# Improvement of Hargreaves–Samani Reference Evapotranspiration Estimates in the Peruvian Altiplano

^{1}

^{2}

^{*}

## Abstract

**:**

_{RS}. The results show modified HS (HSM) ETo estimates at validation after K

_{RS}calibration, revealing evident improvements in accuracy with Nash–Sutcliffe efficiency (NSE) between 0.58 and 0.93, percentage bias (PBIAS) between −0.58 and 1.34%, mean absolute error (MAE) between −0.02 and 0.05 mm/d, and root mean square error (RMSE) between 0.14 and 0.25 mm/d. Consequently, the multiple linear regression (MLR) model was used to regionalize the K

_{RS}for the PA. It is concluded that, in the absence of meteorological data, the HSM equation can be used with the new values of K

_{RS}instead of HS for the PA.

## 1. Introduction

^{2}/d), $G$ is the soil heat flux density (MJ/m

^{2}/d), ${T}_{mean}$ is the mean air temperature (°C), ${U}_{2}$ is the mean wind speed at a height 2 m (m/s), ${e}_{s}$ is the saturation vapor pressure (kPa), ${e}_{a}$ is the actual vapor pressure (kPa), ${e}_{s}-{e}_{a}$ is the vapor pressure deficit (kPa) and $\gamma $ is the psychrometric constant (kPa/°C).

^{−0.5}), and 17.8 is an empirical factor related to the temperature units used in the original formulations.

^{−0.5}for arid and semi-arid regions [16]. According to Allen et al. [13], the ${K}_{RS}$ adjustment coefficient differs for inland or coastal regions, thus, for inland locations where land mass dominates or next to it and where air masses are influenced by a nearby water mass, ${K}_{RS}$ = 0.16 °C

^{−0.5}, while for coastal locations with or adjacent to a large land mass where air masses are influenced by a nearby water mass, ${K}_{RS}$ = 0, 19 °C

^{−0.5}. With ${R}_{a}$ in mm/day and the empirical coefficient ${K}_{RS}$ is normally considered as 0.17 °C

^{−0.5}.

_{RS}of the HS equation, under environmental conditions of the PA, and as specific objectives (1) evaluate the original HS equation, (2) calibrate and validate the radiation coefficient K

_{RS}of the HS equation, and (3) regionalize the radiation coefficient ${K}_{RS}$ for the PA based on the geographic characteristics.

## 2. Materials and Method

#### 2.1. Study Area

#### 2.2. Climatic and Terrain Data

_{10}, m/s) at a height of 10 m and precipitation (P, mm) from nine weather stations distributed in the PA (Figure 1a). The selected stations have data records from 2000 to 2019 (information provided by the National Meteorology and Hydrology Service of Peru (SENAMHI)). The stations considered include the representative stations of the PA, defined on the basis of the aridity index suggested by Huerta and Lavado [32].

_{10}, Sh, T

_{max}, T

_{min}and P were analyzed by means of a visual inspection, and the absolute method. For the absolute method, the nonparametric test of Distribution-free cumulative sum (CUSUM) and rank-sum (RS) were applied independently to the data from each weather station using the TREND program (https://toolkit.ewater.org.au/Tools/TREND: accessed on 10 August 2022). CUSUM, is a nonparametric test of step jump in the mean, while RS is a nonparametric test of difference in the mean of two periods [35]. The null hypothesis for CUSUM and RS suggests that there is no step jump in the mean in the data series, and there is no difference in the mean between two data periods, which accepts the null hypothesis if the maximum deviation for CUSUM and the statistic z for calculated RS are less than the critical value of the statistical table at the 5% significance level and rejects it otherwise. Thus, the doubtful periods of the Rh, U

_{10}, Sh, T

_{max}, T

_{min}and P data series that did not meet the assumption of homogeneity were not considered in subsequent analyses.

_{2}, m/s) was calculated from U

_{10}.

_{2}wind speed (m/s) at a height of 2 m; Sh sun hours (h); ETo reference evapotranspiration estimated by the PM method (mm/d); P annual precipitation (mm/year); AI aridity index; CC climatic classification; SH-h subhumid-humid; SH-d subhumid-dry.

_{RS}radiation coefficient calibrated from the HS equation, the NASA Shuttle Radar Topography Mission (SRTM) digital elevation model (DEM) was obtained from the Google Earth Engine (GEE) platform available at https://earthengine.google.com/ (accessed on 15 July 2022), ID de la imagen CGIAR/SRTM90_V4 [39], with a spatial resolution of ~90 m.

#### 2.3. Evaluation of the Original Hargreaves-Samani Equation for Use in the Peruvian Altiplano

^{2}/d), $n$ is the actual duration of sunlight (h), $N$ is the maximum possible duration of sunlight or hours of sunlight (h), ${a}_{s}$ is the regression constant which expresses the fraction of extraterrestrial radiation that reaches the earth on cloudy days $(n=0$) and ${a}_{s}+{b}_{s}$ is the fraction of extraterrestrial radiation that reaches the earth on clear days $(n=N$). In the absence of real measurements and calibration of solar radiation $\left({R}_{s}\right)$, values suggested by Allen et al. [13] are ${a}_{s}=0.25$ and ${b}_{s}=0.50$. However, these default values should not be applied to high altitude sites, where proper calibration is required [40]. In this regard, values of ${a}_{s}=0.23$ and ${b}_{s}=0.60$ suggested by Chipana et al. [3], were used for high altitude areas (3820 to 3950 m.a.s.l.) estimated for the Bolivian altiplano and for altitudes above the 4660 m.a.s.l was considered with slight modifications of ${a}_{s}=0.29$ and ${b}_{s}=0.55$.

^{2}/min, ${d}_{r}$ is the inverse relative distance to the sun, ${\omega}_{s}$ is the hour angle of sunset (rad), $\xd8$ is the latitude (rad), and $\delta $ is the solar declination (rad).

^{2}/d), $\sigma $ is the Stefan-Boltzmann constant (4.903 × 10

^{−9}MJ/K

^{−4}/m

^{2}/d), ${T}_{max,K}^{4}$ is the maximum absolute temperature during the 24 h period (K = °C + 273), ${T}_{min,K}^{4}$ is the minimum absolute temperature during the 24 h period (K = °C + 273), ${e}_{a}$ is the actual vapor pressure (kPa), ${R}_{s}/{R}_{so}$ is the relative shortwave radiation (limited a ≤ 1.0), ${R}_{s}$ is the calculated solar radiation (MJ/m

^{2}/d) and ${R}_{so}$ is the calculated clear-sky radiation (MJ/m

^{2}/d).

#### 2.4. Statistical Performance Metrics

#### 2.5. Calibration and Validation of the Radiation Coefficient K_{RS} of the HS Equation

#### 2.6. Regionalization of the Radiation Coefficient ${\widehat{K}}_{RS}$

## 3. Results and Discussion

#### 3.1. Evaluation of the Original Hargreaves-Samani Equation

_{RS}considered for the HS equation was 0.17 °C

^{−0.5}for all weather stations. The precision of the ETo values of the original HS equation had significant variations in the SH-h and SH-d climatic zones identified in the study area (Table 1). The scatter diagrams reveal that the values of the correlation coefficient (R) within the two climatic zones oscillated between 0.84 and 0.97, presenting lower values in the ANA and PTN stations and higher values in the PNO and JLC stations. On the other hand, in relation to the values of the NSE vary between −0.57 and 0.87, NSE values lower than 0.50 were presented in the PTN, JUL, JLC and ANA stations, indicating poor performance. Meanwhile, values higher than 0.75 were presented in the of HNE and LAM stations, indicating very good performance according to the discretion of Moriasi et al. [43].

#### 3.2. Calibration and Validation of the Radiation Coefficient K_{RS} of the HS Equation

_{RS}can lead to less precise estimates of ETo in the PA, thus, the calibrated values of the ${\widehat{K}}_{RS}$ range from 0.150 °C

^{−0.5}to 0.199 °C

^{−0.5}. For most of the stations, the values of ${\widehat{K}}_{RS}$ differ from the coefficient suggested by Hargreaves and Samani [15], except that of the LAM station where a very close value was obtained (${\widehat{K}}_{RS}$ = 0.173 °C

^{−0.5}). High values of ${\widehat{K}}_{RS}$ were obtained at the DES, JUL, PNO and HNE stations close to LT, with values of 0.184 °C

^{−0.5}, 0.209 °C

^{−0.5}, 0.186 °C

^{−0.5}and 0.179 °C

^{−0.5}, respectively, while low ${\widehat{K}}_{RS}$ values were obtained at the CHQ, JLC and PTN stations with values of 0.158 °C

^{−0.5}, 0.152 °C

^{−0.5}and 0.150 °C

^{−0.5}respectively, which are the same ones that are far from the LT. The ANA station located at high altitude (4660 m.a.s.l.) and far from the LT registered a high value of ${\widehat{K}}_{RS}$ = 0.179 °C

^{−0.5}. Ref. [30] showed that K

_{RS}values do not decrease with distance from the sea. At the same time, the standard values of 0.16 °C

^{−0.5}and 0.19 °C

^{−0.5}should not be assumed. In this regard, the precision of the ETo estimate improved for each weather station by adopting the calibrated and validated ${\widehat{K}}_{RS}$ values in the HS equation.

#### 3.3. Regionalisation of the Radiation Coefficient ${\widehat{K}}_{RS}$

^{2}= 0.557) (Figure 6b). This study found that the regional equation of ${\widehat{K}}_{RS\left(x\right)}$ could better estimate ETo for regions with limited data within the PA, with the magnitude of R

^{2}being acceptable for estimating ${\widehat{K}}_{RS}$ in the PA from geographic characteristics.

^{−0.5}, by Raziei and Pereira [5] obtained values that oscillate between 0.14 and 0.20 °C

^{−0.5}, while results obtained by Paredes et al. [50] show values that oscillate between 0.14 and 0.25 °C

^{−0.5}, respectively.

## 4. Conclusions

_{RS}with solar radiation, it is preferable to calibrate the values of K

_{RS}from the HS equation and thus obtain an HSM equation. The HSM model performed better than the HS model for each weather station. Better estimates of ETo were obtained with HSM after calibration of the radiation coefficient K

_{RS}of the HS equation, mainly removing biases.

_{RS}. The HS equation, together with the calibrated radiation coefficient K

_{RS}, can significantly improve the estimate of ETo over data-deficient regions within the PA.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Location of the study area and spatial distribution of weather stations and (

**b**) land cover.

**Figure 2.**Variation of climatic factors as a function of latitude and altitude (

**a**) P and ETo (

**b**) T

_{min}and T

_{max}, (

**c**) Rh and (

**d**) Sh and U

_{2}.

**Figure 3.**Flow diagram for the improvement of the Hargreaves–Samani reference evapotranspiration estimates in the PA.

**Figure 4.**Scatter diagrams between values of ETo by the PM method versus original HS at the weather stations.

**Figure 5.**Spatial distribution of the NSE, PBIAS (%), MAE (mm/d) and RMSE (mm/d) from the ETo estimates of HS with original ${K}_{RS}$, HSM with ${\widehat{K}}_{RS}$ in the calibration and validation phase. Study area in gray outline.

**Figure 6.**Regionalization of the calibrated radiation coefficient ${\widehat{K}}_{RS}$ of the HS equation as a function of longitude, latitude and altitude: (

**a**) variation of the original K

_{RS}, ${\widehat{K}}_{RS}$ calibrated, ${\widehat{K}}_{RS\left(x\right)}-residue$ and residuals; (

**b**) scatterplot between ${\widehat{K}}_{RS}$ and ${\widehat{K}}_{RS\left(x\right)}-residue$ interpolated without addition of the residual.

**Table 1.**Coordinates of the weather stations used, average of the meteorological variables, aridity index and climatic classification.

Station | Lat. | Long. | Alt. | T_{max} | T_{min} | Rh | U_{2} | Sh | ETo | P | AI | CC |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Ananea (ANA) | −14.676 | −69.534 | 4660 | 10.5 | −1.9 | 80.6 | 2.0 | 6.3 | 2.7 | 658.4 | 0.67 | SH-h |

Lampa (LAM) | −15.361 | −70.374 | 3866 | 17.1 | −0.3 | 55.0 | 2.4 | 8.1 | 3.6 | 757.1 | 0.60 | SH-d |

Chuquibambilla (CHQ) | −14.788 | −70.728 | 3918 | 16.3 | −2.2 | 61.3 | 2.1 | 7.2 | 3.2 | 787.0 | 0.67 | SH-h |

Putina (PTN) | −14.921 | −69.876 | 3861 | 17.3 | 0.1 | 70.1 | 2.6 | 6.9 | 3.1 | 643.9 | 0.56 | SH-d |

Huancané (HNE) | −15.207 | −69.758 | 3840 | 15.7 | 0.3 | 58.9 | 2.9 | 7.7 | 3.4 | 650.5 | 0.52 | SH-d |

Juliaca (JLC) | −15.444 | −70.208 | 3838 | 17.8 | −0.5 | 75.0 | 1.5 | 7.8 | 3.2 | 624.9 | 0.53 | SH-d |

Puno (PNO) | −15.826 | −70.012 | 3812 | 16.3 | 3.5 | 61.1 | 1.8 | 8.1 | 3.5 | 750.6 | 0.59 | SH-d |

Juli (JUL) | −16.204 | −69.460 | 3830 | 14.1 | 3.0 | 58.3 | 2.4 | 8.4 | 3.5 | 948.3 | 0.78 | SH-h |

Desaguadero (DES) | −16.563 | −69.037 | 3833 | 15.2 | 1.7 | 65.7 | 5.5 | 7.4 | 3.4 | 736.9 | 0.60 | SH-d |

**Table 2.**List of statistical performance metrics used for the calibration and validation of the K

_{RS}of the HS equation for the estimation of ETo.

Statistical Performance | Equation ^{1} | Unit | Optimal Value |
---|---|---|---|

Correlation coefficient (R) | $R=\frac{{{\displaystyle \sum}}_{i=1}^{N}\left(E{T}_{O,PM}-{\overline{ET}}_{O,PM}\right)\left(E{T}_{O,HS}-{\overline{ET}}_{O,HS}\right)}{\sqrt{{{\displaystyle \sum}}_{i=1}^{N}{\left(E{T}_{O,PM}-{\overline{ET}}_{O,PM}\right)}^{2}}\sqrt{{{\displaystyle \sum}}_{i=1}^{N}{\left(E{T}_{O,HS}-{\overline{ET}}_{O,HS}\right)}^{2}}}$ | - | ±1 |

Nash–Sutcliffe efficiency (NSE) | $ENS=1.0-\frac{{{\displaystyle \sum}}_{i=1}^{N}{\left(E{T}_{O,PM}-E{T}_{O,HS}\right)}^{2}}{{{\displaystyle \sum}}_{i=1}^{N}{\left(E{T}_{O,PM}-{\overline{ET}}_{O,PM}\right)}^{2}}$ | - | 1 |

Percent bias (PBIAS) | $PBIAS=\frac{{{\displaystyle \sum}}_{i=1}^{N}\left(E{T}_{O,HS}-E{T}_{O,PM}\right)}{{{\displaystyle \sum}}_{i=1}^{n}E{T}_{O,PM}}\times 100$ | % | 0 |

Mean absolute error (MAE) | $MAE=\frac{1}{N}{{\displaystyle \sum}}_{i=1}^{N}\left|E{T}_{O,HS}-E{T}_{O,PM}\right|$ | mm | 0 |

Root mean square error (RMSE) | $RMSE=\sqrt{\frac{1}{N}{{\displaystyle \sum}}_{i=1}^{N}{\left(E{T}_{O,HS}-E{T}_{O,PM}\right)}^{2}}$ | mm | 0 |

^{1}Variables: $E{T}_{O,PM}$ is the estimated value with PM, $E{T}_{O,HS}$ is the calculated value, ${\overline{ET}}_{O,PM}$ is the average of the estimated value with PM, ${\overline{ET}}_{O,HS}$ is the average of the calculated value, and $N$ is the total number of data.

Station | Recommended Equations |
---|---|

Ananea (ANA) | $E{T}_{O,HSM\left(ANA\right)}=0.0135\times 0.179\times {R}_{a}\times \left({T}_{med}+17.8\right){\left({T}_{max}-{T}_{min}\right)}^{0.5}$ |

Lampa (LAM) | $E{T}_{O,HSM\left(LAM\right)}=0.0135\times 0.173\times {R}_{a}\times \left({T}_{med}+17.8\right){\left({T}_{max}-{T}_{min}\right)}^{0.5}$ |

Chuquibambilla (CHQ) | $E{T}_{O,HSM\left(CHQ\right)}=0.0135\times 0.158\times {R}_{a}\times \left({T}_{med}+17.8\right){\left({T}_{max}-{T}_{min}\right)}^{0.5}$ |

Putina (PTN) | $E{T}_{O,HSM\left(PTN\right)}=0.0135\times 0.150\times {R}_{a}\times \left({T}_{med}+17.8\right){\left({T}_{max}-{T}_{min}\right)}^{0.5}$ |

Huancané (HNE) | $E{T}_{O,HSM\left(HNE\right)}=0.0135\times 0.179\times {R}_{a}\times \left({T}_{med}+17.8\right){\left({T}_{max}-{T}_{min}\right)}^{0.5}$ |

Juliaca (JLC) | $E{T}_{O,HSM\left(JLC\right)}=0.0135\times 0.152\times {R}_{a}\times \left({T}_{med}+17.8\right){\left({T}_{max}-{T}_{min}\right)}^{0.5}$ |

Puno (PNO) | $E{T}_{O,HSM\left(PNO\right)}=0.0135\times 0.186\times {R}_{a}\times \left({T}_{med}+17.8\right){\left({T}_{max}-{T}_{min}\right)}^{0.5}$ |

Juli (JUL) | $E{T}_{O,HSM\left(JUL\right)}=0.0135\times 0.209\times {R}_{a}\times \left({T}_{med}+17.8\right){\left({T}_{max}-{T}_{min}\right)}^{0.5}$ |

Desaguadero (DES) | $E{T}_{O,HSM\left(DES\right)}=0.0135\times 0.184\times {R}_{a}\times \left({T}_{med}+17.8\right){\left({T}_{max}-{T}_{min}\right)}^{0.5}$ |

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## Share and Cite

**MDPI and ACS Style**

Lujano, A.; Sanchez-Delgado, M.; Lujano, E.
Improvement of Hargreaves–Samani Reference Evapotranspiration Estimates in the Peruvian Altiplano. *Water* **2023**, *15*, 1410.
https://doi.org/10.3390/w15071410

**AMA Style**

Lujano A, Sanchez-Delgado M, Lujano E.
Improvement of Hargreaves–Samani Reference Evapotranspiration Estimates in the Peruvian Altiplano. *Water*. 2023; 15(7):1410.
https://doi.org/10.3390/w15071410

**Chicago/Turabian Style**

Lujano, Apolinario, Miguel Sanchez-Delgado, and Efrain Lujano.
2023. "Improvement of Hargreaves–Samani Reference Evapotranspiration Estimates in the Peruvian Altiplano" *Water* 15, no. 7: 1410.
https://doi.org/10.3390/w15071410