Technical Note

Parameter Estimation of the Nonlinear Muskingum ModelUsing Parameter-Setting-Free Harmony Search

Zong Woo Geem1

Abstract: Although phenomenon-mimicking algorithms, such as genetic algorithms, particle swarm optimization, and harmony search,have overcome the disadvantages of mathematical algorithms, such as the nonlinear least-squares method, segmented least-squares method,Lagrange multiplier method, a hybrid of pattern search and local search, and the Broyden-Fletcher-Goldfarb-Shanno technique, thealgorithms have an inherent shortcoming. They require a tedious and skillful parameter-setting process for the algorithm parameters, suchas the crossover rate, mutation rate, acceleration coefficients, harmony memory considering rate, and pitch-adjusting rate. Thus, this studyproposes a novel parameter-setting-free technique interfaced with a harmony search algorithm and applies it to the parameter estimationof the nonlinear Muskingum model, which is an optimization problem with continuous decision variables. Results show that the proposedtechnique found good model parameter values while outperforming a classical harmony search algorithm with fixed algorithm parametervalues. DOI: 10.1061/(ASCE)HE.1943-5584.0000352. © 2011 American Society of Civil Engineers.

CE Database subject headings: Hydrologic models; Parameters; Algorithms.

Author keywords:Muskingum model; Parameter estimation; Harmony search; Parameter-setting-free technique; Phenomenon-mimickingalgorithms.

Introduction

Flood routing is a process for calculating the shape of a flood wavealong a river channel. The Muskingum model is a popular modelfor flood routing (Gill 1978), and its storage depends on the inflowand outflow. McCarthy analyzed the data from the MuskingumRiver in Ohio and found certain relationships: storage is propor-tional to outflow; and storage is also proportional to the differencebetween inflow and outflow.

The original Muskingum model has two parameters (K and χ).However, because it is not uncommon to observe a nonlinearrelationship between storage and discharge (Tung 1985), athree-parameter model, rather than the original two-parameterone, has been also researched:

St ¼ K½χIt þ ð1� χÞOt�m ð1Þ

where St = channel storage (dimension: L3) at time t; It = inflowrate (L3=T) at time t; Ot = outflow rate (L3=T) at time t;K = storage-time constant for the river reach, whose dimensionis L3ð1�mÞTm; χ = weighting factor, commonly varying between0.0 and 0.3 for the river channel (dimensionless); and m = exponentfor nonlinearity consideration (dimensionless). The outflow ratecan be expressed as

Ot ¼�

11� χ

��StK

�1=m

��

χ1� χ

�It ð2Þ

Also, Eq. (2) and the continuity equation together result in theexpression

ΔStΔt

¼ ��

11� χ

��StK

�1=m

þ�

11� χ

�It ð3Þ

The routing procedure consists of the following steps(Geem 2006).1. Assume the values of three parameters (K, χ, and m);2. Calculate the storage amount (St) using Eq. (1), where initial

outflow is assumed to be equal to initial inflow (O0 ¼ I0);3. Calculate the time rate of the storage change using Eq. (3)

(Δt was assumed to be unit time in some previous research);4. Calculate the next storage using Stþ1 ¼ St þΔSt;5. Calculate the next outflow (Otþ1) using Eq. (2) (most previous

studies used It rather than Itþ1 for the equation); and6. Repeat Steps 2 to 5 for all times.

O1 is always identical to O0 in this three-parameter nonlinearMuskingum model (see “Appendix. Determination that O0 ¼ O1”).

Originally, various mathematical techniques, such as thesegmented least-squares method (S-LSM) (Gill 1978), a hybridof pattern search (Hooke-Jeeves) and local search (Davidon-Fletcher-Powell) (HJþ DFP) (Tung 1985), the nonlinear least-squares method (NL-LSM) (Yoon and Padmanabhan 1993), theLagrange multiplier method (LMM) (Das 2004), and theBroyden-Fletcher-Goldfarb-Shanno (BFGS) technique (Geem2006) have been used for estimating the three parameter valuesof the model.

However, these techniques have the drawbacks of complexderivative requirement and/or initial vector assumption. For exam-ple, the BFGS technique (Geem 2006), although it reached the bestsolution ever found, relies heavily on the initial solution vector. Outof 75 cases with different initial vectors (K ¼ f0:01; 0:05; 0:10;0:15; 0:20g, χ ¼ f0:25; 0:275; 0:30g, and m ¼ f1:5; 1:75; 2:0;2:25; 2:5g), only 20 cases successfully reached the optimal solu-tion, and four cases found a near-optimal solutions, but 12 cases

1Environmental Planning and Management Program, Johns HopkinsUniv., 11833 Skylark Rd., Clarksburg, MD 20871. E-mail: geem@jhu.edu

Note. This manuscript was submitted on March 31, 2010; approved onOctober 27, 2010; published online on November 30, 2010. Discussionperiod open until January 1, 2012; separate discussions must be submittedfor individual papers. This technical note is part of the Journal of Hydro-logic Engineering, Vol. 16, No. 8, August 1, 2011. ©ASCE, ISSN 1084-0699/2011/8-684–688/$25.00.

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were not able to compute because they were located at infeasiblesolution area, and 39 cases diverged during the computation.

Thus, more recently researchers have also proposed variousphenomenon-mimicking algorithms (PMAs), such as the geneticalgorithm (GA) (Mohan 1997), harmony search (HS) (Kim et al.2001), particle swarm optimization (PSO) (Chu and Chang 2009),and immune clonal selection algorithm (ICSA) (Luo and Xie 2010)to estimate the three parameter values of the model. The advantagesof these algorithms over mathematical techniques include that thereis (1) no derivative requirement, (2) no initial vector requirement,and 3) a better chance to find the global optimum.

However, one of critical disadvantages of PMAs is that they re-quire algorithm users to skillfully set the values for algorithmparameters, such as the crossover rate and mutation rate in GA;the memory consideration rate and pitch adjustment rate in HS;and the acceleration coefficients in PSO. If an algorithm includesmore complex structures with additional parameters, users are cor-respondingly required to exert more effort to find proper parametervalues.

To eliminate this tedious process, parameter-setting-free (PSF)researches are currently gathering popularity (Gibbs et al. 2010).This study proposes a novel PSF model combined with the HS al-gorithm for optimally estimating three parameters of the nonlinearMuskingum model for the first time.

Classical Harmony Search Algorithm

The HS algorithm, which was inspired by music improvisation, hasbeen applied to various optimization problems, including structuraldesign (Saka 2007), water network design (Geem 2009), ground-water management (Ayvaz 2009), geotechnical stability analysis(Cheng et al. 2008), and project scheduling (Geem 2010).

The HS algorithm starts by generating multiple random vectors(x1;…; xHMS) and then storing them in a harmony memory (HM)matrix as follows:

HM ¼

x11 x12 � � � x1n

x21 x22 � � � x2n

..

. � � � � � � � � �xHMS1 xHMS

2 � � � xHMSn

f ðx1Þf ðx2Þ...

f ðxHMSÞ

2666664

3777775

ð4Þ

where n = number of decision variables (in this study, n = threebecause there are three parameters to be estimated: x1 ¼ K,x2 ¼ χ, and x3 ¼ m); f ð·Þ = objective function (in this study, theobjective function is the sum of the squared residuals betweenthe observed and calculated outflows); and HMS = number of sol-ution vectors stored in the HM.

Once the HM matrix containing the random vectors is prepared,a new solution vector xNew is improvised by using the HM. Whenassigning a new value to xNewi , the classical HS algorithm, whichconsiders discrete decision variables, uses any of three operations,such as random selection, memory consideration, and pitch adjust-ment as follows (Geem et al. 2001; Geem 2008):

xNewi ←

8>><>>:

xiðkÞ ∈ fxið1Þ; xið2Þ;…; xiðKiÞg w:p:RRandom

xiðkÞ ∈ fx1i ; x2i ;…; xHMSi g w:p:RMemory

xiðk � lÞ w:p:RPitch

ð5Þ

where fxið1Þ; xið2Þ;…; xiðKiÞg = entire value set for the ith deci-sion variable; fx1i ; x2i ;…; xHMS

i g = value set stored in the HMmatrix; w.p. is an acronym for “with probability;” l = neighboring

index; RRandom = rate of random selection; RMemory = rate ofharmony memory consideration; and RPitch = rate of pitch adjust-ment (RRandom þ RMemory þ RPitch ¼ 1).

However, because the parameter estimation problem in thisstudy considers continuous variables rather than discrete variables,Eq. (5) is modified as follows:

xNewi ←

8>><>>:

xi ≤ xi ≤ xi w:p:RRandom

xiðkÞ ∈ fx1i ; x2i ;…; xHMSi g w:p:RMemory

xiðkÞ þΔ w:p:RPitch

ð6Þ

where xi and xi = minimum and maximum values for the ithdecision variable; and Δ = amount of pitch adjustment. Δ canbe obtained by using various methods (Lee and Geem 2005;Mahdavi et al. 2007; Geem and Roper 2010).

If the newly generated vector xNew is better than the worst one(xWorst) in the HM with respect to the objective function, the formerreplaces the latter:

xNew ∈ HM ∧ xWorst∉HM ð7Þwhere ∧ = AND operator that performs both tasks before and afterthe symbol. The HS algorithm repeats Eqs. (6) and (7) until itreaches the global optimum or the termination criterion is satisfied.

Parameter-Setting-Free Harmony Search Algorithm

In the classical HS algorithm, algorithm parameters, such as theharmony memory consideration rate (HMCR, RMemory þ RPitch)and the pitch-adjusting rate (PAR, RPitch=ðRMemory þ RPitchÞ), werefixed during the computation or changed based on the current iter-ation number as follows (Mahdavi et al. 2007):

PAR ¼ PARmin þPARmax � PARmin

Max iter:� 1ðIter � 1Þ ð8Þ

where PARmin = starting minimum PAR; PARmax = ending maxi-mum PAR; Iter = current iteration; and Max iter. = maximum iter-ation. The starting and ending PARs are arbitrary values used inMahdavi et al. (2007) that have 0.35 or 0.45 as the startingPAR and 0.99 as the ending PAR. In their study, the PAR growsfrom the starting PAR to the ending PAR as the iterations continue.

However, finding proper parameter values requires a dexterousskill in parameter setting, which can be a wearisome and irksometask, especially for novice researchers. Thus, this study proposes anovel PSF technique for a continuous-variable HS algorithm.

Once the HM that contains multiple random vectors is prepared,the proposed PSF-HS algorithm generates new vectors with fixedinitial parameter values (in this study, the initial HMCR is 0.5 andthe initial PAR is 0.5) while introducing another memory matrixthat keeps the information of each operation as follows:

y11 ¼ Random y12 ¼ Pitch � � � y1n ¼ Memory

y21 ¼ Memory y22 ¼ Memory � � � y2n ¼ Pitch

..

. � � � � � � � � �yHMS1 ¼ Memory yHMS

2 ¼ Random � � � yHMSn ¼ Memory

2666664

3777775

ð9Þ

After certain number of new vector generations (in this study,3 × HMS) with the initial parameter values, the PSF-HS algorithmgenerates new vectors with adaptive parameter values that are cal-culated as follows:

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HMCRi ¼nðyji ¼ Memory ∨ yji ¼ PitchÞ

HMS; i ¼ 1;…; n

ð10Þ

PARi ¼nðyji ¼ PitchÞ

nðyji ¼ Memory ∨ yji ¼ PitchÞ ; i ¼ 1;…; n

ð11Þwhere∨ = OR operator that returns true if either or both operands istrue and returns false otherwise; nð·Þ = count function that returnsthe number of elements that satisfy the specified condition.

If the ideal HMCR is very close to 1.0 or the ideal PAR is veryclose to 0.0, it is difficult to maintain the ideal values using Eqs. (10)and (11) because the HMS has a limited number (normally 30).Thus, to consider the situation, Eqs. (10) and (11) are modifiedby adding a tiny amount of noise as follows:

HMCRi←HMCRi þ δHMCR · uð�1; 1Þ ð12Þ

PARi←PARi þ δPAR · uð�1; 1Þ ð13Þwhere δHMCR = maximum amount of noise for HMCR; δPAR =maximum amount of noise for PAR; and uð�1; 1Þ = uniform ran-dom number function that returns a random number from a uniformdistribution between �1 and 1.

If the new vector using the adaptive HMCR and PAR is betterthan the worst one in the HM matrix, the two memory matricesin Eqs. (4) and (9) are updated until the termination criterion issatisfied.

Example of Hydrologic Parameter Estimation

The PSF-HS model developed in the preceding sections is appliedto the optimal parameter estimation of the nonlinear Muskingummodel, as in Eq. (1), which uses Wilson’s data (Geem 2006) tofairly compare the results. The objective function to be minimizedis the sum of the squared residuals (SSQs) between the observedand calculated outflows as follows:

Minimize SSQ ¼Xt

½Ot � OtðK;χ;mÞ�2 ð14Þ

For this continuous-value estimation, Δ in Eq. (6) can be ex-pressed as follows:

Δ←δFW · uð�1; 1Þ ð15Þwhere δFW = maximum amount of fret width (in thisstudy, δFW ¼ ðxi � xiÞ=1;000).

Fig. 1 shows the results of the PSF-HS model after performing10 runs with different random seeds. The mean of the 10 SSQs is37.6196, the standard deviation is 1.6988, and the SSQs range from36.7680–42.3572. Because the optimal SSQ obtained by BFGS(Geem 2006) is 36.7679, the difference between the optimumand average SSQs is 2.3%. Although BFGS reached the optimalor a near-optimal solution 24 times out of 75 runs (success rate32%), PSF-HS always reached a near-optimal solution.

In Fig. 1, the result of Run #5 appears to be stuck in a localoptimum. Actually, this is the very nature of any optimization al-gorithm. In other words, it does not always guarantee good solu-tions, which is observed in both deterministic and stochasticalgorithms. Nonetheless, the PSF technique is currently emerging

in the algorithm field. Although the approach in this study is notperfect at the current stage, future extensive research will hopefullyaddress those insufficient features.

If the SSQ (42.3572) of Run #5, which appears to be stuck in alocal optimum, is excluded as an outlier, the average SSQ of thePSF-HS increases to 37.0932, and the difference between the op-timum and average SSQs becomes only 0.9%.

Fig. 2 shows the history of the HMCR values from Run #1. Allthree HMCR values finally converged into one. However, theHMCR for parameter m increased right after a certain numberof generations (3 × HMS), but that for K slightly decreased andthat for χ decreased considerably before increasing.

Fig. 3 shows the history of PAR values from Run #1. Two PARvalues for K and m finally converged to zero after fluctuating, and

Fig. 1. Results of PSF-HS model for hydrologic parameter estimation

Fig. 2. History of HMCR values

Fig. 3. History of PAR values

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that for χ ended up with 0.47. This situation can be explained byobserving the final HM. When the final HM, which stores 30 har-monies, was observed, K and m have the fixed values of0.086396966882348 and 1.86770329707861 for all 30 vectors,but χ has fluctuating values ranging from 0.286917936587334–0.286927076351643 with corresponding SSQs of 36.76796576and 36.76796575. This means that PSF-HS tried to enhance theSSQ even at the last minute by locally searching using the param-eter χwhile multiple solutions with nearly equivalent qualities existin the HM.

Tables 1 and 2 show the comparison of the best SSQs and thecorresponding outflows obtained from various techniques, such asNL-LSM, S-LSM, LMM, HJþ DFP, GA, ICSA, HS, the currentPSF-HS, and the BFGS technique. The SSQ from PSF-HS(36.7680) is very close to the optimal SSQ (36.7679) obtained fromBFGS, and PSF-HS is free from infeasible starting vectors andcomputational divergence that were observed in BFGS. Also,the computing time of PSF-HS for 5,000 function evaluations isonly 1 s (HS also performed 5,000 function evaluations in 1 s).

This rapid feature may enable PSF-HS to be applied to more com-plex real systems without any computational load problem.

Because the PSO research used a slightly different Muskingummodel (Chu and Chang 2009) that uses an average inflow(�Itþ1 ¼ ðIt þ Itþ1Þ=2) rather than It for Eq. (2), its result wasnot compared in Table 1.

Conclusions

This study proposed a novel technique for eliminating parametersetting efforts in the continuous-variable HS algorithm by introduc-ing an operation memory matrix and noise terms. Also, the PSF-HSalgorithm uses individual parameter values for each decision var-iable, but the classical HS algorithm uses lump parameter values forall decision variables. The proposed PSF-HS algorithm overcamedrawbacks of mathematical algorithms, such as the initial vectorrequirement, and those of phenomenon-mimicking algorithms,such as the tedious setting of algorithm parameters.

The PSF-HS algorithm was applied to the flood forecastingmodel to optimally estimate three parameter values of the nonlinearmodel. Results showed that the technique successfully foundthe model parameter set that is very close to the optimal parameterset. Because the difference between the optimal SSQ by BFGS andthe average SSQ by 10 PSF-HS runs is 2.3%, the technique appearsto be quite robust as a parameter-estimation technique for real-world hydrologic models.

Future study can include the application of PSF-HS to morecomplicated hydrologic models. Also, user-friendly software canbe developed for helping algorithm users who have little knowl-edge of the algorithm parameters to efficiently manipulate. Moreresearch should be done on by-product algorithm parameters, suchas the initial HMCR and PAR, the length of initial generations, andmaximum noise amounts.

Table 2. Comparison of the Observed and Computed Outflows

Time (h) It (cms) Ot (cms)

Computed outflows (cms) from various methods

NL-LSM S-LSM LMM HJþ DFP GA ICSA HS PSF-HS Optimal

0 22.0 22.0 22.0 22.0 22.0 22.0 22.0 22.0 22.0 22.0 22.0

6 23.0 21.0 22.0 22.0 22.0 22.0 22.0 22.0 22.0 22.0 22.0

12 35.0 21.0 22.8 22.8 22.8 22.4 22.4 22.4 22.4 22.4 22.4

18 71.0 26.0 29.3 29.7 29.6 26.7 26.4 26.6 26.6 26.6 26.6

24 103.0 34.0 37.9 39.3 38.7 34.8 34.2 34.4 34.4 34.5 34.5

30 111.0 44.0 45.7 48.0 47.0 44.7 44.2 44.2 44.1 44.2 44.2

36 109.0 55.0 56.0 58.4 57.8 56.9 57.0 56.9 56.8 56.9 56.9

42 100.0 66.0 65.5 67.5 67.6 67.7 68.2 68.1 68.1 68.1 68.1

48 86.0 75.0 73.5 75.1 75.8 76.3 77.2 77.1 77.1 77.1 77.1

54 71.0 82.0 79.8 80.7 81.9 82.2 83.3 83.3 83.3 83.3 83.3

60 59.0 85.0 83.2 83.5 85.0 84.7 85.7 85.9 85.9 85.9 85.9

66 47.0 84.0 83.2 83.0 84.4 83.5 84.2 84.5 84.5 84.5 84.5

72 39.0 80.0 80.9 80.1 81.4 79.8 80.2 80.5 80.6 80.6 80.6

78 32.0 73.0 75.6 74.5 75.2 73.3 73.3 73.6 73.7 73.7 73.7

84 28.0 64.0 68.5 67.0 67.1 65.5 65.1 65.3 65.4 65.4 65.4

90 24.0 54.0 59.4 57.8 57.2 56.5 55.8 55.9 56.0 56.0 56.0

96 22.0 44.0 49.4 47.6 46.5 47.5 46.7 46.6 46.7 46.7 46.7

102 21.0 36.0 38.7 37.0 35.5 38.7 38.0 37.8 37.8 37.8 37.7

108 20.0 30.0 29.1 27.7 26.4 31.4 30.9 30.5 30.5 30.5 30.5

114 19.0 25.0 22.3 21.6 21.1 25.9 25.7 25.3 25.3 25.2 25.2

120 19.0 22.0 19.1 19.0 19.0 22.1 22.2 21.8 21.8 21.7 21.7

126 18.0 19.0 19.0 19.0 19.0 20.2 20.3 20.0 20.0 20.0 20.0

Table 1. Comparison of Best SSQs from Various Methods

Method K χ m SSQ

NL-LSM 0.0600 0.2700 2.3600 156.4399

S-LSM 0.0100 0.2500 2.3470 145.6945

LMM 0.0753 0.2769 2.2932 130.4872

HJþ DFP 0.0764 0.2677 1.8978 45.6120

GA 0.1033 0.2813 1.8282 38.2363

ICSA 0.0884 0.2862 1.8624 36.8026

HS 0.0883 0.2873 1.8630 36.7829

PSF-HS 0.0864 0.2869 1.8677 36.7680

Optimal 0.0863 0.2869 1.8679 36.7679

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Appendix. Determination that O0 � O1

To obtain O1, first calculate So using Eq. (1). Because O0 ¼ I0, Sobecomes KOm

0 as follows:

S0 ¼ K½χI0 þ ð1� χÞO0�m ¼ K½χO0 þ ð1� χÞO0�m ¼ KOm0

ð16ÞThen, calculate the time rate of the storage change using Eq. (3).

Because O0 ¼ I0, ΔSo becomes zero as follows:

ΔS0Δt

¼ ��

11� χ

��S0K

�1=m

þ�

11� χ

�I0

¼ ��

11� χ

��KOm

0

K

�1=m

þ�

11� χ

�I0

¼ ��

11� χ

�O0 þ

�1

1� χ

�I0

¼ ��

11� χ

�I0 þ

�1

1� χ

�I0

¼ 0 ð17ÞThen, calculate S1 as follows:

S1 ¼ S0 þΔS0 ¼ KOm0 þ 0 ¼ KOm

0 ð18ÞFinally, calculate O1 using Eq. (2). Because O0 ¼ I0, O1 becomesO0 as follows:

O1 ¼�

11� χ

��S1K

�1=m

��

χ1� χ

�I0

¼�

11� χ

��KOm

0

K

�1=m

��

χ1� χ

�I0

¼�

11� χ

�O0 �

�χ

1� χ

�I0

¼�

11� χ

�O0 �

�χ

1� χ

�O0

¼ O0 ð19Þ

References

Ayvaz, M. T. (2009). “Application of harmony search algorithm to the sol-ution of groundwater management models.” Adv. Water Resour., 32(6),916–924.

Cheng, Y. M., Li, L., Lansivaara, T., Chi, S. C., and Sun, Y. J. (2008). “Animproved harmony search minimization algorithm using different slipsurface generation methods for slope stability analysis.” Eng. Optim.,40(2), 95–115.

Chu, H. J., and Chang, L. C. (2009). “Applying particle swarm optimizationto parameter estimation of the nonlinear muskingum model.” J. Hydrol.Eng., 14(9), 1024–1027.

Das, A. (2004). “Parameter estimation for muskingum models.” J. Irrig.Drain Eng., 130(2), 140–147.

Geem, Z. W. (2006). “Parameter estimation for the nonlinear muskingummodel using BFGS technique.” J. Irrig. Drain Eng., 132(5), 474–478.

Geem, Z. W. (2008). “Novel derivative of harmony search algorithm fordiscrete design variables.” Appl. Math. Comput., 199(1), 223–230.

Geem, Z. W. (2009). “Particle-swarm harmony search for water networkdesign.” Eng. Optim., 41(4), 297–311.

Geem, Z. W. (2010). “Multiobjective optimization of time-cost trade-offusing harmony search.” J. Constr. Eng. Manage., 136(6), 711–716.

Geem, Z. W., Kim, J. H., and Loganathan, G. V. (2001). “A new heuristicoptimization algorithm: Harmony search.” Simulation, 76(2), 60–68.

Geem, Z. W., and Roper, W. E. (2010). “Various continuous harmonysearch algorithms for web-based hydrologic parameter optimisation.”Int. J. Math. Modell. Numer. Optim., 1(3), 213–226.

Gibbs, M. S., Maier, H. R., and Dandy, G. C. (2010). “Comparison ofgenetic algorithm parameter setting methods for chlorine injectionoptimization.” J. Water Resour. Plann. Manage., 136(2), 288–291.

Gill, M. A. (1978). “Flood routing by the muskingum method.” J. Hydrol.(Amsterdam), 36, 353–363.

Kim, J. H., Geem, Z. W., and Kim, E. S. (2001). “Parameter estimation ofthe nonlinear muskingum model using harmony search.” J. Am. WaterResour. Assoc., 37(5), 1131–1138.

Lee, K. S., and Geem, Z. W. (2005). “A new meta-heuristic algorithm forcontinuous engineering optimization: Harmony search theory and prac-tice.” Comput. Methods Appl. Mech. Eng., 194(36–38), 3902–3933.

Luo, J., and Xie, J. (2010). “Parameter estimation for nonlinear muskingummodel based on immune clonal selection algorithm.” J. Hydrol. Eng.,15(10), 844–851.

Mahdavi, M., Fesanghary, M., and Damangir, E. (2007). “An improvedharmony search algorithm for solving optimization problems.”Appl. Math. Comput., 188(2), 1567–1579.

Mohan, S. (1997). “Parameter estimation of nonlinear muskingum modelsusing genetic algorithm.” J. Hydraul. Eng., 123(2), 137–142.

Saka, M. P. (2007). “Optimum geometry design of geodesic domes usingharmony search algorithm.” Adv. Struct. Eng., 10(6), 595–606.

Tung, Y.-K. (1985). “River flood routing by nonlinear muskingummethod.” J. Hydraul. Eng., 111(12), 1447–1460.

Yoon, J., and Padmanabhan, G. (1993). “Parameter estimation of linearand nonlinear muskingum models.” J. Water Resour. Plann. Manage.,119(5), 600–610.

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