# An Introduction to the Hyperspace of Hargreaves-Samani Reference Evapotranspiration

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}+ 1) was considered. We know that this equation is parabolic, symmetric along the y-axis, vertex above the x-axis, without real zeros, nonlinear and so forth. All of this important information is about the behaviour of the simple 2D equation regardless of its specific application. Of course, understanding the internal behaviour of 4D domain of HS is much more complex because it is not possible to see it in xyz axes, there is no study area and the focus is on the generic structure of HS and not its response to a specific application. As an introductory study, this paper uses domain discretization to be able to present 2D cross sections of the equations for better understanding. The basic concepts of discretization used in this manuscript are the same as those for any 2D and 3D analysis as explained in Reference [22].

## 2. Materials and Methods

#### 2.1. Hargreaves-Samani ETo Equations

^{0.5},

^{0.5},

^{2}− 0.0433 × TR + 0.4023,

^{2}= 0.70 and is being used in various studies [31]. Samani [25] again addressed the importance of Equation (7) he introduced in 2000, which we labelled as HS00. As noted above, HS85 uses KR = 0.17, which according to Equation (7) corresponds to two TR values: 8.3 and 15.1. This means that for TR values within these two numbers, KR is lower than 0.17 and outside the numbers, it is higher. As such, the use of Equation (7) would compensate for the over and under estimations of ETo by HS85. The average of the two numbers (i.e., TR = 11.7) gives the minimum KR but Samani [21] does not explicitly specify the range of valid TR values for Equation (7). However, the TR numbers on the figure of his paper vary from 5 to 17 with the minimum KR of 0.15 (about 10% lower than 0.17) and maximum of 0.23 (about 35% higher than 0.17).

#### 2.2. Computer Program

#### 2.2.1. Domain Discretization

^{2}) 3D sub-figures or subspaces that can be rotated and zoomed in or out. Furthermore, excluding the three diagonal subspaces, for example, TC-TC-ETo (in x-y-z format), the number of unique 3D sub-figures of interest is reduced to six. In order to present these subdomains, computational points or nodes must be defined along the three variables (RA, TC, TR), hence dividing a variable into a number of segments for domain discretization. For example, there are 27,000 ETo values using 30 nodes for each of the three climatic variables (30

^{3}). As the number of nodes increases, the computational cost grows rapidly.

#### 2.2.2. Settings of Temperature Variables

#### 2.3. Data Information

#### 2.3.1. Thresholds of Variables

- To reveal the maximum feasible region
- To cover almost all conditions in the world over any period.

#### 2.3.2. Nodal Data

## 3. Results and Discussion

#### 3.1. Variations of Temperature

#### 3.2. Performance of HS85

- (a)
- (b)
- In Figure 3(3.3), 33% increase in TC from 15 to 20 would result in an increase of 15% in the high value of ETo (i.e., the high boundary explained above).

^{2}values equal to 99.5% and 98.2%, respectively. The 90% on the Cumulative curve occurs in bin 9, that is, ET values of 4.0 to 4.5.

#### 3.3. Performance of HS00

^{2}equal to 99.1% and 97.4 %, respectively. The 90% on the cumulative curve occurs in bin 11, that is, ET values of 5.0 to 5.5, indicating up to 25% increase relative to HS85. Figure 10 also shows that HS00 produces more cases where ETo is greater than 2.5 and, consequently, higher values of ETo relative to HS85-ETo.

## 4. Conclusions

^{0.5}would give R

^{2}= 0.98, making the choice for the exponent of TR questionable. Radiation has a higher influence on possible ETo values as can be seen from Figure 4. HS85-ETo could reach an absolute theoretical maximum of around 10 but mostly less than about six, that is, HS85 can never calculate to the stated threshold of 12.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Feasible domains of Tmin and Tmax combinations within the given TR and TC thresholds (darker areas correspond to higher TR values) (units in Table 1).

**Figure 2.**Feasible domains of Tmin and Tmax combinations within the given TR and TC thresholds (darker areas correspond to higher TC values) (units in Table 1).

**Figure 3.**ETo as a function of temperature (TC) in the Hargreaves-Samani 1985 equation. The four specific cross-sections (sub-figures) are along the temperature range (TR) with radiation (RA) varying within its thresholds (units in Table 1).

**Figure 4.**ETo as a function of radiation (RA) in the Hargreaves-Samani 1985 equation. The four specific cross-sections (sub-figures) are along the temperature (TC) with temperature range (TR) varying within its thresholds (units in Table 1).

**Figure 5.**ETo as a function of temperature range (TR) in the Hargreaves-Samani 1985 equation. The four specific cross-sections (sub-figures) are along the radiation (RA) with temperature (TC) varying within its thresholds (units in Table 1).

**Figure 6.**ETo histogram and its cumulative frequencies for the Hargreaves-Samani 1985 hyperspace with a bin value of 0.5 mm/day.

**Figure 7.**ETo as a function of temperature (TC) in the modified Hargreaves-Samani equation. The four specific cross-sections (sub-figures) are along the temperature range (TR) with radiation (RA) varying within its thresholds (units in Table 1).

**Figure 8.**ETo as a function of radiation (RA) in the modified Hargreaves-Samani equation. The four specific cross-sections (sub-figures) are along the temperature (TC) with temperature range (TR) varying within its thresholds (units in Table 1).

**Figure 9.**ETo as a function of temperature range (TR) in the modified Hargreaves-Samani equation. The four specific cross-sections (sub-figures) are along the radiation (RA) with temperature (TC) varying within its thresholds (units in Table 1).

**Figure 10.**Percentage point frequency differences between the histograms of HS85 (Figure 6) and HS00.

Symbol | Description | Units |
---|---|---|

ETo | Reference crop evapotranspiration | mm/day |

KH | Coefficient in Hargreaves-Samani equation | -- |

KR | Empirical coefficient for the radiation formula | -- |

RA | Extra-Terrestrial radiation | mm/day |

RS | Global solar radiation at the surface | mm/day |

TC | Mean air temperature | °C |

Tmax | Maximum air temperature | °C |

Tmin | Minimum air temperature | °C |

TR | Temperature range | °C |

**Table 2.**Thresholds of the I/O variables of the Hargreaves-Samani equation (units in Table 1).

Variable | Minimum | Maximum |
---|---|---|

ETo | 0 | 12 |

RA | 1 | 18 |

TC | −5 | 35 |

TR | 1 | 22 |

**Table 3.**Information about nodes and cross sections (cuts = 4) of each variable (units in Table 1).

Variable | Difference between Consecutive Nodes | Total Nodes | Difference between Consecutive Cuts | Nodes of the Cuts |
---|---|---|---|---|

RA | 0.63 | 28 | 5.7 | 1, 10, 19, 28 |

TC | 0.70 | 58 | 13.3 | 1, 20, 39, 58 |

TR | 0.70 | 31 | 7.0 | 1, 11, 21, 31 |

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**MDPI and ACS Style**

Haie, N.; Pereira, R.M.; Yen, H.
An Introduction to the Hyperspace of Hargreaves-Samani Reference Evapotranspiration. *Sustainability* **2018**, *10*, 4277.
https://doi.org/10.3390/su10114277

**AMA Style**

Haie N, Pereira RM, Yen H.
An Introduction to the Hyperspace of Hargreaves-Samani Reference Evapotranspiration. *Sustainability*. 2018; 10(11):4277.
https://doi.org/10.3390/su10114277

**Chicago/Turabian Style**

Haie, Naim, Rui M. Pereira, and Haw Yen.
2018. "An Introduction to the Hyperspace of Hargreaves-Samani Reference Evapotranspiration" *Sustainability* 10, no. 11: 4277.
https://doi.org/10.3390/su10114277