## Abstract

Accurate estimation of reference evapotranspiration (*ET _{o}*) is a major task in hydrology, water resources management, irrigation scheduling and determining crop water requirement. There are many empirical equations suggested by numerous references in literature for calculating

*ET*using meteorological data. Some such equations have been developed for specific climatic conditions while some have been applied universally. The potential for usage of these equations depends on the availability of necessary meteorological parameters for calculating

_{o}*ET*in different climate conditions. The focus of the present study was a global cross-comparison of 20

_{o}*ET*estimation equations using daily meteorological records of 10 weather stations (covering a period of 12 years) in a semi-arid region of Iran. Two data management scenarios, namely local and cross-station scenarios, were adopted for calibrating the applied equations against the standard FAO56-PM model. The obtained results revealed that the cross-station calibration might be a good alternative for local calibration of the

_{o}*ET*models when proper similar stations are used for feeding the calibration matrix.

_{o}## INTRODUCTION

Reference evapotranspiration (*ET _{o}*) denotes the evapotranspiration amount of a hypothesized reference crop (height of 0.12 m, surface resistance of 70 s m

^{−1}and albedo of 0.23) (Allen

*et al.*1998). Accurate information on

*ET*magnitude is of crucial significance for computing crop water requirements, as well as for irrigation scheduling, water resources planning and management, hydrology, environmental issues, and determination of the water budget. Various meteorological parameters, e.g. air temperature, humidity, radiation and wind speed, might affect the

_{o}*ET*amount. There have been lots of attempts so far to derive some mathematical/regression-based relationships between the

_{o}*ET*and meteorological parameters. The standard FAO56-PM model is a physics-based approach for calculating

_{o}*ET*using meteorological variables, and has proven to be the benchmark model for computing this parameter as well as for calibrating other empirical

_{o}*ET*equations (Allen

_{o}*et al.*1998). However, the downside of this model is that it requires a lot of meteorological parameters which are not easily measured. On the other hand, some empirical equations, which require less input parameters, have been developed and validated for calculating

*ET*on a universal scale, e.g. Hargreaves (Hargreaves & Samani 1985), Priestley-Taylor (Priestley & Taylor 1972), Makkink (Makkink 1957), Turc (Turc 1961) among other models. Given a specific climatic condition, those empirical equations can be applied for calculating

_{o}*ET*using meteorological parameters. A crucial drawback of such equations would be the need for a local calibration against lysimetric or standard

_{o}*ET*values (calculated by the FAO56-PM model) at different regions (Landeras

_{o}*et al.*2008).

So far, numerous studies have been conducted to compare the performance accuracy of different *ET _{o}* equations worldwide. Among others, DehghaniSanij

*et al.*(2004) compared Penman, Penman–Monteith, Wright–Penman, Blaney–Criddle, radiation balance and Hargreaves–Samani equations in a semi-arid region against the lysimetric values and found that the Penman–Monteith equation outperformed the other equations used. Using the lysimetric observations, Berengena & Gavilan (2005) found that most of the existing empirical equations showed an under-estimation trend in

*ET*simulation in semi-arid regions. Temesgen

_{o}*et al.*(2005) compared some

*ET*estimation equations in California and introduced the Hargreaves–Samani equation as the best temperature-based method in the studied region. Alexandris

_{o}*et al.*(2006) applied a bilinear surface regression analysis to derive a regression-based

*ET*estimation equation and compared the obtained model with the conventional models. They finally introduced the developed equation as a proper method of routine daily

_{o}*ET*estimations. Zhang

_{o}*et al.*(2008) simulated the variations of a vineyard

*ET*by using Penman–Monteith, Shuttle–Wallace and Clumping models in an arid desert region of China and compared the results with those obtained by the Bowen ratio-energy balance method, which confirmed a good ability of the Bowen ratio method in this regard. Trajkovic & Kolakovic (2010) compared the simplified pan-based equations for an

*ET*simulation. Tabari (2010) examined Makkink, Turc, Priestley–Taylor and Hargreaves–Samani equations in four climate conditions and found the Turc equation to be the best model for cold humid and arid climates, and the Hargreaves–Samani equation as the most accurate model for warm humid and semi-arid climatic contexts. Sabziparvar & Tabari (2010) compared temperature-based and radiation-based

_{o}*ET*equations in the arid and semiarid regions of Iran and found the Hargreaves–Samani to be the best model of all.

_{o}Rahimikhoob *et al.* (2012) compared four temperature/radiation-based equations using data from eight weather stations in a subtropical climate of Iran and confirmed the applicability of Hargreaves–Samani and Priestley–Taylor equations in those regions. Tabari *et al.* (2013) compared different temperature-based, radiation-based and mass transfer-based equations for modeling *ET _{o}* in a humid location of Iran and found that the temperature-based Blaney–Criddle and Hargreaves–Samani equations surpassed the other temperature-based models. Kisi (2014) compared various

*ET*equations in a Mediterranean climate. Bourletsikas

_{o}*et al.*(2017) compared 24 different equations for estimating

*ET*in an evergreen broad leaf forest in Greece and concluded that calibrating mass transfer-based models is necessary for improving their performances.

_{o}This survey of the literature by the authors shows that there is limited work on the comprehensive comparison of different *ET _{o}* equations in different climatic conditions. Furthermore, the external calibration of those equations has been less studied in literature. The present study compares 20

*ET*equations (temperature/radiation/mass transfer-based) using data from a semi-arid environment of Iran. The applied equations are calibrated in both local and cross-station scales to investigate the ability of ancillary data supply techniques in estimating

_{o}*ET*in such regions.

_{o}## MATERIALS AND METHODS

### Data used

*T*), relative humidity (

_{A}*R*), solar radiation (

_{H}*R*) and wind speed (

_{S}*W*). Table 1 summarizes the geographical coordinates as well as the utilized parameters of the studied weather stations. The aridity index (

_{S}*I*) is a numerical indicator of the degree of dryness of the studied region (UNEP 1997), defined as where P is the total annual precipitation (mm) and

_{A}*ET*is the total annual

_{o}*ET*(mm) used for classifying the stations' climatic contexts (Table 1). Analyzing the values of this index in the studied locations revealed that all the stations have similar climatic conditions (semi-arid). The same interpretations might be obtained by analyzing the values of the continentality index (

_{o}*CI*), which showed the similarities of air temperature variations in these locations, where

^{CU}*M*and

_{i}*m*are the maximum and minimum average monthly tempterature (°C), respectively, and θ is the station latitude (degrees). The stations' elevations vary between 736 m (Jolfa) and 1682 m (Sarab). The available patterns were thoroughly screened and checked for any inconsistency and meaningless and missing values were remade. Consequently, the missing values for solar radiation, relative humidity and wind speed were estimated through the algorithms suggested by Allen

_{i}*et al.*(1998) and compared by Shiri (2017). Then the estimated data were used for calculating

*ET*.

_{o}Station | Latitude (^{o}N) | Longitude (^{o}E) | Altitude (m) | I (−) _{A} | CI ^{CU} |
---|---|---|---|---|---|

Ahar | 38°26′ | 47°04′ | 1391 | 0.27 | 0.83 |

Bonab | 37°20′ | 46°04′ | 1290 | 0.25 | 0.96 |

Jolfa | 38°56′ | 45°36′ | 736 | 0.20 | 0.84 |

Kalibar | 38°52′ | 47°01′ | 1210 | 0.43 | 0.62 |

Marageh | 37°21′ | 46°12′ | 1344 | 0.21 | 0.84 |

Marand | 38°25′ | 45°46′ | 1550 | 0.43 | 0.63 |

Mianeh | 37°27′ | 47°42′ | 1110 | 0.25 | 0.97 |

Sahand | 37°56′ | 46°07′ | 1641 | 0.17 | 0.65 |

Sarab | 37°56′ | 47°23′ | 1682 | 0.23 | 1.01 |

Tabriz | 38°07′ | 46°14′ | 1364 | 0.20 | 0.84 |

Mean and coefficient of variation values of used data | |||||

T (°C) _{A} | R (%) _{H} | R (MJ m_{S}^{−2} day^{−1}) | W (m s_{S}^{−1}) | ET (mm day_{o}^{−1}) | |

Ahar | 11.41 | 58.15 | 16.02 | 1.24 | 2.83 |

(0.77) | (0.25) | (0.46) | (0.53) | (0.60) | |

Bonab | 13.56 | 52.05 | 17.49 | 0.90 | 3.12 |

(0.74) | (0.32) | (0.46) | (0.60) | (0.66) | |

Jolfa | 15.25 | 55.78 | 16.30 | 1.08 | 3.28 |

(0.73) | (0.28) | (0.48) | (0.71) | (0.75) | |

Kalibar | 12.31 | 62.20 | 15.02 | 1.31 | 2.73 |

(0.69) | (0.31) | (0.48) | (0.49) | (0.66) | |

Marageh | 13.98 | 48.74 | 17.46 | 1.34 | 3.48 |

(0.72) | (0.37) | (0.45) | (0.47) | (0.67) | |

Marand | 12.24 | 50.77 | 16.61 | 0.96 | 2.95 |

(0.86) | (0.37) | (0.47) | (0.65) | (0.69) | |

Mianeh | 14.56 | 52.56 | 17.21 | 0.82 | 3.12 |

(0.72) | (0.33) | (0.45) | (0.56) | (0.68) | |

Sahand | 12.41 | 50.06 | 17.21 | 1.77 | 3.49 |

(0.82) | (0.38) | (0.45) | (0.63) | (0.69) | |

Sarab | 9.08 | 59.70 | 16.99 | 1.12 | 2.83 |

(1.03) | (0.23) | (0.45) | (0.57) | (0.65) | |

Tabriz | 13.54 | 50.46 | 16.80 | 1.36 | 3.38 |

(0.77) | (0.33) | (0.45) | (0.45) | (0.68) |

Station | Latitude (^{o}N) | Longitude (^{o}E) | Altitude (m) | I (−) _{A} | CI ^{CU} |
---|---|---|---|---|---|

Ahar | 38°26′ | 47°04′ | 1391 | 0.27 | 0.83 |

Bonab | 37°20′ | 46°04′ | 1290 | 0.25 | 0.96 |

Jolfa | 38°56′ | 45°36′ | 736 | 0.20 | 0.84 |

Kalibar | 38°52′ | 47°01′ | 1210 | 0.43 | 0.62 |

Marageh | 37°21′ | 46°12′ | 1344 | 0.21 | 0.84 |

Marand | 38°25′ | 45°46′ | 1550 | 0.43 | 0.63 |

Mianeh | 37°27′ | 47°42′ | 1110 | 0.25 | 0.97 |

Sahand | 37°56′ | 46°07′ | 1641 | 0.17 | 0.65 |

Sarab | 37°56′ | 47°23′ | 1682 | 0.23 | 1.01 |

Tabriz | 38°07′ | 46°14′ | 1364 | 0.20 | 0.84 |

Mean and coefficient of variation values of used data | |||||

T (°C) _{A} | R (%) _{H} | R (MJ m_{S}^{−2} day^{−1}) | W (m s_{S}^{−1}) | ET (mm day_{o}^{−1}) | |

Ahar | 11.41 | 58.15 | 16.02 | 1.24 | 2.83 |

(0.77) | (0.25) | (0.46) | (0.53) | (0.60) | |

Bonab | 13.56 | 52.05 | 17.49 | 0.90 | 3.12 |

(0.74) | (0.32) | (0.46) | (0.60) | (0.66) | |

Jolfa | 15.25 | 55.78 | 16.30 | 1.08 | 3.28 |

(0.73) | (0.28) | (0.48) | (0.71) | (0.75) | |

Kalibar | 12.31 | 62.20 | 15.02 | 1.31 | 2.73 |

(0.69) | (0.31) | (0.48) | (0.49) | (0.66) | |

Marageh | 13.98 | 48.74 | 17.46 | 1.34 | 3.48 |

(0.72) | (0.37) | (0.45) | (0.47) | (0.67) | |

Marand | 12.24 | 50.77 | 16.61 | 0.96 | 2.95 |

(0.86) | (0.37) | (0.47) | (0.65) | (0.69) | |

Mianeh | 14.56 | 52.56 | 17.21 | 0.82 | 3.12 |

(0.72) | (0.33) | (0.45) | (0.56) | (0.68) | |

Sahand | 12.41 | 50.06 | 17.21 | 1.77 | 3.49 |

(0.82) | (0.38) | (0.45) | (0.63) | (0.69) | |

Sarab | 9.08 | 59.70 | 16.99 | 1.12 | 2.83 |

(1.03) | (0.23) | (0.45) | (0.57) | (0.65) | |

Tabriz | 13.54 | 50.46 | 16.80 | 1.36 | 3.38 |

(0.77) | (0.33) | (0.45) | (0.45) | (0.68) |

Notes: the values in the brackets denote the coefficient of variations (*C _{V}*) of daily data set.

*I*: aridity index;

_{A}*CI*: Curey continentality index;

^{CU}*T*: temperature (°C);

_{A}*R*: relative humidity;

_{H}*R*: daily solar radiation (MJ m

_{S}^{−2}day

^{−1}) ;

*W*: ave. 24 h wind speed at 2 m (m s

_{S}^{−1});

*ET*: total annual

_{o}*ET*(mm).

_{o}## METHODS

*ET*values, which is a common practice in similar cases (Allen

_{o}*et al.*1998): where is the reference evapotranspiration (mm day

^{−1}), is the slope of the saturation vapor pressure function (kPa °C

^{−1}), is the psychometric constant (kPa °C

^{−1}),

*R*is the net radiation (MJ m

_{n}^{−2}day

^{−1}),

*G*is the soil heat flux density (MJ m

^{−2}day

^{−1}),

*T*is the mean air temperature (°C),

_{mean}*W*is the average 24 h wind speed at 2 m height (m s

_{S}^{−1}),

*e*is the saturation vapor pressure (kPa),

_{S}*e*is the actual vapor pressure (kPa), and is the latent heat of evaporation (MJ kg

_{a}^{−1}).

A total of 20 *ET _{o}* equations (Tables 2–4) comprising six temperature-based, five radiation-based, and ten mass transfer-based equations were compared in this study. The used temperature/radiation-based equations are the commonly used equations in literature, while the mass-transfer models are less-evaluated equations in arid and semi-arid regions. Initially, all equations were applied at each location using all available data. Then the equations were calibrated using local and cross-station data management scenarios as follows.

ET models _{o} | Meteorological inputs | Expression |
---|---|---|

Blaney & Criddle (1950) | T _{mean} | |

Hargreaves-Samani HS1 (2002) | T [_{mean}, T_{max}, T_{min,}Ra] | |

HS2 (2002) | T [_{mean}, T_{max}, T_{min},Ra] | |

HS3 (2002) | T [_{mean}, T_{max}, T_{min}, P,Ra] | |

HS4 (2007) | T [_{mean}, T_{max}, T_{min},Ra] | |

Schendel (1967) | T _{mean}, R_{H} |

ET models _{o} | Meteorological inputs | Expression |
---|---|---|

Blaney & Criddle (1950) | T _{mean} | |

Hargreaves-Samani HS1 (2002) | T [_{mean}, T_{max}, T_{min,}Ra] | |

HS2 (2002) | T [_{mean}, T_{max}, T_{min},Ra] | |

HS3 (2002) | T [_{mean}, T_{max}, T_{min}, P,Ra] | |

HS4 (2007) | T [_{mean}, T_{max}, T_{min},Ra] | |

Schendel (1967) | T _{mean}, R_{H} |

In these equations: *ET _{o}* = reference evapotranspiration (mm day

^{−1}),

*T*= mean air temperature (°C),

_{mean}*W*= average 24 h wind speed at 2 m height (m s

_{S}^{−1}),

*K*= the mean annual percentage of daytime hours,

*a*and

*b*

*=*coefficient of BC models which are determined by regression analysis,

*R*

_{H}*=*relative humidity (%),

*P*is monthly rainfall (mm),

*R*= extraterrestrial radiation (mm day

_{a}^{−1}),

*T*= maximum air temperature (°C) and

_{max}*T*= minimum air temperature (°C).

_{min}ET models _{o} | Meteorological inputs | Expression |
---|---|---|

Irmak et al. (2003) | T [_{mean},Rs] | |

Jensen & Haise (1963) | T [_{mean},R] _{s} | |

Jones & Ritchie (1990) | T [_{max}, T_{min},R] _{s} | |

Tabari-1 (2013) | T [_{mean},R] _{s} | |

Tabari-2 (2013) | T [_{min}, T_{max},R] _{s} |

ET models _{o} | Meteorological inputs | Expression |
---|---|---|

Irmak et al. (2003) | T [_{mean},Rs] | |

Jensen & Haise (1963) | T [_{mean},R] _{s} | |

Jones & Ritchie (1990) | T [_{max}, T_{min},R] _{s} | |

Tabari-1 (2013) | T [_{mean},R] _{s} | |

Tabari-2 (2013) | T [_{min}, T_{max},R] _{s} |

In these equations: *R _{S}* = daily solar radiation (MJ m

^{−2}day

^{−1}), = latent heat of the evaporation (MJ kg

^{−1}), EL = mean sea level elevation (m), = temperature coefficient, = intercept on the temperature axis (°C), = saturation vapor pressures of water at the mean maximum temperatures, for the warmest month of the year in a given area (KPa), = saturation vapor pressures of water at the mean minimum temperatures, for the warmest month of the year in a given area (KPa).

ET models _{o} | Meteorological inputs | Expression |
---|---|---|

Dalton (1802) | e _{a}, e_{S}, W_{S} | |

Trabert (1896) | e _{a}, e_{S}, W_{S} | |

Meyer (1926) | e _{a}, e_{S}, W_{S} | |

WMO (1966) | e _{a}, e_{S}, W_{S} | |

Mahringer (1970) | e _{a}, e_{S}, W_{S} | |

Penman (1948) | e _{a}, e_{S}, W_{S} | |

Albrecht (1950) | e _{a}, e_{S}, W_{S} | |

Brockamp & Wenner (1963) | e _{a}, e_{S}, W_{S} | |

Rohwer (1931) | e _{a}, e_{S}, W_{S} |

ET models _{o} | Meteorological inputs | Expression |
---|---|---|

Dalton (1802) | e _{a}, e_{S}, W_{S} | |

Trabert (1896) | e _{a}, e_{S}, W_{S} | |

Meyer (1926) | e _{a}, e_{S}, W_{S} | |

WMO (1966) | e _{a}, e_{S}, W_{S} | |

Mahringer (1970) | e _{a}, e_{S}, W_{S} | |

Penman (1948) | e _{a}, e_{S}, W_{S} | |

Albrecht (1950) | e _{a}, e_{S}, W_{S} | |

Brockamp & Wenner (1963) | e _{a}, e_{S}, W_{S} | |

Rohwer (1931) | e _{a}, e_{S}, W_{S} |

In these equations: *e _{s}* = saturation vapor pressure,

*e*= actual vapor pressure (hPa in all the equations except Rohwer and Penman models, where units are in mmHg),

_{a}*W*= average 24 h wind speed at 2 m height (m s

_{s}^{−1}in all the equations except Penman model, where unit are miles day

^{−1}).

### Local assessment (calibration) of the models

The applied equations were calibrated, tested and validated per station using 50%, 25% and 25% of available data records, respectively. The general assessments of the models were conducted based on testing and validation statistics.

### Cross-station assessment (calibration) of the models

*SI*) and the mean absolute error (

*MAE*): where

*ET*and

_{M}*ET*denote the estimated and standard values at the

_{o}*i*

^{th}time step, respectively.

*N*is number of time steps (Shiri

*et al.*2015).

*MAE*ratio of calibrated

*ET*models to the non-calibrated versions (

_{o}*R-MAE*) was applied (Landeras

*et al.*2008):

## RESULTS AND DISCUSSIONS

Table 5 summarizes the *SI* values of the applied *ET _{o}* models in the studied locations during the study period (2005–2016). As can be seen from the table, the performance accuracy of the models fluctuates among the stations. In the case of the temperature-based equations, except Schendel, all the applied equations presented the best accuracy in Marageh (with the lowest latitude), while Jolfa (with the highest latitude and the lowest altitude) presented the worst results, except for with the Blaney-Criddle (BC) equation. The overall performance accuracy of the Schendel equation was lower than those of the other applied temperature-based models in all studied locations, while HS1 and HS4 produced the most accurate results in all locations (except Jolfa). A possible reason for the weak performance of the Schendel equation might be the inclusion of mean air temperature (instead of maximum and minimum temperature values) along with the relative humidity in this equation. As the humidity values were low in the studied regions, the combination of higher/lower wind speed (which would affect the temperature difference,

*T*–

_{max}*T*) with lower humidity could result in a considerable over/underestimation in the

_{min}*ET*values produced by this equation. Similar results after application of this equation were reported by Tabari

_{o}*et al.*(2013) for humid regions.

Temperature-based | Radiation-based | Mass transfer-based | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

BC | Schendel | HS1 | HS2 | HS3 | HS4 | JH | RI | IR | T1 | T2 | DA | TR | ME | RO | PE | AL | BW | WMO | MA | |

Ahar | 0.24 | 0.40 | 0.18 | 0.20 | 0.25 | 0.19 | 0.22 | 0.50 | 0.17 | 0.20 | 0.18 | 0.32 | 0.33 | 0.32 | 0.32 | 0.32 | 0.40 | 0.33 | 0.33 | 0.33 |

Bonab | 0.23 | 0.46 | 0.20 | 0.21 | 0.25 | 0.20 | 0.20 | 0.42 | 0.18 | 0.21 | 0.17 | 0.33 | 0.36 | 0.33 | 0.32 | 0.33 | 0.67 | 0.35 | 0.35 | 0.34 |

Jolfa | 0.21 | 0.35 | 0.30 | 0.31 | 0.37 | 0.30 | 0.21 | 0.42 | 0.26 | 0.28 | 0.23 | 0.28 | 0.33 | 0.29 | 0.28 | 0.31 | 0.45 | 0.32 | 0.33 | 0.33 |

Kalibar | 0.22 | 0.40 | 0.19 | 0.18 | 0.22 | 0.19 | 0.22 | 0.54 | 0.18 | 0.20 | 0.19 | 0.36 | 0.37 | 0.36 | 0.36 | 0.37 | 0.41 | 0.36 | 0.37 | 0.37 |

Marageh | 0.18 | 0.43 | 0.17 | 0.17 | 0.19 | 0.17 | 0.16 | 0.43 | 0.18 | 0.23 | 0.18 | 0.27 | 0.29 | 0.27 | 0.27 | 0.29 | 0.36 | 0.29 | 0.30 | 0.29 |

Marand | 0.24 | 0.51 | 0.18 | 0.18 | 0.22 | 0.18 | 0.21 | 0.56 | 0.20 | 0.23 | 0.20 | 0.34 | 0.40 | 0.34 | 0.35 | 0.37 | 0.48 | 0.39 | 0.39 | 0.40 |

Mianeh | 0.22 | 0.45 | 0.20 | 0.21 | 0.24 | 0.20 | 0.18 | 0.39 | 0.20 | 0.23 | 0.18 | 0.33 | 0.35 | 0.33 | 0.32 | 0.33 | 0.42 | 0.34 | 0.34 | 0.35 |

Sahand | 0.23 | 0.55 | 0.21 | 0.21 | 0.24 | 0.21 | 0.23 | 0.59 | 0.23 | 0.26 | 0.24 | 0.30 | 0.29 | 0.32 | 0.29 | 0.29 | 0.37 | 0.29 | 0.30 | 0.29 |

Sarab | 0.25 | 0.43 | 0.17 | 0.19 | 0.24 | 0.18 | 0.19 | 0.53 | 0.17 | 0.20 | 0.18 | 0.32 | 0.33 | 0.33 | 0.32 | 0.32 | 0.39 | 0.32 | 0.32 | 0.33 |

Tabriz | 0.20 | 0.44 | 0.18 | 0.19 | 0.22 | 0.18 | 0.18 | 0.50 | 0.20 | 0.24 | 0.19 | 0.29 | 0.31 | 0.29 | 0.29 | 0.31 | 0.39 | 0.31 | 0.32 | 0.31 |

Temperature-based | Radiation-based | Mass transfer-based | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

BC | Schendel | HS1 | HS2 | HS3 | HS4 | JH | RI | IR | T1 | T2 | DA | TR | ME | RO | PE | AL | BW | WMO | MA | |

Ahar | 0.24 | 0.40 | 0.18 | 0.20 | 0.25 | 0.19 | 0.22 | 0.50 | 0.17 | 0.20 | 0.18 | 0.32 | 0.33 | 0.32 | 0.32 | 0.32 | 0.40 | 0.33 | 0.33 | 0.33 |

Bonab | 0.23 | 0.46 | 0.20 | 0.21 | 0.25 | 0.20 | 0.20 | 0.42 | 0.18 | 0.21 | 0.17 | 0.33 | 0.36 | 0.33 | 0.32 | 0.33 | 0.67 | 0.35 | 0.35 | 0.34 |

Jolfa | 0.21 | 0.35 | 0.30 | 0.31 | 0.37 | 0.30 | 0.21 | 0.42 | 0.26 | 0.28 | 0.23 | 0.28 | 0.33 | 0.29 | 0.28 | 0.31 | 0.45 | 0.32 | 0.33 | 0.33 |

Kalibar | 0.22 | 0.40 | 0.19 | 0.18 | 0.22 | 0.19 | 0.22 | 0.54 | 0.18 | 0.20 | 0.19 | 0.36 | 0.37 | 0.36 | 0.36 | 0.37 | 0.41 | 0.36 | 0.37 | 0.37 |

Marageh | 0.18 | 0.43 | 0.17 | 0.17 | 0.19 | 0.17 | 0.16 | 0.43 | 0.18 | 0.23 | 0.18 | 0.27 | 0.29 | 0.27 | 0.27 | 0.29 | 0.36 | 0.29 | 0.30 | 0.29 |

Marand | 0.24 | 0.51 | 0.18 | 0.18 | 0.22 | 0.18 | 0.21 | 0.56 | 0.20 | 0.23 | 0.20 | 0.34 | 0.40 | 0.34 | 0.35 | 0.37 | 0.48 | 0.39 | 0.39 | 0.40 |

Mianeh | 0.22 | 0.45 | 0.20 | 0.21 | 0.24 | 0.20 | 0.18 | 0.39 | 0.20 | 0.23 | 0.18 | 0.33 | 0.35 | 0.33 | 0.32 | 0.33 | 0.42 | 0.34 | 0.34 | 0.35 |

Sahand | 0.23 | 0.55 | 0.21 | 0.21 | 0.24 | 0.21 | 0.23 | 0.59 | 0.23 | 0.26 | 0.24 | 0.30 | 0.29 | 0.32 | 0.29 | 0.29 | 0.37 | 0.29 | 0.30 | 0.29 |

Sarab | 0.25 | 0.43 | 0.17 | 0.19 | 0.24 | 0.18 | 0.19 | 0.53 | 0.17 | 0.20 | 0.18 | 0.32 | 0.33 | 0.33 | 0.32 | 0.32 | 0.39 | 0.32 | 0.32 | 0.33 |

Tabriz | 0.20 | 0.44 | 0.18 | 0.19 | 0.22 | 0.18 | 0.18 | 0.50 | 0.20 | 0.24 | 0.19 | 0.29 | 0.31 | 0.29 | 0.29 | 0.31 | 0.39 | 0.31 | 0.32 | 0.31 |

Note: BC: Blaney-Criddle; HS1: Hargreaves-Samani 1; HS2: Hargreaves-Samani 2; HS3: Hargreaves-Samani 3; HS 4: Hargreaves-Samani 4; JS: Jensen-Haise; RI: Ritchie; IR: Irmak; T1: Tabari-1; T2: Tabari-2; DA: Dalton; TR: Trabert; ME: Meyer; RO: Rohwer; PE: Penman; AL: Albrecht; BW: Brockamp-Wenner; MA: Mahringer.

Regarding the radiation-based equations, with some exceptions, the regression-based Irmak (IR) and Tabari (T2) equations produced the most accurate results in all stations followed by the Jensen-Haise (JH), while the Ritchie equation gave the worst simulations. However, some previous studies confirmed a good performance accuracy of this equation in humid locations (Tabari *et al.* 2013). Finally, all the applied mass transfer-based equations showed a similar performance accuracy (in terms of *SI*) in all the studied locations, which might be attributed to the similar temperature and wind speed variations (in terms of *C _{V}*) in these locations.

The comparison of the performance of three categories showed that the overall performance accuracies of the temperature-based and radiation-based models were better than those of the mass transfer-based models, confirming the outcomes reported by Kiafar *et al.* (2017). This may be due to the inclusion of the temperature and wind speed as two parameters affecting the total amount of *ET _{o}*, while the energy component of this phenomenon, which can be interpreted through the radiation parameter, was ignored in mass transfer-based models. Though the effect of an aerodynamic component might be considerably higher than that of an energy component in some regions, these results might dictate the similar (at least) influence of the energy and aerodynamic components on the total

*ET*values in the studied locations. Figures 2 and 3 display the monthly

_{o}*ET*magnitudes produced by the applied equations for the Marageh and Jolfa stations, respectively. The presented monthly

_{o}*ET*variations among the models confirm the previous statements given under the explanations of

_{o}*SI*values in Table 5. Nonetheless, as can be observed, the

*ET*values obtained by all the models of each category are closer to each other in the cold season, while the discrepancy among them is obvious in hot seasons (e.g. June, July, August, etc). This could be due to the higher magnitudes of

_{o}*ET*in these months, which make higher differences when estimated with other models and compared with the target magnitudes.

_{o}### Local assessment of the calibrated models

Figure 4 shows the *R-MAE* values of the locally calibrated models during the validation period. From the figure it is seen that the calibration procedure has generally improved the model performance accuracy in all locations. Among the temperature-based models, the maximum performance improvement belonged to the BC, Schendel and HS4 equations. Also, the JH and Ritchie equations showed the maximum performance improvement by calibration procedure among the radiation-based models. Finally, the WMO equation showed the lowest improvement through the local calibration among the mass transfer-based equations. The comparison of the three categories showed the highest *R-MAE* values in all studied locations of the mass transfer-based equations. Nevertheless, there were some instances of an adverse effect of calibration on modeling accuracy (e.g. HS2 in Tabriz, HS4 in Mianeh). Such adverse effects of calibration on the accuracy of equations have been reported previously in other similar studies (e.g. Shiri *et al.* 2013, 2015).

For a better comparison of the *ET _{o}* estimations at the studied locations, the

*SI*values of the locally calibrated models (calibrated and evaluated at each station) have been plotted in Figure 5 for all studied locations, which confirm the aforementioned statements.

Summarizing, it is seen that the local calibration of the *ET _{o}* equations might be a good option for improving the performance accuracy of the models at the studied locations (and other points with similar conditions). However, it should be noted that this procedure has a major drawback because it requires local patterns for generating the calibrated equations, so its applicability would be limited to the locations with the necessary meteorological data available, which is not the case in most areas, especially in developing countries. Therefore, an external calibration procedure which works without using the local patterns should be adopted to estimate

*ET*in limited data conditions. These statements confirm the results obtained by Landeras

_{o}*et al.*(2017) for arid and semi-arid regions of Africa.

### Cross-station calibration of the models

Figure 6 shows the global *R-MAE* values of the *ET _{o}* equations calibrated using single station records. These values have been computed as an average of the indicators of each equation per station when using different single stations for calibrating the equations. Again, the mass transfer-based equations (except WMO and Mahringer) showed the highest increase in performance accuracy in all the studied stations. A reason behind this might be the similarity of the wind speed and temperature variations in the studied locations (as stated before) which made it easy to extrapolate the

*ET*values using the local patterns of the other stations. Among the radiation-based models, the performance of regression-based Irmak, T1 and T2 equations could not be improved through this calibration procedure, which might be linked to their regression-based principal that makes their application limited to a local scale where the data from each location are applied for calibrating and testing the models. Finally, the

_{o}*ET*simulation produced by the HS equation versions could not be improved by this external calibration, while BC and Schendel showed considerable improvement in terms of

_{o}*MAE*reduction. However, these models showed noticeable improvement through calibration in Jolfa which has the highest

*C*values for the

_{V}*R*,

_{S}*S*and

_{W}*ET*parameters.

_{o}Figure 7 shows the *SI* values of the externally calibrated models (using single station data). From the figure it is clear that the temperature- and radiation-based equations generally have had similar *SI* values (except Schendel, JH and Ritchie), giving more accurate simulations than the mass transfer-based equations. Similar results were also observed in the case of local calibration. Although the studied equations groups utilized different meteorological inputs for estimation of *ET _{o}*, there was a similarity between their performance accuracy when they were calibrated externally. This might be due to the general similarity of the

*ET*values in the studied stations. On the other hand, this might show the importance of temperature parameters on

_{o}*ET*modelling in the studied locations, as well as its superiority (in terms of variable importance) to the radiation parameter. The effect of an aerodynamic component seems to be very low as the mass transfer-based models gave the worst results in the studied cases. However, this statement should be used with caution, because a thorough and detailed analysis should be carried out to identify the portion of each component on the final

_{o}*ET*magnitudes which is beyond the scope of the present study.

_{o}*R-MAE* values of the externally calibrated equations using the pooled data from other stations have been plotted in Figure 8 for all the studied locations. Similar to the previous case, the mass transfer-based models presented the highest improvement magnitudes in terms of *MAE* reduction (except WMO and Mahringer) in all the stations. Among the radiation-based equations, JH and Ritchie presented the highest performance improvement in the studied locations and finally, the temperature-based models showed a fluctuating trend of performance improvement by external calibrating through the data pooled from other stations. While BC and Schendel showed considerable accuracy improvement in all locations, the other models showed lower improvement (except HS4). Figure 9 shows the *SI* values of the externally calibrated models (calibration using the pooled data of other stations).

After comparing two external calibration procedures in terms of *R-MAE*, it is seen that the general trend in the increase/decrease of the performance accuracy is similar for both external calibration cases for all three studied categories in all the stations. This might clarify that the models can enjoy the external patterns for calibration in both local and pooled scenarios for improving their performance by taking into consideration that the amount of data supplied for calibrating the models has not affected the modelling global accuracy in the studied case. On the other hand, the higher the number of patterns used to feed the calibrating regression-based models, the higher the involvement of different variations in the calibration matrix, which would make it easy to extrapolate the *ET _{o}* values. In the present case, however, the equations showed their performance accuracy increasing/decreasing through calibration procedure independent of the data amount. This might be explained by general similarity in the studied stations (Table 1) in terms of

*I*and

_{A}*CI*indices. So, it might be concluded that, when using similar stations, the amount of meteorological variables used for making the calibration matrix cannot affect the general performance accuracy of the calibrated models, as the applied patterns might be redundant so that they do not alter the regression-based calibration equation considerably. However, it should be noted that a suitable feeding station (for calibrating the equations) might be defined for each target (test) station. In this case, similarities between the optimum train station and the testing station from the

^{CU}*ET*, as well as climatic context viewpoints, would be very important. By selecting such an optimum station, the accuracy of the external calibration procedure might eventually be increased as the redundant patterns from other stations would be removed from the calibration matrix. This would be an interesting subject for future studies in this field.

_{o}Nevertheless, it should be noted that a commonly used single linear calibration equation was applied here. Utilizing other (non-linear) calibration equations may produce different results than those obtained here, although it could be anticipated that non-linear calibration equations would give more accurate results than the present one as discussed by Shiri *et al.* (2014).

## CONCLUSIONS

The present study was aimed at comparing 20 *ET _{o}* estimation equations in a semi-arid region. The applied equations were categorized in four classes, namely, combination-based (FAO56-PM standard model which was used for calibrating and assessing the applied equations), temperature-based, radiation-based and mass transfer-based equations. Daily meteorological data from 10 weather stations covering a period of 12 years were utilized here.

At first, the equations were applied per station and compared with the benchmark FAO56-PM model. The results revealed that the radiation-based and temperature-based equations produced almost similar results, while the mass transfer-based equations gave the worst simulations in all studied locations.

The applied equations were then calibrated in both local and cross-station scales. Assessing the calibrated equations showed that calibration procedure had the highest quantitative effect on the performance accuracy of the mass transfer-based equations in both local and cross-station scales. Nevertheless, the cross-station calibration procedure presented comparable results to those of the locally calibrated equations, which is not the case in most of the studies reported here. Although not expected, the results showed that the cross-station calibration (through individual or pooled data application) could improve the equations' performance accuracy to a great extent. This would be criticized by previous reports which emphasized the higher accuracy of local application due to the use of local patterns. However, it should be noted that only 20 of the most common equations were used in the present study for a semi-arid region. Future studies might use more equations and stations as well as other calibration scenarios for assessing these outcomes in various climatic contexts.

## REFERENCES

*.*

FAO Irrigation and Drainage Paper, No. 56