# Simulation of Irrigation Strategy Based on Stochastic Rainfall and Evapotranspiration

^{*}

## Abstract

**:**

_{fc}). For the four full irrigation modes (A1, A2, A3, A4), the lower limits were set at 0.6 s

_{fc}, 0.6 s

_{fc}, 0.5 s

_{fc}, and 0.5 s

_{fc}, respectively. The upper limits were defined at two levels: 0.8 s

_{fc}for A1 and A2 and s

_{fc}for A3 and A4. Similarly, for the four deficit irrigation modes (B1, B2, B3, B4), the lower limits were established at 0.4 s

_{fc}, 0.4 s

_{fc}, 0.3 s

_{fc}, and 0.3 s

_{fc}, respectively, with the upper limits set at two levels: 0.8 s

_{fc}for B1 and B2 and the full s

_{fc}for B3 and B4. To investigate the impact of rainfall and potential evapotranspiration on these irrigation modes under long-term fluctuations, we employed a stochastic framework that probabilistically linked rainfall events and irrigation applications. The Monte Carlo method was employed to simulate a long-term series (4000a) of rainfall parameters and evapotranspiration using 62 years of meteorological data from the Xinxiang region, situated in the southern part of the North China Plain. Results showed that the relative yield and net irrigation water requirement of summer maize decreased with decreasing irrigation lower limits. Additionally, the interannual variation of rainfall parameters and evapotranspiration during the growing season were remarkable, which led to the lowest relative yield of the rainfed mode (E) aligned with a larger interannual difference. According to the simulation results, mode A4 (irrigation lower limit equals 0.5 s

_{fc}, irrigation upper limit equals 0.8 s

_{fc}) could be adopted for adequate water resources. Conversely, mode B2 is more suitable for a lack of water resources.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Method

#### 2.1.1. Rainfall Parameter

_{seas}is the length of the growing season.

#### 2.1.2. Potential Evapotranspiration

_{m}is the potential evapotranspiration, E

_{0}is the reference crop evapotranspiration and K

_{c}is the crop coefficient. Summer maize K

_{c}was taken as 1.07, according to Song et al. [23]. Reference crop evapotranspiration was calculated using the FAO-recommended Penman-Monteith equation [22,24]:

_{n}is the net canopy radiation, G is the soil heat flux, T is the air temperature at 2 m above ground, u

_{2}is the wind speed at 2 m above ground, e

_{s}is the saturation water vapor pressure, e

_{a}is the actual water vapor pressure, Δ is the slope of the saturation water vapor pressure curve, and ζ is the hygrometer constant (kPa/°C).

#### 2.1.3. Soil Moisture Density Function

_{r}of a homogeneous of soil of porosity n. The groundwater depth in the study area is below 5 m. Therefore, the influence of groundwater evapotranspiration is disregarded in this study.

_{fc}is the field water-holding capacity; Δ is the canopy interception capacity.

#### 2.1.4. Net Irrigation Water Requirement and Actual Evapotranspiration

_{m}is the maximum crop yield, which is the highest yield (kg/hm

^{2}) that can be obtained under the climatic conditions of the year without restricting the normal growth of the crop by water, fertilizer, pests, and diseases; K

_{y}is the yield response coefficient, with a value of K

_{y}= 1.16 as determined in the study by Kang [28].

#### 2.2. Data Collection and Simulation Design

^{3}, and the soil exhibits a porosity (n) of 0.44. The field water capacity (θ

_{fc}) is 0.33 (volume), while the wilting coefficient (θ

_{w}) is 0.09 (volume). The root active layer depth for summer maize is 80 cm, and the average canopy interception capacity during the growth period is 1.5 mm [29]. Meteorological data for the Xinxiang area from 1951 to 2012 was sourced from the China Meteorological Science Data Sharing Service Network.

_{fc}, 0.8 s

_{fc}) and four lower irrigation limits (0.6 s

_{fc}, 0.5 s

_{fc}, 0.4 s

_{fc}, 0.3 s

_{fc}), as outlined in Table 1. Probability distribution functions for interannual α, λ, and E

_{p}were fitted using meteorological data from 1951 to 2012 in the study area. The Monte Carlo method was employed to generate a long series (4000 years) of α, λ, and E

_{p}[30]. Subsequently, these values were incorporated into the abovementioned equations to calculate net irrigation water demand, relative evapotranspiration, and relative yield for different water regulation modes. This approach enables an analysis of the effects of various water regulation measures.

## 3. Results

#### 3.1. Characteristics of Interannual Variation of Rainfall Parameters and Potential Evapotranspiration

^{−1}, with a maximum of 0.143 d

^{−1}and a minimum of 0.457 d

^{−1}, leading to a coefficient of variation of 0.170. Unlike the rainfall parameters, potential evapotranspiration (E

_{p}) presented relatively stable values of 3.95 mm/day, 4.86 mm/day, 3.33 mm/day, and 0.076 for the multi-year mean, maximum, minimum, and coefficient of variation, respectively.

_{p}~LogNormal (3.945, 0.076). As α and λ do not obey normal distribution, a nonparametric Spearman rank correlation test was conducted to assess their correlation. The resulting correlation coefficient 0.028 (at a confidence interval of 0.05) suggests a weak correlation between α and λ, indicating that their interannual variations are largely independent.

#### 3.2. Effect of Moisture Regulation on Soil Moisture

_{fc}. In Figure 2b,d, the upper limits of irrigation (ω) for full and deficit irrigation are 0.8 s

_{fc}.

_{fc}, the soil moisture density function curve for the rainfed condition is wider, and the peak of p(s) occurs at s = 0.275. For both full and deficit irrigation, the peak of p(s) occurs at s = s

_{fc}. However, it is worth noting that the peak of p(s) for deficit irrigation, as shown in Figure 2c, is noticeably lower than that for full irrigation in Figure 2a.

_{fc}. Similarly, the peak value of p(s) for deficit irrigation remains substantially lower than that for full irrigation, respectively, as shown in Figure 2b,d.

#### 3.3. Effect of Water Regulation on Net Irrigation Water Requirement and Relative Yield

_{fc}and Scenario II with ω = 0.8 s

_{fc}.

#### 3.4. Irrigation Strategies Analysis

_{p}sample spaces were simulated using a computer pseudo-random number generator, and the irrigation water demand, relative evapotranspiration and relative yield of different moisture regulation patterns were calculated by substituting them into the above equations and listed in Table 2.

## 4. Discussion

## 5. Conclusions

_{fc}, ω = s

_{fc}) and A3 (ε = 0.6 s

_{fc}, ω = s

_{fc}) ensure stable and high summer maize yields but come with a relatively high net irrigation water requirement. Full irrigation A2 (ε = 0.6 s

_{fc}, ω = 0.8 s

_{fc}) and A4 (ε = 0.6 s

_{fc}, ω = 0.8 s

_{fc}) yield the same level of output but with less net irrigation water requirement. On the other hand, deficit irrigation methods B1 (ε = 0.4 s

_{fc}, ω = s

_{fc}) and B2 (ε = 0.4 s

_{fc}, ω = 0.8 s

_{fc}) maintain a slightly decreased yield while demanding less net irrigation water. In contrast, the rainfed mode (E) exhibits lower actual evapotranspiration, greater year-to-year variability, and less stable yields.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Interannual variations of rainfall parameters (α and λ) and potential evapotranspiration (E

_{p}). (

**a**,

**c**,

**e**) are the probability density of α, λ and Ep, respectively; (

**b**,

**d**,

**f**) are the cumulative density of α, λ and Ep, respectively.

**Figure 2.**Probability density functions of soil moisture for eight irrigation strategies controlled by four irrigation lower limits (ε) and two irrigation upper limits (ω). (

**a**,

**b**) are four irrigation strategies; (

**c**,

**d**) are four irrigation strategies.

**Figure 3.**Irrigation requirement (V), relative evapotranspiration, and relative yield (Y/Y

_{m}) as a function of irrigation lower limit (ε) for two irrigation upper limits (ω). (

**a**,

**b**) depicts how alterations in the lower irrigation limit directly affect the net irrigation requirement and relative evapotranspiration, as well as relative yield, respectively, in two sceneries (ω = s

_{fc}and ω = 0.8 s

_{fc}).

**Table 1.**Simulation scheme design for irrigation, featuring full irrigation strategies (A1, A2, A3, A4), deficit irrigation strategies (B1, B2, B3, B4), and rainfed mode (E). The irrigation strategies are controlled by various percentages of the field water-holding capacity (s

_{fc}).

Soil Water Regulation | Irrigation Lower Limit | Irrigation Upper Limit |
---|---|---|

A1 | 0.6 s_{fc} | s_{fc} |

A2 | 0.6 s_{fc} | 0.8 s_{fc} |

A3 | 0.5 s_{fc} | s_{fc} |

A4 | 0.5 s_{fc} | 0.8 s_{fc} |

B1 | 0.4 s_{fc} | s_{fc} |

B2 | 0.4 s_{fc} | 0.8 s_{fc} |

B3 | 0.3 s_{fc} | s_{fc} |

B4 | 0.3 s_{fc} | 0.8 s_{fc} |

E | - | - |

**Table 2.**The mean and coefficient of variation of irrigation requirement (V), relative evapotranspiration (E/E

_{p}), and relative yield (Y/Y

_{m}) for different Irrigation strategies: full irrigation strategy (A1, A2, A3, A4), deficit irrigation strategy (B1, B2, B3, B4), and rainfed mode (E).

Irrigation Strategies | The Upper and Lower Limit | V | E/E_{p} | Y/Ym | ||||
---|---|---|---|---|---|---|---|---|

$\mathit{\epsilon}$ | $\mathit{\omega}$ | Mean | CV | Mean | CV | Mean | CV | |

A1 | 0.6 s_{fc} | s_{fc} | 138.55 | 0.500 | 100% | 0 | 100% | 0 |

A2 | 0.6 s_{fc} | 0.8 s_{fc} | 117.76 | 0.603 | 100% | 0 | 100% | 0 |

A3 | 0.5 s_{fc} | s_{fc} | 129.26 | 0.566 | 100% | 0 | 100% | 0 |

A4 | 0.5 s_{fc} | 0.8 s_{fc} | 113.57 | 0.648 | 100% | 0 | 100% | 0 |

B1 | 0.4 s_{fc} | s_{fc} | 118.51 | 0.626 | 99.14% | 0.005 | 99.01% | 0.005 |

B2 | 0.4 s_{fc} | 0.8 s_{fc} | 105.42 | 0.692 | 98.86% | 0.007 | 98.68% | 0.008 |

B3 | 0.3 s_{fc} | s_{fc} | 96.91 | 0.726 | 95.72% | 0.023 | 95.04% | 0.027 |

B4 | 0.3 s_{fc} | 0.8 s_{fc} | 85.12 | 0.779 | 94.82% | 0.031 | 93.99% | 0.036 |

E | - | - | 0 | 0 | 75.41% | 0.257 | 71.47% | 0.315 |

**Table 3.**Relative frequency of irrigation requirements for full irrigation strategy (A1, A2, A3, A4) and deficit irrigation strategy (B1, B2, B3, B4).

Irrigation Strategies | <50 mm | 50–100 mm | 100–150 mm | 150–200 mm | 200–250 mm | 250–300 mm | >300 mm |
---|---|---|---|---|---|---|---|

A1 | 9.78% | 23.15% | 25.70% | 20.85% | 13.15% | 6.58% | 0.80% |

A2 | 19.40% | 27.20% | 23.05% | 15.05% | 10.23% | 4.53% | 0.55% |

A3 | 15.70% | 23.63% | 23.13% | 18.48% | 12.18% | 6.13% | 0.78% |

A4 | 23.60% | 25.38% | 21.33% | 14.63% | 10.03% | 4.50% | 0.55% |

B1 | 21.80% | 22.83% | 22.38% | 16.45% | 11.38% | 4.73% | 0.45% |

B2 | 28.43% | 24.43% | 20.03% | 14.23% | 9.30% | 3.33% | 0.28% |

B3 | 32.33% | 23.55% | 19.90% | 14.23% | 7.95% | 2.03% | 0.03% |

B4 | 38.33% | 24.68% | 18.05% | 11.95% | 6.03% | 0.98% | 0.00% |

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**MDPI and ACS Style**

Long, T.; Wang, D.; Wu, X.; Chen, X.; Huang, Z.
Simulation of Irrigation Strategy Based on Stochastic Rainfall and Evapotranspiration. *Agronomy* **2023**, *13*, 2849.
https://doi.org/10.3390/agronomy13112849

**AMA Style**

Long T, Wang D, Wu X, Chen X, Huang Z.
Simulation of Irrigation Strategy Based on Stochastic Rainfall and Evapotranspiration. *Agronomy*. 2023; 13(11):2849.
https://doi.org/10.3390/agronomy13112849

**Chicago/Turabian Style**

Long, Tingyuan, Dongqi Wang, Xiaolei Wu, Xinhe Chen, and Zhongdong Huang.
2023. "Simulation of Irrigation Strategy Based on Stochastic Rainfall and Evapotranspiration" *Agronomy* 13, no. 11: 2849.
https://doi.org/10.3390/agronomy13112849