Optimising the convergence of a Sobol’ sensitivity analysis for an environmental model: application of an appropriate estimate for the square of the expectation value and the total variance

Over-parameterisation is a key issue in (integrated) environmental modelling. Therefore, a sensitivity analysis (SA) can assist in the proper application of a complex environmental model, such as the Soil and Water Assessment Tool (SWAT). A very powerful sensitivity analysis technique that is gaining popularity in environmental modelling is the variance-based Sobol’ method. Still, the high number of model evaluations necessary to perform the analysis is a major restriction for the method’s use and, as a consequence, an optimised convergence is of the utmost importance. Therefore, this paper presents a study of the influence of some computational issues on the convergence of the sensitivity measures of the Sobol’ SA, i.e. the sensitivity indices. The latter indices are assessed by means of Monte Carlo integrals. These numerical integrals are also used for the estimation of the square of the expectation value and the total variance of the model output, which are important for the computation of the sensitivity indices. The paper investigates the impact of the use of different formulas for the calculation of the integrals of these statistics. We will also show that the choice of the calculation method highly affects the convergence of the sensitivity indices. Finally, we will demonstrate that the convergence of the first order sensitivity indices is mainly determined by the formula used for the estimation of the square of the expectation value, while the convergence of the total sensitivity indices is mainly affected by the equation used for the estimation of the total variance.