A Novel Parameter Estimation Method for Muskingum Model Using New Newton-Type Trust Region Algorithm

1College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, China 2Department of Electrical and Information Engineering, Hunan Institute of Traffic Engineering, Hengyang, Hunan 421001, China 3School of Information Science and Engineering, Hunan City University, Yiyang, Hunan 413000, China 4Department of Mathematics and Computer Science, Chizhou College, Chizhou, Anhui 247000, China


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: where () represents the channel storage at time ; () and () represent the rates of inflow and outflow at time , respectively.The linear Musikingum model is where  is a storage time constant for the river reach and  is a weighting factor commonly varying between 0.0 and 0.3 for the river channel.
It is worth mentioning that the parameters  and  in (2) are graphically estimated by a trial-and-error procedure [2].If  is obtained, the values of [() + (1 − )()] are calculated by using observed data in both upstream and downstream and plotted against .The particular value which generates the loop is accepted as the best estimation of .The slope of the straight line fits through the loop derives .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 Mathematical Problems in Engineering estimating  and , the following finite difference scheme [3] is used to the resulting ordinary differential equation (1): where and (  ) represents observed outflow discharges at time   , (  ) represents the inflow discharge at time interval   , Δ is the time step, and  is the total time number.Substituting ( 4) into (3), we have where By minimizing the sum of the square of the deviations between observed and calculated outflows, we obtain the following objective function: In the past two decades, in order to obtain the parameters  and , 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  and  are based on (3).We know that the truncation error of (3) is only ((Δ) 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  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 .It is noted that the accuracy of the presented difference schemes to approximate the differential equation ( 1) can be improved to third order, when  = 1/3.In other words, we can get a higher accuracy parameter estimation model, if  = 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.

Truncation Errors
Analysis.Here, we analyze the local truncation error of (13) at time direction [16].Firstly, from (9), we have Then, by using Taylor expansion, (10) can be written as follows: Substituting for d W(  )/d from ( 17) into (18), the truncation errors of generalized trapezoidal formula can be obtained as follows: From (19), it is clear that the order of our presented method in (13) is ((Δ) 2 ) if  ̸ = 1/3.In particular, for  = 1/3, our presented method in ( 13) is third-order accuracy.

Newton-Type Trust Region Algorithm.
As far as we know, the basic trust region algorithm used to solve the following unconstrained optimization problem 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   is the th iterative point,   = (  ),   = ∇(  ), and   is the th iteration of Hesse matrix ∇ 2 (  ); the trust region subproblem, which is at the th iterative step of problem (20), can be formulated as follows: where Δ  is a trust region radius and  ∈   is an iterative step and ‖ ⋅ ‖ denotes the Euclidian norm of vectors or its induced matrix norm.Denote the general trust region algorithm [17] for solving the unconstrained optimization problem (20) is given as follows.
Step 2. Utilize the smoothing Newton method to obtain the solution   of the subproblem (21).
Step 3. Calculate the   of formula (24).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) respectively.Then, the following smoothing Newton algorithm [17] is given to solve the subproblem (21).
Step 4. Let ℎ := ℎ + 1, and go to Step 1.It is proved that the method is of super-linear convergence [22].The flow chart of NN-TTRA is given in Figure 1.

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 Δ = 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 .Furthermore, we get the corresponding parameters  and  for each .From these parameters, we use ( 14) to obtain the calculated outflow Q(  ), where 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  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.
Finally, Table 5 lists the numerical results calculated by using NN-TTRA to solve problem (16), where  = 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.

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.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.

Figure 2 :
Figure 2: Comparison between calculated values and observed values in 1960.

Figure 3 :
Figure 3: Comparison between calculated values and observed values in 1961.

Figure 4 :
Figure 4: between calculated values and observed values in 1964.

Table 1 :
Numerical results with different  for flood routing in 1960.

Table 2 :
Numerical results with different  for flood routing in 1961.

Table 3 :
Numerical results with different  for flood routing in 1964.

Table 4 :
Comparison of performance of the two algorithms.

Table 5 :
Comparison of results via several parameter estimation methods.