# Assessing the Impact of Reference Evapotranspiration Models on Decision Support Systems for Irrigation

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Geisenheim Irrigation Scheduling

#### 2.2. Data

- $\mathrm{FAO}56\text{-}E{T}_{0}$ is reference evapotranspiration ($\mathrm{m}\mathrm{m}\xb7{\mathrm{d}}^{-1}$),
- ${R}_{n}$ net radiation at the crop surface ($\mathrm{M}\mathrm{J}\xb7{\mathrm{m}}^{-2}\xb7{\mathrm{d}}^{-1}$),
- G soil heat flux density ($\mathrm{M}\mathrm{J}\xb7{\mathrm{m}}^{-2}\xb7{\mathrm{d}}^{-1}$),
- T mean daily air temperature at 2 $\mathrm{m}$ height ($\xb0\mathrm{C}$),
- ${u}_{2}$ wind speed at 2 $\mathrm{m}$ height ($\mathrm{m}\xb7{\mathrm{s}}^{-1}$),
- ${e}_{s}$ mean saturation vapor pressure ($\mathrm{k}\mathrm{Pa}$),
- ${e}_{a}$ actual vapor pressure ($\mathrm{k}\mathrm{Pa}$),
- ${e}_{s}-{e}_{a}$ saturation vapor pressure deficit ($\mathrm{k}\mathrm{Pa}$),
- $\Delta $ slope vapor pressure curve ($\mathrm{k}\mathrm{Pa}\xb7{\xb0\mathrm{C}}^{-1}$), and
- $\gamma $ psychrometric constant ($\mathrm{k}\mathrm{Pa}\xb7{\xb0\mathrm{C}}^{-1}$), and:

- ${e}_{a}$ is actual vapor pressure ($\mathrm{k}\mathrm{Pa}$),
- ${e}_{s}$ mean saturation vapor pressure ($\mathrm{k}\mathrm{Pa}$), and
- $R{H}_{mean}$ mean relative air humidity (%).

#### 2.3. Sensitivity Analysis with Random Forest

#### 2.4. Regression Model

#### 2.5. Simulations

#### 2.6. Computer Software

## 3. Results and Discussion

#### 3.1. Regression

#### 3.2. Sensitivity of Random Forest

^{−1}to 15 $\mathrm{M}$$\mathrm{J}$ · $\mathrm{m}$${}^{-2}$ · d

^{−1}were present in both models. These findings suggest that a simple linear scaling factor, as achieved with the regression model, may be capable of adjusting both $E{T}_{0}$s on an equal value. Moreover, a similar order of sensitivity rankings in the main vegetation period was also found for FAO56-$E{T}_{0}$, e.g., by Gong et al. [46], DeJonge et al. [47]. Due to this, the method for P2-$E{T}_{0}$ to FAO56-$E{T}_{0}$ adjustments appears applicable in other regions, as well.

#### 3.3. Simulations of GS

#### 3.3.1. The Precision of the CWB and $E{T}_{0}$ Model

#### 3.3.2. Numerical and Statistical Issues

#### 3.3.3. Practical Considerations

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

AFC | Available field capacity of the soil |

BBCH | Crop phenology index |

CWB | Crop water balance |

GS | Geisenheim Irrigation Scheduling |

$E{T}_{0}$ | Reference evapotranspiration |

$E{T}_{c}$ | Actual crop evapotranspiration |

FAO56 | Reference evapotranspiration based on the FAO56 paper |

$G{S}_{FAO56}$ | GS simulation model with FAO56 |

$G{S}_{P2}$ | GS simulation model with P2 |

ICD | Differences in total counts of irrigation events |

IWD | Differences in total amounts of irrigation water |

P2 | Adjusted Penman reference evapotranspiration |

%IncMSE | Relative increase in mean squared error |

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**Figure 1.**Simple linear regression model (blue line) for P2-$E{T}_{0}$∼ FAO56-$E{T}_{0}$ ($\mathrm{m}\mathrm{m}\xb7{\mathrm{d}}^{-1}$). Data for the year 2009, Geisenheim, Germany.

**Figure 2.**Residuals ($\mathrm{m}\mathrm{m}\xb7{\mathrm{d}}^{-1}$) versus predicted ($\mathrm{m}\mathrm{m}\xb7{\mathrm{d}}^{-1}$) plot of the regression model P2-$E{T}_{0}$∼$E{T}_{0}$: P2-$E{T}_{0}-\widehat{P2}$-$E{T}_{0}$ versus $\widehat{P2}$-$E{T}_{0}$. With the null line (black), residual mean (red line), and polynomial trend line of residuals (blue line, $y=1.07\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{x}^{2}-0.82\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}x-0.04,\alpha =0.01,adj.{R}^{2}=0.03$).

**Figure 3.**Ranks of variable importance (%IncMSE), for the Random Forest Models P2-$E{T}_{0}$ and FAO56-$E{T}_{0}$, using weather data of the years 2000–2009 (Explanation of parameters in Table 1).

**Figure 4.**$E{T}_{0}$ ($\mathrm{m}\mathrm{m}\xb7{\mathrm{d}}^{-1}$) predicted by global radiation ($\mathrm{M}\mathrm{J}\xb7{\mathrm{m}}^{-2}\xb7{\mathrm{d}}^{-1}$), based on functions of random forest models for P2-$E{T}_{0}$ and FAO56-$E{T}_{0}$. The shaded box marks a similar the sudden rise of both $E{T}_{0}$s in response to the global radiation.

**Figure 5.**Boxplot of relative irrigation water differences (IWD %), $G{S}_{P2}$ − $G{S}_{FAO56}$, by GS simulations for the six vegetable crops. Black line = median, Boxes = second and third quantile, points = outliers.

**Figure 6.**Occurrence (%) of irrigation count differences, ICD, of $G{S}_{P2}$ relative to $G{S}_{FAO56}$ ($G{S}_{P2}$ − $G{S}_{FAO56}$) as simulated using the six vegetable crops.

Value | Abbreviation | Unit |
---|---|---|

Penman-P2-$E{T}_{0}$ | P2-$E{T}_{0}$ | $\mathrm{m}\mathrm{m}\xb7{\mathrm{d}}^{-1}$ |

FAO56 Penman–Monteith-$E{T}_{0}$ | FAO56-$E{T}_{0}$ | $\mathrm{m}\mathrm{m}\xb7{\mathrm{d}}^{-1}$ |

Temperature mean (24 h) | Temperature | $\xb0\mathrm{C}$ |

Temperature mean (max/min) | Tmm | $\xb0\mathrm{C}$ |

Temperature maxima | Tmax | $\xb0\mathrm{C}$ |

Temperature minima | Tmin | $\xb0\mathrm{C}$ |

Relative air humidity mean (24 h) | AirHumidity | % |

Global radiation | Radiation | $\mathrm{M}\mathrm{J}\xb7{\mathrm{m}}^{-2}\xb7{\mathrm{d}}^{-1}$ |

Wind speed at 2 $\mathrm{m}$ height mean (24 h) | Windspeed2m | $\mathrm{m}\xb7{\mathrm{s}}^{-1}$ |

**Table 2.**Simulation results comparing $G{S}_{P2}$ and $G{S}_{FAO56}$ ($G{S}_{P2}$−$G{S}_{FAO56}$) with mean irrigation water differences (%) (IWD), mean irrigation counts for both models (IC $G{S}_{P2}$, IC $G{S}_{FAO56}$), and mean irrigation count differences (ICD) and their corresponding standard deviation.

Crop | IWD | ± | IC ${\mathit{GS}}_{\mathit{P}2}$ | ± | IC ${\mathit{GS}}_{\mathit{FAO}56}$ | ± | ICD | ± |
---|---|---|---|---|---|---|---|---|

Broccoli | 1.18 | 3.38 | 11.50 | 2.32 | 11.75 | 2.70 | 0.25 | 0.45 |

Bush bean | 5.45 | 12.57 | 7.46 | 1.80 | 7.75 | 1.86 | 0.29 | 0.46 |

Carrot | 2.84 | 4.19 | 15.98 | 3.83 | 16.36 | 4.01 | 0.39 | 0.54 |

Cauliflower | 4.03 | 4.32 | 16.78 | 1.82 | 17.44 | 2.10 | 0.67 | 0.63 |

Leek | 6.76 | 7.14 | 17.00 | 5.36 | 17.76 | 5.54 | 0.76 | 0.70 |

Onion | 1.87 | 5.88 | 17.25 | 4.58 | 17.58 | 4.74 | 0.33 | 0.65 |

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**MDPI and ACS Style**

Olberz, M.; Kahlen, K.; Zinkernagel, J.
Assessing the Impact of Reference Evapotranspiration Models on Decision Support Systems for Irrigation. *Horticulturae* **2018**, *4*, 49.
https://doi.org/10.3390/horticulturae4040049

**AMA Style**

Olberz M, Kahlen K, Zinkernagel J.
Assessing the Impact of Reference Evapotranspiration Models on Decision Support Systems for Irrigation. *Horticulturae*. 2018; 4(4):49.
https://doi.org/10.3390/horticulturae4040049

**Chicago/Turabian Style**

Olberz, Matthias, Katrin Kahlen, and Jana Zinkernagel.
2018. "Assessing the Impact of Reference Evapotranspiration Models on Decision Support Systems for Irrigation" *Horticulturae* 4, no. 4: 49.
https://doi.org/10.3390/horticulturae4040049