Parameters estimation of Muskingum model is very significative in both exploitation and utilization of water resources and hydrological forecasting. The optimal results of parameters directly affect the accuracy of flood forecasting. This paper considers the parameters estimation problem of Muskingum model from the following two aspects. Firstly, based on the general trapezoid formulas, a class of new discretization methods including a parameter θ to approximate Muskingum model is presented. The accuracy of these methods is second-order, when θ≠1/3. Particularly, if we choose θ=1/3, the accuracy of the presented method can be improved to third-order. Secondly, according to the Newton-type trust region algorithm, a new Newton-type trust region algorithm is given to obtain the parameters of Muskingum model. This method can avoid high dependence on the initial parameters. The average absolute errors (AAE) and the average relative errors (ARE) of the proposed algorithm of parameters estimation for Muskingum model are 8.208122 and 2.462438%, respectively, where θ=1/3. It is shown from these results that the presented algorithm has higher forecasting accuracy and wider practicability than other methods.
1. Introduction
Flood routing in open channels is a very important tool in the design of flood protection measures to estimate how the proposed measures will affect the behavior of flood waves in rivers so that enough protection and economic solutions can be found [1]. In general, flood routing procedures can be classified as either hydrologic or hydraulic. Hydrologic routing method is on the basis of the storage-continuity equation, whereas hydraulic routing method is on the basis of both continuity and momentum equations. One of the hydrologic routing approaches, the Muskingum method, was first developed by McCarthy for flood control studies in the Muskingum river basin in Ohio.
The following continuity and storage equations are used to describe the Muskingum model:
(1)dW(t)dt=I(t)-Q(t),
where W(t) represents the channel storage at time t; I(t) and Q(t) represent the rates of inflow and outflow at time t, respectively. The linear Musikingum model is
(2)W(t)=KxIt+1-xQt,
where K is a storage time constant for the river reach and x is a weighting factor commonly varying between 0.0 and 0.3 for the river channel.
It is worth mentioning that the parameters K and x in (2) are graphically estimated by a trial-and-error procedure [2]. If x is obtained, the values of [xI(t)+(1-x)Q(t)] are calculated by using observed data in both upstream and downstream and plotted against W. The particular value which generates the loop is accepted as the best estimation of x. The slope of the straight line fits through the loop derives K. Although the trial-and-error procedure has been used for many years, it is time consuming and subjective interpretation. Therefore, in order to avoid subjective interpretations of observed data in estimating K and x, the following finite difference scheme [3] is used to the resulting ordinary differential equation (1):
(3)W(ti)-W(ti-1)Δt=I(ti)+i(ti-1)2-Q(ti)+Q(ti-1)2,
where
(4)W(ti)=K[xI(ti)+(1-x)Q(ti)],Wti-1=KxIti-1+1-xQti-1,i=2,…,n,
and Q(ti) represents observed outflow discharges at time ti, I(ti) represents the inflow discharge at time interval ti, Δt is the time step, and n is the total time number. Substituting (4) into (3), we have
(5)Q(ti+1)=c0I(ti+1)+c1I(ti)+c2Q(ti),
where
(6)c0=-Kx+Δt/2K(1-x)+Δt/2,c1=Kx+Δt/2K(1-x)+Δt/2,c2=K(1-x)-Δt/2K(1-x)+Δt/2.
By minimizing the sum of the square of the deviations between observed and calculated outflows, we obtain the following objective function:
(7)minf=∑i=1n-1Q(ti+1)-c0I(ti+1)-c1I(ti)-c2Q(ti)2.
In the past two decades, in order to obtain the parameters K and x, some researchers adopted many optimization methods to solve the above optimization problem (7). Until now, these methods can be classified into traditional optimization methods and intelligent algorithms. The existing traditional methods include nonlinear programming method (NPM) [4], the least-square method (L-SM) [5, 6], method of trial-and-error (TAE) [2], the minimum area method (MAM) [1, 7], the Broydene-Fletchere-Goldfarbe-Shanno (BFGS) technique [8], test-method and least residual square method [9], and Nelder-Mead simplex method [10]. These traditional methods have their own advantages, but there also exist complex calculations and poor generality disadvantages, and some of these methods are related to the selection of initial point, which is easy to fall into local optimum. The existing intelligent algorithms include genetic algorithm [11], harmony search [12], Gray-encoded accelerating genetic algorithm [9], particle swarm optimization [13], immune clonal selection algorithm [14], and differential evolution algorithm [15]. The advantages of those intelligent algorithms are that the global search ability is strong and the program is relatively simple to design. However, these methods have unavoidable disadvantages of slow convergence precision, low precision solution in limited generations, and premature convergence.
In a word, almost all of the above methods used to estimate parameters K and x are based on (3). We know that the truncation error of (3) is only O(Δt)2. Thus, it is very necessary to design a higher accuracy method to discrete differential equation (1). Based on the generalized trapezoidal formula [16], this paper will first develop a class of new difference scheme which contains a parameter θ to approximate the differential equation (1). Then, similar to the above optimization problem (7), we can obtain a class of new unconstrained nonlinear optimization problems which contain parameter θ. It is noted that the accuracy of the presented difference schemes to approximate the differential equation (1) can be improved to third order, when θ=1/3. In other words, we can get a higher accuracy parameter estimation model, if θ=1/3. In addition, based on the Newton-type trust region algorithm [17], this paper will design a new Newton-type trust region algorithm (NN-TTRA) for estimating the Muskingum model parameters. Briefly speaking, this algorithm can avoid high dependence on the initial parameters and find the global optimization solution quickly.
2. A New Parameter Estimation of Muskingum Model2.1. Model Description
At first, (1) can be rewritten as follows:
(8)dW(t)dt=W(t)+P(t),
where P(t)=I(t)-Q(t)-W(t).
Then, applying the generalized trapezoidal formula [16] GTF (θ) of Chawla et al. to (8), we can obtain
(9)W~(ti)=W(ti+1)-ΔtdW(ti+1)dt,(10)Wti+1=W(ti)+Δt21-θdWtidthhhhhhhhhhhh+θdW~(ti)dt+dW(ti+1)dt.
Combining (8), (9), and (10), we get
(11)W~(ti)=(1-Δt)W(ti+1)-ΔtP(ti+1),(12)Wti+1=W(ti)+Δt2P(ti+1)+P(ti)+(1-θ)W(ti)W~hhhhhhhhhhhh+θW~(ti)+W(ti+1).
Substituting (11) into (12), we finally obtain
(13)1-Δt2θWti+1=1-Δt2θW(ti)+Δt2I(ti)-Q(ti)+Δt21-ΔtθI(ti+1)-Q(ti+1).
Obviously, when θ=0, (13) becomes (3).
At last, substituting (4) into (13), we have
(14)Q(ti+1)=a0I(ti+1)+a1I(ti)+a2Q(ti),
where
(15)a0=-Kx(1-Δt/2θ)+Δt/2(1-Δtθ)K(1-x)(1-Δt/2θ)+Δt/2(1-Δtθ),a1=Kx(1-Δt/2θ)+Δt/2K(1-x)(1-Δt/2θ)+Δt/2(1-Δtθ),a2=K(1-x)(1-Δt/2θ)-Δt/2K(1-x)(1-Δt/2θ)+Δt/2(1-Δtθ).
By minimizing the residual sum of squares between observed and calculated outflows, the objective function can be given as follows:
(16)minf=∑i=1n-1Q(ti+1)-a0I(ti+1)-a1I(ti)-a2Q(ti)2.
2.2. Truncation Errors Analysis
Here, we analyze the local truncation error of (13) at time direction [16]. Firstly, from (9), we have
(17)dW~(ti)dt=dWti+1dt-Δtd2Wti+1dt2.
Then, by using Taylor expansion, (10) can be written as follows:
(18)-Δt312d3Wtidt4-Δt424d4Wtidt4+OΔt5=Δtθ2dW~tidt-dWtidt+OΔt5.
Substituting for dW~(ti)/dt from (17) into (18), the truncation errors of generalized trapezoidal formula can be obtained as follows:
(19)E=θ4-112Δt3d3W(ti)dt3+θ6-124Δt4d4W(ti)dt4+OΔt5.
From (19), it is clear that the order of our presented method in (13) is O(Δt)2 if θ≠1/3. In particular, for θ=1/3, our presented method in (13) is third-order accuracy.
3. Algorithm Construction3.1. Newton-Type Trust Region Algorithm
As far as we know, the basic trust region algorithm used to solve the following unconstrained optimization problem
(20)minx∈Rnf(x)
was first presented clearly in [19]. Recently, many researchers have proposed some improved trust region algorithm to solve (20); see for example Esmaeili and Kimiaei [20] and Amini and Ahookhosh [21].
Here, we assume that xj is the jth iterative point, fj=f(xj), gj=∇f(xj), and Bj is the jth iteration of Hesse matrix ∇2f(xj); the trust region subproblem, which is at the jth iterative step of problem (20), can be formulated as follows:
(21)minqj(d)=gjTd+12dTBjd,.s.t...d≤Δj,
where Δj is a trust region radius and d∈Rn is an iterative step and · denotes the Euclidian norm of vectors or its induced matrix norm.
Denote
(22)Δfj=f(xj)-f(xj+dj),(23)Δqj=qj(0)-qj(dj),(24)rj=ΔfjΔqj;
the general trust region algorithm [17] for solving the unconstrained optimization problem (20) is given as follows.
Algorithm 1.
Step 0. Given a starting point x0∈Rn, Δ0>0 is the initial trust region radium, ɛ>0, B0=Hesse(x0). Set j:=0.
Step 1. Calculate gj, if gj≤ɛ, and stop iteration; otherwise, go to Step 2.
Step 2. Utilize the smoothing Newton method to obtain the solution dj of the subproblem (21).
Step 3. Calculate the rj of formula (24).
Step 4. Regulate trust region radius. If rj<0.25, let Δj+1:=0.5Δj; if rj>0.75 and dj=Δj, let Δj+1:=2Δj; otherwise, let Δj+1:=Δj.
Step 5. If rj>0.25, let xj+1:=xj+dj, update Bj to be Bj+1=Hesse(xj+1), let j:=j+1, and go to Step 1; otherwise, let xj+1=xj, j:=j+1, and go to Step 2, where Δ0=1/10g(x0), ɛ=10-6.
Obviously, the above trust region subproblem (21) is an unconstrained optimization problem whose objective function is a quadratic. For the sake of convenience, we first denote
(25)z=μ,λ,d,(26)Y(z):=μλ-d22+Δj2-λ+d22-Δj22+4μ2Bj+λEd-gj,(27)β(z)=γYzmin1,Yz,
respectively. Then, the following smoothing Newton algorithm [17] is given to solve the subproblem (21).
Algorithm 2.
Step 0. Given d0∈Rn, z0=(μ0,λ0,d0), z~0=(μ0,0,0). Select parameters δ,σ,γ∈(0,1), μ0γ<1, and γY(z0)<1; set h:=0.
Step 1. Calculate Y(zh), if Y(zh)=0, and terminate algorithm; otherwise, calculate βh=β(zh).
Step 2. Obtain the solution for equations Y(zh)+Y′(zh)Δzh=βhz~, and the solution is Δzh=(Δμh,Δλh,Δdh).
Step 3. Suppose that mh is the minimum nonnegative integer meeting Y(zh+δmhΔzh)≤[1-σ(1-βμ0)δmh]Y(zh). Let αh:=δmh and zh+1=zh+αhΔzh.
Step 4. Let h:=h+1, and go to Step 1.
3.2. A New Newton-Type Trust Region Algorithm
Optimizing ability of Newton-type trust region algorithm (N-TTRA) highly depends on initial parameters. And the algorithm is short of global optimizing capability although its local optimizing is fast. In order to enhance the global optimization capability and get rid of dependence on initial parameters, we construct a new Newton-type trust region algorithm by combining a new BFGS updating formula with N-TTRA. The basic idea is to update Bj+1 with new BFGS formulas at Step 5 in Algorithm 1. This idea guarantees positive definiteness of Bj+1 to improve the global optimization capability; meanwhile, it gets rid of dependence on the initial parameter selection. New adjustment formula of BFGS is as follows:
(28)Bj+1=Bj+yj*yj*TsjTyj*-BjsjsjTBjsjTBjsj,
where yj*=yjTsj/yjTsjyj, sj=xj+1-xj, and yj=gj+1-gj. Here, yj* guarantees positive definiteness of Bj+1, so the new Newton-type trust region algorithm (NN-TTRA) uses (28) as the new adjustment formula of Bj+1 at Step 5 in Algorithm 1. It is proved that the method is of super-linear convergence [22]. The flow chart of NN-TTRA is given in Figure 1.
Flow chart of NN-TTRA.
4. Numerical Experiments and Results Analysis
In this paper, we provide actual observed data of flood runoff process between Chenggouwan and Linqing segment in Nanyunhe River of Haihe River Basin. (Length of the reach is 83.8 km, where there is no tributary, but a levee control on both sides. There may occur lifting irrigation during the water delivery, and flood water may discharge into the reach when rainfall is high. But these situations have little effect on flood, where the routing time interval Δt=12 h.) The detailed data can be seen in [23]. Here, we will give the numerical experiments from the following three aspects.
Firstly, in order to verify the advantage of our new parameter estimation models (16) by using the generalized trapezoid formula to approximate (1), we use the NN-TTRA to solve the above unconstrained optimization problem (16) by choosing different θ. Furthermore, we get the corresponding parameters K and x for each θ. From these parameters, we use (14) to obtain the calculated outflow Q~(ti), where Q~(t1)=Q(t1). At last, the average absolute errors (AAE) and the average relative errors (ARE) can be given as follows:
(29)AAE=1n∑i=1nQ~(ti)-Q(ti),ARE=1n∑i=1nQ~(ti)-Q(ti)Q(ti).
From (29), Tables 1, 2, and 3 list the AAE and ARE for different θ for flood routing in 1960, 1961, and 1964, respectively. It is shown from Tables 1–3 that the AAE and ARE are the smallest when θ=1/3. In other words, the numerical results given in Tables 1–3 confirm the theoretical analysis presented in Section 2.2.
Numerical results with different θ for flood routing in 1960.
θ
AAE
ARE (%)
θ
AAE
ARE (%)
θ=0
8.208160
2.462451
θ=1/3
8.208122
2.462438
θ=1/10
8.208159
2.462451
θ=1/2
8.208163
2.462452
θ=1/7
8.208154
2.462449
θ=2/3
8.208162
2.462452
θ=1/5
8.208169
2.462454
θ=1
8.208162
2.462451
Numerical results with different θ for flood routing in 1961.
θ
AAE
ARE (%)
θ
AAE
ARE (%)
θ=0
4.011335
0.998421
θ=1/3
4.011335
0.998421
θ=1/10
4.011335
0.998421
θ=1/2
4.011336
0.998421
θ=1/7
4.011336
0.998421
θ=2/3
4.011336
0.998421
θ=1/5
4.011336
0.998421
θ=1
4.011335
0.998421
Numerical results with different θ for flood routing in 1964.
θ
AAE
ARE (%)
θ
AAE
ARE (%)
θ=0
10.949204
2.662713
θ=1/3
10.949203
2.662712
θ=1/10
10.949205
2.662713
θ=1/2
10.949204
2.662713
θ=1/7
10.949208
2.662713
θ=2/3
10.949203
2.662712
θ=1/5
10.949205
2.662713
θ=1
10.949204
2.662712
Secondly, to illustrate the efficiency of the NN-TTRA presented in this paper, for different parameter initial values, we employ the N-TTRA and the NN-TTRA to solve the above unconstrained optimization problem (16), respectively, where the parameter θ is set to 1/3 and a maximum number of iterations of the N-TTRA and the NN-TTRA are set to 150. The numerical results are shown in Table 4. From Table 4, we can indicate that the NN-TTRA not only gets rid of dependence on the initial values of parameters, but also guarantees that the number of iterations is finite. Meanwhile, through this algorithm the global optimum value is obtained. However, the N-TTRA highly depends on initial values of parameters. When the selected initial value is far from the optimal ones, the algorithm hardly finds the global optimum and even cannot complete iterations in finite times.
Comparison of performance of the two algorithms.
Algorithm
Initial value
Iterations
K
x
N-TTRA
(1, 1)
5
0
−1.938700
(2, 2)
—
—
—
(11, 1)
3
11.190740
0.998456
NN-TTRA
(1, 1)
27
11.190740
0.998456
(2, 2)
25
11.190740
0.998456
(11, 1)
4
11.190740
0.998456
Finally, Table 5 lists the numerical results calculated by using NN-TTRA to solve problem (16), where θ=1/3. Meanwhile Table 5 also gives the numerical results obtained by using method of trial-and-error (TAE) [2], the least-square method (L-SM) [6], and direct optimal method (DOM) [18], respectively. Here, the observed data is the outflow of flood runoff between Chenggouwan and Linqing segment in Nanyunhe River of Haihe Basin in 1960. It can be seen from Table 5 that the AAE and ARE of NN-TTRA are all less than those of other estimation methods. Thus, we conclude that it is very effective by using NN-TTRA to estimate the parameters of Muskingum model. In addition, the calculated values and observed values of the flows in 1960, 1961, and 1964 are plotted in Figures 2, 3, and 4, respectively. From Figures 2–4, it can be seen that the calculated flow data via the NN-TTRA highly coincides with the observed flow data. In brief, the accuracy of the method is satisfactory.
Comparison of results via several parameter estimation methods.
Algorithm
K
x
AAE
ARE (%)
TAE [2]
12.400000
0.100000
10.230000
3.150000
L-SM [6]
11.790000
−0.325000
9.800000
2.960000
DOM [18]
12.440000
−0.260000
9.700000
2.920000
NN-TTRA
11.190740
0.998456
8.208122
2.462438
Comparison between calculated values and observed values in 1960.
Comparison between calculated values and observed values in 1961.
Comparison between calculated values and observed values in 1964.
5. Conclusions
By combining a new BFGS adjustment formula with N-TTRA, this paper implements parameter estimation of Muskingum model by selecting different initial values for parameters and comparing it with N-TTRA. The results show that the NN-TTRA can get rid of the dependence on initial value selection for parameters when searching solutions for Muskingum model parameters, which avoids the influence of initial value selection for parameters on the optimization results and averts local optimum. What is more, this algorithm is of application value in flood disaster management and should be generalized. In addition, this algorithm can be extended to other similar parameter estimation problems to help obtain excellent results.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (nos. 11301044, 11261006, and 11161003), the Key Projects of Excellent Young Talents Fund in universities of Anhui Province (2013SQRL095ZD), the Project Supported by Scientific Research Fund of Hunan Provincial Education Department (Grant no. 13C333), the Project Supported by the Science and Technology Research Foundation of Hunan Province (Grant no. 2014GK3043), and the Guangxi Natural Science Foundation (2012GXNSFAA053002). The authors would like to acknowledge Guangxi Colleges and Universities Key Laboratory of Mathematics and Its Applications.
WilsonE. M.1990Macmillan EducationSerranoS. E.The Theis solution in heterogeneous aquifers199735346346710.1111/j.1745-6584.1997.tb00106.x2-s2.0-0031148950KangL.WangC.JiangT. B.A new genetic simulated annealing algorithm for flood routing model2004162233239JinJ.DingJ.2000Sichuan University PressGillM. A.Flood routing by the Muskingum method1978363-435336310.1016/0022-1694(78)90153-12-s2.0-0017933375AldamaA. A.Least-squares parameter estimation for Muskingum flood routing1990116458058610.1061/(asce)0733-9429(1990)116:4(580)2-s2.0-0025412981ChowV. T.1990Macmillan EducationGeemZ. W.Parameter estimation for the nonlinear Muskingum model using the BFGS technique2006132547447810.1061/(ASCE)0733-9437(2006)132:5(474)2-s2.0-33748792293ChenJ.YangX.Optimal parameter estimation for Muskingum model based on Gray-encoded accelerating genetic algorithm200712584985810.1016/j.cnsns.2005.06.0052-s2.0-33845669541BaratiR.Parameter estimation of nonlinear Muskingum models using nelder-mead simplex algorithm2011161194695410.1061/(ASCE)HE.1943-5584.00003792-s2.0-83755173792MohanS.Parameter estimation of nonlinear Muskingum models using genetic algorithm1997123213714210.1061/(asce)0733-9429(1997)123:2(137)2-s2.0-0031080586KimJ. H.GeemZ. W.KimE. S.Parameter estimation of the nonlinear Muskingum model using Harmony Search20013751131113810.1111/j.1752-1688.2001.tb03627.x2-s2.0-0035661395ChuH.-J.ChangL.-C.Applying particle swarm optimization to parameter estimation of the nonlinear muskingum model20091491024102710.1061/(ASCE)HE.1943-5584.00000702-s2.0-69249120304LuoJ.XieJ.Parameter estimation for nonlinear Muskingum model based on immune clonal selection algorithm2010151084485110.1061/(asce)he.1943-5584.00002442-s2.0-77956790627XuD.-M.QiuL.ChenS.-Y.Estimation of nonlinear Muskingum model parameter using differential evolution201217234835310.1061/(asce)he.1943-5584.00004322-s2.0-84863293184ChawlaM. M.Al-ZanaidiM. A.EvansD. J.Generalized trapezoidal formulas for parabolic equations199970342944310.1080/00207169908804765MR17126602-s2.0-0032628859MaC.2010Beijing, ChinaScience PressStephensonD.Direct optimization of Muskingum routing coefficient197936353363PowellM. J. D.A new algorithm for unconstrained optimization1970T.P. 393Oxfordshire, UKAtomic Energy Research EstablishmentEsmaeiliH.KimiaeiM.A new adaptive trust-region method for system of nonlinear equations20143811-123003301510.1016/j.apm.2013.11.023MR32018132-s2.0-84900021465AminiK.AhookhoshM.A hybrid of adjustable trust-region and nonmonotone algorithms for unconstrained optimization2014389-102601261210.1016/j.apm.2013.10.062MR31919542-s2.0-84898543721YuanY.SunW.1999Beijing, ChinaScience PressOuyangA.TangZ.LiK.SallamA.ShaE.Estimating parameters of Muskingum model using an adaptive hybrid PSO algorithm201428129145900310.1142/s02180014145900342-s2.0-84897558166