Extensive uncertainties exist in hydrologic risk analysis. Particularly for interdependent hydrometeorological extremes, the random features in individual variables and their dependence structures may lead to bias and
uncertainty in future risk inferences. In this study, an iterative factorial
copula (IFC) approach is proposed to quantify parameter uncertainties and
further reveal their contributions to predictive uncertainties in risk
inferences. Specifically, an iterative factorial analysis (IFA) approach is
developed to diminish the effect of the sample size and provide reliable
characterization for parameters' contributions to the resulting risk
inferences. The proposed approach is applied to multivariate flood risk
inference for the Wei River basin to demonstrate the applicability of IFC
for tracking the major contributors to resulting uncertainty in a
multivariate risk analysis framework. In detail, the multivariate risk model
associated with flood peak and volume will be established and further
introduced into the proposed iterative factorial analysis framework to
reveal the individual and interactive effects of parameter uncertainties on
the predictive uncertainties in the resulting risk inferences. The results
suggest that uncertainties in risk inferences would mainly be attributed to
some parameters of the marginal distributions, while the parameter of the dependence structure (i.e. copula function) would not produce noticeable
effects. Moreover, compared with traditional factorial analysis (FA), the
proposed IFA approach would produce a more reliable visualization for
parameters' impacts on risk inferences, while the traditional FA would
remarkably overestimate the contribution of parameters' interaction to the
failure probability in AND (i.e. all variables would exceed the
corresponding thresholds) and at the same time underestimate the contribution of parameters' interaction to the failure probabilities in OR
(i.e. one variable would exceed its corresponding threshold) and Kendall
(i.e. the correlated variables would exceed a critical multivariate
threshold).
Introduction
Many hydrological and climatological extremes are highly correlated with each other, and it is desired to explore their interdependence through
multivariate approaches. Examples include sea-level rise and fluvial flood (Moftakhari et al., 2017), drought and heat waves (Sun et al., 2019), and soil
moisture and precipitation (AghaKouchak, 2015). Moreover, even one specific
hydrological extreme may have multiple attributes, such as the peak and
volume for a flood, duration and severity for a drought, and duration and
intensity of a storm (Karmakar and Simonovic, 2009; Kong et al., 2019).
Traditional univariate approaches, mainly focusing on one variable or one
attribute of hydrological extremes (e.g. flood peak), may not be sufficient
to describe those hydrological extremes containing multivariate characteristics. Thus the univariate frequency/risk analysis methods may be
unable to obtain reliable risk inferences for the failure probability or
recurrence intervals of interdependent extreme events (Chebana and Ouarda,
2011; Requena et al., 2013; Salvadori et al., 2016; Sadegh et al., 2017).
Since the introduction of the copula function into hydrology and the geosciences by De Michele and Salvadori (2003), the copula-based approaches have been
widely used for multivariate hydrologic risk analysis. The copula functions
are able to model correlated variables with complex or nonlinear dependence
structures. Also, these kinds of methods are easily implemented since the
marginal distributions and dependence models can be estimated in separate
processes which also give flexibility in the selection of both marginal and dependence models. A large amount of research has been developed for
multivariate hydrologic simulation through copula functions, such as
multivariate flood frequency analysis (Sraj et al., 2014; Xu et al., 2016;
Fan et al., 2018, 2020); drought assessments (Song and Singh, 2010; Kao and Govindaraju, 2010; Ma et al., 2013); storm or rainfall dependence analysis
(Zhang and Singh, 2007; Vandenberghe et al., 2010); streamflow simulation (Lee and Salas, 2011; Kong et al., 2015); and other water and environmental engineering applications (Fan et al., 2017; Huang et al., 2017).
For both univariate and multivariate analyses for hydrometeorological risks,
uncertainty would be one of the unavoidable issues which needs to be well
addressed. The uncertainty in hydrometeorological risk inference mainly
results from stochastic variability of hydrometeorological processes and
incomplete knowledge of the watershed systems (Merz and Thieken, 2005). Many
studies have been proposed to address uncertainty in both univariate and
multivariate hydrological risk analysis (e.g. Merz and Thieken, 2005;
Serinaldi, 2013; Dung et al., 2015; Zhang et al., 2015; Sadegh et al., 2017;
Fan et al., 2018). However, one critical issue in uncertainty quantification
of hydrological inference is how to characterize the major sources for
uncertain risk inference. Qi et al. (2016) employed a subsampling ANOVA
approach (Bosshard et al., 2013) to quantify individual and interactive
impacts of the uncertainties in data, probability distribution functions, and probability distribution parameters on the total cost for flood control in
terms of flood peak flows. Even though the subsampling ANOVA approach is
able to reduce the effect of the biased estimator on quantification of
variance contribution resulting from the traditional ANOVA approach, it
should be noticed that merely subsampling one uncertainty parameter/factor
(referred to as single-subsampling ANOVA), as used in the studies by Bosshard et al. (2013) and Qi et al. (2016a), will lead to (i) an underestimation of
the individual contribution for the factor to be sampled and (ii) overestimation of contributions from those non-sampled factors. Moreover,
few studies have been reported to characterize the individual and
interactive effects of parameter uncertainties in marginal and dependence
models on the multivariate risk inferences.
Consequently, as an extension of previous research, this study aims to
propose an iterative factorial copula (IFC) approach for quantifying and
partitioning uncertainty metrics from different sources in multivariate
hydrologic risk inference. In detail, the parameter uncertainties are
quantified through a Monte Carlo based bootstrap algorithm. The interactions of parameter uncertainties are explored through a multilevel factorial
analysis approach. The contributions of parameter uncertainties are analysed through an iterative factorial analysis (IFA) method, in which all
uncertainty factors will be subsampled to generate more reliable results.
The applicability of the proposed IFC approach will be demonstrated through
case studies of flood risk analysis in the Wei River basin in China.
Methodology
Figure 1 illustrates the framework of the proposed IFC approach. The
framework consists of four modules: (i) selection of marginal distributions,
(ii) identification of copulas, (iii) parameter uncertainty quantification,
and (iv) parameter interaction and sensitivity analysis. In IFC, modules (i) and (ii) are proposed to construct the most appropriate copula-based
hydrologic risk model. In detail, a number of distributions, such as Gamma,
generalized extreme value (GEV), lognormal (LN), Pearson type III (P III),
and log-Pearson type III (LP III) distributions, are usually employed to
describe the probabilistic features of individual random variables (e.g.
flood peak and volume). Also, in order to quantify the dependence structures
of correlated random variables, many copula functions have been proposed,
such as Gaussian copula, Student t copula, and the Archimedean copula family (e.g. Clayton, Gumbel, Frank, and Joe copulas). In the current study, the indices of root mean square error (RMSE) and Akaike information criterion (AIC)
will be employed to identify the most appropriate model for hydrologic risk
inference. Module (iii) quantifies parameter uncertainties in marginal
distributions and copulas. Module (iv) would be the core part of our study to identify the main sources of uncertainties in multivariate risk inference
by the proposed iterative factorial analysis (IFA) approach.
A copula function is a multivariate distribution function with uniform
margins on the interval [0, 1]. Sklar's theorem states that any d-dimensional distribution function F can be formulated through a copula and
its marginal distributions (Nelsen, 2006). In detail, a multivariate copula
function can be expressed as
F(x1,x2,…,xd|γ1,γ2,…,γd,θ)=C(FX1(x1|γ1),FX2(x2|γ2),…,FXd(xd|γd)|θ),
where FX1(x1|γ1),FX2(x2|γ2),…,FXd(xd|γd) are marginal
distributions of the random vector (X1, X2, …,
Xd), with γ1, γ2, …, γd, respectively, being the unknown parameters of the marginal distributions. θ is the parameter in the copula function describing
dependence among the correlated variables. If these marginal distributions
are continuous, then a single copula function C exists, which can be written
as (Nelsen, 2006)
C(u1,u2,…,ud|θ)=F(FX1-1(u1|γ1),FX2-1(u2|γ2),…,FXd-1(ud|γd)),
where u1=FX1(x1|γ1), u2=FX2(x2|γ2), …, ud=FXd(xd|γd). More details on the theoretical background and
properties of various copula families can be found in Nelsen (2006).
If appropriate copula functions are specified to reflect the joint
probabilistic characteristics for a multivariate extreme event, the
conditional, primary, and secondary return periods (RPs) can be obtained. Consider one kind of hydrological extreme (denoted as X) with d attributes
(i.e. X=(x1, x2, …, xd)), and for a specific
extreme event X* with its attributes being X*=
(x1*, x2*, …, xd*),
three categories of multivariate RP can be applied to determine the potential risk of X*.
(i) “OR” case TOR:
TOR={(x1,x2,…,xd)∈Rd:x1>x1*∨x2>x2*∨…∨xd>xd*}=μ1-C(F1(x1|γ1),…,Fd(xd|γd)|θ),
where Rd is a d-dimensional real space; μ denotes the average time
between two adjacent events under consideration. The joint RP in OR (denoted
as TOR) indicates the occurrence probability of the extreme event with
one of its variables, xi, i=1, 2, …, d, exceeding the corresponding threshold xi*.
(ii) “AND” case TAND:
TAND={(x1,x2,…,xd)∈Rd:x1>x1*∧x2>x2*∧…∧xd>xd*}=μC^(F‾1(x1|γ1),F‾2(x2|γ2),…,F‾1(xd|γd)|θ),
where C^ is the multivariate survival function of the
Xi proposed by Salvadori et al. (2013, 2016), and F‾i(xi|γi)=P(X>xi)=1-Fi(xi|γi).
Following Salvadori et al. (2013, 2016) and the inclusion–exclusion principle proposed by Joe (2014), the multivariate survival function
C^ can be obtained by
C^(u)=C‾(1-u)
and
C‾(u)=1-∑i=1dui+∑S∈P(-1)#(S)CS(ui:i∈S),
where #(S) denotes the cardinality of S. The joint RP in AND (denoted as
TAND) of the extreme event indicates the occurrence probability with all
of its variables xi, i=1, 2, …, d, exceeding the corresponding thresholds xi*.
(iii) “Kendall” case: the Kendall RP characterizes the hydrologic disasters exceeding a critical layer as defined by Salvadori et al. (2011):
LtF={x∈Rd:F(x)=t}. The Kendall RP can be
expressed as (Salvadori et al., 2011)
TKendall=μ1-KC(t),
where KC is the Kendall distribution function associated with the
copula C, which can be expressed as
KC(t)=P(C(F1(x1|γ1),…,Fd(xd|γd)|θ)≤t).
In addition to the multivariate RP, failure probability (FP) can be another index to provide more coherent, general, and well-devised tools for
multivariate risk assessment and communication. In general, the failure
probability pM to indicate the occurrence of a critical event for at
least one time in M years of design life can be defined as (Salvadori et al.,
2016)
pM=1-∏j=1M(1-pj)=1-(F(xd))M.
Similar to the multivariate RP concept, the failure probability in a
multivariate context can also be characterized in the “OR”, “AND”, and “Kendall” scenarios expressed by the following equations. For a given
critical threshold x*={x1*,x2*,…,xd*}, the failure probabilities violating this critical
value can be expressed as (Salvadori et al., 2016)
10apTOR=1-(C(F1(x1*|γ1),F1(x2*|γ2),…,Fd(xd*|γd)|θ))T10bpTAND=1-(1-C^(F1‾(x1*|γ1),F2‾(x2*|γ2),…,F1‾(xd*|γd)|θ))T10cpTKendall=1-(P(C(F1(x1*|γ1),F1(x2*|γ2),…,Fd(xd*|γd)|θ)≤t))T
where pTOR, pTAND, and pTKendall, respectively, denote the failure probability in the “AND”, “OR”, and “Kendall” cases. T
indicates the service time of the facilities under consideration.
Focusing on a bivariate case, the joint RP and the associated failure
probability in the “OR”, “AND”, and “Kendall” scenarios can be formulated as (Salvadori et al., 2007, 2011; Graler et al., 2013; Sraj et al., 2014;
Serinaldi, 2015)
11aTu1,u2OR=μ1-CU1U2(u1,u2|θ),11bTu1,u2AND=μ1-u1-u2+CU1U2(u1,u2|θ),11cTu1,u2Kendall=μ1-P(CU1U2(u1*,u2*)≤t),11dpTOR=1-(CU1U2(u1*,u2*|θ))T,11epTAND=1-(u1*+u2*-C^U1U2(u1*,u2*|θ))T,11fpTKendall=1-(P(CU1U2(u1*,u2*|θ)≤t))T,
where u1=F1(x1|γ1), u2=F2(x2|γ2), u1*=F1(x1*|γ1), and u2*=F2(x2*|γ2), (x1*, x2*) define the bivariate threshold.
Uncertainty in the copula-based risk model
Extensive uncertainties may be involved in the parametric estimation of a
copula function due to (i) the inherent uncertainty in the flooding process; (ii) uncertainty in the selection of appropriate marginal functions
and copulas; and (iii) statistical uncertainty or parameter uncertainty within the parameter estimation process (e.g. the availability of samples)
(Zhang et al., 2015). Several methods have been proposed to quantify
parameter uncertainties in copula-based models. For instance, Dung et al. (2015) proposed bootstrap-based methods for quantifying the parameter
uncertainties in bivariate copula models. Zhang et al. (2015) employed a
Bayesian inference approach for evaluating uncertainties in copula-based
hydrologic drought models, in which the component-wise hit-and-run Metropolis algorithm is adopted to estimate the posterior probabilities of
model parameters.
In this study, a bootstrap-based algorithm is applied to quantify parameter uncertainties in the copula-based multivariate risk model. The procedures
describing the bootstrap-based algorithm to derive probabilistic
distributions of the parameters in both marginal and dependence models are
presented as follows.
Predefine a large number of bootstrapping samplings NB.
Implement the resampling with replacement over observed pairs Z= (X,
Y) to obtain Z*= (X*, Y*); Z* has the same size as Z.
Fit the chosen marginal distributions to X* and Y* and estimate the associated parameters (γX,γY).
Fit the chosen copula to Z* and estimate the parameter in the copula function θ.
Repeat steps 2–5 NB times and obtain NB sets of (γX,γY,θ). Moreover, in order to reject those parameters that
lead to bad fits for both marginal and copula models, the A–D test and the
Cramér–von Mises test are introduced in the bootstrap procedure to ensure that the obtained parameters can pass statistical tests for both the
marginal distribution and copula models. Then the kernel method will be
adopted to quantify the probabilistic features for γX, γY, and θ.
In order to derive bivariate uncertainty bands for a predefined quantile
curve (QC) with certain joint RP in “AND”, “OR”, or “Kendall” (denoted as TAND, TOR, and TKendall), sample NB1 sets of (γX,γY,θ) from the obtained NB samples.
Sample a large number (Ns) of xiyj from their marginal
distributions.
For each set of (γX,γY,θ) from NB1, evaluate the joint RPs of (xi, yj) (i=1, 2, …,
Ns; y=1, 2, …, Ns) and store the pairs of (xi, yj) approaching the predefined joint RPs.
Repeat step 8 for NB1 and for each predefined QC and plot the bivariate uncertainty bands for each quantile QC.
Interactive and sensitivity analysis for parameter uncertainties
Due to the uncertainties existing in the unknown values of parameters for a
copula model, the associated risk or the return period for a flooding event
may also be uncertain. Few studies have been reported to analyse the effect of uncertainties in the copula model on evaluating the risk for a flood
event. To address the above issue, an iterative factorial analysis (IFA)
approach will be proposed to reveal the individual and interactive effects
of parameter uncertainties on the predictive uncertainties of different risk
inferences.
Consider a copula-based bivariate risk assessment model which has two
marginal distributions (A and B) and one copula (C). The parameters in the two
marginal distributions are assumed to be, respectively, denoted as γA with a levels and γB with b levels, while the parameter
in the copula is denoted with θC with c levels. The three-factor ANOVA model for such a factorial design in terms of the predictive risk
(denoted as R) in response to the parameters γA, γB, and θC and n replicates can be expressed as
Rijkl=R0+RθiC+RγjA+RγkB+RθiCγjA+RθiCγkB+RγjAγkB+RθiCγjAγkB+εijkli=1,2,…,c,j=1,2,…,a,k=1,2,…,b,l=1,2,…,n,
where R0 denotes the overall mean effect; RθiC,RγjA, and RγkB, respectively, indicate the effect for parameter θC in the copula at the ith level,
parameter γA in the first marginal distribution at the jth
level, and parameter γB in the first marginal distribution at
the kth level; RθiCγjA,RθiCγkB, and RγjAγkB indicate interactions between factors θC and γA, θC and γB, as well as γA and γB,
respectively; RθiCγjAγkB denotes
the interaction of factors θC , γA, and γB; εijkl denotes the random error component.
Based on Eq. (12), the total variability of the predictive risk can be
decomposed into its component parts as follows (Montgomery, 2001):
SST=SSθC+SSγA+SSγB+SSI+SSe
and
13bSST=∑i=1c∑j=1a∑k=1b∑l=1nRijkl2-R….2abcn,13cSSθC=1abn∑i=1cRi…2-R….2abcn,13dSSγA=1bcn∑j=1aR.j..2-R….2abcn,13eSSγB=1acn∑k=1bR..k.2-R….2abcn,13fSSe=∑i=1c∑j=1a∑k=1b∑l=1nRijkl2-1n∑i=1c∑j=1a∑k=1bRijk.2,13gSSI=SSθCγA+SSθCγB+SSγBγA+SSθCγAγB=SST-SSθC-SSγA-SSγB-SSe,
where Rijk.=∑l=1nRijkl, Ri…=∑j=1a∑k=1b∑l=1nRijkl, R.j..=∑i=1c∑k=1b∑l=1nRijkl, R..k.=∑i=1c∑j=1a∑l=1nRijkl, and R….=∑i=1c∑j=1a∑k=1b∑l=1nRijkl. Then the contributions of
parameter uncertainties in marginal distributions and dependence structures
can be calculated as the following.
(3) Contribution of parameters in marginal distributions A and B:
14aηA=SSγA/SST,14bηB=SSγB/SST.
(2) Contribution of the parameter in the dependence structure:
ηC=SSθC/SST.
(3) Contribution of internal variability:
ηe=SSe/SST.
(4) Contribution of parameter interactions:
ηI=1-ηA-ηB-ηC-ηe.
However, one major issue for the ANOVA approach is that the biased variance
estimator in ANOVA would underestimate the variance in small sample size
scenarios (Bosshard et al., 2013). Thus the sample size may significantly
affect the resulting variance contributions expressed in Eqs. (14a)–(14e). A subsampling approach has been advanced by Bosshard et al. (2013) to
diminish the effect of the sample size in ANOVA and has been employed for
uncertainty partitioning in flood design and hydrological simulation (Qi et
al., 2016a, b). In such a subsampling scheme, one factor (denoted as X) with
T levels (these levels can be different values for numerical parameters, or
different types for non-numerical factors, e.g. model type) would choose two levels in each iteration. For T possible levels of X, we can obtain a
total of CT2 possible pairs for X, expressed as a 2×CT2 matrix as follows:
g(h,j)=X1X1⋯X1X2X2⋯XT-2XT-2XT-1X2X3⋯XTX3X4⋯XT-1XTXT.
However, such a subsampling approach is mainly applied to subsample merely
one factor or one parameter (here we refer to this method as
single-subsampling ANOVA) in previous studies (Bosshard et al., 2013; Qi et al., 2016a, b). However, a critical issue for the single-subsampling ANOVA
is that it will lead to an underestimation of the individual contribution for the factor to be sampled and overestimation of contributions for those
non-sampled factors. Consequently, in this study, we will propose an IFA
approach to subsample all the factors to be addressed and then quantify the contribution of each factor to the response variation. In the IFA approach,
all factors under consideration will be subsampled, and the corresponding
sum of squares will be obtained. The contribution of one factor would be
characterized by the mean value of its contribution in each iteration. In
detail, for the three-factor ANOVA model expressed by Eq. (12), the subsampling schemes for the three parameters can be formulated as
16agθC(hC,jC)=θ1Cθ1C⋯θ1Cθ2Cθ2C⋯θc-2Cθc-2Cθc-1Cθ2Cθ3C⋯θcCθ3Cθ4C⋯θc-1CθcCθcC,16bgγA(hA,jA)=γ1Aγ1A⋯γ1Aγ2Aγ2A⋯γa-2Aγa-2Aγa-1Aγ2Aγ3A⋯γaAγ3Aγ4A⋯γa-1AγaAγaA,16cgγB(hB,jB)=γ1Bγ1B⋯γ1Bγ2Bγ2B⋯γb-2Bγb-2Bγb-1Bγ2Bγ3B⋯γbBγ3Bγ4B⋯γb-1BγbBγbB.
Consequently, there are a total number of Cc2Ca2Cb2 iterations in IFA for the three-factor model expressed as Eq. (12).
For each iteration, the sums of squares can be reformulated as
17aSSTj=∑hC=12∑hA=12∑hB=12∑l=1nRgθC(hC,jC)gγA(hA,jA)gγB(hB,jB)l2-RgθC(o,jC)gγA(o,jA)gγB(o,jB).28n,17bSSθCj=14n∑hC=12RgθC(hC,jC)gγA(o,jA)gγB(o,jB).2-RgθC(o,jC)gγA(o,jA)gγB(o,jB).28n,17cSSγAj=14n∑hA=12RgθC(hC,o)gγA(hA,jA)gγB(o,jB).2-RgθC(o,jC)gγA(o,jA)gγB(o,jB).28n,17dSSγBj=14n∑hB=12RgθC(hC,o)gγA(o,jA)gγB(hB,jB).2-RgθC(o,jC)gγA(o,jA)gγB(o,jB).28n,17eSSej=∑hC=12∑hA=12∑hB=12∑l=1nRgθC(hC,jC)gγA(hA,jA)gγB(hB,jB)l2-1n∑hC=12∑hA=12∑hB=12RgθC(hC,jC)gγA(hA,jA)gγB(hB,jB).2,17fSSIj=SSTj-SSθCj-SSγAj-SSγBj-SSej,
where
RgθC(hC,jC)gγA(hA,jA)gγB(hB,jB).=∑l=1nRgθC(hC,jC)gγA(hA,jA)gγB(hB,jB)lRgθC(hC,jC)gγA(o,jA)gγB(o,jB).=∑hA=12∑hB=12∑l=1nRgθC(hC,jC)gγA(hA,jA)gγB(hB,jB)l.
RgθC(o,jC)gγA(hA,jA)gγB(o,jB).=∑hC=12∑hB=12∑l=1nRgθC(hC,jC)gγA(hA,jA)gγB(hB,jB)lRgθC(o,jC)gγA(o,jA)gγB(hB,jB).=∑hC=12∑hA=12∑l=1nRgθC(hC,jC)gγA(hA,jA)gγB(hB,jB)lRgθC(o,jC)gγA(o,jA)gγB(o,jB).=∑hC=12∑hA=12∑hB=12∑l=1nRgθC(hC,jC)gγA(hA,jA)gγB(hB,jB)l.
Also, for each iteration, the corresponding contributions for each factor
can be obtained as
18aηAj=SSγAj/SSTj,18bηBj=SSγBj/SSTj,18cηCj=SSθCj/SSTj,18dηej=SSej/SSTjηIj=1-ηAj-ηBj-ηCj-ηej.
Finally, the individual and interactive contributions for those factors can
be obtained by averaging the corresponding contributions in all iterations,
expressed as
19aηA=1J∑j=1JSSγAj/SSTj,19bηB=1J∑j=1JSSγBj/SSTj,19cηC=1J∑j=1JSSθCj/SSTj,19dηe=1J∑j=1JSSej/Tj,19eηI=1J∑j=1JηIj,
where J=Cc2Ca2Cb2.
Applications
The proposed IFC approach can be applied to various multivariate risk
inference problems. In this study, we will apply IFC for multivariate flood
risk inference at the Wei River basin in China. The Weihe River plays a key
role in the economic development of western China and thus is known regionally as the “Mother River” of the Guanzhong Plain of the southern part
of the Loess Plateau (Song et al., 2007; Zuo et al., 2014; Du et al., 2015, Xu et al., 2016). It originates from the Niaoshu mountain at an elevation of
3485 m above mean sea level in Weiyuan County of Gansu Province (Du et al.,
2015). The Weihe River basin is characterized by a semi-arid and sub-humid
continental monsoon climate, resulting in significant temporal–spatial variations in precipitation, with an annual average precipitation of 559 mm
(Xu et al., 2016). Furthermore, there is a strong decreasing gradient from
south to north in which the southern region experiences a sub-humid climate with annual precipitation ranging from 800 to 1000 mm, whereas the northern
region has a semi-arid climate with annual precipitation ranging from 400 to
700 mm (Xu et al., 2016). Over the entire basin, the mean temperature ranges
from 6 to 14 ∘C, the annual potential evapotranspiration fluctuates
from 660 to 1600, and the annual actual evapotranspiration is about 500 mm
(Du et al., 2015).
Observed daily streamflow data at the Xianyang and Zhangjiashan gauging stations were used for hydrologic risk analysis. Figure 2 shows the locations of these two gauging stations based on the daily streamflow data, the flood peak
applied is defined as the maximum daily flow over a period, and the associated flood volume is considered the cumulative flow during the flood period. In this study, the flood characteristics are obtained based on
an annual scale. This means that one flood event is identified in each year.
The detailed method to identify the flood peak and the associated flood
volume can be found in Yue (2000, 2001). Table 1 shows some descriptive
statistical values for the considered variables (peak discharge, Q;
hydrograph volume, V), in which 47 and 55 flood events are characterized at
the Xianyang and Zhangjiashan stations, respectively.
The location of the studied watersheds. The Wei River is the largest tributary of the Yellow River, with a drainage area of 135 000 km2. The
historical flood data from the Xianyang and Zhangjiashan stations on the Wei River are analysed through the proposed IFC approach.
Flood characteristics for different stations.
Station namePeriodFlood variable Peak (m3 s-1)Volume (m3 s-1 d-1)Xianyang1960–2006Minimum139317Median13502491Maximum12 38017 802Zhangjiashan1958–2012Minimum217303.7Median7751365.3Maximum37307576.1Results analysisModel evaluation and selection
There are a number of potential probabilistic models for modelling
individual flood variables and their dependence structures. In this study,
five alternative distributions, including gamma, generalized extreme value
(GEV), lognormal (LN), Pearson type III (P III), and log-Pearson type III
(LP III) distributions, are employed to describe the probabilistic features
of the chosen flood variables (i.e. peak and volume). Moreover,
goodness-of-fit tests are performed through the indices of
Kolmogorov–Smirnov test (K–S test), root mean square error (RMSE), and Akaike information criterion (AIC) to screen the performance of those potential models. The results are presented in Table 2. The results indicate that all
five parametric distributions can produce satisfactory results, with all
p values larger than 0.05. However, it can be concluded that the GEV and lognormal approaches show the best performance for, respectively, modelling flood peak and volume at both gauging stations.
Statistical test results for marginal distribution estimation: LN
means lognormal distribution, P III means Pearson type III distribution, and LP III means log-Pearson type III distribution. K–S test denotes the
Kolmogorov–Smirnov test.
Station nameFlooding variablesMarginal distributionK–S test RMSEAICTp valueZhangjiashanPeakGamma0.0750.5470.038-323.6GEV0.0720.9150.028-389.4LN0.0810.8400.028-388.3P III0.0890.7390.040-349.3LP III0.0800.8510.032-371.4VolumeGamma0.1460.1740.060-306.3GEV0.1020.5840.037-357.1LN0.0900.7250.036-361.3P III0.1590.1110.074-280.9LP III0.0970.6470.037-357.5XianyangPeakGamma0.1160.5530.037-305.4GEV0.0880.8650.031-321.9LN0.1050.6760.044-290.5P III0.1200.5050.042-292.8LP III0.1320.3850.062-255.9VolumeGamma0.1150.5310.045-287.5GEV0.0540.9980.020-364.3LN0.0670.9750.019-367.4P III0.1010.6910.038-302.0LP III0.0720.9520.031-319.7
In addition, a total number of six copulas, including Gaussian, Student's t, Clayton, Gumbel, Frank, and Joe copulas, are considered as the candidate models for quantifying the dependence structures for flood peak volume at the Xianyang and Zhangjiashan gauging stations. Also, the goodness-of-fit
statistic test is performed based on the Cramér–von Mises statistic proposed by Genest et al. (2009). The indices of RMSE and AIC were employed
to evaluate the performance of the obtained copulas and identify the most
appropriate ones. Table 3 shows statistical test results for the selected
copulas. The results show that, for the Zhangjiashan station, all candidate
copulas except the Joe copula performed well, while all six copulas would be
able to provide satisfactory risk inferences at the Xianyang station.
Moreover, based on the values of RMSE and AIC, the Gumbel copula was chosen
to model the dependence of flood peak and volume at the Zhangjiashan station, while the Joe copula performed best at the Xianyang station, except that although the p value is slightly lower than Gumbel, the overall result favours Joe. Consequently, the Gumbel and Joe copulas were chosen in this study to further characterize the uncertainty in model parameters and the
resulting risks at the Zhangjiashan and Xianyang stations, respectively.
Performance for quantifying the joint distributions between flood
peak and volume through different copulas: CvM is the Cramér–von Mises statistic proposed by Genest et al. (2009), with a p value larger than 0.05
indicating satisfactory performance.
Probabilistic features for parameters in marginal distributions and copula:
for both the Xianyang and Zhangjiashan stations, the GEV (parameters include shape, scale, and location) function would be employed to quantify the distribution of flood peak, while the lognormal distribution (parameters
denoted as meanlog and sdlog) is applied for flood volume. The Gumbel and
Joe copulas (parameter denoted as theta) would be, respectively, adopted to model the dependence between flood peak and volume at the Zhangjiashan and Xianyang stations.
Uncertainty in model parameters and risk inferences
Based on the results in Tables 2 and 3, the multivariate risk inference
model was established, in which the GEV and lognormal distributions would, respectively, be adopted to model the individual flood variables at both
gauging stations, while in comparison, the Gumbel and Joe copulas would, respectively, be employed for the Zhangjiashan and Xianyang stations. Afterward, uncertainties would be characterized based on the bootstrap algorithm
illustrated in Sect. 2.2. In this study, a total number of 5000 samples
was chosen in order to generally visualize the uncertainty features in the model parameters. The probabilistic features for obtained parameter values (i.e. shape, scale, and location for GEV, meanlog, sdlog for LN, and theta
for copula) for each sample scenario would be described by the kernel
method. Figure 3 exhibits the probabilistic distributions for the six unknown parameters in the established multivariate risk inference model. Extensive
uncertainties exist in the parameters for both the marginal distribution and
dependence model. As presented in Fig. 3, each parameter, except the
meanlog in the LN distribution, exhibits noticeable uncertainty. Moreover,
most of the parameter uncertainties are approximately normally distributed, except the shape parameter in GEV for Xianyang.
Uncertainty quantification of the joint RP in “AND”: the red
dash lines indicate the predictive means, the two blue dash lines, respectively, indicate the 5 % and 95 % quantiles, and the grey lines
indicate the predictions under different parameter samples with the same
joint RP of the red and blue dash lines. The cyan lines denote the predictions under different return periods, with the model parameters being their mean values.
Uncertainty quantification of the joint RP in “OR”: the red dash
lines indicate the predictive means, the two blue dash lines, respectively, indicate the 5 % and 95 % quantiles, and the grey lines indicate the
predictions under different parameter samples with the same joint RP of the
red and blue dash lines. The cyan lines denote the predictions under different return periods, with the model parameters being their mean values.
Uncertainty quantification of the joint RP in “Kendall”: the red
dash lines indicate the predictive means, the two blue dash lines, respectively, indicate the 5 % and 95 % quantiles, and the grey lines
indicate the predictions under different parameter samples with the same
joint RP of the red and blue dash lines. The cyan lines denote the predictions under different return periods, with the model parameters being
their mean values.
It is quite apparent that different parameter values in the copula model
would lead to different risk inference results. Consequently, parameter
uncertainties in the marginal distributions and copula functions would
definitely result in uncertainties in multivariate risk inferences. Based on
the copula model, some multivariate risk indices can be easily obtained,
such as the joint return period in OR, AND, and Kendall, as expressed in Eqs. (11a)–(11c). However, due to parameter uncertainties, these
risk indices may also exhibit some degrees of uncertainty. Figures 4–6
describe uncertainties for the joint RP in AND, OR, and Kendall at the two stations. In general, the predictive RP in AND exhibit most significant
uncertainty, followed by the predictive RP in OR and Kendall. However, for
moderate or large flood events, considerable uncertainties can be observed
in the inferences for all three joint RPs. Specifically, noticeable uncertainties exist in the predictive joint RP of AND even for a minor flood
event with a 5-year joint RP. For some large flood events with a joint RP
around 100 years, the predictive RP in AND shows remarkable uncertainty,
ranging from less than 50 years to larger than 200 years. For the joint RP
in OR and Kendall, slight uncertainty may exist for small flood events (e.g.
2- or 5-year joint RP). Nevertheless, apparent uncertainties can be observed in the predictive joint RP even for moderate flood events. As shown in Fig. 5, considerable uncertainties may appear in the predictive joint
RP of OR even for a flood with an actual joint RP of 20 years, while
prediction of the Kendall RP for a 20-year (in Kendall RP) flood event may
range from 10 to 50 years, as presented in Fig. 6.
Individual and interactive effects of parameter uncertainties
It has been observed that parameter uncertainties in the copula-based
multivariate risk model would lead to significantly imprecise risk
predictions. However, one critical issue to be addressed is how the
parameter uncertainties and their interactions would influence the risk
inference. Consequently, a multilevel factorial analysis, based on Eqs. (12) and (13), was proposed to primarily visualize the individual and interactive effects of parameter uncertainties in the marginal and
dependence models on the resulting risk inferences. In this study, a total
number of six parameters (i.e. three from GEV, two from LN, and one from copula) was addressed, and based on probabilistic features of these parameters, three quantile levels (i.e. 0.1, 0.5, and 0.9) were chosen to characterize the resulting risk inferences under different parameter values.
This would finally form a 36 factorial design, which has six factors, with each having three levels. The failure probability denoted as Eq. (10) would be considered the response in this factorial design.
ANOVA table for failure probability in AND: A indicates the shape parameter in
GEV, B indicates the scale parameter of GEV, C indicates the location parameter of GEV, D means the
meanlog of LN, E means the sdlog of LN, and F means the parameter (i.e. theta) in copula.
The main and interactive effects of parameter uncertainties on the failure probabilities in AND are visualized in Fig. 7. It is noticeable that at
the two gauge stations, parameter uncertainties have similar main and interactive effects on the failure probabilities in AND, which indicates
that parameters' effects (individual and interactive) on the failure
probability in AND are independent of the location of gauge stations. More
specifically, variations in the shape parameter in GEV and sdlog parameter in LN
would lead to more changes in the corresponding responses (i.e. failure
probability in AND) than the variations in other parameters. Also, as shown
in Fig. 7, the parameter in the copula function (i.e. Cop_theta), describing dependence of the two flood variables, would not have an
effect on the resulting risk as much as the effects from the parameters
(except the location parameter in GEV) in the marginal distributions. In
terms of parameter interactions, the significance of interactive effects for
different parameters varies. The interactive curves for some parameters
(e.g. GEV_shape and GEV_location) are nearly
parallel at the three levels, indicating an insignificant interaction for
these two parameters on the inferred risk. In comparison, there are also
some interactive curves intersecting each other (e.g. GEV_shape and LN_meanlog), implying a significant interaction
between these two parameters. Table 4 provides the results from an ANOVA
table for the failure probability in AND. It is quite interesting that (i) even though the effect from the parameter in the copula function is not as visible as the effects from the parameters (except the location parameter in
GEV) in the marginal distributions (as shown in Fig. 7), such an effect is
still statistically significant; (ii) the effect from the location parameter
of GEV is statistically insignificant, which also leads to insignificant interactive effects between the location parameter and other parameters;
(iii) the interactions between the parameter in copulas and the parameters in marginal distributions would be more likely statistically insignificant; (iv) the statistical significance (significant or not) for individual and
interactive effects from parameters is almost the same between these two
gauge stations. All these conclusions obtained from Table 4 are consistent
with the implications described in Fig. 7.
Main effects plot and full interactions plot matrix for parameters
on the failure probability in AND at the two gauge stations.
Main effects plot and full interactions plot matrix for parameters
on the failure probability in OR at the two gauge stations.
In terms of the failure probabilities in OR and Kendall, as presented in
Figs. 8 and 9, these have similar patterns to the failure probability in AND (presented in Fig. 7). The individual/main effects from the marginal
distributions (except the location parameter in GEV) are generally more
visible than the parameters in copula functions. Also, some interactive
curves, especially the curves between GEV_location and
others, are parallel, showing insignificant interaction between those
parameters. More detailed characterizations of the main and interactive
effects for the failure probabilities in OR and Kendall are described in the
ANOVA tables in Tables 5 and 6. These two tables show some slight
differences from the conclusions given by Table 4. The location parameter in
GEV also has a statistically significant effect on the failure probabilities
results in OR and Kendall, which also leads to some significant interactions
between this parameter and other model parameters. For the failure
probability in Kendall, the parameter in the copula would have more
interactions with other parameters in marginal distributions than the
interactions in the failure probability in AND and OR. As presented in Table 6, the parameter in the copula would have a statistically significant effect
on the inferred failure probability in Kendall with other parameters except
the location parameter in GEV. These results are also implied in the main
effects plots and full interactions plot matrices in Figs. 8 and 9.
ANOVA table for failure probability in OR: A indicates the shape parameter in GEV,
B indicates the scale parameter of GEV, C indicates the location parameter of GEV, D means the meanlog of LN, E
means the sdlog of LN, and F means the parameter (i.e. theta) in copula.
ANOVA table for failure probability in Kendall: A indicates the
shape parameter in GEV, B indicates the scale parameter of GEV, C indicates the location parameter of GEV, D means the
meanlog of LN, E means the sdlog of LN, and F means the parameter (i.e. theta) in copula.
Main effects plot and full interactions plot matrix for parameters
on the failure probability in Kendall at the two gauge stations.
Based on the three-level factorial analysis, it can be generally concluded
that the parameters in the marginal distributions (except the location
parameter in GEV) would have more individual effects on joint risk inference
than the parameter in the copula. The risk indices (i.e. AND, OR, or
Kendall) would not have significantly influenced the individual effects of
model parameters. However, for the interactive effects among model
parameters, they may exhibit slightly different patterns. Specifically, the
parameter in the copula would have more significant interactions with
parameters in the marginal distributions on the failure risk in Kendall than
the other two risk indices. Moreover, the individual and interactive effects
from the model parameters on risk inferences would not be influenced by the
location of the gauge stations.
Contribution partition of uncertainty sources
As a result of parameter uncertainties, the predictive failure probabilities
exhibit noticeable uncertainties, as shown in Figs. 4–6. The three-level
factorial analysis based on Eq. (13) is able to provide a primary
description and visualization related to the individual and interactive
effects of parameter uncertainty on the inferred failure probabilities.
However, two critical issues to be answered are (i) how much would parameter uncertainties contribute to the variation of the inferred risk
values and (ii) do these contributions change significantly for failure probabilities with different service time scenarios? To address these two
issues and get reliable results, an iterative factorial approach (IFA) has
been proposed, which is formulated as Eqs. (15)–(19). Also, like the
three-level factorial analysis, three quantile levels were selected at 0.1,
0.5, and 0.9. Based on IFA, each parameter at its three quantile values
(0.1, 0.5, 0.9) would be further subsampled into three scenarios of two
quantile values (i.e. (0.1, 0.5), (0.1, 0.9), and (0.5, 0.9)). For this
study, we have a total number of six parameters, with each having three quantile values at 0.1, 0.5, and 0.9, which would lead to a total number of 729 (i.e. 36) two-level factorial designs.
Contributions of parameter uncertainties to predictive failure
probabilities in AND under different design standards (i.e. return periods – RPs) and different service periods.
Table 7 shows the detailed contribution table of the model parameters on
uncertainty in predictive failure probabilities of AND at the two gauge
stations. It can be observed that, even though some discrepancies exist at the Zhangjiashan and Xingshan stations, the detailed contributions for each
parameter and their interaction show quite similar features between these
two stations. In detail, uncertainty in the shape parameter in GEV has the
most significant impact on the failure probability in AND, followed by
sdlog in LN, parameter interaction, meanlog in LN, and scale parameter in GEV. Moreover, the
uncertainty in the parameter in the copula would not lead to a significant
variation in the resulting failure probability predictions in AND, which
merely makes a contribution less than 0.5 %. Such conclusions are also
generally consistent with the ANOVA results presented in Fig. 7 and Table 4. Furthermore, with the increase in service time, the contributions of each
parameter and their interactions do not vary significantly. Some individual
contributions from parameter uncertainties would slightly increase, while other individual contributions may slightly decrease. However, the effect
from parameter interactions would generally increase with the increase in service time. In comparison, the enhancement in design standards for
hydraulic infrastructures would lead to a greater chance of deceases in individual effects and, at the same time, increases in parameter
interactions. For instance, as the flood design standard increases from
200 to 500 years for a hydraulic facility with 30-year service time near the Zhangjishan station, the interactive effect of model parameters would increase from 15.14 % to 18.09 %.
Contributions of parameter uncertainties to predictive failure
probabilities in OR under different design standards (i.e. return periods – RPs) and different service periods.
In terms of the failure probability in OR, the individual and interactive
effects of model parameters on predictive risk uncertainties show similar
patterns with the parameters' effects on the failure probability in AND. As
shown in Table 8, the shape parameter in the GEV distribution and the sdlog in the LN
distribution are the two major sources of uncertainty in failure
probabilities in OR. However, compared with the failure probability in AND,
parameter interaction has a less effect on the resulting uncertainty of risk
inference in OR. As shown in Tables 7 and 8, the effect of parameter
interaction on the risk in AND ranges between 13.96 % and 20.05 %, while
in comparison, the parameters' interactive effect on the risk in OR varies
between 10.25 % and 11.57 %. Apparently, it can also be observed that
some external factors such as the design standard and service time of
hydraulic infrastructures have less influence on the parameters' interaction
on risk in OR than the risk in AND. However, the first contributor (i.e.
shape parameter in GEV) would have a larger contribution on the predictive
uncertainty in the failure probability in OR as the increase in the design
standard, while in comparison, this contributor would have a lower
contribution on the risk in AND. For instance, as the design return period
of flood (i.e. design standard) increases from 200 to 500 years and the
service time of the hydraulic facility is 30 years, the contribution of the
shape parameter in GEV would increase from 47.62 % to 50.64 % for the
failure probability in OR at the Xianyang station, while the parameter's
contribution on the failure probability in AND decreases from 49.26 % to
45.77 %.
Variation of parameters' contributions for different risk
inferences at the Zhangjiashan station for a design standard of 200 years and a service time of 30 years.
Correlation for parameters' contributions to risk inferences at the Zhangjiashan station for a design standard of 200 years and a service time of 30 years: the cross sign indicates the correlation is statistically
insignificant.
Contributions of parameter uncertainties to predictive failure
probabilities in Kendall under different design standards (i.e. return
periods – RPs) and different service periods.
For the failure probability in Kendall, the contributions of model
parameters and their interaction are presented in Table 9. Similar to the
failure probabilities in AND and OR, the shape parameter in the GEV distribution and the
sdlog parameter in the LN distribution are the two major contributors, which can
account for nearly 70 % or more in the predictive uncertainty of the
failure probability in Kendall. Meanwhile, the scale parameter in GEV, meanlog in LN, and
parameters' interaction also have noticeable effects on the risk in Kendall,
ranging from 4.72 % (scale parameter in GEV) to 12.64 % (meanlog in LN). Conversely, the
location parameter in GEV and the dependence parameter in copula merely have relatively minor
individual effects. However, it is noticeable that, although the dependence
parameter has a minor effect (0.78 %, 1.03 %) on the risk in Kendall,
such an effect is much higher than the effect on the risk in AND (less than
0.23 %) and the risk in OR (less than 0.06 %).
Even though the prediction equations for the failure probabilities in AND, OR, and Kendall as presented in Eq. (11) are different, the impacts of
parameter uncertainties show quite similar features, in which the shape parameter in GEV
and the sdlog in LN are the two major contributors to the predictive
uncertainties in risk inferences. Nearly 70 % and more variability in the
uncertainties in risk inferences can be attributed by the uncertainties in
the shape parameter in GEV and sdlog parameter in LN. Also, some external factors such as flood design and facility service time may have different influences on
parameters' effects for different risk indices; such influences are not significant and would not lead to remarkable changes in parameters'
contributions to risk inferences. Parameters' interaction has a greater effect on risk inference in AND than the other two risk indices (i.e. OR,
Kendall), while the contribution from the dependence parameter, even though
not noteworthy, has a larger effect on the risk inference in Kendall.
DiscussionDifferences for the hydrologic risk models at different stations
Different copula functions are applied for different stations, which are
chosen based on the indices of RMSE and AIC. However, the selection of
copula models at different stations may also be related to some key characteristics of the drainage areas for those stations. The Gumbel copula
will be applied for the Zhangjiashan station. It can reflect strong
correlation at high values. However, the Joe copula, which is adopted at the
Xianyang station, can reflect a stronger right-tail positive dependence than the Gumbel copula. Both the Xianyang and Zhangjiashan stations have similar drainage areas. The Xianyang station controls a drainage area of 46 480 km2 (Xu et al., 2016), while the Zhangjiashan station has a drainage
area of 45 412 km2 (Sun et al., 2019). Nevertheless, the major reason
that led to different copula functions at these two stations may be the elevation features for those two drainage areas. The drainage area of the Zhangjiashan station is located in the central part of the Loess Plateau of
China, which is mainly characterized as a mountainous region. In comparison,
even though a large part of the drainage area of the Xianyang station is also located in the mountainous region, the Xianyang station also controls a
large part of the Guanzhong Plain, as indicated in the red part of Fig. 2.
Consequently, the flood hydrograph at the Zhangjiashan station may be sharp, while the flood hydrograph at the Xianyang station is relatively flat and shows a stronger right-tail dependence among flood peak and volume. In fact, the value of Kendall's tau between peak and volume for the top 10 floods at the
Zhangjiashan station is 0.33, while such a value of Kendall's tau at the Xianyang station is 0.6. These facts may explain why the Gumbel copula is applicable for the Zhangjiashan station while the Joe copula is applied for the Xianyang station. However, this is an initial guess and may need to be further
demonstrated through more cases in different areas.
Contribution partition of uncertainty sources through different approaches
In this study, the individual and interactive contributions of parameter
uncertainties are quantified through the developed IFA approach, in which
each parameter has three levels (i.e. 0.1, 0.5, and 0.9 quantiles) to be subsampled. In fact, the parameters' contributions can also be characterized
by the traditional factorial analysis (FA) approach based on Eq. (14)
as well as the IFA approach with more factor levels (e.g. four or five levels for each parameter).
Comparison of parameter contributions to predictive uncertainty
for failure probabilities under different levels of subsampling for the Zhangjiashan station: three (i.e. 0.1, 0.5, 0.9) and four (i.e. 0.1, 0.35,
0.6, 0.85) level quantiles are adopted for subsampling and the design return
period is 200 years.
Table 10 shows the comparison of parameter contributions to predictive
uncertainty for failure probabilities in AND at the Zhangjiashan station for
three and four parameter level scenarios for the design standard of 200 years. The results of Table 10b are obtained through the IFA approach, with each parameter having four levels to be its quantiles at 0.1, 0.35,
0.6, and 0.85. Also, Table 11 presents the parameter contributions to predictive uncertainty in failure probabilities obtained by the traditional FA approach for the Zhangjiashan station with the design standard of 200 years and service time of 30 years.
Contributions of parameter uncertainties obtained by three-level ANOVA to predictive failure probabilities for a design return period of
200 years and a service time of 30 years.
It can be seen that for different subsampling scenarios, the resulting
contributions may be different. However, such a difference would be
tolerable since (1) the variations of parameters' contributions are
relatively small and mainly happen for the first two contributors, (2) the
total contribution of the first two contributors does not change remarkably
(around 70 % in total), (3) the contributions of other factors, especially the parameters' interaction, do not vary significantly, and (4) the rank of
the contributions from different sources does not change for the two
subsampling scenarios. In comparison, as presented in Table 7, the
contribution partition of parameter uncertainties obtained through
traditional FA shows totally different patterns for different risk
inferences. Specifically, the traditional FA approach would significantly
overestimate parameter interactive effects on risk inference in AND; at the same time, it would underestimate the interactive effects on risk inference in OR and
Kendall. Consequently, the contribution rank of parameter uncertainties from
traditional FA is different from the results obtained through the developed
IFA approach.
As shown in Table 10, the proposed IFA approach may lead to slightly
different results for different subsampling schemes (four or five levels).
However, an increase in parameter level would highly increase computational
demand. For instance, if each parameter has four levels, the IFA approach
would lead to a total number of 46 656 (i.e. 66) two-level factorial
designs. Moreover, the subsampling scheme for factors with five levels would
lead to a total number of 1 million (i.e. 106) two-level factorial designs. Consequently, the three-level subsampling scheme would generally be
recommended and also can generate acceptable results.
Correlation among parameter contributions
The proposed IFA approach would generally produce a great number of
two-level factorial designs. For one specific factor (e.g.
GEV_shape), it would have two levels (lower and upper levels) for all factorial
designs. However, the detailed value for the lower or upper level may be
different in different factorial designs. This may finally lead to different
contributions for this factor. Figure 10 presents the variations of
parameters' contributions to the prediction of failure probabilities in AND,
OR, and Kendall. We already concluded that the shape parameter in GEV (i.e.
GEV_shape) and the sdlog in LN (i.e. LN_sdlog) distribution would generally have the most
significant contributions to predictive uncertainties in risk inferences.
However, as shown in Fig. 10, the detailed contributions for these two
parameters would vary remarkably for different level values in different
factorial designs. In comparison, the contributions from other parameters
and their interaction have less fluctuation than the individual
contributions of GEV_shape and LN_sdlog. For instance, although the meanlog in LN (i.e.
LN_meanlog), with an average contribution of more than 10 %, may have some chance to
pose a predominant contribution of more than 50 %, most of its
contribution is positively distributed within [0 %, 25 %]. Also, even though
the parameters' interaction has a noteworthy average contribution larger
than 10 %, all the detailed contributions in different factorial designs
are located within [0 %, 25 %].
It has been observed that the parameters' contributions may vary significantly due to the differences in factor values in different factorial
designs. One potential issue to be addressed is how those individual and interactive contributions correlate with each other. Figure 11 presents
Pearson's correlation among individual and interactive contributions of model parameters to different risk inferences (i.e. failure probabilities in
AND, OR, and Kendall). It is noticeable that the parameters in the LN
distribution (i.e. LN_sdlog, LN_meanlog) are generally negatively correlated with the
parameters in the GEV distribution (i.e. GEV_shape, GEV_scale, and GEV_location). Also, for one marginal
distribution (LN or GEV), its parameters are positively correlated. This
implies that an increase in the contribution of one parameter would lead to
a contribution increase for parameters within the same distribution and at
the same time result in a contribution decrease for all parameters in the
other distribution. Moreover, if statistically significant, the contribution
of the dependent parameter (i.e. parameter in copula) generally has positive
correlation with the contributions from other parameters except
GEV_shape and parameters' interaction. Also, the
contributions from parameters' interactions are generally negatively
correlated with the individual contributions from other parameters if such a correlation is statistically significant.
The proposed IFA approach can generally characterize how parameter
uncertainties would influence the predictive uncertainties in risk
inferences. A large number of two-level factorial designs was produced due to different subsampling procedures and then used to generate different
partition results for parameters' contributions. However, for different risk
inferences (i.e. failure probabilities in AND, OR, and Kendall), these
partition results have similar variation features and also show similar
correlation plots.
Conclusions
Uncertainty quantification is an essential issue for both univariate and
multivariate hydrological risk analyses. A number of research works have
been posed to reveal uncertain features in multivariate hydrological risk
inference. However, it is required to know the major sources/contributors
for predictive uncertainties in multivariate risk inferences. In this study,
an iterative factorial copula approach (IFC) has been proposed for
uncertainty quantification and partition in multivariate hydrologic risk
inference. In IFC, a copula-based multivariate risk model has been developed, and the bootstrap method is adopted to quantify the probabilistic features
for the parameters in both marginal distributions and the dependence model.
An iterative factorial analysis (IFA) approach was finally developed to diminish the effect of the sample size in traditional ANOVA computation and
provided reliable contribution partition for parameter uncertainties in
different risk inferences.
The proposed method has been applied for flood risk inferences at two gauge
stations in the Wei River basin. The results indicate that uncertainties in the parameters of the copula-based model would lead to noticeable uncertainties
in the resulting risk inferences, especially for the joint flood risk in
AND. Noticeable uncertainties exist in the predictive joint RP of AND even for a small flood event. However, the results from IFA suggested that those
uncertainties in risk inferences may mainly be attributed to the
uncertainties in the shape parameter in the GEV distribution and the parameter of sdlog in LN for both the two stations. In comparison, the parameter uncertainty in the copula
function would not have an obvious effect on the resulting uncertainty in risk inferences. Such results indicate that, at least for the Wei River basin, the decision makers need to estimate the values or quantify the uncertainties well for the shape parameter in the GEV distribution and sdlog in the LN distribution, in
order to obtain reliable risk inferences. For other catchments, the proposed
IFC method can be adopted to reveal the major sources for uncertainties in
risk inferences and then provide potential pathways to get reliable risk
inferences.
This study is the first attempt to characterize parameter uncertainties in a
copula-based multivariate hydrological risk model and further reveal their
contributions to predictive uncertainties for different risk inferences. As
an improvement of ANOVA, the developed IFA method can mitigate the effect of
bias variance estimation in ANOVA and generate reliable results. Moreover,
another noteworthy feature of the IFA approach is that it not only characterizes the impacts for continuous factors (e.g. model parameters in this study) but also reveals the impacts of discrete or non-numeric factors. Such a feature can allow the proposed IFA approach to be employed
to further explore the impacts of non-numeric factors (e.g. model
structures, sample size) in hydrologic systems analysis.
Code and data availability
The flooding data for the studied catchments as
well as the associated code for this study can be obtained upon email
request to the corresponding authors.
Author contributions
YF, KH, GH, and YL designed the research. YF and FW carried out the research, developed the model code and performed the
simulations. YF prepared the manuscript with contributions from all the co-authors.
Competing interests
The authors declare that they have no conflict of interest.
Acknowledgements
We are very grateful for the editor's and anonymous reviewers' insightful and constructive comments.
Financial support
This research has been supported by the Brunel University Open Access Publishing Fund, the National Key Research and Development Plan (2016YFC0502800), the National Natural Science Foundation of China (51520105013), and the Natural Sciences and Engineering Research Council of Canada.
Review statement
This paper was edited by Alberto Guadagnini and reviewed by Geoff Pegram and one anonymous referee.
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