Variogram Analysis of Response Surfaces (VARS)


The Watershed Systems Analysis and Modelling Lab is home to the VARS sensitivity and uncertainty analysis framework. VARS is applicable to the full range of computer simulation models, including Earth and Environmental Systems Models (EESMs). VARS is under continuous development and new capabilities and features are forthcoming.


FEATURED: Saman Razavi Explaining Underlying Theory behind VARS


Global Sensitivity Analysis (GSA)

Global sensitivity analysis is a systems theoretic approach to characterizing the overall (average) sensitivity of one or more model responses across the factor space, by attributing the variability of those responses to different controlling (but uncertain) factors (e.g., model parameters, forcings, and boundary and initial conditions).

What is the VARS approach to GSA?

VARS, which stands for Variogram Analysis of Response Surfaces, is a general framework for GSA that utilizes directional variogram and covariogram functions to characterize the full spectrum of sensitivity-related information, thereby providing a comprehensive set of “global” sensitivity metrics with minimal computational cost.

What was the motivation for the development of VARS?

VARS was developed to address two major issues with GSA:

  • Ambiguous Definition of “Global” Sensitivity: different GSA methods are based in different philosophies and theoretical definitions of sensitivity, leading to different, even conflicting, assessments of the underlying sensitivities for a given problem.
  • Computational Cost: the cost of carrying out GSA can be large, even excessive, for high-dimensional problems and/or computationally intensive models, where cost (or “efficiency”) is commonly assessed by of the number of required model runs.

What are the special features of VARS?

  • VARS re-defines GSA by characterizing a comprehensive spectrum of information about the underlying sensitivities of a response surface to its factors, while reducing to well-known and commonly used approaches to GSA as special/limiting cases.
  • VARS generates a new set of sensitivity metrics called IVARS (Integrated Variogram Across a Range of Scales) that summarize the variance of change (rate of variability) in model response at a range of perturbation scales in the factor space.
  • VARS also generates the Sobol (variance-based) total-order effect, the most popular metric for GSA, and the Morris (derivative-based) elementary effects across the full range of step sizes in numerical differencing (theoretical relationship exists).
  • VARS is highly efficient and statistically robust, providing stable results within 1-2 orders of magnitude smaller numbers of sampled points (model runs), compared with alternative GSA approaches, such as the Sobol and Morris approaches.
  • VARS effectively and efficiently handles high-dimensional problems, because of its computational efficiency, which is, in part, due to VARS being based on the information contained in pairs of points, rather than in individual points.
  • VARS is unique in that it characterizes different sensitivity-related properties of response surfaces including local sensitivities and their global distribution, the global distribution of model responses, and the structure of the response surface.
  • VARS tackles the scale issue of sensitivity analysis by providing sensitivity information spanning a range of scales across the factor space, from small-scale features such as roughness/noise to large-scale features such as multi-modality.


  1. Razavi, S., and Gupta, H. V., A new framework for comprehensive, robust, and efficient global sensitivity analysis: II. Application, Water Resources Research, 51 (2016) doi:10.1002/2015WR017559.
  2. Razavi, S., and Gupta, H. V., A new framework for comprehensive, robust, and efficient global sensitivity analysis: I. Theory, Water Resources Research, 51 (2016) doi:10.1002/2015WR017558.
  3. Razavi, S., and H. V. Gupta, What do we mean by sensitivity analysis? The need for comprehensive characterization of “global” sensitivity in Earth and Environmental systems models, Water Resources Research, 51, 3070–3092 (2015) doi:10.1002/2014WR016527.