Journal of Environmental Radioactivity 139 (2015) 91e102

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Journal of Environmental Radioactivity journal homepage: www.elsevier.com/locate/jenvrad

Evaluating variability and uncertainty in radiological impact assessment using SYMBIOSE M. Simon-Cornu a, *, K. Beaugelin-Seiller a, P. Boyer a, P. Calmon b, L. Garcia-Sanchez c, C. Mourlon a, V. Nicoulaud a, M. Sy a, M.A. Gonze a Institut de Radioprotection et de Sûret e Nucl eaire (IRSN), PRP-ENV, SERIS, LM2E, Cadarache, France Institut de Radioprotection et de Sûret e Nucl eaire (IRSN), PRP-ENV, SESURE, LERCM, Cadarache, France c Institut de Radioprotection et de Sûret e Nucl eaire (IRSN), PRP-ENV, SERIS, L2BT, Cadarache, France a

b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 13 February 2014 Received in revised form 9 September 2014 Accepted 22 September 2014 Available online

SYMBIOSE is a modelling platform that accounts for variability and uncertainty in radiological impact assessments, when simulating the environmental fate of radionuclides and assessing doses to human populations. The default database of SYMBIOSE is partly based on parameter values that are summarized within International Atomic Energy Agency (IAEA) documents. To characterize uncertainty on the transfer parameters, 331 Probability Distribution Functions (PDFs) were defined from the summary statistics provided within the IAEA documents (i.e. sample size, minimal and maximum values, arithmetic and geometric means, standard and geometric standard deviations) and are made available as spreadsheet files. The methods used to derive the PDFs without complete data sets, but merely the summary statistics, are presented. Then, a simple case-study illustrates the use of the database in a second-order Monte Carlo calculation, separating parametric uncertainty and inter-individual variability. © 2014 Elsevier Ltd. All rights reserved.

Keywords: Multi-media modelling Probabilistic exposure assessment Transfer factors EMRAS TRS 472 Dosimetric impact

1. Introduction The importance of adequately characterizing variability and uncertainty in exposure assessments for human health risk assessments has been emphasized by several national and international organizations (EPA, 1997; FAO/WHO, 2006; InVS and AFSSET, 2007). Briefly, uncertainty is defined as the assessor's lack of knowledge, whereas variability is defined as the true heterogeneity, not reducible through either study or further measurement (Vose, 2000). The International Atomic Energy Agency (IAEA) has continuously acknowledged the importance to study uncertainty and variability. The IAEA Safety Series 100 “Evaluating the reliability of predictions made using environmental transfers” (IAEA, 1989) was a key document derived from early IAEA modelling programmes (e.g. VAMP and BIOMOVS). Subsequently, many IAEA working groups have dealt with uncertainties and variabilities in the “Environmental Modelling for Radiation Safety” programmes (EMRAS I, 2003e2007 and EMRAS II, 2009e2011), and more

* Corresponding author. E-mail address: [email protected] (M. Simon-Cornu). http://dx.doi.org/10.1016/j.jenvrad.2014.09.014 0265-931X/© 2014 Elsevier Ltd. All rights reserved.

recently the “Modelling and Data for Radiological Impact Assessments” programme (MODARIA, 2012e2015). Additionally, uncertainty and variability are also keywords in the 3 challenges of the Strategic Research Agenda (SRA) recently proposed for the European Radioecology Alliance (Hinton et al., 2013). SYMBIOSE is a simulation platform dedicated to model the fate and transport of radioactive substances in environmental systems, and to assess their risks to humans, accounting for uncertainty and variability (Gonze et al., 2011). This platform can be used in a wide range of situations for assessing risks induced by radioactive releases from nuclear facilities under normal operation, accidental, or decommissioning conditions. Environmental models in SYMBIOSE address atmospheric, terrestrial, freshwater and marine systems, as well as the major transfer processes at their interfaces. As a modular platform, SYMBIOSE hosts many inter-connected submodels, most of which have been described elsewhere. For example, agricultural transfer models are based on those of the former ASTRAL code (Maubert et al., 1997) for most radionuclides, whereas specific modelling approaches are dedicated to tritium (Le s et al., 2013), carbon-14 (Le Dize s et al., 2012), and chlorine-36 Dize (Tamponnet and Gonze, 2011). Freshwater models include those of CASTEAUR (Duchesne et al., 2003), among other approaches. Other

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models deal with atmospheric transport and deposition, marine transfers, or dosimetric calculations. The modelled exposure pathways are external radiation (in the plume, and outside of the plume), and internal contamination (inhalation, percutaneous transfer for tritium, accidental ingestion of sea sand and sea water, ingestion of foodstuffs, including e.g. drinking water, leafy vegetables, potatoes, cow milk, beef, hen eggs, river fish, sea fish, etc.). Probability density functions (PDFs) were defined in the default probabilistic database of SYMBIOSE, on the basis of the Technical Report Series 472 (IAEA, 2009), to describe parametric uncertainty about some of the key element-dependent transfer parameters: partition coefficient (Kd) in sediments and in freshwater, freshwater-fish concentration ratio (CR), soileplant transfer factor (TF), and transfer factor (TF) from feed to animal products. This paper briefly presents the SYMBIOSE database (Section 2); introduces the concept of uncertainty and variability separation, and their analyses via second-order Monte Carlo analyses (Section 3); and then presents a theoretical and simple case-study that illustrates how different sources of variability and uncertainty can be assessed in a second-order Monte Carlo impact assessment, using SYMBIOSE and its database (Section 4). 2. Building the probabilistic database of SYMBIOSE This section describes how default parametric PDFs were defined for the database of SYMBIOSE for a set of elementdependent key parameters: partition coefficients (Kd), concentration ratios (CR) and transfer factors (TF). The data from which the PDFs were derived are presented in the TRS 472 (IAEA, 2009), and more detailed in TECDOC 1616 (IAEA, 2010). The building of PDFs excluded foliar uptake because SYMBIOSE parameterisation was different than that of the TECDOC 1616, and due to the lack of data. 2.1. Objective and scope A SYMBIOSE calculation uses thousands of inputs. These inputs cover both the scenario data (i.e. the information characterizing the source term, the weather, the landscape, and the populations) and the parameters (i.e. the information characterizing the physical, biological, and radioecological mechanisms involved in the calculations). For the parameters, the SYMBIOSE database proposes default values (freely modifiable by the user) whereas the scenario data have to be defined by the user. This database covers 16 types of animals (4 livestock for meat: cattle, chicken, pork, sheep, 3 livestock for milk: dairy cow, ewe, goat; 1 livestock for eggs: hens; 4 river species: macrobenthos, omnivorous fish, planktonivorous fish, zooplankton; 4 marine species: crustacean, mollusc, fish, weed); 14 types of crops (leafy vegetables, root vegetables, fruit vegetables, grass, and 10 annual crops); 25 food products; and last but not least 277 isotopes of 64 elements. This article applies to all isotopes except chlorine-36, tritium and carbon-14 for which a s et al., 2012, 2013; Tamponnet specific approach is used (Le Dize and Gonze, 2011). In this database, each value (either proposed as default or filled by the user) can be replaced by a PDF. Monte Carlo simulations are then based on random sampling from the PDFs. The current version of SYMBIOSE allows the user to choose from six types of continuous parametric PDFs: triangular, uniform, truncated normal, logtriangular, log-uniform, or truncated log-normal. 2.2. Theoretical considerations: how to fit a data set without the data set? Fitting a univariate parametric distribution to a data set is a very common task in statistics. It requires judgement and

expertise and generally needs an iterative three-step process of (i) selecting a parametric probability distribution model (e.g. Normal, Log normal, Uniform, Beta, Exponential, Poisson, Gamma); (ii) estimating the parameter(s) (e.g. the mean), i.e. actually fitting the data set with the chosen distribution; and (iii) assessing the quality of evaluation (goodness-of-fit). These steps are usually based on the full data set, with an appropriate tool, such as proposed in many statistical software, e.g. the R packages MASS or fitdistrplus (Pouillot and Delignette-Muller, 2010). An issue arose in building the SYMBIOSE PDFs because the information used to construct them was taken from TRS 472 (IAEA, 2009), and TECDOC 1616 (IAEA, 2010), where only summary statistics are presented. The IAEA documents only tabulate statistics such as: n (size of the data set), min (minimal value), max (maximal value), AM (arithmetic mean) if n ¼ 2 or GM (geometric mean) if n > 2, SD (standard deviation) if n ¼ 2 or GSD (geometric standard deviation) if n > 2. Nevertheless, as demonstrated below, the three steps presented above for deriving PDFs could still be applied, even using only the summary statistics available in the IAEA documents. The first step, choosing a distribution, is usually based on inspection of data, e.g. on two characteristics of the data set: the skewness (asymmetry coefficient) and the kurtosis (peakedness coefficient) (Cullen and Frey, 1999). This was not feasible in the present situation, as the data sets were not available. An exception is the specific case of freshwater Kd for which goodness-of fit to log-normal distributions could be assessed (Ciffroy et al., 2009). Regardless, it is actually recommended to support the choice of the distribution by understanding the underlying processes that generated the data (Cullen and Frey, 1999). By applying the central limit theorem, it can be hypothesized that the result of the multiplication of positive independent variables tends to a lognormal distribution (Murphy, 1998). As for the ERICA-tool (ERICA, 2007), it was considered that a TF, a CR or a Kd is the result of multiplication of a large number of unknown positive parameters, e.g. each of them defining a mechanism at the cellular or molecular level, and that all of the individual mechanisms are successively involved in the general mechanism of transfer. It was then assumed that the underlying distributions behind all of these PDFs are log normal. For the second step, there are various estimation methods; the one most-widely used is the maximum-likelihood estimation (MLE). Other methods include matching moment estimation, matching quantile estimation, or maximizing the goodness-of-fit estimation (i.e. minimizing distance estimation). In the case of the normal distribution, the maximum-likelihood estimation leads to the same fitted distribution as matching moments. Thus, the mean and variance estimated by maximizing the loglikelihood are the ones that can be directly calculated from the data set. On the basis of the information tabulated in the IAEA documents, the format of the data enabled us only to use the “matching moment” method. Indeed, we matched the geometrical moments, i.e. our “fitted” log-normal distribution has the same geometric mean and geometric standard deviation as the data. This leads to equivalent results, as if obtained by maximizing the log-likelihood to fit log-transformed data with a normal (Gaussian) distribution. Be cautious that it is common in data analysis to use an “unbiased” estimator of the variance, with only (n  1) degrees of freedom (where n is the size of the data set), whereas the variance estimated by maximizing the log-likelihood is the “true” variance, i.e. the “biased” estimator of the variance, with n degrees of freedom. It was unclear whether the so-called “geometric standard deviation” given in the tables of TECDOC 1616 or TRS 472

M. Simon-Cornu et al. / Journal of Environmental Radioactivity 139 (2015) 91e102

was the “true” one (biased estimate), or the unbiased estimate. Regardless, given that the ratio between both variances is n/n  1 (where n is the size of the data set) and given that we excluded low n-values (see below), we assumed that this bias was negligible. The third step is usually based on goodness-of-fit tests, such as shown in TECDOC 1616 and by Ciffroy et al. (2009) for the specific case of freshwater Kds. Visual diagnosis is also recommended. Quantileequantile plots (QQ-plots) can thus be used to visually appreciate the quality of the fit, by plotting the quantiles of the empirical distribution against the estimates of the quantiles for the fitted distribution. From each data set, 2 values were actually available from the IAEA tables: the minimal value, i.e. the (0.5/n)th empirical percentile of the data set, and the maximal value, i.e. the (1  0.5/n)th empirical percentile. Then a QQ-plot with only 2 values could have been obtained for each data set. For one type of input (e.g. the root transfer factor for all elements and all plants), all of these QQ-Plots were superimposed. Adequacy of log-normal fitting was checked through QQeplot-like figures. Fig. 1 provides the results for root uptake transfer factors, the log-normal hypothesis was visually considered appropriate. 2.3. Additional practical considerations: how to define PDF in SYMBIOSE? To avoid extreme values, which might be inappropriate in calculations and unrealistic, all uncertainty distributions derived from empirical values and proposed as default for SYMBIOSE are truncated. When the empirical information is considered sufficient, i.e. more than ten values (Ciffroy et al., 2009), a truncated log normal distribution is used, truncated at the 2.5th percentile and the 97.5th percentile, i.e. with p ¼ 0.95, where p is the truncature

93

parameter, the proportion of log-normal distribution between min and max. When there was not enough information, in practice 2e9 published values, it was assumed that the information was too poor to derive a log normal distribution. However, to keep this information, even insufficient, it was arbitrarily chosen to represent the uncertainty by a log-uniform distribution characterized by its extreme values (min, max). Finally, if the available information was judged extremely poor, in practice when there was 0 or 1 published value in the table, or when min ¼ max, it was assumed that the parameter is so “uncertain” that there is no information about the distribution of uncertainty. If acquiring further information is impossible, the modeller usually has the choice between two easy-to-apply attitudes (Oughton et al., 2008): either to define a very broad distribution of uncertainty, assumed to cover at least the range of possibilities; or to impose a fixed value of this parameter, that is to say, to quit the probabilistic calculation and to accept a deterministic calculation. The choice between these two attitudes is closely linked to the culture of the discipline. In radioecology, it was considered that the second one would be more accepted. Then, no default distribution was proposed to the user in this case. On the basis of the previous considerations, the following general rules have been defined so that the probabilistic database of SYMBIOSE translates, as much as possible, the information available in the TRS 472. It is acknowledged that these rules may seem arbitrary, but they were felt necessary to homogeneously deal with a large number of parameters.  If the TRS 472 provides a geometric mean (GM) and a geometric standard deviation (GSD) on the basis of 10 or more published values, a truncated log-normal distribution is derived. The reparameterisation from (GM, GSD) to (min, max, p) is based on the following equations (with p ¼ 0.95):

min ¼ expðlnðGMÞ  2 lnðGSDÞÞ max ¼ expðlnðGMÞ þ 2 lnðGSDÞÞ  If the TRS 472 reported only 2e9 published values, and/or if data are not available as a GM and a GSD but only as a min and a max, then a log-uniform distribution was chosen, characterized by its extreme values (min, max).  If only 0 or 1 published value exists, or if all published values are equal (min ¼ max), then no probability distribution was derived

2.4. Specific rules and results

Fig. 1. Quantileequantile plots (QQ-plots) for the root transfer factor (all elements and all plants). For each data set (one plant and one element): : the minimal value of the data set versus the fitted (0.5/n)th percentile where n is the size of the data set, þ: the maximal value of the data set versus the fitted (1  0.5/n)th percentile.

The general rules developed in part 2.3 were applied to tables of the TRS 472 (IAEA, 2009), and the TECDOC 1616 (IAEA, 2010), as further detailed below and in Supplementary material. For the freshwater system, this process concerned Kd, and concentration ratios (CR). For concentration ratios, general rules in Part 2.3 were applied to Table 5 pages 478e479 of the TECDOC 1616. The corresponding table in the TRS 472, Table 57 pages 124e125, was not used, as it contains errors (figure 8 turned into figure 9). Given that there are two columns, one for whole fish and one for muscles, we retained systematically in a conservative way the highest value when both were available. In the absence of data in one column or the other, the available information was used. If there is very little data a log uniform was described, parameterized with min and max values, respectively the lowest and the highest values, among all values (muscles or the whole

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Table 1 Default probabilistic database of SYMBIOSE for the concentration ratio CRfish (l kg1 fresh weight) in the biotic freshwater modules. Element

Distribution

Minimum

Geometric mean

Maximum

Ag Am As Au Ba Br Ca Ce Cl Co Cr Cs Eu Fe Hg I La Mn Mo Na Ni Pb Po Pu Ra Rb Ru Sb Se Sr Te Tl U Y Zn Zr

Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log

49 72 72 55 16 30 87 0.35 33 160 53 440 15 4 1800 150 2 28 7 32 16 41 6 7700 0.09 2400 10 1 4000 39 96 130 0.02 6 1400 9

110

250 400 2000 1500 140 850 12,000 1800 220 1000 840 20,000 1500 8100 20,000 2900 890 7200 97 620 310 3330 170 50,000 185 16,000 100 5500 11,000 920 890 6000 20 250 15,000 240

normal truncated uniform normal truncated normal truncated normal truncated normal truncated normal truncated normal truncated normal truncated normal truncated normal truncated normal truncated normal truncated normal truncated normal truncated normal truncated normal truncated normal truncated normal truncated normal truncated normal truncated normal truncated uniform uniform normal truncated normal truncated uniform normal truncated normal truncated normal truncated uniform normal truncated uniform normal truncated normal truncated uniform

380 290 47 160 1000 25 85 400 210 3000 150 170 6100 650 37 450 27 140 71 370

4 6100 71 6800 190 900 40 4700

Table 2 Default probabilistic database of SYMBIOSE for the freshwater partition coefficient Kd (l kg1 dry weight) in the abiotic freshwater modules. Element

Distribution

Minimum

GM

Ag Am Ba Be Ce Cm Co Cs Eu Fe I Mn Np Pm Pu Ra Ru Sb Sr Tc Th U Zn Zr

Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log

1.6  104 3.8  103 1.5  102 3.2  103 2.8  104 10  100 3.0  103 8.3  102 2.0  102 1.0  103 2.2  101 2.4  104 2.0  101 1.0  103 5.5  103 7.7  102 8.9  103 3.4  102 1.6  102 1.0  101 4.1  102 2.0  101 1.0  102 1.0  103

8.5 1.2 2.0 4.2 2.2

normal truncated normal truncated normal truncated normal truncated normal truncated uniform normal truncated normal truncated uniform uniform normal truncated normal truncated uniform uniform normal truncated normal truncated normal truncated normal truncated normal truncated uniform normal truncated uniform uniform uniform

Maximum     

104 105 103 104 105

4.4  104 2.9  104

4.4  103 7.8  104

2.4 7.4 3.2 5.0 1.2

    

105 103 104 103 103

1.8  105

4.5 3.9 2.6 5.4 1.7 7.0 6.3 1.0 8.0 1.0 8.6 2.5 1.0 1.0 1.0 7.1 1.2 7.2 8.7 1.0 7.9 1.0 1.0 1.0

                       

105 106 104 105 106 104 105 106 102 104 105 105 102 104 107 104 105 104 103 102 107 103 103 104

fish all together). Table 1 shows the resulting distributions, for 36 elements. For Kd (freshwater), rules presented in 2.3 were applied to Tables 2e13 pages 452e458 of TECDOC 1616. Corresponding tables in the TRS 472 are tables 53 and 54, pages 119e120 were not used, as they contain errors (figure 8 turned into figure 9). When several sets of Kd values were reviewed by the TECDOC 1616 for a single element (i.e. absorption, desorption, and/or field), the following additional rules were applied: in priority, use Kd from field data (presumably more realistic), otherwise use Kd adsorption (presumably fail-safe). As there were sufficient data for 24 elements, Table 2 and the file in the Supplementary material introduces the 24 resulting PDFs. For agricultural systems, processes of concern were: Kd (soil), root uptake transfer factors, and transfer coefficients to animal products (milk, meat, eggs). When several sets of Kd (soil) values were reviewed by the TRS 472 for a single element (i.e. absorption, desorption, and/or field), the same additional rules as in freshwater were applied: in priority, Kd from field data (presumably more realistic), otherwise use Kd adsorption (presumably fail-safe). Rules were applied without exception for the other

Table 3 Default probabilistic database of SYMBIOSE for the soil partition coefficient Kd (l kg1 dry weight) in the agricultural modules. Element

Distribution

Minimum

Ac Ag Am As Be Bi Br Ca Ce Cm Co Cr Cs Fe I In Mn Mo Na Nb Nd Ni Np Pa Pb Pd Po Pu Ra Rb Rh Ru Sb Se Sm Sn Sr Tc Te Th U Y Zn Zr

Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log

450 36 70 25 240 120 15 0.69 46 644 1.9 0.10 24 166 0.24 240 14 7.0 0.35 110 6.5 5.7 0.94 540 20 55 7.2 46 15 55 0.6 4.1 4.1 18 240 42 1.5 0.0027 180 19 1.4 10 7.9 0.93

uniform uniform normal truncated uniform uniform uniform uniform normal truncated normal truncated normal truncated normal truncated normal truncated normal truncated normal truncated normal truncated uniform normal truncated uniform normal truncated normal truncated normal truncated normal truncated normal truncated uniform normal truncated uniform normal truncated normal truncated normal truncated uniform uniform normal truncated normal truncated normal truncated uniform normal truncated normal truncated normal truncated uniform normal truncated normal truncated uniform normal truncated normal truncated

GM

2600

8 1200 9300 480 40 1200 880 7 1200 3 1500 650 280 35 2000 210 3 1500?? 740 2500

270 62 200 1600 52 0.23 1900 200 950 410

Maximum 5400 15,000 97,000 3000 3000 1500 180 92 31,000 130,000 120,000 16,000 59,000 4700 200 730 110,000 130 33 21,000 65,000 14,000 1300 6600 200,000 670 ?? 12,000 420,000 670 29 18,000 940 2200 3000 62,000 1800 20 790 190,000 29,000 380 110,000 180,000

M. Simon-Cornu et al. / Journal of Environmental Radioactivity 139 (2015) 91e102 Table 4 Default probabilistic database of SYMBIOSE for the soileplant root transfer factor TF (kg dry weight kg1 dry weight) of grass (pasture and fodder) in the agricultural modules. Element Am Ba Ce Cm Co Cs I Mn Ni Np Pb Po Pu Ra Se Sr Tc Th U Zn

Distribution Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log

normal truncated uniform normal truncated normal truncated normal truncated normal truncated normal truncated normal truncated normal truncated normal truncated normal truncated normal truncated normal truncated normal truncated uniform normal truncated normal truncated normal truncated normal truncated normal truncated

Minimum 5

8.9  10 1.2  100 1.5  102 1.7  104 3.3  103 1.5  102 1.0  104 1.8  101 2.5  102 8.4  103 4.0  103 6.8  103 6.1  105 1.2  103 2.0  103 2.7  101 8.4  100 3.3  103 1.6  103 2.8  101

GM

Maximum

1.0  10 4.0 8.3 1.7 1.9 3.7 4.3 1.1 3.4 9.2 1.2 3.0 8.8

           

3

101 104 102 101 103 101 101 102 102 101 104 103

1.1  100 7.6  101 9.9  102 7.2  102 7.8  101

2

2.5  10 3.6  100 9.3  100 5.8  103 6.2  101 4.2  100 1.3  101 2.3  100 1.1  100 4.4  101 2.1  100 2.1  100 5.0  103 4.1  100 2.9  100 6.3  100 6.8  102 3.0  100 1.3  100 3.6  100

inputs. Tables 3e11, and the Supplementary material, introduce the resulting distributions: 44 PDFs for Kd soil, 136 PDFs for root uptake TF, and 91 PDFs for transfer coefficients to animal products. 3. The separation of uncertainty and variability 3.1. Definitions of uncertainty and variability Beyond the apparently simple definitions given in introduction, the concepts of variability and uncertainty are not as obvious as they might seem, and differ subtly from one terminology to another, which may often be a source of misunderstanding. This article is based on the definitions proposed by Cullen and Frey (1999), and adapted to the specificities of

Table 5 Default probabilistic database of SYMBIOSE for the soileplant root transfer factor TF (kg dry weight kg1 dry weight) of leafy vegetables in the agricultural modules. Element Ag Am Cm Co Cs I La Mn Nb Np Pb Po Pu Ra Rb Ru Sb Se Sr Tc Th U Zn

Distribution Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log

uniform normal truncated uniform normal truncated normal truncated normal truncated uniform normal truncated uniform uniform normal truncated normal truncated normal truncated normal truncated uniform uniform uniform uniform normal truncated normal truncated normal truncated normal truncated normal truncated

Minimum 5.9 2.5 2.0 2.3 1.7 4.7 1.1 7.1 8.0 5.0 4.7 1.6 1.1 2.0 6.0 2.0 2.2 1.8 2.1 9.9 3.3 3.8 4.2

                      

GM

Maximum

5

10 105 104 102 103 104 103 102 103 103 104 104 105 103 104 102 105 104 102 101 105 104 101

3

2.7  104 1.7  101 6.0  102 6.5  103 4.1  101

8.0 7.4 8.3 9.1

   

102 103 105 102

7.6  101 1.8  102 1.2  103 2.0  102 2.4  100

1.3  10 2.9  103 8.1  103 1.2  100 2.2  100 8.9  102 1.5  102 2.4  100 2.5  102 8.0  102 1.4  101 3.5  101 6.1  104 4.1  100 1.0  102 2.3  101 2.3  104 1.9  100 2.7  101 3.3  104 4.3  102 1.1  100 1.4  101

95

Table 6 Default probabilistic database of SYMBIOSE for the soileplant root transfer factor TF (kg dry weight kg1 dry weight) of root vegetables and fruit vegetables in the agricultural modules. Element Vegetable Distribution Ag Ag Am Am Cm Cm Co Co Cs Cs I La La Mn Mn Mo Nb Np Np Pb Pb Pm Po Pu Pu Ra Ra Sb Sb Se Sr Sr Tc Th Th U U Zn

Fruit veg. Root veg. Fruit veg. Root veg. Fruit veg. Root veg. Fruit veg. Root veg. Fruit veg. Root veg. Root veg. Fruit veg. Root veg. Fruit veg. Root veg. Root veg. Root veg. Fruit veg. Root veg. Fruit veg. Root veg. Root veg. Root veg. Fruit veg. Root veg. Fruit veg. Root veg. Fruit veg. Root veg. Root veg. Fruit veg. Root veg. Root veg. Fruit veg. Root veg. Fruit veg. Root veg. Fruit veg.

Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log

Minimum

uniform uniform uniform uniform uniform uniform uniform normal truncated normal truncated normal truncated normal truncated uniform uniform uniform normal truncated uniform uniform uniform uniform uniform normal truncated uniform normal truncated uniform uniform normal truncated normal truncated uniform uniform uniform normal truncated normal truncated uniform normal truncated normal truncated normal truncated normal truncated uniform

GM

Maximum

4

2.5  10 5.7  104 2.3  105 2.0  104 3.6  105 2.0  104 5.7  102 2.3  102 1.2  103 4.7  103 8.6  104 5.9  103 4.5  104 1.0  101 1.4  102 2.3  102 8.0  103 4.0  103 5.0  103 1.5  103 5.9  105 3.6  102 3.1  104 6.0  106 7.0  105 2.4  104 8.3  104 1.5  105 4.0  104 2.6  104 1.2  102 4.3  102 1.4  101 1.7  105 4.7  106 8.5  104 2.2  104 1.0  101

1.1 2.1 4.2 7.7

   

101 102 102 103

4.2  101

1.5  102 5.8  103

1.7  102 7.0  102

3.6  101 7.2  101 7.8 8.0 1.5 8.4

   

104 104 102 103

2.0  103 3.9  103 1.9  103 1.7  103 1.4  103 3.9  103 2.3  101 5.3  101 3.5  101 3.8  101 6.9  102 6.0  103 6.0  103 1.5  100 1.3  101 4.2  101 2.5  102 5.7  102 3.6  102 3.9  100 3.8  100 6.0  102 1.1  101 2.0  104 5.8  103 1.2  100 5.9  100 1.6  103 1.6  103 1.0  100 1.1  101 1.2  101 7.9  101 3.6  102 1.4  101 2.6  101 3.2  101 9.5  101

Table 7 Default probabilistic database of SYMBIOSE for the soileplant root transfer factor TF (kg dry weight kg1 dry weight) of cereals in the agricultural modules. Element Am Ce Cm Co Cs I Mn Nb Ni Np Pb Pm Po Pu Ra Ru Sb Se Sr Tc Th U Zn

Distribution Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log

normal truncated normal truncated normal truncated normal truncated normal truncated normal truncated normal truncated uniform normal truncated normal truncated uniform normal truncated uniform normal truncated normal truncated normal truncated normal truncated uniform normal truncated uniform normal truncated normal truncated normal truncated

Minimum 1.8 2.3 2.1 2.8 1.7 1.2 2.6 2.0 3.7 1.2 1.9 3.9 2.2 2.1 1.2 4.4 2.5 4.4 1.5 1.8 1.8 1.0 2.5

                      

7

10 104 106 104 103 104 102 103 103 104 103 104 104 107 104 104 104 102 102 101 104 104 101

GM 2.2 3.1 2.3 8.5 2.9 6.3 2.8

Maximum       

5

10 103 105 103 102 104 101

2.7  102 5.5  104 1.4  102 9.5 1.7 3.0 1.8

   

106 102 103 103

1.1  101 2.1  103 6.2  103 1.8  100

2.7  103 4.2  102 2.5  104 2.6  101 4.9  101 3.3  103 3.0  100 2.5  102 2.0  101 7.3  102 4.8  102 5.0  101 2.6  104 4.3  104 2.4  100 2.0  102 1.3  102 8.0  101 8.0  101 2.4  100 2.4  102 3.7  101 1.3  101

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Table 8 Default probabilistic database of SYMBIOSE for the soileplant root transfer factor TF (kg dry weight kg1 dry weight) of oleaginous plants in the agricultural modules. Element

Distribution

Minimum

GM

Maximum

Am Ce Co Cs I La Mn Np Pb Pm Po Pu Ra Ru Sr Tc Th U Zn

Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log

5.6  105 6.0  103 6.8  103 2.9  103 1.6  104 1.6  104 3.5  102 5.2  103 3.7  105 2.0  102 6.0  105 3.2  105 2.1  104 1.0  102 2.6  101 1.1  100 6.0  106 1.5  105 1.6  101

3.8  104

2.6  103 2.0  102 1.9  101 5.5  101 4.7  101 1.8  103 1.4  100 5.5  102 7.6  101 1.2  100 1.0  103 1.2  104 9.4  101 2.0  102 7.4  100 3.0  101 4.7  102 3.2  101 5.2  100

normal truncated uniform normal truncated normal truncated normal truncated uniform normal truncated normal truncated normal truncated uniform uniform normal truncated normal truncated uniform normal truncated uniform normal truncated normal truncated normal truncated

3.6  102 4.0  102 8.5  103 2.2  101 1.7  102 5.3  103

6.3  105 1.4  102 1.4  100 5.3  104 2.2  103 9.1  101

Table 9 Default probabilistic database of SYMBIOSE for the transfer factor TF from feed to meat products (day kg1 fresh weight). Element Type of meat

Distribution

Minimum

Ba Ba Ca Ca Co Co Co Cs Cs Cs Cs Fe I I I La Mn Mn Na Pb Pb Pu Pu Ru Ru S Sb Se Sr Sr Sr Sr Th U U U Zn Zn Zn Zn

Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log

5.0  105 9.2  103 1.0  103 4.4  102 1.3  104 3.0  102 8.0  103 3.8  103 1.1  100 8.9  102 3.9  102 9.0  103 2.0  103 4.0  103 1.5  102 1.1  104 6.0  104 1.0  103 1.0  102 2.0  104 4.0  103 8.8  108 2.0  105 2.2  103 6.3  104 1.2  100 1.1  103 4.1  100 1.5  104 7.0  103 3.4  104 5.2  104 4.0  105 2.5  104 3.0  101 2.6  102 4.0  102 3.8  101 1.3  101 2.0  102

Beef Chicken Beef Chicken Beef Chicken Sheep Beef Chicken Pork Sheep Beef Beef Chicken Pork Beef Beef Chicken Beef Beef Sheep Beef Sheep Beef Sheep Sheep Beef Chicken Beef Chicken Pork Sheep Beef Beef Chicken Pork Beef Chicken Pork Sheep

uniform uniform uniform uniform uniform uniform uniform normal truncated normal truncated normal truncated normal truncated uniform uniform uniform uniform uniform uniform uniform uniform uniform uniform uniform uniform uniform uniform uniform uniform uniform normal truncated uniform normal truncated normal truncated uniform uniform uniform uniform uniform uniform uniform uniform

GM

2.2  102 2.7  100 2.0  101 1.9  101

1.3  103 2.5  103 1.5  103

Maximum 2.3  104 2.9  102 6.1  101 4.4  102 8.4  104 1.9  100 1.6  102 1.3  101 6.9  100 4.5  101 9.2  101 2.5  102 3.8  102 1.5  102 6.6  102 1.5  104 6.0  104 2.8  103 2.0  102 1.6  103 1.0  102 3.0  104 8.5  105 6.4  103 3.6  103 2.1  100 1.3  103 2.8  101 1.1  102 4.1  102 1.8  102 4.3  103 9.6  104 6.3  104 1.2  100 6.2  102 6.3  101 5.3  101 2.0  101 1.4  101

Table 10 Default probabilistic database of SYMBIOSE for the transfer factor TF from feed to milk (day l1). Element Type Distribution of milk

Minimum

Am Ba Ba Ca Ca Ce Co Co Cr Cs Cs Cs Fe I I I Mn Mo Mo Na Ni Ni Pb Po Po Ra Ru S Sb Se Se Sr Sr Sr Te U W Zn Zr

3.7 2.2 2.1 3.5 2.0 2.0 6.0 1.2 1.0 1.2 1.1 2.3 1.0 9.4 3.0 2.6 7.0 4.3 5.0 5.0 6.5 3.2 1.9 8.9 1.8 7.2 6.7 1.3 2.0 9.1 5.9 4.5 1.3 4.0 5.9 5.0 3.4 1.3 5.5

Goat Cow Goat Cow Goat Cow Cow Ewe Cow Cow Ewe Goat Cow Cow Ewe Goat Cow Cow Goat Cow Cow Goat Cow Cow Goat Cow Cow Goat Cow Cow Goat Cow Ewe Goat Cow Cow Cow Cow Cow

Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log Log

uniform normal truncated uniform normal truncated normal truncated uniform uniform uniform uniform normal truncated normal truncated normal truncated uniform normal truncated uniform normal truncated uniform uniform uniform uniform uniform uniform normal truncated uniform uniform normal truncated uniform normal truncated uniform normal truncated uniform normal truncated uniform normal truncated normal truncated uniform uniform uniform uniform

                                      

GM

106 105 103 103 102 106 105 103 105 103 102 102 105 104 102 102 106 104 103 103 104 103 104 105 103 105 107 102 105 104 102 104 102 103 105 104 105 104 107

Maximum

1.0  105 1.6  104 1.2  103 1.5  101 1.0  102 2.9  102 7.3  102 2.6  101 1.3  104 3.0  104 4.1  103 4.3  103 4.6  103 1.8  102 5.8  102 3.1  101 1.1  101 5.3  101 9.7  105 5.4  103 3.1  102 9.4  101 2.2  101 1.9  100 3.3  104 5.2  103 1.1  102 5.0  102 1.3  103 1.6  101 1.9  104 1.9  104 3.0  104 2.7  103 3.8  104 2.0  103 1.4  104 3.8  102 1.1  101 1.1  104 4.0  103 1.8  102 7.9  102 1.3  103 3.8  103 4.0  102 1.6  102 6.4  102 3.4  104 2.0  103 6.1  103 6.8  104 9.0  103 1.7  105

radiological impact assessments, consistent with Safety Series 100 (IAEA, 1989). Variability is defined in this article as the true heterogeneity (or diversity) within the scenario of concern. The scope of the “scenario of concern” is defined by the prerequisites of an assessment: one nuclear installation, under defined conditions (either planned or existing, either normal or accidental), in one landscape, inhabited by one human population. The main sources

Table 11 Default probabilistic database of SYMBIOSE for the transfer factor F from feed to hen eggs (day kg1 fresh weight). Element

Distribution

Minimum

GM

Maximum

Cs Co Fe I Mn Mo Na Pu Se Sr U Zn

Log Log Log Log Log Log Log Log Log Log Log Log

1.8  101 2.6  102 8.5  101 1.9  100 3.2  102 5.2  101 1.9  100 9.9  106 8.8  100 2.5  101 9.2  101 1.2  100

4.0  101

9.0  101 4.0  102 2.8  100 3.2  100 6.2  102 8.7  101 6.0  100 2.3  103 2.8  101 6.4  101 1.2  100 1.9  100

normal truncated uniform uniform uniform uniform uniform uniform uniform uniform uniform uniform uniform

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of this variability are: (i) spatial variability, within the landscape of concern, (ii) temporal variability, along the period of concern, and (iii) inter-individual variability, between the (human) individuals of the population of concern (or multiple subpopulations). This definition of variability is then confined within the scenario of concern, and is then more restrictive than the common use of the word, as a “worldwide” or “universal” dispersion. Uncertainty is defined in this article as the lack of perfect knowledge about the adequacy of a defined model to reflect the situation of concern for a given impact assessment. Cullen and Frey (1999) defined two types of uncertainty: (i) model uncertainty and (ii) input uncertainty. Model uncertainty is the lack of perfect knowledge about the adequacy of the equations to reflect the mechanisms implied in the situation of concern. It includes uncertainty about the structure of the model, about the scenario, and/or about the level of detail needed (Cullen and Frey, 1999). Input uncertainty is the lack of perfect knowledge about the values of parameters and variables in the equations. It can be due to measurement errors, or sampling errors (particularly in the case of very small sampling sizes), or differing views among experts about the interpretation of data. In this article, we also consider as input uncertainty the lack of perfect knowledge about which parameter value is the most appropriate for the situation of concern, among all the values arising from the true heterogeneity of this parameter between situations. This last type of uncertainty is sometimes called ‘uncertainty due to variability’, or even ‘variability’. However, in this article, variability is given a different meaning, as described above, and consistently with Cullen and Frey (1999) and the SRS 100 (IAEA, 1989). The difference between variability and uncertainty is then subjective: it relies on the point of view of the modeller to decide whether a dispersion reflects either (i) variability inside the scenario of concern or (ii) uncertainty about which of the values existing “worldwide” best applies to the scenario of concern. The SRS 100 (IAEA, 1989) expressed this same distinction as: Type A uncertainty (which we call here variability), defined as “due to stochastic variability with respect to the reference unit of the assessment question” Type B uncertainty (which we call here uncertainty), defined as “due to lack of knowledge about items that are invariant with respect to the reference unit of the assessment question”. 3.2. Separation of uncertainty and variability It has been recommended (EPA, 2001) that appropriate methods should be applied to quantify separately the uncertainty and variability, and that both components should then be propagated through the selected model to estimate the uncertainty and variability in the model result. Some authors proposed in the late 1990s a two-dimensional (or second-order) Monte-Carlo simulation (2D-MC) to estimate the uncertainty in the risk estimates stemming from parameter uncertainty (Cullen and Frey, 1999; EPA, 2001). This 2D-MC is a Monte-Carlo simulation in which the distributions reflecting variability and uncertainty are sampled separately, leading to simulate them separately also in the output (Pouillot and Delignette-Muller, 2010). However, such a separation of variability and uncertainty in the inputs of exposure assessment has not been immediately generalized, a reflection of the fact that this can be a daunting task. Gradually, user-friendly tools were developed that allow 2D-MC simulations to be performed more easily. Some of them are generic, such as

97

Excel add-ons (Crystal Ball, @Risk, ModelRisk), or the opensource R package mc2d (Pouillot and Delignette-Muller, 2010), whereas, others are specifically applied to environmental modelling, such as the dynamic simulation program GoldSim, etc. These methods have now been implemented in various fields of human health risk assessment (e.g. Glorennec, 2006; Ozkaynak et al., 2009; Pouillot et al., 2009; Rimbaud et al., 2010; Wu and Tsang, 2004) but are far from systematic. Whereas Kirchner and Steiner (2008) provided very useful guidelines and recommendations to separate and quantify uncertainty and variability in radioecology, applications remain sparse in this field as well (e.g. Albrecht and Miquel, 2010; Jang et al., 2009; Neptune Company, 2011). 3.3. Parallel between the dimensions defined in SYMBIOSE, and the uncertainty and variability dimensions of 2D-MC Assuming the definitions above, it is illustrated in this section how the platform SYMBIOSE can be used for evaluating variability and uncertainty in radiological impact assessment as in a 2D-MC. A specificity of SYMBIOSE is the concept of dimensions, a dimension being understood as time (a list of dates), space (a list of points, arcs, polygons, etc), or any other list of items (radionuclides, age categories, etc). Inputs and outputs are then either scalar; onedimensional (e.g. time-dependent); two-dimensional (e.g. spatially explicit and time-dependent); or n-dimensional (e.g. spatially explicit, time-dependent, function of the element), without any software limitation neither for the value of n (number of dimensions) nor for the length of the lists (size of the dimensions). Some of these dimensions can intuitively be understood as variability dimensions, i.e. spatial variability, temporal variability, and inter-individual variability. Whatever the scenario and its inputs, the temporal variability is always expressed in the outputs, as they are calculated by numerical solving of non-stationary differential equations. Thus, the structure of SYMBIOSE makes it easy to generalize the 2D-MC “two-dimensional” approach. Indeed, SYMBIOSE allows an “n þ 1-dimensional” simulation, where n is the number of variability dimensions, the n þ 1th dimension of uncertainty being given by the PDFs defining some inputs. An illustration of this structure is given in the case-study below. As discussed in Section 3.1, the difference between variability and uncertainty is subjective and depends on the scope of a scenario. The default PDFs of the SYMBIOSE database are supposed to represent the uncertainty on any “local” average (averaged at the scale of a landscape of interest for a scenario). The spatial variability between landscapes is then considered as a source of uncertainty. 4. Case-study 4.1. Material and methods 4.1.1. Scenario The case study is intentionally simple and theoretical, as it only aims to illustrate the use of the probabilistic database of SYMBIOSE, introduced in Section 2, and the dimensions introduced in Section 3. It does not cover all the potential of SYMBIOSE. Neither is it representative of a realistic radiological impact assessment. For a given and constant activity of one radionuclide (1 Bq L1) in a river stream, the daily ingestion dose rate is assessed for an adult local population who would: - drink tap water (assumed to be filtered water taken from this river stream),

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- consume milk from cows who have drunk tap water, - and, for some of them, consume river fishes taken from this river stream. All inputs are defined in Table 12, as well as the sources of the assumptions. There are two sources of inter-individual variability: the consumption rates of tap water and of (local) river fishes. They are based on French data, and assumed to be known with certainty (no uncertainty). There are three sources of parametric uncertainty: Kd in freshwater, concentration ratio between freshwater and fishes, and equilibrium transfer from water to milk. All other inputs are arbitrarily assumed fixed (neither uncertain nor variable), in most cases at their default values of the SYMBIOSE database (cf. Table 12). All calculations were performed for three radionuclides: 137Cs, 129I, and 238U. 4.1.2. Calculations All calculations were performed with SYMBIOSE V2.1.3, using the simplest modelling choices. To check results, a simplified model, approaching true SYMBIOSE calculations, was also defined and run with R and the package mc2d. The equations of the simplified model are given in Table 12. The code is provided in the Appendix. Note that the approximation with R does not account for radioactive decay and then only applies to long-lived radionuclides. In both calculations, the main output is the daily ingestion dose taking into account all foods (Dose). In the R calculation, there were 1001 iterations in the uncertainty dimension and 1001 in the iterations in the variability dimension. In the SYMBIOSE calculation, 100 adults were defined in the inter-individual dimension, which is then taken as variability

dimension as explained in 3.1. Practically, 100 different diets were defined, with a fixed value for water consumption, an empirical distribution of 100 values for river fish consumption, and 100 values randomly sampled from the log-normal distribution characterizing tap water consumption. Using the Monte Carlo functionality of SYMBIOSE, there were 1000 iterations (samplings in the uncertainty dimension). The variability and uncertainty ratios were calculated for the simplified model according to Ozkaynak et al. (2009), through the function mcratio of mc2d in the R calculations, or by applying the definitions for the SYMBIOSE calculations. These ratios are expressed for a given output (here, the daily dose). All calculations are related to a “central tendency” value (50th uncertainty percentile about the 50th variability percentile) noted A. The variability ratio equals the 50th uncertainty percentile about the 97.5th variability percentile (noted B) divided by A. The higher B/A is, the stronger the variability of the inputs impacts the variability of the output. The uncertainty ratio equals the 97.5th uncertainty percentile about the 50th variability percentile (noted C) divided by A. The higher C/ A is, the stronger the uncertainty of the inputs impacts the uncertainty of the output. The overall uncertainty ratio equals the 97.5th uncertainty percentile about the 97.5th variability percentile (noted D) divided by A.

4.2. Results of the case-study Table 13 provides results of the simplified case-study using SYMBIOSE. Consistency between results with the simplified calculation using R was checked (results not shown).

Table 12 Scenario of the simplified case-study: definitions, values and source of information of each input, definition and approximate calculations of each output. Input

Definition

Unit

U/V/F

U/V distribution or fixed value

Source for the distribution or value

Ariver

Activity in the raw water of the river stream Suspended matter of the river stream Concentration ratio for freshwater fishes

Bq l1

F

1

Assumption of the theoretical scenario

kg dry weight l1

F

3  105

Expert opinion (default value in SYMBIOSE)

1

l kg fresh weight

U

IAEA, this article (Table 1)

Kdwat

Freshwater partition coefficient

l kg1 dry weight

U

TFmilk

Transfer factor for cow milk

day l1

U

Conswat,hum

Local tap water consumption by humans Local tap water consumption by cows Local river fishes consumption by humans Local cow milk consumption by humans Dose coefficient conversion

l day1

V

Cs: Log normal (GM ¼ 3000, GSD ¼ 2.6) I: Log normal (GM ¼ 650, GSD ¼ 2.1) U: Log normal (GM ¼ 2.4, GSD ¼ 11) Cs: Log normal (GM ¼ 29,000, GSD ¼ 5.9) I: Log normal (GM ¼ 4400, GSD ¼ 14) U: Log normal (GM ¼ 50, GSD ¼ 1.6) Cs: Log normal (GM ¼ 0.0046, GSD ¼ 2) I: Log normal (GM ¼ 0.0054, GSD ¼ 2.4) U: Log normal (GM ¼ 0.0018, GSD ¼ 1.9) Log normal (GM ¼ 0.46, GSD ¼ 1.96)

Derived from ANSES data (Pouillot et al., 2013)

l day1

F

100

Expert opinion (default value in SYMBIOSE)

kg fresh weight day1 l day1

V

Empirical (from 0 to 20, 100-g-serving(s) per year) 0.075

Derived from ANSES/InVS (2011)

nSv Bq1

F

MESload CRfish

Conswat,cow Consfish Consmilk DCCing

F

137

IAEA, this article (Supplementary material)

IAEA, this article (Supplementary material)

Expert opinion ECRIN (IRSN, 2003) (default value in SYMBIOSE)

Cs: 13 I: 110 238 U: 45 129

Output

Definition

Unit

U/V

Simplified formula

Atap

Activity in the tap water taken from the river stream Activity in the fishes taken from the river stream Activity in the milk of cows Ingestion exposure (daily) Ingestion dose (daily)

Bq l1

U

Ariver 1þKdwat MESload

Bq kg fresh wieght1

U

Ariver  CRFish

Bq l1 Bq day1 nSv day1

U U&V U&V

Atapwat  TFmilk  Conswat;cow Consfish  Afish þ Conswat,hum  Atap Ing  DCCing

wat

Afish Amilk Ing Dose

wat

þ Consmilk  Amilk

M. Simon-Cornu et al. / Journal of Environmental Radioactivity 139 (2015) 91e102

99

Table 13 Results of the simplified case-study: daily ingestion doses (nSv/day). Best-estimates (50th uncertainty percentiles) of 2.5th, 25th, 50th, 75th and 97.5th variability percentiles and their 95% confidence interval [2.5th uncertainty percentiles, 97.5th uncertainty percentile]. A, B, C and D denote the values used to calculate the variability ratio (B/A), the uncertainty ratio (C/A) and the overall uncertainty ratio (D/A). Radionuclide, food

Daily dose 2.5%

25%

50%

0 [0; 0] 2 [0.3; 4] 3 [0.3; 9] 55 [5; 112] 15 [13; 21]

0.8A, B [0.1; 4C,D] 11A [2; 52C] 3A [0.4; 6C] 13A [4; 54C] 86A [17; 162C] 22A [20; 28C]

137

Cs, Cow milk 137 Cs, Fish 137 Cs, Tap water 137 Cs, all 129 I, all 238 U, all

0 [0; 0] 0.9 [0.1; 1.6] 2 [0.1; 5] 24 [2; 77] 8 [6; 13]

For 137Cs, results are given for each food separately. The daily ingestion dose due to cow milk consumption is not variable as it does not depend on any variable input, i.e. it is the same estimation for any of the 100 adults defined in SYMBIOSE (or any of the 1001 iterations in the variability dimension in R). However, it is uncertain, i.e. the uncertainty in the estimation of this single value reflects uncertainty about Kdwat and TFmilk. The range [2.55th percentile; 97.5th percentile] given in Table 13 is assimilated to 95% confidence interval. The daily ingestion dose due to river fish consumption is both uncertain and variable, i.e. it reflects both parametric uncertainty in the concentration ratio (CRfish) and interindividual variability (consumption of river fishes). Then, beware that results are two dimensional. The best-estimate (i.e. median in the uncertainty dimension, or 50th uncertainty percentile) of the 75th variability percentile (i.e. for an adult frequently consuming river fishes) is 44 nSv per day. The uncertainty about this estimate is expressed through a 95% confidence interval of [9e209] nSv per day. The daily ingestion dose due to tap water consumption is also both uncertain and variable, i.e. it reflects both parametric uncertainty in Kdwat and inter-individual variability (consumption of water). Note that the uncertainty ratio is not very high even if Kdwat of caesium is very uncertain. Considering each food (including water) separately, and comparing the variability ratios (B/A), it appears that the most variable dose is that due to ingestion of river fishes, which is explained by highly variable consumption frequencies. The most uncertain ones, i.e. the doses with the highest uncertainty ratios (C/A) are due to river fishes and cow milk. Note that the uncertainty ratio of dose due to tap water is not very high even if Kdwat of caesium is highly uncertain (GSD ¼ 5.9). In sensitivity analysis terms, it means that the calculation of the dose is not very sensitive to the uncertainty of Kdwat. Considering all foods (incl. water) together, leads to estimating Dose, i.e. the (total) daily ingestion dose, which is also twodimensional. Results are presented for the three tested radionuclides in Table 13. For 137Cs, the large variability is reflected by the value of the variability ratio (B/A ¼ 15) and through the range between the best-estimate of the 2.5th percentile, 2 nSv per day, and the best-estimate of the 97.5th percentile, 206 nSv per day. There is also high uncertainty mostly linked to fish ingestion, and then to CRfish. The overall uncertainty ratio is then quite high (D/ A ¼ 88). They were different for iodine and even more for uranium. In the case of uranium, the CRfish is so low that fish consumption does not much affect the dose, far less than in the case of caesium.

75% 44 [9; 209] 4 [0.6; 8] 48 [12; 213] 138 [50; 338] 31 [28; 36]

97.5%

B/A (V)

C/A (U)

D/A (U&V)

202B [41; 965D] 9B [1; 17D] 205B [45; 967D] 402B [148; 1354D] 63B [60; 68D]

1 18 3 15 4.7 2.8

4.9 4.8 1.8 4 1.9 1.2

4.9 88 5.5 72 16 3

5. Discussion This article has introduced some of the specific features of SYMBIOSE, and more precisely of its database, which enable the user to account for input (or parametric) uncertainty and for interindividual variability. Nevertheless, it was not an exhaustive description of SYMBIOSE, and even not of all specificities which make it fully appropriate to account for all sources of variability and of uncertainty. Temporal variability is also partly accessible for the user. Indeed, all SYMBIOSE calculations involve numerical solving of differential equations, so that most outputs are given on a temporal scale, with calendar dates. Averaging over the whole period of interest, or over smaller periods only, is user-friendly. However, even though calculations are by essence dynamic, many of the equations dealing with transfers in terrestrial and freshwater ecosystems are based on the use of constants, such as Kd, or transfer factors. In these cases, the equilibrium hypothesis is then implicitly assumed at each time step, which should be accounted for as a source of modelling uncertainty. Incorporating more knowledge about the dynamic behaviour of radionuclides into dynamic models is one of the challenges of the SRA in radioecology (Hinton et al., 2013). For these current equilibrium-based factors, this article has proposed default PDFs on the basis of international documents. Tables 1 to 11, and the Excel file provided in the Supplementary material, should provide modellers helpful materialisation of the TRS 472 (IAEA, 2009) and TECDOC 1616 (IAEA, 2010) reports. In the late 1990s and early 2000s, there had been earlier proposals of default PDFs for ECOSYS, COSYMA, or RESRAD, as defined or reviewed by NUREG/CR-6523 (Brown et al., 1997) and by NUREG/CR-6697 (US-NRC, 2000). These PDFs were based on earlier literature reviews, or on elicitation by experts. A whole comparison of the results would be out of the scope of this article. Just as a short illustration, Fig. 2 compares different distributions, focussing on two transfer coefficients to animal products, i.e. for two (element, parameter) combinations: (Cs, cow milk) and (Sr, beef meat). The widest ranges were those proposed by NUREG/CR-6523 (Brown et al., 1997), as an “Aggregate expert Distribution”, on the basis of the combination of 5th, 50th and 95th quantiles elicited by experts. This may be explained by the fact that experts intuitively took margins around the data they were aware of. The distributions proposed by NUREG/CR-6697 (US-NRC, 2000) for RESRAD were based on a report by the National Council on Radiation Protection and Measurements, entitled “Recommended Screening Limits (…)”. This objective of “screening” may explain that the distributions appear conservative (biased to higher values than the other

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Fig. 2. Comparison of 5 cumulative distribution functions (CDFs): the truncated lognormal defined for SYMBIOSE (this study), the same distribution without truncature, the log-uniform distribution defined for ECOSYS as mentioned by NUREG/CR-6523 (Brown et al., 1997), the truncated log-normal defined for RESRAD by NUREG/CR6697 (US-NRC, 2000), and the empirical aggregate expert distribution defined for COSYMA by NUREG/CR-6523, for each of two transfer factors from feed to animal products: (a) transfer of Cs to milk (dairy cows), (b) transfer of Sr to beef meat (cattle).

proposed in this study. There is actually an impact of the truncature on the lowest and highest percentile but it appears moderate and consistent with the wish to exclude extreme values. This moderate impact was also confirmed on the results of the case-study earlier (data not shown). Last, the narrowest distributions are the log-uniform distributions defined for ECOSYS as reviewed by NUREG/CR-6523 (Brown et al., 1997), which may appear, according to this example, insufficient to correctly account for the highest values. The log-normal distributions (defined on at least 10 values) presented in Tables 1e11 do not cover the whole list of elements potentially encountered in a scenario. Indeed, in the TRS 472 and TECDOC 1616 tables, for many combinations (element, parameter), only 0 or 1 value were available (in which case, no distribution was derived), or only a few (in which case, the log-uniform that was derived appears debatable). Then, even for these equilibriumbased parameters, scarcity of data is one of the major sources of uncertainty and there is an obvious need to acquire more data about these poorly documented situations, as also underlined by the SRA of radioecology (Hinton et al., 2013). Beyond this necessary acquisition of data, there are also methodological questions: how to cope with an absence of data? or with few data? Answering these questions will require better exploring expert elicitation methods, such as the possibility approach, defining knowledge in the form of intervals (Baccou et al., 2008), or the Bayesian inference, enabling the modeller to combine prior information and new data into a posterior information (Hosseini et al., 2013). A methodological aspect of interest, but not in the scope of this article, is sensitivity analysis. However, the case-study, for a given scenario, provided some insight to the extent to which a variable and/or uncertain input can influence the variability and/or uncertainty of an output. For example, even if the freshwater Kd is highly uncertain, its impact of this uncertainty on the uncertainty of the tap water contamination, is limited (which is obviously explained considering, in Table 12, how Kdwat is used in Atap wat calculation). On the contrary, when an output directly results from multiplication by an uncertain input, the uncertainty of this input (e.g. river fish concentration factor in the case-study) obviously impacts strongly the uncertainty of this output (e.g. ingestion daily dose due to fish consumption). Variability in the consumption features also directly impacts variability in the estimated doses. As discussed by Albrecht and Miquel (2010), it could be argued such insights could just as well be gained through a careful study of the input parameters and the equations without any probabilistic calculations. This could indeed be feasible for a very easy example, with few equations and parameters, as in this case-study. However, for a realistic multi-source, multi-media, multi-pathway assessment, as performed with SYMBIOSE, understanding how all of the multiple inputs influence the results is incomprehensible to human mind, which makes the development of uncertainty (and variability) analysis very useful for assessors, as introduced in this article, and the on-going development of sensitivity analysis, to be introduced in a next step.

Acknowledgements ones). These distributions are, as the ones proposed in the present study, truncated log-normal distributions, but the truncature was limited (0.01th percentile to 99.9th percentile) whereas the truncature proposed in the present study is between 2.5th percentile and 97.5th percentile. Fig. 2 also illustrates the impact of this truncature on the distributions

The Region PACA (France) is acknowledged for financially supporting the PhD study of M. Sy. EDF is acknowledged for financially supporting the development of SYMBIOSE. We also would like to thank T. Hinton for his careful reading of the manuscript.

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Appendix A. R code for the simplified model in the case of caesium 137 (notations in Table 12). library(mc2d) ndvar(1001) ndunc(1001) Ariver