# Uncertainty in Irrigation Return Flow Estimation: Comparing Conceptual and Physically-Based Parameterization Approaches

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Estimation of Irrigation Return Flow

#### 2.1. Mathematical Representation of Irrigation Return Flow

#### 2.2. Paddy Drainage Model

#### 2.2.1. Conceptual Parameter Approach

#### 2.2.2. Physically-Based Approach

^{3}/s); $\mu $ is a coefficient that represents the combined effects of the discharge and approach velocity coefficients; and $W$ is the weir width perpendicular to the direction of flow (m). The combined coefficient, $\mu $ is determined using

^{2}). In this study, the value of $A$ was estimated to range between 1800 m

^{2}and 5900 m

^{2}, based on 100 paddy plots in the central Korea. PHY assumes that paddy plots in an irrigation block are homogeneous (i.e., the paddy plots have the same parameter values ($A$, $K$, and $W$) and water management practices).

#### 2.3. Estimation of Irrigation Return Flow from Paddy Drainage

- Water is not irrigated (zero $IR$) within a rainfall event ($RAIN$).
- Surface runoff discharge ($PRO$) is always followed by irrigation water discharge ($PRF$).
- Irrigation water ponded in a paddy plot is consumed by $ET$ and $INF$ first, and then water ponded by $RAIN$ is used.

#### 2.4. Parameter Value Range Selection Using Expert Knowledge

^{2}was calculated from the average of 100 paddy plots. Following this, we found a nonlinear relationship between $W$ and $a$ (Figure 2).

#### 2.5. Evaluating Uncertainty and Accuracy

## 3. Study Area and Data

^{2}to 5900 m

^{2}, with irrigation and drainage ditches located along both its sides (Figure 3). The paddy block is irrigated with surface water released from the Idong agricultural reservoir, and RF from the block is drained back to the Jinwi river. We monitored water balance in paddy blocks during the rice growing seasons in 2011 and 2012. Meteorological data including temperature, wind speed, relative humidity, and solar radiation were obtained from the Suwon National Weather station, 20 km away from the study area. Rainfall was observed at 10 min intervals using a rain gauge (tipping bucket type, HOBO Data Logging Rain Gauge, Bourne, US) placed 2 km away from the study area. Water level was measured every 10 min using ultrasonic sensors installed at the head and tail of the ditches. These measurements were converted to discharge using a stage–discharge relationship developed for ditches based on regression analysis [29] (Figure 3). The observations of $DMC$ and $INF$ were provided to the model as the boundary condition to explicitly compare CON and PHY approaches. The seasonal variations in $LH$ and crop coefficients were found in the literature and adopted for this study [2,8,32,37]. Measured drainage was compared to model predictions to evaluate the performance of uncalibrated parameter approaches.

## 4. Results and Discussions

#### 4.1. Uncertainty and Accuracy of Conceptual and Physically-Based Parameterization Approaches

#### 4.2. Potential of Conceptual and Physically-Based Parameterization Approaches

## 5. Conclusions

- The results showed that the physically-based parameterization could effectively regulate the behavior of the RF model and thus produce smaller uncertainty compared to that of the conceptual approach, suggesting and confirming the potential of a physically-based approach as a way to effectively reduce modeling uncertainty without parameter calibration.
- When the value ranges of a conceptual parameter were naively (or simply) defined, the CON produced wide uncertainty in RF estimates.
- The conceptual parameter could be reasonably related to the physical and hydraulic characteristics of a study field, substantially reducing the size of uncertainty in RF estimates to as small as that of PHY.
- Narrow-CON and PHY reproduced $TDR$ at similar and acceptable accuracy, and the performance of the ensemble $RF$ (Narrow-CON-Median and PHY-Median) was comparable to that of the calibrated approaches (CON-BestNSE and PHY-BestNSE).

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Dewandel, B.; Gandolfi, J.-M.; de Condappa, D.; Ahmed, S. An efficient methodology for estimating irrigation return flow coefficients of irrigated crops at watershed and seasonal scale. Hydrol. Process.
**2008**, 22, 1700–1712. [Google Scholar] [CrossRef] - Kang, M.; Park, S. Modeling water flows in a serial irrigation reservoir system considering irrigation return flows and reservoir operations. Agric. Water Manag.
**2014**, 143, 131–141. [Google Scholar] [CrossRef] - Kim, H.K.; Jang, T.I.; Im, S.J.; Park, S.W. Estimation of irrigation return flow from paddy fields considering the soil moisture. Agric. Water Manag.
**2009**, 96, 875–882. [Google Scholar] [CrossRef] - Song, J.-H.; Her, Y.; Park, J.; Lee, K.-D.; Kang, M.-S. Simulink implementation of a hydrologic model: A Tank model case study. Water
**2017**, 9, 639. [Google Scholar] [CrossRef] [Green Version] - Song, J.-H.; Kang, M.S.; Song, I.; Jun, S.M. Water balance in irrigation reservoirs considering flood control and irrigation efficiency variation. J. Irrig. Drain. Eng.
**2016**, 142, 04016003. [Google Scholar] [CrossRef] - Song, J.H.; Song, I.; Kim, J.-T.; Kang, M.S. Characteristics of irrigation return flow in a reservoir irrigated district. J. Korean Soc. Agric. Eng.
**2015**, 57, 69–78. [Google Scholar] - Chien, C.-P.; Fang, W.-T. Modeling irrigation return flow for the return flow reuse system in paddy fields. Paddy Water Environ.
**2012**, 10, 187–196. [Google Scholar] [CrossRef] - Song, J.-H.; Her, Y.; Jun, S.M.; Hwang, S.; Park, J.; Kang, M.-S. Lessons from assessing uncertainty in agricultural water supply estimation for sustainable rice production. Agronomy
**2019**, 9, 662. [Google Scholar] [CrossRef] [Green Version] - Zulu, G.; Toyota, M.; Misawa, S. Characteristics of water reuse and its effects on paddy irrigation system water balance and the riceland ecosystem. Agric. Water Manag.
**1996**, 31, 269–283. [Google Scholar] [CrossRef] - Grafton, R.Q.; Williams, J.; Perry, C.J.; Molle, F.; Ringler, C.; Steduto, P.; Udall, B.; Wheeler, S.A.; Wang, Y.; Garrick, D.; et al. The paradox of irrigation efficiency. Science
**2018**, 361, 748–750. [Google Scholar] [CrossRef] [Green Version] - Tan, X.; Shao, D.; Gu, W. Improving water reuse in paddy field districts with cascaded on-farm ponds using hydrologic model simulations. Water Resour. Manag.
**2018**, 32, 1849–1865. [Google Scholar] [CrossRef] - Choo, T.-H. A study on return flow ratio of irrigation for a paddy field in pumping station by water balance method. J. Korea Water Resour. Assoc.
**2004**, 37, 249–255. [Google Scholar] [CrossRef] [Green Version] - Chung, S.-O.; Park, K.-J. Irrigation return flow measurements and analysis in a small size paddy area. J. Korea Water Resour. Assoc.
**2004**, 37, 517–526. [Google Scholar] [CrossRef] - Kim, T.-C.; Lee, H.-C.; Moon, J.-P. Estimation of return flow rate of irrigation water in Daepyeong pumping district. J. Korean Soc. Agric. Eng.
**2010**, 52, 41–49. [Google Scholar] - Kim, J.-S.; Oh, S.-Y.; Oh, K.-Y.; Cho, J.-W. Delivery management water requirement for irrigation ditches associated with large-sized paddy plots in Korea. Paddy Water Environ.
**2005**, 3, 57–62. [Google Scholar] [CrossRef] - Im, S. Modeling Irrigation Return Flow from Paddy Fields on Agricultural Watersheds; Seoul National University: Seoul, Korea, 2000. [Google Scholar]
- Song, J.-H. Hydrologic Analysis System with Multi-Objective Optimization for Agricultural Watersheds; Seoul National University: Seoul, Korea, 2017. [Google Scholar]
- Chang, Y.-C.; Kan, C.-E.; Lin, G.-F.; Chiu, C.-L.; Lee, Y.-C. Potential benefits of increased application of water to paddy fields in Taiwan. Hydrol. Process.
**2001**, 15, 1515–1524. [Google Scholar] [CrossRef] - Chen, R.-S.; Yang, K.-H. Terraced paddy field rainfall-runoff mechanism and simulation using a revised tank model. Paddy Water Environ.
**2011**, 9, 237–247. [Google Scholar] [CrossRef] [Green Version] - Chen, S.-K.; Chen, R.-S.; Yang, T.-Y. Application of a tank model to assess the flood-control function of a terraced paddy field. Hydrol. Sci. J.
**2014**, 59, 1020–1031. [Google Scholar] [CrossRef] [Green Version] - Yoshinaga, I.; Miura, A.; Hitomi, T.; Hamada, K.; Shiratani, E. Runoff nitrogen from a large sized paddy field during a crop period. Agric. Water Manag.
**2007**, 87, 217–222. [Google Scholar] [CrossRef] - Jang, T.I.; Kim, H.K.; Im, S.J.; Park, S.W. Simulations of storm hydrographs in a mixed-landuse watershed using a modified TR-20 model. Agric. Water Manag.
**2010**, 97, 201–207. [Google Scholar] [CrossRef] - Kang, M.S.; Koo, J.H.; Chun, J.A.; Her, Y.G.; Park, S.W.; Yoo, K. Design of drainage culverts considering critical storm duration. Biosyst. Eng.
**2009**, 104, 425–434. [Google Scholar] [CrossRef] - Odhiambo, L.O.; Murty, V.V.N. Modeling water balance components in relation to field layout in lowland paddy fields. I. Model development. Agric. Water Manag.
**1996**, 30, 185–199. [Google Scholar] [CrossRef] - Wu, R.-S.; Sue, W.-R.; Chien, C.-B.; Chen, C.-H.; Chang, J.-S.; Lin, K.-M. A simulation model for investigating the effects of rice paddy fields on the runoff system. Math. Comput. Model.
**2001**, 33, 649–658. [Google Scholar] [CrossRef] - Gharari, S.; Shafiei, M.; Hrachowitz, M.; Kumar, R.; Fenicia, F.; Gupta, H.V.; Savenije, H.H.G. A constraint-based search algorithm for parameter identification of environmental models. Hydrol. Earth Syst. Sci.
**2014**, 18, 4861–4870. [Google Scholar] [CrossRef] [Green Version] - Gharari, S.; Hrachowitz, M.; Fenicia, F.; Gao, H.; Savenije, H.H.G. Using expert knowledge to increase realism in environmental system models can dramatically reduce the need for calibration. Hydrol. Earth Syst. Sci.
**2015**, 18, 4839–4859. [Google Scholar] [CrossRef] [Green Version] - Hrachowitz, M.; Fovet, O.; Ruiz, L.; Euser, T.; Gharari, S.; Nijzink, R.; Freer, J.; Savenije, H.H.G.; Gascuel-Odoux, C. Process consistency in models: The importance of system signatures, expert knowledge, and process complexity. Water Resour. Res.
**2014**, 50, 7445–7469. [Google Scholar] [CrossRef] [Green Version] - Song, J.-H.; Her, Y.; Hwang, S.; Park, J.; Yoon, K.-S.; Kang, M.S. Evaluating the applicability of drainage routing schemes for paddy fields. J. Irrig. Drain. Eng.
**2020**, in press. [Google Scholar] - Allen, R.G.; Pereira, L.S.; Raes, D.; Smith, M. Crop Evapotranspiration-Guidelines for Computing Crop Water Requirements; Food and Agriculture Organization of the United Nations: Rome, Italy, 1998. [Google Scholar]
- Song, J.-H.; Kang, M.-S.; Song, I.; Hwang, S.-H.; Park, J.; Ahn, J.-H. Surface drainage simulation model for irrigation districts composed of paddy and protected cultivation. J. Korean Soc. Agric. Eng.
**2013**, 55, 63–73. [Google Scholar] - Yoo, S.-H.; Choi, J.-Y.; Lee, S.-H.; Oh, Y.-G.; Yun, D.K. Climate change impacts on water storage requirements of an agricultural reservoir considering changes in land use and rice growing season in Korea. Agric. Water Manag.
**2013**, 117, 43–54. [Google Scholar] - Nash, J.E. The form of the instantaneous unit hydrograph. IAHS Publ.
**1957**, 3, 114–121. [Google Scholar] - Purcell, P.J. Physical Analog of the Linear Reservoir. J. Hydrol. Eng.
**2006**, 11, 184–187. [Google Scholar] [CrossRef] - Song, J.-H.; Her, Y.; Park, J.; Kang, M.-S. Exploring parsimonious daily rainfall-runoff model structure using the hyperbolic tangent function and Tank model. J. Hydrol.
**2019**, 574, 574–587. [Google Scholar] [CrossRef] - Chow, V.T.; Maidment, D.R.; Mays, L.W. Applied Hydrology; McGraw-Hill: New York, NY, USA, 1988. [Google Scholar]
- Kang, M.S.; Park, S.W.; Lee, J.J.; Yoo, K.H. Applying SWAT for TMDL programs to a small watershed containing rice paddy fields. Agric. Water Manag.
**2006**, 79, 72–92. [Google Scholar] [CrossRef] - Nash, J.E.; Sutcliffe, J.V. River flow forecasting through conceptual models part I—A discussion of principles. J. Hydrol.
**1970**, 10, 282–290. [Google Scholar] [CrossRef] - Oudin, L.; Andréassian, V.; Mathevet, T.; Perrin, C.; Michel, C. Dynamic averaging of rainfall-runoff model simulations from complementary model parameterizations. Water Resour. Res.
**2006**, 42. [Google Scholar] [CrossRef] - Gupta, H.V.; Sorooshian, S.; Yapo, P.O. Toward improved calibration of hydrologic models: Multiple and noncommensurable measures of information. Water Resour. Res.
**1998**, 34, 751–763. [Google Scholar] [CrossRef] - Beven, K.J. Rainfall-Runoff Modelling: The Primer; John Wiley & Sons: Chichester, UK, 2001; ISBN 1-119-95101-1. [Google Scholar]
- Vinogradov, Y.B.; Semenova, O.M.; Vinogradova, T.A. An approach to the scaling problem in hydrological modelling: The deterministic modelling hydrological system. Hydrol. Process.
**2011**, 25, 1055–1073. [Google Scholar] [CrossRef] - Moriasi, D.N.; Gitau, M.W.; Daggupati, P. Hydrologic and water quality models: Performance measures and evaluation criteria. Trans. ASABE
**2015**, 58, 1763–1785. [Google Scholar] - Efstratiadis, A.; Koutsoyiannis, D. One decade of multi-objective calibration approaches in hydrological modelling: A review. Hydrol. Sci. J.
**2010**, 55, 58–78. [Google Scholar] [CrossRef] [Green Version] - Khu, S.T.; Madsen, H. Multiobjective calibration with Pareto preference ordering: An application to rainfall-runoff model calibration. Water Resour. Res.
**2005**, 41. [Google Scholar] [CrossRef]

**Figure 1.**Schematic of the water balance in paddy fields; (

**a**) irrigation and drainage block, (

**b**) paddy field, and (

**c**) field outlet [29].

**Figure 2.**Relationship between the weir width ($W$) and conceptual parameter representing the drainage capacity ($a$).

**Figure 4.**Temporal variations of inputs ($RAIN$ and $AWS$), outputs ($DMWR$, $TDR$, $RF$, $PRF$, and $DRF$), and uncertainty (the 95% confidence interval) of output modeling using the Wide-CON approach. Simulated $TDR$ is compared with the observations.

**Figure 5.**Accuracy statistics of predicting $TDR$ and uncertainty ranges of ${R}_{RF}$ calculated using the conceptual and physically-based parameter approaches. The height of a box plot is the interquartile range (IQR) (or the distance between the 25th and 75th percentiles). The ends of whiskers represent the minimum and maximum values. Circles outside the whisker ends are outliers.

**Figure 6.**Temporal variations of inputs ($RAIN$ and $AWS$), outputs ($DMWR$, $TDR$, $RF$, $PRF$, and $DRF$), and uncertainty (the 95% confidence interval) of output modeling using the Narrow-CON approach. Simulated $TDR$ is compared with the observations.

**Figure 7.**Temporal variations of inputs ($RAIN$ and $AWS$), outputs ($DMWR$, $TDR$, $RF$, $PRF$, and $DRF$), and uncertainty (the 95% confidence interval) of output modeling using the PHY approach. Simulated $TDR$ is compared with the observations.

Parameter | P.S. | T.S. | Growing and Harvesting Season (10-days) | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | |||

${K}_{c}$ | 0.78 | 0.78 | 0.78 | 0.97 | 1.07 | 1.16 | 1.28 | 1.45 | 1.5 | 1.58 | 1.46 | 1.45 | 1.25 | 1.01 | 1.01 | 1.01 | 1.01 |

$LH$ (mm) | 66.1 | 80.9 | 74 | 57.3 | 34.6 | 72.9 | 67.2 | 57.7 | 63.4 | 67.2 | 66.1 | 66.1 | 66.1 | 66.1 | 66.1 | 66.1 | 66.1 |

**Table 2.**Overview of the conceptual and physically-based parameterization approaches compared in this study.

Schemes | CON | PHY |
---|---|---|

Description | Conceptual based parameter approach | Physically-based parameter approach |

Routing scheme | Linear reservoir with threshold scheme | Broad-crested weir equation |

Calibration parameter | $a$ | $A$, $K$, $W$ |

Simulation time step | Daily | Hourly |

Scheme | Parameter | Definition | Min. | Max. |
---|---|---|---|---|

Wide-CON | $a$ | Drainage capacity of a paddy outlet (dimensionless) | 0.08 | 1 |

Narrow-CON | $a$ | Drainage capacity of a paddy outlet (dimensionless) | 0.32 | 0.91 |

PHY | $W$ | Weir width perpendicular to the direction of flow (m) | 0.08 | 0.8 |

PHY | $K$ | Parameter representing the overall flow condition (dimensionless) | 1.7 | 2.0 |

PHY | $A$ | Area of a unit paddy field (m^{2}) | 1800 | 5900 |

Performance Measures | CON | CON | Narrow-CON | PHY | PHY | |
---|---|---|---|---|---|---|

$\mathit{a}$ = 1 | Best NSE ^{a} | Median ^{b} | Best NSE ^{a} | Median ^{b} | ||

${R}^{2}$ | 0.79 | 0.84 | 0.83 | 0.86 | 0.86 | |

$NSE$ | 0.69 | 0.84 | 0.83 | 0.85 | 0.83 | |

$NS{E}_{ln}$ | −0.05 | 0.44 | 0.56 | 0.56 | 0.46 | |

$PBIAS$ (%) | 3 | 3 | 6 | 2 | 5 | |

${R}_{RF}$ (%) | 2011 | 86 | 85 | 83 | 84 | 84 |

2012 | 79 | 77 | 75 | 76 | 77 | |

Entire period | 82 | 81 | 80 | 80 | 81 |

^{a}The performance of the parameter set for the highest $NSE$ in the all sampled sets;

^{b}The performance of the median of daily uncertainty intervals.

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**MDPI and ACS Style**

Song, J.-H.; Her, Y.; Hwang, S.; Kang, M.-S.
Uncertainty in Irrigation Return Flow Estimation: Comparing Conceptual and Physically-Based Parameterization Approaches. *Water* **2020**, *12*, 1125.
https://doi.org/10.3390/w12041125

**AMA Style**

Song J-H, Her Y, Hwang S, Kang M-S.
Uncertainty in Irrigation Return Flow Estimation: Comparing Conceptual and Physically-Based Parameterization Approaches. *Water*. 2020; 12(4):1125.
https://doi.org/10.3390/w12041125

**Chicago/Turabian Style**

Song, Jung-Hun, Younggu Her, Soonho Hwang, and Moon-Seong Kang.
2020. "Uncertainty in Irrigation Return Flow Estimation: Comparing Conceptual and Physically-Based Parameterization Approaches" *Water* 12, no. 4: 1125.
https://doi.org/10.3390/w12041125