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)