Introduction
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.
Theoretical framework
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.
Sobol' indices for variance-based GSA and generalized polynomial chaos expansion
We consider a target system state variable, y, which depends on N random
parameters. These are collected in vector x= (x1, x2,
…, xN) and defined in the parameter space Γ=Γx1×Γx2×…ΓxN, Γxi=[xi,min,xi,max] being the support of the ith random variable xi.
Variance-based GSA approaches consider variance as the sole metric to
quantify the contribution of each uncertain parameter to the uncertainty of
y. Iman and Hora (1990) introduce the following index
HIxi=V[y]-EV[y|xi]=VE[y|xi],
where E[-] and V[-], respectively, denote expectation and variance operators. Index
HIxi quantifies the expected reduction of variance due to
knowledge of xi (the notation |xi in Eq. (1) indicates
conditioning on xi). A similar measure is offered by the widely used
Sobol' indices (Sobol, 1993). These have been defined starting from the
Hoeffding–Sobol' decomposition (see, e.g., Sobol, 1993; Le Maître and
Knio, 2010) of y(x) when x is a collection of independent random variables such as
yx=y0+∑xi=1Nyxi(xi)+∑xi<xjyxi,xj(xi,xj)+…+yx1,x2,…,xN(x1,x2,…,xN),
where
y0=∫Γy(x)ρΓxdx,yxi(xi)=∫Γ∼xiy(x)ρΓ∼xidx∼xi-y0,yxi,xj(xi,xj)==∫Γ∼xi,xjy(x)ρΓ∼xi,xjdx∼xi,xj-yxi(xi)-yxj(xj)-y0,
and so on, ρΓx
being the pdf of x. The
integral ∫Γ∼xiy(x)ρΓ∼xidx∼xi in Eq. (3) represents integration of
y(x) over the space of all entries of vector x excluding xi,
ρΓ∼xi being the corresponding pdf. The Sobol' index
Sxi1,xi2,…,xis is associated with
the mixed effect of xi1,…,xis on the
variance of y(x), V[y], and can be computed as
Sxi1,…,xis==1Vy∫Γxi1,…,xisyxi1,…,xis(xi1,…,xis)ρΓxi1,…xisdxi1…dxis.
The principal and total Sobol' indices are, respectively, defined as
Sxi=1Vy∫Γxiyxi(xi)2ρΓxidxi,SxiT=Sxi+∑xjSxi,xj+∑xj,xkSxi,xj,xk+….
Note that Sxi describes the relative contribution to Vy due to variability of only xi. Otherwise, SxiT
quantifies the total contribution of xi to Vy, including
all terms where xi appears. In other words, SxiT also
includes interactions between xi and the remaining uncertain parameters,
collected in vector x∼xi. Note that according to Eqs. (1), (2)
and (5)
Sxi=VE[y|xi]Vy=HIxiVy,
i.e., the principal Sobol' index represents the relative expected reduction
of process variance due to knowledge of (or conditioning on) a parameter.
Sobol' indices are commonly evaluated via Monte Carlo quadrature schemes
that can be markedly demanding in terms of computational time, especially
for complex and highly nonlinear settings. Relying on a gPCE as a surrogate of the full mathematical
model of the system (Ghanem and Spanos, 1991; Xiu and Karniadakis, 2002; Le
Maitre and Knio, 2010; Formaggia et al., 2013; Ciriello et al., 2013; Riva
et al., 2015) allows for reducing the computational burden associated with GSA
techniques. The process y(x) is represented as a linear combination of
multivariate polynomials, ψp(x), i.e.,
y(x)≅β0+∑i=1N∑p∈ℑiβpψp(x)+∑i=1N∑j=1N∑p∈ℑi,jβpψp(x)+…,ψp(x)=∏i=1Nψi,pi(xi),βp=∫Γy(x)ψp(x)ρΓxdx,
where p=p1,…,pN∈NN is a
multi-index expressing the degree of each univariate polynomial, ψi,pi(xi); βp are the gPCE coefficients;
ℑi contains all indices such that only the ith component does not
vanish; ℑi,j contains all indices such that only the ith and jth
components are not zero, and so on. Note that β0≡y0; i.e., β0 is the unconditional mean of y(x). Finally, the Sobol'
indices Eqs. (4)–(5) and the variance of y(x) can be computed from Eq. (8) as
Sxi1,…,xis=1Vy∑p∈ℑi1,…,isβp2,Sxi=1Vy∑p∈ℑiβp2,Vy=∑p∈NNβp2-β02.
New metrics for multiple-moment GSA
We introduce new metrics to quantify the expected relative change of main
features of the pdf of y due to variability of model input parameters. In
contrast with traditional variance-based GSA techniques of the kind
described in Sect. 2.1, we quantify changes in the pdf of y through its first
four statistical moments, i.e., mean, E[y], variance, V[y], skewness,
γ[y], and kurtosis, k[y]. The latter is an indicator of the
behavior of the tails of the pdf of y and is particularly useful in the context
of risk analysis, while γ[y] quantifies the asymmetry of the pdf of y.
The effect of changes of x on the mean of y cannot be systematically analyzed
by the metrics currently available in the literature. We therefore introduce
the following quantity
AMAExi==1y0∫Γxiy0-E[y|xi]ρΓxidxi=1y0Ey0-E[y|xi]ify0≠0∫ΓxiE[y|xi]ρΓxidxi=EE[y|xi]ify0=0,
y0 being defined in Eq. (3). Extension of Eq. (10) to consider the
joint effect of xi1,xi2,…,xis on the mean of y is
straightforward, leading to the following index
AMAExi1,…,xis==1y0∫Γxi1,…,xisy0-E[y|xi1,…,xis]ρΓxi1,…,xisdxi1…dxis=1y0Ey0-E[y|xi1,…,xis]ify0≠0.∫Γxi1,…,xisE[y|xi1,…,xis]ρΓxi1,…,xisdxi1…dxis=EE[y|xi1,…,xis]ify0=0.
Note that index AMAExi quantifies the expected relative variation
of the mean of y due to variations of only xi, while AMAExi1,…,xis also includes all interactions amongst parameters
xi1,xi2,…,xis.
Along the same lines, we introduce the following index
AMAVxi=1Vy∫ΓxiVy-V[y|xi]ρΓxidxi==EVy-Vy|xiVy,
quantifying the relative expected discrepancy between unconditional and
conditional (on xi) process variance. Note that Eq. (12) does not
generally coincide with the principal Sobol' index Sxi in Eq. (2)
that quantifies the expected relative reduction of the variance due to
knowledge of xi (or, in other words, the relative contribution to the
variance arising from uncertainty in xi). Index AMAVxi reduces to Sxi only if the conditional variance, V[y|xi], is always (i.e., for each value of xi) smaller than (or equal to)
its unconditional counterpart Vy. The difference between
AMAVxi and Sxi, as well as advantages of using
AMAVxi, will be elucidated through the numerical examples
illustrated in Sect. 3. Extension of Eq. (12) to consider the joint effect
of xi1,xi2,…,xis reads
AMAVxi1…,xis==1Vy∫Γxi1…,xisVy-V[y|xi1,…,xis]ρΓxi1,…,xisdxi1…dxis=1VyEVy-Vy|xi1,…,xis.
Index AMAVxi1,…,xis quantifies the expected
relative discrepancy between Vy and the variance of the
process conditional to joint knowledge of xi1,xi2,…,xis.
We then quantify the relative expected discrepancy between unconditional,
γ[y], and conditional, γ[y|xi], skewness through
the index
AMAγxi==1γy∫Γxiγy-γ[y|xi]ρΓxidxi=1γyEγy-γ[y|xi]ifγy≠0∫Γxiγ[y|xi]ρΓxidxi=Eγ[y|xi]ifγy=0.
Extension of Eq. (14) to consider the joint effect of xi1,xi2,…,xis gives
AMAγxi1,…,xis==1γy∫Γxi1,…,xisγy-γ[y|xi1,…,xis]ρΓxi1,…,xisdxi1…dxis==1γyEγy-γy|xi1,…,xisifγy≠0∫Γxi1,…,xisγ[y|xi1,…,xis]ρΓxi1,…,xisdxi1…dxis=EE[y|xi1,…,xis]ifγy=0.
The relative variation of the kurtosis of y due to variations of a parameter
xi or of the parameter set xi1,xi2,…,xis
can be, respectively, quantified through
AMAkxi=1ky∫Γxiky-k[y|xi]ρΓxidxi==1kyEky-ky|xi,
AMAkxi1…,xis==1ky∫Γxi1,…,xisky-k[y|xi1,…,xis]ρΓxi1,…,xisdxi1…dxis==1kyEky-ky|xi1,…,xis.
Relying jointly on Eqs. (10)–(17) enables one to perform a comprehensive GSA
of the target process y(x) quantifying the impact of x on the first
four (statistical) moments of the pdf of y(x). This strategy yields
information about the way important elements of the distribution of
y(x), such as mean, spread around the mean, symmetry and
tailedness, are affected by uncertain model parameters collected in the
parameter vector x. This analysis is not feasible through a classical
variance-based GSA.
Calculation of the indices we propose entails evaluation of conditional
moments of y(x). This step can be computationally very demanding.
Along the lines of our discussion about Sobol' indices in Sect. 2.1, the
new metrics Eqs. (10)–(17) can be evaluated via a surrogate model, as we
illustrate through our examples in Sect. 3.
Illustrative examples
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 xi collected in
x are considered as independent and identically distributed, i.i.d., random
variables, each characterized by a uniform distribution within the interval
Γxi=xi,min,xi,max. Note that varying
the pdf of the uncertain model input parameters does not impact the definition
of the GSA indices proposed in Sect. 2. Otherwise, it may affect the
actual results, depending on the test case considered. All results are
grounded on 5×105 Monte Carlo realizations, enabling convergence
of all statistical moments analyzed. Series appearing in the gPCE Eq. (8)
are evaluated up to a given order of truncation in all three examples. Here,
we apply the total-degree rule and construct a polynomial of order w through
a sparse grid technique (see, e.g., Formaggia et al., 2013, and references
therein). We then analyze the way the selected order w influences the
results. Note that the optimal choice of the polynomial ψp(x) in Eq. (8) depends on the pdf of the random variables collected
in x (Xiu and Karniadakis, 2002). In our exemplary settings we use the
multidimensional Legendre polynomials which are orthonormal with respect to
the uniform pdf.
Ishigami function
The nonlinear and non-monotonic Ishigami function
yx=ISHx=sin(2πx1-π)+asin2(2πx2-π)+b2πx3-π4sin(2πx1-π)
is widely used in the literature (e.g., Homma and Saltelli, 1996; Chun et
al., 2000; Borgonovo, 2007; Sudret, 2008; Crestaux et al., 2009; Borgonovo
et al. 2011) to benchmark GSA methods. Here, xi (i= 1, 2, 3) are
i.i.d. random variables uniformly distributed within the interval [0, 1].
Unconditional mean EISH, variance, VISH, skewness, γISH, and kurtosis, kISH, of Eq. (18) can be evaluated analytically as
EISH=a2,VISH=12+a28+bπ415+bπ418,γISH=0,
kISH=12V2ISH34+bπ435+bπ412+3bπ4113+π4b68+32a212+a232+π4b15+π4b18.
Equation (19) reveals that the unconditional pdf of ISH is symmetric with tails
that increase with |b| and decrease with |a|,
as quantified by kISH. The conditional mean EISH|xi, variance VISH|xi,
skewness γISH|xi and kurtosis kISH|xi can be evaluated analytically as
EISH|x1=a2-155+bπ4sin2πx1,EISH|x2=asin22πx2,EISH|x3=a2,
VISH|x1=a28+8b2π82251-cos4πx1,VISH|x2=12+bπ415+b18π4,VISH|x3=a28+121+bπ41-2x342,γISH|x1=-128b3π12sin32πx14875VISH|x13/2,γISH|x2=0,ISHy|x3=0,kISH|x1=1V2ISH|x13128a4+475b2π8sin22πx1a2+18495525b2π8sin22πx1,kISH|x2=12V2ISH|x234+bπ435+bπ412+3bπ4113+168bπ4,kISH|x3=3128V2ISH|x3a4+161+bπ41-2x342a2+1+bπ41-2x342.
For the sole purpose of illustrating our approach, here and in the following
we set a= 5 and b= 0.1, which corresponds to EISH=2.50,
VISH=10.84 and kISH=4.18. Figure 1
depicts the first four moments of ISH conditional to values of x1 (blue
curves), x2 (red curves) and x3 (green curves) within the
parameter space. The corresponding unconditional moments (black curves) are
also depicted for completeness.
Variation of the first four moments of ISH Eq. (18) conditional to
values of x1 (blue curves), x2 (red curves) and x3 (green
curves) within the parameter space: (a) expected value, EISH|xi, (b) variance, VISH|xi,
(c) skewness, γISH|xi, and (d) kurtosis,
kISH|xi, (i= 1, 2, 3). The corresponding
unconditional moments (black curves) are also depicted.
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Comparing Eqs. (19a) and (20), it is seen that EISH|x3 coincides with its unconditional counterpart EISH, indicating that conditioning on any value of x3 does not
impact the mean of ISH. Otherwise, setting x1 or x2 to a given
value clearly affects the mean of ISH in a way which is governed by Eq. (20)
and shown in Fig. 1a. It is clear from Eq. (20) that EISH|x2 has a higher frequency of oscillation within Γx2 than EISH|x1 has within Γx1. The global index in Eq. (10) can be evaluated analytically as
AMAEx1=4aπ1+b5π4,AMAEx2=2πaa,AMAEx3=0.
Note that AMAEx2 does not depend on specific values of a and b.
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
VISH|x2 is always smaller than VISH (compare Eqs. 19a and 21) and does not depend on x2;
i.e., conditioning ISH on x2 reduces the process variance regardless
the conditioning value. Otherwise, VISH|x3 can
be significantly larger or smaller than its unconditional counterpart. Table 1 lists values of AMAVxi (xi=x1, x2,
x3) computed via Eq. (12) with the a and b values selected for our
demonstration. The principal Sobol' indices (Sudret, 2008)
Sx1=5+bπ4250VISH,Sx2=a28VISH,Sx3=0,
are also listed for completeness. As expected, values of AMAVxi listed in Table 1 suggest that conditioning on x3 has the strongest
impact on the variance of ISH, followed by x1 and x2. Note that
Sx3= 0, a result which might be interpreted as a symptom that
ISH is insensitive to x3. The apparent inconsistency between the
conclusions which could be drawn by analyzing AMAVx3 and Sx3 is reconciled by the observation that the function VISH-VISH|x3 can be positive and negative in a
way that its integration over Γx3 vanishes (see also Fig. 1b). Therefore, the mean reduction of the variance of ISH due to knowledge of
(or conditioning on) x3 is zero. We note that this observation
does not imply that the variance of ISH does not vary with x3, as clearly
highlighted by Fig. 1b and quantified by AMAVx3.
Global sensitivity index AMAExi Eq. (10), AMAVxi Eq. (12), AMAγxi Eq. (14) and AMAkxi Eq. (16)
associated with the Ishigami function Eq. (18). Principal Sobol' indices,
Sxi Eq. (7), are also listed; xi=x1, x2,
x3.
AMAExi
AMAVxi
Sxi
AMAγxi
AMAkxi
x1
0.75
0.40
0.40
0.45
0.37
x2
0.64
0.29
0.29
0.00
0.33
x3
0.00
0.84
0.00
0.00
0.53
The symmetry of the pdf of ISH is not affected by conditioning on x2 or
x3, as demonstrated by Eq. (22). Otherwise, γISH|x1 is left (or right) skewed when x1 is smaller (or
larger) than 0.5, as dictated by Eq. (22) and shown in Fig. 1c.
Error ej Eq. (26) versus the total degree w of the gPCE
representation of ISH for j= (a) AMAExi, (b) AMAVxi,
(c) AMAγxi and (d) AMAkxi, with xi=x1
(blue curves), x2 (red curves), x3 (green curves). Note
that AMAEx3 is always smaller than 0.001 %. Average slopes of the
rate of decrease of ej for the largest w values considered are
indicated as a reference in (b–d).
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The conditional kurtosis kISH|x2 does not depend
on the conditioning value x2 (see Eq. 23). We then note that this
conditional moment is always larger than (or equal to) its unconditional
counterpart kISH, regardless of the particular values
assigned to a and b, as we verified through extensive numerical tests. This
result implies that the pdf of ISH conditional on x2 is characterized by
tails which are heavier than those of its unconditional counterpart. Figure 1d reveals that kISH|x1 and kISH|x3 are smaller than kISH for the values of
a and b implemented in this example. Table 1 lists the resulting values of
AMAkxi (xi=x1, x2, x3) for the
selected a and b values.
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
ej=jgPCE-jfullmodeljfullmodelifjfullmodel≠0jgPCE-jfullmodelifjfullmodel=0,
where j=AMAExi, AMAVxi, AMAγxi or
AMAkxi (xi=x1, x2, x3); the subscripts full model and gPCE, respectively, indicate that quantity j is
evaluated via Eq. (18) or through a gPCE surrogate model, constructed as
outlined in Sect. 2.1. Figure 2 depicts Eq. (26) versus the total degree
w of the gPCE. Note that the lower limit of the vertical axis of Fig. 2 is
set to 0.001 % for convenience of graphical representation. Approximation
errors associated with GSA indices related to the mean, AMAExi,
rapidly approach zero as w increases. Note that eAMAEx3 is
smaller than 0.001 % for all tested values of w and it is therefore not
included in Fig. 2a. Values of ej linked to AMAVxi,
AMAγxi and AMAkxi do not show a
consistently decreasing trend until w ≥ 5. Values of
ej associated with the variance, skewness and kurtosis decrease with
approximately the same average linear rate (in log–log scale) for the
largest w considered (Fig. 2b, c and d). This example reinforces the need
for reliably testing the accuracy of a gPCE-based model approximation as a
function of the total degree desired, depending on the statistical moment of
interest. Note that a generalization of our findings about the error (Eq. 26) is
outside the scope of the current study. This would require the derivation of
(a) the analytical format of the pdf of a target model output through its gPCE-based approximation at a given order w (see, e.g., Riva et al., 2015) and
(b) the corresponding pdf resulting from the full system model (e.g., by
formulating and solving exact equations for the target pdf, or its moments,
typically invoking problem-specific assumptions).
Intervals of variations of PeT, J, xw.
Γn=[xn,min-xn,max]
ΓPeT
[0.01–0.1]
ΓJ
[8 × 10-4-2.5 × 10-3]
Γxw
[10–33]
Sketch of the critical pumping scenario taking place within a
coastal aquifer of thickness b′. A constant freshwater (in blue) flux,
qf′, flows from the inland to the coastline (saltwater in red). A
constant discharge, Qw′, is pumped from a fully penetrating well
located at a distance xw′ from the coastline. Color scale
indicating variable concentration of salt is only qualitative for
illustration purposes.
![]()
First four moments of Qc Eq. (27) conditional to values of
PeT (blue curves), J (green curves) and xw (red curves)
within the parameter space: (a) expected value, EQc|xi, (b) variance, VQc|xi,
(c) skewness, γQc|xi, and (d) kurtosis,
kQc|xi, (xi=PeT, J, xw). The corresponding unconditional moments (black curves) are also
depicted. Intervals of variation of PeT, J and xw have been
rescaled between 0 and 1 for graphical representation purposes.
![]()
Critical pumping rate in coastal aquifers
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
b′. The setting is sketched in Fig. 3. A constant discharge, Qw′ [L3 T-1], is pumped from a fully penetrating well located at a
distance xw′ [L] from the coastline, and a constant freshwater flux,
qf′ [L T-1], flowing from the inland to the coastline, is set.
Pool and Carrera (2011) introduced a dimensionless well discharge Qw=Qw′/(b′xw′qf′) and defined the critical pumping
rate Qc as the value of Qw at which a normalized solute
concentration monitored at the well exceeds 0.1 %. A key result of the
study of Pool and Carrera (2011) is that Qc can be approximated
through the following implicit equation:
λD=21-Qcπ1/2+Qcπln1-1-Qc/π1/21+1-Qc/π1/2withλD=Δρ′ρf′1-PeT-1/6xwJ.
Here, xw=xw′/b′; J=qf′/K; PeT=b′/αT′; K [L T-1] is the uniform hydraulic conductivity; αT′ [L] is transverse dispersivity; Δρ′=ρs′-ρf′; ρf′ and ρs′ are
fresh and saltwater densities, respectively. The quantity PeT is a
measure of the intensity of dispersive effects, J is the natural head
gradient of the incoming freshwater and xw is the dimensionless
distance of the well from the coastline. Pool and Carrera (2011)
demonstrated the accuracy of Eq. (27) in predicting the critical pumping
rate when λD∈[0-10]. Additional details about the problem
setting, boundary and initial conditions, as well as geometrical
configuration of the system, can be found in Pool and Carrera (2011). Here,
we focus on the main result of Eq. (27) which represents the complete
mathematical description of the problem we analyze. We perform a sensitivity
analysis of Qc with respect to PeT, J and xw. While the
first two quantities are difficult to assess experimentally in practical
applications, the well location can be considered as an operational/design
variable. Table 2 lists the intervals of variation we consider for PeT, J and xw. These are designed to (a) resemble realistic field values
and (b) obey the above-mentioned constraint about λD.
Numerical evaluation of the first four unconditional statistical moment of
Qc yields a mean value EQc=1.65, variance
VQc=0.17, skewness γQc=-0.30 (which indicates a light asymmetry in the pdf) and kurtosis
kQc=2.51 (i.e., pdf tails decrease faster than those of a
Gaussian distribution). Figure 4 depicts the first four moments of Qc
conditional to values of PeT (blue curves), J (green curves) and
xw (red curves) within the parameter space. The corresponding
unconditional moments (black curves) are also depicted for completeness.
Note that each parameter interval of variation has been normalized to span
the range [0, 1] for graphical representation purposes. Table 3 lists the
values of indices AMAExi AMAVxi, Sxi, AMAγxi and AMAkxi (xi=PeT,
J, xw) associated with Qc. As in our first example, it is clear
that sensitivity of Qcwith respect to PeT, J, xw depends on
the statistical moment of interest.
Global sensitivity index AMAExi Eq. (10), AMAVxi Eq. (12), AMAγxi Eq. (14) and AMAkxi Eq. (16)
associated with the critical pumping rate Qc Eq. (27). Principal Sobol'
indices, Sxi Eq. (7), are also listed; xi=PeT,
J, xw.
AMAExi
AMAVxi
Sxi
AMAγxi
AMAkxi
PeT
0.07
0.14
0.09
0.35
0.09
J
0.14
0.41
0.41
0.88
0.12
xw
0.15
0.48
0.48
0.78
0.11
Error ej Eq. (26) versus total degree w of the gPCE
representation of Qc, for j= (a) AMAExi,
(b) AMAVxi, (c) AMAγxi and (d) AMAkxi; xi=PeT (blue curves), J (red curves), xw (green
curves).
![]()
Inspection of Fig. 4a reveals that the mean of Qc is more sensitive to
conditioning on J or xw than to conditioning on PeT. Note that
increasing PeT, i.e., considering advection-dominated scenarios, leads
to an increase of the mean value of Qc. This is so because the
dispersion of the intruding saltwater wedge is diminished and the travel
time of solutes to the well tends to increase. High values of the natural
head gradient of the incoming freshwater, J, are associated with high mean
values of Qc. This is consistent with the observation that the inland
penetration of the wedge is contrasted by the effect of freshwater which
flows in the opposite direction. As expected, decreasing xw (moving
the pumping well towards the coast) leads to a reduction of the mean value
of Qc. Figure 4a shows that mean Qc varies with xw and J in
a similar way. This outcome is consistent with Eq. (27) where Qc
depends on the product xwJ, i.e., increasing xw or J has the same
effect on Qc.
It can be noted (see Table 3) that AMAEPeT is smaller than
AMAEJ and AMAExw, consistent with Fig. 4a. Figure 4b shows
that the variance of Qc decreases as PeT, J or xw increase.
This trend suggests that the uncertainty of Qc, as quantified by the
variance, decreases as (i) the intruding wedge sharpens or is pushed toward
the seaside boundary by the incoming freshwater or (ii) the well is placed at
increasing distance from the coastline. Inspection of Fig. 4c–d shows
that conditioning on PeT, J or xw causes the pdf of Qc to
become less asymmetric and less tailed than its unconditional counterpart.
This behavior suggests that the relative frequency of occurrence of (high or
low) extreme values of Qc tends to decrease as additional information
about the model parameters become available.
Figure 5 depicts error, ej (Eq. (26)), versus total degree, w, of the
gPCE representation of Qc, for j= (a) AMAExi, (b) AMAVxi, (c) AMAγxi and (d) AMAkxi
(xi=PeT, blue curves; J, red curves; xw,
green curves). These results indicate that (i) ej associated with
AMAExi is negligible (≈ 1 %) even for low w; (ii) eAMAVPeT≈ 10 % for w= 2 and rapidly decreases to
values below 1 % for increasing w; (iii) eAMAVJ and eAMAVxw are always smaller than 1 %; and (iv) the trend of eAMAγxi is similar to that of eAMAkxi
for all xi, with values of the order of 10 % or higher for w= 2
and displaying a decrease with increasing w which then stabilizes around values
smaller than 1 % when w≈ 4 or 5. We note that the absolute
relative error (Eq. 26) for AMAExi with a given value of w is
always lower than errors associated with higher-order moments. Similar to
our results in Sect. 3.1, it is clear from Fig. 5 that attaining a given
level of accuracy for the gPCE-based indices for Qc requires
considering a diverse total order w of the gPCE depending on the order of the
statistical moment considered. As such, following the typical practice of
assessing the reliability of a gPCE surrogate model solely on the basis of
the variance or of a few random model realizations does not guarantee
satisfactory accuracy of the uncertainty analysis of a target model output,
which requires considering higher-order statistical moments.
Solute transport in a laboratory-scale porous medium with zoned
heterogeneity
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, C‾(t) (t denoting time), by
numerically solving within the flow domain the classical
advection–dispersion equation implementing an original and accurate
space–time grid adaptation technique. Unknown longitudinal dispersivities of
the two sands (aL,i, i= 1, 2, respectively, denoting the coarse and
fine sand) were considered as uncertain system parameters to be estimated
against the available experimental solute breakthrough data. To minimize the
computational costs in the model calibration process, Esfandiar et al. (2015) relied on a gPCE approximation of C‾(t). The authors
constructed a gPCE of total degree w= 3 by considering log10aL,i to be two i.i.d. random variables uniformly distributed
within Γlog10aL,i= [-6, -2],
aL,i being expressed in [m]. Further details about the problem setup,
numerical discretization and grid adaptation technique as well of the
construction of the gPCE representation can be found in Esfandiar et al. (2015). Here, we ground the application of our new GSA metrics on the gPCE
surrogate model already constructed by Esfandiar et al. (2015) to
approximate C‾(t).
Sketch of the solute transport setting considered by Esfandiar et
al. (2015).
![]()
Temporal evolution of the unconditional (a) expected value,
EC‾t, (b) variance, VC‾t, (c) skewness, γC‾(t), and (d) kurtosis, kC‾(t), of normalized C‾(t). Vertical lines in (a) correspond
to time steps t0.4, t0.02 and t0.96, i.e., the times at
which EC‾t= 0.02, 0.4 and
0.96, respectively.
![]()
Time evolution of the global sensitivity index (a) AMAExi, (b) AMAVxi and Sxi (dashed curves),
(c) AMAγxi and (d) AMAkxi of C‾(t) (xi=log10(aL,1) (blue) or log10(aL,2)
(red)).
![]()
First four moments of C‾(t=t0.02) conditional on
log10(aL,1) (blue curves) and log10(aL,2) (red
curves), at time t=t0.02: (a) expected value, EC‾(t0.02)log10aL,i, (b) variance, VC‾(t0.02)log10aL,i, (c) skewness, γC‾(t0.02)log10aL,i, and (d) kurtosis, kC‾(t0.02)log10aL,i (i= 1, 2). The
corresponding unconditional moments are also depicted (black curves).
![]()
First four moments of C‾(t=t0.4) conditional on
log10(aL,1) (blue curves) and log10(aL,2) (red
curves), at time t=t0.4: (a) expected value, EC‾(t0.4)log10aL,i, (b) variance, VC‾(t0.4)log10aL,i, (c) skewness, γC‾(t0.4)log10aL,i, and (d) kurtosis, kC‾(t0.4)log10aL,i (i= 1, 2). The
corresponding unconditional moments are also depicted (black curves).
![]()
First four moments of C‾(t=t0.96) conditional on
log10(aL,1) (blue curves) and log10(aL,2) (red
curves), at time t=t0.96: (a) expected value, EC‾(t0.96)log10aL,i, (b) variance, VC‾(t0.96)log10aL,i, (c) skewness, γC‾(t0.96)log10aL,i, and (d) kurtosis, kC‾(t0.96)log10aL,i (i= 1, 2). The
corresponding unconditional moments are also depicted (black curves).
![]()
Figure 7 depicts the temporal evolution of the unconditional expected value,
EC‾t, variance, VC‾t, skewness, γC‾t, and kurtosis, kC‾(t), of normalized C‾(t). Time steps t0.02, t0.4, and t0.96, i.e., the times at which EC‾t= 0.02, 0.4 and 0.96, respectively, are highlighted in
Fig. 7a. Figure 7a reveals a pronounced tailing of EC‾t at late times, the short time mean breakthrough
being associated with a rapid temporal increase of EC‾t. A local minimum at t0.4 and two local
peaks are recognized in VC‾t
(Fig. 7b). The variance peaks at times approximately corresponding to the
largest values of ∂2EC‾t/∂t2. This outcome is consistent with the results of
numerical Monte Carlo (MC) simulations depicted in Fig. 8 of Esfandiar et
al. (2015) where the largest spread of the MC results is observed around
these locations. The local minimum displayed by VC‾t suggests that C‾(t) at observation times
close to t0.4 is mainly driven by advection, consistent with the
observation that advective transport components are the main driver of the
displacement of the center of mass of a solute plume. The late time variance
VC‾t tends to vanish because the
normalized breakthrough curve is always very close to unity irrespective of
the values of aL,1 and aL,2. Joint inspection of Fig. 7c
and d reveals that the pdf of C‾(t) tends to be symmetric around the mean
(Fig. 7c) and characterized by light tails (Fig. 7d) at about t0.4.
Otherwise, the pdfs of C‾(t) tend to display heavy right or left
tails, respectively, for observation times shorter or longer than t0.4.
These observations suggest that the relative frequency of rare events (i.e.,
very low or high solute concentrations, which can be of some concern in the
context of risk assessment) is lowest at intermediate observation times
across the duration of the experiment.
Figure 8 depicts the temporal evolution of (a) AMAExi, (b) AMAVxi,
(c) AMAγxi and (d) AMAkxi (xi=log10(aL,1), log10(aL,2)) of
C‾(t). Results shown in Fig. 8 demonstrate that statistical moments
of C‾(t) are more sensitive to log10(aL,1) than to
log10(aL,2) at early times. The opposite occurs when t > t0.4. Our set of results suggests that the overall early
time pattern of solute breakthrough is mainly dictated by the value of
aL,1; the late time behavior is chiefly influenced by aL,2.
These conclusions are supported by the results of Figs. 9–11, where we
depict the expected value, variance, skewness and kurtosis of C‾(t) conditional to log10(aL,1) (blue curves) and log10(aL,2) (red curves), at times t=t0.02 (Fig. 9), t0.4
(Fig. 10) and t0.96 (Fig. 11). The corresponding unconditional
moments are also depicted (black curves) for ease of comparison. Figure 9
shows that the first four statistical moments of C‾t0.02 are practically insensitive to the value of the fine
sand dispersivity, aL,2. As one could expect by considering the
relative size and geometrical pattern of the two sand zones, Fig. 9a shows
that the average amount of solute reaching the cell outlet at early times
increases with aL,1, because dispersion of solute increases through
the coarse sand which resides in the largest portion of the domain. Figure 9b shows that VC‾(t0.02) is negligible when
aL,1 is known. Consistent with this result, Fig. 9c and d,
respectively, show a reduction in the asymmetry and in the tailing behavior
of the pdf of C‾t0.02 when aL,1 is
fixed. These results are a symptom of a reduced process uncertainty, which
is in line with the observation that the bulk of the domain is filled with
the coarse sand whose dispersive properties become deterministic when
aL,1 is known.
Inspection of the first four unconditional statistical moments of C‾t0.4 (black curves in Fig. 10) indicates that the
unconditional pdf of C‾ at this intermediate time closely
resembles a Gaussian distribution. Conditioning C‾t0.4 on dispersivity causes a variance reduction, an increase
of the tailing and the appearance of a negative (left) or positive (right)
skewness, respectively, when conditioning is performed on aL,1 or
aL,2. This behavior suggests that in the type of experimental
setting analyzed the variability of aL,1 promotes the appearance of
values of C‾t0.4 larger than the mean, the
opposite of what occurs when solely aL,2 is considered as uncertain.
Figure 11 shows that all four statistical moment of C‾t0.96 are chiefly sensitive to the dispersivity of the fine
sand box, which is placed near the cell outlet. One can note that knowledge
of aL,2 yields a diminished variance of C‾t0.96, which drops almost to zero, an increased degree of symmetry and
a reduce tailing of the pdf of C‾t0.96, all symptoms of uncertainty reduction.
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 C‾(t) to aL,1 at the early times of
solute breakthrough is in line with the observation that solute particles
are mainly advected and dispersed through the coarse sand. Both
dispersivities affect the behavior of C‾(t) at intermediate
times, when solute is traveling through both sands. The dispersivity of the
coarse sand plays a minor role at late times because virtually no
concentration gradients arise in this portion of the domain. Otherwise,
concentration gradients persist in the fine sand zone close to the outlet,
and the solute breakthrough is clearly controlled by the dispersive
properties of the fine sand.
Conclusions
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, y. As such, they quantify the expected
relative variation due to the variability in one (or more) model input
parameter(s), of the expected value, variance, skewness and kurtosis of y.
When viewed in the current research trend, our work is intended to bridge
the gap between variance-based and pdf-based GSA approaches since it embeds the
simplicity of the former while allowing for a higher-order description of
how the structure of the pdf of y is affected by variations of uncertain model
parameters. We cope with computational costs, which might be high when
evaluating higher-order moments, by coupling our GSA approach with
techniques approximating the full model response through a surrogate model.
For the sake of our study, we consider the generalized polynomial chaos expansion (gPCE), other model reduction techniques being fully compatible
with our approach. Our new indices can be of interest for application in the
context of current practices and evolving trends in factor fixing
procedures (i.e., assessment of the possibility of fixing a parameter value
on the basis of the associated output sensitivity), experiment design,
uncertainty quantification and environmental risk assessment due to the
role of the key features of a model output pdf in such analyses.
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, y, with respect to a parameter
depends on the selected global sensitivity index, i.e., variability of a
model parameter affects statistical moments of y in different ways and with
different relative importance depending on the statistical moment
considered. Relying on the indices we propose allows us to enhance our ability
to quantify how model parameters affect features of the model output pdf, such
as mean, degree of spread, symmetry and tailedness, in a straightforward and
easily transferrable way.
Joint inspection of our moment-based global sensitivity indices and of the
first four statistical conditional and unconditional moments of y increases
our ability to understand the way the structure of the model output pdf is
controlled by model parameters. As demonstrated in our examples, classical
variance-based GSA methods cannot be used for this purpose, leading, in some
cases, to the unwarranted conclusion that a given parameter has a limited
impact on a target output.
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, y, might require considering a diverse total order
w of the gPCE, depending on the target statistical moment considered in the
GSA of y; and (b) in our examples, the absolute relative error (Eq. 26) for
AMAExi based on a given total degree w of the gPCE approximation is
always lower than its counterpart associated with higher-order moments (see
Figs. 2 and 5).