Alex Driedger

Jennifer Sletten
email: jennifer@anthinst.org

Claire Colegrove
email: claire@anthinst.org

Timothe Vincent
email: timothe@anthinst.org

Virgil Zetterlind
email: virgil@anthinst.org

Joachim Claudet
email: joachim.claudet@cnrs.fr

Barbara Horta E Costa

Guidance on marine protected area protection level assignments when faced with unknown regulatory information

Highlights

• Direct MPA regulations or management plans rarely address the complete set of fishing and activity restrictions in place • Regulation-based MPA evaluation systems can reliably identify fully and highly protected areas, even in the face of unknown information • Navigator's Level of Fishing Protection (LFP) scores can help guide assumptions about unknown information • The ability to assign protection levels (following the RBCS or The MPA guide in the near future) via existing MPA databases, allows for faster global assessments of protection

Introduction

The effectiveness of fully or highly protected marine protected areas (MPAs) as a tool to restore fish populations and ocean health has been well studied and documented [START_REF] Halpern | Marine reserves have rapid and lasting effects[END_REF][START_REF] Grorud-Colvert | Marine Protected Area Networks: Assessing Whether the Whole Is Greater than the Sum of Its Parts[END_REF][START_REF] Sala | Assessing real progress towards effective ocean protection. Assessing real progress towards effective ocean protection[END_REF][START_REF] Zupan | How good is your marine protected area at curbing threats?[END_REF]. Based on this evidence, numerous international goals have been set to promote the adoption of MPA networks in national waters. Although the CBD Aichi Target to implement 10% of effective MPAs by 2020 was not achieved in many regions, post-2020 targets are anticipated to increase ambition and set a target of 30% of effective protection by 2030 [START_REF]Convention on Biological Diversity[END_REF]. This new goal was proposed at the 2016 International Union for the Conservation of Nature (IUCN) World Conservation Congress [START_REF]WCC 2016 Res 050[END_REF] and includes the designation of 30% of global oceans in effective MPAs or Other Effective area-based Conservation Measures (OECMs) by 2030. In January 2021, the Biden administration committed the United States to this goal by executive order. Under the European Biodiversity Strategy and European Green Deal, Europe has already committed to protect 30% of its ocean, with 10% under strict protection [START_REF]Biodiversity strategy for 2030[END_REF].

In order for MPAs to realize their potential, MPAs need to implement strong regulatory protections [START_REF] Campbell | Weak compliance undermines the success of no-take zones in a large government-controlled marine protected area[END_REF][START_REF] Edgar | Global conservation outcomes depend on marine protected areas with five key features[END_REF][START_REF] Grorud-Colvert | The MPA Guide: A framework to achieve global goals for the ocean[END_REF][START_REF] Pieraccini | Beyond marine paper parks? Regulation theory to assess and address environmental non-compliance[END_REF][START_REF] Sletten | Beyond the boundaries: How regulation-centered marine protected area information improves ocean protection assessments[END_REF]. To assess MPA protection levels, the complexity of regulations in each MPA should filter through an evaluation system that uses and verifies the measures in place. One such system is the Regulation-Based Classification System (RBCS) [START_REF] Horta E Costa | A regulation-based classification system for Marine Protected Areas (MPAs)[END_REF], which scores MPAs based on the potential impact of the activities that are allowed within an MPA. The RBCS is part of the Blue Parks awards criteria and is now integrated into the protection levels assessment of the recently developed MPA Guide [START_REF] Grorud-Colvert | The MPA Guide: A framework to achieve global goals for the ocean[END_REF]. Recent endeavors used the RBCS to assess the protection levels different countries and regions were using to meet their international coverage commitments [START_REF] Horta E Costa | Categorizing ocean conservation targets to avoid a potential false sense of protection to society: Portugal as a case-study[END_REF][START_REF] Claudet | Underprotected Marine Protected Areas in a Global Biodiversity Hotspot[END_REF][START_REF] Claudet | Critical gaps in the protection of the second largest exclusive economic zone in the world[END_REF][START_REF] Roessger | Turning the tide on protection illusions: The underprotected MPAs of the 'OSPAR Regional Sea Convention[END_REF].

Standardized levels of protection, based on regulations, make estimating potential MPA effectiveness possible. Previous studies suggest the extent of human activities allowed within MPAs (i.e., protection levels) matters in achieving conservation benefits [START_REF] Lester | Biological effects within no-take marine reserves: a global synthesis[END_REF][START_REF] Sciberras | Evaluating the relative conservation value of fully and partially protected marine areas[END_REF][START_REF] Sala | No-take marine reserves are the most effective protected areas in the ocean[END_REF][START_REF] Zupan | Marine partially protected areas: drivers of ecological effectiveness[END_REF]. Fully, highly, and moderately protected areas (particularly if adjacent to fully protected areas), have been found to provide significant conservation benefits, from which socioeconomic benefits are derived [START_REF] Zupan | Marine partially protected areas: drivers of ecological effectiveness[END_REF][START_REF] Giakoumi | Ecological effects of full and partial protection in the crowded Mediterranean Sea: a regional meta-analysis[END_REF]. Yet, using protection levels as proxies for potential positive outcomes assumes MPAs are implemented and enforced in the water with enabling conditions being met [START_REF] Grorud-Colvert | The MPA Guide: A framework to achieve global goals for the ocean[END_REF]. Enabling conditions, such as enforcement, monitoring of results, long-term political commitment, sustainable financing, community participation, and benefits sharing, must be in place for a MPA to be effective [START_REF] Grorud-Colvert | The MPA Guide: A framework to achieve global goals for the ocean[END_REF]23,[START_REF] Bennett | Advancing Social Equity in and Through Marine Conservation[END_REF]25,26,[START_REF] Franco | Five key attributes can increase marine protected areas performance for small-scale fisheries management[END_REF]. Some of these features are not easily and objectively quantified (e.g., enforcement or compliance) or information is not readily available (e.g., funding), but indicators are needed to ensure MPA enabling conditions and MPA success (e.g., [START_REF] Cardoso-Andrade | Setting Performance Indicators for Coastal Marine Protected Areas: An Expert-Based Methodology[END_REF]).

In theory, MPA quality related to the level of protection can be obtained by documenting 'allowed' and 'prohibited' marine activities. Globally, this would lead to an overall picture of marine protections worldwide [START_REF] Sala | Assessing real progress towards effective ocean protection. Assessing real progress towards effective ocean protection[END_REF][START_REF] Zupan | How good is your marine protected area at curbing threats?[END_REF][START_REF] Horta E Costa | A regulation-based classification system for Marine Protected Areas (MPAs)[END_REF][START_REF] Horta E Costa | Categorizing ocean conservation targets to avoid a potential false sense of protection to society: Portugal as a case-study[END_REF][START_REF] Claudet | Underprotected Marine Protected Areas in a Global Biodiversity Hotspot[END_REF][START_REF] Claudet | Critical gaps in the protection of the second largest exclusive economic zone in the world[END_REF][START_REF] Zupan | Marine partially protected areas: drivers of ecological effectiveness[END_REF]. In practice, however, detailed regulatory information is not readily available for many MPAs, which can be a major roadblock in reaching that goal; often MPAs are only broadly referenced in bylaws or regional/national legislation [START_REF] Sletten | Beyond the boundaries: How regulation-centered marine protected area information improves ocean protection assessments[END_REF][START_REF] Jameson | The three screen doors: can marine "protected" areas be effective?[END_REF]. Researching all the regulatory information needed for a worldwide assessment of marine protections could be extremely time-consuming. Therefore, easier access to regulatory marine protections, as well as more robust methodologies to assess marine protections despite gaps in regulatory information could be helpful.

Here, the adequacy of estimating RBCS protection levels with incomplete input data was explored, by testing different assumptive methods. Californian MPAs and other Marine Managed Areas (MMAs) 1 in the ProtectedSeas' Navigator database [START_REF] Navigator -Protectedseas | ProtectedSeas[END_REF] were used as a case study; they are considered to be implemented and meeting the enabling conditions for success [START_REF] Gleason | Designing a network of marine protected areas in California: Achievements, costs, lessons learned, and challenges ahead[END_REF][START_REF] Hamilton | Incorporating biogeography into evaluations of the Channel Islands marine reserve network[END_REF][START_REF] Gleason | Science-based and stakeholder-driven marine protected area network planning: A successful case study from north central California[END_REF][START_REF] Kay | Collaborative assessment of California spiny lobster population and fishery responses to a marine reserve network[END_REF]35]. It was assessed whether the regulation-based metrics can be robust and reliable using only information readily available from MPA regulations and management plans (referred to in this study as 'direct regulations'). It was also assessed whether Navigator's Level of Fishing Protection (LFP) score [START_REF] Sletten | Beyond the boundaries: How regulation-centered marine protected area information improves ocean protection assessments[END_REF] can guide assumptions about unknown information to provide a reasonable first estimate of the RBCS protection levels when regulation information is incomplete in the management plans or equivalent legal texts. The use of such guidance in protection level estimations would enable MPA assessments to increase, allowing broad comparisons between MPAs, MPA networks, and other managed areas when the required information in bylaws or regional/national legislation is difficult to obtain or non-existent.

Methods

MPA and other MMA management plans rarely address the complete set of fishing and activity restrictions in place, as some activities are often managed through bylaws or regional or national regulations (referred to in this study as 'external regulations'). However, for an accurate assignment of a protection level based on the RBCS, detailed information on allowed activities (e.g., fishing, aquaculture, mining, etc.) is necessary. Therefore, gathering all information to fulfill a complete assessment via the RBCS can be onerous at a large scale or might have to ignore areas where information is not available.

Various world databases document MPAs (e.g., World Database on Protected Areas (WDPA), MPAtlas, MAPAMED, etc.), with regulatory information for some regions (see the MPAtlas with some MPAs classified following the RBCS such as the Blue Parks, and recently The MPA Guide). Further, to facilitate extensive and global assessments in a timely manner, Navigator [START_REF] Navigator -Protectedseas | ProtectedSeas[END_REF], launched in 2015, contains information on activities 'allowed' and 'prohibited' by regulation, as well as regulatory references for all areas in its database (i.e., MPAs and MMAs). Navigator focuses on activity restrictions on marine life extraction, i.e., mainly fishing [START_REF] Sletten | Beyond the boundaries: How regulation-centered marine protected area information improves ocean protection assessments[END_REF], but it distinguishes itself from other databases by summarizing direct regulations and documenting a standardized set of uses for every included area. Navigator's Level of Fishing Protection (LFP) score is assigned based on heuristic rules structured in a decision tree that do not require all individual uses to be known. A simplified system capturing regulations, such as LFP, may serve to guide decisions associated with a more complete system (such as the RBCS or The MPA Guide) when detailed information is not easy to obtain.

To explore the adequacy of estimating the RBCS protection levels with incomplete input data, California's network of MPAs and other MMAs (381 areas from Navigator, as of December 2021) was selected as a sample to test potential assumptions that could be made for unknown information globally. For these areas, all the regulations were reviewed through a resource intensive process (whether from direct or external regulations). This allowed gathering the full set of regulations required to follow the RBCS decision tree [START_REF] Horta E Costa | A regulation-based classification system for Marine Protected Areas (MPAs)[END_REF], and assigning those areas with a 'correct' protection level ('real' case or 'Full RBCS'). We worked at the lowest possible resolution, hence MPAs or zones within a MPA in the case of multi-zone MPAs. All of the areas or zones were treated as individual sites (such as in [START_REF] Claudet | Underprotected Marine Protected Areas in a Global Biodiversity Hotspot[END_REF][START_REF] Claudet | Critical gaps in the protection of the second largest exclusive economic zone in the world[END_REF]) in order to not mask full or high protection in weighted-averaging zone scores. Where no information could be found on fishing gears, anchoring, and/or boating occurring in the assessed region, we considered them not prohibited by regulation (i.e., 'allowed'), as suggested by Horta e Costa et al. [START_REF] Horta E Costa | A regulation-based classification system for Marine Protected Areas (MPAs)[END_REF]. For aquaculture, bottom structures including pipelines and artificial reefs (referred to in this study as 'bottom exploitation'), and mining/oil platforms/sand extraction/detonations (referred to in this study as 'bottom extraction'), data layers were retrieved from the California Department of Fish and Wildlife and the U.S. Bureau of Ocean Energy Management to determine their locations (for pipelines, artificial reefs, and drilling platforms) or active lease locations (for aquaculture 2 and oil/gas). Where an aquaculture, bottom exploitation, or bottom extraction location intersected an area's boundary, that activity was coded as 'allowed', otherwise it was coded as 'prohibited'. These Full RBCS protection levels constituted the baseline for this study.

Comparisons between the protection levels obtained with the full set of regulations (Full RBCS) and those obtained under different scenarios with less information -unknowns -and distinct assumptions allowed validating their performance. Two groups of scenarios were tested (Figure 1, Appendix Table 1): (A) one using only the direct regulations of MPAs/MMAs (i.e., management plans or equivalent) which often do not provide the full set of regulations required to assign the RBCS; and (B) one using only the information gathered in Navigator with more unknowns due to the reduced number of uses included in Navigator (RBCS input fields with corresponding Navigator fields were used with the remaining considered unknown, detailed in Appendix Table 4). The first group of scenarios (A) reflects what is often available to evaluators, so the aim was to understand if gathering only the information readily available in management plans and equivalents would significantly change the final RBCS protection levels when guided by assumptions. The second group of scenarios (B) is also relevant to test because there are already more than 20,000 areas in over 120 countries assessed in Navigator, including all American and European MPAs [36]. These areas could be classified following the RBCS or The MPA Guide if guidance on what to do with the unknowns is clarified scientifically and the performance is satisfactory. The type and number of unknowns were evaluated for both groups of scenarios, comparing the number of unknowns between each scenario and the Full RBCS, to understand common unknowns and their impact in final classifications.

Figure 1: Assumptive methods used in each scenario. See Appendix Table 1 for a full description of input information and assumptive methods for unknown fishing gears and nonfishing activities used in each scenario.

Within each group of scenarios (A and B), eight scenarios, named A1-A8 and B1-B8, with increasing complexity in assumptions pertaining to unknown fishing gears and non-fishing activities were tested. Simplest assumptive methods included assuming all unknown fishing gears and non-fishing activities 'allowed' or all 'prohibited', and more complex methods were tested to understand if the classification results could be improved. These included applying different assumptions to unknown non-fishing activities (informed assumptions), unknowns for different types of areas (LFP-guided assumptions), or a mix of approaches, with some assumptions being applied to unknown fishing gears and others to unknown non-fishing activities.

Informed assumptions, used in scenarios A5-A8 and B5-B8, were applied to unknown nonfishing activities. For aquaculture, bottom exploitation, and bottom extraction, given their environmental impact and weight in the RBCS and because they are commonly absent or incomplete in management plans and regulated by external regulations (e.g., specific bylaws with leases), assumptions related to these activities are important. In addition, because such activities in the United States are generally subject to extensive regulatory requirements such as environmental assessments, public comment periods, and permitting, unknowns in these fields were coded as 'prohibited'. Non-fishing activities that are ordinarily unrestricted unless explicitly stated otherwise, such as boating and anchoring, were coded 'allowed' if unknown.

For the LFP-guided assumptions, Navigator's LFP score was used as a basic/heuristic and first step assessment of an area's protection level based on general information of one of the RBCS dimensions (i.e., fishing). Two versions of the LFP-guided assumptions were tested. In the simple version, unknown fishing gears and non-fishing activities were considered 'prohibited' for those areas consistent with fully and highly protected areas and 'allowed' for the areas consistent with moderately, poorly, and unprotected areas. Simple LFP-guided assumptions for unknown non-fishing activities were used in scenarios A3 and B3 and for unknown fishing gears in scenarios A3, B3, A7, and B7. In the complex version, for unknown non-fishing activities: i) unknown bottom extraction was considered 'prohibited' for the areas consistent with fully, highly, and moderately protected areas and 'allowed' for the areas consistent with poorly and unprotected areas; ii) unknown bottom exploitation and aquaculture were considered 'prohibited' for the areas consistent with fully and highly protected areas and 'allowed' for the areas consistent with moderately, poorly, and unprotected areas, and iii) unknown boating and anchoring were considered 'allowed' for all areas. Unknown fishing gears were considered 'prohibited' for those areas consistent with fully and highly protected areas and 'allowed' for the areas consistent with moderately, poorly, and unprotected areas, except for fishing gears with an impact score of 9 (highest impact) (i.e., bottom towed fishing gears; see the table in [START_REF] Horta E Costa | A regulation-based classification system for Marine Protected Areas (MPAs)[END_REF] for RBCS fishing gear impact scores) in the moderately protected areas, in which case they were considered 'prohibited'. Complex LFP-guided assumptions for unknown non-fishing activities were used in scenarios A4 and B4 and for unknown fishing gears in scenarios A4, B4, A8, and B8.

To support using the LFP scores to guide decisions in the RBCS, a chi-square test of independence was performed between the LFP scores and RBCS protection levels, under the Full RBCS, for the 381 study areas. Results showed a significant and relatively strong association between the Full RBCS protection levels and LFP scores (χ² (16, n = 381) = 657.3, p < 0.001; Cramer's V = 0.66), suggesting LFP scores can help guide assumptions about unknown information. The distribution of the RBCS protection levels and LFP scores are plotted in Appendix Figure 1.

RBCS protection level estimates from each scenario were evaluated by computing the percentage of correctly classified areas with respect to the Full RBCS as well as false positive and negative rates. A false positive indicates an area where the RBCS protection level assigned by a scenario did not match that area's 'correct' RBCS protection level calculated by the Full RBCS. A false negative occurs when an area that scored a given RBCS protection level in the Full RBCS was assigned a different RBCS protection level in a scenario. In basic terms, the false positive rate indicates what percentage of areas assigned a given RBCS protection level are incorrect for that level, while the false negative rate indicates how many areas that should be a given RBCS protection level are 'missed' (i.e., assigned an incorrect protection level). The calculation of RBCS protection levels in each scenario was conducted in Python.

Results

The Full RBCS protection levels for California MPAs and other MMAs, calculated using all available information including activity restrictions from sources other than MPA management documentation, are reported in Figure 2 and Appendix Table 2, constituting the baseline for this study. The analysis of the unknowns (Appendix Table 3) revealed that some fishing gears were only known after searching external regulations (in the Full RBCS scenario). Most often these were drift nets, fish traps, fixed fish traps madragues, lines, longlines (bottom), longlines (pelagic), purse seining (bottom), purse seining (pelagic), spearfishing/diving, traps, and trawl pelagic. These gears were not regulated by direct regulations in 25 to 42% of the California areas. All fishing gears had unknowns in some MPAs when only searching for direct regulations -even those that are more destructive such as bottom trawls. Some gears such as trammel nets, gillnets, or dredges also display some unknowns even after researching the entire set of existing regulations (both direct and external) for an area or region. References to non-fishing activities were even less available in direct regulations with 45, 60, 97, and 98% of areas having no information about anchoring, aquaculture, bottom exploitation, and bottom extraction, respectively. When evaluating average unknowns across fishing gear and non-fishing activity fields by RBCS protection level, the percentage of unknowns increased as protection level decreased, except for unprotected areas which had somewhat less unknowns than poorly protected areas (Table 1). The scenarios within group A included all fields in the RBCS but only used information readily available in direct regulations. Both simple assumptive methods returned poor classification results (Table 2). When all unknowns were assumed 'allowed' (A1), fully to moderately protected areas from the Full RBCS shifted towards being classified as less protected, (Table 2, Appendix Table 5). When all unknowns were assumed 'prohibited' (A2), moderately to unprotected areas from the Full RBCS shifted to being classified as more protected (Table 2, Appendix Table 6). Both simple assumptions were not representative of 'reality'. Table 2 summarizes the assumptions and results for all scenarios in group A. To test the reasonableness of the informed non-fishing activity assumptions in scenarios A5-A8 and B5-B8, which treat unknown aquaculture/bottom exploitation/bottom extraction as 'prohibited' and boat/anchoring as 'allowed', the codings ('allowed' or 'prohibited') of each nonfishing activity for each individual area were compared to their respective codings in the Full RBCS. Over 90% of the informed non-fishing activity assumption codings for each activity matched their respective Full RBCS codings (Table 3), proving the assumptions appropriate for use.

Table 3. Informed non-fishing activity assumption coding match with Full RBCS coding.

Activity Match with Full RBCS (%)

Aquaculture 92

Bottom exploitation 93

Bottom extraction 92

Boating / anchoring 100

In the scenarios of group A that used the informed non-fishing assumptions (A5-A8), the first two options tested for the unknown fishing gears were all 'allowed' (A5) or all 'prohibited' (A6).

Strong matches were obtained for all categories in A5, with each protection level having 90% or higher correct classification compared to the Full RBCS -99% and 100% of correct classifications were obtained for fully and highly protected areas, respectively, with very low percentages of false negatives (Table 2, Appendix Table 9). This means that, in California, commercial non-fishing activities (e.g., aquaculture/bottom exploitation/bottom extraction) which are mostly regulated by external regulations, are not commonly allowed within MPAs/MMAs, and this is adequate to assume them as 'prohibited', while recreational non-fishing activities (e.g., boating/anchoring) are generally allowed so it is adequate to assume that as well.

Regarding fishing gears, this indicated that when not stated in direct regulations, fishing gears are typically allowed. Indeed, poor results were obtained when unknown fishing gears were considered 'prohibited' (A6), with moderately to unprotected areas from the Full RBCS being classified as more protected with high error rates (Table 2, Appendix Table 10).

For scenarios where assumptions were differentiated according to a previously calculated protection level (using LFP scores) (A3 and A4), good results were obtained when unknown activities were assumed 'prohibited' in areas consistent with fully and highly protected areas and 'allowed' in the others (A3; Table 2, Appendix Table 7). However, a slight increase in performance was obtained when considering unknown bottom extraction as 'prohibited' in areas consistent with moderately protected areas, boating/anchoring as 'allowed' in all areas, and unknown fishing gears as 'allowed' in areas consistent with moderately protected areas except for large-impacting fishing gears (i.e., bottom towed fishing gears) (A4; Table 2, Appendix Table 8). These two scenarios showed a high rate of correct classification for almost all protection levels, except for poorly protected areas, with fully, highly, and moderately protected areas showing 99, 96 or 97 and 100% correct classification, respectively. However, the overall performance was lower than scenario A5, with these scenarios (A3 and A4) showing a substantially larger percentage of false negatives for moderately and highly protected areas (in A3 only). Most false negatives corresponded to downgrades (i.e., assumed less protected) compared to the Full RBCS classification.

When mixing the informed non-fishing activity assumptions and the simple LFP-guided assumptions for fishing gears (A7) or the more complex LFP-guided assumptions for fishing gears (A8), the overall performance was better than in previous scenarios, particularly for moderately and poorly protected areas (Table 2, Appendix Tables 11 and12). This improvement seems related to the impacting non-fishing activities which, in the California region, are typically not allowed (according to these results). Hence, better performances are reached when these are considered 'prohibited' in all area types. However, mixed scenarios A7 and A8 resulted in very similar results, suggesting that the better performance in A4 compared to in A3 was likely related to fishing assumptions (prohibiting highly impacting fishing gears in the areas consistent with moderately protected areas in A4 but allowing them in those areas in A3).

When applying the same tests and scenarios of group A to group B, which have more unknowns due to fewer individual activity and gear categories in Navigator, similar poor (though poorer) results were obtained for the simple assumptions (B1 and B2; Table 4, Appendix Tables 12 and13). Table 4 summarizes the assumptions and results for all scenarios in group B. When LFP scores were used to guide assumptions for both the unknown fishing gears and nonfishing activities (B3), a good classification was found for fully and highly protected areas with 92% and 94% of correct classification, respectively, and low percentages of false negatives, compared to the Full RBCS (Table 4, Appendix Table 15). Considering fully and highly protected areas together as a group, correct classification rose to 97%. The percentage of false negatives was high in moderately and poorly protected areas though. The slight improvement in the performance of the less protected levels with the complex LFP-guided assumptions (B4) was due to the fact that if an area is consistent with a moderately protected area and bottom extraction is considered 'allowed' (like in B3), that area would be classified as poorly protected; whereas in B4, since bottom extraction is considered 'prohibited' in areas consistent with moderately protected areas, that area might keep the same class. This means that bottom extraction is likely not allowed in areas consistent with moderately protected areas.

The scenario mixing the informed assumptions for the non-fishing activities and the simple LFPguided assumptions for fishing gears (B7) performed the same as B3, indicating that LFPguided assumptions for non-fishing activities ('allowed' in areas consistent with moderately, poorly, and unprotected areas) or considering all these activities as 'prohibited' (informed), resulted in the same performance and classification (Table 4, Appendix Table 19). The same occurred in scenario B8 (Table 4, Appendix Table 20). This suggests that the non-fishing activities do not significantly impact final protection levels when fishing is weakly regulated (i.e., less fishing gear information is available). However, the large number of unknowns, particularly in the less protected areas, preclude a correct classification. Fully and highly protected areas more often explicitly prohibit the most damaging activities in their direct regulations, and thus are much less subject to assumptions.

Discussion

While it is known that protection levels, together with enforcement and other enabling conditions, matter for MPA effectiveness [START_REF] Grorud-Colvert | The MPA Guide: A framework to achieve global goals for the ocean[END_REF][START_REF] Lester | Biological effects within no-take marine reserves: a global synthesis[END_REF][START_REF] Sciberras | Evaluating the relative conservation value of fully and partially protected marine areas[END_REF][START_REF] Sala | No-take marine reserves are the most effective protected areas in the ocean[END_REF][START_REF] Zupan | Marine partially protected areas: drivers of ecological effectiveness[END_REF]23,[START_REF] Bennett | Advancing Social Equity in and Through Marine Conservation[END_REF]25,26,[START_REF] Franco | Five key attributes can increase marine protected areas performance for small-scale fisheries management[END_REF][START_REF] Cardoso-Andrade | Setting Performance Indicators for Coastal Marine Protected Areas: An Expert-Based Methodology[END_REF], this study shows the impact that the general lack of key regulatory information available in direct regulations can have on protection level assessments. In many of the California MPAs/MMAs, direct regulations did not specify whether the complete set of fishing gears required by the RBCS were 'allowed' or 'prohibited', sometimes not even addressing the more destructive gears such as bottom trawling. Non-fishing activities, such as aquaculture, bottom exploitation, and bottom extraction were rarely addressed in direct regulations. Such results suggest that these areas were not planned to manage non-fishing commercial activities or to work as a tool for wider marine spatial planning [START_REF] Frazão Santos | How sustainable is sustainable marine spatial planning? Part I-Linking the concepts[END_REF][START_REF] Frazão Santos | Major challenges in developing marine spatial planning[END_REF]. Further, besides the reduced quality of direct regulations, these results also expose the lack of external regulations, whether regional or state, about fishing gears with substantial impact in the seabed or in marine resources, such as trammel nets, gillnets, or dredges. According to these results, managers and users do not know if they can use these gears inside or in the region of particular MPAs/MMAs. This is occurring in a particularly well-regulated and managed region of the world that is committed to strong conservation efforts, including good examples of MPA implementation and management [START_REF] Gleason | Designing a network of marine protected areas in California: Achievements, costs, lessons learned, and challenges ahead[END_REF][START_REF] Hamilton | Incorporating biogeography into evaluations of the Channel Islands marine reserve network[END_REF][START_REF] Gleason | Science-based and stakeholder-driven marine protected area network planning: A successful case study from north central California[END_REF][START_REF] Kay | Collaborative assessment of California spiny lobster population and fishery responses to a marine reserve network[END_REF]35]. These findings highlight that direct regulations should be given further attention and importance while being planned and revised, to ensure that they mention all regulations that apply within the area, as good management plans are key for the success of MPAs [START_REF] Franco | Five key attributes can increase marine protected areas performance for small-scale fisheries management[END_REF].

By comparing scenario outcomes, guidance can be provided on how to classify MPAs according to their protection levels when complete information is not available in direct regulations. Since these are the mechanisms that should regulate MPAs and should contain the information about what can be done inside them [START_REF] Franco | Five key attributes can increase marine protected areas performance for small-scale fisheries management[END_REF], they represent the main and central resource evaluators have while assessing protection levels. Before this study, in the absence of sufficient direct regulations, each evaluator would either search for additional information, use local knowledge, use their expertise, or a combination of these when assessing whether fishing gears and nonfishing activities are 'prohibited' or 'allowed' in particular MPAs. However, at a global scale, such individual decisions could compromise comparisons between areas or countries. By testing assumptions and displaying their performance, guidance is offered for coherent and comparable assessments when direct information is incomplete and precludes correct classifications.

According to the results of this study, simple and homogeneous assumptions were not representative of reality.

The best assumptions to follow when using direct regulations (group A scenarios) are those using informed assumptions for non-fishing activities (A5) or those with LFP-guided assumptions for both non-fishing activities and fishing gears (A4) or a mix of both informed (for non-fishing activities) and LPF guided assumptions (for fishing gears) (A7 and A8). All these assumptive methods returned high percentages of correct classification and low percentages of false positives and false negatives. The fact that the informed assumptions for non-fishing activities (i.e., assuming impactful activities as 'prohibited') resulted in a good performance compared to the Full RBCS protection levels (in scenarios A5, A7 and A8), is likely due to these activities being mostly forbidden in the California MPAs/MMAs by external regulations. Since this might not be the case in numerous MPAs [START_REF] Rössger | Turning the tide on protection illusions: the underprotected MPAs of the 'OSPAR Regional Sea Convention[END_REF], the evaluator should select this option with caution and based on knowledge about the region being assessed. Even though these nonfishing activities are mostly prohibited, some cases existed where they were allowed in California MPAs/MMAs. Hence, the overall good performance following these informed assumptions also highlights the stronger role of fishing gears in the RBCS, compared with nonfishing activities. This suggests that fishing gears are very important to know for an accurate assessment and that is why the best assumptions have considered fishing gears carefully.

The best LFP-guided assumptions for fishing would be to assume that all the unknown fishing gears that are typically used in the assessed region are 'allowed' in areas consistent with poorly and unprotected protected areas, whereas unknown fishing gears that are low or medium impact (i.e., excluding bottom towed fishing gears) can also occur in areas consistent with moderately protected areas (such as in scenarios A4 and A8). Although this is not what is assumed when all unknown fishing gears are 'allowed', as in scenario A5 where the performance of assumptions was very good, it results in similar protection levels in the California MPAs/MMAs where areas consistent with fully and highly protected areas had clear direct fisheries regulations without unknowns and fishing gears were not considered 'allowed'. In conclusion, as general guidance, scenario A4 would be best suited for areas where no previous knowledge exists or where MPAs are known to have impacting activities occurring inside them. Scenario A8 would be best for areas where such activities are not typically occurring within MPAs. Both options have very good performance for fully and highly protected areas (both with 99% and 97% of correct assignment and 0% and 2% false negatives for fully and highly protected areas, respectively). Further, both options require a quick, previous assessment of general fishing restrictions to assess their levels of fishing protection (LFP) [START_REF] Sletten | Beyond the boundaries: How regulation-centered marine protected area information improves ocean protection assessments[END_REF], to proceed with the respective guidance.

The best assumptions to follow when using the data already gathered by Navigator (group B scenarios) are also those from scenarios B4 and B8, performing the same. The main difference between these two options is in considering the unknown non-fishing activities as all 'prohibited' (as in B8) or only 'allowed' in areas consistent with less protected areas (as in B4). The fact that they performed the same suggests that when there are several unknown fishing gears, the impact of this on the final classification outweighs the impact of using informed assumptions for non-fishing activities. To demonstrate, areas that had differing Aquaculture or Bottom Exploitation Index scores (step 3 of the RBCS, detailed in [START_REF] Horta E Costa | A regulation-based classification system for Marine Protected Areas (MPAs)[END_REF]) between scenarios B4 and B8 were all consistent with moderately, poorly, and unprotected areas. As such, fishing gear counts (step 1 of the RBCS, detailed in [START_REF] Horta E Costa | A regulation-based classification system for Marine Protected Areas (MPAs)[END_REF]) in these areas ended up being high since all unknown fishing gears were assumed 'allowed', and areas with a fishing gear count of >10 are automatically classified by the RBCS as poorly protected or unprotected without considering the impact of non-fishing activities. Both scenarios B4 and B8 offer good classification performance for fully protected areas (92% of correct assignment and 0% of false negatives) and relatively good performance for highly protected areas (94% of correct assignment but 29% of false negatives), suggesting this approach works, at least, to distinguish these two protection levels from the others. While this result is weaker than the scenarios using full direct regulations (group A), it shows that fully and highly protected areas could still be classified with a reasonably high level of accuracy using the existing data in Navigator.

Assessing existing ocean protection using standardized measures is critical given global goals to protect increasing percentages of ocean and increasing attention on the quality of operational implementations [START_REF] Agardy | Dangerous Targets' revisited: Old dangers in new contexts plague marine protected areas[END_REF][START_REF] Gurney | Biodiversity needs every tool in the box: use OECMs[END_REF][START_REF] Meehan | How far have we come? A review of MPA network performance indicators in reaching qualitative elements of Aichi Target 11[END_REF][START_REF] Wood | Assessing progress towards global marine protection targets: shortfalls in information and action[END_REF][START_REF] Woodley | A bold successor to Aichi Target 11[END_REF]. Studies show that fully and highly protected areas are more effective overall than moderately protected areas [START_REF] Zupan | How good is your marine protected area at curbing threats?[END_REF][START_REF] Sciberras | Evaluating the relative conservation value of fully and partially protected marine areas[END_REF][START_REF] Lester | Biological effects within no-take marine reserves: a global synthesis[END_REF][START_REF] Turnbull | Evaluating the social and ecological effectiveness of partially protected marine areas[END_REF], so being able to accurately classify fully and highly protected areas is of greatest importance, which scenarios A4, A8, B4, and B8 were found to accomplish. The ability to assign protection levels (following the RBCS or The MPA Guide in the near future) via existing MPA databases, allows for faster global assessments of protection. Using the Navigator database to assign RBCS protection levels is supported by this study if it can be done in tandem with data improvements by gathering further direct regulations.

This study provides validated guidance for decisions to make in the face of incomplete regulations. Due to the common lack of information on impacting activities, getting the best possible information for a given region and MPA is an important effort, but missing information should not preclude assigning protection levels to understand the potential quality of protection in place.

Figure 2 .

 2 Figure 2. Map of California MPAs and other MMAs, illustrating their size and Full RBCS protection level. Included are all areas that at least partially overlap California marine waters; each dot represents the centroid of the MPA/MMA boundary.

Figure 1 .

 1 Figure 1. California Areas -Distribution of Full RBCS protection levels and Navigator LFP scores, (n = 381).

Table 1 -

 1 Average percentage of unknown information, by RBCS protection level (n = 381)

		Fishing Gears		Non-fishing Activities
	Protection level	Full RBCS	All other scenarios	Full RBCS	All other scenarios
	(according to Full	(avg. % unknown)	(avg. % unknown)	(avg. % unknown)	(avg. % unknown)
	RBCS)
	Fully Protected	0	0	1.10	11.60
	Highly Protected	0.09	0.21	0.58	9.13
	Moderately	1.67	4.74	0.79	10.87
	Protected
	Poorly Protected	4.84	16.09	6.72	22.05
	Unprotected	3.39	11.71	2.10	8.66

Table 2 .

 2 RBCS classification results summary for all scenarios in group A.

	Scenario Name & Description	Classification Shift	Match with Full RBCS Correctly Classified (%)
	A1: using direct regulations w/ simple assumptions	Shift to less protected	Weak
	A2: using direct regulations w/ simple assumptions	Shift to more protected	Weak
	A3: using direct regulations w/ simple LFP-guided	Minor shift to less protected	Strong
	assumptions

A4: using direct regulations w/ complex LFP-guided assumptions Minor shift to less protected Strong A5: using direct regulations w/ informed non-fishing assumptions, simple fishing assumptions Minor shift to more protected Strong A6: using direct regulations w/ informed non-fishing assumptions, simple fishing assumptions Shift to more protected Weak A7: using direct regulations w/ informed non-fishing assumptions, simple LFP-guided fishing assumptions Minor shift to less protected Strong A8: using direct regulations w/ informed non-fishing assumptions, complex LFP-guided fishing assumptions Minor shift to less protected Strong

Table 4 .

 4 RBCS classification results summary for all scenarios in group B.

	Scenario Name & Description	Classification Shift	Match with Full RBCS	Correctly Classified
				(%)
	B1: using Navigator data w/ simple assumptions	Shift to less protected	Weak	26
	B2: using Navigator data w/ simple assumptions	Shift to more protected	Weak	44
	B3: using Navigator data w/ simple LFP-guided	General shift to less protected	Strong for Fully and	62
	assumptions		Highly Protected, Weak
			for Moderately, Poorly
			and Unprotected
	B4: using Navigator data w/ complex LFP-guided	General shift to less protected	Strong for Fully and	65
	assumptions		Highly Protected, Weak
			for Moderately, Poorly
			and Unprotected
	B5: using Navigator data w/ informed non-fishing	Shift to less protected	Weak	26
	assumptions, simple fishing assumptions
	B6: using Navigator data w/ informed non-fishing	Shift to more protected	Weak	44
	assumptions, simple fishing assumptions
	B7: using Navigator data w/ informed non-fishing	General shift to less protected	Strong for Fully and	62
	assumptions, simple LFP-guided fishing assumptions		Highly Protected, Weak
			for Moderately, Poorly
			and Unprotected
	B8: using Navigator data w/ informed non-fishing	General shift to less protected	Strong for Fully and
	assumptions, complex LFP-guided fishing		Highly Protected, Weak
	assumptions		for Moderately, Poorly
			and Unprotected

Table 2 .

 2 List of California areas with their classification results (Full RBCS).

	Area Name	Area (km 2 )	Number of fishing gears	Highest fishing gear score	Aquaculture or Bottom Exploitation Index	Recreational Access Index	MPA Index	RBCS Protection Level
	Pacific Grove Marine Gardens Fish Refuge	3.98	4	5	0	2	4
	James V Fitzgerald Marine Reserve	1.85	2	5	0	2	4
	Blue Cavern Offshore State Marine Conservation Area	19.93	3	5	0	2	4
	Point Conception to Gaviota Trawl Closure	86.40	19	9	0	2	7
	San Francisco Bay Sturgeon Management Area	162.96	21	9	0	2	8
	Special North Coast District Sturgeon Closure	1783.44	21	9	1	2	8
	Pillar Point Motorized Personal Watercraft Zone	2.30	19	9	0	2	7
	Santa Cruz Motorized Personal Watercraft Zone	16.78	19	9	0	2	7
	Moss Landing Motorized Personal Watercraft Zone	21.01	19	9	0	2	7
	Salmon Closure -Smith River Mouth	62.73	21	9	0	2	8
	Monterey Motorized Personal Watercraft Zone	16.64	19	9	0	2	7
	Pillar Point High Surf Warning Motorized Personal Watercraft Zone	0.72	19	9	0	2	7
	Los Angeles County White Croaker Closure	129.75	21	9	0	2	8
	Central California Gillnet Fishery Closure	8852.08	21	9	2	2	8
	South Caspar Point Sea Urchin closure	3.06	21	9	0	2	8
	Newport Bay Harbor entrance lobster trap closure	0.28	21	9	0	2	8
	Dana Point Harbor entrance lobster trap closure	0.56	21	9	0	2	8
	Oceanside Harbor entrance lobster trap closure	0.47	21	9	0	2	8
	Point Arguello Halibut Trawl Closure	37.98	19	9	0	2	7
	Point Conception Halibut Trawl Closure	32.65	19	9	0	2	7
	California State Waters	14855.32	20	9	2	2	7
	Salmon Closure -Klamath River Mouth	56.90	21	9	0	2	8
	Hueneme Canyon Halibut Trawl Closure	8.40	19	9	0	2	7
	Mugu Canyon Halibut Trawl Closure	18.21	19	9	0	2	7
	Point Conception to Gaviota Halibut Trawl Closure	86.40	19	9	0	2	7
	Santa Barbara to Pitas Point Halibut Trawl Closure	61.94	19	9	1	2	7
	Point Arguello to Point Conception and Hueneme Canyon to Point Mugu Halibut Trawl Closure	106.40	19	9	0	2	7
	Monterey Bay Halibut Trawl Closure	79.37	19	9	0	2	7
	Port San Luis Halibut Trawl Closure	163.19	19	9	0	2	7
	Commercial Rock Crab Closure	1630.27	21	9	0	2	8
	Crescent City Harbor	80.17	21	9	0	2	8
	San Francisco Bay	894.45	21	9	0	2	8
	Salmon Closure -Eel River Mouth	26.06	21	9	0	2	8

Table 3 .

 3 Percentage of areas with unknown information in Full RBCS vs. other scenarios, by activity (n = 381). *coded 'prohibited' if area not coastal

		Full RBCS	All other scenarios	Extra information from external regulations that reduced
		(% unknown)	(% unknown)	unknowns in Full RBCS
	Fishing Gears
	Beach seines*	26	26
	Cast nets*	28	28
	Dredges (bivalves)	29	29
	Drift nets	0	42	Allowed in both state and federal waters
	Fish traps	0	37	Allowed in both state and federal waters
	Fixed fish traps (madrague)	0	38	Allowed in both state and federal waters
	Gillnets	23	43	Prohibited in state waters
	Hand dredges (bivalves)*	22	22
	Intertidal hand captures*	27	27
	Lines (jigs/hook and line/rod/troll)	0	33	Allowed in both state and federal waters
	Longlines (bottom)	0	34	Prohibited in state and federal waters
	Longlines (pelagic)	0	34	Prohibited in state and federal waters
	Purse seining (bottom)	0	31	Allowed in both state and federal waters (although some
				species restrictions may apply)
	Purse seining (pelagic)	0	40	Allowed in both state and federal waters
	Spearfishing/diving	0	38	Allowed in both state and federal waters
	Surrounding nets near shore*	29	29
	Trammel nets	17	38	Prohibited in state waters

Table 4 .

 4 RBCS and Navigator data model comparison. (m) denotes a RBCS input field match with a Navigator field, while (+) denotes a RBCS input field without a match and is considered unknown in the group B scenarios.

	RBCS

Beyond MPAs, other MMAs in Navigator include areas managed for fisheries purposes, such as permanent gear closure areas, that regulate extraction of marine life; these are rarely addressed by global databases focusing on MPAs, but they can be similarly classified based on existing regulations.

Aquaculture data layers from the California Department of Fish and Wildlife only show relative positions and shapes of lease areas and do not provide definitive legal boundaries.

For the scenarios in group B, RBCS input fields with corresponding Navigator fields -marked (m = match) in Table4, representing 10 of the 26 total RBCS inputs -were used with the remaining considered unknown.

Appendix

Table 1. Summary of input information and assumptive methods for unknown fishing gears and non-fishing activities used in each scenario.

Scenario Group Scenario Names

Assumptions for unknown non-fishing activities Assumptions for unknown fishing gears Baseline -Direct and external regulations 1

Full RBCS None: Full information Simple: Assume Allowed for all (that are used in the region)


Bruno Cucco

Claudine Katan

Jacky Even

Mikael Kepenekian

George Volonakis
email: yorgos.volonakis@univ-rennes1.fr

Fine Structure of Excitons in Vacancy Ordered Halide Double Perovskites

Vacancy ordered halide double perovskites (VODP) have been widely explored throughout the past few years as promising lead-free alternatives for optoelectronic applications. Yet, the atomic-scale mechanisms that underlie their optical properties remain elusive. In this work, a throughout investigation of the excitonic properties of key members within the VODP family is presented. We employ state-of-art ab-initio calculations and unveil critical details regarding the role of electron-hole interactions in the electronic and optical properties of VODP.

The materials family is sampled by picking prototypes based on the electronic configuration of the tetravalent metal at the center of the octahedron. Hence, groups with a valence comprised of s, p and d closed-shells are represented by the known materials Cs 2 SnX 6 , Cs 2 TeX 6 and Cs 2 ZrX 6 (with X=Br, I), respectively. The electronic structure is investigated within the G 0 W 0 many-body green's function method, while the Bethe-Salpeter equation is solved to account for electron-hole interactions that play a crucial role in the optical properties of the family. A detailed symmetry analysis unravels the fine structure of excitons for all compounds. The exciton binding energy, excitonic wavefunctions and the dark-bright splitting are also reported for each material. It is shown that these quantities can be tuned over a wide range, form Wannier to Frenkel-type excitons, through for example substitutional engineering. In particular, Te-based materials, which share the electronic valency of cornersharing Pb halide perovskites, are predicted to have exciton binding energies of above 1 eV and a dark-bright splitting of the excitons reaching over 100 meV. Our findings provide a fundamental understanding of the optical properties of the entire family of VODP materials and highlight how these are not in fact suitable Pb-free alternatives to traditional halide perovskites.

Perovskites are some of the most common naturally occurring compounds in nature. These AMX 3 materials are made of three-dimensional networks of corner-sharing octahedra, and have been employed in a broad range of applications, like solar cells [START_REF] Stranks | Metal-halide perovskites for photovoltaic and light-emitting devices[END_REF] , highly-efficient light-emitters 2 , memories [START_REF] Waser | Redox-based resistive switching memoriesnanoionic mechanisms, prospects, and challenges[END_REF] , photo-catalysts [START_REF] Zhang | Stable and highly efficient photocatalysis with lead-free double-perovskite of cs 2 agbibr 6[END_REF] , transistors [START_REF] Xing | Long-range balanced electron-and hole-transport lengths in organic-inorganic ch 3 nh 3 pbi 3[END_REF] and superconductors [START_REF] Reyren | Superconducting interfaces between insulating oxides[END_REF] . A 2 MX 6 vacancy ordered double perovskites (VODP), are a particular sub-family of halide perovskites, which have recently emerged as promising materials for a broad range of applications [START_REF] Maughan | Perspectives and design principles of vacancy-ordered double perovskite halide semiconductors[END_REF][START_REF] Ju | Earth-abundant nontoxic titanium(iv)-based vacancy-ordered double perovskite halides with tunable 1.0 to 1.8 ev bandgaps for photovoltaic applications[END_REF] . In fact, the oxidation of Sn +2 in its tetravalent state Sn +4 leads to a structural transformation from the CsSnI 3 perovskite to the more stable VODP Cs 2 SnI 6 [START_REF] Qiu | From unstable cssni 3 to air-stable cs 2 sni 6 : A lead-free perovskite solar cell light absorber with bandgap of 1.48 ev and high absorption coefficient[END_REF] . To-date, the successful synthesis of many VODP materials have been reported, with tetravalent M-site atoms like Te 10 , Ti [START_REF] Chen | Cesium titanium(iv) bromide thin films based stable lead-free perovskite solar cells[END_REF] , Zr [START_REF] Abfalterer | Colloidal synthesis and optical properties of perovskite-inspired cesium zirconium halide nanocrystals[END_REF] , Hf [START_REF] Liu | Colloidal synthesis and tunable multicolor emission of vacancy-ordered cs 2 hfcl 6 perovskite nanocrystals[END_REF] , Pd [START_REF] Sakai | Solution-processed cesium hexabromopalladate(iv), cs 2 pdbr 6 , for optoelectronic applications[END_REF] , Pt [START_REF] Evans | Hydrogen bonding controls the structural evolution in perovskite-related hybrid platinum(iv) iodides[END_REF] , Se [START_REF] Wernicke | Low temperature luminescence spectra of the d 10 s 2 complexes cs 2 mx 6 (m = se, te and x = cl, br). the jahn-teller effect in the γ -4 ( 3 t 1u ) excited state[END_REF] , Sb [START_REF] Day | Spectra and constitution of antimony(iii) antimony(v) hexahalide salts and related compounds[END_REF] , Sn [START_REF] Lee | Air-stable molecular semiconducting iodosalts for solar cell applications: Cs 2 sni 6 as a hole conductor[END_REF] and Ge [START_REF] Laubengayer | Chlorogermanic acid and the chlorogermanates. properties and crystal structure of cesium hexachlorogermanate[END_REF] . Most of these materials are stable and exhibit tunable electronic and optical properties making them targets of choice for optoelectronic applications, such as solar cells [START_REF] Chen | Cesium titanium(iv) bromide thin films based stable lead-free perovskite solar cells[END_REF][START_REF] Lee | Air-stable molecular semiconducting iodosalts for solar cell applications: Cs 2 sni 6 as a hole conductor[END_REF] and light-emitting devices [START_REF] Abfalterer | Colloidal synthesis and optical properties of perovskite-inspired cesium zirconium halide nanocrystals[END_REF][START_REF] Zhang | Highly stable vacancy-ordered double perovskite rb 2 zrcl 6 with broadband emission for down-conversion white light-emitting diodes[END_REF][START_REF] Holzapfel | Cs 4 cd 1-x mn x bi 2 cl 12 -a vacancy-ordered halide perovskite phosphor with high-efficiency orange-red emission[END_REF][START_REF] Sun | Efficient near-infrared luminescence in lanthanide-doped vacancy-ordered double perovskite cs 2 zrcl 6 phosphors via te 4+ sensitization[END_REF] . Among these properties, there is growing interest for the potential presence of strong excitonic effects in VODP [START_REF] Hoye | The role of dimensionality on the optoelectronic properties of oxide and halide perovskites, and their halide derivatives[END_REF][START_REF] Rieger | Excitons and narrow bands determine the optical properties of cesium bismuth halides[END_REF] , as the materials are made of isolated metal-halide octahedra, which are intrinsically structurally confined [START_REF] Cucco | Electronic structure and stability of cs 2 tix 6 and cs 2 zrx 6 (x = br, i) vacancy ordered double perovskites[END_REF] . Such excitonic properties are critical for the potential application of these photo-active materials, as for example they can limit charge carrier separation, or enhance photo-emission, hence open pathways for new optoelectronic devices that take advantage of such exciton physics. A deep understanding of the excitonic processes is necessary for probing the origins of the tunable and broad photoluminescence that is observed in VODP [START_REF] Liu | Colloidal synthesis and tunable multicolor emission of vacancy-ordered cs 2 hfcl 6 perovskite nanocrystals[END_REF][START_REF] Chen | White-light emission lead-free perovskite phosphor cs 2 zrcl 6 :sb 3+[END_REF] , which is commonly associated to excitons trapping. In fact, the formation of self-trapped excitons due to strong lattice distortions is one of the most discussed and debated underlying mechanisms [START_REF] Liu | Colloidal synthesis and tunable multicolor emission of vacancy-ordered cs 2 hfcl 6 perovskite nanocrystals[END_REF][START_REF] Zhang | Highly stable vacancy-ordered double perovskite rb 2 zrcl 6 with broadband emission for down-conversion white light-emitting diodes[END_REF][START_REF] Zeng | Boosting triplet self-trapped exciton emission in te (iv)-doped cs 2 sncl 6 perovskite variants[END_REF][START_REF] Zheng | Vacancy-ordered double perovskite rb 2 zrcl 6-x br x : Facile synthesis and insight into efficient intrinsic self-trapped emission[END_REF] . To-date, a detailed description of even the most fundamental excitonic features of the VODP is still missing.

In this letter, we analyze the role of the electronic structure including electron-hole (e-h) interactions in the VODP materials by employing state-of-art ab-initio G 0 W 0 and Bethe-Salpeter calculations. We unveil key details of their electronic band structure and show the impact of e-h interactions on their optical properties. Prototypical VODP materials, Cs 2 SnX 6 , Cs 2 TeX 6 and Cs 2 ZrX 6 (with X=Br, I), are categorised based on the electronic configuration of the tetravalent atom at the M-site with a valency comprised of an s, p and d closed-shell, respectively. [START_REF] Maughan | Defect tolerance to intolerance in the vacancy-ordered double perovskite semiconductors cs 2 sni 6 and cs 2 tei 6[END_REF][START_REF] Abfalterer | Colloidal synthesis and optical properties of perovskite-inspired cesium zirconium halide nanocrystals[END_REF][START_REF] Lee | Air-stable molecular semiconducting iodosalts for solar cell applications: Cs 2 sni 6 as a hole conductor[END_REF] .

The halogen atoms are identified as a tuning parameter to control the empty space in between the isolated metal-halide octahedra. We further show the effects of both M-and X-site substitutions on the electronic and optical properties by performing a symmetry analysis of the electronic structures. We find the most critical quantities of the photo-active VODP materials, such as the charge carrier effective masses, exciton binding energies and dark-bright exciton splitting and establish trends which allows the enhancement of these parameters via substitutional engineering. Te-based materials exhibit promising hole transport properties, while Sn-based VODP are most favoring electron transport. In addition, we show that Te-based materials on one hand exhibit promising electronic structure with an active lone pair, similar to Pb halide perovskites, on the other exhibit drastically different excitonic features, which can be explained due to strong electron-hole interactions. Thus, we demonstrate that VODP even when sharing the same electron configuration with Pb halide perovskites, cannot be considered as their Pb-free alternatives. Finally, the absorption onset of all materials are shown to be dominated by electron-hole interactions leading to excitons ranging from Wannier-type to Frenkel-type. In all cases, the investigation of the exciton fine structure reveals that a non-optically active dark state lies below the first bright state. The electron-hole exchange interaction at the origin of the splitting between dark and bright exciton states is particularly strong in the case of Te-based materials. Interestingly, by choosing a metal and a halide, one can browse a range of exciton binding energies greater than 1 eV and of dark-bright splitting greater than 150 meV.

VODP typically crystallize in the Fm 3m face-centered cubic lattice (space group number 225).

The lattice corresponds to a rock salt arrangement of corner-sharing MX 6 and ∆X 6 octahedra with ∆ being the vacancy site, or simply a double perovskite for which one of the two M-sites is vacant, as show on Figure 1a. We sample the VODP family by selecting prototypes based [START_REF] Jong | Anharmonic phonons and phase transitions in the vacancy-ordered double perovskite cs 2 sni 6 from first-principles predictions[END_REF] . As previously shown in the context of d 0 VODP, halogen-halogen p-orbital interactions dictate the empty space formed by the vacancies in these materials [START_REF] Cucco | Electronic structure and stability of cs 2 tix 6 and cs 2 zrx 6 (x = br, i) vacancy ordered double perovskites[END_REF] . Here the same is observed for all VODP regardless of the atom at the M-site, and the volume of the empty space in between the MX 6 octahedra depends solely on the halogen size.

Hence, Br to I substitution can be used as a tuning parameter to control the size of the vacancy from 33 to 40 Å 3 , as defined in Table S1. As a final assessment of the mechanical stability of the optimized structures, we calculate the phonon dispersion for each material. These are shown in Figure S1, where the absence of phonon imaginary modes in all studied compounds indicates mechanical stability. The redshift of the frequency that is observed when going from bromides to iodides is consistent with the increase of halogen's atomic mass and thus damping of the M-X oscillation modes.

Two critical parameters for the accurate calculation of excitonic properties are the band-gaps and charge-carrier effective masses, as both can have strong impacts on the exciton binding energies and the relative position of excitonic peaks. To this aim, we employ G 0 W 0 calculations that take into account quasi-particle effects due to screened electron-electron interactions. Figure 2a shows the electronic band structures for the bromides, while the electronic bands of the iodides can be found in Figure S2a note that within DFT-PBE the indirect band-gap of Cs 2 TeI 6 is incorrectly predicted to be located between Γ-L (see Figure S3). Within G 0 W 0 Te-based compounds exhibit an indirect band-gap between L-W, which we also verified by employing PBE0 hybrid functional. Based on symmetry analysis we identify the first direct dipole allowed transition as being from the VBM (E 1/2,g ) to the CBM (E 1/2,u ) at L for Cs 2 TeX 6 . Within G 0 W 0 these transitions are of 2.83 eV and 3.90 eV for Cs 2 TeI 6 and Cs 2 TeBr 6 , respectively. For the Zr-based we identify the first direct dipole allowed transition as being from the VBM-1 (F 3/2,u ) to CBM (F 3/2,g ). Within G 0 W 0 these transitions are of 5.18 eV and 3.64 eV for Cs 2 ZrBr 6 and Cs 2 ZrI 6 , respectively.

The electronic charge density at the VBM and CBM, are shown in Figure 2b and Figure S2b for bromides and iodides, respectively. The VBM for the Sn-based and Zr-based materials is made of halogen p-orbitals, and there is no contribution of the M-site atoms. This is not the case for Tebased compounds where the halogen p-orbitals interact with the Te s-orbitals forming a VBM that resembles in character traditional Pb-based halide perovskites [START_REF] Volonakis | Route to stable lead-free double perovskites with the electronic structure of ch 3 nh 3 pbi 3 : A case for mixed-cation [cs/ch 3 nh 3 /ch(nh 2 ) 2 ] 2 InBiBr 6[END_REF] . This is of potential interest for the hole transport within this materials, as only the Te-based compounds have a uniform density that expands in three-dimensions at the VBM. The CBM for Sn-based materials is made of Sn s-orbitals hybridized with halogen p-orbitals, hence is similar to the VBM of Te-based materials.

The CBM of Te-based materials is made of Te p-orbitals and halogen p-orbitals, while for the Zr-based materials we find Zr d-orbitals and halogen p-orbitals. In terms of electron transport Snbased and Te-based materials are the most promising as the sand p-orbitals of the M-site atoms are involved in the CBM. Overall, the contribution of the atoms at the M-sites to the band edges is in perfect agreement with their electronic valency at their nominal oxidation state, which was described in the s 0 , p 0 and d 0 families of Figure 1b. Table S2 contains the electronic band-gaps within DFT and G 0 W 0 . To quantify the band dispersion we calculate the charge carrier effective masses at the band edges within G 0 W 0 , as summarized in Table 1. The Sn materials exhibit the lightest and isotropic electron effective masses, which are consistent with the M-site s-orbital character involved in their electronic structure. Te compounds exhibit particularly light masses for both holes and electrons, as these involve M-site s-orbitals and p-orbitals at the valence and conduction band, respectively.

These values confirm the potential good electron carrier transport properties of the Sn-based and Te-based VODP, while for hole transport only Te-based materials would not be limited by heavy masses at the VBM. In fact, both Sn-based and Zr-based materials exhibit light and heavy holes at the VBM, which would challenge their potential performance.

We move to evaluate the materials optical properties by employing the ab-initio Bethe-Salpeter formalism [START_REF] Salpeter | A relativistic equation for bound-state problems[END_REF] starting from our G 0 W 0 calculations (BSE@G 0 W 0 ), which allow taking into account the band-gap renormalization and dispersion corrections over standard DFT. The BSE@G 0 W 0 approach allows including e-h coupling interactions, thus the investigation of bounded excitonic states. Figure 3 shows the absorption coefficients obtained (i) at the BSE@G 0 W 0 , and (ii) without e-h interactions at the independent particle (IP) at the G 0 W 0 level. Due to the inclusion of e-h coupling, all the BSE spectra red-shift and the oscillator strengths are redistributed when compared to the IP calculations. For all materials we find a bright exciton state well within the band-gap, and dark states (not optically active) at lower energies. Furthermore, there is clearly a very strong modification to the spectra when e-h interaction included, which is consistent with the isolated octahedra that make the VODP lattice. To further analyse the optical spectra, we first look at the optical band-gaps. Figure 4a, shows the first dipole-allowed transitions (optical band-gap) with and without e-h interactions, and the experimental data. BSE values are in excellent agreement with available measurements, which we extract by picking the first excitonic peak in the absorption of references 12, 32-34.

Next, we define the exciton binding energy (E b BSE ) as the energy difference between the first allowed transition and the bright exciton peak. As shown in Figure 4b, we report large binding energies ranging from 190 meV to 540 meV for iodine-based and 510 meV to 1240 meV for bromine-based compounds. These values are particularly large as can be expected from the quasi-0D nature of the weekly interacting isolated octahedra [START_REF] Maughan | Perspectives and design principles of vacancy-ordered double perovskite halide semiconductors[END_REF][START_REF] Cucco | Electronic structure and stability of cs 2 tix 6 and cs 2 zrx 6 (x = br, i) vacancy ordered double perovskites[END_REF] . Furthermore, the exciton binding en- ergy is systematically larger for bromide compounds than their iodide counterparts. This effect is related to the connection between the electronic band-gap and the dielectric constant of these materials, as the electron polarizability decreases as the band-gap increases. Hence, the screening of the Coulomb interactions between electrons and holes becomes less efficient leading to tightly bounded excitonic states [START_REF] Marongiu | The role of excitons in 3d and 2d lead halide perovskites[END_REF] . This is consistent with the literature for excitons in AMX 3 halide perovskites [START_REF] Marongiu | The role of excitons in 3d and 2d lead halide perovskites[END_REF][START_REF] Galkowski | Determination of the exciton binding energy and effective masses for methylammonium and formamidinium lead tri-halide perovskite semiconductors[END_REF][START_REF] Jiang | Scaling universality between band gap and exciton binding energy of two-dimensional semiconductors[END_REF] and also A 2 MM'X 6 halide double perovskites [START_REF] Palummo | Optical properties of lead-free double perovskites by ab initio excited-state methods[END_REF][START_REF] Biega | Chemically localized resonant excitons in silver-pnictogen halide double perovskites[END_REF] . A surprising finding is the very large E b BSE for the Te-based materials, which share common optical features to Pb halide perovskites, such as strong optical transitions that involve s (i.e. active lone-pair) to p M-site orbitals.

Yet, Pb halide perovskites exhibit low exciton binding energy. [START_REF] Marongiu | The role of excitons in 3d and 2d lead halide perovskites[END_REF][START_REF] Rana | Temperaturedependent electroabsorption and electrophotoluminescence and exciton binding energy in mapbbr3 perovskite quantum dots[END_REF][START_REF] Miyata | Direct measurement of the exciton binding energy and effective masses for charge carriers in organic-inorganic tri-halide perovskites[END_REF] Such strong e-h interactions observed in the case of the Te-based materials, can be attributed to their intrinsic structural confinement and their indirect electronic structure, that give rise to Frenkel-like excitons. Hence, despite their similar valency Te-based VODP are far from being considered as alternatives to Pb-based halide perovskites.

The exciton wavefunction for the first bright exciton is visualised in Figure 5a, as a spatial distribution given by |Ψ(r h , r e )| 2 . The exciton wavefunction is more localized in the case of Te-based VODP, while Cs 2 SnBr 6 exhibits a significantly delocalized excitonic wave-function which extends over several unit-cells of the lattice. This effect is even stronger for the case of iodides shown in Figure S4, in agreement with the calculated exciton binding energies. In fact, the exciton in Cs 2 SnI 6 is closer to the Wannier-type, which is also confirmed by its E b BSE of 190 meV sizeably smaller than for the other compounds. On the other hand, Cs 2 TeX 6 and Cs 2 ZrX 6 compounds are closer to a Frenkel-type description, exhibiting stronger exciton localization. Over the series of closed-shell VODP, there is an interplay between Wannier and Frenkel-type excitons that is related to the atomic orbital at the M-site, as a clear Wannier-like exciton is formed for Cs 2 SnI 6 and a Frenkel-type for Cs 2 TeBr 6 . This trend can be followed across the series as shown with the projection of the exciton weights |A λ eh | over the band-structure, where |λ = ∑ eh A λ eh |eh shown in Figure S5. The exciton weights in Sn-based materials are close to the band edges (delocalised excitons), less so for the case of Zr and Te iodides (localised), while for Zr and Te bromides these expand over the entire Brillouin zone (strongly localised). The materials order from Wannier-like to Frenkel-type excitons perfectly follows the E b BSE trend. In addition, we compare our ab-initio results with a semi-empirical Wannier-Mott model (see SI for details), and we show that the model fails to qualitatively assess the relationship between the exciton binding energies for all cases but the Sn iodide material.

Finally, we investigate the detailed symmetry and fine structure of the excitons. As shown in Figure S5, the bright exciton of Cs 2 SnBr 6 is formed mainly by a localized transition around Γ from the VBM-1 to CBM, and a small contribution from the optically forbidden transition from VBM to CBM, while for Cs 2 SnI 6 this contribution from the VBM is not observed. The symmetry of the exciton is given by a direct product between the irreducible representations of the corresponding electron-hole pairs, i.e. F 3/2,u ⊗ E 1/2,g = E u ⊕ T 1u ⊕ T 2u multiplied by a s-like envelope function (i.e. for s-like excitons, and hence doesn't change the symmetry). The bright exciton of Cs 2 TeX 6 are mainly composed by transitions VBM to CBM at L but with smaller contributions around the W high-symmetry point. Symmetry analysis at the L point gives

E 1/2,g ⊗ E 1/2,u = E u ⊕ A 2u ⊕ A 1u .
Finally, for Cs 2 ZrX 6 the exciton is formed by transitions from VBM to CBM at the Γ point, giving a symmetry of

F 3/2,u ⊗ F 3/2,g = A 1u ⊕ E u ⊕ 2T 1u ⊕ 2T 2u ⊕ A 2u .
We also observe contributions from the transition VBM to CBM+1 for the bromide compound, due to the CBM and CBM+1 be closer in energy when compared to iodides.

The bright exciton states are in fact formed by electrons and holes with anti-parallel spins and can recombine easily thought the emission of a photon, and thus exhibit fast lifetimes [START_REF] Zhang | Magnetic brightening and control of dark excitons in monolayer wse 2[END_REF] . Yet, a bounded excitonic state with parallel spins can also exist and due to spin momentum conservation these cannot recombine via the direct emission of a photon and therefore exhibit a larger radiative lifetime when compared to bright excitons. These so called dark excitons are not intrinsically optically active, i.e. exhibit zero oscillator strength, and arise from the electron-hole exchange interaction. These states and their energy split from the bright states (∆ db ), are of particular interest for state-of-art technologies such as optically controlled information processing and fast/slow light applications. [START_REF] Zhang | Magnetic brightening and control of dark excitons in monolayer wse 2[END_REF] Figure 4b and Table S4 summarizes the obtained ∆ db values for all compounds within this work, while Figure 5b presents the detailed exciton fine structure for the Sn-and Tebased compounds. From the symmetry and oscillator strength analysis of Cs 2 SnX 6 , we find an E u doublet dark exciton state, which is split from the bright triplet peak T 1u by 6 meV and 4 meV for bromide and iodide, respectively. The splitting between the bright peak T 1u and the optically inactive triplet state T 2u is also observed. For the Te-based compounds, we find an A 1u dark singlet exciton which is split from the bright peak comprised of the optically active doublet E u and singlet A 2u . The dark-bright splitting are of 155 meV and 71 meV for bromide and iodide Te-based compounds. The energy difference between the E u and A 2u is below the resolution of our calculation and as so, are treated as practically a triply degenerated state. For the fine structure of Zr-based compounds, due to the large number of energetically close electronic states near the VBM a strong coupling of the valence bands and also a large number of transition states are expected. This leads to several oscillator strengths in the vicinity of the bright excitonic peak, which do not allow the proper identification of the sixteen A 1u ⊕ E u ⊕ 2T 1u ⊕ 2T 2u ⊕ A 2u states, as shown in Figure S6.

Yet, by analysis of the e-h pairs contributions of Figure S5 we identify the dark state as being originated from a VBM to CBM transition at the Γ point. The dark-bright splitting for these are 34 meV and 24 meV for bromide and iodide, respectively. ∆ db decreases from Te, Zr to Sn, with Te-based compounds exhibiting the larger dark-bright splitting when compounds with the same halogen are compared. Moreover, bromine-based compounds are shown to exhibit a larger splitting than iodine-based ones, especially in the case of Cs 2 TeX 6 . To further explore this, we solve the BSE without including the e-h exchange interaction V vck v c k (see SI). Figure S7 shows that the spectrum renormalization caused by the e-h exchange interaction is always greater for bromides than for iodides. Interestingly, this effect has also been observed for CsPbX 3 halide perovskites nanocrystals [START_REF] Becker | Bright triplet excitons in caesium lead halide perovskites[END_REF] , for which the larger exciton binding energy of bromides is attributed to the larger e-h exchange interactions. The effect of the interaction appears particularly strong in the case of Te compounds. In the case of VODP, both metal and halogen substitution allow tuning ∆ db within a considerable range from 4 meV to 155 meV.

In conclusion, we investigated the structural and optical properties of the VODP family of materials. We show that the volume of the vacant site can be tuned solely by the halogen composition.

Based on their electronic structure we find Te-based compounds to be most interesting for hole transport, while Sn-based compounds are suitable for electron transport. As expected for structurally confined materials, it is shown that electron-hole interactions play a major role on the optical properties of these compounds. Indeed, they exhibit large and tunable exciton binding energies through a range of 0.2-1.2 eV by both M-and X-site composition, with bromides exhibiting larger binding energies than iodides. By employing symmetry analysis of the electronic band structures, we elucidate the bright-exciton symmetries and fine-structure as to obtain the dark-bright exchange splitting and show its tunability across the family. The electron-hole exchange interaction is shown to have major contribution on the dark-bright splitting, in particular for Te-based compounds. Finally, by a joint analysis of the exciton wavefunctions and binding energies we conclude that the VODP family can host both Wannier-and Frenkel-type excitons depending on the choice of Mand X-site species. Our findings provide fundamental understanding of the optical properties of the prominent VODP family of materials. It exhibits intrinsically different excitonic properties with respect to APbX 3 materials, even when sharing electronic configuration. Consequently, due to the large and/or indirect electronic band gaps, heavy effective masses of the charge carriers, and the large exciton binding energies these materials cannot be the base of alternative Pb-free photovoltaic devices, though they present great opportunity for white-light emission applications and quantum optical technologies.

Figure 1 :

 1 Figure 1: Vacancy ordered halide double perovskite family. Representation of the VODP family of materials and chosen M 4+ metal sites. The boxes mark the s 0 , p 0 and d 0 columns for a +4 oxidation state and electronic configuration of selected atoms. The atomic species in green have been synthesized as VODP with the marked atoms at the M 4+ -site.

  of the SI. Sn-based compounds exhibit direct band-gaps at the Γ point. The optical transition from the valence band maximum (VBM) to conduction band minimum (CBM) is given by a transition from the irreducible representation F 3/2,g to E 1/2,g which is parity forbidden.The first direct dipole allowed transition for the Sn-based compounds is from VBM-1 (F 3/2,u ) to CBM (E 1/2,g ), i.e. 300 meV and 350 meV larger than the fundamental band-gap of Cs 2 SnBr 6 and Cs 2 SnI 6 , respectively. Due to the different orbitals involved at the band edges, Te-based compounds have indirect band-gaps between the L-W high symmetry points. Interestingly, the shapes of the Cs 2 TeX 6 valence band and the Cs 2 SnX 6 conduction band are the same, which is consistent with the electronic configurations [Kr]5s 2 and [Kr]5s 0 for Te 4+ and Sn 4+ , respectively. We

Figure 2 :

 2 Figure 2: G 0 W 0 electronic band structures and charge density. (a) G 0 W 0 electronic band structures and charge densities. The irreducible representations and point group symmetries are shown and the valence band maximum (VBM) and conduction band minimum (CBM) are marked with blue and orange circles, respectively. The grey and red arrows represents the fundamental band-gap and first direct allowed transition, respectively. (b) Charge densities calculated at the VBM and CBM for Cs 2 MBr 6 (M=Sn, Te, Zr).

Figure 3 :

 3 Figure 3: Optical absorption. Optical absorption coefficient of the Cs 2 MX 6 (M=Sn, Te, Zr and X=Br, I) vacancy ordered perovskite family within BSE@G 0 W 0 and IP@G 0 W 0 level. The red and blue curves show the BSE (with electron-hole interactions) and IP (without electron-hole interactions) coefficients, respectively. The positions of the dark (D) and bright (B) exciton states are shown. The green dotted lines show the position of the first direct dipole-allowed transition E d g at G 0 W 0 level.

Figure 4 :

 4 Figure 4: Band-gaps, exciton binding energy, and dark-bright splitting: (a) Comparison between the first dipole-allowed transition at G 0 W 0 level, first bright peak of BSE calculation, and experimentally measured optical band-gaps. (b) Binding energies E b and dark-bright splitting ∆ db trend obtained from the BSE calculations. Blue and green curves represent bromides and iodides, respectively.

Figure 5 :

 5 Figure 5: Exciton spatial distribution and fine structure. (a) Exciton spatial distribution |Ψ(r h , r e )| 2 for the bright exciton. The hole position r h is fixed in the vicinity of the halogen atoms. The isovalue is set to represent 85% of the total distribution. (b) Calculated exciton oscillator strength with fine structure shown as inset plot. On the right, an illustration of the dark-bright splitting.

  table that correspond to atoms with s-p-d filled closed-shells at a +4 oxidation state. The atomic species that are highlighted in each closed-shell column were successfully employed to form stable VODP. Here we select period 5 of the periodic table, thus Sn, Te and Zr as the atomic species at the M-site of each closed-shell VODP type, generating 6 different materials Cs 2 TeX 6 , Cs 2 SnX 6 , and Cs 2 ZrX 6 (with X=Br, I). Interestingly, The optimized structures for all compounds are in the Fm 3m space group, except Cs 2 SnI 6 , which exhibits very slightly rotated Sn-I octahedra with small octahedra distortions, in agreement with the work of Jong et al.

	Te 4+ is isoelectronic with the Pb +2 ion, which is the building block of the most efficient perovskite
	materials for optoelectronic applications. As a starting point, the atomic coordinates and lattice
	parameters for these materials are fully optimized using ab-initio calculations based on Density
	Functional Theory (DFT) as detailed in the supplementary information file (SI). Table S1 of the
	SI summarizes the obtained lattice parameters, metal-halogen bond lengths and volumetric size of
	the vacancy.

Table 1 :

 1 Average electron (m e ) and hole (m h ) effective masses at the band edges within G 0 W 0 .Numbers between parenthesis represents the heavy bands effective masses. The complete effective mass components can be found at TableS3of SI.

	Effective masses(m 0 )	Cs 2 SnBr 6	Cs 2 SnI 6	Cs 2 TeBr 6 Cs 2 TeI 6 Cs 2 ZrBr 6	Cs 2 ZrI 6
	m e	0.24	0.29	0.33	0.39	1.80	1.24
	m h	0.55 (1.22) 0.58 (1.08)	0.46	0.29	1.12 (2.82) 0.60 (1.45)

Acknowledgments

We gratefully acknowledge discussions with Prof. Feliciano Giustino, Dr. Joshua Leveillee and Dr. Tan Nguyen. The research leading to these results has received funding from the Chaire de Recherche Rennes Metropole and the European Union's Horizon 2020 program, through a FET Open research and innovation action under the grant agreement No 862656 (DROP-IT). This work was granted access to the HPC resources of TGCC under the allocations 2020-A0100911434 and 2021-A0110907682 made by GENCI. We acknowledge PRACE for awarding us access to the ARCHER2, United Kingdom.

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