Cecile Cazes
email: cecile.cazes@u-bordeaux.fr

Kevin Phelan

Victoire Hubert

Rodrigue Alitanou

Harouna Boubacar

L Izie Bozama

Gilbert Tshibangu Sakubu

Aurelie Beuscart

Cyrille Yao

Delphine Gabillard

Cécile Cazes

Izie Liévin

Bozama

Aurélie Beuscart

Moumouni Kinda

Augustin Augier

Xavier Anglaret

Susan Shepherd

Renaud Becquet

Mrs Cécile Cazes

Cécile Cazes

Simplifying and optimising management of acute malnutrition in children aged 6 to 59 months: study protocol for a community-based individually randomised controlled trial in Kasaï, Democratic Republic of Congo

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

INTRODUCTION

Acute malnutrition (AM) affects an estimated 50 million children under 5 years of age, including 16.6 million severe acute malnutrition (SAM) cases, and is an underlying cause of 800 000 deaths each year worldwide. 1 2 Strengths and limitations of this study

Open access

Other estimation methods based on incidence instead of prevalence suggest the caseload for this major public health concern may be substantially higher. 3 4 One quarter of all acutely malnourished children are in Africa and as many as 2 million severely wasted children were reported in the Democratic Republic of Congo (DRC) in 2018. [START_REF]UNICEF DRC humanitarian situation report March, 2018 -Democratic Republic of the Congo[END_REF] Since the publication of the United Nations (UN) Joint Statement in 2009, millions of children have received treatment for SAM via standard community-based management 6 7 with >80% of children recovering in outpatient care. [START_REF] Collins | Key issues in the success of community-based management of severe malnutrition[END_REF] Despite more than a decade of expanded access to treatment, coverage of nutrition programmes remains remarkably low, as few as 20% of wasted children actually receive treatment 9 10 and the World Bank estimates that it would take an additional US$6.3 billion per year to reach a coverage of 80%. [START_REF] Horton | Scaling up nutrition: what will it cost?. The International bank for reconstruction and development/The World Bank[END_REF] There is, therefore an urgent need to better allocate available resources through the simplification and optimisation of current AM protocols. Adapting programming in three key aspects of case management may allow improved access to treatment for acutely malnourished children.

First, AM is as a continuum condition, but it is arbitrarily divided into moderate acute malnutrition (MAM) and SAM categories. This distinction results in separate programmes overseen by different UN agencies and using different protocols and products-ready-to-use supplementary food (RUSF) or fortified-blended flours for children with MAM and ready-to-use therapeutic food (RUTF) for children with SAM. This complicates supply chain, delivery of care, data collection, and creates confusion and extra work for caregivers and health workers alike. [START_REF] Rogers | Coverage of communitybased management of severe acute malnutrition programmes in twenty-one countries, 2012-2013[END_REF] In practice, only SAM treatment is often available, whereas identification and treatment of children earlier in the wasting process would lead to fewer hospitalisations and deaths. The treatment of MAM has been shown to reduce mortality risk by more than 10%. [START_REF] Isanaka | Cost analysis of the treatment of severe acute malnutrition in West Africa[END_REF] Second, current case definitions to determine programme eligibility are unnecessarily complicated, using two independent anthropometric criteria: midupper arm circumference (MUAC) or weight-for-height Z-score (WHZ). WHZ alone or in combination with MUAC does not offer a clear advantage for identifying children at near-term risk of death over MUAC alone. [START_REF] Briend | Mid-upper arm circumference and weight-for-height to identify high-risk malnourished under-five children[END_REF] MUAC-only programming is expanding as evidence accumulates that weight gain and MUAC gain track each other and respond to treatment in similar ways. [START_REF] Binns | Relationship between mid upper arm circumference and weight changes in children aged 6-59 months[END_REF][START_REF] Roberfroid | Utilization of midupper arm circumference versus weight-for-height in nutritional rehabilitation programmes: a systematic review of evidence[END_REF][START_REF] Burza | Community-based management of severe acute malnutrition in India: new evidence from Bihar[END_REF] MUAC is, therefore, becoming a stand-alone practical tool for all phases of nutrition programming: screening, admission, monitoring recovery and determining discharge. Furthermore, the simplicity of MUAC bracelets makes it possible to screen children not only in the community but also at home. [START_REF] Alé | Mothers screening for malnutrition by mid-upper arm circumference is non-inferior to community health workers: results from a large-scale pragmatic trial in rural niger[END_REF] Third, therapeutic food rations could be allocated more efficiently. The RUTF ration for treating SAM (130-200 kcal/kg/day) is paradoxical. The amount of RUTF prescribed remains constant or even increases with weight gain up until discharge (ie, as the child recovers from AM), even though weight and MUAC gain is maximal during the first 2-3 weeks of supplementation. [START_REF] Goossens | Mid-upper arm circumference based nutrition programming: evidence for a new approach in regions with high burden of acute malnutrition[END_REF] Consequently, gradual dose reduction seems to be a more rational use of RUTF which accounts for a large proportion of total malnutrition programme costs. [START_REF] Shekar | An investment framework for nutrition: reaching the global targets for stunting, anemia, breastfeeding, and wasting[END_REF] Results on integrating SAM and MAM treatment in a single MUAC-based protocol using only RUTF at a gradually reduced dose were first published in 2015. This cluster-randomised trial conducted in Sierra Leone found high recovery rates, a reduced caseload of children with SAM, a better coverage rate and reduced RUTF costs when compared with the standard protocol. [START_REF] Maust | Severe and moderate acute malnutrition can be successfully managed with an integrated protocol in Sierra Leone[END_REF] More recently, a non-inferiority randomised trial in Burkina Faso showed that the reduction of the RUTF dose after the first 2 weeks results in similar weight gain velocity and recovery rates as with the standard dose given throughout SAM treatment. [START_REF] Kangas | Impact of reduced dose of ready-to-use therapeutic foods in children with uncomplicated severe acute malnutrition: a randomised non-inferiority trial in Burkina Faso[END_REF] Our research consortium developed the Optimising treatment for Acute Malnutrition (OptiMA) strategy that simplifies the definition of AM to MUAC <125 mm or the presence of bipedal oedema, and uses a single product for treatment-RUTF-at a gradually reduced dose based on a child's weight and MUAC status (see table 1). The OptiMA strategy was first implemented in a proofof-concept single-arm trial in Burkina Faso and showed a recovery rate that exceeded national and international Sphere [START_REF]Reference: sphere association humanitarian charter and minimum standards in humanitarian response[END_REF] standards for both SAM and MAM (86.3%; 95% CI 85.4% to 87.2%) with excellent health worker adherence to the new RUTF dosage table. [START_REF] Daures | New approach to simplifying and optimising acute malnutrition treatment in children aged 6-59 months: the optima single-arm proof-of-concept trial in Burkina Faso[END_REF] However, recovery of children admitted at MUAC <115 mm was poorer than anticipated (70.4%; 95 % CI 67.5% to 73.5%) which necessitated a randomised controlled trial to compare OptiMA to current national malnutrition protocols. We therefore planned this individually randomised non-inferiority clinical trial to compare the OptiMA strategy to the DRC nutritional standard protocol in children 6-59 m with uncomplicated AM.

We hypothesised that the OptiMA strategy would be as effective as the current DRC national protocol currently in use, in children 6 to 59 years old as judged by the success rate in the treatment of uncomplicated SAM and MAM at 6 months postrandomisation and in terms of recovery rate in the treatment of uncomplicated SAM.

Objectives

The principal objective aims to determine, 6 months after inclusion, whether the OptiMA strategy leads a success rate that is non-inferior to that of the standard DRC protocol in use in the same outpatient health facilities. The definition of 'success' is described in the Study outcomes section.

The main secondary objective is to determine whether the recovery rate of children with uncomplicated SAM according to the current WHO definition [START_REF]Community based management of sever acute malnutrition -A joint statement by the World Health Organization the World Food Programme, the United Nations System Standing Committee on Nutrition and the United Nations Children's Fund[END_REF] (ie, MUAC <115 mm or WHZ <-3 or bilateral oedema) managed under the OptiMA protocol is non-inferior to that of the national standard protocol, anticipating that such children treated under OptiMA will receive a smaller overall Other secondary objectives include description and comparison between the two study arms of the following outcomes: RUTF consumption, relapse after nutritional recovery, outcomes compared with international Sphere references, recovery and success in children who present with both wasting and stunting at inclusion, and to describe the nutritional and clinical status of children hospitalised while enrolled in the study.

METHODS

This protocol follows the Standard Protocol Items: recommendations for Interventional Trials (SPIRIT) guidelines. [START_REF] Chan | Spirit 2013 statement: defining standard protocol items for clinical trials[END_REF]

Design

This study is a non-inferiority individually randomised controlled clinical trial conducted at health centres and in the community. Participants were randomly assigned to either the OptiMA strategy arm (intervention=OptiMA) or the national standard protocol arm (control=Standard).

Study setting

The trial took place in the Kamuesha health zone, Kasai province in DRC and was nested within a medical and nutritional emergency humanitarian project launched by Alliance for International Medical Action (ALIMA) in May 2018. The Kamuesha health zone is located in a tropical forest and is particularly isolated.

ALIMA supports the DRC Ministry of Health (MoH) in 9 of 26 health areas in the Kamuesha health zone to address a nutritional crisis occurring after a 2 years of armed conflict with significant population displacement. [START_REF]DR Congo -situation report N°14: complex emergency in the Kasai region[END_REF] During the trial preparation phase, we visited all health areas covered by ALIMA and selected four as research sites based on demographic, epidemiological and logistical criteria in addition to observations provided by local stakeholders. According to projections from the last census, the four health areas chosen covered a population of 65 000 people, including 12 000 children aged 6 to 59 months, spread over 60 villages. Health facilities included in the study were the four community-based primary health centres of these four health areas in addition to the general hospital of Kamuesha district which was used for referring participants with medical complications. Most (n=49) villages were less than 10 km away from the primary health centres while 11 were more than 15 km away.

Eligibility criteria

Children aged 6-59 months, residing in one of the four health areas and meeting at least one of the three following AM criteria were eligible for inclusion in the trial: MUAC <125 mm and/or bilateral pitting oedema (+/++) and/or WHZ <-3. Children with a medical Open access complication requiring hospitalisation, no appetite, oedema grade +++, with a known allergy to milk, peanuts or RUTFs, with a known chronic pathology (such as sickle cell anaemia, trisomy 21, congenital heart disease, neurological disease), or already enrolled in a malnutrition programme were excluded from the trial. Two categories of children were included in the trial but not eligible for randomisation for ethical concerns. First, children with WHZ <-3 and MUAC >125 mm without oedema were systematically included in the control arm, as they were not RUTF eligible according to the wasting definition of the OptiMA strategy. Second, children with a sibling already enrolled in the study were systematically allocated to the study arm of the index sibling.

Study outcomes

The primary outcome will be judged by a binary composite indicator. Children classified as 'success' fulfil all of the following criteria: alive, not acutely malnourished per the definition applied at inclusion and not having an additional episode of AM throughout the 6-month observation period. All other children are classified as 'unsuccessful'.

The main secondary outcome will be determined among children in both arms of the trial who meet the current WHO definition of SAM at inclusion. For this subgroup, recovery is defined in both arms, after a 4-week minimum duration of treatment as, clinically well (an axillary temperature <37.5°C), absence of bipedal oedema and an MUAC >125 mm for the OptiMA arm or MUAC >125 mm or WHZ >-1.5 for the standard arm, for two consecutive weeks. Additional secondary outcomes are listed in table 2.

Sample size considerations

We determined the sample size for the principal objective using an expected success rate of 55% and for the secondary main objective an expected recovery rate of 85%. Both of these hypotheses are based on recovery, death, default and relapse rates available in the literature for severely and moderately wasted children supplemented with RUTF and/or RUSF in DRC and other African contexts. 28 29 We also considered spontaneous recovery rate documented for MAM children without treatment in the standard arm. [START_REF] James | Children with moderate acute malnutrition with no access to supplementary feeding programmes experience high rates of deterioration and NO improvement: results from a prospective cohort study in rural Ethiopia[END_REF] During the preparation phase, we twice visited the four sites included in the study and found that the standard MAM treatment programme was either non-functional or partially functional.

We assumed a margin of non-inferiority of 10%, the same as the three other non-inferiority trials assessing simplified treatment strategies with RUTF for wasted children. [START_REF] Bailey | Combined protocol for acute malnutrition study (ComPAS) in rural South Sudan and urban Kenya: study protocol for a randomized controlled trial[END_REF][START_REF] Bahwere | Cereals and pulse-based ready-to-use therapeutic food as an alternative to the standard milk-and peanut paste-based formulation for treating severe acute malnutrition: a noninferiority, individually randomized controlled efficacy clinical trial[END_REF][START_REF] Bahwere | Effectiveness of milk whey protein-based ready-to-use therapeutic food in treatment of severe acute malnutrition in malawian under-5 children: a randomised, double-blind, controlled non-inferiority clinical trial[END_REF] To demonstrate non-inferiority of the OptiMA strategy compared with standard protocol, 890 and 490 participants were required for the principal and main secondary objectives, respectively, with 80% power at a unilateral significance level of 2.5% and with an inflation of 15% to account for unexploitable data. We estimated 15% of incomplete data in the light of the challenging reality of this study setting where families are regularly confronted with barriers to receiving healthcare, insecurity and displacement.

Randomisation

The trial statistician established and maintained a confidential randomisation list. Randomisation was done in blocks and was stratified at the study site by the severity of AM according to the WHO definition. This double stratification allows for recruitment of comparable groups of SAM and non-SAM children and for simultaneous randomisation at the four sites. The trial data scientist integrated the list into a randomisation software developed for the trial that was programmed on four tablet devices. This software assigned a treatment arm by sequentially drawing from this list each time a randomisation procedure was completed. Once the sample size for the primary objective was reached, only WHO-SAM cases continued to be enrolled and randomised until the sample size required for the main secondary objective was attained.

Patient and public involvement

To ensure that the trial was understood, accepted and implemented in accordance with local customs and practices, meetings to explain trial eligibility and procedures were organised with community representatives in each village and with nutrition actors prior to recruitment. During the trial's active phase, regular oral presentations were held at the local level. We intend to disseminate the findings through meetings with community representatives and national nutrition actors.

Enrolment and informed consent procedure

Prior to enrolment, all community health workers, nurse research officers and MoH medical staff implicated in the study were trained on the trial protocol, standardised operating procedures and the correct use of tools developed for data collection. Nurse research officers were specifically trained to enrol and randomise study participants, and monitor home visits and outpatient visits at the health centre. Children were recruited through monthly active mass malnutrition screenings in each of the 60 villages. Concomitantly, children attending outpatient consultations at the four study health centres underwent routine passive malnutrition screening and were enrolled in the trial if eligible. In both active and passive screening, a nurse research officer, assisted by a community health worker, collected anthropometric (weight, height, MUAC, WHZ) and clinical (presence of oedema) information on eligible children. Children then underwent a medical consultation to determine whether they met the criteria for study inclusion. A standard operating procedure (see online supplemental file 1) for assessing child anthropometry (weight, height, MUAC and oedema) were observed in each site. For each eligible child, the caretaker was given a study information sheet detailing the study's aims, treatment protocols, study duration, and Open access frequency of clinic visits and community follow-up. This information was explained orally in Tshiluba (local language). Caretakers who agreed to participate in the study indicated their consent by signing (signature or fingerprint) a written consent form. When caretakers were not able to read or write, an impartial witness oversaw the consent process and attested to the caretaker's consent by signing the consent form on her/ his behalf. All caretakers were informed about their right to withdraw from the study at any time without affecting the quality of medical care provided to their children. All medical care and nutritional treatment were provided free of charge, regardless of participation in the study.

Intervention and follow-up

In the intervention arm, all children with a MUAC <125 mm or oedema were supplemented with RUTF, according to the OptiMA dosage In the control arm, children included with MUAC <115 mm and/or WHZ <-3 and/or nutritional oedema were treated with RUTF, according to the DRC national protocol dosage table which is based on a child's weight at each visit (ie, RUTF dosage gradually increases as a child's weight increases). The DRC protocol provides a RUTF daily caloric intake of 150-200 kcal/kg/day during the entire treatment course until recovery in children with SAM and one sachet of RUSF per day (500 kcal/day) in children with MAM.

The differences between the OptiMA and the DRC national standard protocols, summarised in table 1, concern treatment eligibility, definitions of recovery and relapse, the type of therapeutic food provided and RUTF dosage calculation. All other aspects of the standard nutrition protocol were applied to all children in both arms. All cases of SAM according to the WHO definition received amoxicillin 50-100 mg/kg/day for 7 days, and all children received vitamin A and deworming regardless of anthropometry. In both arms, a malaria rapid test was systematically performed at inclusion and at any point during follow-up when a child presented with clinical signs of malaria. If positive, an artemisinin combination treatment was prescribed for the child. At the time of the study, the HIV national programme was not functional in the Kamuesha health zone, hence children with SAM were not offered HIV counselling and testing, as is recommended. [START_REF]Community based management of sever acute malnutrition -A joint statement by the World Health Organization the World Food Programme, the United Nations System Standing Committee on Nutrition and the United Nations Children's Fund[END_REF] All participants were monitored for 6 months after inclusion (see figure 1). While in the treatment phase, children came to the health centres once a week for outpatient consultations (or every 2 weeks for those children living in villages more than 14 km from the health centre). On discharge from treatment, or immediately after inclusion for children with MAM who did not receive RUTF in the standard arm, children were consulted every 2 weeks in their villages until 6 months post-inclusion was completed. During village visits, a nurse research officer assisted by one or two community health workers monitored the anthropometric and clinical status of these children, referring any child who needed nutritional or medical care to either the primary healthcare facility or to the Kamuesha general hospital.

Blinding

This was an unblinded trial. Caretakers and study personnel were aware of study arm allocation for participating children due to differences in anthropometric criteria and RUTF ration. On inclusion, the nurse research officer gave each trial participant's caretaker a card specifying the nutritional strategy being followed by the child. Caretakers were asked to present this card in the event of an unplanned outpatient consultation or hospital admission during the study period. In this way, nurses or doctors at any health facility were able to identify the nutritional treatment being followed by the child.

Data collection and management

After obtaining individual informed consent, all sociodemographic characteristics, anthropometric and clinical data collected during follow-up were documented in the case report forms by the research officers and the healthcare providers trained specifically for this study. All data entered into the database were securely transferred and stored on a server hosted by PAC-CI, a satellite for the French National Agency for Research on AIDS and Viral Hepatitis in Abidjan, Côte d'Ivoire which served as the methodology and coordinating centre for this study. All data and any adverse events, including deaths and hospitalisations, were rigorously monitored both on-site and remotely, according to the data monitoring plan. To ensure patient safety and data integrity, the methodology and coordinating centre continuously supervised data management activities, and the senior scientific project leader and investigators performed field visits on a regular basis. If children failed to attend follow-up visits, the community health worker living in the village visited the household to check on the child, to determine the reasons for absence and to encourage the caretakers to adhere to the planned health centre visit schedule.

Statistical analyses

All statistical analyses will be performed using RStudio Software (Boston, Massachusetts, USA) as specified by the data analysis plan published on ClinicalTrials. gov prior to performing any analysis. Statistical tests will be Open access carried out bilaterally with an alpha risk of 5% and unilaterally with an alpha risk of 2.5% for the non-inferiority analysis. Qualitative variables will be described in terms of numbers, percentages and provided with CIs when relevant. If necessary, comparisons of qualitative variables will be made using χ 2 or Fisher exact tests. Quantitative variables will be described in terms of mean, SD and CI or median, range and IQR. If necessary, comparisons of quantitative variables will be made using Student's, Wilcoxon's or the Kruskal-Wallis tests, depending on the distribution of the variable of interest. Time delay type variables will be described in terms of the incidence of occurrence, and the probability of occurrence over time, estimated by the Kaplan-Meier method. If necessary, probability comparisons will be made using log-rank tests, or proportional risk models, after verification of the assumption of proportionality of risks.

Primary analysis of the primary and secondary main outcomes Descriptive summaries of participant characteristics by arm at inclusion will be tabulated. Participants included in the main analyses will be described according to the diagram defined by CONSORT (Consolidated Standards of Reporting Trials) recommendations. [START_REF] Moher | Consort 2010 explanation and elaboration: updated guidelines for reporting parallel group randomised trials[END_REF] The occurrence of the primary 'success' and the secondary main outcome 'recovery' will be compared between the two randomisation arms. These two comparisons will be performed both by 'intention-to-treat' basis and on a 'per-protocol' basis. Non-inferiority analyses will be performed on these two outcomes. Only if non-inferiority for these two outcomes is demonstrated will the other secondary analyses will be performed, and, if appropriate, as a superiority analysis. Superiority will be particularly sought for cost outcomes related to RUTF cost-efficiency.

Sensitivity analysis of the primary outcome First, rates of 'success' and 'recovery' will be analysed based on all available data. Then a sensitivity analysis will be performed using the maximum bias method. In this analysis, deceased participants, those who have withdrawn their consent, who have been transferred to another health facility or have been lost to follow-up, will be considered to have systematically failed to the allocated treatment.

Safety

The trial was monitored monthly by an international steering committee and quarterly by a national steering committee where any adverse events were presented. No interim analysis was planned before the final analyses. The data safety monitoring board could have requested Open access an interim analysis if it deemed it necessary for patient safety.

A child in intervention arm could be removed from the study for two reasons: (1) if he or she had two episodes of >5% weight loss during the course of treatment and (2) if he or she was not SAM at inclusion but deteriorated to SAM during RUTF supplementation.

Ethics and dissemination

The study was conducted in accordance with the Declaration of Helsinki. We obtained ethical approval with annual renewal from DRC National Ethics Committee (CNES) (94/CNES/BN/PMMF/2018) and from the Ethics Evaluation Committee of the French National Institute for Health and Medical Research (Inserm) (18-545). The final version of the protocol, version 3.1, dated 22 October 2019, is available for sharing on request and included three minor amendments to protocol version 2.0 (dated 6 March 2019) and version 3.0 (dated 18 April 2019) which was shared with national and international ethical committees and did not require a new authorisation. In November 2019, the CNES performed an audit at each site and then renewed approval (152/CNES/BN/ PMMF/2019) for the continuation of the trial. All data were anonymised when entered into the database using unique identification numbers. We intend to disseminate the results regardless of positive or negative findings broadly via peer-reviewed publication, conferences and clinical networks targeting academics, policy-makers, clinicians and caregivers.

DISCUSSION

This trial was anchored in the following foundational principle: the need to merge MAM and SAM nutritional care into a single programme using one anthropometric admission criterion and one type of therapeutic food to simplify care for families, health workers and programme managers. This study will contribute to a developing evidence-base composed of similar studies already published or currently in progress in Sierra Leone, Burkina Faso, South Sudan and Kenya. 31 35 Given that the risk profile associated with low anthropometry is highly contextual, it is important to conduct similar trials in multiple settings. This will be the first such trial in DRC, where we expect a higher proportion of oedematous malnutrition than is typically found in East or West Africa. Furthermore, this is an individually randomised trial that will monitor outcomes for an extended period, 4-5 months, after nutritional recovery. It also includes a group of children rarely studied but common in everyday practice: MAM children who do not receive supplemental food. This study design should allow for robust conclusions not only regarding the non-inferiority of OptiMA compared with a typical standard SAM protocol but also an evaluation of the incidence of SAM among children with MAM. This trial will also document outcomes of sustained health and nutrition status over a longer period than previous trials.

Furthermore, the stratified randomisation of children with SAM according to the WHO definition will shed some light on the clinical effects of the OptiMA strategy for this specific category of children. This inclusion strategy will also help to determine the segment of children with SAM defined by only WHZ with MUAC >125 mm, in Kasaï province where stunting prevalence in children is higher than anywhere else in the DRC. [START_REF] Kismul | Determinants of childhood stunting in the Democratic Republic of Congo: further analysis of demographic and health survey 2013-14[END_REF]

Trial status

The recruitment phase began in July 2019 and ended in 22 January 2020. The estimated completion date for this study is September 2020.

on March 8 ,

 8 2021 by guest. Protected by copyright. http://bmjopen.bmj.com/ BMJ Open: first published as 10.1136/bmjopen-2020-041213 on 2 December 2020. Downloaded from Open access RUTF ration than those treated under the DRC national protocol.

on 8 ,

 8 2021 by guest. Protected by copyright. http://bmjopen.bmj.com/ BMJ Open: first published as 10.1136/bmjopen-2020-041213 on 2 December 2020. Downloaded from Cazes C, et al. BMJ Open 2020;10:e041213. doi:10.1136/bmjopen-2020-041213Open access

on March 8 ,

 8 2021 by guest. Protected by copyright. http://bmjopen.bmj.com/ BMJ Open: first published as 10.1136/bmjopen-2020-041213 on 2 December 2020. Downloaded from

Figure 1

 1 Figure 1 Schedule of enrolment, interventions and assessments OptiMA-DRC overview. *Monthly active screening in 60 villages and passive screening during outpatient visit in four health centres. **Weekly (for those living in villages at 14 km or less from the health centre) or bimonthly (for those living in villages more than 14 km from the health centre) outpatient visits at health centre for participants with ready-to-use therapeutic food (RUTF) supplementation. ***Bimonthly home visits for children without RUTF supplementation. d, day; DRC, Democratic Republic of Congo; m, month; OptiMA, optimising acute malnutrition.

Table 1

 1 Wasting definition, treatment products, calculation of dosage and recovery definition in the DRC national and OptiMA protocol

		National DRC protocol	OptiMA protocol
		SAM	MAM	Acute malnutrition
	Wasting	MUAC <115 mm or	MUAC (115-124 mm)	MUAC <125 mm or bipedal oedema
	definition	WHZ <-3 or bipedal	or -3 <WHZ <-2
		oedema
	Treatment	RUTF 150-200 kcal/	Super cereal plus 200 g/day	MUAC	MUAC (115-	MUAC (120-
	product	kg/day	(~1000 kcal/day)	<115 mm	119 mm)	124 mm)
			or RUSF, one 92 g sachet/day (500	or bipedal
			kcal/day)	oedema
				RUTF 170-200	RUTF 125-190	RUTF 50-166
				(kcal/kg/day)	(kcal/kg/day)	(kcal/kg/day)
	Calculation of	According to the	Fixed amount, regardless of weight	According to MUAC status and weight
	dosage	weight	or MUAC status
	Recovery	MUAC ≥125 mm or	MUAC ≥125 mm or WHZ ≥-1.5	MUAC >125 mm for 2 consecutive weeks
	definition	WHZ ≥-1.5 (for 2	If after recovery from SAM: MUAC
		consecutive weeks)	≥125 mm and WHZ ≥-1.5 and
			discharge after 3 months
		And no oedema for 2 consecutive weeks	And No oedema for 2 consecutive weeks
				And minimum 4 weeks in programme and good
				clinical health

DRC, Democratic Republic of Congo; MAM, moderate acute malnutrition; MUAC, mid-upper arm circumference; OptiMA, Optimising acute MAlnutrition; RUSF, ready-to-use supplementary food; RUTF, ready-to-use therapeutic food ; SAM, sever acute malnutrition; WHZ, weightfor-height Z-score. on March 8, 2021 by guest. Protected by copyright. http://bmjopen.bmj.com/ BMJ Open: first published as 10.1136/bmjopen-2020-041213 on 2 December 2020. Downloaded from

Table 2

 2 Primary and secondary outcomes in the OptiMA-DRC trial

	Primary Success Main secondary Recovery Secondary Consumption of RUTF by children with acute malnutrition at inclusion Consumption of RUTF by children who recovered from SAM Total weight gain and daily weight gain in children who recovered from SAM Total MUAC gain and daily MUAC gain in children who recovered from SAM Total length of RUTF treatment in children who recovered from	Measurement variable Composite: alive, not acutely malnourished per the definition applied at inclusion and no additional episode of acute malnutrition throughout the 6-month observation period Composite: in both arms, a 4-week minimum duration of RUTF treatment, an axillary temperature <37.5°C, absence of bipedal oedema, and for the OptiMA arm a MUAC >125 mm and for the standard arm MUAC >125 mm or WHZ >-1.5, for two consecutives weeks Sachets of RUTF consumed Sachets of RUTF consumed Total weight gain (g) and weight gain in g/kg/day Total MUAC gain in mm and MUAC gain in mm/day Total number of days with RUTF treatment	Denominator Children with MUAC <125 mm or WHZ <-3 or bipedal oedema at inclusion Children with MUAC <115 mm or WHZ <-3 or bipedal oedema at inclusion Children with MUAC <125 mm or WHZ <-3 or bipedal oedema at inclusion Children with MUAC <115 mm or WHZ <-3 or bipedal oedema at inclusion who recovered Children with MUAC <115 mm or WHZ <-3 or bipedal oedema at inclusion Children with MUAC <115 mm or WHZ <-3 or bipedal oedema at inclusion who recovered Children with MUAC <115 mm or WHZ <-3 or bipedal oedema at inclusion who recovered	Method of aggregation Timepoint Proportion 6 months post-inclusion Proportion Throughout the 6 months needed for study completion after RUTF treatment has been initiated at inclusion Median/mean 6 months post-inclusion Median/mean At the visit when recovery is determined Median/mean At the visit when recovery is determined Median/mean At the visit when recovery is determined Median/mean At the visit when recovery is determined	http://bmjopen.bmj.com/ BMJ Open: first published as 10.1136/bmjopen-2020-041213 on 2 December 2020. Downloaded from
	SAM Non-response in children with SAM Relapse to a new episode of SAM (WHO definition)	Absence of recovery status Children with MUAC<115 or WHZ <-3 or oedema after RUTF treatment	who recovered Children with MUAC <115 mm or WHZ <-3 or bipedal oedema at inclusion Children with MUAC <115 mm or WHZ <-3 or bipedal oedema at inclusion recovered	Proportion Proportion	After 12 and 16 weeks of nutritional follow-up of SAM episode at inclusion During 3 months following recovery from SAM episode at inclusion	on March 8, 2021 by guest. Protected by copyright.

MUAC, mid-upper arm circumference; OptiMA, optimizing acute malnutrition; RUTF, ready-to-use therapeutic food; SAM, severe acute malnutrition; WHZ, weight-for-height Z-score.

Cazes C, et al. BMJ Open 2020;10:e041213. doi:10.1136/bmjopen-2020-041213

Acknowledgements We are indebted to the women and children who participate in the study. The authors wish to particularly acknowledge the ALIMA operational team on the ground who managed the study and the nutrition programme on a day-to-day basis. The authors also acknowledge the MoH of the DRC through the commitment of key representatives from the Kamuesha Health Zone and the National Nutrition Programme. We acknowledge the implementing partners of the interagency Nutrition Cluster Kasaï Province, in particular representatives from UNICEF and the World Food Programme. We also acknowledge Maguy Daures (Inserm UMR 1219, Bordeaux), Valérie Journot (Inserm UMR 1219, Bordeaux) and Claire Levy-Marchal (ALIMA, Paris) for having advising the study team on the trial methodology. This study was developed as part of the Clinical and Operational Research Alliance (CORAL). This research platform aims at developing high-quality innovative and transformative research programmes within a partnership between scientists from the research institute Inserm (Institut National de la Santé et la Recherche Médicale, Bordeaux and Abidjan) and the humanitarian organisation ALIMA (Alliance for International Medical Action), primarily on improving maternal and child health outcomes. A board of directors defines the scientific policy of the CORAL partnership alongside with the supervision of research projects and dissemination of results. It consists of senior representatives from both ALIMA and Inserm: Renaud Becquet (Inserm Bordeaux, methodological co-chair), Susan Shepherd (ALIMA, medical co-chair), Augustin Augier (ALIMA), Moumouni Kinda (ALIMA), Marie Jaspard (ALIMA and Inserm Abidjan), Claire Levy-Marchal (ALIMA) and Xavier Anglaret (Inserm Bordeaux and Abidjan). The CORAL research platform meets annually with an external scientific advisory board to review projects and strategic orientation. Finally, we warmly acknowledge the members of the Data Safety Monitoring Board (DSMB) for the OptiMA-DRC trial: Yves Martin-Prevel (DSMB chair, French National Institute for Sustainable Development (IRD), NUTRIPASS team, University of Montpellier, Montpellier, France), Matthew Colderon (Epicentre, New York, USA) and Katia Castetbon (DSMB statistician, School of Public Health, Université Libre de Bruxelles, Bruxelles, Belgium). Contributors SS and RB developed the clinical (SS) and methodological (RB) study concept. RB, CC, XA, DG, AA, KP and SS designed the study methodology and wrote the protocol. CC, VH, HB, RA and MK coordinate the study teams. LIB, GTS coordinate the MoH staff working in the trial. CC, VH, HB, LIB, GTS, AB, CY and RB organise and supervise data collection. CY created the software tool for randomisation and developed the database. CC, VH, DG, XA, SS and RB developed the statistical analysis strategy. CC wrote the first draft of the manuscript. SS and on March 8, 2021 by guest. Protected by copyright. http://bmjopen.bmj.com/ BMJ Open: first published as 10.1136/bmjopen-2020-041213 on 2 December 2020. Downloaded from

Ethics and dissemination Ethics approvals were obtained from the National Health Ethics Committee of the Democratic Republic of Congo and from the Ethics Evaluation Committee of Inserm, the French National Institute for Health and Medical Research draft and had a substantial writing contribution to the development of the final manuscript.

Funding The study was supported by the Innocent Foundation (London, UK), grant number ALIMA-2018-DRC, and received additional funding from European Commission through the European Civil Protection and Humanitarian Aid Operations (ECHO, Brussels, Belgium), grant number ECHO/COD/BUD/2018/91023. The funders had no role in study design, data collection and analysis, decision to publish or preparation of the manuscript. This document covers humanitarian aid activities implemented with the financial assistance of the European Union. The views expressed herein should not be taken, in any way,

Provenance and peer review Not commissioned; externally peer reviewed.

Supplemental material This content has been supplied by the author(s). It has not been vetted by BMJ Publishing Group Limited (BMJ) and may not have been peer-reviewed. Any opinions or recommendations discussed are solely those of the author(s) and are not endorsed by BMJ. BMJ disclaims all liability and responsibility arising from any reliance placed on the content. Where the content includes any translated material, BMJ does not warrant the accuracy and reliability of the translations (including but not limited to local regulations, clinical guidelines, terminology, drug names and drug dosages), and is not responsible for any error and/or omissions arising from translation and adaptation or otherwise.


G García-Sánchez

A M Mancho

S Wiggins

A bridge between invariant dynamical structures and uncertainty quantification

We develop a new quantifier for forward time uncertainty for trajectories that are solutions of models generated from data sets. Our uncertainty quantifier is defined on the phase space in which the trajectories evolve and we show that it has a rich structure that is directly related to phase space structures from dynamical systems theory, such as hyperbolic trajectories and their stable and unstable manifolds. We apply our approach to an ocean data set, as well as standard benchmark models from deterministic dynamical systems theory. A significant application of our results, is that they allow a quantitative comparison of the transport performance described from different ocean data sets. This is particularly interesting nowadays when a wide variety of sources are available, since our methodology provides avenues for assessing the effective use of these data sets in a variety of situations.

Introduction

Uncertainty Quantification (UQ) searches for a quantitative characterization of uncertainties in computational models of real-world applications. Uncertainties associated to these models have been studied for many years. Depending on the context they can arise as a result of noisy or incomplete data, errors introduced by numerical models, or by the lack of complete understanding of the governing physical processes.

In geophysical contexts, uncertainty is a topic of much interest because it is inherent to the equations describing the motion of fluids such as the ocean or the atmosphere. The motivation of this paper is trying to acquire a deeper understanding of uncertainty quantification, that allows a better characterization of its presence in ocean models. Large amount of oceanic data are becoming now available. For instance, the Copernicus Marine Environment Monitoring Service (CMEMS) provides regular and systematic information about the physical state and dynamics of the ocean for the global ocean and the European regional seas. The data cover the current and future state of variables and the provision of retrospective data records (re-analysis). Other oceanic services include ocean currents supplied by altimeter satellites, like AVISO; the HYbrid Coordinate Ocean Model (HYCOM), a consortium, which is a multi-institutional effort sponsored by the National Ocean Partnership Program (NOPP), as part of the U. S. Global Ocean Data Assimilation Experiment (GODAE), to develop and evaluate a data-assimilative hybrid coordinate ocean models, etc.

Ocean models are built on partial differential equations, which are only an approximation of reality. Indeed, typically ocean models approach ocean motion by the Reynolds-averaged Navier-Stokes equations using the hydrostatic and Boussinesq assumptions. In this context inaccurate or inadequate models lead to structural uncertainty. Even if these equations were perfectly accurate models, they contain parameters, such as viscosity or diffusivity, which are not known precisely and cause parameter uncertainty. Boundary conditions on input variables, lead to parametric variability. For instance input variables such as velocities, experience at the surface atmospheric winds, which impose forcings on the velocities, and these atmospheric forcings are not precisely known.

Eventually these models are not solved exactly, but numerically through discretization methods and numerical approximations, which are subjected to numerical errors, and causes algorithmic uncertainty. For all these reasons, the predicted ocean currents are also uncertain. Uncertainties in the solutions, system outputs, due to the uncertainties in the system inputs is referred to as forward uncertainty quantification [START_REF] Sullivan | Introduction to Uncertainty Quantification[END_REF].

In this work, we are interested in evaluating the reliability of the outputs, ocean currents, regarding the transport they produce. This is a perspective slightly different to the classical one now described, in which ocean models adequacy is judged just against the velocity fields. Transport in the ocean surface is produced by fluids parcels that follow trajectories x(t) that evolve according to the dynamical system:

dx dt = v(x, t), (1)
where the position is described in longitude (λ) and latitude (φ) coordinates, that is, x = (λ, φ), and v represents the velocity field, which has two components determined by the zonal (u) and meridional (v) velocities. In longitude/latitude coordinates, the dynamical system in Eq.( 1) can be rewritten as:

       dλ dt = u(λ, φ, t) R cos φ dφ dt = v(λ, φ, t) R ( 2
)
where R is the Earth's radius. The system [START_REF] Sullivan | Introduction to Uncertainty Quantification[END_REF], which is a general expression encompassing the specific problem dealt with in (2), is a nonlinear non-autonomous dynamical system in 2D. Uncertainties in the velocities v(x, t), or more generally in the vector field, produce uncertainties in the solutions x(t). In general, as illustrated in figure 1 x 0 . The final observed state x * at time t * is referred to as the "target". The evolution law for these observations is unknown, but is approached by a model, which in our setting involves the velocities v(x, t) of the system (1). The evolution according to a model, of a neighbourhood of points, X 0 , close to the initial observation, x 0 , is illustrated with the pink color.

This paper is structured as follows. In section 2, we will discuss and develop an approach to uncertainty quantification recently taken for an ocean study case in [START_REF] García-Sánchez | Very high resolution tools for the monitoring and assessment of environmental hazards in coastal areas[END_REF]. We will show evidence of connections between the introduced definitions and the dynamical objects of system (1). In section 3, we will provide formal results to show how, for specific simple examples of the system (1), the definition of uncertainty quantification given in section 2, is able to highlight these dynamical objects. These results support one of the main findings of this article which is that stable invariant manifolds of hyperbolic trajectories provide a structure to forward uncertainty quantification. We

Transport uncertainty quantification in ocean models

In this section, we propose uncertainty quantification metrics. We work in the context of transport in ocean models, starting the discussion from the results presented in [START_REF] García-Sánchez | Very high resolution tools for the monitoring and assessment of environmental hazards in coastal areas[END_REF], which considers the performance of very high resolution tools for the monitoring and assessment of environmental hazards in coastal areas. In particular, this work, following the scheme presented in figure 1, quantifies uncertainties of several high resolution hydrodynamic models in the area of Gran Canaria, by measuring an error between the observations and predictions for the evolution of a diesel fuel spill event, well documented by port authorities and tracked with very high resolution remote sensing products. The pollution event was produced after the collision of the passenger ferry 'Volcán de Tamasite' with the Nelson Mandela dike in La Luz Port on April 2017. After the crash, supply pipes along the dike were broken and diesel fuel poured into the sea. SAR Sentinel 1 images were available in the area approximately one day and a half after the accident from which it was possible to identify the spill. For that period ocean currents in the area were available from two sources. One was the Copernicus Marine Service model for the Iberian-Biscay-Irish region (CMEMS IBI-PHY, IBI hereafter) and other was a very high resolution model of Puerto de la Luz setup by Puertos del Estado currently implemented and running operationally in different Spanish Port Authorities within the SAMOA port forecast system [START_REF] Sotillo | Coastal ocean forecasting in Spanish ports: the SAMOA operational service[END_REF]. In [START_REF] García-Sánchez | Very high resolution tools for the monitoring and assessment of environmental hazards in coastal areas[END_REF], for each model, uncertainties are quantified by measuring an error with respect to a target, a 'ground truth', which should be recovered by the model. In particular, following the scheme proposed in figure 1, figure 2 illustrates how the uncertainty quantifier metrics is proposed. The ferry impact point, the Nelson Mandela dike, which is marked with the red dot, represents an initial observation. The currents represented in the background are the ones obtained from the model set by Puertos del Estado at day t 0 . The elongated red blob represents the spill as identified from satellite images one day and a half after the accident. This is the target observation, the 'ground truth'. In panel (a) a set of initial conditions, X 0 , are selected in the neighbourhood of the initial observation and marked in a rectangular pink domain. This domain is divided into sub-domains and in each one a small circular blob with radius r = 3.5 • 10 -3 ( • ), is evolved in a time interval from the initial time t 0 . This is represented just for one of the sub-domains. The blob distorts while it approaches the target observed spill. A way to measure the 'proximity' between the blob and the target spill at the final time, t * , is to compute the distance between the centroid of the evolved blob, (c m ), and that of the observed spill, (c * g ). Mathematically this can be expressed as:

e(t * ) = c m (t * ) -c * g . (3)
Here, • denotes the modulus of the vector. The position of the centroid of the evolved spill, c m , depends on time t, while the centroid of the ground value slick c * g does not, because it is an observation at a final time, t * . The centroid of a finite set of N points {x k } k∈N ∈ R n is defined as:

c = 1 N N k=1 x k , (4)
where x k are the (lon, lat) coordinates that define the contour of the slick in an equirectangular projection. The solid black line e in panel (a) is the distance between centroids of the modelled and observed blobs. This is the value taken by Eq. ( 3). The procedure is repeated for additional circles centered in different positions of the sub-domains or cells, in the neighbourhood of the accident place. The evolution of each of these circular blobs is different due to the chaotic nature of transport in this setting. Panel (b) in figure 2 provides a visualization on how this calculation changes in each sub-domain. This panel represents a colormap which is placed in the port neighbourhood, at the same position than the pink domain in panel (a). It considers a 6 × 6 mesh-grid, and the colorcode within each cell represents the value of Eq. ( 3) obtained for blobs with initial positions in the center of each cell and initial radius of r = 3.5 • 10 -3• . We observe a nonuniform distribution. to a 20 × 20 mesh-grid and decreases the radius of initial blobs to 3.5 • 10 -5 ( • ). Figure 3b) makes visible an underlying structure which is directly related to the invariant dynamical structures, as we will shown later. In the limit r → 0, expression (3) is rewritten as:

L U Q (t * ) = x(t * ) -c * g . (5)
where x(t) is a trajectory of a fluid parcel, a solution to the system (1), with initial position x 0 at each cell in the grid. Figure 4a) illustrates the results of this calculation in a very fine grid.

One of the goals of this paper is to establish connections between the uncertainty quantifier (5) and invariant dynamical objects that control transport in vector fields. These are geometrical objects that organize particle trajectories schematically into regions corresponding to qualitatively distinct dynamical behaviors. In the context of fluid dynamics these objects are referred to as Lagrangian Coherent Structures (LCS). An essential ingredient of the LCS are hyperbolic trajectories characterized by high contraction and expansion rates. Directions of contraction and expansion define, respectively, stable and unstable directions, which are, respectively, related to the stable and unstable manifolds. In the context of the ocean model and the event described above, Garcia-Sánchez et al. have shown in [START_REF] García-Sánchez | Very high resolution tools for the monitoring and assessment of environmental hazards in coastal areas[END_REF] that there exists a hyperbolic trajectory located on the coastline very close to the accident point, in a detachment configuration, which is related to the phenomena of flow separation. Under this configuration, the stable manifold of the hyperbolic trajectory is aligned with the coast, and the unstable manifold is transversal to it. Figure 5 illustrates, respectively, in blue and red the stable and unstable manifolds of the separation trajectory for this particular case that we studied. Any blob placed in the neighbourhood of the separation trajectory, eventually evolves to become aligned with the unstable manifold, which is an attracting material curve. The configuration of the unstable manifold in Figure 5, is consistent with the observed spill marked in red in Figure 2. Indeed, the observed spill has evolved to be completely aligned with the unstable manifold. The invariant dynamical structures displayed in Figure 5 have been obtained

by means of the Lagrangian Descriptor [START_REF] Madrid | Distinguished trajectories in time dependent vector fields[END_REF][START_REF] Mendoza | The hidden geometry of ocean flows[END_REF][START_REF] Mancho | Lagrangian descriptors: A method for revealing phase space structures of general time dependent dynamical systems[END_REF], which measures arc-length of trajectories:

M = t 0 +τ t 0 -τ dx(t) dt dt. (6)
In [START_REF] Lopesino | A theoretical framework for lagrangian descriptors[END_REF], for some generalizations of the expression [START_REF] Mancho | Lagrangian descriptors: A method for revealing phase space structures of general time dependent dynamical systems[END_REF], is proved that stable and unstable manifolds are aligned with singular features of this function. The forward integration highlights the stable manifold, while the backwards integration highlights the unstable manifold. In this work the notion of singular feature is related to an undefined directional derivative in a direction transverse to the manifold curve. In Figure 5, red and blue features are placed on the singular features of M .

The metric given in (5) posses similarities with the forward definition of the function M :

M = t * =t 0 +τ t 0 dx(t) dt dt. (7)
The analogy between structures obtained from expressions ( 5) and ( 7), is confirmed from figure 4.

Panel a) displays expression (5) results, while panels b) and c) display Eq. ( 7) results for τ = 1.5 and 2 days respectively. Figure 4 highlights singularities both for the forward uncertainty quantifier and for the forward M function. From this figure is clear that Uncertainties displayed in panel a) reach minimum values along the stable manifold. Indeed, the stable manifold is an optimal pathway towards the unstable manifold, which in turn is an attracting material curves towards which all fluid parcels evolve. In this case, as the observed evolution of the spill is aligned with the unstable manifold, it is expected that minimum values of the uncertainty quantifier correspond to the stable manifold. On the other hand indicators of structural uncertainty that show inadequacy of the model, would correspond to situations in which these minimum values are not reached along the stable manifolds. This would have been the case if, for instance, the observed spill would have been found far from the unstable manifold.

This section is focused on illustrating the analogies found in figure 4 between the pattern in panel a), which is related to uncertainty quantification, and those in panels b) and c), which are related to stable invariant manifolds. For the event described in the previous section computations in panel a) are performed with the measure proposed in Eq. ( 5) to quantify uncertainty in a neighbourhood of the accident location. When the calculation is performed on a very fine grid it is found that the uncertainty L U Q has a structure that is related to that of invariant stable manifolds. In order to find answers to the question of why there are singular structures obtained from the expression

(5) that are aligned with invariant stable manifolds of hyperbolic trajectories, we need to assume an explicit expression for the vector field in Equation ( 1). Indeed in the example discussed in the previous section, velocities are given as data-sets and this makes difficult to proceed with exact calculations. In this way, in this section we perform an analysis in simplified planar vector fields with exact explicit expressions. Figure 6 displays such example. In panel a) an initial observation at time t 0 is marked with a red circle. At a later time, t * , this observation is at the target position marked with a red asterisk. We model this evolution with the vector field at the background, which is represented at time t 0 . In order to quantify the uncertainty in the neighbourhood of the initial observation, we display the evaluation of L U Q around it. The colorbar placed at the bottom of the figure, measures the uncertainty in the domain units. This colorbar indicates that uncertainties in this neighbourhood are low. At least lower than those displayed for the experiment in panel b). In this case the initial observation is at a different position, although the target is kept at the same position than in a). In this case uncertainties are larger, indicating that the vector field in the background is not a good model to express the transition between these two observations. Finally panel c) displays L U Q evaluated in the whole domain. This expresses the uncertainty associated to the planar flow, assuming that observations start in different points of the representation domain and that they evolve towards the assumed observed target position at the red asterisk. The panel confirms that model has a better performance for those initial observations placed at the diagonal.

Results presented next in this section are displayed following this type of representation.

Our analysis in this section follows the spirit of the work by [START_REF] Mancho | Lagrangian descriptors: A method for revealing phase space structures of general time dependent dynamical systems[END_REF][START_REF] Lopesino | A theoretical framework for lagrangian descriptors[END_REF][START_REF] Lopesino | Lagrangian descriptors for two dimensional, area preserving, autonomous and nonautonomous maps[END_REF][START_REF] García-Garrido | Detection of phase space structures of the cat map with lagrangian descriptors[END_REF]. We assume the definition of singular features given there, by considering that these are features of L U Q on which the transversal derivative is not defined. We will prove, for simple selected examples, that stable manifolds are aligned with those singular features of L U Q .

Finally, before beginning our discussion the definition ( 5) is generalized as follows:

L U Q (x 0 , t, p, δ) = n i=1 |x i (t) -x * i | p 1 p , p > 1, δ < 1, x 0 ∈ R n . (8)
Here, x 0 is the initial condition of the trajectory (x 1 (t), x 2 (t), .., x n (t)). In the particular case 2 ) An alternative for this expression that we will use is the following:

L U Q (x 0 , t, p) = n i=1 |x i (t) -x * i | p , p ≤ 1, x 0 ∈ R n . (9)

The autonomous saddle point

The first example that we analyze is the vector field that corresponds to the Hamiltonian linear saddle case for which x ∈ R 2 , satisfying x = (x, y) and the equations of motions are:

dx dt = λx, dy dt = -λy, (10)
where λ > 0. For any initial observation (x 0 , y 0 ), we consider the unique solution of this system passing through the condition (x 0 , y 0 ), which is:

   x(t) = x 0 e λt y(t) = y 0 e -λt , λ > 0 (11) W s (0, 0) = {(x, y) ∈ R 2 : x = 0, y = 0}, (12)
W u (0, 0) = {(x, y) ∈ R 2 : x = 0, y = 0}, (13)
For simplicity we assume, without loss of generality, that t 0 = 0 (this is possible for autonomous systems). We consider the target in the position (x * , y * ) and apply [START_REF] Mezic | A method for visualization of invariant sets of dynamical systems based on the ergodic partition[END_REF] to [START_REF] García-Garrido | Detection of phase space structures of the cat map with lagrangian descriptors[END_REF] to obtain:

L U Q (x 0 , t, p) = |x 0 e λt -x * | p + |y 0 e -λt -y * | p .
Regrouping terms, we get

L U Q (x 0 , t, p) = |x 0 | p ω -p |1 -aω| p + |y 0 | p |ω -b| p , where a = x * x 0 , b = y * y 0 . ( 14
)
In this expression, ω = e -λt , which always satisfies ω > 0. We explore separately the first and second terms. For the factor |1 -aω| p there exists a t L such that if t > t L , then aω << 1 and

(1-aω) > 0. This is always the case for a < 0 and is a plausible assumption for a > 0, if a << ω -1 .

We recall that a = x * /x 0 and that therefore such t L exists if x 0 = 0. In this case positiveness is guaranteed for sufficiently large t, i.e, a Taylor series around ω = 0, attained if t 1 and t > t L , is performed for the binomial:

(1 -aω) p = 1 -apω + 1 2 a 2 (p -1)pω 2 - 1 6 ω 3 a 3 (p -2)(p -1)p + O ω 4 .
Therefore, 1 ω p (1 -aω) p = ω -p + apω (1-p) + 1 2 a 2 (p -1)pω (2-p) + O ω (3-p) .

We recall that p ≤ 1 and ω = e -λt . This yields,

|x 0 | p |e λt -a| p = |x 0 | p e λtp + O |x 0 | p ae -λt(1-p) .
We analyse next the second term in Eq.( 14). The sign of (ω -b) depends crucially on the sign of b given that for t 1 the term ω can be as small as we like. Let us consider t L such that if t > t L and b < 0, then (e -λt -b) > 0. The following is valid for t above this lower value:

|ω -b| p = (ω -b) p = (-b) p + pω(-b) p-1 + 1 2 (p -1)pω 2 (-b) p-2 + 1 6 (p -2)(p -1)pω 3 (-b) p-3 + O ω 4
Therefore, when t 1, b < 0 and e -λt 1,

(ω -b) p = (-b) p + O e -λt , e -λt 1 (15)
Now, we consider b > 0 and t > t L satisfying (ω -b) < 0. Therefore:

|ω -b| p = (-ω + b) p = b p + pωb p-1 + 1 2 (p -1)pω 2 b p-2 + 1 6 (p -2)(p -1)pω 3 b p-3 + O ω 4 .
Therefore, when t 1, b > 0 and e -λt 1,

(-ω + b) p = b p + O (ω) . (16)
Finally, we conclude that Thus, we can approximate L U Q as

|ω -b| p = |b| p + O e -λt , ω 1. ( 17
L U Q (x 0 , t, p) ≈ |x 0 e λt | p + |y * | p = |x 0 |e λtp + |y * | p (18)
where the dominant term is |x 0 | p e λtp . Hence, to leading order, the stable manifold at x = 0 is aligned with a singular feature of L U Q for 'sufficiently large' t. This statement however must be considered in the sense that expression expresses a good approximation for L U Q as far as x 0 = 0, and that for any |x 0 | > 0 is valid for a sufficient large t satisfying, t > t L .

We have shown that Eq. ( 9) is able to highlight the stable manifolds for the autonomous saddle point. Next, we analyse what happens when we consider Eq.( 8), and in which cases it provides information about the stable manifold of the autonomous saddle point. When we apply [START_REF] Mezic | A method for visualization of invariant sets of dynamical systems based on the ergodic partition[END_REF] to [START_REF] Lopesino | Lagrangian descriptors for two dimensional, area preserving, autonomous and nonautonomous maps[END_REF], it yields

L U Q (x 0 , t, p) = |x 0 e λt -x * | p + |y 0 e -λt -y * | p 1/p .
We remark that our following calculations will be restricted to p > 1 and integer. Rewriting the

expression for ω = e -λt L U Q (x 0 , t, p) = x 0 ω p |1 -aω| p + |y 0 | p |ω -b| p 1/p = x 0 ω |1 -aω| p + y 0 x 0 p ω p |ω -b| p 1/p (19)
Recalling that p > 1, ω > 0 and that (1 -aω) > 0 for sufficiently small ω, i.e. sufficiently large t, and the case b < 0 and (ω -b) > 0, a Taylor series around ω = 0, for p integer number, for the second factor:

L U Q (x 0 , t, p) = x 0 ω 1 -aω + y 0 x 0 p (-b) p ω p p + O (ω) p+1 (20)
second factor is as follows:

L U Q (x 0 , t, p) = x 0 ω 1 -aω + y 0 x 0 p (b) p ω p p + O (ω) p+1 (21)
Therefore in general:

L U Q (x 0 , t, p) = x 0 ω 1 -aω + y 0 x 0 p |b| p ω p p + O (ω) p+1 = x 0 ω -a|x 0 | + |y 0 | p |x 0 | 1-p |b| p p ω p-1 + O (ω) p (22)
We notice that since p > 1, the terms ω p-1 , ω p have positive exponents and if ω 1 in principle are much smaller than the first term ω -1 and therefore:

L U Q (x 0 , t, p) ∼ 1 p x 0 ω , ω 1 (23)
Hence, the stable manifold at x = 0 is aligned with a singular feature of this approximation to L U Q for 'sufficiently large' t. This statement though, must be considered with care, because in Eq. ( 22)

the terms in ω p-1 , ω p , are multiplied by |x 0 | 1-p which has a negative exponent, and therefore have a singularity at x 0 = 0. For this reason neglecting these terms versus the first one if we are very close to x 0 = 0 would require checking that the products, such as, |x 0 | 1-p ω p-1 are really small. This implies that Eq. ( 23) is correctly approximating L U Q for ω 1 as far as we are sufficiently away from 0 in x 0 , i.e.

|x 0 | 1-p ω p-1 1 =⇒ ω p-1 |x 0 | p-1 .
In practice this condition is satisfied for any grid (x 0 , y 0 ) used in later figures that exclude x 0 = 0. Also we can state that for sufficiently large t, L U Q is very close in almost all the domain to the function given in Eq. ( 23), which possess a "singular feature" aligned with the stable manifold. Fig. 7 a) and b), illustrates how the stable manifold is aligned with singular features of the Lagrangian uncertainty quantifier defined either by Eq.( 8) or ( 9).

The autonomous rotated saddle point

This second case that we explore is the vector field of the rotated linear saddle for which the equations of motion are:

   ẋ = λy ẏ = λx, λ > 0 (24)
The general solution to this system is:

   x(t) = ae λt + be -λt y(t) = ae λt -be -λt , λ > 0 (25) a) b)
Figure 7: A representation of Eq.( 8) and Eq.( 9) for t * = 10 and target (x * , y * ) = (0.5, 0.5) for the Hamiltonian linear saddle vector field. a) p = 2; b) p = 0.1 . It can be appreciated how the stable manifold is aligned with a singular feature.

where a and b depend on the initial conditions x 0 and y 0 as follows:

a = x 0 + y 0 2 , b = x 0 -y 0 2 .
Here, a = 0 corresponds to the stable manifold of the hyperbolic fixed point placed at the origin and b = 0 corresponds to its unstable manifold.

For any initial observation (x 0 , y 0 ) and final target observation (x * , y * ), we introduce the solution (25) into Eq.( 9) obtaining:

L U Q (x, t, p) = |ae λt + be -λt -x * | p + |ae λt -be -λt -y * | p (26)
We expand next the first term in Eq.( 26). In particular, we consider the case in which a > 0 for which it is always possible to find a t > t L in which (ae λt + be -λt -x * ) > 0:

|ae λt + be -λt -x * | p = (ae λt + be -λt -x * ) p = a ω + bω -x * p = ω -p a + bω 2 -x * ω p ,
Here we have used ω = e -λt . The Taylor expansion of the binomial when ω 1, i.e., when t 1, is

ω -p a + bω 2 -x * ω p = ω -p a p -pωx * a p-1 + ω 2 bpa p-1 + 1 2 (p -1)p(x * ) 2 a p-2 - 1 6 ω 3 (p -1)px * a p-3 6ab + p(x * ) 2 -2(x * ) 2 + O ω 4 .
Therefore,

ω -p a + bω 2 -x * ω p = aω -p -pωx * a p-1 + ω (2-p) bpa p-1 + 1 2 (p -1)p(x * ) 2 a p-2 - 1 6 ω (3-p) (p -1)px * a p-3 6ab + p(x * ) 2 -2(x * ) 2 + O ω (4-p)
Since p < 1, ω (n-p) 1 for n > 1 when t 1. Hence,

ω -pt a + bω 2t -x * ω t p = a p ω -pt + O ω (1-p)t .
By removing the change of variable, we obtain

ae λt + be -λt -x * p = a p e λpt + O e -(1-p)λt ≈ a p e λpt , when, t 1. ( 27
)
We consider now the case a < 0 that implies ae λt + be -λt -x * < 0 for some sufficiently large t. The Taylor series is the same, except for a minus sign. It yields,

ae λt + be -λt -x * p = -a p e λpt -O e -(1-p)λt ≈ -a p e λpt , when, t 1. (28)
Finally,

|ae λt + be -λt -x * | p = |a| p e λpt + O e -(1-p)λt ≈ |a| p e λpt , when, t 1. ( 29
)
We analyse next the second term in Eq.( 26). As before, we start considering the case a > 0 and (ae λt -be -λt + y * ) > 0. After making the change of variable ω = e -λt we obtain:

ω -p a -bω 2 -y * ω t p = ω -p a p -pω t y * a p-1 - 1 2 ω 2 pa p-2 2ab -p(y * ) 2 + (y * ) 2 + 1 6 (p -1)pω 3 y * a p-3 6ab -p(y * ) 2 + 2(y * ) 2 + O ω 4 . Since p < 1, ω (n-p) 1 for n > 1 when t 1. Hence, ω -p a -bω 2 + y * ω p = a p ω -p + O ω (1-p) .
By removing the change of variable, we obtain:

ae λt -be -λt -y * p = a p e λpt + O e -(1-p)λt ≈ a p e λpt , when, t 1. ( 30
)
We compute the same term but in the case a < 0 ae λt -be -λt -y * < 0. The Taylor series is the same, except for a minus sign. It yields, ae λt -be -λt -y * p = -a p e λpt -O e -(1-p)λt ≈ -a p e λpt , when, t

Finally,

|ae λt -be -λt -y * | p = |a| p e λpt + O e -(1-p)λt ≈ |a| p e λpt , when, t 1. ( 32
) L U Q = |ae λt + be -λt -x * | p + |ae λt -be -λt + y * | p ≈ |a| p e λpt , when, t 1. (33)
The stable manifold at a = 0 is aligned with a singular feature also at a = 0.

As for Eq.( 8) considering p > 1, p = 1/δ and ω = e -λt leads:

L U Q = |ae λt + be -λt -x * | p + |ae λt -be -λt + y * | p 1/p = ω -1 |a + bω 2 -x * ω| p + |a -bω 2 + y * ω| p 1/p = 2 1 p |a|ω -1 -2 -1+ 1 p (2b + x * -y * ) + 2 -3+ 1 p |a| -1 • (4ab + (-1 + p)(-2b + x * + y * ) 2 )ω + O(ω) 2 (34)
As before the leading term in ω leads to:

L U Q ∼ 2 1 p |a|ω -1 (35)
and similar considerations apply in neglected terms to the ones made for Eq. ( 22), but now regarding to singularities at a = 0. Fig. 8 a) and b), illustrate how the stable manifold is aligned with singular features of the Lagrangian uncertainty quantifier defined either by Eq.( 8) or ( 9). a) b)

Figure 8: A representation of Eq.( 8) and Eq.( 9) for t * = 10 and target (x * , y * ) = (0.5, 0.5) for the rotated linear saddle vector field for a) p = 2; b) p = 0.1. It can be appreciated how the stable manifold is aligned with a singular feature.

Discrete maps

These findings in the previous two subsections can be easily extended to discrete time dynamical systems, which are also useful in applications. Discrete time dynamical systems are defined as maps.

The autonomous saddle point

Consider the following linear, area-preserving autonomous map:

   x n+1 = λx n y n+1 = 1 λ y n , λ > 1. (36)
For an initial condition (x 0 , y 0 ), the unique solution of this system is:

   x n = x 0 λ n y n = y 0 λ -n , λ > 1. (37)
As for the continuous time case, the origin (0, 0) is a hyperbolic fixed point with stable and unstable manifolds:

W s (0, 0) = {(x, y) ∈ R 2 : x = 0, y = 0}, (38)
W u (0, 0) = {(x, y) ∈ R 2 : x = 0, y = 0}, (39)
We apply (37) to ( 9) to obtain:

L U Q (x, n, p) = |x 0 λ n -x * | p + |y 0 λ -n -y * | p . (40)
Regrouping terms, we get

L U Q (x, n, p) = |x 0 | p |λ n -a| p + |y 0 | p |λ -n -b| p , where a = x * x 0 , b = y * y 0 . (41)
Considering that the transformation e λt → λ n can be directly applied to all the calculations performed in the continuous time case, we recover from Eq.( 14):

L U Q ≈ |x 0 | p |λ np | + O |λ| -n(1-p) , when, n 1. (42)
When p < 1, the dominant term is |x 0 | p λ np . Hence, the stable manifold at x = 0 is aligned with singular features of L U Q for 'sufficiently large' iteration n.

The same is applicable to L U Q [START_REF] Lopesino | Lagrangian descriptors for two dimensional, area preserving, autonomous and nonautonomous maps[END_REF]. It yields,

L U Q ≈ |x 0 ||λ n | + O (1) , when, n 1 and pδ < 1. ( 43
)
We note that the stable manifold at x = 0 is aligned with singular features of L U Q for a 'sufficiently large' iteration n. The same issues as before regarding singularities on the manifold position apply. Figure 9: In Figure a) there is a representation for the autonomous saddle when we apply Eq.( 8)

for p = 2. In b), we illustrate the same representation but for Eq.( 9) when p = 0.1. It can be appreciated how the stable manifold is aligned with the singular feature.

The autonomous rotated saddle point

We consider the following discrete dynamical system:

F (x, y) = A x y
, where A = 1 2λ

λ 2 + 1 λ 2 -1 λ 2 -1 λ 2 + 1 , λ > 1. ( 44
)
It is easy to see that the stable and the unstable manifolds are given by the vectors (1, -1) and

(1, 1) respectively. The solution of this system yields to,

   x n = aλ n + bλ -n y n = aλ n -bλ -n , λ > 1 ( 45
)
where

a = x 0 + y 0 2 , b = x 0 -y 0 2 .
Again considering the transformation e λt → λ n and the use of previous results for the continuous time case, we recover from Eq.(33):

L U Q (x, n, p) = |aλ n + bλ -n -x * | p + |aλ n -bλ -n -y * | p .
Therefore, the dominant term is

L U Q (x, n, p) ∼ |a| p |λ| np .
Since a = x 0 +y 0 2 , there is a singular feature at x = -y, i.e., in the subspace generated by (1, -1). Hence, the stable manifold is aligned with a singular feature of L U Q .

The same is applicable to L U Q [START_REF] Lopesino | Lagrangian descriptors for two dimensional, area preserving, autonomous and nonautonomous maps[END_REF]. It yields,

L U Q (x, n, p) ∼ |a||λ| n ,
Hence, the stable manifold is aligned with a singular feature of L U Q . Fig. 10 a) and b), illustrates how the stable manifold is aligned with singular features of the Lagrangian uncertainty quantifier defined either by Eq.( 8) or [START_REF] García-Garrido | Detection of phase space structures of the cat map with lagrangian descriptors[END_REF].

The Duffing equation

Results in the previous sections are generalized to the autonomous nonlinear case by means of the Moser's theorem [START_REF] Moser | The analytic invariants of an area-preserving mapping near a hyperbolic fixed point[END_REF]. This theorem applies to analytic two-dimensional symplectic maps having a hyperbolic fixed point or, similarly, to two-dimensional time-periodic Hamiltonian vector fields having a hyperbolic periodic orbit (which can be reduced to the former case considering a Poincaré map). The case of a Hamiltonian nonlinear autonomous system is a one-parameter family of symplectic maps, and therefore Moser's theorem applies. Following proofs sketched by [START_REF] Lopesino | Lagrangian descriptors for two dimensional, area preserving, autonomous and nonautonomous maps[END_REF][START_REF] Lopesino | A theoretical framework for lagrangian descriptors[END_REF] results may be extended to the case of non-autonomous nonlinear dynamical systems by utilizing results like the Hartman-Groβman theorem.

This section discusses further results on the Lagrangian Uncertainty Quantifier by considering the evaluation of (8) over a vector field obtained from the nonlinear periodically forced Duffing equation:

ẋ = y ẏ = x -x 3 + sin t (46)
Prior to discuss outputs of (8) into Eq. (46), we discuss the structure of invariant manifolds of hyperbolic trajectories in Eq. ( 46) for the case = 0.1 and the persistence versus this time dependent perturbation of tori present in the unforced version of Eq. ( 46), i. e, = 0. In the perturbed case, the hyperbolic fixed point placed at the origin, becomes a hyperbolic periodic trajectory [START_REF] Madrid | Distinguished trajectories in time dependent vector fields[END_REF],

and their stable and unstable manifold can be highlighted by the Equation 6 appeared in Section 2 [START_REF] Mancho | Lagrangian descriptors: A method for revealing phase space structures of general time dependent dynamical systems[END_REF]. Additionally, as discussed in [START_REF] Lopesino | A theoretical framework for lagrangian descriptors[END_REF] a scale factor 1/(2τ ) applied to (6) converts the expression to an average, which in compact Hamiltonian systems like (46) converges for τ → ∞ and when this convergence is observed, level curves correspond to invariant structures of the dynamical system. Convergence of means are computationally verifiable on tori, however on hyperbolic sets as discussed in [START_REF] García-Garrido | Detection of phase space structures of the cat map with lagrangian descriptors[END_REF], rounding computational errors practically prevent convergence. Figure 11 11a) may be split into two figures, by separating the backwards and forwards integration of Eq. ( 6), which displays, respectively, the unstable and stable manifolds. These outputs are depicted, respectively, in Figure 12a) and b).

Figure 13 shows the evaluation of L U Q as in Eq.( 8) with p = 2 for t 0 = 0, t = t 0 + τ = 10 and different targets. In panel a) the chosen target is x * = (0.1, 0.1). This target is within the chaotic region displayed in Fig. 11b), close to the unstable manifold of the Duffing equation. It is observed that minimum values are reached on the stable manifold and that the structure of L U Q is correlated to it. This setting is similar to what was observed for the ocean case described in Section 2, in which the observed oil evolution is aligned with the unstable manifold and minimum values of L U Q are found on the stable manifold that are optimal pathways towards the unstable manifolds. In panel b) we show the results for the target x * = (1, 0) which is within the right tori like structure highlighted in Fig. 11b). Accordingly, uncertainty values are very low for initial observations (x 0 , y 0 ) in the corresponding tori region, but very high for the tori like region at the left side. Indeed initial observations in this region will never go near a final observation in the right tori like structure, and therefore this model is structurally uncertain for those observations, i.e. the model is inadequate to represent those. Panel c) shows the computation of the uncertainty for a target x * = (0, 1) outside the geometry of the unstable invariant manifold displayed in Fig. 12a).

It is remarkable the persistence in all these examples of an structure on the uncertainty field with singular features linked to the stable manifold independently of the target value x * .

Figure 14 shows the evaluation of Eq.( 9) with p = 0.1, and different targets or integration periods. Panels a) and b) have the same integration periods, but different targets. In them the black arrow points out different spurious features that seem singular but do not correspond to any invariant manifold. Figure 14 c) shows the same than panel a) for an integration period of 50. The black arrow marks a region, which from the analysis of Fig. 11, is known to be covered by tori, while the structure attained from Eq.( 9) does not highlight this. Indeed, the ergodic partition theory discussed in [START_REF] Mezic | A method for visualization of invariant sets of dynamical systems based on the ergodic partition[END_REF][START_REF] Susuki | Ergodic partition of phase space in continuous dynamical systems[END_REF] and implemented in [START_REF] Lopesino | A theoretical framework for lagrangian descriptors[END_REF] for Lagrangian Descriptors such as that in Eq. ( 6),

shows that this ability to highlight tori, requires averaging along trajectories, while expressions Eq. ( 8) or ( 9) are not averages, and therefore do not qualify for the application of ergodic principles.

These expressions are not suitable for identifying this kind of invariant objects.

Discussion

There is much interest in uncertainty quantification involving trajectories in ocean data sets. Recent efforts in this direction are [START_REF] Kirwan | Predictability, uncertainty, and hyperbolicity in the ocean[END_REF][START_REF] Huntley | Lagrangian predictability assessed in the east china sea[END_REF][START_REF] Feng | Innovative approaches for geometric uncertainty quantification in an operational oil spill modeling system[END_REF][START_REF] Vieira | Uncertainty quantification of trajectory clustering applied to ocean ensemble forecasts[END_REF]. Vieira et al. [START_REF] Vieira | Uncertainty quantification of trajectory clustering applied to ocean ensemble forecasts[END_REF] have developed a clustering method to partition the space of trajectory data sets into distinct flow regions. This method contains free parameters and they use uncertainty quantification to assess the parameter dependence on the partitions they obtain. The work [START_REF] Feng | Innovative approaches for geometric uncertainty quantification in an operational oil spill modeling system[END_REF] discusses approaches for uncertainty quantification from a geometrical perspective, and uncertainty quantifiers based on distances are proposed, but no connections are proposed with invariant dynamical structures. Similarly in [START_REF] Huntley | Lagrangian predictability assessed in the east china sea[END_REF] uncertainty quantifiers based on a descriptive statistics of distances between modelled trajectories and observations are used, but no relations are presented between these and the invariant dynamical structures. Results

appeared in [START_REF] Kaszás | Universal upper estimate for prediction errors under moderate model uncertainty[END_REF] also discuss along these lines, but in the context of general models, not related to ocean data. In this work authors have established links between uncertainty quantification and invariant manifolds under explicit mathematical assumptions for the "true" model. The results in our work do not require such assumptions. Equations similar to (8) or ( 9) have been used in the literature to highlight Lagrangian structures in oceanic flows. For instance [START_REF] Prants | Numerical simulation of propagation of radioactive pollution in the ocean from the fukushima dai-ichi nuclear power plant[END_REF][START_REF] Prants | Lagrangian study of surface transport in the kuroshio extension area based on simulation of propagation of fukushima-derived radionuclides[END_REF] have done so. Prants in [START_REF] Prants | Numerical simulation of propagation of radioactive pollution in the ocean from the fukushima dai-ichi nuclear power plant[END_REF] proposed to use the arc-length D to study the displacement of particles in the coasts of Japan.

D = (x f -x 0 ) 2 + (y f -y 0 ) 2 . ( 47
)
Here, D represents the relative displacement of a particle from its initial position (x 0 , y 0 ) to certain final one (x f , y f ). This expression is the analogue to Eq.( 8) with p = 2 and target adjusted to each initial condition. However, in this work no connections are established between Eq. (47) and uncertainty quantification.

Conclusions

This article explores the implications of a definition for Uncertainty Quantification recently proposed in oceanic contexts [START_REF] García-Sánchez | Very high resolution tools for the monitoring and assessment of environmental hazards in coastal areas[END_REF]. It is found that for this definition, which is associated to forward Uncertainty Quantification, stable invariant manifolds of hyperbolic trajectories of the underlying flow, provide a structure for it. That is, we found that the proposed Uncertainty Quantifier is a function that contains a very rich structure, which is related to these well known structures from dynamical systems theory. For selected examples this statement has been proven. Also examples are discussed in which singular structures of the Uncertainty Quantifier highlight spurious structures, not aligned with invariant features of the dynamical system. Other invariant structures like tori, are not recovered by the Uncertainty Quantifier. Also, connections between UQ and invariant dynamical structures have been used to provide a framework for discussion on structural uncertainty, which is related to inadequate models. This vision enriches traditional descriptions of UQ on which structure is discussed just in terms of means and/or statistical moments of distributions.

The findings described in this article are particularly interesting because they have important environmental applications. Nowadays, multiple ocean data sources are available and in this context our results allow a quantitative comparison of the transport properties associated to them. In this way, discriminating the level of performance of different data sources will help to gain precision in the description of dispersion of contaminants, determination of waste and plastic sources, etc.

Figure 1 :

 1 Figure 1: A graphical representation of two sequential observations and their evolution according to a mathematical model. The initial observation at time t 0 is expressed by the red initial condition

  will see in detail what is meant by this statement. Section 4 presents a discussion with further examples that illustrate the findings of this work and links uncertainty quantification to Lagrangian Descriptors and other Lagrangian indicators found in the literature. Finally, in section 5, we will provide the conclusions.

Figure 2 :

 2 Figure 2: A graphical representation of the uncertainty quantification in a neighbourhood of the accident location (red dot) with respect to a target (the red blob). a) A schematic representation of Eq. (3) computation for an initial pink blob with radius r and its evolution; b) results of the computation of Eq. (3) for different initial blobs of radius r = 3.5 • 10 -3 ( • ) placed in the center of each cell in the meshgrid. The colormap express in degrees values for the final distances to the target.

Figure 3 Figure 3 :Figure 4 :Figure 5 :

 3345 Figure 3 expands results of Figure 2b) by increasing the number of cells in the neighborhood domain and decreasing the radius of each initial blob. In particular, Figure 3 a) considers a mesh

Figure 6 :

 6 Figure 6: a) and b) two successive observations at time t 0 (red circle) and time t * (red asterisk). The transition between them is modeled by the vector field at the background and L U Q is represented in the neighbourhood of the initial observation. Uncertainties are lower for the case a) as confirmed by the colorbar; c) L U Q evaluated in the whole domain. This expresses the uncertainty associated to the planar flow, assuming observations that start in different points of the representation domain and that evolve towards the red asterisk (target position).

  ) Therefore, since b = y * /y 0 |y 0 | p |e -λt -b| p = |y * | p + O |y 0 | p e -λt , e -λt 1.

Fig. 9 a

 9 Fig.9 a) and b), illustrate how the stable manifold is aligned with singular features of the Lagrangian uncertainty quantifier defined either by Eq.(8) or[START_REF] García-Garrido | Detection of phase space structures of the cat map with lagrangian descriptors[END_REF].

Figure 10 :

 10 Figure 10: In Figure a) there is a representation for the rotated saddle when we apply Eq.(8) for p = 1/δ = 2. In b), we illustrate the same representation but for Eq.(8) when p = 1/δ = 0.1. It can be appreciate how the singular feature is aligned with stable manifold.

Figure 11 :

 11 Figure 11: Evaluation of (6) for the Duffing equation (46) at t 0 = 0. a) τ = 10; b) τ = 50. At this τ the average of M in the smooth regions has converged and the level curve represent and invariant tori.

Figure 12

 12 Figure 12: a) Evaluation of the backwards integration of (6) on the Duffing equation (46) at t 0 = 0 and τ = 10. This represents the unstable manifold; b) Evaluation of the forwards integration of (6) on the Duffing equation (46) at t 0 = 0 and τ = 10. This represents the stable manifold.

Figure 13 :

 13 Figure 13: Evaluation of Eq.(8) on the Duffing equation (46) at t 0 = 0, t = t 0 + τ = 10 with different targets. a) Target x * = (0.1, 0.1); b) target x * = (1, 0); c) target x * = (0, 1).

Figure 14 :

 14 Figure 14: Evaluation of Eq.(9) on the Duffing equation (46) at t 0 = 0, for p = 0.1 and different targets. Black arrows highlight features that for this system are known to be spurious. a) t 0 = 0, t = t 0 + τ = 10 and target x * = (0.1, 0.1); b) t 0 = 0, t = t 0 + τ = 10 and target x * = (-0.75, 0.25); c) t 0 = 0, t = t 0 + τ = 50 and target x * = (0.1, 0.1)

Acknowledgements

GGS and AMM acknowledge support from IMPRESSIVE, a project funded by the European Union's Horizon 2020 research and innovation programme under grant agreement No 821922. SW acknowledges the support of ONR Grant No. N00014-01-1-0769.