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This paper investigates calibration of Hargreaves equation in Xiliaohe Basin. Twelve meteorologicalgauges located within Xiliaohe Basin in Northeast China were monitored during 1970 and 2014 providing continuous records of meteorological data. Taking daily ET_{0} calculated by Penman-Montieth equation as the benchmark, the error of Hargreaves equation for computing ET_{0} was evaluated and the investigation on regional calibration of Hargreaves equation was carried out. Results showed there was an obvious difference between the calculating results of Hargreaves and Penman-Monteith equation. The estimation of the former was obviously higher during June and September while lower during the rest time in a year. The three empirical parameters of the Hargreaves equation were calibrated using the SCE-UA (Shuffled Complex Evolution) method, and the calibrated Hargreaves equation showed an obvious promotion in the accuracy both during the calibration and verification period.

Reference Crop Evapotranspiration (ET_{0}) is an important climatic and hydrological variable, which lays the foundation for the calculation of actual evapotranspiration [_{0} is essential to ecological environment protection and planning as well as water and soil resource management. There are dozens of methods for calculating ET_{0} at present which are different from each other on theoretical basis, complexity and applicable conditions. All these methods can be summarized to empirical formula method (such as Blanney-Criddle and Thornthwaite), moisture diffusion method, energy-balanced method (such as Presley-Tylor) and synthesis method. Penman-Monteith (P-M hereinafter) equation was recommended by FAO due to its rigorous physical basis and high calculating accuracy [_{0} with this equation needs a variety of meteorological data (maximum temperature, minimum temperature, average temperature, sunshine hours, relative humidity, average air pressure and wind speed) thus it is often limited by lack of data

when applied to data-deficiency areas. In this case, empirical methods which are more widely used for its relatively low requirement for data became a realistic choice [

As one of the empirical methods in calculating ET_{0}, Hargreaves (H hereinafter) equation has been put forward and improved by Hargreaves et al. since 1950s-1960s [^{th} and 30^{th} ten-days and calibrated the equation through establishing the linear regression between results of H and P-M equation; Fan’s [_{0} in Lhasa with P-M and H equation which showed a difference in spring and rainy season, the factor of average humidity was introduced to adjust the H equation which gained a relatively accurate result; Hu [

In summary, there are mainly several following methods of calibrating H equation: (1) establishment of a linear regression between P-M and H results, such as reference [_{s}) with Angstrom equation, which haven’t yet been evaluated systematically to be reasonable [_{0} a lot.

The aim of this study is to evaluate the error of H equation using the meteorological data of 12 gauging stations in Xiliaohe Basin and to conduct an investigation into the regional calibration of 3 empirical parameters. The calibration will be achieved with the SCE-UA method [_{0} calculation in similar continental semiarid areas.

This case study utilizes the data obtained from 12 meteorological monitoring of 12 gauges located in Xiliaohe Basin which lies between 116˚16'E and 123˚35'E longitude and between 40˚05'N and 45˚13'N latitude with a drainage area of 13.52 × 10^{4} km^{2}. The basin is characterized by high west and low east. The western part of the basin is hilly while the rest are plains (

Along with the increasing development intensity of water resource, grassland and desert vegetation in some parts of Xiliaohe Basin have been degrading in different degrees recently. The largest sand land in China―Horqin Sandland locates in this area with severe water resources shortage, making this land the most eco-fragile area in Northeast China. Meanwhile, the main rivers in the basin cut off occasionally which make the development of agriculture and industry more and more dependent on the exploit of groundwater. In 2001-2010, the average quantity of annual water supply in Xiliaohe Basin was between 4.8 - 5.5 × 10^{9} m^{3} in which the groundwater accounted for 70% - 80% averagely.

According to research, the data of 12 meteorological gauges in Xiliaohe Basin (distribution of stations as

gauges | longitude | latitude | administrative district | gauges | longitude | latitude | administrative district |
---|---|---|---|---|---|---|---|

Jarud | 123.9 | 44.7 | Jarud | Shuangliao | 123.5 | 43.5 | Shuangliao |

Balinzuo | 119.5 | 43.9 | Balinzuo | Siping | 124.4 | 43.2 | Siping |

Changlin | 124.0 | 44.3 | Changlin | Wengniute | 119.0 | 42.9 | Wengniute |

Linxi | 118.1 | 43.6 | Linxi | Chifeng | 119.0 | 42.3 | Chifeng |

Kailu | 121.2 | 43.6 | Kailu | Fuxin | 121.7 | 42.0 | Fuxin |

Tongliao | 122.2 | 43.6 | Tongliao | Zhaoyang | 120.5 | 41.6 | Zhaoyang |

(1) Monthly radiation data from 1976-2014 for calibrating parameter “a” and “b” in Angstrom equation;

(2) Daily meteorological data (including maximum temperature, minimum temperature, average temperature, average relative humidity, sunshine hours, average air pressure, average wind speed) from Jan 1^{st} 1970 to Dec 31^{st} 2014 for calculating ET_{0} with P-M and H equation.

The data above basically came from “daily dataset of Chinese climate data” and “monthly dataset of Chinese radiation data” in National Meteorological Information Center, a spot of missing data (mainly average wind speed and average relative humidity) were interpolated with the pre and post data.

In this study, daily ET_{0} calculated by P-M equation is set as a standard, the form and calculation steps of P-M equation see reference [

where d_{r} is the solar-terrestrial relativedistance, kPa^{0}C^{−1}; _{a} is the monthly astronomical radiation, MJ/(m^{2}∙d); n, N are actual and theoretical sunshine hours respectively, h/d; a and b are the parameter remains to be calibrated.

H equation was put forward based on the two empirical Equations (3)-(4) [

where ET_{0-H} is the ET_{0} calculated by H equation, mm/d; T_{max}, T_{min}, T are daily maximum, minimum, average temperature respectively, ˚C; K_{RS} is an empirical coefficient; “C”, “E”, “T” are 3 parameters of H equation which are recommended as 0.0023, 0.5, 17.8.

The SCE-UA algorithms which is capable of global optimization is adopted to calibrate “C”, “E”, “T” of each station at Xiliaohe Basin, the calibrating steps are as follows:

(1) Division of the research time. The daily meteorological data (1970-2014) is divided into calibrating and verification period according to the ratio of 5:1, thus the former is from 1970-2005, the latter is from 2006-2014.

(2) Definition of the range of “C”, “E”, “T”. According to analysis, debugging and reference [^{-5}, 0.02], E ∈ [0.02, 2.0], T ∈ [2.0, 75.0].

(3) Definition of the objective function.Maximization of function F (Nash-Sutcliffe efficiency coefficient, see formula (6)) and minimization of function G (total relative error, see formula (7)) are set to be the optimization target of SCE-UA algorithm.

where ET_{0-H}(t), ET_{0-PM}(t) are the t^{th} day’s ET_{0} calculated by H and P-M equation respectively, _{0-PM}(t) in the given period.

(4) Operation of the algorithms. Output of the calibration results of parameters.

This article basically investigates the calculating accuracy of pre-and-post calibration of H equation with the following statistical variables: absolute error (BE, see formula (8)) and relative error (RE, see formula (9)) of each month. Additionally, a wilcoxon test is used to detect whether there is an obvious difference between calculating results of P-M and H equation.

where i is the ordinal number of month, i = 1, 2, 3, ..., 12;_{0} of the i^{st} month calculated by H and P-M equation respectively in the given period, _{0} calculated by H equation.

Parameter “a” and “b” of 12 meteorological gauges are calibrated using the monthly radiation data of Xiliaohe Basin during 1976-2014 (see

According to the daily meteorological data of 12 gauges between 1970-2014, average daily ET_{0} of 12 stations is calculated and compared using the P-M and H equation respectively. The average value of absolute and relative error by H equation in each month can be seen in

In order to demonstrate the error of H equation before calibration, Average daily ET_{0} calculated by P-M and H equation in Jarud can be seen

parameter gauge | a | b | parameter gauge | a | b |
---|---|---|---|---|---|

Jarud | 0.24 | 0.46 | Shuangliao | 0.26 | 0.41 |

Balinzuo | 0.22 | 0.44 | Siping | 0.35 | 0.27 |

Changling | 0.29 | 0.39 | Wengniute | 0.22 | 0.42 |

Linxixian | 0.27 | 0.37 | Chifeng | 0.25 | 0.38 |

Kailu | 0.20 | 0.45 | Fuxin | 0.32 | 0.30 |

Tongliao | 0.25 | 0.39 | Zhaoyang | 0.39 | 0.18 |

Month | BE/mm | RE/% | Month | BE/mm | RE/% |
---|---|---|---|---|---|

Jan | −10.86 | −61.95 | Jul | 21.28 | 16.04 |

Feb | −11.40 | −41.25 | Aug | 21.25 | 18.45 |

Mar | −14.61 | −23.89 | Sep | 9.16 | 10.70 |

Apr | −15.23 | −13.73 | Oct | −6.15 | −9.36 |

May | −5.24 | −3.34 | Nov | −10.06 | −31.32 |

Jun | 14.29 | 10.19 | Dec | −9.48 | −51.62 |

Monthly average ET_{0} of Jarud calculated by H equation and its error can be seen in _{0} calculated by H and P-M equation except in June and September.

With the daily meteorological data during 1970-2005, parameter “C”, “E”, “T” of 12 gauges are calibrated using SCE-UA method. Distribution of the calibrated parameters are shown in _{v} 0.37, 0.28, 0.52. In general, parameter “T” shows the largest discrete degree while “E” shows the smallest.

The average relative error of ET_{0 }(absolute value) in each month within the verification period are drawn in the boxplot type, as is shown in

In order to demonstrate the efficiency of calibration, daily ET_{0} of Jarud calculated by 3 equations (namely calibrated H equation, uncalibrated equation, P-M equation) in calibration and verification period are shown in

The monthly comparison of ET_{0} between P-M and H equation (before & after calibration) during 1970-2014 can be seen in _{0} calculated by the calibrated H equation and P-M equation, thus the calibrated H equation can be used to calculate the monthly ET_{0} in the replacement of P-M equation.

Month | ET_{0-PM }/ mm | ET_{0-H} / mm | BE/mm | RE/% | Wilcoxon test |
---|---|---|---|---|---|

1 | 20.7 | 6.0 | −14.7 | −70.9 | significant |

2 | 32.5 | 16.0 | −16.5 | −50.8 | significant |

3 | 68.7 | 46.9 | −21.8 | −31.7 | significant |

4 | 118.3 | 94.7 | −23.6 | −19.9 | significant |

5 | 163.4 | 146.3 | −17.1 | −10.5 | significant |

6 | 153.8 | 162.2 | 8.4 | 5.5 | insignificant |

7 | 140.1 | 159.9 | 19.8 | 14.1 | significant |

8 | 123.6 | 139.9 | 16.3 | 13.2 | significant |

9 | 101.1 | 100.2 | −0.9 | −0.9 | insignificant |

10 | 74.1 | 56.0 | −18.1 | −24.4 | significant |

11 | 37.5 | 20.9 | −16.6 | −44.3 | significant |

12 | 21.0 | 8.0 | −13.1 | −62.2 | significant |

Sum | 1054.78 | 956.91 | −97.87 | −283.17 | - |

Notes: ET_{0-PM} and ET_{0-H} are ET_{0} calculated by P-M and H equation respectively.

month | ET_{0-PM}/mm | before calibration | after calibration | ||||||
---|---|---|---|---|---|---|---|---|---|

ET_{0-H}/mm | BE/mm | RE/% | Wilcoxon Test | ET_{0-H}/mm | BE /mm | RE /% | Wilcoxon Test | ||

1 | 18.01 | 5.98 | −12.03 | −66.79 | significant | 22.66 | 4.66 | 25.87 | insignificant |

2 | 31.20 | 16.10 | −15.10 | −48.39 | significant | 34.86 | 3.67 | 11.76 | insignificant |

3 | 65.10 | 47.48 | −17.62 | −27.07 | significant | 68.50 | 3.40 | 5.23 | insignificant |

4 | 108.50 | 93.79 | −14.71 | −13.56 | significant | 107.00 | −1.50 | −1.38 | insignificant |

5 | 155.62 | 147.10 | −8.53 | −5.48 | insignificant | 149.27 | −6.36 | −4.08 | insignificant |

6 | 150.21 | 163.60 | 13.39 | 8.91 | significant | 157.32 | 7.11 | 4.74 | insignificant |

7 | 138.77 | 163.29 | 24.52 | 17.67 | significant | 154.29 | 15.52 | 11.18 | insignificant |

8 | 130.95 | 146.79 | 15.84 | 12.09 | significant | 140.45 | 9.50 | 7.25 | insignificant |

9 | 100.73 | 104.26 | 3.52 | 3.50 | insignificant | 103.74 | 3.01 | 2.99 | insignificant |

10 | 68.78 | 57.91 | −10.87 | −15.80 | significant | 66.37 | −2.42 | −3.51 | insignificant |

11 | 34.74 | 21.47 | −13.27 | −38.20 | significant | 32.23 | −2.51 | −7.23 | insignificant |

12 | 18.42 | 7.34 | −11.08 | −60.16 | significant | 20.09 | 1.68 | 9.11 | insignificant |

mean | 85.08 | 81.26 | −3.83 | −19.44 | - | 88.32 | 3.23 | 5.41 | - |

Total | 1021.02 | 975.09 | −45.93 | −233.28 | - | 1059.78 | 38.76 | 64.90 | - |

Based on the meteorological and radiation data of 12 gauges in the Xiliaohe Basin, this study analyzes the error of H equation by setting daily ET_{0} calculated by P-M equation as a standard. The empirical parameters of H equation at each gauge are calibrated with the SCE-UA method and the accuracy characteristics of H equation before and after calibration are compared and evaluated. The results of the study provide conclusions that:

(1) The H and P-M equation show some certain consistence in the variation trend of daily ET_{0 }while a significant difference can be detected between the calculating values of the two equations. To be specific, results of H equation are obviously higher than the standard value during June and September while lower during the rest time of year.

(2) According to calibration results, 3 parameters of all gauges deviate from the recommended value by FAO in which most stations show a lower “C” and “E” than recommend while all the gauges show a higher “T”. A significant advancement in accuracy during Jan-Mar and Nov-Dec can be seen after calibration accompanied by certain-degree advancement during April and October. In a word, a better and more stable accuracy can be obtained to calculate ET_{0} with the calibrated H equation in the replacement of the P-M equation.

The research conclusions above show clearly the necessity and feasibility to calibrate the empirical parameters of H equation. However, in consideration of the significant difference between calibrated parameters of different gauges in the same basin, there is an urgent need to study the regional law of distribution to explore whether this phenomenon is attributed to the different meteorological conditions of each gauge. Also, this issue can be further studied in a larger scale to draw more universal conclusions.

The present study is funded by Natural Science Foundation of China (Grant Number: 51509157), Science and Technology Generalization Program (Grant Number: TG1528) and Science Research Program for Common Wealthy (Grant Number: 201301075, 201501014), Ministry of Water Resources, China. The authors also appreciate National Meteorological Information Center (http://data.cma.gov.cn/) for providing part of the meteorological data.

Leizhi Wang,Qingfang Hu,Yintang Wang,Yong Liu,Lingjie Li,Tingting Cui, (2016) Regional Calibration of Hargreaves Equation in the Xiliaohe Basin. Journal of Geoscience and Environment Protection,04,28-36. doi: 10.4236/gep.2016.47004