dynamic Soil-Plant Deposition Model, introduced in (Gonze et al., submitted to this ... kg the seasonality factor, Φ the phase shift (d), α the power coefficient,.
Sensitivity analysis of a radionuclide transfer model describing contaminated vegetation in Fukushima prefecture, using Morris and Sobol’ indices V. Nicoulaud-Gouin1, L. Garcia-Sanchez2, J.M. Métivier1, M.A. Gonze1 1Institut
de Radioprotection et de Sûreté Nucléaire-PRP-ENV/SERIS/LM2E, 2Institut de Radioprotection et de Sûreté Nucléaire-PRPENV/SERIS/L2BT
INTRODUCTION The increasing spatial and temporal complexity of models (non linearity, threshold effect, strong interactions between physical variables ...) demands methods capable of ranking the influence of their large numbers of parameters. Sensitivity analysis (SA) of model parameters specifically arises in assessment studies on the consequences of the Fukushima accident. Local SA deals with the value of the response, while global SA deals with the variability of the response. Global SA quantifies the influence of uncertainties of the model input variables in the whole range of variations among model responses. Therefore, SA aims at measuring the influence of input variability on the output response. Generally, two main approaches are used (Saltelli, 2000; Iooss, 2011): Screening methods : coarse sorting of the most influential inputs among a large number of input parameters; Measures of importance: In this category, there are regression-based methods, assuming a linear or monotonic response (Pearson coefficient, Spearman coefficient), and variance-based methods, without assumptions on the model but requiring an increasingly prohibitive number of evaluations when the number of parameters increases. They provide quantitative sensitivity indices. SA aims to determine if the model follows well the processes which it simulates. If an input variable is found to be very sensitive, whereas it is known to be non-influential, then that is an indication that the model does not reflect the real process. However, knowledge of the less influential variables is useful in that the modeller can consider them deterministic and use only their expected value in the simulations. Determining interactivity between parameters, via SA, is also a means to better understand the studied phenomenon. The methodology of SA will be illustrated on a radionuclide transfer assessment model: the dynamic Soil-Plant Deposition Model, introduced in (Gonze et al., submitted to this conference). MATERIALS AND METHODS Sampling Global SA methods are methods based on sampling. The performance of sensitivity indices calculus is intimately linked to the number of realisations (i.e., model runs). The fundamental challenge is to choose an adequate length of sampling, minimize computer time and maximize information gained. In the Morris method (Campolongo, 2007; Morris, 1991) only one input parameter is given a new value for each model run. It is the number of design repetitions which is difficult to choose in order to have a good assessment of the indices. The Sobol’
method uses low-discrepancy sampling (Owen, 2003; Saltelli, 2002). Its quasi-Monte Carlo sampling, a deterministic one, consists of building uniformly compact sampling without local accumulation zones of points. It has a weaker discrepancy than random series. Morris method With Morris method, from the elementary effects, two sensitivity estimates are obtained, both based on the calculation of incremental ratios at various points of the input space. The incremental ratio is the ratio between the variation of the model output in two different points of the input space and the amplitude of variation of the parameter itself. Finally, it calculates for each elementary effect, the mean (µ; assessing the overall influence of the factor on the output) and the standard deviation (σ; estimating the totality of the higher-order effects, i.e. non-linearity or interactions with others factors). A large value of µ means large effects and a model sensitive to input variations. A large value of σ means interactions among input parameters and effects depending on value (either of the input itself by non-linear effect or of the other inputs by interaction). But the method does not allow distinguishing the two cases. Sobol’ indices This method quantifies the influence of each input, using first order sensitivity indices, as well quantifies the influence of each input through its interactions in the model, using second order sensitivity indices (Saltelli, 2000). The advantage of this method is to decompose the variance of the model in the presence of correlated variables groups. This decomposition allows defining multidimensional sensitivity indices. The Sobol’ indices measure has nevertheless a large disadvantage: high variability of the estimations; so, a precise estimation requires, in general, a high number of evaluations of the function Y. No less than 104 simulations would be necessary in order to obtain a reliable estimation of sensitivity indices for each input (Iooss, 2011). Model In this model, two foliage pools and a root pool are considered, taking into account weathering processes and translocation transfers. The model also describes foliar biomass growth with a Verhulst subroutine. The soil-to-plant deposition model is described by the following equations: π (𝐽 + 𝜙) 𝑘𝑔 = sin 365
| (
)| , 𝑑𝑡𝑑 𝑚 𝛼
𝑓=𝑘
𝑔
𝑚 𝑐𝑜𝑒𝑓𝑚𝑎𝑙𝑡 1 ‒ 𝐾𝑓 𝑚𝑓 ,
(
)
𝑑 𝐴 = 𝜆𝑎𝐴 ‒ 𝜆𝑡 𝐴 ‒ 𝑇𝐹𝑡𝐴 ‒ 𝜆𝑟𝑎𝑑𝐴 𝑑 𝐴 =‒ 𝜆𝑤 + 𝜆𝑎 𝐴 ‒ 𝜆𝑟𝑎𝑑𝐴 𝑓𝑒 𝑓𝑖 𝑓𝑖, ( ) 𝛽𝑟 𝑟 𝑓𝑒 𝑓𝑒 𝑑𝑡 𝑓𝑖 𝑑𝑡 𝑓𝑒 , 𝐴𝑓 𝑑 𝑇𝐹𝑡 𝑡 𝑟𝑎𝑑 𝑑𝑡𝐴𝑟 = 𝜆 𝐴𝑓𝑖 ‒ 𝛽𝑟 𝐴𝑟 ‒ 𝜆 𝐴𝑟 , 𝐴𝑓 = 𝐴𝑓𝑒 + 𝐴𝑓𝑖 , 𝐶𝑓 = 𝑚𝑓
(
(
)
)
With the parameters: kg the seasonality factor, Φ the phase shift (d), α the power coefficient, K the characteristic biomass at which growth vanishes (carrying capacity) kg.m-2, coefmalt the maximum biomass growth rate under optimal conditions s-1, mf (x; t) the foliar biomass kg.m2, β (x; t) the root-to-shoot biomass ratio (-), λw the net weathering rate s-1, λa the net r adsorption rate s-1, λt the to-root translocation rate s-1, TFt the root-to-shoot partitioning factor λrad
at equilibrium, the decay of radionuclide C through decay chains, initial concentration in plants and 𝑚𝑓0 the initial biomass. RESULTS AND CONCLUSIONS
𝑏𝑓𝐶𝑓𝑒0 = 𝐴𝑓𝑒(0) 𝑚𝑓0
the
Morris indices The Morris indices plot (σ over µ for all variables over time; Fig. 1) identifies the noninfluential variables: Φ, α, λt, TFt, βr. This is confirmed by the structural non-identifiability of these parameters. Therefore, the root transfer characterization is not established. The long term effect of the radioactivity is not qualifiable. The phase shift and the power coefficient of the seasonality factor kg, are not influential on the model output. So we can fix them to a most likely deterministic value for a calibration study. 𝑏𝑓𝐶𝑓𝑒0 = 𝐴𝑓𝑒(0)
𝑚𝑓0 , mf0, coefmalt, The most influential variables are the following parameters: w a λ , λ and K. They have actually a dynamical sensitivity: At the beginning of the fallout, the initial concentration in plants, bfCfe0, which represents dry deposition onto the plant and wet deposition flux onto the ground surface, the maximum biomass growth rate, coefmalt, and the weathering rate λw are the three main sensitive parameters, and thereafter the other parameters increase in sensitivity over the time.
By taking the true credible intervals found with a Bayesian calibration, notably of the initial concentration and the weathering rate (dependent on the climate and the Fukushima event) according to the ten observations sites, we could create spatial variability, and we could analyze a spatial and temporal sensitivity of the parameters with ten new designs. A high spatial variability was observed for the Malthusian coefficient indices (Fig. 2).
Figure 1: Morris indices over time
Figure 2: Morris indices over space
Sobol’ indices At this point, we wanted to better understand the sensitivity of the soil-plant deposition model and its parameters bfCfe0, mf0, λw, λa, λt, coefmalt, K. We estimated the sensitivity indices of first order and total indices using the Saltelli 2002 method (Saltelli, 2002) based on 10000 sampling. We attributed a log-uniform distribution to all parameters, and their range spread up to a factor of 5000. In this scope, we observed a complementary relationship between the curve of Sobol’indices of Malthusian coefficient and the curve of Sobol’indices of initial foliar biomass (Fig. 3 and
Fig. 4). These curves join at the moment where sensitivity of initial foliar activity transitions to a plateau. High interactions are noticed between the influence of weathering rate, initial biomass, Malthusian coefficient and initial foliar concentration, according to the differences between total and first indices. The parameters of net adsorption rate and shoot-to-root translocation rate were not influential. Sobol’ indices are more precise in ordering the most influential parameters than Morris indices: The Malthusian coefficient ranked second while it was ranked third with the Morris method.
Figure 3: First Sobol’ indices over time
Figure 4: Total Sobol’ indices over time
Conclusions The two methods provide good assessments of parameter sensitivities. We established a spatial and temporal analysis on the assumption of different designs according to specific sites of Fukushima prefecture. The two methods are complementary and give a global overview of the most influential parameters of a model, which were the initial concentration in external portions of plants, the initial biomass, the maximum biomass growth rate, and the weathering rate. Consequently, assessment of radionuclides transfer in plants should take into account the spatial variability of these parameters. REFERENCES Campolongo, F., J. Cariboni and A. Saltelli, 2007. An effective screening design for sensitivity. Environmental Modelling & Software., 22:1509–1518. Iooss, B., 2011. Revue sur l’analyse de sensibilité globale de modèles numériques. Journal de la Société Française de Statistique, 152:3–25. Morris, M. D., 1991. Factorial sampling plans for preliminary computational experiments. Technometrics, 33:161–174. Owen, A. B., 2003. Monte Carlo Ray Tracing, pages 69–88. SIGGRAPH. Saltelli, A., 2002. Making best use of model evaluations to compute sensitivity indices. Computer Physics Communications, 145:280–297. Saltelli, A., K. Chan and M. Scott, 2000. Sensitivity analysis. John Wiley and Sons, New York.
