2.4 Sensitivity Indices for the Hierarchical Sensitivity Analysis Framework
The core of variance–based sensitivity analysis framework is the variance decomposition of model outputs (Saltelli et al., 1998, 1999, 2010; Saltelli and Sobol’, 1995). For a model with the form of , where is a model output and is a set of uncertain model inputs, the total variance can be decomposed as:
, (1)
In this equation, the first term on the right–hand side is the partial variance contributed by, the second term represents the partial variance caused by the model inputs except. The first–order sensitivity index thus is defined as: . This index measures the percentage of output uncertainty contributed by and estimates its relative importance compared to other uncertain inputs. This variance decomposition technique has been recursively applied by Dai and Ye (2015) and Dai et al. (2017a) to a three–layer hierarchical uncertainty framework considering three groups of uncertain inputs: scenario, model, and parameters. Following the same methodology, we derived a new sensitivity index system considering the hierarchical framework in this research.
For the uncertainty framework shown in Figure 1a, the total variance in the model outputs can be decomposed based on the greenhouse gas emission scenarios as:
, (2)
where ES is the set of multiple alternative greenhouse gas emission scenarios, GC is the set of multiple global climate models, HM is the set of multiple hydrological models, and is the parameter set for all the models with being the parameters for a given model HMk , ~ESrepresents uncertainty sources excluding ES , which areGC , HM , PA . The subscripts refer to the change in GCM samples, hydrological model and parameter combinations under certain fixed GGESs. The first and second terms on the right–hand side of Eq. (2) represent the partial variances contributed by multiple alternative GGESs and other uncertainty sources, respectively.
The partial variance caused by other uncertainty sources, can be further decomposed based on multiple plausible GCMs as:
, (3)
where the first partial variance term on the right–hand side of this equation represents the uncertainty contributed by multiple plausible climate models. The subscripts and refer to the change of climate models under one emission scenario and the change of hydrological models and parameters under one climate model and emission scenario respectively. The second term represents the within–climate model partial variance which is caused by the hydrological models and parameters.
Following the same procedure, the partial variance can be further decomposed based on multiple hydrological models as:
, (4)
where the first term on the right–hand side of Eq. (4) represents the partial variance contributed by multiple plausible hydrological models. The subscripts and refer to the change of hydrological models under one climate model and one emission scenario and the change of hydrological model parameters under one hydrological model, one climate model and one emission scenario respectively. The second term represents the within–hydrological model partial variance which is caused by the hydrological model parameters. Therefore, the total variance in the model outputs can be decomposed as:
, (5)
where V( PA ) , V( HM ) ,V( GC ) , and V( ES ) in Eq. (5), represent the variances contributed from four sources of input uncertainty: parametric uncertainty, hydrological model uncertainty, GCM uncertainty, and GGES uncertainty. Following the definition of the first–order sensitivity index, the new set of sensitivity indices for our hierarchical framework can be defined as:
, (6)
For the three–layer hierarchical uncertainty quantification framework, the total variance can be decomposed following the same process as:
, (7)
The new set of sensitivity indices can be defined as:
. (8)