# Development and Application of Advanced Muskingum Flood Routing Model Considering Continuous Flow

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Overview

#### 2.2. Advanced Nonlinear Muskingum Model Considering Continuous Flow

_{t}is the amount of outflow (m

^{3}/s) and χ is a weighting factor. S

_{t}is the amount of storage (m

^{3}/s) and m is a parameter accounting for the nonlinearity of flood wave behavior. K is the storage factor, β is the factor for lateral flow, and I

_{t}is the amount of inflow (m

^{3}/s). The current inflow with four parameters (K, χ, m, β) is included in this equation. Additionally, a method for considering the lateral flow with the previous inflow was proposed. The current inflow and the previous inflow are weighted to calculate the inflow. The calculation of weighted inflow is shown in Equation (2).

_{t}is the weighted inflow (m

^{3}/s), θ is the weighted factor, and I

_{t−}

_{1}is the amount of previous inflow (m

^{3}/s). The weighted inflow considering the current and previous inflows has five parameters (K, χ, m, β, θ) for the calculation of outflow. Equation (3) was suggested for the calculation of outflow using the current and previous inflows [14].

_{t}(amount of storage) because the calculation of S

_{t}includes the time interval. Equation (4) shows the equation of state using the continuity equations.

_{t+}

_{1}is the amount of storage when the time is t + 1 (m

^{3}/s·day) and S

_{t}is the amount of storage when the time is t (m

^{3}/s·day). I

_{t}is the amount of inflow (m

^{3}/s) and O

_{t}is the amount of outflow (m

^{3}/s). The length of the investigated reach is considered by the parameters of the weighted inflow. If the length of the investigated reach is long, the effects of inflows at previous time will be decreased. Conversely, if the length of the investigated reach is short, the effects of inflows at previous time will be increased.

_{1}is the weighted factor of the first previous inflow, θ

_{2}is the weighted factor of the second previous inflow, and I

_{t−}

_{2}is the amount of second previous inflow (m

^{3}/s). The weighted inflow including first previous, second previous, and current inflows has six parameters (K, χ, m, β, θ

_{1}, θ

_{2}) for the calculation of outflow. Equation (3) in NLMM-L is also used for the calculation of outflow of ANLMM-L. The original Muskingum flood routing model perfectly preserves mass balance. Conversely, the Muskingum-Cunge (MC) contains a loss of mass because it increases with the flatness of the bed slope, reaching values of 8 to 10% at slopes of 10

^{−4}[15]. The concept of ANLMM-L perfectly contains the mass conservations because it is based on Muskingum flood routing model.

#### 2.3. Numerical Method for Parameter Estimation

_{1}, θ

_{2}) were used as decision variables in the objective function. The objective function in ANLMM-L is shown in Equation (6).

_{obs}is the amount of observed outflow (m

^{3}/s) and O

_{cal}is the amount of calculated outflow (m

^{3}/s). The root mean square error (RMSE) and Nash-Sutcliffe efficiency (NSE) as well as SSQ were added to compare the performance of the different models. The function of RMSE is shown in Equation (7).

_{1}is the amount of observed outflow (m

^{3}/s), x

_{2}is the amount of calculated outflow (m

^{3}/s), and n is the number of data. The function of NSE is shown in Equation (8).

_{1}is the amount of observed outflow (m

^{3}/s), x

_{2}is the amount of calculated outflow (m

^{3}/s), $\overline{x}$ is the amount of average outflow (m

^{3}/s) and n is the number of data. The results of kinematic wave model (KWM), linear Muskingum method (LMM), linear Muskingum method incorporating lateral flow (LMM-L), nonlinear Muskingum method (NLMM), NLMM-L, and ANLMM-L are compared for verifying the effectiveness of ANLMM-L.

#### 2.4. Vision Correction Algorithm

_{j}is the value of MTF of the j-th decision variable and k is the total number of decision variables. dx

_{j}represents the distance ratio between A (x

_{i}− x

_{1}) and B (x

_{n}− x

_{1}) at the j-th decision variable (A: distance from the selected decision variable (x

_{i}) to the best decision variable (x

_{1}); B: distance from the worst decision variable (x

_{n}) to the best decision variable). Each decision variable is adjusted using the MR process as given in Equation (11).

- Generate initial solutions
- Calculate the fitness of solutions using the objective function
- Generate new solution
- Compare new solution with current worst solution
- Determine the replacement between two solutions
- Repeat steps 2–5 if iteration process is not finished.

## 3. Results

#### 3.1. Application of Wilson Flood Data

_{1}, and 0.261739 for θ

_{2}using VCA.

^{3}/s, but the difference appears at 6 h. The values of outflow for LMM-L, NLMM, NLMM-L, and ANLMM-L are 22.1 m

^{3}/s, 22 m

^{3}/s, 21.71 m

^{3}/s, and 21.57 m

^{3}/s when the time is 6 h, respectively. The difference between the observed outflows and those calculated using each Muskingum method is apparent at 42 h although the error is larger or smaller at all times. The outflow for LMM-L, NLMM, NLMM-L, and ANLMM-L is 68.9 m

^{3}/s, 68.1 m

^{3}/s, 66.67 m

^{3}/s, and 66.01 m

^{3}/s when the observed outflow (output) at 42 h is 66 m

^{3}/s, respectively. The difference between the observed outflows and those calculated using LMM-L, NLMM, NLMM-L, and ANLMM-L is 2.9 m

^{3}/s, 2.1 m

^{3}/s, 0.67 m

^{3}/s, and 0.01 m

^{3}/s, respectively. The result obtained using ANLMM-L is better than those obtained using the other methods and this method shows the smallest value of SSQ, RMSE, and NSE. Figure 2 shows the comparison of results for the Wilson flood data.

^{3}/s. This result shows that the ANLMM-L is much better than the previous models when applied to the Wilson flood data.

#### 3.2. Application of Flood Data by Wang et al. (2009)

_{1}, and 0.0384045 for θ

_{2}using VCA. The results of LMM-L, NLMM, NLMM-L, and ANLMM-L are listed in Table 5.

^{3}/s and the initial difference between the observed outflows and those calculated using NLMM-L at 24 h is smaller than the corresponding value for the other methods. However, the magnitude of the difference between the observed and calculated outflows varies at each time. The ANLMM-L generally shows better results than the other methods although there is no significant difference at each time. Figure 3 shows the comparison of results for Wang et al.’s data [10].

^{3}/s. The SSQ of ANLMM-L is better than that of the other methods with Wang et al.’s flood data [10].

#### 3.3. Application of Flood Data for River Wye December in 1960

_{1}, and 0.212095 for θ

_{2}using VCA. The results of LMM-L, NLMM, NLMM-L and ANLMM-L are listed in Table 6.

^{3}/s except for that of NLMM (152 m

^{3}/s). The difference between the observed outflows and those calculated using ANLMM-L at 102 h is noticeably smaller than those of the other methods. The outflow of LMM-L, NLMM, NLMM-L, and ANLMM-L is 642 m

^{3}/s, 834 m

^{3}/s, 859.01 m

^{3}/s, and 884.60 m

^{3}/s when the observed outflow (output) at 102 h is 969 m

^{3}/s, respectively. The difference between the observed and calculated outflows of LMM-L, NLMM, NLMM-L, and ANLMM-L is 327 m

^{3}/s, 135 m

^{3}/s, 109.99 m

^{3}/s, and 84.40 m

^{3}/s, respectively. The result obtained using ANLMM-L is better than those obtained using the other methods and this method shows the smallest value of SSQ, RMSE, and NSE. Figure 4 shows the comparison of results for flood data of River Wye December in 1960.

^{3}/s. The ANLMM-L shows better performance than the other methods with the flood data of River Wye December in 1960.

#### 3.4. Application of Sutculer Flood Data

_{1}, and 0.295127 for θ

_{2}using VCA. The results of KWM, NLMM-L and ANLMM-L are listed in Table 7.

#### 3.5. Application of the Flood Data for River Wyre October in 1982

_{1}, and 0.114425 for θ

_{2}using VCA. The results of LMM-L, NLMM-L and ANLMM-L are listed in Table 8.

^{3}/s, 8.79 m

^{3}/s, and 9.94 m

^{3}/s when the observed outflow (output) at 2 h is 9.9 m

^{3}/s, respectively. The maximum error of ANLMM-L in each time does not exceed 4 unlike LMM-L and NLMM-L. Figure 6 shows the comparison of results for the flood data of River Wyre October in 1982.

^{3}/s, 4.29 m

^{3}/s, 3.45 m

^{3}/s, respectively. The ANLMM-L was applied to the five flood data including the Wilson flood data, Wang et al.’s data [10], and the flood data of River Wye December in 1960, Sutculer flood data, and the flood data of River Wyre October in 1982 for overcoming the shortcoming of NLMM-L. It shows better results than other Muskingum flood routing methods (KWM, LMM, LMM-L, NLMM, and NLMM-L). Accurate Muskingum flood routing using ANLMM-L is possible for various flood data.

#### 3.6. Application for the Prediction in Daechung Flood Data

_{1}, and 0.21141 for θ

_{2}using VCA. The optimal parameters of NLMM-L for the Daechung flood data in April, 2010 were determined to be 15.88463 for K, −0.23059 for χ, 0.68005 for m, 0.16082 for β, and 0.00000 for θ using VCA. Additionally, the ANLMM-L and NLMM-L were applied to the Daechung flood data in April, 2014 for the prediction of outflow. The parameters of ANLMM-L and NLMM-L for the Daechung flood data in April, 2014 were same with those for the Daechung flood data in April, 2010. The results of prediction are shown in Figure 7.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ANLMM-L | Advanced nonlinear Muskingum flood routing model considering continuous flow |

NLMM-L | Nonlinear Muskingum flood routing model incorporating lateral flow |

VCA | Vision correction algorithm |

SSQ | Sum of squares |

DR1 | Division rate 1 |

DR2 | Division rate 2 |

MTF | Modulation transfer function |

CF | Compression factor |

AR | Astigmatic rate |

RMSE | Root mean square error |

NSE | Nash-Sutcliffe efficiency |

KWM | Kinematic wave model |

LMM | Linear Muskingum method |

LMM-L | Linear Muskingum method incorporating lateral flow |

NLMM | Nonlinear Muskingum method |

AF | Astigmatic angle |

CG | Candidate glasses |

NFEs | Number of function evaluations |

CSA | Cuckoo search algorithm |

## References

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**Figure 3.**Comparison of results for Wang et al.’s flood data [10].

**Figure 7.**Results of prediction in Daechung flood data: (

**a**) Flood data in April, 2010; (

**b**) Flood data in April, 2014.

Ranges of Parameters in ANLMM-L | K | χ | m | β | θ_{1} | θ_{2} |
---|---|---|---|---|---|---|

Wilson flood data | 0.01–1.00 | −0.50–0.50 | 1.00–3.00 | −0.10–0.10 | 0.00–1.00 | 0.00–1.00 |

Flood data by Wang et al. [10] | 0.01–1.00 | −1.50–1.50 | 1.00–3.00 | −3.00–3.00 | 0.00–1.00 | 0.00–1.00 |

Flood data for River Wye December in 1960 | 0.01–1.00 | −0.50–0.50 | 1.00–3.00 | −0.10–0.10 | 0.00–1.00 | 0.00–1.00 |

Sutculer flood data | 0.01–1.00 | −0.50–0.50 | 1.00–3.00 | −0.10–0.10 | 0.00–1.00 | 0.00–1.00 |

Flood data for River Wyre October in 1982 | 0.01–10.00 | −0.50–0.50 | 0.00–1.00 | −3.00–3.00 | 0.00–1.00 | 0.00–1.00 |

Daechung flood data | 0.01–100.00 | −0.50–0.50 | 1.00–10.00 | −3.00–3.00 | 0.00–1.00 | 0.00–1.00 |

Objective function f(x), x = (x_{1}, x_{2}, …, x_{d})^{T} |

Generate initial glasses |

While (t < Max number of iterations) |

if (DR1 < rand) |

● Choose an existing glasses/generate new glasses |

if (DR2 < rand) |

● Determine global search direction |

end |

end |

if (new solution < current worst solution) |

● Replace new solution with current worst solution |

end |

● Find the current best solution |

End while |

Measures | LMM-L | NLMM | NLMM-L | ANLMM-L |
---|---|---|---|---|

SSQ | 56 s | 63 s | 64 s | 66 s |

RMSE | 60 s | 67 s | 68 s | 70 s |

NSE | 68 s | 70 s | 72 s | 73 s |

Time (h) | Input (m^{3}/s) | Output (m^{3}/s) | LMM-L (O’Donnell [4]) | NLMM (Karahan [17]) | NLMM-L (Karahan [8]) | ANLMM-L (This Study) |
---|---|---|---|---|---|---|

0 | 22 | 22 | 22 | 22 | 22.00 | 22.00 |

6 | 23 | 21 | 22.1 | 22 | 21.71 | 21.57 |

12 | 35 | 21 | 21.7 | 22.4 | 22.02 | 21.67 |

18 | 71 | 26 | 22.6 | 26.6 | 26.08 | 25.46 |

24 | 103 | 34 | 30.7 | 34.5 | 33.51 | 34.59 |

30 | 111 | 44 | 44.7 | 44.2 | 42.83 | 43.73 |

36 | 109 | 55 | 58.1 | 56.9 | 55.44 | 54.59 |

42 | 100 | 66 | 68.9 | 68.1 | 66.67 | 66.01 |

48 | 86 | 75 | 76.1 | 77.1 | 75.77 | 75.52 |

54 | 71 | 82 | 79.2 | 83.3 | 82.12 | 82.16 |

60 | 59 | 85 | 78.5 | 85.9 | 84.78 | 85.04 |

66 | 47 | 84 | 75.6 | 84.5 | 83.42 | 84.00 |

72 | 39 | 80 | 70.7 | 80.6 | 79.44 | 79.62 |

78 | 32 | 73 | 65.1 | 73.7 | 72.48 | 72.63 |

84 | 28 | 64 | 59.1 | 65.4 | 64.08 | 63.80 |

90 | 24 | 54 | 53.4 | 56 | 54.58 | 54.31 |

96 | 22 | 44 | 47.9 | 46.7 | 45.22 | 44.80 |

102 | 21 | 36 | 43.1 | 37.7 | 36.34 | 36.25 |

108 | 20 | 30 | 38.9 | 30.5 | 29.21 | 29.45 |

114 | 19 | 25 | 35.4 | 25.2 | 24.21 | 24.63 |

120 | 19 | 22 | 32.3 | 21.7 | 20.96 | 21.39 |

126 | 18 | 19 | 29.9 | 20 | 19.41 | 19.81 |

SSQ | - | - | 815.68 | 36.77 | 9.82 | 4.54 |

RMSE | - | - | 6.232327 | 1.330234 | 0.683938 | 0.464948 |

NSE | - | - | 0.965417 | 0.998424 | 0.999584 | 0.999808 |

**Table 5.**Comparison of the outflow hydrographs calculated for Wang et al.’s flood data [10].

Time (12 h) | Input (m^{3}/s) | Output (m^{3}/s) | LMM (Wang et al. [10]) | NLMM (Geem [7]) | NLMM-L (Karahan et al. [8]) | ANLMM-L (This Study) |
---|---|---|---|---|---|---|

1 | 261 | 228 | 228 | 228 | 228.00 | 228.00 |

2 | 389 | 300 | 305.19 | 303.8 | 299.74 | 300.92 |

3 | 462 | 382 | 382 | 382.3 | 382.57 | 381.51 |

4 | 505 | 444 | 442.7 | 442.4 | 442.76 | 443.15 |

5 | 525 | 490 | 483.6 | 482.4 | 482.16 | 482.69 |

6 | 543 | 513 | 513 | 511.2 | 509.89 | 510.09 |

7 | 556 | 528 | 534.29 | 532.3 | 530.72 | 530.66 |

8 | 567 | 543 | 550.44 | 548.5 | 546.77 | 546.62 |

9 | 577 | 553 | 563.53 | 561.7 | 559.96 | 559.77 |

10 | 583 | 564 | 573.16 | 571.6 | 569.94 | 569.80 |

11 | 587 | 573 | 580.02 | 578.7 | 577.07 | 576.95 |

12 | 595 | 581 | 587.32 | 586.2 | 584.39 | 584.22 |

13 | 597 | 588 | 592.14 | 591.2 | 589.68 | 589.60 |

14 | 597 | 594 | 594.59 | 593.9 | 592.34 | 592.30 |

15 | 589 | 592 | 592.02 | 591.8 | 590.33 | 590.34 |

16 | 556 | 584 | 574.89 | 575.7 | 574.68 | 574.86 |

17 | 538 | 566 | 556.85 | 558.5 | 556.41 | 556.23 |

18 | 516 | 550 | 536.93 | 539 | 537.43 | 537.13 |

19 | 486 | 520 | 512.18 | 514.8 | 513.47 | 513.35 |

20 | 505 | 504 | 507.96 | 509.6 | 507.07 | 506.51 |

21 | 477 | 483 | 493.22 | 494.9 | 494.86 | 494.95 |

22 | 429 | 461 | 462.34 | 464.8 | 464.39 | 464.94 |

23 | 379 | 420 | 421.87 | 425.1 | 423.97 | 424.15 |

24 | 320 | 368 | 372.34 | 376.1 | 375.05 | 375.07 |

25 | 263 | 318 | 318.97 | 322.4 | 321.35 | 321.35 |

26 | 220 | 271 | 270.39 | 272.5 | 271.42 | 271.40 |

27 | 182 | 234 | 226.99 | 227.5 | 226.94 | 227.09 |

28 | 167 | 193 | 197.2 | 195.7 | 194.92 | 195.13 |

29 | 152 | 178 | 174.87 | 172.6 | 172.46 | 172.76 |

SSQ | - | - | 1086.84 | 979.96 | 917.06 | 909.35 |

RMSE | - | - | 6.121869 | 5.820949 | 5.623120 | 5.599733 |

NSE | - | - | 0.998180 | 0.998354 | 0.998464 | 0.998477 |

Time (h) | Input (m^{3}/s) | Output (m^{3}/s) | LMM-L (O’Donnell [4]) | NLMM (Karahan et al. [17]) | NLMM-L (Karahan et al. [8]) | ANLMM-L (This Study) |
---|---|---|---|---|---|---|

0 | 154 | 102 | 102 | 154 | 102.00 | 102.00 |

6 | 150 | 140 | 116 | 154 | 149.50 | 146.52 |

12 | 219 | 169 | 120 | 152 | 156.59 | 155.74 |

18 | 182 | 190 | 147 | 181 | 191.40 | 194.41 |

24 | 182 | 209 | 158 | 191 | 200.79 | 194.19 |

30 | 192 | 218 | 165 | 185 | 195.14 | 196.05 |

36 | 165 | 210 | 176 | 187 | 197.46 | 198.35 |

42 | 150 | 194 | 178 | 179 | 188.48 | 186.83 |

48 | 128 | 172 | 176 | 162 | 170.80 | 172.12 |

54 | 168 | 149 | 164 | 141 | 148.10 | 150.37 |

60 | 260 | 136 | 160 | 154 | 162.59 | 167.56 |

66 | 471 | 228 | 167 | 198 | 210.36 | 216.61 |

72 | 717 | 303 | 218 | 264 | 281.58 | 294.27 |

78 | 1092 | 366 | 303 | 344 | 367.75 | 378.29 |

84 | 1145 | 456 | 484 | 416 | 447.65 | 461.17 |

90 | 600 | 615 | 690 | 599 | 629.57 | 612.03 |

96 | 365 | 830 | 700 | 871 | 892.78 | 862.51 |

102 | 277 | 969 | 642 | 834 | 859.01 | 884.60 |

108 | 227 | 665 | 572 | 689 | 719.30 | 737.54 |

114 | 187 | 519 | 505 | 535 | 567.50 | 565.33 |

120 | 161 | 444 | 442 | 397 | 427.85 | 414.97 |

126 | 143 | 321 | 386 | 283 | 308.86 | 297.45 |

132 | 126 | 208 | 338 | 202 | 220.90 | 216.14 |

138 | 115 | 176 | 296 | 152 | 163.64 | 164.43 |

144 | 102 | 148 | 260 | 124 | 131.90 | 134.94 |

150 | 93 | 125 | 228 | 106 | 111.93 | 114.46 |

156 | 88 | 114 | 201 | 94 | 99.28 | 101.24 |

162 | 82 | 106 | 179 | 88 | 92.90 | 94.00 |

168 | 76 | 97 | 160 | 82 | 86.14 | 86.94 |

174 | 73 | 89 | 144 | 75 | 79.34 | 80.13 |

180 | 70 | 81 | 130 | 73 | 76.46 | 76.87 |

186 | 67 | 76 | 118 | 69 | 73.13 | 73.54 |

192 | 63 | 71 | 109 | 66 | 69.85 | 70.23 |

198 | 59 | 66 | 100 | 62 | 65.09 | 65.60 |

SSQ | - | - | 251,802 | 37,944.15 | 25,915.27 | 20,494.98 |

RMSE | - | - | 87.351953 | 33.900478 | 28.023846 | 24.921077 |

NSE | - | - | 0.892983 | 0.983882 | 0.988986 | 0.991290 |

Time (h) | Input (m^{3}/s) | Output (m^{3}/s) | KWM (Karahan and Gurarslan [13]) | NLMM-L (Karahan et al. [8]) | ANLMM-L (This Study) |
---|---|---|---|---|---|

0 | 7.53 | 7 | 7.00 | 7.00 | 7.00 |

1 | 9.06 | 8 | 7.62 | 7.24 | 7.26 |

2 | 28 | 23 | 9.98 | 9.00 | 9.01 |

3 | 79.8 | 25 | 29.16 | 27.35 | 27.35 |

4 | 64.3 | 75 | 73.78 | 74.84 | 74.81 |

5 | 38.2 | 60 | 63.01 | 61.57 | 61.59 |

6 | 41.4 | 40 | 41.98 | 37.40 | 37.41 |

7 | 41.3 | 41 | 41.25 | 39.63 | 39.62 |

8 | 33.8 | 41 | 40.48 | 39.47 | 39.47 |

9 | 32 | 32 | 34.64 | 32.57 | 32.58 |

10 | 29 | 30 | 32.13 | 30.68 | 30.68 |

11 | 35 | 34 | 29.52 | 28.00 | 28.00 |

12 | 63.1 | 35 | 36.16 | 33.93 | 33.93 |

13 | 110 | 60 | 62.98 | 60.62 | 60.62 |

14 | 170 | 105 | 108.45 | 105.25 | 105.25 |

15 | 216 | 160 | 166.29 | 162.06 | 162.07 |

16 | 131 | 206 | 206.02 | 204.11 | 204.11 |

17 | 101 | 128 | 136.31 | 127.58 | 127.61 |

18 | 65 | 97 | 104.68 | 97.14 | 97.10 |

19 | 62.4 | 61 | 67.14 | 63.33 | 63.32 |

20 | 53.8 | 60 | 63.35 | 59.78 | 59.76 |

21 | 36.3 | 50 | 53.18 | 51.51 | 51.50 |

22 | 29.6 | 33 | 37.84 | 35.16 | 35.16 |

23 | 25 | 27 | 30.34 | 28.49 | 28.48 |

24 | 21.3 | 23 | 25.11 | 24.03 | 24.03 |

25 | 19.6 | 19 | 21.68 | 20.49 | 20.49 |

26 | 18 | 18 | 19.71 | 18.81 | 18.81 |

27 | 17.3 | 17 | 18.1 | 17.29 | 17.29 |

28 | 17 | 17 | 17.39 | 16.60 | 16.60 |

29 | 16 | 17 | 16.97 | 16.29 | 16.29 |

SSQ | - | - | 532.62 | 281.11 | 280.95 |

RMSE | - | - | 4.213541 | 3.062067 | 3.060222 |

NSE | - | - | 0.992271 | 0.995918 | 0.995923 |

**Table 8.**Comparison of the outflow hydrographs calculated for the food data of River Wyre October in 1982.

Time (h) | Input (m^{3}/s) | Output (m^{3}/s) | LMM-L (O’Donnell [4]) | NLMM-L (Karahan et al. [8]) | ANLMM-L (This Study) |
---|---|---|---|---|---|

0 | 2.6 | 8.3 | 8.3 | 8.3 | 8.30 |

1 | 4.2 | 9 | 8.2 | 8.51 | 8.52 |

2 | 12.3 | 9.9 | 8.1 | 8.79 | 9.94 |

3 | 25.4 | 10.2 | 12.7 | 10.94 | 12.74 |

4 | 24.1 | 18.9 | 27.9 | 20.28 | 19.71 |

5 | 20.3 | 35.9 | 39.9 | 37.54 | 35.73 |

6 | 23.3 | 51.8 | 45.7 | 49.07 | 48.87 |

7 | 27.7 | 59.4 | 52.2 | 55.11 | 55.95 |

8 | 27.7 | 63.3 | 61.4 | 62.5 | 62.74 |

9 | 26.9 | 69.6 | 68.9 | 71.44 | 71.35 |

10 | 24.8 | 76.7 | 74.7 | 78.03 | 77.95 |

11 | 26.9 | 82 | 77.2 | 82.07 | 82.67 |

12 | 33.7 | 85.3 | 79.8 | 83.72 | 85.27 |

13 | 33.9 | 89 | 87.8 | 87.43 | 88.11 |

14 | 27.8 | 94.6 | 95.5 | 95.49 | 94.74 |

15 | 20.8 | 98.8 | 97.7 | 100.88 | 99.90 |

16 | 15.6 | 98 | 94.4 | 99.29 | 98.87 |

17 | 11.9 | 91.8 | 87.9 | 92.06 | 92.05 |

18 | 9.5 | 82.3 | 79.8 | 82.22 | 82.36 |

19 | 7.8 | 72 | 71.5 | 71.75 | 71.88 |

20 | 6.5 | 61.9 | 63.6 | 61.94 | 61.93 |

21 | 5.8 | 53 | 56.1 | 53.12 | 53.10 |

22 | 5.0 | 45.6 | 49.6 | 45.47 | 45.37 |

23 | 4.8 | 39.2 | 43.7 | 39.14 | 39.04 |

24 | 4.5 | 33.8 | 38.8 | 33.76 | 33.65 |

25 | 4.1 | 29.3 | 34.6 | 29.55 | 29.39 |

26 | 3.7 | 26.2 | 30.9 | 26.12 | 25.96 |

27 | 3.4 | 23.5 | 27.7 | 23.2 | 23.08 |

28 | 3.2 | 21.2 | 24.8 | 20.67 | 20.59 |

29 | 2.9 | 19.2 | 22.3 | 18.52 | 18.44 |

30 | 2.8 | 17.7 | 20.1 | 16.71 | 16.68 |

31 | 2.6 | 16.4 | 18.2 | 15.12 | 15.09 |

SSQ | - | - | 468.84 | 53.66 | 40.16 |

RMSE | - | - | 3.790780 | 1.263563 | 1.120320 |

NSE | - | - | 0.989570 | 0.998842 | 0.999090 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lee, E.H.; Lee, H.M.; Kim, J.H.
Development and Application of Advanced Muskingum Flood Routing Model Considering Continuous Flow. *Water* **2018**, *10*, 760.
https://doi.org/10.3390/w10060760

**AMA Style**

Lee EH, Lee HM, Kim JH.
Development and Application of Advanced Muskingum Flood Routing Model Considering Continuous Flow. *Water*. 2018; 10(6):760.
https://doi.org/10.3390/w10060760

**Chicago/Turabian Style**

Lee, Eui Hoon, Ho Min Lee, and Joong Hoon Kim.
2018. "Development and Application of Advanced Muskingum Flood Routing Model Considering Continuous Flow" *Water* 10, no. 6: 760.
https://doi.org/10.3390/w10060760