# Evaluation of Different Methods on the Estimation of the Daily Crop Coefficient of Winter Wheat

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## Abstract

**:**

_{IA}= 0.88), branch-overwintering stage (RMSE = 0.13, MAE = 0.11, r = 0.44, d

_{IA}= 0.55), and heading-maturity stage (RMSE = 0.16, MAE = 0.13, r = 0.94, d

_{IA}= 0.97), while the cumulative crop coefficient method performed best during the greening-jointing stage (RMSE = 0.16, MAE = 0.13, r = 0.88, d

_{IA}= 0.89). Based on this result, an integrated modelling procedure was proposed by applying the best method in each growth stage, which provides higher simulation precision than any single method. When the best method was adopted in each growth stage, the estimated accuracy of the whole growth process was RMSE = 0.13, MAE = 0.09, r = 0.98, d

_{IA}= 0.99.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Area Profile

#### 2.2. Experimental Facilities and Data Selection

#### 2.3. Division of Growth Stages

#### 2.4. Crop Coefficient, Actual Evapotranspiration, and Reference Evapotranspiration

_{c}is the crop coefficient; ET is the actual evapotranspiration (mm); and ET

_{0}is the reference evapotranspiration (mm).

_{g}is the diving evaporation (mm); P

_{a}is the deep leakage rate (mm); ET is the actual evapotranspiration (mm); R is the volume of runoff (mm); and ΔS is the soil storage variable (mm).

_{0}is the reference evapotranspiration (mm × d

^{−1}); R

_{n}is the net surface radiation (MJ × m

^{−2}× d

^{−1}); G is the soil heat flux (MJ × m

^{−2}× d

^{−1}); T is the average daily temperature (°C); u

_{2}is the average wind speed at 2 m above the ground (m × s

^{−1}); e

_{s}is the saturated vapour pressure (kPa); e

_{a}is the actual vapour pressure (kPa); Δ is the slope of the saturated vapour pressure and temperature curve (kPa × °C

^{−1}); and γ is the dry and wet table constant (kPa × °C

^{−1}). Data are from a high precision weather station.

#### 2.5. Crop Coefficient Estimation Method and Evaluation Indices

#### 2.5.1. Temperature Effect Method

_{c}is the crop coefficient; K

_{0}is the crop coefficient at the optimum temperature; T is the average temperature (°C); T

_{0}is the optimum temperature for physiological and ecological processes such as crop growth and photosynthesis (°C); and β is the parameter to be estimated.

_{upper}is the upper limit temperature (°C), which is 30 °C and T

_{base}indicates the lower limit temperature (°C), which is 3 °C [26,27,28].

_{0}, T

_{0}, and β were determined by SPSS software combined with the least squares method and sequential quadratic programming. First, the logarithm of Equation (4) can be obtained:

#### 2.5.2. Cumulative Crop Coefficient Method

_{c cumulative}is the cumulative crop coefficient value, D is the number of days after seeding, and a, b, c, and d are unknown parameters.

#### 2.5.3. Radiative Soil Temperature Method

_{n}-G) refers to the difference between net radiation and soil heat flux, showing a trend of being smaller at the emergence-branching and branch-overwintering stages and gradually increasing at the greening-jointing and heading-maturity stages (Figure 3). During the whole process of winter wheat growth, the correlation coefficient between the crop coefficient and effective energy reached 0.75, showing a strong correlation. The correlation coefficient between the crop coefficient and soil temperature from 0 to 160 cm decreased from 0.66 to 0.70 with increasing depth, as shown in Figure 4.

_{0}is the surface temperature (°C); D

_{10}is the soil temperature of 10 cm (°C); and m and n are unknown parameters. The meanings of the other symbols are the same as described previously.

#### 2.5.4. Indices of Evaluation

_{IA}), which are used to evaluate the error and consistency between the estimated value and the measured value of each estimation method. See Equation (21) to Equation (24) for the calculation formula of each index.

_{IA}are to 1, the closer the estimated value of the model is to the actual value, and the stronger its estimation ability is.

## 3. Results

#### 3.1. The Differences and Causes of Crop Coefficient Estimation by Different Methods

#### 3.2. Determination of The Best Estimation Method for Each Growth Stage

_{IA}are compared in turn to select the method with the strongest estimation ability. In the third step, if the difference between the correlation coefficient and consistency index is less than 0.03, then the cumulative crop coefficient method should be directly adopted according to the principle of minimum required observations if it is still in the option list. Otherwise, the most appropriate method should be adopted in combination with the graph.

_{IA}= 0.88), followed by the cumulative crop coefficient method, and the radiative soil temperature method was inferior. The temperature effect method during the branch-overwintering stage was the best (RMSE = 0.13, MAE = 0.11, r = 0.44, d

_{IA}= 0.55), followed by the cumulative crop coefficient method, and the radiative soil temperature method was inferior. The cumulative crop coefficient method during the greening-jointing stage was the best (RMSE = 0.16, MAE = 0.13, r = 0.88, d

_{IA}= 0.89), followed by the temperature effect method, and the radiative soil temperature method was inferior. The temperature effect method during the heading-maturity stage was the best (RMSE = 0.16, MAE = 0.13, r = 0.94, d

_{IA}= 0.97), followed by the cumulative crop coefficient method, and the radiative soil temperature method was inferior.

_{IA}= 0.96), followed by the temperature effect method, and the radiative soil temperature method was inferior. When the cumulative crop coefficient method and temperature effect method were used, the correlation coefficient and consistency index of the whole growth process estimation results were both greater than 0.80, which met the accuracy requirements of estimation and could be used for crop coefficient estimation. When the radiative soil temperature method was used, the correlation coefficient and consistency index of the whole growth process estimation results were only 0.50 and 0.69, respectively, which could not meet the requirement of estimation accuracy.

## 4. Conclusions and Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

FAO | Food and Agriculture Organization |

FAO-56 | FAO Irrigation and Drainage Paper No. 56 |

K_{c} | Crop coefficient |

ET | Actual evapotranspiration |

K_{s} | Water stress coefficient |

TCARI | Transformed chlorophyll absorption in reflectance index |

RDVI | Renormalized difference vegetation index |

UAV | Unmanned aerial vehicle |

TSEB | Two-source energy balance |

ET_{0} | Reference evapotranspiration |

SPSS | Statistical Product and Service Solutions |

R_{n}-G | Effective energy |

MATLAB | Matrix Laboratory |

r | Correlation coefficient |

d_{IA} | Consistency index |

RMSE | Root mean square error |

MAE | Mean absolute error |

TE | Temperature effect method |

CCC | Cumulative crop coefficient method |

RST | Radiative soil temperature method |

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**Figure 1.**The variation of the actual evapotranspiration (ET) and reference evapotranspiration (ET

_{0}).

**Figure 4.**Correlation between crop coefficient and soil temperature from 0 to 160 cm(a to h in the figure represent the deepening of soil depth).

Stage of Growth | Emergence-Branching | Branch-Overwintering | Greening-Jointing | Heading-Maturity |
---|---|---|---|---|

Date | 2018/11/11–2018/12/1 | 2018/12/2–2019/2/21 | 2019/2/22–2019/4/19 | 2019/4/20–2019/6/4 |

Number of days | 21 | 82 | 57 | 46 |

ET_{0} at this stage | 21.80 mm | 67.80 mm | 216.48 mm | 220.67 mm |

Proportion of total ET_{0} | 4.14% | 12.87% | 41.10% | 41.89% |

Average daily ET_{0} | 1.04 mm | 0.83 mm | 3.80 mm | 4.80 mm |

Method | Temperature Effect | Cumulative Crop Coefficient | Radiative Soil Temperature | ||||||
---|---|---|---|---|---|---|---|---|---|

Parameter | ${\mathit{K}}_{\mathbf{0}}$ | ${\mathit{T}}_{\mathbf{0}}$ | $\mathit{\beta}$ | $\mathit{a}$ | $\mathit{b}$ | $\mathit{c}$ | $\mathit{d}$ | $\mathit{m}$ | $\mathit{n}$ |

Emergence-branching stage | 1.24 | 3.00 | 18.84 | 158.62 | −7.17 | 0.07 | 6.54 | 4.19 | 5.87 |

Branch-overwintering stage | 1.96 | 3.00 | 12.42 | 216.22 | −79.90 | 0.01 | 60.00 | 0.26 | 3.21 |

Greening-jointing stage | 2.14 | 20.95 | 6.61 | 180.23 | −382.33 | −0.01 | 26.25 | 0.41 | 5.62 |

Heading-maturity stage | 2.39 | 20.37 | 5.16 | 227.91 | 0.02 | −0.05 | 43.52 | 6.16 | 8.81 |

Stage | Emergence-Branching Stage | Branch-Overwintering Stage | Greening-Jointing Stage | Heading-Maturity Stage | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Index | RMSE | MAE | r | dIA | RMSE | MAE | r | dIA | RMSE | MAE | r | dIA | RMSE | MAE | r | dIA |

TE | 0.06 | 0.06 | 0.80 | 0.88 | 0.13 | 0.11 | 0.44 | 0.55 | 0.23 | 0.18 | 0.70 | 0.83 | 0.16 | 0.13 | 0.94 | 0.97 |

CCC | 0.08 | 0.07 | 0.57 | 0.69 | 0.13 | 0.12 | 0.36 | 0.51 | 0.16 | 0.13 | 0.88 | 0.89 | 0.20 | 0.16 | 0.91 | 0.94 |

RST | 0.20 | 0.19 | 0.35 | 0.51 | 0.25 | 0.22 | 0.52 | 0.61 | 0.93 | 0.79 | 0.70 | 0.49 | 1.10 | 0.91 | 0.43 | 0.49 |

Method | Root Mean Square Error | Mean Absolute Error | Correlation Coefficient | Consistency Index |
---|---|---|---|---|

TE | 0.34 | 0.25 | 0.87 | 0.93 |

CCC | 0.25 | 0.20 | 0.93 | 0.96 |

RST | 0.79 | 0.58 | 0.50 | 0.69 |

The best | 0.13 | 0.09 | 0.98 | 0.99 |

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**MDPI and ACS Style**

Fang, J.; Wang, Y.; Jiang, P.; Ju, Q.; Zhou, C.; Lu, Y.; Gao, P.; Sun, B.
Evaluation of Different Methods on the Estimation of the Daily Crop Coefficient of Winter Wheat. *Water* **2023**, *15*, 1395.
https://doi.org/10.3390/w15071395

**AMA Style**

Fang J, Wang Y, Jiang P, Ju Q, Zhou C, Lu Y, Gao P, Sun B.
Evaluation of Different Methods on the Estimation of the Daily Crop Coefficient of Winter Wheat. *Water*. 2023; 15(7):1395.
https://doi.org/10.3390/w15071395

**Chicago/Turabian Style**

Fang, Jingjing, Yining Wang, Peng Jiang, Qin Ju, Chao Zhou, Yiran Lu, Pei Gao, and Bo Sun.
2023. "Evaluation of Different Methods on the Estimation of the Daily Crop Coefficient of Winter Wheat" *Water* 15, no. 7: 1395.
https://doi.org/10.3390/w15071395