587 © IWA Publishing 2014 Water Science & Technology | 69.3 | 2014

Water quality model parameter identification of an open

channel in a long distance water transfer project based on

finite difference, difference evolution and Monte Carlo

Dongguo Shao, Haidong Yang, Yi Xiao and Biyu Liu

ABSTRACT

A new method is proposed based on the finite difference method (FDM), differential evolution

algorithm and Markov Chain Monte Carlo (MCMC) simulation to identify water quality model

parameters of an open channel in a long distance water transfer project. Firstly, this parameter

identification problem is considered as a Bayesian estimation problem and the forward numerical

model is solved by FDM, and the posterior probability density function of the parameters is deduced.

Then these parameters are estimated using a sampling method with differential evolution algorithm

and MCMC simulation. Finally this proposed method is compared with FDM–MCMC by a twin

experiment. The results show that the proposed method can be used to identify water quality model

parameters of an open channel in a long distance water transfer project under different scenarios

better with fewer iterations, higher reliability and anti-noise capability compared with FDM–MCMC.

Therefore, it provides a new idea and method to solve the traceability problem in sudden water

pollution accidents.

doi: 10.2166/wst.2013.753

Dongguo ShaoHaidong Yang (corresponding author)Yi XiaoState Key Laboratory of Water Resources &

Hydropower Engineering Science,Wuhan University,Luojia Hill,Wuhan 430072,ChinaE-mail: yanghd428@gmail.com

Biyu LiuManagement Science & Engineering Department,Economics and Management School,Jiulonghu Campus,Southeast University,Jiangning District,Nanjing 211189,China

Key words | difference evolution, finite difference, Monte Carlo, open channel, parameter

identification

INTRODUCTION

The long distance water transfer project (LDWTP) is animportant strategic project to realize reasonable allocationof water resources in China. It has an important role in alle-

viating the serious water shortage in northern China. Ingeneral, Open channel is one of the important forms inLDWTP, on which there are many hydraulic and crossingstructures, such as in the middle route of the South-to-

North water diversion project, there are 936 different typesof structures, among which are included 44 railways and571 bridges over the channel, so it is highly likely that

sudden water pollution accidents will happen (Lian et al.). Once these accidents occur, transportation and diffu-sion properties of pollutants should be informed to put

forward emergency countermeasures promptly, otherwise itwill cause inestimable consequences. As we all know, waterquality model parameters are the basic data to build waterquality models and predict water quality change (Ferrer

et al. ). So the identification of water quality model par-ameters plays a very important role in predicting pollutanttransport processes in sudden water pollution accidents.

At present, the main methods to identify water qualitymodel parameters include the theoretical formula, empiricalformula and tracer test (Liu et al. ). The former two

methods are used to solve problems usually in a forwardapproach. If the information about water depth, flow vel-ocity and shear flow velocity is known, water qualitymodel parameters can be estimated by either of these two

methods. But in practical application, the velocity distri-bution is very difficult to obtain and parameters varywidely with different conditions, which increases the diffi-

culty to find a unified expression. The third method isused to solve problems in an inverse approach, by whichthe identified parameters’ values can reflect the transform-

ation characteristics of pollutant migration in the channelaccurately (Kowalsky et al. ). And it is mainly basedon optimization and uncertainty analysis (Zhu et al. ).With the development of computer technology, trace tests

based on optimization have been widely used, such as thesimplex method (Xue et al. ), particle swarm optimiz-ation (Afshar et al. ) and genetic algorithm (Liu et al.

588 D. Shao et al. | Water quality model parameter identification based on FDM–DEMCMC Water Science & Technology | 69.3 | 2014

), etc. But there are many strict limit conditions using

these methods caused by algorithm, data and factitious sub-jective factors, and the non-identificability is also increasedwith the parameter number. Regarding the trace test

method based on the uncertainty analysis, even thoughuncertainty factors are considered, to some extent, itavoids the decision risk caused by the distortion of the ‘opti-mal’ parameters; the calculation amount will increase

exponentially for the parameters generated randomly(Chen et al. ; Amirov et al. ). Therefore, either ofthem has some drawbacks in identifying water quality

model parameters in sudden water pollution accidents.However, water quantity during the operation of open

channels in LDWTPs is controlled and scheduled by the

sluice structures, for which its parameter identification pro-blem becomes a complex nonlinear problem (Kowalskyet al. ). Therefore, a new method is proposed to solvethis problem based on the finite difference method (FDM)

and Markov Chain Monte Carlo (MCMC) and differentialevolution method (DE) in this paper. And in order toverify this method’s accuracy, efficiency and anti-noise capa-

bility, a canal in the middle route of the South-to-Northwater diversion project is selected as an application subject,whose longitudinal dispersion coefficient is identified by this

proposed method.

Figure 1 | The finite difference format to the wind of water quality model equation.

DEVELOPING A METHOD TO SOLVE THE PROBLEM

Problem description

The water quality model parameters are identified accordingto input and output measurement information based on a

certain criterion. The relationship between them, therefore,can be described by the general form of the nonlinear model

y ¼ f(x, θ)þ e (1)

where x is the system input, y is the system output, θ is the

unknown parameters, e is the noise, f represents mappingfrom model space to parameter space, which is a functionexpressed by analytic or numerical solution. When pollution

source’ properties (such as position, emission intensity, etc.)are known, the solution to Equation (1) can be expressed by

C(x, t) ¼ C(x, tjθ)þ e (2)

where θ is parameters sets, which control the concentrationdistribution, C(x, t) and C(x, t|θ) are respectively the

measurement values and the prediction values of pollutants

concentration (g L�1).For LDWTP, it is a system made-up of a long water con-

veyance canal (including aqueducts, inverted siphons,

culverts, etc.) and a plurality of gates. And it is also aseries system consisting of many canal pools which aredivided by gates (Liu et al. ). So the law of pollutanttransportation and diffusion in LDWTPs can be described

by one-dimensional solute convection diffusion equations

@(AC)@t

þ @(QC)@x

¼ @

@xAEx

@C@x

� �þWc þ S1 �AKC (3)

where A is the cross-sectional area (m2), C is the cross-

sectional average pollutants concentration (g L�1), Q iscross-sectional flow (m3 s�1), t is the time (s), x is the direc-tion of mean flow velocity (m), Ex is the longitudinaldispersion coefficient (m2 s�1), S1 is internal source and

sink (g m�1 s�1), K is reaction rate (s�1),Wc is the exogenousinput terms (g m�1 s�1).

However, the fluid properties and hydraulic character-

istics in LDWTP vary frequently, for which no analyticalsolution is available. Only the numerical discrete methodcan be adopted to solve the water quality model. In this

paper, the implicit finite difference scheme is used to solvethe model by spatial discretization, which is shown inFigure 1.

So Equation (1) is discretized

@(AC)@t

¼ (AC)kþ1i � (AC)ki

Δt@(QC)@x

¼ (QC)kþ1i � (QC)kþ1

i�1

Δxi�1

@

@xAEx

@C@x

� �¼ (AEx)

kþ1i

Ckþ1iþ1 � Ckþ1

i

(Δxi þ Δxi�1)=2

"

�(AEx)kþ1i�1

Ckþ1i � Ckþ1

i�1

(Δxi�1 þ Δxi�2)=2

#1

Δxi�1

S1 þWc �AKC ¼ (S1 þWc)kþ1i � (AKC)kþ1

i

8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>:

(4)

589 D. Shao et al. | Water quality model parameter identification based on FDM–DEMCMC Water Science & Technology | 69.3 | 2014

Here we use a forward time difference, Ckþ1i is the pollu-

tant concentration of the ith canal pool at the end of kperiod. According to Equation (4), we can obtain the follow-ing equation

Δxi�1(S1 þWc)kþ1i ¼ Ckþ1

iAkþ1

i Δxi�1

ΔtþQkþ1

i

(

þ 2(AEx)

kþ1i

(Δxi þ Δxi�1)þ (AEx)

kþ1i�1

(Δxi�1 þ Δxi�2)

" #

þ Δxi�1(AK)kþ1i

)

� Cki

Aki Δxi�1

Δt

� �� 2(AEx)

kþ1i

(Δxi þ Δxi�1)Ckþ1iþ1

� 2(AEx)

kþ1i�1

(Δxi�1 þ Δxi�2)þQkþ1

i�1

" #Ckþ1i�1

(5)

Let ai ¼ 2(AEx)

kþ1i�1

(Δxi�1 þ Δxi�2)þQkþ1

i�1

" #;Ci�1 ¼ Ckþ1

i�1 ,Ci ¼ Ckþ1i

Cki

!

bi ¼Akþ1

i Δxi�1

ΔtþQkþ1

i

þ 2(AEx)

kþ1i

(Δxi þ Δxi�1)þ (AEx)

kþ1i�1

(Δxi�1 þ Δxi�2)

" #

þΔxi�1(AK)kþ1i , �Ak

i Δxi�1

Δt

�

ci ¼ � 2(AEx)kþ1i

(Δxi þ Δxi�1), Ciþ1 ¼ Ckþ1

iþ1 , di ¼ Δxi�1(S1 þWc)kþ1i

Then Equation (4) can be transformed into

aiCi�1 þ biCi þ ciCiþ1 ¼ di (i ¼ 1, 2, � � � ) (6)

Equation (6) can be solved by combining with theupstream and downstream boundary conditions.

Therefore, according to Equations (2) and (6), the identi-fication problem is equivalent to identifying the parameterswith limited concentration measurement data. In order to

deal with the uncertainty of model structure and measure-ment data, a new method is proposed in this paper.

Method

According to the Bayesian theorem, the Bayesian inferencecan be stated as follows

p(θjy) ¼ p(θ)p(yjθ)p(y)

∝ p(θ)p(yjθ) (7)

where θ is the unknown parameters, y is the observed data,

p(y|θ) is the likelihood function, p(θ|y) is the parameters’posterior probability density function, p(θ) is the parameters’joint a priori probability density function.

For θ¼ (θ1, θ2… θm), if their distribution intervals areknown and independent of each other, then

p(θ) ¼Ymi¼1

p(θi) ¼Qmi¼1

1bi � ai

θi ∈ [ai, bi]

0 else

8<: (8)

where m is the number of model parameters, ai and bi areconstants, bi> ai.

Assuming measurement error and prediction error areexpressed respectively as εj, ej ( j¼ 1,…,M), these errorsare irrelevant and obey a Laplace distribution, and each

point is independent, then

p(yjθ) ¼YMj¼1

p(yjjθ)

∝1PM

j¼1 (2σ f,j þ 2σd,j)exp �

XMj¼1

jCj(x, t)�Cj(x, tjθ)jσ f,j þ σd,j

24

35

(9)

According to Equation (7), then

p(θjy) ¼ L(yjθ) � p(θ)

∝p(θ)PM

j¼1 (2σ f,j þ 2σd,j)exp �

XMj¼1

jCj(x, t)�Cj(x, tjθ)j(σ f,j þ σd,j)

24

35

(10)

Therefore, the model parameter identification problemis equivalent to solving p(θ|y). But for more complex pollu-

tant concentration or larger space dimensions of modelparameters, it is very difficult to solve p(θ|y) only by Baye-sian inference (Dotto et al. ).

In recent decades, the MCMC method combining withBayesian inference has been proved useful as a samplingmethod for such problems and DE is a novel parallel

direct search method (Liu et al. ; McCarthy et al.; Suribabu ). However, analyzing the operatingmechanism of DE and MCMC, DE often appears a ‘prema-ture’ phenomenon or slow convergence, MCMC depends

590 D. Shao et al. | Water quality model parameter identification based on FDM–DEMCMC Water Science & Technology | 69.3 | 2014

largely on the initial sample and the initial covariance. So a

new method integrating DE and MCMC which can avoidthese problems is proposed in this paper; the detailed sol-ving steps of this new method are as follows:

(i) The study area is divided into N sections by spatial dis-cretization and each section has only a little change

over time and space.(ii) Determining the population sizeNP and generating the

initial populations X(i)(1), X(i)(2),…, X(i)(NP) in a

range of parameters randomly according to thenumber of variables θ.

(iii) Setting X(i) (1) as the initial value and calculating thecorresponding pollutant concentration, then getting p

(X(i)) which is the conditional probability density func-tion of X(i).

(iv) Taking two values X(i)(a) and X(i)(b) at the same prob-

ability by DE, then generating new testing parametersaccording to the following equation

Xð�Þ ¼ XðiÞð1Þ þ γðXðiÞðbÞ �XðiÞðaÞÞ þ ε (11)

where a, b∈ [0, NP–1], γ is constant, ε is a randomerror.

(v) Calculating pollutant concentration and conditionalprobability density function corresponding to X(*)

through the water quality model.

(vi) Specification of likelihood function which can reflectthe relationship between the model parameters andthe measurement data.

(vii) Calculating the posterior probability density function p(X(*)) corresponding to X(*).

(viii) Getting the accept probability A(X(i), X(*)) at whichMarkov chain moves from X(i) to X(*) by the following

Figure 2 | A schematic diagram of an open channel.

formulation

A(X(i), X(�)) ¼ min 1,p(X(�))p(X(i)jX(�))p(X(i))p(X(�)jX(i))

� �

¼ min 1,p(X(�))p(X(i))

� �(12)

(ix) Generating a random number R (R∈ [0, 1]) with uni-form distribution and if R<A(X(i), X(*)), thenX(iþ1)¼X(*), otherwise, X(iþ1)¼X(i).

(x) Repeating steps (i)∼ (ix) until it reaches a predeterminednumber of iterations.

RESULTS AND DISCUSSION

Numerical simulation analyses

According to the convection–diffusion equation, the mainparameters of thewater quality model are flow velocity, longi-

tudinal dispersion coefficient and reaction coefficient. Thelongitudinal dispersion coefficient is a fundamental par-ameter in hydraulic modeling of river pollution and an

index tomeasure themixing intensity of the pollutants in natu-ral streams (Tayfur ). It is becoming more and moreimportant in researching the identification problem of a

sudden water pollution accident (Wilkerson & McGahan; Khatua & Patra ). Therefore, an open channelwith 3 km length in the Middle Route of a South-to-Northwater Diversion project in China is taken as an example to

identify the longitudinal dispersion coefficient under differentscenarios based on design date, without considering any bio-logical chemical conditions, as shown in Figure 2. Assuming

591 D. Shao et al. | Water quality model parameter identification based on FDM–DEMCMC Water Science & Technology | 69.3 | 2014

the pollutant concentration of the inflowwater fromupstream

concentration is 1.0 mg/L. The flow field distribution is u¼0.5þ 0.001x.Even though some literature show the longitudi-nal dispersion coefficient is related to the fluid properties,

hydraulic characteristics and geometric parameters (Kashefi-pour & Falconer ). But for the inverted siphons and gatesin LDWTP, the solutions solved by theoretical and empiricalformulations can’t accurately reflect the diffusion properties

Table 1 | Pollutant concentration data for parameter identification mg/L

Longitudinal separation/mt/min 500 1,000 1,500 2,000 2,500 3,000

5 0.2374 0.0669 0.0203 0.0065 0.0022 0.0008

10 0.4082 0.1594 0.0610 0.0235 0.0092 0.0037

15 0.5311 0.2553 0.1157 0.0513 0.0226 0.0101

20 0.6195 0.3437 0.1768 0.0876 0.0426 0.0207

25 0.6831 0.4203 0.2385 0.1292 0.0681 0.0357

30 0.7288 0.4838 0.2966 0.1728 0.0974 0.0544

35 0.7618 0.5352 0.3487 0.2156 0.1287 0.0758

40 0.7854 0.5760 0.3938 0.2558 0.1603 0.0989

45 0.8025 0.6078 0.4318 0.2922 0.1906 0.1224

50 0.8147 0.6323 0.4631 0.3240 0.2187 0.1454

Figure 3 | Iterative curves of unknown parameters respectively based on (a) FDM–DEMCMC a

in the channel. Here the ‘twin experiment’ which is proved

an effective means to identify water quality model parameters(Friedrichs et al. ) is chosen to verify this proposedmethod. According to the principle, by which those segments

with approximately same cross section shape and velocity dis-tribution are classified as a canal pool, the channel can bedispersed into N¼ 6 canal pools whose true values (Ex)r(r¼ 1, 2,…, N) are 50, 70, 90, 110, 130 and 140 m2/s.

Scenario 1: steady non-uniform flow

If the flow is identically equal to 10 m3/s, the pollutant con-centration for parameter identification is obtained in Table 1

by the water quality model.Assuming NP¼ 20 and γ¼ 0.1, Iterative curves of

unknown parameters with error level σ¼ 0.01 by 20,000

iterations are shown in Figure 3. The results of parameteridentification under different noise are shown in Table 2.

Figure 3(a) shows that the values of the Markov Chain

((Ex)1, (Ex)2 and (Ex)3), ((Ex)4 and (Ex)5) are close to the truevalue through 100 and 200 iterations. Figure 3(b) shows thatthe iterations are 400 and 800 when the values of MarkovChain ((Ex)1, (Ex)2 and (Ex)3), ((Ex)4 and (Ex)5) are close to

the true value.

nd (b) FDM–MCMC.

Table 2 | Parameter identification results under different noise by FDM–DEMCMC and FDM–MCMC

σ¼ 0.01 σ¼ 0.05 σ¼ 0.1Identification value(m2/s)

Relative error(%)

Identification value(m2/s)

Relative error(%)

Identification value(m2/s) Relative error (%)

r A B A B A B A B A B A B

1 50.037 49.988 0.07 0.02 50.240 50.094 0.48 0.19 50.355 50.196 0.71 0.39

2 69.971 70.043 0.04 0.06 70.443 69.363 0.63 0.91 71.946 69.048 2.78 1.36

3 89.880 89.968 0.13 0.04 90.685 92.650 0.76 2.94 91.868 98.955 2.08 9.95

4 109.992 110.190 0.01 0.17 109.376 112.619 0.57 2.38 110.672 107.981 0.61 1.84

5 128.691 130.609 1.01 0.47 129.395 139.764 0.47 7.51 122.028 124.902 6.13 3.92

6 141.685 130.705 1.20 6.64 149.569 136.184 6.84 2.73 152.670 155.628 9.05 11.16

Note: A and B denote FDM–DEMCMC and FDM–MCMC, respectively.

592 D. Shao et al. | Water quality model parameter identification based on FDM–DEMCMC Water Science & Technology | 69.3 | 2014

The average relative errors of identification results withdifferent measurement errors by the FDM–DEMCMCmethod are 0.41, 1.62 and 3.56%, respectively. But they are

respectively 1.23, 2.78 and 4.77% by FDM–MCMC, asshown in Table 2.

Scenario 2: unsteady non-uniform flow

If the flow is a function of time Q(t)¼ 10þ 0.001t, the pollu-tant concentration data for parameters identification is

obtained by the FDM method as shown in Table 3. Iterativecurves of unknown parameters with error level σ¼ 0.01 by20,000 iterations are shown in Figure 4. The results of par-

ameter identification under different noise are shown inTable 4.

Figure 4 shows that the values of the Markov Chain

((Ex)1, (Ex)2 and (Ex)3) are close to the true values throughabout 100 and 400 iterations by the two methods, the

Table 3 | Pollutant concentration data for parameter identification mg/L

Longitudinal separation/m

t/min 500 1,000 1,500 2,000 2,500 3,000

5 0.2374 0.0669 0.0203 0.0065 0.0022 0.0008

10 0.4035 0.1568 0.0599 0.0230 0.0090 0.0036

15 0.5199 0.2477 0.1116 0.0493 0.0217 0.0097

20 0.6017 0.3294 0.1681 0.0828 0.0401 0.0195

25 0.6593 0.3985 0.2238 0.1203 0.0631 0.0330

30 0.7000 0.4547 0.2750 0.1587 0.0889 0.0494

35 0.7288 0.4994 0.3200 0.1957 0.1159 0.0679

40 0.7493 0.5342 0.3584 0.2298 0.1425 0.0873

45 0.7641 0.5611 0.3902 0.2600 0.1677 0.1067

50 0.7747 0.5817 0.4161 0.2861 0.1906 0.1254

values of the Markov Chain ((Ex)4 and (Ex)5) are close tothe true values through about 200 and 800 iterations.

The average relative errors with different measurement

error are respectively 0.78, 1.63 and 4.4% by FDM–

DEMCMC, but they are respectively 0.87, 3.97 and 5.39%by FDM–MCMC, as shown in Table 4.

According to Tables 1–4 and Figures 3 and 4, comparing

with FDM–MCMC, the FDM–DEMCMC has the followingadvantages:

(a) Wider applicabilityWhen σ¼ 0.1, the average relative errors obtained by theFDM–DEMCMC are all <10% for both scenarios. Itshows the method has strong applicability. It is not

only applicable to the steady flow scenario, but also tothe unsteady flow scenario.

(b) Higher identification accuracy

The average relative error is decreased 0.82, 1.16 and1.21% under the first scenario and 0.09, 1.29 and 0.9%in the second scenario. So in the same circumstances,

the precision of results obtained by FDM–DEMCMCare higher than by FDM–MCMC.

(c) Stronger anti-noise abilityThe average relative errors obtained by the two methods

are both less than the corresponding measurementerrors. But the results obtained by FDM–DEMCMCare lower.

(d) Less iterationAccording to Figures 3 and 4, about three-quarters of theiterations, through which the values of the Markov

Chain are close to the true value can be reduced.

In summary, it is quicker and better to identify the longi-tudinal dispersion coefficient by the proposed method. And

this method has wide applicability, high identification accu-racy and anti-noise ability.

Figure 4 | Iterative curves of unknown parameters respectively based on (a) FDM–DEMCMC and (b) FDM–MCMC.

Table 4 | Parameter identification results under different noise by FDM–DEMCMC and FDM–MCMC

σ¼ 0.01 σ¼ 0.05 σ¼ 0.1Identification value(m2/s)

Relative error(%)

Identification value(m2/s)

Relative error(%)

Identification value(m2/s) Relative error (%)

r A B A B A B A B A B A B

1 49.943 50.014 0.11 0.03 50.321 50.148 0.64 0.30 50.777 49.967 1.55 0.07

2 69.856 70.046 0.21 0.07 70.507 69.269 0.72 1.04 71.325 69.390 1.89 0.87

3 89.879 90.142 0.13 0.16 90.279 93.348 0.31 3.72 94.274 99.589 4.75 10.65

4 110.147 109.934 0.13 0.06 109.231 110.775 0.70 0.70 111.546 107.939 1.41 1.87

5 131.433 131.426 1.10 1.10 128.017 135.627 1.53 4.33 118.574 125.762 8.79 3.26

6 144.208 134.696 3.01 3.79 148.242 150.404 5.89 7.43 151.992 161.836 8.57 15.60

Note: A and B denote FDM–DEMCMC and FDM–MCMC, respectively.

593 D. Shao et al. | Water quality model parameter identification based on FDM–DEMCMC Water Science & Technology | 69.3 | 2014

CONCLUSIONS

According to the water environment information, based on

the framework of Bayesian theory, FDM, DE and MCMCsampling, an improved method is proposed in this paper.The main conclusions are summarized as follows:

(a) This proposed method can be used to identify the waterquality model parameters better with some certain

known information. In addition, it is not only suitableto be applied in the trapezoidal channel, but also appli-cable to natural rivers. So this proposed method hasgood applicability and scalability.

(b) It is quicker to get close to true values to construct theunknown parameters’ Markov Chain value on thebasis of Bayesian–MCMC and the idea of DE. Which

is to say, it can reduce the iterations effectively by theimproved method. This proposed method transforms a

594 D. Shao et al. | Water quality model parameter identification based on FDM–DEMCMC Water Science & Technology | 69.3 | 2014

probability algorithm stochastically into a relatively defi-

nite algorithm to some extent.(c) From the numerical experiment results, the noise level

of the observation data has a certain influence on the

accuracy of parameter identification. The lower thenoise level, the higher the precision of parameters.

Thus, it can estimate water quality model parametersbetter when the DE method is introduced to the MCMCmethod. On the one hand, it can realize numerical calcu-lations to the high dimensional space without explicit

mathematical expressions for the probability density function;on the other hand it can reduce the iterations compared withFDM–MCMC. In addition, considering the time effect and

using the measurement data to update the pollution sourceinformation will be the future research direction.

ACKNOWLEDGEMENTS

The authors thank the referees and the editor for their valu-able comment and suggestions on improvement of thispaper. The study is supported by the national science andtechnology major special project (No. 2012ZX07205005)

and the public welfare industry research project of theMinistry of Water Resources of the People’s Republic ofChina (No. 201101063).

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First received 3 July 2013; accepted in revised form 6 November 2013. Available online 20 November 2013