Jérôme Jacob
email: j.jacob@brgm.fr

Ulysse Moreau

Frédéric Duré

Jérémy Engevin

Justine Huron

Arnault Lassin

Valérie Guérin

AMD passive treatment using DAS at an old tin mine in France

Keywords: AMD, DAS, clogging, passive treatment

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

Introduction

The treatment of acid mine drainage (AMD) is a complex and challenging issue, especially when the water is very acidic or the flow rate is too high for classical passive treatment methods [START_REF] Trumm | Selection of active and passive treatment systems for AMD-flow charts for New Zealand conditions[END_REF]. Anoxic limestone drain is a common treatment method used for moderately acidic mine drainages, but it may not be effective in treating partially oxidized or aluminiumcontaining AMD [START_REF] Watzlaf | Long-term performance of anoxic limestone drains[END_REF]. With technological advancements in the field of mine drainage treatment, efforts have been made to push the limits of the applicability of passive treatment methods. Moreover, the traditional binary division between passive and active treatments is increasingly being replaced by a more nuanced continuum of treatments, ranging from fully passive to semi-passive [START_REF] Kleinmann | The Evolving Nature of Semi-passive Mine Water Treatment[END_REF]. DAS technology is clearly part of the progress made in this area in recent years [START_REF] Ayora | Acid mine drainage in the Iberian Pyrite Belt: 2. Lessons learned from recent passive remediation experiences[END_REF].

In this study, we focus on the treatment of AMD from a closed Sn open-pit mine using a dispersed alkaline substrate reactor (DAS). The lifespan of a DAS is limited by two phenomena: limestone depletion and clogging. Increasing the proportion of limestone in the packing material can delay its depletion, but accelerates pressure drop by reducing the porosity of the mixture, increasing the specific surface area, and increasing the amount of metals that can precipitate in the packing material. The aim of this study is to determine the optimal design of a full scale treatment plant for Abbaretz AMD with the longest possible service life and to predict its service life before failure, regardless of the cause.

Methods

Site

The tin deposit in the Abbaretz region near Nantes (west of France) has been mined industrially from 1952 to 1957 after producing 3750 tonnes of cassiterite (SnO2). It is located onto a carboniferous quartz vein of the Armorican massif. The mineralisation is in the form of cassiterite associated with other minerals: mispickel (FeAsS), beryl (Be3Al2Si6O18), molybdenite (MoS2) and chalcopyrite (CuFeS2). The mineral is associated with inter-layered quartz in micaschists that form the bedrock of the area. The tailings produce an AMD (pH 2.9) containing high concentrations of Fe (300 mg/L at 85% Fe 2+ ), Al (24 mg/L), Mn (4.3 mg/L), Zn (4.5 mg/L), Ni (2.1 mg/L), Co (1.8 mg/L), As (190 µg/L) and sulphate (1270 mg/L) at a mean flow rate of 5 m 3 /h.

Water treatment system

The components of the pilot scale treatment system are: a DAS reactor, followed by an aeration stage, then a settling stage (fig. 1). The DAS is a downflow PMMA square column (0.3 m x 0.3m x 1.5 m) filled with a 64 L of mixture (porosity 55%) of 63%v/v wood chips (grain size 0-30 mm) and 37%v/v limestone sand (grain size 0-4 mm). The reactor was equipped with 2 sampling ports (at 8 cm and 38 cm) and 2 transparent graduated tubes for measurement of pressure drop at 8 cm and 38 cm (total thickness 73 cm). The outlet pipe was adjusted to maintain 15 cm of supernatant and 30 cm of freeboard to handle the build-up of pressure drop. The aeration step was composed of one, then 2 pseudo-cylindrical PVC reactors (working volume respectively 5.5 L and 3.2 L) supplied with 3 L/min of air each. The settling step consisted of 3 rectangular PP containers in series of 77 L each. A peristaltic pump (Watson-Marlow QDOS120) fed the treatment system at 1.3 L/h. The only modification of operating condition was the addition of a 2 nd aeration reactor after day 60, increasing residence time in the oxidation step from 2.4 h to 6.5 h. Residence time was 36h in the DAS and 7.4 days in the settling stage. The pilot scale treatment system operated for 122 days from October 2022 to March 2023 and was shut down due to excessive clogging.

Figure 1. AMD pilot plant on site

Sampling and analytical techniques

Sampling and measurements were made every weeks at the inlet and at the outlet of the DAS reactor, in the aeration reactors and at the outlet of the 3 rd settling tank. In total, 19 measurement and sampling campaigns were carried out. Redox, pH and temperature, conductivity and dissolved oxygen were measured using a WTW Multi 3620 with Sen Tix 950, Sen Tix ORP-T900, TetraCon 925 and FDO925 probes respectively. Samples were immediately filtered using 0.45 μm filters and acidified at pH 2 with nitric acid (HNO3) 67%. Ferrous iron was analysed on site with Merck ® Spectroquant kits. Dissolved Iron, Ca, Al, Mn, Ni, Co, Zn and Mn were quantified by Agilent 4210 MP-AES (microwave plasma atomic emission spectroscopy). Flow rates were checked using a graduated container and a stopwatch.

Results and discussion

The physico-chemical characteristics of the AMD before and during treatment in the pilot plant, as well as the removal percentage of metals and net acidity, are summarized in Table 1. These are averages over the entire study period. An important increase in pH is observed between the inlet and outlet of the DAS, followed by a more moderate increase in the aeration step likely due to CO2 degassing, and finally a decrease in pH in the settling step, probably due to residual Fe hydroxide precipitation.

Regarding the metals, in the treatment system, there is: a total removal of Al ; only a partial removal of iron (82%), caused by insufficient oxidation of Fe II to Fe III in the aeration step; and various removal for other metals, Zn (71%), Co (23%), and Ni (24%). After the DAS, the soluble Fe is almost exclusively composed of Fe II, as Fe III is insoluble at these pH levels. There is no removal of Mn, probably due to the presence of residual Fe II which is known to prevent Mn precipitation. Metals removal is mainly concentrated in the DAS, except for iron, for which 42% of the removal occurs in the aeration and settling steps. The temporal evolution of Fe and Al in the treatment system is presented in Figure 2. There were relatively minor variations in Fe and Al concentrations during the study period. However, it is worth noting that the addition of a 2 nd aeration reactor on day 60 led to an average decrease in Fe concentration at the pilot outlet from 74 mg/L to 49 mg/L. Additionally, during the last two measurement campaigns, Fe concentrations throughout the system were significantly lower, likely due to a reduction in the treatment load resulting from dilution of the AMD by heavy rainfall. Figure 4 shows the temporal evolution of Ca dissolution along DAS depth as percent of total Ca dissolution. At the beginning of the experiment, the majority of the calcite was dissolved within the first 8 cm of the DAS. However, this zone became less effective over time and starting from day 102, it no longer significantly contributed to the dissolution of calcite. The middle zone (8-38 cm) became predominant starting from day 74 and remained so until the end of the pilot. The deepest zone of the pilot started to produce a significant amount of alkalinity from day 102 onwards, but at the end of the pilot, it remained a minor contributor. The reasons for this shift in activity could be limestone passivation, depletion of the fine fraction of the calcite, and/or partial clogging that reduces water residence time. Limestone depletion can be ruled out since at the end of the pilot study, only 30% of the limestone initially present in the 1 st layer (0 -8 cm) was consumed. Although the decrease in the amount of alkalinity produced is perfectly measured, it is difficult to estimate the lifetime of the DAS in this configuration as this amount has been insufficient (lower than AMD net acidity) since the beginning of the study, despite a high limestone fraction.

Hydraulic conductivity in the upper part of the pilot started to decrease 20 days after the beginning, and continued to decrease steadily during the study period (fig. 5). By the end of the experiment, head loss had reached 300 mm H2O. Surprisingly, hydraulic conductivity remained constant in the middle and lower parts of the DAS.

Figure 5. evolution of hydraulic conductivity

As Fe is the contaminant present in the highest concentration, its precipitation is a major driver of head loss. Initially iron precipitation is concentrated almost exclusively in the first 8 cm of the DAS, then precipitation also takes place in the middle part of the reactor while remaining largely located in the upper part (Fig. 6). At the end of the experiment, there is still no Fe precipitation in the lower part. This explains why the pressure drop is concentrated in the 1 st layer of the DAS and the absence of pressure drop in the 8-38 cm zone indicating that the iron precipitation might be more spread out there.

Conclusions

This DAS configuration has achieved promising acidity and metals removal, but it can still be improved, both in terms of performance and lifespan. The first parameter to modify is obviously the proportion of limestone sand in the filling material. It must be reduced because using such a high limestone fraction not only causes rapid clogging, but also does not help to delay limestone depletion. Indeed, it was found that, after 100 days, there is no more limestone dissolution in the first layer of the DAS (0 to 8 cm) while only 30% of the initially present limestone has been consumed, meaning 70% of the limestone in it unused. A new DAS with much less limestone has recently been started: 18%v/v instead of 37%v/v and packing were added in the aeration reactors to improve Fe II oxidation. To improve the system performance, several other measures could be taken, including increasing the residence time, oxidizing a small portion of Fe before the DAS and adding a 2 nd DAS after the sedimentation step. A geochemical modelling is currently underway to validate the operating hypotheses and limitations of the pilot.

Figure 2 .

 2 Figure 2. dissolved Fe and Al in the pilot plantFigure3shows the evolution of net acidity in the pilot as well as the amount of CaCO3 that has dissolved in the DAS, calculated based on Ca concentration difference between the outlet and the inlet. This calculation assumes no gypsum precipitation, as sulphate concentrations do not decrease between the inlet and outlet of the DAS (data not shown). It is observed that the alkalinity produced in the DAS by calcite dissolution is almost always lower than the net acidity of the AMD, and there is a regular decrease in the amount of dissolved calcite. The net acidity removed averages 473 mg/L (65%) in the DAS and 638 mg/L (88%) in the entire system. The amount of dissolved limestone in the DAS shown in Figure3corresponds to an increase in Ca concentration from 20 to 250 mg/L. In the literature, limestone dissolution can be greater, for example, Ca concentration in DAS increases from 200 to 700 mg/L in the study ofRotting et al. (2008a) and in the paper ofCarabello et al. (2011b), while also experiencing gypsum precipitation. These studies also show much greater reduction in net acidity (from 900 to 1600 mg/L), indicating greater metal removal and hence more clogging. The generation of proton during metal precipitation in the DAS is a factor that promotes the dissolution of limestone, the kinetics of which depend on pH.

Figure 3 .

 3 Figure 3. net acidity and dissolved CaCO3 in the DAS reactor

Figure 4 .

 4 Figure 4. evolution of Ca dissolution in the DAS

Figure 6 .

 6 Figure 6. evolution Fe precipitation in the DAS

Table 1 .

 1 AMD characteristics and treatment performances in the pilot plant

	Parameter		Measured values		Removal rate
		Inlet	DAS	Aeration	Settling	DAS	DAS+Settling
	pH	3.0	6.5	6.8	5.9
	Conductivity (µS/cm) 1766	1739	1612	1621
	Dissolved O2 (mg/L)	8.9	0.8	11.2	7.1
	Temperature (°C)	8.8	9.1	8.5	8.1
	Eh SHE (mV)	627	166	190	347
	Net acidity	726	256		90	65%	88%
	(mg/L eq. CaCO3)
	Fe (mg/L)	279	146		49	48%	82%
	Fe II (mg/L)	235	153	87	51	35%	78%
	Al (mg/L)	24	<0.25		<0.25	>99%	>99%
	Zn (mg/L)	3.19	1.31		0.91	59%	71%
	Co (mg/L)	1.25	1.04		0.97	17%	23%
	Ni (mg/L)	1.42	1.12		1.08	21%	24%
	Mn (mg/L)	3.68	3.93		4.07	-7%	-11%
	Ca (mg/L)	3.19	1.31		0.91

Acknowledgements

Financial support for this study was provided by DPSM within the framework of its mission as delegated post-mining operator on behalf of the French state.


Laura Centemeri

Davide Olori

Critical engagement and prefiguration against defuturing processes

Keywords: Disaster, prefiguration, critical sociology, reconstructive sociology, earthquake, Italy

published or not. The documents may come

Introduction

Sociological research on disasters is strongly linked to public sociology because it is intertwined with public processes of accountability and demands for victim reparation.

Historically, disaster research has emerged as 'policy sociology' [START_REF] Burawoy | Public sociologies: Contradictions, dilemmas, and possibilities[END_REF]) and has progressively focused on the priorities dictated by government agencies in relative isolation from the theoretical debates of 'professional sociology' [START_REF] Tierney | From the margins to the mainstream? Disaster research at the crossroads[END_REF]. With the influence of Laura Centemeri, Davide Olori, 2013, "Public sociology in disaster situations: Critical engagement and prefiguration against defuturing processes" in Lavinia Bifulco and Vando Borghi (eds.) Research Handbook on Public Sociology,Edward Elgar Publishing, Please refer to the published version for quotation, minor changes have occurred between this version and the published one 2 disciplines such as critical geography and anthropology, however, disaster sociology has also seen the development of critical disaster sociology in parallel with the growing relevance of crisis situations related to the intensification of economic globalization, the simultaneous emergence of a 'global risk society' [START_REF] Beck | Living in the world risk society[END_REF] and the deepening of ecological problems, especially climate change [START_REF] Tierney | From the margins to the mainstream? Disaster research at the crossroads[END_REF]. Within this context, the public intellectual and activist Naomi Klein produced a highly debated and influential analysis of the disaster-prone nature of contemporary societies as related to the neoliberal 'shock doctrine' that guides 'disaster capitalism' [START_REF] Klein | The Shock doctrine: The rise of disaster capitalism[END_REF].

Examples abound of sociologists engaged in supporting processes of inquiry and denunciation alongside disaster-affected communities, such as those related to mobilizations for 'environmental justice' [START_REF] Allen | Through a maze of studies: Health questions and 'undone science' in a French industrial region[END_REF][START_REF] Jobin | Our 'good neighbor' Formosa Plastics: Petrochemical damage(s) and the meanings of money[END_REF]. The environmental justice framework was first developed by social movement activists and was then used and reflected on as an analytical tool in academia before returning to the social movements domain enriched with new understandings and perspectives [START_REF] Martinez-Alier | Between activism and science: grassroots concepts for sustainability coined by environmental justice organizations[END_REF]. This example confirms that the path that critical sociological knowledge travels to arrive at public relevance, both as a contribution to the transformation of common sense and as an engagement with specific publics, is non-linear. Activists and other actors engaged on the ground are often at the source (rather than the final recipient) of theoretical innovations [START_REF] Lozano | Reframing the public sociology debate: Towards collaborative and decolonial praxis[END_REF]. The public relevance of sociological knowledge thus results from diverse, interconnected forms of circulation across the boundaries that separate academia and other social spheres of knowledge production.

Based on her personal experience, the sociologist Diane Vaughan [START_REF] Vaughan | NASA Revisited: Theory, analogy, and public sociology[END_REF] reflected on the 'relational complexity' and porous, overlapping nature of the boundaries between the categories that Burawoy (2005a) identified as 'professional sociology', 'critical sociology', 'policy sociology' and traditional and organic 'public sociology'. Vaughan also emphasized the difficulties of working in the interstices between professional and public sociology, especially the 'emotional work' that this liminal position imposes on researchers and the risks for junior researchers' future career prospects. It is nevertheless precisely this interstitial condition that, one, ensures the permeability of the boundaries that separate the different types of sociology and, two, explains how theoretical frameworks can become meaningful Laura Centemeri, Davide Olori, 2013, "Public sociology in disaster situations: Critical engagement and prefiguration against defuturing processes" in Lavinia Bifulco and Vando Borghi (eds.) Research Handbook on Public Sociology,Edward Elgar Publishing, Please refer to the published version for quotation, minor changes have occurred between this version and the published one 3 resources for social actors engaged in the construction of public issues and transformative social processes.

Building on these premises, in the first section ( §2), we discuss different ways in which the sociology of disaster can be analysed as a public sociology engaged in the elaboration of: 1) useful expertise for public action; 2) critical approaches that reveal the social determinants of disasters; 3) socioanthropological approaches focused on sense-making processes. We then argue ( §3) that the worsening of systemic crises as a result of the increasing financialized and globalized nature of contemporary capitalism (The Foundational Economy Collective 2018) today requires a critical and 'reconstructive' (in the sense of Vandenberghe 2018) sociology of disasters that is actively engaged both in denouncing structural inequalities and in collaborating in 'prefigurative' experiments with social movements, affected citizens and 'reflective practitioners' [START_REF] Trainor | On bridging research and practice in disaster science and management: Unified system or impossible mission?[END_REF]. In particular, the public engagement of sociologists and other social scientists in disaster situations is crucial to initiate and sustain collaborations among a heterogenous group of affected actors (including non-human beings) in order to design alternative, place-based pathways to recovery.

To substantiate this last point ( §4), we draw on our experience of conducting research on disasters in the Italian context [START_REF] Centemeri | The Seveso disaster's legacy[END_REF]Mela et al. 2017). In particular, we briefly revisit the activities of the collective, self-managed research group Emidio di Treviri (EdT) in the aftermath of the 2016 earthquake in the Central Apennines. EdT has been conducting research (and disseminating knowledge) on the various dimensions of the post-earthquake recovery in this region. Since its beginnings, it has been oriented towards creating an interstitial space of encounter between theoretical reflections and direct social action alongside the populations most affected by the disaster and more broadly the affected territories and their ecologies [START_REF] Olori | Ricerca, attivismo e trasformazione sociale nel post-sisma: L'esperienza di una ricerca collettiva, pubblica e dal basso tra criticità e prospettive[END_REF]. As we will argue, this initiative shows the difficulties of applying public sociology in 'fragile areas' [START_REF] Osti | Le aree marginali[END_REF], that is, (mainly rural) contexts in which communities have long experienced progressive processes of fragmentation, which in some cases have almost led to their total disappearance. The long-term engagement of sociologists is essential to support the reconstructive processes, including first and foremost collective 'capabilities for voice ' (de Leonardis et al. 2012), which can elaborate and advocate an alternative vision of recovery to that promoted by Laura Centemeri, Davide Olori, 2013, "Public sociology in disaster situations: Critical engagement and prefiguration against defuturing processes" in Lavinia Bifulco and Vando Borghi (eds.) Research Handbook on Public Sociology,Edward Elgar Publishing, Please refer to the published version for quotation, minor changes have occurred between this version and the published one aggressive pro-growth coalitions whose land valorization logics actually increase socioecological vulnerability to old and new socioecological catastrophes.

From external shocks to structural vulnerabilities: competing understandings of disasters in sociology and society

The consolidation of disaster research as a distinctive field in the social sciences is related to a need for expertise in disaster situation management that emerged in the United States in the 1950s. While the very early sociological studies of disasters examined theoretical questions (e.g. the dynamics of social change), the focus soon shifted to questions more directly related to disaster prevention and management. Disasters were understood at the time as events whose common denominator was the abrupt disruption of 'normality' due to a sudden external shock. In the geopolitical scenario of the Cold War, the question of how to respond to such disruptive events was key for the US government, mainly for military defence reasons.

In fact, disasters at the time (no matter their phenomenology) were considered equivalent to war strikes [START_REF] Gilbert | Studying disaster: Changes in the main conceptual tools[END_REF]). The research on disasters thus focused on understanding people's reactions and behaviours in the face of sudden disruptive events with the aim of developing tools and procedures to manage populations 'under attack'. However, this meant sociological knowledge became subordinated to the technical knowledge produced by the so-called 'hard sciences', such as engineering and the natural sciences [START_REF] Cabane | La cause des catastrophes. Concurrences scientifiques et actions politiques dans un monde transnational[END_REF]. Moreover, an excessive proximity with governmental bodies also led to a focus on disaster situations in the so-called 'developed countries'. This understanding of disasters as sporadic, exogenous events has been challenged since the 1970s by a group of young, critical, politically engaged geographers (including Ben Wisner, Phil O'Keefe and Terry Cannon) who have shown how poverty resulting from forms of economic and political domination was the key factor explaining humanitarian crisis situations in the Global South that had supposedly been triggered by 'natural' disasters, such as the Pakistan floods and the Sahel drought [START_REF] Revet | Disasterland: An ethnography of the international disaster community[END_REF]).

These critical approaches, which had their roots in a political economy perspective inspired by Marxism, were also influenced by contemporary developments in systems thinking that provided evidence of the anthropogenic nature of the world ecological crisis. They led to the Laura Centemeri, Davide Olori, 2013, "Public sociology in disaster situations: Critical engagement and prefiguration against defuturing processes" in Lavinia Bifulco and Vando Borghi (eds.) Research Handbook on Public Sociology,Edward Elgar Publishing, Please refer to the published version for quotation, minor changes have occurred between this version and the published one emergence of the notion of vulnerability as a key concept for understanding disasters as socioecological phenomena. The vulnerability approach progressively gained centrality not only in disaster research but in the construction of an international space of disaster governance through the activism of the researchers who had first promoted this perspective (Revet 2020, p. 44).

The vulnerability approach highlighted the observable diversity of intra-and inter-community responses to disaster and explained this diversity through a combination of structural, socioeconomic, political and ecological factors. Disaster was analysed as 'the result of underlying community logic, of an inward and social process' (Gilbert 1998, p. 3). In other words, the understanding of disaster evolved from an isolated event caused by an external agent into the outcome of long-term processes that had generated conditions of vulnerability.

Even in so-called 'developed countries', disasters were shown to have different impacts on a population within the same city, region or nation depending on socioeconomic indicators, including class, gender, age and race [START_REF] Cutter | Societal vulnerability to environmental hazards[END_REF].

Sociologists have contributed to this debate by developing an approach focused on 'social capital' and based on an in-depth exploration of the role that different types of social relationships ('strong ties' and 'weak ties') play in generating or conversely reducing vulnerabilities. This focus on social ties involves recognizing the importance of local knowledge and local actors for an effective disaster management [START_REF] Dynes | Community social capital as the primary basis for resilience[END_REF].

These developments also paved the way for linking disaster research with the movement for environmental justice and against environmental racism. 1 More generally, they saw NGOs playing an increasingly important role in disaster situations according to two main logics of intervention. Some NGOs would concentrate on risk education initiatives and would often be criticized for their lack of sensitivity to the diversity of cultural contexts, while others would promote the building of a 'risk culture' based on local knowledge and practices. Although the NGOs usually took cultural diversity into account, they tended to have a quite romanticized vision of the local community that foreclosed internal diversity, tensions and struggles (Revet and Langumier 2015). Laura Centemeri, Davide Olori, 2013, "Public sociology in disaster situations: Critical engagement and prefiguration against defuturing processes" in Lavinia Bifulco and Vando Borghi (eds.) Research Handbook on Public Sociology, Edward Elgar Publishing, pp. 174-187

Public sociology in the Anthropocene: from resilience to the opening of alternative futures

Please refer to the published version for quotation, minor changes have occurred between this version and the published one

A growing awareness of the social determinants of disasters and evidence of the 'slow' (see [START_REF] Nixon | Slow violence and the environmentalism of the poor[END_REF], pervasive, long-lasting nature of the processes generating risks both for societal or community life and for individuals' environments and their bodies have made it increasingly difficult to clearly define where and when a disaster begins and ends and to distinguish between natural and man-made disasters.

In addition, the emergence in the 1980s of the notion of the 'risk society' in parallel with an increase in crisis situations, such as those related to the HIV epidemic and mad cow disease [START_REF] Gilbert | Divergence or complementarity within the research into crises in France[END_REF], highlighted the questions of risk communication, risk perception and risk acceptability and shifted sociologists' attention to the production of meaning and knowledge in contexts of radical uncertainty.

Disasters were approached at the time as events resulting from the loss of 'key standpoints in common sense, and the difficulty of understanding reality through ordinary mental frameworks' (Gilbert 1998, p. 9). This difficulty was considered to be generated by the growing complexity that characterized the relationship at societal level between human, ecological and technological systems.

Acknowledging radical uncertainty meant paying attention to how a disaster and its consequences were made the object of a variety of sense-making struggles, that is, struggles around the meaning of what had happened and how it had affected the given order of things and the possible future [START_REF] Centemeri | The Seveso disaster's legacy[END_REF][START_REF] Centemeri | Investigating the 'Discrete Memory' of the Seveso Disaster in Italy[END_REF]. These developments also led to a questioning of the normativity implicit in categories such as disaster [START_REF] Calandra | Disasta: Rethinking the notion of disaster in the wake of cyclone Pam[END_REF], risk, damage and recovery (Centemeri et al. 2022), showing the contentious nature of apparently consensual frameworks like 'Building Back Better' 2 . Social scientists have proved the existence of forms of exclusion regarding the types of knowledge considered relevant to public action, as in the contrast between scientific and 'lay' knowledge. This is not simply a cognitive issue, however, but more generally concerns the taking into account of different systems of meaning, saliency and value, that is, a plurality of ways of knowing and being, or 'ontologies' (Leach, Scoones and Wynne 2005).

The relevance of this form of exclusion is today at the heart of the critical anthropology of disasters because crisis situations are increasingly managed in a globalized context through 'frictions' [START_REF] Tsing | Friction: An ethnography of global connection[END_REF]) between local contexts of action and instruments of action designed Laura Centemeri, Davide Olori, 2013, "Public sociology in disaster situations: Critical engagement and prefiguration against defuturing processes" in Lavinia Bifulco and Vando Borghi (eds.) Research Handbook on Public Sociology,Edward Elgar Publishing, Please refer to the published version for quotation, minor changes have occurred between this version and the published one at global level. This question is equally relevant to the sociology of participatory practices insofar as it underlines the normativity implicit in the idea of stakeholder [START_REF] Cheyns | Multi-stakeholder initiatives for sustainable agriculture: Limits of the 'inclusiveness' paradigm[END_REF].

More recently, new types of disasters have emerged from the interplay between the dynamics of global capitalism, the rise of the information society, the proliferation of transboundary crises and the emergence of ecological threats at planetary level. These include climate change, financial collapses, terrorism and pandemics. According to Quarantelli et al. (2018, p. 61), 'we are at another important historical juncture with the emergence of a new distinctive class of disasters and crises not often seen before'. [START_REF] Cabane | La cause des catastrophes. Concurrences scientifiques et actions politiques dans un monde transnational[END_REF] noted that this juncture is marked by a return to centrality of technical solutions and approaches to disasters that are dominated by the natural sciences, especially climate science, and a sidelining of the social sciences, which are confined to the study of local adaptation and resilience capacities.

In this context, the notions of resilience and resilient communities have gained momentum both in the professional sociology of disasters and in public policy discourses and grassroots mobilizations (e.g. the Transition Towns Movement), thus generating potential interstitial areas of public sociology. At the same time, the 'social resilience' framework (Hall and Lamont 2013) promotes a specific normativity that is underpinned by neoliberal policies and narratives and relies on individual and collective capacities to cope with and creatively adapt to unavoidable catastrophes, which are seen as opportunities for change.

In this scenario where the systemic catastrophe is diluted in the ordinariness of increasingly precarious and vulnerable social worlds, resilience traps the sociology of disasters in an 'ancillary' role. When disaster sociology instead adopts a lens of critical analysis, what emerges is the inexorability of the reproduction of domination structures where the only way to avoid a planetary socioecological collapse is to influence ongoing processes through a radical wholescale systemic change. Indeed, despite a growing number of international initiatives aimed at disaster risk reduction, including the International Decade for Disaster Reduction (1990Reduction ( -1999) ) and the Sendai Framework for Disaster Risk Reduction (2015), the number of disasters has increased dramatically over the past two decades in parallel with the global increase in the number of people living in extreme poverty, a proven cause of vulnerability and an accelerator of risk and disaster (Oliver-Smith and Hoffman 2019). The controversial term Anthropocene hypostatizes the vision of the incumbent Catastrophe.

On the one hand, the Anthropocene framework often reinforces (promises of) technocratic solutions to managing complexity and reduces the sociologist's role to promoting the social acceptability of technical fixes. On the other, critical sociologists risk being trapped in the impasses of becoming prophets of doom.

Faced with systemic complexity, the sociological analysis of disasters can adopt an approach that is focused on monitoring long-term 'critical processes' [START_REF] Chateauraynaud | Aux bords de l'irréversible. Sociologie pragmatique des transformations[END_REF], that is, it can follow the evolution of scientific debates, public problems and postdisaster reconstruction processes while engaging in place-based 'experimental inquiries' [START_REF] Pappas | What difference can 'experience' make to pragmatism?[END_REF]. In fact, it is only by examining both the long-term dynamics and the specific contexts that it is possible to identify the factors that can turn certain events into ruptures, emergencies, upsurges, bifurcations or turning points.

Today, renewed forms of engaged public sociology are emerging from the encounter between systemic analysis and situated knowledge in situations of post-disaster recovery, encouraging the creation of 'publics' (in Dewey's sense) that denounce 'defuturing' processesthat is, processes of destroying the future by design [START_REF] Fry | A new design philosophy: An introduction to defuturing[END_REF][START_REF] Fry | Defuturing: A new design philosophy[END_REF]. This implies that an engaged public sociology of disasters should take the form of a research practice that is necessarily collective, collaborative, transdisciplinary, multi-scalar, place-based and inscribed in a longterm dynamic.

This perspective also implies encouraging the involvement of local researchers to study their 'own' disasters (see for example Tomassi and Fiorino 2019;[START_REF] Ciccozzi | Parola di scienza. Il terremoto dell[END_REF]. This point was emphasized by [START_REF] Gaillard | Disaster studies inside out[END_REF] in his reflection on how to 'decolonize the approach to researching disasters'. A critical sociology of disasters must first come to terms with the fact that 'intentionally or not, disaster studies has fuelled an imperialist disaster risk reduction agenda that, in no way, is different to other "sectors" of the broader development agenda' (Gaillard 2019, p.13). Gaillard suggested the following possible directions for renewing the critical scope of disaster research: 'encourage local researchers who know best local contexts to study local disasters'; 'invite non-Western researchers to collaborate in studying disasters in the West'; 'move away from Western sources, concepts, and methodologies' and draw on 'different epistemologies to reflect different local realities' (Gaillard 2019, p. 14-15). Gaillard Laura Centemeri, Davide Olori, 2013, "Public sociology in disaster situations: Critical engagement and prefiguration against defuturing processes" in Lavinia Bifulco and Vando Borghi (eds.) Research Handbook on Public Sociology,Edward Elgar Publishing, Please refer to the published version for quotation, minor changes have occurred between this version and the published one stressed that 'transferring power to local scholars to take the lead in studying disasters should be the first political and symbolic move' (Gaillard 2019, p.15).

Transferring power to local researchers is not in itself a guarantee of critically engaged public sociology, however. The main point is to promote disaster research that does not simply 'capitalize' on catastrophe (Schuller and Maldonado 2016, p. 67) but takes a long-term, place-based approach to the affected organizations, territories and populations. The involvement of local researchers only achieves this goal, however, if it is complemented by a research perspective that takes into account the interconnection of systemic dynamics, sensemaking processes, political cultures and local organizational logics.

More specifically, a critical and committed public sociology of disasters today should engage first and foremost in denouncing a management of emergencies that is almost exclusively based on the stakeholder perspective, which implicitly benefits those with clearly identifiable economic interests while excluding alternative ways of framing the situation in terms of public and common goods. However, there are many obstacles facing such a public sociology, and these vary according to the intervention context. In the next section, we examine some of these obstacles in depth within the context of the post-earthquake recovery processes in Italy.

The public sociology of disasters in Italy and the Emidio di Treviri collective project in the Central Apennine earthquake

In Italy, a critically engaged approach to the study of disasters first emerged in the 1960s alongside the first examples of public sociology and mass popular mobilizations in response to the great flood of Florence and the earthquake in the Belice Valley in Sicily.

In 1966, (mainly young) volunteers from all over Italy gathered in Florence to help in the rescue operations after the disastrous flooding of the Arno river, in particular by clearing the mud that had covered the streets and monuments, earning themselves the nickname the 'mud angels'. This grassroots mobilization was one of the first examples of spontaneous youth mobilizations in Italy.

In 1968, a major earthquake in the Belice Valley in Sicily killed 370 people and displaced more than 70,000. The poor management of this emergency revealed deep social inequalities between Southern Italy and the rest of the country. This was the first time that dramatic Laura Centemeri, Davide Olori, 2013, "Public sociology in disaster situations: Critical engagement and prefiguration against defuturing processes" in Lavinia Bifulco and Vando Borghi (eds.) Research Handbook on Public Sociology, Edward Elgar Publishing, pp. 174-187

Please refer to the published version for quotation, minor changes have occurred between this version and the published one 10 disaster scenes had been broadcast on TV in Italy, and the lack of humanitarian aid, the mobilization of volunteers and the scandals of reconstruction were there for all to see. The deep emotional impact of the event turned the 'post-emergency' recovery phase into a key public issue and resulted in a shift away from the still prominent fatalistic interpretation of disastrous events. The sociologist and nonviolence activist Danilo Dolci played a crucial role in triggering this dynamic through his research, action and denunciation 3 . These grassroots experiences contributed not only to creating the national Protezione Civile (civil protection) system but also to permeating the scientific debate. With the prominence at that time of Marxist-inspired critical approaches in the social sciences, the underlying socioeconomic causes of such events came to the fore in the political, theoretical and scientific debates on disasters and emergencies. A reading of disasters through the lenses of class relations and class conflicts led to a sociological analysis of industrial (Conti 1976) and 'natural' disasters [START_REF] Cavazzani | Potere e calamità 'naturali': il terremoto ad Ancona[END_REF].

While these contributions did not result in a critical sociology of disasters research community, their influence facilitated the local reception of the international sociology of disasters, whose development in Italy was supported by the Disaster Research Center at Delaware University and the Institute of International Sociology of Gorizia as part of an internationalization strategy aimed at broadening the disaster research field.

In the 1990s, the social sciences became increasingly involved in the study of technological risks. At the same time, the study of environmental disasters contributed to the emergence of environmental sociology approaches in Italian sociology (Avallone 2010, p. 225).

After the earthquake that destroyed the city of L'Aquila (regional capital of the Abruzzo region) in 2009, there was a resurgence of forms of critical and engaged public sociology of disaster. The emergency management was characterized by a rigid top-down approach, which along with the numerous corruption scandals [START_REF] Puliafito | Protezione civile Spa. Quando la gestione dell'emergenza si fa business[END_REF]) sparked widespread protests.

Social scientists played a fundamental role in highlighting critical issues in all post-event aspects, including the emergency phase [START_REF] Ciccozzi | Aiuti e miracoli ai margini del terremoto dell[END_REF], the management of the camps and humanitarian interventions [START_REF] Bonaccorsi | Potere assoluto: La protezione civile ai tempi di Bertolaso[END_REF], the reconstruction phase [START_REF] Alexander | An evaluation of the medium-term recovery process after the 6 April 2009 earthquake in L'Aquila, central Italy[END_REF], urban planning (Frisch 2009), land use transformation [START_REF] Olori | L'urgenza dei disastri. Alcune note sulla riconfigurazione spaziale di L'Aquila a dieci anni dal sisma[END_REF]) and demography [START_REF] Calandra | Territorio e democrazia. Un laboratorio di geografia sociale nel doposisma aquilano[END_REF].

Laura Centemeri, Davide Olori, 2013, "Public sociology in disaster situations: Critical engagement and prefiguration against defuturing processes" in Lavinia Bifulco and Vando Borghi (eds.) Research Handbook on Public Sociology,Edward Elgar Publishing,

Please refer to the published version for quotation, minor changes have occurred between this version and the published one

This research activity, however, came mainly from junior, nontenured researchers or small academic groups [START_REF] Calandra | Territorio e democrazia. Un laboratorio di geografia sociale nel doposisma aquilano[END_REF], so the initiatives struggled to consolidate a study that was sufficiently systematic to have an impact on public intervention measures. The creation, on the initiative of the OECD, of the Urban Studies and Regional Science department at the Gran Sasso Science Institute was a late and limited attempt to institutionalize and systematize social sciences' contribution to the analysis of the disaster. This was a missed opportunity to create a research centre that would promote social sciences interventions to support participatory and transformative processes.

This trend was reinforced after the earthquake that hit central Italy on 24 August 2016 causing 299 casualties. Its seismic crater was located at the intersection of the Marche, Umbria, Abruzzo and Lazio regions. Several more seismic events followed, notably those of 26 October and 30 October (6.5 magnitude). This earthquake was unprecedented for its temporal and geographical extent. It In a context where cultural infrastructures are lacking and with no external support (such as that provided by the OECD in the case of L'Aquila), the development of coordinated postearthquake research efforts has been difficult. At the same time, the assemblies of volunteers engaged locally in supporting the affected populations have expressed the need from the outset to develop practices of inquiry in order to elaborate the knowledge generated by their mutualistic practices through a dialogue with scientific research. These mutualistic practices have been a bottom-up response to this emergency through the establishment of a network of self-organized aid (independent of the Protezione Civile system), including self-managed Laura Centemeri, Davide Olori, 2013, "Public sociology in disaster situations: Critical engagement and prefiguration against defuturing processes" in Lavinia Bifulco and Vando Borghi (eds.) Research Handbook on Public Sociology,Edward Elgar Publishing, Please refer to the published version for quotation, minor changes have occurred between this version and the published one camps and psychological and legal support. These initiatives were intended to support forms of collective action and autonomous territorial planning during the recovery phase.

In December 2016, an appeal launched across the grassroots mobilization networks led to the creation of the post-earthquake collective investigation project and research group EdT 4 . Dozens of PhD students, academics and professionals have been involved in this selfmanaged research initiative, which has explored various aspects of the post-earthquake recovery with a strong commitment in terms of research restitution to the affected populations. Since its beginnings, the group has adopted an openly critical but also placebased perspective, collecting data through immersive fieldwork and other methodologies and designing research questions in strong collaboration with local actors in the post-disaster processes.

The research conducted at EdT has revealed a number of limitations and difficulties with respect to applying a public, critical and reconstructive sociology in a context such as that of the 2016 earthquake, that is, in the aree interne of central Italy. While the voluntary engagement of researchers together with the self-organization of the research work has allowed a high level of autonomy in defining research topics and methods, this logic of employing junior researchers without remuneration has been criticized and even outright rejected within the context of a wider debate on the rise of unpaid forms of labour in academia (Coin et al. 2017).

The goal of 'co-research' (conricerca) 5 , a practice based on the co-definition of the analytical framework and research methodology alongside the collective actors engaged in supporting social transformation, turned out to be (at best) an aspiration rather than a realistic objective.

A number of obstacles to achieving this 'co-research' goal have emerged, including the impossibility of prolonged fieldwork owing to the researchers' precarious working conditions, the displacement of the local populations due to the emergency management and last but not least the lockdown periods during the COVID-19 pandemic.

A further difficulty relates to the positivist approach to the role of 'science' widely adopted in the public debate. On the one hand, there is a prevailing understanding of science as neutral and detached from social conflicts, which implies that an engaged science is considered to be non-objective. As a result, fundamental epistemological issues (how science constructs the knowledge we then use, for what purpose and ultimately for whom) are never properly Laura Centemeri, Davide Olori, 2013, "Public sociology in disaster situations: Critical engagement and prefiguration against defuturing processes" in Lavinia Bifulco and Vando Borghi (eds.) Research Handbook on Public Sociology,Edward Elgar Publishing, Please refer to the published version for quotation, minor changes have occurred between this version and the published one addressed. On the other hand, science is often confused in the public debate with expertise, that is, with the production of technical truths. Consequently, efforts to problematize established technical certainties are not taken as contributions to the production of a better understanding of the situation. Both these positions have been difficult to 'deconstruct' [START_REF] Pugliese | L'inchiesta tra ricerca scientifica e pratica sociale[END_REF].

The initiatives intended to facilitate the co-production of knowledge through the joint involvement of scientists, experts, activists and lay people have been partially successful in some of the ecological struggles that have emerged in the region in response to recovery projects with a significant environmental impact. Finding effective ways of opening a dialogue with institutional actorsespecially those in charge of the recovery processto publicly discuss EdT's research results and of denouncing power inequalities among the supposedly equal stakeholders have, however, proved difficult.

Notwithstanding these limitations, EdT has nevertheless been able to interact in a transformative way with the social worlds in which the research interventions have taken place in some situations. On the one hand, a variety of local and non-local actors have commended the quality and relevance of the collective's scientific output. Some of its results such as evidence on delays in emergency management, data on demographic decline, statistics on the medicalization of displaced persons and on land consumptionhave been used as an 'informational basis' [START_REF] Borghi | From knowledge to informational basis: Capability, capacity to aspire and research[END_REF]) by earthquake victims, grassroots movements and journalists to make public authorities accountable for their decisions and to construct public claims. The results have thus been helpful in turning discontent and indignation into forms of collective action.

Many of EdT's research activities in the different local contexts have led to the development of relational networks involving a huge variety of actors. These place-based collaborations have been crucial in terms of defining new research objectives and forms of research practice that are more openly oriented towards supporting 'prefigurative', practice-based processes of 'sustainable materialism' [START_REF] Schlosberg | From postmaterialism to sustainable materialism: The environmental politics of practice-based movements[END_REF]. This has involved a shift away from an approach to the disaster as a catastrophic event to an exploration of the long-term and structural determinants accounting for the socioecological fragility of the aree interne. In particular, there has been an increasing focus on exploring the relatively recent marginalization of the Apennine territories in the Italian history of economic development Laura Centemeri, Davide Olori, 2013, "Public sociology in disaster situations: Critical engagement and prefiguration against defuturing processes" in Lavinia Bifulco and Vando Borghi (eds.) Research Handbook on Public Sociology,Edward Elgar Publishing, Please refer to the published version for quotation, minor changes have occurred between this version and the published one [START_REF] Ciuffetti | Appennino. Economie, culture e spazi sociali dal medioevo all'età contemporanea[END_REF] and the contribution of the agrosilvopastoral culture in the production and reproduction of common goods [START_REF] Ostrom | Governing the Commons: The evolution of institutions for collective action[END_REF]. To this end, EdT has supported grassroots social reactivation initiatives, the organization of training schools and the creation of an archive of independent scientific output on the recovery process and local struggles.

Having uncovered a composite world of practices, identities and interests in the earthquake areas that are sometimes discordant if not openly conflicting, the EdT has identified various directions for action, each with its own specific objectives and strategic paths, thus participating in the creation of a series of local publics that have supported the constitution of partially overlapping collective actors. In this sense, the dynamics triggered by EdT have counteracted the individualization of the experience of the disaster engendered by the displacement measures.

Recovery measures in the region have been tailored to meet the needs specifically of stakeholders who are private property owners, which has contributed to further exacerbating the processes of fragmentation and individualization. In fact, recovery has been framed in terms of the 'revitalization' of the aree interne exclusively in terms of economic growth (see [START_REF] Macchiavelli | Grandi opere per ri-disegnare il territorio terremotato. Il 'QuakeLab Center Vettore', paradigma della strategia d'investimenti[END_REF]. The political and academic endorsements of this vision have prevented any serious discussion of an alternative, bottom-up reconstruction guided by objectives aimed at regenerating the mountain socioecosystems in the 'reinhabiting' sense [START_REF] Centemeri | Rethinking environmentalism in a 'ruined' world. Lessons from the permaculture movement[END_REF].

The EdT initiative shows the importance of a "critical public sociology" of disasters, in the sense of a public sociology that goes in search of "potential and actual counter-hegemonic publics" and tightens "relations with social movements but not forgetting other publics that are less active, less organized, less articulate" (Burawoy 2005b, p. 390). This is only possible if sociologists accept the challenges of transdisciplinarity while integrating the ecological perspective as a source for theoretical renewal and experimentation with new modes of practical engagement.

Concluding remarks

The EdT initiative represents neither best practice nor a model to be imported to other contexts. This was not the motive for this brief discussion of its activities. Our aim was to contribute to the general discussion on the current challenges faced by the public sociology of Laura Centemeri, Davide Olori, 2013, "Public sociology in disaster situations: Critical engagement and prefiguration against defuturing processes" in Lavinia Bifulco and Vando Borghi (eds.) Research Handbook on Public Sociology,Edward Elgar Publishing, Please refer to the published version for quotation, minor changes have occurred between this version and the published one disasters in the Anthropocene era by providing experiential insights from a group of researchers who are trying to produce critical and reconstructive sociological knowledge in a disaster situation with its own specific resources and constraints. This initiative confirms that the public sociology of disasters is necessarily an interstitial practice. In our case, interstitial means first and foremost an interdisciplinary and transdisciplinary research practice as a prerequisite for investigating territorial contexts characterized by complex socioecological interactions at different levels. It is interstitial then in the sense of an openness to collaborating with the actors most directly affected by the disaster. This collaboration is essential if we are to design research interventions that can help sustain critical and reconstructive processes that have some reasonable hope of resonating with local sensibilities.

A successful interstitial positioning requires autonomy in the definition of research objectives and methods and above all the potential for continuous redefinition based on lessons learned in the field. This latter condition is difficult to meet in an academic context that increasingly rewards the pursuit of preformatted research objectives.

The autonomy of the EdT project relies on the voluntary-based engagement of usually junior, nontenured researchers. The self-organized nature of EdT partly explains the problems it has faced in not being taken seriously as a qualified partner in the elaboration of public policies.

Furthermore, the difficulty of ensuring the project's continuity has complicated its active contribution to prefigurative practices. However, prefigurative engagement is today a key component of a critical public sociology of disaster in terms of both reviving the affected populations' capacity to aspire [START_REF] Appadurai | The capacity to aspire: Culture and the terms of recognition[END_REF]) and challenging the 'extractivist' disaster capitalism practices [START_REF] Klein | The Shock doctrine: The rise of disaster capitalism[END_REF]) that rely on the many abstractions of neoliberalism [START_REF] Tsing | Friction: An ethnography of global connection[END_REF]). Prefigurative efforts are nevertheless heavily stymied by the structural fragility of interstitial sociological interventions. They also have to contend with the absence of local publics with whom to concretely prefigure reconstructive processes so as to counteract the dynamics that (re)produce socioecological vulnerability.

What emerges from the contemporary Italian post-earthquake recovery process experience are the specific difficulties of applying a public sociology of disasters in 'fragile areas'. In the aree interne of the Apennines, the dynamics of pauperization, aging and depopulation have contributed to the fragmentation of the social context. An emergency management that was Laura Centemeri, Davide Olori, 2013, "Public sociology in disaster situations: Critical engagement and prefiguration against defuturing processes" in Lavinia Bifulco and Vando Borghi (eds.) Research Handbook on Public Sociology,Edward Elgar Publishing, Please refer to the published version for quotation, minor changes have occurred between this version and the published one 16 almost exclusively based on displacement has reinforced this process, generating further isolation, conflict and territorial dispersion. And yet the dominant logics that guide the recovery plans seem to ignore both the problem of defining who the beneficiaries of the reconstruction should be and the voice of future generations (not to mention ecosystems and non-human beings). Political actors uncritically apply the framework of the economic stakeholders as the only legitimate representative of the collective interest. As such, they have no need to work actively on creating local publics who can contribute to defining a shared vision of the problems and solutions. Problematizing the question of the 'subjects' with whom one rebuilds after a disaster is today one of the main tasks of the public sociology of disasters.

From this point of view, the EdT initiative demonstrates the value of a strategy that is promoted more generally by movements and social networks working to support the need for reasoning in terms of common goods as a way out of a paralyzing state versus market opposition. In our case, this alternative translates as the inclusion of forms of selforganization in the management of common-pool resources within the decision-making mechanisms of recovery. In the Apennine area, in fact, there are still forms of collective ownership of agricultural land, forests and pastures (or "communal tenure" according to [START_REF] Ostrom | Governing the Commons: The evolution of institutions for collective action[END_REF]). These so-called "rural commons" are managed by local bodies that are defined as comunanze agrarie. The comunanza is a legally recognized entity with the task of administering the common resources of a given territory through an administration composed of a president and councilors who are chosen from among the residents, by the residents themselves. The comunanza is an inclusive organizational model that historically has been able to combine collective self-organization and institutional recognition based on the pursuit of social equity and environmental sustainability objectives. These experiences of autonomy in the local governing of populations and resources, when guided by objectives of solidarity and co-responsibility (as in the 'sustainable district' notion discussed by [START_REF] Donolo | Il distretto sostenibile: governare i beni comuni per lo sviluppo[END_REF], can be drawn on to imagine alternative pathways of community (re)construction, to recognize socioecological vulnerabilities, to support decision-making processes that aim at the collectivization of risk and to increase collective preparedness for future crises.

In this sense, the public sociology of disasters has an important role to play both in supporting the capacities of communities to repair and reconfigure broken socioecosystems Laura Centemeri, Davide Olori, 2013, "Public sociology in disaster situations: Critical engagement and prefiguration against defuturing processes" in Lavinia Bifulco and Vando Borghi (eds.) Research Handbook on Public Sociology,Edward Elgar Publishing, Please refer to the published version for quotation, minor changes have occurred between this version and the published one and in countering the defuturing systemic processes that continue to feed fast and slow disasters.

Laura Centemeri, Davide Olori, 2013, "Public sociology in disaster situations: Critical engagement and prefiguration against defuturing processes" in Lavinia Bifulco and Vando Borghi (eds.) Research Handbook on Public Sociology,Edward Elgar Publishing, Please refer to the published version for quotation, minor changes have occurred between this version and the published one Laura Centemeri, Davide Olori, 2013, "Public sociology in disaster situations: Critical engagement and prefiguration against defuturing processes" in Lavinia Bifulco and Vando Borghi (eds.) Research Handbook on Public Sociology,Edward Elgar Publishing, Please refer to the published version for quotation, minor changes have occurred between this version and the published one

  predominantly affected mountain areas that had been classified in Italian legislation as aree interne (inner areas) based on indicators of high levels of social vulnerability (see De Rossi 2020). Social vulnerability goes hand in hand with ecological fragility, and both are related to long-term processes of depopulation, demographic aging and pauperization. These socioeconomic trends have affected the whole Italian mountain region (the Apennines and the Alps) over the last fifty years. The earthquake areas have seen an exponential acceleration in depopulation and fragmentation due to a management of the emergency phase that was based almost exclusively on displacement and accompanied by delays in the recovery phase (Emidio di Treviri 2018) and a lack of economic support.

  LauraCentemeri, Davide Olori, 2013, "Public sociology in disaster situations: Critical engagement and prefiguration against defuturing processes" in Lavinia Bifulco and Vando Borghi (eds.) Research Handbook on Public Sociology, Edward Elgar Publishing, pp. 174-187 Please refer to the published version for quotation, minor changes have occurred between this version and the published one 8

Notes 1 In particular, the response to Hurricane Katrina dramatically exposed the enduring racial divide in disaster relief in the US (see Wright 2011). 2 This catchphrase was coined by the former US president Bill Clinton in the aftermath of the 2004 Indian Ocean earthquake and tsunami and was developed into a list of ten propositions conceived as operational guidelines for the humanitarian post-disaster intervention. 3 Danilo Dolci (1924-1997) was an Italian intellectual and activist and one of the leading figures in the nonviolent movement. Dolci had been engaged in the Belice Valley context since the early 1950s, promoting forms of denunciation and direct action against the extreme poverty that plagued the region. After the earthquake, he engaged in denouncing the public institutions' inefficiency and collusion with the Mafia during the reconstruction phase. 4 For a more in-depth discussion of this initiative, see Olori and Menghi (2020) and [START_REF] Di | Sul fronte del sisma. Per una inchiesta militante sul post-terremoto dell[END_REF]. The name Emidio di Treviri represents a form of 'subversion' of local religious devotion practices because Saint Emidio from Ascoli is a local saint who, according to local tradition, protects against earthquakes. The story is that he was born in Trier, as incidentally was Karl Marx. 5 The term conricerca in the Italian public sociology tradition (see Armano 2020) refers to a methodology in which researchers and social actors share a similar vision of the wider goals of social transformation and a shared theoretically informed frame of understanding of a given problematic.


Youssef Diab

Pascal Venet

Gérard Rojat

Yasser Diab

Gerard Rojat

Comparison of the Different Circuits Used for Balancing the Voltage of Supercapacitors: Studying Performance and Lifetime of Supercapacitors

Keywords: Supercapacitor, overvoltage, balancing circuits, temperature, manufacturing tolerance, aging behavior, life expectancy, efficiency

published or not. The documents may come     Comparison of the Different Circuits Used for Balancing the Voltage of Supercapacitors: Studying Performance and Lifetime of Supercapacitors

I. INTRODUCTION

Supercapacitors offer interesting electrical characterizations and high lifetime cycling compared to that of batteries. They appear well suited to automotive advanced applications to boost the battery especially in a hybrid electric vehicle: They allow to provide the necessary power for acceleration and to recuperate the braking energy.

Due to decomposition voltage of the organic electrolyte of approximately 3V, the maximum cellvoltage of supercapacitors is limited. A supercapacitor module is formed by several serial cells in series to obtain voltages suitable for applications. For different reasons, the internal parameters of supercapacitors (capacitance, ESR, and self-discharge rate) of a module may vary. These variations give raise to unequal voltage distribution during charge-discharge cycles and overvoltage may appear on cells. The causes of supercapacitors parameters tolerances and balancing circuits will be analyzed in detail, in addition to the performance and lifetime of supercapacitors [START_REF] Yang | Korea '' Development of Ultracapacitor Modules for 42 Volt Automotive Electrical Systems[END_REF][START_REF] Schneuwly | BOOSTCAP Double-layer Capacitors for Peak Power Automotive Applications[END_REF][START_REF] Linzen | Analysis and Evaluation of Charge-Balancing Circuits on Performance, Reliability, and Lifetime of Supercapacitor Systems[END_REF].

II. CAUSES OF SUPERCAPACITORS NON IDENDICAL PARAMETRES

The internal parameters of supercapacitors (ESR, capacitance and self-discharge rate) vary in accordance with the following conditions:

A. Manufacturing dispersion in supercapacitors parameters:

Parameters dispersion of supercapacitors produced by the same manufacturer can reach ± 20% for capacitance and ± 25% for ESR. A study was conducted on four components which belong to a series manufacturing (Maxwell 2600F 2.5V), tolerances were appeared on capacitance (5%) and on ESR (3%). Figure 1 shows these dispersions on Nyquist plan [START_REF] Schneuwly | BOOSTCAP Double-layer Capacitors for Peak Power Automotive Applications[END_REF][START_REF] Linzen | Analysis and Evaluation of Charge-Balancing Circuits on Performance, Reliability, and Lifetime of Supercapacitor Systems[END_REF]. The dispersion of supercapacitors self-discharge rate can influence the cells-voltage distribution especially in applications with low duty-cycle. Figure 2 shows that the four aforementioned components did not have the same self-discharge ESR rate, consequently a difference in the cells-voltage has appeared.

B. Temperature influence

The ambient temperature has an effect on the supercapacitor behavior: the increase of ambient temperature causes slight increase in the supercapacitor capacitance, huge decrease in ESR as depicted in figure 3, and decrease in the selfdischarge rate [START_REF] Yang | Korea '' Development of Ultracapacitor Modules for 42 Volt Automotive Electrical Systems[END_REF][START_REF] Lajnef | Ultracapacitor Electrical Modeling Using Temperature Dependent Parameters[END_REF].

C. Ageing behaviour

The supercapacitor aging mechanisms result from electrical, mechanical or environmental stresses. Maximum operating voltage and temperature are therefore important factors on aging behavior of supercapacitors. At higher temperature, the aging processes are accelerated by the higher reactivity of the chemical components. At higher voltage impurities undergo a redox reaction. In addition, organic electrolyte starts to decompose, producing gas product, which can lead to the component destruction. This phenomenon may decrease the capacitance by 20%, and increase ESR by 100% and self-discharge rate [START_REF] Linzen | Analysis and Evaluation of Charge-Balancing Circuits on Performance, Reliability, and Lifetime of Supercapacitor Systems[END_REF][START_REF] Hermann | High Performance Double layer Capacitor for Power Electronic Applications[END_REF].

III. ESTIMATE OF LIFE EXPECTANCY OF SUPERCAPACITORS

Operating voltage and ambient temperature largely influence the life expectancy of the supercapacitors. Therefore the life expectancy can be estimated by an exponential function of these variables (operating voltage and ambient temperature) as given by equation 1 according to figure 4 [3]. The average life expectancy can be estimated by the accumulated useful life (wear) of a supercapacitor in function of a dynamic voltage profile ( ) (t u ). As given by equation 2 t 0 and t 1 are the starting and end time respectively.

) 2 ( ) ), ( ( 1 )
), ( (

1 0 exp 0 1 ∫ - = t t av dt t u T t t t u T θ θ IV.

SUPERCAPACITOR MODEL AND EFFICIENCY OF VOLTAGE BALANCING CIRCUITS

To get appropriate simulations a suitable model of supercapacitor was used. This model was selected because it can be trade-off between the precision and the simplicity of the conducted simulations. This model consists of the immediate and the delayed branches as well as the self-Life expectancy (Years) Voltage (V) discharge components. They include the leakage current (Rf 1 ) and redox reaction (Rf 2 ), as depicted in figure 5 [START_REF] Ding | Modeling of the Supercapacitors and Iinfluence of the Temperature during Self-discharge[END_REF][START_REF] Ricketts | Self-discharge of carbon-based supercapacitors with organic electrolytes[END_REF].

In order to compare the performance of voltage balancing circuits, the efficiency of module is defined as given by equation 3.

) 3 ( sc eq sc W W W -= η with W sc is energy stored in supercapacitors and W eq is energy dissipated in voltage equalization circuits.

V. BALANCING CIRCUITS

In order to simplify the study two supercapacitors in series were employed with a capacity tolerance of (∆C) 20%:

C 1sc1 (F) =1749 +573.U 1 C 1sc2 (F) =1399 +458.U 2
We proposed periodical charge-discharge cycle in following settings: constant current I = 400 A for charge and discharge, interval between chargedischarge 20 ks see figure 6. The charge was to reach an overvoltage value on the supercapacitor SC 2 (lower capacitance) of 2.8 V. This last is the surge voltage of the conducted supercapacitor given by the manufacture.

A. Passive balancing circuits

A resistor is connected in parallel with each supercapacitor as shown in figure 7.a [START_REF] Linzen | Analysis and Evaluation of Charge-Balancing Circuits on Performance, Reliability, and Lifetime of Supercapacitor Systems[END_REF][START_REF] Härri | All-Purpose Circuitry Concept "SAM" Applications and Networking for Supercapacitors[END_REF]. It allows balancing cell-voltage by dissipating the supercapacitor energy manifested as overvoltage on supercapacitor (SC 2 ) by thermal effect. Equation 4represents the estimate of resistor value for passive balancing circuits depending on maximum voltage U scmax and leakage current of supercapacitor I leakage (their corresponding values are U scmax = 2.8 V, I leakage = 5 mA ). This type of balancing reduces the efficiency of the module depending on the value of the resistor and the applied current profile because of continuous leakage current in resistors. For value of low balancing resistor the efficiency can fall to the value of ten percents [START_REF] Barrade | Energy storage system using a series connection of supercapacitors, with an active device for equalizing the voltages[END_REF]. On the other hand, this solution can improve lightly the life expectancy of supercapacitors as shown in table 1 [START_REF] Linzen | Analysis and Evaluation of Charge-Balancing Circuits on Performance, Reliability, and Lifetime of Supercapacitor Systems[END_REF].

B. Switched-resistors balancing circuits

The general concept depends on adding a controlled switch to an equalization resistor to limit the energy dissipation as shown in figure 7.c. The switch turns on when cell-voltage goes beyond operating voltage and turns off when the level is lower than this voltage. Figure 9 shows that the cell-voltage (U sc2 ) takes a few minutes to reach to operating voltage (2.5 V) for a value of resistor 1 Ω and there isn't any more thermal loss in the other cell (SC 1 ) [START_REF] Yang | Korea '' Development of Ultracapacitor Modules for 42 Volt Automotive Electrical Systems[END_REF][START_REF] Linzen | Analysis and Evaluation of Charge-Balancing Circuits on Performance, Reliability, and Lifetime of Supercapacitor Systems[END_REF][START_REF] Härri | All-Purpose Circuitry Concept "SAM" Applications and Networking for Supercapacitors[END_REF].

The efficiency of this solution rises largely more than the solutions mentioned above. Life expectancy of supercapacitors increases according to the value of the equalization resistor as shown in table 1. However the using of these circuits adds extra costs due to the requirement of electronic components.

To obtain a more effective balancing dynamic, we need to use value of resistors less than 1 Ω without important decrease in efficiency. The last is almost independent of the value of the balancing resistor and current profile. There is another dissipative balancing circuit uses the Zener diodes as shown in figure 7.b. This circuit equalizes cells-voltage according to Zenervoltage by dissipating the extra energy in thermal effect [START_REF] Linzen | Analysis and Evaluation of Charge-Balancing Circuits on Performance, Reliability, and Lifetime of Supercapacitor Systems[END_REF][START_REF] Härri | All-Purpose Circuitry Concept "SAM" Applications and Networking for Supercapacitors[END_REF][START_REF] Barrade | Energy storage system using a series connection of supercapacitors, with an active device for equalizing the voltages[END_REF]. Therefore we could not find voltage and current of the Zener diodes suitable for our application.

V.2 NON-DISSIPATIVE BALANCING CIRCUITS

All the solutions mentioned above dissipate the energy in order to equalize the cells-voltage. Another type of voltage balancing circuits has been proposed, which employ buck-boost structures with/without transformer. These circuits transfer instantaneously the exceeding energy of cellsovervoltage to other cells in order to equalize cellsvoltage [START_REF] Härri | All-Purpose Circuitry Concept "SAM" Applications and Networking for Supercapacitors[END_REF][START_REF] Barrade | Energy storage system using a series connection of supercapacitors, with an active device for equalizing the voltages[END_REF].

A. Indirect balancing: current diverter using buck-boost converters

The equivalent circuit is given in figure 7.d. It equalizes the cells-voltage locally by two consecutive supercapacitors, using a structure of buck-boost converter for each pair of supercapacitors. The balancing is realized as soon

U sc1

The requested time for cellsvoltage of component reach to operating voltage with resistor of 1 Ω

Usc (V)
as a difference of cells-voltage appears: the extra energy of supercapacitor SC 2 is stored firstly in the inductor (L = 0.5 µH), then transferred to supercapacitor SC 1 using a switch with frequency (f=10 kHz) and duty cycle (50%) as shown in figure 10. An important balancing current (60 A) had to be chosen to carry out the equalization of voltages in accordance with charging process. The instantaneous transfer of energy related to overvoltage from SC 2 to SC 1 leads to higher efficiency and life expectancy than other solutions as pointed out in table 1. The main disadvantage of these balancing circuits resides in the indirect energy transmission especially for the applications with a huge number of supercapacitors in series. This lowers both efficiency and the lifetime. Although the concept of this circuit is nondissipated, some losses may occur in the electronic components. In addition the use of converters (buck-boost) balancing circuits is costly [START_REF] Barrade | Energy storage system using a series connection of supercapacitors, with an active device for equalizing the voltages[END_REF][START_REF] Barrade | Series Connection of Supercapacitors: Comparative Study of Solutions for the Active equalization of the Voltages[END_REF].

B. Direct balancing: flyback converters

with distributed secondary The aim of using a transformer is to transfer directly the energy: the switch T turns on to allow storing some energy in primary of transformer, and then energy is transferred to distributed secondary, via the diodes as shown in figure 11. Most of this energy is provided to cells-lowest voltage. C. Direct balancing: current diverter using forward converters This solution depends on the direct transmission of energy: the transistor T 2 is switched on as soon as the voltage U sc2 starts to be higher than U sc1 as shown in figure 12. This allows transferring some energy from SC 2 to SC 1 . Both of the previous balancing circuits equalize instantaneously the cells-voltage (see figure 13). For the selected model with two supercapacitors in series, we found the performance and the life expectancy of supercapacitors are close to the solution A (current diverter using buck-boost converters). This result was obtained for ideal transformers, but the use of real transformer causes losses in addition to those of the electronics components. This also means a lower efficiency and lower life expectancy [START_REF] Barrade | Energy storage system using a series connection of supercapacitors, with an active device for equalizing the voltages[END_REF][START_REF] Barrade | Series Connection of Supercapacitors: Comparative Study of Solutions for the Active equalization of the Voltages[END_REF]. circuits are employed to solve this problem: resistance, switched-resistors and static converters. Experimental results show that these circuits differ by their efficiency and their life expectancy: The balancing circuits with simple resistance improve lightly the life expectancy of the supercapacitors, but decrease their efficiency. Switched-resistors allow limiting the energy dissipation and improve significantly the life expectancy. The DC/DC converters balances circuits equalize instantaneously cells-voltage by redistributing the stored energy. This in turn increases the efficiency as well as expectancy of supercapacitors.

The estimation of the tolerance caused by the aging behaviour is underway with an explanation to the aging processes. The used equation estimates the life expectancy of supercapacitors depending on two main factors; the ambient temperature and the cells-voltage taking into account that the current effect was neglected. The direct effect of these factors is not clear on the aging processes but it may make the temperature of the components higher than the surrounding ambiance. We envisage tacking into account the current in the equation in order to achieve a better precision. We will realize this, in addition to a complete validation of the results obtained for balancing circuits in the near future.

C ω 1 Fig. 1 :

 11 Fig. 1: Supercapacitors parameters manufacturing tolerances

Fig. 2 :

 2 Fig. 2 : Self-discharge of supercapacitors

Fig. 3 :

 3 Fig. 3: Variation of supercapacitor parameters with temperature

  and θ are the cell voltage and the cell temperature, respectively. The constant parameters C1 to C3 are derived from the relationship given in figure4.

Fig. 4 :

 4 Fig. 4: Life expectancy of a supercapacitor in function of operating voltage and temperature

Fig. 6 :

 6 Fig.6 : Cells-voltage with switched balancing circuits The tolerance of the supercapacitors parameters distributes cells-voltage unequally during chargedischarge cycles. An overvoltage may appear on the component SC 2 . To redistribute and equalize the cells-voltage, various voltage equalization circuits were employed, which are based on two main concepts: dissipative and non-dissipative [3, 8]: V.1 DISSIPATIVE BALANCING CIRCUITS

U

  The value of the balancing resistors was chosen as 50 Ω in order to balance between the efficiency and the good balancing dynamic of voltage. But it can not equalize the cell-voltage quickly: it takes more than 5 ks to reach to the operating voltage 2.5 V (see figure8).

Fig. 5 : 1 Fig. 7 :Fig. 8 :

 5178 Fig.5: Equivalent circuit model for a supercapacitor Maxwell 2600 F 2.5 V: R 1 = 0.47 mΩ, R 2 =1.96 Ω, C 2 = 176 F, C 3 =88.7 KF, Rf 1 = 1.12 kF, Rf 2 =42.9 Ω, C 1 (F) = 1749 +573 U 1

Fig. 9 :

 9 Fig.9: Cells-voltage with switched-resistors balancing circuits

  requested time for cellvoltage reach to operating voltage with resistor of 50 Ω t (s) U sc2

Fig. 10 :

 10 Fig.10: Cells-voltage with buck-boost converters for balancing current 60 A, switched frequency 10kHZ, inductor 0.5 µH and duty cycle 50%

Fig. 11 :

 11 Fig. 11: Balancing circuits: flyback converters with distributed secondary

Fig. 12 :

 12 Fig. 12: Balancing circuits current diverter using forward converters

Fig 13 :

 13 Fig 13 : Cells-voltage with flyback converter with distributed secondary


Matthieu Muller

Daniele Picone

Mauro Dalla Mura

Magnus O Ulfarsson

MODEL-BASED DEMOSAICKING FOR ACQUISITIONS BY A RGBW COLOR FILTER ARRAY

Keywords: Color Filter Array, Demosaicking, Convex Optimization, Computational Imaging

Microsatellites and drones are often equipped with digital cameras whose sensing system is based on color filter arrays (CFAs), which define a pattern of color filter overlaid over the focal plane. Recent commercial cameras have started implementing RGBW patterns, which include some filters with a wideband spectral response together with the more classical RGB ones. This allows for additional light energy to be captured by the relevant pixels and increases the overall SNR of the acquisition. Demosaicking defines reconstructing a multi-spectral image from the raw image and recovering the full color components for all pixels. However, this operation is often tailored for the most widespread patterns, such as the Bayer pattern. Consequently, less common patterns that are still employed in commercial cameras are often neglected. In this work, we present a generalized framework to represent the image formation model of such cameras. This model is then exploited by our proposed demosaicking algorithm to reconstruct the datacube of interest with a Bayesian approach, using a total variation regularizer as prior. Some preliminary experimental results are also presented, which apply to the reconstruction of acquisitions of various RGBW cameras.

INTRODUCTION

In certain remote sensing setups, constellations of microsatellites or drones are utilized for acquiring data. These platforms have limited payload capacity, which necessitate the use of novel solutions for the onboard instruments and often incorporate miniaturized commercial cameras.

In this configuration, the acquired raw image is transmitted as a downstream data to a ground station, which is responsible for processing it to reconstruct the desired datacube. Cameras based on color filter arrays (CFAs) represent a popular choice for this task. The CFA defines an array of spectral filters overlaid over the sensors, typically arranged in periodic patterns. The incident light is consequently filtered by their unique spectral response of each filter, before being captured by the sensor. The usual image acquisition process uses a predetermined CFA pattern to assign a color to each pixel of the sensor. While the most common pattern is the Bayer pattern Fig. 1a [START_REF] Li | Colour filter array demosaicking using cubic spline interpolation[END_REF][START_REF] Yamaguchi | Image demosaicking via chrominance images with parallel convolutional neural networks[END_REF] other configurations exist. Some alternative designs exist, in which the filters' spectral response does not necessarily cover the RGB wavelengths. Such patterns are known with the more generic term spectral filter arrays (SFAs). For example, this category includes filters in the near infrared/ultra violet wavelength range, alternative color patterns such as cyan/magenta/yellow (CMY), and most relevantly to our work, wideband pixels. In the latter case, known as RGBW SFAs, the relevant sensor typically either does not include a filter, matching the sensor sensitivity response, or just cuts off the near infrared components. In layman terms, those pixels are often just referred as "white", and they allow for higher throughput of light energy which allows for increased brightness. An example of those patterns is the Sparse 3 CFA which is used in the AMICal Sat project [START_REF] Barthelemy | Amical sat: A sparse RGB imager on board a 2u cubesat to study the aurora[END_REF] to study northern lights from a microsatellite. Same goes for the Kodak and Sony patterns which are presented in Fig. 1.

The process of reconstructing the full color image from acquisitions taken with CFA-based cameras is known as demosaicking. As previously mentioned, this is typically done on the ground segment. It can be done in multiple ways: by classic interpolation between the pixels [START_REF] Li | Colour filter array demosaicking using cubic spline interpolation[END_REF][START_REF] Malvar | High-quality linear interpolation for demosaicing of bayer-patterned color images[END_REF], by modelbased methods which relies on a known mathematical model linking the input to the raw acquisition. More recently neural networks strategies appeared [START_REF] Sharif | Beyond joint demosaicking and denoising: An image processing pipeline for a pixel-bin image sensor[END_REF][START_REF] Feng | Mosaic convolution-attention network for demosaicing multispectral filter array images[END_REF], in general they are based on end-to-end training with large data-sets. However, all those methods have drawbacks. The interpolation methods often only work with specific patterns, posing restrictions for our scenario. Model-based approaches need a precise knowledge of the real acquisition conditions. We model the image formation with a mathematical formulation. This simulates the camera's acquisition process, inspired by [START_REF] Picone | Joint demosaicing and fusion of multiresolution coded acquisitions: A unified image formation and reconstruction method[END_REF]. While allowing for remarkable flexibility, the model is very sensitive to inaccurate modeling of the real operations performed by the camera. Finally, data-driven based methods need a huge amount of data to be trained on and are often specialised on specific CFAs, leaving few possibilities for adaptation beside new training. For example, [START_REF] Sharif | Beyond joint demosaicking and denoising: An image processing pipeline for a pixel-bin image sensor[END_REF] proposes an efficient demosaicking neural network for Bayer CFA, but fails at reconstructing RGBW images. So is the case for the methods that focus only on Bayer CFA, like in [START_REF] Malvar | High-quality linear interpolation for demosaicing of bayer-patterned color images[END_REF][START_REF] Yamaguchi | Image demosaicking via chrominance images with parallel convolutional neural networks[END_REF][START_REF] Li | Colour filter array demosaicking using cubic spline interpolation[END_REF].

For those reasons this work proposes a general method able to solve the demosaicking problem for commercial designs of RGBW CFAs. This method relies on a model-based approach and minimizes a regularized convex cost function.

The novel contribution of this work are:

• a forward model capable of simulating the camera's action, with support for the RGBW Sparse 3, Kodak and Sony patterns;

• a model-based method using Chambolle-Pock's algorithm to minimize a convex optimization problem solving the demosaicking challenge for RGBW patterns.

PROPOSED FRAMEWORK

This section presents the acquisition process and the reconstruction. The first step is to model the image acquisition process usually done by the camera itself. This operation is performed by a forward model taking in input the reference image, in order to obtain the raw image. Then the reference image will be reconstructed into a RGB image from the raw image using a model-based method.

In the RGBW case the reference image, representing the incoming electromagnetic radiation from a scene under target is denoted by X ∈ R M ×N ×(C+1) where C is the number of channels (C = 3 for a RGB image), an additional channel is added containing the panchromatic, or "white", information. M and N being the spatial dimensions. The raw image representing the observation is encoded by Y ∈ R M ×N , a scalar matrix coming from the forward model. Finally we denote by X ∈ R M ×N ×C the image reconstructed from the raw image Y .

Forward model

We describe here our proposed framework for image formation, by reinterpreting the model proposed in [START_REF] Picone | Joint demosaicing and fusion of multiresolution coded acquisitions: A unified image formation and reconstruction method[END_REF] for our target. In order to apply a CFA a mask H ∈ R M ×N ×(C+1) is used, encoding the wanted CFA. The output of the CFA operation is obtained with:

Y = C+1 k=1 X k ⊙ H k + N ( 1
)
where ⊙ is the Hadamard product between matrices, X k and H k are matrices which denote the k-th channel of the tensors X and H respectively. Moreover N ∈ R M ×N is a noise tensor representing the physical noise on the sensor. Additionally it models the mismatch between the forward model and the real camera. The tensor H encoding the mask contains a ones in the spectral channel associated to the corresponding spectral filter is kept. The rest is set at zeros, discarding the information at those positions. The image Y is then composed of scalar values, where for each pixel is kept only the information from the selected channels.

The deterministic part of eq. ( 1) is a linear operation, which we denote as A : R M ×N ×(C+1) → R M ×N . We can therefore express Y as:

Y = A(X) + N .
For convenience, we also derive here the adjoint A T (Y ) of A as the tensor Z ∈ R M ×N ×(C+1) whose k-th channel is [START_REF] Picone | Joint demosaicing and fusion of multiresolution coded acquisitions: A unified image formation and reconstruction method[END_REF]:

Z k = H k ⊙ Y , ∀k ∈ {1, ..., C + 1}.

Model-based demosaicking

The demosaicking problem aims to reconstruct a RGB image from the raw acquisition. However this task is ill-posed as there are more unknowns than there is information as the goal is to recover data lost during the acquisition process. This problem can be reformulated as a minimization of a cost function with a regularization term:

X = arg min X ∥A(X) -Y ∥ 2 F + λ∥L(X)∥ 221 (2)
where ∥ • ∥ F is the Frobenius norm, λ∥L(•)∥ 221 is the regularizer: L is the isotropic total variation operator, λ is a tuning parameter, and || • || 221 denotes an ℓ 2 norm in the channel and gradient domains followed by an ℓ 1 norm in the spatial one. This term is derived from [START_REF] Picone | Joint demosaicing and fusion of multiresolution coded acquisitions: A unified image formation and reconstruction method[END_REF] and [START_REF] Wen | Efficient total variation minimization methods for color image restoration[END_REF], where experimentation yielded good results in a similar setup to ours. Note that in this setting X ∈ R M ×N ×(C+1) , but as the method's goal is to reconstruct only the RGB image the panchromatic channel is simply discarded and X is considered as a tensor of R M ×N ×C . The Chambolle-Pock algorithm [START_REF] Esser | A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science[END_REF] is a well-known iterative algorithm used to solve optimization problems represented by the sum of two convex functions. This is the case of the demosaicking problem and it can be re-written in the Chambolle-Pock's framework:

min X f (X) + g(L(X))
where f and g denote respectively the fidelity term ∥A(•) -Y ∥ 2 F and the regularization term λ∥ • ∥ 221 of the previous equation. The primal and dual variables are then updated from an arbitrary initialization with the following iterations [START_REF] Pock | Diagonal preconditioning for first order primal-dual algorithms in convex optimization[END_REF][START_REF] Esser | A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science[END_REF]:

X q+1 = prox τ,f (X q -τ L T (Z q )) Z q+1 = prox σ,g * (Z q + σL(2X q+1 -X q )) (3)
with τ and σ being the proximal steps for f and g * (the Fenchel conjugate of g) respectively. q is the iteration number which goes from 1 to Q, the total number of iterations, X q and X q are the primal and dual variables at the q-th iteration.

EXPERIMENTAL RESULTS

In this section we present the results on the RGBW patterns Sparse 3, Kodak and Sony. The associated code to reproduce the results of this section is publicly available on GitHub 1 . We compare our method with the technique presented in [START_REF] Teledyne E2v | Application note : How to interpolate the sparse 3 onyx sensor[END_REF], which is used on the microsatellite AMICal Sat and was quickly extended to work with Kodak and Sony patterns. The idea of the approach is to extract in one image the luminance (a scalar image with full spatial resolution) of the reference image by interpolating the pixels in the W channel associated with the RGB spectral filters. The colors are extracted in another image (of lower spatial resolution by full spectral resolution), by interpolating the R, G and B channels. Then those two images will be merged into a RGB image of full spectral and spatial resolution.

Concerning our method we need to tune the parameters introduced in eq. ( 2) and eq. ( 3) which are: λ, τ , σ and Q. The algorithm will run through Q = 400 iteration as empirical results showed no real improvement with higher number of iterations. The parameters λ, τ and σ are optimized by the Python module Optuna [START_REF] Akiba | Optuna[END_REF]. 1 At https://github.com/mattmull42/RGBW demo The table of errors and the following figures are computed on the PAirMax remote sensing multispectral dataset [START_REF] Vivone | A benchmarking protocol for pansharpening: Dataset, preprocessing, and quality assessment[END_REF]. All the experiences are done with simulated acquisitions using our simulated forward model. The reference is not used in our reconstruction algorithm. We present in Fig. 2 a zoomed region of the outputs of the forward operator. The CFA patterns are noticeable, especially in the blue region.

Fig. 3 shows the proposed reconstruction along with the pansharpening method on the three CFAs.

Table 1 presents the mean and standard deviation of the mean squared error (MSE) of the reconstructions compared to the references, with and without an additive Gaussian noise of zero mean and standard deviation of 0.05. The pansharpening method has the advantage to be quick to implement and to perform, as the results are obtained in less than a second. However it has two main drawbacks. Firstly it is dependent from the setup as it cannot be used in with a CFA where we do not have a majority of pixels without filter. Moreover this algorithm is dependent on the quality of the interpolations inside the pansharpening process.

On the contrary the proposed method is flexible and running it for 400 iterations is only 20 seconds. It solves problems represented by eq. ( 2), and only the operator A has to be adjusted to reflect a change in the acquisition setup. Our proposed algorithm can hence be applied almost effortlessly on CFA patterns such as the ones presented in this work.

CONCLUSION

In this work, we proposed a flexible image formation model for cameras that implement CFAs with RGBW patterns. This framework can be easily adjusted to adapt to different patterns. We proved that our proposed inversion method, which employs a Chambolle-Pock solver with a TV prior, improves the quality of the image by a factor of 10 in comparison to traditional interpolation techniques.

Fig. 1 :

 1 Fig. 1: Bayer CFA and the three RGBW CFAs tested in the experimental section.

Fig. 2 :

 2 Fig. 2: Example acquisitions with RGBW SFAs cameras. Reference scene (top left), outputs: Sparse 3 (top right), Kodak (bottom left) and Sony (bottom right).

Fig. 3 :

 3 Fig. 3: Reconstruction results. Reference is on top, baseline method on first column, proposed on second column. 2nd line is Sparse 3, 3rd line is Kodak and 4th line is Sony.

Table 1 :

 1 The mean and standard deviation of the MSE of the two methods, with and without noise (best results in bold).

		Baseline [11]	Proposed
		Without noise With noise Without noise	With noise
	Sparse 3	1.35 × 10 -2 ±1.2 × 10 -2	1.84 × 10 -2 ±1.1 × 10 -2	1.95 × 10 -3 ±1.9 × 10 -3	3.05 × 10 -3 ±2.5 × 10 -3
	Kodak	1.29 × 10 -2 ±1.1 × 10 -2	1.47 × 10 -2 ±1.1 × 10 -2	2.63 × 10 -3 ±3.3 × 10 -3	2.11 × 10 -3 ±2.8 × 10 -3
	Sony	1.24 × 10 -2 ±1.1 × 10 -2	1.40 × 10 -2 ±1.1 × 10 -2	2.56 × 10 -3 ±3.2 × 10 -3	2.82 × 10 -3 ±2.6 × 10 -3