Development of a surrogate model and sensitivity analysis for spatio-temporal numerical simulators

To evaluate the consequences on human health of radionuclide releases in the environment, numerical simulators are used to model the radionuclide atmospheric dispersion. These codes can be time consuming and depend on many uncertain variables related to radionuclide, release or weather conditions. These variables are of different kind: scalar, functional and qualitative. Given the uncertain parameters, code provides spatial maps of radionuclide concentration for various moments. The objective is to assess how these uncertainties can affect the code predictions and to perform a global sensitivity analysis of code in order to identify the most influential uncertain parameters.This sensitivity analysis often calls for the estimation of variance-based importance measures, called Sobol’ indices. To estimate these indices, we propose a global methodology combining several advanced statistical techniques which enable to deal with the various natures of the uncertain inputs and the high dimension of model outputs. First, a quantification of input uncertainties is made based on data analysis and expert judgment. Then, an initial realistic sampling design is generated and the corresponding code simulations are performed. Based on this sample, a proper orthogonal decomposition of the spatial output is used and the main decomposition coefficients are modeled with Gaussian process surrogate model. The obtained spatial metamodel is then used to compute spatial maps of Sobol’ indices, yielding the identification of global and local influence of each input variable and the detection of areas with interactions. The impact of uncertainty quantification step on the results is also evaluated.

[1]  H. Akaike.  A new look at the statistical model identification , 1974 .

[2]  J. Tukey.  Mathematics and the Picturing of Data , 1975 .

[3]  B. Efron,et al.  The Jackknife Estimate of Variance , 1981 .

[4]  M. E. Johnson,et al.  Minimax and maximin distance designs , 1990 .

[5]  R. Tibshirani,et al.  Generalized Additive Models , 1991 .

[6]  Henry P. Wynn,et al.  Screening, predicting, and computer experiments , 1992 .

[7]  Ilya M. Sobol,et al.  Sensitivity Estimates for Nonlinear Mathematical Models , 1993 .

[8]  Jon C. Helton,et al.  Uncertainty and sensitivity analysis techniques for use in performance assessment for radioactive waste disposal , 1993 .

[9]  A. Saltelli,et al.  Importance measures in global sensitivity analysis of nonlinear models , 1996 .

[10]  Douglas W. Nychka,et al.  Case Studies in Environmental Statistics , 1998 .

[11]  P. Rousseeuw,et al.  The Bagplot: A Bivariate Boxplot , 1999 .

[12]  J. Chilès,et al.  Geostatistics: Modeling Spatial Uncertainty , 1999 .

[13]  A. Chatterjee.  An introduction to the proper orthogonal decomposition , 2000 .

[14]  Richard J. Beckman,et al.  A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output From a Computer Code , 2000, Technometrics.

[15]  Jeremy E. Oakley,et al.  Bayesian Inference for the Uncertainty Distribution , 2000 .

[16]  I. Sobol.  Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates , 2001 .

[17]  A. O'Hagan,et al.  Bayesian inference for the uncertainty distribution of computer model outputs , 2002 .

[18]  Andrea P. Reverberi,et al.  Statistical sensitivity analysis of packed column reactors for contaminated wastewater , 2003 .

[19]  Thomas J. Santner,et al.  Design and analysis of computer experiments , 1998 .

[20]  Thomas J. Santner,et al.  The Design and Analysis of Computer Experiments , 2003, Springer Series in Statistics.

[21]  Runze Li,et al.  Design and Modeling for Computer Experiments , 2005 .

[22]  Brian J. Williams,et al.  Sensitivity analysis when model outputs are functions , 2006, Reliab. Eng. Syst. Saf..

[23]  J Q Shi,et al.  Gaussian Process Functional Regression Modeling for Batch Data , 2007, Biometrics.

[24]  M. J. Bayarri,et al.  Computer model validation with functional output , 2007, 0711.3271.

[25]  D. Higdon,et al.  Computer Model Calibration Using High-Dimensional Output , 2008 .

[26]  Stefano Tarantola,et al.  Uncertainty in Industrial Practice , 2008 .

[27]  Bertrand Iooss,et al.  An efficient methodology for modeling complex computer codes with Gaussian processes , 2008, Comput. Stat. Data Anal..

[28]  B. Iooss,et al.  Global sensitivity analysis for a numerical model of radionuclide migration from the RRC “Kurchatov Institute” radwaste disposal site , 2008 .

[29]  Bertrand Iooss,et al.  Global sensitivity analysis for models with spatially dependent outputs , 2009, 0911.1189.

[30]  Stefano Tarantola,et al.  Sensitivity analysis of spatial models , 2009, Int. J. Geogr. Inf. Sci..

[31]  Carl E. Rasmussen,et al.  Gaussian processes for machine learning , 2005, Adaptive computation and machine learning.

[32]  Paola Annoni,et al.  Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index , 2010, Comput. Phys. Commun..

[33]  Rob J Hyndman,et al.  Rainbow Plots, Bagplots, and Boxplots for Functional Data , 2010 .

[34]  Patrick Armand,et al.  Presentation of the CERES platform used to evaluate the consequences of the emissions of pollutants in the environment , 2011 .

[35]  David Makowski,et al.  Multivariate sensitivity analysis to measure global contribution of input factors in dynamic models , 2011, Reliab. Eng. Syst. Saf..

[36]  Etienne de Rocquigny,et al.  Modelling Under Risk and Uncertainty: An Introduction to Statistical, Phenomenological and Computational Methods , 2012 .

[37]  Jean-Paul Chilès,et al.  Wiley Series in Probability and Statistics , 2012 .

[38]  Enrico Zio,et al.  Polynomial chaos expansion for global sensitivity analysis applied to a model of radionuclide migration in a randomly heterogeneous aquifer , 2013, Stochastic Environmental Research and Risk Assessment.