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For this case, optimal solutions for the VEP-NLMM-L with different numbers of inflow dividing levels (L = 1–5) are listed in Table 2, and corresponding results by the LMM-L (O'Donnell 1985) and the NLMM2 (Niazkar & Afzali 2014) are also given for comparisons. As is shown in Table 2, the SSQ values by LMM-L and NLMM2 are 815.680 and 36.242, respectively. The value 36.242 is the best existing value for the NLMM2 so far in the literature. The SSQs by estimating the VEP-NLMM-L of different L have obvious reductions, at least (L = 1) 98.71% and 70.91% lower than those obtained by the LMM-L and NLMM2, respectively. Comparing the VEP-NLMM-L with the classical NLMM2, the variation range of the exponent parameter is small and the optimal values of other parameters are close, but the improvement on accuracy of flood routing is excellent. For example, the optimal inflow dividing values are , , , when L = 5.

Table 2

Optimal results by estimating different Muskingum models for the data set from Wilson (1974) 

ModelKwαm/mi (i = 1,…,L)SSQ
LMM-L (O'Donnell 1985) 5.4300 0.1393 0.0235 – 815.680 
NLMM2 (Niazkar & Afzali 2014) 0.6589 0.3399 – 1.8456 36.242 
VEP-NLMM-L (this study) L = 1 0.5362 0.3005 −0.0215 1.8634 10.541 
L = 2 0.4921 0.2978 −0.0195 1.8996, 1.8833 7.475 
L = 3 0.5875 0.3048 −0.0199 1.8583, 1.8446, 1.8361 5.730 
L = 4 0.6176 0.3115 −0.0199 1.8474, 1.8344, 1.8299, 1.8231 4.911 
L = 5 0.5565 0.2938 −0.0192 1.8889, 1.8697, 1.8540, 1.8573, 1.8505 4.535 
ModelKwαm/mi (i = 1,…,L)SSQ
LMM-L (O'Donnell 1985) 5.4300 0.1393 0.0235 – 815.680 
NLMM2 (Niazkar & Afzali 2014) 0.6589 0.3399 – 1.8456 36.242 
VEP-NLMM-L (this study) L = 1 0.5362 0.3005 −0.0215 1.8634 10.541 
L = 2 0.4921 0.2978 −0.0195 1.8996, 1.8833 7.475 
L = 3 0.5875 0.3048 −0.0199 1.8583, 1.8446, 1.8361 5.730 
L = 4 0.6176 0.3115 −0.0199 1.8474, 1.8344, 1.8299, 1.8231 4.911 
L = 5 0.5565 0.2938 −0.0192 1.8889, 1.8697, 1.8540, 1.8573, 1.8505 4.535 

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