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Environ Sci Pollut Res (2017) 24:24284–24296
DOI 10.1007/s11356-017-0030-2
RESEARCH ARTICLE
Coupled Monte Carlo simulation and Copula theory
for uncertainty analysis of multiphase flow simulation models
Xue Jiang 1 & Jin Na 2 & Wenxi Lu 3 & Yu Zhang 4
Received: 27 June 2017 / Accepted: 24 August 2017 / Published online: 9 September 2017
# Springer-Verlag GmbH Germany 2017
Abstract Simulation-optimization techniques are effective in
identifying an optimal remediation strategy. Simulation models
with uncertainty, primarily in the form of parameter uncertainty
with different degrees of correlation, influence the reliability of
the optimal remediation strategy. In this study, a coupled Monte
Carlo simulation and Copula theory is proposed for uncertainty
analysis of a simulation model when parameters are correlated.
Using the self-adaptive weight particle swarm optimization
Kriging method, a surrogate model was constructed to replace
the simulation model and reduce the computational burden and
time consumption resulting from repeated and multiple Monte
Carlo simulations. The Akaike information criterion (AIC) and
the Bayesian information criterion (BIC) were employed to
identify whether the t Copula function or the Gaussian
Copula is the optimal Copula function to match the relevant
structure of the parameters. The results show that both the AIC
and BIC values of the t Copula function are less than those of
the Gaussian Copula function. This indicates that the t Copula
function is the optimal function for matching the relevant
Responsible editor: Marcus Schulz
* Jin Na
na_jin@126.com
1
State Key Laboratory of Biogeology and Environmental Geology
and School of Environmental Studies, China University of
Geosciences, Wuhan 430074, China
2
Institute of Disaster Prevention Science and Technology,
Sanhe 065201, China
3
College of Environment and Resources, Jilin University,
Changchun 130021, China
4
Songliao Institute of Water Environment Science, Songliao River
Basin Water Resources Protection Bureau, Changchun 130021,
China
structure of the parameters. The outputs of the simulation model
when parameter correlation was considered and when it was
ignored were compared. The results show that the amplitude of
the fluctuation interval when parameter correlation was considered is less than the corresponding amplitude when parameter
estimation was ignored. Moreover, it was demonstrated that
considering the correlation among parameters is essential for
uncertainty analysis of a simulation model, and the results of
uncertainty analysis should be incorporated into the remediation strategy optimization process.
Keywords Uncertainty analysis . Copula theory . Surrogate
model . DNAPL . SAPSOKRG
Introduction
The leakage of dense nonaqueous phase liquids (DNAPL) has
caused serious environmental pollution, which has resulted in
worldwide health hazards (Qin et al. 2008; Muff et al. 2016;
Hou et al. 2016). DNAPL, with densities greater than that of
water, are difficult to remove from groundwater (Qin et al.
2007). Surfactant-enhanced aquifer remediation (SEAR), a
chemical enhancement of the conventional pump-and-treat
technique, can considerably increase the removal efficiency
of DNAPL-contaminated groundwater. However, SEAR is
quite expensive. Therefore, it is critical to select a costeffective optimal remediation strategy by simulationoptimization techniques. SEAR simulation models are involved in the simulation-optimization process. When the simulation model involves uncertainty, which is primarily manifested in form of parameter uncertainty, the reliability of a
certain optimal strategy could be jeopardized.
Previously, the uncertainty of model parameters has been
considered in certain studies. For example, He et al. (2008)
Environ Sci Pollut Res (2017) 24:24284–24296
established a nonlinear chance-constrained programming
model to address porosity and intrinsic permeability uncertainty and optimize remediation strategy for a laboratoryscale SEAR system. Neuman et al. (2012) proposed a
multimodel approach for data-worth analysis based on a
Bayesian model averaging framework considering parameter
uncertainty. Zheng et al. (2011) and Wu et al. (2014) used a
probabilistic collocation method to assess parameter uncertainty in watershed water quality models and integrated surface water-groundwater models. Fan et al. (2016a) used a
hybrid sequential data assimilation and probabilistic
collocation method to analyze the uncertainty of stream flow
predictions in the form of parameter uncertainty. Zheng and
Han (2016) used Markov chain Monte Carlo algorithms to
assess the parameter uncertainty of watershed water quality
models. Sepulveda and Doherty (2015) analyzed the parameter uncertainty of east-central Florida groundwater flow
models, demonstrating that there is little justification for failing to accompany model predictions with the uncertainty estimate. Hou et al. (2016) analyzed the uncertainty of the simulation model using a Monte Carlo method, where the model
parameters were assumed independent. Li et al. (2017a)
assessed the performance of two parameter uncertainty analysis techniques (generalized likelihood uncertainty estimation
and Bayesian method) for a topographic hydrologic model.
Aquifer parameters, such as permeability and porosity, may
be statistically correlated (Boving and Grathwohl 2001; Wong
and Yeh 2002; Chen and Huang 2003; McPhee and Yeh 2006;
Morin 2006). Even though neglecting parameter correlation
may influence the reliability of uncertainty analysis, few studies focus on this issue and assume that those parameters are
independent. Therefore, there is an urgent need for a method
of handling parameter correlation in the process of uncertainty
analysis. Copula theory was proposed by Sklar (1959), who
pointed out that a multidimensional joint distribution function
can be decomposed into a corresponding edge distribution
function and a Copula function that determine the correlation
among the variables (Sklar 1959; Possolo 2010; Xu et al.
2010; Haslauer et al. 2012). Copulas have been used for
depicting correlation of data in biometric data studies
(Rukhin and Osmoukhina 2005), financial risk management
(Chai et al. 2008), rainfall data generation (Ghosh 2010; Xu
et al. 2010), bivariate hydrologic risk analysis and hydrologic
data assimilation (Fan et al. 2016b, 2017), structural reliability
analysis (Goda 2010; Jiang et al. 2014; Cui et al. 2017), ocean
platform design (Zhai et al. 2017), and conditional quantile
estimators (Rémillard et al. 2017). However, Copula theory
has not been applied in parameter correlation of SEAR simulation models in the context of remediation of groundwater
contamination.
In this study, a coupled Monte Carlo simulation and
Copula theory is proposed for uncertainty analysis of a
SEAR simulation model considering parameter correlation.
24285
In uncertainty analysis, Monte Carlo simulation is repeatedly involved, which results in enormous computational
burden and time requirements. A surrogate model, that
is, an approximation of the simulation model that has
similar input-output relationship to the simulation model
(Lee and Kang 2007; Raza and Kim 2007; Forrester and
Keane 2009; He et al. 2010; Li et al. 2010; Zhang et al.
2014; Nguyen et al. 2014; Li et al. 2017b), is employed
to reduce the computational burden and time requirements.
Specifically, the tasks of the present study involve the
following: (1) constructing a SEAR simulation model of
a DNAPL-contaminated site; (2) constructing a selfadaptive weight particle swarm optimization Kriging
(SAPSOKRG) surrogate model of the SEAR simulation
model; (3) proposing a t Copula function and a
Gaussian Copula function to describe parameter correlation, where the Akaike information criterion (AIC) and
the Bayesian information criterion (BIC) are used to select
an optimal Copula function for matching the relevant
structure of the parameters; (4) generating 5000 samples
of parameters that obey a multidimensional joint distribution function, and entering them into the SAPSOKRG
surrogate model to obtain and analyze the corresponding
output; and (5) generating 5000 additional independent
samples of parameters and entering them into the same
surrogate model to obtain output that is compared with
the output when parameter correlation was considered.
Methods
SEAR simulation model
A SEAR mathematical model was constructed including a
mass conservation equation and definite conditions (Mason
and Kueper 1996; Coulon et al. 2009; Luo and Lu 2014;
Jiang et al. 2015). The University of Texas Chemical
Compositional Simulator (UTCHEM) was applied to solve
the mathematical model. UTCHEM is a three-dimensional,
multiphase, and multicomponent finite-difference model,
which was originally developed to simulate surfactantenhanced oil recovery but was subsequently modified for applications involving surfactant-enhanced remediation of nonaqueous phase liquids contaminated aquifers (Delshad et al.
1996; Qin et al. 2007).
The mass conservation equation can be expressed as
∼ ∂ ϕC k ρk
∂t
2
0
13
!
→ !
!4 3 @ !
þ ∇ ∑ ρk C kl v l −ϕS l K kl ∇ C kl A5
l¼1
¼ Rk ; ðx; y; zÞ∈Ω; t ≥ 0 ;
ð1Þ
24286
Environ Sci Pollut Res (2017) 24:24284–24296
→
→
where K kl and !
v l are the dispersion tensor (m2 s−1) and Darcy
−1
velocity (m s ) of l, respectively, which can be expressed as
→
→
K kl ¼
Ddkl
aTl !
aLl −aTl
δij þ
vl δij þ
τ
ϕS l
ϕS l
→
→
k rl k
!
vl ¼ −
μl
vli vlj
; ðx; y; zÞ∈Ω; t ≥ 0
!
vl !
!
∇ Pl þ ρl g ∇ z ; ðx; y; zÞ∈Ω; t ≥ 0 ;
ð2Þ
ð3Þ
where k denotes the component index, i.e., 1, 2, 3 for water, oil,
and surfactant, respectively; l denotes the phase index; φ denotes
~ k denotes the overall concentration of k (volume
the porosity; C
fraction); ρk denotes the density of k (kg m−3); Ckl denotes the
concentration of k in l (volume fraction); Sl denotes the saturation
of l; Rk denotes the total source/sink of k (kg m−3 s−1); Ddkl
denotes the molecular diffusion coefficient of k in l (m2 s−1); τ
denotes the tortuosity; δij denotes the Kronecker delta function;
αTl and αLl denote the transverse and longitudinal dispersities of
l, respectively (m); vli and vlj denote thw Darcy velocities of l in
the i and j directions, respectively (m s−1) (Qin et al. 2007); !
k
!rl
denotes the relative permeability of the porous medium to l; k
denotes the intrinsic permeability tensor (m2); Pl denotes the
pressure of l (Pa); ρl denotes the density of l (kg m−3); g denotes
the gravity acceleration (m s−2); and z denotes the vertical distance (m) (Qin et al. 2007; Jiang et al. 2015). The definite conditions are elaborated in the BCase study^ section.
Self-adaptive weight particle swarm optimization Kriging
surrogate modeling method
The Kriging (KRG) method is expressed as.
yðxÞ ¼ f T ðxÞβ þ zðxÞ
ð4Þ
where f(x) = [f1(x), f2(x), ⋯ , fp(x)]T denotes the regression
functions,βdenotes the matrix of regression coefficients to be
estimated from the training samples (Hou et al. 2015), and z(x)
is the stochastic model with zero mean and variance σ2 (Lee
et al. 2002; Sakata et al. 2003; Jiang et al. 2015; Zhao et al.
2016).
For m training samples with input X = [(x1, x2,…xm)]T and
corresponding output response Y = [(Y1, Y2,…Ym)]T, the response for the predicted point x is
^^ þ rT ðxÞR−1 Y − F β
^^
ð5Þ
^yðxÞ ¼ f T ðxÞβ
^ are
where r(x), R, and the scalarβ
rðxÞ ¼ ½Rðx; x1 Þ; Rðx; x2 Þ; ⋯; Rðx; xm ÞT
2
3
Rðx1 ; x1 Þ ⋯ Rðx1 ; xm Þ
5
R¼4 ⋮
⋱
⋮
Rðxm ; x1 Þ ⋯ Rðxm ; xm Þ
^ ¼ F T R−1 F −1 F T R−1 Y
β
ð6Þ
ð7Þ
ð8Þ
where r(x) denotes the correlation vector between m sampling
points and the prediction point x, and F denotes the response
column vector.
T
F ¼ f T ðx1 Þ; f T ðx2 Þ; ⋯; f T ðxm Þ
ð9Þ
In this study, the Gaussian function was selected as a correlation function. The correlation function of the n-dimensional vectors xi and xj is
n 2
R θ; xi ; x j ¼ exp −θ ∑ xi −x j ð10Þ
k¼1
where θ is the parameter of the KRG model to be optimized
(Pan and Zhu 2011; Balesdent et al. 2013; Janusevskis and Le
Riche 2013; Lu et al. 2013).
Particle swarm optimization (PSO) is a random optimization algorithm (Hou et al. 2015). A self-adaptive weight was
employed to balance the local improvement ability and global
search ability of the PSO algorithm (Hou et al. 2015). The
self-adaptive weight can be expressed as
8
*
>
< wmin þ ðwmax −wmin Þ ð f − f min Þ ; f ≤ f
avg
f avg − f min
w¼
>
:
wmax ; f > f avg
ð11Þ
where wmin and wmax denote the minimum and maximum
values of the weight, respectively, f denotes the current value
of the objective function (minimization of the errors) of a
particle, favg and fmin denote the average value and the minimum of all current objective function values, respectively. In
this study, the self-adaptive weight particle swarm optimization (SAPSO) was used to optimize the parameters of KRG,
resulting in the SAPSOKRG method.
The construction of the SAPSOKRG surrogate model
is as follows (Fig. 1). A training set of 100 samples and
a testing set of 20 samples were collected in feasible
regions of the five input variables (parameters of SEAR)
using the Latin hypercube sampling (LHS) method,
which can capture more variability in the sample space
compared with the simple random sampling method (Xu
et al. 2005). The corresponding output (average nitrobenzene removal rates) was obtained from the SEAR
simulation model. Based on the input and output training sample sets, the surrogate model for approximating
the relationship between the input variables and the output in the SEAR simulation model was trained and constructed using the SAPSOKRG method. The testing
sample set was used to verify the accuracy of the surrogate model.
Environ Sci Pollut Res (2017) 24:24284–24296
24287
function that connects F(x1, x2, ⋯ , xn) with F1(x1) , F2(x2) ,
⋯ , Fn(xn) (Nelsen 2006; Tang et al. 2013; Li et al. 2014),
namely,
F ðx1 ; x2 ; ⋯; xn Þ ¼ C ½ F 1 ðx1 Þ; F 2 ðx2 Þ; ⋯; F n ðxn Þ
¼ C ðu1 ; u2 ; ⋯; un Þ
ð12Þ
The Sklar theorem converts the multidimensional joint cumulative distribution function of the random variables into a
one-dimensional edge cumulative distribution function of the
random variables F1(x1) , F2(x2) , ⋯ , Fn(xn) and a Copula
function C(u1, u2, ⋯ , un), describing the correlation among
the variables.
Multidimensional elliptical Copula functions, such as the t
Copula function and the Gaussian function, are used to describe positive and negative parameter correlation. Thus, the t
Copula function and the Gaussian Copula function were
adopted in this study to identify the relationship among the
parameters.
(1) t Copula function
The distribution and density functions of the multidimensional t Copula function are (Li et al. 2012)
Fig. 1 Flowchart of SAPSOKRG surrogate model
Copula theory
F(x1, x2, ⋯xn) is the multidimensional joint cumulative distribution function of the edge of the cumulative distribution
function F 1(x 1) , F2 (x2 ) , ⋯ , F n(x n). There is a Copula
Fig. 2 Flow chart of uncertainty
analysis
C ðu1 ; u2 ; ⋯un ; θÞ
¼ T n T −1 ðu1 Þ; T −1 ðu2 Þ; ⋯; T −1 ðun Þ; θ; ν
− νþn
. ν þ n ν n−1 0
2
Γ 2
1 þ ν1 ξ θ−1 ξ
−1 2 Γ
2
Dðu1 ; u2 ; ⋯un ; θÞ ¼ jθj
n
− νþ1
n 2
Γ νþ1
ξ2
2
∏ 1 þ νi
i¼1
ð13Þ
ð14Þ
24288
Environ Sci Pollut Res (2017) 24:24284–24296
where θ is the n-order symmetric positive-definite matrix
whose diagonal elements are 1; Tn(•, ⋯ , •; θ) is the n-dimensional t distribution function with correlation coefficient matrix θ and νdegrees of freedom; Tν−1(•) is the inverse function
of the one-dimensional t distribution function Tν(•); ξ' =
Tν−1(u1) , Tν−1(u2) , ⋯ , Tν−1(un) is the t distribution variable
(Nelsen 2006; Lei 2008).
(2) Gaussian Copula function
The distribution and density functions of the multidimensional Gaussian Copula function are (Li et al. 2012)
C ðu1 ; u2 ; ⋯un ; θÞ
¼ Φn Φ−1 ðu1 Þ; Φ−1 ðu2 Þ; ⋯; Φ−1 ðun Þ; θ
.
−1 2
1 0 −1 exp − ξ θ −I ξ
Dðu1 ; u2 ; ⋯un ; θÞ ¼ jθj
2
matrix θ; Φ−1(•) is the inverse function of the onedimensional standard normal distribution function Φ(•);
ξ' = Φ−1(u1) , Φ−1(u2) , ⋯ , Φ−1(un) is the standard normal
distribution variable; I is the unit matrix (Nelsen 2006;
Lei 2008; Li et al. 2012).
The Akaike information criterion (AIC) and the
Bayesian information criterion (BIC) were employed in
this study to identify whether the t Copula function or
the Gaussian function is the optimal Copula function for
matching the related structures of the parameters.
N
AIC ¼ ‐2 ∑ lnDðu1i ; u2i ; ⋯; uni ; θÞ þ 2k
ð17Þ
i¼1
N
ð15Þ
BIC ¼ ‐2 ∑ lnDðu1i ; u2i ; ⋯; uni ; θÞ þ klnN
ð16Þ
where ∑ lnDðu1i ; u2i ⋯; uni ; θÞ is the likelihood function of
where θ is the n-order symmetric positive-definite matrix whose diagonal elements are 1; |θ| is the determinant of θ; Φn(•, ⋯ , •; θ) is the n-dimensional standard
normal distribution function with correlation coefficient
the Copula function; D(u1i,u2i,...,uni) is the probability density
function of the edge distribution function; n is the dimension;
N is the number of parameters; θ represents the unknown
parameters, whose maximum likelihood estimate is
^θ ¼ argmaxLðθÞ; k is the number of Copula parameters;
Fig. 3 a Boundary conditions
and b parameter distribution
ð18Þ
i¼1
N
i¼1
Environ Sci Pollut Res (2017) 24:24284–24296
24289
Fig. 4 Contaminant
concentration distribution at the
initial time of the remediation
stage and locations of injection
and extraction wells. a
Concentration distribution at X–Y
plane (10th layer). b Extent of the
plume
(u1i,u2i,...,uni) is the empirical distribution of the original data
(x1i,x2i,...,xni). It can be calculated as follows:
8
rank ðx1i Þ
>
>
;
i ¼ 1; 2; ⋯; N
u1i ¼
>
>
>
N þ1
>
>
<
rank ðx2i Þ
u2i ¼
;
i ¼ 1; 2; ⋯; N
ð19Þ
N þ1
>
>
>
⋮
>
>
>
rank ðxni Þ
>
: uni ¼
;
i ¼ 1; 2; ⋯; N
N þ1
Table 1
Physical and chemical parameters of the case study
Parameters
Values
Water density (g/cm3)
Water viscosity (Pa·s)
Water residual saturation
Nitrobenzene density (g/cm3)
Nitrobenzene viscosity (Pa·s)
Nitrobenzene residual saturation
Nitrobenzene solubility in water (g/L)
Nitrobenzene/water interfacial tension (N/m)
Porosity
Permeability (zoneI) (m2)
Permeability (zoneII) (m2)
Longitudinal dispersivity (m)
Transverse dispersivity (m)
1.000
0.001
0.24
1.205
0.00168
0.17
1.9
0.02566
0.30
5.527 × 10−11
1.776 × 10−11
1.00
0.30
where rank(x1i) denotes the rank of x1i among x1 in an ascending order (Li et al. 2012).
Uncertainty analysis based on Copula theory
The specific steps of uncertainty analysis for the SEAR simulation model based on Copula theory and Monte Carlo simulation are as follows (Fig. 2):
(1) Porosity (n), permeability (k), aqueous phase dispersity
(αw), oil phase dispersity (αo) and microemulsion phase
dispersity (αm) are the main parameters influencing the
output of SEAR. Thus, these five parameters were selected as input variables, with the average nitrobenzene
removal rate as the output variable. The surrogate model
Table 2
The strategy for uncertainty analysis
Surfactant flush
duration (day)
Injection/extraction rate (m3/day)
88.17
In1
In2
In3
In4
Ex1
28.24
44.67
37.16
59.56
169.63
24290
Table 3
Environ Sci Pollut Res (2017) 24:24284–24296
Probability distribution of input variables
Input variables
Probability distribution
Porosity
Normal distribution
Permeability (D)
Logarithmic normal distribution
Aqueous phasedispersity (m)
Oil phase dispersity (m)
Normal distribution
Normal distribution
Microemulsion phase dispersity (m)
1D = 9.8697 × 10−13 m2
Normal distribution
for approximating the relationship between the input variables and the output in the SEAR simulation model was
constructed using the SAPSOKRG method.
(2) The number of Monte Carlo simulations was set to 5000
in the present study. Based on the Copula function and
the corresponding calculation formula, 5000 parameter
samples obeying the multidimensional joint distribution
were generated.
(3) The samples were entered into the SAPSOKRG surrogate model to obtain the corresponding output.
(4) Finally, uncertainty analysis was conducted by analyzing
the output.
has dimensions of 2, 2, and 2 m in the x, y, and z directions,
respectively. The former and border boundaries are no-flux
boundaries, and both left and right boundaries are first-type
boundary condition. The upward boundary is a water exchange
boundary, and the downward boundary is a no-flux boundary
(Fig. 3). The initial contaminant plume is shown in Fig. 4.
Based on the contaminant plume at the initial time of the
remediation stage, a SEAR with sodium lauryl sulfate as the
surfactant was designed with one pumping well and four injection wells (Fig. 4). A 5% surfactant solution (volume fraction) was injected into the injection wells. To maintain hydraulic balance, the total injection rate was set equal to the
pumping rate. The physical and chemical parameters of the
case study are listed in Table 1.
The uncertainty analysis focused on the five parameters (n,
k, αw, αo, αm) mentioned above under a certain strategy
(Table 2). The probability distribution of the five variables
was determined empirically (Table 3).
Results and discussion
Surrogate model
The proposed methods were evaluated by applying them to a
characteristic nitrobenzene-contaminated site. The objective
simulation layer was pore phreatic water. Groundwater flowed
from northeast to southwest. The simulation domain was generalized as a heterogeneous and isotropic 3D multiphase flow
and transport model. It was discretized into 10 layers, and each
layer was discretized into 70 × 30 grid blocks. Each grid block
The removal rates of the testing samples obtained from
the SAPSOKRG surrogate model were compared with
those from the simulation model. The comparison is
shown in Fig. 5.
Figure 5 shows that the results of the surrogate model results are close to those of the simulation model for the test
phase, and the maximum absolute residuals is less than 0.8%,
which indicates that the SAPSOKRG surrogate model accurately approximates the simulation model. Therefore, the
SAPSOKRG surrogate model may replace SEAR for uncertainty analysis.
Fig. 5 Distribution of results and residuals of simulation model vs.
SAPSOKRG surrogate model
Fig. 6 Comparison of Gaussian and t Copula functions
Case study
Environ Sci Pollut Res (2017) 24:24284–24296
24291
Fig. 7 Scatter diagram of
parameter values. X1: porosity;
X2: permeability; X3: aqueous
dispersity. In a, c, correlation was
ignored; in b and d, correlation
was considered
Construction of the multidimensional joint distribution
model of the parameters
The t Copula function and the Gaussian Copula function were
proposed for constructing the multidimensional joint distribution model of the parameters of SEAR in the present study.
The values of AIC and BIC are shown in Fig. 6.
Figure 6 shows that the AIC and BIC values of the t Copula
function are less than those of the Gaussian Copula function,
indicating that the t Copula function is the optimal Copula
function fitting to the related structure of the parameters (n,
k, αw, αo, αm). Therefore, the t Copula function was selected
to handle parameter correlation.
Uncertainty analysis
Uncertainty analysis of the simulation model was conducted
based on coupled Monte Carlo simulation and Copula theory
using the MATLAB software application.
Considering the parameter correlation based on the t
Copula function, 5000 samples were generated and entered
into SAPSOKRG surrogate model of the simulation model.
Fig. 8 Output of the simulation model when a parameter correlation was
considered and b parameter correlation was ignored
24292
Environ Sci Pollut Res (2017) 24:24284–24296
Fig. 9 Frequency histogram and cumulative distribution function of nitrobenzene removal rates when a parameter correlation was considered, b
parameter correlation was ignored. CDF of removal rate is shown in c
Subsequently, the corresponding output (nitrobenzene removal rates) was obtained. Furthermore, 5000 additional independent samples of parameters were extracted and entered into
the same surrogate model, and the output was compared with
the output that was obtained when parameter correlation was
considered. The necessity of considering parameter correlation was demonstrated by analyzing these outputs.
Comparison of considering and ignoring parameter
correlation
The scatter diagrams of three representative correlated parameters based on the t Copula function are shown in Fig. 7. It can
be seen that the distribution of the parameter samples varies
significantly, depending on whether the correlation was considered or not. This variation directly affects uncertainty analysis and remediation strategy optimization.
The outputs of the simulation model when parameter correlation was considered and when it was ignored are shown in
Fig. 8. It can be seen that the output of SEAR is uncertain
when the parameter changes within a reasonable scope regardless of whether parameter correlation was considered. The
nitrobenzene removal rates fluctuate in the mean value.
Therefore, the uncertainty analysis of the simulation model
is of great significance.
In addition, the frequency histogram and cumulative distribution function (CDF) of nitrobenzene removal rates when
parameter correlation was considered and when it was ignored
are shown in Fig. 9.
Figure 9a, b shows that the probability distributions under
the two conditions are different. The mean (82.35) and the
standard deviation (1.51) of the output when parameter correlation was ignored are greater than the mean (81.92) and the
standard deviation (1.11) of the output when parameter correlation was considered. In the former case, the mean and
Fig. 10 a Distribution of fluctuations in different intervals considering and ignoring parameter correlation. b Cumulative distribution function of the
removal rate fluctuation
Environ Sci Pollut Res (2017) 24:24284–24296
24293
Fig. 11 Probability of removal
rates in different intervals: a CDF
of removal rate, b probability in
different intervals
variance of removal rates are overestimated, affecting remediation strategy optimization. Figure 9c shows that the main
distribution interval of nitrobenzene removal rates when parameter correlation was considered is more centralized than
when parameter correlation was ignored. Moreover, the significance test (t test) was conducted using SPSS to verify
whether the difference between the two groups of data is statistically significant. The results indicated that the two conditions have significant difference.
To further analyze the uncertainty analysis results, the distribution of fluctuations in different intervals and the cumulative distribution function of the removal rates fluctuation were
drawn when parameter correlation was considered or ignored
(Fig. 10).
Figure 10a shows that the number of samples in the fluctuation intervals < 1 and 1–2 when parameter correlation was
considered is larger than the corresponding number when parameter correlation was ignored, whereas in the fluctuation
intervals 2–3, 3–4, and > 4, the number of samples is smaller
when parameter correlation was considered than when it was
ignored. Figure 10b shows that when fluctuation is less than 2,
the frequency when parameter correlation was considered is
almost 100%, whereas the frequency when parameter correlation was ignored is slightly greater than 80%. Therefore, Fig.
10 indicates that the amplitude of fluctuation intervals when
parameter correlation was considered is less than the corresponding amplitude when parameter correlation was ignored.
In conclusion, the comparison analysis results indicated
that considering the correlation among parameters is critical
Fig. 12 Probability of fluctuation
in different intervals: a CDF of
fluctuation, b probability in
different intervals
and more in line with reality in the uncertainty analysis of the
SEAR simulation model. If parameter correlation is ignored,
the reliability of uncertainty analysis results and optimal remediation strategy can be influenced.
Uncertainty analysis when parameter correlation was
considered
The probability of removal rates and fluctuation in different
intervals when parameter correlation was considered is shown
in Figs. 11 and 12, respectively.
Figure 11 shows that the probability of removal rates in the
interval 82–83 is the highest, followed by the intervals 81–82,
80–81, and 83–84. The probability of removal rates in the
intervals 78–79, 79–80, and > 84 is considerably low.
Figure 12 shows that the probability of the fluctuation interval
< 1 is the highest, followed by the interval 1–2, and the probability of the fluctuation interval > 2 is considerably low.
Compared with recent work of Hou et al. (2016), who
analyzed the uncertainty of the simulation model ignoring
parameter correlation, the results of uncertainty analysis in this
study, where parameter correlation was considered, are more
in line with reality and provide more insightful information,
such as the probability of removal rates and fluctuations in
different intervals. Therefore, considering parameter correlation is essential for uncertainty analysis, and its results should
be incorporated into SEAR strategy optimization. The
DNAPL removal rate of the optimal SEAR strategy should
24294
achieve the remediation objective with an acceptable probability considering parameter uncertainty and correlation.
Discussion on the relative computational cost
The main computational burden results from repeated numerical simulations. It required nearly 300 s of CPU time to run
the simulation model once. In total, the uncertainty analysis
based on the simulation model would require 10,000 simulation runs (35 days) if the surrogate model was not used. By
contrast, the surrogate-based optimization requires only 120
simulation runs (10 h) for training, testing the surrogate model. Thus, the introduction of a surrogate model considerably
reduces the computational burden and time requirements.
Conclusions
In this study, a coupled Monte Carlo simulation and Copula
theory was proposed for uncertainty analysis of SEAR simulation models. To reduce the enormous computational burden
and time requirements resulting from repeated and multiple
Monte Carlo simulation, the SAPSOKRG method was
employed to establish a surrogate model of the simulation
model. The t Copula function and the Gaussian Copula function were proposed and compared to describe parameter correlation. On this basis, 5000 samples of parameters obeying a
multidimensional joint distribution function were generated.
The corresponding output was obtained by entering the samples into the SAPSOKRG surrogate model. Subsequently, uncertainty analysis of the simulation model was conducted.
Furthermore, 5000 additional independent samples of parameters were extracted and entered into the same surrogate model
to obtain output that was compared with the output when
parameter correlation was considered. The conclusions of this
study are as follows:
(1) The t Copula function is more effective than the
Gaussian Copula function in handling parameter correlation. The application of Copula theory provides a new
tool for depicting the complex nonlinear correlation of
parameters in the SEAR simulation model.
(2) The mean and the standard deviation of nitrobenzene
removal rates when parameter correlation was ignored
were greater by 0.43 and 0.4, respectively, than the corresponding figures when parameter correlation was considered. The amplitude of fluctuation intervals when parameter correlation was considered was less than the corresponding amplitude when parameter correlation was
ignored. Moreover, the significance test indicated that
the two conditions have significant difference.
Therefore, neglecting parameter correlation would affect
the reliability of remediation strategy optimization. In
Environ Sci Pollut Res (2017) 24:24284–24296
conclusion, considering parameter correlation is critical
and more in line with reality in the uncertainty analysis
of the simulation model.
(3) The proposed method provided more insightful information, such as the probability of removal rates and fluctuations in different intervals. Uncertainty analysis based
on the t Copula function showed that the output of the
simulation model was uncertain when the parameter varied within a reasonable scope, the removal rates of nitrobenzene fluctuated, with mean 81.92 and standard deviation 1.11, and the probability of the fluctuation interval
0–2 was 95.52%. Thus, the results of uncertainty analysis should be incorporated into SEAR strategy
optimization.
The proposed method provides tools for effective analysis
of uncertainty problems in complex simulation models involving nonlinear correlation parameters. However, certain limitations are worth noting. Even though the SAPSOKRG surrogate model accurately approximates the simulation model (the
maximum absolute residual is less than 0.8%), the error may
affect the results of uncertainty analysis. Therefore, future
work should focus on surrogate model uncertainty.
Acknowledgements This study was financially supported by Project
funded by China Postdoctoral Science Foundation (No. 2016 M602388),
the National Key Research and Development Program of China (No.
2016YFC0402803-02), and the National Natural Science Foundation of
China (No. 41372237).
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