Alderfer | Family functioning and posttraumatic stress disorder in adolescent survivors of childhood cancer[END_REF] .

In the present study, the mothers were solicited to rate the extent to which each statement described their family functioning on a four-point scale ranging from 1 (strongly agree) to 4 (strongly disagree).

Data analysis

Statistical analyses were performed using SAS software, version 9. Missing data was observed for some variables of interest, namely parental education level and marital status (n=3). For this reason, the analyses described hereafter were performed with the maximum number of observations for each variable included in the analyses and are reported in each table. For the FAD, the total score corresponded to the difference between the sum of the positive worded items and the sum of negative worded items, with higher scores indicating better family functioning (maximum score =18). For the STAI, we computed a total score by

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summing the items of the Trait and the State scales (maximum score for each scale= 80), with lower scores indicating reduced levels of anxiety. For the PIP, Frequency and Difficulty subtotal scores were obtained by summing the items that composed each subscale (maximum total score = 42). In addition, we adjusted on the mean values for each subscale by dividing the sum of responses to each subscale item by the total number of items composing the subscale, which rendered the mean values between subscales comparable. For educational level, we generated a binary variable defining whether the mother's current occupation required the completion of a high school diploma (high) or not (low). We opted to define parental educational level based on the mother's current occupation, as sociodemographic information was collected through the hospital's medical records, containing only the register of the mother's occupation.

T-tests, correlation analyses (Pearson's r) and Generalized Linear Models (GLM) were used to examine differences in the responses to the questionnaires according to socio-demographic and medical characteristics of the mother and the child, as well as differences across the four domains (COM, MEDI, ROLE, EMO) in each subscale (paired T tests). Correlation analyses were also used to examine associations between questionnaires. We considered p < .01 as the level required for statistical significance, although we also present results of marginal significance (p < .05).

RESULTS

Demographic characteristics

Mother and child demographic characteristics (see Table 1 Internal consistency values for these subscales were in the acceptable range (α ranging from .63 to .93, see Table 2), as were the reliability indexes for the subscale scores for Frequency and Difficulty (α=.87 and .93, respectively).

Associations between the PIP (Frequency and Difficulty), the FAD (total score) and the

STAI (Trait and State)

PIP-Frequency was significantly correlated with all other scores, whereas the PIP-Difficulty was only significantly correlated with PIP-Frequency and STAI-State (see Table 3). The correlation between STAI-Trait and STAI-State was strong. The FAD was negatively correlated with the STAI-T.

Subscales in the PIP

As presented in Table 2, adjusted means (on the number of items) for the subscale scores ranged from 2.01 to 3.08 for the responses pertaining to Frequency, and from 3.05 to 3.67 for the responses indicating the degree of Difficulty. The subscale adjusted means for Frequency and Difficulty were 2.48 and 3.36, respectively.

The highest value for the adjusted means was observed for the Emotional functioning domain (EMO), followed by Parental Role (ROLE), Communication (COM) and Medical care (MEDI). The total Difficulty score was significantly higher than the total Frequency score. EMO scores were higher than scores for the other three domains (see table 2 for more details).

Item analysis: the ten major stressors among French mothers

For each subscale (PIP-F, PIP-D), we identified the ten most prominent stressors according to the rank of their means. Table 4 shows that the first eight stressors in each subscale concern URL: http://mc.manuscriptcentral.com/wjpo Email: karen.kayser@louisville.edu

MATERNAL STRESS AND PEDIATRIC BRAIN CANCER

the EMO domain. Five of the first ten stressors were the same in the two subscales. They firstly relate to mothers' worries about the long-term impact of the illness and the future (rank 1 for both scales, ranks 3 and 5 for Frequency and Difficulty, respectively), including the prospect of the child's death (ranks 2 and 3 respectively). Secondly, these stressors pertain to the present situation of the sick child ('feeling helpless', ranks 4 and 6 respectively, 'waiting for results', ranks 8 and 10 respectively).

The other stressors that also ranked in the first ten places (out of 42 items) relate to the ROLE domain ('disciplining the child' ranks 9 for Frequency, 'less time for the family' ranks 9 for Difficulty) and the COMmunication domain ('speaking about the illness' ranks 10 for Difficulty). The domain of medical care does not appear in this ranking.

DISCUSSION

Parental stress has recently been conceptualized as a fundamental dimension of mental health in the recommendations for psychosocial care of parents with children diagnosed with cancer, because of the significant psychological difficulties frequently experienced by parents [START_REF] Dunn | Posttraumatic stress symptoms in parents of children with cancer within six months of diagnosis[END_REF][START_REF] Vrijmoet-Wiersma | Assessment of parental psychological stress in pediatric cancer: a review[END_REF] .

Because parental stress associated with paediatric brain cancer has been shown to have an impact on specific concerns linked to both survival and brain injury, the assessment of stress in the present work was performed on a French population of mothers whose children had been diagnosed with a brain tumour.

Our results indicated that the Frequency and Difficulty subscales of the PIP were significantly correlated, and PIP internal consistency was acceptable. These results corroborate the findings of previous studies performed in other countries [START_REF] Streisand | Childhood illness-related parenting stress: the pediatric inventory for parents[END_REF][START_REF] Vrijmoet-Wiersma | Psychometric qualities of the Dutch version of the Pediatric Inventory for Parents (PIP): a multi-center study[END_REF] (for results with American and Dutch samples, respectively). Interestingly, in our study, focused on children with brain tumours, the Frequency and Difficulty subscale scores tended to be higher than those observed in the abovementioned studies [START_REF] Streisand | Childhood illness-related parenting stress: the pediatric inventory for parents[END_REF][START_REF] Vrijmoet-Wiersma | Psychometric qualities of the Dutch version of the Pediatric Inventory for Parents (PIP): a multi-center study[END_REF] , especially for the Difficulty subscale (although no statistical analyses were performed). This result could suggest that mothers of children with brain URL: http://mc.manuscriptcentral.com/wjpo Email: karen.kayser@louisville.edu

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cancer might indeed experience greater levels of stress due to the severe implications of the brain injury and its treatment, in addition to those of cancer per se, when compared with other cancers. This particular finding underlines the relevance of the PIP as an assessment instrument to evaluate parental stress in paediatric brain cancer, and suggests that it might allow for discrimination between discrete levels of parental stress according to distinct paediatric conditions.

The PIP scores, especially on the Frequency subscale, were correlated with anxiety and family functioning scores, meaning that maternal stress overlaps with anxiety and a lack of general wellbeing in the family. These results are in line with earlier findings [START_REF] Streisand | Childhood illness-related parenting stress: the pediatric inventory for parents[END_REF] and also validate the PIP as a measure of stress in specific disease-related situations. It should be noted that in the present study the FAD was strongly negatively-correlated with the STAI-Trait, suggesting that the general tendency towards anxiety observed among mothers might be linked to family cohesion.

The four domains of stress (communication, medical care, parental role and emotional functioning) did not seem to be affected to the same degree in this sample of mothers of children with malignant brain tumours, which confirms our hypothesis that some stressors may be typical of paediatric brain cancer. More precisely, the emotional domain (EMO) appeared to be the most affected, with maternal concerns about survival and the long-term consequences of the brain injury (as demonstrated by the item analysis). For instance, a young mother of a 2-year-old child related informally how the announcement of her child's brain tumour was so hard that she had a miscarriage at 7 months of pregnancy (this exchange was not part of the questionnaire). Worrying about the child's future and feeling helpless over the child's condition reflects parental concern regarding the child's autonomy and learning, areas that tend to be significantly impacted by brain tumours [START_REF] Chevignard | Core deficits and quality of survival after childhood medulloblastoma: a review[END_REF][START_REF] Radcliffe | Adjustment in childhood brain tumor survival: child, mother, and teacher report[END_REF][START_REF] Bhat | Profile of Daily Life in Children With Brain Tumors: An Assessment of Health-Related Quality of Life[END_REF][START_REF] Schulte | Social competence in childhood brain tumor survivors: a comprehensive review[END_REF]

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coping has also been found to be linked to high level of stress in caregivers of children with brain tumours [START_REF] Bennett | Predicting parenting stress in caregivers of children with brain tumours: Parenting stress in caregivers of children with brain tumours[END_REF] .

Together with the Emotional domain (EMO), parental role (ROLE) appeared to be more affected than the communication and the medical care domains in the Frequency subscale.

The item analysis revealed that maternal concerns were mostly about "time for herself", but also for the other family members. A similar result was also found as the first concern mentioned by parents in paediatric cancers [START_REF] Bhat | Profile of Daily Life in Children With Brain Tumors: An Assessment of Health-Related Quality of Life[END_REF] .

Another subdomain that emerged as a major stressor in the parental role (ROLE) domain was the difficulty educating the child ("disciplining the child"), i.e. maintaining parental function.

This result is congruent with previous studies [START_REF] Schulte | Social competence in childhood brain tumor survivors: a comprehensive review[END_REF] concerning the need for aftercare for paediatric brain tumour survivors, whose families reported difficulties in their ability to encourage autonomy in their child [START_REF] Schulte | Social competence in childhood brain tumor survivors: a comprehensive review[END_REF] and the tendency to be overprotective [START_REF] Long | Family Adjustment to Childhood Cancer: A Systematic Review[END_REF] . This particular finding is also in line with clinical observations regarding some parents who are able to lower their expectations regarding their child's academic success, because they are aware that the brain tumour might impair his/her learning. Results about the parental role as a major stressor are congruent with those about the role of parental cognitions like parental (external) locus of control in caregivers of children with brain tumours [START_REF] Bennett | Predicting parenting stress in caregivers of children with brain tumours: Parenting stress in caregivers of children with brain tumours[END_REF] .

It is important to note that the PIP-Frequency score was linked to maternal age, with older mothers reporting less stress than younger ones, in agreement with previous results [START_REF] Streisand | Childhood illness-related parenting stress: the pediatric inventory for parents[END_REF] . We can hypothesise that older mothers may have better knowledge on the subject of brain tumours, as well as more experience in coping with their child's illnesses. Additionally, these "older mothers" might gain the most benefit from social support networks. In agreement with the abovementioned previous study, the PIP scores were not linked to marital status nor to educational level, but to the child's age at diagnosis, with mothers of younger children reporting more stress than mothers of older children.

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Limitations

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Page 12 of 21

The number of participants was small and we were not able to investigate if the nonrespondent mothers were less educated and less able to answer written instructions. Another limitation concerns the significant variability of time since diagnosis, which should be taken into account. Maternal stress was assessed here for children who were mostly under treatment, some with relapse, and a few in remission. Given that maternal stress may be impacted by the history of the illness, a more uniform patient group (e.g. immediately after the diagnosis, just after the end of the treatment, some years after the end of treatment) would be more relevant in future studies. In addition, the lack of normative data, and of a comparison group, limits the extent to which our results can be interpreted in relation to normative and/or clinical populations.

Clinical implications

The clinical and practical implications of the present results are noteworthy. The subdivision of the PIP subscales into Frequency and Difficulty, each subdivided into four domains, could be useful to tailor psychosocial interventions, backed up by the clinical impression of the therapist/clinician assessing the situation. As the emotional domain (EMO) and the parental role (ROLE) domain appeared to be major stressors for mothers, emotional distress calls for psychological interventions, while parental role difficulties could warrant social interventions.

Previous studies have underlined the fact that psychosocial interventions among parents of children with cancer, especially brain tumour survivors, and aimed at strengthening parental skills in coping with their children's disabilities, have the potential to reduce parental stress [START_REF] Bennett | Predicting parenting stress in caregivers of children with brain tumours: Parenting stress in caregivers of children with brain tumours[END_REF][START_REF] Norberg | Experiences of parents of children surviving brain tumour: a happy ending and a rough beginning[END_REF] and to reinforce family quality of life following the completion of treatment [START_REF] Quast | Health-related quality of life in parents of pediatric brain tumor survivors at the end of tumor-directed therapy[END_REF] . The present study on maternal stress suggests that interventions targeting the reduction of emotional distress might also help mothers during the child's treatment.

Conclusion

Table 4

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The first ten stressors in each subscale of the French PIP.

  ) were examined in relation to the PIP responses by computing t-tests and correlations across variables. The Frequency subscale score on the PIP was associated with maternal age (higher scores for younger mothers: r = -52, p = .011) and with the child's age at diagnosis (higher scores for younger children: r = -.53, p = .009). No other significant differences were observed in the mothers' responses to the URL: http://mc.manuscriptcentral.com/wjpo Email: karen.kayser@louisville.eduJournal of Psychosocial OncologyMATERNAL STRESS AND PEDIATRIC BRAIN CANCERPIP, (either for the Frequency or the Difficulty subscale scores) according to the mother and the child's socio-demographic characteristics.Construct validity of the PIPCorrelations across the four subscales of the PIP varied from .46 to .74 for the Frequency dimension and from .70 to .86 for the Difficulty dimension.

  URL: http://mc.manuscriptcentral.com/wjpo Email: karen.kayser@louisville.edu Page 13 of Journal of Psychosocial Oncology MATERNAL STRESS AND PEDIATRIC BRAIN CANCER The present study confirms the importance of assessing maternal stress in children with malignant brain tumours using an instrument that captures the specific patterns of distress experienced by these caregivers, in terms of emotional functioning and parental role. The French PIP appears to be valid assessment tool to gain insight into stress experienced by mothers in the course of their child's illness. Scores for stress on the PIP could help to develop specific interventions. As evidenced in recent research, psychosocial interventions among parents of children with cancer, especially brain tumour survivors, can strengthen caregiver competence and family functioning following the completion of treatment 45 .

Table caption Journal of Psychosocial Oncology MATERNAL STRESS AND PEDIATRIC BRAIN CANCERTable 1 .

 caption1 Demographic characteristics of mothers and children.

Table 2 .

 2 Adjusted means of the 4 PIP subscales and internal consistency of the 2 subscale scores (Frequency and Difficulty).

Table 3 .

 3 Pearson's correlations between PIP (Frequency, Difficulty), STAI (State, Trait) and FAD.

Table 4 .

 4 The first ten stressors in each of the French PIP subscales.

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Table 1

 1 Journal of Psychosocial Oncology

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	Demographic characteristics of mothers and children.
		M (SD)	N (%)
	Mothers
	Age	39.4 (5.9)	35
	Marital status
	Married/living with partner		29 (91)
	Divorced/widowed/single		3
	Educational level
	High		16 (50)
	Low		16 (50)
	Children
	Gender
	Male		19 (54)
	Female		16 (46)
	Age at assessment (years)	8.9 (5.1)
	Age at diagnosis (years)	7.4 (4.9)
	Time since diagnosis (years)	2.3 (3.0)
	Siblings
	No sibling		0
	1 sibling		6
	2 or more siblings		26 (81)
	Relapse (Yes)		9 (26)
	Under treatment (Yes)		20 (57)
	Motor or cognitive sequelae (presence)		19 (54)

Table 2

 2 Journal of Psychosocial Oncology Frequency; PIP-D: Difficulty; N: number of participants; M: adjusted mean;  : Cronbach's coefficient Results from paired T tests showed that EMO > MEDI, COM, p<.0001; EMO > ROLE, p<.001; ROLE > MEDI, COM, p<.001 for PIP-F; for PIP-D, T tests showed that EMO > MEDI, ROLE, COM, p<.001.

		MATERNAL STRESS AND PEDIATRIC BRAIN CANCER
	Adjusted means of the 4 PIP subscales and internal consistency of the 2 subscale scores
	(Frequency and Difficulty).
			French (N=34)
		PIP-F		PIP-D
		M		M	
	MEDI ROLE EMO PIP-F: COM Total	2.22 2.01 2.59 3.08 2.48	.71 .63 .79 .86 .87	3.22 3.05 3.18 3.67 3.36	.80 .85 .77 .87 .93

20

URL: http://mc.manuscriptcentral.com/wjpo Email: karen.kayser@louisville.edu

Table 3

 3 Journal of Psychosocial OncologyMATERNAL STRESS AND PEDIATRIC BRAIN CANCERPearson's correlations between PIP (Frequency, Difficulty), STAI (State, Trait) and FAD. Pearson's coefficient of correlation.

	Page 20 of
	21

*p < .05; ** p< .01; *** p < .001; N: number of participants; r: results The stressors common to both subscales are in bold. 21 URL: http://mc.manuscriptcentral.com/wjpo Email: karen.kayser@louisville.edu Page 21 of

Journal of Psychosocial Oncology

Acknowledgements

The authors would like to thank the parents for taking the time to complete the questionnaires.

Secondly, they express their thanks to the members of the translation team (Mrs Angela Swaine Verdier) and to the staff of the two hospitals (Dr. Anne Laurent-Vannier, Dr Dominique Brugel, Dr. Hanna Toure, Dr. Hina Simonnet, Carline Bernard, Charlyne Duwyme, Emel Uçak, V. Kieffer). Finally, the authors thank Dr. Randi Streisand for her consent for the translation of the PIP into French.


Shahram Khazaie
email: shahram.khazaie@univ-nantes.fr

Régis Cottereau

Influence of local cubic anisotropy on the transition towards an equipartition regime in a 3D texture-less random elastic medium

Keywords: Random medium, Anisotropy, Diffusion regime, Radiative Transfer Equations, Equipartition time

At long lapse times in randomly fluctuating media with macroscopic isotropy (texture-less media), the energy of elastic waves is equipartitioned between compressional (P) and shear (S) waves. This property is independent of the local isotropy or anisotropy of the heterogeneous constitutive tensor and of the type of source. However the local symmetry of the constitutive tensor does influence the rate of convergence to equipartition and this paper discusses the precise influence of local anisotropy on the time required to reach equipartition. More particularly, a randomlyfluctuating medium is considered, whose behavior is statistically isotropic, and locally cubic. After calculating all the differential and total scattering cross-sections in that case, an analytical formula is derived for the rate of convergence to the equipartition regime, function of the second-order statistics of the mechanical parameter fields (bulk and shear moduli and anisotropy parameter). The local anisotropy is shown to influence strongly that transition rate, with a faster transition when the fluctuations of the anisotropy parameter are positively correlated to those of the shear modulus. A numerical model is constructed to illustrate numerically these results. Since the asymptotic regime of equipartition cannot be simulated directly because it would require too large a computational domain, boundaries are introduced and mechanical properties are chosen so as to minimize their influence on equipartition.

Introduction

Modeling of the multiple scattering and diffusion of elastic waves propagating through randomly heterogeneous media via kinetic approaches has received considerable attention in the last few decades. In the stochastic scattering regime, where the correlation length c of the heterogeneities of the medium is of the same order of magnitude as the dominant wavelength λ ( c /λ ∼ 1), the propagation length L is much larger than the wavelength (L/λ 1), and the variance ν 2 of the heterogeneous properties is small (ν 2 1), the radiative transfer equations (RTE) describe the spatio-temporal evolution of the wave vector dependent energy density of the waves as well as their state of polarization [START_REF] Chandrasekhar | Radiative transfer[END_REF][START_REF] Weaver | Diffusivity of ultrasound in polycrystals[END_REF][START_REF] Zeng | Theory of scattered P-wave and S-wave energy in a random isotropic scattering medium[END_REF][START_REF] Sato | Multiple isotropic scattering model including P-S conversions for the seismogram envelope formation[END_REF][START_REF] Turner | Radiative transfer and multiple scattering of diffuse ultrasound in polycrystalline media[END_REF][START_REF] Ryzhik | Transport equations for elastic and other waves in random media[END_REF][START_REF] Margerin | Monte carlo simulation of multiple scattering of elastic waves[END_REF]. These RTE have been used to model the propagation of waves in geophysical media, polycrystalline media and concrete [START_REF] Turner | Radiative transfer and multiple scattering of diffuse ultrasound in polycrystalline media[END_REF][START_REF] Turner | Elastic wave propagation and scattering in heterogeneous, anisotropic media: Textured polycrystalline materials[END_REF][START_REF] Przybilla | Monte Carlo simulation of radiative energy transfer in continuous elastic random media -three-component envelopes and numerical validation[END_REF][START_REF] Gaebler | Seismic scattering and absorption parameters in the W-Bohemia/Vogtland region from elastic and acoustic radiative transfer theory[END_REF][START_REF] Sanborn | Combined effects of deterministic and statistical structure on high-frequency regional seismograms[END_REF]. At long lapse time, these equations can be shown to converge to diffusion equations, through an equilibration of the energy into the different modes of propagation (P and S plane waves modes in all directions in the case of wave propagation in unbounded elastic media). This property of equipartition actually appears before diffusion sets in [START_REF] Shapiro | The energy partitioning and the diffusive character of the seismic coda[END_REF][START_REF] Hennino | Observation of equipartition of seismic waves[END_REF], and can be used to infer parameter values of the underground [START_REF] Weaver | Ultrasonics without a source: Thermal fluctuation correlations at mhz frequencies[END_REF][START_REF] Campillo | Long-range correlations in the diffuse seismic coda[END_REF][START_REF] Saito | Simulating the envelope of scalar waves in 2d random media having power-law spectra of velocity fluctuation[END_REF][START_REF] Borcea | Coherent interferometric imaging in clutter[END_REF][START_REF] Kawase | The optimal use of horizontal-to-vertical spectral ratios of earthquake motions for velocity inversions based on diffuse-field theory for plane waves[END_REF][START_REF] Sánchez-Sesma | Energy partitions among elastic waves for dynamic surface loads in a semi-infinite solid[END_REF][START_REF] Sato | Seismic Wave Propagation and Scattering in the Heterogeneous Earth: Second Edition[END_REF].

In most of these approaches, the mechanical behavior is assumed locally isotropic, which means that the constitutive tensor C(x) relating stresses and strains in the elastic wave equation is assumed isotropic in every point in space.

It is hence fully characterized by only two parameters (Lamé coefficients for instance). It is also assumed that the material properties fluctuate around an isotropic average, so that the relevant modal energies at the macro-scale are Pand S-energy densities. Nevertheless, this property of local isotropy is not appropriate for many materials, even in the case when the macro-scale behavior might be indeed isotropic. Polycrystals, for instance [START_REF] Grigorev | Propagation of ultrasonic waves in polycrystals of cubic symmetry with allowance for multiple scattering[END_REF][START_REF] Hirsekorn | The scattering of ultrasonic waves by polycrystals[END_REF][START_REF] Stanke | A unified theory for elastic wave propagation in polycrystalline materials[END_REF][START_REF] Guo | Scattering of ultrasonic waves in anisotropic polycrystalline metals[END_REF][START_REF] Weaver | Diffusivity of ultrasound in polycrystals[END_REF][START_REF] Turner | Radiative transfer and multiple scattering of diffuse ultrasound in polycrystalline media[END_REF], are composed of assemblies of grain of monocrystals whose behavior is anisotropic, with an orientation that varies from grain to grain. If the grains have no preferred orientation (uniform distribution of orientations), the polycrystal is said to be texture-less [START_REF] Engler | Introduction to texture analysis: macrotexture, microtexture and orientation mapping[END_REF] and its macroscopic behavior is statistically isotropic. Similar phenomena can be observed in concrete and geophysical materials for example, where individual fractures may induce locally anisotropic behavior while the global behavior would remain isotropic if a dense network of fractures with uniformly random orientations is considered.

In the case of anisotropic materials, Turner [START_REF] Turner | Elastic wave propagation and scattering in heterogeneous, anisotropic media: Textured polycrystalline materials[END_REF] derived the RTEs of elastic waves in a transversely isotropic heterogeneous medium. More recently, Baydoun et al. [START_REF] Baydoun | Kinetic modeling of multiple scattering of elastic waves in heterogeneous anisotropic media[END_REF] derived the RTEs of elastic waves in a general anisotropic case, where both the background and the fluctuations may be anisotropic. Two main difficulties arise when dealing with general anisotropic media: (i) the number of fluctuating parameters increases with respect to the isotropic case (253 (cross-) power spectral densities (PSDs) must be specified in general for triclinic materials), and (ii) the eigenvectors of the acoustic tensor are not known in general in closed form so that analytical formulations of the RTEs are more difficult to handle. The latter issue only exists for globally anisotropic materials and disappears for locally anisotropic materials with isotropic background because the eigenvectors that are required for the derivation of the RTE are those of the background material. Also, the former issue is simplified when stronger symmetries are considered. For instance, cubic anisotropy only introduces one additional parameter with respect to elasticity, so that only 10 PSDs must be specified.

In [START_REF] Ta | Modeling of random anisotropic elastic media and impact on wave propagation[END_REF], numerical simulations seemed to indicate that local anisotropy of a material favored a quicker transition to an equipartition regime, but this was never confirmed theoretically, nor through extensive numerical investigations.

The objective of this paper is to challenge this conclusion for a particular texture-less anisotropic material, for which local behavior is cubic. The equipartition time for such a material is therefore derived analytically (Section 3.3), using a particular parameterization (Section 2) and theoretical results of Baydoun et al. [START_REF] Baydoun | Kinetic modeling of multiple scattering of elastic waves in heterogeneous anisotropic media[END_REF] for scattering cross-sections (Section 3 and Appendix A). The formula obtained is then compared to the results of numerical simulations of wave propagation in a set of random media with different statistical properties, and to results in the isotropic case (Section 4).

Both analytical and numerical results will show that the observations of [START_REF] Ta | Modeling of random anisotropic elastic media and impact on wave propagation[END_REF] were chance observations depending on the particular model of material properties that was chosen then. The rate of convergence towards equipartition will actually be shown to depend directly on the correlation between the anisotropy parameter and the shear parameter, with faster convergence for positive correlation.

Elastic wave propagation in a randomly heterogeneous medium

In this section, we briefly introduce some general concepts related to the propagation of elastic waves in 3D random media. Random models of the mechanical properties are also introduced.

Wave equation and mechanical parameters

In non-dissipative elastic media in which the energy losses due to viscosity, friction and radiation are neglected, the displacement vector u(x, t) is governed by the equation of motion as:

ρ ∂ 2 ∂t 2 u(x, t) -∇ x • (C(x) . . ∇ x ⊗ u(x, t)) = S (x 0 , t), (x, t) ∈ Ω × R + (1)
in which x and t denote the space and time, S is a point source at x 0 and Ω ⊆ R 3 is the propagation medium.

In geological media, the coefficient of variation of the density is observed to be significantly less than that of the phase velocities (or equivalently the elastic moduli) [START_REF] Shiomi | Broad-band power-law spectra of well-log data in Japan[END_REF]. Consequently, the mass density of geological media is often considered as constant [START_REF] Sato | Seismic Wave Propagation and Scattering in the Heterogeneous Earth: Second Edition[END_REF]. For the sake of simplicity, we assume throughout that the density is constant (i.e. ρ(x) = ρ ∈ R + ). The medium is characterized by a local fourth-order elasticity tensor C(x) = C i jk (x) which varies continuously in space. Using Voigt's notation, the fourth-order elasticity tensor can be represented via a 6 × 6 symmetric positive-definite matrix [START_REF] Voigt | Lehrbuch der kristallphysik[END_REF]. The number of independent parameters to fully describe the stiffness matrix depends on the type of anisotropy [START_REF] Chadwick | A new proof that the number of linear elastic symmetries is eight[END_REF]. In the general case, so-called triclinic anisotropy, 21 independent parameters are required to fully describe the elasticity matrix.

Parameterization of the propagation medium

A probabilistic approach will be employed to model the heterogeneous stiffness matrix C(x). The randomly heterogeneous elasticity matrix is decomposed into a (constant) average

C 0 = E[C(x)]
, where E[•] denotes ensemble average, and a small-amplitude rapidly-fluctuating random fluctuation:

C(x) = C 0 + √ ε C 1 x ε , (2)
in which ε is a small parameter and E[C 1 (x)]. The particular scaling ( √ ε for the amplitude and 1/ε for the scale of fluctuation) will be discussed in Section 3. As will become apparent in Section 3, the matrices C 0 and C 1 have different contributions to the wave dynamics in the high-frequency regime, and can belong to different symmetry classes. In most papers (see for instance Ryzhik et al. [START_REF] Ryzhik | Transport equations for elastic and other waves in random media[END_REF]), both of them are assumed isotropic. This simplifies greatly the computations in the upscaling process, but is not always appropriate, as discussed in the introduction. In this paper, we still consider an isotropic background, but anisotropic fluctuations (of cubic type).

Being isotropic and constant, the background medium is represented by the following elasticity tensor, in terms of its positive eigenvalues κ 0 (bulk modulus) and µ 0 (shear modulus):

[C 0 ] i jk = κ 0 δ i j δ k + µ 0 δ ik δ j + δ i δ jk - 2 3 δ i j δ k , (3)
in which δ i j is Kronecker's delta (δ i j = 1 if i = j, and δ i j = 0 if i j). Note that the background is assumed homogeneous for simplicity, but fluctuations that are slow with respect to ε can actually be considered without difficulty [START_REF] Ryzhik | Transport equations for elastic and other waves in random media[END_REF].

The Voigt representation of the average is:

C 0 = 1 3                          3κ 0 + 4µ 0 3κ 0 -2µ 0 3κ 0 -2µ 0 0 0 0 3κ 0 -2µ 0 3κ 0 + 4µ 0 3κ 0 -2µ 0 0 0 0 3κ 0 -2µ 0 3κ 0 -2µ 0 3κ 0 + 4µ 0 0 0 0 0 0 0 3µ 0 0 0 0 0 0 0 3µ 0 0 0 0 0 0 0 3µ 0                          . (4)
Having cubic anisotropy, the fluctuation can be represented by the following tensor [START_REF] Guilleminot | On the statistical dependence for the components of random elasticity tensors exhibiting material symmetry properties[END_REF], using 3 real-valued eigenvalues φ i (x) (i ∈ {1, 2, 3}):

[C 1 (x)] i jk = (φ 1 (x) + 2φ 2 (x))δ i δ jk + (φ 1 (x) -φ 2 (x)) δ i j δ k -δ i δ jk + φ 3 (x)δ ik δ j . (5)
In order to simplify the comparison of cubic anisotropy with isotropy, we adopt the following parameterization of the eigenvalues :

κ 1 (x) = φ 1 (x) (with κ 0 + κ 1 > 0), µ 1 (x) = φ 3 (x) (with µ 0 + µ 1 > 0) and A(x) = 3φ 2 (x) -2φ 3 (x) (with A > -2(µ 0 + µ 1 )). Finally: [C 1 (x)] i jk = (µ 1 + A)δ i δ jk + κ 1 - 2 3 µ 1 - 1 3 A δ i j δ k + µ 1 δ ik δ j , (6)
and the Voigt representation of the fluctuation is:

C 1 (x) = 1 3                          3κ 1 + 4µ 1 + 2A 3κ 1 -2µ 1 -A 3κ 1 -2µ 1 -A 0 0 0 3κ 1 -2µ 1 -A 3κ 1 + 4µ 1 + 2A 3κ 1 -2µ 1 -A 0 0 0 3κ 1 -2µ 1 -A 3κ 1 -2µ 1 -A 3κ 1 + 4µ 1 + 2A 0 0 0 0 0 0 3µ 1 0 0 0 0 0 0 3µ 1 0 0 0 0 0 0 3µ 1                          . ( 7
)
Note that κ 1 and µ 1 can be analyzed as fluctuations (normalized by √ ε) of the bulk and shear moduli around their averages κ 0 and µ 0 and A is a measure of the distance from isotropy. In particular, the behavior of the material is isotropic everywhere whenever A = 0.

The three parameters κ 1 (x), µ 1 (x) and A(x) are modeled as correlated zero-mean random functions. The spatial correlation function (CF) of the different pairs of random functions are denoted as R κκ (x, x ), R κµ (x, x ), R κA (x, x ), R µµ (x, x ), R µA (x, x ) and R AA (x, x ). For instance, the second CF is defined as R κµ (x, x ) = E[κ 1 (x)µ 1 (x )], and the others likewise. It is assumed for simplicity that all CF are statistically isotropic, in the sense that they are all 1D functions of the distance between the two points considered. Classically, these CFs are modeled using kernel functions R, correlation length and variance ν 2 . For instance R κµ (x, x ) = ν 2 κµ Rκµ (|xx |/ κµ ). The correlation length specifies the length scale over which the random function decorrelates significantly and can be seen as the typical size of the heterogeneities. This parameter for the second CF is defined for instance as:

κµ = 2 R κµ (0) R + R κµ (r)dr, (8)
and the other correlation lengths likewise. The (cross-) power spectral density functions (PSDF) Φ(ψ = |p| ) are then defined as the Fourier transform of the kernels of the CFs R(η = |xx |/ ), in which p is the Fourier transform relative of xx .

In the analysis and the numerical tests presented in this paper, we will consider five different examples of correlation kernels. They are representative of the models used in the literature and present a wide range of behaviors, with lower or higher frequency content for the same correlation length (in the sense of Eq. ( 8)) [START_REF] Klimeš | Correlation functions of random media[END_REF][START_REF] Khazaie | Influence of the spatial correlation structure of an elastic random medium on its scattering properties[END_REF]. These examples are summarized in Table 1:

Propagation modes

We will see in the next section that energy is transported by modes of the background medium. Since we consider texture-less materials in this paper, the background medium is isotropic, as in Eq. ( 3). In that case, there are two different propagation modes which will be denoted by {P, S}, for the compressional (P) and shear (S) waves. The phase velocities are v P = (κ 0 + (4/3)µ 0 )/ρ and v S = µ 0 /ρ and the dispersion relations are

ω 2 P (k) = v 2 P k 2 and ω 2 S (k) = v 2 S k 2 .
The polarization of the P wave is k = k/k where k = |k| while that of the S wave spans the two-dimensional plane orthogonal to k. There are therefore actually two S waves.

3. Energy transport and equipartition in texture-free anisotropic media

Radiative transfer equations

In a weakly heterogeneous scattering regime ( c ∼ λ, L λ and ν 2 1), a transport regime occurs when the elastic waves propagate through a typical realization of the random medium described in Section 2.2. In this regime, the spatio-temporal evolution of the ensemble-averaged wave energy densities are described via the so-called radiative transfer equations (RTE, see for instance [START_REF] Bal | Accuracy of transport models for waves in random media[END_REF] for details). The RTEs are energy conservation equations corresponding to each of the propagation modes. They consider the full vector nature of the waves along with all possible mode conversions during scattering processes. Using a kinetic approach, transport equations of elastic waves were first derived in the weakly heterogeneous scattering regime for locally isotropic randomly heterogeneous materials by employing a multiscale expansion of the Wigner transform of the wave field [START_REF] Ryzhik | Transport equations for elastic and other waves in random media[END_REF]. Recently, Baydoun et al. [START_REF] Baydoun | Kinetic modeling of multiple scattering of elastic waves in heterogeneous anisotropic media[END_REF] extended these RTEs to the case of locally anisotropic random media via kinetic modeling of multiple scattering and using the spatio-temporal Wigner transforms and their interpretation in terms of semiclassical operators. The RTEs of elastic waves for texture-less anisotropic random media write:

∂a P (k) ∂t -v P k • ∇ x a P (k) = R 3 σ PP (k, k )a P (k )dk -Σ PP (k)a P (k) + R 3 σ PS (k, k )[a S (k )]dk -Σ SP (k)a P (k) (9a) ∂[a S (k)] ∂t -v S k • ∇ x [a S (k)] = R 3 σ SS (k, k )[a S (k )]dk -Σ SS (k)[a S (k)] + R 3 σ SP (k, k )a P (k )dk -Σ PS (k)[a S (k)] (9b) in which a α E[a α ] (α ∈ {P, S}
) is the projection of the average Wigner transform, or otherwise said, the specific energy density of the wave mode α in the phase space (position × wavevector) and k = k/k. In equations ( 9), the left hand side corresponds to the total derivative of the energy density a α and is related to the transport of energy in the homogeneous background medium C 0 . The contribution of the foreground C 1 is found on the right hand side in terms of the scattering cross-sections (σ αβ , Σ αβ ) that depend on the background velocities as well as on the PSDFs of the fluctuations of the components of C 1 . When a mode with order of multiplicity higher than 1 exists (as for S waves when the background is isotropic), the energy densities are matrices and equations ( 9) form a matrix system. With an isotropic background, the body wave energy densities are described via a coupled system of one scalar transport equation and one 2 × 2 matrix transport equation.

Scattering cross-sections in texture-free cubic media

In the right hand side of equations ( 9), the scattering mechanism is described via differential and total scattering cross-sections σ αβ (k, k ) and Σ αβ (k) (α, β ∈ {P, S}). The rate at which the incident energy of a β-wave propagating with wave vector k is scattered into the energy of an α-wave propagating with wave vector k is represented by the differential scattering cross-sections σ αβ (k, k ) (with unit m 3 /s). For the particular case of a cubic fluctuation around an isotropic background, and homogeneous density, the formulas for the differential scattering cross-sections are derived in Appendix A. For clarity, we report below only the final formulas. The differential scattering cross-section from P-wave to P-wave is

σ cubic PP (k, χ)[I 1 ] = πk 2 3 c 18ρ 2 v 3 P 9ν 2 κ Φ κκ + 4ν 2 µ Φ µµ + ν 2 A Φ AA -12ν κµ Φ κµ -6ν κA Φ κA + 4ν µA Φ µA +3χ 2 -4ν 2 µ Φ µµ -ν 2 A Φ AA + 6ν κµ Φ κµ -4ν µA Φ µA + 3ν κA Φ κA + 9χ 4 4ν 2 µ Φ µµ + ν 2 A Φ AA + 4ν µA Φ µA , (10)
in which I n is the n × n identity matrix and all the PSDFs are evaluated at k 2(1 -χ), where k = ω/v P is the wavenumber corresponding to the scattered P waves and χ is the cosine of the scattering angle, i.e. χ = k • k . The differential scattering cross-section from S-wave to P-wave is such that

σ cubic PS (k, χ)[I 2 ] = πk 2 3 c 2ρ 2 v 3 S χ 2 1 -χ 2 4ν 2 µ Φ µµ + 4ν µA Φ µA + ν 2 A Φ AA , (11)
wherein all the PSDFs are evaluated at k 1 + K 2 -2Kχ, where k is again the wavenumber corresponding to the scattered P waves and K is the ratio between the average P to the average S wave speeds, i.e.

K = v P /v S = κ 0 /µ 0 + 4/3.
Hence, K depends solely on the background medium properties (κ 0 , µ 0 ). Note that the differential scattering crosssection for the P-to-S mode conversion reads:

Tr σ cubic SP (k, k )[I 1 ] = σ cubic PS (k , k)[I 2 ]. (12)
Finally, the differential scattering cross-section from S-wave to S-wave is such that

σ cubic SS (k, χ)[I 2 ] = πν 2 µ k 2 3 c 2ρ 2 v 3 S 4χ 4 -3χ 2 + 1 Φ µµ k 2(1 -χ) I 2 , (13)
in which k = ω/v S is the wavenumber corresponding to the scattered S waves.

The rate at which the incident energy of a β-wave propagating with wave vector k is scattered into the energy of an α-wave propagating in any direction is called the total scattering cross-section Σ αβ (k) (with unit 1/s) and related to the differential cross-section through:

Σ αβ (k) = R 3 σ αβ (k, k )[I β ] dk (2π) 3 ; α, β ∈ {P, S}. ( 14
)
where I P = I 1 and I S = I 2 . Although the integration can be performed numerically in all generality, we choose here to consider a slightly simplified setting, in order to obtain more condensed formula. We therefore assume that all PSDFs follow the same functional form Φ(ψ), and that they differ only through the variances and covariances:

Φ κκ (ψ) = Φ κµ (ψ) = Φ κA (ψ) = Φ µµ (ψ) = Φ µA (ψ) = Φ AA (ψ) = Φ(ψ). ( 15
)
This hypothesis is only required for clarity and simplicity of the exposition. Total scattering cross sections for a material with isotropic background (texture-free) and cubic fluctuations are derived with this hypothesis in Appendix A. We report here the S-to-P cross section, which is the most significant for the rest of the paper: the P-to-S total scattering coefficient is:

Σ cubic PS (k) = (3κ -4µ)(4ν 2 µ + 4ν µA + ν 2 A )k 4 3 c 24πµ 2 √ ρµ +1 -1 χ 2 1 -χ 2 Φ k c 1 + K 2 -2Kχ dχ, (16)
The corresponding total scattering cross-section for an isotropic fluctuation Σ iso PS (k) is obtained for A = 0 so that the integral term is the same for both isotropic and cubic fluctuations. This allows in particular to obtain a very simple expression of the ratio for the S-to-P total scattering cross-sections of cubic and isotropic fluctuations:

Σ cubic PS (k) Σ iso PS (k) = 4ν 2 µ + 4ν µA + ν 2 A 4ν 2 µ . ( 17
)
This ratio is wavenumber-independent and only depends on the components of the covariance matrix of (µ, A). This observation will be used in the upcoming analysis.

Influence of anisotropy on the global equipatition time

At long lapse times and after many scattering events, the waves lose track of their source or initial conditions, and the energy spreads equally over all modes of the system [START_REF] Weaver | On diffuse waves in solid media[END_REF][START_REF] Weaver | Diffusivity of ultrasound in polycrystals[END_REF][START_REF] Papanicolaou | Stability of the P to S energy ratio in the diffusive regime[END_REF][START_REF] Ryzhik | Transport equations for elastic and other waves in random media[END_REF][START_REF] Turner | Scattering and diffusion of seismic waves[END_REF][START_REF] Margerin | Diffusion approximation with polarization and resonance effects for the modelling of seismic waves in strongly scattering small-scale media[END_REF]]. In our system with only two modes, energy is then equipartitioned between P and S waves at a value which is a function only of the ratio of phase velocities, and independent of position:

r = lim t→+∞ E S (x, t) E P (x, t) = 2K 3 = 2 v P v S 3 , (18)
in which E P (x, t) and E S (x, t) are the P and S wave energy densities at point x, defined as:

E P (x, t) = 3κ(x) + 4µ(x) 6 (∇ • u(x, t)) 2 , E S (x, t) = µ(x) 2 (∇ × u(x, t)) 2 . ( 19
)
Although this equipartition is always eventually reached, it will settle in at different rates depending on the particular medium. This paper is concerned with the rate of convergence towards equipartition, and in particular on the influence of anisotropy on that rate of convergence. This rate of convergence can be evaluated through the equipartition time, for which a formula was proposed in Trégourès and van Tiggelen [START_REF] Trégourès | Generalized diffusion equation for multiple scattered elastic waves[END_REF]: The equipartition time is therefore inversely proportional to the mean of the P-S and S-P total scattering cross-sections, which is an indicator of the total mixture between the P and S waves [START_REF] Trégourès | Generalized diffusion equation for multiple scattered elastic waves[END_REF][START_REF] Margerin | Effect of absorption on energy partition of elastic waves in the seismic coda[END_REF][START_REF] Nakahara | Radiative transfer of elastic waves in two-dimensional isotropic scattering media: Semi-analytical approach for isotropic source radiation[END_REF]. Note that in the formula above, k is the wave number of the P wave, which explains the argument k/K for Σ PS , which is the wave number of the S wave at the same frequency. Following [START_REF] Khazaie | Influence of the spatial correlation structure of an elastic random medium on its scattering properties[END_REF], the normalized (adimensional) form of the equipartition time can be defined as teq = v P t eq / c .

t eq (k) = 1 Σ PS (k) + Tr Σ SP (kK) = 1 Σ PS (k) 1 + 1 2K 3 . ( 20
) ζ = |k|ℓc 2π [-]
As an illustrative example, consider a heterogeneous material for which the (isotropic) background properties are

µ 0 = 2 ×
ζ = k c /(2π) (
where k is the wave number for the P wave) for different correlation models (see Table 1) and different values of the coefficients of the correlation matrix of µ and A.

Using Eq. ( 20) and ( 17), the ratio of equipartition times in isotropic and cubic media can then be obtained as:

t iso eq (k) t cubic eq (k) = Σ cubic PS (k) Σ iso PS (k) = 4ν 2 µ + 4ν µA + ν 2 A 4ν 2 µ . (21)
Clearly, this ratio implies that (cubic) anisotropy does not necessarily imply faster convergence to equipartition, contrarily to what could have been guessed based on the numerical observations in [START_REF] Ta | Modeling of random anisotropic elastic media and impact on wave propagation[END_REF]. Indeed, ν µA can be negative, and reaches its minimum (the semi-positive definiteness of the covariance matrix of (µ, A) implies that |ν µA | ≤ ν µ ν A ) at ν µA = -ν A ν µ . In that case, the ratio of equipartition time is (1 -ν A /(2ν µ )) 2 ≤ 1, and convergence is faster with isotropic fluctuations. A similar remark for the positive bound yields:

1 - ν A 2ν µ 2 ≤ t iso eq (k) t cubic eq (k) ≤ 1 + ν A 2ν µ 2 . ( 22
)
These bounds are obtained for a perfect anti-correlation and a perfect correlation of µ and A, respectively. Theory therefore predicts that convergence to equipartition can be either faster or slower when local anisotropy is present.

Next section aims at illustrating this conclusion on through adequately chosen numerical simulations.

Numerical observation of the equipartition regime in a 3D random elastic medium

In this section we investigate the onset of an equipartition regime through the numerical simulations of elastic waves propagating in 3D randomly heterogeneous media in both isotropic and cubic anisotropic materials. These numerical simulations are performed with an implementation of an explicit parallel spectral element solver, which is developed at Institut de Physique du Globe de Paris1 [START_REF] Cupillard | RegSEM: a versatile code based on the spectral element method to compute seismic wave propagation at the regional scale[END_REF]. The solver has been modified to account for anisotropic fluctuations of the elasticity tensor, and includes a random field generation pre-processor [START_REF] Cottereau | Validation of software for 3D propagation of waves in heterogeneous and random media[END_REF][START_REF] Ta | Modeling of random anisotropic elastic media and impact on wave propagation[END_REF].

Equipartition in a randomly-fluctuating medium with weak heterogeneities settles in only for very large times (see Fig 1). Numerical observation is therefore very complicated because it requires to perform numerical simulations over domains that are a priori so large that simulation cannot be performed reasonably (see Khazaie et al. [START_REF] Khazaie | Numerical observation of the equipartition regime in a 3D random elastic medium, and discussion of the limiting parameters[END_REF] for a detailed discussion). On the one hand, using stronger heterogeneities is not an option because strong localization inhibits transport and equipartition is then not attained. On the other hand, using reflecting boundary conditions impacts the equilibrium between P and S waves so that it might hide the transition to equipartition. Most of this section will therefore be focused on constructing a particular numerical model in which the influence of anisotropy on equipartition can be observed unambiguously.

Choice of an initial numerical model

A first numerical model is therefore selected to try and understand the implications of truncating the computa- a low-pass white noise (see Table 1), with correlation length c = 100 m. Realizations of the fluctuating constitutive tensor C 1 can be obtained using various schemes [START_REF] Panunzio | Large scale random fields generation using localized Karhunen-Loève expansion[END_REF], and a classical spectral representation approach was chosen in this work [START_REF] Shinozuka | Simulation of stochastic processes by spectral representation[END_REF]. In that technique, the realizations of the random fields are generated in the spectral domain, as sums of cosine functions with increasing frequency and random phases, and with an amplitude controlled by the power spectral density. The random fields generated that way are Gaussian, so their first-order marginal is transformed by local post-processing to the desired first-order marginal density. One of the advantage of this technique is that its computational efficiency can be drastically improved with Fast Fourier Transform algorithms [START_REF] Shinozuka | Simulation of stochastic processes by spectral representation[END_REF]. For very large domains, the algorithm can also be parallelized [START_REF] De Carvalho Paludo | Scalable parallel scheme for sampling of Gaussian random fields over large domains[END_REF]. A typical realization of the shear modulus of the randomly heterogeneous medium characterized with the aforementioned parameters is depicted in Fig. 2. the order of magnitude of the wavelengths is close to the correlation length, which ensures (along with the weak heterogeneities and the long propagation path) the validity of the theoretical approach derived in the previous section.

Since equipartition is an asymptotic regime independent of the initial source, two different sources will be tested and verified to converge to the same regime. The first type of source is explosive (rotational-free volume force) and introduces at the origin almost only energy in the form of P waves. The second type of source is impulsive (unidirectional volume force pointing downwards), introducing P wave energy along the axis of the force, S-wave energy in all other directions, and on average more S energy than P energy.

Observation of equipartition for the initial numerical model

Based on the simulation of the problem chosen above, local energies can be computed using Eq. ( 19). Global P and S energies E P and E S can then be computed as averages over the entire domain of these local energies. For simplicity of comparison with theoretical values of the normalized equipartition time, hereinafter the temporal variations are represented in terms of the normalized time lapse. The latter is defined similar to the normalized equipartition time as t = v P t/ c where t is the simulation time. As such, in this case we have t = v P t/ c ≈ 17.3t. Fig. 3 As expected, the ratio of energy densities converges in time to the same equipartition constant independently of the source. Note that this convergence is uniform so that it is observed in each realization of a random medium and not only as an average over different realizations [START_REF] Bal | Self-averaging of wigner transforms in random media[END_REF][START_REF] Bal | Accuracy of transport models for waves in random media[END_REF]. And the constant obtained at convergence is the value predicted by theory (see Eq. ( 18), which yields r ≈ 10.36 for the parameters of that case). Note also that the ratio initially depends strongly on the type of source, with r > 1 for the impulsive source and r ≈ 0 for the explosion, and only becomes independent of the source after long propagation path within the heterogeneous medium. The transition towards equipartition is however not simple and sees many local peaks and valleys (indicated by vertical dashed lines in the right plot of Fig. 5). These are due to the interaction of the different wave fronts with the reflecting boundaries and is analyzed in more detail below. computed analytically [START_REF] Graff | Wave motion in elastic solids[END_REF][START_REF] Aki | Quantitative seismology[END_REF]:

Influence of the reflection at the boundaries

248                  R PP = sin(2θ inc ) sin(2θ P ref ) -K 2 cos 2 (2θ P ref ) sin(2θ inc ) sin(2θ P ref ) + K 2 cos 2 (2θ P ref ) R PS = 2 sin(2θ inc ) cos(2θ S ref ) sin(2θ inc ) sin(2θ S ref ) + K 2 cos 2 (2θ S ref ) (23)
When the incoming wave is an S-wave with incidence angle θ inc , a P-wave is reflected at angle sin(θ P ref ) = K sin(θ inc ) and an S-wave is reflected at angle θ S ref = θ inc . The reflection coefficients are [START_REF] Graff | Wave motion in elastic solids[END_REF][START_REF] Aki | Quantitative seismology[END_REF]:

                 R SP = -2K 2 sin(2θ inc ) cos(2θ inc ) sin(2θ inc ) sin(2θ P ref ) + K 2 cos 2 (2θ inc ) R SS = sin(2θ inc ) sin(2θ S ref ) -K 2 cos 2 (2θ S ref ) sin(2θ inc ) sin(2θ S ref ) + K 2 cos 2 (2θ S ref ) (24)
Here the reflection coefficients are defined as the ratio between the amplitudes of the reflected and the incident waves.

For an incident S wave, since θ ref = sin -1 (K sin θ ref ), there is a critical incidence angle beyond which the reflection occurs along the boundary. This critical incidence angle is θ c inc = sin -1 (K -1 ) and thus for instance K = √ 3 yields a critical angle of about 35 • . Fig. 6 shows the influence of the incidence angle and the ratio K on reflection coefficients for the P-to-S and S-to-P mode conversions, depending on the incident angle. Coming back to the analysis of the rise of equipartition in Fig. 5, the influence of the interaction of the different coherent waves with the boundaries can be better understood by comparing the time of the dashed lines and the plots in Fig. 3 and4. For instance, around time t ≈ 0.7 -0.8 s ( t ∈ [12.11, 13.8]), the coherent P waves emitted at the source begin interacting with the boundary, while the coherent S waves arrive at the boundary around t ≈ 1 -1.1 s ( t ∈ [17.3, 19]). While the former implies an increase of the S-to-P energy ratio (through conversion of part of the P-energy into S-energy), the latter marks the beginning of a decrease of the S-to-P energy ratio (through conversion of part of the S-energy into P-energy). Later reflections are less salient because the wave fronts are less coherent (the energy is spread more widely in space and propagation direction), and also because the source is not located at the center of the box, so that interactions do not take place simultaneously on all boundaries.

Design of a new model with minimized mode conversion at the boundary

As the objective of this section is to observe equipartition arise from the heterogeneity of the medium only, and not because of mode conversions at the boundaries, a modification of the numerical model of Section 4.1 is therefore proposed. That modification is chosen so as to minimize mode conversions at the boundaries, hence increasing the relative influence of the heterogeneities on equipartition. For instance, with an explosion that creates an initial majority of P waves, it is desirable to minimize the P-to-S conversion. Thanks to the left plot of Fig. 6, it is then chosen to consider a larger value of the velocity ratio at K = 2.5. The previous numerical model is therefore modified by considering κ 0 = 9.8333 × 10 9 Pa and v P = 2500 m/s. With the rest of the setting unchanged, the new expected value of the equipartition ratio is now r ≈ 31.25, and it is indeed observed numerically in Fig. 7 which shows the variation of the energy density ratio in terms of the normalized simulation time. Compared to Fig. 5, it is clear that the local peaks have vanished, and transition to equipartition is now smoother. In order to investigate the influence of the size of the domain on the time to equipartiton, three different domains are considered, all cubes, with respective sides of 2500 m, 3000 m and 3500 m. As expected, the existence of the boundary increases the rate of mixing of modes so that equipartition takes place faster in a smaller domain, because waves interact more often with the boundaries.

Furthermore, since the theoretical model only discusses the influence on equipartition of the bulk heterogeneities, there is a discrepancy with the numerical model. Hence, the time to equipartition in a bounded domain is expected to be smaller than the theoretical value. In order to investigate further the influence of the boundaries on the results, and in particular to make sure that transition is not controlled by the boundaries but rather by the bulk heterogeneities, we design one last numerical simulation with a homogeneous medium excited with the same explosion source. All parameters are the same as above, except that the fluctuations of the properties vanish. The domain is a cube whose sides are 3000 m long.

Figure 8 shows the variation of the energy ratio in terms of the normalized time lapse. Initially, the explosion source creates the P waves so that the ratio vanishes until approximately a normalized time of 17. After that time, interactions with the boundaries take place (over a long time because the explosion is not centered) and the ratio increases. Not however that, contrarily to previous simulations, the theoretical ratio (close to 31) is not reached. A slow convergence is observed, which indicates, as expected, that the boundaries do influence transition to equipartition. However, that transition provoked by the boundaries seems much smaller than that provoked by the heterogeneities. The proposed design for the numerical model seems therefore appropriate to study numerically the influence of anisotropy on equipartition. For the rest of this section, explosive sources will be considered with this ratio of velocities K = 2.5. For all the simulations to be discussed below, the set of numerical and mechanical parameters will be kept identical to those of this section, except for the anisotropy parameter A that will evolve to analyze the influence of anisotropy on equipartition.

Normalized Time Lapse [-]

Comparison between isotropic and cubic anisotropic media

Finally, we can turn to the numerical confirmation of the influence of anisotropy on the rate of convergence to equipartition. Using the numerical model above, different choices of the anisotropy parameter A are introduced:

1. with A = 2µ 1 , the variances are ν 2 A = 4ν 2 µ and ν µA = 2ν 2 µ and a rapid rise of equipartition is observed, slightly faster than in the isotropic case;

2. with A = µ 1 , the variances are ν 2 A = ν µA = ν 2 µ and equipartition comes in slightly slower than for the previous case, and slightly faster than for the isotropic case;

3. with A = 0, the isotropic case is retrieved;

4. with A = -µ 1 , the variances are ν 2 A = ν 2 µ and ν µA = -ν 2 µ and equipartition is slower than in the isotropic case, as expected;

5. with A = -2µ 1 , the lower bound of Eq. ( 21) is attained with ν 2 A = 4ν 2 µ and ν µA = -2ν 2 µ and equipartition time theoretically diverges.

The ratios of global S-to-P energies are plotted in Fig. 9 in terms of the normalized time lapse t for different cases introduced above. Note that the left and right plots correspond to medium with sides 3000 m and 3500 m, respectively. Qualitatively, the results obtained through numerical experiments do confirm the theoretical results obtained in the previous section. Quantitative comparison cannot be considered because the boundaries do eventually influence the rate of convergence, even though this influence has been minimized. Note that for the last case, which yields theoretically an infinite equipartition time, the numerical simulation was stopped before full convergence although equipartition did seem to eventually arise.

Normalized Time Lapse [-]

Conclusion

This paper discussed the importance of local anisotropy on the onset of the equipartition regime, even when the medium is statistically isotropic (texture-free). For a cubic local anisotropy, the full set of differential and total scattering cross sections have been derived and a formula has been proposed for the time to equipartition. Relatively to the isotropic case, this equipartition time was shown to be strongly influenced by the correlation between the shear modulus µ(x) and the anisotropy parameter A(x) for locally cubic media. Note that a strong hypothesis was introduced in the analysis, which is that all random mechanical parameters have the same correlation structure, even though their variance and cross-correlation differ. This hypothesis was necessary in order to derive simple enough results.

Although numerical issues impeded direct verification of this analytical result in full space, a bounded numerical model was constructed that minimized the influence of boundary conditions on the rise of equipartition. The numerical observation of time to equipartition for different correlations confirms that (i) local anisotropy is influential on the rate of convergence to equipartition; and (ii) positive correlation between the shear modulus µ(x) and the anisotropy parameter A(x) favors the rise of equipartition.

Other types of anisotropy could be studied in the same manner, although parameterization would require to introduce more independent random fields for the constitutive tensor. Alternatively, it would be interesting to consider the influence of the anisotropy of the correlation structure, and/or the influence of an anisotropic background. The former case, which is referred to as anisomery in the dedicated literature [START_REF] Margerin | Attenuation, transport and diffusion of scalar waves in textured random media[END_REF], could be obtained quite easily as an extension of this paper, although parameterization would again have to be addressed. For the latter case, a more profound modification would be necessary because the modes of the background in that case are not simple P and S modes and equipartition takes on a more complex form.

Appendix A. Calculation of differential scattering cross-sections for cubic anisotropic fluctuations

In this appendix, we derive the full set of differential and total scattering cross-sections for the elasticity equation with homogeneous density, isotropic background and cubic fluctuations. The orthonormal eigenvectors of the Christoffel tensor of the background medium define the polarization directions, henceforth denoted as:

p P (k) = k, p S (k) = [ẑ 1 (k), ẑ2 (k)] T , (A.1)
in which the superscript T denotes the transpose operation. Following Baydoun et al. [START_REF] Baydoun | Kinetic modeling of multiple scattering of elastic waves in heterogeneous anisotropic media[END_REF], the differential scattering cross-section operator for the general case of anisotropic material behavior corresponding to the β-to-α mode conversion σ αβ (k, q) reads: δ(0)(2π) 3 σ αβ (k, q)[a β (q)] = π 2ω α (k)ω β (q) E H αβ (k, kq, q)a β (q)H βα (q, qk, k) δ ω α (k) -ω β (q) , (A.2)

in which [a β (q)] is the incident wave energy density of type β ∈ {P, S} in phase space, and δ is the Dirac delta distribution. The Dirac delta in the right hand side is related to the conservation of frequency during the scattering process. By contrast, the Dirac delta in the left hand side arises from the definition of the power spectral density of the mechanical properties and will simplify with its counterpart hidden in the the average of product of functions H (see below, and [START_REF] Baydoun | Kinetic modeling of multiple scattering of elastic waves in heterogeneous anisotropic media[END_REF] for more details). In Eq. (A.2) H αβ is defined by pre and post multiplication of a 3 × 3 matrix H(k, p, q) respectively by the polarization vectors p * α (k) and p β (q), where the star * denotes the conjugate transpose of a complex vector. The matrix H(k, p, q) is defined as [H(k, p, q)] ik = ρ -1 k j q [ C 1 (p)] i jk , wherein the Einstein implicit summation convention is utlilized which implies that H(k, p, q) = H * (q, p, k). When the fluctuation matrix C 1 is cubic, the matrices H(k, p, q) and H αβ (k, p, q) can be obtained as

H(k, p, q) = ρ -1             3 κ(p) -2 µ(p) -A(p) 3       k ⊗ q + µ(p)q ⊗ k + µ(p)(k.q)I 3 + A(p)diag(k 1 q 1 , k 2 q 2 , k 3 q 3 )       (A.3)
and

H αβ (k, p, q) = ρ -1 kq             3 κ(p) -2 µ(p) -A(p) 3       ( k • p α (k)) * ( q • p β (q))
+ µ(p) ( q • p α (k)) * ( k • p β (q)) + ( k • q)(p * α (k) • p β (q)) + A(p)p * α (k)diag(k 1 q 1 , k 2 q 2 , k 3 q 3 )p β (q) . (A. in which ν ΞΨ is the correlation coefficient of the random fields and the correlation function is assumed to have an isotropic structure. Hence, from Eq. (A.7) multiplying term by term, taking the mathematical expectation and making use of (A.10) yields:

σ cubic PP (q, χ)[a P (q)] = πq 2 3 c 18ρ 2 v 3 P 9ν 2 κ Φ κκ + 4ν 2 µ Φ µµ + ν 2 A Φ AA -12ν κµ Φ κµ -6ν κA Φ κA + 4ν µA Φ µA +3χ 2 -4ν 2 µ Φ µµ -ν 2
A Φ AA + 6ν κµ Φ κµ -4ν µA Φ µA + 3ν κA Φ κA + 9χ 4 4ν 2 µ Φ µµ + ν 2 A Φ AA + 4ν µA Φ µA a P (q), (A.11) where we used the identity δ(ax) = δ(x)/|a| (for any non-zero scalar a) and then δ(kq) or k = q implies that the argument of all PSDFs is |k -q| = k 2 + q 2 -2kqχ = q 2(1 -χ) in which q = |q| = ω/v P is the wavenumber of a P mode. Note that we have changed the arguments of the scattering function from (k, q) in Eq. (A.7) to (q, χ) above to insist on the fact that only the norm of the incident (or scattered) vector and the angle between the two vectors are influential.

Selecting A = 0 allows to fall back on the classical P-P differential scattering cross-section for isotropic fluctuations [START_REF] Weaver | Diffusivity of ultrasound in polycrystals[END_REF][START_REF] Ryzhik | Transport equations for elastic and other waves in random media[END_REF][START_REF] Turner | Scattering and diffusion of seismic waves[END_REF][START_REF] Baydoun | Kinetic modeling of multiple scattering of elastic waves in heterogeneous anisotropic media[END_REF]: σ iso PP (q, χ)[a P (q)] = πq 2 3 c 18ρ 2 v 3 P 9ν 2 κ Φ κκ + 4ν 2 µ Φ µµ -12ν κµ Φ κµ + 3χ 2 -4ν 2 µ Φ µµ + 6ν κµ Φ κµ + 36χ 4 ν 2 µ Φ µµ a P (q). (A.12)

7 (a) ν µA = ν 2 A = ν 2 µ 7 (c) ν µA = -ν 2 A = -ν 2 µFigure 1 : 2 A = ν 2 µ

 722722122 Figure 1: Normalized global equipartition time as a function of normalized frequency for different values of the correlation of µ and A:ν µA = ν 2 A = ν 2 µ (left), ν µA = ν A = 0 (isotropic,center) and ν µA = -ν 2 A = -ν 2 µ (right); and for different correlation kernels: exponential (thick solid line), power-law (thin dashed-dotted line), Gaussian (thick dashed line), triangular (thin solid line) and low-pass white noise (thin dashed line).

  10 9 Pa, κ 0 = 3.33 × 10 9 Pa, ρ = 2000 kg/m 3 , and the fluctuations are such that ν 2 µ = (0.1 × µ 0 ) 2 = 4 × 10 16 Pa 2 and c = 100 m. Fig. 1 represents the normalized equipartition time as a function of the normalized frequency

3 .

 3 tional domain. The numerical domain is a rectangular cube Ω = {-1500 m ≤ x, y ≤ 1500 m; -3000 m ≤ z ≤ 0 m}, with homogeneous Neumann boundary conditions. The medium has a constant density ρ = 2000kg/m 3 , and randomly-fluctuating shear and bulk moduli, with Gamma first-order marginal density, mean values µ 0 = 2 × 10 9 Pa and κ 0 = 3.33 × 10 9 Pa, so that the average velocities are v S = µ 0 /ρ = 1000 m/s and v P = (κ 0 + (4/3)µ 0 )/ρ = 1730 m/s, and the ratio K = v P /v S is thus close to √ For simplicity purposes (and because the fluctuation of κ plays little role in the effects we are monitoring), the coefficients of variation of µ and κ are assumed equal to δ = 0.1, so that ν µ = 2 × 10 8 Pa and ν κ = 3.33 × 10 8 Pa. The anisotropy coefficient A is assumed perfectly correlated with µ, with ν µA = 8 × 10 16 Pa 2 and ν A = 4 × 10 8 Pa. The correlation structure of the fluctuations of all mechanical parameters is

Figure 2 :

 2 Figure 2: One realization of the random shear modulus µ(x) with low-pass white noise correlation structure, average µ 0 = 2 × 10 9 Pa, standard deviation ν µ = 2 × 10 8 Pa, and correlation length c = 100m. A point source is introduced at position x 0 = (300, 500, -700) m, described in time by a Ricker function (second derivative of a Gaussian function) with a delay of τ = 0.3 s and a central frequency f c = 10 Hz. This implies that

and 4 display

 4 the evolution at different times of the local P and S energies, respectively, along orthogonal planes cutting the computational domain at the position of the source. They give a global picture of the progressive diffraction of the coherent wave front into incoherent energy spreading throughout the domain. Although we only represent the simulation corresponding to the impulsive source, the explosive source yields qualitatively similar plots.(a) t = 12.8 (b) t = 19 (c) t = 32.4 (d) t = 39.3 (e) t = 45 (f) t = 81.8 Figure 3: S energy density at different normalized times for an impulse source at x 0 = (300, 500, -700) m and for an anisotropic medium where A and µ are perfectly correlated via A = 2 (µ -E(µ)).

Fig. 5

 5 Fig. 5 then displays the evolution of the ratio of domain-averaged energy densities for the two different sources.

Figure 5 :

 5 Figure 5: Evolution of the ratio of energy densities of S and P waves in terms of the normalized time lapse t for K = √ 3 and two different sources: explosion (red line) and impulsive (blue line). The right plot displays a zoom on the first 10 seconds (t max = 173) for the impulsive load. The horizontal dashed line indicates the theoretical value of the equipartition constant, given by Eq. (18). The vertical dashed lines in the right plot indicate local peaks and valleys of the ratio of energy densities.

Figure 6 :

 6 R PS (θ inc ) for different K Incidence Angle θ inc [deg] R SP (θ inc ) for different K Reflection coefficients for the P-to-S and S-to-P mode conversions in terms of the incident angle θ inc for different values of the ratio K: K = 2.5 (black line), K = √ 3 (red line) and K = √ 4/3 (blue line).

Figure 7 :

 7 Figure7: Evolution of the ratio of energy densities of S and P waves for K = 2.5 and an explosive source in terms of the normalized time lapse for three different medium sizes. Cubes with respective sides of 2500 m (red line), 3000 m (black line) and 3500 m (blue line). The horizontal dashed line indicates the theoretical value of the equipartition constant, given by Eq. 18.

Figure 8 :

 8 Figure 8: Evolution of the ratio of energy densities of S and P waves for K = 2.5 and an explosive source in terms of the normalized time lapse for a cube whose sides are 3000 m long.

Figure 9 :

 9 Figure9: Variation of the S to P wave energy ratio for an explosion source and different values of the ratio c = ν A /ν µ in terms of the normalized time lapse: c = 2 (solid red line), c = 1 (dashed red line), c = 0 (isotropic, solid black line), c = -1 (dashed blue line), c = -2 (solid blue line). The horizontal dashed line indicates the theoretical value of the equipartition constant, given by Eq. 18. The left and right plots correspond to the cubes with sides 3000 m and 3500 m, respectively.

4 ) cos θ cos φ cos θ sin φ - 9 )

 49 We denote additionally ( k • q) = cos θ = χ. The PSDF of the mechanical properties can be expressed in terms of the mathematical expectation of the product of Fourier transformations of two homogeneous (stationary) random fields Ξ(x) and Ψ(x) as: E Ξ(p) Ψ(-p) = δ(0)(2π)3 3 c ν ΞΨ Φ ΞΨ (|p|), (A.10)

Table 1 :

 1 Definitions of the normalized correlation models.

	Correlation	Normalized CF	Normalized PSDF
	model	R(η)/R(0)	Φ(ψ)/R(0)
	Exponential	exp(-2η)	1 8π 2 1+ ψ 2 4	2
	Power-law	1 4 1+ π 2 η 2	2	1 π 4 exp -2 ψ π
	Gaussian	exp(-πη 2 )	1 8π 3 exp -ψ 2 4π
	Triangular	12(2-2 cos(2πη)-(2πη) sin(2πη)) (2πη) 4	3 8π 4 1 -ψ 2π H (2π -ψ)
	Low-pass white noise	3(sin( 3π 2 η)-3π 2 η cos( 3π 2 η)) ( 3π 2 η) 3	2 9π 4 H 3π 2 -ψ

http://www.ipgp.fr/ ~paulcup/RegSEM.html

Acknowledgments

This work was performed using HPC resources from the "Mésocentre" computing center of CentraleSupélec and École Normale Supérieure Paris-Saclay and the Centre de Calcul Intensif des Pays de la Loire (CCIPL) in Nantes.

Replacing the definitions for H αβ and particularizing for each mode α and β in Eq. (A.2) yields the definition of the differential scattering cross-section for each pair of modes. Then, the total scattering cross-sections is obtained using

dk (2π) 3 ; α, β ∈ {P, S}, (A. [START_REF] Turner | Radiative transfer and multiple scattering of diffuse ultrasound in polycrystalline media[END_REF] where I n is the n × n identity matrix. In the following subsections, the computations of the differential and total scattering cross-sections are performed for an elastic material with an isotropic background and cubic fluctuations.

Appendix A.1. Calculation of P-P scattering cross-sections

For P-P scattering, we have α = β = P and p P (k) = k. Then, ( k • p P (k)) * = q • p P (q) = 1, ( q

Inserting these relations in (A.4) yields:

Inserting this into Eq. (A.2) yields:

Integration can then be performed following Eq. (A.5) to obtain the total scattering cross section for the P-to-P scattering. Although possible in all generality, we choose here to consider a slightly simplified setting, only to present more condensed formula. We therefore assume that all PSDFs follow the same functional form Φ(ψ), and that they differ only through the variances and covariances:

(A.13) Following Eq. (A.5) with this hypothesis (and I 1 = 1) yields:

Similarly as above, the selection of A = 0 yields the classical P-P total scattering cross-section for isotropic fluctuations:

. Calculation of P-S scattering cross-sections

In this case we have α = P and β = S, and the polarizations are

]. From Eq. (A.4), the 1 × 2 matrix H PS (k, p, q) writes:

and likewise the 2 × 1 matrix H SP (q, p, k) writes:

Eq. (A.2) writes:

E H PS (k, kq, q)A S (q)H SP (q, qk, k) δ kv Pqv S .

(A.18) so that inserting Eq. (A. [START_REF] Saito | Simulating the envelope of scalar waves in 2d random media having power-law spectra of velocity fluctuation[END_REF]) and (A.17) yields:

Using Eq. (A.10) we get the following P-S differential scattering cross-section operator:

in which : denotes the double-dot product and the 2 × 2 matrix M is defined as M i j (k, q) = ( k • ẑi (q))( k • ẑ j (q)). Note that we have kv P = qv S (or q = kK) and thus the argument of the PSDFs is |k -q| = k 1 + K 2 -2Kχ in which k is the wavenumber of the P wave mode and K = v P /v S = κ 0 /µ 0 + 4/3 is the wave speed ratio. Thus, applying the P-S differential scattering cross-section operator to the identity matrix I 2 gives:

which can be written as a function only of k and χ:

Setting A = 0 yields the P-S differential scattering cross-section for materials with isotropic fluctuations [START_REF] Weaver | Diffusivity of ultrasound in polycrystals[END_REF][START_REF] Ryzhik | Transport equations for elastic and other waves in random media[END_REF][START_REF] Turner | Scattering and diffusion of seismic waves[END_REF][START_REF] Baydoun | Kinetic modeling of multiple scattering of elastic waves in heterogeneous anisotropic media[END_REF]:

Following Eq. (A.5) and the hypothesis of equal PSDF, the integration of (A.22) over the wavevector k gives the P-S total scattering cross-section:

for which the classical isotropic S-to-P total scattering cross-section can be retrieved by setting A:

Calculation of S-P scattering cross-sections

Using the same arguments as above, the S-P differential scattering cross-section for cubic anisotropy reads:

so that its application on the identity matrix (unity) gives:

which in terms of k and χ and φ writes:

A comparison between Eqs. (A.21) and (A.28) yields the following relation between P-S and S-P scattering crosssections:

Integrating Eq. (A.28) with respect to the wave vector (knowing that in spherical coordinates the angle φ varies in [0, π]), the diagonal components of the S-P total scattering cross-sections will be equal and the extra-diagonal ones are zero. From Eq. (A.29) one can derive the following relation between P-S and S-P total scattering cross-sections:

. Calculation of S-S scattering cross-sections

The objective of this section is to calculate the quadratic S-S scattering cross-section operator σ SS (k, q). In this case α = S and β = S. We have ( k

] = G(k, q) + ( k • q)T(k, q) and p * S (k)diag(k 1 q 1 , k 2 q 2 , k 3 q 3 )p S (q) = 0 × I 2 . In these equations the 2 × 2 matrices G and T are respectively defined as

) and T i j (k, q) = (ẑ i (k) • ẑ j (q)). It should be pointed out that since in this case the first and the last terms in the right hand side of the Eq. (A.4) are both zero, the influence of the anisotropy parameter vanishes so that the S-S scattering parameters will become identical between cubic and isotropic cases.

Inserting these equations into (A.4) yields:

From Eq. (A.2) we have:

Introducing (A.31) into this equation yields:

µ Φ µµ G(k, q)a S (q)G(q, k) + ( k • q) G(k, q)a S (q)T(q, k) + T(k, q)a S (q)G(q, k)

+( k • q) 2 T(k, q)a S (q)T(q, k) δ(kq), (A. [START_REF] Bal | Accuracy of transport models for waves in random media[END_REF] in which the argument of all the PSDFs is |k -q| = k 2(1 -χ), as for the case of P-P scattering but this time with k = ω/v S the wavenumber of the S mode. Applying the S-S scattering cross-section opertator to the identity matrix I 2

gives the following representation in terms of the wavenumber k and χ as: Following Eq. (A.5) and the hypothesis on the PSDF, the integration of (A.35) over the wavevector k yields the S-S total scattering cross-section for a heterogeneous material with isotropic background and cubic fluctuations: .36)