Coralie Carrier

Azza Habibi
email: azza.habibi@irsn.fr

Laurent Ferreux

Luc Solier

Didier Hebert

Celine Augeray

Maxime Morin

Denis Maro

Lucilla Benedetti

Determination of environmental gaseous 129I trapped in charcoal cartridges by ICP-MS/MS

Keywords: Active charcoal, 129 I, ICP-MS/MS, radiochemical treatment

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Introduction

129 I is the radioisotope of iodine with the longest half-life (t 1/2 = 16.1 x 10 6 y [START_REF] García-Toraño | The half-life of 129 I[END_REF]). This isotope is naturally produced by 238 U spontaneous fission or cosmic-ray spallation of Xe isotopes. It is also a nuclear fission product of 235 U and 239 Pu. This radionuclide is therefore chronically released by the nuclear industries, mainly by fuel reprocessing plants (e.g., Orano La Hague, France), in the marine environment and in the atmosphere. Moreover, gaseous or particulate 129 I can be released from sea to land under many forms [START_REF] Carpenter | Marine iodine emissions in a changing world[END_REF][START_REF] Fuge | Iodine and human health, the role of environmental geochemistry and diet, a review[END_REF]. For environmental monitoring, organic and inorganic gaseous iodine are often sampled using charcoal cartridges [START_REF] Hou | A review on speciation of iodine-129 in the environmental and biological samples[END_REF][START_REF] Aneheim | Affinity of charcoals for different forms of radioactive organic iodine[END_REF], often containing 20 g of charcoal to sample iodine following a filtration of 1500 to 150 000 m 3 of air [START_REF]Bilan de l'état radiologique de l'environnement français de[END_REF]. Most of the trapped iodine in these charcoal samples is inorganic (e.g., I 2 , HOI, …). 129 I is often determined by γ-spectroscopy by direct measurement or after an adapted chemical extraction [START_REF] Osterc | 127 I and 129 I/ 127 I isotopic ratio in marine alga Fucus virsoides from the North Adriatic Sea[END_REF][START_REF] Koarashi | A practical method for monitoring 129 I concentration in airborne release[END_REF][START_REF] Jabbar | A review on 129 I analysis in air[END_REF][START_REF] Llopart-Babot | On the determination of 36 Cl and 129 I in solid materials from nuclear decommissioning activities[END_REF][START_REF] Frechou | Optimisation of the measurement protocols of 129 I and 129 I/ 127 I. Methodology establishment for the measurement in environmental matrices[END_REF]. This measurement technic is efficient and can have a short turnaround time but does not always allow reaching environmental activities of 129 I with a limit of detection (LOD) of 0.5 mBq g -1 [START_REF] Fréchou | 129 I in the environment of the La Hague nuclear fuel reprocessing plant-from sea to land[END_REF] or providing the isotopic ratio ( 129 I/ 127 I) determination. AMS (Accelerator Mass Spectrometry) is also used for 129 I measurement after a chemical treatment of charcoal [START_REF] Francisco | Comparison of two methods to determine 129 I in charcoal cartridge samples by AMS[END_REF]. In fact, this technic allows the quantification of 129 I and 129 I/ 127 I with very low limits of detection (10 -9 Bq and 10 -13 respectively) [START_REF] Jacobsen | AMS measurement of 129 I, 36 Cl and 14 C in underground waters from Mururoa and Fangataufa atolls[END_REF][START_REF] Hou | Analysis of 129 I and its application as environmental tracer[END_REF]. Nevertheless, even if AMS performances no longer have to be demonstrated (excellent sensibility and selectivity), this device is not easily accessible because of its price and its dimensions. ICP-MS (Inductively Coupled Plasma -Mass Spectrometry) is an efficient and affordable alternative for 129 I and 129 I/ 127 I determination [START_REF] Ohno | Determination of 129 I in Fukushima Soil Samples by ICP-MS with an Octopole Reaction System[END_REF][START_REF] Kim | Review of the development in determination of 129 I amount and the isotope ratio of 129I/127I using mass spectrometric measurements[END_REF][START_REF] Chang | Determination of 129 I activities and interference in low level radioactive waste by alkaline fusion coupled with ICP-MS[END_REF]. However, iodine extreme volatility generates memory effect [START_REF] Isnard | Accurate measurements of 129 I concentration by isotope dilution using MC-ICPMS for half-life determination[END_REF][START_REF] Reid | Determination of iodine and molybdenum in milk by quadrupole ICP-MS[END_REF] and spectral and non-spectral interferences disturb the measurement. To apply this method to gaseous iodine measurement, a chemical treatment is essential before the ICP-MS/MS measurement.

For solid samples, as charcoal, an extraction step of iodine is indeed necessary. Oxygen combustion [START_REF] Nottoli | Accurate determination of 129 I concentrations and 129 I/ 137 Cs ratios in spent nuclear resins by Accelerator Mass Spectrometry[END_REF] and microwave digestion [START_REF] Mesko | Iodine determination in food by inductively coupled plasma mass spectrometry after digestion by microwave-induced combustion[END_REF][START_REF] Vanhoe | Effect of solvent type on the determination of total iodine in milk powder and human serum by inductively coupled plasma mass spectrometry[END_REF][START_REF] Fernández-Sánchez | Determination of iodine in human milk and infant formulas[END_REF] are both very efficient, but do not enable the treatment of a large quantity of sample. Even if 40 g of sample can be processed with alkali fusion [START_REF] Maro | Dosage par spectrométrie gamma de l'iode 129 dans les échantillons biologiques marins et terrestres Application à des algues prélevées le long des côtes de la Manche : fucus serratus et laminaria digitata[END_REF][START_REF] Doshi | 129 I in soil and grass samples around a nuclear reprocessing plant[END_REF][START_REF] Fan | Progress on 129 I analysis and its application in environmental and geological researches[END_REF], this method could induce about 30 % iodine loss. Pyrolysis [START_REF] Hou | Determination of ultra-low level 129 I in vegetation using pyrolysis for iodine separation and accelerator mass spectrometry measurements[END_REF][START_REF] Hou | Determination of 129 I in environmental solid samples using pyrolysis separation and accelerator mass spectrometry measurement[END_REF][30] and acid digestion in closed system, to oxidize some forms of iodine [START_REF] Hou | Radiochemical analysis of 41 Ca, 90 Sr, 129 I and[END_REF], are often used, allowing treating 10 to 200 g of sample, and providing high-level extraction recoveries (about 90 to 100 %). Moreover, acid digestion is easily manufacturable in every laboratory.

Moreover, the purification step can ensure the reduction of the limit of detection by the potential concentration of the sample and the elimination of some compounds which could affect the measurement of iodine (e.g. salt or interferents) [START_REF] Yiou | Extraction and AMS measurement of carrier free 129I/127I from seawater[END_REF]. Liquid-liquid extraction is the most used technique but requires the handling of organic solvents such as CCL 4 and TMAH [START_REF] Jabbar | A review on 129 I analysis in air[END_REF][START_REF] Doshi | 129 I in soil and grass samples around a nuclear reprocessing plant[END_REF][START_REF] Jabbar | Investigation of the isotopic ratio 129 I/I in petrified wood[END_REF]. To avoid the use of such solvents, solid phase extraction [START_REF] Decamp | Utilization of a mixed-bed column for the removal of iodine from radioactive process waste solutions[END_REF] and ionic chromatography [START_REF] Zhang | A Novel Approach for the Simultaneous Determination of Iodide, Iodate and Organo-Iodide for 127 I and 129 I in Environmental Samples Using Gas Chromatography-Mass Spectrometry[END_REF][START_REF] Heumann | Accurate determination of element species by on-line coupling of chromatographic systems with ICP-MS using isotope dilution technique[END_REF][START_REF] Wang | Determination of Iodine and Bromine Compounds by Ion Chromatography/Dynamic Reaction Cell Inductively Coupled Plasma Mass Spectrometry[END_REF] are also used. However, the compatibility between the separation and the measurement media in ICP-MS must be previously studied.

The aim of this study is to propose a new method for the chemical treatment of charcoal in order the determine 129 I and 129 I/ 127 I by ICP-MS/MS. This method consists of an acid digestion followed by a solid phase extraction allowing to reach a consistent with 129 I activities observed around Orano La Hague. I 2 is of course formed during the extraction process, so it is important to use a closed system to avoid losses and trap iodine in an adapted solution. The extraction step was optimized by a design of experiments (DoE) approach to minimize the number of experiments while maximizing the number of studied factors. This extraction is followed by a purification using the CL resin (Triskem int.). This resin, usually used in LSC (Liquid Scintillation Counting), was adapted to ICP-MS/MS measurement. Behaviors of 129 I interferences on the resin were also studied.

Then, the method was applied to 129 I determination in charcoal samples sampled in the environment of Orano la Hague and allows to reach a limit of detection of 2 mBq sample -1 . For this study, 131 I standard, I131ELSB45 (636 Bq.kg -1 in 50 µg.g -1 of NaI and 50 µg.g -1 of Na 2 S 2 O 3 solution), was supplied by ORANO LEA (Tricastin, France) to evaluate the chemical yield.

Materials and methods

Reagents HNO

Ar and O 2 gas are from Air Products with a purity of 99.999 %.

Samples

For the method development, water solutions containing 0.4 Bq in 1 mL of 129 I were prepared.

To verify the accuracy of the method, active charcoal samples containing 129 I collected near Orano La Hague reprocessing plant, on the PTILH (49.694°N, -1.8733°W), was analyzed. Organic and inorganic 129 I was sampling thanks to a High Volum Sampler at 60 m 3 h -1 during approximately 15 days. In this study, two different charcoals were used. At the top of the column, extra pure charcoal provided by Merck was used, allowing to trap gaseous inorganic iodine. The TC-30 charcoal, impregnated with TEDA (triethylenediamine) provided by Hi-Q was then placed to trap gaseous organic iodine [START_REF] Aneheim | Affinity of charcoals for different forms of radioactive organic iodine[END_REF].

Instrument

Digestion was performed using a homemade modified BÜCHI speed digester K-425 and scrubber B-414 (BÜCHI Sarl), which is represented in fig. 1.

Fig. 1 speed digester K-425 and scrubber B-414 (BÜCHI Sarl) [38,39] 129 I and 127 I are measured using the Agilent 8900 ICP-QQQ in mass-shift mode (Agilent Technologies) and based on the experimental parameters proposed in a previous work for 129 I and 129 I/ 127 I measurement by ICP-MS/MS in water samples [START_REF] Carrier | Correction: Mass-shift mode to quantify low level 129 I in environmental samples by ICP-MS/MS[END_REF]. Parameters are summarized table 1:

Purification method

For this study, liquid chromatography, more specifically solid phase extraction was selected to purify the obtained solution after acid digestion and to preconcentrate it.

CL resin, commercialized by Triskem international, was used in this study. The resin consists of porous microspherical beads of 100-150 µm, composed of a phosphine sulfide derivative. The CL resin is based on a selective extraction system for platinum group metals, gold and silver in acidic medium. In this protocol, the resin is first loaded with Ag + ions to form R-Ag + . Then, the sample is loaded onto the resin and iodide is retained by forming the AgI compound [START_REF] Zulauf | Characterization of an extraction chromatographic resin for the separation and determination of 36 Cl and 129 I[END_REF].

After several rinses described by Zulauf et al. in their study [START_REF] Zulauf | Characterization of an extraction chromatographic resin for the separation and determination of 36 Cl and 129 I[END_REF], iodine is eluted from the resin with Na 2 S 0.35 M since the solubility product of Ag 2 S is lower than that of AgI. The iodide ions will therefore no longer be retained on the resin and are substituted by sulfide ions S 2-.

To improve this purification step, the end of fixation volume and the resin binding capacity will be studied to obtain an optimized chemical yield. Moreover, tests will be carried out to adapt media at the end of the previous extraction and the purification. Adjustments will be made to improve the purification yield.

Results and discussion

Extraction of iodine a) In-house adaptation of the extraction step

In order to obtain the best limit of detection, a large quantity of charcoal has to be analyzed.

Acid digestion is a very efficient technique to get a sample size > 10 g.

The BÜCHI® company has developed a fast mineralization system which also ensures the neutralization of acid vapors for many applications.

This system, the SpeedDigester K-425, allows a rapid transfer of heat from the infrared resistors to the samples (Fig. 1). The heating time is therefore reduced to less than one hour and the cooling is fast.

Several tubes can be inserted in this device; 6 tubes of 300 ml or 5 tubes of 500 ml, which allows to take a large quantity of sample. Indeed, 5 to 10 g of sample can be introduced in each tube, which means that up to 60 g could be introduced into the system.

To perform the iodine extraction step in this study, the SpeedDigester K-425 system was therefore adapted.

For this purpose, only the suction pump of the Scrubber B-414 and the ferrous sulfate solution container were used to avoid acid vapors induced by the use of nitric acid. The operating principle of the acid digestion is based on the form change of iodine in acidic medium, an oxidizing medium. Indeed, iodine will be oxidized to a gaseous form, I 2 , and will thus be extracted from the sample. To improve the efficiency of this step, the sample/acid mixture is heated with a controlled temperature. In the modified system, the gas will be directed toward Na 2 CO 3 with a pump, where it will be trapped according to the equation 1:

3 I 2 + 3 Na 2 CO 3 → 5 I -+ IO 3 -+ 6 Na + + 3 CO 2 (1)
Vapors containing gaseous iodine are successively directed to a first bubbler, then a second one in order to prevent a possible saturation of the first one and finally toward a ferrous sulfate solution to neutralize the potential nitrous vapors.

The modified system is illustrated in Fig. 2: In order to select the optimal parameters for the best extraction yield, a design of experiments was performed.

Experiments were performed with a synthetic solution containing 50 ml of acid and 400 mBq of 129 I.

Extraction yields were determined by ICP-MS/MS measurement of 129 I in mass-shift mode, with reference solutions prepared in the same medium as the trapping solution (Na 2 CO 3 ) in each experiment.

To set up and operate the experimental design, Minitab ® 20 software was used. A factorial design with 7 factors (presented in Table 2) was carried out with a resolution of IV. This factorial design of resolution IV will help to determine the optimal parameters to perform an acid digestion while limiting the number of experiments carried out. Moreover, in order to avoid any contamination in the digestion system, a blank will be performed before each experiment under the same conditions. The number of experiments will thus be doubled. Once the experiments were performed, Minitab ® 20 was used to analyze the obtained responses. The Pareto plot (Figure 3) was generated to graphically determine the importance of the effects by ranking them from the most to the least significant, following a normal distribution.

Fig. 3 -Pareto plot of standardized effects for iodine extraction yields by acid mineralization

The probability of exceeding a standardized effect equal to 2.042 (the red dotted line) is 97.5%. Beyond this value, the factor is considered as statistically significant.

In the case of the factorial design performed in this study, the factors A (pump flow) and F (digestion temperature) have a significant influence on the extraction yield. Given the chosen values for the other factors (table 2), the influence they have on the extraction yield is negligible.

A contour plot (figure 4) confirmed that flow rates and high temperatures provide a high extraction efficiency, without causing an excitement of the solution.

Fig. 4 -Contour plot of iodine extraction yields by acid digestion

The selected parameters are summarized in table 3: The retained acid volume is 100 mL to minimize waste as well as the risk of reaching the safety zone during boiling (fig. 2). The selected acid concentration is 8 M to maximize the extraction recovery. A short bubbling time (30 min) is enough to recover iodine. The selected Na 2 CO 3 volume was 50 mL to minimize the sample volume during the following purification.

However, the volume is higher than 50 mL after this extraction step (around 70 mL) and the pH of the solution is acid (~ 2). To avoid iodine losses, and to stabilize iodine under I -form, ascorbic acid is added to the solution to have a solution of 1 % (w/v) in the bubbler (100 mM Na 2 CO 3 ). Ascorbic acid is a reducing agent that will help maintaining iodine under the form of I -. Furthermore, pH of the bubbling solution is corrected to be > 7 using ammonia 20 %. Ascorbic acid and the pH adjustment prevent the loss of 50 % of iodine at the end of this step.

When analyzing environmental samples, 131 I is added as a tracer to verify the chemical yield. 131 I is used as a tracer for several reasons: it is indeed easily measured by gamma spectrometry due to its high emission intensity (81.2% at 384.5 KeV [START_REF]Application Nucléide -Lara -Laboratoire National Henri Becquerel[END_REF]), and this radionuclide is not present in environmental samples because of its short half-life of 8 days.

Purification by Solid Phase Extraction (SPE)

After the extraction step, iodine is under I -and IO 3 -forms in 50 mL of 6 mM Na 2 CO 3 solution (~ 0.6% w/v). This measurement medium is not compatible with ICP-MS/MS measurement due to the salt concentration. A purification step is necessary to adapt the medium. Moreover, the purification step could be useful for the elimination of ICP-MS interferents that have never been studied. In addition, this step will allow the preconcentration of the sample as the preconized elution volume is 5 mL.

In order to prevent the clogging of the introduction system and to eliminate interferents before the measurement by ICP-MS/MS, a purification step of the iodine fraction was performed. This purification step will transfer iodine to a suitable medium for ICP-MS/MS measurement and preconcentrate the sample.

The CL resin was used in this work.

The purification on this resin is assisted by a peristaltic pump in order to control the flow rate which was initially fixed at 1.35 mL min -1 (3 rpm).

a) Compatibility between elution and ICP-MS/MS measurement media

Iodine is eluted from the CL resin with Na 2 S. In order to study the compatibility of this medium with ICP-MS/MS measurement, solutions of 100 µg L -1 of stable iodine in different molarities of Na 2 S were measured with on-mass mode. The same solutions of Na 2 S containing iodine were also measured after the purification procedure. Purification yields were calculated to check the efficiency of this separation method and the compatibly of Na 2 S with ICP-MS measurement. The results of these experiments are presented in Fig. 5. According to the results shown in Figure 4, the more Na 2 S concentration increases, the more iodine sensitivity decreases. The purification efficiency starts to exceed 75% for concentrations from 0.06 M. The recommended concentration is 0.35 M [START_REF] Zulauf | Characterization of an extraction chromatographic resin for the separation and determination of 36 Cl and 129 I[END_REF], presenting an optimal purification yield. On the other hand, at this concentration, the sensitivity is very low in ICP-MS. At 0.06 M, the yield is slightly lower but the sensitivity is ~ 10 times higher. This concentration allows then having the best compromise between the yield and the sensitivity. Na 2 S solution at 0.06 M will be used to eluate iodine from the resin in order to measure 129 I by ICP-MS/MS. b) Improvement of the sensitivity In a previously published work, Tween ® 20 has been used to increase sensitivity and decrease memory effect when measuring iodine by ICP-MS [START_REF] Carrier | Correction: Mass-shift mode to quantify low level 129 I in environmental samples by ICP-MS/MS[END_REF].

Tests were carried out by adding this surfactant to the elution fraction. In order to study the response, several solutions of 100 µg L -1 of NaI were prepared in 0.06 M Na 2 S and then measured directly and after purification after addition of various quantities of Tween ® 20 in the elution fraction.

The obtained results are presented in Fig. 6.

Fig. 6 -Variation of sensitivity before and after resin passage when adding different quantities of Tween® 20. The error bars correspond to sensitivity RSD

These results confirm that the addition of Tween ® 20, a non-ionic surfactant, to Na 2 S solution prior to ICP-MS/MS measurement improves sensitivity. The highest sensitivity is obtained for a concentration of 4.7%. Therefore, in this study, this concentration will be selected for measurements after purification by SPE.

c) Optimization of purification time

In order to select the flow rates that provides the best balance between separation time and purification yield, 100 µg L -1 solutions were prepared and purification was performed by varying the flow rate, step by step, from 0.4 to 8.8 mL min -1 (1 rpm steps). For each experiment, the purification yields were calculated to determine the optimal flow rates. The flow rate was varied to select the optimal one to have the best purification yield. Selected flow rates are presented in Table 4. The conditioning and rinsing steps with water and sodium carbonate can be performed with a higher flow rate than the 1.35 mL min -1 used before optimization. In contrast, the ammonium thiocyanate rinse step cannot be performed with a high flow rate (more than 0.4 mL min -1 ), as this increases the risk of substituting AgI with AgSCN and thus eluting I -as NH 4 I. The flow rate for iodine elution from the resin with 0.06 mM Na 2 S is optimal from 0.4 to 1.8 mL min -1 . Above 1.8 mL min -1 , the time is too short to allow efficient ion exchange.

By optimizing the flow rates, the purification time was reduced from 1.5 hours to less than 40 minutes for a 10 mL percolated sample with a purification yield of 96 %.

The purification step optimized for ICP-MS/MS measurement is summarized in Fig. 7. The end of fixation volume for iodine on the CL resin is the sample volume at which iodine is no longer fixed for a resin volume of 2 mL (internal diameter of 7 mm and a length of 39 mm).

It may sometimes be necessary to purify a larger sample volume to fit the extraction volume (previous step) or to have a better limit of detection in some cases (seawater analysis for example).

Tests were performed and concluded that up to 1 L of sample could be introduced into the CL resin without impacting the purification yield but only the purification duration.

Percolation of 1 L and elution with only 5 mL imply a preconcentration by a factor of 200.

Then the limit of detection with this method could be 200 times better than that presented in the previous study [START_REF] Carrier | Correction: Mass-shift mode to quantify low level 129 I in environmental samples by ICP-MS/MS[END_REF], 11 mBq. The new limit of detection after a chemical treatment could then reach 2 mBq.

e) Resin binding capacity

Studies by Zulauf et al. showed that up to 20 mg of iodine could be separated with 2 mL of the CL resin and eluted with 5 mL of a 0.35 M Na 2 S solution [START_REF] Zulauf | Characterization of an extraction chromatographic resin for the separation and determination of 36 Cl and 129 I[END_REF]. The selected concentration, 0.06 M, and the modified flow rates could degrade the elution yield. To verify that the changes do not degrade the elution yield for high quantities of iodine, tests were performed by percolating 10 mL at different concentrations through CL resin. The obtained results are presented in Fig. 8. These results confirm that iodine is well fixed and eluted with 0.06 M Na 2 S, for concentrations up to 500 mg L -1 (50 mg inserted on 2 mL of resin). These results are consistent with those obtained by Zulauf et al. and confirm that the protocol modifications do not degrade performances obtained with the CL resin [START_REF] Zulauf | Characterization of an extraction chromatographic resin for the separation and determination of 36 Cl and 129 I[END_REF].

f) Fixation of iodate ions on the CL resin

The calculated chemical yield after the extraction and the purification steps was initially 10 ± 1%. According to Eq. ( 1), iodate and iodide ions are formed during the extraction step. In order to study the behavior of iodate ions during the following purification with CL resin, tests were performed with solutions of IO 3 -prepared in Na 2 CO 3 100 mM. The purification procedure was applied to each solution and all the fractions were collected so that iodine could be measured by ICP-MS/MS. The results are presented in Fig. 9. These results show that iodate ions are not retained on the CL resin. The solubility products of Ag 2 CO 3 is indeed lower than that of AgIO 3 (Eq. ( 2)), so when a sodium carbonate solution containing iodate ions is inserted into the column filled with CL resin, CO 3 -ions preferentially form AgCO 3 compound and iodate ions are not retained. (2)

Ks AgIO 3 = [Ag + ] x [IO 3 - ] = 10 -8
g) ICP-MS interferences behavior on the CL resin As the CL resin was initially designed for a liquid scintillation measurement, 129 I ICP-MS interferences behaviors during the separation step has never been studied. The polyatomic interferents of 129 I by ICP-MS are presented table 5.

Table 5 -129 I interferents in ICP-MS [START_REF] Llopart-Babot | On the determination of 36 Cl and 129 I in solid materials from nuclear decommissioning activities[END_REF], [START_REF] Jacobsen | AMS measurement of 129 I, 36 Cl and 14 C in underground waters from Mururoa and Fangataufa atolls[END_REF], [34, The distribution coefficients (Kd) have been calculated according to Eq. ( 3) for each interferent during iodine purification steps. This study was performed in batch where 200 mg of resin with Ag + were inserted into a 25 mL beaker. 100 µg L -1 of each interferent were added to 5 mL of Na 2 CO 3 , NH 4 SCN, NaOH and Na 2 S solutions ranging from 0.1 mM to 1 M and water. These solutions were stirred for 19 h. The solutions were then recovered for ICP-MS measurement by eliminating the resin using an empty column containing a filter. The sample, after the extraction step, is loaded on the resin in 6 mM Na 2 CO 3 . At this concentration, Y, Cd and In are not retained on the resin. Mo, on the other hand, is relatively well retained with a Kd > 30 mL g -1 .

Fig. 11-Distribution coefficients of iodine 129 interferents on CL resin in water

The interferents are all slightly retained on the resin in water. This step does not, or only slightly, remove the previously retained interferents, but it does rinse saline load excess of the resin. For this purpose, the treatment of activated charcoals in order to quantify 129 I at low level is an important issue. Tests were carried out by measuring 129 I trapped near Orano La Hague reprocessing plant (as described above) on cartridges containing 30 g of charcoal. Reference activities were determined by a validated and a well-established gamma spectrometry routine method. The newly developed method was applied to these samples. Bias and standardized deviations were calculated according to Eq. ( 4) and ( 5).

(4)

Results are presented in table 6. The obtained results presented in table 5 highlight the efficiency of the ICP-MS/MS method with results consistent with the 129 I measurements by gamma spectrometry. Indeed, the bias are all below 15% and the standardized deviations lower than 1.

Unlike gamma spectroscopy, the method newly implemented by ICP-MS/MS enhance the capability to reach 129 I activities in charcoal 5, 6 and 7 due to its lower limit of detection. The obtained limit of detection (calculated following the currie equation [START_REF] Rucker | Methodologies for the practical determination and use of method detection limits[END_REF]) after the extraction of iodine from 30g of charcoal and the purification step is 2 mBq sample -1 .

Conclusion

A new method for 129 I quantification in complex samples, e.g., charcoals, has been developed. This method relies on extraction, purification and measurement steps. The procedure was designed to ensure that iodine loss by volatilization is avoided. Compatibility of the media between the different analytical steps was indeed carefully studied and adapted.

Iodine was extracted from solid samples by acid digestion. A closed system was set up by modifying a commercial device designed to digest samples. Iodine is trapped by bubbling in a basic solution of Na 2 CO 3 . In order to obtain the best performances, a factorial design of experiments was carried out.

A solid phase extraction was then set up using CL resin. This resin was adapted to the measurement of iodine by ICP-MS/MS by optimizing the Na 2 S concentration to 0.06 M for the elution step.

Furthermore, the purification time has been decreased from 1h30 to 40 min for a 10 mL sample volume.

The behavior of 129 I interferents in ICP-MS/MS was studied on CL resin. This new method can reach limit of detection of 2 mBq sample -1 . This allows to reach lower iodine activities than before with radiometric technics with an optimized turnaround time (e.g., less than a day for the entire protocol).

Fig. 2 -

 2 Fig. 2 -Modified system for acid digestion Many parameters can influence the extraction yield obtained with this method. These parameters are: -acid concentration; -volume of acid; -concentration of Na 2 CO 3 ; -volume of Na 2 CO 3 ; -pump flow rate; -digestion temperature; -digestion time. b) Design of experiments

2

  double points and 3 central points are added to the 32 (2 k-2 , with k number of factors) experiments. The double points were randomly chosen by the software to study the stability of the responses. The use of central points is indeed a very robust way to check the linearity of the model.

Fig. 5 -

 5 Fig. 5 -Variation of the ICP-MS/MS sensitivity with different Na 2 S concentrations. Uncertainties are the RSD (standard deviation / average) for the sensitivity before and after purification. They were determined at k = 2 for the yield.

Fig. 7 -

 7 Fig. 7-Optimized purification step. The flow rates are indicated in blue (mL min -1 ) d) Study of the end of fixation volume

Fig. 8 -

 8 Fig. 8-Purification yield as a function of percolated iodine concentration. The error bars were calculated at k =2.

Fig. 9 -

 9 Fig. 9-Study of the behavior of IO 3 -ions on CL resin

Ks Ag 2

 2 CO 3 = 2[Ag + ] x [CO 3 -] = 10 -12

( 3 )

 3 CPS i and CPS f represent the CPS (Counts Per Second) obtained by ICP-MS before and after passing through the resin. V is the volume of solution (mL) and m is the mass of resin (g).The calculated Kd in Na 2 CO 3 , H 2 O, NH 4 SCN, NaOH and Na 2 S media are presented in the following figures:

Fig. 10 -

 10 Fig. 10-Distribution coefficients of iodine 129 interferents on CL resin as a function of Na 2 CO 3 concentration

Fig. 12 -

 12 Fig. 12 -Distribution coefficients of iodine 129 interferents on CL resin as a function of NH 4 SCN concentration NH 4 SCN concentration during sample purification is 100 mM. At this concentration, Mo and In are retained on the resin with Kd > 2000 mL g -1 .

Fig. 13 -

 13 Fig. 13-Distribution coefficients of iodine 129 interferents on CL resin as a function of NaOH concentrationThe concentration of NaOH during sample purification is 250 mM. At this concentration, Mo is not retained on the resin. Cd and In are retained on the resin, as well as Y, which is strongly retained.

Fig. 14 -

 14 Fig. 14-Distribution coefficients of iodine 129 interferents on CL resin as a function of Na 2 S concentrationAccording to the figure15, Cd and In are not retained in the resin when the concentration of Na 2 S is 10 mM. However, these two elements are retained very well when the concentration is 100 mM. In this study, the concentration of Na 2 S is 60 mM.

Fig. 15 -

 15 Fig. 15-Processing strategy for environmental samples Activated charcoals are often used to trap gaseous organic and inorganic iodine. It is indeed used in environmental studies, environmental monitoring or to reduce the emission of radioactive iodine from nuclear facilities.

  3 , H 2 SO 4 NaOH and NaI used in this study are analytical grade provided by VWR BDH Chemicals. Na 2 CO 3 , NH 4 SCN and AgNO 3 NORMAPUR™ quality are from VWR BDH Prolabo®. Analytical grade 60-64% Na 2 S, Tween® 20 and 60-64% extra pure ferrous sulphate are provided by Acros Organics™. All dilutions are done with ultrapure water (18.2 ΩM cm, Merck Millipore).

Table 1 -

 1 Experimental parameters of the ICP-MS/MS

	RF power	1520 W

Table 2 -

 2 Factors studied for the design of experiments factorial design is a type of experimental design to study effects that several factors may have on a response. When conducting an experiment, varying the levels (values) of all the factors simultaneously instead of one at a time allow studying interactions between the different factors.

	N°	Parameter	Range of values
	1	Pump flow rate	3, 9 and 16 L min -1
	2	Acid volume	100, 200 and 300 mL
	3	HNO 3 concentration	6, 7 and 8 M
	4	Na 2 CO 3 concentration	6, 53 and 100 mM
	5	Na 2 CO 3 volume	50, 70 and 100 mL

A

Table 3 -

 3 Selected parameters for iodine extraction by acid digestion

	Parameter	Value
	Pump flow rate	15 L min -1
	Acid volume	100 mL
	HNO 3 concentration	8 M
	Na 2 CO 3 concentration	6 mM
	Na 2 CO 3 volume	50 mL
	Temperature	410 °C
	Duration	30 min

Table 4 -

 4 Selected flow rates for sample purification with CL resin

	Step	Flow rate (mL min -1 )
	1 -Conditioning	3.5
	2 -Percolation	1.8
	3 -Rinse (H 2 O)	3.5
	4 -Rinse + Cl elution (NH 4 SCN)	0.4
	5 -Rinse (NaOH)	4.4
	6 -Elution I (Na 2 S)	1.8

  ]

	Element	Polyatomic interferent
	Y	89 Y 40 Ar +
	Mo	97 Mo 16 O 2 +
	Cd	113 Cd 16 O +
	In	115 In 14 N + , 113 InO +
	I	127 IH 2 + , 127 ID +

Table 6 -

 6 Activated charcoal after new method treatment

	Sample	Reference activity (mBq); k=2	Measured activity (mBq); k=2	Bias (%)	Standardized deviation
	Charcoal 1	131.00 ± 20.00	127.00 ± 4.92	3.06	0.51
	Charcoal 2	114.00 ± 19.00	101.43 ± 8.48	11.03	0.06
	Charcoal 3	91.00 ± 17.00	95.53 ± 4.66	4.98	0.03
	Charcoal 4	66.00 ± 14.00	75.47 ± 8.25	14.35	0.07
	Charcoal 5	6.00 ± 6.00	5.29 ± 1.98	11.9	0.11
	Charcoal 6	< 10.00	3.21 ± 0.08	/	/
	Charcoal 7	< 7.00	5.11 ± 0.47	/	/
	Charcoal 8	< 8.00	2.56 ± 0.18	/	/
	Charcoal 9	< 5.5	3.02 ± 0.55	/	/

The method was validated with cross analysis by gamma spectroscopy using charcoal cartridges containing gaseous iodine.

Conflicts of interest

Authors have no conflicts of interest to disclose.


Christophe Abraham

An Informative Prior distribution on Functions with Application to Functional Regression

Keywords: constrained Gaussian process, Bayesian regression, scalar-on-function regression, functional predictor, shape constraints, RKHS

We provide a prior distribution for a functional parameter so that its trajectories are smooth and vanish on a given subset. This distribution can be interpreted as the distribution of an initial Gaussian process conditioned to be zero on a given subset. Precisely, we show that the initial Gaussian process is the sum of the conditioned process and an independent process with probability one and that all the processes have the same almost sure regularity. This prior distribution is use to provide an interpretable estimate of the coefficient function in the linear scalar-on-function regression; by interpretable, we mean a smooth function that may possibly be zero on some intervals. We apply our model in a simulation and real case studies with two different priors for the null region of the coefficient function. In one case, the null region is known to be an unknown single interval. In the other case, it can be any unknown unions of intervals.

Introduction

We consider the regression model with a functional predictor x i ∈ L 1 (T ) and a scalar response Y i given by 1

Y i = µ + T x i (t)β(t)dt + ε i , i = 1, . . . , n, (1)
where µ ∈ R, β is a continuous function on T , ε 1 , . . . , ε n are independent random variables with a N (0, σ 2 ) distribution. For instance, the scalar response may be the amount of some agricultural production and the functional covariate may be the rainfall or the temperature history. An issue which arises naturally in many applied contexts is the detection of periods of time which influence the final outcome the most or, equivalently, the periods of time for which the covariate has almost no impact on the response. To this end, we provide in this paper an interpretable estimate of the functional coefficient β. By interpretable, we mean a smooth function that may possibly be equal to zero on some intervals. The value of the functional covariate on these intervals has no effet on the response; hence an interpretability of β.

Model (1), usually called the linear scalar-on-function regression, is very popular in the functional data analysis (fda) literature. Many ideas, techniques and applications of fda can be found in the monographs [START_REF] Ramsay | Functional Data Analysis[END_REF] and [START_REF] Ferraty | Nonparametric Functional Data Analysis. Theory and Practice[END_REF]; the latter being focused on nonparametric methods. A more recent overview of concepts of fda is given in [START_REF] Wang | Functional data analysis[END_REF]. In addition to the linear model (1), the literature on scalar-on-function regression also includes nonlinear and nonparametric models; for a comprehensive review, we refer the reader to [START_REF] Reiss | Methods for scalar-on-function regression[END_REF]. A common general approach to fitting model ( 1) is to expand the coefficient function β onto a basis of functions such as splines, wavelets or data-driven bases and to use a regularization technique (penalties, constraints, prior distributions) to prevent overfitting (e.g., [START_REF] Cardot | Functional linear model[END_REF][START_REF] Brown | Bayesian wavelet regression on curves with application to a spectroscopic calibration problem[END_REF][START_REF] Crainiceanu | Bayeisan functional data analysis using winbugs[END_REF][START_REF] Goldsmith | Functional regression via variational bayes[END_REF][START_REF] Zhao | Wavelet-based lasso in functional linear regression[END_REF].

Such standard methods provide estimates of β which are not exactly equal to zero on some intervals even if there is no relationship between the response and the covariate for large regions of t. Furthermore, the interpretation may be hard because of indesirable flutuations of the estimate. Some authors have proposed new approaches to overcome these problems. In a frequentist framework, [START_REF] James | Functional linear regression that's interpretable[END_REF] discretizes the coefficient function and obtains an interpretable estimate with sparse derivatives of different orders using an L 1 -penalty. With a similar discretization, [START_REF] Tibshirani | Sparsity and smoothness via the fused lasso[END_REF] obtains a sparse estimate with local constancy using the Fused lasso penalties. [START_REF] Zhou | Functional linear model with zerovalue coefficient function at sub-regions[END_REF] proposes a two-stage estimator to simultaneously identify the null region of β and an estimate of β on its support. At stage one, the null region is roughly identified by expanding β onto a B-spline basis and using the Dantzig selector. The second stage refines the estimation of the null region and achieves the estimation of β. [START_REF] Picheny | Interpretable sparse sir for functional data[END_REF] also focuses on the estimation of the support of β by using penalized versions of Sliced Inverse Regression. In a Bayesian framework, [START_REF] Grollemund | Bayesian functional linear regression with sparse step functions[END_REF] proposes estimates of both the support and the coefficient function by restraining β to be a step function equal to zero on some intervals.

In this paper, we address the problem by adopting a Bayesian approach: put a prior distribution on the unkown parameters according to a prior knowledge and compute the posterior. We assume that it is known that the function coefficient is smooth and equal to zero on some, possibly unknown, subset T 0 ⊂ T . Our main contribution will be to construct a prior for β according to this prior knowledge. To this end, Gaussian processes are natural candidates as the smoothness of the trajectories can be controlled by the covariance function [START_REF] Cramér | Stationary and Related Stochastic Processes: Sample Function Properties and Their Applications[END_REF][START_REF] Adler | An introduction to continuity, extrema, and related Topics for general Gaussian processes[END_REF]. Furthermore, the implementation is rather easy thanks to the conjugacy property. As the trajectories of a usual Gaussian process with specific smoothness properties do not vanish on some interval, it is natural to consider such a process conditioned to be zero on some intervals. The conditioned process will share the smoothness properties of the initial processes while being equal to zero on some intervals.

Gaussian process priors appear in the Bayesian literature for several decades, including density estimation [START_REF] Leonard | Density estimation,stochastic processes and prior information[END_REF], binary and normal regression [START_REF] Kimeldorf | A correspondence between bayesian estimation on stochastic processes and smoothing by splines[END_REF][START_REF] Wood | A bayesian approach to robust binary nonparametric regression[END_REF], computer experiments [START_REF] Morris | Exploratory designs for computational experiments[END_REF][START_REF] Gu | Parallel partial gaussian process emulation for computer models with massive output[END_REF], Bayesian numerical analysis [START_REF] Diaconis | Bayesian numerical analysis[END_REF][START_REF] O'hagan | Some bayesian numerical analysis. Dans Bayesian Statistics[END_REF][START_REF] Hennig | Probabilistic numerics and uncertainty in computations[END_REF] and functional analysis of variance [START_REF] Kaufman | Bayesian functional ANOVA modeling using Gaussian process prior distributions[END_REF]. For a recent account on priors on functions, we refer the reader to [START_REF] Ghosal | Fundamentals of Nonparametric Bayesian Inference[END_REF].

As we take a Gaussian process for the prior distribution for β, we will denote β t instead of β(t) in the sequel. With such a prior, it can be seen (Section 2) that ( 1) is a special case of the Gaussian process regression model and can be written as follows:

Y i = µ + F (x i ) + ε i , i = 1, . . . , n,
where the process F = (F (x), x ∈ L 1 (T )) is defined by

F (x) = T x(t)β t dt
(2) and β = (β t , t ∈ T ) is a centred, real-valued Gaussian process. Although Gaussian process regression has received considerable attention (see O 'Hagan, 1978;[START_REF] Rasmussen | Gaussian Processes for Machine Learning[END_REF] for scalar covariates and [START_REF] Shi | Gaussian Process Regression Analysis for Functional Data[END_REF][START_REF] Wang | Gaussian process regression with functional covariates and multivariate response[END_REF][START_REF] Konzen | Gaussian process for functional data analysis: The gpfda package for r[END_REF] for functional covariates and a functional response), to my knowledge, the above special case (2) has never been studied in the literature.

In Section 2, we study the process F and derive the posterior distribution of β. Section 3 contains the main contribution of the paper: given a centred Gaussian process β with specific almost sure regularity and a subset T 0 , we construct a Gaussian process β 0 which can be interpreted as the process β conditioned to be zero on a given subset T 0 . The trajectories of β 0 are equal to zero on a given set T 0 while sharing the same almost sure regularity as β. More precisely, it is shown that β is the sum of β 0 and an independent Gaussian processes β 1 with probability one. Results of Sections 2 and 3 are applied to a simulation study and a real case study in Section 4. In this section, T 0 is unknown and two different priors for T 0 , corresponding to different prior knowledges, are considered. For the former, it is known that T 0 is a single (unknown) interval. In the latter case, it is only known that 'simple' subsets T 0 are more likely than others; in particular, T 0 can be any union of unknown intervals. The proofs and some computational formulas are gathered in the Appendix.

The unconstrained model

In this section, we assumed that µ = 0 and that σ 2 is known; the general case with µ and σ 2 unknown will be studied in Section 4. Precisely, we focus on the following Bayesian model:

Y i |β ∼ N ( T x i (t)β t dt, σ 2 ) β ∼ GP (0, K), (3)
where the random variables Y i are independent given β. The last line of (3) means that β = (β t , t ∈ T ) is a real-valued centred Gaussian process with a continuous covariance fonction K. We say that model ( 3) is unconstrained as it is not assumed that the trajectories of β are zero on some intervals; this latter case will be studied in Section 3. Some technical assumptions on the process are required: we assume that T ⊂ R d is a product of compact intervals of R and that β is separable. Separability is a weak assumption in our context as it is always possible to replace the process by a separable modification [START_REF] Doob | Stochastic processes[END_REF]. As we are interested in random functions with smooth trajectories, we also assume the following entropy condition

∞ 0 log N ( )d( ) < ∞, (4)
where N ( ) is the smallest number of closed d-balls of radius that cover T and d(s, t) = E(β s -β t ) 2 is the canonical metric. Entropy condition (4) is sufficient for the almost sure continuity of centred Gaussian processes. Sufficient conditions for (4) to hold can be found in Adler (1990, pages 13-15 and 106).

We denote by H the Hilbert space generated by the random function β, that is, the closure in L 2 (Ω, A, Pr) of the linear span generated by the random variables β t defined on some probability space (Ω, A, Pr). Note that any random variable of H is centred Gaussian (Janson, 1997, Theorem 1.3). Since the trajectory of β is continuous with probability one, T f (t)β t dt can be defined for all f ∈ L 1 (T ) with probability one as well.

The following Lemma shows that expectation and integration with respect to t can be interchanged and is useful to determine the distribution of

T f (t)β t dt.
Lemma 1 The random function β is measurable and, for all f ∈ L 1 (T ),

T f (t)β t dt is the unique element of H such that E Y T f (t)β t dt = T f (t)E[Y β t ]dt, ( 5
)
for all Y ∈ L 2 (Ω, A, Pr).

Consider the random process F = (F (g), g ∈ L 1 (T )) with L 1 (T ) as index set where F (g) is defined by (2). Since F (g) ∈ H by Lemma 1, any linear combination of such variables for some g ∈ L 1 (T ) is in H and is therefore centred Gaussian. Thus, F is a centred Gaussian process whose the covariance function R is given by:

R(g, f ) = T T K(s, t) g(s) f (t)dsdt, (6)
the above equality being obtained by applying (5) twice in the calculation of E(F (g)F (f )). Therefore, model ( 3) is a special case of the regression Gaussian process model with functional covariates and can be rewritten as follows:

Y i |F ∼ N (F (x i ), σ 2 ) F ∼ GP (0, R). (7)
Although the posterior distribution of F for model ( 7) is a standard result, this is apparently not the case for the posterior distribution of β for model (3). This latter distribution is provided by Proposition 1 below. We denote by L the integral operator with kernel K, that is

Lf (t) = T K(t, s)f (s)ds.
Proposition 1 Under model (3), the posterior distribution of β is a Gaussian process on T , GP (m, K * ), where

m(t) = Lx(t) (Σ + σ 2 I n ) -1 Y, K * (s, t) = K(s, t) -Lx(s) (Σ + σ 2 I n ) -1 Lx(t), Lx(t) = (Lx 1 (t), . . . , Lx n (t)) is the transpose of the column vector Lx(t) with entry Lx i (t), Y is the column vector with entry Y i , Σ is the n × n-matrix with entry Σ ij = R(x i , x j ) and I n denotes the n × n identity matrix. Furthermore, the marginal distribution of Y is multivariate normal N n (0, Σ + σ 2 I n ).
3 The constrained Gaussian process β 0

In this section, we consider the prior information that the functional coefficient β is equal to zero on a given set T 0 ⊂ T . A random process β with such a property will be called a constrained random process. We shall construct a constrained Gaussian process β 0 from the unconstrained Gaussian process β of Section 2 in Theorem 1 and study its smoothness properties in Proposition 2. More precisely, we shall show that, with probability one, β can be decomposed into the sum of two independent Gaussian random functions β 0 and β 1 such that β 0 is zero on T 0 . The important point here is that β 0 and β 1 share the same smoothness properties with β.

Denote by H the reproducing kernel Hilbert space (RKHS) with kernel K. It is well-known that H can be identified with an Hilbert space of real-valued functions on T and that Θ :

H → H, defined by Θ(Z)(t) = E(Zβ t ) is a linear isometry. Note that Θ(β s ) = K s where K s is defined by K s (t) = K(s, t).
If (., .) H denotes the inner product on H, we have the reproducing property: (f, K s ) H = f (s) for all f ∈ H. Classical results on RKHS and Gaussian processes can be found in [START_REF] Neveu | Processus aléatoires gaussiens[END_REF]; [START_REF] Janson | Gaussian Hilbert Spaces[END_REF]; [START_REF] Berlinet | Reproducing Kernel Hilbert Spaces in Probability and Statistics[END_REF]; [START_REF] Van Der Vaart | Reproducing kernel hilbert spaces of gaussian priors[END_REF].

Let H 0 be the subspace of H made up of functions that vanish on T 0 . It is shown in the next theorem that H 0 is closed and we denote by P the orthogonal projection onto H 0 . As Θ is an isometry, we deduce that H 0 = Θ -1 (H 0 ) is a Hilbert subspace of H and that P = Θ -1 PΘ is the orthogonal projection onto H 0 . Finally, we set β 0 t = Pβ t for all t ∈ T . The construction of the random function β 0 = (β 0 t , t ∈ T ) is illustrated by the following commutative diagram.

H Θ / / P H P H 0 Θ / / H 0
Similarly, we set H 1 = (H 0 ) ⊥ for the orthogonal complement of H 0 in H and denote by Q the orthogonal projection onto H 1 . Then, we construct

β 1 t = Qβ t where Q = Θ -1 QΘ. We can deduce that H 1 = Θ -1 (H 1 ) is the orthogonal complement of H 0 in H and that Q is the orthogonal projection onto H 1 .
By construction, for every t ∈ T , we have β t = β 0 t + β 1 t with probability one. It is shown in the next theorem that the null set on which the equality fails is independent of t.

Theorem 1 a) The randon functions β 0 and β 1 are independent. Furthermore, with probability one, the random functions β, β 0 and β 1 are continuous and β t = β 0 t + β 1 t for all t ∈ T . b) β 0 is a centred Gaussian process with covariance function K 0 defined by K 0 (s, t) = PK s (t) and RKHS H 0 . With probability one, β 0 t = 0 for all t ∈ T 0 . c) Similarly, β 1 is a centred Gaussian process with covariance function K 1 defined by K 1 (s, t) = QK s (t) and RKHS H 1 .

An interesting consequence of Theorem 1 is that the distribution of β 0 can be viewed as the distribution of β conditioned by β τ = 0 for all τ ∈ T 0 (although a precise definition of a process conditioned by an event of probability zero is beyond the scope of the paper). Indeed, by noting that, for all f ∈ H and all t ∈ T , f (t) = 0 if and only if (f, K t ) H = 0, it can easily be proved that H 1 , the orthogonal complement of H 0 in H, is the closure in H of the subspace spanned by K τ for all τ ∈ T 0 . Then, given that β τ = 0 for all τ ∈ T 0 , we deduce that K τ (t) = E(β τ β t ) = 0 for all τ ∈ T 0 and t ∈ T and therefore H 1 = {0}. Finally, since K 1 ∈ H 1 , we have K 1 = 0, β 1 = 0 and β = β 0 with probability one.

As K 0 ∈ H, it can be expected that the trajectories of β 0 and β share the same smoothness properties. Proposition 2 shows that this is actually the case when the smoothness properties are expressed by means of a RKHS with some kernel K. Several examples of RKHS of smooth functions are given in Berlinet & Thomas-Agnan (2004, Chapter 7).

Proposition 2 Let K be a continuous positive definite kernel on T × T such that H ⊂ H where H denotes the RKHS with kernel K. If the trajectory of β belongs to H with propability one, then the trajectory of β 0 (and β 1 ) belongs to H with propability one as well.

An immediate consequence of Proposition 2 is that, if T ⊂ R and if the trajectory of β has m -1 absolutely continuous derivatives and a square integrable m-th derivative, then the trajectory of β 0 shares the same smoothness property (see [START_REF] Wahba | Spline Models for Observational Data[END_REF], Chapter 1, for a study of the corresponding space H in this case).

Applications

The general Bayesian model

In this section, we consider the following Bayesian model:

Y i |µ, σ 2 , β 0 ∼ N µ + T x i (t)β 0 t dt, σ 2 β 0 |σ 2 , T 0 ∼ GP (0, σ 2 K 0 ) p(µ, σ 2 , T 0 ) ∝ σ -2 p(T 0 ), ( 8
)
where the random variables Y i are independent given (µ, σ 2 , β 0 ) and p is a generic notation for a density distribution. As defined in Section 3, the kernel K 0 is the orthogonal projection on H 0 of a kernel K and, therefore, does depend on T 0 . Note that the last line means that (µ, σ 2 ) is independent of T 0 and has a non-informative prior distribution with density p(µ, σ 2 ) ∝ 1/σ 2 . Two particular prior distributions for T 0 are proposed in Section 4.2.

The posterior distribution of (T 0 , σ 2 , µ) By applying Proposition 1 with

Y i replaced by Y i -µ and K replaced by σ 2 K 0 , we deduce that the conditional distribution of Y given (µ, σ 2 , T 0 ) is N n (µ1 n , σ 2 M) where 1 n = (1, . . . , 1) , M = Σ 0 + I n and Σ 0 ij = T T K 0 (s, t) x i (s) x j (t)dsdt.
A method for calculating the orthogonal projection in a RKHS and a formula for the computation of Σ 0 using the rectangle method are given in the Appendix. Some classical calculations show that the conditional distribution of (µ, σ 2 ) given (Y, T 0 ) is N IG(S 1Y /S 11 , 1/S 11 , (n -1)/2, b) where

S 11 = 1 n M -1 1 n , S 1Y = 1 n M -1 Y, S Y Y = Y M -1 Y, and b = 0.5(S Y Y -S 2 1Y /S 11
). Simulations from the posterior distribution of (µ, σ 2 , T 0 ) can be obtained by a Metropolis-Hastings-Within-Gibbs algorithm.

An estimate of β 0 By applying Proposition 1 with Y i replaced by Y i -µ and K replaced by σ 2 K 0 , we deduce that the posterior distribution of

β 0 given (µ, σ 2 , T 0 ) is GP (m, K * ) with m(t) = L 0 x(t) M -1 (Y -µ1 n ) and K * (s, t) = σ 2 [K 0 (s, t) -L 0 x(s) M -1 L 0 x(t)] ,
where L 0 is the integral operator with kernel K 0 . We deduce from the above paragraph that E(β 0 t |Y, T 0 ) = L 0 x(t) M -1 (Y -S Y 1 /S 11 1 n ) and we estimate the function β 0 by β0 (t) = E(β 0 t |Y, T 0 ) when T 0 is fixed at its posterior mode T 0 . Note that, contrary to E(β 0 t |Y), β0 vanishes on T 0 and is therefore interpretable. A formula for the computation of β0 on a grid is given in the Appendix. A credible interval of β 0 t for each t ∈ T can be obtained by simulating trajectories from the posterior distribution of β 0 (by simulating first (T 0 , σ 2 , µ) from its posterior distribution).

Setting the hyperparameters Recall that K 0 is the orthogonal projection of K in the space H. Therefore, the choice of the covariance function K can have important consequences in the final estimation of β 0 . This choice is usually decomposed into two steps: first, a parametric family is chosen and then the parameters within the chosen family are set. We refer the reader to [START_REF] Rasmussen | Gaussian Processes for Machine Learning[END_REF] for a presentation of the different methodologies for setting these parameters. We arbitrarily take the popular squared-exponential covariance function

K(s, t) = σ 2 K exp - 1 2 s -t l 2
with length scale l and signal variance σ 2 K . Inferential difficulties may arise in estimating both the two parameters σ 2 K and l [START_REF] Zhang | Inconsistent estimation and asymptotically equal interpolations in model-based geostatistics[END_REF]. Therefore, as the value of σ 2 K have a very low impact on the posterior distribution, we set it to 1.

The choice of l is of particular importance as it controls the flexibility of the random function β 0 . We notice on simulations that, if l is too large, the lack of flexibility of β 0 entails the concentration of the posterior distribution of T 0 around the empty set. On the contrary, if l is small or tends to zero, the consequences on the posterior of T 0 are rather insignificant as, in this situation, the range of T 0 is mainly controlled by the data. We use this robustness property to set the value of l: we put a uniform prior distribution on a coarse grid and estimate l by l the infimum of the 90% highest posterior density (HPD) region. This estimate is lower that the classical estimate defined by the mode of the posterior distribution and, as explained above, prevents from a lack of flexibility of β 0 . The HPD region is obtained by simulating l from its posterior distribution. This can be acheived by including an independent proposal for l in the Metropolis-Hastings step above. Note that a full Bayesian analysis with the additionnal parameter l could also be performed, but this would give a very flat posterior distribution for T 0 for small sample sizes. For this reason, we prefer to use an empirical Bayes strategy using the fixed value l in the applications below.

A simulation study

Simulated data We arbitrarily set T = [-3, 3]. The functional coefficient β 0 and the functional covariates x i are discretized on a regular grid (t l ) with size g = 2 7 . The (true) functional coefficient β 0 true , plotted in Figure 1, is the orthogonal projection of the function t → sin(tπ/4) onto the subspace of functions that vanish on T 0 = [-1, 0.5] in the RKHS with kernel K(s, t) = exp (-2(t -s) 2 ). The functional covariates x i are randomly simulated from a centred Gaussian process with the covariance function K. The corresponding values of Y i are obtained from (1) where β is replaced by β 0 true , µ = 0 and σ 2 = 0.25; the integral being simply approximated by the rectangle method. We fix the sample size at n = 30. Figure 1: True functional coefficient β 0 true (dashed line), estimate β0 (plain line) and 95% credible intervals of β 0 (t) for all t; the corresponding true interval T 0 is indicated by two vertical short lines while the estimate T 0 is indicated by a wide band. Left: Prior 1 (T 0 is a single interval). Right: Prior 2 with α = 4 (T 0 is an union of intervals).

Prior 1 (T 0 is a single interval) From now on, we consider the discretized version of T , namely T = {t 1 , . . . , t g }. We assume that T 0 is a single interval and put a uniform prior on the intervals with at least two elements. As T is discrete, an interval is simply a sequence of consecutive elements of T . The proposal distribution for T 0 encreases or reduces the range of T 0 by 1. We first run the algorithm with a uniform prior for the additional parameter l on a regular 20-point grid from 0.1 to 2 as explained above. The posterior distribution of l is clearly unimodal with a 90% HPD region equal to {0.5, . . . , 1.4}. Then, we set l = 0.5, run the chain with 3 × 10 5 iterations and discard the first twenty thousand iterations. The parameters µ and σ 2 are estimated by their posterior expectation, E(µ|Y) = 0.16 and E(σ 2 |Y) = 0.27, while T 0 is estimated by its posterior mode T 0 = [-0.75, 0.70] whose the posterior probability is 0.22%. The chain visits 2345 different values of T 0 ; the posterior distribution of T 0 is rather flat in a neighborhood of T 0 . The estimate β0 and a credible interval are given in Figure 1.

Prior 2 (T 0 is an union of intervals) We propose a prior for T 0 so that every subset of T has a positive probability but with high probabilities for subsets T 0 associated with a small number of runs; by run, we mean a sequence of consecutive elements of T . Note that a run can be viewed as an interval of the discrete set T . The prior density p(T 0 ) is chosen to be proportional to e -α r(T 0 ) where r(T 0 ) denotes the total number of runs of T 0 and of T \ T 0 . The distribution of r(T 0 ) derived from p(T 0 ) is given in the Appendix. We set α = 4; with this value, the expected number of runs is 3.3 (with a standard deviation of 1.5) which corresponds to our prior belief. Note that, when the number of runs is 3, T 0 reduces to a single interval or a union of two intervals. The proposal distribution for T 0 first chooses a run of T at random (not necessarily in T 0 ) and then, adds or removes an element at the ends of the run with probability 0.5 or removes an element in the interior of the run with probability 0.5; in the latter case, the chosen run is splitted into 3 runs.

We proceed similarly as in the previous paragraph and obtain very similar results: l = 0.5, E(µ|Y) = 0.16, E(σ 2 |Y) = 0.27, T 0 = [-0.70, 0.84] (with a posterior probability of 0.23%) and an estimate β0 very close to that obtained with Prior 1 (Figure 1). The chain visits 14452 different values of T 0 , the posterior expectation of the number of runs is 3.24 (with a standard deviation of 0.45).

A real case study

The data set We apply our method to predict the production of black truffles of an orchard in the south of France given the rainfall curves. The dataset is described in detail in [START_REF] Grollemund | Bayesian functional linear regression with sparse step functions[END_REF] along with the biological questions of interest. For each year from 1986 to 1999, the scalar response is the production of black truffles of the current year and the functional predictor consists of the cumulative rainfalls measured every 10 days from the 1st of January of the previous year to the 31st of March of the current year. Hence, the functional predictor reduces to the vector of the cumulative rainfalls for 45 ten-days periods.

Bayesian inference We use model 8 with Prior 2 with α = 1.35; with this value, the expected number of runs is 10.26 (with a standard deviation of 2.7) which implies a set T 0 composed of 5 disjoint intervals and corresponds to our prior belief. We first adopt the strategy of Section 4.1 to fix the value of l. Contrary to the simulation study, the posterior distribution of l is not clearly unimodal but is almost uniform even for a very large grid range. As it seems impossible to fix l by an empirical Bayes approach we proceed as follows. We remark that the correlation between β s and β t reduces to K(s, t) and that K(s, t) = e -5 ≈ 0 when |s -t| = √ 10l. Therefore, by taking l = 3, β s and β t are independent as soon as |s -t| ≥ 3 √ 10 ≈ 10 which roughly corresponds to our prior belief on β.

We run a chain with 3×10 5 iterations and discard the first thousand iterations. The chain visits 99593 different values of T 0 . The posterior expectation of the number of runs is 11.02 (with a standard deviation of 2.95) which is slightly greater than the prior expectation. The posterior distribution of T 0 is rather flat in the neighborhood of the mode T 0 and several values of T 0 with almost the same posterior probability could have been chosen to construct different estimates β0 . For that reason, we consider another way to estimate (the complement of) T 0 . For each t ∈ {1, . . . , 45}, we compute the posterior probability that β 0 (t) = 0, i.e., that t / ∈ T 0 . The results given in Figure 2 (right panel) are roughly consistent with those of [START_REF] Grollemund | Bayesian functional linear regression with sparse step functions[END_REF] but some differences can be noted: in particular, the central peak obtained with our model is shifted to the right for l = 3. We also run our algorithm with different values of l and note that, for small values(l ∈ {1, 2}), a new peak appears at the beginning of winter of the previous year (t = 0). This peak, associated with a negative value of β0 (t), indicates that the rainfalls at the beginning of winter of the previous year may have a negative impact on the production of truffles. We refer the reader to [START_REF] Grollemund | Bayesian functional linear regression with sparse step functions[END_REF] for a review of the literature on black truflles production and life cycle.

Conclusion

In this paper, we provide a prior distribution of a functional parameter β 0 = (β 0 t , t ∈ T ) so that the trajectories of β 0 are smooth and vanish on a given subset T 0 . We show that this distribution can be interpreted as the conditional distribution of a Gaussian process β = (β t , t ∈ T ) given that β t = 0 for all t ∈ T 0 . Then, this prior is used to estimate an interpretable version of the functional parameter of the linear regression model with functional predictor and scalar response. In the applications, we show that this prior can easily be used with different prior distributions for T 0 : from an unknown single interval to a completly unknown subset of T .

Other shape constraints on functions can be taken into account by using the process β 0 of this paper. For instance, assuming that T = [0, 1] if we integrate β 0 once, twice or third on [0, t], we obtain a process whose the trajectories are constant, linear or quadratic on T 0 respectively with the same almost sure regularity properties as the initial process. In the same spirit, the construction of the conditional distribution of Section 3, namely by taking the orthogonal projection onto a subspace of the RKHS of a Gaussian process, can be generalized to any subspace H 0 to obtain processes with particular almost sure properties. We plan to pursue in such directions for future works.

Appendix A: Proof

Proof of Lemma 1 Since β is continuous with probability one, it is also measurable (that is measurable in the pair of variables (t, ω) ∈ T × Ω) by Doob (1953, Theorem 2.5). Take f ∈ L 1 (T ) and set ν(A) = A f (t)dt. Since ν is a bounded measure on T , we can apply Proposition 3.9 of [START_REF] Neveu | Processus aléatoires gaussiens[END_REF] to the Gaussian random function β; this concludes the proof by noting that

T ν(dt)β t = T f (t)β t dt.
Proof of Proposition 1 In this proof, we shall derive the posterior distribution of β from the posterior distribution of F . The posterior distribution of F in model ( 7) is a Gaussian process with mean function m * F and covariance function R * given by:

m * F (g) = R(g, x) (Σ + σ 2 I n ) -1 Y, R * (f, g) = R(f, g) -R(g, x) (Σ + σ 2 I n ) -1 R(f, x), (9)
with the notation R(g, x) = (R(g, x 1 ), . . . , R(g, x n )) for any g ∈ L 1 (T ). This latter result is standard in the literature on regression Gaussian process model. Given the distribution (9) for F , we aim at finding the distribution of β such that F (g) = T g(t)β t dt. Take m ∈ N and t 1 , . . . , t m in T and consider the random vector v = (v 1 , . . . , v m ) with entry v l = F (η t l ,ε ) where η t l ,ε is defined in Lemma 2. Then, v is multivariate Gaussian with

E(v l ) = m * F (η t l ,ε ) and cov(v l , v k ) = R * (η t l ,ε , η t k ,ε ). By noting that R(η t l ,ε , x i ) = T η t l ,ε (u)Lx i (u)du,
we deduce from Lemma 2 that R(η t l ,ε , x i ) → Lx i (t l ) as ε → 0; the continuity of Lx i being a consequence of the continuity of K. Similarly, we deduce from Lemma 2 that R(η t l ,ε , η t k ,ε ) → K(t l , t k ). Then, it is easily seen that m F (η t l ,ε ) → m(t l ) and R * (η t l ,ε , η t k ,ε ) → K * (t l , t k ). On the other hand, since β is continuous with probability one, we deduce from Lemma 2 that v → (β(t 1 ), . . . , β(t m )) as ε → 0 with probability one and the proof is complete.

The marginal distribution of Y for model ( 7) is a standard result; it can easily be obtained by noting that(F (x 1 ), . . . , F (x n )) ∼ N n (0, Σ).

Lemma 2 For all t ∈ T and ε > 0, let

η t,ε (u) = 1 λ(B(t, ε) ∩ T ) 1 B(t,ε) (u)
where λ denotes the Lebesgue measure on R d , B(t, ε) denotes the closed ball with center t and radius ε and 1 A denotes the indicator function of the set A. Then, for all (t, s) ∈ T 2 and all continuous functions f and g,

T η t,ε (u)f (u)du → f (t) and T η t,ε (u)η s,ε (v)g(u, v)dudv → g(t, s) as ε → 0.
Proof of Lemma 2 First note that, as T is a product of compact intervals, λ(B(t, ε) ∩ T ) > 0 for all t ∈ T and ε > 0. We have that

T η t,ε (u)f (u)du -f (t) = T η t,ε (u)(f (u) -f (t))du ≤ sup u∈B(t,ε) |f (u) -f (t)|,
and this last term tends to 0 by the continuity of f . We proceed similarly for the convergence of T η t,ε (u)η s,ε (v)g(u, v)dudv.

Proof of Theorem 1 Let us first show that H 0 = {f ∈ H, f (t) = 0 for all t ∈ T 0 } is a closed subspace of H. Clearly, it is a linear space. Take f n ∈ H 0 such that f n → f in H. Thanks to the reproducing property, we have

f (t) = (f, K t ) H = lim(f n , K t ) H = lim f n (t) = 0
; hence H 0 is closed and the orthogonal projection onto H 0 does exist. a) By construction, H 0 and H 1 are two orthogonal Gaussian Hilbert subspaces of H. Hence, the sub σ-algebras of A generated by H 0 and H 1 are independent (Neveu, 1968, Proposition 2.4). Since β 0 t ∈ H 0 and β 1 s ∈ H 1 , the sub σ-algebras generated by β 0 and β 1 are independent as well. Actually, it can be seen that H 0 (resp. H 1 ) is exactly the Hilbert space generated by the random function β 0 (resp. β 1 ) and, therefore, the sub σ-algebras generated by β 0 and by H 0 (resp. β 1 and by H 1 ) coincide (Neveu, 1968, Lemma 2.3).

Write d 0 (s, t) = E(β 0 s -β 0 t ) 2 for the canonical metric of β 0 and define d 1 , the canonical metric of β 1 , in a similar way. From the independence of β 0 and β 1 , we deduce that d 2 (s, t) = d 2 0 (s, t) + d 2 1 (s, t). Hence, d(s, t) ≥ d 0 (s, t) and any d-ball is included into a d 0 -ball with same center and same radius. If we denote by N 0 ( ) the smallest number of closed d 0 -balls of radius that cover T , we have that N 0 ( ) ≤ N ( ) and we deduce from (4) that ∞ 0 log N 0 ( )d( ) < ∞. We deduce the almost sure continuity of β 0 (and β) from Adler (1990, Corollary 4.15 page 106). We proceed similarly with β 1 .

By construction, for all t ∈ T , we have that β t = β 0 t + β 1 t with probability one. If we denote by T * a countable, dense subset of T , we have that, with probability one, β t = β 0 t + β 1 t for all t ∈ T * . We conclude that, with probability one, β t = β 0 t + β 1 t for all t ∈ T thanks to the continuity of the random functions. b) By [START_REF] Janson | Gaussian Hilbert Spaces[END_REF]

0 that K 0 (s, t) = E(β 0 s β 0 t ) = (PK s , PK t ) H = (PK s , K t ) H = PK s (t) as P = P 2 is self-adjoint; hence K 0 (s, t) = PK s (t)
. We deduce from this last equality and Berlinet & Thomas-Agnan (2004, Theorem 11) that H 0 is the RKHS with reproducing kernel K 0 .

Let us show that, with probability one, β 0 (t) = 0 for t ∈ T 0 . By the definition of P, PK t (s) = 0 for all s ∈ T 0 . Then, for all t ∈ T 0 , E(β 0 t ) 2 = K 0 (t, t) = PK t (t) = 0. Thus, with probability one, β t = 0 for t in a countable dense subset of T 0 and we conclude by the continuity of β 0 .

The same reasoning applies to prove c).

Proof of Proposition 2 Note that, since K is a positive definite, d 2 K (s, t) = K(s, s) -2 K(s, t) + K(t, t) defines a metric d K on T (Lukić & Beder, 2001, Lemma 4.2). Then, since T is compact and K continuous, H is a separable space of continuous functions [START_REF] Berlinet | Reproducing Kernel Hilbert Spaces in Probability and Statistics[END_REF], Corollary 5 page 36). Finally, as continuity (with respect to any usual norm of R d ) and d K -continuity are equivalent [START_REF] Adler | An introduction to continuity, extrema, and related Topics for general Gaussian processes[END_REF], page 3), the trajectories of β and β 0 are also d K -continuous. We are now in a position to apply Theorem 7.5 of [START_REF] Lukić | Stochastic processes with sample paths in reproducing kernel hilbert spaces[END_REF] to β and β 0 . First, since the trajectories of β belong to H with probability one, we conclude that there exists a (unique, positive, symmetric) nuclear linear operator L KK : H → H whose range is contained in H and such that (f, g) H = (L KK f, g) H for all f ∈ H and g ∈ H, where (f, g) H denotes the inner product in H. Now, set L KK 0 = PL KK where P is the orthogonal projection defined in Section 3. Clearly, L KK 0 is a linear operator whose range is contained in H 0 and such that, for all f ∈ H and g ∈ H 0 , we have

(f, g) H = (L KK f, g) H = (L KK f, Pg) H = (L KK 0 f, g) H (10)
as Pg = g and P is self-adjoint. Since H 0 is a closed subspace of H and K 0 (s, t) = PK s (t), the Hilbert subspace H 0 and the RKHS with kernel K 0 coincide [START_REF] Neveu | Processus aléatoires gaussiens[END_REF], Proposition 3.15 page 59). Thus we have (L KK 0 f, g) H = (L KK 0 f, g) H 0 and, from (10), we deduce that L KK 0 is the dominance operator of H over H 0 defined in Lukić & Beder (2001, Theorem 1.1). It is enough to show that L KK 0 is nuclear to conclude, by Lukić & Beder (2001, Theorem 7.5), that the trajectories of β 0 are in H with probability one. For all f ∈ H, since L KK 0 f ∈ H, note that

(L KK 0 f, f ) H = (L KK 0 f, L KK 0 f ) H = PL KK f H
and similarly that (L KK f, f ) H = L KK f H . Since P : H → H is an orthogonal projection onto H 0 , PL KK f H ≤ L KK f H and we have that (L KK 0 f, f ) H ≤ (L KK f, f ) H . Since L KK 0 and L KK are compact non-negative linear operators from H to H, we conclude by the min-max theorem (Gohberg et al., 2003, Theorem 9.1 page 186), that the eigenvalues of L KK 0 are bounded by the eigenvalues of L KK and, therefore, that L KK 0 is nuclear.

K τ τ . Let t = (t i ) be a regular grid of T . If f t and Pf t denote the column vectors with entry f (t i ) and Pf (t i ) respectively, we have Pf t ≈ f t -K tτ K -1 τ τ f τ . Similarly, we obtain the following approximation for the n×n-matrix K 0 tt with entry K 0 (t i , t j ): K 0 tt ≈ [K tt -K tτ K -1 τ τ K τ t ] where K tt and K τ t = K tτ are the matrices with entry K(t i , t j ) and K(τ i , t j ) respectively. In this paper, the discretization (τ j ) is a subset of the grid (t l ) but it is worth noticing that (τ j ) can be chosen independently of (t l ).

Computation of Σ 0 and β0 We denote by L 0 x t the n × g-matrix with entry L 0 x i (t l ) and by β0

t the column vector with entry β0 (t l ). Using the rectangle method, we have L 0 x t ≈ δK 0 tt X and Σ 0 ≈ δ 2 XK 0 tt X where X is the n × g-matrix with entry X ij = x i (t j ) and δ = t i+1 -t i . If we denote by β 0 t the random vector with entry β 0 t l , the posterior distribution of β 0 t given (µ, σ 2 , T 0 ) is N g (m t , K * tt ) with m t = L 0 x t M -1 (Y -µ1 n ) and K * tt = σ 2 [K 0 tt -L 0 x t M -1 L 0 x t ]. Then, since E(µ|Y, T 0 ) = S 1Y /S 11 , we deduce that E(β t |Y, T 0 ) = L 0 x t M -1 (Y-S 1Y /S 11 1 n ) and β0

t is equal to this last expression when T 0 is fixed at its posterior mode.

The prior distribution of T 0 (Prior 2 ) Remember that T = {t 1 , . . . , t g }, p(T 0 ) ∝ e -α r(T 0 ) where r(T 0 ) is the total number of runs of T 0 and of T \ T 0 . Let us derive the distribution of r(T 0 ) from p(T 0 ). It is convenient to represent T 0 by a vector of {0, 1} g whose l-th coordinate is 1 if t l ∈ T 0 and 0 otherwise. Thus, T 0 is associated with successive sequences of 0 and sequences of 1 (runs). We set S k = {T 0 ⊂ T, r(T 0 ) = k} and we denote by c k the cardinal of S k . Then, we have Pr(r(T 0 ) = k) = T 0 ∈S k p(T 0 ) = T 0 ∈S k e -α r(T 0 ) T 0 ⊂T e -α r(T 0 ) = c k e -αk g k=1 c k e -αk .

We remark that T 0 is determined by the value (0 or 1) of the first run and the locations of the beginnings of the other runs. Since there are 2 possible values for the first run and g-1 k-1 ways of choosing the beginnings of the other runs, we have c k = 2 g-1 k-1 .

Figure 2 :

 2 Figure 2: Left: Estimate β0 and 95% credible intervals of β 0 (t) for all t ∈ {1, . . . , 45}. Right: Posterior probability of {t / ∈ T 0 } for all t ∈ {1, . . . , 45}.

  [Theorem 1.3], every element of H is a centred Gaussian random variable. By construction, β 0 t ∈ H 0 ⊂ H is therefore a centred Gaussian random variable. By noting that Θβ 0 s = PK s , we deduce from the definition of K

Appendix B: Computational issues

Computing an orthogonal projection in a RKHS Since we noticed in Section 3 that H 1 is the closure in H of the subspace spanned by K(τ, .) with τ ∈ T 0 , it is thus natural to approximate any g ∈ H 1 by a finite sum of the form m j=1 a j K τ j for some fixed τ j ∈ T 0 ; the best approximation being obtained by the orthogonal projection onto the subspace spanned by {K τ j , j = 1, . . . , m}. Thus, for any f ∈ H = H 0 ⊕ H 1 , we have Pf ≈ f -m j=1 a j K τ j . Since Pf (τ i ) = 0, we deduce that a = (a 1 , . . . , a m ) is a solution of the linear system f τ = K τ τ a where f τ = (f (τ 1 ), . . . , f (τ m )) and K τ τ is the m×m-matrix with entry K(τ i , τ j ). Note that such a solution exists and is unique when K is positive definite; furthermore, it can be efficiently computed by classical algorithms based on the Choleski decomposition of

List of Figures

Figure 1 True functional coefficient β 0 true (dashed line), estimate β0 (plain line) and 95% credible intervals of β 0 (t) for all t; the corresponding true interval T 0 is indicated by two vertical short lines while the estimate T 0 is indicated by a wide band. Left: Prior 1 (T 0 is a single interval). Right: Prior 2 with α = 4 (T 0 is an union of intervals).

Figure 2 Left: Estimate β0 and 95% credible intervals of β 0 (t) for all t ∈ {1, . . . , 45}. Right: Posterior probability of {t / ∈ T 0 } for all t ∈ {1, . . . , 45}.


Coralie Carrier
email: coralie.carrier@irsn.fr

Azza Habibi

Celine Augeray

Didier Hebert

Denis Maro

Lucilla Benedetti

D Hébert

Development of new low-level iodine 129 analysis method applied to the understanding of iodine's transfer mechanisms in the environment

Keywords: Environment, transfer, iodine, mass spectroscopy

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

129 I is chronically and regulatory released by fuel reprocessing plants e.g. Sellafield (UK) and ORANO La Hague (France) and also released in the atmosphere by nuclear power plants during normal operation process or in case of an accident. Iodine 129 is released by reprocessing plants not only in the atmosphere but also into the marine environment. The objective of this study is to develop a new 129 I quantification method in order to evaluate for the first time the re-emission of iodine from the marine environment to the coast by gas or aerosol emission against direct atmospheric releases. To achieve this objective, two High Volume Sampler (HVS) are installed 2 km north of ORANO La Hague and are surrounded by the ocean except the east side. These HVS are connected to a proportional counter (LB134, Berthold). When 85 Kr is detected, the first HVS is automatically started and the collected iodine is attributed to the reprocessing plant. When 85 Kr is not detected anymore, the other HVS is started and the collected iodine is mainly attributed to reemission from marine environment. HVS contains filters to trap aerosols and coal cartridges to trap gaseous iodine. Filters and coal were selected after a study to be compatible with the analytical procedure. In fact, a new analytical method has been developed to analyse 129 I and 129 I/ 127 I in these complex samples. ICP-MS measurement following a chemical treatment is an excellent alternative to AMS for the determination of 129 I and 129 I/ 127 I isotopic ratio. ICP-MS technique allows reaching LOD of ~ 0,3 mBq/L for 129 I and 10 -8 for 129 I/ 127 I (Honda et al., 2018). Difficulties during purification and measurement are related to the extreme volatility of iodine and to its multiple oxidation degrees. Spectral interferences are mainly due to the isobaric 129 Xe + , present as impurity in the Argon gas and to polyatomic interferences generated by 127 et al., 2014). Non-spectral interferences are related to the matrix and memory effects. In the presented work, the ICP-MS measurements were performed using an ICP-MS/MS (8900 Agilent ® ). The medium was meticulously studied to minimize spectral and non-spectral interferences. The octopole collision/reaction cell and the two quadrupole mass filters allowed minimizing polyatomic interferences. The isobaric interference due to 129 Xe was eliminated after studying different reaction gas (O 2 , N 2 O and CO 2 ) with on-mass and mass-shift detection modes.

Solid phase extraction was developed and adapted to ICP-MS measurement medium. The new method, therefore, allows the measurement of iodine 129 either directly or after chemical treatment. The chemical treatment time has been reduced to 40 min. This step increased the test sample up to 1 L. A gain in sensitivity up to a factor 200 was observed and allowed reducing the detection limit up to 0,06 mBq/L after chemical treatment, which is 100 times lower than some current methods.

Figure 1 :

 1 Figure 1: Iodine's transfer near to Orano La Hague Ežerinskis, Ž., Spolaor, A., Kirchgeorg, T., Cozzi, G., Vallelonga, P., A. Kjaer, H., Šapolaitė, J., Barbante, C., Druteikienė, R., 2014. Determination of 129 I in Arctic snow by a novel analytical approach using IC-ICP-SFMS. J. Anal. At. Spectrom. 29, 1827-1834. Honda, M., Takaku, Y., Shikamori, Y., Matsuzaki, H., Sueki, K., 2018. Preparation of Iodine-129 Standard Solutions for Triple Quadrupole ICP-MS. J. Nucl. Radiochem. Sci. 18, 1-4.

I transfered to coastline 129 I transfered to marine compartment