## Abstract

The Muskingum method is one of hydrological approaches that has been used for flood routing for many years thanks to its simplicity and reasonable accuracy over other methods. In engineering works, the calculation of the Peak section of a flood hydrograph is crucially important. In the present study, using the particle swarm optimization (PSO) algorithm, instead of using a single basic flood, the parameters of the linear Muskingum method (X, K, Δt) are calculated by computed arithmetic and geometric means relevant to two basic floods in the form of eight different models for calculating the downstream hydrograph. The results indicate that if the numerical values of the calculated flood inflow are placed in the interval of the inflow and the basic flood which the parameters X, K, Δt are from, the computation accuracy in approximating the outflow flood related to the peak section of the inflow hydrograph increases for all the mentioned models. In other words, if the arithmetic mean of X, K and the geometric mean of Δt, relevant to the two basic floods, are used instead of using values of X, K, Δt of a single basic flood, the computational accuracy in estimating the flood peak section of the hydrograph in downstream has the highest increase among all the eight models. Thus, the Mean Relative Error (MRE) relevant to the peak section of the inflow hydrograph of the third flood (observational flood) obtained by the first and second basic floods was equal to 4.89% and 2.91%, respectively, while in case of using the arithmetic mean of X and K and the geometric mean of Δt, related to the first and second basic floods (the best models presented in this study), this value is equal to 1.66%.

## HIGHLIGHTS

Using parameters of two baseline floods simultaneously for flood routing using the linear Muskingum method.

In addition to the optimized (X, K), the time interval (Δt) is also optimized using the particle swarm optimization (PSO) algorithm.

Using the arithmetic mean of X, K and the geometric mean of Δt related to two baseline floods is the best state for flood routing.

Increasing the accuracy of flood routing calculations in estimating the flood peak section using parameters of two baseline floods rather than using parameters of a single baseline flood.

Using the presented model in this study compared to the case in which only the first or second baseline flood is used, the Mean Relative Error (MRE) will be decreased by 66% and 43%, respectively.

## INTRODUCTION

Floods are one of the natural disasters, and preventing economic, social, socio-economic and other damage from floods has been a concern for humans (Vafaei & Harati 2010; Farzin *et al.* 2018; Fotovatikhah *et al.* 2018; Vatankhah 2018). The flood analysis and control in watersheds threatened by flooding are important for human lives (Rowshan *et al.* 2007). The flood routing is, in fact, the calculation of the flood hydrograph in the downstream. The flood routing is treated as an important issue in analyzing the effects of structures on flood control (Asiaban *et al.* 2015). In particular, the peak flood calculation is of great importance for the construction of flood control structures and reducing natural hazards and economic and social costs (Wu & Chau 2011; Gholami *et al.* 2015; Reggiani *et al.* 2016). Various hydraulic and hydrological methods are used for flood routing. In other words, the estimation of the flood hydrograph is called ‘flood routing’. (Yadav *et al.* 2015; Formetta *et al.* 2018). Due to longer computations and the need for more data, using hydraulic methods is difficult (Yadav *et al.* 2015), while flood routing using hydrological approaches requires much less data and is easier (Yadav *et al.* 2015). (Tsai 2005) examined hydraulic and hydrologic methods of flood routing and concluded that the former are more accurate than the latter but suffer from complexity in solving relevant equations. The Muskingum flood routing is one of the most important hydrological methods. This model uses continuity equations and relationships between discharge, outflow and flooding values (Meng *et al.* 2017). Niazkar & Afzali (2016) presented a new model for the nonlinear Muskingum method and used the hybrid MHBMO-GRG algorithm, which is a combination of the modified honey bee mating optimization (MHBMO) and the generalized reduced gradient (GRG), to optimize the parameters of the mentioned method. Mohan (1997) developed a model based on the genetic algorithm (GA) to estimate the parameters of the nonlinear Muskingum. Furthermore, Barati (2011) investigated the parameters of the nonlinear Muskingum model using Nelder–Mead Simplex (NMS) and compared the performance of this algorithm with other techniques for evaluating the Muskingum parameters. (Zhang *et al.* 2017) presented a new model for the nonlinear Muskingum method entitled ‘VEP-NLMM-L’ and used the improved real coded adaptive genetic algorithm (RAGA) to estimate the parameters of the model. (Choudhury *et al.* 2002) developed a method to investigate flood routing in river networks. In this method, multiple inflows are substituted by an equivalent single inflow by means of finite-difference, correlation equation and momentum methods. To solve the equation, the Levenberg-Marquardt algorithm in SPSS software is used. The comparison between this and methods previously proposed by Gill and Wilson confirms the high accuracy of this method. Barati (2013) applied GRG and Evolutionary Solver methods in Microsoft Excel 2010 to optimize the Muskingum parameters. Hirpurkar & Ghare (2014) analyzed three different nonlinear forms of the parameter (m) in the nonlinear Muskingum method using Microsoft Excel. Their results confirms higher accuracy of the nonlinear Muskingum when using the form presented by Chow (1959) a S = K[XI + (1 − X)O]^{m}. The particle swarm optimization (PSO) algorithm is a population-based evolutionary algorithm and is applicable to civil engineering and water resource optimization issues. These include reservoir operations (Nagesh Kumar & Janga Reddy 2007), water quality management (Lu *et al.* 2002; Chau 2005; Afshar *et al.* 2011), water resources management in the basin (Shourian *et al.* 2008), flood control management (Meraji 2004). Chu & Chang (2009) optimized the parameters in the nonlinear Muskingum method using the PSO algorithm. The comparison between this method and previous ones including harmony search (HS), linear regression (LR) and GA indicates the high accuracy and speed of the PSO algorithm in estimating the parameters of the nonlinear Muskingum. (Moghaddam *et al.* 2016) proposed a new four-parameter model for the nonlinear Muskingum method that was used for four flood routings. The results indicated that the PSO algorithm optimized the four parameters of the presented model with high accuracy and fast convergence rate. Also, to enhance the linearization process and increase the accuracy of the linear Muskingum method, Bazargan & Norouzi (2018) divided the inflow hydrograph into the start, peak and end sections and optimized the parameters(X, K, Δt) of each section using the PSO algorithm. Owing to low computational time, algorithms are very capable of optimizing the parameters of the Muskingum method. Increasing the number of the parameters of the Muskingum method, causes the increase of the algorithm calculation time, while the accuracy of results does not change significantly (Farahani *et al.* 2018).

In the present study, as the inflow discharge variation related to the third flood (computational flood) is in the limit of inflow discharge variation related to the first and second floods (basic floods), rather than using values of X, K, Δt related to a single basic flood, the arithmetic and geometric means of respective parameters of the basic floods are used in eight different models to improve the accuracy of the linear Muskingum method in estimating the flood peak section of the outflow hydrograph. It should be noted that the values of X, K, Δt in each basic flood are estimated through the PSO.

## MATERIALS AND METHODS

### Study area

In this study, the recorded data for the Mollasani hydrometric station (station no. 21–308, 48°53′ E, 31°35′N) at the upstream and the Ahwaz hydrometric station (station no. 21–309, 48°40′E, 31°20′N) (Distance 60.5 Km) at the downstream of the study reach are used, both related to the Karun River in Iran (Figure 1). The flood data (Table 1) with the variation of the inflow from 221(m^{3}/s) to 565(m^{3}/s) are utilized as the first basic flood and the flood data (Table 1) with the variation of the inflow from 316(m^{3}/s) to 490(m^{3}/s) are employed as the second basic flood for computing the parameters of linear Muskingum method (X, K, Δt). The main advantage of the Muskingum method is that it can be used to route any other floods that have occurred during the study period (to obtain the characteristics of the flood at downstream)., provided that the morphology of the river has not changed, using the basic flood parameters (floods whose inflow and outflow values have been recorded). For this reason, in the present study, the parameters obtained from the basic floods (first and second floods) have been used to calculate the outflow hydrograph of the third flood (Table 1) with the range of changes in the inflow from 222 m^{3}/s to 494 m^{3}/s. It is worth mentioning that all three floods are related to the mentioned range (Mollasani hydrometric station at upstream and Ahwaz station at downstream) of the Karun River.

. | First flood . | Second flood . | Third flood . | |||
---|---|---|---|---|---|---|

Time(h) . | Inflow (m^{3}/s)
. | Observed outflow (m^{3}/s)
. | Inflow (m^{3}/s)
. | Observed outflow (m^{3}/s)
. | Inflow (m^{3}/s)
. | Observed outflow (m^{3}/s)
. |

0 | 221 | 198 | 366 | 290 | 287 | 243 |

1 | 224 | 198 | 376 | 296 | 287 | 244 |

2 | 231 | 198 | 387 | 299 | 292 | 250 |

3 | 235 | 198 | 396 | 303 | 292 | 250 |

4 | 242 | 198 | 407 | 310 | 294 | 257 |

5 | 246 | 200 | 416 | 314 | 296 | 257 |

6 | 255 | 200 | 427 | 317 | 298 | 257 |

7 | 263 | 203 | 436 | 325 | 300 | 260 |

8 | 272 | 203 | 446 | 328 | 305 | 263 |

9 | 283 | 206 | 453 | 336 | 307 | 263 |

10 | 296 | 209 | 459 | 339 | 311 | 266 |

11 | 309 | 214 | 469 | 347 | 316 | 270 |

12 | 320 | 217 | 473 | 354 | 320 | 273 |

13 | 331 | 223 | 478 | 362 | 325 | 276 |

14 | 342 | 229 | 483 | 366 | 329 | 276 |

15 | 351 | 235 | 485 | 373 | 334 | 280 |

16 | 362 | 238 | 490 | 381 | 340 | 283 |

17 | 374 | 247 | 490 | 389 | 349 | 286 |

18 | 385 | 254 | 488 | 397 | 356 | 286 |

19 | 396 | 263 | 488 | 401 | 367 | 290 |

20 | 410 | 270 | 485 | 405 | 380 | 293 |

21 | 423 | 280 | 481 | 413 | 392 | 293 |

22 | 437 | 293 | 478 | 417 | 407 | 297 |

23 | 450 | 300 | 473 | 426 | 421 | 304 |

24 | 464 | 311 | 469 | 426 | 434 | 307 |

25 | 478 | 322 | 459 | 426 | 450 | 311 |

26 | 491 | 333 | 455 | 426 | 462 | 318 |

27 | 503 | 344 | 448 | 430 | 471 | 329 |

28 | 512 | 359 | 441 | 434 | 478 | 336 |

29 | 521 | 371 | 436 | 434 | 482 | 344 |

30 | 530 | 383 | 430 | 434 | 487 | 356 |

31 | 537 | 400 | 423 | 434 | 489 | 363 |

32 | 542 | 412 | 416 | 434 | 491 | 375 |

33 | 549 | 425 | 409 | 434 | 494 | 387 |

34 | 553 | 438 | 401 | 434 | 494 | 396 |

35 | 558 | 451 | 394 | 430 | 491 | 404 |

36 | 560 | 460 | 387 | 430 | 491 | 412 |

37 | 563 | 474 | 381 | 426 | 489 | 421 |

38 | 563 | 483 | 374 | 421 | 489 | 429 |

39 | 565 | 493 | 366 | 421 | 487 | 438 |

40 | 565 | 497 | 361 | 413 | 485 | 438 |

41 | 563 | 507 | 353 | 409 | 482 | 447 |

42 | 563 | 517 | 349 | 401 | 478 | 443 |

43 | 560 | 521 | 343 | 397 | 475 | 447 |

44 | 558 | 526 | 336 | 393 | 473 | 451 |

45 | 556 | 531 | 330 | 389 | 469 | 451 |

46 | 553 | 531 | 326 | 381 | 464 | 451 |

47 | 551 | 536 | 320 | 377 | 459 | 451 |

48 | 549 | 536 | 316 | 373 | 455 | 451 |

49 | 542 | 541 | 450 | 451 | ||

50 | 540 | 541 | 446 | 451 | ||

51 | 535 | 541 | 441 | 451 | ||

52 | 530 | 546 | 434 | 451 | ||

53 | 526 | 546 | 430 | 451 | ||

54 | 521 | 546 | 423 | 451 | ||

55 | 514 | 546 | 419 | 447 | ||

56 | 510 | 546 | 412 | 447 | ||

57 | 503 | 551 | 405 | 443 | ||

58 | 498 | 546 | 398 | 443 | ||

59 | 494 | 546 | 392 | 438 | ||

60 | 487 | 541 | 383 | 434 | ||

61 | 482 | 541 | 376 | 429 | ||

62 | 475 | 541 | 369 | 429 | ||

63 | 471 | 531 | 367 | 421 | ||

64 | 464 | 531 | 356 | 417 | ||

65 | 457 | 526 | 349 | 412 | ||

66 | 453 | 521 | 342 | 408 | ||

67 | 446 | 517 | 338 | 400 | ||

68 | 439 | 507 | 331 | 396 | ||

69 | 432 | 502 | 327 | 391 | ||

70 | 428 | 497 | 320 | 387 | ||

71 | 421 | 493 | 316 | 375 | ||

72 | 414 | 488 | 311 | 371 | ||

73 | 410 | 483 | 307 | 367 | ||

74 | 405 | 479 | 303 | 363 | ||

75 | 401 | 474 | 298 | 356 | ||

76 | 394 | 465 | 292 | 352 | ||

77 | 389 | 460 | 289 | 344 | ||

78 | 385 | 456 | 285 | 340 | ||

79 | 383 | 451 | 279 | 333 | ||

80 | 378 | 443 | 274 | 329 | ||

81 | 371 | 438 | 270 | 325 | ||

82 | 367 | 434 | 265 | 322 | ||

83 | 365 | 429 | 263 | 318 | ||

84 | 360 | 421 | 259 | 314 | ||

85 | 356 | 417 | 255 | 311 | ||

86 | 351 | 412 | 250 | 304 | ||

87 | 349 | 404 | 246 | 300 | ||

88 | 347 | 400 | 242 | 293 | ||

89 | 342 | 396 | 239 | 290 | ||

90 | 342 | 391 | 235 | 286 | ||

91 | 338 | 387 | 233 | 283 | ||

92 | 336 | 383 | 231 | 280 | ||

93 | 334 | 379 | 229 | 276 | ||

94 | 331 | 375 | 226 | 273 | ||

95 | 329 | 375 | 224 | 266 | ||

96 | 327 | 371 | 222 | 263 | ||

97 | 322 | 367 | ||||

98 | 322 | 363 | ||||

99 | 320 | 359 | ||||

100 | 320 | 356 | ||||

101 | 318 | 356 | ||||

102 | 316 | 352 | ||||

103 | 314 | 348 | ||||

104 | 311 | 344 | ||||

105 | 309 | 340 | ||||

106 | 309 | 340 | ||||

107 | 307 | 336 | ||||

108 | 305 | 333 | ||||

109 | 305 | 333 | ||||

110 | 303 | 333 | ||||

111 | 300 | 329 | ||||

112 | 300 | 325 | ||||

113 | 298 | 325 | ||||

114 | 298 | 322 | ||||

115 | 296 | 318 | ||||

116 | 294 | 318 | ||||

117 | 296 | 318 | ||||

118 | 294 | 314 | ||||

119 | 294 | 311 | ||||

120 | 294 | 311 |

. | First flood . | Second flood . | Third flood . | |||
---|---|---|---|---|---|---|

Time(h) . | Inflow (m^{3}/s)
. | Observed outflow (m^{3}/s)
. | Inflow (m^{3}/s)
. | Observed outflow (m^{3}/s)
. | Inflow (m^{3}/s)
. | Observed outflow (m^{3}/s)
. |

0 | 221 | 198 | 366 | 290 | 287 | 243 |

1 | 224 | 198 | 376 | 296 | 287 | 244 |

2 | 231 | 198 | 387 | 299 | 292 | 250 |

3 | 235 | 198 | 396 | 303 | 292 | 250 |

4 | 242 | 198 | 407 | 310 | 294 | 257 |

5 | 246 | 200 | 416 | 314 | 296 | 257 |

6 | 255 | 200 | 427 | 317 | 298 | 257 |

7 | 263 | 203 | 436 | 325 | 300 | 260 |

8 | 272 | 203 | 446 | 328 | 305 | 263 |

9 | 283 | 206 | 453 | 336 | 307 | 263 |

10 | 296 | 209 | 459 | 339 | 311 | 266 |

11 | 309 | 214 | 469 | 347 | 316 | 270 |

12 | 320 | 217 | 473 | 354 | 320 | 273 |

13 | 331 | 223 | 478 | 362 | 325 | 276 |

14 | 342 | 229 | 483 | 366 | 329 | 276 |

15 | 351 | 235 | 485 | 373 | 334 | 280 |

16 | 362 | 238 | 490 | 381 | 340 | 283 |

17 | 374 | 247 | 490 | 389 | 349 | 286 |

18 | 385 | 254 | 488 | 397 | 356 | 286 |

19 | 396 | 263 | 488 | 401 | 367 | 290 |

20 | 410 | 270 | 485 | 405 | 380 | 293 |

21 | 423 | 280 | 481 | 413 | 392 | 293 |

22 | 437 | 293 | 478 | 417 | 407 | 297 |

23 | 450 | 300 | 473 | 426 | 421 | 304 |

24 | 464 | 311 | 469 | 426 | 434 | 307 |

25 | 478 | 322 | 459 | 426 | 450 | 311 |

26 | 491 | 333 | 455 | 426 | 462 | 318 |

27 | 503 | 344 | 448 | 430 | 471 | 329 |

28 | 512 | 359 | 441 | 434 | 478 | 336 |

29 | 521 | 371 | 436 | 434 | 482 | 344 |

30 | 530 | 383 | 430 | 434 | 487 | 356 |

31 | 537 | 400 | 423 | 434 | 489 | 363 |

32 | 542 | 412 | 416 | 434 | 491 | 375 |

33 | 549 | 425 | 409 | 434 | 494 | 387 |

34 | 553 | 438 | 401 | 434 | 494 | 396 |

35 | 558 | 451 | 394 | 430 | 491 | 404 |

36 | 560 | 460 | 387 | 430 | 491 | 412 |

37 | 563 | 474 | 381 | 426 | 489 | 421 |

38 | 563 | 483 | 374 | 421 | 489 | 429 |

39 | 565 | 493 | 366 | 421 | 487 | 438 |

40 | 565 | 497 | 361 | 413 | 485 | 438 |

41 | 563 | 507 | 353 | 409 | 482 | 447 |

42 | 563 | 517 | 349 | 401 | 478 | 443 |

43 | 560 | 521 | 343 | 397 | 475 | 447 |

44 | 558 | 526 | 336 | 393 | 473 | 451 |

45 | 556 | 531 | 330 | 389 | 469 | 451 |

46 | 553 | 531 | 326 | 381 | 464 | 451 |

47 | 551 | 536 | 320 | 377 | 459 | 451 |

48 | 549 | 536 | 316 | 373 | 455 | 451 |

49 | 542 | 541 | 450 | 451 | ||

50 | 540 | 541 | 446 | 451 | ||

51 | 535 | 541 | 441 | 451 | ||

52 | 530 | 546 | 434 | 451 | ||

53 | 526 | 546 | 430 | 451 | ||

54 | 521 | 546 | 423 | 451 | ||

55 | 514 | 546 | 419 | 447 | ||

56 | 510 | 546 | 412 | 447 | ||

57 | 503 | 551 | 405 | 443 | ||

58 | 498 | 546 | 398 | 443 | ||

59 | 494 | 546 | 392 | 438 | ||

60 | 487 | 541 | 383 | 434 | ||

61 | 482 | 541 | 376 | 429 | ||

62 | 475 | 541 | 369 | 429 | ||

63 | 471 | 531 | 367 | 421 | ||

64 | 464 | 531 | 356 | 417 | ||

65 | 457 | 526 | 349 | 412 | ||

66 | 453 | 521 | 342 | 408 | ||

67 | 446 | 517 | 338 | 400 | ||

68 | 439 | 507 | 331 | 396 | ||

69 | 432 | 502 | 327 | 391 | ||

70 | 428 | 497 | 320 | 387 | ||

71 | 421 | 493 | 316 | 375 | ||

72 | 414 | 488 | 311 | 371 | ||

73 | 410 | 483 | 307 | 367 | ||

74 | 405 | 479 | 303 | 363 | ||

75 | 401 | 474 | 298 | 356 | ||

76 | 394 | 465 | 292 | 352 | ||

77 | 389 | 460 | 289 | 344 | ||

78 | 385 | 456 | 285 | 340 | ||

79 | 383 | 451 | 279 | 333 | ||

80 | 378 | 443 | 274 | 329 | ||

81 | 371 | 438 | 270 | 325 | ||

82 | 367 | 434 | 265 | 322 | ||

83 | 365 | 429 | 263 | 318 | ||

84 | 360 | 421 | 259 | 314 | ||

85 | 356 | 417 | 255 | 311 | ||

86 | 351 | 412 | 250 | 304 | ||

87 | 349 | 404 | 246 | 300 | ||

88 | 347 | 400 | 242 | 293 | ||

89 | 342 | 396 | 239 | 290 | ||

90 | 342 | 391 | 235 | 286 | ||

91 | 338 | 387 | 233 | 283 | ||

92 | 336 | 383 | 231 | 280 | ||

93 | 334 | 379 | 229 | 276 | ||

94 | 331 | 375 | 226 | 273 | ||

95 | 329 | 375 | 224 | 266 | ||

96 | 327 | 371 | 222 | 263 | ||

97 | 322 | 367 | ||||

98 | 322 | 363 | ||||

99 | 320 | 359 | ||||

100 | 320 | 356 | ||||

101 | 318 | 356 | ||||

102 | 316 | 352 | ||||

103 | 314 | 348 | ||||

104 | 311 | 344 | ||||

105 | 309 | 340 | ||||

106 | 309 | 340 | ||||

107 | 307 | 336 | ||||

108 | 305 | 333 | ||||

109 | 305 | 333 | ||||

110 | 303 | 333 | ||||

111 | 300 | 329 | ||||

112 | 300 | 325 | ||||

113 | 298 | 325 | ||||

114 | 298 | 322 | ||||

115 | 296 | 318 | ||||

116 | 294 | 318 | ||||

117 | 296 | 318 | ||||

118 | 294 | 314 | ||||

119 | 294 | 311 | ||||

120 | 294 | 311 |

### Introducing linear Muskingum method

The Muskingum model hydrologic flood-routing method was introduced by McCarthy (1938) in studies of floods of the Muskingum River in Ohio in the United States.

*t*= rate of change of storage during a time interval Δ

*t*; K = storage: time constant for the river reach; and X = dimensionless weighting factor representing the inflow outflow effects on storage. X ranges between 0 and 0.5 for reservoir storage and 0 and 0.3 for stream channels (Mohan 1997).

If I_{1}, I_{2}, O_{1} are initially known, O_{2} can be calculated using Equation (4) provided that K and X are determined. Then for a new time step, next discharge in the inflow hydrograph is considered as I_{2}; and I_{2} and O_{2} as new I_{1} and O_{1}.

### Determination of outflow hydrograph

*Q*,

_{i}*O*are observational outflow discharge and routed outflow discharge (computational), respectively.

_{i}To calculate the arithmetic and geometric means of K and Δt, the same equations are used presented for X.

The flowchart used in this study for optimizing the parameters of the Muskingum method using the PSO algorithm and the cost function SAD is illustrated in Figure 2.

Some previous researchers considered the parameter Δt equal to the time interval of reading inflow and outflow hydrographs. However, the optimization results for the first and second floods (basic floods) are not the same. In other words, the optimization of the parameter Δt leads to increasing the accuracy of the linear Muskingum method in estimating the outflow hydrograph. Subramanya (1994) has emphasized the selection of the optimal time interval (Δt).

In the present study, the parameters of the linear Muskingum method (*X*, *K*, Δt) are calculated using the PSO method. In general, this study is developed as follows:

- 1.
Optimization of the linear Muskingum parameters (

*X*,*K*, Δt) using the first basic flood in which inflow discharge variation is between 221 (m^{3}/s) and 565 (m^{3}/s). - 2.
Optimization of linear Muskingum parameters (

*X*,*K*, Δt) using the second basic flood in which inflow discharge variation is between 316 (m^{3}/s) and 490 (m^{3}/s). - 3.
Calculation of arithmetic and geometric means of the estimated parameters from steps 1 and 2.

- 4.
Calculation of different models for those parameters according to Tables 2–5.

- 5.
Using the values of

*X*,*K*, Δt, estimated in the above steps to calculate the third flood outflow hydrograph (computational flood) in which inflow discharge variation is between 222 (m^{3}/s) and 494 (m^{3}/s).

Basic flood . | X . | K (h) . | Δt (h) . | MRE % . | MRE % (peak section, T = 27 to T = 45(h)) . |
---|---|---|---|---|---|

First flood | 0.401 | 14.709 | 1.102 | 4.69 | 4.89 |

Second flood | 0.078 | 11.957 | 0.675 | 5.37 | 2.91 |

Basic flood . | X . | K (h) . | Δt (h) . | MRE % . | MRE % (peak section, T = 27 to T = 45(h)) . |
---|---|---|---|---|---|

First flood | 0.401 | 14.709 | 1.102 | 4.69 | 4.89 |

Second flood | 0.078 | 11.957 | 0.675 | 5.37 | 2.91 |

Muskingum parameters models . | X
. | K
. | Δt . | MRE % . | MRE % (peak section, T = 27 to T = 45(h)) . |
---|---|---|---|---|---|

Δt & X (AM^{a}), K (GM^{b}) | 0.2395 | 13.2618 | 0.8885 | 4.68 | 2.19 |

Δt & K (AM), X (GM) | 0.1768 | 13.333 | 0.8885 | 4.64 | 2.25 |

Δt (AM), X & K (GM) | 0.1768 | 13.2618 | 0.8885 | 4.64 | 2.33 |

Muskingum parameters models . | X
. | K
. | Δt . | MRE % . | MRE % (peak section, T = 27 to T = 45(h)) . |
---|---|---|---|---|---|

Δt & X (AM^{a}), K (GM^{b}) | 0.2395 | 13.2618 | 0.8885 | 4.68 | 2.19 |

Δt & K (AM), X (GM) | 0.1768 | 13.333 | 0.8885 | 4.64 | 2.25 |

Δt (AM), X & K (GM) | 0.1768 | 13.2618 | 0.8885 | 4.64 | 2.33 |

^{a}Arithmetic mean.

^{b}Geometric mean.

Muskingum parameters models . | X
. | K
. | Δt . | MRE % . | MRE % (peak section, T = 27 to T = 45(h)) . |
---|---|---|---|---|---|

X, K, Δt (Geometric mean) | 0.1768 | 13.2618 | 0.8624 | 4.67 | 2 |

X, K, Δt (Arithmetic mean) | 0.2395 | 13.333 | 0.8885 | 4.68 | 2.11 |

Muskingum parameters models . | X
. | K
. | Δt . | MRE % . | MRE % (peak section, T = 27 to T = 45(h)) . |
---|---|---|---|---|---|

X, K, Δt (Geometric mean) | 0.1768 | 13.2618 | 0.8624 | 4.67 | 2 |

X, K, Δt (Arithmetic mean) | 0.2395 | 13.333 | 0.8885 | 4.68 | 2.11 |

Muskingum parameters models . | X
. | K
. | Δt . | MRE % . | MRE % (peak section, T = 27 to T = 45(h)) . |
---|---|---|---|---|---|

X & K (AM), Δt (GM) | 0.2395 | 13.333 | 0.8624 | 4.71 | 1.66 |

X (AM^{a}), K & Δt (GM^{b}) | 0.2395 | 13.2618 | 0.8624 | 4.70 | 1.73 |

K (AM), X & Δt (GM) | 0.1768 | 13.333 | 0.8624 | 4.68 | 1.95 |

Muskingum parameters models . | X
. | K
. | Δt . | MRE % . | MRE % (peak section, T = 27 to T = 45(h)) . |
---|---|---|---|---|---|

X & K (AM), Δt (GM) | 0.2395 | 13.333 | 0.8624 | 4.71 | 1.66 |

X (AM^{a}), K & Δt (GM^{b}) | 0.2395 | 13.2618 | 0.8624 | 4.70 | 1.73 |

K (AM), X & Δt (GM) | 0.1768 | 13.333 | 0.8624 | 4.68 | 1.95 |

^{a}Arithmetic mean.

^{b}Geometric mean.

The investigations show that values of *X*, *K*, Δt in any river reach depend upon numerical values of inflow and outflow hydrographs or upon the extent of the flooding section in different cross-sections of the river during the given reach. In other words, by increasing the peak values of inflow hydrograph and extending the flooding section in the river, the values of *X*, *K*, Δt will change. Therefore, if values of inflow discharge of the computational flood are closer to values of inflow discharge of the observational flood, the accuracy of the linear Muskingum method in estimating outflow discharge will increase, and if values of inflow discharge of the computational flood (the third flood) are placed in the limit of two observational basic floods, it would be expected that the passing flow behavior of the computational flood is placed in the limit of the behavior of the first and second basic floods. Therefore, in the case of using different models for estimating *X*, *K*, and Δt from two basic floods (Tables 2–5), it is more likely that the accuracy of the linear Muskingum will increase.

## RESULTS AND DISCUSSION

As can be seen in Table 1, variations of the third flood's inflow discharge (computational flood) are approximately in the limit of variations in inflow discharge of the first and second floods. For this reason, rather than using parameters of a single basic flood to calculate the outflow hydrograph of the third flood, the developed models in Tables 2–5 are used for calculating the mentioned parameters.

The parameters of the linear Muskingum method (*X*, *K*, Δt) for each of the first and second floods (basic floods) which have been optimized by the PSO algorithm are given in Table 2. Also, if the parameters of each of the base floods are used alone to calculate the hydrograph of the third flood (computational flood), the values for the Mean Relative Error (MRE) of the total flood and the MRE of the peak flood section are presented in Table 2.

If the arithmetic mean of the parameter Δt for the two baseline floods is used in three possible models, the values for the parameters X, K, and Δt of each model and the error values obtained for the third flood routing (computational flood) are given in Table 3.

If the geometric and arithmetic means of all three parameters *X*, *K*, and Δt are used to calculate the computational flood outflow hydrograph, the values related to the mentioned parameters as well as the error values obtained for calculating the computational flood are presented in Table 4.

The error values obtained for the routing of the third flood (computational), as well as the values for the parameters *X*, *K*, and Δt, are given in Table 5 in the case where the geometric mean of the Δt parameter is used in three possible models.

As can be seen in Tables 2–5, the models presented in this study, in all cases, have increased the accuracy of the linear Muskingum method in estimating the outflow associated with the third flood (computational flood) peak section of the inflow hydrograph. According to Table 5, when using the three models presented in the geometric mean of the parameter Δt to calculate the third flood (computational flood) outflow hydrograph, the highest accuracy is obtained for the linear Muskingum method in estimating the outflow hydrograph of the peak flood section.

As it is presented in Tables 2–5, the MRE of the total outflow hydrograph has not been significantly reduced in the case of using different models for values of X, K and Δt, compared to the case where only the first and second basic flood data are used. However, the values related to the MRE of the peak flood section using the models presented in the present study are reduced in all cases. In other words, the results of the present study indicate that, if the range of the inflow variations of the computational flood (third flood) is within the range of the variations of the inflow of the basic floods (first and second floods), in terms of using the geometric mean parameter Δt and the arithmetic mean of the parameters *X*, *K*, the linear Muskingum method is more accurate in estimating the hydrograph of the flood peak section (whether using a single base flood alone or when using other models presented in the present study). So the MRE related to the peak flood hydrograph section of the third flood in the case of using the mentioned model is obtained to be 1.66%. While this value is obtained equal to 4.89% and 2.91%, respectively, when only the first and second basic flood parameters are used. Since the flood hydrograph peak section is more important in engineering, the use of this model in estimating the outflow hydrograph is more efficient.

Observed inflow and outflow hydrographs related to the third flood as well as computational values of the outflow hydrograph in the case of using the parameters of the first and second basic floods alone, and also the best model presented in the present study (using the geometric mean of the parameter Δt and the arithmetic mean of the parameters X, K), are shown in Figure 3.

## CONCLUSION

In the present study, the amplitude of the variations of the third flood inflow, which was used as a computational flood, is within the range of the inflow of the first and second floods (basic floods), so that its minimum value was close to the inflow of the first basic flood and its maximum value was close to the inflow of the second basic flood. Instead of using the parameters obtained from one basic flood, the use of the arithmetic and geometric means related to the two basic floods in different modes presented in Tables 3–5 was examined. For this reason, the use of the parameters obtained in the above cases, instead of using the usual linear Muskingum method (using the parameters of one basic flood) in all cases, increased the accuracy of the linear Muskingum method in estimating the peak section of the computational flood inflow hydrograph.

The results suggested that MRE for the peak section of the hydrograph in the case of using the first and second basic flood was 4.89 and 2.91 percent, respectively. While this value was reduced to 1.66 percent in the case of using the arithmetic mean for *X*, *K* and the geometric mean for the parameter Δt using the first and second basic floods. In other words, in the case of using the presented model compared to the case where only the first or second basic floods are used, the computational error decreased 66% and 43%, respectively.

## ACKNOWLEDGEMENT

The authors gratefully acknowledge the research department of Iran Water Resources Management Co. for its support in collecting and providing data required in this project.

## SUPPLEMENTARY MATERIAL

The Supplementary Material for this paper is available online at https://dx.doi.org/10.2166/ws.2020.099.