Catherine Jeandel

Derek Vance

New tools, new discoveries in Marine Geochemistry

Keywords: Ocean, Trace elements and isotopes, sampling and detecting challenges, oceanic (un)constrained budgets 94

Progress in the development of seawater sampling systems, analytical procedures and mass spectrometry have allowed measurements of trace elements and their isotopes in the ocean at spatial resolutions and at concentrations never achieved before. Marine geochemists are now exploiting these new opportunities to measure, for the first time, the stable isotopes of trace metals that are essential for marine life (e.g: Fe, Cu, Ni, Mo, Zn). The new data have already produced new insights into the interaction of the ocean's life with dissolved trace constituents, and the sources and sinks of these trace elements.

INTRODUCTION

The Earth's ocean is large (371x10 6 km 2 , corresponding to 70.8% of the Earth's surface), deep (3800 m on average), and represents the largest reservoir of water on Earth (1.4x10 9 km 3 ). It is a remote and often hostile environment. Exploring the chemistry of this ocean is a challenge, the first constraint being the requirement for specialized oceanographic research vessels to collect samples. Beyond this primary and obvious challenge, there are three main obstacles that make sampling and analysis for trace elements and their isotopes ("TEI" hereafter) difficult: pressure, which increases by 1 bar every 10m, the fact that the water is very salty and the fact that most of the reactive species of interest to the marine geochemist occur at very low concentrations, often ranging from nanomolar (10 -9 mol/L) to atomolar (10 -18 mol/L). In other words, marine geochemists measure very rare substances: They are trying to isolate something like 1 gram of Fe from 1250 tons of salt.

The immediate consequence is that research in marine geochemistry is strongly dependent on very clean sampling and analytical tools, as well as their continuous technical development. With this article we wish to underline how progress in understanding ocean chemistry is closely tied to the development of such methodologies. In the last decade the GEOTRACES programme that has been instrumental in driving such progress, on the sea and back home in the laboratory. We cite, as an example, a recent compilation describing the collection and chemical analysis of suspended and sinking particles (McDonnell et al 2015;[START_REF] Lam | Methods for analyzing the concentration and speciation of major and trace elements in marine particles[END_REF].

In addition, because the GEOTRACES program is generating an open source data base, it has developed rigorous quality control of the data (Bowie and Tagliabue, toolkit section, this issue). Finally the availability of superb sample sets has driven the isotope community to perfect the complicated and rigorous laboratory approaches required for isotope analysis, and new disciplines have opened up as a consequence.

As a result, we have today a large spectrum of tracers that allow the description, understanding and quantification of many processes in the ocean. An example is the pioneering measurements of abyssal

TRACING RARE METALS IN THE SEA: THE CHALLENGES

Sampling: Filling a bottle deep in the sea

Collecting seawater under pressures that vary from 1 to 600 bars (up to 1000 bars in deep ocean trenches), and preserving the integrity of the sample from the collection depth to the surface -i.e., avoiding leaks -remains a challenge. The reliability of sampling bottles has improved with time: today, multiple bottles are mounted on a "rosette" (cover figure) and operated using an electro-mechanical Kevlar cable. The low concentrations of the TEIs impose the further requirement that these sampling systems be "clean", in other words that there is no contamination of the samples from the local environment. On a metal-rich and rusty ship, this is a major issue.

In addition to the dissolved pool, particle concentrations and composition are crucial parameters for understanding ocean chemistry, including interaction with life in the ocean and transfer to sediment. Sinking particles transfer solid material and associated chemical species from surface to deep, contributing to the sequestration of these elements and species to sediment. In order to sample this vertical flux, sediment traps have been developed and moored since the late 1970s, despite the pressure difficulty on the one hand, the requirement for cleanliness on the other, and the risk of tilting by currents. However, this sinking material represents less than 20% of the particle stock, the remainder being composed of small suspended particles, which exchange chemical elements with the dissolved pool. Suspended particles are rare and precious sources of information in the open ocean: their concentrations vary from 10 to 50 µg/L. It rapidly became clear that filtering large volumes of seawater was necessary in order to obtain large enough suspended particulate samples for the analysis of TEIs. This requires the deployment of in situ pumping systems that are also able to work at high pressure, and to filter ~ 1000 L in a few hours. With the exception of a first pump developed by our Indian colleagues in the 1970s [START_REF] Krishnaswami | Large-volume in-situ filtration of deep Pacific waters: Mineralogical and radioisotope studies[END_REF], these in situ pumping systems have been fully commercialized only since the late 1980s.

The other main constraint is imposed by the requirement for dense geographical coverage of a huge reservoir in a limited time. An oceanographic cruise typically lasts 50 days, presenting the challenge of documenting TEI distributions with the highest possible resolution in the time available. The classical tools described above are still time limited: cables ascend and descend at 1m/s, bottles are classically 12 L in volume -so that several casts at the same location are often required to satisfy all the TEI measurement requirements. In situ pumping system casts often require 5 to 6 hours to filter large volumes of water at 8 depths only. Today, this requirement of high resolution/high frequency sampling is driving the development of new tools: for example, chemical sensors that can be mounted on gliders or floats. However, while these newest tools can deliver nutrient and oxygen concentrations, they are still a long way from being able to detect the low concentrations that characterize TEI distributions [START_REF] Fassbender | Perspectives on Chemical Oceanography in the 21st century: Participants of the COME ABOARD Meeting examine aspects of the field in the context of 40 years of DISCO[END_REF].

Detection: finding a trace of Fe in tons of salt.

Another route to increasing the number of samples is to reduce analytical detection limits, to reduce the volume required for a given element -in effect reducing the "cable time" and improving the sampling density.

This field has undergone the greatest progress over the past 20 years, mostly with the laboratory-based development of high performance resins and the availability of extremely sensitive mass spectrometers.

The main analytical challenge is to concentrate the desired chemical element(s) from the very dilute seawater solution. Seawater is a complex matrix characterized by high concentrations of major ions composing the "salt" (e.g; Na + , Cl -, K+, Ca ++ , Mg ++ , SO 4 --, etc…). These ions can interfere with trace metal analysis. Moreover, salt is a real poison for the analytical devices because it rapidly clogs their inlet systems, resulting in a critical drop in their sensitivity. Given that one liter of seawater contains 35 g of salt and only a few nanograms, picograms or femtograms of the target element, a concentration step that does not introduce contamination prior to detection is challenging. Many pre-concentration methods have been developed: organic solvent extraction [START_REF] Bruland | Sampling and analytical methods for the determination of copper, cadmium, zinc, and nickel[END_REF], co-precipitation with magnesium hydroxide [START_REF] Wu | Low blank preconcentration technique for the determination of lead, copper, and cadmium in small-volume seawater samples by isotope dilution ICPMS[END_REF], solid phase extraction 3 using various chelating resins either in columns (Sohrin et al, 2010) or as batch extraction onto resin beads [START_REF]Analysis of trace metals (Cu, Cd, Pb, and Fe) in seawater using single batch Nitrilotriacetate resin extraction and isotope dilution inductively coupled plasma mass spectrometry[END_REF]. In this domain, again, considerable improvement of the chelating resin capacities by commercial chemical companies has yielded progressively better recoveries and more efficient elimination of salt. The end result has been a significant decrease in the required sample volume and analyst effort to perform any TEI extraction. Chemical protocol improvement, together with the development of mass spectrometers coupled to a plasma source (ICP-MS), operating at high mass resolution (HR-ICPMS) or with multiple collectors (MC-ICPMS), has helped to propel marine geochemistry, technologically, into the 21 st century.

High resolution sampling: less means more

In the late 70s, measuring marine Co, Cd, Zn and Ni concentrations required passing 4 l of seawater through a Chelex-100 column (a tube filled with tiny resin beads designed to separate trace elements from a concentrated matrix), followed by a flameless atomic absorption analysis [START_REF] Bruland | Sampling and analytical methods for the determination of copper, cadmium, zinc, and nickel[END_REF]. Today, the same set of tracers and others like Fe, Mn, Pb and Co can be simultaneously extracted and analyzed using only 12ml

of seawater [START_REF] Milne | Determination of Mn, Fe, Co, Ni, Cu, Zn, Cd and Pb in seawater using high resolution magnetic sector inductively coupled mass spectrometry (HR-ICP-MS)[END_REF]. Such extremely powerful multi-element analysis results from the coupling of a state of the art chelating resin to analysis by ICP-MS. This allows the documentation of oceanic sections for trace metals at geographical resolutions that have never reached before, revealing: for example, uncovering contrasting structures in the distributions of two similar lithogenic elements such as Al and Fe (Fig. 1).

Significant progress has also been made in the analysis of the concentrations of the rare radioactive isotopes of thorium and protactinium in seawater, both quantitative tracers of particle dynamics (Geibert, this Detailed ocean cross sections such as these, being published now, reveal more precisely than ever the particle dynamics that transport not only TEIs but also carbon and nutrients, allowing us to better quantify how elements are transferred to depth through processing by organisms, the so-called "biological pump "(Lohan and Tagliabue, this issue, Hatje et al, this issue).

NEW TRACERS, NEW IDEAS

In this section we highlight some of the new tracers and new ideas that have recently emerged in marine geochemistry, exploiting the technical advances summarized above. This brief survey cannot be exhaustive but by focusing on selected examples offered by the new metal stable isotope tool set we convey an impression of the excitement in this field.

Trace metal isotope budgeting of sinks: the example of molybdenum

Amongst the new tools that have driven significant recent progress in ocean chemistry is our ability to determine the relative abundancies of metal stable isotopes -which we could not measure to the required precision before the turn of the century. These include the isotopes of trace metals essential for marine life (e.g., Fe, Zn, Mo, Ni). The fluxes of these metals in and out of the dissolved pool are important to quantify for a number of reasons. First, the input fluxes and residence time determine the inventory that is available to phytoplankton and other organisms today. Second, the controls on the input and output fluxes reflect important planetary processes, e.g., chemical weathering, biological uptake, burial of organic carbon, planetary oxidation, whose rate and pattern on the modern and past Earth have much to tell us about the operation of the Earth System. Quantifying these budgets using elemental inventories alone is, however, difficult. For example, though trace metal concentrations have been measured in the dissolved load of many rivers, the impact of the massive change in chemistry, from fresh to salt water, at the continent-ocean interface is still poorly constrained. There is debate over whether the dissolved load is the dominant input of trace metals to the oceans, or whether dissolution of metals from the particulate load can impact seawater chemistry (e.g., [START_REF] Jeandel | The influence of terrigeneous particulate material dissolution on ocean chemistry and global element cycles[END_REF]. Such a mechanism is well illustrated and quantified by the distribution and fate of 10 Be/ 9 Be ratio between rivers, estuaries and Open Ocean (von [START_REF] Blanckenburg | River fluxes to the sea from the ocean's 10 Be/ 9 Be ratio[END_REF].

Characterisation of sinks from the dissolved load of the oceans is also difficult, requiring a large number of measurements of sediments in different oceanographic settings, a good knowledge of the geographic distribution of those settings and, importantly, an understanding of the diagenetic processes that may remobilise trace metals back into the aqueous phase. Isotope-based characterisation of the oceanic dissolved pool, as well as these inputs and outputs, can provide a better route to understanding the ocean, and this tool has received renewed impetus through GEOTRACES.

We first illustrate such an approach with a relatively well-understood element and isotope system, molybdenum (Mo), and two of its stable isotopes: 98 Mo and 95 Mo. Fig. 3 shows a summary of the modern oceanic budget of Mo from [START_REF] Kendall | The stable isotope geochemistry of molybdenum[END_REF], and the narrative below is based on that synthesis. Mo, at a concentration of about 105 nM, is the most abundant transition metal in the oceans and is well mixed and homogenised due to its long residence time of about 450kyr. The continental crust is a principal source, via rivers, with hydrothermal fluids contributing a more minor input. Molybdenum exhibits two distinct behaviours in seawater, and key to this system is that the resulting pathways are associated with distinct stable isotope signatures. Under the oxidised conditions that predominate in the modern ocean, Mo occurs as a highly inert and soluble molybdate ion (MoO 4 2 ), and in these conditions the main sink is via slow adsorption to particulate Fe-Mn oxides. There is isotope fractionation associated with this slow sink:

98 Mo/ 95 Mo on the Fe-Mn oxide surface is 3 permil lower than that of the MoO 4 2-dissolved in seawater. In contrast, under sulphidic conditions such as in the modern Black Sea, dissolved Mo is progressively sulphidised, ultimately to MoS 4 2-. These species are much more reactive and removed to sediment. Because this process is often almost quantitative, there is little isotope fractionation associated with it. If we now know the isotope composition of the oceans' Mo inputs, the Mo composition of the seawater, and the Mo composition of the sinks, we can quantify the partitioning into oxic sinks (associated with isotope fractionation) and anoxic sinks (no isotope fractionation).

The Mo isotope composition of the dissolved pool of the oceans is homogeneous (as expected from its long residence time) and distinctly heavier in Mo isotopes than either the upper continental crust at  98 Mo ~ +0.3‰, or the dissolved load of rivers at about +0.7‰ (see Fig. 3 for details of notation). This difference is due to the preferential removal of light isotopes from seawater onto Fe-Mn oxide particulates. The other principal sink, that mediated by the conversion to much more reactive sulphide species in euxinic basins, results in sediment with the  98 Mo of seawater (Fig. 3), as there is no isotope fractionation associated with its complete precipitation. Using an isotope mass-balance, we find that the oxic pool obtains 1x10 8 moles yr -1

, and the euxinic pool 0.2x10 8 moles yr -1 (Fig. 3). However, a complication arose with the realisation that there are isotope fractionations associated with other important sinks, namely those deposited at productive, upwelling, ocean margins. Revised Mo isotope budgeting, in fact, puts upwelling margins at about 50% of the total output (Fig. 3). This example illustrates how the isotopic fingerprinting of the main inputs and outputs serves to place constraints on oceanic budgets.

of diagenetic processing and remobilisation of sedimentary trace metals from seafloor sediment, and the metals contained on the particulate load deposited by rivers, including the isotope fractionation involved with particle-seawater exchange and during redox shifts. In other words, the particle-solution interactions at the land-sea and air-sea interfaces may be key. A further example concerns the interactions between the biological realm and oceanic TEIs, which remains an open field and is also associated with substantial isotope fractionation: how do living species assimilate TEIs; what is the role of the bacteria in TE speciation? We suggest that progress in understanding these processes is essential to support increasing applications of these systems in Earth history. One of the next frontiers for ocean exploration lies in the development of in situ sensors able to detect trace elements and automated clean sample collection systems.

  issue; Chase et al., this issue). Early and seminal papers on the distributions of 230 Th and 231 Pa in the open ocean required samples of 1000 liters because of the very low abundance of these tracers (femtomoles per liter or less) andthe sensitivity constraint imposed by the detection method used at that time, alpha-counting(Nozaki et al, 1981, Bacon and[START_REF] Bacon | Distribution of thorium isotopes between dissolved and particulate forms in the deep sea[END_REF]. In the early 2000s, the advent of high resolution ICP-MS, revolutionized the laborious data acquisition involved. The required volume dropped to 1-20 l (GEOTRACES cookbook, 2017), resulting in the ability to produce oceanic sections at unprecedented vertical and horizontal resolution (e.g.Hayes et al, 2015). In the North Atlantic, for example, these sections revealed the locations of intensified scavenging (benthic layers, in the vicinity of continental margins, or in the hydrothermal plume emanating from the Mid Atlantic Ridge, Fig.2). Because the residence time of 231 Pa is longer than that of230 Th, it is transported horizontally in deep waters until it encounters stronger particle fluxes. The relative roles of these twin processes of horizontal advection versus element scavenging have also been better deciphered by these data(Hayes et al, 2015).

Figure 1 :

 1 Figure 1: Contrasting distributions of dissolved Al (dAl, top) and Fe (dFe, bottom) along the meridional GEOTRACES section GA02 in the Atlantic Ocean. Note the contrasting patterns despite the fact that both elements are of lithogenic origin: Dissolved Al distribution maps the main water masses encountered in the Atlantic Ocean: with North Atlantic Deep Waters (NADW) enriched by inputs from the continents versus depleted water masses originating from the remote Southern Ocean (Antarctic Intermediate Water AAIW, and Antarctic Bottom Water AABW). High surface values in the tropics reflect Saharan dust inputs. Dissolved Fe distributions are patchier, mostly reflecting the micro-nutrient properties of this tracer: depleted at the surface due to biological uptake and enriched at depth due to redissolution.

Figure 2 :

 2 Figure 2: Distribution of the radioactive 230 Th isotope formed by decay of 238 U in the Atlantic Ocean,

Figure 3 :

 3 Figure 3: A simplified oceanic budget for Mo and its isotopes showing the main inputs and outputs to and from the dissolved pool (after Kendall et al., 2017). The numbers below each sink and source are elemental fluxes in units of 10 8 moles yr -1 (top) and isotopic composition of that flux.    is the 98 Mo/ 95 Mo ratio expressed

Figure 4 :

 4 Figure 4: Simplified oceanic budget for Ni and its isotopes, showing the main inputs and outputs to and from the dissolved pool (after Ciscato et al., 2018).   Ni is the 60 Ni/ 58 Ni ratio expressed relative to a value of 0‰ for NIST SRM 986. Unlike Mo, at our present level of understanding this budget appears not to be balanced. The apparent imbalance is represented on the diagram as a missing input. But an alternative explanation of the imbalance is that the output fluxes shown here do not represent real burial fluxes, quantitatively or isotopically: i.e., that there is diagenetic remobilisation within sediment and a poorly quantified back flux to the water column.

C, requested in the early 1960s by Henry Stommel from geochemists able to perform these difficult analyses, like Wally Broecker at Columbia University, USA. The result was the first estimate of the average mixing time of the ocean, a major step forward. As detailed in this and the other papers in this issue, TEIs are the sources of information on an array of processes, from biological productivity to past ocean circulation.

Acknowledgements

Catherine Jeandel conducts her research at LEGOS in Toulouse, and she is supported by Centre National de la Recherche Scientifique. Derek Vance is supported by ETH Zürich and the Swiss National Science Foundation.

We are grateful to Elena Masferrer-Dodas, Bob Anderson, Friedhelm von Blanckenburg as well as an anonymous reviewer who made useful comments on the manuscript and figures and to Reiner Schlitzer who kindly redraw Figure 2.

There are bonuses in this story: because global ocean  98 Mo is homogenous, and because ancient euxinic sediments record the  98 Mo of seawater from which they precipitated, records from such sediments have allowed the reconstruction of  98 Mo of ancient seawater. Consequently, many studies have sought to exploit Mo isotopes to reconstruct secular trends in ocean redox state throughout Earth history.

Trace metal isotope budgeting of sources and diagenesis: nickel

Micronutrients such as Zn, Cd, Fe and Ni, have strong links with ocean biology (Lohan and Tagliabue, this issue).

Unlike Mo, they have short ocean residence times, partly a result of removal into cells but also in part because of their insoluble nature in the oxidised form (e.g. Fe). The resulting spatial heterogeneity has made establishing their elemental and isotope distributions more difficult, but has also offered exciting new research avenues in that their isotope distribution patterns disclose the intensity of, for example, internal biological cycling. Their isotope distributions have also uncovered imbalances in the oceanic budgets of some of these metals.

We illustrate this with the example of nickel and its stable isotopes 60 Ni and 58 Ni. Nickel is a biologically-active element that shows a concentration profile with depth similar to the major nutrients, depleted in the photic zone due to biological uptake and enriched at depth where organic matter is re-oxidised by bacterial respiration (e.g. [START_REF] Bruland | Sampling and analytical methods for the determination of copper, cadmium, zinc, and nickel[END_REF]. In the modern ocean, Ni is an essential element in at least seven enzymes involved in the biological cycling of carbon, nitrogen and sulphur, and is also an essential element for one of the most ancient organisms on Earth, methanogens [START_REF] Ragsdale | Nickel-based enzyme systems[END_REF]. New data show that uptake of Ni in the photic zone preferentially removes the light isotopes, leaving the photic zone dissolved pool heavy at 

4).

In contrast to Mo the oceanic budget of Ni and its isotopes is not well balanced (Fig. 4). The average  60 Ni of oceanic dissolved Ni is much higher than external inputs -which are at 0.8‰ maximum. But, the sinks are even heavier than the ocean (e.g. [START_REF] Gall | Nickel isotopic compositions of ferromanganese crusts and the constancy of deep ocean inputs and continental weathering effects over the Cenozoic[END_REF]Ciscato et al, 2018, Fig.4). Low  60 Ni sources and high  60 Ni sinks constitute an isotope imbalance that would drive the ocean to lighter isotope compositions through time. Since there is no evidence for such secular change (e.g. [START_REF] Gall | Nickel isotopic compositions of ferromanganese crusts and the constancy of deep ocean inputs and continental weathering effects over the Cenozoic[END_REF], geochemists are seeking an as yet unidentified heavier source or a light and efficient sink. The biological output of light Ni does not solve the problem, being too small and too close in  60 Ni to the ocean. The same is true for euxinic settings like the Black Sea (Fig. 4). Mn oxides are known as efficient scavengers of the trace metals, and the nature of this sink is often characterized using Fe-Mn encrustations on the sea floor (e.g. [START_REF] Gall | Nickel isotopic compositions of ferromanganese crusts and the constancy of deep ocean inputs and continental weathering effects over the Cenozoic[END_REF]. It is the huge size and the high  60

Ni of this heavy sink on Fig. 4 that creates the budgetary problem. An intriguing possibility has been suggested very recently. Recent experiments suggest that up to 50% of adsorbed Ni is released during diagenetic transformation of Mn oxide in sediment during diagenesis [START_REF] Atkins | Release of Ni from birnessite during transformation of birnessite to todorokite: implications for Ni cycling in marine sediments[END_REF]. The isotope effect of this process has not yet been fully evaluated, but low  60 Ni values that have been measured in Mn-rich hemipelagic settings provide a promising explanation [START_REF] Little | Metal isotope signatures in Mn-rich pelagic sediments: Implications for oceanic budgets[END_REF].

Whatever the solution to this conundrum, the clear isotope fractionation of Ni isotopes by biology, as well as the drivers of an isotopically heavy ocean, hold much potential for discoveries in marine biochemistry and paleoceanography.

SUMMARY OF PROGRESS AND WHAT REMAINS TO BE DONE

Recent progresses based on isotopes have opened up new questions -as illustrated here with the apparent imbalances in oceanic budgets of some trace elements. Identifying the missing sources and poorly understood sinks is of prime importance. Constraining ocean budgets with high accuracy will rely on quantifying the impact


Julien Renet

Lisa Leprêtre

Jocelyn Champagnon

Philippe Lambret

Monitoring amphibian species with complex chromatophore patterns: a noninvasive approach with an evaluation of software effectiveness and reliability

Monitoring amphibian species with complex chromatophore patterns: a non-invasive approach with an evaluation of software effectiveness and reliability

.IntroductIon

The assessment of demographic parameters using standardised methods is a key part of conservation biology, especially to understand population trends and make reliable predictions about their viability [START_REF] Griffiths | Science, statistics and surveys: a herpetological perspective[END_REF]. Capture -mark -recapture models (CMR) that estimate population size, survival and detectability are useful tools for obtaining data for conservation [START_REF] Mccrea | Analysis of Capture-Recapture Data[END_REF]. This method requires that individual marks are perennial in order to allow individual recognition over time [START_REF] Chao | Estimating population size for sparse data in capture-recapture experiments[END_REF][START_REF] Nichols | Capture-recapture models[END_REF]. For amphibians, marking techniques can be either invasive or non-invasive. For some species, evident and consistent colour patterns such as spots or stripes can be used as natural marks for individual recognition [START_REF] Arntzen | Cost comparison of marking techniques in long-term population studies: PIT-tags versus pattern maps[END_REF][START_REF] Wengert | Using chin spot patterns to identify individual mountain yellow-legged frogs[END_REF][START_REF] Ferner | Measuring and marking post-metamorphic amphibians[END_REF]. Although visual matching can be an efficient method for monitoring a small population over a limited time, this technique becomes tedious and time-consuming (and thus sensitive to mistakes) when used to monitor large populations over several years [START_REF] Morrison | Estimating survival in photographic capturerecapture studies: overcoming misidentification error[END_REF][START_REF] Cruickshank | Error rates and variation between observers are reduced with the use of photographic matching software for capture-recapture studies[END_REF].

The development of algorithms for pattern recognition allows pattern mapping, and therefore pattern matching, to be automated [START_REF] Sacchi | Digital identification and analysis[END_REF]. However, two types of error are likely to induce a bias in the assessment of demographic parameters [START_REF] Yoshizaki | Modeling misidentification errors in capturerecapture studies using photographic identification of evolving marks[END_REF][START_REF] Morrison | Estimating survival in photographic capturerecapture studies: overcoming misidentification error[END_REF]. The first, namely False Acceptance (FA), consists of assigning the same identification to two different individuals. The second, False Rejection (FR), consists of the failure of software to match two pictures of the same individual [START_REF] Jain | Biometric recognition[END_REF]. The rates at which these two types of error occur (respectively FAR and FRR) can be used to assess software effectiveness, the latter being more commonly used [START_REF] Bolger | A computer-assisted system for photographic mark-recapture analysis[END_REF].

There are three types of software that can be distinguished: (1) software, such as I3S series [START_REF] Van Tienhoven | A computer-aided program for pattern-matching of natural marks on the spotted raggedtooth shark Carcharias taurus[END_REF] and AmphIdent [START_REF] Matthé | Computergestützte Bildanalyse von auchfleckenmustern des Kammmolchs Triturus cristatus[END_REF], that require laborious and time consuming pre-processing treatment, which consists of tracing (i.e. with the computer's mouse) the pattern outlines on each picture, (2) software not requiring any pre-processing of pictures (except the recommended cropping of pictures), and consequently timesaving, such as Wild-ID [START_REF] Bolger | A computer-assisted system for photographic mark-recapture analysis[END_REF] and Hotspotter [START_REF] Crall | Hotspotter-patterned species instance recognition[END_REF], and (3) intermediate software, such as APHIS [START_REF] Moya | APHIS: A new software for photo-matching in ecological studies[END_REF], which has been recently developed especially for herpetofauna, i.e., with a reduced preprocessing approach consisting of identifying two landmarks on each picture that the software can use as reference points (e.g. the base of left and right hind legs). Type 1 software is only used for species with gross body patterns (e.g. Triturus carnifex, Paramesotriton hongkongensis) because patterns need to be characterised easily [START_REF] Fu | Breeding Dynamics, Diet, and Body Condition of the Hong Kong Newt Paramesotriton hongkongensis[END_REF][START_REF] Sannolo | Photo-identification in amphibian studies: a test of I3S Pattern[END_REF]. Type 2 requires a good quality of individual pictures, otherwise the accuracy of recognition may not be reliable [START_REF] Morrison | Individual Identification of the Endangered Wyoming Toad Anaxyrus baxteri and Implications for Monitoring Species Recovery[END_REF]. While there is most probably a continuum rather than a true dichotomy in the specific pattern complexity, the user-friendly software Wild-ID is generally used for the identification of amphibians showing large and/or contrasted spots (e.g. [START_REF] Caorsi | Clip or Snap? An evaluation of toe-clipping and photo-identification methods for identifying individual Southern red-bellied toads, Melanophryniscus cambaraensis[END_REF][START_REF] Elgue | Validity of photo-identification technique to analyze natural markings in Melanophryniscus montevidensis (Anura: Bufonidae)[END_REF][START_REF] Morrison | Individual Identification of the Endangered Wyoming Toad Anaxyrus baxteri and Implications for Monitoring Species Recovery[END_REF][START_REF] Romiti | Photographic Identification Method (PIM) using natural body marks: a simple tool to make a long story short[END_REF], but has been assessed only once with a complex colour (i.e. fine spotted) pattern, using a neotenic salamander [START_REF] Bendik | Computer-assisted photo identification outperforms visible implant elastomers in an endangered salamander, Eurycea tonkawae[END_REF]. Also, the lowest FRRs were obtained using pictures taken with a "digital single-lens reflex" camera (DSLR) and in highly standardised conditions (e.g. cropped pictures, laboratory conditions) [START_REF] Mettouris | A newt does notchange its spots: Using pattern mapping for the Identification of individuals in large populations of newt species[END_REF]. Since APHIS benefits from two reference spots indicated by hand, it theoretically allows for a lower standardisation level compared to fully automated software and could be a useful compromise between type 1 and 2 software in terms of time and handling needed to process the pictures. Yet to date, APHIS has rarely been used for amphibian monitoring (but see [START_REF] Romano | Conservation of salamanders in managed forests: methods and costs of monitoring abundance and habitat selection[END_REF], 2018 with Salamandrina perspicillata).

In this study, we used Wild-ID and APHIS to identify the French cave salamanders Hydromantes strinatii. This species bears a complex chromatophore network resulting in a stippled colour pattern and thus provides a useful study species for testing software effectiveness and reliability. To assess software effectiveness (i.e. the FRR values) and the reliability of the entire photographic identification process (i.e. the FRR variability), we took pictures in field conditions with a "point and shoot" camera; in other words, we rejected the option of using a heavy-handling set-up in the field. We considered two different regions of the salamander body (the chest and the cloaca) to identify differences in FRR between body sites. We first compared the respective FRR and, secondly, we assessed the effect of the lowest FRR on the estimation of population size, as demographic parameters are the ultimate goals of CMR. This allowed us to determine whether the best combination of software and body region lead to acceptable errors in this estimation.

MaterIals and Methods

Study species

Hydromantes strinatii is a salamandrid endemic to south-eastern France (Alpes-Maritimes and Alpes-de-Haute-Provence departments) and north-western Italy (Ligure region). It has a complex chromatophore pattern with a discrete phenotypic variability in its design and thus, it is a good model for the assessment of non-invasive identification methods, such as photo-identification, used to compute the demographic parameters needed for population viability analyses (Fig. 1). Further, it is important to study this species because it has been evaluated as "Near Threatened" and is therefore of conservation interest [START_REF] Temple | European Red List of Amphibians[END_REF][START_REF] Renet | Vers la mise en place d'une stratégie conservatoire en faveur du Spéléomante de Strinati Speleomantes strinatii (Aellen, 1958) dans le sud-est de la France[END_REF]. Its snoutvent length does not exceed ca. 75-80 mm, while maximal total length is 123 mm [START_REF] Lanza | Morphologic and genetic studies on the European plethodontid salamanders: taxonomic inferences (genus Hydromantes)[END_REF]. Individuals shy away from daylight in natural and artificial cavities. During the night, they come out and can be found in epigeous rocky habitats.

Population sampling and individual marking

During April 2015, we sampled a population in an epigeic habitat south of Roquebillière city (43°59'50.05''N, 7°18'41.57''E, Alpes-Maritimes department). We searched for individuals on six occasions with an equal sampling effort along a 94 m section of retaining walls, during the maximal activity period of the species, i.e., during nights with relative humidity ranging from 70 % to 100 % and temperature ranging from 5 °C to 15 °C [START_REF] Lanza | A review of systematics, taxonomy, genetics, biogeography and natural history of the genus Speleomantes dubois, 1984 (Amphibia Caudata Plethodontidae)[END_REF]. As we repeatedly sampled the same section of wall, some individuals were captured on more than one sampling occasion (i.e. recaptured). Pattern resolution (i.e. scale) is likely to vary with body size and hence with individual growth. In order to avoid this potential bias, we captured adults only and avoided juveniles. Each individual was stored in a 5×10×10 cm box, with holes so that it could breathe, until it could be photographed. At the end of each sampling session, the animals were moved to the nearby photography setup. This consisted of a box lined on the bottom with polypropylene foam and equipped with a swinging glass plate. The animals were placed one by one upside down on the foam and held in place with the glass plate so that their ventral surface was flattened against the glass. For each individual, we took 4-5 pictures of the chest and 4-5 of the cloaca (Fig. 1). They were released immediately after photography at the exact location of capture, which was marked on each storage box. Manipulation of each individual did not exceed 5 minutes and total storage duration did not exceed 2 hours. Permission for this programme was issued by order of the Prefet , according to French law.

Pictures were taken with a TG-3 Olympus© digital camera "point and shoot" (sensor: BSI CMOS 16 Mpx; zoom: 4 × 25-100mm f/2-4.9). We used the camera under the automatic "super-macro" mode with an additional LED LG-1 ring, which allowed the pictures to be taken 1 cm from the individuals. Pictures were taken with a 3456 × 4608 px resolution. No further processing was done, not even cropping; however, we also assessed whether cropping improved Wild-ID effectiveness (see below).

For each combination of sampling occasion x individual × body region, we selected the best picture among the 4-5 we had taken, thereby obtaining n1 = 314 pictures for each body region. We then allocated each pair of chest and cloaca pictures, among which some were capture pictures and others were recapture pictures, to an individual animal by visual matching, which we assume to be 100 % effective in discriminating each individual (i.e. FRR = 0). Indeed, the 314 pairs of chest and cloaca pictures were examined by two naive and independent observers who both recognised the same 253 individuals. There were 314 -253 = 61 recaptures of 52 individuals; 61 -52 = 9 and 52 -(61 -52) = 43 individuals were, respectively, recaptured twice and once, and therefore photographed at three and two occasions; 314 -(43 × 2 + 9 × 3) = 201 individuals were captured and photographed once. We accordingly attributed an identification number (Id) to each of the 52 recaptured individuals. In this study, we used the ITM process, which requires the definition of two landmarks on the body of each individual; we used the armpit and the vent for the chest and the cloaca regions respectively (Fig. 1: arrows). From there, the software creates a rectangle divided into six areas, which are analysed independently [START_REF] Moya | APHIS: A new software for photo-matching in ecological studies[END_REF].

Wild-ID (ver. 1.0.1; [START_REF] Bolger | A computer-assisted system for photographic mark-recapture analysis[END_REF] is available for free from http://software.dartmouth.edu/Macintosh/ Academic/Wild-ID_1.0.0.zip. It uses the SIFT algorithm (Scale Invariant Feature Transform; [START_REF] Lowe | Distinctive image features from scale-invariant key-points[END_REF] to extract the body pattern, and then compares the geometric arrangement of the SIFT features for each couple of pictures (i.e. matches) without pre-processing.

APHIS and Wild-ID offer, respectively, 100 and 20 "top-ranked" matches following a decreasing calculated score provided by their algorithms (i.e. the picture proposed at the 1st rank is that from the dataset showing the highest score with the submitted picture). One has to then search visually for the correct individual among the software proposals.

We assessed effectiveness of each software for each body region. For this, we determined, for each of the 61 recaptures, the rank at which the software proposed the correct picture (i.e. the 100 % true matched picture, identified from the visual assessment) among the top-ranked matches. Hence, we were able to state whether the software programme failed to rank each of the 61 recapture pictures within the Top 1, 5 or 10 matches; in other words, failed to correctly identify the 313 pictures of both 1st captures and recaptures. This means that the size of the dataset each new recapture picture is compared to increases for each subsequent picture. When assessing the FRR according to this method, i.e. by comparing one picture only to other pictures that had been processed so far rather than against all other existing pictures (all capture and recapture picture), the false rejection assignation, and therefore the FRR, may be strongly affected by the order of processing of the pictures. To take this potential bias into account, we decided to assess the software reliability, i.e. the variability of the FRR by repeating the process five times (i.e. five replicates). For each replicate the recapture pictures were processed in a different random order. Each of the three Top rank datasets therefore contained 1220 data points (5 replicates × 2 softwares × 2 body regions × 61 recapture pictures).

Wild-ID manual recommends to crop pictures so as to remove the background as much as possible, however, this can be a time consuming and laborious process. Thus, we assessed whether cropping was require to improve effectiveness. For this, we cropped cloaca pictures using a 1700 × 4608 px frame (Fig. 1). We then randomly selected 30 recapture pictures and compared each of these to a database composed by the 253 + 60 = 313 pictures of both 1st captures and recaptures; for each comparison, the database was therefore renewed and slightly different from the previous (i.e. by one picture out of 313). This was done with the cropped pictures and with the original pictures, using the same 30 randomly selected individuals.

Statistical analysis

Failures, i.e. false rejections, were coded as 1 and successes coded as 0 in a new data frame, with software, body region and replicate as co-variables. We used generalised linear mixed models to test differences of FRR (dependent variable) between software and between body regions (explanatory variables, i.e. fixed factors). Mixed models were used to account for (1) the pseudo-replication induced by the fact that some individuals were recaptured and photographed more than once [START_REF] Hurlbert | Pseudoreplication and the design of ecological field experiments[END_REF] and (2) the replication scheme of the FRR computing. "Id" and "Replicate" were thus included as random factors. To assess whether FRR was influenced by the software and/or body region, we used each Top rank match and analysed the datasets using a binomial error using the following function:

where Soft is the software, Reg is the body region and Soft*Reg is their interaction, Id is this individual and Repl is the replicate. Contrast analyses were performed in order to assess whether the differences between the best combination of software and body region was significantly different from other combinations or not. To assess the variability of the FRR regarding the order in which recapture pictures were processed in, we computed the FRR for each replicate independently using the following function:

where i is the replicate number. We then computed the standard deviation (SD) of the 5 FRR we obtained. Results are given as FRR estimated by the models together with their 95 % CI. Significance of fixed factors is given by a range of corresponding estimates ± standard error which does not overlap zero (i.e. meaning a null effect of the fixed factor). All analyses described above were performed in R 3.4.4 (R Core Team, 2018) using packages 'lme4' [START_REF] Bates | lme4: Linear mixed-effects models using Eigen and S4. R package version 1[END_REF], 'lmerTest' [START_REF] Kuznetsova | Tests in linear mixed effects models[END_REF] and 'effects' [START_REF] Fox | Effect displays for linear, generalised linear, and other models[END_REF].

To assess the bias on the estimation of population size induced by the best software and body region FRRs obtained for each of the Top 1 and 10 matches, 1000 CMR datasets were simulated to which we applied the FRRs obtained from the generalised linear mixed model. The datasets were created assuming a population of 600 individuals, six occasions of sampling with a mean capture probability of 0.1 varying randomly for each sampling occasion. Those assumptions were based on the results of Mt standard closed capture model [START_REF] Otis | Statistical inference from capture data on closed animal populations[END_REF] applied to the real dataset with a time effect on detection probability and run in a Bayesian framework (population size = 543 ± 44 SD; mean detection probability = 0.098 ± 0.050 SD). The same Mt closed capture model was then applied to the 1000 simulated datasets with respectively no FRR, Top10 and Top1 FRRs. MCMC was applied using JAGS [START_REF] Plummer | JAGS: A Program for Analysis of Bayesian Graphical Models Using Gibbs Sampling[END_REF] via the package R2jags [START_REF] Su | R2jags: A Package for Running jags from R. R Package Version 0[END_REF] in R (R Core Team, 2018). We used a burn-in of 500 iterations, three chains, a thinning rate of two and 2000 iterations for each posterior distribution. Convergence was reached and mixing of the chains was good. Table 1. Software effectiveness: False Rejection Rate (FRR) when matching individual pictures of the French cave salamander Hydromantes strinatii, comparing software programmes and body regions. Contrast analyses were made between (1) the smallest FRR and others,and (2) between APHIS FRRs: ns = not significant (p > 0.05), * = significant (p < 0.05), *** = highly significant (p < 0.001).

Results

Fixed effects driving False Rejection Rate

False Rejection Rate and its variability: the effects of software and body region

Wild-ID produced the smaller FRR for Top 1, Top 5 and Top 10 matches i.e. 0.082, 0.050 and 0.042, respectively (Table 1). These FRR values were significantly smaller than those of APHIS, for both chest and cloaca. Wild-ID FRR values were not significantly different for the two body regions (Table 1). In contrast, APHIS FRRs were significantly smaller for the chest than the cloaca region for Top 1, Top 5 and Top 10 matches. The standard deviation of the FRR across the five replicates ranged from 0.004 to 0.057 (Fig. 2). The FRR variability regarding the order of processing of the recapture pictures (i.e. across replicates) was greater with APHIS (SD range: 0.022-0.057) than with Wild-ID (SD range: 0.004-0.016).

Influence of cropping of pictures on Wild-ID effectiveness

Cropping did not significantly affect Wild-ID effectiveness: removing the background reduced the rank at which Wild-ID proposed the correct picture in only two recapture pictures out of 30: 7th instead of 8th rank and 18th instead of >20th rank. Hence, when coding these data as binary (false rejection or success), the two new data frames for cropped and original pictures were absolutely similar for the Top 1, Top 5 and Top 10 matches.

Influence of error in Wild-ID on the estimation of population size

With the selected software programme (Wild-ID) used on chest, FRR had a weak influence on the estimation of population size of Hydromantes strinatii (Table 3). Mean estimated population size for Control, Top 10 and Top 1 matches (the latter being the highest FRR and therefore inducing the strongest bias during the estimation of demographic parameters) all included the known population size of 600 individuals within their 95% confidence interval. Mean population size estimated from simulated data without FRR was underestimated by 2.0% [-16.6;14.5] while mean estimated population sizes were overestimated by 2.7% [-12.9;18.7] using Top 10 FRR, and by 9.0% [-7.7;25.8] using the Top 1 FRR.

DiscussIon

Software effectiveness in recapture identification

Our results, obtained in field conditions, showed that Wild-ID was more effective and reliable than APHIS in matching pictures of a complex chromatophore pattern salamander (Table 1). Despite the field conditions and the use of a point and shoot camera, our Top 10 Wild-ID FRR values are among the lowest ever obtained and our more stringent Top 1 Wild-ID FRR values are close to the first quartile (i.e. among the lowest) of those found throughout the literature (Table 2). These low values are undoubtedly influenced by the small size of our database (N = 253 captures + 61 recaptures = 314) but there is probably also some influence of technological progress in the quality of sensor and optic lenses since the other studies were conducted, which helps to cope with poorly-standardised conditions. Even in our study, software effectiveness could have been improved further with even higher quality pictures produced by (i) holding individuals manually or using a shutter with a hole without glass (i.e. allowing a direct view on a selected body region) in order to avoid bubbles and impurities, (ii) improving lighting, and therefore the depth of field, for instance using a powerful flash, and (iii) holding the camera still with a tripod. Also, cropping pictures is supposed to improve software effectiveness. By comparing here FRRs obtained with uncropped pictures (Wild-ID) vs pictures where we restricted the searching algorithm to the focus area by generating a set of landmarks (APHIS), we potentially skewed our results in favour of APHIS. However, the little improvement cropping pictures provided in Wild-ID picture ranking had no effect on the computing of FRR values. If it had, it only had strengthened the dominance of Wild-ID over APHIS. Using APHIS with the ITM procedure on chest pictures of a salamander, [START_REF] Moya | APHIS: A new software for photo-matching in ecological studies[END_REF] detected 100 % of recaptures (i.e. FRR = 0, n = 305). In contrast, we obtained with APHIS poor FRR values (i.e. lower effectiveness), which were in addition highly variable (i.e. lower reliability) (Fig. 2). This is probably due to our low level of standardisation (e.g. body distortion, variable lighting), for which the software cannot fully compensate despite preprocessing of images, combined with human error since preprocessing is executed by hand during each replicate by positioning two new reference points on the same pictures.

Wild-ID was as effective with pictures of the chest as with those of cloaca. On the other hand, APHIS appeared to be less effective with cloaca pictures than with chest from such stable body regions with all software. Though I3S straighten software copes with body contortion, it requires a manual pre-processing approach (Den [START_REF] Hartog | Interactive individual identification system (I3S)[END_REF], which is time consuming. pictures. When flattened against the glass, the soft cloaca can vary in shape, therefore modifying patterns. The higher sensitivity of APHIS to distortion is most probably once again due to the pre-processing conducted by hand. Pictures of the less mobile and more stable body regions such as the chest produced results at least as good as the cloaca pictures in our study, so we suggest using pictures from such stable body regions with all software.Though 135 straighten software copes with boydy contortion, it requires a manual pre-processing approach (Den [START_REF] Hartog | Interactive individual identification system (I3S)[END_REF], witch is time consuming. A regression spline smoother with 95% confidence intervals was added for each group of data to help with visual interpretation. The smoother was fitted using the mgcv package [START_REF] Wood | Generalized additive models: an introduction with R[END_REF] and explains respectively 10%, 12% and 17% of the variation in the estimated population sizes for the Control, Top10 and Top1 datasets.

Consequences for the estimation of population size

One assumption necessary for valid inference from capture-mark-recapture studies is that marks are correctly read [START_REF] Lindberg | A review of designs for capture-mark-recapture studies in discrete time[END_REF]. In case of non-invasive sampling, errors in the identification of individuals maylead to bias in estimation of demographic parameters [START_REF] Creel | Population size estimation in Yellowstone wolves with error-prone noninvasive microsatellite genotypes[END_REF]. Here we showed that considering first Wild-ID proposals (Top 1 high-score matches) leads to an overestimation of population size of 9%. Underestimation of population size without FRR was due to the low detection probability found in this species (<0.1, see Methods section) and applied to the simulated dataset. Indeed, datasets created from simulations that implied lower number of recaptures induced higher uncertainty and underestimation of the population size (Fig. 3). As a consequence, we found wide 95 % confidence intervals in the estimated population size (Table 3). A 9% overestimation of population size is however twenty times less than the overestimation of population size found by [START_REF] Morrison | Individual Identification of the Endangered Wyoming Toad Anaxyrus baxteri and Implications for Monitoring Species Recovery[END_REF] using Wild-ID. This is consistent with the much higher FRR (0.47 for Top 20) found by [START_REF] Morrison | Individual Identification of the Endangered Wyoming Toad Anaxyrus baxteri and Implications for Monitoring Species Recovery[END_REF] on Wyoming toad (Anaxyrus baxteri) than those found in this study. In order to limit bias in estimation of population size when individuals are misidentified, closed capture models incorporating misidentification as implemented in the programme MARK may be used but, as a consequence in such case, confidence intervals would become much wider [START_REF] Lukacs | Estimating population size from DNA-based closed capture-recapture data incorporating genotyping error[END_REF]. When a long-term monitoring programme is launched, a further important point to consider is the possibility that chromatophore patterns vary over time in amphibia [START_REF] Church | Iteroparity in the variable environment of the salamander Ambystoma tigrinum[END_REF][START_REF] Drechsler | Genetic fingerprinting proves cross-correlated automatic photoidentification of individuals as highly efficient in large capture-mark-recapture studies[END_REF][START_REF] Balogova | Changes in dorsal spotpattern in adult Salamandra salamandra (Linnaeus, 1758)[END_REF].

Management implications

The assessment of bias in individual recognition is an essential step when software is used to estimate the demographic parameters, especially in a threatened species. Indeed, these values can greatly influence the decisions regarding conservation strategies. To limit this bias without spending too much time reviewing all available pictures, especially with huge sample sizes (i.e. n > 1000), we recommend considering the Top 10 high-score matches proposed by Wild-ID, at least for H. strinatii and most certainly for several other urodela species that have a similar pigment pattern. Our study suggests that in the conditions described here (low detection probability), overestimation of the population size is 3 %, and true population size falls within the 95% confidence interval.

Although Wild-ID was initially developed for large terrestrial mammals, we show that the software is able to cope effectively with a small sized amphibian with a complex chromatophore pattern. We have also shown that it is effective in field conditions and without pre-processing treatment of pictures. Therefore, it can work well even with limited equipment and time. This also means stress to the animals and the cost of the study may be reduced using Wild-ID than other software -two aspects which are nowadays of primary concern for reasons of ethics and efficiency. The low FRR values obtained through the present study in H. strinatii provide encouraging prospects for using Wild-ID to identify and monitor species with complex pigment patterns. Even if the FRR may increase with the size of the dataset, this rate can be reduced by increasing the quality of the pictures [START_REF] Gamble | Multi-scale features for identifying individuals in large biological databases: an application of pattern recognition technology to the marbled salamander Ambystoma opacum[END_REF][START_REF] Matthé | Comparison of photo-matching algorithms commonly used for photographic capture-recapture studies[END_REF].

Figure 1 .

 1 Figure 1. Variability of the complex chromatophore pattern of the French cave salamander Hydromantes strinatii, photographed at the two body regions considered in this study: chest (b) and cloaca (c). Picture b1 and b2 concern an individual during capture and recapture, respectively, the latter being identified as the first rank matching picture by Wild-ID during the five replicates despite minimal-standardisation conditions (see impurities and air bubbles indicated by circles). Black arrows indicate the armpit (b1) and the groin (c1) which we used as reference spots required by APHIS ITM approach. Red lines indicate the frame (1700 × 4608 px) used during the cropping test. Software characteristics and effectiveness assessment APHIS (ver. 1.0; Moya et al., 2015) is available for free from https://imedea.uib-csic.es/bc/gep/docs/aphis/ APHISPROGRAM/program/. It offers two individual matching processes: the Spots Pattern Matching (SPM) and the Image Template Matching (ITM). The former is based on I3S algorithm (Van Tienhoven et al., 2007) while the latter is based on algorithms developed by OpenCV (Open source Computer Vision).In this study, we used the ITM process, which requires the definition of two landmarks on the body of each individual; we used the armpit and the vent for the chest and the cloaca regions respectively (Fig.1: arrows). From there, the software creates a rectangle divided into six areas, which are analysed independently[START_REF] Moya | APHIS: A new software for photo-matching in ecological studies[END_REF].

  For all three types of Top rank matches (n1 = n2 = n3 = 1220), the FRR was significantly influenced by software (Top 1: |estimate| = 1.640 ± 0.234; Top 5: |estimate| = 1.909 ± 0.273; Top 10: |estimate| = 1.891 ± 0.286) and body region (Top 1: |estimate| = 0.826 ± 0.192; Top 5: |estimate| = 0.868 ± 0.196; Top 10: |estimate| = 0.914 ± 0.200) and their interaction (Top 1: |estimate|=0.961±0.324;Top5:|estimate|=0.596±0.360; Top 10: |estimate| = 0.914 ± 0.390). Hence, we hereafter provide the FRR of each combination of software and body region separately.

Figure 2 .

 2 Figure 2. Software reliability: variability for each combination of software, body region and Top rank matches of the False Rejection Rate (FRR) across five replicates (r1-r5) of the matching of individual recapture pictures of the French cave salamander Hydromantes strinatii. Grey columns show the estimated mean FRR and black bars their 95 % CI.

Figure 3 .

 3 Figure 3. Impact of false rejections on estimation of abundance of a simulated population of 600 French cave salamanders. Capture-recapture data were collected over six closed capture events with mean detection probability of 0.1. Estimated population size from simulated data with no error are presented with open circles, from simulated data with False Rejection Rate (FRR) corresponding to the use of Top 10 Wild-ID on the chest are presented with crosses and with FRR corresponding to the use of Top1 Wild-ID on the chest are presented with black triangles.A regression spline smoother with 95% confidence intervals was added for each group of data to help with visual interpretation. The smoother was fitted using the mgcv package[START_REF] Wood | Generalized additive models: an introduction with R[END_REF] and explains respectively 10%, 12% and 17% of the variation in the estimated population sizes for the Control, Top10 and Top1 datasets.

Table 2 .

 2 Review of FRRs obtained with Wild-ID when monitoring amphibia in different studies. PAS = "point and shoot" camera; DSLR = "digital single lens reflex" camera; CC = Camcorder

Table 3 .

 3 Population size estimated from a simulated French cave salamander population of 600 individuals. 1000 datasets with randomly varying detection probabilities over six closed capture events (mean detection probability of 0.1) were simulated. False Rejection Rates (FRRs) obtained from Table 1 (chest Wild-ID Top 10 and Top 1 matches) were applied to simulated datasets. Population sizes were estimated applying standard closed capture model with a time effect on detection probability.

Acknowledgments

This study was partly funded by the Direction Régionale de l'Environnement, de l'Aménagement et du Logement de Provence Alpes-Cote-d'Azur as part of a project "Agenda 21" of the Métropole Nice Côte d'Azur, we thank Pascale Babillot and Samuel Pauvert for their support. We are indebted to Julien Banus (Direction Départementale des Territoires et de la Mer des Alpes-Maritimes) for his help in obtaining the sampling permit issued by the Préfecture des Alpes-Maritimes (Arrêté préfectoral n°2015-227). We are grateful to Olivier Gerriet (Muséum d'Histoire naturelle de Nice), Aude Lavalle, Pauline Chevalier (Métropole Nice Côte d'Azur), Ghislaine Dusfour (CEN PACA) and Olivier Borde for their assistance in the field and to Nathan Bendik (University of Texas Arlington), Andrea Costa (Università degli Studi di Genova), Giacomo Tavecchia (Population Ecology Group, IMEDEA -Spanish National Council for Research-UIB), Onoufrios Mettouris (Department of Biology, University of Patras), Laurent Tatin (CEN PACA), Jean-Marie Ballouard (CRCC-SOPTOM) and Óscar Moya (Fundación BIT) for providing us with helpful information. We also warmly thank Dr. Patrick Duncan and Dr. Nigel Taylor for improving the English language and providing useful comments. Two anonymous referees helped us to strengthen an earlier version of the manuscript.


Zannier

ON THE BETTI MAP ASSOCIATED TO ABELIAN LOGARITHMS

Keywords:

PROBLEM AND MAIN RESULTS.

1.1. Let A be a complex abelian variety of dimension g. We write Λ for the its period lattice H 1 (A(C), Z), identified with the kernel of the exponential map exp A : Lie A → A.

We fix a complex point σ of A, and let Λ σ be the lattice formed by the pairs ( ∈ Lie A, m ∈ Z) such that exp A ( ) = mσ. It sits in an extension

(1.1) 0 → Λ → Λ σ ( ,m) →m → Zσ → 0,
and we fix a splitting λ : Zσ → Λ σ (i.e. a branch of the "abelian logarithm"). On the other hand, let us consider the first projection Λ σ ( ,m) → → Lie A and its R-linear extension Λ σ ⊗ Z R → Lie A . Since the composed map

(1.2) Λ ⊗ Z R → Λ σ ⊗ Z R → Lie A
is an isomorphism, one has a canonical projection Λ σ ⊗ Z R → Λ ⊗ Z R, and the image of λ(σ) in Λ ⊗ Z R can be expressed, in terms of a basis of Λ, by 2g real coordinates: the Betti coordinates 1 of σ.

1.2. We are actually interested in the relative setting. Let S be a smooth connected complex algebraic variety. Let A f → S be an abelian scheme of relative dimension g. Its Lie algebra Lie A is a rank g vector bundle on S. For our purpose, there will be no 1 the terminology seems due to D. Bertrand and refers to the fact that Λ σ is nothing but the Betti realization of the 1-motive [Z 1 →σ → A] in the sense of [D2, X]. Note that the Betti coordinates depend not only on σ but also on the auxiliary choice of a splitting λ and of a basis of Λ. loss of generality2 in assuming, as we shall do, that S is affine and that this vector bundle is trivial, i.e. ΓLie A is a free ΓO S -module of rank g.

Let σ : S → A be a section of f . The above constructions extend to this relative setting as follows. Let S be a universal covering of S(C), with its canonical structure of complex analytic manifold. The kernel of exp A is a locally constant sheaf on S(C), which can be viewed as a constant lattice Λ on S. Similarly, one constructs an exact sequence of lattices on S

(1.3) 0 → Λ → Λ σ ( ,m) →m → Zσ → 0
and fixes a splitting λ : Zσ → Λ σ .

Let Sreal be S viewed as a real-analytic manifold and let O R ( Sreal ) (resp. O( Sreal )) be the ring of real-valued (resp. complex valued) real-analytic functions on S. The composed map

(1.4) Λ ⊗ Z O R ( Sreal ) → Λ σ ⊗ Z O R ( Sreal ) → ΓLie A ⊗ ΓO S O( Sreal ) is an isomorphism of free O R ( Sreal )-modules. One thus has a canonical projection Λ σ ⊗ Z O R ( Sreal ) → Λ ⊗ Z O R ( Sreal )
, and the image of λ(σ) in Λ ⊗ Z O R ( Sreal ) can be expressed, in terms of a basis of the lattice Λ, by a real-analytic map (the Betti map)

(1.5) β = β(σ, λ, ) : S → R 2g .
1.3. We assume for convenience that A → S is principally polarized, which allows to consider to the moduli map µ : S → A g . We denote by A η the geometric generic fiber, and we set (1.6) d = dim µ(S).

We are interested in the range of β. It is easy to see that the its dimension is ≤ min (2d, 2g). Under what condition is the range of β maximal, i.e. of dimension min(2d, 2g)?

This may not be the case for various trivial reasons (e.g. when the σ is a torsion section, or when A f → S is the pull-back of an abelian scheme defined over a base of dimension < d ≤ g).

1.4.

Here is a sample of concrete results [not our final results, hopefully!]:

1.4.1. Theorem. Assume that g ≤ min(3, d) and that the section σ is not contained in a proper subgroup scheme of A. Then the range of β has real dimension 2g. More precisely, the restriction of β to the inverse image of a dense Zariski-open subset of S is a submersion.

The proof relies on an analysis of the Kodaira-Spencer map.

1.4.2. Theorem. Assume that σ is not contained in a proper subgroup scheme of A. Assume moreover that d ≤ min(2, g). Then β has real image 2d. More precisely, the restriction of β to the inverse image of a dense Zariski-open subset of S is an immersion. 2.1. Our approach is infinitesimal. In fact, the problem we address is the following:

under what condition is β almost everywhere an immersion, resp. a submersion (according to whether d ≤ g or g ≤ d) ?

2.1.1. Proposition. For every s ∈ S, Ker β(s) is a complex vector subspace of Lie A s ∼ = R 2g . Moreover the fibers of β are complex-analytic subspaces of S, and all non-empty fibers have dimension ≥ 2d -r, where r is the generic rank of β.

Proof. -[cf. your prop 2.1 + 2.2].
2.2. We may first replace S by a finite etale covering and assume that for some n ≥ 3, the n-torsion sections are defined over S, then replace it by its image in the moduli space A g,n of principally polarized abelian varieties of dimension g with level n-structure. We thus consider the rank of a jacobian matrix of size (2d, 2g), and for this purpose, we introduce the following notation:

(ω 1 , . . . , ω g , η 1 , . . . , η g ) is a symplectic basis of

H 1 dR (A/S) with ω 1 , . . . , ω g ∈ Ω A , (γ 1 , . . . , γ g ) is a symplectic basis of Λ, Ω 1 := γ i ω j i,j=1,...g , Ω 2 := γ i+g ω j i,j=1,...g , Z := Ω 1 • Ω -1 2 = t Z (which takes values in H g ), L := σ ω j j=1,...g • Ω -1 2 .
Let r be min(d, g). In the next step, we transform the vanishing of a 2r-2r-minor involving real-analytic functions by the vanishing of a r-r-minor of a related matrix involving only complex-analytic functions on S. In order to do this, we take advantage of the monodromy, using a theorem of linear independence of abelian logarithms with respect to periods [A1, th. 2] (cf. also [A2, §1]).

At the end, we come along with the following translation of the problem (under extra assumptions): 2.2.1. Proposition. (with S ⊂ A g,n ). Assume that σ is not contained in a proper sub-group scheme. Then for any s ∈ S, β is an immersion, resp. a submersion (according to whether d ≤ g or g ≤ d) at s if and only if for some (λ 1 , . . . , λ g ) ∈ C g , the matrix

(2.1) ((∂ i L)(s) + k=1,...,g λ k (∂ i Z kj )(s)) i=1,...,d, j=1,...g
has maximal rank (where the ∂ i form a basis of the tangent space T S (s)). This is already the case if for some (λ 1 , . . . , λ g ) ∈ C g , the matrix

(2.2) ( j=1,...,g λ k (∂ i Z kj )(s)) i=1,...,d, j=1,...g has maximal rank. Proof. ... [your §3.5].
The second assertion follows from the first, putting an arbitrary coefficient λ 0 in front of (∂ i L)(s) and letting λ 0 tend to 0.

Remark. For d = g, the negation of the conclusion leads to a special case of "Dieudonné's problem", which is notoriously difficult: describe vector spaces of singular matrices Z, resp. singular symmetric matrices, cf. e.g.

[L] 3 .

ROLE OF THE KODAIRA-SPENCER MAP.

3.1. We know consider the following condition: (3.1) For every s in an dense open affine subset of S, the matrix (2.2) has maximal rank.

We shall translate it in terms of the Kodaira-Spencer map

θ : T S ⊗ Ω A → Ω ∨ A = Lie A,
where Ω A denotes the locally free O S -module of invariant 1-forms on A. We refer to [A3, 1.4, 2.1] for background about the Kodaira-Spencer map of a polarized abelian scheme A of relative dimension g over a smooth Cscheme S. For any ∂ ∈ ΓT S , we denote by θ ∂ the corresponding map

Ω A → Ω ∨ A , which belongs to Hom sym (Ω A , Ω ∨ A ).
Let us recall that in terms of the symplectic basis ω 1 , . . . , ω g , η 1 , . . . , η g , the matrix of θ ∂ is given by the lower left quadrant T ∂ of the Gauss-Manin connection written in the form

∂ Ω 1 N 1 Ω 2 N 2 = Ω 1 N 1 Ω 2 N 2 R ∂ S ∂ T ∂ U ∂ , where R ∂ , S ∂ , T ∂ , U ∂ ∈ M g (O(S)
). Moreover T ∂ is a symmetric matrix.

3 we are grateful to M. Brion for this reference.

3.1.1. Lemma. Assume that d = g. Then condition (3.1) is equivalent to:

(3.2) ∃ω ∈ ΓΩ A , ∀∂ ∈ ΓT S \ 0, θ ∂ .ω = 0.
Proof. One has

∂Z = (∂Ω 1 -Ω 1 Ω -1 2 ∂Ω 2 )Ω -1 2 = ((Ω 1 R ∂ +N 1 T ∂ )-Ω 1 Ω -1 2 (Ω 2 R ∂ +N 2 T ∂ ))Ω -1 2 = (N 1 -Ω 1 Ω -1 2 N 2 )T ∂ Ω -1 2 = -Ω -1 2 ( t Ω 1 N 2 -t Ω 2 N 1 )T ∂ Ω -1 2 = -2πiΩ -1 2 T ∂ Ω -1 2 . If µ = -2πi λΩ -1 2 , one has det( k=1,...,g λ k (∂ i Z kj )(s)) = det( k=1,...,g µ k (T ∂ i ) kj (s)) • (det Ω 2 ) -1 . Setting ω = µ k ω k , one has µ k (T ∂ i ) kj = θ ∂ i (ω) • η j . Thus (3.1) is equivalent to: there is a dense open affine subset S ⊂ S such that ∀s ∈ S (C), ∃ω s ∈ Ω 1 (A s ), ∀∂ s ∈ T S,s \ 0, (θ(s) ∂s )(ω s ) = 0.
It is easy to see that this condition is equivalent in turn to (3.2) (for the direct implication, any ω lifting ω s for some s will do). Proof. Let θ ω : T S → Ω ∨ A the contraction of θ by ω ∈ Ω A . Condition (3.2) is equivalent to the injectivity of θ ω for some ω. Since d = g, this is equivalent to the generic surjectivity of θ ω for some ω, which implies the generic surjectivity of θ, which is in turn equivalent to (3.3).

In [A3]

Question. Does the converse of the lemma holds?

3.2.4. Remark. Let η be the generic point of S and P (Ω Aη ) (resp. P (T η )) be the projective space of lines on Ω Aη ) (resp. T S ). We denote by ω (resp. ∂) the class of ω in P (Ω Aη ) (resp. ∂ in P (T η )).

One could try to answer the question in general by explaining the "logical seesaw" between quantifiers from (3.3) to (3.2) through a "geometric seesaw" provided by the incidence variety Σ ⊂ P (Ω Aη ) × P (T η ) formed by pairs (ω, ∂) such that θ ∂ • ω = 0, and let p 1 , p 2 be the projections on the factors. Let further Ph ⊂ P (T η ) be the locally closed subvariety formed by elements ∂ such that θ ∂,η has rank h. One has

p 2 (Σ) ⊂ h=g-1 h=1 Ph ⊂ P (T η ) = h=g h=1
Ph ,

and dim p -1 2 {∂} = g -1 -h if ∂ ∈ Ph . By (3.3), Pg is non empty and in fact dense (the condition is open in ∂), hence dim p 2 (Σ) ≤ g -2.
The key problem is whether one has

(3.4) dim Ph-1 < dim Ph
(which is equivalent to dim Ph = ∅ since Ph ⊃ Ph-1 ) for e < h < g, where e is the first index such that Pe = ∅.

If so, then dim Ph ≤ h -1 and dim Σ ≤ g -2, so p 1 is not dominant, and any ω outside the image of p 1 will do.

We have only partial answers to 3.2.3.

3.2.5. Proposition. If d = g ≤ 3, (3.3) ⇒ (3.2).
Proof. We use the previous remark: the case g ≤ 2 is trivial ((3.4) is satisfied). For g = 3, dim P 2 = 1, and it suffices to show that P 1 is of dimension 0 or empty. Let us identify P (T η ) with a plane in the projective space of S 2 Lie A η . Let ∆ h ⊂ P (S 2 Lie A η ) be the locus of matrices of rank h. Then ∆ 1 may be identified with the Veronese surface V ⊂ P 5 , since its elements are the symmetric matrices of the form

  x 2 0 x 0 x 1 x 0 x 2 x 0 x 1 x 2 1 x 1 x 2 x 0 x 2 x 1 x 2 x 2 2   ,
and P (T η ) ∩ V may be identified with P 1 . It is enough to show that there any plane P ⊂ P 5 such that P ∩ V is a curve is contained in ∆ 2 . Such a plane would be cut by three hyperplanes, which would correspond to three non collinear quadratic forms q 0 , q 1 , q 2 in x 0 , x 1 , x 2 . If P ∩ V is a curve, then one could choose the q i 's of the form .x i , where is a linear form in x 0 , x 1 , x 2 ; by change of coordinates, may be taken to be x 0 , and it is then clear that P ⊂ ∆ 2 .

3.2.6. Proposition. If µ(S) is a sufficiently general subvariety of dimension g of a special subvariety S 1 of A g of restricted PEL type (cf. [A3, 4.2

.1]), then (3.3) ⇒ (3.2).
Proof. Let D := End A η (assumed to be a division algebra), E a maximal subfield of D containing F (and chosen to be CM except if D is a totally real field) and E + the maximal totally real subfield of E. One has a decomposition Ω A = ⊕ ν:E→C Ω A ⊗ E⊗C,ν C. Fix ω ∈ Ω A with non-zero components ω ν . Fix s ∈ S(C) such that ω ν,s remains non-zero. With the notation of [A3, rem. 4.2.3 i)], the tangent space T S 1 ,s decomposes as ⊕ ν:E→C T S 1 ,s,ν , and the universal Kodaira-Spencer map on the family of PEL type induces an isomorphism T S 1 ,s,ν ∼ = S 2 Ω ∨ A,s,ν or T S 1 ,s,ν ∼ = ⊗ 2 Ω ∨ A,s,ν (according to whether D is of type I or II, or of type III or IV). In particular the contraction (θ ων 1 ) s : T S 1 ,s,ν → Ω ∨ A,s,ν is surjective, hence also (θ ω 1 ) s : T S 1 ,s → Ω ∨ A,s . Since S is sufficiently general in S 1 of dimension g, then T S is transverse to the kernel of θ ω 1 on a suitable dense open subset of S [details should be checked].

A counterexample.

We indicate here what seems to be an example with smallest g > 0 of a subvariety (resp. a special subvariety) of dimension d of A g,n with d = g, such that the restriction of the universal abelian scheme does not satisfy condition (3.3) (hence not (3.1); in this example the Betti map is nowhere a submersion). In fact d = g = 5 (resp. 6).

Let us first treat the modular example. Let E + be a real quadratic field and E a totally imaginary quadratic extension fo E + . Let A → S be a principally polarized abelian scheme of relative dimension 6, with level n ≥ 3 structure, such that End S A = O E , and with Shimura type (r ν , s ν ) = (1, 2), (1, 2) (for the two embeddings ν 1 , ν 2 of E + in C). According to [S], there exists such a family, with base S a special subvariety of dimension r ν • s ν = 6 of A 6,n . By functoriality, θ commutes with the O L -action, hence respects the decomposition H 1 dR (A/S) = ⊕ ρ:E→C H ρ . For every ρ, and general ∂, the restriction of the Kodaira-Spencer map θ ∂ to (Ω A ) ρ has rank 1 (either the source or the target is of rank 1), hence θ ∂ has rank 4 < 6.

The non modular example consists in taking a general hypersuface section in a modular example with d = 6, g = 5. Here E is a quadratic imaginary field, and End S A = O E with Shimura type (r, s) = (2, 3). For each complex embedding ρ of E, the restriction of the Kodaira-Spencer map θ ∂ to (Ω A ) ρ has rank 2 (either the source or the target is of rank 2), hence θ ∂ has rank 4 < 5.

3.3.1. Remark. It would be interesting to go beyond, and settle (3.2) fro instance in the case of maximal monodromy Sp 2g . Let A → S be the jacobian of the universal hyperelliptic curve of genus g > 0. Here S is the affine space M 0,2g+2 ∼ = (P 1 \ {0, 1, ∞}) 2g-1 . By Torelli's theorem, one has dim µ(S) = 2g -1. On the other hand, according to [M2, th. 5.6], EndA η = Z (in fact, it is knwon that the monodromy representation, which is a specialization of the Burau representation of the braid group, has Zariski dense image in Sp 2g , cf. [aC]). Hence (3.2) whould imply that, after replacing (P 1 \ {0, 1, ∞}) 2g-1 by a suitable dense Zariski-open affine subset, β is a submersion. [There are explicit calculations (without detail) by Bryant of θ in this case on MathOverflow (tag: hyperelliptic) which should settle the matter] 3.4. Let us turn to the proof of th. 1.4.1. [sketch] One may replace S by a finite etale covering and assume that A/S has a level n structure for some n ≥ 3, then replace S by a smooth dense affine open subset of its image in A g,n . By assumption, d = dim µ(S) ≥ g. Taking a sufficiently general locally closed affine subvariety S ⊂ S of dimension g, the restriction to S of the abelian scheme and of the section satisfy the same assumption. Moreover, one checks that the condition d ≥ g implies that we are in the restricted PE case4 . Remplacing S by S , one is in the situation d = g. One can then combine prop. 2.2.1, 3.2.5 and th. 3.2.1. [A similar deduction holds with prop. 3.2.6 in place of 3.2.5].

4. PROOF OF THEOREM 1.5.1.

[your §4 and 5].

1. 5 .

 5 [Motivation for the problem, and application of the results (if any)... General conjecture ....] 2. AN INFINITESIMAL APPROACH, AND ITS HOLOMORPHIC TRANSLATION.

  , a condition similar to (3.2) is studied (cf.[A3, lemma 1.4.5]), where the quantifiers are interchanged:(3.3) ∀∂ ∈ ΓT S \ 0, ∃ω ∈ ΓΩ A , θ ∂ .ω = 0. 3.2.1. Theorem. [A3, def. 4.2.1, th. 4.2.2] In the restricted PE case, condition (3.3) is satisfied. 3.2.2. Lemma. If d = g, (3.2) ⇒ (3.3).

since S can be covered by such affine open subsets.

here one needs a generalization of the notion of restricted PE to the case of an abelian scheme whose generic fiber is not necessarily simple. This does not pose problem.