José Seco

José C Xavier

Andrew S Brierley

Paco Bustamante

João P Coelho

Susan Gregory

Sophie Fielding

Miguel A Pardal

Bárbara Pereira

Gabriele Stowasser

Geraint A Tarling

Eduarda Pereira

Mercury levels in Southern Ocean Squid: variability over the last decade

Keywords: Organic mercury, Muscle, Gills, Digestive gland, Tissue allocation, Temporal trends

The concentrations of total and proportions of organic mercury were measured in tissues of 355 individuals of 8 species of Southern Ocean squid (Alluroteuthis antarcticus, Bathyteuthis abyssicola, Filippovia knipovitchi, Galiteuthis glacialis, Gonatus antarcticus, Kondakovia longimana, Psychroteuthis glacialis and Slosarczykovia circumantarctica). Squid were caught around South Georgia (Scotia Sea) during 5 cruises, between the austral summers of 2006/07 to 2016/17 to evaluate temporal changes in bioaccumulation and tissue partitioning. Total mercury concentrations varied between 4 ng g -1 and 804 ng g -1 among all tissues. Net accumulation of mercury in muscle with size was observed in A. antarcticus, B. abyssicola and P. glacialis, but no relationship was found for S. circumantarctica and lower concentrations were observed in larger individuals of G. glacialis. Muscle tissues had the highest mercury concentrations in the majority of species, except for F. knipovitchi for which the digestive gland contained highest concentrations. In terms of the percentage of organic mercury relative to total mercury in tissues, muscle always contained the highest values (67% to 97%), followed by the digestive gland (22% to 38%). Lowest organic mercury percentages were found consistently in the gills (9% to 19%), suggesting only low levels of incorporation through the dissolved pathway and/or a limited redistribution of dietary organic mercury towards this tissue. Overall, results are indicative of a decreasing trend of mercury concentrations in the majority of analysed species over the last decade. As cephalopods are an important Southern Ocean trophic link between primary consumers and top predators, these changes suggest decreasing mercury levels in lower trophic levels (i.e. squid prey) and an alleviation of the mercury burden on higher predators that consume squid.

Introduction

Evidence suggest that coleoid cephalopods, such as squid and octopods, are one of the groups that are benefitting from environmental change [START_REF] Halpern | A global map of human impact on marine ecosystems[END_REF], with some cephalopod populations being on the rise at a variety of locations worldwide [START_REF] Arkhipkin | World Squid Fisheries[END_REF][START_REF] Doubleday | Global proliferation of cephalopods[END_REF]. Within the Southern Ocean, cephalopods are important links between primary consumers and top predators [START_REF] Clarke | The role of cephalopods in the world's oceans: An introduction[END_REF][START_REF] Collins | Southern Ocean Cephalopods[END_REF][START_REF] Seco | Distribution, habitat and trophic ecology of Antarctic squid Kondakovia longimana and Moroteuthis knipovitchi: inferences from predators and stable isotopes[END_REF]. They prey mainly on crustaceans [START_REF] Kear | The Diet of Antarctic Squid -Comparison of Conventional and Serological Gut Contents Analyses[END_REF][START_REF] Xavier | A review on the biodiversity, distribution and trophic role of cephalopods in the Arctic and Antarctic marine ecosystems under a changing ocean[END_REF] and are eaten by a wide range of predators including fish, penguins, albatrosses, seals and whales [START_REF] Mikhalev | To the problem of the feeding of sperm whales from the Southern Hemisphere[END_REF][START_REF] Split | The diet of southern elephant seals (Mirounga leonina) from Heard Island[END_REF][START_REF] Xavier | Cephalopod beak guide for the Southern Ocean[END_REF]. In the Southern Ocean, the waters around South Georgia host a variety of squid species. Galiteuthis glacialis, Gonatus antarcticus, Filippovia knipovitchi, Kondakovia longimana and Psychroteuthis glacialis are oceanic species and some of the most important prey for several predators (such as seabirds, seals and whales [START_REF] Xavier | Cephalopod beak guide for the Southern Ocean[END_REF]), both by number and by mass [START_REF] Xavier | A review on the biodiversity, distribution and trophic role of cephalopods in the Arctic and Antarctic marine ecosystems under a changing ocean[END_REF]. Alluroteuthis antarcticus and Slosarczykovia circumantarctica are taken by a wide range of predators although not in high numbers [START_REF] Collins | Southern Ocean Cephalopods[END_REF][START_REF] Xavier | Cephalopod beak guide for the Southern Ocean[END_REF]. Bathyteuthis abyssicola is a deep-sea squid which is rarely found in the diet of predators. Despite their important role in the Antarctic ecosystem, there is still a lack of knowledge about their ecology [START_REF] Clarke | Cephalopod Biomass -Estimation from Predation[END_REF][START_REF] Collins | Southern Ocean Cephalopods[END_REF][START_REF] Xavier | A review on the biodiversity, distribution and trophic role of cephalopods in the Arctic and Antarctic marine ecosystems under a changing ocean[END_REF]. Moreover, very few studies have focused on the ecotoxicological aspects of Southern Ocean cephalopods [START_REF] Anderson | Influence of trophic position and foraging range on mercury levels within a seabird community[END_REF][START_REF] Bustamante | Cephalopods as a vector for the transfer of cadmium to top marine predators in the north-east Atlantic Ocean[END_REF][START_REF] Cipro | Trace elements in invertebrates and fish from Kerguelen waters, southern Indian Ocean[END_REF]. Their focal position within Southern Ocean food webs means that cephalopods are likely to be vectors of contaminants and could be valuable bioindicators of ecosystem contamination.

Mercury is one of the contaminants that has been acknowledged as a global toxicity problem [START_REF] Selin | Global Biogeochemical Cycling of Mercury: A Review[END_REF]UNEP, 2013). Due to its high affinity to proteins [START_REF] Bloom | On the chemical form of mercury in edible fish and marine invertebrate tissue[END_REF], mercury is highly bioaccumulative, becoming toxic for marine organisms higher up the food chain [START_REF] Ackerman | Mercury in birds of San Francisco Bay-Delta, California: trophic pathways, bioaccumulation, and ecotoxicological risk to avian reproduction[END_REF][START_REF] Coelho | Mercury bioaccumulation in the spotted dogfish (Scyliorhinus canicula) from the Atlantic Ocean[END_REF][START_REF] Dehn | Trophic relationships in an Arctic food web and implications for trace metal transfer[END_REF]. Furthermore, it biomagnifies along food webs, putting long-lived top predators particularly at risk (e.g. [START_REF] Goutte | Demographic responses to mercury exposure in two closely related Antarctic top predators[END_REF][START_REF] Tartu | Age-Related Mercury Contamination and Relationship with Luteinizing Hormone in a Long-Lived Antarctic Bird[END_REF][START_REF] Tavares | Influence of age, sex and breeding status on mercury accumulation patterns in the wandering albatross Diomedea exulans[END_REF]. Indeed, there is already evidence that major cephalopod predators have high levels of mercury in their tissues (e.g. Bustamante et al., 2003;Fontaine et al., 2014;[START_REF] Tavares | Influence of age, sex and breeding status on mercury accumulation patterns in the wandering albatross Diomedea exulans[END_REF]. In terms of bioavailability within foodwebs, organic mercury (due to its high affinity to Sulphur-based protein groups) is a particularly toxic form of this element which demands further attention.

In a warming world, in which Antarctica is one of the most rapidly changing and vulnerable areas [START_REF] Ipcc ; Stocker | Summary for Policymakers[END_REF][START_REF] Rintoul | Choosing the future of Antarctica[END_REF][START_REF] Turner | Antarctic climate change and the environment: an update[END_REF], mercury is becoming more bioavailable due to the combined influence of increased temperature and depletion of oxygen, which favour methylation of the element by microorganisms [START_REF] Cossa | Marine biogeochemistry: Methylmercury manufacture[END_REF]. It is therefore important to evaluate the impact that these changes have on the bioavailability of mercury in the Southern Ocean, along with any potential temporal trends in bioaccumulation. Fast growing, short-lived (i.e. generally 1-2 year life cycles) organisms such as squid [START_REF] Arkhipkin | Diversity in growth and longevity in short-lived animals: squid of the suborder Oegopsina[END_REF][START_REF] Boyle | Cephalopods ecology and fisheries[END_REF] are likely to be responsive bioindicators of contaminant variability over time and space.

Cephalopods bioaccumulate mercury from two main sources: seawater and food. Some mercury uptake can occur through the gills during respiration, from seawater, but mercury in prey is considered to be the main intake pathway [START_REF] Lacoue-Labarthe | Bioaccumulation of inorganic Hg by the juvenile cuttlefish Sepia officinalis exposed to 203Hg radiolabelled seawater and food[END_REF]. To assess the relative importance of the two pathways in the bioaccumulation of mercury, we analysed mercury in three different tissues: 1) muscle, which represents most of the body weight of the animal and is expected to accumulate high levels of organic mercury [START_REF] Bustamante | Total and organic Hg concentrations in cephalopods from the North Eastern Atlantic waters: Influence of geographical origin and feeding ecology[END_REF]; 2) gills, responsible for respiration and subjected to a constant water flow, so likely to be a pathway of incorporation of dissolved Hg from seawater; and 3) digestive gland, which is most affected by the dietary pathway [START_REF] Bustamante | Total and organic Hg concentrations in cephalopods from the North Eastern Atlantic waters: Influence of geographical origin and feeding ecology[END_REF][START_REF] Penicaud | Metal bioaccumulation and detoxification processes in cephalopods: A review[END_REF][START_REF] Pierce | Geographic, seasonal and ontogenetic variation in cadmium and mercury concentrations in squid (Cephalopoda: Teuthoidea) from UK waters[END_REF] and plays a major role in both the metabolism and detoxification of contaminants such as mercury [START_REF] Bustamante | Total and organic Hg concentrations in cephalopods from the North Eastern Atlantic waters: Influence of geographical origin and feeding ecology[END_REF].

In this study, we evaluate the concentrations of total and organic mercury in 8 different squid species (A. antarcticus, B. abyssicola, F. knipovitchi, G. glacialis, G. antarcticus, K. longimana, P. glacialis, S. circumantarctica) from South Georgia between the austral summers of 2006/07 and 2016/17. These species were selected due to their different ecological roles in the Southern Ocean ecosystem [START_REF] Xavier | A review on the biodiversity, distribution and trophic role of cephalopods in the Arctic and Antarctic marine ecosystems under a changing ocean[END_REF]. The specific objectives of this study are: 1) to evaluate the accumulation pattern along with size (a proxy of age) of Antarctic squid, 2) to understand the partitioning of total and organic mercury in different tissues (muscles, gills and digestive gland) and 3) to assess variability and trends in total and organic mercury concentrations of these species over a 10 year period (2006/07 to 2016/17).

Material and methods

Sampling

South Georgia is a sub-Antarctic island located in the southwest Atlantic (Figure 1). Water temperatures around this area vary from -0.95 ºC in winter to 1.75 ºC in summer. South Georgia is a highly productive area of the Southern Ocean, therefore it holds large populations of seabirds, marine mammals and it is one of the most important Southern Ocean fishing areas [START_REF] Collins | Cephalopods of the South Georgia slope[END_REF][START_REF] Murphy | Spatial and temporal operation of the Scotia Sea ecosystem: a review of large-scale links in a krill centred food web[END_REF].

The samples were collected from the Scotia Sea, around South Georgia (Figure 1 On board the RRS JCR, samples were collected using an 8 or 25 m² mouthopening Rectangular Midwater Trawl (RMT8 -mesh size reducing from 4.5 mm to 2.5 mm in the cod end; RMT25 -mesh size reducing from 8 mm to 4.5 mm in the cod end [START_REF] Roe | A new multiple rectangular midwater trawl (RMT 1+8M) and some modifications to the institute of oceanographic sciences' RMT 1+8[END_REF]). The nets were rigged with two opening/closing nets that could be remotely opened and closed at different depths. Samples were collected from 1000m deep to surface. Cephalopods were identified [START_REF] Nesis | Cephalopods of the World[END_REF][START_REF] Xavier | Cephalopod beak guide for the Southern Ocean[END_REF], measured and weighed on board. Samples were preserved individually in separate ziplock bags at -20 ºC for later laboratory analyses.

The samples collected by FV Sil were obtained from bottom trawls using an FP120 trawl net with a standard steel bobbin rig, conducted at tow speeds between 3.1 and 4.1 knots over a distance of between 1.25 and 2.1 nautical miles, dependent on the prevailing sea conditions and bottom topography. Whenever possible, samples were identified on board but, in some cases, identification was not possible. In each case, individuals were frozen at -20 ºC for later laboratory processing.

Laboratory procedures

All samples were checked for identification (using cephalopod beaks to confirm identification where there was any doubt [START_REF] Xavier | Cephalopod beak guide for the Southern Ocean[END_REF])), measured and weighed again. When the measurement of the mantle length (ML) of the individual was not possible, allometric equations were used, based on beak size [START_REF] Xavier | Cephalopod beak guide for the Southern Ocean[END_REF]). An effort was made to collect samples of muscle, gills and digestive gland in all individuals, although the digestive gland was destroyed in some specimens.

After being dissected, the collected tissues were frozen in sterilised decontaminated plastic vials and freeze-dried for at least 48 hours. Dried tissues were homogenized and analysed for total mercury by thermal decomposition atomic absorption spectrometry with gold amalgamation, using a LECO AMA-254 (Advanced mercury analyser) following [START_REF] Coelho | Pattern and pathways for mercury lifespan bioaccumulation in Carcinus maenas[END_REF]. Organic mercury was determined through digestion with a mixture of 18 % potassium bromide (KBr) in 5 % sulphuric acid (H2SO4), followed by extraction of organic mercury into toluene (C7H8) and backextraction with an aqueous solution of thiosulphate (Na2S2O3), as described in [START_REF] Válega | Determination of Organic Mercury in Biota, Plants and Contaminated Sediments Using a Thermal Atomic Absorption Spectrometry Technique[END_REF]. Where there was low individual mass (less than 200 mg), samples were aggregated in groups of the same species, with similar sizes and collected from the same location. Analytical quality control was performed using three certified reference materials (CRM): NIST 2976 mussel tissue, ERM-CE278K mussel tissues and TORT-3 lobster hepatopancreas. The obtained values (mean ± SD) for the whole of the CRM analyses ranged from 82 to 105 % (NIST 2976: 85 ± 7%; ERM-CE278K: 92 ± 5%; TORT-3: 93 ± 8 %). Analyses were performed in duplicate, and the coefficient of variation between replicates never exceeded 10%. TORT-3 was also used to validate organic mercury analyses, with an extraction efficiency of 95 ± 10 %. The limit of detection for this analytical method was 0.01 ng of absolute mercury. All concentration data are expressed as a function of dry weight (dw).

Statistical analysis

All analyses were performed using the R software version 3.4.2 (R Core Team, 2013). Correlations were determined between the mercury concentration and the mantle length. Hg levels were tested for normality using Shappiro-Wilk normality test and homogeneity using a Bartlett's test. All values are presented as mean ± SD. The significance level for statistical analyses was always set at α = 5%.

Results

Total mercury concentrations in Antarctic squid muscle

Differential tissue accumulation of total mercury in Antarctic squid

Analysis of mercury accumulation in different tissues was possible in four of the species (A. antarcticus, F. knipovitchi, K. longimana and P. glacialis). F.

knipovitchi was the only species where there were significant differences between the tissues (Friedman test: χ 2 (2) = 7.6, p= 0.024), with the digestive gland having a concentration 3 times higher than the muscle and gills. In the other three species, no statistical differences were observed between the mercury concentrations of the three tissues investigated (Friedman test: χ 2 (2) = 3, p= 0.5; χ 2 (2) = 3, p= 0.5; χ 2 (2) = 3, p= 0.5; A. antarcticus, K. longimana and P. glacialis, respectively; Figure 3). For G. antarcticus, only the muscle and the gills were analysed, and no differences between these two tissues were found (Wilcoxon Signed Ranks Test; Z= -1.826, p= 0.125).

Temporal trends of total mercury concentrations in muscle of Antarctic squid

Over the 10 year study period, there was a suggestive decreasing trend in mercury concentrations in the analysed species (Figure 4). There were no differences in body size between the years for all the analysed species (A.

Organic mercury concentrations and proportions in Antarctic squid

Regarding the percentage of organic mercury relative to total mercury in the different tissues of the five squid species sampled in 2013 (SG13 samples; Table 1), muscle had the highest values (67% to 97%) in all the analysed species, followed by the digestive gland (22% to 38%) and the gills (9% to 19%).

Organic mercury concentrations were also analysed in the muscle of three species (B. abyssicola, G. glacialis and S. circumantarctica, Table 2) across sampling years. Concentrations varied from 2 ng g -1 to 84 ng g -1 , constituting between 25% to 77% of total mercury. S. circumantarctica was the only species where organic mercury was lower than 50% (40% in 2007/08; 47% in 2008/09; 25% in 2016/17). Although organic mercury proportions differed significantly between species, there were no statistical differences between years.

Discussion

While some data exist on mercury concentrations in Antarctic cephalopods [START_REF] Anderson | Influence of trophic position and foraging range on mercury levels within a seabird community[END_REF][START_REF] Cipro | Trace elements in invertebrates and fish from Kerguelen waters, southern Indian Ocean[END_REF][START_REF] Mcarthur | Mercury in the marine food chain in the Southern Ocean at Macquarie Island: an analysis of a top predator, Patagonian toothfish (Dissostichus eleginoides) and a mid-trophic species, the warty squid (Moroteuthis ingens)[END_REF], to the best of our knowledge, this is the first study to measure total and organic mercury concentrations, and to determine the percentage of organic mercury relative to total mercury in different tissues, across a range of Southern Ocean squid species.

Total mercury concentrations according to size

Biological and environmental factors such as size, sex, prey preferences and habitat are drivers for mercury concentrations in cephalopods [START_REF] Bustamante | Total and organic Hg concentrations in cephalopods from the North Eastern Atlantic waters: Influence of geographical origin and feeding ecology[END_REF][START_REF] Monteiro | Inter-and intra-specific variation of mercury levels in muscle of cephalopods from the Azores 13-22[END_REF][START_REF] Storelli | Cadmium and total mercury in some cephalopods from the South Adriatic Sea (Italy)[END_REF]. When looking into the relationship between size and contamination level in squid from South Georgia (Figure 3), it is possible to identify three different patterns: 1) A. antarcticus and P. glacialis had a positive correlation between contamination level and size, 2) The correlation for G. glacialis was negative, 3) Mercury concentration did not appear to be related to size in B. abyssicola or S. circumantarctica.

Significant variation and opposite patterns between size and mercury concentrations in cephalopods have been reported previously. Mercury concentrations increased with size in the veined squid Loligo forbesi [START_REF] Chouvelon | Inter-specific and ontogenic differences in δ13C and δ15N values and Hg and Cd concentrations in cephalopods[END_REF][START_REF] Monteiro | Inter-and intra-specific variation of mercury levels in muscle of cephalopods from the Azores 13-22[END_REF][START_REF] Pierce | Geographic, seasonal and ontogenetic variation in cadmium and mercury concentrations in squid (Cephalopoda: Teuthoidea) from UK waters[END_REF], the neon flying squid

Ommastrephes bartramii [START_REF] Monteiro | Inter-and intra-specific variation of mercury levels in muscle of cephalopods from the Azores 13-22[END_REF], the orange-back flying squid

Sthenoteuthis pteropus [START_REF] Lischka | High cadmium and mercury concentrations in the tissues of the orange-back flying squid, Sthenoteuthis pteropus, from the tropical Eastern Atlantic[END_REF], the common octopus Octopus vulgaris [START_REF] Rjeibi | Interspecific and geographical variations of trace metal concentrations in cephalopods from Tunisian waters[END_REF], the curled octopus Eledone cirrhosa [START_REF] Rossi | Mercury in Eledone cirrhosa from the Northern Tyrrhenian Sea: contents and relations with life cycle[END_REF], the common cuttlefish Sepia officinalis and the European squid Loligo vulgaris [START_REF] Chouvelon | Inter-specific and ontogenic differences in δ13C and δ15N values and Hg and Cd concentrations in cephalopods[END_REF]. However, in O. vulgaris [START_REF] Raimundo | Geographical variation and partition of metals in tissues of Octopus vulgaris along the Portuguese coast[END_REF]2010a) and the greater hooked Onykia ingens [START_REF] Mcarthur | Mercury in the marine food chain in the Southern Ocean at Macquarie Island: an analysis of a top predator, Patagonian toothfish (Dissostichus eleginoides) and a mid-trophic species, the warty squid (Moroteuthis ingens)[END_REF], no relationships were found, while in the spider octopus Octopus salutii [START_REF] Storelli | Cadmium and total mercury in some cephalopods from the South Adriatic Sea (Italy)[END_REF], O. vulgaris and Loligo vulgaris [START_REF] Rjeibi | Interspecific and geographical variations of trace metal concentrations in cephalopods from Tunisian waters[END_REF] there were negative relationships. Hence, the effect of size on mercury bioaccumulation in cephalopods is variable [START_REF] Penicaud | Metal bioaccumulation and detoxification processes in cephalopods: A review[END_REF],

and is likely to be sensitive to fluctuating environmental conditions (see below).

Tissue allocation of mercury in Antarctic squid

For four species (A. antarcticus, G. antarcticus K. longimana and P. glacialis), there were no differences in total mercury concentrations between tissues. However, the fraction of organic mercury was always higher in muscle (from 67% to 97%) than in the digestive gland (from 22% to 38%) and gills (9% to 19%). While presently the reasons for the mercury distribution patterns are still a matter of debate, the partitioning of total mercury among tissues is consistent with previous studies on oceanic squid (Cranchidae, Histioteuthidae and Ommastrephidae [START_REF] Bustamante | Total and organic Hg concentrations in cephalopods from the North Eastern Atlantic waters: Influence of geographical origin and feeding ecology[END_REF]), suggesting some degree of equilibrium between contaminant accumulation and excretion rates, as well as efficient redistribution of total mercury between tissues.

In most studies of cephalopod species however, the digestive gland had higher total mercury concentrations than other tissues [START_REF] Bustamante | Total and organic Hg concentrations in cephalopods from the North Eastern Atlantic waters: Influence of geographical origin and feeding ecology[END_REF][START_REF] Pierce | Geographic, seasonal and ontogenetic variation in cadmium and mercury concentrations in squid (Cephalopoda: Teuthoidea) from UK waters[END_REF]Raimundo et al., 2010b), as we observed in F. knipovitchi (Figure 4). The high rates of absorption and assimilation of trace metals in the digestive gland [START_REF] Bustamante | Cadmium detoxification processes in the digestive gland of cephalopods in relation to accumulated cadmium concentrations[END_REF][START_REF] Miramand | Concentration and distribution of heavy metals in tissues of two cephalopods, Eledone cirrhosa and Sepia officinalis, from the French coast of the English Channel[END_REF][START_REF] Raimundo | Geographical variation and partition of metals in tissues of Octopus vulgaris along the Portuguese coast[END_REF]) make this organ a major pathway of incorporation for contaminants into cephalopods [START_REF] Bustamante | Cadmium detoxification processes in the digestive gland of cephalopods in relation to accumulated cadmium concentrations[END_REF], and reflects the predominant role of the dietary pathway for mercury bioaccumulation. The lower concentrations of organic mercury in the digestive gland is in accordance with the detoxification of organic mercury by demethylation that is suspected to occur in this organ [START_REF] Bustamante | Total and organic Hg concentrations in cephalopods from the North Eastern Atlantic waters: Influence of geographical origin and feeding ecology[END_REF][START_REF] Penicaud | Metal bioaccumulation and detoxification processes in cephalopods: A review[END_REF]. As this form of mercury is incorporated from the diet, a fraction is likely to be demethylated in the digestive gland, to allow its excretion [START_REF] Lacoue-Labarthe | Bioaccumulation of inorganic Hg by the juvenile cuttlefish Sepia officinalis exposed to 203Hg radiolabelled seawater and food[END_REF], and a portion will be circulated, trapped and stored in muscle tissues. Because organic mercury has a strong affinity for the sulphydryl groups of proteins, its accumulation in muscle tissues is favoured [START_REF] Bloom | On the chemical form of mercury in edible fish and marine invertebrate tissue[END_REF][START_REF] Leaner | Methylmercury uptake and distribution kinetics in sheepshead minnows, Cyprinodon variegatus, after exposure to CH3Hg-spiked food[END_REF]. In fish muscles, for instance, it may be tightly bound by carbon-mercury and sulphydryl linkages [START_REF] Ruelas-Inzunza | Methylmercury and total mercury distribution in tissues of gray whales (Eschrichtius robustus) and spinner dolphins (Stenella longirostris) stranded along the lower Gulf of California, Mexico[END_REF]. As muscles comprise the highest mass fraction of cephalopods, and given the known affinity of organic mercury to muscular proteins [START_REF] Bloom | On the chemical form of mercury in edible fish and marine invertebrate tissue[END_REF], high concentrations and proportions of organic mercury are to be expected in this tissue. The present results also highlight the high bioavailability of mercury within the Southern Ocean food web [START_REF] Chouvelon | Enhanced bioaccumulation of mercury in deep-sea fauna from the Bay of Biscay (north-east Atlantic) in relation to trophic positions identified by analysis of carbon and nitrogen stable isotopes[END_REF][START_REF] Clarkson | Mercury: Major Issues in Environmental Health[END_REF][START_REF] Dehn | Trophic relationships in an Arctic food web and implications for trace metal transfer[END_REF].

Concentrations and proportions of organic mercury also show a considerable variation between species. The high intra-species variation may be explained by the dietary ontogenetic shift that occurs in cephalopods [START_REF] Xavier | A review on the biodiversity, distribution and trophic role of cephalopods in the Arctic and Antarctic marine ecosystems under a changing ocean[END_REF] when they change from feeding on small crustaceans (lower percentage of organic mercury (Seco et al., 2019)) to prey at higher trophic levels (higher percentage of organic mercury [START_REF] Chouvelon | Inter-specific and ontogenic differences in δ13C and δ15N values and Hg and Cd concentrations in cephalopods[END_REF]), however this can not be applied to G. glacialis, as mercury in this species decreases in bigger individuals. Similarly, interspecific differences are probably due to variations in diet [START_REF] Boyle | Cephalopods ecology and fisheries[END_REF], although the trophic ecology of these cephalopods is still poorly known [START_REF] Collins | Southern Ocean Cephalopods[END_REF].

Temporal trends of mercury concentrations in Antarctic squid

Our study analysed samples obtained over a 10 year period (between 2006/07 and 2016/17), although only two species were captured in all sampling seasons (G.

glacialis, S. circumantarctica) and the number of individuals captured for some species was relatively low despite the sustained sampling effort. Squid have a highly developed sensory system and can swim fast, while sampling nets are relatively small and slow, so most adult squid can avoid capture [START_REF] Clarke | Beaks, nets and numbers[END_REF][START_REF] Xavier | Distribution of cephalopods recorded in the diet of the Patagonian toothfish (Dissostichus eleginoides) around South Georgia[END_REF].

Squid are considered an r-selected species [START_REF] Pianka | On r and K selection[END_REF]: they have a fast growth rate, are semelparous (reproduce only once) and are short lived (~< 1-2 years) [START_REF] Arkhipkin | Diversity in growth and longevity in short-lived animals: squid of the suborder Oegopsina[END_REF][START_REF] Boyle | Cephalopods ecology and fisheries[END_REF][START_REF] Xavier | A review on the biodiversity, distribution and trophic role of cephalopods in the Arctic and Antarctic marine ecosystems under a changing ocean[END_REF], although, some large squid species, like K. longimana may live longer [START_REF] Jarre | Re-examination of growth estimates in oceanic squids: the case of Kondakovia longimana (Onychoteuthidae)[END_REF][START_REF] Lynnes | A big mouthful for predators: The largest recorded specimen of Kondakovia longimana (Cephalopoda : Onychoteuthidae)[END_REF]. These characteristics make them responsive bioindicators with which would be possible to monitor trends in mercury concentrations over time, as they will reflect rapidly any changes in the bioavailability of this contaminant.

Data from the squid species A. antarcticus, B abyssicola, G. glacialis and S.

circumantarctica appears to indicate a decreasing trend of mercury concentration along time, although further monitoring is required to confirm this pattern. This suggestive pattern of declining concentrations of mercury in the majority of species is consistent with the decreasing trend of atmospheric mercury over the last decade [START_REF] Soerensen | Multi-decadal decline of mercury in the North Atlantic atmosphere explained by changing subsurface seawater concentrations[END_REF] as a consequence of the reduction of worldwide anthropogenic emissions of mercury [START_REF] Streets | Total Mercury Released to the Environment by Human Activities[END_REF][START_REF] Zhang | Observed decrease in atmospheric mercury explained by global decline in anthropogenic emissions[END_REF]. The decrease of mercury in the global atmosphere could also mean a reduction in mercury deposition levels in our study area. Comparing our results with a previous study in the same region [START_REF] Anderson | Influence of trophic position and foraging range on mercury levels within a seabird community[END_REF] The pattern of mercury bioaccumulation in species is influenced by specific traits such as dietary preference, ingestion, excretion, and growth rate. Prevailing environmental conditions may also enhance or reduce contaminant bioavailability to cephalopods. Habitat use has a major effect on mercury accumulation in organisms, as sites contaminated by mercury will likely have higher bioavailability of this toxic element. All of our samples were collected in the Scotia Sea, around South Georgia (Figure 1), which is known to be a fairly stable ecosystem, which should mean lower mercury variation between sampling sites.

The majority of our study species have a wide range of vertical distributions:

the depth of occurrence of A. antarcticus is normally at 800-900 m [START_REF] Rodhouse | Distribution of the neoteuthid squid Alluroteuthis antarcticus odhner in the Atlantic sector ot the Southern Ocean[END_REF], G. glacialis, at 600-1000 m [START_REF] Roper | Vertical Distribution of pelagic cephalopods[END_REF], G. antarcticus, at 250-928 m [START_REF] Collins | Cephalopods of the South Georgia slope[END_REF][START_REF] Roper | FAO Species Catalogue. Volume 3: Cephalopods of the World. An Annotated and Illustrated Catalogue of Species of Interest to Fisheries[END_REF], F. knipovitchi, at 480-760 m [START_REF] Collins | Cephalopods of the South Georgia slope[END_REF], K. longimana, at 300-900 m [START_REF] Collins | Cephalopods of the South Georgia slope[END_REF][START_REF] Xavier | Distribution of cephalopods recorded in the diet of the Patagonian toothfish (Dissostichus eleginoides) around South Georgia[END_REF], P.

glacialis, at 275-928 m [START_REF] Collins | Cephalopods of the South Georgia slope[END_REF][START_REF] Xavier | Distribution of cephalopods recorded in the diet of the Patagonian toothfish (Dissostichus eleginoides) around South Georgia[END_REF] and S.

circumantarctica, at 487-928 m [START_REF] Collins | Cephalopods of the South Georgia slope[END_REF]). An outlier amongst these is B.

abyssicola, which is a deep-sea squid and occurs at a depth of 1000-1500 m [START_REF] Roper | Vertical Distribution of pelagic cephalopods[END_REF]. Higher mercury levels were expected in species from deep habitats as increasing depth raises mercury concentrations in fish [START_REF] Choy | The influence of depth on mercury levels in pelagic fishes and their prey[END_REF][START_REF] Bourg | Effect of body length, trophic position and habitat use on mercury concentrations of sharks from contrasted ecosystems in the southwestern Indian Ocean[END_REF][START_REF] Monteiro | Mercury concentrations in prey fish indicate enhanced bioaccumulation in mesopelagic environments[END_REF]. However, that was not the case in our study, as mercury concentrations in the tissues of the deep-sea squid B. abyssicola were within the same range as the majority of other analysed species (see Figure 4). The lack of difference between species may be explained by the frequent diel vertical migration associated with cephalopods [START_REF] Norman | Cephalopods of the World. an Annotated and Illustrated Catalogue of Cephalopod Species Known to Date. Volume 3 Octopods and Vampire Squids[END_REF] in order to feed and to avoid predation. The differences in mercury concentrations between years may be partially explained by shifts in the abundance and contaminant loads of prey items [START_REF] Chouvelon | Inter-specific and ontogenic differences in δ13C and δ15N values and Hg and Cd concentrations in cephalopods[END_REF][START_REF] Paiva | The Influence of Diet on Mercury Intake by Little Tern Chicks[END_REF]. However, there is still a lack of data on the diets of the studied species [START_REF] Collins | Southern Ocean Cephalopods[END_REF][START_REF] Xavier | A review on the biodiversity, distribution and trophic role of cephalopods in the Arctic and Antarctic marine ecosystems under a changing ocean[END_REF][START_REF] Xavier | Cephalopod beak guide for the Southern Ocean[END_REF]) and it is presently not possible to evaluate the effect of diet on mercury bioaccumulation in our study species.

  ), during scientific cruises during the austral summer of 2006/07 (on board of the Royal Research Ship (RRS) James Clark Ross (JCR): cruise JR161 [October -December 2006], 2007/08 -JR177 [December 2007 -February 2008], 2008/09 -JR200 [March -April 2009], Fishing Vessel (FV) Sil research survey SG13 [13 January 2013]) and RRS JCR cruise JR16003 (December 2016 -January 2017), 2016/17.

Figure 1 .

 1 Figure 1. Sampling sites and distribution of species captured around South Georgia across all sampling years. Lines represent 1000m isobath.

  Friedman tests were used to compare Hg values between the different tissues (muscle, gills, digestive gland) and a Wilcoxon Signed Ranks Test for the tissues of G. antarcticus (muscle, gills). Wilcoxon rank test and Kruskal-Wallis test were used to compare Hg in muscle tissue between different years, followed by Dunn's test multiple comparison test.

Figure 2 .

 2 Figure 2. Total mercury concentration (ng g -1 dw) versus mantle length (ML; mm) in individual Antarctic squid collected around South Georgia in the austral summer of 2007/08 (except Psychroteuthis glacialis 2016/17). Regression equations are given in the text.

Figure 3 .

 3 Figure 3. Total mercury concentrations (Mean ± SD, ng g -1 dw) in different tissues (muscle, gills and digestive gland) of Antarctic squid collected around South Georgia in the austral summer of 2012/2013. No digestive gland tissue was available for G. antarcticus.

Figure 4 .

 4 Figure 4. Total mercury concentrations (Mean ± SD, ng g -1 dw) in the muscles of Antarctic squid from the Southern Ocean collected around South Georgia in the austral summers of 2006/07, 2007/08, 2008/09 and 2016/17.

  antarcticus, Mann Whitney test, U = 9, p = 0.889; B. abyssicola, Kruskal-Wallis H= 1.754, p = 0.442; G. glacialis, Kruskal-Wallis H= 10.84, p = 0.442; P.

	from 2006/07 had statistical higher concentration than the other 3 sampling years
	(Kruskal-Wallis H= 60.08, p< 0.001).
	glacialis, Mann Whitney test, U = 4.5, p = 0.8; S. circumantarctica, Kruskal-Wallis H=
	65.46, p = 0.156). A. antarcticus and P. glacialis were each only caught in two of the
	austral seasons sampled. For A. antarcticus, total mercury was similar between the
	years 2007/08 (90  60 ng g -1 ) and 2008/09 (30  3 ng g -1 ; Wilcoxon rank test; W =
	14, p = 0.19), as was P. glacialis between 2008/09 (10  2 ng g -1 ) and 2016/17 (40 
	20 ng g -1 ; Wilcoxon rank test; W = 10, p= 0.057). B. abyssicola were caught in three
	years, 2006/07 (150  20 ng g -1 ), 2007/08 (80  10 ng g -1 ) and 2016/17 (80  1 ng g -
	caught on the four sampling cruises, 2006/07 (100  60 ng g -1 ), 20007/08 (150  90
	ng g -1 ), 2008/09 (20  10 ng g -1 ) and 2016/17 (20  7 ng g -1 ). Mercury concentrations
	for this species were similar between the years 2006/07 and 2007/08 (Dunn's test Q
	= -8.77, p = 0.52) and also similar between the years 2008/09 and 2016/17 (Dunn's
	test Q = 0.057, p > 0.99), but were different between the two similarity groups
	(2006/07, 2007/08 and 2008/09, 2016/17 (Kruskal-Wallis H= 28.69, p< 0.001). S.
	circumantarctica was also caught in all the sampling years: 2006/07 (70  10 ng g -1 ),
	2007/08 (20  6 ng g -1 ), 2008/09 (10  6 ng g -1 ) and 2016/17 (10  7 ng g -1 ); individuals

1

), with individuals from 2006/07 having statistically higher concentrations of mercury than 2007/08 and 2016/17 (Kruskal-Wallis H= 6.709, p= 0.013; Figure

4

). There were no differences between samples collected on the other two years. G. glacialis were

Table 1 .

 1 Total mercury (THg) and organic mercury (OHg) concentrations (ng g -1 dw) and percentage of OHg relative to THg in different tissues (digestive gland, gills and muscle) of Antarctic squid collected around South Georgia in the austral summer of 2013/14.

Table 2 .

 2 Total mercury (THg) and organic mercury (OHg) concentrations (ng g -1 dw) and percentage of OHg relative to THg in the muscles of Antarctic squid collected around South Georgia in the austral summers of 2006/07, 2007/08, 2008/09 and 2015/16.

Acknowledgments

We thank the crew, officers and scientists aboard the RSS James Clark Ross during the cruises JR161, JR177, JR200 and JR16003, and on board the FV Sil during research survey SG13, for their assistance in collecting samples, namely Mark Belchier. We also thank Peter Enderlein, Carson McAfee, Daniel Ashurst and Scott Polfrey for assembling and maintaining the net We acknowledge the Portuguese Foundation for the Science and Technology (FCT) through a PhD grant to José Seco (SRFH/PD/BD/113487). Acknowledgments are due also to the Integrated Program of SR&TD 'Smart Valorization of Endogenous Marine Biological Resources Under a Changing Climate' (reference Centro-01-0145-FEDER-000018), co-funded by Centro 2020 program, Portugal 2020, European Union, through the European Regional Development Fund, for personal funding to J.P.Coelho. The IUF (Institut Universitaire de France) is acknowledged for its support to P. Bustamante as a Senior Member. This research was also within José Xavier strategic program of MARE (MARE -UID/ MAR/04292/2013). GAT, GS and SF were supported by the Ecosystems programme at the British Antarctic Survey.

Our results suggest a decreasing trend of total mercury concentration in most of the South Georgia squid species analysed over the last decade, with a stable proportion of organic mercury. Considering that cephalopods are a major link between primary consumer and top predators, these changes possibly reflect a drop in mercury bioavailability in lower trophic levels and suggests that mercury intake by squid predators may have decreased.


Niclas Börlin
email: niclas.borlin@cs.umu.se

Arnadi Murtiyoso
email: arnadi.murtiyoso@insa-strasbourg.fr

Pierre Grussenmeyer
email: pierre.grussenmeyer@insa-strasbourg.fr

Fabio Menna
email: fmenna@fbk.eu

Erica Nocerino
email: erica.nocerino@univ-amu.fr

Flexible Photogrammetric Computations using Modular Bundle Adjustment The chain rule and the collinearity equations

The main purpose of this paper is to show that photogrammetric bundle adjustment computations can be sequentially organized into modules. Furthermore, the chain rule can be used to simplify the computation of the analytical Jacobians needed by the adjustment. Novel projection models can be flexibly evaluated by inserting, modifying, or swapping the order of selected modules.

As a proof of concept, two variants of the pin-hole projection model with Brown lens distortion were implemented in the open-source Damped Bundle Adjustment Toolbox (DBAT) and applied to simulated and calibration data for a non-conventional lens system. The results show a significant difference for the simulated, error-free, data but not for the real calibration data.

The current flexible implementation incurs a performance loss. However, in cases where flexibility is more important, the modular formulation should be a useful tool to investigate novel sensors, data processing techniques, and refractive models.

Introduction 1.Background

In Photogrammetry and Computer Vision, Bundle Adjustment (BA) is a fine-tuning estimation process where some or all parameters that describe a scene are estimated simultaneously. Bundle adjustment was introduced to photogrammetry some sixty years ago [START_REF] Brown | A solution to the general problem of multiple station analytical stereotriangulation RCA Data reduction technical report 43[END_REF]. The technique became popular in the Computer Vision community after the paper by [START_REF] Triggs | Bundle adjustment -A modern synthesis Vision Algorithms: Theory and Practice[END_REF].

Photogrammetric BA uses the collinearity equations to describe the world-to-camera projection. The typical presentation of the function and its linearization has been as a single, compact computation, see, e.g., Kraus (1993, Sec. 5.3.2), Wolf and Dewitt (2000, App. D-4), Mikhail et al. (2001, App. C.3), andLuhmann et al. (2014, Sec. 4.2.2). In contrast, the standard presentation of the world-to-camera projection in the Computer Vision community has been as a sequence of small operations [START_REF] Tsai | A versatile camera calibration technique for high-accuracy 3d machine vision metrology using off-the-shelf tv cameras and lenses[END_REF] with the Jacobians described using the chain rule [START_REF] Brown | Recognising panoramas, 9th IEEE International Conference on Computer Vision (ICCV 2003) 14-17[END_REF][START_REF] Fusiello | Visione computazionale. Tecniche di ricostruzione tridimensionale[END_REF]. In both presentations, the lens distortion has been treated separately. Although both research communities refer to the same [START_REF] Brown | Close-range camera calibration[END_REF] paper, the details of the application has differed.

Since its introduction, BA has evolved to be a central algorithm in photogrammetry and computer vision [START_REF] Brown | The bundle adjustment -progress and prospects[END_REF][START_REF] Triggs | Bundle adjustment -A modern synthesis Vision Algorithms: Theory and Practice[END_REF][START_REF] Luhmann | Close-range photogrammetry and 3D Imaging 2nd edition[END_REF]. Originally, the procedure was used by a small research community that understood the algorithm in detail. With a broadening user base, the fraction of users with an intimate understanding of the inner workings of BA has declined. This development has further increased with the inclusion of BA into commercial, black-box, software.

Related work

The Damped Bundle Adjustment Toolbox (DBAT) was originally developed to illustrate the behaviour of different damping schemes known from non-linear optimization [START_REF] Börlin | Bundle adjustment with and without damping[END_REF]2014). Later work has focused on the use to validate commercial black-box photogrammetric software, e.g., PhotoModeler and Photoscan [START_REF] Börlin | External verification of the bundle adjustment in photogrammetric software using the damped bundle adjustment toolbox[END_REF], especially to provide detailed diagnostics of the photogrammetric network [START_REF] Dall'asta | Network design and quality checks in automatic orientation of close-range photogrammetric blocks[END_REF][START_REF] Murtiyoso | Reprocessing close range terrestrial and UAV photogrammetric projects with the DBAT toolbox for independent verification and quality control[END_REF][START_REF] Börlin | Modular bundle adjustment for photogrammeric computations[END_REF]. Recently, the differential observation weighting capabilities of DBAT was used by [START_REF] Menna | Improving underwater accuracy by empirical weighting of image observations[END_REF] to mitigate peripheral image quality degradation in underwater photogrammetry.

The chain rule has been used to simplify geometrical computations in e.g., computer vision [START_REF] Brown | Recognising panoramas, 9th IEEE International Conference on Computer Vision (ICCV 2003) 14-17[END_REF][START_REF] Fusiello | Visione computazionale. Tecniche di ricostruzione tridimensionale[END_REF], computer graphics [START_REF] Barr | Global and local deformations of solid primitives[END_REF][START_REF] Piponi | Automatic differentiation, C++ templates, and photogrammetry[END_REF], imaging [START_REF] Gallego | A compact formula for the derivative of a 3-d rotation in exponential coordinates[END_REF], robotics [START_REF] Koditschek | Robot navigation functions on manifolds with boundary[END_REF], and registration of TLS point clouds [START_REF] Rabbani | An integrated approach for modelling and global registration of point clouds[END_REF].

In photogrammetry, the chain rule has been used for, e.g., modelling of reflected ray paths [START_REF] Rupnik | Sinusoidal wave estimation using photogrammetry and short video sequences[END_REF] and relative pose estimation [START_REF] Cheng | Relative camera pose estimation method using optimization on the manifold[END_REF]. To the authors' knowledge, the chain rule has not been used to express the photogrammetric world-to-camera projection model, including lens distortion, before.

Aim

The aim of this paper is to apply the sequential computation to the whole photogrammetric projection model, including lens distortion, and to use the chain rule to modularize the computations. Furthermore, to illustrate the flexibility of the modular approach, a non-conventional optical system -a tilt-shift lens -will be investigated.

A first version of this approach was presented in [START_REF] Börlin | Modular bundle adjustment for photogrammeric computations[END_REF]. This paper is an extended version with an added emphasis on performance vs. flexibility.

Disassembling collinearity

The collinearity equations

The mathematical problem that is solved by the BA process includes a residual between two components. One residual component is the simulated projection of an object point according to the camera model at hand. The other component is computed from the corresponding image measurement. As a typical presentation, consider the collinearity equations as presented by Luhmann et al. (2014, equation (4.94)):

𝑥 ′ = 𝑥 0 ′ + 𝑧 ′ 𝑟 11 (𝑋 -𝑋 0 ) + 𝑟 21 (𝑌 -𝑌 0 ) + 𝑟 31 (𝑍 -𝑍 0 ) 𝑟 13 (𝑋 -𝑋 0 ) + 𝑟 23 (𝑌 -𝑌 0 ) + 𝑟 33 (𝑍 -𝑍 0 ) + ∆𝑥 ′ , 𝑦 ′ = y 0 ′ + 𝑧 ′ 𝑟 12 (𝑋 -𝑋 0 ) + 𝑟 22 (𝑌 -𝑌 0 ) + 𝑟 32 (𝑍 -𝑍 0 ) 𝑟 13 (𝑋 -𝑋 0 ) + 𝑟 23 (𝑌 -𝑌 0 ) + 𝑟 33 (𝑍 -𝑍 0 ) + ∆y ′ .

(1)

Ignoring ∆x ′ and ∆y ′ , equation (1) describes the projection of an object point at (𝑋, 𝑌, 𝑍) through a pinhole camera with principal point (𝑥 0 ′ , 𝑦 0 ′ ) and camera constant 𝑧 ′ placed at (𝑋 0 , 𝑌 0 , 𝑍 0 ) and with an orientation given by the matrix 𝑅.

Another requirement for the bundle adjustment process is the linearization of equation ( 1) with respect to any parameter that is to be estimated, see, e.g., Kraus (1993, Sec. 5.3.2), Wolf and Dewitt (2000, App. D-4), or Mikhail et al. (2001, App. C.3). Some partial derivatives of equation ( 1

where 𝑘 𝑋 and 𝑘 𝑌 are the respective numerators of equation ( 1), and 𝑁 is the denominator.

Basic functions

If we group the parameters as

𝑝 = ( 𝑋 𝑌 𝑍 ) , 𝑝 0 = ( 𝑋 0 𝑌 0 𝑍 0 ) , 𝑞 = ( 𝑘 𝑋 𝑘 𝑌 𝑁 ) , 𝑢 = ( 𝑥 ′ 𝑦 ′ ) , 𝑢 0 = ( 𝑥 0 ′ 𝑦 0 ′ ) , ∆𝑢 = ( ∆𝑥 ′ ∆𝑦 ′ ) , (3)
and introduce the basic functions 𝑇 3 (𝑝, 𝑝 0 ) = 𝑝 + 𝑝 0 , 3D translation 𝐿(𝑅, 𝑣) = 𝑅𝑣, 3D rotation

𝐻(𝑞) = 1 𝑞 3 ( 𝑞 1 𝑞 2 ) , Pinhole projection 𝑆(𝑐, 𝑚) = 𝑐𝑚, Scaling 𝑇 2 (𝑛, 𝑢 0 ) = 𝑛 + 𝑢 0 , 2D translation (4)
where 𝑐 is scalar and 𝑚 and 𝑛 are 2-vectors, we may write equation (1) as the following composition: 𝑢 = 𝑔(𝑝, 𝑝 0 , 𝑅, 𝑧 ′ , 𝑢 0 ) ≡ 𝑇 2 (𝑆 (𝑧 ′ , 𝐻 (𝐿(𝑅 𝑇 , 𝑇 3 (𝑝, -𝑝 0 )))) , 𝑢 0 ).

(5)

In the definition of the functions in equation ( 4), generality has been favoured over typical usage to simplify future extensions. For instance, the function 𝐿(𝑅, 𝑣) describes an arbitrary 3D linear transformation whereas the collinearity equations assumes 𝑅 to be a rotation matrix. Similarly, the translation 𝑇 3 (𝑝, 𝑝 0 ) is defined with a positive sign on 𝑝 0 whereas the collinearity equations use 𝑇 3 (𝑝, -𝑝 0 ).

The algorithm corresponding to equation ( 5) is shown in Algorithm 1 with a graphical representation in Figure 1.

Linearization

The linearization of function compositions is governed by the chain rule, see, e.g., Magnus and Neudecker (2007). For example, the Jacobian of the projection function 𝑔 in equation ( 5) with respect to the object point coordinates 𝑝 is the matrix product of five Jacobians

[ 𝑑𝑔 𝑑𝑝 ] = [ 𝑑𝑇 2 𝑑𝑛 ] 𝑛=𝑆(… ) [ 𝑑𝑆 𝑑𝑚 ] 𝑚=𝐻(… ) [ 𝑑𝐻 𝑑𝑞 ] 𝑞=𝐿(… ) [ 𝑑𝐿 𝑑𝑣 ] 𝑣=𝑇 3 (… ) [ 𝑑𝑇 3 𝑑𝑝 ], (6)
where the brackets indicate a matrix and the subscript indicate the point at which the respective Jacobians are evaluated. A detailed inspection of equation ( 6) reveals that it is identical to equation

(2).

A function as a module

If we require that each basic function of equation ( 4) can compute both the function value and the Jacobians with respect to all its parameters, we may similarly extend Algorithm 1 to compute both its function value and its Jacobians. The extended Algorithm 2 illustrates the computation of equation ( 5) and the Jacobian of equation [START_REF] Barr | Global and local deformations of solid primitives[END_REF]. In this way we can modularize the residual computations and use the modules as building blocks. Furthermore, as each module is selfcontained, it is possible to validate the analytical Jacobian of each module independently.

Other photogrammetric modules

In addition to the functions listed in equation ( 4), we add the computation of the rotation matrix 𝑅 from the 𝜔 -𝜙 -𝜅 Euler angles (Förstner & Wrobel, 2004, equations (2.128)-(2.130))

𝑅(𝜔, 𝜙, 𝜅) = 𝑅 3 (𝜅)𝑅 2 (𝜙)𝑅 1 (𝜔), (7)
the [START_REF] Brown | Close-range camera calibration[END_REF] lens distortion model 𝐷(𝑢, 𝐾, 𝑃) = 𝑢 + 𝑑 𝑟 (𝑢, 𝐾) + 𝑑 𝑡 (𝑢, 𝑃),

and the 2D affine transformation

𝐴 2 (𝑢, 𝑏) = ( 1 + 𝑏 1 𝑏 2 0 1 ) 𝑢. (9)
In equation ( 8), the vectors 𝐾 and 𝑃 contain the radial and tangential distortion coefficients, respectively. In equation ( 9), the scalar 𝑏 1 controls the aspect ratio and the scalar 𝑏 2 controls the skew. For more details and derivation of the Jacobians, see (Börlin et .al (2018, App. B).

3 Reassembling collinearity

The Damped Bundle Adjustment Toolbox (DBAT)

The modular technique has been used in the open-source Damped Bundle Adjustment Toolbox (DBAT)1 package for Matlab [START_REF] Börlin | Bundle adjustment with and without damping[END_REF]2016). For instance, the modules presented in equation ( 4) and ( 7)-( 9) were combined differently to implement two bundle pipelines that used different adaptations of the [START_REF] Brown | Close-range camera calibration[END_REF] lens distortion model. In the Photogrammetric formulation, the Brown polynomials are used to "correct" for the effect of lens distortion on the measured image coordinates. In contrast, the formulation largely adopted by the Computer Vision community uses the same polynomials to "add" lens distortion to the ideal projection of object points [START_REF] Tsai | A versatile camera calibration technique for high-accuracy 3d machine vision metrology using off-the-shelf tv cameras and lenses[END_REF][START_REF] Heikkilä | A four-step camera calibration procedure with implicit image correction[END_REF][START_REF] Zhang | A flexible new technique for camera calibration[END_REF][START_REF] Drap | An exact formula for calculating inverse radial lens distortions[END_REF]. The two pipelines are visually contrasted in figures Figure 2 and Figure 3.

DBAT has the possibility to post-process files from two popular software: EOS Photomodeler and Agisoft Photoscan. DBAT version 0.8 was used in this paper.

Extended pinhole models

For this paper, we have used the modular technique used in DBAT to implement two extensions of the photogrammetric pipeline of Figure 2. The reference implementation (called Model 2) has no affinity. In Model 3, the affinity is applied before lens distortion. In Model 4, the affinity is applied after. The corresponding functions are:

𝑟 2 = 𝐷(𝑇 2 (𝑆(𝑠, 𝑢), -𝑢 0 ), 𝐾, 𝑃), 𝑟 3 = 𝐷(𝐴 2 (𝑇 2 (𝑆(𝑠, 𝑢), -𝑢 0 ), 𝑏), 𝐾, 𝑃), 𝑟 4 = 𝐴 2 (𝐷(𝑇 2 (𝑆(𝑠, 𝑢), -𝑢 0 ), 𝐾, 𝑃), 𝑏), ( 10
)
where 𝑠 is the pixel size and 𝑢 the measured image coordinates. The models are visually illustrated in Figure 4. The difference in the computation of the Jacobian with respect to the principal point 𝑢 0 was limited to inserting the Jacobian [ 𝑑𝐴 2 𝑑𝑢 ] at the proper place in the matrix product (in addition to a change in the evaluation points):

[ 𝑑𝑟 2 𝑑𝑢 0 ] = -[ 𝑑𝐷 𝑑𝑢 ] 𝑢=𝑇 2 (… ) [ 𝑑𝑇 2 𝑑𝑢 0 ] , [ 𝑑𝑟 3 𝑑𝑢 0 ] = -[ 𝑑𝐷 𝑑𝑢 ] 𝑢=𝐴 2 (… ) [ 𝑑𝐴 2 𝑑𝑢 ] 𝑢=𝑇 2 (… ) [ 𝑑𝑇 2 𝑑𝑢 0 ] [ 𝑑𝑟 4 𝑑𝑢 0 ] = -[ 𝑑𝐴 2 𝑑𝑢 ] 𝑢=𝐷(… ) [ 𝑑𝐷 𝑑𝑢 ] 𝑢=𝑇 2 (… ) [ 𝑑𝑇 2 𝑑𝑢 0 ] , , (11)

Image-variant estimation

A recent addition to DBAT allows IO parameters to be declared as block-invariant or image-variant [START_REF] Moniwa | The concept of photo-variant self-calibration and its application in block adjustment with bundles[END_REF] on an individual basis. Thus, some parameters can be the same for the full block whereas others can be individual for an image.

4 Experiments and results

Test case: The tilt-shift lens

A tilt-shift lens is a non-conventional optical system that allows the following lens movements [START_REF] Ray | Applied photographic optics 3rd edition[END_REF] (Figure 5):

 a tilt, i.e., a rotation of the optical axis about either the exit pupil or the centre of the sensor plane,  a shift, i.e., a translation of the optical axis, and  a rotation, i.e., a rotation about the optical axis.

The projection of a tilt-shift camera lens system is not completely modelled by the classic photogrammetric projection model (pin-hole with Brown lens distortion) available in regular photogrammetric software, including DBAT. In order to illustrate the modularity of DBAT, we chose to investigate whether the ordering of the affine and lens distortion steps, described in Section 3.2, could improve the projection model.

One simulation experiment, two calibration experiments, and one performance experiment were designed. The purpose of the simulation experiment was to investigate whether the variants in the application of the affinity had any measurable effect. The purpose of the calibration experiments was to investigate whether any of the affinity variants could improve an actual calibration of the tiltshift lens, with or without image-variant camera parameters. Finally, the purpose of the performance experiment was to investigate the relative performance of the DBAT models.

Simulation experiment

The first experiment was constructed to investigate the effect, if any, of the difference in the affinelens distortion ordering. Two sets of synthetic data were generated, where simulated error-free image observations were generated by back-projection of known 3D object coordinates and with known exterior orientation parameters (EO), and camera calibration parameters (IO) of a strong self-calibration network. The network consisted of 24 cameras at varying roll angles. About 100 targets were simulated in a 3D configuration with a largest dimension of 1000 mm (Figure 6).

Generation of synthetic data

The following algorithms were used to simulate measurements generated by Model 3 and Model 4.

Note that the sequence of the steps below is reversed compared to Figure 4 as we are building image observations.

Model 3

1. Apply the collinearity equations (3D translation, 3D rotation, pinhole projection and optical scaling).

Introduce lens distortion (iterative, [mm]).

3. Convert from mm to pixels using a non-square pixel size (𝑏 1 = 0.01218).

Introduce the principal point [pixels].

Model 4

1. Apply the collinearity equations (3D translation, 3D rotation, pinhole projection and optical scaling).

2. Apply an affine transformation to the image coordinates (𝑏 1 = 0.01218).

Introduce lens distortion (iterative, [mm]).

4. Convert from mm to pixels using a square pixel size.

Introduce the principal point [pixel].

Both simulations used a skew (shear) parameter of 𝑏 2 = 0. The algorithms were implemented in software developed in-house at FBK and not by DBAT.

Analysis of synthetic data

Each synthetic data set was analysed by a self-calibration bundle adjustment using DBAT models 2, 3, and 4. No control points were used. Instead, the datum problem was solved by fixing seven EO parameters. The prior weight for the image observations corresponded to a sigma of 0.1 pixels. The following parameters were estimated; the camera constant 𝑐, the principal point (𝑥 0 , 𝑦 0 ), the radial distortion parameters 𝐾 1 -𝐾 3 , and the tangential distortion parameters 𝑃 1 -𝑃 2 . For models 3 and 4, the affine parameters 𝑏 1 -𝑏 2 were also estimated. The quality of each analysis was evaluated in image space (𝜎 0 and 2D image point RMS) and object space (3D RMSE between the true and estimated OP coordinates). The results are given in Table 1. When the correct estimation model was used, the simulated 𝑏 1 value was recovered to the number of available decimals and the internal and external residuals were effectively zero. When the wrong model was used, the residuals were significantly higher.

Camera calibration

In the second experiment, models 2, 3 and 4 were used to analyse calibration data for a camera with a tilt-shift lens. The data set from [START_REF] Nocerino | Experiments on calibrating tilt-shift lenses for close-range photogrammetry[END_REF] was used for the calibration. The dataset was acquired in a controlled environment, with stable environmental conditions, consequently reducing the influences of uncontrolled factors on the calibration. Furthermore, high accuracy reference coordinates were available: A point precision vector length of 2.4 μm and an RMSE with respect to another photogrammetric system of 6 μm. The calibration target was a 3D photogrammetric calibration test object (Figure 7) with about 170 coded targets with a largest dimension of 900 mm. The used camera was a Nikon D750 full-frame DSLR camera with a PC-E Micro NIKKOR 45mm f/2.8D ED tilt-shift lens (Figure 5) in two different configurations:

 NORMAL: The normal configuration where neither tilt nor shift was applied.

 TILTED: The lens was tilted in the vertical plane by 4 degrees.

A high redundancy photogrammetric camera network was realized, consisting of up to 48 convergent images with a diversity of camera roll angles to enhance the determinability of the IO parameters [START_REF] Fraser | Photogrammetric camera component calibration: A review of analytical techniques Calibration and Orientation of Cameras in Computer Vision[END_REF]. The ground sample distance (GSD) was about 0.13 mm. The image measurements were performed by PhotoModeler Scanner v.6 and exported as a text file that was imported to DBAT.

The NORMAL data set was analysed and used as a reference for the processing of the TILTED data set. The TILTED data set was analysed by a self-calibration bundle adjustment in DBAT using models 2, 3 or 4. The datum and prior weights were the same as in the synthetic experiment. Furthermore, the parameters 𝑏 1 , 𝑏 2 , 𝑃 1 , and 𝑃 2 were individually included or excluded from the estimation (𝑃 1 and 𝑃 2 were always estimated together). The quality of the estimation was evaluated as in the synthetic experiment with the results of the NORMAL data set used as the reference. The results are given in Table 2.

From Table 2, we see that the difference between models 3 and 4 on this data is small. The internal and external residuals were small when the aspect parameter 𝑏 1 and the decentering distortion parameters 𝑃 1 -𝑃 2 were included in the estimation. In those cases, the 3D RMSE was about 0.09 GSD and the estimated 𝑏 1 value was about 0.0019. This value is equal to the value in Nocerino et al.

(2016, Table 3, column T-4S0R90) that was estimated by other software. In contrast, the inclusion of 𝑏 2 resulted in an estimated value of 𝑏 2 was about 10 -5 and had a negligible effect on the residuals.

Image-variant calibration

The best 𝑏 2 -free setup for model 3 from the second experiment was used as the reference for the third experiment. Three different sets of IO parameters were allowed to be image-variant. The first set consisted of the principal distance and point. The second set added the radial and tangential lens distortion parameter. The final set consisted of all IO parameters, including the aspect parameter 𝑏 1 .

In order to avoid too weak networks, prior observations of the image-variant parameters were added [START_REF] Tecklenburg | Camera modelling with image-variant parameters and finite elements Optical 3-D measurement techniques V[END_REF]. As the prior values, the posterior estimated values from the reference experiment were used. The prior weights were chosen to correspond to twice the posterior standard deviation from the reference experiment. The results are given in Table 3.

The internal assessment suggests a slight improvement in accuracy. However, the comparison with the reference 3D coordinates instead suggests that the accuracy slightly worsens for all imagevariant experiments.

Performance

To investigate the relative performance between the models, the St-Pierre-le-Jeune data set from [START_REF] Murtiyoso | Reprocessing close range terrestrial and UAV photogrammetric projects with the DBAT toolbox for independent verification and quality control[END_REF] was used. The data set consisted of 239 images and about 18000 object points. DBAT model 1, a previous, more optimized implementation of Model 2, was used as a performance reference. A self-calibration bundle adjustment process was applied to the St-Pierre data set. The timings were performed using an HP EliteDesk 800 with Quad Intel Core i7-4790 CPU at 3.60GHz with 32 GB RAM running Matlab R2018b under Ubuntu Linux 14.04.5. In all cases, the models required the same number of iterations and produced the same results to the reported number of digits. The average bundle execution time over five repeated runs was about 13 seconds for Model 1. The corresponding times for models 2-4 were 53-56% higher.

Discussion and conclusion

The primary goal of this paper was to apply the sequential presentation to the whole photogrammetric projection model, including lens distortion, and to use the chain rule to modularize the computations. Previous presentations have left out the camera-internal operations, notably lens distortion. The presentation in this paper does show that the residual computations used by bundle adjustment can be split into modules. If each module is responsible for computing Jacobians with respect to each parameter, in addition to the function value proper, the chain rule can be used to combine the analytical Jacobians of the simple operations into analytical Jacobians of complex expressions. Furthermore, the presentation illustrates how the [START_REF] Brown | Close-range camera calibration[END_REF] lens distortion model is treated differently by the photogrammetry and computer vision communities.

The tilt-shift lens is a complex design whose projection model is not yet rigorously supported by regular photogrammetric software, including DBAT. In a previous paper, it was found that some of the deviation of the tilt-shift lens from the classic photogrammetric projection model (pin-hole with Brown lens distortion) could be mitigated by including the affine distortion parameter 𝑏 1 in the estimation [START_REF] Nocerino | Experiments on calibrating tilt-shift lenses for close-range photogrammetry[END_REF]. The modularity of DBAT presented an opportunity to investigate further modifications of the projection model. As such, two models with different placement of the affine transformation compared to lens distortion were implemented with minimal effort.

An experiment on synthetic image observations verified that the relative placement of the affine transformation and lens distortion can be detected by DBAT on synthetic, error-free data. A calibration experiment was constructed to test the hypothesis that either ordering could explain more of the tilt-shift distortion of a real tilt-shift lens. Furthermore, an image-variant bundle adjustment [START_REF] Moniwa | The concept of photo-variant self-calibration and its application in block adjustment with bundles[END_REF][START_REF] Tecklenburg | Camera modelling with image-variant parameters and finite elements Optical 3-D measurement techniques V[END_REF] was tested to verify if it would be possible to better model the distortions introduced by gravity effects and the tilt angle. The results did not support either hypothesis. Compared to other, possibly unmodelled effects, the ordering does not significantly contribute to the observed residuals and the addition of image-variant camera parameters did not improve the results. Instead, the results support the conclusion in Nocerino et al.

(2016) that the aspect parameter 𝑏 1 and the decenter parameters 𝑃 1 -𝑃 2 are significant but the skew parameter 𝑏 2 is not.

From the results, we may conclude that the standard pinhole camera model with Brown lens distortion cannot fit the tilted data better, even with the image-variant approach. However, if an RMSE of 0.09 times the GSD is considered satisfactory, standard photogrammetric formulations and software can be used in 3D modelling application that require the tilt-shift lens system. We believe that to improve on those results, a proper mathematical projection model of the tilt-shift lens is required. In relation to the main goals of this paper, we conclude that the modularity of DBAT can indeed be used to test novel projection models with minimal implementation effort.

The performance results in this paper show an increase of about 50% in computation time on a medium-sized problem for the flexible implementation compared to the original implementation in DBAT. A potential explanation is if the computation of the composed Jacobian as a chained multiplication of simpler Jacobians is inefficient. Another possible explanation is if some of the computations that depend on the same parameters are distributed over multiple sub-expressions. In that case, the expressions may need to be jointly considered in order to be fully optimized. If true, either case would highlight a limitation of the chain rule approach.

The reported increase in execution time would hardly be acceptable for production code, where speed is essential. However, the driving force behind the development of DBAT has been to provide an open-source, statistically rigorous, reference implementation of Bundle Adjustment. Performance has been important, but never the primary focus. Furthermore, the code for the original projection model has gone through several optimization iterations whereas the flexible chain rule code has not.

It remains to be seen how much performance can be recovered in the future.

In some cases, flexibility is more important than performance. The flexibility of the chain rule formulation should allow researchers to investigate novel and non-standard sensors or data processing techniques. Furthermore, several new photogrammetric applications, including underwater applications, need refined refractive models. In teaching, the flexibility should allow students to experiment with both existing and novel computations.

In summary, the presentation and results in this paper show that it is possible to use the chain rule to modularize the bundle adjustment computations, including lens distortion. Currently, there is a performance loss associated with the modular implementation. In cases where flexibility is more important than performance, we argue that DBAT with the chain rule formulation can be a useful tool to investigate novel sensors, data processing techniques, and refractive models.

Future work include optimization of the flexible models, complementing the pin-hole module with a fish-eye module, complementing the omega-phi-kappa rotation model by, e.g. azimuth-tilt-swing, and extensions to refractive models to handle the tilt-shift lens or multimedia projection models in, e.g., underwater photogrammetry.

Algorithms

Algorithm 1 The pinhole projection ( 5) is evaluated by a sequence of function calls.

procedure PinholeNoJac(𝑝, 𝑝 0 , 𝑅, 𝑐, 𝑢 0 )

 𝑎 1 ← 𝑇 3 (𝑝, -𝑝 0 )

 𝑎 2 ← 𝐿(𝑅 𝑇 , 𝑎 1 )

 𝑎 3 ← 𝐻(𝑎 2 )
 𝑎 4 ← 𝑆(𝑐, 𝑎 3 ) Figure 2: The computational chain implemented in DBAT. In the photogrammetric formulation, the optical scaling in the camera results in image coordinates expressed in physical units, e.g., mm. The image coordinates are scaled from pixels to mm before the (Brown D. C., 1971) polynomials are used to "correct" the measured image coordinates for lens distortion. The residual (thick circle) is computed as the difference between the projected ideal point and the corrected point.


Figure 3: The DBAT implementation of the Computer Vision formulation of the (Brown D. C., 1971) lens distortion model. The Brown polynomials are used to introduce lens distortion to the projected ideal point in normalized units before the coordinates are scaled directly to pixels. The residual (thick circle) is computed between the measured point and the "distorted" projected point. The same modules have been used as in Figure 2. 9 Tables

Figure 1 :

 1 Figure 1: The computational chain corresponding to equation (5). The grey rectangles indicate bundle adjustment parameters. The white circles indicate component functions from equation (4). The arrows indicate how the parameters and results are propagated. The arrow labels indicate the name of the formal (input) parameter of the corresponding function.

Figure 4 :

 4 Figure 4: Two versions of the right-hand side of Figure2with different relative placement of the affine transformation with respect to lens distortion. In Model 3 (left), the affine distortion was applied before lens distortion removal. In Model 4 (right), the affine distortion was applied after lens distortion removal. In both cases, the digital scaling uses square pixel sizes. Any non-unit aspect ratio is handled by the affine step.

Figure 5 :

 5 Figure 5: The Nikon D750 DSLR camera with the PC-E Micro NIKKOR 45 mm f/2.8D ED tilt-shift lens used in this paper.

Figure 6 :

 6 Figure 6: The synthetic network consisted of 24 cameras at varying roll angles. The simulated targets were in a 3D configuration with a largest dimension of 1000 mm.

  𝑎 5 ← 𝑇 2 (𝑎 4 , 𝑢 0 )

	8 Figures
	 return 𝑎 5
	end procedure
	 (𝑎 1 , 𝐽 𝑝 ) ← 𝑇 3 (𝑝, -𝑝 0 ),	𝐽 𝑝 = [	𝑑𝑇 3 𝑑𝑝 ],
	 (𝑎 2 , 𝐽 𝑣 ) ← 𝐿(𝑅 𝑇 , 𝑎 1 ),	𝐽 𝑣 = [	𝑑𝐿 𝑑𝑣 ] 𝑣=𝑎 1	,
	 (𝑎 3 , 𝐽 𝑞 ) ← 𝐻(𝑎 2 ),	𝐽 𝑞 = [	𝑑𝐻 𝑑𝑞 ] 𝑞=𝑎 2	,
	 (𝑎 4 , 𝐽 𝑚 ) ← 𝑆(𝑐, 𝑎 3 ),	𝐽 𝑚 = [	𝑑𝑆 𝑑𝑚 ] 𝑚=𝑎 3	,
	 (𝑎 5 , 𝐽 𝑛 ) ← 𝑇 2 (𝑎 4 , 𝑢 0 ),	𝐽 𝑛 = [	𝑑𝑇 2 𝑑𝑛 ]	𝑛=𝑎 4	,
	 𝐽 ← 𝐽 𝑛 𝐽 𝑚 𝐽 𝑞 𝐽 𝑣 𝐽 𝑝 ,	𝐽 = [	𝑑𝑔 𝑑𝑝 ],
	 return (𝑎 5 , 𝐽)
	end procedure

Algorithm 2: The pinhole projection and one of its Jacobians computed by the same sequence of function calls as in Algorithm 1. Each called function computes both the function value and the necessary Jacobians. The complete analytical Jacobian 𝐽 is formed by multiplying the component Jacobians.

procedure PinholeWithJac(𝑝, 𝑝 0 , 𝑅, 𝑐, 𝑢 0 )

Table 1 :

 1 Assessment of the analysis of the synthetic data sets. The IO parameters columns indicate what IO parameters andhow many (𝑛) were included in the estimation. The RMS is the point residual over all image observations. The RMSE is the 3D error over all object points. The 𝑏 1̂ column shows the estimated 𝑏 1 value. The rows with the smallest residuals are indicated in grey.

	Simulated DBAT	IO parameters		Internal/external assessment
	Dataset model parameters	𝑛	𝑏 1 ̂	𝜎 0 ̂	RMS (px) RMSE (μm)
	3	2	𝑏 1 𝑏 2 𝑃	8	0	34.2847	4.56	620.60
		3	𝑏 1 𝑏 2 𝑃	10 0.01218 00.0003	0.00	000.01
		4	𝑏 1 𝑏 2 𝑃	10 0.01223 00.3773	0.05	008.26
	4	2	𝑏 1 𝑏 2 𝑃	8	0	34.1330	4.54	616.59
		3	𝑏 1 𝑏 2 𝑃	10 0.01212 00.3697	0.05	008.33
		4	𝑏 1 𝑏 2 𝑃	10 0.01218 00.0003	0.00	000.01

Table 2 :

 2 Assessment of DBAT estimation models 2, 3, and 4 on the TILTED data set. The IO parameters and assessment are as in Table1, except that the target coordinates computed from the NORMAL data set were used as the true data. The redundancy (number of observation minus number of parameters) was around 7800. The smallest residuals were obtained when 𝑏 1 and P were estimated (gray rows).

	DBAT	IO parameters	Internal/external assessment
	Model	𝑏 1 𝑏 2 𝑃 𝑛	𝜎 0	RMS (px) RMSE (μm)
	2		𝑃 8	5.9	0.79	82.3
			6	5.9	0.80	92.9
	3	𝑏 1	𝑃 9	1.1	0.15	11.9
		𝑏 1 𝑏 2 𝑃 10 1.1	0.15	11.9
		𝑏 1	7	3.0	0.41	72.9
		𝑏 1 𝑏 2	8	3.0	0.41	73.2
		𝑏 2	7	5.9	0.80	92.6
		𝑏 2 𝑃 9	5.9	0.79	82.3
	4	𝑏 1	𝑃 9	1.1	0.15	11.9
		𝑏 1 𝑏 2 𝑃 10 1.1	0.15	11.9
		𝑏 1	7	3.0	0.41	73.3
		𝑏 1 𝑏 2	8	3.0	0.41	73.3
		𝑏 2 𝑃 9	5.9	0.79	82.3
		𝑏 2	7	5.9	0.80	92.8

Table 3 :

 3 Results of the image-variant experiments for DBAT model 3 without 𝑏 2 with three different sets of image-variantparameters. The number of IO parameters increase from 𝑛 = 9 for the fully block-invariant case to 𝑛 = 432 for the fully image-variant case. Since a corresponding number of prior observations are added, the redundancy remains around 7800.The grey row corresponds to the first model 3 row of Table2.

	IO parameters		Internal/external assessment
	Image-variant params	𝑛	𝜎 0 ̂	RMS (px) RMSE (μm)
	None	9	1.1	0.15	11.9
	𝑐, 𝑥 0 , 𝑦 0	150 1.0	0.14	12.0
	𝑐, 𝑥 0 , 𝑦 0 , 𝐾 𝑖 , 𝑃 𝑖	385 1.0	0.13	12.1
	𝑐, 𝑥 0 , 𝑦 0 , 𝐾 𝑖 , 𝑃 𝑖 , 𝑏 1	423 1.0	0.13	12.1

https://github.com/niclasborlin/dbat


domain: sde chim

text:
I Jovovic

V Grossi

P Adam

F Gelin

I Antheaume

M Ader

P Cartigny

A NEW METHOD FOR ORGANIC AND MINERAL SULFUR MULTI-ISOTOPE ANALYSIS OF SEDIMENTS AND PETROLEUM SOURCE ROCKS

Introduction

Sulfur (S) is a key element of biogeochemical cycles. Its four stable isotopes ( 32 S, 34 S, 33 S, 36 S, in order of decreasing abundance) are subject to equilibrium and kinetic fractionations, which can be used as tracers to understand processes involved in the S cycle. However, the analytical procedure for the determination of the S multi-isotope composition requires the quantitative extraction and purification of S and its transformation into Ag2S, which can then be analyzed by Fluorination-GC-IRMS. If the specific extraction and purification of the various mineral S pools (sulfides, sulfates and elemental sulfur, S 0 ) are well-known procedures, the quantitative extraction and purification of organic sulfur (Sorg) still requires further development. Indeed, existing procedures for the extraction of Sorg based on either Raney Nickel desulfurization (Oduro et al., 2011), or Parr bomb oxidation (e.g. Marcano et al., 2013), are not well adapted since the former is restricted to specific S species and is sensitive to steric hindrance, whereas the latter is technically restrictive.

Here we propose a new stepwise chemical procedure for the quantitative recovery and multiisotope analysis of Sorg from both solvent soluble and insoluble fractions, which further allows the multi-isotope analysis of inorganic S species in the same samples. Its application to different well-known petroleum source rocks and to an oil demonstrates its great interest for disentangling the S biogeochemical cycle in these different settings.

Results

The new procedure that was developed (Fig. 1) allows the quantitative and sequential recovery of all the mineral (sulfides, sulfates and S 0 ) and organic (Sorg from soluble and insoluble organic matter, OM) S pools as analyzable Ag2S from the same sample (Jovovic et al., in prep.). Erreur ! Source du renvoi introuvable.The extraction of Sorg is based on the quantitative oxidation of S bound either to soluble or insoluble OM into sulfates using hypochlorite solutions. Ag2S for multi-isotopic analysis can then be generated from these sulfates via classical procedures. The kinetics of the oxidative digestion of OM and its possible influence on the S isotopic composition of Sorg were investigated, allowing optimal conditions to be set up.The comparison of our new procedure with previous ones based on Raney Nickel desulfurization (Oduro et al., 2011) showed extraction yields 3 to 200 times higher with the new procedure when applied to soluble and insoluble OM.

This procedure was validated on various samples with different S contents comprising recent sediments from an euxinic crater lake, petroleum source rocks from the Green River and Monterey formations and an oil (Rozel Point). We were able to purify and analyse the abovementioned S pools from these samples with a good replicability, and to reconstruct their S mass-balance states. These mass-balances matched perfectly with the S-content and δ 34 S evaluated by elemental analysis of the same samples, demonstrating their reliability.

Figure 1 Workflow for the sequential recovery and multi-isotope analysis of organic and mineral sulfur pools from sediments/oils.

The δ 34 S values of mineral S species retrieved from the different samples investigated showed large ranges of variation (e.g. +2.5‰ <δ 34 Ssulfides<+35‰) but were compatible with data previously published in the literature (when available). A 34 S-enrichment of Sorg relative to Ssulfides was observed in some samples, also matching literature data (e.g. δ 34 Sorgδ 34 Ssulfides= +3‰ in the Monterey Fm; Zaback and Pratt, 1992). However, unprecedented 34 Senrichments of Sorg were observed in Green River Fm samples, with a difference between δ 34 Sorg and δ 34 Ssulfides ranging from -20 to -25‰ instead of the classical +5‰ shifts reported previously (e.g. Tuttle and Goldhaber, 1993). Such differences might be explained by high heterogeneities of Sorg in some source rocks. However, since our Sorg recovery yields were significantly higher than those reported in previous studies for equivalent samples, it may be that earlier procedures used for determination of δ 34 Sorg were not quantitative and that they led to isotopic fractionations (and thus to incorrect δ 34 S values). Interestingly, similar shifts between the δ 34 S of soluble and insoluble Sorg and the δ 34 S of the different mineral pools (sulfides, sulfates and S 0 ) were observed in sediments from an euxinic modern analogue of lacustrine source rocks, supporting the reliability and geochemical significance of such unusual δ 34 S differences.

Conclusions

Our new procedure allows to recover separately, specifically and quantitatively all the S pools present in various types of samples, and to analyze them for S multi-isotopes. In addition to demonstrating the interest of an S multi-pool approach, some data lead to the reconsideration and to a new understanding of the S cycle in complex biogeochemical systems including important petroleum source rocks.

th International Meeting on Organic Geochemistry (IMOG) 1-6 September 2019, Gothenburg, Sweden