We propose new metrics to assist global sensitivity analysis, GSA, of
hydrological and Earth systems. Our approach allows assessing the impact of
uncertain parameters on main features of the probability density function,
pdf, of a target model output,

Our improved understanding of physical–chemical mechanisms governing hydrological processes on multiple scales of space and time and the ever increasing power of modern computational resources are at the heart of the formulation of conceptual models which are frequently characterized by marked levels of sophistication and complexity. This is evident when one considers the spectrum of mathematical formulations and ensuing level of model parametrization rendering our conceptual understanding of given environmental scenarios (Willmann et al., 2006; Grauso et al., 2008; Koutsoyiannis, 2010; Wagener et al., 2010; Elshorbagy et al., 2010a, b; Wagener and Montanari, 2011; Hartmann et al., 2013; Herman et al., 2013; Förster et al., 2014; Paniconi and Putti, 2015). Model complexity can in turn exacerbate challenges associated with the need to quantify the way uncertainties associated with parameters of a given model propagate to target state variables.

In this context, approaches based on rigorous sensitivity analysis are valuable tools to improve our ability to (i) quantify uncertainty, (ii) enhance our understanding of the relationships between model input and outputs and (iii) tackle the challenges of model- and data-driven design of experiments. These also offer insights to guide model simplification, for example, by identifying model input parameters that have negligible effects on a target output. The variety of available sensitivity methodologies can be roughly subdivided into two broad categories, i.e., local and global approaches. Local sensitivity analyses consider the variation of a model output against variations of model input solely in the neighborhood of a given set of parameter values. Otherwise, global sensitivity analysis (GSA) quantifies model sensitivity across the complete support within which model parameters can vary. Error measurements and/or lack of knowledge about parameters can be naturally accommodated in a GSA by specifying appropriate parameter intervals and evaluating sensitivity over the complete parameter space. Recent studies and reviews on available sensitivity analysis and approaches are offered by, e.g., Pianosi et al. (2016), Sarrazin et al. (2016) and Razavi and Gupta (2015).

Our study is framed in the context of GSA methods. A broadly recognized strategy to quantify global sensitivity of uncertain model parameters to model outputs relies on the evaluation of the Sobol' indices (Sobol, 1993). These are typically referred to as variance-based sensitivity measures because the output variance is taken as the metric upon which sensitivity is quantified. A key limitation of a variance-based GSA is that the uncertainty of the output is implicitly considered to be fully characterized by its variance. Relying solely on this criterion can provide an incomplete picture of a system response to model parameters, also considering that probability densities of typical hydrological quantities can be characterized by highly skewed and tailed distributions (e.g., Borgonovo et al., 2011). Recent studies (e.g., Krykacz-Hausmann, 2001; Borgonovo, 2007; Borgonovo et al., 2011) introduced a sensitivity metric grounded on the complete probability density function, pdf, of the model output. These so-called moment-independent analyses may suffer from operational constraints because a robust evaluation of the complete pdf may require a number of model runs which is computationally unaffordable. The PAWN method developed by Pianosi and Wagener (2015) attempts to overcome this limitation introducing a sensitivity metric based on the cumulative density function, which can potentially be estimated more robustly than its associated pdf for a given sample size.

It is clear that while a variance-based GSA can be favored for its conceptual simplicity and ease of implementation, and variance can be considered in some cases as an adequate proxy of the spread around the mean, it does not yield a forthright quantification of the way variations of a parameter can affect the structure of the pdf of a target model output. Otherwise, moment-independent methodologies condense sensitivity of the entire pdf in only one index, somehow shadowing our understanding of how the structure of the pdf is affected by variations of each uncertain model parameter. Here, our distinctive objective is to contribute to bridge the gap between these two types of GSA. We do so by introducing theoretical elements and an implementation strategy which enable us to appraise parameter sensitivity through the joint use of sensitivity indices based on four (statistical) moments of the pdf of the model output: expected value, variance, skewness and kurtosis. The key idea at the basis of this strategy is that linking parameter sensitivity to multiple statistical moments leads to improved understanding of the way a given uncertain parameter can govern key features of the shape of the pdf of desired model outputs, which is of interest in modern applications of hydrological and Earth sciences.

Variance-based GSA has also been applied (a) to guide reduction of model complexity, e.g., by setting the value of a parameter which is deemed as uninfluential to the variance of a target model output (e.g., Fu et al., 2012; Chu et al., 2015; Punzo et al., 2015) and (b) in the context of uncertainty quantification (Saltelli et al., 2008; Pianosi et al., 2016; Colombo et al., 2016). Only limited attention has been devoted to assessing the relative effects of uncertain model parameters to the first four statistical moments of the target model output. This information would complement a model complexity analysis by introducing a quantification of the impact that conditioning the process on prescribed parameter values would have on the first four statistical moments of the output. Our approach is based on the joint use of multiple (statistical) moments for GSA. It enables us to address the following critical questions: when can the variance be considered as a reliable proxy for characterizing model output uncertainty? Which model parameter most affects asymmetry and/or the tailing behavior of a model output pdf? Does a given model parameter have a marked role in controlling some of the first four statistical moments of the model output, while being uninfluential to others? Addressing these questions would contribute to prioritizing our efforts to characterize model parameters that are most relevant in affecting important aspects of model prediction uncertainty.

Even as the richness of information content that a GSA grounded on the first four statistical moments might carry can be a significant added value to our system understanding, it may sometimes be challenging to obtain robust and stable evaluation of the proposed metrics for complex and computationally demanding models. This can be especially true when considering higher-order moments such as skewness and kurtosis. To overcome this difficulty, we cast the problem within a computationally tractable framework and rely on the use of surrogate models that mimic the full model response with a reduced computational burden. Amongst the diverse available techniques to construct a surrogate model (see, e.g., Razavi et al., 2012a, b), we exemplify our approach by considering the generalized polynomial chaos expansion (gPCE) that has been successfully applied to a variety of complex environmental problems (Sudret, 2008; Ciriello et al., 2013; Formaggia et al., 2013; Riva et al., 2015; Gläser et al., 2016), other model reduction techniques being fully compatible with our GSA framework. In this context, we also investigate the error associated with the evaluation of the sensitivity metrics we propose by replacing the original (full) system model with the selected surrogate model. We consider three test cases in our analysis. These include a widely employed analytical benchmark, a pumping scenario in a coastal aquifers and a laboratory-scale transport setting. The remainder of the work is organized as follows. Section 2 presents our theoretical framework and developments. Section 3 illustrates our results for the three test cases indicated above, and conclusions are drawn in Sect. 4.

We start by recalling the widely used variance-based GSA metrics in Sect. 2.1. These allow quantifying the contribution of each uncertain parameter to the total variance of a state variable of interest. We also provide a brief overview of the gPCE technique, which we use to construct a surrogate of the full system model. We then illustrate in Sect. 2.2 the theoretical developments underlying our approach and introduce novel GSA indices.

We consider a target system state variable,

We introduce new metrics to quantify the expected relative change of main
features of the pdf of

The effect of changes of

Along the same lines, we introduce the following index

We then quantify the relative expected discrepancy between unconditional,

Calculation of the indices we propose entails evaluation of conditional
moments of

The theoretical framework introduced in Sect. 2 is here applied to three diverse test beds: (a) the Ishigami function, which constitutes an analytical benchmark typically employed in GSA studies; (b) a pumping scenario in a coastal aquifer, where the state variable of interest is the critical pumping rate, i.e., the largest admissible pumping rate to ensure that the extraction well is still not contaminated by seawater; and (c) a laboratory-scale setting associated with non-reactive transport in porous media. In the first two examples the relatively low computational costs associated with the complete mathematical description of the target outputs enables us to assess the error associated with the evaluation of indices Eqs. (10), (12), (14) and (16) through a gPCE representation of the output. In the third case, due to the complexity of the problem and the associated computational costs, we rely on the gPCE representation for the target quantity of interest. We emphasize that the use of a gPCE as a surrogate model is here considered only as an example, as our GSA approach is fully compatible with any full model and/or model order reduction technique. A critical limiting factor to our and any GSA approach could be the associated computational burden. This is expected to increase according to the following two features, which are mainly associated with the conceptual and mathematical model used to describe the target variables of interest: (a) the complexity of the hydrological system (in terms of, e.g., hydrogeological heterogeneity, nonlinearity and/or transient effects) and/or (b) the number of uncertain model input parameters considered. According to the relative weight of these features, some computational constraints might arise limiting our ability to (i) perform the analysis by relying exclusively on the full system model or (ii) construct a sufficiently accurate surrogate model through a number of full model runs that can be affordable in terms of available computational resources. Application of our GSA methodology to scenarios of increased level of complexity will be the subject of a future study.

In all of the above scenarios, uncertain parameters

The nonlinear and non-monotonic Ishigami function

Variation of the first four moments of ISH Eq. (18) conditional to
values of

Comparing Eqs. (19a) and (20), it is seen that

Equation (21) shows that all random model parameters influence the variance
of ISH, albeit to different extents, as also illustrated in Fig. 1b. Note that

Global sensitivity index AMAE

The symmetry of the pdf of ISH is not affected by conditioning on

Error

The conditional kurtosis

We close this part of the study by investigating the error which would arise
when one evaluates our GSA indices by replacing ISH through a gPCE surrogate
model. We do so on the basis of the absolute relative error

Intervals of variations of

Sketch of the critical pumping scenario taking place within a
coastal aquifer of thickness

First four moments of

The example we consider here is taken from the study of Pool and Carrera (2011) related to the analysis of salt water contamination of a pumping well
operating in a homogenous confined coastal aquifer of uniform thickness

Numerical evaluation of the first four unconditional statistical moment of

Global sensitivity index AMAE

Error

Inspection of Fig. 4a reveals that the mean of

It can be noted (see Table 3) that AMAE

Figure 5 depicts error,

As a last exemplary showcase, we consider the laboratory-scale experimental analysis of nonreactive chemical transport illustrated by Esfandiar et al. (2015). These authors consider tracer transport within a rectangular flow cell filled with two types of uniform sands. These were characterized by diverse porosity and permeability values, which were measured through separate, standard laboratory tests. A sketch of the experimental setup displaying the geometry of the two uniform zones, respectively, formed by coarse and fine sand is illustrated in Fig. 6.

After establishing fully saturated steady-state flow, a solution containing a
constant tracer concentration is injected as a step input at the cell inlet.
The tracer breakthrough curve is then defined in terms of the temporal
variation of the spatial mean of the concentration detected along the flow
cell outlet. Esfandiar et al. (2015) modeled the temporal evolution of
normalized (with respect to the solute concentration of the injected fluid)
concentration at the outlet,

Sketch of the solute transport setting considered by Esfandiar et al. (2015).

Temporal evolution of the unconditional

Time evolution of the global sensitivity index

First four moments of

First four moments of

First four moments of

Figure 7 depicts the temporal evolution of the unconditional expected value,

Figure 8 depicts the temporal evolution of (a) AMAE

Inspection of the first four unconditional statistical moments of

Figure 11 shows that all four statistical moment of

Results depicted in Figs. 9–11 and our earlier observations about Fig. 7 are
consistent with the expected behavior of transport in the system and the
relative role of the dispersivities of the two sand regions. The high level
of sensitivity of

We introduce a set of new indices to be employed in the context of global
sensitivity analysis, GSA, of hydrological and Earth systems. These indices
consider the first four (statistical) moments of the probability density
function, pdf, of a desired model output,

We exemplify our methodology on three test beds: (a) the Ishigami function, which is widely employed to test sensitivity analysis techniques, (b) the evaluation of the critical pumping rate to avoid salinization of a pumping well in a coastal aquifer and (c) a laboratory-scale nonreactive transport experiment. Our theoretical analyses and application examples lead to the following major conclusions.

The calculated sensitivity of a model output,

Joint inspection of our moment-based global sensitivity indices and of the
first four statistical conditional and unconditional moments of

Analysis of the errors associated with the use of a surrogate model for the
evaluation of our moment-based sensitivity indices suggests that (a) attaining a given level of accuracy for the gPCE-based indices associated
with a target variable,

All data used in the paper will be retained by the authors for at least 5 years after publication and will be available to the readers upon request.

The authors declare that they have no conflict of interest.

Funding from the Italian Ministry of Education, University and Research, Water JPI, WaterWorks 2014, project: WE-NEED (WatEr NEEDs, availability, quality and sustainability) and from the European Union's Horizon 2020 Research and Innovation program (project: Furthering the knowledge Base for Reducing the Environmental Footprint of Shale Gas Development – FRACRISK, grant agreement 636811) is acknowledged. Edited by: Bill Hu Reviewed by: Holger Class and Jacob Bensabat