Global sensitivity analysis: A flexible and efficient framework with an example from stochastic hydrogeology S. Oladyshkin a,⇑ , F.P.J. de Barros b , W. Nowak a a SRC Simulation Technology, Institute of Hydraulic Engineering (LH2), University of Stuttgart, Pfaffenwaldring 61, 70569 Stuttgart, Germany b Dept. of Geotechnical Engineering and Geo-sciences, Universitat Politecnica de Catalunya-BarcelonaTech, Jordi Girona 1-3, 08034 Barcelona, Spain article info Article history: Received 20 April 2011 Received in revised form 2 November 2011 Accepted 4 November 2011 Available online 22 November 2011 Keywords: Global sensitivity analysis Arbitrary polynomial chaos expansion Orthonormal basis Response surface methods Stochastic hydrogeology abstract When investigating, modeling or operating uncertain systems, sensitivity analysis with respect to uncer- tain model parameters yields valuable information. It helps to quantify the relevance of parameters, to estimate their individual contributions to prediction uncertainty, and to better direct further data acqui- sition. In this work, we propose a response surface method for global sensitivity analysis (based on the arbitrary polynomial chaos expansion, aPC). The key advantages of our proposed technique are: (1) aPC alleviates the computational burden associated with conventional global sensitivity analysis meth- ods that require many evaluations of a simulation model; (2) the proposed method incorporates arbitrary independent probability densities or weighting functions for the investigated model parameters, thus generalizing several existing methods to reflect the expected relevance of parameters values within their allowable ranges; and (3) our framework allows to incorporate this information while requiring only a finite number of statistical moments for the investigated parameters. We generalize the polynomial- based computation of Sobol indices to arbitrary distributions, and suggest an associated complementary new sensitivity measure based on polynomial representation, which allows both univariate and multivar- iate global analysis. Compared to Sobol indices, our new weighted measure is absolute rather than relative, and converges faster with increasing order of expansion. We use analytical and hybrid analyti- cal-numerical formulations that further improve computational efficiency. Altogether, we can conduct global sensitivity analysis at computational costs that are almost as low as those of local sensitivity analyses. We illustrate our approach for a 3D groundwater quality and human health risk problem in heterogeneous porous media. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction Understanding the general role of parameters in models and the impact of varying model parameters on the response of prediction models is a relevant subject in various fields of science and engi- neering. Characterizing the impact of parameter variations is known as sensitivity analysis and can be subdivided into local and global analysis [1–3]. In many cases of practical interest, we wish to perform a global sensitivity analysis (GSA) in order to ana- lyze a model as such or to investigate, quantify and rank the effects of parameter variation or parameter uncertainty on the overall model uncertainty. GSA can also be used to: (1) quantify the relative importance of each individual input parameter in the final predic- tion [1,4,5]; (2) aid engineers to produce more robust designs; and finally (3) help decision makers to allocate financial resources towards better uncertainty reduction, e.g., in problems related to risk assessment [6,7]. An important recommendation to keep in mind is that GSA should be global not only in the sense of looking at the entire range of possible parameter variations. It should also be used to assess the importance of parameters on a global, final model output or post-processing result that is relevant to generate new insight, or relevant for final decisions. GSA should not merely be applied to model-internal quantities that are of secondary importance for the scientific or application task at hand [2]. For example, the field of subsurface contaminant hydrology requires uncertainty estimates due to the ubiquitous lack of parameter knowledge caused by spatial heterogeneity of hydraulic properties in combination with incomplete characterization [8,9]. For such reasons, we need to rely on probabilistic tools to predict contaminant levels and their overall health effects, and to quantify the corresponding uncertainties. Having efficient computational approaches to estimate uncertainty and to perform GSA in hydro- geological applications (and many other fields of science and engi- neering that feature uncertain dynamic or distributed systems) is desirable. It can inform modelers about the relevance of processes or parameters in the models they compile, and can inform engi- neers and decision makers about which parameters require most 0309-1708/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.advwatres.2011.11.001 ⇑ Corresponding author. E-mail address: sergey.oladyshkin@iws.uni-stuttgart.de (S. Oladyshkin). Advances in Water Resources 37 (2012) 10–22 Contents lists available at SciVerse ScienceDirect Advances in Water Resources journal homepage: www.elsevier.com/locate/advwatres