Kedzierski Mikaël
email: mikael.kedzierski@univ-ubs.fr

Falcou-Préfol Mathilde

Kerros Marie Emmanuelle

Henry Maryvonne

Pedrotti Maria Luiza

Bruzaud Stéphane

Mikaël Kedzierski

A machine learning algorithm for high throughput identification of FTIR spectra: Application on microplastics collected in the Mediterranean Sea

Keywords: Microplastic, Tara Mediterranean campaign, FTIR spectra, Machine learning, k-nearest neighbor classification

The development of methods to automatically determine the chemical nature of microplastics by FTIR-ATR spectra is an important challenge. A machine learning method, named k-nearest neighbors classification, has been applied on spectra of microplastics collected during Tara Expedition in the Mediterranean Sea (2014). To realize these tests, a learning database composed of 969 microplastic spectra has been created. Results show that the machine learning process is very efficient to identify spectra of classical polymers such as poly(ethylene), but also that the learning database must be enhanced with less common microplastic spectra. Finally, this method has been applied on more than 4000 spectra of unidentified microplastics. The verification protocol showed less than 10% difference in the results between the proposed automated method and a human expertise, 75% of which can be very easily corrected.

Highlights

► A machine learning algorithm was developed to determine the chemical nature of microplastics. ► This method allows a fast and reliable automated identification even when several thousand of FTIR spectra have to be studied.► This method is the first part of a software dedicated to the study of microplastics: POSEIDON.

INTRODUCTION

Pollution of the marine environment by plastics, and especially by microplastics [START_REF] Thompson | Lost at sea: where is all the plastic?[END_REF], is the subject of an increasing number of studies [START_REF] Barboza | Microplastics in the marine environment: Current trends and future perspectives[END_REF]. Many expeditions have been carried out at sea [START_REF] Brach | Anticyclonic eddies increase accumulation of microplastic in the North Atlantic subtropical gyre[END_REF][START_REF] Dussud | Evidence of niche partitioning among bacteria living on plastics, organic particles and surrounding seawaters[END_REF][START_REF] Kanhai | Microplastic abundance, distribution and composition along a latitudinal gradient in the Atlantic Ocean[END_REF][START_REF] Pedrotti | TARA Mediterranean Expedition: Assessing the Impact of Microplastics on Mediterranean Ecosystem[END_REF][START_REF] Pedrotti | Changes in the Floating Plastic Pollution of the Mediterranean Sea in Relation to the Distance to Land[END_REF], on lakes [START_REF] Eriksen | Microplastic pollution in the surface waters of the Laurentian Great Lakes[END_REF][START_REF] Free | High-levels of microplastic pollution in a large, remote, mountain lake[END_REF]Imhof et al., 2013) or in the ice pack [START_REF] Obbard | Global warming releases microplastic legacy frozen in Arctic Sea ice[END_REF][START_REF] Peeken | Arctic sea ice is an important temporal sink and means of transport for microplastic[END_REF] to determine the state of contamination of the environment. Among the parameters studied on these particles, the chemical nature of microplastics is increasingly determined (Gewert et al., 2017;Imhof et al., 2017;Löder and Gerdts, 2015;[START_REF] Qiu | Extraction, enumeration and identification methods for monitoring microplastics in the environment[END_REF][START_REF] Wang | Microplastics in the surface M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 28 sediments from the Beijiang River littoral zone: Composition, abundance, surface textures and interaction with heavy metals[END_REF]. As the number of particles to be analyzed can range from several hundred to several thousand [START_REF] Falcou-Préfol | Statistical Methodology for Identifying Microplastic Samples Collected During TARA Mediterranean Campaign[END_REF][START_REF] Pedrotti | TARA Mediterranean Expedition: Assessing the Impact of Microplastics on Mediterranean Ecosystem[END_REF][START_REF] Pedrotti | Changes in the Floating Plastic Pollution of the Mediterranean Sea in Relation to the Distance to Land[END_REF], it is necessary to develop methods for efficient microplastics analysis [START_REF] Kedzierski | Microplastics in Mediterranean Sea: A protocol to robustly assess contamination characteristics[END_REF]. When determining the chemical nature of samples, methods based on thermal degradation as pyrolysis-gas chromatography in combination with mass spectrometry (GC/MS) are fast and efficient [START_REF] Dümichen | Fast identification of microplastics in complex environmental samples by a thermal degradation method[END_REF][START_REF] Dümichen | Analysis of polyethylene microplastics in environmental samples, using a thermal decomposition method[END_REF]Gimeno et al., 2015). Nevertheless, this type of method results in the destruction of the analyzed sample [START_REF] Primpke | Reference database design for the automated analysis of microplastic samples based on Fourier transform infrared (FTIR) spectroscopy[END_REF] which is considered as a limitation in the study of microplastics [START_REF] Rocha-Santos | A critical overview of the analytical approaches to the occurrence, the fate and the behavior of microplastics in the environment[END_REF]. Indeed, with this type of analysis, important parameters such as the size distribution, particle mass or the plastisphere are lost (Amaral- [START_REF] Albertsson | Degradation product pattern and morphology changes as means to differentiate abiotically and biotically aged degradable polyethylene[END_REF][START_REF] Dussud | Evidence of niche partitioning among bacteria living on plastics, organic particles and surrounding seawaters[END_REF][START_REF] Frère | Microplastic bacterial communities in the Bay of Brest: Influence of polymer type and size[END_REF][START_REF] Primpke | Reference database design for the automated analysis of microplastic samples based on Fourier transform infrared (FTIR) spectroscopy[END_REF][START_REF] Zettler | Life in the "plastisphere": Microbial communities on plastic marine debris[END_REF]. Furthermore, in the case of large sampling campaigns, the samples, taken at a given location and time, are unique and must be

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3 preserved over the long term in library. Thus, destructive analyses must be kept to a strict minimum.

Other methods, based on Raman (Frère et al., 2016;Imhof et al., 2017Imhof et al., , 2012;;[START_REF] Lenz | A critical assessment of visual identification of marine microplastic using Raman spectroscopy for analysis improvement[END_REF][START_REF] Zhao | An approach for extraction, characterization and quantitation of microplastic in natural marine snow using Raman microscopy[END_REF] or Infrared (IR) spectroscopy [START_REF] Ioakeimidis | The degradation potential of PET bottles in the marine environment: An ATR-FTIR based approach[END_REF]Löder and Gerdts, 2015;[START_REF] Primpke | Reference database design for the automated analysis of microplastic samples based on Fourier transform infrared (FTIR) spectroscopy[END_REF][START_REF] Primpke | An automated approach for microplastics analysis using focal plane array (FPA) FTIR microscopy and image analysis[END_REF], have shown their efficiency for the identification of the chemical nature of microplastics. However, the analysis of the spectra is time-consuming and can be a source of misinterpretation [START_REF] Elert | Comparison of different methods for MP detection: What can we learn from them, and why asking the right question before measurements matters?[END_REF][START_REF] Käppler | Analysis of environmental microplastics by vibrational microspectroscopy: FTIR, Raman or both? Anal[END_REF]Löder et al., 2015;[START_REF] Primpke | Reference database design for the automated analysis of microplastic samples based on Fourier transform infrared (FTIR) spectroscopy[END_REF]. It is indeed often necessary to compare one by one the different spectra obtained with reference spectra. Moreover, a significant part of misinterpretations is due to the fact that, during the aging of the plastics, additional bands appear on spectra [START_REF] Albertsson | Degradation product pattern and morphology changes as means to differentiate abiotically and biotically aged degradable polyethylene[END_REF][START_REF] Andrady | The plastic in microplastics: A review[END_REF][START_REF] Fotopoulou | Surface properties of beached plastic pellets[END_REF][START_REF] Küpper | Attenuated total reflection infrared spectroscopy for micro-domain analysis of polyethylene samples after accelerated ageing within weathering chambers[END_REF]. This phenomenon also become noticeable because of the fouling present on particles (Howe et al., 2002;[START_REF] Maquelin | Identification of medically relevant microorganisms by vibrational spectroscopy[END_REF][START_REF] Sudhakar | Biofouling and biodegradation of polyolefins in ocean waters[END_REF].

However, the reference spectra are made on new and clean plastics. Thus, an aged plastic is sometimes matched to the spectrum of a reference plastic due to the fact that this reference plastic has characteristic bands similar to those which appear during plastic aging. So, the greater the weight of the bands associated with aging and fouling, the more automated interpretation based on new and clean reference plastics may diverge from the actual nature of the polymer.

The development of methods to automatically determine the spectra of microplastics is therefore an important challenge of the coming years. Very recent publications began to take an interest in the issue and proposed answers. In 2017, an automated method based on the identification of the most relevant bands was tested [START_REF] Renner | Data preprocessing & evaluation used in the microplastics identification process: A critical review & practical guide[END_REF][START_REF] Renner | A New Chemometric Approach for Automatic Identification of Microplastics from Environmental Compartments Based on FT-IR Spectroscopy[END_REF]. In 2018, an automated identification method based on hierarchical cluster analysis was published [START_REF] Primpke | Reference database design for the automated analysis of microplastic samples based on Fourier transform infrared (FTIR) spectroscopy[END_REF]. More recently a new method based on Random Decision Forest was proposed (Hufnagl et al., 2019).

These three methods have shown a high degree of spectrum identification. However, other approaches are possible and have not been yet tested for the recognition of microplastic spectra. This is the case, for example, of methods belonging to the machine learning family which are quite

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commonly used in the field of spectrometry and are known to give good results [START_REF] Wehrens | Chemometrics with R Multivariate Data Analysis in the Natural Sciences and Life Sciences[END_REF].

Recently, a machine learning process has been successfully used on the determination of the chemical nature of microplastics (Hufnagl et al., 2019). Among these methods, one of them, called K-Nearest Neighbors (KNN), is known for its simplicity and efficiency in automated spectrum recognition [START_REF] Ripley | Pattern recognition and neural networks[END_REF][START_REF] Venables | Modern applied statistics with S, 4th ed, Statistics and computing[END_REF][START_REF] Wehrens | Chemometrics with R Multivariate Data Analysis in the Natural Sciences and Life Sciences[END_REF]. The originality of this work is the test of this machine learning method in the context of the study of microplastics. This project is part of the Tara Expedition in the Mediterranean Sea (2014). Tara Foundation is a French non-profit organization acting for the environment since 2003. In 2014, Tara Expedition conducted sampling for 7 months across the Mediterranean Sea. The objective of this expedition was to characterize microplastic contamination in order to better describe the effects of plastic litter on marine ecosystems. In particular, the Tara Mediterranean consortium is trying to evaluate the spatial distribution of floating debris in the Mediterranean Sea, to chemically characterize the different types of plastics and to characterize bacteria communities [START_REF] Dussud | Evidence of niche partitioning among bacteria living on plastics, organic particles and surrounding seawaters[END_REF] as well as the microand macro-organisms fixed on these plastic particles [START_REF] Pedrotti | TARA Mediterranean Expedition: Assessing the Impact of Microplastics on Mediterranean Ecosystem[END_REF]. The scientific knowledge provided by this project will allow in the long run to better target the actions to be taken to reduce contamination by plastic waste in the Mediterranean Sea. In this context, the development of an automated spectrum identification tool would facilitate the analysis of microplastics collected during the expedition. This study therefore attempted to answer the following questions: can k-nearest neighbors classification method be used for the automated identification of the chemical nature of the FTIR spectra of microplastics? What is the origin of the misinterpretations and how could they be reduced? Finally, this method was applied for the analysis of more than 4,000 microplastic spectra.

MATERIALS AND METHODS

Sample collection

Microplastic samples were collected from Mediterranean Sea waters during the Tara Expedition which was conducted between May and November 2014. Sampling was conducted using a 4.4 m long manta net (mesh size: 333 µm; net opening: 16 x 60 cm), from 120 sites which were selected based on ocean color satellite images supplied by ACRI-ST and analyzed with the Mercator circulation model (Fig. 1). Geographical coordinates and dates of sampling analyzed are available at Pangea Data Publisher http://www.pangaea. At each site, the manta net was towed on the sea surface for ca. 60 min behind the boat at an average speed of 2.5 knots, enabling thus the filtration of about 507 m 3 of seawater.

Fig. 1. Sampling effort during the Tara campaign in the Mediterranean Sea. Ocean Data View [START_REF] Schlitzer | Data Analysis and Visualization with Ocean Data View[END_REF].

Laboratory preparations and subsampling

Under dissecting microscope, preserved organic material was removed from microplastics. Particles were then counted and measured using the ZooScan image analysis method at the Laboratoire d'Océanographie de Villefranche-sur-Mer (LOV, Villefranche-sur-Mer, France) [START_REF] Pedrotti | Changes in the Floating Plastic Pollution of the Mediterranean Sea in Relation to the Distance to Land[END_REF].

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6 A total of 13,374 particles, larger than 315 µm from 42 mantas trawls, were then transferred to 96well microplates and named with a unique identifier at the Institut de Recherche Dupuy de Lôme (IRDL, Lorient, France) [START_REF] Kedzierski | Microplastics in Mediterranean Sea: A protocol to robustly assess contamination characteristics[END_REF]. To perform the tests of the machine learning method, 3,090 particles were randomly selected. For samples M14 (767 particles) and M23 (207 particles), the particles were entirely analyzed during a previous study [START_REF] Falcou-Préfol | Statistical Methodology for Identifying Microplastic Samples Collected During TARA Mediterranean Campaign[END_REF][START_REF] Kedzierski | Microplastics in Mediterranean Sea: A protocol to robustly assess contamination characteristics[END_REF]. Therefore, a total of 4,064 spectra, performed on the samples of Tara Expedition, were used in the study. The particles chosen for analysis came mainly from M14 (762 particles; 18.9%), M8 (289 particles; 7.1%), M209 (272 particles; 6.7%), M25 (212 particles; 5.2%) and M23 (207 particles; 5.1%), all sampled in the Ligurian sea (Fig. 2.A).

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The preparation step was performed in an area dedicated to the treatment of Tara Mediterranean Sea samples in controlled conditions (chemical laboratory). Contamination risks were avoided during the sample preparation stage by cleaning the different parts of the apparatus, especially glassware, with distilled water, ethanol and/or acetone. The use of plastic apparatus was avoided as far as possible. If this was impossible, the Fourier-transform infrared spectroscopy (FTIR) spectra from these materials were obtained to check whether potential contamination of the samples had occurred.

Fourier-transform infrared spectroscopy (FTIR)

The spectra of the particles were acquired using a non-destructive Attenuated Total Reflection Fourier Transform Infrared spectrometer (ATR-FTIR Vertex70v, Bruker). All spectra were recorded in absorbance mode in the 4,000-600 cm -1 region with 4 cm -1 resolution and 16 scans. Each particle was placed onto the germanium diamond cell (ATR Golden Gate) used to obtain a better quality spectrum. After each analysis by ATR-FTIR, the sample holder was cleaned with ethanol or acetone.

The sample chamber was also cleaned out with a vacuum cleaner after every sixty analyses. A total of 4,064 particles was analyzed.

Machine learning database

The learning database consists of 969 spectra, with 64.4% of them came from the M14 and M23 sites already analysed during a previous study [START_REF] Falcou-Préfol | Statistical Methodology for Identifying Microplastic Samples Collected During TARA Mediterranean Campaign[END_REF][START_REF] Kedzierski | Microplastics in Mediterranean Sea: A protocol to robustly assess contamination characteristics[END_REF]

(Fig. 2.B).
5.6% of the database's spectra were acquired from cigarette filters, based on cellulose acetate, collected on the beach of Toulhars (Larmor-Plage, France). The latter data were not part of the Tara Mediterranean campaign, but were used to complete and to test the learning database. The remaining 30.0% came from 22 different manta samples (Fig. 2.B). The spectra of the learning database were selected to represent the diversity present in the initial sample. In order to facilitate automated learning, "rare" spectra were systematically kept for the machine learning database while

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ACCEPTED MANUSCRIPT 8 more "common" spectra (mostly poly(ethylene) and poly(propylene)) were often discarded so as not unnecessarily slow down the learning process. As a result, "rare" spectra are over-represented in the machine learning database compared to more "common" spectra. The diversity of the database is currently representative of the diversity of plastic pieces found on the surface of the Mediterranean Sea. The database must be gradually enriched with new spectra such as PVC, PET, paints.

Each of the 969 spectra were then compared with the OMNIC specta 2.0 software (Thermo Scientific). The reference database was the Hummel Polymer Sample Library. This library, developed by Professor Dieter Hummel of the Institute of Physics and Chemistry at the University of Cologne (Köln), included about 2,000 spectra of polymers, polymer impurities and additives. The software interpretation was then systematically validated or rejected and reinterpreted by a polymer expert.

Thus, 17 classes were created (Fig. 2.C). The best represented class was poly(ethylene) (23.5%). It was closely associated with two other classes: "poly(ethylene) + fouling" (15.3%) and "poly(ethylene) like" (2.6%). The "poly(ethylene) + fouling" class was characterized by bands more or less intense between about 900 and 1,100 cm -1 , as well as around 1,641 cm -1 , which are not normally present on poly(ethylene) spectra. These bands can be associated with the presence of fouling on the plastic.

The "poly(ethylene) like" class was characterized by the presence of one or more additional bands compared to a poly(ethylene) spectrum. The presence of these additional bands casted some doubts on the precise identification of the spectrum. It was therefore decided not to associate them directly with the "poly(ethylene)" class. The "poly(propylene)" class was the second well represented class with 20.1% of microplastics. This class was very similar to that of "poly(propylene) like" since it included spectra with the same bands as those of "poly(propylene)", but with supernumerary bands.

The classes of "poly(styrene)", "cellulose acetate", "ethylene-propylene rubber", "poly(ethylene-vinyl acetate)", "poly(methyl methacrylate)", "poly(amide)" and "poly(urethane)" were composed with spectra of the corresponding polymers. It should be noted here that ethylene-propylene rubber is a copolymer which has a spectrum sharing similar characteristics to those of the poly(ethylene) and

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9 poly(propylene). If the ethylene/propylene ratio is low or very high, the distinction between poly(ethylene) or poly(propylene) and this copolymer can be confusing. Two classes were named morphotype 1 and 2. A "morphotype" was defined in this context as a set of spectra, numbering ten or more, with strong similarities, but for which the identification of the precise chemical nature remained unclear. This choice was made to describe and name new spectra, despite a chemical nature that remains not clearly identified at the time of the study. The last class, called "cellulose like" grouped all spectra showing strong similarities with cellulose. These spectra could therefore correspond to poorly sorted plant debris or plastics with a largely colonized surface.

Spectra pre-processing and machine learning process

All the programming was done using the software R i386 3.1.2 which is a free and open source software (The R Core Team, 2019). The pre-processing procedure of the infrared spectra was classic [START_REF] Renner | Data preprocessing & evaluation used in the microplastics identification process: A critical review & practical guide[END_REF][START_REF] Wehrens | Chemometrics with R Multivariate Data Analysis in the Natural Sciences and Life Sciences[END_REF]. Firstly, the baseline of the spectra was corrected using function "baseline()" (library "baseline") and more particularly with the 4S Peak Filling baseline estimation procedure [START_REF] Liland | 4S Peak Filling -baseline estimation by iterative mean suppression[END_REF]. This procedure, based on an iterative spectrum suppression consisting of a moving window minimum replacement, was chosen due to the fact that it is particularly adapted for non-linear baseline. The suggested starting values of the four parameters necessary for the function were respected [START_REF] Liland | 4S Peak Filling -baseline estimation by iterative mean suppression[END_REF]. The second derivative penalty for smoothing (lambda) was equal to 4 (centered in noise band). The number of buckets for subsampling (int) was equal to 176 (about 1/10 of the number of wavelengths). The initial half width of windows used for suppression (hwi) was equal to 25. The number of iteration for suppression (it) was equal to 10. Finally, after the baseline correction, spectra were normalized.

To test the process, the machine learning database was randomly divided: two third for the learning process and one third to test the process. The sampling was performed using the package "base" version 3.1.2 and the "sample()" function [START_REF] Becker | The New S Language: A Programming Environment for Data Analysis and Graphics[END_REF][START_REF] Ripley | Stochastic Simulation[END_REF]. The machine learning process was performed using k-nearest neighbor classification [START_REF] Ripley | Pattern recognition and neural networks[END_REF][START_REF] Venables | Modern applied statistics with S, 4th ed, Statistics and computing[END_REF] [START_REF] Wehrens | Chemometrics with R Multivariate Data Analysis in the Natural Sciences and Life Sciences[END_REF]. The number of neighbor (k=10) was determined by recurrence. If the entire k-nearest neighbor belongs to the same class, the spectrum is directly identified; else a majority of vote is performed. If there are less than 3 votes, the spectrum is classified in the "unknown" category. Then, the model inspection was made with a confusion matrix ("confusionMatrix()" function; library "caret). The number of spectra well categorized (true positive; TP) or placed by the model in the wrong class (false negative; FN) were counted, as well as the number of spectra misclassified (false positive; FP) and well-classified (true negative; TN) in the predicted category. The sensitivity and the specificity were then calculated [START_REF] Fawcett | An introduction to ROC analysis[END_REF]:

ݕݐ݅ݒ݅ݐ݅ݏ݊݁ܵ = ் ்ାிே (eq. 1) ݕݐ݂݅ܿ݅݅ܿ݁ܵ = ி ிା்ே (eq. 2)
In order to test the reliability of the proposed methodology, the automated learning was carried out 1,000 times. Each time, a new set of training data was randomly drawn, each time leading to a slightly different training and therefore to results that may differ. The results presented are the averages with standard deviations calculated from these thousand tests.

In order to highlight the origin of classification error, a representation of the confusion matrix was made using the "alluvial()" function (library "alluvial") and a synthetic version was designed using Adobe Illustrator CS5.

Identification of FTIR spectra with k-nearest neighbors classification and verification of the results

At this stage, 3,095 FTIR spectra still needed to be analysed. For this purpose, the KNN method and the learning database were used.

In order to test the accuracy of the final classification, a verification step was performed. This step was carried out for each of the identified classes in two sub-steps. First, a hierarchical cluster analysis was performed on all spectra of a same class ("hclust" function, centroid method). The number of

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11 sub-clusters was estimated visually. A principal component analysis (PCA; "prcomp" function) was used to check whether or not the estimated number of sub-clusters was relevant. The objective here was to identify possible outliers. In a second step, the average spectrum of each subcluster generated was then calculated and verified. If the average spectra of a subcluster did not match the correct cluster, the spectrum or the spectra group was manually identified and possibly reallocated to another class.

POSEIDON

The computer programs developed on R software in this publication are integrated into a more global software called POSEIDON (Plastic pOllutionS ExtractIon, DetectiOn and aNalysis). POSEIDON is an under development (Beta version, 2018(Beta version, -2022) ) free and open source software (Supplementary material). It aims to integrate in a single tool functionalities developed on R software to help in the study of pollution by plastics.

RESULTS AND DISCUSSION

Test of the machine learning process

The machine learning process test showed efficient learning (Fig. 3). Thus, based on the classification proposed by [START_REF] Renner | Data preprocessing & evaluation used in the microplastics identification process: A critical review & practical guide[END_REF], nine classes were placed in the "working well" category (sensitivity greater than 75%). These included "poly(amide)", "poly(propylene)", "poly(styrene)" and "cellulose acetate" classes for which sensitivity and specificity were higher than 95%. For morphotypes 1 and 2, as well as for the "poly(ethylene)+fouling" class, the results were also very good with a sensitivity ranging between 87 and 90%. The results for the "ethylene-propylene rubber" class were slightly lower but the sensitivity remained on average above 75%. Four classes fell into the "working" category: "PEVA", "poly(ethylene) like", "cellulose like" and "poly(propylene) like". Finally, the machine learning process did not work actually for four categories: "PMMA", "poly(urethane)", "poly(vinylchloride)" and "unknown".

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12 These results for "poly(ethylene)", "poly(propylene)" and "polystyrene" were very similar to those obtained by [START_REF] Renner | A New Chemometric Approach for Automatic Identification of Microplastics from Environmental Compartments Based on FT-IR Spectroscopy[END_REF], with sensitivity values close to 100%. However, the KNN method showed higher results (100%) than [START_REF] Renner | A New Chemometric Approach for Automatic Identification of Microplastics from Environmental Compartments Based on FT-IR Spectroscopy[END_REF]87.5%) and [START_REF] Primpke | Reference database design for the automated analysis of microplastic samples based on Fourier transform infrared (FTIR) spectroscopy[END_REF]91%) for "polyamide" class. Nonetheless, the results of sensitivity for [START_REF] Renner | A New Chemometric Approach for Automatic Identification of Microplastics from Environmental Compartments Based on FT-IR Spectroscopy[END_REF]50%), and especially for [START_REF] Primpke | Reference database design for the automated analysis of microplastic samples based on Fourier transform infrared (FTIR) spectroscopy[END_REF]99%) were better for "poly(urethane)" class.

These differences are mainly due to the difference in methodology and database. For example, in the particular case of "poly(urethane)", it is clearly a lack of spectra in the learning database that explain these poor results. The addition of poly(urethane) spectra collected at sea will quickly improve the results. However, this type of microplastic was relatively rare in the samples collected by Tara Expedition in the studied campaign. The contribution of other teams, that have already acquired this type of FTIR spectra, could be a solution to quickly improve the quality of the database. Fig. 3. Sensitivity (see eq. 1) as a function of 1-specificity (see eq. 2).

Sources of errors

The false negative (spectra placed in the wrong category) of the classes "poly(ethylene)+fouling", "ethylene-propylene rubber", and "poly(ethylene vinyl acetate)" were essentially related to spectra placed by the model in the category "poly(ethylene)" (Fig. 4). This type of error was also observed between the "poly(ethyelene vinyl acetate)" and "poly(ethylene)+fouling classes", as well as

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13 between the "poly(ethylene) like" and "poly(ethyelene vinyl acetate)" classes. Thus, it was possible to observe that most of these errors corresponded to misclassified spectra of classes whose chemical nature is very marked by ethylene. The same observation could be made for propylene and vinyl. Thus, spectra of the "ethylene-propylene rubber" and "poly(propylene) like" classes tended to be placed by the model in the "poly(propylene)" class. Most of the spectra of the "poly(vinyl chloride)" class were classified by the model in the "poly(ethylene vinyl acetate)" class. Two reasons can explain this type of errors. First, it was possible that the spectra had too many similarities and so the model failed to distinguish them correctly. This was probably the case, for example, between the "poly(ethylene)+fouling" and "poly(ethylene)" classes, where the differences are only related to the intensity of the band at 1,015 cm -1 . It was also the case for the classes of "poly(ethylene)", "poly(propylene)" and "ethylene-propylene rubber", for which there are no clear limits. The copolymer ethylene propylene rubber (EPR) is made from two different monomers (ethylene and propylene), which are also used in the manufacture of poly(ethylene) (PE) and poly(propylene) (PP).

Therefore, EPR being a mixture of monomers of ethylene and propylene, it has a transition spectrum between its two "pure" chemical poles, PE and PP. Depending on the desired properties, the proportions between ethylene and propylene are adjusted during manufacture. There are therefore EPRs containing high levels of propylene and others of low levels. These levels will give spectra similar to those of a poly(propylene) if the propylene content in the EPR is high. On the contrary, if it is low, the spectrum is close to that of a PE. When the mixture is balanced, the EPR spectrum has clearly identifiable characteristic bands of ethylene, as well as propylene. It is thus possible to gradually move from the "pure" PE pole to the "pure" PP pole by a set of transition spectra corresponding to different mixtures of PE and PP and where only the intensity of certain bands varied.

In some cases, spectra could belong to both classes at the same time. However, it was still necessary to place it, when creating the learning database, in a class rather than in the two. If the model in the machine learning test makes a different choice, it will automatically be counted as an error, but from

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14 the expert's point of view it is not. Thus, the sensitivity of the classes "poly(propylene) like" (51%), "poly(ethylene) like" (69%) or "ethylene-propylene rubber" (76%) was probably reduced by 20 to 30% because of transfers of these transition spectra. Thus, 35.4% of the spectra of the "poly(propylene) like" class was transferred to the "poly(propylene) class".

A FTIR spectra of a weathered microplastic is composed by three types of bands. The first one is the bands characteristic of the polymer, the second one is linked to the aging of the polymer, and the last one to the presence of fouling. Thus, microplastic spectra may have the same bands related to the presence of organisms on their surface or related to relatively similar aging process. This is probably the case, for example, of certain bands (between 1,650 and 1,700 cm -1 ) observed in some poly(ethylene) and which could be linked to the presence of proteins on their surface [START_REF] Maquelin | Identification of medically relevant microorganisms by vibrational spectroscopy[END_REF]. The impact of biofouling on the spectra of plastics should be better studied. On the one hand, it may be possible to obtain information about living organisms from the surface of microplastics and on the other hand, it is important to better control the risks in terms of misinterpretation that these additional bands can generate. The bands between 1,700 and 1,760 cm - 1 are another example. These bands, linked to the formation of carbonyl bonds, appear classically when the polymer ages [START_REF] Andrady | Microplastics in the marine environment[END_REF]. The aging of the polymer may vary according to its environment and it is possible that the spectra may not be exactly the same as a function of the history of the microplastic. These small potential variations would be worth studying on large databases.

Another source of error was the number of spectra in the database. For some classes, the database did not have enough spectra to allow an effective learning by the model. This was the case for the classes "poly(methyl methacrylate)", "poly(urethane)", "poly(vinyl chloride)" and "unknown", which do not currently have enough spectra to allow learning by the model. In these cases, the progressive enrichment of the database should make it possible to achieve learning outcomes close to those of "poly(ethylene)" or "poly(amide)".

Analysis of the full spectra database and verification

After validation of the KNN method, this method was then applied to the remaining 3,095 spectra to be identified. Spectra were classified into 13 classes, 171 sub-clusters whose average spectra, verified during the verification step, were then created. Thus, 9.5% of the average spectra differ in terms of interpretation between the model and the human expert. These differences were explained

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16 by the presence of a significant number of poly(ethylene) spectra in "morphotype 1" class. This error, which was very easily identified by the expert, corresponded to 45% of the errors made by the model. The addition of spectra in the "morphotype 1" class, in future versions of the database, should enable to reduce this kind of error. The second main error done by the model was the attribution to the "poly(propylene)" class of many spectra belonging rather to the "poly(propylene) like" class. This error was more complex to determine by the expert, because it involved spectra that varied subtly to few wavelengths. It represented about 30% of the errors made by the model. The addition of spectra in the "poly(propylene) like" class could be a small improvement in the learning.

However, it can be assumed that this class will always be at the origin of more or less significant variation in interpretation between human expertise and the KNN model. In fact, the presence in this class of transition spectra and the significant disparity in the variations observed in the spectra of this class make the machine learning more difficult. This will probably also be the case for ethylene based materials classes. Errors made, related to spectra exchange between these different classes, represented about 11% of the differences in interpretation between the model and the expert.

Finally, all other errors represented about 15% of the differences in interpretation.

With a success rate of 90.5%, the automated learning method proposed in this study is slightly inferior to the method proposed by [START_REF] Renner | A New Chemometric Approach for Automatic Identification of Microplastics from Environmental Compartments Based on FT-IR Spectroscopy[END_REF]96.1%). However, our method has been tested on 3,000 microplastic spectra compared to 300 microplastics previously and by adding a step of rapid visualization and validation of the results proposed by the model. A user can then quickly correct the most obvious model errors and increase the effects results rate to nearly 97%.

After verification and validation of the data, the 4,064 microplastics collected and analyzed by IR spectrophotometry consisted mainly of poly(ethylene) (65.5%), 10.6% of which showed traces of significant fouling and poly(propylene) (21.2%) (Fig. 5). The low represented classes where polystryrene (3%), ethylene-propylene rubber (2.3%) and PEVA (1.5%). Therefore, these results do not stand as the real microplastics type distribution observed in the Mediterranean Sea. The application of the statistical method previously developed [START_REF] Falcou-Préfol | Statistical Methodology for Identifying Microplastic Samples Collected During TARA Mediterranean Campaign[END_REF][START_REF] Kedzierski | Microplastics in Mediterranean Sea: A protocol to robustly assess contamination characteristics[END_REF] to these data will enable to refine the obtained results and determine the percentages of the various polymers in the North-West Mediterranean basin, as well as manta by manta.

Strengths and areas for improvement

POSEIDON has several advantages over the previously developed tools. First of all, it is a collaborative tool written with R software. The code can therefore be consulted and modified. New R programs facilitating the analysis of microplastic can be integrated into POSEIDON. POSEIDON is therefore a tool designed to evolve over the long term.

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

In the particular case of modules developed for the automated determination of microplastic spectra, several strengths appear. First of all, the tool allows to process and determine a large number of spectra at once (up to 50,000 currently) which is a real advantage compared to the spectrum-by spectrum determination offered by many specialized software. The work carried out in the study is based on the analysis of 4,064 different particles, which is one of the largest databases on FTIR spectra of microplastics available to date. The learning process is carried out on microplastic spectra collected at sea, which is more realistic than methods based on spectra of virgin sample.

Working from the spectra of plastics aged at sea makes it possible to better take into account the fouling and aging of plastics. Finally, through the process proposed in this work, which combines hierarchical clustering and PCA, the user can easily identify if, in the interpretation, there are outliers. This identification is quick and allows the user to judge the interpretations made by the machine. The KNN method applied to IR spectra could also be applied to data obtained using automated methods such as focal plane array (FPA) detector combined with BaF2 window or, with another learning database, to data from Raman spectrometry [START_REF] Cabernard | Comparison of Raman and Fourier Transform Infrared Spectroscopy for the Quantification of Microplastics in the Aquatic Environment[END_REF]Hufnagl et al., 2019;[START_REF] Primpke | Automated identification and quantification of microfibres and microplastics[END_REF]. Raman spectrometry is indeed often used for the analysis of microplastics and is likely to provide additional information (Frère et al., 2017;Hahn et al., 1997;Hiejima et al., 2018;[START_REF] Zhao | An approach for extraction, characterization and quantitation of microplastic in natural marine snow using Raman microscopy[END_REF]. In addition, it has recently been shown that Raman spectrometry can identify microplastics slightly more efficiently than with FTIR analysis [START_REF] Cabernard | Comparison of Raman and Fourier Transform Infrared Spectroscopy for the Quantification of Microplastics in the Aquatic Environment[END_REF].

However, there is some evolution that could improve the program over the time. First of all, the program cannot identify plastics that it does not know or does not know well. This is the case here, for example, of poly(urethane) or PVC. However, it will be possible to identify in the interpretation results an outlier corresponding to a new spectrum and therefore misinterpreted by POSEIDON. To improve learning, it will therefore be necessary to improve the learning database with new spectra.

The user can also do this himself if he has additional spectra by adding them in the file "D4_4_publication.csv" (Poseidon_files_V0.1.1.1\Data\IR_References). However, the characteristics of the new spectra must be the same as those of the reference spectra. Another solution may be to 19 send them to authors with associated metadata, so that they can be added to other reference spectra and distributed to the community. The improvement of the learning database is one of the main research approaches to improve the results. Nevertheless, the addition of new spectra will slow down the learning process as the KNN method, although very efficient, is very computationally intensive. The identification and removal of the least informative spectrum's areas could be a way to compensate in the future the slowdown associated with the addition of new spectra. This research approach has already been explored and seems to yield promising results (Hufnagl et al., 2019;[START_REF] Renner | Data preprocessing & evaluation used in the microplastics identification process: A critical review & practical guide[END_REF][START_REF] Renner | A New Chemometric Approach for Automatic Identification of Microplastics from Environmental Compartments Based on FT-IR Spectroscopy[END_REF]. Another limitation is the use of a programming language that is not necessarily mastered by the user. This first version of POSEIDON attempts to facilitate its use by limiting the use of computer code by the user. A fully interfaced version of POSEIDON could be developed. It would make it easier to process data and share results in connection with the study of pollution by microplastics without any particular knowledge of a programming language. To this end, adequate resources must be made available to develop and maintain this future version of POSEIDON.

CONCLUSION

Sampling microplastics at sea sometimes involves collecting a very large number of particles that are then analysed with spectrophotometry methods. In the case of the Tara Mediterranean Sea campaign, more than 4,000 FTIR spectra have been made. The analysis of these spectra is time consuming and skill dependent. To solve this problem, an automated learning method based on the KNN method has been implemented. The method test showed a high efficiency of this method on certain polymers (e.g. poly(ethylene), poly(propylene), poly(amide)). For others, however, such as poly(vinyl chloride), the learning database lacked spectra for the machine learning to be effective.

Additional spectra need to be added to the learning database in the near future to strengthen the results. In view of the FTIR analyses carried out on microplastics by other studies, other FTIR spectrum databases exist. Sharing them would make it possible to quickly and significantly increase Another research approach would be to integrate the various programs developed during this study, as well as those previously developed [START_REF] Kedzierski | Microplastics in Mediterranean Sea: A protocol to robustly assess contamination characteristics[END_REF][START_REF] Kedzierski | Threat of plastic ageing in marine environment. Adsorption/desorption of micropollutants[END_REF][START_REF] Kedzierski | Microplastics elutriation system. Part A: Numerical modeling[END_REF]

Fig. 2 .

 2 Fig. 2. Characteristics of the data used. A) Origin of the spectra available for the study (%). B) Origin

  the "knn()" function (library "class"). This technique is based on the calculation of the distances between objects

Fig. 4 .

 4 Fig. 4. Percentage of false negatives (greater than 5%) of the different classes in the learning

Fig. 5 .

 5 Fig. 5. Distribution of the different classes constituting the database of 4,064 spectra (learning

  . Nevertheless, the application of this method on more than 4,000 spectra confirmed its high efficiency. The synthetic verification step, based on different clustering methods, further improves the good results ratio.

  , into the POSEIDON software. Thus, POSEIDON would allow people working on microplastics to use a set of digital tools to facilitate their work.

ACCEPTED MANUSCRIPT

Acknowledgments

We thank the commitment of the following institutions, persons and sponsors: CNRS, UPMC, LOV, Genoscope/CEA, the Tara Expeditions Foundation and its founders: agnès b. ® , Etienne Bourgois, Romain Troublé, the Veolia Environment Foundation, Lorient Agglomeration, Serge Ferrari, the Foundation Prince Albert II de Monaco, IDEC, the "Tara" schooner and teams. We thank MERCATOR-CORIOLIS and ACRI-ST for providing daily satellite data during the expedition. We are also grateful to the French Ministry of Foreign Affairs for supporting the expedition and to the countries that graciously granted sampling permission. This article is contribution number 4 of Tara Mediterranean.

 Paul-Pont, I., Moreau, J., Soudant, P., Lambert, C., Huvet, A., Rinnert, E., 2016


domain: sde spi

text:
Michel Hathout

| Laurent Peyras

Claudio Carvajal

Youssef Diab

| Marc Vuillet

Development of an approach to evaluate the failure probabilities of river levees based on expert judgement: Application to a case study

Keywords: bias, calibration, debiasing, elicitation, expert judgement, failure probability, levees

Studies carried out to analyse the risks of levees must include an evaluation of the probabilities of occurrence of different failure mechanisms. The probabilistic quantitative evaluation of these mechanisms remains difficult due to often insufficient data, the natural variability of the materials, structures that are very long, and the unavailability of mechanical models for certain failure mechanisms. This makes it necessary to call for expert judgement to evaluate the probabilities of failure. However, expert judgement generally has qualitative and subjective dimensions, and it includes biases that are liable to impair the capacities of an expert to elicit their evaluations. This article proposes an approach to processing expert judgement that includes the modalities of Individual expert Elicitation, Calibration, Aggregation, and Debiasing of expert judgement (IeCAD). This IeCAD approach has been developed for river levees in view to correcting biased expert evaluations in the case of evaluating the failure probability of structures.

| INTRODUCTION

River levees are structures that are raised above the natural level of the land in view to protecting naturally floodable areas [START_REF] Peyras | Development of an expert-led GIS-based approach for assessing the performance of river levees: The Digsure method and tool[END_REF]. Evaluating the reliability of levees is a major challenge for the managers of these structures in order to predict a risk of failure [START_REF] Kolen | The impacts of storm Xynthia February 27-28, 2010 in France: Lessons for flood risk management[END_REF]. Moreover, many regulations require the evaluation of the reliability of structures in a probabilistic framework, in order to demonstrate levee failure risks [START_REF] Ciria | The international levee handbook[END_REF].

Many levees are old structures and there is limited knowledge on their initial construction and further reinforcements over time, and information on their behaviour is seldom available [START_REF] Ciria | The international levee handbook[END_REF][START_REF] Tourment | Small dams and levees-General report[END_REF]. So evaluating the reliability of river levees in a probabilistic framework is difficult due to the scarcity of available data, related to the material composition of the levee throughout its length, their geotechnical properties, hydraulic and mechanical behaviour of levees as leakages, pore pressures, and displacements. Contrary to other hydraulic structures such as dams for which large amounts of data linked to their construction and monitoring are available [START_REF] Mouyeaux | Probabilistic stability analysis of an earth dam by SFEM based on field data[END_REF], the lack of information on the composition of levees often makes difficult their evaluation using probabilistic approaches.

Several studies have proposed methods to quantify the reliability of levees. The International Levee Handbook [START_REF] Ciria | The international levee handbook[END_REF] and the FloodProBE project [START_REF] Floodprobe | Combining information for urban levee assessment[END_REF] highlight existing methods for analysing the reliability of levees: (a) Expert assessment based on previous experiences using index-based methods; (b) Index-based methods in which a number of performance features are assessed to determine reliability; (c) Empirical models for levee failure mechanisms where rules can be established to assess performance; and (d) Physical models to determine the reliability of a levee based solely on physics [START_REF] Floodprobe | Combining information for urban levee assessment[END_REF].

According to this classification, different levee management organisations have developed methods and tools to conduct levee assessments. [START_REF] Serre | Evaluation of hydraulic works in civil genius[END_REF]; [START_REF] Serre | Levee performance assessment methods integrated in a GIS to support planning maintenance actions[END_REF] developed a method to assess levees based on data from visual inspections. [START_REF] Vuillet | Levees performance evaluation based on subjective probability[END_REF] and [START_REF] Peyras | Development of an expert-led GIS-based approach for assessing the performance of river levees: The Digsure method and tool[END_REF] developed a levee assessment method aimed at estimating the performance of levees using probability distributions for levee performance indicators. Reliability analysis tools can be used to determine the overall reliability of a levee, such as the Risk Assessment for Strategic Planning method in the UK [START_REF] Gouldby | A methodology for regional-scale flood risk assessment[END_REF], and the flood early warning system in the Netherlands [START_REF] Knoeff | Automated flood risk management[END_REF].

Due to complex levee failure mechanisms [START_REF] Simm | The significance of failure modes in the design and management of levees-A perspective from the International Levee Handbook team[END_REF], the different sources of data for levee assessments [START_REF] Van Der Meij | Combining different data sources for assessment of urban flood defences[END_REF], and the uncertainty on the data available [START_REF] Vuillet | Levees performance evaluation based on subjective probability[END_REF], a high level of expertise is required to determine the probabilities of the levee failure mechanisms [START_REF] Mériaux | Surveillance, maintenance and diagnosis of flood protection dikes: A Practical handbook for owners and operators[END_REF]. The objective of this study is to propose a probabilistic approach to evaluate the reliability of levees based on expert judgement. The study proposes specific elicitation, calibration, and debiasing methods to process expert evaluations and thus reduce the biases intrinsic to expert judgement. In the end, the objective is to propose methods based on expert judgement for estimating the levee failure probability (the estimated or the subjective probability of failure), constituting an expert estimate of the actual levee failure probability.

In the scientific literature, expert judgement is the expression of an opinion based on knowledge and experience that the expert makes in response to a question [START_REF] Ortiz | Use of expert judgment in NUREG-1150[END_REF]. Despite their competences and in-depth knowledge, the opinions expressed by the expert contain the biases specific to opinions and judgements [START_REF] O'hagan | Uncetains judgments: Eliciting experts' probabilities[END_REF]. Indeed, there is always a transformation that can cause a difference between the knowledge held and the way it is presented and expressed [START_REF] Bateson | Mind and nature: A necessary unity[END_REF]. The cognitive literature gives a definition of biases linked to expert judgement [START_REF] Yachanin | The effect of the thematic content on cognitive strategies in the four-card selection task[END_REF]: biases are distortions of expert reasoning that impair the validity of its inferences and conclusions [START_REF] Kahneman | Judgement under uncertainty, heuristics and Biaises[END_REF].

The methods of treating expert judgement are mainly dealt with in the statistical literature, with applications in economics [START_REF] Lichtendahl | Is it better to average probabilities or quantiles?[END_REF], insurance [START_REF] Goulet | Modeling without data using expert opinion[END_REF], energy [START_REF] Shlyakhter | Quantifying the credibility of energy rojections from trends in past data: The U.S. energy sector[END_REF], and industry [START_REF] Clemen | Debiasing expert overconfidence: A Bayesian calibration model[END_REF][START_REF] Cooke | Experts in uncertainty: Opinion and subjective probability in science[END_REF]. They have been developed to reduce biases when employing expert judgement and consist of three main modalities for dealing with bias: (a) elicitation of expert opinion, by focusing on developing the best modalities for collecting opinions in view to eliciting the most precise expert judgement possible [START_REF] Vick | Degrees of belief: Subjective probability and engineering judgment[END_REF], [START_REF] Soll | Overconfidence in intervalle estimates[END_REF], [START_REF] Koehler | Blackwell handbook of judgment & decision making[END_REF]; (b) calibration of expert judgements [START_REF] Cooke | Experts in uncertainty: Opinion and subjective probability in science[END_REF], including the weighting of opinions as a function of their pertinence, and their aggregation [START_REF] Lichtendahl | Is it better to average probabilities or quantiles?[END_REF]; (c) debiasing expert judgement, aimed at applying mathematical corrections to expert evaluations [START_REF] Clemen | Debiasing expert overconfidence: A Bayesian calibration model[END_REF].

Regarding civil engineering, there are few works that focus on treating expert judgement and reducing bias. [START_REF] Peyras | Dam ageing diagnosis and risk analysis: Development of methods to support expert judgment[END_REF] suggested employing expert judgement to measure the occurrence of function loss. Nonetheless, the subjective probabilities provided by the experts were not subjected to any specific protocol relating to their collection or treatment. [START_REF] Vuillet | Levees performance evaluation based on subjective probability[END_REF] proposed modalities for eliciting expert judgement aimed at reducing biases, though they did not lead to the explicit reduction of cognitive biases.

This article first presents the development of a protocol for eliciting expert opinions in the form of failure mechanisms probabilities. Then, a calibration and aggregation model is proposed for the opinions expressed by a panel of experts. Lastly, this article presents the development of an approach used to debias the opinion of the expert panel. The approach developed is applied to the case of an existing levee.

| APPROACH TO EVALUATING FAILURE PROBABILITIES BY EXPERT JUDGEMENT

| Principles-general overview

The principle of the approach developed to evaluate the failure probabilities of levees is based on the comparison between:

• The calibration variables whose true values are known through numerical calculation, and which permit evaluating the capacity of the experts to provide a pertinent and precise estimation, and then calibrate and debias their evaluation, • The variables of interest whose values are searched by expert judgement.

The calibration variables are determined by expert judgement on the one hand, and by numerical calculation on the other. The comparison between the expert and numerical evaluation allows determining the calibration weighting given to the expert opinions of each expert of the panel and the correction coefficients given to the expert biases.

The proposed IeCAD (Individual elicitation, Calibration, Aggregation, and Debiasing) approach comprises three steps of treating expert judgement (Figure 1):

• The elicitation by expert judgement of the failure probabilities of levee failure mechanisms; • The calibration and aggregation of the opinions of different experts in view to aggregating them; • The debiaising of expert opinions in order to process the biases of over-and under-confidence liable to impair the calibrated and aggregated expert evaluations:

| Identification of the calibration variables and the variables of interest

In our study, the variables of interest are the failure probability of cross-sections of an existing levee with respect to the sliding, internal erosion, and scouring failure mechanisms. Concerning the sliding mechanism, the probability of failure can be assessed by a reliability analysis. Concerning internal erosion and scouring, these both failure mechanisms are subject to many stability criteria that can be found in the literature according to the type of erosion, materials, and levee profiles [START_REF] Ciria | The international levee handbook[END_REF]Vrijiling, Vrijling, 2001;Apel and[START_REF] Apel | A probabilistic modelling system for assessing flood risks[END_REF]Blöschl, 2006, Mazzoleni, Barontini, Ranzi, &[START_REF] Mazzoleni | An innovative probabilistic methodology for evaluating the reliability of discrete levee reaches owing to piping[END_REF][START_REF] Vorogushyn | A new methodology for flood hazard assessment considering dike breaches[END_REF]. So for internal erosion and scouring mechanisms, it is difficult to make formulation choices given the large number of physical laws and parameters potentially combinable as well as uncertainties resulting from these modelling assumptions and from the natural variability of the properties of the material composition of the levee throughout its length. In the end, there is not a really state of the art consensus for the limit state conditions for internal erosion and scouring in the national regulations, standards and professional recommendations. Failure probabilities of levee with respect to the internal erosion and scouring failure mechanisms cannot be obtained easily with mechanical and modelling approaches, as evidenced by the important research carried out on the subject such as in the ICOLD regional club "European Working Group on Internal Erosion of Dams, Dikes and Levees and their Foundations" (https://internal-erosion.irstea.fr/). So, we are looking for the levee failure probability with respect to internal erosion and scouring mechanisms using expert judgement, and the variables of interest that we consider in our study are the failure probability of levee cross-sections with respect to these mechanisms.

The calibration variables must be of the same nature as the variables of interest to permit calibration and debiasing [START_REF] Cooke | Experts in uncertainty: Opinion and subjective probability in science[END_REF]. We adopted for the calibration variables the failure probability of the cross-sections of levees with respect to the sliding failure mechanism. Indeed, the sliding mechanism has a precise formulation of its limit conditions and so permits calculating a failure probability using quantitative approaches [START_REF] Peyras | Study on a semi-probabilistic method for embankment hydraulic works -Application to sliding mechanism[END_REF]. To determine the calibration variables by numerical calculation, we search the failure probability of the sliding failure mechanism using a mechanical-probabilistic model based on the limit equilibrium and in which the resistive properties of the materials are modelled by probabilistic laws [START_REF] Mouyeaux | Probabilistic stability analysis of an earth dam by SFEM based on field data[END_REF]. It is then possible to use Monte Carlo simulations to obtain a probabilistic distribution of the safety factor and calculate the failure probability of the cross-section studied. Figure 2 illustrates the probabilistic distributions for friction and cohesion angle on a levee cross-section. With Monte Carlo simulations and considering a Morgenstern-Price method for sliding criteria, the probabilistic distribution of the safety factor obtained (Figure 2), and then the failure probability of the levee cross-section can be evaluated considering the integral of safety factor distribution less than 1.

For the sliding failure mechanism, we study inner slope instability in considering deep sliding circle into the embankment and levee foundations, in order to study slope instability conducting to the complete levee failure (see the slope circle considered in Figure 3).

| Approach to elicit expert opinions for river levees

Implementing the approach to eliciting expert opinions starts with identifying a panel with several experts whose

F I G U R E 2
Example of a probabilistic model of the resistive properties of materials for a levee cross-section activities are directly linked to river levees [START_REF] Koehler | Blackwell handbook of judgment & decision making[END_REF].

Each of the experts of the panel is questioned individually on the failure probabilities of several levee crosssections to be evaluated with respect to different failure mechanisms. The experts are given enough information and time to fully understand the issue before being questioned. At this purpose, a questionnaire form used to collect the expert judgement was developed to this end. For each calibration variable and variable of interest to be elicited, the form contains three main items (Figure 3): the question, the information, and the response:

• The question asked to the experts concerns the sliding failure mechanism: "Given all the information available, what is the probability P f that the levee will fail due to sliding of the downstream slope if a flood occurs that reaches the crest of the levee?." Each expert estimates the probability P f related to the sliding failure probability in the hypothetic situation where the flood reaches the crest of the levee. These estimates (variables of interest) will be after compared to the calculated probabilities (calibration variables) obtained in the identical hydraulic situation. • The information are the geometric, hydraulic, and geotechnical data, the standard cross-section to be evaluated, the probability laws of the geotechnical characteristics of the cross-section, the data from the deterministic analysis represented by the safety factor SF related to the sliding failure mechanism. The approach to calibrating the panel experts' opinions is based on [START_REF] Cooke | Experts in uncertainty: Opinion and subjective probability in science[END_REF] model. It consists in evaluating and weighting the panel experts' opinions in relation to calibration variables whose real values are known. [START_REF] Cooke | Experts in uncertainty: Opinion and subjective probability in science[END_REF] model allows calculating an individual calibration weight w e for each expert and the relative calibration weight w 0 e in relation to all the experts of the panel, since the closer this expert's elicitations are to reality, the higher the calibration weight assigned to them is.

The calibration weight is determined on the basis of the calibration score C e and the entropy score K e according to the following formulas [START_REF] Cooke | Experts in uncertainty: Opinion and subjective probability in science[END_REF]:

w e = C e :K e and w 0 e = C e :K e P z e = 1 C e :K e , ð1Þ
where w e is the calibration weight of the expert e, w 0 e is the calibration weight relative to the expert e in a panel of experts, C e is the expert e's calibration score e, K e is the expert e's entropy score e, and z is the number of experts.

The calibration score C e permits evaluating the accuracy of the information given by the expert for the calibration variables, by comparing between a distribution obtained by the expert and a distribution calculated numerically. It is determined by an error probability evaluated with the χ 2 statistical test (interdependence test). The calibration score C e is evaluated by the following formula [START_REF] Cooke | Experts in uncertainty: Opinion and subjective probability in science[END_REF]:

C e = 1-P 2*n*I e c, p ð Þ ð Þ = 1-X 2 2*n*I e c,p ð Þ ð Þ , ð2Þ
where P() is the probability of a random variable following a χ 2 law; X 2 ðÞ is the distribution function of a random variable following the χ 2 law; n is the number of calibration variables; c is the calibration vector representing the portion of true values included within each inter-quantile interval c = {c 1 , c 2 , …, c j }; p is the theoretical probabilities vector representing each inter-quantile interval p = {p 1 , p 2 , …, p j } ; I e c,p ð Þ=

P j i = 1 c i ln c i p i : relative information
between the theoretical probabilities p and the calibration vector c.

The entropy score K e permits measuring the quantity of information contained in the probabilistic distributions given by the experts. It is based on a measure of the distances between the vector of subjective probabilities s and the vector of theoretical probabilities p. The entropy score K e is evaluated by the following equation [START_REF] Cooke | Experts in uncertainty: Opinion and subjective probability in science[END_REF]:

K e = 1 n X n i = 1 I e,i p, s ð Þ, ð3Þ
where s is the vector of subjective probability representing the subjective probabilities of each interquantile interval. The probabilities of s are determined by the percentage of each inter-quantile interval relative to the probabilistic scale of the calibration variable s = q 5%q 0% q 100%q 0% , q 50%q 5% q 100%q 0% , q 95%q 50% q 100%q 0% , q 100%q 95% q 100%q 0% n o ; and

I e p,s ð Þ= P j i = 1 p i ln p i s i
is the relative information between the vector of subjective probabilities s and the vector of theoretical probabilities p.

| Aggregation of expert opinions

We propose applying the aggregation of the panel experts' opinions by way of the median of the quantiles. The aim is to conserve the initial form of the expert elicitations given in a probabilistic format, with a value considered as the most likely, and an uncertainty interval [START_REF] Lichtendahl | Is it better to average probabilities or quantiles?[END_REF]. This approach corresponds to the weighted sum of the calibrated quantiles of the panel experts' opinions. The aggregated quantiles (q 5% , q 50% , q 95% ) are evaluated by the following formulas:

q 5% = X w 0 e :q 5% e ð Þ ; q 50% = X w 0 e :q 50% e ð Þ ;q 95% = X w 0 e :q 95% e ð Þ, ð4Þ
where q 5% , q 50% , q 95% are the aggregated quantile corresponding to a probability, respectively of 5, 50, and 95%. At the end of the expert opinion calibration and aggregation phase, we now have a single evaluation of the expert panel that takes into account the relative weight of each expert in the panel. This calibrated and aggregated evaluation is the response of the expert panel to the question of evaluating the failure probabilities of levee for a given failure mechanism.

| Expert judgement debiasing approach for river levees

The aim is to apply mathematical corrections to the panel's expert evaluations, in order to get as close as possible to the calibration variables. The objective is to quantitatively identify the trend of the expert panel's opinions for under-or over-confidence which impair the estimation of the most likely value and the uncertainty interval, and then to apply a mathematical correction. We propose to implement an expert opinion debiasing approach based on the model of [START_REF] Clemen | Debiasing expert overconfidence: A Bayesian calibration model[END_REF], consisting in determining three correction coefficients by iterative calculation.

The first coefficient β is intended to correct the most likely value corresponding to the 50% quantile:

R * i = β R i , ð5Þ
where R * i is the most likely debiased value; β is the correction coefficient of the most likely value; R i is the calibrated and aggregated expert elicitation of the most likely value (q50%).

The second coefficient α L is aimed at correcting the value elicited of the lower bound of the uncertainty interval, corresponding to the 5% quantile:

L * i = β -α L ð ÞR i + α L L i , ð6Þ
where L * i is the debiased lower bound of the uncertainty interval; α L is the correction coefficient of the lower bound of the uncertainty interval; L i is the calibrated; and aggregated expert elicitation of the lower bound of the uncertainty interval (q5%).

The purpose of the third coefficient α U is to correct the elicited value of the upper bound of the uncertainty interval, corresponding to the 95% quantile:

U * i = β -α U ð ÞR i + α U U i , ð7Þ
where U * i is the debiased upper bound of the uncertainty interval; α U is the correction coefficient of the upper bound of the uncertainty interval; U i is the calibrated; and aggregated elicitation of the upper bound of the uncertainty interval (q95%).

| APPLICATION AND RESULTS OF THE APPROACH DEVELOPED

The levee studied is an earth-fill levee 5,500 m long raised to protect a town in France (Figure 4). The height of the levee varies from 1 to 6 m (Figure 5).

The approach developed was implemented by a panel of six engineers having different professional backgrounds and experiences (geotechnics, river hydraulics, civil engineering, and hydrology). The calibration variables (variables no. 1-30) correspond to failure probabilities associated with the sliding mechanism for 30 cross-sections: these variables are obtained by numerical mechanical-probabilistic calculation and permit a robust statistical analysis.

The study was applied to 30 variables of interest (variables no. 31-60; Table 1) obtained by expert judgement:

• Variables of interest no. 31-40 correspond to failure probabilities of the sliding mechanism. For these 10 variables of interest, we have expert judgement evaluation but also numerical calculation evaluation, so we will be able to compare the results from IeCAD approach to the true values of the variables, • Variables of interest no. 41-60 correspond to failure probabilities with respect to internal erosion and scouring mechanisms.

| Application and results of the expert opinion elicitation phase

Figure 6 presents the elicitations of expert no. 1 relating to failure probabilities (P f ) regarding three failure mechanisms. The number of the variable related to its cross section is given in the vertical axis, the expert values (quantiles 5, 50, 95%) are the black squares and the true values obtained by calculations are the grey dots.

Table 2 shows the distribution the true known values of the calibration variables in the interquantile intervals elicited by the six experts for the cross-sections no. 1 to 30:

The percentage of the true values contained in the range [5%, 95%] varies from 40% (17% + 23%) for expert no. 3 to 66% (33% + 33%) for expert no.2, which is considerably lower than the target percentage of 90% (90% corresponds to the interquantile interval between quantiles 5 and 95%). The distribution of the median of the elicitations of expert opinions contains 55% of the true values in the uncertainty interval [5%, 95%], which is also considerably lower than the target percentage of 90%. This means that a large number of true values lie outside the uncertainty intervals elicited by the panel experts, reflecting a trend towards overconfidence or underconfidence in the expert judgement.

The distribution of the median of the elicitations of expert opinions contains 26% of true values in the interval [0%, 5%], which is higher than target percentage of 5%. This means that a high percentage of the expert evaluations led to overestimated values for the failure probability, reflecting a trend towards under-confidence, and expressing caution in the experts' evaluations. These overestimated evaluations tend towards safety, but can also lead to decisions to carry out expensive levee reinforcement works in excess of their real necessity. On the other hand, the distribution of the mean of the elicitations of expert opinions contains 19% of true values in the interval [95%, 100%], which is also higher than the target percentage of 5%. This means that a high percentage of the expert evaluations also gave underestimated failure probability values, reflecting a trend towards overconfidence. These underestimated evaluations do not tend towards safety and lead to overestimating the levee's resistance. The combined presence of large numbers of biases of over-and under-confidence (with nonetheless a greater bias towards underconfidence tending towards safety) demonstrate in a general way the biases intrinsic to expert evaluations.

F I G U R E 4 Aerial view of the levee studied (scale: 1/25,000 in A3 format)

| Application and results of the expert opinion calibration phase

The result of the expert opinion calibration is given with the mean of the calibration scores, the entropy scores and relative calibration weighting (Table 3), obtained from the 30 calibration variables (no. 1-30).

Since elicitation is performed on the q 5% , q 50% and q 95% quantiles, the theoretical probability vector p associated with the inter-quantile intervals is p = {0.05, 0.45, 0.45, 0.05}. The calibration vectors c can be obtained directly from Table 2 (for expert 1, c = {0.30, 0.37, 0.23, 0.10}). The calibration score C e presented in Table 3 is then calculated using the relative information between c and p vectors, I e (c,p) according Equation 2(for expert 1: I e (c,p) = 0.38, which gives a score C e = 0.79 by using Equation 2).

For the calculation of the entropy score, as the elicited failure probability values are expressed in the format (10 -x ), we change the variable in order to conserve a constant difference in absolute value between (10 -x ) and (10 -(x -1) ), which amounts to applying the common logarithm to the elicited probabilities (log 10 (P f )). For example, expert 1 elicits the probability values of 10 -4 , 10 -2 and 10 -1 for the quantiles 5, 50 and 95%, respectively, for the first calibration variable. With the variable change to base log 10 , we thus obtain q 5% = -4, q 50% = -2 and q 95% = -1, which makes it possible to evaluate the subjective probability vector s 1 = {0.24, 0.39, 0.19, 0.18} and the relative information I e,1 (p,s) = 0.31 with Equation 3. The entropy score K e is finally evaluated with Equation 3, and corresponds to the average of the relative information I e,i (p,s) evaluated for the 30 calibration variables.

The results obtained from the calibration of the expert opinions show that the values of the calibration score (C e ) vary between 3 and 91%. The highest value was obtained by expert no.2, indicating that their elicitations contained the highest number of true calibration values in comparison to the elicitations of the other panel experts. On the contrary, the lowest calibration score was obtained by expert no.3, indicating that their elicitations contained the lowest number of true calibration values.

Regarding the entropy score, the values varied from 0.52 to 1.04. The highest value was obtained by expert no.4, indicating that the uncertainty intervals elicited by expert no. 4 were more precise than those given by the other experts of the panel. On the contrary, the lowest score was obtained by expert no.5, indicating that the uncertainty intervals elicited by this expert were wider than those given by the other experts of the panel.

Finally, the calibration phase allowed assigning a relative calibration weighting to each expert, taking into account both the pertinence and the precision of their evaluations. In our application, it turned out that expert no. 2 had the best relative calibration weighting (40%) in the panel, contrary to expert no.5 (3%).

| Application and results of the expert opinion aggregation phase

The aggregation phase corresponds to a weighted sum performed for each of the quantiles q 5% , q 50% and q 95% (using Equation 4) and for each of the variables. For the levee section corresponding to variable n 1, the results obtained are q 5% = -3.34, q 50% = -1.93 and q 95% = -0.72 (in log 10 [P f ] scale), which correspond respectively to q 5% = 4.6E-04, q 50% = 1.2E-02 and q 95% = 1.9E-01 (P f ).

Figure 7 illustrates the result for the 30 calibration variables associated with the theoretical levee crosssections (continuous lines). Figure 7 can be interpreted using Table 4 below, which presents the distribution of the true values of the calibration variables in the inter-quantile intervals at the end of the expert opinion aggregation phase:

At the end of the calibration and aggregation phase, we observed that the distributions of the calibrated and aggregated expert opinions contained 60% of true values in the uncertainty interval [5%, 95%]. Thus, the aggregation and the calibration of expert opinions led F I G U R E 6 Elicitations of expert no. 1 relating to calibration variables to a higher number of true values in the uncertainty interval [5%, 95%] in comparison to the raw elicitations comprising 55% of true values in this interval (Table 2). This calibration and aggregation phase therefore improved the quality of the expert evaluations, despite the fact that the correction did not allow reaching the percentage of the ideal target interval (90%), leaving a still considerable bias of overconfidence or underconfidence following the calibration and aggregation phase.

More specifically, the results obtained following the calibration and aggregation phase showed a substantial reduction of the true values in the overconfidence interval [95%; 100%], falling from 19% (Table 2) to 10% (Table 4), which was a significant improvement and thus tending to increase the reliability of the levees. The evolutions in the interval [5%, 95%] were slight, changing from 26% (Table 2) to 30% (Table 4), expressing a small evolution in processing under-confidence during the calibration and aggregation phase.

| Application and results of the expert panel opinion debiasing phase

The correction coefficients defined in the debiasing phase were obtained by an iterative calculation, starting with the coefficient β (Equation 5): until obtaining 50% of the true values in the interquartile interval [0%,50%] (and therefore 50% in the interquartile interval [50%,100%]). Then, the correction coefficients α L and α U (Equations ( 6) and ( 7) respectively) are also obtained by iterative calculation: until obtaining respectively 5% of the true values in the interquantile interval [0%, 5%] and 5% of the true values in the inter-quantile interval [95%, 100%]. The correction coefficients obtained by these iterative calculations are: (a) coefficient β = 1.22 for the correction of quantile q 50% , (b) coefficient α L = 1.51 for the correction of quantile q 5% and (c) coefficient α U = 1.83 for the correction of quantile q 95% .

• The correction coefficient (β = 1.22 ) shows that panel of experts tended to reduce its estimations regarding the most likely central variables (R i ). Thus, a coefficient β must be applied to the central values resulting from the aggregation phase to obtain the value of the 50% quantile of the calibration variable; • The correction coefficient ( α L = 1.51 ) reflects that the distances estimated between (L i ) and (R i ) resulting from the aggregation phase tend to be lower than those they should have been. Thus, it was necessary to apply a coefficient α L to obtain the debiased distance between the 5% and the 50% quantile; • The correction coefficient ( α U = 1.83 ) reflects that the distances estimated between (R i ) and (U i ) resulting from the aggregation phase tend to be lower than they should have been. Thus, a coefficient α U must be applied to obtain the debiased distance between the 50% and the 95% quantile.

F I G U R
For the levee section corresponding to variable n 1, with these correction coefficient values, and using Equations ( 5)-( 7), the values of the debiased quantiles are q 5% = -4.48, q 50% = -2.35 and q 95% = -0.14 (in log 10 (P f ) scale), which correspond, respectively, to q 5% = 3.3E-05, q 50% = 4.4E-03 and q 95% = 7.2E-01 (P f ). Figure 8 shows the debiased opinions of the expert panel resulting from the application of correction coefficients to the aggregated opinions of the panel of experts.

Figure 8 is interpreted using Table 5 below, which presents the distribution of the true values of the calibration variables in the interquantile intervals following the debiasing phase.

At the end of the debiasing phase, we observed that the distributions of the opinions of the expert panel contained 94% of true values in the uncertainty interval [5%, 95%]. Thus, the debiasing phase of the opinions of the expert panel permitted reaching the percentage of the ideal target.

The results obtained after the debiasing phase show an optimal reduction of true values in the overconfidence interval [0%; 5%], falling from 30% (Table 4) to 3% (Table 5), indicating a considerable improvement in the I G U R E 8 Debiased calibration variables of the panel of experts quality of the evaluation of the structures. The evolutions in the interval [95%, 100%] were also considerable, decreasing from 10% (Table 4) to 3% (Table 5), expressing a marked evolution in the treatment of overconfidence bias during the expert opinion debiasing phase, tending towards an improvement in the reliability of the structures.

| Synthesis of the IeCAD approach to processing expert judgement

Figure 9 summarises the evolution of the quality of information provided by expert judgement in the sequence of the different phases of processing the IeCAD approach:

• interval [5%, 95%]: the distributions of the expert panel opinions contained 55% of true values following the elicitation phase, then 60% following the calibration phase, then 94% following the debiasing phase; • interval [0%, 5%]: the distributions of the expert panel opinions contained 26% of true values following the elicitation phase, then 30% following the calibration phase, then 3% following the debiasing phase; • interval [95%, 100%]: the distributions of the expert panel opinions contained 19% of true values following the elicitation phase, then 10% following the calibration phase, then 3% following the debiasing phase.

| Application to the variables of interest

The expert judgement calibration and debiasing approach was applied to the variables of interest. Figure 2 presents The correction provided by debiasing therefore appears significant, demonstrating the advantage of applying a full treatment including calibrationaggregation and debiasing to the evaluations performed by the panel of experts for failure probabilities. By way of illustration, Figure 10 shows the treatment of the variables of interest for the internal erosion (no.41-50) and scouring (no.51-60) failure mechanisms for which no limit state models were available.

| Discussion

The results obtained in the case study show that the calibration-aggregation phase as well as the debiasing phase of the developed approach allow improving the results obtained from expert elicitations.

The quality of the final result is logically conditioned by the quality of the experts. Logically more experienced experts can result in better evaluation of the levee failure probability, statistically. This would be reflected in particular in the width of the uncertainty intervals ([5%, 95%] in the case study): the better the experts, the narrower the interval (leading to a more precise result); on the contrary, with less experienced experts, the interval obtained would be wider (leading to an imprecise result).

In the case study, the panel was composed of six experts, which is a rather high number for a risk analysis study of river levee [START_REF] Peyras | Probability-based assessment of dam safety using combined risk analysis and reliability methods: Application to hazards studies[END_REF]. These experts had different experiences in the concerned disciples (geotechnics, civil engineering, hydraulics) and also different levels of qualification (junior, confirmed and senior). Thus, there is necessarily a dispersion in their assessments.

This article did not study the influence of the number of experts on the results. In the case study, a panel of six experts was selected in order to reproduce a panel of experts as implemented in the river dike risk analysis studies (typically 3-6 experts-see [START_REF] Peyras | Probability-based assessment of dam safety using combined risk analysis and reliability methods: Application to hazards studies[END_REF]. However, an increase in the number of experts would not necessarily lead to an improvement in results: for example, it can be expected that the quality of results would be better if the number of experts were reduced by keeping the most qualified experts. Thus, the professional qualification of experts would have a greater influence on the results in relation to the number of experts.

| CONCLUSION

Regarding the field of river levees, the lack of data on structures, the uncertainties that impair them and the lack of consensus in the state of the art for the limit states related to internal erosion and scouring failure mechanisms, reduce the opportunity of using statistical and probabilistic methods for determining their failure probability using mechanistic-probabilistic approaches. This makes expert judgement essential for evaluating the failure probabilities of levees with respect to different failure mechanisms.

However, the presence of biases, which will impair the opinions of experts, is a considerable drawback when calling on expert judgement in a risk analysis study. Consequently, our study proposed an approach for evaluating the failure probabilities of levees based on expert judgement and which includes the treatment of the latter.

The approach proposed permits eliciting, calibrating, aggregating and debiasing expert evaluations. The application of this methodological approach provided significant advantages:

• combining expert opinions based on calibration and entropy scores; • estimating a quantitative uncertainty on the final failure probability obtained using an uncertainty interval; • identifying the best expert elicitations and assigning a calibration weighting according to their pertinence and precision; • treating the presence of over-and under-confidence bias quantitatively in order to obtain a final debiased failure probability.

At the end, the methods proposed use the expert judgement for estimating the levee failure probability (the estimated or the subjective probability of failure), constituting an expert estimate of the actual levee failure probability.

On the basis of this work, our research will be continued by testing our method to several dike case studies so as to continue validating the robustness of the approach. At the end, we plan to use the IeCAD approach in an operational framework to evaluate the probabilities of failure mechanisms (internal erosion and scouring) in risk analysis studies.

Otherwise, the approach developed for river levees can be adapted and applied to other areas where recourse to expert judgement is the only means available for obtaining usable information in the framework of carrying out a reliability analysis. In particular, the approach applied to levees can be extended to other hydraulic structures by making specific adjustments for each category of civil engineering structure, as small dams often poorly documented or large linear channels.

F

  I G U R E 1 Explanatory chart of the Individual expert Elicitation, Calibration, Aggregation, and Debiasing (IeCAD) approach developed

•

  The response of the expert elicitations in the form of an uncertainty interval and a most likely value: • An uncertainty interval [quantile 5%, quantile 95%]. Civil engineers are used to working with such quantiles since they are the same in semiprobabilistic methods such as Eurocodes (Vuillet et al., 2013), • The most likely value contained in the uncertainty interval elicited.

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  I G U R E 3 Example of a levee cross-section sheet taken from the questionnaire form 2.4 | Expert opinion calibration and aggregation approach for river levees 2.4.1 | Calibration of expert opinions

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  I G U R E 5 Standard cross-section T A B L E 1

  Table summarising the calibration variables and the variables of interest used in the approach

		Calibration variables		Variables of interest
	Failure mechanism	Number of variables	No. of variable	Number of variables	No. of variable
	Sliding	30	No.1-30	10	No. 31-40
	Internal erosion	-	-	10	No. 41-50
	Scouring	-	-	10	No. 51-60

  Results of the expert opinion calibration phase in comparison to the calibration variables

	T A B L E 2 Distribution of the true values of the calibration variables		Calibration variable (no. 1-30)
	(no. 1-30) in the interquantile intervals	Inter-quantile interval	[0%, 5%]	[5%, 50%]	[50%, 95%]	[95%, 100%]
	elicited by the experts and the median		Expert no. 1
	of the elicitations		(P f ) (no. 1-30)	n	9	1 1	7	3
					%	30%	37%	23%	10%
				Expert no. 2
				(P f ) (no. 1-30)	n	9	1 0	1 0	1
					%	30%	33%	33%	3%
				Expert no. 3
				(P f ) (no. 1-30)	n	10	5	7	8
					%	33%	17%	23%	27%
				Expert no. 4
				(P f ) (no. 1-30)	n	10	8	8	4
					%	33%	27%	27%	13%
				Expert no. 5
				(P f ) (no. 1-30)	n	0	4	15	11
					%	0%	13%	50%	37%
				Expert no. 6
				(P f ) (no. 1-30)	n	8	8	6	8
					%	27%	27%	20%	27%
				Median of elicitations
				(P f ) (no. 1-30)	n	7.67	7.67	8.83	5.83
					%	26%	26%	29%	19%
	T A B L E 3 No. expert	Calibration score Entropy score log 10 (P f ) Weight of individual calibration Weight of relative calibration C e K e w e w 0 e
	1	0.79	0.74	0.59			0.28
	2	0.91	0.91	0.83			0.40
	3	0.03	0.79	0.02			0.01
	4	0.46	1.04	0.48			0.23
	5	0.14	0.44	0.06			0.03
	6	0.19	0.52	0.10			0.05
				Total		1

Table 6 -

 6 Distribution of the true values of the calibration variables (no. 1-30) in the debiased interquantile intervals calibration and aggregation phase) to 10% (Table 6-debiasing phase), which is a considerable improvement, tending towards increased reliability of the structures. The evolutions in the interval [95%, 100%] are equally considerable, dropping from 10% (Table6calibration and aggregation phase) to 0% (Table6-debiasing phase), reflecting a substantial change in the treatment of the overconfidence bias during the expert opinion debiasing phase. These evolutions of overconfidence bias are positive since they tend towards greater reliability for the structure.

	Inter-quantile interval	[0%, 5%]	[5%, 50%]	[50%, 95%]	[95%, 100%]
	(P f ) (no. 1-30)	n	1	1 4	1 4	1
		%	3%	47%	47%	3%

T A B L E 5 F I G U R E 9 Evolution of the quality of information in the sequence of processing the Individual expert Elicitation, Calibration, Aggregation, and Debiasing (IeCAD) approach (

DATA AVAILABILITY STATEMENT Data available on request from the authors: The data that support the findings of this study are available from the corresponding author upon reasonable request.

the results on 30 variables of interest corresponding to 10 levee cross-sections studied for three failure mechanisms (sliding, internal erosion, and scouring).

For variables of interest no. 31-40 relating to the sliding mechanism, Table 6 shows the distribution of the true values in the interquantile intervals, calibrated and aggregated on the one hand, and aggregated and debiased on the other.

At the end of the debiasing phase, we observed that the distributions of the debiased expert panel opinions contained 90% of true values in the uncertainty interval [5%, 95%]. Thus, the expert panel opinion debiasing phase permitted reaching the ideal target interval.

In detail, the results obtained after the debiasing phase show a reduction of the true values in the underconfidence interval [0%; 5%], falling from 20% F I G U R E 1 0 Debiased variables of interest associated with their confidence intervals